Recover the contra lemmas
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5 changed files with 110 additions and 28 deletions
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@ -72,7 +72,9 @@ Module Par.
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| ConstCong k :
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R (Const k) (Const k)
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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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| BotCong :
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R Bot Bot.
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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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@ -134,6 +136,7 @@ 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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Qed.
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Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
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@ -213,6 +216,7 @@ Module Par.
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by asimpl.
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- move => n k m ξ []//=. hauto l:on.
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- move => n i n0 ξ []//=. hauto l:on.
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- hauto q:on inv:Tm ctrs:R.
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Qed.
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End Par.
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@ -277,7 +281,7 @@ End Pars.
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Definition var_or_const {n} (a : Tm n) :=
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match a with
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| VarTm _ => true
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| Const _ => true
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| Bot => true
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| _ => false
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end.
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@ -326,7 +330,9 @@ Module RPar.
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| ConstCong k :
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R (Const k) (Const k)
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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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| BotCong :
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R Bot Bot.
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Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
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@ -394,6 +400,7 @@ 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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Qed.
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Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
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@ -533,6 +540,7 @@ 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 []//=; hauto lq:on ctrs:R.
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Qed.
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End RPar.
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@ -573,7 +581,9 @@ Module RPar'.
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| ConstCong k :
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R (Const k) (Const k)
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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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| BotCong :
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R Bot Bot.
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Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
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@ -639,6 +649,7 @@ 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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Qed.
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Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
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@ -758,6 +769,7 @@ 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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- hauto q:on inv:Tm ctrs:R.
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Qed.
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End RPar'.
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@ -896,7 +908,9 @@ Module EPar.
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| ConstCong k :
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R (Const k) (Const k)
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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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| BotCong :
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R Bot Bot.
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Lemma refl n (a : Tm n) : EPar.R a a.
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Proof.
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@ -941,6 +955,7 @@ Module EPar.
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- hauto l:on ctrs:R use:renaming inv:option.
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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 a0 a1 (b0 b1 : Tm n) :
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@ -1344,6 +1359,7 @@ Proof.
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- hauto lq:on inv:RPar.R ctrs:EPar.R, rtc use:RPars.BindCong.
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- hauto l:on ctrs:EPar.R inv:RPar.R.
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- hauto l:on ctrs:EPar.R inv:RPar.R.
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- hauto l:on ctrs:EPar.R inv:RPar.R.
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Qed.
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Lemma commutativity1 n (a b0 b1 : Tm n) :
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@ -1457,6 +1473,20 @@ Lemma Const_EPar' n k (u : Tm n) :
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- hauto l:on ctrs:OExp.R.
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Qed.
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Lemma Bot_EPar' n (u : Tm n) :
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EPar.R (Bot) u ->
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rtc OExp.R (Bot) u.
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move E : (Bot) => t h.
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move : E. elim : n t u /h => //=.
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- move => n a0 a1 h ih ?. subst.
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specialize ih with (1 := eq_refl).
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hauto lq:on ctrs:OExp.R use:rtc_r.
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- move => n a0 a1 h ih ?. subst.
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specialize ih with (1 := eq_refl).
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hauto lq:on ctrs:OExp.R use:rtc_r.
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- hauto l:on ctrs:OExp.R.
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Qed.
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Lemma Univ_EPar' n i (u : Tm n) :
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EPar.R (Univ i) u ->
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rtc OExp.R (Univ i) u.
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@ -1520,6 +1550,7 @@ Proof.
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exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
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- qauto use:Const_EPar', EPar.refl.
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- qauto use:Univ_EPar', EPar.refl.
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- qauto use:Bot_EPar', EPar.refl.
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Qed.
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Function tstar {n} (a : Tm n) :=
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@ -1537,6 +1568,7 @@ Function tstar {n} (a : Tm n) :=
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| TBind p a b => TBind p (tstar a) (tstar b)
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| Const k => Const k
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| Univ i => Univ i
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| Bot => Bot
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end.
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Lemma RPar_triangle n (a : Tm n) : forall b, RPar.R a b -> RPar.R b (tstar a).
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@ -1555,6 +1587,7 @@ Proof.
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- hauto lq:on inv:RPar.R ctrs:RPar.R.
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- hauto lq:on inv:RPar.R ctrs:RPar.R.
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- hauto lq:on inv:RPar.R ctrs:RPar.R.
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- hauto lq:on inv:RPar.R ctrs:RPar.R.
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Qed.
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Function tstar' {n} (a : Tm n) :=
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@ -1569,6 +1602,7 @@ Function tstar' {n} (a : Tm n) :=
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| TBind p a b => TBind p (tstar' a) (tstar' b)
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| Const k => Const k
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| Univ i => Univ i
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| Bot => Bot
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end.
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Lemma RPar'_triangle n (a : Tm n) : forall b, RPar'.R a b -> RPar'.R b (tstar' a).
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@ -1585,6 +1619,7 @@ Proof.
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- hauto lq:on inv:RPar'.R ctrs:RPar'.R.
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- hauto lq:on inv:RPar'.R ctrs:RPar'.R.
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- hauto lq:on inv:RPar'.R ctrs:RPar'.R.
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- hauto lq:on inv:RPar'.R ctrs:RPar'.R.
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Qed.
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Lemma RPar_diamond n (c a1 b1 : Tm n) :
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@ -1651,7 +1686,9 @@ Inductive prov {n} : Tm n -> Tm n -> Prop :=
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| P_Var i :
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prov (VarTm i) (VarTm i)
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| P_Univ i :
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prov (Univ i) (Univ i).
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prov (Univ i) (Univ i)
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| P_Bot :
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prov Bot Bot.
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Lemma ERed_EPar n (a b : Tm n) : ERed.R a b -> EPar.R a b.
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Proof.
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@ -1671,6 +1708,7 @@ Proof.
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- eauto using EReds.BindCong.
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- auto using rtc_refl.
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- auto using rtc_refl.
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- auto using rtc_refl.
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Qed.
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Lemma EPar_Par n (a b : Tm n) : EPar.R a b -> Par.R a b.
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@ -1723,6 +1761,7 @@ Proof.
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- hauto lq:on ctrs:prov inv:RPar.R.
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- hauto l:on ctrs:RPar.R inv:RPar.R.
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- hauto l:on ctrs:RPar.R inv:RPar.R.
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- hauto l:on ctrs:RPar.R inv:RPar.R.
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Qed.
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@ -1777,6 +1816,7 @@ Proof.
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- hauto lq:on inv:ERed.R, prov ctrs:prov.
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- hauto lq:on inv:ERed.R, prov ctrs:prov.
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- hauto lq:on inv:ERed.R, prov ctrs:prov.
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- hauto lq:on inv:ERed.R, prov ctrs:prov.
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Qed.
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Lemma prov_ereds n (u : Tm n) a b : prov u a -> rtc ERed.R a b -> prov u b.
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@ -1787,13 +1827,14 @@ Qed.
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Fixpoint extract {n} (a : Tm n) : Tm n :=
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match a with
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| TBind p A B => TBind p A B
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| Abs a => subst_Tm (scons (Const TPi) VarTm) (extract a)
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| Abs a => subst_Tm (scons Bot VarTm) (extract a)
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| App a b => extract a
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| Pair a b => extract a
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| Proj p a => extract a
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| Const k => Const k
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| VarTm i => VarTm i
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| Univ i => Univ i
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| Bot => Bot
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end.
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Lemma ren_extract n m (a : Tm n) (ξ : fin n -> fin m) :
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@ -1810,6 +1851,7 @@ Proof.
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- hauto q:on.
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- sfirstorder.
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- sfirstorder.
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- sfirstorder.
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Qed.
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Lemma ren_morphing n m (a : Tm n) (ρ : fin n -> Tm m) :
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@ -1828,7 +1870,7 @@ Proof.
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Qed.
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Lemma ren_subst_bot n (a : Tm (S n)) :
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extract (subst_Tm (scons (Const TPi) VarTm) a) = subst_Tm (scons (Const TPi) VarTm) (extract a).
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extract (subst_Tm (scons Bot VarTm) a) = subst_Tm (scons Bot VarTm) (extract a).
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Proof.
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apply ren_morphing. destruct i as [i|] => //=.
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Qed.
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@ -1839,6 +1881,7 @@ Definition prov_extract_spec {n} u (a : Tm n) :=
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| Univ i => extract a = Univ i
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| VarTm i => extract a = VarTm i
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| (Const i) => extract a = (Const i)
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| Bot => extract a = Bot
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| _ => True
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end.
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@ -1851,24 +1894,28 @@ Proof.
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- move => h a ha ih.
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case : h ha ih => //=.
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+ move => i ha ih.
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move /(_ (Const TPi)) in ih.
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move /(_ Bot) in ih.
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rewrite -ih.
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by rewrite ren_subst_bot.
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+ move => p A B h ih.
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move /(_ (Const TPi)) : ih => [A0][B0][h0][h1]h2.
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move /(_ Bot) : ih => [A0][B0][h0][h1]h2.
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rewrite ren_subst_bot in h0.
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rewrite h0.
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eauto.
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+ move => p _ /(_ (Const TPi)).
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+ move => p _ /(_ Bot).
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by rewrite ren_subst_bot.
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+ move => i h /(_ (Const TPi)).
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+ move => i h /(_ Bot).
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by rewrite ren_subst_bot => ->.
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+ move /(_ Bot).
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move => h /(_ Bot).
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by rewrite ren_subst_bot.
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- hauto lq:on.
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- hauto lq:on.
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- hauto lq:on.
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- case => //=.
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- sfirstorder.
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- sfirstorder.
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- sfirstorder.
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Qed.
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Definition union {A : Type} (R0 R1 : A -> A -> Prop) a b :=
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@ -2072,6 +2119,7 @@ Proof.
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- sfirstorder use:ERPars.BindCong.
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- sfirstorder.
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- sfirstorder.
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- sfirstorder.
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Qed.
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Lemma Pars_ERPar n (a b : Tm n) : rtc Par.R a b -> rtc ERPar.R a b.
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@ -2371,18 +2419,19 @@ Fixpoint ne {n} (a : Tm n) :=
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| Proj _ a => ne a
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| Pair _ _ => false
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| Const _ => false
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| Bot => true
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end
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with nf {n} (a : Tm n) :=
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match a with
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| VarTm i => true
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| TBind _ A B => nf A && nf B
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| (Const TPi) => true
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| App a b => ne a && nf b
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| Abs a => nf a
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| Univ _ => true
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| Proj _ a => ne a
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| Pair a b => nf a && nf b
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| Const _ => true
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| Bot => true
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end.
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Lemma ne_nf n a : @ne n a -> nf a.
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@ -2469,10 +2518,10 @@ Proof.
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Qed.
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Lemma ext_wn n (a : Tm n) :
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wn (App a (Const TPi)) ->
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wn (App a Bot) ->
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wn a.
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Proof.
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move E : (App a (Const TPi)) => a0 [v [hr hv]].
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move E : (App a (Bot)) => a0 [v [hr hv]].
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move : a E.
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move : hv.
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elim : a0 v / hr.
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@ -2483,9 +2532,9 @@ Proof.
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case : a0 hr0=>// => b0 b1.
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elim /RPar'.inv=>// _.
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+ move => a0 a3 b2 b3 ? ? [? ?] ? [? ?]. subst.
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have ? : b3 = (Const TPi) by hauto lq:on inv:RPar'.R. subst.
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have ? : b3 = (Bot) by hauto lq:on inv:RPar'.R. subst.
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suff : wn (Abs a3) by hauto lq:on ctrs:RPar'.R, rtc unfold:wn.
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have : wn (subst_Tm (scons (Const TPi) VarTm) a3) by sfirstorder.
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have : wn (subst_Tm (scons (Bot) VarTm) a3) by sfirstorder.
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move => h. apply wn_abs.
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move : h. apply wn_antirenaming.
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hauto lq:on rew:off inv:option.
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