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Prove join univ pi contra

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Yiyun Liu 2024-12-25 21:11:58 -05:00
parent e2702ed277
commit 80d8b13e49
4 changed files with 202 additions and 8 deletions
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21
theories/Autosubst2/syntax.v
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@ -26,7 +26,8 @@ Inductive Tm (n_Tm : nat) : Type :=
| 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
| Bot : Tm n_Tm.
| Bot : Tm n_Tm
| Univ : nat -> 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.
@ -71,6 +72,12 @@ Proof.
exact (eq_refl).
Qed.
Lemma congr_Univ {m_Tm : nat} {s0 : nat} {t0 : nat} (H0 : s0 = t0) :
Univ m_Tm s0 = Univ m_Tm t0.
Proof.
exact (eq_trans eq_refl (ap (fun x => Univ m_Tm x) H0)).
Qed.
Lemma upRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n) :
fin (S m) -> fin (S n).
Proof.
@ -93,6 +100,7 @@ Fixpoint ren_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
| 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
| Univ _ s0 => Univ n_Tm s0
end.
Lemma up_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm) :
@ -119,6 +127,7 @@ Fixpoint subst_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
| Pi _ s0 s1 =>
Pi n_Tm (subst_Tm sigma_Tm s0) (subst_Tm (up_Tm_Tm sigma_Tm) s1)
| Bot _ => Bot n_Tm
| Univ _ s0 => Univ n_Tm s0
end.
Lemma upId_Tm_Tm {m_Tm : nat} (sigma : fin m_Tm -> Tm m_Tm)
@ -158,6 +167,7 @@ subst_Tm sigma_Tm s = s :=
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
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
Lemma upExtRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n)
@ -201,6 +211,7 @@ Fixpoint extRen_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
(extRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upExtRen_Tm_Tm _ _ Eq_Tm) s1)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
Lemma upExt_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm)
@ -245,6 +256,7 @@ Fixpoint ext_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
(ext_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm) (upExt_Tm_Tm _ _ Eq_Tm)
s1)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
Lemma up_ren_ren_Tm_Tm {k : nat} {l : nat} {m : nat} (xi : fin k -> fin l)
@ -289,6 +301,7 @@ Fixpoint compRenRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
(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
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
Lemma up_ren_subst_Tm_Tm {k : nat} {l : nat} {m_Tm : nat}
@ -343,6 +356,7 @@ Fixpoint compRenSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
(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
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
Lemma up_subst_ren_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
@ -418,6 +432,7 @@ ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
(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
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
Lemma up_subst_subst_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
@ -494,6 +509,7 @@ subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
(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
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
Lemma renRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
@ -609,6 +625,7 @@ Fixpoint rinst_inst_Tm {m_Tm : nat} {n_Tm : nat}
(rinst_inst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm sigma_Tm)
(rinstInst_up_Tm_Tm _ _ Eq_Tm) s1)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
Lemma rinstInst'_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
@ -807,6 +824,8 @@ Core.
Arguments VarTm {n_Tm}.
Arguments Univ {n_Tm}.
Arguments Bot {n_Tm}.
Arguments Pi {n_Tm}.