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Need to tweak the definition of Prov

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Yiyun Liu 2024-12-24 15:31:50 -05:00
parent c6edc1b0be
commit cbe9941046
3 changed files with 95 additions and 10 deletions
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20
theories/Autosubst2/syntax.v
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@ -25,7 +25,8 @@ Inductive Tm (n_Tm : nat) : Type :=
| App : Tm n_Tm -> Tm n_Tm -> Tm n_Tm
| Pair : Tm n_Tm -> Tm n_Tm -> Tm n_Tm
| Proj : PTag -> Tm n_Tm -> Tm n_Tm
| Pi : Tm n_Tm -> Tm (S n_Tm) -> Tm n_Tm.
| Pi : Tm n_Tm -> Tm (S n_Tm) -> Tm n_Tm
| Bot : Tm n_Tm.
Lemma congr_Abs {m_Tm : nat} {s0 : Tm (S m_Tm)} {t0 : Tm (S m_Tm)}
(H0 : s0 = t0) : Abs m_Tm s0 = Abs m_Tm t0.
@ -65,6 +66,11 @@ exact (eq_trans (eq_trans eq_refl (ap (fun x => Pi m_Tm x s1) H0))
(ap (fun x => Pi m_Tm t0 x) H1)).
Qed.
Lemma congr_Bot {m_Tm : nat} : Bot m_Tm = Bot m_Tm.
Proof.
exact (eq_refl).
Qed.
Lemma upRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n) :
fin (S m) -> fin (S n).
Proof.
@ -86,6 +92,7 @@ Fixpoint ren_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
| Pair _ s0 s1 => Pair n_Tm (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
| Proj _ s0 s1 => Proj n_Tm s0 (ren_Tm xi_Tm s1)
| Pi _ s0 s1 => Pi n_Tm (ren_Tm xi_Tm s0) (ren_Tm (upRen_Tm_Tm xi_Tm) s1)
| Bot _ => Bot n_Tm
end.
Lemma up_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm) :
@ -111,6 +118,7 @@ Fixpoint subst_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
| Proj _ s0 s1 => Proj n_Tm s0 (subst_Tm sigma_Tm s1)
| Pi _ s0 s1 =>
Pi n_Tm (subst_Tm sigma_Tm s0) (subst_Tm (up_Tm_Tm sigma_Tm) s1)
| Bot _ => Bot n_Tm
end.
Lemma upId_Tm_Tm {m_Tm : nat} (sigma : fin m_Tm -> Tm m_Tm)
@ -149,6 +157,7 @@ subst_Tm sigma_Tm s = s :=
| Pi _ s0 s1 =>
congr_Pi (idSubst_Tm sigma_Tm Eq_Tm s0)
(idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s1)
| Bot _ => congr_Bot
end.
Lemma upExtRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n)
@ -191,6 +200,7 @@ Fixpoint extRen_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
congr_Pi (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
(extRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upExtRen_Tm_Tm _ _ Eq_Tm) s1)
| Bot _ => congr_Bot
end.
Lemma upExt_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm)
@ -234,6 +244,7 @@ Fixpoint ext_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
congr_Pi (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
(ext_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm) (upExt_Tm_Tm _ _ Eq_Tm)
s1)
| Bot _ => congr_Bot
end.
Lemma up_ren_ren_Tm_Tm {k : nat} {l : nat} {m : nat} (xi : fin k -> fin l)
@ -277,6 +288,7 @@ Fixpoint compRenRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
congr_Pi (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
(compRenRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s1)
| Bot _ => congr_Bot
end.
Lemma up_ren_subst_Tm_Tm {k : nat} {l : nat} {m_Tm : nat}
@ -330,6 +342,7 @@ Fixpoint compRenSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
congr_Pi (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
(compRenSubst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm tau_Tm)
(up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s1)
| Bot _ => congr_Bot
end.
Lemma up_subst_ren_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
@ -404,6 +417,7 @@ ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
congr_Pi (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
(compSubstRen_Tm (up_Tm_Tm sigma_Tm) (upRen_Tm_Tm zeta_Tm)
(up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s1)
| Bot _ => congr_Bot
end.
Lemma up_subst_subst_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
@ -479,6 +493,7 @@ subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
congr_Pi (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
(compSubstSubst_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm)
(up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s1)
| Bot _ => congr_Bot
end.
Lemma renRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
@ -593,6 +608,7 @@ Fixpoint rinst_inst_Tm {m_Tm : nat} {n_Tm : nat}
congr_Pi (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
(rinst_inst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm sigma_Tm)
(rinstInst_up_Tm_Tm _ _ Eq_Tm) s1)
| Bot _ => congr_Bot
end.
Lemma rinstInst'_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
@ -791,6 +807,8 @@ Core.
Arguments VarTm {n_Tm}.
Arguments Bot {n_Tm}.
Arguments Pi {n_Tm}.
Arguments Proj {n_Tm}.