Require Export fintype.
Require Import Vectors.Vector Lia.
Require Import Utils.Various_utils.
Import VectorNotations.
Section ptypentype.
Context {w : nat} .
Inductive ptype (nntype : nat) : Type :=
| top : ptype (nntype)
| bAllN : (Vector.t (ntype (nntype)) w) -> ntype ((S) nntype) -> ptype (nntype)
with ntype (nntype : nat) : Type :=
| var_ntype : fin w -> (fin) (nntype) -> ntype (nntype)
| uAllN : ptype (S nntype) -> ntype (nntype).
Inductive ctx : nat -> Type :=
| empty : ctx 0
| add : forall {m : nat}, Vector.t (ntype m) w -> ctx m -> ctx (S m).
Definition nonEmptyCtx {m : nat} (Γ : ctx (S m)) : forall (P : ctx (S m) -> Type) (H : forall h t, P (add h t)), P Γ
:= match Γ with
| add h t => fun P H => H h t
| _ => fun x => False_rect (@IDProp) x
end.
| top : ptype (nntype)
| bAllN : (Vector.t (ntype (nntype)) w) -> ntype ((S) nntype) -> ptype (nntype)
with ntype (nntype : nat) : Type :=
| var_ntype : fin w -> (fin) (nntype) -> ntype (nntype)
| uAllN : ptype (S nntype) -> ntype (nntype).
Inductive ctx : nat -> Type :=
| empty : ctx 0
| add : forall {m : nat}, Vector.t (ntype m) w -> ctx m -> ctx (S m).
Definition nonEmptyCtx {m : nat} (Γ : ctx (S m)) : forall (P : ctx (S m) -> Type) (H : forall h t, P (add h t)), P Γ
:= match Γ with
| add h t => fun P H => H h t
| _ => fun x => False_rect (@IDProp) x
end.
Fixpoint psize {m} (t : @ptype m) : nat := match t with
| top _ => 1
| bAllN _ v s => 1 + vsum (map nsize v) + nsize s
end
with nsize {m} (s : @ntype m) : nat := match s with
| var_ntype _ _ _ => 1
| uAllN _ t => 1 + psize t
end.
Lemma ptype_ntype_size_ind (P : forall m, @ptype m -> Prop) (Q : forall m, @ntype m -> Prop) :
(forall m t, (forall m' s, nsize s < psize t -> Q m' s) -> P m t) ->
(forall m t, (forall m' s, psize s < nsize t -> P m' s) -> Q m t) ->
forall m, (forall t, P m t) /\ (forall s, Q m s).
Proof. intros Hp Hq. split; intros; [apply Hp | apply Hq].
-assert (H : forall n m s, nsize s < n -> Q m s).
{intro. induction n; intros. exfalso. lia. apply Hq. intros. apply Hp. intros. apply IHn. lia. }
apply H.
-assert (H : forall n m s, psize s < n -> P m s).
{intro. induction n; intros. exfalso. lia. apply Hp. intros. apply Hq. intros. apply IHn. lia. }
apply H.
Qed.
Fact in_ns {m} a t s: In a t -> @nsize m a < psize (bAllN _ t s).
Proof. intro. cbn.
assert (In (nsize a) (map nsize t)). { induction H; simpl. apply In_cons_hd. now apply In_cons_tl. }
pose (in_smaller _ _ H0). lia.
Qed.
Lemma ptype_ntype_mutind : forall (P : forall m : nat, @ptype m -> Prop)
(Q : forall m : nat, @ntype m -> Prop),
(forall m : nat, P m (@top m)) ->
(forall (m : nat) (t : Vector.t (@ntype m) w) (n : @ntype (S m)),
Q (S m) n -> @Forall (@ntype m) (Q m) _ t -> P m (@bAllN m t n)) ->
(forall (m : nat) (t0 : fin m) (t : fin w), Q m (@var_ntype m t t0)) ->
(forall (m : nat) (p : @ptype (S m)), P (S m) p -> Q m (@uAllN m p)) ->
forall m : nat, (forall p : @ptype m, P m p) /\ (forall n : @ntype m, Q m n).
Proof. intros P Q Htop HbAllN Hvar HuAllN. eapply ptype_ntype_size_ind.
-intros. destruct t. apply Htop. apply HbAllN. apply H. cbn. lia.
apply Forall_forall. intros. apply H. now apply in_ns.
-intros. destruct t. apply Hvar. apply HuAllN, H. cbn. lia.
Qed.
Modified Autosubst output
Renamings and substitutions have an extra argument to access the vectors, this argument remains unchanged when the renaming or substitution is modifiedLemma congr_top { mntype : nat } : top (mntype) = top (mntype) .
Proof. congruence. Qed.
Lemma congr_bAllN { mntype : nat } { s0 : (Vector.t (ntype (mntype)) w) } { s1 : ntype ((S) mntype) } { t0 : (Vector.t (ntype (mntype)) w) } { t1 : ntype ((S) mntype) } (H1 : s0 = t0) (H2 : s1 = t1) : bAllN (mntype) s0 s1 = bAllN (mntype) t0 t1 .
Proof. congruence. Qed.
Lemma congr_uAllN { mntype : nat } { s0 : ptype (S mntype) } { t0 : ptype (S mntype) } (H1 : s0 = t0) : uAllN (mntype) s0 = uAllN (mntype) t0 .
Proof. congruence. Qed.
Definition upRen_ntype_ntype { m : nat } { n' : nat } (xi : (fin) (m) -> (fin) (n')) : (fin) ((S) (m)) -> (fin) ((S) (n')) := (up_ren) (xi).
Fixpoint ren_ptype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (s : ptype (mntype)) : ptype (nntype) :=
match s return ptype (nntype) with
| top (_) => top (nntype)
| bAllN (_) s0 s1 => bAllN (nntype) (Vector.map (ren_ntype xintype) s0) ((ren_ntype (upRen_ntype_ntype xintype)) s1)
end
with ren_ntype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (s : ntype (mntype)) : ntype (nntype) :=
match s return ntype (nntype) with
| var_ntype (_) i s => (var_ntype (nntype)) i (xintype s)
| uAllN (_) s0 => uAllN (nntype) ((ren_ptype (upRen_ntype_ntype xintype)) s0)
end.
Definition ctx_at {m} (Γ : ctx m) (i : fin w) (j : fin m) : ntype m.
Proof. induction Γ. contradiction. apply (ren_ntype ↑). destruct j. apply IHΓ, f. exact (nth' t i).
Defined.
Definition up_ntype_ntype { m : nat } { nntype : nat } (sigma : fin w -> (fin) (m) -> ntype (nntype)) : fin w -> (fin) ((S) (m)) -> ntype ((S) nntype) :=
fun i => (scons) ((var_ntype ((S) nntype) i) (var_zero)) ((funcomp) (ren_ntype (shift)) (sigma i)).
Fixpoint subst_ptype { mntype : nat } { nntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (nntype)) (s : ptype (mntype)) : ptype (nntype) :=
(match s return ptype (nntype) with
| top (_) => top (nntype)
| bAllN (_) s0 s1 => bAllN (nntype) (Vector.map (subst_ntype sigmantype) s0) ((subst_ntype (up_ntype_ntype sigmantype)) s1)
end)
with subst_ntype { mntype : nat } { nntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (nntype)) (s : ntype (mntype)) : ntype (nntype) :=
match s return ntype (nntype) with
| var_ntype (_) i s => sigmantype i s
| uAllN (_) s0 => uAllN (nntype) ((subst_ptype (up_ntype_ntype sigmantype)) s0)
end.
(* Instantiations *)
Definition ninst {m} (v : Vector.t (ntype m) w) (s : ntype (S m)) : ntype m
:= subst_ntype (fun i : fin w => nth' v i .; var_ntype m i) s.
Definition pinst {m} (v : Vector.t (ntype m) w) (s : ptype (S m)) : ptype m
:= subst_ptype (fun i : fin w => nth' v i .; var_ntype m i) s.
Definition upId_ntype_ntype { mntype : nat } (sigma : fin w -> (fin) (mntype) -> ntype (mntype)) (Eq : forall i x, sigma i x = (var_ntype (mntype) i) x) : forall i x, (up_ntype_ntype sigma i) x = (var_ntype ((S) mntype) i) x :=
fun i n => match n with
| Some fin_n => (ap) (ren_ntype (shift)) (Eq i fin_n)
| None => eq_refl
end.
(* Fixpoint idSubst_ptype { mntype : nat } (sigmantype : fin n -> (fin) (mntype) -> ntype (mntype)) (Eqntype : forall i x, sigmantype i x = (var_ntype (mntype) i) x) (s : ptype (mntype)) : subst_ptype sigmantype s = s :=
match s return subst_ptype sigmantype s = s with
| top (_) => congr_top
| bAllN (_) s0 s1 => congr_bAllN ((idSubst_ntype sigmantype Eqntype) s0) ((idSubst_ntype (up_ntype_ntype sigmantype) (upId_ntype_ntype (_) Eqntype)) s1)
end
with idSubst_ntype { mntype : nat } (sigmantype : fin n -> (fin) (mntype) -> ntype (mntype)) (Eqntype : forall i x, sigmantype i x = (var_ntype (mntype) i) x) (s : ntype (mntype)) : subst_ntype sigmantype s = s :=
match s return subst_ntype sigmantype s = s with
| var_ntype (_) s => Eqntype s
| uAllN (_) s0 => congr_uAllN ((idSubst_ptype sigmantype Eqntype) s0)
end. *)
In general each lemma is proved for both positive and negative types simultaneously with the mutual induction principle
Lemma idSubst_ptype_ntype : forall m,
(forall s (sigmantype : fin w -> fin m -> ntype m) (Eq : forall i x, sigmantype i x = var_ntype m i x), subst_ptype sigmantype s = s)
/\ (forall s (sigmantype : fin w -> fin m -> ntype m) (Eq : forall i x, sigmantype i x = var_ntype m i x), subst_ntype sigmantype s = s).
Proof. eapply ptype_ntype_mutind; intros.
-apply congr_top.
-apply congr_bAllN.
+rewrite (map_ext_in _ _ _ id). apply map_id. unfold id. apply Forall_forall. apply (Forall_impl _ (fun n0 : ntype m => forall sigmantype : fin w -> fin m -> ntype m, (forall i x, sigmantype i x = var_ntype m i x) -> subst_ntype sigmantype n0 = n0)); auto.
+apply H. intros. apply (upId_ntype_ntype _ Eq).
-apply Eq.
-apply congr_uAllN. apply H. intros. apply (upId_ntype_ntype _ Eq).
Qed.
(forall s (sigmantype : fin w -> fin m -> ntype m) (Eq : forall i x, sigmantype i x = var_ntype m i x), subst_ptype sigmantype s = s)
/\ (forall s (sigmantype : fin w -> fin m -> ntype m) (Eq : forall i x, sigmantype i x = var_ntype m i x), subst_ntype sigmantype s = s).
Proof. eapply ptype_ntype_mutind; intros.
-apply congr_top.
-apply congr_bAllN.
+rewrite (map_ext_in _ _ _ id). apply map_id. unfold id. apply Forall_forall. apply (Forall_impl _ (fun n0 : ntype m => forall sigmantype : fin w -> fin m -> ntype m, (forall i x, sigmantype i x = var_ntype m i x) -> subst_ntype sigmantype n0 = n0)); auto.
+apply H. intros. apply (upId_ntype_ntype _ Eq).
-apply Eq.
-apply congr_uAllN. apply H. intros. apply (upId_ntype_ntype _ Eq).
Qed.
We can now define the lemma for positive and negative types individually
Definition idSubst_ptype { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (mntype)) (Eqntype : forall i x, sigmantype i x = (var_ntype (mntype) i) x) (s : ptype (mntype)) : subst_ptype sigmantype s = s.
Proof. destruct (idSubst_ptype_ntype mntype). now apply H. Defined.
Definition idSubst_ntype { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (mntype)) (Eqntype : forall i x, sigmantype i x = (var_ntype (mntype) i) x) (s : ntype (mntype)) : subst_ntype sigmantype s = s.
Proof. destruct (idSubst_ptype_ntype mntype). now apply H0. Defined.
Definition upExtRen_ntype_ntype { m : nat } { n' : nat } (xi : (fin) (m) -> (fin) (n')) (zeta : (fin) (m) -> (fin) (n')) (Eq : forall x, xi x = zeta x) : forall x, (upRen_ntype_ntype xi) x = (upRen_ntype_ntype zeta) x :=
fun n => match n with
| Some fin_n => (ap) (shift) (Eq fin_n)
| None => eq_refl
end.
(* Fixpoint extRen_ptype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (zetantype : (fin) (mntype) -> (fin) (nntype)) (Eqntype : forall x, xintype x = zetantype x) (s : ptype (mntype)) : ren_ptype xintype s = ren_ptype zetantype s :=
match s return ren_ptype xintype s = ren_ptype zetantype s with
| top (_) => congr_top
| bAllN (_) s0 s1 => congr_bAllN ((extRen_ntype xintype zetantype Eqntzype) s0) ((extRen_ntype (upRen_ntype_ntype xintype) (upRen_ntype_ntype zetantype) (upExtRen_ntype_ntype (_) (_) Eqntype)) s1)
end
with extRen_ntype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (zetantype : (fin) (mntype) -> (fin) (nntype)) (Eqntype : forall x, xintype x = zetantype x) (s : ntype (mntype)) : ren_ntype xintype s = ren_ntype zetantype s :=
match s return ren_ntype xintype s = ren_ntype zetantype s with
| var_ntype (_) s => (ap) (var_ntype (nntype)) (Eqntype s)
| uAllN (_) s0 => congr_uAllN ((extRen_ptype xintype zetantype Eqntype) s0)
end. *)
Lemma extRen_ptype_ntype : forall m,
(forall s l (xintype : fin m -> fin l) (zetantype : fin m -> fin l) (Eqntype : forall x, xintype x = zetantype x), ren_ptype xintype s = ren_ptype zetantype s)
/\ (forall s l (xintype : fin m -> fin l) (zetantype : fin m -> fin l) (Eqntype : forall x, xintype x = zetantype x), ren_ntype xintype s = ren_ntype zetantype s).
Proof. eapply ptype_ntype_mutind; intros.
-apply congr_top.
-apply congr_bAllN.
+apply map_ext_in. apply Forall_forall. apply (Forall_impl _ (fun n : ntype m => forall (l : nat) (xintype zetantype : fin m -> fin l), (forall x : fin m, xintype x = zetantype x) -> ren_ntype xintype n = ren_ntype zetantype n)); auto.
+apply H. intros. apply (upExtRen_ntype_ntype _ _ Eqntype).
-apply (ap (var_ntype l t)). apply Eqntype.
-apply congr_uAllN. apply H. intros. apply (upExtRen_ntype_ntype _ _ Eqntype).
Qed.
Definition extRen_ptype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (zetantype : (fin) (mntype) -> (fin) (nntype)) (Eqntype : forall x, xintype x = zetantype x) (s : ptype (mntype)) : ren_ptype xintype s = ren_ptype zetantype s.
Proof. destruct (extRen_ptype_ntype mntype). now apply H. Defined.
Definition extRen_ntype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (zetantype : (fin) (mntype) -> (fin) (nntype)) (Eqntype : forall x, xintype x = zetantype x) (s : ntype (mntype)) : ren_ntype xintype s = ren_ntype zetantype s.
Proof. destruct (extRen_ptype_ntype mntype). now apply H0. Defined.
Definition upExt_ntype_ntype { m : nat } { nntype : nat } (sigma : fin w -> (fin) (m) -> ntype (nntype)) (tau : fin w -> (fin) (m) -> ntype (nntype)) (Eq : forall i x, sigma i x = tau i x) : forall i x, (up_ntype_ntype sigma) i x = (up_ntype_ntype tau) i x :=
fun i n => match n with
| Some fin_n => (ap) (ren_ntype (shift)) (Eq i fin_n)
| None => eq_refl
end.
(* Fixpoint ext_ptype { mntype : nat } { nntype : nat } (sigmantype : (fin) (mntype) -> ntype (nntype)) (tauntype : (fin) (mntype) -> ntype (nntype)) (Eqntype : forall x, sigmantype x = tauntype x) (s : ptype (mntype)) : subst_ptype sigmantype s = subst_ptype tauntype s :=
match s return subst_ptype sigmantype s = subst_ptype tauntype s with
| top (_) => congr_top
| bAllN (_) s0 s1 => congr_bAllN ((ext_ntype sigmantype tauntype Eqntype) s0) ((ext_ntype (up_ntype_ntype sigmantype) (up_ntype_ntype tauntype) (upExt_ntype_ntype (_) (_) Eqntype)) s1)
end
with ext_ntype { mntype : nat } { nntype : nat } (sigmantype : (fin) (mntype) -> ntype (nntype)) (tauntype : (fin) (mntype) -> ntype (nntype)) (Eqntype : forall x, sigmantype x = tauntype x) (s : ntype (mntype)) : subst_ntype sigmantype s = subst_ntype tauntype s :=
match s return subst_ntype sigmantype s = subst_ntype tauntype s with
| var_ntype (_) s => Eqntype s
| uAllN (_) s0 => congr_uAllN ((ext_ptype sigmantype tauntype Eqntype) s0)
end. *)
Lemma ext_ptype_ntype : forall m,
(forall s l (sig : fin w -> fin m -> ntype l) (tau : fin w -> fin m -> ntype l) (Eq : forall i x, sig i x = tau i x),
subst_ptype sig s = subst_ptype tau s)
/\ (forall s l (sig : fin w -> fin m -> ntype l) (tau : fin w -> fin m -> ntype l) (Eq : forall i x, sig i x = tau i x),
subst_ntype sig s = subst_ntype tau s).
Proof. eapply ptype_ntype_mutind; intros.
-apply congr_top.
-apply congr_bAllN.
+apply map_ext_in. apply Forall_forall. apply (Forall_impl _ (fun n0 : ntype m => forall (l : nat) (sig tau : fin w -> fin m -> ntype l), (forall (i : fin w) (x : fin m), sig i x = tau i x) -> subst_ntype sig n0 = subst_ntype tau n0)); auto.
+apply H. intros. apply (upExt_ntype_ntype _ _ Eq).
-apply Eq.
-apply congr_uAllN. apply H. intros. apply (upExt_ntype_ntype _ _ Eq).
Qed.
Definition ext_ptype { mntype : nat } { nntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (nntype)) (tauntype : fin w -> (fin) (mntype) -> ntype (nntype)) (Eqntype : forall i x, sigmantype i x = tauntype i x) (s : ptype (mntype)) : subst_ptype sigmantype s = subst_ptype tauntype s.
Proof. destruct (ext_ptype_ntype mntype). now apply H. Defined.
Definition ext_ntype { mntype : nat } { nntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (nntype)) (tauntype : fin w -> (fin) (mntype) -> ntype (nntype)) (Eqntype : forall i x, sigmantype i x = tauntype i x) (s : ntype (mntype)) : subst_ntype sigmantype s = subst_ntype tauntype s.
Proof. destruct (ext_ptype_ntype mntype). now apply H0. Defined.
Definition up_ren_ren_ntype_ntype { k : nat } { l : nat } { m : nat } (xi : (fin) (k) -> (fin) (l)) (tau : (fin) (l) -> (fin) (m)) (theta : (fin) (k) -> (fin) (m)) (Eq : forall x, ((funcomp) (tau) (xi)) x = theta x) : forall x, ((funcomp) (upRen_ntype_ntype tau) (upRen_ntype_ntype xi)) x = (upRen_ntype_ntype theta) x :=
up_ren_ren (xi) (tau) (theta) (Eq).
(* Fixpoint compRenRen_ptype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (rhontype : (fin) (mntype) -> (fin) (lntype)) (Eqntype : forall x, ((funcomp) zetantype xintype) x = rhontype x) (s : ptype (mntype)) : ren_ptype zetantype (ren_ptype xintype s) = ren_ptype rhontype s :=
match s return ren_ptype zetantype (ren_ptype xintype s) = ren_ptype rhontype s with
| top (_) => congr_top
| bAllN (_) s0 s1 => congr_bAllN ((compRenRen_ntype xintype zetantype rhontype Eqntype) s0) ((compRenRen_ntype (upRen_ntype_ntype xintype) (upRen_ntype_ntype zetantype) (upRen_ntype_ntype rhontype) (up_ren_ren (_) (_) (_) Eqntype)) s1)
end
with compRenRen_ntype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (rhontype : (fin) (mntype) -> (fin) (lntype)) (Eqntype : forall x, ((funcomp) zetantype xintype) x = rhontype x) (s : ntype (mntype)) : ren_ntype zetantype (ren_ntype xintype s) = ren_ntype rhontype s :=
match s return ren_ntype zetantype (ren_ntype xintype s) = ren_ntype rhontype s with
| var_ntype (_) s => (ap) (var_ntype (lntype)) (Eqntype s)
| uAllN (_) s0 => congr_uAllN ((compRenRen_ptype xintype zetantype rhontype Eqntype) s0)
end. *)
Lemma compRenRen_ptype_ntype : forall m,
(forall s k l (xi : fin m -> fin k) (zet : fin k -> fin l) (rho : fin m -> fin l)
(Eq : forall x, funcomp (zet) (xi) x = rho x),
ren_ptype zet (ren_ptype xi s) = ren_ptype rho s)
/\ (forall s k l (xi : fin m -> fin k) (zet : fin k -> fin l) (rho : fin m -> fin l)
(Eq : forall x, funcomp (zet) (xi) x = rho x),
ren_ntype zet (ren_ntype xi s) = ren_ntype rho s).
Proof. eapply ptype_ntype_mutind; intros.
-apply congr_top.
-apply congr_bAllN.
+rewrite map_map. apply map_ext_in. apply Forall_forall. apply (Forall_impl _ (fun n : ntype m =>
forall (k l : nat) (xi : fin m -> fin k) (zet : fin k -> fin l) (rho : fin m -> fin l),
(forall x : fin m, (xi >> zet) x = rho x) -> ren_ntype zet (ren_ntype xi n) = ren_ntype rho n)); auto.
+apply H. intros. apply (up_ren_ren _ _ _ Eq).
-apply (ap (var_ntype l t)). apply Eq.
-apply congr_uAllN. apply H. intros. apply (up_ren_ren _ _ _ (Eq)).
Qed.
Definition compRenRen_ptype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (rhontype : (fin) (mntype) -> (fin) (lntype)) (Eqntype : forall x, ((funcomp) (zetantype) (xintype)) x = rhontype x) (s : ptype (mntype)) : ren_ptype zetantype (ren_ptype xintype s) = ren_ptype rhontype s.
Proof. destruct (compRenRen_ptype_ntype mntype). now apply H. Defined.
Definition compRenRen_ntype { kntype : nat } { lntype : nat } { mntype : nat }(xintype : (fin) (mntype) -> (fin) (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (rhontype : (fin) (mntype) -> (fin) (lntype)) (Eqntype : forall x, ((funcomp) (zetantype) (xintype)) x = rhontype x) (s : ntype (mntype)) : ren_ntype zetantype (ren_ntype xintype s) = ren_ntype rhontype s.
Proof. destruct (compRenRen_ptype_ntype mntype). now apply H0. Defined.
Definition up_ren_subst_ntype_ntype { k : nat } { l : nat } { mntype : nat } (xi : (fin) (k) -> (fin) (l)) (tau : fin w -> (fin) (l) -> ntype (mntype)) (theta : fin w -> (fin) (k) -> ntype (mntype)) (Eq : forall i x, ((funcomp) (tau i) (xi)) x = theta i x) : forall i x, ((funcomp) (up_ntype_ntype tau i) (upRen_ntype_ntype xi)) x = (up_ntype_ntype theta) i x :=
fun i n => match n with
| Some fin_n => (ap) (ren_ntype (shift)) (Eq i fin_n)
| None => eq_refl
end.
(* Fixpoint compRenSubst_ptype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (tauntype : (fin) (kntype) -> ntype (lntype)) (thetantype : (fin) (mntype) -> ntype (lntype)) (Eqntype : forall x, ((funcomp) tauntype xintype) x = thetantype x) (s : ptype (mntype)) : subst_ptype tauntype (ren_ptype xintype s) = subst_ptype thetantype s :=
match s return subst_ptype tauntype (ren_ptype xintype s) = subst_ptype thetantype s with
| top (_) => congr_top
| bAllN (_) s0 s1 => congr_bAllN ((compRenSubst_ntype xintype tauntype thetantype Eqntype) s0) ((compRenSubst_ntype (upRen_ntype_ntype xintype) (up_ntype_ntype tauntype) (up_ntype_ntype thetantype) (up_ren_subst_ntype_ntype (_) (_) (_) Eqntype)) s1)
end
with compRenSubst_ntype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (tauntype : (fin) (kntype) -> ntype (lntype)) (thetantype : (fin) (mntype) -> ntype (lntype)) (Eqntype : forall x, ((funcomp) tauntype xintype) x = thetantype x) (s : ntype (mntype)) : subst_ntype tauntype (ren_ntype xintype s) = subst_ntype thetantype s :=
match s return subst_ntype tauntype (ren_ntype xintype s) = subst_ntype thetantype s with
| var_ntype (_) s => Eqntype s
| uAllN (_) s0 => congr_uAllN ((compRenSubst_ptype xintype tauntype thetantype Eqntype) s0)
end. *)
Lemma compRenSubst_ptype_ntype : forall m,
(forall s k l (xi : fin m -> fin k) (tau : fin w -> fin k -> ntype l) (theta : fin w -> fin m -> ntype l)
(Eq : forall i x, (xi >> (tau i)) x = theta i x), subst_ptype tau (ren_ptype xi s) = subst_ptype theta s)
/\ (forall s k l (xi : fin m -> fin k) (tau : fin w -> fin k -> ntype l) (theta : fin w -> fin m -> ntype l)
(Eq : forall i x, (xi >> (tau i)) x = theta i x), subst_ntype tau (ren_ntype xi s) = subst_ntype theta s).
Proof. eapply ptype_ntype_mutind; intros.
-apply congr_top.
-apply congr_bAllN.
+rewrite map_map. apply map_ext_in. apply Forall_forall. apply (Forall_impl _ (fun n0 : ntype m =>
forall (k l : nat) (xi : fin m -> fin k) (tau : fin w -> fin k -> ntype l)
(theta : fin w -> fin m -> ntype l),
(forall (i : fin w) (x : fin m), (xi >> tau i) x = theta i x) ->
subst_ntype tau (ren_ntype xi n0) = subst_ntype theta n0)); auto.
+apply H. intros. apply (up_ren_subst_ntype_ntype _ _ _ Eq).
-apply Eq.
-apply congr_uAllN. apply H. intros. apply (up_ren_subst_ntype_ntype _ _ _ Eq).
Qed.
Definition compRenSubst_ptype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) (thetantype : fin w -> (fin) (mntype) -> ntype (lntype)) (Eqntype : forall i x, ((funcomp) (tauntype i) (xintype)) x = thetantype i x) (s : ptype (mntype)) : subst_ptype tauntype (ren_ptype xintype s) = subst_ptype thetantype s.
Proof. destruct (compRenSubst_ptype_ntype mntype). now apply H. Defined.
Definition compRenSubst_ntype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) (thetantype : fin w -> (fin) (mntype) -> ntype (lntype)) (Eqntype : forall i x, ((funcomp) (tauntype i) (xintype)) x = thetantype i x) (s : ntype (mntype)) : subst_ntype tauntype (ren_ntype xintype s) = subst_ntype thetantype s.
Proof. destruct (compRenSubst_ptype_ntype mntype). now apply H0. Defined.
Definition up_subst_ren_ntype_ntype { k : nat } { lntype : nat } { mntype : nat } (sigma : fin w -> (fin) (k) -> ntype (lntype)) (zetantype : (fin) (lntype) -> (fin) (mntype)) (theta : fin w -> (fin) (k) -> ntype (mntype)) (Eq : forall i x, ((funcomp) (ren_ntype zetantype) (sigma i)) x = theta i x) : forall i x, ((funcomp) (ren_ntype (upRen_ntype_ntype zetantype)) (up_ntype_ntype sigma i)) x = (up_ntype_ntype theta) i x :=
fun i n => match n with
| Some fin_n => (eq_trans) (compRenRen_ntype (shift) (upRen_ntype_ntype zetantype) ((funcomp) (shift) zetantype) (fun x => eq_refl) (sigma i fin_n)) ((eq_trans) ((eq_sym) (compRenRen_ntype zetantype (shift) ((funcomp) (shift) zetantype) (fun x => eq_refl) (sigma i fin_n))) ((ap) (ren_ntype (shift)) (Eq i fin_n)))
| None => eq_refl
end.
(* Definition up_subst_ren_ntype_ntype { k : nat } { lntype : nat } { mntype : nat } (sigma : fin n -> (fin) (k) -> ntype (lntype)) (zetantype : (fin) (lntype) -> (fin) (mntype)) (theta : fin n -> (fin) (k) -> ntype (mntype)) (Eq : forall i x, ((funcomp) (ren_ntype zetantype) (sigma i)) x = theta i x) : forall i x, ((funcomp) (ren_ntype (upRen_ntype_ntype zetantype)) (up_ntype_ntype sigma i)) x = (up_ntype_ntype theta) i x
:= fun i x => match x as o return ((up_ntype_ntype sigma i >> ren_ntype (upRen_ntype_ntype zetantype)) o = up_ntype_ntype theta i o) with
| Some fin_n => eq_trans (compRenRen_ntype (↑) (upRen_ntype_ntype zetantype)
(fun (i0 : fin n) (x0 : fin lntype) => ↑ (zetantype i0 x0))
(fun (i0 : fin n) (x0 : fin lntype) => eq_refl)
(sigma i fin_n))
(eq_trans (eq_sym (compRenRen_ntype zetantype (fun _ : fin n => ↑)
(fun (i0 : fin n) (x0 : fin lntype) => ↑ (zetantype i0 x0))
(fun (i0 : fin n) (x0 : fin lntype) => eq_refl)
(sigma i fin_n)))
(ap (ren_ntype (fun _ : fin n => ↑)) (Eq i fin_n)))
| None => eq_refl
end. *)
(* Fixpoint compSubstRen_ptype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : (fin) (mntype) -> ntype (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (thetantype : (fin) (mntype) -> ntype (lntype)) (Eqntype : forall x, ((funcomp) (ren_ntype zetantype) sigmantype) x = thetantype x) (s : ptype (mntype)) : ren_ptype zetantype (subst_ptype sigmantype s) = subst_ptype thetantype s :=
match s return ren_ptype zetantype (subst_ptype sigmantype s) = subst_ptype thetantype s with
| top (_) => congr_top
| bAllN (_) s0 s1 => congr_bAllN ((compSubstRen_ntype sigmantype zetantype thetantype Eqntype) s0) ((compSubstRen_ntype (up_ntype_ntype sigmantype) (upRen_ntype_ntype zetantype) (up_ntype_ntype thetantype) (up_subst_ren_ntype_ntype (_) (_) (_) Eqntype)) s1)
end
with compSubstRen_ntype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : (fin) (mntype) -> ntype (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (thetantype : (fin) (mntype) -> ntype (lntype)) (Eqntype : forall x, ((funcomp) (ren_ntype zetantype) sigmantype) x = thetantype x) (s : ntype (mntype)) : ren_ntype zetantype (subst_ntype sigmantype s) = subst_ntype thetantype s :=
match s return ren_ntype zetantype (subst_ntype sigmantype s) = subst_ntype thetantype s with
| var_ntype (_) s => Eqntype s
| uAllN (_) s0 => congr_uAllN ((compSubstRen_ptype sigmantype zetantype thetantype Eqntype) s0)
end. *)
Lemma compSubstRen_ptype_ntype : forall m,
(forall s k l (sig : fin w -> fin m -> ntype k) (zeta : fin k -> fin l) (theta : fin w -> fin m -> ntype l)
(Eq : forall i x, ((sig i) >> (ren_ntype zeta)) x = theta i x), ren_ptype zeta (subst_ptype sig s) = subst_ptype theta s)
/\ (forall s k l (sig : fin w -> fin m -> ntype k) (zeta : fin k -> fin l) (theta : fin w -> fin m -> ntype l)
(Eq : forall i x, ((sig i) >> (ren_ntype zeta)) x = theta i x), ren_ntype zeta (subst_ntype sig s) = subst_ntype theta s).
Proof. eapply ptype_ntype_mutind; intros.
-apply congr_top.
-apply congr_bAllN.
+rewrite map_map. apply map_ext_in. apply Forall_forall. apply (Forall_impl _ (fun n0 : ntype m => forall (k l : nat) (sig : fin w -> fin m -> ntype k) (zeta : fin k -> fin l) (theta : fin w -> fin m -> ntype l), (forall (i : fin w) (x : fin m), (sig i >> ren_ntype zeta) x = theta i x) -> ren_ntype zeta (subst_ntype sig n0) = subst_ntype theta n0)); auto.
+apply H. intros. apply (up_subst_ren_ntype_ntype _ _ _ Eq).
-apply Eq.
-apply congr_uAllN. apply H. intros. apply (up_subst_ren_ntype_ntype _ _ _ Eq).
Qed.
Definition compSubstRen_ptype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (thetantype : fin w -> (fin) (mntype) -> ntype (lntype)) (Eqntype : forall i x, ((funcomp) (ren_ntype zetantype) (sigmantype i)) x = thetantype i x) (s : ptype (mntype)) : ren_ptype zetantype (subst_ptype sigmantype s) = subst_ptype thetantype s.
Proof. destruct (compSubstRen_ptype_ntype mntype). now apply H. Defined.
Definition compSubstRen_ntype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (thetantype : fin w -> (fin) (mntype) -> ntype (lntype)) (Eqntype : forall i x, ((funcomp) (ren_ntype zetantype) (sigmantype i)) x = thetantype i x) (s : ntype (mntype)) : ren_ntype zetantype (subst_ntype sigmantype s) = subst_ntype thetantype s.
Proof. destruct (compSubstRen_ptype_ntype mntype). now apply H0. Defined.
(* Definition up_subst_subst_ntype_ntype { k : nat } { lntype : nat } { mntype : nat } (sigma : (fin) (k) -> ntype (lntype)) (tauntype : (fin) (lntype) -> ntype (mntype)) (theta : (fin) (k) -> ntype (mntype)) (Eq : forall x, ((funcomp) (subst_ntype tauntype) sigma) x = theta x) : forall x, ((funcomp) (subst_ntype (up_ntype_ntype tauntype)) (up_ntype_ntype sigma)) x = (up_ntype_ntype theta) x :=
fun n => match n with
| Some fin_n => (eq_trans) (compRenSubst_ntype (shift) (up_ntype_ntype tauntype) ((funcomp) (up_ntype_ntype tauntype) (shift)) (fun x => eq_refl) (sigma fin_n)) ((eq_trans) ((eq_sym) (compSubstRen_ntype tauntype (shift) ((funcomp) (ren_ntype (shift)) tauntype) (fun x => eq_refl) (sigma fin_n))) ((ap) (ren_ntype (shift)) (Eq fin_n)))
| None => eq_refl
end. *)
Definition up_subst_subst_ntype_ntype { k : nat } { lntype : nat } { mntype : nat } (sigma : fin w -> (fin) (k) -> ntype (lntype)) (tauntype : fin w -> (fin) (lntype) -> ntype (mntype)) (theta : fin w -> (fin) (k) -> ntype (mntype)) (Eq : forall i x, ((funcomp) (subst_ntype tauntype) (sigma i)) x = theta i x) : forall i x, ((funcomp) (subst_ntype (up_ntype_ntype tauntype)) (up_ntype_ntype sigma i)) x = (up_ntype_ntype theta) i x
:= fun i x => match x as o return ((up_ntype_ntype sigma i >> subst_ntype (up_ntype_ntype tauntype)) o = up_ntype_ntype theta i o) with
| Some fin_n => eq_trans
(compRenSubst_ntype (↑) (up_ntype_ntype tauntype)
(fun (i0 : fin w) (x0 : fin lntype) =>
up_ntype_ntype tauntype i0 (↑ x0))
(fun (i0 : fin w) (x0 : fin lntype) => eq_refl)
(sigma i fin_n))
(eq_trans (eq_sym (compSubstRen_ntype tauntype (↑)
(fun (i0 : fin w) (x0 : fin lntype) =>
ren_ntype (↑) (tauntype i0 x0))
(fun (i0 : fin w) (x0 : fin lntype) => eq_refl)
(sigma i fin_n)))
(ap (ren_ntype (↑)) (Eq i fin_n)))
| None => eq_refl
end.
(* Fixpoint compSubstSubst_ptype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : (fin) (mntype) -> ntype (kntype)) (tauntype : (fin) (kntype) -> ntype (lntype)) (thetantype : (fin) (mntype) -> ntype (lntype)) (Eqntype : forall x, ((funcomp) (subst_ntype tauntype) sigmantype) x = thetantype x) (s : ptype (mntype)) : subst_ptype tauntype (subst_ptype sigmantype s) = subst_ptype thetantype s :=
match s return subst_ptype tauntype (subst_ptype sigmantype s) = subst_ptype thetantype s with
| top (_) => congr_top
| bAllN (_) s0 s1 => congr_bAllN ((compSubstSubst_ntype sigmantype tauntype thetantype Eqntype) s0) ((compSubstSubst_ntype (up_ntype_ntype sigmantype) (up_ntype_ntype tauntype) (up_ntype_ntype thetantype) (up_subst_subst_ntype_ntype (_) (_) (_) Eqntype)) s1)
end
with compSubstSubst_ntype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : (fin) (mntype) -> ntype (kntype)) (tauntype : (fin) (kntype) -> ntype (lntype)) (thetantype : (fin) (mntype) -> ntype (lntype)) (Eqntype : forall x, ((funcomp) (subst_ntype tauntype) sigmantype) x = thetantype x) (s : ntype (mntype)) : subst_ntype tauntype (subst_ntype sigmantype s) = subst_ntype thetantype s :=
match s return subst_ntype tauntype (subst_ntype sigmantype s) = subst_ntype thetantype s with
| var_ntype (_) s => Eqntype s
| uAllN (_) s0 => congr_uAllN ((compSubstSubst_ptype sigmantype tauntype thetantype Eqntype) s0)
end. *)
Lemma compSubstSubst_ptype_ntype : forall m,
(forall s k l (sig : fin w -> fin m -> ntype k) (tau : fin w -> fin k -> ntype l) (theta : fin w -> fin m -> ntype l)
(Eq : forall i x, (sig i >> subst_ntype tau) x = theta i x), subst_ptype tau (subst_ptype sig s) = subst_ptype theta s)
/\ (forall s k l (sig : fin w -> fin m -> ntype k) (tau : fin w -> fin k -> ntype l) (theta : fin w -> fin m -> ntype l)
(Eq : forall i x, (sig i >> subst_ntype tau) x = theta i x), subst_ntype tau (subst_ntype sig s) = subst_ntype theta s).
Proof. eapply ptype_ntype_mutind; intros.
-apply congr_top.
-apply congr_bAllN.
+rewrite map_map. apply map_ext_in,Forall_forall. apply (Forall_impl _ (fun n0 : ntype m => forall (k l : nat) (sig : fin w -> fin m -> ntype k) (tau : fin w -> fin k -> ntype l) (theta : fin w -> fin m -> ntype l), (forall (i : fin w) (x : fin m), (sig i >> subst_ntype tau) x = theta i x) -> subst_ntype tau (subst_ntype sig n0) = subst_ntype theta n0)); auto.
+apply H. intros. apply (up_subst_subst_ntype_ntype _ _ _ Eq).
-apply Eq.
-apply congr_uAllN. apply H. intros. apply (up_subst_subst_ntype_ntype _ _ _ Eq).
Qed.
Definition compSubstSubst_ptype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) (thetantype : fin w -> (fin) (mntype) -> ntype (lntype)) (Eqntype : forall i x, ((funcomp) (subst_ntype tauntype) (sigmantype i)) x = thetantype i x) (s : ptype (mntype)) : subst_ptype tauntype (subst_ptype sigmantype s) = subst_ptype thetantype s.
Proof. destruct (compSubstSubst_ptype_ntype mntype). now apply H. Defined.
Definition compSubstSubst_ntype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) (thetantype : fin w -> (fin) (mntype) -> ntype (lntype)) (Eqntype : forall i x, ((funcomp) (subst_ntype tauntype) (sigmantype i)) x = thetantype i x) (s : ntype (mntype)) : subst_ntype tauntype (subst_ntype sigmantype s) = subst_ntype thetantype s.
Proof. destruct (compSubstSubst_ptype_ntype mntype). now apply H0. Defined.
Definition rinstInst_up_ntype_ntype { m : nat } { nntype : nat } (xi : (fin) (m) -> (fin) (nntype)) (sigma : fin w -> (fin) (m) -> ntype (nntype)) (Eq : forall i x, ((funcomp) (var_ntype (nntype) i) (xi)) x = sigma i x) : forall i x, ((funcomp) (var_ntype ((S) nntype) i) (upRen_ntype_ntype xi)) x = (up_ntype_ntype sigma) i x :=
fun i n => match n with
| Some fin_n => (ap) (ren_ntype (shift)) (Eq i fin_n)
| None => eq_refl
end.
(* Fixpoint rinst_inst_ptype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (sigmantype : (fin) (mntype) -> ntype (nntype)) (Eqntype : forall x, ((funcomp) (var_ntype (nntype)) xintype) x = sigmantype x) (s : ptype (mntype)) : ren_ptype xintype s = subst_ptype sigmantype s :=
match s return ren_ptype xintype s = subst_ptype sigmantype s with
| top (_) => congr_top
| bAllN (_) s0 s1 => congr_bAllN ((rinst_inst_ntype xintype sigmantype Eqntype) s0) ((rinst_inst_ntype (upRen_ntype_ntype xintype) (up_ntype_ntype sigmantype) (rinstInst_up_ntype_ntype (_) (_) Eqntype)) s1)
end
with rinst_inst_ntype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (sigmantype : (fin) (mntype) -> ntype (nntype)) (Eqntype : forall x, ((funcomp) (var_ntype (nntype)) xintype) x = sigmantype x) (s : ntype (mntype)) : ren_ntype xintype s = subst_ntype sigmantype s :=
match s return ren_ntype xintype s = subst_ntype sigmantype s with
| var_ntype (_) s => Eqntype s
| uAllN (_) s0 => congr_uAllN ((rinst_inst_ptype xintype sigmantype Eqntype) s0)
end. *)
Lemma rinst_inst_ptype_ntype : forall m,
(forall s l (xi : fin m -> fin l) (sig : fin w -> fin m -> ntype l) (Eq : forall i x, var_ntype l i (xi x) = sig i x),
ren_ptype xi s = subst_ptype sig s)
/\ (forall s l (xi : fin m -> fin l) (sig : fin w -> fin m -> ntype l) (Eq : forall i x, var_ntype l i (xi x) = sig i x),
ren_ntype xi s = subst_ntype sig s).
Proof. eapply ptype_ntype_mutind; intros.
-apply congr_top.
-apply congr_bAllN.
+apply map_ext_in,Forall_forall. apply (Forall_impl _ (fun n0 : ntype m => forall (l : nat) (xi : fin m -> fin l) (sig : fin w -> fin m -> ntype l), (forall (i : fin w) (x : fin m), var_ntype l i (xi x) = sig i x) -> ren_ntype xi n0 = subst_ntype sig n0)); auto.
+apply H. intros. apply (rinstInst_up_ntype_ntype _ _ Eq).
-apply Eq.
-apply congr_uAllN. apply H. intros. apply (rinstInst_up_ntype_ntype _ _ Eq).
Qed.
Definition rinst_inst_ptype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (sigmantype : fin w -> (fin) (mntype) -> ntype (nntype)) (Eqntype : forall i x, ((funcomp) (var_ntype (nntype) i) (xintype)) x = sigmantype i x) (s : ptype (mntype)) : ren_ptype xintype s = subst_ptype sigmantype s.
Proof. destruct (rinst_inst_ptype_ntype mntype). now apply H. Defined.
Definition rinst_inst_ntype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (sigmantype : fin w -> (fin) (mntype) -> ntype (nntype)) (Eqntype : forall i x, ((funcomp) (var_ntype (nntype) i) (xintype)) x = sigmantype i x) (s : ntype (mntype)) : ren_ntype xintype s = subst_ntype sigmantype s.
Proof. destruct (rinst_inst_ptype_ntype mntype). now apply H0. Defined.
Lemma rinstInst_ptype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) : ren_ptype xintype = subst_ptype (fun i => (funcomp) (var_ntype (nntype) i) (xintype)).
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => rinst_inst_ptype xintype (_) (fun i n => eq_refl) x)). Qed.
Lemma rinstInst_ntype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) : ren_ntype xintype = subst_ntype (fun i => (funcomp) (var_ntype (nntype) i) (xintype)) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => rinst_inst_ntype xintype (_) (fun _ n => eq_refl) x)). Qed.
Lemma instId_ptype { mntype : nat } : subst_ptype (fun i => var_ntype (mntype) i) = id .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => idSubst_ptype (var_ntype (mntype)) (fun _ n => eq_refl) ((id) x))). Qed.
Lemma instId_ntype { mntype : nat } : subst_ntype (fun i => var_ntype (mntype) i) = id .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => idSubst_ntype (var_ntype (mntype)) (fun _ n => eq_refl) ((id) x))). Qed.
Lemma rinstId_ptype { mntype : nat } : @ren_ptype (mntype) (mntype) (id) = id .
Proof. exact ((eq_trans) (rinstInst_ptype ((id) (_))) instId_ptype). Qed.
Lemma rinstId_ntype { mntype : nat } : @ren_ntype (mntype) (mntype) (id) = id .
Proof. exact ((eq_trans) (rinstInst_ntype ((id) (_))) instId_ntype). Qed.
Lemma varL_ntype { mntype : nat } { nntype : nat } {i : fin w} (sigmantype : fin w -> (fin) (mntype) -> ntype (nntype)) : (fun i => funcomp (subst_ntype sigmantype) (var_ntype (mntype) i)) = sigmantype .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => eq_refl)). Qed.
Lemma varLRen_ntype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) : (fun i => funcomp (ren_ntype xintype) (var_ntype (mntype) i)) = (fun i => funcomp (var_ntype (nntype) i) (xintype)).
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => eq_refl)). Qed.
Lemma compComp_ptype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) (s : ptype (mntype)) : subst_ptype tauntype (subst_ptype sigmantype s) = subst_ptype (fun i => (funcomp) (subst_ntype tauntype) (sigmantype i)) s .
Proof. exact (compSubstSubst_ptype sigmantype tauntype (_) (fun _ n => eq_refl) s). Qed.
Lemma compComp_ntype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) (s : ntype (mntype)) : subst_ntype tauntype (subst_ntype sigmantype s) = subst_ntype (fun i => (funcomp) (subst_ntype tauntype) (sigmantype i)) s .
Proof. exact (compSubstSubst_ntype sigmantype tauntype (_) (fun _ n => eq_refl) s). Qed.
Lemma compComp'_ptype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) : (funcomp) (subst_ptype tauntype) (subst_ptype sigmantype) = subst_ptype (fun i => (funcomp) (subst_ntype tauntype) (sigmantype i)) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => compComp_ptype sigmantype tauntype n)). Qed.
Lemma compComp'_ntype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) : (funcomp) (subst_ntype tauntype) (subst_ntype sigmantype) = subst_ntype (fun i => (funcomp) (subst_ntype tauntype) (sigmantype i)) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => compComp_ntype sigmantype tauntype n)). Qed.
Lemma compRen_ptype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (s : ptype (mntype)) : ren_ptype zetantype (subst_ptype sigmantype s) = subst_ptype (fun i => (funcomp) (ren_ntype zetantype) (sigmantype i)) s .
Proof. exact (compSubstRen_ptype sigmantype zetantype (_) (fun _ n => eq_refl) s). Qed.
Lemma compRen_ntype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (s : ntype (mntype)) : ren_ntype zetantype (subst_ntype sigmantype s) = subst_ntype (fun i => (funcomp) (ren_ntype zetantype) (sigmantype i)) s .
Proof. exact (compSubstRen_ntype sigmantype zetantype (_) (fun _ n => eq_refl) s). Qed.
Lemma compRen'_ptype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) : (funcomp) (ren_ptype zetantype) (subst_ptype sigmantype) = subst_ptype (fun i => (funcomp) (ren_ntype zetantype) (sigmantype i)) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => compRen_ptype sigmantype zetantype n)). Qed.
Lemma compRen'_ntype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) : (funcomp) (ren_ntype zetantype) (subst_ntype sigmantype) = subst_ntype (fun i => (funcomp) (ren_ntype zetantype) (sigmantype i)) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => compRen_ntype sigmantype zetantype n)). Qed.
Lemma renComp_ptype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) (s : ptype (mntype)) : subst_ptype tauntype (ren_ptype xintype s) = subst_ptype (fun i => (funcomp) (tauntype i) (xintype)) s .
Proof. exact (compRenSubst_ptype xintype tauntype (_) (fun _ n => eq_refl) s). Qed.
Lemma renComp_ntype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) (s : ntype (mntype)) : subst_ntype tauntype (ren_ntype xintype s) = subst_ntype (fun i => (funcomp) (tauntype i) (xintype)) s .
Proof. exact (compRenSubst_ntype xintype tauntype (_) (fun _ n => eq_refl) s). Qed.
Lemma renComp'_ptype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) : (funcomp) (subst_ptype tauntype) (ren_ptype xintype) = subst_ptype (fun i => (funcomp) (tauntype i) (xintype )) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => renComp_ptype xintype tauntype n)). Qed.
Lemma renComp'_ntype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) : (funcomp) (subst_ntype tauntype) (ren_ntype xintype) = subst_ntype (fun i => (funcomp) (tauntype i) (xintype)) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => renComp_ntype xintype tauntype n)). Qed.
Lemma renRen_ptype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (s : ptype (mntype)) : ren_ptype zetantype (ren_ptype xintype s) = ren_ptype ((funcomp) (zetantype) (xintype)) s .
Proof. exact (compRenRen_ptype xintype zetantype (_) (fun n => eq_refl) s). Qed.
Lemma renRen_ntype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (zetantype :(fin) (kntype) -> (fin) (lntype)) (s : ntype (mntype)) : ren_ntype zetantype (ren_ntype xintype s) = ren_ntype ((funcomp) (zetantype) (xintype)) s .
Proof. exact (compRenRen_ntype xintype zetantype (_) (fun n => eq_refl) s). Qed.
Lemma renRen'_ptype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) : (funcomp) (ren_ptype zetantype) (ren_ptype xintype) = ren_ptype ((funcomp) (zetantype) (xintype)) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => renRen_ptype xintype zetantype n)). Qed.
Lemma renRen'_ntype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) : (funcomp) (ren_ntype zetantype) (ren_ntype xintype) = ren_ntype ((funcomp) (zetantype) (xintype)) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => renRen_ntype xintype zetantype n)). Qed.
End ptypentype.
Arguments top {w} {nntype}.
Arguments bAllN {w} {nntype}.
Arguments var_ntype {w} {nntype}.
Arguments uAllN {w} {nntype}.
(*
Global Instance Subst_ptype { mntype : nat } { nntype : nat } : Subst1 ((fin) (mntype) -> ntype (nntype)) (ptype (mntype)) (ptype (nntype)) := @subst_ptype (mntype) (nntype) .
Global Instance Subst_ntype { mntype : nat } { nntype : nat } : Subst1 ((fin) (mntype) -> ntype (nntype)) (ntype (mntype)) (ntype (nntype)) := @subst_ntype (mntype) (nntype) .
Global Instance Ren_ptype { mntype : nat } { nntype : nat } : Ren1 ((fin) (mntype) -> (fin) (nntype)) (ptype (mntype)) (ptype (nntype)) := @ren_ptype (mntype) (nntype) .
Global Instance Ren_ntype { mntype : nat } { nntype : nat } : Ren1 ((fin) (mntype) -> (fin) (nntype)) (ntype (mntype)) (ntype (nntype)) := @ren_ntype (mntype) (nntype) .
Global Instance VarInstance_ntype { mntype : nat } : Var ((fin) (mntype)) (ntype (mntype)) := @var_ntype (mntype) .
Notation "x '__ntype'" := (var_ntype x) (at level 5, format "x __ntype") : subst_scope.
Notation "x '__ntype'" := (@ids (_) (_) VarInstance_ntype x) (at level 5, only printing, format "x __ntype") : subst_scope.
Notation "'var'" := (var_ntype) (only printing, at level 1) : subst_scope.
Class Up_ntype X Y := up_ntype : X -> Y.
Notation "↑__ntype" := (up_ntype) (only printing) : subst_scope.
Notation "↑__ntype" := (up_ntype_ntype) (only printing) : subst_scope.
Global Instance Up_ntype_ntype { m : nat } { nntype : nat } : Up_ntype (_) (_) := @up_ntype_ntype (m) (nntype) .
*)
(* Notation "s sigmantype " := (subst_ptype sigmantype s) (at level 7, left associativity, only printing) : subst_scope. *)
(* Notation " sigmantype " := (subst_ptype sigmantype) (at level 1, left associativity, only printing) : fscope. *)
(* Notation "s ⟨ xintype ⟩" := (ren_ptype xintype s) (at level 7, left associativity, only printing) : subst_scope. *)
(* Notation "⟨ xintype ⟩" := (ren_ptype xintype) (at level 1, left associativity, only printing) : fscope. *)
(* Notation "s sigmantype " := (subst_ntype sigmantype s) (at level 7, left associativity, only printing) : subst_scope. *)
(* Notation " sigmantype " := (subst_ntype sigmantype) (at level 1, left associativity, only printing) : fscope. *)
(* Notation "s ⟨ xintype ⟩" := (ren_ntype xintype s) (at level 7, left associativity, only printing) : subst_scope. *)
(* Notation "⟨ xintype ⟩" := (ren_ntype xintype) (at level 1, left associativity, only printing) : fscope. *)
Ltac auto_unfold := repeat unfold subst1, subst2, Subst1, Subst2, ids, ren1, ren2, Ren1, Ren2(*, Subst_ptype, Subst_ntype, Ren_ptype, Ren_ntype, VarInstance_ntype *).
Tactic Notation "auto_unfold" "in" "*" := repeat unfold subst1, subst2, Subst1, Subst2, ids, ren1, ren2, Ren1, Ren2(* , Subst_ptype, Subst_ntype, Ren_ptype, Ren_ntype, VarInstance_ntype *) in *.
Ltac asimpl' := repeat first [progress rewrite ?instId_ptype| progress rewrite ?compComp_ptype| progress rewrite ?compComp'_ptype| progress rewrite ?instId_ntype| progress rewrite ?compComp_ntype| progress rewrite ?compComp'_ntype| progress rewrite ?rinstId_ptype| progress rewrite ?compRen_ptype| progress rewrite ?compRen'_ptype| progress rewrite ?renComp_ptype| progress rewrite ?renComp'_ptype| progress rewrite ?renRen_ptype| progress rewrite ?renRen'_ptype| progress rewrite ?rinstId_ntype| progress rewrite ?compRen_ntype| progress rewrite ?compRen'_ntype| progress rewrite ?renComp_ntype| progress rewrite ?renComp'_ntype| progress rewrite ?renRen_ntype| progress rewrite ?renRen'_ntype| progress rewrite ?varL_ntype| progress rewrite ?varLRen_ntype| progress (unfold up_ren, upRen_ntype_ntype, up_ntype_ntype)| progress (cbn [subst_ptype subst_ntype ren_ptype ren_ntype])| fsimpl].
Ltac asimpl := repeat try unfold_funcomp; auto_unfold in *; asimpl'; repeat try unfold_funcomp.
Tactic Notation "asimpl" "in" hyp(J) := revert J; asimpl; intros J.
Tactic Notation "auto_case" := auto_case (asimpl; cbn; eauto).
Tactic Notation "asimpl" "in" "*" := auto_unfold in *; repeat first [progress rewrite ?instId_ptype in *| progress rewrite ?compComp_ptype in *| progress rewrite ?compComp'_ptype in *| progress rewrite ?instId_ntype in *| progress rewrite ?compComp_ntype in *| progress rewrite ?compComp'_ntype in *| progress rewrite ?rinstId_ptype in *| progress rewrite ?compRen_ptype in *| progress rewrite ?compRen'_ptype in *| progress rewrite ?renComp_ptype in *| progress rewrite ?renComp'_ptype in *| progress rewrite ?renRen_ptype in *| progress rewrite ?renRen'_ptype in *| progress rewrite ?rinstId_ntype in *| progress rewrite ?compRen_ntype in *| progress rewrite ?compRen'_ntype in *| progress rewrite ?renComp_ntype in *| progress rewrite ?renComp'_ntype in *| progress rewrite ?renRen_ntype in *| progress rewrite ?renRen'_ntype in *| progress rewrite ?varL_ntype in *| progress rewrite ?varLRen_ntype in *| progress (unfold up_ren, upRen_ntype_ntype, up_ntype_ntype in *)| progress (cbn [subst_ptype subst_ntype ren_ptype ren_ntype] in *)| fsimpl in *].
Ltac substify := auto_unfold; try repeat (erewrite rinstInst_ptype); try repeat (erewrite rinstInst_ntype).
Ltac renamify := auto_unfold; try repeat (erewrite <- rinstInst_ptype); try repeat (erewrite <- rinstInst_ntype).
Notation "⊤" := (top).
Notation "∀+ S « T »" := (bAllN S T).
Notation "∀- « T »" := (uAllN T).
Proof. destruct (idSubst_ptype_ntype mntype). now apply H. Defined.
Definition idSubst_ntype { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (mntype)) (Eqntype : forall i x, sigmantype i x = (var_ntype (mntype) i) x) (s : ntype (mntype)) : subst_ntype sigmantype s = s.
Proof. destruct (idSubst_ptype_ntype mntype). now apply H0. Defined.
Definition upExtRen_ntype_ntype { m : nat } { n' : nat } (xi : (fin) (m) -> (fin) (n')) (zeta : (fin) (m) -> (fin) (n')) (Eq : forall x, xi x = zeta x) : forall x, (upRen_ntype_ntype xi) x = (upRen_ntype_ntype zeta) x :=
fun n => match n with
| Some fin_n => (ap) (shift) (Eq fin_n)
| None => eq_refl
end.
(* Fixpoint extRen_ptype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (zetantype : (fin) (mntype) -> (fin) (nntype)) (Eqntype : forall x, xintype x = zetantype x) (s : ptype (mntype)) : ren_ptype xintype s = ren_ptype zetantype s :=
match s return ren_ptype xintype s = ren_ptype zetantype s with
| top (_) => congr_top
| bAllN (_) s0 s1 => congr_bAllN ((extRen_ntype xintype zetantype Eqntzype) s0) ((extRen_ntype (upRen_ntype_ntype xintype) (upRen_ntype_ntype zetantype) (upExtRen_ntype_ntype (_) (_) Eqntype)) s1)
end
with extRen_ntype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (zetantype : (fin) (mntype) -> (fin) (nntype)) (Eqntype : forall x, xintype x = zetantype x) (s : ntype (mntype)) : ren_ntype xintype s = ren_ntype zetantype s :=
match s return ren_ntype xintype s = ren_ntype zetantype s with
| var_ntype (_) s => (ap) (var_ntype (nntype)) (Eqntype s)
| uAllN (_) s0 => congr_uAllN ((extRen_ptype xintype zetantype Eqntype) s0)
end. *)
Lemma extRen_ptype_ntype : forall m,
(forall s l (xintype : fin m -> fin l) (zetantype : fin m -> fin l) (Eqntype : forall x, xintype x = zetantype x), ren_ptype xintype s = ren_ptype zetantype s)
/\ (forall s l (xintype : fin m -> fin l) (zetantype : fin m -> fin l) (Eqntype : forall x, xintype x = zetantype x), ren_ntype xintype s = ren_ntype zetantype s).
Proof. eapply ptype_ntype_mutind; intros.
-apply congr_top.
-apply congr_bAllN.
+apply map_ext_in. apply Forall_forall. apply (Forall_impl _ (fun n : ntype m => forall (l : nat) (xintype zetantype : fin m -> fin l), (forall x : fin m, xintype x = zetantype x) -> ren_ntype xintype n = ren_ntype zetantype n)); auto.
+apply H. intros. apply (upExtRen_ntype_ntype _ _ Eqntype).
-apply (ap (var_ntype l t)). apply Eqntype.
-apply congr_uAllN. apply H. intros. apply (upExtRen_ntype_ntype _ _ Eqntype).
Qed.
Definition extRen_ptype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (zetantype : (fin) (mntype) -> (fin) (nntype)) (Eqntype : forall x, xintype x = zetantype x) (s : ptype (mntype)) : ren_ptype xintype s = ren_ptype zetantype s.
Proof. destruct (extRen_ptype_ntype mntype). now apply H. Defined.
Definition extRen_ntype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (zetantype : (fin) (mntype) -> (fin) (nntype)) (Eqntype : forall x, xintype x = zetantype x) (s : ntype (mntype)) : ren_ntype xintype s = ren_ntype zetantype s.
Proof. destruct (extRen_ptype_ntype mntype). now apply H0. Defined.
Definition upExt_ntype_ntype { m : nat } { nntype : nat } (sigma : fin w -> (fin) (m) -> ntype (nntype)) (tau : fin w -> (fin) (m) -> ntype (nntype)) (Eq : forall i x, sigma i x = tau i x) : forall i x, (up_ntype_ntype sigma) i x = (up_ntype_ntype tau) i x :=
fun i n => match n with
| Some fin_n => (ap) (ren_ntype (shift)) (Eq i fin_n)
| None => eq_refl
end.
(* Fixpoint ext_ptype { mntype : nat } { nntype : nat } (sigmantype : (fin) (mntype) -> ntype (nntype)) (tauntype : (fin) (mntype) -> ntype (nntype)) (Eqntype : forall x, sigmantype x = tauntype x) (s : ptype (mntype)) : subst_ptype sigmantype s = subst_ptype tauntype s :=
match s return subst_ptype sigmantype s = subst_ptype tauntype s with
| top (_) => congr_top
| bAllN (_) s0 s1 => congr_bAllN ((ext_ntype sigmantype tauntype Eqntype) s0) ((ext_ntype (up_ntype_ntype sigmantype) (up_ntype_ntype tauntype) (upExt_ntype_ntype (_) (_) Eqntype)) s1)
end
with ext_ntype { mntype : nat } { nntype : nat } (sigmantype : (fin) (mntype) -> ntype (nntype)) (tauntype : (fin) (mntype) -> ntype (nntype)) (Eqntype : forall x, sigmantype x = tauntype x) (s : ntype (mntype)) : subst_ntype sigmantype s = subst_ntype tauntype s :=
match s return subst_ntype sigmantype s = subst_ntype tauntype s with
| var_ntype (_) s => Eqntype s
| uAllN (_) s0 => congr_uAllN ((ext_ptype sigmantype tauntype Eqntype) s0)
end. *)
Lemma ext_ptype_ntype : forall m,
(forall s l (sig : fin w -> fin m -> ntype l) (tau : fin w -> fin m -> ntype l) (Eq : forall i x, sig i x = tau i x),
subst_ptype sig s = subst_ptype tau s)
/\ (forall s l (sig : fin w -> fin m -> ntype l) (tau : fin w -> fin m -> ntype l) (Eq : forall i x, sig i x = tau i x),
subst_ntype sig s = subst_ntype tau s).
Proof. eapply ptype_ntype_mutind; intros.
-apply congr_top.
-apply congr_bAllN.
+apply map_ext_in. apply Forall_forall. apply (Forall_impl _ (fun n0 : ntype m => forall (l : nat) (sig tau : fin w -> fin m -> ntype l), (forall (i : fin w) (x : fin m), sig i x = tau i x) -> subst_ntype sig n0 = subst_ntype tau n0)); auto.
+apply H. intros. apply (upExt_ntype_ntype _ _ Eq).
-apply Eq.
-apply congr_uAllN. apply H. intros. apply (upExt_ntype_ntype _ _ Eq).
Qed.
Definition ext_ptype { mntype : nat } { nntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (nntype)) (tauntype : fin w -> (fin) (mntype) -> ntype (nntype)) (Eqntype : forall i x, sigmantype i x = tauntype i x) (s : ptype (mntype)) : subst_ptype sigmantype s = subst_ptype tauntype s.
Proof. destruct (ext_ptype_ntype mntype). now apply H. Defined.
Definition ext_ntype { mntype : nat } { nntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (nntype)) (tauntype : fin w -> (fin) (mntype) -> ntype (nntype)) (Eqntype : forall i x, sigmantype i x = tauntype i x) (s : ntype (mntype)) : subst_ntype sigmantype s = subst_ntype tauntype s.
Proof. destruct (ext_ptype_ntype mntype). now apply H0. Defined.
Definition up_ren_ren_ntype_ntype { k : nat } { l : nat } { m : nat } (xi : (fin) (k) -> (fin) (l)) (tau : (fin) (l) -> (fin) (m)) (theta : (fin) (k) -> (fin) (m)) (Eq : forall x, ((funcomp) (tau) (xi)) x = theta x) : forall x, ((funcomp) (upRen_ntype_ntype tau) (upRen_ntype_ntype xi)) x = (upRen_ntype_ntype theta) x :=
up_ren_ren (xi) (tau) (theta) (Eq).
(* Fixpoint compRenRen_ptype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (rhontype : (fin) (mntype) -> (fin) (lntype)) (Eqntype : forall x, ((funcomp) zetantype xintype) x = rhontype x) (s : ptype (mntype)) : ren_ptype zetantype (ren_ptype xintype s) = ren_ptype rhontype s :=
match s return ren_ptype zetantype (ren_ptype xintype s) = ren_ptype rhontype s with
| top (_) => congr_top
| bAllN (_) s0 s1 => congr_bAllN ((compRenRen_ntype xintype zetantype rhontype Eqntype) s0) ((compRenRen_ntype (upRen_ntype_ntype xintype) (upRen_ntype_ntype zetantype) (upRen_ntype_ntype rhontype) (up_ren_ren (_) (_) (_) Eqntype)) s1)
end
with compRenRen_ntype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (rhontype : (fin) (mntype) -> (fin) (lntype)) (Eqntype : forall x, ((funcomp) zetantype xintype) x = rhontype x) (s : ntype (mntype)) : ren_ntype zetantype (ren_ntype xintype s) = ren_ntype rhontype s :=
match s return ren_ntype zetantype (ren_ntype xintype s) = ren_ntype rhontype s with
| var_ntype (_) s => (ap) (var_ntype (lntype)) (Eqntype s)
| uAllN (_) s0 => congr_uAllN ((compRenRen_ptype xintype zetantype rhontype Eqntype) s0)
end. *)
Lemma compRenRen_ptype_ntype : forall m,
(forall s k l (xi : fin m -> fin k) (zet : fin k -> fin l) (rho : fin m -> fin l)
(Eq : forall x, funcomp (zet) (xi) x = rho x),
ren_ptype zet (ren_ptype xi s) = ren_ptype rho s)
/\ (forall s k l (xi : fin m -> fin k) (zet : fin k -> fin l) (rho : fin m -> fin l)
(Eq : forall x, funcomp (zet) (xi) x = rho x),
ren_ntype zet (ren_ntype xi s) = ren_ntype rho s).
Proof. eapply ptype_ntype_mutind; intros.
-apply congr_top.
-apply congr_bAllN.
+rewrite map_map. apply map_ext_in. apply Forall_forall. apply (Forall_impl _ (fun n : ntype m =>
forall (k l : nat) (xi : fin m -> fin k) (zet : fin k -> fin l) (rho : fin m -> fin l),
(forall x : fin m, (xi >> zet) x = rho x) -> ren_ntype zet (ren_ntype xi n) = ren_ntype rho n)); auto.
+apply H. intros. apply (up_ren_ren _ _ _ Eq).
-apply (ap (var_ntype l t)). apply Eq.
-apply congr_uAllN. apply H. intros. apply (up_ren_ren _ _ _ (Eq)).
Qed.
Definition compRenRen_ptype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (rhontype : (fin) (mntype) -> (fin) (lntype)) (Eqntype : forall x, ((funcomp) (zetantype) (xintype)) x = rhontype x) (s : ptype (mntype)) : ren_ptype zetantype (ren_ptype xintype s) = ren_ptype rhontype s.
Proof. destruct (compRenRen_ptype_ntype mntype). now apply H. Defined.
Definition compRenRen_ntype { kntype : nat } { lntype : nat } { mntype : nat }(xintype : (fin) (mntype) -> (fin) (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (rhontype : (fin) (mntype) -> (fin) (lntype)) (Eqntype : forall x, ((funcomp) (zetantype) (xintype)) x = rhontype x) (s : ntype (mntype)) : ren_ntype zetantype (ren_ntype xintype s) = ren_ntype rhontype s.
Proof. destruct (compRenRen_ptype_ntype mntype). now apply H0. Defined.
Definition up_ren_subst_ntype_ntype { k : nat } { l : nat } { mntype : nat } (xi : (fin) (k) -> (fin) (l)) (tau : fin w -> (fin) (l) -> ntype (mntype)) (theta : fin w -> (fin) (k) -> ntype (mntype)) (Eq : forall i x, ((funcomp) (tau i) (xi)) x = theta i x) : forall i x, ((funcomp) (up_ntype_ntype tau i) (upRen_ntype_ntype xi)) x = (up_ntype_ntype theta) i x :=
fun i n => match n with
| Some fin_n => (ap) (ren_ntype (shift)) (Eq i fin_n)
| None => eq_refl
end.
(* Fixpoint compRenSubst_ptype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (tauntype : (fin) (kntype) -> ntype (lntype)) (thetantype : (fin) (mntype) -> ntype (lntype)) (Eqntype : forall x, ((funcomp) tauntype xintype) x = thetantype x) (s : ptype (mntype)) : subst_ptype tauntype (ren_ptype xintype s) = subst_ptype thetantype s :=
match s return subst_ptype tauntype (ren_ptype xintype s) = subst_ptype thetantype s with
| top (_) => congr_top
| bAllN (_) s0 s1 => congr_bAllN ((compRenSubst_ntype xintype tauntype thetantype Eqntype) s0) ((compRenSubst_ntype (upRen_ntype_ntype xintype) (up_ntype_ntype tauntype) (up_ntype_ntype thetantype) (up_ren_subst_ntype_ntype (_) (_) (_) Eqntype)) s1)
end
with compRenSubst_ntype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (tauntype : (fin) (kntype) -> ntype (lntype)) (thetantype : (fin) (mntype) -> ntype (lntype)) (Eqntype : forall x, ((funcomp) tauntype xintype) x = thetantype x) (s : ntype (mntype)) : subst_ntype tauntype (ren_ntype xintype s) = subst_ntype thetantype s :=
match s return subst_ntype tauntype (ren_ntype xintype s) = subst_ntype thetantype s with
| var_ntype (_) s => Eqntype s
| uAllN (_) s0 => congr_uAllN ((compRenSubst_ptype xintype tauntype thetantype Eqntype) s0)
end. *)
Lemma compRenSubst_ptype_ntype : forall m,
(forall s k l (xi : fin m -> fin k) (tau : fin w -> fin k -> ntype l) (theta : fin w -> fin m -> ntype l)
(Eq : forall i x, (xi >> (tau i)) x = theta i x), subst_ptype tau (ren_ptype xi s) = subst_ptype theta s)
/\ (forall s k l (xi : fin m -> fin k) (tau : fin w -> fin k -> ntype l) (theta : fin w -> fin m -> ntype l)
(Eq : forall i x, (xi >> (tau i)) x = theta i x), subst_ntype tau (ren_ntype xi s) = subst_ntype theta s).
Proof. eapply ptype_ntype_mutind; intros.
-apply congr_top.
-apply congr_bAllN.
+rewrite map_map. apply map_ext_in. apply Forall_forall. apply (Forall_impl _ (fun n0 : ntype m =>
forall (k l : nat) (xi : fin m -> fin k) (tau : fin w -> fin k -> ntype l)
(theta : fin w -> fin m -> ntype l),
(forall (i : fin w) (x : fin m), (xi >> tau i) x = theta i x) ->
subst_ntype tau (ren_ntype xi n0) = subst_ntype theta n0)); auto.
+apply H. intros. apply (up_ren_subst_ntype_ntype _ _ _ Eq).
-apply Eq.
-apply congr_uAllN. apply H. intros. apply (up_ren_subst_ntype_ntype _ _ _ Eq).
Qed.
Definition compRenSubst_ptype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) (thetantype : fin w -> (fin) (mntype) -> ntype (lntype)) (Eqntype : forall i x, ((funcomp) (tauntype i) (xintype)) x = thetantype i x) (s : ptype (mntype)) : subst_ptype tauntype (ren_ptype xintype s) = subst_ptype thetantype s.
Proof. destruct (compRenSubst_ptype_ntype mntype). now apply H. Defined.
Definition compRenSubst_ntype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) (thetantype : fin w -> (fin) (mntype) -> ntype (lntype)) (Eqntype : forall i x, ((funcomp) (tauntype i) (xintype)) x = thetantype i x) (s : ntype (mntype)) : subst_ntype tauntype (ren_ntype xintype s) = subst_ntype thetantype s.
Proof. destruct (compRenSubst_ptype_ntype mntype). now apply H0. Defined.
Definition up_subst_ren_ntype_ntype { k : nat } { lntype : nat } { mntype : nat } (sigma : fin w -> (fin) (k) -> ntype (lntype)) (zetantype : (fin) (lntype) -> (fin) (mntype)) (theta : fin w -> (fin) (k) -> ntype (mntype)) (Eq : forall i x, ((funcomp) (ren_ntype zetantype) (sigma i)) x = theta i x) : forall i x, ((funcomp) (ren_ntype (upRen_ntype_ntype zetantype)) (up_ntype_ntype sigma i)) x = (up_ntype_ntype theta) i x :=
fun i n => match n with
| Some fin_n => (eq_trans) (compRenRen_ntype (shift) (upRen_ntype_ntype zetantype) ((funcomp) (shift) zetantype) (fun x => eq_refl) (sigma i fin_n)) ((eq_trans) ((eq_sym) (compRenRen_ntype zetantype (shift) ((funcomp) (shift) zetantype) (fun x => eq_refl) (sigma i fin_n))) ((ap) (ren_ntype (shift)) (Eq i fin_n)))
| None => eq_refl
end.
(* Definition up_subst_ren_ntype_ntype { k : nat } { lntype : nat } { mntype : nat } (sigma : fin n -> (fin) (k) -> ntype (lntype)) (zetantype : (fin) (lntype) -> (fin) (mntype)) (theta : fin n -> (fin) (k) -> ntype (mntype)) (Eq : forall i x, ((funcomp) (ren_ntype zetantype) (sigma i)) x = theta i x) : forall i x, ((funcomp) (ren_ntype (upRen_ntype_ntype zetantype)) (up_ntype_ntype sigma i)) x = (up_ntype_ntype theta) i x
:= fun i x => match x as o return ((up_ntype_ntype sigma i >> ren_ntype (upRen_ntype_ntype zetantype)) o = up_ntype_ntype theta i o) with
| Some fin_n => eq_trans (compRenRen_ntype (↑) (upRen_ntype_ntype zetantype)
(fun (i0 : fin n) (x0 : fin lntype) => ↑ (zetantype i0 x0))
(fun (i0 : fin n) (x0 : fin lntype) => eq_refl)
(sigma i fin_n))
(eq_trans (eq_sym (compRenRen_ntype zetantype (fun _ : fin n => ↑)
(fun (i0 : fin n) (x0 : fin lntype) => ↑ (zetantype i0 x0))
(fun (i0 : fin n) (x0 : fin lntype) => eq_refl)
(sigma i fin_n)))
(ap (ren_ntype (fun _ : fin n => ↑)) (Eq i fin_n)))
| None => eq_refl
end. *)
(* Fixpoint compSubstRen_ptype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : (fin) (mntype) -> ntype (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (thetantype : (fin) (mntype) -> ntype (lntype)) (Eqntype : forall x, ((funcomp) (ren_ntype zetantype) sigmantype) x = thetantype x) (s : ptype (mntype)) : ren_ptype zetantype (subst_ptype sigmantype s) = subst_ptype thetantype s :=
match s return ren_ptype zetantype (subst_ptype sigmantype s) = subst_ptype thetantype s with
| top (_) => congr_top
| bAllN (_) s0 s1 => congr_bAllN ((compSubstRen_ntype sigmantype zetantype thetantype Eqntype) s0) ((compSubstRen_ntype (up_ntype_ntype sigmantype) (upRen_ntype_ntype zetantype) (up_ntype_ntype thetantype) (up_subst_ren_ntype_ntype (_) (_) (_) Eqntype)) s1)
end
with compSubstRen_ntype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : (fin) (mntype) -> ntype (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (thetantype : (fin) (mntype) -> ntype (lntype)) (Eqntype : forall x, ((funcomp) (ren_ntype zetantype) sigmantype) x = thetantype x) (s : ntype (mntype)) : ren_ntype zetantype (subst_ntype sigmantype s) = subst_ntype thetantype s :=
match s return ren_ntype zetantype (subst_ntype sigmantype s) = subst_ntype thetantype s with
| var_ntype (_) s => Eqntype s
| uAllN (_) s0 => congr_uAllN ((compSubstRen_ptype sigmantype zetantype thetantype Eqntype) s0)
end. *)
Lemma compSubstRen_ptype_ntype : forall m,
(forall s k l (sig : fin w -> fin m -> ntype k) (zeta : fin k -> fin l) (theta : fin w -> fin m -> ntype l)
(Eq : forall i x, ((sig i) >> (ren_ntype zeta)) x = theta i x), ren_ptype zeta (subst_ptype sig s) = subst_ptype theta s)
/\ (forall s k l (sig : fin w -> fin m -> ntype k) (zeta : fin k -> fin l) (theta : fin w -> fin m -> ntype l)
(Eq : forall i x, ((sig i) >> (ren_ntype zeta)) x = theta i x), ren_ntype zeta (subst_ntype sig s) = subst_ntype theta s).
Proof. eapply ptype_ntype_mutind; intros.
-apply congr_top.
-apply congr_bAllN.
+rewrite map_map. apply map_ext_in. apply Forall_forall. apply (Forall_impl _ (fun n0 : ntype m => forall (k l : nat) (sig : fin w -> fin m -> ntype k) (zeta : fin k -> fin l) (theta : fin w -> fin m -> ntype l), (forall (i : fin w) (x : fin m), (sig i >> ren_ntype zeta) x = theta i x) -> ren_ntype zeta (subst_ntype sig n0) = subst_ntype theta n0)); auto.
+apply H. intros. apply (up_subst_ren_ntype_ntype _ _ _ Eq).
-apply Eq.
-apply congr_uAllN. apply H. intros. apply (up_subst_ren_ntype_ntype _ _ _ Eq).
Qed.
Definition compSubstRen_ptype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (thetantype : fin w -> (fin) (mntype) -> ntype (lntype)) (Eqntype : forall i x, ((funcomp) (ren_ntype zetantype) (sigmantype i)) x = thetantype i x) (s : ptype (mntype)) : ren_ptype zetantype (subst_ptype sigmantype s) = subst_ptype thetantype s.
Proof. destruct (compSubstRen_ptype_ntype mntype). now apply H. Defined.
Definition compSubstRen_ntype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (thetantype : fin w -> (fin) (mntype) -> ntype (lntype)) (Eqntype : forall i x, ((funcomp) (ren_ntype zetantype) (sigmantype i)) x = thetantype i x) (s : ntype (mntype)) : ren_ntype zetantype (subst_ntype sigmantype s) = subst_ntype thetantype s.
Proof. destruct (compSubstRen_ptype_ntype mntype). now apply H0. Defined.
(* Definition up_subst_subst_ntype_ntype { k : nat } { lntype : nat } { mntype : nat } (sigma : (fin) (k) -> ntype (lntype)) (tauntype : (fin) (lntype) -> ntype (mntype)) (theta : (fin) (k) -> ntype (mntype)) (Eq : forall x, ((funcomp) (subst_ntype tauntype) sigma) x = theta x) : forall x, ((funcomp) (subst_ntype (up_ntype_ntype tauntype)) (up_ntype_ntype sigma)) x = (up_ntype_ntype theta) x :=
fun n => match n with
| Some fin_n => (eq_trans) (compRenSubst_ntype (shift) (up_ntype_ntype tauntype) ((funcomp) (up_ntype_ntype tauntype) (shift)) (fun x => eq_refl) (sigma fin_n)) ((eq_trans) ((eq_sym) (compSubstRen_ntype tauntype (shift) ((funcomp) (ren_ntype (shift)) tauntype) (fun x => eq_refl) (sigma fin_n))) ((ap) (ren_ntype (shift)) (Eq fin_n)))
| None => eq_refl
end. *)
Definition up_subst_subst_ntype_ntype { k : nat } { lntype : nat } { mntype : nat } (sigma : fin w -> (fin) (k) -> ntype (lntype)) (tauntype : fin w -> (fin) (lntype) -> ntype (mntype)) (theta : fin w -> (fin) (k) -> ntype (mntype)) (Eq : forall i x, ((funcomp) (subst_ntype tauntype) (sigma i)) x = theta i x) : forall i x, ((funcomp) (subst_ntype (up_ntype_ntype tauntype)) (up_ntype_ntype sigma i)) x = (up_ntype_ntype theta) i x
:= fun i x => match x as o return ((up_ntype_ntype sigma i >> subst_ntype (up_ntype_ntype tauntype)) o = up_ntype_ntype theta i o) with
| Some fin_n => eq_trans
(compRenSubst_ntype (↑) (up_ntype_ntype tauntype)
(fun (i0 : fin w) (x0 : fin lntype) =>
up_ntype_ntype tauntype i0 (↑ x0))
(fun (i0 : fin w) (x0 : fin lntype) => eq_refl)
(sigma i fin_n))
(eq_trans (eq_sym (compSubstRen_ntype tauntype (↑)
(fun (i0 : fin w) (x0 : fin lntype) =>
ren_ntype (↑) (tauntype i0 x0))
(fun (i0 : fin w) (x0 : fin lntype) => eq_refl)
(sigma i fin_n)))
(ap (ren_ntype (↑)) (Eq i fin_n)))
| None => eq_refl
end.
(* Fixpoint compSubstSubst_ptype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : (fin) (mntype) -> ntype (kntype)) (tauntype : (fin) (kntype) -> ntype (lntype)) (thetantype : (fin) (mntype) -> ntype (lntype)) (Eqntype : forall x, ((funcomp) (subst_ntype tauntype) sigmantype) x = thetantype x) (s : ptype (mntype)) : subst_ptype tauntype (subst_ptype sigmantype s) = subst_ptype thetantype s :=
match s return subst_ptype tauntype (subst_ptype sigmantype s) = subst_ptype thetantype s with
| top (_) => congr_top
| bAllN (_) s0 s1 => congr_bAllN ((compSubstSubst_ntype sigmantype tauntype thetantype Eqntype) s0) ((compSubstSubst_ntype (up_ntype_ntype sigmantype) (up_ntype_ntype tauntype) (up_ntype_ntype thetantype) (up_subst_subst_ntype_ntype (_) (_) (_) Eqntype)) s1)
end
with compSubstSubst_ntype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : (fin) (mntype) -> ntype (kntype)) (tauntype : (fin) (kntype) -> ntype (lntype)) (thetantype : (fin) (mntype) -> ntype (lntype)) (Eqntype : forall x, ((funcomp) (subst_ntype tauntype) sigmantype) x = thetantype x) (s : ntype (mntype)) : subst_ntype tauntype (subst_ntype sigmantype s) = subst_ntype thetantype s :=
match s return subst_ntype tauntype (subst_ntype sigmantype s) = subst_ntype thetantype s with
| var_ntype (_) s => Eqntype s
| uAllN (_) s0 => congr_uAllN ((compSubstSubst_ptype sigmantype tauntype thetantype Eqntype) s0)
end. *)
Lemma compSubstSubst_ptype_ntype : forall m,
(forall s k l (sig : fin w -> fin m -> ntype k) (tau : fin w -> fin k -> ntype l) (theta : fin w -> fin m -> ntype l)
(Eq : forall i x, (sig i >> subst_ntype tau) x = theta i x), subst_ptype tau (subst_ptype sig s) = subst_ptype theta s)
/\ (forall s k l (sig : fin w -> fin m -> ntype k) (tau : fin w -> fin k -> ntype l) (theta : fin w -> fin m -> ntype l)
(Eq : forall i x, (sig i >> subst_ntype tau) x = theta i x), subst_ntype tau (subst_ntype sig s) = subst_ntype theta s).
Proof. eapply ptype_ntype_mutind; intros.
-apply congr_top.
-apply congr_bAllN.
+rewrite map_map. apply map_ext_in,Forall_forall. apply (Forall_impl _ (fun n0 : ntype m => forall (k l : nat) (sig : fin w -> fin m -> ntype k) (tau : fin w -> fin k -> ntype l) (theta : fin w -> fin m -> ntype l), (forall (i : fin w) (x : fin m), (sig i >> subst_ntype tau) x = theta i x) -> subst_ntype tau (subst_ntype sig n0) = subst_ntype theta n0)); auto.
+apply H. intros. apply (up_subst_subst_ntype_ntype _ _ _ Eq).
-apply Eq.
-apply congr_uAllN. apply H. intros. apply (up_subst_subst_ntype_ntype _ _ _ Eq).
Qed.
Definition compSubstSubst_ptype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) (thetantype : fin w -> (fin) (mntype) -> ntype (lntype)) (Eqntype : forall i x, ((funcomp) (subst_ntype tauntype) (sigmantype i)) x = thetantype i x) (s : ptype (mntype)) : subst_ptype tauntype (subst_ptype sigmantype s) = subst_ptype thetantype s.
Proof. destruct (compSubstSubst_ptype_ntype mntype). now apply H. Defined.
Definition compSubstSubst_ntype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) (thetantype : fin w -> (fin) (mntype) -> ntype (lntype)) (Eqntype : forall i x, ((funcomp) (subst_ntype tauntype) (sigmantype i)) x = thetantype i x) (s : ntype (mntype)) : subst_ntype tauntype (subst_ntype sigmantype s) = subst_ntype thetantype s.
Proof. destruct (compSubstSubst_ptype_ntype mntype). now apply H0. Defined.
Definition rinstInst_up_ntype_ntype { m : nat } { nntype : nat } (xi : (fin) (m) -> (fin) (nntype)) (sigma : fin w -> (fin) (m) -> ntype (nntype)) (Eq : forall i x, ((funcomp) (var_ntype (nntype) i) (xi)) x = sigma i x) : forall i x, ((funcomp) (var_ntype ((S) nntype) i) (upRen_ntype_ntype xi)) x = (up_ntype_ntype sigma) i x :=
fun i n => match n with
| Some fin_n => (ap) (ren_ntype (shift)) (Eq i fin_n)
| None => eq_refl
end.
(* Fixpoint rinst_inst_ptype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (sigmantype : (fin) (mntype) -> ntype (nntype)) (Eqntype : forall x, ((funcomp) (var_ntype (nntype)) xintype) x = sigmantype x) (s : ptype (mntype)) : ren_ptype xintype s = subst_ptype sigmantype s :=
match s return ren_ptype xintype s = subst_ptype sigmantype s with
| top (_) => congr_top
| bAllN (_) s0 s1 => congr_bAllN ((rinst_inst_ntype xintype sigmantype Eqntype) s0) ((rinst_inst_ntype (upRen_ntype_ntype xintype) (up_ntype_ntype sigmantype) (rinstInst_up_ntype_ntype (_) (_) Eqntype)) s1)
end
with rinst_inst_ntype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (sigmantype : (fin) (mntype) -> ntype (nntype)) (Eqntype : forall x, ((funcomp) (var_ntype (nntype)) xintype) x = sigmantype x) (s : ntype (mntype)) : ren_ntype xintype s = subst_ntype sigmantype s :=
match s return ren_ntype xintype s = subst_ntype sigmantype s with
| var_ntype (_) s => Eqntype s
| uAllN (_) s0 => congr_uAllN ((rinst_inst_ptype xintype sigmantype Eqntype) s0)
end. *)
Lemma rinst_inst_ptype_ntype : forall m,
(forall s l (xi : fin m -> fin l) (sig : fin w -> fin m -> ntype l) (Eq : forall i x, var_ntype l i (xi x) = sig i x),
ren_ptype xi s = subst_ptype sig s)
/\ (forall s l (xi : fin m -> fin l) (sig : fin w -> fin m -> ntype l) (Eq : forall i x, var_ntype l i (xi x) = sig i x),
ren_ntype xi s = subst_ntype sig s).
Proof. eapply ptype_ntype_mutind; intros.
-apply congr_top.
-apply congr_bAllN.
+apply map_ext_in,Forall_forall. apply (Forall_impl _ (fun n0 : ntype m => forall (l : nat) (xi : fin m -> fin l) (sig : fin w -> fin m -> ntype l), (forall (i : fin w) (x : fin m), var_ntype l i (xi x) = sig i x) -> ren_ntype xi n0 = subst_ntype sig n0)); auto.
+apply H. intros. apply (rinstInst_up_ntype_ntype _ _ Eq).
-apply Eq.
-apply congr_uAllN. apply H. intros. apply (rinstInst_up_ntype_ntype _ _ Eq).
Qed.
Definition rinst_inst_ptype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (sigmantype : fin w -> (fin) (mntype) -> ntype (nntype)) (Eqntype : forall i x, ((funcomp) (var_ntype (nntype) i) (xintype)) x = sigmantype i x) (s : ptype (mntype)) : ren_ptype xintype s = subst_ptype sigmantype s.
Proof. destruct (rinst_inst_ptype_ntype mntype). now apply H. Defined.
Definition rinst_inst_ntype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) (sigmantype : fin w -> (fin) (mntype) -> ntype (nntype)) (Eqntype : forall i x, ((funcomp) (var_ntype (nntype) i) (xintype)) x = sigmantype i x) (s : ntype (mntype)) : ren_ntype xintype s = subst_ntype sigmantype s.
Proof. destruct (rinst_inst_ptype_ntype mntype). now apply H0. Defined.
Lemma rinstInst_ptype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) : ren_ptype xintype = subst_ptype (fun i => (funcomp) (var_ntype (nntype) i) (xintype)).
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => rinst_inst_ptype xintype (_) (fun i n => eq_refl) x)). Qed.
Lemma rinstInst_ntype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) : ren_ntype xintype = subst_ntype (fun i => (funcomp) (var_ntype (nntype) i) (xintype)) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => rinst_inst_ntype xintype (_) (fun _ n => eq_refl) x)). Qed.
Lemma instId_ptype { mntype : nat } : subst_ptype (fun i => var_ntype (mntype) i) = id .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => idSubst_ptype (var_ntype (mntype)) (fun _ n => eq_refl) ((id) x))). Qed.
Lemma instId_ntype { mntype : nat } : subst_ntype (fun i => var_ntype (mntype) i) = id .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => idSubst_ntype (var_ntype (mntype)) (fun _ n => eq_refl) ((id) x))). Qed.
Lemma rinstId_ptype { mntype : nat } : @ren_ptype (mntype) (mntype) (id) = id .
Proof. exact ((eq_trans) (rinstInst_ptype ((id) (_))) instId_ptype). Qed.
Lemma rinstId_ntype { mntype : nat } : @ren_ntype (mntype) (mntype) (id) = id .
Proof. exact ((eq_trans) (rinstInst_ntype ((id) (_))) instId_ntype). Qed.
Lemma varL_ntype { mntype : nat } { nntype : nat } {i : fin w} (sigmantype : fin w -> (fin) (mntype) -> ntype (nntype)) : (fun i => funcomp (subst_ntype sigmantype) (var_ntype (mntype) i)) = sigmantype .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => eq_refl)). Qed.
Lemma varLRen_ntype { mntype : nat } { nntype : nat } (xintype : (fin) (mntype) -> (fin) (nntype)) : (fun i => funcomp (ren_ntype xintype) (var_ntype (mntype) i)) = (fun i => funcomp (var_ntype (nntype) i) (xintype)).
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun x => eq_refl)). Qed.
Lemma compComp_ptype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) (s : ptype (mntype)) : subst_ptype tauntype (subst_ptype sigmantype s) = subst_ptype (fun i => (funcomp) (subst_ntype tauntype) (sigmantype i)) s .
Proof. exact (compSubstSubst_ptype sigmantype tauntype (_) (fun _ n => eq_refl) s). Qed.
Lemma compComp_ntype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) (s : ntype (mntype)) : subst_ntype tauntype (subst_ntype sigmantype s) = subst_ntype (fun i => (funcomp) (subst_ntype tauntype) (sigmantype i)) s .
Proof. exact (compSubstSubst_ntype sigmantype tauntype (_) (fun _ n => eq_refl) s). Qed.
Lemma compComp'_ptype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) : (funcomp) (subst_ptype tauntype) (subst_ptype sigmantype) = subst_ptype (fun i => (funcomp) (subst_ntype tauntype) (sigmantype i)) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => compComp_ptype sigmantype tauntype n)). Qed.
Lemma compComp'_ntype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) : (funcomp) (subst_ntype tauntype) (subst_ntype sigmantype) = subst_ntype (fun i => (funcomp) (subst_ntype tauntype) (sigmantype i)) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => compComp_ntype sigmantype tauntype n)). Qed.
Lemma compRen_ptype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (s : ptype (mntype)) : ren_ptype zetantype (subst_ptype sigmantype s) = subst_ptype (fun i => (funcomp) (ren_ntype zetantype) (sigmantype i)) s .
Proof. exact (compSubstRen_ptype sigmantype zetantype (_) (fun _ n => eq_refl) s). Qed.
Lemma compRen_ntype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (s : ntype (mntype)) : ren_ntype zetantype (subst_ntype sigmantype s) = subst_ntype (fun i => (funcomp) (ren_ntype zetantype) (sigmantype i)) s .
Proof. exact (compSubstRen_ntype sigmantype zetantype (_) (fun _ n => eq_refl) s). Qed.
Lemma compRen'_ptype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) : (funcomp) (ren_ptype zetantype) (subst_ptype sigmantype) = subst_ptype (fun i => (funcomp) (ren_ntype zetantype) (sigmantype i)) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => compRen_ptype sigmantype zetantype n)). Qed.
Lemma compRen'_ntype { kntype : nat } { lntype : nat } { mntype : nat } (sigmantype : fin w -> (fin) (mntype) -> ntype (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) : (funcomp) (ren_ntype zetantype) (subst_ntype sigmantype) = subst_ntype (fun i => (funcomp) (ren_ntype zetantype) (sigmantype i)) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => compRen_ntype sigmantype zetantype n)). Qed.
Lemma renComp_ptype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) (s : ptype (mntype)) : subst_ptype tauntype (ren_ptype xintype s) = subst_ptype (fun i => (funcomp) (tauntype i) (xintype)) s .
Proof. exact (compRenSubst_ptype xintype tauntype (_) (fun _ n => eq_refl) s). Qed.
Lemma renComp_ntype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) (s : ntype (mntype)) : subst_ntype tauntype (ren_ntype xintype s) = subst_ntype (fun i => (funcomp) (tauntype i) (xintype)) s .
Proof. exact (compRenSubst_ntype xintype tauntype (_) (fun _ n => eq_refl) s). Qed.
Lemma renComp'_ptype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) : (funcomp) (subst_ptype tauntype) (ren_ptype xintype) = subst_ptype (fun i => (funcomp) (tauntype i) (xintype )) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => renComp_ptype xintype tauntype n)). Qed.
Lemma renComp'_ntype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (tauntype : fin w -> (fin) (kntype) -> ntype (lntype)) : (funcomp) (subst_ntype tauntype) (ren_ntype xintype) = subst_ntype (fun i => (funcomp) (tauntype i) (xintype)) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => renComp_ntype xintype tauntype n)). Qed.
Lemma renRen_ptype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) (s : ptype (mntype)) : ren_ptype zetantype (ren_ptype xintype s) = ren_ptype ((funcomp) (zetantype) (xintype)) s .
Proof. exact (compRenRen_ptype xintype zetantype (_) (fun n => eq_refl) s). Qed.
Lemma renRen_ntype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (zetantype :(fin) (kntype) -> (fin) (lntype)) (s : ntype (mntype)) : ren_ntype zetantype (ren_ntype xintype s) = ren_ntype ((funcomp) (zetantype) (xintype)) s .
Proof. exact (compRenRen_ntype xintype zetantype (_) (fun n => eq_refl) s). Qed.
Lemma renRen'_ptype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) : (funcomp) (ren_ptype zetantype) (ren_ptype xintype) = ren_ptype ((funcomp) (zetantype) (xintype)) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => renRen_ptype xintype zetantype n)). Qed.
Lemma renRen'_ntype { kntype : nat } { lntype : nat } { mntype : nat } (xintype : (fin) (mntype) -> (fin) (kntype)) (zetantype : (fin) (kntype) -> (fin) (lntype)) : (funcomp) (ren_ntype zetantype) (ren_ntype xintype) = ren_ntype ((funcomp) (zetantype) (xintype)) .
Proof. exact ((FunctionalExtensionality.functional_extensionality _ _ ) (fun n => renRen_ntype xintype zetantype n)). Qed.
End ptypentype.
Arguments top {w} {nntype}.
Arguments bAllN {w} {nntype}.
Arguments var_ntype {w} {nntype}.
Arguments uAllN {w} {nntype}.
(*
Global Instance Subst_ptype { mntype : nat } { nntype : nat } : Subst1 ((fin) (mntype) -> ntype (nntype)) (ptype (mntype)) (ptype (nntype)) := @subst_ptype (mntype) (nntype) .
Global Instance Subst_ntype { mntype : nat } { nntype : nat } : Subst1 ((fin) (mntype) -> ntype (nntype)) (ntype (mntype)) (ntype (nntype)) := @subst_ntype (mntype) (nntype) .
Global Instance Ren_ptype { mntype : nat } { nntype : nat } : Ren1 ((fin) (mntype) -> (fin) (nntype)) (ptype (mntype)) (ptype (nntype)) := @ren_ptype (mntype) (nntype) .
Global Instance Ren_ntype { mntype : nat } { nntype : nat } : Ren1 ((fin) (mntype) -> (fin) (nntype)) (ntype (mntype)) (ntype (nntype)) := @ren_ntype (mntype) (nntype) .
Global Instance VarInstance_ntype { mntype : nat } : Var ((fin) (mntype)) (ntype (mntype)) := @var_ntype (mntype) .
Notation "x '__ntype'" := (var_ntype x) (at level 5, format "x __ntype") : subst_scope.
Notation "x '__ntype'" := (@ids (_) (_) VarInstance_ntype x) (at level 5, only printing, format "x __ntype") : subst_scope.
Notation "'var'" := (var_ntype) (only printing, at level 1) : subst_scope.
Class Up_ntype X Y := up_ntype : X -> Y.
Notation "↑__ntype" := (up_ntype) (only printing) : subst_scope.
Notation "↑__ntype" := (up_ntype_ntype) (only printing) : subst_scope.
Global Instance Up_ntype_ntype { m : nat } { nntype : nat } : Up_ntype (_) (_) := @up_ntype_ntype (m) (nntype) .
*)
(* Notation "s sigmantype " := (subst_ptype sigmantype s) (at level 7, left associativity, only printing) : subst_scope. *)
(* Notation " sigmantype " := (subst_ptype sigmantype) (at level 1, left associativity, only printing) : fscope. *)
(* Notation "s ⟨ xintype ⟩" := (ren_ptype xintype s) (at level 7, left associativity, only printing) : subst_scope. *)
(* Notation "⟨ xintype ⟩" := (ren_ptype xintype) (at level 1, left associativity, only printing) : fscope. *)
(* Notation "s sigmantype " := (subst_ntype sigmantype s) (at level 7, left associativity, only printing) : subst_scope. *)
(* Notation " sigmantype " := (subst_ntype sigmantype) (at level 1, left associativity, only printing) : fscope. *)
(* Notation "s ⟨ xintype ⟩" := (ren_ntype xintype s) (at level 7, left associativity, only printing) : subst_scope. *)
(* Notation "⟨ xintype ⟩" := (ren_ntype xintype) (at level 1, left associativity, only printing) : fscope. *)
Ltac auto_unfold := repeat unfold subst1, subst2, Subst1, Subst2, ids, ren1, ren2, Ren1, Ren2(*, Subst_ptype, Subst_ntype, Ren_ptype, Ren_ntype, VarInstance_ntype *).
Tactic Notation "auto_unfold" "in" "*" := repeat unfold subst1, subst2, Subst1, Subst2, ids, ren1, ren2, Ren1, Ren2(* , Subst_ptype, Subst_ntype, Ren_ptype, Ren_ntype, VarInstance_ntype *) in *.
Ltac asimpl' := repeat first [progress rewrite ?instId_ptype| progress rewrite ?compComp_ptype| progress rewrite ?compComp'_ptype| progress rewrite ?instId_ntype| progress rewrite ?compComp_ntype| progress rewrite ?compComp'_ntype| progress rewrite ?rinstId_ptype| progress rewrite ?compRen_ptype| progress rewrite ?compRen'_ptype| progress rewrite ?renComp_ptype| progress rewrite ?renComp'_ptype| progress rewrite ?renRen_ptype| progress rewrite ?renRen'_ptype| progress rewrite ?rinstId_ntype| progress rewrite ?compRen_ntype| progress rewrite ?compRen'_ntype| progress rewrite ?renComp_ntype| progress rewrite ?renComp'_ntype| progress rewrite ?renRen_ntype| progress rewrite ?renRen'_ntype| progress rewrite ?varL_ntype| progress rewrite ?varLRen_ntype| progress (unfold up_ren, upRen_ntype_ntype, up_ntype_ntype)| progress (cbn [subst_ptype subst_ntype ren_ptype ren_ntype])| fsimpl].
Ltac asimpl := repeat try unfold_funcomp; auto_unfold in *; asimpl'; repeat try unfold_funcomp.
Tactic Notation "asimpl" "in" hyp(J) := revert J; asimpl; intros J.
Tactic Notation "auto_case" := auto_case (asimpl; cbn; eauto).
Tactic Notation "asimpl" "in" "*" := auto_unfold in *; repeat first [progress rewrite ?instId_ptype in *| progress rewrite ?compComp_ptype in *| progress rewrite ?compComp'_ptype in *| progress rewrite ?instId_ntype in *| progress rewrite ?compComp_ntype in *| progress rewrite ?compComp'_ntype in *| progress rewrite ?rinstId_ptype in *| progress rewrite ?compRen_ptype in *| progress rewrite ?compRen'_ptype in *| progress rewrite ?renComp_ptype in *| progress rewrite ?renComp'_ptype in *| progress rewrite ?renRen_ptype in *| progress rewrite ?renRen'_ptype in *| progress rewrite ?rinstId_ntype in *| progress rewrite ?compRen_ntype in *| progress rewrite ?compRen'_ntype in *| progress rewrite ?renComp_ntype in *| progress rewrite ?renComp'_ntype in *| progress rewrite ?renRen_ntype in *| progress rewrite ?renRen'_ntype in *| progress rewrite ?varL_ntype in *| progress rewrite ?varLRen_ntype in *| progress (unfold up_ren, upRen_ntype_ntype, up_ntype_ntype in *)| progress (cbn [subst_ptype subst_ntype ren_ptype ren_ntype] in *)| fsimpl in *].
Ltac substify := auto_unfold; try repeat (erewrite rinstInst_ptype); try repeat (erewrite rinstInst_ntype).
Ltac renamify := auto_unfold; try repeat (erewrite <- rinstInst_ptype); try repeat (erewrite <- rinstInst_ntype).
Notation "⊤" := (top).
Notation "∀+ S « T »" := (bAllN S T).
Notation "∀- « T »" := (uAllN T).