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APG.v
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(* Coq Formalization of Algebraic Property Graphs, by Ryan Wisnesky.
See project at https://github.com/CategoricalData/ApgCoq *)
Open Scope type_scope.
Set Implicit Arguments.
Require Import JMeq FunctionalExtensionality
ProofIrrelevance Logic.ClassicalEpsilon Setoid.
(* Helper functions ****************************************)
Lemma FnHelper {X Y} (x:X) {P Q:X->Y}
(pf: (fun x => P x) = (fun x => Q x)) : P x = Q x.
Proof.
refine (match pf in _ = y return P x = y x with
| refl_equal => refl_equal
end).
Qed.
Lemma ExPfIrr {X} {Q:X->Prop} (h1 h2 : { x | Q x } )
: (proj1_sig h1 = proj1_sig h2) -> h1 = h2.
Proof.
intros pf. destruct h1,h2;simpl in *. subst. f_equal.
apply proof_irrelevance.
Qed.
Lemma ExistTHelper {A T} a t a0 t0 :
a = a0 -> JMeq t t0 -> @existT A T a t = @existT A T a0 t0.
Proof.
intros pf. subst. intros pf. f_equal. auto. apply JMeq_eq. auto.
Qed.
Lemma EqDecAx {X} (x1 x2 : X) : {x1 = x2} + {x1 <> x2}.
Proof.
apply excluded_middle_informative.
Qed.
Record Quotient {T} {R: T->T->Prop} (pfA : equivalence T R) := mkQuo {
Quo : Type;
eqc : T -> Quo;
repr: Quo -> T;
pfResp1 : forall {a b}, R a b -> eqc a = eqc b;
pfResp2 : forall {p q}, eqc p = eqc q -> R p q;
pfSurj : forall q, eqc (repr q) = q;
pfMake : forall {X} (f: T -> X) (pf: forall t1 t2, R t1 t2 -> f t1 = f t2), Quo -> X ;
pfProp : forall {X} (f: T -> X) (pf: forall t1 t2, R t1 t2 -> f t1 = f t2)
(x:T) , (pfMake f pf (eqc x)) = f x;
}.
Definition mapQ {T1 T2 R1 R2} {pf1: equivalence T1 R1} {pf2: equivalence T2 R2}
(Q1 : Quotient pf1) (Q2 : Quotient pf2) (f: T1 -> T2)
(pf: forall x y, R1 x y -> R2 (f x) (f y)) : Quo Q1 -> Quo Q2.
apply (pfMake Q1 (fun x => eqc Q2 (f x)) (fun x y pf0 => pfResp1 Q2 (pf x y pf0) )).
Defined.
Axiom axiomQ : forall {T} {R:T->T->Prop} {pf: equivalence T R}, Quotient pf.
(* Category Theory ************************************************* *)
Record Category : Type := newCategory {
ob : Type;
hom : ob -> ob -> Type;
Id : forall {x}, hom x x;
Comp : forall {x y z}, hom x y -> hom y z -> hom x z;
catId1 : forall {x y} (f : hom x y), (Comp Id f) = f;
catId2 : forall {x y} (f : hom x y), Comp f Id = f;
catComp : forall {w x y z}
(f: hom w x) g (h: hom y z), Comp f (Comp g h) = Comp (Comp f g) h;
}.
Definition Ob {C} := ob C.
Definition Hom {C} x y := hom C x y.
Definition id {C} x := @Id C x.
Definition comp {C} {x y z : ob C} (f: Hom x y) (g: Hom y z) := Comp C f g.
Definition terminal {C} (x: ob C) := forall a, exists! (f: Hom a x), True.
Definition initial {C} (x: ob C) := forall a, exists! (f: Hom x a), True.
Definition product {C} (X1 X2 X: ob C) (p1 : Hom X X1) (p2 : Hom X X2) :=
forall Y (f1 : Hom Y X1) (f2 : Hom Y X2),
exists! (f: Hom Y X), comp f p1 = f1 /\ comp f p2 = f2.
Definition coproduct {C} (X1 X2 X: ob C) (i1 : Hom X1 X) (i2 : Hom X2 X) :=
forall Y (f1 : Hom X1 Y) (f2 : Hom X2 Y),
exists! (f: Hom X Y), comp i1 f = f1 /\ comp i2 f = f2.
Definition coequalizer {C} (X Y: ob C) (f g: Hom X Y) Q (q : Hom Y Q) := comp f q = comp g q /\
forall Q0 (q0 : Hom Y Q0), comp f q0 = comp g q0 ->
exists! (u: Hom Q Q0), comp q u = q0.
Definition equalizer {C} (X Y: ob C) (f g: Hom Y X) Q (q : Hom Q Y) := comp q f = comp q g /\
forall Q0 (q0 : Hom Q0 Y), comp q0 f = comp q0 g ->
exists! (u: Hom Q0 Q), comp u q = q0.
(* Functors *****************************)
Record Functor C D := newFunctor {
ApplyO : ob C -> ob D;
ApplyF : forall {x y}, hom C x y -> hom D (ApplyO x) (ApplyO y);
funId : forall {x}, ApplyF (id x) = id (ApplyO x);
funComp : forall {x y z} (f: hom C x y) (g: hom C y z),
ApplyF (comp f g) = comp (ApplyF f) (ApplyF g);
}.
Definition applyF {C D} (F: Functor C D) {x y} := @ApplyF C D F x y.
Definition applyO {C D} (F: Functor C D) := @ApplyO C D F .
Definition IdFunctor C : Functor C C. refine (
newFunctor C C (fun x => x) (fun x y f => f) _ _); auto.
Defined.
Definition CompFunctor {C D E} (F: Functor C D) (G: Functor D E) : Functor C E.
refine (newFunctor C E (fun x => applyO G (applyO F x))
(fun _ _ f => applyF G (applyF F f)) _ _).
destruct F,G; compute in *; congruence.
destruct F,G; compute in *; congruence.
Defined.
(* Natural Transformations *****************************)
Record Transform {C D} (F G : Functor C D) := newTransform {
component : forall x, Hom (applyO F x) (applyO G x);
natural : forall {x y} (f : hom C x y),
comp (component x) (applyF G f) = comp (applyF F f) (component y);
}.
Definition Component {C D} {F G : Functor C D} x := @component C D F G x.
Lemma TransPfIrr {C D} {F G : Functor C D} (h1 h2 : Transform F G)
: (Component h1 = Component h2) -> h1 = h2.
Proof.
intros pf. destruct h1,h2;simpl in *. subst. f_equal. apply proof_irrelevance.
Qed.
Definition IdTrans {C D} (F: Functor C D) : Transform F F.
refine (newTransform F F (fun c => id (applyO F c)) _). intros.
unfold comp. rewrite (@catId1 D). rewrite (@catId2 D). constructor.
Defined.
Definition CompTrans {C D} {F G H: @Functor C D}
(h1 : Transform F G) (h2: Transform G H) : Transform F H.
refine (newTransform F H (fun c => comp (Component h1 c) (Component h2 c)) _).
intros. unfold comp.
rewrite (@catComp D). rewrite <- (natural h1).
rewrite <- (catComp D). rewrite (natural h2). rewrite (@catComp D).
constructor.
Defined.
(* Examples *****************************)
Definition SET : Category. refine (newCategory (fun x y => x -> y)
(fun x => fun y:x => y) (fun x y z f g a => g (f a)) _ _ _).
constructor. constructor. constructor.
Defined.
Definition CSet (C: Category) : Category.
refine (@newCategory (Functor C SET) (Transform) IdTrans (@CompTrans C SET) _ _ _);
intros; try apply TransPfIrr; try apply PI; simpl in *;
apply functional_extensionality_dep ; intros;
apply functional_extensionality_dep; intros; auto.
Defined.
Theorem SetInitial : @initial SET Empty_set.
Proof.
intros a. exists (fun x : Empty_set => match x with end).
split; auto. intros. apply functional_extensionality_dep.
intros x; elim x; auto.
Qed.
Theorem SetTerminal : @terminal SET unit.
Proof.
intros a. exists (fun x => tt).
split; auto. intros. apply functional_extensionality_dep.
intros x; destruct (x' x); auto.
Qed.
Theorem SetProduct {X1 X2} : @product SET X1 X2 (X1 * X2) fst snd.
Proof.
intros X. intros. exists (fun x => (f1 x, f2 x)).
unfold unique. split; simpl; auto. intros f pf.
apply functional_extensionality_dep.
intros x. destruct pf. subst. destruct (f x). simpl. auto.
Qed.
Theorem SetCoProduct {X1 X2} : @coproduct SET X1 X2 (X1 + X2) (@inl X1 X2) (@inr X1 X2).
Proof.
intros X. intros. exists (fun x => match x with inl y => f1 y | inr y => f2 y end).
unfold unique. split; simpl; auto. intros f pf. destruct pf.
apply functional_extensionality_dep.
intros x. subst. destruct x; auto.
Qed.
Definition EqualizerHelper {Y Z} (f : Y -> Z) (g : Y -> Z) Q0
(q0 : Q0 -> Y) : forall
(pf:(fun a : Q0 => f (q0 a)) = (fun a : Q0 => g (q0 a))) (u : Q0),
{y : Y | f y = g y}.
intros pf q. exists (q0 q). simpl. refine (
match pf in _ = y return f (q0 q) = y q with
| refl_equal => refl_equal
end
).
Defined.
Theorem SetEqualizer {X Y} (f g : Hom Y X)
: @equalizer SET X Y f g { y:Y | f y = g y} (fun x => proj1_sig x).
Proof.
split. simpl. apply functional_extensionality_dep. intros. destruct x; auto.
intros. simpl in *.
exists (EqualizerHelper f g q0 H). split. split. intros u pf.
compute. apply functional_extensionality_dep. intros q0'.
apply ExPfIrr. simpl. subst. auto.
Qed.
Inductive Lift {S T:Type} (f g : S -> T) : T -> T -> Prop :=
| LiftRefl : forall {t}, Lift f g t t
| LiftSym : forall {t1 t2}, Lift f g t1 t2 -> Lift f g t2 t1
| LiftTrans: forall {t1 t2 t3}, Lift f g t1 t2 -> Lift f g t2 t3 -> Lift f g t1 t3
| LiftInj : forall {s}, Lift f g (f s) (g s).
Lemma LiftEquiv {S T:Type} (f g : S -> T) : equivalence _ (Lift f g).
Proof.
split; compute; intros. constructor. exact (LiftTrans H H0). exact (LiftSym H).
Qed.
Definition quotient {S T} (f g : S -> T) := Quo (@axiomQ _ (Lift f g) (LiftEquiv f g)).
Definition Eqc {S T} (f g : S -> T) := eqc (@axiomQ _ (Lift f g) (LiftEquiv f g)).
Definition Repr {S T} (f g : S -> T) := repr (@axiomQ _ (Lift f g) (LiftEquiv f g)).
Theorem SetCoEq {X Y} f g
: @coequalizer SET X Y f g (quotient f g) (Eqc f g).
Proof.
split. apply functional_extensionality_dep.
intros x. compute in *. destruct (axiomQ ). apply pfResp3. apply LiftInj.
intros. exists ( @comp SET _ _ _ (Repr f g) q0 ).
split.
apply functional_extensionality_dep. intros y.
rewrite (catComp SET). simpl in *.
unfold Repr,Eqc.
destruct (axiomQ). simpl.
induction (@pfResp4 (repr0 (eqc0 y)) y).
auto. etransitivity; eauto. congruence.
symmetry. change ((fun a : X => q0 (g a)) s = q0 (f s)).
rewrite <- H. reflexivity. auto.
intros u pf. subst. simpl in *.
unfold Repr,Eqc,quotient in *. compute in *. destruct (axiomQ ).
apply functional_extensionality_dep. intros q.
f_equal. apply pfSurj0.
Qed.
(* Algebraic Property Graphs ******************************************)
Section APG.
Variable B : Type.
Variable P : B -> Type.
Inductive Ty L :=
| Zero : Ty L
| One : Ty L
| Plus : Ty L -> Ty L -> Ty L
| Times : Ty L -> Ty L -> Ty L
| Base : B -> Ty L
| Label : L -> Ty L.
Inductive Term {E L} (lam : E -> L) : Ty L -> Type :=
| ID : forall e, Term lam (Label (lam e))
| Prim: forall b (c: P b), Term lam (Base _ b)
| Tt : Term lam (One _)
| Pair: forall {t t'}, Term lam t -> Term lam t' -> Term lam (Times t t')
| Inl : forall {t t'}, Term lam t -> Term lam (Plus t t')
| Inr : forall {t t'}, Term lam t'-> Term lam (Plus t t').
Fixpoint MapTy {L1 L2} (f: L1 -> @Ty L2) (ty: @Ty L1) : @Ty L2 :=
match ty with
| Zero _ => Zero _
| One _ => One _
| Plus a b => Plus (MapTy f a) (MapTy f b)
| Times a b => Times (MapTy f a) (MapTy f b)
| Label l => f l
| Base _ b => Base _ b
end.
Fixpoint MapTerm {E1 L1} {lam1 : E1 -> L1} {E2 L2} {lam2 : E2 -> L2}
(g : L1 -> Ty L2) (h : forall (e:E1), Term lam2 (g (lam1 e)))
{ty} (term: Term lam1 ty) : Term lam2 (MapTy g ty) :=
match term with
| ID _ e => h e
| Prim _ c => Prim lam2 c
| Tt _ => Tt _
| Pair x y => Pair (MapTerm g h x) (MapTerm g h y)
| Inl x => Inl (MapTerm g h x)
| Inr y => Inr (MapTerm g h y)
end.
Definition mapTy {L1 L2} (f: L1 -> L2) (ty: @Ty L1) : @Ty L2
:= MapTy (fun x => Label (f x)) ty.
Definition mapTerm {E1 L1} {lam1 : E1 -> L1} {E2 L2} {lam2 : E2 -> L2}
(f : L1 -> L2) (g : E1 -> E2) (pfNat : forall e, f (lam1 e) = lam2 (g e))
: forall {ty} (x:Term lam1 ty) , Term lam2 (mapTy f ty).
refine (fix mapTerm {ty} term := match term with
| ID _ e => _
| Prim _ c => Prim lam2 c
| Tt _ => Tt _
| Pair x y => Pair (mapTerm x) (mapTerm y)
| Inl x => Inl (mapTerm x)
| Inr y => Inr (mapTerm y)
end). simpl. pose (ID lam2 (g e)). rewrite <- (pfNat e) in t. exact t.
Defined.
Record APG := newAPG {
L : Type;
sigma : L -> Ty L;
E : Type;
lam : E -> L;
phi : forall e, Term lam (sigma (lam e));
}.
(* Morphisms of APGs **********************************************)
Record APGMorphism (G1 G2: APG) := newAPGMorphism {
lMap : L G1 -> L G2 ;
eMap : E G1 -> E G2 ;
pfNat : forall e, lMap (lam G1 e) = lam G2 (eMap e);
}.
Lemma APGMorphismEq (G1 G2: APG) (h1 h2 : APGMorphism G1 G2) :
lMap h1 = lMap h2 -> eMap h1 = eMap h2 -> h1 = h2.
Proof.
intros pf1 pf2. destruct h1,h2; simpl in *. subst. f_equal. apply proof_irrelevance.
Qed.
Definition TransHelper1 G H (j k:APGMorphism G H) e0 : eMap j e0 =
eMap k e0 -> lMap j (lam G e0) =
lMap k (lam G e0).
intros pf. rewrite (pfNat j e0) . rewrite (pfNat k e0) .
rewrite pf. auto.
Defined.
Definition IdApgMorphism G : APGMorphism G G.
refine (newAPGMorphism G G (fun l => l) (fun e => e) (fun e => refl_equal _)).
Defined.
Definition ApgMorphismCompose {G1 G2 G3} (F: APGMorphism G1 G2) (G: APGMorphism G2 G3) : APGMorphism G1 G3.
refine (newAPGMorphism G1 G3 (fun l => lMap G (lMap F l)) (fun e => eMap G (eMap F e)) _).
intros e. destruct F,G. simpl. rewrite pfNat0. rewrite pfNat1. reflexivity.
Defined.
Definition APGInst : Category.
refine (@newCategory APG APGMorphism IdApgMorphism (@ApgMorphismCompose) _ _ _).
intros. unfold IdApgMorphism,ApgMorphismCompose. destruct f. simpl. f_equal. apply proof_irrelevance.
intros. unfold IdApgMorphism,ApgMorphismCompose. destruct f. simpl. f_equal. apply proof_irrelevance.
intros. unfold IdApgMorphism,ApgMorphismCompose. destruct f,g,h. simpl in *. f_equal. apply proof_irrelevance.
Defined.
Lemma lemma1 {X Y}
(f g : Hom Y X)
(Q0 : ob APGInst)
(q0 : Hom Q0 Y)
(pf : comp q0 f = comp q0 g)
(l : L Q0)
(pfNq : lMap f (lMap q0 l) <>
lMap g (lMap q0 l)) : False.
Proof.
destruct f,g; simpl in *. unfold ApgMorphismCompose in *. simpl in *.
inversion pf. pose (FnHelper l H0). contradiction.
Qed.
Lemma lemma2 {X Y}
(f g : Hom Y X)
(Q0 : ob APGInst)
(q0 : Hom Q0 Y)
(pf : comp q0 f = comp q0 g)
(e : E Q0)
(pfNq : eMap f (eMap q0 e) <>
eMap g (eMap q0 e)) : False.
Proof.
destruct f,g; simpl in *. unfold ApgMorphismCompose in *. simpl in *.
inversion pf. pose (FnHelper e H1). contradiction.
Qed.
(* Constructions on APGs ******************************************)
Definition InitialAPG : APG := @newAPG Empty_set (fun x => match x with end)
Empty_set (fun x => match x with end) (fun x => match x with end).
Definition InitialAPGMorphism (G: APG) : APGMorphism InitialAPG G.
refine (newAPGMorphism InitialAPG G
(fun x => match x with end) (fun x => match x with end) _).
intros. elim e.
Defined.
Theorem APGInstInitial : @initial APGInst InitialAPG.
Proof.
intros a. exists (InitialAPGMorphism a).
split; auto. intros. apply APGMorphismEq.
apply functional_extensionality_dep. intros x; elim x.
apply functional_extensionality_dep. intros x; elim x.
Qed.
Definition TerminalAPG : APG := @newAPG unit (fun x => One _)
unit (fun x => tt) (fun x => Tt _).
Definition TerminalAPGMorphism (G: APG) : APGMorphism G TerminalAPG.
refine (newAPGMorphism G TerminalAPG (fun x => tt) (fun x => tt) _).
intros. auto.
Defined.
Theorem APGInstTerminal : @terminal APGInst TerminalAPG.
Proof.
intros a. exists (TerminalAPGMorphism a).
split; auto. intros h. intros pf; clear pf. apply APGMorphismEq.
apply functional_extensionality_dep. intros l.
unfold TerminalAPG,TerminalAPGMorphism;simpl.
match goal with [ _ : _ |- tt = ?x ] => destruct x end. auto.
unfold TerminalAPG,TerminalAPGMorphism;simpl.
apply functional_extensionality_dep. intros e.
match goal with [ _ : _ |- tt = ?x ] => destruct x end. auto.
Qed.
Definition CoProductAPG (G1 G2 : APG) : APG.
refine (@newAPG
(L G1 + L G2)
(fun l => match l with
| inl a => mapTy inl (sigma G1 a)
| inr b => mapTy inr (sigma G2 b)
end)
(E G1 + E G2)
(fun l => match l with
| inl a => inl (lam G1 a)
| inr b => inr (lam G2 b)
end)
(fun e => match e with
| inl a => mapTerm inl inl _ (phi G1 a)
| inr b => mapTerm inr inr _ (phi G2 b)
end)); auto.
Defined.
Definition InlAPGMorphism (G1 G2: APG)
: APGMorphism G1 (CoProductAPG G1 G2).
refine (newAPGMorphism G1 (CoProductAPG G1 G2) (inl) (inl) _).
intros. auto.
Defined.
Definition InrAPGMorphism (G1 G2: APG)
: APGMorphism G2 (CoProductAPG G1 G2).
refine (newAPGMorphism G2 (CoProductAPG G1 G2) (inr) (inr) _).
intros. auto.
Defined.
Definition APGCoproduct (X1 X2: APG)
Y (f1 : APGMorphism X1 Y) (f2 : APGMorphism X2 Y)
: APGMorphism (CoProductAPG X1 X2) Y.
refine (newAPGMorphism (CoProductAPG X1 X2) Y (fun l => match l with
| inl a => lMap f1 a
| inr b => lMap f2 b
end) (fun e => match e with
| inl a => eMap f1 a
| inr b => eMap f2 b
end) _). intros. destruct e; simpl. destruct f1. auto. destruct f2. auto.
Defined.
Theorem APGHasCoProducts {X1 X2} : @coproduct APGInst X1 X2 (CoProductAPG X1 X2) (InlAPGMorphism X1 X2) (InrAPGMorphism X1 X2).
intros X. intros. exists (@APGCoproduct X1 X2 X f1 f2).
split. split.
apply APGMorphismEq. auto. auto.
apply APGMorphismEq. auto. auto.
intros f pf. destruct pf as [pf1 pf2].
subst. simpl. destruct f,X1,X2,X; compute in *.
apply APGMorphismEq. simpl in *. apply functional_extensionality_dep. intros x; destruct x; auto.
simpl in *. apply functional_extensionality_dep. intros x; destruct x; auto.
Qed.
Definition ProductAPG (G1 G2 : APG) : APG.
refine (@newAPG
(L G1 * L G2)
(fun l => match l with
| (a, b) => Times (mapTy (fun x => (x, b)) (sigma G1 a))
(mapTy (fun x => (a, x)) (sigma G2 b))
end)
(E G1 * E G2)
(fun l => match l with
| (a, b) => (lam G1 a, lam G2 b)
end)
(fun e => match e with
| (a, b) => Pair
(mapTerm _ (fun x => (x, b)) _ (phi G1 a))
(mapTerm _ (fun x => (a, x)) _ (phi G2 b))
end)); auto.
Defined.
Definition FstAPGMorphism (G1 G2: APG)
: APGMorphism (ProductAPG G1 G2) G1.
refine (newAPGMorphism (ProductAPG G1 G2) G1 fst fst _).
intros. destruct e. compute. auto.
Defined.
Definition SndAPGMorphism (G1 G2: APG)
: APGMorphism (ProductAPG G1 G2) G2.
refine (newAPGMorphism (ProductAPG G1 G2) G2 snd snd _).
intros. destruct e. compute. auto.
Defined.
Definition APGProduct (X1 X2: APG)
Y (f1 : APGMorphism Y X1) (f2 : APGMorphism Y X2)
: APGMorphism Y (ProductAPG X1 X2).
refine (newAPGMorphism Y (ProductAPG X1 X2)
(fun l => (lMap f1 l, lMap f2 l))
(fun e => (eMap f1 e, eMap f2 e))
_). intros. simpl. f_equal. destruct f1. simpl. auto.
destruct f2. simpl. auto.
Defined.
Theorem APGHasProducts {X1 X2} : @product APGInst X1 X2 (ProductAPG X1 X2) (FstAPGMorphism X1 X2) (SndAPGMorphism X1 X2).
Proof.
intros X. intros. exists (@APGProduct X1 X2 X f1 f2).
split. split.
apply APGMorphismEq. auto. auto.
apply APGMorphismEq. auto. auto.
intros f pf. destruct pf as [pf1 pf2].
subst. simpl. destruct f,X1,X2,X; compute in *.
apply APGMorphismEq. simpl in *. apply functional_extensionality_dep.
intros x. destruct (lMap0 x). auto.
simpl in *. apply functional_extensionality_dep. intros x.
destruct (eMap0 x); auto.
Qed.
Definition EqualizerAPG {G H : APG} (j k : APGMorphism G H) : APG.
refine (@newAPG
{ l : L G | lMap j l = lMap k l }
(fun l => match l with
| exist _ l0 pf => MapTy (fun l1 =>
match EqDecAx (lMap j l1) (lMap k l1) with
| left pfEq => Plus (One _) (Label (exist _ l1 pfEq))
| right pfEq => One _
end) (sigma G l0)
end)
{ e : E G | eMap j e = eMap k e }
(fun e => match e with
| exist _ e0 pf => exist _ (lam G e0) (TransHelper1 j k e0 pf)
end)
(fun e => match e with
| exist _ e0 pf => MapTerm (fun l1 =>
match EqDecAx (lMap j l1) (lMap k l1) with
| left pfEq => Plus (One _) (Label (exist _ l1 pfEq))
| right pfEq => One _
end)
(fun e1 => match EqDecAx (eMap j e1) (eMap k e1) with
| left pfEq => _
| right pfEq => _ end ) (phi G e0)
end)
).
pose (TransHelper1 j k e1 pfEq).
pose (ID (fun
e2 : {e2 : E G |
eMap j e2 = eMap k e2}
=>
let (e3, pf0) := e2 in
exist
(fun l : L G =>
lMap j l = lMap k l)
(lam G e3)
(TransHelper1 j k e3 pf0)) (exist _ e1 pfEq)).
destruct ( EqDecAx (lMap j (lam G e1))
(lMap k (lam G e1)) ).
assert ((TransHelper1 j k e1 pfEq) = e3). apply proof_irrelevance. rewrite <- H0. exact (Inr t).
contradiction.
destruct ( EqDecAx (lMap j (lam G e1))
(lMap k (lam G e1)) ). apply Inl. apply Tt. apply Tt.
Defined.
Definition EqualizerAPGMorphism {G H : APG} (j k : APGMorphism G H)
: APGMorphism (EqualizerAPG j k) G.
refine (newAPGMorphism (EqualizerAPG j k) G
(fun l => proj1_sig l) (fun e => proj1_sig e) _).
intros. destruct e. simpl in *. auto.
Defined.
Definition ApgEqualizerHelper {X Y} (f g : Hom Y X)
(Q0 : ob APGInst) (q0 : Hom Q0 Y) (pf : comp q0 f = comp q0 g)
: Hom Q0 (EqualizerAPG f g).
refine (newAPGMorphism Q0 (EqualizerAPG f g)
(fun l => match EqDecAx (lMap f (lMap q0 l)) (lMap g (lMap q0 l)) with
| left pfEq => exist _ (lMap q0 l) pfEq
| right pfNq => match lemma1 pf l pfNq with end
end) (fun e => match EqDecAx (eMap f (eMap q0 e)) (eMap g (eMap q0 e)) with
| left pfEq => exist _ (eMap q0 e) pfEq
| right pfNq => match lemma2 pf e pfNq with end
end) _).
intros. destruct f,g; simpl in *.
unfold ApgMorphismCompose in *. simpl in *.
inversion pf. simpl in *.
destruct X,Y,q0,Q0; compute in *.
pose (FnHelper e H1). pose (FnHelper (lam2 e) H0).
simpl in *. destruct ( EqDecAx (lMap0 (lMap2 (lam2 e)))
(lMap1 (lMap2 (lam2 e))) ). simpl in *.
apply ExPfIrr. simpl. destruct (EqDecAx (eMap0 (eMap2 e))
(eMap1 (eMap2 e))). simpl. auto.
simpl. contradiction. simpl. contradiction.
Defined.
Theorem APGHasEqualizers {X Y} f g : @equalizer APGInst X Y f g
(EqualizerAPG f g) (EqualizerAPGMorphism f g).
Proof.
split. simpl. unfold EqualizerAPGMorphism,ApgMorphismCompose.
apply APGMorphismEq. simpl. apply functional_extensionality.
intros l0; destruct l0; simpl in *; auto.
simpl. apply functional_extensionality.
intros l0; destruct l0; simpl in *; auto.
intros Q0 q0 pf. exists (ApgEqualizerHelper pf).
destruct f,g. unfold ApgMorphismCompose in *.
unfold comp in *. unfold Comp in *. simpl in *. inversion pf. split.
apply APGMorphismEq. simpl. apply functional_extensionality.
intros l. pose (FnHelper l H0).
simpl in *. destruct ( EqDecAx (lMap0 (lMap q0 l))
(lMap1 (lMap q0 l)) ). simpl in *. auto.
contradiction. simpl in *.
apply functional_extensionality.
intros e. pose (FnHelper e H1). destruct ( EqDecAx (eMap0 (eMap q0 e))
(eMap1 (eMap q0 e)) ). simpl in *. auto.
simpl in *. contradiction.
intros u pf2. apply APGMorphismEq. simpl. apply functional_extensionality.
intros l. pose (FnHelper l H0).
simpl in *. destruct ( EqDecAx (lMap0 (lMap q0 l))
(lMap1 (lMap q0 l)) ). apply ExPfIrr. simpl.
subst. simpl in *. auto. contradiction.
simpl in *. apply functional_extensionality.
intros e. pose (FnHelper e H1). destruct ( EqDecAx (eMap0 (eMap q0 e))
(eMap1 (eMap q0 e)) ). simpl in *. apply ExPfIrr. simpl in *.
subst. simpl in *. auto. contradiction.
Qed.
(*******************************************************************)
(* For co-equalizers, we specialize the definitions. *)
Section coeq.
Variable sL : Type.
Variable ssigma : sL -> Ty sL.
Record sAPG := newsAPG {
sE : Type;
slam : sE -> sL;
sphi : forall e, Term slam (ssigma (slam e));
}.
Record sAPGMorphism (G1 G2: sAPG) := newsAPGMorphism {
seMap : sE G1 -> sE G2 ;
spfNat : forall e, (slam G1 e) = slam G2 (seMap e)
}.
Lemma sAPGMorphismEq (G1 G2: sAPG) (h1 h2 : sAPGMorphism G1 G2) :
seMap h1 = seMap h2 -> h1 = h2.
Proof.
intros pf1. destruct h1,h2; simpl in *. subst. f_equal. apply proof_irrelevance.
Qed.
Definition IdsApgMorphism G : sAPGMorphism G G.
refine (newsAPGMorphism G G (fun e => e) (fun e => refl_equal _)).
Defined.
Definition sApgMorphismCompose {G1 G2 G3} (F: sAPGMorphism G1 G2) (G: sAPGMorphism G2 G3) : sAPGMorphism G1 G3.
refine (newsAPGMorphism G1 G3 (fun e => seMap G (seMap F e)) _).
intros e. destruct F,G. simpl. rewrite spfNat0. rewrite spfNat1. reflexivity.
Defined.
Definition sAPGInst : Category.
refine (@newCategory sAPG sAPGMorphism IdsApgMorphism (@sApgMorphismCompose) _ _ _).
intros. unfold IdsApgMorphism,sApgMorphismCompose. destruct f. simpl. f_equal. apply proof_irrelevance.
intros. unfold IdsApgMorphism,sApgMorphismCompose. destruct f. simpl. f_equal. apply proof_irrelevance.
intros. unfold IdsApgMorphism,sApgMorphismCompose. destruct f,g,h. simpl in *. f_equal. apply proof_irrelevance.
Defined.
Definition mapTermAlt {E1} {lam1 : E1 -> sL} {E2}
(g : E1 -> E2) : forall
{ty} (x:Term lam1 ty) (lam2 : E2 -> sL)
(pfNat : forall e, (lam1 e) = lam2 (g e)) , Term lam2 ty.
refine (fix mapTermAlt {ty} term lam2 pf := match term with
| ID _ e => _
| Prim _ c => Prim _ c
| Tt _ => Tt _
| Pair x y => Pair (mapTermAlt x lam2 pf) (mapTermAlt y lam2 pf)
| Inl x => Inl (mapTermAlt x lam2 pf)
| Inr y => Inr (mapTermAlt y lam2 pf)
end). simpl. pose (ID lam2 (g e)). rewrite <- (pf e) in t. exact t.
Defined.
Definition CoEqualizersAPG {G1 G2} (j k : sAPGMorphism G1 G2)
(pf: forall x y, Lift (seMap j) (seMap k) x y -> slam G2 x = slam G2 y) : sAPG.
refine (@newsAPG (quotient (seMap j) (seMap k))
(fun e => slam G2 (Repr _ _ e)) (fun e => _ (sphi G2 (Repr _ _ e)))).
intros a.
apply (@mapTermAlt (sE G2) (slam G2) (quotient (seMap j) (seMap k)) (fun x => Eqc _ _ x) (ssigma (slam G2 (Repr _ _ e))) a (fun e => slam G2 (Repr _ _ e))).
intros x. apply pf. unfold Repr,Eqc,quotient in *.
destruct ( (axiomQ
)). simpl in *. apply pfResp4. auto.
Defined.
Definition CoEqualizersAPGMorphism {G1 G2} (j k : sAPGMorphism G1 G2)
(pf: forall x y, Lift (seMap j) (seMap k) x y -> slam G2 x = slam G2 y)
: sAPGMorphism G2 (CoEqualizersAPG j k pf).
destruct j,k,G2. refine (newsAPGMorphism _ (CoEqualizersAPG
{|
seMap := seMap0;
spfNat := spfNat0 |}
{|
seMap := seMap1;
spfNat := spfNat1 |} pf) (fun e => Eqc _ _ e) _).
compute in *.
destruct ( (axiomQ )). simpl in *.
intros e. apply pf. auto.
Defined.
Definition ApgCoEqualizerHelper
{G1 G2}
(j k : sAPGMorphism G1 G2)
(pf : forall x y : sE G2,
Lift (seMap j)
(seMap k) x y ->
slam G2 x = slam G2 y)
{Q0}
(q0 : sAPGMorphism G2 Q0)
(H : sApgMorphismCompose j q0 =
sApgMorphismCompose k q0)
: hom sAPGInst (CoEqualizersAPG j k pf) Q0.
refine (newsAPGMorphism (CoEqualizersAPG j k pf) Q0 (fun e => seMap q0 (Repr _ _ e)) _).
destruct j,k,q0,Q0. compute in *. destruct ( (axiomQ )).
intros q. apply spfNat2.
Defined.
Theorem APGHasCoEqualizers {G1 G2}
(j k : sAPGMorphism G1 G2)
(pf3: forall x y, Lift (seMap j) (seMap k) x y -> slam G2 x = slam G2 y)
: @coequalizer sAPGInst _ _ j k
(CoEqualizersAPG j k pf3) (CoEqualizersAPGMorphism j k pf3).
Proof.
split.
apply sAPGMorphismEq. apply functional_extensionality. intros e.
destruct j,k,G1,G2. compute in *.
destruct ( (axiomQ )).
apply pfResp3. apply LiftInj.
simpl in *. intros. exists (@ApgCoEqualizerHelper G1 G2 j k pf3 Q0 q0 H). split.
destruct j,k,q0,Q0,G1,G2. compute in *.
destruct ( (axiomQ )).
compute in *. apply sAPGMorphismEq. simpl in *.
inversion H; clear H. apply functional_extensionality_dep. intros e.
induction (pfResp4 e (repr0 (eqc0 e))). auto. auto. congruence. symmetry. apply (FnHelper s H1). auto.
intros u pf. destruct j,k,q0,Q0,G1,G2,u. compute in *.
destruct ( (axiomQ )).
inversion H; clear H.
inversion pf; clear pf. subst.
apply sAPGMorphismEq. simpl in *.
apply functional_extensionality_dep. intros q.
rewrite (pfSurj0 q). auto.
Qed.
End coeq.
(* The Commutative Square Category ********************************* *)
Inductive CD_O : Type := EE | TT | LL | VV.
Inductive CD_A : CD_O -> CD_O -> Type := | EEEE : CD_A EE EE | EEVV : CD_A EE VV | TTTT : CD_A TT TT | LLLL : CD_A LL LL | VVVV : CD_A VV VV | EELL : CD_A EE LL | LLTT : CD_A LL TT | VVTT : CD_A VV TT | EETT : CD_A EE TT.
Definition CD_id x: CD_A x x := match x as x return CD_A x x with | EE => EEEE | TT => TTTT | LL => LLLL | VV => VVVV end.
Definition CD_o1 x : forall z y (f: CD_A x y) (g: CD_A y z), CD_A x z.
refine (match x as x return forall z y (f: CD_A x y) (g: CD_A y z), CD_A x z with
| EE => fun z => match z as z return forall y (f: CD_A EE y) (g: CD_A y z), CD_A EE z with | EE => fun y f g => EEEE | TT => fun y f g => EETT | LL => fun y f g => EELL | VV => fun y f g => EEVV end
| TT => fun z => match z as z return forall y (f: CD_A TT y) (g: CD_A y z), CD_A TT z with | EE => fun y f g => _ | TT => fun y f g => TTTT | LL => fun y f g => _ | VV => fun y f g => _ end
| LL => fun z => match z as z return forall y (f: CD_A LL y) (g: CD_A y z), CD_A LL z with | EE => fun y f g => _ | TT => fun y f g => LLTT | LL => fun y f g => LLLL | VV => fun y f g => _ end
| VV => fun z => match z as z return forall y (f: CD_A VV y) (g: CD_A y z), CD_A VV z with | EE => fun y f g => _ | TT => fun y f g => VVTT | LL => fun y f g => _ | VV => fun y f g => VVVV end
end); destruct y; inversion f; try auto; inversion g; try auto.
Defined.
Definition CD_o x y z (f: CD_A x y) (g: CD_A y z) : CD_A x z := CD_o1 f g.
Definition CDCat : Category. refine (@newCategory CD_O CD_A CD_id CD_o _ _ _); intros.
destruct f; auto.
destruct f; auto.
inversion f; inversion g; inversion h;
subst; try auto; try discriminate.
Defined.
Lemma lEEEE (g : CD_A EE EE) : g = EEEE. Proof. refine (match g with | EEEE => _ | EEVV => _ | TTTT => _ | LLLL => _ | VVVV => _ | EELL => _ | LLTT => _ | VVTT => _ | EETT => _ end); compute; auto. Qed.
Lemma lLLLL (g : CD_A LL LL) : g = LLLL. Proof. refine (match g with | EEEE => _ | EEVV => _ | TTTT => _ | LLLL => _ | VVVV => _ | EELL => _ | LLTT => _ | VVTT => _ | EETT => _ end); compute; auto. Qed.
Lemma lTTTT (g : CD_A TT TT) : g = TTTT. Proof. refine (match g with | EEEE => _ | EEVV => _ | TTTT => _ | LLLL => _ | VVVV => _ | EELL => _ | LLTT => _ | VVTT => _ | EETT => _ end); compute; auto. Qed.
Lemma lVVVV (g : CD_A VV VV) : g = VVVV. Proof. refine (match g with | EEEE => _ | EEVV => _ | TTTT => _ | LLLL => _ | VVVV => _ | EELL => _ | LLTT => _ | VVTT => _ | EETT => _ end); compute; auto. Qed.
Lemma lEELL (g : CD_A EE LL) : g = EELL. Proof. refine (match g with | EEEE => _ | EEVV => _ | TTTT => _ | LLLL => _ | VVVV => _ | EELL => _ | LLTT => _ | VVTT => _ | EETT => _ end); compute; auto. Qed.
Lemma lEETT (g : CD_A EE TT) : g = EETT. Proof. refine (match g with | EEEE => _ | EEVV => _ | TTTT => _ | LLLL => _ | VVVV => _ | EELL => _ | LLTT => _ | VVTT => _ | EETT => _ end); compute; auto. Qed.
Lemma lVVTT (g : CD_A VV TT) : g = VVTT. Proof. refine (match g with | EEEE => _ | EEVV => _ | TTTT => _ | LLLL => _ | VVVV => _ | EELL => _ | LLTT => _ | VVTT => _ | EETT => _ end); compute; auto. Qed.
Lemma lLLTT (g : CD_A LL TT) : g = LLTT. Proof. refine (match g with | EEEE => _ | EEVV => _ | TTTT => _ | LLLL => _ | VVVV => _ | EELL => _ | LLTT => _ | VVTT => _ | EETT => _ end); compute; auto. Qed.
Lemma lEEVV (g : CD_A EE VV) : g = EEVV. Proof. refine (match g with | EEEE => _ | EEVV => _ | TTTT => _ | LLLL => _ | VVVV => _ | EELL => _ | LLTT => _ | VVTT => _ | EETT => _ end); compute; auto. Qed.
Lemma lVVEE (g : CD_A VV EE) : False. Proof. refine (match g with | EEEE => _ | EEVV => _ | TTTT => _ | LLLL => _ | VVVV => _ | EELL => _ | LLTT => _ | VVTT => _ | EETT => _ end); compute; auto. Qed.
Lemma lVVLL (g : CD_A VV LL) : False. Proof. refine (match g with | EEEE => _ | EEVV => _ | TTTT => _ | LLLL => _ | VVVV => _ | EELL => _ | LLTT => _ | VVTT => _ | EETT => _ end); compute; auto. Qed.
Lemma lTTEE (g : CD_A TT EE) : False. Proof. refine (match g with | EEEE => _ | EEVV => _ | TTTT => _ | LLLL => _ | VVVV => _ | EELL => _ | LLTT => _ | VVTT => _ | EETT => _ end); compute; auto. Qed.
Lemma lTTLL (g : CD_A TT LL) : False. Proof. refine (match g with | EEEE => _ | EEVV => _ | TTTT => _ | LLLL => _ | VVVV => _ | EELL => _ | LLTT => _ | VVTT => _ | EETT => _ end); compute; auto. Qed.
Lemma lLLEE (g : CD_A LL EE) : False. Proof. refine (match g with | EEEE => _ | EEVV => _ | TTTT => _ | LLLL => _ | VVVV => _ | EELL => _ | LLTT => _ | VVTT => _ | EETT => _ end); compute; auto. Qed.
Lemma lTTVV (g : CD_A TT VV) : False. Proof. refine (match g with | EEEE => _ | EEVV => _ | TTTT => _ | LLLL => _ | VVVV => _ | EELL => _ | LLTT => _ | VVTT => _ | EETT => _ end); compute; auto. Qed.
Lemma lLLVV (g : CD_A LL VV) : False. Proof. refine (match g with | EEEE => _ | EEVV => _ | TTTT => _ | LLLL => _ | VVVV => _ | EELL => _ | LLTT => _ | VVTT => _ | EETT => _ end); compute; auto. Qed.
(* APG Morphisms as Natural Transformations *************************)
Definition APGtoFunctor (G:APG) : Functor CDCat SET. refine (
newFunctor CDCat SET
(fun x => match x with
| EE => E G
| TT => Ty (L G)
| LL => L G
| VV => sigT (Term (lam G))
end )
(fun z y f => match f with
| EEEE => fun x => x
| TTTT => fun x => x
| LLLL => fun x => x
| VVVV => fun x => x
| EEVV => fun x => existT _ _ (phi G x)
| EELL => fun x => lam G x
| LLTT => fun x => sigma G x
| VVTT => fun x => projT1 x
| EETT => fun x => sigma G (lam G x)
end) _ _).
intros. destruct x; auto.
intros. destruct x,y,z;simpl;
try (rewrite ( lEEEE g)); try (rewrite ( lLLLL g)); try (rewrite ( lTTTT g)); try (rewrite ( lVVVV g)); try (rewrite ( lEELL g)); try (rewrite ( lEETT g)); try (rewrite ( lVVTT g)); try (rewrite ( lLLTT g)); try (rewrite ( lEEVV g)); try (rewrite ( lVVEE g)); try (rewrite ( lVVLL g)); try (rewrite ( lTTEE g)); try (rewrite ( lTTLL g)); try (rewrite ( lLLEE g)); try (rewrite ( lTTVV g)); try (rewrite ( lLLVV g)); try (rewrite ( lEEEE f)); try (rewrite ( lLLLL f)); try (rewrite ( lTTTT f)); try (rewrite ( lVVVV f)); try (rewrite ( lEELL f)); try (rewrite ( lEETT f)); try (rewrite ( lVVTT f)); try (rewrite ( lLLTT f)); try (rewrite ( lEEVV f)); try (rewrite ( lVVEE f)); try (rewrite ( lVVLL f)); try (rewrite ( lTTEE f)); try (rewrite ( lTTLL f)); try (rewrite ( lLLEE f)); try (rewrite ( lTTVV f)); try (rewrite ( lLLVV f)) ; auto;
try (elim ( lEEEE g)); try (elim ( lLLLL g)); try (elim ( lTTTT g)); try (elim ( lVVVV g)); try (elim ( lEELL g)); try (elim ( lEETT g)); try (elim ( lVVTT g)); try (elim ( lLLTT g)); try (elim ( lEEVV g)); try (elim ( lVVEE g)); try (elim ( lVVLL g)); try (elim ( lTTEE g)); try (elim ( lTTLL g)); try (elim ( lLLEE g)); try (elim ( lTTVV g)); try (elim ( lLLVV g)); try (elim ( lEEEE f)); try (elim ( lLLLL f)); try (elim ( lTTTT f)); try (elim ( lVVVV f)); try (elim ( lEELL f)); try (elim ( lEETT f)); try (elim ( lVVTT f)); try (elim ( lLLTT f)); try (elim ( lEEVV f)); try (elim ( lVVEE f)); try (elim ( lVVLL f)); try (elim ( lTTEE f)); try (elim ( lTTLL f)); try (elim ( lLLEE f)); try (elim ( lTTVV f)); try (elim ( lLLVV f)).
Defined.
Definition APGInstBig : Category.
refine (
@newCategory APG
(fun x y => Transform (APGtoFunctor x) (APGtoFunctor y) )
(fun x => IdTrans (APGtoFunctor x) )
(fun x y z f g => CompTrans f g ) _ _ _ ).
intros; try apply TransPfIrr; try apply PI; simpl in *;
apply functional_extensionality_dep ; intros;
apply functional_extensionality_dep; intros; auto.
intros; try apply TransPfIrr; try apply PI; simpl in *;
apply functional_extensionality_dep ; intros;
apply functional_extensionality_dep; intros; auto.
intros; try apply TransPfIrr; try apply PI; simpl in *;
apply functional_extensionality_dep ; intros;
apply functional_extensionality_dep; intros; auto.
Defined.
(* Natural Transformations *************************************************)
Lemma ExistTHelper2 {A T} a t a0 t0 :
a = a0 -> (a = a0 -> JMeq t t0) -> @existT A T a t = @existT A T a0 t0.
Proof.
intros pf. subst. intros pf. f_equal. pose (pf (refl_equal _)).
clearbody j. clear pf. auto. apply JMeq_eq. auto.
Qed.
Lemma TyInvOne {L E} {lam1 : E -> L}
(x : Term lam1 (One L)) : x = (Tt lam1).
Proof.
refine (match x as x0 in Term _ z return z = One L -> JMeq x x0 -> x = Tt lam1 with
| ID _ e => fun pf pf2 => _
| Prim _ c => fun pf pf2 => _
| Tt _ => fun pf pf2 => _
| Pair x y => fun pf pf2 => _
| Inl x => fun pf pf2 => _
| Inr y => fun pf pf2 => _
end ( refl_equal _ ) (JMeq_refl _)); try discriminate; try auto.
apply JMeq_eq. auto.
Qed.
Lemma TyInvZero {L E} {lam1 : E -> L}
(x : Term lam1 (Zero L)) : False.
Proof.
refine (match x as x0 in Term _ z return z = Zero L -> JMeq x x0 -> False with
| ID _ e => fun pf pf2 => _
| Prim _ c => fun pf pf2 => _
| Tt _ => fun pf pf2 => _
| Pair x y => fun pf pf2 => _
| Inl x => fun pf pf2 => _
| Inr y => fun pf pf2 => _
end ( refl_equal _ ) (JMeq_refl _)); try discriminate; try auto.
Qed.
Lemma TyInvPair {L E} {lam1 : E -> L} {t1 t2}
(x : Term lam1 (Times t1 t2)) : exists a, exists b,
x = Pair a b.
Proof.
refine (match x as x0 in Term _ z return z = Times t1 t2 -> JMeq x x0 -> exists a, exists b,
x = Pair a b with
| ID _ e => fun pf pf2 => _
| Prim _ c => fun pf pf2 => _
| Tt _ => fun pf pf2 => _
| Pair x y => fun pf pf2 => _
| Inl x => fun pf pf2 => _
| Inr y => fun pf pf2 => _
end ( refl_equal _ ) (JMeq_refl _)); try discriminate; try auto.
assert (t0 = t2). injection pf. intros pf1. subst. auto.
subst. assert (t = t1). injection pf. auto. subst. clear pf.
exists x. exists y. apply JMeq_eq. auto.
Qed.
Lemma TyInvPlus {L E} {lam1 : E -> L} {t1 t2}
(x : Term lam1 (Plus t1 t2)) : (exists a, x = Inl a ) \/ (exists b, x = Inr b).
Proof.
refine (match x as x0 in Term _ z return z = Plus t1 t2 -> JMeq x x0 ->
(exists a, x = Inl a ) \/ (exists b, x = Inr b) with
| ID _ e => fun pf pf2 => _
| Prim _ c => fun pf pf2 => _
| Tt _ => fun pf pf2 => _
| Pair x y => fun pf pf2 => _
| Inl x => fun pf pf2 => _
| Inr y => fun pf pf2 => _
end ( refl_equal _ ) (JMeq_refl _)); try discriminate; try auto.
assert (t0 = t2). injection pf. intros pf1. subst. auto.
subst. assert (t = t1). injection pf. auto. subst. clear pf.
left. exists x. apply JMeq_eq. auto. right. assert (t0 = t2). injection pf. intros pf1. subst. auto.
subst. assert (t = t1). injection pf. auto. subst. clear pf.
exists y. apply JMeq_eq. auto.
Qed.
Definition APGMorphismToTransform {G1 G2:APG} (h: APGMorphism G1 G2)
(pfNatL : forall l, sigma G2 (lMap h l) = mapTy (lMap h) (sigma G1 l))
(pfNatE : forall e, mapTerm (lMap h) (eMap h) (pfNat h) (phi G1 e) =
match pfNatL (lam G1 e) in _ = y return Term (lam G2)
y with
| refl_equal => match symmetry (pfNat h e) in _ = z
return Term (lam G2) (sigma G2 z) with
| refl_equal => phi G2 (eMap h e)
end
end) : Transform (APGtoFunctor G1) (APGtoFunctor G2).
refine (@newTransform CDCat SET (APGtoFunctor G1) (APGtoFunctor G2)
(fun x => match x with
| EE => eMap h
| TT => mapTy (lMap h)
| LL => lMap h
| VV => fun v => existT _ (mapTy (lMap h) (projT1 v)) (mapTerm (lMap h) (eMap h) (pfNat h) (projT2 v) )
end) _).
intros;
destruct G1,G2,h,f; simpl in *; auto.
Focus 2. apply functional_extensionality_dep. auto.
Focus 2. apply functional_extensionality_dep. intros w. rewrite (pfNatL w). auto.
Focus 2. apply functional_extensionality_dep. intros w. rewrite <- (pfNat0 w). auto.
apply functional_extensionality_dep. intros w.
rewrite (pfNatE w). clear pfNatE. rewrite <- (pfNatL (lam0 w)).
rewrite (pfNat0 w). f_equal.
Qed.
Definition Gx1 : APG := @newAPG unit (fun x => Zero _)
Empty_set (fun x => match x with end) (fun x => match x with end).
Definition Gx2 : APG := @newAPG unit (fun x => One _)
Empty_set (fun x => match x with end) (fun x => match x with end).
Definition hx : APGMorphism Gx1 Gx2.
refine (newAPGMorphism Gx1 Gx2 (fun l => l) (fun e => match e with end) _).
intros. compute in *. elim e.
Defined.
Lemma ApgMorphismsNotNatural : exists l,
sigma Gx2 (lMap hx l) <> mapTy (lMap hx) (sigma Gx1 l).
Proof.
intros. exists tt. intros pf.
simpl in *. unfold mapTy in pf. simpl in pf. discriminate.
Qed.
(* Free BCCC ***************************************************)
Inductive term {L} {lam : L -> Ty L} : Ty L -> Ty L -> Type :=
| ELEM: forall l, term (Label l) (lam l)
| IDEN: forall {t}, term t t
| COMP: forall {s t u}, term s t -> term t u -> term s u
| PRIM: forall {b} (c: P b), term (One _) (Base _ b)
| tT : forall {t}, term t (One _)
| fF : forall {t}, term (Zero _) t
| PAIR: forall {s t t'}, term s t -> term s t'-> term s (Times t t')
| CASE: forall {s s' t }, term s t -> term s' t -> term (Plus s s') t
| INL : forall {s t }, term s (Plus s t)
| INR : forall {s t }, term t (Plus s t)
| FST : forall {s t }, term (Times s t) s
| SND : forall {s t }, term (Times s t) t.
Inductive termEq {L} {lam : L -> Ty L} : forall {a b}, @term L lam a b -> term a b -> Prop :=
| termRefl : forall {a b} (t: term a b), termEq t t
| termSym : forall {a b} (t1 t2: term a b),
termEq t1 t2 -> termEq t2 t1
| termTrans : forall {a b} (t1 t2 t3: term a b),
termEq t1 t2 -> termEq t2 t3 -> termEq t1 t3
| termCongComp : forall {a b c} (f f0 : term a b) (g g0 : term b c),
termEq f f0 -> termEq g g0 -> termEq (COMP f g) (COMP f0 g0)
| termCongPair : forall {a b c} (f f0 : term a b) (g g0 : term a c),
termEq f f0 -> termEq g g0 -> termEq (PAIR f g) (PAIR f0 g0)
| termCongCase : forall {a b c} (f f0 : term a c) (g g0 : term b c),
termEq f f0 -> termEq g g0 -> termEq (CASE f g) (CASE f0 g0)
| termId1 : forall {a b} (f: term a b), termEq (COMP IDEN f) f
| termId2 : forall {a b} (f: term a b), termEq (COMP f IDEN) f
| termAssoc : forall {a b c d} (f: term a b) (g: term b c) (h: term c d),
termEq (COMP (COMP f g) h) (COMP f (COMP g h))
| termOne : forall {a} (t: term a (One _)), termEq t (tT)
| termZero : forall {a} (t: term (Zero _) a), termEq t (fF )
| termTimesBeta1 : forall {a b c} (f: term a b) (g: term a c),
termEq (COMP (PAIR f g) FST) f
| termTimesBeta2 : forall {a b c} (f: term a b) (g: term a c),
termEq (COMP (PAIR f g) SND) g
| termPlusBeta1 : forall {a b c} (f: term b c) (g: term a c),
termEq (COMP INL (CASE f g)) f
| termPlusBeta2 : forall {a b c} (f: term b c) (g: term a c),
termEq (COMP INR (CASE f g)) g
| termTimesEta : forall {a b},
termEq (PAIR (@FST L lam a b) SND) IDEN