Add some admits to work on later
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2 changed files with 31 additions and 7 deletions
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@ -92,6 +92,15 @@ Proof.
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move => ->. eauto using T_Pair.
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Qed.
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Lemma T_Ind' s Γ P (a : PTm s) b c i U :
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U = subst_PTm (scons a VarPTm) P ->
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funcomp (ren_PTm shift) (scons PNat Γ) ⊢ P ∈ PUniv i ->
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Γ ⊢ a ∈ PNat ->
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Γ ⊢ b ∈ subst_PTm (scons PZero VarPTm) P ->
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funcomp (ren_PTm shift)(scons P (funcomp (ren_PTm shift) (scons PNat Γ))) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) ->
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Γ ⊢ PInd P a b c ∈ U.
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Proof. move =>->. apply T_Ind. Qed.
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Lemma T_Proj2' n Γ (a : PTm n) A B U :
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U = subst_PTm (scons (PProj PL a) VarPTm) B ->
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Γ ⊢ a ∈ PBind PSig A B ->
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@ -103,9 +112,7 @@ Lemma E_Proj2' n Γ i (a b : PTm n) A B U :
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Γ ⊢ PBind PSig A B ∈ (PUniv i) ->
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Γ ⊢ a ≡ b ∈ PBind PSig A B ->
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Γ ⊢ PProj PR a ≡ PProj PR b ∈ U.
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Proof.
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move => ->. apply E_Proj2.
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Qed.
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Proof. move => ->. apply E_Proj2. Qed.
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Lemma E_Bind' n Γ i p (A0 A1 : PTm n) B0 B1 :
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Γ ⊢ A0 ∈ PUniv i ->
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@ -184,6 +191,7 @@ Proof.
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- move => n Γ a A b B i hA ihA hB ihB hS ihS m Δ ξ hξ hΔ.
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eapply T_Pair' with (U := ren_PTm ξ (subst_PTm (scons a VarPTm) B));eauto. by asimpl.
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- move => n Γ a A B ha iha m Δ ξ hΔ hξ. apply : T_Proj2'; eauto. by asimpl.
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- admit.
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- hauto lq:on ctrs:Wt,LEq.
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- hauto lq:on ctrs:Eq.
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- hauto lq:on rew:off use:E_Bind', Wff_Cons, renaming_up.
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@ -199,6 +207,7 @@ Proof.
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move : ihb hΔ hξ. repeat move/[apply].
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by asimpl.
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- move => *. apply : E_Proj2'; eauto. by asimpl.
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- admit.
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- qauto l:on ctrs:Eq, LEq.
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- move => n Γ a b A B i hP ihP hb ihb ha iha m Δ ξ hΔ hξ.
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move : ihP (hξ) (hΔ). repeat move/[apply].
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@ -216,6 +225,8 @@ Proof.
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- move => n Γ a b A B i hP ihP ha iha hb ihb m Δ ξ hΔ hξ.
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apply : E_ProjPair2'; eauto. by asimpl.
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move : ihb hξ hΔ; repeat move/[apply]. by asimpl.
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- admit.
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- admit.
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- move => *.
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apply : E_AppEta'; eauto. by asimpl.
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- qauto l:on use:E_PairEta.
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@ -228,7 +239,7 @@ Proof.
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- qauto l:on ctrs:LEq.
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- move => *; apply : Su_Pi_Proj2'; eauto; by asimpl.
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- move => *. apply : Su_Sig_Proj2'; eauto; by asimpl.
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Qed.
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Admitted.
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Definition morphing_ok {n m} Δ Γ (ρ : fin n -> PTm m) :=
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forall i, Δ ⊢ ρ i ∈ subst_PTm ρ (Γ i).
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@ -342,6 +353,10 @@ Proof.
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- move => *. apply : T_Proj2'; eauto. by asimpl.
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- hauto lq:on ctrs:Wt,LEq.
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- qauto l:on ctrs:Wt.
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- qauto l:on ctrs:Wt.
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- qauto l:on ctrs:Wt.
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- admit.
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- qauto l:on ctrs:Wt.
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- hauto lq:on ctrs:Eq.
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- hauto lq:on ctrs:Eq.
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- hauto lq:on ctrs:Eq.
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@ -359,6 +374,7 @@ Proof.
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by asimpl.
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- hauto q:on ctrs:Eq.
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- move => *. apply : E_Proj2'; eauto. by asimpl.
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- admit.
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- qauto l:on ctrs:Eq, LEq.
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- move => n Γ a b A B i hP ihP hb ihb ha iha m Δ ρ hΔ hρ.
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move : ihP (hρ) (hΔ). repeat move/[apply].
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@ -376,6 +392,8 @@ Proof.
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- move => n Γ a b A B i hP ihP ha iha hb ihb m Δ ρ hΔ hρ.
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apply : E_ProjPair2'; eauto. by asimpl.
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move : ihb hρ hΔ; repeat move/[apply]. by asimpl.
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- admit.
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- admit.
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- move => *.
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apply : E_AppEta'; eauto. by asimpl.
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- qauto l:on use:E_PairEta.
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@ -388,7 +406,7 @@ Proof.
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- qauto l:on ctrs:LEq.
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- move => *; apply : Su_Pi_Proj2'; eauto; by asimpl.
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- move => *. apply : Su_Sig_Proj2'; eauto; by asimpl.
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Qed.
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Admitted.
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Lemma morphing_wt : forall n Γ (a A : PTm n), Γ ⊢ a ∈ A -> forall m Δ (ρ : fin n -> PTm m), ⊢ Δ -> morphing_ok Δ Γ ρ -> Δ ⊢ subst_PTm ρ a ∈ subst_PTm ρ A.
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Proof. sfirstorder use:morphing. Qed.
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@ -505,6 +523,7 @@ Proof.
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exists j. have : Γ ⊢ PProj PL a ∈ A by qauto use:T_Proj1.
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move : substing_wt h1; repeat move/[apply].
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by asimpl.
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- admit.
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- sfirstorder.
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- sfirstorder.
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- sfirstorder.
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@ -535,9 +554,12 @@ Proof.
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eauto using bind_inst.
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move /T_Proj1 in iha.
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hauto lq:on ctrs:Wt,Eq,LEq use:Bind_Inv, substing_wt.
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- admit.
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- hauto lq:on ctrs:Wt.
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- hauto q:on use:substing_wt db:wt.
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- hauto l:on use:bind_inst db:wt.
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- admit.
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- admit.
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- move => n Γ b A B i hΓ ihΓ hP _ hb [i0 ihb].
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repeat split => //=; eauto with wt.
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have {}hb : funcomp (ren_PTm shift) (scons A Γ) ⊢ ren_PTm shift b ∈ ren_PTm shift (PBind PPi A B)
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@ -603,7 +625,7 @@ Proof.
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+ apply Cumulativity with (i := i1); eauto.
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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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Admitted.
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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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