diff --git a/syntax.sig b/syntax.sig index e63c3b4..5448432 100644 --- a/syntax.sig +++ b/syntax.sig @@ -14,11 +14,9 @@ PAbs : (bind PTm in PTm) -> PTm PApp : PTm -> PTm -> PTm PPair : PTm -> PTm -> PTm PProj : PTag -> PTm -> PTm -PConst : TTag -> PTm -PUniv : nat -> PTm -PBot : PTm +PConst : nat -> PTm -Abs : (bind Tm in Tm) -> Tm +Abs : Tm -> (bind Tm in Tm) -> Tm App : Tm -> Tm -> Tm Pair : Tm -> Tm -> Tm Proj : PTag -> Tm -> Tm diff --git a/theories/Autosubst2/syntax.v b/theories/Autosubst2/syntax.v index 659d8b0..26f4c0d 100644 --- a/theories/Autosubst2/syntax.v +++ b/theories/Autosubst2/syntax.v @@ -19,29 +19,13 @@ Proof. exact (eq_refl). Qed. -Inductive TTag : Type := - | TPi : TTag - | TSig : TTag. - -Lemma congr_TPi : TPi = TPi. -Proof. -exact (eq_refl). -Qed. - -Lemma congr_TSig : TSig = TSig. -Proof. -exact (eq_refl). -Qed. - Inductive PTm : Type := | VarPTm : nat -> PTm | PAbs : PTm -> PTm | PApp : PTm -> PTm -> PTm | PPair : PTm -> PTm -> PTm | PProj : PTag -> PTm -> PTm - | PConst : TTag -> PTm - | PUniv : nat -> PTm - | PBot : PTm. + | PConst : nat -> PTm. Lemma congr_PAbs {s0 : PTm} {t0 : PTm} (H0 : s0 = t0) : PAbs s0 = PAbs t0. Proof. @@ -69,22 +53,12 @@ exact (eq_trans (eq_trans eq_refl (ap (fun x => PProj x s1) H0)) (ap (fun x => PProj t0 x) H1)). Qed. -Lemma congr_PConst {s0 : TTag} {t0 : TTag} (H0 : s0 = t0) : +Lemma congr_PConst {s0 : nat} {t0 : nat} (H0 : s0 = t0) : PConst s0 = PConst t0. Proof. exact (eq_trans eq_refl (ap (fun x => PConst x) H0)). Qed. -Lemma congr_PUniv {s0 : nat} {t0 : nat} (H0 : s0 = t0) : PUniv s0 = PUniv t0. -Proof. -exact (eq_trans eq_refl (ap (fun x => PUniv x) H0)). -Qed. - -Lemma congr_PBot : PBot = PBot. -Proof. -exact (eq_refl). -Qed. - Lemma upRen_PTm_PTm (xi : nat -> nat) : nat -> nat. Proof. exact (up_ren xi). @@ -98,8 +72,6 @@ Fixpoint ren_PTm (xi_PTm : nat -> nat) (s : PTm) {struct s} : PTm := | PPair s0 s1 => PPair (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1) | PProj s0 s1 => PProj s0 (ren_PTm xi_PTm s1) | PConst s0 => PConst s0 - | PUniv s0 => PUniv s0 - | PBot => PBot end. Lemma up_PTm_PTm (sigma : nat -> PTm) : nat -> PTm. @@ -115,8 +87,6 @@ Fixpoint subst_PTm (sigma_PTm : nat -> PTm) (s : PTm) {struct s} : PTm := | PPair s0 s1 => PPair (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1) | PProj s0 s1 => PProj s0 (subst_PTm sigma_PTm s1) | PConst s0 => PConst s0 - | PUniv s0 => PUniv s0 - | PBot => PBot end. Lemma upId_PTm_PTm (sigma : nat -> PTm) (Eq : forall x, sigma x = VarPTm x) : @@ -145,8 +115,6 @@ subst_PTm sigma_PTm s = s := (idSubst_PTm sigma_PTm Eq_PTm s1) | PProj s0 s1 => congr_PProj (eq_refl s0) (idSubst_PTm sigma_PTm Eq_PTm s1) | PConst s0 => congr_PConst (eq_refl s0) - | PUniv s0 => congr_PUniv (eq_refl s0) - | PBot => congr_PBot end. Lemma upExtRen_PTm_PTm (xi : nat -> nat) (zeta : nat -> nat) @@ -177,8 +145,6 @@ ren_PTm xi_PTm s = ren_PTm zeta_PTm s := | PProj s0 s1 => congr_PProj (eq_refl s0) (extRen_PTm xi_PTm zeta_PTm Eq_PTm s1) | PConst s0 => congr_PConst (eq_refl s0) - | PUniv s0 => congr_PUniv (eq_refl s0) - | PBot => congr_PBot end. Lemma upExt_PTm_PTm (sigma : nat -> PTm) (tau : nat -> PTm) @@ -210,8 +176,6 @@ subst_PTm sigma_PTm s = subst_PTm tau_PTm s := | PProj s0 s1 => congr_PProj (eq_refl s0) (ext_PTm sigma_PTm tau_PTm Eq_PTm s1) | PConst s0 => congr_PConst (eq_refl s0) - | PUniv s0 => congr_PUniv (eq_refl s0) - | PBot => congr_PBot end. Lemma up_ren_ren_PTm_PTm (xi : nat -> nat) (zeta : nat -> nat) @@ -242,8 +206,6 @@ Fixpoint compRenRen_PTm (xi_PTm : nat -> nat) (zeta_PTm : nat -> nat) congr_PProj (eq_refl s0) (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1) | PConst s0 => congr_PConst (eq_refl s0) - | PUniv s0 => congr_PUniv (eq_refl s0) - | PBot => congr_PBot end. Lemma up_ren_subst_PTm_PTm (xi : nat -> nat) (tau : nat -> PTm) @@ -278,8 +240,6 @@ Fixpoint compRenSubst_PTm (xi_PTm : nat -> nat) (tau_PTm : nat -> PTm) congr_PProj (eq_refl s0) (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1) | PConst s0 => congr_PConst (eq_refl s0) - | PUniv s0 => congr_PUniv (eq_refl s0) - | PBot => congr_PBot end. Lemma up_subst_ren_PTm_PTm (sigma : nat -> PTm) (zeta_PTm : nat -> nat) @@ -325,8 +285,6 @@ ren_PTm zeta_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s := congr_PProj (eq_refl s0) (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1) | PConst s0 => congr_PConst (eq_refl s0) - | PUniv s0 => congr_PUniv (eq_refl s0) - | PBot => congr_PBot end. Lemma up_subst_subst_PTm_PTm (sigma : nat -> PTm) (tau_PTm : nat -> PTm) @@ -373,8 +331,6 @@ subst_PTm tau_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s := congr_PProj (eq_refl s0) (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1) | PConst s0 => congr_PConst (eq_refl s0) - | PUniv s0 => congr_PUniv (eq_refl s0) - | PBot => congr_PBot end. Lemma renRen_PTm (xi_PTm : nat -> nat) (zeta_PTm : nat -> nat) (s : PTm) : @@ -464,8 +420,6 @@ Fixpoint rinst_inst_PTm (xi_PTm : nat -> nat) (sigma_PTm : nat -> PTm) | PProj s0 s1 => congr_PProj (eq_refl s0) (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1) | PConst s0 => congr_PConst (eq_refl s0) - | PUniv s0 => congr_PUniv (eq_refl s0) - | PBot => congr_PBot end. Lemma rinstInst'_PTm (xi_PTm : nat -> nat) (s : PTm) : @@ -527,9 +481,23 @@ Proof. exact (fun x => eq_refl). Qed. +Inductive TTag : Type := + | TPi : TTag + | TSig : TTag. + +Lemma congr_TPi : TPi = TPi. +Proof. +exact (eq_refl). +Qed. + +Lemma congr_TSig : TSig = TSig. +Proof. +exact (eq_refl). +Qed. + Inductive Tm : Type := | VarTm : nat -> Tm - | Abs : Tm -> Tm + | Abs : Tm -> Tm -> Tm | App : Tm -> Tm -> Tm | Pair : Tm -> Tm -> Tm | Proj : PTag -> Tm -> Tm @@ -539,9 +507,11 @@ Inductive Tm : Type := | Bool : Tm | If : Tm -> Tm -> Tm -> Tm. -Lemma congr_Abs {s0 : Tm} {t0 : Tm} (H0 : s0 = t0) : Abs s0 = Abs t0. +Lemma congr_Abs {s0 : Tm} {s1 : Tm} {t0 : Tm} {t1 : Tm} (H0 : s0 = t0) + (H1 : s1 = t1) : Abs s0 s1 = Abs t0 t1. Proof. -exact (eq_trans eq_refl (ap (fun x => Abs x) H0)). +exact (eq_trans (eq_trans eq_refl (ap (fun x => Abs x s1) H0)) + (ap (fun x => Abs t0 x) H1)). Qed. Lemma congr_App {s0 : Tm} {s1 : Tm} {t0 : Tm} {t1 : Tm} (H0 : s0 = t0) @@ -606,7 +576,7 @@ Defined. Fixpoint ren_Tm (xi_Tm : nat -> nat) (s : Tm) {struct s} : Tm := match s with | VarTm s0 => VarTm (xi_Tm s0) - | Abs s0 => Abs (ren_Tm (upRen_Tm_Tm xi_Tm) s0) + | Abs s0 s1 => Abs (ren_Tm xi_Tm s0) (ren_Tm (upRen_Tm_Tm xi_Tm) s1) | App s0 s1 => App (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1) | Pair s0 s1 => Pair (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1) | Proj s0 s1 => Proj s0 (ren_Tm xi_Tm s1) @@ -626,7 +596,7 @@ Defined. Fixpoint subst_Tm (sigma_Tm : nat -> Tm) (s : Tm) {struct s} : Tm := match s with | VarTm s0 => sigma_Tm s0 - | Abs s0 => Abs (subst_Tm (up_Tm_Tm sigma_Tm) s0) + | Abs s0 s1 => Abs (subst_Tm sigma_Tm s0) (subst_Tm (up_Tm_Tm sigma_Tm) s1) | App s0 s1 => App (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1) | Pair s0 s1 => Pair (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1) | Proj s0 s1 => Proj s0 (subst_Tm sigma_Tm s1) @@ -654,8 +624,9 @@ Fixpoint idSubst_Tm (sigma_Tm : nat -> Tm) subst_Tm sigma_Tm s = s := match s with | VarTm s0 => Eq_Tm s0 - | Abs s0 => - congr_Abs (idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s0) + | Abs s0 s1 => + congr_Abs (idSubst_Tm sigma_Tm Eq_Tm s0) + (idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s1) | App s0 s1 => congr_App (idSubst_Tm sigma_Tm Eq_Tm s0) (idSubst_Tm sigma_Tm Eq_Tm s1) | Pair s0 s1 => @@ -688,10 +659,10 @@ Fixpoint extRen_Tm (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat) ren_Tm xi_Tm s = ren_Tm zeta_Tm s := match s with | VarTm s0 => ap (VarTm) (Eq_Tm s0) - | Abs s0 => - congr_Abs + | Abs s0 s1 => + congr_Abs (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0) (extRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm) - (upExtRen_Tm_Tm _ _ Eq_Tm) s0) + (upExtRen_Tm_Tm _ _ Eq_Tm) s1) | App s0 s1 => congr_App (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1) @@ -727,10 +698,10 @@ Fixpoint ext_Tm (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm) subst_Tm sigma_Tm s = subst_Tm tau_Tm s := match s with | VarTm s0 => Eq_Tm s0 - | Abs s0 => - congr_Abs + | Abs s0 s1 => + congr_Abs (ext_Tm sigma_Tm tau_Tm Eq_Tm s0) (ext_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm) (upExt_Tm_Tm _ _ Eq_Tm) - s0) + s1) | App s0 s1 => congr_App (ext_Tm sigma_Tm tau_Tm Eq_Tm s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1) @@ -762,10 +733,10 @@ Fixpoint compRenRen_Tm (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat) (s : Tm) {struct s} : ren_Tm zeta_Tm (ren_Tm xi_Tm s) = ren_Tm rho_Tm s := match s with | VarTm s0 => ap (VarTm) (Eq_Tm s0) - | Abs s0 => - congr_Abs + | Abs s0 s1 => + congr_Abs (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0) (compRenRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm) - (upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s0) + (upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s1) | App s0 s1 => congr_App (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0) (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1) @@ -804,10 +775,10 @@ Fixpoint compRenSubst_Tm (xi_Tm : nat -> nat) (tau_Tm : nat -> Tm) subst_Tm tau_Tm (ren_Tm xi_Tm s) = subst_Tm theta_Tm s := match s with | VarTm s0 => Eq_Tm s0 - | Abs s0 => - congr_Abs + | Abs s0 s1 => + congr_Abs (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0) (compRenSubst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm tau_Tm) - (up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s0) + (up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s1) | App s0 s1 => congr_App (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0) (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1) @@ -860,10 +831,10 @@ Fixpoint compSubstRen_Tm (sigma_Tm : nat -> Tm) (zeta_Tm : nat -> nat) ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s := match s with | VarTm s0 => Eq_Tm s0 - | Abs s0 => - congr_Abs + | Abs s0 s1 => + congr_Abs (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0) (compSubstRen_Tm (up_Tm_Tm sigma_Tm) (upRen_Tm_Tm zeta_Tm) - (up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s0) + (up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s1) | App s0 s1 => congr_App (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0) (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1) @@ -916,10 +887,10 @@ Fixpoint compSubstSubst_Tm (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm) subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s := match s with | VarTm s0 => Eq_Tm s0 - | Abs s0 => - congr_Abs + | Abs s0 s1 => + congr_Abs (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0) (compSubstSubst_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm) - (up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s0) + (up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s1) | App s0 s1 => congr_App (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0) (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1) @@ -1013,10 +984,10 @@ Fixpoint rinst_inst_Tm (xi_Tm : nat -> nat) (sigma_Tm : nat -> Tm) : ren_Tm xi_Tm s = subst_Tm sigma_Tm s := match s with | VarTm s0 => Eq_Tm s0 - | Abs s0 => - congr_Abs + | Abs s0 s1 => + congr_Abs (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0) (rinst_inst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm sigma_Tm) - (rinstInst_up_Tm_Tm _ _ Eq_Tm) s0) + (rinstInst_up_Tm_Tm _ _ Eq_Tm) s1) | App s0 s1 => congr_App (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0) (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1) diff --git a/theories/compile.v b/theories/compile.v index e481d71..73da296 100644 --- a/theories/compile.v +++ b/theories/compile.v @@ -1,47 +1,47 @@ -Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax fp_red. +Require Import Autosubst2.core Autosubst2.unscoped Autosubst2.syntax fp_red. Require Import ssreflect ssrbool. From Hammer Require Import Tactics. From stdpp Require Import relations (rtc(..)). Module Compile. - Fixpoint F {n} (a : Tm n) : Tm n := + Definition compileTag p := if p is TPi then 0 else 1. + + Fixpoint F (a : Tm) : PTm := match a with - | TBind p A B => Pair (Pair (Const p) (F A)) (Abs (F B)) - | Const k => Const k - | Univ i => Univ i - | Abs a => Abs (F a) - | App a b => App (F a) (F b) - | VarTm i => VarTm i - | Pair a b => Pair (F a) (F b) - | Proj t a => Proj t (F a) - | Bot => Bot - | If a b c => App (App (F a) (F b)) (F c) - | BVal b => if b then (Abs (Abs (VarTm (shift var_zero)))) else (Abs (Abs (VarTm var_zero))) - | Bool => Bool + | TBind p A B => PPair (PPair (PConst (compileTag p)) (F A)) (PAbs (F B)) + | Univ i => PConst (3 + i) + | Abs _ a => PAbs (F a) + | App a b => PApp (F a) (F b) + | VarTm i => VarPTm i + | Pair a b => PPair (F a) (F b) + | Proj t a => PProj t (F a) + | If a b c => PApp (PApp (F a) (F b)) (F c) + | BVal b => if b then (PAbs (PAbs (VarPTm (shift var_zero)))) else (PAbs (PAbs (VarPTm var_zero))) + | Bool => PConst 2 end. - Lemma renaming n m (a : Tm n) (ξ : fin n -> fin m) : - F (ren_Tm ξ a)= ren_Tm ξ (F a). - Proof. move : m ξ. elim : n / a => //=; hauto lq:on. Qed. + Lemma renaming (a : Tm) (ξ : nat -> nat) : + F (ren_Tm ξ a)= ren_PTm ξ (F a). + Proof. move : ξ. elim : a => //=; hauto lq:on. Qed. #[local]Hint Rewrite Compile.renaming : compile. - Lemma morphing n m (a : Tm n) (ρ0 ρ1 : fin n -> Tm m) : + Lemma morphing (a : Tm) ρ0 ρ1 : (forall i, ρ0 i = F (ρ1 i)) -> - subst_Tm ρ0 (F a) = F (subst_Tm ρ1 a). + subst_PTm ρ0 (F a) = F (subst_Tm ρ1 a). Proof. - move : m ρ0 ρ1. elim : n / a => n//=. - - hauto lq:on inv:option rew:db:compile unfold:funcomp. + move : ρ0 ρ1. elim : a =>//=. + - hauto lq:on inv:nat rew:db:compile unfold:funcomp. - hauto lq:on rew:off. - hauto lq:on rew:off. - hauto lq:on. - - hauto lq:on inv:option rew:db:compile unfold:funcomp. + - hauto lq:on inv:nat rew:db:compile unfold:funcomp. - hauto lq:on rew:off. - hauto lq:on rew:off. Qed. - Lemma substing n b (a : Tm (S n)) : - subst_Tm (scons (F b) VarTm) (F a) = F (subst_Tm (scons b VarTm) a). + Lemma substing b (a : Tm) : + subst_PTm (scons (F b) VarPTm) (F a) = F (subst_Tm (scons b VarTm) a). Proof. apply morphing. case => //=. @@ -53,38 +53,55 @@ End Compile. Module Join. - Definition R {n} (a b : Tm n) := join (Compile.F a) (Compile.F b). + Definition R (a b : Tm) := join (Compile.F a) (Compile.F b). - Lemma BindInj n p0 p1 (A0 A1 : Tm n) B0 B1 : + Lemma compileTagInj p0 p1 : + Compile.compileTag p0 = Compile.compileTag p1 -> p0 = p1. + Proof. + case : p0 ; case : p1 => //. + Qed. + + Lemma BindInj p0 p1 (A0 A1 : Tm) B0 B1 : R (TBind p0 A0 B0) (TBind p1 A1 B1) -> p0 = p1 /\ R A0 A1 /\ R B0 B1. Proof. rewrite /R /= !join_pair_inj. - move => [[/join_const_inj h0 h1] h2]. + move => [[/join_const_inj /compileTagInj h0 h1] h2]. apply abs_eq in h2. - evar (t : Tm (S n)). - have : join (App (ren_Tm shift (Abs (Compile.F B1))) (VarTm var_zero)) t by + evar (t : PTm ). + have : join (PApp (ren_PTm shift (PAbs (Compile.F B1))) (VarPTm var_zero)) t by apply Join.FromPar; apply Par.AppAbs; auto using Par.refl. - subst t. rewrite -/ren_Tm. - move : h2. move /join_transitive => /[apply]. asimpl => h2. + subst t. rewrite -/ren_PTm. + move : h2. move /join_transitive => /[apply]. asimpl; rewrite subst_id => h2. tauto. Qed. - Lemma UnivInj n i j : R (Univ i : Tm n) (Univ j) -> i = j. - Proof. hauto l:on use:join_univ_inj. Qed. + Lemma BindCong p A0 A1 B0 B1 : + R A0 A1 -> + R B0 B1 -> + R (TBind p A0 B0) (TBind p A1 B1). + Proof. + move => h0 h1. rewrite /R /=. + apply join_pair_inj. + split. apply join_pair_inj. split. apply join_refl. done. + by apply Join.AbsCong. + Qed. - Lemma transitive n (a b c : Tm n) : + Lemma UnivInj i j : R (Univ i : Tm) (Univ j) -> i = j. + Proof. hauto l:on use:join_const_inj. Qed. + + Lemma transitive (a b c : Tm) : R a b -> R b c -> R a c. Proof. hauto l:on use:join_transitive unfold:R. Qed. - Lemma symmetric n (a b : Tm n) : + Lemma symmetric (a b : Tm) : R a b -> R b a. Proof. hauto l:on use:join_symmetric. Qed. - Lemma reflexive n (a : Tm n) : + Lemma reflexive (a : Tm) : R a a. Proof. hauto l:on use:join_refl. Qed. - Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) : + Lemma substing (a b : Tm) (ρ : nat -> Tm) : R a b -> R (subst_Tm ρ a) (subst_Tm ρ b). Proof. rewrite /R. @@ -95,92 +112,3 @@ Module Join. Qed. End Join. - -Module Equiv. - Inductive R {n} : Tm n -> Tm n -> Prop := - (***************** Beta ***********************) - | AppAbs a b : - R (App (Abs a) b) (subst_Tm (scons b VarTm) a) - | ProjPair p a b : - R (Proj p (Pair a b)) (if p is PL then a else b) - - (****************** Eta ***********************) - | AppEta a : - R a (Abs (App (ren_Tm shift a) (VarTm var_zero))) - | PairEta a : - R a (Pair (Proj PL a) (Proj PR a)) - - (*************** Congruence ********************) - | Var i : R (VarTm i) (VarTm i) - | AbsCong a b : - R a b -> - R (Abs a) (Abs b) - | AppCong a0 a1 b0 b1 : - R a0 a1 -> - R b0 b1 -> - R (App a0 b0) (App a1 b1) - | PairCong a0 a1 b0 b1 : - R a0 a1 -> - R b0 b1 -> - R (Pair a0 b0) (Pair a1 b1) - | ProjCong p a0 a1 : - R a0 a1 -> - R (Proj p a0) (Proj p a1) - | BindCong p A0 A1 B0 B1: - R A0 A1 -> - R B0 B1 -> - R (TBind p A0 B0) (TBind p A1 B1) - | UnivCong i : - R (Univ i) (Univ i). -End Equiv. - -Module EquivJoin. - Lemma FromEquiv n (a b : Tm n) : Equiv.R a b -> Join.R a b. - Proof. - move => h. elim : n a b /h => n. - - move => a b. - rewrite /Join.R /join /=. - eexists. split. apply relations.rtc_once. - apply Par.AppAbs; auto using Par.refl. - rewrite Compile.substing. - apply relations.rtc_refl. - - move => p a b. - apply Join.FromPar. - simpl. apply : Par.ProjPair'; auto using Par.refl. - case : p => //=. - - move => a. apply Join.FromPar => /=. - apply : Par.AppEta'; auto using Par.refl. - by autorewrite with compile. - - move => a. apply Join.FromPar => /=. - apply : Par.PairEta; auto using Par.refl. - - hauto l:on use:Join.FromPar, Par.Var. - - hauto lq:on use:Join.AbsCong. - - qauto l:on use:Join.AppCong. - - qauto l:on use:Join.PairCong. - - qauto use:Join.ProjCong. - - rewrite /Join.R => p A0 A1 B0 B1 _ hA _ hB /=. - sfirstorder use:Join.PairCong,Join.AbsCong,Join.FromPar,Par.ConstCong. - - hauto l:on. - Qed. -End EquivJoin. - -Lemma compile_rpar n (a b : Tm n) : RPar'.R a b -> RPar'.R (Compile.F a) (Compile.F b). -Proof. - move => h. elim : n a b /h. - - move => n a0 a1 b0 b1 ha iha hb ihb /=. - rewrite -Compile.substing. - apply RPar'.AppAbs => //. - - hauto q:on use:RPar'.ProjPair'. - - qauto ctrs:RPar'.R. - - hauto lq:on ctrs:RPar'.R. - - hauto lq:on ctrs:RPar'.R. - - hauto lq:on ctrs:RPar'.R. - - hauto lq:on ctrs:RPar'.R. - - hauto lq:on ctrs:RPar'.R. - - hauto lq:on ctrs:RPar'.R. - - hauto lq:on ctrs:RPar'.R. - - hauto lq:on ctrs:RPar'.R. -Qed. - -Lemma compile_rpars n (a b : Tm n) : rtc RPar'.R a b -> rtc RPar'.R (Compile.F a) (Compile.F b). -Proof. induction 1; hauto lq:on ctrs:rtc use:compile_rpar. Qed. diff --git a/theories/fp_red.v b/theories/fp_red.v index f9abc08..7bb5f2e 100644 --- a/theories/fp_red.v +++ b/theories/fp_red.v @@ -66,11 +66,7 @@ Module Par. R a0 a1 -> R (PProj p a0) (PProj p a1) | ConstCong k : - R (PConst k) (PConst k) - | Univ i : - R (PUniv i) (PUniv i) - | Bot : - R PBot PBot. + R (PConst k) (PConst k). Lemma refl (a : PTm) : R a a. elim : a; hauto ctrs:R. @@ -130,8 +126,6 @@ Module Par. - qauto l:on ctrs:R. - qauto l:on ctrs:R. - hauto l:on inv:option ctrs:R use:renaming. - - qauto l:on ctrs:R. - - qauto l:on ctrs:R. Qed. Lemma substing (a b : PTm) (ρ : nat -> PTm) : @@ -204,26 +198,24 @@ Module Par. eexists. split. by apply ProjCong; eauto. done. - hauto q:on inv:PTm ctrs:R. - - hauto q:on inv:PTm ctrs:R. - - hauto q:on inv:PTm ctrs:R. Qed. End Par. Module Pars. - Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) : + Lemma renaming (a b : PTm) (ξ : nat -> nat) : rtc Par.R a b -> rtc Par.R (ren_PTm ξ a) (ren_PTm ξ b). Proof. induction 1; hauto lq:on ctrs:rtc use:Par.renaming. Qed. - Lemma substing n m (a b : PTm n) (ρ : fin n -> PTm m) : + Lemma substing (a b : PTm) (ρ : nat -> PTm) : rtc Par.R a b -> rtc Par.R (subst_PTm ρ a) (subst_PTm ρ b). induction 1; hauto l:on ctrs:rtc use:Par.substing. Qed. - Lemma antirenaming n m (a : PTm n) (b : PTm m) (ξ : fin n -> fin m) : + Lemma antirenaming (a : PTm) (b : PTm) (ξ : nat -> nat) : rtc Par.R (ren_PTm ξ a) b -> exists b0, rtc Par.R a b0 /\ ren_PTm ξ b0 = b. Proof. move E :(ren_PTm ξ a) => u h. @@ -243,41 +235,33 @@ Module Pars. #[local]Ltac solve_s := repeat (induction 1; last by solve_s_rec); apply rtc_refl. - Lemma ProjCong n p (a0 a1 : PTm n) : + Lemma ProjCong p (a0 a1 : PTm) : rtc Par.R a0 a1 -> rtc Par.R (PProj p a0) (PProj p a1). Proof. solve_s. Qed. - Lemma PairCong n (a0 a1 b0 b1 : PTm n) : + Lemma PairCong (a0 a1 b0 b1 : PTm) : rtc Par.R a0 a1 -> rtc Par.R b0 b1 -> rtc Par.R (PPair a0 b0) (PPair a1 b1). Proof. solve_s. Qed. - Lemma AppCong n (a0 a1 b0 b1 : PTm n) : + Lemma AppCong (a0 a1 b0 b1 : PTm) : rtc Par.R a0 a1 -> rtc Par.R b0 b1 -> rtc Par.R (PApp a0 b0) (PApp a1 b1). Proof. solve_s. Qed. - Lemma AbsCong n (a b : PTm (S n)) : + Lemma AbsCong (a b : PTm) : rtc Par.R a b -> rtc Par.R (PAbs a) (PAbs b). Proof. solve_s. Qed. End Pars. -Definition var_or_const {n} (a : PTm n) := - match a with - | VarPTm _ => true - | PBot => true - | _ => false - end. - - (***************** Beta rules only ***********************) Module RPar. - Inductive R {n} : PTm n -> PTm n -> Prop := + Inductive R : PTm -> PTm -> Prop := (***************** Beta ***********************) | AppAbs a0 a1 b0 b1 : R a0 a1 -> @@ -314,64 +298,60 @@ Module RPar. R a0 a1 -> R (PProj p a0) (PProj p a1) | ConstCong k : - R (PConst k) (PConst k) - | Univ i : - R (PUniv i) (PUniv i) - | Bot : - R PBot PBot. + R (PConst k) (PConst k). - Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop. + Derive Dependent Inversion inv with (forall (a b : PTm), R a b) Sort Prop. - Lemma refl n (a : PTm n) : R a a. + Lemma refl (a : PTm) : R a a. Proof. induction a; hauto lq:on ctrs:R. Qed. - Lemma AppAbs' n a0 a1 (b0 b1 t : PTm n) : + Lemma AppAbs' a0 a1 (b0 b1 t : PTm) : t = subst_PTm (scons b1 VarPTm) a1 -> R a0 a1 -> R b0 b1 -> R (PApp (PAbs a0) b0) t. Proof. move => ->. apply AppAbs. Qed. - Lemma ProjPair' n p (a0 a1 b0 b1 : PTm n) t : + Lemma ProjPair' p (a0 a1 b0 b1 : PTm) t : t = (if p is PL then a1 else b1) -> R a0 a1 -> R b0 b1 -> R (PProj p (PPair a0 b0)) t. Proof. move => > ->. apply ProjPair. Qed. - Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) : + Lemma renaming (a b : PTm) (ξ : nat -> nat) : R a b -> R (ren_PTm ξ a) (ren_PTm ξ b). Proof. - move => h. move : m ξ. - elim : n a b /h. + move => h. move : ξ. + elim : a b /h. move => *; apply : AppAbs'; eauto; by asimpl. all : qauto ctrs:R use:ProjPair'. Qed. - Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> PTm m) (ξ : fin m -> fin p) : + Lemma morphing_ren (ρ0 ρ1 : nat -> PTm) (ξ : nat -> nat) : (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 : + Lemma morphing_ext (ρ0 ρ1 : nat -> PTm) 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:nat. Qed. - Lemma morphing_up n m (ρ0 ρ1 : fin n -> PTm m) : + Lemma morphing_up (ρ0 ρ1 : nat -> PTm) : (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) : + Lemma morphing (a b : PTm) (ρ0 ρ1 : nat -> PTm) : (forall i, R (ρ0 i) (ρ1 i)) -> R a b -> R (subst_PTm ρ0 a) (subst_PTm ρ1 b). Proof. - move => + h. move : m ρ0 ρ1. - elim : n a b /h. + move => + h. move : ρ0 ρ1. + elim : a b /h. - move => *. apply : AppAbs'; eauto using morphing_up. by asimpl. @@ -384,144 +364,80 @@ Module RPar. - hauto lq:on ctrs:R. - hauto lq:on ctrs:R. - hauto lq:on ctrs:R use:morphing_up. - - hauto lq:on ctrs:R. - - hauto lq:on ctrs:R. Qed. - Lemma substing n m (a b : PTm n) (ρ : fin n -> PTm m) : + Lemma substing (a b : PTm) (ρ : nat -> PTm) : R a b -> R (subst_PTm ρ a) (subst_PTm ρ b). Proof. hauto l:on use:morphing, refl. Qed. - Lemma cong n (a b : PTm (S n)) c d : + Lemma cong (a b : PTm) c d : R a b -> R c d -> R (subst_PTm (scons c VarPTm) a) (subst_PTm (scons d VarPTm) b). Proof. move => h0 h1. apply morphing => //=. - qauto l:on ctrs:R inv:nat. + destruct i as [|i]. + - done. + - simpl. apply Var. Qed. - Lemma var_or_const_imp {n} (a b : PTm n) : - var_or_const a -> - a = b -> ~~ var_or_const b -> False. - Proof. - hauto lq:on inv:PTm. - Qed. - Lemma var_or_const_up n m (ρ : fin n -> PTm m) : - (forall i, var_or_const (ρ i)) -> - (forall i, var_or_const (up_PTm_PTm ρ i)). - Proof. - move => h /= [i|]. - - asimpl. - move /(_ i) in h. - rewrite /funcomp. - move : (ρ i) h. - case => //=. - - sfirstorder. - 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 + | nat -> nat => (ltac1:(case;hauto q:on depth:2 ctrs:R)) + | nat -> PTm => (ltac1:(case;hauto q:on depth:2 ctrs:R)) + | _ => solve_anti_ren () + end. - Local Ltac antiimp := qauto l:on use:var_or_const_imp. + Ltac solve_anti_ren := ltac2:(Control.enter solve_anti_ren). - Lemma antirenaming n m (a : PTm n) (b : PTm m) (ρ : fin n -> PTm m) : - (forall i, var_or_const (ρ i)) -> - R (subst_PTm ρ a) b -> exists b0, R a b0 /\ subst_PTm ρ b0 = b. + Lemma antirenaming (a : PTm) (b : PTm) (ρ : nat -> nat) : + R (ren_PTm ρ a) b -> exists b0, R a b0 /\ ren_PTm ρ b0 = b. Proof. - move E : (subst_PTm ρ a) => u hρ h. - move : n ρ hρ a E. elim : m u b/h. - - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=; - first by antiimp. + move E : (ren_PTm ρ a) => u h. + move : ρ a E. elim : u b/h; try solve_anti_ren. + - move => a0 a1 b0 b1 ha iha hb ihb ρ []//=. move => c c0 [+ ?]. subst. - case : c => //=; first by antiimp. + case : c => //=. move => c [?]. subst. spec_refl. - have /var_or_const_up hρ' := hρ. - move : iha hρ' => /[apply] iha. - move : ihb hρ => /[apply] ihb. - spec_refl. move : iha => [c1][ih0]?. subst. move : ihb => [c2][ih1]?. subst. eexists. split. apply AppAbs; eauto. by asimpl. - - move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc m ρ hρ. - move => []//=; - first by antiimp. - move => []//=; first by antiimp. - move => t t0 t1 [*]. subst. - have {}/iha := hρ => iha. - have {}/ihb := hρ => ihb. - have {}/ihc := hρ => ihc. + - move => a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc ρ. + move => []//=. + move => []//=. + move => p p0 p1 [*]. subst. spec_refl. move : iha => [? [*]]. move : ihb => [? [*]]. - move : ihc => [? [*]]. + move : ihc => [? [*]]. subst. eexists. split. - apply AppPair; hauto. subst. + apply AppPair; hauto. by asimpl. - - move => n p a0 a1 ha iha m ρ hρ []//=; - first by antiimp. - move => p0 []//= t [*]; first by antiimp. subst. - have /var_or_const_up {}/iha := hρ => iha. + - move => p a0 a1 ha iha ρ []//=. + move => p0 []//= t [*]. subst. spec_refl. move : iha => [b0 [? ?]]. subst. eexists. split. apply ProjAbs; eauto. by asimpl. - - move => n p a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=; - first by antiimp. - move => p0 []//=; first by antiimp. move => t t0[*]. + - move => p a0 a1 b0 b1 ha iha hb ihb ρ []//=. + move => p0 []//=. move => t t0[*]. subst. - have {}/iha := (hρ) => iha. - have {}/ihb := (hρ) => ihb. spec_refl. move : iha => [b0 [? ?]]. move : ihb => [c0 [? ?]]. subst. eexists. split. by eauto using ProjPair. hauto q:on. - - move => n i m ρ hρ []//=. - hauto l:on. - - move => n a0 a1 ha iha m ρ hρ []//=; first by antiimp. - move => t [*]. subst. - have /var_or_const_up {}/iha := hρ => iha. - spec_refl. - move :iha => [b0 [? ?]]. subst. - eexists. split. by apply AbsCong; eauto. - by asimpl. - - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=; - first by antiimp. - move => t t0 [*]. subst. - have {}/iha := (hρ) => iha. - have {}/ihb := (hρ) => ihb. - spec_refl. - move : iha => [b0 [? ?]]. subst. - move : ihb => [c0 [? ?]]. subst. - eexists. split. by apply AppCong; eauto. - done. - - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=; - first by antiimp. - move => t t0[*]. subst. - have {}/iha := (hρ) => iha. - have {}/ihb := (hρ) => ihb. - spec_refl. - move : iha => [b0 [? ?]]. subst. - move : ihb => [c0 [? ?]]. subst. - eexists. split. by apply PairCong; eauto. - by asimpl. - - move => n p a0 a1 ha iha m ρ hρ []//=; - first by antiimp. - move => p0 t [*]. subst. - have {}/iha := (hρ) => iha. - spec_refl. - move : iha => [b0 [? ?]]. subst. - eexists. split. apply ProjCong; eauto. reflexivity. - - hauto q:on ctrs:R inv:PTm. - - hauto q:on ctrs:R inv:PTm. - - hauto q:on ctrs:R inv:PTm. Qed. End RPar. (***************** Beta rules only ***********************) Module RPar'. - Inductive R {n} : PTm n -> PTm n -> Prop := + Inductive R : PTm -> PTm -> Prop := (***************** Beta ***********************) | AppAbs a0 a1 b0 b1 : R a0 a1 -> @@ -550,64 +466,60 @@ Module RPar'. R a0 a1 -> R (PProj p a0) (PProj p a1) | ConstCong k : - R (PConst k) (PConst k) - | UnivCong i : - R (PUniv i) (PUniv i) - | BotCong : - R PBot PBot. + R (PConst k) (PConst k). - Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop. + Derive Inversion inv with (forall (a b : PTm), R a b) Sort Prop. - Lemma refl n (a : PTm n) : R a a. + Lemma refl (a : PTm) : R a a. Proof. induction a; hauto lq:on ctrs:R. Qed. - Lemma AppAbs' n a0 a1 (b0 b1 t : PTm n) : + Lemma AppAbs' a0 a1 (b0 b1 t : PTm) : t = subst_PTm (scons b1 VarPTm) a1 -> R a0 a1 -> R b0 b1 -> R (PApp (PAbs a0) b0) t. Proof. move => ->. apply AppAbs. Qed. - Lemma ProjPair' n p (a0 a1 b0 b1 : PTm n) t : + Lemma ProjPair' p (a0 a1 b0 b1 : PTm) t : t = (if p is PL then a1 else b1) -> R a0 a1 -> R b0 b1 -> R (PProj p (PPair a0 b0)) t. Proof. move => > ->. apply ProjPair. Qed. - Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) : + Lemma renaming (a b : PTm) (ξ : nat -> nat) : R a b -> R (ren_PTm ξ a) (ren_PTm ξ b). Proof. - move => h. move : m ξ. - elim : n a b /h. + move => h. move : ξ. + elim : a b /h. move => *; apply : AppAbs'; eauto; by asimpl. all : qauto ctrs:R use:ProjPair'. Qed. - Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> PTm m) (ξ : fin m -> fin p) : + Lemma morphing_ren (ρ0 ρ1 : nat -> PTm) (ξ : nat -> nat) : (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 : + Lemma morphing_ext (ρ0 ρ1 : nat -> PTm) 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:nat. Qed. - Lemma morphing_up n m (ρ0 ρ1 : fin n -> PTm m) : + Lemma morphing_up (ρ0 ρ1 : nat -> PTm) : (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) : + Lemma morphing (a b : PTm) (ρ0 ρ1 : nat -> PTm) : (forall i, R (ρ0 i) (ρ1 i)) -> R a b -> R (subst_PTm ρ0 a) (subst_PTm ρ1 b). Proof. - move => + h. move : m ρ0 ρ1. - elim : n a b /h. + move => + h. move : ρ0 ρ1. + elim : a b /h. - move => *. apply : AppAbs'; eauto using morphing_up. by asimpl. @@ -618,123 +530,60 @@ Module RPar'. - hauto lq:on ctrs:R. - hauto lq:on ctrs:R. - hauto l:on ctrs:R use:morphing_up. - - hauto lq:on ctrs:R. - - hauto lq:on ctrs:R. Qed. - Lemma substing n m (a b : PTm n) (ρ : fin n -> PTm m) : + Lemma substing (a b : PTm) (ρ : nat -> PTm) : R a b -> R (subst_PTm ρ a) (subst_PTm ρ b). Proof. hauto l:on use:morphing, refl. Qed. - Lemma cong n (a b : PTm (S n)) c d : + Lemma cong (a b : PTm) c d : R a b -> R c d -> R (subst_PTm (scons c VarPTm) a) (subst_PTm (scons d VarPTm) b). Proof. move => h0 h1. apply morphing => //=. - qauto l:on ctrs:R inv:nat. + hauto l:on ctrs:R inv:nat. Qed. - Lemma var_or_const_imp {n} (a b : PTm n) : - var_or_const a -> - a = b -> ~~ var_or_const b -> False. - Proof. - hauto lq:on inv:PTm. - 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 + | nat -> nat => (ltac1:(case;hauto q:on depth:2 ctrs:R)) + | nat -> PTm => (ltac1:(case;hauto q:on depth:2 ctrs:R)) + | _ => solve_anti_ren () + end. - Lemma var_or_const_up n m (ρ : fin n -> PTm m) : - (forall i, var_or_const (ρ i)) -> - (forall i, var_or_const (up_PTm_PTm ρ i)). - Proof. - move => h /= [i|]. - - asimpl. - move /(_ i) in h. - rewrite /funcomp. - move : (ρ i) h. - case => //=. - - sfirstorder. - Qed. + Ltac solve_anti_ren := ltac2:(Control.enter solve_anti_ren). - Local Ltac antiimp := qauto l:on use:var_or_const_imp. - - Lemma antirenaming n m (a : PTm n) (b : PTm m) (ρ : fin n -> PTm m) : - (forall i, var_or_const (ρ i)) -> - R (subst_PTm ρ a) b -> exists b0, R a b0 /\ subst_PTm ρ b0 = b. + Lemma antirenaming (a : PTm) (b : PTm) (ρ : nat -> nat) : + R (ren_PTm ρ a) b -> exists b0, R a b0 /\ ren_PTm ρ b0 = b. Proof. - move E : (subst_PTm ρ a) => u hρ h. - move : n ρ hρ a E. elim : m u b/h. - - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=; - first by antiimp. - move => c c0 [+ ?]. subst. - case : c => //=; first by antiimp. - move => c [?]. subst. - spec_refl. - have /var_or_const_up hρ' := hρ. - move : iha hρ' => /[apply] iha. - move : ihb hρ => /[apply] ihb. + move E : (ren_PTm ρ a) => u h. + move : ρ a E. elim : u b/h; try solve_anti_ren. + - move => a0 a1 b0 b1 ha iha hb ihb ρ []//=. + move => []//=. + move => p p0 [*]. subst. spec_refl. move : iha => [c1][ih0]?. subst. move : ihb => [c2][ih1]?. subst. eexists. split. apply AppAbs; eauto. by asimpl. - - move => n p a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=; - first by antiimp. - move => p0 []//=; first by antiimp. move => t t0[*]. + - move => p a0 a1 b0 b1 ha iha hb ihb ρ []//=. + move => p0 []//=. move => t t0[*]. subst. - have {}/iha := (hρ) => iha. - have {}/ihb := (hρ) => ihb. spec_refl. move : iha => [b0 [? ?]]. move : ihb => [c0 [? ?]]. subst. eexists. split. by eauto using ProjPair. hauto q:on. - - move => n i m ρ hρ []//=. - hauto l:on. - - move => n a0 a1 ha iha m ρ hρ []//=; first by antiimp. - move => t [*]. subst. - have /var_or_const_up {}/iha := hρ => iha. - spec_refl. - move :iha => [b0 [? ?]]. subst. - eexists. split. by apply AbsCong; eauto. - by asimpl. - - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=; - first by antiimp. - move => t t0 [*]. subst. - have {}/iha := (hρ) => iha. - have {}/ihb := (hρ) => ihb. - spec_refl. - move : iha => [b0 [? ?]]. subst. - move : ihb => [c0 [? ?]]. subst. - eexists. split. by apply AppCong; eauto. - done. - - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=; - first by antiimp. - move => t t0[*]. subst. - have {}/iha := (hρ) => iha. - have {}/ihb := (hρ) => ihb. - spec_refl. - move : iha => [b0 [? ?]]. subst. - move : ihb => [c0 [? ?]]. subst. - eexists. split. by apply PairCong; eauto. - by asimpl. - - move => n p a0 a1 ha iha m ρ hρ []//=; - first by antiimp. - move => p0 t [*]. subst. - have {}/iha := (hρ) => iha. - spec_refl. - move : iha => [b0 [? ?]]. subst. - eexists. split. apply ProjCong; eauto. reflexivity. - - hauto q:on ctrs:R inv:PTm. - - move => n i n0 ρ hρ []//=; first by antiimp. - hauto l:on. - - hauto q:on inv:PTm ctrs:R. Qed. End RPar'. Module ERed. - Inductive R {n} : PTm n -> PTm n -> Prop := + Inductive R : PTm -> PTm -> Prop := (****************** Eta ***********************) | AppEta a : R a (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero))) @@ -761,30 +610,30 @@ Module ERed. R a0 a1 -> R (PProj p a0) (PProj p a1). - Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop. + Derive Inversion inv with (forall (a b : PTm), R a b) Sort Prop. - Lemma AppEta' n a (u : PTm n) : + Lemma AppEta' a (u : PTm) : u = (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero))) -> R a u. Proof. move => ->. apply AppEta. Qed. - Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) : + Lemma renaming (a b : PTm) (ξ : nat -> nat) : R a b -> R (ren_PTm ξ a) (ren_PTm ξ b). Proof. - move => h. move : m ξ. - elim : n a b /h. + move => h. move : ξ. + elim : a b /h. - move => n a m ξ. + move => a ξ. apply AppEta'. by asimpl. all : qauto ctrs:R. Qed. - Lemma substing n m (a : PTm n) b (ρ : fin n -> PTm m) : + Lemma substing (a : PTm) b (ρ : nat -> PTm) : 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 /=. + move => h. move : ρ. elim : a b / h. + move => a ρ /=. apply : AppEta'; eauto. by asimpl. all : hauto ctrs:R inv:nat use:renaming. Qed. @@ -800,31 +649,31 @@ Module EReds. #[local]Ltac solve_s := repeat (induction 1; last by solve_s_rec); apply rtc_refl. - Lemma AbsCong n (a b : PTm (S n)) : + Lemma AbsCong (a b : PTm) : 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) : + Lemma AppCong (a0 a1 b0 b1 : PTm) : 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) : + Lemma PairCong (a0 a1 b0 b1 : PTm) : 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) : + Lemma ProjCong p (a0 a1 : PTm) : rtc ERed.R a0 a1 -> rtc ERed.R (PProj p a0) (PProj p a1). Proof. solve_s. Qed. End EReds. Module EPar. - Inductive R {n} : PTm n -> PTm n -> Prop := + Inductive R : PTm -> PTm -> Prop := (****************** Eta ***********************) | AppEta a0 a1 : R a0 a1 -> @@ -850,45 +699,41 @@ Module EPar. R a0 a1 -> R (PProj p a0) (PProj p a1) | ConstCong k : - R (PConst k) (PConst k) - | UnivCong i : - R (PUniv i) (PUniv i) - | BotCong : - R PBot PBot. + R (PConst k) (PConst k). - Lemma refl n (a : PTm n) : EPar.R a a. + Lemma refl (a : PTm) : EPar.R a a. Proof. induction a; hauto lq:on ctrs:EPar.R. Qed. - Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) : + Lemma renaming (a b : PTm) (ξ : nat -> nat) : R a b -> R (ren_PTm ξ a) (ren_PTm ξ b). Proof. - move => h. move : m ξ. - elim : n a b /h. + move => h. move : ξ. + elim : a b /h. - move => n a0 a1 ha iha m ξ /=. - move /(_ _ ξ) /AppEta : iha. + move => a0 a1 ha iha ξ /=. + move /(_ ξ) /AppEta : iha. by asimpl. all : qauto ctrs:R. Qed. - Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop. + Derive Inversion inv with (forall (a b : PTm), R a b) Sort Prop. - Lemma AppEta' n (a0 a1 b : PTm n) : + Lemma AppEta' (a0 a1 b : PTm) : b = (PAbs (PApp (ren_PTm shift a1) (VarPTm var_zero))) -> R a0 a1 -> R a0 b. Proof. move => ->; apply AppEta. Qed. - Lemma morphing n m (a b : PTm n) (ρ0 ρ1 : fin n -> PTm m) : + Lemma morphing (a b : PTm) (ρ0 ρ1 : nat -> PTm) : R a b -> (forall i, R (ρ0 i) (ρ1 i)) -> R (subst_PTm ρ0 a) (subst_PTm ρ1 b). Proof. - move => h. move : m ρ0 ρ1. elim : n a b / h => n. - - move => a0 a1 ha iha m ρ0 ρ1 hρ /=. + move => h. move : ρ0 ρ1. elim : a b / h. + - move => a0 a1 ha iha ρ0 ρ1 hρ /=. apply : AppEta'; eauto. by asimpl. - hauto lq:on ctrs:R. - hauto lq:on ctrs:R. @@ -897,11 +742,9 @@ Module EPar. - hauto q:on ctrs:R. - hauto q:on ctrs:R. - hauto l:on ctrs:R use:renaming inv:nat. - - hauto lq:on ctrs:R. - - hauto lq:on ctrs:R. Qed. - Lemma substing n a0 a1 (b0 b1 : PTm n) : + Lemma substing a0 a1 (b0 b1 : PTm) : R a0 a1 -> R b0 b1 -> R (subst_PTm (scons b0 VarPTm) a0) (subst_PTm (scons b1 VarPTm) a1). @@ -914,14 +757,14 @@ End EPar. Module OExp. - Inductive R {n} : PTm n -> PTm n -> Prop := + Inductive R : PTm -> PTm -> Prop := (****************** Eta ***********************) | AppEta a : R a (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero))) | PairEta a : R a (PPair (PProj PL a) (PProj PR a)). - Lemma merge n (t a b : PTm n) : + Lemma merge (t a b : PTm) : rtc R a b -> EPar.R t a -> EPar.R t b. @@ -931,7 +774,7 @@ Module OExp. - hauto q:on ctrs:EPar.R inv:R. Qed. - Lemma commutativity n (a b c : PTm n) : + Lemma commutativity (a b c : PTm) : EPar.R a b -> R a c -> exists d, R b d /\ EPar.R c d. Proof. move => h. @@ -940,7 +783,7 @@ Module OExp. - hauto lq:on ctrs:EPar.R, R. Qed. - Lemma commutativity0 n (a b c : PTm n) : + Lemma commutativity0 (a b c : PTm) : EPar.R a b -> rtc R a c -> exists d, rtc R b d /\ EPar.R c d. Proof. move => + h. move : b. @@ -965,29 +808,29 @@ Module RPars. #[local]Ltac solve_s := repeat (induction 1; last by solve_s_rec); apply rtc_refl. - Lemma AbsCong n (a b : PTm (S n)) : + Lemma AbsCong (a b : PTm) : rtc RPar.R a b -> rtc RPar.R (PAbs a) (PAbs b). Proof. solve_s. Qed. - Lemma AppCong n (a0 a1 b0 b1 : PTm n) : + Lemma AppCong (a0 a1 b0 b1 : PTm) : rtc RPar.R a0 a1 -> rtc RPar.R b0 b1 -> rtc RPar.R (PApp a0 b0) (PApp a1 b1). Proof. solve_s. Qed. - Lemma PairCong n (a0 a1 b0 b1 : PTm n) : + Lemma PairCong (a0 a1 b0 b1 : PTm) : rtc RPar.R a0 a1 -> rtc RPar.R b0 b1 -> rtc RPar.R (PPair a0 b0) (PPair a1 b1). Proof. solve_s. Qed. - Lemma ProjCong n p (a0 a1 : PTm n) : + Lemma ProjCong p (a0 a1 : PTm) : rtc RPar.R a0 a1 -> rtc RPar.R (PProj p a0) (PProj p a1). Proof. solve_s. Qed. - Lemma renaming n (a0 a1 : PTm n) m (ξ : fin n -> fin m) : + Lemma renaming (a0 a1 : PTm) (ξ : nat -> nat) : rtc RPar.R a0 a1 -> rtc RPar.R (ren_PTm ξ a0) (ren_PTm ξ a1). Proof. @@ -996,12 +839,12 @@ Module RPars. - eauto using RPar.renaming, rtc_l. Qed. - Lemma weakening n (a0 a1 : PTm n) : + Lemma weakening (a0 a1 : PTm) : rtc RPar.R a0 a1 -> rtc RPar.R (ren_PTm shift a0) (ren_PTm shift a1). Proof. apply renaming. Qed. - Lemma Abs_inv n (a : PTm (S n)) b : + Lemma Abs_inv (a : PTm) b : rtc RPar.R (PAbs a) b -> exists a', b = PAbs a' /\ rtc RPar.R a a'. Proof. move E : (PAbs a) => b0 h. move : a E. @@ -1010,27 +853,25 @@ Module RPars. - hauto lq:on ctrs:rtc inv:RPar.R, rtc. Qed. - Lemma morphing n m (a b : PTm n) (ρ : fin n -> PTm m) : + Lemma morphing (a b : PTm) (ρ : nat -> PTm) : rtc RPar.R a b -> rtc RPar.R (subst_PTm ρ a) (subst_PTm ρ b). Proof. induction 1; qauto l:on ctrs:rtc use:RPar.substing. Qed. - Lemma substing n (a b : PTm (S n)) c : + Lemma substing (a b : PTm) c : rtc RPar.R a b -> rtc RPar.R (subst_PTm (scons c VarPTm) a) (subst_PTm (scons c VarPTm) b). Proof. hauto lq:on use:morphing inv:nat. Qed. - Lemma antirenaming n m (a : PTm n) (b : PTm m) (ρ : fin n -> PTm m) : - (forall i, var_or_const (ρ i)) -> - rtc RPar.R (subst_PTm ρ a) b -> exists b0, rtc RPar.R a b0 /\ subst_PTm ρ b0 = b. + Lemma antirenaming (a : PTm) (b : PTm) (ρ : nat -> nat) : + rtc RPar.R (ren_PTm ρ a) b -> exists b0, rtc RPar.R a b0 /\ ren_PTm ρ b0 = b. Proof. - move E :(subst_PTm ρ a) => u hρ h. + move E :(ren_PTm ρ a) => u h. move : a E. elim : u b /h. - sfirstorder. - move => a b c h0 h1 ih1 a0 ?. subst. move /RPar.antirenaming : h0. - move /(_ hρ). move => [b0 [h2 ?]]. subst. hauto lq:on rew:off ctrs:rtc. Qed. @@ -1046,29 +887,29 @@ Module RPars'. #[local]Ltac solve_s := repeat (induction 1; last by solve_s_rec); apply rtc_refl. - Lemma AbsCong n (a b : PTm (S n)) : + Lemma AbsCong (a b : PTm) : rtc RPar'.R a b -> rtc RPar'.R (PAbs a) (PAbs b). Proof. solve_s. Qed. - Lemma AppCong n (a0 a1 b0 b1 : PTm n) : + Lemma AppCong (a0 a1 b0 b1 : PTm) : rtc RPar'.R a0 a1 -> rtc RPar'.R b0 b1 -> rtc RPar'.R (PApp a0 b0) (PApp a1 b1). Proof. solve_s. Qed. - Lemma PairCong n (a0 a1 b0 b1 : PTm n) : + Lemma PairCong (a0 a1 b0 b1 : PTm) : rtc RPar'.R a0 a1 -> rtc RPar'.R b0 b1 -> rtc RPar'.R (PPair a0 b0) (PPair a1 b1). Proof. solve_s. Qed. - Lemma ProjCong n p (a0 a1 : PTm n) : + Lemma ProjCong p (a0 a1 : PTm) : rtc RPar'.R a0 a1 -> rtc RPar'.R (PProj p a0) (PProj p a1). Proof. solve_s. Qed. - Lemma renaming n (a0 a1 : PTm n) m (ξ : fin n -> fin m) : + Lemma renaming (a0 a1 : PTm) (ξ : nat -> nat) : rtc RPar'.R a0 a1 -> rtc RPar'.R (ren_PTm ξ a0) (ren_PTm ξ a1). Proof. @@ -1077,12 +918,12 @@ Module RPars'. - eauto using RPar'.renaming, rtc_l. Qed. - Lemma weakening n (a0 a1 : PTm n) : + Lemma weakening (a0 a1 : PTm) : rtc RPar'.R a0 a1 -> rtc RPar'.R (ren_PTm shift a0) (ren_PTm shift a1). Proof. apply renaming. Qed. - Lemma Abs_inv n (a : PTm (S n)) b : + Lemma Abs_inv (a : PTm) b : rtc RPar'.R (PAbs a) b -> exists a', b = PAbs a' /\ rtc RPar'.R a a'. Proof. move E : (PAbs a) => b0 h. move : a E. @@ -1091,34 +932,41 @@ Module RPars'. - hauto lq:on ctrs:rtc inv:RPar'.R, rtc. Qed. - Lemma morphing n m (a b : PTm n) (ρ : fin n -> PTm m) : + Lemma morphing (a b : PTm) (ρ : nat -> PTm) : rtc RPar'.R a b -> rtc RPar'.R (subst_PTm ρ a) (subst_PTm ρ b). Proof. induction 1; qauto l:on ctrs:rtc use:RPar'.substing. Qed. - Lemma substing n (a b : PTm (S n)) c : + Lemma substing (a b : PTm ) c : rtc RPar'.R a b -> rtc RPar'.R (subst_PTm (scons c VarPTm) a) (subst_PTm (scons c VarPTm) b). Proof. hauto lq:on use:morphing inv:nat. Qed. - Lemma antirenaming n m (a : PTm n) (b : PTm m) (ρ : fin n -> PTm m) : - (forall i, var_or_const (ρ i)) -> - rtc RPar'.R (subst_PTm ρ a) b -> exists b0, rtc RPar'.R a b0 /\ subst_PTm ρ b0 = b. + Lemma antirenaming (a : PTm) (b : PTm) (ρ : nat -> nat) : + rtc RPar'.R (ren_PTm ρ a) b -> exists b0, rtc RPar'.R a b0 /\ ren_PTm ρ b0 = b. Proof. - move E :(subst_PTm ρ a) => u hρ h. + move E :(ren_PTm ρ a) => u h. move : a E. elim : u b /h. - sfirstorder. - move => a b c h0 h1 ih1 a0 ?. subst. move /RPar'.antirenaming : h0. - move /(_ hρ). move => [b0 [h2 ?]]. subst. hauto lq:on rew:off ctrs:rtc. Qed. End RPars'. -Lemma Abs_EPar n a (b : PTm n) : +Lemma subst_id: forall d : PTm, subst_PTm (scons (VarPTm 0) (funcomp VarPTm shift)) d = d. +Proof. + intros d. + have h : d = subst_PTm VarPTm d by asimpl. + rewrite {2}h. + apply ext_PTm. + destruct x => //=. +Qed. + +Lemma Abs_EPar a (b : PTm) : EPar.R (PAbs a) b -> (exists d, EPar.R a d /\ rtc RPar.R (PApp (ren_PTm shift b) (VarPTm var_zero)) d) /\ @@ -1128,8 +976,8 @@ Lemma Abs_EPar n a (b : PTm n) : Proof. move E : (PAbs a) => u h. move : a E. - elim : n u b /h => //=. - - move => n a0 a1 ha iha b ?. subst. + elim : u b /h => //=. + - move => a0 a1 ha iha b ?. subst. specialize iha with (1 := eq_refl). move : iha => [[d [ih0 ih1]] _]. split; exists d. @@ -1144,7 +992,7 @@ Proof. apply : RPar.ProjAbs. by apply RPar.refl. eauto using RPars.ProjCong, RPars.AbsCong. - - move => n ? a1 ha iha a0 ?. subst. specialize iha with (1 := eq_refl). + - move => ? a1 ha iha a0 ?. subst. specialize iha with (1 := eq_refl). move : iha => [_ [d [ih0 ih1]]]. split. + exists (PPair (PProj PL d) (PProj PR d)). @@ -1158,19 +1006,20 @@ Proof. by eauto using RPars.AppCong, rtc_refl. apply relations.rtc_once => /=. apply : RPar.AppAbs'; eauto using RPar.refl. - by asimpl. + simpl. f_equal. asimpl. clear. + by rewrite subst_id. + exists d. repeat split => //. move => p. apply : rtc_l; eauto. hauto q:on use:RPar.ProjPair', RPar.refl. - - move => n a0 a1 ha _ ? [*]. subst. + - move => a0 a1 ha _ ? [*]. subst. split. + exists a1. split => //. - apply rtc_once. apply : RPar.AppAbs'; eauto using RPar.refl. by asimpl. + apply rtc_once. apply : RPar.AppAbs'; eauto using RPar.refl. by asimpl; rewrite subst_id. + exists a1. split => // p. apply rtc_once. apply : RPar.ProjAbs; eauto using RPar.refl. Qed. -Lemma Pair_EPar n (a b c : PTm n) : +Lemma Pair_EPar (a b c : PTm) : EPar.R (PPair a b) c -> (forall p, exists d, rtc RPar.R (PProj p c) d /\ EPar.R (if p is PL then a else b) d) /\ (exists d0 d1, rtc RPar.R (PApp (ren_PTm shift c) (VarPTm var_zero)) @@ -1178,8 +1027,8 @@ Lemma Pair_EPar n (a b c : PTm n) : EPar.R a d0 /\ EPar.R b d1). Proof. move E : (PPair a b) => u h. move : a b E. - elim : n u c /h => //=. - - move => n a0 a1 ha iha a b ?. subst. + elim : u c /h => //=. + - move => a0 a1 ha iha a b ?. subst. specialize iha with (1 := eq_refl). move : iha => [_ [d0 [d1 [ih0 [ih1 ih2]]]]]. split. @@ -1196,7 +1045,7 @@ Proof. repeat split => //. apply : rtc_l. apply : RPar.AppAbs'; eauto using RPar.refl => //=. by asimpl; renamify. - - move => n a0 a1 ha iha a b ?. subst. specialize iha with (1 := eq_refl). + - move => a0 a1 ha iha a b ?. subst. specialize iha with (1 := eq_refl). split => [p|]. + move : iha => [/(_ p) [d [ih0 ih1]] _]. exists d. split=>//. @@ -1211,7 +1060,7 @@ Proof. repeat split => //=. apply : rtc_l. apply RPar.AppPair; eauto using RPar.refl. apply RPars.PairCong; apply RPars.AppCong; eauto using rtc_refl. - - move => n a0 a1 b0 b1 ha _ hb _ a b [*]. subst. + - move => a0 a1 b0 b1 ha _ hb _ a b [*]. subst. split. + move => p. exists (if p is PL then a1 else b1). @@ -1223,30 +1072,30 @@ Proof. split => //. Qed. -Lemma commutativity0 n (a b0 b1 : PTm n) : +Lemma commutativity0 (a b0 b1 : PTm) : EPar.R a b0 -> RPar.R a b1 -> exists c, rtc RPar.R b0 c /\ EPar.R b1 c. Proof. move => h. move : b1. - elim : n a b0 / h. - - move => n a b0 ha iha b1 hb. + elim : a b0 / h. + - move => a b0 ha iha b1 hb. move : iha (hb) => /[apply]. move => [c [ih0 ih1]]. exists (PAbs (PApp (ren_PTm shift c) (VarPTm var_zero))). split. + hauto lq:on ctrs:rtc use:RPars.AbsCong, RPars.AppCong, RPars.renaming. + hauto lq:on ctrs:EPar.R use:EPar.refl, EPar.renaming. - - move => n a b0 hb0 ihb0 b1 /[dup] hb1 {}/ihb0. + - move => a b0 hb0 ihb0 b1 /[dup] hb1 {}/ihb0. move => [c [ih0 ih1]]. exists (PPair (PProj PL c) (PProj PR c)). split. + apply RPars.PairCong; by apply RPars.ProjCong. + hauto lq:on ctrs:EPar.R use:EPar.refl, EPar.renaming. - hauto l:on ctrs:rtc inv:RPar.R. - - move => n a0 a1 h ih b1. + - move => a0 a1 h ih b1. elim /RPar.inv => //= _. move => a2 a3 ? [*]. subst. hauto lq:on ctrs:rtc, RPar.R, EPar.R use:RPars.AbsCong. - - move => n a0 a1 b0 b1 ha iha hb ihb b2. + - move => a0 a1 b0 b1 ha iha hb ihb b2. elim /RPar.inv => //= _. + move => a2 a3 b3 b4 h0 h1 [*]. subst. move /(_ _ ltac:(by eauto)) : ihb => [b [ihb0 ihb1]]. @@ -1274,7 +1123,7 @@ Proof. apply EPar.PairCong; by apply EPar.AppCong. + hauto lq:on ctrs:EPar.R use:RPars.AppCong. - hauto lq:on ctrs:EPar.R inv:RPar.R use:RPars.PairCong. - - move => n p a b0 h0 ih0 b1. + - move => p a b0 h0 ih0 b1. elim /RPar.inv => //= _. + move => ? a0 a1 h [*]. subst. move /(_ _ ltac:(by eauto using RPar.AbsCong)) : ih0 => [c [ih0 ih1]]. @@ -1288,11 +1137,9 @@ Proof. hauto lq:on use:RPars.ProjCong, relations.rtc_transitive. + hauto lq:on ctrs:EPar.R use:RPars.ProjCong. - hauto l:on ctrs:EPar.R inv:RPar.R. - - hauto l:on ctrs:EPar.R inv:RPar.R. - - hauto l:on ctrs:EPar.R inv:RPar.R. Qed. -Lemma commutativity1 n (a b0 b1 : PTm n) : +Lemma commutativity1 (a b0 b1 : PTm) : EPar.R a b0 -> rtc RPar.R a b1 -> exists c, rtc RPar.R b0 c /\ EPar.R b1 c. Proof. move => + h. move : b0. @@ -1301,7 +1148,7 @@ Proof. - qauto l:on use:relations.rtc_transitive, commutativity0. Qed. -Lemma commutativity n (a b0 b1 : PTm n) : +Lemma commutativity (a b0 b1 : PTm) : rtc EPar.R a b0 -> rtc RPar.R a b1 -> exists c, rtc RPar.R b0 c /\ rtc EPar.R b1 c. move => h. move : b1. elim : a b0 /h. - sfirstorder. @@ -1310,124 +1157,96 @@ Lemma commutativity n (a b0 b1 : PTm n) : hauto q:on ctrs:rtc. Qed. -Lemma Abs_EPar' n a (b : PTm n) : +Lemma Abs_EPar' a (b : PTm) : EPar.R (PAbs a) b -> (exists d, EPar.R a d /\ rtc OExp.R (PAbs d) b). Proof. move E : (PAbs a) => u h. move : a E. - elim : n u b /h => //=. - - move => n a0 a1 ha iha a ?. subst. + elim : u b /h => //=. + - move => a0 a1 ha iha a ?. subst. specialize iha with (1 := eq_refl). hauto lq:on ctrs:OExp.R use:rtc_r. - - move => n a0 a1 ha iha a ?. subst. + - move => a0 a1 ha iha a ?. subst. specialize iha with (1 := eq_refl). hauto lq:on ctrs:OExp.R use:rtc_r. - hauto l:on ctrs:OExp.R. Qed. -Lemma Proj_EPar' n p a (b : PTm n) : +Lemma Proj_EPar' p a (b : PTm) : EPar.R (PProj p a) b -> (exists d, EPar.R a d /\ rtc OExp.R (PProj p d) b). Proof. move E : (PProj p a) => u h. move : p a E. - elim : n u b /h => //=. - - move => n a0 a1 ha iha a p ?. subst. + elim : u b /h => //=. + - move => a0 a1 ha iha a p ?. subst. specialize iha with (1 := eq_refl). hauto lq:on ctrs:OExp.R use:rtc_r. - - move => n a0 a1 ha iha a p ?. subst. + - move => a0 a1 ha iha a p ?. subst. specialize iha with (1 := eq_refl). hauto lq:on ctrs:OExp.R use:rtc_r. - hauto l:on ctrs:OExp.R. Qed. -Lemma App_EPar' n (a b u : PTm n) : +Lemma App_EPar' (a b u : PTm) : EPar.R (PApp a b) u -> (exists a0 b0, EPar.R a a0 /\ EPar.R b b0 /\ rtc OExp.R (PApp a0 b0) u). Proof. move E : (PApp a b) => t h. - move : a b E. elim : n t u /h => //=. - - move => n a0 a1 ha iha a b ?. subst. + move : a b E. elim : t u /h => //=. + - move => a0 a1 ha iha a b ?. subst. specialize iha with (1 := eq_refl). hauto lq:on ctrs:OExp.R use:rtc_r. - - move => n a0 a1 ha iha a b ?. subst. + - move => a0 a1 ha iha a b ?. subst. specialize iha with (1 := eq_refl). hauto lq:on ctrs:OExp.R use:rtc_r. - hauto l:on ctrs:OExp.R. Qed. -Lemma Pair_EPar' n (a b u : PTm n) : +Lemma Pair_EPar' (a b u : PTm) : EPar.R (PPair a b) u -> exists a0 b0, EPar.R a a0 /\ EPar.R b b0 /\ rtc OExp.R (PPair a0 b0) u. Proof. move E : (PPair a b) => t h. - move : a b E. elim : n t u /h => //=. - - move => n a0 a1 ha iha a b ?. subst. + move : a b E. elim : t u /h => //=. + - move => a0 a1 ha iha a b ?. subst. specialize iha with (1 := eq_refl). hauto lq:on ctrs:OExp.R use:rtc_r. - - move => n a0 a1 ha iha a b ?. subst. + - move => a0 a1 ha iha a b ?. subst. specialize iha with (1 := eq_refl). hauto lq:on ctrs:OExp.R use:rtc_r. - hauto l:on ctrs:OExp.R. Qed. -Lemma Const_EPar' n k (u : PTm n) : +Lemma Const_EPar' k (u : PTm) : EPar.R (PConst k) u -> rtc OExp.R (PConst k) u. move E : (PConst k) => t h. - move : k E. elim : n t u /h => //=. - - move => n a0 a1 h ih k ?. subst. + move : k E. elim : t u /h => //=. + - move => a0 a1 h ih k ?. subst. specialize ih with (1 := eq_refl). hauto lq:on ctrs:OExp.R use:rtc_r. - - move => n a0 a1 h ih k ?. subst. + - move => a0 a1 h ih k ?. subst. specialize ih with (1 := eq_refl). hauto lq:on ctrs:OExp.R use:rtc_r. - hauto l:on ctrs:OExp.R. Qed. -Lemma Bot_EPar' n (u : PTm n) : - EPar.R (PBot) u -> - rtc OExp.R (PBot) u. - move E : (PBot) => t h. - move : E. elim : n t u /h => //=. - - move => n a0 a1 h ih ?. subst. - specialize ih with (1 := eq_refl). - hauto lq:on ctrs:OExp.R use:rtc_r. - - move => n a0 a1 h ih ?. subst. - specialize ih with (1 := eq_refl). - hauto lq:on ctrs:OExp.R use:rtc_r. - - hauto l:on ctrs:OExp.R. -Qed. - -Lemma Univ_EPar' n i (u : PTm n) : - EPar.R (PUniv i) u -> - rtc OExp.R (PUniv i) u. - move E : (PUniv i) => t h. - move : E. elim : n t u /h => //=. - - move => n a0 a1 h ih ?. subst. - specialize ih with (1 := eq_refl). - hauto lq:on ctrs:OExp.R use:rtc_r. - - move => n a0 a1 h ih ?. subst. - specialize ih with (1 := eq_refl). - hauto lq:on ctrs:OExp.R use:rtc_r. - - hauto l:on ctrs:OExp.R. -Qed. - -Lemma EPar_diamond n (c a1 b1 : PTm n) : +Lemma EPar_diamond (c a1 b1 : PTm) : EPar.R c a1 -> EPar.R c b1 -> exists d2, EPar.R a1 d2 /\ EPar.R b1 d2. Proof. - move => h. move : b1. elim : n c a1 / h. - - move => n c a1 ha iha b1 /iha [d2 [hd0 hd1]]. + move => h. move : b1. elim : c a1 / h. + - move => c a1 ha iha b1 /iha [d2 [hd0 hd1]]. exists(PAbs (PApp (ren_PTm shift d2) (VarPTm var_zero))). hauto lq:on ctrs:EPar.R use:EPar.renaming. - hauto lq:on rew:off ctrs:EPar.R. - hauto lq:on use:EPar.refl. - - move => n a0 a1 ha iha a2. + - move => a0 a1 ha iha a2. move /Abs_EPar' => [d [hd0 hd1]]. move : iha hd0; repeat move/[apply]. move => [d2 [h0 h1]]. @@ -1435,21 +1254,21 @@ Proof. move : OExp.commutativity0 hd1; repeat move/[apply]. move => [d1 [hd1 hd2]]. exists d1. hauto lq:on ctrs:EPar.R use:OExp.merge. - - move => n a0 a1 b0 b1 ha iha hb ihb c. + - move => a0 a1 b0 b1 ha iha hb ihb c. move /App_EPar' => [a2][b2][/iha [a3 h0]][/ihb [b3 h1]]h2 {iha ihb}. have : EPar.R (PApp a2 b2)(PApp a3 b3) by hauto l:on use:EPar.AppCong. move : OExp.commutativity0 h2; repeat move/[apply]. move => [d h]. exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge. - - move => n a0 a1 b0 b1 ha iha hb ihb c. + - move => a0 a1 b0 b1 ha iha hb ihb c. move /Pair_EPar' => [a2][b2][/iha [a3 h0]][/ihb [b3 h1]]h2 {iha ihb}. have : EPar.R (PPair a2 b2)(PPair a3 b3) by hauto l:on use:EPar.PairCong. move : OExp.commutativity0 h2; repeat move/[apply]. move => [d h]. exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge. - - move => n p a0 a1 ha iha b. + - move => p a0 a1 ha iha b. move /Proj_EPar' => [d [/iha [d2 h] h1]] {iha}. have : EPar.R (PProj p d) (PProj p d2) by hauto l:on use:EPar.ProjCong. @@ -1457,11 +1276,9 @@ Proof. move => [d1 h1]. exists d1. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge. - qauto use:Const_EPar', EPar.refl. - - qauto use:Univ_EPar', EPar.refl. - - qauto use:Bot_EPar', EPar.refl. Qed. -Function tstar {n} (a : PTm n) := +Function tstar (a : PTm) := match a with | VarPTm i => a | PAbs a => PAbs (tstar a) @@ -1474,13 +1291,11 @@ Function tstar {n} (a : PTm n) := | PProj p (PAbs a) => (PAbs (PProj p (tstar a))) | PProj p a => PProj p (tstar a) | PConst k => PConst k - | PUniv i => PUniv i - | PBot => PBot end. -Lemma RPar_triangle n (a : PTm n) : forall b, RPar.R a b -> RPar.R b (tstar a). +Lemma RPar_triangle (a : PTm) : forall b, RPar.R a b -> RPar.R b (tstar a). Proof. - apply tstar_ind => {n a}. + apply tstar_ind => {a}. - hauto lq:on inv:RPar.R ctrs:RPar.R. - hauto lq:on inv:RPar.R ctrs:RPar.R. - hauto lq:on use:RPar.cong, RPar.refl ctrs:RPar.R inv:RPar.R. @@ -1492,11 +1307,9 @@ Proof. - hauto lq:on inv:RPar.R ctrs:RPar.R. - hauto lq:on inv:RPar.R ctrs:RPar.R. - hauto lq:on inv:RPar.R ctrs:RPar.R. - - hauto lq:on inv:RPar.R ctrs:RPar.R. - - hauto lq:on inv:RPar.R ctrs:RPar.R. Qed. -Function tstar' {n} (a : PTm n) := +Function tstar' (a : PTm) := match a with | VarPTm i => a | PAbs a => PAbs (tstar' a) @@ -1506,13 +1319,11 @@ Function tstar' {n} (a : PTm n) := | PProj p (PPair a b) => if p is PL then (tstar' a) else (tstar' b) | PProj p a => PProj p (tstar' a) | PConst k => PConst k - | PUniv i => PUniv i - | PBot => PBot end. -Lemma RPar'_triangle n (a : PTm n) : forall b, RPar'.R a b -> RPar'.R b (tstar' a). +Lemma RPar'_triangle (a : PTm) : forall b, RPar'.R a b -> RPar'.R b (tstar' a). Proof. - apply tstar'_ind => {n a}. + apply tstar'_ind => {a}. - hauto lq:on inv:RPar'.R ctrs:RPar'.R. - hauto lq:on inv:RPar'.R ctrs:RPar'.R. - hauto lq:on use:RPar'.cong, RPar'.refl ctrs:RPar'.R inv:RPar'.R. @@ -1522,23 +1333,21 @@ Proof. - hauto drew:off inv:RPar'.R use:RPar'.refl, RPar'.ProjPair'. - hauto lq:on inv:RPar'.R ctrs:RPar'.R. - hauto lq:on inv:RPar'.R ctrs:RPar'.R. - - hauto lq:on inv:RPar'.R ctrs:RPar'.R. - - hauto lq:on inv:RPar'.R ctrs:RPar'.R. Qed. -Lemma RPar_diamond n (c a1 b1 : PTm n) : +Lemma RPar_diamond (c a1 b1 : PTm) : RPar.R c a1 -> RPar.R c b1 -> exists d2, RPar.R a1 d2 /\ RPar.R b1 d2. Proof. hauto l:on use:RPar_triangle. Qed. -Lemma RPar'_diamond n (c a1 b1 : PTm n) : +Lemma RPar'_diamond (c a1 b1 : PTm) : RPar'.R c a1 -> RPar'.R c b1 -> exists d2, RPar'.R a1 d2 /\ RPar'.R b1 d2. Proof. hauto l:on use:RPar'_triangle. Qed. -Lemma RPar_confluent n (c a1 b1 : PTm n) : +Lemma RPar_confluent (c a1 b1 : PTm) : rtc RPar.R c a1 -> rtc RPar.R c b1 -> exists d2, rtc RPar.R a1 d2 /\ rtc RPar.R b1 d2. @@ -1546,7 +1355,7 @@ Proof. sfirstorder use:relations.diamond_confluent, RPar_diamond. Qed. -Lemma EPar_confluent n (c a1 b1 : PTm n) : +Lemma EPar_confluent (c a1 b1 : PTm) : rtc EPar.R c a1 -> rtc EPar.R c b1 -> exists d2, rtc EPar.R a1 d2 /\ rtc EPar.R b1 d2. @@ -1554,7 +1363,7 @@ Proof. sfirstorder use:relations.diamond_confluent, EPar_diamond. Qed. -Inductive prov {n} : PTm n -> PTm n -> Prop := +Inductive prov : PTm -> PTm -> Prop := | P_Abs h a : (forall b, prov h (subst_PTm (scons b VarPTm) a)) -> prov h (PAbs a) @@ -1571,20 +1380,16 @@ Inductive prov {n} : PTm n -> PTm n -> Prop := | P_Const k : prov (PConst k) (PConst k) | P_Var i : - prov (VarPTm i) (VarPTm i) -| P_Univ i : - prov (PUniv i) (PUniv i) -| P_Bot : - prov PBot PBot. + prov (VarPTm i) (VarPTm i). -Lemma ERed_EPar n (a b : PTm n) : ERed.R a b -> EPar.R a b. +Lemma ERed_EPar (a b : PTm) : ERed.R a b -> EPar.R a b. Proof. induction 1; hauto lq:on ctrs:EPar.R use:EPar.refl. Qed. -Lemma EPar_ERed n (a b : PTm n) : EPar.R a b -> rtc ERed.R a b. +Lemma EPar_ERed (a b : PTm) : EPar.R a b -> rtc ERed.R a b. Proof. - move => h. elim : n a b /h. + move => h. elim : a b /h. - eauto using rtc_r, ERed.AppEta. - eauto using rtc_r, ERed.PairEta. - auto using rtc_refl. @@ -1593,31 +1398,29 @@ Proof. - eauto using EReds.PairCong. - eauto using EReds.ProjCong. - auto using rtc_refl. - - auto using rtc_refl. - - auto using rtc_refl. Qed. -Lemma EPar_Par n (a b : PTm n) : EPar.R a b -> Par.R a b. +Lemma EPar_Par (a b : PTm) : EPar.R a b -> Par.R a b. Proof. - move => h. elim : n a b /h; qauto ctrs:Par.R. + move => h. elim : a b /h; qauto ctrs:Par.R. Qed. -Lemma RPar_Par n (a b : PTm n) : RPar.R a b -> Par.R a b. +Lemma RPar_Par (a b : PTm) : RPar.R a b -> Par.R a b. Proof. - move => h. elim : n a b /h; hauto lq:on ctrs:Par.R. + move => h. elim : a b /h; hauto lq:on ctrs:Par.R. Qed. -Lemma rtc_idem n (R : PTm n -> PTm n -> Prop) (a b : PTm n) : rtc (rtc R) a b -> rtc R a b. +Lemma rtc_idem (R : PTm -> PTm -> Prop) (a b : PTm) : rtc (rtc R) a b -> rtc R a b. Proof. induction 1; hauto l:on use:@relations.rtc_transitive, @rtc_r. Qed. -Lemma EPars_EReds {n} (a b : PTm n) : rtc EPar.R a b <-> rtc ERed.R a b. +Lemma EPars_EReds (a b : PTm) : rtc EPar.R a b <-> rtc ERed.R a b. Proof. sfirstorder use:@relations.rtc_subrel, EPar_ERed, rtc_idem, ERed_EPar. Qed. -Lemma prov_rpar n (u : PTm n) a b : prov u a -> RPar.R a b -> prov u b. +Lemma prov_rpar (u : PTm) a b : prov u a -> RPar.R a b -> prov u b. Proof. move => h. move : b. @@ -1646,24 +1449,22 @@ Proof. + hauto lq:on ctrs:prov. - hauto lq:on ctrs:prov inv:RPar.R. - hauto l:on ctrs:RPar.R inv:RPar.R. - - hauto l:on ctrs:RPar.R inv:RPar.R. - - hauto l:on ctrs:RPar.R inv:RPar.R. Qed. -Lemma prov_lam n (u : PTm n) a : prov u a <-> prov u (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero))). +Lemma prov_lam (u : PTm) a : prov u a <-> prov u (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero))). Proof. split. move => h. constructor. move => b. asimpl. by constructor. inversion 1; subst. - specialize H2 with (b := PBot). + specialize H2 with (b := (VarPTm var_zero)). move : H2. asimpl. inversion 1; subst. done. Qed. -Lemma prov_pair n (u : PTm n) a : prov u a <-> prov u (PPair (PProj PL a) (PProj PR a)). +Lemma prov_pair (u : PTm) a : prov u a <-> prov u (PPair (PProj PL a) (PProj PR a)). Proof. hauto lq:on inv:prov ctrs:prov. Qed. -Lemma prov_ered n (u : PTm n) a b : prov u a -> ERed.R a b -> prov u b. +Lemma prov_ered (u : PTm) a b : prov u a -> ERed.R a b -> prov u b. Proof. move => h. move : b. @@ -1691,74 +1492,66 @@ Proof. - hauto lq:on inv:ERed.R, prov ctrs:prov. - hauto lq:on inv:ERed.R, prov ctrs:prov. - hauto lq:on inv:ERed.R, prov ctrs:prov. - - hauto lq:on inv:ERed.R, prov ctrs:prov. - - hauto lq:on inv:ERed.R, prov ctrs:prov. Qed. -Lemma prov_ereds n (u : PTm n) a b : prov u a -> rtc ERed.R a b -> prov u b. +Lemma prov_ereds (u : PTm) a b : prov u a -> rtc ERed.R a b -> prov u b. Proof. induction 2; sfirstorder use:prov_ered. Qed. -Fixpoint extract {n} (a : PTm n) : PTm n := +Fixpoint extract (a : PTm) : PTm := match a with - | PAbs a => subst_PTm (scons PBot VarPTm) (extract a) + | PAbs a => subst_PTm (scons (PConst 0) VarPTm) (extract a) | PApp a b => extract a | PPair a b => extract a | PProj p a => extract a | PConst k => PConst k | VarPTm i => VarPTm i - | PUniv i => PUniv i - | PBot => PBot end. -Lemma ren_extract n m (a : PTm n) (ξ : fin n -> fin m) : +Lemma ren_extract (a : PTm) (ξ : nat -> nat) : extract (ren_PTm ξ a) = ren_PTm ξ (extract a). Proof. - move : m ξ. elim : n/a. + move : ξ. elim : a. - sfirstorder. - - move => n a ih m ξ /=. + - move => a ih ξ /=. rewrite ih. by asimpl. - hauto q:on. - hauto q:on. - hauto q:on. - hauto q:on. - - sfirstorder. - - sfirstorder. Qed. -Lemma ren_morphing n m (a : PTm n) (ρ : fin n -> PTm m) : +Lemma ren_morphing (a : PTm) (ρ : nat -> PTm) : (forall i, ρ i = extract (ρ i)) -> extract (subst_PTm ρ a) = subst_PTm ρ (extract a). Proof. - move : m ρ. - elim : n /a => n //=. - move => a ha m ρ hi. + move : ρ. + elim : a => //=. + move => a ha ρ hi. rewrite ha. - - destruct i as [i|] => //. + - destruct i as [|i] => //. rewrite ren_extract. rewrite -hi. by asimpl. - by asimpl. Qed. -Lemma ren_subst_bot n (a : PTm (S n)) : - extract (subst_PTm (scons PBot VarPTm) a) = subst_PTm (scons PBot VarPTm) (extract a). +Lemma ren_subst_bot (a : PTm) : + extract (subst_PTm (scons (PConst 0) VarPTm) a) = subst_PTm (scons (PConst 0) VarPTm) (extract a). Proof. - apply ren_morphing. destruct i as [i|] => //=. + apply ren_morphing. destruct i => //=. Qed. -Definition prov_extract_spec {n} u (a : PTm n) := +Definition prov_extract_spec u (a : PTm) := match u with - | PUniv i => extract a = PUniv i | VarPTm i => extract a = VarPTm i | (PConst i) => extract a = (PConst i) - | PBot => extract a = PBot | _ => True end. -Lemma prov_extract n u (a : PTm n) : +Lemma prov_extract u (a : PTm) : prov u a -> prov_extract_spec u a. Proof. move => h. @@ -1766,42 +1559,35 @@ Proof. - move => h a ha ih. case : h ha ih => //=. + move => i ha ih. - move /(_ PBot) in ih. + move /(_ (PConst 0)) in ih. rewrite -ih. by rewrite ren_subst_bot. - + move => p _ /(_ PBot). - by rewrite ren_subst_bot. - + move => i h /(_ PBot). - by rewrite ren_subst_bot => ->. - + move /(_ PBot). - move => h /(_ PBot). + + move => p _ /(_ (PConst 0)). by rewrite ren_subst_bot. - hauto lq:on. - hauto lq:on. - hauto lq:on. - case => //=. - sfirstorder. - - sfirstorder. - - sfirstorder. Qed. Definition union {A : Type} (R0 R1 : A -> A -> Prop) a b := R0 a b \/ R1 a b. Module ERPar. - Definition R {n} (a b : PTm n) := union RPar.R EPar.R a b. - Lemma RPar {n} (a b : PTm n) : RPar.R a b -> R a b. + Definition R (a b : PTm) := union RPar.R EPar.R a b. + Lemma RPar (a b : PTm) : RPar.R a b -> R a b. Proof. sfirstorder. Qed. - Lemma EPar {n} (a b : PTm n) : EPar.R a b -> R a b. + Lemma EPar (a b : PTm) : EPar.R a b -> R a b. Proof. sfirstorder. Qed. - Lemma refl {n} ( a : PTm n) : ERPar.R a a. + Lemma refl ( a : PTm) : ERPar.R a a. Proof. sfirstorder use:RPar.refl, EPar.refl. Qed. - Lemma ProjCong n p (a0 a1 : PTm n) : + Lemma ProjCong p (a0 a1 : PTm) : R a0 a1 -> rtc R (PProj p a0) (PProj p a1). Proof. @@ -1816,7 +1602,7 @@ Module ERPar. by apply EPar.ProjCong. Qed. - Lemma AbsCong n (a0 a1 : PTm (S n)) : + Lemma AbsCong (a0 a1 : PTm) : R a0 a1 -> rtc R (PAbs a0) (PAbs a1). Proof. @@ -1831,7 +1617,7 @@ Module ERPar. by apply EPar.AbsCong. Qed. - Lemma AppCong n (a0 a1 b0 b1 : PTm n) : + Lemma AppCong (a0 a1 b0 b1 : PTm) : R a0 a1 -> R b0 b1 -> rtc R (PApp a0 b0) (PApp a1 b1). @@ -1851,7 +1637,7 @@ Module ERPar. - sfirstorder use:EPar.AppCong, @rtc_once. Qed. - Lemma PairCong n (a0 a1 b0 b1 : PTm n) : + Lemma PairCong (a0 a1 b0 b1 : PTm) : R a0 a1 -> R b0 b1 -> rtc R (PPair a0 b0) (PPair a1 b1). @@ -1871,7 +1657,7 @@ Module ERPar. - sfirstorder use:EPar.PairCong, @rtc_once. Qed. - Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) : + Lemma renaming (a b : PTm) (ξ : nat -> nat) : R a b -> R (ren_PTm ξ a) (ren_PTm ξ b). Proof. sfirstorder use:EPar.renaming, RPar.renaming. @@ -1888,29 +1674,29 @@ Module ERPars. #[local]Ltac solve_s := repeat (induction 1; last by solve_s_rec); apply rtc_refl. - Lemma AppCong n (a0 a1 b0 b1 : PTm n) : + Lemma AppCong (a0 a1 b0 b1 : PTm) : rtc ERPar.R a0 a1 -> rtc ERPar.R b0 b1 -> rtc ERPar.R (PApp a0 b0) (PApp a1 b1). Proof. solve_s. Qed. - Lemma AbsCong n (a0 a1 : PTm (S n)) : + Lemma AbsCong (a0 a1 : PTm) : rtc ERPar.R a0 a1 -> rtc ERPar.R (PAbs a0) (PAbs a1). Proof. solve_s. Qed. - Lemma PairCong n (a0 a1 b0 b1 : PTm n) : + Lemma PairCong (a0 a1 b0 b1 : PTm) : rtc ERPar.R a0 a1 -> rtc ERPar.R b0 b1 -> rtc ERPar.R (PPair a0 b0) (PPair a1 b1). Proof. solve_s. Qed. - Lemma ProjCong n p (a0 a1 : PTm n) : + Lemma ProjCong p (a0 a1 : PTm) : rtc ERPar.R a0 a1 -> rtc ERPar.R (PProj p a0) (PProj p a1). Proof. solve_s. Qed. - Lemma renaming n (a0 a1 : PTm n) m (ξ : fin n -> fin m) : + Lemma renaming (a0 a1 : PTm) (ξ : nat -> nat) : rtc ERPar.R a0 a1 -> rtc ERPar.R (ren_PTm ξ a0) (ren_PTm ξ a1). Proof. @@ -1921,35 +1707,35 @@ Module ERPars. End ERPars. -Lemma ERPar_Par n (a b : PTm n) : ERPar.R a b -> Par.R a b. +Lemma ERPar_Par (a b : PTm) : ERPar.R a b -> Par.R a b. Proof. sfirstorder use:EPar_Par, RPar_Par. Qed. -Lemma Par_ERPar n (a b : PTm n) : Par.R a b -> rtc ERPar.R a b. +Lemma Par_ERPar (a b : PTm) : Par.R a b -> rtc ERPar.R a b. Proof. - move => h. elim : n a b /h. - - move => n a0 a1 b0 b1 ha iha hb ihb. + move => h. elim : a b /h. + - move => a0 a1 b0 b1 ha iha hb ihb. suff ? : rtc ERPar.R (PApp (PAbs a0) b0) (PApp (PAbs a1) b1). apply : relations.rtc_transitive; eauto. apply rtc_once. apply ERPar.RPar. by apply RPar.AppAbs; eauto using RPar.refl. eauto using ERPars.AppCong,ERPars.AbsCong. - - move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc. + - move => a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc. apply : rtc_l. apply ERPar.RPar. apply RPar.AppPair; eauto using RPar.refl. sfirstorder use:ERPars.AppCong, ERPars.PairCong. - - move => n p a0 a1 ha iha. + - move => p a0 a1 ha iha. apply : rtc_l. apply ERPar.RPar. apply RPar.ProjAbs; eauto using RPar.refl. sfirstorder use:ERPars.AbsCong, ERPars.ProjCong. - - move => n p a0 a1 b0 b1 ha iha hb ihb. + - move => p a0 a1 b0 b1 ha iha hb ihb. apply : rtc_l. apply ERPar.RPar. apply RPar.ProjPair; eauto using RPar.refl. hauto lq:on. - - move => n a0 a1 ha iha. + - move => a0 a1 ha iha. apply : rtc_l. apply ERPar.EPar. apply EPar.AppEta; eauto using EPar.refl. hauto lq:on ctrs:rtc use:ERPars.AppCong, ERPars.AbsCong, ERPars.renaming. - - move => n a0 a1 ha iha. + - move => a0 a1 ha iha. apply : rtc_l. apply ERPar.EPar. apply EPar.PairEta; eauto using EPar.refl. sfirstorder use:ERPars.PairCong, ERPars.ProjCong. - sfirstorder. @@ -1958,48 +1744,46 @@ Proof. - sfirstorder use:ERPars.PairCong. - sfirstorder use:ERPars.ProjCong. - sfirstorder. - - sfirstorder. - - sfirstorder. Qed. -Lemma Pars_ERPar n (a b : PTm n) : rtc Par.R a b -> rtc ERPar.R a b. +Lemma Pars_ERPar (a b : PTm) : rtc Par.R a b -> rtc ERPar.R a b. Proof. induction 1; hauto l:on use:Par_ERPar, @relations.rtc_transitive. Qed. -Lemma Par_ERPar_iff n (a b : PTm n) : rtc Par.R a b <-> rtc ERPar.R a b. +Lemma Par_ERPar_iff (a b : PTm) : rtc Par.R a b <-> rtc ERPar.R a b. Proof. split. sfirstorder use:Pars_ERPar, @relations.rtc_subrel. sfirstorder use:ERPar_Par, @relations.rtc_subrel. Qed. -Lemma RPar_ERPar n (a b : PTm n) : rtc RPar.R a b -> rtc ERPar.R a b. +Lemma RPar_ERPar (a b : PTm) : rtc RPar.R a b -> rtc ERPar.R a b. Proof. sfirstorder use:@relations.rtc_subrel. Qed. -Lemma EPar_ERPar n (a b : PTm n) : rtc EPar.R a b -> rtc ERPar.R a b. +Lemma EPar_ERPar (a b : PTm) : rtc EPar.R a b -> rtc ERPar.R a b. Proof. sfirstorder use:@relations.rtc_subrel. Qed. Module Type HindleyRosen. - Parameter A : nat -> Type. - Parameter R0 R1 : forall n, A n -> A n -> Prop. - Axiom diamond_R0 : forall n, relations.diamond (R0 n). - Axiom diamond_R1 : forall n, relations.diamond (R1 n). - Axiom commutativity : forall n, - forall a b c, R0 n a b -> R1 n a c -> exists d, R1 n b d /\ R0 n c d. + Parameter A : Type. + Parameter R0 R1 : A -> A -> Prop. + Axiom diamond_R0 : relations.diamond R0. + Axiom diamond_R1 : relations.diamond R1. + Axiom commutativity : + forall a b c, R0 a b -> R1 a c -> exists d, R1 b d /\ R0 c d. End HindleyRosen. Module HindleyRosenFacts (M : HindleyRosen). Import M. Lemma R0_comm : - forall n a b c, R0 n a b -> rtc (union (R0 n) (R1 n)) a c -> - exists d, rtc (union (R0 n) (R1 n)) b d /\ R0 n c d. + forall a b c, R0 a b -> rtc (union R0 R1) a c -> + exists d, rtc (union R0 R1) b d /\ R0 c d. Proof. - move => n a + c + h. + move => a + c + h. elim : a c /h. - sfirstorder. - move => a0 a1 a2 ha ha0 ih b h. @@ -2011,10 +1795,10 @@ Module HindleyRosenFacts (M : HindleyRosen). Qed. Lemma R1_comm : - forall n a b c, R1 n a b -> rtc (union (R0 n) (R1 n)) a c -> - exists d, rtc (union (R0 n) (R1 n)) b d /\ R1 n c d. + forall a b c, R1 a b -> rtc (union (R0) (R1)) a c -> + exists d, rtc (union (R0) (R1)) b d /\ R1 c d. Proof. - move => n a + c + h. + move => a + c + h. elim : a c /h. - sfirstorder. - move => a0 a1 a2 ha ha0 ih b h. @@ -2026,17 +1810,17 @@ Module HindleyRosenFacts (M : HindleyRosen). Qed. Lemma U_comm : - forall n a b c, (union (R0 n) (R1 n)) a b -> rtc (union (R0 n) (R1 n)) a c -> - exists d, rtc (union (R0 n) (R1 n)) b d /\ (union (R0 n) (R1 n)) c d. + forall a b c, (union R0 R1) a b -> rtc (union R0 R1) a c -> + exists d, rtc (union R0 R1) b d /\ (union R0 R1) c d. Proof. hauto lq:on use:R0_comm, R1_comm. Qed. Lemma U_comms : - forall n a b c, rtc (union (R0 n) (R1 n)) a b -> rtc (union (R0 n) (R1 n)) a c -> - exists d, rtc (union (R0 n) (R1 n)) b d /\ rtc (union (R0 n) (R1 n)) c d. + forall a b c, rtc (union R0 R1) a b -> rtc (union R0 R1) a c -> + exists d, rtc (union R0 R1) b d /\ rtc (union R0 R1) c d. Proof. - move => n a b + h. + move => a b + h. elim : a b /h. - sfirstorder. - hecrush ctrs:rtc use:U_comm. @@ -2046,16 +1830,15 @@ End HindleyRosenFacts. Module HindleyRosenER <: HindleyRosen. Definition A := PTm. - Definition R0 n := rtc (@RPar.R n). - Definition R1 n := rtc (@EPar.R n). - Lemma diamond_R0 : forall n, relations.diamond (R0 n). + Definition R0 := rtc (@RPar.R ). + Definition R1 := rtc (@EPar.R ). + Lemma diamond_R0 : relations.diamond (R0). sfirstorder use:RPar_confluent. Qed. - Lemma diamond_R1 : forall n, relations.diamond (R1 n). + Lemma diamond_R1 : relations.diamond (R1). sfirstorder use:EPar_confluent. Qed. - Lemma commutativity : forall n, - forall a b c, R0 n a b -> R1 n a c -> exists d, R1 n b d /\ R0 n c d. + Lemma commutativity : forall a b c, R0 a b -> R1 a c -> exists d, R1 b d /\ R0 c d. Proof. hauto l:on use:commutativity. Qed. @@ -2063,7 +1846,7 @@ End HindleyRosenER. Module ERFacts := HindleyRosenFacts HindleyRosenER. -Lemma rtc_union n (a b : PTm n) : +Lemma rtc_union (a b : PTm) : rtc (union RPar.R EPar.R) a b <-> rtc (union (rtc RPar.R) (rtc EPar.R)) a b. Proof. @@ -2085,7 +1868,7 @@ Proof. sfirstorder. Qed. -Lemma prov_erpar n (u : PTm n) a b : prov u a -> ERPar.R a b -> prov u b. +Lemma prov_erpar (u : PTm) a b : prov u a -> ERPar.R a b -> prov u b. Proof. move => h []. - sfirstorder use:prov_rpar. @@ -2093,7 +1876,7 @@ Proof. sfirstorder use:prov_ereds. Qed. -Lemma prov_pars n (u : PTm n) a b : prov u a -> rtc Par.R a b -> prov u b. +Lemma prov_pars (u : PTm) a b : prov u a -> rtc Par.R a b -> prov u b. Proof. move => h /Pars_ERPar. move => h0. @@ -2103,51 +1886,41 @@ Proof. - hauto lq:on use:prov_erpar. Qed. -Lemma Par_confluent n (a b c : PTm n) : +Lemma Par_confluent (a b c : PTm) : rtc Par.R a b -> rtc Par.R a c -> exists d, rtc Par.R b d /\ rtc Par.R c d. Proof. - move : n a b c. - suff : forall (n : nat) (a b c : PTm n), + move : a b c. + suff : forall (a b c : PTm), rtc ERPar.R a b -> - rtc ERPar.R a c -> exists d : PTm n, rtc ERPar.R b d /\ rtc ERPar.R c d. - move => h n a b c h0 h1. + rtc ERPar.R a c -> exists d : PTm, rtc ERPar.R b d /\ rtc ERPar.R c d. + move => h a b c h0 h1. apply Par_ERPar_iff in h0, h1. move : h h0 h1; repeat move/[apply]. hauto lq:on use:Par_ERPar_iff. have h := ERFacts.U_comms. - move => n a b c. + move => a b c. rewrite /HindleyRosenER.R0 /HindleyRosenER.R1 in h. - specialize h with (n := n). rewrite /HindleyRosenER.A in h. rewrite /ERPar.R. - have eq : (fun a0 b0 : PTm n => union RPar.R EPar.R a0 b0) = union RPar.R EPar.R by reflexivity. + have eq : (fun a0 b0 : PTm => union RPar.R EPar.R a0 b0) = union RPar.R EPar.R by reflexivity. rewrite !{}eq. move /rtc_union => + /rtc_union. move : h; repeat move/[apply]. hauto lq:on use:rtc_union. Qed. -Lemma pars_univ_inv n i (c : PTm n) : - rtc Par.R (PUniv i) c -> - extract c = PUniv i. -Proof. - have : prov (PUniv i) (PUniv i : PTm n) by sfirstorder. - move : prov_pars. repeat move/[apply]. - apply prov_extract. -Qed. - -Lemma pars_const_inv n i (c : PTm n) : +Lemma pars_const_inv i (c : PTm) : rtc Par.R (PConst i) c -> extract c = PConst i. Proof. - have : prov (PConst i) (PConst i : PTm n) by sfirstorder. + have : prov (PConst i) (PConst i : PTm) by sfirstorder. move : prov_pars. repeat move/[apply]. apply prov_extract. Qed. -Lemma pars_var_inv n (i : fin n) C : +Lemma pars_var_inv (i : nat) C : rtc Par.R (VarPTm i) C -> extract C = VarPTm i. Proof. @@ -2156,15 +1929,7 @@ Proof. apply prov_extract. Qed. -Lemma pars_univ_inj n i j (C : PTm n) : - rtc Par.R (PUniv i) C -> - rtc Par.R (PUniv j) C -> - i = j. -Proof. - sauto l:on use:pars_univ_inv. -Qed. - -Lemma pars_const_inj n i j (C : PTm n) : +Lemma pars_const_inj i j (C : PTm) : rtc Par.R (PConst i) C -> rtc Par.R (PConst j) C -> i = j. @@ -2172,10 +1937,10 @@ Proof. sauto l:on use:pars_const_inv. Qed. -Definition join {n} (a b : PTm n) := +Definition join (a b : PTm) := exists c, rtc Par.R a c /\ rtc Par.R b c. -Lemma join_transitive n (a b c : PTm n) : +Lemma join_transitive (a b c : PTm) : join a b -> join b c -> join a c. Proof. rewrite /join. @@ -2185,100 +1950,90 @@ Proof. eauto using relations.rtc_transitive. Qed. -Lemma join_symmetric n (a b : PTm n) : +Lemma join_symmetric (a b : PTm) : join a b -> join b a. Proof. sfirstorder unfold:join. Qed. -Lemma join_refl n (a : PTm n) : join a a. +Lemma join_refl (a : PTm) : join a a. Proof. hauto lq:on ctrs:rtc unfold:join. Qed. -Lemma join_univ_inj n i j : - join (PUniv i : PTm n) (PUniv j) -> i = j. -Proof. - sfirstorder use:pars_univ_inj. -Qed. - -Lemma join_const_inj n i j : - join (PConst i : PTm n) (PConst j) -> i = j. +Lemma join_const_inj i j : + join (PConst i : PTm) (PConst j) -> i = j. Proof. sfirstorder use:pars_const_inj. Qed. -Lemma join_substing n m (a b : PTm n) (ρ : fin n -> PTm m) : +Lemma join_substing (a b : PTm) (ρ : nat -> PTm) : join a b -> join (subst_PTm ρ a) (subst_PTm ρ b). Proof. hauto lq:on unfold:join use:Pars.substing. Qed. -Fixpoint ne {n} (a : PTm n) := +Fixpoint ne (a : PTm) := match a with | VarPTm i => true | PApp a b => ne a && nf b | PAbs a => false - | PUniv _ => false | PProj _ a => ne a | PPair _ _ => false | PConst _ => false - | PBot => true end -with nf {n} (a : PTm n) := +with nf (a : PTm) := match a with | VarPTm i => true | PApp a b => ne a && nf b | PAbs a => nf a - | PUniv _ => true | PProj _ a => ne a | PPair a b => nf a && nf b | PConst _ => true - | PBot => true end. -Lemma ne_nf n a : @ne n a -> nf a. +Lemma ne_nf a : ne a -> nf a. Proof. elim : a => //=. Qed. -Definition wn {n} (a : PTm n) := exists b, rtc RPar'.R a b /\ nf b. -Definition wne {n} (a : PTm n) := exists b, rtc RPar'.R a b /\ ne b. +Definition wn (a : PTm) := exists b, rtc RPar'.R a b /\ nf b. +Definition wne (a : PTm) := exists b, rtc RPar'.R a b /\ ne b. (* Weakly neutral implies weakly normal *) -Lemma wne_wn n a : @wne n a -> wn a. +Lemma wne_wn a : @wne a -> wn a. Proof. sfirstorder use:ne_nf. Qed. (* Normal implies weakly normal *) -Lemma nf_wn n v : @nf n v -> wn v. +Lemma nf_wn v : @nf v -> wn v. Proof. sfirstorder ctrs:rtc. Qed. -Lemma nf_refl n (a b : PTm n) (h : RPar'.R a b) : (nf a -> b = a) /\ (ne a -> b = a). +Lemma nf_refl (a b : PTm) (h : RPar'.R a b) : (nf a -> b = a) /\ (ne a -> b = a). Proof. elim : a b /h => //=; solve [hauto b:on]. Qed. -Lemma ne_nf_ren n m (a : PTm n) (ξ : fin n -> fin m) : +Lemma ne_nf_ren (a : PTm) (ξ : nat -> nat) : (ne a <-> ne (ren_PTm ξ a)) /\ (nf a <-> nf (ren_PTm ξ a)). Proof. - move : m ξ. elim : n / a => //=; solve [hauto b:on]. + move : ξ. elim : a => //=; solve [hauto b:on]. Qed. -Lemma wne_app n (a b : PTm n) : +Lemma wne_app (a b : PTm) : wne a -> wn b -> wne (PApp a b). Proof. move => [a0 [? ?]] [b0 [? ?]]. exists (PApp a0 b0). hauto b:on drew:off use:RPars'.AppCong. Qed. -Lemma wn_abs n a (h : wn a) : @wn n (PAbs a). +Lemma wn_abs a (h : wn a) : @wn (PAbs a). Proof. move : h => [v [? ?]]. exists (PAbs v). eauto using RPars'.AbsCong. Qed. -Lemma wn_pair n (a b : PTm n) : wn a -> wn b -> wn (PPair a b). +Lemma wn_pair (a b : PTm) : wn a -> wn b -> wn (PPair a b). Proof. move => [a0 [? ?]] [b0 [? ?]]. exists (PPair a0 b0). hauto lqb:on use:RPars'.PairCong. Qed. -Lemma wne_proj n p (a : PTm n) : wne a -> wne (PProj p a). +Lemma wne_proj p (a : PTm) : wne a -> wne (PProj p a). Proof. move => [a0 [? ?]]. exists (PProj p a0). hauto lqb:on use:RPars'.ProjCong. @@ -2287,31 +2042,30 @@ Qed. Create HintDb nfne. #[export]Hint Resolve nf_wn ne_nf wne_wn nf_refl : nfne. -Lemma ne_nf_antiren n m (a : PTm n) (ρ : fin n -> PTm m) : - (forall i, var_or_const (ρ i)) -> - (ne (subst_PTm ρ a) -> ne a) /\ (nf (subst_PTm ρ a) -> nf a). +Lemma ne_nf_antiren (a : PTm) (ρ : nat -> nat) : + (ne (ren_PTm ρ a) -> ne a) /\ (nf (ren_PTm ρ a) -> nf a). Proof. - move : m ρ. elim : n / a => //; - hauto b:on drew:off use:RPar.var_or_const_up. + move : ρ. elim : a => //; + hauto b:on drew:off . Qed. -Lemma wn_antirenaming n m a (ρ : fin n -> PTm m) : - (forall i, var_or_const (ρ i)) -> - wn (subst_PTm ρ a) -> wn a. +Lemma wn_antirenaming a (ρ : nat -> nat) : + wn (ren_PTm ρ a) -> wn a. Proof. - rewrite /wn => hρ. + rewrite /wn. move => [v [rv nfv]]. move /RPars'.antirenaming : rv. - move /(_ hρ) => [b [hb ?]]. subst. + move => [b [hb ?]]. subst. exists b. split => //=. move : nfv. by eapply ne_nf_antiren. Qed. -Lemma ext_wn n (a : PTm n) : - wn (PApp a PBot) -> +Lemma ext_wn (a : PTm) : + wn (PApp a (VarPTm var_zero)) -> wn a. Proof. + set PBot := VarPTm var_zero. move E : (PApp a (PBot)) => a0 [v [hr hv]]. move : a E. move : hv. @@ -2325,49 +2079,51 @@ Proof. + move => a0 a3 b2 b3 ? ? [? ?] ? [? ?]. subst. have ? : b3 = (PBot) by hauto lq:on inv:RPar'.R. subst. suff : wn (PAbs a3) by hauto lq:on ctrs:RPar'.R, rtc unfold:wn. - have : wn (subst_PTm (scons (PBot) VarPTm) a3) by sfirstorder. + have : wn (subst_PTm (scons (VarPTm var_zero) VarPTm) a3) by sfirstorder. + asimpl. move => h. apply wn_abs. - move : h. apply wn_antirenaming. - hauto lq:on rew:off inv:nat. + move : h. + have -> : subst_PTm (scons (VarPTm var_zero) VarPTm) a3 = ren_PTm (scons var_zero id) a3 by substify; asimpl. + apply wn_antirenaming. + hauto q:on inv:RPar'.R ctrs:rtc b:on. Qed. Module Join. - Lemma ProjCong p n (a0 a1 : PTm n) : + Lemma ProjCong p (a0 a1 : PTm) : join a0 a1 -> join (PProj p a0) (PProj p a1). Proof. hauto lq:on use:Pars.ProjCong unfold:join. Qed. - Lemma PairCong n (a0 a1 b0 b1 : PTm n) : + Lemma PairCong (a0 a1 b0 b1 : PTm) : join a0 a1 -> join b0 b1 -> join (PPair a0 b0) (PPair a1 b1). Proof. hauto lq:on use:Pars.PairCong unfold:join. Qed. - Lemma AppCong n (a0 a1 b0 b1 : PTm n) : + Lemma AppCong (a0 a1 b0 b1 : PTm) : join a0 a1 -> join b0 b1 -> join (PApp a0 b0) (PApp a1 b1). Proof. hauto lq:on use:Pars.AppCong. Qed. - Lemma AbsCong n (a b : PTm (S n)) : + Lemma AbsCong (a b : PTm) : join a b -> join (PAbs a) (PAbs b). Proof. hauto lq:on use:Pars.AbsCong. Qed. - Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) : + Lemma renaming (a b : PTm) (ξ : nat -> nat) : join a b -> join (ren_PTm ξ a) (ren_PTm ξ b). Proof. induction 1; hauto lq:on use:Pars.renaming. Qed. - Lemma weakening n (a b : PTm n) : + Lemma weakening (a b : PTm) : join a b -> join (ren_PTm shift a) (ren_PTm shift b). Proof. apply renaming. Qed. - Lemma FromPar n (a b : PTm n) : + Lemma FromPar (a b : PTm) : Par.R a b -> join a b. Proof. @@ -2375,7 +2131,7 @@ Module Join. Qed. End Join. -Lemma abs_eq n a (b : PTm n) : +Lemma abs_eq a (b : PTm) : join (PAbs a) b <-> join a (PApp (ren_PTm shift b) (VarPTm var_zero)). Proof. split. @@ -2385,13 +2141,27 @@ Proof. simpl. move => ?. apply : join_transitive; eauto. apply join_symmetric. apply Join.FromPar. - apply : Par.AppAbs'; eauto using Par.refl. by asimpl. + apply : Par.AppAbs'; eauto using Par.refl. by asimpl; rewrite subst_id. - move /Join.AbsCong. move /join_transitive. apply. apply join_symmetric. apply Join.FromPar. apply Par.AppEta. apply Par.refl. Qed. -Lemma pair_eq n (a0 a1 b : PTm n) : +(* Lemma abs_inj a b : *) +(* join a b <-> join (PAbs a) (PAbs b). *) +(* Proof. *) +(* split. *) + +(* transitivity (join a (PApp (ren_PTm shift (PAbs b)) (VarPTm var_zero))); last by rewrite abs_eq. *) +(* have h : RPar.R (PApp (ren_PTm shift (PAbs b)) (VarPTm var_zero)) (subst_PTm (scons (VarPTm var_zero) VarPTm) (ren_PTm (upRen_PTm_PTm shift) b)). *) +(* apply RPar.AppAbs. rewrite -/ren_PTm. asimpl. substify. asimpl. apply RPar.refl. apply RPar.refl. *) +(* split. *) +(* move => h1. apply : join_transitive; eauto. *) +(* apply join_symmetric. *) +(* apply *) + + +Lemma pair_eq (a0 a1 b : PTm) : join (PPair a0 a1) b <-> join a0 (PProj PL b) /\ join a1 (PProj PR b). Proof. split. @@ -2407,7 +2177,7 @@ Proof. apply Join.FromPar. hauto lq:on ctrs:Par.R use:Par.refl. Qed. -Lemma join_pair_inj n (a0 a1 b0 b1 : PTm n) : +Lemma join_pair_inj (a0 a1 b0 b1 : PTm) : join (PPair a0 a1) (PPair b0 b1) <-> join a0 b0 /\ join a1 b1. Proof. split; last by hauto lq:on use:Join.PairCong. diff --git a/theories/typing.v b/theories/typing.v index d41facd..ac3ed0f 100644 --- a/theories/typing.v +++ b/theories/typing.v @@ -1,251 +1,93 @@ -Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax. +Require Import Autosubst2.core Autosubst2.unscoped compile Autosubst2.syntax ssreflect. +From Hammer Require Import Tactics. Reserved Notation "Γ ⊢ a ∈ A" (at level 70). -Reserved Notation "Γ ⊢ a ≡ b ∈ A" (at level 70). -Reserved Notation "Γ ⊢ A ≲ B" (at level 70). Reserved Notation "⊢ Γ" (at level 70). -Inductive Wt : list PTm -> PTm -> PTm -> Prop := +Inductive lookup : nat -> list Tm -> Tm -> Prop := +| here A Γ : lookup 0 (cons A Γ) (ren_Tm shift A) +| there i Γ A B : + lookup i Γ A -> + lookup (S i) (cons B Γ) (ren_Tm shift A). + +Lemma lookup_deter i Γ A B : + lookup i Γ A -> + lookup i Γ B -> + A = B. +Proof. move => h. move : B. induction h; hauto lq:on inv:lookup. Qed. + +Lemma here' A Γ U : U = ren_Tm shift A -> lookup 0 (A :: Γ) U. +Proof. move => ->. apply here. Qed. + +Lemma there' i Γ A B U : U = ren_Tm shift A -> lookup i Γ A -> + lookup (S i) (cons B Γ) U. +Proof. move => ->. apply there. Qed. + +Derive Inversion lookup_inv with (forall i Γ A, lookup i Γ A). + + +Inductive Wt : list Tm -> Tm -> Tm -> Prop := | T_Var i Γ A : ⊢ Γ -> lookup i Γ A -> - Γ ⊢ VarPTm i ∈ A + Γ ⊢ VarTm i ∈ A -| T_Bind Γ i p (A : PTm) (B : PTm) : - Γ ⊢ A ∈ PUniv i -> - cons A Γ ⊢ B ∈ PUniv i -> - Γ ⊢ PBind p A B ∈ PUniv i +| T_Bind Γ i p A B : + Γ ⊢ A ∈ Univ i -> + cons A Γ ⊢ B ∈ Univ i -> + Γ ⊢ TBind p A B ∈ Univ i -| T_Abs Γ (a : PTm) A B i : - Γ ⊢ PBind PPi A B ∈ (PUniv i) -> +| T_Abs Γ a A B i : + Γ ⊢ TBind TPi A B ∈ (Univ i) -> (cons A Γ) ⊢ a ∈ B -> - Γ ⊢ PAbs a ∈ PBind PPi A B + Γ ⊢ Abs A a ∈ TBind TPi A B -| T_App Γ (b a : PTm) A B : - Γ ⊢ b ∈ PBind PPi A B -> +| T_App Γ b a A B : + Γ ⊢ b ∈ TBind TPi A B -> Γ ⊢ a ∈ A -> - Γ ⊢ PApp b a ∈ subst_PTm (scons a VarPTm) B + Γ ⊢ App b a ∈ subst_Tm (scons a VarTm) B -| T_Pair Γ (a b : PTm) A B i : - Γ ⊢ PBind PSig A B ∈ (PUniv i) -> +| T_Pair Γ (a b : Tm) A B i : + Γ ⊢ TBind TSig A B ∈ (Univ i) -> Γ ⊢ a ∈ A -> - Γ ⊢ b ∈ subst_PTm (scons a VarPTm) B -> - Γ ⊢ PPair a b ∈ PBind PSig A B + Γ ⊢ b ∈ subst_Tm (scons a VarTm) B -> + Γ ⊢ Pair a b ∈ TBind TSig A B -| T_Proj1 Γ (a : PTm) A B : - Γ ⊢ a ∈ PBind PSig A B -> - Γ ⊢ PProj PL a ∈ A +| T_Proj1 Γ (a : Tm) A B : + Γ ⊢ a ∈ TBind TSig A B -> + Γ ⊢ Proj PL a ∈ A -| T_Proj2 Γ (a : PTm) A B : - Γ ⊢ a ∈ PBind PSig A B -> - Γ ⊢ PProj PR a ∈ subst_PTm (scons (PProj PL a) VarPTm) B +| T_Proj2 Γ (a : Tm) A B : + Γ ⊢ a ∈ TBind TSig A B -> + Γ ⊢ Proj PR a ∈ subst_Tm (scons (Proj PL a) VarTm) B | T_Univ Γ i : ⊢ Γ -> - Γ ⊢ PUniv i ∈ PUniv (S i) + Γ ⊢ Univ i ∈ Univ (S i) -| T_Nat Γ i : - ⊢ Γ -> - Γ ⊢ PNat ∈ PUniv i - -| T_Zero Γ : - ⊢ Γ -> - Γ ⊢ PZero ∈ PNat - -| T_Suc Γ (a : PTm) : - Γ ⊢ a ∈ PNat -> - Γ ⊢ PSuc a ∈ PNat - -| T_Ind Γ P (a : PTm) b c i : - cons PNat Γ ⊢ P ∈ PUniv i -> - Γ ⊢ a ∈ PNat -> - Γ ⊢ b ∈ subst_PTm (scons PZero VarPTm) P -> - (cons P (cons PNat Γ)) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) -> - Γ ⊢ PInd P a b c ∈ subst_PTm (scons a VarPTm) P - -| T_Conv Γ (a : PTm) A B : +| T_Conv Γ (a : Tm) A B i : Γ ⊢ a ∈ A -> - Γ ⊢ A ≲ B -> + Γ ⊢ B ∈ Univ i -> + Join.R A B -> Γ ⊢ a ∈ B -with Eq : list PTm -> PTm -> PTm -> PTm -> Prop := -(* Structural *) -| E_Refl Γ (a : PTm ) A : - Γ ⊢ a ∈ A -> - Γ ⊢ a ≡ a ∈ A - -| E_Symmetric Γ (a b : PTm) A : - Γ ⊢ a ≡ b ∈ A -> - Γ ⊢ b ≡ a ∈ A - -| E_Transitive Γ (a b c : PTm) A : - Γ ⊢ a ≡ b ∈ A -> - Γ ⊢ b ≡ c ∈ A -> - Γ ⊢ a ≡ c ∈ A - -(* Congruence *) -| E_Bind Γ i p (A0 A1 : PTm) B0 B1 : - Γ ⊢ A0 ∈ PUniv i -> - Γ ⊢ A0 ≡ A1 ∈ PUniv i -> - (cons A0 Γ) ⊢ B0 ≡ B1 ∈ PUniv i -> - Γ ⊢ PBind p A0 B0 ≡ PBind p A1 B1 ∈ PUniv i - -| E_Abs Γ (a b : PTm) A B i : - Γ ⊢ PBind PPi A B ∈ (PUniv i) -> - (cons A Γ) ⊢ a ≡ b ∈ B -> - Γ ⊢ PAbs a ≡ PAbs b ∈ PBind PPi A B - -| E_App Γ i (b0 b1 a0 a1 : PTm) A B : - Γ ⊢ PBind PPi A B ∈ (PUniv i) -> - Γ ⊢ b0 ≡ b1 ∈ PBind PPi A B -> - Γ ⊢ a0 ≡ a1 ∈ A -> - Γ ⊢ PApp b0 a0 ≡ PApp b1 a1 ∈ subst_PTm (scons a0 VarPTm) B - -| E_Pair Γ (a0 a1 b0 b1 : PTm) A B i : - Γ ⊢ PBind PSig A B ∈ (PUniv i) -> - Γ ⊢ a0 ≡ a1 ∈ A -> - Γ ⊢ b0 ≡ b1 ∈ subst_PTm (scons a0 VarPTm) B -> - Γ ⊢ PPair a0 b0 ≡ PPair a1 b1 ∈ PBind PSig A B - -| E_Proj1 Γ (a b : PTm) A B : - Γ ⊢ a ≡ b ∈ PBind PSig A B -> - Γ ⊢ PProj PL a ≡ PProj PL b ∈ A - -| E_Proj2 Γ i (a b : PTm) A B : - Γ ⊢ PBind PSig A B ∈ (PUniv i) -> - Γ ⊢ a ≡ b ∈ PBind PSig A B -> - Γ ⊢ PProj PR a ≡ PProj PR b ∈ subst_PTm (scons (PProj PL a) VarPTm) B - -| E_IndCong Γ P0 P1 (a0 a1 : PTm) b0 b1 c0 c1 i : - (cons PNat Γ) ⊢ P0 ∈ PUniv i -> - (cons PNat Γ) ⊢ P0 ≡ P1 ∈ PUniv i -> - Γ ⊢ a0 ≡ a1 ∈ PNat -> - Γ ⊢ b0 ≡ b1 ∈ subst_PTm (scons PZero VarPTm) P0 -> - (cons P0 ((cons PNat Γ))) ⊢ c0 ≡ c1 ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P0) -> - Γ ⊢ PInd P0 a0 b0 c0 ≡ PInd P1 a1 b1 c1 ∈ subst_PTm (scons a0 VarPTm) P0 - -| E_SucCong Γ (a b : PTm) : - Γ ⊢ a ≡ b ∈ PNat -> - Γ ⊢ PSuc a ≡ PSuc b ∈ PNat - -| E_Conv Γ (a b : PTm) A B : - Γ ⊢ a ≡ b ∈ A -> - Γ ⊢ A ≲ B -> - Γ ⊢ a ≡ b ∈ B - -(* Beta *) -| E_AppAbs Γ (a : PTm) b A B i: - Γ ⊢ PBind PPi A B ∈ PUniv i -> - Γ ⊢ b ∈ A -> - (cons A Γ) ⊢ a ∈ B -> - Γ ⊢ PApp (PAbs a) b ≡ subst_PTm (scons b VarPTm) a ∈ subst_PTm (scons b VarPTm ) B - -| E_ProjPair1 Γ (a b : PTm) A B i : - Γ ⊢ PBind PSig A B ∈ (PUniv i) -> - Γ ⊢ a ∈ A -> - Γ ⊢ b ∈ subst_PTm (scons a VarPTm) B -> - Γ ⊢ PProj PL (PPair a b) ≡ a ∈ A - -| E_ProjPair2 Γ (a b : PTm) A B i : - Γ ⊢ PBind PSig A B ∈ (PUniv i) -> - Γ ⊢ a ∈ A -> - Γ ⊢ b ∈ subst_PTm (scons a VarPTm) B -> - Γ ⊢ PProj PR (PPair a b) ≡ b ∈ subst_PTm (scons a VarPTm) B - -| E_IndZero Γ P i (b : PTm) c : - (cons PNat Γ) ⊢ P ∈ PUniv i -> - Γ ⊢ b ∈ subst_PTm (scons PZero VarPTm) P -> - (cons P (cons PNat Γ)) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) -> - Γ ⊢ PInd P PZero b c ≡ b ∈ subst_PTm (scons PZero VarPTm) P - -| E_IndSuc Γ P (a : PTm) b c i : - (cons PNat Γ) ⊢ P ∈ PUniv i -> - Γ ⊢ a ∈ PNat -> - Γ ⊢ b ∈ subst_PTm (scons PZero VarPTm) P -> - (cons P (cons PNat Γ)) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) -> - Γ ⊢ PInd P (PSuc a) b c ≡ (subst_PTm (scons (PInd P a b c) (scons a VarPTm)) c) ∈ subst_PTm (scons (PSuc a) VarPTm) P - -(* Eta *) -| E_AppEta Γ (b : PTm) A B i : - Γ ⊢ PBind PPi A B ∈ (PUniv i) -> - Γ ⊢ b ∈ PBind PPi A B -> - Γ ⊢ PAbs (PApp (ren_PTm shift b) (VarPTm var_zero)) ≡ b ∈ PBind PPi A B - -| E_PairEta Γ (a : PTm ) A B i : - Γ ⊢ PBind PSig A B ∈ (PUniv i) -> - Γ ⊢ a ∈ PBind PSig A B -> - Γ ⊢ a ≡ PPair (PProj PL a) (PProj PR a) ∈ PBind PSig A B - -with LEq : list PTm -> PTm -> PTm -> Prop := -(* Structural *) -| Su_Transitive Γ (A B C : PTm) : - Γ ⊢ A ≲ B -> - Γ ⊢ B ≲ C -> - Γ ⊢ A ≲ C - -(* Congruence *) -| Su_Univ Γ i j : - ⊢ Γ -> - i <= j -> - Γ ⊢ PUniv i ≲ PUniv j - -| Su_Pi Γ (A0 A1 : PTm) B0 B1 i : - Γ ⊢ A0 ∈ PUniv i -> - Γ ⊢ A1 ≲ A0 -> - (cons A0 Γ) ⊢ B0 ≲ B1 -> - Γ ⊢ PBind PPi A0 B0 ≲ PBind PPi A1 B1 - -| Su_Sig Γ (A0 A1 : PTm) B0 B1 i : - Γ ⊢ A1 ∈ PUniv i -> - Γ ⊢ A0 ≲ A1 -> - (cons A1 Γ) ⊢ B0 ≲ B1 -> - Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A1 B1 - -(* Injecting from equalities *) -| Su_Eq Γ (A : PTm) B i : - Γ ⊢ A ≡ B ∈ PUniv i -> - Γ ⊢ A ≲ B - -(* Projection axioms *) -| Su_Pi_Proj1 Γ (A0 A1 : PTm) B0 B1 : - Γ ⊢ PBind PPi A0 B0 ≲ PBind PPi A1 B1 -> - Γ ⊢ A1 ≲ A0 - -| Su_Sig_Proj1 Γ (A0 A1 : PTm) B0 B1 : - Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A1 B1 -> - Γ ⊢ A0 ≲ A1 - -| Su_Pi_Proj2 Γ (a0 a1 A0 A1 : PTm ) B0 B1 : - Γ ⊢ PBind PPi A0 B0 ≲ PBind PPi A1 B1 -> - Γ ⊢ a0 ≡ a1 ∈ A1 -> - Γ ⊢ subst_PTm (scons a0 VarPTm) B0 ≲ subst_PTm (scons a1 VarPTm) B1 - -| Su_Sig_Proj2 Γ (a0 a1 A0 A1 : PTm) B0 B1 : - Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A1 B1 -> - Γ ⊢ a0 ≡ a1 ∈ A0 -> - Γ ⊢ subst_PTm (scons a0 VarPTm) B0 ≲ subst_PTm (scons a1 VarPTm) B1 - -with Wff : list PTm -> Prop := +with Wff : list Tm -> Prop := | Wff_Nil : ⊢ nil -| Wff_Cons Γ (A : PTm) i : +| Wff_Cons Γ (A : Tm) i : ⊢ Γ -> - Γ ⊢ A ∈ PUniv i -> + Γ ⊢ A ∈ Univ i -> (* -------------------------------- *) ⊢ (cons A Γ) where -"Γ ⊢ a ∈ A" := (Wt Γ a A) and "⊢ Γ" := (Wff Γ) and "Γ ⊢ a ≡ b ∈ A" := (Eq Γ a b A) and "Γ ⊢ A ≲ B" := (LEq Γ A B). +"Γ ⊢ a ∈ A" := (Wt Γ a A) and "⊢ Γ" := (Wff Γ). Scheme wf_ind := Induction for Wff Sort Prop - with wt_ind := Induction for Wt Sort Prop - with eq_ind := Induction for Eq Sort Prop - with le_ind := Induction for LEq Sort Prop. + with wt_ind := Induction for Wt Sort Prop. -Combined Scheme wt_mutual from wf_ind, wt_ind, eq_ind, le_ind. +Combined Scheme wt_mutual from wf_ind, wt_ind. (* Lemma lem : *) -(* (forall n (Γ : fin n -> PTm n), ⊢ Γ -> ...) /\ *) -(* (forall n Γ (a A : PTm n), Γ ⊢ a ∈ A -> ...) /\ *) -(* (forall n Γ (a b A : PTm n), Γ ⊢ a ≡ b ∈ A -> ...) /\ *) -(* (forall n Γ (A B : PTm n), Γ ⊢ A ≲ B -> ...). *) +(* (forall n (Γ : fin n -> Tm n), ⊢ Γ -> ...) /\ *) +(* (forall n Γ (a A : Tm n), Γ ⊢ a ∈ A -> ...) /\ *) (* Proof. apply wt_mutual. ... *) diff --git a/theories/typing_properties.v b/theories/typing_properties.v new file mode 100644 index 0000000..b8defe3 --- /dev/null +++ b/theories/typing_properties.v @@ -0,0 +1,134 @@ +Require Import Autosubst2.core Autosubst2.unscoped compile Autosubst2.syntax ssreflect typing. +From Hammer Require Import Tactics. + +Lemma Bind_Inv Γ p A B U : + Γ ⊢ TBind p A B ∈ U -> + exists i, Γ ⊢ A ∈ Univ i /\ cons A Γ ⊢ B ∈ Univ i /\ Join.R (Univ i) U. +Proof. + move E : (TBind p A B) => u hu. + move : p A B E. + elim : Γ u U / hu => //=; hauto lq:on ctrs:Wt use:Join.reflexive, Join.transitive. +Qed. + +Lemma Univ_Inv Γ i U : + Γ ⊢ Univ i ∈ U -> + Γ ⊢ Univ i ∈ Univ (S i) /\ Join.R (Univ (S i)) U. +Proof. + move E : (Univ i) => u hu. + move : i E. + elim : Γ u U / hu => //=; hauto lq:on ctrs:Wt use:Join.reflexive, Join.transitive. +Qed. + +Lemma App_Inv Γ b a U : + Γ ⊢ App b a ∈ U -> + exists A B, Γ ⊢ b ∈ TBind TPi A B /\ Γ ⊢ a ∈ A /\ Join.R (subst_Tm (scons a VarTm) B) U. +Proof. + move E : (App b a) => u hu. + move : b a E. + elim : Γ u U / hu => //=; hauto lq:on ctrs:Wt use:Join.reflexive, Join.transitive. +Qed. + +Lemma Abs_Inv Γ A a U : + Γ ⊢ Abs A a ∈ U -> + exists B, cons A Γ ⊢ a ∈ B /\ Join.R (TBind TPi A B) U. +Proof. + move E : (Abs A a) => u hu. + move : A a E. + elim : Γ u U / hu => //=; hauto lq:on ctrs:Wt use:Join.reflexive, Join.transitive. +Qed. + +Lemma Var_Inv Γ i U : + Γ ⊢ VarTm i ∈ U -> + exists A, lookup i Γ A /\ Join.R A U. +Proof. + move E : (VarTm i) => u hu. + move : i E. + elim : Γ u U / hu => //=; hauto lq:on ctrs:Wt use:Join.reflexive, Join.transitive. +Qed. + +Lemma Pair_Inv Γ a b U : + Γ ⊢ Pair a b ∈ U -> + exists A B, Γ ⊢ a ∈ A /\ Γ ⊢ b ∈ subst_Tm (scons a VarTm) B /\ Join.R (TBind TSig A B) U. +Proof. + move E : (Pair a b ) => u hu. + move : a b E. + elim : Γ u U / hu => //=; hauto lq:on ctrs:Wt use:Join.reflexive, Join.transitive. +Qed. + +Lemma ProjL_Inv Γ a U : + Γ ⊢ Proj PL a ∈ U -> + exists A B, Γ ⊢ a ∈ TBind TSig A B /\ Join.R A U. +Proof. + move E : (Proj PL a) => u hu. + move : a E. + elim : Γ u U / hu => //=; hauto lq:on ctrs:Wt use:Join.reflexive, Join.transitive. +Qed. + +Lemma ProjR_Inv Γ a U : + Γ ⊢ Proj PR a ∈ U -> + exists A B, Γ ⊢ a ∈ TBind TSig A B /\ Join.R (subst_Tm (scons (Proj PL a) VarTm) B) U. +Proof. + move E : (Proj PR a) => u hu. + move : a E. + elim : Γ u U / hu => //=; hauto lq:on ctrs:Wt use:Join.reflexive, Join.transitive. +Qed. + +Lemma ctx_wff_mutual : + (forall Γ, ⊢ Γ -> True) /\ + (forall Γ a A, Γ ⊢ a ∈ A -> ⊢ Γ). +Proof. apply wt_mutual => //=. Qed. + +Lemma lookup_deter i Γ A A0 : + lookup i Γ A -> + lookup i Γ A0 -> A = A0. +Proof. + move => h. move : A0. elim : i Γ A / h; hauto lq:on inv:lookup. +Qed. + +Lemma wt_unique : + (forall Γ, ⊢ Γ -> True) /\ + (forall Γ a A, Γ ⊢ a ∈ A -> forall B, Γ ⊢ a ∈ B -> Join.R A B). +Proof. + apply wt_mutual => //=. + - move => i Γ A hΓ _ hl B. + move /Var_Inv. + move => [A0 [h0 h1]]. + move : hl h0. + move : lookup_deter; repeat move/[apply]. move => ?. by subst. + - move => Γ i p A B hA ihA hB ihB U. + move /Bind_Inv => [j][ih0][ih1]ih2. + apply ihB in ih1. + move /Join.UnivInj in ih1. by subst. + - move => Γ a A B i hP ihP ha iha U. + move /Abs_Inv => [B0][ha']hJ. + move /iha in ha' => {iha}. + apply : Join.transitive; eauto. + apply Join.BindCong; eauto using Join.reflexive. + - move => Γ b a A B hb ihb ha iha U. + move /App_Inv. move => [A0][B0][hb'][ha']hU. + apply ihb in hb' => {ihb}. + move /Join.BindInj : hb'. + move => [_][hJ0]hJ1. + apply : Join.transitive; eauto. + by apply Join.substing. + - move => Γ a b A B i hS ihS ha iha hb ihb U. + move /Pair_Inv. + move => [A0][B0][{}/iha ha'][{}/ihb hb']hJ. + apply : Join.transitive; eauto. + apply Join.BindCong; eauto. + admit. + - move => Γ a A B ha iha U. + move /ProjL_Inv. + move => [A0][B0][{}/iha ha0]hU. + apply Join.BindInj in ha0. + decompose record ha0. + eauto using Join.transitive. + - move => Γ a A B ha iha U /ProjR_Inv [A0][B0][{}/iha /Join.BindInj ha']. + decompose record ha'. + move => h. apply : Join.transitive; eauto. + by apply Join.substing. + - move => Γ i hΓ _ B /Univ_Inv. tauto. + - move => Γ a A B i ha iha hb ihb. + move => h0 B0 {}/iha ha'. + eauto using Join.symmetric, Join.transitive. +Admitted.