From b750ea4095c9c2195a056c9987bac03b5d23c01d Mon Sep 17 00:00:00 2001
From: Yiyun Liu <liuyiyun@seas.upenn.edu>
Date: Fri, 10 Jan 2025 16:51:49 -0500
Subject: [PATCH 01/13] Initial commit (still tweaking the rules)
---
syntax.sig | 4 ++
theories/Autosubst2/syntax.v | 86 +++++++++++++++++++++++++++++++++++-
theories/fp_red.v | 21 ++++++++-
3 files changed, 109 insertions(+), 2 deletions(-)
diff --git a/syntax.sig b/syntax.sig
index 4549f7f..d2354e2 100644
--- a/syntax.sig
+++ b/syntax.sig
@@ -2,6 +2,7 @@ nat : Type
Tm(VarTm) : Type
PTag : Type
TTag : Type
+bool : Type
TPi : TTag
TSig : TTag
@@ -14,3 +15,6 @@ 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
\ No newline at end of file
diff --git a/theories/Autosubst2/syntax.v b/theories/Autosubst2/syntax.v
index b972476..ffb7c7c 100644
--- a/theories/Autosubst2/syntax.v
+++ b/theories/Autosubst2/syntax.v
@@ -41,7 +41,10 @@ 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.
+ | 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.
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.
@@ -94,6 +97,27 @@ 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.
@@ -118,6 +142,10 @@ 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) :
@@ -145,6 +173,11 @@ 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)
@@ -185,6 +218,11 @@ 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)
@@ -229,6 +267,11 @@ 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)
@@ -274,6 +317,11 @@ 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)
@@ -319,6 +367,12 @@ 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}
@@ -375,6 +429,12 @@ 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}
@@ -452,6 +512,12 @@ 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}
@@ -530,6 +596,12 @@ 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}
@@ -646,6 +718,12 @@ 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)
@@ -844,6 +922,12 @@ 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}.
diff --git a/theories/fp_red.v b/theories/fp_red.v
index d2fbdf6..9424da6 100644
--- a/theories/fp_red.v
+++ b/theories/fp_red.v
@@ -33,6 +33,8 @@ Module Par.
R b0 b1 ->
R c0 c1 ->
R (App (Pair a0 b0) c0) (Pair (App a1 c1) (App b1 c1))
+ | AppBool b a :
+ R (App (BVal b) a) (BVal b)
| ProjAbs p a0 a1 :
R a0 a1 ->
R (Proj p (Abs a0)) (Abs (Proj p a1))
@@ -40,7 +42,24 @@ Module Par.
R a0 a1 ->
R b0 b1 ->
R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1)
-
+ | ProjBool p b :
+ R (Proj p (BVal b)) (BVal b)
+ | IfAbs a0 a1 b0 b1 c0 c1 d0 d1 :
+ R a0 a1 ->
+ R b0 b1 ->
+ R c0 c1 ->
+ R d0 d1 ->
+ R (If (Abs a0) b0 c0) (Abs (If a0 (ren_Tm shift b0) (ren_Tm shift c0)))
+ | IfPair a0 a1 b0 b1 c0 c1 d0 d1 :
+ R a0 a1 ->
+ R b0 b1 ->
+ R c0 c1 ->
+ R d0 d1 ->
+ R (If (Pair a0 b0) c0 d0) (Pair (If a0 c0 d0) (If b0 c0 d0))
+ | IfBool a b0 b1 c0 c1 :
+ R b0 b1 ->
+ R c0 c1 ->
+ R (If (BVal a) b0 c0) (if a then b1 else c1)
(****************** Eta ***********************)
| AppEta a0 a1 :
R a0 a1 ->
From bec803949f4da3522031a185cdfcd522be675b82 Mon Sep 17 00:00:00 2001
From: Yiyun Liu <liuyiyun@seas.upenn.edu>
Date: Fri, 10 Jan 2025 20:51:59 -0500
Subject: [PATCH 02/13] Finish Par and RPar lemmas
---
theories/fp_red.v | 117 +++++++++++++++++++++++++++++++++++++---------
1 file changed, 96 insertions(+), 21 deletions(-)
diff --git a/theories/fp_red.v b/theories/fp_red.v
index 9424da6..c8076e1 100644
--- a/theories/fp_red.v
+++ b/theories/fp_red.v
@@ -33,8 +33,8 @@ Module Par.
R b0 b1 ->
R c0 c1 ->
R (App (Pair a0 b0) c0) (Pair (App a1 c1) (App b1 c1))
- | AppBool b a :
- R (App (BVal b) a) (BVal b)
+ (* | AppBool b a : *)
+ (* R (App (BVal b) a) (BVal b) *)
| ProjAbs p a0 a1 :
R a0 a1 ->
R (Proj p (Abs a0)) (Abs (Proj p a1))
@@ -42,20 +42,20 @@ Module Par.
R a0 a1 ->
R b0 b1 ->
R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1)
- | ProjBool p b :
- R (Proj p (BVal b)) (BVal b)
- | IfAbs a0 a1 b0 b1 c0 c1 d0 d1 :
- R a0 a1 ->
- R b0 b1 ->
- R c0 c1 ->
- R d0 d1 ->
- R (If (Abs a0) b0 c0) (Abs (If a0 (ren_Tm shift b0) (ren_Tm shift c0)))
- | IfPair a0 a1 b0 b1 c0 c1 d0 d1 :
- R a0 a1 ->
- R b0 b1 ->
- R c0 c1 ->
- R d0 d1 ->
- R (If (Pair a0 b0) c0 d0) (Pair (If a0 c0 d0) (If b0 c0 d0))
+ (* | ProjBool p b : *)
+ (* R (Proj p (BVal b)) (BVal b) *)
+ (* | IfAbs a0 a1 b0 b1 c0 c1 d0 d1 : *)
+ (* R a0 a1 -> *)
+ (* R b0 b1 -> *)
+ (* R c0 c1 -> *)
+ (* R d0 d1 -> *)
+ (* R (If (Abs a0) b0 c0) (Abs (If a0 (ren_Tm shift b0) (ren_Tm shift c0))) *)
+ (* | IfPair a0 a1 b0 b1 c0 c1 d0 d1 : *)
+ (* R a0 a1 -> *)
+ (* R b0 b1 -> *)
+ (* R c0 c1 -> *)
+ (* R d0 d1 -> *)
+ (* R (If (Pair a0 b0) c0 d0) (Pair (If a0 c0 d0) (If b0 c0 d0)) *)
| IfBool a b0 b1 c0 c1 :
R b0 b1 ->
R c0 c1 ->
@@ -92,7 +92,16 @@ Module Par.
| BotCong :
R Bot Bot
| UnivCong i :
- R (Univ i) (Univ i).
+ R (Univ i) (Univ i)
+ | BoolCong :
+ R Bool Bool
+ | BValCong b :
+ R (BVal b) (BVal b)
+ | IfCong a0 a1 b0 b1 c0 c1 :
+ R a0 a1 ->
+ R b0 b1 ->
+ R c0 c1 ->
+ R (If a0 b0 c0) (If a1 b1 c1).
Lemma refl n (a : Tm n) : R a a.
elim : n /a; hauto ctrs:R.
@@ -112,6 +121,13 @@ Module Par.
R (Proj p (Pair a0 b0)) t.
Proof. move => > ->. apply ProjPair. Qed.
+ Lemma IfBool' n a b0 b1 c0 c1 u :
+ u = (if a then (b1 : Tm n) else c1) ->
+ R b0 b1 ->
+ R c0 c1 ->
+ R (If (BVal a) b0 c0) u.
+ Proof. move => ->. apply IfBool. Qed.
+
Lemma AppEta' n (a0 a1 b : Tm n) :
b = (Abs (App (ren_Tm shift a1) (VarTm var_zero))) ->
R a0 a1 ->
@@ -126,7 +142,7 @@ Module Par.
move => *; apply : AppAbs'; eauto; by asimpl.
all : match goal with
| [ |- context[var_zero]] => move => *; apply : AppEta'; eauto; by asimpl
- | _ => qauto ctrs:R use:ProjPair'
+ | _ => qauto ctrs:R use:ProjPair', IfBool'
end.
Qed.
@@ -143,6 +159,7 @@ Module Par.
- hauto lq:on rew:off ctrs:R.
- hauto l:on inv:option use:renaming ctrs:R.
- hauto lq:on use:ProjPair'.
+ - hauto q:on use:IfBool'.
- move => n a0 a1 ha iha m ρ0 ρ1 hρ /=.
apply : AppEta'; eauto. by asimpl.
- hauto lq:on ctrs:R.
@@ -154,6 +171,9 @@ Module Par.
- hauto l:on inv:option ctrs:R use:renaming.
- sfirstorder.
- sfirstorder.
+ - sfirstorder.
+ - sfirstorder.
+ - hauto lq:on ctrs:R.
Qed.
Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
@@ -193,6 +213,12 @@ Module Par.
move : ihb => [c0 [? ?]]. subst.
eexists. split. by eauto using ProjPair.
hauto q:on.
+ - move => n a b0 b1 c0 c1 hb ihb hc ihc m ξ []//= p0 p1 p2 [h *]. subst.
+ spec_refl.
+ move : ihb => [b0 [? ?]].
+ move : ihc => [c0 [? ?]]. subst.
+ case : p0 h => //=.
+ hauto q:on use:IfBool'.
- move => n a0 a1 ha iha m ξ a ?. subst.
spec_refl. move : iha => [a0 [? ?]]. subst.
eexists. split. apply AppEta; eauto.
@@ -233,6 +259,11 @@ Module Par.
by asimpl.
- move => n n0 ξ []//=. hauto l:on.
- move => n i n0 ξ []//=. hauto l:on.
+ - move => n m ξ []//=. hauto l:on.
+ - move => n b m ξ []//=. hauto l:on.
+ - move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc m ξ []//= t t0 t1[*]. subst.
+ spec_refl.
+ qauto l:on ctrs:R.
Qed.
End Par.
@@ -321,6 +352,10 @@ Module RPar.
R a0 a1 ->
R b0 b1 ->
R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1)
+ | IfBool a b0 b1 c0 c1 :
+ R b0 b1 ->
+ R c0 c1 ->
+ R (If (BVal a) b0 c0) (if a then b1 else c1)
(*************** Congruence ********************)
@@ -346,7 +381,16 @@ Module RPar.
| BotCong :
R Bot Bot
| UnivCong i :
- R (Univ i) (Univ i).
+ R (Univ i) (Univ i)
+ | BoolCong :
+ R Bool Bool
+ | BValCong b :
+ R (BVal b) (BVal b)
+ | IfCong a0 a1 b0 b1 c0 c1 :
+ R a0 a1 ->
+ R b0 b1 ->
+ R c0 c1 ->
+ R (If a0 b0 c0) (If a1 b1 c1).
Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
@@ -369,13 +413,21 @@ Module RPar.
R (Proj p (Pair a0 b0)) t.
Proof. move => > ->. apply ProjPair. Qed.
+ Lemma IfBool' n a b0 b1 c0 c1 u :
+ u = (if a then (b1 : Tm n) else c1) ->
+ R b0 b1 ->
+ R c0 c1 ->
+ R (If (BVal a) b0 c0) u.
+ Proof. move => ->. apply IfBool. Qed.
+
+
Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
Proof.
move => h. move : m ξ.
elim : n a b /h.
move => *; apply : AppAbs'; eauto; by asimpl.
- all : qauto ctrs:R use:ProjPair'.
+ all : qauto ctrs:R use:ProjPair', IfBool'.
Qed.
Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> Tm m) (ξ : fin m -> fin p) :
@@ -406,6 +458,7 @@ Module RPar.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R use:morphing_up.
- hauto lq:on ctrs:R use:ProjPair' use:morphing_up.
+ - hauto lq:on ctrs:R use:IfBool' use:morphing_up.
- hauto lq:on ctrs:R use:morphing_up.
- hauto lq:on ctrs:R use:morphing_up.
- hauto lq:on ctrs:R use:morphing_up.
@@ -414,6 +467,9 @@ Module RPar.
- hauto lq:on ctrs:R use:morphing_up.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R.
+ - hauto lq:on ctrs:R.
+ - hauto lq:on ctrs:R.
+ - hauto lq:on ctrs:R.
Qed.
Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
@@ -434,7 +490,7 @@ Module RPar.
var_or_bot a ->
a = b -> ~~ var_or_bot b -> False.
Proof.
- hauto lq:on inv:Tm.
+ hauto l:on inv:Tm.
Qed.
Lemma var_or_bot_up n m (ρ : fin n -> Tm m) :
@@ -504,6 +560,14 @@ Module RPar.
move : ihb => [c0 [? ?]]. subst.
eexists. split. by eauto using ProjPair.
hauto q:on.
+ - move => n a b0 b1 c0 c1 hb ihb hc ihc m ρ hρ []//=; first by antiimp.
+ move => t t0 t1 [h *]. subst.
+ case : t h => //=; first by antiimp.
+ move => b [*]. subst.
+ have {}/ihb := (hρ) => ihb.
+ have {}/ihc := (hρ) => ihc.
+ spec_refl.
+ hauto q:on use:IfBool.
- move => n i m ρ hρ []//=.
hauto l:on.
- move => n a0 a1 ha iha m ρ hρ []//=; first by antiimp.
@@ -553,6 +617,17 @@ Module RPar.
- hauto q:on ctrs:R inv:Tm.
- move => n i n0 ρ hρ []//=; first by antiimp.
hauto l:on.
+ - move => n m ρ hρ []//=; first by antiimp.
+ hauto lq:on ctrs:R.
+ - move => n b m ρ hρ []//; first by antiimp.
+ hauto lq:on ctrs:R.
+ - move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc m ρ hρ []//=; first by antiimp.
+ move => t t0 t1 [*]. subst.
+ have {}/iha := (hρ) => iha.
+ have {}/ihb := (hρ) => ihb.
+ have {}/ihc := (hρ) => ihc.
+ spec_refl.
+ hauto lq:on ctrs:R.
Qed.
End RPar.
From 53bef034ad74a93ff6b98e330b48c4139e4f3962 Mon Sep 17 00:00:00 2001
From: Yiyun Liu <liuyiyun@seas.upenn.edu>
Date: Fri, 10 Jan 2025 22:28:53 -0500
Subject: [PATCH 03/13] Add the "junk" rules for boolean
---
theories/fp_red.v | 897 ++++++++++++++++++++++++++--------------------
1 file changed, 509 insertions(+), 388 deletions(-)
diff --git a/theories/fp_red.v b/theories/fp_red.v
index c8076e1..a8c8a3a 100644
--- a/theories/fp_red.v
+++ b/theories/fp_red.v
@@ -33,8 +33,8 @@ Module Par.
R b0 b1 ->
R c0 c1 ->
R (App (Pair a0 b0) c0) (Pair (App a1 c1) (App b1 c1))
- (* | AppBool b a : *)
- (* R (App (BVal b) a) (BVal b) *)
+ | AppBool b a :
+ R (App (BVal b) a) (BVal b)
| ProjAbs p a0 a1 :
R a0 a1 ->
R (Proj p (Abs a0)) (Abs (Proj p a1))
@@ -42,20 +42,19 @@ Module Par.
R a0 a1 ->
R b0 b1 ->
R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1)
- (* | ProjBool p b : *)
- (* R (Proj p (BVal b)) (BVal b) *)
- (* | IfAbs a0 a1 b0 b1 c0 c1 d0 d1 : *)
- (* R a0 a1 -> *)
- (* R b0 b1 -> *)
- (* R c0 c1 -> *)
- (* R d0 d1 -> *)
- (* R (If (Abs a0) b0 c0) (Abs (If a0 (ren_Tm shift b0) (ren_Tm shift c0))) *)
- (* | IfPair a0 a1 b0 b1 c0 c1 d0 d1 : *)
- (* R a0 a1 -> *)
- (* R b0 b1 -> *)
- (* R c0 c1 -> *)
- (* R d0 d1 -> *)
- (* R (If (Pair a0 b0) c0 d0) (Pair (If a0 c0 d0) (If b0 c0 d0)) *)
+ | ProjBool p b :
+ R (Proj p (BVal b)) (BVal b)
+ | IfAbs (a0 a1 : Tm (S n)) b0 b1 c0 c1 :
+ R a0 a1 ->
+ R b0 b1 ->
+ R c0 c1 ->
+ R (If (Abs a0) b0 c0) (Abs (If a1 (ren_Tm shift b1) (ren_Tm shift c1)))
+ | IfPair a0 a1 b0 b1 c0 c1 d0 d1 :
+ R a0 a1 ->
+ R b0 b1 ->
+ R c0 c1 ->
+ R d0 d1 ->
+ R (If (Pair a0 b0) c0 d0) (Pair (If a1 c1 d1) (If b1 c1 d1))
| IfBool a b0 b1 c0 c1 :
R b0 b1 ->
R c0 c1 ->
@@ -134,6 +133,14 @@ Module Par.
R a0 b.
Proof. move => ->; apply AppEta. Qed.
+ Lemma IfAbs' n (a0 a1 : Tm (S n)) (b0 b1 c0 c1 : Tm n) u :
+ u = (Abs (If a1 (ren_Tm shift b1) (ren_Tm shift c1))) ->
+ @R (S n) a0 a1 ->
+ R b0 b1 ->
+ R c0 c1 ->
+ R (If (Abs a0) b0 c0) u.
+ Proof. move => ->. apply IfAbs. Qed.
+
Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
Proof.
@@ -142,11 +149,11 @@ Module Par.
move => *; apply : AppAbs'; eauto; by asimpl.
all : match goal with
| [ |- context[var_zero]] => move => *; apply : AppEta'; eauto; by asimpl
+ | [ |- context[ren_Tm]] => move => * /=; apply : IfAbs'; eauto; by asimpl
| _ => qauto ctrs:R use:ProjPair', IfBool'
end.
Qed.
-
Lemma morphing n m (a b : Tm n) (ρ0 ρ1 : fin n -> Tm m) :
(forall i, R (ρ0 i) (ρ1 i)) ->
R a b -> R (subst_Tm ρ0 a) (subst_Tm ρ1 b).
@@ -157,8 +164,14 @@ Module Par.
by asimpl.
hauto l:on use:renaming inv:option.
- hauto lq:on rew:off ctrs:R.
+ - hauto lq:on rew:off ctrs:R.
- hauto l:on inv:option use:renaming ctrs:R.
- hauto lq:on use:ProjPair'.
+ - hauto lq:on rew:off ctrs:R.
+ - move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc m ρ0 ρ1 hρ /=.
+ eapply IfAbs' with (a1 := (subst_Tm (up_Tm_Tm ρ1) a1)); eauto. by asimpl.
+ hauto l:on use:renaming inv:option.
+ - qauto l:on ctrs:R.
- hauto q:on use:IfBool'.
- move => n a0 a1 ha iha m ρ0 ρ1 hρ /=.
apply : AppEta'; eauto. by asimpl.
@@ -204,6 +217,8 @@ Module Par.
eexists. split.
apply AppPair; hauto. subst.
by asimpl.
+ - move => n b a m ξ []//=.
+ hauto q:on ctrs:R inv:Tm.
- move => n p a0 a1 ha iha m ξ []//= p0 []//= t [*]. subst.
spec_refl. move : iha => [b0 [? ?]]. subst.
eexists. split. apply ProjAbs; eauto. by asimpl.
@@ -213,6 +228,23 @@ Module Par.
move : ihb => [c0 [? ?]]. subst.
eexists. split. by eauto using ProjPair.
hauto q:on.
+ - move => n p b m ξ []//=.
+ move => + []//=.
+ hauto lq:on ctrs:R.
+ - move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc m ξ []//= t t0 t1 [h *]. subst.
+ case : t h => // t [*]. subst.
+ spec_refl.
+ move : iha => [a2 [iha ?]]. subst.
+ move : ihb => [b2 [ihb ?]]. subst.
+ move : ihc => [c2 [ihc ?]]. subst.
+ exists (Abs (If a2 (ren_Tm shift b2) (ren_Tm shift c2))).
+ split; last by asimpl.
+ hauto lq:on ctrs:R.
+ - move => n a0 a1 b0 b1 c0 c1 d0 d1 ha iha hb ihb hc ihc hd ihd m ξ []//=.
+ move => a2 b2 c2 [h *]. subst.
+ case : a2 h => //= a3 b3 [*]. subst.
+ spec_refl.
+ hauto lq:on ctrs:R.
- move => n a b0 b1 c0 c1 hb ihb hc ihc m ξ []//= p0 p1 p2 [h *]. subst.
spec_refl.
move : ihb => [b0 [? ?]].
@@ -345,6 +377,8 @@ Module RPar.
R b0 b1 ->
R c0 c1 ->
R (App (Pair a0 b0) c0) (Pair (App a1 c1) (App b1 c1))
+ | AppBool b a :
+ R (App (BVal b) a) (BVal b)
| ProjAbs p a0 a1 :
R a0 a1 ->
R (Proj p (Abs a0)) (Abs (Proj p a1))
@@ -352,6 +386,19 @@ Module RPar.
R a0 a1 ->
R b0 b1 ->
R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1)
+ | ProjBool p b :
+ R (Proj p (BVal b)) (BVal b)
+ | IfAbs (a0 a1 : Tm (S n)) b0 b1 c0 c1 :
+ R a0 a1 ->
+ R b0 b1 ->
+ R c0 c1 ->
+ R (If (Abs a0) b0 c0) (Abs (If a1 (ren_Tm shift b1) (ren_Tm shift c1)))
+ | IfPair a0 a1 b0 b1 c0 c1 d0 d1 :
+ R a0 a1 ->
+ R b0 b1 ->
+ R c0 c1 ->
+ R d0 d1 ->
+ R (If (Pair a0 b0) c0 d0) (Pair (If a1 c1 d1) (If b1 c1 d1))
| IfBool a b0 b1 c0 c1 :
R b0 b1 ->
R c0 c1 ->
@@ -421,13 +468,20 @@ Module RPar.
Proof. move => ->. apply IfBool. Qed.
+ Lemma IfAbs' n (a0 a1 : Tm (S n)) (b0 b1 c0 c1 : Tm n) u :
+ u = (Abs (If a1 (ren_Tm shift b1) (ren_Tm shift c1))) ->
+ @R (S n) a0 a1 ->
+ R b0 b1 ->
+ R c0 c1 ->
+ R (If (Abs a0) b0 c0) u.
+ Proof. move => ->. apply IfAbs. Qed.
+
Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
Proof.
move => h. move : m ξ.
- elim : n a b /h.
- move => *; apply : AppAbs'; eauto; by asimpl.
- all : qauto ctrs:R use:ProjPair', IfBool'.
+ elim : n a b /h; try solve [(move => *; apply : AppAbs'; eauto; by asimpl) | (move => *; apply : IfAbs'; eauto; by asimpl)].
+ all : try qauto ctrs:R use:ProjPair', IfBool'.
Qed.
Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> Tm m) (ξ : fin m -> fin p) :
@@ -456,8 +510,14 @@ Module RPar.
apply : AppAbs'; eauto using morphing_up.
by asimpl.
- hauto lq:on ctrs:R.
+ - hauto lq:on ctrs:R.
- hauto lq:on ctrs:R use:morphing_up.
- hauto lq:on ctrs:R use:ProjPair' use:morphing_up.
+ - sfirstorder.
+ - move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc m ρ0 ρ1 hρ /=.
+ apply : IfAbs'; eauto using morphing_up.
+ by asimpl.
+ - hauto lq:on ctrs:R.
- hauto lq:on ctrs:R use:IfBool' use:morphing_up.
- hauto lq:on ctrs:R use:morphing_up.
- hauto lq:on ctrs:R use:morphing_up.
@@ -543,6 +603,9 @@ Module RPar.
eexists. split.
apply AppPair; hauto. subst.
by asimpl.
+ - move => n b a m ρ hρ []//=; first by antiimp.
+ case => //=; first by antiimp.
+ hauto lq:on ctrs:R.
- move => n p a0 a1 ha iha m ρ hρ []//=;
first by antiimp.
move => p0 []//= t [*]; first by antiimp. subst.
@@ -560,6 +623,31 @@ Module RPar.
move : ihb => [c0 [? ?]]. subst.
eexists. split. by eauto using ProjPair.
hauto q:on.
+ - move => n p b m ρ hρ []//=; first by antiimp.
+ move => p0 + []//=. case => //=; first by antiimp.
+ hauto lq:on ctrs:R.
+ - move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc m ρ hρ []//=; first by antiimp.
+ move => + b2 c2 [+ *]. subst. case => //=; first by antiimp.
+ move => a [*]. subst.
+ have /var_or_bot_up hρ' := hρ.
+ move : iha hρ' => /[apply] iha.
+ move : ihb (hρ) => /[apply] ihb.
+ move : ihc hρ => /[apply] ihc.
+ spec_refl.
+ move : iha => [a0 [ha0 ?]]. subst.
+ move : ihb => [b0 [hb0 ?]]. subst.
+ move : ihc => [c0 [hc0 ?]]. subst.
+ exists (Abs (If a0 (ren_Tm shift b0) (ren_Tm shift c0))). split.
+ hauto lq:on ctrs:R.
+ by asimpl.
+ - move => n a0 a1 b0 b1 c0 c1 d0 d1 ha iha hb ihb hc ihc hd ihd m ρ hρ []//=; first by antiimp.
+ move => a2 b2 c2 [+ *]. subst. case : a2 => //=; first by antiimp.
+ move => a3 b3 [*]. subst.
+ have {}/iha := (hρ) => iha.
+ have {}/ihb := (hρ) => ihb.
+ have {}/ihc := (hρ) => ihc.
+ spec_refl.
+ hauto lq:on ctrs:R.
- move => n a b0 b1 c0 c1 hb ihb hc ihc m ρ hρ []//=; first by antiimp.
move => t t0 t1 [h *]. subst.
case : t h => //=; first by antiimp.
@@ -631,332 +719,332 @@ Module RPar.
Qed.
End RPar.
-(***************** Beta rules only ***********************)
-Module RPar'.
- Inductive R {n} : Tm n -> Tm n -> Prop :=
- (***************** Beta ***********************)
- | AppAbs a0 a1 b0 b1 :
- R a0 a1 ->
- R b0 b1 ->
- R (App (Abs a0) b0) (subst_Tm (scons b1 VarTm) a1)
- | ProjPair p a0 a1 b0 b1 :
- R a0 a1 ->
- R b0 b1 ->
- R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1)
+(* (***************** Beta rules only ***********************) *)
+(* Module RPar'. *)
+(* Inductive R {n} : Tm n -> Tm n -> Prop := *)
+(* (***************** Beta ***********************) *)
+(* | AppAbs a0 a1 b0 b1 : *)
+(* R a0 a1 -> *)
+(* R b0 b1 -> *)
+(* R (App (Abs a0) b0) (subst_Tm (scons b1 VarTm) a1) *)
+(* | ProjPair p a0 a1 b0 b1 : *)
+(* R a0 a1 -> *)
+(* R b0 b1 -> *)
+(* R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1) *)
- (*************** Congruence ********************)
- | Var i : R (VarTm i) (VarTm i)
- | AbsCong a0 a1 :
- R a0 a1 ->
- R (Abs a0) (Abs a1)
- | AppCong a0 a1 b0 b1 :
- R a0 a1 ->
- R b0 b1 ->
- R (App a0 b0) (App a1 b1)
- | PairCong a0 a1 b0 b1 :
- R a0 a1 ->
- R b0 b1 ->
- R (Pair a0 b0) (Pair a1 b1)
- | ProjCong p a0 a1 :
- R a0 a1 ->
- R (Proj p a0) (Proj p a1)
- | BindCong p A0 A1 B0 B1:
- R A0 A1 ->
- R B0 B1 ->
- R (TBind p A0 B0) (TBind p A1 B1)
- | BotCong :
- R Bot Bot
- | UnivCong i :
- R (Univ i) (Univ i).
+(* (*************** Congruence ********************) *)
+(* | Var i : R (VarTm i) (VarTm i) *)
+(* | AbsCong a0 a1 : *)
+(* R a0 a1 -> *)
+(* R (Abs a0) (Abs a1) *)
+(* | AppCong a0 a1 b0 b1 : *)
+(* R a0 a1 -> *)
+(* R b0 b1 -> *)
+(* R (App a0 b0) (App a1 b1) *)
+(* | PairCong a0 a1 b0 b1 : *)
+(* R a0 a1 -> *)
+(* R b0 b1 -> *)
+(* R (Pair a0 b0) (Pair a1 b1) *)
+(* | ProjCong p a0 a1 : *)
+(* R a0 a1 -> *)
+(* R (Proj p a0) (Proj p a1) *)
+(* | BindCong p A0 A1 B0 B1: *)
+(* R A0 A1 -> *)
+(* R B0 B1 -> *)
+(* R (TBind p A0 B0) (TBind p A1 B1) *)
+(* | BotCong : *)
+(* R Bot Bot *)
+(* | UnivCong i : *)
+(* R (Univ i) (Univ i). *)
- Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
+(* Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop. *)
- Lemma refl n (a : Tm n) : R a a.
- Proof.
- induction a; hauto lq:on ctrs:R.
- Qed.
+(* Lemma refl n (a : Tm n) : R a a. *)
+(* Proof. *)
+(* induction a; hauto lq:on ctrs:R. *)
+(* Qed. *)
- Lemma AppAbs' n a0 a1 (b0 b1 t : Tm n) :
- t = subst_Tm (scons b1 VarTm) a1 ->
- R a0 a1 ->
- R b0 b1 ->
- R (App (Abs a0) b0) t.
- Proof. move => ->. apply AppAbs. Qed.
+(* Lemma AppAbs' n a0 a1 (b0 b1 t : Tm n) : *)
+(* t = subst_Tm (scons b1 VarTm) a1 -> *)
+(* R a0 a1 -> *)
+(* R b0 b1 -> *)
+(* R (App (Abs a0) b0) t. *)
+(* Proof. move => ->. apply AppAbs. Qed. *)
- Lemma ProjPair' n p (a0 a1 b0 b1 : Tm n) t :
- t = (if p is PL then a1 else b1) ->
- R a0 a1 ->
- R b0 b1 ->
- R (Proj p (Pair a0 b0)) t.
- Proof. move => > ->. apply ProjPair. Qed.
+(* Lemma ProjPair' n p (a0 a1 b0 b1 : Tm n) t : *)
+(* t = (if p is PL then a1 else b1) -> *)
+(* R a0 a1 -> *)
+(* R b0 b1 -> *)
+(* R (Proj p (Pair a0 b0)) t. *)
+(* Proof. move => > ->. apply ProjPair. Qed. *)
- Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
- R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
- Proof.
- move => h. move : m ξ.
- elim : n a b /h.
- move => *; apply : AppAbs'; eauto; by asimpl.
- all : qauto ctrs:R use:ProjPair'.
- Qed.
+(* Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) : *)
+(* R a b -> R (ren_Tm ξ a) (ren_Tm ξ b). *)
+(* Proof. *)
+(* move => h. move : m ξ. *)
+(* elim : n a b /h. *)
+(* move => *; apply : AppAbs'; eauto; by asimpl. *)
+(* all : qauto ctrs:R use:ProjPair'. *)
+(* Qed. *)
- Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> Tm m) (ξ : fin m -> fin p) :
- (forall i, R (ρ0 i) (ρ1 i)) ->
- (forall i, R ((funcomp (ren_Tm ξ) ρ0) i) ((funcomp (ren_Tm ξ) ρ1) i)).
- Proof. eauto using renaming. Qed.
+(* Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> Tm m) (ξ : fin m -> fin p) : *)
+(* (forall i, R (ρ0 i) (ρ1 i)) -> *)
+(* (forall i, R ((funcomp (ren_Tm ξ) ρ0) i) ((funcomp (ren_Tm ξ) ρ1) i)). *)
+(* Proof. eauto using renaming. Qed. *)
- Lemma morphing_ext n m (ρ0 ρ1 : fin n -> Tm m) a b :
- R a b ->
- (forall i, R (ρ0 i) (ρ1 i)) ->
- (forall i, R ((scons a ρ0) i) ((scons b ρ1) i)).
- Proof. hauto q:on inv:option. Qed.
+(* Lemma morphing_ext n m (ρ0 ρ1 : fin n -> Tm m) a b : *)
+(* R a b -> *)
+(* (forall i, R (ρ0 i) (ρ1 i)) -> *)
+(* (forall i, R ((scons a ρ0) i) ((scons b ρ1) i)). *)
+(* Proof. hauto q:on inv:option. Qed. *)
- Lemma morphing_up n m (ρ0 ρ1 : fin n -> Tm m) :
- (forall i, R (ρ0 i) (ρ1 i)) ->
- (forall i, R (up_Tm_Tm ρ0 i) (up_Tm_Tm ρ1 i)).
- Proof. hauto l:on ctrs:R use:morphing_ext, morphing_ren unfold:up_Tm_Tm. Qed.
+(* Lemma morphing_up n m (ρ0 ρ1 : fin n -> Tm m) : *)
+(* (forall i, R (ρ0 i) (ρ1 i)) -> *)
+(* (forall i, R (up_Tm_Tm ρ0 i) (up_Tm_Tm ρ1 i)). *)
+(* Proof. hauto l:on ctrs:R use:morphing_ext, morphing_ren unfold:up_Tm_Tm. Qed. *)
- Lemma morphing n m (a b : Tm n) (ρ0 ρ1 : fin n -> Tm m) :
- (forall i, R (ρ0 i) (ρ1 i)) ->
- R a b -> R (subst_Tm ρ0 a) (subst_Tm ρ1 b).
- Proof.
- move => + h. move : m ρ0 ρ1.
- elim : n a b /h.
- - move => *.
- apply : AppAbs'; eauto using morphing_up.
- by asimpl.
- - hauto lq:on ctrs:R use:ProjPair' use:morphing_up.
- - hauto lq:on ctrs:R use:morphing_up.
- - hauto lq:on ctrs:R use:morphing_up.
- - hauto lq:on ctrs:R use:morphing_up.
- - hauto lq:on ctrs:R.
- - hauto lq:on ctrs:R.
- - hauto lq:on ctrs:R use:morphing_up.
- - hauto lq:on ctrs:R.
- - hauto lq:on ctrs:R.
- Qed.
+(* Lemma morphing n m (a b : Tm n) (ρ0 ρ1 : fin n -> Tm m) : *)
+(* (forall i, R (ρ0 i) (ρ1 i)) -> *)
+(* R a b -> R (subst_Tm ρ0 a) (subst_Tm ρ1 b). *)
+(* Proof. *)
+(* move => + h. move : m ρ0 ρ1. *)
+(* elim : n a b /h. *)
+(* - move => *. *)
+(* apply : AppAbs'; eauto using morphing_up. *)
+(* by asimpl. *)
+(* - hauto lq:on ctrs:R use:ProjPair' use:morphing_up. *)
+(* - hauto lq:on ctrs:R use:morphing_up. *)
+(* - hauto lq:on ctrs:R use:morphing_up. *)
+(* - hauto lq:on ctrs:R use:morphing_up. *)
+(* - hauto lq:on ctrs:R. *)
+(* - hauto lq:on ctrs:R. *)
+(* - hauto lq:on ctrs:R use:morphing_up. *)
+(* - hauto lq:on ctrs:R. *)
+(* - hauto lq:on ctrs:R. *)
+(* Qed. *)
- Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
- R a b ->
- R (subst_Tm ρ a) (subst_Tm ρ b).
- Proof. hauto l:on use:morphing, refl. Qed.
+(* Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) : *)
+(* R a b -> *)
+(* R (subst_Tm ρ a) (subst_Tm ρ b). *)
+(* Proof. hauto l:on use:morphing, refl. Qed. *)
- Lemma cong n (a b : Tm (S n)) c d :
- R a b ->
- R c d ->
- R (subst_Tm (scons c VarTm) a) (subst_Tm (scons d VarTm) b).
- Proof.
- move => h0 h1. apply morphing => //=.
- qauto l:on ctrs:R inv:option.
- Qed.
+(* Lemma cong n (a b : Tm (S n)) c d : *)
+(* R a b -> *)
+(* R c d -> *)
+(* R (subst_Tm (scons c VarTm) a) (subst_Tm (scons d VarTm) b). *)
+(* Proof. *)
+(* move => h0 h1. apply morphing => //=. *)
+(* qauto l:on ctrs:R inv:option. *)
+(* Qed. *)
- Lemma var_or_bot_imp {n} (a b : Tm n) :
- var_or_bot a ->
- a = b -> ~~ var_or_bot b -> False.
- Proof.
- hauto lq:on inv:Tm.
- Qed.
+(* Lemma var_or_bot_imp {n} (a b : Tm n) : *)
+(* var_or_bot a -> *)
+(* a = b -> ~~ var_or_bot b -> False. *)
+(* Proof. *)
+(* hauto lq:on inv:Tm. *)
+(* Qed. *)
- Lemma var_or_bot_up n m (ρ : fin n -> Tm m) :
- (forall i, var_or_bot (ρ i)) ->
- (forall i, var_or_bot (up_Tm_Tm ρ i)).
- Proof.
- move => h /= [i|].
- - asimpl.
- move /(_ i) in h.
- rewrite /funcomp.
- move : (ρ i) h.
- case => //=.
- - sfirstorder.
- Qed.
+(* Lemma var_or_bot_up n m (ρ : fin n -> Tm m) : *)
+(* (forall i, var_or_bot (ρ i)) -> *)
+(* (forall i, var_or_bot (up_Tm_Tm ρ i)). *)
+(* Proof. *)
+(* move => h /= [i|]. *)
+(* - asimpl. *)
+(* move /(_ i) in h. *)
+(* rewrite /funcomp. *)
+(* move : (ρ i) h. *)
+(* case => //=. *)
+(* - sfirstorder. *)
+(* Qed. *)
- Local Ltac antiimp := qauto l:on use:var_or_bot_imp.
+(* Local Ltac antiimp := qauto l:on use:var_or_bot_imp. *)
- Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
- (forall i, var_or_bot (ρ i)) ->
- R (subst_Tm ρ a) b -> exists b0, R a b0 /\ subst_Tm ρ b0 = b.
- Proof.
- move E : (subst_Tm ρ a) => u hρ h.
- move : n ρ hρ a E. elim : m u b/h.
- - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
- first by antiimp.
- move => c c0 [+ ?]. subst.
- case : c => //=; first by antiimp.
- move => c [?]. subst.
- spec_refl.
- have /var_or_bot_up hρ' := hρ.
- move : iha hρ' => /[apply] iha.
- move : ihb hρ => /[apply] ihb.
- spec_refl.
- move : iha => [c1][ih0]?. subst.
- move : ihb => [c2][ih1]?. subst.
- eexists. split.
- apply AppAbs; eauto.
- by asimpl.
- - move => n p a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
- first by antiimp.
- move => p0 []//=; first by antiimp. move => t t0[*].
- subst.
- have {}/iha := (hρ) => iha.
- have {}/ihb := (hρ) => ihb.
- spec_refl.
- move : iha => [b0 [? ?]].
- move : ihb => [c0 [? ?]]. subst.
- eexists. split. by eauto using ProjPair.
- hauto q:on.
- - move => n i m ρ hρ []//=.
- hauto l:on.
- - move => n a0 a1 ha iha m ρ hρ []//=; first by antiimp.
- move => t [*]. subst.
- have /var_or_bot_up {}/iha := hρ => iha.
- spec_refl.
- move :iha => [b0 [? ?]]. subst.
- eexists. split. by apply AbsCong; eauto.
- by asimpl.
- - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
- first by antiimp.
- move => t t0 [*]. subst.
- have {}/iha := (hρ) => iha.
- have {}/ihb := (hρ) => ihb.
- spec_refl.
- move : iha => [b0 [? ?]]. subst.
- move : ihb => [c0 [? ?]]. subst.
- eexists. split. by apply AppCong; eauto.
- done.
- - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
- first by antiimp.
- move => t t0[*]. subst.
- have {}/iha := (hρ) => iha.
- have {}/ihb := (hρ) => ihb.
- spec_refl.
- move : iha => [b0 [? ?]]. subst.
- move : ihb => [c0 [? ?]]. subst.
- eexists. split. by apply PairCong; eauto.
- by asimpl.
- - move => n p a0 a1 ha iha m ρ hρ []//=;
- first by antiimp.
- move => p0 t [*]. subst.
- have {}/iha := (hρ) => iha.
- spec_refl.
- move : iha => [b0 [? ?]]. subst.
- eexists. split. apply ProjCong; eauto. reflexivity.
- - move => n p A0 A1 B0 B1 ha iha hB ihB m ρ hρ []//=;
- first by antiimp.
- move => ? t t0 [*]. subst.
- have {}/iha := (hρ) => iha.
- have /var_or_bot_up {}/ihB := (hρ) => ihB.
- spec_refl.
- move : iha => [b0 [? ?]].
- move : ihB => [c0 [? ?]]. subst.
- eexists. split. by apply BindCong; eauto.
- by asimpl.
- - hauto q:on ctrs:R inv:Tm.
- - move => n i n0 ρ hρ []//=; first by antiimp.
- hauto l:on.
- Qed.
-End RPar'.
+(* Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) : *)
+(* (forall i, var_or_bot (ρ i)) -> *)
+(* R (subst_Tm ρ a) b -> exists b0, R a b0 /\ subst_Tm ρ b0 = b. *)
+(* Proof. *)
+(* move E : (subst_Tm ρ a) => u hρ h. *)
+(* move : n ρ hρ a E. elim : m u b/h. *)
+(* - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=; *)
+(* first by antiimp. *)
+(* move => c c0 [+ ?]. subst. *)
+(* case : c => //=; first by antiimp. *)
+(* move => c [?]. subst. *)
+(* spec_refl. *)
+(* have /var_or_bot_up hρ' := hρ. *)
+(* move : iha hρ' => /[apply] iha. *)
+(* move : ihb hρ => /[apply] ihb. *)
+(* spec_refl. *)
+(* move : iha => [c1][ih0]?. subst. *)
+(* move : ihb => [c2][ih1]?. subst. *)
+(* eexists. split. *)
+(* apply AppAbs; eauto. *)
+(* by asimpl. *)
+(* - move => n p a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=; *)
+(* first by antiimp. *)
+(* move => p0 []//=; first by antiimp. move => t t0[*]. *)
+(* subst. *)
+(* have {}/iha := (hρ) => iha. *)
+(* have {}/ihb := (hρ) => ihb. *)
+(* spec_refl. *)
+(* move : iha => [b0 [? ?]]. *)
+(* move : ihb => [c0 [? ?]]. subst. *)
+(* eexists. split. by eauto using ProjPair. *)
+(* hauto q:on. *)
+(* - move => n i m ρ hρ []//=. *)
+(* hauto l:on. *)
+(* - move => n a0 a1 ha iha m ρ hρ []//=; first by antiimp. *)
+(* move => t [*]. subst. *)
+(* have /var_or_bot_up {}/iha := hρ => iha. *)
+(* spec_refl. *)
+(* move :iha => [b0 [? ?]]. subst. *)
+(* eexists. split. by apply AbsCong; eauto. *)
+(* by asimpl. *)
+(* - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=; *)
+(* first by antiimp. *)
+(* move => t t0 [*]. subst. *)
+(* have {}/iha := (hρ) => iha. *)
+(* have {}/ihb := (hρ) => ihb. *)
+(* spec_refl. *)
+(* move : iha => [b0 [? ?]]. subst. *)
+(* move : ihb => [c0 [? ?]]. subst. *)
+(* eexists. split. by apply AppCong; eauto. *)
+(* done. *)
+(* - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=; *)
+(* first by antiimp. *)
+(* move => t t0[*]. subst. *)
+(* have {}/iha := (hρ) => iha. *)
+(* have {}/ihb := (hρ) => ihb. *)
+(* spec_refl. *)
+(* move : iha => [b0 [? ?]]. subst. *)
+(* move : ihb => [c0 [? ?]]. subst. *)
+(* eexists. split. by apply PairCong; eauto. *)
+(* by asimpl. *)
+(* - move => n p a0 a1 ha iha m ρ hρ []//=; *)
+(* first by antiimp. *)
+(* move => p0 t [*]. subst. *)
+(* have {}/iha := (hρ) => iha. *)
+(* spec_refl. *)
+(* move : iha => [b0 [? ?]]. subst. *)
+(* eexists. split. apply ProjCong; eauto. reflexivity. *)
+(* - move => n p A0 A1 B0 B1 ha iha hB ihB m ρ hρ []//=; *)
+(* first by antiimp. *)
+(* move => ? t t0 [*]. subst. *)
+(* have {}/iha := (hρ) => iha. *)
+(* have /var_or_bot_up {}/ihB := (hρ) => ihB. *)
+(* spec_refl. *)
+(* move : iha => [b0 [? ?]]. *)
+(* move : ihB => [c0 [? ?]]. subst. *)
+(* eexists. split. by apply BindCong; eauto. *)
+(* by asimpl. *)
+(* - hauto q:on ctrs:R inv:Tm. *)
+(* - move => n i n0 ρ hρ []//=; first by antiimp. *)
+(* hauto l:on. *)
+(* Qed. *)
+(* End RPar'. *)
-Module ERed.
- Inductive R {n} : Tm n -> Tm n -> Prop :=
- (****************** Eta ***********************)
- | AppEta a :
- R a (Abs (App (ren_Tm shift a) (VarTm var_zero)))
- | PairEta a :
- R a (Pair (Proj PL a) (Proj PR a))
+(* Module ERed. *)
+(* Inductive R {n} : Tm n -> Tm n -> Prop := *)
+(* (****************** Eta ***********************) *)
+(* | AppEta a : *)
+(* R a (Abs (App (ren_Tm shift a) (VarTm var_zero))) *)
+(* | PairEta a : *)
+(* R a (Pair (Proj PL a) (Proj PR a)) *)
- (*************** Congruence ********************)
- | AbsCong a0 a1 :
- R a0 a1 ->
- R (Abs a0) (Abs a1)
- | AppCong0 a0 a1 b :
- R a0 a1 ->
- R (App a0 b) (App a1 b)
- | AppCong1 a b0 b1 :
- R b0 b1 ->
- R (App a b0) (App a b1)
- | PairCong0 a0 a1 b :
- R a0 a1 ->
- R (Pair a0 b) (Pair a1 b)
- | PairCong1 a b0 b1 :
- R b0 b1 ->
- R (Pair a b0) (Pair a b1)
- | ProjCong p a0 a1 :
- R a0 a1 ->
- R (Proj p a0) (Proj p a1)
- | BindCong0 p A0 A1 B:
- R A0 A1 ->
- R (TBind p A0 B) (TBind p A1 B)
- | BindCong1 p A B0 B1:
- R B0 B1 ->
- R (TBind p A B0) (TBind p A B1).
+(* (*************** Congruence ********************) *)
+(* | AbsCong a0 a1 : *)
+(* R a0 a1 -> *)
+(* R (Abs a0) (Abs a1) *)
+(* | AppCong0 a0 a1 b : *)
+(* R a0 a1 -> *)
+(* R (App a0 b) (App a1 b) *)
+(* | AppCong1 a b0 b1 : *)
+(* R b0 b1 -> *)
+(* R (App a b0) (App a b1) *)
+(* | PairCong0 a0 a1 b : *)
+(* R a0 a1 -> *)
+(* R (Pair a0 b) (Pair a1 b) *)
+(* | PairCong1 a b0 b1 : *)
+(* R b0 b1 -> *)
+(* R (Pair a b0) (Pair a b1) *)
+(* | ProjCong p a0 a1 : *)
+(* R a0 a1 -> *)
+(* R (Proj p a0) (Proj p a1) *)
+(* | BindCong0 p A0 A1 B: *)
+(* R A0 A1 -> *)
+(* R (TBind p A0 B) (TBind p A1 B) *)
+(* | BindCong1 p A B0 B1: *)
+(* R B0 B1 -> *)
+(* R (TBind p A B0) (TBind p A B1). *)
- Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
+(* Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop. *)
- Lemma AppEta' n a (u : Tm n) :
- u = (Abs (App (ren_Tm shift a) (VarTm var_zero))) ->
- R a u.
- Proof. move => ->. apply AppEta. Qed.
+(* Lemma AppEta' n a (u : Tm n) : *)
+(* u = (Abs (App (ren_Tm shift a) (VarTm var_zero))) -> *)
+(* R a u. *)
+(* Proof. move => ->. apply AppEta. Qed. *)
- Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
- R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
- Proof.
- move => h. move : m ξ.
- elim : n a b /h.
+(* Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) : *)
+(* R a b -> R (ren_Tm ξ a) (ren_Tm ξ b). *)
+(* Proof. *)
+(* move => h. move : m ξ. *)
+(* elim : n a b /h. *)
- move => n a m ξ.
- apply AppEta'. by asimpl.
- all : qauto ctrs:R.
- Qed.
+(* move => n a m ξ. *)
+(* apply AppEta'. by asimpl. *)
+(* all : qauto ctrs:R. *)
+(* Qed. *)
- Lemma substing n m (a : Tm n) b (ρ : fin n -> Tm m) :
- R a b ->
- R (subst_Tm ρ a) (subst_Tm ρ b).
- Proof.
- move => h. move : m ρ. elim : n a b / h => n.
- move => a m ρ /=.
- apply : AppEta'; eauto. by asimpl.
- all : hauto ctrs:R inv:option use:renaming.
- Qed.
+(* Lemma substing n m (a : Tm n) b (ρ : fin n -> Tm m) : *)
+(* R a b -> *)
+(* R (subst_Tm ρ a) (subst_Tm ρ b). *)
+(* Proof. *)
+(* move => h. move : m ρ. elim : n a b / h => n. *)
+(* move => a m ρ /=. *)
+(* apply : AppEta'; eauto. by asimpl. *)
+(* all : hauto ctrs:R inv:option use:renaming. *)
+(* Qed. *)
-End ERed.
+(* End ERed. *)
-Module EReds.
+(* Module EReds. *)
- #[local]Ltac solve_s_rec :=
- move => *; eapply rtc_l; eauto;
- hauto lq:on ctrs:ERed.R.
+(* #[local]Ltac solve_s_rec := *)
+(* move => *; eapply rtc_l; eauto; *)
+(* hauto lq:on ctrs:ERed.R. *)
- #[local]Ltac solve_s :=
- repeat (induction 1; last by solve_s_rec); apply rtc_refl.
+(* #[local]Ltac solve_s := *)
+(* repeat (induction 1; last by solve_s_rec); apply rtc_refl. *)
- Lemma AbsCong n (a b : Tm (S n)) :
- rtc ERed.R a b ->
- rtc ERed.R (Abs a) (Abs b).
- Proof. solve_s. Qed.
+(* Lemma AbsCong n (a b : Tm (S n)) : *)
+(* rtc ERed.R a b -> *)
+(* rtc ERed.R (Abs a) (Abs b). *)
+(* Proof. solve_s. Qed. *)
- Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
- rtc ERed.R a0 a1 ->
- rtc ERed.R b0 b1 ->
- rtc ERed.R (App a0 b0) (App a1 b1).
- Proof. solve_s. Qed.
+(* Lemma AppCong n (a0 a1 b0 b1 : Tm n) : *)
+(* rtc ERed.R a0 a1 -> *)
+(* rtc ERed.R b0 b1 -> *)
+(* rtc ERed.R (App a0 b0) (App a1 b1). *)
+(* Proof. solve_s. Qed. *)
- Lemma BindCong n p (a0 a1 : Tm n) b0 b1 :
- rtc ERed.R a0 a1 ->
- rtc ERed.R b0 b1 ->
- rtc ERed.R (TBind p a0 b0) (TBind p a1 b1).
- Proof. solve_s. Qed.
+(* Lemma BindCong n p (a0 a1 : Tm n) b0 b1 : *)
+(* rtc ERed.R a0 a1 -> *)
+(* rtc ERed.R b0 b1 -> *)
+(* rtc ERed.R (TBind p a0 b0) (TBind p a1 b1). *)
+(* Proof. solve_s. Qed. *)
- Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
- rtc ERed.R a0 a1 ->
- rtc ERed.R b0 b1 ->
- rtc ERed.R (Pair a0 b0) (Pair a1 b1).
- Proof. solve_s. Qed.
+(* Lemma PairCong n (a0 a1 b0 b1 : Tm n) : *)
+(* rtc ERed.R a0 a1 -> *)
+(* rtc ERed.R b0 b1 -> *)
+(* rtc ERed.R (Pair a0 b0) (Pair a1 b1). *)
+(* Proof. solve_s. Qed. *)
- Lemma ProjCong n p (a0 a1 : Tm n) :
- rtc ERed.R a0 a1 ->
- rtc ERed.R (Proj p a0) (Proj p a1).
- Proof. solve_s. Qed.
-End EReds.
+(* Lemma ProjCong n p (a0 a1 : Tm n) : *)
+(* rtc ERed.R a0 a1 -> *)
+(* rtc ERed.R (Proj p a0) (Proj p a1). *)
+(* Proof. solve_s. Qed. *)
+(* End EReds. *)
Module EPar.
Inductive R {n} : Tm n -> Tm n -> Prop :=
@@ -991,7 +1079,16 @@ Module EPar.
| BotCong :
R Bot Bot
| UnivCong i :
- R (Univ i) (Univ i).
+ R (Univ i) (Univ i)
+ | BoolCong :
+ R Bool Bool
+ | BValCong b :
+ R (BVal b) (BVal b)
+ | IfCong a0 a1 b0 b1 c0 c1 :
+ R a0 a1 ->
+ R b0 b1 ->
+ R c0 c1 ->
+ R (If a0 b0 c0) (If a1 b1 c1).
Lemma refl n (a : Tm n) : EPar.R a a.
Proof.
@@ -1036,6 +1133,9 @@ Module EPar.
- hauto l:on ctrs:R use:renaming inv:option.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R.
+ - hauto lq:on ctrs:R.
+ - hauto lq:on ctrs:R.
+ - hauto lq:on ctrs:R.
Qed.
Lemma substing n a0 a1 (b0 b1 : Tm n) :
@@ -1125,6 +1225,13 @@ Module RPars.
rtc RPar.R (Pair a0 b0) (Pair a1 b1).
Proof. solve_s. Qed.
+ Lemma IfCong n (a0 a1 b0 b1 c0 c1 : Tm n) :
+ rtc RPar.R a0 a1 ->
+ rtc RPar.R b0 b1 ->
+ rtc RPar.R c0 c1 ->
+ rtc RPar.R (If a0 b0 c0) (If a1 b1 c1).
+ Proof. solve_s. Qed.
+
Lemma ProjCong n p (a0 a1 : Tm n) :
rtc RPar.R a0 a1 ->
rtc RPar.R (Proj p a0) (Proj p a1).
@@ -1180,92 +1287,92 @@ Module RPars.
End RPars.
-Module RPars'.
+(* Module RPars'. *)
- #[local]Ltac solve_s_rec :=
- move => *; eapply rtc_l; eauto;
- hauto lq:on ctrs:RPar'.R use:RPar'.refl.
+(* #[local]Ltac solve_s_rec := *)
+(* move => *; eapply rtc_l; eauto; *)
+(* hauto lq:on ctrs:RPar'.R use:RPar'.refl. *)
- #[local]Ltac solve_s :=
- repeat (induction 1; last by solve_s_rec); apply rtc_refl.
+(* #[local]Ltac solve_s := *)
+(* repeat (induction 1; last by solve_s_rec); apply rtc_refl. *)
- Lemma AbsCong n (a b : Tm (S n)) :
- rtc RPar'.R a b ->
- rtc RPar'.R (Abs a) (Abs b).
- Proof. solve_s. Qed.
+(* Lemma AbsCong n (a b : Tm (S n)) : *)
+(* rtc RPar'.R a b -> *)
+(* rtc RPar'.R (Abs a) (Abs b). *)
+(* Proof. solve_s. Qed. *)
- Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
- rtc RPar'.R a0 a1 ->
- rtc RPar'.R b0 b1 ->
- rtc RPar'.R (App a0 b0) (App a1 b1).
- Proof. solve_s. Qed.
+(* Lemma AppCong n (a0 a1 b0 b1 : Tm n) : *)
+(* rtc RPar'.R a0 a1 -> *)
+(* rtc RPar'.R b0 b1 -> *)
+(* rtc RPar'.R (App a0 b0) (App a1 b1). *)
+(* Proof. solve_s. Qed. *)
- Lemma BindCong n p (a0 a1 : Tm n) b0 b1 :
- rtc RPar'.R a0 a1 ->
- rtc RPar'.R b0 b1 ->
- rtc RPar'.R (TBind p a0 b0) (TBind p a1 b1).
- Proof. solve_s. Qed.
+(* Lemma BindCong n p (a0 a1 : Tm n) b0 b1 : *)
+(* rtc RPar'.R a0 a1 -> *)
+(* rtc RPar'.R b0 b1 -> *)
+(* rtc RPar'.R (TBind p a0 b0) (TBind p a1 b1). *)
+(* Proof. solve_s. Qed. *)
- Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
- rtc RPar'.R a0 a1 ->
- rtc RPar'.R b0 b1 ->
- rtc RPar'.R (Pair a0 b0) (Pair a1 b1).
- Proof. solve_s. Qed.
+(* Lemma PairCong n (a0 a1 b0 b1 : Tm n) : *)
+(* rtc RPar'.R a0 a1 -> *)
+(* rtc RPar'.R b0 b1 -> *)
+(* rtc RPar'.R (Pair a0 b0) (Pair a1 b1). *)
+(* Proof. solve_s. Qed. *)
- Lemma ProjCong n p (a0 a1 : Tm n) :
- rtc RPar'.R a0 a1 ->
- rtc RPar'.R (Proj p a0) (Proj p a1).
- Proof. solve_s. Qed.
+(* Lemma ProjCong n p (a0 a1 : Tm n) : *)
+(* rtc RPar'.R a0 a1 -> *)
+(* rtc RPar'.R (Proj p a0) (Proj p a1). *)
+(* Proof. solve_s. Qed. *)
- Lemma renaming n (a0 a1 : Tm n) m (ξ : fin n -> fin m) :
- rtc RPar'.R a0 a1 ->
- rtc RPar'.R (ren_Tm ξ a0) (ren_Tm ξ a1).
- Proof.
- induction 1.
- - apply rtc_refl.
- - eauto using RPar'.renaming, rtc_l.
- Qed.
+(* Lemma renaming n (a0 a1 : Tm n) m (ξ : fin n -> fin m) : *)
+(* rtc RPar'.R a0 a1 -> *)
+(* rtc RPar'.R (ren_Tm ξ a0) (ren_Tm ξ a1). *)
+(* Proof. *)
+(* induction 1. *)
+(* - apply rtc_refl. *)
+(* - eauto using RPar'.renaming, rtc_l. *)
+(* Qed. *)
- Lemma weakening n (a0 a1 : Tm n) :
- rtc RPar'.R a0 a1 ->
- rtc RPar'.R (ren_Tm shift a0) (ren_Tm shift a1).
- Proof. apply renaming. Qed.
+(* Lemma weakening n (a0 a1 : Tm n) : *)
+(* rtc RPar'.R a0 a1 -> *)
+(* rtc RPar'.R (ren_Tm shift a0) (ren_Tm shift a1). *)
+(* Proof. apply renaming. Qed. *)
- Lemma Abs_inv n (a : Tm (S n)) b :
- rtc RPar'.R (Abs a) b -> exists a', b = Abs a' /\ rtc RPar'.R a a'.
- Proof.
- move E : (Abs a) => b0 h. move : a E.
- elim : b0 b / h.
- - hauto lq:on ctrs:rtc.
- - hauto lq:on ctrs:rtc inv:RPar'.R, rtc.
- Qed.
+(* Lemma Abs_inv n (a : Tm (S n)) b : *)
+(* rtc RPar'.R (Abs a) b -> exists a', b = Abs a' /\ rtc RPar'.R a a'. *)
+(* Proof. *)
+(* move E : (Abs a) => b0 h. move : a E. *)
+(* elim : b0 b / h. *)
+(* - hauto lq:on ctrs:rtc. *)
+(* - hauto lq:on ctrs:rtc inv:RPar'.R, rtc. *)
+(* Qed. *)
- Lemma morphing n m (a b : Tm n) (ρ : fin n -> Tm m) :
- rtc RPar'.R a b ->
- rtc RPar'.R (subst_Tm ρ a) (subst_Tm ρ b).
- Proof. induction 1; qauto l:on ctrs:rtc use:RPar'.substing. Qed.
+(* Lemma morphing n m (a b : Tm n) (ρ : fin n -> Tm m) : *)
+(* rtc RPar'.R a b -> *)
+(* rtc RPar'.R (subst_Tm ρ a) (subst_Tm ρ b). *)
+(* Proof. induction 1; qauto l:on ctrs:rtc use:RPar'.substing. Qed. *)
- Lemma substing n (a b : Tm (S n)) c :
- rtc RPar'.R a b ->
- rtc RPar'.R (subst_Tm (scons c VarTm) a) (subst_Tm (scons c VarTm) b).
- Proof. hauto lq:on use:morphing inv:option. Qed.
+(* Lemma substing n (a b : Tm (S n)) c : *)
+(* rtc RPar'.R a b -> *)
+(* rtc RPar'.R (subst_Tm (scons c VarTm) a) (subst_Tm (scons c VarTm) b). *)
+(* Proof. hauto lq:on use:morphing inv:option. Qed. *)
- Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
- (forall i, var_or_bot (ρ i)) ->
- rtc RPar'.R (subst_Tm ρ a) b -> exists b0, rtc RPar'.R a b0 /\ subst_Tm ρ b0 = b.
- Proof.
- move E :(subst_Tm ρ a) => u hρ h.
- move : a E.
- elim : u b /h.
- - sfirstorder.
- - move => a b c h0 h1 ih1 a0 ?. subst.
- move /RPar'.antirenaming : h0.
- move /(_ hρ).
- move => [b0 [h2 ?]]. subst.
- hauto lq:on rew:off ctrs:rtc.
- Qed.
+(* Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) : *)
+(* (forall i, var_or_bot (ρ i)) -> *)
+(* rtc RPar'.R (subst_Tm ρ a) b -> exists b0, rtc RPar'.R a b0 /\ subst_Tm ρ b0 = b. *)
+(* Proof. *)
+(* move E :(subst_Tm ρ a) => u hρ h. *)
+(* move : a E. *)
+(* elim : u b /h. *)
+(* - sfirstorder. *)
+(* - move => a b c h0 h1 ih1 a0 ?. subst. *)
+(* move /RPar'.antirenaming : h0. *)
+(* move /(_ hρ). *)
+(* move => [b0 [h2 ?]]. subst. *)
+(* hauto lq:on rew:off ctrs:rtc. *)
+(* Qed. *)
-End RPars'.
+(* End RPars'. *)
Lemma Abs_EPar n a (b : Tm n) :
EPar.R (Abs a) b ->
@@ -1421,6 +1528,7 @@ Proof.
split.
hauto lq:on use:relations.rtc_transitive, RPars.AppCong.
apply EPar.PairCong; by apply EPar.AppCong.
+ + admit.
+ hauto lq:on ctrs:EPar.R use:RPars.AppCong.
- hauto lq:on ctrs:EPar.R inv:RPar.R use:RPars.PairCong.
- move => n p a b0 h0 ih0 b1.
@@ -1435,11 +1543,21 @@ Proof.
move /Pair_EPar : ih1 => [/(_ p)[d [ihd ihd']] _].
exists d. split => //.
hauto lq:on use:RPars.ProjCong, relations.rtc_transitive.
+ + admit.
+ hauto lq:on ctrs:EPar.R use:RPars.ProjCong.
- hauto lq:on inv:RPar.R ctrs:EPar.R, rtc use:RPars.BindCong.
- hauto l:on ctrs:EPar.R inv:RPar.R.
- hauto l:on ctrs:EPar.R inv:RPar.R.
-Qed.
+ - hauto l:on ctrs:EPar.R inv:RPar.R.
+ - hauto l:on ctrs:EPar.R inv:RPar.R.
+ - move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc u.
+ elim /RPar.inv => //= _.
+ + admit.
+ + admit.
+ + admit.
+ + (* Need the rule that if commutes with abs *)
+ admit.
+Admitted.
Lemma commutativity1 n (a b0 b1 : Tm n) :
EPar.R a b0 -> rtc RPar.R a b1 -> exists c, rtc RPar.R b0 c /\ EPar.R b1 c.
@@ -1615,7 +1733,10 @@ Proof.
exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
- qauto use:Bot_EPar', EPar.refl.
- qauto use:Univ_EPar', EPar.refl.
-Qed.
+ - admit.
+ - admit.
+ - admit.
+Admitted.
Function tstar {n} (a : Tm n) :=
match a with
From d0760cd9dbb99910918cebd24a29b9abe6ddb985 Mon Sep 17 00:00:00 2001
From: Yiyun Liu <liuyiyun@seas.upenn.edu>
Date: Sat, 11 Jan 2025 00:07:30 -0500
Subject: [PATCH 04/13] Stuck at commutativity (trouble with if pair)
---
theories/fp_red.v | 48 ++++++++++++++++++++++++++++++++++++++++++++---
1 file changed, 45 insertions(+), 3 deletions(-)
diff --git a/theories/fp_red.v b/theories/fp_red.v
index a8c8a3a..5094c57 100644
--- a/theories/fp_red.v
+++ b/theories/fp_red.v
@@ -1374,6 +1374,11 @@ End RPars.
(* End RPars'. *)
+(* (forall p, exists d, rtc RPar.R (Proj p c) d /\ EPar.R (if p is PL then a else b) d) *)
+(* (exists d0 d1, rtc RPar.R (App (ren_Tm shift c) (VarTm var_zero)) *)
+(* (Pair (App (ren_Tm shift d0) (VarTm var_zero))(App (ren_Tm shift d1) (VarTm var_zero))) /\ *)
+(* EPar.R a d0 /\ EPar.R b d1) *)
+
Lemma Abs_EPar n a (b : Tm n) :
EPar.R (Abs a) b ->
(exists d, EPar.R a d /\
@@ -1426,6 +1431,32 @@ Proof.
apply rtc_once. apply : RPar.ProjAbs; eauto using RPar.refl.
Qed.
+Lemma Abs_EPar_If n (a : Tm (S n)) q :
+ EPar.R (Abs a) q ->
+ exists d, EPar.R a d /\
+ forall b c, rtc RPar.R (If q b c) (Abs (If d (ren_Tm shift b) (ren_Tm shift c))).
+Proof.
+ move E : (Abs a) => u h.
+ move : a E.
+ elim : n u q /h => //= n.
+ - move => a0 a1 ha _ b ?. subst.
+ move /Abs_EPar : ha.
+ move => [[d [h0 h1]] _].
+ exists d.
+ split => // b0 c.
+ apply : rtc_l.
+ apply RPar.IfAbs; auto using RPar.refl.
+ apply RPars.AbsCong.
+ apply RPars.IfCong; auto using rtc_refl.
+ - move => a0 a1 ha _ a ?. subst.
+ move /Abs_EPar : ha => [_ [d [h0 h1]]].
+ exists d. split => //.
+ move => b c.
+ apply : rtc_l.
+ apply RPar.IfPair; auto using RPar.refl.
+Admitted.
+
+
Lemma Pair_EPar n (a b c : Tm n) :
EPar.R (Pair a b) c ->
(forall p, exists d, rtc RPar.R (Proj p c) d /\ EPar.R (if p is PL then a else b) d) /\
@@ -1528,7 +1559,9 @@ Proof.
split.
hauto lq:on use:relations.rtc_transitive, RPars.AppCong.
apply EPar.PairCong; by apply EPar.AppCong.
- + admit.
+ + case : a0 ha iha => //=.
+ move => b ha hc b3 a0 [*]. subst.
+ admit.
+ hauto lq:on ctrs:EPar.R use:RPars.AppCong.
- hauto lq:on ctrs:EPar.R inv:RPar.R use:RPars.PairCong.
- move => n p a b0 h0 ih0 b1.
@@ -1543,7 +1576,8 @@ Proof.
move /Pair_EPar : ih1 => [/(_ p)[d [ihd ihd']] _].
exists d. split => //.
hauto lq:on use:RPars.ProjCong, relations.rtc_transitive.
- + admit.
+ + move => p0 b2 [*]. subst.
+ admit.
+ hauto lq:on ctrs:EPar.R use:RPars.ProjCong.
- hauto lq:on inv:RPar.R ctrs:EPar.R, rtc use:RPars.BindCong.
- hauto l:on ctrs:EPar.R inv:RPar.R.
@@ -1552,7 +1586,15 @@ Proof.
- hauto l:on ctrs:EPar.R inv:RPar.R.
- move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc u.
elim /RPar.inv => //= _.
- + admit.
+ + move => a2 a3 b2 b3 c2 c3 ha2 hb2 hc2 [*]. subst.
+ move /(_ _ ltac:(by eauto using RPar.AbsCong)) : iha => [c [ih0 ih1]].
+ move /Abs_EPar : ih1 => [[d [ih2 ih3]] _].
+ have {}/ihb := hb2. move => [b4 [ihb0 ihb1]].
+ have {}/ihc := hc2. move => [c4 [ihc0 ihc1]].
+
+ exists (Abs (If d (ren_Tm shift b4) (ren_Tm shift c4))).
+ split. admit.
+ hauto lq:on ctrs:EPar.R use:EPar.renaming.
+ admit.
+ admit.
+ (* Need the rule that if commutes with abs *)
From f3718707f2b4668ee61ec48531a6576dbf29b123 Mon Sep 17 00:00:00 2001
From: Yiyun Liu <liuyiyun@seas.upenn.edu>
Date: Mon, 20 Jan 2025 13:56:40 -0500
Subject: [PATCH 05/13] Add constants to the reduction semantics
---
syntax.sig | 6 +-
theories/Autosubst2/syntax.v | 29 +++---
theories/fp_red.v | 169 ++++++++++++-----------------------
3 files changed, 74 insertions(+), 130 deletions(-)
diff --git a/syntax.sig b/syntax.sig
index 4549f7f..d268994 100644
--- a/syntax.sig
+++ b/syntax.sig
@@ -3,14 +3,14 @@ Tm(VarTm) : Type
PTag : Type
TTag : Type
-TPi : TTag
-TSig : TTag
PL : PTag
PR : PTag
+TPi : TTag
+TSig : TTag
Abs : (bind Tm in Tm) -> Tm
App : Tm -> Tm -> Tm
Pair : Tm -> Tm -> Tm
Proj : PTag -> Tm -> Tm
TBind : TTag -> Tm -> (bind Tm in Tm) -> Tm
-Bot : Tm
+Const : TTag -> Tm
Univ : nat -> Tm
diff --git a/theories/Autosubst2/syntax.v b/theories/Autosubst2/syntax.v
index b972476..05fb668 100644
--- a/theories/Autosubst2/syntax.v
+++ b/theories/Autosubst2/syntax.v
@@ -40,7 +40,7 @@ Inductive Tm (n_Tm : nat) : Type :=
| Pair : Tm n_Tm -> Tm n_Tm -> Tm n_Tm
| Proj : PTag -> Tm n_Tm -> Tm n_Tm
| TBind : TTag -> Tm n_Tm -> Tm (S n_Tm) -> Tm n_Tm
- | Bot : Tm n_Tm
+ | Const : TTag -> Tm n_Tm
| Univ : nat -> Tm n_Tm.
Lemma congr_Abs {m_Tm : nat} {s0 : Tm (S m_Tm)} {t0 : Tm (S m_Tm)}
@@ -83,9 +83,10 @@ exact (eq_trans
(ap (fun x => TBind m_Tm t0 t1 x) H2)).
Qed.
-Lemma congr_Bot {m_Tm : nat} : Bot m_Tm = Bot m_Tm.
+Lemma congr_Const {m_Tm : nat} {s0 : TTag} {t0 : TTag} (H0 : s0 = t0) :
+ Const m_Tm s0 = Const m_Tm t0.
Proof.
-exact (eq_refl).
+exact (eq_trans eq_refl (ap (fun x => Const m_Tm x) H0)).
Qed.
Lemma congr_Univ {m_Tm : nat} {s0 : nat} {t0 : nat} (H0 : s0 = t0) :
@@ -116,7 +117,7 @@ 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)
| TBind _ s0 s1 s2 =>
TBind n_Tm s0 (ren_Tm xi_Tm s1) (ren_Tm (upRen_Tm_Tm xi_Tm) s2)
- | Bot _ => Bot n_Tm
+ | Const _ s0 => Const n_Tm s0
| Univ _ s0 => Univ n_Tm s0
end.
@@ -143,7 +144,7 @@ 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)
| TBind _ s0 s1 s2 =>
TBind n_Tm s0 (subst_Tm sigma_Tm s1) (subst_Tm (up_Tm_Tm sigma_Tm) s2)
- | Bot _ => Bot n_Tm
+ | Const _ s0 => Const n_Tm s0
| Univ _ s0 => Univ n_Tm s0
end.
@@ -183,7 +184,7 @@ subst_Tm sigma_Tm s = s :=
| TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
(idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s2)
- | Bot _ => congr_Bot
+ | Const _ s0 => congr_Const (eq_refl s0)
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
@@ -227,7 +228,7 @@ Fixpoint extRen_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
congr_TBind (eq_refl s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
(extRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upExtRen_Tm_Tm _ _ Eq_Tm) s2)
- | Bot _ => congr_Bot
+ | Const _ s0 => congr_Const (eq_refl s0)
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
@@ -272,7 +273,7 @@ Fixpoint ext_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
congr_TBind (eq_refl s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
(ext_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm) (upExt_Tm_Tm _ _ Eq_Tm)
s2)
- | Bot _ => congr_Bot
+ | Const _ s0 => congr_Const (eq_refl s0)
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
@@ -317,7 +318,7 @@ Fixpoint compRenRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
congr_TBind (eq_refl s0) (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
(compRenRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s2)
- | Bot _ => congr_Bot
+ | Const _ s0 => congr_Const (eq_refl s0)
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
@@ -373,7 +374,7 @@ Fixpoint compRenSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
(compRenSubst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm tau_Tm)
(up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s2)
- | Bot _ => congr_Bot
+ | Const _ s0 => congr_Const (eq_refl s0)
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
@@ -450,7 +451,7 @@ ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
(compSubstRen_Tm (up_Tm_Tm sigma_Tm) (upRen_Tm_Tm zeta_Tm)
(up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s2)
- | Bot _ => congr_Bot
+ | Const _ s0 => congr_Const (eq_refl s0)
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
@@ -528,7 +529,7 @@ subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
(compSubstSubst_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm)
(up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s2)
- | Bot _ => congr_Bot
+ | Const _ s0 => congr_Const (eq_refl s0)
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
@@ -644,7 +645,7 @@ Fixpoint rinst_inst_Tm {m_Tm : nat} {n_Tm : nat}
congr_TBind (eq_refl s0) (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
(rinst_inst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm sigma_Tm)
(rinstInst_up_Tm_Tm _ _ Eq_Tm) s2)
- | Bot _ => congr_Bot
+ | Const _ s0 => congr_Const (eq_refl s0)
| Univ _ s0 => congr_Univ (eq_refl s0)
end.
@@ -846,7 +847,7 @@ Arguments VarTm {n_Tm}.
Arguments Univ {n_Tm}.
-Arguments Bot {n_Tm}.
+Arguments Const {n_Tm}.
Arguments TBind {n_Tm}.
diff --git a/theories/fp_red.v b/theories/fp_red.v
index d2fbdf6..30f3354 100644
--- a/theories/fp_red.v
+++ b/theories/fp_red.v
@@ -69,9 +69,8 @@ Module Par.
R A0 A1 ->
R B0 B1 ->
R (TBind p A0 B0) (TBind p A1 B1)
- (* Bot is useful for making the prov function computable *)
- | BotCong :
- R Bot Bot
+ | ConstCong k :
+ R (Const k) (Const k)
| UnivCong i :
R (Univ i) (Univ i).
@@ -212,7 +211,7 @@ Module Par.
move : ihB => [c0 [? ?]]. subst.
eexists. split. by apply BindCong; eauto.
by asimpl.
- - move => n n0 ξ []//=. hauto l:on.
+ - move => n k m ξ []//=. hauto l:on.
- move => n i n0 ξ []//=. hauto l:on.
Qed.
@@ -275,10 +274,10 @@ Module Pars.
End Pars.
-Definition var_or_bot {n} (a : Tm n) :=
+Definition var_or_const {n} (a : Tm n) :=
match a with
| VarTm _ => true
- | Bot => true
+ | Const _ => true
| _ => false
end.
@@ -324,8 +323,8 @@ Module RPar.
R A0 A1 ->
R B0 B1 ->
R (TBind p A0 B0) (TBind p A1 B1)
- | BotCong :
- R Bot Bot
+ | ConstCong k :
+ R (Const k) (Const k)
| UnivCong i :
R (Univ i) (Univ i).
@@ -571,8 +570,8 @@ Module RPar'.
R A0 A1 ->
R B0 B1 ->
R (TBind p A0 B0) (TBind p A1 B1)
- | BotCong :
- R Bot Bot
+ | ConstCong k :
+ R (Const k) (Const k)
| UnivCong i :
R (Univ i) (Univ i).
@@ -656,16 +655,16 @@ Module RPar'.
qauto l:on ctrs:R inv:option.
Qed.
- Lemma var_or_bot_imp {n} (a b : Tm n) :
- var_or_bot a ->
- a = b -> ~~ var_or_bot b -> False.
+ Lemma var_or_const_imp {n} (a b : Tm n) :
+ var_or_const a ->
+ a = b -> ~~ var_or_const b -> False.
Proof.
hauto lq:on inv:Tm.
Qed.
- Lemma var_or_bot_up n m (ρ : fin n -> Tm m) :
- (forall i, var_or_bot (ρ i)) ->
- (forall i, var_or_bot (up_Tm_Tm ρ i)).
+ Lemma var_or_const_up n m (ρ : fin n -> Tm m) :
+ (forall i, var_or_const (ρ i)) ->
+ (forall i, var_or_const (up_Tm_Tm ρ i)).
Proof.
move => h /= [i|].
- asimpl.
@@ -676,10 +675,10 @@ Module RPar'.
- sfirstorder.
Qed.
- Local Ltac antiimp := qauto l:on use:var_or_bot_imp.
+ Local Ltac antiimp := qauto l:on use:var_or_const_imp.
Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
- (forall i, var_or_bot (ρ i)) ->
+ (forall i, var_or_const (ρ i)) ->
R (subst_Tm ρ a) b -> exists b0, R a b0 /\ subst_Tm ρ b0 = b.
Proof.
move E : (subst_Tm ρ a) => u hρ h.
@@ -690,7 +689,7 @@ Module RPar'.
case : c => //=; first by antiimp.
move => c [?]. subst.
spec_refl.
- have /var_or_bot_up hρ' := hρ.
+ have /var_or_const_up hρ' := hρ.
move : iha hρ' => /[apply] iha.
move : ihb hρ => /[apply] ihb.
spec_refl.
@@ -714,7 +713,7 @@ Module RPar'.
hauto l:on.
- move => n a0 a1 ha iha m ρ hρ []//=; first by antiimp.
move => t [*]. subst.
- have /var_or_bot_up {}/iha := hρ => iha.
+ have /var_or_const_up {}/iha := hρ => iha.
spec_refl.
move :iha => [b0 [? ?]]. subst.
eexists. split. by apply AbsCong; eauto.
@@ -750,7 +749,7 @@ Module RPar'.
first by antiimp.
move => ? t t0 [*]. subst.
have {}/iha := (hρ) => iha.
- have /var_or_bot_up {}/ihB := (hρ) => ihB.
+ have /var_or_const_up {}/ihB := (hρ) => ihB.
spec_refl.
move : iha => [b0 [? ?]].
move : ihB => [c0 [? ?]]. subst.
@@ -894,8 +893,8 @@ Module EPar.
R A0 A1 ->
R B0 B1 ->
R (TBind p A0 B0) (TBind p A1 B1)
- | BotCong :
- R Bot Bot
+ | ConstCong k :
+ R (Const k) (Const k)
| UnivCong i :
R (Univ i) (Univ i).
@@ -1070,7 +1069,7 @@ Module RPars.
Proof. hauto lq:on use:morphing inv:option. Qed.
Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
- (forall i, var_or_bot (ρ i)) ->
+ (forall i, var_or_const (ρ i)) ->
rtc RPar.R (subst_Tm ρ a) b -> exists b0, rtc RPar.R a b0 /\ subst_Tm ρ b0 = b.
Proof.
move E :(subst_Tm ρ a) => u hρ h.
@@ -1157,7 +1156,7 @@ Module RPars'.
Proof. hauto lq:on use:morphing inv:option. Qed.
Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
- (forall i, var_or_bot (ρ i)) ->
+ (forall i, var_or_const (ρ i)) ->
rtc RPar'.R (subst_Tm ρ a) b -> exists b0, rtc RPar'.R a b0 /\ subst_Tm ρ b0 = b.
Proof.
move E :(subst_Tm ρ a) => u hρ h.
@@ -1444,15 +1443,15 @@ Proof.
- hauto l:on ctrs:OExp.R.
Qed.
-Lemma Bot_EPar' n (u : Tm n) :
- EPar.R Bot u ->
- rtc OExp.R Bot u.
- move E : Bot => t h.
- move : E. elim : n t u /h => //=.
- - move => n a0 a1 h ih ?. subst.
+Lemma Const_EPar' n k (u : Tm n) :
+ EPar.R (Const k) u ->
+ rtc OExp.R (Const k) u.
+ move E : (Const k) => t h.
+ move : k E. elim : n t u /h => //=.
+ - move => n a0 a1 h ih k ?. subst.
specialize ih with (1 := eq_refl).
hauto lq:on ctrs:OExp.R use:rtc_r.
- - move => n a0 a1 h ih ?. subst.
+ - move => n a0 a1 h ih k ?. subst.
specialize ih with (1 := eq_refl).
hauto lq:on ctrs:OExp.R use:rtc_r.
- hauto l:on ctrs:OExp.R.
@@ -1519,7 +1518,7 @@ Proof.
move : OExp.commutativity0 h2; repeat move/[apply].
move => [d h].
exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
- - qauto use:Bot_EPar', EPar.refl.
+ - qauto use:Const_EPar', EPar.refl.
- qauto use:Univ_EPar', EPar.refl.
Qed.
@@ -1536,7 +1535,7 @@ Function tstar {n} (a : Tm n) :=
| Proj p (Abs a) => (Abs (Proj p (tstar a)))
| Proj p a => Proj p (tstar a)
| TBind p a b => TBind p (tstar a) (tstar b)
- | Bot => Bot
+ | Const k => Const k
| Univ i => Univ i
end.
@@ -1568,7 +1567,7 @@ Function tstar' {n} (a : Tm n) :=
| Proj p (Pair a b) => if p is PL then (tstar' a) else (tstar' b)
| Proj p a => Proj p (tstar' a)
| TBind p a b => TBind p (tstar' a) (tstar' b)
- | Bot => Bot
+ | Const k => Const k
| Univ i => Univ i
end.
@@ -1616,63 +1615,6 @@ Proof.
sfirstorder use:relations.diamond_confluent, EPar_diamond.
Qed.
-Fixpoint depth_tm {n} (a : Tm n) :=
- match a with
- | VarTm _ => 1
- | TBind _ A B => 1 + max (depth_tm A) (depth_tm B)
- | Abs a => 1 + depth_tm a
- | App a b => 1 + max (depth_tm a) (depth_tm b)
- | Proj p a => 1 + depth_tm a
- | Pair a b => 1 + max (depth_tm a) (depth_tm b)
- | Bot => 1
- | Univ i => 1
- end.
-
-Lemma depth_ren n m (ξ: fin n -> fin m) a :
- depth_tm a = depth_tm (ren_Tm ξ a).
-Proof.
- move : m ξ. elim : n / a; scongruence.
-Qed.
-
-Lemma depth_subst n m (ρ : fin n -> Tm m) a :
- (forall i, depth_tm (ρ i) = 1) ->
- depth_tm a = depth_tm (subst_Tm ρ a).
-Proof.
- move : m ρ. elim : n / a.
- - sfirstorder.
- - move => n a iha m ρ hρ.
- simpl.
- f_equal. apply iha.
- destruct i as [i|].
- + simpl.
- by rewrite -depth_ren.
- + by simpl.
- - hauto lq:on rew:off.
- - hauto lq:on rew:off.
- - hauto lq:on rew:off.
- - move => n p a iha b ihb m ρ hρ.
- simpl. f_equal.
- f_equal.
- by apply iha.
- apply ihb.
- destruct i as [i|].
- + simpl.
- by rewrite -depth_ren.
- + by simpl.
- - sfirstorder.
- - sfirstorder.
-Qed.
-
-Lemma depth_subst_bool n (a : Tm (S n)) :
- depth_tm a = depth_tm (subst_Tm (scons Bot VarTm) a).
-Proof.
- apply depth_subst.
- destruct i as [i|] => //=.
-Qed.
-
-Local Ltac prov_tac := sfirstorder use:depth_ren.
-Local Ltac extract_tac := rewrite -?depth_subst_bool;hauto use:depth_subst_bool.
-
Definition prov_bind {n} p0 A0 B0 (a : Tm n) :=
match a with
| TBind p A B => p = p0 /\ rtc Par.R A A0 /\ rtc Par.R B B0
@@ -1704,8 +1646,8 @@ Inductive prov {n} : Tm n -> Tm n -> Prop :=
| P_Proj h p a :
prov h a ->
prov h (Proj p a)
-| P_Bot :
- prov Bot Bot
+| P_Const k :
+ prov (Const k) (Const k)
| P_Var i :
prov (VarTm i) (VarTm i)
| P_Univ i :
@@ -1789,7 +1731,7 @@ Proof.
split.
move => h. constructor. move => b. asimpl. by constructor.
inversion 1; subst.
- specialize H2 with (b := Bot).
+ specialize H2 with (b := Const TPi).
move : H2. asimpl. inversion 1; subst. done.
Qed.
@@ -1845,11 +1787,11 @@ Qed.
Fixpoint extract {n} (a : Tm n) : Tm n :=
match a with
| TBind p A B => TBind p A B
- | Abs a => subst_Tm (scons Bot VarTm) (extract a)
+ | Abs a => subst_Tm (scons (Const TPi) VarTm) (extract a)
| App a b => extract a
| Pair a b => extract a
| Proj p a => extract a
- | Bot => Bot
+ | Const k => Const k
| VarTm i => VarTm i
| Univ i => Univ i
end.
@@ -1886,7 +1828,7 @@ Proof.
Qed.
Lemma ren_subst_bot n (a : Tm (S n)) :
- extract (subst_Tm (scons Bot VarTm) a) = subst_Tm (scons Bot VarTm) (extract a).
+ extract (subst_Tm (scons (Const TPi) VarTm) a) = subst_Tm (scons (Const TPi) VarTm) (extract a).
Proof.
apply ren_morphing. destruct i as [i|] => //=.
Qed.
@@ -1896,7 +1838,7 @@ Definition prov_extract_spec {n} u (a : Tm n) :=
| TBind p A B => exists A0 B0, extract a = TBind p A0 B0 /\ rtc Par.R A A0 /\ rtc Par.R B B0
| Univ i => extract a = Univ i
| VarTm i => extract a = VarTm i
- | Bot => extract a = Bot
+ | (Const TPi) => extract a = (Const TPi)
| _ => True
end.
@@ -1909,22 +1851,22 @@ Proof.
- move => h a ha ih.
case : h ha ih => //=.
+ move => i ha ih.
- move /(_ Bot) in ih.
+ move /(_ (Const TPi)) in ih.
rewrite -ih.
by rewrite ren_subst_bot.
+ move => p A B h ih.
- move /(_ Bot) : ih => [A0][B0][h0][h1]h2.
+ move /(_ (Const TPi)) : ih => [A0][B0][h0][h1]h2.
rewrite ren_subst_bot in h0.
rewrite h0.
eauto.
- + move => _ /(_ Bot).
+ + move => p _ /(_ (Const TPi)).
by rewrite ren_subst_bot.
- + move => i h /(_ Bot).
+ + move => i h /(_ (Const TPi)).
by rewrite ren_subst_bot => ->.
- hauto lq:on.
- hauto lq:on.
- hauto lq:on.
- - sfirstorder.
+ - case => //=.
- sfirstorder.
- sfirstorder.
Qed.
@@ -2400,23 +2342,24 @@ Fixpoint ne {n} (a : Tm n) :=
match a with
| VarTm i => true
| TBind _ A B => false
- | Bot => true
| App a b => ne a && nf b
| Abs a => false
| Univ _ => false
| Proj _ a => ne a
| Pair _ _ => false
+ | Const _ => false
end
with nf {n} (a : Tm n) :=
match a with
| VarTm i => true
| TBind _ A B => nf A && nf B
- | Bot => true
+ | (Const TPi) => true
| App a b => ne a && nf b
| Abs a => nf a
| Univ _ => true
| Proj _ a => ne a
| Pair a b => nf a && nf b
+ | Const _ => true
end.
Lemma ne_nf n a : @ne n a -> nf a.
@@ -2482,15 +2425,15 @@ Create HintDb nfne.
#[export]Hint Resolve nf_wn ne_nf wne_wn nf_refl : nfne.
Lemma ne_nf_antiren n m (a : Tm n) (ρ : fin n -> Tm m) :
- (forall i, var_or_bot (ρ i)) ->
+ (forall i, var_or_const (ρ i)) ->
(ne (subst_Tm ρ a) -> ne a) /\ (nf (subst_Tm ρ a) -> nf a).
Proof.
move : m ρ. elim : n / a => //;
- hauto b:on drew:off use:RPar.var_or_bot_up.
+ hauto b:on drew:off use:RPar.var_or_const_up.
Qed.
Lemma wn_antirenaming n m a (ρ : fin n -> Tm m) :
- (forall i, var_or_bot (ρ i)) ->
+ (forall i, var_or_const (ρ i)) ->
wn (subst_Tm ρ a) -> wn a.
Proof.
rewrite /wn => hρ.
@@ -2503,10 +2446,10 @@ Proof.
Qed.
Lemma ext_wn n (a : Tm n) :
- wn (App a Bot) ->
+ wn (App a (Const TPi)) ->
wn a.
Proof.
- move E : (App a Bot) => a0 [v [hr hv]].
+ move E : (App a (Const TPi)) => a0 [v [hr hv]].
move : a E.
move : hv.
elim : a0 v / hr.
@@ -2517,9 +2460,9 @@ Proof.
case : a0 hr0=>// => b0 b1.
elim /RPar'.inv=>// _.
+ move => a0 a3 b2 b3 ? ? [? ?] ? [? ?]. subst.
- have ? : b3 = Bot by hauto lq:on inv:RPar'.R. subst.
+ have ? : b3 = (Const TPi) by hauto lq:on inv:RPar'.R. subst.
suff : wn (Abs a3) by hauto lq:on ctrs:RPar'.R, rtc unfold:wn.
- have : wn (subst_Tm (scons Bot VarTm) a3) by sfirstorder.
+ have : wn (subst_Tm (scons (Const TPi) VarTm) a3) by sfirstorder.
move => h. apply wn_abs.
move : h. apply wn_antirenaming.
hauto lq:on rew:off inv:option.
From d68df5d0bc84617900727ed91a1948c360ec066b Mon Sep 17 00:00:00 2001
From: Yiyun Liu <liuyiyun@seas.upenn.edu>
Date: Mon, 20 Jan 2025 15:33:46 -0500
Subject: [PATCH 06/13] Add compile
---
theories/compile.v | 136 +++++++++++++++++++++++++++++++++++++++++++++
theories/fp_red.v | 49 +++++++++++-----
2 files changed, 172 insertions(+), 13 deletions(-)
create mode 100644 theories/compile.v
diff --git a/theories/compile.v b/theories/compile.v
new file mode 100644
index 0000000..d9d6ca3
--- /dev/null
+++ b/theories/compile.v
@@ -0,0 +1,136 @@
+Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax fp_red.
+Require Import ssreflect ssrbool.
+From Hammer Require Import Tactics.
+
+Module Compile.
+ Fixpoint F {n} (a : Tm n) : Tm n :=
+ match a with
+ | TBind p A B => Pair (Pair (Const p) (F A)) (Abs (F B))
+ | Const k => Const k
+ | Univ i => Univ i
+ | Abs a => Abs (F a)
+ | App a b => App (F a) (F b)
+ | VarTm i => VarTm i
+ | Pair a b => Pair (F a) (F b)
+ | Proj t a => Proj t (F a)
+ 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.
+
+ #[local]Hint Rewrite Compile.renaming : compile.
+
+ Lemma morphing n m (a : Tm n) (ρ0 ρ1 : fin n -> Tm m) :
+ (forall i, ρ0 i = F (ρ1 i)) ->
+ subst_Tm ρ0 (F a) = F (subst_Tm ρ1 a).
+ Proof.
+ move : m ρ0 ρ1. elim : n / a => n//=.
+ - hauto lq:on inv:option rew:db:compile unfold:funcomp.
+ - hauto lq:on rew:off.
+ - hauto lq:on rew:off.
+ - hauto lq:on.
+ - hauto lq:on inv:option rew:db:compile unfold:funcomp.
+ Qed.
+
+ Lemma substing n b (a : Tm (S n)) :
+ subst_Tm (scons (F b) VarTm) (F a) = F (subst_Tm (scons b VarTm) a).
+ Proof.
+ apply morphing.
+ case => //=.
+ Qed.
+
+End Compile.
+
+#[export] Hint Rewrite Compile.renaming Compile.morphing : compile.
+
+
+Module Join.
+ Definition R {n} (a b : Tm n) := join (Compile.F a) (Compile.F b).
+
+ Lemma BindInj n p0 p1 (A0 A1 : Tm n) B0 B1 :
+ R (TBind p0 A0 B0) (TBind p1 A1 B1) -> p0 = p1 /\ R A0 A1 /\ R B0 B1.
+ Proof.
+ rewrite /R /= !join_pair_inj.
+ move => [[/join_const_inj h0 h1] h2].
+ apply abs_eq in h2.
+ evar (t : Tm (S n)).
+ have : join (App (ren_Tm shift (Abs (Compile.F B1))) (VarTm var_zero)) t by
+ apply Join.FromPar; apply Par.AppAbs; auto using Par.refl.
+ subst t. rewrite -/ren_Tm.
+ move : h2. move /join_transitive => /[apply]. asimpl => h2.
+ tauto.
+ Qed.
+
+ Lemma UnivInj n i j : R (Univ i : Tm n) (Univ j) -> i = j.
+ Proof. hauto l:on use:join_univ_inj. Qed.
+
+End Join.
+
+Module Equiv.
+ Inductive R {n} : Tm n -> Tm n -> Prop :=
+ (***************** Beta ***********************)
+ | AppAbs a b :
+ R (App (Abs a) b) (subst_Tm (scons b VarTm) a)
+ | ProjPair p a b :
+ R (Proj p (Pair a b)) (if p is PL then a else b)
+
+ (****************** Eta ***********************)
+ | AppEta a :
+ R a (Abs (App (ren_Tm shift a) (VarTm var_zero)))
+ | PairEta a :
+ R a (Pair (Proj PL a) (Proj PR a))
+
+ (*************** Congruence ********************)
+ | Var i : R (VarTm i) (VarTm i)
+ | AbsCong a b :
+ R a b ->
+ R (Abs a) (Abs b)
+ | AppCong a0 a1 b0 b1 :
+ R a0 a1 ->
+ R b0 b1 ->
+ R (App a0 b0) (App a1 b1)
+ | PairCong a0 a1 b0 b1 :
+ R a0 a1 ->
+ R b0 b1 ->
+ R (Pair a0 b0) (Pair a1 b1)
+ | ProjCong p a0 a1 :
+ R a0 a1 ->
+ R (Proj p a0) (Proj p a1)
+ | BindCong p A0 A1 B0 B1:
+ R A0 A1 ->
+ R B0 B1 ->
+ R (TBind p A0 B0) (TBind p A1 B1)
+ | UnivCong i :
+ R (Univ i) (Univ i).
+End Equiv.
+
+Module EquivJoin.
+ Lemma FromEquiv n (a b : Tm n) : Equiv.R a b -> Join.R a b.
+ Proof.
+ move => h. elim : n a b /h => n.
+ - move => a b.
+ rewrite /Join.R /join /=.
+ eexists. split. apply relations.rtc_once.
+ apply Par.AppAbs; auto using Par.refl.
+ rewrite Compile.substing.
+ apply relations.rtc_refl.
+ - move => p a b.
+ apply Join.FromPar.
+ simpl. apply : Par.ProjPair'; auto using Par.refl.
+ case : p => //=.
+ - move => a. apply Join.FromPar => /=.
+ apply : Par.AppEta'; auto using Par.refl.
+ by autorewrite with compile.
+ - move => a. apply Join.FromPar => /=.
+ apply : Par.PairEta; auto using Par.refl.
+ - hauto l:on use:Join.FromPar, Par.Var.
+ - hauto lq:on use:Join.AbsCong.
+ - qauto l:on use:Join.AppCong.
+ - qauto l:on use:Join.PairCong.
+ - qauto use:Join.ProjCong.
+ - rewrite /Join.R => p A0 A1 B0 B1 _ hA _ hB /=.
+ sfirstorder use:Join.PairCong,Join.AbsCong,Join.FromPar,Par.ConstCong.
+ - hauto l:on.
+ Qed.
+End EquivJoin.
diff --git a/theories/fp_red.v b/theories/fp_red.v
index 30f3354..adb05ea 100644
--- a/theories/fp_red.v
+++ b/theories/fp_red.v
@@ -410,16 +410,16 @@ Module RPar.
qauto l:on ctrs:R inv:option.
Qed.
- Lemma var_or_bot_imp {n} (a b : Tm n) :
- var_or_bot a ->
- a = b -> ~~ var_or_bot b -> False.
+ Lemma var_or_const_imp {n} (a b : Tm n) :
+ var_or_const a ->
+ a = b -> ~~ var_or_const b -> False.
Proof.
hauto lq:on inv:Tm.
Qed.
- Lemma var_or_bot_up n m (ρ : fin n -> Tm m) :
- (forall i, var_or_bot (ρ i)) ->
- (forall i, var_or_bot (up_Tm_Tm ρ i)).
+ Lemma var_or_const_up n m (ρ : fin n -> Tm m) :
+ (forall i, var_or_const (ρ i)) ->
+ (forall i, var_or_const (up_Tm_Tm ρ i)).
Proof.
move => h /= [i|].
- asimpl.
@@ -430,10 +430,10 @@ Module RPar.
- sfirstorder.
Qed.
- Local Ltac antiimp := qauto l:on use:var_or_bot_imp.
+ Local Ltac antiimp := qauto l:on use:var_or_const_imp.
Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
- (forall i, var_or_bot (ρ i)) ->
+ (forall i, var_or_const (ρ i)) ->
R (subst_Tm ρ a) b -> exists b0, R a b0 /\ subst_Tm ρ b0 = b.
Proof.
move E : (subst_Tm ρ a) => u hρ h.
@@ -444,7 +444,7 @@ Module RPar.
case : c => //=; first by antiimp.
move => c [?]. subst.
spec_refl.
- have /var_or_bot_up hρ' := hρ.
+ have /var_or_const_up hρ' := hρ.
move : iha hρ' => /[apply] iha.
move : ihb hρ => /[apply] ihb.
spec_refl.
@@ -470,7 +470,7 @@ Module RPar.
- move => n p a0 a1 ha iha m ρ hρ []//=;
first by antiimp.
move => p0 []//= t [*]; first by antiimp. subst.
- have /var_or_bot_up {}/iha := hρ => iha.
+ have /var_or_const_up {}/iha := hρ => iha.
spec_refl. move : iha => [b0 [? ?]]. subst.
eexists. split. apply ProjAbs; eauto. by asimpl.
- move => n p a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
@@ -488,7 +488,7 @@ Module RPar.
hauto l:on.
- move => n a0 a1 ha iha m ρ hρ []//=; first by antiimp.
move => t [*]. subst.
- have /var_or_bot_up {}/iha := hρ => iha.
+ have /var_or_const_up {}/iha := hρ => iha.
spec_refl.
move :iha => [b0 [? ?]]. subst.
eexists. split. by apply AbsCong; eauto.
@@ -524,7 +524,7 @@ Module RPar.
first by antiimp.
move => ? t t0 [*]. subst.
have {}/iha := (hρ) => iha.
- have /var_or_bot_up {}/ihB := (hρ) => ihB.
+ have /var_or_const_up {}/ihB := (hρ) => ihB.
spec_refl.
move : iha => [b0 [? ?]].
move : ihB => [c0 [? ?]]. subst.
@@ -1838,7 +1838,7 @@ Definition prov_extract_spec {n} u (a : Tm n) :=
| TBind p A B => exists A0 B0, extract a = TBind p A0 B0 /\ rtc Par.R A A0 /\ rtc Par.R B B0
| Univ i => extract a = Univ i
| VarTm i => extract a = VarTm i
- | (Const TPi) => extract a = (Const TPi)
+ | (Const i) => extract a = (Const i)
| _ => True
end.
@@ -2250,6 +2250,15 @@ Proof.
apply prov_extract.
Qed.
+Lemma pars_const_inv n i (c : Tm n) :
+ rtc Par.R (Const i) c ->
+ extract c = Const i.
+Proof.
+ have : prov (Const i) (Const i : Tm n) by sfirstorder.
+ move : prov_pars. repeat move/[apply].
+ apply prov_extract.
+Qed.
+
Lemma pars_pi_inv n p (A : Tm n) B C :
rtc Par.R (TBind p A B) C ->
exists A0 B0, extract C = TBind p A0 B0 /\
@@ -2277,6 +2286,14 @@ Proof.
sauto l:on use:pars_univ_inv.
Qed.
+Lemma pars_const_inj n i j (C : Tm n) :
+ rtc Par.R (Const i) C ->
+ rtc Par.R (Const j) C ->
+ i = j.
+Proof.
+ sauto l:on use:pars_const_inv.
+Qed.
+
Lemma pars_pi_inj n p0 p1 (A0 A1 : Tm n) B0 B1 C :
rtc Par.R (TBind p0 A0 B0) C ->
rtc Par.R (TBind p1 A1 B1) C ->
@@ -2314,6 +2331,12 @@ Proof.
sfirstorder use:pars_univ_inj.
Qed.
+Lemma join_const_inj n i j :
+ join (Const i : Tm n) (Const j) -> i = j.
+Proof.
+ sfirstorder use:pars_const_inj.
+Qed.
+
Lemma join_pi_inj n p0 p1 (A0 A1 : Tm n) B0 B1 :
join (TBind p0 A0 B0) (TBind p1 A1 B1) ->
p0 = p1 /\ join A0 A1 /\ join B0 B1.
From 9c9ce52b63ed216299646e519e38dfb419b31ed7 Mon Sep 17 00:00:00 2001
From: Yiyun Liu <liuyiyun@seas.upenn.edu>
Date: Mon, 20 Jan 2025 16:28:44 -0500
Subject: [PATCH 07/13] Recover the contra lemmas
---
syntax.sig | 1 +
theories/Autosubst2/syntax.v | 20 ++++++++-
theories/compile.v | 1 +
theories/fp_red.v | 83 ++++++++++++++++++++++++++++--------
theories/logrel.v | 33 +++++++++-----
5 files changed, 110 insertions(+), 28 deletions(-)
diff --git a/syntax.sig b/syntax.sig
index d268994..3101cab 100644
--- a/syntax.sig
+++ b/syntax.sig
@@ -14,3 +14,4 @@ Proj : PTag -> Tm -> Tm
TBind : TTag -> Tm -> (bind Tm in Tm) -> Tm
Const : TTag -> Tm
Univ : nat -> Tm
+Bot : Tm
diff --git a/theories/Autosubst2/syntax.v b/theories/Autosubst2/syntax.v
index 05fb668..a5cb002 100644
--- a/theories/Autosubst2/syntax.v
+++ b/theories/Autosubst2/syntax.v
@@ -41,7 +41,8 @@ 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
| Const : TTag -> Tm n_Tm
- | Univ : nat -> Tm n_Tm.
+ | Univ : nat -> 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.
@@ -95,6 +96,11 @@ Proof.
exact (eq_trans eq_refl (ap (fun x => Univ m_Tm x) H0)).
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.
@@ -119,6 +125,7 @@ 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)
| Const _ s0 => Const n_Tm s0
| Univ _ s0 => Univ n_Tm s0
+ | Bot _ => Bot n_Tm
end.
Lemma up_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm) :
@@ -146,6 +153,7 @@ 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)
| Const _ s0 => Const n_Tm s0
| Univ _ s0 => Univ n_Tm s0
+ | Bot _ => Bot n_Tm
end.
Lemma upId_Tm_Tm {m_Tm : nat} (sigma : fin m_Tm -> Tm m_Tm)
@@ -186,6 +194,7 @@ subst_Tm sigma_Tm s = s :=
(idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s2)
| Const _ s0 => congr_Const (eq_refl s0)
| Univ _ s0 => congr_Univ (eq_refl s0)
+ | Bot _ => congr_Bot
end.
Lemma upExtRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n)
@@ -230,6 +239,7 @@ Fixpoint extRen_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
(upExtRen_Tm_Tm _ _ Eq_Tm) s2)
| Const _ s0 => congr_Const (eq_refl s0)
| Univ _ s0 => congr_Univ (eq_refl s0)
+ | Bot _ => congr_Bot
end.
Lemma upExt_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm)
@@ -275,6 +285,7 @@ Fixpoint ext_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
s2)
| Const _ s0 => congr_Const (eq_refl s0)
| Univ _ s0 => congr_Univ (eq_refl s0)
+ | Bot _ => congr_Bot
end.
Lemma up_ren_ren_Tm_Tm {k : nat} {l : nat} {m : nat} (xi : fin k -> fin l)
@@ -320,6 +331,7 @@ Fixpoint compRenRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
(upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s2)
| Const _ s0 => congr_Const (eq_refl s0)
| Univ _ s0 => congr_Univ (eq_refl s0)
+ | Bot _ => congr_Bot
end.
Lemma up_ren_subst_Tm_Tm {k : nat} {l : nat} {m_Tm : nat}
@@ -376,6 +388,7 @@ 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)
| Const _ s0 => congr_Const (eq_refl s0)
| Univ _ s0 => congr_Univ (eq_refl s0)
+ | Bot _ => congr_Bot
end.
Lemma up_subst_ren_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
@@ -453,6 +466,7 @@ 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)
| Const _ s0 => congr_Const (eq_refl s0)
| Univ _ s0 => congr_Univ (eq_refl s0)
+ | Bot _ => congr_Bot
end.
Lemma up_subst_subst_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
@@ -531,6 +545,7 @@ 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)
| Const _ s0 => congr_Const (eq_refl s0)
| Univ _ s0 => congr_Univ (eq_refl s0)
+ | Bot _ => congr_Bot
end.
Lemma renRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
@@ -647,6 +662,7 @@ Fixpoint rinst_inst_Tm {m_Tm : nat} {n_Tm : nat}
(rinstInst_up_Tm_Tm _ _ Eq_Tm) s2)
| Const _ s0 => congr_Const (eq_refl s0)
| Univ _ s0 => congr_Univ (eq_refl s0)
+ | Bot _ => congr_Bot
end.
Lemma rinstInst'_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
@@ -845,6 +861,8 @@ Core.
Arguments VarTm {n_Tm}.
+Arguments Bot {n_Tm}.
+
Arguments Univ {n_Tm}.
Arguments Const {n_Tm}.
diff --git a/theories/compile.v b/theories/compile.v
index d9d6ca3..3eaaf42 100644
--- a/theories/compile.v
+++ b/theories/compile.v
@@ -13,6 +13,7 @@ Module Compile.
| VarTm i => VarTm i
| Pair a b => Pair (F a) (F b)
| Proj t a => Proj t (F a)
+ | Bot => Bot
end.
Lemma renaming n m (a : Tm n) (ξ : fin n -> fin m) :
diff --git a/theories/fp_red.v b/theories/fp_red.v
index adb05ea..6932f84 100644
--- a/theories/fp_red.v
+++ b/theories/fp_red.v
@@ -72,7 +72,9 @@ Module Par.
| ConstCong k :
R (Const k) (Const k)
| UnivCong i :
- R (Univ i) (Univ i).
+ R (Univ i) (Univ i)
+ | BotCong :
+ R Bot Bot.
Lemma refl n (a : Tm n) : R a a.
elim : n /a; hauto ctrs:R.
@@ -134,6 +136,7 @@ Module Par.
- hauto l:on inv:option ctrs:R use:renaming.
- sfirstorder.
- sfirstorder.
+ - sfirstorder.
Qed.
Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
@@ -213,6 +216,7 @@ Module Par.
by asimpl.
- move => n k m ξ []//=. hauto l:on.
- move => n i n0 ξ []//=. hauto l:on.
+ - hauto q:on inv:Tm ctrs:R.
Qed.
End Par.
@@ -277,7 +281,7 @@ End Pars.
Definition var_or_const {n} (a : Tm n) :=
match a with
| VarTm _ => true
- | Const _ => true
+ | Bot => true
| _ => false
end.
@@ -326,7 +330,9 @@ Module RPar.
| ConstCong k :
R (Const k) (Const k)
| UnivCong i :
- R (Univ i) (Univ i).
+ R (Univ i) (Univ i)
+ | BotCong :
+ R Bot Bot.
Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
@@ -394,6 +400,7 @@ Module RPar.
- hauto lq:on ctrs:R use:morphing_up.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R.
+ - hauto lq:on ctrs:R.
Qed.
Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
@@ -533,6 +540,7 @@ Module RPar.
- hauto q:on ctrs:R inv:Tm.
- move => n i n0 ρ hρ []//=; first by antiimp.
hauto l:on.
+ - move => n m ρ hρ []//=; hauto lq:on ctrs:R.
Qed.
End RPar.
@@ -573,7 +581,9 @@ Module RPar'.
| ConstCong k :
R (Const k) (Const k)
| UnivCong i :
- R (Univ i) (Univ i).
+ R (Univ i) (Univ i)
+ | BotCong :
+ R Bot Bot.
Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
@@ -639,6 +649,7 @@ Module RPar'.
- hauto lq:on ctrs:R use:morphing_up.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R.
+ - hauto lq:on ctrs:R.
Qed.
Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
@@ -758,6 +769,7 @@ Module RPar'.
- hauto q:on ctrs:R inv:Tm.
- move => n i n0 ρ hρ []//=; first by antiimp.
hauto l:on.
+ - hauto q:on inv:Tm ctrs:R.
Qed.
End RPar'.
@@ -896,7 +908,9 @@ Module EPar.
| ConstCong k :
R (Const k) (Const k)
| UnivCong i :
- R (Univ i) (Univ i).
+ R (Univ i) (Univ i)
+ | BotCong :
+ R Bot Bot.
Lemma refl n (a : Tm n) : EPar.R a a.
Proof.
@@ -941,6 +955,7 @@ Module EPar.
- hauto l:on ctrs:R use:renaming inv:option.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R.
+ - hauto lq:on ctrs:R.
Qed.
Lemma substing n a0 a1 (b0 b1 : Tm n) :
@@ -1344,6 +1359,7 @@ Proof.
- hauto lq:on inv:RPar.R ctrs:EPar.R, rtc use:RPars.BindCong.
- hauto l:on ctrs:EPar.R inv:RPar.R.
- hauto l:on ctrs:EPar.R inv:RPar.R.
+ - hauto l:on ctrs:EPar.R inv:RPar.R.
Qed.
Lemma commutativity1 n (a b0 b1 : Tm n) :
@@ -1457,6 +1473,20 @@ Lemma Const_EPar' n k (u : Tm n) :
- hauto l:on ctrs:OExp.R.
Qed.
+Lemma Bot_EPar' n (u : Tm n) :
+ EPar.R (Bot) u ->
+ rtc OExp.R (Bot) u.
+ move E : (Bot) => t h.
+ move : E. elim : n t u /h => //=.
+ - move => n a0 a1 h ih ?. subst.
+ specialize ih with (1 := eq_refl).
+ hauto lq:on ctrs:OExp.R use:rtc_r.
+ - move => n a0 a1 h ih ?. subst.
+ specialize ih with (1 := eq_refl).
+ hauto lq:on ctrs:OExp.R use:rtc_r.
+ - hauto l:on ctrs:OExp.R.
+Qed.
+
Lemma Univ_EPar' n i (u : Tm n) :
EPar.R (Univ i) u ->
rtc OExp.R (Univ i) u.
@@ -1520,6 +1550,7 @@ Proof.
exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
- qauto use:Const_EPar', EPar.refl.
- qauto use:Univ_EPar', EPar.refl.
+ - qauto use:Bot_EPar', EPar.refl.
Qed.
Function tstar {n} (a : Tm n) :=
@@ -1537,6 +1568,7 @@ Function tstar {n} (a : Tm n) :=
| TBind p a b => TBind p (tstar a) (tstar b)
| Const k => Const k
| Univ i => Univ i
+ | Bot => Bot
end.
Lemma RPar_triangle n (a : Tm n) : forall b, RPar.R a b -> RPar.R b (tstar a).
@@ -1555,6 +1587,7 @@ Proof.
- hauto lq:on inv:RPar.R ctrs:RPar.R.
- hauto lq:on inv:RPar.R ctrs:RPar.R.
- hauto lq:on inv:RPar.R ctrs:RPar.R.
+ - hauto lq:on inv:RPar.R ctrs:RPar.R.
Qed.
Function tstar' {n} (a : Tm n) :=
@@ -1569,6 +1602,7 @@ Function tstar' {n} (a : Tm n) :=
| TBind p a b => TBind p (tstar' a) (tstar' b)
| Const k => Const k
| Univ i => Univ i
+ | Bot => Bot
end.
Lemma RPar'_triangle n (a : Tm n) : forall b, RPar'.R a b -> RPar'.R b (tstar' a).
@@ -1585,6 +1619,7 @@ Proof.
- hauto lq:on inv:RPar'.R ctrs:RPar'.R.
- hauto lq:on inv:RPar'.R ctrs:RPar'.R.
- hauto lq:on inv:RPar'.R ctrs:RPar'.R.
+ - hauto lq:on inv:RPar'.R ctrs:RPar'.R.
Qed.
Lemma RPar_diamond n (c a1 b1 : Tm n) :
@@ -1651,7 +1686,9 @@ Inductive prov {n} : Tm n -> Tm n -> Prop :=
| P_Var i :
prov (VarTm i) (VarTm i)
| P_Univ i :
- prov (Univ i) (Univ i).
+ prov (Univ i) (Univ i)
+| P_Bot :
+ prov Bot Bot.
Lemma ERed_EPar n (a b : Tm n) : ERed.R a b -> EPar.R a b.
Proof.
@@ -1671,6 +1708,7 @@ Proof.
- eauto using EReds.BindCong.
- auto using rtc_refl.
- auto using rtc_refl.
+ - auto using rtc_refl.
Qed.
Lemma EPar_Par n (a b : Tm n) : EPar.R a b -> Par.R a b.
@@ -1723,6 +1761,7 @@ Proof.
- hauto lq:on ctrs:prov inv:RPar.R.
- hauto l:on ctrs:RPar.R inv:RPar.R.
- hauto l:on ctrs:RPar.R inv:RPar.R.
+ - hauto l:on ctrs:RPar.R inv:RPar.R.
Qed.
@@ -1777,6 +1816,7 @@ Proof.
- hauto lq:on inv:ERed.R, prov ctrs:prov.
- hauto lq:on inv:ERed.R, prov ctrs:prov.
- hauto lq:on inv:ERed.R, prov ctrs:prov.
+ - hauto lq:on inv:ERed.R, prov ctrs:prov.
Qed.
Lemma prov_ereds n (u : Tm n) a b : prov u a -> rtc ERed.R a b -> prov u b.
@@ -1787,13 +1827,14 @@ Qed.
Fixpoint extract {n} (a : Tm n) : Tm n :=
match a with
| TBind p A B => TBind p A B
- | Abs a => subst_Tm (scons (Const TPi) VarTm) (extract a)
+ | Abs a => subst_Tm (scons Bot VarTm) (extract a)
| App a b => extract a
| Pair a b => extract a
| Proj p a => extract a
| Const k => Const k
| VarTm i => VarTm i
| Univ i => Univ i
+ | Bot => Bot
end.
Lemma ren_extract n m (a : Tm n) (ξ : fin n -> fin m) :
@@ -1810,6 +1851,7 @@ Proof.
- hauto q:on.
- sfirstorder.
- sfirstorder.
+ - sfirstorder.
Qed.
Lemma ren_morphing n m (a : Tm n) (ρ : fin n -> Tm m) :
@@ -1828,7 +1870,7 @@ Proof.
Qed.
Lemma ren_subst_bot n (a : Tm (S n)) :
- extract (subst_Tm (scons (Const TPi) VarTm) a) = subst_Tm (scons (Const TPi) VarTm) (extract a).
+ extract (subst_Tm (scons Bot VarTm) a) = subst_Tm (scons Bot VarTm) (extract a).
Proof.
apply ren_morphing. destruct i as [i|] => //=.
Qed.
@@ -1839,6 +1881,7 @@ Definition prov_extract_spec {n} u (a : Tm n) :=
| Univ i => extract a = Univ i
| VarTm i => extract a = VarTm i
| (Const i) => extract a = (Const i)
+ | Bot => extract a = Bot
| _ => True
end.
@@ -1851,24 +1894,28 @@ Proof.
- move => h a ha ih.
case : h ha ih => //=.
+ move => i ha ih.
- move /(_ (Const TPi)) in ih.
+ move /(_ Bot) in ih.
rewrite -ih.
by rewrite ren_subst_bot.
+ move => p A B h ih.
- move /(_ (Const TPi)) : ih => [A0][B0][h0][h1]h2.
+ move /(_ Bot) : ih => [A0][B0][h0][h1]h2.
rewrite ren_subst_bot in h0.
rewrite h0.
eauto.
- + move => p _ /(_ (Const TPi)).
+ + move => p _ /(_ Bot).
by rewrite ren_subst_bot.
- + move => i h /(_ (Const TPi)).
+ + move => i h /(_ Bot).
by rewrite ren_subst_bot => ->.
+ + move /(_ Bot).
+ move => h /(_ Bot).
+ by rewrite ren_subst_bot.
- hauto lq:on.
- hauto lq:on.
- hauto lq:on.
- case => //=.
- sfirstorder.
- sfirstorder.
+ - sfirstorder.
Qed.
Definition union {A : Type} (R0 R1 : A -> A -> Prop) a b :=
@@ -2072,6 +2119,7 @@ Proof.
- sfirstorder use:ERPars.BindCong.
- sfirstorder.
- sfirstorder.
+ - sfirstorder.
Qed.
Lemma Pars_ERPar n (a b : Tm n) : rtc Par.R a b -> rtc ERPar.R a b.
@@ -2371,18 +2419,19 @@ Fixpoint ne {n} (a : Tm n) :=
| Proj _ a => ne a
| Pair _ _ => false
| Const _ => false
+ | Bot => true
end
with nf {n} (a : Tm n) :=
match a with
| VarTm i => true
| TBind _ A B => nf A && nf B
- | (Const TPi) => true
| App a b => ne a && nf b
| Abs a => nf a
| Univ _ => true
| Proj _ a => ne a
| Pair a b => nf a && nf b
| Const _ => true
+ | Bot => true
end.
Lemma ne_nf n a : @ne n a -> nf a.
@@ -2469,10 +2518,10 @@ Proof.
Qed.
Lemma ext_wn n (a : Tm n) :
- wn (App a (Const TPi)) ->
+ wn (App a Bot) ->
wn a.
Proof.
- move E : (App a (Const TPi)) => a0 [v [hr hv]].
+ move E : (App a (Bot)) => a0 [v [hr hv]].
move : a E.
move : hv.
elim : a0 v / hr.
@@ -2483,9 +2532,9 @@ Proof.
case : a0 hr0=>// => b0 b1.
elim /RPar'.inv=>// _.
+ move => a0 a3 b2 b3 ? ? [? ?] ? [? ?]. subst.
- have ? : b3 = (Const TPi) by hauto lq:on inv:RPar'.R. subst.
+ have ? : b3 = (Bot) by hauto lq:on inv:RPar'.R. subst.
suff : wn (Abs a3) by hauto lq:on ctrs:RPar'.R, rtc unfold:wn.
- have : wn (subst_Tm (scons (Const TPi) VarTm) a3) by sfirstorder.
+ have : wn (subst_Tm (scons (Bot) VarTm) a3) by sfirstorder.
move => h. apply wn_abs.
move : h. apply wn_antirenaming.
hauto lq:on rew:off inv:option.
diff --git a/theories/logrel.v b/theories/logrel.v
index a4b15d2..a421cda 100644
--- a/theories/logrel.v
+++ b/theories/logrel.v
@@ -1,5 +1,4 @@
-Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax.
-Require Import fp_red.
+Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax fp_red compile.
From Hammer Require Import Tactics.
From Equations Require Import Equations.
Require Import ssreflect ssrbool.
@@ -295,9 +294,9 @@ Proof.
- hauto lq:on ctrs:prov.
- hauto lq:on rew:off ctrs:prov b:on.
- hauto lq:on ctrs:prov.
- - move => n.
+ - move => n k.
have : @prov n Bot Bot by auto using P_Bot.
- tauto.
+ eauto.
Qed.
Lemma ne_pars_inv n (a b : Tm n) :
@@ -311,23 +310,37 @@ Lemma ne_pars_extract n (a b : Tm n) :
ne a -> rtc Par.R a b -> (exists i, extract b = (VarTm i)) \/ extract b = Bot.
Proof. hauto lq:on rew:off use:ne_pars_inv, prov_extract. Qed.
+Lemma const_pars_extract n k b :
+ rtc Par.R (Const k : Tm n) b -> extract b = Const k.
+Proof. hauto l:on use:pars_const_inv, prov_extract. Qed.
+
+Lemma compile_ne n (a : Tm n) :
+ ne a = ne (Compile.F a) /\ nf a = nf (Compile.F a).
+Proof.
+ elim : n / a => //=; sfirstorder b:on.
+Qed.
+
Lemma join_bind_ne_contra n p (A : Tm n) B C :
ne C ->
- join (TBind p A B) C -> False.
+ Join.R (TBind p A B) C -> False.
Proof.
- move => hC [D [h0 h1]].
- move /pars_pi_inv : h0 => [A0 [B0 [h2 [h3 h4]]]].
+ rewrite /Join.R. move => hC /=.
+ rewrite !pair_eq.
+ move => [[[D [h0 h1]] _] _].
+ have {}hC : ne (Compile.F C) by hauto lq:on use:compile_ne.
+ have {}hC : ne (Proj PL (Proj PL (Compile.F C))) by scongruence.
have : (exists i, extract D = (VarTm i)) \/ extract D = Bot by eauto using ne_pars_extract.
+ have : extract D = Const p by eauto using const_pars_extract.
sfirstorder.
Qed.
Lemma join_univ_ne_contra n i C :
ne C ->
- join (Univ i : Tm n) C -> False.
+ Join.R (Univ i : Tm n) C -> False.
Proof.
move => hC [D [h0 h1]].
move /pars_univ_inv : h0 => ?.
- have : (exists i, extract D = (VarTm i)) \/ extract D = Bot by eauto using ne_pars_extract.
+ have : (exists i, extract D = (VarTm i)) \/ exists k, extract D =(Const k) by hauto q:on use:ne_pars_extract, compile_ne.
sfirstorder.
Qed.
@@ -336,7 +349,7 @@ Qed.
Lemma InterpExt_Join n i I (A B : Tm n) PA PB :
⟦ A ⟧ i ;; I ↘ PA ->
⟦ B ⟧ i ;; I ↘ PB ->
- join A B ->
+ Join.R A B ->
PA = PB.
Proof.
move => h. move : B PB. elim : A PA /h.
From 78503149353745d8dd838169b9dbb6dbc3387172 Mon Sep 17 00:00:00 2001
From: Yiyun Liu <liuyiyun@seas.upenn.edu>
Date: Mon, 20 Jan 2025 16:49:47 -0500
Subject: [PATCH 08/13] Add stub for par compile
---
theories/compile.v | 15 +++++++++++++++
theories/logrel.v | 2 +-
2 files changed, 16 insertions(+), 1 deletion(-)
diff --git a/theories/compile.v b/theories/compile.v
index 3eaaf42..f063b16 100644
--- a/theories/compile.v
+++ b/theories/compile.v
@@ -66,6 +66,18 @@ Module Join.
Lemma UnivInj n i j : R (Univ i : Tm n) (Univ j) -> i = j.
Proof. hauto l:on use:join_univ_inj. Qed.
+ Lemma transitive n (a b c : Tm n) :
+ R a b -> R b c -> R a c.
+ Proof. hauto l:on use:join_transitive unfold:R. Qed.
+
+ Lemma symmetric n (a b : Tm n) :
+ R a b -> R b a.
+ Proof. hauto l:on use:join_symmetric. Qed.
+
+ Lemma reflexive n (a : Tm n) :
+ R a a.
+ Proof. hauto l:on use:join_refl. Qed.
+
End Join.
Module Equiv.
@@ -135,3 +147,6 @@ Module EquivJoin.
- hauto l:on.
Qed.
End EquivJoin.
+
+Module CompilePar.
+End CompilePar.
diff --git a/theories/logrel.v b/theories/logrel.v
index a421cda..c822d06 100644
--- a/theories/logrel.v
+++ b/theories/logrel.v
@@ -344,7 +344,7 @@ Proof.
sfirstorder.
Qed.
-#[export]Hint Resolve join_univ_ne_contra join_bind_ne_contra join_univ_pi_contra join_symmetric join_transitive : join.
+#[export]Hint Resolve join_univ_ne_contra join_bind_ne_contra join_univ_pi_contra Join.symmetric Join.transitive : join.
Lemma InterpExt_Join n i I (A B : Tm n) PA PB :
⟦ A ⟧ i ;; I ↘ PA ->
From 36427daa61ba60fffd0308e1ec452c938ac4be43 Mon Sep 17 00:00:00 2001
From: Yiyun Liu <liuyiyun@seas.upenn.edu>
Date: Mon, 20 Jan 2025 20:11:51 -0500
Subject: [PATCH 09/13] Refactor the equational theory to use the compile
function
---
theories/compile.v | 33 +++++++++++++++++++++++++++--
theories/fp_red.v | 10 ---------
theories/logrel.v | 52 ++++++++++++++++++++++++++++++++--------------
3 files changed, 67 insertions(+), 28 deletions(-)
diff --git a/theories/compile.v b/theories/compile.v
index f063b16..05bc9a8 100644
--- a/theories/compile.v
+++ b/theories/compile.v
@@ -1,6 +1,7 @@
Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax fp_red.
Require Import ssreflect ssrbool.
From Hammer Require Import Tactics.
+From stdpp Require Import relations (rtc(..)).
Module Compile.
Fixpoint F {n} (a : Tm n) : Tm n :=
@@ -78,6 +79,16 @@ Module Join.
R a a.
Proof. hauto l:on use:join_refl. Qed.
+ Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
+ R a b -> R (subst_Tm ρ a) (subst_Tm ρ b).
+ Proof.
+ rewrite /R.
+ rewrite /join.
+ move => [C [h0 h1]].
+ repeat (rewrite <- Compile.morphing with (ρ0 := funcomp Compile.F ρ); last by reflexivity).
+ hauto lq:on use:join_substing.
+ Qed.
+
End Join.
Module Equiv.
@@ -148,5 +159,23 @@ Module EquivJoin.
Qed.
End EquivJoin.
-Module CompilePar.
-End CompilePar.
+Lemma compile_rpar n (a b : Tm n) : RPar'.R a b -> RPar'.R (Compile.F a) (Compile.F b).
+Proof.
+ move => h. elim : n a b /h.
+ - move => n a0 a1 b0 b1 ha iha hb ihb /=.
+ rewrite -Compile.substing.
+ apply RPar'.AppAbs => //.
+ - hauto q:on use:RPar'.ProjPair'.
+ - qauto ctrs:RPar'.R.
+ - hauto lq:on ctrs:RPar'.R.
+ - hauto lq:on ctrs:RPar'.R.
+ - hauto lq:on ctrs:RPar'.R.
+ - hauto lq:on ctrs:RPar'.R.
+ - hauto lq:on ctrs:RPar'.R.
+ - hauto lq:on ctrs:RPar'.R.
+ - hauto lq:on ctrs:RPar'.R.
+ - hauto lq:on ctrs:RPar'.R.
+Qed.
+
+Lemma compile_rpars n (a b : Tm n) : rtc RPar'.R a b -> rtc RPar'.R (Compile.F a) (Compile.F b).
+Proof. induction 1; hauto lq:on ctrs:rtc use:compile_rpar. Qed.
diff --git a/theories/fp_red.v b/theories/fp_red.v
index 6932f84..e9b5591 100644
--- a/theories/fp_red.v
+++ b/theories/fp_red.v
@@ -2394,16 +2394,6 @@ Proof.
sfirstorder unfold:join.
Qed.
-Lemma join_univ_pi_contra n p (A : Tm n) B i :
- join (TBind p A B) (Univ i) -> False.
-Proof.
- rewrite /join.
- move => [c [h0 h1]].
- move /pars_univ_inv : h1.
- move /pars_pi_inv : h0.
- hauto l:on.
-Qed.
-
Lemma join_substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
join a b ->
join (subst_Tm ρ a) (subst_Tm ρ b).
diff --git a/theories/logrel.v b/theories/logrel.v
index c822d06..b9c854d 100644
--- a/theories/logrel.v
+++ b/theories/logrel.v
@@ -262,8 +262,12 @@ Lemma RPar's_Pars n (A B : Tm n) :
Proof. hauto lq:on use:RPar'_Par, rtc_subrel. Qed.
Lemma RPar's_join n (A B : Tm n) :
- rtc RPar'.R A B -> join A B.
-Proof. hauto lq:on ctrs:rtc use:RPar's_Pars. Qed.
+ rtc RPar'.R A B -> Join.R A B.
+Proof.
+ rewrite /Join.R => h.
+ have {}h : rtc RPar'.R (Compile.F A) (Compile.F B) by eauto using compile_rpars.
+ rewrite /join. eauto using RPar's_Pars, rtc_refl.
+Qed.
Lemma bindspace_iff n p (PA : Tm n -> Prop) PF PF0 b :
(forall (a : Tm n) (PB PB0 : Tm n -> Prop), PF a PB -> PF0 a PB0 -> PB = PB0) ->
@@ -320,6 +324,19 @@ Proof.
elim : n / a => //=; sfirstorder b:on.
Qed.
+Lemma join_univ_pi_contra n p (A : Tm n) B i :
+ Join.R (TBind p A B) (Univ i) -> False.
+Proof.
+ rewrite /Join.R /=.
+ rewrite !pair_eq.
+ move => [[h _ ]_ ].
+ move : h => [C [h0 h1]].
+ have ? : extract C = Const p by hauto l:on use:pars_const_inv.
+ have h : prov (Univ i : Tm n) (Proj PL (Proj PL (Univ i))) by hauto lq:on ctrs:prov.
+ have {h} : prov (Univ i) C by eauto using prov_pars.
+ move /prov_extract=>/=. congruence.
+Qed.
+
Lemma join_bind_ne_contra n p (A : Tm n) B C :
ne C ->
Join.R (TBind p A B) C -> False.
@@ -370,9 +387,9 @@ Proof.
move /InterpExt_Bind_inv : hPi.
move => [PA0][PF0][hPA0][hTot0][hRes0]?. subst.
move => hjoin.
- have{}hr : join U (TBind p0 A0 B0) by auto using RPar's_join.
- have hj : join (TBind p A B) (TBind p0 A0 B0) by eauto using join_transitive.
- have {hj} : p0 = p /\ join A A0 /\ join B B0 by hauto l:on use:join_pi_inj.
+ have{}hr : Join.R U (TBind p0 A0 B0) by auto using RPar's_join.
+ have hj : Join.R (TBind p A B) (TBind p0 A0 B0) by eauto using Join.transitive.
+ have {hj} : p0 = p /\ Join.R A A0 /\ Join.R B B0 by hauto l:on use:Join.BindInj.
move => [? [h0 h1]]. subst.
have ? : PA0 = PA by hauto l:on. subst.
rewrite /ProdSpace.
@@ -381,13 +398,15 @@ Proof.
apply bindspace_iff; eauto.
move => a PB PB0 hPB hPB0.
apply : ihPF; eauto.
- by apply join_substing.
+ rewrite /Join.R.
+ rewrite -!Compile.substing.
+ hauto l:on use:join_substing.
+ move => j ?. subst.
move => [h0 h1] h.
- have ? : join U (Univ j) by eauto using RPar's_join.
- have : join (TBind p A B) (Univ j) by eauto using join_transitive.
+ have ? : Join.R U (Univ j) by eauto using RPar's_join.
+ have : Join.R (TBind p A B) (Univ j) by eauto using Join.transitive.
move => ?. exfalso.
- eauto using join_univ_pi_contra.
+ hauto l: on use: join_univ_pi_contra.
+ move => A0 ? ? [/RPar's_join ?]. subst.
move => _ ?. exfalso. eauto with join.
- move => j ? B PB /InterpExtInv.
@@ -397,19 +416,19 @@ Proof.
move /RPar's_join => *.
exfalso. eauto with join.
+ move => m _ [/RPar's_join h0 + h1].
- have /join_univ_inj {h0 h1} ? : join (Univ j : Tm n) (Univ m) by eauto using join_transitive.
+ have /join_univ_inj {h0 h1} ? : Join.R (Univ j : Tm n) (Univ m) by eauto using Join.transitive.
subst.
move /InterpExt_Univ_inv. firstorder.
+ move => A ? ? [/RPar's_join] *. subst. exfalso. eauto with join.
- move => A A0 PA h.
- have /join_symmetric {}h : join A A0 by hauto lq:on ctrs:rtc use:RPar'_Par, relations.rtc_once.
- eauto using join_transitive.
+ have /Join.symmetric {}h : Join.R A A0 by hauto lq:on ctrs:rtc use:RPar's_join, relations.rtc_once.
+ eauto with join.
Qed.
Lemma InterpUniv_Join n i (A B : Tm n) PA PB :
⟦ A ⟧ i ↘ PA ->
⟦ B ⟧ i ↘ PB ->
- join A B ->
+ Join.R A B ->
PA = PB.
Proof. hauto l:on use:InterpExt_Join rew:db:InterpUniv. Qed.
@@ -442,7 +461,7 @@ Proof. hauto use:InterpExt_Functional rew:db:InterpUniv. Qed.
Lemma InterpUniv_Join' n i j (A B : Tm n) PA PB :
⟦ A ⟧ i ↘ PA ->
⟦ B ⟧ j ↘ PB ->
- join A B ->
+ Join.R A B ->
PA = PB.
Proof.
have [? ?] : i <= max i j /\ j <= max i j by lia.
@@ -749,12 +768,12 @@ Qed.
Lemma ST_Conv n Γ (a : Tm n) A B i :
Γ ⊨ a ∈ A ->
Γ ⊨ B ∈ Univ i ->
- join A B ->
+ Join.R A B ->
Γ ⊨ a ∈ B.
Proof.
move => ha /SemWt_Univ h h0.
move => ρ hρ.
- have {}h0 : join (subst_Tm ρ A) (subst_Tm ρ B) by eauto using join_substing.
+ have {}h0 : Join.R (subst_Tm ρ A) (subst_Tm ρ B) by eauto using Join.substing.
move /ha : (hρ){ha} => [m [PA [h1 h2]]].
move /h : (hρ){h} => [S hS].
have ? : PA = S by eauto using InterpUniv_Join'. subst.
@@ -909,6 +928,7 @@ Fixpoint size_tm {n} (a : Tm n) :=
| Proj p a => 1 + size_tm a
| Pair a b => 1 + Nat.add (size_tm a) (size_tm b)
| Bot => 1
+ | Const _ => 1
| Univ _ => 1
end.
From 1f7460fd11d9dc166856288ab771c8b285097c17 Mon Sep 17 00:00:00 2001
From: Yiyun Liu <liuyiyun@seas.upenn.edu>
Date: Mon, 20 Jan 2025 20:42:40 -0500
Subject: [PATCH 10/13] Add new syntax for booleans
---
syntax.sig | 4 ++
theories/Autosubst2/syntax.v | 86 +++++++++++++++++++++++++++++++++++-
theories/compile.v | 7 ++-
3 files changed, 95 insertions(+), 2 deletions(-)
diff --git a/syntax.sig b/syntax.sig
index 3101cab..9e04431 100644
--- a/syntax.sig
+++ b/syntax.sig
@@ -2,6 +2,7 @@ nat : Type
Tm(VarTm) : Type
PTag : Type
TTag : Type
+bool : Type
PL : PTag
PR : PTag
@@ -15,3 +16,6 @@ TBind : TTag -> Tm -> (bind Tm in Tm) -> Tm
Const : TTag -> Tm
Univ : nat -> Tm
Bot : Tm
+BVal : bool -> Tm
+Bool : Tm
+If : Tm -> Tm -> Tm -> Tm
diff --git a/theories/Autosubst2/syntax.v b/theories/Autosubst2/syntax.v
index a5cb002..5b06281 100644
--- a/theories/Autosubst2/syntax.v
+++ b/theories/Autosubst2/syntax.v
@@ -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}.
diff --git a/theories/compile.v b/theories/compile.v
index 05bc9a8..e481d71 100644
--- a/theories/compile.v
+++ b/theories/compile.v
@@ -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)) :
From 255bd4acbf81260957643466a8edc6f70129ba70 Mon Sep 17 00:00:00 2001
From: Yiyun Liu <yliu8e5f7@protonmail.com>
Date: Fri, 24 Jan 2025 14:52:35 -0700
Subject: [PATCH 11/13] Rename the term constructors
---
syntax.sig | 12 +-
theories/Autosubst2/syntax.v | 855 +++++++++++++++++++++++--
theories/diagram.txt | 22 +
theories/fp_red.v | 1132 +++++++++++++++-------------------
4 files changed, 1323 insertions(+), 698 deletions(-)
diff --git a/syntax.sig b/syntax.sig
index 9e04431..e63c3b4 100644
--- a/syntax.sig
+++ b/syntax.sig
@@ -1,4 +1,5 @@
nat : Type
+PTm(VarPTm) : Type
Tm(VarTm) : Type
PTag : Type
TTag : Type
@@ -8,14 +9,21 @@ PL : PTag
PR : PTag
TPi : TTag
TSig : TTag
+
+PAbs : (bind PTm in PTm) -> PTm
+PApp : PTm -> PTm -> PTm
+PPair : PTm -> PTm -> PTm
+PProj : PTag -> PTm -> PTm
+PConst : TTag -> PTm
+PUniv : nat -> PTm
+PBot : PTm
+
Abs : (bind Tm in Tm) -> Tm
App : Tm -> Tm -> Tm
Pair : Tm -> Tm -> Tm
Proj : PTag -> Tm -> Tm
TBind : TTag -> Tm -> (bind Tm in Tm) -> Tm
-Const : TTag -> Tm
Univ : nat -> Tm
-Bot : Tm
BVal : bool -> Tm
Bool : Tm
If : Tm -> Tm -> Tm -> Tm
diff --git a/theories/Autosubst2/syntax.v b/theories/Autosubst2/syntax.v
index 5b06281..75ef645 100644
--- a/theories/Autosubst2/syntax.v
+++ b/theories/Autosubst2/syntax.v
@@ -33,6 +33,674 @@ Proof.
exact (eq_refl).
Qed.
+Inductive PTm (n_PTm : nat) : Type :=
+ | VarPTm : fin n_PTm -> PTm n_PTm
+ | PAbs : PTm (S n_PTm) -> PTm n_PTm
+ | PApp : PTm n_PTm -> PTm n_PTm -> PTm n_PTm
+ | PPair : PTm n_PTm -> PTm n_PTm -> PTm n_PTm
+ | PProj : PTag -> PTm n_PTm -> PTm n_PTm
+ | PConst : TTag -> PTm n_PTm
+ | PUniv : nat -> PTm n_PTm
+ | PBot : PTm n_PTm.
+
+Lemma congr_PAbs {m_PTm : nat} {s0 : PTm (S m_PTm)} {t0 : PTm (S m_PTm)}
+ (H0 : s0 = t0) : PAbs m_PTm s0 = PAbs m_PTm t0.
+Proof.
+exact (eq_trans eq_refl (ap (fun x => PAbs m_PTm x) H0)).
+Qed.
+
+Lemma congr_PApp {m_PTm : nat} {s0 : PTm m_PTm} {s1 : PTm m_PTm}
+ {t0 : PTm m_PTm} {t1 : PTm m_PTm} (H0 : s0 = t0) (H1 : s1 = t1) :
+ PApp m_PTm s0 s1 = PApp m_PTm t0 t1.
+Proof.
+exact (eq_trans (eq_trans eq_refl (ap (fun x => PApp m_PTm x s1) H0))
+ (ap (fun x => PApp m_PTm t0 x) H1)).
+Qed.
+
+Lemma congr_PPair {m_PTm : nat} {s0 : PTm m_PTm} {s1 : PTm m_PTm}
+ {t0 : PTm m_PTm} {t1 : PTm m_PTm} (H0 : s0 = t0) (H1 : s1 = t1) :
+ PPair m_PTm s0 s1 = PPair m_PTm t0 t1.
+Proof.
+exact (eq_trans (eq_trans eq_refl (ap (fun x => PPair m_PTm x s1) H0))
+ (ap (fun x => PPair m_PTm t0 x) H1)).
+Qed.
+
+Lemma congr_PProj {m_PTm : nat} {s0 : PTag} {s1 : PTm m_PTm} {t0 : PTag}
+ {t1 : PTm m_PTm} (H0 : s0 = t0) (H1 : s1 = t1) :
+ PProj m_PTm s0 s1 = PProj m_PTm t0 t1.
+Proof.
+exact (eq_trans (eq_trans eq_refl (ap (fun x => PProj m_PTm x s1) H0))
+ (ap (fun x => PProj m_PTm t0 x) H1)).
+Qed.
+
+Lemma congr_PConst {m_PTm : nat} {s0 : TTag} {t0 : TTag} (H0 : s0 = t0) :
+ PConst m_PTm s0 = PConst m_PTm t0.
+Proof.
+exact (eq_trans eq_refl (ap (fun x => PConst m_PTm x) H0)).
+Qed.
+
+Lemma congr_PUniv {m_PTm : nat} {s0 : nat} {t0 : nat} (H0 : s0 = t0) :
+ PUniv m_PTm s0 = PUniv m_PTm t0.
+Proof.
+exact (eq_trans eq_refl (ap (fun x => PUniv m_PTm x) H0)).
+Qed.
+
+Lemma congr_PBot {m_PTm : nat} : PBot m_PTm = PBot m_PTm.
+Proof.
+exact (eq_refl).
+Qed.
+
+Lemma upRen_PTm_PTm {m : nat} {n : nat} (xi : fin m -> fin n) :
+ fin (S m) -> fin (S n).
+Proof.
+exact (up_ren xi).
+Defined.
+
+Lemma upRen_list_PTm_PTm (p : nat) {m : nat} {n : nat} (xi : fin m -> fin n)
+ : fin (plus p m) -> fin (plus p n).
+Proof.
+exact (upRen_p p xi).
+Defined.
+
+Fixpoint ren_PTm {m_PTm : nat} {n_PTm : nat}
+(xi_PTm : fin m_PTm -> fin n_PTm) (s : PTm m_PTm) {struct s} : PTm n_PTm :=
+ match s with
+ | VarPTm _ s0 => VarPTm n_PTm (xi_PTm s0)
+ | PAbs _ s0 => PAbs n_PTm (ren_PTm (upRen_PTm_PTm xi_PTm) s0)
+ | PApp _ s0 s1 => PApp n_PTm (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
+ | PPair _ s0 s1 => PPair n_PTm (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
+ | PProj _ s0 s1 => PProj n_PTm s0 (ren_PTm xi_PTm s1)
+ | PConst _ s0 => PConst n_PTm s0
+ | PUniv _ s0 => PUniv n_PTm s0
+ | PBot _ => PBot n_PTm
+ end.
+
+Lemma up_PTm_PTm {m : nat} {n_PTm : nat} (sigma : fin m -> PTm n_PTm) :
+ fin (S m) -> PTm (S n_PTm).
+Proof.
+exact (scons (VarPTm (S n_PTm) var_zero) (funcomp (ren_PTm shift) sigma)).
+Defined.
+
+Lemma up_list_PTm_PTm (p : nat) {m : nat} {n_PTm : nat}
+ (sigma : fin m -> PTm n_PTm) : fin (plus p m) -> PTm (plus p n_PTm).
+Proof.
+exact (scons_p p (funcomp (VarPTm (plus p n_PTm)) (zero_p p))
+ (funcomp (ren_PTm (shift_p p)) sigma)).
+Defined.
+
+Fixpoint subst_PTm {m_PTm : nat} {n_PTm : nat}
+(sigma_PTm : fin m_PTm -> PTm n_PTm) (s : PTm m_PTm) {struct s} : PTm n_PTm
+:=
+ match s with
+ | VarPTm _ s0 => sigma_PTm s0
+ | PAbs _ s0 => PAbs n_PTm (subst_PTm (up_PTm_PTm sigma_PTm) s0)
+ | PApp _ s0 s1 =>
+ PApp n_PTm (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1)
+ | PPair _ s0 s1 =>
+ PPair n_PTm (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1)
+ | PProj _ s0 s1 => PProj n_PTm s0 (subst_PTm sigma_PTm s1)
+ | PConst _ s0 => PConst n_PTm s0
+ | PUniv _ s0 => PUniv n_PTm s0
+ | PBot _ => PBot n_PTm
+ end.
+
+Lemma upId_PTm_PTm {m_PTm : nat} (sigma : fin m_PTm -> PTm m_PTm)
+ (Eq : forall x, sigma x = VarPTm m_PTm x) :
+ forall x, up_PTm_PTm sigma x = VarPTm (S m_PTm) x.
+Proof.
+exact (fun n =>
+ match n with
+ | Some fin_n => ap (ren_PTm shift) (Eq fin_n)
+ | None => eq_refl
+ end).
+Qed.
+
+Lemma upId_list_PTm_PTm {p : nat} {m_PTm : nat}
+ (sigma : fin m_PTm -> PTm m_PTm) (Eq : forall x, sigma x = VarPTm m_PTm x)
+ : forall x, up_list_PTm_PTm p sigma x = VarPTm (plus p m_PTm) x.
+Proof.
+exact (fun n =>
+ scons_p_eta (VarPTm (plus p m_PTm))
+ (fun n => ap (ren_PTm (shift_p p)) (Eq n)) (fun n => eq_refl)).
+Qed.
+
+Fixpoint idSubst_PTm {m_PTm : nat} (sigma_PTm : fin m_PTm -> PTm m_PTm)
+(Eq_PTm : forall x, sigma_PTm x = VarPTm m_PTm x) (s : PTm m_PTm) {struct s}
+ : subst_PTm sigma_PTm s = s :=
+ match s with
+ | VarPTm _ s0 => Eq_PTm s0
+ | PAbs _ s0 =>
+ congr_PAbs
+ (idSubst_PTm (up_PTm_PTm sigma_PTm) (upId_PTm_PTm _ Eq_PTm) s0)
+ | PApp _ s0 s1 =>
+ congr_PApp (idSubst_PTm sigma_PTm Eq_PTm s0)
+ (idSubst_PTm sigma_PTm Eq_PTm s1)
+ | PPair _ s0 s1 =>
+ congr_PPair (idSubst_PTm sigma_PTm Eq_PTm s0)
+ (idSubst_PTm sigma_PTm Eq_PTm s1)
+ | PProj _ s0 s1 =>
+ congr_PProj (eq_refl s0) (idSubst_PTm sigma_PTm Eq_PTm s1)
+ | PConst _ s0 => congr_PConst (eq_refl s0)
+ | PUniv _ s0 => congr_PUniv (eq_refl s0)
+ | PBot _ => congr_PBot
+ end.
+
+Lemma upExtRen_PTm_PTm {m : nat} {n : nat} (xi : fin m -> fin n)
+ (zeta : fin m -> fin n) (Eq : forall x, xi x = zeta x) :
+ forall x, upRen_PTm_PTm xi x = upRen_PTm_PTm zeta x.
+Proof.
+exact (fun n =>
+ match n with
+ | Some fin_n => ap shift (Eq fin_n)
+ | None => eq_refl
+ end).
+Qed.
+
+Lemma upExtRen_list_PTm_PTm {p : nat} {m : nat} {n : nat}
+ (xi : fin m -> fin n) (zeta : fin m -> fin n)
+ (Eq : forall x, xi x = zeta x) :
+ forall x, upRen_list_PTm_PTm p xi x = upRen_list_PTm_PTm p zeta x.
+Proof.
+exact (fun n =>
+ scons_p_congr (fun n => eq_refl) (fun n => ap (shift_p p) (Eq n))).
+Qed.
+
+Fixpoint extRen_PTm {m_PTm : nat} {n_PTm : nat}
+(xi_PTm : fin m_PTm -> fin n_PTm) (zeta_PTm : fin m_PTm -> fin n_PTm)
+(Eq_PTm : forall x, xi_PTm x = zeta_PTm x) (s : PTm m_PTm) {struct s} :
+ren_PTm xi_PTm s = ren_PTm zeta_PTm s :=
+ match s with
+ | VarPTm _ s0 => ap (VarPTm n_PTm) (Eq_PTm s0)
+ | PAbs _ s0 =>
+ congr_PAbs
+ (extRen_PTm (upRen_PTm_PTm xi_PTm) (upRen_PTm_PTm zeta_PTm)
+ (upExtRen_PTm_PTm _ _ Eq_PTm) s0)
+ | PApp _ s0 s1 =>
+ congr_PApp (extRen_PTm xi_PTm zeta_PTm Eq_PTm s0)
+ (extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
+ | PPair _ s0 s1 =>
+ congr_PPair (extRen_PTm xi_PTm zeta_PTm Eq_PTm s0)
+ (extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
+ | PProj _ s0 s1 =>
+ congr_PProj (eq_refl s0) (extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
+ | PConst _ s0 => congr_PConst (eq_refl s0)
+ | PUniv _ s0 => congr_PUniv (eq_refl s0)
+ | PBot _ => congr_PBot
+ end.
+
+Lemma upExt_PTm_PTm {m : nat} {n_PTm : nat} (sigma : fin m -> PTm n_PTm)
+ (tau : fin m -> PTm n_PTm) (Eq : forall x, sigma x = tau x) :
+ forall x, up_PTm_PTm sigma x = up_PTm_PTm tau x.
+Proof.
+exact (fun n =>
+ match n with
+ | Some fin_n => ap (ren_PTm shift) (Eq fin_n)
+ | None => eq_refl
+ end).
+Qed.
+
+Lemma upExt_list_PTm_PTm {p : nat} {m : nat} {n_PTm : nat}
+ (sigma : fin m -> PTm n_PTm) (tau : fin m -> PTm n_PTm)
+ (Eq : forall x, sigma x = tau x) :
+ forall x, up_list_PTm_PTm p sigma x = up_list_PTm_PTm p tau x.
+Proof.
+exact (fun n =>
+ scons_p_congr (fun n => eq_refl)
+ (fun n => ap (ren_PTm (shift_p p)) (Eq n))).
+Qed.
+
+Fixpoint ext_PTm {m_PTm : nat} {n_PTm : nat}
+(sigma_PTm : fin m_PTm -> PTm n_PTm) (tau_PTm : fin m_PTm -> PTm n_PTm)
+(Eq_PTm : forall x, sigma_PTm x = tau_PTm x) (s : PTm m_PTm) {struct s} :
+subst_PTm sigma_PTm s = subst_PTm tau_PTm s :=
+ match s with
+ | VarPTm _ s0 => Eq_PTm s0
+ | PAbs _ s0 =>
+ congr_PAbs
+ (ext_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
+ (upExt_PTm_PTm _ _ Eq_PTm) s0)
+ | PApp _ s0 s1 =>
+ congr_PApp (ext_PTm sigma_PTm tau_PTm Eq_PTm s0)
+ (ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
+ | PPair _ s0 s1 =>
+ congr_PPair (ext_PTm sigma_PTm tau_PTm Eq_PTm s0)
+ (ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
+ | PProj _ s0 s1 =>
+ congr_PProj (eq_refl s0) (ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
+ | PConst _ s0 => congr_PConst (eq_refl s0)
+ | PUniv _ s0 => congr_PUniv (eq_refl s0)
+ | PBot _ => congr_PBot
+ end.
+
+Lemma up_ren_ren_PTm_PTm {k : nat} {l : nat} {m : nat} (xi : fin k -> fin l)
+ (zeta : fin l -> fin m) (rho : fin k -> fin m)
+ (Eq : forall x, funcomp zeta xi x = rho x) :
+ forall x,
+ funcomp (upRen_PTm_PTm zeta) (upRen_PTm_PTm xi) x = upRen_PTm_PTm rho x.
+Proof.
+exact (up_ren_ren xi zeta rho Eq).
+Qed.
+
+Lemma up_ren_ren_list_PTm_PTm {p : nat} {k : nat} {l : nat} {m : nat}
+ (xi : fin k -> fin l) (zeta : fin l -> fin m) (rho : fin k -> fin m)
+ (Eq : forall x, funcomp zeta xi x = rho x) :
+ forall x,
+ funcomp (upRen_list_PTm_PTm p zeta) (upRen_list_PTm_PTm p xi) x =
+ upRen_list_PTm_PTm p rho x.
+Proof.
+exact (up_ren_ren_p Eq).
+Qed.
+
+Fixpoint compRenRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
+(xi_PTm : fin m_PTm -> fin k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm)
+(rho_PTm : fin m_PTm -> fin l_PTm)
+(Eq_PTm : forall x, funcomp zeta_PTm xi_PTm x = rho_PTm x) (s : PTm m_PTm)
+{struct s} : ren_PTm zeta_PTm (ren_PTm xi_PTm s) = ren_PTm rho_PTm s :=
+ match s with
+ | VarPTm _ s0 => ap (VarPTm l_PTm) (Eq_PTm s0)
+ | PAbs _ s0 =>
+ congr_PAbs
+ (compRenRen_PTm (upRen_PTm_PTm xi_PTm) (upRen_PTm_PTm zeta_PTm)
+ (upRen_PTm_PTm rho_PTm) (up_ren_ren _ _ _ Eq_PTm) s0)
+ | PApp _ s0 s1 =>
+ congr_PApp (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s0)
+ (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
+ | PPair _ s0 s1 =>
+ congr_PPair (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s0)
+ (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
+ | PProj _ s0 s1 =>
+ congr_PProj (eq_refl s0)
+ (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
+ | PConst _ s0 => congr_PConst (eq_refl s0)
+ | PUniv _ s0 => congr_PUniv (eq_refl s0)
+ | PBot _ => congr_PBot
+ end.
+
+Lemma up_ren_subst_PTm_PTm {k : nat} {l : nat} {m_PTm : nat}
+ (xi : fin k -> fin l) (tau : fin l -> PTm m_PTm)
+ (theta : fin k -> PTm m_PTm) (Eq : forall x, funcomp tau xi x = theta x) :
+ forall x,
+ funcomp (up_PTm_PTm tau) (upRen_PTm_PTm xi) x = up_PTm_PTm theta x.
+Proof.
+exact (fun n =>
+ match n with
+ | Some fin_n => ap (ren_PTm shift) (Eq fin_n)
+ | None => eq_refl
+ end).
+Qed.
+
+Lemma up_ren_subst_list_PTm_PTm {p : nat} {k : nat} {l : nat} {m_PTm : nat}
+ (xi : fin k -> fin l) (tau : fin l -> PTm m_PTm)
+ (theta : fin k -> PTm m_PTm) (Eq : forall x, funcomp tau xi x = theta x) :
+ forall x,
+ funcomp (up_list_PTm_PTm p tau) (upRen_list_PTm_PTm p xi) x =
+ up_list_PTm_PTm p theta x.
+Proof.
+exact (fun n =>
+ eq_trans (scons_p_comp' _ _ _ n)
+ (scons_p_congr (fun z => scons_p_head' _ _ z)
+ (fun z =>
+ eq_trans (scons_p_tail' _ _ (xi z))
+ (ap (ren_PTm (shift_p p)) (Eq z))))).
+Qed.
+
+Fixpoint compRenSubst_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
+(xi_PTm : fin m_PTm -> fin k_PTm) (tau_PTm : fin k_PTm -> PTm l_PTm)
+(theta_PTm : fin m_PTm -> PTm l_PTm)
+(Eq_PTm : forall x, funcomp tau_PTm xi_PTm x = theta_PTm x) (s : PTm m_PTm)
+{struct s} : subst_PTm tau_PTm (ren_PTm xi_PTm s) = subst_PTm theta_PTm s :=
+ match s with
+ | VarPTm _ s0 => Eq_PTm s0
+ | PAbs _ s0 =>
+ congr_PAbs
+ (compRenSubst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm tau_PTm)
+ (up_PTm_PTm theta_PTm) (up_ren_subst_PTm_PTm _ _ _ Eq_PTm) s0)
+ | PApp _ s0 s1 =>
+ congr_PApp (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s0)
+ (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
+ | PPair _ s0 s1 =>
+ congr_PPair (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s0)
+ (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
+ | PProj _ s0 s1 =>
+ congr_PProj (eq_refl s0)
+ (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
+ | PConst _ s0 => congr_PConst (eq_refl s0)
+ | PUniv _ s0 => congr_PUniv (eq_refl s0)
+ | PBot _ => congr_PBot
+ end.
+
+Lemma up_subst_ren_PTm_PTm {k : nat} {l_PTm : nat} {m_PTm : nat}
+ (sigma : fin k -> PTm l_PTm) (zeta_PTm : fin l_PTm -> fin m_PTm)
+ (theta : fin k -> PTm m_PTm)
+ (Eq : forall x, funcomp (ren_PTm zeta_PTm) sigma x = theta x) :
+ forall x,
+ funcomp (ren_PTm (upRen_PTm_PTm zeta_PTm)) (up_PTm_PTm sigma) x =
+ up_PTm_PTm theta x.
+Proof.
+exact (fun n =>
+ match n with
+ | Some fin_n =>
+ eq_trans
+ (compRenRen_PTm shift (upRen_PTm_PTm zeta_PTm)
+ (funcomp shift zeta_PTm) (fun x => eq_refl) (sigma fin_n))
+ (eq_trans
+ (eq_sym
+ (compRenRen_PTm zeta_PTm shift (funcomp shift zeta_PTm)
+ (fun x => eq_refl) (sigma fin_n)))
+ (ap (ren_PTm shift) (Eq fin_n)))
+ | None => eq_refl
+ end).
+Qed.
+
+Lemma up_subst_ren_list_PTm_PTm {p : nat} {k : nat} {l_PTm : nat}
+ {m_PTm : nat} (sigma : fin k -> PTm l_PTm)
+ (zeta_PTm : fin l_PTm -> fin m_PTm) (theta : fin k -> PTm m_PTm)
+ (Eq : forall x, funcomp (ren_PTm zeta_PTm) sigma x = theta x) :
+ forall x,
+ funcomp (ren_PTm (upRen_list_PTm_PTm p zeta_PTm)) (up_list_PTm_PTm p sigma)
+ x = up_list_PTm_PTm p theta x.
+Proof.
+exact (fun n =>
+ eq_trans (scons_p_comp' _ _ _ n)
+ (scons_p_congr
+ (fun x => ap (VarPTm (plus p m_PTm)) (scons_p_head' _ _ x))
+ (fun n =>
+ eq_trans
+ (compRenRen_PTm (shift_p p) (upRen_list_PTm_PTm p zeta_PTm)
+ (funcomp (shift_p p) zeta_PTm)
+ (fun x => scons_p_tail' _ _ x) (sigma n))
+ (eq_trans
+ (eq_sym
+ (compRenRen_PTm zeta_PTm (shift_p p)
+ (funcomp (shift_p p) zeta_PTm) (fun x => eq_refl)
+ (sigma n))) (ap (ren_PTm (shift_p p)) (Eq n)))))).
+Qed.
+
+Fixpoint compSubstRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
+(sigma_PTm : fin m_PTm -> PTm k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm)
+(theta_PTm : fin m_PTm -> PTm l_PTm)
+(Eq_PTm : forall x, funcomp (ren_PTm zeta_PTm) sigma_PTm x = theta_PTm x)
+(s : PTm m_PTm) {struct s} :
+ren_PTm zeta_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
+ match s with
+ | VarPTm _ s0 => Eq_PTm s0
+ | PAbs _ s0 =>
+ congr_PAbs
+ (compSubstRen_PTm (up_PTm_PTm sigma_PTm) (upRen_PTm_PTm zeta_PTm)
+ (up_PTm_PTm theta_PTm) (up_subst_ren_PTm_PTm _ _ _ Eq_PTm) s0)
+ | PApp _ s0 s1 =>
+ congr_PApp (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s0)
+ (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
+ | PPair _ s0 s1 =>
+ congr_PPair (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s0)
+ (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
+ | PProj _ s0 s1 =>
+ congr_PProj (eq_refl s0)
+ (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
+ | PConst _ s0 => congr_PConst (eq_refl s0)
+ | PUniv _ s0 => congr_PUniv (eq_refl s0)
+ | PBot _ => congr_PBot
+ end.
+
+Lemma up_subst_subst_PTm_PTm {k : nat} {l_PTm : nat} {m_PTm : nat}
+ (sigma : fin k -> PTm l_PTm) (tau_PTm : fin l_PTm -> PTm m_PTm)
+ (theta : fin k -> PTm m_PTm)
+ (Eq : forall x, funcomp (subst_PTm tau_PTm) sigma x = theta x) :
+ forall x,
+ funcomp (subst_PTm (up_PTm_PTm tau_PTm)) (up_PTm_PTm sigma) x =
+ up_PTm_PTm theta x.
+Proof.
+exact (fun n =>
+ match n with
+ | Some fin_n =>
+ eq_trans
+ (compRenSubst_PTm shift (up_PTm_PTm tau_PTm)
+ (funcomp (up_PTm_PTm tau_PTm) shift) (fun x => eq_refl)
+ (sigma fin_n))
+ (eq_trans
+ (eq_sym
+ (compSubstRen_PTm tau_PTm shift
+ (funcomp (ren_PTm shift) tau_PTm) (fun x => eq_refl)
+ (sigma fin_n))) (ap (ren_PTm shift) (Eq fin_n)))
+ | None => eq_refl
+ end).
+Qed.
+
+Lemma up_subst_subst_list_PTm_PTm {p : nat} {k : nat} {l_PTm : nat}
+ {m_PTm : nat} (sigma : fin k -> PTm l_PTm)
+ (tau_PTm : fin l_PTm -> PTm m_PTm) (theta : fin k -> PTm m_PTm)
+ (Eq : forall x, funcomp (subst_PTm tau_PTm) sigma x = theta x) :
+ forall x,
+ funcomp (subst_PTm (up_list_PTm_PTm p tau_PTm)) (up_list_PTm_PTm p sigma) x =
+ up_list_PTm_PTm p theta x.
+Proof.
+exact (fun n =>
+ eq_trans
+ (scons_p_comp' (funcomp (VarPTm (plus p l_PTm)) (zero_p p)) _ _ n)
+ (scons_p_congr
+ (fun x => scons_p_head' _ (fun z => ren_PTm (shift_p p) _) x)
+ (fun n =>
+ eq_trans
+ (compRenSubst_PTm (shift_p p) (up_list_PTm_PTm p tau_PTm)
+ (funcomp (up_list_PTm_PTm p tau_PTm) (shift_p p))
+ (fun x => eq_refl) (sigma n))
+ (eq_trans
+ (eq_sym
+ (compSubstRen_PTm tau_PTm (shift_p p) _
+ (fun x => eq_sym (scons_p_tail' _ _ x)) (sigma n)))
+ (ap (ren_PTm (shift_p p)) (Eq n)))))).
+Qed.
+
+Fixpoint compSubstSubst_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
+(sigma_PTm : fin m_PTm -> PTm k_PTm) (tau_PTm : fin k_PTm -> PTm l_PTm)
+(theta_PTm : fin m_PTm -> PTm l_PTm)
+(Eq_PTm : forall x, funcomp (subst_PTm tau_PTm) sigma_PTm x = theta_PTm x)
+(s : PTm m_PTm) {struct s} :
+subst_PTm tau_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
+ match s with
+ | VarPTm _ s0 => Eq_PTm s0
+ | PAbs _ s0 =>
+ congr_PAbs
+ (compSubstSubst_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
+ (up_PTm_PTm theta_PTm) (up_subst_subst_PTm_PTm _ _ _ Eq_PTm) s0)
+ | PApp _ s0 s1 =>
+ congr_PApp (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s0)
+ (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
+ | PPair _ s0 s1 =>
+ congr_PPair (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s0)
+ (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
+ | PProj _ s0 s1 =>
+ congr_PProj (eq_refl s0)
+ (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
+ | PConst _ s0 => congr_PConst (eq_refl s0)
+ | PUniv _ s0 => congr_PUniv (eq_refl s0)
+ | PBot _ => congr_PBot
+ end.
+
+Lemma renRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
+ (xi_PTm : fin m_PTm -> fin k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm)
+ (s : PTm m_PTm) :
+ ren_PTm zeta_PTm (ren_PTm xi_PTm s) = ren_PTm (funcomp zeta_PTm xi_PTm) s.
+Proof.
+exact (compRenRen_PTm xi_PTm zeta_PTm _ (fun n => eq_refl) s).
+Qed.
+
+Lemma renRen'_PTm_pointwise {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
+ (xi_PTm : fin m_PTm -> fin k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm) :
+ pointwise_relation _ eq (funcomp (ren_PTm zeta_PTm) (ren_PTm xi_PTm))
+ (ren_PTm (funcomp zeta_PTm xi_PTm)).
+Proof.
+exact (fun s => compRenRen_PTm xi_PTm zeta_PTm _ (fun n => eq_refl) s).
+Qed.
+
+Lemma renSubst_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
+ (xi_PTm : fin m_PTm -> fin k_PTm) (tau_PTm : fin k_PTm -> PTm l_PTm)
+ (s : PTm m_PTm) :
+ subst_PTm tau_PTm (ren_PTm xi_PTm s) = subst_PTm (funcomp tau_PTm xi_PTm) s.
+Proof.
+exact (compRenSubst_PTm xi_PTm tau_PTm _ (fun n => eq_refl) s).
+Qed.
+
+Lemma renSubst_PTm_pointwise {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
+ (xi_PTm : fin m_PTm -> fin k_PTm) (tau_PTm : fin k_PTm -> PTm l_PTm) :
+ pointwise_relation _ eq (funcomp (subst_PTm tau_PTm) (ren_PTm xi_PTm))
+ (subst_PTm (funcomp tau_PTm xi_PTm)).
+Proof.
+exact (fun s => compRenSubst_PTm xi_PTm tau_PTm _ (fun n => eq_refl) s).
+Qed.
+
+Lemma substRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
+ (sigma_PTm : fin m_PTm -> PTm k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm)
+ (s : PTm m_PTm) :
+ ren_PTm zeta_PTm (subst_PTm sigma_PTm s) =
+ subst_PTm (funcomp (ren_PTm zeta_PTm) sigma_PTm) s.
+Proof.
+exact (compSubstRen_PTm sigma_PTm zeta_PTm _ (fun n => eq_refl) s).
+Qed.
+
+Lemma substRen_PTm_pointwise {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
+ (sigma_PTm : fin m_PTm -> PTm k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm) :
+ pointwise_relation _ eq (funcomp (ren_PTm zeta_PTm) (subst_PTm sigma_PTm))
+ (subst_PTm (funcomp (ren_PTm zeta_PTm) sigma_PTm)).
+Proof.
+exact (fun s => compSubstRen_PTm sigma_PTm zeta_PTm _ (fun n => eq_refl) s).
+Qed.
+
+Lemma substSubst_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
+ (sigma_PTm : fin m_PTm -> PTm k_PTm) (tau_PTm : fin k_PTm -> PTm l_PTm)
+ (s : PTm m_PTm) :
+ subst_PTm tau_PTm (subst_PTm sigma_PTm s) =
+ subst_PTm (funcomp (subst_PTm tau_PTm) sigma_PTm) s.
+Proof.
+exact (compSubstSubst_PTm sigma_PTm tau_PTm _ (fun n => eq_refl) s).
+Qed.
+
+Lemma substSubst_PTm_pointwise {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
+ (sigma_PTm : fin m_PTm -> PTm k_PTm) (tau_PTm : fin k_PTm -> PTm l_PTm) :
+ pointwise_relation _ eq (funcomp (subst_PTm tau_PTm) (subst_PTm sigma_PTm))
+ (subst_PTm (funcomp (subst_PTm tau_PTm) sigma_PTm)).
+Proof.
+exact (fun s => compSubstSubst_PTm sigma_PTm tau_PTm _ (fun n => eq_refl) s).
+Qed.
+
+Lemma rinstInst_up_PTm_PTm {m : nat} {n_PTm : nat} (xi : fin m -> fin n_PTm)
+ (sigma : fin m -> PTm n_PTm)
+ (Eq : forall x, funcomp (VarPTm n_PTm) xi x = sigma x) :
+ forall x,
+ funcomp (VarPTm (S n_PTm)) (upRen_PTm_PTm xi) x = up_PTm_PTm sigma x.
+Proof.
+exact (fun n =>
+ match n with
+ | Some fin_n => ap (ren_PTm shift) (Eq fin_n)
+ | None => eq_refl
+ end).
+Qed.
+
+Lemma rinstInst_up_list_PTm_PTm {p : nat} {m : nat} {n_PTm : nat}
+ (xi : fin m -> fin n_PTm) (sigma : fin m -> PTm n_PTm)
+ (Eq : forall x, funcomp (VarPTm n_PTm) xi x = sigma x) :
+ forall x,
+ funcomp (VarPTm (plus p n_PTm)) (upRen_list_PTm_PTm p xi) x =
+ up_list_PTm_PTm p sigma x.
+Proof.
+exact (fun n =>
+ eq_trans (scons_p_comp' _ _ (VarPTm (plus p n_PTm)) n)
+ (scons_p_congr (fun z => eq_refl)
+ (fun n => ap (ren_PTm (shift_p p)) (Eq n)))).
+Qed.
+
+Fixpoint rinst_inst_PTm {m_PTm : nat} {n_PTm : nat}
+(xi_PTm : fin m_PTm -> fin n_PTm) (sigma_PTm : fin m_PTm -> PTm n_PTm)
+(Eq_PTm : forall x, funcomp (VarPTm n_PTm) xi_PTm x = sigma_PTm x)
+(s : PTm m_PTm) {struct s} : ren_PTm xi_PTm s = subst_PTm sigma_PTm s :=
+ match s with
+ | VarPTm _ s0 => Eq_PTm s0
+ | PAbs _ s0 =>
+ congr_PAbs
+ (rinst_inst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm sigma_PTm)
+ (rinstInst_up_PTm_PTm _ _ Eq_PTm) s0)
+ | PApp _ s0 s1 =>
+ congr_PApp (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s0)
+ (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
+ | PPair _ s0 s1 =>
+ congr_PPair (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s0)
+ (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
+ | PProj _ s0 s1 =>
+ congr_PProj (eq_refl s0) (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
+ | PConst _ s0 => congr_PConst (eq_refl s0)
+ | PUniv _ s0 => congr_PUniv (eq_refl s0)
+ | PBot _ => congr_PBot
+ end.
+
+Lemma rinstInst'_PTm {m_PTm : nat} {n_PTm : nat}
+ (xi_PTm : fin m_PTm -> fin n_PTm) (s : PTm m_PTm) :
+ ren_PTm xi_PTm s = subst_PTm (funcomp (VarPTm n_PTm) xi_PTm) s.
+Proof.
+exact (rinst_inst_PTm xi_PTm _ (fun n => eq_refl) s).
+Qed.
+
+Lemma rinstInst'_PTm_pointwise {m_PTm : nat} {n_PTm : nat}
+ (xi_PTm : fin m_PTm -> fin n_PTm) :
+ pointwise_relation _ eq (ren_PTm xi_PTm)
+ (subst_PTm (funcomp (VarPTm n_PTm) xi_PTm)).
+Proof.
+exact (fun s => rinst_inst_PTm xi_PTm _ (fun n => eq_refl) s).
+Qed.
+
+Lemma instId'_PTm {m_PTm : nat} (s : PTm m_PTm) :
+ subst_PTm (VarPTm m_PTm) s = s.
+Proof.
+exact (idSubst_PTm (VarPTm m_PTm) (fun n => eq_refl) s).
+Qed.
+
+Lemma instId'_PTm_pointwise {m_PTm : nat} :
+ pointwise_relation _ eq (subst_PTm (VarPTm m_PTm)) id.
+Proof.
+exact (fun s => idSubst_PTm (VarPTm m_PTm) (fun n => eq_refl) s).
+Qed.
+
+Lemma rinstId'_PTm {m_PTm : nat} (s : PTm m_PTm) : ren_PTm id s = s.
+Proof.
+exact (eq_ind_r (fun t => t = s) (instId'_PTm s) (rinstInst'_PTm id s)).
+Qed.
+
+Lemma rinstId'_PTm_pointwise {m_PTm : nat} :
+ pointwise_relation _ eq (@ren_PTm m_PTm m_PTm id) id.
+Proof.
+exact (fun s =>
+ eq_ind_r (fun t => t = s) (instId'_PTm s) (rinstInst'_PTm id s)).
+Qed.
+
+Lemma varL'_PTm {m_PTm : nat} {n_PTm : nat}
+ (sigma_PTm : fin m_PTm -> PTm n_PTm) (x : fin m_PTm) :
+ subst_PTm sigma_PTm (VarPTm m_PTm x) = sigma_PTm x.
+Proof.
+exact (eq_refl).
+Qed.
+
+Lemma varL'_PTm_pointwise {m_PTm : nat} {n_PTm : nat}
+ (sigma_PTm : fin m_PTm -> PTm n_PTm) :
+ pointwise_relation _ eq (funcomp (subst_PTm sigma_PTm) (VarPTm m_PTm))
+ sigma_PTm.
+Proof.
+exact (fun x => eq_refl).
+Qed.
+
+Lemma varLRen'_PTm {m_PTm : nat} {n_PTm : nat}
+ (xi_PTm : fin m_PTm -> fin n_PTm) (x : fin m_PTm) :
+ ren_PTm xi_PTm (VarPTm m_PTm x) = VarPTm n_PTm (xi_PTm x).
+Proof.
+exact (eq_refl).
+Qed.
+
+Lemma varLRen'_PTm_pointwise {m_PTm : nat} {n_PTm : nat}
+ (xi_PTm : fin m_PTm -> fin n_PTm) :
+ pointwise_relation _ eq (funcomp (ren_PTm xi_PTm) (VarPTm m_PTm))
+ (funcomp (VarPTm n_PTm) xi_PTm).
+Proof.
+exact (fun x => eq_refl).
+Qed.
+
Inductive Tm (n_Tm : nat) : Type :=
| VarTm : fin n_Tm -> Tm n_Tm
| Abs : Tm (S n_Tm) -> Tm n_Tm
@@ -40,9 +708,7 @@ Inductive Tm (n_Tm : nat) : Type :=
| Pair : Tm n_Tm -> Tm n_Tm -> Tm n_Tm
| Proj : PTag -> Tm n_Tm -> Tm n_Tm
| 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
| BVal : bool -> Tm n_Tm
| Bool : Tm n_Tm
| If : Tm n_Tm -> Tm n_Tm -> Tm n_Tm -> Tm n_Tm.
@@ -87,23 +753,12 @@ exact (eq_trans
(ap (fun x => TBind m_Tm t0 t1 x) H2)).
Qed.
-Lemma congr_Const {m_Tm : nat} {s0 : TTag} {t0 : TTag} (H0 : s0 = t0) :
- Const m_Tm s0 = Const m_Tm t0.
-Proof.
-exact (eq_trans eq_refl (ap (fun x => Const m_Tm x) H0)).
-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 congr_Bot {m_Tm : nat} : Bot m_Tm = Bot 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.
@@ -147,9 +802,7 @@ 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)
| TBind _ s0 s1 s2 =>
TBind n_Tm s0 (ren_Tm xi_Tm s1) (ren_Tm (upRen_Tm_Tm xi_Tm) s2)
- | 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 =>
@@ -179,9 +832,7 @@ 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)
| TBind _ s0 s1 s2 =>
TBind n_Tm s0 (subst_Tm sigma_Tm s1) (subst_Tm (up_Tm_Tm sigma_Tm) s2)
- | 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 =>
@@ -225,9 +876,7 @@ subst_Tm sigma_Tm s = s :=
| TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
(idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s2)
- | 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 =>
@@ -275,9 +924,7 @@ Fixpoint extRen_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
congr_TBind (eq_refl s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
(extRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upExtRen_Tm_Tm _ _ Eq_Tm) s2)
- | 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 =>
@@ -326,9 +973,7 @@ Fixpoint ext_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
congr_TBind (eq_refl s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
(ext_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm) (upExt_Tm_Tm _ _ Eq_Tm)
s2)
- | 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 =>
@@ -377,9 +1022,7 @@ Fixpoint compRenRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
congr_TBind (eq_refl s0) (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
(compRenRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s2)
- | 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 =>
@@ -440,9 +1083,7 @@ Fixpoint compRenSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
(compRenSubst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm tau_Tm)
(up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s2)
- | 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 =>
@@ -524,9 +1165,7 @@ ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
(compSubstRen_Tm (up_Tm_Tm sigma_Tm) (upRen_Tm_Tm zeta_Tm)
(up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s2)
- | 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 =>
@@ -609,9 +1248,7 @@ subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
(compSubstSubst_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm)
(up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s2)
- | 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 =>
@@ -732,9 +1369,7 @@ Fixpoint rinst_inst_Tm {m_Tm : nat} {n_Tm : nat}
congr_TBind (eq_refl s0) (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
(rinst_inst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm sigma_Tm)
(rinstInst_up_Tm_Tm _ _ Eq_Tm) s2)
- | 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 =>
@@ -809,6 +1444,9 @@ Qed.
Class Up_Tm X Y :=
up_Tm : X -> Y.
+Class Up_PTm X Y :=
+ up_PTm : X -> Y.
+
#[global]
Instance Subst_Tm {m_Tm n_Tm : nat}: (Subst1 _ _ _) := (@subst_Tm m_Tm n_Tm).
@@ -821,6 +1459,19 @@ Instance Ren_Tm {m_Tm n_Tm : nat}: (Ren1 _ _ _) := (@ren_Tm m_Tm n_Tm).
#[global]
Instance VarInstance_Tm {n_Tm : nat}: (Var _ _) := (@VarTm n_Tm).
+#[global]
+Instance Subst_PTm {m_PTm n_PTm : nat}: (Subst1 _ _ _) :=
+ (@subst_PTm m_PTm n_PTm).
+
+#[global]
+Instance Up_PTm_PTm {m n_PTm : nat}: (Up_PTm _ _) := (@up_PTm_PTm m n_PTm).
+
+#[global]
+Instance Ren_PTm {m_PTm n_PTm : nat}: (Ren1 _ _ _) := (@ren_PTm m_PTm n_PTm).
+
+#[global]
+Instance VarInstance_PTm {n_PTm : nat}: (Var _ _) := (@VarPTm n_PTm).
+
Notation "[ sigma_Tm ]" := (subst_Tm sigma_Tm)
( at level 1, left associativity, only printing) : fscope.
@@ -845,6 +1496,30 @@ Notation "x '__Tm'" := (@ids _ _ VarInstance_Tm x)
Notation "x '__Tm'" := (VarTm x) ( at level 5, format "x __Tm") :
subst_scope.
+Notation "[ sigma_PTm ]" := (subst_PTm sigma_PTm)
+( at level 1, left associativity, only printing) : fscope.
+
+Notation "s [ sigma_PTm ]" := (subst_PTm sigma_PTm s)
+( at level 7, left associativity, only printing) : subst_scope.
+
+Notation "↑__PTm" := up_PTm (only printing) : subst_scope.
+
+Notation "↑__PTm" := up_PTm_PTm (only printing) : subst_scope.
+
+Notation "⟨ xi_PTm ⟩" := (ren_PTm xi_PTm)
+( at level 1, left associativity, only printing) : fscope.
+
+Notation "s ⟨ xi_PTm ⟩" := (ren_PTm xi_PTm s)
+( at level 7, left associativity, only printing) : subst_scope.
+
+Notation "'var'" := VarPTm ( at level 1, only printing) : subst_scope.
+
+Notation "x '__PTm'" := (@ids _ _ VarInstance_PTm x)
+( at level 5, format "x __PTm", only printing) : subst_scope.
+
+Notation "x '__PTm'" := (VarPTm x) ( at level 5, format "x __PTm") :
+subst_scope.
+
#[global]
Instance subst_Tm_morphism {m_Tm : nat} {n_Tm : nat}:
(Proper (respectful (pointwise_relation _ eq) (respectful eq eq))
@@ -881,15 +1556,56 @@ Proof.
exact (fun f_Tm g_Tm Eq_Tm s => extRen_Tm f_Tm g_Tm Eq_Tm s).
Qed.
+#[global]
+Instance subst_PTm_morphism {m_PTm : nat} {n_PTm : nat}:
+ (Proper (respectful (pointwise_relation _ eq) (respectful eq eq))
+ (@subst_PTm m_PTm n_PTm)).
+Proof.
+exact (fun f_PTm g_PTm Eq_PTm s t Eq_st =>
+ eq_ind s (fun t' => subst_PTm f_PTm s = subst_PTm g_PTm t')
+ (ext_PTm f_PTm g_PTm Eq_PTm s) t Eq_st).
+Qed.
+
+#[global]
+Instance subst_PTm_morphism2 {m_PTm : nat} {n_PTm : nat}:
+ (Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
+ (@subst_PTm m_PTm n_PTm)).
+Proof.
+exact (fun f_PTm g_PTm Eq_PTm s => ext_PTm f_PTm g_PTm Eq_PTm s).
+Qed.
+
+#[global]
+Instance ren_PTm_morphism {m_PTm : nat} {n_PTm : nat}:
+ (Proper (respectful (pointwise_relation _ eq) (respectful eq eq))
+ (@ren_PTm m_PTm n_PTm)).
+Proof.
+exact (fun f_PTm g_PTm Eq_PTm s t Eq_st =>
+ eq_ind s (fun t' => ren_PTm f_PTm s = ren_PTm g_PTm t')
+ (extRen_PTm f_PTm g_PTm Eq_PTm s) t Eq_st).
+Qed.
+
+#[global]
+Instance ren_PTm_morphism2 {m_PTm : nat} {n_PTm : nat}:
+ (Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
+ (@ren_PTm m_PTm n_PTm)).
+Proof.
+exact (fun f_PTm g_PTm Eq_PTm s => extRen_PTm f_PTm g_PTm Eq_PTm s).
+Qed.
+
Ltac auto_unfold := repeat
- unfold VarInstance_Tm, Var, ids, Ren_Tm, Ren1, ren1,
- Up_Tm_Tm, Up_Tm, up_Tm, Subst_Tm, Subst1, subst1.
+ unfold VarInstance_PTm, Var, ids, Ren_PTm, Ren1, ren1,
+ Up_PTm_PTm, Up_PTm, up_PTm, Subst_PTm, Subst1, subst1,
+ VarInstance_Tm, Var, ids, Ren_Tm, Ren1, ren1, Up_Tm_Tm,
+ Up_Tm, up_Tm, Subst_Tm, Subst1, subst1.
Tactic Notation "auto_unfold" "in" "*" := repeat
- unfold VarInstance_Tm, Var, ids,
- Ren_Tm, Ren1, ren1, Up_Tm_Tm,
- Up_Tm, up_Tm, Subst_Tm, Subst1,
- subst1 in *.
+ unfold VarInstance_PTm, Var, ids,
+ Ren_PTm, Ren1, ren1, Up_PTm_PTm,
+ Up_PTm, up_PTm, Subst_PTm,
+ Subst1, subst1, VarInstance_Tm,
+ Var, ids, Ren_Tm, Ren1, ren1,
+ Up_Tm_Tm, Up_Tm, up_Tm, Subst_Tm,
+ Subst1, subst1 in *.
Ltac asimpl' := repeat (first
[ progress setoid_rewrite substSubst_Tm_pointwise
@@ -900,6 +1616,14 @@ Ltac asimpl' := repeat (first
| progress setoid_rewrite renSubst_Tm
| progress setoid_rewrite renRen'_Tm_pointwise
| progress setoid_rewrite renRen_Tm
+ | progress setoid_rewrite substSubst_PTm_pointwise
+ | progress setoid_rewrite substSubst_PTm
+ | progress setoid_rewrite substRen_PTm_pointwise
+ | progress setoid_rewrite substRen_PTm
+ | progress setoid_rewrite renSubst_PTm_pointwise
+ | progress setoid_rewrite renSubst_PTm
+ | progress setoid_rewrite renRen'_PTm_pointwise
+ | progress setoid_rewrite renRen_PTm
| progress setoid_rewrite varLRen'_Tm_pointwise
| progress setoid_rewrite varLRen'_Tm
| progress setoid_rewrite varL'_Tm_pointwise
@@ -908,27 +1632,42 @@ Ltac asimpl' := repeat (first
| progress setoid_rewrite rinstId'_Tm
| progress setoid_rewrite instId'_Tm_pointwise
| progress setoid_rewrite instId'_Tm
+ | progress setoid_rewrite varLRen'_PTm_pointwise
+ | progress setoid_rewrite varLRen'_PTm
+ | progress setoid_rewrite varL'_PTm_pointwise
+ | progress setoid_rewrite varL'_PTm
+ | progress setoid_rewrite rinstId'_PTm_pointwise
+ | progress setoid_rewrite rinstId'_PTm
+ | progress setoid_rewrite instId'_PTm_pointwise
+ | progress setoid_rewrite instId'_PTm
| progress
unfold up_list_Tm_Tm, up_Tm_Tm, upRen_list_Tm_Tm,
- upRen_Tm_Tm, up_ren
- | progress cbn[subst_Tm ren_Tm]
+ upRen_Tm_Tm, up_list_PTm_PTm, up_PTm_PTm,
+ upRen_list_PTm_PTm, upRen_PTm_PTm, up_ren
+ | progress cbn[subst_Tm ren_Tm subst_PTm ren_PTm]
| progress fsimpl ]).
Ltac asimpl := check_no_evars;
repeat
- unfold VarInstance_Tm, Var, ids, Ren_Tm, Ren1, ren1,
- Up_Tm_Tm, Up_Tm, up_Tm, Subst_Tm, Subst1, subst1 in *;
- asimpl'; minimize.
+ unfold VarInstance_PTm, Var, ids, Ren_PTm, Ren1, ren1,
+ Up_PTm_PTm, Up_PTm, up_PTm, Subst_PTm, Subst1, subst1,
+ VarInstance_Tm, Var, ids, Ren_Tm, Ren1, ren1, Up_Tm_Tm,
+ Up_Tm, up_Tm, Subst_Tm, Subst1, subst1 in *; asimpl';
+ minimize.
Tactic Notation "asimpl" "in" hyp(J) := revert J; asimpl; intros J.
Tactic Notation "auto_case" := auto_case ltac:(asimpl; cbn; eauto).
Ltac substify := auto_unfold; try setoid_rewrite rinstInst'_Tm_pointwise;
- try setoid_rewrite rinstInst'_Tm.
+ try setoid_rewrite rinstInst'_Tm;
+ try setoid_rewrite rinstInst'_PTm_pointwise;
+ try setoid_rewrite rinstInst'_PTm.
Ltac renamify := auto_unfold; try setoid_rewrite_left rinstInst'_Tm_pointwise;
- try setoid_rewrite_left rinstInst'_Tm.
+ try setoid_rewrite_left rinstInst'_Tm;
+ try setoid_rewrite_left rinstInst'_PTm_pointwise;
+ try setoid_rewrite_left rinstInst'_PTm.
End Core.
@@ -945,12 +1684,8 @@ Arguments Bool {n_Tm}.
Arguments BVal {n_Tm}.
-Arguments Bot {n_Tm}.
-
Arguments Univ {n_Tm}.
-Arguments Const {n_Tm}.
-
Arguments TBind {n_Tm}.
Arguments Proj {n_Tm}.
@@ -961,10 +1696,30 @@ Arguments App {n_Tm}.
Arguments Abs {n_Tm}.
+Arguments VarPTm {n_PTm}.
+
+Arguments PBot {n_PTm}.
+
+Arguments PUniv {n_PTm}.
+
+Arguments PConst {n_PTm}.
+
+Arguments PProj {n_PTm}.
+
+Arguments PPair {n_PTm}.
+
+Arguments PApp {n_PTm}.
+
+Arguments PAbs {n_PTm}.
+
#[global]Hint Opaque subst_Tm: rewrite.
#[global]Hint Opaque ren_Tm: rewrite.
+#[global]Hint Opaque subst_PTm: rewrite.
+
+#[global]Hint Opaque ren_PTm: rewrite.
+
End Extra.
Module interface.
diff --git a/theories/diagram.txt b/theories/diagram.txt
index 2cf16e8..ab47f3d 100644
--- a/theories/diagram.txt
+++ b/theories/diagram.txt
@@ -18,3 +18,25 @@ a0 >> a1
| |
v v
b0 >> b1
+
+
+prov x (x, x)
+
+prov x b
+
+
+a => b
+
+prov x a
+
+prov y b
+
+prov x c
+prov y c
+
+extract c = x
+extract c = y
+
+prov x b
+
+pr
diff --git a/theories/fp_red.v b/theories/fp_red.v
index e9b5591..3482f45 100644
--- a/theories/fp_red.v
+++ b/theories/fp_red.v
@@ -22,86 +22,82 @@ Ltac spec_refl := ltac2:(spec_refl ()).
(* Trying my best to not write C style module_funcname *)
Module Par.
- Inductive R {n} : Tm n -> Tm n -> Prop :=
+ Inductive R {n} : PTm n -> PTm n -> Prop :=
(***************** Beta ***********************)
| AppAbs a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (App (Abs a0) b0) (subst_Tm (scons b1 VarTm) a1)
+ R (PApp (PAbs a0) b0) (subst_PTm (scons b1 VarPTm) a1)
| AppPair a0 a1 b0 b1 c0 c1:
R a0 a1 ->
R b0 b1 ->
R c0 c1 ->
- R (App (Pair a0 b0) c0) (Pair (App a1 c1) (App b1 c1))
+ R (PApp (PPair a0 b0) c0) (PPair (PApp a1 c1) (PApp b1 c1))
| ProjAbs p a0 a1 :
R a0 a1 ->
- R (Proj p (Abs a0)) (Abs (Proj p a1))
+ R (PProj p (PAbs a0)) (PAbs (PProj p a1))
| ProjPair p a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1)
+ R (PProj p (PPair a0 b0)) (if p is PL then a1 else b1)
(****************** Eta ***********************)
| AppEta a0 a1 :
R a0 a1 ->
- R a0 (Abs (App (ren_Tm shift a1) (VarTm var_zero)))
+ R a0 (PAbs (PApp (ren_PTm shift a1) (VarPTm var_zero)))
| PairEta a0 a1 :
R a0 a1 ->
- R a0 (Pair (Proj PL a1) (Proj PR a1))
+ R a0 (PPair (PProj PL a1) (PProj PR a1))
(*************** Congruence ********************)
- | Var i : R (VarTm i) (VarTm i)
+ | Var i : R (VarPTm i) (VarPTm i)
| AbsCong a0 a1 :
R a0 a1 ->
- R (Abs a0) (Abs a1)
+ R (PAbs a0) (PAbs a1)
| AppCong a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (App a0 b0) (App a1 b1)
+ R (PApp a0 b0) (PApp a1 b1)
| PairCong a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (Pair a0 b0) (Pair a1 b1)
+ R (PPair a0 b0) (PPair a1 b1)
| ProjCong p a0 a1 :
R a0 a1 ->
- R (Proj p a0) (Proj p a1)
- | BindCong p A0 A1 B0 B1:
- R A0 A1 ->
- R B0 B1 ->
- R (TBind p A0 B0) (TBind p A1 B1)
+ R (PProj p a0) (PProj p a1)
| ConstCong k :
- R (Const k) (Const k)
- | UnivCong i :
- R (Univ i) (Univ i)
- | BotCong :
- R Bot Bot.
+ R (PConst k) (PConst k)
+ | Univ i :
+ R (PUniv i) (PUniv i)
+ | Bot :
+ R PBot PBot.
- Lemma refl n (a : Tm n) : R a a.
+ Lemma refl n (a : PTm n) : R a a.
elim : n /a; hauto ctrs:R.
Qed.
- Lemma AppAbs' n a0 a1 (b0 b1 t : Tm n) :
- t = subst_Tm (scons b1 VarTm) a1 ->
+ Lemma AppAbs' n a0 a1 (b0 b1 t : PTm n) :
+ t = subst_PTm (scons b1 VarPTm) a1 ->
R a0 a1 ->
R b0 b1 ->
- R (App (Abs a0) b0) t.
+ R (PApp (PAbs a0) b0) t.
Proof. move => ->. apply AppAbs. Qed.
- Lemma ProjPair' n p (a0 a1 b0 b1 : Tm n) t :
+ Lemma ProjPair' n p (a0 a1 b0 b1 : PTm n) t :
t = (if p is PL then a1 else b1) ->
R a0 a1 ->
R b0 b1 ->
- R (Proj p (Pair a0 b0)) t.
+ R (PProj p (PPair a0 b0)) t.
Proof. move => > ->. apply ProjPair. Qed.
- Lemma AppEta' n (a0 a1 b : Tm n) :
- b = (Abs (App (ren_Tm shift a1) (VarTm var_zero))) ->
+ Lemma AppEta' n (a0 a1 b : PTm n) :
+ b = (PAbs (PApp (ren_PTm shift a1) (VarPTm var_zero))) ->
R a0 a1 ->
R a0 b.
Proof. move => ->; apply AppEta. Qed.
- Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
- R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
+ Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
+ R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
Proof.
move => h. move : m ξ.
elim : n a b /h.
@@ -113,13 +109,13 @@ Module Par.
Qed.
- Lemma morphing n m (a b : Tm n) (ρ0 ρ1 : fin n -> Tm m) :
+ Lemma morphing n m (a b : PTm n) (ρ0 ρ1 : fin n -> PTm m) :
(forall i, R (ρ0 i) (ρ1 i)) ->
- R a b -> R (subst_Tm ρ0 a) (subst_Tm ρ1 b).
+ R a b -> R (subst_PTm ρ0 a) (subst_PTm ρ1 b).
Proof.
move => + h. move : m ρ0 ρ1. elim : n a b/h.
- move => n a0 a1 b0 b1 ha iha hb ihb m ρ0 ρ1 hρ /=.
- eapply AppAbs' with (a1 := subst_Tm (up_Tm_Tm ρ1) a1); eauto.
+ eapply AppAbs' with (a1 := subst_PTm (up_PTm_PTm ρ1) a1); eauto.
by asimpl.
hauto l:on use:renaming inv:option.
- hauto lq:on rew:off ctrs:R.
@@ -134,19 +130,18 @@ Module Par.
- qauto l:on ctrs:R.
- qauto l:on ctrs:R.
- hauto l:on inv:option ctrs:R use:renaming.
- - sfirstorder.
- - sfirstorder.
- - sfirstorder.
+ - qauto l:on ctrs:R.
+ - qauto l:on ctrs:R.
Qed.
- Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
- R a b -> R (subst_Tm ρ a) (subst_Tm ρ b).
+ Lemma substing n m (a b : PTm n) (ρ : fin n -> PTm m) :
+ R a b -> R (subst_PTm ρ a) (subst_PTm ρ b).
Proof. hauto l:on use:morphing, refl. Qed.
- Lemma antirenaming n m (a : Tm n) (b : Tm m) (ξ : fin n -> fin m) :
- R (ren_Tm ξ a) b -> exists b0, R a b0 /\ ren_Tm ξ b0 = b.
+ Lemma antirenaming n m (a : PTm n) (b : PTm m) (ξ : fin n -> fin m) :
+ R (ren_PTm ξ a) b -> exists b0, R a b0 /\ ren_PTm ξ b0 = b.
Proof.
- move E : (ren_Tm ξ a) => u h.
+ move E : (ren_PTm ξ a) => u h.
move : n ξ a E. elim : m u b/h.
- move => n a0 a1 b0 b1 ha iha hb ihb m ξ []//=.
move => c c0 [+ ?]. subst.
@@ -190,7 +185,7 @@ Module Par.
spec_refl.
move :iha => [b0 [? ?]]. subst.
eexists. split. by apply AbsCong; eauto.
- by asimpl.
+ done.
- move => n a0 a1 b0 b1 ha iha hb ihb m ξ []//= t t0 [*]. subst.
spec_refl.
move : iha => [b0 [? ?]]. subst.
@@ -201,43 +196,37 @@ Module Par.
spec_refl.
move : iha => [b0 [? ?]]. subst.
move : ihb => [c0 [? ?]]. subst.
- eexists. split. by apply PairCong; eauto.
- by asimpl.
+ eexists. split=>/=. by apply PairCong; eauto.
+ done.
- move => n p a0 a1 ha iha m ξ []//= p0 t [*]. subst.
spec_refl.
move : iha => [b0 [? ?]]. subst.
eexists. split. by apply ProjCong; eauto.
- by asimpl.
- - move => n p A0 A1 B0 B1 ha iha hB ihB m ξ []//= ? t t0 [*]. subst.
- spec_refl.
- move : iha => [b0 [? ?]].
- move : ihB => [c0 [? ?]]. subst.
- eexists. split. by apply BindCong; eauto.
- by asimpl.
- - move => n k m ξ []//=. hauto l:on.
- - move => n i n0 ξ []//=. hauto l:on.
- - hauto q:on inv:Tm ctrs:R.
+ done.
+ - hauto q:on inv:PTm ctrs:R.
+ - hauto q:on inv:PTm ctrs:R.
+ - hauto q:on inv:PTm ctrs:R.
Qed.
End Par.
Module Pars.
- Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
- rtc Par.R a b -> rtc Par.R (ren_Tm ξ a) (ren_Tm ξ b).
+ Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
+ rtc Par.R a b -> rtc Par.R (ren_PTm ξ a) (ren_PTm ξ b).
Proof.
induction 1; hauto lq:on ctrs:rtc use:Par.renaming.
Qed.
- Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
+ Lemma substing n m (a b : PTm n) (ρ : fin n -> PTm m) :
rtc Par.R a b ->
- rtc Par.R (subst_Tm ρ a) (subst_Tm ρ b).
+ rtc Par.R (subst_PTm ρ a) (subst_PTm ρ b).
induction 1; hauto l:on ctrs:rtc use:Par.substing.
Qed.
- Lemma antirenaming n m (a : Tm n) (b : Tm m) (ξ : fin n -> fin m) :
- rtc Par.R (ren_Tm ξ a) b -> exists b0, rtc Par.R a b0 /\ ren_Tm ξ b0 = b.
+ Lemma antirenaming n m (a : PTm n) (b : PTm m) (ξ : fin n -> fin m) :
+ rtc Par.R (ren_PTm ξ a) b -> exists b0, rtc Par.R a b0 /\ ren_PTm ξ b0 = b.
Proof.
- move E :(ren_Tm ξ a) => u h.
+ move E :(ren_PTm ξ a) => u h.
move : a E.
elim : u b /h.
- sfirstorder.
@@ -254,109 +243,106 @@ Module Pars.
#[local]Ltac solve_s :=
repeat (induction 1; last by solve_s_rec); apply rtc_refl.
- Lemma ProjCong n p (a0 a1 : Tm n) :
+ Lemma ProjCong n p (a0 a1 : PTm n) :
rtc Par.R a0 a1 ->
- rtc Par.R (Proj p a0) (Proj p a1).
+ rtc Par.R (PProj p a0) (PProj p a1).
Proof. solve_s. Qed.
- Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
+ Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
rtc Par.R a0 a1 ->
rtc Par.R b0 b1 ->
- rtc Par.R (Pair a0 b0) (Pair a1 b1).
+ rtc Par.R (PPair a0 b0) (PPair a1 b1).
Proof. solve_s. Qed.
- Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
+ Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
rtc Par.R a0 a1 ->
rtc Par.R b0 b1 ->
- rtc Par.R (App a0 b0) (App a1 b1).
+ rtc Par.R (PApp a0 b0) (PApp a1 b1).
Proof. solve_s. Qed.
- Lemma AbsCong n (a b : Tm (S n)) :
+ Lemma AbsCong n (a b : PTm (S n)) :
rtc Par.R a b ->
- rtc Par.R (Abs a) (Abs b).
+ rtc Par.R (PAbs a) (PAbs b).
Proof. solve_s. Qed.
End Pars.
-Definition var_or_const {n} (a : Tm n) :=
+Definition var_or_const {n} (a : PTm n) :=
match a with
- | VarTm _ => true
- | Bot => true
+ | VarPTm _ => true
+ | PBot => true
| _ => false
end.
+
(***************** Beta rules only ***********************)
Module RPar.
- Inductive R {n} : Tm n -> Tm n -> Prop :=
+ Inductive R {n} : PTm n -> PTm n -> Prop :=
(***************** Beta ***********************)
| AppAbs a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (App (Abs a0) b0) (subst_Tm (scons b1 VarTm) a1)
+ R (PApp (PAbs a0) b0) (subst_PTm (scons b1 VarPTm) a1)
| AppPair a0 a1 b0 b1 c0 c1:
R a0 a1 ->
R b0 b1 ->
R c0 c1 ->
- R (App (Pair a0 b0) c0) (Pair (App a1 c1) (App b1 c1))
+ R (PApp (PPair a0 b0) c0) (PPair (PApp a1 c1) (PApp b1 c1))
| ProjAbs p a0 a1 :
R a0 a1 ->
- R (Proj p (Abs a0)) (Abs (Proj p a1))
+ R (PProj p (PAbs a0)) (PAbs (PProj p a1))
| ProjPair p a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1)
+ R (PProj p (PPair a0 b0)) (if p is PL then a1 else b1)
(*************** Congruence ********************)
- | Var i : R (VarTm i) (VarTm i)
+ | Var i : R (VarPTm i) (VarPTm i)
| AbsCong a0 a1 :
R a0 a1 ->
- R (Abs a0) (Abs a1)
+ R (PAbs a0) (PAbs a1)
| AppCong a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (App a0 b0) (App a1 b1)
+ R (PApp a0 b0) (PApp a1 b1)
| PairCong a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (Pair a0 b0) (Pair a1 b1)
+ R (PPair a0 b0) (PPair a1 b1)
| ProjCong p a0 a1 :
R a0 a1 ->
- R (Proj p a0) (Proj p a1)
- | BindCong p A0 A1 B0 B1:
- R A0 A1 ->
- R B0 B1 ->
- R (TBind p A0 B0) (TBind p A1 B1)
+ R (PProj p a0) (PProj p a1)
| ConstCong k :
- R (Const k) (Const k)
- | UnivCong i :
- R (Univ i) (Univ i)
- | BotCong :
- R Bot Bot.
+ R (PConst k) (PConst k)
+ | Univ i :
+ R (PUniv i) (PUniv i)
+ | Bot :
+ R PBot PBot.
- Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
+ Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop.
- Lemma refl n (a : Tm n) : R a a.
+ Lemma refl n (a : PTm n) : R a a.
Proof.
induction a; hauto lq:on ctrs:R.
Qed.
- Lemma AppAbs' n a0 a1 (b0 b1 t : Tm n) :
- t = subst_Tm (scons b1 VarTm) a1 ->
+ Lemma AppAbs' n a0 a1 (b0 b1 t : PTm n) :
+ t = subst_PTm (scons b1 VarPTm) a1 ->
R a0 a1 ->
R b0 b1 ->
- R (App (Abs a0) b0) t.
+ R (PApp (PAbs a0) b0) t.
Proof. move => ->. apply AppAbs. Qed.
- Lemma ProjPair' n p (a0 a1 b0 b1 : Tm n) t :
+ Lemma ProjPair' n p (a0 a1 b0 b1 : PTm n) t :
t = (if p is PL then a1 else b1) ->
R a0 a1 ->
R b0 b1 ->
- R (Proj p (Pair a0 b0)) t.
+ R (PProj p (PPair a0 b0)) t.
Proof. move => > ->. apply ProjPair. Qed.
- Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
- R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
+ Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
+ R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
Proof.
move => h. move : m ξ.
elim : n a b /h.
@@ -364,25 +350,25 @@ Module RPar.
all : qauto ctrs:R use:ProjPair'.
Qed.
- Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> Tm m) (ξ : fin m -> fin p) :
+ Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> PTm m) (ξ : fin m -> fin p) :
(forall i, R (ρ0 i) (ρ1 i)) ->
- (forall i, R ((funcomp (ren_Tm ξ) ρ0) i) ((funcomp (ren_Tm ξ) ρ1) i)).
+ (forall i, R ((funcomp (ren_PTm ξ) ρ0) i) ((funcomp (ren_PTm ξ) ρ1) i)).
Proof. eauto using renaming. Qed.
- Lemma morphing_ext n m (ρ0 ρ1 : fin n -> Tm m) a b :
+ Lemma morphing_ext n m (ρ0 ρ1 : fin n -> PTm m) a b :
R a b ->
(forall i, R (ρ0 i) (ρ1 i)) ->
(forall i, R ((scons a ρ0) i) ((scons b ρ1) i)).
Proof. hauto q:on inv:option. Qed.
- Lemma morphing_up n m (ρ0 ρ1 : fin n -> Tm m) :
+ Lemma morphing_up n m (ρ0 ρ1 : fin n -> PTm m) :
(forall i, R (ρ0 i) (ρ1 i)) ->
- (forall i, R (up_Tm_Tm ρ0 i) (up_Tm_Tm ρ1 i)).
- Proof. hauto l:on ctrs:R use:morphing_ext, morphing_ren unfold:up_Tm_Tm. Qed.
+ (forall i, R (up_PTm_PTm ρ0 i) (up_PTm_PTm ρ1 i)).
+ Proof. hauto l:on ctrs:R use:morphing_ext, morphing_ren unfold:up_PTm_PTm. Qed.
- Lemma morphing n m (a b : Tm n) (ρ0 ρ1 : fin n -> Tm m) :
+ Lemma morphing n m (a b : PTm n) (ρ0 ρ1 : fin n -> PTm m) :
(forall i, R (ρ0 i) (ρ1 i)) ->
- R a b -> R (subst_Tm ρ0 a) (subst_Tm ρ1 b).
+ R a b -> R (subst_PTm ρ0 a) (subst_PTm ρ1 b).
Proof.
move => + h. move : m ρ0 ρ1.
elim : n a b /h.
@@ -400,33 +386,32 @@ Module RPar.
- hauto lq:on ctrs:R use:morphing_up.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R.
- - hauto lq:on ctrs:R.
Qed.
- Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
+ Lemma substing n m (a b : PTm n) (ρ : fin n -> PTm m) :
R a b ->
- R (subst_Tm ρ a) (subst_Tm ρ b).
+ R (subst_PTm ρ a) (subst_PTm ρ b).
Proof. hauto l:on use:morphing, refl. Qed.
- Lemma cong n (a b : Tm (S n)) c d :
+ Lemma cong n (a b : PTm (S n)) c d :
R a b ->
R c d ->
- R (subst_Tm (scons c VarTm) a) (subst_Tm (scons d VarTm) b).
+ R (subst_PTm (scons c VarPTm) a) (subst_PTm (scons d VarPTm) b).
Proof.
move => h0 h1. apply morphing => //=.
qauto l:on ctrs:R inv:option.
Qed.
- Lemma var_or_const_imp {n} (a b : Tm n) :
+ Lemma var_or_const_imp {n} (a b : PTm n) :
var_or_const a ->
a = b -> ~~ var_or_const b -> False.
Proof.
- hauto lq:on inv:Tm.
+ hauto lq:on inv:PTm.
Qed.
- Lemma var_or_const_up n m (ρ : fin n -> Tm m) :
+ Lemma var_or_const_up n m (ρ : fin n -> PTm m) :
(forall i, var_or_const (ρ i)) ->
- (forall i, var_or_const (up_Tm_Tm ρ i)).
+ (forall i, var_or_const (up_PTm_PTm ρ i)).
Proof.
move => h /= [i|].
- asimpl.
@@ -439,11 +424,11 @@ Module RPar.
Local Ltac antiimp := qauto l:on use:var_or_const_imp.
- Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
+ Lemma antirenaming n m (a : PTm n) (b : PTm m) (ρ : fin n -> PTm m) :
(forall i, var_or_const (ρ i)) ->
- R (subst_Tm ρ a) b -> exists b0, R a b0 /\ subst_Tm ρ b0 = b.
+ R (subst_PTm ρ a) b -> exists b0, R a b0 /\ subst_PTm ρ b0 = b.
Proof.
- move E : (subst_Tm ρ a) => u hρ h.
+ move E : (subst_PTm ρ a) => u hρ h.
move : n ρ hρ a E. elim : m u b/h.
- move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
first by antiimp.
@@ -460,7 +445,8 @@ Module RPar.
eexists. split.
apply AppAbs; eauto.
by asimpl.
- - move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc m ρ hρ []//=;
+ - move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc m ρ hρ.
+ move => []//=;
first by antiimp.
move => []//=; first by antiimp.
move => t t0 t1 [*]. subst.
@@ -527,87 +513,72 @@ Module RPar.
spec_refl.
move : iha => [b0 [? ?]]. subst.
eexists. split. apply ProjCong; eauto. reflexivity.
- - move => n p A0 A1 B0 B1 ha iha hB ihB m ρ hρ []//=;
- first by antiimp.
- move => ? t t0 [*]. subst.
- have {}/iha := (hρ) => iha.
- have /var_or_const_up {}/ihB := (hρ) => ihB.
- spec_refl.
- move : iha => [b0 [? ?]].
- move : ihB => [c0 [? ?]]. subst.
- eexists. split. by apply BindCong; eauto.
- by asimpl.
- - hauto q:on ctrs:R inv:Tm.
- - move => n i n0 ρ hρ []//=; first by antiimp.
- hauto l:on.
- - move => n m ρ hρ []//=; hauto lq:on ctrs:R.
+ - hauto q:on ctrs:R inv:PTm.
+ - hauto q:on ctrs:R inv:PTm.
+ - hauto q:on ctrs:R inv:PTm.
Qed.
End RPar.
(***************** Beta rules only ***********************)
Module RPar'.
- Inductive R {n} : Tm n -> Tm n -> Prop :=
+ Inductive R {n} : PTm n -> PTm n -> Prop :=
(***************** Beta ***********************)
| AppAbs a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (App (Abs a0) b0) (subst_Tm (scons b1 VarTm) a1)
+ R (PApp (PAbs a0) b0) (subst_PTm (scons b1 VarPTm) a1)
| ProjPair p a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1)
+ R (PProj p (PPair a0 b0)) (if p is PL then a1 else b1)
(*************** Congruence ********************)
- | Var i : R (VarTm i) (VarTm i)
+ | Var i : R (VarPTm i) (VarPTm i)
| AbsCong a0 a1 :
R a0 a1 ->
- R (Abs a0) (Abs a1)
+ R (PAbs a0) (PAbs a1)
| AppCong a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (App a0 b0) (App a1 b1)
+ R (PApp a0 b0) (PApp a1 b1)
| PairCong a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (Pair a0 b0) (Pair a1 b1)
+ R (PPair a0 b0) (PPair a1 b1)
| ProjCong p a0 a1 :
R a0 a1 ->
- R (Proj p a0) (Proj p a1)
- | BindCong p A0 A1 B0 B1:
- R A0 A1 ->
- R B0 B1 ->
- R (TBind p A0 B0) (TBind p A1 B1)
+ R (PProj p a0) (PProj p a1)
| ConstCong k :
- R (Const k) (Const k)
+ R (PConst k) (PConst k)
| UnivCong i :
- R (Univ i) (Univ i)
+ R (PUniv i) (PUniv i)
| BotCong :
- R Bot Bot.
+ R PBot PBot.
- Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
+ Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop.
- Lemma refl n (a : Tm n) : R a a.
+ Lemma refl n (a : PTm n) : R a a.
Proof.
induction a; hauto lq:on ctrs:R.
Qed.
- Lemma AppAbs' n a0 a1 (b0 b1 t : Tm n) :
- t = subst_Tm (scons b1 VarTm) a1 ->
+ Lemma AppAbs' n a0 a1 (b0 b1 t : PTm n) :
+ t = subst_PTm (scons b1 VarPTm) a1 ->
R a0 a1 ->
R b0 b1 ->
- R (App (Abs a0) b0) t.
+ R (PApp (PAbs a0) b0) t.
Proof. move => ->. apply AppAbs. Qed.
- Lemma ProjPair' n p (a0 a1 b0 b1 : Tm n) t :
+ Lemma ProjPair' n p (a0 a1 b0 b1 : PTm n) t :
t = (if p is PL then a1 else b1) ->
R a0 a1 ->
R b0 b1 ->
- R (Proj p (Pair a0 b0)) t.
+ R (PProj p (PPair a0 b0)) t.
Proof. move => > ->. apply ProjPair. Qed.
- Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
- R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
+ Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
+ R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
Proof.
move => h. move : m ξ.
elim : n a b /h.
@@ -615,25 +586,25 @@ Module RPar'.
all : qauto ctrs:R use:ProjPair'.
Qed.
- Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> Tm m) (ξ : fin m -> fin p) :
+ Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> PTm m) (ξ : fin m -> fin p) :
(forall i, R (ρ0 i) (ρ1 i)) ->
- (forall i, R ((funcomp (ren_Tm ξ) ρ0) i) ((funcomp (ren_Tm ξ) ρ1) i)).
+ (forall i, R ((funcomp (ren_PTm ξ) ρ0) i) ((funcomp (ren_PTm ξ) ρ1) i)).
Proof. eauto using renaming. Qed.
- Lemma morphing_ext n m (ρ0 ρ1 : fin n -> Tm m) a b :
+ Lemma morphing_ext n m (ρ0 ρ1 : fin n -> PTm m) a b :
R a b ->
(forall i, R (ρ0 i) (ρ1 i)) ->
(forall i, R ((scons a ρ0) i) ((scons b ρ1) i)).
Proof. hauto q:on inv:option. Qed.
- Lemma morphing_up n m (ρ0 ρ1 : fin n -> Tm m) :
+ Lemma morphing_up n m (ρ0 ρ1 : fin n -> PTm m) :
(forall i, R (ρ0 i) (ρ1 i)) ->
- (forall i, R (up_Tm_Tm ρ0 i) (up_Tm_Tm ρ1 i)).
- Proof. hauto l:on ctrs:R use:morphing_ext, morphing_ren unfold:up_Tm_Tm. Qed.
+ (forall i, R (up_PTm_PTm ρ0 i) (up_PTm_PTm ρ1 i)).
+ Proof. hauto l:on ctrs:R use:morphing_ext, morphing_ren unfold:up_PTm_PTm. Qed.
- Lemma morphing n m (a b : Tm n) (ρ0 ρ1 : fin n -> Tm m) :
+ Lemma morphing n m (a b : PTm n) (ρ0 ρ1 : fin n -> PTm m) :
(forall i, R (ρ0 i) (ρ1 i)) ->
- R a b -> R (subst_Tm ρ0 a) (subst_Tm ρ1 b).
+ R a b -> R (subst_PTm ρ0 a) (subst_PTm ρ1 b).
Proof.
move => + h. move : m ρ0 ρ1.
elim : n a b /h.
@@ -646,36 +617,35 @@ Module RPar'.
- hauto lq:on ctrs:R use:morphing_up.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R.
- - hauto lq:on ctrs:R use:morphing_up.
- - hauto lq:on ctrs:R.
+ - hauto l:on ctrs:R use:morphing_up.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R.
Qed.
- Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
+ Lemma substing n m (a b : PTm n) (ρ : fin n -> PTm m) :
R a b ->
- R (subst_Tm ρ a) (subst_Tm ρ b).
+ R (subst_PTm ρ a) (subst_PTm ρ b).
Proof. hauto l:on use:morphing, refl. Qed.
- Lemma cong n (a b : Tm (S n)) c d :
+ Lemma cong n (a b : PTm (S n)) c d :
R a b ->
R c d ->
- R (subst_Tm (scons c VarTm) a) (subst_Tm (scons d VarTm) b).
+ R (subst_PTm (scons c VarPTm) a) (subst_PTm (scons d VarPTm) b).
Proof.
move => h0 h1. apply morphing => //=.
qauto l:on ctrs:R inv:option.
Qed.
- Lemma var_or_const_imp {n} (a b : Tm n) :
+ Lemma var_or_const_imp {n} (a b : PTm n) :
var_or_const a ->
a = b -> ~~ var_or_const b -> False.
Proof.
- hauto lq:on inv:Tm.
+ hauto lq:on inv:PTm.
Qed.
- Lemma var_or_const_up n m (ρ : fin n -> Tm m) :
+ Lemma var_or_const_up n m (ρ : fin n -> PTm m) :
(forall i, var_or_const (ρ i)) ->
- (forall i, var_or_const (up_Tm_Tm ρ i)).
+ (forall i, var_or_const (up_PTm_PTm ρ i)).
Proof.
move => h /= [i|].
- asimpl.
@@ -688,11 +658,11 @@ Module RPar'.
Local Ltac antiimp := qauto l:on use:var_or_const_imp.
- Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
+ Lemma antirenaming n m (a : PTm n) (b : PTm m) (ρ : fin n -> PTm m) :
(forall i, var_or_const (ρ i)) ->
- R (subst_Tm ρ a) b -> exists b0, R a b0 /\ subst_Tm ρ b0 = b.
+ R (subst_PTm ρ a) b -> exists b0, R a b0 /\ subst_PTm ρ b0 = b.
Proof.
- move E : (subst_Tm ρ a) => u hρ h.
+ move E : (subst_PTm ρ a) => u hρ h.
move : n ρ hρ a E. elim : m u b/h.
- move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
first by antiimp.
@@ -756,66 +726,50 @@ Module RPar'.
spec_refl.
move : iha => [b0 [? ?]]. subst.
eexists. split. apply ProjCong; eauto. reflexivity.
- - move => n p A0 A1 B0 B1 ha iha hB ihB m ρ hρ []//=;
- first by antiimp.
- move => ? t t0 [*]. subst.
- have {}/iha := (hρ) => iha.
- have /var_or_const_up {}/ihB := (hρ) => ihB.
- spec_refl.
- move : iha => [b0 [? ?]].
- move : ihB => [c0 [? ?]]. subst.
- eexists. split. by apply BindCong; eauto.
- by asimpl.
- - hauto q:on ctrs:R inv:Tm.
+ - hauto q:on ctrs:R inv:PTm.
- move => n i n0 ρ hρ []//=; first by antiimp.
hauto l:on.
- - hauto q:on inv:Tm ctrs:R.
+ - hauto q:on inv:PTm ctrs:R.
Qed.
End RPar'.
Module ERed.
- Inductive R {n} : Tm n -> Tm n -> Prop :=
+ Inductive R {n} : PTm n -> PTm n -> Prop :=
(****************** Eta ***********************)
| AppEta a :
- R a (Abs (App (ren_Tm shift a) (VarTm var_zero)))
+ R a (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero)))
| PairEta a :
- R a (Pair (Proj PL a) (Proj PR a))
+ R a (PPair (PProj PL a) (PProj PR a))
(*************** Congruence ********************)
| AbsCong a0 a1 :
R a0 a1 ->
- R (Abs a0) (Abs a1)
+ R (PAbs a0) (PAbs a1)
| AppCong0 a0 a1 b :
R a0 a1 ->
- R (App a0 b) (App a1 b)
+ R (PApp a0 b) (PApp a1 b)
| AppCong1 a b0 b1 :
R b0 b1 ->
- R (App a b0) (App a b1)
+ R (PApp a b0) (PApp a b1)
| PairCong0 a0 a1 b :
R a0 a1 ->
- R (Pair a0 b) (Pair a1 b)
+ R (PPair a0 b) (PPair a1 b)
| PairCong1 a b0 b1 :
R b0 b1 ->
- R (Pair a b0) (Pair a b1)
+ R (PPair a b0) (PPair a b1)
| ProjCong p a0 a1 :
R a0 a1 ->
- R (Proj p a0) (Proj p a1)
- | BindCong0 p A0 A1 B:
- R A0 A1 ->
- R (TBind p A0 B) (TBind p A1 B)
- | BindCong1 p A B0 B1:
- R B0 B1 ->
- R (TBind p A B0) (TBind p A B1).
+ R (PProj p a0) (PProj p a1).
- Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
+ Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop.
- Lemma AppEta' n a (u : Tm n) :
- u = (Abs (App (ren_Tm shift a) (VarTm var_zero))) ->
+ Lemma AppEta' n a (u : PTm n) :
+ u = (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero))) ->
R a u.
Proof. move => ->. apply AppEta. Qed.
- Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
- R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
+ Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
+ R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
Proof.
move => h. move : m ξ.
elim : n a b /h.
@@ -825,9 +779,9 @@ Module ERed.
all : qauto ctrs:R.
Qed.
- Lemma substing n m (a : Tm n) b (ρ : fin n -> Tm m) :
+ Lemma substing n m (a : PTm n) b (ρ : fin n -> PTm m) :
R a b ->
- R (subst_Tm ρ a) (subst_Tm ρ b).
+ R (subst_PTm ρ a) (subst_PTm ρ b).
Proof.
move => h. move : m ρ. elim : n a b / h => n.
move => a m ρ /=.
@@ -846,79 +800,69 @@ Module EReds.
#[local]Ltac solve_s :=
repeat (induction 1; last by solve_s_rec); apply rtc_refl.
- Lemma AbsCong n (a b : Tm (S n)) :
+ Lemma AbsCong n (a b : PTm (S n)) :
rtc ERed.R a b ->
- rtc ERed.R (Abs a) (Abs b).
+ rtc ERed.R (PAbs a) (PAbs b).
Proof. solve_s. Qed.
- Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
+ Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
rtc ERed.R a0 a1 ->
rtc ERed.R b0 b1 ->
- rtc ERed.R (App a0 b0) (App a1 b1).
+ rtc ERed.R (PApp a0 b0) (PApp a1 b1).
Proof. solve_s. Qed.
- Lemma BindCong n p (a0 a1 : Tm n) b0 b1 :
+ Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
rtc ERed.R a0 a1 ->
rtc ERed.R b0 b1 ->
- rtc ERed.R (TBind p a0 b0) (TBind p a1 b1).
+ rtc ERed.R (PPair a0 b0) (PPair a1 b1).
Proof. solve_s. Qed.
- Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
+ Lemma ProjCong n p (a0 a1 : PTm n) :
rtc ERed.R a0 a1 ->
- rtc ERed.R b0 b1 ->
- rtc ERed.R (Pair a0 b0) (Pair a1 b1).
- Proof. solve_s. Qed.
-
- Lemma ProjCong n p (a0 a1 : Tm n) :
- rtc ERed.R a0 a1 ->
- rtc ERed.R (Proj p a0) (Proj p a1).
+ rtc ERed.R (PProj p a0) (PProj p a1).
Proof. solve_s. Qed.
End EReds.
Module EPar.
- Inductive R {n} : Tm n -> Tm n -> Prop :=
+ Inductive R {n} : PTm n -> PTm n -> Prop :=
(****************** Eta ***********************)
| AppEta a0 a1 :
R a0 a1 ->
- R a0 (Abs (App (ren_Tm shift a1) (VarTm var_zero)))
+ R a0 (PAbs (PApp (ren_PTm shift a1) (VarPTm var_zero)))
| PairEta a0 a1 :
R a0 a1 ->
- R a0 (Pair (Proj PL a1) (Proj PR a1))
+ R a0 (PPair (PProj PL a1) (PProj PR a1))
(*************** Congruence ********************)
- | Var i : R (VarTm i) (VarTm i)
+ | Var i : R (VarPTm i) (VarPTm i)
| AbsCong a0 a1 :
R a0 a1 ->
- R (Abs a0) (Abs a1)
+ R (PAbs a0) (PAbs a1)
| AppCong a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (App a0 b0) (App a1 b1)
+ R (PApp a0 b0) (PApp a1 b1)
| PairCong a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (Pair a0 b0) (Pair a1 b1)
+ R (PPair a0 b0) (PPair a1 b1)
| ProjCong p a0 a1 :
R a0 a1 ->
- R (Proj p a0) (Proj p a1)
- | BindCong p A0 A1 B0 B1:
- R A0 A1 ->
- R B0 B1 ->
- R (TBind p A0 B0) (TBind p A1 B1)
+ R (PProj p a0) (PProj p a1)
| ConstCong k :
- R (Const k) (Const k)
+ R (PConst k) (PConst k)
| UnivCong i :
- R (Univ i) (Univ i)
+ R (PUniv i) (PUniv i)
| BotCong :
- R Bot Bot.
+ R PBot PBot.
- Lemma refl n (a : Tm n) : EPar.R a a.
+ Lemma refl n (a : PTm n) : EPar.R a a.
Proof.
induction a; hauto lq:on ctrs:EPar.R.
Qed.
- Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
- R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
+ Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
+ R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
Proof.
move => h. move : m ξ.
elim : n a b /h.
@@ -930,18 +874,18 @@ Module EPar.
all : qauto ctrs:R.
Qed.
- Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
+ Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop.
- Lemma AppEta' n (a0 a1 b : Tm n) :
- b = (Abs (App (ren_Tm shift a1) (VarTm var_zero))) ->
+ Lemma AppEta' n (a0 a1 b : PTm n) :
+ b = (PAbs (PApp (ren_PTm shift a1) (VarPTm var_zero))) ->
R a0 a1 ->
R a0 b.
Proof. move => ->; apply AppEta. Qed.
- Lemma morphing n m (a b : Tm n) (ρ0 ρ1 : fin n -> Tm m) :
+ Lemma morphing n m (a b : PTm n) (ρ0 ρ1 : fin n -> PTm m) :
R a b ->
(forall i, R (ρ0 i) (ρ1 i)) ->
- R (subst_Tm ρ0 a) (subst_Tm ρ1 b).
+ R (subst_PTm ρ0 a) (subst_PTm ρ1 b).
Proof.
move => h. move : m ρ0 ρ1. elim : n a b / h => n.
- move => a0 a1 ha iha m ρ0 ρ1 hρ /=.
@@ -955,13 +899,12 @@ Module EPar.
- hauto l:on ctrs:R use:renaming inv:option.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R.
- - hauto lq:on ctrs:R.
Qed.
- Lemma substing n a0 a1 (b0 b1 : Tm n) :
+ Lemma substing n a0 a1 (b0 b1 : PTm n) :
R a0 a1 ->
R b0 b1 ->
- R (subst_Tm (scons b0 VarTm) a0) (subst_Tm (scons b1 VarTm) a1).
+ R (subst_PTm (scons b0 VarPTm) a0) (subst_PTm (scons b1 VarPTm) a1).
Proof.
move => h0 h1. apply morphing => //.
hauto lq:on ctrs:R inv:option.
@@ -971,14 +914,14 @@ End EPar.
Module OExp.
- Inductive R {n} : Tm n -> Tm n -> Prop :=
+ Inductive R {n} : PTm n -> PTm n -> Prop :=
(****************** Eta ***********************)
| AppEta a :
- R a (Abs (App (ren_Tm shift a) (VarTm var_zero)))
+ R a (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero)))
| PairEta a :
- R a (Pair (Proj PL a) (Proj PR a)).
+ R a (PPair (PProj PL a) (PProj PR a)).
- Lemma merge n (t a b : Tm n) :
+ Lemma merge n (t a b : PTm n) :
rtc R a b ->
EPar.R t a ->
EPar.R t b.
@@ -988,7 +931,7 @@ Module OExp.
- hauto q:on ctrs:EPar.R inv:R.
Qed.
- Lemma commutativity n (a b c : Tm n) :
+ Lemma commutativity n (a b c : PTm n) :
EPar.R a b -> R a c -> exists d, R b d /\ EPar.R c d.
Proof.
move => h.
@@ -997,7 +940,7 @@ Module OExp.
- hauto lq:on ctrs:EPar.R, R.
Qed.
- Lemma commutativity0 n (a b c : Tm n) :
+ Lemma commutativity0 n (a b c : PTm n) :
EPar.R a b -> rtc R a c -> exists d, rtc R b d /\ EPar.R c d.
Proof.
move => + h. move : b.
@@ -1022,72 +965,66 @@ Module RPars.
#[local]Ltac solve_s :=
repeat (induction 1; last by solve_s_rec); apply rtc_refl.
- Lemma AbsCong n (a b : Tm (S n)) :
+ Lemma AbsCong n (a b : PTm (S n)) :
rtc RPar.R a b ->
- rtc RPar.R (Abs a) (Abs b).
+ rtc RPar.R (PAbs a) (PAbs b).
Proof. solve_s. Qed.
- Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
+ Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
rtc RPar.R a0 a1 ->
rtc RPar.R b0 b1 ->
- rtc RPar.R (App a0 b0) (App a1 b1).
+ rtc RPar.R (PApp a0 b0) (PApp a1 b1).
Proof. solve_s. Qed.
- Lemma BindCong n p (a0 a1 : Tm n) b0 b1 :
+ Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
rtc RPar.R a0 a1 ->
rtc RPar.R b0 b1 ->
- rtc RPar.R (TBind p a0 b0) (TBind p a1 b1).
+ rtc RPar.R (PPair a0 b0) (PPair a1 b1).
Proof. solve_s. Qed.
- Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
+ Lemma ProjCong n p (a0 a1 : PTm n) :
rtc RPar.R a0 a1 ->
- rtc RPar.R b0 b1 ->
- rtc RPar.R (Pair a0 b0) (Pair a1 b1).
+ rtc RPar.R (PProj p a0) (PProj p a1).
Proof. solve_s. Qed.
- Lemma ProjCong n p (a0 a1 : Tm n) :
+ Lemma renaming n (a0 a1 : PTm n) m (ξ : fin n -> fin m) :
rtc RPar.R a0 a1 ->
- rtc RPar.R (Proj p a0) (Proj p a1).
- Proof. solve_s. Qed.
-
- Lemma renaming n (a0 a1 : Tm n) m (ξ : fin n -> fin m) :
- rtc RPar.R a0 a1 ->
- rtc RPar.R (ren_Tm ξ a0) (ren_Tm ξ a1).
+ rtc RPar.R (ren_PTm ξ a0) (ren_PTm ξ a1).
Proof.
induction 1.
- apply rtc_refl.
- eauto using RPar.renaming, rtc_l.
Qed.
- Lemma weakening n (a0 a1 : Tm n) :
+ Lemma weakening n (a0 a1 : PTm n) :
rtc RPar.R a0 a1 ->
- rtc RPar.R (ren_Tm shift a0) (ren_Tm shift a1).
+ rtc RPar.R (ren_PTm shift a0) (ren_PTm shift a1).
Proof. apply renaming. Qed.
- Lemma Abs_inv n (a : Tm (S n)) b :
- rtc RPar.R (Abs a) b -> exists a', b = Abs a' /\ rtc RPar.R a a'.
+ Lemma Abs_inv n (a : PTm (S n)) b :
+ rtc RPar.R (PAbs a) b -> exists a', b = PAbs a' /\ rtc RPar.R a a'.
Proof.
- move E : (Abs a) => b0 h. move : a E.
+ move E : (PAbs a) => b0 h. move : a E.
elim : b0 b / h.
- hauto lq:on ctrs:rtc.
- hauto lq:on ctrs:rtc inv:RPar.R, rtc.
Qed.
- Lemma morphing n m (a b : Tm n) (ρ : fin n -> Tm m) :
+ Lemma morphing n m (a b : PTm n) (ρ : fin n -> PTm m) :
rtc RPar.R a b ->
- rtc RPar.R (subst_Tm ρ a) (subst_Tm ρ b).
+ rtc RPar.R (subst_PTm ρ a) (subst_PTm ρ b).
Proof. induction 1; qauto l:on ctrs:rtc use:RPar.substing. Qed.
- Lemma substing n (a b : Tm (S n)) c :
+ Lemma substing n (a b : PTm (S n)) c :
rtc RPar.R a b ->
- rtc RPar.R (subst_Tm (scons c VarTm) a) (subst_Tm (scons c VarTm) b).
+ rtc RPar.R (subst_PTm (scons c VarPTm) a) (subst_PTm (scons c VarPTm) b).
Proof. hauto lq:on use:morphing inv:option. Qed.
- Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
+ Lemma antirenaming n m (a : PTm n) (b : PTm m) (ρ : fin n -> PTm m) :
(forall i, var_or_const (ρ i)) ->
- rtc RPar.R (subst_Tm ρ a) b -> exists b0, rtc RPar.R a b0 /\ subst_Tm ρ b0 = b.
+ rtc RPar.R (subst_PTm ρ a) b -> exists b0, rtc RPar.R a b0 /\ subst_PTm ρ b0 = b.
Proof.
- move E :(subst_Tm ρ a) => u hρ h.
+ move E :(subst_PTm ρ a) => u hρ h.
move : a E.
elim : u b /h.
- sfirstorder.
@@ -1109,72 +1046,66 @@ Module RPars'.
#[local]Ltac solve_s :=
repeat (induction 1; last by solve_s_rec); apply rtc_refl.
- Lemma AbsCong n (a b : Tm (S n)) :
+ Lemma AbsCong n (a b : PTm (S n)) :
rtc RPar'.R a b ->
- rtc RPar'.R (Abs a) (Abs b).
+ rtc RPar'.R (PAbs a) (PAbs b).
Proof. solve_s. Qed.
- Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
+ Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
rtc RPar'.R a0 a1 ->
rtc RPar'.R b0 b1 ->
- rtc RPar'.R (App a0 b0) (App a1 b1).
+ rtc RPar'.R (PApp a0 b0) (PApp a1 b1).
Proof. solve_s. Qed.
- Lemma BindCong n p (a0 a1 : Tm n) b0 b1 :
+ Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
rtc RPar'.R a0 a1 ->
rtc RPar'.R b0 b1 ->
- rtc RPar'.R (TBind p a0 b0) (TBind p a1 b1).
+ rtc RPar'.R (PPair a0 b0) (PPair a1 b1).
Proof. solve_s. Qed.
- Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
+ Lemma ProjCong n p (a0 a1 : PTm n) :
rtc RPar'.R a0 a1 ->
- rtc RPar'.R b0 b1 ->
- rtc RPar'.R (Pair a0 b0) (Pair a1 b1).
+ rtc RPar'.R (PProj p a0) (PProj p a1).
Proof. solve_s. Qed.
- Lemma ProjCong n p (a0 a1 : Tm n) :
+ Lemma renaming n (a0 a1 : PTm n) m (ξ : fin n -> fin m) :
rtc RPar'.R a0 a1 ->
- rtc RPar'.R (Proj p a0) (Proj p a1).
- Proof. solve_s. Qed.
-
- Lemma renaming n (a0 a1 : Tm n) m (ξ : fin n -> fin m) :
- rtc RPar'.R a0 a1 ->
- rtc RPar'.R (ren_Tm ξ a0) (ren_Tm ξ a1).
+ rtc RPar'.R (ren_PTm ξ a0) (ren_PTm ξ a1).
Proof.
induction 1.
- apply rtc_refl.
- eauto using RPar'.renaming, rtc_l.
Qed.
- Lemma weakening n (a0 a1 : Tm n) :
+ Lemma weakening n (a0 a1 : PTm n) :
rtc RPar'.R a0 a1 ->
- rtc RPar'.R (ren_Tm shift a0) (ren_Tm shift a1).
+ rtc RPar'.R (ren_PTm shift a0) (ren_PTm shift a1).
Proof. apply renaming. Qed.
- Lemma Abs_inv n (a : Tm (S n)) b :
- rtc RPar'.R (Abs a) b -> exists a', b = Abs a' /\ rtc RPar'.R a a'.
+ Lemma Abs_inv n (a : PTm (S n)) b :
+ rtc RPar'.R (PAbs a) b -> exists a', b = PAbs a' /\ rtc RPar'.R a a'.
Proof.
- move E : (Abs a) => b0 h. move : a E.
+ move E : (PAbs a) => b0 h. move : a E.
elim : b0 b / h.
- hauto lq:on ctrs:rtc.
- hauto lq:on ctrs:rtc inv:RPar'.R, rtc.
Qed.
- Lemma morphing n m (a b : Tm n) (ρ : fin n -> Tm m) :
+ Lemma morphing n m (a b : PTm n) (ρ : fin n -> PTm m) :
rtc RPar'.R a b ->
- rtc RPar'.R (subst_Tm ρ a) (subst_Tm ρ b).
+ rtc RPar'.R (subst_PTm ρ a) (subst_PTm ρ b).
Proof. induction 1; qauto l:on ctrs:rtc use:RPar'.substing. Qed.
- Lemma substing n (a b : Tm (S n)) c :
+ Lemma substing n (a b : PTm (S n)) c :
rtc RPar'.R a b ->
- rtc RPar'.R (subst_Tm (scons c VarTm) a) (subst_Tm (scons c VarTm) b).
+ rtc RPar'.R (subst_PTm (scons c VarPTm) a) (subst_PTm (scons c VarPTm) b).
Proof. hauto lq:on use:morphing inv:option. Qed.
- Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
+ Lemma antirenaming n m (a : PTm n) (b : PTm m) (ρ : fin n -> PTm m) :
(forall i, var_or_const (ρ i)) ->
- rtc RPar'.R (subst_Tm ρ a) b -> exists b0, rtc RPar'.R a b0 /\ subst_Tm ρ b0 = b.
+ rtc RPar'.R (subst_PTm ρ a) b -> exists b0, rtc RPar'.R a b0 /\ subst_PTm ρ b0 = b.
Proof.
- move E :(subst_Tm ρ a) => u hρ h.
+ move E :(subst_PTm ρ a) => u hρ h.
move : a E.
elim : u b /h.
- sfirstorder.
@@ -1187,15 +1118,15 @@ Module RPars'.
End RPars'.
-Lemma Abs_EPar n a (b : Tm n) :
- EPar.R (Abs a) b ->
+Lemma Abs_EPar n a (b : PTm n) :
+ EPar.R (PAbs a) b ->
(exists d, EPar.R a d /\
- rtc RPar.R (App (ren_Tm shift b) (VarTm var_zero)) d) /\
+ rtc RPar.R (PApp (ren_PTm shift b) (VarPTm var_zero)) d) /\
(exists d,
EPar.R a d /\ forall p,
- rtc RPar.R (Proj p b) (Abs (Proj p d))).
+ rtc RPar.R (PProj p b) (PAbs (PProj p d))).
Proof.
- move E : (Abs a) => u h.
+ move E : (PAbs a) => u h.
move : a E.
elim : n u b /h => //=.
- move => n a0 a1 ha iha b ?. subst.
@@ -1216,14 +1147,14 @@ Proof.
- move => n ? a1 ha iha a0 ?. subst. specialize iha with (1 := eq_refl).
move : iha => [_ [d [ih0 ih1]]].
split.
- + exists (Pair (Proj PL d) (Proj PR d)).
+ + exists (PPair (PProj PL d) (PProj PR d)).
split; first by apply EPar.PairEta.
apply : rtc_l.
apply RPar.AppPair; eauto using RPar.refl.
- suff h : forall p, rtc RPar.R (App (Proj p (ren_Tm shift a1)) (VarTm var_zero)) (Proj p d) by
+ suff h : forall p, rtc RPar.R (PApp (PProj p (ren_PTm shift a1)) (VarPTm var_zero)) (PProj p d) by
sfirstorder use:RPars.PairCong.
move => p. move /(_ p) /RPars.weakening in ih1.
- apply relations.rtc_transitive with (y := App (ren_Tm shift (Abs (Proj p d))) (VarTm var_zero)).
+ apply relations.rtc_transitive with (y := PApp (ren_PTm shift (PAbs (PProj p d))) (VarPTm var_zero)).
by eauto using RPars.AppCong, rtc_refl.
apply relations.rtc_once => /=.
apply : RPar.AppAbs'; eauto using RPar.refl.
@@ -1239,21 +1170,21 @@ Proof.
apply rtc_once. apply : RPar.ProjAbs; eauto using RPar.refl.
Qed.
-Lemma Pair_EPar n (a b c : Tm n) :
- EPar.R (Pair a b) c ->
- (forall p, exists d, rtc RPar.R (Proj p c) d /\ EPar.R (if p is PL then a else b) d) /\
- (exists d0 d1, rtc RPar.R (App (ren_Tm shift c) (VarTm var_zero))
- (Pair (App (ren_Tm shift d0) (VarTm var_zero))(App (ren_Tm shift d1) (VarTm var_zero))) /\
+Lemma Pair_EPar n (a b c : PTm n) :
+ EPar.R (PPair a b) c ->
+ (forall p, exists d, rtc RPar.R (PProj p c) d /\ EPar.R (if p is PL then a else b) d) /\
+ (exists d0 d1, rtc RPar.R (PApp (ren_PTm shift c) (VarPTm var_zero))
+ (PPair (PApp (ren_PTm shift d0) (VarPTm var_zero))(PApp (ren_PTm shift d1) (VarPTm var_zero))) /\
EPar.R a d0 /\ EPar.R b d1).
Proof.
- move E : (Pair a b) => u h. move : a b E.
+ move E : (PPair a b) => u h. move : a b E.
elim : n u c /h => //=.
- move => n a0 a1 ha iha a b ?. subst.
specialize iha with (1 := eq_refl).
move : iha => [_ [d0 [d1 [ih0 [ih1 ih2]]]]].
split.
+ move => p.
- exists (Abs (App (ren_Tm shift (if p is PL then d0 else d1)) (VarTm var_zero))).
+ exists (PAbs (PApp (ren_PTm shift (if p is PL then d0 else d1)) (VarPTm var_zero))).
split.
* apply : relations.rtc_transitive.
** apply RPars.ProjCong. apply RPars.AbsCong. eassumption.
@@ -1271,7 +1202,7 @@ Proof.
exists d. split=>//.
apply : rtc_l. apply RPar.ProjPair; eauto using RPar.refl.
set q := (X in rtc RPar.R X d).
- by have -> : q = Proj p a1 by hauto lq:on.
+ by have -> : q = PProj p a1 by hauto lq:on.
+ move :iha => [iha _].
move : (iha PL) => [d0 [ih0 ih0']].
move : (iha PR) => [d1 [ih1 ih1']] {iha}.
@@ -1292,7 +1223,7 @@ Proof.
split => //.
Qed.
-Lemma commutativity0 n (a b0 b1 : Tm n) :
+Lemma commutativity0 n (a b0 b1 : PTm n) :
EPar.R a b0 -> RPar.R a b1 -> exists c, rtc RPar.R b0 c /\ EPar.R b1 c.
Proof.
move => h. move : b1.
@@ -1300,13 +1231,13 @@ Proof.
- move => n a b0 ha iha b1 hb.
move : iha (hb) => /[apply].
move => [c [ih0 ih1]].
- exists (Abs (App (ren_Tm shift c) (VarTm var_zero))).
+ exists (PAbs (PApp (ren_PTm shift c) (VarPTm var_zero))).
split.
+ hauto lq:on ctrs:rtc use:RPars.AbsCong, RPars.AppCong, RPars.renaming.
+ hauto lq:on ctrs:EPar.R use:EPar.refl, EPar.renaming.
- move => n a b0 hb0 ihb0 b1 /[dup] hb1 {}/ihb0.
move => [c [ih0 ih1]].
- exists (Pair (Proj PL c) (Proj PR c)). split.
+ exists (PPair (PProj PL c) (PProj PR c)). split.
+ apply RPars.PairCong;
by apply RPars.ProjCong.
+ hauto lq:on ctrs:EPar.R use:EPar.refl, EPar.renaming.
@@ -1319,9 +1250,9 @@ Proof.
elim /RPar.inv => //= _.
+ move => a2 a3 b3 b4 h0 h1 [*]. subst.
move /(_ _ ltac:(by eauto)) : ihb => [b [ihb0 ihb1]].
- have {}/iha : RPar.R (Abs a2) (Abs a3) by hauto lq:on ctrs:RPar.R.
+ have {}/iha : RPar.R (PAbs a2) (PAbs a3) by hauto lq:on ctrs:RPar.R.
move => [c [ih0 /Abs_EPar [[d [ih1 ih2]] _]]].
- exists (subst_Tm (scons b VarTm) d).
+ exists (subst_PTm (scons b VarPTm) d).
split.
(* By substitution *)
* move /RPars.substing : ih2.
@@ -1337,7 +1268,7 @@ Proof.
move /Pair_EPar : ihc1 => [_ [d0 [d1 [ih0 [ih1 ih2]]]]].
move /RPars.substing : ih0. move /(_ d).
asimpl => h.
- exists (Pair (App d0 d) (App d1 d)).
+ exists (PPair (PApp d0 d) (PApp d1 d)).
split.
hauto lq:on use:relations.rtc_transitive, RPars.AppCong.
apply EPar.PairCong; by apply EPar.AppCong.
@@ -1348,7 +1279,7 @@ Proof.
+ move => ? a0 a1 h [*]. subst.
move /(_ _ ltac:(by eauto using RPar.AbsCong)) : ih0 => [c [ih0 ih1]].
move /Abs_EPar : ih1 => [_ [d [ih1 ih2]]].
- exists (Abs (Proj p d)).
+ exists (PAbs (PProj p d)).
qauto l:on ctrs:EPar.R use:RPars.ProjCong, @relations.rtc_transitive.
+ move => p0 a0 a1 b2 b3 h1 h2 [*]. subst.
move /(_ _ ltac:(by eauto using RPar.PairCong)) : ih0 => [c [ih0 ih1]].
@@ -1356,13 +1287,12 @@ Proof.
exists d. split => //.
hauto lq:on use:RPars.ProjCong, relations.rtc_transitive.
+ hauto lq:on ctrs:EPar.R use:RPars.ProjCong.
- - hauto lq:on inv:RPar.R ctrs:EPar.R, rtc use:RPars.BindCong.
- hauto l:on ctrs:EPar.R inv:RPar.R.
- hauto l:on ctrs:EPar.R inv:RPar.R.
- hauto l:on ctrs:EPar.R inv:RPar.R.
Qed.
-Lemma commutativity1 n (a b0 b1 : Tm n) :
+Lemma commutativity1 n (a b0 b1 : PTm n) :
EPar.R a b0 -> rtc RPar.R a b1 -> exists c, rtc RPar.R b0 c /\ EPar.R b1 c.
Proof.
move => + h. move : b0.
@@ -1371,7 +1301,7 @@ Proof.
- qauto l:on use:relations.rtc_transitive, commutativity0.
Qed.
-Lemma commutativity n (a b0 b1 : Tm n) :
+Lemma commutativity n (a b0 b1 : PTm n) :
rtc EPar.R a b0 -> rtc RPar.R a b1 -> exists c, rtc RPar.R b0 c /\ rtc EPar.R b1 c.
move => h. move : b1. elim : a b0 /h.
- sfirstorder.
@@ -1380,12 +1310,12 @@ Lemma commutativity n (a b0 b1 : Tm n) :
hauto q:on ctrs:rtc.
Qed.
-Lemma Abs_EPar' n a (b : Tm n) :
- EPar.R (Abs a) b ->
+Lemma Abs_EPar' n a (b : PTm n) :
+ EPar.R (PAbs a) b ->
(exists d, EPar.R a d /\
- rtc OExp.R (Abs d) b).
+ rtc OExp.R (PAbs d) b).
Proof.
- move E : (Abs a) => u h.
+ move E : (PAbs a) => u h.
move : a E.
elim : n u b /h => //=.
- move => n a0 a1 ha iha a ?. subst.
@@ -1397,12 +1327,12 @@ Proof.
- hauto l:on ctrs:OExp.R.
Qed.
-Lemma Proj_EPar' n p a (b : Tm n) :
- EPar.R (Proj p a) b ->
+Lemma Proj_EPar' n p a (b : PTm n) :
+ EPar.R (PProj p a) b ->
(exists d, EPar.R a d /\
- rtc OExp.R (Proj p d) b).
+ rtc OExp.R (PProj p d) b).
Proof.
- move E : (Proj p a) => u h.
+ move E : (PProj p a) => u h.
move : p a E.
elim : n u b /h => //=.
- move => n a0 a1 ha iha a p ?. subst.
@@ -1414,11 +1344,11 @@ Proof.
- hauto l:on ctrs:OExp.R.
Qed.
-Lemma App_EPar' n (a b u : Tm n) :
- EPar.R (App a b) u ->
- (exists a0 b0, EPar.R a a0 /\ EPar.R b b0 /\ rtc OExp.R (App a0 b0) u).
+Lemma App_EPar' n (a b u : PTm n) :
+ EPar.R (PApp a b) u ->
+ (exists a0 b0, EPar.R a a0 /\ EPar.R b b0 /\ rtc OExp.R (PApp a0 b0) u).
Proof.
- move E : (App a b) => t h.
+ move E : (PApp a b) => t h.
move : a b E. elim : n t u /h => //=.
- move => n a0 a1 ha iha a b ?. subst.
specialize iha with (1 := eq_refl).
@@ -1429,11 +1359,11 @@ Proof.
- hauto l:on ctrs:OExp.R.
Qed.
-Lemma Bind_EPar' n p (a : Tm n) b u :
- EPar.R (TBind p a b) u ->
- (exists a0 b0, EPar.R a a0 /\ EPar.R b b0 /\ rtc OExp.R (TBind p a0 b0) u).
+Lemma Pair_EPar' n (a b u : PTm n) :
+ EPar.R (PPair a b) u ->
+ exists a0 b0, EPar.R a a0 /\ EPar.R b b0 /\ rtc OExp.R (PPair a0 b0) u.
Proof.
- move E : (TBind p a b) => t h.
+ move E : (PPair a b) => t h.
move : a b E. elim : n t u /h => //=.
- move => n a0 a1 ha iha a b ?. subst.
specialize iha with (1 := eq_refl).
@@ -1444,25 +1374,10 @@ Proof.
- hauto l:on ctrs:OExp.R.
Qed.
-Lemma Pair_EPar' n (a b u : Tm n) :
- EPar.R (Pair a b) u ->
- exists a0 b0, EPar.R a a0 /\ EPar.R b b0 /\ rtc OExp.R (Pair a0 b0) u.
-Proof.
- move E : (Pair a b) => t h.
- move : a b E. elim : n t u /h => //=.
- - move => n a0 a1 ha iha a b ?. subst.
- specialize iha with (1 := eq_refl).
- hauto lq:on ctrs:OExp.R use:rtc_r.
- - move => n a0 a1 ha iha a b ?. subst.
- specialize iha with (1 := eq_refl).
- hauto lq:on ctrs:OExp.R use:rtc_r.
- - hauto l:on ctrs:OExp.R.
-Qed.
-
-Lemma Const_EPar' n k (u : Tm n) :
- EPar.R (Const k) u ->
- rtc OExp.R (Const k) u.
- move E : (Const k) => t h.
+Lemma Const_EPar' n k (u : PTm n) :
+ EPar.R (PConst k) u ->
+ rtc OExp.R (PConst k) u.
+ move E : (PConst k) => t h.
move : k E. elim : n t u /h => //=.
- move => n a0 a1 h ih k ?. subst.
specialize ih with (1 := eq_refl).
@@ -1473,10 +1388,10 @@ Lemma Const_EPar' n k (u : Tm n) :
- hauto l:on ctrs:OExp.R.
Qed.
-Lemma Bot_EPar' n (u : Tm n) :
- EPar.R (Bot) u ->
- rtc OExp.R (Bot) u.
- move E : (Bot) => t h.
+Lemma Bot_EPar' n (u : PTm n) :
+ EPar.R (PBot) u ->
+ rtc OExp.R (PBot) u.
+ move E : (PBot) => t h.
move : E. elim : n t u /h => //=.
- move => n a0 a1 h ih ?. subst.
specialize ih with (1 := eq_refl).
@@ -1487,10 +1402,10 @@ Lemma Bot_EPar' n (u : Tm n) :
- hauto l:on ctrs:OExp.R.
Qed.
-Lemma Univ_EPar' n i (u : Tm n) :
- EPar.R (Univ i) u ->
- rtc OExp.R (Univ i) u.
- move E : (Univ i) => t h.
+Lemma Univ_EPar' n i (u : PTm n) :
+ EPar.R (PUniv i) u ->
+ rtc OExp.R (PUniv i) u.
+ move E : (PUniv i) => t h.
move : E. elim : n t u /h => //=.
- move => n a0 a1 h ih ?. subst.
specialize ih with (1 := eq_refl).
@@ -1501,14 +1416,14 @@ Lemma Univ_EPar' n i (u : Tm n) :
- hauto l:on ctrs:OExp.R.
Qed.
-Lemma EPar_diamond n (c a1 b1 : Tm n) :
+Lemma EPar_diamond n (c a1 b1 : PTm n) :
EPar.R c a1 ->
EPar.R c b1 ->
exists d2, EPar.R a1 d2 /\ EPar.R b1 d2.
Proof.
move => h. move : b1. elim : n c a1 / h.
- move => n c a1 ha iha b1 /iha [d2 [hd0 hd1]].
- exists(Abs (App (ren_Tm shift d2) (VarTm var_zero))).
+ exists(PAbs (PApp (ren_PTm shift d2) (VarPTm var_zero))).
hauto lq:on ctrs:EPar.R use:EPar.renaming.
- hauto lq:on rew:off ctrs:EPar.R.
- hauto lq:on use:EPar.refl.
@@ -1516,62 +1431,54 @@ Proof.
move /Abs_EPar' => [d [hd0 hd1]].
move : iha hd0; repeat move/[apply].
move => [d2 [h0 h1]].
- have : EPar.R (Abs d) (Abs d2) by eauto using EPar.AbsCong.
+ have : EPar.R (PAbs d) (PAbs d2) by eauto using EPar.AbsCong.
move : OExp.commutativity0 hd1; repeat move/[apply].
move => [d1 [hd1 hd2]].
exists d1. hauto lq:on ctrs:EPar.R use:OExp.merge.
- move => n a0 a1 b0 b1 ha iha hb ihb c.
move /App_EPar' => [a2][b2][/iha [a3 h0]][/ihb [b3 h1]]h2 {iha ihb}.
- have : EPar.R (App a2 b2)(App a3 b3)
+ have : EPar.R (PApp a2 b2)(PApp a3 b3)
by hauto l:on use:EPar.AppCong.
move : OExp.commutativity0 h2; repeat move/[apply].
move => [d h].
exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
- move => n a0 a1 b0 b1 ha iha hb ihb c.
move /Pair_EPar' => [a2][b2][/iha [a3 h0]][/ihb [b3 h1]]h2 {iha ihb}.
- have : EPar.R (Pair a2 b2)(Pair a3 b3)
+ have : EPar.R (PPair a2 b2)(PPair a3 b3)
by hauto l:on use:EPar.PairCong.
move : OExp.commutativity0 h2; repeat move/[apply].
move => [d h].
exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
- move => n p a0 a1 ha iha b.
move /Proj_EPar' => [d [/iha [d2 h] h1]] {iha}.
- have : EPar.R (Proj p d) (Proj p d2)
+ have : EPar.R (PProj p d) (PProj p d2)
by hauto l:on use:EPar.ProjCong.
move : OExp.commutativity0 h1; repeat move/[apply].
move => [d1 h1].
exists d1. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
- - move => n p a0 a1 b0 b1 ha iha hb ihb c.
- move /Bind_EPar' => [a2][b2][/iha [a3 h0]][/ihb [b3 h1]]h2 {iha ihb}.
- have : EPar.R (TBind p a2 b2)(TBind p a3 b3)
- by hauto l:on use:EPar.BindCong.
- move : OExp.commutativity0 h2; repeat move/[apply].
- move => [d h].
- exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
- qauto use:Const_EPar', EPar.refl.
- qauto use:Univ_EPar', EPar.refl.
- qauto use:Bot_EPar', EPar.refl.
Qed.
-Function tstar {n} (a : Tm n) :=
+Function tstar {n} (a : PTm n) :=
match a with
- | VarTm i => a
- | Abs a => Abs (tstar a)
- | App (Abs a) b => subst_Tm (scons (tstar b) VarTm) (tstar a)
- | App (Pair a b) c =>
- Pair (App (tstar a) (tstar c)) (App (tstar b) (tstar c))
- | App a b => App (tstar a) (tstar b)
- | Pair a b => Pair (tstar a) (tstar b)
- | Proj p (Pair a b) => if p is PL then (tstar a) else (tstar b)
- | Proj p (Abs a) => (Abs (Proj p (tstar a)))
- | Proj p a => Proj p (tstar a)
- | TBind p a b => TBind p (tstar a) (tstar b)
- | Const k => Const k
- | Univ i => Univ i
- | Bot => Bot
+ | VarPTm i => a
+ | PAbs a => PAbs (tstar a)
+ | PApp (PAbs a) b => subst_PTm (scons (tstar b) VarPTm) (tstar a)
+ | PApp (PPair a b) c =>
+ PPair (PApp (tstar a) (tstar c)) (PApp (tstar b) (tstar c))
+ | PApp a b => PApp (tstar a) (tstar b)
+ | PPair a b => PPair (tstar a) (tstar b)
+ | PProj p (PPair a b) => if p is PL then (tstar a) else (tstar b)
+ | PProj p (PAbs a) => (PAbs (PProj p (tstar a)))
+ | PProj p a => PProj p (tstar a)
+ | PConst k => PConst k
+ | PUniv i => PUniv i
+ | PBot => PBot
end.
-Lemma RPar_triangle n (a : Tm n) : forall b, RPar.R a b -> RPar.R b (tstar a).
+Lemma RPar_triangle n (a : PTm n) : forall b, RPar.R a b -> RPar.R b (tstar a).
Proof.
apply tstar_ind => {n a}.
- hauto lq:on inv:RPar.R ctrs:RPar.R.
@@ -1587,25 +1494,23 @@ Proof.
- hauto lq:on inv:RPar.R ctrs:RPar.R.
- hauto lq:on inv:RPar.R ctrs:RPar.R.
- hauto lq:on inv:RPar.R ctrs:RPar.R.
- - hauto lq:on inv:RPar.R ctrs:RPar.R.
Qed.
-Function tstar' {n} (a : Tm n) :=
+Function tstar' {n} (a : PTm n) :=
match a with
- | VarTm i => a
- | Abs a => Abs (tstar' a)
- | App (Abs a) b => subst_Tm (scons (tstar' b) VarTm) (tstar' a)
- | App a b => App (tstar' a) (tstar' b)
- | Pair a b => Pair (tstar' a) (tstar' b)
- | Proj p (Pair a b) => if p is PL then (tstar' a) else (tstar' b)
- | Proj p a => Proj p (tstar' a)
- | TBind p a b => TBind p (tstar' a) (tstar' b)
- | Const k => Const k
- | Univ i => Univ i
- | Bot => Bot
+ | VarPTm i => a
+ | PAbs a => PAbs (tstar' a)
+ | PApp (PAbs a) b => subst_PTm (scons (tstar' b) VarPTm) (tstar' a)
+ | PApp a b => PApp (tstar' a) (tstar' b)
+ | PPair a b => PPair (tstar' a) (tstar' b)
+ | PProj p (PPair a b) => if p is PL then (tstar' a) else (tstar' b)
+ | PProj p a => PProj p (tstar' a)
+ | PConst k => PConst k
+ | PUniv i => PUniv i
+ | PBot => PBot
end.
-Lemma RPar'_triangle n (a : Tm n) : forall b, RPar'.R a b -> RPar'.R b (tstar' a).
+Lemma RPar'_triangle n (a : PTm n) : forall b, RPar'.R a b -> RPar'.R b (tstar' a).
Proof.
apply tstar'_ind => {n a}.
- hauto lq:on inv:RPar'.R ctrs:RPar'.R.
@@ -1619,22 +1524,21 @@ Proof.
- hauto lq:on inv:RPar'.R ctrs:RPar'.R.
- hauto lq:on inv:RPar'.R ctrs:RPar'.R.
- hauto lq:on inv:RPar'.R ctrs:RPar'.R.
- - hauto lq:on inv:RPar'.R ctrs:RPar'.R.
Qed.
-Lemma RPar_diamond n (c a1 b1 : Tm n) :
+Lemma RPar_diamond n (c a1 b1 : PTm n) :
RPar.R c a1 ->
RPar.R c b1 ->
exists d2, RPar.R a1 d2 /\ RPar.R b1 d2.
Proof. hauto l:on use:RPar_triangle. Qed.
-Lemma RPar'_diamond n (c a1 b1 : Tm n) :
+Lemma RPar'_diamond n (c a1 b1 : PTm n) :
RPar'.R c a1 ->
RPar'.R c b1 ->
exists d2, RPar'.R a1 d2 /\ RPar'.R b1 d2.
Proof. hauto l:on use:RPar'_triangle. Qed.
-Lemma RPar_confluent n (c a1 b1 : Tm n) :
+Lemma RPar_confluent n (c a1 b1 : PTm n) :
rtc RPar.R c a1 ->
rtc RPar.R c b1 ->
exists d2, rtc RPar.R a1 d2 /\ rtc RPar.R b1 d2.
@@ -1642,7 +1546,7 @@ Proof.
sfirstorder use:relations.diamond_confluent, RPar_diamond.
Qed.
-Lemma EPar_confluent n (c a1 b1 : Tm n) :
+Lemma EPar_confluent n (c a1 b1 : PTm n) :
rtc EPar.R c a1 ->
rtc EPar.R c b1 ->
exists d2, rtc EPar.R a1 d2 /\ rtc EPar.R b1 d2.
@@ -1650,52 +1554,35 @@ Proof.
sfirstorder use:relations.diamond_confluent, EPar_diamond.
Qed.
-Definition prov_bind {n} p0 A0 B0 (a : Tm n) :=
- match a with
- | TBind p A B => p = p0 /\ rtc Par.R A A0 /\ rtc Par.R B B0
- | _ => False
- end.
-
-Definition prov_univ {n} i0 (a : Tm n) :=
- match a with
- | Univ i => i = i0
- | _ => False
- end.
-
-
-Inductive prov {n} : Tm n -> Tm n -> Prop :=
-| P_Bind p A A0 B B0 :
- rtc Par.R A A0 ->
- rtc Par.R B B0 ->
- prov (TBind p A B) (TBind p A0 B0)
+Inductive prov {n} : PTm n -> PTm n -> Prop :=
| P_Abs h a :
- (forall b, prov h (subst_Tm (scons b VarTm) a)) ->
- prov h (Abs a)
+ (forall b, prov h (subst_PTm (scons b VarPTm) a)) ->
+ prov h (PAbs a)
| P_App h a b :
prov h a ->
- prov h (App a b)
+ prov h (PApp a b)
| P_Pair h a b :
prov h a ->
prov h b ->
- prov h (Pair a b)
+ prov h (PPair a b)
| P_Proj h p a :
prov h a ->
- prov h (Proj p a)
+ prov h (PProj p a)
| P_Const k :
- prov (Const k) (Const k)
+ prov (PConst k) (PConst k)
| P_Var i :
- prov (VarTm i) (VarTm i)
+ prov (VarPTm i) (VarPTm i)
| P_Univ i :
- prov (Univ i) (Univ i)
+ prov (PUniv i) (PUniv i)
| P_Bot :
- prov Bot Bot.
+ prov PBot PBot.
-Lemma ERed_EPar n (a b : Tm n) : ERed.R a b -> EPar.R a b.
+Lemma ERed_EPar n (a b : PTm n) : ERed.R a b -> EPar.R a b.
Proof.
induction 1; hauto lq:on ctrs:EPar.R use:EPar.refl.
Qed.
-Lemma EPar_ERed n (a b : Tm n) : EPar.R a b -> rtc ERed.R a b.
+Lemma EPar_ERed n (a b : PTm n) : EPar.R a b -> rtc ERed.R a b.
Proof.
move => h. elim : n a b /h.
- eauto using rtc_r, ERed.AppEta.
@@ -1705,57 +1592,56 @@ Proof.
- eauto using EReds.AppCong.
- eauto using EReds.PairCong.
- eauto using EReds.ProjCong.
- - eauto using EReds.BindCong.
- auto using rtc_refl.
- auto using rtc_refl.
- auto using rtc_refl.
Qed.
-Lemma EPar_Par n (a b : Tm n) : EPar.R a b -> Par.R a b.
+Lemma EPar_Par n (a b : PTm n) : EPar.R a b -> Par.R a b.
Proof.
move => h. elim : n a b /h; qauto ctrs:Par.R.
Qed.
-Lemma RPar_Par n (a b : Tm n) : RPar.R a b -> Par.R a b.
+Lemma RPar_Par n (a b : PTm n) : RPar.R a b -> Par.R a b.
Proof.
move => h. elim : n a b /h; hauto lq:on ctrs:Par.R.
Qed.
-Lemma rtc_idem n (R : Tm n -> Tm n -> Prop) (a b : Tm n) : rtc (rtc R) a b -> rtc R a b.
+Lemma rtc_idem n (R : PTm n -> PTm n -> Prop) (a b : PTm n) : rtc (rtc R) a b -> rtc R a b.
Proof.
induction 1; hauto l:on use:@relations.rtc_transitive, @rtc_r.
Qed.
-Lemma EPars_EReds {n} (a b : Tm n) : rtc EPar.R a b <-> rtc ERed.R a b.
+Lemma EPars_EReds {n} (a b : PTm n) : rtc EPar.R a b <-> rtc ERed.R a b.
Proof.
sfirstorder use:@relations.rtc_subrel, EPar_ERed, rtc_idem, ERed_EPar.
Qed.
-Lemma prov_rpar n (u : Tm n) a b : prov u a -> RPar.R a b -> prov u b.
+Lemma prov_rpar n (u : PTm n) a b : prov u a -> RPar.R a b -> prov u b.
Proof.
move => h.
move : b.
elim : u a / h.
- - qauto l:on ctrs:prov inv:RPar.R use:@rtc_r, RPar_Par.
+ (* - qauto l:on ctrs:prov inv:RPar.R use:@rtc_r, RPar_Par. *)
- hauto lq:on ctrs:prov inv:RPar.R use:RPar.substing.
- move => h a b ha iha b0.
elim /RPar.inv => //= _.
+ move => a0 a1 b1 b2 h0 h1 [*]. subst.
- have {}iha : prov h (Abs a1) by hauto lq:on ctrs:RPar.R.
+ have {}iha : prov h (PAbs a1) by hauto lq:on ctrs:RPar.R.
hauto lq:on inv:prov use:RPar.substing.
+ move => a0 a1 b1 b2 c0 c1.
move => h0 h1 h2 [*]. subst.
- have {}iha : prov h (Pair a1 b2) by hauto lq:on ctrs:RPar.R.
+ have {}iha : prov h (PPair a1 b2) by hauto lq:on ctrs:RPar.R.
hauto lq:on inv:prov ctrs:prov.
+ hauto lq:on ctrs:prov.
- hauto lq:on ctrs:prov inv:RPar.R.
- move => h p a ha iha b.
elim /RPar.inv => //= _.
+ move => p0 a0 a1 h0 [*]. subst.
- have {iha} : prov h (Abs a1) by hauto lq:on ctrs:RPar.R.
+ have {iha} : prov h (PAbs a1) by hauto lq:on ctrs:RPar.R.
hauto lq:on ctrs:prov inv:prov use:RPar.substing.
+ move => p0 a0 a1 b0 b1 h0 h1 [*]. subst.
- have {iha} : prov h (Pair a1 b1) by hauto lq:on ctrs:RPar.R.
+ have {iha} : prov h (PPair a1 b1) by hauto lq:on ctrs:RPar.R.
qauto l:on inv:prov.
+ hauto lq:on ctrs:prov.
- hauto lq:on ctrs:prov inv:RPar.R.
@@ -1765,33 +1651,23 @@ Proof.
Qed.
-Lemma prov_lam n (u : Tm n) a : prov u a <-> prov u (Abs (App (ren_Tm shift a) (VarTm var_zero))).
+Lemma prov_lam n (u : PTm n) a : prov u a <-> prov u (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero))).
Proof.
split.
move => h. constructor. move => b. asimpl. by constructor.
inversion 1; subst.
- specialize H2 with (b := Const TPi).
+ specialize H2 with (b := PBot).
move : H2. asimpl. inversion 1; subst. done.
Qed.
-Lemma prov_pair n (u : Tm n) a : prov u a <-> prov u (Pair (Proj PL a) (Proj PR a)).
+Lemma prov_pair n (u : PTm n) a : prov u a <-> prov u (PPair (PProj PL a) (PProj PR a)).
Proof. hauto lq:on inv:prov ctrs:prov. Qed.
-Lemma prov_ered n (u : Tm n) a b : prov u a -> ERed.R a b -> prov u b.
+Lemma prov_ered n (u : PTm n) a b : prov u a -> ERed.R a b -> prov u b.
Proof.
move => h.
move : b.
elim : u a / h.
- - move => p A A0 B B0 hA hB b.
- elim /ERed.inv => // _.
- + move => a0 *. subst.
- rewrite -prov_lam.
- by constructor.
- + move => a0 *. subst.
- rewrite -prov_pair.
- by constructor.
- + qauto l:on ctrs:prov use:@rtc_r, ERed_EPar, EPar_Par.
- + qauto l:on ctrs:prov use:@rtc_r, ERed_EPar, EPar_Par.
- move => h a ha iha b.
elim /ERed.inv => // _.
+ move => a0 *. subst.
@@ -1819,26 +1695,25 @@ Proof.
- hauto lq:on inv:ERed.R, prov ctrs:prov.
Qed.
-Lemma prov_ereds n (u : Tm n) a b : prov u a -> rtc ERed.R a b -> prov u b.
+Lemma prov_ereds n (u : PTm n) a b : prov u a -> rtc ERed.R a b -> prov u b.
Proof.
induction 2; sfirstorder use:prov_ered.
Qed.
-Fixpoint extract {n} (a : Tm n) : Tm n :=
+Fixpoint extract {n} (a : PTm n) : PTm n :=
match a with
- | TBind p A B => TBind p A B
- | Abs a => subst_Tm (scons Bot VarTm) (extract a)
- | App a b => extract a
- | Pair a b => extract a
- | Proj p a => extract a
- | Const k => Const k
- | VarTm i => VarTm i
- | Univ i => Univ i
- | Bot => Bot
+ | PAbs a => subst_PTm (scons PBot VarPTm) (extract a)
+ | PApp a b => extract a
+ | PPair a b => extract a
+ | PProj p a => extract a
+ | PConst k => PConst k
+ | VarPTm i => VarPTm i
+ | PUniv i => PUniv i
+ | PBot => PBot
end.
-Lemma ren_extract n m (a : Tm n) (ξ : fin n -> fin m) :
- extract (ren_Tm ξ a) = ren_Tm ξ (extract a).
+Lemma ren_extract n m (a : PTm n) (ξ : fin n -> fin m) :
+ extract (ren_PTm ξ a) = ren_PTm ξ (extract a).
Proof.
move : m ξ. elim : n/a.
- sfirstorder.
@@ -1851,12 +1726,11 @@ Proof.
- hauto q:on.
- sfirstorder.
- sfirstorder.
- - sfirstorder.
Qed.
-Lemma ren_morphing n m (a : Tm n) (ρ : fin n -> Tm m) :
+Lemma ren_morphing n m (a : PTm n) (ρ : fin n -> PTm m) :
(forall i, ρ i = extract (ρ i)) ->
- extract (subst_Tm ρ a) = subst_Tm ρ (extract a).
+ extract (subst_PTm ρ a) = subst_PTm ρ (extract a).
Proof.
move : m ρ.
elim : n /a => n //=.
@@ -1869,45 +1743,38 @@ Proof.
- by asimpl.
Qed.
-Lemma ren_subst_bot n (a : Tm (S n)) :
- extract (subst_Tm (scons Bot VarTm) a) = subst_Tm (scons Bot VarTm) (extract a).
+Lemma ren_subst_bot n (a : PTm (S n)) :
+ extract (subst_PTm (scons PBot VarPTm) a) = subst_PTm (scons PBot VarPTm) (extract a).
Proof.
apply ren_morphing. destruct i as [i|] => //=.
Qed.
-Definition prov_extract_spec {n} u (a : Tm n) :=
+Definition prov_extract_spec {n} u (a : PTm n) :=
match u with
- | TBind p A B => exists A0 B0, extract a = TBind p A0 B0 /\ rtc Par.R A A0 /\ rtc Par.R B B0
- | Univ i => extract a = Univ i
- | VarTm i => extract a = VarTm i
- | (Const i) => extract a = (Const i)
- | Bot => extract a = Bot
+ | PUniv i => extract a = PUniv i
+ | VarPTm i => extract a = VarPTm i
+ | (PConst i) => extract a = (PConst i)
+ | PBot => extract a = PBot
| _ => True
end.
-Lemma prov_extract n u (a : Tm n) :
+Lemma prov_extract n u (a : PTm n) :
prov u a -> prov_extract_spec u a.
Proof.
move => h.
elim : u a /h.
- - sfirstorder.
- move => h a ha ih.
case : h ha ih => //=.
+ move => i ha ih.
- move /(_ Bot) in ih.
+ move /(_ PBot) in ih.
rewrite -ih.
by rewrite ren_subst_bot.
- + move => p A B h ih.
- move /(_ Bot) : ih => [A0][B0][h0][h1]h2.
- rewrite ren_subst_bot in h0.
- rewrite h0.
- eauto.
- + move => p _ /(_ Bot).
+ + move => p _ /(_ PBot).
by rewrite ren_subst_bot.
- + move => i h /(_ Bot).
+ + move => i h /(_ PBot).
by rewrite ren_subst_bot => ->.
- + move /(_ Bot).
- move => h /(_ Bot).
+ + move /(_ PBot).
+ move => h /(_ PBot).
by rewrite ren_subst_bot.
- hauto lq:on.
- hauto lq:on.
@@ -1922,21 +1789,21 @@ Definition union {A : Type} (R0 R1 : A -> A -> Prop) a b :=
R0 a b \/ R1 a b.
Module ERPar.
- Definition R {n} (a b : Tm n) := union RPar.R EPar.R a b.
- Lemma RPar {n} (a b : Tm n) : RPar.R a b -> R a b.
+ Definition R {n} (a b : PTm n) := union RPar.R EPar.R a b.
+ Lemma RPar {n} (a b : PTm n) : RPar.R a b -> R a b.
Proof. sfirstorder. Qed.
- Lemma EPar {n} (a b : Tm n) : EPar.R a b -> R a b.
+ Lemma EPar {n} (a b : PTm n) : EPar.R a b -> R a b.
Proof. sfirstorder. Qed.
- Lemma refl {n} ( a : Tm n) : ERPar.R a a.
+ Lemma refl {n} ( a : PTm n) : ERPar.R a a.
Proof.
sfirstorder use:RPar.refl, EPar.refl.
Qed.
- Lemma ProjCong n p (a0 a1 : Tm n) :
+ Lemma ProjCong n p (a0 a1 : PTm n) :
R a0 a1 ->
- rtc R (Proj p a0) (Proj p a1).
+ rtc R (PProj p a0) (PProj p a1).
Proof.
move => [].
- move => h.
@@ -1949,9 +1816,9 @@ Module ERPar.
by apply EPar.ProjCong.
Qed.
- Lemma AbsCong n (a0 a1 : Tm (S n)) :
+ Lemma AbsCong n (a0 a1 : PTm (S n)) :
R a0 a1 ->
- rtc R (Abs a0) (Abs a1).
+ rtc R (PAbs a0) (PAbs a1).
Proof.
move => [].
- move => h.
@@ -1964,10 +1831,10 @@ Module ERPar.
by apply EPar.AbsCong.
Qed.
- Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
+ Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
R a0 a1 ->
R b0 b1 ->
- rtc R (App a0 b0) (App a1 b1).
+ rtc R (PApp a0 b0) (PApp a1 b1).
Proof.
move => [] + [].
- sfirstorder use:RPar.AppCong, @rtc_once.
@@ -1984,30 +1851,10 @@ Module ERPar.
- sfirstorder use:EPar.AppCong, @rtc_once.
Qed.
- Lemma BindCong n p (a0 a1 : Tm n) b0 b1:
+ Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
R a0 a1 ->
R b0 b1 ->
- rtc R (TBind p a0 b0) (TBind p a1 b1).
- Proof.
- move => [] + [].
- - sfirstorder use:RPar.BindCong, @rtc_once.
- - move => h0 h1.
- apply : rtc_l.
- left. apply RPar.BindCong; eauto; apply RPar.refl.
- apply rtc_once.
- hauto l:on use:EPar.BindCong, EPar.refl.
- - move => h0 h1.
- apply : rtc_l.
- left. apply RPar.BindCong; eauto; apply RPar.refl.
- apply rtc_once.
- hauto l:on use:EPar.BindCong, EPar.refl.
- - sfirstorder use:EPar.BindCong, @rtc_once.
- Qed.
-
- Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
- R a0 a1 ->
- R b0 b1 ->
- rtc R (Pair a0 b0) (Pair a1 b1).
+ rtc R (PPair a0 b0) (PPair a1 b1).
Proof.
move => [] + [].
- sfirstorder use:RPar.PairCong, @rtc_once.
@@ -2024,15 +1871,15 @@ Module ERPar.
- sfirstorder use:EPar.PairCong, @rtc_once.
Qed.
- Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
- R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
+ Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
+ R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
Proof.
sfirstorder use:EPar.renaming, RPar.renaming.
Qed.
End ERPar.
-Hint Resolve ERPar.AppCong ERPar.refl ERPar.AbsCong ERPar.PairCong ERPar.ProjCong ERPar.BindCong : erpar.
+Hint Resolve ERPar.AppCong ERPar.refl ERPar.AbsCong ERPar.PairCong ERPar.ProjCong : erpar.
Module ERPars.
#[local]Ltac solve_s_rec :=
@@ -2041,37 +1888,31 @@ Module ERPars.
#[local]Ltac solve_s :=
repeat (induction 1; last by solve_s_rec); apply rtc_refl.
- Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
+ Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
rtc ERPar.R a0 a1 ->
rtc ERPar.R b0 b1 ->
- rtc ERPar.R (App a0 b0) (App a1 b1).
+ rtc ERPar.R (PApp a0 b0) (PApp a1 b1).
Proof. solve_s. Qed.
- Lemma AbsCong n (a0 a1 : Tm (S n)) :
+ Lemma AbsCong n (a0 a1 : PTm (S n)) :
rtc ERPar.R a0 a1 ->
- rtc ERPar.R (Abs a0) (Abs a1).
+ rtc ERPar.R (PAbs a0) (PAbs a1).
Proof. solve_s. Qed.
- Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
+ Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
rtc ERPar.R a0 a1 ->
rtc ERPar.R b0 b1 ->
- rtc ERPar.R (Pair a0 b0) (Pair a1 b1).
+ rtc ERPar.R (PPair a0 b0) (PPair a1 b1).
Proof. solve_s. Qed.
- Lemma ProjCong n p (a0 a1 : Tm n) :
+ Lemma ProjCong n p (a0 a1 : PTm n) :
rtc ERPar.R a0 a1 ->
- rtc ERPar.R (Proj p a0) (Proj p a1).
+ rtc ERPar.R (PProj p a0) (PProj p a1).
Proof. solve_s. Qed.
- Lemma BindCong n p (a0 a1 : Tm n) b0 b1:
+ Lemma renaming n (a0 a1 : PTm n) m (ξ : fin n -> fin m) :
rtc ERPar.R a0 a1 ->
- rtc ERPar.R b0 b1 ->
- rtc ERPar.R (TBind p a0 b0) (TBind p a1 b1).
- Proof. solve_s. Qed.
-
- Lemma renaming n (a0 a1 : Tm n) m (ξ : fin n -> fin m) :
- rtc ERPar.R a0 a1 ->
- rtc ERPar.R (ren_Tm ξ a0) (ren_Tm ξ a1).
+ rtc ERPar.R (ren_PTm ξ a0) (ren_PTm ξ a1).
Proof.
induction 1.
- apply rtc_refl.
@@ -2080,16 +1921,16 @@ Module ERPars.
End ERPars.
-Lemma ERPar_Par n (a b : Tm n) : ERPar.R a b -> Par.R a b.
+Lemma ERPar_Par n (a b : PTm n) : ERPar.R a b -> Par.R a b.
Proof.
sfirstorder use:EPar_Par, RPar_Par.
Qed.
-Lemma Par_ERPar n (a b : Tm n) : Par.R a b -> rtc ERPar.R a b.
+Lemma Par_ERPar n (a b : PTm n) : Par.R a b -> rtc ERPar.R a b.
Proof.
move => h. elim : n a b /h.
- move => n a0 a1 b0 b1 ha iha hb ihb.
- suff ? : rtc ERPar.R (App (Abs a0) b0) (App (Abs a1) b1).
+ suff ? : rtc ERPar.R (PApp (PAbs a0) b0) (PApp (PAbs a1) b1).
apply : relations.rtc_transitive; eauto.
apply rtc_once. apply ERPar.RPar.
by apply RPar.AppAbs; eauto using RPar.refl.
@@ -2116,30 +1957,29 @@ Proof.
- sfirstorder use:ERPars.AppCong.
- sfirstorder use:ERPars.PairCong.
- sfirstorder use:ERPars.ProjCong.
- - sfirstorder use:ERPars.BindCong.
- sfirstorder.
- sfirstorder.
- sfirstorder.
Qed.
-Lemma Pars_ERPar n (a b : Tm n) : rtc Par.R a b -> rtc ERPar.R a b.
+Lemma Pars_ERPar n (a b : PTm n) : rtc Par.R a b -> rtc ERPar.R a b.
Proof.
induction 1; hauto l:on use:Par_ERPar, @relations.rtc_transitive.
Qed.
-Lemma Par_ERPar_iff n (a b : Tm n) : rtc Par.R a b <-> rtc ERPar.R a b.
+Lemma Par_ERPar_iff n (a b : PTm n) : rtc Par.R a b <-> rtc ERPar.R a b.
Proof.
split.
sfirstorder use:Pars_ERPar, @relations.rtc_subrel.
sfirstorder use:ERPar_Par, @relations.rtc_subrel.
Qed.
-Lemma RPar_ERPar n (a b : Tm n) : rtc RPar.R a b -> rtc ERPar.R a b.
+Lemma RPar_ERPar n (a b : PTm n) : rtc RPar.R a b -> rtc ERPar.R a b.
Proof.
sfirstorder use:@relations.rtc_subrel.
Qed.
-Lemma EPar_ERPar n (a b : Tm n) : rtc EPar.R a b -> rtc ERPar.R a b.
+Lemma EPar_ERPar n (a b : PTm n) : rtc EPar.R a b -> rtc ERPar.R a b.
Proof.
sfirstorder use:@relations.rtc_subrel.
Qed.
@@ -2205,7 +2045,7 @@ Module HindleyRosenFacts (M : HindleyRosen).
End HindleyRosenFacts.
Module HindleyRosenER <: HindleyRosen.
- Definition A := Tm.
+ Definition A := PTm.
Definition R0 n := rtc (@RPar.R n).
Definition R1 n := rtc (@EPar.R n).
Lemma diamond_R0 : forall n, relations.diamond (R0 n).
@@ -2223,7 +2063,7 @@ End HindleyRosenER.
Module ERFacts := HindleyRosenFacts HindleyRosenER.
-Lemma rtc_union n (a b : Tm n) :
+Lemma rtc_union n (a b : PTm n) :
rtc (union RPar.R EPar.R) a b <->
rtc (union (rtc RPar.R) (rtc EPar.R)) a b.
Proof.
@@ -2245,7 +2085,7 @@ Proof.
sfirstorder.
Qed.
-Lemma prov_erpar n (u : Tm n) a b : prov u a -> ERPar.R a b -> prov u b.
+Lemma prov_erpar n (u : PTm n) a b : prov u a -> ERPar.R a b -> prov u b.
Proof.
move => h [].
- sfirstorder use:prov_rpar.
@@ -2253,7 +2093,7 @@ Proof.
sfirstorder use:prov_ereds.
Qed.
-Lemma prov_pars n (u : Tm n) a b : prov u a -> rtc Par.R a b -> prov u b.
+Lemma prov_pars n (u : PTm n) a b : prov u a -> rtc Par.R a b -> prov u b.
Proof.
move => h /Pars_ERPar.
move => h0.
@@ -2263,15 +2103,15 @@ Proof.
- hauto lq:on use:prov_erpar.
Qed.
-Lemma Par_confluent n (a b c : Tm n) :
+Lemma Par_confluent n (a b c : PTm n) :
rtc Par.R a b ->
rtc Par.R a c ->
exists d, rtc Par.R b d /\ rtc Par.R c d.
Proof.
move : n a b c.
- suff : forall (n : nat) (a b c : Tm n),
+ suff : forall (n : nat) (a b c : PTm n),
rtc ERPar.R a b ->
- rtc ERPar.R a c -> exists d : Tm n, rtc ERPar.R b d /\ rtc ERPar.R c d.
+ rtc ERPar.R a c -> exists d : PTm n, rtc ERPar.R b d /\ rtc ERPar.R c d.
move => h n a b c h0 h1.
apply Par_ERPar_iff in h0, h1.
move : h h0 h1; repeat move/[apply].
@@ -2282,32 +2122,32 @@ Proof.
specialize h with (n := n).
rewrite /HindleyRosenER.A in h.
rewrite /ERPar.R.
- have eq : (fun a0 b0 : Tm n => union RPar.R EPar.R a0 b0) = union RPar.R EPar.R by reflexivity.
+ have eq : (fun a0 b0 : PTm n => union RPar.R EPar.R a0 b0) = union RPar.R EPar.R by reflexivity.
rewrite !{}eq.
move /rtc_union => + /rtc_union.
move : h; repeat move/[apply].
hauto lq:on use:rtc_union.
Qed.
-Lemma pars_univ_inv n i (c : Tm n) :
- rtc Par.R (Univ i) c ->
- extract c = Univ i.
+Lemma pars_univ_inv n i (c : PTm n) :
+ rtc Par.R (PUniv i) c ->
+ extract c = PUniv i.
Proof.
- have : prov (Univ i) (Univ i : Tm n) by sfirstorder.
+ have : prov (PUniv i) (Univ i : PTm n) by sfirstorder.
move : prov_pars. repeat move/[apply].
apply prov_extract.
Qed.
-Lemma pars_const_inv n i (c : Tm n) :
+Lemma pars_const_inv n i (c : PTm n) :
rtc Par.R (Const i) c ->
extract c = Const i.
Proof.
- have : prov (Const i) (Const i : Tm n) by sfirstorder.
+ have : prov (Const i) (Const i : PTm n) by sfirstorder.
move : prov_pars. repeat move/[apply].
apply prov_extract.
Qed.
-Lemma pars_pi_inv n p (A : Tm n) B C :
+Lemma pars_pi_inv n p (A : PTm n) B C :
rtc Par.R (TBind p A B) C ->
exists A0 B0, extract C = TBind p A0 B0 /\
rtc Par.R A A0 /\ rtc Par.R B B0.
@@ -2318,15 +2158,15 @@ Proof.
Qed.
Lemma pars_var_inv n (i : fin n) C :
- rtc Par.R (VarTm i) C ->
- extract C = VarTm i.
+ rtc Par.R (VarPTm i) C ->
+ extract C = VarPTm i.
Proof.
- have : prov (VarTm i) (VarTm i) by hauto lq:on ctrs:prov, rtc.
+ have : prov (VarPTm i) (VarPTm i) by hauto lq:on ctrs:prov, rtc.
move : prov_pars. repeat move/[apply].
apply prov_extract.
Qed.
-Lemma pars_univ_inj n i j (C : Tm n) :
+Lemma pars_univ_inj n i j (C : PTm n) :
rtc Par.R (Univ i) C ->
rtc Par.R (Univ j) C ->
i = j.
@@ -2334,7 +2174,7 @@ Proof.
sauto l:on use:pars_univ_inv.
Qed.
-Lemma pars_const_inj n i j (C : Tm n) :
+Lemma pars_const_inj n i j (C : PTm n) :
rtc Par.R (Const i) C ->
rtc Par.R (Const j) C ->
i = j.
@@ -2342,7 +2182,7 @@ Proof.
sauto l:on use:pars_const_inv.
Qed.
-Lemma pars_pi_inj n p0 p1 (A0 A1 : Tm n) B0 B1 C :
+Lemma pars_pi_inj n p0 p1 (A0 A1 : PTm n) B0 B1 C :
rtc Par.R (TBind p0 A0 B0) C ->
rtc Par.R (TBind p1 A1 B1) C ->
exists A2 B2, p1 = p0 /\ rtc Par.R A0 A2 /\ rtc Par.R A1 A2 /\
@@ -2353,10 +2193,10 @@ Proof.
exists A2, B2. hauto l:on.
Qed.
-Definition join {n} (a b : Tm n) :=
+Definition join {n} (a b : PTm n) :=
exists c, rtc Par.R a c /\ rtc Par.R b c.
-Lemma join_transitive n (a b c : Tm n) :
+Lemma join_transitive n (a b c : PTm n) :
join a b -> join b c -> join a c.
Proof.
rewrite /join.
@@ -2366,26 +2206,26 @@ Proof.
eauto using relations.rtc_transitive.
Qed.
-Lemma join_symmetric n (a b : Tm n) :
+Lemma join_symmetric n (a b : PTm n) :
join a b -> join b a.
Proof. sfirstorder unfold:join. Qed.
-Lemma join_refl n (a : Tm n) : join a a.
+Lemma join_refl n (a : PTm n) : join a a.
Proof. hauto lq:on ctrs:rtc unfold:join. Qed.
Lemma join_univ_inj n i j :
- join (Univ i : Tm n) (Univ j) -> i = j.
+ join (Univ i : PTm n) (Univ j) -> i = j.
Proof.
sfirstorder use:pars_univ_inj.
Qed.
Lemma join_const_inj n i j :
- join (Const i : Tm n) (Const j) -> i = j.
+ join (Const i : PTm n) (Const j) -> i = j.
Proof.
sfirstorder use:pars_const_inj.
Qed.
-Lemma join_pi_inj n p0 p1 (A0 A1 : Tm n) B0 B1 :
+Lemma join_pi_inj n p0 p1 (A0 A1 : PTm n) B0 B1 :
join (TBind p0 A0 B0) (TBind p1 A1 B1) ->
p0 = p1 /\ join A0 A1 /\ join B0 B1.
Proof.
@@ -2394,14 +2234,14 @@ Proof.
sfirstorder unfold:join.
Qed.
-Lemma join_substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
+Lemma join_substing n m (a b : PTm n) (ρ : fin n -> PTm m) :
join a b ->
- join (subst_Tm ρ a) (subst_Tm ρ b).
+ join (subst_PTm ρ a) (subst_PTm ρ b).
Proof. hauto lq:on unfold:join use:Pars.substing. Qed.
-Fixpoint ne {n} (a : Tm n) :=
+Fixpoint ne {n} (a : PTm n) :=
match a with
- | VarTm i => true
+ | VarPTm i => true
| TBind _ A B => false
| App a b => ne a && nf b
| Abs a => false
@@ -2411,9 +2251,9 @@ Fixpoint ne {n} (a : Tm n) :=
| Const _ => false
| Bot => true
end
-with nf {n} (a : Tm n) :=
+with nf {n} (a : PTm n) :=
match a with
- | VarTm i => true
+ | VarPTm i => true
| TBind _ A B => nf A && nf B
| App a b => ne a && nf b
| Abs a => nf a
@@ -2427,8 +2267,8 @@ end.
Lemma ne_nf n a : @ne n a -> nf a.
Proof. elim : a => //=. Qed.
-Definition wn {n} (a : Tm n) := exists b, rtc RPar'.R a b /\ nf b.
-Definition wne {n} (a : Tm n) := exists b, rtc RPar'.R a b /\ ne b.
+Definition wn {n} (a : PTm n) := exists b, rtc RPar'.R a b /\ nf b.
+Definition wne {n} (a : PTm n) := exists b, rtc RPar'.R a b /\ ne b.
(* Weakly neutral implies weakly normal *)
Lemma wne_wn n a : @wne n a -> wn a.
@@ -2438,18 +2278,18 @@ Proof. sfirstorder use:ne_nf. Qed.
Lemma nf_wn n v : @nf n v -> wn v.
Proof. sfirstorder ctrs:rtc. Qed.
-Lemma nf_refl n (a b : Tm n) (h : RPar'.R a b) : (nf a -> b = a) /\ (ne a -> b = a).
+Lemma nf_refl n (a b : PTm n) (h : RPar'.R a b) : (nf a -> b = a) /\ (ne a -> b = a).
Proof.
elim : a b /h => //=; solve [hauto b:on].
Qed.
-Lemma ne_nf_ren n m (a : Tm n) (ξ : fin n -> fin m) :
- (ne a <-> ne (ren_Tm ξ a)) /\ (nf a <-> nf (ren_Tm ξ a)).
+Lemma ne_nf_ren n m (a : PTm n) (ξ : fin n -> fin m) :
+ (ne a <-> ne (ren_PTm ξ a)) /\ (nf a <-> nf (ren_PTm ξ a)).
Proof.
move : m ξ. elim : n / a => //=; solve [hauto b:on].
Qed.
-Lemma wne_app n (a b : Tm n) :
+Lemma wne_app n (a b : PTm n) :
wne a -> wn b -> wne (App a b).
Proof.
move => [a0 [? ?]] [b0 [? ?]].
@@ -2470,14 +2310,14 @@ Proof.
hauto lqb:on use:RPars'.BindCong.
Qed.
-Lemma wn_pair n (a b : Tm n) : wn a -> wn b -> wn (Pair a b).
+Lemma wn_pair n (a b : PTm n) : wn a -> wn b -> wn (Pair a b).
Proof.
move => [a0 [? ?]] [b0 [? ?]].
exists (Pair a0 b0).
hauto lqb:on use:RPars'.PairCong.
Qed.
-Lemma wne_proj n p (a : Tm n) : wne a -> wne (Proj p a).
+Lemma wne_proj n p (a : PTm n) : wne a -> wne (Proj p a).
Proof.
move => [a0 [? ?]].
exists (Proj p a0). hauto lqb:on use:RPars'.ProjCong.
@@ -2486,17 +2326,17 @@ Qed.
Create HintDb nfne.
#[export]Hint Resolve nf_wn ne_nf wne_wn nf_refl : nfne.
-Lemma ne_nf_antiren n m (a : Tm n) (ρ : fin n -> Tm m) :
+Lemma ne_nf_antiren n m (a : PTm n) (ρ : fin n -> PTm m) :
(forall i, var_or_const (ρ i)) ->
- (ne (subst_Tm ρ a) -> ne a) /\ (nf (subst_Tm ρ a) -> nf a).
+ (ne (subst_PTm ρ a) -> ne a) /\ (nf (subst_PTm ρ a) -> nf a).
Proof.
move : m ρ. elim : n / a => //;
hauto b:on drew:off use:RPar.var_or_const_up.
Qed.
-Lemma wn_antirenaming n m a (ρ : fin n -> Tm m) :
+Lemma wn_antirenaming n m a (ρ : fin n -> PTm m) :
(forall i, var_or_const (ρ i)) ->
- wn (subst_Tm ρ a) -> wn a.
+ wn (subst_PTm ρ a) -> wn a.
Proof.
rewrite /wn => hρ.
move => [v [rv nfv]].
@@ -2507,7 +2347,7 @@ Proof.
by eapply ne_nf_antiren.
Qed.
-Lemma ext_wn n (a : Tm n) :
+Lemma ext_wn n (a : PTm n) :
wn (App a Bot) ->
wn a.
Proof.
@@ -2515,7 +2355,7 @@ Proof.
move : a E.
move : hv.
elim : a0 v / hr.
- - hauto q:on inv:Tm ctrs:rtc b:on db: nfne.
+ - hauto q:on inv:PTm ctrs:rtc b:on db: nfne.
- move => a0 a1 a2 hr0 hr1 ih hnfa2.
move /(_ hnfa2) in ih.
move => a.
@@ -2524,7 +2364,7 @@ Proof.
+ move => a0 a3 b2 b3 ? ? [? ?] ? [? ?]. subst.
have ? : b3 = (Bot) by hauto lq:on inv:RPar'.R. subst.
suff : wn (Abs a3) by hauto lq:on ctrs:RPar'.R, rtc unfold:wn.
- have : wn (subst_Tm (scons (Bot) VarTm) a3) by sfirstorder.
+ have : wn (subst_PTm (scons (Bot) VarPTm) a3) by sfirstorder.
move => h. apply wn_abs.
move : h. apply wn_antirenaming.
hauto lq:on rew:off inv:option.
@@ -2532,41 +2372,41 @@ Proof.
Qed.
Module Join.
- Lemma ProjCong p n (a0 a1 : Tm n) :
+ Lemma ProjCong p n (a0 a1 : PTm n) :
join a0 a1 ->
join (Proj p a0) (Proj p a1).
Proof. hauto lq:on use:Pars.ProjCong unfold:join. Qed.
- Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
+ Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
join a0 a1 ->
join b0 b1 ->
join (Pair a0 b0) (Pair a1 b1).
Proof. hauto lq:on use:Pars.PairCong unfold:join. Qed.
- Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
+ Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
join a0 a1 ->
join b0 b1 ->
join (App a0 b0) (App a1 b1).
Proof. hauto lq:on use:Pars.AppCong. Qed.
- Lemma AbsCong n (a b : Tm (S n)) :
+ Lemma AbsCong n (a b : PTm (S n)) :
join a b ->
join (Abs a) (Abs b).
Proof. hauto lq:on use:Pars.AbsCong. Qed.
- Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
- join a b -> join (ren_Tm ξ a) (ren_Tm ξ b).
+ Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
+ join a b -> join (ren_PTm ξ a) (ren_PTm ξ b).
Proof.
induction 1; hauto lq:on use:Pars.renaming.
Qed.
- Lemma weakening n (a b : Tm n) :
- join a b -> join (ren_Tm shift a) (ren_Tm shift b).
+ Lemma weakening n (a b : PTm n) :
+ join a b -> join (ren_PTm shift a) (ren_PTm shift b).
Proof.
apply renaming.
Qed.
- Lemma FromPar n (a b : Tm n) :
+ Lemma FromPar n (a b : PTm n) :
Par.R a b ->
join a b.
Proof.
@@ -2574,12 +2414,12 @@ Module Join.
Qed.
End Join.
-Lemma abs_eq n a (b : Tm n) :
- join (Abs a) b <-> join a (App (ren_Tm shift b) (VarTm var_zero)).
+Lemma abs_eq n a (b : PTm n) :
+ join (Abs a) b <-> join a (App (ren_PTm shift b) (VarPTm var_zero)).
Proof.
split.
- move => /Join.weakening h.
- have {h} : join (App (ren_Tm shift (Abs a)) (VarTm var_zero)) (App (ren_Tm shift b) (VarTm var_zero))
+ have {h} : join (App (ren_PTm shift (Abs a)) (VarPTm var_zero)) (App (ren_PTm shift b) (VarPTm var_zero))
by hauto l:on use:Join.AppCong, join_refl.
simpl.
move => ?. apply : join_transitive; eauto.
@@ -2590,7 +2430,7 @@ Proof.
apply join_symmetric. apply Join.FromPar. apply Par.AppEta. apply Par.refl.
Qed.
-Lemma pair_eq n (a0 a1 b : Tm n) :
+Lemma pair_eq n (a0 a1 b : PTm n) :
join (Pair a0 a1) b <-> join a0 (Proj PL b) /\ join a1 (Proj PR b).
Proof.
split.
@@ -2606,7 +2446,7 @@ Proof.
apply Join.FromPar. hauto lq:on ctrs:Par.R use:Par.refl.
Qed.
-Lemma join_pair_inj n (a0 a1 b0 b1 : Tm n) :
+Lemma join_pair_inj n (a0 a1 b0 b1 : PTm n) :
join (Pair a0 a1) (Pair b0 b1) <-> join a0 b0 /\ join a1 b1.
Proof.
split; last by hauto lq:on use:Join.PairCong.
From 398a18d7709e8dc9ff0933c9ceb8dc5a0ed1bd3f Mon Sep 17 00:00:00 2001
From: Yiyun Liu <yliu8e5f7@protonmail.com>
Date: Fri, 24 Jan 2025 14:58:35 -0700
Subject: [PATCH 12/13] Finish renaming
---
theories/fp_red.v | 135 +++++++++++++++++-----------------------------
1 file changed, 48 insertions(+), 87 deletions(-)
diff --git a/theories/fp_red.v b/theories/fp_red.v
index 3482f45..bbe58d9 100644
--- a/theories/fp_red.v
+++ b/theories/fp_red.v
@@ -2133,26 +2133,16 @@ Lemma pars_univ_inv n i (c : PTm n) :
rtc Par.R (PUniv i) c ->
extract c = PUniv i.
Proof.
- have : prov (PUniv i) (Univ i : PTm n) by sfirstorder.
+ have : prov (PUniv i) (PUniv i : PTm n) by sfirstorder.
move : prov_pars. repeat move/[apply].
apply prov_extract.
Qed.
Lemma pars_const_inv n i (c : PTm n) :
- rtc Par.R (Const i) c ->
- extract c = Const i.
+ rtc Par.R (PConst i) c ->
+ extract c = PConst i.
Proof.
- have : prov (Const i) (Const i : PTm n) by sfirstorder.
- move : prov_pars. repeat move/[apply].
- apply prov_extract.
-Qed.
-
-Lemma pars_pi_inv n p (A : PTm n) B C :
- rtc Par.R (TBind p A B) C ->
- exists A0 B0, extract C = TBind p A0 B0 /\
- rtc Par.R A A0 /\ rtc Par.R B B0.
-Proof.
- have : prov (TBind p A B) (TBind p A B) by hauto lq:on ctrs:prov, rtc.
+ have : prov (PConst i) (PConst i : PTm n) by sfirstorder.
move : prov_pars. repeat move/[apply].
apply prov_extract.
Qed.
@@ -2167,32 +2157,21 @@ Proof.
Qed.
Lemma pars_univ_inj n i j (C : PTm n) :
- rtc Par.R (Univ i) C ->
- rtc Par.R (Univ j) C ->
+ rtc Par.R (PUniv i) C ->
+ rtc Par.R (PUniv j) C ->
i = j.
Proof.
sauto l:on use:pars_univ_inv.
Qed.
Lemma pars_const_inj n i j (C : PTm n) :
- rtc Par.R (Const i) C ->
- rtc Par.R (Const j) C ->
+ rtc Par.R (PConst i) C ->
+ rtc Par.R (PConst j) C ->
i = j.
Proof.
sauto l:on use:pars_const_inv.
Qed.
-Lemma pars_pi_inj n p0 p1 (A0 A1 : PTm n) B0 B1 C :
- rtc Par.R (TBind p0 A0 B0) C ->
- rtc Par.R (TBind p1 A1 B1) C ->
- exists A2 B2, p1 = p0 /\ rtc Par.R A0 A2 /\ rtc Par.R A1 A2 /\
- rtc Par.R B0 B2 /\ rtc Par.R B1 B2.
-Proof.
- move /pars_pi_inv => [A2 [B2 [? [h0 h1]]]].
- move /pars_pi_inv => [A3 [B3 [? [h2 h3]]]].
- exists A2, B2. hauto l:on.
-Qed.
-
Definition join {n} (a b : PTm n) :=
exists c, rtc Par.R a c /\ rtc Par.R b c.
@@ -2214,26 +2193,17 @@ Lemma join_refl n (a : PTm n) : join a a.
Proof. hauto lq:on ctrs:rtc unfold:join. Qed.
Lemma join_univ_inj n i j :
- join (Univ i : PTm n) (Univ j) -> i = j.
+ join (PUniv i : PTm n) (PUniv j) -> i = j.
Proof.
sfirstorder use:pars_univ_inj.
Qed.
Lemma join_const_inj n i j :
- join (Const i : PTm n) (Const j) -> i = j.
+ join (PConst i : PTm n) (PConst j) -> i = j.
Proof.
sfirstorder use:pars_const_inj.
Qed.
-Lemma join_pi_inj n p0 p1 (A0 A1 : PTm n) B0 B1 :
- join (TBind p0 A0 B0) (TBind p1 A1 B1) ->
- p0 = p1 /\ join A0 A1 /\ join B0 B1.
-Proof.
- move => [c []].
- move : pars_pi_inj; repeat move/[apply].
- sfirstorder unfold:join.
-Qed.
-
Lemma join_substing n m (a b : PTm n) (ρ : fin n -> PTm m) :
join a b ->
join (subst_PTm ρ a) (subst_PTm ρ b).
@@ -2242,26 +2212,24 @@ Proof. hauto lq:on unfold:join use:Pars.substing. Qed.
Fixpoint ne {n} (a : PTm n) :=
match a with
| VarPTm i => true
- | TBind _ A B => false
- | App a b => ne a && nf b
- | Abs a => false
- | Univ _ => false
- | Proj _ a => ne a
- | Pair _ _ => false
- | Const _ => false
- | Bot => true
+ | PApp a b => ne a && nf b
+ | PAbs a => false
+ | PUniv _ => false
+ | PProj _ a => ne a
+ | PPair _ _ => false
+ | PConst _ => false
+ | PBot => true
end
with nf {n} (a : PTm n) :=
match a with
| VarPTm i => true
- | TBind _ A B => nf A && nf B
- | App a b => ne a && nf b
- | Abs a => nf a
- | Univ _ => true
- | Proj _ a => ne a
- | Pair a b => nf a && nf b
- | Const _ => true
- | Bot => true
+ | PApp a b => ne a && nf b
+ | PAbs a => nf a
+ | PUniv _ => true
+ | PProj _ a => ne a
+ | PPair a b => nf a && nf b
+ | PConst _ => true
+ | PBot => true
end.
Lemma ne_nf n a : @ne n a -> nf a.
@@ -2290,37 +2258,30 @@ Proof.
Qed.
Lemma wne_app n (a b : PTm n) :
- wne a -> wn b -> wne (App a b).
+ wne a -> wn b -> wne (PApp a b).
Proof.
move => [a0 [? ?]] [b0 [? ?]].
- exists (App a0 b0). hauto b:on drew:off use:RPars'.AppCong.
+ exists (PApp a0 b0). hauto b:on drew:off use:RPars'.AppCong.
Qed.
-Lemma wn_abs n a (h : wn a) : @wn n (Abs a).
+Lemma wn_abs n a (h : wn a) : @wn n (PAbs a).
Proof.
move : h => [v [? ?]].
- exists (Abs v).
+ exists (PAbs v).
eauto using RPars'.AbsCong.
Qed.
-Lemma wn_bind n p A B : wn A -> wn B -> wn (@TBind n p A B).
-Proof.
- move => [A0 [? ?]] [B0 [? ?]].
- exists (TBind p A0 B0).
- hauto lqb:on use:RPars'.BindCong.
-Qed.
-
-Lemma wn_pair n (a b : PTm n) : wn a -> wn b -> wn (Pair a b).
+Lemma wn_pair n (a b : PTm n) : wn a -> wn b -> wn (PPair a b).
Proof.
move => [a0 [? ?]] [b0 [? ?]].
- exists (Pair a0 b0).
+ exists (PPair a0 b0).
hauto lqb:on use:RPars'.PairCong.
Qed.
-Lemma wne_proj n p (a : PTm n) : wne a -> wne (Proj p a).
+Lemma wne_proj n p (a : PTm n) : wne a -> wne (PProj p a).
Proof.
move => [a0 [? ?]].
- exists (Proj p a0). hauto lqb:on use:RPars'.ProjCong.
+ exists (PProj p a0). hauto lqb:on use:RPars'.ProjCong.
Qed.
Create HintDb nfne.
@@ -2348,10 +2309,10 @@ Proof.
Qed.
Lemma ext_wn n (a : PTm n) :
- wn (App a Bot) ->
+ wn (PApp a PBot) ->
wn a.
Proof.
- move E : (App a (Bot)) => a0 [v [hr hv]].
+ move E : (PApp a (PBot)) => a0 [v [hr hv]].
move : a E.
move : hv.
elim : a0 v / hr.
@@ -2362,9 +2323,9 @@ Proof.
case : a0 hr0=>// => b0 b1.
elim /RPar'.inv=>// _.
+ move => a0 a3 b2 b3 ? ? [? ?] ? [? ?]. subst.
- have ? : b3 = (Bot) by hauto lq:on inv:RPar'.R. subst.
- suff : wn (Abs a3) by hauto lq:on ctrs:RPar'.R, rtc unfold:wn.
- have : wn (subst_PTm (scons (Bot) VarPTm) a3) by sfirstorder.
+ have ? : b3 = (PBot) by hauto lq:on inv:RPar'.R. subst.
+ suff : wn (PAbs a3) by hauto lq:on ctrs:RPar'.R, rtc unfold:wn.
+ have : wn (subst_PTm (scons (PBot) VarPTm) a3) by sfirstorder.
move => h. apply wn_abs.
move : h. apply wn_antirenaming.
hauto lq:on rew:off inv:option.
@@ -2374,24 +2335,24 @@ Qed.
Module Join.
Lemma ProjCong p n (a0 a1 : PTm n) :
join a0 a1 ->
- join (Proj p a0) (Proj p a1).
+ join (PProj p a0) (PProj p a1).
Proof. hauto lq:on use:Pars.ProjCong unfold:join. Qed.
Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
join a0 a1 ->
join b0 b1 ->
- join (Pair a0 b0) (Pair a1 b1).
+ join (PPair a0 b0) (PPair a1 b1).
Proof. hauto lq:on use:Pars.PairCong unfold:join. Qed.
Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
join a0 a1 ->
join b0 b1 ->
- join (App a0 b0) (App a1 b1).
+ join (PApp a0 b0) (PApp a1 b1).
Proof. hauto lq:on use:Pars.AppCong. Qed.
Lemma AbsCong n (a b : PTm (S n)) :
join a b ->
- join (Abs a) (Abs b).
+ join (PAbs a) (PAbs b).
Proof. hauto lq:on use:Pars.AbsCong. Qed.
Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
@@ -2415,11 +2376,11 @@ Module Join.
End Join.
Lemma abs_eq n a (b : PTm n) :
- join (Abs a) b <-> join a (App (ren_PTm shift b) (VarPTm var_zero)).
+ join (PAbs a) b <-> join a (PApp (ren_PTm shift b) (VarPTm var_zero)).
Proof.
split.
- move => /Join.weakening h.
- have {h} : join (App (ren_PTm shift (Abs a)) (VarPTm var_zero)) (App (ren_PTm shift b) (VarPTm var_zero))
+ have {h} : join (PApp (ren_PTm shift (PAbs a)) (VarPTm var_zero)) (PApp (ren_PTm shift b) (VarPTm var_zero))
by hauto l:on use:Join.AppCong, join_refl.
simpl.
move => ?. apply : join_transitive; eauto.
@@ -2431,12 +2392,12 @@ Proof.
Qed.
Lemma pair_eq n (a0 a1 b : PTm n) :
- join (Pair a0 a1) b <-> join a0 (Proj PL b) /\ join a1 (Proj PR b).
+ join (PPair a0 a1) b <-> join a0 (PProj PL b) /\ join a1 (PProj PR b).
Proof.
split.
- move => h.
have /Join.ProjCong {}h := h.
- have h0 : forall p, join (if p is PL then a0 else a1) (Proj p (Pair a0 a1))
+ have h0 : forall p, join (if p is PL then a0 else a1) (PProj p (PPair a0 a1))
by hauto lq:on use:join_symmetric, Join.FromPar, Par.ProjPair', Par.refl.
hauto lq:on rew:off use:join_transitive, join_symmetric.
- move => [h0 h1].
@@ -2447,11 +2408,11 @@ Proof.
Qed.
Lemma join_pair_inj n (a0 a1 b0 b1 : PTm n) :
- join (Pair a0 a1) (Pair b0 b1) <-> join a0 b0 /\ join a1 b1.
+ join (PPair a0 a1) (PPair b0 b1) <-> join a0 b0 /\ join a1 b1.
Proof.
split; last by hauto lq:on use:Join.PairCong.
move /pair_eq => [h0 h1].
- have : join (Proj PL (Pair b0 b1)) b0 by hauto lq:on use:Join.FromPar, Par.refl, Par.ProjPair'.
- have : join (Proj PR (Pair b0 b1)) b1 by hauto lq:on use:Join.FromPar, Par.refl, Par.ProjPair'.
+ have : join (PProj PL (PPair b0 b1)) b0 by hauto lq:on use:Join.FromPar, Par.refl, Par.ProjPair'.
+ have : join (PProj PR (PPair b0 b1)) b1 by hauto lq:on use:Join.FromPar, Par.refl, Par.ProjPair'.
eauto using join_transitive.
Qed.
From 9c17ec5cac4f702a69146c86cc244669a82b739e Mon Sep 17 00:00:00 2001
From: Yiyun Liu <yliu8e5f7@protonmail.com>
Date: Tue, 1 Apr 2025 23:21:39 -0400
Subject: [PATCH 13/13] Start refactoring to unscoped syntax
---
Makefile | 8 +-
theories/Autosubst2/fintype.v | 419 ----------
theories/Autosubst2/syntax.v | 1362 ++++++++++++--------------------
theories/Autosubst2/unscoped.v | 213 +++++
theories/fp_red.v | 84 +-
theories/typing.v | 251 ++++++
6 files changed, 994 insertions(+), 1343 deletions(-)
delete mode 100644 theories/Autosubst2/fintype.v
create mode 100644 theories/Autosubst2/unscoped.v
create mode 100644 theories/typing.v
diff --git a/Makefile b/Makefile
index ef5b76f..c56e771 100644
--- a/Makefile
+++ b/Makefile
@@ -6,7 +6,7 @@ theories: $(COQMAKEFILE) FORCE
validate: $(COQMAKEFILE) FORCE
$(MAKE) -f $(COQMAKEFILE) validate
-$(COQMAKEFILE) : theories/Autosubst2/syntax.v theories/Autosubst2/core.v theories/Autosubst2/fintype.v
+$(COQMAKEFILE) : theories/Autosubst2/syntax.v theories/Autosubst2/core.v theories/Autosubst2/unscoped.v
$(COQBIN)coq_makefile -f _CoqProject -o $(COQMAKEFILE)
install: $(COQMAKEFILE)
@@ -15,13 +15,13 @@ install: $(COQMAKEFILE)
uninstall: $(COQMAKEFILE)
$(MAKE) -f $(COQMAKEFILE) uninstall
-theories/Autosubst2/syntax.v theories/Autosubst2/core.v theories/Autosubst2/fintype.v : syntax.sig
- autosubst -f -v ge813 -s coq -o theories/Autosubst2/syntax.v syntax.sig
+theories/Autosubst2/syntax.v theories/Autosubst2/core.v theories/Autosubst2/unscoped.v : syntax.sig
+ autosubst -f -v ge813 -s ucoq -o theories/Autosubst2/syntax.v syntax.sig
.PHONY: clean FORCE
clean:
test ! -f $(COQMAKEFILE) || ( $(MAKE) -f $(COQMAKEFILE) clean && rm $(COQMAKEFILE) )
- rm -f theories/Autosubst2/syntax.v theories/Autosubst2/core.v theories/Autosubst2/fintype.v
+ rm -f theories/Autosubst2/syntax.v theories/Autosubst2/core.v theories/Autosubst2/fintype.v theories/Autosubst2/unscoped.v
FORCE:
diff --git a/theories/Autosubst2/fintype.v b/theories/Autosubst2/fintype.v
deleted file mode 100644
index 99508b6..0000000
--- a/theories/Autosubst2/fintype.v
+++ /dev/null
@@ -1,419 +0,0 @@
-(** * Autosubst Header for Scoped Syntax
- Our development utilises well-scoped de Bruijn syntax. This means that the de Bruijn indices are taken from finite types. As a consequence, any kind of substitution or environment used in conjunction with well-scoped syntax takes the form of a mapping from some finite type _I^n_. In particular, _renamings_ are mappings _I^n -> I^m_. Here we develop the theory of how these parts interact.
-
-Version: December 11, 2019.
-*)
-Require Import core.
-Require Import Setoid Morphisms Relation_Definitions.
-
-Set Implicit Arguments.
-Unset Strict Implicit.
-
-Definition ap {X Y} (f : X -> Y) {x y : X} (p : x = y) : f x = f y :=
- match p with eq_refl => eq_refl end.
-
-Definition apc {X Y} {f g : X -> Y} {x y : X} (p : f = g) (q : x = y) : f x = g y :=
- match q with eq_refl => match p with eq_refl => eq_refl end end.
-
-(** ** Primitives of the Sigma Calculus
- We implement the finite type with _n_ elements, _I^n_, as the _n_-fold iteration of the Option Type. _I^0_ is implemented as the empty type.
-*)
-
-Fixpoint fin (n : nat) : Type :=
- match n with
- | 0 => False
- | S m => option (fin m)
- end.
-
-(** Renamings and Injective Renamings
- _Renamings_ are mappings between finite types.
-*)
-Definition ren (m n : nat) : Type := fin m -> fin n.
-
-Definition id {X} := @Datatypes.id X.
-
-Definition idren {k: nat} : ren k k := @Datatypes.id (fin k).
-
-(** We give a special name, to the newest element in a non-empty finite type, as it usually corresponds to a freshly bound variable. *)
-Definition var_zero {n : nat} : fin (S n) := None.
-
-Definition null {T} (i : fin 0) : T := match i with end.
-
-Definition shift {n : nat} : ren n (S n) :=
- Some.
-
-(** Extension of Finite Mappings
- Assume we are given a mapping _f_ from _I^n_ to some type _X_, then we can _extend_ this mapping with a new value from _x : X_ to a mapping from _I^n+1_ to _X_. We denote this operation by _x . f_ and define it as follows:
-*)
-Definition scons {X : Type} {n : nat} (x : X) (f : fin n -> X) (m : fin (S n)) : X :=
- match m with
- | None => x
- | Some i => f i
- end.
-
-#[ export ]
-Hint Opaque scons : rewrite.
-
-(** ** Type Class Instances for Notation *)
-
-(** *** Type classes for renamings. *)
-
-Class Ren1 (X1 : Type) (Y Z : Type) :=
- ren1 : X1 -> Y -> Z.
-
-Class Ren2 (X1 X2 : Type) (Y Z : Type) :=
- ren2 : X1 -> X2 -> Y -> Z.
-
-Class Ren3 (X1 X2 X3 : Type) (Y Z : Type) :=
- ren3 : X1 -> X2 -> X3 -> Y -> Z.
-
-Class Ren4 (X1 X2 X3 X4 : Type) (Y Z : Type) :=
- ren4 : X1 -> X2 -> X3 -> X4 -> Y -> Z.
-
-Class Ren5 (X1 X2 X3 X4 X5 : Type) (Y Z : Type) :=
- ren5 : X1 -> X2 -> X3 -> X4 -> X5 -> Y -> Z.
-
-Module RenNotations.
- Notation "s ⟨ xi1 ⟩" := (ren1 xi1 s) (at level 7, left associativity, format "s ⟨ xi1 ⟩") : subst_scope.
-
- Notation "s ⟨ xi1 ; xi2 ⟩" := (ren2 xi1 xi2 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ⟩") : subst_scope.
-
- Notation "s ⟨ xi1 ; xi2 ; xi3 ⟩" := (ren3 xi1 xi2 xi3 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ; xi3 ⟩") : subst_scope.
-
- Notation "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ⟩" := (ren4 xi1 xi2 xi3 xi4 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ⟩") : subst_scope.
-
- Notation "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ; xi5 ⟩" := (ren5 xi1 xi2 xi3 xi4 xi5 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ; xi5 ⟩") : subst_scope.
-
- Notation "⟨ xi ⟩" := (ren1 xi) (at level 1, left associativity, format "⟨ xi ⟩") : fscope.
-
- Notation "⟨ xi1 ; xi2 ⟩" := (ren2 xi1 xi2) (at level 1, left associativity, format "⟨ xi1 ; xi2 ⟩") : fscope.
-End RenNotations.
-
-(** *** Type Classes for Substiution *)
-
-Class Subst1 (X1 : Type) (Y Z: Type) :=
- subst1 : X1 -> Y -> Z.
-
-Class Subst2 (X1 X2 : Type) (Y Z: Type) :=
- subst2 : X1 -> X2 -> Y -> Z.
-
-Class Subst3 (X1 X2 X3 : Type) (Y Z: Type) :=
- subst3 : X1 -> X2 -> X3 -> Y -> Z.
-
-Class Subst4 (X1 X2 X3 X4: Type) (Y Z: Type) :=
- subst4 : X1 -> X2 -> X3 -> X4 -> Y -> Z.
-
-Class Subst5 (X1 X2 X3 X4 X5 : Type) (Y Z: Type) :=
- subst5 : X1 -> X2 -> X3 -> X4 -> X5 -> Y -> Z.
-
-Module SubstNotations.
- Notation "s [ sigma ]" := (subst1 sigma s) (at level 7, left associativity, format "s '/' [ sigma ]") : subst_scope.
-
- Notation "s [ sigma ; tau ]" := (subst2 sigma tau s) (at level 7, left associativity, format "s '/' [ sigma ; '/' tau ]") : subst_scope.
-End SubstNotations.
-
-(** ** Type Class for Variables *)
-Class Var X Y :=
- ids : X -> Y.
-
-
-(** ** Proofs for substitution primitives *)
-
-(** Forward Function Composition
- Substitutions represented as functions are ubiquitious in this development and we often have to compose them, without talking about their pointwise behaviour.
- That is, we are interested in the forward compostion of functions, _f o g_, for which we introduce a convenient notation, "f >> g". The direction of the arrow serves as a reminder of the _forward_ nature of this composition, that is first apply _f_, then _g_. *)
-
-Arguments funcomp {X Y Z} (g)%fscope (f)%fscope.
-
-Module CombineNotations.
- Notation "f >> g" := (funcomp g f) (at level 50) : fscope.
-
- Notation "s .: sigma" := (scons s sigma) (at level 55, sigma at next level, right associativity) : subst_scope.
-
- #[ global ]
- Open Scope fscope.
- #[ global ]
- Open Scope subst_scope.
-End CombineNotations.
-
-Import CombineNotations.
-
-
-(** Generic lifting operation for renamings *)
-Definition up_ren {m n} (xi : ren m n) : ren (S m) (S n) :=
- var_zero .: xi >> shift.
-
-(** Generic proof that lifting of renamings composes. *)
-Lemma up_ren_ren k l m (xi: ren k l) (zeta : ren l m) (rho: ren k m) (E: forall x, (xi >> zeta) x = rho x) :
- forall x, (up_ren xi >> up_ren zeta) x = up_ren rho x.
-Proof.
- intros [x|].
- - cbn. unfold funcomp. now rewrite <- E.
- - reflexivity.
-Qed.
-
-Arguments up_ren_ren {k l m} xi zeta rho E.
-
-Lemma fin_eta {X} (f g : fin 0 -> X) :
- pointwise_relation _ eq f g.
-Proof. intros []. Qed.
-
-(** Eta laws *)
-Lemma scons_eta' {T} {n : nat} (f : fin (S n) -> T) :
- pointwise_relation _ eq (f var_zero .: (funcomp f shift)) f.
-Proof. intros x. destruct x; reflexivity. Qed.
-
-Lemma scons_eta_id' {n : nat} :
- pointwise_relation (fin (S n)) eq (var_zero .: shift) id.
-Proof. intros x. destruct x; reflexivity. Qed.
-
-Lemma scons_comp' {T:Type} {U} {m} (s: T) (sigma: fin m -> T) (tau: T -> U) :
- pointwise_relation _ eq (funcomp tau (s .: sigma)) ((tau s) .: (funcomp tau sigma)).
-Proof. intros x. destruct x; reflexivity. Qed.
-
-(* Lemma scons_tail'_pointwise {X} {n} (s : X) (f : fin n -> X) : *)
-(* pointwise_relation _ eq (funcomp (scons s f) shift) f. *)
-(* Proof. *)
-(* reflexivity. *)
-(* Qed. *)
-
-(* Lemma scons_tail' {X} {n} (s : X) (f : fin n -> X) x : *)
-(* (scons s f (shift x)) = f x. *)
-(* Proof. *)
-(* reflexivity. *)
-(* Qed. *)
-
-(* Morphism for Setoid Rewriting. The only morphism that can be defined statically. *)
-#[export] Instance scons_morphism {X: Type} {n:nat} :
- Proper (eq ==> pointwise_relation _ eq ==> pointwise_relation _ eq) (@scons X n).
-Proof.
- intros t t' -> sigma tau H.
- intros [x|].
- cbn. apply H.
- reflexivity.
-Qed.
-
-#[export] Instance scons_morphism2 {X: Type} {n: nat} :
- Proper (eq ==> pointwise_relation _ eq ==> eq ==> eq) (@scons X n).
-Proof.
- intros ? t -> sigma tau H ? x ->.
- destruct x as [x|].
- cbn. apply H.
- reflexivity.
-Qed.
-
-(** ** Variadic Substitution Primitives *)
-
-Fixpoint shift_p (p : nat) {n} : ren n (p + n) :=
- fun n => match p with
- | 0 => n
- | S p => Some (shift_p p n)
- end.
-
-Fixpoint scons_p {X: Type} {m : nat} : forall {n} (f : fin m -> X) (g : fin n -> X), fin (m + n) -> X.
-Proof.
- destruct m.
- - intros n f g. exact g.
- - intros n f g. cbn. apply scons.
- + exact (f var_zero).
- + apply scons_p.
- * intros z. exact (f (Some z)).
- * exact g.
-Defined.
-
-#[export] Hint Opaque scons_p : rewrite.
-
-#[export] Instance scons_p_morphism {X: Type} {m n:nat} :
- Proper (pointwise_relation _ eq ==> pointwise_relation _ eq ==> pointwise_relation _ eq) (@scons_p X m n).
-Proof.
- intros sigma sigma' Hsigma tau tau' Htau.
- intros x.
- induction m.
- - cbn. apply Htau.
- - cbn. change (fin (S m + n)) with (fin (S (m + n))) in x.
- destruct x as [x|].
- + cbn. apply IHm.
- intros ?. apply Hsigma.
- + cbn. apply Hsigma.
-Qed.
-
-#[export] Instance scons_p_morphism2 {X: Type} {m n:nat} :
- Proper (pointwise_relation _ eq ==> pointwise_relation _ eq ==> eq ==> eq) (@scons_p X m n).
-Proof.
- intros sigma sigma' Hsigma tau tau' Htau ? x ->.
- induction m.
- - cbn. apply Htau.
- - cbn. change (fin (S m + n)) with (fin (S (m + n))) in x.
- destruct x as [x|].
- + cbn. apply IHm.
- intros ?. apply Hsigma.
- + cbn. apply Hsigma.
-Qed.
-
-Definition zero_p {m : nat} {n} : fin m -> fin (m + n).
-Proof.
- induction m.
- - intros [].
- - intros [x|].
- + exact (shift_p 1 (IHm x)).
- + exact var_zero.
-Defined.
-
-Lemma scons_p_head' {X} {m n} (f : fin m -> X) (g : fin n -> X) z:
- (scons_p f g) (zero_p z) = f z.
-Proof.
- induction m.
- - inversion z.
- - destruct z.
- + simpl. simpl. now rewrite IHm.
- + reflexivity.
-Qed.
-
-Lemma scons_p_head_pointwise {X} {m n} (f : fin m -> X) (g : fin n -> X) :
- pointwise_relation _ eq (funcomp (scons_p f g) zero_p) f.
-Proof.
- intros z.
- unfold funcomp.
- induction m.
- - inversion z.
- - destruct z.
- + simpl. now rewrite IHm.
- + reflexivity.
-Qed.
-
-Lemma scons_p_tail' X m n (f : fin m -> X) (g : fin n -> X) z :
- scons_p f g (shift_p m z) = g z.
-Proof. induction m; cbn; eauto. Qed.
-
-Lemma scons_p_tail_pointwise X m n (f : fin m -> X) (g : fin n -> X) :
- pointwise_relation _ eq (funcomp (scons_p f g) (shift_p m)) g.
-Proof. intros z. induction m; cbn; eauto. Qed.
-
-Lemma destruct_fin {m n} (x : fin (m + n)):
- (exists x', x = zero_p x') \/ exists x', x = shift_p m x'.
-Proof.
- induction m; simpl in *.
- - right. eauto.
- - destruct x as [x|].
- + destruct (IHm x) as [[x' ->] |[x' ->]].
- * left. now exists (Some x').
- * right. eauto.
- + left. exists None. eauto.
-Qed.
-
-Lemma scons_p_comp' X Y m n (f : fin m -> X) (g : fin n -> X) (h : X -> Y) :
- pointwise_relation _ eq (funcomp h (scons_p f g)) (scons_p (f >> h) (g >> h)).
-Proof.
- intros x.
- destruct (destruct_fin x) as [[x' ->]|[x' ->]].
- (* TODO better way to solve this? *)
- - revert x'.
- apply pointwise_forall.
- change (fun x => (scons_p f g >> h) (zero_p x)) with (zero_p >> scons_p f g >> h).
- now setoid_rewrite scons_p_head_pointwise.
- - revert x'.
- apply pointwise_forall.
- change (fun x => (scons_p f g >> h) (shift_p m x)) with (shift_p m >> scons_p f g >> h).
- change (fun x => scons_p (f >> h) (g >> h) (shift_p m x)) with (shift_p m >> scons_p (f >> h) (g >> h)).
- now rewrite !scons_p_tail_pointwise.
-Qed.
-
-
-Lemma scons_p_congr {X} {m n} (f f' : fin m -> X) (g g': fin n -> X) z:
- (forall x, f x = f' x) -> (forall x, g x = g' x) -> scons_p f g z = scons_p f' g' z.
-Proof. intros H1 H2. induction m; eauto. cbn. destruct z; eauto. Qed.
-
-(** Generic n-ary lifting operation. *)
-Definition upRen_p p { m : nat } { n : nat } (xi : (fin) (m) -> (fin) (n)) : fin (p + m) -> fin (p + n) :=
- scons_p (zero_p ) (xi >> shift_p _).
-
-Arguments upRen_p p {m n} xi.
-
-(** Generic proof for composition of n-ary lifting. *)
-Lemma up_ren_ren_p p k l m (xi: ren k l) (zeta : ren l m) (rho: ren k m) (E: forall x, (xi >> zeta) x = rho x) :
- forall x, (upRen_p p xi >> upRen_p p zeta) x = upRen_p p rho x.
-Proof.
- intros x. destruct (destruct_fin x) as [[? ->]|[? ->]].
- - unfold upRen_p. unfold funcomp. now repeat rewrite scons_p_head'.
- - unfold upRen_p. unfold funcomp. repeat rewrite scons_p_tail'.
- now rewrite <- E.
-Qed.
-
-
-Arguments zero_p m {n}.
-Arguments scons_p {X} m {n} f g.
-
-Lemma scons_p_eta {X} {m n} {f : fin m -> X}
- {g : fin n -> X} (h: fin (m + n) -> X) {z: fin (m + n)}:
- (forall x, g x = h (shift_p m x)) -> (forall x, f x = h (zero_p m x)) -> scons_p m f g z = h z.
-Proof.
- intros H1 H2. destruct (destruct_fin z) as [[? ->] |[? ->]].
- - rewrite scons_p_head'. eauto.
- - rewrite scons_p_tail'. eauto.
-Qed.
-
-Arguments scons_p_eta {X} {m n} {f g} h {z}.
-Arguments scons_p_congr {X} {m n} {f f'} {g g'} {z}.
-
-(** ** Notations for Scoped Syntax *)
-
-Module ScopedNotations.
- Include RenNotations.
- Include SubstNotations.
- Include CombineNotations.
-
-(* Notation "s , sigma" := (scons s sigma) (at level 60, format "s , sigma", right associativity) : subst_scope. *)
-
- Notation "s '..'" := (scons s ids) (at level 1, format "s ..") : subst_scope.
-
- Notation "↑" := (shift) : subst_scope.
-
- #[global]
- Open Scope fscope.
- #[global]
- Open Scope subst_scope.
-End ScopedNotations.
-
-
-(** ** Tactics for Scoped Syntax *)
-
-Tactic Notation "auto_case" tactic(t) := (match goal with
- | [|- forall (i : fin 0), _] => intros []; t
- | [|- forall (i : fin (S (S (S (S _))))), _] => intros [[[[|]|]|]|]; t
- | [|- forall (i : fin (S (S (S _)))), _] => intros [[[|]|]|]; t
- | [|- forall (i : fin (S (S _))), _] => intros [[?|]|]; t
- | [|- forall (i : fin (S _)), _] => intros [?|]; t
- end).
-
-#[export] Hint Rewrite @scons_p_head' @scons_p_tail' : FunctorInstances.
-
-(** Generic fsimpl tactic: simplifies the above primitives in a goal. *)
-Ltac fsimpl :=
- repeat match goal with
- | [|- context[id >> ?f]] => change (id >> f) with f (* AsimplCompIdL *)
- | [|- context[?f >> id]] => change (f >> id) with f (* AsimplCompIdR *)
- | [|- context [id ?s]] => change (id s) with s
- | [|- context[(?f >> ?g) >> ?h]] => change ((f >> g) >> h) with (f >> (g >> h)) (* AsimplComp *)
- (* | [|- zero_p >> scons_p ?f ?g] => rewrite scons_p_head *)
- | [|- context[(?s .: ?sigma) var_zero]] => change ((s.:sigma) var_zero) with s
- | [|- context[(?s .: ?sigma) (shift ?m)]] => change ((s.:sigma) (shift m)) with (sigma m)
- (* first [progress setoid_rewrite scons_tail' | progress setoid_rewrite scons_tail'_pointwise ] *)
- | [|- context[idren >> ?f]] => change (idren >> f) with f
- | [|- context[?f >> idren]] => change (f >> idren) with f
- | [|- context[?f >> (?x .: ?g)]] => change (f >> (x .: g)) with g (* f should evaluate to shift *)
- | [|- context[?x2 .: (funcomp ?f shift)]] => change (scons x2 (funcomp f shift)) with (scons (f var_zero) (funcomp f shift)); setoid_rewrite (@scons_eta' _ _ f); eta_reduce
- | [|- context[?f var_zero .: ?g]] => change (scons (f var_zero) g) with (scons (f var_zero) (funcomp f shift)); setoid_rewrite scons_eta'; eta_reduce
- | [|- _ = ?h (?f ?s)] => change (h (f s)) with ((f >> h) s)
- | [|- ?h (?f ?s) = _] => change (h (f s)) with ((f >> h) s)
- | [|- context[funcomp _ (scons _ _)]] => setoid_rewrite scons_comp'; eta_reduce
- | [|- context[funcomp _ (scons_p _ _ _)]] => setoid_rewrite scons_p_comp'; eta_reduce
- | [|- context[scons (@var_zero _) shift]] => setoid_rewrite scons_eta_id'; eta_reduce
- (* | _ => progress autorewrite with FunctorInstances *)
- | [|- context[funcomp (scons_p _ _ _) (zero_p _)]] =>
- first [progress setoid_rewrite scons_p_head_pointwise | progress setoid_rewrite scons_p_head' ]
- | [|- context[scons_p _ _ _ (zero_p _ _)]] => setoid_rewrite scons_p_head'
- | [|- context[funcomp (scons_p _ _ _) (shift_p _)]] =>
- first [progress setoid_rewrite scons_p_tail_pointwise | progress setoid_rewrite scons_p_tail' ]
- | [|- context[scons_p _ _ _ (shift_p _ _)]] => setoid_rewrite scons_p_tail'
- | _ => first [progress minimize | progress cbn [shift scons scons_p] ]
- end.
diff --git a/theories/Autosubst2/syntax.v b/theories/Autosubst2/syntax.v
index 75ef645..659d8b0 100644
--- a/theories/Autosubst2/syntax.v
+++ b/theories/Autosubst2/syntax.v
@@ -1,4 +1,4 @@
-Require Import core fintype.
+Require Import core unscoped.
Require Import Setoid Morphisms Relation_Definitions.
@@ -33,345 +33,257 @@ Proof.
exact (eq_refl).
Qed.
-Inductive PTm (n_PTm : nat) : Type :=
- | VarPTm : fin n_PTm -> PTm n_PTm
- | PAbs : PTm (S n_PTm) -> PTm n_PTm
- | PApp : PTm n_PTm -> PTm n_PTm -> PTm n_PTm
- | PPair : PTm n_PTm -> PTm n_PTm -> PTm n_PTm
- | PProj : PTag -> PTm n_PTm -> PTm n_PTm
- | PConst : TTag -> PTm n_PTm
- | PUniv : nat -> PTm n_PTm
- | PBot : PTm n_PTm.
+Inductive PTm : Type :=
+ | VarPTm : nat -> PTm
+ | PAbs : PTm -> PTm
+ | PApp : PTm -> PTm -> PTm
+ | PPair : PTm -> PTm -> PTm
+ | PProj : PTag -> PTm -> PTm
+ | PConst : TTag -> PTm
+ | PUniv : nat -> PTm
+ | PBot : PTm.
-Lemma congr_PAbs {m_PTm : nat} {s0 : PTm (S m_PTm)} {t0 : PTm (S m_PTm)}
- (H0 : s0 = t0) : PAbs m_PTm s0 = PAbs m_PTm t0.
+Lemma congr_PAbs {s0 : PTm} {t0 : PTm} (H0 : s0 = t0) : PAbs s0 = PAbs t0.
Proof.
-exact (eq_trans eq_refl (ap (fun x => PAbs m_PTm x) H0)).
+exact (eq_trans eq_refl (ap (fun x => PAbs x) H0)).
Qed.
-Lemma congr_PApp {m_PTm : nat} {s0 : PTm m_PTm} {s1 : PTm m_PTm}
- {t0 : PTm m_PTm} {t1 : PTm m_PTm} (H0 : s0 = t0) (H1 : s1 = t1) :
- PApp m_PTm s0 s1 = PApp m_PTm t0 t1.
+Lemma congr_PApp {s0 : PTm} {s1 : PTm} {t0 : PTm} {t1 : PTm} (H0 : s0 = t0)
+ (H1 : s1 = t1) : PApp s0 s1 = PApp t0 t1.
Proof.
-exact (eq_trans (eq_trans eq_refl (ap (fun x => PApp m_PTm x s1) H0))
- (ap (fun x => PApp m_PTm t0 x) H1)).
+exact (eq_trans (eq_trans eq_refl (ap (fun x => PApp x s1) H0))
+ (ap (fun x => PApp t0 x) H1)).
Qed.
-Lemma congr_PPair {m_PTm : nat} {s0 : PTm m_PTm} {s1 : PTm m_PTm}
- {t0 : PTm m_PTm} {t1 : PTm m_PTm} (H0 : s0 = t0) (H1 : s1 = t1) :
- PPair m_PTm s0 s1 = PPair m_PTm t0 t1.
+Lemma congr_PPair {s0 : PTm} {s1 : PTm} {t0 : PTm} {t1 : PTm} (H0 : s0 = t0)
+ (H1 : s1 = t1) : PPair s0 s1 = PPair t0 t1.
Proof.
-exact (eq_trans (eq_trans eq_refl (ap (fun x => PPair m_PTm x s1) H0))
- (ap (fun x => PPair m_PTm t0 x) H1)).
+exact (eq_trans (eq_trans eq_refl (ap (fun x => PPair x s1) H0))
+ (ap (fun x => PPair t0 x) H1)).
Qed.
-Lemma congr_PProj {m_PTm : nat} {s0 : PTag} {s1 : PTm m_PTm} {t0 : PTag}
- {t1 : PTm m_PTm} (H0 : s0 = t0) (H1 : s1 = t1) :
- PProj m_PTm s0 s1 = PProj m_PTm t0 t1.
+Lemma congr_PProj {s0 : PTag} {s1 : PTm} {t0 : PTag} {t1 : PTm}
+ (H0 : s0 = t0) (H1 : s1 = t1) : PProj s0 s1 = PProj t0 t1.
Proof.
-exact (eq_trans (eq_trans eq_refl (ap (fun x => PProj m_PTm x s1) H0))
- (ap (fun x => PProj m_PTm t0 x) H1)).
+exact (eq_trans (eq_trans eq_refl (ap (fun x => PProj x s1) H0))
+ (ap (fun x => PProj t0 x) H1)).
Qed.
-Lemma congr_PConst {m_PTm : nat} {s0 : TTag} {t0 : TTag} (H0 : s0 = t0) :
- PConst m_PTm s0 = PConst m_PTm t0.
+Lemma congr_PConst {s0 : TTag} {t0 : TTag} (H0 : s0 = t0) :
+ PConst s0 = PConst t0.
Proof.
-exact (eq_trans eq_refl (ap (fun x => PConst m_PTm x) H0)).
+exact (eq_trans eq_refl (ap (fun x => PConst x) H0)).
Qed.
-Lemma congr_PUniv {m_PTm : nat} {s0 : nat} {t0 : nat} (H0 : s0 = t0) :
- PUniv m_PTm s0 = PUniv m_PTm t0.
+Lemma congr_PUniv {s0 : nat} {t0 : nat} (H0 : s0 = t0) : PUniv s0 = PUniv t0.
Proof.
-exact (eq_trans eq_refl (ap (fun x => PUniv m_PTm x) H0)).
+exact (eq_trans eq_refl (ap (fun x => PUniv x) H0)).
Qed.
-Lemma congr_PBot {m_PTm : nat} : PBot m_PTm = PBot m_PTm.
+Lemma congr_PBot : PBot = PBot.
Proof.
exact (eq_refl).
Qed.
-Lemma upRen_PTm_PTm {m : nat} {n : nat} (xi : fin m -> fin n) :
- fin (S m) -> fin (S n).
+Lemma upRen_PTm_PTm (xi : nat -> nat) : nat -> nat.
Proof.
exact (up_ren xi).
Defined.
-Lemma upRen_list_PTm_PTm (p : nat) {m : nat} {n : nat} (xi : fin m -> fin n)
- : fin (plus p m) -> fin (plus p n).
-Proof.
-exact (upRen_p p xi).
-Defined.
-
-Fixpoint ren_PTm {m_PTm : nat} {n_PTm : nat}
-(xi_PTm : fin m_PTm -> fin n_PTm) (s : PTm m_PTm) {struct s} : PTm n_PTm :=
+Fixpoint ren_PTm (xi_PTm : nat -> nat) (s : PTm) {struct s} : PTm :=
match s with
- | VarPTm _ s0 => VarPTm n_PTm (xi_PTm s0)
- | PAbs _ s0 => PAbs n_PTm (ren_PTm (upRen_PTm_PTm xi_PTm) s0)
- | PApp _ s0 s1 => PApp n_PTm (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
- | PPair _ s0 s1 => PPair n_PTm (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
- | PProj _ s0 s1 => PProj n_PTm s0 (ren_PTm xi_PTm s1)
- | PConst _ s0 => PConst n_PTm s0
- | PUniv _ s0 => PUniv n_PTm s0
- | PBot _ => PBot n_PTm
+ | VarPTm s0 => VarPTm (xi_PTm s0)
+ | PAbs s0 => PAbs (ren_PTm (upRen_PTm_PTm xi_PTm) s0)
+ | PApp s0 s1 => PApp (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
+ | PPair s0 s1 => PPair (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
+ | PProj s0 s1 => PProj s0 (ren_PTm xi_PTm s1)
+ | PConst s0 => PConst s0
+ | PUniv s0 => PUniv s0
+ | PBot => PBot
end.
-Lemma up_PTm_PTm {m : nat} {n_PTm : nat} (sigma : fin m -> PTm n_PTm) :
- fin (S m) -> PTm (S n_PTm).
+Lemma up_PTm_PTm (sigma : nat -> PTm) : nat -> PTm.
Proof.
-exact (scons (VarPTm (S n_PTm) var_zero) (funcomp (ren_PTm shift) sigma)).
+exact (scons (VarPTm var_zero) (funcomp (ren_PTm shift) sigma)).
Defined.
-Lemma up_list_PTm_PTm (p : nat) {m : nat} {n_PTm : nat}
- (sigma : fin m -> PTm n_PTm) : fin (plus p m) -> PTm (plus p n_PTm).
-Proof.
-exact (scons_p p (funcomp (VarPTm (plus p n_PTm)) (zero_p p))
- (funcomp (ren_PTm (shift_p p)) sigma)).
-Defined.
-
-Fixpoint subst_PTm {m_PTm : nat} {n_PTm : nat}
-(sigma_PTm : fin m_PTm -> PTm n_PTm) (s : PTm m_PTm) {struct s} : PTm n_PTm
-:=
+Fixpoint subst_PTm (sigma_PTm : nat -> PTm) (s : PTm) {struct s} : PTm :=
match s with
- | VarPTm _ s0 => sigma_PTm s0
- | PAbs _ s0 => PAbs n_PTm (subst_PTm (up_PTm_PTm sigma_PTm) s0)
- | PApp _ s0 s1 =>
- PApp n_PTm (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1)
- | PPair _ s0 s1 =>
- PPair n_PTm (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1)
- | PProj _ s0 s1 => PProj n_PTm s0 (subst_PTm sigma_PTm s1)
- | PConst _ s0 => PConst n_PTm s0
- | PUniv _ s0 => PUniv n_PTm s0
- | PBot _ => PBot n_PTm
+ | VarPTm s0 => sigma_PTm s0
+ | PAbs s0 => PAbs (subst_PTm (up_PTm_PTm sigma_PTm) s0)
+ | PApp s0 s1 => PApp (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1)
+ | PPair s0 s1 => PPair (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1)
+ | PProj s0 s1 => PProj s0 (subst_PTm sigma_PTm s1)
+ | PConst s0 => PConst s0
+ | PUniv s0 => PUniv s0
+ | PBot => PBot
end.
-Lemma upId_PTm_PTm {m_PTm : nat} (sigma : fin m_PTm -> PTm m_PTm)
- (Eq : forall x, sigma x = VarPTm m_PTm x) :
- forall x, up_PTm_PTm sigma x = VarPTm (S m_PTm) x.
+Lemma upId_PTm_PTm (sigma : nat -> PTm) (Eq : forall x, sigma x = VarPTm x) :
+ forall x, up_PTm_PTm sigma x = VarPTm x.
Proof.
exact (fun n =>
match n with
- | Some fin_n => ap (ren_PTm shift) (Eq fin_n)
- | None => eq_refl
+ | S n' => ap (ren_PTm shift) (Eq n')
+ | O => eq_refl
end).
Qed.
-Lemma upId_list_PTm_PTm {p : nat} {m_PTm : nat}
- (sigma : fin m_PTm -> PTm m_PTm) (Eq : forall x, sigma x = VarPTm m_PTm x)
- : forall x, up_list_PTm_PTm p sigma x = VarPTm (plus p m_PTm) x.
-Proof.
-exact (fun n =>
- scons_p_eta (VarPTm (plus p m_PTm))
- (fun n => ap (ren_PTm (shift_p p)) (Eq n)) (fun n => eq_refl)).
-Qed.
-
-Fixpoint idSubst_PTm {m_PTm : nat} (sigma_PTm : fin m_PTm -> PTm m_PTm)
-(Eq_PTm : forall x, sigma_PTm x = VarPTm m_PTm x) (s : PTm m_PTm) {struct s}
- : subst_PTm sigma_PTm s = s :=
+Fixpoint idSubst_PTm (sigma_PTm : nat -> PTm)
+(Eq_PTm : forall x, sigma_PTm x = VarPTm x) (s : PTm) {struct s} :
+subst_PTm sigma_PTm s = s :=
match s with
- | VarPTm _ s0 => Eq_PTm s0
- | PAbs _ s0 =>
+ | VarPTm s0 => Eq_PTm s0
+ | PAbs s0 =>
congr_PAbs
(idSubst_PTm (up_PTm_PTm sigma_PTm) (upId_PTm_PTm _ Eq_PTm) s0)
- | PApp _ s0 s1 =>
+ | PApp s0 s1 =>
congr_PApp (idSubst_PTm sigma_PTm Eq_PTm s0)
(idSubst_PTm sigma_PTm Eq_PTm s1)
- | PPair _ s0 s1 =>
+ | PPair s0 s1 =>
congr_PPair (idSubst_PTm sigma_PTm Eq_PTm s0)
(idSubst_PTm sigma_PTm Eq_PTm s1)
- | PProj _ s0 s1 =>
- congr_PProj (eq_refl s0) (idSubst_PTm sigma_PTm Eq_PTm s1)
- | PConst _ s0 => congr_PConst (eq_refl s0)
- | PUniv _ s0 => congr_PUniv (eq_refl s0)
- | PBot _ => congr_PBot
+ | PProj s0 s1 => congr_PProj (eq_refl s0) (idSubst_PTm sigma_PTm Eq_PTm s1)
+ | PConst s0 => congr_PConst (eq_refl s0)
+ | PUniv s0 => congr_PUniv (eq_refl s0)
+ | PBot => congr_PBot
end.
-Lemma upExtRen_PTm_PTm {m : nat} {n : nat} (xi : fin m -> fin n)
- (zeta : fin m -> fin n) (Eq : forall x, xi x = zeta x) :
+Lemma upExtRen_PTm_PTm (xi : nat -> nat) (zeta : nat -> nat)
+ (Eq : forall x, xi x = zeta x) :
forall x, upRen_PTm_PTm xi x = upRen_PTm_PTm zeta x.
Proof.
-exact (fun n =>
- match n with
- | Some fin_n => ap shift (Eq fin_n)
- | None => eq_refl
- end).
+exact (fun n => match n with
+ | S n' => ap shift (Eq n')
+ | O => eq_refl
+ end).
Qed.
-Lemma upExtRen_list_PTm_PTm {p : nat} {m : nat} {n : nat}
- (xi : fin m -> fin n) (zeta : fin m -> fin n)
- (Eq : forall x, xi x = zeta x) :
- forall x, upRen_list_PTm_PTm p xi x = upRen_list_PTm_PTm p zeta x.
-Proof.
-exact (fun n =>
- scons_p_congr (fun n => eq_refl) (fun n => ap (shift_p p) (Eq n))).
-Qed.
-
-Fixpoint extRen_PTm {m_PTm : nat} {n_PTm : nat}
-(xi_PTm : fin m_PTm -> fin n_PTm) (zeta_PTm : fin m_PTm -> fin n_PTm)
-(Eq_PTm : forall x, xi_PTm x = zeta_PTm x) (s : PTm m_PTm) {struct s} :
+Fixpoint extRen_PTm (xi_PTm : nat -> nat) (zeta_PTm : nat -> nat)
+(Eq_PTm : forall x, xi_PTm x = zeta_PTm x) (s : PTm) {struct s} :
ren_PTm xi_PTm s = ren_PTm zeta_PTm s :=
match s with
- | VarPTm _ s0 => ap (VarPTm n_PTm) (Eq_PTm s0)
- | PAbs _ s0 =>
+ | VarPTm s0 => ap (VarPTm) (Eq_PTm s0)
+ | PAbs s0 =>
congr_PAbs
(extRen_PTm (upRen_PTm_PTm xi_PTm) (upRen_PTm_PTm zeta_PTm)
(upExtRen_PTm_PTm _ _ Eq_PTm) s0)
- | PApp _ s0 s1 =>
+ | PApp s0 s1 =>
congr_PApp (extRen_PTm xi_PTm zeta_PTm Eq_PTm s0)
(extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
- | PPair _ s0 s1 =>
+ | PPair s0 s1 =>
congr_PPair (extRen_PTm xi_PTm zeta_PTm Eq_PTm s0)
(extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
- | PProj _ s0 s1 =>
+ | PProj s0 s1 =>
congr_PProj (eq_refl s0) (extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
- | PConst _ s0 => congr_PConst (eq_refl s0)
- | PUniv _ s0 => congr_PUniv (eq_refl s0)
- | PBot _ => congr_PBot
+ | PConst s0 => congr_PConst (eq_refl s0)
+ | PUniv s0 => congr_PUniv (eq_refl s0)
+ | PBot => congr_PBot
end.
-Lemma upExt_PTm_PTm {m : nat} {n_PTm : nat} (sigma : fin m -> PTm n_PTm)
- (tau : fin m -> PTm n_PTm) (Eq : forall x, sigma x = tau x) :
+Lemma upExt_PTm_PTm (sigma : nat -> PTm) (tau : nat -> PTm)
+ (Eq : forall x, sigma x = tau x) :
forall x, up_PTm_PTm sigma x = up_PTm_PTm tau x.
Proof.
exact (fun n =>
match n with
- | Some fin_n => ap (ren_PTm shift) (Eq fin_n)
- | None => eq_refl
+ | S n' => ap (ren_PTm shift) (Eq n')
+ | O => eq_refl
end).
Qed.
-Lemma upExt_list_PTm_PTm {p : nat} {m : nat} {n_PTm : nat}
- (sigma : fin m -> PTm n_PTm) (tau : fin m -> PTm n_PTm)
- (Eq : forall x, sigma x = tau x) :
- forall x, up_list_PTm_PTm p sigma x = up_list_PTm_PTm p tau x.
-Proof.
-exact (fun n =>
- scons_p_congr (fun n => eq_refl)
- (fun n => ap (ren_PTm (shift_p p)) (Eq n))).
-Qed.
-
-Fixpoint ext_PTm {m_PTm : nat} {n_PTm : nat}
-(sigma_PTm : fin m_PTm -> PTm n_PTm) (tau_PTm : fin m_PTm -> PTm n_PTm)
-(Eq_PTm : forall x, sigma_PTm x = tau_PTm x) (s : PTm m_PTm) {struct s} :
+Fixpoint ext_PTm (sigma_PTm : nat -> PTm) (tau_PTm : nat -> PTm)
+(Eq_PTm : forall x, sigma_PTm x = tau_PTm x) (s : PTm) {struct s} :
subst_PTm sigma_PTm s = subst_PTm tau_PTm s :=
match s with
- | VarPTm _ s0 => Eq_PTm s0
- | PAbs _ s0 =>
+ | VarPTm s0 => Eq_PTm s0
+ | PAbs s0 =>
congr_PAbs
(ext_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
(upExt_PTm_PTm _ _ Eq_PTm) s0)
- | PApp _ s0 s1 =>
+ | PApp s0 s1 =>
congr_PApp (ext_PTm sigma_PTm tau_PTm Eq_PTm s0)
(ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
- | PPair _ s0 s1 =>
+ | PPair s0 s1 =>
congr_PPair (ext_PTm sigma_PTm tau_PTm Eq_PTm s0)
(ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
- | PProj _ s0 s1 =>
+ | PProj s0 s1 =>
congr_PProj (eq_refl s0) (ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
- | PConst _ s0 => congr_PConst (eq_refl s0)
- | PUniv _ s0 => congr_PUniv (eq_refl s0)
- | PBot _ => congr_PBot
+ | PConst s0 => congr_PConst (eq_refl s0)
+ | PUniv s0 => congr_PUniv (eq_refl s0)
+ | PBot => congr_PBot
end.
-Lemma up_ren_ren_PTm_PTm {k : nat} {l : nat} {m : nat} (xi : fin k -> fin l)
- (zeta : fin l -> fin m) (rho : fin k -> fin m)
- (Eq : forall x, funcomp zeta xi x = rho x) :
+Lemma up_ren_ren_PTm_PTm (xi : nat -> nat) (zeta : nat -> nat)
+ (rho : nat -> nat) (Eq : forall x, funcomp zeta xi x = rho x) :
forall x,
funcomp (upRen_PTm_PTm zeta) (upRen_PTm_PTm xi) x = upRen_PTm_PTm rho x.
Proof.
exact (up_ren_ren xi zeta rho Eq).
Qed.
-Lemma up_ren_ren_list_PTm_PTm {p : nat} {k : nat} {l : nat} {m : nat}
- (xi : fin k -> fin l) (zeta : fin l -> fin m) (rho : fin k -> fin m)
- (Eq : forall x, funcomp zeta xi x = rho x) :
- forall x,
- funcomp (upRen_list_PTm_PTm p zeta) (upRen_list_PTm_PTm p xi) x =
- upRen_list_PTm_PTm p rho x.
-Proof.
-exact (up_ren_ren_p Eq).
-Qed.
-
-Fixpoint compRenRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
-(xi_PTm : fin m_PTm -> fin k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm)
-(rho_PTm : fin m_PTm -> fin l_PTm)
-(Eq_PTm : forall x, funcomp zeta_PTm xi_PTm x = rho_PTm x) (s : PTm m_PTm)
-{struct s} : ren_PTm zeta_PTm (ren_PTm xi_PTm s) = ren_PTm rho_PTm s :=
+Fixpoint compRenRen_PTm (xi_PTm : nat -> nat) (zeta_PTm : nat -> nat)
+(rho_PTm : nat -> nat)
+(Eq_PTm : forall x, funcomp zeta_PTm xi_PTm x = rho_PTm x) (s : PTm) {struct
+ s} : ren_PTm zeta_PTm (ren_PTm xi_PTm s) = ren_PTm rho_PTm s :=
match s with
- | VarPTm _ s0 => ap (VarPTm l_PTm) (Eq_PTm s0)
- | PAbs _ s0 =>
+ | VarPTm s0 => ap (VarPTm) (Eq_PTm s0)
+ | PAbs s0 =>
congr_PAbs
(compRenRen_PTm (upRen_PTm_PTm xi_PTm) (upRen_PTm_PTm zeta_PTm)
(upRen_PTm_PTm rho_PTm) (up_ren_ren _ _ _ Eq_PTm) s0)
- | PApp _ s0 s1 =>
+ | PApp s0 s1 =>
congr_PApp (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s0)
(compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
- | PPair _ s0 s1 =>
+ | PPair s0 s1 =>
congr_PPair (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s0)
(compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
- | PProj _ s0 s1 =>
+ | PProj s0 s1 =>
congr_PProj (eq_refl s0)
(compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
- | PConst _ s0 => congr_PConst (eq_refl s0)
- | PUniv _ s0 => congr_PUniv (eq_refl s0)
- | PBot _ => congr_PBot
+ | PConst s0 => congr_PConst (eq_refl s0)
+ | PUniv s0 => congr_PUniv (eq_refl s0)
+ | PBot => congr_PBot
end.
-Lemma up_ren_subst_PTm_PTm {k : nat} {l : nat} {m_PTm : nat}
- (xi : fin k -> fin l) (tau : fin l -> PTm m_PTm)
- (theta : fin k -> PTm m_PTm) (Eq : forall x, funcomp tau xi x = theta x) :
+Lemma up_ren_subst_PTm_PTm (xi : nat -> nat) (tau : nat -> PTm)
+ (theta : nat -> PTm) (Eq : forall x, funcomp tau xi x = theta x) :
forall x,
funcomp (up_PTm_PTm tau) (upRen_PTm_PTm xi) x = up_PTm_PTm theta x.
Proof.
exact (fun n =>
match n with
- | Some fin_n => ap (ren_PTm shift) (Eq fin_n)
- | None => eq_refl
+ | S n' => ap (ren_PTm shift) (Eq n')
+ | O => eq_refl
end).
Qed.
-Lemma up_ren_subst_list_PTm_PTm {p : nat} {k : nat} {l : nat} {m_PTm : nat}
- (xi : fin k -> fin l) (tau : fin l -> PTm m_PTm)
- (theta : fin k -> PTm m_PTm) (Eq : forall x, funcomp tau xi x = theta x) :
- forall x,
- funcomp (up_list_PTm_PTm p tau) (upRen_list_PTm_PTm p xi) x =
- up_list_PTm_PTm p theta x.
-Proof.
-exact (fun n =>
- eq_trans (scons_p_comp' _ _ _ n)
- (scons_p_congr (fun z => scons_p_head' _ _ z)
- (fun z =>
- eq_trans (scons_p_tail' _ _ (xi z))
- (ap (ren_PTm (shift_p p)) (Eq z))))).
-Qed.
-
-Fixpoint compRenSubst_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
-(xi_PTm : fin m_PTm -> fin k_PTm) (tau_PTm : fin k_PTm -> PTm l_PTm)
-(theta_PTm : fin m_PTm -> PTm l_PTm)
-(Eq_PTm : forall x, funcomp tau_PTm xi_PTm x = theta_PTm x) (s : PTm m_PTm)
-{struct s} : subst_PTm tau_PTm (ren_PTm xi_PTm s) = subst_PTm theta_PTm s :=
+Fixpoint compRenSubst_PTm (xi_PTm : nat -> nat) (tau_PTm : nat -> PTm)
+(theta_PTm : nat -> PTm)
+(Eq_PTm : forall x, funcomp tau_PTm xi_PTm x = theta_PTm x) (s : PTm) {struct
+ s} : subst_PTm tau_PTm (ren_PTm xi_PTm s) = subst_PTm theta_PTm s :=
match s with
- | VarPTm _ s0 => Eq_PTm s0
- | PAbs _ s0 =>
+ | VarPTm s0 => Eq_PTm s0
+ | PAbs s0 =>
congr_PAbs
(compRenSubst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm tau_PTm)
(up_PTm_PTm theta_PTm) (up_ren_subst_PTm_PTm _ _ _ Eq_PTm) s0)
- | PApp _ s0 s1 =>
+ | PApp s0 s1 =>
congr_PApp (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s0)
(compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
- | PPair _ s0 s1 =>
+ | PPair s0 s1 =>
congr_PPair (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s0)
(compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
- | PProj _ s0 s1 =>
+ | PProj s0 s1 =>
congr_PProj (eq_refl s0)
(compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
- | PConst _ s0 => congr_PConst (eq_refl s0)
- | PUniv _ s0 => congr_PUniv (eq_refl s0)
- | PBot _ => congr_PBot
+ | PConst s0 => congr_PConst (eq_refl s0)
+ | PUniv s0 => congr_PUniv (eq_refl s0)
+ | PBot => congr_PBot
end.
-Lemma up_subst_ren_PTm_PTm {k : nat} {l_PTm : nat} {m_PTm : nat}
- (sigma : fin k -> PTm l_PTm) (zeta_PTm : fin l_PTm -> fin m_PTm)
- (theta : fin k -> PTm m_PTm)
+Lemma up_subst_ren_PTm_PTm (sigma : nat -> PTm) (zeta_PTm : nat -> nat)
+ (theta : nat -> PTm)
(Eq : forall x, funcomp (ren_PTm zeta_PTm) sigma x = theta x) :
forall x,
funcomp (ren_PTm (upRen_PTm_PTm zeta_PTm)) (up_PTm_PTm sigma) x =
@@ -379,72 +291,46 @@ Lemma up_subst_ren_PTm_PTm {k : nat} {l_PTm : nat} {m_PTm : nat}
Proof.
exact (fun n =>
match n with
- | Some fin_n =>
+ | S n' =>
eq_trans
(compRenRen_PTm shift (upRen_PTm_PTm zeta_PTm)
- (funcomp shift zeta_PTm) (fun x => eq_refl) (sigma fin_n))
+ (funcomp shift zeta_PTm) (fun x => eq_refl) (sigma n'))
(eq_trans
(eq_sym
(compRenRen_PTm zeta_PTm shift (funcomp shift zeta_PTm)
- (fun x => eq_refl) (sigma fin_n)))
- (ap (ren_PTm shift) (Eq fin_n)))
- | None => eq_refl
+ (fun x => eq_refl) (sigma n')))
+ (ap (ren_PTm shift) (Eq n')))
+ | O => eq_refl
end).
Qed.
-Lemma up_subst_ren_list_PTm_PTm {p : nat} {k : nat} {l_PTm : nat}
- {m_PTm : nat} (sigma : fin k -> PTm l_PTm)
- (zeta_PTm : fin l_PTm -> fin m_PTm) (theta : fin k -> PTm m_PTm)
- (Eq : forall x, funcomp (ren_PTm zeta_PTm) sigma x = theta x) :
- forall x,
- funcomp (ren_PTm (upRen_list_PTm_PTm p zeta_PTm)) (up_list_PTm_PTm p sigma)
- x = up_list_PTm_PTm p theta x.
-Proof.
-exact (fun n =>
- eq_trans (scons_p_comp' _ _ _ n)
- (scons_p_congr
- (fun x => ap (VarPTm (plus p m_PTm)) (scons_p_head' _ _ x))
- (fun n =>
- eq_trans
- (compRenRen_PTm (shift_p p) (upRen_list_PTm_PTm p zeta_PTm)
- (funcomp (shift_p p) zeta_PTm)
- (fun x => scons_p_tail' _ _ x) (sigma n))
- (eq_trans
- (eq_sym
- (compRenRen_PTm zeta_PTm (shift_p p)
- (funcomp (shift_p p) zeta_PTm) (fun x => eq_refl)
- (sigma n))) (ap (ren_PTm (shift_p p)) (Eq n)))))).
-Qed.
-
-Fixpoint compSubstRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
-(sigma_PTm : fin m_PTm -> PTm k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm)
-(theta_PTm : fin m_PTm -> PTm l_PTm)
+Fixpoint compSubstRen_PTm (sigma_PTm : nat -> PTm) (zeta_PTm : nat -> nat)
+(theta_PTm : nat -> PTm)
(Eq_PTm : forall x, funcomp (ren_PTm zeta_PTm) sigma_PTm x = theta_PTm x)
-(s : PTm m_PTm) {struct s} :
+(s : PTm) {struct s} :
ren_PTm zeta_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
match s with
- | VarPTm _ s0 => Eq_PTm s0
- | PAbs _ s0 =>
+ | VarPTm s0 => Eq_PTm s0
+ | PAbs s0 =>
congr_PAbs
(compSubstRen_PTm (up_PTm_PTm sigma_PTm) (upRen_PTm_PTm zeta_PTm)
(up_PTm_PTm theta_PTm) (up_subst_ren_PTm_PTm _ _ _ Eq_PTm) s0)
- | PApp _ s0 s1 =>
+ | PApp s0 s1 =>
congr_PApp (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s0)
(compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
- | PPair _ s0 s1 =>
+ | PPair s0 s1 =>
congr_PPair (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s0)
(compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
- | PProj _ s0 s1 =>
+ | PProj s0 s1 =>
congr_PProj (eq_refl s0)
(compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
- | PConst _ s0 => congr_PConst (eq_refl s0)
- | PUniv _ s0 => congr_PUniv (eq_refl s0)
- | PBot _ => congr_PBot
+ | PConst s0 => congr_PConst (eq_refl s0)
+ | PUniv s0 => congr_PUniv (eq_refl s0)
+ | PBot => congr_PBot
end.
-Lemma up_subst_subst_PTm_PTm {k : nat} {l_PTm : nat} {m_PTm : nat}
- (sigma : fin k -> PTm l_PTm) (tau_PTm : fin l_PTm -> PTm m_PTm)
- (theta : fin k -> PTm m_PTm)
+Lemma up_subst_subst_PTm_PTm (sigma : nat -> PTm) (tau_PTm : nat -> PTm)
+ (theta : nat -> PTm)
(Eq : forall x, funcomp (subst_PTm tau_PTm) sigma x = theta x) :
forall x,
funcomp (subst_PTm (up_PTm_PTm tau_PTm)) (up_PTm_PTm sigma) x =
@@ -452,649 +338,501 @@ Lemma up_subst_subst_PTm_PTm {k : nat} {l_PTm : nat} {m_PTm : nat}
Proof.
exact (fun n =>
match n with
- | Some fin_n =>
+ | S n' =>
eq_trans
(compRenSubst_PTm shift (up_PTm_PTm tau_PTm)
(funcomp (up_PTm_PTm tau_PTm) shift) (fun x => eq_refl)
- (sigma fin_n))
+ (sigma n'))
(eq_trans
(eq_sym
(compSubstRen_PTm tau_PTm shift
(funcomp (ren_PTm shift) tau_PTm) (fun x => eq_refl)
- (sigma fin_n))) (ap (ren_PTm shift) (Eq fin_n)))
- | None => eq_refl
+ (sigma n'))) (ap (ren_PTm shift) (Eq n')))
+ | O => eq_refl
end).
Qed.
-Lemma up_subst_subst_list_PTm_PTm {p : nat} {k : nat} {l_PTm : nat}
- {m_PTm : nat} (sigma : fin k -> PTm l_PTm)
- (tau_PTm : fin l_PTm -> PTm m_PTm) (theta : fin k -> PTm m_PTm)
- (Eq : forall x, funcomp (subst_PTm tau_PTm) sigma x = theta x) :
- forall x,
- funcomp (subst_PTm (up_list_PTm_PTm p tau_PTm)) (up_list_PTm_PTm p sigma) x =
- up_list_PTm_PTm p theta x.
-Proof.
-exact (fun n =>
- eq_trans
- (scons_p_comp' (funcomp (VarPTm (plus p l_PTm)) (zero_p p)) _ _ n)
- (scons_p_congr
- (fun x => scons_p_head' _ (fun z => ren_PTm (shift_p p) _) x)
- (fun n =>
- eq_trans
- (compRenSubst_PTm (shift_p p) (up_list_PTm_PTm p tau_PTm)
- (funcomp (up_list_PTm_PTm p tau_PTm) (shift_p p))
- (fun x => eq_refl) (sigma n))
- (eq_trans
- (eq_sym
- (compSubstRen_PTm tau_PTm (shift_p p) _
- (fun x => eq_sym (scons_p_tail' _ _ x)) (sigma n)))
- (ap (ren_PTm (shift_p p)) (Eq n)))))).
-Qed.
-
-Fixpoint compSubstSubst_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
-(sigma_PTm : fin m_PTm -> PTm k_PTm) (tau_PTm : fin k_PTm -> PTm l_PTm)
-(theta_PTm : fin m_PTm -> PTm l_PTm)
+Fixpoint compSubstSubst_PTm (sigma_PTm : nat -> PTm) (tau_PTm : nat -> PTm)
+(theta_PTm : nat -> PTm)
(Eq_PTm : forall x, funcomp (subst_PTm tau_PTm) sigma_PTm x = theta_PTm x)
-(s : PTm m_PTm) {struct s} :
+(s : PTm) {struct s} :
subst_PTm tau_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
match s with
- | VarPTm _ s0 => Eq_PTm s0
- | PAbs _ s0 =>
+ | VarPTm s0 => Eq_PTm s0
+ | PAbs s0 =>
congr_PAbs
(compSubstSubst_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
(up_PTm_PTm theta_PTm) (up_subst_subst_PTm_PTm _ _ _ Eq_PTm) s0)
- | PApp _ s0 s1 =>
+ | PApp s0 s1 =>
congr_PApp (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s0)
(compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
- | PPair _ s0 s1 =>
+ | PPair s0 s1 =>
congr_PPair (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s0)
(compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
- | PProj _ s0 s1 =>
+ | PProj s0 s1 =>
congr_PProj (eq_refl s0)
(compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
- | PConst _ s0 => congr_PConst (eq_refl s0)
- | PUniv _ s0 => congr_PUniv (eq_refl s0)
- | PBot _ => congr_PBot
+ | PConst s0 => congr_PConst (eq_refl s0)
+ | PUniv s0 => congr_PUniv (eq_refl s0)
+ | PBot => congr_PBot
end.
-Lemma renRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
- (xi_PTm : fin m_PTm -> fin k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm)
- (s : PTm m_PTm) :
+Lemma renRen_PTm (xi_PTm : nat -> nat) (zeta_PTm : nat -> nat) (s : PTm) :
ren_PTm zeta_PTm (ren_PTm xi_PTm s) = ren_PTm (funcomp zeta_PTm xi_PTm) s.
Proof.
exact (compRenRen_PTm xi_PTm zeta_PTm _ (fun n => eq_refl) s).
Qed.
-Lemma renRen'_PTm_pointwise {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
- (xi_PTm : fin m_PTm -> fin k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm) :
+Lemma renRen'_PTm_pointwise (xi_PTm : nat -> nat) (zeta_PTm : nat -> nat) :
pointwise_relation _ eq (funcomp (ren_PTm zeta_PTm) (ren_PTm xi_PTm))
(ren_PTm (funcomp zeta_PTm xi_PTm)).
Proof.
exact (fun s => compRenRen_PTm xi_PTm zeta_PTm _ (fun n => eq_refl) s).
Qed.
-Lemma renSubst_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
- (xi_PTm : fin m_PTm -> fin k_PTm) (tau_PTm : fin k_PTm -> PTm l_PTm)
- (s : PTm m_PTm) :
+Lemma renSubst_PTm (xi_PTm : nat -> nat) (tau_PTm : nat -> PTm) (s : PTm) :
subst_PTm tau_PTm (ren_PTm xi_PTm s) = subst_PTm (funcomp tau_PTm xi_PTm) s.
Proof.
exact (compRenSubst_PTm xi_PTm tau_PTm _ (fun n => eq_refl) s).
Qed.
-Lemma renSubst_PTm_pointwise {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
- (xi_PTm : fin m_PTm -> fin k_PTm) (tau_PTm : fin k_PTm -> PTm l_PTm) :
+Lemma renSubst_PTm_pointwise (xi_PTm : nat -> nat) (tau_PTm : nat -> PTm) :
pointwise_relation _ eq (funcomp (subst_PTm tau_PTm) (ren_PTm xi_PTm))
(subst_PTm (funcomp tau_PTm xi_PTm)).
Proof.
exact (fun s => compRenSubst_PTm xi_PTm tau_PTm _ (fun n => eq_refl) s).
Qed.
-Lemma substRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
- (sigma_PTm : fin m_PTm -> PTm k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm)
- (s : PTm m_PTm) :
+Lemma substRen_PTm (sigma_PTm : nat -> PTm) (zeta_PTm : nat -> nat) (s : PTm)
+ :
ren_PTm zeta_PTm (subst_PTm sigma_PTm s) =
subst_PTm (funcomp (ren_PTm zeta_PTm) sigma_PTm) s.
Proof.
exact (compSubstRen_PTm sigma_PTm zeta_PTm _ (fun n => eq_refl) s).
Qed.
-Lemma substRen_PTm_pointwise {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
- (sigma_PTm : fin m_PTm -> PTm k_PTm) (zeta_PTm : fin k_PTm -> fin l_PTm) :
+Lemma substRen_PTm_pointwise (sigma_PTm : nat -> PTm) (zeta_PTm : nat -> nat)
+ :
pointwise_relation _ eq (funcomp (ren_PTm zeta_PTm) (subst_PTm sigma_PTm))
(subst_PTm (funcomp (ren_PTm zeta_PTm) sigma_PTm)).
Proof.
exact (fun s => compSubstRen_PTm sigma_PTm zeta_PTm _ (fun n => eq_refl) s).
Qed.
-Lemma substSubst_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
- (sigma_PTm : fin m_PTm -> PTm k_PTm) (tau_PTm : fin k_PTm -> PTm l_PTm)
- (s : PTm m_PTm) :
+Lemma substSubst_PTm (sigma_PTm : nat -> PTm) (tau_PTm : nat -> PTm)
+ (s : PTm) :
subst_PTm tau_PTm (subst_PTm sigma_PTm s) =
subst_PTm (funcomp (subst_PTm tau_PTm) sigma_PTm) s.
Proof.
exact (compSubstSubst_PTm sigma_PTm tau_PTm _ (fun n => eq_refl) s).
Qed.
-Lemma substSubst_PTm_pointwise {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
- (sigma_PTm : fin m_PTm -> PTm k_PTm) (tau_PTm : fin k_PTm -> PTm l_PTm) :
+Lemma substSubst_PTm_pointwise (sigma_PTm : nat -> PTm)
+ (tau_PTm : nat -> PTm) :
pointwise_relation _ eq (funcomp (subst_PTm tau_PTm) (subst_PTm sigma_PTm))
(subst_PTm (funcomp (subst_PTm tau_PTm) sigma_PTm)).
Proof.
exact (fun s => compSubstSubst_PTm sigma_PTm tau_PTm _ (fun n => eq_refl) s).
Qed.
-Lemma rinstInst_up_PTm_PTm {m : nat} {n_PTm : nat} (xi : fin m -> fin n_PTm)
- (sigma : fin m -> PTm n_PTm)
- (Eq : forall x, funcomp (VarPTm n_PTm) xi x = sigma x) :
- forall x,
- funcomp (VarPTm (S n_PTm)) (upRen_PTm_PTm xi) x = up_PTm_PTm sigma x.
+Lemma rinstInst_up_PTm_PTm (xi : nat -> nat) (sigma : nat -> PTm)
+ (Eq : forall x, funcomp (VarPTm) xi x = sigma x) :
+ forall x, funcomp (VarPTm) (upRen_PTm_PTm xi) x = up_PTm_PTm sigma x.
Proof.
exact (fun n =>
match n with
- | Some fin_n => ap (ren_PTm shift) (Eq fin_n)
- | None => eq_refl
+ | S n' => ap (ren_PTm shift) (Eq n')
+ | O => eq_refl
end).
Qed.
-Lemma rinstInst_up_list_PTm_PTm {p : nat} {m : nat} {n_PTm : nat}
- (xi : fin m -> fin n_PTm) (sigma : fin m -> PTm n_PTm)
- (Eq : forall x, funcomp (VarPTm n_PTm) xi x = sigma x) :
- forall x,
- funcomp (VarPTm (plus p n_PTm)) (upRen_list_PTm_PTm p xi) x =
- up_list_PTm_PTm p sigma x.
-Proof.
-exact (fun n =>
- eq_trans (scons_p_comp' _ _ (VarPTm (plus p n_PTm)) n)
- (scons_p_congr (fun z => eq_refl)
- (fun n => ap (ren_PTm (shift_p p)) (Eq n)))).
-Qed.
-
-Fixpoint rinst_inst_PTm {m_PTm : nat} {n_PTm : nat}
-(xi_PTm : fin m_PTm -> fin n_PTm) (sigma_PTm : fin m_PTm -> PTm n_PTm)
-(Eq_PTm : forall x, funcomp (VarPTm n_PTm) xi_PTm x = sigma_PTm x)
-(s : PTm m_PTm) {struct s} : ren_PTm xi_PTm s = subst_PTm sigma_PTm s :=
+Fixpoint rinst_inst_PTm (xi_PTm : nat -> nat) (sigma_PTm : nat -> PTm)
+(Eq_PTm : forall x, funcomp (VarPTm) xi_PTm x = sigma_PTm x) (s : PTm)
+{struct s} : ren_PTm xi_PTm s = subst_PTm sigma_PTm s :=
match s with
- | VarPTm _ s0 => Eq_PTm s0
- | PAbs _ s0 =>
+ | VarPTm s0 => Eq_PTm s0
+ | PAbs s0 =>
congr_PAbs
(rinst_inst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm sigma_PTm)
(rinstInst_up_PTm_PTm _ _ Eq_PTm) s0)
- | PApp _ s0 s1 =>
+ | PApp s0 s1 =>
congr_PApp (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s0)
(rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
- | PPair _ s0 s1 =>
+ | PPair s0 s1 =>
congr_PPair (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s0)
(rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
- | PProj _ s0 s1 =>
+ | PProj s0 s1 =>
congr_PProj (eq_refl s0) (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
- | PConst _ s0 => congr_PConst (eq_refl s0)
- | PUniv _ s0 => congr_PUniv (eq_refl s0)
- | PBot _ => congr_PBot
+ | PConst s0 => congr_PConst (eq_refl s0)
+ | PUniv s0 => congr_PUniv (eq_refl s0)
+ | PBot => congr_PBot
end.
-Lemma rinstInst'_PTm {m_PTm : nat} {n_PTm : nat}
- (xi_PTm : fin m_PTm -> fin n_PTm) (s : PTm m_PTm) :
- ren_PTm xi_PTm s = subst_PTm (funcomp (VarPTm n_PTm) xi_PTm) s.
+Lemma rinstInst'_PTm (xi_PTm : nat -> nat) (s : PTm) :
+ ren_PTm xi_PTm s = subst_PTm (funcomp (VarPTm) xi_PTm) s.
Proof.
exact (rinst_inst_PTm xi_PTm _ (fun n => eq_refl) s).
Qed.
-Lemma rinstInst'_PTm_pointwise {m_PTm : nat} {n_PTm : nat}
- (xi_PTm : fin m_PTm -> fin n_PTm) :
+Lemma rinstInst'_PTm_pointwise (xi_PTm : nat -> nat) :
pointwise_relation _ eq (ren_PTm xi_PTm)
- (subst_PTm (funcomp (VarPTm n_PTm) xi_PTm)).
+ (subst_PTm (funcomp (VarPTm) xi_PTm)).
Proof.
exact (fun s => rinst_inst_PTm xi_PTm _ (fun n => eq_refl) s).
Qed.
-Lemma instId'_PTm {m_PTm : nat} (s : PTm m_PTm) :
- subst_PTm (VarPTm m_PTm) s = s.
+Lemma instId'_PTm (s : PTm) : subst_PTm (VarPTm) s = s.
Proof.
-exact (idSubst_PTm (VarPTm m_PTm) (fun n => eq_refl) s).
+exact (idSubst_PTm (VarPTm) (fun n => eq_refl) s).
Qed.
-Lemma instId'_PTm_pointwise {m_PTm : nat} :
- pointwise_relation _ eq (subst_PTm (VarPTm m_PTm)) id.
+Lemma instId'_PTm_pointwise : pointwise_relation _ eq (subst_PTm (VarPTm)) id.
Proof.
-exact (fun s => idSubst_PTm (VarPTm m_PTm) (fun n => eq_refl) s).
+exact (fun s => idSubst_PTm (VarPTm) (fun n => eq_refl) s).
Qed.
-Lemma rinstId'_PTm {m_PTm : nat} (s : PTm m_PTm) : ren_PTm id s = s.
+Lemma rinstId'_PTm (s : PTm) : ren_PTm id s = s.
Proof.
exact (eq_ind_r (fun t => t = s) (instId'_PTm s) (rinstInst'_PTm id s)).
Qed.
-Lemma rinstId'_PTm_pointwise {m_PTm : nat} :
- pointwise_relation _ eq (@ren_PTm m_PTm m_PTm id) id.
+Lemma rinstId'_PTm_pointwise : pointwise_relation _ eq (@ren_PTm id) id.
Proof.
exact (fun s =>
eq_ind_r (fun t => t = s) (instId'_PTm s) (rinstInst'_PTm id s)).
Qed.
-Lemma varL'_PTm {m_PTm : nat} {n_PTm : nat}
- (sigma_PTm : fin m_PTm -> PTm n_PTm) (x : fin m_PTm) :
- subst_PTm sigma_PTm (VarPTm m_PTm x) = sigma_PTm x.
+Lemma varL'_PTm (sigma_PTm : nat -> PTm) (x : nat) :
+ subst_PTm sigma_PTm (VarPTm x) = sigma_PTm x.
Proof.
exact (eq_refl).
Qed.
-Lemma varL'_PTm_pointwise {m_PTm : nat} {n_PTm : nat}
- (sigma_PTm : fin m_PTm -> PTm n_PTm) :
- pointwise_relation _ eq (funcomp (subst_PTm sigma_PTm) (VarPTm m_PTm))
- sigma_PTm.
+Lemma varL'_PTm_pointwise (sigma_PTm : nat -> PTm) :
+ pointwise_relation _ eq (funcomp (subst_PTm sigma_PTm) (VarPTm)) sigma_PTm.
Proof.
exact (fun x => eq_refl).
Qed.
-Lemma varLRen'_PTm {m_PTm : nat} {n_PTm : nat}
- (xi_PTm : fin m_PTm -> fin n_PTm) (x : fin m_PTm) :
- ren_PTm xi_PTm (VarPTm m_PTm x) = VarPTm n_PTm (xi_PTm x).
+Lemma varLRen'_PTm (xi_PTm : nat -> nat) (x : nat) :
+ ren_PTm xi_PTm (VarPTm x) = VarPTm (xi_PTm x).
Proof.
exact (eq_refl).
Qed.
-Lemma varLRen'_PTm_pointwise {m_PTm : nat} {n_PTm : nat}
- (xi_PTm : fin m_PTm -> fin n_PTm) :
- pointwise_relation _ eq (funcomp (ren_PTm xi_PTm) (VarPTm m_PTm))
- (funcomp (VarPTm n_PTm) xi_PTm).
+Lemma varLRen'_PTm_pointwise (xi_PTm : nat -> nat) :
+ pointwise_relation _ eq (funcomp (ren_PTm xi_PTm) (VarPTm))
+ (funcomp (VarPTm) xi_PTm).
Proof.
exact (fun x => eq_refl).
Qed.
-Inductive Tm (n_Tm : nat) : Type :=
- | VarTm : fin n_Tm -> Tm n_Tm
- | Abs : Tm (S n_Tm) -> Tm n_Tm
- | 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
- | TBind : TTag -> Tm n_Tm -> Tm (S n_Tm) -> Tm n_Tm
- | Univ : nat -> 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.
+Inductive Tm : Type :=
+ | VarTm : nat -> Tm
+ | Abs : Tm -> Tm
+ | App : Tm -> Tm -> Tm
+ | Pair : Tm -> Tm -> Tm
+ | Proj : PTag -> Tm -> Tm
+ | TBind : TTag -> Tm -> Tm -> Tm
+ | Univ : nat -> Tm
+ | BVal : bool -> Tm
+ | Bool : Tm
+ | If : Tm -> Tm -> Tm -> 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.
+Lemma congr_Abs {s0 : Tm} {t0 : Tm} (H0 : s0 = t0) : Abs s0 = Abs t0.
Proof.
-exact (eq_trans eq_refl (ap (fun x => Abs m_Tm x) H0)).
+exact (eq_trans eq_refl (ap (fun x => Abs x) H0)).
Qed.
-Lemma congr_App {m_Tm : nat} {s0 : Tm m_Tm} {s1 : Tm m_Tm} {t0 : Tm m_Tm}
- {t1 : Tm m_Tm} (H0 : s0 = t0) (H1 : s1 = t1) :
- App m_Tm s0 s1 = App m_Tm t0 t1.
+Lemma congr_App {s0 : Tm} {s1 : Tm} {t0 : Tm} {t1 : Tm} (H0 : s0 = t0)
+ (H1 : s1 = t1) : App s0 s1 = App t0 t1.
Proof.
-exact (eq_trans (eq_trans eq_refl (ap (fun x => App m_Tm x s1) H0))
- (ap (fun x => App m_Tm t0 x) H1)).
+exact (eq_trans (eq_trans eq_refl (ap (fun x => App x s1) H0))
+ (ap (fun x => App t0 x) H1)).
Qed.
-Lemma congr_Pair {m_Tm : nat} {s0 : Tm m_Tm} {s1 : Tm m_Tm} {t0 : Tm m_Tm}
- {t1 : Tm m_Tm} (H0 : s0 = t0) (H1 : s1 = t1) :
- Pair m_Tm s0 s1 = Pair m_Tm t0 t1.
+Lemma congr_Pair {s0 : Tm} {s1 : Tm} {t0 : Tm} {t1 : Tm} (H0 : s0 = t0)
+ (H1 : s1 = t1) : Pair s0 s1 = Pair t0 t1.
Proof.
-exact (eq_trans (eq_trans eq_refl (ap (fun x => Pair m_Tm x s1) H0))
- (ap (fun x => Pair m_Tm t0 x) H1)).
+exact (eq_trans (eq_trans eq_refl (ap (fun x => Pair x s1) H0))
+ (ap (fun x => Pair t0 x) H1)).
Qed.
-Lemma congr_Proj {m_Tm : nat} {s0 : PTag} {s1 : Tm m_Tm} {t0 : PTag}
- {t1 : Tm m_Tm} (H0 : s0 = t0) (H1 : s1 = t1) :
- Proj m_Tm s0 s1 = Proj m_Tm t0 t1.
+Lemma congr_Proj {s0 : PTag} {s1 : Tm} {t0 : PTag} {t1 : Tm} (H0 : s0 = t0)
+ (H1 : s1 = t1) : Proj s0 s1 = Proj t0 t1.
Proof.
-exact (eq_trans (eq_trans eq_refl (ap (fun x => Proj m_Tm x s1) H0))
- (ap (fun x => Proj m_Tm t0 x) H1)).
+exact (eq_trans (eq_trans eq_refl (ap (fun x => Proj x s1) H0))
+ (ap (fun x => Proj t0 x) H1)).
Qed.
-Lemma congr_TBind {m_Tm : nat} {s0 : TTag} {s1 : Tm m_Tm} {s2 : Tm (S m_Tm)}
- {t0 : TTag} {t1 : Tm m_Tm} {t2 : Tm (S m_Tm)} (H0 : s0 = t0) (H1 : s1 = t1)
- (H2 : s2 = t2) : TBind m_Tm s0 s1 s2 = TBind m_Tm t0 t1 t2.
+Lemma congr_TBind {s0 : TTag} {s1 : Tm} {s2 : Tm} {t0 : TTag} {t1 : Tm}
+ {t2 : Tm} (H0 : s0 = t0) (H1 : s1 = t1) (H2 : s2 = t2) :
+ TBind s0 s1 s2 = TBind t0 t1 t2.
Proof.
exact (eq_trans
- (eq_trans (eq_trans eq_refl (ap (fun x => TBind m_Tm x s1 s2) H0))
- (ap (fun x => TBind m_Tm t0 x s2) H1))
- (ap (fun x => TBind m_Tm t0 t1 x) H2)).
+ (eq_trans (eq_trans eq_refl (ap (fun x => TBind x s1 s2) H0))
+ (ap (fun x => TBind t0 x s2) H1))
+ (ap (fun x => TBind t0 t1 x) H2)).
Qed.
-Lemma congr_Univ {m_Tm : nat} {s0 : nat} {t0 : nat} (H0 : s0 = t0) :
- Univ m_Tm s0 = Univ m_Tm t0.
+Lemma congr_Univ {s0 : nat} {t0 : nat} (H0 : s0 = t0) : Univ s0 = Univ t0.
Proof.
-exact (eq_trans eq_refl (ap (fun x => Univ m_Tm x) H0)).
+exact (eq_trans eq_refl (ap (fun x => Univ x) H0)).
Qed.
-Lemma congr_BVal {m_Tm : nat} {s0 : bool} {t0 : bool} (H0 : s0 = t0) :
- BVal m_Tm s0 = BVal m_Tm t0.
+Lemma congr_BVal {s0 : bool} {t0 : bool} (H0 : s0 = t0) : BVal s0 = BVal t0.
Proof.
-exact (eq_trans eq_refl (ap (fun x => BVal m_Tm x) H0)).
+exact (eq_trans eq_refl (ap (fun x => BVal x) H0)).
Qed.
-Lemma congr_Bool {m_Tm : nat} : Bool m_Tm = Bool m_Tm.
+Lemma congr_Bool : Bool = Bool.
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.
+Lemma congr_If {s0 : Tm} {s1 : Tm} {s2 : Tm} {t0 : Tm} {t1 : Tm} {t2 : Tm}
+ (H0 : s0 = t0) (H1 : s1 = t1) (H2 : s2 = t2) : If s0 s1 s2 = If 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)).
+ (eq_trans (eq_trans eq_refl (ap (fun x => If x s1 s2) H0))
+ (ap (fun x => If t0 x s2) H1)) (ap (fun x => If t0 t1 x) H2)).
Qed.
-Lemma upRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n) :
- fin (S m) -> fin (S n).
+Lemma upRen_Tm_Tm (xi : nat -> nat) : nat -> nat.
Proof.
exact (up_ren xi).
Defined.
-Lemma upRen_list_Tm_Tm (p : nat) {m : nat} {n : nat} (xi : fin m -> fin n) :
- fin (plus p m) -> fin (plus p n).
-Proof.
-exact (upRen_p p xi).
-Defined.
-
-Fixpoint ren_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
-(s : Tm m_Tm) {struct s} : Tm n_Tm :=
+Fixpoint ren_Tm (xi_Tm : nat -> nat) (s : Tm) {struct s} : Tm :=
match s with
- | VarTm _ s0 => VarTm n_Tm (xi_Tm s0)
- | Abs _ s0 => Abs n_Tm (ren_Tm (upRen_Tm_Tm xi_Tm) s0)
- | App _ s0 s1 => App n_Tm (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
- | 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)
- | TBind _ s0 s1 s2 =>
- TBind n_Tm s0 (ren_Tm xi_Tm s1) (ren_Tm (upRen_Tm_Tm xi_Tm) s2)
- | Univ _ s0 => Univ n_Tm s0
- | 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)
+ | VarTm s0 => VarTm (xi_Tm s0)
+ | Abs s0 => Abs (ren_Tm (upRen_Tm_Tm xi_Tm) s0)
+ | App s0 s1 => App (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
+ | Pair s0 s1 => Pair (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
+ | Proj s0 s1 => Proj s0 (ren_Tm xi_Tm s1)
+ | TBind s0 s1 s2 =>
+ TBind s0 (ren_Tm xi_Tm s1) (ren_Tm (upRen_Tm_Tm xi_Tm) s2)
+ | Univ s0 => Univ s0
+ | BVal s0 => BVal s0
+ | Bool => Bool
+ | If s0 s1 s2 => If (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) :
- fin (S m) -> Tm (S n_Tm).
+Lemma up_Tm_Tm (sigma : nat -> Tm) : nat -> Tm.
Proof.
-exact (scons (VarTm (S n_Tm) var_zero) (funcomp (ren_Tm shift) sigma)).
+exact (scons (VarTm var_zero) (funcomp (ren_Tm shift) sigma)).
Defined.
-Lemma up_list_Tm_Tm (p : nat) {m : nat} {n_Tm : nat}
- (sigma : fin m -> Tm n_Tm) : fin (plus p m) -> Tm (plus p n_Tm).
-Proof.
-exact (scons_p p (funcomp (VarTm (plus p n_Tm)) (zero_p p))
- (funcomp (ren_Tm (shift_p p)) sigma)).
-Defined.
-
-Fixpoint subst_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
-(s : Tm m_Tm) {struct s} : Tm n_Tm :=
+Fixpoint subst_Tm (sigma_Tm : nat -> Tm) (s : Tm) {struct s} : Tm :=
match s with
- | VarTm _ s0 => sigma_Tm s0
- | Abs _ s0 => Abs n_Tm (subst_Tm (up_Tm_Tm sigma_Tm) s0)
- | App _ s0 s1 => App n_Tm (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
- | Pair _ s0 s1 => Pair n_Tm (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
- | Proj _ s0 s1 => Proj n_Tm s0 (subst_Tm sigma_Tm s1)
- | TBind _ s0 s1 s2 =>
- TBind n_Tm s0 (subst_Tm sigma_Tm s1) (subst_Tm (up_Tm_Tm sigma_Tm) s2)
- | Univ _ s0 => Univ n_Tm s0
- | 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)
+ | VarTm s0 => sigma_Tm s0
+ | Abs s0 => Abs (subst_Tm (up_Tm_Tm sigma_Tm) s0)
+ | App s0 s1 => App (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
+ | Pair s0 s1 => Pair (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
+ | Proj s0 s1 => Proj s0 (subst_Tm sigma_Tm s1)
+ | TBind s0 s1 s2 =>
+ TBind s0 (subst_Tm sigma_Tm s1) (subst_Tm (up_Tm_Tm sigma_Tm) s2)
+ | Univ s0 => Univ s0
+ | BVal s0 => BVal s0
+ | Bool => Bool
+ | If s0 s1 s2 =>
+ If (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)
- (Eq : forall x, sigma x = VarTm m_Tm x) :
- forall x, up_Tm_Tm sigma x = VarTm (S m_Tm) x.
+Lemma upId_Tm_Tm (sigma : nat -> Tm) (Eq : forall x, sigma x = VarTm x) :
+ forall x, up_Tm_Tm sigma x = VarTm x.
Proof.
exact (fun n =>
match n with
- | Some fin_n => ap (ren_Tm shift) (Eq fin_n)
- | None => eq_refl
+ | S n' => ap (ren_Tm shift) (Eq n')
+ | O => eq_refl
end).
Qed.
-Lemma upId_list_Tm_Tm {p : nat} {m_Tm : nat} (sigma : fin m_Tm -> Tm m_Tm)
- (Eq : forall x, sigma x = VarTm m_Tm x) :
- forall x, up_list_Tm_Tm p sigma x = VarTm (plus p m_Tm) x.
-Proof.
-exact (fun n =>
- scons_p_eta (VarTm (plus p m_Tm))
- (fun n => ap (ren_Tm (shift_p p)) (Eq n)) (fun n => eq_refl)).
-Qed.
-
-Fixpoint idSubst_Tm {m_Tm : nat} (sigma_Tm : fin m_Tm -> Tm m_Tm)
-(Eq_Tm : forall x, sigma_Tm x = VarTm m_Tm x) (s : Tm m_Tm) {struct s} :
+Fixpoint idSubst_Tm (sigma_Tm : nat -> Tm)
+(Eq_Tm : forall x, sigma_Tm x = VarTm x) (s : Tm) {struct s} :
subst_Tm sigma_Tm s = s :=
match s with
- | VarTm _ s0 => Eq_Tm s0
- | Abs _ s0 =>
+ | VarTm s0 => Eq_Tm s0
+ | Abs s0 =>
congr_Abs (idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s0)
- | App _ s0 s1 =>
+ | App s0 s1 =>
congr_App (idSubst_Tm sigma_Tm Eq_Tm s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
- | Pair _ s0 s1 =>
+ | Pair s0 s1 =>
congr_Pair (idSubst_Tm sigma_Tm Eq_Tm s0)
(idSubst_Tm sigma_Tm Eq_Tm s1)
- | Proj _ s0 s1 => congr_Proj (eq_refl s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
- | TBind _ s0 s1 s2 =>
+ | Proj s0 s1 => congr_Proj (eq_refl s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
+ | TBind s0 s1 s2 =>
congr_TBind (eq_refl s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
(idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s2)
- | Univ _ s0 => congr_Univ (eq_refl s0)
- | BVal _ s0 => congr_BVal (eq_refl s0)
- | Bool _ => congr_Bool
- | If _ s0 s1 s2 =>
+ | Univ s0 => congr_Univ (eq_refl s0)
+ | 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)
- (zeta : fin m -> fin n) (Eq : forall x, xi x = zeta x) :
+Lemma upExtRen_Tm_Tm (xi : nat -> nat) (zeta : nat -> nat)
+ (Eq : forall x, xi x = zeta x) :
forall x, upRen_Tm_Tm xi x = upRen_Tm_Tm zeta x.
Proof.
-exact (fun n =>
- match n with
- | Some fin_n => ap shift (Eq fin_n)
- | None => eq_refl
- end).
+exact (fun n => match n with
+ | S n' => ap shift (Eq n')
+ | O => eq_refl
+ end).
Qed.
-Lemma upExtRen_list_Tm_Tm {p : nat} {m : nat} {n : nat} (xi : fin m -> fin n)
- (zeta : fin m -> fin n) (Eq : forall x, xi x = zeta x) :
- forall x, upRen_list_Tm_Tm p xi x = upRen_list_Tm_Tm p zeta x.
-Proof.
-exact (fun n =>
- scons_p_congr (fun n => eq_refl) (fun n => ap (shift_p p) (Eq n))).
-Qed.
-
-Fixpoint extRen_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
-(zeta_Tm : fin m_Tm -> fin n_Tm) (Eq_Tm : forall x, xi_Tm x = zeta_Tm x)
-(s : Tm m_Tm) {struct s} : ren_Tm xi_Tm s = ren_Tm zeta_Tm s :=
+Fixpoint extRen_Tm (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat)
+(Eq_Tm : forall x, xi_Tm x = zeta_Tm x) (s : Tm) {struct s} :
+ren_Tm xi_Tm s = ren_Tm zeta_Tm s :=
match s with
- | VarTm _ s0 => ap (VarTm n_Tm) (Eq_Tm s0)
- | Abs _ s0 =>
+ | VarTm s0 => ap (VarTm) (Eq_Tm s0)
+ | Abs s0 =>
congr_Abs
(extRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upExtRen_Tm_Tm _ _ Eq_Tm) s0)
- | App _ s0 s1 =>
+ | App s0 s1 =>
congr_App (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
(extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
- | Pair _ s0 s1 =>
+ | Pair s0 s1 =>
congr_Pair (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
(extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
- | Proj _ s0 s1 =>
- congr_Proj (eq_refl s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
- | TBind _ s0 s1 s2 =>
+ | Proj s0 s1 => congr_Proj (eq_refl s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
+ | TBind s0 s1 s2 =>
congr_TBind (eq_refl s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
(extRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upExtRen_Tm_Tm _ _ Eq_Tm) s2)
- | Univ _ s0 => congr_Univ (eq_refl s0)
- | BVal _ s0 => congr_BVal (eq_refl s0)
- | Bool _ => congr_Bool
- | If _ s0 s1 s2 =>
+ | Univ s0 => congr_Univ (eq_refl s0)
+ | 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)
- (tau : fin m -> Tm n_Tm) (Eq : forall x, sigma x = tau x) :
+Lemma upExt_Tm_Tm (sigma : nat -> Tm) (tau : nat -> Tm)
+ (Eq : forall x, sigma x = tau x) :
forall x, up_Tm_Tm sigma x = up_Tm_Tm tau x.
Proof.
exact (fun n =>
match n with
- | Some fin_n => ap (ren_Tm shift) (Eq fin_n)
- | None => eq_refl
+ | S n' => ap (ren_Tm shift) (Eq n')
+ | O => eq_refl
end).
Qed.
-Lemma upExt_list_Tm_Tm {p : nat} {m : nat} {n_Tm : nat}
- (sigma : fin m -> Tm n_Tm) (tau : fin m -> Tm n_Tm)
- (Eq : forall x, sigma x = tau x) :
- forall x, up_list_Tm_Tm p sigma x = up_list_Tm_Tm p tau x.
-Proof.
-exact (fun n =>
- scons_p_congr (fun n => eq_refl)
- (fun n => ap (ren_Tm (shift_p p)) (Eq n))).
-Qed.
-
-Fixpoint ext_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
-(tau_Tm : fin m_Tm -> Tm n_Tm) (Eq_Tm : forall x, sigma_Tm x = tau_Tm x)
-(s : Tm m_Tm) {struct s} : subst_Tm sigma_Tm s = subst_Tm tau_Tm s :=
+Fixpoint ext_Tm (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm)
+(Eq_Tm : forall x, sigma_Tm x = tau_Tm x) (s : Tm) {struct s} :
+subst_Tm sigma_Tm s = subst_Tm tau_Tm s :=
match s with
- | VarTm _ s0 => Eq_Tm s0
- | Abs _ s0 =>
+ | VarTm s0 => Eq_Tm s0
+ | Abs s0 =>
congr_Abs
(ext_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm) (upExt_Tm_Tm _ _ Eq_Tm)
s0)
- | App _ s0 s1 =>
+ | App s0 s1 =>
congr_App (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
(ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
- | Pair _ s0 s1 =>
+ | Pair s0 s1 =>
congr_Pair (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
(ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
- | Proj _ s0 s1 => congr_Proj (eq_refl s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
- | TBind _ s0 s1 s2 =>
+ | Proj s0 s1 => congr_Proj (eq_refl s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
+ | TBind s0 s1 s2 =>
congr_TBind (eq_refl s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
(ext_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm) (upExt_Tm_Tm _ _ Eq_Tm)
s2)
- | Univ _ s0 => congr_Univ (eq_refl s0)
- | BVal _ s0 => congr_BVal (eq_refl s0)
- | Bool _ => congr_Bool
- | If _ s0 s1 s2 =>
+ | Univ s0 => congr_Univ (eq_refl s0)
+ | 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)
- (zeta : fin l -> fin m) (rho : fin k -> fin m)
- (Eq : forall x, funcomp zeta xi x = rho x) :
+Lemma up_ren_ren_Tm_Tm (xi : nat -> nat) (zeta : nat -> nat)
+ (rho : nat -> nat) (Eq : forall x, funcomp zeta xi x = rho x) :
forall x, funcomp (upRen_Tm_Tm zeta) (upRen_Tm_Tm xi) x = upRen_Tm_Tm rho x.
Proof.
exact (up_ren_ren xi zeta rho Eq).
Qed.
-Lemma up_ren_ren_list_Tm_Tm {p : nat} {k : nat} {l : nat} {m : nat}
- (xi : fin k -> fin l) (zeta : fin l -> fin m) (rho : fin k -> fin m)
- (Eq : forall x, funcomp zeta xi x = rho x) :
- forall x,
- funcomp (upRen_list_Tm_Tm p zeta) (upRen_list_Tm_Tm p xi) x =
- upRen_list_Tm_Tm p rho x.
-Proof.
-exact (up_ren_ren_p Eq).
-Qed.
-
-Fixpoint compRenRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
-(xi_Tm : fin m_Tm -> fin k_Tm) (zeta_Tm : fin k_Tm -> fin l_Tm)
-(rho_Tm : fin m_Tm -> fin l_Tm)
-(Eq_Tm : forall x, funcomp zeta_Tm xi_Tm x = rho_Tm x) (s : Tm m_Tm) {struct
- s} : ren_Tm zeta_Tm (ren_Tm xi_Tm s) = ren_Tm rho_Tm s :=
+Fixpoint compRenRen_Tm (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat)
+(rho_Tm : nat -> nat) (Eq_Tm : forall x, funcomp zeta_Tm xi_Tm x = rho_Tm x)
+(s : Tm) {struct s} : ren_Tm zeta_Tm (ren_Tm xi_Tm s) = ren_Tm rho_Tm s :=
match s with
- | VarTm _ s0 => ap (VarTm l_Tm) (Eq_Tm s0)
- | Abs _ s0 =>
+ | VarTm s0 => ap (VarTm) (Eq_Tm s0)
+ | Abs s0 =>
congr_Abs
(compRenRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s0)
- | App _ s0 s1 =>
+ | App s0 s1 =>
congr_App (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
(compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
- | Pair _ s0 s1 =>
+ | Pair s0 s1 =>
congr_Pair (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
(compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
- | Proj _ s0 s1 =>
+ | Proj s0 s1 =>
congr_Proj (eq_refl s0) (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
- | TBind _ s0 s1 s2 =>
+ | TBind s0 s1 s2 =>
congr_TBind (eq_refl s0) (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
(compRenRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s2)
- | Univ _ s0 => congr_Univ (eq_refl s0)
- | BVal _ s0 => congr_BVal (eq_refl s0)
- | Bool _ => congr_Bool
- | If _ s0 s1 s2 =>
+ | Univ s0 => congr_Univ (eq_refl s0)
+ | 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}
- (xi : fin k -> fin l) (tau : fin l -> Tm m_Tm) (theta : fin k -> Tm m_Tm)
- (Eq : forall x, funcomp tau xi x = theta x) :
+Lemma up_ren_subst_Tm_Tm (xi : nat -> nat) (tau : nat -> Tm)
+ (theta : nat -> Tm) (Eq : forall x, funcomp tau xi x = theta x) :
forall x, funcomp (up_Tm_Tm tau) (upRen_Tm_Tm xi) x = up_Tm_Tm theta x.
Proof.
exact (fun n =>
match n with
- | Some fin_n => ap (ren_Tm shift) (Eq fin_n)
- | None => eq_refl
+ | S n' => ap (ren_Tm shift) (Eq n')
+ | O => eq_refl
end).
Qed.
-Lemma up_ren_subst_list_Tm_Tm {p : nat} {k : nat} {l : nat} {m_Tm : nat}
- (xi : fin k -> fin l) (tau : fin l -> Tm m_Tm) (theta : fin k -> Tm m_Tm)
- (Eq : forall x, funcomp tau xi x = theta x) :
- forall x,
- funcomp (up_list_Tm_Tm p tau) (upRen_list_Tm_Tm p xi) x =
- up_list_Tm_Tm p theta x.
-Proof.
-exact (fun n =>
- eq_trans (scons_p_comp' _ _ _ n)
- (scons_p_congr (fun z => scons_p_head' _ _ z)
- (fun z =>
- eq_trans (scons_p_tail' _ _ (xi z))
- (ap (ren_Tm (shift_p p)) (Eq z))))).
-Qed.
-
-Fixpoint compRenSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
-(xi_Tm : fin m_Tm -> fin k_Tm) (tau_Tm : fin k_Tm -> Tm l_Tm)
-(theta_Tm : fin m_Tm -> Tm l_Tm)
-(Eq_Tm : forall x, funcomp tau_Tm xi_Tm x = theta_Tm x) (s : Tm m_Tm) {struct
- s} : subst_Tm tau_Tm (ren_Tm xi_Tm s) = subst_Tm theta_Tm s :=
+Fixpoint compRenSubst_Tm (xi_Tm : nat -> nat) (tau_Tm : nat -> Tm)
+(theta_Tm : nat -> Tm)
+(Eq_Tm : forall x, funcomp tau_Tm xi_Tm x = theta_Tm x) (s : Tm) {struct s} :
+subst_Tm tau_Tm (ren_Tm xi_Tm s) = subst_Tm theta_Tm s :=
match s with
- | VarTm _ s0 => Eq_Tm s0
- | Abs _ s0 =>
+ | VarTm s0 => Eq_Tm s0
+ | Abs s0 =>
congr_Abs
(compRenSubst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm tau_Tm)
(up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s0)
- | App _ s0 s1 =>
+ | App s0 s1 =>
congr_App (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
- | Pair _ s0 s1 =>
+ | Pair s0 s1 =>
congr_Pair (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
- | Proj _ s0 s1 =>
+ | Proj s0 s1 =>
congr_Proj (eq_refl s0)
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
- | TBind _ s0 s1 s2 =>
+ | TBind s0 s1 s2 =>
congr_TBind (eq_refl s0)
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
(compRenSubst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm tau_Tm)
(up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s2)
- | Univ _ s0 => congr_Univ (eq_refl s0)
- | BVal _ s0 => congr_BVal (eq_refl s0)
- | Bool _ => congr_Bool
- | If _ s0 s1 s2 =>
+ | Univ s0 => congr_Univ (eq_refl s0)
+ | 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}
- (sigma : fin k -> Tm l_Tm) (zeta_Tm : fin l_Tm -> fin m_Tm)
- (theta : fin k -> Tm m_Tm)
+Lemma up_subst_ren_Tm_Tm (sigma : nat -> Tm) (zeta_Tm : nat -> nat)
+ (theta : nat -> Tm)
(Eq : forall x, funcomp (ren_Tm zeta_Tm) sigma x = theta x) :
forall x,
funcomp (ren_Tm (upRen_Tm_Tm zeta_Tm)) (up_Tm_Tm sigma) x =
@@ -1102,341 +840,257 @@ Lemma up_subst_ren_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
Proof.
exact (fun n =>
match n with
- | Some fin_n =>
+ | S n' =>
eq_trans
(compRenRen_Tm shift (upRen_Tm_Tm zeta_Tm)
- (funcomp shift zeta_Tm) (fun x => eq_refl) (sigma fin_n))
+ (funcomp shift zeta_Tm) (fun x => eq_refl) (sigma n'))
(eq_trans
(eq_sym
(compRenRen_Tm zeta_Tm shift (funcomp shift zeta_Tm)
- (fun x => eq_refl) (sigma fin_n)))
- (ap (ren_Tm shift) (Eq fin_n)))
- | None => eq_refl
+ (fun x => eq_refl) (sigma n')))
+ (ap (ren_Tm shift) (Eq n')))
+ | O => eq_refl
end).
Qed.
-Lemma up_subst_ren_list_Tm_Tm {p : nat} {k : nat} {l_Tm : nat} {m_Tm : nat}
- (sigma : fin k -> Tm l_Tm) (zeta_Tm : fin l_Tm -> fin m_Tm)
- (theta : fin k -> Tm m_Tm)
- (Eq : forall x, funcomp (ren_Tm zeta_Tm) sigma x = theta x) :
- forall x,
- funcomp (ren_Tm (upRen_list_Tm_Tm p zeta_Tm)) (up_list_Tm_Tm p sigma) x =
- up_list_Tm_Tm p theta x.
-Proof.
-exact (fun n =>
- eq_trans (scons_p_comp' _ _ _ n)
- (scons_p_congr
- (fun x => ap (VarTm (plus p m_Tm)) (scons_p_head' _ _ x))
- (fun n =>
- eq_trans
- (compRenRen_Tm (shift_p p) (upRen_list_Tm_Tm p zeta_Tm)
- (funcomp (shift_p p) zeta_Tm)
- (fun x => scons_p_tail' _ _ x) (sigma n))
- (eq_trans
- (eq_sym
- (compRenRen_Tm zeta_Tm (shift_p p)
- (funcomp (shift_p p) zeta_Tm) (fun x => eq_refl)
- (sigma n))) (ap (ren_Tm (shift_p p)) (Eq n)))))).
-Qed.
-
-Fixpoint compSubstRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
-(sigma_Tm : fin m_Tm -> Tm k_Tm) (zeta_Tm : fin k_Tm -> fin l_Tm)
-(theta_Tm : fin m_Tm -> Tm l_Tm)
-(Eq_Tm : forall x, funcomp (ren_Tm zeta_Tm) sigma_Tm x = theta_Tm x)
-(s : Tm m_Tm) {struct s} :
+Fixpoint compSubstRen_Tm (sigma_Tm : nat -> Tm) (zeta_Tm : nat -> nat)
+(theta_Tm : nat -> Tm)
+(Eq_Tm : forall x, funcomp (ren_Tm zeta_Tm) sigma_Tm x = theta_Tm x)
+(s : Tm) {struct s} :
ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
match s with
- | VarTm _ s0 => Eq_Tm s0
- | Abs _ s0 =>
+ | VarTm s0 => Eq_Tm s0
+ | Abs s0 =>
congr_Abs
(compSubstRen_Tm (up_Tm_Tm sigma_Tm) (upRen_Tm_Tm zeta_Tm)
(up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s0)
- | App _ s0 s1 =>
+ | App s0 s1 =>
congr_App (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
- | Pair _ s0 s1 =>
+ | Pair s0 s1 =>
congr_Pair (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
- | Proj _ s0 s1 =>
+ | Proj s0 s1 =>
congr_Proj (eq_refl s0)
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
- | TBind _ s0 s1 s2 =>
+ | TBind s0 s1 s2 =>
congr_TBind (eq_refl s0)
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
(compSubstRen_Tm (up_Tm_Tm sigma_Tm) (upRen_Tm_Tm zeta_Tm)
(up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s2)
- | Univ _ s0 => congr_Univ (eq_refl s0)
- | BVal _ s0 => congr_BVal (eq_refl s0)
- | Bool _ => congr_Bool
- | If _ s0 s1 s2 =>
+ | Univ s0 => congr_Univ (eq_refl s0)
+ | 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}
- (sigma : fin k -> Tm l_Tm) (tau_Tm : fin l_Tm -> Tm m_Tm)
- (theta : fin k -> Tm m_Tm)
+Lemma up_subst_subst_Tm_Tm (sigma : nat -> Tm) (tau_Tm : nat -> Tm)
+ (theta : nat -> Tm)
(Eq : forall x, funcomp (subst_Tm tau_Tm) sigma x = theta x) :
forall x,
funcomp (subst_Tm (up_Tm_Tm tau_Tm)) (up_Tm_Tm sigma) x = up_Tm_Tm theta x.
Proof.
exact (fun n =>
match n with
- | Some fin_n =>
+ | S n' =>
eq_trans
(compRenSubst_Tm shift (up_Tm_Tm tau_Tm)
(funcomp (up_Tm_Tm tau_Tm) shift) (fun x => eq_refl)
- (sigma fin_n))
+ (sigma n'))
(eq_trans
(eq_sym
(compSubstRen_Tm tau_Tm shift
(funcomp (ren_Tm shift) tau_Tm) (fun x => eq_refl)
- (sigma fin_n))) (ap (ren_Tm shift) (Eq fin_n)))
- | None => eq_refl
+ (sigma n'))) (ap (ren_Tm shift) (Eq n')))
+ | O => eq_refl
end).
Qed.
-Lemma up_subst_subst_list_Tm_Tm {p : nat} {k : nat} {l_Tm : nat} {m_Tm : nat}
- (sigma : fin k -> Tm l_Tm) (tau_Tm : fin l_Tm -> Tm m_Tm)
- (theta : fin k -> Tm m_Tm)
- (Eq : forall x, funcomp (subst_Tm tau_Tm) sigma x = theta x) :
- forall x,
- funcomp (subst_Tm (up_list_Tm_Tm p tau_Tm)) (up_list_Tm_Tm p sigma) x =
- up_list_Tm_Tm p theta x.
-Proof.
-exact (fun n =>
- eq_trans
- (scons_p_comp' (funcomp (VarTm (plus p l_Tm)) (zero_p p)) _ _ n)
- (scons_p_congr
- (fun x => scons_p_head' _ (fun z => ren_Tm (shift_p p) _) x)
- (fun n =>
- eq_trans
- (compRenSubst_Tm (shift_p p) (up_list_Tm_Tm p tau_Tm)
- (funcomp (up_list_Tm_Tm p tau_Tm) (shift_p p))
- (fun x => eq_refl) (sigma n))
- (eq_trans
- (eq_sym
- (compSubstRen_Tm tau_Tm (shift_p p) _
- (fun x => eq_sym (scons_p_tail' _ _ x)) (sigma n)))
- (ap (ren_Tm (shift_p p)) (Eq n)))))).
-Qed.
-
-Fixpoint compSubstSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
-(sigma_Tm : fin m_Tm -> Tm k_Tm) (tau_Tm : fin k_Tm -> Tm l_Tm)
-(theta_Tm : fin m_Tm -> Tm l_Tm)
+Fixpoint compSubstSubst_Tm (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm)
+(theta_Tm : nat -> Tm)
(Eq_Tm : forall x, funcomp (subst_Tm tau_Tm) sigma_Tm x = theta_Tm x)
-(s : Tm m_Tm) {struct s} :
+(s : Tm) {struct s} :
subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
match s with
- | VarTm _ s0 => Eq_Tm s0
- | Abs _ s0 =>
+ | VarTm s0 => Eq_Tm s0
+ | Abs s0 =>
congr_Abs
(compSubstSubst_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm)
(up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s0)
- | App _ s0 s1 =>
+ | App s0 s1 =>
congr_App (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
- | Pair _ s0 s1 =>
+ | Pair s0 s1 =>
congr_Pair (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
- | Proj _ s0 s1 =>
+ | Proj s0 s1 =>
congr_Proj (eq_refl s0)
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
- | TBind _ s0 s1 s2 =>
+ | TBind s0 s1 s2 =>
congr_TBind (eq_refl s0)
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
(compSubstSubst_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm)
(up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s2)
- | Univ _ s0 => congr_Univ (eq_refl s0)
- | BVal _ s0 => congr_BVal (eq_refl s0)
- | Bool _ => congr_Bool
- | If _ s0 s1 s2 =>
+ | Univ s0 => congr_Univ (eq_refl s0)
+ | 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}
- (xi_Tm : fin m_Tm -> fin k_Tm) (zeta_Tm : fin k_Tm -> fin l_Tm)
- (s : Tm m_Tm) :
+Lemma renRen_Tm (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat) (s : Tm) :
ren_Tm zeta_Tm (ren_Tm xi_Tm s) = ren_Tm (funcomp zeta_Tm xi_Tm) s.
Proof.
exact (compRenRen_Tm xi_Tm zeta_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma renRen'_Tm_pointwise {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
- (xi_Tm : fin m_Tm -> fin k_Tm) (zeta_Tm : fin k_Tm -> fin l_Tm) :
+Lemma renRen'_Tm_pointwise (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat) :
pointwise_relation _ eq (funcomp (ren_Tm zeta_Tm) (ren_Tm xi_Tm))
(ren_Tm (funcomp zeta_Tm xi_Tm)).
Proof.
exact (fun s => compRenRen_Tm xi_Tm zeta_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma renSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
- (xi_Tm : fin m_Tm -> fin k_Tm) (tau_Tm : fin k_Tm -> Tm l_Tm) (s : Tm m_Tm)
- : subst_Tm tau_Tm (ren_Tm xi_Tm s) = subst_Tm (funcomp tau_Tm xi_Tm) s.
+Lemma renSubst_Tm (xi_Tm : nat -> nat) (tau_Tm : nat -> Tm) (s : Tm) :
+ subst_Tm tau_Tm (ren_Tm xi_Tm s) = subst_Tm (funcomp tau_Tm xi_Tm) s.
Proof.
exact (compRenSubst_Tm xi_Tm tau_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma renSubst_Tm_pointwise {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
- (xi_Tm : fin m_Tm -> fin k_Tm) (tau_Tm : fin k_Tm -> Tm l_Tm) :
+Lemma renSubst_Tm_pointwise (xi_Tm : nat -> nat) (tau_Tm : nat -> Tm) :
pointwise_relation _ eq (funcomp (subst_Tm tau_Tm) (ren_Tm xi_Tm))
(subst_Tm (funcomp tau_Tm xi_Tm)).
Proof.
exact (fun s => compRenSubst_Tm xi_Tm tau_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma substRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
- (sigma_Tm : fin m_Tm -> Tm k_Tm) (zeta_Tm : fin k_Tm -> fin l_Tm)
- (s : Tm m_Tm) :
+Lemma substRen_Tm (sigma_Tm : nat -> Tm) (zeta_Tm : nat -> nat) (s : Tm) :
ren_Tm zeta_Tm (subst_Tm sigma_Tm s) =
subst_Tm (funcomp (ren_Tm zeta_Tm) sigma_Tm) s.
Proof.
exact (compSubstRen_Tm sigma_Tm zeta_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma substRen_Tm_pointwise {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
- (sigma_Tm : fin m_Tm -> Tm k_Tm) (zeta_Tm : fin k_Tm -> fin l_Tm) :
+Lemma substRen_Tm_pointwise (sigma_Tm : nat -> Tm) (zeta_Tm : nat -> nat) :
pointwise_relation _ eq (funcomp (ren_Tm zeta_Tm) (subst_Tm sigma_Tm))
(subst_Tm (funcomp (ren_Tm zeta_Tm) sigma_Tm)).
Proof.
exact (fun s => compSubstRen_Tm sigma_Tm zeta_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma substSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
- (sigma_Tm : fin m_Tm -> Tm k_Tm) (tau_Tm : fin k_Tm -> Tm l_Tm)
- (s : Tm m_Tm) :
+Lemma substSubst_Tm (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm) (s : Tm) :
subst_Tm tau_Tm (subst_Tm sigma_Tm s) =
subst_Tm (funcomp (subst_Tm tau_Tm) sigma_Tm) s.
Proof.
exact (compSubstSubst_Tm sigma_Tm tau_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma substSubst_Tm_pointwise {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
- (sigma_Tm : fin m_Tm -> Tm k_Tm) (tau_Tm : fin k_Tm -> Tm l_Tm) :
+Lemma substSubst_Tm_pointwise (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm) :
pointwise_relation _ eq (funcomp (subst_Tm tau_Tm) (subst_Tm sigma_Tm))
(subst_Tm (funcomp (subst_Tm tau_Tm) sigma_Tm)).
Proof.
exact (fun s => compSubstSubst_Tm sigma_Tm tau_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma rinstInst_up_Tm_Tm {m : nat} {n_Tm : nat} (xi : fin m -> fin n_Tm)
- (sigma : fin m -> Tm n_Tm)
- (Eq : forall x, funcomp (VarTm n_Tm) xi x = sigma x) :
- forall x, funcomp (VarTm (S n_Tm)) (upRen_Tm_Tm xi) x = up_Tm_Tm sigma x.
+Lemma rinstInst_up_Tm_Tm (xi : nat -> nat) (sigma : nat -> Tm)
+ (Eq : forall x, funcomp (VarTm) xi x = sigma x) :
+ forall x, funcomp (VarTm) (upRen_Tm_Tm xi) x = up_Tm_Tm sigma x.
Proof.
exact (fun n =>
match n with
- | Some fin_n => ap (ren_Tm shift) (Eq fin_n)
- | None => eq_refl
+ | S n' => ap (ren_Tm shift) (Eq n')
+ | O => eq_refl
end).
Qed.
-Lemma rinstInst_up_list_Tm_Tm {p : nat} {m : nat} {n_Tm : nat}
- (xi : fin m -> fin n_Tm) (sigma : fin m -> Tm n_Tm)
- (Eq : forall x, funcomp (VarTm n_Tm) xi x = sigma x) :
- forall x,
- funcomp (VarTm (plus p n_Tm)) (upRen_list_Tm_Tm p xi) x =
- up_list_Tm_Tm p sigma x.
-Proof.
-exact (fun n =>
- eq_trans (scons_p_comp' _ _ (VarTm (plus p n_Tm)) n)
- (scons_p_congr (fun z => eq_refl)
- (fun n => ap (ren_Tm (shift_p p)) (Eq n)))).
-Qed.
-
-Fixpoint rinst_inst_Tm {m_Tm : nat} {n_Tm : nat}
-(xi_Tm : fin m_Tm -> fin n_Tm) (sigma_Tm : fin m_Tm -> Tm n_Tm)
-(Eq_Tm : forall x, funcomp (VarTm n_Tm) xi_Tm x = sigma_Tm x) (s : Tm m_Tm)
-{struct s} : ren_Tm xi_Tm s = subst_Tm sigma_Tm s :=
+Fixpoint rinst_inst_Tm (xi_Tm : nat -> nat) (sigma_Tm : nat -> Tm)
+(Eq_Tm : forall x, funcomp (VarTm) xi_Tm x = sigma_Tm x) (s : Tm) {struct s}
+ : ren_Tm xi_Tm s = subst_Tm sigma_Tm s :=
match s with
- | VarTm _ s0 => Eq_Tm s0
- | Abs _ s0 =>
+ | VarTm s0 => Eq_Tm s0
+ | Abs s0 =>
congr_Abs
(rinst_inst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm sigma_Tm)
(rinstInst_up_Tm_Tm _ _ Eq_Tm) s0)
- | App _ s0 s1 =>
+ | App s0 s1 =>
congr_App (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
(rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
- | Pair _ s0 s1 =>
+ | Pair s0 s1 =>
congr_Pair (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
(rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
- | Proj _ s0 s1 =>
+ | Proj s0 s1 =>
congr_Proj (eq_refl s0) (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
- | TBind _ s0 s1 s2 =>
+ | TBind s0 s1 s2 =>
congr_TBind (eq_refl s0) (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
(rinst_inst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm sigma_Tm)
(rinstInst_up_Tm_Tm _ _ Eq_Tm) s2)
- | Univ _ s0 => congr_Univ (eq_refl s0)
- | BVal _ s0 => congr_BVal (eq_refl s0)
- | Bool _ => congr_Bool
- | If _ s0 s1 s2 =>
+ | Univ s0 => congr_Univ (eq_refl s0)
+ | 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)
- (s : Tm m_Tm) : ren_Tm xi_Tm s = subst_Tm (funcomp (VarTm n_Tm) xi_Tm) s.
+Lemma rinstInst'_Tm (xi_Tm : nat -> nat) (s : Tm) :
+ ren_Tm xi_Tm s = subst_Tm (funcomp (VarTm) xi_Tm) s.
Proof.
exact (rinst_inst_Tm xi_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma rinstInst'_Tm_pointwise {m_Tm : nat} {n_Tm : nat}
- (xi_Tm : fin m_Tm -> fin n_Tm) :
- pointwise_relation _ eq (ren_Tm xi_Tm)
- (subst_Tm (funcomp (VarTm n_Tm) xi_Tm)).
+Lemma rinstInst'_Tm_pointwise (xi_Tm : nat -> nat) :
+ pointwise_relation _ eq (ren_Tm xi_Tm) (subst_Tm (funcomp (VarTm) xi_Tm)).
Proof.
exact (fun s => rinst_inst_Tm xi_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma instId'_Tm {m_Tm : nat} (s : Tm m_Tm) : subst_Tm (VarTm m_Tm) s = s.
+Lemma instId'_Tm (s : Tm) : subst_Tm (VarTm) s = s.
Proof.
-exact (idSubst_Tm (VarTm m_Tm) (fun n => eq_refl) s).
+exact (idSubst_Tm (VarTm) (fun n => eq_refl) s).
Qed.
-Lemma instId'_Tm_pointwise {m_Tm : nat} :
- pointwise_relation _ eq (subst_Tm (VarTm m_Tm)) id.
+Lemma instId'_Tm_pointwise : pointwise_relation _ eq (subst_Tm (VarTm)) id.
Proof.
-exact (fun s => idSubst_Tm (VarTm m_Tm) (fun n => eq_refl) s).
+exact (fun s => idSubst_Tm (VarTm) (fun n => eq_refl) s).
Qed.
-Lemma rinstId'_Tm {m_Tm : nat} (s : Tm m_Tm) : ren_Tm id s = s.
+Lemma rinstId'_Tm (s : Tm) : ren_Tm id s = s.
Proof.
exact (eq_ind_r (fun t => t = s) (instId'_Tm s) (rinstInst'_Tm id s)).
Qed.
-Lemma rinstId'_Tm_pointwise {m_Tm : nat} :
- pointwise_relation _ eq (@ren_Tm m_Tm m_Tm id) id.
+Lemma rinstId'_Tm_pointwise : pointwise_relation _ eq (@ren_Tm id) id.
Proof.
exact (fun s => eq_ind_r (fun t => t = s) (instId'_Tm s) (rinstInst'_Tm id s)).
Qed.
-Lemma varL'_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
- (x : fin m_Tm) : subst_Tm sigma_Tm (VarTm m_Tm x) = sigma_Tm x.
+Lemma varL'_Tm (sigma_Tm : nat -> Tm) (x : nat) :
+ subst_Tm sigma_Tm (VarTm x) = sigma_Tm x.
Proof.
exact (eq_refl).
Qed.
-Lemma varL'_Tm_pointwise {m_Tm : nat} {n_Tm : nat}
- (sigma_Tm : fin m_Tm -> Tm n_Tm) :
- pointwise_relation _ eq (funcomp (subst_Tm sigma_Tm) (VarTm m_Tm)) sigma_Tm.
+Lemma varL'_Tm_pointwise (sigma_Tm : nat -> Tm) :
+ pointwise_relation _ eq (funcomp (subst_Tm sigma_Tm) (VarTm)) sigma_Tm.
Proof.
exact (fun x => eq_refl).
Qed.
-Lemma varLRen'_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
- (x : fin m_Tm) : ren_Tm xi_Tm (VarTm m_Tm x) = VarTm n_Tm (xi_Tm x).
+Lemma varLRen'_Tm (xi_Tm : nat -> nat) (x : nat) :
+ ren_Tm xi_Tm (VarTm x) = VarTm (xi_Tm x).
Proof.
exact (eq_refl).
Qed.
-Lemma varLRen'_Tm_pointwise {m_Tm : nat} {n_Tm : nat}
- (xi_Tm : fin m_Tm -> fin n_Tm) :
- pointwise_relation _ eq (funcomp (ren_Tm xi_Tm) (VarTm m_Tm))
- (funcomp (VarTm n_Tm) xi_Tm).
+Lemma varLRen'_Tm_pointwise (xi_Tm : nat -> nat) :
+ pointwise_relation _ eq (funcomp (ren_Tm xi_Tm) (VarTm))
+ (funcomp (VarTm) xi_Tm).
Proof.
exact (fun x => eq_refl).
Qed.
@@ -1447,30 +1101,22 @@ Class Up_Tm X Y :=
Class Up_PTm X Y :=
up_PTm : X -> Y.
-#[global]
-Instance Subst_Tm {m_Tm n_Tm : nat}: (Subst1 _ _ _) := (@subst_Tm m_Tm n_Tm).
+#[global] Instance Subst_Tm : (Subst1 _ _ _) := @subst_Tm.
+
+#[global] Instance Up_Tm_Tm : (Up_Tm _ _) := @up_Tm_Tm.
+
+#[global] Instance Ren_Tm : (Ren1 _ _ _) := @ren_Tm.
+
+#[global] Instance VarInstance_Tm : (Var _ _) := @VarTm.
+
+#[global] Instance Subst_PTm : (Subst1 _ _ _) := @subst_PTm.
+
+#[global] Instance Up_PTm_PTm : (Up_PTm _ _) := @up_PTm_PTm.
+
+#[global] Instance Ren_PTm : (Ren1 _ _ _) := @ren_PTm.
#[global]
-Instance Up_Tm_Tm {m n_Tm : nat}: (Up_Tm _ _) := (@up_Tm_Tm m n_Tm).
-
-#[global]
-Instance Ren_Tm {m_Tm n_Tm : nat}: (Ren1 _ _ _) := (@ren_Tm m_Tm n_Tm).
-
-#[global]
-Instance VarInstance_Tm {n_Tm : nat}: (Var _ _) := (@VarTm n_Tm).
-
-#[global]
-Instance Subst_PTm {m_PTm n_PTm : nat}: (Subst1 _ _ _) :=
- (@subst_PTm m_PTm n_PTm).
-
-#[global]
-Instance Up_PTm_PTm {m n_PTm : nat}: (Up_PTm _ _) := (@up_PTm_PTm m n_PTm).
-
-#[global]
-Instance Ren_PTm {m_PTm n_PTm : nat}: (Ren1 _ _ _) := (@ren_PTm m_PTm n_PTm).
-
-#[global]
-Instance VarInstance_PTm {n_PTm : nat}: (Var _ _) := (@VarPTm n_PTm).
+Instance VarInstance_PTm : (Var _ _) := @VarPTm.
Notation "[ sigma_Tm ]" := (subst_Tm sigma_Tm)
( at level 1, left associativity, only printing) : fscope.
@@ -1521,9 +1167,9 @@ Notation "x '__PTm'" := (VarPTm x) ( at level 5, format "x __PTm") :
subst_scope.
#[global]
-Instance subst_Tm_morphism {m_Tm : nat} {n_Tm : nat}:
+Instance subst_Tm_morphism :
(Proper (respectful (pointwise_relation _ eq) (respectful eq eq))
- (@subst_Tm m_Tm n_Tm)).
+ (@subst_Tm)).
Proof.
exact (fun f_Tm g_Tm Eq_Tm s t Eq_st =>
eq_ind s (fun t' => subst_Tm f_Tm s = subst_Tm g_Tm t')
@@ -1531,17 +1177,16 @@ exact (fun f_Tm g_Tm Eq_Tm s t Eq_st =>
Qed.
#[global]
-Instance subst_Tm_morphism2 {m_Tm : nat} {n_Tm : nat}:
+Instance subst_Tm_morphism2 :
(Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
- (@subst_Tm m_Tm n_Tm)).
+ (@subst_Tm)).
Proof.
exact (fun f_Tm g_Tm Eq_Tm s => ext_Tm f_Tm g_Tm Eq_Tm s).
Qed.
#[global]
-Instance ren_Tm_morphism {m_Tm : nat} {n_Tm : nat}:
- (Proper (respectful (pointwise_relation _ eq) (respectful eq eq))
- (@ren_Tm m_Tm n_Tm)).
+Instance ren_Tm_morphism :
+ (Proper (respectful (pointwise_relation _ eq) (respectful eq eq)) (@ren_Tm)).
Proof.
exact (fun f_Tm g_Tm Eq_Tm s t Eq_st =>
eq_ind s (fun t' => ren_Tm f_Tm s = ren_Tm g_Tm t')
@@ -1549,17 +1194,17 @@ exact (fun f_Tm g_Tm Eq_Tm s t Eq_st =>
Qed.
#[global]
-Instance ren_Tm_morphism2 {m_Tm : nat} {n_Tm : nat}:
+Instance ren_Tm_morphism2 :
(Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
- (@ren_Tm m_Tm n_Tm)).
+ (@ren_Tm)).
Proof.
exact (fun f_Tm g_Tm Eq_Tm s => extRen_Tm f_Tm g_Tm Eq_Tm s).
Qed.
#[global]
-Instance subst_PTm_morphism {m_PTm : nat} {n_PTm : nat}:
+Instance subst_PTm_morphism :
(Proper (respectful (pointwise_relation _ eq) (respectful eq eq))
- (@subst_PTm m_PTm n_PTm)).
+ (@subst_PTm)).
Proof.
exact (fun f_PTm g_PTm Eq_PTm s t Eq_st =>
eq_ind s (fun t' => subst_PTm f_PTm s = subst_PTm g_PTm t')
@@ -1567,17 +1212,16 @@ exact (fun f_PTm g_PTm Eq_PTm s t Eq_st =>
Qed.
#[global]
-Instance subst_PTm_morphism2 {m_PTm : nat} {n_PTm : nat}:
+Instance subst_PTm_morphism2 :
(Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
- (@subst_PTm m_PTm n_PTm)).
+ (@subst_PTm)).
Proof.
exact (fun f_PTm g_PTm Eq_PTm s => ext_PTm f_PTm g_PTm Eq_PTm s).
Qed.
#[global]
-Instance ren_PTm_morphism {m_PTm : nat} {n_PTm : nat}:
- (Proper (respectful (pointwise_relation _ eq) (respectful eq eq))
- (@ren_PTm m_PTm n_PTm)).
+Instance ren_PTm_morphism :
+ (Proper (respectful (pointwise_relation _ eq) (respectful eq eq)) (@ren_PTm)).
Proof.
exact (fun f_PTm g_PTm Eq_PTm s t Eq_st =>
eq_ind s (fun t' => ren_PTm f_PTm s = ren_PTm g_PTm t')
@@ -1585,9 +1229,9 @@ exact (fun f_PTm g_PTm Eq_PTm s t Eq_st =>
Qed.
#[global]
-Instance ren_PTm_morphism2 {m_PTm : nat} {n_PTm : nat}:
+Instance ren_PTm_morphism2 :
(Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
- (@ren_PTm m_PTm n_PTm)).
+ (@ren_PTm)).
Proof.
exact (fun f_PTm g_PTm Eq_PTm s => extRen_PTm f_PTm g_PTm Eq_PTm s).
Qed.
@@ -1641,9 +1285,8 @@ Ltac asimpl' := repeat (first
| progress setoid_rewrite instId'_PTm_pointwise
| progress setoid_rewrite instId'_PTm
| progress
- unfold up_list_Tm_Tm, up_Tm_Tm, upRen_list_Tm_Tm,
- upRen_Tm_Tm, up_list_PTm_PTm, up_PTm_PTm,
- upRen_list_PTm_PTm, upRen_PTm_PTm, up_ren
+ unfold up_Tm_Tm, upRen_Tm_Tm, up_PTm_PTm, upRen_PTm_PTm,
+ up_ren
| progress cbn[subst_Tm ren_Tm subst_PTm ren_PTm]
| progress fsimpl ]).
@@ -1673,52 +1316,15 @@ End Core.
Module Extra.
-Import
-Core.
+Import Core.
-Arguments VarTm {n_Tm}.
+#[global] Hint Opaque subst_Tm: rewrite.
-Arguments If {n_Tm}.
+#[global] Hint Opaque ren_Tm: rewrite.
-Arguments Bool {n_Tm}.
+#[global] Hint Opaque subst_PTm: rewrite.
-Arguments BVal {n_Tm}.
-
-Arguments Univ {n_Tm}.
-
-Arguments TBind {n_Tm}.
-
-Arguments Proj {n_Tm}.
-
-Arguments Pair {n_Tm}.
-
-Arguments App {n_Tm}.
-
-Arguments Abs {n_Tm}.
-
-Arguments VarPTm {n_PTm}.
-
-Arguments PBot {n_PTm}.
-
-Arguments PUniv {n_PTm}.
-
-Arguments PConst {n_PTm}.
-
-Arguments PProj {n_PTm}.
-
-Arguments PPair {n_PTm}.
-
-Arguments PApp {n_PTm}.
-
-Arguments PAbs {n_PTm}.
-
-#[global]Hint Opaque subst_Tm: rewrite.
-
-#[global]Hint Opaque ren_Tm: rewrite.
-
-#[global]Hint Opaque subst_PTm: rewrite.
-
-#[global]Hint Opaque ren_PTm: rewrite.
+#[global] Hint Opaque ren_PTm: rewrite.
End Extra.
diff --git a/theories/Autosubst2/unscoped.v b/theories/Autosubst2/unscoped.v
new file mode 100644
index 0000000..55f8172
--- /dev/null
+++ b/theories/Autosubst2/unscoped.v
@@ -0,0 +1,213 @@
+(** * Autosubst Header for Unnamed Syntax
+
+Version: December 11, 2019.
+ *)
+
+(* Adrian:
+ I changed this library a bit to work better with my generated code.
+ 1. I use nat directly instead of defining fin to be nat and using Some/None as S/O
+ 2. I removed the "s, sigma" notation for scons because it interacts with dependent function types "forall x, A"*)
+Require Import core.
+Require Import Setoid Morphisms Relation_Definitions.
+
+Definition ap {X Y} (f : X -> Y) {x y : X} (p : x = y) : f x = f y :=
+ match p with eq_refl => eq_refl end.
+
+Definition apc {X Y} {f g : X -> Y} {x y : X} (p : f = g) (q : x = y) : f x = g y :=
+ match q with eq_refl => match p with eq_refl => eq_refl end end.
+
+(** ** Primitives of the Sigma Calculus. *)
+
+Definition shift := S.
+
+Definition var_zero := 0.
+
+Definition id {X} := @Datatypes.id X.
+
+Definition scons {X: Type} (x : X) (xi : nat -> X) :=
+ fun n => match n with
+ | 0 => x
+ | S n => xi n
+ end.
+
+#[ export ]
+Hint Opaque scons : rewrite.
+
+(** ** Type Class Instances for Notation
+Required to make notation work. *)
+
+(** *** Type classes for renamings. *)
+
+Class Ren1 (X1 : Type) (Y Z : Type) :=
+ ren1 : X1 -> Y -> Z.
+
+Class Ren2 (X1 X2 : Type) (Y Z : Type) :=
+ ren2 : X1 -> X2 -> Y -> Z.
+
+Class Ren3 (X1 X2 X3 : Type) (Y Z : Type) :=
+ ren3 : X1 -> X2 -> X3 -> Y -> Z.
+
+Class Ren4 (X1 X2 X3 X4 : Type) (Y Z : Type) :=
+ ren4 : X1 -> X2 -> X3 -> X4 -> Y -> Z.
+
+Class Ren5 (X1 X2 X3 X4 X5 : Type) (Y Z : Type) :=
+ ren5 : X1 -> X2 -> X3 -> X4 -> X5 -> Y -> Z.
+
+Module RenNotations.
+ Notation "s ⟨ xi1 ⟩" := (ren1 xi1 s) (at level 7, left associativity, format "s ⟨ xi1 ⟩") : subst_scope.
+
+ Notation "s ⟨ xi1 ; xi2 ⟩" := (ren2 xi1 xi2 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ⟩") : subst_scope.
+
+ Notation "s ⟨ xi1 ; xi2 ; xi3 ⟩" := (ren3 xi1 xi2 xi3 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ; xi3 ⟩") : subst_scope.
+
+ Notation "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ⟩" := (ren4 xi1 xi2 xi3 xi4 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ⟩") : subst_scope.
+
+ Notation "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ; xi5 ⟩" := (ren5 xi1 xi2 xi3 xi4 xi5 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ; xi5 ⟩") : subst_scope.
+
+ Notation "⟨ xi ⟩" := (ren1 xi) (at level 1, left associativity, format "⟨ xi ⟩") : fscope.
+
+ Notation "⟨ xi1 ; xi2 ⟩" := (ren2 xi1 xi2) (at level 1, left associativity, format "⟨ xi1 ; xi2 ⟩") : fscope.
+End RenNotations.
+
+(** *** Type Classes for Substiution *)
+
+Class Subst1 (X1 : Type) (Y Z: Type) :=
+ subst1 : X1 -> Y -> Z.
+
+Class Subst2 (X1 X2 : Type) (Y Z: Type) :=
+ subst2 : X1 -> X2 -> Y -> Z.
+
+Class Subst3 (X1 X2 X3 : Type) (Y Z: Type) :=
+ subst3 : X1 -> X2 -> X3 -> Y -> Z.
+
+Class Subst4 (X1 X2 X3 X4: Type) (Y Z: Type) :=
+ subst4 : X1 -> X2 -> X3 -> X4 -> Y -> Z.
+
+Class Subst5 (X1 X2 X3 X4 X5 : Type) (Y Z: Type) :=
+ subst5 : X1 -> X2 -> X3 -> X4 -> X5 -> Y -> Z.
+
+Module SubstNotations.
+ Notation "s [ sigma ]" := (subst1 sigma s) (at level 7, left associativity, format "s '/' [ sigma ]") : subst_scope.
+
+ Notation "s [ sigma ; tau ]" := (subst2 sigma tau s) (at level 7, left associativity, format "s '/' [ sigma ; '/' tau ]") : subst_scope.
+End SubstNotations.
+
+(** *** Type Class for Variables *)
+
+Class Var X Y :=
+ ids : X -> Y.
+
+#[export] Instance idsRen : Var nat nat := id.
+
+(** ** Proofs for the substitution primitives. *)
+
+Arguments funcomp {X Y Z} (g)%fscope (f)%fscope.
+
+Module CombineNotations.
+ Notation "f >> g" := (funcomp g f) (at level 50) : fscope.
+
+ Notation "s .: sigma" := (scons s sigma) (at level 55, sigma at next level, right associativity) : subst_scope.
+
+ #[ global ]
+ Open Scope fscope.
+ #[ global ]
+ Open Scope subst_scope.
+End CombineNotations.
+
+Import CombineNotations.
+
+
+(** A generic lifting of a renaming. *)
+Definition up_ren (xi : nat -> nat) :=
+ 0 .: (xi >> S).
+
+(** A generic proof that lifting of renamings composes. *)
+Lemma up_ren_ren (xi: nat -> nat) (zeta : nat -> nat) (rho: nat -> nat) (E: forall x, (xi >> zeta) x = rho x) :
+ forall x, (up_ren xi >> up_ren zeta) x = up_ren rho x.
+Proof.
+ intros [|x].
+ - reflexivity.
+ - unfold up_ren. cbn. unfold funcomp. f_equal. apply E.
+Qed.
+
+(** Eta laws. *)
+Lemma scons_eta' {T} (f : nat -> T) :
+ pointwise_relation _ eq (f var_zero .: (funcomp f shift)) f.
+Proof. intros x. destruct x; reflexivity. Qed.
+
+Lemma scons_eta_id' :
+ pointwise_relation _ eq (var_zero .: shift) id.
+Proof. intros x. destruct x; reflexivity. Qed.
+
+Lemma scons_comp' (T: Type) {U} (s: T) (sigma: nat -> T) (tau: T -> U) :
+ pointwise_relation _ eq (funcomp tau (s .: sigma)) ((tau s) .: (funcomp tau sigma)).
+Proof. intros x. destruct x; reflexivity. Qed.
+
+(* Morphism for Setoid Rewriting. The only morphism that can be defined statically. *)
+#[export] Instance scons_morphism {X: Type} :
+ Proper (eq ==> pointwise_relation _ eq ==> pointwise_relation _ eq) (@scons X).
+Proof.
+ intros ? t -> sigma tau H.
+ intros [|x].
+ cbn. reflexivity.
+ apply H.
+Qed.
+
+#[export] Instance scons_morphism2 {X: Type} :
+ Proper (eq ==> pointwise_relation _ eq ==> eq ==> eq) (@scons X).
+Proof.
+ intros ? t -> sigma tau H ? x ->.
+ destruct x as [|x].
+ cbn. reflexivity.
+ apply H.
+Qed.
+
+(** ** Generic lifting of an allfv predicate *)
+Definition up_allfv (p: nat -> Prop) : nat -> Prop := scons True p.
+
+(** ** Notations for unscoped syntax *)
+Module UnscopedNotations.
+ Include RenNotations.
+ Include SubstNotations.
+ Include CombineNotations.
+
+ (* Notation "s , sigma" := (scons s sigma) (at level 60, format "s , sigma", right associativity) : subst_scope. *)
+
+ Notation "s '..'" := (scons s ids) (at level 1, format "s ..") : subst_scope.
+
+ Notation "↑" := (shift) : subst_scope.
+
+ #[global]
+ Open Scope fscope.
+ #[global]
+ Open Scope subst_scope.
+End UnscopedNotations.
+
+(** ** Tactics for unscoped syntax *)
+
+(** Automatically does a case analysis on a natural number, useful for proofs with context renamings/context morphisms. *)
+Tactic Notation "auto_case" tactic(t) := (match goal with
+ | [|- forall (i : nat), _] => intros []; t
+ end).
+
+
+(** Generic fsimpl tactic: simplifies the above primitives in a goal. *)
+Ltac fsimpl :=
+ repeat match goal with
+ | [|- context[id >> ?f]] => change (id >> f) with f (* AsimplCompIdL *)
+ | [|- context[?f >> id]] => change (f >> id) with f (* AsimplCompIdR *)
+ | [|- context [id ?s]] => change (id s) with s
+ | [|- context[(?f >> ?g) >> ?h]] => change ((f >> g) >> h) with (f >> (g >> h))
+ | [|- context[(?v .: ?g) var_zero]] => change ((v .: g) var_zero) with v
+ | [|- context[(?v .: ?g) 0]] => change ((v .: g) 0) with v
+ | [|- context[(?v .: ?g) (S ?n)]] => change ((v .: g) (S n)) with (g n)
+ | [|- context[?f >> (?x .: ?g)]] => change (f >> (x .: g)) with g (* f should evaluate to shift *)
+ | [|- context[var_zero]] => change var_zero with 0
+ | [|- context[?x2 .: (funcomp ?f shift)]] => change (scons x2 (funcomp f shift)) with (scons (f var_zero) (funcomp f shift)); setoid_rewrite (@scons_eta' _ _ f)
+ | [|- context[?f var_zero .: ?g]] => change (scons (f var_zero) g) with (scons (f var_zero) (funcomp f shift)); rewrite scons_eta'
+ | [|- _ = ?h (?f ?s)] => change (h (f s)) with ((f >> h) s)
+ | [|- ?h (?f ?s) = _] => change (h (f s)) with ((f >> h) s)
+ (* DONE had to put an underscore as the last argument to scons. This might be an argument against unfolding funcomp *)
+ | [|- context[funcomp _ (scons _ _)]] => setoid_rewrite scons_comp'; eta_reduce
+ | [|- context[scons var_zero shift]] => setoid_rewrite scons_eta_id'; eta_reduce
+ end.
\ No newline at end of file
diff --git a/theories/fp_red.v b/theories/fp_red.v
index bbe58d9..f9abc08 100644
--- a/theories/fp_red.v
+++ b/theories/fp_red.v
@@ -7,7 +7,7 @@ Require Import Arith.Wf_nat.
Require Import Psatz.
From stdpp Require Import relations (rtc (..), rtc_once, rtc_r).
From Hammer Require Import Tactics.
-Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax.
+Require Import Autosubst2.core Autosubst2.unscoped Autosubst2.syntax.
Ltac2 spec_refl () :=
List.iter
@@ -22,7 +22,7 @@ Ltac spec_refl := ltac2:(spec_refl ()).
(* Trying my best to not write C style module_funcname *)
Module Par.
- Inductive R {n} : PTm n -> PTm n -> Prop :=
+ Inductive R : PTm -> PTm -> Prop :=
(***************** Beta ***********************)
| AppAbs a0 a1 b0 b1 :
R a0 a1 ->
@@ -72,35 +72,35 @@ Module Par.
| Bot :
R PBot PBot.
- Lemma refl n (a : PTm n) : R a a.
- elim : n /a; hauto ctrs:R.
+ Lemma refl (a : PTm) : R a a.
+ elim : a; hauto ctrs:R.
Qed.
- Lemma AppAbs' n a0 a1 (b0 b1 t : PTm n) :
+ Lemma AppAbs' a0 a1 (b0 b1 t : PTm) :
t = subst_PTm (scons b1 VarPTm) a1 ->
R a0 a1 ->
R b0 b1 ->
R (PApp (PAbs a0) b0) t.
Proof. move => ->. apply AppAbs. Qed.
- Lemma ProjPair' n p (a0 a1 b0 b1 : PTm n) t :
+ Lemma ProjPair' p (a0 a1 b0 b1 : PTm) t :
t = (if p is PL then a1 else b1) ->
R a0 a1 ->
R b0 b1 ->
R (PProj p (PPair a0 b0)) t.
Proof. move => > ->. apply ProjPair. Qed.
- Lemma AppEta' n (a0 a1 b : PTm n) :
+ Lemma AppEta' (a0 a1 b : PTm) :
b = (PAbs (PApp (ren_PTm shift a1) (VarPTm var_zero))) ->
R a0 a1 ->
R a0 b.
Proof. move => ->; apply AppEta. Qed.
- Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
+ Lemma renaming (a b : PTm) (ξ : nat -> nat) :
R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
Proof.
- move => h. move : m ξ.
- elim : n a b /h.
+ move => h. move : ξ.
+ elim : a b /h.
move => *; apply : AppAbs'; eauto; by asimpl.
all : match goal with
| [ |- context[var_zero]] => move => *; apply : AppEta'; eauto; by asimpl
@@ -109,23 +109,23 @@ Module Par.
Qed.
- Lemma morphing n m (a b : PTm n) (ρ0 ρ1 : fin n -> PTm m) :
+ Lemma morphing (a b : PTm) (ρ0 ρ1 : nat -> PTm) :
(forall i, R (ρ0 i) (ρ1 i)) ->
R a b -> R (subst_PTm ρ0 a) (subst_PTm ρ1 b).
Proof.
- move => + h. move : m ρ0 ρ1. elim : n a b/h.
- - move => n a0 a1 b0 b1 ha iha hb ihb m ρ0 ρ1 hρ /=.
+ move => + h. move : ρ0 ρ1. elim : a b/h.
+ - move => a0 a1 b0 b1 ha iha hb ihb ρ0 ρ1 hρ /=.
eapply AppAbs' with (a1 := subst_PTm (up_PTm_PTm ρ1) a1); eauto.
by asimpl.
- hauto l:on use:renaming inv:option.
+ hauto l:on use:renaming inv:nat.
- hauto lq:on rew:off ctrs:R.
- - hauto l:on inv:option use:renaming ctrs:R.
+ - hauto l:on inv:nat use:renaming ctrs:R.
- hauto lq:on use:ProjPair'.
- - move => n a0 a1 ha iha m ρ0 ρ1 hρ /=.
+ - move => a0 a1 ha iha ρ0 ρ1 hρ /=.
apply : AppEta'; eauto. by asimpl.
- hauto lq:on ctrs:R.
- sfirstorder.
- - hauto l:on inv:option ctrs:R use:renaming.
+ - hauto l:on inv:nat ctrs:R use:renaming.
- hauto q:on ctrs:R.
- qauto l:on ctrs:R.
- qauto l:on ctrs:R.
@@ -134,16 +134,16 @@ Module Par.
- qauto l:on ctrs:R.
Qed.
- Lemma substing n m (a b : PTm n) (ρ : fin n -> PTm m) :
+ Lemma substing (a b : PTm) (ρ : nat -> PTm) :
R a b -> R (subst_PTm ρ a) (subst_PTm ρ b).
Proof. hauto l:on use:morphing, refl. Qed.
- Lemma antirenaming n m (a : PTm n) (b : PTm m) (ξ : fin n -> fin m) :
+ Lemma antirenaming (a : PTm) (b : PTm) (ξ : nat -> nat) :
R (ren_PTm ξ a) b -> exists b0, R a b0 /\ ren_PTm ξ b0 = b.
Proof.
move E : (ren_PTm ξ a) => u h.
- move : n ξ a E. elim : m u b/h.
- - move => n a0 a1 b0 b1 ha iha hb ihb m ξ []//=.
+ move : ξ a E. elim : u b/h.
+ - move => a0 a1 b0 b1 ha iha hb ihb ξ []//=.
move => c c0 [+ ?]. subst.
case : c => //=.
move => c [?]. subst.
@@ -153,7 +153,7 @@ Module Par.
eexists. split.
apply AppAbs; eauto.
by asimpl.
- - move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc m ξ []//=.
+ - move => a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc ξ []//=.
move => []//= t t0 t1 [*]. subst.
spec_refl.
move : iha => [? [*]].
@@ -162,43 +162,43 @@ Module Par.
eexists. split.
apply AppPair; hauto. subst.
by asimpl.
- - move => n p a0 a1 ha iha m ξ []//= p0 []//= t [*]. subst.
+ - move => p a0 a1 ha iha ξ []//= p0 []//= t [*]. subst.
spec_refl. move : iha => [b0 [? ?]]. subst.
eexists. split. apply ProjAbs; eauto. by asimpl.
- - move => n p a0 a1 b0 b1 ha iha hb ihb m ξ []//= p0 []//= t t0[*].
+ - move => p a0 a1 b0 b1 ha iha hb ihb ξ []//= p0 []//= t t0[*].
subst. spec_refl.
move : iha => [b0 [? ?]].
move : ihb => [c0 [? ?]]. subst.
eexists. split. by eauto using ProjPair.
hauto q:on.
- - move => n a0 a1 ha iha m ξ a ?. subst.
+ - move => a0 a1 ha iha ξ a ?. subst.
spec_refl. move : iha => [a0 [? ?]]. subst.
eexists. split. apply AppEta; eauto.
by asimpl.
- - move => n a0 a1 ha iha m ξ a ?. subst.
+ - move => a0 a1 ha iha ξ a ?. subst.
spec_refl. move : iha => [b0 [? ?]]. subst.
eexists. split. apply PairEta; eauto.
by asimpl.
- - move => n i m ξ []//=.
+ - move => i ξ []//=.
hauto l:on.
- - move => n a0 a1 ha iha m ξ []//= t [*]. subst.
+ - move => a0 a1 ha iha ξ []//= t [*]. subst.
spec_refl.
move :iha => [b0 [? ?]]. subst.
eexists. split. by apply AbsCong; eauto.
done.
- - move => n a0 a1 b0 b1 ha iha hb ihb m ξ []//= t t0 [*]. subst.
+ - move => a0 a1 b0 b1 ha iha hb ihb ξ []//= t t0 [*]. subst.
spec_refl.
move : iha => [b0 [? ?]]. subst.
move : ihb => [c0 [? ?]]. subst.
eexists. split. by apply AppCong; eauto.
done.
- - move => n a0 a1 b0 b1 ha iha hb ihb m ξ []//= t t0[*]. subst.
+ - move => a0 a1 b0 b1 ha iha hb ihb ξ []//= t t0[*]. subst.
spec_refl.
move : iha => [b0 [? ?]]. subst.
move : ihb => [c0 [? ?]]. subst.
eexists. split=>/=. by apply PairCong; eauto.
done.
- - move => n p a0 a1 ha iha m ξ []//= p0 t [*]. subst.
+ - move => p a0 a1 ha iha ξ []//= p0 t [*]. subst.
spec_refl.
move : iha => [b0 [? ?]]. subst.
eexists. split. by apply ProjCong; eauto.
@@ -359,7 +359,7 @@ Module RPar.
R a b ->
(forall i, R (ρ0 i) (ρ1 i)) ->
(forall i, R ((scons a ρ0) i) ((scons b ρ1) i)).
- Proof. hauto q:on inv:option. Qed.
+ Proof. hauto q:on inv:nat. Qed.
Lemma morphing_up n m (ρ0 ρ1 : fin n -> PTm m) :
(forall i, R (ρ0 i) (ρ1 i)) ->
@@ -399,7 +399,7 @@ Module RPar.
R (subst_PTm (scons c VarPTm) a) (subst_PTm (scons d VarPTm) b).
Proof.
move => h0 h1. apply morphing => //=.
- qauto l:on ctrs:R inv:option.
+ qauto l:on ctrs:R inv:nat.
Qed.
Lemma var_or_const_imp {n} (a b : PTm n) :
@@ -595,7 +595,7 @@ Module RPar'.
R a b ->
(forall i, R (ρ0 i) (ρ1 i)) ->
(forall i, R ((scons a ρ0) i) ((scons b ρ1) i)).
- Proof. hauto q:on inv:option. Qed.
+ Proof. hauto q:on inv:nat. Qed.
Lemma morphing_up n m (ρ0 ρ1 : fin n -> PTm m) :
(forall i, R (ρ0 i) (ρ1 i)) ->
@@ -633,7 +633,7 @@ Module RPar'.
R (subst_PTm (scons c VarPTm) a) (subst_PTm (scons d VarPTm) b).
Proof.
move => h0 h1. apply morphing => //=.
- qauto l:on ctrs:R inv:option.
+ qauto l:on ctrs:R inv:nat.
Qed.
Lemma var_or_const_imp {n} (a b : PTm n) :
@@ -786,7 +786,7 @@ Module ERed.
move => h. move : m ρ. elim : n a b / h => n.
move => a m ρ /=.
apply : AppEta'; eauto. by asimpl.
- all : hauto ctrs:R inv:option use:renaming.
+ all : hauto ctrs:R inv:nat use:renaming.
Qed.
End ERed.
@@ -892,11 +892,11 @@ Module EPar.
apply : AppEta'; eauto. by asimpl.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R.
- - hauto l:on ctrs:R use:renaming inv:option.
+ - hauto l:on ctrs:R use:renaming inv:nat.
- hauto q:on ctrs:R.
- hauto q:on ctrs:R.
- hauto q:on ctrs:R.
- - hauto l:on ctrs:R use:renaming inv:option.
+ - hauto l:on ctrs:R use:renaming inv:nat.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R.
Qed.
@@ -907,7 +907,7 @@ Module EPar.
R (subst_PTm (scons b0 VarPTm) a0) (subst_PTm (scons b1 VarPTm) a1).
Proof.
move => h0 h1. apply morphing => //.
- hauto lq:on ctrs:R inv:option.
+ hauto lq:on ctrs:R inv:nat.
Qed.
End EPar.
@@ -1018,7 +1018,7 @@ Module RPars.
Lemma substing n (a b : PTm (S n)) c :
rtc RPar.R a b ->
rtc RPar.R (subst_PTm (scons c VarPTm) a) (subst_PTm (scons c VarPTm) b).
- Proof. hauto lq:on use:morphing inv:option. Qed.
+ Proof. hauto lq:on use:morphing inv:nat. Qed.
Lemma antirenaming n m (a : PTm n) (b : PTm m) (ρ : fin n -> PTm m) :
(forall i, var_or_const (ρ i)) ->
@@ -1099,7 +1099,7 @@ Module RPars'.
Lemma substing n (a b : PTm (S n)) c :
rtc RPar'.R a b ->
rtc RPar'.R (subst_PTm (scons c VarPTm) a) (subst_PTm (scons c VarPTm) b).
- Proof. hauto lq:on use:morphing inv:option. Qed.
+ Proof. hauto lq:on use:morphing inv:nat. Qed.
Lemma antirenaming n m (a : PTm n) (b : PTm m) (ρ : fin n -> PTm m) :
(forall i, var_or_const (ρ i)) ->
@@ -2328,7 +2328,7 @@ Proof.
have : wn (subst_PTm (scons (PBot) VarPTm) a3) by sfirstorder.
move => h. apply wn_abs.
move : h. apply wn_antirenaming.
- hauto lq:on rew:off inv:option.
+ hauto lq:on rew:off inv:nat.
+ hauto q:on inv:RPar'.R ctrs:rtc b:on.
Qed.
diff --git a/theories/typing.v b/theories/typing.v
new file mode 100644
index 0000000..d41facd
--- /dev/null
+++ b/theories/typing.v
@@ -0,0 +1,251 @@
+Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax.
+
+Reserved Notation "Γ ⊢ a ∈ A" (at level 70).
+Reserved Notation "Γ ⊢ a ≡ b ∈ A" (at level 70).
+Reserved Notation "Γ ⊢ A ≲ B" (at level 70).
+Reserved Notation "⊢ Γ" (at level 70).
+Inductive Wt : list PTm -> PTm -> PTm -> Prop :=
+| T_Var i Γ A :
+ ⊢ Γ ->
+ lookup i Γ A ->
+ Γ ⊢ VarPTm i ∈ A
+
+| T_Bind Γ i p (A : PTm) (B : PTm) :
+ Γ ⊢ A ∈ PUniv i ->
+ cons A Γ ⊢ B ∈ PUniv i ->
+ Γ ⊢ PBind p A B ∈ PUniv i
+
+| T_Abs Γ (a : PTm) A B i :
+ Γ ⊢ PBind PPi A B ∈ (PUniv i) ->
+ (cons A Γ) ⊢ a ∈ B ->
+ Γ ⊢ PAbs a ∈ PBind PPi A B
+
+| T_App Γ (b a : PTm) A B :
+ Γ ⊢ b ∈ PBind PPi A B ->
+ Γ ⊢ a ∈ A ->
+ Γ ⊢ PApp b a ∈ subst_PTm (scons a VarPTm) B
+
+| T_Pair Γ (a b : PTm) A B i :
+ Γ ⊢ PBind PSig A B ∈ (PUniv i) ->
+ Γ ⊢ a ∈ A ->
+ Γ ⊢ b ∈ subst_PTm (scons a VarPTm) B ->
+ Γ ⊢ PPair a b ∈ PBind PSig A B
+
+| T_Proj1 Γ (a : PTm) A B :
+ Γ ⊢ a ∈ PBind PSig A B ->
+ Γ ⊢ PProj PL a ∈ A
+
+| T_Proj2 Γ (a : PTm) A B :
+ Γ ⊢ a ∈ PBind PSig A B ->
+ Γ ⊢ PProj PR a ∈ subst_PTm (scons (PProj PL a) VarPTm) B
+
+| T_Univ Γ i :
+ ⊢ Γ ->
+ Γ ⊢ PUniv i ∈ PUniv (S i)
+
+| T_Nat Γ i :
+ ⊢ Γ ->
+ Γ ⊢ PNat ∈ PUniv i
+
+| T_Zero Γ :
+ ⊢ Γ ->
+ Γ ⊢ PZero ∈ PNat
+
+| T_Suc Γ (a : PTm) :
+ Γ ⊢ a ∈ PNat ->
+ Γ ⊢ PSuc a ∈ PNat
+
+| T_Ind Γ P (a : PTm) b c i :
+ cons PNat Γ ⊢ P ∈ PUniv i ->
+ Γ ⊢ a ∈ PNat ->
+ Γ ⊢ b ∈ subst_PTm (scons PZero VarPTm) P ->
+ (cons P (cons PNat Γ)) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) ->
+ Γ ⊢ PInd P a b c ∈ subst_PTm (scons a VarPTm) P
+
+| T_Conv Γ (a : PTm) A B :
+ Γ ⊢ a ∈ A ->
+ Γ ⊢ A ≲ B ->
+ Γ ⊢ a ∈ B
+
+with Eq : list PTm -> PTm -> PTm -> PTm -> Prop :=
+(* Structural *)
+| E_Refl Γ (a : PTm ) A :
+ Γ ⊢ a ∈ A ->
+ Γ ⊢ a ≡ a ∈ A
+
+| E_Symmetric Γ (a b : PTm) A :
+ Γ ⊢ a ≡ b ∈ A ->
+ Γ ⊢ b ≡ a ∈ A
+
+| E_Transitive Γ (a b c : PTm) A :
+ Γ ⊢ a ≡ b ∈ A ->
+ Γ ⊢ b ≡ c ∈ A ->
+ Γ ⊢ a ≡ c ∈ A
+
+(* Congruence *)
+| E_Bind Γ i p (A0 A1 : PTm) B0 B1 :
+ Γ ⊢ A0 ∈ PUniv i ->
+ Γ ⊢ A0 ≡ A1 ∈ PUniv i ->
+ (cons A0 Γ) ⊢ B0 ≡ B1 ∈ PUniv i ->
+ Γ ⊢ PBind p A0 B0 ≡ PBind p A1 B1 ∈ PUniv i
+
+| E_Abs Γ (a b : PTm) A B i :
+ Γ ⊢ PBind PPi A B ∈ (PUniv i) ->
+ (cons A Γ) ⊢ a ≡ b ∈ B ->
+ Γ ⊢ PAbs a ≡ PAbs b ∈ PBind PPi A B
+
+| E_App Γ i (b0 b1 a0 a1 : PTm) A B :
+ Γ ⊢ PBind PPi A B ∈ (PUniv i) ->
+ Γ ⊢ b0 ≡ b1 ∈ PBind PPi A B ->
+ Γ ⊢ a0 ≡ a1 ∈ A ->
+ Γ ⊢ PApp b0 a0 ≡ PApp b1 a1 ∈ subst_PTm (scons a0 VarPTm) B
+
+| E_Pair Γ (a0 a1 b0 b1 : PTm) A B i :
+ Γ ⊢ PBind PSig A B ∈ (PUniv i) ->
+ Γ ⊢ a0 ≡ a1 ∈ A ->
+ Γ ⊢ b0 ≡ b1 ∈ subst_PTm (scons a0 VarPTm) B ->
+ Γ ⊢ PPair a0 b0 ≡ PPair a1 b1 ∈ PBind PSig A B
+
+| E_Proj1 Γ (a b : PTm) A B :
+ Γ ⊢ a ≡ b ∈ PBind PSig A B ->
+ Γ ⊢ PProj PL a ≡ PProj PL b ∈ A
+
+| E_Proj2 Γ i (a b : PTm) A B :
+ Γ ⊢ PBind PSig A B ∈ (PUniv i) ->
+ Γ ⊢ a ≡ b ∈ PBind PSig A B ->
+ Γ ⊢ PProj PR a ≡ PProj PR b ∈ subst_PTm (scons (PProj PL a) VarPTm) B
+
+| E_IndCong Γ P0 P1 (a0 a1 : PTm) b0 b1 c0 c1 i :
+ (cons PNat Γ) ⊢ P0 ∈ PUniv i ->
+ (cons PNat Γ) ⊢ P0 ≡ P1 ∈ PUniv i ->
+ Γ ⊢ a0 ≡ a1 ∈ PNat ->
+ Γ ⊢ b0 ≡ b1 ∈ subst_PTm (scons PZero VarPTm) P0 ->
+ (cons P0 ((cons PNat Γ))) ⊢ c0 ≡ c1 ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P0) ->
+ Γ ⊢ PInd P0 a0 b0 c0 ≡ PInd P1 a1 b1 c1 ∈ subst_PTm (scons a0 VarPTm) P0
+
+| E_SucCong Γ (a b : PTm) :
+ Γ ⊢ a ≡ b ∈ PNat ->
+ Γ ⊢ PSuc a ≡ PSuc b ∈ PNat
+
+| E_Conv Γ (a b : PTm) A B :
+ Γ ⊢ a ≡ b ∈ A ->
+ Γ ⊢ A ≲ B ->
+ Γ ⊢ a ≡ b ∈ B
+
+(* Beta *)
+| E_AppAbs Γ (a : PTm) b A B i:
+ Γ ⊢ PBind PPi A B ∈ PUniv i ->
+ Γ ⊢ b ∈ A ->
+ (cons A Γ) ⊢ a ∈ B ->
+ Γ ⊢ PApp (PAbs a) b ≡ subst_PTm (scons b VarPTm) a ∈ subst_PTm (scons b VarPTm ) B
+
+| E_ProjPair1 Γ (a b : PTm) A B i :
+ Γ ⊢ PBind PSig A B ∈ (PUniv i) ->
+ Γ ⊢ a ∈ A ->
+ Γ ⊢ b ∈ subst_PTm (scons a VarPTm) B ->
+ Γ ⊢ PProj PL (PPair a b) ≡ a ∈ A
+
+| E_ProjPair2 Γ (a b : PTm) A B i :
+ Γ ⊢ PBind PSig A B ∈ (PUniv i) ->
+ Γ ⊢ a ∈ A ->
+ Γ ⊢ b ∈ subst_PTm (scons a VarPTm) B ->
+ Γ ⊢ PProj PR (PPair a b) ≡ b ∈ subst_PTm (scons a VarPTm) B
+
+| E_IndZero Γ P i (b : PTm) c :
+ (cons PNat Γ) ⊢ P ∈ PUniv i ->
+ Γ ⊢ b ∈ subst_PTm (scons PZero VarPTm) P ->
+ (cons P (cons PNat Γ)) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) ->
+ Γ ⊢ PInd P PZero b c ≡ b ∈ subst_PTm (scons PZero VarPTm) P
+
+| E_IndSuc Γ P (a : PTm) b c i :
+ (cons PNat Γ) ⊢ P ∈ PUniv i ->
+ Γ ⊢ a ∈ PNat ->
+ Γ ⊢ b ∈ subst_PTm (scons PZero VarPTm) P ->
+ (cons P (cons PNat Γ)) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) ->
+ Γ ⊢ PInd P (PSuc a) b c ≡ (subst_PTm (scons (PInd P a b c) (scons a VarPTm)) c) ∈ subst_PTm (scons (PSuc a) VarPTm) P
+
+(* Eta *)
+| E_AppEta Γ (b : PTm) A B i :
+ Γ ⊢ PBind PPi A B ∈ (PUniv i) ->
+ Γ ⊢ b ∈ PBind PPi A B ->
+ Γ ⊢ PAbs (PApp (ren_PTm shift b) (VarPTm var_zero)) ≡ b ∈ PBind PPi A B
+
+| E_PairEta Γ (a : PTm ) A B i :
+ Γ ⊢ PBind PSig A B ∈ (PUniv i) ->
+ Γ ⊢ a ∈ PBind PSig A B ->
+ Γ ⊢ a ≡ PPair (PProj PL a) (PProj PR a) ∈ PBind PSig A B
+
+with LEq : list PTm -> PTm -> PTm -> Prop :=
+(* Structural *)
+| Su_Transitive Γ (A B C : PTm) :
+ Γ ⊢ A ≲ B ->
+ Γ ⊢ B ≲ C ->
+ Γ ⊢ A ≲ C
+
+(* Congruence *)
+| Su_Univ Γ i j :
+ ⊢ Γ ->
+ i <= j ->
+ Γ ⊢ PUniv i ≲ PUniv j
+
+| Su_Pi Γ (A0 A1 : PTm) B0 B1 i :
+ Γ ⊢ A0 ∈ PUniv i ->
+ Γ ⊢ A1 ≲ A0 ->
+ (cons A0 Γ) ⊢ B0 ≲ B1 ->
+ Γ ⊢ PBind PPi A0 B0 ≲ PBind PPi A1 B1
+
+| Su_Sig Γ (A0 A1 : PTm) B0 B1 i :
+ Γ ⊢ A1 ∈ PUniv i ->
+ Γ ⊢ A0 ≲ A1 ->
+ (cons A1 Γ) ⊢ B0 ≲ B1 ->
+ Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A1 B1
+
+(* Injecting from equalities *)
+| Su_Eq Γ (A : PTm) B i :
+ Γ ⊢ A ≡ B ∈ PUniv i ->
+ Γ ⊢ A ≲ B
+
+(* Projection axioms *)
+| Su_Pi_Proj1 Γ (A0 A1 : PTm) B0 B1 :
+ Γ ⊢ PBind PPi A0 B0 ≲ PBind PPi A1 B1 ->
+ Γ ⊢ A1 ≲ A0
+
+| Su_Sig_Proj1 Γ (A0 A1 : PTm) B0 B1 :
+ Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A1 B1 ->
+ Γ ⊢ A0 ≲ A1
+
+| Su_Pi_Proj2 Γ (a0 a1 A0 A1 : PTm ) B0 B1 :
+ Γ ⊢ PBind PPi A0 B0 ≲ PBind PPi A1 B1 ->
+ Γ ⊢ a0 ≡ a1 ∈ A1 ->
+ Γ ⊢ subst_PTm (scons a0 VarPTm) B0 ≲ subst_PTm (scons a1 VarPTm) B1
+
+| Su_Sig_Proj2 Γ (a0 a1 A0 A1 : PTm) B0 B1 :
+ Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A1 B1 ->
+ Γ ⊢ a0 ≡ a1 ∈ A0 ->
+ Γ ⊢ subst_PTm (scons a0 VarPTm) B0 ≲ subst_PTm (scons a1 VarPTm) B1
+
+with Wff : list PTm -> Prop :=
+| Wff_Nil :
+ ⊢ nil
+| Wff_Cons Γ (A : PTm) i :
+ ⊢ Γ ->
+ Γ ⊢ A ∈ PUniv i ->
+ (* -------------------------------- *)
+ ⊢ (cons A Γ)
+
+where
+"Γ ⊢ a ∈ A" := (Wt Γ a A) and "⊢ Γ" := (Wff Γ) and "Γ ⊢ a ≡ b ∈ A" := (Eq Γ a b A) and "Γ ⊢ A ≲ B" := (LEq Γ A B).
+
+Scheme wf_ind := Induction for Wff Sort Prop
+ with wt_ind := Induction for Wt Sort Prop
+ with eq_ind := Induction for Eq Sort Prop
+ with le_ind := Induction for LEq Sort Prop.
+
+Combined Scheme wt_mutual from wf_ind, wt_ind, eq_ind, le_ind.
+
+(* Lemma lem : *)
+(* (forall n (Γ : fin n -> PTm n), ⊢ Γ -> ...) /\ *)
+(* (forall n Γ (a A : PTm n), Γ ⊢ a ∈ A -> ...) /\ *)
+(* (forall n Γ (a b A : PTm n), Γ ⊢ a ≡ b ∈ A -> ...) /\ *)
+(* (forall n Γ (A B : PTm n), Γ ⊢ A ≲ B -> ...). *)
+(* Proof. apply wt_mutual. ... *)