825 lines
29 KiB
Coq
825 lines
29 KiB
Coq
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Require Import core fintype.
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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 Ty : Type :=
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| Fun : Ty -> Ty -> Ty
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| Prod : Ty -> Ty -> Ty
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| Void : Ty.
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Lemma congr_Fun {s0 : Ty} {s1 : Ty} {t0 : Ty} {t1 : Ty} (H0 : s0 = t0)
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(H1 : s1 = t1) : Fun s0 s1 = Fun t0 t1.
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Proof.
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exact (eq_trans (eq_trans eq_refl (ap (fun x => Fun x s1) H0))
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(ap (fun x => Fun t0 x) H1)).
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Qed.
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Lemma congr_Prod {s0 : Ty} {s1 : Ty} {t0 : Ty} {t1 : Ty} (H0 : s0 = t0)
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(H1 : s1 = t1) : Prod s0 s1 = Prod t0 t1.
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Proof.
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exact (eq_trans (eq_trans eq_refl (ap (fun x => Prod x s1) H0))
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(ap (fun x => Prod t0 x) H1)).
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Qed.
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Lemma congr_Void : Void = Void.
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Proof.
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exact (eq_refl).
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Qed.
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Inductive PTm (n_PTm : nat) : Type :=
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| VarPTm : fin n_PTm -> PTm n_PTm
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| PAbs : Ty -> PTm (S n_PTm) -> PTm n_PTm
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| PApp : PTm n_PTm -> PTm n_PTm -> PTm n_PTm
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| PPair : PTm n_PTm -> PTm n_PTm -> PTm n_PTm
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| PProj : PTag -> PTm n_PTm -> PTm n_PTm.
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Lemma congr_PAbs {m_PTm : nat} {s0 : Ty} {s1 : PTm (S m_PTm)} {t0 : Ty}
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{t1 : PTm (S m_PTm)} (H0 : s0 = t0) (H1 : s1 = t1) :
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PAbs m_PTm s0 s1 = PAbs m_PTm t0 t1.
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Proof.
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exact (eq_trans (eq_trans eq_refl (ap (fun x => PAbs m_PTm x s1) H0))
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(ap (fun x => PAbs m_PTm t0 x) H1)).
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Qed.
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Lemma congr_PApp {m_PTm : nat} {s0 : PTm m_PTm} {s1 : PTm m_PTm}
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{t0 : PTm m_PTm} {t1 : PTm m_PTm} (H0 : s0 = t0) (H1 : s1 = t1) :
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PApp m_PTm s0 s1 = PApp m_PTm t0 t1.
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Proof.
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exact (eq_trans (eq_trans eq_refl (ap (fun x => PApp m_PTm x s1) H0))
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(ap (fun x => PApp m_PTm t0 x) H1)).
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Qed.
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Lemma congr_PPair {m_PTm : nat} {s0 : PTm m_PTm} {s1 : PTm m_PTm}
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{t0 : PTm m_PTm} {t1 : PTm m_PTm} (H0 : s0 = t0) (H1 : s1 = t1) :
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PPair m_PTm s0 s1 = PPair m_PTm t0 t1.
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Proof.
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exact (eq_trans (eq_trans eq_refl (ap (fun x => PPair m_PTm x s1) H0))
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(ap (fun x => PPair m_PTm t0 x) H1)).
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Qed.
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Lemma congr_PProj {m_PTm : nat} {s0 : PTag} {s1 : PTm m_PTm} {t0 : PTag}
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{t1 : PTm m_PTm} (H0 : s0 = t0) (H1 : s1 = t1) :
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PProj m_PTm s0 s1 = PProj m_PTm t0 t1.
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Proof.
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exact (eq_trans (eq_trans eq_refl (ap (fun x => PProj m_PTm x s1) H0))
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(ap (fun x => PProj m_PTm t0 x) H1)).
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Qed.
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Lemma upRen_PTm_PTm {m : nat} {n : nat} (xi : fin m -> fin n) :
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fin (S m) -> fin (S n).
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Proof.
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exact (up_ren xi).
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Defined.
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Lemma upRen_list_PTm_PTm (p : nat) {m : nat} {n : nat} (xi : fin m -> fin n)
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: fin (plus p m) -> fin (plus p n).
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Proof.
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exact (upRen_p p xi).
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Defined.
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Fixpoint ren_PTm {m_PTm : nat} {n_PTm : nat}
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(xi_PTm : fin m_PTm -> fin n_PTm) (s : PTm m_PTm) {struct s} : PTm n_PTm :=
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match s with
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| VarPTm _ s0 => VarPTm n_PTm (xi_PTm s0)
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| PAbs _ s0 s1 => PAbs n_PTm s0 (ren_PTm (upRen_PTm_PTm xi_PTm) s1)
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| PApp _ s0 s1 => PApp n_PTm (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
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| PPair _ s0 s1 => PPair n_PTm (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
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| PProj _ s0 s1 => PProj n_PTm s0 (ren_PTm xi_PTm s1)
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end.
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Lemma up_PTm_PTm {m : nat} {n_PTm : nat} (sigma : fin m -> PTm n_PTm) :
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fin (S m) -> PTm (S n_PTm).
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Proof.
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exact (scons (VarPTm (S n_PTm) var_zero) (funcomp (ren_PTm shift) sigma)).
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Defined.
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Lemma up_list_PTm_PTm (p : nat) {m : nat} {n_PTm : nat}
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(sigma : fin m -> PTm n_PTm) : fin (plus p m) -> PTm (plus p n_PTm).
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Proof.
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exact (scons_p p (funcomp (VarPTm (plus p n_PTm)) (zero_p p))
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(funcomp (ren_PTm (shift_p p)) sigma)).
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Defined.
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Fixpoint subst_PTm {m_PTm : nat} {n_PTm : nat}
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(sigma_PTm : fin m_PTm -> PTm n_PTm) (s : PTm m_PTm) {struct s} : PTm n_PTm
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:=
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match s with
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| VarPTm _ s0 => sigma_PTm s0
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| PAbs _ s0 s1 => PAbs n_PTm s0 (subst_PTm (up_PTm_PTm sigma_PTm) s1)
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| PApp _ s0 s1 =>
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PApp n_PTm (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1)
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| PPair _ s0 s1 =>
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PPair n_PTm (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1)
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| PProj _ s0 s1 => PProj n_PTm s0 (subst_PTm sigma_PTm s1)
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end.
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Lemma upId_PTm_PTm {m_PTm : nat} (sigma : fin m_PTm -> PTm m_PTm)
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(Eq : forall x, sigma x = VarPTm m_PTm x) :
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forall x, up_PTm_PTm sigma x = VarPTm (S m_PTm) x.
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Proof.
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exact (fun n =>
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match n with
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| Some fin_n => ap (ren_PTm shift) (Eq fin_n)
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| None => eq_refl
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end).
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Qed.
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Lemma upId_list_PTm_PTm {p : nat} {m_PTm : nat}
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(sigma : fin m_PTm -> PTm m_PTm) (Eq : forall x, sigma x = VarPTm m_PTm x)
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: forall x, up_list_PTm_PTm p sigma x = VarPTm (plus p m_PTm) x.
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Proof.
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exact (fun n =>
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scons_p_eta (VarPTm (plus p m_PTm))
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(fun n => ap (ren_PTm (shift_p p)) (Eq n)) (fun n => eq_refl)).
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Qed.
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Fixpoint idSubst_PTm {m_PTm : nat} (sigma_PTm : fin m_PTm -> PTm m_PTm)
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(Eq_PTm : forall x, sigma_PTm x = VarPTm m_PTm x) (s : PTm m_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 s1 =>
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congr_PAbs (eq_refl s0)
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(idSubst_PTm (up_PTm_PTm sigma_PTm) (upId_PTm_PTm _ Eq_PTm) s1)
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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 =>
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congr_PProj (eq_refl s0) (idSubst_PTm sigma_PTm Eq_PTm s1)
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end.
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Lemma upExtRen_PTm_PTm {m : nat} {n : nat} (xi : fin m -> fin n)
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(zeta : fin m -> fin n) (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 =>
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match n with
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| Some fin_n => ap shift (Eq fin_n)
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| None => eq_refl
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end).
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Qed.
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Lemma upExtRen_list_PTm_PTm {p : nat} {m : nat} {n : nat}
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(xi : fin m -> fin n) (zeta : fin m -> fin n)
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(Eq : forall x, xi x = zeta x) :
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forall x, upRen_list_PTm_PTm p xi x = upRen_list_PTm_PTm p zeta x.
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Proof.
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exact (fun n =>
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scons_p_congr (fun n => eq_refl) (fun n => ap (shift_p p) (Eq n))).
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Qed.
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Fixpoint extRen_PTm {m_PTm : nat} {n_PTm : nat}
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(xi_PTm : fin m_PTm -> fin n_PTm) (zeta_PTm : fin m_PTm -> fin n_PTm)
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(Eq_PTm : forall x, xi_PTm x = zeta_PTm x) (s : PTm m_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 n_PTm) (Eq_PTm s0)
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| PAbs _ s0 s1 =>
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congr_PAbs (eq_refl s0)
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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) s1)
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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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end.
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Lemma upExt_PTm_PTm {m : nat} {n_PTm : nat} (sigma : fin m -> PTm n_PTm)
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(tau : fin m -> PTm n_PTm) (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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| Some fin_n => ap (ren_PTm shift) (Eq fin_n)
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| None => eq_refl
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end).
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Qed.
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Lemma upExt_list_PTm_PTm {p : nat} {m : nat} {n_PTm : nat}
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(sigma : fin m -> PTm n_PTm) (tau : fin m -> PTm n_PTm)
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(Eq : forall x, sigma x = tau x) :
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forall x, up_list_PTm_PTm p sigma x = up_list_PTm_PTm p tau x.
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Proof.
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exact (fun n =>
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scons_p_congr (fun n => eq_refl)
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(fun n => ap (ren_PTm (shift_p p)) (Eq n))).
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Qed.
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Fixpoint ext_PTm {m_PTm : nat} {n_PTm : nat}
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(sigma_PTm : fin m_PTm -> PTm n_PTm) (tau_PTm : fin m_PTm -> PTm n_PTm)
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(Eq_PTm : forall x, sigma_PTm x = tau_PTm x) (s : PTm m_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 s1 =>
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congr_PAbs (eq_refl s0)
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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) s1)
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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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end.
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Lemma up_ren_ren_PTm_PTm {k : nat} {l : nat} {m : nat} (xi : fin k -> fin l)
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(zeta : fin l -> fin m) (rho : fin k -> fin m)
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(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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Lemma up_ren_ren_list_PTm_PTm {p : nat} {k : nat} {l : nat} {m : nat}
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(xi : fin k -> fin l) (zeta : fin l -> fin m) (rho : fin k -> fin m)
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(Eq : forall x, funcomp zeta xi x = rho x) :
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forall x,
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funcomp (upRen_list_PTm_PTm p zeta) (upRen_list_PTm_PTm p xi) x =
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upRen_list_PTm_PTm p rho x.
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Proof.
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exact (up_ren_ren_p Eq).
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Qed.
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Fixpoint compRenRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
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(xi_PTm : fin m_PTm -> fin k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm)
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(rho_PTm : fin m_PTm -> fin l_PTm)
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(Eq_PTm : forall x, funcomp zeta_PTm xi_PTm x = rho_PTm x) (s : PTm m_PTm)
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{struct 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 l_PTm) (Eq_PTm s0)
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| PAbs _ s0 s1 =>
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congr_PAbs (eq_refl s0)
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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) s1)
|
||
|
|
| PApp _ s0 s1 =>
|
||
|
|
congr_PApp (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s0)
|
||
|
|
(compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
|
||
|
|
| PPair _ s0 s1 =>
|
||
|
|
congr_PPair (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s0)
|
||
|
|
(compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
|
||
|
|
| PProj _ s0 s1 =>
|
||
|
|
congr_PProj (eq_refl s0)
|
||
|
|
(compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
|
||
|
|
end.
|
||
|
|
|
||
|
|
Lemma up_ren_subst_PTm_PTm {k : nat} {l : nat} {m_PTm : nat}
|
||
|
|
(xi : fin k -> fin l) (tau : fin l -> PTm m_PTm)
|
||
|
|
(theta : fin k -> PTm m_PTm) (Eq : forall x, funcomp tau xi x = theta x) :
|
||
|
|
forall x,
|
||
|
|
funcomp (up_PTm_PTm tau) (upRen_PTm_PTm xi) x = up_PTm_PTm theta x.
|
||
|
|
Proof.
|
||
|
|
exact (fun n =>
|
||
|
|
match n with
|
||
|
|
| Some fin_n => ap (ren_PTm shift) (Eq fin_n)
|
||
|
|
| None => eq_refl
|
||
|
|
end).
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Lemma up_ren_subst_list_PTm_PTm {p : nat} {k : nat} {l : nat} {m_PTm : nat}
|
||
|
|
(xi : fin k -> fin l) (tau : fin l -> PTm m_PTm)
|
||
|
|
(theta : fin k -> PTm m_PTm) (Eq : forall x, funcomp tau xi x = theta x) :
|
||
|
|
forall x,
|
||
|
|
funcomp (up_list_PTm_PTm p tau) (upRen_list_PTm_PTm p xi) x =
|
||
|
|
up_list_PTm_PTm p theta x.
|
||
|
|
Proof.
|
||
|
|
exact (fun n =>
|
||
|
|
eq_trans (scons_p_comp' _ _ _ n)
|
||
|
|
(scons_p_congr (fun z => scons_p_head' _ _ z)
|
||
|
|
(fun z =>
|
||
|
|
eq_trans (scons_p_tail' _ _ (xi z))
|
||
|
|
(ap (ren_PTm (shift_p p)) (Eq z))))).
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Fixpoint compRenSubst_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
|
||
|
|
(xi_PTm : fin m_PTm -> fin k_PTm) (tau_PTm : fin k_PTm -> PTm l_PTm)
|
||
|
|
(theta_PTm : fin m_PTm -> PTm l_PTm)
|
||
|
|
(Eq_PTm : forall x, funcomp tau_PTm xi_PTm x = theta_PTm x) (s : PTm m_PTm)
|
||
|
|
{struct s} : subst_PTm tau_PTm (ren_PTm xi_PTm s) = subst_PTm theta_PTm s :=
|
||
|
|
match s with
|
||
|
|
| VarPTm _ s0 => Eq_PTm s0
|
||
|
|
| PAbs _ s0 s1 =>
|
||
|
|
congr_PAbs (eq_refl s0)
|
||
|
|
(compRenSubst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm tau_PTm)
|
||
|
|
(up_PTm_PTm theta_PTm) (up_ren_subst_PTm_PTm _ _ _ Eq_PTm) s1)
|
||
|
|
| PApp _ s0 s1 =>
|
||
|
|
congr_PApp (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s0)
|
||
|
|
(compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
|
||
|
|
| PPair _ s0 s1 =>
|
||
|
|
congr_PPair (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s0)
|
||
|
|
(compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
|
||
|
|
| PProj _ s0 s1 =>
|
||
|
|
congr_PProj (eq_refl s0)
|
||
|
|
(compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
|
||
|
|
end.
|
||
|
|
|
||
|
|
Lemma up_subst_ren_PTm_PTm {k : nat} {l_PTm : nat} {m_PTm : nat}
|
||
|
|
(sigma : fin k -> PTm l_PTm) (zeta_PTm : fin l_PTm -> fin m_PTm)
|
||
|
|
(theta : fin k -> PTm m_PTm)
|
||
|
|
(Eq : forall x, funcomp (ren_PTm zeta_PTm) sigma x = theta x) :
|
||
|
|
forall x,
|
||
|
|
funcomp (ren_PTm (upRen_PTm_PTm zeta_PTm)) (up_PTm_PTm sigma) x =
|
||
|
|
up_PTm_PTm theta x.
|
||
|
|
Proof.
|
||
|
|
exact (fun n =>
|
||
|
|
match n with
|
||
|
|
| Some fin_n =>
|
||
|
|
eq_trans
|
||
|
|
(compRenRen_PTm shift (upRen_PTm_PTm zeta_PTm)
|
||
|
|
(funcomp shift zeta_PTm) (fun x => eq_refl) (sigma fin_n))
|
||
|
|
(eq_trans
|
||
|
|
(eq_sym
|
||
|
|
(compRenRen_PTm zeta_PTm shift (funcomp shift zeta_PTm)
|
||
|
|
(fun x => eq_refl) (sigma fin_n)))
|
||
|
|
(ap (ren_PTm shift) (Eq fin_n)))
|
||
|
|
| None => eq_refl
|
||
|
|
end).
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Lemma up_subst_ren_list_PTm_PTm {p : nat} {k : nat} {l_PTm : nat}
|
||
|
|
{m_PTm : nat} (sigma : fin k -> PTm l_PTm)
|
||
|
|
(zeta_PTm : fin l_PTm -> fin m_PTm) (theta : fin k -> PTm m_PTm)
|
||
|
|
(Eq : forall x, funcomp (ren_PTm zeta_PTm) sigma x = theta x) :
|
||
|
|
forall x,
|
||
|
|
funcomp (ren_PTm (upRen_list_PTm_PTm p zeta_PTm)) (up_list_PTm_PTm p sigma)
|
||
|
|
x = up_list_PTm_PTm p theta x.
|
||
|
|
Proof.
|
||
|
|
exact (fun n =>
|
||
|
|
eq_trans (scons_p_comp' _ _ _ n)
|
||
|
|
(scons_p_congr
|
||
|
|
(fun x => ap (VarPTm (plus p m_PTm)) (scons_p_head' _ _ x))
|
||
|
|
(fun n =>
|
||
|
|
eq_trans
|
||
|
|
(compRenRen_PTm (shift_p p) (upRen_list_PTm_PTm p zeta_PTm)
|
||
|
|
(funcomp (shift_p p) zeta_PTm)
|
||
|
|
(fun x => scons_p_tail' _ _ x) (sigma n))
|
||
|
|
(eq_trans
|
||
|
|
(eq_sym
|
||
|
|
(compRenRen_PTm zeta_PTm (shift_p p)
|
||
|
|
(funcomp (shift_p p) zeta_PTm) (fun x => eq_refl)
|
||
|
|
(sigma n))) (ap (ren_PTm (shift_p p)) (Eq n)))))).
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Fixpoint compSubstRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
|
||
|
|
(sigma_PTm : fin m_PTm -> PTm k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm)
|
||
|
|
(theta_PTm : fin m_PTm -> PTm l_PTm)
|
||
|
|
(Eq_PTm : forall x, funcomp (ren_PTm zeta_PTm) sigma_PTm x = theta_PTm x)
|
||
|
|
(s : PTm m_PTm) {struct s} :
|
||
|
|
ren_PTm zeta_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
|
||
|
|
match s with
|
||
|
|
| VarPTm _ s0 => Eq_PTm s0
|
||
|
|
| PAbs _ s0 s1 =>
|
||
|
|
congr_PAbs (eq_refl s0)
|
||
|
|
(compSubstRen_PTm (up_PTm_PTm sigma_PTm) (upRen_PTm_PTm zeta_PTm)
|
||
|
|
(up_PTm_PTm theta_PTm) (up_subst_ren_PTm_PTm _ _ _ Eq_PTm) s1)
|
||
|
|
| PApp _ s0 s1 =>
|
||
|
|
congr_PApp (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s0)
|
||
|
|
(compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
|
||
|
|
| PPair _ s0 s1 =>
|
||
|
|
congr_PPair (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s0)
|
||
|
|
(compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
|
||
|
|
| PProj _ s0 s1 =>
|
||
|
|
congr_PProj (eq_refl s0)
|
||
|
|
(compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
|
||
|
|
end.
|
||
|
|
|
||
|
|
Lemma up_subst_subst_PTm_PTm {k : nat} {l_PTm : nat} {m_PTm : nat}
|
||
|
|
(sigma : fin k -> PTm l_PTm) (tau_PTm : fin l_PTm -> PTm m_PTm)
|
||
|
|
(theta : fin k -> PTm m_PTm)
|
||
|
|
(Eq : forall x, funcomp (subst_PTm tau_PTm) sigma x = theta x) :
|
||
|
|
forall x,
|
||
|
|
funcomp (subst_PTm (up_PTm_PTm tau_PTm)) (up_PTm_PTm sigma) x =
|
||
|
|
up_PTm_PTm theta x.
|
||
|
|
Proof.
|
||
|
|
exact (fun n =>
|
||
|
|
match n with
|
||
|
|
| Some fin_n =>
|
||
|
|
eq_trans
|
||
|
|
(compRenSubst_PTm shift (up_PTm_PTm tau_PTm)
|
||
|
|
(funcomp (up_PTm_PTm tau_PTm) shift) (fun x => eq_refl)
|
||
|
|
(sigma fin_n))
|
||
|
|
(eq_trans
|
||
|
|
(eq_sym
|
||
|
|
(compSubstRen_PTm tau_PTm shift
|
||
|
|
(funcomp (ren_PTm shift) tau_PTm) (fun x => eq_refl)
|
||
|
|
(sigma fin_n))) (ap (ren_PTm shift) (Eq fin_n)))
|
||
|
|
| None => eq_refl
|
||
|
|
end).
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Lemma up_subst_subst_list_PTm_PTm {p : nat} {k : nat} {l_PTm : nat}
|
||
|
|
{m_PTm : nat} (sigma : fin k -> PTm l_PTm)
|
||
|
|
(tau_PTm : fin l_PTm -> PTm m_PTm) (theta : fin k -> PTm m_PTm)
|
||
|
|
(Eq : forall x, funcomp (subst_PTm tau_PTm) sigma x = theta x) :
|
||
|
|
forall x,
|
||
|
|
funcomp (subst_PTm (up_list_PTm_PTm p tau_PTm)) (up_list_PTm_PTm p sigma) x =
|
||
|
|
up_list_PTm_PTm p theta x.
|
||
|
|
Proof.
|
||
|
|
exact (fun n =>
|
||
|
|
eq_trans
|
||
|
|
(scons_p_comp' (funcomp (VarPTm (plus p l_PTm)) (zero_p p)) _ _ n)
|
||
|
|
(scons_p_congr
|
||
|
|
(fun x => scons_p_head' _ (fun z => ren_PTm (shift_p p) _) x)
|
||
|
|
(fun n =>
|
||
|
|
eq_trans
|
||
|
|
(compRenSubst_PTm (shift_p p) (up_list_PTm_PTm p tau_PTm)
|
||
|
|
(funcomp (up_list_PTm_PTm p tau_PTm) (shift_p p))
|
||
|
|
(fun x => eq_refl) (sigma n))
|
||
|
|
(eq_trans
|
||
|
|
(eq_sym
|
||
|
|
(compSubstRen_PTm tau_PTm (shift_p p) _
|
||
|
|
(fun x => eq_sym (scons_p_tail' _ _ x)) (sigma n)))
|
||
|
|
(ap (ren_PTm (shift_p p)) (Eq n)))))).
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Fixpoint compSubstSubst_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
|
||
|
|
(sigma_PTm : fin m_PTm -> PTm k_PTm) (tau_PTm : fin k_PTm -> PTm l_PTm)
|
||
|
|
(theta_PTm : fin m_PTm -> PTm l_PTm)
|
||
|
|
(Eq_PTm : forall x, funcomp (subst_PTm tau_PTm) sigma_PTm x = theta_PTm x)
|
||
|
|
(s : PTm m_PTm) {struct s} :
|
||
|
|
subst_PTm tau_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
|
||
|
|
match s with
|
||
|
|
| VarPTm _ s0 => Eq_PTm s0
|
||
|
|
| PAbs _ s0 s1 =>
|
||
|
|
congr_PAbs (eq_refl s0)
|
||
|
|
(compSubstSubst_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
|
||
|
|
(up_PTm_PTm theta_PTm) (up_subst_subst_PTm_PTm _ _ _ Eq_PTm) s1)
|
||
|
|
| PApp _ s0 s1 =>
|
||
|
|
congr_PApp (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s0)
|
||
|
|
(compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
|
||
|
|
| PPair _ s0 s1 =>
|
||
|
|
congr_PPair (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s0)
|
||
|
|
(compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
|
||
|
|
| PProj _ s0 s1 =>
|
||
|
|
congr_PProj (eq_refl s0)
|
||
|
|
(compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
|
||
|
|
end.
|
||
|
|
|
||
|
|
Lemma renRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
|
||
|
|
(xi_PTm : fin m_PTm -> fin k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm)
|
||
|
|
(s : PTm m_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 {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
|
||
|
|
(xi_PTm : fin m_PTm -> fin k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm) :
|
||
|
|
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 {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
|
||
|
|
(xi_PTm : fin m_PTm -> fin k_PTm) (tau_PTm : fin k_PTm -> PTm l_PTm)
|
||
|
|
(s : PTm m_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 {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
|
||
|
|
(xi_PTm : fin m_PTm -> fin k_PTm) (tau_PTm : fin k_PTm -> PTm l_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 {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
|
||
|
|
(sigma_PTm : fin m_PTm -> PTm k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm)
|
||
|
|
(s : PTm m_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 {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
|
||
|
|
(sigma_PTm : fin m_PTm -> PTm k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm) :
|
||
|
|
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 {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
|
||
|
|
(sigma_PTm : fin m_PTm -> PTm k_PTm) (tau_PTm : fin k_PTm -> PTm l_PTm)
|
||
|
|
(s : PTm m_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 {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
|
||
|
|
(sigma_PTm : fin m_PTm -> PTm k_PTm) (tau_PTm : fin k_PTm -> PTm l_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 {m : nat} {n_PTm : nat} (xi : fin m -> fin n_PTm)
|
||
|
|
(sigma : fin m -> PTm n_PTm)
|
||
|
|
(Eq : forall x, funcomp (VarPTm n_PTm) xi x = sigma x) :
|
||
|
|
forall x,
|
||
|
|
funcomp (VarPTm (S n_PTm)) (upRen_PTm_PTm xi) x = up_PTm_PTm sigma x.
|
||
|
|
Proof.
|
||
|
|
exact (fun n =>
|
||
|
|
match n with
|
||
|
|
| Some fin_n => ap (ren_PTm shift) (Eq fin_n)
|
||
|
|
| None => eq_refl
|
||
|
|
end).
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Lemma rinstInst_up_list_PTm_PTm {p : nat} {m : nat} {n_PTm : nat}
|
||
|
|
(xi : fin m -> fin n_PTm) (sigma : fin m -> PTm n_PTm)
|
||
|
|
(Eq : forall x, funcomp (VarPTm n_PTm) xi x = sigma x) :
|
||
|
|
forall x,
|
||
|
|
funcomp (VarPTm (plus p n_PTm)) (upRen_list_PTm_PTm p xi) x =
|
||
|
|
up_list_PTm_PTm p sigma x.
|
||
|
|
Proof.
|
||
|
|
exact (fun n =>
|
||
|
|
eq_trans (scons_p_comp' _ _ (VarPTm (plus p n_PTm)) n)
|
||
|
|
(scons_p_congr (fun z => eq_refl)
|
||
|
|
(fun n => ap (ren_PTm (shift_p p)) (Eq n)))).
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Fixpoint rinst_inst_PTm {m_PTm : nat} {n_PTm : nat}
|
||
|
|
(xi_PTm : fin m_PTm -> fin n_PTm) (sigma_PTm : fin m_PTm -> PTm n_PTm)
|
||
|
|
(Eq_PTm : forall x, funcomp (VarPTm n_PTm) xi_PTm x = sigma_PTm x)
|
||
|
|
(s : PTm m_PTm) {struct s} : ren_PTm xi_PTm s = subst_PTm sigma_PTm s :=
|
||
|
|
match s with
|
||
|
|
| VarPTm _ s0 => Eq_PTm s0
|
||
|
|
| PAbs _ s0 s1 =>
|
||
|
|
congr_PAbs (eq_refl s0)
|
||
|
|
(rinst_inst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm sigma_PTm)
|
||
|
|
(rinstInst_up_PTm_PTm _ _ Eq_PTm) s1)
|
||
|
|
| 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)
|
||
|
|
end.
|
||
|
|
|
||
|
|
Lemma rinstInst'_PTm {m_PTm : nat} {n_PTm : nat}
|
||
|
|
(xi_PTm : fin m_PTm -> fin n_PTm) (s : PTm m_PTm) :
|
||
|
|
ren_PTm xi_PTm s = subst_PTm (funcomp (VarPTm n_PTm) xi_PTm) s.
|
||
|
|
Proof.
|
||
|
|
exact (rinst_inst_PTm xi_PTm _ (fun n => eq_refl) s).
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Lemma rinstInst'_PTm_pointwise {m_PTm : nat} {n_PTm : nat}
|
||
|
|
(xi_PTm : fin m_PTm -> fin n_PTm) :
|
||
|
|
pointwise_relation _ eq (ren_PTm xi_PTm)
|
||
|
|
(subst_PTm (funcomp (VarPTm n_PTm) xi_PTm)).
|
||
|
|
Proof.
|
||
|
|
exact (fun s => rinst_inst_PTm xi_PTm _ (fun n => eq_refl) s).
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Lemma instId'_PTm {m_PTm : nat} (s : PTm m_PTm) :
|
||
|
|
subst_PTm (VarPTm m_PTm) s = s.
|
||
|
|
Proof.
|
||
|
|
exact (idSubst_PTm (VarPTm m_PTm) (fun n => eq_refl) s).
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Lemma instId'_PTm_pointwise {m_PTm : nat} :
|
||
|
|
pointwise_relation _ eq (subst_PTm (VarPTm m_PTm)) id.
|
||
|
|
Proof.
|
||
|
|
exact (fun s => idSubst_PTm (VarPTm m_PTm) (fun n => eq_refl) s).
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Lemma rinstId'_PTm {m_PTm : nat} (s : PTm m_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 {m_PTm : nat} :
|
||
|
|
pointwise_relation _ eq (@ren_PTm m_PTm m_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 {m_PTm : nat} {n_PTm : nat}
|
||
|
|
(sigma_PTm : fin m_PTm -> PTm n_PTm) (x : fin m_PTm) :
|
||
|
|
subst_PTm sigma_PTm (VarPTm m_PTm x) = sigma_PTm x.
|
||
|
|
Proof.
|
||
|
|
exact (eq_refl).
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Lemma varL'_PTm_pointwise {m_PTm : nat} {n_PTm : nat}
|
||
|
|
(sigma_PTm : fin m_PTm -> PTm n_PTm) :
|
||
|
|
pointwise_relation _ eq (funcomp (subst_PTm sigma_PTm) (VarPTm m_PTm))
|
||
|
|
sigma_PTm.
|
||
|
|
Proof.
|
||
|
|
exact (fun x => eq_refl).
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Lemma varLRen'_PTm {m_PTm : nat} {n_PTm : nat}
|
||
|
|
(xi_PTm : fin m_PTm -> fin n_PTm) (x : fin m_PTm) :
|
||
|
|
ren_PTm xi_PTm (VarPTm m_PTm x) = VarPTm n_PTm (xi_PTm x).
|
||
|
|
Proof.
|
||
|
|
exact (eq_refl).
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Lemma varLRen'_PTm_pointwise {m_PTm : nat} {n_PTm : nat}
|
||
|
|
(xi_PTm : fin m_PTm -> fin n_PTm) :
|
||
|
|
pointwise_relation _ eq (funcomp (ren_PTm xi_PTm) (VarPTm m_PTm))
|
||
|
|
(funcomp (VarPTm n_PTm) xi_PTm).
|
||
|
|
Proof.
|
||
|
|
exact (fun x => eq_refl).
|
||
|
|
Qed.
|
||
|
|
|
||
|
|
Class Up_PTm X Y :=
|
||
|
|
up_PTm : X -> Y.
|
||
|
|
|
||
|
|
#[global]
|
||
|
|
Instance Subst_PTm {m_PTm n_PTm : nat}: (Subst1 _ _ _) :=
|
||
|
|
(@subst_PTm m_PTm n_PTm).
|
||
|
|
|
||
|
|
#[global]
|
||
|
|
Instance Up_PTm_PTm {m n_PTm : nat}: (Up_PTm _ _) := (@up_PTm_PTm m n_PTm).
|
||
|
|
|
||
|
|
#[global]
|
||
|
|
Instance Ren_PTm {m_PTm n_PTm : nat}: (Ren1 _ _ _) := (@ren_PTm m_PTm n_PTm).
|
||
|
|
|
||
|
|
#[global]
|
||
|
|
Instance VarInstance_PTm {n_PTm : nat}: (Var _ _) := (@VarPTm n_PTm).
|
||
|
|
|
||
|
|
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_PTm_morphism {m_PTm : nat} {n_PTm : nat}:
|
||
|
|
(Proper (respectful (pointwise_relation _ eq) (respectful eq eq))
|
||
|
|
(@subst_PTm m_PTm n_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 {m_PTm : nat} {n_PTm : nat}:
|
||
|
|
(Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
|
||
|
|
(@subst_PTm m_PTm n_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 {m_PTm : nat} {n_PTm : nat}:
|
||
|
|
(Proper (respectful (pointwise_relation _ eq) (respectful eq eq))
|
||
|
|
(@ren_PTm m_PTm n_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 {m_PTm : nat} {n_PTm : nat}:
|
||
|
|
(Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
|
||
|
|
(@ren_PTm m_PTm n_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.
|
||
|
|
|
||
|
|
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 in *.
|
||
|
|
|
||
|
|
Ltac asimpl' := repeat (first
|
||
|
|
[ 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'_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_list_PTm_PTm, up_PTm_PTm, upRen_list_PTm_PTm,
|
||
|
|
upRen_PTm_PTm, up_ren
|
||
|
|
| progress cbn[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
|
||
|
|
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'_PTm_pointwise;
|
||
|
|
try setoid_rewrite rinstInst'_PTm.
|
||
|
|
|
||
|
|
Ltac renamify := auto_unfold;
|
||
|
|
try setoid_rewrite_left rinstInst'_PTm_pointwise;
|
||
|
|
try setoid_rewrite_left rinstInst'_PTm.
|
||
|
|
|
||
|
|
End Core.
|
||
|
|
|
||
|
|
Module Extra.
|
||
|
|
|
||
|
|
Import
|
||
|
|
Core.
|
||
|
|
|
||
|
|
Arguments VarPTm {n_PTm}.
|
||
|
|
|
||
|
|
Arguments PProj {n_PTm}.
|
||
|
|
|
||
|
|
Arguments PPair {n_PTm}.
|
||
|
|
|
||
|
|
Arguments PApp {n_PTm}.
|
||
|
|
|
||
|
|
Arguments PAbs {n_PTm}.
|
||
|
|
|
||
|
|
#[global]Hint Opaque subst_PTm: rewrite.
|
||
|
|
|
||
|
|
#[global]Hint Opaque ren_PTm: rewrite.
|
||
|
|
|
||
|
|
End Extra.
|
||
|
|
|
||
|
|
Module interface.
|
||
|
|
|
||
|
|
Export Core.
|
||
|
|
|
||
|
|
Export Extra.
|
||
|
|
|
||
|
|
End interface.
|
||
|
|
|
||
|
|
Export interface.
|
||
|
|
|