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2 changed files with 32 additions and 32 deletions
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@ -1,5 +1,5 @@
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Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax
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common typing preservation admissible fp_red.
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common typing preservation admissible fp_red structural.
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From Hammer Require Import Tactics.
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Require Import ssreflect ssrbool.
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Require Import Psatz.
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@ -140,22 +140,6 @@ Scheme
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Combined Scheme coqeq_mutual from coqeq_neu_ind, coqeq_ind, coqeq_r_ind.
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Lemma Var_Inv n Γ (i : fin n) A :
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Γ ⊢ VarPTm i ∈ A ->
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⊢ Γ /\ Γ ⊢ Γ i ≲ A.
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Proof.
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move E : (VarPTm i) => u hu.
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move : i E.
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elim : n Γ u A / hu=>//=.
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- move => n Γ i hΓ i0 [?]. subst.
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repeat split => //=.
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have h : Γ ⊢ VarPTm i ∈ Γ i by eauto using T_Var.
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eapply structural.regularity in h.
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move : h => [i0]?.
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apply : Su_Eq. apply E_Refl; eassumption.
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- sfirstorder use:Su_Transitive.
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Qed.
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Lemma coqeq_sound_mutual : forall n,
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(forall (a b : PTm n), a ∼ b -> forall Γ A B, Γ ⊢ a ∈ A -> Γ ⊢ b ∈ B -> exists C,
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Γ ⊢ C ≲ A /\ Γ ⊢ C ≲ B /\ Γ ⊢ a ≡ b ∈ C) /\
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@ -193,7 +177,7 @@ Proof.
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move => [C][ih0][ih1]ih.
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have [A2 [B2 [{}ih0 [{}ih1 {}ih]]]] : exists A2 B2, Γ ⊢ PBind PSig A2 B2 ≲ PBind PSig A0 B0 /\ Γ ⊢ PBind PSig A2 B2 ≲ PBind PSig A1 B1 /\ Γ ⊢ u0 ≡ u1 ∈ PBind PSig A2 B2 by admit.
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exists (subst_PTm (scons (PProj PL u0) VarPTm) B2).
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have [? ?] : Γ ⊢ u0 ∈ PBind PSig A2 B2 /\ Γ ⊢ u1 ∈ PBind PSig A2 B2 by hauto l:on use:structural.regularity.
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have [? ?] : Γ ⊢ u0 ∈ PBind PSig A2 B2 /\ Γ ⊢ u1 ∈ PBind PSig A2 B2 by hauto l:on use:regularity.
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repeat split => //=.
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apply : Su_Transitive ;eauto.
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apply : Su_Sig_Proj2; eauto.
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@ -223,13 +207,13 @@ Proof.
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have [h2 h3] : Γ ⊢ A2 ≲ A0 /\ Γ ⊢ A2 ≲ A1 by hauto l:on use:Su_Pi_Proj1.
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apply E_Conv with (A := PBind PPi A2 B2); cycle 1.
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eauto using E_Symmetric, Su_Eq.
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apply : E_Abs; eauto. hauto l:on use:structural.regularity.
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apply : E_Abs; eauto. hauto l:on use:regularity.
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apply iha.
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eapply structural.ctx_eq_subst_one with (A0 := A0); eauto.
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eapply ctx_eq_subst_one with (A0 := A0); eauto.
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admit.
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admit.
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admit.
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eapply structural.ctx_eq_subst_one with (A0 := A1); eauto.
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eapply ctx_eq_subst_one with (A0 := A1); eauto.
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admit.
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admit.
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admit.
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@ -243,21 +227,21 @@ Proof.
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have /regularity_sub0 [i' hPi0] := hPi.
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have : Γ ⊢ PAbs (PApp (ren_PTm shift u) (VarPTm var_zero)) ≡ u ∈ PBind PPi A2 B2.
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apply : E_AppEta; eauto.
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sfirstorder use:structural.wff_mutual.
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hauto l:on use:structural.regularity.
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sfirstorder use:wff_mutual.
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hauto l:on use:regularity.
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apply T_Conv with (A := A);eauto.
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eauto using Su_Eq.
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move => ?.
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suff : Γ ⊢ PAbs a ≡ PAbs (PApp (ren_PTm shift u) (VarPTm var_zero)) ∈ PBind PPi A2 B2
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by eauto using E_Transitive.
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apply : E_Abs; eauto. hauto l:on use:structural.regularity.
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apply : E_Abs; eauto. hauto l:on use:regularity.
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apply iha.
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admit.
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eapply structural.T_App' with (A := ren_PTm shift A2) (B := ren_PTm (upRen_PTm_PTm shift) B2). by asimpl.
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eapply structural.weakening_wt' with (a := u) (A := PBind PPi A2 B2). reflexivity. by asimpl.
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eapply T_App' with (A := ren_PTm shift A2) (B := ren_PTm (upRen_PTm_PTm shift) B2). by asimpl.
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eapply weakening_wt' with (a := u) (A := PBind PPi A2 B2). reflexivity. by asimpl.
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admit.
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apply : T_Conv; eauto. apply : Su_Eq; eauto.
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apply T_Var. apply : structural.Wff_Cons'; eauto.
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apply T_Var. apply : Wff_Cons'; eauto.
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admit.
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(* Mirrors the last case *)
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- admit.
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@ -266,7 +250,7 @@ Proof.
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move /Pair_Inv => [A1][B1][h10][h11]h12.
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have [i[A2[B2 h2]]] : exists i A2 B2, Γ ⊢ A ≡ PBind PSig A2 B2 ∈ PUniv i by admit.
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apply E_Conv with (A := PBind PSig A2 B2); last by eauto using E_Symmetric, Su_Eq.
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apply : E_Pair; eauto. hauto l:on use:structural.regularity.
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apply : E_Pair; eauto. hauto l:on use:regularity.
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apply iha. admit. admit.
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admit.
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- move => n a0 a1 u neu h0 ih0 h1 ih1 Γ A ha hu.
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@ -275,17 +259,17 @@ Proof.
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have hA' : Γ ⊢ PBind PSig A2 B2 ≲ A by eauto using E_Symmetric, Su_Eq.
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move /E_Conv : (hA'). apply.
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apply : E_Transitive; last (apply E_Symmetric; apply : E_PairEta).
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apply : E_Pair; eauto. hauto l:on use:structural.regularity.
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apply : E_Pair; eauto. hauto l:on use:regularity.
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apply ih0. admit. admit.
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apply ih1. admit. admit.
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hauto l:on use:structural.regularity.
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hauto l:on use:regularity.
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apply : T_Conv; eauto using Su_Eq.
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(* Same as before *)
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- admit.
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- sfirstorder use:E_Refl.
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- move => n p A0 A1 B0 B1 hA ihA hB ihB Γ A hA0 hA1.
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move /structural.Bind_Inv : hA0 => [i][hA0][hB0]hU.
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move /structural.Bind_Inv : hA1 => [j][hA1][hB1]hU1.
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move /Bind_Inv : hA0 => [i][hA0][hB0]hU.
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move /Bind_Inv : hA1 => [j][hA1][hB1]hU1.
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have [l [k hk]] : exists l k, Γ ⊢ A ≡ PUniv k ∈ PUniv l by admit.
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apply E_Conv with (A := PUniv k); last by eauto using Su_Eq, E_Symmetric.
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apply E_Bind. admit.
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@ -604,3 +604,19 @@ Proof.
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have : Γ ⊢ a1 ∈ A1 by eauto using T_Conv.
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move : substing_wt ih1';repeat move/[apply]. by asimpl.
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Qed.
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Lemma Var_Inv n Γ (i : fin n) A :
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Γ ⊢ VarPTm i ∈ A ->
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⊢ Γ /\ Γ ⊢ Γ i ≲ A.
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Proof.
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move E : (VarPTm i) => u hu.
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move : i E.
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elim : n Γ u A / hu=>//=.
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- move => n Γ i hΓ i0 [?]. subst.
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repeat split => //=.
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have h : Γ ⊢ VarPTm i ∈ Γ i by eauto using T_Var.
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eapply regularity in h.
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move : h => [i0]?.
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apply : Su_Eq. apply E_Refl; eassumption.
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- sfirstorder use:Su_Transitive.
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Qed.
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