Add new syntax for booleans
This commit is contained in:
parent
d9d96d2c8b
commit
1f7460fd11
3 changed files with 95 additions and 2 deletions
|
|
@ -42,7 +42,10 @@ Inductive Tm (n_Tm : nat) : Type :=
|
|||
| TBind : TTag -> Tm n_Tm -> Tm (S n_Tm) -> Tm n_Tm
|
||||
| Const : TTag -> Tm n_Tm
|
||||
| Univ : nat -> Tm n_Tm
|
||||
| Bot : Tm n_Tm.
|
||||
| Bot : Tm n_Tm
|
||||
| BVal : bool -> Tm n_Tm
|
||||
| Bool : Tm n_Tm
|
||||
| If : Tm n_Tm -> Tm n_Tm -> Tm n_Tm -> 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.
|
||||
|
|
@ -101,6 +104,27 @@ 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_Bool {m_Tm : nat} : Bool m_Tm = Bool m_Tm.
|
||||
Proof.
|
||||
exact (eq_refl).
|
||||
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.
|
||||
|
|
@ -126,6 +150,10 @@ Fixpoint ren_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
|
|||
| Const _ s0 => Const n_Tm s0
|
||||
| Univ _ s0 => Univ n_Tm s0
|
||||
| Bot _ => Bot n_Tm
|
||||
| BVal _ s0 => BVal n_Tm s0
|
||||
| Bool _ => Bool n_Tm
|
||||
| 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) :
|
||||
|
|
@ -154,6 +182,11 @@ Fixpoint subst_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
|
|||
| Const _ s0 => Const n_Tm s0
|
||||
| Univ _ s0 => Univ n_Tm s0
|
||||
| Bot _ => Bot n_Tm
|
||||
| BVal _ s0 => BVal n_Tm s0
|
||||
| Bool _ => Bool n_Tm
|
||||
| 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)
|
||||
|
|
@ -195,6 +228,11 @@ subst_Tm sigma_Tm s = s :=
|
|||
| Const _ s0 => congr_Const (eq_refl s0)
|
||||
| Univ _ s0 => congr_Univ (eq_refl s0)
|
||||
| Bot _ => congr_Bot
|
||||
| BVal _ s0 => congr_BVal (eq_refl s0)
|
||||
| Bool _ => congr_Bool
|
||||
| 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)
|
||||
|
|
@ -240,6 +278,11 @@ Fixpoint extRen_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
|
|||
| Const _ s0 => congr_Const (eq_refl s0)
|
||||
| Univ _ s0 => congr_Univ (eq_refl s0)
|
||||
| Bot _ => congr_Bot
|
||||
| BVal _ s0 => congr_BVal (eq_refl s0)
|
||||
| Bool _ => congr_Bool
|
||||
| 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)
|
||||
|
|
@ -286,6 +329,11 @@ Fixpoint ext_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
|
|||
| Const _ s0 => congr_Const (eq_refl s0)
|
||||
| Univ _ s0 => congr_Univ (eq_refl s0)
|
||||
| Bot _ => congr_Bot
|
||||
| BVal _ s0 => congr_BVal (eq_refl s0)
|
||||
| Bool _ => congr_Bool
|
||||
| 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)
|
||||
|
|
@ -332,6 +380,12 @@ Fixpoint compRenRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
|
|||
| Const _ s0 => congr_Const (eq_refl s0)
|
||||
| Univ _ s0 => congr_Univ (eq_refl s0)
|
||||
| Bot _ => congr_Bot
|
||||
| BVal _ s0 => congr_BVal (eq_refl s0)
|
||||
| Bool _ => congr_Bool
|
||||
| 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}
|
||||
|
|
@ -389,6 +443,12 @@ Fixpoint compRenSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
|
|||
| Const _ s0 => congr_Const (eq_refl s0)
|
||||
| Univ _ s0 => congr_Univ (eq_refl s0)
|
||||
| Bot _ => congr_Bot
|
||||
| BVal _ s0 => congr_BVal (eq_refl s0)
|
||||
| Bool _ => congr_Bool
|
||||
| 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}
|
||||
|
|
@ -467,6 +527,12 @@ ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
|
|||
| Const _ s0 => congr_Const (eq_refl s0)
|
||||
| Univ _ s0 => congr_Univ (eq_refl s0)
|
||||
| Bot _ => congr_Bot
|
||||
| BVal _ s0 => congr_BVal (eq_refl s0)
|
||||
| Bool _ => congr_Bool
|
||||
| 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}
|
||||
|
|
@ -546,6 +612,12 @@ subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
|
|||
| Const _ s0 => congr_Const (eq_refl s0)
|
||||
| Univ _ s0 => congr_Univ (eq_refl s0)
|
||||
| Bot _ => congr_Bot
|
||||
| BVal _ s0 => congr_BVal (eq_refl s0)
|
||||
| Bool _ => congr_Bool
|
||||
| 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}
|
||||
|
|
@ -663,6 +735,12 @@ Fixpoint rinst_inst_Tm {m_Tm : nat} {n_Tm : nat}
|
|||
| Const _ s0 => congr_Const (eq_refl s0)
|
||||
| Univ _ s0 => congr_Univ (eq_refl s0)
|
||||
| Bot _ => congr_Bot
|
||||
| BVal _ s0 => congr_BVal (eq_refl s0)
|
||||
| Bool _ => congr_Bool
|
||||
| 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)
|
||||
|
|
@ -861,6 +939,12 @@ Core.
|
|||
|
||||
Arguments VarTm {n_Tm}.
|
||||
|
||||
Arguments If {n_Tm}.
|
||||
|
||||
Arguments Bool {n_Tm}.
|
||||
|
||||
Arguments BVal {n_Tm}.
|
||||
|
||||
Arguments Bot {n_Tm}.
|
||||
|
||||
Arguments Univ {n_Tm}.
|
||||
|
|
|
|||
|
|
@ -15,11 +15,14 @@ Module Compile.
|
|||
| Pair a b => Pair (F a) (F b)
|
||||
| Proj t a => Proj t (F a)
|
||||
| Bot => Bot
|
||||
| If a b c => App (App (F a) (F b)) (F c)
|
||||
| BVal b => if b then (Abs (Abs (VarTm (shift var_zero)))) else (Abs (Abs (VarTm var_zero)))
|
||||
| Bool => Bool
|
||||
end.
|
||||
|
||||
Lemma renaming n m (a : Tm n) (ξ : fin n -> fin m) :
|
||||
F (ren_Tm ξ a)= ren_Tm ξ (F a).
|
||||
Proof. move : m ξ. elim : n / a => //=; scongruence. Qed.
|
||||
Proof. move : m ξ. elim : n / a => //=; hauto lq:on. Qed.
|
||||
|
||||
#[local]Hint Rewrite Compile.renaming : compile.
|
||||
|
||||
|
|
@ -33,6 +36,8 @@ Module Compile.
|
|||
- hauto lq:on rew:off.
|
||||
- hauto lq:on.
|
||||
- hauto lq:on inv:option rew:db:compile unfold:funcomp.
|
||||
- hauto lq:on rew:off.
|
||||
- hauto lq:on rew:off.
|
||||
Qed.
|
||||
|
||||
Lemma substing n b (a : Tm (S n)) :
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue