517 lines
16 KiB
Coq
517 lines
16 KiB
Coq
Require Import core unscoped.
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Require Import Setoid Morphisms Relation_Definitions.
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Module Core.
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Inductive Tm : Type :=
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| VarTm : nat -> Tm
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| Abs : Tm -> Tm
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| App : Tm -> Tm -> Tm.
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Lemma congr_Abs {s0 : Tm} {t0 : Tm} (H0 : s0 = t0) : Abs s0 = Abs t0.
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Proof.
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exact (eq_trans eq_refl (ap (fun x => Abs x) H0)).
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Qed.
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Lemma congr_App {s0 : Tm} {s1 : Tm} {t0 : Tm} {t1 : Tm} (H0 : s0 = t0)
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(H1 : s1 = t1) : App s0 s1 = App t0 t1.
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Proof.
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exact (eq_trans (eq_trans eq_refl (ap (fun x => App x s1) H0))
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(ap (fun x => App t0 x) H1)).
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Qed.
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Lemma upRen_Tm_Tm (xi : nat -> nat) : nat -> nat.
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Proof.
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exact (up_ren xi).
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Defined.
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Fixpoint ren_Tm (xi_Tm : nat -> nat) (s : Tm) {struct s} : Tm :=
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match s with
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| VarTm s0 => VarTm (xi_Tm s0)
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| Abs s0 => Abs (ren_Tm (upRen_Tm_Tm xi_Tm) s0)
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| App s0 s1 => App (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
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end.
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Lemma up_Tm_Tm (sigma : nat -> Tm) : nat -> Tm.
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Proof.
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exact (scons (VarTm var_zero) (funcomp (ren_Tm shift) sigma)).
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Defined.
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Fixpoint subst_Tm (sigma_Tm : nat -> Tm) (s : Tm) {struct s} : Tm :=
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match s with
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| VarTm s0 => sigma_Tm s0
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| Abs s0 => Abs (subst_Tm (up_Tm_Tm sigma_Tm) s0)
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| App s0 s1 => App (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
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end.
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Lemma upId_Tm_Tm (sigma : nat -> Tm) (Eq : forall x, sigma x = VarTm x) :
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forall x, up_Tm_Tm sigma x = VarTm x.
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Proof.
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exact (fun n =>
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match n with
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| S n' => ap (ren_Tm shift) (Eq n')
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| O => eq_refl
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end).
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Qed.
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Fixpoint idSubst_Tm (sigma_Tm : nat -> Tm)
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(Eq_Tm : forall x, sigma_Tm x = VarTm x) (s : Tm) {struct s} :
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subst_Tm sigma_Tm s = s :=
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match s with
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| VarTm s0 => Eq_Tm s0
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| Abs s0 =>
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congr_Abs (idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s0)
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| App s0 s1 =>
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congr_App (idSubst_Tm sigma_Tm Eq_Tm s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
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end.
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Lemma upExtRen_Tm_Tm (xi : nat -> nat) (zeta : nat -> nat)
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(Eq : forall x, xi x = zeta x) :
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forall x, upRen_Tm_Tm xi x = upRen_Tm_Tm zeta x.
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Proof.
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exact (fun n => match n with
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| S n' => ap shift (Eq n')
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| O => eq_refl
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end).
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Qed.
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Fixpoint extRen_Tm (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat)
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(Eq_Tm : forall x, xi_Tm x = zeta_Tm x) (s : Tm) {struct s} :
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ren_Tm xi_Tm s = ren_Tm zeta_Tm s :=
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match s with
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| VarTm s0 => ap (VarTm) (Eq_Tm s0)
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| Abs s0 =>
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congr_Abs
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(extRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
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(upExtRen_Tm_Tm _ _ Eq_Tm) s0)
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| App s0 s1 =>
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congr_App (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
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(extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
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end.
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Lemma upExt_Tm_Tm (sigma : nat -> Tm) (tau : nat -> Tm)
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(Eq : forall x, sigma x = tau x) :
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forall x, up_Tm_Tm sigma x = up_Tm_Tm tau x.
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Proof.
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exact (fun n =>
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match n with
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| S n' => ap (ren_Tm shift) (Eq n')
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| O => eq_refl
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end).
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Qed.
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Fixpoint ext_Tm (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm)
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(Eq_Tm : forall x, sigma_Tm x = tau_Tm x) (s : Tm) {struct s} :
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subst_Tm sigma_Tm s = subst_Tm tau_Tm s :=
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match s with
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| VarTm s0 => Eq_Tm s0
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| Abs s0 =>
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congr_Abs
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(ext_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm) (upExt_Tm_Tm _ _ Eq_Tm)
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s0)
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| App s0 s1 =>
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congr_App (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
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(ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
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end.
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Lemma up_ren_ren_Tm_Tm (xi : nat -> nat) (zeta : nat -> nat)
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(rho : nat -> nat) (Eq : forall x, funcomp zeta xi x = rho x) :
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forall x, funcomp (upRen_Tm_Tm zeta) (upRen_Tm_Tm xi) x = upRen_Tm_Tm rho x.
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Proof.
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exact (up_ren_ren xi zeta rho Eq).
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Qed.
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Fixpoint compRenRen_Tm (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat)
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(rho_Tm : nat -> nat) (Eq_Tm : forall x, funcomp zeta_Tm xi_Tm x = rho_Tm x)
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(s : Tm) {struct s} : ren_Tm zeta_Tm (ren_Tm xi_Tm s) = ren_Tm rho_Tm s :=
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match s with
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| VarTm s0 => ap (VarTm) (Eq_Tm s0)
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| Abs s0 =>
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congr_Abs
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(compRenRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
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(upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s0)
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| App s0 s1 =>
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congr_App (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
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(compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
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end.
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Lemma up_ren_subst_Tm_Tm (xi : nat -> nat) (tau : nat -> Tm)
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(theta : nat -> Tm) (Eq : forall x, funcomp tau xi x = theta x) :
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forall x, funcomp (up_Tm_Tm tau) (upRen_Tm_Tm xi) x = up_Tm_Tm theta x.
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Proof.
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exact (fun n =>
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match n with
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| S n' => ap (ren_Tm shift) (Eq n')
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| O => eq_refl
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end).
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Qed.
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Fixpoint compRenSubst_Tm (xi_Tm : nat -> nat) (tau_Tm : nat -> Tm)
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(theta_Tm : nat -> Tm)
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(Eq_Tm : forall x, funcomp tau_Tm xi_Tm x = theta_Tm x) (s : Tm) {struct s} :
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subst_Tm tau_Tm (ren_Tm xi_Tm s) = subst_Tm theta_Tm s :=
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match s with
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| VarTm s0 => Eq_Tm s0
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| Abs s0 =>
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congr_Abs
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(compRenSubst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm tau_Tm)
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(up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s0)
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| App s0 s1 =>
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congr_App (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
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(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
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end.
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Lemma up_subst_ren_Tm_Tm (sigma : nat -> Tm) (zeta_Tm : nat -> nat)
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(theta : nat -> Tm)
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(Eq : forall x, funcomp (ren_Tm zeta_Tm) sigma x = theta x) :
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forall x,
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funcomp (ren_Tm (upRen_Tm_Tm zeta_Tm)) (up_Tm_Tm sigma) x =
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up_Tm_Tm theta x.
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Proof.
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exact (fun n =>
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match n with
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| S n' =>
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eq_trans
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(compRenRen_Tm shift (upRen_Tm_Tm zeta_Tm)
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(funcomp shift zeta_Tm) (fun x => eq_refl) (sigma n'))
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(eq_trans
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(eq_sym
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(compRenRen_Tm zeta_Tm shift (funcomp shift zeta_Tm)
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(fun x => eq_refl) (sigma n')))
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(ap (ren_Tm shift) (Eq n')))
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| O => eq_refl
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end).
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Qed.
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Fixpoint compSubstRen_Tm (sigma_Tm : nat -> Tm) (zeta_Tm : nat -> nat)
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(theta_Tm : nat -> Tm)
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(Eq_Tm : forall x, funcomp (ren_Tm zeta_Tm) sigma_Tm x = theta_Tm x)
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(s : Tm) {struct s} :
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ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
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match s with
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| VarTm s0 => Eq_Tm s0
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| Abs s0 =>
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congr_Abs
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(compSubstRen_Tm (up_Tm_Tm sigma_Tm) (upRen_Tm_Tm zeta_Tm)
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(up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s0)
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| App s0 s1 =>
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congr_App (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
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(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
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end.
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Lemma up_subst_subst_Tm_Tm (sigma : nat -> Tm) (tau_Tm : nat -> Tm)
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(theta : nat -> Tm)
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(Eq : forall x, funcomp (subst_Tm tau_Tm) sigma x = theta x) :
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forall x,
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funcomp (subst_Tm (up_Tm_Tm tau_Tm)) (up_Tm_Tm sigma) x = up_Tm_Tm theta x.
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Proof.
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exact (fun n =>
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match n with
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| S n' =>
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eq_trans
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(compRenSubst_Tm shift (up_Tm_Tm tau_Tm)
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(funcomp (up_Tm_Tm tau_Tm) shift) (fun x => eq_refl)
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(sigma n'))
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(eq_trans
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(eq_sym
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(compSubstRen_Tm tau_Tm shift
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(funcomp (ren_Tm shift) tau_Tm) (fun x => eq_refl)
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(sigma n'))) (ap (ren_Tm shift) (Eq n')))
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| O => eq_refl
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end).
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Qed.
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Fixpoint compSubstSubst_Tm (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm)
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(theta_Tm : nat -> Tm)
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(Eq_Tm : forall x, funcomp (subst_Tm tau_Tm) sigma_Tm x = theta_Tm x)
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(s : Tm) {struct s} :
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subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
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match s with
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| VarTm s0 => Eq_Tm s0
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| Abs s0 =>
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congr_Abs
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(compSubstSubst_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm)
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(up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s0)
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| App s0 s1 =>
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congr_App (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
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(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
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end.
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Lemma renRen_Tm (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat) (s : Tm) :
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ren_Tm zeta_Tm (ren_Tm xi_Tm s) = ren_Tm (funcomp zeta_Tm xi_Tm) s.
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Proof.
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exact (compRenRen_Tm xi_Tm zeta_Tm _ (fun n => eq_refl) s).
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Qed.
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Lemma renRen'_Tm_pointwise (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat) :
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pointwise_relation _ eq (funcomp (ren_Tm zeta_Tm) (ren_Tm xi_Tm))
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(ren_Tm (funcomp zeta_Tm xi_Tm)).
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Proof.
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exact (fun s => compRenRen_Tm xi_Tm zeta_Tm _ (fun n => eq_refl) s).
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Qed.
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Lemma renSubst_Tm (xi_Tm : nat -> nat) (tau_Tm : nat -> Tm) (s : Tm) :
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subst_Tm tau_Tm (ren_Tm xi_Tm s) = subst_Tm (funcomp tau_Tm xi_Tm) s.
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Proof.
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exact (compRenSubst_Tm xi_Tm tau_Tm _ (fun n => eq_refl) s).
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Qed.
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Lemma renSubst_Tm_pointwise (xi_Tm : nat -> nat) (tau_Tm : nat -> Tm) :
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pointwise_relation _ eq (funcomp (subst_Tm tau_Tm) (ren_Tm xi_Tm))
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(subst_Tm (funcomp tau_Tm xi_Tm)).
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Proof.
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exact (fun s => compRenSubst_Tm xi_Tm tau_Tm _ (fun n => eq_refl) s).
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Qed.
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Lemma substRen_Tm (sigma_Tm : nat -> Tm) (zeta_Tm : nat -> nat) (s : Tm) :
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ren_Tm zeta_Tm (subst_Tm sigma_Tm s) =
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subst_Tm (funcomp (ren_Tm zeta_Tm) sigma_Tm) s.
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Proof.
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exact (compSubstRen_Tm sigma_Tm zeta_Tm _ (fun n => eq_refl) s).
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Qed.
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Lemma substRen_Tm_pointwise (sigma_Tm : nat -> Tm) (zeta_Tm : nat -> nat) :
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pointwise_relation _ eq (funcomp (ren_Tm zeta_Tm) (subst_Tm sigma_Tm))
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(subst_Tm (funcomp (ren_Tm zeta_Tm) sigma_Tm)).
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Proof.
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exact (fun s => compSubstRen_Tm sigma_Tm zeta_Tm _ (fun n => eq_refl) s).
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Qed.
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Lemma substSubst_Tm (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm) (s : Tm) :
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subst_Tm tau_Tm (subst_Tm sigma_Tm s) =
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subst_Tm (funcomp (subst_Tm tau_Tm) sigma_Tm) s.
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Proof.
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exact (compSubstSubst_Tm sigma_Tm tau_Tm _ (fun n => eq_refl) s).
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Qed.
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Lemma substSubst_Tm_pointwise (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm) :
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pointwise_relation _ eq (funcomp (subst_Tm tau_Tm) (subst_Tm sigma_Tm))
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(subst_Tm (funcomp (subst_Tm tau_Tm) sigma_Tm)).
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Proof.
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exact (fun s => compSubstSubst_Tm sigma_Tm tau_Tm _ (fun n => eq_refl) s).
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Qed.
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Lemma rinstInst_up_Tm_Tm (xi : nat -> nat) (sigma : nat -> Tm)
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(Eq : forall x, funcomp (VarTm) xi x = sigma x) :
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forall x, funcomp (VarTm) (upRen_Tm_Tm xi) x = up_Tm_Tm sigma x.
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Proof.
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exact (fun n =>
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match n with
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| S n' => ap (ren_Tm shift) (Eq n')
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| O => eq_refl
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end).
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Qed.
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Fixpoint rinst_inst_Tm (xi_Tm : nat -> nat) (sigma_Tm : nat -> Tm)
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(Eq_Tm : forall x, funcomp (VarTm) xi_Tm x = sigma_Tm x) (s : Tm) {struct s}
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: ren_Tm xi_Tm s = subst_Tm sigma_Tm s :=
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match s with
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| VarTm s0 => Eq_Tm s0
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| Abs s0 =>
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congr_Abs
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(rinst_inst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm sigma_Tm)
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(rinstInst_up_Tm_Tm _ _ Eq_Tm) s0)
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| App s0 s1 =>
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congr_App (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
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(rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
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end.
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Lemma rinstInst'_Tm (xi_Tm : nat -> nat) (s : Tm) :
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ren_Tm xi_Tm s = subst_Tm (funcomp (VarTm) xi_Tm) s.
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Proof.
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exact (rinst_inst_Tm xi_Tm _ (fun n => eq_refl) s).
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Qed.
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Lemma rinstInst'_Tm_pointwise (xi_Tm : nat -> nat) :
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pointwise_relation _ eq (ren_Tm xi_Tm) (subst_Tm (funcomp (VarTm) xi_Tm)).
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Proof.
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exact (fun s => rinst_inst_Tm xi_Tm _ (fun n => eq_refl) s).
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Qed.
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Lemma instId'_Tm (s : Tm) : subst_Tm (VarTm) s = s.
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Proof.
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exact (idSubst_Tm (VarTm) (fun n => eq_refl) s).
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Qed.
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Lemma instId'_Tm_pointwise : pointwise_relation _ eq (subst_Tm (VarTm)) id.
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Proof.
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exact (fun s => idSubst_Tm (VarTm) (fun n => eq_refl) s).
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Qed.
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Lemma rinstId'_Tm (s : Tm) : ren_Tm id s = s.
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Proof.
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exact (eq_ind_r (fun t => t = s) (instId'_Tm s) (rinstInst'_Tm id s)).
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Qed.
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Lemma rinstId'_Tm_pointwise : pointwise_relation _ eq (@ren_Tm id) id.
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Proof.
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exact (fun s => eq_ind_r (fun t => t = s) (instId'_Tm s) (rinstInst'_Tm id s)).
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Qed.
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Lemma varL'_Tm (sigma_Tm : nat -> Tm) (x : nat) :
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subst_Tm sigma_Tm (VarTm x) = sigma_Tm x.
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Proof.
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exact (eq_refl).
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Qed.
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Lemma varL'_Tm_pointwise (sigma_Tm : nat -> Tm) :
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pointwise_relation _ eq (funcomp (subst_Tm sigma_Tm) (VarTm)) sigma_Tm.
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Proof.
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exact (fun x => eq_refl).
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Qed.
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Lemma varLRen'_Tm (xi_Tm : nat -> nat) (x : nat) :
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ren_Tm xi_Tm (VarTm x) = VarTm (xi_Tm x).
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Proof.
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exact (eq_refl).
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Qed.
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Lemma varLRen'_Tm_pointwise (xi_Tm : nat -> nat) :
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pointwise_relation _ eq (funcomp (ren_Tm xi_Tm) (VarTm))
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(funcomp (VarTm) xi_Tm).
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Proof.
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exact (fun x => eq_refl).
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Qed.
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Class Up_Tm X Y :=
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up_Tm : X -> Y.
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#[global] Instance Subst_Tm : (Subst1 _ _ _) := @subst_Tm.
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#[global] Instance Up_Tm_Tm : (Up_Tm _ _) := @up_Tm_Tm.
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#[global] Instance Ren_Tm : (Ren1 _ _ _) := @ren_Tm.
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#[global]
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Instance VarInstance_Tm : (Var _ _) := @VarTm.
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Notation "[ sigma_Tm ]" := (subst_Tm sigma_Tm)
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( at level 1, left associativity, only printing) : fscope.
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Notation "s [ sigma_Tm ]" := (subst_Tm sigma_Tm s)
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( at level 7, left associativity, only printing) : subst_scope.
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Notation "↑__Tm" := up_Tm (only printing) : subst_scope.
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Notation "↑__Tm" := up_Tm_Tm (only printing) : subst_scope.
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Notation "⟨ xi_Tm ⟩" := (ren_Tm xi_Tm)
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( at level 1, left associativity, only printing) : fscope.
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Notation "s ⟨ xi_Tm ⟩" := (ren_Tm xi_Tm s)
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( at level 7, left associativity, only printing) : subst_scope.
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Notation "'var'" := VarTm ( at level 1, only printing) : subst_scope.
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Notation "x '__Tm'" := (@ids _ _ VarInstance_Tm x)
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( at level 5, format "x __Tm", only printing) : subst_scope.
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Notation "x '__Tm'" := (VarTm x) ( at level 5, format "x __Tm") :
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subst_scope.
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#[global]
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Instance subst_Tm_morphism :
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(Proper (respectful (pointwise_relation _ eq) (respectful eq eq))
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(@subst_Tm)).
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Proof.
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exact (fun f_Tm g_Tm Eq_Tm s t Eq_st =>
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eq_ind s (fun t' => subst_Tm f_Tm s = subst_Tm g_Tm t')
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(ext_Tm f_Tm g_Tm Eq_Tm s) t Eq_st).
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Qed.
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#[global]
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Instance subst_Tm_morphism2 :
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(Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
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(@subst_Tm)).
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Proof.
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exact (fun f_Tm g_Tm Eq_Tm s => ext_Tm f_Tm g_Tm Eq_Tm s).
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Qed.
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#[global]
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Instance ren_Tm_morphism :
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(Proper (respectful (pointwise_relation _ eq) (respectful eq eq)) (@ren_Tm)).
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Proof.
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exact (fun f_Tm g_Tm Eq_Tm s t Eq_st =>
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eq_ind s (fun t' => ren_Tm f_Tm s = ren_Tm g_Tm t')
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(extRen_Tm f_Tm g_Tm Eq_Tm s) t Eq_st).
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Qed.
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#[global]
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Instance ren_Tm_morphism2 :
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(Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
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(@ren_Tm)).
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Proof.
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exact (fun f_Tm g_Tm Eq_Tm s => extRen_Tm f_Tm g_Tm Eq_Tm s).
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Qed.
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Ltac auto_unfold := repeat
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unfold VarInstance_Tm, Var, ids, Ren_Tm, Ren1, ren1,
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Up_Tm_Tm, Up_Tm, up_Tm, Subst_Tm, Subst1, subst1.
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Tactic Notation "auto_unfold" "in" "*" := repeat
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unfold VarInstance_Tm, Var, ids,
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Ren_Tm, Ren1, ren1, Up_Tm_Tm,
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Up_Tm, up_Tm, Subst_Tm, Subst1,
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subst1 in *.
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Ltac asimpl' := repeat (first
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[ progress setoid_rewrite substSubst_Tm_pointwise
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| progress setoid_rewrite substSubst_Tm
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| progress setoid_rewrite substRen_Tm_pointwise
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| progress setoid_rewrite substRen_Tm
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| progress setoid_rewrite renSubst_Tm_pointwise
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| progress setoid_rewrite renSubst_Tm
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| progress setoid_rewrite renRen'_Tm_pointwise
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| progress setoid_rewrite renRen_Tm
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| progress setoid_rewrite varLRen'_Tm_pointwise
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| progress setoid_rewrite varLRen'_Tm
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| progress setoid_rewrite varL'_Tm_pointwise
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| progress setoid_rewrite varL'_Tm
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| progress setoid_rewrite rinstId'_Tm_pointwise
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| progress setoid_rewrite rinstId'_Tm
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| progress setoid_rewrite instId'_Tm_pointwise
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| progress setoid_rewrite instId'_Tm
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| progress unfold up_Tm_Tm, upRen_Tm_Tm, up_ren
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| progress cbn[subst_Tm ren_Tm]
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| progress fsimpl ]).
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Ltac asimpl := check_no_evars;
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repeat
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|
unfold VarInstance_Tm, Var, ids, Ren_Tm, Ren1, ren1,
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|
Up_Tm_Tm, Up_Tm, up_Tm, Subst_Tm, Subst1, subst1 in *;
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asimpl'; minimize.
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Tactic Notation "asimpl" "in" hyp(J) := revert J; asimpl; intros J.
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Tactic Notation "auto_case" := auto_case ltac:(asimpl; cbn; eauto).
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Ltac substify := auto_unfold; try setoid_rewrite rinstInst'_Tm_pointwise;
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try setoid_rewrite rinstInst'_Tm.
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Ltac renamify := auto_unfold; try setoid_rewrite_left rinstInst'_Tm_pointwise;
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|
try setoid_rewrite_left rinstInst'_Tm.
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|
|
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End Core.
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Module Extra.
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Import Core.
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#[global] Hint Opaque subst_Tm: rewrite.
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#[global] Hint Opaque ren_Tm: rewrite.
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End Extra.
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Module interface.
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Export Core.
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Export Extra.
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End interface.
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Export interface.
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