diff --git a/theories/algorithmic.v b/theories/algorithmic.v index 8a9146a..ab8fe97 100644 --- a/theories/algorithmic.v +++ b/theories/algorithmic.v @@ -1050,6 +1050,12 @@ Proof. Qed. +Lemma hreds_nf_refl a b : + HRed.nf a -> + rtc HRed.R a b -> + a = b. +Proof. inversion 2; sfirstorder. Qed. + Lemma lored_nsteps_app_cong k (a0 a1 b : PTm) : nsteps LoRed.R k a0 a1 -> ishne a0 -> diff --git a/theories/executable.v b/theories/executable.v index 0586a41..156e691 100644 --- a/theories/executable.v +++ b/theories/executable.v @@ -486,3 +486,8 @@ Lemma check_equal_conf a b nfa nfb nfab : Proof. destruct a; destruct b => //=. Qed. #[export]Hint Rewrite check_equal_abs_abs check_equal_abs_neu check_equal_neu_abs check_equal_pair_pair check_equal_pair_neu check_equal_neu_pair check_equal_bind_bind check_equal_hredl check_equal_hredr check_equal_nfnf check_equal_conf : ce_prop. + +Scheme algo_ind := Induction for algo_dom Sort Prop + with algor_ind := Induction for algo_dom_r Sort Prop. + +Combined Scheme algo_dom_mutual from algo_ind, algor_ind. diff --git a/theories/executable_correct.v b/theories/executable_correct.v index 0720d03..fdfee5a 100644 --- a/theories/executable_correct.v +++ b/theories/executable_correct.v @@ -4,12 +4,6 @@ Require Import ssreflect ssrbool. From stdpp Require Import relations (rtc(..)). From Hammer Require Import Tactics. -Scheme algo_ind := Induction for algo_dom Sort Prop - with algor_ind := Induction for algo_dom_r Sort Prop. - -Combined Scheme algo_dom_mutual from algo_ind, algor_ind. - - Lemma coqeqr_no_hred a b : a ∼ b -> HRed.nf a /\ HRed.nf b. Proof. induction 1; sauto lq:on unfold:HRed.nf. Qed. @@ -18,7 +12,6 @@ Proof. induction 1; qauto inv:HRed.R use:coqeqr_no_hred, hne_no_hred unfold:HRed.nf. Qed. - Lemma coqeq_neuneu u0 u1 : ishne u0 -> ishne u1 -> u0 ↔ u1 -> u0 ∼ u1. Proof. @@ -88,12 +81,6 @@ Proof. sauto lq:on rew:off. Qed. -Lemma hreds_nf_refl a b : - HRed.nf a -> - rtc HRed.R a b -> - a = b. -Proof. inversion 2; sfirstorder. Qed. - Ltac ce_solv := move => *; simp ce_prop in *; hauto qb:on rew:off inv:CoqEq, CoqEq_Neu. Lemma check_equal_complete : diff --git a/theories/termination.v b/theories/termination.v index 61f1e1d..53474b6 100644 --- a/theories/termination.v +++ b/theories/termination.v @@ -1,14 +1,266 @@ Require Import common Autosubst2.core Autosubst2.unscoped Autosubst2.syntax algorithmic fp_red executable. From Hammer Require Import Tactics. Require Import ssreflect ssrbool. -From stdpp Require Import relations (nsteps (..)). +From stdpp Require Import relations (nsteps (..), rtc(..)). +Import Psatz. + +Lemma tm_conf_sym a b : tm_conf a b = tm_conf b a. +Proof. case : a; case : b => //=. Qed. + +#[export]Hint Constructors algo_dom algo_dom_r : adom. + +Lemma algo_dom_r_inv a b : + algo_dom_r a b -> exists a0 b0, algo_dom a0 b0 /\ rtc HRed.R a a0 /\ rtc HRed.R b b0. +Proof. + induction 1; hauto lq:on ctrs:rtc. +Qed. + +Lemma A_HRedsL a a' b : + rtc HRed.R a a' -> + algo_dom_r a' b -> + algo_dom_r a b. + induction 1; sfirstorder use:A_HRedL. +Qed. + +Lemma A_HRedsR a b b' : + HRed.nf a -> + rtc HRed.R b b' -> + algo_dom a b' -> + algo_dom_r a b. +Proof. induction 2; sauto. Qed. + +Lemma algo_dom_sym : + (forall a b (h : algo_dom a b), algo_dom b a) /\ + (forall a b (h : algo_dom_r a b), algo_dom_r b a). +Proof. + apply algo_dom_mutual; try qauto use:tm_conf_sym db:adom. + move => a a' b hr h ih. + move /algo_dom_r_inv : ih => [a0][b0][ih0][ih1]ih2. + have nfa0 : HRed.nf a0 by sfirstorder use:algo_dom_no_hred. + sauto use:A_HRedsL, A_HRedsR. +Qed. Definition term_metric k (a b : PTm) := exists i j va vb, nsteps LoRed.R i a va /\ nsteps LoRed.R j b vb /\ nf va /\ nf vb /\ size_PTm va + size_PTm vb + i + j <= k. +Lemma term_metric_sym k (a b : PTm) : + term_metric k a b -> term_metric k b a. +Proof. hauto lq:on unfold:term_metric solve+:lia. Qed. + +Lemma term_metric_case k (a b : PTm) : + term_metric k a b -> + (ishf a \/ ishne a) \/ exists k' a', HRed.R a a' /\ term_metric k' a' b /\ k' < k. +Proof. + move=>[i][j][va][vb][h0] [h1][h2][h3]h4. + case : a h0 => //=; try firstorder. + - inversion h0 as [|A B C D E F]; subst. + hauto qb:on use:ne_hne. + inversion E; subst => /=. + + hauto lq:on use:HRed.AppAbs unfold:term_metric solve+:lia. + + hauto q:on ctrs:HRed.R use: hf_hred_lored unfold:term_metric solve+:lia. + + sfirstorder qb:on use:ne_hne. + - inversion h0 as [|A B C D E F]; subst. + hauto qb:on use:ne_hne. + inversion E; subst => /=. + + hauto lq:on use:HRed.ProjPair unfold:term_metric solve+:lia. + + hauto q:on ctrs:HRed.R use: hf_hred_lored unfold:term_metric solve+:lia. + - inversion h0 as [|A B C D E F]; subst. + hauto qb:on use:ne_hne. + inversion E; subst => /=. + + hauto lq:on use:HRed.IndZero unfold:term_metric solve+:lia. + + hauto lq:on ctrs:HRed.R use:hf_hred_lored unfold:term_metric solve+:lia. + + sfirstorder use:ne_hne. + + hauto lq:on ctrs:HRed.R use:hf_hred_lored unfold:term_metric solve+:lia. + + sfirstorder use:ne_hne. + + sfirstorder use:ne_hne. +Qed. + +Lemma A_Conf' a b : + ishf a \/ ishne a -> + ishf b \/ ishne b -> + tm_conf a b -> + algo_dom_r a b. +Proof. + move => ha hb. + have {}ha : HRed.nf a by sfirstorder use:hf_no_hred, hne_no_hred. + have {}hb : HRed.nf b by sfirstorder use:hf_no_hred, hne_no_hred. + move => ?. + apply A_NfNf. + by apply A_Conf. +Qed. + +Lemma term_metric_abs : forall k a b, + term_metric k (PAbs a) (PAbs b) -> + exists k', k' < k /\ term_metric k' a b. +Proof. + move => k a b [i][j][va][vb][hva][hvb][nfa][nfb]h. + apply lored_nsteps_abs_inv in hva, hvb. + move : hva => [a'][hva]?. subst. + move : hvb => [b'][hvb]?. subst. + simpl in *. exists (k - 1). + hauto lq:on unfold:term_metric solve+:lia. +Qed. + +Lemma term_metric_pair : forall k a0 b0 a1 b1, + term_metric k (PPair a0 b0) (PPair a1 b1) -> + exists k', k' < k /\ term_metric k' a0 a1 /\ term_metric k' b0 b1. +Proof. + move => k a0 b0 a1 b1 [i][j][va][vb][hva][hvb][nfa][nfb]h. + apply lored_nsteps_pair_inv in hva, hvb. + move : hva => [?][?][?][?][?][?][?][?]?. + move : hvb => [?][?][?][?][?][?][?][?]?. subst. + simpl in *. exists (k - 1). + hauto lqb:on solve+:lia. +Qed. + +Lemma term_metric_bind : forall k p0 a0 b0 p1 a1 b1, + term_metric k (PBind p0 a0 b0) (PBind p1 a1 b1) -> + exists k', k' < k /\ term_metric k' a0 a1 /\ term_metric k' b0 b1. +Proof. + move => k p0 a0 b0 p1 a1 b1 [i][j][va][vb][hva][hvb][nfa][nfb]h. + apply lored_nsteps_bind_inv in hva, hvb. + move : hva => [?][?][?][?][?][?][?][?]?. + move : hvb => [?][?][?][?][?][?][?][?]?. subst. + simpl in *. exists (k - 1). + hauto lqb:on solve+:lia. +Qed. + +Lemma term_metric_suc : forall k a b, + term_metric k (PSuc a) (PSuc b) -> + exists k', k' < k /\ term_metric k' a b. +Proof. + move => k a b [i][j][va][vb][hva][hvb][nfa][nfb]h. + apply lored_nsteps_suc_inv in hva, hvb. + move : hva => [a'][hva]?. subst. + move : hvb => [b'][hvb]?. subst. + simpl in *. exists (k - 1). + hauto lq:on unfold:term_metric solve+:lia. +Qed. + +Lemma term_metric_abs_neu k (a0 : PTm) u : + term_metric k (PAbs a0) u -> + ishne u -> + exists j, j < k /\ term_metric j a0 (PApp (ren_PTm shift u) (VarPTm var_zero)). +Proof. + move => [i][j][va][vb][h0][h1][h2][h3]h4 neu. + have neva : ne vb by hauto lq:on use:hne_nf_ne, loreds_hne_preservation, @relations.rtc_nsteps. + move /lored_nsteps_abs_inv : h0 => [a1][h01]?. subst. + exists (k - 1). + simpl in *. split. lia. + exists i,j,a1,(PApp (ren_PTm shift vb) (VarPTm var_zero)). + repeat split => //=. + apply lored_nsteps_app_cong. + by apply lored_nsteps_renaming. + by rewrite ishne_ren. + rewrite Bool.andb_true_r. + sfirstorder use:ne_nf_ren. + rewrite size_PTm_ren. lia. +Qed. + +Lemma term_metric_pair_neu k (a0 b0 : PTm) u : + term_metric k (PPair a0 b0) u -> + ishne u -> + exists j, j < k /\ term_metric j (PProj PL u) a0 /\ term_metric j (PProj PR u) b0. +Proof. + move => [i][j][va][vb][h0][h1][h2][h3]h4 neu. + have neva : ne vb by hauto lq:on use:hne_nf_ne, loreds_hne_preservation, @relations.rtc_nsteps. + move /lored_nsteps_pair_inv : h0 => [i0][j0][a1][b1][?][?][?][?]?. subst. + exists (k-1). sauto qb:on use:lored_nsteps_proj_cong unfold:term_metric solve+:lia. +Qed. + +Lemma term_metric_app k (a0 b0 a1 b1 : PTm) : + term_metric k (PApp a0 b0) (PApp a1 b1) -> + ishne a0 -> + ishne a1 -> + exists j, j < k /\ term_metric j a0 a1 /\ term_metric j b0 b1. +Proof. + move => [i][j][va][vb][h0][h1][h2][h3]h4. + move => hne0 hne1. + move : lored_nsteps_app_inv h0 (hne0);repeat move/[apply]. + move => [i0][i1][a2][b2][?][?][?][ha02]hb02. subst. + move : lored_nsteps_app_inv h1 (hne1);repeat move/[apply]. + move => [j0][j1][a3][b3][?][?][?][ha13]hb13. subst. + simpl in *. exists (k - 1). hauto lqb:on use:lored_nsteps_app_cong, ne_nf unfold:term_metric solve+:lia. +Qed. + +Lemma term_metric_proj k p0 p1 (a0 a1 : PTm) : + term_metric k (PProj p0 a0) (PProj p1 a1) -> + ishne a0 -> + ishne a1 -> + exists j, j < k /\ term_metric j a0 a1. +Proof. + move => [i][j][va][vb][h0][h1][h2][h3]h4 hne0 hne1. + move : lored_nsteps_proj_inv h0 (hne0);repeat move/[apply]. + move => [i0][a2][hi][?]ha02. subst. + move : lored_nsteps_proj_inv h1 (hne1);repeat move/[apply]. + move => [i1][a3][hj][?]ha13. subst. + exists (k- 1). hauto q:on use:ne_nf solve+:lia. +Qed. + +Lemma term_metric_ind k P0 (a0 : PTm ) b0 c0 P1 a1 b1 c1 : + term_metric k (PInd P0 a0 b0 c0) (PInd P1 a1 b1 c1) -> + ishne a0 -> + ishne a1 -> + exists j, j < k /\ term_metric j P0 P1 /\ term_metric j a0 a1 /\ + term_metric j b0 b1 /\ term_metric j c0 c1. +Proof. + move => [i][j][va][vb][h0][h1][h2][h3]h4 hne0 hne1. + move /lored_nsteps_ind_inv /(_ hne0) : h0. + move =>[iP][ia][ib][ic][P2][a2][b2][c2][?][?][?][?][?][?][?][?]?. subst. + move /lored_nsteps_ind_inv /(_ hne1) : h1. + move =>[iP0][ia0][ib0][ic0][P3][a3][b3][c3][?][?][?][?][?][?][?][?]?. subst. + exists (k -1). simpl in *. + hauto lq:on rew:off use:ne_nf b:on solve+:lia. +Qed. + + +Lemma algo_dom_r_algo_dom : forall a b, HRed.nf a -> HRed.nf b -> algo_dom_r a b -> algo_dom a b. +Proof. hauto l:on use:algo_dom_r_inv, hreds_nf_refl. Qed. + Lemma term_metric_algo_dom : forall k a b, term_metric k a b -> algo_dom_r a b. Proof. + move => [:hneL]. elim /Wf_nat.lt_wf_ind. move => n ih a b h. - case : (fancy_hred a); cycle 1. - move => [a' ha']. apply : A_HRedL; eauto. + case /term_metric_case : (h); cycle 1. + move => [k'][a'][h0][h1]h2. + by apply : A_HRedL; eauto. + case /term_metric_sym /term_metric_case : (h); cycle 1. + move => [k'][b'][hb][/term_metric_sym h0]h1. + move => ha. have {}ha : HRed.nf a by sfirstorder use:hf_no_hred, hne_no_hred. + by apply : A_HRedR; eauto. + move => /[swap]. + case => hfa; case => hfb. + - move : hfa hfb h. + case : a; case : b => //=; eauto 5 using A_Conf' with adom. + + hauto lq:on use:term_metric_abs db:adom. + + hauto lq:on use:term_metric_pair db:adom. + + hauto lq:on use:term_metric_bind db:adom. + + hauto lq:on use:term_metric_suc db:adom. + - abstract : hneL n ih a b h hfa hfb. + case : a hfa h => //=. + + hauto lq:on use:term_metric_abs_neu db:adom. + + scrush use:term_metric_pair_neu db:adom. + + case : b hfb => //=; eauto 5 using A_Conf' with adom. + + case : b hfb => //=; eauto 5 using A_Conf' with adom. + + case : b hfb => //=; eauto 5 using A_Conf' with adom. + + case : b hfb => //=; eauto 5 using A_Conf' with adom. + + case : b hfb => //=; eauto 5 using A_Conf' with adom. + - hauto q:on use:algo_dom_sym, term_metric_sym. + - move {hneL}. + case : b hfa hfb h => //=; case a => //=; eauto 5 using A_Conf' with adom. + + move => a0 b0 a1 b1 nfa0 nfa1. + move /term_metric_app /(_ nfa0 nfa1) => [j][hj][ha]hb. + apply A_NfNf. apply A_AppCong => //; eauto. + have {}nfa0 : HRed.nf a0 by sfirstorder use:hne_no_hred. + have {}nfb0 : HRed.nf a1 by sfirstorder use:hne_no_hred. + eauto using algo_dom_r_algo_dom. + + move => p0 A0 p1 A1 neA0 neA1. + have {}nfa0 : HRed.nf A0 by sfirstorder use:hne_no_hred. + have {}nfb0 : HRed.nf A1 by sfirstorder use:hne_no_hred. + hauto lq:on use:term_metric_proj, algo_dom_r_algo_dom db:adom. + + move => P0 a0 b0 c0 P1 a1 b1 c1 nea0 nea1. + have {}nfa0 : HRed.nf a0 by sfirstorder use:hne_no_hred. + have {}nfb0 : HRed.nf a1 by sfirstorder use:hne_no_hred. + hauto lq:on use:term_metric_ind, algo_dom_r_algo_dom db:adom. +Qed.