theories/algorithmic.v

@@ -1050,12 +1050,6 @@ 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

@@ -212,67 +212,6 @@ with algo_dom_r : PTm  -> PTm -> Prop :=
   (* ----------------------- *)
   algo_dom_r a b.
 
-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.
-
-
-(* Inductive salgo_dom : PTm -> PTm -> Prop := *)
-(* | S_UnivCong i j : *)
-(*   (* -------------------------- *) *)
-(*   salgo_dom (PUniv i) (PUniv j) *)
-
-(* | S_PiCong A0 A1 B0 B1 : *)
-(*   salgo_dom_r A1 A0 -> *)
-(*   salgo_dom_r B0 B1 -> *)
-(*   (* ---------------------------- *) *)
-(*   salgo_dom (PBind PPi A0 B0) (PBind PPi A1 B1) *)
-
-(* | S_SigCong A0 A1 B0 B1 : *)
-(*   salgo_dom_r A0 A1 -> *)
-(*   salgo_dom_r B0 B1 -> *)
-(*   (* ---------------------------- *) *)
-(*   salgo_dom (PBind PSig A0 B0) (PBind PSig A1 B1) *)
-
-(* | S_NatCong : *)
-(*   salgo_dom PNat PNat *)
-
-(* | S_NeuNeu a b : *)
-(*   ishne a -> *)
-(*   ishne b -> *)
-(*   algo_dom a b -> *)
-(*   (* ------------------- *) *)
-(*   salgo_dom *)
-
-(* | S_Conf a b : *)
-(*   HRed.nf a -> *)
-(*   HRed.nf b -> *)
-(*   tm_conf a b -> *)
-(*   salgo_dom a b *)
-
-(* with salgo_dom_r : PTm  -> PTm -> Prop := *)
-(* | S_NfNf a b : *)
-(*   salgo_dom a b -> *)
-(*   salgo_dom_r a b *)
-
-(* | S_HRedL a a' b  : *)
-(*   HRed.R a a' -> *)
-(*   salgo_dom_r a' b -> *)
-(*   (* ----------------------- *) *)
-(*   salgo_dom_r a b *)
-
-(* | S_HRedR a b b'  : *)
-(*   HRed.nf a -> *)
-(*   HRed.R b b' -> *)
-(*   salgo_dom_r a b' -> *)
-(*   (* ----------------------- *) *)
-(*   salgo_dom_r a b. *)
-
-(* Scheme salgo_ind := Induction for salgo_dom Sort Prop *)
-(*   with algor_ind := Induction for salgo_dom_r Sort Prop. *)
-
-
 Lemma hf_no_hred (a b : PTm) :
   ishf a ->
   HRed.R a b ->
@@ -412,6 +351,7 @@ Ltac solve_check_equal :=
         | inr b' := check_equal_r a (proj1_sig b') _;
                  | inl h'' := check_equal a b _.
 
+
 Next Obligation.
   intros.
   inversion h; subst => //=.
@@ -546,115 +486,3 @@ 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.
-
-Obligation Tactic := try solve [check_equal_triv | sfirstorder].
-
-Program Fixpoint check_sub (q : bool) (a b : PTm) (h : algo_dom a b) {struct h} :=
-  match a, b with
-  | PBind PPi A0 B0, PBind PPi A1 B1 =>
-      check_sub_r (negb q) A0 A1 _ && check_sub_r q B0 B1 _
-  | PBind PSig A0 B0, PBind PSig A1 B1 =>
-      check_sub_r q A0 A1 _ && check_sub_r q B0 B1 _
-  | PUniv i, PUniv j =>
-      if q then PeanoNat.Nat.leb i j else PeanoNat.Nat.leb j i
-  | PNat, PNat => true
-  | _ ,_ => ishne a && ishne b && check_equal a b h
-  end
-with check_sub_r (q : bool) (a b : PTm) (h : algo_dom_r a b) {struct h} :=
-  match fancy_hred a with
-  | inr a' => check_sub_r q (proj1_sig a') b _
-  | inl ha' => match fancy_hred b with
-            | inr b' => check_sub_r q a (proj1_sig b') _
-            | inl hb' => check_sub q a b _
-            end
-  end.
-Next Obligation.
-  simpl. intros. clear Heq_anonymous.  destruct a' as [a' ha']. simpl.
-  inversion h; subst => //=.
-  exfalso. sfirstorder use:algo_dom_no_hred.
-  assert (a' = a'0) by eauto using hred_deter. by subst.
-  exfalso. sfirstorder.
-Defined.
-
-Next Obligation.
-  simpl. intros. clear Heq_anonymous Heq_anonymous0.
-  destruct b' as [b' hb']. simpl.
-  inversion h; subst.
-  - exfalso.
-    sfirstorder use:algo_dom_no_hred.
-  - exfalso.
-    sfirstorder.
-  - assert (b' = b'0) by eauto using hred_deter. by subst.
-Defined.
-
-(* Need to avoid ssreflect tactics since they generate terms that make the termination checker upset *)
-Next Obligation.
-  move => _ /= a b hdom ha _ hb _.
-  inversion hdom; subst.
-  - assumption.
-  - exfalso; sfirstorder.
-  - exfalso; sfirstorder.
-Defined.
-
-Lemma check_sub_pi_pi q A0 B0  A1 B1 h0 h1 :
-  check_sub q (PBind PPi A0 B0) (PBind PPi A1 B1) (A_BindCong PPi PPi A0 A1 B0 B1 h0 h1) =
-    check_sub_r (~~ q) A0 A1 h0 && check_sub_r q B0 B1 h1.
-Proof. reflexivity. Qed.
-
-Lemma check_sub_sig_sig q A0 B0  A1 B1 h0 h1 :
-  check_sub q (PBind PSig A0 B0) (PBind PSig A1 B1) (A_BindCong PSig PSig A0 A1 B0 B1 h0 h1) =
-    check_sub_r q A0 A1 h0 && check_sub_r q B0 B1 h1.
-  reflexivity. Qed.
-
-Lemma check_sub_univ_univ i j :
-  check_sub true (PUniv i) (PUniv j) (A_UnivCong i j) = PeanoNat.Nat.leb i j.
-Proof. reflexivity. Qed.
-
-Lemma check_sub_univ_univ' i j :
-  check_sub false (PUniv i) (PUniv j) (A_UnivCong i j) = PeanoNat.Nat.leb j i.
-  reflexivity. Qed.
-
-Lemma check_sub_nfnf q a b dom : check_sub_r q a b (A_NfNf a b dom) = check_sub q a b dom.
-Proof.
-  have [h0 h1] : HRed.nf a /\ HRed.nf b by hauto l:on use:algo_dom_no_hred.
-  have [h3 h4] : hred a = None /\ hred b = None by sfirstorder use:hf_no_hred, hne_no_hred, hred_none.
-  simpl.
-  destruct (fancy_hred b)=>//=.
-  destruct (fancy_hred a) =>//=.
-  destruct s as [a' ha']. simpl.
-  hauto l:on use:hred_complete.
-  destruct s as [b' hb']. simpl.
-  hauto l:on use:hred_complete.
-Qed.
-
-Lemma check_sub_hredl q a b a' ha doma :
-  check_sub_r q a b (A_HRedL a a' b ha doma) = check_sub_r q a' b doma.
-Proof.
-  simpl.
-  destruct (fancy_hred a).
-  - hauto q:on unfold:HRed.nf.
-  - destruct s as [x ?].
-    have ? : x = a' by eauto using hred_deter. subst.
-    simpl.
-    f_equal.
-    apply PropExtensionality.proof_irrelevance.
-Qed.
-
-Lemma check_sub_hredr q a b b' hu r a0 :
-  check_sub_r q a b (A_HRedR a b b' hu r a0) =
-    check_sub_r q a b' a0.
-Proof.
-  simpl.
-  destruct (fancy_hred a).
-  - simpl.
-    destruct (fancy_hred b) as [|[b'' hb']].
-    + hauto lq:on unfold:HRed.nf.
-    + simpl.
-      have ? : (b'' = b') by eauto using hred_deter. subst.
-      f_equal.
-      apply PropExtensionality.proof_irrelevance.
-  - exfalso.
-    sfirstorder use:hne_no_hred, hf_no_hred.
-Qed.
-
-#[export]Hint Rewrite check_sub_hredl check_sub_hredr check_sub_nfnf check_sub_univ_univ' check_sub_univ_univ check_sub_pi_pi check_sub_sig_sig : ce_prop.

theories/executable_correct.v

@@ -4,6 +4,12 @@ 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.
 
@@ -12,10 +18,6 @@ Proof. induction 1;
          qauto inv:HRed.R use:coqeqr_no_hred, hne_no_hred unfold:HRed.nf.
 Qed.
 
-Lemma coqleq_no_hred a b : a ⋖ b -> HRed.nf a /\ HRed.nf b.
-Proof.
-  induction 1; qauto inv:HRed.R use:coqeqr_no_hred, hne_no_hred, coqeqr_no_hred unfold:HRed.nf.
-Qed.
 
 Lemma coqeq_neuneu u0 u1 :
   ishne u0 -> ishne u1 -> u0 ↔ u1 -> u0 ∼ u1.
@@ -86,6 +88,12 @@ 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 :
@@ -169,134 +177,3 @@ Proof.
       sfirstorder unfold:HRed.nf.
     + sauto lq:on use:hred_deter.
 Qed.
-
-Ltac simp_sub := with_strategy opaque [check_equal] simpl in *.
-
-(* Reusing algo_dom results in an inefficient proof, but i'll brute force it so i can get the result more quickly *)
-Lemma check_sub_sound :
-  (forall a b (h : algo_dom a b), forall q, check_sub q a b h -> if q then a ⋖ b else b ⋖ a) /\
-    (forall a b (h : algo_dom_r a b), forall q, check_sub_r q a b h -> if q then a ≪ b else b ≪ a).
-Proof.
-  apply algo_dom_mutual; try done.
-  - move => a [] //=; hauto qb:on.
-  - move => a0 a1 []//=; hauto qb:on.
-  - simpl. move => i j [];
-    sauto lq:on use:Reflect.Nat_leb_le.
-  - move => p0 p1 A0 A1 B0 B1 a iha b ihb q.
-    case : p0; case : p1; try done; simp ce_prop.
-    sauto lqb:on.
-    sauto lqb:on.
-  - hauto l:on.
-  - move => i j q h.
-    have {}h : nat_eqdec i j by sfirstorder.
-    case : nat_eqdec h => //=; sauto lq:on.
-  - simp_sub.
-    sauto lqb:on use:coqeq_symmetric_mutual, coqeq_neuneu, check_equal_sound.
-  - simp_sub.
-    sauto lqb:on use:coqeq_symmetric_mutual, coqeq_neuneu, check_equal_sound.
-  - simp_sub.
-    sauto lqb:on use:coqeq_symmetric_mutual, coqeq_neuneu, check_equal_sound.
-  - move => a b n n0 i q h.
-    exfalso.
-    destruct a, b; try done; simp_sub; hauto lb:on use:check_equal_conf.
-  - move => a b doma ihdom q.
-    simp ce_prop. sauto lq:on.
-  - move => a a' b hr doma ihdom q.
-    simp ce_prop. move : ihdom => /[apply]. move {doma}.
-    sauto lq:on.
-  - move => a b b' n r dom ihdom q.
-    simp ce_prop.
-    move : ihdom => /[apply].
-    move {dom}.
-    sauto lq:on rew:off.
-Qed.
-
-Lemma check_sub_complete :
-  (forall a b (h : algo_dom a b), forall q, check_sub q a b h = false -> if q then ~ a ⋖ b else ~ b ⋖ a) /\
-    (forall a b (h : algo_dom_r a b), forall q, check_sub_r q a b h = false -> if q then ~ a ≪ b else ~ b ≪ a).
-Proof.
-  apply algo_dom_mutual; try first [done | hauto depth:4 lq:on inv:CoqLEq, CoqEq_Neu].
-  - move => i j q /=.
-    sauto lq:on rew:off use:PeanoNat.Nat.leb_le.
-  - move => p0 p1 A0 A1 B0 B1 a iha b ihb [].
-    + move => + h. inversion h; subst; simp ce_prop.
-      * move /nandP.
-        case.
-        by move => /negbTE {}/iha.
-        by move => /negbTE {}/ihb.
-      * move /nandP.
-        case.
-        by move => /negbTE {}/iha.
-        by move => /negbTE {}/ihb.
-      * inversion H.
-    + move => + h. inversion h; subst; simp ce_prop.
-      * move /nandP.
-        case.
-        by move => /negbTE {}/iha.
-        by move => /negbTE {}/ihb.
-      * move /nandP.
-        case.
-        by move => /negbTE {}/iha.
-        by move => /negbTE {}/ihb.
-      * inversion H.
-  - simp_sub.
-    sauto lq:on use:check_equal_complete.
-  - simp_sub.
-    move => p0 p1 u0 u1 i i0 a iha q.
-    move /nandP.
-    case.
-    move /nandP.
-    case => //.
-    by move /negP.
-    by move /negP.
-    move /negP.
-    move => h. eapply check_equal_complete in h.
-    sauto use:coqeq_symmetric_mutual inv:CoqLEq lq:on.
-  - move => u0 u1 a0 a1 i i0 a hdom ihdom hdom' ihdom'.
-    simp_sub.
-    move /nandP.
-    case.
-    move/nandP.
-    case.
-    by move/negP.
-    by move/negP.
-    move /negP.
-    move => h. eapply check_equal_complete in h.
-    sauto use:coqeq_symmetric_mutual inv:CoqLEq lq:on.
-  - move => P0 P1 u0 u1 b0 b1 c0 c1 i i0 dom ihdom dom' ihdom' dom'' ihdom'' dom''' ihdom''' q.
-    move /nandP.
-    case.
-    move /nandP.
-    case => //=.
-    by move/negP.
-    by move/negP.
-    move /negP => h. eapply check_equal_complete in h.
-    sauto use:coqeq_symmetric_mutual inv:CoqLEq lq:on.
-  - move => a b h ih q. simp ce_prop => {}/ih.
-    case : q => h0;
-    inversion 1; subst; hauto l:on use:algo_dom_no_hred, hreds_nf_refl.
-  - move => a a' b r dom ihdom q.
-    simp ce_prop => {}/ihdom.
-    case : q => h0.
-    inversion 1; subst.
-    inversion H0; subst. sfirstorder use:coqleq_no_hred.
-    have ? : a' = y by eauto using hred_deter. subst.
-    sauto lq:on.
-    inversion 1; subst.
-    inversion H1; subst. sfirstorder use:coqleq_no_hred.
-    have ? : a' = y by eauto using hred_deter. subst.
-    sauto lq:on.
-  - move => a b b' n r hr ih q.
-    simp ce_prop => {}/ih.
-    case : q.
-    + move => h. inversion 1; subst.
-      inversion H1; subst.
-      sfirstorder use:coqleq_no_hred.
-      have ? : b' = y by eauto using hred_deter. subst.
-      sauto lq:on.
-    + move => h. inversion 1; subst.
-      inversion H0; subst.
-      sfirstorder use:coqleq_no_hred.
-      have ? : b' = y by eauto using hred_deter. subst.
-      sauto lq:on.
-Qed.

theories/termination.v

@@ -1,282 +1,14 @@
 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 (..), 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.
+From stdpp Require Import relations (nsteps (..)).
 
 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.
-  decompose record hva => {hva}.
-  decompose record hvb => {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.
-  decompose record hva => {hva}.
-  decompose record hvb => {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 /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.
-
-Lemma sn_term_metric (a b : PTm) : SN a -> SN b -> exists k, term_metric k a b.
-Proof.
-  move /LoReds.FromSN => [va [ha0 ha1]].
-  move /LoReds.FromSN => [vb [hb0 hb1]].
-  eapply relations.rtc_nsteps in ha0.
-  eapply relations.rtc_nsteps in hb0.
-  hauto lq:on unfold:term_metric solve+:lia.
-Qed.
-
-Lemma sn_algo_dom a b : SN a -> SN b -> algo_dom_r a b.
-Proof.
-  move : sn_term_metric; repeat move/[apply].
-  move => [k]+.
-  eauto using term_metric_algo_dom.
-Qed.
+  case : (fancy_hred a); cycle 1.
+  move => [a' ha']. apply : A_HRedL; eauto.