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Prove the case for pair and neutral

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Yiyun Liu 2025-02-12 21:13:47 -05:00
parent ba77752329
commit 1d3920fce1
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39
theories/algorithmic.v
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@ -291,22 +291,51 @@ Proof.
move /Pair_Inv => [A1][B1][h10][h11]h12.
have [i[A2[B2 h2]]] : exists i A2 B2, Γ ⊢ A ≡ PBind PSig A2 B2 ∈ PUniv i by admit.
apply E_Conv with (A := PBind PSig A2 B2); last by eauto using E_Symmetric, Su_Eq.
have h0 : Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A2 B2 by eauto using Su_Transitive, Su_Eq, E_Symmetric.
have h1 : Γ ⊢ PBind PSig A1 B1 ≲ PBind PSig A2 B2 by eauto using Su_Transitive, Su_Eq, E_Symmetric.
have /Su_Sig_Proj1 h0' := h0.
have /Su_Sig_Proj1 h1' := h1.
move => [:eqa].
apply : E_Pair; eauto. hauto l:on use:regularity.
apply iha. admit. admit.
admit.
abstract : eqa. apply iha; eauto using T_Conv.
apply ihb.
+ apply T_Conv with (A := subst_PTm (scons a0 VarPTm) B0); eauto.
have : Γ ⊢ a0 ≡ a0 ∈A0 by eauto using E_Refl.
hauto l:on use:Su_Sig_Proj2.
+ apply T_Conv with (A := subst_PTm (scons a1 VarPTm) B2); eauto; cycle 1.
move /E_Symmetric in eqa.
have ? : Γ ⊢ PBind PSig A2 B2 ∈ PUniv i by hauto use:regularity.
apply:bind_inst; eauto.
apply : T_Conv; eauto.
have : Γ ⊢ a1 ≡ a1 ∈ A1 by eauto using E_Refl.
hauto l:on use:Su_Sig_Proj2.
- move => {hAppL}.
abstract : hPairL.
move => {hAppL}.
move => n a0 a1 u neu h0 ih0 h1 ih1 Γ A ha hu.
move /Pair_Inv : ha => [A0][B0][ha0][ha1]ha.
have [i [A2 [B2 hA]]] : exists i A2 B2, Γ ⊢ A ≡ PBind PSig A2 B2 ∈ PUniv i by admit.
have hA' : Γ ⊢ PBind PSig A2 B2 ≲ A by eauto using E_Symmetric, Su_Eq.
move /E_Conv : (hA'). apply.
have hSig : Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A2 B2 by eauto using E_Symmetric, Su_Eq, Su_Transitive.
have hA02 : Γ ⊢ A0 ≲ A2 by sfirstorder use:Su_Sig_Proj1.
have hu' : Γ ⊢ u ∈ PBind PSig A2 B2 by eauto using T_Conv_E.
move => [:ih0'].
apply : E_Transitive; last (apply E_Symmetric; apply : E_PairEta).
apply : E_Pair; eauto. hauto l:on use:regularity.
apply ih0. admit. admit.
apply ih1. admit. admit.
abstract : ih0'.
apply ih0. apply : T_Conv; eauto.
by eauto using T_Proj1.
apply ih1. apply : T_Conv; eauto.
move /E_Refl in ha0.
hauto l:on use:Su_Sig_Proj2.
move /T_Proj2 in hu'.
apply : T_Conv; eauto.
move /E_Symmetric in ih0'.
move /regularity_sub0 in hA'.
hauto l:on use:bind_inst.
hauto l:on use:regularity.
apply : T_Conv; eauto using Su_Eq.
eassumption.
(* Same as before *)
- move {hAppL}.
move => *. apply E_Symmetric. apply : hPairL;