Restructure proof to use hindley rosen
This commit is contained in:
Yiyun Liu
parent
9d3c3726dd
commit
21bb2944a3
1 changed files with 302 additions and 78 deletions
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@ -9,7 +9,7 @@ From stdpp Require Import relations (rtc (..), rtc_once, rtc_r, sn).
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From Hammer Require Import Tactics.
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Require Import Autosubst2.core Autosubst2.unscoped Autosubst2.syntax.
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Module βηPar.
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Module ηPar.
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Inductive R : Tm -> Tm -> Prop :=
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| VarCong i :
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R (VarTm i) (VarTm i)
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@ -20,10 +20,6 @@ Module βηPar.
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| AbsCong a0 a1 :
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R a0 a1 ->
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R (Abs a0) (Abs a1)
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| AppAbs b0 b1 a0 a1 :
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R b0 b1 ->
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R a0 a1 ->
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R (App (Abs b0) a0) (subst_Tm (scons a1 VarTm) b1)
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| AbsEta b0 b1 :
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R b0 b1 ->
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R b0 (Abs (App (ren_Tm shift b1) (VarTm var_zero))).
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@ -32,13 +28,6 @@ Module βηPar.
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Derive Inversion inv with (forall a b, R a b).
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Lemma AppAbs' b0 b1 a0 a1 u :
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u = (subst_Tm (scons a1 VarTm) b1) ->
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R b0 b1 ->
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R a0 a1 ->
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R (App (Abs b0) a0) u.
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Proof. move => ->. apply AppAbs. Qed.
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Lemma AbsEta' b0 b1 u :
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u = (Abs (App (ren_Tm shift b1) (VarTm var_zero))) ->
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R b0 b1 ->
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@ -54,7 +43,6 @@ Module βηPar.
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Proof.
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move => h. move : ρ. elim : a b /h => /=; eauto with βηPar.
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- eauto using refl.
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- move => *; apply : AppAbs'; eauto. by asimpl.
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- move => *; apply : AbsEta'; eauto. by asimpl.
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Qed.
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@ -94,33 +82,286 @@ Module βηPar.
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Proof.
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move => h. move : ρ0 ρ1.
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elim : a b /h => //=; eauto using morphing_up with βηPar.
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- move => * /=.
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apply : AppAbs'; eauto using morphing_up. by asimpl.
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- move => * /=.
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apply : AbsEta'; eauto using morphing_up. by asimpl.
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Qed.
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End βηPar.
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End ηPar.
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Module βPar.
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Inductive R : Tm -> Tm -> Prop :=
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| VarCong i :
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R (VarTm i) (VarTm i)
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| AppCong b0 b1 a0 a1 :
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R b0 b1 ->
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R a0 a1 ->
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R (App b0 a0) (App b1 a1)
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| AbsCong a0 a1 :
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R a0 a1 ->
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R (Abs a0) (Abs a1)
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| AppAbs b0 b1 a0 a1 :
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R b0 b1 ->
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R a0 a1 ->
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R (App (Abs b0) a0) (subst_Tm (scons a1 VarTm) b1).
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#[export]Hint Constructors R : βηPar.
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Derive Inversion inv with (forall a b, R a b).
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Lemma AppAbs' b0 b1 a0 a1 u :
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u = (subst_Tm (scons a1 VarTm) b1) ->
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R b0 b1 ->
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R a0 a1 ->
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R (App (Abs b0) a0) u.
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Proof. move => ->. apply AppAbs. Qed.
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Lemma refl a : R a a.
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Proof. elim : a => //=; eauto with βηPar. Qed.
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Lemma morphing a b ρ :
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R a b ->
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R (subst_Tm ρ a) (subst_Tm ρ b).
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Proof.
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move => h. move : ρ. elim : a b /h => /=; eauto with βηPar.
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- eauto using refl.
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- move => *; apply : AppAbs'; eauto. by asimpl.
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Qed.
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Lemma renaming a b ξ :
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R a b ->
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R (ren_Tm ξ a) (ren_Tm ξ b).
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Proof. substify. apply morphing. Qed.
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Definition morphing2_ok ρ0 ρ1 := forall (i : nat), R (ρ0 i) (ρ1 i).
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Lemma morphing2_ren (ξ : nat -> nat) ρ0 ρ1 :
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morphing2_ok ρ0 ρ1 ->
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morphing2_ok (funcomp (ren_Tm ξ) ρ0) (funcomp (ren_Tm ξ) ρ1).
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Proof. rewrite /morphing2_ok; eauto using renaming. Qed.
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Lemma morphing2_ext a b ρ0 ρ1 :
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R a b ->
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morphing2_ok ρ0 ρ1 ->
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morphing2_ok (scons a ρ0) (scons b ρ1).
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Proof.
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move => * [|i] //=.
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Qed.
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Lemma morphing_up ρ0 ρ1 :
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morphing2_ok ρ0 ρ1 ->
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morphing2_ok (up_Tm_Tm ρ0) (up_Tm_Tm ρ1).
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Proof.
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move => h.
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apply morphing2_ext. apply VarCong.
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by apply morphing2_ren.
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Qed.
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Lemma morphing2 a b ρ0 ρ1 :
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R a b ->
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morphing2_ok ρ0 ρ1 ->
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R (subst_Tm ρ0 a) (subst_Tm ρ1 b).
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Proof.
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move => h. move : ρ0 ρ1.
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elim : a b /h => //=; eauto using morphing_up with βηPar.
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- move => * /=.
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apply : AppAbs'; eauto using morphing_up. by asimpl.
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Qed.
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End βPar.
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Module ηexp.
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Inductive R : Tm -> Tm -> Prop :=
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| AppCong0 b0 b1 a :
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R b0 b1 ->
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R (App b0 a) (App b1 a)
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| AppCong1 b a0 a1 :
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R a0 a1 ->
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R (App b a0) (App b a1)
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| AbsCong a0 a1 :
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R a0 a1 ->
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R (Abs a0) (Abs a1)
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| AbsEta a :
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R a (Abs (App (ren_Tm shift a) (VarTm var_zero))).
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Derive Inversion inv with (forall a b, R a b).
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Lemma AbsEta' a u :
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u = (Abs (App (ren_Tm shift a) (VarTm var_zero))) ->
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R a u.
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Proof. move => ->. apply AbsEta. Qed.
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Lemma morphing a b ρ :
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R a b ->
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R (subst_Tm ρ a) (subst_Tm ρ b).
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Proof.
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move => h. move : ρ. elim : a b /h => //=; try qauto ctrs:R.
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- move => *; apply : AbsEta'; eauto. by asimpl.
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Qed.
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End ηexp.
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Module ηexps.
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#[local]Ltac solve_s_rec :=
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move => *; eapply rtc_l; eauto;
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hauto lq:on ctrs:ηexp.R.
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#[local]Ltac solve_s :=
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repeat (induction 1; last by solve_s_rec); apply rtc_refl.
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Lemma AbsCong (a b : Tm) :
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rtc ηexp.R a b ->
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rtc ηexp.R (Abs a) (Abs b).
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Proof. solve_s. Qed.
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Lemma AppCong (a0 a1 b0 b1 : Tm) :
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rtc ηexp.R a0 a1 ->
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rtc ηexp.R b0 b1 ->
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rtc ηexp.R (App a0 b0) (App a1 b1).
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Proof. solve_s. Qed.
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Lemma morphing a b ρ :
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rtc ηexp.R a b ->
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rtc ηexp.R (subst_Tm ρ a) (subst_Tm ρ b).
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Proof.
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induction 1; hauto l:on use:ηexp.morphing ctrs:rtc.
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Qed.
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Lemma renaming a b ξ :
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rtc ηexp.R a b ->
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rtc ηexp.R (ren_Tm ξ a) (ren_Tm ξ b).
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Proof. substify. apply morphing. Qed.
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Definition lifting_ok ρ0 ρ1 := forall (i : nat), rtc ηexp.R (ρ0 i) (ρ1 i).
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Lemma lifting_ren (ξ : nat -> nat) ρ0 ρ1 :
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lifting_ok ρ0 ρ1 ->
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lifting_ok (funcomp (ren_Tm ξ) ρ0) (funcomp (ren_Tm ξ) ρ1).
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Proof. rewrite /lifting_ok; eauto using renaming. Qed.
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Lemma lifting_ext a b ρ0 ρ1 :
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rtc ηexp.R a b ->
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lifting_ok ρ0 ρ1 ->
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lifting_ok (scons a ρ0) (scons b ρ1).
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Proof.
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move => * [|i] //=.
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Qed.
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Lemma lifting_up ρ0 ρ1 :
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lifting_ok ρ0 ρ1 ->
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lifting_ok (up_Tm_Tm ρ0) (up_Tm_Tm ρ1).
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Proof.
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move => h.
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apply lifting_ext. apply rtc_refl.
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by apply lifting_ren.
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Qed.
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Lemma lifting a ρ0 ρ1 :
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lifting_ok ρ0 ρ1 ->
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rtc ηexp.R (subst_Tm ρ0 a) (subst_Tm ρ1 a).
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Proof.
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move : ρ0 ρ1.
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elim : a => //=;
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qauto use:lifting_up, ηexps.AbsCong,ηexps.AppCong .
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Qed.
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End ηexps.
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Lemma subst_id b : subst_Tm (scons (VarTm var_zero) (funcomp VarTm shift)) b = b.
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symmetry. have h : b = subst_Tm VarTm b by asimpl.
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rewrite {1}h.
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apply ext_Tm.
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case => //=.
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Qed.
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Lemma βη_commute0 a b c :
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βPar.R a b ->
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ηexp.R a c ->
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exists d, rtc ηexp.R b d /\ βPar.R c d.
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Proof.
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move => h. move : c.
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elim :a b /h.
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- move => i c. elim /ηexp.inv => //=_.
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move => *. subst.
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eexists. split; last by apply βPar.refl.
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apply rtc_once. constructor.
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- move => b0 b1 a0 a1 hb ihb ha iha u.
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elim /ηexp.inv => //=_.
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+ move => b2 b3 a2 hb' [*]. subst.
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move : ihb hb' => /[apply]. move=> [b2 [ihb0 ihb1]].
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clear iha.
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exists (App b2 a1).
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split.
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sfirstorder use:ηexps.AppCong, rtc_refl.
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hauto lq:on ctrs:βPar.R use:βPar.refl.
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+ move => b2 ? a2 + [*]. subst.
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move => {}/iha {ihb} [a12 [ih0 ih1]].
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exists (App b1 a12).
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split.
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sfirstorder use:ηexps.AppCong, rtc_refl.
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hauto lq:on ctrs:βPar.R use:βPar.refl.
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+ move => *. subst. move {ihb iha}.
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eexists. split.
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apply rtc_once. apply ηexp.AbsEta. simpl.
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hauto lq:on ctrs:βPar.R use:βPar.renaming.
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- move => a0 a1 ha iha u.
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elim /ηexp.inv => //=_.
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+ qauto l:on ctrs:βPar.R use:ηexps.AbsCong.
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+ move => *. subst. move {iha}.
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eexists.
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split.
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apply rtc_once. apply ηexp.AbsEta.
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hauto lq:on ctrs:βPar.R use:βPar.renaming.
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- move => b0 b1 a0 a1 hb ihb ha iha u.
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elim /ηexp.inv => //=_.
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+ move => b2 b3 a2 + [*]. subst.
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elim /ηexp.inv => //=_.
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* move => a2 a3 + [*]. subst.
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move => /[dup] hba.
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move => {}/ihb {iha}.
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move => [bd [ih0 ih1]].
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exists (subst_Tm (scons a1 VarTm ) bd).
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split.
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** sfirstorder use:ηexps.morphing.
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** constructor; eauto.
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* move => *. subst. move {ihb iha}.
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exists (subst_Tm (scons a1 VarTm) b1).
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split. apply rtc_refl.
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constructor.
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apply : βPar.AppAbs'; cycle 1. sfirstorder use:βPar.renaming.
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constructor. by asimpl; rewrite subst_id.
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eauto.
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+ move => b2 a2 a3 + [*]. subst.
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move => {}/iha {ihb}.
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move => [a13 [ih0 ih1]].
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exists (subst_Tm (scons a13 VarTm) b1).
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split; last by constructor.
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apply ηexps.lifting.
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case => //=. eauto using rtc_refl.
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+ move => *. subst. move {ihb iha}.
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exists (Abs (App (ren_Tm shift (subst_Tm (scons a1 VarTm) b1)) (VarTm var_zero))). split.
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* apply rtc_once. constructor.
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* constructor. constructor; last by apply βPar.refl.
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apply : βPar.AppAbs'; eauto using βPar.renaming.
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by asimpl.
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Qed.
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Module IPar.
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Inductive R : Tm -> Tm -> Prop :=
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| VarCong i :
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R (VarTm i) (VarTm i)
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| AppCong b0 b1 a0 a1 :
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βηPar.R b0 b1 ->
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βηPar.R a0 a1 ->
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ηPar.R b0 b1 ->
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ηPar.R a0 a1 ->
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R (App b0 a0) (App b1 a1)
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| AbsCong a0 a1 :
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βηPar.R a0 a1 ->
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R (Abs a0) (Abs a1)
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| AppAbs b0 b1 a0 a1 :
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βηPar.R b0 b1 ->
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βηPar.R a0 a1 ->
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R (App (Abs b0) a0) (subst_Tm (scons a1 VarTm) b1).
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ηPar.R a0 a1 ->
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R (Abs a0) (Abs a1).
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Derive Inversion inv with (forall a b, R a b).
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Lemma ToβηPar a b : R a b -> βηPar.R a b.
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Proof. induction 1; hauto lq:on ctrs:βηPar.R. Qed.
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Lemma ToηPar a b : R a b -> ηPar.R a b.
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Proof. induction 1; hauto lq:on ctrs:ηPar.R. Qed.
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End IPar.
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Module OExp.
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@ -131,22 +372,22 @@ Module OExp.
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Derive Inversion inv with (forall a b, R a b).
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End OExp.
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Lemma βηO_commute a b c :
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βηPar.R a b -> OExp.R a c ->
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exists d, OExp.R b d /\ βηPar.R c d.
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Lemma ηO_commute a b c :
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ηPar.R a b -> OExp.R a c ->
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exists d, OExp.R b d /\ ηPar.R c d.
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Proof.
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hauto lq:on inv:OExp.R ctrs:OExp.R,βηPar.R use:βηPar.renaming, βηPar.refl.
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hauto lq:on inv:OExp.R ctrs:OExp.R,ηPar.R use:ηPar.renaming, ηPar.refl.
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Qed.
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Lemma βηO_commute0 a b c :
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βηPar.R a b -> rtc OExp.R a c ->
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exists d, rtc OExp.R b d /\ βηPar.R c d.
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Lemma ηO_commute0 a b c :
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ηPar.R a b -> rtc OExp.R a c ->
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exists d, rtc OExp.R b d /\ ηPar.R c d.
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Proof.
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move => + h. move : b. induction h; hauto lq:on ctrs:rtc use:βηO_commute.
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move => + h. move : b. induction h; hauto lq:on ctrs:rtc use:ηO_commute.
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Qed.
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Lemma IO_factorization a c :
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βηPar.R a c -> exists b, IPar.R a b /\ rtc OExp.R b c.
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ηPar.R a c -> exists b, IPar.R a b /\ rtc OExp.R b c.
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Proof.
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move => h. elim : a c /h.
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- move => i. exists (VarTm i).
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@ -159,9 +400,6 @@ Proof.
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- move => a0 a1 ha [a' [iha0 iha1]].
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exists (Abs a1). split. by apply IPar.AbsCong.
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apply rtc_refl.
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- move => b0 b1 a0 a1 hb [b' [ihb0 ihb1]] ha [a' [iha0 iha1]].
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eexists. split. apply IPar.AppAbs; eauto.
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apply rtc_refl.
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- move => b0 b1 hb [b' [ihb0 ihb1]].
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exists b'. split => //.
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apply : rtc_r; eauto.
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@ -169,41 +407,15 @@ Proof.
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Qed.
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Lemma IO_merge a b c :
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βηPar.R a b -> OExp.R b c -> βηPar.R a c.
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Proof. hauto lq:on inv:OExp.R ctrs:βηPar.R. Qed.
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ηPar.R a b -> OExp.R b c -> ηPar.R a c.
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Proof. hauto lq:on inv:OExp.R ctrs:ηPar.R. Qed.
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Lemma IO_merge' a b c :
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βηPar.R a b -> rtc OExp.R b c -> βηPar.R a c.
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ηPar.R a b -> rtc OExp.R b c -> ηPar.R a c.
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Proof. induction 2; hauto l:on use:IO_merge. Qed.
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(* Lemma AppAbsEta b0 a0 b1 a1 : *)
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(* βηPar.R b0 (Abs b1) -> *)
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(* βηPar.R a0 a1 -> *)
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(* βηPar.R *)
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Lemma diamond a b c : IPar.R a b -> IPar.R a c -> exists d, IPar.R b d /\ IPar.R c d.
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Proof.
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elim : a b c.
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- hauto lq:on inv:IPar.R ctrs:IPar.R.
|
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- move => a iha b c.
|
||||
elim /IPar.inv => //=_.
|
||||
move => a0 a1 + [?]?. subst.
|
||||
move => /[swap]. elim /IPar.inv => //=_.
|
||||
move => a0 a2 + [?]?. subst.
|
||||
move /IO_factorization.
|
||||
move => [a3 [h0 h1]].
|
||||
move /IO_factorization.
|
||||
move => [a4 [h2 h3]].
|
||||
move : iha h0 h2. repeat move/[apply].
|
||||
move => [d [h2 h4]].
|
||||
move :
|
||||
|
||||
|
||||
|
||||
|
||||
Lemma diamond a b0 b1 :
|
||||
βηPar.R a b0 -> βηPar.R a b1 -> exists c, βηPar.R b0 c /\ βηPar.R b1 c.
|
||||
ηPar.R a b0 -> ηPar.R a b1 -> exists c, ηPar.R b0 c /\ ηPar.R b1 c.
|
||||
Proof.
|
||||
move => h. move : b1.
|
||||
elim : a b0 / h.
|
||||
|
|
@ -213,7 +425,7 @@ Proof.
|
|||
split => //=.
|
||||
apply : IO_merge'; eauto.
|
||||
constructor.
|
||||
apply βηPar.refl.
|
||||
apply ηPar.refl.
|
||||
- move => b0 b1 a0 a1 hb ihb ha iha b2 /IO_factorization.
|
||||
move => [u [h0 h1]].
|
||||
elim /IPar.inv : h0=>//_.
|
||||
|
|
@ -221,15 +433,27 @@ Proof.
|
|||
have {}/ihb [b14 [ihb0 ihb1]] := hb'.
|
||||
have {}/iha [a13 [iha0 iha1]] := ha'.
|
||||
set q := (App _ _) in h1.
|
||||
have : βηPar.R q (App b14 a13) by constructor.
|
||||
move : βηO_commute0 h1. subst q. repeat move/[apply].
|
||||
have : ηPar.R q (App b14 a13) by constructor.
|
||||
move : ηO_commute0 h1. subst q. repeat move/[apply].
|
||||
move => [d [h0 h1]].
|
||||
exists d.
|
||||
split => //.
|
||||
apply : IO_merge'; eauto using βηPar.AppCong.
|
||||
+ move => b3 b4 a2 a3 hb' ha' [*]. subst.
|
||||
move /IO_factorization : hb.
|
||||
move => [b []].
|
||||
elim /IPar.inv => //=_ a2 b5 hb'' [?]? ho. subst.
|
||||
move /βηPar.AbsCong : hb'. move => {}/ihb. move => [b14 [ihb0 ihb1]].
|
||||
move : iha ha' => /[apply]. move => [a13 [iha0 iha1]].
|
||||
apply : IO_merge'; eauto using ηPar.AppCong.
|
||||
- move => a0 a1 ha iha u /IO_factorization.
|
||||
move => [v [ih0 ih1]].
|
||||
elim /IPar.inv : ih0 => //= _.
|
||||
move => a2 a3 + [*]. subst.
|
||||
move => {}/iha. move => [a2 [h0 h1]].
|
||||
move /ηPar.AbsCong in h1.
|
||||
move : ηO_commute0 ih1 h1; repeat move/[apply].
|
||||
move => [d [h1 h2]].
|
||||
exists d.
|
||||
split => //.
|
||||
apply : IO_merge'; eauto.
|
||||
by constructor.
|
||||
- move => b0 b1 hb ihb b2 {}/ihb.
|
||||
move => [c [h0 h1]].
|
||||
eexists. split; cycle 1.
|
||||
apply ηPar.AbsEta; eauto.
|
||||
hauto lq:on ctrs:ηPar.R use:ηPar.renaming.
|
||||
Qed.
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue