49 lines
1.7 KiB
Coq
49 lines
1.7 KiB
Coq
Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax common typing.
|
|
From Hammer Require Import Tactics.
|
|
Require Import ssreflect.
|
|
|
|
|
|
Lemma lem :
|
|
(forall n (Γ : fin n -> PTm n), ⊢ Γ -> True) /\
|
|
(forall n Γ (a A : PTm n), Γ ⊢ a ∈ A -> ) /\
|
|
(forall n Γ (a b A : PTm n), Γ ⊢ a ≡ b ∈ A -> ...).
|
|
Proof. apply wt_mutual. ...
|
|
|
|
|
|
Lemma wff_mutual :
|
|
(forall n (Γ : fin n -> PTm n), ⊢ Γ -> True) /\
|
|
(forall n Γ (a A : PTm n), Γ ⊢ a ∈ A -> ⊢ Γ) /\
|
|
(forall n Γ (a b A : PTm n), Γ ⊢ a ≡ b ∈ A -> ⊢ Γ).
|
|
Proof. apply wt_mutual; eauto. Qed.
|
|
|
|
#[export]Hint Constructors Wt Wff Eq : wt.
|
|
|
|
Lemma renaming_up n m (ξ : fin n -> fin m) Δ Γ A :
|
|
renaming_ok Δ Γ ξ ->
|
|
renaming_ok (funcomp (ren_PTm shift) (scons (ren_PTm ξ A) Δ)) (funcomp (ren_PTm shift) (scons A Γ)) (upRen_PTm_PTm ξ) .
|
|
Proof.
|
|
move => h i.
|
|
destruct i as [i|].
|
|
asimpl. rewrite /renaming_ok in h.
|
|
rewrite /funcomp. rewrite -h.
|
|
by asimpl.
|
|
by asimpl.
|
|
Qed.
|
|
|
|
Lemma renaming :
|
|
(forall n (Γ : fin n -> PTm n), ⊢ Γ -> True) /\
|
|
(forall n Γ (a A : PTm n), Γ ⊢ a ∈ A -> forall m Δ (ξ : fin n -> fin m), ⊢ Δ -> renaming_ok Δ Γ ξ ->
|
|
Δ ⊢ ren_PTm ξ a ∈ ren_PTm ξ A) /\
|
|
(forall n Γ (a b A : PTm n), Γ ⊢ a ≡ b ∈ A -> forall m Δ (ξ : fin n -> fin m), ⊢ Δ -> renaming_ok Δ Γ ξ ->
|
|
Δ ⊢ ren_PTm ξ a ≡ ren_PTm ξ b ∈ ren_PTm ξ A).
|
|
Proof.
|
|
apply wt_mutual => //=; eauto 3 with wt.
|
|
- move => n Γ i hΓ _ m Δ ξ hΔ hξ.
|
|
rewrite hξ.
|
|
by apply T_Var.
|
|
- hauto lq:on rew:off ctrs:Wt, Wff use:renaming_up.
|
|
- move => n Γ a A B i hP ihP ha iha m Δ ξ hΔ hξ.
|
|
apply : T_Abs; eauto.
|
|
apply iha; last by apply renaming_up.
|
|
econstructor; eauto.
|
|
by apply renaming_up.
|