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One case left for nat

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Yiyun Liu 2025-02-24 01:25:10 -05:00
parent 5544e401a2
commit 2a24700f9a
1 changed files with 10 additions and 64 deletions
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74
theories/logrel.v
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@ -1415,7 +1415,6 @@ Proof.
have snc : SN (subst_PTm (up_PTm_PTm (up_PTm_PTm ρ)) c)
by apply sn_bot_up2; move : hb hρbb => /[apply]; hauto lq:on use:adequacy.
elim : u /hu.
+ exists m, PA. split.
* move : ha0. by asimpl.
@ -1432,7 +1431,7 @@ Proof.
+ move => a ha [j][S][h0]h1.
have hρ' : ρ_ok (funcomp (ren_PTm shift) (scons PNat Γ)) (scons (PSuc a) ρ)by
apply : ρ_ok_cons; eauto using (InterpUniv_Nat 0); hauto ctrs:SNat.
move /SemWt_Univ : (hA) hρ' => /[apply].
move /SemWt_Univ : (hA) (hρ') => /[apply].
move => [S0 hS0].
exists i, S0. split => //.
apply : InterpUniv_back_clos; eauto.
@ -1440,68 +1439,15 @@ Proof.
move : (PInd (subst_PTm (up_PTm_PTm ρ) P) a b
(subst_PTm (up_PTm_PTm (up_PTm_PTm ρ)) c)) h1.
move => r hr.
have :
move /[swap] => ->.
+ move => ? a0 ? ih c hc ha. subst.
move /(_ a0 ltac:(apply rtc_refl) ha) : ih => [m0][PA1][hPA1]hr.
have hρ' : ρ_ok (tNat :: Γ) (a0 .: ρ).
{
apply : ρ_ok_cons; auto.
apply InterpUnivN_Nat.
hauto lq:on ctrs:rtc.
}
have : ρ_ok (A :: tNat :: Γ) ((tInd a[ρ] bs a0) .: (a0 .: ρ))
by eauto using ρ_ok_cons.
move /hb => {hb} [m1][PA2][hPA2]h.
exists m1, PA2.
split.
* move : hPA2. asimpl.
move /InterpUnivN_back_preservation_star. apply.
qauto l:on use:Pars_morphing_star,good_Pars_morphing_ext ctrs:rtc.
* move : h.
move /InterpUnivN_back_clos_star. apply; eauto.
subst bs.
apply : P_IndSuc_star'; eauto.
by asimpl.
+ move => a0 ? <- _ a1 *.
have ? : wne a1 by hauto lq:on.
suff /hA : ρ_ok (tNat :: Γ) (a1 .: ρ).
move => [S hS].
exists l, S. split=>//.
suff ? : wn bs.
have ? : wn a[ρ] by sfirstorder use:adequacy.
have : wne (tInd a[ρ] bs a1) by auto using wne_ind.
eapply adequacy; eauto.
subst bs.
rewrite /SemWt in hb.
have /hA : ρ_ok (tNat :: Γ) (var_tm 0 .: ρ).
{
apply : ρ_ok_cons; auto.
apply InterpUnivN_Nat.
hauto lq:on ctrs:rtc.
}
move => [S1 hS1].
have /hb : ρ_ok (A :: tNat :: Γ) (var_tm 0 .: (var_tm 0 .: ρ)).
{
apply : ρ_ok_cons; cycle 2; eauto.
apply : ρ_ok_cons; cycle 2; eauto.
apply InterpUnivN_Nat.
hauto lq:on ctrs:rtc.
hauto q:on ctrs:rtc use:adequacy.
}
move =>[m0][PA1][h1]h2.
have : wn b[var_tm 0 .: (var_tm 0 .: ρ)] by hauto q:on use:adequacy.
clear => h.
apply wn_antirenaming with (ξ := var_zero .: (var_zero .: id)).
by asimpl.
apply : ρ_ok_cons; auto.
apply InterpUnivN_Nat.
hauto lq:on use:adequacy db:nfne.
have hρ'' : ρ_ok
(funcomp (ren_PTm shift) (scons P (funcomp (ren_PTm shift) (scons PNat Γ)))) (scons r (scons a ρ)) by
eauto using ρ_ok_cons, (InterpUniv_Nat 0).
move : hb hρ'' => /[apply].
move => [k][PA1][h2]h3.
move : h2. asimpl => ?.
have ? : PA1 = S0 by eauto using InterpUniv_Functional'.
by subst.
+
Admitted.
Lemma SSu_Univ n Γ i j :