1340 lines
45 KiB
Coq
1340 lines
45 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 PTag : Type :=
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| PL : PTag
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| PR : PTag.
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Lemma congr_PL : PL = PL.
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Proof.
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exact (eq_refl).
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Qed.
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Lemma congr_PR : PR = PR.
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Proof.
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exact (eq_refl).
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Qed.
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Inductive TTag : Type :=
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| TPi : TTag
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| TSig : TTag.
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Lemma congr_TPi : TPi = TPi.
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Proof.
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exact (eq_refl).
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Qed.
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Lemma congr_TSig : TSig = TSig.
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Proof.
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exact (eq_refl).
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Qed.
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Inductive PTm : Type :=
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| VarPTm : nat -> PTm
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| PAbs : PTm -> PTm
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| PApp : PTm -> PTm -> PTm
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| PPair : PTm -> PTm -> PTm
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| PProj : PTag -> PTm -> PTm
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| PConst : TTag -> PTm
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| PUniv : nat -> PTm
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| PBot : PTm.
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Lemma congr_PAbs {s0 : PTm} {t0 : PTm} (H0 : s0 = t0) : PAbs s0 = PAbs t0.
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Proof.
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exact (eq_trans eq_refl (ap (fun x => PAbs x) H0)).
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Qed.
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Lemma congr_PApp {s0 : PTm} {s1 : PTm} {t0 : PTm} {t1 : PTm} (H0 : s0 = t0)
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(H1 : s1 = t1) : PApp s0 s1 = PApp t0 t1.
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Proof.
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exact (eq_trans (eq_trans eq_refl (ap (fun x => PApp x s1) H0))
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(ap (fun x => PApp t0 x) H1)).
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Qed.
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Lemma congr_PPair {s0 : PTm} {s1 : PTm} {t0 : PTm} {t1 : PTm} (H0 : s0 = t0)
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(H1 : s1 = t1) : PPair s0 s1 = PPair t0 t1.
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Proof.
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exact (eq_trans (eq_trans eq_refl (ap (fun x => PPair x s1) H0))
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(ap (fun x => PPair t0 x) H1)).
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Qed.
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Lemma congr_PProj {s0 : PTag} {s1 : PTm} {t0 : PTag} {t1 : PTm}
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(H0 : s0 = t0) (H1 : s1 = t1) : PProj s0 s1 = PProj t0 t1.
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Proof.
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exact (eq_trans (eq_trans eq_refl (ap (fun x => PProj x s1) H0))
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(ap (fun x => PProj t0 x) H1)).
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Qed.
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Lemma congr_PConst {s0 : TTag} {t0 : TTag} (H0 : s0 = t0) :
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PConst s0 = PConst t0.
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Proof.
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exact (eq_trans eq_refl (ap (fun x => PConst x) H0)).
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Qed.
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Lemma congr_PUniv {s0 : nat} {t0 : nat} (H0 : s0 = t0) : PUniv s0 = PUniv t0.
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Proof.
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exact (eq_trans eq_refl (ap (fun x => PUniv x) H0)).
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Qed.
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Lemma congr_PBot : PBot = PBot.
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Proof.
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exact (eq_refl).
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Qed.
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Lemma upRen_PTm_PTm (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_PTm (xi_PTm : nat -> nat) (s : PTm) {struct s} : PTm :=
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match s with
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| VarPTm s0 => VarPTm (xi_PTm s0)
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| PAbs s0 => PAbs (ren_PTm (upRen_PTm_PTm xi_PTm) s0)
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| PApp s0 s1 => PApp (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
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| PPair s0 s1 => PPair (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
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| PProj s0 s1 => PProj s0 (ren_PTm xi_PTm s1)
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| PConst s0 => PConst s0
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| PUniv s0 => PUniv s0
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| PBot => PBot
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end.
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Lemma up_PTm_PTm (sigma : nat -> PTm) : nat -> PTm.
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Proof.
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exact (scons (VarPTm var_zero) (funcomp (ren_PTm shift) sigma)).
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Defined.
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Fixpoint subst_PTm (sigma_PTm : nat -> PTm) (s : PTm) {struct s} : PTm :=
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match s with
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| VarPTm s0 => sigma_PTm s0
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| PAbs s0 => PAbs (subst_PTm (up_PTm_PTm sigma_PTm) s0)
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| PApp s0 s1 => PApp (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1)
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| PPair s0 s1 => PPair (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1)
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| PProj s0 s1 => PProj s0 (subst_PTm sigma_PTm s1)
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| PConst s0 => PConst s0
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| PUniv s0 => PUniv s0
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| PBot => PBot
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end.
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Lemma upId_PTm_PTm (sigma : nat -> PTm) (Eq : forall x, sigma x = VarPTm x) :
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forall x, up_PTm_PTm sigma x = VarPTm 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_PTm 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_PTm (sigma_PTm : nat -> PTm)
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(Eq_PTm : forall x, sigma_PTm x = VarPTm x) (s : PTm) {struct s} :
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subst_PTm sigma_PTm s = s :=
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match s with
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| VarPTm s0 => Eq_PTm s0
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| PAbs s0 =>
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congr_PAbs
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(idSubst_PTm (up_PTm_PTm sigma_PTm) (upId_PTm_PTm _ Eq_PTm) s0)
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| PApp s0 s1 =>
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congr_PApp (idSubst_PTm sigma_PTm Eq_PTm s0)
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(idSubst_PTm sigma_PTm Eq_PTm s1)
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| PPair s0 s1 =>
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congr_PPair (idSubst_PTm sigma_PTm Eq_PTm s0)
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(idSubst_PTm sigma_PTm Eq_PTm s1)
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| PProj s0 s1 => congr_PProj (eq_refl s0) (idSubst_PTm sigma_PTm Eq_PTm s1)
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| PConst s0 => congr_PConst (eq_refl s0)
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| PUniv s0 => congr_PUniv (eq_refl s0)
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| PBot => congr_PBot
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end.
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Lemma upExtRen_PTm_PTm (xi : nat -> nat) (zeta : nat -> nat)
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(Eq : forall x, xi x = zeta x) :
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forall x, upRen_PTm_PTm xi x = upRen_PTm_PTm 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_PTm (xi_PTm : nat -> nat) (zeta_PTm : nat -> nat)
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(Eq_PTm : forall x, xi_PTm x = zeta_PTm x) (s : PTm) {struct s} :
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ren_PTm xi_PTm s = ren_PTm zeta_PTm s :=
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match s with
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| VarPTm s0 => ap (VarPTm) (Eq_PTm s0)
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| PAbs s0 =>
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congr_PAbs
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(extRen_PTm (upRen_PTm_PTm xi_PTm) (upRen_PTm_PTm zeta_PTm)
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(upExtRen_PTm_PTm _ _ Eq_PTm) s0)
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| PApp s0 s1 =>
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congr_PApp (extRen_PTm xi_PTm zeta_PTm Eq_PTm s0)
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(extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
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| PPair s0 s1 =>
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congr_PPair (extRen_PTm xi_PTm zeta_PTm Eq_PTm s0)
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(extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
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| PProj s0 s1 =>
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congr_PProj (eq_refl s0) (extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
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| PConst s0 => congr_PConst (eq_refl s0)
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| PUniv s0 => congr_PUniv (eq_refl s0)
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| PBot => congr_PBot
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end.
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Lemma upExt_PTm_PTm (sigma : nat -> PTm) (tau : nat -> PTm)
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(Eq : forall x, sigma x = tau x) :
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forall x, up_PTm_PTm sigma x = up_PTm_PTm 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_PTm 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_PTm (sigma_PTm : nat -> PTm) (tau_PTm : nat -> PTm)
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(Eq_PTm : forall x, sigma_PTm x = tau_PTm x) (s : PTm) {struct s} :
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subst_PTm sigma_PTm s = subst_PTm tau_PTm s :=
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match s with
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| VarPTm s0 => Eq_PTm s0
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| PAbs s0 =>
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congr_PAbs
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(ext_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
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(upExt_PTm_PTm _ _ Eq_PTm) s0)
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| PApp s0 s1 =>
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congr_PApp (ext_PTm sigma_PTm tau_PTm Eq_PTm s0)
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(ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
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| PPair s0 s1 =>
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congr_PPair (ext_PTm sigma_PTm tau_PTm Eq_PTm s0)
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(ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
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| PProj s0 s1 =>
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congr_PProj (eq_refl s0) (ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
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| PConst s0 => congr_PConst (eq_refl s0)
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| PUniv s0 => congr_PUniv (eq_refl s0)
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| PBot => congr_PBot
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end.
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Lemma up_ren_ren_PTm_PTm (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,
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funcomp (upRen_PTm_PTm zeta) (upRen_PTm_PTm xi) x = upRen_PTm_PTm 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_PTm (xi_PTm : nat -> nat) (zeta_PTm : nat -> nat)
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(rho_PTm : nat -> nat)
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(Eq_PTm : forall x, funcomp zeta_PTm xi_PTm x = rho_PTm x) (s : PTm) {struct
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s} : ren_PTm zeta_PTm (ren_PTm xi_PTm s) = ren_PTm rho_PTm s :=
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match s with
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| VarPTm s0 => ap (VarPTm) (Eq_PTm s0)
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| PAbs s0 =>
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congr_PAbs
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(compRenRen_PTm (upRen_PTm_PTm xi_PTm) (upRen_PTm_PTm zeta_PTm)
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(upRen_PTm_PTm rho_PTm) (up_ren_ren _ _ _ Eq_PTm) s0)
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| PApp s0 s1 =>
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congr_PApp (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s0)
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(compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
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| PPair s0 s1 =>
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congr_PPair (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s0)
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(compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
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| PProj s0 s1 =>
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congr_PProj (eq_refl s0)
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(compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
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| PConst s0 => congr_PConst (eq_refl s0)
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| PUniv s0 => congr_PUniv (eq_refl s0)
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| PBot => congr_PBot
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end.
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Lemma up_ren_subst_PTm_PTm (xi : nat -> nat) (tau : nat -> PTm)
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(theta : nat -> PTm) (Eq : forall x, funcomp tau xi x = theta x) :
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forall x,
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funcomp (up_PTm_PTm tau) (upRen_PTm_PTm xi) x = up_PTm_PTm 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_PTm 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_PTm (xi_PTm : nat -> nat) (tau_PTm : nat -> PTm)
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(theta_PTm : nat -> PTm)
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(Eq_PTm : forall x, funcomp tau_PTm xi_PTm x = theta_PTm x) (s : PTm) {struct
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s} : subst_PTm tau_PTm (ren_PTm xi_PTm s) = subst_PTm theta_PTm s :=
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match s with
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| VarPTm s0 => Eq_PTm s0
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| PAbs s0 =>
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congr_PAbs
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(compRenSubst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm tau_PTm)
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(up_PTm_PTm theta_PTm) (up_ren_subst_PTm_PTm _ _ _ Eq_PTm) s0)
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| PApp s0 s1 =>
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congr_PApp (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s0)
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(compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
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| PPair s0 s1 =>
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congr_PPair (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s0)
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(compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
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| PProj s0 s1 =>
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congr_PProj (eq_refl s0)
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(compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
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| PConst s0 => congr_PConst (eq_refl s0)
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| PUniv s0 => congr_PUniv (eq_refl s0)
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| PBot => congr_PBot
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end.
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Lemma up_subst_ren_PTm_PTm (sigma : nat -> PTm) (zeta_PTm : nat -> nat)
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(theta : nat -> PTm)
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(Eq : forall x, funcomp (ren_PTm zeta_PTm) sigma x = theta x) :
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forall x,
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funcomp (ren_PTm (upRen_PTm_PTm zeta_PTm)) (up_PTm_PTm sigma) x =
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up_PTm_PTm 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_PTm shift (upRen_PTm_PTm zeta_PTm)
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(funcomp shift zeta_PTm) (fun x => eq_refl) (sigma n'))
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(eq_trans
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(eq_sym
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(compRenRen_PTm zeta_PTm shift (funcomp shift zeta_PTm)
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(fun x => eq_refl) (sigma n')))
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(ap (ren_PTm 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_PTm (sigma_PTm : nat -> PTm) (zeta_PTm : nat -> nat)
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(theta_PTm : nat -> PTm)
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(Eq_PTm : forall x, funcomp (ren_PTm zeta_PTm) sigma_PTm x = theta_PTm x)
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(s : PTm) {struct s} :
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ren_PTm zeta_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
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match s with
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| VarPTm s0 => Eq_PTm s0
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| PAbs s0 =>
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congr_PAbs
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(compSubstRen_PTm (up_PTm_PTm sigma_PTm) (upRen_PTm_PTm zeta_PTm)
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(up_PTm_PTm theta_PTm) (up_subst_ren_PTm_PTm _ _ _ Eq_PTm) s0)
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| PApp s0 s1 =>
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congr_PApp (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s0)
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(compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
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| PPair s0 s1 =>
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congr_PPair (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s0)
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(compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
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| PProj s0 s1 =>
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congr_PProj (eq_refl s0)
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(compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
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| PConst s0 => congr_PConst (eq_refl s0)
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| PUniv s0 => congr_PUniv (eq_refl s0)
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| PBot => congr_PBot
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end.
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Lemma up_subst_subst_PTm_PTm (sigma : nat -> PTm) (tau_PTm : nat -> PTm)
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(theta : nat -> PTm)
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(Eq : forall x, funcomp (subst_PTm tau_PTm) sigma x = theta x) :
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forall x,
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funcomp (subst_PTm (up_PTm_PTm tau_PTm)) (up_PTm_PTm sigma) x =
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up_PTm_PTm 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_PTm shift (up_PTm_PTm tau_PTm)
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(funcomp (up_PTm_PTm tau_PTm) 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_PTm tau_PTm shift
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(funcomp (ren_PTm shift) tau_PTm) (fun x => eq_refl)
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(sigma n'))) (ap (ren_PTm 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_PTm (sigma_PTm : nat -> PTm) (tau_PTm : nat -> PTm)
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(theta_PTm : nat -> PTm)
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(Eq_PTm : forall x, funcomp (subst_PTm tau_PTm) sigma_PTm x = theta_PTm x)
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(s : PTm) {struct s} :
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subst_PTm tau_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
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match s with
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| VarPTm s0 => Eq_PTm s0
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| PAbs s0 =>
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congr_PAbs
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(compSubstSubst_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
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(up_PTm_PTm theta_PTm) (up_subst_subst_PTm_PTm _ _ _ Eq_PTm) s0)
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| PApp s0 s1 =>
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congr_PApp (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s0)
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(compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
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| PPair s0 s1 =>
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congr_PPair (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s0)
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(compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
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| PProj s0 s1 =>
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congr_PProj (eq_refl s0)
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(compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
|
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| PConst s0 => congr_PConst (eq_refl s0)
|
|
| PUniv s0 => congr_PUniv (eq_refl s0)
|
|
| PBot => congr_PBot
|
|
end.
|
|
|
|
Lemma renRen_PTm (xi_PTm : nat -> nat) (zeta_PTm : nat -> nat) (s : PTm) :
|
|
ren_PTm zeta_PTm (ren_PTm xi_PTm s) = ren_PTm (funcomp zeta_PTm xi_PTm) s.
|
|
Proof.
|
|
exact (compRenRen_PTm xi_PTm zeta_PTm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma renRen'_PTm_pointwise (xi_PTm : nat -> nat) (zeta_PTm : nat -> nat) :
|
|
pointwise_relation _ eq (funcomp (ren_PTm zeta_PTm) (ren_PTm xi_PTm))
|
|
(ren_PTm (funcomp zeta_PTm xi_PTm)).
|
|
Proof.
|
|
exact (fun s => compRenRen_PTm xi_PTm zeta_PTm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma renSubst_PTm (xi_PTm : nat -> nat) (tau_PTm : nat -> PTm) (s : PTm) :
|
|
subst_PTm tau_PTm (ren_PTm xi_PTm s) = subst_PTm (funcomp tau_PTm xi_PTm) s.
|
|
Proof.
|
|
exact (compRenSubst_PTm xi_PTm tau_PTm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma renSubst_PTm_pointwise (xi_PTm : nat -> nat) (tau_PTm : nat -> PTm) :
|
|
pointwise_relation _ eq (funcomp (subst_PTm tau_PTm) (ren_PTm xi_PTm))
|
|
(subst_PTm (funcomp tau_PTm xi_PTm)).
|
|
Proof.
|
|
exact (fun s => compRenSubst_PTm xi_PTm tau_PTm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma substRen_PTm (sigma_PTm : nat -> PTm) (zeta_PTm : nat -> nat) (s : PTm)
|
|
:
|
|
ren_PTm zeta_PTm (subst_PTm sigma_PTm s) =
|
|
subst_PTm (funcomp (ren_PTm zeta_PTm) sigma_PTm) s.
|
|
Proof.
|
|
exact (compSubstRen_PTm sigma_PTm zeta_PTm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma substRen_PTm_pointwise (sigma_PTm : nat -> PTm) (zeta_PTm : nat -> nat)
|
|
:
|
|
pointwise_relation _ eq (funcomp (ren_PTm zeta_PTm) (subst_PTm sigma_PTm))
|
|
(subst_PTm (funcomp (ren_PTm zeta_PTm) sigma_PTm)).
|
|
Proof.
|
|
exact (fun s => compSubstRen_PTm sigma_PTm zeta_PTm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma substSubst_PTm (sigma_PTm : nat -> PTm) (tau_PTm : nat -> PTm)
|
|
(s : PTm) :
|
|
subst_PTm tau_PTm (subst_PTm sigma_PTm s) =
|
|
subst_PTm (funcomp (subst_PTm tau_PTm) sigma_PTm) s.
|
|
Proof.
|
|
exact (compSubstSubst_PTm sigma_PTm tau_PTm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma substSubst_PTm_pointwise (sigma_PTm : nat -> PTm)
|
|
(tau_PTm : nat -> PTm) :
|
|
pointwise_relation _ eq (funcomp (subst_PTm tau_PTm) (subst_PTm sigma_PTm))
|
|
(subst_PTm (funcomp (subst_PTm tau_PTm) sigma_PTm)).
|
|
Proof.
|
|
exact (fun s => compSubstSubst_PTm sigma_PTm tau_PTm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma rinstInst_up_PTm_PTm (xi : nat -> nat) (sigma : nat -> PTm)
|
|
(Eq : forall x, funcomp (VarPTm) xi x = sigma x) :
|
|
forall x, funcomp (VarPTm) (upRen_PTm_PTm xi) x = up_PTm_PTm sigma x.
|
|
Proof.
|
|
exact (fun n =>
|
|
match n with
|
|
| S n' => ap (ren_PTm shift) (Eq n')
|
|
| O => eq_refl
|
|
end).
|
|
Qed.
|
|
|
|
Fixpoint rinst_inst_PTm (xi_PTm : nat -> nat) (sigma_PTm : nat -> PTm)
|
|
(Eq_PTm : forall x, funcomp (VarPTm) xi_PTm x = sigma_PTm x) (s : PTm)
|
|
{struct s} : ren_PTm xi_PTm s = subst_PTm sigma_PTm s :=
|
|
match s with
|
|
| VarPTm s0 => Eq_PTm s0
|
|
| PAbs s0 =>
|
|
congr_PAbs
|
|
(rinst_inst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm sigma_PTm)
|
|
(rinstInst_up_PTm_PTm _ _ Eq_PTm) s0)
|
|
| PApp s0 s1 =>
|
|
congr_PApp (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s0)
|
|
(rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
|
|
| PPair s0 s1 =>
|
|
congr_PPair (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s0)
|
|
(rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
|
|
| PProj s0 s1 =>
|
|
congr_PProj (eq_refl s0) (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
|
|
| PConst s0 => congr_PConst (eq_refl s0)
|
|
| PUniv s0 => congr_PUniv (eq_refl s0)
|
|
| PBot => congr_PBot
|
|
end.
|
|
|
|
Lemma rinstInst'_PTm (xi_PTm : nat -> nat) (s : PTm) :
|
|
ren_PTm xi_PTm s = subst_PTm (funcomp (VarPTm) xi_PTm) s.
|
|
Proof.
|
|
exact (rinst_inst_PTm xi_PTm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma rinstInst'_PTm_pointwise (xi_PTm : nat -> nat) :
|
|
pointwise_relation _ eq (ren_PTm xi_PTm)
|
|
(subst_PTm (funcomp (VarPTm) xi_PTm)).
|
|
Proof.
|
|
exact (fun s => rinst_inst_PTm xi_PTm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma instId'_PTm (s : PTm) : subst_PTm (VarPTm) s = s.
|
|
Proof.
|
|
exact (idSubst_PTm (VarPTm) (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma instId'_PTm_pointwise : pointwise_relation _ eq (subst_PTm (VarPTm)) id.
|
|
Proof.
|
|
exact (fun s => idSubst_PTm (VarPTm) (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma rinstId'_PTm (s : PTm) : ren_PTm id s = s.
|
|
Proof.
|
|
exact (eq_ind_r (fun t => t = s) (instId'_PTm s) (rinstInst'_PTm id s)).
|
|
Qed.
|
|
|
|
Lemma rinstId'_PTm_pointwise : pointwise_relation _ eq (@ren_PTm id) id.
|
|
Proof.
|
|
exact (fun s =>
|
|
eq_ind_r (fun t => t = s) (instId'_PTm s) (rinstInst'_PTm id s)).
|
|
Qed.
|
|
|
|
Lemma varL'_PTm (sigma_PTm : nat -> PTm) (x : nat) :
|
|
subst_PTm sigma_PTm (VarPTm x) = sigma_PTm x.
|
|
Proof.
|
|
exact (eq_refl).
|
|
Qed.
|
|
|
|
Lemma varL'_PTm_pointwise (sigma_PTm : nat -> PTm) :
|
|
pointwise_relation _ eq (funcomp (subst_PTm sigma_PTm) (VarPTm)) sigma_PTm.
|
|
Proof.
|
|
exact (fun x => eq_refl).
|
|
Qed.
|
|
|
|
Lemma varLRen'_PTm (xi_PTm : nat -> nat) (x : nat) :
|
|
ren_PTm xi_PTm (VarPTm x) = VarPTm (xi_PTm x).
|
|
Proof.
|
|
exact (eq_refl).
|
|
Qed.
|
|
|
|
Lemma varLRen'_PTm_pointwise (xi_PTm : nat -> nat) :
|
|
pointwise_relation _ eq (funcomp (ren_PTm xi_PTm) (VarPTm))
|
|
(funcomp (VarPTm) xi_PTm).
|
|
Proof.
|
|
exact (fun x => eq_refl).
|
|
Qed.
|
|
|
|
Inductive Tm : Type :=
|
|
| VarTm : nat -> Tm
|
|
| Abs : Tm -> Tm
|
|
| App : Tm -> Tm -> Tm
|
|
| Pair : Tm -> Tm -> Tm
|
|
| Proj : PTag -> Tm -> Tm
|
|
| TBind : TTag -> Tm -> Tm -> Tm
|
|
| Univ : nat -> Tm
|
|
| BVal : bool -> Tm
|
|
| Bool : Tm
|
|
| If : Tm -> Tm -> Tm -> Tm.
|
|
|
|
Lemma congr_Abs {s0 : Tm} {t0 : Tm} (H0 : s0 = t0) : Abs s0 = Abs t0.
|
|
Proof.
|
|
exact (eq_trans eq_refl (ap (fun x => Abs x) H0)).
|
|
Qed.
|
|
|
|
Lemma congr_App {s0 : Tm} {s1 : Tm} {t0 : Tm} {t1 : Tm} (H0 : s0 = t0)
|
|
(H1 : s1 = t1) : App s0 s1 = App t0 t1.
|
|
Proof.
|
|
exact (eq_trans (eq_trans eq_refl (ap (fun x => App x s1) H0))
|
|
(ap (fun x => App t0 x) H1)).
|
|
Qed.
|
|
|
|
Lemma congr_Pair {s0 : Tm} {s1 : Tm} {t0 : Tm} {t1 : Tm} (H0 : s0 = t0)
|
|
(H1 : s1 = t1) : Pair s0 s1 = Pair t0 t1.
|
|
Proof.
|
|
exact (eq_trans (eq_trans eq_refl (ap (fun x => Pair x s1) H0))
|
|
(ap (fun x => Pair t0 x) H1)).
|
|
Qed.
|
|
|
|
Lemma congr_Proj {s0 : PTag} {s1 : Tm} {t0 : PTag} {t1 : Tm} (H0 : s0 = t0)
|
|
(H1 : s1 = t1) : Proj s0 s1 = Proj t0 t1.
|
|
Proof.
|
|
exact (eq_trans (eq_trans eq_refl (ap (fun x => Proj x s1) H0))
|
|
(ap (fun x => Proj t0 x) H1)).
|
|
Qed.
|
|
|
|
Lemma congr_TBind {s0 : TTag} {s1 : Tm} {s2 : Tm} {t0 : TTag} {t1 : Tm}
|
|
{t2 : Tm} (H0 : s0 = t0) (H1 : s1 = t1) (H2 : s2 = t2) :
|
|
TBind s0 s1 s2 = TBind t0 t1 t2.
|
|
Proof.
|
|
exact (eq_trans
|
|
(eq_trans (eq_trans eq_refl (ap (fun x => TBind x s1 s2) H0))
|
|
(ap (fun x => TBind t0 x s2) H1))
|
|
(ap (fun x => TBind t0 t1 x) H2)).
|
|
Qed.
|
|
|
|
Lemma congr_Univ {s0 : nat} {t0 : nat} (H0 : s0 = t0) : Univ s0 = Univ t0.
|
|
Proof.
|
|
exact (eq_trans eq_refl (ap (fun x => Univ x) H0)).
|
|
Qed.
|
|
|
|
Lemma congr_BVal {s0 : bool} {t0 : bool} (H0 : s0 = t0) : BVal s0 = BVal t0.
|
|
Proof.
|
|
exact (eq_trans eq_refl (ap (fun x => BVal x) H0)).
|
|
Qed.
|
|
|
|
Lemma congr_Bool : Bool = Bool.
|
|
Proof.
|
|
exact (eq_refl).
|
|
Qed.
|
|
|
|
Lemma congr_If {s0 : Tm} {s1 : Tm} {s2 : Tm} {t0 : Tm} {t1 : Tm} {t2 : Tm}
|
|
(H0 : s0 = t0) (H1 : s1 = t1) (H2 : s2 = t2) : If s0 s1 s2 = If t0 t1 t2.
|
|
Proof.
|
|
exact (eq_trans
|
|
(eq_trans (eq_trans eq_refl (ap (fun x => If x s1 s2) H0))
|
|
(ap (fun x => If t0 x s2) H1)) (ap (fun x => If t0 t1 x) H2)).
|
|
Qed.
|
|
|
|
Lemma upRen_Tm_Tm (xi : nat -> nat) : nat -> nat.
|
|
Proof.
|
|
exact (up_ren xi).
|
|
Defined.
|
|
|
|
Fixpoint ren_Tm (xi_Tm : nat -> nat) (s : Tm) {struct s} : Tm :=
|
|
match s with
|
|
| VarTm s0 => VarTm (xi_Tm s0)
|
|
| Abs s0 => Abs (ren_Tm (upRen_Tm_Tm xi_Tm) s0)
|
|
| App s0 s1 => App (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
|
|
| Pair s0 s1 => Pair (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
|
|
| Proj s0 s1 => Proj s0 (ren_Tm xi_Tm s1)
|
|
| TBind s0 s1 s2 =>
|
|
TBind s0 (ren_Tm xi_Tm s1) (ren_Tm (upRen_Tm_Tm xi_Tm) s2)
|
|
| Univ s0 => Univ s0
|
|
| BVal s0 => BVal s0
|
|
| Bool => Bool
|
|
| If s0 s1 s2 => If (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1) (ren_Tm xi_Tm s2)
|
|
end.
|
|
|
|
Lemma up_Tm_Tm (sigma : nat -> Tm) : nat -> Tm.
|
|
Proof.
|
|
exact (scons (VarTm var_zero) (funcomp (ren_Tm shift) sigma)).
|
|
Defined.
|
|
|
|
Fixpoint subst_Tm (sigma_Tm : nat -> Tm) (s : Tm) {struct s} : Tm :=
|
|
match s with
|
|
| VarTm s0 => sigma_Tm s0
|
|
| Abs s0 => Abs (subst_Tm (up_Tm_Tm sigma_Tm) s0)
|
|
| App s0 s1 => App (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
|
|
| Pair s0 s1 => Pair (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
|
|
| Proj s0 s1 => Proj s0 (subst_Tm sigma_Tm s1)
|
|
| TBind s0 s1 s2 =>
|
|
TBind s0 (subst_Tm sigma_Tm s1) (subst_Tm (up_Tm_Tm sigma_Tm) s2)
|
|
| Univ s0 => Univ s0
|
|
| BVal s0 => BVal s0
|
|
| Bool => Bool
|
|
| If s0 s1 s2 =>
|
|
If (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1) (subst_Tm sigma_Tm s2)
|
|
end.
|
|
|
|
Lemma upId_Tm_Tm (sigma : nat -> Tm) (Eq : forall x, sigma x = VarTm x) :
|
|
forall x, up_Tm_Tm sigma x = VarTm x.
|
|
Proof.
|
|
exact (fun n =>
|
|
match n with
|
|
| S n' => ap (ren_Tm shift) (Eq n')
|
|
| O => eq_refl
|
|
end).
|
|
Qed.
|
|
|
|
Fixpoint idSubst_Tm (sigma_Tm : nat -> Tm)
|
|
(Eq_Tm : forall x, sigma_Tm x = VarTm x) (s : Tm) {struct s} :
|
|
subst_Tm sigma_Tm s = s :=
|
|
match s with
|
|
| VarTm s0 => Eq_Tm s0
|
|
| Abs s0 =>
|
|
congr_Abs (idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s0)
|
|
| App s0 s1 =>
|
|
congr_App (idSubst_Tm sigma_Tm Eq_Tm s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
|
|
| Pair s0 s1 =>
|
|
congr_Pair (idSubst_Tm sigma_Tm Eq_Tm s0)
|
|
(idSubst_Tm sigma_Tm Eq_Tm s1)
|
|
| Proj s0 s1 => congr_Proj (eq_refl s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
|
|
| TBind s0 s1 s2 =>
|
|
congr_TBind (eq_refl s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
|
|
(idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s2)
|
|
| Univ s0 => congr_Univ (eq_refl s0)
|
|
| BVal s0 => congr_BVal (eq_refl s0)
|
|
| Bool => congr_Bool
|
|
| If s0 s1 s2 =>
|
|
congr_If (idSubst_Tm sigma_Tm Eq_Tm s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
|
|
(idSubst_Tm sigma_Tm Eq_Tm s2)
|
|
end.
|
|
|
|
Lemma upExtRen_Tm_Tm (xi : nat -> nat) (zeta : nat -> nat)
|
|
(Eq : forall x, xi x = zeta x) :
|
|
forall x, upRen_Tm_Tm xi x = upRen_Tm_Tm zeta x.
|
|
Proof.
|
|
exact (fun n => match n with
|
|
| S n' => ap shift (Eq n')
|
|
| O => eq_refl
|
|
end).
|
|
Qed.
|
|
|
|
Fixpoint extRen_Tm (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat)
|
|
(Eq_Tm : forall x, xi_Tm x = zeta_Tm x) (s : Tm) {struct s} :
|
|
ren_Tm xi_Tm s = ren_Tm zeta_Tm s :=
|
|
match s with
|
|
| VarTm s0 => ap (VarTm) (Eq_Tm s0)
|
|
| Abs s0 =>
|
|
congr_Abs
|
|
(extRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
|
|
(upExtRen_Tm_Tm _ _ Eq_Tm) s0)
|
|
| App s0 s1 =>
|
|
congr_App (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
|
|
(extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
|
|
| Pair s0 s1 =>
|
|
congr_Pair (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
|
|
(extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
|
|
| Proj s0 s1 => congr_Proj (eq_refl s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
|
|
| TBind s0 s1 s2 =>
|
|
congr_TBind (eq_refl s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
|
|
(extRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
|
|
(upExtRen_Tm_Tm _ _ Eq_Tm) s2)
|
|
| Univ s0 => congr_Univ (eq_refl s0)
|
|
| BVal s0 => congr_BVal (eq_refl s0)
|
|
| Bool => congr_Bool
|
|
| If s0 s1 s2 =>
|
|
congr_If (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
|
|
(extRen_Tm xi_Tm zeta_Tm Eq_Tm s1) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s2)
|
|
end.
|
|
|
|
Lemma upExt_Tm_Tm (sigma : nat -> Tm) (tau : nat -> Tm)
|
|
(Eq : forall x, sigma x = tau x) :
|
|
forall x, up_Tm_Tm sigma x = up_Tm_Tm tau x.
|
|
Proof.
|
|
exact (fun n =>
|
|
match n with
|
|
| S n' => ap (ren_Tm shift) (Eq n')
|
|
| O => eq_refl
|
|
end).
|
|
Qed.
|
|
|
|
Fixpoint ext_Tm (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm)
|
|
(Eq_Tm : forall x, sigma_Tm x = tau_Tm x) (s : Tm) {struct s} :
|
|
subst_Tm sigma_Tm s = subst_Tm tau_Tm s :=
|
|
match s with
|
|
| VarTm s0 => Eq_Tm s0
|
|
| Abs s0 =>
|
|
congr_Abs
|
|
(ext_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm) (upExt_Tm_Tm _ _ Eq_Tm)
|
|
s0)
|
|
| App s0 s1 =>
|
|
congr_App (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
|
|
(ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
|
|
| Pair s0 s1 =>
|
|
congr_Pair (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
|
|
(ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
|
|
| Proj s0 s1 => congr_Proj (eq_refl s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
|
|
| TBind s0 s1 s2 =>
|
|
congr_TBind (eq_refl s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
|
|
(ext_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm) (upExt_Tm_Tm _ _ Eq_Tm)
|
|
s2)
|
|
| Univ s0 => congr_Univ (eq_refl s0)
|
|
| BVal s0 => congr_BVal (eq_refl s0)
|
|
| Bool => congr_Bool
|
|
| If s0 s1 s2 =>
|
|
congr_If (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
|
|
(ext_Tm sigma_Tm tau_Tm Eq_Tm s1) (ext_Tm sigma_Tm tau_Tm Eq_Tm s2)
|
|
end.
|
|
|
|
Lemma up_ren_ren_Tm_Tm (xi : nat -> nat) (zeta : nat -> nat)
|
|
(rho : nat -> nat) (Eq : forall x, funcomp zeta xi x = rho x) :
|
|
forall x, funcomp (upRen_Tm_Tm zeta) (upRen_Tm_Tm xi) x = upRen_Tm_Tm rho x.
|
|
Proof.
|
|
exact (up_ren_ren xi zeta rho Eq).
|
|
Qed.
|
|
|
|
Fixpoint compRenRen_Tm (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat)
|
|
(rho_Tm : nat -> nat) (Eq_Tm : forall x, funcomp zeta_Tm xi_Tm x = rho_Tm x)
|
|
(s : Tm) {struct s} : ren_Tm zeta_Tm (ren_Tm xi_Tm s) = ren_Tm rho_Tm s :=
|
|
match s with
|
|
| VarTm s0 => ap (VarTm) (Eq_Tm s0)
|
|
| Abs s0 =>
|
|
congr_Abs
|
|
(compRenRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
|
|
(upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s0)
|
|
| App s0 s1 =>
|
|
congr_App (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
|
|
(compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
|
|
| Pair s0 s1 =>
|
|
congr_Pair (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
|
|
(compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
|
|
| Proj s0 s1 =>
|
|
congr_Proj (eq_refl s0) (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
|
|
| TBind s0 s1 s2 =>
|
|
congr_TBind (eq_refl s0) (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
|
|
(compRenRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
|
|
(upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s2)
|
|
| Univ s0 => congr_Univ (eq_refl s0)
|
|
| BVal s0 => congr_BVal (eq_refl s0)
|
|
| Bool => congr_Bool
|
|
| If s0 s1 s2 =>
|
|
congr_If (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
|
|
(compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
|
|
(compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s2)
|
|
end.
|
|
|
|
Lemma up_ren_subst_Tm_Tm (xi : nat -> nat) (tau : nat -> Tm)
|
|
(theta : nat -> Tm) (Eq : forall x, funcomp tau xi x = theta x) :
|
|
forall x, funcomp (up_Tm_Tm tau) (upRen_Tm_Tm xi) x = up_Tm_Tm theta x.
|
|
Proof.
|
|
exact (fun n =>
|
|
match n with
|
|
| S n' => ap (ren_Tm shift) (Eq n')
|
|
| O => eq_refl
|
|
end).
|
|
Qed.
|
|
|
|
Fixpoint compRenSubst_Tm (xi_Tm : nat -> nat) (tau_Tm : nat -> Tm)
|
|
(theta_Tm : nat -> Tm)
|
|
(Eq_Tm : forall x, funcomp tau_Tm xi_Tm x = theta_Tm x) (s : Tm) {struct s} :
|
|
subst_Tm tau_Tm (ren_Tm xi_Tm s) = subst_Tm theta_Tm s :=
|
|
match s with
|
|
| VarTm s0 => Eq_Tm s0
|
|
| Abs s0 =>
|
|
congr_Abs
|
|
(compRenSubst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm tau_Tm)
|
|
(up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s0)
|
|
| App s0 s1 =>
|
|
congr_App (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
|
|
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
|
|
| Pair s0 s1 =>
|
|
congr_Pair (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
|
|
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
|
|
| Proj s0 s1 =>
|
|
congr_Proj (eq_refl s0)
|
|
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
|
|
| TBind s0 s1 s2 =>
|
|
congr_TBind (eq_refl s0)
|
|
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
|
|
(compRenSubst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm tau_Tm)
|
|
(up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s2)
|
|
| Univ s0 => congr_Univ (eq_refl s0)
|
|
| BVal s0 => congr_BVal (eq_refl s0)
|
|
| Bool => congr_Bool
|
|
| If s0 s1 s2 =>
|
|
congr_If (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
|
|
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
|
|
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s2)
|
|
end.
|
|
|
|
Lemma up_subst_ren_Tm_Tm (sigma : nat -> Tm) (zeta_Tm : nat -> nat)
|
|
(theta : nat -> Tm)
|
|
(Eq : forall x, funcomp (ren_Tm zeta_Tm) sigma x = theta x) :
|
|
forall x,
|
|
funcomp (ren_Tm (upRen_Tm_Tm zeta_Tm)) (up_Tm_Tm sigma) x =
|
|
up_Tm_Tm theta x.
|
|
Proof.
|
|
exact (fun n =>
|
|
match n with
|
|
| S n' =>
|
|
eq_trans
|
|
(compRenRen_Tm shift (upRen_Tm_Tm zeta_Tm)
|
|
(funcomp shift zeta_Tm) (fun x => eq_refl) (sigma n'))
|
|
(eq_trans
|
|
(eq_sym
|
|
(compRenRen_Tm zeta_Tm shift (funcomp shift zeta_Tm)
|
|
(fun x => eq_refl) (sigma n')))
|
|
(ap (ren_Tm shift) (Eq n')))
|
|
| O => eq_refl
|
|
end).
|
|
Qed.
|
|
|
|
Fixpoint compSubstRen_Tm (sigma_Tm : nat -> Tm) (zeta_Tm : nat -> nat)
|
|
(theta_Tm : nat -> Tm)
|
|
(Eq_Tm : forall x, funcomp (ren_Tm zeta_Tm) sigma_Tm x = theta_Tm x)
|
|
(s : Tm) {struct s} :
|
|
ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
|
|
match s with
|
|
| VarTm s0 => Eq_Tm s0
|
|
| Abs s0 =>
|
|
congr_Abs
|
|
(compSubstRen_Tm (up_Tm_Tm sigma_Tm) (upRen_Tm_Tm zeta_Tm)
|
|
(up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s0)
|
|
| App s0 s1 =>
|
|
congr_App (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
|
|
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
|
|
| Pair s0 s1 =>
|
|
congr_Pair (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
|
|
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
|
|
| Proj s0 s1 =>
|
|
congr_Proj (eq_refl s0)
|
|
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
|
|
| TBind s0 s1 s2 =>
|
|
congr_TBind (eq_refl s0)
|
|
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
|
|
(compSubstRen_Tm (up_Tm_Tm sigma_Tm) (upRen_Tm_Tm zeta_Tm)
|
|
(up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s2)
|
|
| Univ s0 => congr_Univ (eq_refl s0)
|
|
| BVal s0 => congr_BVal (eq_refl s0)
|
|
| Bool => congr_Bool
|
|
| If s0 s1 s2 =>
|
|
congr_If (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
|
|
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
|
|
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s2)
|
|
end.
|
|
|
|
Lemma up_subst_subst_Tm_Tm (sigma : nat -> Tm) (tau_Tm : nat -> Tm)
|
|
(theta : nat -> Tm)
|
|
(Eq : forall x, funcomp (subst_Tm tau_Tm) sigma x = theta x) :
|
|
forall x,
|
|
funcomp (subst_Tm (up_Tm_Tm tau_Tm)) (up_Tm_Tm sigma) x = up_Tm_Tm theta x.
|
|
Proof.
|
|
exact (fun n =>
|
|
match n with
|
|
| S n' =>
|
|
eq_trans
|
|
(compRenSubst_Tm shift (up_Tm_Tm tau_Tm)
|
|
(funcomp (up_Tm_Tm tau_Tm) shift) (fun x => eq_refl)
|
|
(sigma n'))
|
|
(eq_trans
|
|
(eq_sym
|
|
(compSubstRen_Tm tau_Tm shift
|
|
(funcomp (ren_Tm shift) tau_Tm) (fun x => eq_refl)
|
|
(sigma n'))) (ap (ren_Tm shift) (Eq n')))
|
|
| O => eq_refl
|
|
end).
|
|
Qed.
|
|
|
|
Fixpoint compSubstSubst_Tm (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm)
|
|
(theta_Tm : nat -> Tm)
|
|
(Eq_Tm : forall x, funcomp (subst_Tm tau_Tm) sigma_Tm x = theta_Tm x)
|
|
(s : Tm) {struct s} :
|
|
subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
|
|
match s with
|
|
| VarTm s0 => Eq_Tm s0
|
|
| Abs s0 =>
|
|
congr_Abs
|
|
(compSubstSubst_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm)
|
|
(up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s0)
|
|
| App s0 s1 =>
|
|
congr_App (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
|
|
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
|
|
| Pair s0 s1 =>
|
|
congr_Pair (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
|
|
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
|
|
| Proj s0 s1 =>
|
|
congr_Proj (eq_refl s0)
|
|
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
|
|
| TBind s0 s1 s2 =>
|
|
congr_TBind (eq_refl s0)
|
|
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
|
|
(compSubstSubst_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm)
|
|
(up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s2)
|
|
| Univ s0 => congr_Univ (eq_refl s0)
|
|
| BVal s0 => congr_BVal (eq_refl s0)
|
|
| Bool => congr_Bool
|
|
| If s0 s1 s2 =>
|
|
congr_If (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
|
|
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
|
|
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s2)
|
|
end.
|
|
|
|
Lemma renRen_Tm (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat) (s : Tm) :
|
|
ren_Tm zeta_Tm (ren_Tm xi_Tm s) = ren_Tm (funcomp zeta_Tm xi_Tm) s.
|
|
Proof.
|
|
exact (compRenRen_Tm xi_Tm zeta_Tm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma renRen'_Tm_pointwise (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat) :
|
|
pointwise_relation _ eq (funcomp (ren_Tm zeta_Tm) (ren_Tm xi_Tm))
|
|
(ren_Tm (funcomp zeta_Tm xi_Tm)).
|
|
Proof.
|
|
exact (fun s => compRenRen_Tm xi_Tm zeta_Tm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma renSubst_Tm (xi_Tm : nat -> nat) (tau_Tm : nat -> Tm) (s : Tm) :
|
|
subst_Tm tau_Tm (ren_Tm xi_Tm s) = subst_Tm (funcomp tau_Tm xi_Tm) s.
|
|
Proof.
|
|
exact (compRenSubst_Tm xi_Tm tau_Tm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma renSubst_Tm_pointwise (xi_Tm : nat -> nat) (tau_Tm : nat -> Tm) :
|
|
pointwise_relation _ eq (funcomp (subst_Tm tau_Tm) (ren_Tm xi_Tm))
|
|
(subst_Tm (funcomp tau_Tm xi_Tm)).
|
|
Proof.
|
|
exact (fun s => compRenSubst_Tm xi_Tm tau_Tm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma substRen_Tm (sigma_Tm : nat -> Tm) (zeta_Tm : nat -> nat) (s : Tm) :
|
|
ren_Tm zeta_Tm (subst_Tm sigma_Tm s) =
|
|
subst_Tm (funcomp (ren_Tm zeta_Tm) sigma_Tm) s.
|
|
Proof.
|
|
exact (compSubstRen_Tm sigma_Tm zeta_Tm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma substRen_Tm_pointwise (sigma_Tm : nat -> Tm) (zeta_Tm : nat -> nat) :
|
|
pointwise_relation _ eq (funcomp (ren_Tm zeta_Tm) (subst_Tm sigma_Tm))
|
|
(subst_Tm (funcomp (ren_Tm zeta_Tm) sigma_Tm)).
|
|
Proof.
|
|
exact (fun s => compSubstRen_Tm sigma_Tm zeta_Tm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma substSubst_Tm (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm) (s : Tm) :
|
|
subst_Tm tau_Tm (subst_Tm sigma_Tm s) =
|
|
subst_Tm (funcomp (subst_Tm tau_Tm) sigma_Tm) s.
|
|
Proof.
|
|
exact (compSubstSubst_Tm sigma_Tm tau_Tm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma substSubst_Tm_pointwise (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm) :
|
|
pointwise_relation _ eq (funcomp (subst_Tm tau_Tm) (subst_Tm sigma_Tm))
|
|
(subst_Tm (funcomp (subst_Tm tau_Tm) sigma_Tm)).
|
|
Proof.
|
|
exact (fun s => compSubstSubst_Tm sigma_Tm tau_Tm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma rinstInst_up_Tm_Tm (xi : nat -> nat) (sigma : nat -> Tm)
|
|
(Eq : forall x, funcomp (VarTm) xi x = sigma x) :
|
|
forall x, funcomp (VarTm) (upRen_Tm_Tm xi) x = up_Tm_Tm sigma x.
|
|
Proof.
|
|
exact (fun n =>
|
|
match n with
|
|
| S n' => ap (ren_Tm shift) (Eq n')
|
|
| O => eq_refl
|
|
end).
|
|
Qed.
|
|
|
|
Fixpoint rinst_inst_Tm (xi_Tm : nat -> nat) (sigma_Tm : nat -> Tm)
|
|
(Eq_Tm : forall x, funcomp (VarTm) xi_Tm x = sigma_Tm x) (s : Tm) {struct s}
|
|
: ren_Tm xi_Tm s = subst_Tm sigma_Tm s :=
|
|
match s with
|
|
| VarTm s0 => Eq_Tm s0
|
|
| Abs s0 =>
|
|
congr_Abs
|
|
(rinst_inst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm sigma_Tm)
|
|
(rinstInst_up_Tm_Tm _ _ Eq_Tm) s0)
|
|
| App s0 s1 =>
|
|
congr_App (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
|
|
(rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
|
|
| Pair s0 s1 =>
|
|
congr_Pair (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
|
|
(rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
|
|
| Proj s0 s1 =>
|
|
congr_Proj (eq_refl s0) (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
|
|
| TBind s0 s1 s2 =>
|
|
congr_TBind (eq_refl s0) (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
|
|
(rinst_inst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm sigma_Tm)
|
|
(rinstInst_up_Tm_Tm _ _ Eq_Tm) s2)
|
|
| Univ s0 => congr_Univ (eq_refl s0)
|
|
| BVal s0 => congr_BVal (eq_refl s0)
|
|
| Bool => congr_Bool
|
|
| If s0 s1 s2 =>
|
|
congr_If (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
|
|
(rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
|
|
(rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s2)
|
|
end.
|
|
|
|
Lemma rinstInst'_Tm (xi_Tm : nat -> nat) (s : Tm) :
|
|
ren_Tm xi_Tm s = subst_Tm (funcomp (VarTm) xi_Tm) s.
|
|
Proof.
|
|
exact (rinst_inst_Tm xi_Tm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma rinstInst'_Tm_pointwise (xi_Tm : nat -> nat) :
|
|
pointwise_relation _ eq (ren_Tm xi_Tm) (subst_Tm (funcomp (VarTm) xi_Tm)).
|
|
Proof.
|
|
exact (fun s => rinst_inst_Tm xi_Tm _ (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma instId'_Tm (s : Tm) : subst_Tm (VarTm) s = s.
|
|
Proof.
|
|
exact (idSubst_Tm (VarTm) (fun n => eq_refl) s).
|
|
Qed.
|
|
|
|
Lemma instId'_Tm_pointwise : pointwise_relation _ eq (subst_Tm (VarTm)) id.
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|
Proof.
|
|
exact (fun s => idSubst_Tm (VarTm) (fun n => eq_refl) s).
|
|
Qed.
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|
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Lemma rinstId'_Tm (s : Tm) : ren_Tm id s = s.
|
|
Proof.
|
|
exact (eq_ind_r (fun t => t = s) (instId'_Tm s) (rinstInst'_Tm id s)).
|
|
Qed.
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|
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Lemma rinstId'_Tm_pointwise : pointwise_relation _ eq (@ren_Tm id) id.
|
|
Proof.
|
|
exact (fun s => eq_ind_r (fun t => t = s) (instId'_Tm s) (rinstInst'_Tm id s)).
|
|
Qed.
|
|
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Lemma varL'_Tm (sigma_Tm : nat -> Tm) (x : nat) :
|
|
subst_Tm sigma_Tm (VarTm x) = sigma_Tm x.
|
|
Proof.
|
|
exact (eq_refl).
|
|
Qed.
|
|
|
|
Lemma varL'_Tm_pointwise (sigma_Tm : nat -> Tm) :
|
|
pointwise_relation _ eq (funcomp (subst_Tm sigma_Tm) (VarTm)) sigma_Tm.
|
|
Proof.
|
|
exact (fun x => eq_refl).
|
|
Qed.
|
|
|
|
Lemma varLRen'_Tm (xi_Tm : nat -> nat) (x : nat) :
|
|
ren_Tm xi_Tm (VarTm x) = VarTm (xi_Tm x).
|
|
Proof.
|
|
exact (eq_refl).
|
|
Qed.
|
|
|
|
Lemma varLRen'_Tm_pointwise (xi_Tm : nat -> nat) :
|
|
pointwise_relation _ eq (funcomp (ren_Tm xi_Tm) (VarTm))
|
|
(funcomp (VarTm) xi_Tm).
|
|
Proof.
|
|
exact (fun x => eq_refl).
|
|
Qed.
|
|
|
|
Class Up_Tm X Y :=
|
|
up_Tm : X -> Y.
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|
|
|
Class Up_PTm X Y :=
|
|
up_PTm : X -> Y.
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|
|
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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.
|
|
|
|
#[global] Instance VarInstance_Tm : (Var _ _) := @VarTm.
|
|
|
|
#[global] Instance Subst_PTm : (Subst1 _ _ _) := @subst_PTm.
|
|
|
|
#[global] Instance Up_PTm_PTm : (Up_PTm _ _) := @up_PTm_PTm.
|
|
|
|
#[global] Instance Ren_PTm : (Ren1 _ _ _) := @ren_PTm.
|
|
|
|
#[global]
|
|
Instance VarInstance_PTm : (Var _ _) := @VarPTm.
|
|
|
|
Notation "[ sigma_Tm ]" := (subst_Tm sigma_Tm)
|
|
( at level 1, left associativity, only printing) : fscope.
|
|
|
|
Notation "s [ sigma_Tm ]" := (subst_Tm sigma_Tm s)
|
|
( at level 7, left associativity, only printing) : subst_scope.
|
|
|
|
Notation "↑__Tm" := up_Tm (only printing) : subst_scope.
|
|
|
|
Notation "↑__Tm" := up_Tm_Tm (only printing) : subst_scope.
|
|
|
|
Notation "⟨ xi_Tm ⟩" := (ren_Tm xi_Tm)
|
|
( at level 1, left associativity, only printing) : fscope.
|
|
|
|
Notation "s ⟨ xi_Tm ⟩" := (ren_Tm xi_Tm s)
|
|
( at level 7, left associativity, only printing) : subst_scope.
|
|
|
|
Notation "'var'" := VarTm ( at level 1, only printing) : subst_scope.
|
|
|
|
Notation "x '__Tm'" := (@ids _ _ VarInstance_Tm x)
|
|
( at level 5, format "x __Tm", only printing) : subst_scope.
|
|
|
|
Notation "x '__Tm'" := (VarTm x) ( at level 5, format "x __Tm") :
|
|
subst_scope.
|
|
|
|
Notation "[ sigma_PTm ]" := (subst_PTm sigma_PTm)
|
|
( at level 1, left associativity, only printing) : fscope.
|
|
|
|
Notation "s [ sigma_PTm ]" := (subst_PTm sigma_PTm s)
|
|
( at level 7, left associativity, only printing) : subst_scope.
|
|
|
|
Notation "↑__PTm" := up_PTm (only printing) : subst_scope.
|
|
|
|
Notation "↑__PTm" := up_PTm_PTm (only printing) : subst_scope.
|
|
|
|
Notation "⟨ xi_PTm ⟩" := (ren_PTm xi_PTm)
|
|
( at level 1, left associativity, only printing) : fscope.
|
|
|
|
Notation "s ⟨ xi_PTm ⟩" := (ren_PTm xi_PTm s)
|
|
( at level 7, left associativity, only printing) : subst_scope.
|
|
|
|
Notation "'var'" := VarPTm ( at level 1, only printing) : subst_scope.
|
|
|
|
Notation "x '__PTm'" := (@ids _ _ VarInstance_PTm x)
|
|
( at level 5, format "x __PTm", only printing) : subst_scope.
|
|
|
|
Notation "x '__PTm'" := (VarPTm x) ( at level 5, format "x __PTm") :
|
|
subst_scope.
|
|
|
|
#[global]
|
|
Instance subst_Tm_morphism :
|
|
(Proper (respectful (pointwise_relation _ eq) (respectful eq eq))
|
|
(@subst_Tm)).
|
|
Proof.
|
|
exact (fun f_Tm g_Tm Eq_Tm s t Eq_st =>
|
|
eq_ind s (fun t' => subst_Tm f_Tm s = subst_Tm g_Tm t')
|
|
(ext_Tm f_Tm g_Tm Eq_Tm s) t Eq_st).
|
|
Qed.
|
|
|
|
#[global]
|
|
Instance subst_Tm_morphism2 :
|
|
(Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
|
|
(@subst_Tm)).
|
|
Proof.
|
|
exact (fun f_Tm g_Tm Eq_Tm s => ext_Tm f_Tm g_Tm Eq_Tm s).
|
|
Qed.
|
|
|
|
#[global]
|
|
Instance ren_Tm_morphism :
|
|
(Proper (respectful (pointwise_relation _ eq) (respectful eq eq)) (@ren_Tm)).
|
|
Proof.
|
|
exact (fun f_Tm g_Tm Eq_Tm s t Eq_st =>
|
|
eq_ind s (fun t' => ren_Tm f_Tm s = ren_Tm g_Tm t')
|
|
(extRen_Tm f_Tm g_Tm Eq_Tm s) t Eq_st).
|
|
Qed.
|
|
|
|
#[global]
|
|
Instance ren_Tm_morphism2 :
|
|
(Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
|
|
(@ren_Tm)).
|
|
Proof.
|
|
exact (fun f_Tm g_Tm Eq_Tm s => extRen_Tm f_Tm g_Tm Eq_Tm s).
|
|
Qed.
|
|
|
|
#[global]
|
|
Instance subst_PTm_morphism :
|
|
(Proper (respectful (pointwise_relation _ eq) (respectful eq eq))
|
|
(@subst_PTm)).
|
|
Proof.
|
|
exact (fun f_PTm g_PTm Eq_PTm s t Eq_st =>
|
|
eq_ind s (fun t' => subst_PTm f_PTm s = subst_PTm g_PTm t')
|
|
(ext_PTm f_PTm g_PTm Eq_PTm s) t Eq_st).
|
|
Qed.
|
|
|
|
#[global]
|
|
Instance subst_PTm_morphism2 :
|
|
(Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
|
|
(@subst_PTm)).
|
|
Proof.
|
|
exact (fun f_PTm g_PTm Eq_PTm s => ext_PTm f_PTm g_PTm Eq_PTm s).
|
|
Qed.
|
|
|
|
#[global]
|
|
Instance ren_PTm_morphism :
|
|
(Proper (respectful (pointwise_relation _ eq) (respectful eq eq)) (@ren_PTm)).
|
|
Proof.
|
|
exact (fun f_PTm g_PTm Eq_PTm s t Eq_st =>
|
|
eq_ind s (fun t' => ren_PTm f_PTm s = ren_PTm g_PTm t')
|
|
(extRen_PTm f_PTm g_PTm Eq_PTm s) t Eq_st).
|
|
Qed.
|
|
|
|
#[global]
|
|
Instance ren_PTm_morphism2 :
|
|
(Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
|
|
(@ren_PTm)).
|
|
Proof.
|
|
exact (fun f_PTm g_PTm Eq_PTm s => extRen_PTm f_PTm g_PTm Eq_PTm s).
|
|
Qed.
|
|
|
|
Ltac auto_unfold := repeat
|
|
unfold VarInstance_PTm, Var, ids, Ren_PTm, Ren1, ren1,
|
|
Up_PTm_PTm, Up_PTm, up_PTm, Subst_PTm, Subst1, subst1,
|
|
VarInstance_Tm, Var, ids, Ren_Tm, Ren1, ren1, Up_Tm_Tm,
|
|
Up_Tm, up_Tm, Subst_Tm, Subst1, subst1.
|
|
|
|
Tactic Notation "auto_unfold" "in" "*" := repeat
|
|
unfold VarInstance_PTm, Var, ids,
|
|
Ren_PTm, Ren1, ren1, Up_PTm_PTm,
|
|
Up_PTm, up_PTm, Subst_PTm,
|
|
Subst1, subst1, VarInstance_Tm,
|
|
Var, ids, Ren_Tm, Ren1, ren1,
|
|
Up_Tm_Tm, Up_Tm, up_Tm, Subst_Tm,
|
|
Subst1, subst1 in *.
|
|
|
|
Ltac asimpl' := repeat (first
|
|
[ progress setoid_rewrite substSubst_Tm_pointwise
|
|
| progress setoid_rewrite substSubst_Tm
|
|
| progress setoid_rewrite substRen_Tm_pointwise
|
|
| progress setoid_rewrite substRen_Tm
|
|
| progress setoid_rewrite renSubst_Tm_pointwise
|
|
| progress setoid_rewrite renSubst_Tm
|
|
| progress setoid_rewrite renRen'_Tm_pointwise
|
|
| progress setoid_rewrite renRen_Tm
|
|
| progress setoid_rewrite substSubst_PTm_pointwise
|
|
| progress setoid_rewrite substSubst_PTm
|
|
| progress setoid_rewrite substRen_PTm_pointwise
|
|
| progress setoid_rewrite substRen_PTm
|
|
| progress setoid_rewrite renSubst_PTm_pointwise
|
|
| progress setoid_rewrite renSubst_PTm
|
|
| progress setoid_rewrite renRen'_PTm_pointwise
|
|
| progress setoid_rewrite renRen_PTm
|
|
| progress setoid_rewrite varLRen'_Tm_pointwise
|
|
| progress setoid_rewrite varLRen'_Tm
|
|
| progress setoid_rewrite varL'_Tm_pointwise
|
|
| progress setoid_rewrite varL'_Tm
|
|
| progress setoid_rewrite rinstId'_Tm_pointwise
|
|
| progress setoid_rewrite rinstId'_Tm
|
|
| progress setoid_rewrite instId'_Tm_pointwise
|
|
| progress setoid_rewrite instId'_Tm
|
|
| progress setoid_rewrite varLRen'_PTm_pointwise
|
|
| progress setoid_rewrite varLRen'_PTm
|
|
| progress setoid_rewrite varL'_PTm_pointwise
|
|
| progress setoid_rewrite varL'_PTm
|
|
| progress setoid_rewrite rinstId'_PTm_pointwise
|
|
| progress setoid_rewrite rinstId'_PTm
|
|
| progress setoid_rewrite instId'_PTm_pointwise
|
|
| progress setoid_rewrite instId'_PTm
|
|
| progress
|
|
unfold up_Tm_Tm, upRen_Tm_Tm, up_PTm_PTm, upRen_PTm_PTm,
|
|
up_ren
|
|
| progress cbn[subst_Tm ren_Tm subst_PTm ren_PTm]
|
|
| progress fsimpl ]).
|
|
|
|
Ltac asimpl := check_no_evars;
|
|
repeat
|
|
unfold VarInstance_PTm, Var, ids, Ren_PTm, Ren1, ren1,
|
|
Up_PTm_PTm, Up_PTm, up_PTm, Subst_PTm, Subst1, subst1,
|
|
VarInstance_Tm, Var, ids, Ren_Tm, Ren1, ren1, Up_Tm_Tm,
|
|
Up_Tm, up_Tm, Subst_Tm, Subst1, subst1 in *; asimpl';
|
|
minimize.
|
|
|
|
Tactic Notation "asimpl" "in" hyp(J) := revert J; asimpl; intros J.
|
|
|
|
Tactic Notation "auto_case" := auto_case ltac:(asimpl; cbn; eauto).
|
|
|
|
Ltac substify := auto_unfold; try setoid_rewrite rinstInst'_Tm_pointwise;
|
|
try setoid_rewrite rinstInst'_Tm;
|
|
try setoid_rewrite rinstInst'_PTm_pointwise;
|
|
try setoid_rewrite rinstInst'_PTm.
|
|
|
|
Ltac renamify := auto_unfold; try setoid_rewrite_left rinstInst'_Tm_pointwise;
|
|
try setoid_rewrite_left rinstInst'_Tm;
|
|
try setoid_rewrite_left rinstInst'_PTm_pointwise;
|
|
try setoid_rewrite_left rinstInst'_PTm.
|
|
|
|
End Core.
|
|
|
|
Module Extra.
|
|
|
|
Import Core.
|
|
|
|
#[global] Hint Opaque subst_Tm: rewrite.
|
|
|
|
#[global] Hint Opaque ren_Tm: rewrite.
|
|
|
|
#[global] Hint Opaque subst_PTm: rewrite.
|
|
|
|
#[global] Hint Opaque ren_PTm: rewrite.
|
|
|
|
End Extra.
|
|
|
|
Module interface.
|
|
|
|
Export Core.
|
|
|
|
Export Extra.
|
|
|
|
End interface.
|
|
|
|
Export interface.
|
|
|