phanerozoic/Coq-CoLoR · Datasets at Hugging Face

fact stringlengths 6 103k	type stringclasses 14 values	library stringclasses 13 values	imports listlengths 1 10	filename stringclasses 157 values	symbolic_name stringlengths 1 54	docstring stringclasses 185 values
rev_tac_condFs_ok := match goal with \| \|- is_unary _ => is_unary Fs_ok \| \|- rules_preserve_vars _ => rules_preserve_vars \| \|- WF _ => idtac end.	Ltac	Conversion	[ "From CoLoR Require Import Srs ATrs SReverse ATerm_of_String String_of_ATerm", "From CoLoR Require Import AVariables" ]	Conversion/AReverse.v	rev_tac_cond	null
rev_tacFs_ok := (rewrite <- WF_red_rev_eq \|\| rewrite <- WF_red_mod_rev_eq); rev_tac_cond Fs_ok.	Ltac	Conversion	[ "From CoLoR Require Import Srs ATrs SReverse ATerm_of_String String_of_ATerm", "From CoLoR Require Import AVariables" ]	Conversion/AReverse.v	rev_tac	null
ar(_ : letter) := 1. Import ATrs.	Definition	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	ar	corresponding algebraic signature
ASig_of_SSig:= mkSignature ar (@VSignature.beq_symb_ok SSig). Notation ASig := ASig_of_SSig. Notation term := (term ASig). Notation terms := (vector term). Notation context := (context ASig). Import AUnary.	Definition	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	ASig_of_SSig	null
is_unary_sig: is_unary ASig. Proof. intro. refl. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	is_unary_sig	null
arity:= arity1 is_unary_sig.	Ltac	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	arity	null
is_unary:= try (exact is_unary_sig). Notation Fun1 := (Fun1 is_unary_sig). Notation Cont1 := (Cont1 is_unary_sig). Notation cont := (cont is_unary_sig). Notation term_ind_forall := (term_ind_forall is_unary_sig).	Ltac	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	is_unary	null
term_of_string(s : string) : term := match s with \| List.nil => Var 0 \| a :: w => @Fun ASig a (Vcons (term_of_string w) Vnil) end.	Fixpoint	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	term_of_string	conversion string <-> term
var_term_of_string: forall s, var (term_of_string s) = 0. Proof. induction s; auto. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	var_term_of_string	null
reset_term_of_string: forall s, reset (term_of_string s) = term_of_string s. Proof. induction s. refl. unfold reset. simpl. apply args_eq. fold (reset (term_of_string s)). rewrite IHs. refl. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	reset_term_of_string	null
string_of_term(t : term) : string := match t with \| Var _ => List.nil \| Fun f ts => f :: Vmap_first List.nil string_of_term ts end.	Fixpoint	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	string_of_term	null
string_of_term_epi: forall s, string_of_term (term_of_string s) = s. Proof. induction s; simpl; intros. refl. rewrite IHs. refl. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	string_of_term_epi	null
term_of_string_epi: forall t, maxvar t = 0 -> term_of_string (string_of_term t) = t. Proof. intro t; pattern t; apply term_ind_forall; clear t. simpl in *. intros. subst. refl. intros f t IH. unfold AUnary.Fun1. simpl. rewrite (Vcast_cons (is_unary_sig f)), Vcast_refl. simpl. rewrite max_0_r. intro h. rewrite (IH h). refl. Qed. Arguments term_of_string_epi [t] _.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	term_of_string_epi	null
string_of_term_sub: forall s l, string_of_term (sub s l) = string_of_term l ++ string_of_term (s (var l)). Proof. intro s. apply term_ind_forall. refl. intros f t IH. rewrite sub_fun1. simpl. rewrite !(Vcast_cons (is_unary_sig f)). simpl. rewrite IH. refl. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	string_of_term_sub	null
string_of_cont(c : context) : string := match c with \| Hole => List.nil \| @Cont _ f _ _ _ _ d _ => f :: string_of_cont d end.	Fixpoint	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	string_of_cont	conversion string <-> context
string_of_cont_cont: forall t, string_of_cont (cont t) = string_of_term t. Proof. apply term_ind_forall. refl. intros f t IH. simpl. rewrite !(Vcast_cons (is_unary_sig f)). simpl. rewrite IH. refl. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	string_of_cont_cont	null
string_of_term_fill: forall t c, string_of_term (fill c t) = string_of_cont c ++ string_of_term t. Proof. induction c. refl. simpl. rewrite Vmap_first_cast. arity. simpl. rewrite IHc. refl. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	string_of_term_fill	null
cont_of_string(s : string) : context := match s with \| List.nil => Hole \| a :: w => Cont1 a (cont_of_string w) end.	Fixpoint	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	cont_of_string	null
sub_of_strings := single 0 (term_of_string s).	Definition	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	sub_of_string	conversion string <-> substitution
term_of_string_fill: forall c s, term_of_string (SContext.fill c s) = fill (cont_of_string (lft c)) (sub (sub_of_string (rgt c)) (term_of_string s)). Proof. intros [l r] s. elim l; unfold SContext.fill; simpl. elim s. refl. intros. simpl. rewrite H. refl. intros. rewrite H, (Vcast_cons (cont_aux is_unary_sig a)), Vcast_refl. refl. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	term_of_string_fill	null
term_of_string_app: forall s1 s2, term_of_string (s1 ++ s2) = sub (sub_of_string s2) (term_of_string s1). Proof. induction s1. refl. intro. simpl. apply args_eq. rewrite IHs1. refl. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	term_of_string_app	null
rule_of_srule(x : srule) := mkRule (term_of_string (Srs.lhs x)) (term_of_string (Srs.rhs x)).	Definition	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	rule_of_srule	rules
trs_of_srsR := List.map rule_of_srule R.	Definition	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	trs_of_srs	null
rules_preserve_vars_trs_of_srs: forall R, rules_preserve_vars (trs_of_srs R). Proof. unfold rules_preserve_vars. induction R; simpl; intuition. unfold rule_of_srule in H0. destruct a. simpl in H0. inversion H0. rewrite !vars_var; is_unary. rewrite !var_term_of_string. unfold incl; simpl; intuition. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	rules_preserve_vars_trs_of_srs	null
preserve_vars:= try (apply rules_preserve_vars_trs_of_srs).	Ltac	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	preserve_vars	null
red_of_sred: forall x y, Srs.red R x y -> red (trs_of_srs R) (term_of_string x) (term_of_string y). Proof. intros. do 3 destruct H. decomp H. subst x. subst y. rewrite !term_of_string_fill. apply red_rule. change (List.In (rule_of_srule (Srs.mkRule x0 x1)) (trs_of_srs R)). unfold trs_of_srs. apply in_map. exact H0. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	red_of_sred	null
rtc_red_of_sred: forall x y, Srs.red R # x y -> red (trs_of_srs R) # (term_of_string x) (term_of_string y). Proof. intros. elim H; intros. apply rt_step. apply red_of_sred. exact H0. apply rt_refl. eapply rt_trans. apply H1. exact H3. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	rtc_red_of_sred	null
sred_of_red: forall t u, red (trs_of_srs R) t u -> Srs.red R (string_of_term t) (string_of_term u). Proof. intros. redtac. subst. rewrite !string_of_term_fill, !string_of_term_sub, (rules_preserve_vars_var is_unary_sig (rules_preserve_vars_trs_of_srs R) lr). set (c' := mkContext (string_of_cont c) (string_of_term (s (var l)))). change (Srs.red R (SContext.fill c' (string_of_term l)) (SContext.fill c' (string_of_term r))). apply Srs.red_rule. clear c'. destruct (in_map_elim lr). destruct H. destruct x. unfold rule_of_srule in H0. simpl in H0. inversion H0. rewrite !string_of_term_epi. hyp. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	sred_of_red	null
rtc_sred_of_red: forall t u, red (trs_of_srs R) # t u -> Srs.red R # (string_of_term t) (string_of_term u). Proof. induction 1. apply rt_step. apply sred_of_red. hyp. apply rt_refl. apply rt_trans with (string_of_term y); hyp. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	rtc_sred_of_red	null
red_mod_of_sred_mod: forall x y, Srs.red_mod E R x y -> red_mod (trs_of_srs E) (trs_of_srs R) (term_of_string x) (term_of_string y). Proof. intros. do 2 destruct H. exists (term_of_string x0). split. apply rtc_red_of_sred. exact H. apply red_of_sred. exact H0. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	red_mod_of_sred_mod	null
WF_red_mod_of_WF_sred_mod: WF (red_mod (trs_of_srs E) (trs_of_srs R)) -> WF (Srs.red_mod E R). Proof. unfold WF. intro H. cut (forall t s, t = term_of_string s -> SN (Srs.red_mod E R) s). intros. apply H0 with (term_of_string x). refl. intro t. gen (H t). induction 1. intros. apply SN_intro. intros. apply H1 with (term_of_string y). 2: refl. subst x. apply red_mod_of_sred_mod. exact H3. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	WF_red_mod_of_WF_sred_mod	null
sred_mod_of_red_mod: forall t u, red_mod (trs_of_srs E) (trs_of_srs R) t u -> Srs.red_mod E R (string_of_term t) (string_of_term u). Proof. intros. do 2 destruct H. exists (string_of_term x); split. apply rtc_sred_of_red. hyp. apply sred_of_red. hyp. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	sred_mod_of_red_mod	preservation of termination
WF_sred_mod_of_WF_red_mod: WF (Srs.red_mod E R) -> WF (red_mod (trs_of_srs E) (trs_of_srs R)). Proof. intro. rewrite WF_red_mod0; is_unary; preserve_vars. cut (forall s, SN (Srs.red_mod E R) s -> forall t, maxvar t = 0 -> t = term_of_string s -> SN (red_mod0 (trs_of_srs E) (trs_of_srs R)) t). intros. intro t. destruct (eq_nat_dec (maxvar t) 0). 2: apply SN_intro; intros ? []; lia. apply H0 with (string_of_term t). apply H. hyp. rewrite term_of_string_epi. refl. hyp. induction 1; intros. apply SN_intro; intros. assert (h : maxvar y = 0). apply (red_mod0_maxvar (rules_preserve_vars_trs_of_srs E) (rules_preserve_vars_trs_of_srs R) H4). rewrite <- (term_of_string_epi h). apply H1 with (string_of_term y). 3: refl. 2: rewrite (term_of_string_epi h); hyp. rewrite <- (string_of_term_epi x). apply sred_mod_of_red_mod. subst. eapply incl_elim with (R := red_mod0 (trs_of_srs E) (trs_of_srs R)). apply red_mod0_incl_red_mod. hyp. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	WF_sred_mod_of_WF_red_mod	null
WF_red_mod: WF (Srs.red_mod E R) <-> WF (red_mod (trs_of_srs E) (trs_of_srs R)). Proof. split; intro. apply WF_sred_mod_of_WF_red_mod. hyp. apply WF_red_mod_of_WF_sred_mod. hyp. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	WF_red_mod	null
WF_red: WF (Srs.red R) <-> WF (red (trs_of_srs R)). Proof. rewrite <- Srs.red_mod_empty, <- red_mod_empty. apply WF_red_mod. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	WF_red	null
as_trs:= rewrite WF_red_mod \|\| rewrite WF_red.	Ltac	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]	Conversion/ATerm_of_String.v	as_trs	tactics for Rainbow
beq_statuss1 s2 := match s1, s2 with \| Lex, Lex \| Mul, Mul => true \| _, _ => false end.	Definition	Conversion	[ "From CoLoR Require Import LogicUtil ATerm VecUtil", "From CoLoR Require Import term_spec EqUtil", "From CoLoR Require Import NatUtil", "From Stdlib Require Import List Relations", "From CoLoR Require Import term SN ASubstitution", "From CoLoR Require Import rpo rpo_extension", "From CoLoR Require Import ARedPair ARelation RelUtil BoolUtil" ]	Conversion/Coccinelle.v	beq_status	decide compatibility of statuses wrt precedences
beq_status_ok: forall s1 s2, beq_status s1 s2 = true <-> s1 = s2. Proof. beq_symb_ok. Qed.	Lemma	Conversion	[ "From CoLoR Require Import LogicUtil ATerm VecUtil", "From CoLoR Require Import term_spec EqUtil", "From CoLoR Require Import NatUtil", "From Stdlib Require Import List Relations", "From CoLoR Require Import term SN ASubstitution", "From CoLoR Require Import rpo rpo_extension", "From CoLoR Require Import ARedPair ARelation RelUtil BoolUtil" ]	Conversion/Coccinelle.v	beq_status_ok	null
prec_eq_statuss p o := apply (bprec_eq_status_ok s p o); check_eq \|\| fail 10 "statuses incompatible with precedences".	Ltac	Conversion	[ "From CoLoR Require Import LogicUtil ATerm VecUtil", "From CoLoR Require Import term_spec EqUtil", "From CoLoR Require Import NatUtil", "From Stdlib Require Import List Relations", "From CoLoR Require Import term SN ASubstitution", "From CoLoR Require Import rpo rpo_extension", "From CoLoR Require Import ARedPair ARelation RelUtil BoolUtil" ]	Conversion/Coccinelle.v	prec_eq_status	null
as_srs_condFs_ok := match goal with \| \|- is_unary _ => is_unary Fs_ok \| \|- rules_preserve_vars _ => rules_preserve_vars \| \|- WF _ => idtac end.	Ltac	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN ListUtil Srs ATrs AUnary VecUtil", "From CoLoR Require Import AVariables" ]	Conversion/String_of_ATerm.v	as_srs_cond	null
as_srsFs_ok := (rewrite WF_red_mod \|\| rewrite WF_red); as_srs_cond Fs_ok.	Ltac	Conversion	[ "From CoLoR Require Import LogicUtil RelUtil SN ListUtil Srs ATrs AUnary VecUtil", "From CoLoR Require Import AVariables" ]	Conversion/String_of_ATerm.v	as_srs	null
co_scc:= check_eq \|\| fail 10 "not a co_scc". From CoLoR Require Import AVariables.	Ltac	DP	[ "From CoLoR Require Import AGraph Union ATrs RelUtil ListUtil LogicUtil SN", "From CoLoR Require Import AVariables" ]	DP/ADecomp.v	co_scc	tactics
graph_decompSig f d := apply WF_decomp_co_scc with (approx := f) (cs := d); [idtac \| rules_preserve_vars \| incl_rule Sig \|\| fail 10 "the decomposition does not contain all DPs" \| incl_rule Sig \|\| fail 10 "the decomposition contains a rule that is not a DP" \| check_eq \|\| fail 10 "the decomposition is not valid" \| unfold lforall; repeat split].	Ltac	DP	[ "From CoLoR Require Import AGraph Union ATrs RelUtil ListUtil LogicUtil SN", "From CoLoR Require Import AVariables" ]	DP/ADecomp.v	graph_decomp	null
negb_subterm(t u : term) := negb (bsubterm u t).	Definition	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	negb_subterm	definition of dependancy pairs
mkdp(S : rules) : rules := match S with \| nil => nil \| a :: S' => let (l, r) := a in map (mkRule l) (filter (negb_subterm l) (calls R r)) ++ mkdp S' end.	Fixpoint	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	mkdp	null
dp:= mkdp R.	Definition	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	dp	null
mkdp_app: forall l1 l2, mkdp (l1 ++ l2) = mkdp l1 ++ mkdp l2. Proof. induction l1; simpl; intros. refl. destruct a as [l r]. rewrite app_ass, IHl1. refl. Qed.	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	mkdp_app	null
mkdp_elim: forall l t S, In (mkRule l t) (mkdp S) -> exists r, In (mkRule l r) S /\ In t (calls R r) /\ negb_subterm l t = true. Proof. induction S; simpl; intros. contr. destruct a. rewrite in_app in H. destruct H. destruct (in_map_elim H). destruct H0. injection H1. intros. subst x. subst lhs. clear H1. rewrite filter_In in H0. destruct H0. exists rhs. intuition. destruct (IHS H). destruct H0. destruct H1. exists x. intuition. Qed. Arguments mkdp_elim [l t S] _.	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	mkdp_elim	null
dp_intro: forall l r t, In (mkRule l r) R -> In t (calls R r) -> negb_subterm l t = true -> In (mkRule l t) dp. Proof. intros. ded (in_elim H). do 2 destruct H2. ded (in_elim H0). do 2 destruct H3. unfold dp. rewrite H2, mkdp_app. simpl. rewrite H3, filter_app, map_app. simpl. rewrite H1. apply in_appr. apply in_appl. apply in_appr. apply in_eq. Qed.	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	dp_intro	basic properties
dp_elim: forall l t, In (mkRule l t) dp -> exists r, In (mkRule l r) R /\ In t (calls R r) /\ negb_subterm l t = true. Proof. intros. apply mkdp_elim. hyp. Qed. Arguments dp_elim [l t] _.	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	dp_elim	null
in_calls_hd_red_dp: forall l r t s, In (mkRule l r) R -> In t (calls R r) -> negb_subterm l t = true -> hd_red dp (sub s l) (sub s t). Proof. intros. exists l. exists t. exists s. intuition. eapply dp_intro. apply H. hyp. hyp. Qed.	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	in_calls_hd_red_dp	null
lhs_dp: incl (map lhs dp) (map lhs R). Proof. unfold incl. intros. ded (in_map_elim H). do 2 destruct H0. subst a. destruct x as [l r]. ded (dp_elim H0). do 2 destruct H1. change (In (lhs (mkRule l x)) (map lhs R)). apply in_map. exact H1. Qed. (*FIXME: to be finished	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	lhs_dp	null
is_notvar_lhs_dp_aux: forall S, forallb (@is_notvar_lhs Sig) (mkdp S) = true. Proof. induction S; simpl; intros. refl. destruct a. simpl. rewrite forallb_app. Qed.	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	is_notvar_lhs_dp_aux	null
is_notvar_lhs_dp: forallb (@is_notvar_lhs Sig) dp = true. Proof. unfold dp. Qed.*)	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	is_notvar_lhs_dp	null
chain:= int_red R # @ hd_red dp.	Definition	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	chain	dependency chains
in_calls_chain: forall l r t s, In (mkRule l r) R -> In t (calls R r) -> negb_subterm l t = true -> chain (sub s l) (sub s t). Proof. intros. unfold chain, compose. exists (sub s l). split. apply rt_refl. eapply in_calls_hd_red_dp. apply H. hyp. hyp. Qed.	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	in_calls_chain	null
gt_chain: forall f ts us v, Vrel1 (red R) ts us -> chain (Fun f us) v -> chain (Fun f ts) v. Proof. unfold chain, compose. intros. do 2 destruct H0. exists x. split. apply rt_trans with (y := Fun f us). apply rt_step. apply Vgt_prod_fun. exact H. exact H0. exact H1. Qed.	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	gt_chain	null
chain_mins t := chain s t /\ lforall (SN (red R)) (direct_subterms s) /\ lforall (SN (red R)) (direct_subterms t).	Definition	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	chain_min	minimal dependency chain (subterms are terminating)
chain_min_incl_chain: chain_min << chain. Proof. red. intros x y cmin. elim cmin. auto. Qed.	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	chain_min_incl_chain	null
gt_chain_min: forall f ts us v, Vrel1 (red R) ts us -> Vforall (SN (red R)) ts -> chain_min (Fun f us) v -> chain_min (Fun f ts) v. Proof. intros f ts us v gt_ts_us sn_ts chain_min_fus_v. unfold chain_min in chain_min_fus_v. destruct chain_min_fus_v as [chain_fus_v sn]. destruct sn as [esn_us esn_ts]. unfold chain_min. split. apply gt_chain with us; trivial. split. simpl. apply Vforall_lforall in sn_ts. trivial. trivial. Qed. Variable hyp1 : forallb (@is_notvar_lhs Sig) R = true. Variable hyp2 : rules_preserve_vars R. Arguments hyp2 [l r] _ _ _.	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	gt_chain_min	null
dp_elim_vars: forall l t, In (mkRule l t) dp -> exists r, In (mkRule l r) R /\ In t (calls R r) /\ negb_subterm l t = true /\ incl (vars t) (vars l). Proof. intros. destruct (dp_elim H). decomp H0. exists x. intuition. trans (vars x). unfold incl. intros. eapply subterm_eq_vars. eapply in_calls_subterm. apply H3. hyp. apply hyp2. hyp. Qed. Arguments dp_elim_vars [l t] _.	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	dp_elim_vars	dp preserves variables
dp_preserve_vars: rules_preserve_vars dp. Proof. unfold rules_preserve_vars. intros. destruct (dp_elim_vars H). intuition. Qed. Notation capa := (capa R). Notation cap := (cap R). Notation alien_sub := (alien_sub R). Notation SNR := (SN (red R)).	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	dp_preserve_vars	null
chain_min_fun: forall f, defined f R = true -> forall ts, SN chain_min (Fun f ts) -> Vforall SNR ts -> SNR (Fun f ts). Proof. cut (forall t, SN chain_min t -> forall f, defined f R = true -> forall ts, t = Fun f ts -> Vforall SNR ts -> SNR t). intros. apply H with (f := f) (ts := ts); (hyp \|\| refl). intros t H. elim H. clear t H. intros t H IH f H0 ts H1 Hsnts. assert (SN (Vrel1 (red R)) ts). apply Vforall_SN_rel1. hyp. gen IH. rewrite H1. elim H2. clear IH ts H1 Hsnts H2. intros ts H1 IH1 IH2. assert (Hsnts : Vforall SNR ts). apply SN_rel1_forall. apply SN_intro. hyp. clear H1. assert (H1 : forall y, Vrel1 (red R) ts y -> SNR (Fun f y)). intros. apply IH1. hyp. intros. eapply IH2. eapply gt_chain_min. apply H1. trivial. apply H2. apply H3. apply H4. hyp. clear IH1. apply SN_intro. intros u H2. redtac. destruct c; simpl in xl, yr. destruct (fun_eq_sub xl). destruct e as [ls]. rewrite H2, sub_fun in xl. Funeqtac. assert (Hsnsx : forall x, In x (vars l) -> SNR (s x)). intros. eapply sub_fun_sn with (f := f). subst l. apply H4. rewrite H3 in Hsnts. exact Hsnts. subst u. assert (r = sub (alien_sub r) (cap r)). apply sym_eq. apply (alien_sub_cap R). rewrite H4, sub_sub. apply no_call_sub_sn. hyp. apply calls_cap. intros. ded (vars_cap R H5). case (le_lt_dec x (maxvar r)); intro; unfold sub_comp, ACap.alien_sub. ded (vars_cap_inf R H5 l0). ded (hyp2 lr _ H7). rewrite fsub_inf. simpl. apply Hsnsx. hyp. hyp. rewrite (fsub_nth (aliens (capa r)) l0 H6). set (a := Vnth (aliens (capa r)) (lt_pm (k:=projT1 (capa r)) l0 H6)). assert (Fun f ts = sub s l). rewrite H3, H2. refl. assert (In a (calls R r)). apply aliens_incl_calls. unfold a. apply Vnth_in. ded (in_calls H8). destruct H9 as [g]. destruct H9 as [vs]. destruct H9. eapply calls_sn with (r := r). hyp. intros. apply Hsnsx. apply (hyp2 lr _ H11). intros h ws H13 H14. case_eq (negb_subterm l (Fun h ws)); intros. rename H11 into z. apply IH2 with (y := Fun h (Vmap (sub s) ws)) (f := h) (ts := Vmap (sub s) ws). unfold chain_min. split. rewrite H7, <- sub_fun. eapply in_calls_chain. apply lr. hyp. hyp. split. simpl. apply Vforall_lforall. trivial. simpl. apply Vforall_lforall. trivial. eapply in_calls_defined. apply H13. refl. hyp. revert H11. unfold negb_subterm. rewrite negb_lr. simpl negb. rewrite bsubterm_ok. intro H11. rewrite H2 in H11. destruct (subterm_fun_elim H11). destruct H12. apply subterm_eq_sn with (t := sub s x0). eapply Vforall_in. apply Hsnts. rewrite H3. apply Vin_map_intro. hyp. apply subterm_eq_sub. hyp. hyp. decomp e. subst l. is_var_lhs. Funeqtac. subst u. apply H1. rewrite H2. apply Vrel1_cast_intro. apply Vrel1_app_intro_l. apply Vrel1_cons_intro. left. split. eapply red_rule. hyp. refl. Qed.	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	chain_min_fun	null
chain_fun: forall f, defined f R = true -> forall ts, SN chain (Fun f ts) -> Vforall SNR ts -> SNR (Fun f ts). Proof. intros f defined_f ts sn_chain_f_ts sn_ts. apply chain_min_fun; auto. apply (SN_incl chain). apply chain_min_incl_chain. hyp. Qed.	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	chain_fun	null
WF_chain: WF chain -> WF (red R). Proof. intro Hwf. unfold WF. apply term_ind_forall. apply sn_var. apply hyp1. intro f. case_eq (defined f R); intros. apply chain_fun. hyp. apply Hwf. hyp. apply sn_args_sn_fun; auto. Qed.	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	WF_chain	null
chain_hd_red_mod: chain << hd_red_mod R dp. Proof. unfold chain, hd_red_mod. comp. apply rtc_incl. apply int_red_incl_red. Qed.	Lemma	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	chain_hd_red_mod	null
chain:= no_relative_rules; apply WF_chain; [ check_eq \|\| fail 10 "a LHS is a variable" \| rules_preserve_vars \| idtac].	Ltac	DP	[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]	DP/ADP.v	chain	null
hyp': rules_preserve_vars DP. Proof. apply dp_preserve_vars. exact hyp. Qed.	Lemma	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	hyp'	null
dp_grapha1 a2 := In a1 DP /\ In a2 DP /\ exists p, exists s, int_red R # (sub s (rhs a1)) (sub s (shift p (lhs a2))).	Definition	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	dp_graph	dependancy pairs graph
restricted_dp_graph: is_restricted dp_graph DP. Proof. unfold is_restricted, dp_graph, inclusion. intros. intuition. Qed.	Lemma	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	restricted_dp_graph	null
chain_dpa t u := In a DP /\ exists s, int_red R # t (sub s (lhs a)) /\ u = sub s (rhs a).	Definition	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	chain_dp	chain relation for a given dependency pair
chain_dp_chain: forall a, chain_dp a << Chain. Proof. unfold inclusion. intros. destruct H. do 2 destruct H0. exists (sub x0 (lhs a)). intuition. subst y. destruct a. simpl. apply hd_red_rule. exact H. Qed.	Lemma	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	chain_dp_chain	null
chain_chain_dp: forall t u, Chain t u -> exists a, chain_dp a t u. Proof. intros. do 2 destruct H. redtac. subst x. subst u. exists (mkRule l r). unfold chain_dp. simpl. intuition. exists s. intuition. Qed.	Lemma	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	chain_chain_dp	null
chain_dps(a : rule) (l : rules) : relation term := match l with \| nil => chain_dp a \| b :: m => chain_dp a @ chain_dps b m end.	Fixpoint	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	chain_dps	iteration of chain steps given a list of dependency pairs
chain_dps_app: forall l a b m, chain_dps a (l ++ b :: m) << chain_dps a l @ chain_dps b m. Proof. induction l; simpl; intros. refl. assoc. comp. apply IHl. Qed. Arguments chain_dps_app [l a b m] _ _ _.	Lemma	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	chain_dps_app	null
chain_dps_app': forall a l m b p, a :: l = m ++ b :: p -> chain_dps a l << chain_dps a (tail m) % @ chain_dps b p. Proof. intros. destruct m; simpl in H; injection H; unfold inclusion; intros. subst p. subst b. exists x. unfold clos_refl. intuition auto with . subst r. subst l. simpl. ded (chain_dps_app _ _ H2). do 2 destruct H0. exists x0. unfold clos_refl. intuition auto with . Qed. Arguments chain_dps_app' [a l m b p] _ [x y] _.	Lemma	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	chain_dps_app'	null
chain_dps_iter_chain: forall l a, chain_dps a l << iter Chain (length l). Proof. induction l; simpl; intros. apply chain_dp_chain. comp. apply chain_dp_chain. apply IHl. Qed.	Lemma	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	chain_dps_iter_chain	null
iter_chain_chain_dps: forall n t u, iter Chain n t u -> exists a, exists l, length l = n /\ chain_dps a l t u. Proof. induction n; simpl; intros. ded (chain_chain_dp H). destruct H0. exists x. exists nil. intuition. do 2 destruct H. ded (chain_chain_dp H). destruct H1. exists x0. ded (IHn _ _ H0). do 3 destruct H2. exists (x1 :: x2). simpl. intuition. exists x. intuition. Qed. Arguments iter_chain_chain_dps [n t u] _.	Lemma	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	iter_chain_chain_dps	null
chain_dp2_dp_graph: forall a1 a2 t u v, chain_dp a1 t u -> chain_dp a2 u v -> dp_graph a1 a2. Proof. intros. destruct a1 as [l0 r0]. destruct a2 as [l1 r1]. destruct H. simpl in H1. do 2 destruct H1. subst u. rename x into s0. destruct H0. simpl in H2. do 2 destruct H2. subst v. rename x into s1. unfold dp_graph. intuition. simpl. set (p := maxvar l0 + 1). exists p. (* take the union of s0 (restricted to [vars l0]) and [comp s1 (shift_inv_sub p l1)] (restricted to [vars (shift p l1)] *) set (s0' := restrict s0 (vars l0)). set (s1' := restrict (sub_comp s1 (shift_inv_sub p l1)) (vars (shift p l1))). set (s := ASubstitution.union s0' s1'). exists s. assert (compat s0' s1' (vars l0) (vars (shift p l1))). unfold compat. intros. ded (vars_max H3). ded (in_vars_shift_min H4). unfold p in H6. absurd (x <= maxvar l0). lia. hyp. assert (dom_incl s0' (vars l0)). unfold s0'. apply dom_incl_restrict. assert (dom_incl s1' (vars (shift p l1))). unfold s1'. apply dom_incl_restrict. assert (sub s r0 = sub s0' r0). unfold s. eapply sub_union1. apply H5. apply H3. apply hyp'. hyp. rewrite H6. assert (sub s0' r0 = sub s0 r0). unfold s0'. sym. apply sub_restrict_incl. apply hyp'. hyp. rewrite H7. assert (sub s (shift p l1) = sub s1' (shift p l1)). unfold s. eapply sub_union2. apply H4. apply H3. refl. rewrite H8. assert (sub s1' (shift p l1) = sub s1 l1). unfold s1'. rewrite <- sub_restrict, <- sub_shift. refl. rewrite H9. hyp. Qed.	Lemma	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	chain_dp2_dp_graph	two consecutive chain steps provide a dp_graph step
chain_dps_path_dp_graph: forall l a b t u, chain_dps a (l ++ b :: nil) t u -> path dp_graph a b l. Proof. induction l; simpl; intros; do 2 destruct H. eapply chain_dp2_dp_graph. apply H. apply H0. ded (IHl _ _ _ _ H0). intuition. destruct l; simpl in H0; do 2 destruct H0. eapply chain_dp2_dp_graph. apply H. apply H0. eapply chain_dp2_dp_graph. apply H. apply H0. Qed. Arguments chain_dps_path_dp_graph [l a b t u] _. (** hypotheses of the criterion based on cycles using the same reduction pair for every cycle *) Import ACompat. Variables (succ succ_eq : relation term) (Hredord : weak_rewrite_ordering succ succ_eq) (Hreword : rewrite_ordering succ_eq) (Habsorb : absorbs_left succ succ_eq) (HcompR : compat succ_eq R) (HcompDP : compat succ_eq (dp R)) (Hwf : WF succ) (Hcycle : forall a l, length l < length DP -> cycle_min dp_graph a l -> exists b, In b (a :: l) /\ succ (lhs b) (rhs b)). Notation eq_dec := (@eq_rule_dec Sig). Notation occur := (occur eq_dec).	Lemma	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	chain_dps_path_dp_graph	null
chain_dp_hd_red_mod: forall a, chain_dp a << hd_red_mod R (a::nil). Proof. unfold inclusion. intros. destruct H. do 2 destruct H0. subst y. destruct a. simpl. simpl in H0. exists (sub x0 lhs). split. apply incl_elim with (R := int_red R #). apply rtc_incl. apply int_red_incl_red. exact H0. apply hd_red_rule. simpl. auto. Qed.	Lemma	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	chain_dp_hd_red_mod	compatibility properties of chains
compat_chain_dp: forall a, chain_dp a << succ_eq#. Proof. intros. incl_trans (red_mod R DP). apply incl_trans with Chain; try incl_refl. apply chain_dp_chain. incl_trans (hd_red_mod R DP). apply chain_hd_red_mod. apply hd_red_mod_incl_red_mod. incl_trans (succ_eq!). apply compat_red_mod_tc; hyp. apply tc_incl_rtc. Qed.	Lemma	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	compat_chain_dp	null
compat_chain_dps: forall l a, chain_dps a l << succ_eq#. Proof. induction l; simpl; intros. apply compat_chain_dp. incl_trans (succ_eq# @ succ_eq#). comp. apply compat_chain_dp. apply IHl. apply comp_rtc_idem. Qed.	Lemma	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	compat_chain_dps	null
compat_chain_dp_strict: forall a, succ (lhs a) (rhs a) -> chain_dp a << succ. Proof. unfold inclusion. intros. destruct H0. do 2 destruct H1. subst y. apply (absorbs_left_rtc Habsorb). exists (sub x0 (lhs a)). split. apply incl_elim with (R := int_red R #). 2: exact H1. apply rtc_incl. incl_trans (red R). apply int_red_incl_red. apply compat_red; hyp. destruct Hredord. apply H2. exact H. Qed.	Lemma	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	compat_chain_dp_strict	null
WF_cycles: WF (chain R). Proof. set (n := length DP). apply WF_iter with (S n). intro. assert (Hwf' : WF (succ @ succ_eq#)). apply absorb_WF_modulo_r; hyp. gen (Hwf' x). induction 1. apply SN_intro; intros. apply H0. ded (iter_chain_chain_dps H1). do 3 destruct H2. assert (length x1 > 0). lia. ded (last_intro H4). do 3 destruct H5. clear H4. rewrite H5 in H3. ded (chain_dps_path_dp_graph H3). set (l' := x0 :: x2 ++ x3 :: nil). assert (exists z, occur z l' >= 2). unfold l'. eapply long_path_occur. apply restricted_dp_graph. apply H4. rewrite H6, H2. unfold n. lia. assert (exists l, exists a, exists m, l' = l ++ a :: m /\ succ (lhs a) (rhs a)). destruct H7. set (p := occur x4 l' - 2). assert (occur x4 l' = S (S p)). unfold p. lia. ded (occur_S eq_dec H8). do 3 destruct H9. destruct H10. clear H8. rewrite H9. ded (occur_S eq_dec H11). do 3 destruct H8. destruct H12. clear H11. clear H13. clear p. rewrite H8. rewrite H8 in H9. clear H8. clear H10. assert (cycle dp_graph x4 x7). unfold cycle. eapply sub_path. unfold l' in H9. apply H9. exact H4. ded (cycle_min_intro eq_dec H8). decomp H10. assert (length l' = n+2). unfold l'. simpl. rewrite <- H5. lia. rewrite H9 in H10. change (length (x5++(x4::x7)++x4::x8) = n+2) in H10. rewrite H11 in H10. repeat (rewrite length_app in H10; simpl in H10). unfold n in H10. assert (h : length x11 < length DP). unfold cycle_min in H14. destruct H14. assert (length (x10::x11) <= length DP). apply nodup_incl_length. apply eq_dec. exact H14. apply incl_appr_incl with (x10::nil). simpl. eapply restricted_path_incl. apply restricted_dp_graph. exact H13. simpl in H15. lia. clear H10. ded (Hcycle h H14). do 2 destruct H10. ded (in_elim H10). do 2 destruct H15. exists (x5++x9++x14). exists x13. exists (x15++x12++x4::x8). intuition. rewrite !app_ass. trans (x5++(x4::x7)++x4::x8). refl. rewrite H11, app_ass. trans (x5++x9++(x10::x11)++x12++x4::x8). simpl. rewrite app_ass. refl. rewrite H15, app_ass. refl. do 4 destruct H8. unfold l' in H8. ded (chain_dps_app' H8 H3). do 2 destruct H10. assert (succ_eq# x x7). destruct H10. subst x7. intuition auto with . eapply incl_elim. eapply compat_chain_dps. apply H10. assert ((succ @ succ_eq#) x7 y). destruct x6; simpl in H11. ded (compat_chain_dp_strict H9 H11). exists y. intuition auto with . do 2 destruct H11. exists x8. split. eapply incl_elim. apply (compat_chain_dp_strict H9). exact H11. eapply incl_elim. eapply compat_chain_dps. apply H13. do 2 destruct H13. exists x8. intuition. eapply incl_elim with (R := succ_eq# @ succ). apply absorbs_left_rtc. intuition. exists x7. intuition. Qed.	Lemma	DP	[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]	DP/ADPGraph.v	WF_cycles	proof of the criterion based on cycles
dpg_unif_N_correct:= (apply dpg_unif_N_mark_correct \|\| apply dpg_unif_N_correct); (check_eq \|\| fail 10 "a LHS is a variable").	Ltac	DP	[ "From Stdlib Require Import Lia Compare_dec", "From CoLoR Require Import ADecomp AUnif ARenCap ATrs ListUtil RelUtil AGraph" ]	DP/ADPUnif.v	dpg_unif_N_correct	tactics
dp_trans:= chain; eapply WF_incl; [apply hd_red_Mod_of_chain \| idtac].	Ltac	DP	[ "From CoLoR Require Import LogicUtil ATrs RelUtil RelSub SN AShift ADPGraph", "From CoLoR Require Export ADP", "From Stdlib Require Import List Lia" ]	DP/AGraph.v	dp_trans	tactics
hder1 r2 := In r1 D /\ In r2 D /\ match rhs r1 with \| Var _ => True \| Fun f _ => match lhs r2 with \| Var _ => True \| Fun g _ => f=g end end.	Definition	DP	[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]	DP/AHDE.v	hde	null
hde_restricted: is_restricted hde D. Proof. unfold is_restricted. intros. unfold hde in H; tauto. Qed. Notation eq_rule_dec := (@eq_rule_dec Sig). Notation eq_symb_dec := (@eq_symb_dec Sig).	Lemma	DP	[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]	DP/AHDE.v	hde_restricted	null
hde_dec: forall r1 r2, {hde r1 r2} + {~hde r1 r2}. Proof. intros. unfold hde. destruct (In_dec eq_rule_dec r1 D); try tauto. destruct (In_dec eq_rule_dec r2 D); try tauto. destruct (rhs r1). left; auto. destruct (lhs r2); auto. destruct (eq_symb_dec f f0); try tauto. Defined.	Lemma	DP	[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]	DP/AHDE.v	hde_dec	null
int_red_hd_rules_graph_incl_hde: hd_rules_graph (int_red R #) D << hde D. Proof. unfold inclusion. intros. destruct x. destruct y. destruct H. destruct H0. do 2 destruct H1. unfold hde. destruct lhs0; destruct rhs; simpl; auto. intuition; auto. ded (int_red_rtc_preserve_hd H1). destruct H2. simpl in H2. inversion H2; auto. do 4 destruct H2. inversion H2. inversion H3. congruence. Qed.	Lemma	DP	[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]	DP/AHDE.v	int_red_hd_rules_graph_incl_hde	null
dup_hd_rules_graph_incl_hde: hd_rules_graph (red R #) D << hde D. Proof. unfold inclusion. intros. apply (@int_red_hd_rules_graph_incl_hde _ R D x y). destruct H. decomp H0. rewrite forallb_forall in hd_hyp. ded (hd_hyp _ H). compute in H0. destruct x. destruct rhs. discr. destruct f. unfold hd_rules_graph. intuition. exists x0. exists x1. simpl rhs. rewrite sub_fun. apply dup_int_rules_int_red_rtc; hyp. discr. Qed.	Lemma	DP	[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]	DP/AHDE.v	dup_hd_rules_graph_incl_hde	null
hd_eq(u v : term Sig) := match u with \| Var _ => true \| Fun f _ => match v with \| Var _ => true \| Fun g _ => beq_symb f g end end. Notation mem := (mem (@beq_rule Sig)). Notation mem_ok := (mem_ok (@beq_rule_ok Sig)).	Definition	DP	[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]	DP/AHDE.v	hd_eq	null
hde_boolr1 r2 := mem r1 D && mem r2 D && hd_eq (rhs r1) (lhs r2).	Definition	DP	[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]	DP/AHDE.v	hde_bool	null
hde_bool_correct_aux: forall x y, hde D x y <-> Graph hde_bool x y. Proof. intros x y. unfold hde, hde_bool, hd_eq, Graph; simpl. pose proof (fun s x y => proj1 (@beq_symb_ok s x y)). pose proof (fun s x y => proj2 (@beq_symb_ok s x y)). rewrite <- !mem_ok. destruct (rhs x); bool; intuition auto with core; repeat match goal with \| H : _ && _ = true \|- _ => apply andb_elim in H; case H as [] \| H : _ = true \|- _ => rewrite H; cbn [andb] \| H : match ?x in term _ return _ with _ => _ end \|- _ => case x eqn:? in ; subst \| H : match ?x in term _ return _ with _ => _ end = true \|- _ => case x eqn:? in ; subst end; auto. Qed.	Lemma	DP	[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]	DP/AHDE.v	hde_bool_correct_aux	null
hde_bool_correct: hd_rules_graph (int_red R #) D << Graph (hde_bool D). Proof. incl_trans (hde D). apply int_red_hd_rules_graph_incl_hde. intros x y. rewrite hde_bool_correct_aux. auto. Qed. Notation Sig' := (dup_sig Sig). Notation R' := (dup_int_rules R). Variable hyp : forallb (@is_notvar_lhs Sig') R' = true.	Lemma	DP	[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]	DP/AHDE.v	hde_bool_correct	null
hde_bool_mark_correct: hd_rules_graph (red (dup_int_rules R) #) (dup_hd_rules D) << Graph (hde_bool (dup_hd_rules D)). Proof. incl_trans (hde (dup_hd_rules D)). 2:{ intros x y. rewrite hde_bool_correct_aux. auto. } intros x y h. destruct h. decomp H0. unfold hde. intuition. destruct (in_map_elim H). destruct H0. destruct x2. unfold dup_hd_rule in H3. simpl in H3. subst x. simpl in . destruct (in_map_elim H1). destruct H3. destruct x. unfold dup_hd_rule in H4. simpl in H4. subst y. simpl in . destruct rhs; simpl. trivial. destruct lhs0; simpl. trivial. simpl dup_hd_term in H2. unfold shift in H2. rewrite !sub_fun in H2. destruct (rtc_red_dup_int_hd_symb hyp H2). Funeqtac. auto. Qed.	Lemma	DP	[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]	DP/AHDE.v	hde_bool_mark_correct	null
hde_bool_correct:= (apply hde_bool_mark_correct \|\| apply hde_bool_correct); (check_eq \|\| fail 10 "a LHS is a variable").	Ltac	DP	[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]	DP/AHDE.v	hde_bool_correct	tactics
use_SCC_tagh M S R t := let x := fresh in (set (x := SCC_tag_fast h M t); norm_in x (SCC_tag_fast h M t); match eval compute in x with \| Some ?X1 => let Hi := fresh in (assert (Hi : X1 < length R); [ norm (length R); lia \| let L := fresh in (set (L := SCC_list_fast R M Hi); norm_in L (SCC_list_fast R M Hi); assert (WF (hd_red_Mod S L)); subst L; clear Hi; clear x)]) end).	Ltac	DP	[ "From Stdlib Require Import Permutation Multiset Setoid PermutSetoid", "From CoLoR Require Import SCCTopoOrdering AGraph ATrs RelUtil RelSub BoundNat" ]	DP/ASCCUnion.v	use_SCC_tag	tactics
use_SCC_hypM l := let b := fresh "b" in (set (b := Vnth (Vnth M l) l); norm_in b (Vnth (Vnth M l) l); match eval compute in b with \| false => apply WF_hd_red_Mod_SCC_fast_trivial with (Hi:=l); eauto \| true => eapply WF_incl; [eapply hd_red_Mod_SCC'_hd_red_Mod_fast with (Hi:=l); auto \| auto] end).	Ltac	DP	[ "From Stdlib Require Import Permutation Multiset Setoid PermutSetoid", "From CoLoR Require Import SCCTopoOrdering AGraph ATrs RelUtil RelSub BoundNat" ]	DP/ASCCUnion.v	use_SCC_hyp	null

Ltac

Conversion

[ "From CoLoR Require Import Srs ATrs SReverse ATerm_of_String String_of_ATerm", "From CoLoR Require Import AVariables" ]

Conversion/AReverse.v

rev_tac_cond

null

rev_tacFs_ok := (rewrite <- WF_red_rev_eq || rewrite <- WF_red_mod_rev_eq); rev_tac_cond Fs_ok.

Ltac

Conversion

[ "From CoLoR Require Import Srs ATrs SReverse ATerm_of_String String_of_ATerm", "From CoLoR Require Import AVariables" ]

Conversion/AReverse.v

rev_tac

null

ar(_ : letter) := 1. Import ATrs.

Definition

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

ar

corresponding algebraic signature

ASig_of_SSig:= mkSignature ar (@VSignature.beq_symb_ok SSig). Notation ASig := ASig_of_SSig. Notation term := (term ASig). Notation terms := (vector term). Notation context := (context ASig). Import AUnary.

Definition

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

ASig_of_SSig

null

is_unary_sig: is_unary ASig. Proof. intro. refl. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

is_unary_sig

null

arity:= arity1 is_unary_sig.

Ltac

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

arity

null

is_unary:= try (exact is_unary_sig). Notation Fun1 := (Fun1 is_unary_sig). Notation Cont1 := (Cont1 is_unary_sig). Notation cont := (cont is_unary_sig). Notation term_ind_forall := (term_ind_forall is_unary_sig).

Ltac

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

is_unary

null

term_of_string(s : string) : term := match s with | List.nil => Var 0 | a :: w => @Fun ASig a (Vcons (term_of_string w) Vnil) end.

Fixpoint

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

term_of_string

conversion string <-> term

var_term_of_string: forall s, var (term_of_string s) = 0. Proof. induction s; auto. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

var_term_of_string

null

reset_term_of_string: forall s, reset (term_of_string s) = term_of_string s. Proof. induction s. refl. unfold reset. simpl. apply args_eq. fold (reset (term_of_string s)). rewrite IHs. refl. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

reset_term_of_string

null

string_of_term(t : term) : string := match t with | Var _ => List.nil | Fun f ts => f :: Vmap_first List.nil string_of_term ts end.

Fixpoint

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

string_of_term

null

string_of_term_epi: forall s, string_of_term (term_of_string s) = s. Proof. induction s; simpl; intros. refl. rewrite IHs. refl. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

string_of_term_epi

null

term_of_string_epi: forall t, maxvar t = 0 -> term_of_string (string_of_term t) = t. Proof. intro t; pattern t; apply term_ind_forall; clear t. simpl in *. intros. subst. refl. intros f t IH. unfold AUnary.Fun1. simpl. rewrite (Vcast_cons (is_unary_sig f)), Vcast_refl. simpl. rewrite max_0_r. intro h. rewrite (IH h). refl. Qed. Arguments term_of_string_epi [t] _.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

term_of_string_epi

null

string_of_term_sub: forall s l, string_of_term (sub s l) = string_of_term l ++ string_of_term (s (var l)). Proof. intro s. apply term_ind_forall. refl. intros f t IH. rewrite sub_fun1. simpl. rewrite !(Vcast_cons (is_unary_sig f)). simpl. rewrite IH. refl. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

string_of_term_sub

null

string_of_cont(c : context) : string := match c with | Hole => List.nil | @Cont _ f _ _ _ _ d _ => f :: string_of_cont d end.

Fixpoint

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

string_of_cont

conversion string <-> context

string_of_cont_cont: forall t, string_of_cont (cont t) = string_of_term t. Proof. apply term_ind_forall. refl. intros f t IH. simpl. rewrite !(Vcast_cons (is_unary_sig f)). simpl. rewrite IH. refl. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

string_of_cont_cont

null

string_of_term_fill: forall t c, string_of_term (fill c t) = string_of_cont c ++ string_of_term t. Proof. induction c. refl. simpl. rewrite Vmap_first_cast. arity. simpl. rewrite IHc. refl. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

string_of_term_fill

null

cont_of_string(s : string) : context := match s with | List.nil => Hole | a :: w => Cont1 a (cont_of_string w) end.

Fixpoint

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

cont_of_string

null

sub_of_strings := single 0 (term_of_string s).

Definition

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

sub_of_string

conversion string <-> substitution

term_of_string_fill: forall c s, term_of_string (SContext.fill c s) = fill (cont_of_string (lft c)) (sub (sub_of_string (rgt c)) (term_of_string s)). Proof. intros [l r] s. elim l; unfold SContext.fill; simpl. elim s. refl. intros. simpl. rewrite H. refl. intros. rewrite H, (Vcast_cons (cont_aux is_unary_sig a)), Vcast_refl. refl. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

term_of_string_fill

null

term_of_string_app: forall s1 s2, term_of_string (s1 ++ s2) = sub (sub_of_string s2) (term_of_string s1). Proof. induction s1. refl. intro. simpl. apply args_eq. rewrite IHs1. refl. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

term_of_string_app

null

rule_of_srule(x : srule) := mkRule (term_of_string (Srs.lhs x)) (term_of_string (Srs.rhs x)).

Definition

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

rule_of_srule

rules

trs_of_srsR := List.map rule_of_srule R.

Definition

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

trs_of_srs

null

rules_preserve_vars_trs_of_srs: forall R, rules_preserve_vars (trs_of_srs R). Proof. unfold rules_preserve_vars. induction R; simpl; intuition. unfold rule_of_srule in H0. destruct a. simpl in H0. inversion H0. rewrite !vars_var; is_unary. rewrite !var_term_of_string. unfold incl; simpl; intuition. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

rules_preserve_vars_trs_of_srs

null

preserve_vars:= try (apply rules_preserve_vars_trs_of_srs).

Ltac

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

preserve_vars

null

red_of_sred: forall x y, Srs.red R x y -> red (trs_of_srs R) (term_of_string x) (term_of_string y). Proof. intros. do 3 destruct H. decomp H. subst x. subst y. rewrite !term_of_string_fill. apply red_rule. change (List.In (rule_of_srule (Srs.mkRule x0 x1)) (trs_of_srs R)). unfold trs_of_srs. apply in_map. exact H0. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

red_of_sred

null

rtc_red_of_sred: forall x y, Srs.red R # x y -> red (trs_of_srs R) # (term_of_string x) (term_of_string y). Proof. intros. elim H; intros. apply rt_step. apply red_of_sred. exact H0. apply rt_refl. eapply rt_trans. apply H1. exact H3. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

rtc_red_of_sred

null

sred_of_red: forall t u, red (trs_of_srs R) t u -> Srs.red R (string_of_term t) (string_of_term u). Proof. intros. redtac. subst. rewrite !string_of_term_fill, !string_of_term_sub, (rules_preserve_vars_var is_unary_sig (rules_preserve_vars_trs_of_srs R) lr). set (c' := mkContext (string_of_cont c) (string_of_term (s (var l)))). change (Srs.red R (SContext.fill c' (string_of_term l)) (SContext.fill c' (string_of_term r))). apply Srs.red_rule. clear c'. destruct (in_map_elim lr). destruct H. destruct x. unfold rule_of_srule in H0. simpl in H0. inversion H0. rewrite !string_of_term_epi. hyp. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

sred_of_red

null

rtc_sred_of_red: forall t u, red (trs_of_srs R) # t u -> Srs.red R # (string_of_term t) (string_of_term u). Proof. induction 1. apply rt_step. apply sred_of_red. hyp. apply rt_refl. apply rt_trans with (string_of_term y); hyp. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

rtc_sred_of_red

null

red_mod_of_sred_mod: forall x y, Srs.red_mod E R x y -> red_mod (trs_of_srs E) (trs_of_srs R) (term_of_string x) (term_of_string y). Proof. intros. do 2 destruct H. exists (term_of_string x0). split. apply rtc_red_of_sred. exact H. apply red_of_sred. exact H0. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

red_mod_of_sred_mod

null

WF_red_mod_of_WF_sred_mod: WF (red_mod (trs_of_srs E) (trs_of_srs R)) -> WF (Srs.red_mod E R). Proof. unfold WF. intro H. cut (forall t s, t = term_of_string s -> SN (Srs.red_mod E R) s). intros. apply H0 with (term_of_string x). refl. intro t. gen (H t). induction 1. intros. apply SN_intro. intros. apply H1 with (term_of_string y). 2: refl. subst x. apply red_mod_of_sred_mod. exact H3. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

WF_red_mod_of_WF_sred_mod

null

sred_mod_of_red_mod: forall t u, red_mod (trs_of_srs E) (trs_of_srs R) t u -> Srs.red_mod E R (string_of_term t) (string_of_term u). Proof. intros. do 2 destruct H. exists (string_of_term x); split. apply rtc_sred_of_red. hyp. apply sred_of_red. hyp. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

sred_mod_of_red_mod

preservation of termination

WF_sred_mod_of_WF_red_mod: WF (Srs.red_mod E R) -> WF (red_mod (trs_of_srs E) (trs_of_srs R)). Proof. intro. rewrite WF_red_mod0; is_unary; preserve_vars. cut (forall s, SN (Srs.red_mod E R) s -> forall t, maxvar t = 0 -> t = term_of_string s -> SN (red_mod0 (trs_of_srs E) (trs_of_srs R)) t). intros. intro t. destruct (eq_nat_dec (maxvar t) 0). 2: apply SN_intro; intros ? []; lia. apply H0 with (string_of_term t). apply H. hyp. rewrite term_of_string_epi. refl. hyp. induction 1; intros. apply SN_intro; intros. assert (h : maxvar y = 0). apply (red_mod0_maxvar (rules_preserve_vars_trs_of_srs E) (rules_preserve_vars_trs_of_srs R) H4). rewrite <- (term_of_string_epi h). apply H1 with (string_of_term y). 3: refl. 2: rewrite (term_of_string_epi h); hyp. rewrite <- (string_of_term_epi x). apply sred_mod_of_red_mod. subst. eapply incl_elim with (R := red_mod0 (trs_of_srs E) (trs_of_srs R)). apply red_mod0_incl_red_mod. hyp. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

WF_sred_mod_of_WF_red_mod

null

WF_red_mod: WF (Srs.red_mod E R) <-> WF (red_mod (trs_of_srs E) (trs_of_srs R)). Proof. split; intro. apply WF_sred_mod_of_WF_red_mod. hyp. apply WF_red_mod_of_WF_sred_mod. hyp. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

WF_red_mod

null

WF_red: WF (Srs.red R) <-> WF (red (trs_of_srs R)). Proof. rewrite <- Srs.red_mod_empty, <- red_mod_empty. apply WF_red_mod. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

WF_red

null

as_trs:= rewrite WF_red_mod || rewrite WF_red.

Ltac

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN VecUtil ListUtil NatUtil" ]

Conversion/ATerm_of_String.v

as_trs

tactics for Rainbow

beq_statuss1 s2 := match s1, s2 with | Lex, Lex | Mul, Mul => true | _, _ => false end.

Definition

Conversion

[ "From CoLoR Require Import LogicUtil ATerm VecUtil", "From CoLoR Require Import term_spec EqUtil", "From CoLoR Require Import NatUtil", "From Stdlib Require Import List Relations", "From CoLoR Require Import term SN ASubstitution", "From CoLoR Require Import rpo rpo_extension", "From CoLoR Require Import ARedPair ARelation RelUtil BoolUtil" ]

Conversion/Coccinelle.v

beq_status

decide compatibility of statuses wrt precedences

beq_status_ok: forall s1 s2, beq_status s1 s2 = true <-> s1 = s2. Proof. beq_symb_ok. Qed.

Lemma

Conversion

[ "From CoLoR Require Import LogicUtil ATerm VecUtil", "From CoLoR Require Import term_spec EqUtil", "From CoLoR Require Import NatUtil", "From Stdlib Require Import List Relations", "From CoLoR Require Import term SN ASubstitution", "From CoLoR Require Import rpo rpo_extension", "From CoLoR Require Import ARedPair ARelation RelUtil BoolUtil" ]

Conversion/Coccinelle.v

beq_status_ok

null

prec_eq_statuss p o := apply (bprec_eq_status_ok s p o); check_eq || fail 10 "statuses incompatible with precedences".

Ltac

Conversion

[ "From CoLoR Require Import LogicUtil ATerm VecUtil", "From CoLoR Require Import term_spec EqUtil", "From CoLoR Require Import NatUtil", "From Stdlib Require Import List Relations", "From CoLoR Require Import term SN ASubstitution", "From CoLoR Require Import rpo rpo_extension", "From CoLoR Require Import ARedPair ARelation RelUtil BoolUtil" ]

Conversion/Coccinelle.v

prec_eq_status

null

Ltac

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN ListUtil Srs ATrs AUnary VecUtil", "From CoLoR Require Import AVariables" ]

Conversion/String_of_ATerm.v

as_srs_cond

null

as_srsFs_ok := (rewrite WF_red_mod || rewrite WF_red); as_srs_cond Fs_ok.

Ltac

Conversion

[ "From CoLoR Require Import LogicUtil RelUtil SN ListUtil Srs ATrs AUnary VecUtil", "From CoLoR Require Import AVariables" ]

Conversion/String_of_ATerm.v

as_srs

null

co_scc:= check_eq || fail 10 "not a co_scc". From CoLoR Require Import AVariables.

Ltac

DP

[ "From CoLoR Require Import AGraph Union ATrs RelUtil ListUtil LogicUtil SN", "From CoLoR Require Import AVariables" ]

DP/ADecomp.v

co_scc

tactics

graph_decompSig f d := apply WF_decomp_co_scc with (approx := f) (cs := d); [idtac | rules_preserve_vars | incl_rule Sig || fail 10 "the decomposition does not contain all DPs" | incl_rule Sig || fail 10 "the decomposition contains a rule that is not a DP" | check_eq || fail 10 "the decomposition is not valid" | unfold lforall; repeat split].

Ltac

DP

[ "From CoLoR Require Import AGraph Union ATrs RelUtil ListUtil LogicUtil SN", "From CoLoR Require Import AVariables" ]

DP/ADecomp.v

graph_decomp

null

negb_subterm(t u : term) := negb (bsubterm u t).

Definition

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

negb_subterm

definition of dependancy pairs

mkdp(S : rules) : rules := match S with | nil => nil | a :: S' => let (l, r) := a in map (mkRule l) (filter (negb_subterm l) (calls R r)) ++ mkdp S' end.

Fixpoint

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

mkdp

null

dp:= mkdp R.

Definition

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

dp

null

mkdp_app: forall l1 l2, mkdp (l1 ++ l2) = mkdp l1 ++ mkdp l2. Proof. induction l1; simpl; intros. refl. destruct a as [l r]. rewrite app_ass, IHl1. refl. Qed.

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

mkdp_app

null

mkdp_elim: forall l t S, In (mkRule l t) (mkdp S) -> exists r, In (mkRule l r) S /\ In t (calls R r) /\ negb_subterm l t = true. Proof. induction S; simpl; intros. contr. destruct a. rewrite in_app in H. destruct H. destruct (in_map_elim H). destruct H0. injection H1. intros. subst x. subst lhs. clear H1. rewrite filter_In in H0. destruct H0. exists rhs. intuition. destruct (IHS H). destruct H0. destruct H1. exists x. intuition. Qed. Arguments mkdp_elim [l t S] _.

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

mkdp_elim

null

dp_intro: forall l r t, In (mkRule l r) R -> In t (calls R r) -> negb_subterm l t = true -> In (mkRule l t) dp. Proof. intros. ded (in_elim H). do 2 destruct H2. ded (in_elim H0). do 2 destruct H3. unfold dp. rewrite H2, mkdp_app. simpl. rewrite H3, filter_app, map_app. simpl. rewrite H1. apply in_appr. apply in_appl. apply in_appr. apply in_eq. Qed.

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

dp_intro

basic properties

dp_elim: forall l t, In (mkRule l t) dp -> exists r, In (mkRule l r) R /\ In t (calls R r) /\ negb_subterm l t = true. Proof. intros. apply mkdp_elim. hyp. Qed. Arguments dp_elim [l t] _.

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

dp_elim

null

in_calls_hd_red_dp: forall l r t s, In (mkRule l r) R -> In t (calls R r) -> negb_subterm l t = true -> hd_red dp (sub s l) (sub s t). Proof. intros. exists l. exists t. exists s. intuition. eapply dp_intro. apply H. hyp. hyp. Qed.

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

in_calls_hd_red_dp

null

lhs_dp: incl (map lhs dp) (map lhs R). Proof. unfold incl. intros. ded (in_map_elim H). do 2 destruct H0. subst a. destruct x as [l r]. ded (dp_elim H0). do 2 destruct H1. change (In (lhs (mkRule l x)) (map lhs R)). apply in_map. exact H1. Qed. (*FIXME: to be finished

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

lhs_dp

null

is_notvar_lhs_dp_aux: forall S, forallb (@is_notvar_lhs Sig) (mkdp S) = true. Proof. induction S; simpl; intros. refl. destruct a. simpl. rewrite forallb_app. Qed.

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

is_notvar_lhs_dp_aux

null

is_notvar_lhs_dp: forallb (@is_notvar_lhs Sig) dp = true. Proof. unfold dp. Qed.*)

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

is_notvar_lhs_dp

null

chain:= int_red R # @ hd_red dp.

Definition

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

chain

dependency chains

in_calls_chain: forall l r t s, In (mkRule l r) R -> In t (calls R r) -> negb_subterm l t = true -> chain (sub s l) (sub s t). Proof. intros. unfold chain, compose. exists (sub s l). split. apply rt_refl. eapply in_calls_hd_red_dp. apply H. hyp. hyp. Qed.

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

in_calls_chain

null

gt_chain: forall f ts us v, Vrel1 (red R) ts us -> chain (Fun f us) v -> chain (Fun f ts) v. Proof. unfold chain, compose. intros. do 2 destruct H0. exists x. split. apply rt_trans with (y := Fun f us). apply rt_step. apply Vgt_prod_fun. exact H. exact H0. exact H1. Qed.

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

gt_chain

null

chain_mins t := chain s t /\ lforall (SN (red R)) (direct_subterms s) /\ lforall (SN (red R)) (direct_subterms t).

Definition

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

chain_min

minimal dependency chain (subterms are terminating)

chain_min_incl_chain: chain_min << chain. Proof. red. intros x y cmin. elim cmin. auto. Qed.

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

chain_min_incl_chain

null

gt_chain_min: forall f ts us v, Vrel1 (red R) ts us -> Vforall (SN (red R)) ts -> chain_min (Fun f us) v -> chain_min (Fun f ts) v. Proof. intros f ts us v gt_ts_us sn_ts chain_min_fus_v. unfold chain_min in chain_min_fus_v. destruct chain_min_fus_v as [chain_fus_v sn]. destruct sn as [esn_us esn_ts]. unfold chain_min. split. apply gt_chain with us; trivial. split. simpl. apply Vforall_lforall in sn_ts. trivial. trivial. Qed. Variable hyp1 : forallb (@is_notvar_lhs Sig) R = true. Variable hyp2 : rules_preserve_vars R. Arguments hyp2 [l r] _ _ _.

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

gt_chain_min

null

dp_elim_vars: forall l t, In (mkRule l t) dp -> exists r, In (mkRule l r) R /\ In t (calls R r) /\ negb_subterm l t = true /\ incl (vars t) (vars l). Proof. intros. destruct (dp_elim H). decomp H0. exists x. intuition. trans (vars x). unfold incl. intros. eapply subterm_eq_vars. eapply in_calls_subterm. apply H3. hyp. apply hyp2. hyp. Qed. Arguments dp_elim_vars [l t] _.

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

dp_elim_vars

dp preserves variables

dp_preserve_vars: rules_preserve_vars dp. Proof. unfold rules_preserve_vars. intros. destruct (dp_elim_vars H). intuition. Qed. Notation capa := (capa R). Notation cap := (cap R). Notation alien_sub := (alien_sub R). Notation SNR := (SN (red R)).

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

dp_preserve_vars

null

chain_min_fun: forall f, defined f R = true -> forall ts, SN chain_min (Fun f ts) -> Vforall SNR ts -> SNR (Fun f ts). Proof. cut (forall t, SN chain_min t -> forall f, defined f R = true -> forall ts, t = Fun f ts -> Vforall SNR ts -> SNR t). intros. apply H with (f := f) (ts := ts); (hyp || refl). intros t H. elim H. clear t H. intros t H IH f H0 ts H1 Hsnts. assert (SN (Vrel1 (red R)) ts). apply Vforall_SN_rel1. hyp. gen IH. rewrite H1. elim H2. clear IH ts H1 Hsnts H2. intros ts H1 IH1 IH2. assert (Hsnts : Vforall SNR ts). apply SN_rel1_forall. apply SN_intro. hyp. clear H1. assert (H1 : forall y, Vrel1 (red R) ts y -> SNR (Fun f y)). intros. apply IH1. hyp. intros. eapply IH2. eapply gt_chain_min. apply H1. trivial. apply H2. apply H3. apply H4. hyp. clear IH1. apply SN_intro. intros u H2. redtac. destruct c; simpl in xl, yr. destruct (fun_eq_sub xl). destruct e as [ls]. rewrite H2, sub_fun in xl. Funeqtac. assert (Hsnsx : forall x, In x (vars l) -> SNR (s x)). intros. eapply sub_fun_sn with (f := f). subst l. apply H4. rewrite H3 in Hsnts. exact Hsnts. subst u. assert (r = sub (alien_sub r) (cap r)). apply sym_eq. apply (alien_sub_cap R). rewrite H4, sub_sub. apply no_call_sub_sn. hyp. apply calls_cap. intros. ded (vars_cap R H5). case (le_lt_dec x (maxvar r)); intro; unfold sub_comp, ACap.alien_sub. ded (vars_cap_inf R H5 l0). ded (hyp2 lr _ H7). rewrite fsub_inf. simpl. apply Hsnsx. hyp. hyp. rewrite (fsub_nth (aliens (capa r)) l0 H6). set (a := Vnth (aliens (capa r)) (lt_pm (k:=projT1 (capa r)) l0 H6)). assert (Fun f ts = sub s l). rewrite H3, H2. refl. assert (In a (calls R r)). apply aliens_incl_calls. unfold a. apply Vnth_in. ded (in_calls H8). destruct H9 as [g]. destruct H9 as [vs]. destruct H9. eapply calls_sn with (r := r). hyp. intros. apply Hsnsx. apply (hyp2 lr _ H11). intros h ws H13 H14. case_eq (negb_subterm l (Fun h ws)); intros. rename H11 into z. apply IH2 with (y := Fun h (Vmap (sub s) ws)) (f := h) (ts := Vmap (sub s) ws). unfold chain_min. split. rewrite H7, <- sub_fun. eapply in_calls_chain. apply lr. hyp. hyp. split. simpl. apply Vforall_lforall. trivial. simpl. apply Vforall_lforall. trivial. eapply in_calls_defined. apply H13. refl. hyp. revert H11. unfold negb_subterm. rewrite negb_lr. simpl negb. rewrite bsubterm_ok. intro H11. rewrite H2 in H11. destruct (subterm_fun_elim H11). destruct H12. apply subterm_eq_sn with (t := sub s x0). eapply Vforall_in. apply Hsnts. rewrite H3. apply Vin_map_intro. hyp. apply subterm_eq_sub. hyp. hyp. decomp e. subst l. is_var_lhs. Funeqtac. subst u. apply H1. rewrite H2. apply Vrel1_cast_intro. apply Vrel1_app_intro_l. apply Vrel1_cons_intro. left. split. eapply red_rule. hyp. refl. Qed.

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

chain_min_fun

null

chain_fun: forall f, defined f R = true -> forall ts, SN chain (Fun f ts) -> Vforall SNR ts -> SNR (Fun f ts). Proof. intros f defined_f ts sn_chain_f_ts sn_ts. apply chain_min_fun; auto. apply (SN_incl chain). apply chain_min_incl_chain. hyp. Qed.

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

chain_fun

null

WF_chain: WF chain -> WF (red R). Proof. intro Hwf. unfold WF. apply term_ind_forall. apply sn_var. apply hyp1. intro f. case_eq (defined f R); intros. apply chain_fun. hyp. apply Hwf. hyp. apply sn_args_sn_fun; auto. Qed.

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

WF_chain

null

chain_hd_red_mod: chain << hd_red_mod R dp. Proof. unfold chain, hd_red_mod. comp. apply rtc_incl. apply int_red_incl_red. Qed.

Lemma

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

chain_hd_red_mod

null

chain:= no_relative_rules; apply WF_chain; [ check_eq || fail 10 "a LHS is a variable" | rules_preserve_vars | idtac].

Ltac

DP

[ "From CoLoR Require Import LogicUtil ATrs ACalls ACap ASN RelUtil ListUtil", "From CoLoR Require Import AVariables" ]

DP/ADP.v

chain

null

hyp': rules_preserve_vars DP. Proof. apply dp_preserve_vars. exact hyp. Qed.

Lemma

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

hyp'

null

dp_grapha1 a2 := In a1 DP /\ In a2 DP /\ exists p, exists s, int_red R # (sub s (rhs a1)) (sub s (shift p (lhs a2))).

Definition

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

dp_graph

dependancy pairs graph

restricted_dp_graph: is_restricted dp_graph DP. Proof. unfold is_restricted, dp_graph, inclusion. intros. intuition. Qed.

Lemma

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

restricted_dp_graph

null

chain_dpa t u := In a DP /\ exists s, int_red R # t (sub s (lhs a)) /\ u = sub s (rhs a).

Definition

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

chain_dp

chain relation for a given dependency pair

chain_dp_chain: forall a, chain_dp a << Chain. Proof. unfold inclusion. intros. destruct H. do 2 destruct H0. exists (sub x0 (lhs a)). intuition. subst y. destruct a. simpl. apply hd_red_rule. exact H. Qed.

Lemma

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

chain_dp_chain

null

chain_chain_dp: forall t u, Chain t u -> exists a, chain_dp a t u. Proof. intros. do 2 destruct H. redtac. subst x. subst u. exists (mkRule l r). unfold chain_dp. simpl. intuition. exists s. intuition. Qed.

Lemma

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

chain_chain_dp

null

chain_dps(a : rule) (l : rules) : relation term := match l with | nil => chain_dp a | b :: m => chain_dp a @ chain_dps b m end.

Fixpoint

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

chain_dps

iteration of chain steps given a list of dependency pairs

chain_dps_app: forall l a b m, chain_dps a (l ++ b :: m) << chain_dps a l @ chain_dps b m. Proof. induction l; simpl; intros. refl. assoc. comp. apply IHl. Qed. Arguments chain_dps_app [l a b m] _ _ _.

Lemma

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

chain_dps_app

null

chain_dps_app': forall a l m b p, a :: l = m ++ b :: p -> chain_dps a l << chain_dps a (tail m) % @ chain_dps b p. Proof. intros. destruct m; simpl in H; injection H; unfold inclusion; intros. subst p. subst b. exists x. unfold clos_refl. intuition auto with *. subst r. subst l. simpl. ded (chain_dps_app _ _ H2). do 2 destruct H0. exists x0. unfold clos_refl. intuition auto with *. Qed. Arguments chain_dps_app' [a l m b p] _ [x y] _.

Lemma

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

chain_dps_app'

null

chain_dps_iter_chain: forall l a, chain_dps a l << iter Chain (length l). Proof. induction l; simpl; intros. apply chain_dp_chain. comp. apply chain_dp_chain. apply IHl. Qed.

Lemma

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

chain_dps_iter_chain

null

iter_chain_chain_dps: forall n t u, iter Chain n t u -> exists a, exists l, length l = n /\ chain_dps a l t u. Proof. induction n; simpl; intros. ded (chain_chain_dp H). destruct H0. exists x. exists nil. intuition. do 2 destruct H. ded (chain_chain_dp H). destruct H1. exists x0. ded (IHn _ _ H0). do 3 destruct H2. exists (x1 :: x2). simpl. intuition. exists x. intuition. Qed. Arguments iter_chain_chain_dps [n t u] _.

Lemma

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

iter_chain_chain_dps

null

chain_dp2_dp_graph: forall a1 a2 t u v, chain_dp a1 t u -> chain_dp a2 u v -> dp_graph a1 a2. Proof. intros. destruct a1 as [l0 r0]. destruct a2 as [l1 r1]. destruct H. simpl in H1. do 2 destruct H1. subst u. rename x into s0. destruct H0. simpl in H2. do 2 destruct H2. subst v. rename x into s1. unfold dp_graph. intuition. simpl. set (p := maxvar l0 + 1). exists p. (* take the union of s0 (restricted to [vars l0]) and [comp s1 (shift_inv_sub p l1)] (restricted to [vars (shift p l1)] *) set (s0' := restrict s0 (vars l0)). set (s1' := restrict (sub_comp s1 (shift_inv_sub p l1)) (vars (shift p l1))). set (s := ASubstitution.union s0' s1'). exists s. assert (compat s0' s1' (vars l0) (vars (shift p l1))). unfold compat. intros. ded (vars_max H3). ded (in_vars_shift_min H4). unfold p in H6. absurd (x <= maxvar l0). lia. hyp. assert (dom_incl s0' (vars l0)). unfold s0'. apply dom_incl_restrict. assert (dom_incl s1' (vars (shift p l1))). unfold s1'. apply dom_incl_restrict. assert (sub s r0 = sub s0' r0). unfold s. eapply sub_union1. apply H5. apply H3. apply hyp'. hyp. rewrite H6. assert (sub s0' r0 = sub s0 r0). unfold s0'. sym. apply sub_restrict_incl. apply hyp'. hyp. rewrite H7. assert (sub s (shift p l1) = sub s1' (shift p l1)). unfold s. eapply sub_union2. apply H4. apply H3. refl. rewrite H8. assert (sub s1' (shift p l1) = sub s1 l1). unfold s1'. rewrite <- sub_restrict, <- sub_shift. refl. rewrite H9. hyp. Qed.

Lemma

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

chain_dp2_dp_graph

two consecutive chain steps provide a dp_graph step

chain_dps_path_dp_graph: forall l a b t u, chain_dps a (l ++ b :: nil) t u -> path dp_graph a b l. Proof. induction l; simpl; intros; do 2 destruct H. eapply chain_dp2_dp_graph. apply H. apply H0. ded (IHl _ _ _ _ H0). intuition. destruct l; simpl in H0; do 2 destruct H0. eapply chain_dp2_dp_graph. apply H. apply H0. eapply chain_dp2_dp_graph. apply H. apply H0. Qed. Arguments chain_dps_path_dp_graph [l a b t u] _. (** hypotheses of the criterion based on cycles using the same reduction pair for every cycle *) Import ACompat. Variables (succ succ_eq : relation term) (Hredord : weak_rewrite_ordering succ succ_eq) (Hreword : rewrite_ordering succ_eq) (Habsorb : absorbs_left succ succ_eq) (HcompR : compat succ_eq R) (HcompDP : compat succ_eq (dp R)) (Hwf : WF succ) (Hcycle : forall a l, length l < length DP -> cycle_min dp_graph a l -> exists b, In b (a :: l) /\ succ (lhs b) (rhs b)). Notation eq_dec := (@eq_rule_dec Sig). Notation occur := (occur eq_dec).

Lemma

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

chain_dps_path_dp_graph

null

chain_dp_hd_red_mod: forall a, chain_dp a << hd_red_mod R (a::nil). Proof. unfold inclusion. intros. destruct H. do 2 destruct H0. subst y. destruct a. simpl. simpl in H0. exists (sub x0 lhs). split. apply incl_elim with (R := int_red R #). apply rtc_incl. apply int_red_incl_red. exact H0. apply hd_red_rule. simpl. auto. Qed.

Lemma

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

chain_dp_hd_red_mod

compatibility properties of chains

compat_chain_dp: forall a, chain_dp a << succ_eq#. Proof. intros. incl_trans (red_mod R DP). apply incl_trans with Chain; try incl_refl. apply chain_dp_chain. incl_trans (hd_red_mod R DP). apply chain_hd_red_mod. apply hd_red_mod_incl_red_mod. incl_trans (succ_eq!). apply compat_red_mod_tc; hyp. apply tc_incl_rtc. Qed.

Lemma

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

compat_chain_dp

null

compat_chain_dps: forall l a, chain_dps a l << succ_eq#. Proof. induction l; simpl; intros. apply compat_chain_dp. incl_trans (succ_eq# @ succ_eq#). comp. apply compat_chain_dp. apply IHl. apply comp_rtc_idem. Qed.

Lemma

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

compat_chain_dps

null

compat_chain_dp_strict: forall a, succ (lhs a) (rhs a) -> chain_dp a << succ. Proof. unfold inclusion. intros. destruct H0. do 2 destruct H1. subst y. apply (absorbs_left_rtc Habsorb). exists (sub x0 (lhs a)). split. apply incl_elim with (R := int_red R #). 2: exact H1. apply rtc_incl. incl_trans (red R). apply int_red_incl_red. apply compat_red; hyp. destruct Hredord. apply H2. exact H. Qed.

Lemma

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

compat_chain_dp_strict

null

WF_cycles: WF (chain R). Proof. set (n := length DP). apply WF_iter with (S n). intro. assert (Hwf' : WF (succ @ succ_eq#)). apply absorb_WF_modulo_r; hyp. gen (Hwf' x). induction 1. apply SN_intro; intros. apply H0. ded (iter_chain_chain_dps H1). do 3 destruct H2. assert (length x1 > 0). lia. ded (last_intro H4). do 3 destruct H5. clear H4. rewrite H5 in H3. ded (chain_dps_path_dp_graph H3). set (l' := x0 :: x2 ++ x3 :: nil). assert (exists z, occur z l' >= 2). unfold l'. eapply long_path_occur. apply restricted_dp_graph. apply H4. rewrite H6, H2. unfold n. lia. assert (exists l, exists a, exists m, l' = l ++ a :: m /\ succ (lhs a) (rhs a)). destruct H7. set (p := occur x4 l' - 2). assert (occur x4 l' = S (S p)). unfold p. lia. ded (occur_S eq_dec H8). do 3 destruct H9. destruct H10. clear H8. rewrite H9. ded (occur_S eq_dec H11). do 3 destruct H8. destruct H12. clear H11. clear H13. clear p. rewrite H8. rewrite H8 in H9. clear H8. clear H10. assert (cycle dp_graph x4 x7). unfold cycle. eapply sub_path. unfold l' in H9. apply H9. exact H4. ded (cycle_min_intro eq_dec H8). decomp H10. assert (length l' = n+2). unfold l'. simpl. rewrite <- H5. lia. rewrite H9 in H10. change (length (x5++(x4::x7)++x4::x8) = n+2) in H10. rewrite H11 in H10. repeat (rewrite length_app in H10; simpl in H10). unfold n in H10. assert (h : length x11 < length DP). unfold cycle_min in H14. destruct H14. assert (length (x10::x11) <= length DP). apply nodup_incl_length. apply eq_dec. exact H14. apply incl_appr_incl with (x10::nil). simpl. eapply restricted_path_incl. apply restricted_dp_graph. exact H13. simpl in H15. lia. clear H10. ded (Hcycle h H14). do 2 destruct H10. ded (in_elim H10). do 2 destruct H15. exists (x5++x9++x14). exists x13. exists (x15++x12++x4::x8). intuition. rewrite !app_ass. trans (x5++(x4::x7)++x4::x8). refl. rewrite H11, app_ass. trans (x5++x9++(x10::x11)++x12++x4::x8). simpl. rewrite app_ass. refl. rewrite H15, app_ass. refl. do 4 destruct H8. unfold l' in H8. ded (chain_dps_app' H8 H3). do 2 destruct H10. assert (succ_eq# x x7). destruct H10. subst x7. intuition auto with *. eapply incl_elim. eapply compat_chain_dps. apply H10. assert ((succ @ succ_eq#) x7 y). destruct x6; simpl in H11. ded (compat_chain_dp_strict H9 H11). exists y. intuition auto with *. do 2 destruct H11. exists x8. split. eapply incl_elim. apply (compat_chain_dp_strict H9). exact H11. eapply incl_elim. eapply compat_chain_dps. apply H13. do 2 destruct H13. exists x8. intuition. eapply incl_elim with (R := succ_eq# @ succ). apply absorbs_left_rtc. intuition. exists x7. intuition. Qed.

Lemma

DP

[ "From Stdlib Require Import Lia", "From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil RelSub Path ARelation", "From CoLoR Require Export ADP" ]

DP/ADPGraph.v

WF_cycles

proof of the criterion based on cycles

dpg_unif_N_correct:= (apply dpg_unif_N_mark_correct || apply dpg_unif_N_correct); (check_eq || fail 10 "a LHS is a variable").

Ltac

DP

[ "From Stdlib Require Import Lia Compare_dec", "From CoLoR Require Import ADecomp AUnif ARenCap ATrs ListUtil RelUtil AGraph" ]

DP/ADPUnif.v

dpg_unif_N_correct

tactics

dp_trans:= chain; eapply WF_incl; [apply hd_red_Mod_of_chain | idtac].

Ltac

DP

[ "From CoLoR Require Import LogicUtil ATrs RelUtil RelSub SN AShift ADPGraph", "From CoLoR Require Export ADP", "From Stdlib Require Import List Lia" ]

DP/AGraph.v

dp_trans

tactics

hder1 r2 := In r1 D /\ In r2 D /\ match rhs r1 with | Var _ => True | Fun f _ => match lhs r2 with | Var _ => True | Fun g _ => f=g end end.

Definition

DP

[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]

DP/AHDE.v

hde

null

hde_restricted: is_restricted hde D. Proof. unfold is_restricted. intros. unfold hde in H; tauto. Qed. Notation eq_rule_dec := (@eq_rule_dec Sig). Notation eq_symb_dec := (@eq_symb_dec Sig).

Lemma

DP

[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]

DP/AHDE.v

hde_restricted

null

hde_dec: forall r1 r2, {hde r1 r2} + {~hde r1 r2}. Proof. intros. unfold hde. destruct (In_dec eq_rule_dec r1 D); try tauto. destruct (In_dec eq_rule_dec r2 D); try tauto. destruct (rhs r1). left; auto. destruct (lhs r2); auto. destruct (eq_symb_dec f f0); try tauto. Defined.

Lemma

DP

[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]

DP/AHDE.v

hde_dec

null

int_red_hd_rules_graph_incl_hde: hd_rules_graph (int_red R #) D << hde D. Proof. unfold inclusion. intros. destruct x. destruct y. destruct H. destruct H0. do 2 destruct H1. unfold hde. destruct lhs0; destruct rhs; simpl; auto. intuition; auto. ded (int_red_rtc_preserve_hd H1). destruct H2. simpl in H2. inversion H2; auto. do 4 destruct H2. inversion H2. inversion H3. congruence. Qed.

Lemma

DP

[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]

DP/AHDE.v

int_red_hd_rules_graph_incl_hde

null

dup_hd_rules_graph_incl_hde: hd_rules_graph (red R #) D << hde D. Proof. unfold inclusion. intros. apply (@int_red_hd_rules_graph_incl_hde _ R D x y). destruct H. decomp H0. rewrite forallb_forall in hd_hyp. ded (hd_hyp _ H). compute in H0. destruct x. destruct rhs. discr. destruct f. unfold hd_rules_graph. intuition. exists x0. exists x1. simpl rhs. rewrite sub_fun. apply dup_int_rules_int_red_rtc; hyp. discr. Qed.

Lemma

DP

[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]

DP/AHDE.v

dup_hd_rules_graph_incl_hde

null

hd_eq(u v : term Sig) := match u with | Var _ => true | Fun f _ => match v with | Var _ => true | Fun g _ => beq_symb f g end end. Notation mem := (mem (@beq_rule Sig)). Notation mem_ok := (mem_ok (@beq_rule_ok Sig)).

Definition

DP

[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]

DP/AHDE.v

hd_eq

null

hde_boolr1 r2 := mem r1 D && mem r2 D && hd_eq (rhs r1) (lhs r2).

Definition

DP

[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]

DP/AHDE.v

hde_bool

null

hde_bool_correct_aux: forall x y, hde D x y <-> Graph hde_bool x y. Proof. intros x y. unfold hde, hde_bool, hd_eq, Graph; simpl. pose proof (fun s x y => proj1 (@beq_symb_ok s x y)). pose proof (fun s x y => proj2 (@beq_symb_ok s x y)). rewrite <- !mem_ok. destruct (rhs x); bool; intuition auto with core; repeat match goal with | H : _ && _ = true |- _ => apply andb_elim in H; case H as [] | H : _ = true |- _ => rewrite H; cbn [andb] | H : match ?x in term _ return _ with _ => _ end |- _ => case x eqn:? in *; subst | H : match ?x in term _ return _ with _ => _ end = true |- _ => case x eqn:? in *; subst end; auto. Qed.

Lemma

DP

[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]

DP/AHDE.v

hde_bool_correct_aux

null

hde_bool_correct: hd_rules_graph (int_red R #) D << Graph (hde_bool D). Proof. incl_trans (hde D). apply int_red_hd_rules_graph_incl_hde. intros x y. rewrite hde_bool_correct_aux. auto. Qed. Notation Sig' := (dup_sig Sig). Notation R' := (dup_int_rules R). Variable hyp : forallb (@is_notvar_lhs Sig') R' = true.

Lemma

DP

[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]

DP/AHDE.v

hde_bool_correct

null

hde_bool_mark_correct: hd_rules_graph (red (dup_int_rules R) #) (dup_hd_rules D) << Graph (hde_bool (dup_hd_rules D)). Proof. incl_trans (hde (dup_hd_rules D)). 2:{ intros x y. rewrite hde_bool_correct_aux. auto. } intros x y h. destruct h. decomp H0. unfold hde. intuition. destruct (in_map_elim H). destruct H0. destruct x2. unfold dup_hd_rule in H3. simpl in H3. subst x. simpl in *. destruct (in_map_elim H1). destruct H3. destruct x. unfold dup_hd_rule in H4. simpl in H4. subst y. simpl in *. destruct rhs; simpl. trivial. destruct lhs0; simpl. trivial. simpl dup_hd_term in H2. unfold shift in H2. rewrite !sub_fun in H2. destruct (rtc_red_dup_int_hd_symb hyp H2). Funeqtac. auto. Qed.

Lemma

DP

[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]

DP/AHDE.v

hde_bool_mark_correct

null

hde_bool_correct:= (apply hde_bool_mark_correct || apply hde_bool_correct); (check_eq || fail 10 "a LHS is a variable").

Ltac

DP

[ "From CoLoR Require Import ADecomp ADuplicateSymb ATrs ListUtil RelSub RelUtil" ]

DP/AHDE.v

hde_bool_correct

tactics

use_SCC_tagh M S R t := let x := fresh in (set (x := SCC_tag_fast h M t); norm_in x (SCC_tag_fast h M t); match eval compute in x with | Some ?X1 => let Hi := fresh in (assert (Hi : X1 < length R); [ norm (length R); lia | let L := fresh in (set (L := SCC_list_fast R M Hi); norm_in L (SCC_list_fast R M Hi); assert (WF (hd_red_Mod S L)); subst L; clear Hi; clear x)]) end).

Ltac

DP

[ "From Stdlib Require Import Permutation Multiset Setoid PermutSetoid", "From CoLoR Require Import SCCTopoOrdering AGraph ATrs RelUtil RelSub BoundNat" ]

DP/ASCCUnion.v

use_SCC_tag

tactics

use_SCC_hypM l := let b := fresh "b" in (set (b := Vnth (Vnth M l) l); norm_in b (Vnth (Vnth M l) l); match eval compute in b with | false => apply WF_hd_red_Mod_SCC_fast_trivial with (Hi:=l); eauto | true => eapply WF_incl; [eapply hd_red_Mod_SCC'_hd_red_Mod_fast with (Hi:=l); auto | auto] end).

Ltac

DP

[ "From Stdlib Require Import Permutation Multiset Setoid PermutSetoid", "From CoLoR Require Import SCCTopoOrdering AGraph ATrs RelUtil RelSub BoundNat" ]

DP/ASCCUnion.v

use_SCC_hyp

null

Property	Value
Total Entries	3,304
Files Processed	264

Type	Count
Lemma	2,015
Definition	540
Parameter	223
Fixpoint	172
Ltac	157
Inductive	101
Theorem	27
Program	25
Record	19
Let	11
Instance	7
Axiom	4
Coercion	2
Remark	1

Column	Type	Description
`fact`	string	Declaration body (name, signature, proof)
`type`	string	Declaration type
`library`	string	Library component
`imports`	list[string]	Import statements
`filename`	string	Source file path
`symbolic_name`	string	Declaration identifier
`docstring`	string	Documentation comment (if present)

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phanerozoic
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Coq-CoLoR

Coq-CoLoR

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