Move projection axioms to the subtyping relation

theories/structural.v

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+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.