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Add SE_FunExt

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Yiyun Liu 2025-04-16 23:57:00 -04:00
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14
theories/logrel.v
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@ -1516,6 +1516,20 @@ Proof.
rewrite /DJoin.R. hauto lq:on ctrs:rtc,RERed.R. rewrite /DJoin.R. hauto lq:on ctrs:rtc,RERed.R.
Qed. Qed.
Lemma SE_FunExt Γ (a b : PTm) A B i :
Γ ⊨ PBind PPi A B ∈ PUniv i ->
Γ ⊨ a ∈ PBind PPi A B ->
Γ ⊨ b ∈ PBind PPi A B ->
A :: Γ ⊨ PApp (ren_PTm shift a) (VarPTm var_zero) ≡ PApp (ren_PTm shift b) (VarPTm var_zero) ∈ B ->
Γ ⊨ a ≡ b ∈ PBind PPi A B.
Proof.
move => hpi ha hb he.
move : SE_Abs (hpi) he. repeat move/[apply]. move => he.
have /SE_Symmetric : Γ ⊨ PAbs (PApp (ren_PTm shift a) (VarPTm var_zero)) ≡ a ∈ PBind PPi A B by eauto using SE_AppEta.
have : Γ ⊨ PAbs (PApp (ren_PTm shift b) (VarPTm var_zero)) ≡ b ∈ PBind PPi A B by eauto using SE_AppEta.
eauto using SE_Transitive.
Qed.
Lemma SE_Nat Γ (a b : PTm) : Lemma SE_Nat Γ (a b : PTm) :
Γ ⊨ a ≡ b ∈ PNat -> Γ ⊨ a ≡ b ∈ PNat ->
Γ ⊨ PSuc a ≡ PSuc b ∈ PNat. Γ ⊨ PSuc a ≡ PSuc b ∈ PNat.