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Add more cases to the soundness proof

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Yiyun Liu 2025-02-12 20:18:12 -05:00
parent 48adb34946
commit ba77752329
2 changed files with 34 additions and 15 deletions
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33
theories/algorithmic.v
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@ -232,8 +232,8 @@ Proof.
move /Abs_Inv : h0 => [A0][B0][h0]h0'. move /Abs_Inv : h0 => [A0][B0][h0]h0'.
move /Abs_Inv : h1 => [A1][B1][h1]h1'. move /Abs_Inv : h1 => [A1][B1][h1]h1'.
have [i [A2 [B2 h]]] : exists i A2 B2, Γ ⊢ A ≡ PBind PPi A2 B2 ∈ PUniv i by admit. 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 hp0 : 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 hp1 : 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 [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 [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 [l ?] : exists j, Γ ⊢ A2 ∈ PUniv j by hauto lq:on rew:off use:regularity, Bind_Inv.
@ -242,21 +242,21 @@ Proof.
eauto using E_Symmetric, Su_Eq. eauto using E_Symmetric, Su_Eq.
apply : E_Abs; eauto. hauto l:on use:regularity. apply : E_Abs; eauto. hauto l:on use:regularity.
apply iha. apply iha.
move /Su_Pi_Proj2_Var in hp0.
apply : T_Conv; eauto.
eapply ctx_eq_subst_one with (A0 := A0); eauto. eapply ctx_eq_subst_one with (A0 := A0); eauto.
admit. move /Su_Pi_Proj2_Var in hp1.
admit. apply : T_Conv; eauto.
admit.
eapply ctx_eq_subst_one with (A0 := A1); eauto. eapply ctx_eq_subst_one with (A0 := A1); eauto.
admit.
admit.
admit.
(* Need to use the fundamental lemma to show that U normalizes to a Pi type *)
- abstract : hAppL. - abstract : hAppL.
move => n a u hneu ha iha Γ A wta wtu. move => n a u hneu ha iha Γ A wta wtu.
move /Abs_Inv : wta => [A0][B0][wta]hPi. move /Abs_Inv : wta => [A0][B0][wta]hPi.
have [i [A2 [B2 h]]] : exists i A2 B2, Γ ⊢ A ≡ PBind PPi A2 B2 ∈ PUniv i by admit. have [i [A2 [B2 h]]] : exists i A2 B2, Γ ⊢ A ≡ PBind PPi A2 B2 ∈ PUniv i by admit.
have hPi'' : Γ ⊢ PBind PPi A2 B2 ≲ A by eauto using Su_Eq, Su_Transitive, E_Symmetric. have hPi'' : Γ ⊢ PBind PPi A2 B2 ≲ A by eauto using Su_Eq, Su_Transitive, E_Symmetric.
have [j0 ?] : exists j0, Γ ⊢ A0 ∈ PUniv j0 by move /regularity_sub0 in hPi; hauto lq:on use:Bind_Inv.
have [j2 ?] : exists j0, Γ ⊢ A2 ∈ PUniv j0 by move /regularity_sub0 in hPi''; hauto lq:on use:Bind_Inv.
have hPi' : Γ ⊢ PBind PPi A0 B0 ≲ PBind PPi A2 B2 by eauto using Su_Eq, Su_Transitive. have hPi' : Γ ⊢ PBind PPi A0 B0 ≲ PBind PPi A2 B2 by eauto using Su_Eq, Su_Transitive.
have hPidup := hPi'.
apply E_Conv with (A := PBind PPi A2 B2); eauto. apply E_Conv with (A := PBind PPi A2 B2); eauto.
have /regularity_sub0 [i' hPi0] := hPi. have /regularity_sub0 [i' hPi0] := hPi.
have : Γ ⊢ PAbs (PApp (ren_PTm shift u) (VarPTm var_zero)) ≡ u ∈ PBind PPi A2 B2. have : Γ ⊢ PAbs (PApp (ren_PTm shift u) (VarPTm var_zero)) ≡ u ∈ PBind PPi A2 B2.
@ -270,13 +270,16 @@ Proof.
by eauto using E_Transitive. by eauto using E_Transitive.
apply : E_Abs; eauto. hauto l:on use:regularity. apply : E_Abs; eauto. hauto l:on use:regularity.
apply iha. apply iha.
admit. move /Su_Pi_Proj2_Var in hPi'.
apply : T_Conv; eauto.
eapply ctx_eq_subst_one with (A0 := A0); eauto.
sfirstorder use:Su_Pi_Proj1.
(* move /Su_Pi_Proj2_Var in hPidup. *)
(* apply : T_Conv; eauto. *)
eapply T_App' with (A := ren_PTm shift A2) (B := ren_PTm (upRen_PTm_PTm shift) B2). by asimpl. eapply T_App' with (A := ren_PTm shift A2) (B := ren_PTm (upRen_PTm_PTm shift) B2). by asimpl.
eapply weakening_wt' with (a := u) (A := PBind PPi A2 B2). reflexivity. by asimpl. eapply weakening_wt' with (a := u) (A := PBind PPi A2 B2);eauto.
admit. by eauto using T_Conv_E. apply T_Var. apply : Wff_Cons'; eauto.
apply : T_Conv; eauto. apply : Su_Eq; eauto.
apply T_Var. apply : Wff_Cons'; eauto.
admit.
(* Mirrors the last case *) (* Mirrors the last case *)
- move => n a u hu ha iha Γ A hu0 ha0. - move => n a u hu ha iha Γ A hu0 ha0.
apply E_Symmetric. apply E_Symmetric.

16
theories/structural.v
View file

@ -658,3 +658,19 @@ Proof.
by asimpl. by asimpl.
by asimpl. by asimpl.
Qed. Qed.
Lemma Su_Sig_Proj2_Var n Γ (A0 A1 : PTm n) B0 B1 :
Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A1 B1 ->
funcomp (ren_PTm shift) (scons A0 Γ) ⊢ B0 ≲ B1.
Proof.
move => h.
have /Su_Sig_Proj1 h1 := h.
have /regularity_sub0 [i h2] := h1.
move /weakening_su : (h) h2. move => /[apply].
move => h2.
apply : Su_Sig_Proj2'; try eassumption; rewrite -?/ren_PTm; cycle 2.
apply E_Refl. apply T_Var' with (i := var_zero); eauto.
sfirstorder use:wff_mutual.
by asimpl.
by asimpl.
Qed.