Prove termination

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 ->

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.

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 :

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.
+  case /term_metric_case : (h); cycle 1.
-  move => [a' ha']. apply : A_HRedL; eauto.
+  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.