282 lines
10 KiB
Coq
282 lines
10 KiB
Coq
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.
|
|
|
|
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.
|