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Finish the pair pair case

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Yiyun Liu 2025-02-16 19:51:08 -05:00
parent 60a4eb886f
commit 49a254fbef
1 changed files with 32 additions and 5 deletions
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37
theories/algorithmic.v
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@ -945,12 +945,39 @@ Proof.
sfirstorder. sfirstorder.
+ case : b => //=; try qauto depth:1 use:T_AbsPair_Imp, T_PairBind_Imp, T_PairUniv_Imp. + case : b => //=; try qauto depth:1 use:T_AbsPair_Imp, T_PairBind_Imp, T_PairUniv_Imp.
move => a1 b1 a0 b0 h _ _ Γ A hu0 hu1. move => a1 b1 a0 b0 h _ _ Γ A hu0 hu1.
have [sn0 sn1] : SN (PPair a0 b0) /\ SN (PPair a1 b1)
by qauto l:on use:fundamental_theorem, logrel.SemWt_SN.
apply : CE_HRed; eauto using rtc_refl. apply : CE_HRed; eauto using rtc_refl.
admit. move /Pair_Inv : hu0 => [A0][B0][ha0][hb0]hSu0.
(* suff : exists l, l < n /\ algo_metric l a0 b0 /\ algo_metric l a1 b1. *) move /Pair_Inv : hu1 => [A1][B1][ha1][hb1]hSu1.
(* move => [l [hl [hal0 hal1]]]. *) move /Sub_Bind_InvR : (hSu0).
(* apply CE_PairPair. eapply ih; eauto. *) move => [i][A2][B2]hE.
(* by eapply ih; eauto. *) have hSu12 : Γ ⊢ PBind PSig A1 B1 ≲ PBind PSig A2 B2
by eauto using Su_Transitive, Su_Eq.
have hSu02 : Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A2 B2
by eauto using Su_Transitive, Su_Eq.
have hA02 : Γ ⊢ A0 ≲ A2 by eauto using Su_Sig_Proj1.
have hA12 : Γ ⊢ A1 ≲ A2 by eauto using Su_Sig_Proj1.
have ha0A2 : Γ ⊢ a0 ∈ A2 by eauto using T_Conv.
have ha1A2 : Γ ⊢ a1 ∈ A2 by eauto using T_Conv.
move /algo_metric_pair : h sn0 sn1; repeat move/[apply].
move => [j][hj][ih0 ih1].
have haE : a0 ⇔ a1 by hauto l:on.
apply CE_PairPair => //.
have {}haE : Γ ⊢ a0 ≡ a1 ∈ A2
by hauto l:on use:coqeq_sound_mutual.
have {}hb1 : Γ ⊢ b1 ∈ subst_PTm (scons a1 VarPTm) B2.
apply : T_Conv; eauto.
move /E_Refl in ha1. hauto l:on use:Su_Sig_Proj2.
eapply ih; cycle -1; eauto.
apply : T_Conv; eauto.
apply Su_Transitive with (B := subst_PTm (scons a0 VarPTm) B2).
move /E_Refl in ha0. hauto l:on use:Su_Sig_Proj2.
move : hE haE. clear.
move => h.
eapply regularity in h.
move : h => [_ [hB _]].
eauto using bind_inst.
+ case : b => //=. + case : b => //=.
* hauto lq:on use:T_AbsBind_Imp. * hauto lq:on use:T_AbsBind_Imp.
* hauto lq:on rew:off use:T_PairBind_Imp. * hauto lq:on rew:off use:T_PairBind_Imp.