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Add semantic rules for function beta and eta

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Yiyun Liu 2025-02-09 16:12:57 -05:00
parent 4396786701
commit ab1bd8eef8
3 changed files with 132 additions and 16 deletions
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56
theories/logrel.v
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@ -755,6 +755,13 @@ Proof.
asimpl. hauto lq:on.
Qed.
Lemma ST_App' n Γ (b a : PTm n) A B U :
U = subst_PTm (scons a VarPTm) B ->
Γ ⊨ b ∈ PBind PPi A B ->
Γ ⊨ a ∈ A ->
Γ ⊨ PApp b a ∈ U.
Proof. move => ->. apply ST_App. Qed.
Lemma ST_Pair n Γ (a b : PTm n) A B i :
Γ ⊨ PBind PSig A B ∈ (PUniv i) ->
Γ ⊨ a ∈ A ->
@ -1010,6 +1017,48 @@ Proof.
apply SemWt_SemEq; eauto using DJoin.AbsCong, ST_Abs.
Qed.
Lemma SBind_inv1 n Γ i p (A : PTm n) B :
Γ ⊨ PBind p A B ∈ PUniv i ->
Γ ⊨ A ∈ PUniv i.
move /SemWt_Univ => h. apply SemWt_Univ.
hauto lq:on rew:off use:InterpUniv_Bind_inv.
Qed.
Lemma SE_AppEta n Γ (b : PTm n) A B i :
⊨ Γ ->
Γ ⊨ PBind PPi A B ∈ (PUniv i) ->
Γ ⊨ b ∈ PBind PPi A B ->
Γ ⊨ PAbs (PApp (ren_PTm shift b) (VarPTm var_zero)) ∈ PBind PPi A B.
Proof.
move => hΓ h0 h1. apply : ST_Abs; eauto.
have hA : Γ ⊨ A ∈ PUniv i by eauto using SBind_inv1.
eapply ST_App' with (A := ren_PTm shift A)(B:= ren_PTm (upRen_PTm_PTm shift) B). by asimpl.
2 : {
apply ST_Var.
eauto using SemWff_cons.
}
change (PBind PPi (ren_PTm shift A) (ren_PTm (upRen_PTm_PTm shift) B)) with
(ren_PTm shift (PBind PPi A B)).
apply : weakening_Sem; eauto.
Qed.
Lemma SE_AppAbs n Γ (a : PTm (S n)) b A B i:
Γ ⊨ PBind PPi A B ∈ PUniv i ->
Γ ⊨ b ∈ A ->
funcomp (ren_PTm shift) (scons A Γ) ⊨ a ∈ B ->
Γ ⊨ PApp (PAbs a) b ≡ subst_PTm (scons b VarPTm) a ∈ subst_PTm (scons b VarPTm ) B.
Proof.
move => h h0 h1. apply SemWt_SemEq; eauto using ST_App, ST_Abs.
move => ρ hρ.
have {}/h0 := hρ.
move => [k][PA][hPA]hb.
move : ρ_ok_cons hPA hb (hρ); repeat move/[apply].
move => {}/h1.
by asimpl.
apply DJoin.FromRRed0.
apply RRed.AppAbs.
Qed.
Lemma SE_Conv' n Γ (a b : PTm n) A B i :
Γ ⊨ a ≡ b ∈ A ->
Γ ⊨ B ∈ PUniv i ->
@ -1094,13 +1143,6 @@ Proof.
apply : ST_Conv_E; eauto using ST_App, DJoin.cong, DJoin.symmetric, SBind_inst.
Qed.
Lemma SBind_inv1 n Γ i p (A : PTm n) B :
Γ ⊨ PBind p A B ∈ PUniv i ->
Γ ⊨ A ∈ PUniv i.
move /SemWt_Univ => h. apply SemWt_Univ.
hauto lq:on rew:off use:InterpUniv_Bind_inv.
Qed.
Lemma SSu_Eq n Γ (A B : PTm n) i :
Γ ⊨ A ≡ B ∈ PUniv i ->
Γ ⊨ A ≲ B.