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Add simplified projection lemma

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Yiyun Liu 2025-02-12 19:53:20 -05:00
parent d053f93100
commit 48adb34946
3 changed files with 41 additions and 22 deletions
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22
theories/algorithmic.v
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@ -58,25 +58,6 @@ Lemma E_Conv_E n Γ (a b : PTm n) A B i :
Γ ⊢ a ≡ b ∈ B.
Proof. qauto use:E_Conv, Su_Eq, E_Symmetric. Qed.
Lemma renaming_su' : forall n m Δ Γ (ξ : fin n -> fin m) (A B : PTm n) u U ,
u = ren_PTm ξ A ->
U = ren_PTm ξ B ->
Γ ⊢ A ≲ B ->
⊢ Δ -> renaming_ok Δ Γ ξ ->
Δ ⊢ u ≲ U.
Proof. move => > -> ->. hauto l:on use:renaming. Qed.
Lemma weakening_su : forall n Γ (A0 A1 B : PTm n) i,
Γ ⊢ B ∈ PUniv i ->
Γ ⊢ A0 ≲ A1 ->
funcomp (ren_PTm shift) (scons B Γ) ⊢ ren_PTm shift A0 ≲ ren_PTm shift A1.
Proof.
move => n Γ A0 A1 B i hB hlt.
apply : renaming_su'; eauto.
apply : Wff_Cons'; eauto.
apply : renaming_shift; eauto.
Qed.
(* Coquand's algorithm with subtyping *)
Reserved Notation "a ∼ b" (at level 70).
Reserved Notation "a ↔ b" (at level 70).
@ -253,6 +234,9 @@ Proof.
have [i [A2 [B2 h]]] : exists i A2 B2, Γ ⊢ A ≡ PBind PPi A2 B2 ∈ PUniv i by admit.
have ? : Γ ⊢ PBind PPi A0 B0 ≲ PBind PPi A2 B2 by eauto using Su_Transitive, Su_Eq.
have ? : Γ ⊢ PBind PPi A1 B1 ≲ PBind PPi A2 B2 by eauto using Su_Transitive, Su_Eq.
have [j ?] : exists j, Γ ⊢ A0 ∈ PUniv j by qauto l:on use:Wff_Cons_Inv, wff_mutual.
have [k ?] : exists j, Γ ⊢ A1 ∈ PUniv j by qauto l:on use:Wff_Cons_Inv, wff_mutual.
have [l ?] : exists j, Γ ⊢ A2 ∈ PUniv j by hauto lq:on rew:off use:regularity, Bind_Inv.
have [h2 h3] : Γ ⊢ A2 ≲ A0 /\ Γ ⊢ A2 ≲ A1 by hauto l:on use:Su_Pi_Proj1.
apply E_Conv with (A := PBind PPi A2 B2); cycle 1.
eauto using E_Symmetric, Su_Eq.