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yiyunliu:cred
yiyunliu:commute-rules
yiyunliu:boolean-nojunk
yiyunliu:boolean
yiyunliu:main
yiyunliu:norm
yiyunliu:norm-take3
yiyunliu:erase
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4
syntax.sig
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@ -2,7 +2,6 @@ nat : Type
Tm(VarTm) : Type
PTag : Type
TTag : Type
bool : Type
TPi : TTag
TSig : TTag
@ -15,6 +14,3 @@ Proj : PTag -> Tm -> Tm
TBind : TTag -> Tm -> (bind Tm in Tm) -> Tm
Bot : Tm
Univ : nat -> Tm
Bool : Tm
BVal : bool -> Tm
If : Tm -> Tm -> Tm -> Tm

86
theories/Autosubst2/syntax.v
View file

@ -41,10 +41,7 @@ Inductive Tm (n_Tm : nat) : Type :=
| Proj : PTag -> Tm n_Tm -> Tm n_Tm
| TBind : TTag -> Tm n_Tm -> Tm (S n_Tm) -> Tm n_Tm
| Bot : Tm n_Tm
| Univ : nat -> Tm n_Tm
| Bool : Tm n_Tm
| BVal : bool -> Tm n_Tm
| If : Tm n_Tm -> Tm n_Tm -> Tm n_Tm -> 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.
@ -97,27 +94,6 @@ Proof.
exact (eq_trans eq_refl (ap (fun x => Univ m_Tm x) H0)).
Qed.
Lemma congr_Bool {m_Tm : nat} : Bool m_Tm = Bool m_Tm.
Proof.
exact (eq_refl).
Qed.
Lemma congr_BVal {m_Tm : nat} {s0 : bool} {t0 : bool} (H0 : s0 = t0) :
BVal m_Tm s0 = BVal m_Tm t0.
Proof.
exact (eq_trans eq_refl (ap (fun x => BVal m_Tm x) H0)).
Qed.
Lemma congr_If {m_Tm : nat} {s0 : Tm m_Tm} {s1 : Tm m_Tm} {s2 : Tm m_Tm}
{t0 : Tm m_Tm} {t1 : Tm m_Tm} {t2 : Tm m_Tm} (H0 : s0 = t0) (H1 : s1 = t1)
(H2 : s2 = t2) : If m_Tm s0 s1 s2 = If m_Tm t0 t1 t2.
Proof.
exact (eq_trans
(eq_trans (eq_trans eq_refl (ap (fun x => If m_Tm x s1 s2) H0))
(ap (fun x => If m_Tm t0 x s2) H1))
(ap (fun x => If m_Tm t0 t1 x) H2)).
Qed.
Lemma upRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n) :
fin (S m) -> fin (S n).
Proof.
@ -142,10 +118,6 @@ Fixpoint ren_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
TBind n_Tm s0 (ren_Tm xi_Tm s1) (ren_Tm (upRen_Tm_Tm xi_Tm) s2)
| Bot _ => Bot n_Tm
| Univ _ s0 => Univ n_Tm s0
| Bool _ => Bool n_Tm
| BVal _ s0 => BVal n_Tm s0
| If _ s0 s1 s2 =>
If n_Tm (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1) (ren_Tm xi_Tm s2)
end.
Lemma up_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm) :
@ -173,11 +145,6 @@ Fixpoint subst_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
TBind n_Tm s0 (subst_Tm sigma_Tm s1) (subst_Tm (up_Tm_Tm sigma_Tm) s2)
| Bot _ => Bot n_Tm
| Univ _ s0 => Univ n_Tm s0
| Bool _ => Bool n_Tm
| BVal _ s0 => BVal n_Tm s0
| If _ s0 s1 s2 =>
If n_Tm (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
(subst_Tm sigma_Tm s2)
end.
Lemma upId_Tm_Tm {m_Tm : nat} (sigma : fin m_Tm -> Tm m_Tm)
@ -218,11 +185,6 @@ subst_Tm sigma_Tm s = s :=
(idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s2)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
| Bool _ => congr_Bool
| BVal _ s0 => congr_BVal (eq_refl s0)
| If _ s0 s1 s2 =>
congr_If (idSubst_Tm sigma_Tm Eq_Tm s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
(idSubst_Tm sigma_Tm Eq_Tm s2)
end.
Lemma upExtRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n)
@ -267,11 +229,6 @@ Fixpoint extRen_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
(upExtRen_Tm_Tm _ _ Eq_Tm) s2)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
| Bool _ => congr_Bool
| BVal _ s0 => congr_BVal (eq_refl s0)
| If _ s0 s1 s2 =>
congr_If (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
(extRen_Tm xi_Tm zeta_Tm Eq_Tm s1) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s2)
end.
Lemma upExt_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm)
@ -317,11 +274,6 @@ Fixpoint ext_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
s2)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
| Bool _ => congr_Bool
| BVal _ s0 => congr_BVal (eq_refl s0)
| If _ s0 s1 s2 =>
congr_If (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
(ext_Tm sigma_Tm tau_Tm Eq_Tm s1) (ext_Tm sigma_Tm tau_Tm Eq_Tm s2)
end.
Lemma up_ren_ren_Tm_Tm {k : nat} {l : nat} {m : nat} (xi : fin k -> fin l)
@ -367,12 +319,6 @@ Fixpoint compRenRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
(upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s2)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
| Bool _ => congr_Bool
| BVal _ s0 => congr_BVal (eq_refl s0)
| If _ s0 s1 s2 =>
congr_If (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
(compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
(compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s2)
end.
Lemma up_ren_subst_Tm_Tm {k : nat} {l : nat} {m_Tm : nat}
@ -429,12 +375,6 @@ Fixpoint compRenSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
(up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s2)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
| Bool _ => congr_Bool
| BVal _ s0 => congr_BVal (eq_refl s0)
| If _ s0 s1 s2 =>
congr_If (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s2)
end.
Lemma up_subst_ren_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
@ -512,12 +452,6 @@ ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
(up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s2)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
| Bool _ => congr_Bool
| BVal _ s0 => congr_BVal (eq_refl s0)
| If _ s0 s1 s2 =>
congr_If (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s2)
end.
Lemma up_subst_subst_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
@ -596,12 +530,6 @@ subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
(up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s2)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
| Bool _ => congr_Bool
| BVal _ s0 => congr_BVal (eq_refl s0)
| If _ s0 s1 s2 =>
congr_If (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s2)
end.
Lemma renRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
@ -718,12 +646,6 @@ Fixpoint rinst_inst_Tm {m_Tm : nat} {n_Tm : nat}
(rinstInst_up_Tm_Tm _ _ Eq_Tm) s2)
| Bot _ => congr_Bot
| Univ _ s0 => congr_Univ (eq_refl s0)
| Bool _ => congr_Bool
| BVal _ s0 => congr_BVal (eq_refl s0)
| If _ s0 s1 s2 =>
congr_If (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
(rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
(rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s2)
end.
Lemma rinstInst'_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
@ -922,12 +844,6 @@ Core.
Arguments VarTm {n_Tm}.
Arguments If {n_Tm}.
Arguments BVal {n_Tm}.
Arguments Bool {n_Tm}.
Arguments Univ {n_Tm}.
Arguments Bot {n_Tm}.

1011
theories/fp_red.v
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