3559 lines
107 KiB
Coq
3559 lines
107 KiB
Coq
From Ltac2 Require Ltac2.
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Import Ltac2.Notations.
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Import Ltac2.Control.
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Require Import ssreflect ssrbool.
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Require Import FunInd.
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Require Import Arith.Wf_nat (well_founded_lt_compat).
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Require Import Psatz.
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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.fintype Autosubst2.syntax common.
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Require Import Btauto.
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Require Import Cdcl.Itauto.
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Ltac2 spec_refl () :=
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List.iter
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(fun a => match a with
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| (i, _, _) =>
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let h := Control.hyp i in
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try (specialize $h with (1 := eq_refl))
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end) (Control.hyps ()).
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Ltac spec_refl := ltac2:(Control.enter spec_refl).
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Module EPar.
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Inductive R {n} : PTm n -> PTm n -> Prop :=
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(****************** Eta ***********************)
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| AppEta a0 a1 :
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R a0 a1 ->
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R (PAbs (PApp (ren_PTm shift a0) (VarPTm var_zero))) a1
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| PairEta a0 a1 :
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R a0 a1 ->
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R (PPair (PProj PL a0) (PProj PR a0)) a1
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(*************** Congruence ********************)
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| AbsCong a0 a1 :
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R a0 a1 ->
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R (PAbs a0) (PAbs a1)
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| AppCong a0 a1 b0 b1 :
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R a0 a1 ->
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R b0 b1 ->
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R (PApp a0 b0) (PApp a1 b1)
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| PairCong a0 a1 b0 b1 :
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R a0 a1 ->
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R b0 b1 ->
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R (PPair a0 b0) (PPair a1 b1)
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| ProjCong p a0 a1 :
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R a0 a1 ->
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R (PProj p a0) (PProj p a1)
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| VarTm i :
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R (VarPTm i) (VarPTm i)
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| Univ i :
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R (PUniv i) (PUniv i)
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| BindCong p A0 A1 B0 B1 :
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R A0 A1 ->
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R B0 B1 ->
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R (PBind p A0 B0) (PBind p A1 B1)
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| BotCong :
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R PBot PBot
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| NatCong :
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R PNat PNat
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| IndCong P0 P1 a0 a1 b0 b1 c0 c1 :
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R P0 P1 ->
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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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(* ----------------------- *)
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R (PInd P0 a0 b0 c0) (PInd P1 a1 b1 c1)
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| ZeroCong :
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R PZero PZero
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| SucCong a0 a1 :
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R a0 a1 ->
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(* ------------ *)
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R (PSuc a0) (PSuc a1).
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Lemma refl n (a : PTm n) : R a a.
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Proof.
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elim : n / a; hauto lq:on ctrs:R.
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Qed.
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Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop.
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Lemma AppEta' n a0 a1 (u : PTm n) :
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u = (PAbs (PApp (ren_PTm shift a0) (VarPTm var_zero))) ->
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R a0 a1 ->
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R u a1.
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Proof. move => ->. apply AppEta. Qed.
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Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
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R a b -> R (ren_PTm ξ a) (ren_PTm 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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all : try qauto ctrs:R.
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move => n a0 a1 ha iha m ξ /=.
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eapply AppEta'; eauto. by asimpl.
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Qed.
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Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> PTm m) (ξ : fin m -> fin p) :
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(forall i, R (ρ0 i) (ρ1 i)) ->
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(forall i, R ((funcomp (ren_PTm ξ) ρ0) i) ((funcomp (ren_PTm ξ) ρ1) i)).
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Proof. eauto using renaming. Qed.
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Lemma morphing_ext n m (ρ0 ρ1 : fin n -> PTm m) a b :
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R a b ->
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(forall i, R (ρ0 i) (ρ1 i)) ->
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(forall i, R ((scons a ρ0) i) ((scons b ρ1) i)).
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Proof. hauto q:on inv:option. Qed.
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Lemma morphing_up n m (ρ0 ρ1 : fin n -> PTm m) :
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(forall i, R (ρ0 i) (ρ1 i)) ->
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(forall i, R (up_PTm_PTm ρ0 i) (up_PTm_PTm ρ1 i)).
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Proof. hauto l:on ctrs:R use:morphing_ext, morphing_ren unfold:up_PTm_PTm. Qed.
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Lemma morphing n m (a b : PTm n) (ρ0 ρ1 : fin n -> PTm m) :
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(forall i, R (ρ0 i) (ρ1 i)) ->
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R a b -> R (subst_PTm ρ0 a) (subst_PTm ρ1 b).
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Proof.
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move => + h. move : m ρ0 ρ1. elim : n a b / h => n.
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move => a0 a1 ha iha m ρ0 ρ1 hρ /=.
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eapply AppEta'; eauto. by asimpl.
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all : hauto lq:on ctrs:R use:morphing_up.
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Qed.
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Lemma substing n m (a b : PTm n) (ρ : fin n -> PTm m) :
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R a b -> R (subst_PTm ρ a) (subst_PTm ρ b).
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Proof. hauto l:on use:morphing, refl. Qed.
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End EPar.
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Inductive SNe {n} : PTm n -> Prop :=
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| N_Var i :
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SNe (VarPTm i)
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| N_App a b :
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SNe a ->
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SN b ->
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SNe (PApp a b)
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| N_Proj p a :
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SNe a ->
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SNe (PProj p a)
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| N_Bot :
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SNe PBot
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| N_Ind P a b c :
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SN P ->
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SNe a ->
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SN b ->
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SN c ->
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SNe (PInd P a b c)
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with SN {n} : PTm n -> Prop :=
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| N_Pair a b :
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SN a ->
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SN b ->
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SN (PPair a b)
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| N_Abs a :
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SN a ->
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SN (PAbs a)
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| N_SNe a :
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SNe a ->
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SN a
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| N_Exp a b :
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TRedSN a b ->
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SN b ->
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SN a
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| N_Bind p A B :
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SN A ->
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SN B ->
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SN (PBind p A B)
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| N_Univ i :
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SN (PUniv i)
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| N_Nat :
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SN PNat
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| N_Zero :
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SN PZero
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| N_Suc a :
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SN a ->
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SN (PSuc a)
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with TRedSN {n} : PTm n -> PTm n -> Prop :=
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| N_β a b :
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SN b ->
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TRedSN (PApp (PAbs a) b) (subst_PTm (scons b VarPTm) a)
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| N_AppL a0 a1 b :
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SN b ->
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TRedSN a0 a1 ->
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TRedSN (PApp a0 b) (PApp a1 b)
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| N_ProjPairL a b :
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SN b ->
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TRedSN (PProj PL (PPair a b)) a
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| N_ProjPairR a b :
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SN a ->
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TRedSN (PProj PR (PPair a b)) b
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| N_ProjCong p a b :
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TRedSN a b ->
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TRedSN (PProj p a) (PProj p b)
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| N_IndZero P b c :
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SN P ->
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SN b ->
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SN c ->
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TRedSN (PInd P PZero b c) b
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| N_IndSuc P a b c :
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SN P ->
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SN a ->
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SN b ->
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SN c ->
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TRedSN (PInd P (PSuc a) b c) (subst_PTm (scons (PInd P a b c) (scons a VarPTm)) c)
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| N_IndCong P a0 a1 b c :
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SN P ->
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SN b ->
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SN c ->
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TRedSN a0 a1 ->
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TRedSN (PInd P a0 b c) (PInd P a1 b c).
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Derive Dependent Inversion tred_inv with (forall n (a b : PTm n), TRedSN a b) Sort Prop.
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Inductive SNat {n} : PTm n -> Prop :=
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| S_Zero : SNat PZero
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| S_Neu a : SNe a -> SNat a
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| S_Suc a : SNat a -> SNat (PSuc a)
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| S_Red a b : TRedSN a b -> SNat b -> SNat a.
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Lemma PProj_imp n p a :
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@ishf n a ->
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~~ ispair a ->
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~ SN (PProj p a).
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Proof.
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move => + + h. move E : (PProj p a) h => u h.
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move : p a E.
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elim : n u / h => //=.
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hauto lq:on inv:SNe,PTm.
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hauto lq:on inv:TRedSN.
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Qed.
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Lemma PApp_imp n a b :
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@ishf n a ->
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~~ isabs a ->
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~ SN (PApp a b).
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Proof.
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move => + + h. move E : (PApp a b) h => u h.
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move : a b E. elim : n u /h=>//=.
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hauto lq:on inv:SNe,PTm.
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hauto lq:on inv:TRedSN.
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Qed.
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Lemma PInd_imp n P (a : PTm n) b c :
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ishf a ->
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~~ iszero a ->
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~~ issuc a ->
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~ SN (PInd P a b c).
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Proof.
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move => + + + h. move E : (PInd P a b c) h => u h.
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move : P a b c E.
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elim : n u /h => //=.
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hauto lq:on inv:SNe,PTm.
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hauto lq:on inv:TRedSN.
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Qed.
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Lemma PProjAbs_imp n p (a : PTm (S n)) :
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~ SN (PProj p (PAbs a)).
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Proof.
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sfirstorder use:PProj_imp.
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Qed.
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Lemma PAppPair_imp n (a b0 b1 : PTm n ) :
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~ SN (PApp (PPair b0 b1) a).
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Proof.
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sfirstorder use:PApp_imp.
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Qed.
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Lemma PAppBind_imp n p (A : PTm n) B b :
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~ SN (PApp (PBind p A B) b).
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Proof.
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sfirstorder use:PApp_imp.
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Qed.
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Lemma PProjBind_imp n p p' (A : PTm n) B :
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~ SN (PProj p (PBind p' A B)).
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Proof.
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move E :(PProj p (PBind p' A B)) => u hu.
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move : p p' A B E.
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elim : n u /hu=>//=.
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hauto lq:on inv:SNe.
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hauto lq:on inv:TRedSN.
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Qed.
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Scheme sne_ind := Induction for SNe Sort Prop
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with sn_ind := Induction for SN Sort Prop
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with sred_ind := Induction for TRedSN Sort Prop.
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Combined Scheme sn_mutual from sne_ind, sn_ind, sred_ind.
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Fixpoint ne {n} (a : PTm n) :=
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match a with
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| VarPTm i => true
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| PApp a b => ne a && nf b
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| PAbs a => false
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| PPair _ _ => false
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| PProj _ a => ne a
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| PUniv _ => false
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| PBind _ _ _ => false
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| PBot => true
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| PInd P a b c => nf P && ne a && nf b && nf c
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| PNat => false
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| PSuc a => false
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| PZero => false
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end
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with nf {n} (a : PTm n) :=
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match a with
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| VarPTm i => true
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| PApp a b => ne a && nf b
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| PAbs a => nf a
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| PPair a b => nf a && nf b
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| PProj _ a => ne a
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| PUniv _ => true
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| PBind _ A B => nf A && nf B
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| PBot => true
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| PInd P a b c => nf P && ne a && nf b && nf c
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| PNat => true
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| PSuc a => nf a
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| PZero => true
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end.
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Lemma ne_nf n a : @ne n a -> nf a.
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Proof. elim : a => //=. Qed.
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Lemma ne_nf_ren n m (a : PTm n) (ξ : fin n -> fin m) :
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(ne a <-> ne (ren_PTm ξ a)) /\ (nf a <-> nf (ren_PTm ξ a)).
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Proof.
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move : m ξ. elim : n / a => //=; solve [hauto b:on].
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Qed.
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Inductive TRedSN' {n} (a : PTm n) : PTm n -> Prop :=
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| T_Refl :
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TRedSN' a a
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| T_Once b :
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TRedSN a b ->
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TRedSN' a b.
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Lemma SN_Proj n p (a : PTm n) :
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SN (PProj p a) -> SN a.
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Proof.
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move E : (PProj p a) => u h.
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move : a E.
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elim : n u / h => n //=; sauto.
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Qed.
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Lemma N_β' n a (b : PTm n) u :
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u = (subst_PTm (scons b VarPTm) a) ->
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SN b ->
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TRedSN (PApp (PAbs a) b) u.
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Proof. move => ->. apply N_β. Qed.
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Lemma N_IndSuc' n P a b c u :
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u = (subst_PTm (scons (PInd P a b c) (scons a VarPTm)) c) ->
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SN P ->
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@SN n a ->
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SN b ->
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SN c ->
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TRedSN (PInd P (PSuc a) b c) u.
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Proof. move => ->. apply N_IndSuc. Qed.
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Lemma sn_renaming n :
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(forall (a : PTm n) (s : SNe a), forall m (ξ : fin n -> fin m), SNe (ren_PTm ξ a)) /\
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(forall (a : PTm n) (s : SN a), forall m (ξ : fin n -> fin m), SN (ren_PTm ξ a)) /\
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(forall (a b : PTm n) (_ : TRedSN a b), forall m (ξ : fin n -> fin m), TRedSN (ren_PTm ξ a) (ren_PTm ξ b)).
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Proof.
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move : n. apply sn_mutual => n; try qauto ctrs:SN, SNe, TRedSN depth:1.
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move => a b ha iha m ξ /=.
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apply N_β'. by asimpl. eauto.
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move => * /=.
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apply N_IndSuc';eauto. by asimpl.
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Qed.
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Lemma ne_nf_embed n (a : PTm n) :
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(ne a -> SNe a) /\ (nf a -> SN a).
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Proof.
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elim : n / a => //=; hauto qb:on ctrs:SNe, SN.
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Qed.
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#[export]Hint Constructors SN SNe TRedSN : sn.
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Ltac2 rec solve_anti_ren () :=
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let x := Fresh.in_goal (Option.get (Ident.of_string "x")) in
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intro $x;
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lazy_match! Constr.type (Control.hyp x) with
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| fin _ -> _ _ => (ltac1:(case;qauto depth:2 db:sn))
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| _ => solve_anti_ren ()
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end.
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Ltac solve_anti_ren := ltac2:(Control.enter solve_anti_ren).
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Lemma sn_antirenaming n :
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(forall (a : PTm n) (s : SNe a), forall m (ξ : fin m -> fin n) b, a = ren_PTm ξ b -> SNe b) /\
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(forall (a : PTm n) (s : SN a), forall m (ξ : fin m -> fin n) b, a = ren_PTm ξ b -> SN b) /\
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(forall (a b : PTm n) (_ : TRedSN a b), forall m (ξ : fin m -> fin n) a0,
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a = ren_PTm ξ a0 -> exists b0, TRedSN a0 b0 /\ b = ren_PTm ξ b0).
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Proof.
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move : n. apply sn_mutual => n; try solve_anti_ren.
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move => a b ha iha m ξ []//= u u0 [+ ?]. subst.
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case : u => //= => u [*]. subst.
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spec_refl. eexists. split. apply N_β=>//. by asimpl.
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move => a b hb ihb m ξ[]//= p p0 [? +]. subst.
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case : p0 => //= p p0 [*]. subst.
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spec_refl. by eauto with sn.
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move => a b ha iha m ξ[]//= u u0 [? ]. subst.
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case : u0 => //=. move => p p0 [*]. subst.
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spec_refl. by eauto with sn.
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move => P b c ha iha hb ihb hc ihc m ξ []//= P0 a0 b0 c0 [?]. subst.
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case : a0 => //= _ *. subst.
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spec_refl. by eauto with sn.
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move => P a b c hP ihP ha iha hb ihb hc ihc m ξ []//= P0 a0 b0 c0 [?]. subst.
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case : a0 => //= a0 [*]. subst.
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spec_refl. eexists; repeat split; eauto with sn.
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by asimpl. Qed.
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Lemma sn_unmorphing n :
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(forall (a : PTm n) (s : SNe a), forall m (ρ : fin m -> PTm n) b, a = subst_PTm ρ b -> SNe b) /\
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(forall (a : PTm n) (s : SN a), forall m (ρ : fin m -> PTm n) b, a = subst_PTm ρ b -> SN b) /\
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(forall (a b : PTm n) (_ : TRedSN a b), forall m (ρ : fin m -> PTm n) a0,
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a = subst_PTm ρ a0 -> (exists b0, b = subst_PTm ρ b0 /\ TRedSN a0 b0) \/ SNe a0).
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Proof.
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move : n. apply sn_mutual => n; try solve_anti_ren.
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- move => a b ha iha m ξ b0.
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case : b0 => //=.
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+ hauto lq:on rew:off db:sn.
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+ move => p p0 [+ ?]. subst.
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case : p => //=.
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hauto lq:on db:sn.
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move => p [?]. subst.
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asimpl. left.
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spec_refl.
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eexists. split; last by eauto using N_β.
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by asimpl.
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- move => a0 a1 b hb ihb ha iha m ρ []//=.
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+ hauto lq:on rew:off db:sn.
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+ move => t0 t1 [*]. subst.
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spec_refl.
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case : iha.
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* move => [u [? hu]]. subst.
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left. eexists. split; eauto using N_AppL.
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reflexivity.
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* move => h.
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right.
|
||
apply N_App => //.
|
||
- move => a b hb ihb m ρ []//=.
|
||
+ hauto l:on ctrs:TRedSN.
|
||
+ move => p p0 [?]. subst.
|
||
case : p0 => //=.
|
||
* hauto lq:on rew:off db:sn.
|
||
* move => p p0 [*]. subst.
|
||
hauto lq:on db:sn.
|
||
- move => a b ha iha m ρ []//=; first by hauto l:on db:sn.
|
||
case => //=. move => []//=.
|
||
+ hauto lq:on db:sn.
|
||
+ hauto lq:on db:sn.
|
||
- move => p a b ha iha m ρ []//=; first by hauto l:on db:sn.
|
||
move => t0 t1 [*]. subst.
|
||
spec_refl.
|
||
case : iha.
|
||
+ move => [b0 [? h]]. subst.
|
||
left. eexists. split; last by eauto with sn.
|
||
reflexivity.
|
||
+ hauto lq:on db:sn.
|
||
- move => P b c hP ihP hb ihb hc ihc m ρ []//=.
|
||
+ hauto lq:on db:sn.
|
||
+ move => p []//=.
|
||
* hauto lq:on db:sn.
|
||
* hauto q:on db:sn.
|
||
- move => P a b c hP ihP ha iha hb ihb hc ihc m ρ []//=.
|
||
+ hauto lq:on db:sn.
|
||
+ move => P0 a0 b0 c0 [?]. subst.
|
||
case : a0 => //=.
|
||
* hauto lq:on db:sn.
|
||
* move => a0 [*]. subst.
|
||
spec_refl.
|
||
left. eexists. split; last by eauto with sn.
|
||
by asimpl.
|
||
- move => P a0 a1 b c hP ihP hb ihb hc ihc ha iha m ρ[]//=.
|
||
+ hauto lq:on db:sn.
|
||
+ move => P1 a2 b2 c2 [*]. subst.
|
||
spec_refl.
|
||
case : iha.
|
||
* move => [a3][?]h. subst.
|
||
left. eexists; split; last by eauto with sn.
|
||
asimpl. eauto with sn.
|
||
* hauto lq:on db:sn.
|
||
Qed.
|
||
|
||
Lemma SN_AppInv : forall n (a b : PTm n), SN (PApp a b) -> SN a /\ SN b.
|
||
Proof.
|
||
move => n a b. move E : (PApp a b) => u hu. move : a b E.
|
||
elim : n u /hu=>//=.
|
||
hauto lq:on rew:off inv:SNe db:sn.
|
||
move => n a b ha hb ihb a0 b0 ?. subst.
|
||
inversion ha; subst.
|
||
move {ihb}.
|
||
hecrush use:sn_unmorphing.
|
||
hauto lq:on db:sn.
|
||
Qed.
|
||
|
||
Lemma SN_ProjInv : forall n p (a : PTm n), SN (PProj p a) -> SN a.
|
||
Proof.
|
||
move => n p a. move E : (PProj p a) => u hu.
|
||
move : p a E.
|
||
elim : n u / hu => //=.
|
||
hauto lq:on rew:off inv:SNe db:sn.
|
||
hauto lq:on rew:off inv:TRedSN db:sn.
|
||
Qed.
|
||
|
||
Lemma SN_IndInv : forall n P (a : PTm n) b c, SN (PInd P a b c) -> SN P /\ SN a /\ SN b /\ SN c.
|
||
Proof.
|
||
move => n P a b c. move E : (PInd P a b c) => u hu. move : P a b c E.
|
||
elim : n u / hu => //=.
|
||
hauto lq:on rew:off inv:SNe db:sn.
|
||
hauto lq:on rew:off inv:TRedSN db:sn.
|
||
Qed.
|
||
|
||
Lemma epar_sn_preservation n :
|
||
(forall (a : PTm n) (s : SNe a), forall b, EPar.R a b -> SNe b) /\
|
||
(forall (a : PTm n) (s : SN a), forall b, EPar.R a b -> SN b) /\
|
||
(forall (a b : PTm n) (_ : TRedSN a b), forall c, EPar.R a c -> exists d, TRedSN' c d /\ EPar.R b d).
|
||
Proof.
|
||
move : n. apply sn_mutual => n.
|
||
- sauto lq:on.
|
||
- sauto lq:on.
|
||
- sauto lq:on.
|
||
- sauto lq:on.
|
||
- sauto lq:on.
|
||
- move => a b ha iha hb ihb b0.
|
||
inversion 1; subst.
|
||
+ have /iha : (EPar.R (PProj PL a0) (PProj PL b0)) by sauto lq:on.
|
||
sfirstorder use:SN_Proj.
|
||
+ sauto lq:on.
|
||
- move => a ha iha b.
|
||
inversion 1; subst.
|
||
+ have : EPar.R (PApp (ren_PTm shift a0) (VarPTm var_zero)) (PApp (ren_PTm shift b) (VarPTm var_zero)).
|
||
apply EPar.AppCong; eauto using EPar.refl.
|
||
sfirstorder use:EPar.renaming.
|
||
move /iha.
|
||
move /SN_AppInv => [+ _].
|
||
hauto l:on use:sn_antirenaming.
|
||
+ sauto lq:on.
|
||
- sauto lq:on.
|
||
- sauto lq:on.
|
||
- sauto lq:on.
|
||
- sauto lq:on.
|
||
- sauto lq:on.
|
||
- sauto lq:on.
|
||
- sauto lq:on.
|
||
- move => a b ha iha c h0.
|
||
inversion h0; subst.
|
||
inversion H1; subst.
|
||
+ exists (PApp a1 b1). split. sfirstorder.
|
||
asimpl.
|
||
sauto lq:on.
|
||
+ have {}/iha := H3 => iha.
|
||
exists (subst_PTm (scons b1 VarPTm) a2).
|
||
split.
|
||
sauto lq:on.
|
||
hauto lq:on use:EPar.morphing, EPar.refl inv:option.
|
||
- sauto.
|
||
- move => a b hb ihb c.
|
||
elim /EPar.inv => //= _.
|
||
move => p a0 a1 ha [*]. subst.
|
||
elim /EPar.inv : ha => //= _.
|
||
+ move => a0 a2 ha' [*]. subst.
|
||
exists (PProj PL a1).
|
||
split. sauto.
|
||
sauto lq:on.
|
||
+ sauto lq:on rew:off.
|
||
- move => a b ha iha c.
|
||
elim /EPar.inv => //=_.
|
||
move => p a0 a1 + [*]. subst.
|
||
elim /EPar.inv => //=_.
|
||
+ move => a0 a2 h1 [*]. subst.
|
||
exists (PProj PR a1).
|
||
split. sauto.
|
||
sauto lq:on.
|
||
+ sauto lq:on.
|
||
- sauto.
|
||
- sauto q:on.
|
||
- move => P a b c hP ihP ha iha hb ihb hc ihc u.
|
||
elim /EPar.inv => //=_.
|
||
move => P0 P1 a0 a1 b0 b1 c0 c1 hP0 ha0 hb0 hc0 [*]. subst.
|
||
elim /EPar.inv : ha0 => //=_.
|
||
move => a0 a2 ha0 [*]. subst.
|
||
eexists. split. apply T_Once. apply N_IndSuc; eauto.
|
||
hauto q:on ctrs:EPar.R use:EPar.morphing, EPar.refl inv:option.
|
||
- sauto q:on.
|
||
Qed.
|
||
|
||
Module RRed.
|
||
Inductive R {n} : PTm n -> PTm n -> Prop :=
|
||
(****************** Beta ***********************)
|
||
| AppAbs a b :
|
||
R (PApp (PAbs a) b) (subst_PTm (scons b VarPTm) a)
|
||
|
||
| ProjPair p a b :
|
||
R (PProj p (PPair a b)) (if p is PL then a else b)
|
||
|
||
| IndZero P b c :
|
||
R (PInd P PZero b c) b
|
||
|
||
| IndSuc P a b c :
|
||
R (PInd P (PSuc a) b c) (subst_PTm (scons (PInd P a b c) (scons a VarPTm)) c)
|
||
(*************** Congruence ********************)
|
||
| AbsCong a0 a1 :
|
||
R a0 a1 ->
|
||
R (PAbs a0) (PAbs a1)
|
||
| AppCong0 a0 a1 b :
|
||
R a0 a1 ->
|
||
R (PApp a0 b) (PApp a1 b)
|
||
| AppCong1 a b0 b1 :
|
||
R b0 b1 ->
|
||
R (PApp a b0) (PApp a b1)
|
||
| PairCong0 a0 a1 b :
|
||
R a0 a1 ->
|
||
R (PPair a0 b) (PPair a1 b)
|
||
| PairCong1 a b0 b1 :
|
||
R b0 b1 ->
|
||
R (PPair a b0) (PPair a b1)
|
||
| ProjCong p a0 a1 :
|
||
R a0 a1 ->
|
||
R (PProj p a0) (PProj p a1)
|
||
| BindCong0 p A0 A1 B :
|
||
R A0 A1 ->
|
||
R (PBind p A0 B) (PBind p A1 B)
|
||
| BindCong1 p A B0 B1 :
|
||
R B0 B1 ->
|
||
R (PBind p A B0) (PBind p A B1)
|
||
| IndCong0 P0 P1 a b c :
|
||
R P0 P1 ->
|
||
R (PInd P0 a b c) (PInd P1 a b c)
|
||
| IndCong1 P a0 a1 b c :
|
||
R a0 a1 ->
|
||
R (PInd P a0 b c) (PInd P a1 b c)
|
||
| IndCong2 P a b0 b1 c :
|
||
R b0 b1 ->
|
||
R (PInd P a b0 c) (PInd P a b1 c)
|
||
| IndCong3 P a b c0 c1 :
|
||
R c0 c1 ->
|
||
R (PInd P a b c0) (PInd P a b c1)
|
||
| SucCong a0 a1 :
|
||
R a0 a1 ->
|
||
R (PSuc a0) (PSuc a1).
|
||
|
||
Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop.
|
||
|
||
Lemma AppAbs' n a (b : PTm n) u :
|
||
u = (subst_PTm (scons b VarPTm) a) ->
|
||
R (PApp (PAbs a) b) u.
|
||
Proof.
|
||
move => ->. by apply AppAbs. Qed.
|
||
|
||
Lemma IndSuc' n P a b c u :
|
||
u = (subst_PTm (scons (PInd P a b c) (scons a VarPTm)) c) ->
|
||
R (@PInd n P (PSuc a) b c) u.
|
||
Proof. move => ->. apply IndSuc. Qed.
|
||
|
||
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.
|
||
|
||
all : try qauto ctrs:R.
|
||
move => n a b m ξ /=.
|
||
apply AppAbs'. by asimpl.
|
||
move => */=; apply IndSuc'; eauto. by asimpl.
|
||
Qed.
|
||
|
||
Ltac2 rec solve_anti_ren () :=
|
||
let x := Fresh.in_goal (Option.get (Ident.of_string "x")) in
|
||
intro $x;
|
||
lazy_match! Constr.type (Control.hyp x) with
|
||
| fin _ -> _ _ => (ltac1:(case;hauto q:on depth:2 ctrs:RRed.R))
|
||
| _ => solve_anti_ren ()
|
||
end.
|
||
|
||
Ltac solve_anti_ren := ltac2:(Control.enter solve_anti_ren).
|
||
|
||
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_PTm ξ a) => u h.
|
||
move : n ξ a E. elim : m u b/h; try solve_anti_ren.
|
||
- move => n a b m ξ []//=. move => []//= t t0 [*]. subst.
|
||
eexists. split. apply AppAbs. by asimpl.
|
||
- move => n p a b m ξ []//=.
|
||
move => p0 []//=. hauto q:on ctrs:R.
|
||
- move => n p b c m ξ []//= P a0 b0 c0 [*]. subst.
|
||
destruct a0 => //=.
|
||
hauto lq:on ctrs:R.
|
||
- move => n P a b c m ξ []//=.
|
||
move => p p0 p1 p2 [?]. subst.
|
||
case : p0 => //=.
|
||
move => p0 [?] *. subst. eexists. split; eauto using IndSuc.
|
||
by asimpl.
|
||
Qed.
|
||
|
||
Lemma nf_imp n (a b : PTm n) :
|
||
nf a ->
|
||
R a b -> False.
|
||
Proof. move/[swap]. induction 1; hauto qb:on inv:PTm. Qed.
|
||
|
||
Lemma FromRedSN n (a b : PTm n) :
|
||
TRedSN a b ->
|
||
RRed.R a b.
|
||
Proof. induction 1; hauto lq:on ctrs:RRed.R. Qed.
|
||
|
||
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.
|
||
move => h. move : m ρ. elim : n a b / h => n.
|
||
|
||
all : try hauto lq:on ctrs:R.
|
||
move => a b m ρ /=.
|
||
eapply AppAbs'; eauto; cycle 1. by asimpl.
|
||
move => */=; apply : IndSuc'; eauto. by asimpl.
|
||
Qed.
|
||
|
||
End RRed.
|
||
|
||
Module RPar.
|
||
Inductive R {n} : PTm n -> PTm n -> Prop :=
|
||
(****************** Beta ***********************)
|
||
| AppAbs a0 a1 b0 b1 :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (PApp (PAbs a0) b0) (subst_PTm (scons b1 VarPTm) a1)
|
||
|
||
| ProjPair p a0 a1 b0 b1 :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (PProj p (PPair a0 b0)) (if p is PL then a1 else b1)
|
||
|
||
| IndZero P b0 b1 c :
|
||
R b0 b1 ->
|
||
R (PInd P PZero b0 c) b1
|
||
|
||
| IndSuc P0 P1 a0 a1 b0 b1 c0 c1 :
|
||
R P0 P1 ->
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R c0 c1 ->
|
||
(* ----------------------------- *)
|
||
R (PInd P0 (PSuc a0) b0 c0) (subst_PTm (scons (PInd P1 a1 b1 c1) (scons a1 VarPTm)) c1)
|
||
|
||
(*************** Congruence ********************)
|
||
| AbsCong a0 a1 :
|
||
R a0 a1 ->
|
||
R (PAbs a0) (PAbs a1)
|
||
| AppCong a0 a1 b0 b1 :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (PApp a0 b0) (PApp a1 b1)
|
||
| PairCong a0 a1 b0 b1 :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (PPair a0 b0) (PPair a1 b1)
|
||
| ProjCong p a0 a1 :
|
||
R a0 a1 ->
|
||
R (PProj p a0) (PProj p a1)
|
||
| VarTm i :
|
||
R (VarPTm i) (VarPTm i)
|
||
| Univ i :
|
||
R (PUniv i) (PUniv i)
|
||
| BindCong p A0 A1 B0 B1 :
|
||
R A0 A1 ->
|
||
R B0 B1 ->
|
||
R (PBind p A0 B0) (PBind p A1 B1)
|
||
| BotCong :
|
||
R PBot PBot
|
||
| NatCong :
|
||
R PNat PNat
|
||
| IndCong P0 P1 a0 a1 b0 b1 c0 c1 :
|
||
R P0 P1 ->
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R c0 c1 ->
|
||
(* ----------------------- *)
|
||
R (PInd P0 a0 b0 c0) (PInd P1 a1 b1 c1)
|
||
| ZeroCong :
|
||
R PZero PZero
|
||
| SucCong a0 a1 :
|
||
R a0 a1 ->
|
||
(* ------------ *)
|
||
R (PSuc a0) (PSuc a1).
|
||
|
||
Lemma refl n (a : PTm n) : R a a.
|
||
Proof.
|
||
elim : n / a; hauto lq:on ctrs:R.
|
||
Qed.
|
||
|
||
Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop.
|
||
|
||
Lemma AppAbs' n a0 a1 (b0 b1 : PTm n) u :
|
||
u = (subst_PTm (scons b1 VarPTm) a1) ->
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (PApp (PAbs a0) b0) u.
|
||
Proof. move => ->. apply AppAbs. Qed.
|
||
|
||
Lemma ProjPair' n p u (a0 a1 b0 b1 : PTm n) :
|
||
u = (if p is PL then a1 else b1) ->
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (PProj p (PPair a0 b0)) u.
|
||
Proof. move => ->. apply ProjPair. Qed.
|
||
|
||
Lemma IndSuc' n P0 P1 (a0 a1 : PTm n) b0 b1 c0 c1 u :
|
||
u = (subst_PTm (scons (PInd P1 a1 b1 c1) (scons a1 VarPTm)) c1) ->
|
||
R P0 P1 ->
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R c0 c1 ->
|
||
(* ----------------------------- *)
|
||
R (PInd P0 (PSuc a0) b0 c0) u.
|
||
Proof. move => ->. apply IndSuc. Qed.
|
||
|
||
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.
|
||
|
||
all : try qauto ctrs:R use:ProjPair'.
|
||
|
||
move => n a0 a1 b0 b1 ha iha hb ihb m ξ /=.
|
||
eapply AppAbs'; eauto. by asimpl.
|
||
|
||
move => * /=. apply : IndSuc'; eauto. by asimpl.
|
||
Qed.
|
||
|
||
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_PTm ξ) ρ0) i) ((funcomp (ren_PTm ξ) ρ1) i)).
|
||
Proof. eauto using renaming. Qed.
|
||
|
||
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 -> PTm m) :
|
||
(forall i, R (ρ0 i) (ρ1 i)) ->
|
||
(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 : PTm n) (ρ0 ρ1 : fin n -> PTm m) :
|
||
(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 => n.
|
||
all : try hauto lq:on ctrs:R use:morphing_up, ProjPair'.
|
||
move => a0 a1 b0 b1 ha iha hb ihb m ρ0 ρ1 hρ /=.
|
||
eapply AppAbs'; eauto; cycle 1. sfirstorder use:morphing_up. by asimpl.
|
||
move => */=; eapply IndSuc'; eauto; cycle 1.
|
||
sfirstorder use:morphing_up.
|
||
sfirstorder use:morphing_up.
|
||
by asimpl.
|
||
Qed.
|
||
|
||
Lemma substing n m (a : PTm n) b (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 cong n (a0 a1 : PTm (S n)) b0 b1 :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (subst_PTm (scons b0 VarPTm) a0) (subst_PTm (scons b1 VarPTm) a1).
|
||
Proof.
|
||
move => h0 h1. apply morphing=>//.
|
||
hauto q:on inv:option ctrs:R.
|
||
Qed.
|
||
|
||
Lemma FromRRed n (a b : PTm n) :
|
||
RRed.R a b -> RPar.R a b.
|
||
Proof.
|
||
induction 1; qauto l:on use:RPar.refl ctrs:RPar.R.
|
||
Qed.
|
||
|
||
Function tstar {n} (a : PTm n) :=
|
||
match a with
|
||
| 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)
|
||
| PUniv i => PUniv i
|
||
| PBind p A B => PBind p (tstar A) (tstar B)
|
||
| PBot => PBot
|
||
| PNat => PNat
|
||
| PZero => PZero
|
||
| PSuc a => PSuc (tstar a)
|
||
| PInd P PZero b c => tstar b
|
||
| PInd P (PSuc a) b c =>
|
||
(subst_PTm (scons (PInd (tstar P) (tstar a) (tstar b) (tstar c)) (scons (tstar a) VarPTm)) (tstar c))
|
||
| PInd P a b c => PInd (tstar P) (tstar a) (tstar b) (tstar c)
|
||
end.
|
||
|
||
Lemma triangle n (a b : PTm n) :
|
||
RPar.R a b -> RPar.R b (tstar a).
|
||
Proof.
|
||
move : b.
|
||
apply tstar_ind => {}n{}a.
|
||
- hauto lq:on ctrs:R inv:R.
|
||
- hauto lq:on ctrs:R inv:R.
|
||
- hauto lq:on rew:off inv:R use:cong ctrs:R.
|
||
- hauto lq:on ctrs:R inv:R.
|
||
- hauto lq:on ctrs:R inv:R.
|
||
- move => p a0 b ? ? ih b0. subst.
|
||
elim /inv => //=_.
|
||
+ move => p a1 a2 b1 b2 h0 h1[*]. subst.
|
||
by apply ih.
|
||
+ move => p a1 a2 ha [*]. subst.
|
||
elim /inv : ha => //=_.
|
||
move => a1 a3 b0 b1 h0 h1 [*]. subst.
|
||
apply : ProjPair'; eauto using refl.
|
||
- move => p a0 b ? p0 ?. subst.
|
||
case : p0 => //= _.
|
||
move => ih b0. elim /inv => //=_.
|
||
+ hauto l:on.
|
||
+ move => p a1 a2 ha [*]. subst.
|
||
elim /inv : ha => //=_ > ? ? [*]. subst.
|
||
apply : ProjPair'; eauto using refl.
|
||
- hauto lq:on ctrs:R inv:R.
|
||
- hauto lq:on ctrs:R inv:R.
|
||
- hauto lq:on ctrs:R inv:R.
|
||
- hauto lq:on ctrs:R inv:R.
|
||
- hauto lq:on ctrs:R inv:R.
|
||
- hauto lq:on ctrs:R inv:R.
|
||
- hauto lq:on ctrs:R inv:R.
|
||
- move => P b c ?. subst.
|
||
move => h0. inversion 1; subst. hauto lq:on ctrs:R. qauto l:on inv:R ctrs:R.
|
||
- move => P a0 b c ? hP ihP ha iha hb ihb u. subst.
|
||
elim /inv => //= _.
|
||
+ move => P0 P1 a1 a2 b0 b1 c0 c1 hP' ha' hb' hc' [*]. subst.
|
||
apply morphing. hauto lq:on ctrs:R inv:option.
|
||
eauto.
|
||
+ sauto q:on ctrs:R.
|
||
- sauto lq:on.
|
||
Qed.
|
||
|
||
Lemma diamond n (a b c : PTm n) :
|
||
R a b -> R a c -> exists d, R b d /\ R c d.
|
||
Proof. eauto using triangle. Qed.
|
||
End RPar.
|
||
|
||
Lemma red_sn_preservation n :
|
||
(forall (a : PTm n) (s : SNe a), forall b, RPar.R a b -> SNe b) /\
|
||
(forall (a : PTm n) (s : SN a), forall b, RPar.R a b -> SN b) /\
|
||
(forall (a b : PTm n) (_ : TRedSN a b), forall c, RPar.R a c -> exists d, TRedSN' c d /\ RPar.R b d).
|
||
Proof.
|
||
move : n. apply sn_mutual => n.
|
||
- hauto l:on inv:RPar.R.
|
||
- qauto l:on inv:RPar.R,SNe,SN ctrs:SNe.
|
||
- hauto lq:on inv:RPar.R, SNe ctrs:SNe.
|
||
- hauto lq:on inv:RPar.R, SNe ctrs:SNe.
|
||
- qauto l:on inv:RPar.R, SN,SNe ctrs:SNe.
|
||
- qauto l:on ctrs:SN inv:RPar.R.
|
||
- hauto lq:on ctrs:SN inv:RPar.R.
|
||
- hauto lq:on ctrs:SN.
|
||
- hauto q:on ctrs:SN inv:SN, TRedSN'.
|
||
- hauto lq:on ctrs:SN inv:RPar.R.
|
||
- hauto lq:on ctrs:SN inv:RPar.R.
|
||
- hauto l:on inv:RPar.R.
|
||
- hauto l:on inv:RPar.R.
|
||
- hauto lq:on ctrs:SN inv:RPar.R.
|
||
- move => a b ha iha hb ihb.
|
||
elim /RPar.inv : ihb => //=_.
|
||
+ move => a0 a1 b0 b1 ha0 hb0 [*]. subst.
|
||
eauto using RPar.cong, T_Refl.
|
||
+ move => a0 a1 b0 b1 h0 h1 [*]. subst.
|
||
elim /RPar.inv : h0 => //=_.
|
||
move => a0 a2 h [*]. subst.
|
||
eexists. split. apply T_Once. hauto lq:on ctrs:TRedSN.
|
||
eauto using RPar.cong.
|
||
- move => a0 a1 b hb ihb ha iha c.
|
||
elim /RPar.inv => //=_.
|
||
+ qauto l:on inv:TRedSN.
|
||
+ move => a2 a3 b0 b1 h0 h1 [*]. subst.
|
||
have {}/iha := h0.
|
||
move => [d [iha0 iha1]].
|
||
hauto lq:on rew:off inv:TRedSN' ctrs:TRedSN, RPar.R, TRedSN'.
|
||
- hauto lq:on inv:RPar.R ctrs:RPar.R, TRedSN', TRedSN.
|
||
- hauto lq:on inv:RPar.R ctrs:RPar.R, TRedSN', TRedSN.
|
||
- sauto.
|
||
- sauto.
|
||
- move => P a b c hP ihP ha iha hb ihb hc ihc u.
|
||
elim /RPar.inv => //=_.
|
||
+ move => P0 P1 a0 a1 b0 b1 c0 c1 hP' ha' hb' hc' [*]. subst.
|
||
eexists. split; first by apply T_Refl.
|
||
apply RPar.morphing=>//. hauto lq:on ctrs:RPar.R inv:option.
|
||
+ move => P0 P1 a0 a1 b0 b1 c0 c1 hP' ha' hb' hc' [*]. subst.
|
||
elim /RPar.inv : ha' => //=_.
|
||
move => a0 a2 ha' [*]. subst.
|
||
eexists. split. apply T_Once.
|
||
apply N_IndSuc; eauto.
|
||
hauto q:on use:RPar.morphing ctrs:RPar.R inv:option.
|
||
- sauto q:on.
|
||
Qed.
|
||
|
||
Module RReds.
|
||
|
||
#[local]Ltac solve_s_rec :=
|
||
move => *; eapply rtc_l; eauto;
|
||
hauto lq:on ctrs:RRed.R.
|
||
|
||
#[local]Ltac solve_s :=
|
||
repeat (induction 1; last by solve_s_rec); apply rtc_refl.
|
||
|
||
Lemma AbsCong n (a b : PTm (S n)) :
|
||
rtc RRed.R a b ->
|
||
rtc RRed.R (PAbs a) (PAbs b).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
|
||
rtc RRed.R a0 a1 ->
|
||
rtc RRed.R b0 b1 ->
|
||
rtc RRed.R (PApp a0 b0) (PApp a1 b1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
|
||
rtc RRed.R a0 a1 ->
|
||
rtc RRed.R b0 b1 ->
|
||
rtc RRed.R (PPair a0 b0) (PPair a1 b1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma ProjCong n p (a0 a1 : PTm n) :
|
||
rtc RRed.R a0 a1 ->
|
||
rtc RRed.R (PProj p a0) (PProj p a1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma SucCong n (a0 a1 : PTm n) :
|
||
rtc RRed.R a0 a1 ->
|
||
rtc RRed.R (PSuc a0) (PSuc a1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma IndCong n P0 P1 (a0 a1 : PTm n) b0 b1 c0 c1 :
|
||
rtc RRed.R P0 P1 ->
|
||
rtc RRed.R a0 a1 ->
|
||
rtc RRed.R b0 b1 ->
|
||
rtc RRed.R c0 c1 ->
|
||
rtc RRed.R (PInd P0 a0 b0 c0) (PInd P1 a1 b1 c1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma BindCong n p (A0 A1 : PTm n) B0 B1 :
|
||
rtc RRed.R A0 A1 ->
|
||
rtc RRed.R B0 B1 ->
|
||
rtc RRed.R (PBind p A0 B0) (PBind p A1 B1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
|
||
rtc RRed.R a b -> rtc RRed.R (ren_PTm ξ a) (ren_PTm ξ b).
|
||
Proof.
|
||
move => h. move : m ξ. elim : a b /h; hauto lq:on ctrs:rtc use:RRed.renaming.
|
||
Qed.
|
||
|
||
Lemma FromRPar n (a b : PTm n) (h : RPar.R a b) :
|
||
rtc RRed.R a b.
|
||
Proof.
|
||
elim : n a b /h; eauto using AbsCong, AppCong, PairCong, ProjCong, rtc_refl, BindCong, IndCong, SucCong.
|
||
move => n a0 a1 b0 b1 ha iha hb ihb.
|
||
apply : rtc_r; last by apply RRed.AppAbs.
|
||
by eauto using AppCong, AbsCong.
|
||
|
||
move => n p a0 a1 b0 b1 ha iha hb ihb.
|
||
apply : rtc_r; last by apply RRed.ProjPair.
|
||
by eauto using PairCong, ProjCong.
|
||
|
||
hauto lq:on ctrs:RRed.R, rtc.
|
||
|
||
move => *.
|
||
apply : rtc_r; last by apply RRed.IndSuc.
|
||
by eauto using SucCong, IndCong.
|
||
Qed.
|
||
|
||
Lemma RParIff n (a b : PTm n) :
|
||
rtc RRed.R a b <-> rtc RPar.R a b.
|
||
Proof.
|
||
split.
|
||
induction 1; hauto l:on ctrs:rtc use:RPar.FromRRed, @relations.rtc_transitive.
|
||
induction 1; hauto l:on ctrs:rtc use:FromRPar, @relations.rtc_transitive.
|
||
Qed.
|
||
|
||
Lemma nf_refl n (a b : PTm n) :
|
||
rtc RRed.R a b -> nf a -> a = b.
|
||
Proof. induction 1; sfirstorder use:RRed.nf_imp. Qed.
|
||
|
||
Lemma FromRedSNs n (a b : PTm n) :
|
||
rtc TRedSN a b ->
|
||
rtc RRed.R a b.
|
||
Proof. induction 1; hauto lq:on ctrs:rtc use:RRed.FromRedSN. Qed.
|
||
|
||
End RReds.
|
||
|
||
|
||
Module NeEPar.
|
||
Inductive R_nonelim {n} : PTm n -> PTm n -> Prop :=
|
||
(****************** Eta ***********************)
|
||
| AppEta a0 a1 :
|
||
~~ ishf a0 ->
|
||
R_elim a0 a1 ->
|
||
R_nonelim (PAbs (PApp (ren_PTm shift a0) (VarPTm var_zero))) a1
|
||
| PairEta a0 a1 :
|
||
~~ ishf a0 ->
|
||
R_elim a0 a1 ->
|
||
R_nonelim (PPair (PProj PL a0) (PProj PR a0)) a1
|
||
(*************** Congruence ********************)
|
||
| AbsCong a0 a1 :
|
||
R_nonelim a0 a1 ->
|
||
R_nonelim (PAbs a0) (PAbs a1)
|
||
| AppCong a0 a1 b0 b1 :
|
||
R_elim a0 a1 ->
|
||
R_nonelim b0 b1 ->
|
||
R_nonelim (PApp a0 b0) (PApp a1 b1)
|
||
| PairCong a0 a1 b0 b1 :
|
||
R_nonelim a0 a1 ->
|
||
R_nonelim b0 b1 ->
|
||
R_nonelim (PPair a0 b0) (PPair a1 b1)
|
||
| ProjCong p a0 a1 :
|
||
R_elim a0 a1 ->
|
||
R_nonelim (PProj p a0) (PProj p a1)
|
||
| VarTm i :
|
||
R_nonelim (VarPTm i) (VarPTm i)
|
||
| Univ i :
|
||
R_nonelim (PUniv i) (PUniv i)
|
||
| BindCong p A0 A1 B0 B1 :
|
||
R_nonelim A0 A1 ->
|
||
R_nonelim B0 B1 ->
|
||
R_nonelim (PBind p A0 B0) (PBind p A1 B1)
|
||
| BotCong :
|
||
R_nonelim PBot PBot
|
||
| NatCong :
|
||
R_nonelim PNat PNat
|
||
| IndCong P0 P1 a0 a1 b0 b1 c0 c1 :
|
||
R_nonelim P0 P1 ->
|
||
R_elim a0 a1 ->
|
||
R_nonelim b0 b1 ->
|
||
R_nonelim c0 c1 ->
|
||
(* ----------------------- *)
|
||
R_nonelim (PInd P0 a0 b0 c0) (PInd P1 a1 b1 c1)
|
||
| ZeroCong :
|
||
R_nonelim PZero PZero
|
||
| SucCong a0 a1 :
|
||
R_nonelim a0 a1 ->
|
||
(* ------------ *)
|
||
R_nonelim (PSuc a0) (PSuc a1)
|
||
with R_elim {n} : PTm n -> PTm n -> Prop :=
|
||
| NAbsCong a0 a1 :
|
||
R_nonelim a0 a1 ->
|
||
R_elim (PAbs a0) (PAbs a1)
|
||
| NAppCong a0 a1 b0 b1 :
|
||
R_elim a0 a1 ->
|
||
R_nonelim b0 b1 ->
|
||
R_elim (PApp a0 b0) (PApp a1 b1)
|
||
| NPairCong a0 a1 b0 b1 :
|
||
R_nonelim a0 a1 ->
|
||
R_nonelim b0 b1 ->
|
||
R_elim (PPair a0 b0) (PPair a1 b1)
|
||
| NProjCong p a0 a1 :
|
||
R_elim a0 a1 ->
|
||
R_elim (PProj p a0) (PProj p a1)
|
||
| NVarTm i :
|
||
R_elim (VarPTm i) (VarPTm i)
|
||
| NUniv i :
|
||
R_elim (PUniv i) (PUniv i)
|
||
| NBindCong p A0 A1 B0 B1 :
|
||
R_nonelim A0 A1 ->
|
||
R_nonelim B0 B1 ->
|
||
R_elim (PBind p A0 B0) (PBind p A1 B1)
|
||
| NBotCong :
|
||
R_elim PBot PBot
|
||
| NNatCong :
|
||
R_elim PNat PNat
|
||
| NIndCong P0 P1 a0 a1 b0 b1 c0 c1 :
|
||
R_nonelim P0 P1 ->
|
||
R_elim a0 a1 ->
|
||
R_nonelim b0 b1 ->
|
||
R_nonelim c0 c1 ->
|
||
(* ----------------------- *)
|
||
R_elim (PInd P0 a0 b0 c0) (PInd P1 a1 b1 c1)
|
||
| NZeroCong :
|
||
R_elim PZero PZero
|
||
| NSucCong a0 a1 :
|
||
R_nonelim a0 a1 ->
|
||
(* ------------ *)
|
||
R_elim (PSuc a0) (PSuc a1).
|
||
|
||
Scheme epar_elim_ind := Induction for R_elim Sort Prop
|
||
with epar_nonelim_ind := Induction for R_nonelim Sort Prop.
|
||
|
||
Combined Scheme epar_mutual from epar_elim_ind, epar_nonelim_ind.
|
||
|
||
Lemma R_elim_nf n :
|
||
(forall (a b : PTm n), R_elim a b -> nf b -> nf a) /\
|
||
(forall (a b : PTm n), R_nonelim a b -> nf b -> nf a).
|
||
Proof.
|
||
move : n. apply epar_mutual => n //=.
|
||
- move => a0 a1 b0 b1 h ih h' ih' /andP [h0 h1].
|
||
have hb0 : nf b0 by eauto.
|
||
suff : ne a0 by qauto b:on.
|
||
hauto q:on inv:R_elim.
|
||
- hauto lb:on.
|
||
- hauto lq:on inv:R_elim.
|
||
- hauto b:on.
|
||
- hauto lqb:on inv:R_elim.
|
||
- move => a0 a1 /negP ha' ha ih ha1.
|
||
have {ih} := ih ha1.
|
||
move => ha0.
|
||
suff : ne a0 by hauto lb:on drew:off use:ne_nf_ren.
|
||
inversion ha; subst => //=.
|
||
- move => a0 a1 /negP ha' ha ih ha1.
|
||
have {}ih := ih ha1.
|
||
have : ne a0 by hauto lq:on inv:PTm.
|
||
qauto lb:on.
|
||
- move => a0 a1 b0 b1 ha iha hb ihb /andP [h0 h1].
|
||
have {}ihb := ihb h1.
|
||
have {}iha := iha ltac:(eauto using ne_nf).
|
||
suff : ne a0 by hauto lb:on.
|
||
move : ha h0. hauto lq:on inv:R_elim.
|
||
- hauto lb: on drew: off.
|
||
- hauto lq:on rew:off inv:R_elim.
|
||
- sfirstorder b:on.
|
||
- hauto lqb:on inv:R_elim.
|
||
Qed.
|
||
|
||
Lemma R_nonelim_nothf n (a b : PTm n) :
|
||
R_nonelim a b ->
|
||
~~ ishf a ->
|
||
R_elim a b.
|
||
Proof.
|
||
move => h. elim : n a b /h=>//=; hauto lq:on ctrs:R_elim.
|
||
Qed.
|
||
|
||
Lemma R_elim_nonelim n (a b : PTm n) :
|
||
R_elim a b ->
|
||
R_nonelim a b.
|
||
move => h. elim : n a b /h=>//=; hauto lq:on ctrs:R_nonelim.
|
||
Qed.
|
||
|
||
Lemma ToEPar : forall n, (forall (a b : PTm n), R_elim a b -> EPar.R a b) /\
|
||
(forall (a b : PTm n), R_nonelim a b -> EPar.R a b).
|
||
Proof.
|
||
apply epar_mutual; qauto l:on ctrs:EPar.R.
|
||
Qed.
|
||
|
||
End NeEPar.
|
||
|
||
Module Type NoForbid.
|
||
Parameter P : forall n, PTm n -> Prop.
|
||
Arguments P {n}.
|
||
|
||
Axiom P_EPar : forall n (a b : PTm n), EPar.R a b -> P a -> P b.
|
||
Axiom P_RRed : forall n (a b : PTm n), RRed.R a b -> P a -> P b.
|
||
(* Axiom P_AppPair : forall n (a b c : PTm n), ~ P (PApp (PPair a b) c). *)
|
||
(* Axiom P_ProjAbs : forall n p (a : PTm (S n)), ~ P (PProj p (PAbs a)). *)
|
||
(* Axiom P_ProjBind : forall n p p' (A : PTm n) B, ~ P (PProj p (PBind p' A B)). *)
|
||
(* Axiom P_AppBind : forall n p (A : PTm n) B b, ~ P (PApp (PBind p A B) b). *)
|
||
Axiom PApp_imp : forall n a b, @ishf n a -> ~~ isabs a -> ~ P (PApp a b).
|
||
Axiom PProj_imp : forall n p a, @ishf n a -> ~~ ispair a -> ~ P (PProj p a).
|
||
Axiom PInd_imp : forall n Q (a : PTm n) b c,
|
||
ishf a ->
|
||
~~ iszero a ->
|
||
~~ issuc a -> ~ P (PInd Q a b c).
|
||
|
||
Axiom P_AppInv : forall n (a b : PTm n), P (PApp a b) -> P a /\ P b.
|
||
Axiom P_AbsInv : forall n (a : PTm (S n)), P (PAbs a) -> P a.
|
||
Axiom P_SucInv : forall n (a : PTm n), P (PSuc a) -> P a.
|
||
Axiom P_BindInv : forall n p (A : PTm n) B, P (PBind p A B) -> P A /\ P B.
|
||
Axiom P_IndInv : forall n Q (a : PTm n) b c, P (PInd Q a b c) -> P Q /\ P a /\ P b /\ P c.
|
||
|
||
Axiom P_PairInv : forall n (a b : PTm n), P (PPair a b) -> P a /\ P b.
|
||
Axiom P_ProjInv : forall n p (a : PTm n), P (PProj p a) -> P a.
|
||
Axiom P_renaming : forall n m (ξ : fin n -> fin m) a , P (ren_PTm ξ a) <-> P a.
|
||
|
||
End NoForbid.
|
||
|
||
Module Type NoForbid_FactSig (M : NoForbid).
|
||
|
||
Axiom P_EPars : forall n (a b : PTm n), rtc EPar.R a b -> M.P a -> M.P b.
|
||
|
||
Axiom P_RReds : forall n (a b : PTm n), rtc RRed.R a b -> M.P a -> M.P b.
|
||
|
||
End NoForbid_FactSig.
|
||
|
||
Module NoForbid_Fact (M : NoForbid) : NoForbid_FactSig M.
|
||
Import M.
|
||
|
||
Lemma P_EPars : forall n (a b : PTm n), rtc EPar.R a b -> P a -> P b.
|
||
Proof.
|
||
induction 1; eauto using P_EPar, rtc_l, rtc_refl.
|
||
Qed.
|
||
|
||
Lemma P_RReds : forall n (a b : PTm n), rtc RRed.R a b -> P a -> P b.
|
||
Proof.
|
||
induction 1; eauto using P_RRed, rtc_l, rtc_refl.
|
||
Qed.
|
||
End NoForbid_Fact.
|
||
|
||
Module SN_NoForbid <: NoForbid.
|
||
Definition P := @SN.
|
||
Arguments P {n}.
|
||
|
||
Lemma P_EPar : forall n (a b : PTm n), EPar.R a b -> P a -> P b.
|
||
Proof. sfirstorder use:epar_sn_preservation. Qed.
|
||
|
||
Lemma P_RRed : forall n (a b : PTm n), RRed.R a b -> P a -> P b.
|
||
Proof. hauto q:on use:red_sn_preservation, RPar.FromRRed. Qed.
|
||
|
||
Lemma PApp_imp : forall n a b, @ishf n a -> ~~ isabs a -> ~ P (PApp a b).
|
||
sfirstorder use:fp_red.PApp_imp. Qed.
|
||
Lemma PProj_imp : forall n p a, @ishf n a -> ~~ ispair a -> ~ P (PProj p a).
|
||
sfirstorder use:fp_red.PProj_imp. Qed.
|
||
|
||
Lemma PInd_imp : forall n Q (a : PTm n) b c,
|
||
ishf a ->
|
||
~~ iszero a ->
|
||
~~ issuc a -> ~ P (PInd Q a b c).
|
||
Proof. sfirstorder use: fp_red.PInd_imp. Qed.
|
||
|
||
Lemma P_AppInv : forall n (a b : PTm n), P (PApp a b) -> P a /\ P b.
|
||
Proof. sfirstorder use:SN_AppInv. Qed.
|
||
|
||
Lemma P_PairInv : forall n (a b : PTm n), P (PPair a b) -> P a /\ P b.
|
||
move => n a b. move E : (PPair a b) => u h.
|
||
move : a b E. elim : n u / h; sauto lq:on rew:off. Qed.
|
||
|
||
Lemma P_ProjInv : forall n p (a : PTm n), P (PProj p a) -> P a.
|
||
Proof. sfirstorder use:SN_ProjInv. Qed.
|
||
|
||
Lemma P_BindInv : forall n p (A : PTm n) B, P (PBind p A B) -> P A /\ P B.
|
||
Proof.
|
||
move => n p A B.
|
||
move E : (PBind p A B) => u hu.
|
||
move : p A B E. elim : n u /hu=>//=;sauto lq:on rew:off.
|
||
Qed.
|
||
|
||
Lemma P_SucInv : forall n (a : PTm n), P (PSuc a) -> P a.
|
||
Proof. sauto lq:on. Qed.
|
||
|
||
Lemma P_AbsInv : forall n (a : PTm (S n)), P (PAbs a) -> P a.
|
||
Proof.
|
||
move => n a. move E : (PAbs a) => u h.
|
||
move : E. move : a.
|
||
induction h; sauto lq:on rew:off.
|
||
Qed.
|
||
|
||
Lemma P_renaming : forall n m (ξ : fin n -> fin m) a , P (ren_PTm ξ a) <-> P a.
|
||
Proof. hauto lq:on use:sn_antirenaming, sn_renaming. Qed.
|
||
|
||
Lemma P_ProjBind : forall n p p' (A : PTm n) B, ~ P (PProj p (PBind p' A B)).
|
||
Proof. sfirstorder use:PProjBind_imp. Qed.
|
||
|
||
Lemma P_AppBind : forall n p (A : PTm n) B b, ~ P (PApp (PBind p A B) b).
|
||
Proof. sfirstorder use:PAppBind_imp. Qed.
|
||
|
||
Lemma P_IndInv : forall n Q (a : PTm n) b c, P (PInd Q a b c) -> P Q /\ P a /\ P b /\ P c.
|
||
Proof. sfirstorder use:SN_IndInv. Qed.
|
||
|
||
End SN_NoForbid.
|
||
|
||
Module NoForbid_FactSN := NoForbid_Fact SN_NoForbid.
|
||
|
||
Module UniqueNF (M : NoForbid) (MFacts : NoForbid_FactSig M).
|
||
Import M MFacts.
|
||
#[local]Hint Resolve P_EPar P_RRed PApp_imp PProj_imp : forbid.
|
||
|
||
Lemma η_split n (a0 a1 : PTm n) :
|
||
EPar.R a0 a1 ->
|
||
P a0 ->
|
||
exists b, rtc RRed.R a0 b /\ NeEPar.R_nonelim b a1.
|
||
Proof.
|
||
move => h. elim : n a0 a1 /h .
|
||
- move => n a0 a1 ha ih /[dup] hP.
|
||
move /P_AbsInv /P_AppInv => [/P_renaming ha0 _].
|
||
have {ih} := ih ha0.
|
||
move => [b [ih0 ih1]].
|
||
case /orP : (orNb (ishf b)).
|
||
exists (PAbs (PApp (ren_PTm shift b) (VarPTm var_zero))).
|
||
split. apply RReds.AbsCong. apply RReds.AppCong; auto using rtc_refl.
|
||
by eauto using RReds.renaming.
|
||
apply NeEPar.AppEta=>//.
|
||
sfirstorder use:NeEPar.R_nonelim_nothf.
|
||
case /orP : (orbN (isabs b)).
|
||
+ case : b ih0 ih1 => //= p ih0 ih1 _ _.
|
||
set q := PAbs _.
|
||
suff : rtc RRed.R q (PAbs p) by sfirstorder.
|
||
subst q.
|
||
apply : rtc_r.
|
||
apply RReds.AbsCong. apply RReds.AppCong.
|
||
by eauto using RReds.renaming.
|
||
apply rtc_refl.
|
||
apply : RRed.AbsCong => /=.
|
||
apply RRed.AppAbs'. by asimpl.
|
||
(* violates SN *)
|
||
+ move /P_AbsInv in hP.
|
||
have {}hP : P (PApp (ren_PTm shift b) (VarPTm var_zero))
|
||
by sfirstorder use:P_RReds, RReds.AppCong, @rtc_refl, RReds.renaming.
|
||
move => ? ?.
|
||
have ? : ~~ isabs (ren_PTm shift b) by scongruence use:isabs_ren.
|
||
have ? : ishf (ren_PTm shift b) by scongruence use:ishf_ren.
|
||
exfalso.
|
||
sfirstorder use:PApp_imp.
|
||
- move => n a0 a1 h ih /[dup] hP.
|
||
move /P_PairInv => [/P_ProjInv + _].
|
||
move : ih => /[apply].
|
||
move => [b [ih0 ih1]].
|
||
case /orP : (orNb (ishf b)).
|
||
exists (PPair (PProj PL b) (PProj PR b)).
|
||
split. sfirstorder use:RReds.PairCong,RReds.ProjCong.
|
||
hauto lq:on ctrs:NeEPar.R_nonelim use:NeEPar.R_nonelim_nothf.
|
||
case /orP : (orbN (ispair b)).
|
||
+ case : b ih0 ih1 => //=.
|
||
move => t0 t1 ih0 h1 _ _.
|
||
exists (PPair t0 t1).
|
||
split => //=.
|
||
apply RReds.PairCong.
|
||
apply : rtc_r; eauto using RReds.ProjCong.
|
||
apply RRed.ProjPair.
|
||
apply : rtc_r; eauto using RReds.ProjCong.
|
||
apply RRed.ProjPair.
|
||
+ move => ? ?. exfalso.
|
||
move/P_PairInv : hP=>[hP _].
|
||
have : rtc RRed.R (PProj PL a0) (PProj PL b)
|
||
by eauto using RReds.ProjCong.
|
||
move : P_RReds hP. repeat move/[apply] => /=.
|
||
sfirstorder use:PProj_imp.
|
||
- hauto lq:on ctrs:NeEPar.R_nonelim use:RReds.AbsCong, P_AbsInv.
|
||
- move => n a0 a1 b0 b1 ha iha hb ihb.
|
||
move => /[dup] hP /P_AppInv [hP0 hP1].
|
||
have {iha} [a2 [iha0 iha1]] := iha hP0.
|
||
have {ihb} [b2 [ihb0 ihb1]] := ihb hP1.
|
||
case /orP : (orNb (ishf a2)) => [h|].
|
||
+ exists (PApp a2 b2). split; first by eauto using RReds.AppCong.
|
||
hauto lq:on ctrs:NeEPar.R_nonelim use:NeEPar.R_nonelim_nothf.
|
||
+ case /orP : (orbN (isabs a2)).
|
||
(* case : a2 iha0 iha1 => //=. *)
|
||
* case : a2 iha0 iha1 => //= p h0 h1 _ _.
|
||
inversion h1; subst.
|
||
** exists (PApp a2 b2).
|
||
split.
|
||
apply : rtc_r.
|
||
apply RReds.AppCong; eauto.
|
||
apply RRed.AppAbs'. by asimpl.
|
||
hauto lq:on ctrs:NeEPar.R_nonelim.
|
||
** hauto lq:on ctrs:NeEPar.R_nonelim,NeEPar.R_elim use:RReds.AppCong.
|
||
(* Impossible *)
|
||
* move =>*. exfalso.
|
||
have : P (PApp a2 b0) by sfirstorder use:RReds.AppCong, @rtc_refl, P_RReds.
|
||
sfirstorder use:PApp_imp.
|
||
- hauto lq:on ctrs:NeEPar.R_nonelim use:RReds.PairCong, P_PairInv.
|
||
- move => n p a0 a1 ha ih /[dup] hP /P_ProjInv.
|
||
move : ih => /[apply]. move => [a2 [iha0 iha1]].
|
||
case /orP : (orNb (ishf a2)) => [h|].
|
||
exists (PProj p a2).
|
||
split. eauto using RReds.ProjCong.
|
||
qauto l:on ctrs:NeEPar.R_nonelim, NeEPar.R_elim use:NeEPar.R_nonelim_nothf.
|
||
|
||
case /orP : (orNb (ispair a2)).
|
||
+ move => *. exfalso.
|
||
have : rtc RRed.R (PProj p a0) (PProj p a2)
|
||
by sfirstorder use:RReds.ProjCong ctrs:rtc.
|
||
move : P_RReds hP. repeat move/[apply].
|
||
sfirstorder use:PProj_imp.
|
||
+ case : a2 iha0 iha1 => //= u0 u1 iha0 iha1 _ _.
|
||
inversion iha1; subst.
|
||
* exists (PProj p a2). split.
|
||
apply : rtc_r.
|
||
apply RReds.ProjCong; eauto.
|
||
clear. hauto l:on inv:PTag.
|
||
hauto lq:on ctrs:NeEPar.R_nonelim.
|
||
* hauto lq:on ctrs:NeEPar.R_nonelim,NeEPar.R_elim use:RReds.ProjCong.
|
||
- hauto lq:on ctrs:rtc, NeEPar.R_nonelim.
|
||
- hauto l:on.
|
||
- hauto lq:on ctrs:NeEPar.R_nonelim, rtc use:RReds.BindCong, P_BindInv.
|
||
- hauto lq:on ctrs:NeEPar.R_nonelim, rtc use:RReds.BindCong, P_BindInv.
|
||
- hauto l:on ctrs:NeEPar.R_nonelim.
|
||
- move => n P0 P1 a0 a1 b0 b1 c0 c1 hP ihP ha iha hb ihb hc ihc /[dup] hInd /P_IndInv.
|
||
move => []. move : ihP => /[apply].
|
||
move => [P01][ihP0]ihP1.
|
||
move => []. move : iha => /[apply].
|
||
move => [a01][iha0]iha1.
|
||
move => []. move : ihb => /[apply].
|
||
move => [b01][ihb0]ihb1.
|
||
move : ihc => /[apply].
|
||
move => [c01][ihc0]ihc1.
|
||
case /orP : (orNb (ishf a01)) => [h|].
|
||
+ eexists. split. by eauto using RReds.IndCong.
|
||
hauto q:on ctrs:NeEPar.R_nonelim use:NeEPar.R_nonelim_nothf.
|
||
+ move => h.
|
||
case /orP : (orNb (issuc a01 || iszero a01)).
|
||
* move /norP.
|
||
have : P (PInd P01 a01 b01 c01) by eauto using P_RReds, RReds.IndCong.
|
||
hauto lq:on use:PInd_imp.
|
||
* move => ha01.
|
||
eexists. split. eauto using RReds.IndCong.
|
||
apply NeEPar.IndCong; eauto.
|
||
move : iha1 ha01. clear.
|
||
inversion 1; subst => //=; hauto lq:on ctrs:NeEPar.R_elim.
|
||
- hauto l:on.
|
||
- hauto lq:on ctrs:NeEPar.R_nonelim use:RReds.SucCong, P_SucInv.
|
||
Qed.
|
||
|
||
Lemma eta_postponement n a b c :
|
||
@P n a ->
|
||
EPar.R a b ->
|
||
RRed.R b c ->
|
||
exists d, rtc RRed.R a d /\ EPar.R d c.
|
||
Proof.
|
||
move => + h.
|
||
move : c.
|
||
elim : n a b /h => //=.
|
||
- move => n a0 a1 ha iha c /[dup] hP /P_AbsInv /P_AppInv [/P_renaming hP' _] hc.
|
||
move : iha (hP') (hc); repeat move/[apply].
|
||
move => [d [h0 h1]].
|
||
exists (PAbs (PApp (ren_PTm shift d) (VarPTm var_zero))).
|
||
split. hauto lq:on rew:off ctrs:rtc use:RReds.AbsCong, RReds.AppCong, RReds.renaming.
|
||
hauto lq:on ctrs:EPar.R.
|
||
- move => n a0 a1 ha iha c /P_PairInv [/P_ProjInv + _].
|
||
move /iha => /[apply].
|
||
move => [d [h0 h1]].
|
||
exists (PPair (PProj PL d) (PProj PR d)).
|
||
hauto lq:on ctrs:EPar.R use:RReds.PairCong, RReds.ProjCong.
|
||
- move => n a0 a1 ha iha c /P_AbsInv /[swap].
|
||
elim /RRed.inv => //=_.
|
||
move => a2 a3 + [? ?]. subst.
|
||
move : iha; repeat move/[apply].
|
||
hauto lq:on use:RReds.AbsCong ctrs:EPar.R.
|
||
- move => n a0 a1 b0 b1 ha iha hb ihb c hP.
|
||
elim /RRed.inv => //= _.
|
||
+ move => a2 b2 [*]. subst.
|
||
have [hP' hP''] : P a0 /\ P b0 by sfirstorder use:P_AppInv.
|
||
move {iha ihb}.
|
||
move /η_split /(_ hP') : ha.
|
||
move => [b [h0 h1]].
|
||
inversion h1; subst.
|
||
* inversion H0; subst.
|
||
exists (subst_PTm (scons b0 VarPTm) a3).
|
||
split; last by scongruence use:EPar.morphing.
|
||
apply : relations.rtc_transitive.
|
||
apply RReds.AppCong.
|
||
eassumption.
|
||
apply rtc_refl.
|
||
apply : rtc_l.
|
||
apply RRed.AppCong0. apply RRed.AbsCong. simpl. apply RRed.AppAbs.
|
||
asimpl.
|
||
apply rtc_once.
|
||
apply RRed.AppAbs.
|
||
* exfalso.
|
||
move : hP h0. clear => hP h0.
|
||
have : rtc RRed.R (PApp a0 b0) (PApp (PPair (PProj PL a1) (PProj PR a1)) b0)
|
||
by qauto l:on ctrs:rtc use:RReds.AppCong.
|
||
move : P_RReds hP. repeat move/[apply].
|
||
sfirstorder use:PApp_imp.
|
||
* exists (subst_PTm (scons b0 VarPTm) a1).
|
||
split.
|
||
apply : rtc_r; last by apply RRed.AppAbs.
|
||
hauto lq:on ctrs:rtc use:RReds.AppCong.
|
||
hauto l:on inv:option use:EPar.morphing,NeEPar.ToEPar.
|
||
+ move => a2 a3 b2 ha2 [*]. subst.
|
||
move : iha (ha2) {ihb} => /[apply].
|
||
have : P a0 by sfirstorder use:P_AppInv.
|
||
move /[swap]/[apply].
|
||
move => [d [h0 h1]].
|
||
exists (PApp d b0).
|
||
hauto lq:on ctrs:EPar.R, rtc use:RReds.AppCong.
|
||
+ move => a2 b2 b3 hb2 [*]. subst.
|
||
move {iha}.
|
||
have : P b0 by sfirstorder use:P_AppInv.
|
||
move : ihb hb2; repeat move /[apply].
|
||
hauto lq:on rew:off ctrs:EPar.R, rtc use:RReds.AppCong.
|
||
- move => n a0 a1 b0 b1 ha iha hb ihb c /P_PairInv [hP hP'].
|
||
elim /RRed.inv => //=_;
|
||
hauto lq:on rew:off ctrs:EPar.R, rtc use:RReds.PairCong.
|
||
- move => n p a0 a1 ha iha c /[dup] hP /P_ProjInv hP'.
|
||
elim / RRed.inv => //= _.
|
||
+ move => p0 a2 b0 [*]. subst.
|
||
move : η_split hP' ha; repeat move/[apply].
|
||
move => [a1 [h0 h1]].
|
||
inversion h1; subst.
|
||
* sauto q:on ctrs:rtc use:RReds.ProjCong, PProj_imp, P_RReds.
|
||
* inversion H0; subst.
|
||
exists (if p is PL then a1 else b1).
|
||
split; last by scongruence use:NeEPar.ToEPar.
|
||
apply : relations.rtc_transitive.
|
||
apply RReds.ProjCong; eauto.
|
||
apply : rtc_l.
|
||
apply RRed.ProjCong.
|
||
apply RRed.PairCong0.
|
||
apply RRed.ProjPair.
|
||
apply : rtc_l.
|
||
apply RRed.ProjCong.
|
||
apply RRed.PairCong1.
|
||
apply RRed.ProjPair.
|
||
apply rtc_once. apply RRed.ProjPair.
|
||
* exists (if p is PL then a3 else b1).
|
||
split; last by hauto lq:on use:NeEPar.ToEPar.
|
||
apply : relations.rtc_transitive.
|
||
eauto using RReds.ProjCong.
|
||
apply rtc_once.
|
||
apply RRed.ProjPair.
|
||
+ move => p0 a2 a3 h0 [*]. subst.
|
||
move : iha hP' h0;repeat move/[apply].
|
||
hauto lq:on ctrs:rtc, EPar.R use:RReds.ProjCong.
|
||
- hauto lq:on inv:RRed.R.
|
||
- hauto lq:on inv:RRed.R ctrs:rtc.
|
||
- sauto lq:on ctrs:EPar.R, rtc use:RReds.BindCong, P_BindInv, @relations.rtc_transitive.
|
||
- hauto lq:on inv:RRed.R ctrs:rtc.
|
||
- hauto q:on ctrs:rtc inv:RRed.R.
|
||
- move => n P0 P1 a0 a1 b0 b1 c0 c1 hP ihP ha iha hb ihb hc ihc u.
|
||
move => /[dup] hInd.
|
||
move /P_IndInv.
|
||
move => [pP0][pa0][pb0]pc0.
|
||
elim /RRed.inv => //= _.
|
||
+ move => P2 b2 c2 [*]. subst.
|
||
move /η_split : pa0 ha; repeat move/[apply].
|
||
move => [a1][h0]h1 {iha}.
|
||
inversion h1; subst.
|
||
* exfalso.
|
||
have :P (PInd P0 (PAbs (PApp (ren_PTm shift a2) (VarPTm var_zero))) b0 c0) by eauto using RReds.IndCong, rtc_refl, P_RReds.
|
||
clear. hauto lq:on use:PInd_imp.
|
||
* have :P (PInd P0 (PPair (PProj PL a2) (PProj PR a2)) b0 c0) by eauto using RReds.IndCong, rtc_refl, P_RReds.
|
||
clear. hauto lq:on use:PInd_imp.
|
||
* eexists. split; eauto.
|
||
apply : rtc_r.
|
||
apply : RReds.IndCong; eauto; eauto using rtc_refl.
|
||
apply RRed.IndZero.
|
||
+ move => P2 a2 b2 c [*]. subst.
|
||
move /η_split /(_ pa0) : ha.
|
||
move => [a1][h0]h1.
|
||
inversion h1; subst.
|
||
* have :P (PInd P0 (PAbs (PApp (ren_PTm shift a3) (VarPTm var_zero))) b0 c0) by eauto using RReds.IndCong, rtc_refl, P_RReds.
|
||
clear. hauto q:on use:PInd_imp.
|
||
* have :P (PInd P0 (PPair (PProj PL a3) (PProj PR a3)) b0 c0) by eauto using RReds.IndCong, rtc_refl, P_RReds.
|
||
clear. hauto q:on use:PInd_imp.
|
||
* eexists. split.
|
||
apply : rtc_r.
|
||
apply RReds.IndCong; eauto; eauto using rtc_refl.
|
||
apply RRed.IndSuc.
|
||
apply EPar.morphing;eauto. hauto lq:on ctrs:EPar.R inv:option use:NeEPar.ToEPar.
|
||
+ move => P2 P3 a2 b2 c hP0 [*]. subst.
|
||
move : ihP hP0 pP0. repeat move/[apply].
|
||
move => [P2][h0]h1.
|
||
exists (PInd P2 a0 b0 c0).
|
||
sfirstorder use:RReds.IndCong, @rtc_refl, EPar.IndCong.
|
||
+ move => P2 a2 a3 b2 c + [*]. subst.
|
||
move : iha pa0; repeat move/[apply].
|
||
move => [a2][*].
|
||
exists (PInd P0 a2 b0 c0).
|
||
sfirstorder use:RReds.IndCong, @rtc_refl, EPar.IndCong.
|
||
+ move => P2 a2 b2 b3 c + [*]. subst.
|
||
move : ihb pb0; repeat move/[apply].
|
||
move => [b2][*].
|
||
exists (PInd P0 a0 b2 c0).
|
||
sfirstorder use:RReds.IndCong, @rtc_refl, EPar.IndCong.
|
||
+ move => P2 a2 b2 b3 c + [*]. subst.
|
||
move : ihc pc0; repeat move/[apply].
|
||
move => [c2][*].
|
||
exists (PInd P0 a0 b0 c2).
|
||
sfirstorder use:RReds.IndCong, @rtc_refl, EPar.IndCong.
|
||
- hauto lq:on inv:RRed.R ctrs:rtc, EPar.R.
|
||
- move => n a0 a1 ha iha u /P_SucInv ha0.
|
||
elim /RRed.inv => //= _ a2 a3 h [*]. subst.
|
||
move : iha (ha0) (h); repeat move/[apply].
|
||
move => [a2 [ih0 ih1]].
|
||
exists (PSuc a2).
|
||
split. by apply RReds.SucCong.
|
||
by apply EPar.SucCong.
|
||
Qed.
|
||
|
||
Lemma η_postponement_star n a b c :
|
||
@P n a ->
|
||
EPar.R a b ->
|
||
rtc RRed.R b c ->
|
||
exists d, rtc RRed.R a d /\ EPar.R d c.
|
||
Proof.
|
||
move => + + h. move : a.
|
||
elim : b c / h.
|
||
- sfirstorder.
|
||
- move => a0 a1 a2 ha ha' iha u hu hu'.
|
||
move : eta_postponement (hu) ha hu'; repeat move/[apply].
|
||
move => [d [h0 h1]].
|
||
have : P d by sfirstorder use:P_RReds.
|
||
move : iha h1; repeat move/[apply].
|
||
sfirstorder use:@relations.rtc_transitive.
|
||
Qed.
|
||
|
||
Lemma η_postponement_star' n a b c :
|
||
@P n a ->
|
||
EPar.R a b ->
|
||
rtc RRed.R b c ->
|
||
exists d, rtc RRed.R a d /\ NeEPar.R_nonelim d c.
|
||
Proof.
|
||
move => h0 h1 h2.
|
||
have : exists d, rtc RRed.R a d /\ EPar.R d c by eauto using η_postponement_star.
|
||
move => [d [h3 /η_split]].
|
||
move /(_ ltac:(eauto using P_RReds)).
|
||
sfirstorder use:@relations.rtc_transitive.
|
||
Qed.
|
||
|
||
End UniqueNF.
|
||
|
||
Module SN_UniqueNF := UniqueNF SN_NoForbid NoForbid_FactSN.
|
||
|
||
Module ERed.
|
||
Inductive R {n} : PTm n -> PTm n -> Prop :=
|
||
|
||
(****************** Eta ***********************)
|
||
| AppEta a :
|
||
R (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero))) a
|
||
| PairEta a :
|
||
R (PPair (PProj PL a) (PProj PR a)) a
|
||
|
||
(*************** Congruence ********************)
|
||
| AbsCong a0 a1 :
|
||
R a0 a1 ->
|
||
R (PAbs a0) (PAbs a1)
|
||
| AppCong0 a0 a1 b :
|
||
R a0 a1 ->
|
||
R (PApp a0 b) (PApp a1 b)
|
||
| AppCong1 a b0 b1 :
|
||
R b0 b1 ->
|
||
R (PApp a b0) (PApp a b1)
|
||
| PairCong0 a0 a1 b :
|
||
R a0 a1 ->
|
||
R (PPair a0 b) (PPair a1 b)
|
||
| PairCong1 a b0 b1 :
|
||
R b0 b1 ->
|
||
R (PPair a b0) (PPair a b1)
|
||
| ProjCong p a0 a1 :
|
||
R a0 a1 ->
|
||
R (PProj p a0) (PProj p a1)
|
||
| BindCong0 p A0 A1 B :
|
||
R A0 A1 ->
|
||
R (PBind p A0 B) (PBind p A1 B)
|
||
| BindCong1 p A B0 B1 :
|
||
R B0 B1 ->
|
||
R (PBind p A B0) (PBind p A B1)
|
||
| IndCong0 P0 P1 a b c :
|
||
R P0 P1 ->
|
||
R (PInd P0 a b c) (PInd P1 a b c)
|
||
| IndCong1 P a0 a1 b c :
|
||
R a0 a1 ->
|
||
R (PInd P a0 b c) (PInd P a1 b c)
|
||
| IndCong2 P a b0 b1 c :
|
||
R b0 b1 ->
|
||
R (PInd P a b0 c) (PInd P a b1 c)
|
||
| IndCong3 P a b c0 c1 :
|
||
R c0 c1 ->
|
||
R (PInd P a b c0) (PInd P a b c1)
|
||
| SucCong a0 a1 :
|
||
R a0 a1 ->
|
||
R (PSuc a0) (PSuc a1).
|
||
|
||
Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop.
|
||
|
||
Lemma ToEPar n (a b : PTm n) :
|
||
ERed.R a b -> EPar.R a b.
|
||
Proof.
|
||
induction 1; hauto lq:on use:EPar.refl ctrs:EPar.R.
|
||
Qed.
|
||
|
||
Ltac2 rec solve_anti_ren () :=
|
||
let x := Fresh.in_goal (Option.get (Ident.of_string "x")) in
|
||
intro $x;
|
||
lazy_match! Constr.type (Control.hyp x) with
|
||
| fin _ -> _ _ => (ltac1:(case;hauto q:on depth:2 ctrs:ERed.R))
|
||
| _ => solve_anti_ren ()
|
||
end.
|
||
|
||
Ltac solve_anti_ren := ltac2:(Control.enter solve_anti_ren).
|
||
|
||
(* Definition down n m (ξ : fin n -> fin m) (a : fin (S n)) : fin m. *)
|
||
(* destruct a. *)
|
||
(* exact (ξ f). *)
|
||
|
||
Lemma AppEta' n a u :
|
||
u = (@PApp (S n) (ren_PTm shift a) (VarPTm var_zero)) ->
|
||
R (PAbs u) a.
|
||
Proof. move => ->. apply AppEta. Qed.
|
||
|
||
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.
|
||
|
||
move => n a m ξ /=.
|
||
apply AppEta'; eauto. by asimpl.
|
||
all : qauto ctrs:R.
|
||
Qed.
|
||
|
||
(* Characterization of variable freeness conditions *)
|
||
Definition tm_i_free {n} a (i : fin n) := exists m (ξ ξ0 : fin n -> fin m), ξ i <> ξ0 i /\ ren_PTm ξ a = ren_PTm ξ0 a.
|
||
|
||
Lemma subst_differ_one_ren_up n m i (ξ0 ξ1 : fin n -> fin m) :
|
||
(forall j, i <> j -> ξ0 j = ξ1 j) ->
|
||
(forall j, shift i <> j -> upRen_PTm_PTm ξ0 j = upRen_PTm_PTm ξ1 j).
|
||
Proof.
|
||
move => hξ.
|
||
destruct j. asimpl.
|
||
move => h. have : i<> f by hauto lq:on rew:off inv:option.
|
||
move /hξ. by rewrite /funcomp => ->.
|
||
done.
|
||
Qed.
|
||
|
||
Lemma tm_free_ren_any n a i :
|
||
tm_i_free a i ->
|
||
forall m (ξ0 ξ1 : fin n -> fin m), (forall j, i <> j -> ξ0 j = ξ1 j) ->
|
||
ren_PTm ξ0 a = ren_PTm ξ1 a.
|
||
Proof.
|
||
rewrite /tm_i_free.
|
||
move => [+ [+ [+ +]]].
|
||
move : i.
|
||
elim : n / a => n.
|
||
- hauto q:on.
|
||
- move => a iha i m ρ0 ρ1 [] => h [] h' m' ξ0 ξ1 hξ /=.
|
||
f_equal. move /subst_differ_one_ren_up in hξ.
|
||
move /(_ (shift i)) in iha.
|
||
move : iha hξ; move/[apply].
|
||
apply=>//. split; eauto.
|
||
asimpl. rewrite /funcomp. hauto l:on.
|
||
- hauto lq:on rew:off.
|
||
- hauto lq:on rew:off.
|
||
- hauto lq:on rew:off.
|
||
- move => p A ihA a iha i m ρ0 ρ1 [] ? h m' ξ0 ξ1 hξ /=.
|
||
f_equal. hauto lq:on rew:off.
|
||
move /subst_differ_one_ren_up in hξ.
|
||
move /(_ (shift i)) in iha.
|
||
move : iha hξ. repeat move/[apply].
|
||
move /(_ _ (upRen_PTm_PTm ρ0) (upRen_PTm_PTm ρ1)).
|
||
apply. hauto l:on.
|
||
- hauto lq:on rew:off.
|
||
- hauto lq:on rew:off.
|
||
- hauto lq:on rew:off.
|
||
- hauto lq:on rew:off.
|
||
- hauto lq:on rew:off.
|
||
- admit.
|
||
Admitted.
|
||
|
||
Lemma antirenaming n m (a : PTm n) (b : PTm m) (ξ : fin n -> fin m) :
|
||
(forall i j, ξ i = ξ j -> i = j) ->
|
||
R (ren_PTm ξ a) b -> exists b0, R a b0 /\ ren_PTm ξ b0 = b.
|
||
Proof.
|
||
move => hξ.
|
||
move E : (ren_PTm ξ a) => u hu.
|
||
move : n ξ a hξ E.
|
||
elim : m u b / hu; try solve_anti_ren.
|
||
- move => n a m ξ []//=.
|
||
move => u hξ [].
|
||
case : u => //=.
|
||
move => u0 u1 [].
|
||
case : u1 => //=.
|
||
move => i /[swap] [].
|
||
case : i => //= _ h.
|
||
have : exists p, ren_PTm shift p = u0 by admit.
|
||
move => [p ?]. subst.
|
||
move : h. asimpl.
|
||
replace (ren_PTm (funcomp shift ξ) p) with
|
||
(ren_PTm shift (ren_PTm ξ p)); last by asimpl.
|
||
move /ren_injective.
|
||
move /(_ ltac:(hauto l:on)).
|
||
move => ?. subst.
|
||
exists p. split=>//. apply AppEta.
|
||
- move => n a m ξ [] //=.
|
||
move => u u0 hξ [].
|
||
case : u => //=.
|
||
case : u0 => //=.
|
||
move => p p0 p1 p2 [? ?] [? h]. subst.
|
||
have ? : p0 = p2 by eauto using ren_injective. subst.
|
||
hauto l:on.
|
||
- move => n a0 a1 ha iha m ξ []//= p hξ [?]. subst.
|
||
sauto lq:on use:up_injective.
|
||
- move => n p A B0 B1 hB ihB m ξ + hξ.
|
||
case => //= p' A2 B2 [*]. subst.
|
||
have : (forall i j, (upRen_PTm_PTm ξ) i = (upRen_PTm_PTm ξ) j -> i = j) by sauto.
|
||
move => {}/ihB => ihB.
|
||
spec_refl.
|
||
sauto lq:on.
|
||
Admitted.
|
||
|
||
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.
|
||
move => h. move : m ρ. elim : n a b /h => n.
|
||
move => a m ρ /=.
|
||
eapply AppEta'; eauto. by asimpl.
|
||
all : hauto lq:on ctrs:R.
|
||
Qed.
|
||
|
||
Lemma nf_preservation n (a b : PTm n) :
|
||
ERed.R a b -> (nf a -> nf b) /\ (ne a -> ne b).
|
||
Proof.
|
||
move => h.
|
||
elim : n a b /h => //=;
|
||
hauto lqb:on use:ne_nf_ren,ne_nf.
|
||
Qed.
|
||
|
||
End ERed.
|
||
|
||
Module EReds.
|
||
|
||
#[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.
|
||
|
||
Lemma AbsCong n (a b : PTm (S n)) :
|
||
rtc ERed.R a b ->
|
||
rtc ERed.R (PAbs a) (PAbs b).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
|
||
rtc ERed.R a0 a1 ->
|
||
rtc ERed.R b0 b1 ->
|
||
rtc ERed.R (PApp a0 b0) (PApp a1 b1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
|
||
rtc ERed.R a0 a1 ->
|
||
rtc ERed.R b0 b1 ->
|
||
rtc ERed.R (PPair a0 b0) (PPair a1 b1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma ProjCong n p (a0 a1 : PTm n) :
|
||
rtc ERed.R a0 a1 ->
|
||
rtc ERed.R (PProj p a0) (PProj p a1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma BindCong n p (A0 A1 : PTm n) B0 B1 :
|
||
rtc ERed.R A0 A1 ->
|
||
rtc ERed.R B0 B1 ->
|
||
rtc ERed.R (PBind p A0 B0) (PBind p A1 B1).
|
||
Proof. solve_s. Qed.
|
||
|
||
|
||
Lemma SucCong n (a0 a1 : PTm n) :
|
||
rtc ERed.R a0 a1 ->
|
||
rtc ERed.R (PSuc a0) (PSuc a1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma IndCong n P0 P1 (a0 a1 : PTm n) b0 b1 c0 c1 :
|
||
rtc ERed.R P0 P1 ->
|
||
rtc ERed.R a0 a1 ->
|
||
rtc ERed.R b0 b1 ->
|
||
rtc ERed.R c0 c1 ->
|
||
rtc ERed.R (PInd P0 a0 b0 c0) (PInd P1 a1 b1 c1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
|
||
rtc ERed.R a b -> rtc ERed.R (ren_PTm ξ a) (ren_PTm ξ b).
|
||
Proof. induction 1; hauto l:on use:ERed.renaming ctrs:rtc. Qed.
|
||
|
||
Lemma FromEPar n (a b : PTm n) :
|
||
EPar.R a b ->
|
||
rtc ERed.R a b.
|
||
Proof.
|
||
move => h. elim : n a b /h; eauto using AbsCong, AppCong, PairCong, ProjCong, rtc_refl, BindCong, IndCong, SucCong.
|
||
- move => n a0 a1 _ h.
|
||
have {}h : rtc ERed.R (ren_PTm shift a0) (ren_PTm shift a1) by apply renaming.
|
||
apply : rtc_r. apply AbsCong. apply AppCong; eauto. apply rtc_refl.
|
||
apply ERed.AppEta.
|
||
- move => n a0 a1 _ h.
|
||
apply : rtc_r.
|
||
apply PairCong; eauto using ProjCong.
|
||
apply ERed.PairEta.
|
||
Qed.
|
||
|
||
Lemma FromEPars n (a b : PTm n) :
|
||
rtc EPar.R a b ->
|
||
rtc ERed.R a b.
|
||
Proof. induction 1; hauto l:on use:FromEPar, @relations.rtc_transitive. Qed.
|
||
|
||
Lemma substing n m (a b : PTm n) (ρ : fin n -> PTm m) :
|
||
rtc ERed.R a b -> rtc ERed.R (subst_PTm ρ a) (subst_PTm ρ b).
|
||
Proof.
|
||
induction 1; hauto lq:on ctrs:rtc use:ERed.substing.
|
||
Qed.
|
||
|
||
Lemma app_inv n (a0 b0 C : PTm n) :
|
||
rtc ERed.R (PApp a0 b0) C ->
|
||
exists a1 b1, C = PApp a1 b1 /\
|
||
rtc ERed.R a0 a1 /\
|
||
rtc ERed.R b0 b1.
|
||
Proof.
|
||
move E : (PApp a0 b0) => u hu.
|
||
move : a0 b0 E.
|
||
elim : u C / hu.
|
||
- hauto lq:on ctrs:rtc.
|
||
- move => a b c ha ha' iha a0 b0 ?. subst.
|
||
hauto lq:on rew:off ctrs:rtc, ERed.R inv:ERed.R.
|
||
Qed.
|
||
|
||
Lemma proj_inv n p (a C : PTm n) :
|
||
rtc ERed.R (PProj p a) C ->
|
||
exists c, C = PProj p c /\ rtc ERed.R a c.
|
||
Proof.
|
||
move E : (PProj p a) => u hu.
|
||
move : p a E.
|
||
elim : u C /hu;
|
||
scrush ctrs:rtc,ERed.R inv:ERed.R.
|
||
Qed.
|
||
|
||
Lemma bind_inv n p (A : PTm n) B C :
|
||
rtc ERed.R (PBind p A B) C ->
|
||
exists A0 B0, C = PBind p A0 B0 /\ rtc ERed.R A A0 /\ rtc ERed.R B B0.
|
||
Proof.
|
||
move E : (PBind p A B) => u hu.
|
||
move : p A B E.
|
||
elim : u C / hu.
|
||
hauto lq:on ctrs:rtc.
|
||
hauto lq:on rew:off ctrs:rtc, ERed.R inv:ERed.R, rtc.
|
||
Qed.
|
||
|
||
Lemma suc_inv n (a : PTm n) C :
|
||
rtc ERed.R (PSuc a) C ->
|
||
exists b, rtc ERed.R a b /\ C = PSuc b.
|
||
Proof.
|
||
move E : (PSuc a) => u hu.
|
||
move : a E.
|
||
elim : u C / hu=>//=.
|
||
- hauto l:on.
|
||
- hauto lq:on rew:off ctrs:rtc, ERed.R inv:ERed.R, rtc.
|
||
Qed.
|
||
|
||
Lemma ind_inv n P (a : PTm n) b c C :
|
||
rtc ERed.R (PInd P a b c) C ->
|
||
exists P0 a0 b0 c0, rtc ERed.R P P0 /\ rtc ERed.R a a0 /\
|
||
rtc ERed.R b b0 /\ rtc ERed.R c c0 /\
|
||
C = PInd P0 a0 b0 c0.
|
||
Proof.
|
||
move E : (PInd P a b c) => u hu.
|
||
move : P a b c E.
|
||
elim : u C / hu.
|
||
- hauto lq:on ctrs:rtc.
|
||
- hauto lq:on rew:off ctrs:rtc, ERed.R inv:ERed.R, rtc.
|
||
Qed.
|
||
|
||
End EReds.
|
||
|
||
#[export]Hint Constructors ERed.R RRed.R EPar.R : red.
|
||
|
||
Module EJoin.
|
||
Definition R {n} (a b : PTm n) := exists c, rtc ERed.R a c /\ rtc ERed.R b c.
|
||
|
||
Lemma hne_app_inj n (a0 b0 a1 b1 : PTm n) :
|
||
R (PApp a0 b0) (PApp a1 b1) ->
|
||
R a0 a1 /\ R b0 b1.
|
||
Proof.
|
||
hauto lq:on use:EReds.app_inv.
|
||
Qed.
|
||
|
||
Lemma hne_proj_inj n p0 p1 (a0 a1 : PTm n) :
|
||
R (PProj p0 a0) (PProj p1 a1) ->
|
||
p0 = p1 /\ R a0 a1.
|
||
Proof.
|
||
hauto lq:on rew:off use:EReds.proj_inv.
|
||
Qed.
|
||
|
||
Lemma bind_inj n p0 p1 (A0 A1 : PTm n) B0 B1 :
|
||
R (PBind p0 A0 B0) (PBind p1 A1 B1) ->
|
||
p0 = p1 /\ R A0 A1 /\ R B0 B1.
|
||
Proof.
|
||
hauto lq:on rew:off use:EReds.bind_inv.
|
||
Qed.
|
||
|
||
Lemma suc_inj n (A0 A1 : PTm n) :
|
||
R (PSuc A0) (PSuc A1) ->
|
||
R A0 A1.
|
||
Proof.
|
||
hauto lq:on rew:off use:EReds.suc_inv.
|
||
Qed.
|
||
|
||
Lemma ind_inj n P0 P1 (a0 a1 : PTm n) b0 b1 c0 c1 :
|
||
R (PInd P0 a0 b0 c0) (PInd P1 a1 b1 c1) ->
|
||
R P0 P1 /\ R a0 a1 /\ R b0 b1 /\ R c0 c1.
|
||
Proof. hauto q:on use:EReds.ind_inv. Qed.
|
||
|
||
End EJoin.
|
||
|
||
Module RERed.
|
||
Inductive R {n} : PTm n -> PTm n -> Prop :=
|
||
(****************** Beta ***********************)
|
||
| AppAbs a b :
|
||
R (PApp (PAbs a) b) (subst_PTm (scons b VarPTm) a)
|
||
|
||
| ProjPair p a b :
|
||
R (PProj p (PPair a b)) (if p is PL then a else b)
|
||
|
||
| IndZero P b c :
|
||
R (PInd P PZero b c) b
|
||
|
||
| IndSuc P a b c :
|
||
R (PInd P (PSuc a) b c) (subst_PTm (scons (PInd P a b c) (scons a VarPTm)) c)
|
||
|
||
(****************** Eta ***********************)
|
||
| AppEta a :
|
||
R (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero))) a
|
||
| PairEta a :
|
||
R (PPair (PProj PL a) (PProj PR a)) a
|
||
|
||
(*************** Congruence ********************)
|
||
| AbsCong a0 a1 :
|
||
R a0 a1 ->
|
||
R (PAbs a0) (PAbs a1)
|
||
| AppCong0 a0 a1 b :
|
||
R a0 a1 ->
|
||
R (PApp a0 b) (PApp a1 b)
|
||
| AppCong1 a b0 b1 :
|
||
R b0 b1 ->
|
||
R (PApp a b0) (PApp a b1)
|
||
| PairCong0 a0 a1 b :
|
||
R a0 a1 ->
|
||
R (PPair a0 b) (PPair a1 b)
|
||
| PairCong1 a b0 b1 :
|
||
R b0 b1 ->
|
||
R (PPair a b0) (PPair a b1)
|
||
| ProjCong p a0 a1 :
|
||
R a0 a1 ->
|
||
R (PProj p a0) (PProj p a1)
|
||
| BindCong0 p A0 A1 B :
|
||
R A0 A1 ->
|
||
R (PBind p A0 B) (PBind p A1 B)
|
||
| BindCong1 p A B0 B1 :
|
||
R B0 B1 ->
|
||
R (PBind p A B0) (PBind p A B1)
|
||
| IndCong0 P0 P1 a b c :
|
||
R P0 P1 ->
|
||
R (PInd P0 a b c) (PInd P1 a b c)
|
||
| IndCong1 P a0 a1 b c :
|
||
R a0 a1 ->
|
||
R (PInd P a0 b c) (PInd P a1 b c)
|
||
| IndCong2 P a b0 b1 c :
|
||
R b0 b1 ->
|
||
R (PInd P a b0 c) (PInd P a b1 c)
|
||
| IndCong3 P a b c0 c1 :
|
||
R c0 c1 ->
|
||
R (PInd P a b c0) (PInd P a b c1)
|
||
| SucCong a0 a1 :
|
||
R a0 a1 ->
|
||
R (PSuc a0) (PSuc a1).
|
||
|
||
Lemma ToBetaEta n (a b : PTm n) :
|
||
R a b ->
|
||
ERed.R a b \/ RRed.R a b.
|
||
Proof. induction 1; hauto lq:on db:red. Qed.
|
||
|
||
Lemma FromBeta n (a b : PTm n) :
|
||
RRed.R a b -> RERed.R a b.
|
||
Proof. induction 1; qauto l:on ctrs:R. Qed.
|
||
|
||
Lemma FromEta n (a b : PTm n) :
|
||
ERed.R a b -> RERed.R a b.
|
||
Proof. induction 1; qauto l:on ctrs:R. Qed.
|
||
|
||
Lemma ToBetaEtaPar n (a b : PTm n) :
|
||
R a b ->
|
||
EPar.R a b \/ RRed.R a b.
|
||
Proof.
|
||
hauto q:on use:ERed.ToEPar, ToBetaEta.
|
||
Qed.
|
||
|
||
Lemma sn_preservation n (a b : PTm n) :
|
||
R a b ->
|
||
SN a ->
|
||
SN b.
|
||
Proof. hauto q:on use:ToBetaEtaPar, epar_sn_preservation, red_sn_preservation, RPar.FromRRed. Qed.
|
||
|
||
Lemma bind_preservation n (a b : PTm n) :
|
||
R a b -> isbind a -> isbind b.
|
||
Proof. hauto q:on inv:R. Qed.
|
||
|
||
Lemma univ_preservation n (a b : PTm n) :
|
||
R a b -> isuniv a -> isuniv b.
|
||
Proof. hauto q:on inv:R. Qed.
|
||
|
||
Lemma sne_preservation n (a b : PTm n) :
|
||
R a b -> SNe a -> SNe b.
|
||
Proof.
|
||
hauto q:on use:ToBetaEtaPar, RPar.FromRRed use:red_sn_preservation, epar_sn_preservation.
|
||
Qed.
|
||
|
||
Lemma substing n m (a b : PTm n) (ρ : fin n -> PTm m) :
|
||
RERed.R a b -> RERed.R (subst_PTm ρ a) (subst_PTm ρ b).
|
||
Proof.
|
||
hauto q:on use:ToBetaEta, FromBeta, FromEta, RRed.substing, ERed.substing.
|
||
Qed.
|
||
|
||
Lemma hne_preservation n (a b : PTm n) :
|
||
RERed.R a b -> ishne a -> ishne b.
|
||
Proof. induction 1; sfirstorder. Qed.
|
||
|
||
End RERed.
|
||
|
||
Module REReds.
|
||
Lemma hne_preservation n (a b : PTm n) :
|
||
rtc RERed.R a b -> ishne a -> ishne b.
|
||
Proof. induction 1; eauto using RERed.hne_preservation, rtc_refl, rtc. Qed.
|
||
|
||
Lemma sn_preservation n (a b : PTm n) :
|
||
rtc RERed.R a b ->
|
||
SN a ->
|
||
SN b.
|
||
Proof. induction 1; eauto using RERed.sn_preservation. Qed.
|
||
|
||
Lemma FromRReds n (a b : PTm n) :
|
||
rtc RRed.R a b ->
|
||
rtc RERed.R a b.
|
||
Proof. induction 1; hauto lq:on ctrs:rtc use:RERed.FromBeta. Qed.
|
||
|
||
Lemma FromEReds n (a b : PTm n) :
|
||
rtc ERed.R a b ->
|
||
rtc RERed.R a b.
|
||
Proof. induction 1; hauto lq:on ctrs:rtc use:RERed.FromEta. Qed.
|
||
|
||
#[local]Ltac solve_s_rec :=
|
||
move => *; eapply rtc_l; eauto;
|
||
hauto lq:on ctrs:RERed.R.
|
||
|
||
#[local]Ltac solve_s :=
|
||
repeat (induction 1; last by solve_s_rec); apply rtc_refl.
|
||
|
||
Lemma AbsCong n (a b : PTm (S n)) :
|
||
rtc RERed.R a b ->
|
||
rtc RERed.R (PAbs a) (PAbs b).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
|
||
rtc RERed.R a0 a1 ->
|
||
rtc RERed.R b0 b1 ->
|
||
rtc RERed.R (PApp a0 b0) (PApp a1 b1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
|
||
rtc RERed.R a0 a1 ->
|
||
rtc RERed.R b0 b1 ->
|
||
rtc RERed.R (PPair a0 b0) (PPair a1 b1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma ProjCong n p (a0 a1 : PTm n) :
|
||
rtc RERed.R a0 a1 ->
|
||
rtc RERed.R (PProj p a0) (PProj p a1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma SucCong n (a0 a1 : PTm n) :
|
||
rtc RERed.R a0 a1 ->
|
||
rtc RERed.R (PSuc a0) (PSuc a1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma IndCong n P0 P1 (a0 a1 : PTm n) b0 b1 c0 c1 :
|
||
rtc RERed.R P0 P1 ->
|
||
rtc RERed.R a0 a1 ->
|
||
rtc RERed.R b0 b1 ->
|
||
rtc RERed.R c0 c1 ->
|
||
rtc RERed.R (PInd P0 a0 b0 c0) (PInd P1 a1 b1 c1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma BindCong n p (A0 A1 : PTm n) B0 B1 :
|
||
rtc RERed.R A0 A1 ->
|
||
rtc RERed.R B0 B1 ->
|
||
rtc RERed.R (PBind p A0 B0) (PBind p A1 B1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma bind_preservation n (a b : PTm n) :
|
||
rtc RERed.R a b -> isbind a -> isbind b.
|
||
Proof. induction 1; qauto l:on ctrs:rtc use:RERed.bind_preservation. Qed.
|
||
|
||
Lemma univ_preservation n (a b : PTm n) :
|
||
rtc RERed.R a b -> isuniv a -> isuniv b.
|
||
Proof. induction 1; qauto l:on ctrs:rtc use:RERed.univ_preservation. Qed.
|
||
|
||
Lemma sne_preservation n (a b : PTm n) :
|
||
rtc RERed.R a b -> SNe a -> SNe b.
|
||
Proof. induction 1; qauto l:on ctrs:rtc use:RERed.sne_preservation. Qed.
|
||
|
||
Lemma bind_inv n p A B C :
|
||
rtc (@RERed.R n) (PBind p A B) C ->
|
||
exists A0 B0, C = PBind p A0 B0 /\ rtc RERed.R A A0 /\ rtc RERed.R B B0.
|
||
Proof.
|
||
move E : (PBind p A B) => u hu.
|
||
move : p A B E.
|
||
elim : u C / hu; sauto lq:on rew:off.
|
||
Qed.
|
||
|
||
Lemma univ_inv n i C :
|
||
rtc (@RERed.R n) (PUniv i) C ->
|
||
C = PUniv i.
|
||
Proof.
|
||
move E : (PUniv i) => u hu.
|
||
move : i E. elim : u C / hu=>//=.
|
||
hauto lq:on rew:off ctrs:rtc inv:RERed.R.
|
||
Qed.
|
||
|
||
Lemma var_inv n (i : fin n) C :
|
||
rtc RERed.R (VarPTm i) C ->
|
||
C = VarPTm i.
|
||
Proof.
|
||
move E : (VarPTm i) => u hu.
|
||
move : i E. elim : u C /hu; hauto lq:on rew:off inv:RERed.R.
|
||
Qed.
|
||
|
||
Lemma hne_app_inv n (a0 b0 C : PTm n) :
|
||
rtc RERed.R (PApp a0 b0) C ->
|
||
ishne a0 ->
|
||
exists a1 b1, C = PApp a1 b1 /\
|
||
rtc RERed.R a0 a1 /\
|
||
rtc RERed.R b0 b1.
|
||
Proof.
|
||
move E : (PApp a0 b0) => u hu.
|
||
move : a0 b0 E.
|
||
elim : u C / hu.
|
||
- hauto lq:on ctrs:rtc.
|
||
- move => a b c ha ha' iha a0 b0 ?. subst.
|
||
hauto lq:on rew:off ctrs:rtc, RERed.R use:RERed.hne_preservation inv:RERed.R.
|
||
Qed.
|
||
|
||
Lemma hne_proj_inv n p (a C : PTm n) :
|
||
rtc RERed.R (PProj p a) C ->
|
||
ishne a ->
|
||
exists c, C = PProj p c /\ rtc RERed.R a c.
|
||
Proof.
|
||
move E : (PProj p a) => u hu.
|
||
move : p a E.
|
||
elim : u C /hu => //=;
|
||
scrush ctrs:rtc,RERed.R use:RERed.hne_preservation inv:RERed.R.
|
||
Qed.
|
||
|
||
Lemma hne_ind_inv n P a b c (C : PTm n) :
|
||
rtc RERed.R (PInd P a b c) C -> ishne a ->
|
||
exists P0 a0 b0 c0, C = PInd P0 a0 b0 c0 /\
|
||
rtc RERed.R P P0 /\
|
||
rtc RERed.R a a0 /\
|
||
rtc RERed.R b b0 /\
|
||
rtc RERed.R c c0.
|
||
Proof.
|
||
move E : (PInd P a b c) => u hu.
|
||
move : P a b c E.
|
||
elim : u C / hu => //=;
|
||
scrush ctrs:rtc,RERed.R use:RERed.hne_preservation inv:RERed.R.
|
||
Qed.
|
||
|
||
Lemma substing n m (a b : PTm n) (ρ : fin n -> PTm m) :
|
||
rtc RERed.R a b -> rtc RERed.R (subst_PTm ρ a) (subst_PTm ρ b).
|
||
Proof.
|
||
induction 1; hauto lq:on ctrs:rtc use:RERed.substing.
|
||
Qed.
|
||
|
||
|
||
Lemma cong_up n m (ρ0 ρ1 : fin n -> PTm m) :
|
||
(forall i, rtc RERed.R (ρ0 i) (ρ1 i)) ->
|
||
(forall i, rtc RERed.R (up_PTm_PTm ρ0 i) (up_PTm_PTm ρ1 i)).
|
||
Proof. move => h i. destruct i as [i|].
|
||
simpl. rewrite /funcomp.
|
||
substify. by apply substing.
|
||
apply rtc_refl.
|
||
Qed.
|
||
|
||
Lemma cong_up2 n m (ρ0 ρ1 : fin n -> PTm m) :
|
||
(forall i, rtc RERed.R (ρ0 i) (ρ1 i)) ->
|
||
(forall i, rtc RERed.R (up_PTm_PTm (up_PTm_PTm ρ0) i) (up_PTm_PTm (up_PTm_PTm ρ1) i)).
|
||
Proof. hauto l:on use:cong_up. Qed.
|
||
|
||
Lemma cong n m (a : PTm n) (ρ0 ρ1 : fin n -> PTm m) :
|
||
(forall i, rtc RERed.R (ρ0 i) (ρ1 i)) ->
|
||
rtc RERed.R (subst_PTm ρ0 a) (subst_PTm ρ1 a).
|
||
Proof.
|
||
move : m ρ0 ρ1. elim : n / a => /=;
|
||
eauto 20 using AppCong, AbsCong, BindCong, ProjCong, PairCong, cong_up, rtc_refl, IndCong, SucCong, cong_up2.
|
||
Qed.
|
||
|
||
Lemma cong' n m (a b : PTm n) (ρ0 ρ1 : fin n -> PTm m) :
|
||
rtc RERed.R a b ->
|
||
(forall i, rtc RERed.R (ρ0 i) (ρ1 i)) ->
|
||
rtc RERed.R (subst_PTm ρ0 a) (subst_PTm ρ1 b).
|
||
Proof.
|
||
move => h0 h1.
|
||
have : rtc RERed.R (subst_PTm ρ0 a) (subst_PTm ρ1 a) by eauto using cong.
|
||
move => ?. apply : relations.rtc_transitive; eauto.
|
||
hauto l:on use:substing.
|
||
Qed.
|
||
|
||
Lemma ToEReds n (a b : PTm n) :
|
||
nf a ->
|
||
rtc RERed.R a b -> rtc ERed.R a b.
|
||
Proof.
|
||
move => + h.
|
||
induction h; hauto lq:on rew:off ctrs:rtc use:RERed.ToBetaEta, RReds.nf_refl, @rtc_once, ERed.nf_preservation.
|
||
Qed.
|
||
|
||
Lemma suc_inv n (a : PTm n) C :
|
||
rtc RERed.R (PSuc a) C ->
|
||
exists b, rtc RERed.R a b /\ C = PSuc b.
|
||
Proof.
|
||
move E : (PSuc a) => u hu.
|
||
move : a E.
|
||
elim : u C / hu=>//=.
|
||
- hauto l:on.
|
||
- hauto lq:on rew:off ctrs:rtc, RERed.R inv:RERed.R, rtc.
|
||
Qed.
|
||
|
||
End REReds.
|
||
|
||
Module LoRed.
|
||
Inductive R {n} : PTm n -> PTm n -> Prop :=
|
||
(****************** Beta ***********************)
|
||
| AppAbs a b :
|
||
R (PApp (PAbs a) b) (subst_PTm (scons b VarPTm) a)
|
||
|
||
| ProjPair p a b :
|
||
R (PProj p (PPair a b)) (if p is PL then a else b)
|
||
|
||
| IndZero P b c :
|
||
R (PInd P PZero b c) b
|
||
|
||
| IndSuc P a b c :
|
||
R (PInd P (PSuc a) b c) (subst_PTm (scons (PInd P a b c) (scons a VarPTm)) c)
|
||
|
||
|
||
(*************** Congruence ********************)
|
||
| AbsCong a0 a1 :
|
||
R a0 a1 ->
|
||
R (PAbs a0) (PAbs a1)
|
||
| AppCong0 a0 a1 b :
|
||
~~ ishf a0 ->
|
||
R a0 a1 ->
|
||
R (PApp a0 b) (PApp a1 b)
|
||
| AppCong1 a b0 b1 :
|
||
ne a ->
|
||
R b0 b1 ->
|
||
R (PApp a b0) (PApp a b1)
|
||
| PairCong0 a0 a1 b :
|
||
R a0 a1 ->
|
||
R (PPair a0 b) (PPair a1 b)
|
||
| PairCong1 a b0 b1 :
|
||
nf a ->
|
||
R b0 b1 ->
|
||
R (PPair a b0) (PPair a b1)
|
||
| ProjCong p a0 a1 :
|
||
~~ ishf a0 ->
|
||
R a0 a1 ->
|
||
R (PProj p a0) (PProj p a1)
|
||
| BindCong0 p A0 A1 B :
|
||
R A0 A1 ->
|
||
R (PBind p A0 B) (PBind p A1 B)
|
||
| BindCong1 p A B0 B1 :
|
||
nf A ->
|
||
R B0 B1 ->
|
||
R (PBind p A B0) (PBind p A B1)
|
||
| IndCong0 P0 P1 a b c :
|
||
ne a ->
|
||
R P0 P1 ->
|
||
R (PInd P0 a b c) (PInd P1 a b c)
|
||
| IndCong1 P a0 a1 b c :
|
||
~~ ishf a0 ->
|
||
R a0 a1 ->
|
||
R (PInd P a0 b c) (PInd P a1 b c)
|
||
| IndCong2 P a b0 b1 c :
|
||
nf P ->
|
||
ne a ->
|
||
R b0 b1 ->
|
||
R (PInd P a b0 c) (PInd P a b1 c)
|
||
| IndCong3 P a b c0 c1 :
|
||
nf P ->
|
||
ne a ->
|
||
nf b ->
|
||
R c0 c1 ->
|
||
R (PInd P a b c0) (PInd P a b c1)
|
||
| SucCong a0 a1 :
|
||
R a0 a1 ->
|
||
R (PSuc a0) (PSuc a1).
|
||
|
||
Lemma hf_preservation n (a b : PTm n) :
|
||
LoRed.R a b ->
|
||
ishf a ->
|
||
ishf b.
|
||
Proof.
|
||
move => h. elim : n a b /h=>//=.
|
||
Qed.
|
||
|
||
Lemma ToRRed n (a b : PTm n) :
|
||
LoRed.R a b ->
|
||
RRed.R a b.
|
||
Proof. induction 1; hauto lq:on ctrs:RRed.R. Qed.
|
||
|
||
Lemma AppAbs' n a (b : PTm n) u :
|
||
u = (subst_PTm (scons b VarPTm) a) ->
|
||
R (PApp (PAbs a) b) u.
|
||
Proof. move => ->. apply AppAbs. Qed.
|
||
|
||
Lemma IndSuc' n P a b c u :
|
||
u = (subst_PTm (scons (PInd P a b c) (scons a VarPTm)) c) ->
|
||
R (@PInd n P (PSuc a) b c) u.
|
||
Proof. move => ->. apply IndSuc. Qed.
|
||
|
||
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.
|
||
|
||
move => n a b m ξ /=.
|
||
apply AppAbs'. by asimpl.
|
||
all : try qauto ctrs:R use:ne_nf_ren, ishf_ren.
|
||
move => * /=; apply IndSuc'. by asimpl.
|
||
Qed.
|
||
|
||
End LoRed.
|
||
|
||
Module LoReds.
|
||
Lemma hf_preservation n (a b : PTm n) :
|
||
rtc LoRed.R a b ->
|
||
ishf a ->
|
||
ishf b.
|
||
Proof.
|
||
induction 1; eauto using LoRed.hf_preservation.
|
||
Qed.
|
||
|
||
Lemma hf_ne_imp n (a b : PTm n) :
|
||
rtc LoRed.R a b ->
|
||
ne b ->
|
||
~~ ishf a.
|
||
Proof.
|
||
move : hf_preservation. repeat move/[apply].
|
||
case : a; case : b => //=; itauto.
|
||
Qed.
|
||
|
||
#[local]Ltac solve_s_rec :=
|
||
move => *; eapply rtc_l; eauto;
|
||
hauto lq:on ctrs:LoRed.R, rtc use:hf_ne_imp.
|
||
|
||
#[local]Ltac solve_s :=
|
||
repeat (induction 1; last by solve_s_rec); (move => *; apply rtc_refl).
|
||
|
||
Lemma AbsCong n (a b : PTm (S n)) :
|
||
rtc LoRed.R a b ->
|
||
rtc LoRed.R (PAbs a) (PAbs b).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
|
||
rtc LoRed.R a0 a1 ->
|
||
rtc LoRed.R b0 b1 ->
|
||
ne a1 ->
|
||
rtc LoRed.R (PApp a0 b0) (PApp a1 b1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
|
||
rtc LoRed.R a0 a1 ->
|
||
rtc LoRed.R b0 b1 ->
|
||
nf a1 ->
|
||
rtc LoRed.R (PPair a0 b0) (PPair a1 b1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma ProjCong n p (a0 a1 : PTm n) :
|
||
rtc LoRed.R a0 a1 ->
|
||
ne a1 ->
|
||
rtc LoRed.R (PProj p a0) (PProj p a1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma BindCong n p (A0 A1 : PTm n) B0 B1 :
|
||
rtc LoRed.R A0 A1 ->
|
||
rtc LoRed.R B0 B1 ->
|
||
nf A1 ->
|
||
rtc LoRed.R (PBind p A0 B0) (PBind p A1 B1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma IndCong n P0 P1 (a0 a1 : PTm n) b0 b1 c0 c1 :
|
||
rtc LoRed.R a0 a1 ->
|
||
rtc LoRed.R P0 P1 ->
|
||
rtc LoRed.R b0 b1 ->
|
||
rtc LoRed.R c0 c1 ->
|
||
ne a1 ->
|
||
nf P1 ->
|
||
nf b1 ->
|
||
rtc LoRed.R (PInd P0 a0 b0 c0) (PInd P1 a1 b1 c1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma SucCong n (a0 a1 : PTm n) :
|
||
rtc LoRed.R a0 a1 ->
|
||
rtc LoRed.R (PSuc a0) (PSuc a1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Local Ltac triv := simpl in *; itauto.
|
||
|
||
Lemma FromSN_mutual : forall n,
|
||
(forall (a : PTm n) (_ : SNe a), exists v, rtc LoRed.R a v /\ ne v) /\
|
||
(forall (a : PTm n) (_ : SN a), exists v, rtc LoRed.R a v /\ nf v) /\
|
||
(forall (a b : PTm n) (_ : TRedSN a b), LoRed.R a b).
|
||
Proof.
|
||
apply sn_mutual.
|
||
- hauto lq:on ctrs:rtc.
|
||
- hauto lq:on rew:off use:LoReds.AppCong solve+:triv.
|
||
- hauto l:on use:LoReds.ProjCong solve+:triv.
|
||
- hauto lq:on ctrs:rtc.
|
||
- hauto lq:on use:LoReds.IndCong solve+:triv.
|
||
- hauto q:on use:LoReds.PairCong solve+:triv.
|
||
- hauto q:on use:LoReds.AbsCong solve+:triv.
|
||
- sfirstorder use:ne_nf.
|
||
- hauto lq:on ctrs:rtc.
|
||
- hauto lq:on use:LoReds.BindCong solve+:triv.
|
||
- hauto lq:on ctrs:rtc.
|
||
- hauto lq:on ctrs:rtc.
|
||
- hauto lq:on ctrs:rtc.
|
||
- hauto l:on use:SucCong.
|
||
- qauto ctrs:LoRed.R.
|
||
- move => n a0 a1 b hb ihb h.
|
||
have : ~~ ishf a0 by inversion h.
|
||
hauto lq:on ctrs:LoRed.R.
|
||
- qauto ctrs:LoRed.R.
|
||
- qauto ctrs:LoRed.R.
|
||
- move => n p a b h.
|
||
have : ~~ ishf a by inversion h.
|
||
hauto lq:on ctrs:LoRed.R.
|
||
- sfirstorder.
|
||
- sfirstorder.
|
||
- move => n P a0 a1 b c hP ihP hb ihb hc ihc hr.
|
||
have : ~~ ishf a0 by inversion hr.
|
||
hauto q:on ctrs:LoRed.R.
|
||
Qed.
|
||
|
||
Lemma FromSN : forall n a, @SN n a -> exists v, rtc LoRed.R a v /\ nf v.
|
||
Proof. firstorder using FromSN_mutual. Qed.
|
||
|
||
Lemma ToRReds : forall n (a b : PTm n), rtc LoRed.R a b -> rtc RRed.R a b.
|
||
Proof. induction 1; hauto lq:on ctrs:rtc use:LoRed.ToRRed. Qed.
|
||
|
||
End LoReds.
|
||
|
||
|
||
Fixpoint size_PTm {n} (a : PTm n) :=
|
||
match a with
|
||
| VarPTm _ => 1
|
||
| PAbs a => 3 + size_PTm a
|
||
| PApp a b => 1 + Nat.add (size_PTm a) (size_PTm b)
|
||
| PProj p a => 1 + size_PTm a
|
||
| PPair a b => 3 + Nat.add (size_PTm a) (size_PTm b)
|
||
| PUniv _ => 3
|
||
| PBind p A B => 3 + Nat.add (size_PTm A) (size_PTm B)
|
||
| PInd P a b c => 3 + size_PTm P + size_PTm a + size_PTm b + size_PTm c
|
||
| PNat => 3
|
||
| PSuc a => 3 + size_PTm a
|
||
| PZero => 3
|
||
| PBot => 1
|
||
end.
|
||
|
||
Lemma size_PTm_ren n m (ξ : fin n -> fin m) a : size_PTm (ren_PTm ξ a) = size_PTm a.
|
||
Proof.
|
||
move : m ξ. elim : n / a => //=; scongruence.
|
||
Qed.
|
||
|
||
#[export]Hint Rewrite size_PTm_ren : sizetm.
|
||
|
||
Lemma ered_size {n} (a b : PTm n) :
|
||
ERed.R a b ->
|
||
size_PTm b < size_PTm a.
|
||
Proof.
|
||
move => h. elim : n a b /h; hauto l:on rew:db:sizetm.
|
||
Qed.
|
||
|
||
Lemma ered_sn n (a : PTm n) : sn ERed.R a.
|
||
Proof.
|
||
hauto lq:on rew:off use:size_PTm_ren, ered_size,
|
||
well_founded_lt_compat unfold:well_founded.
|
||
Qed.
|
||
|
||
Lemma ered_local_confluence n (a b c : PTm n) :
|
||
ERed.R a b ->
|
||
ERed.R a c ->
|
||
exists d, rtc ERed.R b d /\ rtc ERed.R c d.
|
||
Proof.
|
||
move => h. move : c.
|
||
elim : n a b / h => n.
|
||
- move => a c.
|
||
elim /ERed.inv => //= _.
|
||
+ move => a0 [+ ?]. subst => h.
|
||
apply f_equal with (f := subst_PTm (scons (PAbs (VarPTm var_zero)) VarPTm)) in h.
|
||
move : h. asimpl => ?. subst.
|
||
eauto using rtc_refl.
|
||
+ move => a0 a1 ha [*]. subst.
|
||
elim /ERed.inv : ha => //= _.
|
||
* move => a0 a2 b0 ha [*]. subst. rename a2 into a1.
|
||
move /ERed.antirenaming : ha.
|
||
move /(_ ltac:(hauto lq:on)) => [a' [h0 h1]]. subst.
|
||
hauto lq:on ctrs:rtc, ERed.R.
|
||
* hauto q:on ctrs:rtc, ERed.R inv:ERed.R.
|
||
- move => a c ha.
|
||
elim /ERed.inv : ha => //= _.
|
||
+ hauto l:on.
|
||
+ move => a0 a1 b0 ha [*]. subst.
|
||
elim /ERed.inv : ha => //= _.
|
||
move => p a0 a2 ha [*]. subst.
|
||
hauto q:on ctrs:rtc, ERed.R.
|
||
+ move => a0 b0 b1 ha [*]. subst.
|
||
elim /ERed.inv : ha => //= _.
|
||
move => p a0 a2 ha [*]. subst.
|
||
hauto q:on ctrs:rtc, ERed.R.
|
||
- move => a0 a1 ha iha c.
|
||
elim /ERed.inv => //= _.
|
||
+ move => a2 [*]. subst.
|
||
elim /ERed.inv : ha => //=_.
|
||
* move => a0 a2 b0 ha [*] {iha}. subst.
|
||
have [a0 [h0 h1]] : exists a0, ERed.R c a0 /\ a2 = ren_PTm shift a0 by hauto lq:on use:ERed.antirenaming. subst.
|
||
exists a0. split; last by apply relations.rtc_once.
|
||
apply relations.rtc_once. apply ERed.AppEta.
|
||
* hauto q:on inv:ERed.R.
|
||
+ hauto lq:on use:EReds.AbsCong.
|
||
- move => a0 a1 b ha iha c.
|
||
elim /ERed.inv => //= _.
|
||
+ hauto lq:on ctrs:rtc use:EReds.AppCong.
|
||
+ hauto lq:on use:@relations.rtc_once ctrs:ERed.R.
|
||
- move => a b0 b1 hb ihb c.
|
||
elim /ERed.inv => //=_.
|
||
+ move => a0 a1 a2 ha [*]. subst.
|
||
move {ihb}. exists (PApp a1 b1).
|
||
hauto lq:on use:@relations.rtc_once ctrs:ERed.R.
|
||
+ hauto lq:on ctrs:rtc use:EReds.AppCong.
|
||
- move => a0 a1 b ha iha c.
|
||
elim /ERed.inv => //= _.
|
||
+ sauto lq:on.
|
||
+ hauto lq:on ctrs:rtc use:EReds.PairCong.
|
||
+ hauto lq:on ctrs:ERed.R use:@relations.rtc_once.
|
||
- move => a b0 b1 hb hc c. elim /ERed.inv => //= _.
|
||
+ move => ? [*]. subst.
|
||
sauto lq:on.
|
||
+ hauto lq:on ctrs:ERed.R use:@relations.rtc_once.
|
||
+ hauto lq:on ctrs:rtc use:EReds.PairCong.
|
||
- qauto l:on inv:ERed.R use:EReds.ProjCong.
|
||
- move => p A0 A1 B hA ihA u.
|
||
elim /ERed.inv => //=_;
|
||
hauto lq:on ctrs:rtc use:EReds.BindCong.
|
||
- move => p A B0 B1 hB ihB u.
|
||
elim /ERed.inv => //=_;
|
||
hauto lq:on ctrs:rtc use:EReds.BindCong.
|
||
- move => P0 P1 a b c hP ihP u.
|
||
elim /ERed.inv => //=_.
|
||
+ move => P2 P3 a0 b0 c0 hP' [*]. subst.
|
||
move : ihP hP' => /[apply]. move => [P4][hP0]hP1.
|
||
eauto using EReds.IndCong, rtc_refl.
|
||
+ move => P2 a0 a1 b0 c0 + [*]. subst.
|
||
eauto 20 using rtc_refl, EReds.IndCong, rtc_l.
|
||
+ move => P2 a0 b0 b1 c0 hb [*] {ihP}. subst.
|
||
eauto 20 using rtc_refl, EReds.IndCong, rtc_l.
|
||
+ move => P2 a0 b0 c0 c1 h [*] {ihP}. subst.
|
||
eauto 20 using rtc_refl, EReds.IndCong, rtc_l.
|
||
- move => P a0 a1 b c ha iha u.
|
||
elim /ERed.inv => //=_;
|
||
try solve [move => P0 P1 a2 b0 c0 hP[*]; subst;
|
||
eauto 20 using rtc_refl, EReds.IndCong, rtc_l].
|
||
move => P0 P1 a2 b0 c0 hP[*]. subst.
|
||
move : iha hP => /[apply].
|
||
move => [? [*]].
|
||
eauto 20 using rtc_refl, EReds.IndCong, rtc_l.
|
||
- move => P a b0 b1 c hb ihb u.
|
||
elim /ERed.inv => //=_;
|
||
try solve [
|
||
move => P0 P1 a0 b2 c0 hP [*]; subst;
|
||
eauto 20 using rtc_refl, EReds.IndCong, rtc_l].
|
||
move => P0 a0 b2 b3 c0 h [*]. subst.
|
||
move : ihb h => /[apply]. move => [b2 [*]].
|
||
eauto 20 using rtc_refl, EReds.IndCong, rtc_l.
|
||
- move => P a b c0 c1 hc ihc u.
|
||
elim /ERed.inv => //=_;
|
||
try solve [
|
||
move => P0 P1 a0 b0 c hP [*]; subst;
|
||
eauto 20 using rtc_refl, EReds.IndCong, rtc_l].
|
||
move => P0 a0 b0 c2 c3 h [*]. subst.
|
||
move : ihc h => /[apply]. move => [c2][*].
|
||
eauto 20 using rtc_refl, EReds.IndCong, rtc_l.
|
||
- qauto l:on inv:ERed.R ctrs:rtc use:EReds.SucCong.
|
||
Qed.
|
||
|
||
Lemma ered_confluence n (a b c : PTm n) :
|
||
rtc ERed.R a b ->
|
||
rtc ERed.R a c ->
|
||
exists d, rtc ERed.R b d /\ rtc ERed.R c d.
|
||
Proof.
|
||
sfirstorder use:relations.locally_confluent_confluent, ered_sn, ered_local_confluence.
|
||
Qed.
|
||
|
||
Lemma red_confluence n (a b c : PTm n) :
|
||
rtc RRed.R a b ->
|
||
rtc RRed.R a c ->
|
||
exists d, rtc RRed.R b d /\ rtc RRed.R c d.
|
||
suff : rtc RPar.R a b -> rtc RPar.R a c -> exists d : PTm n, rtc RPar.R b d /\ rtc RPar.R c d
|
||
by hauto lq:on use:RReds.RParIff.
|
||
apply relations.diamond_confluent.
|
||
rewrite /relations.diamond.
|
||
eauto using RPar.diamond.
|
||
Qed.
|
||
|
||
Lemma red_uniquenf n (a b c : PTm n) :
|
||
rtc RRed.R a b ->
|
||
rtc RRed.R a c ->
|
||
nf b ->
|
||
nf c ->
|
||
b = c.
|
||
Proof.
|
||
move : red_confluence; repeat move/[apply].
|
||
move => [d [h0 h1]].
|
||
move => *.
|
||
suff [] : b = d /\ c = d by congruence.
|
||
sfirstorder use:RReds.nf_refl.
|
||
Qed.
|
||
|
||
Module NeEPars.
|
||
Lemma R_nonelim_nf n (a b : PTm n) :
|
||
rtc NeEPar.R_nonelim a b ->
|
||
nf b ->
|
||
nf a.
|
||
Proof. induction 1; sfirstorder use:NeEPar.R_elim_nf. Qed.
|
||
|
||
Lemma ToEReds : forall n, (forall (a b : PTm n), rtc NeEPar.R_nonelim a b -> rtc ERed.R a b).
|
||
Proof.
|
||
induction 1; hauto l:on use:NeEPar.ToEPar, EReds.FromEPar, @relations.rtc_transitive.
|
||
Qed.
|
||
End NeEPars.
|
||
|
||
|
||
Lemma rered_standardization n (a c : PTm n) :
|
||
SN a ->
|
||
rtc RERed.R a c ->
|
||
exists b, rtc RRed.R a b /\ rtc NeEPar.R_nonelim b c.
|
||
Proof.
|
||
move => + h. elim : a c /h.
|
||
by eauto using rtc_refl.
|
||
move => a b c.
|
||
move /RERed.ToBetaEtaPar.
|
||
case.
|
||
- move => h0 h1 ih hP.
|
||
have : SN b by qauto use:epar_sn_preservation.
|
||
move => {}/ih [b' [ihb0 ihb1]].
|
||
hauto lq:on ctrs:rtc use:SN_UniqueNF.η_postponement_star'.
|
||
- hauto lq:on ctrs:rtc use:red_sn_preservation, RPar.FromRRed.
|
||
Qed.
|
||
|
||
Lemma rered_confluence n (a b c : PTm n) :
|
||
SN a ->
|
||
rtc RERed.R a b ->
|
||
rtc RERed.R a c ->
|
||
exists d, rtc RERed.R b d /\ rtc RERed.R c d.
|
||
Proof.
|
||
move => hP hb hc.
|
||
have [] : SN b /\ SN c by qauto use:REReds.sn_preservation.
|
||
move => /LoReds.FromSN [bv [/LoReds.ToRReds /REReds.FromRReds hbv hbv']].
|
||
move => /LoReds.FromSN [cv [/LoReds.ToRReds /REReds.FromRReds hcv hcv']].
|
||
have [] : SN b /\ SN c by sfirstorder use:REReds.sn_preservation.
|
||
move : rered_standardization hbv; repeat move/[apply]. move => [bv' [hb0 hb1]].
|
||
move : rered_standardization hcv; repeat move/[apply]. move => [cv' [hc0 hc1]].
|
||
|
||
have [] : rtc RERed.R a bv' /\ rtc RERed.R a cv'
|
||
by sfirstorder use:@relations.rtc_transitive, REReds.FromRReds.
|
||
move : rered_standardization (hP). repeat move/[apply]. move => [bv'' [hb3 hb4]].
|
||
move : rered_standardization (hP). repeat move/[apply]. move => [cv'' [hc3 hc4]].
|
||
have hb2 : rtc NeEPar.R_nonelim bv'' bv by hauto lq:on use:@relations.rtc_transitive.
|
||
have hc2 : rtc NeEPar.R_nonelim cv'' cv by hauto lq:on use:@relations.rtc_transitive.
|
||
have [hc5 hb5] : nf cv'' /\ nf bv'' by sfirstorder use:NeEPars.R_nonelim_nf.
|
||
have ? : bv'' = cv'' by sfirstorder use:red_uniquenf. subst.
|
||
apply NeEPars.ToEReds in hb2, hc2.
|
||
move : ered_confluence (hb2) (hc2); repeat move/[apply].
|
||
move => [v [hv hv']].
|
||
exists v. split.
|
||
move /NeEPars.ToEReds /REReds.FromEReds : hb1.
|
||
move /REReds.FromRReds : hb0. move /REReds.FromEReds : hv. eauto using relations.rtc_transitive.
|
||
move /NeEPars.ToEReds /REReds.FromEReds : hc1.
|
||
move /REReds.FromRReds : hc0. move /REReds.FromEReds : hv'. eauto using relations.rtc_transitive.
|
||
Qed.
|
||
|
||
(* Beta joinability *)
|
||
Module BJoin.
|
||
Definition R {n} (a b : PTm n) := exists c, rtc RRed.R a c /\ rtc RRed.R b c.
|
||
Lemma refl n (a : PTm n) : R a a.
|
||
Proof. sfirstorder use:@rtc_refl unfold:R. Qed.
|
||
|
||
Lemma symmetric n (a b : PTm n) : R a b -> R b a.
|
||
Proof. sfirstorder unfold:R. Qed.
|
||
|
||
Lemma transitive n (a b c : PTm n) : R a b -> R b c -> R a c.
|
||
Proof.
|
||
rewrite /R.
|
||
move => [ab [ha +]] [bc [+ hc]].
|
||
move : red_confluence; repeat move/[apply].
|
||
move => [v [h0 h1]].
|
||
exists v. sfirstorder use:@relations.rtc_transitive.
|
||
Qed.
|
||
|
||
(* Lemma AbsCong n (a b : PTm (S n)) : *)
|
||
(* R a b -> *)
|
||
(* R (PAbs a) (PAbs b). *)
|
||
(* Proof. hauto lq:on use:RReds.AbsCong unfold:R. Qed. *)
|
||
|
||
(* Lemma AppCong n (a0 a1 b0 b1 : PTm n) : *)
|
||
(* R a0 a1 -> *)
|
||
(* R b0 b1 -> *)
|
||
(* R (PApp a0 b0) (PApp a1 b1). *)
|
||
(* Proof. hauto lq:on use:RReds.AppCong unfold:R. Qed. *)
|
||
End BJoin.
|
||
|
||
Module DJoin.
|
||
Definition R {n} (a b : PTm n) := exists c, rtc RERed.R a c /\ rtc RERed.R b c.
|
||
|
||
Lemma refl n (a : PTm n) : R a a.
|
||
Proof. sfirstorder use:@rtc_refl unfold:R. Qed.
|
||
|
||
Lemma FromEJoin n (a b : PTm n) : EJoin.R a b -> DJoin.R a b.
|
||
Proof. hauto q:on use:REReds.FromEReds. Qed.
|
||
|
||
Lemma ToEJoin n (a b : PTm n) : nf a -> nf b -> DJoin.R a b -> EJoin.R a b.
|
||
Proof. hauto q:on use:REReds.ToEReds. Qed.
|
||
|
||
Lemma symmetric n (a b : PTm n) : R a b -> R b a.
|
||
Proof. sfirstorder unfold:R. Qed.
|
||
|
||
Lemma transitive n (a b c : PTm n) : SN b -> R a b -> R b c -> R a c.
|
||
Proof.
|
||
rewrite /R.
|
||
move => + [ab [ha +]] [bc [+ hc]].
|
||
move : rered_confluence; repeat move/[apply].
|
||
move => [v [h0 h1]].
|
||
exists v. sfirstorder use:@relations.rtc_transitive.
|
||
Qed.
|
||
|
||
Lemma AbsCong n (a b : PTm (S n)) :
|
||
R a b ->
|
||
R (PAbs a) (PAbs b).
|
||
Proof. hauto lq:on use:REReds.AbsCong unfold:R. Qed.
|
||
|
||
Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (PApp a0 b0) (PApp a1 b1).
|
||
Proof. hauto lq:on use:REReds.AppCong unfold:R. Qed.
|
||
|
||
Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (PPair a0 b0) (PPair a1 b1).
|
||
Proof. hauto q:on use:REReds.PairCong. Qed.
|
||
|
||
Lemma ProjCong n p (a0 a1 : PTm n) :
|
||
R a0 a1 ->
|
||
R (PProj p a0) (PProj p a1).
|
||
Proof. hauto q:on use:REReds.ProjCong. Qed.
|
||
|
||
Lemma BindCong n p (A0 A1 : PTm n) B0 B1 :
|
||
R A0 A1 ->
|
||
R B0 B1 ->
|
||
R (PBind p A0 B0) (PBind p A1 B1).
|
||
Proof. hauto q:on use:REReds.BindCong. Qed.
|
||
|
||
Lemma FromRedSNs n (a b : PTm n) :
|
||
rtc TRedSN a b ->
|
||
R a b.
|
||
Proof.
|
||
move /RReds.FromRedSNs /REReds.FromRReds.
|
||
sfirstorder use:@rtc_refl unfold:R.
|
||
Qed.
|
||
|
||
Lemma sne_bind_noconf n (a b : PTm n) :
|
||
R a b -> SNe a -> isbind b -> False.
|
||
Proof.
|
||
move => [c [? ?]] *.
|
||
have : SNe c /\ isbind c by sfirstorder use:REReds.sne_preservation, REReds.bind_preservation.
|
||
qauto l:on inv:SNe.
|
||
Qed.
|
||
|
||
Lemma sne_univ_noconf n (a b : PTm n) :
|
||
R a b -> SNe a -> isuniv b -> False.
|
||
Proof.
|
||
hauto q:on use:REReds.sne_preservation,
|
||
REReds.univ_preservation inv:SNe.
|
||
Qed.
|
||
|
||
Lemma bind_univ_noconf n (a b : PTm n) :
|
||
R a b -> isbind a -> isuniv b -> False.
|
||
Proof.
|
||
move => [c [h0 h1]] h2 h3.
|
||
have {h0 h1 h2 h3} : isbind c /\ isuniv c by
|
||
hauto l:on use:REReds.bind_preservation,
|
||
REReds.univ_preservation.
|
||
case : c => //=; itauto.
|
||
Qed.
|
||
|
||
Lemma hne_univ_noconf n (a b : PTm n) :
|
||
R a b -> ishne a -> isuniv b -> False.
|
||
Proof.
|
||
move => [c [h0 h1]] h2 h3.
|
||
have {h0 h1 h2 h3} : ishne c /\ isuniv c by
|
||
hauto l:on use:REReds.hne_preservation,
|
||
REReds.univ_preservation.
|
||
move => [].
|
||
case : c => //=.
|
||
Qed.
|
||
|
||
Lemma hne_bind_noconf n (a b : PTm n) :
|
||
R a b -> ishne a -> isbind b -> False.
|
||
Proof.
|
||
move => [c [h0 h1]] h2 h3.
|
||
have {h0 h1 h2 h3} : ishne c /\ isbind c by
|
||
hauto l:on use:REReds.hne_preservation,
|
||
REReds.bind_preservation.
|
||
move => [].
|
||
case : c => //=.
|
||
Qed.
|
||
|
||
Lemma bind_inj n p0 p1 (A0 A1 : PTm n) B0 B1 :
|
||
DJoin.R (PBind p0 A0 B0) (PBind p1 A1 B1) ->
|
||
p0 = p1 /\ DJoin.R A0 A1 /\ DJoin.R B0 B1.
|
||
Proof.
|
||
rewrite /R.
|
||
hauto lq:on rew:off use:REReds.bind_inv.
|
||
Qed.
|
||
|
||
Lemma var_inj n (i j : fin n) :
|
||
R (VarPTm i) (VarPTm j) -> i = j.
|
||
Proof. sauto lq:on rew:off use:REReds.var_inv unfold:R. Qed.
|
||
|
||
Lemma univ_inj n i j :
|
||
@R n (PUniv i) (PUniv j) -> i = j.
|
||
Proof.
|
||
sauto lq:on rew:off use:REReds.univ_inv.
|
||
Qed.
|
||
|
||
Lemma suc_inj n (A0 A1 : PTm n) :
|
||
R (PSuc A0) (PSuc A1) ->
|
||
R A0 A1.
|
||
Proof.
|
||
hauto lq:on rew:off use:REReds.suc_inv.
|
||
Qed.
|
||
|
||
Lemma hne_app_inj n (a0 b0 a1 b1 : PTm n) :
|
||
R (PApp a0 b0) (PApp a1 b1) ->
|
||
ishne a0 ->
|
||
ishne a1 ->
|
||
R a0 a1 /\ R b0 b1.
|
||
Proof.
|
||
hauto lq:on use:REReds.hne_app_inv.
|
||
Qed.
|
||
|
||
Lemma hne_proj_inj n p0 p1 (a0 a1 : PTm n) :
|
||
R (PProj p0 a0) (PProj p1 a1) ->
|
||
ishne a0 ->
|
||
ishne a1 ->
|
||
p0 = p1 /\ R a0 a1.
|
||
Proof.
|
||
sauto lq:on use:REReds.hne_proj_inv.
|
||
Qed.
|
||
|
||
Lemma FromRRed0 n (a b : PTm n) :
|
||
RRed.R a b -> R a b.
|
||
Proof.
|
||
hauto lq:on ctrs:rtc use:RERed.FromBeta unfold:R.
|
||
Qed.
|
||
|
||
Lemma FromRRed1 n (a b : PTm n) :
|
||
RRed.R b a -> R a b.
|
||
Proof.
|
||
hauto lq:on ctrs:rtc use:RERed.FromBeta unfold:R.
|
||
Qed.
|
||
|
||
Lemma FromRReds n (a b : PTm n) :
|
||
rtc RRed.R b a -> R a b.
|
||
Proof.
|
||
hauto lq:on ctrs:rtc use:REReds.FromRReds unfold:R.
|
||
Qed.
|
||
|
||
Lemma FromBJoin n (a b : PTm n) :
|
||
BJoin.R a b -> R a b.
|
||
Proof.
|
||
hauto lq:on ctrs:rtc use:REReds.FromRReds unfold:R.
|
||
Qed.
|
||
|
||
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 lq:on rew:off ctrs:rtc unfold:R use:REReds.substing.
|
||
Qed.
|
||
|
||
Lemma renaming n m (a b : PTm n) (ρ : fin n -> fin m) :
|
||
R a b -> R (ren_PTm ρ a) (ren_PTm ρ b).
|
||
Proof. substify. apply substing. Qed.
|
||
|
||
Lemma weakening n (a b : PTm n) :
|
||
R a b -> R (ren_PTm shift a) (ren_PTm shift b).
|
||
Proof. apply renaming. Qed.
|
||
|
||
Lemma cong n m (a : PTm (S n)) c d (ρ : fin n -> PTm m) :
|
||
R c d -> R (subst_PTm (scons c ρ) a) (subst_PTm (scons d ρ) a).
|
||
Proof.
|
||
rewrite /R. move => [cd [h0 h1]].
|
||
exists (subst_PTm (scons cd ρ) a).
|
||
hauto q:on ctrs:rtc inv:option use:REReds.cong.
|
||
Qed.
|
||
|
||
Lemma pair_inj n (a0 a1 b0 b1 : PTm n) :
|
||
SN (PPair a0 b0) ->
|
||
SN (PPair a1 b1) ->
|
||
R (PPair a0 b0) (PPair a1 b1) ->
|
||
R a0 a1 /\ R b0 b1.
|
||
Proof.
|
||
move => sn0 sn1.
|
||
have [? [? [? ?]]] : SN a0 /\ SN b0 /\ SN a1 /\ SN b1 by sfirstorder use:SN_NoForbid.P_PairInv.
|
||
have ? : SN (PProj PL (PPair a0 b0)) by hauto lq:on db:sn.
|
||
have ? : SN (PProj PR (PPair a0 b0)) by hauto lq:on db:sn.
|
||
have ? : SN (PProj PL (PPair a1 b1)) by hauto lq:on db:sn.
|
||
have ? : SN (PProj PR (PPair a1 b1)) by hauto lq:on db:sn.
|
||
have h0L : RRed.R (PProj PL (PPair a0 b0)) a0 by eauto with red.
|
||
have h0R : RRed.R (PProj PR (PPair a0 b0)) b0 by eauto with red.
|
||
have h1L : RRed.R (PProj PL (PPair a1 b1)) a1 by eauto with red.
|
||
have h1R : RRed.R (PProj PR (PPair a1 b1)) b1 by eauto with red.
|
||
move => h2.
|
||
move /ProjCong in h2.
|
||
have h2L := h2 PL.
|
||
have {h2}h2R := h2 PR.
|
||
move /FromRRed1 in h0L.
|
||
move /FromRRed0 in h1L.
|
||
move /FromRRed1 in h0R.
|
||
move /FromRRed0 in h1R.
|
||
split; eauto using transitive.
|
||
Qed.
|
||
|
||
Lemma ejoin_pair_inj n (a0 a1 b0 b1 : PTm n) :
|
||
nf a0 -> nf b0 -> nf a1 -> nf b1 ->
|
||
EJoin.R (PPair a0 b0) (PPair a1 b1) ->
|
||
EJoin.R a0 a1 /\ EJoin.R b0 b1.
|
||
Proof.
|
||
move => h0 h1 h2 h3 /FromEJoin.
|
||
have [? ?] : SN (PPair a0 b0) /\ SN (PPair a1 b1) by sauto lqb:on rew:off use:ne_nf_embed.
|
||
move => ?.
|
||
have : R a0 a1 /\ R b0 b1 by hauto l:on use:pair_inj.
|
||
hauto l:on use:ToEJoin.
|
||
Qed.
|
||
|
||
Lemma abs_inj n (a0 a1 : PTm (S n)) :
|
||
SN a0 -> SN a1 ->
|
||
R (PAbs a0) (PAbs a1) ->
|
||
R a0 a1.
|
||
Proof.
|
||
move => sn0 sn1.
|
||
move /weakening => /=.
|
||
move /AppCong. move /(_ (VarPTm var_zero) (VarPTm var_zero)).
|
||
move => /(_ ltac:(sfirstorder use:refl)).
|
||
move => h.
|
||
have /FromRRed1 h0 : RRed.R (PApp (PAbs (ren_PTm (upRen_PTm_PTm shift) a0)) (VarPTm var_zero)) a0 by apply RRed.AppAbs'; asimpl.
|
||
have /FromRRed0 h1 : RRed.R (PApp (PAbs (ren_PTm (upRen_PTm_PTm shift) a1)) (VarPTm var_zero)) a1 by apply RRed.AppAbs'; asimpl.
|
||
have sn0' := sn0.
|
||
have sn1' := sn1.
|
||
eapply sn_renaming with (ξ := (upRen_PTm_PTm shift)) in sn0.
|
||
eapply sn_renaming with (ξ := (upRen_PTm_PTm shift)) in sn1.
|
||
apply : transitive; try apply h0.
|
||
apply : N_Exp. apply N_β. sauto.
|
||
by asimpl.
|
||
apply : transitive; try apply h1.
|
||
apply : N_Exp. apply N_β. sauto.
|
||
by asimpl.
|
||
eauto.
|
||
Qed.
|
||
|
||
Lemma ejoin_abs_inj n (a0 a1 : PTm (S n)) :
|
||
nf a0 -> nf a1 ->
|
||
EJoin.R (PAbs a0) (PAbs a1) ->
|
||
EJoin.R a0 a1.
|
||
Proof.
|
||
move => h0 h1.
|
||
have [? [? [? ?]]] : SN a0 /\ SN a1 /\ SN (PAbs a0) /\ SN (PAbs a1) by sauto lqb:on rew:off use:ne_nf_embed.
|
||
move /FromEJoin.
|
||
move /abs_inj.
|
||
hauto l:on use:ToEJoin.
|
||
Qed.
|
||
|
||
Lemma standardization n (a b : PTm n) :
|
||
SN a -> SN b -> R a b ->
|
||
exists va vb, rtc RRed.R a va /\ rtc RRed.R b vb /\ nf va /\ nf vb /\ EJoin.R va vb.
|
||
Proof.
|
||
move => h0 h1 [ab [hv0 hv1]].
|
||
have hv : SN ab by sfirstorder use:REReds.sn_preservation.
|
||
have : exists v, rtc RRed.R ab v /\ nf v by hauto q:on use:LoReds.FromSN, LoReds.ToRReds.
|
||
move => [v [hv2 hv3]].
|
||
have : rtc RERed.R a v by hauto q:on use:@relations.rtc_transitive, REReds.FromRReds.
|
||
have : rtc RERed.R b v by hauto q:on use:@relations.rtc_transitive, REReds.FromRReds.
|
||
move : h0 h1 hv3. clear.
|
||
move => h0 h1 hv hbv hav.
|
||
move : rered_standardization (h0) hav. repeat move/[apply].
|
||
move => [a1 [ha0 ha1]].
|
||
move : rered_standardization (h1) hbv. repeat move/[apply].
|
||
move => [b1 [hb0 hb1]].
|
||
have [*] : nf a1 /\ nf b1 by sfirstorder use:NeEPars.R_nonelim_nf.
|
||
hauto q:on use:NeEPars.ToEReds unfold:EJoin.R.
|
||
Qed.
|
||
|
||
Lemma standardization_lo n (a b : PTm n) :
|
||
SN a -> SN b -> R a b ->
|
||
exists va vb, rtc LoRed.R a va /\ rtc LoRed.R b vb /\ nf va /\ nf vb /\ EJoin.R va vb.
|
||
Proof.
|
||
move => /[dup] sna + /[dup] snb.
|
||
move : standardization; repeat move/[apply].
|
||
move => [va][vb][hva][hvb][nfva][nfvb]hj.
|
||
move /LoReds.FromSN : sna => [va' [hva' hva'0]].
|
||
move /LoReds.FromSN : snb => [vb' [hvb' hvb'0]].
|
||
exists va', vb'.
|
||
repeat split => //=.
|
||
have : va = va' /\ vb = vb' by sfirstorder use:red_uniquenf, LoReds.ToRReds.
|
||
case; congruence.
|
||
Qed.
|
||
End DJoin.
|
||
|
||
|
||
Module Sub1.
|
||
Inductive R {n} : PTm n -> PTm n -> Prop :=
|
||
| Refl a :
|
||
R a a
|
||
| Univ i j :
|
||
i <= j ->
|
||
R (PUniv i) (PUniv j)
|
||
| PiCong A0 A1 B0 B1 :
|
||
R A1 A0 ->
|
||
R B0 B1 ->
|
||
R (PBind PPi A0 B0) (PBind PPi A1 B1)
|
||
| SigCong A0 A1 B0 B1 :
|
||
R A0 A1 ->
|
||
R B0 B1 ->
|
||
R (PBind PSig A0 B0) (PBind PSig A1 B1).
|
||
|
||
Lemma transitive0 n (A B C : PTm n) :
|
||
R A B -> (R B C -> R A C) /\ (R C A -> R C B).
|
||
Proof.
|
||
move => h. move : C.
|
||
elim : n A B /h; hauto lq:on ctrs:R inv:R solve+:lia.
|
||
Qed.
|
||
|
||
Lemma transitive n (A B C : PTm n) :
|
||
R A B -> R B C -> R A C.
|
||
Proof. hauto q:on use:transitive0. Qed.
|
||
|
||
Lemma refl n (A : PTm n) : R A A.
|
||
Proof. sfirstorder. Qed.
|
||
|
||
Lemma commutativity0 n (A B C : PTm n) :
|
||
R A B ->
|
||
(RERed.R A C ->
|
||
exists D, RERed.R B D /\ R C D) /\
|
||
(RERed.R B C ->
|
||
exists D, RERed.R A D /\ R D C).
|
||
Proof.
|
||
move => h. move : C.
|
||
elim : n A B / h; hauto lq:on ctrs:RERed.R, R inv:RERed.R.
|
||
Qed.
|
||
|
||
Lemma commutativity_Ls n (A B C : PTm n) :
|
||
R A B ->
|
||
rtc RERed.R A C ->
|
||
exists D, rtc RERed.R B D /\ R C D.
|
||
Proof.
|
||
move => + h. move : B. induction h; sauto lq:on use:commutativity0.
|
||
Qed.
|
||
|
||
Lemma commutativity_Rs n (A B C : PTm n) :
|
||
R A B ->
|
||
rtc RERed.R B C ->
|
||
exists D, rtc RERed.R A D /\ R D C.
|
||
Proof.
|
||
move => + h. move : A. induction h; sauto lq:on use:commutativity0.
|
||
Qed.
|
||
|
||
Lemma sn_preservation : forall n,
|
||
(forall (a : PTm n) (s : SNe a), forall b, R a b \/ R b a -> a = b) /\
|
||
(forall (a : PTm n) (s : SN a), forall b, R a b \/ R b a -> SN b) /\
|
||
(forall (a b : PTm n) (_ : TRedSN a b), forall c, R a c \/ R c a -> a = c).
|
||
Proof.
|
||
apply sn_mutual; hauto lq:on inv:R ctrs:SN.
|
||
Qed.
|
||
|
||
Lemma bind_preservation n (a b : PTm n) :
|
||
R a b -> isbind a = isbind b.
|
||
Proof. hauto q:on inv:R. Qed.
|
||
|
||
Lemma univ_preservation n (a b : PTm n) :
|
||
R a b -> isuniv a = isuniv b.
|
||
Proof. hauto q:on inv:R. Qed.
|
||
|
||
Lemma sne_preservation n (a b : PTm n) :
|
||
R a b -> SNe a <-> SNe b.
|
||
Proof. hfcrush use:sn_preservation. Qed.
|
||
|
||
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; hauto lq:on ctrs:R.
|
||
Qed.
|
||
|
||
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. move => h. move : m ρ. elim : n a b /h; hauto lq:on ctrs:R. Qed.
|
||
|
||
Lemma hne_refl n (a b : PTm n) :
|
||
ishne a \/ ishne b -> R a b -> a = b.
|
||
Proof. hauto q:on inv:R. Qed.
|
||
|
||
End Sub1.
|
||
|
||
Module ESub.
|
||
Definition R {n} (a b : PTm n) := exists c0 c1, rtc ERed.R a c0 /\ rtc ERed.R b c1 /\ Sub1.R c0 c1.
|
||
|
||
Lemma pi_inj n (A0 A1 : PTm n) B0 B1 :
|
||
R (PBind PPi A0 B0) (PBind PPi A1 B1) ->
|
||
R A1 A0 /\ R B0 B1.
|
||
Proof.
|
||
move => [u0 [u1 [h0 [h1 h2]]]].
|
||
move /EReds.bind_inv : h0 => [A2][B2][?][h3]h4. subst.
|
||
move /EReds.bind_inv : h1 => [A3][B3][?][h5]h6. subst.
|
||
sauto lq:on rew:off inv:Sub1.R.
|
||
Qed.
|
||
|
||
Lemma sig_inj n (A0 A1 : PTm n) B0 B1 :
|
||
R (PBind PSig A0 B0) (PBind PSig A1 B1) ->
|
||
R A0 A1 /\ R B0 B1.
|
||
Proof.
|
||
move => [u0 [u1 [h0 [h1 h2]]]].
|
||
move /EReds.bind_inv : h0 => [A2][B2][?][h3]h4. subst.
|
||
move /EReds.bind_inv : h1 => [A3][B3][?][h5]h6. subst.
|
||
sauto lq:on rew:off inv:Sub1.R.
|
||
Qed.
|
||
|
||
Lemma suc_inj n (a b : PTm n) :
|
||
R (PSuc a) (PSuc b) ->
|
||
R a b.
|
||
Proof.
|
||
sauto lq:on use:EReds.suc_inv inv:Sub1.R.
|
||
Qed.
|
||
|
||
End ESub.
|
||
|
||
Module Sub.
|
||
Definition R {n} (a b : PTm n) := exists c d, rtc RERed.R a c /\ rtc RERed.R b d /\ Sub1.R c d.
|
||
|
||
Lemma refl n (a : PTm n) : R a a.
|
||
Proof. sfirstorder use:@rtc_refl unfold:R. Qed.
|
||
|
||
Lemma ToJoin n (a b : PTm n) :
|
||
ishne a \/ ishne b ->
|
||
R a b ->
|
||
DJoin.R a b.
|
||
Proof.
|
||
move => h [c][d][h0][h1]h2.
|
||
have : ishne c \/ ishne d by hauto q:on use:REReds.hne_preservation.
|
||
hauto lq:on rew:off use:Sub1.hne_refl.
|
||
Qed.
|
||
|
||
Lemma transitive n (a b c : PTm n) : SN b -> R a b -> R b c -> R a c.
|
||
Proof.
|
||
rewrite /R.
|
||
move => h [a0][b0][ha][hb]ha0b0 [b1][c0][hb'][hc]hb1c0.
|
||
move : hb hb'.
|
||
move : rered_confluence h. repeat move/[apply].
|
||
move => [b'][hb0]hb1.
|
||
have [a' ?] : exists a', rtc RERed.R a0 a' /\ Sub1.R a' b' by hauto l:on use:Sub1.commutativity_Rs.
|
||
have [c' ?] : exists a', rtc RERed.R c0 a' /\ Sub1.R b' a' by hauto l:on use:Sub1.commutativity_Ls.
|
||
exists a',c'; hauto l:on use:Sub1.transitive, @relations.rtc_transitive.
|
||
Qed.
|
||
|
||
Lemma FromJoin n (a b : PTm n) : DJoin.R a b -> R a b.
|
||
Proof. sfirstorder. Qed.
|
||
|
||
Lemma PiCong n (A0 A1 : PTm n) B0 B1 :
|
||
R A1 A0 ->
|
||
R B0 B1 ->
|
||
R (PBind PPi A0 B0) (PBind PPi A1 B1).
|
||
Proof.
|
||
rewrite /R.
|
||
move => [A][A'][h0][h1]h2.
|
||
move => [B][B'][h3][h4]h5.
|
||
exists (PBind PPi A' B), (PBind PPi A B').
|
||
repeat split; eauto using REReds.BindCong, Sub1.PiCong.
|
||
Qed.
|
||
|
||
Lemma SigCong n (A0 A1 : PTm n) B0 B1 :
|
||
R A0 A1 ->
|
||
R B0 B1 ->
|
||
R (PBind PSig A0 B0) (PBind PSig A1 B1).
|
||
Proof.
|
||
rewrite /R.
|
||
move => [A][A'][h0][h1]h2.
|
||
move => [B][B'][h3][h4]h5.
|
||
exists (PBind PSig A B), (PBind PSig A' B').
|
||
repeat split; eauto using REReds.BindCong, Sub1.SigCong.
|
||
Qed.
|
||
|
||
Lemma UnivCong n i j :
|
||
i <= j ->
|
||
@R n (PUniv i) (PUniv j).
|
||
Proof. hauto lq:on ctrs:Sub1.R, rtc. Qed.
|
||
|
||
Lemma sne_bind_noconf n (a b : PTm n) :
|
||
R a b -> SNe a -> isbind b -> False.
|
||
Proof.
|
||
rewrite /R.
|
||
move => [c[d] [? []]] *.
|
||
have : SNe c /\ isbind c by
|
||
hauto l:on use:REReds.sne_preservation, REReds.bind_preservation, Sub1.sne_preservation, Sub1.bind_preservation.
|
||
qauto l:on inv:SNe.
|
||
Qed.
|
||
|
||
Lemma hne_bind_noconf n (a b : PTm n) :
|
||
R a b -> ishne a -> isbind b -> False.
|
||
Proof.
|
||
rewrite /R.
|
||
move => [c[d] [? []]] h0 h1 h2 h3.
|
||
have : ishne c by eauto using REReds.hne_preservation.
|
||
have : isbind d by eauto using REReds.bind_preservation.
|
||
move : h1. clear. inversion 1; subst => //=.
|
||
clear. case : d => //=.
|
||
Qed.
|
||
|
||
Lemma bind_sne_noconf n (a b : PTm n) :
|
||
R a b -> SNe b -> isbind a -> False.
|
||
Proof.
|
||
rewrite /R.
|
||
move => [c[d] [? []]] *.
|
||
have : SNe c /\ isbind c by
|
||
hauto l:on use:REReds.sne_preservation, REReds.bind_preservation, Sub1.sne_preservation, Sub1.bind_preservation.
|
||
qauto l:on inv:SNe.
|
||
Qed.
|
||
|
||
Lemma sne_univ_noconf n (a b : PTm n) :
|
||
R a b -> SNe a -> isuniv b -> False.
|
||
Proof.
|
||
hauto l:on use:REReds.sne_preservation,
|
||
REReds.univ_preservation, Sub1.sne_preservation, Sub1.univ_preservation inv:SNe.
|
||
Qed.
|
||
|
||
Lemma univ_sne_noconf n (a b : PTm n) :
|
||
R a b -> SNe b -> isuniv a -> False.
|
||
Proof.
|
||
move => [c[d] [? []]] *.
|
||
have ? : SNe d by eauto using REReds.sne_preservation.
|
||
have : SNe c by sfirstorder use:Sub1.sne_preservation.
|
||
have : isuniv c by sfirstorder use:REReds.univ_preservation.
|
||
clear. case : c => //=. inversion 2.
|
||
Qed.
|
||
|
||
Lemma bind_univ_noconf n (a b : PTm n) :
|
||
R a b -> isbind a -> isuniv b -> False.
|
||
Proof.
|
||
move => [c[d] [h0 [h1 h1']]] h2 h3.
|
||
have : isbind c /\ isuniv c by
|
||
hauto l:on use:REReds.bind_preservation,
|
||
REReds.univ_preservation, Sub1.bind_preservation, Sub1.univ_preservation.
|
||
move : h2 h3. clear.
|
||
case : c => //=; itauto.
|
||
Qed.
|
||
|
||
Lemma univ_bind_noconf n (a b : PTm n) :
|
||
R a b -> isbind b -> isuniv a -> False.
|
||
Proof.
|
||
move => [c[d] [h0 [h1 h1']]] h2 h3.
|
||
have : isbind c /\ isuniv c by
|
||
hauto l:on use:REReds.bind_preservation,
|
||
REReds.univ_preservation, Sub1.bind_preservation, Sub1.univ_preservation.
|
||
move : h2 h3. clear.
|
||
case : c => //=; itauto.
|
||
Qed.
|
||
|
||
Lemma bind_inj n p0 p1 (A0 A1 : PTm n) B0 B1 :
|
||
R (PBind p0 A0 B0) (PBind p1 A1 B1) ->
|
||
p0 = p1 /\ (if p0 is PPi then R A1 A0 else R A0 A1) /\ R B0 B1.
|
||
Proof.
|
||
rewrite /R.
|
||
move => [u][v][h0][h1]h.
|
||
move /REReds.bind_inv : h0 => [A2][B2][?][h00]h01. subst.
|
||
move /REReds.bind_inv : h1 => [A3][B3][?][h10]h11. subst.
|
||
inversion h; subst; sauto lq:on.
|
||
Qed.
|
||
|
||
Lemma univ_inj n i j :
|
||
@R n (PUniv i) (PUniv j) -> i <= j.
|
||
Proof.
|
||
sauto lq:on rew:off use:REReds.univ_inv.
|
||
Qed.
|
||
|
||
Lemma suc_inj n (A0 A1 : PTm n) :
|
||
R (PSuc A0) (PSuc A1) ->
|
||
R A0 A1.
|
||
Proof.
|
||
sauto q:on use:REReds.suc_inv.
|
||
Qed.
|
||
|
||
|
||
Lemma cong n m (a b : PTm (S n)) c d (ρ : fin n -> PTm m) :
|
||
R a b -> DJoin.R c d -> R (subst_PTm (scons c ρ) a) (subst_PTm (scons d ρ) b).
|
||
Proof.
|
||
rewrite /R.
|
||
move => [a0][b0][h0][h1]h2.
|
||
move => [cd][h3]h4.
|
||
exists (subst_PTm (scons cd ρ) a0), (subst_PTm (scons cd ρ) b0).
|
||
repeat split.
|
||
hauto l:on use:REReds.cong' inv:option.
|
||
hauto l:on use:REReds.cong' inv:option.
|
||
eauto using Sub1.substing.
|
||
Qed.
|
||
|
||
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.
|
||
rewrite /R.
|
||
move => [a0][b0][h0][h1]h2.
|
||
hauto ctrs:rtc use:REReds.cong', Sub1.substing.
|
||
Qed.
|
||
|
||
Lemma ToESub n (a b : PTm n) : nf a -> nf b -> R a b -> ESub.R a b.
|
||
Proof. hauto q:on use:REReds.ToEReds. Qed.
|
||
|
||
Lemma standardization n (a b : PTm n) :
|
||
SN a -> SN b -> R a b ->
|
||
exists va vb, rtc RRed.R a va /\ rtc RRed.R b vb /\ nf va /\ nf vb /\ ESub.R va vb.
|
||
Proof.
|
||
move => h0 h1 hS.
|
||
have : exists v, rtc RRed.R a v /\ nf v by sfirstorder use:LoReds.FromSN, LoReds.ToRReds.
|
||
move => [v [hv2 hv3]].
|
||
have : exists v, rtc RRed.R b v /\ nf v by sfirstorder use:LoReds.FromSN, LoReds.ToRReds.
|
||
move => [v' [hv2' hv3']].
|
||
move : (hv2) (hv2') => *.
|
||
apply DJoin.FromRReds in hv2, hv2'.
|
||
move/DJoin.symmetric in hv2'.
|
||
apply FromJoin in hv2, hv2'.
|
||
have hv : R v v' by eauto using transitive.
|
||
have {}hv : ESub.R v v' by hauto l:on use:ToESub.
|
||
hauto lq:on.
|
||
Qed.
|
||
|
||
Lemma standardization_lo n (a b : PTm n) :
|
||
SN a -> SN b -> R a b ->
|
||
exists va vb, rtc LoRed.R a va /\ rtc LoRed.R b vb /\ nf va /\ nf vb /\ ESub.R va vb.
|
||
Proof.
|
||
move => /[dup] sna + /[dup] snb.
|
||
move : standardization; repeat move/[apply].
|
||
move => [va][vb][hva][hvb][nfva][nfvb]hj.
|
||
move /LoReds.FromSN : sna => [va' [hva' hva'0]].
|
||
move /LoReds.FromSN : snb => [vb' [hvb' hvb'0]].
|
||
exists va', vb'.
|
||
repeat split => //=.
|
||
have : va = va' /\ vb = vb' by sfirstorder use:red_uniquenf, LoReds.ToRReds.
|
||
case; congruence.
|
||
Qed.
|
||
|
||
End Sub.
|