Finish Par and RPar lemmas
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1 changed files with 96 additions and 21 deletions
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@ -33,8 +33,8 @@ Module Par.
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R b0 b1 ->
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R c0 c1 ->
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R (App (Pair a0 b0) c0) (Pair (App a1 c1) (App b1 c1))
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| AppBool b a :
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R (App (BVal b) a) (BVal b)
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(* | AppBool b a : *)
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(* R (App (BVal b) a) (BVal b) *)
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| ProjAbs p a0 a1 :
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R a0 a1 ->
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R (Proj p (Abs a0)) (Abs (Proj p a1))
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@ -42,20 +42,20 @@ Module Par.
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R a0 a1 ->
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R b0 b1 ->
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R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1)
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| ProjBool p b :
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R (Proj p (BVal b)) (BVal b)
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| IfAbs a0 a1 b0 b1 c0 c1 d0 d1 :
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R a0 a1 ->
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R b0 b1 ->
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R c0 c1 ->
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R d0 d1 ->
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R (If (Abs a0) b0 c0) (Abs (If a0 (ren_Tm shift b0) (ren_Tm shift c0)))
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| IfPair a0 a1 b0 b1 c0 c1 d0 d1 :
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R a0 a1 ->
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R b0 b1 ->
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R c0 c1 ->
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R d0 d1 ->
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R (If (Pair a0 b0) c0 d0) (Pair (If a0 c0 d0) (If b0 c0 d0))
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(* | ProjBool p b : *)
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(* R (Proj p (BVal b)) (BVal b) *)
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(* | IfAbs a0 a1 b0 b1 c0 c1 d0 d1 : *)
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(* R a0 a1 -> *)
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(* R b0 b1 -> *)
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(* R c0 c1 -> *)
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(* R d0 d1 -> *)
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(* R (If (Abs a0) b0 c0) (Abs (If a0 (ren_Tm shift b0) (ren_Tm shift c0))) *)
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(* | IfPair a0 a1 b0 b1 c0 c1 d0 d1 : *)
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(* R a0 a1 -> *)
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(* R b0 b1 -> *)
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(* R c0 c1 -> *)
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(* R d0 d1 -> *)
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(* R (If (Pair a0 b0) c0 d0) (Pair (If a0 c0 d0) (If b0 c0 d0)) *)
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| IfBool a b0 b1 c0 c1 :
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R b0 b1 ->
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R c0 c1 ->
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@ -92,7 +92,16 @@ Module Par.
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| BotCong :
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R Bot Bot
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| UnivCong i :
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R (Univ i) (Univ i).
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R (Univ i) (Univ i)
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| BoolCong :
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R Bool Bool
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| BValCong b :
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R (BVal b) (BVal b)
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| IfCong a0 a1 b0 b1 c0 c1 :
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R a0 a1 ->
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R b0 b1 ->
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R c0 c1 ->
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R (If a0 b0 c0) (If a1 b1 c1).
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Lemma refl n (a : Tm n) : R a a.
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elim : n /a; hauto ctrs:R.
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@ -112,6 +121,13 @@ Module Par.
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R (Proj p (Pair a0 b0)) t.
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Proof. move => > ->. apply ProjPair. Qed.
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Lemma IfBool' n a b0 b1 c0 c1 u :
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u = (if a then (b1 : Tm n) else c1) ->
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R b0 b1 ->
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R c0 c1 ->
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R (If (BVal a) b0 c0) u.
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Proof. move => ->. apply IfBool. Qed.
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Lemma AppEta' n (a0 a1 b : Tm n) :
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b = (Abs (App (ren_Tm shift a1) (VarTm var_zero))) ->
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R a0 a1 ->
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@ -126,7 +142,7 @@ Module Par.
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move => *; apply : AppAbs'; eauto; by asimpl.
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all : match goal with
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| [ |- context[var_zero]] => move => *; apply : AppEta'; eauto; by asimpl
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| _ => qauto ctrs:R use:ProjPair'
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| _ => qauto ctrs:R use:ProjPair', IfBool'
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end.
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Qed.
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@ -143,6 +159,7 @@ Module Par.
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- hauto lq:on rew:off ctrs:R.
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- hauto l:on inv:option use:renaming ctrs:R.
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- hauto lq:on use:ProjPair'.
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- hauto q:on use:IfBool'.
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- move => n a0 a1 ha iha m ρ0 ρ1 hρ /=.
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apply : AppEta'; eauto. by asimpl.
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- hauto lq:on ctrs:R.
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@ -154,6 +171,9 @@ Module Par.
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- hauto l:on inv:option ctrs:R use:renaming.
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- sfirstorder.
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- sfirstorder.
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- sfirstorder.
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- sfirstorder.
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- hauto lq:on ctrs:R.
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Qed.
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Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
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@ -193,6 +213,12 @@ Module Par.
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move : ihb => [c0 [? ?]]. subst.
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eexists. split. by eauto using ProjPair.
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hauto q:on.
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- move => n a b0 b1 c0 c1 hb ihb hc ihc m ξ []//= p0 p1 p2 [h *]. subst.
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spec_refl.
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move : ihb => [b0 [? ?]].
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move : ihc => [c0 [? ?]]. subst.
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case : p0 h => //=.
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hauto q:on use:IfBool'.
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- move => n a0 a1 ha iha m ξ a ?. subst.
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spec_refl. move : iha => [a0 [? ?]]. subst.
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eexists. split. apply AppEta; eauto.
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@ -233,6 +259,11 @@ Module Par.
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by asimpl.
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- move => n n0 ξ []//=. hauto l:on.
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- move => n i n0 ξ []//=. hauto l:on.
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- move => n m ξ []//=. hauto l:on.
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- move => n b m ξ []//=. hauto l:on.
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- move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc m ξ []//= t t0 t1[*]. subst.
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spec_refl.
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qauto l:on ctrs:R.
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Qed.
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End Par.
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@ -321,6 +352,10 @@ Module RPar.
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R a0 a1 ->
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R b0 b1 ->
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R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1)
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| IfBool a b0 b1 c0 c1 :
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R b0 b1 ->
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R c0 c1 ->
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R (If (BVal a) b0 c0) (if a then b1 else c1)
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(*************** Congruence ********************)
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@ -346,7 +381,16 @@ Module RPar.
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| BotCong :
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R Bot Bot
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| UnivCong i :
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R (Univ i) (Univ i).
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R (Univ i) (Univ i)
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| BoolCong :
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R Bool Bool
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| BValCong b :
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R (BVal b) (BVal b)
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| IfCong a0 a1 b0 b1 c0 c1 :
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R a0 a1 ->
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R b0 b1 ->
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R c0 c1 ->
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R (If a0 b0 c0) (If a1 b1 c1).
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Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
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@ -369,13 +413,21 @@ Module RPar.
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R (Proj p (Pair a0 b0)) t.
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Proof. move => > ->. apply ProjPair. Qed.
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Lemma IfBool' n a b0 b1 c0 c1 u :
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u = (if a then (b1 : Tm n) else c1) ->
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R b0 b1 ->
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R c0 c1 ->
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R (If (BVal a) b0 c0) u.
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Proof. move => ->. apply IfBool. Qed.
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Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
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R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
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Proof.
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move => h. move : m ξ.
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elim : n a b /h.
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move => *; apply : AppAbs'; eauto; by asimpl.
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all : qauto ctrs:R use:ProjPair'.
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all : qauto ctrs:R use:ProjPair', IfBool'.
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Qed.
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Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> Tm m) (ξ : fin m -> fin p) :
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@ -406,6 +458,7 @@ Module RPar.
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- hauto lq:on ctrs:R.
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- hauto lq:on ctrs:R use:morphing_up.
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- hauto lq:on ctrs:R use:ProjPair' use:morphing_up.
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- hauto lq:on ctrs:R use:IfBool' use:morphing_up.
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- hauto lq:on ctrs:R use:morphing_up.
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- hauto lq:on ctrs:R use:morphing_up.
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- hauto lq:on ctrs:R use:morphing_up.
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@ -414,6 +467,9 @@ Module RPar.
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- hauto lq:on ctrs:R use:morphing_up.
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- hauto lq:on ctrs:R.
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- hauto lq:on ctrs:R.
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- hauto lq:on ctrs:R.
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- hauto lq:on ctrs:R.
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- hauto lq:on ctrs:R.
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Qed.
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Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
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@ -434,7 +490,7 @@ Module RPar.
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var_or_bot a ->
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a = b -> ~~ var_or_bot b -> False.
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Proof.
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hauto lq:on inv:Tm.
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hauto l:on inv:Tm.
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Qed.
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Lemma var_or_bot_up n m (ρ : fin n -> Tm m) :
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@ -504,6 +560,14 @@ Module RPar.
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move : ihb => [c0 [? ?]]. subst.
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eexists. split. by eauto using ProjPair.
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hauto q:on.
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- move => n a b0 b1 c0 c1 hb ihb hc ihc m ρ hρ []//=; first by antiimp.
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move => t t0 t1 [h *]. subst.
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case : t h => //=; first by antiimp.
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move => b [*]. subst.
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have {}/ihb := (hρ) => ihb.
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have {}/ihc := (hρ) => ihc.
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spec_refl.
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hauto q:on use:IfBool.
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- move => n i m ρ hρ []//=.
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hauto l:on.
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- move => n a0 a1 ha iha m ρ hρ []//=; first by antiimp.
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@ -553,6 +617,17 @@ Module RPar.
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- hauto q:on ctrs:R inv:Tm.
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- move => n i n0 ρ hρ []//=; first by antiimp.
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hauto l:on.
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- move => n m ρ hρ []//=; first by antiimp.
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hauto lq:on ctrs:R.
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- move => n b m ρ hρ []//; first by antiimp.
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hauto lq:on ctrs:R.
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- move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc m ρ hρ []//=; first by antiimp.
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move => t t0 t1 [*]. subst.
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have {}/iha := (hρ) => iha.
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have {}/ihb := (hρ) => ihb.
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have {}/ihc := (hρ) => ihc.
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spec_refl.
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hauto lq:on ctrs:R.
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Qed.
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End RPar.
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