From d8629792be271c01e44e9d87bb050811f0ff179a Mon Sep 17 00:00:00 2001 From: =?utf8?q?Anders=20M=C3=B6rtberg?= Date: Thu, 7 Jul 2016 15:57:52 +0200 Subject: [PATCH] rename Id to Path --- examples/aim.ctt | 186 ++++----- examples/binnat.ctt | 82 ++-- examples/bool.ctt | 132 +++--- examples/category.ctt | 784 ++++++++++++++++++------------------ examples/circle.ctt | 34 +- examples/collection.ctt | 20 +- examples/csystem.ctt | 216 +++++----- examples/demo.ctt | 88 ++-- examples/discor.ctt | 26 +- examples/equiv.ctt | 78 ++-- examples/groupoidTrunc.ctt | 84 ++-- examples/hedberg.ctt | 50 +-- examples/helix.ctt | 224 +++++------ examples/hnat.ctt | 12 +- examples/hz.ctt | 26 +- examples/implicit_point.ctt | 16 +- examples/injective.ctt | 58 +-- examples/int.ctt | 14 +- examples/integer.ctt | 24 +- examples/interval.ctt | 12 +- examples/list.ctt | 14 +- examples/multS1.ctt | 54 +-- examples/nat.ctt | 44 +- examples/ordinal.ctt | 26 +- examples/pi.ctt | 28 +- examples/prelude.ctt | 160 ++++---- examples/prop.ctt | 32 +- examples/quotient.ctt | 18 +- examples/retract.ctt | 20 +- examples/setquot.ctt | 176 ++++---- examples/sigma.ctt | 124 +++--- examples/subset.ctt | 56 +-- examples/susp.ctt | 66 +-- examples/torsor.ctt | 662 +++++++++++++++--------------- examples/torus.ctt | 34 +- examples/univalence.ctt | 130 +++--- 36 files changed, 1905 insertions(+), 1905 deletions(-) diff --git a/examples/aim.ctt b/examples/aim.ctt index 9608695..4ec7ca1 100644 --- a/examples/aim.ctt +++ b/examples/aim.ctt @@ -48,7 +48,7 @@ not : bool -> bool = split true -> false {- - Identity types, names and formulas + Path types, names and formulas -------------------------------------------------------------------------- An element in ID A B is a line in the universe connecting A and B: @@ -59,30 +59,30 @@ An element in ID A B is a line in the universe connecting A and B: |- ID A B Type -IdP is heterogeneous equality: +PathP is heterogeneous equality: |- P : ID A B |- a : A |- b : B ----------------------------------------------- - |- IdP P a b Type + |- PathP P a b Type -} --- The usual identity can be seen a special case of IdP: -Id (A : U) (a b : A) : U = IdP ( A) a b +-- The constant Path type can be seen a special case of PathP: +Path (A : U) (a b : A) : U = PathP ( A) a b -- "" abstracts the name/color/dimension i: -refl (A : U) (a : A) : Id A a a = a +refl (A : U) (a : A) : Path A a a = a {- -A proof "p : Id A a b" is thought of as a line between a and b: +A proof "p : Path A a b" is thought of as a line between a and b: p a ------> b -A proof "sq : Id (Id A a b) p q" is thought of as a square between p and q: +A proof "sq : Path (Path A a b) p q" is thought of as a square between p and q: q @@ -100,19 +100,19 @@ And so on... -- It is possible to take the face of a path to obtain its endpoints: -face0 (A : U) (a b : A) (p : Id A a b) : Id A a a = refl A (p @ 0) -face1 (A : U) (a b : A) (p : Id A a b) : Id A b b = refl A (p @ 1) +face0 (A : U) (a b : A) (p : Path A a b) : Path A a a = refl A (p @ 0) +face1 (A : U) (a b : A) (p : Path A a b) : Path A b b = refl A (p @ 1) -- By applying a path to a name i, "p @ i", it is seen as a path "in -- dimension i" connecting "p @ 0" to "p @ 1". This way we get a -- simple proof of cong: -cong (A B : U) (f : A -> B) (a b : A) (p : Id A a b) : Id B (f a) (f b) = +cong (A B : U) (f : A -> B) (a b : A) (p : Path A a b) : Path B (f a) (f b) = f (p @ i) -- This also gives a short proof of function extensionality: funExt (A : U) (B : A -> U) (f g : (x : A) -> B x) - (p : (x : A) -> Id (B x) (f x) (g x)) : Id ((y : A) -> B y) f g = + (p : (x : A) -> Path (B x) (f x) (g x)) : Path ((y : A) -> B y) f g = \(x : A) -> (p x) @ i {- @@ -151,11 +151,11 @@ min(i,j), i \/ j is max(i,j) and -i is 1 - i. -} -- Applying a path to a negated name inverts it: -sym (A : U) (a b : A) (p : Id A a b) : Id A b a = p @ -i +sym (A : U) (a b : A) (p : Path A a b) : Path A b a = p @ -i -- This operation is an involution: -symK (A : U) (a b : A) (p : Id A a b) : Id (Id A a b) p p = - refl (Id A a b) (sym A b a (sym A a b p)) +symK (A : U) (a b : A) (p : Path A a b) : Path (Path A a b) p p = + refl (Path A a b) (sym A b a (sym A a b p)) {- Connections: @@ -196,10 +196,10 @@ And a disjunction gives: -- This gives a simple proof that singletons are contractible: -singl (A : U) (a : A) : U = (x : A) * Id A a x +singl (A : U) (a : A) : U = (x : A) * Path A a x -contrSingl (A : U) (a b : A) (p : Id A a b) : - Id (singl A a) (a,refl A a) (b,p) = +contrSingl (A : U) (a b : A) (p : Path A a b) : + Path (singl A a) (a,refl A a) (b,p) = (p @ i, p @ i /\ j) @@ -208,28 +208,28 @@ contrSingl (A : U) (a b : A) (p : Id A a b) : Note that formulas does not form a boolean algebra, but a de Morgan algebra. This means that "p @ 0" and "p @ i /\ -i" are different! -testDeMorganAlgebra (A : U) (a b : A) (p : Id A a b) : - Id (Id A a a) (<_> p @ 0) (<_> p @ 0) = - refl (Id A a a) ( p @ i /\ -i) +testDeMorganAlgebra (A : U) (a b : A) (p : Path A a b) : + Path (Path A a a) (<_> p @ 0) (<_> p @ 0) = + refl (Path A a a) ( p @ i /\ -i) This is clear with the intuition that /\ correspond to min. More about this later... -} -testConj (A : U) (a b : A) (p : Id A a b) : Id A a a = p @ i /\ -i -testDisj (A : U) (a b : A) (p : Id A a b) : Id A b b = p @ i \/ -i +testConj (A : U) (a b : A) (p : Path A a b) : Path A a a = p @ i /\ -i +testDisj (A : U) (a b : A) (p : Path A a b) : Path A b b = p @ i \/ -i {- -So far we have: MLTT + Id, , @, i/\j... +So far we have: MLTT + Path, , @, i/\j... We can prove that singletons are contractible. But, we don't have transport: -transportf (A : U) (P : A -> U) (a b : A) (p : Id A a b) : P a -> P b +transportf (A : U) (P : A -> U) (a b : A) (p : Path A a b) : P a -> P b Which is what is needed to get J as: @@ -253,7 +253,7 @@ a system describing the rest of the shape, and produces the missing side (opposite of the principal side). -Transitivity of Id is obtained from a composition of this open square: +Transitivity of Path is obtained from a composition of this open square: i : I, j : I |- @@ -269,7 +269,7 @@ i : I, j : I |- p @ i The composition computes the dashed line at the top of the square. - ( (<_>a)@i ) = ( (<_>a)@j) : Id (Id A a a) ( a) ( a) + ( (<_>a)@i ) = ( (<_>a)@j) : Path (Path A a a) ( a) ( a) p = p @ i @@ -279,21 +279,21 @@ i : I |- (<_>a) @i -} -pathscomp0 (A : U) (a b c : A) (p : Id A a b) (q : Id A b c) : Id A a c = +pathscomp0 (A : U) (a b c : A) (p : Path A a b) (q : Path A b c) : Path A a c = comp (<_> A) (p @ i) [ (i = 0) -> <_> a , (i = 1) -> q ] -- The first two nonzero h-levels are propositions and sets: -prop (A : U) : U = (a b : A) -> Id A a b -set (A : U) : U = (a b : A) -> prop (Id A a b) +prop (A : U) : U = (a b : A) -> Path A a b +set (A : U) : U = (a b : A) -> prop (Path A a b) -- Using compositions we get a short proof that any prop is a set. To -- understand this proof one should draw an open cube with the back -- face being a constant square in a and the sides given by the system -- below. propSet (A : U) (h : prop A) : set A = - \(a b : A) (p q : Id A a b) -> + \(a b : A) (p q : Path A a b) -> comp (<_> A) a [ (j=0) -> h a a , (j=1) -> h a b , (i=0) -> h a (p @ j) @@ -319,11 +319,11 @@ We can also define the type of squares: v -} -Square (A : U) (a0 a1 b0 b1 : A) (u : Id A a0 a1) (v : Id A b0 b1) - (r0 : Id A a0 b0) (r1 : Id A a1 b1) : U - = IdP ( (IdP ( A) (u @ i) (v @ i))) r0 r1 +Square (A : U) (a0 a1 b0 b1 : A) (u : Path A a0 a1) (v : Path A b0 b1) + (r0 : Path A a0 b0) (r1 : Path A a1 b1) : U + = PathP ( (PathP ( A) (u @ i) (v @ i))) r0 r1 -constSquare (A : U) (a : A) (p : Id A a a) : Square A a a a a p p p p = +constSquare (A : U) (a : A) (p : Path A a a) : Square A a a a a p p p p = comp (<_> A) a [ (i = 0) -> p @ j \/ - k , (i = 1) -> p @ j /\ k , (j = 0) -> p @ i \/ - k @@ -335,17 +335,17 @@ constSquare (A : U) (a : A) (p : Id A a a) : Square A a a a a p p p p = -- have only considered constant lines.. -- "Kan composition": -kan (A : U) (a b c d : A) (p : Id A a b) - (q : Id A a c) (r : Id A b d) : Id A c d = +kan (A : U) (a b c d : A) (p : Path A a b) + (q : Path A a c) (r : Path A b d) : Path A c d = comp (<_> A) (p @ i) [ (i = 0) -> q, (i = 1) -> r ] -- Generalized Kan composition: -kan' (A B : U) (P : Id U A B) (a b : A) (c d : B) (p : Id A a b) - (q : IdP P a c) (r : IdP P b d) : Id B c d = +kan' (A B : U) (P : Path U A B) (a b : A) (c d : B) (p : Path A a b) + (q : PathP P a c) (r : PathP P b d) : Path B c d = comp P (p @ i) [ (i = 0) -> q, (i = 1) -> r ] -kan'' (A : U) (a b c d : A) (p : Id A a b) - (q : Id A a c) (r : Id A b d) : Id A c d = +kan'' (A : U) (a b c d : A) (p : Path A a b) + (q : Path A a c) (r : Path A b d) : Path A c d = kan' A A (<_> A) a b c d p q r @@ -365,16 +365,16 @@ add (a : nat) : nat -> nat = split suc b -> suc (add a b) -- Exercise (for solution see bottom of the file): -addZero : (a : nat) -> Id nat (add zero a) a = undefined +addZero : (a : nat) -> Path nat (add zero a) a = undefined -addSuc (a : nat) : (b : nat) -> Id nat (add (suc a) b) (suc (add a b)) = split +addSuc (a : nat) : (b : nat) -> Path nat (add (suc a) b) (suc (add a b)) = split zero -> suc a suc b' -> suc (addSuc a b' @ i) -- Exercise (can be done with a composition): -addCom (a : nat) : (b : nat) -> Id nat (add a b) (add b a) = undefined +addCom (a : nat) : (b : nat) -> Path nat (add a b) (add b a) = undefined -addAssoc (a b : nat) : (c : nat) -> Id nat (add a (add b c)) (add (add a b) c) = split +addAssoc (a b : nat) : (c : nat) -> Path nat (add a (add b c)) (add (add a b) c) = split zero -> add a b suc c' -> suc (addAssoc a b c' @ i) @@ -388,41 +388,41 @@ addAssoc (a b : nat) : (c : nat) -> Id nat (add a (add b c)) (add (add a b) c) = Transport takes a path in the universe between A and B and produces a function from A to B: - transport : Id U A B -> A -> B + transport : Path U A B -> A -> B -} -- This gives a simple proof of subst (called transportf above and in -- UniMath): -subst (A : U) (P : A -> U) (a b : A) (p : Id A a b) (e : P a) : P b = +subst (A : U) (P : A -> U) (a b : A) (p : Path A a b) (e : P a) : P b = transport (cong A U P a b p) e -- Transport is defined in the Haskell code as a composition with an -- empty cube/system. It can also be directly defined as: -trans (A B : U) (p : Id U A B) (a : A) : B = comp p a [] +trans (A B : U) (p : Path U A B) (a : A) : B = comp p a [] -- However these are not exactly the same because the lack of implicit -- arguments. -- subst with the fact that singletons are contractible this gives -- the J eliminator: -J (A : U) (a : A) (C : (x : A) -> Id A a x -> U) - (d : C a (refl A a)) (x : A) (p : Id A a x) : C x p = +J (A : U) (a : A) (C : (x : A) -> Path A a x -> U) + (d : C a (refl A a)) (x : A) (p : Path A a x) : C x p = subst (singl A a) T (a,refl A a) (x,p) (contrSingl A a x p) d where T (bp : singl A a) : U = C bp.1 bp.2 -- Note: Transporting with refl is not the identity function: --- transRefl (A : U) (a : A) : Id A a a = refl A (transport (refl U A) a) +-- transRefl (A : U) (a : A) : Path A a a = refl A (transport (refl U A) a) -- This implies that the elimination rule for J does not hold definitonally: --- defEqJ (A : U) (a : A) (C : (x : A) -> Id A a x -> U) (d : C a (refl A a)) : --- Id (C a (refl A a)) (J A a C d a (refl A a)) d = refl (C a (refl A a)) d +-- defEqJ (A : U) (a : A) (C : (x : A) -> Path A a x -> U) (d : C a (refl A a)) : +-- Path (C a (refl A a)) (J A a C d a (refl A a)) d = refl (C a (refl A a)) d -- We can get the equality between a and the transport of a by filling: -transFill (A : U) (a : A) : Id A a (transport (<_> A) a) = +transFill (A : U) (a : A) : Path A a (transport (<_> A) a) = fill (<_> A) a [] @@ -437,13 +437,13 @@ transFill (A : U) (a : A) : Id A a (transport (<_> A) a) = -- composition in the universe). -- A "strange" term showing what happens without regularity --- strange (A : U) (a b c : A) (p : Id A a b) (q : Id A b c) : --- Id A (comp (<_> A) a []) c = +-- strange (A : U) (a b c : A) (p : Path A a b) (q : Path A b c) : +-- Path A (comp (<_> A) a []) c = -- comp (<_> A) (p @ i) [ (i = 1) -> q ] --- It seems like we can recover a new Id type with the correct +-- It seems like we can recover a new Path type with the correct -- definitional equalities using ideas from awfs of Andrew -- Swan... Simon has implemented this in the branch "defeq". @@ -457,7 +457,7 @@ transFill (A : U) (a : A) : Id A a (transport (<_> A) a) = {- -So far we have: MLTT + Id, , @, i/\j, compositions, transport +So far we have: MLTT + Path, , @, i/\j, compositions, transport With all of this we get J, but so far without def. eq. @@ -480,10 +480,10 @@ a map from Equiv(A,B) to A = B. -} -isContr (A : U) : U = (x : A) * ((y : A) -> Id A x y) +isContr (A : U) : U = (x : A) * ((y : A) -> Path A x y) fiber (A B : U) (f : A -> B) (y : B) : U = - (x : A) * Id B y (f x) + (x : A) * Path B y (f x) isEquiv (A B : U) (f : A -> B) : U = (y : B) -> isContr (fiber A B f y) @@ -494,18 +494,18 @@ idIsEquiv (A : U) : isEquiv A A (idfun A) = -- Using the grad lemma we can transform isomorphisms into equalities. lemIso (A B : U) (f : A -> B) (g : B -> A) - (s : (y : B) -> Id B (f (g y)) y) - (t : (x : A) -> Id A (g (f x)) x) - (y : B) (x0 x1 : A) (p0 : Id B y (f x0)) (p1 : Id B y (f x1)) : - Id (fiber A B f y) (x0,p0) (x1,p1) = (p @ i,sq1 @ i) + (s : (y : B) -> Path B (f (g y)) y) + (t : (x : A) -> Path A (g (f x)) x) + (y : B) (x0 x1 : A) (p0 : Path B y (f x0)) (p1 : Path B y (f x1)) : + Path (fiber A B f y) (x0,p0) (x1,p1) = (p @ i,sq1 @ i) where - rem0 : Id A (g y) x0 = + rem0 : Path A (g y) x0 = comp (<_> A) (g (p0 @ i)) [ (i = 1) -> t x0, (i = 0) -> <_> g y ] - rem1 : Id A (g y) x1 = + rem1 : Path A (g y) x1 = comp (<_> A) (g (p1 @ i)) [ (i = 1) -> t x1, (i = 0) -> <_> g y ] - p : Id A x0 x1 = + p : Path A x0 x1 = comp (<_> A) (g y) [ (i = 0) -> rem0 , (i = 1) -> rem1 ] @@ -543,15 +543,15 @@ lemIso (A B : U) (f : A -> B) (g : B -> A) , (j = 0) -> s y ] gradLemma (A B : U) (f : A -> B) (g : B -> A) - (s : (y : B) -> Id B (f (g y)) y) - (t : (x : A) -> Id A (g (f x)) x) : isEquiv A B f = + (s : (y : B) -> Path B (f (g y)) y) + (t : (x : A) -> Path A (g (f x)) x) : isEquiv A B f = \ (y:B) -> ((g y,s y@-i),\ (z:fiber A B f y) -> lemIso A B f g s t y (g y) z.1 (s y@-i) z.2) -isoId (A B : U) (f : A -> B) (g : B -> A) - (s : (y : B) -> Id B (f (g y)) y) - (t : (x : A) -> Id A (g (f x)) x) : Id U A B = +isoPath (A B : U) (f : A -> B) (g : B -> A) + (s : (y : B) -> Path B (f (g y)) y) + (t : (x : A) -> Path A (g (f x)) x) : Path U A B = glue B [ (i = 0) -> (A,f,gradLemma A B f g s t) , (i = 1) -> (B,idfun B,idIsEquiv B) ] @@ -565,12 +565,12 @@ isoId (A B : U) (f : A -> B) (g : B -> A) --------------------------------------------------------- -- Not is involutive: -notK : (b : bool) -> Id bool (not (not b)) b = split +notK : (b : bool) -> Path bool (not (not b)) b = split false -> false true -> true -- This defines a non-trivial equality between bool and bool: -notEq : Id U bool bool = isoId bool bool not not notK notK +notEq : Path U bool bool = isoPath bool bool not not notK notK -- We can transport true along this non-trivial equality: testFalse : bool = transport notEq true @@ -619,30 +619,30 @@ predZ : Z -> Z = split zero -> inl zero suc n -> inr n -sucpredZ : (x : Z) -> Id Z (sucZ (predZ x)) x = split +sucpredZ : (x : Z) -> Path Z (sucZ (predZ x)) x = split inl u -> inl u inr v -> lem v where - lem : (u : nat) -> Id Z (sucZ (predZ (inr u))) (inr u) = split + lem : (u : nat) -> Path Z (sucZ (predZ (inr u))) (inr u) = split zero -> inr zero suc n -> inr (suc n) -predsucZ : (x : Z) -> Id Z (predZ (sucZ x)) x = split +predsucZ : (x : Z) -> Path Z (predZ (sucZ x)) x = split inl u -> lem u where - lem : (u : nat) -> Id Z (predZ (sucZ (inl u))) (inl u) = split + lem : (u : nat) -> Path Z (predZ (sucZ (inl u))) (inl u) = split zero -> inl zero suc n -> inl (suc n) inr v -> inr v -sucIdZ : Id U Z Z = isoId Z Z sucZ predZ sucpredZ predsucZ +sucPathZ : Path U Z Z = isoPath Z Z sucZ predZ sucpredZ predsucZ -- We can transport along the proof forward and backwards: -testOneZ : Z = transport sucIdZ zeroZ -testNOneZ : Z = transport ( sucIdZ @ - i) zeroZ +testOneZ : Z = transport sucPathZ zeroZ +testNOneZ : Z = transport ( sucPathZ @ - i) zeroZ --- transport sucIdZ = sucZ ? +-- transport sucPathZ = sucZ ? @@ -687,7 +687,7 @@ data int = pos (n : nat) , (i = 1) -> neg zero ] -- Nice version of the zero constructor: -zeroPath : Id int (pos zero) (neg zero) = zeroP {int} @ i +zeroPath : Path int (pos zero) (neg zero) = zeroP {int} @ i sucInt : int -> int = split pos n -> pos (suc n) @@ -709,19 +709,19 @@ fromZ : Z -> int = split inl n -> neg (suc n) inr n -> pos n -toZK : (a : Z) -> Id Z (toZ (fromZ a)) a = split +toZK : (a : Z) -> Path Z (toZ (fromZ a)) a = split inl n -> inl n inr n -> inr n -fromZK : (a : int) -> Id int (fromZ (toZ a)) a = split +fromZK : (a : int) -> Path int (fromZ (toZ a)) a = split pos n -> pos n neg n -> rem n - where rem : (n : nat) -> Id int (fromZ (toZ (neg n))) (neg n) = split + where rem : (n : nat) -> Path int (fromZ (toZ (neg n))) (neg n) = split zero -> zeroPath suc m -> neg (suc m) zeroP @ i -> zeroPath @ i /\ j -- A connection makes this proof easy! -isoIntZ : Id U Z int = isoId Z int fromZ toZ fromZK toZK +isoIntZ : Path U Z int = isoPath Z int fromZ toZ fromZK toZK @@ -732,7 +732,7 @@ isoIntZ : Id U Z int = isoId Z int fromZ toZ fromZK toZK data S1 = base | loop [ (i = 0) -> base, (i = 1) -> base] -loopS1 : U = Id S1 base base +loopS1 : U = Path S1 base base -- The loop constructor loop1 : loopS1 = loop{S1} @ i @@ -741,7 +741,7 @@ invLoop : loopS1 = loop1 @ -i helix : S1 -> U = split base -> Z - loop @ i -> sucIdZ @ i + loop @ i -> sucPathZ @ i -- The winding number: winding (p : loopS1) : Z = transport ( helix (p @ i)) zeroZ @@ -756,7 +756,7 @@ loopZ3 : Z = winding (compS1 loop1 (compS1 loop1 loop1)) loopZN1 : Z = winding invLoop loopZ0 : Z = winding (compS1 loop1 invLoop) -mLoop : (x : S1) -> Id S1 x x = split +mLoop : (x : S1) -> Path S1 x x = split base -> loop1 loop @ i -> constSquare S1 base loop1 @ i @@ -777,7 +777,7 @@ loopZ8 : Z = winding (doubleLoop (doubleLoop (compS1 loop1 loop1))) -- A nice example of a homotopy on the circle. The path going halfway -- around the circle and then back is contractible: -hmtpy : Id loopS1 ( base) ( loop{S1} @ (i /\ -i)) = +hmtpy : Path loopS1 ( base) ( loop{S1} @ (i /\ -i)) = loop{S1} @ j /\ i /\ -i @@ -813,11 +813,11 @@ hmtpy : Id loopS1 ( base) ( loop{S1} @ (i /\ -i)) = {- Solutions: -addZero : (a : nat) -> Id nat (add zero a) a = split +addZero : (a : nat) -> Path nat (add zero a) a = split zero -> zero suc a' -> suc (addZero a' @ i) -addCom (a : nat) : (b : nat) -> Id nat (add a b) (add b a) = split +addCom (a : nat) : (b : nat) -> Path nat (add a b) (add b a) = split zero -> addZero a @ -i suc b' -> comp (<_> nat) (suc (addCom a b' @ i)) [ (i = 0) -> suc (add a b') , (i = 1) -> addSuc b' a @ -j ] diff --git a/examples/binnat.ctt b/examples/binnat.ctt index 5f0818d..aee16ef 100644 --- a/examples/binnat.ctt +++ b/examples/binnat.ctt @@ -30,15 +30,15 @@ posInd (P : pos -> U) (h1 : P pos1) (hS : (p : pos) -> P p -> P (sucPos p)) (p : x1 p -> hS (x0 p) (posInd (\(p : pos) -> P (x0 p)) (hS pos1 h1) H p) in f p -sucPosSuc : (p : pos) -> Id nat (PosToN (sucPos p)) (suc (PosToN p)) = split +sucPosSuc : (p : pos) -> Path nat (PosToN (sucPos p)) (suc (PosToN p)) = split pos1 -> <_> suc (suc zero) x0 p -> <_> suc (doubleN (PosToN p)) x1 p -> doubleN (sucPosSuc p @ i) -zeronPosToN (p : pos) : neg (Id nat zero (PosToN p)) = - posInd (\(p : pos) -> neg (Id nat zero (PosToN p))) (\(prf : Id nat zero one) -> znots zero prf) hS p +zeronPosToN (p : pos) : neg (Path nat zero (PosToN p)) = + posInd (\(p : pos) -> neg (Path nat zero (PosToN p))) (\(prf : Path nat zero one) -> znots zero prf) hS p where - hS (p : pos) (_ : neg (Id nat zero (PosToN p))) (prf : Id nat zero (PosToN (sucPos p))) : N0 = + hS (p : pos) (_ : neg (Path nat zero (PosToN p))) (prf : Path nat zero (PosToN (sucPos p))) : N0 = znots (PosToN p) ( comp ( nat) (prf @ i) [ (i=0) -> <_> zero, (i = 1) -> sucPosSuc p ]) -- Inverse of PosToN: @@ -50,37 +50,37 @@ NtoPos : nat -> pos = split zero -> pos1 suc n -> NtoPos' n -PosToNK : (n : nat) -> Id nat (PosToN (NtoPos (suc n))) (suc n) = split +PosToNK : (n : nat) -> Path nat (PosToN (NtoPos (suc n))) (suc n) = split zero -> <_> (suc zero) suc n -> - let ih : Id nat (suc (PosToN (NtoPos' n))) (suc (suc n)) = suc (PosToNK n @ i) - in compId nat (PosToN (NtoPos (suc (suc n)))) (suc (PosToN (NtoPos' n))) + let ih : Path nat (suc (PosToN (NtoPos' n))) (suc (suc n)) = suc (PosToNK n @ i) + in compPath nat (PosToN (NtoPos (suc (suc n)))) (suc (PosToN (NtoPos' n))) (suc (suc n)) (sucPosSuc (NtoPos' n)) ih -NtoPosSuc : (n : nat) -> neg (Id nat zero n) -> Id pos (NtoPos (suc n)) (sucPos (NtoPos n)) = split - zero -> \(p : neg (Id nat zero zero)) -> efq (Id pos (NtoPos (suc zero)) (sucPos (NtoPos zero))) (p (<_> zero)) - suc n -> \(_ : neg (Id nat zero (suc n))) -> <_> (sucPos (NtoPos' n)) +NtoPosSuc : (n : nat) -> neg (Path nat zero n) -> Path pos (NtoPos (suc n)) (sucPos (NtoPos n)) = split + zero -> \(p : neg (Path nat zero zero)) -> efq (Path pos (NtoPos (suc zero)) (sucPos (NtoPos zero))) (p (<_> zero)) + suc n -> \(_ : neg (Path nat zero (suc n))) -> <_> (sucPos (NtoPos' n)) -NtoPosK (p:pos) : Id pos (NtoPos (PosToN p)) p - = posInd (\(p:pos) -> Id pos (NtoPos (PosToN p)) p) (refl pos pos1) hS p +NtoPosK (p:pos) : Path pos (NtoPos (PosToN p)) p + = posInd (\(p:pos) -> Path pos (NtoPos (PosToN p)) p) (refl pos pos1) hS p where - hS (p : pos) (hp: Id pos (NtoPos (PosToN p)) p) : Id pos (NtoPos (PosToN (sucPos p))) (sucPos p) + hS (p : pos) (hp: Path pos (NtoPos (PosToN p)) p) : Path pos (NtoPos (PosToN (sucPos p))) (sucPos p) = - let H1 : Id pos (NtoPos (PosToN (sucPos p))) (NtoPos (suc (PosToN p))) + let H1 : Path pos (NtoPos (PosToN (sucPos p))) (NtoPos (suc (PosToN p))) = mapOnPath nat pos NtoPos (PosToN (sucPos p)) (suc (PosToN p)) (sucPosSuc p) - H2 : Id pos (NtoPos (suc (PosToN p))) (sucPos (NtoPos (PosToN p))) + H2 : Path pos (NtoPos (suc (PosToN p))) (sucPos (NtoPos (PosToN p))) = NtoPosSuc (PosToN p) (zeronPosToN p) - H3 : Id pos (sucPos (NtoPos (PosToN p))) (sucPos p) + H3 : Path pos (sucPos (NtoPos (PosToN p))) (sucPos p) = mapOnPath pos pos sucPos (NtoPos (PosToN p)) p hp - in compId pos (NtoPos (PosToN (sucPos p))) (sucPos (NtoPos (PosToN p))) (sucPos p) - (compId pos (NtoPos (PosToN (sucPos p))) (NtoPos (suc (PosToN p))) (sucPos (NtoPos (PosToN p))) H1 H2) + in compPath pos (NtoPos (PosToN (sucPos p))) (sucPos (NtoPos (PosToN p))) (sucPos p) + (compPath pos (NtoPos (PosToN (sucPos p))) (NtoPos (suc (PosToN p))) (sucPos (NtoPos (PosToN p))) H1 H2) H3 posToNinj : injective pos nat PosToN - = \ (p0 p1:pos) (h:Id nat (PosToN p0) (PosToN p1)) -> + = \ (p0 p1:pos) (h:Path nat (PosToN p0) (PosToN p1)) -> comp (<_>pos) (NtoPos (h@i)) [(i=0) -> NtoPosK p0,(i=1) -> NtoPosK p1] -- Binary natural numbers @@ -94,33 +94,33 @@ BinNtoN : binN -> nat = split binN0 -> zero binNpos p -> PosToN p -NtoBinNK : (n:nat) -> Id nat (BinNtoN (NtoBinN n)) n = split +NtoBinNK : (n:nat) -> Path nat (BinNtoN (NtoBinN n)) n = split zero -> refl nat zero suc n -> PosToNK n -rem (p : pos) : Id binN (NtoBinN (PosToN (sucPos p))) (binNpos (sucPos p)) = - compId binN (NtoBinN (PosToN (sucPos p))) (binNpos (NtoPos (suc (PosToN p)))) +rem (p : pos) : Path binN (NtoBinN (PosToN (sucPos p))) (binNpos (sucPos p)) = + compPath binN (NtoBinN (PosToN (sucPos p))) (binNpos (NtoPos (suc (PosToN p)))) (binNpos (sucPos p)) rem1 rem2 where - rem1 : Id binN (NtoBinN (PosToN (sucPos p))) (binNpos (NtoPos (suc (PosToN p)))) + rem1 : Path binN (NtoBinN (PosToN (sucPos p))) (binNpos (NtoPos (suc (PosToN p)))) = mapOnPath nat binN NtoBinN (PosToN (sucPos p)) (suc (PosToN p)) (sucPosSuc p) - rem2 : Id binN (binNpos (NtoPos (suc (PosToN p)))) (binNpos (sucPos p)) + rem2 : Path binN (binNpos (NtoPos (suc (PosToN p)))) (binNpos (sucPos p)) = binNpos - (compId pos (NtoPos (suc (PosToN p))) (sucPos (NtoPos (PosToN p))) (sucPos p) + (compPath pos (NtoPos (suc (PosToN p))) (sucPos (NtoPos (PosToN p))) (sucPos p) (NtoPosSuc (PosToN p) (zeronPosToN p)) (mapOnPath pos pos sucPos (NtoPos (PosToN p)) p (NtoPosK p))@i) -lem1 (p : pos) : Id binN (NtoBinN (PosToN p)) (binNpos p) - = posInd (\ (p:pos) -> Id binN (NtoBinN (PosToN p)) (binNpos p)) (refl binN (binNpos pos1)) - (\ (p:pos) (_:Id binN (NtoBinN (PosToN p)) (binNpos p)) -> rem p) p +lem1 (p : pos) : Path binN (NtoBinN (PosToN p)) (binNpos p) + = posInd (\ (p:pos) -> Path binN (NtoBinN (PosToN p)) (binNpos p)) (refl binN (binNpos pos1)) + (\ (p:pos) (_:Path binN (NtoBinN (PosToN p)) (binNpos p)) -> rem p) p -BinNtoNK : (b:binN) -> Id binN (NtoBinN (BinNtoN b)) b = split -- retract binN N BinNtoN NtoBinN = split +BinNtoNK : (b:binN) -> Path binN (NtoBinN (BinNtoN b)) b = split -- retract binN N BinNtoN NtoBinN = split binN0 -> refl binN binN0 binNpos p -> lem1 p -IdbinNN : Id U binN nat - = isoId binN nat BinNtoN NtoBinN NtoBinNK BinNtoNK +PathbinNN : Path U binN nat + = isoPath binN nat BinNtoN NtoBinN NtoBinNK BinNtoNK -- Doubling @@ -175,7 +175,7 @@ iter (A : U) : nat -> (A -> A) -> A -> A = split -- 2^10 * e = 2^5 * (2^5 * e) propDouble (D : Double) : U - = Id (carrier D) (iter (carrier D) (doubleN five) (double D) (elt D)) + = Path (carrier D) (iter (carrier D) (doubleN five) (double D) (elt D)) (iter (carrier D) five (double D) (iter (carrier D) five (double D) (elt D))) -- The property we want to prove that takes long to typecheck: @@ -194,29 +194,29 @@ doubleBinN' (b:binN) : binN DoubleBinN' : Double = D binN doubleBinN' (NtoBinN n1024) -eqDouble1 : Id Double DoubleN DoubleBinN' - = elimIsIso nat (\(B : U) (f : nat -> B) (g : B -> nat) -> Id Double DoubleN (D B (\(b : B) -> f (doubleN (g b))) (f n1024))) +eqDouble1 : Path Double DoubleN DoubleBinN' + = elimIsIso nat (\(B : U) (f : nat -> B) (g : B -> nat) -> Path Double DoubleN (D B (\(b : B) -> f (doubleN (g b))) (f n1024))) (refl Double DoubleN) binN NtoBinN BinNtoN BinNtoNK NtoBinNK -eqDouble2 : Id Double DoubleBinN' DoubleBinN +eqDouble2 : Path Double DoubleBinN' DoubleBinN = mapOnPath (binN -> binN) Double F doubleBinN' doubleBinN rem where F (d:binN -> binN) : Double = D binN d (NtoBinN n1024) - rem : Id (binN -> binN) doubleBinN' doubleBinN + rem : Path (binN -> binN) doubleBinN' doubleBinN = funExt binN (\(x:binN) -> binN) doubleBinN' doubleBinN rem1 where - rem1 : (n : binN) -> Id binN (doubleBinN' n) (doubleBinN n) + rem1 : (n : binN) -> Path binN (doubleBinN' n) (doubleBinN n) = split binN0 -> refl binN binN0 binNpos p -> - let p1 : Id binN (NtoBinN (doubleN (PosToN p))) (NtoBinN (PosToN (x0 p))) + let p1 : Path binN (NtoBinN (doubleN (PosToN p))) (NtoBinN (PosToN (x0 p))) = mapOnPath nat binN NtoBinN (doubleN (PosToN p)) (PosToN (x0 p)) (refl nat (doubleN (PosToN p))) - in compId binN (NtoBinN (doubleN (PosToN p))) (NtoBinN (PosToN (x0 p))) (binNpos (x0 p)) p1 (lem1 (x0 p)) + in compPath binN (NtoBinN (doubleN (PosToN p))) (NtoBinN (PosToN (x0 p))) (binNpos (x0 p)) p1 (lem1 (x0 p)) -eqDouble : Id Double DoubleN DoubleBinN - = compId Double DoubleN DoubleBinN' DoubleBinN eqDouble1 eqDouble2 +eqDouble : Path Double DoubleN DoubleBinN + = compPath Double DoubleN DoubleBinN' DoubleBinN eqDouble1 eqDouble2 -- opaque doubleN diff --git a/examples/bool.ctt b/examples/bool.ctt index 5edda04..eb159f9 100644 --- a/examples/bool.ctt +++ b/examples/bool.ctt @@ -10,55 +10,55 @@ caseBool (A : U) (f t : A) : bool -> A = split false -> f true -> t -falseNeqTrue : neg (Id bool false true) = - \(h : Id bool false true) -> subst bool (caseBool U bool N0) false true h false +falseNeqTrue : neg (Path bool false true) = + \(h : Path bool false true) -> subst bool (caseBool U bool N0) false true h false -trueNeqFalse : neg (Id bool true false) = - \(h : Id bool true false) -> subst bool (caseBool U N0 bool) true false h true +trueNeqFalse : neg (Path bool true false) = + \(h : Path bool true false) -> subst bool (caseBool U N0 bool) true false h true -boolDec : (b1 b2 : bool) -> dec (Id bool b1 b2) = split +boolDec : (b1 b2 : bool) -> dec (Path bool b1 b2) = split false -> rem where - rem : (b : bool) -> dec (Id bool false b) = split + rem : (b : bool) -> dec (Path bool false b) = split false -> inl ( false) true -> inr falseNeqTrue true -> rem where - rem : (b : bool) -> dec (Id bool true b) = split + rem : (b : bool) -> dec (Path bool true b) = split false -> inr trueNeqFalse true -> inl ( true) setbool' : set bool = hedberg bool boolDec -- Direct proof that bool is a set: -lem1 : (y:bool) (p:Id bool true y) -> Id bool true y = split - false -> \ (p : Id bool true false) -> p - true -> \ (p : Id bool true true) -> true +lem1 : (y:bool) (p:Path bool true y) -> Path bool true y = split + false -> \ (p : Path bool true false) -> p + true -> \ (p : Path bool true true) -> true -lem2 : (x y :bool) (p:Id bool true x) (q:Id bool true y) -> Id bool x y = split - false -> \ (y:bool) (p:Id bool true false) (q:Id bool true y) -> efq (Id bool false y) (trueNeqFalse p) - true -> \ (y:bool) (p:Id bool true true) (q:Id bool true y) -> lem1 y q +lem2 : (x y :bool) (p:Path bool true x) (q:Path bool true y) -> Path bool x y = split + false -> \ (y:bool) (p:Path bool true false) (q:Path bool true y) -> efq (Path bool false y) (trueNeqFalse p) + true -> \ (y:bool) (p:Path bool true true) (q:Path bool true y) -> lem1 y q -lem3 : prop (Id bool true true) = - \ (p q:Id bool true true) -> lem2 (p@i) (q@i) (p@i/\k) (q@i/\k) @ j +lem3 : prop (Path bool true true) = + \ (p q:Path bool true true) -> lem2 (p@i) (q@i) (p@i/\k) (q@i/\k) @ j -lem4 : (y:bool) (p:Id bool false y) -> Id bool false y = split - false -> \ (p : Id bool false false) -> false - true -> \ (p : Id bool false true) -> p +lem4 : (y:bool) (p:Path bool false y) -> Path bool false y = split + false -> \ (p : Path bool false false) -> false + true -> \ (p : Path bool false true) -> p -lem5 : (x y :bool) (p:Id bool false x) (q:Id bool false y) -> Id bool x y = split - false -> \ (y:bool) (p:Id bool false false) (q:Id bool false y) -> lem4 y q - true -> \ (y:bool) (p:Id bool false true) (q:Id bool false y) -> efq (Id bool true y) (falseNeqTrue p) +lem5 : (x y :bool) (p:Path bool false x) (q:Path bool false y) -> Path bool x y = split + false -> \ (y:bool) (p:Path bool false false) (q:Path bool false y) -> lem4 y q + true -> \ (y:bool) (p:Path bool false true) (q:Path bool false y) -> efq (Path bool true y) (falseNeqTrue p) -lem6 : prop (Id bool false false) = - \ (p q:Id bool false false) -> lem5 (p@i) (q@i) (p@i/\k) (q@i/\k) @ j +lem6 : prop (Path bool false false) = + \ (p q:Path bool false false) -> lem5 (p@i) (q@i) (p@i/\k) (q@i/\k) @ j -lem7 : (y:bool) (p:Id bool false y) (q:Id bool false y) -> Id (Id bool false y) p q = split +lem7 : (y:bool) (p:Path bool false y) (q:Path bool false y) -> Path (Path bool false y) p q = split false -> lem6 - true -> \ (p:Id bool false true) (q:Id bool false true) -> efq (Id (Id bool false true) p q) (falseNeqTrue p) + true -> \ (p:Path bool false true) (q:Path bool false true) -> efq (Path (Path bool false true) p q) (falseNeqTrue p) -lem8 : (y:bool) (p:Id bool true y) (q:Id bool true y) -> Id (Id bool true y) p q = split - false -> \ (p:Id bool true false) (q:Id bool true false) -> efq (Id (Id bool true false) p q) (trueNeqFalse p) +lem8 : (y:bool) (p:Path bool true y) (q:Path bool true y) -> Path (Path bool true y) p q = split + false -> \ (p:Path bool true false) (q:Path bool true false) -> efq (Path (Path bool true false) p q) (trueNeqFalse p) true -> lem3 setbool : set bool = split @@ -75,13 +75,13 @@ negBool : bool -> bool = split true -> false -- negBool is involutive: -negBoolK : (b : bool) -> Id bool (negBool (negBool b)) b = split +negBoolK : (b : bool) -> Path bool (negBool (negBool b)) b = split false -> refl bool false true -> refl bool true -- This defines a non-trivial equality between bool and bool: -negBoolEq : Id U bool bool = - isoId bool bool negBool negBool negBoolK negBoolK +negBoolEq : Path U bool bool = + isoPath bool bool negBool negBool negBoolK negBoolK -- We can transport true along this non-trivial equality: testFalse : bool = transport negBoolEq true @@ -100,93 +100,93 @@ boolToF2 : bool -> F2 = split false -> zeroF2 true -> oneF2 -f2ToBoolK : (x : F2) -> Id F2 (boolToF2 (f2ToBool x)) x = split +f2ToBoolK : (x : F2) -> Path F2 (boolToF2 (f2ToBool x)) x = split zeroF2 -> refl F2 zeroF2 oneF2 -> refl F2 oneF2 -boolToF2K : (b : bool) -> Id bool (f2ToBool (boolToF2 b)) b = split +boolToF2K : (b : bool) -> Path bool (f2ToBool (boolToF2 b)) b = split false -> refl bool false true -> refl bool true -boolEqF2 : Id U bool F2 = - isoId bool F2 boolToF2 f2ToBool f2ToBoolK boolToF2K +boolEqF2 : Path U bool F2 = + isoPath bool F2 boolToF2 f2ToBool f2ToBoolK boolToF2K negF2 : F2 -> F2 = subst U (\(X : U) -> (X -> X)) bool F2 boolEqF2 negBool --- lemTest (A : U) : (B : U) (p : Id U A B) (a : A) -> IdP p a (transport p a) = --- J U A (\(B : U) (p : Id U A B) -> (a : A) -> IdP p a (transport p a)) (refl A) +-- lemTest (A : U) : (B : U) (p : Path U A B) (a : A) -> PathP p a (transport p a) = +-- J U A (\(B : U) (p : Path U A B) -> (a : A) -> PathP p a (transport p a)) (refl A) --- test : IdP boolEqF2 true oneF2 = glueElem oneF2 [ (i = 0) -> true ] --- test1 : IdP boolEqF2 true oneF2 = lemTest bool F2 boolEqF2 true +-- test : PathP boolEqF2 true oneF2 = glueElem oneF2 [ (i = 0) -> true ] +-- test1 : PathP boolEqF2 true oneF2 = lemTest bool F2 boolEqF2 true -F2EqBool : Id U F2 bool = inv U bool F2 boolEqF2 +F2EqBool : Path U F2 bool = inv U bool F2 boolEqF2 negBool' : bool -> bool = subst U (\(X : U) -> (X -> X)) F2 bool F2EqBool negF2 -F2EqBoolComp : Id U F2 bool = - compId U F2 bool bool F2EqBool negBoolEq +F2EqBoolComp : Path U F2 bool = + compPath U F2 bool bool F2EqBool negBoolEq test2 : bool = trans F2 bool F2EqBoolComp oneF2 -negNegEq : Id U bool bool = - compId U bool bool bool negBoolEq negBoolEq +negNegEq : Path U bool bool = + compPath U bool bool bool negBoolEq negBoolEq test3 : bool = trans bool bool negNegEq true -test4 : Id U bool bool = negNegEq @ i +test4 : Path U bool bool = negNegEq @ i -kanBool : Id U bool bool = +kanBool : Path U bool bool = kan U bool bool bool bool negBoolEq negBoolEq negBoolEq squareBoolF2 : Square U bool bool bool F2 (refl U bool) boolEqF2 (refl U bool) boolEqF2 = boolEqF2 @ i /\ j -test5 : IdP boolEqF2 true oneF2 = +test5 : PathP boolEqF2 true oneF2 = comp ( boolEqF2 @ i /\ j) true [] -test6 : Id bool true true = +test6 : Path bool true true = comp ( F2EqBool @ i \/ j) (test5 @ - i) [] -test7 : Id U F2 F2 = - subst U (\(X:U) -> Id U X X) bool F2 boolEqF2 negNegEq +test7 : Path U F2 F2 = + subst U (\(X:U) -> Path U X X) bool F2 boolEqF2 negNegEq -test8 : Id U F2 F2 = - subst U (\(X:U) -> Id U X X) bool F2 boolEqF2 (refl U bool) +test8 : Path U F2 F2 = + subst U (\(X:U) -> Path U X X) bool F2 boolEqF2 (refl U bool) -test9 : Id U F2 F2 = - comp ( Id U (boolEqF2 @ i) (boolEqF2 @ i)) (refl U bool) [] +test9 : Path U F2 F2 = + comp ( Path U (boolEqF2 @ i) (boolEqF2 @ i)) (refl U bool) [] -p : Id U F2 bool = comp (<_> U) bool [ (i = 0) -> boolEqF2, (i = 1) -> <_> bool] -q : Id U F2 F2 = p @ (i /\ - i) +p : Path U F2 bool = comp (<_> U) bool [ (i = 0) -> boolEqF2, (i = 1) -> <_> bool] +q : Path U F2 F2 = p @ (i /\ - i) -- A small test of univalence using boolean negation isEquivNegBool : isEquiv bool bool negBool = gradLemma bool bool negBool negBool negBoolK negBoolK -eqBool : Id U bool bool = - trans (equiv bool bool) (Id U bool bool) +eqBool : Path U bool bool = + trans (equiv bool bool) (Path U bool bool) ( corrUniv bool bool @ -i) (negBool,isEquivNegBool) testf : bool = trans bool bool eqBool true testt : bool = trans bool bool eqBool false -- Some tests of normal forms of univalence: -testUniv1 (A : U) : Id U A A = - trans (equiv A A) (Id U A A) ( corrUniv A A @ -i) (idEquiv A) +testUniv1 (A : U) : Path U A A = + trans (equiv A A) (Path U A A) ( corrUniv A A @ -i) (idEquiv A) -- obtained by normal form -ntestUniv1 (A:U) : Id U A A = +ntestUniv1 (A:U) : Path U A A = comp (<_>U) (comp (<_>U) A - [ (i = 0) -> comp (<_>U) A [ (l = 0) -> glue A [ (i = 0) -> (A,(\(x : A) -> x,\(a : A) -> ((a, a),\(z : ((x : A) * IdP ( A) a x)) -> (z.2 @ l, z.2 @ (l /\ i))))), (i = 1) -> (A,(comp ( A -> A) (\(x : A) -> x) [],comp ( (y : A) -> - ((x : ((x : A) * IdP ( A) y (comp ( A) (comp ( A) x []) [ (l = 0) -> comp ( A) x [ (j = 1) -> x ] ]))) * (y0 : ((x0 : A) * IdP ( A) y (comp ( A) (comp ( A) x0 []) [ (l = 0) -> comp ( A) x0 [ (j = 1) -> x0 ] ]))) -> IdP ( ((x0 : A) * IdP ( A) y (comp ( A) (comp ( A) x0 []) [ (l = 0) -> comp ( A) x0 [ (j = 1) -> x0 ] ]))) x y0)) (\(a : A) -> ((a, a),\(z : ((x : A) * IdP ( A) a x)) -> (z.2 @ l, z.2 @ (l /\ i)))) [])) ], (l = 1) -> glue A [ (i = 0) -> (A,(\(x : A) -> x,\(a : A) -> ((a, a),\(z : ((x : A) * IdP ( A) a x)) -> (z.2 @ l, z.2 @ (l /\ i))))), (i = 1) -> (A,(comp ( A -> A) (\(x : A) -> x) [],comp ( (y : A) -> - ((x : ((x : A) * IdP ( A) y (comp ( A) (comp ( A) x []) [ (l = 0) -> comp ( A) x [ (j = 1) -> x ] ]))) * (y0 : ((x0 : A) * IdP ( A) y (comp ( A) (comp ( A) x0 []) [ (l = 0) -> comp ( A) x0 [ (j = 1) -> x0 ] ]))) -> IdP ( ((x0 : A) * IdP ( A) y (comp ( A) (comp ( A) x0 []) [ (l = 0) -> comp ( A) x0 [ (j = 1) -> x0 ] ]))) x y0)) (\(a : A) -> ((a, a),\(z : ((x : A) * IdP ( A) a x)) -> (z.2 @ l, z.2 @ (l /\ i)))) [])) ] ], (i = 1) -> A ]) [ (i = 0) -> A, (i = 1) -> A ] + [ (i = 0) -> comp (<_>U) A [ (l = 0) -> glue A [ (i = 0) -> (A,(\(x : A) -> x,\(a : A) -> ((a, a),\(z : ((x : A) * PathP ( A) a x)) -> (z.2 @ l, z.2 @ (l /\ i))))), (i = 1) -> (A,(comp ( A -> A) (\(x : A) -> x) [],comp ( (y : A) -> + ((x : ((x : A) * PathP ( A) y (comp ( A) (comp ( A) x []) [ (l = 0) -> comp ( A) x [ (j = 1) -> x ] ]))) * (y0 : ((x0 : A) * PathP ( A) y (comp ( A) (comp ( A) x0 []) [ (l = 0) -> comp ( A) x0 [ (j = 1) -> x0 ] ]))) -> PathP ( ((x0 : A) * PathP ( A) y (comp ( A) (comp ( A) x0 []) [ (l = 0) -> comp ( A) x0 [ (j = 1) -> x0 ] ]))) x y0)) (\(a : A) -> ((a, a),\(z : ((x : A) * PathP ( A) a x)) -> (z.2 @ l, z.2 @ (l /\ i)))) [])) ], (l = 1) -> glue A [ (i = 0) -> (A,(\(x : A) -> x,\(a : A) -> ((a, a),\(z : ((x : A) * PathP ( A) a x)) -> (z.2 @ l, z.2 @ (l /\ i))))), (i = 1) -> (A,(comp ( A -> A) (\(x : A) -> x) [],comp ( (y : A) -> + ((x : ((x : A) * PathP ( A) y (comp ( A) (comp ( A) x []) [ (l = 0) -> comp ( A) x [ (j = 1) -> x ] ]))) * (y0 : ((x0 : A) * PathP ( A) y (comp ( A) (comp ( A) x0 []) [ (l = 0) -> comp ( A) x0 [ (j = 1) -> x0 ] ]))) -> PathP ( ((x0 : A) * PathP ( A) y (comp ( A) (comp ( A) x0 []) [ (l = 0) -> comp ( A) x0 [ (j = 1) -> x0 ] ]))) x y0)) (\(a : A) -> ((a, a),\(z : ((x : A) * PathP ( A) a x)) -> (z.2 @ l, z.2 @ (l /\ i)))) [])) ] ], (i = 1) -> A ]) [ (i = 0) -> A, (i = 1) -> A ] testUniv2 : bool = trans bool bool (ntestUniv1 bool) true ntestUniv2 : bool = - comp ( comp (<_>U) (comp (<_>U) bool [ (i = 0) -> comp (<_>U) bool [ (j = 0) -> glue bool [ (i = 0) -> (bool,(\(x : bool) -> x,\(a : bool) -> ((a, a),\(z : ((x : bool) * IdP ( bool) a x)) -> (z.2 @ i, z.2 @ (i /\ j))))), (i = 1) -> (bool,(comp ( bool -> bool) (\(x : bool) -> x) [],comp ( (y : bool) -> - ((x : ((x : bool) * IdP ( bool) y (comp ( bool) (comp ( bool) x []) [ (i = 0) -> comp ( bool) x [ (j = 1) -> x ] ]))) * (y0 : ((x0 : bool) * IdP ( bool) y (comp ( bool) (comp ( bool) x0 []) [ (i = 0) -> comp ( bool) x0 [ (j = 1) -> x0 ] ]))) -> IdP ( ((x0 : bool) * IdP ( bool) y (comp ( bool) (comp ( bool) x0 []) [ (i = 0) -> comp ( bool) x0 [ (j = 1) -> x0 ] ]))) x y0)) (\(a : bool) -> ((a, a),\(z : ((x : bool) * IdP ( bool) a x)) -> (z.2 @ i, z.2 @ (i /\ j)))) [])) ], (j = 1) -> glue bool [ (i = 0) -> (bool,(\(x : bool) -> x,\(a : bool) -> ((a, a),\(z : ((x : bool) * IdP ( bool) a x)) -> (z.2 @ i, z.2 @ (i /\ j))))), (i = 1) -> (bool,(comp ( bool -> bool) (\(x : bool) -> x) [],comp ( (y : bool) -> - ((x : ((x : bool) * IdP ( bool) y (comp ( bool) (comp ( bool) x []) [ (i = 0) -> comp ( bool) x [ (j = 1) -> x ] ]))) * (y0 : ((x0 : bool) * IdP ( bool) y (comp ( bool) (comp ( bool) x0 []) [ (i = 0) -> comp ( bool) x0 [ (j = 1) -> x0 ] ]))) -> IdP ( ((x0 : bool) * IdP ( bool) y (comp ( bool) (comp ( bool) x0 []) [ (i = 0) -> comp ( bool) x0 [ (j = 1) -> x0 ] ]))) x y0)) (\(a : bool) -> ((a, a),\(z : ((x : bool) * IdP ( bool) a x)) -> (z.2 @ i, z.2 @ (i /\ j)))) [])) ] ], (i = 1) -> bool ]) [ (i = 0) -> bool, (i = 1) -> bool ]) true [] + comp ( comp (<_>U) (comp (<_>U) bool [ (i = 0) -> comp (<_>U) bool [ (j = 0) -> glue bool [ (i = 0) -> (bool,(\(x : bool) -> x,\(a : bool) -> ((a, a),\(z : ((x : bool) * PathP ( bool) a x)) -> (z.2 @ i, z.2 @ (i /\ j))))), (i = 1) -> (bool,(comp ( bool -> bool) (\(x : bool) -> x) [],comp ( (y : bool) -> + ((x : ((x : bool) * PathP ( bool) y (comp ( bool) (comp ( bool) x []) [ (i = 0) -> comp ( bool) x [ (j = 1) -> x ] ]))) * (y0 : ((x0 : bool) * PathP ( bool) y (comp ( bool) (comp ( bool) x0 []) [ (i = 0) -> comp ( bool) x0 [ (j = 1) -> x0 ] ]))) -> PathP ( ((x0 : bool) * PathP ( bool) y (comp ( bool) (comp ( bool) x0 []) [ (i = 0) -> comp ( bool) x0 [ (j = 1) -> x0 ] ]))) x y0)) (\(a : bool) -> ((a, a),\(z : ((x : bool) * PathP ( bool) a x)) -> (z.2 @ i, z.2 @ (i /\ j)))) [])) ], (j = 1) -> glue bool [ (i = 0) -> (bool,(\(x : bool) -> x,\(a : bool) -> ((a, a),\(z : ((x : bool) * PathP ( bool) a x)) -> (z.2 @ i, z.2 @ (i /\ j))))), (i = 1) -> (bool,(comp ( bool -> bool) (\(x : bool) -> x) [],comp ( (y : bool) -> + ((x : ((x : bool) * PathP ( bool) y (comp ( bool) (comp ( bool) x []) [ (i = 0) -> comp ( bool) x [ (j = 1) -> x ] ]))) * (y0 : ((x0 : bool) * PathP ( bool) y (comp ( bool) (comp ( bool) x0 []) [ (i = 0) -> comp ( bool) x0 [ (j = 1) -> x0 ] ]))) -> PathP ( ((x0 : bool) * PathP ( bool) y (comp ( bool) (comp ( bool) x0 []) [ (i = 0) -> comp ( bool) x0 [ (j = 1) -> x0 ] ]))) x y0)) (\(a : bool) -> ((a, a),\(z : ((x : bool) * PathP ( bool) a x)) -> (z.2 @ i, z.2 @ (i /\ j)))) [])) ] ], (i = 1) -> bool ]) [ (i = 0) -> bool, (i = 1) -> bool ]) true [] diff --git a/examples/category.ctt b/examples/category.ctt index 0bf48de..b1e4d23 100644 --- a/examples/category.ctt +++ b/examples/category.ctt @@ -5,57 +5,57 @@ import equiv import nat import univalence -lemReflComp (A : U) (a b : A) (p : Id A a b) : Id (Id A a b) (compId A a a b (<_> a) p) p = +lemReflComp (A : U) (a b : A) (p : Path A a b) : Path (Path A a b) (compPath A a a b (<_> a) p) p = comp ( A) (p @ i /\ j) [(i=0) -> <_> a, (j=1) -> <_> p @ i, (i=1) -> p @ k \/ j ] opaque lemReflComp -lemReflComp' (A : U) (a b : A) (p : Id A a b) : Id (Id A a b) (compId A a b b p (<_> b)) p = +lemReflComp' (A : U) (a b : A) (p : Path A a b) : Path (Path A a b) (compPath A a b b p (<_> b)) p = comp ( A) (p @ i) [(i=0) -> <_> a, (j=1) -> <_> p @ i, (i=1) -> <_> b ] opaque lemReflComp' setPi (A : U) (B : A -> U) (h : (x : A) -> set (B x)) (f0 f1 : (x : A) -> B x) - (p1 p2 : Id ((x : A) -> B x) f0 f1) - : Id (Id ((x : A) -> B x) f0 f1) p1 p2 + (p1 p2 : Path ((x : A) -> B x) f0 f1) + : Path (Path ((x : A) -> B x) f0 f1) p1 p2 = \(x : A) -> (h x (f0 x) (f1 x) ( (p1@i) x) ( (p2@i) x)) @ i @ j opaque setPi -lemIdPProp (A B : U) (AProp : prop A) (p : Id U A B) : (x : A) -> (y : B) -> IdP p x y - = J U A (\(B : U) -> \(p : Id U A B) -> (x : A) -> (y : B) -> IdP p x y) AProp B p -opaque lemIdPProp +lemPathPProp (A B : U) (AProp : prop A) (p : Path U A B) : (x : A) -> (y : B) -> PathP p x y + = J U A (\(B : U) -> \(p : Path U A B) -> (x : A) -> (y : B) -> PathP p x y) AProp B p +opaque lemPathPProp -lemIdPSet (A B : U) (ASet : set A) (p : Id U A B) : (x : A) (y : B) (s t : IdP p x y) -> Id (IdP p x y) s t - = J U A (\(B : U) -> \(p : Id U A B) -> (x : A) (y : B) (s t : IdP p x y) -> Id (IdP p x y) s t) ASet B p -opaque lemIdPSet +lemPathPSet (A B : U) (ASet : set A) (p : Path U A B) : (x : A) (y : B) (s t : PathP p x y) -> Path (PathP p x y) s t + = J U A (\(B : U) -> \(p : Path U A B) -> (x : A) (y : B) (s t : PathP p x y) -> Path (PathP p x y) s t) ASet B p +opaque lemPathPSet -lemIdPSet2 (A B : U) (ASet : set A) (p1 : Id U A B) - : (p2 : Id U A B) -> (p : Id (Id U A B) p1 p2) -> - (x : A) -> (y : B) -> (s : IdP p1 x y) -> (t : IdP p2 x y) -> IdP ( (IdP (p @ i) x y)) s t - = J (Id U A B) p1 (\(p2 : Id U A B) -> \(p : Id (Id U A B) p1 p2) -> (x : A) -> (y : B) -> (s : IdP p1 x y) -> (t : IdP p2 x y) -> IdP ( (IdP (p @ i) x y)) s t) - (lemIdPSet A B ASet p1) -opaque lemIdPSet2 +lemPathPSet2 (A B : U) (ASet : set A) (p1 : Path U A B) + : (p2 : Path U A B) -> (p : Path (Path U A B) p1 p2) -> + (x : A) -> (y : B) -> (s : PathP p1 x y) -> (t : PathP p2 x y) -> PathP ( (PathP (p @ i) x y)) s t + = J (Path U A B) p1 (\(p2 : Path U A B) -> \(p : Path (Path U A B) p1 p2) -> (x : A) -> (y : B) -> (s : PathP p1 x y) -> (t : PathP p2 x y) -> PathP ( (PathP (p @ i) x y)) s t) + (lemPathPSet A B ASet p1) +opaque lemPathPSet2 -substIdP (A B : U) (p : Id U A B) (x : A) (y : B) (q : Id B (transport p x) y) : IdP p x y - = transport ( IdP p x (q@i)) hole +substPathP (A B : U) (p : Path U A B) (x : A) (y : B) (q : Path B (transport p x) y) : PathP p x y + = transport ( PathP p x (q@i)) hole where - hole : IdP p x (transport p x) = comp ( p @ (i /\ j)) x [(i=0) -> <_> x] -opaque substIdP + hole : PathP p x (transport p x) = comp ( p @ (i /\ j)) x [(i=0) -> <_> x] +opaque substPathP -transRefl (A : U) (a : A) : Id A (transport (<_> A) a) a = comp (<_> A) a [(i=1) -> <_>a] +transRefl (A : U) (a : A) : Path A (transport (<_> A) a) a = comp (<_> A) a [(i=1) -> <_>a] opaque transRefl -isContrProp (A : U) (p : isContr A) (x y : A) : Id A x y = compId A x p.1 y ( p.2 x @ -i) (p.2 y) +isContrProp (A : U) (p : isContr A) (x y : A) : Path A x y = compPath A x p.1 y ( p.2 x @ -i) (p.2 y) opaque isContrProp equivProp (A B : U) (AProp : prop A) (BProp : prop B) (F : A -> B) (G : B -> A) : equiv A B = (F, gradLemma A B F G (\(y : B) -> BProp (F (G y)) y) (\(x : A) -> AProp (G (F x)) x)) opaque equivProp -idProp (A B : U) (AProp : prop A) (BProp : prop B) (F : A -> B) (G : B -> A) : Id U A B - = equivId A B (equivProp A B AProp BProp F G).1 (equivProp A B AProp BProp F G).2 +idProp (A B : U) (AProp : prop A) (BProp : prop B) (F : A -> B) (G : B -> A) : Path U A B + = equivPath A B (equivProp A B AProp BProp F G).1 (equivProp A B AProp BProp F G).2 opaque idProp -lemTransEquiv (A B:U) (e:Id U A B) (x:A) : Id B (transport e x) ((transport ( equiv (e@-i) B) (idEquiv B)).1 x) +lemTransEquiv (A B:U) (e:Path U A B) (x:A) : Path B (transport e x) ((transport ( equiv (e@-i) B) (idEquiv B)).1 x) = transRefl B (transport e x)@-i opaque lemTransEquiv @@ -67,9 +67,9 @@ isPrecategory2 (C : categoryData) (id : (x : C.1) -> C.2 x x) (c : (x y z : C.1) = let A : U = C.1 hom : A -> A -> U = C.2 in (homSet : (x y : A) -> set (hom x y)) - * (cIdL : (x y : A) -> (f : hom x y) -> Id (hom x y) (c x x y (id x) f) f) - * (cIdR : (x y : A) -> (f : hom x y) -> Id (hom x y) (c x y y f (id y)) f) - * ( (x y z w : A) -> (f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Id (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) + * (cPathL : (x y : A) -> (f : hom x y) -> Path (hom x y) (c x x y (id x) f) f) + * (cPathR : (x y : A) -> (f : hom x y) -> Path (hom x y) (c x y y f (id y)) f) + * ( (x y z w : A) -> (f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Path (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) propIsPrecategory2 (C : categoryData) (id : (x : C.1) -> C.2 x x) (c : (x y z : C.1) -> C.2 x y -> C.2 y z -> C.2 x z) : prop (isPrecategory2 C id c) @@ -77,34 +77,34 @@ propIsPrecategory2 (C : categoryData) (id : (x : C.1) -> C.2 x x) (c : (x y z : propSig ((x y : A) -> set (hom x y)) (\(_:(x y : A) -> set (hom x y))-> - ((cIdL : (x y : A) -> (f : hom x y) -> Id (hom x y) (c x x y (id x) f) f) - * (cIdR : (x y : A) -> (f : hom x y) -> Id (hom x y) (c x y y f (id y)) f) - * ( (x y z w : A) -> (f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Id (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))))) + ((cPathL : (x y : A) -> (f : hom x y) -> Path (hom x y) (c x x y (id x) f) f) + * (cPathR : (x y : A) -> (f : hom x y) -> Path (hom x y) (c x y y f (id y)) f) + * ( (x y z w : A) -> (f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Path (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))))) (propPi A (\(x:A)->(y : A) -> set (hom x y)) (\(x:A)->propPi A (\(y : A) -> set (hom x y)) (\(y:A)->setIsProp (hom x y)))) (\(hset:(x y : A) -> set (hom x y))-> (propAnd - ((x y : A) -> (f : hom x y) -> Id (hom x y) (c x x y (id x) f) f) - ((cIdR : (x y : A) -> (f : hom x y) -> Id (hom x y) (c x y y f (id y)) f) - * ( (x y z w : A) -> (f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Id (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h)))) - (propPi A (\(x:A)->(y : A) -> (f : hom x y) -> Id (hom x y) (c x x y (id x) f) f) - (\(x:A)->propPi A (\(y : A) -> (f : hom x y) -> Id (hom x y) (c x x y (id x) f) f) - (\(y:A)->propPi (hom x y) (\(f : hom x y) -> Id (hom x y) (c x x y (id x) f) f) + ((x y : A) -> (f : hom x y) -> Path (hom x y) (c x x y (id x) f) f) + ((cPathR : (x y : A) -> (f : hom x y) -> Path (hom x y) (c x y y f (id y)) f) + * ( (x y z w : A) -> (f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Path (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h)))) + (propPi A (\(x:A)->(y : A) -> (f : hom x y) -> Path (hom x y) (c x x y (id x) f) f) + (\(x:A)->propPi A (\(y : A) -> (f : hom x y) -> Path (hom x y) (c x x y (id x) f) f) + (\(y:A)->propPi (hom x y) (\(f : hom x y) -> Path (hom x y) (c x x y (id x) f) f) (\(f:hom x y)->hset x y (c x x y (id x) f) f)))) (propAnd - ((x y : A) -> (f : hom x y) -> Id (hom x y) (c x y y f (id y)) f) - ((x y z w : A) -> (f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Id (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) - (propPi A (\(x:A)->(y : A) -> (f : hom x y) -> Id (hom x y) (c x y y f (id y)) f) - (\(x:A)->propPi A (\(y : A) -> (f : hom x y) -> Id (hom x y) (c x y y f (id y)) f) - (\(y:A)->propPi (hom x y) (\(f : hom x y) -> Id (hom x y) (c x y y f (id y)) f) + ((x y : A) -> (f : hom x y) -> Path (hom x y) (c x y y f (id y)) f) + ((x y z w : A) -> (f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Path (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) + (propPi A (\(x:A)->(y : A) -> (f : hom x y) -> Path (hom x y) (c x y y f (id y)) f) + (\(x:A)->propPi A (\(y : A) -> (f : hom x y) -> Path (hom x y) (c x y y f (id y)) f) + (\(y:A)->propPi (hom x y) (\(f : hom x y) -> Path (hom x y) (c x y y f (id y)) f) (\(f:hom x y)->hset x y (c x y y f (id y)) f)))) - (propPi A (\(x:A)->(y z w : A) -> (f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Id (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) - (\(x:A)->propPi A (\(y:A)->(z w : A) -> (f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Id (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) - (\(y:A)->propPi A (\(z:A)->(w : A) -> (f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Id (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) - (\(z:A)->propPi A (\(w:A)-> (f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Id (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) - (\(w:A)->propPi (hom x y) (\(f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Id (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) - (\(f:hom x y)->propPi (hom y z) (\(g : hom y z) -> (h : hom z w) -> Id (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) - (\(g:hom y z)->propPi (hom z w) (\(h : hom z w) -> Id (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) + (propPi A (\(x:A)->(y z w : A) -> (f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Path (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) + (\(x:A)->propPi A (\(y:A)->(z w : A) -> (f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Path (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) + (\(y:A)->propPi A (\(z:A)->(w : A) -> (f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Path (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) + (\(z:A)->propPi A (\(w:A)-> (f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Path (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) + (\(w:A)->propPi (hom x y) (\(f : hom x y) -> (g : hom y z) -> (h : hom z w) -> Path (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) + (\(f:hom x y)->propPi (hom y z) (\(g : hom y z) -> (h : hom z w) -> Path (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) + (\(g:hom y z)->propPi (hom z w) (\(h : hom z w) -> Path (hom x w) (c x z w (c x y z f g) h) (c x y w f (c y z w g h))) (\(h:hom z w)->hset x w (c x z w (c x y z f g) h) (c x y w f (c y z w g h))))))))))))) isPrecategory (C : categoryData) : U = @@ -118,34 +118,34 @@ cA (C : precategory) : U = C.1.1 cH (C : precategory) (a b : cA C) : U = C.1.2 a b cHSet (C : precategory) (a b : cA C) : set (cH C a b) = C.2.2.2.1 a b cC (C : precategory) (x y z : cA C) (f : cH C x y) (g : cH C y z) : cH C x z = C.2.2.1 x y z f g -cId (C : precategory) (x : cA C) : cH C x x = C.2.1 x -cIdL (C : precategory) (x y : cA C) (f : cH C x y) - : Id (cH C x y) (cC C x x y (cId C x) f) f = C.2.2.2.2.1 x y f -cIdR (C : precategory) (x y : cA C) (f : cH C x y) - : Id (cH C x y) (cC C x y y f (cId C y)) f = C.2.2.2.2.2.1 x y f -cIdC (C : precategory) (x y z w : cA C) (f : cH C x y) (g : cH C y z) (h : cH C z w) - : Id (cH C x w) (cC C x z w (cC C x y z f g) h) (cC C x y w f (cC C y z w g h)) = C.2.2.2.2.2.2 x y z w f g h - -catIso3 (A B : precategory) : U = (e1 : Id U (cA A) (cA B)) - * (e2 : IdP ((x y : e1@i)->U) (cH A) (cH B)) - * (_ : IdP ( (x:e1@i)->(e2@i) x x) (cId A) (cId B)) - * (IdP ( (x y z:e1@i)->(e2@i) x y->(e2@i) y z->(e2@i) x z) (cC A) (cC B)) - -eCatIso3 (A B : precategory) : Id U (Id precategory A B) (catIso3 A B) - = isoId (Id precategory A B) (catIso3 A B) +cPath (C : precategory) (x : cA C) : cH C x x = C.2.1 x +cPathL (C : precategory) (x y : cA C) (f : cH C x y) + : Path (cH C x y) (cC C x x y (cPath C x) f) f = C.2.2.2.2.1 x y f +cPathR (C : precategory) (x y : cA C) (f : cH C x y) + : Path (cH C x y) (cC C x y y f (cPath C y)) f = C.2.2.2.2.2.1 x y f +cPathC (C : precategory) (x y z w : cA C) (f : cH C x y) (g : cH C y z) (h : cH C z w) + : Path (cH C x w) (cC C x z w (cC C x y z f g) h) (cC C x y w f (cC C y z w g h)) = C.2.2.2.2.2.2 x y z w f g h + +catIso3 (A B : precategory) : U = (e1 : Path U (cA A) (cA B)) + * (e2 : PathP ((x y : e1@i)->U) (cH A) (cH B)) + * (_ : PathP ( (x:e1@i)->(e2@i) x x) (cPath A) (cPath B)) + * (PathP ( (x y z:e1@i)->(e2@i) x y->(e2@i) y z->(e2@i) x z) (cC A) (cC B)) + +eCatIso3 (A B : precategory) : Path U (Path precategory A B) (catIso3 A B) + = isoPath (Path precategory A B) (catIso3 A B) F G FG GF where - F (e:Id precategory A B):catIso3 A B = ((e@i).1.1, (e@i).1.2, (e@i).2.1, (e@i).2.2.1) - G (e:catIso3 A B):Id precategory A B = ((e.1@i, e.2.1@i), e.2.2.1@i, e.2.2.2@i - ,lemIdPProp (isPrecategory2 A.1 A.2.1 A.2.2.1) + F (e:Path precategory A B):catIso3 A B = ((e@i).1.1, (e@i).1.2, (e@i).2.1, (e@i).2.2.1) + G (e:catIso3 A B):Path precategory A B = ((e.1@i, e.2.1@i), e.2.2.1@i, e.2.2.2@i + ,lemPathPProp (isPrecategory2 A.1 A.2.1 A.2.2.1) (isPrecategory2 B.1 B.2.1 B.2.2.1) (propIsPrecategory2 A.1 A.2.1 A.2.2.1) (isPrecategory2 (e.1@i, e.2.1@i) (e.2.2.1@i) (e.2.2.2@i)) A.2.2.2 B.2.2.2 @ i) - FG(e:catIso3 A B):Id (catIso3 A B) (F (G e)) e=<_>e - GF(e:Id precategory A B):Id (Id precategory A B) (G (F e)) e + FG(e:catIso3 A B):Path (catIso3 A B) (F (G e)) e=<_>e + GF(e:Path precategory A B):Path (Path precategory A B) (G (F e)) e =((e@j).1, (e@j).2.1, (e@j).2.2.1 - ,lemIdPSet (isPrecategory2 A.1 A.2.1 A.2.2.1) + ,lemPathPSet (isPrecategory2 A.1 A.2.1 A.2.2.1) (isPrecategory2 B.1 B.2.1 B.2.2.1) (propSet (isPrecategory2 A.1 A.2.1 A.2.2.1) (propIsPrecategory2 A.1 A.2.1 A.2.2.1)) (isPrecategory2 (e@i).1 (e@i).2.1 (e@i).2.2.1) @@ -153,12 +153,12 @@ eCatIso3 (A B : precategory) : Id U (Id precategory A B) (catIso3 A B) ( (G (F e)@j).2.2.2) ((e@j).2.2.2) @ i @ j) -catIso30 (A B : precategory) : U = (e1 : Id U (cA A) (cA B)) - * (e2 : Id ((x y : cA A)->U) (cH A) (\(x y:cA A)->cH B (transport e1 x) (transport e1 y))) - * (_ : Id ((x:cA A)->cH A x x) - (cId A) - (\(x : cA A) -> transport ((e2@-j) x x) (cId B (transport e1 x)))) - * (Id ((x y z:cA A)->cH A x y->cH A y z->cH A x z) +catIso30 (A B : precategory) : U = (e1 : Path U (cA A) (cA B)) + * (e2 : Path ((x y : cA A)->U) (cH A) (\(x y:cA A)->cH B (transport e1 x) (transport e1 y))) + * (_ : Path ((x:cA A)->cH A x x) + (cPath A) + (\(x : cA A) -> transport ((e2@-j) x x) (cPath B (transport e1 x)))) + * (Path ((x y z:cA A)->cH A x y->cH A y z->cH A x z) (\(x y z:cA A)(f:cH A x y)(g:cH A y z)->cC A x y z f g) (\(x y z:cA A)(f:cH A x y)(g:cH A y z)-> transport ((e2@-i) x z) @@ -166,15 +166,15 @@ catIso30 (A B : precategory) : U = (e1 : Id U (cA A) (cA B)) (transport ((e2@i) x y) f) (transport ((e2@i) y z) g)))) -eCatIso30 (A B : precategory) : Id U (catIso3 A B) (catIso30 A B) - = (e1 : Id U (cA A) (cA B)) +eCatIso30 (A B : precategory) : Path U (catIso3 A B) (catIso30 A B) + = (e1 : Path U (cA A) (cA B)) * (let e21 : (x y : e1@-i) -> U = comp (<_> (x y : e1@-i) -> U) (\(x y : e1@-i) -> cH B (transport (e1@-i\/j) x) (transport (e1@-i\/j) y)) [(i=1)-><_>\(x y:cA A) -> cH B (transport e1 x) (transport e1 y) ,(i=0)->\(x y:cA B) -> cH B (transRefl (cA B) x @ j) (transRefl (cA B) y @ j) ] - f21 : Id ((x y : e1@-i) -> U) + f21 : Path ((x y : e1@-i) -> U) (\(x y : e1@-i) -> cH B (transport (e1@-i\/j) x) (transport (e1@-i\/j) y)) e21 = fill (<_> (x y : e1@-i) -> U) @@ -183,25 +183,25 @@ eCatIso30 (A B : precategory) : Id U (catIso3 A B) (catIso30 A B) ,(i=0)->\(x y:cA B) -> cH B (transRefl (cA B) x @ j) (transRefl (cA B) y @ j) ] in - (e2 : IdP ( (x y : e1@j/\-i) -> U) (cH A) e21) - * (_ : IdP ( (x : e1@j/\-i) -> (e2@j/\-i) x x) - (cId A) + (e2 : PathP ( (x y : e1@j/\-i) -> U) (cH A) e21) + * (_ : PathP ( (x : e1@j/\-i) -> (e2@j/\-i) x x) + (cPath A) (comp (<_> (x:e1@-i)->(e2@-i) x x) - (\(x : e1@-i) -> transport ((e2@-i\/-j) x x) (transport ((f21@j) x x) (cId B (transport (e1@-i\/j) x)))) - [(i=1)->\(x:cA A) -> transport ((e2@-j) x x) (transRefl ((f21@0) x x) (cId B (transport e1 x)) @ j) + (\(x : e1@-i) -> transport ((e2@-i\/-j) x x) (transport ((f21@j) x x) (cPath B (transport (e1@-i\/j) x)))) + [(i=1)->\(x:cA A) -> transport ((e2@-j) x x) (transRefl ((f21@0) x x) (cPath B (transport e1 x)) @ j) ,(i=0)->\(x:cA B) -> transRefl ((e2@1) x x) - (compId (cH B x x) + (compPath (cH B x x) (transport (cH B (transRefl (cA B) x@k) (transRefl (cA B) x@k)) - (cId B (transport (<_> cA B) x))) - (transport (<_>cH B x x) (cId B x)) - (cId B x) + (cPath B (transport (<_> cA B) x))) + (transport (<_>cH B x x) (cPath B x)) + (cPath B x) (transport (cH B (transRefl (cA B) x@j\/k) (transRefl (cA B) x@j\/k)) - (cId B (transRefl (cA B) x @ j))) - (transRefl (cH B x x) (cId B x)) + (cPath B (transRefl (cA B) x @ j))) + (transRefl (cH B x x) (cPath B x)) @ j) @ j ])) - * (IdP ( (x y z : e1@j/\-i) (f : (e2@j/\-i) x y) (g : (e2@j/\-i) y z) -> (e2@j/\-i) x z) + * (PathP ( (x y z : e1@j/\-i) (f : (e2@j/\-i) x y) (g : (e2@j/\-i) y z) -> (e2@j/\-i) x z) (cC A) (comp (<_> (x y z : e1@-i) (f : (e2@-i) x y) (g : (e2@-i) y z) -> (e2@-i) x z) (\(x y z : e1@-i) (f : (e2@-i) x y) (g : (e2@-i) y z) -> @@ -216,7 +216,7 @@ eCatIso30 (A B : precategory) : Id U (catIso3 A B) (catIso30 A B) (transRefl ((f21@0) y z) (transport ((e2@j) y z) g) @ j)) @ j) ,(i=0)->\(x y z : cA B) (f : cH B x y) (g : cH B y z) -> transRefl ((e2@1) x z) - (compId (cH B x z) + (compPath (cH B x z) (transport (cH B (transRefl (cA B) x@k) (transRefl (cA B) z@k)) (cC B (transport (<_>cA B) x) (transport (<_>cA B) y) (transport (<_>cA B) z) (transport (cH B (transRefl (cA B) x@-k) (transRefl (cA B) y@-k)) (transport ((e2@1) x y) f)) @@ -234,40 +234,40 @@ eCatIso30 (A B : precategory) : Id U (catIso3 A B) (catIso30 A B) @ j) @ j ]))) -catIso311 (A B : precategory) (e1:Id U (cA A) (cA B)) : U - = Id ((x y : cA A)->U) (cH A) (\(x y:cA A)->cH B (transport e1 x) (transport e1 y)) -catIso312 (A B : precategory) (e1:Id U (cA A) (cA B)) (e2:catIso311 A B e1) : U - = Id ((x:cA A)->cH B (transport e1 x) (transport e1 x)) - (\(x:cA A)->transport ( (e2@i) x x) (cId A x)) - (\(x : cA A) -> cId B (transport e1 x)) -catIso313 (A B : precategory) (e1:Id U (cA A) (cA B)) (e2:catIso311 A B e1) : U - = Id ((x y z:cA A)->cH A x y->cH A y z->cH B (transport e1 x) (transport e1 z)) +catIso311 (A B : precategory) (e1:Path U (cA A) (cA B)) : U + = Path ((x y : cA A)->U) (cH A) (\(x y:cA A)->cH B (transport e1 x) (transport e1 y)) +catIso312 (A B : precategory) (e1:Path U (cA A) (cA B)) (e2:catIso311 A B e1) : U + = Path ((x:cA A)->cH B (transport e1 x) (transport e1 x)) + (\(x:cA A)->transport ( (e2@i) x x) (cPath A x)) + (\(x : cA A) -> cPath B (transport e1 x)) +catIso313 (A B : precategory) (e1:Path U (cA A) (cA B)) (e2:catIso311 A B e1) : U + = Path ((x y z:cA A)->cH A x y->cH A y z->cH B (transport e1 x) (transport e1 z)) (\(x y z:cA A)(f:cH A x y)(g:cH A y z)->transport ((e2@i)x z) (cC A x y z f g)) (\(x y z:cA A)(f:cH A x y)(g:cH A y z)-> cC B (transport e1 x) (transport e1 y) (transport e1 z) (transport ((e2@i) x y) f) (transport ((e2@i) y z) g)) -catIso31 (A B : precategory) : U = (e1 : Id U (cA A) (cA B)) +catIso31 (A B : precategory) : U = (e1 : Path U (cA A) (cA B)) * (e2 : catIso311 A B e1) * (_ : catIso312 A B e1 e2) * ( catIso313 A B e1 e2) -eCatIso31 (A B : precategory) : Id U (catIso30 A B) (catIso31 A B) - = (e1 : Id U (cA A) (cA B)) - * (e2 : Id ((x y : cA A)->U) (cH A) (\(x y:cA A)->cH B (transport e1 x) (transport e1 y))) +eCatIso31 (A B : precategory) : Path U (catIso30 A B) (catIso31 A B) + = (e1 : Path U (cA A) (cA B)) + * (e2 : Path ((x y : cA A)->U) (cH A) (\(x y:cA A)->cH B (transport e1 x) (transport e1 y))) * (_:comp (<_> U) - (Id ((x:cA A)->(e2@i) x x) - (\(x:cA A)->transport ( (e2@i/\j) x x) (cId A x)) - (\(x:cA A)->transport ( (e2@i\/-j) x x) (cId B (transport e1 x)))) - [(i=0)->Id ((x:cA A)->cH A x x) - (\(x:cA A)->transRefl (cH A x x) (cId A x) @ k) - (\(x:cA A)->transport ( (e2@-j) x x) (cId B (transport e1 x))) - ,(i=1)->Id ((x:cA A)->(e2@i) x x) - (\(x:cA A)->transport ( (e2@j) x x) (cId A x)) - (\(x:cA A)->transRefl ((e2@1) x x) (cId B (transport e1 x)) @ k) + (Path ((x:cA A)->(e2@i) x x) + (\(x:cA A)->transport ( (e2@i/\j) x x) (cPath A x)) + (\(x:cA A)->transport ( (e2@i\/-j) x x) (cPath B (transport e1 x)))) + [(i=0)->Path ((x:cA A)->cH A x x) + (\(x:cA A)->transRefl (cH A x x) (cPath A x) @ k) + (\(x:cA A)->transport ( (e2@-j) x x) (cPath B (transport e1 x))) + ,(i=1)->Path ((x:cA A)->(e2@i) x x) + (\(x:cA A)->transport ( (e2@j) x x) (cPath A x)) + (\(x:cA A)->transRefl ((e2@1) x x) (cPath B (transport e1 x)) @ k) ]) * (comp (<_> U) - (Id ((x y z:cA A)->cH A x y->cH A y z->(e2@i) x z) + (Path ((x y z:cA A)->cH A x y->cH A y z->(e2@i) x z) (\(x y z:cA A)(f:cH A x y)(g:cH A y z)-> transport ( (e2@i/\j) x z) (cC A x y z f g)) (\(x y z:cA A)(f:cH A x y)(g:cH A y z)-> @@ -275,7 +275,7 @@ eCatIso31 (A B : precategory) : Id U (catIso30 A B) (catIso31 A B) (cC B (transport e1 x) (transport e1 y) (transport e1 z) (transport ((e2@j) x y) f) (transport ((e2@j) y z) g)))) - [(i=0)->Id ((x y z:cA A)->cH A x y->cH A y z->cH A x z) + [(i=0)->Path ((x y z:cA A)->cH A x y->cH A y z->cH A x z) (\(x y z:cA A)(f:cH A x y)(g:cH A y z)-> transRefl ((e2@0) x z) (cC A x y z f g) @ k) (\(x y z:cA A)(f:cH A x y)(g:cH A y z)-> @@ -283,7 +283,7 @@ eCatIso31 (A B : precategory) : Id U (catIso30 A B) (catIso31 A B) (cC B (transport e1 x) (transport e1 y) (transport e1 z) (transport ((e2@j) x y) f) (transport ((e2@j) y z) g))) - ,(i=1)->Id ((x y z:cA A)->cH A x y->cH A y z->(e2@1) x z) + ,(i=1)->Path ((x y z:cA A)->cH A x y->cH A y z->(e2@1) x z) (\(x y z:cA A)(f:cH A x y)(g:cH A y z)-> transport ((e2@j) x z) (cC A x y z f g)) (\(x y z:cA A)(f:cH A x y)(g:cH A y z)-> @@ -294,10 +294,10 @@ eCatIso31 (A B : precategory) : Id U (catIso30 A B) (catIso31 A B) ]) lemEquivCoh (A B:U) (e:equiv A B) (x:A) - : Id (Id B (e.1 x) (e.1 (invEq A B e (e.1 x)))) + : Path (Path B (e.1 x) (e.1 (invEq A B e (e.1 x)))) (e.1(secEq A B e x@-j)) (retEq A B e (e.1 x)@-j) - = comp ( Id B (e.1 x) (e.1 (secEq A B e x @ -i /\ -j))) + = comp ( Path B (e.1 x) (e.1 (secEq A B e x @ -i /\ -j))) (((e.2 (e.1 x)).2 (x,<_>e.1 x)@-i).2) [(i=0)->e.1 (secEq A B e x@-k\/-j) ,(i=1)->retEq A B e (e.1 x)@-k @@ -311,19 +311,19 @@ sigEquivLem (A A':U) (B:A->U) (B':A'->U) where F (x:Sigma A B):Sigma A' B' = (e.1 x.1, (f x.1).1 x.2) G (x:Sigma A' B'):Sigma A B = ((e.2 x.1).1.1, ((f (e.2 x.1).1.1).2 (transport ( B' ((e.2 x.1).1.2 @ i)) x.2)).1.1) - FG (x:Sigma A' B'):Id (Sigma A' B') (F (G x)) x + FG (x:Sigma A' B'):Path (Sigma A' B') (F (G x)) x = ((e.2 x.1).1.2 @ -i ,comp (<_> B' ((e.2 x.1).1.2 @ -i)) (transport ( B' ((e.2 x.1).1.2 @ j/\-i)) x.2) [(i=0)->((f (e.2 x.1).1.1).2 (transport ( B' ((e.2 x.1).1.2 @ i)) x.2)).1.2 @ j ,(i=1)->transRefl (B' ((e.2 x.1).1.2 @ 0)) x.2@j ]) - GF (x:Sigma A B):Id (Sigma A B) (G (F x)) x + GF (x:Sigma A B):Path (Sigma A B) (G (F x)) x = (secEq A A' e x.1 @ i ,comp (<_> B (secEq A A' e x.1 @ i)) ((f (secEq A A' e x.1 @ i)).2 (transport ( B' (e.1 (secEq A A' e x.1 @ i\/-j))) ((f x.1).1 x.2))).1.1 [(i=0)->((f (secEq A A' e x.1 @ 0)).2 (transport ( B' (lemEquivCoh A A' e x.1@k@i)) ((f x.1).1 x.2))).1.1 - ,(i=1)->compId (B x.1) + ,(i=1)->compPath (B x.1) ((f x.1).2 (transport ( B' (e.1 x.1)) ((f x.1).1 x.2))).1.1 ((f x.1).2 ((f x.1).1 x.2)).1.1 x.2 @@ -333,32 +333,32 @@ sigEquivLem (A A':U) (B:A->U) (B':A'->U) sigEquivLem' (A A':U) (B:A->U) (B':A'->U) (e:equiv A A') - (f:(x:A)->Id U (B x) (B' (e.1 x))) - : Id U (Sigma A B) (Sigma A' B') - = transEquivToId (Sigma A B) (Sigma A' B') (sigEquivLem A A' B B' e (\(x:A)->transEquiv' (B' (e.1 x)) (B x) (f x))) - -catIso321 (A B : precategory) (e1:Id U (cA A) (cA B)) : U - = (x y:cA A) -> Id U (cH A x y) (cH B (transport e1 x) (transport e1 y)) -catIso322 (A B : precategory) (e1:Id U (cA A) (cA B)) (e2:catIso321 A B e1) : U - = (x:cA A) -> Id (cH B (transport e1 x) (transport e1 x)) - (transport (e2 x x) (cId A x)) - (cId B (transport e1 x)) -catIso323 (A B : precategory) (e1:Id U (cA A) (cA B)) (e2:catIso321 A B e1) : U + (f:(x:A)->Path U (B x) (B' (e.1 x))) + : Path U (Sigma A B) (Sigma A' B') + = transEquivToPath (Sigma A B) (Sigma A' B') (sigEquivLem A A' B B' e (\(x:A)->transEquiv' (B' (e.1 x)) (B x) (f x))) + +catIso321 (A B : precategory) (e1:Path U (cA A) (cA B)) : U + = (x y:cA A) -> Path U (cH A x y) (cH B (transport e1 x) (transport e1 y)) +catIso322 (A B : precategory) (e1:Path U (cA A) (cA B)) (e2:catIso321 A B e1) : U + = (x:cA A) -> Path (cH B (transport e1 x) (transport e1 x)) + (transport (e2 x x) (cPath A x)) + (cPath B (transport e1 x)) +catIso323 (A B : precategory) (e1:Path U (cA A) (cA B)) (e2:catIso321 A B e1) : U = (x y z:cA A)(f:cH A x y)(g:cH A y z)-> - Id (cH B (transport e1 x) (transport e1 z)) + Path (cH B (transport e1 x) (transport e1 z)) (transport (e2 x z) (cC A x y z f g)) (cC B (transport e1 x) (transport e1 y) (transport e1 z) (transport (e2 x y) f) (transport (e2 y z) g)) -catIso32 (A B : precategory) : U = (e1 : Id U (cA A) (cA B)) +catIso32 (A B : precategory) : U = (e1 : Path U (cA A) (cA B)) * (e2 : catIso321 A B e1) * (_ : catIso322 A B e1 e2) * ( catIso323 A B e1 e2) -eCatIso32 (A B : precategory) : Id U (catIso31 A B) (catIso32 A B) - = (e1 : Id U (cA A) (cA B)) +eCatIso32 (A B : precategory) : Path U (catIso31 A B) (catIso32 A B) + = (e1 : Path U (cA A) (cA B)) * (let F(e2:catIso311 A B e1):catIso321 A B e1 = \(x y:cA A)->(e2@i) x y in - transEquivToId + transEquivToPath ((e2 : catIso311 A B e1) * (_ : catIso312 A B e1 e2) * ( catIso313 A B e1 e2)) ((e2 : catIso321 A B e1) * (_ : catIso322 A B e1 e2) * ( catIso323 A B e1 e2)) (sigEquivLem (catIso311 A B e1) (catIso321 A B e1) @@ -386,55 +386,55 @@ eCatIso32 (A B : precategory) : Id U (catIso31 A B) (catIso32 A B) iso (C : precategory) (A B : cA C) : U = (f : cH C A B) * (g : cH C B A) - * (_ : Id (cH C A A) (cC C A B A f g) (cId C A)) - * (Id (cH C B B) (cC C B A B g f) (cId C B)) + * (_ : Path (cH C A A) (cC C A B A f g) (cPath C A)) + * (Path (cH C B B) (cC C B A B g f) (cPath C B)) -isoEq (C : precategory) (A B : cA C) (f g : iso C A B) (p1 : Id (cH C A B) f.1 g.1) (p2 : Id (cH C B A) f.2.1 g.2.1) - : Id (iso C A B) f g +isoEq (C : precategory) (A B : cA C) (f g : iso C A B) (p1 : Path (cH C A B) f.1 g.1) (p2 : Path (cH C B A) f.2.1 g.2.1) + : Path (iso C A B) f g = (p1@i,p2@i - ,lemIdPProp (Id (cH C A A) (cC C A B A (p1@0) (p2@0)) (cId C A)) - (Id (cH C A A) (cC C A B A (p1@1) (p2@1)) (cId C A)) - (cHSet C A A (cC C A B A (p1@0) (p2@0)) (cId C A)) - (Id (cH C A A) (cC C A B A (p1@i) (p2@i)) (cId C A)) + ,lemPathPProp (Path (cH C A A) (cC C A B A (p1@0) (p2@0)) (cPath C A)) + (Path (cH C A A) (cC C A B A (p1@1) (p2@1)) (cPath C A)) + (cHSet C A A (cC C A B A (p1@0) (p2@0)) (cPath C A)) + (Path (cH C A A) (cC C A B A (p1@i) (p2@i)) (cPath C A)) f.2.2.1 g.2.2.1 @ i - ,lemIdPProp (Id (cH C B B) (cC C B A B (p2@0) (p1@0)) (cId C B)) - (Id (cH C B B) (cC C B A B (p2@1) (p1@1)) (cId C B)) - (cHSet C B B (cC C B A B (p2@0) (p1@0)) (cId C B)) - (Id (cH C B B) (cC C B A B (p2@i) (p1@i)) (cId C B)) + ,lemPathPProp (Path (cH C B B) (cC C B A B (p2@0) (p1@0)) (cPath C B)) + (Path (cH C B B) (cC C B A B (p2@1) (p1@1)) (cPath C B)) + (cHSet C B B (cC C B A B (p2@0) (p1@0)) (cPath C B)) + (Path (cH C B B) (cC C B A B (p2@i) (p1@i)) (cPath C B)) f.2.2.2 g.2.2.2 @ i) opaque isoEq invIso (C : precategory) (A B : cA C) (f : iso C A B) : iso C B A = (f.2.1, f.1, f.2.2.2, f.2.2.1) -idIso (C : precategory) (A : cA C) : iso C A A = (cId C A, cId C A, cIdL C A A (cId C A), cIdL C A A (cId C A)) +idIso (C : precategory) (A : cA C) : iso C A A = (cPath C A, cPath C A, cPathL C A A (cPath C A), cPathL C A A (cPath C A)) invIsoEquiv (C : precategory) (A B : cA C) : isEquiv (iso C A B) (iso C B A) (invIso C A B) = gradLemma (iso C A B) (iso C B A) (invIso C A B) (invIso C B A) (\(x : iso C B A) -> <_> x) (\(x : iso C A B) -> <_> x) -invIsoEq (C : precategory) (A B : cA C) : Id U (iso C A B) (iso C B A) - = equivId (iso C A B) (iso C B A) (invIso C A B) (invIsoEquiv C A B) +invIsoEq (C : precategory) (A B : cA C) : Path U (iso C A B) (iso C B A) + = equivPath (iso C A B) (iso C B A) (invIso C A B) (invIsoEquiv C A B) compIsoHelper (X : precategory) (A B C : cA X) (f : iso X A B) (g : iso X B C) - : Id (cH X A A) (cC X A C A (cC X A B C f.1 g.1) (cC X C B A g.2.1 f.2.1)) (cId X A) - = compId (cH X A A) + : Path (cH X A A) (cC X A C A (cC X A B C f.1 g.1) (cC X C B A g.2.1 f.2.1)) (cPath X A) + = compPath (cH X A A) (cC X A C A (cC X A B C f.1 g.1) (cC X C B A g.2.1 f.2.1)) (cC X A B A f.1 (cC X B C A g.1 (cC X C B A g.2.1 f.2.1))) - (cId X A) - (cIdC X A B C A f.1 g.1 (cC X C B A g.2.1 f.2.1)) - (compId (cH X A A) + (cPath X A) + (cPathC X A B C A f.1 g.1 (cC X C B A g.2.1 f.2.1)) + (compPath (cH X A A) (cC X A B A f.1 (cC X B C A g.1 (cC X C B A g.2.1 f.2.1))) (cC X A B A f.1 (cC X B B A (cC X B C B g.1 g.2.1) f.2.1)) - (cId X A) - (cC X A B A f.1 (cIdC X B C B A g.1 g.2.1 f.2.1 @ -i)) - (compId (cH X A A) + (cPath X A) + (cC X A B A f.1 (cPathC X B C B A g.1 g.2.1 f.2.1 @ -i)) + (compPath (cH X A A) (cC X A B A f.1 (cC X B B A (cC X B C B g.1 g.2.1) f.2.1)) - (cC X A B A f.1 (cC X B B A (cId X B) f.2.1)) - (cId X A) + (cC X A B A f.1 (cC X B B A (cPath X B) f.2.1)) + (cPath X A) (cC X A B A f.1 (cC X B B A (g.2.2.1 @ i) f.2.1)) - (compId (cH X A A) - (cC X A B A f.1 (cC X B B A (cId X B) f.2.1)) + (compPath (cH X A A) + (cC X A B A f.1 (cC X B B A (cPath X B) f.2.1)) (cC X A B A f.1 f.2.1) - (cId X A) - (cC X A B A f.1 (cIdL X B A f.2.1 @ i)) + (cPath X A) + (cC X A B A f.1 (cPathL X B A f.2.1 @ i)) f.2.2.1))) opaque compIsoHelper @@ -443,25 +443,25 @@ compIso (X : precategory) (A B C : cA X) (f : iso X A B) (g : iso X B C) : iso X ,compIsoHelper X A B C f g ,compIsoHelper X C B A (invIso X B C g) (invIso X A B f)) -IdLIso (C : precategory) (A B : cA C) (f : iso C A B) : Id (iso C A B) (compIso C A A B (idIso C A) f) f - = isoEq C A B (compIso C A A B (idIso C A) f) f (cIdL C A B f.1) (cIdR C B A f.2.1) -opaque IdLIso -IdRIso (C : precategory) (A B : cA C) (f : iso C A B) : Id (iso C A B) (compIso C A B B f (idIso C B)) f - = isoEq C A B (compIso C A B B f (idIso C B)) f (cIdR C A B f.1) (cIdL C B A f.2.1) -opaque IdRIso -IdCIso (X : precategory) (A B C D : cA X) (f : iso X A B) (g : iso X B C) (h : iso X C D) - : Id (iso X A D) (compIso X A C D (compIso X A B C f g) h) (compIso X A B D f (compIso X B C D g h)) +PathLIso (C : precategory) (A B : cA C) (f : iso C A B) : Path (iso C A B) (compIso C A A B (idIso C A) f) f + = isoEq C A B (compIso C A A B (idIso C A) f) f (cPathL C A B f.1) (cPathR C B A f.2.1) +opaque PathLIso +PathRIso (C : precategory) (A B : cA C) (f : iso C A B) : Path (iso C A B) (compIso C A B B f (idIso C B)) f + = isoEq C A B (compIso C A B B f (idIso C B)) f (cPathR C A B f.1) (cPathL C B A f.2.1) +opaque PathRIso +PathCIso (X : precategory) (A B C D : cA X) (f : iso X A B) (g : iso X B C) (h : iso X C D) + : Path (iso X A D) (compIso X A C D (compIso X A B C f g) h) (compIso X A B D f (compIso X B C D g h)) = isoEq X A D (compIso X A C D (compIso X A B C f g) h) (compIso X A B D f (compIso X B C D g h)) - (cIdC X A B C D f.1 g.1 h.1) (cIdC X D C B A h.2.1 g.2.1 f.2.1@-i) -opaque IdCIso -IdInvLIso (X : precategory) (A B : cA X) (f : iso X A B) : Id (iso X B B) (compIso X B A B (invIso X A B f) f) (idIso X B) + (cPathC X A B C D f.1 g.1 h.1) (cPathC X D C B A h.2.1 g.2.1 f.2.1@-i) +opaque PathCIso +PathInvLIso (X : precategory) (A B : cA X) (f : iso X A B) : Path (iso X B B) (compIso X B A B (invIso X A B f) f) (idIso X B) = isoEq X B B (compIso X B A B (invIso X A B f) f) (idIso X B) f.2.2.2 f.2.2.2 -opaque IdInvLIso +opaque PathInvLIso opaque compIso -eqToIso (C : precategory) (A B : cA C) (p : Id (cA C) A B) : iso C A B - = J (cA C) A (\(B : cA C) -> \(p : Id (cA C) A B) -> iso C A B) (idIso C A) B p +eqToIso (C : precategory) (A B : cA C) (p : Path (cA C) A B) : iso C A B + = J (cA C) A (\(B : cA C) -> \(p : Path (cA C) A B) -> iso C A B) (idIso C A) B p isCategory (C : precategory) : U = (A : cA C) -> isContr ((B : cA C) * iso C A B) propIsCategory (C : precategory) : prop (isCategory C) @@ -469,14 +469,14 @@ propIsCategory (C : precategory) : prop (isCategory C) (\(A : cA C) -> propIsContr ((B : cA C) * iso C A B)) category : U = (C:precategory)*isCategory C -lemIsCategory (C : precategory) (isC : isCategory C) (A B : cA C) (e : iso C A B) : Id ((B : cA C) * iso C A B) (A, idIso C A) (B, e) +lemIsCategory (C : precategory) (isC : isCategory C) (A B : cA C) (e : iso C A B) : Path ((B : cA C) * iso C A B) (A, idIso C A) (B, e) = (isContrProp ((B : cA C) * iso C A B) (isC A) (A, idIso C A) (B, e) @ i) -lemIsCategory2 (C : precategory) (isC : isCategory C) (A B : cA C) : isEquiv (Id (cA C) A B) (iso C A B) (eqToIso C A B) - = equivFunFib (cA C) (Id (cA C) A) (iso C A) (eqToIso C A) (lemSinglContr (cA C) A) (isC A) B +lemIsCategory2 (C : precategory) (isC : isCategory C) (A B : cA C) : isEquiv (Path (cA C) A B) (iso C A B) (eqToIso C A B) + = equivFunFib (cA C) (Path (cA C) A) (iso C A) (eqToIso C A) (lemSinglContr (cA C) A) (isC A) B -lemIsCategory3 (C : precategory) (isC : isCategory C) (A B : cA C) : Id U (Id (cA C) A B) (iso C A B) - = equivId (Id (cA C) A B) (iso C A B) (eqToIso C A B) (lemIsCategory2 C isC A B) +lemIsCategory3 (C : precategory) (isC : isCategory C) (A B : cA C) : Path U (Path (cA C) A B) (iso C A B) + = equivPath (Path (cA C) A B) (iso C A B) (eqToIso C A B) (lemIsCategory2 C isC A B) -- @@ -496,7 +496,7 @@ structure (X : precategory) : U = (P : cA X -> U) * (H : (x y : cA X) -> (a : P x) -> (b : P y) -> (f : cH X x y) -> U) * (propH : (x y : cA X) -> (a : P x) -> (b : P y) -> (f : cH X x y) -> prop (H x y a b f)) - * (Hid : (x : cA X) -> (a : P x) -> H x x a a (cId X x)) + * (Hid : (x : cA X) -> (a : P x) -> H x x a a (cPath X x)) * ((x y z : cA X) -> (a : P x) -> (b : P y) -> (c : P z) -> (f : cH X x y) -> (g : cH X y z) -> H x y a b f -> H y z b c g -> H x z a c (cC X x y z f g)) @@ -505,7 +505,7 @@ sP (X : precategory) (S : structure X) : cA X -> U = S.1 sH (X : precategory) (S : structure X) : (x y : cA X) -> (a : sP X S x) -> (b : sP X S y) -> (f : cH X x y) -> U = S.2.1 sHProp (X : precategory) (S : structure X) : (x y : cA X) -> (a : sP X S x) -> (b : sP X S y) -> (f : cH X x y) -> prop (sH X S x y a b f) = S.2.2.1 -sHId (X : precategory) (S : structure X) : (x : cA X) -> (a : sP X S x) -> sH X S x x a a (cId X x) = S.2.2.2.1 +sHPath (X : precategory) (S : structure X) : (x : cA X) -> (a : sP X S x) -> sH X S x x a a (cPath X x) = S.2.2.2.1 sHComp (X : precategory) (S : structure X) : (x y z : cA X) -> (a : sP X S x) -> (b : sP X S y) -> (c : sP X S z) -> (f : cH X x y) -> (g : cH X y z) -> @@ -514,8 +514,8 @@ sHComp (X : precategory) (S : structure X) isStandardStructure (X : precategory) (S : structure X) : U = (x : cA X) -> (a b : sP X S x) -> - sH X S x x a b (cId X x) -> sH X S x x b a (cId X x) -> - Id (sP X S x) a b + sH X S x x a b (cPath X x) -> sH X S x x b a (cPath X x) -> + Path (sP X S x) a b sipPrecategory (C : precategory) (S : structure C) : precategory = ((ob, hom), (id, cmp, homSet, idL, idR, idC)) where @@ -524,32 +524,32 @@ sipPrecategory (C : precategory) (S : structure C) : precategory = ((ob, hom), ( homSet (X Y : ob) : set (hom X Y) = setSig (cH C X.1 Y.1) (sH C S X.1 Y.1 X.2 Y.2) (cHSet C X.1 Y.1) (\(f : cH C X.1 Y.1) -> propSet (sH C S X.1 Y.1 X.2 Y.2 f) (sHProp C S X.1 Y.1 X.2 Y.2 f)) - id (X : ob) : hom X X = (cId C X.1, sHId C S X.1 X.2) + id (X : ob) : hom X X = (cPath C X.1, sHPath C S X.1 X.2) cmp (x y z : ob) (f : hom x y) (g : hom y z) : hom x z = (cC C x.1 y.1 z.1 f.1 g.1, sHComp C S x.1 y.1 z.1 x.2 y.2 z.2 f.1 g.1 f.2 g.2) - idL (x y : ob) (f : hom x y) : Id (hom x y) (cmp x x y (id x) f) f - = (cIdL C x.1 y.1 f.1 @ i - ,lemIdPProp (sH C S x.1 y.1 x.2 y.2 (cmp x x y (id x) f).1) + idL (x y : ob) (f : hom x y) : Path (hom x y) (cmp x x y (id x) f) f + = (cPathL C x.1 y.1 f.1 @ i + ,lemPathPProp (sH C S x.1 y.1 x.2 y.2 (cmp x x y (id x) f).1) (sH C S x.1 y.1 x.2 y.2 f.1) (sHProp C S x.1 y.1 x.2 y.2 (cmp x x y (id x) f).1) - (sH C S x.1 y.1 x.2 y.2 (cIdL C x.1 y.1 f.1 @ i)) + (sH C S x.1 y.1 x.2 y.2 (cPathL C x.1 y.1 f.1 @ i)) (cmp x x y (id x) f).2 f.2 @ i) - idR (x y : ob) (f : hom x y) : Id (hom x y) (cmp x y y f (id y)) f - = (cIdR C x.1 y.1 f.1 @ i - ,lemIdPProp (sH C S x.1 y.1 x.2 y.2 (cmp x y y f (id y)).1) + idR (x y : ob) (f : hom x y) : Path (hom x y) (cmp x y y f (id y)) f + = (cPathR C x.1 y.1 f.1 @ i + ,lemPathPProp (sH C S x.1 y.1 x.2 y.2 (cmp x y y f (id y)).1) (sH C S x.1 y.1 x.2 y.2 f.1) (sHProp C S x.1 y.1 x.2 y.2 (cmp x y y f (id y)).1) - (sH C S x.1 y.1 x.2 y.2 (cIdR C x.1 y.1 f.1 @ i)) + (sH C S x.1 y.1 x.2 y.2 (cPathR C x.1 y.1 f.1 @ i)) (cmp x y y f (id y)).2 f.2 @ i) - idC (x y z w : ob) (f : hom x y) (g : hom y z) (h : hom z w) : Id (hom x w) (cmp x z w (cmp x y z f g) h) (cmp x y w f (cmp y z w g h)) - = (cIdC C x.1 y.1 z.1 w.1 f.1 g.1 h.1 @ i - ,lemIdPProp (sH C S x.1 w.1 x.2 w.2 (cmp x z w (cmp x y z f g) h).1) + idC (x y z w : ob) (f : hom x y) (g : hom y z) (h : hom z w) : Path (hom x w) (cmp x z w (cmp x y z f g) h) (cmp x y w f (cmp y z w g h)) + = (cPathC C x.1 y.1 z.1 w.1 f.1 g.1 h.1 @ i + ,lemPathPProp (sH C S x.1 w.1 x.2 w.2 (cmp x z w (cmp x y z f g) h).1) (sH C S x.1 w.1 x.2 w.2 (cmp x y w f (cmp y z w g h)).1) (sHProp C S x.1 w.1 x.2 w.2 (cmp x z w (cmp x y z f g) h).1) - (sH C S x.1 w.1 x.2 w.2 (cIdC C x.1y.1 z.1 w.1 f.1 g.1 h.1 @ i)) + (sH C S x.1 w.1 x.2 w.2 (cPathC C x.1y.1 z.1 w.1 f.1 g.1 h.1 @ i)) (cmp x z w (cmp x y z f g) h).2 (cmp x y w f (cmp y z w g h)).2 @ i) -isContrProp (A : U) (p : isContr A) (x y : A) : Id A x y = compId A x p.1 y ( p.2 x @ -i) (p.2 y) +isContrProp (A : U) (p : isContr A) (x y : A) : Path A x y = compPath A x p.1 y ( p.2 x @ -i) (p.2 y) isContrSig (A:U) (B:A-> U) (cA:isContr A) (cB : (x:A) -> isContr (B x)) : isContr (Sigma A B) = ((cA.1, (cB cA.1).1), \(x:Sigma A B) -> propSig A B (isContrProp A cA) (\(x:A)->isContrProp (B x) (cB x)) (cA.1, (cB cA.1).1) x) @@ -559,10 +559,10 @@ sip (X : precategory) (isC : isCategory X) (S : structure X) (isS : isStandardSt -- where -- D : precategory = sipPrecategory X S -- eq1 (A : cA D) - -- : Id U ((B : cA D) * iso D A B) + -- : Path U ((B : cA D) * iso D A B) -- ((C : (B1 : cA X) * ( iso X A.1 B1)) -- * (B2 : sP X S C.1) * (_ : sH X S A.1 C.1 A.2 B2 C.2.1) * sH X S C.1 A.1 B2 A.2 C.2.2.1) - -- = isoId + -- = isoPath -- ((B : cA D) * iso D A B) -- ((C : (B1 : cA X) * ( iso X A.1 B1)) -- * (B2 : sP X S C.1) * (_ : sH X S A.1 C.1 A.2 B2 C.2.1) * sH X S C.1 A.1 B2 A.2 C.2.2.1) @@ -578,45 +578,45 @@ sip (X : precategory) (isC : isCategory X) (S : structure X) (isS : isStandardSt -- : (B : cA D) * iso D A B -- = ((C.1.1, C.2.1), (C.1.2.1, C.2.2.1), (C.1.2.2.1, C.2.2.2) -- , (C.1.2.2.2.1 @ i - -- ,lemIdPProp (sH X S A.1 A.1 A.2 A.2 (cC X A.1 C.1.1 A.1 C.1.2.1 C.1.2.2.1)) - -- (sH X S A.1 A.1 A.2 A.2 (cId X A.1)) + -- ,lemPathPProp (sH X S A.1 A.1 A.2 A.2 (cC X A.1 C.1.1 A.1 C.1.2.1 C.1.2.2.1)) + -- (sH X S A.1 A.1 A.2 A.2 (cPath X A.1)) -- (sHProp X S A.1 A.1 A.2 A.2 (cC X A.1 C.1.1 A.1 C.1.2.1 C.1.2.2.1)) -- (sH X S A.1 A.1 A.2 A.2 (C.1.2.2.2.1 @ i)) -- (sHComp X S A.1 C.1.1 A.1 A.2 C.2.1 A.2 C.1.2.1 C.1.2.2.1 C.2.2.1 C.2.2.2) - -- (sHId X S A.1 A.2) @ i) + -- (sHPath X S A.1 A.2) @ i) -- , (C.1.2.2.2.2 @ i - -- ,lemIdPProp (sH X S C.1.1 C.1.1 C.2.1 C.2.1 (cC X C.1.1 A.1 C.1.1 C.1.2.2.1 C.1.2.1)) - -- (sH X S C.1.1 C.1.1 C.2.1 C.2.1 (cId X C.1.1)) + -- ,lemPathPProp (sH X S C.1.1 C.1.1 C.2.1 C.2.1 (cC X C.1.1 A.1 C.1.1 C.1.2.2.1 C.1.2.1)) + -- (sH X S C.1.1 C.1.1 C.2.1 C.2.1 (cPath X C.1.1)) -- (sHProp X S C.1.1 C.1.1 C.2.1 C.2.1 (cC X C.1.1 A.1 C.1.1 C.1.2.2.1 C.1.2.1)) -- (sH X S C.1.1 C.1.1 C.2.1 C.2.1 (C.1.2.2.2.2 @ i)) -- (sHComp X S C.1.1 A.1 C.1.1 C.2.1 A.2 C.2.1 C.1.2.2.1 C.1.2.1 C.2.2.2 C.2.2.1) - -- (sHId X S C.1.1 C.2.1) @ i)) + -- (sHPath X S C.1.1 C.2.1) @ i)) -- FG (C : ((C : (B1 : cA X) * ( iso X A.1 B1)) -- * (B2 : sP X S C.1) * (_ : sH X S A.1 C.1 A.2 B2 C.2.1) * sH X S C.1 A.1 B2 A.2 C.2.2.1)) - -- : Id ((C : (B1 : cA X) * ( iso X A.1 B1)) + -- : Path ((C : (B1 : cA X) * ( iso X A.1 B1)) -- * (B2 : sP X S C.1) * (_ : sH X S A.1 C.1 A.2 B2 C.2.1) * sH X S C.1 A.1 B2 A.2 C.2.2.1) -- (F (G C)) C -- = <_> C - -- GF (C : (B : cA D) * iso D A B) : Id ((B : cA D) * iso D A B) (G (F C)) C + -- GF (C : (B : cA D) * iso D A B) : Path ((B : cA D) * iso D A B) (G (F C)) C -- = (C.1, C.2.1, C.2.2.1 -- , ((C.2.2.2.1 @ j).1 - -- ,lemIdPSet (sH X S A.1 A.1 A.2 A.2 (cC X A.1 C.1.1 A.1 C.2.1.1 C.2.2.1.1)) - -- (sH X S A.1 A.1 A.2 A.2 (cId X A.1)) + -- ,lemPathPSet (sH X S A.1 A.1 A.2 A.2 (cC X A.1 C.1.1 A.1 C.2.1.1 C.2.2.1.1)) + -- (sH X S A.1 A.1 A.2 A.2 (cPath X A.1)) -- (propSet (sH X S A.1 A.1 A.2 A.2 (cC X A.1 C.1.1 A.1 C.2.1.1 C.2.2.1.1)) -- (sHProp X S A.1 A.1 A.2 A.2 (cC X A.1 C.1.1 A.1 C.2.1.1 C.2.2.1.1))) -- (sH X S A.1 A.1 A.2 A.2 (C.2.2.2.1 @ j).1) -- (sHComp X S A.1 C.1.1 A.1 A.2 C.1.2 A.2 C.2.1.1 C.2.2.1.1 C.2.1.2 C.2.2.1.2) - -- (sHId X S A.1 A.2) + -- (sHPath X S A.1 A.2) -- (((G (F C)).2.2.2.1 @ j).2) -- ((C.2.2.2.1 @ j).2) @i@j) -- , ((C.2.2.2.2 @ j).1 - -- ,lemIdPSet (sH X S C.1.1 C.1.1 C.1.2 C.1.2 (cC X C.1.1 A.1 C.1.1 C.2.2.1.1 C.2.1.1)) - -- (sH X S C.1.1 C.1.1 C.1.2 C.1.2 (cId X C.1.1)) + -- ,lemPathPSet (sH X S C.1.1 C.1.1 C.1.2 C.1.2 (cC X C.1.1 A.1 C.1.1 C.2.2.1.1 C.2.1.1)) + -- (sH X S C.1.1 C.1.1 C.1.2 C.1.2 (cPath X C.1.1)) -- (propSet (sH X S C.1.1 C.1.1 C.1.2 C.1.2 (cC X C.1.1 A.1 C.1.1 C.2.2.1.1 C.2.1.1)) -- (sHProp X S C.1.1 C.1.1 C.1.2 C.1.2 (cC X C.1.1 A.1 C.1.1 C.2.2.1.1 C.2.1.1))) -- (sH X S C.1.1 C.1.1 C.1.2 C.1.2 (C.2.2.2.2 @ j).1) -- (sHComp X S C.1.1 A.1 C.1.1 C.1.2 A.2 C.1.2 C.2.2.1.1 C.2.1.1 C.2.2.1.2 C.2.1.2) - -- (sHId X S C.1.1 C.1.2) + -- (sHPath X S C.1.1 C.1.2) -- (((G (F C)).2.2.2.2 @ j).2) -- ((C.2.2.2.2 @ j).2) @i@j)) -- hole0 (A : cA D) @@ -627,23 +627,23 @@ sip (X : precategory) (isC : isCategory X) (S : structure X) (isS : isStandardSt -- (B2 : sP X S C.1) * (_ : sH X S A.1 C.1 A.2 B2 C.2.1) * sH X S C.1 A.1 B2 A.2 C.2.2.1) -- (isC A.1) -- (\(C : (B1 : cA X) * ( iso X A.1 B1)) -> - -- let p : Id ((B1 : cA X) * ( iso X A.1 B1)) (A.1, idIso X A.1) C + -- let p : Path ((B1 : cA X) * ( iso X A.1 B1)) (A.1, idIso X A.1) C -- = lemIsCategory X isC A.1 C.1 C.2 -- in transport -- ( isContr ((B2 : sP X S (p@i).1) * (_ : sH X S A.1 (p@i).1 A.2 B2 (p@i).2.1) * sH X S (p@i).1 A.1 B2 A.2 (p@i).2.2.1)) - -- ((A.2,sHId X S A.1 A.2,sHId X S A.1 A.2) - -- ,\(Y : (B2 : sP X S A.1) * (_ : sH X S A.1 A.1 A.2 B2 (cId X A.1)) * sH X S A.1 A.1 B2 A.2 (cId X A.1)) -> + -- ((A.2,sHPath X S A.1 A.2,sHPath X S A.1 A.2) + -- ,\(Y : (B2 : sP X S A.1) * (_ : sH X S A.1 A.1 A.2 B2 (cPath X A.1)) * sH X S A.1 A.1 B2 A.2 (cPath X A.1)) -> -- (isS A.1 A.2 Y.1 Y.2.1 Y.2.2 @ i - -- ,lemIdPProp (sH X S A.1 A.1 A.2 A.2 (cId X A.1)) - -- (sH X S A.1 A.1 A.2 Y.1 (cId X A.1)) - -- (sHProp X S A.1 A.1 A.2 A.2 (cId X A.1)) - -- (sH X S A.1 A.1 A.2 (isS A.1 A.2 Y.1 Y.2.1 Y.2.2 @ i) (cId X A.1)) - -- (sHId X S A.1 A.2) Y.2.1 @ i - -- ,lemIdPProp (sH X S A.1 A.1 A.2 A.2 (cId X A.1)) - -- (sH X S A.1 A.1 Y.1 A.2 (cId X A.1)) - -- (sHProp X S A.1 A.1 A.2 A.2 (cId X A.1)) - -- (sH X S A.1 A.1 (isS A.1 A.2 Y.1 Y.2.1 Y.2.2 @ i) A.2 (cId X A.1)) - -- (sHId X S A.1 A.2) Y.2.2 @ i))) + -- ,lemPathPProp (sH X S A.1 A.1 A.2 A.2 (cPath X A.1)) + -- (sH X S A.1 A.1 A.2 Y.1 (cPath X A.1)) + -- (sHProp X S A.1 A.1 A.2 A.2 (cPath X A.1)) + -- (sH X S A.1 A.1 A.2 (isS A.1 A.2 Y.1 Y.2.1 Y.2.2 @ i) (cPath X A.1)) + -- (sHPath X S A.1 A.2) Y.2.1 @ i + -- ,lemPathPProp (sH X S A.1 A.1 A.2 A.2 (cPath X A.1)) + -- (sH X S A.1 A.1 Y.1 A.2 (cPath X A.1)) + -- (sHProp X S A.1 A.1 A.2 A.2 (cPath X A.1)) + -- (sH X S A.1 A.1 (isS A.1 A.2 Y.1 Y.2.1 Y.2.2 @ i) A.2 (cPath X A.1)) + -- (sHPath X S A.1 A.2) Y.2.2 @ i))) -- hole (A : cA D) : isContr ((B : cA D) * iso D A B) = transport (isContr(eq1 A@-i)) (hole0 A) opaque sip @@ -652,26 +652,26 @@ opaque sip functor (A B : precategory) : U = (Fob : cA A -> cA B) * (Fmor : (x y : cA A) -> cH A x y -> cH B (Fob x) (Fob y)) - * (Fid : (x : cA A) -> Id (cH B (Fob x) (Fob x)) (Fmor x x (cId A x)) (cId B (Fob x))) - * ((x y z : cA A) -> (f : cH A x y) -> (g : cH A y z) -> Id (cH B (Fob x) (Fob z)) (Fmor x z (cC A x y z f g)) (cC B (Fob x) (Fob y) (Fob z) (Fmor x y f) (Fmor y z g))) + * (Fid : (x : cA A) -> Path (cH B (Fob x) (Fob x)) (Fmor x x (cPath A x)) (cPath B (Fob x))) + * ((x y z : cA A) -> (f : cH A x y) -> (g : cH A y z) -> Path (cH B (Fob x) (Fob z)) (Fmor x z (cC A x y z f g)) (cC B (Fob x) (Fob y) (Fob z) (Fmor x y f) (Fmor y z g))) idFunctor (A : precategory) : functor A A = (\(x : cA A) -> x ,\(x y : cA A) (h : cH A x y) -> h - ,\(x : cA A) -> <_> cId A x + ,\(x : cA A) -> <_> cPath A x ,\(x y z : cA A) (f : cH A x y) (g : cH A y z) -> <_> cC A x y z f g) compFunctor (A B C : precategory) (F : functor A B) (G : functor B C) : functor A C = (\(x : cA A) -> G.1 (F.1 x) ,\(x y : cA A) (h : cH A x y) -> G.2.1 (F.1 x) (F.1 y) (F.2.1 x y h) - ,\(x : cA A) -> compId (cH C (G.1 (F.1 x)) (G.1 (F.1 x))) - (G.2.1 (F.1 x) (F.1 x) (F.2.1 x x (cId A x))) - (G.2.1 (F.1 x) (F.1 x) (cId B (F.1 x))) - (cId C (G.1 (F.1 x))) + ,\(x : cA A) -> compPath (cH C (G.1 (F.1 x)) (G.1 (F.1 x))) + (G.2.1 (F.1 x) (F.1 x) (F.2.1 x x (cPath A x))) + (G.2.1 (F.1 x) (F.1 x) (cPath B (F.1 x))) + (cPath C (G.1 (F.1 x))) (G.2.1 (F.1 x) (F.1 x) (F.2.2.1 x @ i)) (G.2.2.1 (F.1 x)) ,\(x y z : cA A) (f : cH A x y) (g : cH A y z) -> - compId (cH C (G.1 (F.1 x)) (G.1 (F.1 z))) + compPath (cH C (G.1 (F.1 x)) (G.1 (F.1 z))) (G.2.1 (F.1 x) (F.1 z) (F.2.1 x z (cC A x y z f g))) (G.2.1 (F.1 x) (F.1 z) (cC B (F.1 x) (F.1 y) (F.1 z) (F.2.1 x y f) (F.2.1 y z g))) (cC C (G.1 (F.1 x)) (G.1 (F.1 y)) (G.1 (F.1 z)) (G.2.1 (F.1 x) (F.1 y) (F.2.1 x y f)) (G.2.1 (F.1 y) (F.1 z) (F.2.1 y z g))) @@ -682,109 +682,109 @@ compFunctor (A B C : precategory) (F : functor A B) (G : functor B C) : functor ffFunctor (A B : precategory) (F : functor A B) : U = (a b : cA A) -> isEquiv (cH A a b) (cH B (F.1 a) (F.1 b)) (F.2.1 a b) ffEq (A B : precategory) (F : functor A B) (ff : ffFunctor A B F) (a b : cA A) - : Id U (cH A a b) (cH B (F.1 a) (F.1 b)) - = equivId (cH A a b) (cH B (F.1 a) (F.1 b)) (F.2.1 a b) (ff a b) + : Path U (cH A a b) (cH B (F.1 a) (F.1 b)) + = equivPath (cH A a b) (cH B (F.1 a) (F.1 b)) (F.2.1 a b) (ff a b) propFFFunctor (A B : precategory) (F : functor A B) : prop (ffFunctor A B F) = propPi (cA A) (\(a : cA A) -> (b : cA A) -> isEquiv (cH A a b) (cH B (F.1 a) (F.1 b)) (F.2.1 a b)) (\(a : cA A) -> propPi (cA A) (\(b : cA A) -> isEquiv (cH A a b) (cH B (F.1 a) (F.1 b)) (F.2.1 a b)) (\(b : cA A) -> propIsEquiv (cH A a b) (cH B (F.1 a) (F.1 b)) (F.2.1 a b))) -lem10 (A B : U) (e : equiv A B) (x y : B) (p : Id A (e.2 x).1.1 (e.2 y).1.1) : Id B x y +lem10 (A B : U) (e : equiv A B) (x y : B) (p : Path A (e.2 x).1.1 (e.2 y).1.1) : Path B x y = transport - (compId U (Id B (e.1 (e.2 x).1.1) (e.1 (e.2 y).1.1)) (Id B x (e.1 (e.2 y).1.1)) (Id B x y) - ( Id B (retEq A B e x @ i) (e.1 (e.2 y).1.1)) ( Id B x (retEq A B e y @ i))) + (compPath U (Path B (e.1 (e.2 x).1.1) (e.1 (e.2 y).1.1)) (Path B x (e.1 (e.2 y).1.1)) (Path B x y) + ( Path B (retEq A B e x @ i) (e.1 (e.2 y).1.1)) ( Path B x (retEq A B e y @ i))) (mapOnPath A B e.1 (e.2 x).1.1 (e.2 y).1.1 p) opaque lem10 -lem10' (A B : U) (e : equiv A B) (x y : A) (p : Id B (e.1 x) (e.1 y)) : Id A x y +lem10' (A B : U) (e : equiv A B) (x y : A) (p : Path B (e.1 x) (e.1 y)) : Path A x y = transport - (compId U (Id A (e.2 (e.1 x)).1.1 (e.2 (e.1 y)).1.1) (Id A x (e.2 (e.1 y)).1.1) (Id A x y) - ( Id A (secEq A B e x @ i) (e.2 (e.1 y)).1.1) ( Id A x (secEq A B e y @ i)) + (compPath U (Path A (e.2 (e.1 x)).1.1 (e.2 (e.1 y)).1.1) (Path A x (e.2 (e.1 y)).1.1) (Path A x y) + ( Path A (secEq A B e x @ i) (e.2 (e.1 y)).1.1) ( Path A x (secEq A B e y @ i)) ) (mapOnPath B A (invEq A B e) (e.1 x) (e.1 y) p) opaque lem10' lemFF (A B : precategory) (F : functor A B) (ff : ffFunctor A B F) (x y : cA A) - : Id U (iso A x y) (iso B (F.1 x) (F.1 y)) + : Path U (iso A x y) (iso B (F.1 x) (F.1 y)) = hole where F0 (f : iso A x y) : iso B (F.1 x) (F.1 y) = (F.2.1 x y f.1, F.2.1 y x f.2.1 - ,compId (cH B (F.1 x) (F.1 x)) + ,compPath (cH B (F.1 x) (F.1 x)) (cC B (F.1 x) (F.1 y) (F.1 x) (F.2.1 x y f.1) (F.2.1 y x f.2.1)) (F.2.1 x x (cC A x y x f.1 f.2.1)) - (cId B (F.1 x)) + (cPath B (F.1 x)) (F.2.2.2 x y x f.1 f.2.1 @-i) - (compId (cH B (F.1 x) (F.1 x)) + (compPath (cH B (F.1 x) (F.1 x)) (F.2.1 x x (cC A x y x f.1 f.2.1)) - (F.2.1 x x (cId A x)) - (cId B (F.1 x)) + (F.2.1 x x (cPath A x)) + (cPath B (F.1 x)) (F.2.1 x x (f.2.2.1@i)) (F.2.2.1 x)) - ,compId (cH B (F.1 y) (F.1 y)) + ,compPath (cH B (F.1 y) (F.1 y)) (cC B (F.1 y) (F.1 x) (F.1 y) (F.2.1 y x f.2.1) (F.2.1 x y f.1)) (F.2.1 y y (cC A y x y f.2.1 f.1)) - (cId B (F.1 y)) + (cPath B (F.1 y)) (F.2.2.2 y x y f.2.1 f.1 @-i) - (compId (cH B (F.1 y) (F.1 y)) + (compPath (cH B (F.1 y) (F.1 y)) (F.2.1 y y (cC A y x y f.2.1 f.1)) - (F.2.1 y y (cId A y)) - (cId B (F.1 y)) + (F.2.1 y y (cPath A y)) + (cPath B (F.1 y)) (F.2.1 y y (f.2.2.2@i)) (F.2.2.1 y))) G0 (g : iso B (F.1 x) (F.1 y)) : iso A x y = ((ff x y g.1).1.1 ,(ff y x g.2.1).1.1 - ,lem10' (cH A x x) (cH B (F.1 x) (F.1 x)) (F.2.1 x x, ff x x) (cC A x y x (ff x y g.1).1.1 (ff y x g.2.1).1.1) (cId A x) - (compId (cH B (F.1 x) (F.1 x)) + ,lem10' (cH A x x) (cH B (F.1 x) (F.1 x)) (F.2.1 x x, ff x x) (cC A x y x (ff x y g.1).1.1 (ff y x g.2.1).1.1) (cPath A x) + (compPath (cH B (F.1 x) (F.1 x)) (F.2.1 x x (cC A x y x (ff x y g.1).1.1 (ff y x g.2.1).1.1)) (cC B (F.1 x) (F.1 y) (F.1 x) (F.2.1 x y (ff x y g.1).1.1) (F.2.1 y x (ff y x g.2.1).1.1)) - (F.2.1 x x (cId A x)) + (F.2.1 x x (cPath A x)) (F.2.2.2 x y x (ff x y g.1).1.1 (ff y x g.2.1).1.1) - (compId (cH B (F.1 x) (F.1 x)) + (compPath (cH B (F.1 x) (F.1 x)) (cC B (F.1 x) (F.1 y) (F.1 x) (F.2.1 x y (ff x y g.1).1.1) (F.2.1 y x (ff y x g.2.1).1.1)) (cC B (F.1 x) (F.1 y) (F.1 x) g.1 g.2.1) - (F.2.1 x x (cId A x)) + (F.2.1 x x (cPath A x)) (cC B (F.1 x) (F.1 y) (F.1 x) ((ff x y g.1).1.2@-i) ((ff y x g.2.1).1.2@-i)) - (compId (cH B (F.1 x) (F.1 x)) + (compPath (cH B (F.1 x) (F.1 x)) (cC B (F.1 x) (F.1 y) (F.1 x) g.1 g.2.1) - (cId B (F.1 x)) - (F.2.1 x x (cId A x)) + (cPath B (F.1 x)) + (F.2.1 x x (cPath A x)) g.2.2.1 (F.2.2.1 x @ -i)))) - ,lem10' (cH A y y) (cH B (F.1 y) (F.1 y)) (F.2.1 y y, ff y y) (cC A y x y (ff y x g.2.1).1.1 (ff x y g.1).1.1) (cId A y) - (compId (cH B (F.1 y) (F.1 y)) + ,lem10' (cH A y y) (cH B (F.1 y) (F.1 y)) (F.2.1 y y, ff y y) (cC A y x y (ff y x g.2.1).1.1 (ff x y g.1).1.1) (cPath A y) + (compPath (cH B (F.1 y) (F.1 y)) (F.2.1 y y (cC A y x y (ff y x g.2.1).1.1 (ff x y g.1).1.1)) (cC B (F.1 y) (F.1 x) (F.1 y) (F.2.1 y x (ff y x g.2.1).1.1) (F.2.1 x y (ff x y g.1).1.1)) - (F.2.1 y y (cId A y)) + (F.2.1 y y (cPath A y)) (F.2.2.2 y x y (ff y x g.2.1).1.1 (ff x y g.1).1.1) - (compId (cH B (F.1 y) (F.1 y)) + (compPath (cH B (F.1 y) (F.1 y)) (cC B (F.1 y) (F.1 x) (F.1 y) (F.2.1 y x (ff y x g.2.1).1.1) (F.2.1 x y (ff x y g.1).1.1)) (cC B (F.1 y) (F.1 x) (F.1 y) g.2.1 g.1) - (F.2.1 y y (cId A y)) + (F.2.1 y y (cPath A y)) (cC B (F.1 y) (F.1 x) (F.1 y) ((ff y x g.2.1).1.2@-i) ((ff x y g.1).1.2@-i)) - (compId (cH B (F.1 y) (F.1 y)) + (compPath (cH B (F.1 y) (F.1 y)) (cC B (F.1 y) (F.1 x) (F.1 y) g.2.1 g.1) - (cId B (F.1 y)) - (F.2.1 y y (cId A y)) + (cPath B (F.1 y)) + (F.2.1 y y (cPath A y)) g.2.2.2 (F.2.2.1 y @ -i)))) ) - FG (g : iso B (F.1 x) (F.1 y)) : Id (iso B (F.1 x) (F.1 y)) (F0 (G0 g)) g + FG (g : iso B (F.1 x) (F.1 y)) : Path (iso B (F.1 x) (F.1 y)) (F0 (G0 g)) g = isoEq B (F.1 x) (F.1 y) (F0 (G0 g)) g ((ff x y g.1).1.2@-i) ((ff y x g.2.1).1.2@-i) - GF (f : iso A x y) : Id (iso A x y) (G0 (F0 f)) f + GF (f : iso A x y) : Path (iso A x y) (G0 (F0 f)) f = isoEq A x y (G0 (F0 f)) f ( ((ff x y (F.2.1 x y f.1)).2 (f.1,F.2.1 x y f.1)@i).1) ( ((ff y x (F.2.1 y x f.2.1)).2 (f.2.1,F.2.1 y x f.2.1)@i).1) - hole : Id U (iso A x y) (iso B (F.1 x) (F.1 y)) = isoId (iso A x y) (iso B (F.1 x) (F.1 y)) F0 G0 FG GF + hole : Path U (iso A x y) (iso B (F.1 x) (F.1 y)) = isoPath (iso A x y) (iso B (F.1 x) (F.1 y)) F0 G0 FG GF opaque lemFF F12 (A B : precategory) (isC : isCategory A) (F : functor A B) (p1 : ffFunctor A B F) (x : cA A) : isContr ((y : cA A) * iso B (F.1 y) (F.1 x)) = transport (isContr ((y : cA A) * (invIsoEq B (F.1 x) (F.1 y)@i))) hole where - hole2 (y : cA A) : Id U (iso A x y) (iso B (F.1 x) (F.1 y)) + hole2 (y : cA A) : Path U (iso A x y) (iso B (F.1 x) (F.1 y)) = lemFF A B F p1 x y hole : isContr ((y : cA A) * iso B (F.1 x) (F.1 y)) = transport ( isContr ((y : cA A) * (hole2 y @ i))) (isC x) @@ -792,37 +792,37 @@ opaque F12 F23 (A B : precategory) (F : functor A B) (p2 : (x : cA A) -> isContr ((y : cA A) * iso B (F.1 y) (F.1 x))) (x : cA B) - (a b : (y : cA A) * iso B (F.1 y) x) : Id ((y : cA A) * iso B (F.1 y) x) a b + (a b : (y : cA A) * iso B (F.1 y) x) : Path ((y : cA A) * iso B (F.1 y) x) a b = undefined -- = transport p hole3 -- where --- hole2 : Id ((y : cA A) * iso B (F.1 y) (F.1 a.1)) +-- hole2 : Path ((y : cA A) * iso B (F.1 y) (F.1 a.1)) -- (a.1, idIso B (F.1 a.1)) (b.1, compIso B (F.1 b.1) x (F.1 a.1) b.2 (invIso B (F.1 a.1) x a.2)) -- = isContrProp ((y : cA A) * iso B (F.1 y) (F.1 a.1)) (p2 a.1) -- (a.1, idIso B (F.1 a.1)) (b.1, compIso B (F.1 b.1) x (F.1 a.1) b.2 (invIso B (F.1 a.1) x a.2)) -- opaque hole2 --- hole3 : Id ((y : cA A) * iso B (F.1 y) x) +-- hole3 : Path ((y : cA A) * iso B (F.1 y) x) -- (a.1, compIso B (F.1 a.1) (F.1 a.1) x (idIso B (F.1 a.1)) a.2) -- (b.1, compIso B (F.1 b.1) (F.1 a.1) x (compIso B (F.1 b.1) x (F.1 a.1) b.2 (invIso B (F.1 a.1) x a.2)) a.2) -- = ((hole2@i).1, compIso B (F.1 (hole2@i).1) (F.1 a.1) x (hole2@i).2 a.2) -- opaque hole3 --- p : Id U (Id ((y : cA A) * iso B (F.1 y) x) +-- p : Path U (Path ((y : cA A) * iso B (F.1 y) x) -- (a.1, compIso B (F.1 a.1) (F.1 a.1) x (idIso B (F.1 a.1)) a.2) -- (b.1, compIso B (F.1 b.1) (F.1 a.1) x (compIso B (F.1 b.1) x (F.1 a.1) b.2 (invIso B (F.1 a.1) x a.2)) a.2)) --- (Id ((y : cA A) * iso B (F.1 y) x) a b) --- = (Id ((y : cA A) * iso B (F.1 y) x) --- (a.1, IdLIso B (F.1 a.1) x a.2 @ i) --- (b.1, compId (iso B (F.1 b.1) x) +-- (Path ((y : cA A) * iso B (F.1 y) x) a b) +-- = (Path ((y : cA A) * iso B (F.1 y) x) +-- (a.1, PathLIso B (F.1 a.1) x a.2 @ i) +-- (b.1, compPath (iso B (F.1 b.1) x) -- (compIso B (F.1 b.1) (F.1 a.1) x (compIso B (F.1 b.1) x (F.1 a.1) b.2 (invIso B (F.1 a.1) x a.2)) a.2) -- (compIso B (F.1 b.1) x x b.2 (compIso B x (F.1 a.1) x (invIso B (F.1 a.1) x a.2) a.2)) -- b.2 --- (IdCIso B (F.1 b.1) x (F.1 a.1) x b.2 (invIso B (F.1 a.1) x a.2) a.2) --- (compId (iso B (F.1 b.1) x) +-- (PathCIso B (F.1 b.1) x (F.1 a.1) x b.2 (invIso B (F.1 a.1) x a.2) a.2) +-- (compPath (iso B (F.1 b.1) x) -- (compIso B (F.1 b.1) x x b.2 (compIso B x (F.1 a.1) x (invIso B (F.1 a.1) x a.2) a.2)) -- (compIso B (F.1 b.1) x x b.2 (idIso B x)) -- b.2 --- (compIso B (F.1 b.1) x x b.2 (IdInvLIso B (F.1 a.1) x a.2 @ i)) --- (IdRIso B (F.1 b.1) x b.2)) +-- (compIso B (F.1 b.1) x x b.2 (PathInvLIso B (F.1 a.1) x a.2 @ i)) +-- (PathRIso B (F.1 b.1) x b.2)) -- @ i)) opaque F23 @@ -835,10 +835,10 @@ catPropIsEquiv (A B : precategory) (isC : isCategory A) (F : functor A B) : prop (propFFFunctor A B F) (\(ff : ffFunctor A B F) -> propPi (cA B) (\(b : cA B) -> (a : cA A) * iso B (F.1 a) b) (\(b : cA B) -> F23 A B F (F12 A B isC F ff) b)) -catIdIsEquiv (A : precategory) : catIsEquiv A A (idFunctor A) +catPathIsEquiv (A : precategory) : catIsEquiv A A (idFunctor A) = (\(a b : cA A) -> idIsEquiv (cH A a b) ,\(b:cA A) -> (b, idIso A b)) -catIdEquiv (A : precategory) : catEquiv A A = (idFunctor A, catIdIsEquiv A) +catPathEquiv (A : precategory) : catEquiv A A = (idFunctor A, catPathIsEquiv A) catIsIso (A B : precategory) (F : functor A B) : U = (_ : ffFunctor A B F) * isEquiv (cA A) (cA B) F.1 @@ -851,12 +851,12 @@ catIso (A B : precategory) : U = (F : functor A B) * catIsIso A B F catIso21 (A B : precategory) (e1:equiv (cA A) (cA B)) : U = (x y:cA A) -> equiv (cH A x y) (cH B (e1.1 x) (e1.1 y)) catIso22 (A B : precategory) (e1:equiv (cA A) (cA B)) (e2:catIso21 A B e1) : U - = (x:cA A) -> Id (cH B (e1.1 x) (e1.1 x)) - ((e2 x x).1 (cId A x)) - (cId B (e1.1 x)) + = (x:cA A) -> Path (cH B (e1.1 x) (e1.1 x)) + ((e2 x x).1 (cPath A x)) + (cPath B (e1.1 x)) catIso23 (A B : precategory) (e1:equiv (cA A) (cA B)) (e2:catIso21 A B e1) : U = (x y z:cA A)(f:cH A x y)(g:cH A y z)-> - Id (cH B (e1.1 x) (e1.1 z)) + Path (cH B (e1.1 x) (e1.1 z)) ((e2 x z).1 (cC A x y z f g)) (cC B (e1.1 x) (e1.1 y) (e1.1 z) ((e2 x y).1 f) ((e2 y z).1 g)) catIso2 (A B : precategory) : U = (e1 : equiv (cA A) (cA B)) @@ -864,22 +864,22 @@ catIso2 (A B : precategory) : U = (e1 : equiv (cA A) (cA B)) * (_ : catIso22 A B e1 e2) * ( catIso23 A B e1 e2) -eCatIso (A B : precategory) : Id U (catIso A B) (catIso2 A B) - = transEquivToId (catIso A B) (catIso2 A B) +eCatIso (A B : precategory) : Path U (catIso A B) (catIso2 A B) + = transEquivToPath (catIso A B) (catIso2 A B) (F, gradLemma (catIso A B) (catIso2 A B) F G (\(e:catIso2 A B)-><_>e) (\(e:catIso A B)-><_>e)) where F(e:catIso A B):catIso2 A B=((e.1.1, e.2.2),\(x y:cA A)->(e.1.2.1 x y, e.2.1 x y),e.1.2.2.1,e.1.2.2.2) G(e:catIso2 A B):catIso A B=((e.1.1,\(x y:cA A)->(e.2.1 x y).1,e.2.2.1,e.2.2.2),\(x y:cA A)->(e.2.1 x y).2,e.1.2) -catIso21' (A B : precategory) (e1:Id U (cA A) (cA B)) : U +catIso21' (A B : precategory) (e1:Path U (cA A) (cA B)) : U = (x y:cA A) -> equiv (cH A x y) (cH B (transport e1 x) (transport e1 y)) -catIso22' (A B : precategory) (e1:Id U (cA A) (cA B)) (e2:catIso21' A B e1) : U - = (x:cA A) -> Id (cH B (transport e1 x) (transport e1 x)) - ((e2 x x).1 (cId A x)) - (cId B (transport e1 x)) -catIso23' (A B : precategory) (e1:Id U (cA A) (cA B)) (e2:catIso21' A B e1) : U +catIso22' (A B : precategory) (e1:Path U (cA A) (cA B)) (e2:catIso21' A B e1) : U + = (x:cA A) -> Path (cH B (transport e1 x) (transport e1 x)) + ((e2 x x).1 (cPath A x)) + (cPath B (transport e1 x)) +catIso23' (A B : precategory) (e1:Path U (cA A) (cA B)) (e2:catIso21' A B e1) : U = (x y z:cA A)(f:cH A x y)(g:cH A y z)-> - Id (cH B (transport e1 x) (transport e1 z)) + Path (cH B (transport e1 x) (transport e1 z)) ((e2 x z).1 (cC A x y z f g)) (cC B (transport e1 x) (transport e1 y) (transport e1 z) ((e2 x y).1 f) ((e2 y z).1 g)) @@ -891,20 +891,20 @@ equivPi (A:U) (B0 B1:A->U) (e:(a:A)->equiv (B0 a) (B1 a)) : equiv ((a:A)->B0 a) ) where F(f:(a:A)->B0 a)(a:A):B1 a= (e a).1 (f a) -eCatIso2 (A B:precategory):Id U (catIso32 A B) (catIso2 A B) - = sigEquivLem' (Id U (cA A) (cA B)) (equiv (cA A) (cA B)) - (\(e1:Id U (cA A) (cA B)) -> (e2:catIso321 A B e1)*(_:catIso322 A B e1 e2)*(catIso323 A B e1 e2)) +eCatIso2 (A B:precategory):Path U (catIso32 A B) (catIso2 A B) + = sigEquivLem' (Path U (cA A) (cA B)) (equiv (cA A) (cA B)) + (\(e1:Path U (cA A) (cA B)) -> (e2:catIso321 A B e1)*(_:catIso322 A B e1 e2)*(catIso323 A B e1 e2)) (\(e1:equiv (cA A) (cA B)) -> (e2:catIso21 A B e1)*(_:catIso22 A B e1 e2)*(catIso23 A B e1 e2)) (corrUniv' (cA A) (cA B)) - (\(e1:Id U (cA A) (cA B)) -> let e1' : equiv (cA A) (cA B) = transEquiv' (cA B) (cA A) e1 - p0 (x:cA A):Id (cA B) (transport e1 x) (e1'.1 x)=lemTransEquiv (cA A) (cA B) e1 x + (\(e1:Path U (cA A) (cA B)) -> let e1' : equiv (cA A) (cA B) = transEquiv' (cA B) (cA A) e1 + p0 (x:cA A):Path (cA B) (transport e1 x) (e1'.1 x)=lemTransEquiv (cA A) (cA B) e1 x in transport - ( Id U ((e2:catIso321 A B e1)*(_:catIso322 A B e1 e2)*(catIso323 A B e1 e2)) + ( Path U ((e2:catIso321 A B e1)*(_:catIso322 A B e1 e2)*(catIso323 A B e1 e2)) ((e2 : (x y:cA A) -> equiv (cH A x y) (cH B (p0 x@i) (p0 y@i))) - * (_ : (x:cA A) -> Id (cH B (p0 x@i) (p0 x@i)) ((e2 x x).1 (cId A x)) (cId B (p0 x@i))) + * (_ : (x:cA A) -> Path (cH B (p0 x@i) (p0 x@i)) ((e2 x x).1 (cPath A x)) (cPath B (p0 x@i))) * ((x y z:cA A)(f:cH A x y)(g:cH A y z)-> - Id (cH B (p0 x@i) (p0 z@i)) + Path (cH B (p0 x@i) (p0 z@i)) ((e2 x z).1 (cC A x y z f g)) (cC B (p0 x@i) (p0 y@i) (p0 z@i) ((e2 x y).1 f) ((e2 y z).1 g)))) ) @@ -912,20 +912,20 @@ eCatIso2 (A B:precategory):Id U (catIso32 A B) (catIso2 A B) (\(e2:catIso321 A B e1) -> (_:catIso322 A B e1 e2)*(catIso323 A B e1 e2)) (\(e2:catIso21' A B e1) -> (_:catIso22' A B e1 e2)*(catIso23' A B e1 e2)) (equivPi (cA A) - (\(x:cA A) -> (y:cA A) -> Id U (cH A x y) (cH B (transport e1 x) (transport e1 y))) + (\(x:cA A) -> (y:cA A) -> Path U (cH A x y) (cH B (transport e1 x) (transport e1 y))) (\(x:cA A) -> (y:cA A) -> equiv (cH A x y) (cH B (transport e1 x) (transport e1 y))) (\(x:cA A) -> equivPi (cA A) - (\(y:cA A) -> Id U (cH A x y) (cH B (transport e1 x) (transport e1 y))) + (\(y:cA A) -> Path U (cH A x y) (cH B (transport e1 x) (transport e1 y))) (\(y:cA A) -> equiv (cH A x y) (cH B (transport e1 x) (transport e1 y))) (\(y:cA A) -> corrUniv' (cH A x y) (cH B (transport e1 x) (transport e1 y))))) (\(e2:catIso321 A B e1) -> let e2' (x y:cA A) : equiv (cH A x y) (cH B (transport e1 x) (transport e1 y)) = transEquiv' (cH B (transport e1 x) (transport e1 y)) (cH A x y) (e2 x y) - p2 (x y:cA A)(f:cH A x y):Id (cH B (transport e1 x) (transport e1 y)) (transport (e2 x y) f) ((e2' x y).1 f) + p2 (x y:cA A)(f:cH A x y):Path (cH B (transport e1 x) (transport e1 y)) (transport (e2 x y) f) ((e2' x y).1 f) = lemTransEquiv (cH A x y) (cH B (transport e1 x) (transport e1 y)) (e2 x y) f in - (_:(x:cA A) -> Id (cH B (transport e1 x) (transport e1 x)) (p2 x x (cId A x)@i) (cId B (transport e1 x))) + (_:(x:cA A) -> Path (cH B (transport e1 x) (transport e1 x)) (p2 x x (cPath A x)@i) (cPath B (transport e1 x))) * ((x y z:cA A)(f:cH A x y)(g:cH A y z)-> - Id (cH B (transport e1 x) (transport e1 z)) + Path (cH B (transport e1 x) (transport e1 z)) (p2 x z (cC A x y z f g)@i) (cC B (transport e1 x) (transport e1 y) (transport e1 z) (p2 x y f@i) (p2 y z g@i))) ))) @@ -934,8 +934,8 @@ eCatIso2 (A B:precategory):Id U (catIso32 A B) (catIso2 A B) catIsoIsEquiv (A B : precategory) (F : functor A B) (e : catIsIso A B F) : catIsEquiv A B F = (e.1,\(b:cA B)->((e.2 b).1.1, eqToIso B (F.1 (e.2 b).1.1) b ((e.2 b).1.2@-i))) -invEquiv (A:U) (a b:A) : Id U (Id A a b) (Id A b a) - = equivId (Id A a b) (Id A b a) (inv A a b) (gradLemma (Id A a b) (Id A b a) (inv A a b) (inv A b a) (\(x:Id A b a) -> <_> x) (\(x:Id A a b) -> <_> x)) +invEquiv (A:U) (a b:A) : Path U (Path A a b) (Path A b a) + = equivPath (Path A a b) (Path A b a) (inv A a b) (gradLemma (Path A a b) (Path A b a) (inv A a b) (inv A b a) (\(x:Path A b a) -> <_> x) (\(x:Path A a b) -> <_> x)) catEquivIsIso (A B : precategory) (isCA : isCategory A) (isCB : isCategory B) (F : functor A B) (e : catIsEquiv A B F) : catIsIso A B F @@ -953,31 +953,31 @@ catIsEquivEqIso (A B : precategory) (isCA : isCategory A) (isCB : isCategory B) = equivProp (catIsEquiv A B F) (catIsIso A B F) (catPropIsEquiv A B isCA F) (catPropIsIso A B F) (catEquivIsIso A B isCA isCB F) (catIsoIsEquiv A B F) -catEquivEqIso (A B : precategory) (isCA : isCategory A) (isCB : isCategory B) : Id U (catEquiv A B) (catIso A B) - = (F : functor A B) * (transEquivToId (catIsEquiv A B F) (catIsIso A B F) (catIsEquivEqIso A B isCA isCB F) @ i) +catEquivEqIso (A B : precategory) (isCA : isCategory A) (isCB : isCategory B) : Path U (catEquiv A B) (catIso A B) + = (F : functor A B) * (transEquivToPath (catIsEquiv A B F) (catIsIso A B F) (catIsEquivEqIso A B isCA isCB F) @ i) -catIsoEqId (A B : precategory) : Id U (catIso A B) (Id precategory A B) - = compId U (catIso A B) (catIso2 A B) (Id precategory A B) +catIsoEqPath (A B : precategory) : Path U (catIso A B) (Path precategory A B) + = compPath U (catIso A B) (catIso2 A B) (Path precategory A B) (eCatIso A B) - (compId U (catIso2 A B) (catIso32 A B) (Id precategory A B) + (compPath U (catIso2 A B) (catIso32 A B) (Path precategory A B) (eCatIso2 A B@-i) - (compId U (catIso32 A B) (catIso31 A B) (Id precategory A B) + (compPath U (catIso32 A B) (catIso31 A B) (Path precategory A B) (eCatIso32 A B@-i) - (compId U (catIso31 A B) (catIso30 A B) (Id precategory A B) + (compPath U (catIso31 A B) (catIso30 A B) (Path precategory A B) (eCatIso31 A B@-i) - (compId U (catIso30 A B) (catIso3 A B) (Id precategory A B) + (compPath U (catIso30 A B) (catIso3 A B) (Path precategory A B) (eCatIso30 A B@-i) (eCatIso3 A B@-i))))) -catEquivEqId (A B : precategory) (isCA : isCategory A) (isCB : isCategory B) : Id U (catEquiv A B) (Id precategory A B) - = compId U (catEquiv A B) (catIso A B) (Id precategory A B) (catEquivEqIso A B isCA isCB) (catIsoEqId A B) +catEquivEqPath (A B : precategory) (isCA : isCategory A) (isCB : isCategory B) : Path U (catEquiv A B) (Path precategory A B) + = compPath U (catEquiv A B) (catIso A B) (Path precategory A B) (catEquivEqIso A B isCA isCB) (catIsoEqPath A B) -catEquivEqId' (A : category) : isContr ((B : category) * catEquiv A.1 B.1) - = transport ( isContr ((B : category) * catEquivEqId A.1 B.1 A.2 B.2@-i)) +catEquivEqPath' (A : category) : isContr ((B : category) * catEquiv A.1 B.1) + = transport ( isContr ((B : category) * catEquivEqPath A.1 B.1 A.2 B.2@-i)) ((A, <_> A.1) - ,\(BB:((B : category) * Id precategory A.1 B.1))-> + ,\(BB:((B : category) * Path precategory A.1 B.1))-> ((BB.2@i - ,lemIdPProp (isCategory A.1) + ,lemPathPProp (isCategory A.1) (isCategory BB.1.1) (propIsCategory A.1) ( isCategory (BB.2@i)) @@ -998,45 +998,45 @@ cospan (C : precategory) : U = (X : cA C) * (_ : homTo C X) * homTo C X hasCospanCone (C : precategory) (D : cospan C) (W : cA C) : U = (f : cH C W D.2.1.1) * (g : cH C W D.2.2.1) - * Id (cH C W D.1) + * Path (cH C W D.1) (cC C W D.2.1.1 D.1 f D.2.1.2) (cC C W D.2.2.1 D.1 g D.2.2.2) cospanCone (C : precategory) (D : cospan C) : U = (W : cA C) * hasCospanCone C D W isCospanConeHom (C : precategory) (D : cospan C) (E1 E2 : cospanCone C D) (h : cH C E1.1 E2.1) : U - = (_ : Id (cH C E1.1 D.2.1.1) (cC C E1.1 E2.1 D.2.1.1 h E2.2.1) E1.2.1) - * (Id (cH C E1.1 D.2.2.1) (cC C E1.1 E2.1 D.2.2.1 h E2.2.2.1) E1.2.2.1) + = (_ : Path (cH C E1.1 D.2.1.1) (cC C E1.1 E2.1 D.2.1.1 h E2.2.1) E1.2.1) + * (Path (cH C E1.1 D.2.2.1) (cC C E1.1 E2.1 D.2.2.1 h E2.2.2.1) E1.2.2.1) isCospanConeHomProp (C : precategory) (D : cospan C) (E1 E2 : cospanCone C D) (h : cH C E1.1 E2.1) : prop (isCospanConeHom C D E1 E2 h) - = propAnd (Id (cH C E1.1 D.2.1.1) (cC C E1.1 E2.1 D.2.1.1 h E2.2.1) E1.2.1) - (Id (cH C E1.1 D.2.2.1) (cC C E1.1 E2.1 D.2.2.1 h E2.2.2.1) E1.2.2.1) + = propAnd (Path (cH C E1.1 D.2.1.1) (cC C E1.1 E2.1 D.2.1.1 h E2.2.1) E1.2.1) + (Path (cH C E1.1 D.2.2.1) (cC C E1.1 E2.1 D.2.2.1 h E2.2.2.1) E1.2.2.1) (cHSet C E1.1 D.2.1.1 (cC C E1.1 E2.1 D.2.1.1 h E2.2.1) E1.2.1) (cHSet C E1.1 D.2.2.1 (cC C E1.1 E2.1 D.2.2.1 h E2.2.2.1) E1.2.2.1) cospanConeHom (C : precategory) (D : cospan C) (E1 E2 : cospanCone C D) : U = (h : cH C E1.1 E2.1) * isCospanConeHom C D E1 E2 h -cospanConeId (C : precategory) (D : cospan C) (E : cospanCone C D) : cospanConeHom C D E E - = (cId C E.1, cIdL C E.1 D.2.1.1 E.2.1, cIdL C E.1 D.2.2.1 E.2.2.1) +cospanConePath (C : precategory) (D : cospan C) (E : cospanCone C D) : cospanConeHom C D E E + = (cPath C E.1, cPathL C E.1 D.2.1.1 E.2.1, cPathL C E.1 D.2.2.1 E.2.2.1) cospanConeComp (C : precategory) (D : cospan C) (X Y Z : cospanCone C D) (F : cospanConeHom C D X Y) (G : cospanConeHom C D Y Z) : cospanConeHom C D X Z = (cC C X.1 Y.1 Z.1 F.1 G.1 - ,compId (cH C X.1 D.2.1.1) + ,compPath (cH C X.1 D.2.1.1) (cC C X.1 Z.1 D.2.1.1 (cC C X.1 Y.1 Z.1 F.1 G.1) Z.2.1) (cC C X.1 Y.1 D.2.1.1 F.1 (cC C Y.1 Z.1 D.2.1.1 G.1 Z.2.1)) X.2.1 - (cIdC C X.1 Y.1 Z.1 D.2.1.1 F.1 G.1 Z.2.1) - (compId (cH C X.1 D.2.1.1) + (cPathC C X.1 Y.1 Z.1 D.2.1.1 F.1 G.1 Z.2.1) + (compPath (cH C X.1 D.2.1.1) (cC C X.1 Y.1 D.2.1.1 F.1 (cC C Y.1 Z.1 D.2.1.1 G.1 Z.2.1)) (cC C X.1 Y.1 D.2.1.1 F.1 Y.2.1) X.2.1 ( cC C X.1 Y.1 D.2.1.1 F.1 (G.2.1 @ i)) F.2.1) - ,compId (cH C X.1 D.2.2.1) + ,compPath (cH C X.1 D.2.2.1) (cC C X.1 Z.1 D.2.2.1 (cC C X.1 Y.1 Z.1 F.1 G.1) Z.2.2.1) (cC C X.1 Y.1 D.2.2.1 F.1 (cC C Y.1 Z.1 D.2.2.1 G.1 Z.2.2.1)) X.2.2.1 - (cIdC C X.1 Y.1 Z.1 D.2.2.1 F.1 G.1 Z.2.2.1) - (compId (cH C X.1 D.2.2.1) + (cPathC C X.1 Y.1 Z.1 D.2.2.1 F.1 G.1 Z.2.2.1) + (compPath (cH C X.1 D.2.2.1) (cC C X.1 Y.1 D.2.2.1 F.1 (cC C Y.1 Z.1 D.2.2.1 G.1 Z.2.2.1)) (cC C X.1 Y.1 D.2.2.1 F.1 Y.2.2.1) X.2.2.1 @@ -1054,7 +1054,7 @@ cospanConeStructure (C : precategory) (D : cospan C) : structure C = (hasCospanCone C D ,\(x y : cA C) (a : hasCospanCone C D x) (b : hasCospanCone C D y) -> isCospanConeHom C D (x, a) (y, b) ,\(x y : cA C) (a : hasCospanCone C D x) (b : hasCospanCone C D y) -> isCospanConeHomProp C D (x, a) (y, b) - ,\(x : cA C) (a : hasCospanCone C D x) -> (cospanConeId C D (x, a)).2 + ,\(x : cA C) (a : hasCospanCone C D x) -> (cospanConePath C D (x, a)).2 ,\(x y z : cA C) (a : hasCospanCone C D x) (b : hasCospanCone C D y) (c : hasCospanCone C D z) (f : cH C x y) (g : cH C y z) (Hf : isCospanConeHom C D (x, a) (y, b) f) @@ -1069,18 +1069,18 @@ isCategoryCospanCone (C : precategory) (D : cospan C) (isC : isCategory C) : isC where hole : isStandardStructure C (cospanConeStructure C D) = \(x : cA C) (a b : hasCospanCone C D x) - (c : isCospanConeHom C D (x, a) (x, b) (cId C x)) - (d : isCospanConeHom C D (x, b) (x, a) (cId C x)) -> - (compId (cH C x D.2.1.1) a.1 (cC C x x D.2.1.1 (cId C x) a.1) b.1 (cIdL C x D.2.1.1 a.1 @-i) d.1 @ i - ,compId (cH C x D.2.2.1) a.2.1 (cC C x x D.2.2.1 (cId C x) a.2.1) b.2.1 (cIdL C x D.2.2.1 a.2.1 @-i) d.2 @ i - ,lemIdPProp (Id (cH C x D.1) (cC C x D.2.1.1 D.1 a.1 D.2.1.2) (cC C x D.2.2.1 D.1 a.2.1 D.2.2.2)) - (Id (cH C x D.1) (cC C x D.2.1.1 D.1 b.1 D.2.1.2) (cC C x D.2.2.1 D.1 b.2.1 D.2.2.2)) + (c : isCospanConeHom C D (x, a) (x, b) (cPath C x)) + (d : isCospanConeHom C D (x, b) (x, a) (cPath C x)) -> + (compPath (cH C x D.2.1.1) a.1 (cC C x x D.2.1.1 (cPath C x) a.1) b.1 (cPathL C x D.2.1.1 a.1 @-i) d.1 @ i + ,compPath (cH C x D.2.2.1) a.2.1 (cC C x x D.2.2.1 (cPath C x) a.2.1) b.2.1 (cPathL C x D.2.2.1 a.2.1 @-i) d.2 @ i + ,lemPathPProp (Path (cH C x D.1) (cC C x D.2.1.1 D.1 a.1 D.2.1.2) (cC C x D.2.2.1 D.1 a.2.1 D.2.2.2)) + (Path (cH C x D.1) (cC C x D.2.1.1 D.1 b.1 D.2.1.2) (cC C x D.2.2.1 D.1 b.2.1 D.2.2.2)) (cHSet C x D.1 (cC C x D.2.1.1 D.1 a.1 D.2.1.2) (cC C x D.2.2.1 D.1 a.2.1 D.2.2.2)) - (Id (cH C x D.1) - (cC C x D.2.1.1 D.1 (compId (cH C x D.2.1.1) a.1 (cC C x x D.2.1.1 (cId C x) a.1) b.1 - (cIdL C x D.2.1.1 a.1 @-i) d.1 @ i) D.2.1.2) - (cC C x D.2.2.1 D.1 (compId (cH C x D.2.2.1) a.2.1 (cC C x x D.2.2.1 (cId C x) a.2.1) b.2.1 - (cIdL C x D.2.2.1 a.2.1 @-i) d.2 @ i) D.2.2.2)) + (Path (cH C x D.1) + (cC C x D.2.1.1 D.1 (compPath (cH C x D.2.1.1) a.1 (cC C x x D.2.1.1 (cPath C x) a.1) b.1 + (cPathL C x D.2.1.1 a.1 @-i) d.1 @ i) D.2.1.2) + (cC C x D.2.2.1 D.1 (compPath (cH C x D.2.2.1) a.2.1 (cC C x x D.2.2.1 (cPath C x) a.2.1) b.2.1 + (cPathL C x D.2.2.1 a.2.1 @-i) d.2 @ i) D.2.2.2)) a.2.2 b.2.2 @ i) isFinal (C : precategory) (A : cA C) : U = (B : cA C) -> isContr (cH C B A) @@ -1094,10 +1094,10 @@ hasFinalProp (C : precategory) (isC : isCategory C) : prop (hasFinal C) = \(x y : hasFinal C) -> let p : iso C x.1 y.1 = ((y.2 x.1).1, (x.2 y.1).1 - , isContrProp (cH C x.1 x.1) (x.2 x.1) (cC C x.1 y.1 x.1 (y.2 x.1).1 (x.2 y.1).1) (cId C x.1) - , isContrProp (cH C y.1 y.1) (y.2 y.1) (cC C y.1 x.1 y.1 (x.2 y.1).1 (y.2 x.1).1) (cId C y.1)) + , isContrProp (cH C x.1 x.1) (x.2 x.1) (cC C x.1 y.1 x.1 (y.2 x.1).1 (x.2 y.1).1) (cPath C x.1) + , isContrProp (cH C y.1 y.1) (y.2 y.1) (cC C y.1 x.1 y.1 (x.2 y.1).1 (y.2 x.1).1) (cPath C y.1)) in ( (lemIsCategory C isC x.1 y.1 p @ i).1 - , lemIdPProp (isFinal C x.1) + , lemPathPProp (isFinal C x.1) (isFinal C y.1) (isFinalProp C x.1) ( isFinal C (lemIsCategory C isC x.1 y.1 p @ i).1) @@ -1112,7 +1112,7 @@ isCommutative (CA : precategory) (A B C D : cA CA) (f : cH CA A B) (g : cH CA C D) (h : cH CA A C) (i : cH CA B D) - : U = Id (cH CA A D) (cC CA A C D h g) (cC CA A B D f i) + : U = Path (cH CA A D) (cC CA A C D h g) (cC CA A B D f i) pullbackPasting (CA : precategory) @@ -1127,78 +1127,78 @@ pullbackPasting (pb3 : isPullback CA (F, (D, cC CA D E F h i), (C, l)) (A, j, cC CA A B C f g, cc3)) : isPullback CA (E, (D, h), (B, k)) (A, j, f, cc1) = \(c : cospanCone CA (E, (D, h), (B, k))) -> - let hole : Id (cH CA c.1 F) + let hole : Path (cH CA c.1 F) (cC CA c.1 D F c.2.1 (cC CA D E F h i)) (cC CA c.1 C F (cC CA c.1 B C c.2.2.1 g) l) - = compId (cH CA c.1 F) + = compPath (cH CA c.1 F) (cC CA c.1 D F c.2.1 (cC CA D E F h i)) (cC CA c.1 E F (cC CA c.1 D E c.2.1 h) i) (cC CA c.1 C F (cC CA c.1 B C c.2.2.1 g) l) - (cIdC CA c.1 D E F c.2.1 h i@-n) - (compId (cH CA c.1 F) + (cPathC CA c.1 D E F c.2.1 h i@-n) + (compPath (cH CA c.1 F) (cC CA c.1 E F (cC CA c.1 D E c.2.1 h) i) (cC CA c.1 E F (cC CA c.1 B E c.2.2.1 k) i) (cC CA c.1 C F (cC CA c.1 B C c.2.2.1 g) l) (cC CA c.1 E F (c.2.2.2@n) i) - (compId (cH CA c.1 F) + (compPath (cH CA c.1 F) (cC CA c.1 E F (cC CA c.1 B E c.2.2.1 k) i) (cC CA c.1 B F c.2.2.1 (cC CA B E F k i)) (cC CA c.1 C F (cC CA c.1 B C c.2.2.1 g) l) - (cIdC CA c.1 B E F c.2.2.1 k i) - (compId (cH CA c.1 F) + (cPathC CA c.1 B E F c.2.2.1 k i) + (compPath (cH CA c.1 F) (cC CA c.1 B F c.2.2.1 (cC CA B E F k i)) (cC CA c.1 B F c.2.2.1 (cC CA B C F g l)) (cC CA c.1 C F (cC CA c.1 B C c.2.2.1 g) l) (cC CA c.1 B F c.2.2.1 (cc2@n)) - (cIdC CA c.1 B C F c.2.2.1 g l@-n)))) + (cPathC CA c.1 B C F c.2.2.1 g l@-n)))) c' : cospanCone CA (F, (D, cC CA D E F h i), (C, l)) = (c.1 ,c.2.1 ,cC CA c.1 B C c.2.2.1 g ,hole) - hole2 : Id (cH CA c.1 F) + hole2 : Path (cH CA c.1 F) (cC CA c.1 E F (cC CA c.1 D E c.2.1 h) i) (cC CA c.1 C F (cC CA c.1 B C c.2.2.1 g) l) - = compId (cH CA c.1 F) + = compPath (cH CA c.1 F) (cC CA c.1 E F (cC CA c.1 D E c.2.1 h) i) (cC CA c.1 D F c.2.1 (cC CA D E F h i)) (cC CA c.1 C F (cC CA c.1 B C c.2.2.1 g) l) - (cIdC CA c.1 D E F c.2.1 h i) + (cPathC CA c.1 D E F c.2.1 h i) hole cc : cospanCone CA (F, (E, i), (C, l)) = (c.1, cC CA c.1 D E c.2.1 h, c'.2.2.1, hole2) - p0 (h' : cH CA c.1 A) (p : Id (cH CA c.1 D) (cC CA c.1 A D h' j) c.2.1) - : Id U (Id (cH CA c.1 B) (cC CA c.1 A B h' f) c.2.2.1) - (Id (cH CA c.1 C) (cC CA c.1 A C h' (cC CA A B C f g)) c'.2.2.1) + p0 (h' : cH CA c.1 A) (p : Path (cH CA c.1 D) (cC CA c.1 A D h' j) c.2.1) + : Path U (Path (cH CA c.1 B) (cC CA c.1 A B h' f) c.2.2.1) + (Path (cH CA c.1 C) (cC CA c.1 A C h' (cC CA A B C f g)) c'.2.2.1) = transport - ( Id U (Id (cH CA c.1 B) (cC CA c.1 A B h' f) c.2.2.1) - (Id (cH CA c.1 C) (cIdC CA c.1 A B C h' f g @ i) c'.2.2.1)) - (idProp (Id (cH CA c.1 B) (cC CA c.1 A B h' f) c.2.2.1) - (Id (cH CA c.1 C) (cIdC CA c.1 A B C h' f g @ 0) c'.2.2.1) + ( Path U (Path (cH CA c.1 B) (cC CA c.1 A B h' f) c.2.2.1) + (Path (cH CA c.1 C) (cPathC CA c.1 A B C h' f g @ i) c'.2.2.1)) + (idProp (Path (cH CA c.1 B) (cC CA c.1 A B h' f) c.2.2.1) + (Path (cH CA c.1 C) (cPathC CA c.1 A B C h' f g @ 0) c'.2.2.1) (cHSet CA c.1 B (cC CA c.1 A B h' f) c.2.2.1) - (cHSet CA c.1 C (cIdC CA c.1 A B C h' f g @ 0) c'.2.2.1) - (\(p:Id (cH CA c.1 B) (cC CA c.1 A B h' f) c.2.2.1) -> + (cHSet CA c.1 C (cPathC CA c.1 A B C h' f g @ 0) c'.2.2.1) + (\(p:Path (cH CA c.1 B) (cC CA c.1 A B h' f) c.2.2.1) -> cC CA c.1 B C (p@i) g) - (\(p1:Id (cH CA c.1 C) (cIdC CA c.1 A B C h' f g @ 0) c'.2.2.1) -> + (\(p1:Path (cH CA c.1 C) (cPathC CA c.1 A B C h' f g @ 0) c'.2.2.1) -> let h1 : cospanConeHom CA (F, (E, i), (C, l)) cc (B, k, g, cc2) = (cC CA c.1 A B h' f - ,compId (cH CA c.1 E) + ,compPath (cH CA c.1 E) (cC CA c.1 B E (cC CA c.1 A B h' f) k) (cC CA c.1 A E h' (cC CA A B E f k)) (cC CA c.1 D E c.2.1 h) - (cIdC CA c.1 A B E h' f k) - (compId (cH CA c.1 E) + (cPathC CA c.1 A B E h' f k) + (compPath (cH CA c.1 E) (cC CA c.1 A E h' (cC CA A B E f k)) (cC CA c.1 A E h' (cC CA A D E j h)) (cC CA c.1 D E c.2.1 h) (cC CA c.1 A E h' (cc1@-i)) - (compId (cH CA c.1 E) + (compPath (cH CA c.1 E) (cC CA c.1 A E h' (cC CA A D E j h)) (cC CA c.1 D E (cC CA c.1 A D h' j) h) (cC CA c.1 D E c.2.1 h) - ( cIdC CA c.1 A D E h' j h @ -i) + ( cPathC CA c.1 A D E h' j h @ -i) ( cC CA c.1 D E (p@i) h))) ,p1) h2 : cospanConeHom CA (F, (E, i), (C, l)) cc (B, k, g, cc2) @@ -1208,9 +1208,9 @@ pullbackPasting in (isContrProp (cospanConeHom CA (F, (E, i), (C, l)) cc (B, k, g, cc2)) (pb2 cc) h1 h2 @ n).1 )) - p : Id U (cospanConeHom CA (F, (D, cC CA D E F h i), (C, l)) c' (A, j, cC CA A B C f g, cc3)) + p : Path U (cospanConeHom CA (F, (D, cC CA D E F h i), (C, l)) c' (A, j, cC CA A B C f g, cc3)) (cospanConeHom CA (E, (D, h), (B, k)) c (A, j, f, cc1)) = (h : cH CA c.1 A) - * (p : Id (cH CA c.1 D) (cC CA c.1 A D h j) c.2.1) + * (p : Path (cH CA c.1 D) (cC CA c.1 A D h j) c.2.1) * (p0 h p @ -i) in transport ( isContr (p@i)) (pb3 c') diff --git a/examples/circle.ctt b/examples/circle.ctt index d9863aa..5748481 100644 --- a/examples/circle.ctt +++ b/examples/circle.ctt @@ -10,7 +10,7 @@ import int data S1 = base | loop [(i=0) -> base, (i=1) -> base] -loopS1 : U = Id S1 base base +loopS1 : U = Path S1 base base loop1 : loopS1 = loop{S1} @ i @@ -22,18 +22,18 @@ moebius : S1 -> U = split helix : S1 -> U = split base -> Z - loop @ i -> sucIdZ @ i + loop @ i -> sucPathZ @ i winding (p : loopS1) : Z = trans Z Z rem zeroZ where - rem : Id U Z Z = helix (p @ i) + rem : Path U Z Z = helix (p @ i) -compS1 : loopS1 -> loopS1 -> loopS1 = compId S1 base base base +compS1 : loopS1 -> loopS1 -> loopS1 = compPath S1 base base base -- All of these should be equal to "posZ (suc zero)": loop2 : loopS1 = compS1 loop1 loop1 -loop2' : loopS1 = compId' S1 base base base loop1 loop1 -loop2'' : loopS1 = compId'' S1 base base loop1 base loop1 +loop2' : loopS1 = compPath' S1 base base base loop1 loop1 +loop2'' : loopS1 = compPath'' S1 base base loop1 base loop1 -- More examples: loopZ1 : Z = winding loop1 @@ -42,7 +42,7 @@ loopZ3 : Z = winding (compS1 loop1 (compS1 loop1 loop1)) loopZN1 : Z = winding invLoop loopZ0 : Z = winding (compS1 loop1 invLoop) -mLoop : (x : S1) -> Id S1 x x = split +mLoop : (x : S1) -> Path S1 x x = split base -> loop1 loop @ i -> constSquare S1 base loop1 @ i @@ -62,15 +62,15 @@ triv : loopS1 = base -- A nice example of a homotopy on the circle. The path going halfway -- around the circle and then back is contractible: -hmtpy : Id loopS1 ( base) ( loop{S1} @ (i /\ -i)) = +hmtpy : Path loopS1 ( base) ( loop{S1} @ (i /\ -i)) = loop{S1} @ j /\ i /\ -i -circleelim (X : U) (x : X) (p : Id X x x) : S1 -> X = split +circleelim (X : U) (x : X) (p : Path X x x) : S1 -> X = split base -> x loop @ i -> p @ i -apcircleelim (A B : U) (x : A) (p : Id A x x) (f : A -> B) : - (z : S1) -> Id B (f (circleelim A x p z)) +apcircleelim (A B : U) (x : A) (p : Path A x x) (f : A -> B) : + (z : S1) -> Path B (f (circleelim A x p z)) (circleelim B (f x) ( f (p @ i)) z) = split base -> <_> f x loop @ i -> <_> f (p @ i) @@ -78,7 +78,7 @@ apcircleelim (A B : U) (x : A) (p : Id A x x) (f : A -> B) : -- a special case, Lemmas 6.2.5-6.2.9 in the book -aLoop (A:U) : U = (a:A) * Id A a a +aLoop (A:U) : U = (a:A) * Path A a a phi (A:U) (al : aLoop A) : S1 -> A = split base -> al.1 @@ -86,18 +86,18 @@ phi (A:U) (al : aLoop A) : S1 -> A = split psi (A:U) (f:S1 -> A) : aLoop A = (f base,f (loop1@i)) -rem (A:U) (f : S1 -> A) : (u : S1) -> Id A (phi A (psi A f) u) (f u) = split +rem (A:U) (f : S1 -> A) : (u : S1) -> Path A (phi A (psi A f) u) (f u) = split base -> refl A (f base) loop @ i -> f (loop1@i) -lem (A:U) (f : S1 -> A) : Id (S1 -> A) (phi A (psi A f)) f = +lem (A:U) (f : S1 -> A) : Path (S1 -> A) (phi A (psi A f)) f = \ (x:S1) -> (rem A f x) @ i -thm (A:U) : Id U (aLoop A) (S1 -> A) = isoId T0 T1 f g t s +thm (A:U) : Path U (aLoop A) (S1 -> A) = isoPath T0 T1 f g t s where T0 : U = aLoop A T1 : U = S1 -> A f : T0 -> T1 = phi A g : T1 -> T0 = psi A - s (x:T0) : Id T0 (g (f x)) x = refl T0 x - t : (y:T1) -> Id T1 (f (g y)) y = lem A + s (x:T0) : Path T0 (g (f x)) x = refl T0 x + t : (y:T1) -> Path T1 (f (g y)) y = lem A diff --git a/examples/collection.ctt b/examples/collection.ctt index bb751dd..61963fa 100644 --- a/examples/collection.ctt +++ b/examples/collection.ctt @@ -9,27 +9,27 @@ setFun (A B : U) (sB : set B) : set (A -> B) = setPi A (\(x : A) -> B) (\(x : A) -> sB) eqEquivFst (A B : U) : (t u : equiv A B) -> - Id U (Id (equiv A B) t u) (Id (A -> B) t.1 u.1) + Path U (Path (equiv A B) t u) (Path (A -> B) t.1 u.1) = lemSigProp (A -> B) (isEquiv A B) (propIsEquiv A B) -- groupoidFun (A B : U) (gB:groupoid B) : groupoid (A -> B) = -- groupoidPi A (\(x : A) -> B) (\(x : A) -> gB) --- lem5 (A B : U) (gB:groupoid B) (t u:equiv A B) : set (Id (equiv A B) t u) --- = substInv U set (Id (equiv A B) t u) (Id (A -> B) t.1 u.1) +-- lem5 (A B : U) (gB:groupoid B) (t u:equiv A B) : set (Path (equiv A B) t u) +-- = substInv U set (Path (equiv A B) t u) (Path (A -> B) t.1 u.1) -- (eqEquivFst A B t u) (groupoidFun A B gB t.1 u.1) -setId (A B : U) (sB : set B) : set (Id U A B) = - substInv U set (Id U A B) (equiv A B) (corrUniv A B) (rem A B sB) +setPath (A B : U) (sB : set B) : set (Path U A B) = + substInv U set (Path U A B) (equiv A B) (corrUniv A B) (rem A B sB) where - rem (A B : U) (sB:set B) (t u:equiv A B) : prop (Id (equiv A B) t u) - = substInv U prop (Id (equiv A B) t u) (Id (A -> B) t.1 u.1) + rem (A B : U) (sB:set B) (t u:equiv A B) : prop (Path (equiv A B) t u) + = substInv U prop (Path (equiv A B) t u) (Path (A -> B) t.1 u.1) (eqEquivFst A B t u) (setFun A B sB t.1 u.1) -- the collection of all sets is a groupoid groupoidSET : groupoid SET = \(A B : SET) -> - let rem : set (Id U A.1 B.1) = setId A.1 B.1 B.2 - rem1 : Id U (Id SET A B) (Id U A.1 B.1) = + let rem : set (Path U A.1 B.1) = setPath A.1 B.1 B.2 + rem1 : Path U (Path SET A B) (Path U A.1 B.1) = lemSigProp U set setIsProp A B - in substInv U set (Id SET A B) (Id U A.1 B.1) rem1 rem + in substInv U set (Path SET A B) (Path U A.1 B.1) rem1 rem diff --git a/examples/csystem.ctt b/examples/csystem.ctt index 7fd2875..05e2819 100644 --- a/examples/csystem.ctt +++ b/examples/csystem.ctt @@ -17,27 +17,27 @@ isC0System (ob : nat -> U) (hom : Sigma nat ob -> Sigma nat ob -> U) (isC : isPr * (ft0 : (n : nat) -> ob (suc n) -> ob n) * (let ft (x : A) : A = mkFt ob ft0 x.1 x.2 in - (p : (x y : A) -> Id A (ft x) y -> hom x y) + (p : (x y : A) -> Path A (ft x) y -> hom x y) * (sq : (n : nat) -> (X : ob (suc n)) -> (Y : A) -> (f : hom Y (n, ft0 n X)) -> - (f_star : ob (suc Y.1)) * (ftf : Id A (Y.1, ft0 Y.1 f_star) Y) + (f_star : ob (suc Y.1)) * (ftf : Path A (Y.1, ft0 Y.1 f_star) Y) * (q : hom (suc Y.1, f_star) (suc n, X)) - * Id (hom (suc Y.1, f_star) (n, ft0 n X)) + * Path (hom (suc Y.1, f_star) (n, ft0 n X)) (cC C (suc Y.1, f_star) Y (n, ft0 n X) (p (suc Y.1, f_star) Y ftf) f) (cC C (suc Y.1, f_star) (suc n, X) (n, ft0 n X) q (p (suc n, X) (n, ft0 n X) (<_> (n, ft0 n X)))) ) - * (sqId : (n : nat) -> (X : ob (suc n)) -> - (p0 : Id A (suc n, (sq n X (n, ft0 n X) (cId C (n, ft0 n X))).1) (suc n, X)) - * (IdP (cH C (p0@i) (suc n, X)) (sq n X (n, ft0 n X) (cId C (n, ft0 n X))).2.2.1 (cId C (suc n, X))) + * (sqPath : (n : nat) -> (X : ob (suc n)) -> + (p0 : Path A (suc n, (sq n X (n, ft0 n X) (cPath C (n, ft0 n X))).1) (suc n, X)) + * (PathP (cH C (p0@i) (suc n, X)) (sq n X (n, ft0 n X) (cPath C (n, ft0 n X))).2.2.1 (cPath C (suc n, X))) ) * ((n : nat) -> (X : ob (suc n)) -> (Y : A) -> (f : hom Y (n, ft0 n X)) -> (Z : A) -> (g : hom Z Y) -> (let f_star : ob (suc Y.1) = (sq n X Y f).1 g' : hom Z (Y.1, ft0 Y.1 f_star) = transport (cH C Z ((sq n X Y f).2.1@-i)) g - in (p0 : Id A (suc Z.1, (sq Y.1 f_star Z g').1) + in (p0 : Path A (suc Z.1, (sq Y.1 f_star Z g').1) (suc Z.1, (sq n X Z (cC C Z Y (n, ft0 n X) g f)).1)) - * IdP ( cH C (p0@i) (suc n, X)) + * PathP ( cH C (p0@i) (suc n, X)) (cC C (suc Z.1, (sq Y.1 f_star Z g').1) (suc Y.1, f_star) (suc n, X) (sq Y.1 f_star Z g').2.2.1 (sq n X Y f).2.2.1) (sq n X Z (cC C Z Y (n, ft0 n X) g f)).2.2.1))) @@ -50,7 +50,7 @@ c0C (C : C0System) : precategory = ((Sigma nat C.1, C.2.1), C.2.2.1) c0ASet (C : C0System) : set (cA (c0C C)) = setSig nat C.1 natSet C.2.2.2.1 c0Ft (C : C0System) (x : cA (c0C C)) : cA (c0C C) = mkFt C.1 C.2.2.2.2.1 x.1 x.2 -c0P (C : C0System) : (x y : cA (c0C C)) -> Id (cA (c0C C)) (c0Ft C x) y -> C.2.1 x y = C.2.2.2.2.2.1 +c0P (C : C0System) : (x y : cA (c0C C)) -> Path (cA (c0C C)) (c0Ft C x) y -> C.2.1 x y = C.2.2.2.2.2.1 c0CanSq (C : C0System) : U = (n : nat) * (X : C.1 (suc n)) * (Y : cA (c0C C)) * (C.2.1 Y (n, C.2.2.2.2.1 n X)) @@ -58,7 +58,7 @@ c0CanSq (C : C0System) : U c0Star (C : C0System) (T : c0CanSq C) : cA (c0C C) = (suc T.2.2.1.1, (C.2.2.2.2.2.2.1 T.1 T.2.1 T.2.2.1 T.2.2.2).1) c0FtStar (C : C0System) (T : c0CanSq C) - : Id (cA (c0C C)) (c0Ft C (c0Star C T)) T.2.2.1 + : Path (cA (c0C C)) (c0Ft C (c0Star C T)) T.2.2.1 = (C.2.2.2.2.2.2.1 T.1 T.2.1 T.2.2.1 T.2.2.2).2.1 c0Q (C : C0System) (T : c0CanSq C) : C.2.1 (c0Star C T) (suc T.1, T.2.1) @@ -66,7 +66,7 @@ c0Q (C : C0System) (T : c0CanSq C) : C.2.1 (c0Star C T) (suc T.1, T.2.1) c0Square (C : C0System) (T : c0CanSq C) : (let X : cA (c0C C) = (suc T.1, T.2.1) in - Id (C.2.1 (c0Star C T) (c0Ft C X)) + Path (C.2.1 (c0Star C T) (c0Ft C X)) (cC (c0C C) (c0Star C T) T.2.2.1 (c0Ft C X) (c0P C (c0Star C T) T.2.2.1 (c0FtStar C T)) T.2.2.2) (cC (c0C C) (c0Star C T) X (c0Ft C X) (c0Q C T) (c0P C X (c0Ft C X) (<_> (c0Ft C X))))) = (C.2.2.2.2.2.2.1 T.1 T.2.1 T.2.2.1 T.2.2.2).2.2.2 @@ -108,20 +108,20 @@ uc : U ucEquiv (A B : uc) : U = (e : catEquiv A.1 B.1) - * (V : Id (cA B.1) (e.1.1 A.2.2.2.1) B.2.2.2.1) - * (VT : Id (cA B.1) (e.1.1 A.2.2.2.2.1) B.2.2.2.2.1) - * (IdP (cH B.1 (VT@i) (V@i)) (e.1.2.1 A.2.2.2.2.1 A.2.2.2.1 A.2.2.2.2.2.1) B.2.2.2.2.2.1) + * (V : Path (cA B.1) (e.1.1 A.2.2.2.1) B.2.2.2.1) + * (VT : Path (cA B.1) (e.1.1 A.2.2.2.2.1) B.2.2.2.2.1) + * (PathP (cH B.1 (VT@i) (V@i)) (e.1.2.1 A.2.2.2.2.1 A.2.2.2.1 A.2.2.2.2.2.1) B.2.2.2.2.2.1) -ucEquivId (A B : uc) (e : ucEquiv A B) : Id uc A B - = let p : Id ((C:category)*catEquiv A.1 C.1) ((A.1, A.2.1), catIdEquiv A.1) ((B.1, B.2.1), e.1) - = isContrProp ((C:category)*catEquiv A.1 C.1) (catEquivEqId' (A.1, A.2.1)) ((A.1, A.2.1), catIdEquiv A.1) ((B.1, B.2.1), e.1) - pV : IdP ( cA (p@i).1.1) A.2.2.2.1 B.2.2.2.1 +ucEquivPath (A B : uc) (e : ucEquiv A B) : Path uc A B + = let p : Path ((C:category)*catEquiv A.1 C.1) ((A.1, A.2.1), catPathEquiv A.1) ((B.1, B.2.1), e.1) + = isContrProp ((C:category)*catEquiv A.1 C.1) (catEquivEqPath' (A.1, A.2.1)) ((A.1, A.2.1), catPathEquiv A.1) ((B.1, B.2.1), e.1) + pV : PathP ( cA (p@i).1.1) A.2.2.2.1 B.2.2.2.1 = comp (<_> cA (p@i).1.1) ((p@i).2.1.1 A.2.2.2.1) [(i=0)-><_>A.2.2.2.1 ,(i=1)->e.2.1@k] - pVT : IdP ( cA (p@i).1.1) A.2.2.2.2.1 B.2.2.2.2.1 + pVT : PathP ( cA (p@i).1.1) A.2.2.2.2.1 B.2.2.2.2.1 = comp (<_> cA (p@i).1.1) ((p@i).2.1.1 A.2.2.2.2.1) [(i=0)-><_>A.2.2.2.2.1 ,(i=1)->e.2.2.1@k] - pP : IdP ( cH (p@i).1.1 (pVT@i) (pV@i)) A.2.2.2.2.2.1 B.2.2.2.2.2.1 + pP : PathP ( cH (p@i).1.1 (pVT@i) (pV@i)) A.2.2.2.2.2.1 B.2.2.2.2.2.1 = comp ( cH (p@i).1.1 (fill (<_> cA (p@i).1.1) ((p@i).2.1.1 A.2.2.2.2.1) [(i=0)-><_>A.2.2.2.2.1 ,(i=1)->e.2.2.1@k]@k) (fill (<_> cA (p@i).1.1) ((p@i).2.1.1 A.2.2.2.1) [(i=0)-><_>A.2.2.2.1 ,(i=1)->e.2.1@k]@k)) @@ -129,13 +129,13 @@ ucEquivId (A B : uc) (e : ucEquiv A B) : Id uc A B [(i=0)-><_>A.2.2.2.2.2.1, (i=1)->e.2.2.2@k] in ((p@i).1.1 ,(p@i).1.2 - ,lemIdPProp (hasFinal A.1) + ,lemPathPProp (hasFinal A.1) (hasFinal B.1) (hasFinalProp A.1 A.2.1) (hasFinal (p@i).1.1) A.2.2.1 B.2.2.1 @ i ,pV@i, pVT@i, pP@i - ,lemIdPProp ((f : homTo A.1 (pV@0)) -> hasPullback A.1 (pV@0, f, pVT@0, pP@0)) + ,lemPathPProp ((f : homTo A.1 (pV@0)) -> hasPullback A.1 (pV@0, f, pVT@0, pP@0)) ((f : homTo B.1 (pV@1)) -> hasPullback B.1 (pV@1, f, pVT@1, pP@1)) (propPi (homTo A.1 (pV@0)) (\(f : homTo A.1 (pV@0)) -> hasPullback A.1 (pV@0, f, pVT@0, pP@0)) (\(f : homTo A.1 (pV@0)) -> hasPullbackProp A.1 A.2.1 (pV@0, f, pVT@0, pP@0))) @@ -165,34 +165,34 @@ ucToC0 (C : uc) : C0System = hole intD (x : obD) : cA C.1 = int x.1 x.2 homD (a b : obD) : U = C.1.1.2 (intD a) (intD b) homDSet (a b : obD) : set (homD a b) = cHSet C.1 (intD a) (intD b) - DId (a : obD) : homD a a = cId C.1 (intD a) + DPath (a : obD) : homD a a = cPath C.1 (intD a) DC (a b c : obD) (f : homD a b) (g : homD b c) : homD a c = cC C.1 (intD a) (intD b) (intD c) f g - DIdL (a b : obD) (g : homD a b) : Id (homD a b) (DC a a b (DId a) g) g = C.1.2.2.2.2.1 (intD a) (intD b) g - DIdR (a b : obD) (g : homD a b) : Id (homD a b) (DC a b b g (DId b)) g = C.1.2.2.2.2.2.1 (intD a) (intD b) g - DIdC (a b c d : obD) (f : homD a b) (g : homD b c) (h : homD c d) - : Id (homD a d) (DC a c d (DC a b c f g) h) (DC a b d f (DC b c d g h)) + DPathL (a b : obD) (g : homD a b) : Path (homD a b) (DC a a b (DPath a) g) g = C.1.2.2.2.2.1 (intD a) (intD b) g + DPathR (a b : obD) (g : homD a b) : Path (homD a b) (DC a b b g (DPath b)) g = C.1.2.2.2.2.2.1 (intD a) (intD b) g + DPathC (a b c d : obD) (f : homD a b) (g : homD b c) (h : homD c d) + : Path (homD a d) (DC a c d (DC a b c f g) h) (DC a b d f (DC b c d g h)) = C.1.2.2.2.2.2.2 (intD a) (intD b) (intD c) (intD d) f g h DD : categoryData = (obD, homD) D : isPrecategory DD - = (DId, DC, homDSet, DIdL, DIdR, DIdC) + = (DPath, DC, homDSet, DPathL, DPathR, DPathC) DC : precategory = (DD, D) ft0 (n : nat) (x : ob (suc n)) : ob n = x.1 ft (x : obD) : obD = mkFt ob ft0 x.1 x.2 p0 : (n : nat) -> (x : ob n) -> homD (n, x) (ft (n, x)) = split - zero -> \(A:Unit) -> DId (zero, A) + zero -> \(A:Unit) -> DPath (zero, A) suc n -> \(X:ob (suc n)) -> (pb n X).1.2.1 - pD (x y : obD) (p : Id obD (ft x) y) : homD x y = transport (homD x (p@i)) (p0 x.1 x.2) + pD (x y : obD) (p : Path obD (ft x) y) : homD x y = transport (homD x (p@i)) (p0 x.1 x.2) fstar (n : nat) (X : ob (suc n)) (Y : obD) (f : homD Y (n, X.1)) : obD = (suc Y.1, Y.2, cC C.1 (intD Y) (int n X.1) V f X.2) q_ (n : nat) (X : ob (suc n)) (Y : obD) (f : homD Y (n, X.1)) : (q : homD (fstar n X Y f) (suc n, X)) - * (qSq : Id (homD (fstar n X Y f) (n, X.1)) + * (qSq : Path (homD (fstar n X Y f) (n, X.1)) (cC DC (fstar n X Y f) Y (n, X.1) (p0 (suc Y.1) (fstar n X Y f).2) f) (cC DC (fstar n X Y f) (suc n, X) (n, X.1) q (p0 (suc n) X))) - * (_ : Id (cH C.1 (intD (fstar n X Y f)) VT) + * (_ : Path (cH C.1 (intD (fstar n X Y f)) VT) (cC C.1 (intD (fstar n X Y f)) (int (suc n) X) VT q (pb n X).1.2.2.1) (pb Y.1 (Y.2, cC C.1 (intD Y) (int n X.1) V f X.2)).1.2.2.1) * isPullback DC ((n, X.1), (Y, f), ((suc n, X), p0 (suc n) X)) (fstar n X Y f, p0 (suc Y.1) (fstar n X Y f).2, q, qSq) @@ -201,23 +201,23 @@ ucToC0 (C : uc) : C0System = hole gF : cH C.1 (intD Y) V = cC C.1 (intD Y) (int n X.1) V f X.2 qq : cH C.1 if_star VT = (pb Y.1 (Y.2, gF)).1.2.2.1 pg : cH C.1 if_star (int n X.1) = cC C.1 if_star (intD Y) (int n X.1) (p0 (suc Y.1) f_star.2) f - hole0 : Id (cH C.1 if_star V) + hole0 : Path (cH C.1 if_star V) (cC C.1 if_star (int n X.1) V pg X.2) (cC C.1 if_star VT V qq p) - = compId (cH C.1 if_star V) + = compPath (cH C.1 if_star V) (cC C.1 if_star (int n X.1) V pg X.2) (cC C.1 if_star (intD Y) V (pb Y.1 (Y.2, gF)).1.2.1 gF) (cC C.1 if_star VT V qq p) - (cIdC C.1 if_star (intD Y) (int n X.1) V (pb Y.1 (Y.2, gF)).1.2.1 f X.2) + (cPathC C.1 if_star (intD Y) (int n X.1) V (pb Y.1 (Y.2, gF)).1.2.1 f X.2) (pb Y.1 (Y.2, gF)).1.2.2.2 pp : cospanCone C.1 (cs n X) = (if_star, pg, qq, hole0) q : homD (fstar n X Y f) (suc n, X) = ((pb n X).2 pp).1.1 - hole1 : Id (cH C.1 if_star V) + hole1 : Path (cH C.1 if_star V) (cC C.1 if_star (intD Y) V (pb Y.1 (Y.2, gF)).1.2.1 gF) (cC C.1 if_star VT V (cC C.1 if_star (int (suc n) X) VT q (pb n X).1.2.2.1) p) = transport - ( Id (cH C.1 if_star V) + ( Path (cH C.1 if_star V) (cC C.1 if_star (intD Y) V (pb Y.1 (Y.2, gF)).1.2.1 gF) (cC C.1 if_star VT V (((pb n X).2 pp).1.2.2 @ -i) p)) (pb Y.1 (Y.2, gF)).1.2.2.2 @@ -238,14 +238,14 @@ ucToC0 (C : uc) : C0System = hole ((pb Y.1 f_star.2).1.1 ,(pb Y.1 f_star.2).1.2.1 ,((pb n X).2 pp).1.2.2@-i - ,lemIdPProp - (Id (cH C.1 if_star V) (cC C.1 if_star (intD Y) V (pb Y.1 (Y.2, gF)).1.2.1 gF) + ,lemPathPProp + (Path (cH C.1 if_star V) (cC C.1 if_star (intD Y) V (pb Y.1 (Y.2, gF)).1.2.1 gF) (cC C.1 if_star VT V (((pb n X).2 pp).1.2.2@1) p)) - (Id (cH C.1 if_star V) (cC C.1 if_star (intD Y) V (pb Y.1 (Y.2, gF)).1.2.1 gF) + (Path (cH C.1 if_star V) (cC C.1 if_star (intD Y) V (pb Y.1 (Y.2, gF)).1.2.1 gF) (cC C.1 if_star VT V (((pb n X).2 pp).1.2.2@0) p)) (cHSet C.1 if_star V (cC C.1 if_star (intD Y) V (pb Y.1 (Y.2, gF)).1.2.1 gF) (cC C.1 if_star VT V (((pb n X).2 pp).1.2.2@1) p)) - (Id (cH C.1 if_star V) (cC C.1 if_star (intD Y) V (pb Y.1 (Y.2, gF)).1.2.1 gF) + (Path (cH C.1 if_star V) (cC C.1 if_star (intD Y) V (pb Y.1 (Y.2, gF)).1.2.1 gF) (cC C.1 if_star VT V (((pb n X).2 pp).1.2.2@-i) p)) (pb Y.1 f_star.2).1.2.2.2 hole1 @ i )) @@ -258,76 +258,76 @@ ucToC0 (C : uc) : C0System = hole ) sqD (n : nat) (X : ob (suc n)) (Y : obD) (f : homD Y (n, X.1)) - : (f_star : ob (suc Y.1)) * (ftf : Id obD (Y.1, ft0 Y.1 f_star) Y) + : (f_star : ob (suc Y.1)) * (ftf : Path obD (Y.1, ft0 Y.1 f_star) Y) * (q : homD (suc Y.1, f_star) (suc n, X)) - * Id (homD (suc Y.1, f_star) (n, X.1)) + * Path (homD (suc Y.1, f_star) (n, X.1)) (cC DC (suc Y.1, f_star) Y (n, X.1) (pD (suc Y.1, f_star) Y ftf) f) (cC DC (suc Y.1, f_star) (suc n, X) (n, X.1) q (pD (suc n, X) (n, X.1) (<_> (n, X.1)))) = ((fstar n X Y f).2, <_> Y, (q_ n X Y f).1, transport - ( Id (homD (fstar n X Y f) (n, X.1)) + ( Path (homD (fstar n X Y f) (n, X.1)) (cC DC (fstar n X Y f) Y (n, X.1) (transRefl (cH DC (fstar n X Y f) Y) (p0 (suc Y.1) (fstar n X Y f).2) @ -i) f) (cC DC (fstar n X Y f) (suc n, X) (n, X.1) (q_ n X Y f).1 (transRefl (cH DC (suc n, X) (n, X.1)) (p0 (suc n) X) @ -i))) (q_ n X Y f).2.1) - qId (n : nat) (X : ob (suc n)) : - (p0 : Id obD (fstar n X (n, X.1) (cId DC (n, X.1))) (suc n, X)) - * (IdP (cH DC (p0@i) (suc n, X)) (q_ n X (n, X.1) (cId DC (n, X.1))).1 (cId DC (suc n, X))) - = let f_star : obD = fstar n X (n, X.1) (cId DC (n, X.1)) - p1 : Id obD f_star (suc n, X) = (suc n, X.1, cIdL C.1 (int n X.1) V X.2 @ i) + qPath (n : nat) (X : ob (suc n)) : + (p0 : Path obD (fstar n X (n, X.1) (cPath DC (n, X.1))) (suc n, X)) + * (PathP (cH DC (p0@i) (suc n, X)) (q_ n X (n, X.1) (cPath DC (n, X.1))).1 (cPath DC (suc n, X))) + = let f_star : obD = fstar n X (n, X.1) (cPath DC (n, X.1)) + p1 : Path obD f_star (suc n, X) = (suc n, X.1, cPathL C.1 (int n X.1) V X.2 @ i) if_star : cA C.1 = intD f_star - gF : cH C.1 (int n X.1) V = cC C.1 (int n X.1) (int n X.1) V (cId DC (n, X.1)) X.2 + gF : cH C.1 (int n X.1) V = cC C.1 (int n X.1) (int n X.1) V (cPath DC (n, X.1)) X.2 qq : cH C.1 if_star VT = (pb n (X.1, gF)).1.2.2.1 pg : cH C.1 if_star (int n X.1) - = cC C.1 if_star (int n X.1) (int n X.1) (p0 (suc n) f_star.2) (cId DC (n, X.1)) - hole0 : Id (cH C.1 if_star V) + = cC C.1 if_star (int n X.1) (int n X.1) (p0 (suc n) f_star.2) (cPath DC (n, X.1)) + hole0 : Path (cH C.1 if_star V) (cC C.1 if_star (int n X.1) V pg X.2) (cC C.1 if_star VT V qq p) - = compId (cH C.1 if_star V) + = compPath (cH C.1 if_star V) (cC C.1 if_star (int n X.1) V pg X.2) (cC C.1 if_star (int n X.1) V (pb n (X.1, gF)).1.2.1 gF) (cC C.1 if_star VT V qq p) - (cIdC C.1 if_star (int n X.1) (int n X.1) V (pb n (X.1, gF)).1.2.1 (cId DC (n, X.1)) X.2) + (cPathC C.1 if_star (int n X.1) (int n X.1) V (pb n (X.1, gF)).1.2.1 (cPath DC (n, X.1)) X.2) (pb n (X.1, gF)).1.2.2.2 pp : cospanCone C.1 (cs n X) = (if_star, pg, qq, hole0) - ppId : Id (cospanCone C.1 (cs n X)) pp (pb n X).1 + ppPath : Path (cospanCone C.1 (cs n X)) pp (pb n X).1 = (intD (p1@i), - cIdR C.1 (intD (p1@i)) (int n X.1) (p0 (suc n) (p1@i).2) @ i, - (pb n (X.1, cIdL C.1 (int n X.1) V X.2 @ i)).1.2.2.1, - lemIdPProp (Id (cH C.1 if_star V) (cC C.1 if_star (int n X.1) V pg X.2) (cC C.1 if_star VT V qq p)) - (Id (cH C.1 (int (suc n) X) V) + cPathR C.1 (intD (p1@i)) (int n X.1) (p0 (suc n) (p1@i).2) @ i, + (pb n (X.1, cPathL C.1 (int n X.1) V X.2 @ i)).1.2.2.1, + lemPathPProp (Path (cH C.1 if_star V) (cC C.1 if_star (int n X.1) V pg X.2) (cC C.1 if_star VT V qq p)) + (Path (cH C.1 (int (suc n) X) V) (cC C.1 (int (suc n) X) (int n X.1) V (p0 (suc n) X) X.2) (cC C.1 (int (suc n) X) VT V (pb n X).1.2.2.1 p)) (cHSet C.1 if_star V (cC C.1 if_star (int n X.1) V pg X.2) (cC C.1 if_star VT V qq p)) - (Id (cH C.1 (intD (p1@i)) V) - (cC C.1 (intD (p1@i)) (int n X.1) V (cIdR C.1 (intD (p1@i)) (int n X.1) (p0 (suc n) (p1@i).2) @ i) X.2) - (cC C.1 (intD (p1@i)) VT V ((pb n (X.1, cIdL C.1 (int n X.1) V X.2 @ i)).1.2.2.1) p)) + (Path (cH C.1 (intD (p1@i)) V) + (cC C.1 (intD (p1@i)) (int n X.1) V (cPathR C.1 (intD (p1@i)) (int n X.1) (p0 (suc n) (p1@i).2) @ i) X.2) + (cC C.1 (intD (p1@i)) VT V ((pb n (X.1, cPathL C.1 (int n X.1) V X.2 @ i)).1.2.2.1) p)) hole0 (pb n X).1.2.2.2 @ i ) pph : cospanConeHom C.1 (cs n X) pp (pb n X).1 - = transport ( cospanConeHom C.1 (cs n X) (ppId@-i) (pb n X).1) (cospanConeId C.1 (cs n X) (pb n X).1) - pphId : Id (cospanConeHom C.1 (cs n X) pp (pb n X).1) ((pb n X).2 pp).1 pph + = transport ( cospanConeHom C.1 (cs n X) (ppPath@-i) (pb n X).1) (cospanConePath C.1 (cs n X) (pb n X).1) + pphPath : Path (cospanConeHom C.1 (cs n X) pp (pb n X).1) ((pb n X).2 pp).1 pph = ((pb n X).2 pp).2 pph - qId : Id (cH DC f_star (suc n, X)) ((pb n X).2 pp).1.1 (transport ( cH C.1 (ppId@-i).1 (int (suc n) X)) (cId DC (suc n, X))) - = (pphId@i).1 + qPath : Path (cH DC f_star (suc n, X)) ((pb n X).2 pp).1.1 (transport ( cH C.1 (ppPath@-i).1 (int (suc n) X)) (cPath DC (suc n, X))) + = (pphPath@i).1 in ( p1 - , substIdP + , substPathP (cH DC (p1@1) (suc n, X)) (cH DC (p1@0) (suc n, X)) - (cH DC (p1@-i) (suc n, X)) (cId DC (suc n, X)) (q_ n X (n, X.1) (cId DC (n, X.1))).1 - (qId@-i) + (cH DC (p1@-i) (suc n, X)) (cPath DC (suc n, X)) (q_ n X (n, X.1) (cPath DC (n, X.1))).1 + (qPath@-i) @ -i ) qComp (n : nat) (X : ob (suc n)) (Y : obD) (f : homD Y (n, X.1)) (Z : obD) (g : homD Z Y) - : (p0 : Id obD (fstar Y.1 (fstar n X Y f).2 Z g) + : (p0 : Path obD (fstar Y.1 (fstar n X Y f).2 Z g) (fstar n X Z (cC DC Z Y (n, X.1) g f))) - * IdP ( cH DC (p0@i) (suc n, X)) + * PathP ( cH DC (p0@i) (suc n, X)) (cC DC (fstar Y.1 (fstar n X Y f).2 Z g) (fstar n X Y f) (suc n, X) (q_ Y.1 (fstar n X Y f).2 Z g).1 (q_ n X Y f).1) (q_ n X Z (cC DC Z Y (n, X.1) g f)).1 @@ -337,14 +337,14 @@ ucToC0 (C : uc) : C0System = hole gF : cH C.1 (intD Z) V = cC C.1 (intD Z) (int n X.1) V F X.2 qq : cH C.1 if_star VT = (pb Z.1 (Z.2, gF)).1.2.2.1 pg : cH C.1 if_star (int n X.1) = cC C.1 if_star (intD Z) (int n X.1) (p0 (suc Z.1) f_star.2) F - hole0 : Id (cH C.1 if_star V) + hole0 : Path (cH C.1 if_star V) (cC C.1 if_star (int n X.1) V pg X.2) (cC C.1 if_star VT V qq p) - = compId (cH C.1 if_star V) + = compPath (cH C.1 if_star V) (cC C.1 if_star (int n X.1) V pg X.2) (cC C.1 if_star (intD Z) V (pb Z.1 (Z.2, gF)).1.2.1 gF) (cC C.1 if_star VT V qq p) - (cIdC C.1 if_star (intD Z) (int n X.1) V (pb Z.1 (Z.2, gF)).1.2.1 F X.2) + (cPathC C.1 if_star (intD Z) (int n X.1) V (pb Z.1 (Z.2, gF)).1.2.1 F X.2) (pb Z.1 (Z.2, gF)).1.2.2.2 pp : cospanCone C.1 (cs n X) = (if_star, pg, qq, hole0) @@ -355,35 +355,35 @@ ucToC0 (C : uc) : C0System = hole = cC DC f_star2 Z (n,X.1) (p0 (suc Z.1) f_star2.2) F qq2 : cH C.1 if_star2 VT = (pb Z.1 (Z.2, cC C.1 (intD Z) (intD Y) V g (cC C.1 (intD Y) (int n X.1) V f X.2))).1.2.2.1 - p1 : Id obD f_star2 f_star - = (suc Z.1, Z.2, cIdC C.1 (intD Z) (intD Y) (int n X.1) V g f X.2 @ -i) + p1 : Path obD f_star2 f_star + = (suc Z.1, Z.2, cPathC C.1 (intD Z) (intD Y) (int n X.1) V g f X.2 @ -i) pp2 : cospanCone C.1 (cs n X) = (if_star2, pg2, qq2, - transport (Id (cH C.1 (intD(p1@-i)) V) + transport (Path (cH C.1 (intD(p1@-i)) V) (cC C.1 (intD(p1@-i)) (int n X.1) V (cC DC (p1@-i) Z (n,X.1) (p0 (suc Z.1) (p1@-i).2) F) X.2) - (cC C.1 (intD(p1@-i)) VT V (pb Z.1 (Z.2, cIdC C.1 (intD Z) (intD Y) (int n X.1) V g f X.2 @ i)).1.2.2.1 p)) + (cC C.1 (intD(p1@-i)) VT V (pb Z.1 (Z.2, cPathC C.1 (intD Z) (intD Y) (int n X.1) V g f X.2 @ i)).1.2.2.1 p)) hole0) - ppId : Id (cospanCone C.1 (cs n X)) pp2 pp + ppPath : Path (cospanCone C.1 (cs n X)) pp2 pp = (intD (p1@i), cC DC (p1@i) Z (n,X.1) (p0 (suc Z.1) (p1@i).2) F, - (pb Z.1 (Z.2, cIdC C.1 (intD Z) (intD Y) (int n X.1) V g f X.2 @ -i)).1.2.2.1, - lemIdPProp - (Id (cH C.1 if_star2 V) (cC C.1 if_star2 (int n X.1) V pg2 X.2) (cC C.1 if_star2 VT V qq2 p)) - (Id (cH C.1 if_star V) (cC C.1 if_star (int n X.1) V pg X.2) (cC C.1 if_star VT V qq p)) + (pb Z.1 (Z.2, cPathC C.1 (intD Z) (intD Y) (int n X.1) V g f X.2 @ -i)).1.2.2.1, + lemPathPProp + (Path (cH C.1 if_star2 V) (cC C.1 if_star2 (int n X.1) V pg2 X.2) (cC C.1 if_star2 VT V qq2 p)) + (Path (cH C.1 if_star V) (cC C.1 if_star (int n X.1) V pg X.2) (cC C.1 if_star VT V qq p)) (cHSet C.1 if_star2 V (cC C.1 if_star2 (int n X.1) V pg2 X.2) (cC C.1 if_star2 VT V qq2 p)) - (Id (cH C.1 (intD(p1@i)) V) + (Path (cH C.1 (intD(p1@i)) V) (cC C.1 (intD(p1@i)) (int n X.1) V (cC DC (p1@i) Z (n,X.1) (p0 (suc Z.1) (p1@i).2) F) X.2) - (cC C.1 (intD(p1@i)) VT V (pb Z.1 (Z.2, cIdC C.1 (intD Z) (intD Y) (int n X.1) V g f X.2 @ -i)).1.2.2.1 p)) + (cC C.1 (intD(p1@i)) VT V (pb Z.1 (Z.2, cPathC C.1 (intD Z) (intD Y) (int n X.1) V g f X.2 @ -i)).1.2.2.1 p)) pp2.2.2.2 hole0 @ i) - hole3 : Id (cH C.1 if_star2 (int n X.1)) + hole3 : Path (cH C.1 if_star2 (int n X.1)) (cC DC f_star2 (suc n, X) (n,X.1) (cC C.1 if_star2 (intD (fstar n X Y f)) (int (suc n) X) q2 (q_ n X Y f).1) (p0 (suc n) X) ) (cC DC f_star2 Z (n,X.1) (p0 (suc Z.1) f_star2.2) F) = undefined - -- = compId (cH C.1 if_star2 (int n X.1)) + -- = compPath (cH C.1 if_star2 (int n X.1)) -- (cC DC f_star2 (suc n, X) (n,X.1) -- (cC DC f_star2 (fstar n X Y f) (suc n, X) q2 (q_ n X Y f).1) -- (p0 (suc n) X)) @@ -391,8 +391,8 @@ ucToC0 (C : uc) : C0System = hole -- q2 -- (cC DC (fstar n X Y f) (suc n, X) (n, X.1) (q_ n X Y f).1 (p0 (suc n) X))) -- (cC DC f_star2 Z (n,X.1) (p0 (suc Z.1) f_star2.2) F) - -- (cIdC DC f_star2 (fstar n X Y f) (suc n, X) (n,X.1) q2 (q_ n X Y f).1 (p0 (suc n) X)) - -- (compId (cH C.1 if_star2 (int n X.1)) + -- (cPathC DC f_star2 (fstar n X Y f) (suc n, X) (n,X.1) q2 (q_ n X Y f).1 (p0 (suc n) X)) + -- (compPath (cH C.1 if_star2 (int n X.1)) -- (cC DC f_star2 (fstar n X Y f) (n,X.1) -- q2 -- (cC DC (fstar n X Y f) (suc n, X) (n, X.1) (q_ n X Y f).1 (p0 (suc n) X))) @@ -402,7 +402,7 @@ ucToC0 (C : uc) : C0System = hole -- (cC DC f_star2 Z (n,X.1) (p0 (suc Z.1) f_star2.2) F) -- ( cC DC f_star2 (fstar n X Y f) (n,X.1) q2 -- ((q_ n X Y f).2.1@-i)) - -- (compId (cH C.1 if_star2 (int n X.1)) + -- (compPath (cH C.1 if_star2 (int n X.1)) -- (cC DC f_star2 (fstar n X Y f) (n,X.1) -- q2 -- (cC DC (fstar n X Y f) Y (n, X.1) (p0 (suc Y.1) (fstar n X Y f).2) f)) @@ -410,8 +410,8 @@ ucToC0 (C : uc) : C0System = hole -- (cC DC f_star2 (fstar n X Y f) Y q2 (p0 (suc Y.1) (fstar n X Y f).2)) -- f) -- (cC DC f_star2 Z (n,X.1) (p0 (suc Z.1) f_star2.2) F) - -- (cIdC DC f_star2 (fstar n X Y f) Y (n, X.1) q2 (p0 (suc Y.1) (fstar n X Y f).2) f@-i) - -- (compId (cH C.1 if_star2 (int n X.1)) + -- (cPathC DC f_star2 (fstar n X Y f) Y (n, X.1) q2 (p0 (suc Y.1) (fstar n X Y f).2) f@-i) + -- (compPath (cH C.1 if_star2 (int n X.1)) -- (cC DC f_star2 Y (n,X.1) -- (cC DC f_star2 (fstar n X Y f) Y q2 (p0 (suc Y.1) (fstar n X Y f).2)) -- f) @@ -421,16 +421,16 @@ ucToC0 (C : uc) : C0System = hole -- (cC DC f_star2 Z (n,X.1) (p0 (suc Z.1) f_star2.2) F) -- (cC DC f_star2 Y (n,X.1) -- ((q_ Y.1 (fstar n X Y f).2 Z g).2.1@-i) f) - -- (cIdC DC f_star2 Z Y (n, X.1) (p0 (suc Z.1) f_star2.2) g f)))) + -- (cPathC DC f_star2 Z Y (n, X.1) (p0 (suc Z.1) f_star2.2) g f)))) -- opaque hole3 - hole4 : Id (cH C.1 if_star2 VT) + hole4 : Path (cH C.1 if_star2 VT) (cC C.1 if_star2 (int (suc n) X) VT (cC C.1 if_star2 (intD (fstar n X Y f)) (int (suc n) X) q2 (q_ n X Y f).1) (pb n X).1.2.2.1) (pb Z.1 (Z.2, cC C.1 (intD Z) (intD Y) V g (cC C.1 (intD Y) (int n X.1) V f X.2))).1.2.2.1 = undefined - -- = compId (cH C.1 if_star2 VT) + -- = compPath (cH C.1 if_star2 VT) -- (cC C.1 if_star2 (int (suc n) X) VT -- (cC C.1 if_star2 (intD (fstar n X Y f)) (int (suc n) X) q2 (q_ n X Y f).1) -- (pb n X).1.2.2.1) @@ -438,8 +438,8 @@ ucToC0 (C : uc) : C0System = hole -- q2 -- (cC C.1 (intD (fstar n X Y f)) (int (suc n) X) VT (q_ n X Y f).1 (pb n X).1.2.2.1)) -- (pb Z.1 (Z.2, cC C.1 (intD Z) (intD Y) V g (cC C.1 (intD Y) (int n X.1) V f X.2))).1.2.2.1 - -- (cIdC C.1 if_star2 (intD (fstar n X Y f)) (int (suc n) X) VT q2 (q_ n X Y f).1 (pb n X).1.2.2.1) - -- (compId (cH C.1 if_star2 VT) + -- (cPathC C.1 if_star2 (intD (fstar n X Y f)) (int (suc n) X) VT q2 (q_ n X Y f).1 (pb n X).1.2.2.1) + -- (compPath (cH C.1 if_star2 VT) -- (cC C.1 if_star2 (intD (fstar n X Y f)) VT -- q2 -- (cC C.1 (intD (fstar n X Y f)) (int (suc n) X) VT (q_ n X Y f).1 (pb n X).1.2.2.1)) @@ -453,43 +453,43 @@ ucToC0 (C : uc) : C0System = hole pph : cospanConeHom C.1 (cs n X) pp (pb n X).1 = transport - (cospanConeHom C.1 (cs n X) (ppId@i) (pb n X).1) + (cospanConeHom C.1 (cs n X) (ppPath@i) (pb n X).1) ( cC C.1 if_star2 (intD (fstar n X Y f)) (int (suc n) X) q2 (q_ n X Y f).1 , hole3 , hole4 ) - pphId : Id (cospanConeHom C.1 (cs n X) pp (pb n X).1) ((pb n X).2 pp).1 pph + pphPath : Path (cospanConeHom C.1 (cs n X) pp (pb n X).1) ((pb n X).2 pp).1 pph = ((pb n X).2 pp).2 pph - qId : Id (cH C.1 if_star (int (suc n) X)) + qPath : Path (cH C.1 if_star (int (suc n) X)) (transport (cH C.1 (intD(p1@i)) (int (suc n) X)) (cC C.1 if_star2 (intD (fstar n X Y f)) (int (suc n) X) q2 (q_ n X Y f).1)) (q_ n X Z (cC DC Z Y (n, X.1) g f)).1 - = (pphId@-i).1 + = (pphPath@-i).1 in (p1, - substIdP (cH DC (p1@0) (suc n, X)) (cH DC (p1@1) (suc n, X)) + substPathP (cH DC (p1@0) (suc n, X)) (cH DC (p1@1) (suc n, X)) ( cH DC (p1@i) (suc n, X)) (cC C.1 if_star2 (intD (fstar n X Y f)) (int (suc n) X) q2 (q_ n X Y f).1) (q_ n X Z (cC DC Z Y (n, X.1) g f)).1 - qId) + qPath) qComp' (n : nat) (X : ob (suc n)) (Y : obD) (f : homD Y (n, X.1)) (Z : obD) (g : homD Z Y) : (let g' : homD Z Y = transport (<_> homD Z Y) g in - (p0 : Id obD (fstar Y.1 (fstar n X Y f).2 Z g') + (p0 : Path obD (fstar Y.1 (fstar n X Y f).2 Z g') (fstar n X Z (cC DC Z Y (n, X.1) g f))) - * IdP ( cH DC (p0@i) (suc n, X)) + * PathP ( cH DC (p0@i) (suc n, X)) (cC DC (fstar Y.1 (fstar n X Y f).2 Z g') (fstar n X Y f) (suc n, X) (q_ Y.1 (fstar n X Y f).2 Z g').1 (q_ n X Y f).1) (q_ n X Z (cC DC Z Y (n, X.1) g f)).1) = transport - ( (p0 : Id obD (fstar Y.1 (fstar n X Y f).2 Z (transRefl (homD Z Y) g @ -j)) + ( (p0 : Path obD (fstar Y.1 (fstar n X Y f).2 Z (transRefl (homD Z Y) g @ -j)) (fstar n X Z (cC DC Z Y (n, X.1) g f))) - * IdP ( cH DC (p0@i) (suc n, X)) + * PathP ( cH DC (p0@i) (suc n, X)) (cC DC (fstar Y.1 (fstar n X Y f).2 Z (transRefl (homD Z Y) g @ -j)) (fstar n X Y f) (suc n, X) (q_ Y.1 (fstar n X Y f).2 Z (transRefl (homD Z Y) g @ -j)).1 (q_ n X Y f).1) (q_ n X Z (cC DC Z Y (n, X.1) g f)).1) (qComp n X Y f Z g) hole : C0System - = (ob, homD, D, obSet, ft0, pD, sqD, qId, qComp') + = (ob, homD, D, obSet, ft0, pD, sqD, qPath, qComp') diff --git a/examples/demo.ctt b/examples/demo.ctt index f2f8e78..a40fe3d 100644 --- a/examples/demo.ctt +++ b/examples/demo.ctt @@ -56,31 +56,31 @@ An element in ID A B is a line in the universe connecting A and B: |- ID A B Type -IdP is heterogeneous equality: +PathP is heterogeneous equality: |- P : ID A B |- a : A |- b : B ----------------------------------------------- - |- IdP P a b Type + |- PathP P a b Type -} --- The usual identity can be seen a special case of IdP: -Id (A : U) (a b : A) : U = IdP ( A) a b +-- The usual identity can be seen a special case of PathP: +Path (A : U) (a b : A) : U = PathP ( A) a b -- "" abstracts the name/color/dimension i: -refl (A : U) (a : A) : Id A a a = a +refl (A : U) (a : A) : Path A a a = a {- -A proof "p : Id A a b" is thought of as a line between a and b: +A proof "p : Path A a b" is thought of as a line between a and b: p a ------> b -A proof "sq : Id (Id A a b) p q" is thought of as a square between p and q: +A proof "sq : Path (Path A a b) p q" is thought of as a square between p and q: q @@ -98,19 +98,19 @@ And so on... -- It is possible to take the face of a path to obtain its endpoints: -face0 (A : U) (a b : A) (p : Id A a b) : Id A a a = refl A (p @ 0) -face1 (A : U) (a b : A) (p : Id A a b) : Id A b b = refl A (p @ 1) +face0 (A : U) (a b : A) (p : Path A a b) : Path A a a = refl A (p @ 0) +face1 (A : U) (a b : A) (p : Path A a b) : Path A b b = refl A (p @ 1) -- By applying a path to a name i, "p @ i", it is seen as a path "in -- dimension i" connecting "p @ 0" to "p @ 1". This way we get a -- simple proof of cong: -cong (A B : U) (f : A -> B) (a b : A) (p : Id A a b) : Id B (f a) (f b) = +cong (A B : U) (f : A -> B) (a b : A) (p : Path A a b) : Path B (f a) (f b) = f (p @ i) -- This also gives a short proof of function extensionality: funExt (A : U) (B : A -> U) (f g : (x : A) -> B x) - (p : (x : A) -> Id (B x) (f x) (g x)) : Id ((y : A) -> B y) f g = + (p : (x : A) -> Path (B x) (f x) (g x)) : Path ((y : A) -> B y) f g = \(a : A) -> (p a) @ i {- @@ -149,11 +149,11 @@ min(i,j), i \/ j is max(i,j) and -i is 1 - i. -} -- Applying a path to a negated name inverts it: -sym (A : U) (a b : A) (p : Id A a b) : Id A b a = p @ -i +sym (A : U) (a b : A) (p : Path A a b) : Path A b a = p @ -i -- This operation is an involution: -symK (A : U) (a b : A) (p : Id A a b) : Id (Id A a b) p p = - refl (Id A a b) (sym A b a (sym A a b p)) +symK (A : U) (a b : A) (p : Path A a b) : Path (Path A a b) p p = + refl (Path A a b) (sym A b a (sym A a b p)) {- Connections: @@ -194,10 +194,10 @@ And a disjunction gives: -- This gives a simple proof that singletons are contractible: -singl (A : U) (a : A) : U = (x : A) * Id A a x +singl (A : U) (a : A) : U = (x : A) * Path A a x -contrSingl (A : U) (a b : A) (p : Id A a b) : - Id (singl A a) (a,refl A a) (b,p) = +contrSingl (A : U) (a b : A) (p : Path A a b) : + Path (singl A a) (a,refl A a) (b,p) = (p @ i, p @ i /\ j) @@ -206,17 +206,17 @@ contrSingl (A : U) (a b : A) (p : Id A a b) : Note that formulas does not form a boolean algebra, but a de Morgan algebra. This means that "p @ 0" and "p @ i /\ -i" are different! -testDeMorganAlgebra (A : U) (a b : A) (p : Id A a b) : - Id (Id A a a) (<_> p @ 0) (<_> p @ 0) = - refl (Id A a a) ( p @ i /\ -i) +testDeMorganAlgebra (A : U) (a b : A) (p : Path A a b) : + Path (Path A a a) (<_> p @ 0) (<_> p @ 0) = + refl (Path A a a) ( p @ i /\ -i) This is clear with the intuition that /\ correspond to min. More about this later... -} -testConj (A : U) (a b : A) (p : Id A a b) : Id A a a = p @ i /\ -i -testDisj (A : U) (a b : A) (p : Id A a b) : Id A b b = p @ i \/ -i +testConj (A : U) (a b : A) (p : Path A a b) : Path A a a = p @ i /\ -i +testDisj (A : U) (a b : A) (p : Path A a b) : Path A b b = p @ i \/ -i {- @@ -232,7 +232,7 @@ a system describing the rest of the shape, and produces the missing side (opposite of the principal side). -Transitivity of Id is obtained from a composition of this open square: +Transitivity of Path is obtained from a composition of this open square: a - - - - - - - - > c @@ -249,25 +249,25 @@ Transitivity of Id is obtained from a composition of this open square: The composition computes the dashed line at the top of the square. -} -trans (A : U) (a b c : A) (p : Id A a b) (q : Id A b c) : Id A a c = +trans (A : U) (a b c : A) (p : Path A a b) (q : Path A b c) : Path A a c = comp (<_> A) (p @ i) [ (i = 0) -> a, (i = 1) -> q ] -- "Kan composition": -kan (A : U) (a b c d : A) (p : Id A a b) - (q : Id A a c) (r : Id A b d) : Id A c d = +kan (A : U) (a b c d : A) (p : Path A a b) + (q : Path A a c) (r : Path A b d) : Path A c d = comp (<_> A) (p @ i) [ (i = 0) -> q, (i = 1) -> r ] -- The first two nonzero h-levels are propositions and sets: -prop (A : U) : U = (a b : A) -> Id A a b -set (A : U) : U = (a b : A) -> prop (Id A a b) +prop (A : U) : U = (a b : A) -> Path A a b +set (A : U) : U = (a b : A) -> prop (Path A a b) -- Using compositions we get a short proof that any prop is a set. To -- understand this proof one should draw an open cube with the back -- face being a constant square in a and the sides given by the system -- below. propSet (A : U) (h : prop A) : set A = - \(a b : A) (p q : Id A a b) -> + \(a b : A) (p q : Path A a b) -> comp (<_> A) a [ (j=0) -> h a a , (j=1) -> h a b , (i=0) -> h a (p @ j) @@ -293,11 +293,11 @@ We can also define the type of squares: v -} -Square (A : U) (a0 a1 b0 b1 : A) (u : Id A a0 a1) (v : Id A b0 b1) - (r0 : Id A a0 b0) (r1 : Id A a1 b1) : U - = IdP ( (IdP ( A) (u @ i) (v @ i))) r0 r1 +Square (A : U) (a0 a1 b0 b1 : A) (u : Path A a0 a1) (v : Path A b0 b1) + (r0 : Path A a0 b0) (r1 : Path A a1 b1) : U + = PathP ( (PathP ( A) (u @ i) (v @ i))) r0 r1 -constSquare (A : U) (a : A) (p : Id A a a) : Square A a a a a p p p p = +constSquare (A : U) (a : A) (p : Path A a a) : Square A a a a a p p p p = comp (<_> A) a [ (i = 0) -> p @ j \/ - k , (i = 1) -> p @ j /\ k , (j = 0) -> p @ i \/ - k @@ -318,19 +318,19 @@ add (a : nat) : nat -> nat = split zero -> a suc b -> suc (add a b) -addZero : (a : nat) -> Id nat (add zero a) a = split +addZero : (a : nat) -> Path nat (add zero a) a = split zero -> zero suc a' -> suc (addZero a' @ i) -addSuc (a : nat) : (b : nat) -> Id nat (add (suc a) b) (suc (add a b)) = split +addSuc (a : nat) : (b : nat) -> Path nat (add (suc a) b) (suc (add a b)) = split zero -> suc a suc b' -> suc (addSuc a b' @ i) -addCom (a : nat) : (b : nat) -> Id nat (add a b) (add b a) = split +addCom (a : nat) : (b : nat) -> Path nat (add a b) (add b a) = split zero -> addZero a @ -i suc b' -> comp (<_> nat) (suc (addCom a b' @ i)) [ (i = 0) -> suc (add a b') , (i = 1) -> addSuc b' a @ -j ] -addAssoc (a b : nat) : (c : nat) -> Id nat (add a (add b c)) (add (add a b) c) = split +addAssoc (a b : nat) : (c : nat) -> Path nat (add a (add b c)) (add (add a b) c) = split zero -> add a b suc c' -> suc (addAssoc a b c' @ i) @@ -342,31 +342,31 @@ addAssoc (a b : nat) : (c : nat) -> Id nat (add a (add b c)) (add (add a b) c) = Transport takes a path in the universe between A and B and produces a function from A to B: - transport : Id U A B -> A -> B + transport : Path U A B -> A -> B It is defined internally as a composition with an empty system. -} -- This gives a simple proof of subst: -subst (A : U) (P : A -> U) (a b : A) (p : Id A a b) (e : P a) : P b = +subst (A : U) (P : A -> U) (a b : A) (p : Path A a b) (e : P a) : P b = transport (cong A U P a b p) e -- Combined with the fact that singletons are contractible this gives -- the J eliminator: -J (A : U) (a : A) (C : (x : A) -> Id A a x -> U) - (d : C a (refl A a)) (x : A) (p : Id A a x) : C x p = +J (A : U) (a : A) (C : (x : A) -> Path A a x -> U) + (d : C a (refl A a)) (x : A) (p : Path A a x) : C x p = subst (singl A a) T (a,refl A a) (x,p) (contrSingl A a x p) d where T (bp : singl A a) : U = C bp.1 bp.2 -- Note: Transporting with refl is not the identity function: --- transRefl (A : U) (a : A) : Id A a a = refl A (transport (refl U A) a) +-- transRefl (A : U) (a : A) : Path A a a = refl A (transport (refl U A) a) -- This implies that the elimination rule for J does not hold definitonally: --- defEqJ (A : U) (a : A) (C : (x : A) -> Id A a x -> U) (d : C a (refl A a)) : --- Id (C a (refl A a)) (J A a C d a (refl A a)) d = refl (C a (refl A a)) d +-- defEqJ (A : U) (a : A) (C : (x : A) -> Path A a x -> U) (d : C a (refl A a)) : +-- Path (C a (refl A a)) (J A a C d a (refl A a)) d = refl (C a (refl A a)) d -- This might not be too bad though. Nils-Anders Danielsson has -- verified in Agda that many of the consequences of univalence can be diff --git a/examples/discor.ctt b/examples/discor.ctt index e983ff1..5b39f91 100644 --- a/examples/discor.ctt +++ b/examples/discor.ctt @@ -2,49 +2,49 @@ module discor where import prelude -inlNotinr (A B:U) (a:A) (b:B) (h: Id (or A B) (inl a) (inr b)) : N0 = +inlNotinr (A B:U) (a:A) (b:B) (h: Path (or A B) (inl a) (inr b)) : N0 = subst (or A B) T (inl a) (inr b) h tt where T : or A B -> U = split inl _ -> Unit inr _ -> N0 -inrNotinl (A B:U) (a:A) (b:B) (h : Id (or A B) (inr b) (inl a)) : N0 = +inrNotinl (A B:U) (a:A) (b:B) (h : Path (or A B) (inr b) (inl a)) : N0 = subst (or A B) T (inr b) (inl a) h tt where T : or A B -> U = split inl _ -> N0 inr _ -> Unit -injInl (A B :U) (x0 x1:A) (h : Id (or A B) (inl x0) (inl x1)) : Id A x0 x1 = +injInl (A B :U) (x0 x1:A) (h : Path (or A B) (inl x0) (inl x1)) : Path A x0 x1 = subst (or A B) T (inl x0) (inl x1) h (refl A x0) where T : or A B -> U = split - inl x -> Id A x0 x + inl x -> Path A x0 x inr _ -> N0 -injInr (A B :U) (x0 x1:B) (h: Id (or A B) (inr x0) (inr x1)) : Id B x0 x1 = +injInr (A B :U) (x0 x1:B) (h: Path (or A B) (inr x0) (inr x1)) : Path B x0 x1 = subst (or A B) T (inr x0) (inr x1) h (refl B x0) where T : or A B -> U = split inl _ -> N0 - inr x -> Id B x0 x + inr x -> Path B x0 x -- If A and B are discrete then "A or B" is discrete orDisc (A B : U) (dA : discrete A) (dB : discrete B) : - (z z1 : or A B) -> dec (Id (or A B) z z1) = split + (z z1 : or A B) -> dec (Path (or A B) z z1) = split inl a -> rem1 - where rem1 : (z1:or A B) -> dec (Id (or A B) (inl a) z1) = split + where rem1 : (z1:or A B) -> dec (Path (or A B) (inl a) z1) = split inl a1 -> rem (dA a a1) - where rem : dec (Id A a a1) -> dec (Id (or A B) (inl a) (inl a1)) = split + where rem : dec (Path A a a1) -> dec (Path (or A B) (inl a) (inl a1)) = split inl p -> inl ( inl (p @ i)) - inr h -> inr (\ (p:Id (or A B) (inl a) (inl a1)) -> h (injInl A B a a1 p)) + inr h -> inr (\ (p:Path (or A B) (inl a) (inl a1)) -> h (injInl A B a a1 p)) inr b -> inr (inlNotinr A B a b) inr b -> rem1 - where rem1 : (z1:or A B) -> dec (Id (or A B) (inr b) z1) = split + where rem1 : (z1:or A B) -> dec (Path (or A B) (inr b) z1) = split inl a -> inr (inrNotinl A B a b) inr b1 -> rem (dB b b1) - where rem : dec (Id B b b1) -> dec (Id (or A B) (inr b) (inr b1)) = split + where rem : dec (Path B b b1) -> dec (Path (or A B) (inr b) (inr b1)) = split inl p -> inl ( inr (p @ i)) - inr h -> inr (\ (p:Id (or A B) (inr b) (inr b1)) -> h (injInr A B b b1 p)) + inr h -> inr (\ (p:Path (or A B) (inr b) (inr b1)) -> h (injInr A B b b1 p)) diff --git a/examples/equiv.ctt b/examples/equiv.ctt index d8280fe..e23bce3 100644 --- a/examples/equiv.ctt +++ b/examples/equiv.ctt @@ -5,7 +5,7 @@ module equiv where import prelude fiber (A B : U) (f : A -> B) (y : B) : U = - (x : A) * Id B y (f x) + (x : A) * Path B y (f x) isEquiv (A B : U) (f : A -> B) : U = (y : B) -> isContr (fiber A B f y) @@ -17,62 +17,62 @@ propIsEquiv (A B : U) (f : A -> B) \ (y:B) -> propIsContr (fiber A B f y) (u0 y) (u1 y) @ i equivLemma (A B : U) - : (v w : equiv A B) -> Id (A -> B) v.1 w.1 -> Id (equiv A B) v w + : (v w : equiv A B) -> Path (A -> B) v.1 w.1 -> Path (equiv A B) v w = lemSig (A -> B) (isEquiv A B) (propIsEquiv A B) idIsEquiv (A : U) : isEquiv A A (idfun A) = \(a : A) -> ((a, refl A a) ,\(z : fiber A A (idfun A) a) -> contrSingl A a z.1 z.2) -equivId (T A : U) (f : T -> A) (p : isEquiv T A f) : Id U T A = +equivPath (T A : U) (f : T -> A) (p : isEquiv T A f) : Path U T A = glue A [ (i=0) -> (T,f,p), (i=1) -> (A,idfun A, idIsEquiv A)] -- for univalence invEq (A B:U) (w:equiv A B) (y:B) : A = (w.2 y).1.1 -retEq (A B:U) (w:equiv A B) (y:B) : Id B (w.1 (invEq A B w y)) y = +retEq (A B:U) (w:equiv A B) (y:B) : Path B (w.1 (invEq A B w y)) y = (w.2 y).1.2@-i -secEq (A B:U) (w:equiv A B) (x:A) : Id A (invEq A B w (w.1 x)) x = +secEq (A B:U) (w:equiv A B) (x:A) : Path A (invEq A B w (w.1 x)) x = ((w.2 (w.1 x)).2 (x,w.1 x)@i).1 -transEquiv (A B:U) (p:Id U A B) : equiv A B = (f,p) +transEquiv (A B:U) (p:Path U A B) : equiv A B = (f,p) where f (x:A) : B = comp p x [] g (y:B) : A = comp (p@-i) y [] - lem1 (x:A) : IdP p x (f x) = comp (p@(i/\j)) x [(i=0) -> x] - lem2 (y:B) : IdP p (g y) y = comp (p@(i\/-j)) y [(i=1) -> y] - lem3 (y:B) : Id B y (f (g y)) = comp p (g y) [(i=0) -> lem2 y,(i=1) -> lem1 (g y)] - lem4 (y:B) : IdP (Id (p@i) (lem2 y@i) (lem1 (g y)@i)) (g y) (lem3 y) = + lem1 (x:A) : PathP p x (f x) = comp (p@(i/\j)) x [(i=0) -> x] + lem2 (y:B) : PathP p (g y) y = comp (p@(i\/-j)) y [(i=1) -> y] + lem3 (y:B) : Path B y (f (g y)) = comp p (g y) [(i=0) -> lem2 y,(i=1) -> lem1 (g y)] + lem4 (y:B) : PathP (Path (p@i) (lem2 y@i) (lem1 (g y)@i)) (g y) (lem3 y) = fill p (g y) [(i=0) -> lem2 y,(i=1) -> lem1 (g y)] @ j - lem5 (y:B) (x:A) (q:Id B y (f x)) : Id A (g y) x = + lem5 (y:B) (x:A) (q:Path B y (f x)) : Path A (g y) x = comp (p@-j) (q@i) [(i=0) -> lem2 y@-j,(i=1) -> lem1 x@-j] - lem6 (y:B) (x:A) (q:Id B y (f x)) : IdP (Id (p@i) (lem2 y@i) (lem1 x@i)) (lem5 y x q) q = + lem6 (y:B) (x:A) (q:Path B y (f x)) : PathP (Path (p@i) (lem2 y@i) (lem1 x@i)) (lem5 y x q) q = fill (p@-j) (q@i) [(i=0) -> lem2 y@-k,(i=1) -> lem1 x@-k] @ -j - lem7 (y:B) (x:A) (q:Id B y (f x)) : IdP (Id B y (f (lem5 y x q@i))) (lem3 y) q = + lem7 (y:B) (x:A) (q:Path B y (f x)) : PathP (Path B y (f (lem5 y x q@i))) (lem3 y) q = comp p (lem5 y x q@i/\j) [(i=0) -> lem2 y, (i=1) -> lem1 (lem5 y x q@j),(j=0) -> lem4 y@k@i,(j=1) -> lem6 y x q@k@i] lem8 (y:B) : fiber A B f y = (g y,lem3 y) - lem9 (y:B) (z:fiber A B f y) : Id (fiber A B f y) (lem8 y) z = + lem9 (y:B) (z:fiber A B f y) : Path (fiber A B f y) (lem8 y) z = (lem5 y z.1 z.2@i,lem7 y z.1 z.2@i) p (y:B) : isContr (fiber A B f y) = (lem8 y,lem9 y) -lemSinglContr (A:U) (a:A) : isContr ((x:A) * Id A a x) = - ((a,refl A a),\ (z:(x:A) * Id A a x) -> contrSingl A a z.1 z.2) +lemSinglContr (A:U) (a:A) : isContr ((x:A) * Path A a x) = + ((a,refl A a),\ (z:(x:A) * Path A a x) -> contrSingl A a z.1 z.2) idEquiv (A:U) : equiv A A = (\ (x:A) -> x, lemSinglContr A) -transEquiv (A X:U) (p:Id U A X) : equiv A X = +transEquiv (A X:U) (p:Path U A X) : equiv A X = substTrans U (equiv A) A X p (idEquiv A) transDelta (A:U) : equiv A A = transEquiv A A (A) -transEquivToId (A B : U) (w : equiv A B) : Id U A B = +transEquivToPath (A B : U) (w : equiv A B) : Path U A B = glue B [ (i = 1) -> (B,eB) , (i = 0) -> (A,w) ] where eB : equiv B B = transDelta B -eqToEq (A B : U) (p : Id U A B) - : Id (Id U A B) (transEquivToId A B (transEquiv A B p)) p +eqToEq (A B : U) (p : Path U A B) + : Path (Path U A B) (transEquivToPath A B (transEquiv A B p)) p = let e : equiv A B = transEquiv A B p f : equiv B B = transDelta B Ai : U = p@i @@ -82,36 +82,36 @@ eqToEq (A B : U) (p : Id U A B) , (i = 1) -> (B,f) , (j = 1) -> (p@i,g)] -transIdFun (A B : U) (w : equiv A B) - : Id (A -> B) w.1 (transEquiv A B (transEquivToId A B w)).1 = +transPathFun (A B : U) (w : equiv A B) + : Path (A -> B) w.1 (transEquiv A B (transEquivToPath A B w)).1 = \ (a:A) -> let b : B = w.1 a u : A = comp (A) a [] - q : Id B (w.1 u) b = w.1 (comp (A) a [(i=1) -> a]) + q : Path B (w.1 u) b = w.1 (comp (A) a [(i=1) -> a]) in comp ( B) (comp ( B) (comp ( B) (comp ( B) (w.1 u) [(i=0)->q]) [(i=0)->b]) [(i=0)->b]) [(i=0)->b] -idToId (A B : U) (w : equiv A B) - : Id (equiv A B) (transEquiv A B (transEquivToId A B w)) w - = equivLemma A B (transEquiv A B (transEquivToId A B w)) w - (transIdFun A B w@-i) +idToPath (A B : U) (w : equiv A B) + : Path (equiv A B) (transEquiv A B (transEquivToPath A B w)) w + = equivLemma A B (transEquiv A B (transEquivToPath A B w)) w + (transPathFun A B w@-i) -- The grad lemma lemIso (A B : U) (f : A -> B) (g : B -> A) - (s : (y : B) -> Id B (f (g y)) y) - (t : (x : A) -> Id A (g (f x)) x) - (y : B) (x0 x1 : A) (p0 : Id B y (f x0)) (p1 : Id B y (f x1)) : - Id (fiber A B f y) (x0,p0) (x1,p1) = (p @ i,sq1 @ i) + (s : (y : B) -> Path B (f (g y)) y) + (t : (x : A) -> Path A (g (f x)) x) + (y : B) (x0 x1 : A) (p0 : Path B y (f x0)) (p1 : Path B y (f x1)) : + Path (fiber A B f y) (x0,p0) (x1,p1) = (p @ i,sq1 @ i) where - rem0 : Id A (g y) x0 = + rem0 : Path A (g y) x0 = comp ( A) (g (p0 @ i)) [ (i = 1) -> t x0, (i = 0) -> g y ] - rem1 : Id A (g y) x1 = + rem1 : Path A (g y) x1 = comp ( A) (g (p1 @ i)) [ (i = 1) -> t x1, (i = 0) -> g y ] - p : Id A x0 x1 = + p : Path A x0 x1 = comp ( A) (g y) [ (i = 0) -> rem0 , (i = 1) -> rem1 ] @@ -149,14 +149,14 @@ lemIso (A B : U) (f : A -> B) (g : B -> A) , (j = 0) -> s y ] gradLemma (A B : U) (f : A -> B) (g : B -> A) - (s : (y : B) -> Id B (f (g y)) y) - (t : (x : A) -> Id A (g (f x)) x) : isEquiv A B f = + (s : (y : B) -> Path B (f (g y)) y) + (t : (x : A) -> Path A (g (f x)) x) : isEquiv A B f = \(y:B) -> ((g y,s y@-i),\ (z:fiber A B f y) -> lemIso A B f g s t y (g y) z.1 (s y@-i) z.2) -isoId (A B : U) (f : A -> B) (g : B -> A) - (s : (y : B) -> Id B (f (g y)) y) - (t : (x : A) -> Id A (g (f x)) x) : Id U A B = +isoPath (A B : U) (f : A -> B) (g : B -> A) + (s : (y : B) -> Path B (f (g y)) y) + (t : (x : A) -> Path A (g (f x)) x) : Path U A B = glue B [ (i = 0) -> (A,f,gradLemma A B f g s t) , (i = 1) -> (B,idfun B,idIsEquiv B) ] diff --git a/examples/groupoidTrunc.ctt b/examples/groupoidTrunc.ctt index 6ea3f25..e3149bf 100644 --- a/examples/groupoidTrunc.ctt +++ b/examples/groupoidTrunc.ctt @@ -5,8 +5,8 @@ import sigma data gTrunc (A : U) = inc (a : A) - | squashC (a b : gTrunc A) (p q : Id (gTrunc A) a b) - (r s: Id (Id (gTrunc A) a b) p q) + | squashC (a b : gTrunc A) (p q : Path (gTrunc A) a b) + (r s: Path (Path (gTrunc A) a b) p q) [ (i=0) -> r @ j @ k , (i=1) -> s @ j @ k , (j=0) -> p @ k @@ -14,9 +14,9 @@ data gTrunc (A : U) , (k=0) -> a , (k=1) -> b] -gTr (A:U) (a b : gTrunc A) (p q : Id (gTrunc A) a b) - (r s: Id (Id (gTrunc A) a b) p q) : - Id (Id (Id (gTrunc A) a b) p q) r s = +gTr (A:U) (a b : gTrunc A) (p q : Path (gTrunc A) a b) + (r s: Path (Path (gTrunc A) a b) p q) : + Path (Path (Path (gTrunc A) a b) p q) r s = squashC{gTrunc A} a b p q r s@ i @ j @k gTruncRec (A B : U) (bG : groupoid B) (f : A -> B) : gTrunc A -> B = split @@ -29,43 +29,43 @@ gTruncRec (A B : U) (bG : groupoid B) (f : A -> B) : gTrunc A -> B = split ( gTruncRec A B bG f (s @ m @ n)) @ i @ j @ k lem1 (A:U) (P:A -> U) (gP:(x:A) -> groupoid (P x)) (a :A) : - (s:Id (Id A a a) (refl A a) (refl A a)) - (t:Id (Id (Id A a a) (refl A a) (refl A a)) (refl (Id A a a) (refl A a)) s) - (a1 b1:P a) (p1 q1: Id (P a) a1 b1) - (r1 : Id (Id (P a) a1 b1) p1 q1) (s1 : IdP (IdP (P (s@i@j)) a1 b1) p1 q1) -> - IdP (IdP (IdP (P (t@i@j@k)) a1 b1) p1 q1) r1 s1 = - J (Id (Id A a a) (refl A a) (refl A a)) (refl (Id A a a) (refl A a)) - (\ (s:Id (Id A a a) (refl A a) (refl A a)) - (t:Id (Id (Id A a a) (refl A a) (refl A a)) (refl (Id A a a) (refl A a)) s) -> - (a1 b1 :P a) (p1 q1: Id (P a) a1 b1) - (r1 : Id (Id (P a) a1 b1) p1 q1) (s1 : IdP (IdP (P (s@i@j)) a1 b1) p1 q1) -> - IdP (IdP (IdP (P (t@i@j@k)) a1 b1) p1 q1) r1 s1) (gP a) + (s:Path (Path A a a) (refl A a) (refl A a)) + (t:Path (Path (Path A a a) (refl A a) (refl A a)) (refl (Path A a a) (refl A a)) s) + (a1 b1:P a) (p1 q1: Path (P a) a1 b1) + (r1 : Path (Path (P a) a1 b1) p1 q1) (s1 : PathP (PathP (P (s@i@j)) a1 b1) p1 q1) -> + PathP (PathP (PathP (P (t@i@j@k)) a1 b1) p1 q1) r1 s1 = + J (Path (Path A a a) (refl A a) (refl A a)) (refl (Path A a a) (refl A a)) + (\ (s:Path (Path A a a) (refl A a) (refl A a)) + (t:Path (Path (Path A a a) (refl A a) (refl A a)) (refl (Path A a a) (refl A a)) s) -> + (a1 b1 :P a) (p1 q1: Path (P a) a1 b1) + (r1 : Path (Path (P a) a1 b1) p1 q1) (s1 : PathP (PathP (P (s@i@j)) a1 b1) p1 q1) -> + PathP (PathP (PathP (P (t@i@j@k)) a1 b1) p1 q1) r1 s1) (gP a) lem (A:U) (P:A -> U) (gP:(x:A) -> groupoid (P x)) - (a :A) : (b:A) (p q:Id A a b) (r s:Id (Id A a b) p q) (t:Id (Id (Id A a b) p q) r s) - (a1:P a) (b1:P b) (p1: IdP (P (p@i)) a1 b1) (q1: IdP (P (q@i)) a1 b1) - (r1 : IdP (IdP (P (r@i@j)) a1 b1) p1 q1) (s1 : IdP (IdP (P (s@i@j)) a1 b1) p1 q1) -> - IdP (IdP (IdP (P (t@i@j@k)) a1 b1) p1 q1) r1 s1 = - J A a (\ (b:A) (p :Id A a b) -> (q:Id A a b) (r s:Id (Id A a b) p q) (t:Id (Id (Id A a b) p q) r s) - (a1:P a) (b1:P b) (p1: IdP (P (p@i)) a1 b1) (q1: IdP (P (q@i)) a1 b1) - (r1 : IdP (IdP (P (r@i@j)) a1 b1) p1 q1) (s1 : IdP (IdP (P (s@i@j)) a1 b1) p1 q1) -> - IdP (IdP (IdP (P (t@i@j@k)) a1 b1) p1 q1) r1 s1) rem + (a :A) : (b:A) (p q:Path A a b) (r s:Path (Path A a b) p q) (t:Path (Path (Path A a b) p q) r s) + (a1:P a) (b1:P b) (p1: PathP (P (p@i)) a1 b1) (q1: PathP (P (q@i)) a1 b1) + (r1 : PathP (PathP (P (r@i@j)) a1 b1) p1 q1) (s1 : PathP (PathP (P (s@i@j)) a1 b1) p1 q1) -> + PathP (PathP (PathP (P (t@i@j@k)) a1 b1) p1 q1) r1 s1 = + J A a (\ (b:A) (p :Path A a b) -> (q:Path A a b) (r s:Path (Path A a b) p q) (t:Path (Path (Path A a b) p q) r s) + (a1:P a) (b1:P b) (p1: PathP (P (p@i)) a1 b1) (q1: PathP (P (q@i)) a1 b1) + (r1 : PathP (PathP (P (r@i@j)) a1 b1) p1 q1) (s1 : PathP (PathP (P (s@i@j)) a1 b1) p1 q1) -> + PathP (PathP (PathP (P (t@i@j@k)) a1 b1) p1 q1) r1 s1) rem where - rem : (q:Id A a a) (r s:Id (Id A a a) (refl A a) q) (t:Id (Id (Id A a a) (refl A a) q) r s) - (a1:P a) (b1:P a) (p1: Id (P a) a1 b1) (q1: IdP (P (q@i)) a1 b1) - (r1 : IdP (IdP (P (r@i@j)) a1 b1) p1 q1) (s1 : IdP (IdP (P (s@i@j)) a1 b1) p1 q1) -> - IdP (IdP (IdP (P (t@i@j@k)) a1 b1) p1 q1) r1 s1 = - J (Id A a a) (refl A a) (\ (q:Id A a a) (r:Id (Id A a a) (refl A a) q) -> - (s:Id (Id A a a) (refl A a) q) (t:Id (Id (Id A a a) (refl A a) q) r s) - (a1:P a) (b1:P a) (p1: Id (P a) a1 b1) (q1: IdP (P (q@i)) a1 b1) - (r1 : IdP (IdP (P (r@i@j)) a1 b1) p1 q1) (s1 : IdP (IdP (P (s@i@j)) a1 b1) p1 q1) -> - IdP (IdP (IdP (P (t@i@j@k)) a1 b1) p1 q1) r1 s1) (lem1 A P gP a) + rem : (q:Path A a a) (r s:Path (Path A a a) (refl A a) q) (t:Path (Path (Path A a a) (refl A a) q) r s) + (a1:P a) (b1:P a) (p1: Path (P a) a1 b1) (q1: PathP (P (q@i)) a1 b1) + (r1 : PathP (PathP (P (r@i@j)) a1 b1) p1 q1) (s1 : PathP (PathP (P (s@i@j)) a1 b1) p1 q1) -> + PathP (PathP (PathP (P (t@i@j@k)) a1 b1) p1 q1) r1 s1 = + J (Path A a a) (refl A a) (\ (q:Path A a a) (r:Path (Path A a a) (refl A a) q) -> + (s:Path (Path A a a) (refl A a) q) (t:Path (Path (Path A a a) (refl A a) q) r s) + (a1:P a) (b1:P a) (p1: Path (P a) a1 b1) (q1: PathP (P (q@i)) a1 b1) + (r1 : PathP (PathP (P (r@i@j)) a1 b1) p1 q1) (s1 : PathP (PathP (P (s@i@j)) a1 b1) p1 q1) -> + PathP (PathP (PathP (P (t@i@j@k)) a1 b1) p1 q1) r1 s1) (lem1 A P gP a) T:U = (A:U) (P:A -> U) (gP:(x:A) -> groupoid (P x)) - (a b:A) (p q:Id A a b) (r s:Id (Id A a b) p q) (t:Id (Id (Id A a b) p q) r s) - (a1:P a) (b1:P b) (p1: IdP (P (p@i)) a1 b1) (q1: IdP (P (q@i)) a1 b1) - (r1 : IdP (IdP (P (r@i@j)) a1 b1) p1 q1) (s1 : IdP (IdP (P (s@i@j)) a1 b1) p1 q1) -> - IdP (IdP (IdP (P (t@i@j@k)) a1 b1) p1 q1) r1 s1 + (a b:A) (p q:Path A a b) (r s:Path (Path A a b) p q) (t:Path (Path (Path A a b) p q) r s) + (a1:P a) (b1:P b) (p1: PathP (P (p@i)) a1 b1) (q1: PathP (P (q@i)) a1 b1) + (r1 : PathP (PathP (P (r@i@j)) a1 b1) p1 q1) (s1 : PathP (PathP (P (s@i@j)) a1 b1) p1 q1) -> + PathP (PathP (PathP (P (t@i@j@k)) a1 b1) p1 q1) r1 s1 gTruncElim1 (lem:T) (A : U) (B : (gTrunc A) -> U) @@ -86,17 +86,17 @@ gTruncElim : (A : U) (bG : (z:gTrunc A) -> groupoid (B z)) (f : (x:A) -> B (inc x)) (z:gTrunc A) -> B z = gTruncElim1 lem -univG (A B:U) (bG:groupoid B) : Id U ((gTrunc A) -> B) (A -> B) = - isoId (gTrunc A -> B) (A -> B) F G s t +univG (A B:U) (bG:groupoid B) : Path U ((gTrunc A) -> B) (A -> B) = + isoPath (gTrunc A -> B) (A -> B) F G s t where F (h : gTrunc A -> B) (a: A) : B = h (inc a) G : (A -> B) -> gTrunc A -> B = gTruncRec A B bG - s (h : A -> B) : Id (A -> B) (F (G h)) h = \ (x:A) -> h x + s (h : A -> B) : Path (A -> B) (F (G h)) h = \ (x:A) -> h x - t (h : gTrunc A -> B) : Id (gTrunc A -> B) (G (F h)) h = \ (z:gTrunc A) -> rem z @ i + t (h : gTrunc A -> B) : Path (gTrunc A -> B) (G (F h)) h = \ (z:gTrunc A) -> rem z @ i where - P (z:gTrunc A) : U = Id B (G (F h) z) (h z) + P (z:gTrunc A) : U = Path B (G (F h) z) (h z) tP (z : gTrunc A) : groupoid (P z) = setGroupoid (P z) (bG (G (F h) z) (h z)) diff --git a/examples/hedberg.ctt b/examples/hedberg.ctt index 816256d..9a57b92 100644 --- a/examples/hedberg.ctt +++ b/examples/hedberg.ctt @@ -2,61 +2,61 @@ module hedberg where import prelude -hedbergLemma (A: U) (a b:A) (f : (x : A) -> Id A a x -> Id A a x) (p : Id A a b) : +hedbergLemma (A: U) (a b:A) (f : (x : A) -> Path A a x -> Path A a x) (p : Path A a b) : Square A a a a b (refl A a) p (f a (refl A a)) (f b p) = comp ( Square A a a a (p @ i) (<_> a) ( p @ i /\ j) (f a (<_> a)) (f (p @ i) ( p @ i /\ j))) ( f a (<_> a)) [] -hedbergStable (A : U) (a b : A) (h : (x : A) -> stable (Id A a x)) - (p q : Id A a b) : Id (Id A a b) p q = +hedbergStable (A : U) (a b : A) (h : (x : A) -> stable (Path A a x)) + (p q : Path A a b) : Path (Path A a b) p q = comp (<_> A) a [ (j = 0) -> rem2 @ i , (j = 1) -> rem3 @ i , (i = 0) -> r , (i = 1) -> rem4 @ j] where - ra : Id A a a = <_> a - rem1 (x : A) : exConst (Id A a x) = stableConst (Id A a x) (h x) - f (x : A) : Id A a x -> Id A a x = (rem1 x).1 - fIsConst (x : A) : const (Id A a x) (f x) = (rem1 x).2 + ra : Path A a a = <_> a + rem1 (x : A) : exConst (Path A a x) = stableConst (Path A a x) (h x) + f (x : A) : Path A a x -> Path A a x = (rem1 x).1 + fIsConst (x : A) : const (Path A a x) (f x) = (rem1 x).2 rem4 : Square A a a b b ra (refl A b) (f b p) (f b q) = fIsConst b p q - r : Id A a a = f a ra + r : Path A a a = f a ra rem2 : Square A a a a b ra p r (f b p) = hedbergLemma A a b f p rem3 : Square A a a a b ra q r (f b q) = hedbergLemma A a b f q -hedbergS (A:U) (h : (a x:A) -> stable (Id A a x)) : set A = +hedbergS (A:U) (h : (a x:A) -> stable (Path A a x)) : set A = \(a b : A) -> hedbergStable A a b (h a) hedberg (A : U) (h : discrete A) : set A = - \(a b : A) -> hedbergStable A a b (\(b : A) -> decStable (Id A a b) (h a b)) + \(a b : A) -> hedbergStable A a b (\(b : A) -> decStable (Path A a b) (h a b)) -- Alternative version: --- hedbergLemma (A: U) (f : (a b : A) -> Id A a b -> Id A a b) (a :A) : --- (b : A) (p : Id A a b) -> --- Id (Id A a b) (compId A a a b (f a a (refl A a)) p) (f a b p) = --- J A a (\ (b:A) (p:Id A a b) -> Id (Id A a b) (compId A a a b (f a a (refl A a)) p) (f a b p)) --- (refl (Id A a a) (f a a (refl A a))) +-- hedbergLemma (A: U) (f : (a b : A) -> Path A a b -> Path A a b) (a :A) : +-- (b : A) (p : Path A a b) -> +-- Path (Path A a b) (compPath A a a b (f a a (refl A a)) p) (f a b p) = +-- J A a (\ (b:A) (p:Path A a b) -> Path (Path A a b) (compPath A a a b (f a a (refl A a)) p) (f a b p)) +-- (refl (Path A a a) (f a a (refl A a))) --- hedberg (A : U) (h : discrete A) : set A = \(a b : A) (p q : Id A a b) -> --- let rem1 (x y : A) : exConst (Id A x y) = decConst (Id A x y) (h x y) +-- hedberg (A : U) (h : discrete A) : set A = \(a b : A) (p q : Path A a b) -> +-- let rem1 (x y : A) : exConst (Path A x y) = decConst (Path A x y) (h x y) --- f (x y : A) : Id A x y -> Id A x y = (rem1 x y).1 +-- f (x y : A) : Path A x y -> Path A x y = (rem1 x y).1 --- fIsConst (x y : A) : const (Id A x y) (f x y) = (rem1 x y).2 +-- fIsConst (x y : A) : const (Path A x y) (f x y) = (rem1 x y).2 --- r : Id A a a = f a a (refl A a) +-- r : Path A a a = f a a (refl A a) --- rem2 : Id (Id A a b) (compId A a a b r p) (f a b p) = hedbergLemma A f a b p +-- rem2 : Path (Path A a b) (compPath A a a b r p) (f a b p) = hedbergLemma A f a b p --- rem3 : Id (Id A a b) (compId A a a b r q) (f a b q) = hedbergLemma A f a b q +-- rem3 : Path (Path A a b) (compPath A a a b r q) (f a b q) = hedbergLemma A f a b q --- rem4 : Id (Id A a b) (f a b p) (f a b q) = fIsConst a b p q +-- rem4 : Path (Path A a b) (f a b p) (f a b q) = fIsConst a b p q --- rem5 : Id (Id A a b) (compId A a a b r p) (compId A a a b r q) = --- compDown (Id A a b) (compId A a a b r p) (f a b p) (compId A a a b r q) +-- rem5 : Path (Path A a b) (compPath A a a b r p) (compPath A a a b r q) = +-- compDown (Path A a b) (compPath A a a b r p) (f a b p) (compPath A a a b r q) -- (f a b q) rem2 rem3 rem4 -- in lemSimpl A a a b r p q rem5 diff --git a/examples/helix.ctt b/examples/helix.ctt index 8a66280..5e0e4be 100644 --- a/examples/helix.ctt +++ b/examples/helix.ctt @@ -8,16 +8,16 @@ oneTurn (l: loopS1) : loopS1 = compS1 l loop1 backTurn (l: loopS1) : loopS1 = compS1 l invLoop -compInv (A:U) (a:A) : (x:A) (p:Id A a x) -> Id (Id A x x) (<_>x) (compId A x a x (p@-i) p) = - J A a (\ (x:A) (p:Id A a x) -> Id (Id A x x) (<_>x) (compId A x a x (p@-i) p)) rem - where rem : Id (Id A a a) (<_>a) (comp (<_>A) a [(i=0) -> <_>a,(i=1) -> <_>a]) = +compInv (A:U) (a:A) : (x:A) (p:Path A a x) -> Path (Path A x x) (<_>x) (compPath A x a x (p@-i) p) = + J A a (\ (x:A) (p:Path A a x) -> Path (Path A x x) (<_>x) (compPath A x a x (p@-i) p)) rem + where rem : Path (Path A a a) (<_>a) (comp (<_>A) a [(i=0) -> <_>a,(i=1) -> <_>a]) = comp (<_>A) a [(j=0) -> <_>a,(i=0) -> <_>a,(i=1) -> <_>a] -compInvS1 : Id loopS1 (refl S1 base) (compS1 invLoop loop1) = compInv S1 base base loop1 +compInvS1 : Path loopS1 (refl S1 base) (compS1 invLoop loop1) = compInv S1 base base loop1 -compInv (A:U) (a b:A) (q:Id A a b) : (x:A) (p:Id A b x) -> Id (Id A a b) q (compId A a x b (compId A a b x q p) (p@-i)) = - J A b (\ (x:A) (p:Id A b x) -> Id (Id A a b) q (compId A a x b (compId A a b x q p) (p@-i))) rem - where rem : Id (Id A a b) q +compInv (A:U) (a b:A) (q:Path A a b) : (x:A) (p:Path A b x) -> Path (Path A a b) q (compPath A a x b (compPath A a b x q p) (p@-i)) = + J A b (\ (x:A) (p:Path A b x) -> Path (Path A a b) q (compPath A a x b (compPath A a b x q p) (p@-i))) rem + where rem : Path (Path A a b) q (comp (<_>A) (comp (<_>A) (q@i) [(i=0) -> <_>a,(i=1) -> <_>b]) [(i=0) -> <_>a,(i=1) -> <_>b]) = comp (<_>A) (comp (<_>A) (q@i) [(j=0) -> <_>q@i,(i=0) -> <_>a,(i=1) -> <_>b]) [(j=0) -> <_>q@i,(i=0) -> <_>a,(i=1) -> <_>b] @@ -27,52 +27,52 @@ transC (A:U) (a:A) : A = comp (<_>A) a [] -- it is equal to the identity function -lemTransC (A:U) (a:A) : Id A (transC A a) a = comp (<_>A) a [(i=1) -> <_>a] +lemTransC (A:U) (a:A) : Path A (transC A a) a = comp (<_>A) a [(i=1) -> <_>a] lemFib1 (A:U) (F G : A -> U) (a:A) (fa : F a -> G a) : - (x:A) (p : Id A a x) -> (fx : F x -> G x) -> - Id U (Id (F a -> G x) (\ (u:F a) -> subst A G a x p (fa u)) (\ (u:F a) -> fx (subst A F a x p u))) - (IdP (F (p@i) -> G (p@i)) fa fx) = - J A a (\ (x:A) (p : Id A a x) -> (fx : F x -> G x) -> - Id U (Id (F a -> G x) (\ (u:F a) -> subst A G a x p (fa u)) (\ (u:F a) -> fx (subst A F a x p u))) - (IdP (F (p@i) -> G (p@i)) fa fx)) rem + (x:A) (p : Path A a x) -> (fx : F x -> G x) -> + Path U (Path (F a -> G x) (\ (u:F a) -> subst A G a x p (fa u)) (\ (u:F a) -> fx (subst A F a x p u))) + (PathP (F (p@i) -> G (p@i)) fa fx) = + J A a (\ (x:A) (p : Path A a x) -> (fx : F x -> G x) -> + Path U (Path (F a -> G x) (\ (u:F a) -> subst A G a x p (fa u)) (\ (u:F a) -> fx (subst A F a x p u))) + (PathP (F (p@i) -> G (p@i)) fa fx)) rem where - rem (ga : F a -> G a) : Id U (Id (F a -> G a) (\ (u:F a) -> transC (G a) (fa u)) (\ (u:F a) -> ga (transC (F a) u))) - (Id (F a -> G a) fa ga) = - Id (F a -> G a) (\ (u:F a) -> lemTransC (G a) (fa u)@j) (\ (u : F a) -> ga (lemTransC (F a) u@j)) + rem (ga : F a -> G a) : Path U (Path (F a -> G a) (\ (u:F a) -> transC (G a) (fa u)) (\ (u:F a) -> ga (transC (F a) u))) + (Path (F a -> G a) fa ga) = + Path (F a -> G a) (\ (u:F a) -> lemTransC (G a) (fa u)@j) (\ (u : F a) -> ga (lemTransC (F a) u@j)) -- special case -corFib1 (A:U) (F G : A -> U) (a:A) (fa ga : F a -> G a) (p:Id A a a) - (h : (u:F a) -> Id (G a) (subst A G a a p (fa u)) (ga (subst A F a a p u))) : IdP (F (p@i) -> G (p@i)) fa ga = +corFib1 (A:U) (F G : A -> U) (a:A) (fa ga : F a -> G a) (p:Path A a a) + (h : (u:F a) -> Path (G a) (subst A G a a p (fa u)) (ga (subst A F a a p u))) : PathP (F (p@i) -> G (p@i)) fa ga = comp (lemFib1 A F G a fa a p ga) (\ (u:F a) -> h u@i) [] -compIdL (A:U) (a b:A) (p : Id A a b) : Id (Id A a b) p (compId A a b b p (<_>b)) = +compPathL (A:U) (a b:A) (p : Path A a b) : Path (Path A a b) p (compPath A a b b p (<_>b)) = comp (<_>A) (p @ i) [(i=0) -> <_> a, (i = 1) -> <_>b, (j=0) -> <_>(p@i) ] -lemFib2 (A:U) (F:A->U) (a b:A) (p:Id A a b) (u:F a) : - (c:A) (q:Id A b c) -> Id (F c) (subst A F b c q (subst A F a b p u)) (subst A F a c (compId A a b c p q) u) = - J A b (\ (c:A) (q:Id A b c) -> Id (F c) (subst A F b c q (subst A F a b p u)) (subst A F a c (compId A a b c p q) u)) +lemFib2 (A:U) (F:A->U) (a b:A) (p:Path A a b) (u:F a) : + (c:A) (q:Path A b c) -> Path (F c) (subst A F b c q (subst A F a b p u)) (subst A F a c (compPath A a b c p q) u) = + J A b (\ (c:A) (q:Path A b c) -> Path (F c) (subst A F b c q (subst A F a b p u)) (subst A F a c (compPath A a b c p q) u)) rem where - rem1 : Id (F b) (subst A F a b p u) (subst A F b b (<_>b) (subst A F a b p u)) = lemTransC (F b) (subst A F a b p u)@-i - rem2 : Id (F b) (subst A F a b p u) (subst A F a b (compId A a b b p (<_>b)) u) = - subst A F a b (compIdL A a b p@i) u - rem : Id (F b) (subst A F b b (<_>b) (subst A F a b p u)) (subst A F a b (compId A a b b p (<_>b)) u) = - comp (Id (F b) (rem1@i) (rem2@i)) (<_>subst A F a b p u) [] - -testIsoId (A B : U) (f : A -> B) (g : B -> A) - (s : (y : B) -> Id B (f (g y)) y) - (t : (x : A) -> Id A (g (f x)) x) (a:A) : Id B (f a) (trans A B (isoId A B f g s t) a) = + rem1 : Path (F b) (subst A F a b p u) (subst A F b b (<_>b) (subst A F a b p u)) = lemTransC (F b) (subst A F a b p u)@-i + rem2 : Path (F b) (subst A F a b p u) (subst A F a b (compPath A a b b p (<_>b)) u) = + subst A F a b (compPathL A a b p@i) u + rem : Path (F b) (subst A F b b (<_>b) (subst A F a b p u)) (subst A F a b (compPath A a b b p (<_>b)) u) = + comp (Path (F b) (rem1@i) (rem2@i)) (<_>subst A F a b p u) [] + +testIsoPath (A B : U) (f : A -> B) (g : B -> A) + (s : (y : B) -> Path B (f (g y)) y) + (t : (x : A) -> Path A (g (f x)) x) (a:A) : Path B (f a) (trans A B (isoPath A B f g s t) a) = comp (<_>B) (comp (<_>B) (f a) [(i=0) -> <_> f a]) [(i=0) -> <_> f a] testH (n:Z) : Z = subst S1 helix base base loop1 n -testHelix : Id (Z->Z) sucZ (subst S1 helix base base loop1) = - \(n:Z) -> testIsoId Z Z sucZ predZ sucpredZ predsucZ n@i +testHelix : Path (Z->Z) sucZ (subst S1 helix base base loop1) = + \(n:Z) -> testIsoPath Z Z sucZ predZ sucpredZ predsucZ n@i -encode (x:S1) (p:Id S1 base x) : helix x = subst S1 helix base x p zeroZ +encode (x:S1) (p:Path S1 base x) : helix x = subst S1 helix base x p zeroZ itLoop : nat -> loopS1 = split zero -> triv @@ -86,94 +86,94 @@ loopIt : Z -> loopS1 = split inl n -> itLoopNeg n inr n -> itLoop n -lemItNeg : (n:nat) -> Id loopS1 (transport (Id S1 base (loop{S1} @ i)) (loopIt (inl n))) (loopIt (sucZ (inl n))) = split +lemItNeg : (n:nat) -> Path loopS1 (transport (Path S1 base (loop{S1} @ i)) (loopIt (inl n))) (loopIt (sucZ (inl n))) = split zero -> lemInv S1 base base loop1 suc n -> lemCompInv S1 base base base (itLoopNeg n) invLoop l0 : loopS1 = base l1 : loopS1 = oneTurn l0 -test1ItPos (n:nat) : U = Id loopS1 (loopIt (sucZ (inr n))) (oneTurn ((loopIt (inr n)))) +test1ItPos (n:nat) : U = Path loopS1 (loopIt (sucZ (inr n))) (oneTurn ((loopIt (inr n)))) -lem1ItPos : (n:nat) -> Id loopS1 (loopIt (sucZ (inr n))) (oneTurn ((loopIt (inr n)))) = split +lem1ItPos : (n:nat) -> Path loopS1 (loopIt (sucZ (inr n))) (oneTurn ((loopIt (inr n)))) = split zero -> refl loopS1 l1 suc p -> oneTurn (lem1ItPos p@i) -test1ItNeg (n:nat) : U = Id loopS1 (loopIt (sucZ (inl n))) (oneTurn ((loopIt (inl n)))) +test1ItNeg (n:nat) : U = Path loopS1 (loopIt (sucZ (inl n))) (oneTurn ((loopIt (inl n)))) -lem1ItNeg : (n:nat) -> Id loopS1 (loopIt (sucZ (inl n))) (oneTurn (loopIt (inl n))) = split +lem1ItNeg : (n:nat) -> Path loopS1 (loopIt (sucZ (inl n))) (oneTurn (loopIt (inl n))) = split zero -> compInvS1 suc p -> compInv S1 base base (loopIt (inl p)) base invLoop -lem1It : (n:Z) -> Id loopS1 (loopIt (sucZ n)) (oneTurn (loopIt n)) = split +lem1It : (n:Z) -> Path loopS1 (loopIt (sucZ n)) (oneTurn (loopIt n)) = split inl n -> lem1ItNeg n inr n -> lem1ItPos n -decode : (x:S1) -> helix x -> Id S1 base x = split +decode : (x:S1) -> helix x -> Path S1 base x = split base -> loopIt loop @ i -> rem @ i where T : U = Z -> loopS1 - G (x:S1) : U = Id S1 base x - p : Id U T T = helix (loop1@j) -> Id S1 base (loop1@j) - rem2 (n:Z) : Id loopS1 (oneTurn (loopIt n)) (loopIt (sucZ n)) = lem1It n@-i - rem1 (n:Z) : Id loopS1 (subst S1 G base base loop1 (loopIt n)) (loopIt (subst S1 helix base base loop1 n)) = - comp ( Id loopS1 (oneTurn (loopIt n)) (loopIt (testHelix@i n))) (rem2 n) [] - rem : IdP p loopIt loopIt = corFib1 S1 helix G base loopIt loopIt loop1 rem1 - -encodeDecode (x:S1) (p : Id S1 base x) : Id (Id S1 base x) (decode x (encode x p)) p = - transport (Id (Id S1 base (p@i)) (decode (p@i) (encode (p@i) (p@(i/\j)))) (p@(i/\j))) (refl loopS1 triv) - --- encodeDecode : (c : S1) (p : Id S1 base c) -> Id (Id S1 base c) (decode c (encode c p)) p = --- J S1 base (\ (c : S1) (p : Id S1 base c) -> Id (Id S1 base c) (decode c (encode c p)) p) + G (x:S1) : U = Path S1 base x + p : Path U T T = helix (loop1@j) -> Path S1 base (loop1@j) + rem2 (n:Z) : Path loopS1 (oneTurn (loopIt n)) (loopIt (sucZ n)) = lem1It n@-i + rem1 (n:Z) : Path loopS1 (subst S1 G base base loop1 (loopIt n)) (loopIt (subst S1 helix base base loop1 n)) = + comp ( Path loopS1 (oneTurn (loopIt n)) (loopIt (testHelix@i n))) (rem2 n) [] + rem : PathP p loopIt loopIt = corFib1 S1 helix G base loopIt loopIt loop1 rem1 + +encodeDecode (x:S1) (p : Path S1 base x) : Path (Path S1 base x) (decode x (encode x p)) p = + transport (Path (Path S1 base (p@i)) (decode (p@i) (encode (p@i) (p@(i/\j)))) (p@(i/\j))) (refl loopS1 triv) + +-- encodeDecode : (c : S1) (p : Path S1 base c) -> Path (Path S1 base c) (decode c (encode c p)) p = +-- J S1 base (\ (c : S1) (p : Path S1 base c) -> Path (Path S1 base c) (decode c (encode c p)) p) -- (<_> (<_> base)) -lemTransOneTurn (n:nat) : Id Z (transport (helix (loop1@i)) (inr n)) (inr (suc n)) = +lemTransOneTurn (n:nat) : Path Z (transport (helix (loop1@i)) (inr n)) (inr (suc n)) = inr (suc (comp (<_>nat) (comp (<_>nat) n [(i=1) -> <_>n]) [(i=1) -> <_>n])) -lemTransBackTurn (n:nat) : Id Z (transport (helix (loop1@-i)) (inl n)) (inl (suc n)) = +lemTransBackTurn (n:nat) : Path Z (transport (helix (loop1@-i)) (inl n)) (inl (suc n)) = inl (suc (comp (<_>nat) (comp (<_>nat) n [(i=1) -> <_>n]) [(i=1) -> <_>n])) -corFib2 (u:Z) (l:loopS1) : Id Z (transport (helix (oneTurn l@i)) u) +corFib2 (u:Z) (l:loopS1) : Path Z (transport (helix (oneTurn l@i)) u) (transport (helix (loop1@i)) (transport (helix (l@i)) u)) = lemFib2 S1 helix base base l u base loop1@-i -corFib3 (u:Z) (l:loopS1) : Id Z (transport (helix (backTurn l@i)) u) +corFib3 (u:Z) (l:loopS1) : Path Z (transport (helix (backTurn l@i)) u) (transport (helix (loop1@-i)) (transport (helix (l@i)) u)) = lemFib2 S1 helix base base l u base (loop1@-j)@-i -decodeEncodeBasePos : (n : nat) -> Id Z (transport ( helix (itLoop n @ x)) (inr zero)) (inr n) = split +decodeEncodeBasePos : (n : nat) -> Path Z (transport ( helix (itLoop n @ x)) (inr zero)) (inr n) = split zero -> <_> inr zero - suc n -> comp (Id Z (transport (helix (oneTurn l@i)) (inr zero)) (lemTransOneTurn n@j)) rem3 [] + suc n -> comp (Path Z (transport (helix (oneTurn l@i)) (inr zero)) (lemTransOneTurn n@j)) rem3 [] where l : loopS1 = itLoop n - rem1 : Id Z (transport ( helix (l@i)) (inr zero)) (inr n) = decodeEncodeBasePos n - rem2 : Id Z (transport (helix (oneTurn l@i)) (inr zero)) + rem1 : Path Z (transport ( helix (l@i)) (inr zero)) (inr n) = decodeEncodeBasePos n + rem2 : Path Z (transport (helix (oneTurn l@i)) (inr zero)) (transport (helix (loop1@i)) (transport (helix (l@i)) (inr zero))) = corFib2 (inr zero) l - rem3 : Id Z (transport (helix (oneTurn l@i)) (inr zero)) + rem3 : Path Z (transport (helix (oneTurn l@i)) (inr zero)) (transport (helix (loop1@i)) (inr n)) = - comp (Id Z (transport (helix (oneTurn l@i)) (inr zero)) + comp (Path Z (transport (helix (oneTurn l@i)) (inr zero)) (transport (helix (loop1@i)) (rem1@j))) rem2 [] -decodeEncodeBaseNeg : (n : nat) -> Id Z (transport ( helix (itLoopNeg n @ x)) (inr zero)) (inl n) = split +decodeEncodeBaseNeg : (n : nat) -> Path Z (transport ( helix (itLoopNeg n @ x)) (inr zero)) (inl n) = split zero -> <_> inl zero - suc n -> comp (Id Z (transport (helix (backTurn l@i)) (inr zero)) (lemTransBackTurn n@j)) rem3 [] + suc n -> comp (Path Z (transport (helix (backTurn l@i)) (inr zero)) (lemTransBackTurn n@j)) rem3 [] where l : loopS1 = itLoopNeg n - rem1 : Id Z (transport ( helix (l@i)) (inr zero)) (inl n) = decodeEncodeBaseNeg n - rem2 : Id Z (transport (helix (backTurn l@i)) (inr zero)) + rem1 : Path Z (transport ( helix (l@i)) (inr zero)) (inl n) = decodeEncodeBaseNeg n + rem2 : Path Z (transport (helix (backTurn l@i)) (inr zero)) (transport (helix (loop1@-i)) (transport (helix (l@i)) (inr zero))) = corFib3 (inr zero) l - rem3 : Id Z (transport (helix (backTurn l@i)) (inr zero)) + rem3 : Path Z (transport (helix (backTurn l@i)) (inr zero)) (transport (helix (loop1@-i)) (inl n)) = - comp (Id Z (transport (helix (backTurn l@i)) (inr zero)) + comp (Path Z (transport (helix (backTurn l@i)) (inr zero)) (transport (helix (loop1@-i)) (rem1@j))) rem2 [] -decodeEncodeBase : (n : Z) -> Id Z (encode base (decode base n)) n = split +decodeEncodeBase : (n : Z) -> Path Z (encode base (decode base n)) n = split inl n -> decodeEncodeBaseNeg n inr n -> decodeEncodeBasePos n -- the loop space of the circle is equal to Z -loopS1equalsZ : Id U loopS1 Z = - isoId loopS1 Z (encode base) (decode base) decodeEncodeBase (encodeDecode base) +loopS1equalsZ : Path U loopS1 Z = + isoPath loopS1 Z (encode base) (decode base) decodeEncodeBase (encodeDecode base) setLoop : set loopS1 = substInv U set loopS1 Z loopS1equalsZ ZSet @@ -187,25 +187,25 @@ helixSet : (x:S1) -> set (helix x) = lemPropFib (\ (x:S1) -> set (helix x)) rem -- S1 is a groupoid isGroupoidS1 : groupoid S1 = lem where - lem2 : (y : S1) -> set (Id S1 base y) - = lemPropFib (\ (y:S1) -> set (Id S1 base y)) (\ (y:S1) -> setIsProp (Id S1 base y)) setLoop + lem2 : (y : S1) -> set (Path S1 base y) + = lemPropFib (\ (y:S1) -> set (Path S1 base y)) (\ (y:S1) -> setIsProp (Path S1 base y)) setLoop - lem : (x y : S1) -> set (Id S1 x y) - = lemPropFib (\ (x:S1) -> (y : S1) -> set (Id S1 x y)) pP lem2 + lem : (x y : S1) -> set (Path S1 x y) + = lemPropFib (\ (x:S1) -> (y : S1) -> set (Path S1 x y)) pP lem2 where - pP (x:S1) : prop ((y:S1) -> set (Id S1 x y)) = - propPi S1 (\ (y:S1) -> set (Id S1 x y)) (\ (y:S1) -> setIsProp (Id S1 x y)) + pP (x:S1) : prop ((y:S1) -> set (Path S1 x y)) = + propPi S1 (\ (y:S1) -> set (Path S1 x y)) (\ (y:S1) -> setIsProp (Path S1 x y)) -substInv (A : U) (P : A -> U) (a x : A) (p : Id A a x) : P x -> P a = +substInv (A : U) (P : A -> U) (a x : A) (p : Path A a x) : P x -> P a = subst A P x a (p @ -i) -funDepTr (A0 A1:U) (p:Id U A0 A1) (u0:A0) (u1:A1) : - Id U (IdP p u0 u1) (Id A1 (transport p u0) u1) = - IdP (p @ (i\/l)) (comp (p @ (i/\l)) u0 [(i=0) -> <_>u0]) u1 +funDepTr (A0 A1:U) (p:Path U A0 A1) (u0:A0) (u1:A1) : + Path U (PathP p u0 u1) (Path A1 (transport p u0) u1) = + PathP (p @ (i\/l)) (comp (p @ (i/\l)) u0 [(i=0) -> <_>u0]) u1 lemSetTorus (E : S1 -> S1 -> U) (sE : set (E base base)) (f : (y:S1) -> E base y) (g : (x:S1) -> E x base) - (efg : Id (E base base) (f base) (g base)) : (x y:S1) -> E x y = split + (efg : Path (E base base) (f base) (g base)) : (x y:S1) -> E x y = split base -> f loop @ i -> lem2 @ i where @@ -213,60 +213,60 @@ lemSetTorus (E : S1 -> S1 -> U) (sE : set (E base base)) G (y x:S1) : U = E x y - lem1 : (y:S1) -> IdS S1 (G y) base base loop1 (f y) (f y) = lemPropFib P pP bP + lem1 : (y:S1) -> PathS S1 (G y) base base loop1 (f y) (f y) = lemPropFib P pP bP where - P (y:S1) : U = IdS S1 (G y) base base loop1 (f y) (f y) + P (y:S1) : U = PathS S1 (G y) base base loop1 (f y) (f y) sbE : (y : S1) -> set (E base y) = lemPropFib (\ (y:S1) -> set (E base y)) (\ (y:S1) -> setIsProp (E base y)) sE pP (y:S1) : prop (P y) = rem3 where - rem1 : Id U (P y) (Id (E base y) (subst S1 (G y) base base loop1 (f y)) (f y)) + rem1 : Path U (P y) (Path (E base y) (subst S1 (G y) base base loop1 (f y)) (f y)) = funDepTr (G y base) (G y base) (G y (loop{S1} @ j)) (f y) (f y) - rem2 : prop (Id (E base y) (subst S1 (G y) base base loop1 (f y)) (f y)) + rem2 : prop (Path (E base y) (subst S1 (G y) base base loop1 (f y)) (f y)) = sbE y (subst S1 (G y) base base loop1 (f y)) (f y) rem3 : prop (P y) - = substInv U prop (P y) (Id (E base y) (subst S1 (G y) base base loop1 (f y)) (f y)) rem1 rem2 + = substInv U prop (P y) (Path (E base y) (subst S1 (G y) base base loop1 (f y)) (f y)) rem1 rem2 - lem2 : IdS S1 (G base) base base loop1 (g base) (g base) + lem2 : PathS S1 (G base) base base loop1 (g base) (g base) = g (loop1 @ j) bP : P base - = substInv (E base base) (\ (u:E base base) -> IdS S1 (G base) base base loop1 u u) (f base) (g base) efg lem2 + = substInv (E base base) (\ (u:E base base) -> PathS S1 (G base) base base loop1 u u) (f base) (g base) efg lem2 - lem2 : IdS S1 F base base loop1 f f = \ (y:S1) -> (lem1 y) @ j + lem2 : PathS S1 F base base loop1 f f = \ (y:S1) -> (lem1 y) @ j -- commutativity of mult, at last -idL : (x : S1) -> Id S1 (mult base x) x = split +idL : (x : S1) -> Path S1 (mult base x) x = split base -> refl S1 base loop @ i -> loop1 @ i -multCom : (x y : S1) -> Id S1 (mult x y) (mult y x) = +multCom : (x y : S1) -> Path S1 (mult x y) (mult y x) = lemSetTorus E sE idL g efg where - E (x y: S1) : U = Id S1 (mult x y) (mult y x) + E (x y: S1) : U = Path S1 (mult x y) (mult y x) sE : set (E base base) = isGroupoidS1 base base g (x : S1) : E x base = inv S1 (mult base x) (mult x base) (idL x) - efg : Id (E base base) (idL base) (g base) = refl (E base base) (idL base) + efg : Path (E base base) (idL base) (g base) = refl (E base base) (idL base) -- associativity -multAssoc (x :S1) : (y z : S1) -> Id S1 (mult x (mult y z)) (mult (mult x y) z) = +multAssoc (x :S1) : (y z : S1) -> Path S1 (mult x (mult y z)) (mult (mult x y) z) = lemSetTorus E sE f g efg where - E (y z : S1) : U = Id S1 (mult x (mult y z)) (mult (mult x y) z) + E (y z : S1) : U = Path S1 (mult x (mult y z)) (mult (mult x y) z) sE : set (E base base) = isGroupoidS1 x x f (z : S1) : E base z = rem where - rem1 : Id S1 (mult base z) z = multCom base z + rem1 : Path S1 (mult base z) z = multCom base z - rem : Id S1 (mult x (mult base z)) (mult x z) = mult x (rem1 @ i) + rem : Path S1 (mult x (mult base z)) (mult x z) = mult x (rem1 @ i) g (y : S1) : E y base = refl S1 (mult x y) - efg : Id (E base base) (f base) (g base) = refl (E base base) (f base) + efg : Path (E base base) (f base) (g base) = refl (E base base) (f base) -- inverse law @@ -276,27 +276,27 @@ lemPropRel (P:S1 -> S1 -> U) (pP:(x y:S1) -> prop (P x y)) (bP:P base base) : (x (lemPropFib (P base) (pP base) bP) invLaw : (x y : S1) -> - Id (Id S1 (mult x y) (mult x y)) (refl S1 (mult x y)) - (compId S1 (mult x y) (mult y x) (mult x y) (multCom x y) (multCom y x)) = lemPropRel P pP bP + Path (Path S1 (mult x y) (mult x y)) (refl S1 (mult x y)) + (compPath S1 (mult x y) (mult y x) (mult x y) (multCom x y) (multCom y x)) = lemPropRel P pP bP where P (x y : S1) : U - = Id (Id S1 (mult x y) (mult x y)) (refl S1 (mult x y)) - (compId S1 (mult x y) (mult y x) (mult x y) (multCom x y) (multCom y x)) + = Path (Path S1 (mult x y) (mult x y)) (refl S1 (mult x y)) + (compPath S1 (mult x y) (mult y x) (mult x y) (multCom x y) (multCom y x)) pP (x y : S1) : prop (P x y) = isGroupoidS1 (mult x y) (mult x y) (refl S1 (mult x y)) - (compId S1 (mult x y) (mult y x) (mult x y) (multCom x y) (multCom y x)) + (compPath S1 (mult x y) (mult y x) (mult x y) (multCom x y) (multCom y x)) bP : P base base = - comp (Id (Id S1 base base) (refl S1 base) (comp (<_>S1) base [(i=0) -> refl S1 base, (j=0) -> refl S1 base, (j=1) -> refl S1 base])) - (refl (Id S1 base base) (refl S1 base)) [] + comp (Path (Path S1 base base) (refl S1 base) (comp (<_>S1) base [(i=0) -> refl S1 base, (j=0) -> refl S1 base, (j=1) -> refl S1 base])) + (refl (Path S1 base base) (refl S1 base)) [] -- the multiplication is invertible multIsEquiv : (x:S1) -> isEquiv S1 S1 (mult x) = lemPropFib P pP bP where P (x:S1) : U = isEquiv S1 S1 (mult x) pP (x:S1) : prop (P x) = propIsEquiv S1 S1 (mult x) - rem : Id (S1 -> S1) (idfun S1) (mult base) = \ (x:S1) -> idL x @ -i + rem : Path (S1 -> S1) (idfun S1) (mult base) = \ (x:S1) -> idL x @ -i bP : P base = subst (S1->S1) (isEquiv S1 S1) (idfun S1) (mult base) rem (idIsEquiv S1) -- inverse of multiplication by x @@ -305,11 +305,11 @@ invMult (x y:S1) : S1 = (multIsEquiv x y).1.1 invS1 (x:S1) : S1 = invMult x base -lemInvS1 : Id S1 (invS1 base) base = comp (<_>S1) (comp (<_>S1) base [(i=1) -> refl S1 base]) [(i=1) -> refl S1 base] +lemInvS1 : Path S1 (invS1 base) base = comp (<_>S1) (comp (<_>S1) base [(i=1) -> refl S1 base]) [(i=1) -> refl S1 base] -loopInvS1 : U = Id S1 (invS1 base) (invS1 base) +loopInvS1 : U = Path S1 (invS1 base) (invS1 base) -rePar (l: loopInvS1) : loopS1 = transport (Id S1 (lemInvS1@i) (lemInvS1@i)) l +rePar (l: loopInvS1) : loopS1 = transport (Path S1 (lemInvS1@i) (lemInvS1@i)) l test2 : Z = winding (rePar (invS1 (loop2@i))) -- EVAL: inl (suc zero) Time: 1m26.400s diff --git a/examples/hnat.ctt b/examples/hnat.ctt index 005ed28..723de51 100644 --- a/examples/hnat.ctt +++ b/examples/hnat.ctt @@ -13,7 +13,7 @@ test0 : hnat = comp (<_> hnat) nzero [] -- This reduces to "zero" test1 : nat = comp (<_> nat) zero [] -test2 : Id hnat nzero (comp (<_> hnat) nzero []) = +test2 : Path hnat nzero (comp (<_> hnat) nzero []) = fill (<_> hnat) nzero [] toNat : hnat -> nat = split @@ -24,16 +24,16 @@ fromNat : nat -> hnat = split zero -> nzero suc n -> nsuc (fromNat n) -toNatK : (n : hnat) -> Id hnat (fromNat (toNat n)) n = split +toNatK : (n : hnat) -> Path hnat (fromNat (toNat n)) n = split nzero -> <_> nzero nsuc n -> nsuc (toNatK n @ i) -fromNatK : (n : nat) -> Id nat (toNat (fromNat n)) n = split +fromNatK : (n : nat) -> Path nat (toNat (fromNat n)) n = split zero -> <_> zero suc n -> suc (fromNatK n @ i) -hnatEqNat : Id U hnat nat = - isoId hnat nat toNat fromNat fromNatK toNatK +hnatEqNat : Path U hnat nat = + isoPath hnat nat toNat fromNat fromNatK toNatK -- This is zero test3 : nat = trans hnat nat hnatEqNat test0 @@ -42,7 +42,7 @@ test3 : nat = trans hnat nat hnatEqNat test0 test4 : hnat = trans nat hnat ( hnatEqNat @ -i) zero -- This is "hComp (hnat) (hComp (hnat) nzero []) []" -test5 : hnat = trans hnat hnat (compId U hnat nat hnat hnatEqNat ( hnatEqNat @ -i)) nzero +test5 : hnat = trans hnat hnat (compPath U hnat nat hnat hnatEqNat ( hnatEqNat @ -i)) nzero hnatSet : set hnat = subst U set nat hnat ( hnatEqNat @ -i) natSet diff --git a/examples/hz.ctt b/examples/hz.ctt index 782b58c..b6d819a 100644 --- a/examples/hz.ctt +++ b/examples/hz.ctt @@ -8,8 +8,8 @@ import setquot -- shorthand for nat x nat nat2 : U = and nat nat -natlemma (a b c d : nat) : Id nat (add (add a b) (add c d)) (add (add a d) (add c b)) = - let rem : Id nat (add a (add b (add c d))) (add a (add d (add c b))) = +natlemma (a b c d : nat) : Path nat (add (add a b) (add c d)) (add (add a d) (add c b)) = + let rem : Path nat (add a (add b (add c d))) (add a (add d (add c b))) = add a (add_comm3 b c d @ i) in comp (<_> nat) (rem @ i) [ (i = 0) -> assocAdd a b (add c d) , (i = 1) -> assocAdd a d (add c b) ] @@ -17,35 +17,35 @@ natlemma (a b c d : nat) : Id nat (add (add a b) (add c d)) (add (add a d) (add rel : eqrel nat2 = (r,rem) where r : hrel nat2 = \(x y : nat2) -> - (Id nat (add x.1 y.2) (add x.2 y.1),natSet (add x.1 y.2) (add x.2 y.1)) + (Path nat (add x.1 y.2) (add x.2 y.1),natSet (add x.1 y.2) (add x.2 y.1)) rem : iseqrel nat2 r = ((rem1,rem2),rem3) where rem1 : istrans nat2 r = \(x y z : nat2) - (h1 : Id nat (add x.1 y.2) (add x.2 y.1)) - (h2 : Id nat (add y.1 z.2) (add y.2 z.1)) -> - let rem : Id nat (add (add x.1 y.2) (add y.1 z.2)) (add (add x.2 y.1) (add y.2 z.1)) = + (h1 : Path nat (add x.1 y.2) (add x.2 y.1)) + (h2 : Path nat (add y.1 z.2) (add y.2 z.1)) -> + let rem : Path nat (add (add x.1 y.2) (add y.1 z.2)) (add (add x.2 y.1) (add y.2 z.1)) = add (h1 @ i) (h2 @ i) - rem1 : Id nat (add (add x.1 y.2) (add y.1 z.2)) (add (add x.1 z.2) (add y.1 y.2)) = + rem1 : Path nat (add (add x.1 y.2) (add y.1 z.2)) (add (add x.1 z.2) (add y.1 y.2)) = natlemma x.1 y.2 y.1 z.2 - rem2 : Id nat (add (add x.2 y.1) (add y.2 z.1)) (add (add x.2 z.1) (add y.2 y.1)) = + rem2 : Path nat (add (add x.2 y.1) (add y.2 z.1)) (add (add x.2 z.1) (add y.2 y.1)) = natlemma x.2 y.1 y.2 z.1 - rem3 : Id nat (add (add x.2 z.1) (add y.2 y.1)) (add (add x.2 z.1) (add y.1 y.2)) = + rem3 : Path nat (add (add x.2 z.1) (add y.2 y.1)) (add (add x.2 z.1) (add y.1 y.2)) = add (add x.2 z.1) (add_comm y.2 y.1 @ i) - rem4 : Id nat (add (add x.2 y.1) (add y.2 z.1)) (add (add x.2 z.1) (add y.1 y.2)) = + rem4 : Path nat (add (add x.2 y.1) (add y.2 z.1)) (add (add x.2 z.1) (add y.1 y.2)) = comp (<_> nat) (add (add x.2 z.1) (add y.2 y.1)) [ (i = 0) -> rem2 @ -j , (i = 1) -> rem3 ] - rem5 : Id nat (add (add x.1 z.2) (add y.1 y.2)) (add (add x.2 z.1) (add y.1 y.2)) = + rem5 : Path nat (add (add x.1 z.2) (add y.1 y.2)) (add (add x.2 z.1) (add y.1 y.2)) = comp (<_> nat) (rem @ i) [ (i = 0) -> rem1, (i = 1) -> rem4 ] in natcancelr (add x.1 z.2) (add x.2 z.1) (add y.1 y.2) rem5 rem2 : isrefl nat2 r = \(x : nat2) -> add_comm x.1 x.2 - rem3 : issymm nat2 r = \(x y : nat2) (h : Id nat (add x.1 y.2) (add x.2 y.1)) -> - let rem : Id nat (add x.2 y.1) (add y.2 x.1) = + rem3 : issymm nat2 r = \(x y : nat2) (h : Path nat (add x.1 y.2) (add x.2 y.1)) -> + let rem : Path nat (add x.2 y.1) (add y.2 x.1) = comp (<_> nat) (add x.1 y.2) [ (i = 0) -> h , (i = 1) -> add_comm x.1 y.2 ] in comp (<_> nat) (add x.2 y.1) [ (i = 0) -> add_comm x.2 y.1 diff --git a/examples/implicit_point.ctt b/examples/implicit_point.ctt index 32a78c8..80d8405 100644 --- a/examples/implicit_point.ctt +++ b/examples/implicit_point.ctt @@ -6,14 +6,14 @@ data NoPoints = p [] propNoPoints : prop NoPoints = split - p @ i -> let rem : (b : NoPoints) -> Id NoPoints (p{NoPoints} @ i) b = split + p @ i -> let rem : (b : NoPoints) -> Path NoPoints (p{NoPoints} @ i) b = split p @ j -> p{NoPoints} @ (i /\ -k) \/ (j /\ k) in rem point0 : NoPoints = p{NoPoints} @ 0 point1 : NoPoints = p{NoPoints} @ 1 -p' : Id NoPoints point0 point1 = p{NoPoints} @ i +p' : Path NoPoints point0 point1 = p{NoPoints} @ i f1 : NoPoints -> Unit = split p @ i -> tt @@ -21,15 +21,15 @@ f1 : NoPoints -> Unit = split f2 : Unit -> NoPoints = split tt -> point0 -test : Id U NoPoints Unit = - isoId NoPoints Unit f1 f2 rem1 rem2 +test : Path U NoPoints Unit = + isoPath NoPoints Unit f1 f2 rem1 rem2 where - rem1 : (y : Unit) -> Id Unit (f1 (f2 y)) y = split + rem1 : (y : Unit) -> Path Unit (f1 (f2 y)) y = split tt -> tt - rem2 : (x : NoPoints) -> Id NoPoints (f2 (f1 x)) x = split + rem2 : (x : NoPoints) -> Path NoPoints (f2 (f1 x)) x = split p @ i -> p{NoPoints} @ j /\ i -fext (A B : U) (f g : A -> B) (h : (x : A) -> Id B (f x) (g x)) : - Id (A -> B) f g = (\(x : A) -> htpy x (p{NoPoints} @ j)) +fext (A B : U) (f g : A -> B) (h : (x : A) -> Path B (f x) (g x)) : + Path (A -> B) f g = (\(x : A) -> htpy x (p{NoPoints} @ j)) where htpy (x : A) : NoPoints -> B = split p @ i -> h x @ i diff --git a/examples/injective.ctt b/examples/injective.ctt index 5721337..bd339d4 100644 --- a/examples/injective.ctt +++ b/examples/injective.ctt @@ -5,25 +5,25 @@ import prop -- First definition of injectivity, informally: if two elements f a0, f a1 are -- equal in B, then a0, a1 must be equal in A. inj0 (A B : U) (f : A -> B) (sA : set A) (sB : set B) : U - = (a0 a1 : A) -> Id B (f a0) (f a1) -> Id A a0 a1 + = (a0 a1 : A) -> Path B (f a0) (f a1) -> Path A a0 a1 -- Second definition of injectivity, informally: for any b in B, there are -- only one elment a in A such that f a is equal to b. inj1 (A B : U) (f : A -> B) (sA : set A) (sB : set B) : U - = (b : B) -> prop ((a : A) * Id B (f a) b) + = (b : B) -> prop ((a : A) * Path B (f a) b) -- A map from the first to the second definition. inj01 (A B : U) (f : A -> B) (sA : set A) (sB : set B) : inj0 A B f sA sB -> inj1 A B f sA sB - = \ (i0 : inj0 A B f sA sB) (b : B) (c d : (a : A) * Id B (f a) b) -> let + = \ (i0 : inj0 A B f sA sB) (b : B) (c d : (a : A) * Path B (f a) b) -> let F (a : A) : U - = Id B (f a) b + = Path B (f a) b pF (a : A) : prop (F a) = sB (f a) b - p : Id B (f c.1) (f d.1) + p : Path B (f c.1) (f d.1) = comp ( B) (c.2 @ i) [ (i = 0) -> f c.1 , (i = 1) -> d.2 @ -j ] - q : Id A c.1 d.1 + q : Path A c.1 d.1 = i0 c.1 d.1 p in lemSig A F pF c d q @@ -31,15 +31,15 @@ inj01 (A B : U) (f : A -> B) (sA : set A) (sB : set B) : inj0 A B f sA sB -> -- A map from the second to the first definition. inj10 (A B : U) (f : A -> B) (sA : set A) (sB : set B) : inj1 A B f sA sB -> inj0 A B f sA sB - = \ (i1 : inj1 A B f sA sB) (a0 a1 : A) (p : Id B (f a0) (f a1)) -> let - c : (a : A) * Id B (f a) (f a1) + = \ (i1 : inj1 A B f sA sB) (a0 a1 : A) (p : Path B (f a0) (f a1)) -> let + c : (a : A) * Path B (f a) (f a1) = (a0, p) - d : (a : A) * Id B (f a) (f a1) + d : (a : A) * Path B (f a) (f a1) = (a1, f a1) - q : Id ((a : A) * Id B (f a) (f a1)) c d + q : Path ((a : A) * Path B (f a) (f a1)) c d = i1 (f a1) c d - fst : ((a : A) * Id B (f a) (f a1)) -> A - = \ (x : (a : A) * Id B (f a) (f a1)) -> x.1 + fst : ((a : A) * Path B (f a) (f a1)) -> A + = \ (x : (a : A) * Path B (f a) (f a1)) -> x.1 in fst (q @ i) @@ -47,18 +47,18 @@ inj10 (A B : U) (f : A -> B) (sA : set A) (sB : set B) : inj1 A B f sA sB -> prop_inj0 (A B : U) (f : A -> B) (sA : set A) (sB : set B) : prop (inj0 A B f sA sB) = let - c (a0 a1 : A) : prop (Id B (f a0) (f a1) -> Id A a0 a1) + c (a0 a1 : A) : prop (Path B (f a0) (f a1) -> Path A a0 a1) = let - P : Id B (f a0) (f a1) -> U - = \ (_ : Id B (f a0) (f a1)) -> Id A a0 a1 - h : (x : Id B (f a0) (f a1)) -> prop (P x) - = \ (_ : Id B (f a0) (f a1)) -> sA a0 a1 + P : Path B (f a0) (f a1) -> U + = \ (_ : Path B (f a0) (f a1)) -> Path A a0 a1 + h : (x : Path B (f a0) (f a1)) -> prop (P x) + = \ (_ : Path B (f a0) (f a1)) -> sA a0 a1 in - propPi (Id B (f a0) (f a1)) P h - d (a0 : A) : prop ((a1 : A) -> Id B (f a0) (f a1) -> Id A a0 a1) + propPi (Path B (f a0) (f a1)) P h + d (a0 : A) : prop ((a1 : A) -> Path B (f a0) (f a1) -> Path A a0 a1) = let P : A -> U - = \ (a1 : A) -> ( Id B (f a0) (f a1) -> Id A a0 a1 ) + = \ (a1 : A) -> ( Path B (f a0) (f a1) -> Path A a0 a1 ) h : (a1 : A) -> prop (P a1) = \ (a1 : A) -> c a0 a1 in @@ -66,8 +66,8 @@ prop_inj0 (A B : U) (f : A -> B) (sA : set A) (sB : set B) e : prop (inj0 A B f sA sB) = let P : A -> U - = \ (a0 : A) -> ( (a1 : A) -> Id B (f a0) (f a1) -> Id A a0 a1 ) - h : (a0 : A) -> prop ( (a1 : A) -> Id B (f a0) (f a1) -> Id A a0 a1 ) + = \ (a0 : A) -> ( (a1 : A) -> Path B (f a0) (f a1) -> Path A a0 a1 ) + h : (a0 : A) -> prop ( (a1 : A) -> Path B (f a0) (f a1) -> Path A a0 a1 ) = \ (a0 : A) -> d a0 in propPi A P h @@ -79,7 +79,7 @@ prop_inj1 (A B : U) (f : A -> B) (sA : set A) (sB : set B) : prop (inj1 A B f sA sB) = let P : B -> U - = \ (b : B) -> (a : A) * Id B (f a) b + = \ (b : B) -> (a : A) * Path B (f a) b Q : B -> U = \ (b : B) -> prop (P b) h : (b : B) -> prop (Q b) @@ -89,14 +89,14 @@ prop_inj1 (A B : U) (f : A -> B) (sA : set A) (sB : set B) : -- A proof that two propositions with maps between them can be identified with -- each other -propId (A B : U) (f : A -> B) (g : B -> A) (pA : prop A) (pB : prop B) : - Id U A B - = isoId A B f g (\ (b : B) -> pB (f (g b)) b) (\ (a : A) -> pA (g (f a)) a) +propPath (A B : U) (f : A -> B) (g : B -> A) (pA : prop A) (pB : prop B) : + Path U A B + = isoPath A B f g (\ (b : B) -> pB (f (g b)) b) (\ (a : A) -> pA (g (f a)) a) -- A proof that the two definitions of injectivity can be identified with each -- other -injId (A B : U) (f : A -> B) (sA : set A) (sB : set B) : - Id U (inj0 A B f sA sB) (inj1 A B f sA sB) - = propId (inj0 A B f sA sB) (inj1 A B f sA sB) +injPath (A B : U) (f : A -> B) (sA : set A) (sB : set B) : + Path U (inj0 A B f sA sB) (inj1 A B f sA sB) + = propPath (inj0 A B f sA sB) (inj1 A B f sA sB) (inj01 A B f sA sB) (inj10 A B f sA sB) (prop_inj0 A B f sA sB) (prop_inj1 A B f sA sB) \ No newline at end of file diff --git a/examples/int.ctt b/examples/int.ctt index 5ff3ccd..1bd6891 100644 --- a/examples/int.ctt +++ b/examples/int.ctt @@ -38,27 +38,27 @@ predZ : Z -> Z = split zero -> inl zero suc n -> inr n -sucpredZ : (x : Z) -> Id Z (sucZ (predZ x)) x = split +sucpredZ : (x : Z) -> Path Z (sucZ (predZ x)) x = split inl u -> refl Z (inl u) inr v -> lem v where - lem : (u : nat) -> Id Z (sucZ (predZ (inr u))) (inr u) = split + lem : (u : nat) -> Path Z (sucZ (predZ (inr u))) (inr u) = split zero -> refl Z (inr zero) suc n -> refl Z (inr (suc n)) -predsucZ : (x : Z) -> Id Z (predZ (sucZ x)) x = split +predsucZ : (x : Z) -> Path Z (predZ (sucZ x)) x = split inl u -> lem u where - lem : (u : nat) -> Id Z (predZ (sucZ (inl u))) (inl u) = split + lem : (u : nat) -> Path Z (predZ (sucZ (inl u))) (inl u) = split zero -> refl Z (inl zero) suc n -> refl Z (inl (suc n)) inr v -> refl Z (inr v) -sucIdZ : Id U Z Z = isoId Z Z sucZ predZ sucpredZ predsucZ +sucPathZ : Path U Z Z = isoPath Z Z sucZ predZ sucpredZ predsucZ -- We can transport along the proof forward and backwards: -testOneZ : Z = transport sucIdZ zeroZ -testNOneZ : Z = transport ( sucIdZ @ - i) zeroZ +testOneZ : Z = transport sucPathZ zeroZ +testNOneZ : Z = transport ( sucPathZ @ - i) zeroZ ZSet : set Z = hedberg Z (orDisc nat nat natDec natDec) diff --git a/examples/integer.ctt b/examples/integer.ctt index 948def4..4886fa5 100644 --- a/examples/integer.ctt +++ b/examples/integer.ctt @@ -11,7 +11,7 @@ data int = pos (n : nat) | zeroP [(i=0) -> pos zero, (i=1) -> neg zero] -- Nice version of the zero constructor: -zeroZ : Id int (pos zero) (neg zero) = zeroP {int} @ i +zeroZ : Path int (pos zero) (neg zero) = zeroP {int} @ i sucInt : int -> int = split pos n -> pos (suc n) @@ -41,36 +41,36 @@ fromZ : Z -> int = split inl n -> neg (suc n) inr n -> pos n -toZK : (a : Z) -> Id Z (toZ (fromZ a)) a = split +toZK : (a : Z) -> Path Z (toZ (fromZ a)) a = split inl n -> refl Z (inl n) inr n -> refl Z (inr n) -fromZK : (a : int) -> Id int (fromZ (toZ a)) a = split +fromZK : (a : int) -> Path int (fromZ (toZ a)) a = split pos n -> refl int (pos n) neg n -> rem n - where rem : (n : nat) -> Id int (fromZ (toZ (neg n))) (neg n) = split + where rem : (n : nat) -> Path int (fromZ (toZ (neg n))) (neg n) = split zero -> zeroZ suc m -> refl int (neg (suc m)) zeroP @ i -> zeroZ @ i /\ j -isoIntZ : Id U Z int = isoId Z int fromZ toZ fromZK toZK +isoIntZ : Path U Z int = isoPath Z int fromZ toZ fromZK toZK intSet : set int = subst U set Z int isoIntZ ZSet -- a concrete instance -T : U = Id int (pos zero) (pos zero) +T : U = Path int (pos zero) (pos zero) p0 : T = refl int (pos zero) -p1 : T = compId int (pos zero) (neg zero) (pos zero) zeroZ (zeroZ@-i) +p1 : T = compPath int (pos zero) (neg zero) (pos zero) zeroZ (zeroZ@-i) -test0 : Id (Id Z (inr zero) (inr zero)) (refl Z (inr zero)) (refl Z (inr zero)) = +test0 : Path (Path Z (inr zero) (inr zero)) (refl Z (inr zero)) (refl Z (inr zero)) = ZSet (inr zero) (inr zero) (refl Z (inr zero)) (refl Z (inr zero)) -- Tests for normal forms: -test1 : Id T p0 p1 = intSet (pos zero) (pos zero) p0 p1 -test2 : Id T p0 p0 = intSet (pos zero) (pos zero) p0 p0 +test1 : Path T p0 p1 = intSet (pos zero) (pos zero) p0 p1 +test2 : Path T p0 p0 = intSet (pos zero) (pos zero) p0 p0 -ntest1 : Id T p0 p1 = comp (<_> int) (pos zero) [ (i1 = 0) -> pos zero, (i1 = 1) -> comp (<_> int) (zeroP {int} @ (i2 /\ i3)) [ (i2 = 0) -> pos zero, (i2 = 1) -> zeroP {int} @ (-i4 /\ i3), (i3 = 0) -> pos zero, (i3 = 1) -> comp (<_> int) (zeroP {int} @ i2) [ (i2 = 0) -> pos zero, (i2 = 1) -> zeroP {int} @ (-i4 \/ -i5), (i4 = 0) -> zeroP {int} @ i2 ] ], (i2 = 0) -> pos zero, (i2 = 1) -> pos zero ] +ntest1 : Path T p0 p1 = comp (<_> int) (pos zero) [ (i1 = 0) -> pos zero, (i1 = 1) -> comp (<_> int) (zeroP {int} @ (i2 /\ i3)) [ (i2 = 0) -> pos zero, (i2 = 1) -> zeroP {int} @ (-i4 /\ i3), (i3 = 0) -> pos zero, (i3 = 1) -> comp (<_> int) (zeroP {int} @ i2) [ (i2 = 0) -> pos zero, (i2 = 1) -> zeroP {int} @ (-i4 \/ -i5), (i4 = 0) -> zeroP {int} @ i2 ] ], (i2 = 0) -> pos zero, (i2 = 1) -> pos zero ] -ntest2 : Id T p0 p0 = comp (<_> int) (pos zero) [ (i1 = 0) -> pos zero, (i1 = 1) -> pos zero, (i2 = 0) -> pos zero, (i2 = 1) -> pos zero ] +ntest2 : Path T p0 p0 = comp (<_> int) (pos zero) [ (i1 = 0) -> pos zero, (i1 = 1) -> pos zero, (i2 = 0) -> pos zero, (i2 = 1) -> pos zero ] diff --git a/examples/interval.ctt b/examples/interval.ctt index 9029372..a0d3645 100644 --- a/examples/interval.ctt +++ b/examples/interval.ctt @@ -7,8 +7,8 @@ import equiv data I = zero | one | seg [(i = 0) -> zero, (i = 1) -> one] -- Proof of funext from the interval -fext (A B : U) (f g : A -> B) (p : (x : A) -> Id B (f x) (g x)) : - Id (A -> B) f g = (\(x : A) -> htpy x (seg{I} @ j)) +fext (A B : U) (f g : A -> B) (p : (x : A) -> Path B (f x) (g x)) : + Path (A -> B) f g = (\(x : A) -> htpy x (seg{I} @ j)) where htpy (x : A) : I -> B = split zero -> f x one -> g x @@ -23,21 +23,21 @@ toUnit : I -> Unit = split fromUnit : Unit -> I = split tt -> zero -toUnitK : (a : Unit) -> Id Unit (toUnit (fromUnit a)) a = split +toUnitK : (a : Unit) -> Path Unit (toUnit (fromUnit a)) a = split tt -> tt -fromUnitK : (a : I) -> Id I (fromUnit (toUnit a)) a = split +fromUnitK : (a : I) -> Path I (fromUnit (toUnit a)) a = split zero -> zero one -> seg {I} @ i seg @ i -> seg {I} @ i /\ j -unitEqI : Id U Unit I = isoId Unit I fromUnit toUnit fromUnitK toUnitK +unitEqI : Path U Unit I = isoPath Unit I fromUnit toUnit fromUnitK toUnitK propI : prop I = subst U prop Unit I unitEqI propUnit setI : set I = subst U set Unit I unitEqI setUnit -T : U = Id I zero zero +T : U = Path I zero zero p0 : T = refl I zero test : T = propI zero zero diff --git a/examples/list.ctt b/examples/list.ctt index cba3326..83fdac8 100644 --- a/examples/list.ctt +++ b/examples/list.ctt @@ -17,13 +17,13 @@ map (A B:U) (f:A->B) : list A -> list B = split cons x xs -> cons (f x) (map A B f xs) lem (A B C:U) (f:A->B) (g:B -> C) : - (xs:list A) -> Id (list C) (map B C g (map A B f xs)) (map A C (\ (x:A) -> g (f x)) xs) = split + (xs:list A) -> Path (list C) (map B C g (map A B f xs)) (map A C (\ (x:A) -> g (f x)) xs) = split nil -> nil cons x xs -> cons (g (f x)) (lem A B C f g xs@i) -funId (A:U) (x:A) : A = x +funPath (A:U) (x:A) : A = x -lem1 (A:U) : (xs:list A) -> Id (list A) (map A A (funId A) xs) xs = split +lem1 (A:U) : (xs:list A) -> Path (list A) (map A A (funPath A) xs) xs = split nil -> nil cons x xs -> cons x (lem1 A xs@i) @@ -31,22 +31,22 @@ reverse (A : U) : list A -> list A = split nil -> nil cons x xs -> append A (reverse A xs) (cons x nil) -lem2 (A:U) : (xs:list A) -> Id (list A) (append A xs nil) xs = split +lem2 (A:U) : (xs:list A) -> Path (list A) (append A xs nil) xs = split nil -> nil cons x xs -> cons x (lem2 A xs@i) -assoc (A:U) : (xs ys zs : list A) -> Id (list A) (append A (append A xs ys) zs) (append A xs (append A ys zs)) = split +assoc (A:U) : (xs ys zs : list A) -> Path (list A) (append A (append A xs ys) zs) (append A xs (append A ys zs)) = split nil -> \ (ys zs:list A) -> append A ys zs cons x xs -> \ (ys zs:list A) -> cons x (assoc A xs ys zs@i) {- -lem4 (A:U) : (xs ys:list A) -> Id (list A) (reverse A (append A xs ys)) (append A (reverse A ys) (reverse A xs)) = split +lem4 (A:U) : (xs ys:list A) -> Path (list A) (reverse A (append A xs ys)) (append A (reverse A ys) (reverse A xs)) = split nil -> \ (ys:list A) -> lem2 A (reverse A ys)@-i cons x xs -> \ (ys:list A) -> comp (list A) (append A (lem4 A xs ys@i) (cons x nil)) [(i=1) -> assoc A (reverse A ys) (reverse A xs) (cons x nil)] -lem5 (A:U) : (xs:list A) -> Id (list A) (reverse A (reverse A xs)) xs = split +lem5 (A:U) : (xs:list A) -> Path (list A) (reverse A (reverse A xs)) xs = split nil -> nil cons x xs -> comp (list A) (lem4 A (reverse A xs) (cons x nil)@i) [(i=1) -> cons x (lem5 A xs@j)] diff --git a/examples/multS1.ctt b/examples/multS1.ctt index d7d3a3e..c6bf3ef 100644 --- a/examples/multS1.ctt +++ b/examples/multS1.ctt @@ -10,14 +10,14 @@ lemPropFib (P:S1 -> U) (pP:(x:S1) -> prop (P x)) (bP: P base) : (x:S1) -> P x = base -> bP loop @ i -> (lemPropF S1 P pP base base loop1 bP bP) @ i -idL : (x : S1) -> Id S1 (mult base x) x = split +idL : (x : S1) -> Path S1 (mult base x) x = split base -> refl S1 base loop @ i -> loop1 @ i multIsEquiv : (x:S1) -> isEquiv S1 S1 (mult x) = lemPropFib P pP bP where P (x:S1) : U = isEquiv S1 S1 (mult x) pP (x:S1) : prop (P x) = propIsEquiv S1 S1 (mult x) - rem : Id (S1 -> S1) (idfun S1) (mult base) = \ (x:S1) -> idL x @ -i + rem : Path (S1 -> S1) (idfun S1) (mult base) = \ (x:S1) -> idL x @ -i bP : P base = subst (S1->S1) (isEquiv S1 S1) (idfun S1) (mult base) rem (idIsEquiv S1) -- inverse of multiplication by x @@ -29,14 +29,14 @@ invS1 (x:S1) : S1 = invMult x base pt0 : S1 = mapOnPath S1 S1 invS1 base base loop2@0 test1 : S1 = mapOnPath S1 S1 invS1 base base loop2@1 -test2 : Id S1 pt0 pt0 = mapOnPath S1 S1 invS1 base base loop2 +test2 : Path S1 pt0 pt0 = mapOnPath S1 S1 invS1 base base loop2 -ntest2 : Id S1 pt0 pt0 = +ntest2 : Path S1 pt0 pt0 = comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) []) [ (i = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [ (j = 1) -> comp (<_>S1) (comp (<_>S1) base []) [] ], (i = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base [ (j = 0) -> comp (<_>S1) base [ (k = 0) -> base ], (j = 1) -> comp (<_>S1) base [ (k = 0) -> base ] ]) [ (j = 0) -> comp (<_>S1) (comp (<_>S1) base []) [ (k = 0) -> comp (<_>S1) base [] ], (j = 1) -> comp (<_>S1) (comp (<_>S1) base []) [ (k = 0) -> comp (<_>S1) base [] ] ]) [ (j = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [ (k = 0) -> comp (<_>S1) (comp (<_>S1) base []) [] ], (j = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [ (k = 0) -> comp (<_>S1) (comp (<_>S1) base []) [] ] ]) [ (j = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [], (j = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) []) [ (k = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [] ] ] ]) [ (i = 0) -> comp (<_>S1) (comp (<_>S1) base []) [ (j = 1) -> comp (<_>S1) base [] ], (i = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) [ (j = 1) -> comp (<_>S1) base [] ]) []) [] ]) [ (i = 0) -> comp (<_>S1) (comp (<_>S1) base []) [ (j = 0) -> comp (<_>S1) base [] ], (i = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (loop {S1} @ -j) []) []) []) [ (j = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [] ] ]) [ (i = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [ (j = 0) -> comp (<_>S1) (comp (<_>S1) base []) [] ], (i = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base [ (j = 0) -> comp (<_>S1) base [ (k = 0) -> base ], (j = 1) -> comp (<_>S1) base [ (k = 0) -> base ] ]) [ (j = 0) -> comp (<_>S1) (comp (<_>S1) base []) [ (k = 0) -> comp (<_>S1) base [] ], (j = 1) -> comp (<_>S1) (comp (<_>S1) base []) [ (k = 0) -> comp (<_>S1) base [] ] ]) [ (j = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [ (k = 0) -> comp (<_>S1) (comp (<_>S1) base []) [] ], (j = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [ (k = 0) -> comp (<_>S1) (comp (<_>S1) base []) [] ] ]) [ (j = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) []) [ (k = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [] ], (j = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) []) [ (k = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [] ] ]) [ (j = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) []) [], (j = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) []) []) [ (k = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) []) [] ] ] ]) [ (i = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) []) [ (j = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [] ], (i = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) []) []) [ (j = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) []) [] ] ]) [ (i = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) [ (j = 1) -> comp (<_>S1) base [] ]) []) [], (i = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) []) [ (j = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [] ] ]) [ (i = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base [ (j = 1) -> base ]) []) [], (i = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base [ (j = 1) -> base ]) []) []) [ (j = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [ (k = 1) -> comp (<_>S1) (comp (<_>S1) base []) [] ], (j = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base [ (k = 0) -> comp (<_>S1) base [ (l = 0) -> base ], (k = 1) -> comp (<_>S1) base [ (l = 0) -> base ] ]) [ (k = 0) -> comp (<_>S1) (comp (<_>S1) base []) [ (l = 0) -> comp (<_>S1) base [] ], (k = 1) -> comp (<_>S1) (comp (<_>S1) base []) [ (l = 0) -> comp (<_>S1) base [] ] ]) [ (k = 0) -> comp (<_>S1) (comp (<_>S1) base []) [], (k = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [ (l = 0) -> comp (<_>S1) (comp (<_>S1) base []) [] ] ] ]) [ (j = 0) -> comp (<_>S1) (comp (<_>S1) base []) [ (k = 1) -> comp (<_>S1) base [] ], (j = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base [ (k = 1) -> base ]) []) [] ]) [ (j = 0) -> comp (<_>S1) (comp (<_>S1) base []) [ (k = 0) -> comp (<_>S1) base [] ], (j = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (loop {S1} @ -k) []) []) [ (k = 0) -> comp (<_>S1) (comp (<_>S1) base []) [] ] ]) [ (j = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [ (k = 0) -> comp (<_>S1) (comp (<_>S1) base []) [] ], (j = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base [ (k = 0) -> comp (<_>S1) base [ (l = 0) -> base ], (k = 1) -> comp (<_>S1) base [ (l = 0) -> base ] ]) [ (k = 0) -> comp (<_>S1) (comp (<_>S1) base []) [ (l = 0) -> comp (<_>S1) base [] ], (k = 1) -> comp (<_>S1) (comp (<_>S1) base []) [ (l = 0) -> comp (<_>S1) base [] ] ]) [ (k = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [ (l = 0) -> comp (<_>S1) (comp (<_>S1) base []) [] ], (k = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [ (l = 0) -> comp (<_>S1) (comp (<_>S1) base []) [] ] ]) [ (k = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [], (k = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) []) [ (l = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [] ] ] ]) [ (j = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) []) [ (k = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [] ], (j = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) []) [ (k = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [] ] ]) [ (j = 0) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) [ (k = 1) -> comp (<_>S1) base [] ]) []) [], (j = 1) -> comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) [ (k = 1) -> comp (<_>S1) (comp (<_>S1) base []) [] ] ] ] -- invsone : sone -> sone = subst U (\ (X:U) -> X -> X) <_>S1 sone (s1EqCircle@-i) inv<_>S1 --- cSone : Id U sone sone = <_>sone +-- cSone : Path U sone sone = <_>sone -- pt1 : sone = -- transport cSone @@ -50,7 +50,7 @@ ntest2 : Id S1 pt0 pt0 = -- (transport cSone -- (transport cSone (transport cSone north)))))))))) --- lemPt1 : Id sone north pt1 = +-- lemPt1 : Path sone north pt1 = -- comp cSone -- (comp cSone -- (comp cSone @@ -62,8 +62,8 @@ ntest2 : Id S1 pt0 pt0 = -- (comp cSone -- (comp cSone (comp cSone north [(i=0)-><_>north]) [(i=0)-><_>north]) [(i=0)-><_>north]) [(i=0)-><_>north]) [(i=0)-><_>north]) [(i=0)-><_>north]) [(i=0)-><_>north]) [(i=0)-><_>north]) [(i=0)-><_>north]) [(i=0)-><_>north]) [(i=0)-><_>north] --- transpSone (l:Id sone pt1 pt1) : Id sone north north = --- compId sone north pt1 north lemPt1 (compId sone pt1 pt1 north l (lemPt1@-i)) +-- transpSone (l:Path sone pt1 pt1) : Path sone north north = +-- compPath sone north pt1 north lemPt1 (compPath sone pt1 pt1 north l (lemPt1@-i)) -- -- take a lot of time and memory @@ -72,9 +72,9 @@ ntest2 : Id S1 pt0 pt0 = -- test5 : Z = windingS (transpSone (invsone (loop2S@i))) {- take a lot of time to type-check -loopM2 : Id <_>S1 pt0 test1 = mapOnPath S1 S1 invS1 base base loop2 +loopM2 : Path <_>S1 pt0 test1 = mapOnPath S1 S1 invS1 base base loop2 -loopM0 : Id S1 pt0 pt0 = invMult (loop2@i) (loop2@i) +loopM0 : Path S1 pt0 pt0 = invMult (loop2@i) (loop2@i) -} @@ -89,38 +89,38 @@ invS : S1 -> S1 = split loop @ i -> loop1 @ -i test (x:S1) : S1 = mult x (invS x) -testL : Id S1 base base = test (loop{S1} @ i) +testL : Path S1 base base = test (loop{S1} @ i) test1 (x:S1) : S1 = mult (invS x) x -- test0S : Z = winding (mult (loop2@i) (invS (loop2@i))) -- loop4 : loopS1 = compS1 loop2 loop2 -- test00S : Z = winding (mult (loop4@i) (invS (loop4@i))) --- lemInv1 : (x:S1) -> Id S1 (mult (invS x) x) base = split +-- lemInv1 : (x:S1) -> Path S1 (mult (invS x) x) base = split -- base -> refl S1 base -- loop @ i -> base -- loop1 @ -i\/i\/j test2 : loopS1 = mult (loop1@i) (loop1@-i) --- test1 : IdP (Id S1 (mult (loop1@i) base) (mult (loop1@i) base)) loop1 loop1 = mult (loop1@i) (loop1@j) +-- test1 : PathP (Path S1 (mult (loop1@i) base) (mult (loop1@i) base)) loop1 loop1 = mult (loop1@i) (loop1@j) test3 : loopS1 = mult (loop1@i) (loop1@i) --- test3 : Id loopS1 test2 (compS1 loop1 loop1) = refl loopS1 test2 +-- test3 : Path loopS1 test2 (compS1 loop1 loop1) = refl loopS1 test2 --- lemBase (y:S1) : Id S1 (mult base (mult base y)) y = --- compId S1 (mult base (mult base y)) (mult base y) y (idL (mult base y)) (idL y) +-- lemBase (y:S1) : Path S1 (mult base (mult base y)) y = +-- compPath S1 (mult base (mult base y)) (mult base y) y (idL (mult base y)) (idL y) --- corrInv (x y : S1) : Id S1 (mult x (mult (invS x) y)) y = --- compId S1 (mult x (mult (invS x) y)) (mult (mult x (invS x)) y) y rem1 rem --- where rem1 : Id S1 (mult x (mult (invS x) y)) (mult (mult x (invS x)) y) = multAssoc x (invS x) y +-- corrInv (x y : S1) : Path S1 (mult x (mult (invS x) y)) y = +-- compPath S1 (mult x (mult (invS x) y)) (mult (mult x (invS x)) y) y rem1 rem +-- where rem1 : Path S1 (mult x (mult (invS x) y)) (mult (mult x (invS x)) y) = multAssoc x (invS x) y --- rem : Id S1 (mult (mult x (invS x)) y) y = --- compId S1 (mult (mult x (invS x)) y) (mult base y) y (mult (lemInv x @i) y) (idL y) +-- rem : Path S1 (mult (mult x (invS x)) y) y = +-- compPath S1 (mult (mult x (invS x)) y) (mult base y) y (mult (lemInv x @i) y) (idL y) --- corrInv1 (x y : S1) : Id S1 (mult (invS x) (mult x y)) y = --- compId S1 (mult (invS x) (mult x y)) (mult (mult (invS x) x) y) y rem1 rem --- where rem1 : Id S1 (mult (invS x) (mult x y)) (mult (mult (invS x) x) y) = multAssoc (invS x) x y +-- corrInv1 (x y : S1) : Path S1 (mult (invS x) (mult x y)) y = +-- compPath S1 (mult (invS x) (mult x y)) (mult (mult (invS x) x) y) y rem1 rem +-- where rem1 : Path S1 (mult (invS x) (mult x y)) (mult (mult (invS x) x) y) = multAssoc (invS x) x y --- rem : Id S1 (mult (mult (invS x) x) y) y = --- compId S1 (mult (mult (invS x) x) y) (mult base y) y (mult (lemInv1 x@i) y) (idL y) +-- rem : Path S1 (mult (mult (invS x) x) y) y = +-- compPath S1 (mult (mult (invS x) x) y) (mult base y) y (mult (lemInv1 x@i) y) (idL y) --- eqS1 (x:S1) : Id U S1 S1 = isoId S1 S1 (mult x) (mult (invS x)) (corrInv x) (corrInv1 x) +-- eqS1 (x:S1) : Path U S1 S1 = isoPath S1 S1 (mult x) (mult (invS x)) (corrInv x) (corrInv1 x) diff --git a/examples/nat.ctt b/examples/nat.ctt index d38334a..b98f38c 100644 --- a/examples/nat.ctt +++ b/examples/nat.ctt @@ -16,21 +16,21 @@ add (m : nat) : nat -> nat = split zero -> m suc n -> suc (add m n) -add_zero : (n : nat) -> Id nat (add zero n) n = split +add_zero : (n : nat) -> Path nat (add zero n) n = split zero -> zero suc n -> suc (add_zero n @ i) -add_suc (a:nat) : (n : nat) -> Id nat (add (suc a) n) (suc (add a n)) = split +add_suc (a:nat) : (n : nat) -> Path nat (add (suc a) n) (suc (add a n)) = split zero -> suc a suc m -> suc (add_suc a m @ i) -add_comm (a : nat) : (n : nat) -> Id nat (add a n) (add n a) = split +add_comm (a : nat) : (n : nat) -> Path nat (add a n) (add n a) = split zero -> add_zero a @ -i suc m -> comp (<_> nat) (suc (add_comm a m @ i)) [ (i = 0) -> suc (add a m) , (i = 1) -> add_suc m a @ -j ] -assocAdd (a b:nat) : (c:nat) -> Id nat (add a (add b c)) (add (add a b) c) = split +assocAdd (a b:nat) : (c:nat) -> Path nat (add a (add b c)) (add (add a b) c) = split zero -> add a b suc c1 -> suc (assocAdd a b c1@i) @@ -38,27 +38,27 @@ add' : nat -> nat -> nat = split zero -> \(x : nat) -> x suc n -> \(x : nat) -> suc (add' n x) -sucInj (n m : nat) (p : Id nat (suc n) (suc m)) : Id nat n m = +sucInj (n m : nat) (p : Path nat (suc n) (suc m)) : Path nat n m = pred (p @ i) -add_comm3 (a b c : nat) : Id nat (add a (add b c)) (add c (add b a)) = - let rem : Id nat (add a (add b c)) (add a (add c b)) = add a (add_comm b c @ i) - rem1 : Id nat (add a (add c b)) (add (add c b) a) = add_comm a (add c b) - rem2 : Id nat (add (add c b) a) (add c (add b a)) = assocAdd c b a @ -i +add_comm3 (a b c : nat) : Path nat (add a (add b c)) (add c (add b a)) = + let rem : Path nat (add a (add b c)) (add a (add c b)) = add a (add_comm b c @ i) + rem1 : Path nat (add a (add c b)) (add (add c b) a) = add_comm a (add c b) + rem2 : Path nat (add (add c b) a) (add c (add b a)) = assocAdd c b a @ -i in comp (<_> nat) (rem1 @ i) [ (i = 0) -> rem @ -j, (i = 1) -> rem2 ] -natcancelr (a b : nat) : (x : nat) -> Id nat (add a x) (add b x) -> Id nat a b = split - zero -> \(h : Id nat a b) -> h - suc x' -> \(h : Id nat (suc (add a x')) (suc (add b x'))) -> +natcancelr (a b : nat) : (x : nat) -> Path nat (add a x) (add b x) -> Path nat a b = split + zero -> \(h : Path nat a b) -> h + suc x' -> \(h : Path nat (suc (add a x')) (suc (add b x'))) -> natcancelr a b x' (sucInj (add a x') (add b x') h) idnat : nat -> nat = split zero -> zero suc n -> suc (idnat n) -test : Id (nat -> nat) idnat (idfun nat) = funExt nat (\(_ : nat) -> nat) idnat (idfun nat) rem +test : Path (nat -> nat) idnat (idfun nat) = funExt nat (\(_ : nat) -> nat) idnat (idfun nat) rem where - rem : (n : nat) -> Id nat (idnat n) n = split + rem : (n : nat) -> Path nat (idnat n) n = split zero -> refl nat zero suc n -> mapOnPath nat nat (\(x : nat) -> suc x) (idnat n) n (rem n) @@ -71,16 +71,16 @@ caseDNat (P:nat -> U) (a0 :P zero) (aS : (n:nat) -> P (suc n)) zero -> a0 suc n -> aS n -znots (n : nat) : neg (Id nat zero (suc n)) = - \ (h:Id nat zero (suc n)) -> subst nat (caseNat U nat N0) zero (suc n) h zero +znots (n : nat) : neg (Path nat zero (suc n)) = + \ (h:Path nat zero (suc n)) -> subst nat (caseNat U nat N0) zero (suc n) h zero -snotz (n : nat) : neg (Id nat (suc n) zero) = - \ (h:Id nat (suc n) zero) -> znots n (inv nat (suc n) zero h) +snotz (n : nat) : neg (Path nat (suc n) zero) = + \ (h:Path nat (suc n) zero) -> znots n (inv nat (suc n) zero h) -natDec : (n m:nat) -> dec (Id nat n m) = split - zero -> caseDNat (\ (m:nat) -> dec (Id nat zero m)) (inl (refl nat zero)) (\ (m:nat) -> inr (znots m)) - suc n -> caseDNat (\ (m:nat) -> dec (Id nat (suc n) m)) (inr (snotz n)) - (\ (m:nat) -> decEqCong (Id nat n m) (Id nat (suc n) (suc m)) (\ (p:Id nat n m) -> suc (p @ i)) +natDec : (n m:nat) -> dec (Path nat n m) = split + zero -> caseDNat (\ (m:nat) -> dec (Path nat zero m)) (inl (refl nat zero)) (\ (m:nat) -> inr (znots m)) + suc n -> caseDNat (\ (m:nat) -> dec (Path nat (suc n) m)) (inr (snotz n)) + (\ (m:nat) -> decEqCong (Path nat n m) (Path nat (suc n) (suc m)) (\ (p:Path nat n m) -> suc (p @ i)) (sucInj n m) (natDec n m)) natSet : set nat = hedberg nat natDec diff --git a/examples/ordinal.ctt b/examples/ordinal.ctt index 3e2721a..a5fe863 100644 --- a/examples/ordinal.ctt +++ b/examples/ordinal.ctt @@ -31,7 +31,7 @@ O2 (n:nat) : ord2 -> U = split zero -> Unit succ z -> O2 n z lim f -> (p:nat) -> O2 n (f p) - lim2 f -> (x:ord) -> and (O2 n (f x)) (Id ord (G2 (f x) n) (G2 (f (inj0 (G1 x n))) n)) + lim2 f -> (x:ord) -> and (O2 n (f x)) (Path ord (G2 (f x) n) (G2 (f (inj0 (G1 x n))) n)) inj12 : ord -> ord2 = split zero -> zero @@ -50,15 +50,15 @@ H2 : ord2 -> ord -> ord = split lim2 f -> \ (x:ord) -> H2 (f x) x collapsing (n:nat) : - (x:ord2) (y:ord) -> O2 n x -> Id nat (G1 (H2 x y) n) (H1 (G2 x n) (G1 y n)) = split + (x:ord2) (y:ord) -> O2 n x -> Path nat (G1 (H2 x y) n) (H1 (G2 x n) (G1 y n)) = split zero -> \ (y:ord) (h:O2 n zero) -> G1 y n succ z -> \ (y:ord) (h:O2 n (succ z)) -> collapsing n z (succ y) h lim f -> \ (y:ord) (h:O2 n (lim f)) -> collapsing n (f n) y (h n) lim2 f -> \ (y:ord) (h:O2 n (lim2 f)) -> let - rem : Id ord (G2 (f y) n) (G2 (f (inj0 (G1 y n))) n) = (h y).2 - rem1 : Id nat (G1 (H2 (f y) y) n) (H1 (G2 (f y) n) (G1 y n)) = collapsing n (f y) y (h y).1 - in comp (Id nat (G1 (H2 (f y) y) n) (H1 (rem@i) (G1 y n))) rem1 [] + rem : Path ord (G2 (f y) n) (G2 (f (inj0 (G1 y n))) n) = (h y).2 + rem1 : Path nat (G1 (H2 (f y) y) n) (H1 (G2 (f y) n) (G1 y n)) = collapsing n (f y) y (h y).1 + in comp (Path nat (G1 (H2 (f y) y) n) (H1 (rem@i) (G1 y n))) rem1 [] -- an application @@ -71,33 +71,33 @@ lemOmega1 (n:nat) : O2 n omega1 = \ (x:ord) -> (rem x,rem1 x) zero -> tt succ z -> rem z lim f -> \ (p:nat) -> rem (f p) - rem1 : (x:ord) -> Id ord (G2 (inj12 x) n) (G2 (inj12 (inj0 (G1 x n))) n) = split + rem1 : (x:ord) -> Path ord (G2 (inj12 x) n) (G2 (inj12 (inj0 (G1 x n))) n) = split zero -> zero succ z -> succ ((rem1 z)@i) lim f -> rem1 (f n) -corr1 (n:nat) : Id nat (G1 (H2 omega1 omega) n) (H1 (G2 omega1 n) (G1 omega n)) = +corr1 (n:nat) : Path nat (G1 (H2 omega1 omega) n) (H1 (G2 omega1 n) (G1 omega n)) = collapsing n omega1 omega (lemOmega1 n) -lem : (n p:nat) -> Id nat (G1 (inj0 n) p) n = split +lem : (n p:nat) -> Path nat (G1 (inj0 n) p) n = split zero -> \ (p:nat) -> zero succ q -> \ (p:nat) -> succ (lem q p@i) -lem1 (n:nat) : Id nat (G1 omega n) n = lem n n +lem1 (n:nat) : Path nat (G1 omega n) n = lem n n -lem2 : (n p:nat) -> Id ord (G2 (inj12 (inj0 n)) p) (inj0 n) = split +lem2 : (n p:nat) -> Path ord (G2 (inj12 (inj0 n)) p) (inj0 n) = split zero -> \ (p:nat) -> inj0 zero succ q -> \ (p:nat) -> succ (lem2 q p@i) test (n:nat) : ord = G2 omega1 n -lem3 (n:nat) : Id ord (G2 (inj12 (inj0 n)) n) (inj0 n) = lem2 n n +lem3 (n:nat) : Path ord (G2 (inj12 (inj0 n)) n) (inj0 n) = lem2 n n -lem4 (n:nat) : Id nat (H1 (G2 omega1 n) n) (H1 omega n) = +lem4 (n:nat) : Path nat (H1 (G2 omega1 n) n) (H1 omega n) = H1 (lem3 n@i) n -- the G1 and H1 hierarchy coincides: G1 (H2 omega1 omega) and H1 omega are the same function -corr2 : Id (nat -> nat) (G1 (H2 omega1 omega)) (H1 omega) = +corr2 : Path (nat -> nat) (G1 (H2 omega1 omega)) (H1 omega) = \ (n:nat) -> comp (<_>nat) (H1 (G2 omega1 n) ((lem1 n)@i)) [(i=0) -> corr1 n@-j,(i=1) -> lem4 n] diff --git a/examples/pi.ctt b/examples/pi.ctt index 0a92598..99aa68e 100644 --- a/examples/pi.ctt +++ b/examples/pi.ctt @@ -8,29 +8,29 @@ import equiv pi (A:U) (P:A->U) : U = (x:A) -> P x -idPi (A:U) (B:A->U) (f g : pi A B) : Id U (Id (pi A B) f g) ((x:A) -> Id (B x) (f x) (g x)) = - isoId (Id (pi A B) f g) ((x:A) -> Id (B x) (f x) (g x)) F G S T - where T0 : U = Id (pi A B) f g - T1 : U = (x:A) -> Id (B x) (f x) (g x) +idPi (A:U) (B:A->U) (f g : pi A B) : Path U (Path (pi A B) f g) ((x:A) -> Path (B x) (f x) (g x)) = + isoPath (Path (pi A B) f g) ((x:A) -> Path (B x) (f x) (g x)) F G S T + where T0 : U = Path (pi A B) f g + T1 : U = (x:A) -> Path (B x) (f x) (g x) F (p:T0) : T1 = \ (x:A) -> p@i x G (p:T1) : T0 = \ (x:A) -> p x @ i - S (p:T1) : Id T1 (F (G p)) p = refl T1 p - T (p:T0) : Id T0 (G (F p)) p = refl T0 p + S (p:T1) : Path T1 (F (G p)) p = refl T1 p + T (p:T0) : Path T0 (G (F p)) p = refl T0 p -setPi (A:U) (B:A -> U) (h:(x:A) -> set (B x)) (f g:pi A B) : prop (Id (pi A B) f g) = +setPi (A:U) (B:A -> U) (h:(x:A) -> set (B x)) (f g:pi A B) : prop (Path (pi A B) f g) = rem where - T : U = (x:A) -> Id (B x) (f x) (g x) + T : U = (x:A) -> Path (B x) (f x) (g x) rem1 : prop T = \ (p q : T) -> \ (x:A) -> h x (f x) (g x) (p x) (q x)@i - rem : prop (Id (pi A B) f g) = - subst U prop T (Id (pi A B) f g) (idPi A B f g@-i) rem1 + rem : prop (Path (pi A B) f g) = + subst U prop T (Path (pi A B) f g) (idPi A B f g@-i) rem1 -groupoidPi (A:U) (B:A -> U) (h:(x:A) -> groupoid (B x)) (f g:pi A B) : set (Id (pi A B) f g) = - subst U set T (Id (pi A B) f g) (idPi A B f g@-i) rem1 +groupoidPi (A:U) (B:A -> U) (h:(x:A) -> groupoid (B x)) (f g:pi A B) : set (Path (pi A B) f g) = + subst U set T (Path (pi A B) f g) (idPi A B f g@-i) rem1 where - T : U = (x:A) -> Id (B x) (f x) (g x) - rem1 : set T = setPi A (\ (x:A) -> Id (B x) (f x) (g x)) (\ (x:A) -> h x (f x) (g x)) + T : U = (x:A) -> Path (B x) (f x) (g x) + rem1 : set T = setPi A (\ (x:A) -> Path (B x) (f x) (g x)) (\ (x:A) -> h x (f x) (g x)) diff --git a/examples/prelude.ctt b/examples/prelude.ctt index dd1fc1b..ef24e35 100644 --- a/examples/prelude.ctt +++ b/examples/prelude.ctt @@ -2,98 +2,98 @@ module prelude where -- Identity types -Id (A : U) (a0 a1 : A) : U = IdP ( A) a0 a1 +Path (A : U) (a0 a1 : A) : U = PathP ( A) a0 a1 -refl (A : U) (a : A) : Id A a a = a +refl (A : U) (a : A) : Path A a a = a -testEta (A : U) (a b : A) (p : Id A a b) : Id (Id A a b) p p = refl (Id A a b) ( p @ i) +testEta (A : U) (a b : A) (p : Path A a b) : Path (Path A a b) p p = refl (Path A a b) ( p @ i) mapOnPath (A B : U) (f : A -> B) (a b : A) - (p : Id A a b) : Id B (f a) (f b) = f (p @ i) + (p : Path A a b) : Path B (f a) (f b) = f (p @ i) funExt (A : U) (B : A -> U) (f g : (x : A) -> B x) - (p : (x : A) -> Id (B x) (f x) (g x)) : - Id ((y : A) -> B y) f g = \(a : A) -> (p a) @ i + (p : (x : A) -> Path (B x) (f x) (g x)) : + Path ((y : A) -> B y) f g = \(a : A) -> (p a) @ i -- Transport can be defined using comp -trans (A B : U) (p : Id U A B) (a : A) : B = comp p a [] +trans (A B : U) (p : Path U A B) (a : A) : B = comp p a [] -- subst can be defined using trans: -substTrans (A : U) (P : A -> U) (a b : A) (p : Id A a b) (e : P a) : P b = +substTrans (A : U) (P : A -> U) (a b : A) (p : Path A a b) (e : P a) : P b = trans (P a) (P b) (mapOnPath A U P a b p) e -subst (A : U) (P : A -> U) (a b : A) (p : Id A a b) (e : P a) : P b = +subst (A : U) (P : A -> U) (a b : A) (p : Path A a b) (e : P a) : P b = transport (mapOnPath A U P a b p) e substEq (A : U) (P : A -> U) (a : A) (e : P a) - : Id (P a) e (subst A P a a (refl A a) e) = + : Path (P a) e (subst A P a a (refl A a) e) = fill ( P a) e [] -substInv (A : U) (P : A -> U) (a b : A) (p : Id A a b) : P b -> P a = +substInv (A : U) (P : A -> U) (a b : A) (p : Path A a b) : P b -> P a = subst A P b a ( p @ -i) -singl (A : U) (a : A) : U = (x : A) * Id A a x +singl (A : U) (a : A) : U = (x : A) * Path A a x -contrSingl (A : U) (a b : A) (p : Id A a b) : - Id (singl A a) (a,refl A a) (b,p) = (p @ i, p @ i/\j) +contrSingl (A : U) (a b : A) (p : Path A a b) : + Path (singl A a) (a,refl A a) (b,p) = (p @ i, p @ i/\j) -J (A : U) (a : A) (C : (x : A) -> Id A a x -> U) - (d : C a (refl A a)) (x : A) (p : Id A a x) : C x p = +J (A : U) (a : A) (C : (x : A) -> Path A a x -> U) + (d : C a (refl A a)) (x : A) (p : Path A a x) : C x p = subst (singl A a) T (a, refl A a) (x, p) (contrSingl A a x p) d where T (z : singl A a) : U = C (z.1) (z.2) -JEq (A : U) (a : A) (C : (x : A) -> Id A a x -> U) (d : C a (refl A a)) - : Id (C a (refl A a)) d (J A a C d a (refl A a)) = +JEq (A : U) (a : A) (C : (x : A) -> Path A a x -> U) (d : C a (refl A a)) + : Path (C a (refl A a)) d (J A a C d a (refl A a)) = substEq (singl A a) T (a, refl A a) d where T (z : singl A a) : U = C (z.1) (z.2) -inv (A : U) (a b : A) (p : Id A a b) : Id A b a = p @ -i +inv (A : U) (a b : A) (p : Path A a b) : Path A b a = p @ -i -compId (A : U) (a b c : A) (p : Id A a b) (q : Id A b c) : Id A a c = +compPath (A : U) (a b c : A) (p : Path A a b) (q : Path A b c) : Path A a c = comp (A) (p @ i) [ (i = 1) -> q, (i=0) -> a ] -compId' (A : U) (a b c : A) (p : Id A a b) (q : Id A b c) : Id A a c = - subst A (Id A a) b c q p +compPath' (A : U) (a b c : A) (p : Path A a b) (q : Path A b c) : Path A a c = + subst A (Path A a) b c q p -compId'' (A : U) (a b : A) (p : Id A a b) : (c : A) -> (q : Id A b c) -> Id A a c = - J A a ( \ (b : A) (p : Id A a b) -> (c : A) -> (q : Id A b c) -> Id A a c) rem b p - where rem (c : A) (p : Id A a c) : Id A a c = p +compPath'' (A : U) (a b : A) (p : Path A a b) : (c : A) -> (q : Path A b c) -> Path A a c = + J A a ( \ (b : A) (p : Path A a b) -> (c : A) -> (q : Path A b c) -> Path A a c) rem b p + where rem (c : A) (p : Path A a c) : Path A a c = p compUp (A : U) (a a' b b' : A) - (p : Id A a a') (q : Id A b b') (r : Id A a b) : Id A a' b' = + (p : Path A a a') (q : Path A b b') (r : Path A a b) : Path A a' b' = comp (A) (r @ i) [(i = 0) -> p, (i = 1) -> q] compDown (A : U) (a a' b b' : A) - (p : Id A a a') (q: Id A b b') : Id A a' b' -> Id A a b = + (p : Path A a a') (q: Path A b b') : Path A a' b' -> Path A a b = compUp A a' a b' b (inv A a a' p) (inv A b b' q) -lemCompInv (A:U) (a b c:A) (p:Id A a b) (q:Id A b c) - : Id (Id A a b) (compId A a c b (compId A a b c p q) (inv A b c q)) p = +lemCompInv (A:U) (a b c:A) (p:Path A a b) (q:Path A b c) + : Path (Path A a b) (compPath A a c b (compPath A a b c p q) (inv A b c q)) p = comp (A) ((fill (A) (p @ i) [(i=0) -> a, (i=1) -> q]) @ -j) [ (i=0) -> a , (i=1) -> q @ - (j \/ k) - , (j=0) -> fill (A) ((compId A a b c p q @ i)) + , (j=0) -> fill (A) ((compPath A a b c p q @ i)) [(i=0) -> a, (i=1) -> q @ -k ] , (j=1) -> p @ i ] -lemInv (A:U) (a b:A) (p:Id A a b) : Id (Id A b b) (compId A b a b (inv A a b p) p) (refl A b) = +lemInv (A:U) (a b:A) (p:Path A a b) : Path (Path A b b) (compPath A b a b (inv A a b p) p) (refl A b) = comp (A) (p @ (-i \/ j)) [(i=0) -> b, (j=1) -> b, (i=1) -> p @ (j \/ k)] -test0 (A : U) (a b : A) (p : Id A a b) : Id A a a = refl A (p @ 0) -test1 (A : U) (a b : A) (p : Id A a b) : Id A b b = refl A (p @ 1) +test0 (A : U) (a b : A) (p : Path A a b) : Path A a a = refl A (p @ 0) +test1 (A : U) (a b : A) (p : Path A a b) : Path A b b = refl A (p @ 1) --- compEmpty (A : U) (a b : A) (p : Id A a b) : Id A a b = +-- compEmpty (A : U) (a b : A) (p : Path A a b) : Path A a b = -- comp A (p @ i) [ ] -kan (A : U) (a b c d : A) (p : Id A a b) (q : Id A a c) - (r : Id A b d) : Id A c d = +kan (A : U) (a b c d : A) (p : Path A a b) (q : Path A a c) + (r : Path A b d) : Path A c d = comp (A) (p @ i) [ (i = 0) -> q, (i = 1) -> r ] -lemSimpl (A : U) (a b c : A) (p : Id A a b) (q q' : Id A b c) - (s : Id (Id A a c) (compId A a b c p q) (compId A a b c p q')) : - Id (Id A b c) q q' = +lemSimpl (A : U) (a b c : A) (p : Path A a b) (q q' : Path A b c) + (s : Path (Path A a c) (compPath A a b c p q) (compPath A a b c p q')) : + Path (Path A b c) q q' = comp ( A) a [ (j = 0) -> comp ( A) (p @ i) [ (k = 0) -> p @ i @@ -106,8 +106,8 @@ lemSimpl (A : U) (a b c : A) (p : Id A a b) (q q' : Id A b c) , (k = 0) -> p , (k = 1) -> s @ j ] -IdPathTest1 (A : U) (a b : A) (p : Id A a b) : - Id (Id A a b) p ( comp ( A) (p @ i) [(i=0) -> a,(i=1) -> b]) = +PathPathTest1 (A : U) (a b : A) (p : Path A a b) : + Path (Path A a b) p ( comp ( A) (p @ i) [(i=0) -> a,(i=1) -> b]) = fill ( A) (p @ i) [(i=0) -> a,(i=1) -> b] @ j idfun (A : U) (a : A) : A = a @@ -121,26 +121,26 @@ idfun (A : U) (a : A) : A = a -- b0 -----> b1 -- v Square (A : U) (a0 a1 b0 b1 : A) - (u : Id A a0 a1) (v : Id A b0 b1) - (r0 : Id A a0 b0) (r1 : Id A a1 b1) : U - = IdP ( (IdP ( A) (u @ i) (v @ i))) r0 r1 + (u : Path A a0 a1) (v : Path A b0 b1) + (r0 : Path A a0 b0) (r1 : Path A a1 b1) : U + = PathP ( (PathP ( A) (u @ i) (v @ i))) r0 r1 -constSquare (A : U) (a : A) (p : Id A a a) : Square A a a a a p p p p = +constSquare (A : U) (a : A) (p : Path A a a) : Square A a a a a p p p p = comp (A) a [(i = 0) -> p @ (j \/ - k), (i = 1) -> p @ (j /\ k), (j = 0) -> p @ (i \/ - k), (j = 1) -> p @ (i /\ k)] -prop (A : U) : U = (a b : A) -> Id A a b -set (A : U) : U = (a b : A) -> prop (Id A a b) -groupoid (A : U) : U = (a b : A) -> set (Id A a b) +prop (A : U) : U = (a b : A) -> Path A a b +set (A : U) : U = (a b : A) -> prop (Path A a b) +groupoid (A : U) : U = (a b : A) -> set (Path A a b) -- the collection of all sets SET : U = (X:U) * set X propSet (A : U) (h : prop A) : set A = - \(a b : A) (p q : Id A a b) -> + \(a b : A) (p q : Path A a b) -> comp (A) a [ (i=0) -> h a a , (i=1) -> h a b , (j=0) -> h a (p @ i) @@ -152,67 +152,67 @@ propIsProp (A : U) : prop (prop A) = setIsProp (A : U) : prop (set A) = \(f g : set A) -> \(a b :A) -> - propIsProp (Id A a b) (f a b) (g a b) @ i + propIsProp (Path A a b) (f a b) (g a b) @ i -IdS (A : U) (P : A -> U) (a0 a1 : A) - (p : Id A a0 a1) (u0 : P a0) (u1 : P a1) : U = - IdP ( P (p @ i)) u0 u1 +PathS (A : U) (P : A -> U) (a0 a1 : A) + (p : Path A a0 a1) (u0 : P a0) (u1 : P a1) : U = + PathP ( P (p @ i)) u0 u1 lemProp (A : U) (h : A -> prop A) : prop A = \(a : A) -> h a a propPi (A : U) (B : A -> U) (h : (x : A) -> prop (B x)) - (f0 f1 : (x : A) -> B x) : Id ((x : A) -> B x) f0 f1 + (f0 f1 : (x : A) -> B x) : Path ((x : A) -> B x) f0 f1 = \ (x:A) -> (h x (f0 x) (f1 x)) @ i lemPropF (A : U) (P : A -> U) (pP : (x : A) -> prop (P x)) (a0 a1 :A) - (p : Id A a0 a1) (b0 : P a0) (b1 : P a1) : IdP (P (p@i)) b0 b1 = + (p : Path A a0 a1) (b0 : P a0) (b1 : P a1) : PathP (P (p@i)) b0 b1 = pP (p@i) (comp (P (p@i/\j)) b0 [(i=0) -> <_>b0]) (comp (P (p@i\/-j)) b1 [(i=1) -> <_>b1])@i -- other proof -- lemPropF (A : U) (P : A -> U) (pP : (x : A) -> prop (P x)) (a :A) : --- (a1 : A) (p : Id A a a1) (b0 : P a) (b1 : P a1) -> IdP (P (p@i)) b0 b1 = --- J A a (\ (a1 : A) (p : Id A a a1) -> --- (b0 : P a) (b1 : P a1) -> IdP (P (p@i)) b0 b1) +-- (a1 : A) (p : Path A a a1) (b0 : P a) (b1 : P a1) -> PathP (P (p@i)) b0 b1 = +-- J A a (\ (a1 : A) (p : Path A a a1) -> +-- (b0 : P a) (b1 : P a1) -> PathP (P (p@i)) b0 b1) -- rem --- where rem : (b0 b1:P a) -> Id (P a) b0 b1 = pP a +-- where rem : (b0 b1:P a) -> Path (P a) b0 b1 = pP a Sigma (A : U) (B : A -> U) : U = (x : A) * B x lemSig (A : U) (B : A -> U) (pB : (x : A) -> prop (B x)) - (u v : (x:A) * B x) (p : Id A u.1 v.1) : - Id ((x:A) * B x) u v = + (u v : (x:A) * B x) (p : Path A u.1 v.1) : + Path ((x:A) * B x) u v = (p@i,(lemPropF A B pB u.1 v.1 p u.2 v.2)@i) propSig (A : U) (B : A -> U) (pA : prop A) (pB : (x : A) -> prop (B x)) (t u : (x:A) * B x) : - Id ((x:A) * B x) t u = + Path ((x:A) * B x) t u = lemSig A B pB t u (pA t.1 u.1) -isContr (A : U) : U = (x : A) * ((y : A) -> Id A x y) +isContr (A : U) : U = (x : A) * ((y : A) -> Path A x y) propIsContr (A : U) : prop (isContr A) = lemProp (isContr A) rem where rem (t : isContr A) : prop (isContr A) = propSig A T pA pB where - T (x : A) : U = (y : A) -> Id A x y - pA (x y : A) : Id A x y = compId A x t.1 y ( t.2 x @ -i) (t.2 y) + T (x : A) : U = (y : A) -> Path A x y + pA (x y : A) : Path A x y = compPath A x t.1 y ( t.2 x @ -i) (t.2 y) pB (x : A) : prop (T x) = - propPi A (\ (y : A) -> Id A x y) (propSet A pA x) + propPi A (\ (y : A) -> Path A x y) (propSet A pA x) -- Alternative proof: --- propIsContr (A:U) (z0 z1:isContr A) : Id (isContr A) z0 z1 = +-- propIsContr (A:U) (z0 z1:isContr A) : Path (isContr A) z0 z1 = -- (p0 a1@j, -- \ (x:A) -> -- comp (<_>A) (lem1 x@i@j) [ (i=0) -> p0 a1@j, (i=1) -> p0 x@(j\/k), -- (j=0) -> p0 x@(i/\k), (j=1) -> p1 x@i ]) -- where -- a0 : A = z0.1 --- p0 : (x:A) -> Id A a0 x = z0.2 +-- p0 : (x:A) -> Path A a0 x = z0.2 -- a1 : A = z1.1 --- p1 : (x:A) -> Id A a1 x = z1.2 --- lem1 (x:A) : IdP (Id A a0 (p1 x@i)) (p0 a1) (p0 x) = p0 (p1 x@i) @ j +-- p1 : (x:A) -> Path A a1 x = z1.2 +-- lem1 (x:A) : PathP (Path A a0 (p1 x@i)) (p0 a1) (p0 x) = p0 (p1 x@i) @ j -- Basic data types @@ -225,7 +225,7 @@ neg (A : U) : U = A -> N0 data Unit = tt propUnit : prop Unit = split - tt -> split@((x:Unit) -> Id Unit tt x) with + tt -> split@((x:Unit) -> Path Unit tt x) with tt -> tt setUnit : set Unit = propSet Unit propUnit @@ -234,20 +234,20 @@ data or (A B : U) = inl (a : A) | inr (b : B) propOr (A B : U) (hA : prop A) (hB : prop B) (h : A -> neg B) : prop (or A B) = split - inl a' -> split@((b : or A B) -> Id (or A B) (inl a') b) with + inl a' -> split@((b : or A B) -> Path (or A B) (inl a') b) with inl b' -> inl (hA a' b' @ i) - inr b' -> efq (Id (or A B) (inl a') (inr b')) (h a' b') - inr a' -> split@((b : or A B) -> Id (or A B) (inr a') b) with - inl b' -> efq (Id (or A B) (inr a') (inl b')) (h b' a') + inr b' -> efq (Path (or A B) (inl a') (inr b')) (h a' b') + inr a' -> split@((b : or A B) -> Path (or A B) (inr a') b) with + inl b' -> efq (Path (or A B) (inr a') (inl b')) (h b' a') inr b' -> inr (hB a' b' @ i) stable (A:U) : U = neg (neg A) -> A -const (A : U) (f : A -> A) : U = (x y : A) -> Id A (f x) (f y) +const (A : U) (f : A -> A) : U = (x y : A) -> Path A (f x) (f y) exConst (A : U) : U = (f:A -> A) * const A f -propN0 : prop N0 = \ (x y:N0) -> efq (Id N0 x y) x +propN0 : prop N0 = \ (x y:N0) -> efq (Path N0 x y) x propNeg (A:U) : prop (neg A) = \ (f g:neg A) -> \(x:A) -> (propN0 (f x) (g x))@i @@ -268,15 +268,15 @@ decStable (A:U) : dec A -> stable A = split decConst (A : U) : dec A -> exConst A = split inl a -> (\ (x:A) -> a, \ (x y:A) -> refl A a) - inr h -> (\ (x:A) -> x, \ (x y:A) -> efq (Id A x y) (h x)) + inr h -> (\ (x:A) -> x, \ (x y:A) -> efq (Path A x y) (h x)) stableConst (A : U) (sA: stable A) : exConst A = (\ (x:A) -> sA (dNeg A x),\ (x y:A) -> sA (propNeg (neg A) (dNeg A x) (dNeg A y) @ i)) -discrete (A : U) : U = (a b : A) -> dec (Id A a b) +discrete (A : U) : U = (a b : A) -> dec (Path A a b) injective (A B : U) (f : A -> B) : U = - (a0 a1 : A) -> Id B (f a0) (f a1) -> Id A a0 a1 + (a0 a1 : A) -> Path B (f a0) (f a1) -> Path A a0 a1 and (A B : U) : U = (_ : A) * B diff --git a/examples/prop.ctt b/examples/prop.ctt index e00387d..a700d76 100644 --- a/examples/prop.ctt +++ b/examples/prop.ctt @@ -4,16 +4,16 @@ import prelude import equiv lemProp (A B : U) (f : A -> B) (g : B -> A) - (s : (y:B) -> Id B (f (g y)) y) - (t : (x:A) -> Id A (g (f x)) x) (pA:prop A) : prop B = - \ (b0 b1:B) -> subst U prop A B (isoId A B f g s t) pA b0 b1 + (s : (y:B) -> Path B (f (g y)) y) + (t : (x:A) -> Path A (g (f x)) x) (pA:prop A) : prop B = + \ (b0 b1:B) -> subst U prop A B (isoPath A B f g s t) pA b0 b1 {- normal form \(A B : U) -> \(f : A -> B) -> \(g : B -> A) -> - \(s : (y : B) -> IdP ( B) (f (g y)) y) -> - \(t : (x : A) -> IdP ( A) (g (f x)) x) -> - \(pA : (a : A) -> (b : A) -> IdP ( A) a b) -> + \(s : (y : B) -> PathP ( B) (f (g y)) y) -> + \(t : (x : A) -> PathP ( A) (g (f x)) x) -> + \(pA : (a : A) -> (b : A) -> PathP ( A) a b) -> \(b0 b1 : B) -> comp B (f (pA (g b0) (g b1) @ !1)) [ (!1 = 0) -> comp B b0 @@ -25,27 +25,27 @@ lemProp (A B : U) (f : A -> B) (g : B -> A) [ (!3 = 0) -> s b1 @ !4, (!3 = 1) -> s (f (g b1)) @ !4 ] ] ] \(A B : U) -> \(f : A -> B) -> \(g : B -> A) -> - \(s : (y : B) -> IdP ( B) (f (g y)) y) -> - \(t : (x : A) -> IdP ( A) (g (f x)) x) -> - \(pA : (a b : A) -> IdP ( A) a b) -> + \(s : (y : B) -> PathP ( B) (f (g y)) y) -> + \(t : (x : A) -> PathP ( A) (g (f x)) x) -> + \(pA : (a b : A) -> PathP ( A) a b) -> \(b0 b1 : B) -> comp B (f (pA (g b0) (g b1) @ !1)) [ (!1 = 0) -> comp B b0 [ (!2 = 0) -> comp B (f (comp A (g b0) [ (!3 = 1) -> t (g b0) @ -!4 ])) [ (!3 = 0) -> s b0 @ !4, (!3 = 1) -> s (f (g b0)) @ !4 ] ], (!1 = 1) -> comp B b1 [ (!2 = 0) -> comp B (f (comp A (g b1) [ (!3 = 1) -> t (g b1) @ -!4 ])) [ (!3 = 0) -> s b1 @ !4, (!3 = 1) -> s (f (g b1)) @ !4 ] ] ] -} lemSet (A B : U) (f : A -> B) (g : B -> A) - (s : (y:B) -> Id B (f (g y)) y) - (t : (x:A) -> Id A (g (f x)) x) (sA:set A) : set B = - \ (b0 b1:B) (p q : Id B b0 b1) -> subst U set A B (isoId A B f g s t) sA b0 b1 p q + (s : (y:B) -> Path B (f (g y)) y) + (t : (x:A) -> Path A (g (f x)) x) (sA:set A) : set B = + \ (b0 b1:B) (p q : Path B b0 b1) -> subst U set A B (isoPath A B f g s t) sA b0 b1 p q {- normal form \(A B : U) -> \(f : A -> B) -> \(g : B -> A) -> - \(s : (y : B) -> IdP ( B) (f (g y)) y) -> - \(t : (x : A) -> IdP ( A) (g (f x)) x) -> - \(sA : (a b : A) -> (a0 b0 : IdP ( A) a b) -> IdP ( IdP ( A) a b) a0 b0) -> + \(s : (y : B) -> PathP ( B) (f (g y)) y) -> + \(t : (x : A) -> PathP ( A) (g (f x)) x) -> + \(sA : (a b : A) -> (a0 b0 : PathP ( A) a b) -> PathP ( PathP ( A) a b) a0 b0) -> \(b0 b1 : B) -> - \(p q : IdP ( B) b0 b1) -> + \(p q : PathP ( B) b0 b1) -> comp B (comp B (f (sA (g b0) (g b1) ( comp A (g (p @ !1)) [ (!1 = 0) -> comp A (g (comp B b0 [ (!2 = 1) -> comp B (f (comp A (g b0) [ (!3 = 1) -> t (g b0) @ -!4 ])) [ (!3 = 0) -> s b0 @ !4, (!3 = 1) -> s (f (g b0)) @ !4 ] ])) [ (!2 = 1) -> t (g b0) @ !3 ], (!1 = 1) -> comp A (g (comp B b1 [ (!2 = 1) -> comp B (f (comp A (g b1) [ (!3 = 1) -> t (g b1) @ -!4 ])) [ (!3 = 0) -> s b1 @ !4, (!3 = 1) -> s (f (g b1)) @ !4 ] ])) [ (!2 = 1) -> t (g b1) @ !3 ] ]) ( comp A (g (q @ !1)) [ (!1 = 0) -> comp A (g (comp B b0 [ (!2 = 1) -> comp B (f (comp A (g b0) [ (!3 = 1) -> t (g b0) @ -!4 ])) [ (!3 = 0) -> s b0 @ !4, (!3 = 1) -> s (f (g b0)) @ !4 ] ])) [ (!2 = 1) -> t (g b0) @ !3 ], (!1 = 1) -> comp A (g (comp B b1 [ (!2 = 1) -> comp B (f (comp A (g b1) [ (!3 = 1) -> t (g b1) @ -!4 ])) [ (!3 = 0) -> s b1 @ !4, (!3 = 1) -> s (f (g b1)) @ !4 ] ])) [ (!2 = 1) -> t (g b1) @ !3 ] ]) @ !1 @ !2)) [ (!2 = 0) -> comp B b0 [ (!3 = 0) -> comp B (f (comp A (g b0) [ (!4 = 1) -> t (g b0) @ -!5 ])) [ (!4 = 0) -> s b0 @ !5, (!4 = 1) -> s (f (g b0)) @ !5 ] ], (!2 = 1) -> comp B b1 [ (!3 = 0) -> comp B (f (comp A (g b1) [ (!4 = 1) -> t (g b1) @ -!5 ])) [ (!4 = 0) -> s b1 @ !5, (!4 = 1) -> s (f (g b1)) @ !5 ] ] ]) [ (!1 = 0) -> comp B (comp B (comp B (comp B (p @ !2) [ (!3 = 0) -> comp B (f (comp A (g (p @ !2)) [ (!4 = 1) -> t (g (p @ !2)) @ -!5 ])) [ (!4 = 0) -> s (p @ !2) @ !5, (!4 = 1) -> s (f (g (p @ !2))) @ !5 ] ]) [ (!2 = 0) -> comp B (comp B b0 [ (!3 = 0)(!4 = 1) -> comp B (f (comp A (g b0) [ (!5 = 1) -> t (g b0) @ -!6 ])) [ (!5 = 0) -> s b0 @ !6, (!5 = 1) -> s (f (g b0)) @ !6 ] ]) [ (!3 = 0) -> comp B (f (comp A (comp A (g (comp B b0 [ (!4 = 1) -> comp B (f (comp A (g b0) [ (!5 = 1) -> t (g b0) @ -!6 ])) [ (!5 = 0) -> s b0 @ !6, (!5 = 1) -> s (f (g b0)) @ !6 ] ])) [ (!4 = 1) -> t (g b0) @ (!5 /\ !6) ]) [ (!4 = 1) -> t (g b0) @ (!5 /\ -!6), (!5 = 1) -> t (comp A (g (comp B b0 [ (!4 = 1) -> comp B (f (comp A (g b0) [ (!5 = 1) -> t (g b0) @ -!6 ])) [ (!5 = 0) -> s b0 @ !6, (!5 = 1) -> s (f (g b0)) @ !6 ] ])) [ (!4 = 1) -> t (g b0) @ !5 ]) @ -!6 ])) [ (!4 = 1) -> s (f (g b0)) @ !6, (!5 = 0) -> s (comp B b0 [ (!4 = 1) -> comp B (f (comp A (g b0) [ (!5 = 1) -> t (g b0) @ -!6 ])) [ (!5 = 0) -> s b0 @ !6, (!5 = 1) -> s (f (g b0)) @ !6 ] ]) @ !6, (!5 = 1) -> s (f (comp A (g (comp B b0 [ (!4 = 1) -> comp B (f (comp A (g b0) [ (!5 = 1) -> t (g b0) @ -!6 ])) [ (!5 = 0) -> s b0 @ !6, (!5 = 1) -> s (f (g b0)) @ !6 ] ])) [ (!4 = 1) -> t (g b0) @ !5 ])) @ !6 ] ], (!2 = 1) -> comp B (comp B b1 [ (!3 = 0)(!4 = 1) -> comp B (f (comp A (g b1) [ (!5 = 1) -> t (g b1) @ -!6 ])) [ (!5 = 0) -> s b1 @ !6, (!5 = 1) -> s (f (g b1)) @ !6 ] ]) [ (!3 = 0) -> comp B (f (comp A (comp A (g (comp B b1 [ (!4 = 1) -> comp B (f (comp A (g b1) [ (!5 = 1) -> t (g b1) @ -!6 ])) [ (!5 = 0) -> s b1 @ !6, (!5 = 1) -> s (f (g b1)) @ !6 ] ])) [ (!4 = 1) -> t (g b1) @ (!5 /\ !6) ]) [ (!4 = 1) -> t (g b1) @ (!5 /\ -!6), (!5 = 1) -> t (comp A (g (comp B b1 [ (!4 = 1) -> comp B (f (comp A (g b1) [ (!5 = 1) -> t (g b1) @ -!6 ])) [ (!5 = 0) -> s b1 @ !6, (!5 = 1) -> s (f (g b1)) @ !6 ] ])) [ (!4 = 1) -> t (g b1) @ !5 ]) @ -!6 ])) [ (!4 = 1) -> s (f (g b1)) @ !6, (!5 = 0) -> s (comp B b1 [ (!4 = 1) -> comp B (f (comp A (g b1) [ (!5 = 1) -> t (g b1) @ -!6 ])) [ (!5 = 0) -> s b1 @ !6, (!5 = 1) -> s (f (g b1)) @ !6 ] ]) @ !6, (!5 = 1) -> s (f (comp A (g (comp B b1 [ (!4 = 1) -> comp B (f (comp A (g b1) [ (!5 = 1) -> t (g b1) @ -!6 ])) [ (!5 = 0) -> s b1 @ !6, (!5 = 1) -> s (f (g b1)) @ !6 ] ])) [ (!4 = 1) -> t (g b1) @ !5 ])) @ !6 ] ] ]) [ (!3 = 0) -> comp B (f (g (p @ !2))) [ (!2 = 0) -> f (comp A (g (comp B b0 [ (!5 = 1) -> comp B (f (comp A (g b0) [ (!6 = 1) -> t (g b0) @ -!7 ])) [ (!6 = 0) -> s b0 @ !7, (!6 = 1) -> s (f (g b0)) @ !7 ] ])) [ (!5 = 1) -> t (g b0) @ !6 ]), (!2 = 1) -> f (comp A (g (comp B b1 [ (!5 = 1) -> comp B (f (comp A (g b1) [ (!6 = 1) -> t (g b1) @ -!7 ])) [ (!6 = 0) -> s b1 @ !7, (!6 = 1) -> s (f (g b1)) @ !7 ] ])) [ (!5 = 1) -> t (g b1) @ !6 ]), (!4 = 0) -> comp B (f (g (p @ !2))) [ (!2 = 0) -> f (comp A (g (comp B b0 [ (!5 = 1)(!6 = 1) -> comp B (f (comp A (g b0) [ (!7 = 1) -> t (g b0) @ -!8 ])) [ (!7 = 0) -> s b0 @ !8, (!7 = 1) -> s (f (g b0)) @ !8 ] ])) [ (!5 = 1)(!6 = 1) -> t (g b0) @ !7 ]), (!2 = 1) -> f (comp A (g (comp B b1 [ (!5 = 1)(!6 = 1) -> comp B (f (comp A (g b1) [ (!7 = 1) -> t (g b1) @ -!8 ])) [ (!7 = 0) -> s b1 @ !8, (!7 = 1) -> s (f (g b1)) @ !8 ] ])) [ (!5 = 1)(!6 = 1) -> t (g b1) @ !7 ]) ], (!4 = 1) -> f (comp A (g (p @ !2)) [ (!2 = 0) -> comp A (g (comp B b0 [ (!5 = 1)(!6 = 1) -> comp B (f (comp A (g b0) [ (!7 = 1) -> t (g b0) @ -!8 ])) [ (!7 = 0) -> s b0 @ !8, (!7 = 1) -> s (f (g b0)) @ !8 ] ])) [ (!5 = 1)(!6 = 1) -> t (g b0) @ !7 ], (!2 = 1) -> comp A (g (comp B b1 [ (!5 = 1)(!6 = 1) -> comp B (f (comp A (g b1) [ (!7 = 1) -> t (g b1) @ -!8 ])) [ (!7 = 0) -> s b1 @ !8, (!7 = 1) -> s (f (g b1)) @ !8 ] ])) [ (!5 = 1)(!6 = 1) -> t (g b1) @ !7 ] ]) ] ]) [ (!2 = 0) -> comp B b0 [ (!3 = 0)(!4 = 0) -> comp B (f (comp A (g b0) [ (!5 = 1) -> t (g b0) @ -!6 ])) [ (!5 = 0) -> s b0 @ !6, (!5 = 1) -> s (f (g b0)) @ !6 ] ], (!2 = 1) -> comp B b1 [ (!3 = 0)(!4 = 0) -> comp B (f (comp A (g b1) [ (!5 = 1) -> t (g b1) @ -!6 ])) [ (!5 = 0) -> s b1 @ !6, (!5 = 1) -> s (f (g b1)) @ !6 ] ] ], (!1 = 1) -> comp B (comp B (comp B (comp B (q @ !2) [ (!3 = 0) -> comp B (f (comp A (g (q @ !2)) [ (!4 = 1) -> t (g (q @ !2)) @ -!5 ])) [ (!4 = 0) -> s (q @ !2) @ !5, (!4 = 1) -> s (f (g (q @ !2))) @ !5 ] ]) [ (!2 = 0) -> comp B (comp B b0 [ (!3 = 0)(!4 = 1) -> comp B (f (comp A (g b0) [ (!5 = 1) -> t (g b0) @ -!6 ])) [ (!5 = 0) -> s b0 @ !6, (!5 = 1) -> s (f (g b0)) @ !6 ] ]) [ (!3 = 0) -> comp B (f (comp A (comp A (g (comp B b0 [ (!4 = 1) -> comp B (f (comp A (g b0) [ (!5 = 1) -> t (g b0) @ -!6 ])) [ (!5 = 0) -> s b0 @ !6, (!5 = 1) -> s (f (g b0)) @ !6 ] ])) [ (!4 = 1) -> t (g b0) @ (!5 /\ !6) ]) [ (!4 = 1) -> t (g b0) @ (!5 /\ -!6), (!5 = 1) -> t (comp A (g (comp B b0 [ (!4 = 1) -> comp B (f (comp A (g b0) [ (!5 = 1) -> t (g b0) @ -!6 ])) [ (!5 = 0) -> s b0 @ !6, (!5 = 1) -> s (f (g b0)) @ !6 ] ])) [ (!4 = 1) -> t (g b0) @ !5 ]) @ -!6 ])) [ (!4 = 1) -> s (f (g b0)) @ !6, (!5 = 0) -> s (comp B b0 [ (!4 = 1) -> comp B (f (comp A (g b0) [ (!5 = 1) -> t (g b0) @ -!6 ])) [ (!5 = 0) -> s b0 @ !6, (!5 = 1) -> s (f (g b0)) @ !6 ] ]) @ !6, (!5 = 1) -> s (f (comp A (g (comp B b0 [ (!4 = 1) -> comp B (f (comp A (g b0) [ (!5 = 1) -> t (g b0) @ -!6 ])) [ (!5 = 0) -> s b0 @ !6, (!5 = 1) -> s (f (g b0)) @ !6 ] ])) [ (!4 = 1) -> t (g b0) @ !5 ])) @ !6 ] ], (!2 = 1) -> comp B (comp B b1 [ (!3 = 0)(!4 = 1) -> comp B (f (comp A (g b1) [ (!5 = 1) -> t (g b1) @ -!6 ])) [ (!5 = 0) -> s b1 @ !6, (!5 = 1) -> s (f (g b1)) @ !6 ] ]) [ (!3 = 0) -> comp B (f (comp A (comp A (g (comp B b1 [ (!4 = 1) -> comp B (f (comp A (g b1) [ (!5 = 1) -> t (g b1) @ -!6 ])) [ (!5 = 0) -> s b1 @ !6, (!5 = 1) -> s (f (g b1)) @ !6 ] ])) [ (!4 = 1) -> t (g b1) @ (!5 /\ !6) ]) [ (!4 = 1) -> t (g b1) @ (!5 /\ -!6), (!5 = 1) -> t (comp A (g (comp B b1 [ (!4 = 1) -> comp B (f (comp A (g b1) [ (!5 = 1) -> t (g b1) @ -!6 ])) [ (!5 = 0) -> s b1 @ !6, (!5 = 1) -> s (f (g b1)) @ !6 ] ])) [ (!4 = 1) -> t (g b1) @ !5 ]) @ -!6 ])) [ (!4 = 1) -> s (f (g b1)) @ !6, (!5 = 0) -> s (comp B b1 [ (!4 = 1) -> comp B (f (comp A (g b1) [ (!5 = 1) -> t (g b1) @ -!6 ])) [ (!5 = 0) -> s b1 @ !6, (!5 = 1) -> s (f (g b1)) @ !6 ] ]) @ !6, (!5 = 1) -> s (f (comp A (g (comp B b1 [ (!4 = 1) -> comp B (f (comp A (g b1) [ (!5 = 1) -> t (g b1) @ -!6 ])) [ (!5 = 0) -> s b1 @ !6, (!5 = 1) -> s (f (g b1)) @ !6 ] ])) [ (!4 = 1) -> t (g b1) @ !5 ])) @ !6 ] ] ]) [ (!3 = 0) -> comp B (f (g (q @ !2))) [ (!2 = 0) -> f (comp A (g (comp B b0 [ (!5 = 1) -> comp B (f (comp A (g b0) [ (!6 = 1) -> t (g b0) @ -!7 ])) [ (!6 = 0) -> s b0 @ !7, (!6 = 1) -> s (f (g b0)) @ !7 ] ])) [ (!5 = 1) -> t (g b0) @ !6 ]), (!2 = 1) -> f (comp A (g (comp B b1 [ (!5 = 1) -> comp B (f (comp A (g b1) [ (!6 = 1) -> t (g b1) @ -!7 ])) [ (!6 = 0) -> s b1 @ !7, (!6 = 1) -> s (f (g b1)) @ !7 ] ])) [ (!5 = 1) -> t (g b1) @ !6 ]), (!4 = 0) -> comp B (f (g (q @ !2))) [ (!2 = 0) -> f (comp A (g (comp B b0 [ (!5 = 1)(!6 = 1) -> comp B (f (comp A (g b0) [ (!7 = 1) -> t (g b0) @ -!8 ])) [ (!7 = 0) -> s b0 @ !8, (!7 = 1) -> s (f (g b0)) @ !8 ] ])) [ (!5 = 1)(!6 = 1) -> t (g b0) @ !7 ]), (!2 = 1) -> f (comp A (g (comp B b1 [ (!5 = 1)(!6 = 1) -> comp B (f (comp A (g b1) [ (!7 = 1) -> t (g b1) @ -!8 ])) [ (!7 = 0) -> s b1 @ !8, (!7 = 1) -> s (f (g b1)) @ !8 ] ])) [ (!5 = 1)(!6 = 1) -> t (g b1) @ !7 ]) ], (!4 = 1) -> f (comp A (g (q @ !2)) [ (!2 = 0) -> comp A (g (comp B b0 [ (!5 = 1)(!6 = 1) -> comp B (f (comp A (g b0) [ (!7 = 1) -> t (g b0) @ -!8 ])) [ (!7 = 0) -> s b0 @ !8, (!7 = 1) -> s (f (g b0)) @ !8 ] ])) [ (!5 = 1)(!6 = 1) -> t (g b0) @ !7 ], (!2 = 1) -> comp A (g (comp B b1 [ (!5 = 1)(!6 = 1) -> comp B (f (comp A (g b1) [ (!7 = 1) -> t (g b1) @ -!8 ])) [ (!7 = 0) -> s b1 @ !8, (!7 = 1) -> s (f (g b1)) @ !8 ] ])) [ (!5 = 1)(!6 = 1) -> t (g b1) @ !7 ] ]) ] ]) [ (!2 = 0) -> comp B b0 [ (!3 = 0)(!4 = 0) -> comp B (f (comp A (g b0) [ (!5 = 1) -> t (g b0) @ -!6 ])) [ (!5 = 0) -> s b0 @ !6, (!5 = 1) -> s (f (g b0)) @ !6 ] ], (!2 = 1) -> comp B b1 [ (!3 = 0)(!4 = 0) -> comp B (f (comp A (g b1) [ (!5 = 1) -> t (g b1) @ -!6 ])) [ (!5 = 0) -> s b1 @ !6, (!5 = 1) -> s (f (g b1)) @ !6 ] ] ] ] -} diff --git a/examples/quotient.ctt b/examples/quotient.ctt index 3480eac..7669e2d 100644 --- a/examples/quotient.ctt +++ b/examples/quotient.ctt @@ -10,7 +10,7 @@ data Quot (A : U) (R : A -> A -> U) = [ (i = 0) -> inj a, (i = 1) -> inj b ] quoteq' (A : U) (R : A -> A -> U) (a b : A) (r : R a b) - : Id (Quot A R) (inj a) (inj b) = quoteq {Quot A R} a b r @ i + : Path (Quot A R) (inj a) (inj b) = quoteq {Quot A R} a b r @ i -- Test to define circle as a quotient of unit @@ -25,16 +25,16 @@ f2 : S1 -> s1quot = split base -> inj tt loop @ i -> quoteq{s1quot} tt tt tt @ i -rem3 : (a : Unit) -> Id s1quot (inj tt) (inj a) = split +rem3 : (a : Unit) -> Path s1quot (inj tt) (inj a) = split tt -> inj tt -test : Id U s1quot S1 = - isoId s1quot S1 f1 f2 rem1 rem2 +test : Path U s1quot S1 = + isoPath s1quot S1 f1 f2 rem1 rem2 where - rem1 : (y : S1) -> Id S1 (f1 (f2 y)) y = split + rem1 : (y : S1) -> Path S1 (f1 (f2 y)) y = split base -> base loop @ i -> loop1 @ i - rem2 : (x : s1quot) -> Id s1quot (f2 (f1 x)) x = split + rem2 : (x : s1quot) -> Path s1quot (f2 (f1 x)) x = split inj a -> rem3 a quoteq a b r @ i -> ? @@ -43,7 +43,7 @@ test : Id U s1quot S1 = data setquot (A : U) (R : A -> A -> U) = quot (a : A) | identification (a b : A) (r : R a b) [ (i = 0) -> quot a, (i = 1) -> quot b ] - | setTruncation (a b : setquot A R) (p q : Id (setquot A R) a b) + | setTruncation (a b : setquot A R) (p q : Path (setquot A R) a b) [ (i = 0) -> p @ j , (i = 1) -> q @ j , (j = 0) -> a @@ -57,8 +57,8 @@ data setquot (A : U) (R : A -> A -> U) = -} identsetquot (A : U) (R : A -> A -> U) (a b : A) (r : R a b) - : Id (setquot A R) (quot a) (quot b) = identification {setquot A R} a b r @ i + : Path (setquot A R) (quot a) (quot b) = identification {setquot A R} a b r @ i setsetquot (A : U) (R : A -> A -> U) : set (setquot A R) = - \(a b : setquot A R) (p q : Id (setquot A R) a b) -> + \(a b : setquot A R) (p q : Path (setquot A R) a b) -> setTruncation {setquot A R} a b p q @ i @ j diff --git a/examples/retract.ctt b/examples/retract.ctt index 7bfdbe6..e11548f 100644 --- a/examples/retract.ctt +++ b/examples/retract.ctt @@ -2,33 +2,33 @@ module retract where import prelude -section (A B : U) (f : A -> B) (g : B -> A) :U = (b : B) -> Id B (f (g b)) b +section (A B : U) (f : A -> B) (g : B -> A) :U = (b : B) -> Path B (f (g b)) b retract (A B : U) (f : A -> B) (g : B -> A) : U = section B A g f lemRetract (A B : U) (f : A -> B) (g : B -> A) (rfg : retract A B f g) (x y:A) : - Id A (g (f x)) (g (f y)) -> Id A x y + Path A (g (f x)) (g (f y)) -> Path A x y = compUp A (g (f x)) x (g (f y)) y (rfg x) (rfg y) retractProp (A B : U) (f : A -> B) (g : B -> A) (rfg : retract A B f g) (pB :prop B) (x y:A) - : Id A x y = lemRetract A B f g rfg x y ( g (pB (f x) (f y) @ i)) + : Path A x y = lemRetract A B f g rfg x y ( g (pB (f x) (f y) @ i)) retractInv (A B : U) (f : A -> B) (g : B -> A) (rfg : retract A B f g) - (x y : A) (q: Id B (f x) (f y)) : Id A x y = + (x y : A) (q: Path B (f x) (f y)) : Path A x y = compUp A (g (f x)) x (g (f y)) y (rfg x) (rfg y) ( (g (q @ i))) --- lemRSquare (A B : U) (f : A -> B) (g : B -> A) (rfg: retract A B f g)(x y:A) (p : Id A x y) : +-- lemRSquare (A B : U) (f : A -> B) (g : B -> A) (rfg: retract A B f g)(x y:A) (p : Path A x y) : -- Square A (g (f x)) (g (f y)) ( g (f (p @ i))) x y -- (retractInv A B f g rfg x y ( f (p@ i))) (rfg x) (rfg y) = -- comp A (g (f (p @ j))) [(j=0) -> (rfg x) @ (i/\l), (j=1) -> (rfg y) @ (i/\l)] --- retractId (A B : U)(f : A -> B) (g : B -> A) (rfg : retract A B f g) (x y :A) (p:Id A x y) : --- Id (Id A x y) (retractInv A B f g rfg x y ( f (p@ i))) p = +-- retractPath (A B : U)(f : A -> B) (g : B -> A) (rfg : retract A B f g) (x y :A) (p:Path A x y) : +-- Path (Path A x y) (retractInv A B f g rfg x y ( f (p@ i))) p = -- comp A (g (f (p @ j))) [(j=0) -> rfg x,(j=1) -> rfg y, -- (i=0) -> (lemRSquare A B f g rfg x y p) @ j,(i=1) -> rfg (p @ j)] -- retractSet (A B : U) (f : A -> B) (g : B -> A) (rfg : retract A B f g) --- (sB : set B) (x y : A) : prop (Id A x y) = --- retractProp (Id A x y) (Id B (f x) (f y)) (mapOnPath A B f x y) --- (retractInv A B f g rfg x y) (retractId A B f g rfg x y) (sB (f x) (f y)) +-- (sB : set B) (x y : A) : prop (Path A x y) = +-- retractProp (Path A x y) (Path B (f x) (f y)) (mapOnPath A B f x y) +-- (retractInv A B f g rfg x y) (retractPath A B f g rfg x y) (sB (f x) (f y)) diff --git a/examples/setquot.ctt b/examples/setquot.ctt index e0fe8f2..24dbafd 100644 --- a/examples/setquot.ctt +++ b/examples/setquot.ctt @@ -7,10 +7,10 @@ import pi import univalence subtypeEquality (A : U) (B : A -> U) (pB : (x : A) -> prop (B x)) - (s t : (x : A) * B x) : Id A s.1 t.1 -> Id (Sigma A B) s t = - trans (Id A s.1 t.1) (Id (Sigma A B) s t) rem + (s t : (x : A) * B x) : Path A s.1 t.1 -> Path (Sigma A B) s t = + trans (Path A s.1 t.1) (Path (Sigma A B) s t) rem where - rem : Id U (Id A s.1 t.1) (Id (Sigma A B) s t) = + rem : Path U (Path A s.1 t.1) (Path (Sigma A B) s t) = lemSigProp A B pB s t @ -i -- (* Propositions *) @@ -44,43 +44,43 @@ isEquivprop (A B : U) (f : A -> B) (g : B -> A) (pA : prop A) (pB : prop B) : is rem (y : B) : isContr (fiber A B f y) = (s,t) where s : fiber A B f y = (g y,pB y (f (g y))) - t (w : fiber A B f y) : Id ((x : A) * Id B y (f x)) s w = - subtypeEquality A (\(x : A) -> Id B y (f x)) pb s w r1 + t (w : fiber A B f y) : Path ((x : A) * Path B y (f x)) s w = + subtypeEquality A (\(x : A) -> Path B y (f x)) pb s w r1 where - pb (x : A) : (a b : Id B y (f x)) -> Id (Id B y (f x)) a b = propSet B pB y (f x) - r1 : Id A s.1 w.1 = pA s.1 w.1 + pb (x : A) : (a b : Path B y (f x)) -> Path (Path B y (f x)) a b = propSet B pB y (f x) + r1 : Path A s.1 w.1 = pA s.1 w.1 equivhProp (P P' : hProp) (f : P.1 -> P'.1) (g : P'.1 -> P.1) : equiv P.1 P'.1 = (f,isEquivprop P.1 P'.1 f g P.2 P'.2) -- Proof of uahp using full univalence -uahp' (P P' : hProp) (f : P.1 -> P'.1) (g : P'.1 -> P.1) : Id hProp P P' = +uahp' (P P' : hProp) (f : P.1 -> P'.1) (g : P'.1 -> P.1) : Path hProp P P' = subtypeEquality U prop propIsProp P P' rem where - rem : Id U P.1 P'.1 = transport ( corrUniv P.1 P'.1 @ -i) (equivhProp P P' f g) + rem : Path U P.1 P'.1 = transport ( corrUniv P.1 P'.1 @ -i) (equivhProp P P' f g) -- Direct proof of uahp -uahp (P P' : hProp) (f : P.1 -> P'.1) (g : P'.1 -> P.1) : Id hProp P P' = +uahp (P P' : hProp) (f : P.1 -> P'.1) (g : P'.1 -> P.1) : Path hProp P P' = subtypeEquality U prop propIsProp P P' rem where - rem : Id U P.1 P'.1 = isoId P.1 P'.1 f g s t - where s (y : P'.1) : Id P'.1 (f (g y)) y = P'.2 (f (g y)) y - t (x : P.1) : Id P.1 (g (f x)) x = P.2 (g (f x)) x + rem : Path U P.1 P'.1 = isoPath P.1 P'.1 f g s t + where s (y : P'.1) : Path P'.1 (f (g y)) y = P'.2 (f (g y)) y + t (x : P.1) : Path P.1 (g (f x)) x = P.2 (g (f x)) x -- A short proof that hProp form a set using univalence: (this is not needed!) -propequiv (X Y : U) (H : prop Y) (f g : equiv X Y) : Id (equiv X Y) f g = +propequiv (X Y : U) (H : prop Y) (f g : equiv X Y) : Path (equiv X Y) f g = equivLemma X Y f g ( \(x : X) -> H (f.1 x) (g.1 x) @ i) -propidU (X Y : U) : Id U X Y -> prop Y -> prop X = substInv U prop X Y +propidU (X Y : U) : Path U X Y -> prop Y -> prop X = substInv U prop X Y -sethProp (P P' : hProp) : prop (Id hProp P P') = - propidU (Id hProp P P') (equiv P.1 P'.1) rem (propequiv P.1 P'.1 P'.2) +sethProp (P P' : hProp) : prop (Path hProp P P') = + propidU (Path hProp P P') (equiv P.1 P'.1) rem (propequiv P.1 P'.1 P'.2) where - rem1 : Id U (Id hProp P P') (Id U P.1 P'.1) = lemSigProp U prop propIsProp P P' - rem2 : Id U (Id U P.1 P'.1) (equiv P.1 P'.1) = corrUniv P.1 P'.1 - rem : Id U (Id hProp P P') (equiv P.1 P'.1) = - compId U (Id hProp P P') (Id U P.1 P'.1) (equiv P.1 P'.1) rem1 rem2 + rem1 : Path U (Path hProp P P') (Path U P.1 P'.1) = lemSigProp U prop propIsProp P P' + rem2 : Path U (Path U P.1 P'.1) (equiv P.1 P'.1) = corrUniv P.1 P'.1 + rem : Path U (Path hProp P P') (equiv P.1 P'.1) = + compPath U (Path hProp P P') (Path U P.1 P'.1) (equiv P.1 P'.1) rem1 rem2 -- (* Sets *) @@ -118,12 +118,12 @@ propiseqclass (X : U) (R : hrel X) (A : hsubtypes X) : prop (iseqclass X R A) = -- This proof is quite cool, but it looks ugly... p2 (f g : (x1 x2 : X) -> (R x1 x2).1 -> (A x1).1 -> (A x2).1) : - Id ((x1 x2 : X) -> (R x1 x2).1 -> (A x1).1 -> (A x2).1) f g = + Path ((x1 x2 : X) -> (R x1 x2).1 -> (A x1).1 -> (A x2).1) f g = \(x1 x2 : X) (h1 : (R x1 x2).1) (h2 : (A x1).1) -> (A x2).2 (f x1 x2 h1 h2) (g x1 x2 h1 h2) @ i p3 (f g : (x1 x2 : X) -> (A x1).1 -> (A x2).1 -> (R x1 x2).1) : - Id ((x1 x2 : X) -> (A x1).1 -> (A x2).1 -> (R x1 x2).1) f g = + Path ((x1 x2 : X) -> (A x1).1 -> (A x2).1 -> (R x1 x2).1) f g = \(x1 x2 : X) (h1 : (A x1).1) (h2 : (A x2).1) -> (R x1 x2).2 (f x1 x2 h1 h2) (g x1 x2 h1 h2) @ i @@ -160,11 +160,11 @@ setquotpr (X : U) (R : eqrel X) (X0 : X) : setquot X R.1 = (A,((p1,p2),p3)) p3 (x1 x2 : X) (X1 : (A x1).1) (X2 : (A x2).1) : (R.1 x1 x2).1 = tax x1 X0 x2 (sax X0 x1 X1) X2 setquotl0 (X : U) (R : eqrel X) (c : setquot X R.1) (x : carrier X c.1) : - Id (setquot X R.1) (setquotpr X R x.1) c = subtypeEquality (hsubtypes X) (iseqclass X R.1) p (setquotpr X R x.1) c rem + Path (setquot X R.1) (setquotpr X R x.1) c = subtypeEquality (hsubtypes X) (iseqclass X R.1) p (setquotpr X R x.1) c rem where p (A : hsubtypes X) : prop (iseqclass X R.1 A) = propiseqclass X R.1 A - rem : Id (hsubtypes X) (setquotpr X R x.1).1 c.1 = \(x : X) -> (rem' x) @ i -- inlined use of funext - where rem' (a : X) : Id hProp ((setquotpr X R x.1).1 a) (c.1 a) = + rem : Path (hsubtypes X) (setquotpr X R x.1).1 c.1 = \(x : X) -> (rem' x) @ i -- inlined use of funext + where rem' (a : X) : Path hProp ((setquotpr X R x.1).1 a) (c.1 a) = uahp' ((setquotpr X R x.1).1 a) (c.1 a) l2r r2l -- This is where uahp appears where l2r (r : ((setquotpr X R x.1).1 a).1) : (c.1 a).1 = eqax1 X R.1 c.1 c.2 x.1 a r x.2 @@ -174,7 +174,7 @@ setquotunivprop (X : U) (R : eqrel X) (P : setquot X R.1 -> hProp) (ps : (x : X) -> (P (setquotpr X R x)).1) (c : setquot X R.1) : (P c).1 = hinhuniv (carrier X c.1) (P c) f rem where f (x : carrier X c.1) : (P c).1 = - let e : Id (setquot X R.1) (setquotpr X R x.1) c = setquotl0 X R c x + let e : Path (setquot X R.1) (setquotpr X R x.1) c = setquotl0 X R c x in subst (setquot X R.1) (\(w : setquot X R.1) -> (P w).1) (setquotpr X R x.1) c e (ps x.1) rem : (ishinh (carrier X c.1)).1 = eqax0 X R.1 c.1 c.2 @@ -192,14 +192,14 @@ setsetquot (X : U) (R : hrel X) : set (setquot X R) = sB (x : hsubtypes X) : set (iseqclass X R x) = propSet (iseqclass X R x) (propiseqclass X R x) iscompsetquotpr (X : U) (R : eqrel X) (x x' : X) (a : (R.1 x x').1) : - Id (setquot X R.1) (setquotpr X R x) (setquotpr X R x') = + Path (setquot X R.1) (setquotpr X R x) (setquotpr X R x') = subtypeEquality (hsubtypes X) (iseqclass X R.1) rem1 (setquotpr X R x) (setquotpr X R x') rem2 where rem1 (x : hsubtypes X) : prop (iseqclass X R.1 x) = propiseqclass X R.1 x - rem2 : Id (hsubtypes X) (setquotpr X R x).1 (setquotpr X R x').1 = + rem2 : Path (hsubtypes X) (setquotpr X R x).1 (setquotpr X R x').1 = \(x0 : X) -> rem x0 @ i where - rem (x0 : X) : Id hProp (R.1 x x0) (R.1 x' x0) = uahp' (R.1 x x0) (R.1 x' x0) f g + rem (x0 : X) : Path hProp (R.1 x x0) (R.1 x' x0) = uahp' (R.1 x x0) (R.1 x' x0) f g where f (r0 : (R.1 x x0).1) : (R.1 x' x0).1 = eqreltrans X R x' x x0 (eqrelsymm X R x x' a) r0 @@ -207,24 +207,24 @@ iscompsetquotpr (X : U) (R : eqrel X) (x x' : X) (a : (R.1 x x').1) : eqreltrans X R x x' x0 a r0 weqpathsinsetquot (X : U) (R : eqrel X) (x x' : X) : - equiv (R.1 x x').1 (Id (setquot X R.1) (setquotpr X R x) (setquotpr X R x')) = + equiv (R.1 x x').1 (Path (setquot X R.1) (setquotpr X R x) (setquotpr X R x')) = (iscompsetquotpr X R x x',rem) where rem : isEquiv (R.1 x x').1 - (Id (setquot X R.1) (setquotpr X R x) (setquotpr X R x')) + (Path (setquot X R.1) (setquotpr X R x) (setquotpr X R x')) (iscompsetquotpr X R x x') = isEquivprop (R.1 x x').1 - (Id (setquot X R.1) (setquotpr X R x) (setquotpr X R x')) + (Path (setquot X R.1) (setquotpr X R x) (setquotpr X R x')) (iscompsetquotpr X R x x') g pA pB - where g (e : Id (setquot X R.1) (setquotpr X R x) (setquotpr X R x')) : + where g (e : Path (setquot X R.1) (setquotpr X R x) (setquotpr X R x')) : (R.1 x x').1 = transport ( (rem1 @ i).1) rem where rem : (R.1 x' x').1 = eqrelrefl X R x' - rem2 : Id hProp (R.1 x x') (R.1 x' x') = (e @ i).1 x' - rem1 : Id hProp (R.1 x' x') (R.1 x x') = rem2 @ -i + rem2 : Path hProp (R.1 x x') (R.1 x' x') = (e @ i).1 x' + rem1 : Path hProp (R.1 x' x') (R.1 x x') = rem2 @ -i pA : prop (R.1 x x').1 = (R.1 x x').2 - pB : prop (Id (setquot X R.1) (setquotpr X R x) (setquotpr X R x')) = + pB : prop (Path (setquot X R.1) (setquotpr X R x) (setquotpr X R x')) = setsetquot X R.1 (setquotpr X R x) (setquotpr X R x') isdecprop (X : U) : U = and (prop X) (dec X) @@ -235,12 +235,12 @@ propisdecprop (X : U): prop (isdecprop X) = rem1 : prop (prop X) = propIsProp X rem2 : prop X -> prop (dec X) = propDec X -isdeceqif (X : U) (h : (x x' : X) -> isdecprop (Id X x x')) : discrete X = +isdeceqif (X : U) (h : (x x' : X) -> isdecprop (Path X x x')) : discrete X = \(x x' : X) -> (h x x').2 propEquiv (X Y : U) (w : equiv X Y) : prop X -> prop Y = subst U prop X Y rem where - rem : Id U X Y = transport ( corrUniv X Y @ -i) w + rem : Path U X Y = transport ( corrUniv X Y @ -i) w isdecpropweqf (X Y : U) (w : equiv X Y) (hX : isdecprop X) : isdecprop Y = (rem1,rem2 hX.2) where @@ -252,18 +252,18 @@ isdecpropweqf (X Y : U) (w : equiv X Y) (hX : isdecprop X) : isdecprop Y = (rem1 isdiscretesetquot (X : U) (R : eqrel X) (is : (x x' : X) -> isdecprop (R.1 x x').1) : discrete (setquot X R.1) = isdeceqif (setquot X R.1) rem where - rem : (x x' : setquot X R.1) -> isdecprop (Id (setquot X R.1) x x') = + rem : (x x' : setquot X R.1) -> isdecprop (Path (setquot X R.1) x x') = setquotuniv2prop X R - (\(x0 x0' : setquot X R.1) -> (isdecprop (Id (setquot X R.1) x0 x0'), - propisdecprop (Id (setquot X R.1) x0 x0'))) rem' + (\(x0 x0' : setquot X R.1) -> (isdecprop (Path (setquot X R.1) x0 x0'), + propisdecprop (Path (setquot X R.1) x0 x0'))) rem' where - rem' (x0 x0' : X) : isdecprop (Id (setquot X R.1) (setquotpr X R x0) (setquotpr X R x0')) = - isdecpropweqf (R.1 x0 x0').1 (Id (setquot X R.1) (setquotpr X R x0) (setquotpr X R x0')) + rem' (x0 x0' : X) : isdecprop (Path (setquot X R.1) (setquotpr X R x0) (setquotpr X R x0')) = + isdecpropweqf (R.1 x0 x0').1 (Path (setquot X R.1) (setquotpr X R x0) (setquotpr X R x0')) (weqpathsinsetquot X R x0 x0') (is x0 x0') discretetobool (X : U) (h : discrete X) (x y : X) : bool = rem (h x y) where - rem : dec (Id X x y) -> bool = split + rem : dec (Path X x y) -> bool = split inl _ -> true inr _ -> false @@ -271,86 +271,86 @@ discretetobool (X : U) (h : discrete X) (x y : X) : bool = rem (h x y) R : eqrel bool = (r1,r2) where - r1 : hrel bool = \(x y : bool) -> (Id bool x y,setbool x y) - r2 : iseqrel bool r1 = ((compId bool,refl bool),inv bool) + r1 : hrel bool = \(x y : bool) -> (Path bool x y,setbool x y) + r2 : iseqrel bool r1 = ((compPath bool,refl bool),inv bool) bool' : U = setquot bool R.1 true' : bool' = setquotpr bool R true false' : bool' = setquotpr bool R false P' (t : bool') : hProp = - hdisj (Id bool' t true') (Id bool' t false') + hdisj (Path bool' t true') (Path bool' t false') K' (t : bool') : (P' t).1 = setquotunivprop bool R P' ps t where ps : (x : bool) -> (P' (setquotpr bool R x)).1 = split - false -> hdisj_in2 (Id bool' false' true') - (Id bool' false' false') (<_> false') - true -> hdisj_in1 (Id bool' true' true') - (Id bool' true' false') (<_> true') + false -> hdisj_in2 (Path bool' false' true') + (Path bool' false' false') (<_> false') + true -> hdisj_in1 (Path bool' true' true') + (Path bool' true' false') (<_> true') -test : (P' true').1 = hdisj_in1 (Id bool' true' true') - (Id bool' true' false') (<_> true') +test : (P' true').1 = hdisj_in1 (Path bool' true' true') + (Path bool' true' false') (<_> true') test' : (P' true').1 = K' true' --- test'' : Id (P' true').1 test test' = (P' true').2 test test' +-- test'' : Path (P' true').1 test test' = (P' true').2 test test' -- These two terms are not convertible: --- test' : Id (P' true').1 (K' true') --- (hdisj_in1 (Id (setquot bool R.1) true' true') (Id (setquot bool R.1) true' false') (<_> true')) = +-- test' : Path (P' true').1 (K' true') +-- (hdisj_in1 (Path (setquot bool R.1) true' true') (Path (setquot bool R.1) true' false') (<_> true')) = -- <_> K' true' -- Another test: -true'neqfalse' (H : Id bool' true' false') : N0 = falseNeqTrue rem1 +true'neqfalse' (H : Path bool' true' false') : N0 = falseNeqTrue rem1 where - rem : Id U (Id bool true true) (Id bool false true) = ((H @ i).1 true).1 - rem1 : Id bool false true = comp rem (<_> true) [] + rem : Path U (Path bool true true) (Path bool false true) = ((H @ i).1 true).1 + rem1 : Path bool false true = comp rem (<_> true) [] -test1 (x : bool') (H1 : Id bool' x true') (H2 : Id bool' x false') : N0 = true'neqfalse' rem +test1 (x : bool') (H1 : Path bool' x true') (H2 : Path bool' x false') : N0 = true'neqfalse' rem where - rem : Id bool' true' false' = comp (<_> bool') x [(i = 0) -> H1, (i = 1) -> H2] + rem : Path bool' true' false' = comp (<_> bool') x [(i = 0) -> H1, (i = 1) -> H2] -test2 (x : bool') (p1 : (ishinh (Id bool' x true')).1) - (p2 : (ishinh (Id bool' x false')).1) : N0 = - hinhuniv (Id bool' x true') (N0,propN0) rem p1 +test2 (x : bool') (p1 : (ishinh (Path bool' x true')).1) + (p2 : (ishinh (Path bool' x false')).1) : N0 = + hinhuniv (Path bool' x true') (N0,propN0) rem p1 where - rem (H1 : Id bool' x true') : N0 = - hinhuniv (Id bool' x false') (N0,propN0) - (\(H2 : Id bool' x false') -> test1 x H1 H2) p2 + rem (H1 : Path bool' x true') : N0 = + hinhuniv (Path bool' x false') (N0,propN0) + (\(H2 : Path bool' x false') -> test1 x H1 H2) p2 -- shorthand for this big type -T (x : bool') : U = or (ishinh (Id bool' x true')).1 (ishinh (Id bool' x false')).1 +T (x : bool') : U = or (ishinh (Path bool' x true')).1 (ishinh (Path bool' x false')).1 -- test3 (x : bool') : prop (T x) -test3 (x : bool') : (a b : T x) -> Id (T x) a b = split +test3 (x : bool') : (a b : T x) -> Path (T x) a b = split inl a' -> rem where - rem : (b : T x) -> Id (T x) (inl a') b = split - inl b' -> inl (propishinh (Id bool' x true') a' b' @ i) - inr b' -> efq (Id (T x) (inl a') (inr b')) (test2 x a' b') + rem : (b : T x) -> Path (T x) (inl a') b = split + inl b' -> inl (propishinh (Path bool' x true') a' b' @ i) + inr b' -> efq (Path (T x) (inl a') (inr b')) (test2 x a' b') inr a' -> rem where - rem : (b : T x) -> Id (T x) (inr a') b = split - inl b' -> efq (Id (T x) (inr a') (inl b')) (test2 x b' a') - inr b' -> inr (propishinh (Id bool' x false') a' b' @ i) + rem : (b : T x) -> Path (T x) (inr a') b = split + inl b' -> efq (Path (T x) (inr a') (inl b')) (test2 x b' a') + inr b' -> inr (propishinh (Path bool' x false') a' b' @ i) -f (x : bool') : or (ishinh (Id bool' x true')).1 (ishinh (Id bool' x false')).1 -> bool = split +f (x : bool') : or (ishinh (Path bool' x true')).1 (ishinh (Path bool' x false')).1 -> bool = split inl _ -> true inr _ -> false -bar (x : bool') : or (Id bool' x true') (Id bool' x false') -> - or (ishinh (Id bool' x true')).1 (ishinh (Id bool' x false')).1 = split - inl p -> inl (hinhpr (Id bool' x true') p) - inr p -> inr (hinhpr (Id bool' x false') p) +bar (x : bool') : or (Path bool' x true') (Path bool' x false') -> + or (ishinh (Path bool' x true')).1 (ishinh (Path bool' x false')).1 = split + inl p -> inl (hinhpr (Path bool' x true') p) + inr p -> inr (hinhpr (Path bool' x false') p) -- finally the map: foo (x : bool') (x' : (P' x).1) : bool = f x rem where - rem : or (ishinh (Id bool' x true')).1 (ishinh (Id bool' x false')).1 = - hinhuniv (or (Id bool' x true') (Id bool' x false')) - (or (ishinh (Id bool' x true')).1 (ishinh (Id bool' x false')).1,test3 x) + rem : or (ishinh (Path bool' x true')).1 (ishinh (Path bool' x false')).1 = + hinhuniv (or (Path bool' x true') (Path bool' x false')) + (or (ishinh (Path bool' x true')).1 (ishinh (Path bool' x false')).1,test3 x) (bar x) x' -- > :n testfoo @@ -358,15 +358,15 @@ foo (x : bool') (x' : (P' x).1) : bool = f x rem -- Time: 0m0.490s testfoo : bool = foo true' (K' true') -testfoo' : Id bool (foo true' (K' true')) true = foo true' (K' true') +testfoo' : Path bool (foo true' (K' true')) true = foo true' (K' true') -- Tests of checking normal forms: -ntrue' : bool' = (\(x : bool) -> (IdP ( bool) true x,lem8 x),((\(P : Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b)) -> \(f : (Sigma bool (\(x : bool) -> IdP ( bool) true x)) -> P.1) -> f ((true, true)),\(x1 x2 : bool) -> \(X1 : IdP ( bool) x1 x2) -> \(X2 : IdP ( bool) true x1) -> comp ( bool) (X2 @ i0) [ (i0 = 0) -> true, (i0 = 1) -> X1 @ i1 ]),\(x1 x2 : bool) -> \(X1 : IdP ( bool) true x1) -> \(X2 : IdP ( bool) true x2) -> comp ( bool) (X1 @ -i0) [ (i0 = 0) -> x1, (i0 = 1) -> X2 @ i1 ])) +ntrue' : bool' = (\(x : bool) -> (PathP ( bool) true x,lem8 x),((\(P : Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b)) -> \(f : (Sigma bool (\(x : bool) -> PathP ( bool) true x)) -> P.1) -> f ((true, true)),\(x1 x2 : bool) -> \(X1 : PathP ( bool) x1 x2) -> \(X2 : PathP ( bool) true x1) -> comp ( bool) (X2 @ i0) [ (i0 = 0) -> true, (i0 = 1) -> X1 @ i1 ]),\(x1 x2 : bool) -> \(X1 : PathP ( bool) true x1) -> \(X2 : PathP ( bool) true x2) -> comp ( bool) (X1 @ -i0) [ (i0 = 0) -> x1, (i0 = 1) -> X2 @ i1 ])) nhdisj_in1 : (P Q : U) (X : P) -> (hdisj P Q).1 = - \(P Q : U) -> \(X : P) -> \(P0 : Sigma U (\(X0 : U) -> (a b : X0) -> IdP ( X0) a b)) -> \(f : or P Q -> P0.1) -> f (inl X) + \(P Q : U) -> \(X : P) -> \(P0 : Sigma U (\(X0 : U) -> (a b : X0) -> PathP ( X0) a b)) -> \(f : or P Q -> P0.1) -> f (inl X) -ntest : (P' true').1 = \(P : Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b)) -> \(f : or (IdP ( Sigma (bool -> (Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b))) (\(A : bool -> (Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b))) -> Sigma (Sigma ((P0 : Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b)) -> ((Sigma bool (\(x : bool) -> (A x).1)) -> P0.1) -> P0.1) (\(_ : (P0 : Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b)) -> ((Sigma bool (\(x : bool) -> (A x).1)) -> P0.1) -> P0.1) -> (x1 x2 : bool) -> (IdP ( bool) x1 x2) -> ((A x1).1 -> (A x2).1))) (\(_ : Sigma ((P0 : Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b)) -> ((Sigma bool (\(x : bool) -> (A x).1)) -> P0.1) -> P0.1) (\(_ : (P0 : Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b)) -> ((Sigma bool (\(x : bool) -> (A x).1)) -> P0.1) -> P0.1) -> (x1 x2 : bool) -> (IdP ( bool) x1 x2) -> ((A x1).1 -> (A x2).1))) -> (x1 x2 : bool) -> (A x1).1 -> ((A x2).1 -> (IdP ( bool) x1 x2))))) ((\(x : bool) -> (IdP ( bool) true x,lem8 x),((\(P0 : Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b)) -> \(f : (Sigma bool (\(x : bool) -> IdP ( bool) true x)) -> P0.1) -> f ((true, true)),\(x1 x2 : bool) -> \(X1 : IdP ( bool) x1 x2) -> \(X2 : IdP ( bool) true x1) -> comp ( bool) (X2 @ !0) [ (!0 = 0) -> true, (!0 = 1) -> X1 @ !1 ]),\(x1 x2 : bool) -> \(X1 : IdP ( bool) true x1) -> \(X2 : IdP ( bool) true x2) -> comp ( bool) (X1 @ -!0) [ (!0 = 0) -> x1, (!0 = 1) -> X2 @ !1 ]))) ((\(x : bool) -> (IdP ( bool) true x,lem8 x),((\(P0 : Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b)) -> \(f : (Sigma bool (\(x : bool) -> IdP ( bool) true x)) -> P0.1) -> f ((true, true)),\(x1 x2 : bool) -> \(X1 : IdP ( bool) x1 x2) -> \(X2 : IdP ( bool) true x1) -> comp ( bool) (X2 @ !0) [ (!0 = 0) -> true, (!0 = 1) -> X1 @ !1 ]),\(x1 x2 : bool) -> \(X1 : IdP ( bool) true x1) -> \(X2 : IdP ( bool) true x2) -> comp ( bool) (X1 @ -!0) [ (!0 = 0) -> x1, (!0 = 1) -> X2 @ !1 ])))) IdP ( Sigma (bool -> (Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b))) (\(A : bool -> (Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b))) -> Sigma (Sigma ((P0 : Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b)) -> ((Sigma bool (\(x : bool) -> (A x).1)) -> P0.1) -> P0.1) (\(_ : (P0 : Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b)) -> ((Sigma bool (\(x : bool) -> (A x).1)) -> P0.1) -> P0.1) -> (x1 x2 : bool) -> (IdP ( bool) x1 x2) -> ((A x1).1 -> (A x2).1))) (\(_ : Sigma ((P0 : Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b)) -> ((Sigma bool (\(x : bool) -> (A x).1)) -> P0.1) -> P0.1) (\(_ : (P0 : Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b)) -> ((Sigma bool (\(x : bool) -> (A x).1)) -> P0.1) -> P0.1) -> (x1 x2 : bool) -> (IdP ( bool) x1 x2) -> ((A x1).1 -> (A x2).1))) -> (x1 x2 : bool) -> (A x1).1 -> ((A x2).1 -> (IdP ( bool) x1 x2))))) ((\(x : bool) -> (IdP ( bool) true x,lem8 x),((\(P0 : Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b)) -> \(f : (Sigma bool (\(x : bool) -> IdP ( bool) true x)) -> P0.1) -> f ((true, true)),\(x1 x2 : bool) -> \(X1 : IdP ( bool) x1 x2) -> \(X2 : IdP ( bool) true x1) -> comp ( bool) (X2 @ !0) [ (!0 = 0) -> true, (!0 = 1) -> X1 @ !1 ]),\(x1 x2 : bool) -> \(X1 : IdP ( bool) true x1) -> \(X2 : IdP ( bool) true x2) -> comp ( bool) (X1 @ -!0) [ (!0 = 0) -> x1, (!0 = 1) -> X2 @ !1 ]))) ((\(x : bool) -> (IdP ( bool) false x,lem7 x),((\(P0 : Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b)) -> \(f : (Sigma bool (\(x : bool) -> IdP ( bool) false x)) -> P0.1) -> f ((false, false)),\(x1 x2 : bool) -> \(X1 : IdP ( bool) x1 x2) -> \(X2 : IdP ( bool) false x1) -> comp ( bool) (X2 @ !0) [ (!0 = 0) -> false, (!0 = 1) -> X1 @ !1 ]),\(x1 x2 : bool) -> \(X1 : IdP ( bool) false x1) -> \(X2 : IdP ( bool) false x2) -> comp ( bool) (X1 @ -!0) [ (!0 = 0) -> x1, (!0 = 1) -> X2 @ !1 ]))) -> P.1) -> f (inl ( (\(x : bool) -> (IdP ( bool) true x,lem8 x),((\(P0 : Sigma U (\(X : U) -> (a b : X) -> IdP ( X) a b)) -> \(f0 : (Sigma bool (\(x : bool) -> IdP ( bool) true x)) -> P0.1) -> f0 ((true, true)),\(x1 x2 : bool) -> \(X1 : IdP ( bool) x1 x2) -> \(X2 : IdP ( bool) true x1) -> comp ( bool) (X2 @ !0) [ (!0 = 0) -> true, (!0 = 1) -> X1 @ !1 ]),\(x1 x2 : bool) -> \(X1 : IdP ( bool) true x1) -> \(X2 : IdP ( bool) true x2) -> comp ( bool) (X1 @ -!0) [ (!0 = 0) -> x1, (!0 = 1) -> X2 @ !1 ])))) +ntest : (P' true').1 = \(P : Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b)) -> \(f : or (PathP ( Sigma (bool -> (Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b))) (\(A : bool -> (Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b))) -> Sigma (Sigma ((P0 : Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b)) -> ((Sigma bool (\(x : bool) -> (A x).1)) -> P0.1) -> P0.1) (\(_ : (P0 : Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b)) -> ((Sigma bool (\(x : bool) -> (A x).1)) -> P0.1) -> P0.1) -> (x1 x2 : bool) -> (PathP ( bool) x1 x2) -> ((A x1).1 -> (A x2).1))) (\(_ : Sigma ((P0 : Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b)) -> ((Sigma bool (\(x : bool) -> (A x).1)) -> P0.1) -> P0.1) (\(_ : (P0 : Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b)) -> ((Sigma bool (\(x : bool) -> (A x).1)) -> P0.1) -> P0.1) -> (x1 x2 : bool) -> (PathP ( bool) x1 x2) -> ((A x1).1 -> (A x2).1))) -> (x1 x2 : bool) -> (A x1).1 -> ((A x2).1 -> (PathP ( bool) x1 x2))))) ((\(x : bool) -> (PathP ( bool) true x,lem8 x),((\(P0 : Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b)) -> \(f : (Sigma bool (\(x : bool) -> PathP ( bool) true x)) -> P0.1) -> f ((true, true)),\(x1 x2 : bool) -> \(X1 : PathP ( bool) x1 x2) -> \(X2 : PathP ( bool) true x1) -> comp ( bool) (X2 @ !0) [ (!0 = 0) -> true, (!0 = 1) -> X1 @ !1 ]),\(x1 x2 : bool) -> \(X1 : PathP ( bool) true x1) -> \(X2 : PathP ( bool) true x2) -> comp ( bool) (X1 @ -!0) [ (!0 = 0) -> x1, (!0 = 1) -> X2 @ !1 ]))) ((\(x : bool) -> (PathP ( bool) true x,lem8 x),((\(P0 : Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b)) -> \(f : (Sigma bool (\(x : bool) -> PathP ( bool) true x)) -> P0.1) -> f ((true, true)),\(x1 x2 : bool) -> \(X1 : PathP ( bool) x1 x2) -> \(X2 : PathP ( bool) true x1) -> comp ( bool) (X2 @ !0) [ (!0 = 0) -> true, (!0 = 1) -> X1 @ !1 ]),\(x1 x2 : bool) -> \(X1 : PathP ( bool) true x1) -> \(X2 : PathP ( bool) true x2) -> comp ( bool) (X1 @ -!0) [ (!0 = 0) -> x1, (!0 = 1) -> X2 @ !1 ])))) PathP ( Sigma (bool -> (Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b))) (\(A : bool -> (Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b))) -> Sigma (Sigma ((P0 : Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b)) -> ((Sigma bool (\(x : bool) -> (A x).1)) -> P0.1) -> P0.1) (\(_ : (P0 : Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b)) -> ((Sigma bool (\(x : bool) -> (A x).1)) -> P0.1) -> P0.1) -> (x1 x2 : bool) -> (PathP ( bool) x1 x2) -> ((A x1).1 -> (A x2).1))) (\(_ : Sigma ((P0 : Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b)) -> ((Sigma bool (\(x : bool) -> (A x).1)) -> P0.1) -> P0.1) (\(_ : (P0 : Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b)) -> ((Sigma bool (\(x : bool) -> (A x).1)) -> P0.1) -> P0.1) -> (x1 x2 : bool) -> (PathP ( bool) x1 x2) -> ((A x1).1 -> (A x2).1))) -> (x1 x2 : bool) -> (A x1).1 -> ((A x2).1 -> (PathP ( bool) x1 x2))))) ((\(x : bool) -> (PathP ( bool) true x,lem8 x),((\(P0 : Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b)) -> \(f : (Sigma bool (\(x : bool) -> PathP ( bool) true x)) -> P0.1) -> f ((true, true)),\(x1 x2 : bool) -> \(X1 : PathP ( bool) x1 x2) -> \(X2 : PathP ( bool) true x1) -> comp ( bool) (X2 @ !0) [ (!0 = 0) -> true, (!0 = 1) -> X1 @ !1 ]),\(x1 x2 : bool) -> \(X1 : PathP ( bool) true x1) -> \(X2 : PathP ( bool) true x2) -> comp ( bool) (X1 @ -!0) [ (!0 = 0) -> x1, (!0 = 1) -> X2 @ !1 ]))) ((\(x : bool) -> (PathP ( bool) false x,lem7 x),((\(P0 : Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b)) -> \(f : (Sigma bool (\(x : bool) -> PathP ( bool) false x)) -> P0.1) -> f ((false, false)),\(x1 x2 : bool) -> \(X1 : PathP ( bool) x1 x2) -> \(X2 : PathP ( bool) false x1) -> comp ( bool) (X2 @ !0) [ (!0 = 0) -> false, (!0 = 1) -> X1 @ !1 ]),\(x1 x2 : bool) -> \(X1 : PathP ( bool) false x1) -> \(X2 : PathP ( bool) false x2) -> comp ( bool) (X1 @ -!0) [ (!0 = 0) -> x1, (!0 = 1) -> X2 @ !1 ]))) -> P.1) -> f (inl ( (\(x : bool) -> (PathP ( bool) true x,lem8 x),((\(P0 : Sigma U (\(X : U) -> (a b : X) -> PathP ( X) a b)) -> \(f0 : (Sigma bool (\(x : bool) -> PathP ( bool) true x)) -> P0.1) -> f0 ((true, true)),\(x1 x2 : bool) -> \(X1 : PathP ( bool) x1 x2) -> \(X2 : PathP ( bool) true x1) -> comp ( bool) (X2 @ !0) [ (!0 = 0) -> true, (!0 = 1) -> X1 @ !1 ]),\(x1 x2 : bool) -> \(X1 : PathP ( bool) true x1) -> \(X2 : PathP ( bool) true x2) -> comp ( bool) (X1 @ -!0) [ (!0 = 0) -> x1, (!0 = 1) -> X2 @ !1 ])))) diff --git a/examples/sigma.ctt b/examples/sigma.ctt index 1f101b6..9494866 100644 --- a/examples/sigma.ctt +++ b/examples/sigma.ctt @@ -2,106 +2,106 @@ module sigma where import equiv -lemIdSig (A:U) (B : A -> U) (t u : Sigma A B) : - Id U (Id (Sigma A B) t u) ((p : Id A t.1 u.1) * IdP ( B (p @ i)) t.2 u.2) = - isoId T0 T1 f g s t where - T0 : U = Id (Sigma A B) t u - T1 : U = (p:Id A t.1 u.1) * IdP ( B (p@i)) t.2 u.2 +lemPathSig (A:U) (B : A -> U) (t u : Sigma A B) : + Path U (Path (Sigma A B) t u) ((p : Path A t.1 u.1) * PathP ( B (p @ i)) t.2 u.2) = + isoPath T0 T1 f g s t where + T0 : U = Path (Sigma A B) t u + T1 : U = (p:Path A t.1 u.1) * PathP ( B (p@i)) t.2 u.2 f (q:T0) : T1 = ( (q@i).1, (q@i).2) g (z:T1) : T0 = (z.1 @i,z.2 @i) - s (z:T1) : Id T1 (f (g z)) z = refl T1 z - t (q:T0) : Id T0 (g (f q)) q = refl T0 q + s (z:T1) : Path T1 (f (g z)) z = refl T1 z + t (q:T0) : Path T0 (g (f q)) q = refl T0 q -lemIdAnd (A B : U) (t u : and A B) : - Id U (Id (and A B) t u) (and (Id A t.1 u.1) (Id B t.2 u.2)) = lemIdSig A (\(_ : A) -> B) t u +lemPathAnd (A B : U) (t u : and A B) : + Path U (Path (and A B) t u) (and (Path A t.1 u.1) (Path B t.2 u.2)) = lemPathSig A (\(_ : A) -> B) t u -lemTransp (A:U) (a:A) : Id A a (transport (<_>A) a) = fill (<_>A) a [] +lemTransp (A:U) (a:A) : Path A a (transport (<_>A) a) = fill (<_>A) a [] -funDepTr (A:U) (P:A->U) (a0 a1 :A) (p:Id A a0 a1) (u0:P a0) (u1:P a1) : - Id U (IdP ( P (p@i)) u0 u1) (Id (P a1) (transport ( P (p@i)) u0) u1) = - IdP (P (p@j\/i)) (comp (P (p@j/\i)) u0 [(j=0)-><_>u0]) u1 +funDepTr (A:U) (P:A->U) (a0 a1 :A) (p:Path A a0 a1) (u0:P a0) (u1:P a1) : + Path U (PathP ( P (p@i)) u0 u1) (Path (P a1) (transport ( P (p@i)) u0) u1) = + PathP (P (p@j\/i)) (comp (P (p@j/\i)) u0 [(j=0)-><_>u0]) u1 -lem2 (A:U) (B:A-> U) (t u : Sigma A B) (p:Id A t.1 u.1) : - Id U (IdP (B (p@i)) t.2 u.2) (Id (B u.1) (transport (B (p@i)) t.2) u.2) = +lem2 (A:U) (B:A-> U) (t u : Sigma A B) (p:Path A t.1 u.1) : + Path U (PathP (B (p@i)) t.2 u.2) (Path (B u.1) (transport (B (p@i)) t.2) u.2) = funDepTr A B t.1 u.1 p t.2 u.2 -corSigProp (A:U) (B:A-> U) (pB : (x:A) -> prop (B x)) (t u : Sigma A B) (p:Id A t.1 u.1) : - prop (IdP (B (p@i)) t.2 u.2) = substInv U prop T0 T1 rem rem1 - where P : Id U (B t.1) (B u.1) = B (p@i) - T0 : U = IdP P t.2 u.2 - T1 : U = Id (B u.1) (transport P t.2) u.2 - rem : Id U T0 T1 = lem2 A B t u p -- funDepTr (B t.1) (B u.1) P t.2 u.2 +corSigProp (A:U) (B:A-> U) (pB : (x:A) -> prop (B x)) (t u : Sigma A B) (p:Path A t.1 u.1) : + prop (PathP (B (p@i)) t.2 u.2) = substInv U prop T0 T1 rem rem1 + where P : Path U (B t.1) (B u.1) = B (p@i) + T0 : U = PathP P t.2 u.2 + T1 : U = Path (B u.1) (transport P t.2) u.2 + rem : Path U T0 T1 = lem2 A B t u p -- funDepTr (B t.1) (B u.1) P t.2 u.2 v2 : B u.1 = transport P t.2 rem1 : prop T1 = propSet (B u.1) (pB u.1) v2 u.2 -corSigSet (A:U) (B:A-> U) (sB : (x:A) -> set (B x)) (t u : Sigma A B) (p:Id A t.1 u.1) : - prop (IdP (B (p@i)) t.2 u.2) = substInv U prop T0 T1 rem rem1 - where P : Id U (B t.1) (B u.1) = B (p@i) - T0 : U = IdP P t.2 u.2 - T1 : U = Id (B u.1) (transport P t.2) u.2 - rem : Id U T0 T1 = lem2 A B t u p -- funDepTr (B t.1) (B u.1) P t.2 u.2 +corSigSet (A:U) (B:A-> U) (sB : (x:A) -> set (B x)) (t u : Sigma A B) (p:Path A t.1 u.1) : + prop (PathP (B (p@i)) t.2 u.2) = substInv U prop T0 T1 rem rem1 + where P : Path U (B t.1) (B u.1) = B (p@i) + T0 : U = PathP P t.2 u.2 + T1 : U = Path (B u.1) (transport P t.2) u.2 + rem : Path U T0 T1 = lem2 A B t u p -- funDepTr (B t.1) (B u.1) P t.2 u.2 v2 : B u.1 = transport P t.2 rem1 : prop T1 = sB u.1 v2 u.2 -setSig (A:U) (B:A-> U) (sA: set A) (sB : (x:A) -> set (B x)) (t u : Sigma A B) : prop (Id (Sigma A B) t u) = - substInv U prop (Id (Sigma A B) t u) ((p:T) * C p) rem3 rem2 +setSig (A:U) (B:A-> U) (sA: set A) (sB : (x:A) -> set (B x)) (t u : Sigma A B) : prop (Path (Sigma A B) t u) = + substInv U prop (Path (Sigma A B) t u) ((p:T) * C p) rem3 rem2 where - T : U = Id A t.1 u.1 - C (p:T) : U = IdP ( B (p@i)) t.2 u.2 + T : U = Path A t.1 u.1 + C (p:T) : U = PathP ( B (p@i)) t.2 u.2 rem (p : T) : prop (C p) = corSigSet A B sB t u p rem1 : prop T = sA t.1 u.1 rem2 : prop ((p:T) * C p) = propSig T C rem1 rem - rem3 : Id U (Id (Sigma A B) t u) ((p:T) * C p) = lemIdSig A B t u - -corSigGroupoid (A:U) (B:A-> U) (gB : (x:A) -> groupoid (B x)) (t u : Sigma A B) (p:Id A t.1 u.1) : - set (IdP (B (p@i)) t.2 u.2) = substInv U set T0 T1 rem rem1 - where P : Id U (B t.1) (B u.1) = B (p@i) - T0 : U = IdP P t.2 u.2 - T1 : U = Id (B u.1) (transport P t.2) u.2 - rem : Id U T0 T1 = lem2 A B t u p -- funDepTr (B t.1) (B u.1) P t.2 u.2 + rem3 : Path U (Path (Sigma A B) t u) ((p:T) * C p) = lemPathSig A B t u + +corSigGroupoid (A:U) (B:A-> U) (gB : (x:A) -> groupoid (B x)) (t u : Sigma A B) (p:Path A t.1 u.1) : + set (PathP (B (p@i)) t.2 u.2) = substInv U set T0 T1 rem rem1 + where P : Path U (B t.1) (B u.1) = B (p@i) + T0 : U = PathP P t.2 u.2 + T1 : U = Path (B u.1) (transport P t.2) u.2 + rem : Path U T0 T1 = lem2 A B t u p -- funDepTr (B t.1) (B u.1) P t.2 u.2 v2 : B u.1 = transport P t.2 rem1 : set T1 = gB u.1 v2 u.2 -groupoidSig (A:U) (B:A-> U) (gA: groupoid A) (gB : (x:A) -> groupoid (B x)) (t u : Sigma A B) : set (Id (Sigma A B) t u) = - substInv U set (Id (Sigma A B) t u) ((p:T) * C p) rem3 rem2 +groupoidSig (A:U) (B:A-> U) (gA: groupoid A) (gB : (x:A) -> groupoid (B x)) (t u : Sigma A B) : set (Path (Sigma A B) t u) = + substInv U set (Path (Sigma A B) t u) ((p:T) * C p) rem3 rem2 where - T : U = Id A t.1 u.1 - C (p:T) : U = IdP ( B (p@i)) t.2 u.2 + T : U = Path A t.1 u.1 + C (p:T) : U = PathP ( B (p@i)) t.2 u.2 rem (p : T) : set (C p) = corSigGroupoid A B gB t u p rem1 : set T = gA t.1 u.1 rem2 : set ((p:T) * C p) = setSig T C rem1 rem - rem3 : Id U (Id (Sigma A B) t u) ((p:T) * C p) = lemIdSig A B t u + rem3 : Path U (Path (Sigma A B) t u) ((p:T) * C p) = lemPathSig A B t u lemContr (A:U) (pA:prop A) (a:A) : isContr A = (a,rem) - where rem (y:A) : Id A a y = pA a y - -lem3 (A:U) (B:A-> U) (pB : (x:A) -> prop (B x)) (t u : Sigma A B) (p:Id A t.1 u.1) : - isContr (IdP (B (p@i)) t.2 u.2) = lemContr T0 (substInv U prop T0 T1 rem rem1) rem2 - where P : Id U (B t.1) (B u.1) = B (p@i) - T0 : U = IdP P t.2 u.2 - T1 : U = Id (B u.1) (transport P t.2) u.2 - rem : Id U T0 T1 = lem2 A B t u p + where rem (y:A) : Path A a y = pA a y + +lem3 (A:U) (B:A-> U) (pB : (x:A) -> prop (B x)) (t u : Sigma A B) (p:Path A t.1 u.1) : + isContr (PathP (B (p@i)) t.2 u.2) = lemContr T0 (substInv U prop T0 T1 rem rem1) rem2 + where P : Path U (B t.1) (B u.1) = B (p@i) + T0 : U = PathP P t.2 u.2 + T1 : U = Path (B u.1) (transport P t.2) u.2 + rem : Path U T0 T1 = lem2 A B t u p v2 : B u.1 = transport P t.2 rem1 : prop T1 = propSet (B u.1) (pB u.1) v2 u.2 rem2 : T0 = transport (rem@-i) (pB u.1 v2 u.2) -lem6 (A:U) (P:A-> U) (cA:(x:A) -> isContr (P x)) : Id U ((x:A)*P x) A = isoId T A f g t s +lem6 (A:U) (P:A-> U) (cA:(x:A) -> isContr (P x)) : Path U ((x:A)*P x) A = isoPath T A f g t s where T : U = (x:A) * P x f (z:T) : A = z.1 g (x:A) : T = (x,(cA x).1) - s (z:T) : Id T (g (f z)) z = (z.1,((cA z.1).2 z.2)@ i) - t (x:A) : Id A (f (g x)) x = refl A x + s (z:T) : Path T (g (f z)) z = (z.1,((cA z.1).2 z.2)@ i) + t (x:A) : Path A (f (g x)) x = refl A x -lemSigProp (A:U) (B:A-> U) (pB : (x:A) -> prop (B x)) (t u : Sigma A B) : Id U (Id (Sigma A B) t u) (Id A t.1 u.1) = - compId U (Id (Sigma A B) t u) ((p:Id A t.1 u.1) * IdP ( B (p@i)) t.2 u.2) (Id A t.1 u.1) rem2 rem1 +lemSigProp (A:U) (B:A-> U) (pB : (x:A) -> prop (B x)) (t u : Sigma A B) : Path U (Path (Sigma A B) t u) (Path A t.1 u.1) = + compPath U (Path (Sigma A B) t u) ((p:Path A t.1 u.1) * PathP ( B (p@i)) t.2 u.2) (Path A t.1 u.1) rem2 rem1 where - T : U = Id A t.1 u.1 - C (p:T) : U = IdP ( B (p@i)) t.2 u.2 + T : U = Path A t.1 u.1 + C (p:T) : U = PathP ( B (p@i)) t.2 u.2 rem (p : T) : isContr (C p) = lem3 A B pB t u p - rem1 : Id U ((p:T) * C p) T = lem6 T C rem - rem2 : Id U (Id (Sigma A B) t u) ((p:T) * C p) = lemIdSig A B t u + rem1 : Path U ((p:T) * C p) T = lem6 T C rem + rem2 : Path U (Path (Sigma A B) t u) ((p:T) * C p) = lemPathSig A B t u -setGroupoid (A:U) (sA:set A) (a0 a1:A) : set (Id A a0 a1) = propSet (Id A a0 a1) (sA a0 a1) +setGroupoid (A:U) (sA:set A) (a0 a1:A) : set (Path A a0 a1) = propSet (Path A a0 a1) (sA a0 a1) propGroupoid (A:U) (pA: prop A) : groupoid A = setGroupoid A (propSet A pA) diff --git a/examples/subset.ctt b/examples/subset.ctt index 694f537..1e279f0 100644 --- a/examples/subset.ctt +++ b/examples/subset.ctt @@ -28,14 +28,14 @@ subset01 (A : U) (sA : set A) : subset0 A sA -> subset1 A sA = s.2.2.2 -- Construct a proposition to tag the element a with X : U - = (b : B) * (Id A (f b) a) + = (b : B) * (Path A (f b) a) pX : prop X = i a in (X, pX) -lem (A:U) (P:A->U) (pP:(x:A) -> prop (P x)) (u v:(x:A) * P x) (p:Id A u.1 v.1) : - Id ((x:A)*P x) u v = (p@i,(lemPropF A P pP u.1 v.1 p u.2 v.2)@i) +lem (A:U) (P:A->U) (pP:(x:A) -> prop (P x)) (u v:(x:A) * P x) (p:Path A u.1 v.1) : + Path ((x:A)*P x) u v = (p@i,(lemPropF A P pP u.1 v.1 p u.2 v.2)@i) -- A map from the second to the first definition of subsets. subset10 (A : U) (sA : set A) @@ -59,14 +59,14 @@ subset10 (A : U) (sA : set A) = \ (b : B) -> b.1 -- Show that f is injective. inj : inj1 B A f sB sA - = \ (a : A) (c d : (b : B) * Id A (f b) a) -> let - p : Id A c.1.1 d.1.1 + = \ (a : A) (c d : (b : B) * Path A (f b) a) -> let + p : Path A c.1.1 d.1.1 = comp ( A) (c.2 @ i) [ (i = 0) -> c.1.1 , (i = 1) -> d.2 @ -j ] - q : Id B c.1 d.1 + q : Path B c.1 d.1 = lem A Q (\ (x : A) -> (P x).2) c.1 d.1 p - r : Id ((b : B) * Id A (f b) a) c d - = lem B (\(b : B) -> Id A (f b) a) (\ (b : B) -> sA (f b) a) c d q + r : Path ((b : B) * Path A (f b) a) c d + = lem B (\(b : B) -> Path A (f b) a) (\ (b : B) -> sA (f b) a) c d q in r in @@ -77,7 +77,7 @@ opaque subst -- Show that subset10 ∘ subset01 can be identified with the identity function subsetIso0 (A : U) (sA : set A) : (s0 : subset0 A sA) -> - Id (subset0 A sA) (subset10 A sA (subset01 A sA s0)) s0 + Path (subset0 A sA) (subset10 A sA (subset01 A sA s0)) s0 = \ (s0 : subset0 A sA) -> let s0' : subset0 A sA @@ -104,32 +104,32 @@ subsetIso0 (A : U) (sA : set A) : (s0 : subset0 A sA) -> = (f b, b, f b) g' (b' : B') : B = b'.2.1 - s (x : B) : Id B (g' (g x)) x + s (x : B) : Path B (g' (g x)) x = x - t (x : B') : Id B' (g (g' x)) x + t (x : B') : Path B' (g (g' x)) x = (x.2.2 @ i, x.2.1, x.2.2 @ i /\ j) -- Compute a path between B' and B, as well as a path between f'∘g∘g' and f P (X : U) (h: X -> B) : U - = (p : Id U X B) * (IdP ( p @ i -> A) (\ (x : X) -> f' (g (h x))) f) + = (p : Path U X B) * (PathP ( p @ i -> A) (\ (x : X) -> f' (g (h x))) f) q : P B' g' = elimEquiv B P ( B, f) B' (g', gradLemma B' B g' g s t) - idB : Id U B' B + idB : Path U B' B = q.1 -- Show that sB can be identified with sB' - idsB : IdP ( set (idB @ i)) sB' sB + idsB : PathP ( set (idB @ i)) sB' sB = lemPropF U set setIsProp B' B idB sB' sB -- Show that f' can be identified with f. This follows from g∘g' ⇔ \x -> x -- and that there is a path q.2 between f'∘g∘g' and f - idf : IdP ( idB @ i -> A) f' f + idf : PathP ( idB @ i -> A) f' f = let Q (h : B' -> B') : U - = IdP ( q.1 @ i -> A) (\ (x : B') -> f' (h x)) f + = PathP ( q.1 @ i -> A) (\ (x : B') -> f' (h x)) f a : B' -> B' = \ (x : B') -> g (g' x) b : B' -> B' = \ (x : B') -> x - p : Id (B' -> B') a b = \ (x : B') -> (t x) @ i + p : Path (B' -> B') a b = \ (x : B') -> (t x) @ i in subst (B' -> B') Q a b p q.2 -- Show that inj can be identified with inj' - idinj : IdP ( inj1 (idB @ i) A (idf @ i) (idsB @ i) sA) inj' inj + idinj : PathP ( inj1 (idB @ i) A (idf @ i) (idsB @ i) sA) inj' inj = let T : U = (X : U) * (_ : X -> A) * (set X) @@ -141,7 +141,7 @@ subsetIso0 (A : U) (sA : set A) : (s0 : subset0 A sA) -> = (B', f', sB') t1 : T = (B, f, sB) - idT : Id T t0 t1 + idT : Path T t0 t1 = (idB @ i, idf @ i, idsB @ i) in lemPropF T P pP t0 t1 idT inj' inj in @@ -149,13 +149,13 @@ subsetIso0 (A : U) (sA : set A) : (s0 : subset0 A sA) -> -- Show that subset10 ∘ subset01 can be identified with the identity function subsetIso1 (A : U) (sA : set A) : (s1 : subset1 A sA) -> - Id (subset1 A sA) (subset01 A sA (subset10 A sA s1)) s1 + Path (subset1 A sA) (subset01 A sA (subset10 A sA s1)) s1 = \ (s1 : subset1 A sA) -> let -- Construct the second subset s1' from s1. s1' : subset1 A sA = subset01 A sA (subset10 A sA s1) -- Show that s1' and s1 produces the same result for all a : A - ids1 : (a : A) -> Id ((X : U) * (prop X)) (s1' a) (s1 a) + ids1 : (a : A) -> Path ((X : U) * (prop X)) (s1' a) (s1 a) = \ (a : A) -> let -- Construct isomorphism between (s1' a).1 and (s1 a).1 to show that -- (s1' a).1 can be identified with (s1 a).1 @@ -163,15 +163,15 @@ subsetIso1 (A : U) (sA : set A) : (s1 : subset1 A sA) -> = \ (x : (s1' a).1) -> subst A (\(a : A) -> (s1 a).1) x.1.1 a x.2 x.1.2 g : (s1 a).1 -> (s1' a).1 = \ (x : (s1 a).1) -> ((a, x), a) - s : (x : (s1 a).1) -> Id (s1 a).1 (f (g x)) x + s : (x : (s1 a).1) -> Path (s1 a).1 (f (g x)) x = \ (x : (s1 a).1) -> (s1 a).2 (f (g x)) x - t : (x : (s1' a).1) -> Id (s1' a).1 (g (f x)) x + t : (x : (s1' a).1) -> Path (s1' a).1 (g (f x)) x = \ (x : (s1' a).1) -> (s1' a).2 (g (f x)) x - p : Id U (s1' a).1 (s1 a).1 - = isoId (s1' a).1 (s1 a).1 f g s t + p : Path U (s1' a).1 (s1 a).1 + = isoPath (s1' a).1 (s1 a).1 f g s t -- Show that for x : prop (s1' a).1, y : prop (s1 a).1, -- x can be identified with y. - q : IdP ( prop (p @ i)) (s1' a).2 (s1 a).2 + q : PathP ( prop (p @ i)) (s1' a).2 (s1 a).2 = lemPropF U prop propIsProp (s1' a).1 (s1 a).1 p (s1' a).2 (s1 a).2 in (p @ i, q @ i) @@ -179,6 +179,6 @@ subsetIso1 (A : U) (sA : set A) : (s1 : subset1 A sA) -> funExt A (\ (_ : A) -> (X : U) * (prop X)) s1' s1 ids1 -- Show that we can identify the two definitions of subsets with each other -subsetId (A : U) (sA : set A) : Id U (subset0 A sA) (subset1 A sA) - = isoId (subset0 A sA) (subset1 A sA) (subset01 A sA) (subset10 A sA) +subsetPath (A : U) (sA : set A) : Path U (subset0 A sA) (subset1 A sA) + = isoPath (subset0 A sA) (subset1 A sA) (subset01 A sA) (subset10 A sA) (subsetIso1 A sA) (subsetIso0 A sA) \ No newline at end of file diff --git a/examples/susp.ctt b/examples/susp.ctt index b6050d8..b051f5b 100644 --- a/examples/susp.ctt +++ b/examples/susp.ctt @@ -15,7 +15,7 @@ sphere : nat -> U = split -- (Similar to HoTT Book, Lemma 6.5.1) sone : U = sphere one -path : bool -> Id S1 base base = split +path : bool -> Path S1 base base = split false -> loop1 true -> refl S1 base @@ -24,41 +24,41 @@ s1ToCircle : sone -> S1 = split south -> base merid b @ i -> path b @ i -m0 : Id sone north south = merid{sone} false @ i +m0 : Path sone north south = merid{sone} false @ i -m1 : Id sone north south = merid{sone} true @ i +m1 : Path sone north south = merid{sone} true @ i -invm1 : Id sone south north = inv sone north south m1 +invm1 : Path sone south north = inv sone north south m1 circleToS1 : S1 -> sone = split base -> north - loop @ i -> compId sone north south north m0 invm1 @ i + loop @ i -> compPath sone north south north m0 invm1 @ i -merid1 (b:bool) : Id sone north south = merid{sone} b @ i +merid1 (b:bool) : Path sone north south = merid{sone} b @ i co (x: sone) : sone = circleToS1 (s1ToCircle x) -lemSquare (A:U) (a b : A) (m0 m1 : Id A a b) : - Square A a a a b (compId A a b a m0 (inv A a b m1)) m0 (refl A a) m1 = +lemSquare (A:U) (a b : A) (m0 m1 : Path A a b) : + Square A a a a b (compPath A a b a m0 (inv A a b m1)) m0 (refl A a) m1 = comp (<_>A) (m0 @ i) [(i=1) -> m1 @ (j \/ -k), (i=0) -> <_>a, (j=1) -> <_>m0@i, (j=0) -> comp (<_>A) (m0 @ i) [(k=0) -> <_>m0@i, (i=0) -> <_>a, (i=1) -> m1 @ (-k \/ -l)]] -coid : (x : sone) -> Id sone (co x) x = split +coid : (x : sone) -> Path sone (co x) x = split north -> refl sone north south -> m1 merid b @ i -> ind b @ i where - F (x:sone) : U = Id sone (co x) x + F (x:sone) : U = Path sone (co x) x - ind : (b:bool) -> IdS sone F north south (merid1 b) (refl sone north) m1 = split + ind : (b:bool) -> PathS sone F north south (merid1 b) (refl sone north) m1 = split false -> lemSquare sone north south m0 m1 true -> m1 @ (j /\ k) oc (x:S1) : S1 = s1ToCircle (circleToS1 x) -ocid : (x : S1) -> Id S1 (oc x) x = +ocid : (x : S1) -> Path S1 (oc x) x = split base -> refl S1 base loop @ i -> comp (<_>S1) (loop1@i) [(i=0) -> <_>base,(i=1) -> <_>base,(j=1) -> <_>loop1@i, @@ -66,11 +66,11 @@ ocid : (x : S1) -> Id S1 (oc x) x = -s1EqCircle : Id U sone S1 = isoId sone S1 s1ToCircle circleToS1 ocid coid +s1EqCircle : Path U sone S1 = isoPath sone S1 s1ToCircle circleToS1 ocid coid -s1EqS1 : Id U S1 S1 = compId U S1 sone S1 (inv U sone S1 s1EqCircle) s1EqCircle +s1EqS1 : Path U S1 S1 = compPath U S1 sone S1 (inv U sone S1 s1EqCircle) s1EqCircle -lem (A:U) (a:A) : Id A (comp (<_>A) (comp (<_>A) (comp (<_>A) a []) []) []) a = +lem (A:U) (a:A) : Path A (comp (<_>A) (comp (<_>A) (comp (<_>A) a []) []) []) a = comp (<_>A) (comp (<_>A) (comp (<_>A) a [(i=1) -> <_>a]) [(i=1) -> <_>a]) [(i=1) -> <_>a] @@ -79,45 +79,45 @@ lem (A:U) (a:A) : Id A (comp (<_>A) (comp (<_>A) (comp (<_>A) a []) []) []) a = ptU : U = (X : U) * X -lemPt (A :U) (B:U) (p:Id U A B) (a:A) : Id ptU (A,a) (B,transport p a) = +lemPt (A :U) (B:U) (p:Path U A B) (a:A) : Path ptU (A,a) (B,transport p a) = (p @ i,comp ( p @ (i/\j)) a [(i=0) -> <_>a]) -Omega (X:ptU) : ptU = (Id X.1 X.2 X.2,refl X.1 X.2) +Omega (X:ptU) : ptU = (Path X.1 X.2 X.2,refl X.1 X.2) -lem (A:U) (a:A) : Id A (comp (<_>A) (comp (<_>A) (comp (<_>A) a []) []) []) a = +lem (A:U) (a:A) : Path A (comp (<_>A) (comp (<_>A) (comp (<_>A) a []) []) []) a = comp (<_>A) (comp (<_>A) (comp (<_>A) a [(i=1) -> <_>a]) [(i=1) -> <_>a]) [(i=1) -> <_>a] -lem1 (A:U) (a:A) : Id ptU (A,comp (<_>A) (comp (<_>A) (comp (<_>A) a []) []) []) (A,a) = +lem1 (A:U) (a:A) : Path ptU (A,comp (<_>A) (comp (<_>A) (comp (<_>A) a []) []) []) (A,a) = (A,lem A a@i) --- s1PtCircle : Id ptU (sone,north) (S1,base) = --- compId ptU (sone,north) (S1,comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) []) (S1,base) (lemPt sone S1 s1EqCircle north) (lem1 S1 base) +-- s1PtCircle : Path ptU (sone,north) (S1,base) = +-- compPath ptU (sone,north) (S1,comp (<_>S1) (comp (<_>S1) (comp (<_>S1) base []) []) []) (S1,base) (lemPt sone S1 s1EqCircle north) (lem1 S1 base) --- windingS : Id sone north north -> Z = rem1 +-- windingS : Path sone north north -> Z = rem1 -- where -- G (X:ptU) : U = (Omega X).1 -> Z -- rem : G (S1,base) = winding -- rem1 : G (sone,north) = subst ptU G (S1,base) (sone,north) ( s1PtCircle @ -i) rem --- s1ToCId (p: Id sone north north) : Id S1 base base = transport s1EqCircle (p @ i) +-- s1ToCPath (p: Path sone north north) : Path S1 base base = transport s1EqCircle (p @ i) --- s1ToCIdInv (p : Id S1 base base) : Id sone north north = (transport ( s1EqCircle @ -j) (p @ i)) +-- s1ToCPathInv (p : Path S1 base base) : Path sone north north = (transport ( s1EqCircle @ -j) (p @ i)) -loop1S : Id sone north north = compId sone north south north m0 invm1 +loop1S : Path sone north north = compPath sone north south north m0 invm1 -loop2S : Id sone north north = compId sone north north north loop1S loop1S +loop2S : Path sone north north = compPath sone north north north loop1S loop1S -- test0S : Z = windingS (refl sone north) -- test2S : Z = windingS loop2S --- test4S : Z = windingS (compId sone north north north loop2S loop2S) +-- test4S : Z = windingS (compPath sone north north north loop2S loop2S) -- indSusp: -suspOf (A X : U) : U = (u:X) * (v:X) * (A -> Id X u v) +suspOf (A X : U) : U = (u:X) * (v:X) * (A -> Path X u v) funToL (A X:U) (f:susp A -> X) : suspOf A X = (f north,f south,\ (a:A) -> f (merid{susp A} a@i)) @@ -127,16 +127,16 @@ lToFun (A X:U) (z:suspOf A X) : susp A -> X = split south -> z.2.1 merid a @ i-> z.2.2 a @ i -test1 (A X:U) (z:suspOf A X) : Id (suspOf A X) (funToL A X (lToFun A X z)) z +test1 (A X:U) (z:suspOf A X) : Path (suspOf A X) (funToL A X (lToFun A X z)) z = refl (suspOf A X) z -rem (A X:U) (f:susp A ->X) : (u:susp A) -> Id X (lToFun A X (funToL A X f) u) (f u) = split +rem (A X:U) (f:susp A ->X) : (u:susp A) -> Path X (lToFun A X (funToL A X f) u) (f u) = split north -> refl X (f north) south -> refl X (f south) merid a @ i -> refl X (f (merid{susp A} a @ i)) -test2 (A X:U) (f:susp A ->X) : Id (susp A ->X) (lToFun A X (funToL A X f)) f +test2 (A X:U) (f:susp A ->X) : Path (susp A ->X) (lToFun A X (funToL A X f)) f = \ (u:susp A) -> rem A X f u @ i -funSusp (A X:U) : Id U (susp A -> X) (suspOf A X) = - isoId (susp A -> X) (suspOf A X) (funToL A X) (lToFun A X) (test1 A X) (test2 A X) +funSusp (A X:U) : Path U (susp A -> X) (suspOf A X) = + isoPath (susp A -> X) (suspOf A X) (funToL A X) (lToFun A X) (test1 A X) (test2 A X) diff --git a/examples/torsor.ctt b/examples/torsor.ctt index 2983ab1..0193c6a 100644 --- a/examples/torsor.ctt +++ b/examples/torsor.ctt @@ -10,33 +10,33 @@ import univalence isEquivComp (A B C : U) (F : A -> B) (G : B -> C) (ef : isEquiv A B F) (eg : isEquiv B C G) : isEquiv A C (\(x : A) -> G (F x)) = gradLemma A C (\(x : A) -> G (F x)) (\(x : C) -> (ef (eg x).1.1).1.1) - (\(x : C) -> compId C (G (F (ef (eg x).1.1).1.1)) (G (eg x).1.1) x + (\(x : C) -> compPath C (G (F (ef (eg x).1.1).1.1)) (G (eg x).1.1) x ( G (retEq A B (F, ef) (eg x).1.1 @ i)) (retEq B C (G, eg) x)) - (\(x : A) -> compId A ((ef (eg (G (F x))).1.1).1.1) (ef (F x)).1.1 x + (\(x : A) -> compPath A ((ef (eg (G (F x))).1.1).1.1) (ef (F x)).1.1 x ( (ef (secEq B C (G, eg) (F x) @ i)).1.1) (secEq A B (F, ef) x)) isEquivComp' (A B C : U) (F : A -> B) (G : C -> B) (ef : isEquiv A B F) (eg : isEquiv C B G) : isEquiv A C (\(x : A) -> (eg (F x)).1.1) = gradLemma A C (\(x : A) -> (eg (F x)).1.1) (\(x : C) -> (ef (G x)).1.1) - (\(x : C) -> compId C (eg (F (ef (G x)).1.1)).1.1 (eg (G x)).1.1 x + (\(x : C) -> compPath C (eg (F (ef (G x)).1.1)).1.1 (eg (G x)).1.1 x ( (eg (retEq A B (F, ef) (G x) @ i)).1.1) (secEq C B (G, eg) x)) - (\(x : A) -> compId A ((ef (G (eg (F x)).1.1)).1.1) (ef (F x)).1.1 x + (\(x : A) -> compPath A ((ef (G (eg (F x)).1.1)).1.1) (ef (F x)).1.1 x ( (ef (retEq C B (G, eg) (F x) @ i)).1.1) (secEq A B (F, ef) x)) -lemHcomp (x : loopS1) : Id loopS1 ( comp (<_>S1) (x@i) [(i=0)-><_>base,(i=1)-><_>base]) x +lemHcomp (x : loopS1) : Path loopS1 ( comp (<_>S1) (x@i) [(i=0)-><_>base,(i=1)-><_>base]) x = comp (<_>S1) (x@i) [(i=0)-><_>base,(j=1)-><_>x@i,(i=1)-><_>base] opaque lemHcomp lemHcomp3 (x : loopS1) - : Id loopS1 + : Path loopS1 ( comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (x@i) [(i=0)-><_>base,(i=1)-><_>base]) [(i=0)-><_>base,(i=1)-><_>base]) [(i=0)-><_>base,(i=1)-><_>base]) x - = compId loopS1 + = compPath loopS1 ( comp (<_>S1) (comp (<_>S1) (comp (<_>S1) (x@i) [(i=0)-><_>base,(i=1)-><_>base]) [(i=0)-><_>base,(i=1)-><_>base]) [(i=0)-><_>base,(i=1)-><_>base]) ( comp (<_>S1) (comp (<_>S1) (x@i) [(i=0)-><_>base,(i=1)-><_>base]) [(i=0)-><_>base,(i=1)-><_>base]) x ( comp (<_>S1) (comp (<_>S1) (lemHcomp x@j@i) [(i=0)-><_>base,(i=1)-><_>base]) [(i=0)-><_>base,(i=1)-><_>base]) - (compId loopS1 + (compPath loopS1 ( comp (<_>S1) (comp (<_>S1) (x@i) [(i=0)-><_>base,(i=1)-><_>base]) [(i=0)-><_>base,(i=1)-><_>base]) ( comp (<_>S1) (x@i) [(i=0)-><_>base,(i=1)-><_>base]) x @@ -44,35 +44,35 @@ lemHcomp3 (x : loopS1) (lemHcomp x)) opaque lemHcomp3 -lemEquiv1 (A B : U) (F : A -> B) (G : A -> B) (e : isEquiv A B G) (p : (x : A) -> Id A (e (F x)).1.1 x) : Id (A -> B) F G - = \(x : A) -> transport ( Id B (retEq A B (G, e) (F x) @ i) (G x)) (mapOnPath A B G (e (F x)).1.1 x (p x)) @ i +lemEquiv1 (A B : U) (F : A -> B) (G : A -> B) (e : isEquiv A B G) (p : (x : A) -> Path A (e (F x)).1.1 x) : Path (A -> B) F G + = \(x : A) -> transport ( Path B (retEq A B (G, e) (F x) @ i) (G x)) (mapOnPath A B G (e (F x)).1.1 x (p x)) @ i -transRefl (A : U) (a : A) : Id A (transport (<_> A) a) a = comp (<_> A) a [(i=1) -> <_>a] -lemReflComp (A : U) (a b : A) (p : Id A a b) : Id (Id A a b) (compId A a a b (<_> a) p) p = +transRefl (A : U) (a : A) : Path A (transport (<_> A) a) a = comp (<_> A) a [(i=1) -> <_>a] +lemReflComp (A : U) (a b : A) (p : Path A a b) : Path (Path A a b) (compPath A a a b (<_> a) p) p = comp ( A) (p @ i /\ j) [(i=0) -> <_> a, (j=1) -> <_> p @ i, (i=1) -> p @ k \/ j ] -lemReflComp' (A : U) (a b : A) (p : Id A a b) : Id (Id A a b) (compId A a b b p (<_> b)) p = +lemReflComp' (A : U) (a b : A) (p : Path A a b) : Path (Path A a b) (compPath A a b b p (<_> b)) p = comp ( A) (p @ i) [(i=0) -> <_> a, (j=1) -> <_> p @ i, (i=1) -> <_> b ] -lem2ItPos : (n:nat) -> Id loopS1 (loopIt (predZ (inr n))) (backTurn (loopIt (inr n))) = split - zero -> transport ( Id loopS1 invLoop (lemReflComp S1 base base invLoop @ -i)) (<_> invLoop) +lem2ItPos : (n:nat) -> Path loopS1 (loopIt (predZ (inr n))) (backTurn (loopIt (inr n))) = split + zero -> transport ( Path loopS1 invLoop (lemReflComp S1 base base invLoop @ -i)) (<_> invLoop) suc p -> compInv S1 base base (loopIt (inr p)) base loop1 -lem2It : (n:Z) -> Id loopS1 (loopIt (predZ n)) (backTurn (loopIt n)) = split +lem2It : (n:Z) -> Path loopS1 (loopIt (predZ n)) (backTurn (loopIt n)) = split inl n -> <_> loopIt (inl (suc n)) inr n -> lem2ItPos n -transportCompId (a b c : U) (p1 : Id U a b) (p2 : Id U b c) (x : a) - : Id c (transport (compId U a b c p1 p2) x) (transport p2 (transport p1 x)) - = J U b (\(c : U) -> \(p2 : Id U b c) -> Id c (comp (compId U a b c p1 p2) x []) (comp p2 (transport p1 x) [])) +transportCompPath (a b c : U) (p1 : Path U a b) (p2 : Path U b c) (x : a) + : Path c (transport (compPath U a b c p1 p2) x) (transport p2 (transport p1 x)) + = J U b (\(c : U) -> \(p2 : Path U b c) -> Path c (comp (compPath U a b c p1 p2) x []) (comp p2 (transport p1 x) [])) hole c p2 where - hole : Id b (comp (compId U a b b p1 (<_> b)) x []) (comp (<_> b) (transport p1 x) []) = - compId b (comp (compId U a b b p1 (<_> b)) x []) (transport p1 x) (comp (<_> b) (transport p1 x) []) + hole : Path b (comp (compPath U a b b p1 (<_> b)) x []) (comp (<_> b) (transport p1 x) []) = + compPath b (comp (compPath U a b b p1 (<_> b)) x []) (transport p1 x) (comp (<_> b) (transport p1 x) []) ( comp (lemReflComp' U a b p1 @ i) x []) ( transRefl b (transport p1 x) @ -i) -lemRevCompId (A : U) (a b c : A) (p1 : Id A a b) (p2 : Id A b c) - : Id (Id A c a) ( compId A a b c p1 p2 @ -i) (compId A c b a ( p2@-i) ( p1@-i)) +lemRevCompPath (A : U) (a b c : A) (p1 : Path A a b) (p2 : Path A b c) + : Path (Path A c a) ( compPath A a b c p1 p2 @ -i) (compPath A c b a ( p2@-i) ( p1@-i)) = comp ( A) (hole1 @ i @ j) [(i=0) -> p2 @ k \/ j, (i=1) -> p1 @ -k /\ j] where hole1 : Square A b a c b ( p1@-i) ( p2@-i) p2 p1 @@ -80,87 +80,87 @@ lemRevCompId (A : U) (a b c : A) (p1 : Id A a b) (p2 : Id A b c) setPi (A : U) (B : A -> U) (h : (x : A) -> set (B x)) (f0 f1 : (x : A) -> B x) - (p1 p2 : Id ((x : A) -> B x) f0 f1) - : Id (Id ((x : A) -> B x) f0 f1) p1 p2 + (p1 p2 : Path ((x : A) -> B x) f0 f1) + : Path (Path ((x : A) -> B x) f0 f1) p1 p2 = \(x : A) -> (h x (f0 x) (f1 x) ( (p1@i) x) ( (p2@i) x)) @ i @ j -lemIdPProp (A B : U) (AProp : prop A) (p : Id U A B) : (x : A) -> (y : B) -> IdP p x y - = J U A (\(B : U) -> \(p : Id U A B) -> (x : A) -> (y : B) -> IdP p x y) AProp B p +lemPathPProp (A B : U) (AProp : prop A) (p : Path U A B) : (x : A) -> (y : B) -> PathP p x y + = J U A (\(B : U) -> \(p : Path U A B) -> (x : A) -> (y : B) -> PathP p x y) AProp B p -lemIdPSet (A B : U) (ASet : set A) (p : Id U A B) : (x : A) (y : B) (s t : IdP p x y) -> Id (IdP p x y) s t - = J U A (\(B : U) -> \(p : Id U A B) -> (x : A) (y : B) (s t : IdP p x y) -> Id (IdP p x y) s t) ASet B p +lemPathPSet (A B : U) (ASet : set A) (p : Path U A B) : (x : A) (y : B) (s t : PathP p x y) -> Path (PathP p x y) s t + = J U A (\(B : U) -> \(p : Path U A B) -> (x : A) (y : B) (s t : PathP p x y) -> Path (PathP p x y) s t) ASet B p -lemIdPSet2 (A B : U) (ASet : set A) (p1 : Id U A B) - : (p2 : Id U A B) -> (p : Id (Id U A B) p1 p2) -> - (x : A) -> (y : B) -> (s : IdP p1 x y) -> (t : IdP p2 x y) -> IdP ( (IdP (p @ i) x y)) s t - = J (Id U A B) p1 (\(p2 : Id U A B) -> \(p : Id (Id U A B) p1 p2) -> (x : A) -> (y : B) -> (s : IdP p1 x y) -> (t : IdP p2 x y) -> IdP ( (IdP (p @ i) x y)) s t) - (lemIdPSet A B ASet p1) +lemPathPSet2 (A B : U) (ASet : set A) (p1 : Path U A B) + : (p2 : Path U A B) -> (p : Path (Path U A B) p1 p2) -> + (x : A) -> (y : B) -> (s : PathP p1 x y) -> (t : PathP p2 x y) -> PathP ( (PathP (p @ i) x y)) s t + = J (Path U A B) p1 (\(p2 : Path U A B) -> \(p : Path (Path U A B) p1 p2) -> (x : A) -> (y : B) -> (s : PathP p1 x y) -> (t : PathP p2 x y) -> PathP ( (PathP (p @ i) x y)) s t) + (lemPathPSet A B ASet p1) -isEquivId (A : U) : isEquiv A A (\(x : A) -> x) = gradLemma A A (\(x : A) -> x) (\(x : A) -> x) (\(x : A) -> <_> x) (\(x : A) -> <_> x) +isEquivPath (A : U) : isEquiv A A (\(x : A) -> x) = gradLemma A A (\(x : A) -> x) (\(x : A) -> x) (\(x : A) -> <_> x) (\(x : A) -> <_> x) -lem10 (A B : U) (e : equiv A B) (x y : B) (p : Id A (e.2 x).1.1 (e.2 y).1.1) : Id B x y +lem10 (A B : U) (e : equiv A B) (x y : B) (p : Path A (e.2 x).1.1 (e.2 y).1.1) : Path B x y = transport - (compId U (Id B (e.1 (e.2 x).1.1) (e.1 (e.2 y).1.1)) (Id B x (e.1 (e.2 y).1.1)) (Id B x y) - ( Id B (retEq A B e x @ i) (e.1 (e.2 y).1.1)) ( Id B x (retEq A B e y @ i))) + (compPath U (Path B (e.1 (e.2 x).1.1) (e.1 (e.2 y).1.1)) (Path B x (e.1 (e.2 y).1.1)) (Path B x y) + ( Path B (retEq A B e x @ i) (e.1 (e.2 y).1.1)) ( Path B x (retEq A B e y @ i))) (mapOnPath A B e.1 (e.2 x).1.1 (e.2 y).1.1 p) -lem10' (A B : U) (e : equiv A B) (x y : A) (p : Id B (e.1 x) (e.1 y)) : Id A x y +lem10' (A B : U) (e : equiv A B) (x y : A) (p : Path B (e.1 x) (e.1 y)) : Path A x y = transport - (compId U (Id A (e.2 (e.1 x)).1.1 (e.2 (e.1 y)).1.1) (Id A x (e.2 (e.1 y)).1.1) (Id A x y) - ( Id A (secEq A B e x @ i) (e.2 (e.1 y)).1.1) ( Id A x (secEq A B e y @ i)) + (compPath U (Path A (e.2 (e.1 x)).1.1 (e.2 (e.1 y)).1.1) (Path A x (e.2 (e.1 y)).1.1) (Path A x y) + ( Path A (secEq A B e x @ i) (e.2 (e.1 y)).1.1) ( Path A x (secEq A B e y @ i)) ) (mapOnPath B A (invEq A B e) (e.1 x) (e.1 y) p) -lem11 (A : U) : Id (Id U A A) (univalence A A (\(x : A) -> x, isEquivId A)).1.1 (<_> A) = hole +lem11 (A : U) : Path (Path U A A) (univalence A A (\(x : A) -> x, isEquivPath A)).1.1 (<_> A) = hole where - hole0 : Id (equiv A A) - (\(x : A) -> x, isEquivId A) + hole0 : Path (equiv A A) + (\(x : A) -> x, isEquivPath A) (transEquiv' A A (<_> A)) = ( (transRefl (A->A) (\(x : A) -> x) @ -i) - , lemIdPProp (isEquiv A A (\(x : A) -> x)) (isEquiv A A (transEquiv' A A (<_> A)).1) + , lemPathPProp (isEquiv A A (\(x : A) -> x)) (isEquiv A A (transEquiv' A A (<_> A)).1) (propIsEquiv A A (\(x : A) -> x)) ( isEquiv A A (transRefl (A->A) (\(x : A) -> x) @ -j)) - (isEquivId A) (transEquiv' A A (<_> A)).2 @ i + (isEquivPath A) (transEquiv' A A (<_> A)).2 @ i ) - hole1 : Id (equiv A A) (transEquiv' A A (univalence A A (\(x : A) -> x, isEquivId A)).1.1) (\(x : A) -> x, isEquivId A) - = retEq (Id U A A) (equiv A A) (corrUniv' A A) (\(x : A) -> x, isEquivId A) - hole : Id (Id U A A) (univalence A A (\(x : A) -> x, isEquivId A)).1.1 (<_> A) - = lem10' (Id U A A) (equiv A A) (corrUniv' A A) - (univalence A A (\(x : A) -> x, isEquivId A)).1.1 (<_> A) - (compId (equiv A A) (transEquiv' A A (univalence A A (\(x : A) -> x, isEquivId A)).1.1) (\(x : A) -> x, isEquivId A) (transEquiv' A A (<_> A)) hole1 hole0) + hole1 : Path (equiv A A) (transEquiv' A A (univalence A A (\(x : A) -> x, isEquivPath A)).1.1) (\(x : A) -> x, isEquivPath A) + = retEq (Path U A A) (equiv A A) (corrUniv' A A) (\(x : A) -> x, isEquivPath A) + hole : Path (Path U A A) (univalence A A (\(x : A) -> x, isEquivPath A)).1.1 (<_> A) + = lem10' (Path U A A) (equiv A A) (corrUniv' A A) + (univalence A A (\(x : A) -> x, isEquivPath A)).1.1 (<_> A) + (compPath (equiv A A) (transEquiv' A A (univalence A A (\(x : A) -> x, isEquivPath A)).1.1) (\(x : A) -> x, isEquivPath A) (transEquiv' A A (<_> A)) hole1 hole0) opaque lem11 -substIdP (A B : U) (p : Id U A B) (x : A) (y : B) (q : Id B (transport p x) y) : IdP p x y - = transport ( IdP p x (q@i)) hole +substPathP (A B : U) (p : Path U A B) (x : A) (y : B) (q : Path B (transport p x) y) : PathP p x y + = transport ( PathP p x (q@i)) hole where - hole : IdP p x (transport p x) = comp ( p @ (i /\ j)) x [(i=0) -> <_> x] + hole : PathP p x (transport p x) = comp ( p @ (i /\ j)) x [(i=0) -> <_> x] -lemIdSig (A:U) (B : A -> U) (t u : Sigma A B) : - Id U (Id (Sigma A B) t u) ((p : Id A t.1 u.1) * IdP ( B (p @ i)) t.2 u.2) = - isoId T0 T1 f g s t where - T0 : U = Id (Sigma A B) t u - T1 : U = (p:Id A t.1 u.1) * IdP ( B (p@i)) t.2 u.2 +lemPathSig (A:U) (B : A -> U) (t u : Sigma A B) : + Path U (Path (Sigma A B) t u) ((p : Path A t.1 u.1) * PathP ( B (p @ i)) t.2 u.2) = + isoPath T0 T1 f g s t where + T0 : U = Path (Sigma A B) t u + T1 : U = (p:Path A t.1 u.1) * PathP ( B (p@i)) t.2 u.2 f (q:T0) : T1 = ( (q@i).1, (q@i).2) g (z:T1) : T0 = (z.1 @i,z.2 @i) - s (z:T1) : Id T1 (f (g z)) z = refl T1 z - t (q:T0) : Id T0 (g (f q)) q = refl T0 q + s (z:T1) : Path T1 (f (g z)) z = refl T1 z + t (q:T0) : Path T0 (g (f q)) q = refl T0 q transConstN (A : U) (a : A) : (n : nat) -> A = split zero -> a suc n -> transport (<_> A) (transConstN A a n) -transConstNRefl (A : U) (a : A) : (n : nat) -> Id A (transConstN A a n) a = split +transConstNRefl (A : U) (a : A) : (n : nat) -> Path A (transConstN A a n) a = split zero -> <_> a - suc n -> compId A (transport (<_> A) (transConstN A a n)) (transConstN A a n) a + suc n -> compPath A (transport (<_> A) (transConstN A a n)) (transConstN A a n) a (transRefl A (transConstN A a n)) (transConstNRefl A a n) lem0 (A B : U) (f : A -> B) (e : isEquiv A B f) (x : A) - : Id B (transport (univalence B A (f, e)).1.1 x) (f x) + : Path B (transport (univalence B A (f, e)).1.1 x) (f x) = transConstNRefl B (f x) (suc (suc (suc (suc (suc (suc (suc (suc (suc (suc (suc zero))))))))))) lem1 (A B : U) (f : A -> B) (e : isEquiv A B f) (x : B) - : Id A (transport ( (univalence B A (f, e)).1.1 @ -i) x) (e x).1.1 - = compId A + : Path A (transport ( (univalence B A (f, e)).1.1 @ -i) x) (e x).1.1 + = compPath A (transConstN A (e (transConstN B x (suc (suc (suc (suc (suc (suc (suc (suc zero)))))))))).1.1 (suc (suc (suc zero)))) (transConstN A (e x).1.1 (suc (suc (suc zero)))) (e x).1.1 @@ -188,94 +188,94 @@ inhPropContr (A : U) (x : prop A) (y : ishinh_UU A) : isContr A = y (isContr A, -- P1 <-> P2 <-> P3 -P1 (A B : U) (F : A -> B) : U = (x y : A) -> isEquiv (Id A x y) (Id B (F x) (F y)) (mapOnPath A B F x y) -P2 (A B : U) (F : A -> B) : U = (x : A) -> isContr ((y : A) * Id B (F x) (F y)) -P3 (A B : U) (F : A -> B) : U = (x : B) -> prop ((y : A) * Id B x (F y)) +P1 (A B : U) (F : A -> B) : U = (x y : A) -> isEquiv (Path A x y) (Path B (F x) (F y)) (mapOnPath A B F x y) +P2 (A B : U) (F : A -> B) : U = (x : A) -> isContr ((y : A) * Path B (F x) (F y)) +P3 (A B : U) (F : A -> B) : U = (x : B) -> prop ((y : A) * Path B x (F y)) propP1 (A B : U) (F : A -> B) : prop (P1 A B F) - = propPi A (\(x : A) -> (y : A) -> isEquiv (Id A x y) (Id B (F x) (F y)) (mapOnPath A B F x y)) - (\(x : A) -> propPi A (\(y : A) -> isEquiv (Id A x y) (Id B (F x) (F y)) (mapOnPath A B F x y)) - (\(y : A) -> propIsEquiv (Id A x y) (Id B (F x) (F y)) (mapOnPath A B F x y))) + = propPi A (\(x : A) -> (y : A) -> isEquiv (Path A x y) (Path B (F x) (F y)) (mapOnPath A B F x y)) + (\(x : A) -> propPi A (\(y : A) -> isEquiv (Path A x y) (Path B (F x) (F y)) (mapOnPath A B F x y)) + (\(y : A) -> propIsEquiv (Path A x y) (Path B (F x) (F y)) (mapOnPath A B F x y))) propP2 (A B : U) (F : A -> B) : prop (P2 A B F) - = propPi A (\(x : A) -> isContr ((y : A) * Id B (F x) (F y))) - (\(x : A) -> propIsContr ((y : A) * Id B (F x) (F y))) + = propPi A (\(x : A) -> isContr ((y : A) * Path B (F x) (F y))) + (\(x : A) -> propIsContr ((y : A) * Path B (F x) (F y))) propP3 (A B : U) (F : A -> B) : prop (P3 A B F) - = propPi B (\(x : B) -> prop ((y : A) * Id B x (F y))) - (\(x : B) -> propIsProp ((y : A) * Id B x (F y))) + = propPi B (\(x : B) -> prop ((y : A) * Path B x (F y))) + (\(x : B) -> propIsProp ((y : A) * Path B x (F y))) -isoIdProp (A B : U) (AProp : prop A) (BProp : prop B) (F : A -> B) (G : B -> A) : Id U A B = - isoId A B F G (\(y : B) -> BProp (F (G y)) y) (\(x : A) -> AProp (G (F x)) x) +isoPathProp (A B : U) (AProp : prop A) (BProp : prop B) (F : A -> B) (G : B -> A) : Path U A B = + isoPath A B F G (\(y : B) -> BProp (F (G y)) y) (\(x : A) -> AProp (G (F x)) x) equivProp (A B : U) (AProp : prop A) (BProp : prop B) (F : A -> B) (G : B -> A) : equiv A B = (F, gradLemma A B F G (\(y : B) -> BProp (F (G y)) y) (\(x : A) -> AProp (G (F x)) x)) -isContrProp (A : U) (p : isContr A) (x y : A) : Id A x y = compId A x p.1 y ( p.2 x @ -i) (p.2 y) +isContrProp (A : U) (p : isContr A) (x y : A) : Path A x y = compPath A x p.1 y ( p.2 x @ -i) (p.2 y) -lem2 (A : U) (x : A) : isContr ((y : A) * Id A x y) +lem2 (A : U) (x : A) : isContr ((y : A) * Path A x y) = ( (x, refl A x) - , \(z : (y : A) * Id A x y) -> - J A x (\(y : A) -> \(p : Id A x y) -> Id ((y : A) * Id A x y) (x, refl A x) (y, p)) - (refl ((y : A) * Id A x y) (x, refl A x)) z.1 z.2 + , \(z : (y : A) * Path A x y) -> + J A x (\(y : A) -> \(p : Path A x y) -> Path ((y : A) * Path A x y) (x, refl A x) (y, p)) + (refl ((y : A) * Path A x y) (x, refl A x)) z.1 z.2 ) -lem31192 (A : U) (P : A -> U) (aC : isContr A) : Id U (Sigma A P) (P aC.1) = - isoId (Sigma A P) (P aC.1) F G FG GF +lem31192 (A : U) (P : A -> U) (aC : isContr A) : Path U (Sigma A P) (P aC.1) = + isoPath (Sigma A P) (P aC.1) F G FG GF where F (a : Sigma A P) : P aC.1 = transport ( P ((aC.2 a.1) @ -i)) a.2 G (a : P aC.1) : Sigma A P = (aC.1, a) - FG (a : P aC.1) : Id (P aC.1) (transport ( P ((aC.2 aC.1) @ -i)) a) a = hole + FG (a : P aC.1) : Path (P aC.1) (transport ( P ((aC.2 aC.1) @ -i)) a) a = hole where - prf : Id (Id A aC.1 aC.1) (aC.2 aC.1) (<_> aC.1) = propSet A (isContrProp A aC) aC.1 aC.1 (aC.2 aC.1) (<_> aC.1) - hole1 : Id (P aC.1) (transport (<_> P aC.1) a) a = transRefl (P aC.1) a - hole : Id (P aC.1) (transport ( P ((aC.2 aC.1) @ -i)) a) a - = transport ( Id (P aC.1) (transport ( P ((prf @ -i) @ -j)) a) a) hole1 - GF (a : Sigma A P) : Id (Sigma A P) (aC.1, transport ( P ((aC.2 a.1) @ -i)) a.2) a = hole + prf : Path (Path A aC.1 aC.1) (aC.2 aC.1) (<_> aC.1) = propSet A (isContrProp A aC) aC.1 aC.1 (aC.2 aC.1) (<_> aC.1) + hole1 : Path (P aC.1) (transport (<_> P aC.1) a) a = transRefl (P aC.1) a + hole : Path (P aC.1) (transport ( P ((aC.2 aC.1) @ -i)) a) a + = transport ( Path (P aC.1) (transport ( P ((prf @ -i) @ -j)) a) a) hole1 + GF (a : Sigma A P) : Path (Sigma A P) (aC.1, transport ( P ((aC.2 a.1) @ -i)) a.2) a = hole where - hole2 : Id A aC.1 a.1 = aC.2 a.1 - hole1 : IdP ( P (hole2 @ i)) (transport ( P ((aC.2 a.1) @ -i)) a.2) a.2 + hole2 : Path A aC.1 a.1 = aC.2 a.1 + hole1 : PathP ( P (hole2 @ i)) (transport ( P ((aC.2 a.1) @ -i)) a.2) a.2 = comp ( P (hole2 @ i \/ -j)) a.2 [(i=1) -> <_> a.2] - hole : Id (Sigma A P) (aC.1, transport ( P ((aC.2 a.1) @ -i)) a.2) a - = transport ( (lemIdSig A P (aC.1, transport ( P ((aC.2 a.1) @ -i)) a.2) a) @ -i) (hole2, hole1) + hole : Path (Sigma A P) (aC.1, transport ( P ((aC.2 a.1) @ -i)) a.2) a + = transport ( (lemPathSig A P (aC.1, transport ( P ((aC.2 a.1) @ -i)) a.2) a) @ -i) (hole2, hole1) total (A : U) (P Q : A -> U) (f : (x : A) -> P x -> Q x) (a : (x : A) * P x) : (x : A) * Q x = (a.1, f a.1 a.2) -ex210 (A : U) (B : A -> U) (C : (x : A) -> B x -> U) : Id U ((x : A) * (y : B x) * C x y) ((x : Sigma A B) * C x.1 x.2) - = isoId ((x : A) * (y : B x) * C x y) ((x : Sigma A B) * C x.1 x.2) F G FG GF +ex210 (A : U) (B : A -> U) (C : (x : A) -> B x -> U) : Path U ((x : A) * (y : B x) * C x y) ((x : Sigma A B) * C x.1 x.2) + = isoPath ((x : A) * (y : B x) * C x y) ((x : Sigma A B) * C x.1 x.2) F G FG GF where F (a : (x : A) * (y : B x) * C x y) : ((x : Sigma A B) * C x.1 x.2) = ((a.1, a.2.1), a.2.2) G (a : (x : Sigma A B) * C x.1 x.2) : ((x : A) * (y : B x) * C x y) = (a.1.1, (a.1.2, a.2)) - FG (a : (x : Sigma A B) * C x.1 x.2) : Id ((x : Sigma A B) * C x.1 x.2) (F (G a)) a = <_> a - GF (a : (x : A) * (y : B x) * C x y) : Id ((x : A) * (y : B x) * C x y) (G (F a)) a = <_> a + FG (a : (x : Sigma A B) * C x.1 x.2) : Path ((x : Sigma A B) * C x.1 x.2) (F (G a)) a = <_> a + GF (a : (x : A) * (y : B x) * C x y) : Path ((x : A) * (y : B x) * C x y) (G (F a)) a = <_> a -cSigma (A : U) (B : U) (C : A -> B -> U) : Id U ((x : A) * (y : B) * C x y) ((y : B) * (x : A) * C x y) = - isoId ((x : A) * (y : B) * C x y) ((y : B) * (x : A) * C x y) +cSigma (A : U) (B : U) (C : A -> B -> U) : Path U ((x : A) * (y : B) * C x y) ((y : B) * (x : A) * C x y) = + isoPath ((x : A) * (y : B) * C x y) ((y : B) * (x : A) * C x y) (\(a : (x : A) * (y : B) * C x y) -> (a.2.1, (a.1, a.2.2))) (\(a : (y : B) * (x : A) * C x y) -> (a.2.1, (a.1, a.2.2))) (\(a : (y : B) * (x : A) * C x y) -> <_> a) (\(a : (x : A) * (y : B) * C x y) -> <_> a) th476 (A : U) (P Q : A -> U) (f : (x : A) -> P x -> Q x) (x : A) (v : Q x) - : Id U (fiber (Sigma A P) (Sigma A Q) (total A P Q f) (x, v)) (fiber (P x) (Q x) (f x) v) + : Path U (fiber (Sigma A P) (Sigma A Q) (total A P Q f) (x, v)) (fiber (P x) (Q x) (f x) v) = hole where - A1 : U = (w : Sigma A P) * Id (Sigma A Q) (x, v) (total A P Q f w) - A2 : U = (a : A) * (u : P a) * Id (Sigma A Q) (x, v) (a, f a u) - A3 : U = (a : A) * (u : P a) * (p : Id A x a) * IdP ( Q (p @ i)) v (f a u) - A4 : U = (a : A) * (p : Id A x a) * (u : P a) * IdP ( Q (p @ i)) v (f a u) - A5 : U = (w : (a : A) * Id A x a) * (u : P w.1) * IdP ( Q (w.2 @ i)) v (f w.1 u) - A6 : U = (u : P x) * Id (Q x) v (f x u) - E12 : Id U A1 A2 = (ex210 A P (\(a : A) -> \(b : P a) -> Id (Sigma A Q) (x, v) (a, f a b))) @ -i - E23 : Id U A2 A3 = (a : A) * (u : P a) * (lemIdSig A Q (x, v) (a, f a u)) @ i - E34 : Id U A3 A4 = (a : A) * (cSigma (P a) (Id A x a) (\(u : P a) -> \(p : Id A x a) -> IdP ( Q (p @ j)) v (f a u))) @ i - E45 : Id U A4 A5 = ex210 A (Id A x) (\(a : A) -> \(p : Id A x a) -> (u : P a) * IdP ( Q (p @ i)) v (f a u)) - E56 : Id U A5 A6 = lem31192 ((a : A) * Id A x a) (\(w : (a : A) * Id A x a) -> (u : P w.1) * IdP ( Q (w.2 @ i)) v (f w.1 u)) + A1 : U = (w : Sigma A P) * Path (Sigma A Q) (x, v) (total A P Q f w) + A2 : U = (a : A) * (u : P a) * Path (Sigma A Q) (x, v) (a, f a u) + A3 : U = (a : A) * (u : P a) * (p : Path A x a) * PathP ( Q (p @ i)) v (f a u) + A4 : U = (a : A) * (p : Path A x a) * (u : P a) * PathP ( Q (p @ i)) v (f a u) + A5 : U = (w : (a : A) * Path A x a) * (u : P w.1) * PathP ( Q (w.2 @ i)) v (f w.1 u) + A6 : U = (u : P x) * Path (Q x) v (f x u) + E12 : Path U A1 A2 = (ex210 A P (\(a : A) -> \(b : P a) -> Path (Sigma A Q) (x, v) (a, f a b))) @ -i + E23 : Path U A2 A3 = (a : A) * (u : P a) * (lemPathSig A Q (x, v) (a, f a u)) @ i + E34 : Path U A3 A4 = (a : A) * (cSigma (P a) (Path A x a) (\(u : P a) -> \(p : Path A x a) -> PathP ( Q (p @ j)) v (f a u))) @ i + E45 : Path U A4 A5 = ex210 A (Path A x) (\(a : A) -> \(p : Path A x a) -> (u : P a) * PathP ( Q (p @ i)) v (f a u)) + E56 : Path U A5 A6 = lem31192 ((a : A) * Path A x a) (\(w : (a : A) * Path A x a) -> (u : P w.1) * PathP ( Q (w.2 @ i)) v (f w.1 u)) (lem2 A x) - hole : Id U A1 A6 = compId U A1 A2 A6 E12 (compId U A2 A3 A6 E23 (compId U A3 A4 A6 E34 (compId U A4 A5 A6 E45 E56))) + hole : Path U A1 A6 = compPath U A1 A2 A6 E12 (compPath U A2 A3 A6 E23 (compPath U A3 A4 A6 E34 (compPath U A4 A5 A6 E45 E56))) thm477 (A : U) (P Q : A -> U) (f : (x : A) -> P x -> Q x) - : Id U ((x : A) -> isEquiv (P x) (Q x) (f x)) (isEquiv ((x : A) * P x) ((x : A) * Q x) (total A P Q f)) + : Path U ((x : A) -> isEquiv (P x) (Q x) (f x)) (isEquiv ((x : A) * P x) ((x : A) * Q x) (total A P Q f)) = hole where AProp : prop ((x : A) -> isEquiv (P x) (Q x) (f x)) @@ -285,56 +285,56 @@ thm477 (A : U) (P Q : A -> U) (f : (x : A) -> P x -> Q x) = transport ( isContr (th476 A P Q f y.1 y.2 @ -i)) (a y.1 y.2) G (a : isEquiv ((x : A) * P x) ((x : A) * Q x) (total A P Q f)) (x : A) (y : Q x) : isContr (fiber (P x) (Q x) (f x) y) = transport ( isContr (th476 A P Q f x y @ i)) (a (x, y)) - hole : Id U ((x : A) -> isEquiv (P x) (Q x) (f x)) (isEquiv ((x : A) * P x) ((x : A) * Q x) (total A P Q f)) - = isoIdProp ((x : A) -> isEquiv (P x) (Q x) (f x)) (isEquiv ((x : A) * P x) ((x : A) * Q x) (total A P Q f)) AProp BProp F G + hole : Path U ((x : A) -> isEquiv (P x) (Q x) (f x)) (isEquiv ((x : A) * P x) ((x : A) * Q x) (total A P Q f)) + = isoPathProp ((x : A) -> isEquiv (P x) (Q x) (f x)) (isEquiv ((x : A) * P x) ((x : A) * Q x) (total A P Q f)) AProp BProp F G -F12 (A B : U) (F : A -> B) (p1 : P1 A B F) (x : A) : isContr ((y : A) * Id B (F x) (F y)) = hole +F12 (A B : U) (F : A -> B) (p1 : P1 A B F) (x : A) : isContr ((y : A) * Path B (F x) (F y)) = hole where - hole3 : ((y : A) * Id A x y) -> ((y : A) * Id B (F x) (F y)) - = total A (\(y : A) -> Id A x y) (\(y : A) -> Id B (F x) (F y)) (mapOnPath A B F x) - hole2 : isEquiv ((y : A) * Id A x y) ((y : A) * Id B (F x) (F y)) hole3 - = transport (thm477 A (\(y : A) -> Id A x y) (\(y : A) -> Id B (F x) (F y)) (mapOnPath A B F x)) (p1 x) - hole4 : Id U ((y : A) * Id A x y) ((y : A) * Id B (F x) (F y)) = equivId ((y : A) * Id A x y) ((y : A) * Id B (F x) (F y)) hole3 hole2 - hole : isContr ((y : A) * Id B (F x) (F y)) = transport ( isContr (hole4@i)) (lem2 A x) + hole3 : ((y : A) * Path A x y) -> ((y : A) * Path B (F x) (F y)) + = total A (\(y : A) -> Path A x y) (\(y : A) -> Path B (F x) (F y)) (mapOnPath A B F x) + hole2 : isEquiv ((y : A) * Path A x y) ((y : A) * Path B (F x) (F y)) hole3 + = transport (thm477 A (\(y : A) -> Path A x y) (\(y : A) -> Path B (F x) (F y)) (mapOnPath A B F x)) (p1 x) + hole4 : Path U ((y : A) * Path A x y) ((y : A) * Path B (F x) (F y)) = equivPath ((y : A) * Path A x y) ((y : A) * Path B (F x) (F y)) hole3 hole2 + hole : isContr ((y : A) * Path B (F x) (F y)) = transport ( isContr (hole4@i)) (lem2 A x) contrEquiv (A B : U) (aC : isContr A) (bC : isContr B) : equiv A B = (\(_ : A) -> bC.1, gradLemma A B (\(_ : A) -> bC.1) (\(_ : B) -> aC.1) (\(x : B) -> bC.2 x) (\(x : A) -> aC.2 x)) -F21 (A B : U) (F : A -> B) (p2 : P2 A B F) (x y : A) : isEquiv (Id A x y) (Id B (F x) (F y)) (mapOnPath A B F x y) = hole3 y +F21 (A B : U) (F : A -> B) (p2 : P2 A B F) (x y : A) : isEquiv (Path A x y) (Path B (F x) (F y)) (mapOnPath A B F x y) = hole3 y where - hole0 : ((y : A) * Id A x y) -> ((y : A) * Id B (F x) (F y)) - = total A (\(y : A) -> Id A x y) (\(y : A) -> Id B (F x) (F y)) (mapOnPath A B F x) - hole2 : isEquiv ((y : A) * Id A x y) ((y : A) * Id B (F x) (F y)) hole0 - = (equivProp ((y : A) * Id A x y) ((y : A) * Id B (F x) (F y)) - (isContrProp ((y : A) * Id A x y) (lem2 A x)) - (isContrProp ((y : A) * Id B (F x) (F y)) (p2 x)) - hole0 (\(_ : (y : A) * Id B (F x) (F y)) -> (x, <_> x))).2 - hole4 : Id U ((y : A) -> isEquiv (Id A x y) (Id B (F x) (F y)) (mapOnPath A B F x y)) (isEquiv ((y : A) * Id A x y) ((y : A) * Id B (F x) (F y)) hole0) - = thm477 A (\(y : A) -> Id A x y) (\(y : A) -> Id B (F x) (F y)) (mapOnPath A B F x) - hole3 : (y : A) -> isEquiv (Id A x y) (Id B (F x) (F y)) (mapOnPath A B F x y) + hole0 : ((y : A) * Path A x y) -> ((y : A) * Path B (F x) (F y)) + = total A (\(y : A) -> Path A x y) (\(y : A) -> Path B (F x) (F y)) (mapOnPath A B F x) + hole2 : isEquiv ((y : A) * Path A x y) ((y : A) * Path B (F x) (F y)) hole0 + = (equivProp ((y : A) * Path A x y) ((y : A) * Path B (F x) (F y)) + (isContrProp ((y : A) * Path A x y) (lem2 A x)) + (isContrProp ((y : A) * Path B (F x) (F y)) (p2 x)) + hole0 (\(_ : (y : A) * Path B (F x) (F y)) -> (x, <_> x))).2 + hole4 : Path U ((y : A) -> isEquiv (Path A x y) (Path B (F x) (F y)) (mapOnPath A B F x y)) (isEquiv ((y : A) * Path A x y) ((y : A) * Path B (F x) (F y)) hole0) + = thm477 A (\(y : A) -> Path A x y) (\(y : A) -> Path B (F x) (F y)) (mapOnPath A B F x) + hole3 : (y : A) -> isEquiv (Path A x y) (Path B (F x) (F y)) (mapOnPath A B F x y) = transport ( hole4 @ -i) hole2 -F32 (A B : U) (F : A -> B) (p3 : P3 A B F) (x : A) : isContr ((y : A) * Id B (F x) (F y)) - = ((x, refl B (F x)), \(q : ((y : A) * Id B (F x) (F y))) -> p3 (F x) (x, refl B (F x)) q) +F32 (A B : U) (F : A -> B) (p3 : P3 A B F) (x : A) : isContr ((y : A) * Path B (F x) (F y)) + = ((x, refl B (F x)), \(q : ((y : A) * Path B (F x) (F y))) -> p3 (F x) (x, refl B (F x)) q) -F23 (A B : U) (F : A -> B) (p2 : P2 A B F) (x : B) (a b : (y : A) * Id B x (F y)) : Id ((y : A) * Id B x (F y)) a b = hole +F23 (A B : U) (F : A -> B) (p2 : P2 A B F) (x : B) (a b : (y : A) * Path B x (F y)) : Path ((y : A) * Path B x (F y)) a b = hole where - hole2 : Id ((y : A) * Id B (F a.1) (F y)) (a.1, refl B (F a.1)) (b.1, compId B (F a.1) x (F b.1) ( a.2 @ -i) b.2) - = isContrProp ((y : A) * Id B (F a.1) (F y)) (p2 (a.1)) (a.1, refl B (F a.1)) (b.1, compId B (F a.1) x (F b.1) ( a.2 @ -i) b.2) - hole3 : (Id ((y : A) * Id B x (F y)) a (b.1, compId B x x (F b.1) (<_> x) b.2)) + hole2 : Path ((y : A) * Path B (F a.1) (F y)) (a.1, refl B (F a.1)) (b.1, compPath B (F a.1) x (F b.1) ( a.2 @ -i) b.2) + = isContrProp ((y : A) * Path B (F a.1) (F y)) (p2 (a.1)) (a.1, refl B (F a.1)) (b.1, compPath B (F a.1) x (F b.1) ( a.2 @ -i) b.2) + hole3 : (Path ((y : A) * Path B x (F y)) a (b.1, compPath B x x (F b.1) (<_> x) b.2)) = transport - ( Id ((y : A) * Id B (a.2 @ -i) (F y)) (a.1, a.2 @ -i \/ j) (b.1, compId B (a.2 @ -i) x (F b.1) ( a.2 @ -i /\ -j) b.2)) + ( Path ((y : A) * Path B (a.2 @ -i) (F y)) (a.1, a.2 @ -i \/ j) (b.1, compPath B (a.2 @ -i) x (F b.1) ( a.2 @ -i /\ -j) b.2)) hole2 - hole : Id ((y : A) * Id B x (F y)) a b + hole : Path ((y : A) * Path B x (F y)) a b = transport - ( Id ((y : A) * Id B x (F y)) a (b.1, (lemReflComp B x (F b.1) b.2) @ i)) + ( Path ((y : A) * Path B x (F y)) a (b.1, (lemReflComp B x (F b.1) b.2) @ i)) hole3 -E12 (A B : U) (F : A -> B) : Id U (P1 A B F) (P2 A B F) = isoIdProp (P1 A B F) (P2 A B F) (propP1 A B F) (propP2 A B F) (F12 A B F) (F21 A B F) +E12 (A B : U) (F : A -> B) : Path U (P1 A B F) (P2 A B F) = isoPathProp (P1 A B F) (P2 A B F) (propP1 A B F) (propP2 A B F) (F12 A B F) (F21 A B F) opaque E12 -E23 (A B : U) (F : A -> B) : Id U (P2 A B F) (P3 A B F) = isoIdProp (P2 A B F) (P3 A B F) (propP2 A B F) (propP3 A B F) (F23 A B F) (F32 A B F) +E23 (A B : U) (F : A -> B) : Path U (P2 A B F) (P3 A B F) = isoPathProp (P2 A B F) (P3 A B F) (propP2 A B F) (propP3 A B F) (F23 A B F) (F32 A B F) opaque E23 -E13 (A B : U) (F : A -> B) : Id U (P1 A B F) (P3 A B F) = compId U (P1 A B F) (P2 A B F) (P3 A B F) (E12 A B F) (E23 A B F) +E13 (A B : U) (F : A -> B) : Path U (P1 A B F) (P3 A B F) = compPath U (P1 A B F) (P2 A B F) (P3 A B F) (E12 A B F) (E23 A B F) opaque E13 -- torsor @@ -353,11 +353,11 @@ actionEquiv (A : U) (shift : equiv A A) : (y : Z) -> isEquiv A A (\(x : A) -> ac inl n -> hole n where hole : (n : nat) -> isEquiv A A (\(x : A) -> action A shift x (inl n)) = split zero -> hole0 - where hole0 : isEquiv A A (\(x : A) -> (shift.2 x).1.1) = isEquivComp' A A A (\(x : A) -> x) shift.1 (isEquivId A) shift.2 + where hole0 : isEquiv A A (\(x : A) -> (shift.2 x).1.1) = isEquivComp' A A A (\(x : A) -> x) shift.1 (isEquivPath A) shift.2 suc n -> isEquivComp' A A A (\(x : A) -> action A shift x (inl n)) shift.1 (hole n) shift.2 inr n -> hole n where hole : (n : nat) -> isEquiv A A (\(x : A) -> action A shift x (inr n)) = split - zero -> isEquivId A + zero -> isEquivPath A suc n -> isEquivComp A A A (\(x : A) -> action A shift x (inr n)) shift.1 (hole n) shift.2 BZ : U = (A : U) * (a : ishinh_UU A) @@ -372,10 +372,10 @@ BZS (A : BZ) : BZSet A -> BZSet A = (BZShift A).1 BZP (A : BZ) (a : BZSet A) : BZSet A = ((BZShift A).2 a).1.1 BZEquiv (A : BZ) : (x : BZSet A) -> isEquiv Z (BZSet A) (BZAction A x) = A.2.2.2 BZASet (x : BZ) : set (BZSet x) = BZNE x (set (BZSet x), setIsProp (BZSet x)) - (\(a : BZSet x) -> transport ( set (equivId Z (BZSet x) (BZAction x a) (BZEquiv x a) @ i)) ZSet) + (\(a : BZSet x) -> transport ( set (equivPath Z (BZSet x) (BZAction x a) (BZEquiv x a) @ i)) ZSet) -- Two Z-torsors are equal if their underlying sets are equal in a way compatible with the actions -lemBZ (BA BB : BZ) : equiv ((p : Id U (BZSet BA) (BZSet BB)) * IdP ( p@i -> p@i) (BZS BA) (BZS BB)) (Id BZ BA BB) = hole +lemBZ (BA BB : BZ) : equiv ((p : Path U (BZSet BA) (BZSet BB)) * PathP ( p@i -> p@i) (BZS BA) (BZS BB)) (Path BZ BA BB) = hole where A : U = BA.1 a : ishinh_UU A = BA.2.1 @@ -385,44 +385,44 @@ lemBZ (BA BB : BZ) : equiv ((p : Id U (BZSet BA) (BZSet BB)) * IdP ( p@i -> p b : ishinh_UU B = BB.2.1 BShift : equiv B B = BB.2.2.1 BEquiv : (x : B) -> isEquiv Z B (BZAction BB x) = BB.2.2.2 - F (w : (p : Id U A B) * IdP ( p@i -> p@i) (BZS BA) (BZS BB)) : Id BZ BA BB = hole + F (w : (p : Path U A B) * PathP ( p@i -> p@i) (BZS BA) (BZS BB)) : Path BZ BA BB = hole where - p : Id U A B = w.1 - pS : IdP ( p@i -> p@i) (BZS BA) (BZS BB) = w.2 - pNE : IdP ( ishinh_UU (p@i)) a b - = lemIdPProp (ishinh_UU A) (ishinh_UU B) (propishinh A) ( ishinh_UU (p@i)) a b - pShift : IdP ( equiv (p@i) (p@i)) AShift BShift = + p : Path U A B = w.1 + pS : PathP ( p@i -> p@i) (BZS BA) (BZS BB) = w.2 + pNE : PathP ( ishinh_UU (p@i)) a b + = lemPathPProp (ishinh_UU A) (ishinh_UU B) (propishinh A) ( ishinh_UU (p@i)) a b + pShift : PathP ( equiv (p@i) (p@i)) AShift BShift = ( pS @ i - , (lemIdPProp + , (lemPathPProp (isEquiv A A AShift.1) (isEquiv B B BShift.1) (propIsEquiv A A AShift.1) ( isEquiv (p@j) (p@j) (pS@j)) AShift.2 BShift.2) @ i ) - pEquiv : IdP ( (x : p@i) -> isEquiv Z (p@i) (action (p@i) (pShift@i) x)) AEquiv BEquiv - = lemIdPProp + pEquiv : PathP ( (x : p@i) -> isEquiv Z (p@i) (action (p@i) (pShift@i) x)) AEquiv BEquiv + = lemPathPProp ((x : A) -> isEquiv Z A (action A AShift x)) ((x : B) -> isEquiv Z B (action B BShift x)) (propPi A (\(x : A) -> isEquiv Z A (action A AShift x)) (\(x : A) -> propIsEquiv Z A (action A AShift x))) ( (x : p@i) -> isEquiv Z (p@i) (action (p@i) (pShift@i) x)) AEquiv BEquiv - hole : Id BZ BA BB = (p@i, (pNE@i, (pShift@i, pEquiv@i))) - G (q : Id BZ BA BB) : (p : Id U A B) * IdP ( p@i -> p@i) (BZS BA) (BZS BB) = ( BZSet (q @ i), (BZShift (q@i)).1) - GF (w : (p : Id U A B) * IdP ( p@i -> p@i) (BZS BA) (BZS BB)) - : Id ((p : Id U A B) * IdP ( p@i -> p@i) (BZS BA) (BZS BB)) ( BZSet (F w @ i), (BZShift (F w @ i)).1) w + hole : Path BZ BA BB = (p@i, (pNE@i, (pShift@i, pEquiv@i))) + G (q : Path BZ BA BB) : (p : Path U A B) * PathP ( p@i -> p@i) (BZS BA) (BZS BB) = ( BZSet (q @ i), (BZShift (q@i)).1) + GF (w : (p : Path U A B) * PathP ( p@i -> p@i) (BZS BA) (BZS BB)) + : Path ((p : Path U A B) * PathP ( p@i -> p@i) (BZS BA) (BZS BB)) ( BZSet (F w @ i), (BZShift (F w @ i)).1) w = <_> w - FG (q : Id BZ BA BB) : Id (Id BZ BA BB) (F ( BZSet (q@i), (BZShift (q@i)).1)) q = hole + FG (q : Path BZ BA BB) : Path (Path BZ BA BB) (F ( BZSet (q@i), (BZShift (q@i)).1)) q = hole where - p : Id U A B = BZSet (q@i) - p2 : IdP ( p@i -> p@i) (BZS BA) (BZS BB) = (BZShift (q@i)).1 - q2 : Id BZ BA BB = F (p, p2) - pNE : Id (IdP ( ishinh_UU (p@i)) a b) ( BZNE (q2@i)) ( BZNE (q@i)) - = lemIdPSet (ishinh_UU A) (ishinh_UU B) (propSet (ishinh_UU A) (propishinh A)) ( ishinh_UU (p@i)) a b ( BZNE (q2@i)) ( BZNE (q@i)) - pShift : Id (IdP ( equiv (p@i) (p@i)) AShift BShift) ( BZShift (q2@i)) ( BZShift (q@i)) = + p : Path U A B = BZSet (q@i) + p2 : PathP ( p@i -> p@i) (BZS BA) (BZS BB) = (BZShift (q@i)).1 + q2 : Path BZ BA BB = F (p, p2) + pNE : Path (PathP ( ishinh_UU (p@i)) a b) ( BZNE (q2@i)) ( BZNE (q@i)) + = lemPathPSet (ishinh_UU A) (ishinh_UU B) (propSet (ishinh_UU A) (propishinh A)) ( ishinh_UU (p@i)) a b ( BZNE (q2@i)) ( BZNE (q@i)) + pShift : Path (PathP ( equiv (p@i) (p@i)) AShift BShift) ( BZShift (q2@i)) ( BZShift (q@i)) = ( p2 @ i - , (lemIdPSet + , (lemPathPSet (isEquiv A A AShift.1) (isEquiv B B BShift.1) (propSet (isEquiv A A AShift.1) (propIsEquiv A A AShift.1)) @@ -430,8 +430,8 @@ lemBZ (BA BB : BZ) : equiv ((p : Id U (BZSet BA) (BZSet BB)) * IdP ( p@i -> p AShift.2 BShift.2 ( (BZShift (q2@i)).2) ( (BZShift (q@i)).2)) @ j @ i ) - pEquiv : IdP ( IdP ( (x : p@i) -> isEquiv Z (p@i) (action (p@i) (pShift@j@i) x)) AEquiv BEquiv) ( BZEquiv (q2@i)) ( BZEquiv (q@i)) - = lemIdPSet2 + pEquiv : PathP ( PathP ( (x : p@i) -> isEquiv Z (p@i) (action (p@i) (pShift@j@i) x)) AEquiv BEquiv) ( BZEquiv (q2@i)) ( BZEquiv (q@i)) + = lemPathPSet2 ((x : A) -> isEquiv Z A (action A AShift x)) ((x : B) -> isEquiv Z B (action B BShift x)) (propSet ((x : A) -> isEquiv Z A (action A AShift x)) @@ -441,31 +441,31 @@ lemBZ (BA BB : BZ) : equiv ((p : Id U (BZSet BA) (BZSet BB)) * IdP ( p@i -> p ( (x : p@i) -> isEquiv Z (p@i) (action (p@i) (pShift@1@i) x)) ( (x : p@i) -> isEquiv Z (p@i) (action (p@i) (pShift@j@i) x)) AEquiv BEquiv ( BZEquiv (q2@i)) ( BZEquiv (q@i)) - hole : Id (Id BZ BA BB) q2 q = (p@i, (pNE@j@i, (pShift@j@i, pEquiv@j@i))) - hole : equiv ((p : Id U (BZSet BA) (BZSet BB)) * IdP ( p@i -> p@i) (BZS BA) (BZS BB)) (Id BZ BA BB) - = (F, gradLemma ((p : Id U (BZSet BA) (BZSet BB)) * IdP ( p@i -> p@i) (BZS BA) (BZS BB)) (Id BZ BA BB) F G FG GF) + hole : Path (Path BZ BA BB) q2 q = (p@i, (pNE@j@i, (pShift@j@i, pEquiv@j@i))) + hole : equiv ((p : Path U (BZSet BA) (BZSet BB)) * PathP ( p@i -> p@i) (BZS BA) (BZS BB)) (Path BZ BA BB) + = (F, gradLemma ((p : Path U (BZSet BA) (BZSet BB)) * PathP ( p@i -> p@i) (BZS BA) (BZS BB)) (Path BZ BA BB) F G FG GF) -lem2 (A : U) (shift : equiv A A) (a : A) : (x : Z) -> Id A (shift.1 (action A shift a x)) (action A shift a (sucZ x)) = split +lem2 (A : U) (shift : equiv A A) (a : A) : (x : Z) -> Path A (shift.1 (action A shift a x)) (action A shift a (sucZ x)) = split inl n -> lem2Aux n where - lem2Aux : (n : nat) -> Id A (shift.1 (action A shift a (inl n))) (action A shift a (sucZ (inl n))) = split + lem2Aux : (n : nat) -> Path A (shift.1 (action A shift a (inl n))) (action A shift a (sucZ (inl n))) = split zero -> retEq A A shift a suc n -> retEq A A shift (action A shift a (inl n)) inr n -> lem2Aux n where - lem2Aux : (n : nat) -> Id A (shift.1 (action A shift a (inr n))) (action A shift a (sucZ (inr n))) = split + lem2Aux : (n : nat) -> Path A (shift.1 (action A shift a (inr n))) (action A shift a (sucZ (inr n))) = split zero -> <_> shift.1 a suc n -> <_> shift.1 (action A shift a (inr (suc n))) -lem2' (A : U) (shift : equiv A A) (a : A) : (x : Z) -> Id A (shift.2 (action A shift a x)).1.1 (action A shift a (predZ x)) = split +lem2' (A : U) (shift : equiv A A) (a : A) : (x : Z) -> Path A (shift.2 (action A shift a x)).1.1 (action A shift a (predZ x)) = split inl n -> lem2Aux n where - lem2Aux : (n : nat) -> Id A (shift.2 (action A shift a (inl n))).1.1 (action A shift a (predZ (inl n))) = split + lem2Aux : (n : nat) -> Path A (shift.2 (action A shift a (inl n))).1.1 (action A shift a (predZ (inl n))) = split zero -> <_> action A shift a (predZ (inl zero)) suc n -> <_> action A shift a (predZ (inl (suc n))) inr n -> lem2Aux n where - lem2Aux : (n : nat) -> Id A (shift.2 (action A shift a (inr n))).1.1 (action A shift a (predZ (inr n))) = split + lem2Aux : (n : nat) -> Path A (shift.2 (action A shift a (inr n))).1.1 (action A shift a (predZ (inr n))) = split zero -> <_> action A shift a (inl zero) suc n -> secEq A A shift (action A shift a (inr n)) @@ -473,229 +473,229 @@ ZBZ : BZ = (Z, (hinhpr Z zeroZ, (ZShift, ZEquiv))) where ZShift : equiv Z Z = (sucZ, gradLemma Z Z sucZ predZ sucpredZ predsucZ) plus : Z -> Z -> Z = action Z ZShift - plusCommZero : (y : Z) -> Id Z (plus zeroZ y) (plus y zeroZ) = split + plusCommZero : (y : Z) -> Path Z (plus zeroZ y) (plus y zeroZ) = split inl u -> hole u - where hole : (n : nat) -> Id Z (plus zeroZ (inl n)) (inl n) = split + where hole : (n : nat) -> Path Z (plus zeroZ (inl n)) (inl n) = split zero -> <_> inl zero suc n -> predZ (hole n @ i) inr u -> hole u - where hole : (n : nat) -> Id Z (plus zeroZ (inr n)) (inr n) = split + where hole : (n : nat) -> Path Z (plus zeroZ (inr n)) (inr n) = split zero -> <_> inr zero suc n -> sucZ (plusCommZero (inr n) @ i) - plusCommSuc (x : Z) : (y : Z) -> Id Z (plus (sucZ x) y) (plus x (sucZ y)) = split + plusCommSuc (x : Z) : (y : Z) -> Path Z (plus (sucZ x) y) (plus x (sucZ y)) = split inl u -> hole u - where hole : (n : nat) -> Id Z (plus (sucZ x) (inl n)) (plus x (sucZ (inl n))) = split + where hole : (n : nat) -> Path Z (plus (sucZ x) (inl n)) (plus x (sucZ (inl n))) = split zero -> predsucZ x suc n -> hole0 n - where hole0 : (n : nat) -> Id Z (predZ (plus (sucZ x) (inl n))) (plus x (sucZ (predZ (inl n)))) = split + where hole0 : (n : nat) -> Path Z (predZ (plus (sucZ x) (inl n))) (plus x (sucZ (predZ (inl n)))) = split zero -> predZ (predsucZ x @ i) - suc n -> compId Z (predZ (predZ (plus (sucZ x) (inl n)))) + suc n -> compPath Z (predZ (predZ (plus (sucZ x) (inl n)))) (predZ (plus x (inl n))) (predZ (plus x (sucZ (predZ (inl n))))) ( predZ (hole0 n @ i)) ( predZ (plus x (sucpredZ (inl n) @ -i))) inr u -> hole u - where hole : (n : nat) -> Id Z (plus (sucZ x) (inr n)) (plus x (inr (suc n))) = split + where hole : (n : nat) -> Path Z (plus (sucZ x) (inr n)) (plus x (inr (suc n))) = split zero -> <_> sucZ x suc n -> sucZ (hole n @ i) - plusCommPred (x y : Z) : Id Z (plus (predZ x) y) (plus x (predZ y)) - = transport ( Id Z (plus (predZ x) (sucpredZ y @ i)) (plus (sucpredZ x @ i) (predZ y))) + plusCommPred (x y : Z) : Path Z (plus (predZ x) y) (plus x (predZ y)) + = transport ( Path Z (plus (predZ x) (sucpredZ y @ i)) (plus (sucpredZ x @ i) (predZ y))) ( plusCommSuc (predZ x) (predZ y) @ -i) - plusComm (x : Z) : (y : Z) -> Id Z (plus y x) (plus x y) = split + plusComm (x : Z) : (y : Z) -> Path Z (plus y x) (plus x y) = split inl u -> hole u - where hole : (n : nat) -> Id Z (plus (inl n) x) (plus x (inl n)) = split - zero -> compId Z (plus (inl zero) x) (plus (inr zero) (predZ x)) (predZ x) + where hole : (n : nat) -> Path Z (plus (inl n) x) (plus x (inl n)) = split + zero -> compPath Z (plus (inl zero) x) (plus (inr zero) (predZ x)) (predZ x) (plusCommPred (inr zero) x) (plusCommZero (predZ x)) - suc n -> compId Z (plus (inl (suc n)) x) (plus (inl n) (predZ x)) (predZ (plus x (inl n))) + suc n -> compPath Z (plus (inl (suc n)) x) (plus (inl n) (predZ x)) (predZ (plus x (inl n))) (plusCommPred (inl n) x) - (compId Z (plus (inl n) (predZ x)) (predZ (plus (inl n) x)) (predZ (plus x (inl n))) + (compPath Z (plus (inl n) (predZ x)) (predZ (plus (inl n) x)) (predZ (plus x (inl n))) ( lem2' Z ZShift (inl n) x @ -i) ( predZ (hole n @ i))) inr u -> hole u - where hole : (n : nat) -> Id Z (plus (inr n) x) (plus x (inr n)) = split + where hole : (n : nat) -> Path Z (plus (inr n) x) (plus x (inr n)) = split zero -> plusCommZero x - suc n -> compId Z (plus (inr (suc n)) x) (plus (inr n) (sucZ x)) (sucZ (plus x (inr n))) + suc n -> compPath Z (plus (inr (suc n)) x) (plus (inr n) (sucZ x)) (sucZ (plus x (inr n))) (plusCommSuc (inr n) x) - (compId Z (plus (inr n) (sucZ x)) (sucZ (plus (inr n) x)) (sucZ (plus x (inr n))) + (compPath Z (plus (inr n) (sucZ x)) (sucZ (plus (inr n) x)) (sucZ (plus x (inr n))) ( lem2 Z ZShift (inr n) x @ -i) ( sucZ (hole n @ i))) ZEquiv (x : Z) : isEquiv Z Z (plus x) = transport ( isEquiv Z Z (\(y : Z) -> plusComm x y @ i)) (actionEquiv Z ZShift x) -loopBZ : U = Id BZ ZBZ ZBZ -compBZ : loopBZ -> loopBZ -> loopBZ = compId BZ ZBZ ZBZ ZBZ +loopBZ : U = Path BZ ZBZ ZBZ +compBZ : loopBZ -> loopBZ -> loopBZ = compPath BZ ZBZ ZBZ ZBZ -transportIsoId (A B : U) (f : A -> B) (g : B -> A) - (s : (y : B) -> Id B (f (g y)) y) - (t : (x : A) -> Id A (g (f x)) x) +transportIsoPath (A B : U) (f : A -> B) (g : B -> A) + (s : (y : B) -> Path B (f (g y)) y) + (t : (x : A) -> Path A (g (f x)) x) (x : A) - : Id B (transport (isoId A B f g s t) x) (f x) - = compId B (transport (<_> B) (transport (<_> B) (f x))) (transport (<_> B) (f x)) (f x) + : Path B (transport (isoPath A B f g s t) x) (f x) + = compPath B (transport (<_> B) (transport (<_> B) (f x))) (transport (<_> B) (f x)) (f x) ( transport (<_> B) (transRefl B (f x) @ i)) (transRefl B (f x)) -transportIsoId' (A B : U) (f : A -> B) (g : B -> A) - (s : (y : B) -> Id B (f (g y)) y) - (t : (x : A) -> Id A (g (f x)) x) +transportIsoPath' (A B : U) (f : A -> B) (g : B -> A) + (s : (y : B) -> Path B (f (g y)) y) + (t : (x : A) -> Path A (g (f x)) x) (x : B) - : Id A (transport ( isoId A B f g s t @ -i) x) (g x) - = compId A (transport (<_> A) (g (transport (<_> B) x))) (transport (<_> A) (g x)) (g x) + : Path A (transport ( isoPath A B f g s t @ -i) x) (g x) + = compPath A (transport (<_> A) (g (transport (<_> B) x))) (transport (<_> A) (g x)) (g x) ( transport (<_> A) (g (transRefl B x @ i))) (transRefl A (g x)) -loopA (A : BZ) : Id BZ A A = (lemBZ A A).1 (shiftId, hole) +loopA (A : BZ) : Path BZ A A = (lemBZ A A).1 (shiftPath, hole) where AS : U = BZSet A - shiftId : Id U AS AS = equivId AS AS (BZShift A).1 (BZShift A).2 - hole1 (x : AS) : Id AS (transport (<_>AS) (transport (<_> AS) (BZS A (BZS A (transport (<_> AS) (BZP A (transport (<_>AS) x))))))) (BZS A (BZS A (BZP A x))) - = compId AS (transport (<_>AS) (transport (<_> AS) (BZS A (BZS A (transport (<_> AS) (BZP A (transport (<_>AS) x))))))) + shiftPath : Path U AS AS = equivPath AS AS (BZShift A).1 (BZShift A).2 + hole1 (x : AS) : Path AS (transport (<_>AS) (transport (<_> AS) (BZS A (BZS A (transport (<_> AS) (BZP A (transport (<_>AS) x))))))) (BZS A (BZS A (BZP A x))) + = compPath AS (transport (<_>AS) (transport (<_> AS) (BZS A (BZS A (transport (<_> AS) (BZP A (transport (<_>AS) x))))))) (BZS A (BZS A (transport (<_> AS) (BZP A (transport (<_>AS) x))))) (BZS A (BZS A (BZP A x))) (transConstNRefl AS (BZS A (BZS A (transport (<_> AS) (BZP A (transport (<_>AS) x))))) (suc (suc zero))) - (compId AS (BZS A (BZS A (transport (<_> AS) (BZP A (transport (<_>AS) x))))) + (compPath AS (BZS A (BZS A (transport (<_> AS) (BZP A (transport (<_>AS) x))))) (BZS A (BZS A (BZP A (transport (<_>AS) x)))) (BZS A (BZS A (BZP A x))) ( BZS A (BZS A (transRefl AS (BZP A (transport (<_>AS) x)) @ i))) ( BZS A (BZS A (BZP A (transRefl AS x @ i)))) ) - hole2 (x : AS) : Id AS (BZS A (BZS A (BZP A x))) (BZS A x) = BZS A (retEq AS AS (BZShift A) x @ i) - hole3 : Id (AS -> AS) (transport ( (shiftId@i) -> (shiftId@i)) (BZS A)) (BZS A) - = \(x : AS) -> compId AS (transport shiftId (BZS A (transport ( shiftId @ -j) x))) (BZS A (BZS A (BZP A x))) (BZS A x) + hole2 (x : AS) : Path AS (BZS A (BZS A (BZP A x))) (BZS A x) = BZS A (retEq AS AS (BZShift A) x @ i) + hole3 : Path (AS -> AS) (transport ( (shiftPath@i) -> (shiftPath@i)) (BZS A)) (BZS A) + = \(x : AS) -> compPath AS (transport shiftPath (BZS A (transport ( shiftPath @ -j) x))) (BZS A (BZS A (BZP A x))) (BZS A x) (hole1 x) (hole2 x) @ i - hole : IdP ( (shiftId@i) -> (shiftId@i)) (BZS A) (BZS A) - = substIdP (AS->AS) (AS->AS) ( (shiftId@i) -> (shiftId@i)) (BZS A) (BZS A) hole3 + hole : PathP ( (shiftPath@i) -> (shiftPath@i)) (BZS A) (BZS A) + = substPathP (AS->AS) (AS->AS) ( (shiftPath@i) -> (shiftPath@i)) (BZS A) (BZS A) hole3 loopZ : loopBZ = loopA ZBZ invLoopZ : loopBZ = loopZ @ -i -- loopBZ = Z = loopS1 -encodeZ (A : BZ) (p : Id BZ ZBZ A) : BZSet A = transport ( BZSet (p@i)) zeroZ +encodeZ (A : BZ) (p : Path BZ ZBZ A) : BZSet A = transport ( BZSet (p@i)) zeroZ -decodeZP (A : BZ) (a : BZSet A) : Id U (BZSet A) Z = (univalence (BZSet A) Z (BZAction A a, BZEquiv A a)).1.1 @ -i +decodeZP (A : BZ) (a : BZSet A) : Path U (BZSet A) Z = (univalence (BZSet A) Z (BZAction A a, BZEquiv A a)).1.1 @ -i -decodeZ1 (A : BZ) (a : BZSet A) (x : Z) : Id Z (transport (decodeZP A a) (BZS A (transport ( (decodeZP A a) @ -i) x))) +decodeZ1 (A : BZ) (a : BZSet A) (x : Z) : Path Z (transport (decodeZP A a) (BZS A (transport ( (decodeZP A a) @ -i) x))) (BZEquiv A a (BZS A (BZAction A a x))).1.1 - = compId Z (transport (decodeZP A a) (BZS A (transport ( (decodeZP A a) @ -i) x))) + = compPath Z (transport (decodeZP A a) (BZS A (transport ( (decodeZP A a) @ -i) x))) (transport (decodeZP A a) (BZS A (BZAction A a x))) (BZEquiv A a (BZS A (BZAction A a x))).1.1 ( transport (decodeZP A a) (BZS A (lem0 Z (BZSet A) (BZAction A a) (BZEquiv A a) x @ i))) (lem1 Z (BZSet A) (BZAction A a) (BZEquiv A a) (BZS A (BZAction A a x))) opaque decodeZ1 -decodeZ2 (A : BZ) (a : BZSet A) (x : Z) : Id Z (BZEquiv A a (BZS A (BZAction A a x))).1.1 (sucZ x) - = compId Z (BZEquiv A a (BZS A (BZAction A a x))).1.1 (BZEquiv A a (BZAction A a (sucZ x))).1.1 (sucZ x) +decodeZ2 (A : BZ) (a : BZSet A) (x : Z) : Path Z (BZEquiv A a (BZS A (BZAction A a x))).1.1 (sucZ x) + = compPath Z (BZEquiv A a (BZS A (BZAction A a x))).1.1 (BZEquiv A a (BZAction A a (sucZ x))).1.1 (sucZ x) ( (BZEquiv A a (lem2 (BZSet A) (BZShift A) a x @ i)).1.1) (secEq Z (BZSet A) (BZAction A a, BZEquiv A a) (sucZ x)) opaque decodeZ2 -decodeZ3 (A : BZ) (a : BZSet A) : Id (Z->Z) (\(x : Z) -> transport (decodeZP A a) (BZS A (transport ( (decodeZP A a) @ -i) x))) sucZ - = \(x : Z) -> compId Z (transport (decodeZP A a) (BZS A (transport ( (decodeZP A a) @ -i) x))) (BZEquiv A a (BZS A (BZAction A a x))).1.1 (sucZ x) (decodeZ1 A a x) (decodeZ2 A a x) @ i +decodeZ3 (A : BZ) (a : BZSet A) : Path (Z->Z) (\(x : Z) -> transport (decodeZP A a) (BZS A (transport ( (decodeZP A a) @ -i) x))) sucZ + = \(x : Z) -> compPath Z (transport (decodeZP A a) (BZS A (transport ( (decodeZP A a) @ -i) x))) (BZEquiv A a (BZS A (BZAction A a x))).1.1 (sucZ x) (decodeZ1 A a x) (decodeZ2 A a x) @ i opaque decodeZ3 -decodeZQ (A : BZ) (a : BZSet A) : IdP ( ((decodeZP A a)@-i) -> ((decodeZP A a)@-i)) sucZ (BZS A) - = substIdP (BZSet A -> BZSet A) (Z -> Z) ( ((decodeZP A a)@i) -> ((decodeZP A a)@i)) (BZS A) sucZ (decodeZ3 A a) @ -i +decodeZQ (A : BZ) (a : BZSet A) : PathP ( ((decodeZP A a)@-i) -> ((decodeZP A a)@-i)) sucZ (BZS A) + = substPathP (BZSet A -> BZSet A) (Z -> Z) ( ((decodeZP A a)@i) -> ((decodeZP A a)@i)) (BZS A) sucZ (decodeZ3 A a) @ -i opaque decodeZQ -decodeZ (A : BZ) (a : BZSet A) : Id BZ ZBZ A = (lemBZ ZBZ A).1 ( decodeZP A a @ -i, decodeZQ A a) +decodeZ (A : BZ) (a : BZSet A) : Path BZ ZBZ A = (lemBZ ZBZ A).1 ( decodeZP A a @ -i, decodeZQ A a) AA : U = (X : U) * (X -> X) -BZ' : U = (X : AA) * ishinh_UU (Id AA X (Z, sucZ)) -BZequivBZ' : Id U BZ BZ' = isoId BZ BZ' F G FG GF +BZ' : U = (X : AA) * ishinh_UU (Path AA X (Z, sucZ)) +BZequivBZ' : Path U BZ BZ' = isoPath BZ BZ' F G FG GF where - F (A : BZ) : BZ' = ((BZSet A, BZS A), BZNE A (ishinh (Id AA (BZSet A, BZS A) (Z, sucZ))) - (\(a : BZSet A) -> hinhpr (Id AA (BZSet A, BZS A) (Z, sucZ)) ( (decodeZP A a @ i, decodeZQ A a @ -i)))) + F (A : BZ) : BZ' = ((BZSet A, BZS A), BZNE A (ishinh (Path AA (BZSet A, BZS A) (Z, sucZ))) + (\(a : BZSet A) -> hinhpr (Path AA (BZSet A, BZS A) (Z, sucZ)) ( (decodeZP A a @ i, decodeZQ A a @ -i)))) G (A : BZ') : BZ = (A.1.1, - (A.2 (ishinh A.1.1) (\(a : Id AA A.1 (Z, sucZ)) -> hinhpr A.1.1 (transport ( (a @ -i).1) zeroZ)), + (A.2 (ishinh A.1.1) (\(a : Path AA A.1 (Z, sucZ)) -> hinhpr A.1.1 (transport ( (a @ -i).1) zeroZ)), ((A.1.2, SHIFT), EQUIV))) where SHIFT : isEquiv A.1.1 A.1.1 A.1.2 = A.2 (isEquiv A.1.1 A.1.1 A.1.2, propIsEquiv A.1.1 A.1.1 A.1.2) - (\(a : Id AA A.1 (Z, sucZ)) -> transport ( isEquiv (a@-i).1 (a@-i).1 (a@-i).2) (gradLemma Z Z sucZ predZ sucpredZ predsucZ)) + (\(a : Path AA A.1 (Z, sucZ)) -> transport ( isEquiv (a@-i).1 (a@-i).1 (a@-i).2) (gradLemma Z Z sucZ predZ sucpredZ predsucZ)) ZEquiv : (x : Z) -> isEquiv Z Z (action Z (sucZ, gradLemma Z Z sucZ predZ sucpredZ predsucZ) x) = BZEquiv ZBZ - hole (a : Id AA A.1 (Z, sucZ)) - : IdP ( isEquiv (a@-i).1 (a@-i).1 (a@-i).2) (gradLemma Z Z sucZ predZ sucpredZ predsucZ) SHIFT - = lemIdPProp (isEquiv Z Z sucZ) (isEquiv A.1.1 A.1.1 A.1.2) (propIsEquiv Z Z sucZ) + hole (a : Path AA A.1 (Z, sucZ)) + : PathP ( isEquiv (a@-i).1 (a@-i).1 (a@-i).2) (gradLemma Z Z sucZ predZ sucpredZ predsucZ) SHIFT + = lemPathPProp (isEquiv Z Z sucZ) (isEquiv A.1.1 A.1.1 A.1.2) (propIsEquiv Z Z sucZ) ( isEquiv (a@-i).1 (a@-i).1 (a@-i).2) (gradLemma Z Z sucZ predZ sucpredZ predsucZ) SHIFT - ZEquivEq (a : Id AA A.1 (Z, sucZ)) - : Id U ((x : Z) -> isEquiv Z Z (action Z (sucZ, gradLemma Z Z sucZ predZ sucpredZ predsucZ) x)) + ZEquivEq (a : Path AA A.1 (Z, sucZ)) + : Path U ((x : Z) -> isEquiv Z Z (action Z (sucZ, gradLemma Z Z sucZ predZ sucpredZ predsucZ) x)) ((x : A.1.1) -> isEquiv Z A.1.1 (action A.1.1 (A.1.2, SHIFT) x)) = (x : (a@-i).1) -> isEquiv Z (a@-i).1 (action (a@-i).1 ((a@-i).2, hole a @ i) x) EQUIV : (x : A.1.1) -> isEquiv Z A.1.1 (action A.1.1 (A.1.2, SHIFT) x) = A.2 ((x : A.1.1) -> isEquiv Z A.1.1 (action A.1.1 (A.1.2, SHIFT) x) , propPi A.1.1 (\(x : A.1.1) -> isEquiv Z A.1.1 (action A.1.1 (A.1.2, SHIFT) x)) (\(x : A.1.1) -> propIsEquiv Z A.1.1 (action A.1.1 (A.1.2, SHIFT) x))) - (\(a : Id AA A.1 (Z, sucZ)) -> transport (ZEquivEq a) ZEquiv) - FG (A : BZ') : Id BZ' (F (G A)) A = ((A.1.1, A.1.2), propishinh (Id AA (A.1.1, A.1.2) (Z, sucZ)) (F (G A)).2 A.2 @ i) - GF (A : BZ) : Id BZ (G (F A)) A = (lemBZ (G (F A)) A).1 (<_> BZSet A, <_> BZS A) + (\(a : Path AA A.1 (Z, sucZ)) -> transport (ZEquivEq a) ZEquiv) + FG (A : BZ') : Path BZ' (F (G A)) A = ((A.1.1, A.1.2), propishinh (Path AA (A.1.1, A.1.2) (Z, sucZ)) (F (G A)).2 A.2 @ i) + GF (A : BZ) : Path BZ (G (F A)) A = (lemBZ (G (F A)) A).1 (<_> BZSet A, <_> BZS A) -prf : Id (equiv Z Z) (BZAction ZBZ zeroZ, BZEquiv ZBZ zeroZ) (\(x : Z) -> x, isEquivId Z) = hole +prf : Path (equiv Z Z) (BZAction ZBZ zeroZ, BZEquiv ZBZ zeroZ) (\(x : Z) -> x, isEquivPath Z) = hole where - hole0 : (x : Z) -> Id Z (BZAction ZBZ zeroZ x) x = split + hole0 : (x : Z) -> Path Z (BZAction ZBZ zeroZ x) x = split inl n -> hole1 n where - hole1 : (n : nat) -> Id Z (BZAction ZBZ zeroZ (inl n)) (inl n) = split + hole1 : (n : nat) -> Path Z (BZAction ZBZ zeroZ (inl n)) (inl n) = split zero -> <_> inl zero suc n -> mapOnPath Z Z predZ (BZAction ZBZ zeroZ (inl n)) (inl n) (hole1 n) inr n -> hole1 n where - hole1 : (n : nat) -> Id Z (BZAction ZBZ zeroZ (inr n)) (inr n) = split + hole1 : (n : nat) -> Path Z (BZAction ZBZ zeroZ (inr n)) (inr n) = split zero -> <_> inr zero suc n -> mapOnPath Z Z sucZ (BZAction ZBZ zeroZ (inr n)) (inr n) (hole1 n) - hole : Id (equiv Z Z) (BZAction ZBZ zeroZ, BZEquiv ZBZ zeroZ) (\(x : Z) -> x, isEquivId Z) + hole : Path (equiv Z Z) (BZAction ZBZ zeroZ, BZEquiv ZBZ zeroZ) (\(x : Z) -> x, isEquivPath Z) = (\(x : Z) -> hole0 x @ i, - lemIdPProp + lemPathPProp (isEquiv Z Z (BZAction ZBZ zeroZ)) (isEquiv Z Z (\(x : Z) -> x)) (propIsEquiv Z Z (BZAction ZBZ zeroZ)) ( isEquiv Z Z (\(x : Z) -> hole0 x @ j)) - (BZEquiv ZBZ zeroZ) (isEquivId Z) @ i + (BZEquiv ZBZ zeroZ) (isEquivPath Z) @ i ) opaque prf decodeEncodeZRefl0 - : Id (Id U Z Z) (univalence Z Z (BZAction ZBZ zeroZ, BZEquiv ZBZ zeroZ)).1.1 (<_> Z) - = transport ( Id (Id U Z Z) (univalence Z Z (prf @ -i)).1.1 (<_> Z)) (lem11 Z) + : Path (Path U Z Z) (univalence Z Z (BZAction ZBZ zeroZ, BZEquiv ZBZ zeroZ)).1.1 (<_> Z) + = transport ( Path (Path U Z Z) (univalence Z Z (prf @ -i)).1.1 (<_> Z)) (lem11 Z) opaque decodeEncodeZRefl0 decodeEncodeZRefl1 - : IdP ( (IdP ( decodeEncodeZRefl0@j@i -> decodeEncodeZRefl0@j@i) sucZ sucZ)) + : PathP ( (PathP ( decodeEncodeZRefl0@j@i -> decodeEncodeZRefl0@j@i) sucZ sucZ)) ( (BZShift (decodeZ ZBZ zeroZ@i)).1) (<_> sucZ) - = lemIdPSet2 (Z->Z) (Z->Z) (setPi Z (\(_ : Z) -> Z) (\(_ : Z) -> ZSet)) + = lemPathPSet2 (Z->Z) (Z->Z) (setPi Z (\(_ : Z) -> Z) (\(_ : Z) -> ZSet)) ( decodeEncodeZRefl0@0@j -> decodeEncodeZRefl0@0@j) ( decodeEncodeZRefl0@1@j -> decodeEncodeZRefl0@1@j) ( decodeEncodeZRefl0@i@j -> decodeEncodeZRefl0@i@j) sucZ sucZ ( (BZShift (decodeZ ZBZ zeroZ@i)).1) (<_> sucZ) opaque decodeEncodeZRefl1 -decodeEncodeZRefl2 : Id ((p : Id U Z Z) * IdP ( p@i -> p@i) sucZ sucZ) ((lemBZ ZBZ ZBZ).2 (decodeZ ZBZ zeroZ)).1.1 ((lemBZ ZBZ ZBZ).2 (<_> ZBZ)).1.1 +decodeEncodeZRefl2 : Path ((p : Path U Z Z) * PathP ( p@i -> p@i) sucZ sucZ) ((lemBZ ZBZ ZBZ).2 (decodeZ ZBZ zeroZ)).1.1 ((lemBZ ZBZ ZBZ).2 (<_> ZBZ)).1.1 = (decodeEncodeZRefl0 @ i, decodeEncodeZRefl1 @ i) opaque decodeEncodeZRefl2 -decodeEncodeZRefl : Id loopBZ (decodeZ ZBZ zeroZ) (<_> ZBZ) - = lem10 ((p : Id U Z Z) * IdP ( p@i -> p@i) sucZ sucZ) loopBZ (lemBZ ZBZ ZBZ) (decodeZ ZBZ zeroZ) (<_> ZBZ) decodeEncodeZRefl2 +decodeEncodeZRefl : Path loopBZ (decodeZ ZBZ zeroZ) (<_> ZBZ) + = lem10 ((p : Path U Z Z) * PathP ( p@i -> p@i) sucZ sucZ) loopBZ (lemBZ ZBZ ZBZ) (decodeZ ZBZ zeroZ) (<_> ZBZ) decodeEncodeZRefl2 opaque decodeEncodeZRefl -decodeEncodeZ : (A : BZ) -> (p : Id BZ ZBZ A) -> Id (Id BZ ZBZ A) (decodeZ A (encodeZ A p)) p - = J BZ ZBZ (\(A : BZ) -> \(p : Id BZ ZBZ A) -> Id (Id BZ ZBZ A) (decodeZ A (encodeZ A p)) p) decodeEncodeZRefl +decodeEncodeZ : (A : BZ) -> (p : Path BZ ZBZ A) -> Path (Path BZ ZBZ A) (decodeZ A (encodeZ A p)) p + = J BZ ZBZ (\(A : BZ) -> \(p : Path BZ ZBZ A) -> Path (Path BZ ZBZ A) (decodeZ A (encodeZ A p)) p) decodeEncodeZRefl opaque decodeEncodeZ -encodeDecodeZ (A : BZ) (p : BZSet A) : Id (BZSet A) (transport (univalence (BZSet A) Z (BZAction A p, BZEquiv A p)).1.1 zeroZ) p +encodeDecodeZ (A : BZ) (p : BZSet A) : Path (BZSet A) (transport (univalence (BZSet A) Z (BZAction A p, BZEquiv A p)).1.1 zeroZ) p = lem0 Z (BZSet A) (BZAction A p) (BZEquiv A p) zeroZ opaque encodeDecodeZ -encodeLoop (x : loopBZ) : Id Z (encodeZ ZBZ (compBZ x loopZ)) (sucZ (encodeZ ZBZ x)) - = J BZ ZBZ (\(A : BZ) -> \(x : Id BZ ZBZ A) -> Id (BZSet A) (encodeZ A (compId BZ ZBZ A A x (loopA A))) (BZS A (encodeZ A x))) +encodeLoop (x : loopBZ) : Path Z (encodeZ ZBZ (compBZ x loopZ)) (sucZ (encodeZ ZBZ x)) + = J BZ ZBZ (\(A : BZ) -> \(x : Path BZ ZBZ A) -> Path (BZSet A) (encodeZ A (compPath BZ ZBZ A A x (loopA A))) (BZS A (encodeZ A x))) (<_> inr (suc zero)) ZBZ x opaque encodeLoop -encodeInvLoop (x : loopBZ) : Id Z (encodeZ ZBZ (compBZ x invLoopZ)) (predZ (encodeZ ZBZ x)) - = J BZ ZBZ (\(A : BZ) -> \(x : Id BZ ZBZ A) -> Id (BZSet A) (encodeZ A (compId BZ ZBZ A A x ( loopA A @ -i))) (BZP A (encodeZ A x))) +encodeInvLoop (x : loopBZ) : Path Z (encodeZ ZBZ (compBZ x invLoopZ)) (predZ (encodeZ ZBZ x)) + = J BZ ZBZ (\(A : BZ) -> \(x : Path BZ ZBZ A) -> Path (BZSet A) (encodeZ A (compPath BZ ZBZ A A x ( loopA A @ -i))) (BZP A (encodeZ A x))) (<_> inl zero) ZBZ x opaque encodeInvLoop -decodeLoop (z : Z) : Id loopBZ (compBZ (decodeZ ZBZ z) loopZ) (decodeZ ZBZ (sucZ z)) - = transport ( Id loopBZ (decodeEncodeZ ZBZ (compBZ (decodeZ ZBZ z) loopZ) @ i) (decodeZ ZBZ (sucZ (encodeDecodeZ ZBZ z @ i)))) +decodeLoop (z : Z) : Path loopBZ (compBZ (decodeZ ZBZ z) loopZ) (decodeZ ZBZ (sucZ z)) + = transport ( Path loopBZ (decodeEncodeZ ZBZ (compBZ (decodeZ ZBZ z) loopZ) @ i) (decodeZ ZBZ (sucZ (encodeDecodeZ ZBZ z @ i)))) ( decodeZ ZBZ (encodeLoop (decodeZ ZBZ z) @ i)) opaque decodeLoop -decodeInvLoop (z : Z) : Id (Id BZ ZBZ ZBZ) (compBZ (decodeZ ZBZ z) invLoopZ) (decodeZ ZBZ (predZ z)) - = transport ( Id loopBZ (decodeEncodeZ ZBZ (compBZ (decodeZ ZBZ z) invLoopZ) @ i) (decodeZ ZBZ (predZ (encodeDecodeZ ZBZ z @ i)))) +decodeInvLoop (z : Z) : Path (Path BZ ZBZ ZBZ) (compBZ (decodeZ ZBZ z) invLoopZ) (decodeZ ZBZ (predZ z)) + = transport ( Path loopBZ (decodeEncodeZ ZBZ (compBZ (decodeZ ZBZ z) invLoopZ) @ i) (decodeZ ZBZ (predZ (encodeDecodeZ ZBZ z @ i)))) ( decodeZ ZBZ (encodeInvLoop (decodeZ ZBZ z) @ i)) opaque decodeInvLoop @@ -703,45 +703,45 @@ multZ (A B : BZ) : BZ = (T, (TNE, (TShift A B, TEquiv))) where opaque decodeZ opaque loopA - T : U = Id BZ A B + T : U = Path BZ A B TNE : ishinh_UU T = BZNE A (ishinh T) (\(a : BZSet A) -> BZNE B (ishinh T) (\(b : BZSet B) -> - hinhpr T (compId BZ A ZBZ B ( decodeZ A a @ -i) (decodeZ B b)))) - F (A B : BZ) (x : (Id BZ A B)) : (Id BZ A B) = compId BZ A B B x (loopA B) - G (A B : BZ) (x : (Id BZ A B)) : (Id BZ A B) = compId BZ A B B x ( loopA B @ -i) - FG (A B : BZ) (x : (Id BZ A B)) : Id (Id BZ A B) (F A B (G A B x)) x = lemCompInv BZ A B B x ( loopA B @ -i) + hinhpr T (compPath BZ A ZBZ B ( decodeZ A a @ -i) (decodeZ B b)))) + F (A B : BZ) (x : (Path BZ A B)) : (Path BZ A B) = compPath BZ A B B x (loopA B) + G (A B : BZ) (x : (Path BZ A B)) : (Path BZ A B) = compPath BZ A B B x ( loopA B @ -i) + FG (A B : BZ) (x : (Path BZ A B)) : Path (Path BZ A B) (F A B (G A B x)) x = lemCompInv BZ A B B x ( loopA B @ -i) opaque FG - GF (A B : BZ) (x : (Id BZ A B)) : Id (Id BZ A B) (G A B (F A B x)) x = lemCompInv BZ A B B x (loopA B) + GF (A B : BZ) (x : (Path BZ A B)) : Path (Path BZ A B) (G A B (F A B x)) x = lemCompInv BZ A B B x (loopA B) opaque GF - TShift (A B : BZ) : equiv (Id BZ A B) (Id BZ A B) = (F A B, gradLemma (Id BZ A B) (Id BZ A B) (F A B) (G A B) (FG A B) (GF A B)) - hole : (y : Z) -> Id (Id BZ ZBZ ZBZ) (action (Id BZ ZBZ ZBZ) (TShift ZBZ ZBZ) (<_> ZBZ) y) (decodeZ ZBZ y) = split + TShift (A B : BZ) : equiv (Path BZ A B) (Path BZ A B) = (F A B, gradLemma (Path BZ A B) (Path BZ A B) (F A B) (G A B) (FG A B) (GF A B)) + hole : (y : Z) -> Path (Path BZ ZBZ ZBZ) (action (Path BZ ZBZ ZBZ) (TShift ZBZ ZBZ) (<_> ZBZ) y) (decodeZ ZBZ y) = split inl u -> hole0 u - where hole0 : (n : nat) -> Id (Id BZ ZBZ ZBZ) (action (Id BZ ZBZ ZBZ) (TShift ZBZ ZBZ) (<_> ZBZ) (inl n)) (decodeZ ZBZ (inl n)) = split + where hole0 : (n : nat) -> Path (Path BZ ZBZ ZBZ) (action (Path BZ ZBZ ZBZ) (TShift ZBZ ZBZ) (<_> ZBZ) (inl n)) (decodeZ ZBZ (inl n)) = split zero -> hole1 - where hole1 : Id (Id BZ ZBZ ZBZ) (compBZ (<_> ZBZ) invLoopZ) (decodeZ ZBZ (inl zero)) - = compId loopBZ (compBZ (<_> ZBZ) invLoopZ) (compBZ (decodeZ ZBZ (inr zero)) invLoopZ) (decodeZ ZBZ (inl zero)) + where hole1 : Path (Path BZ ZBZ ZBZ) (compBZ (<_> ZBZ) invLoopZ) (decodeZ ZBZ (inl zero)) + = compPath loopBZ (compBZ (<_> ZBZ) invLoopZ) (compBZ (decodeZ ZBZ (inr zero)) invLoopZ) (decodeZ ZBZ (inl zero)) ( compBZ (decodeEncodeZRefl @ -i) invLoopZ) (decodeInvLoop (inr zero)) suc n -> hole1 - where hole1 : Id (Id BZ ZBZ ZBZ) (compBZ (action loopBZ (TShift ZBZ ZBZ) (<_> ZBZ) (inl n)) invLoopZ) (decodeZ ZBZ (inl (suc n))) - = compId (Id BZ ZBZ ZBZ) (compBZ (action (Id BZ ZBZ ZBZ) (TShift ZBZ ZBZ) (<_> ZBZ) (inl n)) invLoopZ) + where hole1 : Path (Path BZ ZBZ ZBZ) (compBZ (action loopBZ (TShift ZBZ ZBZ) (<_> ZBZ) (inl n)) invLoopZ) (decodeZ ZBZ (inl (suc n))) + = compPath (Path BZ ZBZ ZBZ) (compBZ (action (Path BZ ZBZ ZBZ) (TShift ZBZ ZBZ) (<_> ZBZ) (inl n)) invLoopZ) (compBZ (decodeZ ZBZ (inl n)) invLoopZ) (decodeZ ZBZ (inl (suc n))) ( compBZ (hole0 n @ i) invLoopZ) (decodeInvLoop (inl n)) inr u -> hole0 u - where hole0 : (n : nat) -> Id (Id BZ ZBZ ZBZ) (action (Id BZ ZBZ ZBZ) (TShift ZBZ ZBZ) (<_> ZBZ) (inr n)) (decodeZ ZBZ (inr n)) = split + where hole0 : (n : nat) -> Path (Path BZ ZBZ ZBZ) (action (Path BZ ZBZ ZBZ) (TShift ZBZ ZBZ) (<_> ZBZ) (inr n)) (decodeZ ZBZ (inr n)) = split zero -> decodeEncodeZRefl @ -i - suc n -> compId (Id BZ ZBZ ZBZ) (compBZ (action (Id BZ ZBZ ZBZ) (TShift ZBZ ZBZ) (<_> ZBZ) (inr n)) loopZ) + suc n -> compPath (Path BZ ZBZ ZBZ) (compBZ (action (Path BZ ZBZ ZBZ) (TShift ZBZ ZBZ) (<_> ZBZ) (inr n)) loopZ) (compBZ (decodeZ ZBZ (inr n)) loopZ) (decodeZ ZBZ (inr (suc n))) ( compBZ (hole0 n @ i) loopZ) (decodeLoop (inr n)) visible decodeZ - TEquiv''' : isEquiv Z (Id BZ ZBZ ZBZ) (action (Id BZ ZBZ ZBZ) (TShift ZBZ ZBZ) (<_> ZBZ)) + TEquiv''' : isEquiv Z (Path BZ ZBZ ZBZ) (action (Path BZ ZBZ ZBZ) (TShift ZBZ ZBZ) (<_> ZBZ)) = transport ( isEquiv Z loopBZ (\(y : Z) -> hole y @ -i)) (gradLemma Z loopBZ (decodeZ ZBZ) (encodeZ ZBZ) (decodeEncodeZ ZBZ) (encodeDecodeZ ZBZ)) - TEquiv'' (b : BZSet B) (x : Id BZ ZBZ B) : isEquiv Z (Id BZ ZBZ B) (action (Id BZ ZBZ B) (TShift ZBZ B) x) - = J BZ ZBZ (\(B : BZ) -> \(x : Id BZ ZBZ B) -> isEquiv Z (Id BZ ZBZ B) (action (Id BZ ZBZ B) (TShift ZBZ B) x)) + TEquiv'' (b : BZSet B) (x : Path BZ ZBZ B) : isEquiv Z (Path BZ ZBZ B) (action (Path BZ ZBZ B) (TShift ZBZ B) x) + = J BZ ZBZ (\(B : BZ) -> \(x : Path BZ ZBZ B) -> isEquiv Z (Path BZ ZBZ B) (action (Path BZ ZBZ B) (TShift ZBZ B) x)) TEquiv''' B x TEquiv' (a : BZSet A) (b : BZSet B) : (x : T) -> isEquiv Z T (action T (TShift A B) x) - = J BZ ZBZ (\(A : BZ) -> \(p : Id BZ ZBZ A) -> (x : Id BZ A B) -> isEquiv Z (Id BZ A B) (action (Id BZ A B) (TShift A B) x)) + = J BZ ZBZ (\(A : BZ) -> \(p : Path BZ ZBZ A) -> (x : Path BZ A B) -> isEquiv Z (Path BZ A B) (action (Path BZ A B) (TShift A B) x)) (TEquiv'' b) A (decodeZ A a) TEquiv (x : T) : isEquiv Z T (action T (TShift A B) x) = BZNE A (isEquiv Z T (action T (TShift A B) x), propIsEquiv Z T (action T (TShift A B) x)) @@ -749,9 +749,9 @@ multZ (A B : BZ) : BZ = (T, (TNE, (TShift A B, TEquiv))) (\(b : BZSet B) -> TEquiv' a b x)) visible decodeZ -loopBZequalsZ : Id U loopBZ Z = isoId loopBZ Z (encodeZ ZBZ) (decodeZ ZBZ) (encodeDecodeZ ZBZ) (decodeEncodeZ ZBZ) +loopBZequalsZ : Path U loopBZ Z = isoPath loopBZ Z (encodeZ ZBZ) (decodeZ ZBZ) (encodeDecodeZ ZBZ) (decodeEncodeZ ZBZ) -loopS1equalsLoopBZ : Id U loopS1 loopBZ = compId U loopS1 Z loopBZ loopS1equalsZ ( loopBZequalsZ @ -i) +loopS1equalsLoopBZ : Path U loopS1 loopBZ = compPath U loopS1 Z loopBZ loopS1equalsZ ( loopBZequalsZ @ -i) loopS1equalsLoopBZ' : equiv loopS1 loopBZ = transEquiv' loopBZ loopS1 loopS1equalsLoopBZ -- BZ = S1 @@ -762,86 +762,86 @@ F : S1 -> BZ = split -- mapOnPath S1 BZ F base base = loopS1equalsLoopBZ'.1 : same left inverse and loopS1equalsLoopBZ'.1 is an equivalence -G : (y : S1) -> Id BZ ZBZ (F y) -> Id S1 base y = split +G : (y : S1) -> Path BZ ZBZ (F y) -> Path S1 base y = split base -> \(x : loopBZ) -> (loopS1equalsLoopBZ'.2 x).1.1 loop @ i -> hole @ i where - hole4 (x : Z) : Id loopS1 (loopIt (predZ x)) (compS1 (loopIt x) invLoop) + hole4 (x : Z) : Path loopS1 (loopIt (predZ x)) (compS1 (loopIt x) invLoop) = lem2It x hole3 (x : loopBZ) - : Id loopS1 + : Path loopS1 (compS1 (decode base (encodeZ ZBZ (compBZ x invLoopZ))) loop1) (decode base (encodeZ ZBZ x)) - = compId loopS1 + = compPath loopS1 (compS1 (decode base (encodeZ ZBZ (compBZ x invLoopZ))) loop1) (compS1 (decode base (predZ (encodeZ ZBZ x))) loop1) (decode base (encodeZ ZBZ x)) ( compS1 (decode base (encodeInvLoop x @ i)) loop1) - (compId loopS1 + (compPath loopS1 (compS1 (decode base (predZ (encodeZ ZBZ x))) loop1) (compS1 (compS1 (decode base (encodeZ ZBZ x)) invLoop) loop1) (decode base (encodeZ ZBZ x)) ( compS1 (hole4 (encodeZ ZBZ x) @ i) loop1) ( compInv S1 base base (decode base (encodeZ ZBZ x)) base invLoop @ -i)) - hole6 (x : loopBZ) : Id loopS1 (loopS1equalsLoopBZ'.2 x).1.1 (decode base (encodeZ ZBZ x)) - = compId loopS1 (loopS1equalsLoopBZ'.2 x).1.1 (loopIt (transConstN Z (encodeZ ZBZ x) (suc (suc (suc (suc (suc zero))))))) (decode base (encodeZ ZBZ x)) + hole6 (x : loopBZ) : Path loopS1 (loopS1equalsLoopBZ'.2 x).1.1 (decode base (encodeZ ZBZ x)) + = compPath loopS1 (loopS1equalsLoopBZ'.2 x).1.1 (loopIt (transConstN Z (encodeZ ZBZ x) (suc (suc (suc (suc (suc zero))))))) (decode base (encodeZ ZBZ x)) (lemHcomp3 (loopIt (transConstN Z (encodeZ ZBZ x) (suc (suc (suc (suc (suc zero)))))))) ( decode base (transConstNRefl Z (encodeZ ZBZ x) (suc (suc (suc (suc (suc zero))))) @ i)) - hole1 : Id (loopBZ -> loopS1) + hole1 : Path (loopBZ -> loopS1) (\(x : loopBZ) -> compS1 ((loopS1equalsLoopBZ'.2 (compBZ x invLoopZ)).1.1) loop1) (\(x : loopBZ) -> (loopS1equalsLoopBZ'.2 x).1.1) = \(x : loopBZ) -> - transport ( Id loopS1 (compS1 (hole6 (compBZ x invLoopZ) @ -i) loop1) (loopS1equalsLoopBZ'.2 x).1.1) - (transport ( Id loopS1 (compS1 (decode base (encodeZ ZBZ (compBZ x invLoopZ))) loop1) (hole6 x @ -i)) + transport ( Path loopS1 (compS1 (hole6 (compBZ x invLoopZ) @ -i) loop1) (loopS1equalsLoopBZ'.2 x).1.1) + (transport ( Path loopS1 (compS1 (decode base (encodeZ ZBZ (compBZ x invLoopZ))) loop1) (hole6 x @ -i)) (hole3 x)) @ j - hole : IdP ( Id BZ ZBZ (F (loop1 @ i)) -> Id S1 base (loop1 @ i)) + hole : PathP ( Path BZ ZBZ (F (loop1 @ i)) -> Path S1 base (loop1 @ i)) (\(x : loopBZ) -> (loopS1equalsLoopBZ'.2 x).1.1) (\(x : loopBZ) -> (loopS1equalsLoopBZ'.2 x).1.1) - = substIdP (loopBZ -> loopS1) (loopBZ -> loopS1) - ( Id BZ ZBZ (F (loop1 @ i)) -> Id S1 base (loop1 @ i)) + = substPathP (loopBZ -> loopS1) (loopBZ -> loopS1) + ( Path BZ ZBZ (F (loop1 @ i)) -> Path S1 base (loop1 @ i)) (\(x : loopBZ) -> (loopS1equalsLoopBZ'.2 x).1.1) (\(x : loopBZ) -> (loopS1equalsLoopBZ'.2 x).1.1) hole1 -GF : (y : S1) -> (x : Id S1 base y) -> Id (Id S1 base y) (G y (mapOnPath S1 BZ F base y x)) x - = J S1 base (\(y : S1) -> \(x : Id S1 base y) -> Id (Id S1 base y) (G y (mapOnPath S1 BZ F base y x)) x) +GF : (y : S1) -> (x : Path S1 base y) -> Path (Path S1 base y) (G y (mapOnPath S1 BZ F base y x)) x + = J S1 base (\(y : S1) -> \(x : Path S1 base y) -> Path (Path S1 base y) (G y (mapOnPath S1 BZ F base y x)) x) (lemHcomp3 (<_> base)) opaque GF -F_fullyFaithful0 : Id (loopS1 -> loopBZ) (mapOnPath S1 BZ F base base) loopS1equalsLoopBZ'.1 +F_fullyFaithful0 : Path (loopS1 -> loopBZ) (mapOnPath S1 BZ F base base) loopS1equalsLoopBZ'.1 = lemEquiv1 loopS1 loopBZ (mapOnPath S1 BZ F base base) loopS1equalsLoopBZ'.1 loopS1equalsLoopBZ'.2 (GF base) opaque F_fullyFaithful0 -F_fullyFaithful : (x y : S1) -> isEquiv (Id S1 x y) (Id BZ (F x) (F y)) (mapOnPath S1 BZ F x y) - = lemPropFib (\(x : S1) -> (y : S1) -> isEquiv (Id S1 x y) (Id BZ (F x) (F y)) (mapOnPath S1 BZ F x y)) - (\(x : S1) -> propPi S1 (\(y : S1) -> isEquiv (Id S1 x y) (Id BZ (F x) (F y)) (mapOnPath S1 BZ F x y)) - (\(y : S1) -> propIsEquiv (Id S1 x y) (Id BZ (F x) (F y)) (mapOnPath S1 BZ F x y))) - (lemPropFib (\(y : S1) -> isEquiv (Id S1 base y) (Id BZ ZBZ (F y)) (mapOnPath S1 BZ F base y)) - (\(y : S1) -> propIsEquiv (Id S1 base y) (Id BZ ZBZ (F y)) (mapOnPath S1 BZ F base y)) +F_fullyFaithful : (x y : S1) -> isEquiv (Path S1 x y) (Path BZ (F x) (F y)) (mapOnPath S1 BZ F x y) + = lemPropFib (\(x : S1) -> (y : S1) -> isEquiv (Path S1 x y) (Path BZ (F x) (F y)) (mapOnPath S1 BZ F x y)) + (\(x : S1) -> propPi S1 (\(y : S1) -> isEquiv (Path S1 x y) (Path BZ (F x) (F y)) (mapOnPath S1 BZ F x y)) + (\(y : S1) -> propIsEquiv (Path S1 x y) (Path BZ (F x) (F y)) (mapOnPath S1 BZ F x y))) + (lemPropFib (\(y : S1) -> isEquiv (Path S1 base y) (Path BZ ZBZ (F y)) (mapOnPath S1 BZ F base y)) + (\(y : S1) -> propIsEquiv (Path S1 base y) (Path BZ ZBZ (F y)) (mapOnPath S1 BZ F base y)) hole) where hole : isEquiv loopS1 loopBZ (mapOnPath S1 BZ F base base) = transport ( isEquiv loopS1 loopBZ (F_fullyFaithful0 @ -i)) loopS1equalsLoopBZ'.2 opaque F_fullyFaithful -F_essentiallySurjective (y : BZ) : (x : S1) * Id BZ y (F x) = hole +F_essentiallySurjective (y : BZ) : (x : S1) * Path BZ y (F x) = hole where - hInh (y : BZ) : ishinh_UU ((x : S1) * Id BZ y (F x)) = hole + hInh (y : BZ) : ishinh_UU ((x : S1) * Path BZ y (F x)) = hole where - hole2 (a : BZSet y) : (x : S1) * Id BZ y (F x) = (base, decodeZ y a @ -i) - hole1 (a : BZSet y) : ishinh_UU ((x : S1) * Id BZ y (F x)) = hinhpr ((x : S1) * Id BZ y (F x)) (hole2 a) - hole : ishinh_UU ((x : S1) * Id BZ y (F x)) = BZNE y (ishinh_UU ((x : S1) * Id BZ y (F x)), propishinh ((x : S1) * Id BZ y (F x))) hole1 - hProp : prop ((x : S1) * Id BZ y (F x)) = transport (E13 S1 BZ F) (F_fullyFaithful) y - hContr : isContr ((x : S1) * Id BZ y (F x)) = inhPropContr ((x : S1) * Id BZ y (F x)) hProp (hInh y) - hole : (x : S1) * Id BZ y (F x) = hContr.1 - -S1equalsBZ : Id U S1 BZ = hole + hole2 (a : BZSet y) : (x : S1) * Path BZ y (F x) = (base, decodeZ y a @ -i) + hole1 (a : BZSet y) : ishinh_UU ((x : S1) * Path BZ y (F x)) = hinhpr ((x : S1) * Path BZ y (F x)) (hole2 a) + hole : ishinh_UU ((x : S1) * Path BZ y (F x)) = BZNE y (ishinh_UU ((x : S1) * Path BZ y (F x)), propishinh ((x : S1) * Path BZ y (F x))) hole1 + hProp : prop ((x : S1) * Path BZ y (F x)) = transport (E13 S1 BZ F) (F_fullyFaithful) y + hContr : isContr ((x : S1) * Path BZ y (F x)) = inhPropContr ((x : S1) * Path BZ y (F x)) hProp (hInh y) + hole : (x : S1) * Path BZ y (F x) = hContr.1 + +S1equalsBZ : Path U S1 BZ = hole where G (y : BZ) : S1 = (F_essentiallySurjective y).1 - FG (y : BZ) : Id BZ (F (G y)) y = (F_essentiallySurjective y).2 @ -i - GF (x : S1) : Id S1 (G (F x)) x = (F_fullyFaithful (G (F x)) x (FG (F x))).1.1 - hole : Id U S1 BZ = isoId S1 BZ F G FG GF + FG (y : BZ) : Path BZ (F (G y)) y = (F_essentiallySurjective y).2 @ -i + GF (x : S1) : Path S1 (G (F x)) x = (F_fullyFaithful (G (F x)) x (FG (F x))).1.1 + hole : Path U S1 BZ = isoPath S1 BZ F G FG GF invBZ : BZ -> BZ = transport ( S1equalsBZ@i -> S1equalsBZ@i) invS1 doubleLoopBZ : loopBZ -> loopBZ = transport ( loopS1equalsLoopBZ@i -> loopS1equalsLoopBZ@i) doubleLoop @@ -850,7 +850,7 @@ doubleLoopBZ : loopBZ -> loopBZ = transport ( loopS1equalsLoopBZ@i -> loopS1e -- EVAL: inr (suc (suc zero)) -- Time: 0m25.191s --- > let visible_all in let doubleLoop : Id BZ (multZ ZBZ ZBZ) (multZ ZBZ ZBZ) = multZ (loopZ@-i) (loopZ@i) in transport ( BZSet (transport ( BZSet (doubleLoop @ i)) (<_> ZBZ) @ j)) zeroZ +-- > let visible_all in let doubleLoop : Path BZ (multZ ZBZ ZBZ) (multZ ZBZ ZBZ) = multZ (loopZ@-i) (loopZ@i) in transport ( BZSet (transport ( BZSet (doubleLoop @ i)) (<_> ZBZ) @ j)) zeroZ -- too slow -- > let visible_all in transport ( BZSet (transport ( BZSet (multZ (loopZ@-i) (loopZ@i))) (<_> ZBZ) @ j)) zeroZ diff --git a/examples/torus.ctt b/examples/torus.ctt index 7fc9cae..08925fc 100644 --- a/examples/torus.ctt +++ b/examples/torus.ctt @@ -44,12 +44,12 @@ c2t_base : S1 -> Torus = split loop @ x -> pathTwoT{Torus} @ x -- branch for loop -c2t_loop' : (c : S1) -> IdP (<_>Torus) (c2t_base c) (c2t_base c) = split +c2t_loop' : (c : S1) -> PathP (<_>Torus) (c2t_base c) (c2t_base c) = split base -> < x > pathOneT{Torus} @ x loop @ y -> < x > squareT{Torus} @ y @ x -- use funext to exchange the interval variable and the S1 variable -c2t_loop : IdP (<_>S1 -> Torus) c2t_base c2t_base = +c2t_loop : PathP (<_>S1 -> Torus) c2t_base c2t_base = \(c : S1) -> c2t_loop' c @ y c2t' : S1 -> S1 -> Torus = split @@ -61,7 +61,7 @@ c2t (cs : and S1 S1) : Torus = c2t' cs.1 cs.2 ------------------------------------------------------------------------ -- first composite: induct and reflexivity! -t2c2t : (t : Torus) -> IdP (<_> Torus) (c2t (t2c t)) t = split +t2c2t : (t : Torus) -> PathP (<_> Torus) (c2t (t2c t)) t = split ptT -> <_> ptT pathOneT @ y -> <_> pathOneT{Torus} @ y pathTwoT @ x -> <_> pathTwoT{Torus} @ x @@ -72,7 +72,7 @@ t2c2t : (t : Torus) -> IdP (<_> Torus) (c2t (t2c t)) t = split -- except we need to use the same tricks as in c2t to do the inner -- induction -c2t2c_base : (c2 : S1) -> IdP (<_> and S1 S1) (t2c (c2t_base c2)) (base,c2) = split +c2t2c_base : (c2 : S1) -> PathP (<_> and S1 S1) (t2c (c2t_base c2)) (base,c2) = split base -> <_> (base,base) loop @ y -> <_> (base,loop1 @ y) @@ -80,13 +80,13 @@ c2t2c_base : (c2 : S1) -> IdP (<_> and S1 S1) (t2c (c2t_base c2)) (base,c2) = sp -- two binders exchanged. -- c2t' (loop @ y) c2 = (c2t_loop @ y c2) = c2t_loop' c2 @ y c2t2c_loop' : (c2 : S1) -> - IdP ( IdP (<_> and S1 S1) (t2c (c2t_loop @ y c2)) (loop1 @ y , c2)) + PathP ( PathP (<_> and S1 S1) (t2c (c2t_loop @ y c2)) (loop1 @ y , c2)) (c2t2c_base c2) (c2t2c_base c2) = split base -> <_> (loop1 @ y, base) loop @ x -> <_> (loop1 @ y, loop1 @ x) -c2t2c : (c1 : S1) (c2 : S1) -> IdP (<_> and S1 S1) (t2c (c2t' c1 c2)) (c1,c2) = split +c2t2c : (c1 : S1) (c2 : S1) -> PathP (<_> and S1 S1) (t2c (c2t' c1 c2)) (c1,c2) = split base -> c2t2c_base -- again, I shouldn't need to do funext here; -- I should be able to do a split inside of an interval binding @@ -96,25 +96,25 @@ c2t2c : (c1 : S1) (c2 : S1) -> IdP (<_> and S1 S1) (t2c (c2t' c1 c2)) (c1,c2) = ------------------------------------------------------------------------ -- combine everything to get that Torus = S1 * S1 -S1S1equalsTorus : Id U (and S1 S1) Torus = isoId (and S1 S1) Torus c2t t2c t2c2t rem +S1S1equalsTorus : Path U (and S1 S1) Torus = isoPath (and S1 S1) Torus c2t t2c t2c2t rem where - rem (c:and S1 S1) : Id (and S1 S1) (t2c (c2t c)) c = c2t2c c.1 c.2 + rem (c:and S1 S1) : Path (and S1 S1) (t2c (c2t c)) c = c2t2c c.1 c.2 -TorusEqualsS1S1 : Id U Torus (and S1 S1) = S1S1equalsTorus @ -i +TorusEqualsS1S1 : Path U Torus (and S1 S1) = S1S1equalsTorus @ -i -loopT : U = Id Torus ptT ptT +loopT : U = Path Torus ptT ptT --- funDep (A0 A1 :U) (p:Id U A0 A1) (u0:A0) (u1:A1) : --- Id U (Id A0 u0 (transport (p@-i) u1)) (Id A1 (transport p u0) u1) = --- Id (p @ i) (transport ( p @ (i/\l)) u0) (transport ( p @ (i\/-l)) u1) +-- funDep (A0 A1 :U) (p:Path U A0 A1) (u0:A0) (u1:A1) : +-- Path U (Path A0 u0 (transport (p@-i) u1)) (Path A1 (transport p u0) u1) = +-- Path (p @ i) (transport ( p @ (i/\l)) u0) (transport ( p @ (i\/-l)) u1) --- loopTorusEqualsZZ : Id U loopT (and Z Z) = comp U (rem @ i) [(i = 1) -> rem1] +-- loopTorusEqualsZZ : Path U loopT (and Z Z) = comp U (rem @ i) [(i = 1) -> rem1] -- where --- rem : Id U loopT (Id (and S1 S1) (base,base) (base,base)) = +-- rem : Path U loopT (Path (and S1 S1) (base,base) (base,base)) = -- funDep Torus (and S1 S1) TorusEqualsS1S1 ptT (base,base) --- rem1 : Id U (Id (and S1 S1) (base,base) (base,base)) (and Z Z) = --- comp U (lemIdAnd S1 S1 (base,base) (base,base) @ i) +-- rem1 : Path U (Path (and S1 S1) (base,base) (base,base)) (and Z Z) = +-- comp U (lemPathAnd S1 S1 (base,base) (base,base) @ i) -- [(i = 1) -> and (loopS1equalsZ @ j) (loopS1equalsZ @ j)] diff --git a/examples/univalence.ctt b/examples/univalence.ctt index 97cf7f2..82a1b7a 100644 --- a/examples/univalence.ctt +++ b/examples/univalence.ctt @@ -11,36 +11,36 @@ import equiv ------------------------------------------------------------------------------ -- Proof of univalence using unglue: -retIsContr (A B :U) (f:A->B) (g:B->A) (h : (x:A) -> Id A (g (f x)) x) (v:isContr B) +retIsContr (A B :U) (f:A->B) (g:B->A) (h : (x:A) -> Path A (g (f x)) x) (v:isContr B) : isContr A = (g b,p) where b : B = v.1 - q : (y:B) -> Id B b y = v.2 - p (x:A) : Id A (g b) x = comp (A) (g (q (f x)@i)) [(i=0) -> g b,(i=1) -> h x] + q : (y:B) -> Path B b y = v.2 + p (x:A) : Path A (g b) x = comp (A) (g (q (f x)@i)) [(i=0) -> g b,(i=1) -> h x] sigIsContr (A:U) (B:A->U) (u:isContr A) (q:(x:A) -> isContr (B x)) : isContr ((x:A)*B x) = ((a,g a),r) where a : A = u.1 - p : (x:A) -> Id A a x = u.2 + p : (x:A) -> Path A a x = u.2 g (x:A) : B x = (q x).1 - h (x:A) : (y:B x) -> Id (B x) (g x) y = (q x).2 + h (x:A) : (y:B x) -> Path (B x) (g x) y = (q x).2 C : U = (x:A) * B x - r (z:C) : Id C (a,g a) z = + r (z:C) : Path C (a,g a) z = (p z.1@i,h (p z.1@i) (comp (B (p z.1@i\/-j)) z.2 [(i=1)->z.2])@i) -isPathContr (A:U) (cA:isContr A) (x y:A) : isContr (Id A x y) = (p0,q) +isPathContr (A:U) (cA:isContr A) (x y:A) : isContr (Path A x y) = (p0,q) where a : A = cA.1 - f : (x:A) -> Id A a x = cA.2 - p0 : Id A x y = comp (A) a [(i=0) -> f x,(i=1) -> f y] - q (p:Id A x y) : Id (Id A x y) p0 p = + f : (x:A) -> Path A a x = cA.2 + p0 : Path A x y = comp (A) a [(i=0) -> f x,(i=1) -> f y] + q (p:Path A x y) : Path (Path A x y) p0 p = comp (A) a [(i=0) -> f x,(i=1) -> f y, (j=0) -> comp (A) a [(k=0) -> a,(i=0) -> f x@k/\l,(i=1) -> f y@k/\l], (j=1) -> f (p@i)] isEquivContr (A B:U) (cA:isContr A) (cB:isContr B) (f:A->B) : isEquiv A B f = - \ (y:B) -> sigIsContr A (\ (x:A) -> Id B y (f x)) cA (\ (x:A) -> isPathContr B cB y (f x)) + \ (y:B) -> sigIsContr A (\ (x:A) -> Path B y (f x)) cA (\ (x:A) -> isPathContr B cB y (f x)) totalFun (A:U) (B C : A->U) (f : (x:A) -> B x -> C x) (w:Sigma A B) : Sigma A C = (w.1,f (w.1) (w.2)) @@ -53,17 +53,17 @@ funFib2 (A:U) (B C : A->U) (f : (x:A) -> B x -> C x) (x0:A) (z0:C x0) where x : A = w.1.1 b : B x = w.1.2 - p : Id A x0 x = (w.2@i).1 - q : IdP (C (p@i)) z0 (f x b) = (w.2@i).2 + p : Path A x0 x = (w.2@i).1 + q : PathP (C (p@i)) z0 (f x b) = (w.2@i).2 b0 : B x0 = comp (B (p@-i)) b [] - r : IdP (B (p@-i)) b b0 = comp (B (p@-j\/-i)) b [(i=0) -> b] - s : Id (C x0) z0 (f x0 b0) = comp (C (p@(i/\-j))) (q@i) [(i=0) -> z0,(i=1) -> f (p@-k) (r@k)] + r : PathP (B (p@-i)) b b0 = comp (B (p@-j\/-i)) b [(i=0) -> b] + s : Path (C x0) z0 (f x0 b0) = comp (C (p@(i/\-j))) (q@i) [(i=0) -> z0,(i=1) -> f (p@-k) (r@k)] compFunFib (A:U) (B C : A->U) (f : (x:A) -> B x -> C x) (x0:A) (z0:C x0) (u:fiber (B x0) (C x0) (f x0) z0) : fiber (B x0) (C x0) (f x0) z0 = funFib2 A B C f x0 z0 (funFib1 A B C f x0 z0 u) retFunFib (A:U) (B C : A->U) (f : (x:A) -> B x -> C x) (x0:A) (z0:C x0) (u:fiber (B x0) (C x0) (f x0) z0) : - Id (fiber (B x0) (C x0) (f x0) z0) (funFib2 A B C f x0 z0 (funFib1 A B C f x0 z0 u)) u = + Path (fiber (B x0) (C x0) (f x0) z0) (funFib2 A B C f x0 z0 (funFib1 A B C f x0 z0 u)) u = (comp ( B x0) u.1 [(l=1) -> u.1], comp ( C x0) (u.2 @ i) [ (l=1) -> u.2 @ i, (i = 0) -> z0, @@ -78,7 +78,7 @@ equivFunFib (A:U) (B C : A->U) (f : (x:A) -> B x -> C x) (cB : isContr (Sigma A (isEquivContr (Sigma A B) (Sigma A C) cB cC (totalFun A B C f) (x,z)) -- test normal form --- nequivFunFib : (A:U) (B C : A->U) (f : (x:A) -> B x -> C x) (cB : isContr (Sigma A B)) (cC : isContr (Sigma A C)) (x:A) ->isEquiv (B x) (C x) (f x) = \(A : U) -> \(B C : A -> U) -> \(f : (x : A) -> (B x) -> (C x)) -> \(cB : Sigma (Sigma A (\(x : A) -> B x)) (\(x : Sigma A (\(x : A) -> B x)) -> (y : Sigma A (\(x0 : A) -> B x0)) -> IdP ( Sigma A (\(x0 : A) -> B x0)) x y)) -> \(cC : Sigma (Sigma A (\(x : A) -> C x)) (\(x : Sigma A (\(x : A) -> C x)) -> (y : Sigma A (\(x0 : A) -> C x0)) -> IdP ( Sigma A (\(x0 : A) -> C x0)) x y)) -> \(x : A) -> \(z : C x) -> ((comp ( B (comp ( A) cC.1.1 [ (!0 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !1).1, (!0 = 1) -> (cC.2 ((x,z)) @ !1).1 ])) cB.1.2 [], comp ( C (comp ( A) cC.1.1 [ (!0 = 0) -> (cC.2 ((x,z)) @ !2).1, (!0 = 1)(!1 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !2).1, (!1 = 1) -> (cC.2 ((x,z)) @ !2).1 ])) (comp ( C (comp ( A) cC.1.1 [ (!0 = 0) -> (cC.2 ((x,z)) @ (!1 /\ !2)).1, (!0 = 1) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!1 /\ !2)).1, (!1 = 0) -> cC.1.1 ])) cC.1.2 [ (!0 = 0) -> (cC.2 ((x,z)) @ !1).2, (!0 = 1) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !1).2 ]) [ (!0 = 0) -> z, (!0 = 1) -> f (comp ( A) cC.1.1 [ (!1 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !2).1, (!1 = 1) -> (cC.2 ((x,z)) @ !2).1 ]) (comp ( B (comp ( A) cC.1.1 [ (!1 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !3).1, (!1 = 1)(!2 = 1) -> (cC.2 ((x,z)) @ !3).1, (!2 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !3).1 ])) cB.1.2 [ (!1 = 0) -> cB.1.2 ]) ]),\(x0 : Sigma (B x) (\(x0 : B x) -> IdP ( C x) z (f x x0))) -> (comp ( B x) (comp ( B (comp ( A) cC.1.1 [ (!0 = 0) -> comp ( A) cC.1.1 [ (!1 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!2 /\ !3)).1, (!1 = 1) -> (cC.2 ((x,z)) @ (!2 /\ !3)).1, (!2 = 0) -> cC.1.1 ], (!0 = 1) -> (cC.2 ((x,x0.2 @ -!1)) @ !2).1, (!1 = 0) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ !2).1, (!1 = 1) -> (cC.2 ((x,z)) @ !2).1 ])) (cB.2 ((x,x0.1)) @ !0).2 []) [ (!0 = 0) -> comp ( B (comp ( A) cC.1.1 [ (!0 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !1).1, (!0 = 1) -> (cC.2 ((x,z)) @ !1).1 ])) cB.1.2 [], (!0 = 1) -> comp ( B x) x0.1 [ (!1 = 1) -> x0.1 ] ], comp ( C x) (comp ( C (comp ( A) cC.1.1 [ (!0 = 0) -> comp ( A) cC.1.1 [ (!2 = 0) -> (cC.2 ((x,z)) @ (!4 /\ !5)).1, (!2 = 1)(!3 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!4 /\ !5)).1, (!3 = 1) -> (cC.2 ((x,z)) @ (!4 /\ !5)).1, (!4 = 0) -> cC.1.1 ], (!0 = 1) -> (cC.2 ((x,x0.2 @ (!2 /\ -!3))) @ !4).1, (!2 = 0) -> (cC.2 ((x,z)) @ !4).1, (!2 = 1)(!3 = 0) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ !4).1, (!3 = 1) -> (cC.2 ((x,z)) @ !4).1 ])) (comp ( C (comp ( A) cC.1.1 [ (!0 = 0) -> comp ( A) cC.1.1 [ (!2 = 0) -> (cC.2 ((x,z)) @ (!3 /\ !4 /\ !5)).1, (!2 = 1) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!3 /\ !4 /\ !5)).1, (!3 = 0) -> cC.1.1, (!4 = 0) -> cC.1.1 ], (!0 = 1) -> (cC.2 ((x,x0.2 @ !2)) @ (!3 /\ !4)).1, (!2 = 0) -> (cC.2 ((x,z)) @ (!3 /\ !4)).1, (!2 = 1) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ (!3 /\ !4)).1, (!3 = 0) -> cC.1.1 ])) cC.1.2 [ (!0 = 0) -> comp ( C (comp ( A) cC.1.1 [ (!2 = 0) -> (cC.2 ((x,z)) @ (!3 /\ !4 /\ !5)).1, (!2 = 1) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!3 /\ !4 /\ !5)).1, (!3 = 0) -> cC.1.1, (!4 = 0) -> cC.1.1 ])) cC.1.2 [ (!2 = 0) -> (cC.2 ((x,z)) @ (!3 /\ !4)).2, (!2 = 1) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!3 /\ !4)).2, (!3 = 0) -> cC.1.2 ], (!0 = 1) -> (cC.2 ((x,x0.2 @ !2)) @ !3).2, (!2 = 0) -> (cC.2 ((x,z)) @ !3).2, (!2 = 1) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ !3).2 ]) [ (!2 = 0) -> z, (!2 = 1) -> f (comp ( A) cC.1.1 [ (!0 = 0) -> comp ( A) cC.1.1 [ (!3 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!4 /\ !5)).1, (!3 = 1) -> (cC.2 ((x,z)) @ (!4 /\ !5)).1, (!4 = 0) -> cC.1.1 ], (!0 = 1) -> (cC.2 ((x,x0.2 @ -!3)) @ !4).1, (!3 = 0) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ !4).1, (!3 = 1) -> (cC.2 ((x,z)) @ !4).1 ]) (comp ( B (comp ( A) cC.1.1 [ (!0 = 0) -> comp ( A) cC.1.1 [ (!3 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!5 /\ !6)).1, (!3 = 1)(!4 = 1) -> (cC.2 ((x,z)) @ (!5 /\ !6)).1, (!4 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!5 /\ !6)).1, (!5 = 0) -> cC.1.1 ], (!0 = 1) -> (cC.2 ((x,x0.2 @ (-!3 \/ -!4))) @ !5).1, (!3 = 0) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ !5).1, (!3 = 1)(!4 = 1) -> (cC.2 ((x,z)) @ !5).1, (!4 = 0) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ !5).1 ])) (cB.2 ((x,x0.1)) @ !0).2 [ (!3 = 0) -> (cB.2 ((x,x0.1)) @ !0).2 ]) ]) [ (!0 = 0) -> comp ( C (comp ( A) cC.1.1 [ (!2 = 0) -> (cC.2 ((x,z)) @ !4).1, (!2 = 1)(!3 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !4).1, (!3 = 1) -> (cC.2 ((x,z)) @ !4).1 ])) (comp ( C (comp ( A) cC.1.1 [ (!2 = 0) -> (cC.2 ((x,z)) @ (!3 /\ !4)).1, (!2 = 1) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!3 /\ !4)).1, (!3 = 0) -> cC.1.1 ])) cC.1.2 [ (!2 = 0) -> (cC.2 ((x,z)) @ !3).2, (!2 = 1) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !3).2 ]) [ (!2 = 0) -> z, (!2 = 1) -> f (comp ( A) cC.1.1 [ (!3 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !4).1, (!3 = 1) -> (cC.2 ((x,z)) @ !4).1 ]) (comp ( B (comp ( A) cC.1.1 [ (!3 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !5).1, (!3 = 1)(!4 = 1) -> (cC.2 ((x,z)) @ !5).1, (!4 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !5).1 ])) cB.1.2 [ (!3 = 0) -> cB.1.2 ]) ], (!0 = 1) -> comp ( C x) (x0.2 @ !2) [ (!2 = 0) -> z, (!2 = 1) -> f x (comp ( B x) x0.1 [ (!3 = 1) -> x0.1, (!4 = 0) -> x0.1 ]), (!3 = 1) -> x0.2 @ !2 ], (!2 = 0) -> z, (!2 = 1) -> f x (comp ( B x) (comp ( B (comp ( A) cC.1.1 [ (!0 = 0) -> comp ( A) cC.1.1 [ (!1 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!2 /\ !3)).1, (!1 = 1) -> (cC.2 ((x,z)) @ (!2 /\ !3)).1, (!2 = 0) -> cC.1.1 ], (!0 = 1) -> (cC.2 ((x,x0.2 @ -!1)) @ !2).1, (!1 = 0) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ !2).1, (!1 = 1) -> (cC.2 ((x,z)) @ !2).1 ])) (cB.2 ((x,x0.1)) @ !0).2 []) [ (!0 = 0) -> comp ( B (comp ( A) cC.1.1 [ (!0 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !1).1, (!0 = 1) -> (cC.2 ((x,z)) @ !1).1 ])) cB.1.2 [], (!0 = 1) -> comp ( B x) x0.1 [ (!3 = 1)(!4 = 1) -> x0.1 ], (!3 = 0) -> comp ( B (comp ( A) cC.1.1 [ (!0 = 0) -> comp ( A) cC.1.1 [ (!1 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!2 /\ !3)).1, (!1 = 1) -> (cC.2 ((x,z)) @ (!2 /\ !3)).1, (!2 = 0) -> cC.1.1 ], (!0 = 1) -> (cC.2 ((x,x0.2 @ -!1)) @ !2).1, (!1 = 0) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ !2).1, (!1 = 1) -> (cC.2 ((x,z)) @ !2).1 ])) (cB.2 ((x,x0.1)) @ !0).2 [] ]) ])) +-- nequivFunFib : (A:U) (B C : A->U) (f : (x:A) -> B x -> C x) (cB : isContr (Sigma A B)) (cC : isContr (Sigma A C)) (x:A) ->isEquiv (B x) (C x) (f x) = \(A : U) -> \(B C : A -> U) -> \(f : (x : A) -> (B x) -> (C x)) -> \(cB : Sigma (Sigma A (\(x : A) -> B x)) (\(x : Sigma A (\(x : A) -> B x)) -> (y : Sigma A (\(x0 : A) -> B x0)) -> PathP ( Sigma A (\(x0 : A) -> B x0)) x y)) -> \(cC : Sigma (Sigma A (\(x : A) -> C x)) (\(x : Sigma A (\(x : A) -> C x)) -> (y : Sigma A (\(x0 : A) -> C x0)) -> PathP ( Sigma A (\(x0 : A) -> C x0)) x y)) -> \(x : A) -> \(z : C x) -> ((comp ( B (comp ( A) cC.1.1 [ (!0 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !1).1, (!0 = 1) -> (cC.2 ((x,z)) @ !1).1 ])) cB.1.2 [], comp ( C (comp ( A) cC.1.1 [ (!0 = 0) -> (cC.2 ((x,z)) @ !2).1, (!0 = 1)(!1 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !2).1, (!1 = 1) -> (cC.2 ((x,z)) @ !2).1 ])) (comp ( C (comp ( A) cC.1.1 [ (!0 = 0) -> (cC.2 ((x,z)) @ (!1 /\ !2)).1, (!0 = 1) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!1 /\ !2)).1, (!1 = 0) -> cC.1.1 ])) cC.1.2 [ (!0 = 0) -> (cC.2 ((x,z)) @ !1).2, (!0 = 1) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !1).2 ]) [ (!0 = 0) -> z, (!0 = 1) -> f (comp ( A) cC.1.1 [ (!1 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !2).1, (!1 = 1) -> (cC.2 ((x,z)) @ !2).1 ]) (comp ( B (comp ( A) cC.1.1 [ (!1 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !3).1, (!1 = 1)(!2 = 1) -> (cC.2 ((x,z)) @ !3).1, (!2 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !3).1 ])) cB.1.2 [ (!1 = 0) -> cB.1.2 ]) ]),\(x0 : Sigma (B x) (\(x0 : B x) -> PathP ( C x) z (f x x0))) -> (comp ( B x) (comp ( B (comp ( A) cC.1.1 [ (!0 = 0) -> comp ( A) cC.1.1 [ (!1 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!2 /\ !3)).1, (!1 = 1) -> (cC.2 ((x,z)) @ (!2 /\ !3)).1, (!2 = 0) -> cC.1.1 ], (!0 = 1) -> (cC.2 ((x,x0.2 @ -!1)) @ !2).1, (!1 = 0) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ !2).1, (!1 = 1) -> (cC.2 ((x,z)) @ !2).1 ])) (cB.2 ((x,x0.1)) @ !0).2 []) [ (!0 = 0) -> comp ( B (comp ( A) cC.1.1 [ (!0 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !1).1, (!0 = 1) -> (cC.2 ((x,z)) @ !1).1 ])) cB.1.2 [], (!0 = 1) -> comp ( B x) x0.1 [ (!1 = 1) -> x0.1 ] ], comp ( C x) (comp ( C (comp ( A) cC.1.1 [ (!0 = 0) -> comp ( A) cC.1.1 [ (!2 = 0) -> (cC.2 ((x,z)) @ (!4 /\ !5)).1, (!2 = 1)(!3 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!4 /\ !5)).1, (!3 = 1) -> (cC.2 ((x,z)) @ (!4 /\ !5)).1, (!4 = 0) -> cC.1.1 ], (!0 = 1) -> (cC.2 ((x,x0.2 @ (!2 /\ -!3))) @ !4).1, (!2 = 0) -> (cC.2 ((x,z)) @ !4).1, (!2 = 1)(!3 = 0) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ !4).1, (!3 = 1) -> (cC.2 ((x,z)) @ !4).1 ])) (comp ( C (comp ( A) cC.1.1 [ (!0 = 0) -> comp ( A) cC.1.1 [ (!2 = 0) -> (cC.2 ((x,z)) @ (!3 /\ !4 /\ !5)).1, (!2 = 1) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!3 /\ !4 /\ !5)).1, (!3 = 0) -> cC.1.1, (!4 = 0) -> cC.1.1 ], (!0 = 1) -> (cC.2 ((x,x0.2 @ !2)) @ (!3 /\ !4)).1, (!2 = 0) -> (cC.2 ((x,z)) @ (!3 /\ !4)).1, (!2 = 1) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ (!3 /\ !4)).1, (!3 = 0) -> cC.1.1 ])) cC.1.2 [ (!0 = 0) -> comp ( C (comp ( A) cC.1.1 [ (!2 = 0) -> (cC.2 ((x,z)) @ (!3 /\ !4 /\ !5)).1, (!2 = 1) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!3 /\ !4 /\ !5)).1, (!3 = 0) -> cC.1.1, (!4 = 0) -> cC.1.1 ])) cC.1.2 [ (!2 = 0) -> (cC.2 ((x,z)) @ (!3 /\ !4)).2, (!2 = 1) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!3 /\ !4)).2, (!3 = 0) -> cC.1.2 ], (!0 = 1) -> (cC.2 ((x,x0.2 @ !2)) @ !3).2, (!2 = 0) -> (cC.2 ((x,z)) @ !3).2, (!2 = 1) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ !3).2 ]) [ (!2 = 0) -> z, (!2 = 1) -> f (comp ( A) cC.1.1 [ (!0 = 0) -> comp ( A) cC.1.1 [ (!3 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!4 /\ !5)).1, (!3 = 1) -> (cC.2 ((x,z)) @ (!4 /\ !5)).1, (!4 = 0) -> cC.1.1 ], (!0 = 1) -> (cC.2 ((x,x0.2 @ -!3)) @ !4).1, (!3 = 0) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ !4).1, (!3 = 1) -> (cC.2 ((x,z)) @ !4).1 ]) (comp ( B (comp ( A) cC.1.1 [ (!0 = 0) -> comp ( A) cC.1.1 [ (!3 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!5 /\ !6)).1, (!3 = 1)(!4 = 1) -> (cC.2 ((x,z)) @ (!5 /\ !6)).1, (!4 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!5 /\ !6)).1, (!5 = 0) -> cC.1.1 ], (!0 = 1) -> (cC.2 ((x,x0.2 @ (-!3 \/ -!4))) @ !5).1, (!3 = 0) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ !5).1, (!3 = 1)(!4 = 1) -> (cC.2 ((x,z)) @ !5).1, (!4 = 0) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ !5).1 ])) (cB.2 ((x,x0.1)) @ !0).2 [ (!3 = 0) -> (cB.2 ((x,x0.1)) @ !0).2 ]) ]) [ (!0 = 0) -> comp ( C (comp ( A) cC.1.1 [ (!2 = 0) -> (cC.2 ((x,z)) @ !4).1, (!2 = 1)(!3 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !4).1, (!3 = 1) -> (cC.2 ((x,z)) @ !4).1 ])) (comp ( C (comp ( A) cC.1.1 [ (!2 = 0) -> (cC.2 ((x,z)) @ (!3 /\ !4)).1, (!2 = 1) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!3 /\ !4)).1, (!3 = 0) -> cC.1.1 ])) cC.1.2 [ (!2 = 0) -> (cC.2 ((x,z)) @ !3).2, (!2 = 1) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !3).2 ]) [ (!2 = 0) -> z, (!2 = 1) -> f (comp ( A) cC.1.1 [ (!3 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !4).1, (!3 = 1) -> (cC.2 ((x,z)) @ !4).1 ]) (comp ( B (comp ( A) cC.1.1 [ (!3 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !5).1, (!3 = 1)(!4 = 1) -> (cC.2 ((x,z)) @ !5).1, (!4 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !5).1 ])) cB.1.2 [ (!3 = 0) -> cB.1.2 ]) ], (!0 = 1) -> comp ( C x) (x0.2 @ !2) [ (!2 = 0) -> z, (!2 = 1) -> f x (comp ( B x) x0.1 [ (!3 = 1) -> x0.1, (!4 = 0) -> x0.1 ]), (!3 = 1) -> x0.2 @ !2 ], (!2 = 0) -> z, (!2 = 1) -> f x (comp ( B x) (comp ( B (comp ( A) cC.1.1 [ (!0 = 0) -> comp ( A) cC.1.1 [ (!1 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!2 /\ !3)).1, (!1 = 1) -> (cC.2 ((x,z)) @ (!2 /\ !3)).1, (!2 = 0) -> cC.1.1 ], (!0 = 1) -> (cC.2 ((x,x0.2 @ -!1)) @ !2).1, (!1 = 0) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ !2).1, (!1 = 1) -> (cC.2 ((x,z)) @ !2).1 ])) (cB.2 ((x,x0.1)) @ !0).2 []) [ (!0 = 0) -> comp ( B (comp ( A) cC.1.1 [ (!0 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ !1).1, (!0 = 1) -> (cC.2 ((x,z)) @ !1).1 ])) cB.1.2 [], (!0 = 1) -> comp ( B x) x0.1 [ (!3 = 1)(!4 = 1) -> x0.1 ], (!3 = 0) -> comp ( B (comp ( A) cC.1.1 [ (!0 = 0) -> comp ( A) cC.1.1 [ (!1 = 0) -> (cC.2 ((cB.1.1,f cB.1.1 cB.1.2)) @ (!2 /\ !3)).1, (!1 = 1) -> (cC.2 ((x,z)) @ (!2 /\ !3)).1, (!2 = 0) -> cC.1.1 ], (!0 = 1) -> (cC.2 ((x,x0.2 @ -!1)) @ !2).1, (!1 = 0) -> (cC.2 (((cB.2 ((x,x0.1)) @ !0).1,f (cB.2 ((x,x0.1)) @ !0).1 (cB.2 ((x,x0.1)) @ !0).2)) @ !2).1, (!1 = 1) -> (cC.2 ((x,z)) @ !2).1 ])) (cB.2 ((x,x0.1)) @ !0).2 [] ]) ])) lem1 (B:U) : isContr ((X:U) * equiv X B) = @@ -100,7 +100,7 @@ lem1 (B:U) : isContr ((X:U) * equiv X B) = ,fill ( B) b [(i=0) -> b ,(i=1) -> (w.2.2 b).1.2]) contr (v : fiber glueB B unglueElemB b) - : Id (fiber glueB B unglueElemB b) center v + : Path (fiber glueB B unglueElemB b) center v = (glueElem (comp ( B) b [(i=0) -> v.2 @ (j /\ k) ,(i=1) -> ((w.2.2 b).2 v @ j).2 ,(j=0) -> fill ( B) b [(i=0) -> b @@ -115,31 +115,31 @@ lem1 (B:U) : isContr ((X:U) * equiv X B) = ,(j=1) -> v.2]) in (center,contr))) --- nlem1 (B:U) : isContr ((X:U) * equiv X B) = ((B,(\(x : B) -> x,\(a : B) -> ((a, a),\(z : Sigma B (\(x : B) -> IdP ( B) a x)) -> (z.2 @ !0, z.2 @ (!0 /\ !1))))),\(w : Sigma U (\(X : U) -> Sigma (X -> B) (\(f : X -> B) -> (y : B) -> Sigma (Sigma X (\(x : X) -> IdP ( B) y (f x))) (\(x : Sigma X (\(x : X) -> IdP ( B) y (f x))) -> (y0 : Sigma X (\(x0 : X) -> IdP ( B) y (f x0))) -> IdP ( Sigma X (\(x0 : X) -> IdP ( B) y (f x0))) x y0)))) -> (glue B [ (!0 = 0) -> (B,(\(x : B) -> x,\(a : B) -> ((a, a),\(z : Sigma B (\(x : B) -> IdP ( B) a x)) -> (z.2 @ !0, z.2 @ (!0 /\ !1))))), (!0 = 1) -> w ],(\(g : glue B [ (!0 = 0) -> (B,(\(x : B) -> x,\(a : B) -> ((a, a),\(z : Sigma B (\(x : B) -> IdP ( B) a x)) -> (z.2 @ !0, z.2 @ (!0 /\ !1))))), (!0 = 1) -> w ]) -> unglueElem g [ (!0 = 0) -> (B,(\(x : B) -> x,\(a : B) -> ((a, a),\(z : Sigma B (\(x : B) -> IdP ( B) a x)) -> (z.2 @ !0, z.2 @ (!0 /\ !1))))), (!0 = 1) -> w ],\(b : B) -> ((glueElem (comp ( B) b [ (!0 = 0) -> b, (!0 = 1) -> (w.2.2 b).1.2 @ !1 ]) [ (!0 = 0) -> b, (!0 = 1) -> (w.2.2 b).1.1 ], comp ( B) b [ (!0 = 0) -> b, (!0 = 1) -> (w.2.2 b).1.2 @ (!1 /\ !2), (!1 = 0) -> b ]),\(v : Sigma (glue B [ (!0 = 0) -> (B,(\(x : B) -> x,\(a : B) -> ((a, a),\(z : Sigma B (\(x : B) -> IdP ( B) a x)) -> (z.2 @ !0, z.2 @ (!0 /\ !1))))), (!0 = 1) -> w ]) (\(x : glue B [ (!0 = 0) -> (B,(\(x : B) -> x,\(a : B) -> ((a, a),\(z : Sigma B (\(x : B) -> IdP ( B) a x)) -> (z.2 @ !0, z.2 @ (!0 /\ !1))))), (!0 = 1) -> w ]) -> IdP ( B) b (unglueElem x [ (!0 = 0) -> (B,(\(x0 : B) -> x0,\(a : B) -> ((a, a),\(z : Sigma B (\(x0 : B) -> IdP ( B) a x0)) -> (z.2 @ !0, z.2 @ (!0 /\ !1))))), (!0 = 1) -> w ]))) -> (glueElem (comp ( B) b [ (!0 = 0) -> v.2 @ (!1 /\ !2), (!0 = 1) -> ((w.2.2 b).2 v @ !1).2 @ !2, (!1 = 0) -> comp ( B) b [ (!0 = 0) -> b, (!0 = 1) -> (w.2.2 b).1.2 @ (!2 /\ !3), (!2 = 0) -> b ], (!1 = 1) -> v.2 @ !2 ]) [ (!0 = 0) -> v.2 @ !1, (!0 = 1) -> ((w.2.2 b).2 v @ !1).1 ], comp ( B) b [ (!0 = 0) -> v.2 @ (!1 /\ !2 /\ !3), (!0 = 1) -> ((w.2.2 b).2 v @ !1).2 @ (!2 /\ !3), (!1 = 0) -> comp ( B) b [ (!0 = 0) -> b, (!0 = 1) -> (w.2.2 b).1.2 @ (!2 /\ !3 /\ !4), (!2 = 0) -> b, (!3 = 0) -> b ], (!1 = 1) -> v.2 @ (!2 /\ !3), (!2 = 0) -> b ]))))) +-- nlem1 (B:U) : isContr ((X:U) * equiv X B) = ((B,(\(x : B) -> x,\(a : B) -> ((a, a),\(z : Sigma B (\(x : B) -> PathP ( B) a x)) -> (z.2 @ !0, z.2 @ (!0 /\ !1))))),\(w : Sigma U (\(X : U) -> Sigma (X -> B) (\(f : X -> B) -> (y : B) -> Sigma (Sigma X (\(x : X) -> PathP ( B) y (f x))) (\(x : Sigma X (\(x : X) -> PathP ( B) y (f x))) -> (y0 : Sigma X (\(x0 : X) -> PathP ( B) y (f x0))) -> PathP ( Sigma X (\(x0 : X) -> PathP ( B) y (f x0))) x y0)))) -> (glue B [ (!0 = 0) -> (B,(\(x : B) -> x,\(a : B) -> ((a, a),\(z : Sigma B (\(x : B) -> PathP ( B) a x)) -> (z.2 @ !0, z.2 @ (!0 /\ !1))))), (!0 = 1) -> w ],(\(g : glue B [ (!0 = 0) -> (B,(\(x : B) -> x,\(a : B) -> ((a, a),\(z : Sigma B (\(x : B) -> PathP ( B) a x)) -> (z.2 @ !0, z.2 @ (!0 /\ !1))))), (!0 = 1) -> w ]) -> unglueElem g [ (!0 = 0) -> (B,(\(x : B) -> x,\(a : B) -> ((a, a),\(z : Sigma B (\(x : B) -> PathP ( B) a x)) -> (z.2 @ !0, z.2 @ (!0 /\ !1))))), (!0 = 1) -> w ],\(b : B) -> ((glueElem (comp ( B) b [ (!0 = 0) -> b, (!0 = 1) -> (w.2.2 b).1.2 @ !1 ]) [ (!0 = 0) -> b, (!0 = 1) -> (w.2.2 b).1.1 ], comp ( B) b [ (!0 = 0) -> b, (!0 = 1) -> (w.2.2 b).1.2 @ (!1 /\ !2), (!1 = 0) -> b ]),\(v : Sigma (glue B [ (!0 = 0) -> (B,(\(x : B) -> x,\(a : B) -> ((a, a),\(z : Sigma B (\(x : B) -> PathP ( B) a x)) -> (z.2 @ !0, z.2 @ (!0 /\ !1))))), (!0 = 1) -> w ]) (\(x : glue B [ (!0 = 0) -> (B,(\(x : B) -> x,\(a : B) -> ((a, a),\(z : Sigma B (\(x : B) -> PathP ( B) a x)) -> (z.2 @ !0, z.2 @ (!0 /\ !1))))), (!0 = 1) -> w ]) -> PathP ( B) b (unglueElem x [ (!0 = 0) -> (B,(\(x0 : B) -> x0,\(a : B) -> ((a, a),\(z : Sigma B (\(x0 : B) -> PathP ( B) a x0)) -> (z.2 @ !0, z.2 @ (!0 /\ !1))))), (!0 = 1) -> w ]))) -> (glueElem (comp ( B) b [ (!0 = 0) -> v.2 @ (!1 /\ !2), (!0 = 1) -> ((w.2.2 b).2 v @ !1).2 @ !2, (!1 = 0) -> comp ( B) b [ (!0 = 0) -> b, (!0 = 1) -> (w.2.2 b).1.2 @ (!2 /\ !3), (!2 = 0) -> b ], (!1 = 1) -> v.2 @ !2 ]) [ (!0 = 0) -> v.2 @ !1, (!0 = 1) -> ((w.2.2 b).2 v @ !1).1 ], comp ( B) b [ (!0 = 0) -> v.2 @ (!1 /\ !2 /\ !3), (!0 = 1) -> ((w.2.2 b).2 v @ !1).2 @ (!2 /\ !3), (!1 = 0) -> comp ( B) b [ (!0 = 0) -> b, (!0 = 1) -> (w.2.2 b).1.2 @ (!2 /\ !3 /\ !4), (!2 = 0) -> b, (!3 = 0) -> b ], (!1 = 1) -> v.2 @ (!2 /\ !3), (!2 = 0) -> b ]))))) -contrSingl' (A : U) (a b : A) (p : Id A a b) : - Id ((x:A) * Id A x b) (b,refl A b) (a,p) = (p @ -i, p @ -i\/j) +contrSingl' (A : U) (a b : A) (p : Path A a b) : + Path ((x:A) * Path A x b) (b,refl A b) (a,p) = (p @ -i, p @ -i\/j) -lemSinglContr' (A:U) (a:A) : isContr ((x:A) * Id A x a) = - ((a,refl A a),\ (z:(x:A) * Id A x a) -> contrSingl' A z.1 a z.2) +lemSinglContr' (A:U) (a:A) : isContr ((x:A) * Path A x a) = + ((a,refl A a),\ (z:(x:A) * Path A x a) -> contrSingl' A z.1 a z.2) -thmUniv (t : (A X : U) -> Id U X A -> equiv X A) (A : U) : - (X : U) -> isEquiv (Id U X A) (equiv X A) (t A X) = - equivFunFib U (\(X : U) -> Id U X A) (\(X : U) -> equiv X A) +thmUniv (t : (A X : U) -> Path U X A -> equiv X A) (A : U) : + (X : U) -> isEquiv (Path U X A) (equiv X A) (t A X) = + equivFunFib U (\(X : U) -> Path U X A) (\(X : U) -> equiv X A) (t A) (lemSinglContr' U A) (lem1 A) -transEquiv' (A X : U) (p : Id U X A) : equiv X A = +transEquiv' (A X : U) (p : Path U X A) : equiv X A = substTrans U (\(Y : U) -> equiv Y A) A X ( p @ -i) (idEquiv A) -- The univalence axiom -univalence (A X : U) : isEquiv (Id U X A) (equiv X A) (transEquiv' A X) = +univalence (A X : U) : isEquiv (Path U X A) (equiv X A) (transEquiv' A X) = thmUniv transEquiv' A X -- The standard formulation of univalence whose normal can be computed: -corrUniv (A B : U) : Id U (Id U A B) (equiv A B) = - equivId (Id U A B) (equiv A B) (transEquiv' B A) (univalence B A) +corrUniv (A B : U) : Path U (Path U A B) (equiv A B) = + equivPath (Path U A B) (equiv A B) (transEquiv' B A) (univalence B A) -corrUniv' (A B : U) : equiv (Id U A B) (equiv A B) = +corrUniv' (A B : U) : equiv (Path U A B) (equiv A B) = (transEquiv' B A,univalence B A) @@ -149,59 +149,59 @@ corrUniv' (A B : U) : equiv (Id U A B) (equiv A B) = -- transEquiv is an equiv transEquivIsEquiv (A B : U) - : isEquiv (Id U A B) (equiv A B) (transEquiv A B) - = gradLemma (Id U A B) (equiv A B) (transEquiv A B) - (transEquivToId A B) (idToId A B) (eqToEq A B) + : isEquiv (Path U A B) (equiv A B) (transEquiv A B) + = gradLemma (Path U A B) (equiv A B) (transEquiv A B) + (transEquivToPath A B) (idToPath A B) (eqToEq A B) -- Univalence proved using transEquiv -- univalenceTrans takes extremely much memory when normalizing -univalenceTrans (A B:U) : Id U (Id U A B) (equiv A B) = - isoId (Id U A B) (equiv A B) (transEquiv A B) - (transEquivToId A B) (idToId A B) (eqToEq A B) +univalenceTrans (A B:U) : Path U (Path U A B) (equiv A B) = + isoPath (Path U A B) (equiv A B) (transEquiv A B) + (transEquivToPath A B) (idToPath A B) (eqToEq A B) -univalenceTrans' (A B : U) : equiv (Id U A B) (equiv A B) = +univalenceTrans' (A B : U) : equiv (Path U A B) (equiv A B) = (transEquiv A B,transEquivIsEquiv A B) -- This also takes too long to normalize: -slowtest (A : U) : Id (equiv A A) - (transEquiv A A (transEquivToId A A (idEquiv A))) (idEquiv A) = - idToId A A (idEquiv A) +slowtest (A : U) : Path (equiv A A) + (transEquiv A A (transEquivToPath A A (idEquiv A))) (idEquiv A) = + idToPath A A (idEquiv A) -- ------------------------------------------------------------------------------ -- -- TODO: Adapt this to new definition of equiv -- -- The canonical map defined using J --- -- canIdToEquiv (A : U) : (B : U) -> Id U A B -> equiv A B = --- -- J U A (\ (B : U) (_ : Id U A B) -> equiv A B) (idEquiv A) +-- -- canPathToEquiv (A : U) : (B : U) -> Path U A B -> equiv A B = +-- -- J U A (\ (B : U) (_ : Path U A B) -> equiv A B) (idEquiv A) --- -- canDiagTrans (A : U) : Id (A -> A) (canIdToEquiv A A (<_> A)).1 (idfun A) = +-- -- canDiagTrans (A : U) : Path (A -> A) (canPathToEquiv A A (<_> A)).1 (idfun A) = -- -- fill (<_> A -> A) (idfun A) [] @ -i --- -- transDiagTrans (A : U) : Id (A -> A) (idfun A) (trans A A (<_> A)) = +-- -- transDiagTrans (A : U) : Path (A -> A) (idfun A) (trans A A (<_> A)) = -- -- \ (a : A) -> fill (<_> A) a [] @ i --- -- canIdToEquivLem (A : U) : (B : U) (p : Id U A B) -> --- -- Id (A -> B) (canIdToEquiv A B p).1 (transEquiv A B p).1 +-- -- canPathToEquivLem (A : U) : (B : U) (p : Path U A B) -> +-- -- Path (A -> B) (canPathToEquiv A B p).1 (transEquiv A B p).1 -- -- = J U A --- -- (\ (B : U) (p : Id U A B) -> --- -- Id (A -> B) (canIdToEquiv A B p).1 (transEquiv A B p).1) --- -- (compId (A -> A) --- -- (canIdToEquiv A A (<_> A)).1 (idfun A) (trans A A (<_> A)) +-- -- (\ (B : U) (p : Path U A B) -> +-- -- Path (A -> B) (canPathToEquiv A B p).1 (transEquiv A B p).1) +-- -- (compPath (A -> A) +-- -- (canPathToEquiv A A (<_> A)).1 (idfun A) (trans A A (<_> A)) -- -- (canDiagTrans A) (transDiagTrans A)) --- -- canIdToEquivIsTransEquiv (A B : U) : --- -- Id (Id U A B -> equiv A B) (canIdToEquiv A B) (transEquiv A B) = --- -- \ (p : Id U A B) -> --- -- equivLemma A B (canIdToEquiv A B p) (transEquiv A B p) --- -- (canIdToEquivLem A B p) @ i +-- -- canPathToEquivIsTransEquiv (A B : U) : +-- -- Path (Path U A B -> equiv A B) (canPathToEquiv A B) (transEquiv A B) = +-- -- \ (p : Path U A B) -> +-- -- equivLemma A B (canPathToEquiv A B p) (transEquiv A B p) +-- -- (canPathToEquivLem A B p) @ i -- -- -- The canonical map is an equivalence --- -- univalenceJ (A B : U) : isEquiv (Id U A B) (equiv A B) (canIdToEquiv A B) = --- -- substInv (Id U A B -> equiv A B) --- -- (isEquiv (Id U A B) (equiv A B)) --- -- (canIdToEquiv A B) (transEquiv A B) --- -- (canIdToEquivIsTransEquiv A B) +-- -- univalenceJ (A B : U) : isEquiv (Path U A B) (equiv A B) (canPathToEquiv A B) = +-- -- substInv (Path U A B -> equiv A B) +-- -- (isEquiv (Path U A B) (equiv A B)) +-- -- (canPathToEquiv A B) (transEquiv A B) +-- -- (canPathToEquivIsTransEquiv A B) -- -- (transEquivIsEquiv A B) @@ -211,21 +211,21 @@ slowtest (A : U) : Id (equiv A A) isContrProp (A : U) (h : isContr A) : prop A = p where - p (a b : A) : Id A a b = comp (<_> A) h.1 [ (i = 0) -> h.2 a, (i = 1) -> h.2 b ] + p (a b : A) : Path A a b = comp (<_> A) h.1 [ (i = 0) -> h.2 a, (i = 1) -> h.2 b ] idEquiv (A : U) : equiv A A = (idfun A,idIsEquiv A) contrSinglEquiv (A B : U) (f : equiv A B) : - Id ((X : U) * equiv X B) (B,idEquiv B) (A,f) = rem + Path ((X : U) * equiv X B) (B,idEquiv B) (A,f) = rem where rem1 : prop ((X : U) * equiv X B) = isContrProp ((X : U) * equiv X B) (lem1 B) - rem : Id ((X : U) * equiv X B) (B,idEquiv B) (A,f) = rem1 (B,idEquiv B) (A,f) + rem : Path ((X : U) * equiv X B) (B,idEquiv B) (A,f) = rem1 (B,idEquiv B) (A,f) elimEquiv (B : U) (P : (A : U) -> (A -> B) -> U) (d : P B (idfun B)) (A : U) (f : equiv A B) : P A f.1 = rem where T (z : (X : U) * equiv X B) : U = P z.1 z.2.1 - rem1 : Id ((X : U) * equiv X B) (B,idEquiv B) (A,f) = contrSinglEquiv A B f + rem1 : Path ((X : U) * equiv X B) (B,idEquiv B) (A,f) = contrSinglEquiv A B f rem : P A f.1 = subst ((X : U) * equiv X B) T (B,idEquiv B) (A,f) rem1 d elimIso (B : U) (Q : (A : U) -> (A -> B) -> (B -> A) -> U) -- 2.34.1