Require terminaison. Require Relations. Require term. Require List. Require equational_theory. Require rpo_extension. Require equational_extension. Require closure_extension. Require term_extension. Require dp. Require Inclusion. Require or_ext_generated. Require ZArith. Require ring_extention. Require Zwf. Require Inverse_Image. Require matrix. Require more_list_extention. Import List. Import ZArith. Set Implicit Arguments. Module algebra. Module F <:term.Signature. Inductive symb : Set := (* id_g *) | id_g : symb (* id_f' *) | id_f' : symb (* id_foldf *) | id_foldf : symb (* id_B *) | id_B : symb (* id_f'' *) | id_f'' : symb (* id_cons *) | id_cons : symb (* id_A *) | id_A : symb (* id_triple *) | id_triple : symb (* id_nil *) | id_nil : symb (* id_C *) | id_C : symb (* id_f *) | id_f : symb . Definition symb_eq_bool (f1 f2:symb) : bool := match f1,f2 with | id_g,id_g => true | id_f',id_f' => true | id_foldf,id_foldf => true | id_B,id_B => true | id_f'',id_f'' => true | id_cons,id_cons => true | id_A,id_A => true | id_triple,id_triple => true | id_nil,id_nil => true | id_C,id_C => true | id_f,id_f => true | _,_ => false end. (* Proof of decidability of equality over symb *) Definition symb_eq_bool_ok(f1 f2:symb) : match symb_eq_bool f1 f2 with | true => f1 = f2 | false => f1 <> f2 end. Proof. intros f1 f2. refine match f1 as u1,f2 as u2 return match symb_eq_bool u1 u2 return Prop with | true => u1 = u2 | false => u1 <> u2 end with | id_g,id_g => refl_equal _ | id_f',id_f' => refl_equal _ | id_foldf,id_foldf => refl_equal _ | id_B,id_B => refl_equal _ | id_f'',id_f'' => refl_equal _ | id_cons,id_cons => refl_equal _ | id_A,id_A => refl_equal _ | id_triple,id_triple => refl_equal _ | id_nil,id_nil => refl_equal _ | id_C,id_C => refl_equal _ | id_f,id_f => refl_equal _ | _,_ => _ end;intros abs;discriminate. Defined. Definition arity (f:symb) := match f with | id_g => term.Free 1 | id_f' => term.Free 2 | id_foldf => term.Free 2 | id_B => term.Free 0 | id_f'' => term.Free 1 | id_cons => term.Free 2 | id_A => term.Free 0 | id_triple => term.Free 3 | id_nil => term.Free 0 | id_C => term.Free 0 | id_f => term.Free 2 end. Definition symb_order (f1 f2:symb) : bool := match f1,f2 with | id_g,id_g => true | id_g,id_f' => false | id_g,id_foldf => false | id_g,id_B => false | id_g,id_f'' => false | id_g,id_cons => false | id_g,id_A => false | id_g,id_triple => false | id_g,id_nil => false | id_g,id_C => false | id_g,id_f => false | id_f',id_g => true | id_f',id_f' => true | id_f',id_foldf => false | id_f',id_B => false | id_f',id_f'' => false | id_f',id_cons => false | id_f',id_A => false | id_f',id_triple => false | id_f',id_nil => false | id_f',id_C => false | id_f',id_f => false | id_foldf,id_g => true | id_foldf,id_f' => true | id_foldf,id_foldf => true | id_foldf,id_B => false | id_foldf,id_f'' => false | id_foldf,id_cons => false | id_foldf,id_A => false | id_foldf,id_triple => false | id_foldf,id_nil => false | id_foldf,id_C => false | id_foldf,id_f => false | id_B,id_g => true | id_B,id_f' => true | id_B,id_foldf => true | id_B,id_B => true | id_B,id_f'' => false | id_B,id_cons => false | id_B,id_A => false | id_B,id_triple => false | id_B,id_nil => false | id_B,id_C => false | id_B,id_f => false | id_f'',id_g => true | id_f'',id_f' => true | id_f'',id_foldf => true | id_f'',id_B => true | id_f'',id_f'' => true | id_f'',id_cons => false | id_f'',id_A => false | id_f'',id_triple => false | id_f'',id_nil => false | id_f'',id_C => false | id_f'',id_f => false | id_cons,id_g => true | id_cons,id_f' => true | id_cons,id_foldf => true | id_cons,id_B => true | id_cons,id_f'' => true | id_cons,id_cons => true | id_cons,id_A => false | id_cons,id_triple => false | id_cons,id_nil => false | id_cons,id_C => false | id_cons,id_f => false | id_A,id_g => true | id_A,id_f' => true | id_A,id_foldf => true | id_A,id_B => true | id_A,id_f'' => true | id_A,id_cons => true | id_A,id_A => true | id_A,id_triple => false | id_A,id_nil => false | id_A,id_C => false | id_A,id_f => false | id_triple,id_g => true | id_triple,id_f' => true | id_triple,id_foldf => true | id_triple,id_B => true | id_triple,id_f'' => true | id_triple,id_cons => true | id_triple,id_A => true | id_triple,id_triple => true | id_triple,id_nil => false | id_triple,id_C => false | id_triple,id_f => false | id_nil,id_g => true | id_nil,id_f' => true | id_nil,id_foldf => true | id_nil,id_B => true | id_nil,id_f'' => true | id_nil,id_cons => true | id_nil,id_A => true | id_nil,id_triple => true | id_nil,id_nil => true | id_nil,id_C => false | id_nil,id_f => false | id_C,id_g => true | id_C,id_f' => true | id_C,id_foldf => true | id_C,id_B => true | id_C,id_f'' => true | id_C,id_cons => true | id_C,id_A => true | id_C,id_triple => true | id_C,id_nil => true | id_C,id_C => true | id_C,id_f => false | id_f,id_g => true | id_f,id_f' => true | id_f,id_foldf => true | id_f,id_B => true | id_f,id_f'' => true | id_f,id_cons => true | id_f,id_A => true | id_f,id_triple => true | id_f,id_nil => true | id_f,id_C => true | id_f,id_f => true end. Module Symb. Definition A := symb. Definition eq_A := @eq A. Definition eq_proof : equivalence A eq_A. Proof. constructor. red ;reflexivity . red ;intros ;transitivity y ;assumption. red ;intros ;symmetry ;assumption. Defined. Add Relation A eq_A reflexivity proved by (@equiv_refl _ _ eq_proof) symmetry proved by (@equiv_sym _ _ eq_proof) transitivity proved by (@equiv_trans _ _ eq_proof) as EQA . Definition eq_bool := symb_eq_bool. Definition eq_bool_ok := symb_eq_bool_ok. End Symb. Export Symb. End F. Module Alg := term.Make'(F)(term_extension.IntVars). Module Alg_ext := term_extension.Make(Alg). Module EQT := equational_theory.Make(Alg). Module EQT_ext := equational_extension.Make(EQT). End algebra. Module R_xml_0_deep_rew. Inductive R_xml_0_rules : algebra.Alg.term ->algebra.Alg.term ->Prop := (* g(A) -> A *) | R_xml_0_rule_0 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_A nil) (algebra.Alg.Term algebra.F.id_g ((algebra.Alg.Term algebra.F.id_A nil)::nil)) (* g(B) -> A *) | R_xml_0_rule_1 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_A nil) (algebra.Alg.Term algebra.F.id_g ((algebra.Alg.Term algebra.F.id_B nil)::nil)) (* g(B) -> B *) | R_xml_0_rule_2 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_B nil) (algebra.Alg.Term algebra.F.id_g ((algebra.Alg.Term algebra.F.id_B nil)::nil)) (* g(C) -> A *) | R_xml_0_rule_3 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_A nil) (algebra.Alg.Term algebra.F.id_g ((algebra.Alg.Term algebra.F.id_C nil)::nil)) (* g(C) -> B *) | R_xml_0_rule_4 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_B nil) (algebra.Alg.Term algebra.F.id_g ((algebra.Alg.Term algebra.F.id_C nil)::nil)) (* g(C) -> C *) | R_xml_0_rule_5 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_C nil) (algebra.Alg.Term algebra.F.id_g ((algebra.Alg.Term algebra.F.id_C nil)::nil)) (* foldf(x_,nil) -> x_ *) | R_xml_0_rule_6 : R_xml_0_rules (algebra.Alg.Var 1) (algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Var 1):: (algebra.Alg.Term algebra.F.id_nil nil)::nil)) (* foldf(x_,cons(y_,z_)) -> f(foldf(x_,z_),y_) *) | R_xml_0_rule_7 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Var 1):: (algebra.Alg.Var 3)::nil))::(algebra.Alg.Var 2)::nil)) (algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Var 1):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 2):: (algebra.Alg.Var 3)::nil))::nil)) (* f(t_,x_) -> f'(t_,g(x_)) *) | R_xml_0_rule_8 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_f' ((algebra.Alg.Var 4):: (algebra.Alg.Term algebra.F.id_g ((algebra.Alg.Var 1)::nil))::nil)) (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Var 4):: (algebra.Alg.Var 1)::nil)) (* f'(triple(a_,b_,c_),C) -> triple(a_,b_,cons(C,c_)) *) | R_xml_0_rule_9 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Var 5)::(algebra.Alg.Var 6):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_C nil)::(algebra.Alg.Var 7)::nil))::nil)) (algebra.Alg.Term algebra.F.id_f' ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Var 5)::(algebra.Alg.Var 6):: (algebra.Alg.Var 7)::nil))::(algebra.Alg.Term algebra.F.id_C nil)::nil)) (* f'(triple(a_,b_,c_),B) -> f(triple(a_,b_,c_),A) *) | R_xml_0_rule_10 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Var 5):: (algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil)):: (algebra.Alg.Term algebra.F.id_A nil)::nil)) (algebra.Alg.Term algebra.F.id_f' ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Var 5)::(algebra.Alg.Var 6):: (algebra.Alg.Var 7)::nil))::(algebra.Alg.Term algebra.F.id_B nil)::nil)) (* f'(triple(a_,b_,c_),A) -> f''(foldf(triple(cons(A,a_),nil,c_),b_)) *) | R_xml_0_rule_11 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_f'' ((algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_A nil)::(algebra.Alg.Var 5)::nil)):: (algebra.Alg.Term algebra.F.id_nil nil):: (algebra.Alg.Var 7)::nil)):: (algebra.Alg.Var 6)::nil))::nil)) (algebra.Alg.Term algebra.F.id_f' ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Var 5)::(algebra.Alg.Var 6):: (algebra.Alg.Var 7)::nil))::(algebra.Alg.Term algebra.F.id_A nil)::nil)) (* f''(triple(a_,b_,c_)) -> foldf(triple(a_,b_,nil),c_) *) | R_xml_0_rule_12 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Var 5):: (algebra.Alg.Var 6)::(algebra.Alg.Term algebra.F.id_nil nil)::nil))::(algebra.Alg.Var 7)::nil)) (algebra.Alg.Term algebra.F.id_f'' ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Var 5)::(algebra.Alg.Var 6):: (algebra.Alg.Var 7)::nil))::nil)) . Definition R_xml_0_rule_as_list_0 := ((algebra.Alg.Term algebra.F.id_g ((algebra.Alg.Term algebra.F.id_A nil)::nil)),(algebra.Alg.Term algebra.F.id_A nil))::nil. Definition R_xml_0_rule_as_list_1 := ((algebra.Alg.Term algebra.F.id_g ((algebra.Alg.Term algebra.F.id_B nil)::nil)),(algebra.Alg.Term algebra.F.id_A nil)):: R_xml_0_rule_as_list_0. Definition R_xml_0_rule_as_list_2 := ((algebra.Alg.Term algebra.F.id_g ((algebra.Alg.Term algebra.F.id_B nil)::nil)),(algebra.Alg.Term algebra.F.id_B nil)):: R_xml_0_rule_as_list_1. Definition R_xml_0_rule_as_list_3 := ((algebra.Alg.Term algebra.F.id_g ((algebra.Alg.Term algebra.F.id_C nil)::nil)),(algebra.Alg.Term algebra.F.id_A nil)):: R_xml_0_rule_as_list_2. Definition R_xml_0_rule_as_list_4 := ((algebra.Alg.Term algebra.F.id_g ((algebra.Alg.Term algebra.F.id_C nil)::nil)),(algebra.Alg.Term algebra.F.id_B nil)):: R_xml_0_rule_as_list_3. Definition R_xml_0_rule_as_list_5 := ((algebra.Alg.Term algebra.F.id_g ((algebra.Alg.Term algebra.F.id_C nil)::nil)),(algebra.Alg.Term algebra.F.id_C nil)):: R_xml_0_rule_as_list_4. Definition R_xml_0_rule_as_list_6 := ((algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Var 1):: (algebra.Alg.Term algebra.F.id_nil nil)::nil)),(algebra.Alg.Var 1)):: R_xml_0_rule_as_list_5. Definition R_xml_0_rule_as_list_7 := ((algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Var 1):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 2):: (algebra.Alg.Var 3)::nil))::nil)), (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Var 1)::(algebra.Alg.Var 3)::nil)):: (algebra.Alg.Var 2)::nil)))::R_xml_0_rule_as_list_6. Definition R_xml_0_rule_as_list_8 := ((algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Var 4):: (algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_f' ((algebra.Alg.Var 4):: (algebra.Alg.Term algebra.F.id_g ((algebra.Alg.Var 1)::nil))::nil))):: R_xml_0_rule_as_list_7. Definition R_xml_0_rule_as_list_9 := ((algebra.Alg.Term algebra.F.id_f' ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil)):: (algebra.Alg.Term algebra.F.id_C nil)::nil)), (algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Var 5):: (algebra.Alg.Var 6)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_C nil):: (algebra.Alg.Var 7)::nil))::nil)))::R_xml_0_rule_as_list_8. Definition R_xml_0_rule_as_list_10 := ((algebra.Alg.Term algebra.F.id_f' ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil)):: (algebra.Alg.Term algebra.F.id_B nil)::nil)), (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil)):: (algebra.Alg.Term algebra.F.id_A nil)::nil)))::R_xml_0_rule_as_list_9. Definition R_xml_0_rule_as_list_11 := ((algebra.Alg.Term algebra.F.id_f' ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil)):: (algebra.Alg.Term algebra.F.id_A nil)::nil)), (algebra.Alg.Term algebra.F.id_f'' ((algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_A nil):: (algebra.Alg.Var 5)::nil))::(algebra.Alg.Term algebra.F.id_nil nil):: (algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 6)::nil))::nil))):: R_xml_0_rule_as_list_10. Definition R_xml_0_rule_as_list_12 := ((algebra.Alg.Term algebra.F.id_f'' ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Var 5)::(algebra.Alg.Var 6):: (algebra.Alg.Var 7)::nil))::nil)), (algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Var 5)::(algebra.Alg.Var 6):: (algebra.Alg.Term algebra.F.id_nil nil)::nil)):: (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_11. Definition R_xml_0_rule_as_list := R_xml_0_rule_as_list_12. Lemma R_xml_0_rules_included : forall l r, R_xml_0_rules r l <-> In (l,r) R_xml_0_rule_as_list. Proof. intros l r. constructor. intros H. case H;clear H; (apply (more_list.mem_impl_in (@eq (algebra.Alg.term*algebra.Alg.term))); [tauto|idtac]); match goal with | |- _ _ _ ?t ?l => let u := fresh "u" in (generalize (more_list.mem_bool_ok _ _ algebra.Alg_ext.eq_term_term_bool_ok t l); set (u:=more_list.mem_bool algebra.Alg_ext.eq_term_term_bool t l) in *; vm_compute in u|-;unfold u in *;clear u;intros H;refine H) end . intros H. vm_compute in H|-. rewrite <- or_ext_generated.or14_equiv in H|-. case H;clear H;intros H. injection H;intros ;subst;constructor 13. injection H;intros ;subst;constructor 12. injection H;intros ;subst;constructor 11. injection H;intros ;subst;constructor 10. injection H;intros ;subst;constructor 9. injection H;intros ;subst;constructor 8. injection H;intros ;subst;constructor 7. injection H;intros ;subst;constructor 6. injection H;intros ;subst;constructor 5. injection H;intros ;subst;constructor 4. injection H;intros ;subst;constructor 3. injection H;intros ;subst;constructor 2. injection H;intros ;subst;constructor 1. elim H. Qed. Lemma R_xml_0_non_var : forall x t, ~R_xml_0_rules t (algebra.EQT.T.Var x). Proof. intros x t H. inversion H. Qed. Lemma R_xml_0_reg : forall s t, (R_xml_0_rules s t) -> forall x, In x (algebra.Alg.var_list s) ->In x (algebra.Alg.var_list t). Proof. intros s t H. inversion H;intros x Hx; (apply (more_list.mem_impl_in (@eq algebra.Alg.variable));[tauto|idtac]); apply (more_list.in_impl_mem (@eq algebra.Alg.variable)) in Hx; vm_compute in Hx|-*;tauto. Qed. Inductive and_6 (x9 x10 x11 x12 x13 x14:Prop) : Prop := | conj_6 : x9->x10->x11->x12->x13->x14->and_6 x9 x10 x11 x12 x13 x14 . Lemma are_constuctors_of_R_xml_0 : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> and_6 (t = (algebra.Alg.Term algebra.F.id_B nil) -> t' = (algebra.Alg.Term algebra.F.id_B nil)) (forall x10 x12, t = (algebra.Alg.Term algebra.F.id_cons (x10::x12::nil)) -> exists x9, exists x11, t' = (algebra.Alg.Term algebra.F.id_cons (x9::x11::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12)) (t = (algebra.Alg.Term algebra.F.id_A nil) -> t' = (algebra.Alg.Term algebra.F.id_A nil)) (forall x10 x12 x14, t = (algebra.Alg.Term algebra.F.id_triple (x10::x12::x14::nil)) -> exists x9, exists x11, exists x13, t' = (algebra.Alg.Term algebra.F.id_triple (x9::x11::x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)) (t = (algebra.Alg.Term algebra.F.id_nil nil) -> t' = (algebra.Alg.Term algebra.F.id_nil nil)) (t = (algebra.Alg.Term algebra.F.id_C nil) -> t' = (algebra.Alg.Term algebra.F.id_C nil)). Proof. intros t t' H. induction H as [|y IH z z_to_y] using closure_extension.refl_trans_clos_ind2. constructor 1. intros H;intuition;constructor 1. intros x10 x12 H;exists x10;exists x12;intuition;constructor 1. intros H;intuition;constructor 1. intros x10 x12 x14 H;exists x10;exists x12;exists x14;intuition; constructor 1. intros H;intuition;constructor 1. intros H;intuition;constructor 1. inversion z_to_y as [t1 t2 H H0 H1|f l1 l2 H0 H H2];clear z_to_y;subst. inversion H as [t1 t2 sigma H2 H1 H0];clear H IH;subst;inversion H2; clear ;constructor;try (intros until 0 );clear ;intros abs; discriminate abs. destruct IH as [H_id_B H_id_cons H_id_A H_id_triple H_id_nil H_id_C]. constructor. clear H_id_cons H_id_A H_id_triple H_id_nil H_id_C;intros H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). clear H_id_B H_id_A H_id_triple H_id_nil H_id_C;intros x10 x12 H; injection H;clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x10 |- _ => destruct (H_id_cons y x12 (refl_equal _)) as [x9 [x11]];intros ; intuition;exists x9;exists x11;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x12 |- _ => destruct (H_id_cons x10 y (refl_equal _)) as [x9 [x11]];intros ; intuition;exists x9;exists x11;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_B H_id_cons H_id_triple H_id_nil H_id_C;intros H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). clear H_id_B H_id_cons H_id_A H_id_nil H_id_C;intros x10 x12 x14 H; injection H;clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x10 |- _ => destruct (H_id_triple y x12 x14 (refl_equal _)) as [x9 [x11 [x13]]]; intros ;intuition;exists x9;exists x11;exists x13;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x12 |- _ => destruct (H_id_triple x10 y x14 (refl_equal _)) as [x9 [x11 [x13]]]; intros ;intuition;exists x9;exists x11;exists x13;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_triple x10 x12 y (refl_equal _)) as [x9 [x11 [x13]]]; intros ;intuition;exists x9;exists x11;exists x13;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_B H_id_cons H_id_A H_id_triple H_id_C;intros H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). clear H_id_B H_id_cons H_id_A H_id_triple H_id_nil;intros H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). Qed. Lemma id_B_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> t = (algebra.Alg.Term algebra.F.id_B nil) -> t' = (algebra.Alg.Term algebra.F.id_B nil). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_cons_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x10 x12, t = (algebra.Alg.Term algebra.F.id_cons (x10::x12::nil)) -> exists x9, exists x11, t' = (algebra.Alg.Term algebra.F.id_cons (x9::x11::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_A_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> t = (algebra.Alg.Term algebra.F.id_A nil) -> t' = (algebra.Alg.Term algebra.F.id_A nil). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_triple_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x10 x12 x14, t = (algebra.Alg.Term algebra.F.id_triple (x10::x12::x14::nil)) -> exists x9, exists x11, exists x13, t' = (algebra.Alg.Term algebra.F.id_triple (x9::x11::x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x11 x12)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_nil_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> t = (algebra.Alg.Term algebra.F.id_nil nil) -> t' = (algebra.Alg.Term algebra.F.id_nil nil). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_C_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> t = (algebra.Alg.Term algebra.F.id_C nil) -> t' = (algebra.Alg.Term algebra.F.id_C nil). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Ltac impossible_star_reduction_R_xml_0 := match goal with | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_B nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_B_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; impossible_star_reduction_R_xml_0 )) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_cons (?x10::?x9::nil)) |- _ => let x10 := fresh "x" in (let x9 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_cons_is_R_xml_0_constructor H (refl_equal _)) as [x10 [x9 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_A nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_A_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; impossible_star_reduction_R_xml_0 )) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_triple (?x11::?x10::?x9::nil)) |- _ => let x11 := fresh "x" in (let x10 := fresh "x" in (let x9 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_triple_is_R_xml_0_constructor H (refl_equal _)) as [x11 [x10 [x9 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_nil nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_nil_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; impossible_star_reduction_R_xml_0 )) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_C nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_C_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; impossible_star_reduction_R_xml_0 )) end . Ltac simplify_star_reduction_R_xml_0 := match goal with | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_B nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_B_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; try (simplify_star_reduction_R_xml_0 ))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_cons (?x10::?x9::nil)) |- _ => let x10 := fresh "x" in (let x9 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_cons_is_R_xml_0_constructor H (refl_equal _)) as [x10 [x9 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_A nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_A_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; try (simplify_star_reduction_R_xml_0 ))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_triple (?x11::?x10::?x9::nil)) |- _ => let x11 := fresh "x" in (let x10 := fresh "x" in (let x9 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_triple_is_R_xml_0_constructor H (refl_equal _)) as [x11 [x10 [x9 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_nil nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_nil_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; try (simplify_star_reduction_R_xml_0 ))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_C nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_C_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; try (simplify_star_reduction_R_xml_0 ))) end . End R_xml_0_deep_rew. Module InterpGen := interp.Interp(algebra.EQT). Module ddp := dp.MakeDP(algebra.EQT). Module SymbType. Definition A := algebra.Alg.F.Symb.A. End SymbType. Module Symb_more_list := more_list_extention.Make(SymbType)(algebra.Alg.F.Symb). Module SymbSet := list_set.Make(algebra.F.Symb). Module Interp. Section S. Require Import interp. Hypothesis A : Type. Hypothesis Ale Alt Aeq : A -> A -> Prop. Hypothesis Aop : interp.ordering_pair Aeq Alt Ale. Hypothesis A0 : A. Notation Local "a <= b" := (Ale a b). Hypothesis P_id_g : A ->A. Hypothesis P_id_f' : A ->A ->A. Hypothesis P_id_foldf : A ->A ->A. Hypothesis P_id_B : A. Hypothesis P_id_f'' : A ->A. Hypothesis P_id_cons : A ->A ->A. Hypothesis P_id_A : A. Hypothesis P_id_triple : A ->A ->A ->A. Hypothesis P_id_nil : A. Hypothesis P_id_C : A. Hypothesis P_id_f : A ->A ->A. Hypothesis P_id_g_monotonic : forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_g x10 <= P_id_g x9. Hypothesis P_id_f'_monotonic : forall x12 x10 x9 x11, (A0 <= x12)/\ (x12 <= x11) -> (A0 <= x10)/\ (x10 <= x9) ->P_id_f' x10 x12 <= P_id_f' x9 x11. Hypothesis P_id_foldf_monotonic : forall x12 x10 x9 x11, (A0 <= x12)/\ (x12 <= x11) -> (A0 <= x10)/\ (x10 <= x9) ->P_id_foldf x10 x12 <= P_id_foldf x9 x11. Hypothesis P_id_f''_monotonic : forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_f'' x10 <= P_id_f'' x9. Hypothesis P_id_cons_monotonic : forall x12 x10 x9 x11, (A0 <= x12)/\ (x12 <= x11) -> (A0 <= x10)/\ (x10 <= x9) ->P_id_cons x10 x12 <= P_id_cons x9 x11. Hypothesis P_id_triple_monotonic : forall x12 x10 x14 x9 x13 x11, (A0 <= x14)/\ (x14 <= x13) -> (A0 <= x12)/\ (x12 <= x11) -> (A0 <= x10)/\ (x10 <= x9) -> P_id_triple x10 x12 x14 <= P_id_triple x9 x11 x13. Hypothesis P_id_f_monotonic : forall x12 x10 x9 x11, (A0 <= x12)/\ (x12 <= x11) -> (A0 <= x10)/\ (x10 <= x9) ->P_id_f x10 x12 <= P_id_f x9 x11. Hypothesis P_id_g_bounded : forall x9, (A0 <= x9) ->A0 <= P_id_g x9. Hypothesis P_id_f'_bounded : forall x10 x9, (A0 <= x9) ->(A0 <= x10) ->A0 <= P_id_f' x10 x9. Hypothesis P_id_foldf_bounded : forall x10 x9, (A0 <= x9) ->(A0 <= x10) ->A0 <= P_id_foldf x10 x9. Hypothesis P_id_B_bounded : A0 <= P_id_B . Hypothesis P_id_f''_bounded : forall x9, (A0 <= x9) ->A0 <= P_id_f'' x9. Hypothesis P_id_cons_bounded : forall x10 x9, (A0 <= x9) ->(A0 <= x10) ->A0 <= P_id_cons x10 x9. Hypothesis P_id_A_bounded : A0 <= P_id_A . Hypothesis P_id_triple_bounded : forall x10 x9 x11, (A0 <= x9) ->(A0 <= x10) ->(A0 <= x11) ->A0 <= P_id_triple x11 x10 x9. Hypothesis P_id_nil_bounded : A0 <= P_id_nil . Hypothesis P_id_C_bounded : A0 <= P_id_C . Hypothesis P_id_f_bounded : forall x10 x9, (A0 <= x9) ->(A0 <= x10) ->A0 <= P_id_f x10 x9. Fixpoint measure t { struct t } := match t with | (algebra.Alg.Term algebra.F.id_g (x9::nil)) => P_id_g (measure x9) | (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) => P_id_f' (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_foldf (x10::x9::nil)) => P_id_foldf (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_B nil) => P_id_B | (algebra.Alg.Term algebra.F.id_f'' (x9::nil)) => P_id_f'' (measure x9) | (algebra.Alg.Term algebra.F.id_cons (x10::x9::nil)) => P_id_cons (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_A nil) => P_id_A | (algebra.Alg.Term algebra.F.id_triple (x11::x10::x9::nil)) => P_id_triple (measure x11) (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_C nil) => P_id_C | (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) => P_id_f (measure x10) (measure x9) | _ => A0 end. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_g (x9::nil)) => P_id_g (measure x9) | (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) => P_id_f' (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_foldf (x10::x9::nil)) => P_id_foldf (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_B nil) => P_id_B | (algebra.Alg.Term algebra.F.id_f'' (x9::nil)) => P_id_f'' (measure x9) | (algebra.Alg.Term algebra.F.id_cons (x10::x9::nil)) => P_id_cons (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_A nil) => P_id_A | (algebra.Alg.Term algebra.F.id_triple (x11::x10:: x9::nil)) => P_id_triple (measure x11) (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_C nil) => P_id_C | (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) => P_id_f (measure x10) (measure x9) | _ => A0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Definition Pols f : InterpGen.Pol_type A (InterpGen.get_arity f) := match f with | algebra.F.id_g => P_id_g | algebra.F.id_f' => P_id_f' | algebra.F.id_foldf => P_id_foldf | algebra.F.id_B => P_id_B | algebra.F.id_f'' => P_id_f'' | algebra.F.id_cons => P_id_cons | algebra.F.id_A => P_id_A | algebra.F.id_triple => P_id_triple | algebra.F.id_nil => P_id_nil | algebra.F.id_C => P_id_C | algebra.F.id_f => P_id_f end. Lemma same_measure : forall t, measure t = InterpGen.measure A0 Pols t. Proof. fix 1 . intros [a| f l]. simpl in |-*. unfold eq_rect_r, eq_rect, sym_eq in |-*. reflexivity . refine match f with | algebra.F.id_g => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_f' => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_foldf => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_B => match l with | nil => _ | _::_ => _ end | algebra.F.id_f'' => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_cons => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_A => match l with | nil => _ | _::_ => _ end | algebra.F.id_triple => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_nil => match l with | nil => _ | _::_ => _ end | algebra.F.id_C => match l with | nil => _ | _::_ => _ end | algebra.F.id_f => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end end;simpl in |-*;unfold eq_rect_r, eq_rect, sym_eq in |-*; try (reflexivity );f_equal ;auto. Qed. Lemma measure_bounded : forall t, A0 <= measure t. Proof. intros t. rewrite same_measure in |-*. apply (InterpGen.measure_bounded Aop). intros f. case f. vm_compute in |-*;intros ;apply P_id_g_bounded;assumption. vm_compute in |-*;intros ;apply P_id_f'_bounded;assumption. vm_compute in |-*;intros ;apply P_id_foldf_bounded;assumption. vm_compute in |-*;intros ;apply P_id_B_bounded;assumption. vm_compute in |-*;intros ;apply P_id_f''_bounded;assumption. vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption. vm_compute in |-*;intros ;apply P_id_A_bounded;assumption. vm_compute in |-*;intros ;apply P_id_triple_bounded;assumption. vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption. vm_compute in |-*;intros ;apply P_id_C_bounded;assumption. vm_compute in |-*;intros ;apply P_id_f_bounded;assumption. Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Hypothesis rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. intros . do 2 (rewrite same_measure in |-*). apply InterpGen.measure_star_monotonic with (1:=Aop) (Pols:=Pols) (rules:=R_xml_0_deep_rew.R_xml_0_rules). intros f. case f. vm_compute in |-*;intros ;apply P_id_g_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_f'_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_foldf_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_f''_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_cons_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_triple_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_f_monotonic;assumption. intros f. case f. vm_compute in |-*;intros ;apply P_id_g_bounded;assumption. vm_compute in |-*;intros ;apply P_id_f'_bounded;assumption. vm_compute in |-*;intros ;apply P_id_foldf_bounded;assumption. vm_compute in |-*;intros ;apply P_id_B_bounded;assumption. vm_compute in |-*;intros ;apply P_id_f''_bounded;assumption. vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption. vm_compute in |-*;intros ;apply P_id_A_bounded;assumption. vm_compute in |-*;intros ;apply P_id_triple_bounded;assumption. vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption. vm_compute in |-*;intros ;apply P_id_C_bounded;assumption. vm_compute in |-*;intros ;apply P_id_f_bounded;assumption. intros . do 2 (rewrite <- same_measure in |-*). apply rules_monotonic;assumption. assumption. Qed. Hypothesis P_id_F : A ->A ->A. Hypothesis P_id_G : A ->A. Hypothesis P_id_F' : A ->A ->A. Hypothesis P_id_FOLDF : A ->A ->A. Hypothesis P_id_F'' : A ->A. Hypothesis P_id_F_monotonic : forall x12 x10 x9 x11, (A0 <= x12)/\ (x12 <= x11) -> (A0 <= x10)/\ (x10 <= x9) ->P_id_F x10 x12 <= P_id_F x9 x11. Hypothesis P_id_G_monotonic : forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_G x10 <= P_id_G x9. Hypothesis P_id_F'_monotonic : forall x12 x10 x9 x11, (A0 <= x12)/\ (x12 <= x11) -> (A0 <= x10)/\ (x10 <= x9) ->P_id_F' x10 x12 <= P_id_F' x9 x11. Hypothesis P_id_FOLDF_monotonic : forall x12 x10 x9 x11, (A0 <= x12)/\ (x12 <= x11) -> (A0 <= x10)/\ (x10 <= x9) ->P_id_FOLDF x10 x12 <= P_id_FOLDF x9 x11. Hypothesis P_id_F''_monotonic : forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_F'' x10 <= P_id_F'' x9. Definition marked_measure t := match t with | (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) => P_id_F (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_g (x9::nil)) => P_id_G (measure x9) | (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) => P_id_F' (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_foldf (x10::x9::nil)) => P_id_FOLDF (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_f'' (x9::nil)) => P_id_F'' (measure x9) | _ => measure t end. Definition Marked_pols : forall f, (algebra.EQT.defined R_xml_0_deep_rew.R_xml_0_rules f) -> InterpGen.Pol_type A (InterpGen.get_arity f). Proof. intros f H. apply ddp.defined_list_complete with (1:=R_xml_0_deep_rew.R_xml_0_rules_included) in H . apply (Symb_more_list.change_in algebra.F.symb_order) in H . set (u := (Symb_more_list.qs algebra.F.symb_order (Symb_more_list.XSet.remove_red (ddp.defined_list R_xml_0_deep_rew.R_xml_0_rule_as_list)))) in * . vm_compute in u . unfold u in * . clear u . unfold more_list.mem_bool in H . match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x10 x9;apply (P_id_F x10 x9). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x9;apply (P_id_F'' x9). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x10 x9;apply (P_id_FOLDF x10 x9). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x10 x9;apply (P_id_F' x10 x9). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x9;apply (P_id_G x9). discriminate H. Defined. Lemma same_marked_measure : forall t, marked_measure t = InterpGen.marked_measure A0 Pols Marked_pols (ddp.defined_dec _ _ R_xml_0_deep_rew.R_xml_0_rules_included) t. Proof. intros [a| f l]. simpl in |-*. unfold eq_rect_r, eq_rect, sym_eq in |-*. reflexivity . refine match f with | algebra.F.id_g => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_f' => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_foldf => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_B => match l with | nil => _ | _::_ => _ end | algebra.F.id_f'' => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_cons => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_A => match l with | nil => _ | _::_ => _ end | algebra.F.id_triple => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_nil => match l with | nil => _ | _::_ => _ end | algebra.F.id_C => match l with | nil => _ | _::_ => _ end | algebra.F.id_f => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end end. vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . Qed. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) => P_id_F (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_g (x9::nil)) => P_id_G (measure x9) | (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) => P_id_F' (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_foldf (x10:: x9::nil)) => P_id_FOLDF (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_f'' (x9::nil)) => P_id_F'' (measure x9) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. intros . do 2 (rewrite same_marked_measure in |-*). apply InterpGen.marked_measure_star_monotonic with (1:=Aop) (Pols:= Pols) (rules:=R_xml_0_deep_rew.R_xml_0_rules). clear f. intros f. case f. vm_compute in |-*;intros ;apply P_id_g_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_f'_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_foldf_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_f''_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_cons_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_triple_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_f_monotonic;assumption. clear f. intros f. case f. vm_compute in |-*;intros ;apply P_id_g_bounded;assumption. vm_compute in |-*;intros ;apply P_id_f'_bounded;assumption. vm_compute in |-*;intros ;apply P_id_foldf_bounded;assumption. vm_compute in |-*;intros ;apply P_id_B_bounded;assumption. vm_compute in |-*;intros ;apply P_id_f''_bounded;assumption. vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption. vm_compute in |-*;intros ;apply P_id_A_bounded;assumption. vm_compute in |-*;intros ;apply P_id_triple_bounded;assumption. vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption. vm_compute in |-*;intros ;apply P_id_C_bounded;assumption. vm_compute in |-*;intros ;apply P_id_f_bounded;assumption. intros . do 2 (rewrite <- same_measure in |-*). apply rules_monotonic;assumption. clear f. intros f. clear H. intros H. generalize H. apply ddp.defined_list_complete with (1:=R_xml_0_deep_rew.R_xml_0_rules_included) in H . apply (Symb_more_list.change_in algebra.F.symb_order) in H . set (u := (Symb_more_list.qs algebra.F.symb_order (Symb_more_list.XSet.remove_red (ddp.defined_list R_xml_0_deep_rew.R_xml_0_rule_as_list)))) in * . vm_compute in u . unfold u in * . clear u . unfold more_list.mem_bool in H . match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_F_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_F''_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_FOLDF_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_F'_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_G_monotonic;assumption. discriminate H. assumption. Qed. End S. End Interp. Module InterpZ. Section S. Open Scope Z_scope. Hypothesis min_value : Z. Import ring_extention. Notation Local "'Alt'" := (Zwf.Zwf min_value). Notation Local "'Ale'" := Zle. Notation Local "'Aeq'" := (@eq Z). Notation Local "a <= b" := (Ale a b). Notation Local "a < b" := (Alt a b). Hypothesis P_id_g : Z ->Z. Hypothesis P_id_f' : Z ->Z ->Z. Hypothesis P_id_foldf : Z ->Z ->Z. Hypothesis P_id_B : Z. Hypothesis P_id_f'' : Z ->Z. Hypothesis P_id_cons : Z ->Z ->Z. Hypothesis P_id_A : Z. Hypothesis P_id_triple : Z ->Z ->Z ->Z. Hypothesis P_id_nil : Z. Hypothesis P_id_C : Z. Hypothesis P_id_f : Z ->Z ->Z. Hypothesis P_id_g_monotonic : forall x10 x9, (min_value <= x10)/\ (x10 <= x9) ->P_id_g x10 <= P_id_g x9. Hypothesis P_id_f'_monotonic : forall x12 x10 x9 x11, (min_value <= x12)/\ (x12 <= x11) -> (min_value <= x10)/\ (x10 <= x9) ->P_id_f' x10 x12 <= P_id_f' x9 x11. Hypothesis P_id_foldf_monotonic : forall x12 x10 x9 x11, (min_value <= x12)/\ (x12 <= x11) -> (min_value <= x10)/\ (x10 <= x9) -> P_id_foldf x10 x12 <= P_id_foldf x9 x11. Hypothesis P_id_f''_monotonic : forall x10 x9, (min_value <= x10)/\ (x10 <= x9) ->P_id_f'' x10 <= P_id_f'' x9. Hypothesis P_id_cons_monotonic : forall x12 x10 x9 x11, (min_value <= x12)/\ (x12 <= x11) -> (min_value <= x10)/\ (x10 <= x9) -> P_id_cons x10 x12 <= P_id_cons x9 x11. Hypothesis P_id_triple_monotonic : forall x12 x10 x14 x9 x13 x11, (min_value <= x14)/\ (x14 <= x13) -> (min_value <= x12)/\ (x12 <= x11) -> (min_value <= x10)/\ (x10 <= x9) -> P_id_triple x10 x12 x14 <= P_id_triple x9 x11 x13. Hypothesis P_id_f_monotonic : forall x12 x10 x9 x11, (min_value <= x12)/\ (x12 <= x11) -> (min_value <= x10)/\ (x10 <= x9) ->P_id_f x10 x12 <= P_id_f x9 x11. Hypothesis P_id_g_bounded : forall x9, (min_value <= x9) ->min_value <= P_id_g x9. Hypothesis P_id_f'_bounded : forall x10 x9, (min_value <= x9) ->(min_value <= x10) ->min_value <= P_id_f' x10 x9. Hypothesis P_id_foldf_bounded : forall x10 x9, (min_value <= x9) ->(min_value <= x10) ->min_value <= P_id_foldf x10 x9. Hypothesis P_id_B_bounded : min_value <= P_id_B . Hypothesis P_id_f''_bounded : forall x9, (min_value <= x9) ->min_value <= P_id_f'' x9. Hypothesis P_id_cons_bounded : forall x10 x9, (min_value <= x9) ->(min_value <= x10) ->min_value <= P_id_cons x10 x9. Hypothesis P_id_A_bounded : min_value <= P_id_A . Hypothesis P_id_triple_bounded : forall x10 x9 x11, (min_value <= x9) -> (min_value <= x10) -> (min_value <= x11) ->min_value <= P_id_triple x11 x10 x9. Hypothesis P_id_nil_bounded : min_value <= P_id_nil . Hypothesis P_id_C_bounded : min_value <= P_id_C . Hypothesis P_id_f_bounded : forall x10 x9, (min_value <= x9) ->(min_value <= x10) ->min_value <= P_id_f x10 x9. Definition measure := Interp.measure min_value P_id_g P_id_f' P_id_foldf P_id_B P_id_f'' P_id_cons P_id_A P_id_triple P_id_nil P_id_C P_id_f. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_g (x9::nil)) => P_id_g (measure x9) | (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) => P_id_f' (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_foldf (x10::x9::nil)) => P_id_foldf (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_B nil) => P_id_B | (algebra.Alg.Term algebra.F.id_f'' (x9::nil)) => P_id_f'' (measure x9) | (algebra.Alg.Term algebra.F.id_cons (x10::x9::nil)) => P_id_cons (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_A nil) => P_id_A | (algebra.Alg.Term algebra.F.id_triple (x11::x10:: x9::nil)) => P_id_triple (measure x11) (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_C nil) => P_id_C | (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) => P_id_f (measure x10) (measure x9) | _ => min_value end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, min_value <= measure t. Proof. unfold measure in |-*. apply Interp.measure_bounded with Alt Aeq; (apply interp.o_Z)|| (cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;auto with zarith). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Hypothesis rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply Interp.measure_star_monotonic with Alt Aeq. (apply interp.o_Z)|| (cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;auto with zarith). intros ;apply P_id_g_monotonic;assumption. intros ;apply P_id_f'_monotonic;assumption. intros ;apply P_id_foldf_monotonic;assumption. intros ;apply P_id_f''_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_triple_monotonic;assumption. intros ;apply P_id_f_monotonic;assumption. intros ;apply P_id_g_bounded;assumption. intros ;apply P_id_f'_bounded;assumption. intros ;apply P_id_foldf_bounded;assumption. intros ;apply P_id_B_bounded;assumption. intros ;apply P_id_f''_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_A_bounded;assumption. intros ;apply P_id_triple_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_C_bounded;assumption. intros ;apply P_id_f_bounded;assumption. apply rules_monotonic. Qed. Hypothesis P_id_F : Z ->Z ->Z. Hypothesis P_id_G : Z ->Z. Hypothesis P_id_F' : Z ->Z ->Z. Hypothesis P_id_FOLDF : Z ->Z ->Z. Hypothesis P_id_F'' : Z ->Z. Hypothesis P_id_F_monotonic : forall x12 x10 x9 x11, (min_value <= x12)/\ (x12 <= x11) -> (min_value <= x10)/\ (x10 <= x9) ->P_id_F x10 x12 <= P_id_F x9 x11. Hypothesis P_id_G_monotonic : forall x10 x9, (min_value <= x10)/\ (x10 <= x9) ->P_id_G x10 <= P_id_G x9. Hypothesis P_id_F'_monotonic : forall x12 x10 x9 x11, (min_value <= x12)/\ (x12 <= x11) -> (min_value <= x10)/\ (x10 <= x9) ->P_id_F' x10 x12 <= P_id_F' x9 x11. Hypothesis P_id_FOLDF_monotonic : forall x12 x10 x9 x11, (min_value <= x12)/\ (x12 <= x11) -> (min_value <= x10)/\ (x10 <= x9) -> P_id_FOLDF x10 x12 <= P_id_FOLDF x9 x11. Hypothesis P_id_F''_monotonic : forall x10 x9, (min_value <= x10)/\ (x10 <= x9) ->P_id_F'' x10 <= P_id_F'' x9. Definition marked_measure := Interp.marked_measure min_value P_id_g P_id_f' P_id_foldf P_id_B P_id_f'' P_id_cons P_id_A P_id_triple P_id_nil P_id_C P_id_f P_id_F P_id_G P_id_F' P_id_FOLDF P_id_F''. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) => P_id_F (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_g (x9::nil)) => P_id_G (measure x9) | (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) => P_id_F' (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_foldf (x10:: x9::nil)) => P_id_FOLDF (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_f'' (x9::nil)) => P_id_F'' (measure x9) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply Interp.marked_measure_star_monotonic with Alt Aeq. (apply interp.o_Z)|| (cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;auto with zarith). intros ;apply P_id_g_monotonic;assumption. intros ;apply P_id_f'_monotonic;assumption. intros ;apply P_id_foldf_monotonic;assumption. intros ;apply P_id_f''_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_triple_monotonic;assumption. intros ;apply P_id_f_monotonic;assumption. intros ;apply P_id_g_bounded;assumption. intros ;apply P_id_f'_bounded;assumption. intros ;apply P_id_foldf_bounded;assumption. intros ;apply P_id_B_bounded;assumption. intros ;apply P_id_f''_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_A_bounded;assumption. intros ;apply P_id_triple_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_C_bounded;assumption. intros ;apply P_id_f_bounded;assumption. apply rules_monotonic. intros ;apply P_id_F_monotonic;assumption. intros ;apply P_id_G_monotonic;assumption. intros ;apply P_id_F'_monotonic;assumption. intros ;apply P_id_FOLDF_monotonic;assumption. intros ;apply P_id_F''_monotonic;assumption. Qed. End S. End InterpZ. Module WF_R_xml_0_deep_rew. Inductive DP_R_xml_0 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_0 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_cons (x2::x3::nil)) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_foldf (x1::x3::nil))::x2::nil)) (algebra.Alg.Term algebra.F.id_foldf (x10::x9::nil)) (* *) | DP_R_xml_0_1 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_cons (x2::x3::nil)) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_foldf (x1::x3::nil)) (algebra.Alg.Term algebra.F.id_foldf (x10::x9::nil)) (* *) | DP_R_xml_0_2 : forall x4 x10 x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_f' (x4::(algebra.Alg.Term algebra.F.id_g (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) (* *) | DP_R_xml_0_3 : forall x4 x10 x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_g (x1::nil)) (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) (* *) | DP_R_xml_0_4 : forall x10 x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_B nil) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil))::(algebra.Alg.Term algebra.F.id_A nil)::nil)) (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) (* *) | DP_R_xml_0_5 : forall x10 x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_A nil) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_f'' ((algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_A nil)::x5::nil))::(algebra.Alg.Term algebra.F.id_nil nil)::x7::nil))::x6::nil))::nil)) (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) (* *) | DP_R_xml_0_6 : forall x10 x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_A nil) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_A nil)::x5::nil)):: (algebra.Alg.Term algebra.F.id_nil nil)::x7::nil)):: x6::nil)) (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) (* *) | DP_R_xml_0_7 : forall x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Term algebra.F.id_triple (x5::x6::(algebra.Alg.Term algebra.F.id_nil nil)::nil))::x7::nil)) (algebra.Alg.Term algebra.F.id_f'' (x9::nil)) . Module ddp := dp.MakeDP(algebra.EQT). Lemma R_xml_0_dp_step_spec : forall x y, (ddp.dp_step R_xml_0_deep_rew.R_xml_0_rules x y) -> exists f, exists l1, exists l2, y = algebra.Alg.Term f l2/\ (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2)/\ (ddp.dp R_xml_0_deep_rew.R_xml_0_rules x (algebra.Alg.Term f l1)). Proof. intros x y H. induction H. inversion H. subst. destruct t0. refine ((False_ind) _ _). refine (R_xml_0_deep_rew.R_xml_0_non_var H0). simpl in H|-*. exists a. exists ((List.map) (algebra.Alg.apply_subst sigma) l). exists ((List.map) (algebra.Alg.apply_subst sigma) l). repeat (constructor). assumption. exists f. exists l2. exists l1. constructor. constructor. constructor. constructor. rewrite <- closure.rwr_list_trans_clos_one_step_list. assumption. assumption. Qed. Ltac included_dp_tac H := injection H;clear H;intros;subst; repeat (match goal with | H: closure.refl_trans_clos (closure.one_step_list _) (_::_) _ |- _=> let x := fresh "x" in let l := fresh "l" in let h1 := fresh "h" in let h2 := fresh "h" in let h3 := fresh "h" in destruct (@algebra.EQT_ext.one_step_list_star_decompose_cons _ _ _ _ H) as [x [l[h1[h2 h3]]]];clear H;subst | H: closure.refl_trans_clos (closure.one_step_list _) nil _ |- _ => rewrite (@algebra.EQT_ext.one_step_list_star_decompose_nil _ _ H) in *;clear H end );simpl; econstructor eassumption . Ltac dp_concl_tac h2 h cont_tac t := match t with | False => let h' := fresh "a" in (set (h':=t) in *;cont_tac h'; repeat ( let e := type of h in (match e with | ?t => unfold t in h|-; (case h; [abstract (clear h;intros h;injection h; clear h;intros ;subst; included_dp_tac h2)| clear h;intros h;clear t]) | ?t => unfold t in h|-;elim h end ) )) | or ?a ?b => let cont_tac h' := let h'' := fresh "a" in (set (h'':=or a h') in *;cont_tac h'') in (dp_concl_tac h2 h cont_tac b) end . Module WF_DP_R_xml_0. Inductive DP_R_xml_0_non_scc_1 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_1_0 : forall x4 x10 x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x9) -> DP_R_xml_0_non_scc_1 (algebra.Alg.Term algebra.F.id_g (x1::nil)) (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) . Lemma acc_DP_R_xml_0_non_scc_1 : forall x y, (DP_R_xml_0_non_scc_1 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_scc_2 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_2_0 : forall x4 x10 x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x9) -> DP_R_xml_0_scc_2 (algebra.Alg.Term algebra.F.id_f' (x4:: (algebra.Alg.Term algebra.F.id_g (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) (* *) | DP_R_xml_0_scc_2_1 : forall x10 x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_B nil) x9) -> DP_R_xml_0_scc_2 (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)):: (algebra.Alg.Term algebra.F.id_A nil)::nil)) (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) (* *) | DP_R_xml_0_scc_2_2 : forall x10 x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_A nil) x9) -> DP_R_xml_0_scc_2 (algebra.Alg.Term algebra.F.id_f'' ((algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_A nil)::x5::nil)):: (algebra.Alg.Term algebra.F.id_nil nil):: x7::nil))::x6::nil))::nil)) (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) (* *) | DP_R_xml_0_scc_2_3 : forall x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x9) -> DP_R_xml_0_scc_2 (algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Term algebra.F.id_triple (x5::x6:: (algebra.Alg.Term algebra.F.id_nil nil)::nil)):: x7::nil)) (algebra.Alg.Term algebra.F.id_f'' (x9::nil)) (* *) | DP_R_xml_0_scc_2_4 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_cons (x2::x3::nil)) x9) -> DP_R_xml_0_scc_2 (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_foldf (x1::x3::nil))::x2::nil)) (algebra.Alg.Term algebra.F.id_foldf (x10::x9::nil)) (* *) | DP_R_xml_0_scc_2_5 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_cons (x2::x3::nil)) x9) -> DP_R_xml_0_scc_2 (algebra.Alg.Term algebra.F.id_foldf (x1::x3::nil)) (algebra.Alg.Term algebra.F.id_foldf (x10::x9::nil)) (* *) | DP_R_xml_0_scc_2_6 : forall x10 x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_A nil) x9) -> DP_R_xml_0_scc_2 (algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_A nil)::x5::nil)):: (algebra.Alg.Term algebra.F.id_nil nil):: x7::nil))::x6::nil)) (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) . Module WF_DP_R_xml_0_scc_2. Inductive DP_R_xml_0_scc_2_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_2_large_0 : forall x4 x10 x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x9) -> DP_R_xml_0_scc_2_large (algebra.Alg.Term algebra.F.id_f' (x4:: (algebra.Alg.Term algebra.F.id_g (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) (* *) | DP_R_xml_0_scc_2_large_1 : forall x10 x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_B nil) x9) -> DP_R_xml_0_scc_2_large (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_triple (x5:: x6::x7::nil))::(algebra.Alg.Term algebra.F.id_A nil)::nil)) (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) (* *) | DP_R_xml_0_scc_2_large_2 : forall x10 x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_A nil) x9) -> DP_R_xml_0_scc_2_large (algebra.Alg.Term algebra.F.id_f'' ((algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_A nil):: x5::nil))::(algebra.Alg.Term algebra.F.id_nil nil)::x7::nil)):: x6::nil))::nil)) (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) (* *) | DP_R_xml_0_scc_2_large_3 : forall x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x9) -> DP_R_xml_0_scc_2_large (algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Term algebra.F.id_triple (x5:: x6::(algebra.Alg.Term algebra.F.id_nil nil)::nil))::x7::nil)) (algebra.Alg.Term algebra.F.id_f'' (x9::nil)) (* *) | DP_R_xml_0_scc_2_large_4 : forall x10 x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_A nil) x9) -> DP_R_xml_0_scc_2_large (algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_A nil):: x5::nil))::(algebra.Alg.Term algebra.F.id_nil nil)::x7::nil))::x6::nil)) (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) . Inductive DP_R_xml_0_scc_2_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_2_strict_0 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_cons (x2::x3::nil)) x9) -> DP_R_xml_0_scc_2_strict (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_foldf (x1:: x3::nil))::x2::nil)) (algebra.Alg.Term algebra.F.id_foldf (x10::x9::nil)) (* *) | DP_R_xml_0_scc_2_strict_1 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_cons (x2::x3::nil)) x9) -> DP_R_xml_0_scc_2_strict (algebra.Alg.Term algebra.F.id_foldf (x1:: x3::nil)) (algebra.Alg.Term algebra.F.id_foldf (x10::x9::nil)) . Module WF_DP_R_xml_0_scc_2_large. Inductive DP_R_xml_0_scc_2_large_non_scc_1 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_2_large_non_scc_1_0 : forall x10 x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_A nil) x9) -> DP_R_xml_0_scc_2_large_non_scc_1 (algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_A nil)::x5::nil)):: (algebra.Alg.Term algebra.F.id_nil nil):: x7::nil))::x6::nil)) (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) . Lemma acc_DP_R_xml_0_scc_2_large_non_scc_1 : forall x y, (DP_R_xml_0_scc_2_large_non_scc_1 x y) -> Acc WF_DP_R_xml_0_scc_2.DP_R_xml_0_scc_2_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_scc_2_large_non_scc_2 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_2_large_non_scc_2_0 : forall x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x9) -> DP_R_xml_0_scc_2_large_non_scc_2 (algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Term algebra.F.id_triple (x5::x6:: (algebra.Alg.Term algebra.F.id_nil nil)::nil)):: x7::nil)) (algebra.Alg.Term algebra.F.id_f'' (x9::nil)) . Lemma acc_DP_R_xml_0_scc_2_large_non_scc_2 : forall x y, (DP_R_xml_0_scc_2_large_non_scc_2 x y) -> Acc WF_DP_R_xml_0_scc_2.DP_R_xml_0_scc_2_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_scc_2_large_non_scc_3 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_2_large_non_scc_3_0 : forall x10 x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_A nil) x9) -> DP_R_xml_0_scc_2_large_non_scc_3 (algebra.Alg.Term algebra.F.id_f'' ((algebra.Alg.Term algebra.F.id_foldf ((algebra.Alg.Term algebra.F.id_triple ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_A nil)::x5::nil)):: (algebra.Alg.Term algebra.F.id_nil nil):: x7::nil))::x6::nil))::nil)) (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) . Lemma acc_DP_R_xml_0_scc_2_large_non_scc_3 : forall x y, (DP_R_xml_0_scc_2_large_non_scc_3 x y) -> Acc WF_DP_R_xml_0_scc_2.DP_R_xml_0_scc_2_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_2_large_non_scc_2; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_2_large_scc_4 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_2_large_scc_4_0 : forall x10 x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_B nil) x9) -> DP_R_xml_0_scc_2_large_scc_4 (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_triple (x5::x6:: x7::nil))::(algebra.Alg.Term algebra.F.id_A nil)::nil)) (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) (* *) | DP_R_xml_0_scc_2_large_scc_4_1 : forall x4 x10 x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x9) -> DP_R_xml_0_scc_2_large_scc_4 (algebra.Alg.Term algebra.F.id_f' (x4::(algebra.Alg.Term algebra.F.id_g (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) . Module WF_DP_R_xml_0_scc_2_large_scc_4. Inductive DP_R_xml_0_scc_2_large_scc_4_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_2_large_scc_4_large_0 : forall x4 x10 x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x9) -> DP_R_xml_0_scc_2_large_scc_4_large (algebra.Alg.Term algebra.F.id_f' (x4:: (algebra.Alg.Term algebra.F.id_g (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) . Inductive DP_R_xml_0_scc_2_large_scc_4_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_2_large_scc_4_strict_0 : forall x10 x6 x9 x5 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_triple (x5::x6::x7::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_B nil) x9) -> DP_R_xml_0_scc_2_large_scc_4_strict (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_triple (x5::x6:: x7::nil))::(algebra.Alg.Term algebra.F.id_A nil)::nil)) (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) . Module WF_DP_R_xml_0_scc_2_large_scc_4_large. Inductive DP_R_xml_0_scc_2_large_scc_4_large_non_scc_1 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_2_large_scc_4_large_non_scc_1_0 : forall x4 x10 x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x9) -> DP_R_xml_0_scc_2_large_scc_4_large_non_scc_1 (algebra.Alg.Term algebra.F.id_f' (x4:: (algebra.Alg.Term algebra.F.id_g (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) . Lemma acc_DP_R_xml_0_scc_2_large_scc_4_large_non_scc_1 : forall x y, (DP_R_xml_0_scc_2_large_scc_4_large_non_scc_1 x y) -> Acc WF_DP_R_xml_0_scc_2_large_scc_4.DP_R_xml_0_scc_2_large_scc_4_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Lemma wf : well_founded WF_DP_R_xml_0_scc_2_large_scc_4.DP_R_xml_0_scc_2_large_scc_4_large . Proof. constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_2_large_scc_4_large_non_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_2_large_scc_4_large_non_scc_0; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail))). Qed. End WF_DP_R_xml_0_scc_2_large_scc_4_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_g (x9:Z) := 1* x9. Definition P_id_f' (x9:Z) (x10:Z) := 0. Definition P_id_foldf (x9:Z) (x10:Z) := 2* x9. Definition P_id_B := 1. Definition P_id_f'' (x9:Z) := 0. Definition P_id_cons (x9:Z) (x10:Z) := 0. Definition P_id_A := 0. Definition P_id_triple (x9:Z) (x10:Z) (x11:Z) := 0. Definition P_id_nil := 0. Definition P_id_C := 1. Definition P_id_f (x9:Z) (x10:Z) := 1* x9. Lemma P_id_g_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_g x10 <= P_id_g x9. Proof. intros x10 x9. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_f'_monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_f' x10 x12 <= P_id_f' x9 x11. Proof. intros x12 x11 x10 x9. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_foldf_monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_foldf x10 x12 <= P_id_foldf x9 x11. Proof. intros x12 x11 x10 x9. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_f''_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_f'' x10 <= P_id_f'' x9. Proof. intros x10 x9. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_cons_monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_cons x10 x12 <= P_id_cons x9 x11. Proof. intros x12 x11 x10 x9. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_triple_monotonic : forall x12 x10 x14 x9 x13 x11, (0 <= x14)/\ (x14 <= x13) -> (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) -> P_id_triple x10 x12 x14 <= P_id_triple x9 x11 x13. Proof. intros x14 x13 x12 x11 x10 x9. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_f_monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_f x10 x12 <= P_id_f x9 x11. Proof. intros x12 x11 x10 x9. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_g_bounded : forall x9, (0 <= x9) ->0 <= P_id_g x9. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_f'_bounded : forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id_f' x10 x9. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_foldf_bounded : forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id_foldf x10 x9. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_B_bounded : 0 <= P_id_B . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_f''_bounded : forall x9, (0 <= x9) ->0 <= P_id_f'' x9. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_cons_bounded : forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id_cons x10 x9. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_bounded : 0 <= P_id_A . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_triple_bounded : forall x10 x9 x11, (0 <= x9) ->(0 <= x10) ->(0 <= x11) ->0 <= P_id_triple x11 x10 x9. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_C_bounded : 0 <= P_id_C . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_f_bounded : forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id_f x10 x9. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_g P_id_f' P_id_foldf P_id_B P_id_f'' P_id_cons P_id_A P_id_triple P_id_nil P_id_C P_id_f. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_g (x9::nil)) => P_id_g (measure x9) | (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) => P_id_f' (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_foldf (x10::x9::nil)) => P_id_foldf (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_B nil) => P_id_B | (algebra.Alg.Term algebra.F.id_f'' (x9::nil)) => P_id_f'' (measure x9) | (algebra.Alg.Term algebra.F.id_cons (x10::x9::nil)) => P_id_cons (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_A nil) => P_id_A | (algebra.Alg.Term algebra.F.id_triple (x11::x10:: x9::nil)) => P_id_triple (measure x11) (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_C nil) => P_id_C | (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) => P_id_f (measure x10) (measure x9) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_g_monotonic;assumption. intros ;apply P_id_f'_monotonic;assumption. intros ;apply P_id_foldf_monotonic;assumption. intros ;apply P_id_f''_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_triple_monotonic;assumption. intros ;apply P_id_f_monotonic;assumption. intros ;apply P_id_g_bounded;assumption. intros ;apply P_id_f'_bounded;assumption. intros ;apply P_id_foldf_bounded;assumption. intros ;apply P_id_B_bounded;assumption. intros ;apply P_id_f''_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_A_bounded;assumption. intros ;apply P_id_triple_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_C_bounded;assumption. intros ;apply P_id_f_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_F (x9:Z) (x10:Z) := 2* x10. Definition P_id_G (x9:Z) := 0. Definition P_id_F' (x9:Z) (x10:Z) := 2* x10. Definition P_id_FOLDF (x9:Z) (x10:Z) := 0. Definition P_id_F'' (x9:Z) := 0. Lemma P_id_F_monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_F x10 x12 <= P_id_F x9 x11. Proof. intros x12 x11 x10 x9. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_G_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_G x10 <= P_id_G x9. Proof. intros x10 x9. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_F'_monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_F' x10 x12 <= P_id_F' x9 x11. Proof. intros x12 x11 x10 x9. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_FOLDF_monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_FOLDF x10 x12 <= P_id_FOLDF x9 x11. Proof. intros x12 x11 x10 x9. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_F''_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_F'' x10 <= P_id_F'' x9. Proof. intros x10 x9. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_g P_id_f' P_id_foldf P_id_B P_id_f'' P_id_cons P_id_A P_id_triple P_id_nil P_id_C P_id_f P_id_F P_id_G P_id_F' P_id_FOLDF P_id_F''. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_f (x10:: x9::nil)) => P_id_F (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_g (x9::nil)) => P_id_G (measure x9) | (algebra.Alg.Term algebra.F.id_f' (x10:: x9::nil)) => P_id_F' (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_foldf (x10:: x9::nil)) => P_id_FOLDF (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_f'' (x9::nil)) => P_id_F'' (measure x9) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_g_monotonic;assumption. intros ;apply P_id_f'_monotonic;assumption. intros ;apply P_id_foldf_monotonic;assumption. intros ;apply P_id_f''_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_triple_monotonic;assumption. intros ;apply P_id_f_monotonic;assumption. intros ;apply P_id_g_bounded;assumption. intros ;apply P_id_f'_bounded;assumption. intros ;apply P_id_foldf_bounded;assumption. intros ;apply P_id_B_bounded;assumption. intros ;apply P_id_f''_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_A_bounded;assumption. intros ;apply P_id_triple_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_C_bounded;assumption. intros ;apply P_id_f_bounded;assumption. apply rules_monotonic. intros ;apply P_id_F_monotonic;assumption. intros ;apply P_id_G_monotonic;assumption. intros ;apply P_id_F'_monotonic;assumption. intros ;apply P_id_FOLDF_monotonic;assumption. intros ;apply P_id_F''_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_2_large_scc_4_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_2_large_scc_4_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_2_large_scc_4_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_2_large_scc_4_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_2_large_scc_4_large := WF_DP_R_xml_0_scc_2_large_scc_4_large.wf. Lemma wf : well_founded WF_DP_R_xml_0_scc_2_large.DP_R_xml_0_scc_2_large_scc_4. Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_2_large_scc_4_large). clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx';apply DP_R_xml_0_scc_2_large_scc_4_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_2_large_scc_4_large_in_le; econstructor eassumption])). apply wf_DP_R_xml_0_scc_2_large_scc_4_large. Qed. End WF_DP_R_xml_0_scc_2_large_scc_4. Definition wf_DP_R_xml_0_scc_2_large_scc_4 := WF_DP_R_xml_0_scc_2_large_scc_4.wf. Lemma acc_DP_R_xml_0_scc_2_large_scc_4 : forall x y, (DP_R_xml_0_scc_2_large_scc_4 x y) -> Acc WF_DP_R_xml_0_scc_2.DP_R_xml_0_scc_2_large x. Proof. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_2_large_scc_4). intros x' _ Hrec y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply Hrec;econstructor eassumption)|| ((eapply acc_DP_R_xml_0_scc_2_large_non_scc_3; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_2_large_non_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))). apply wf_DP_R_xml_0_scc_2_large_scc_4. Qed. Lemma wf : well_founded WF_DP_R_xml_0_scc_2.DP_R_xml_0_scc_2_large. Proof. constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_2_large_non_scc_3; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_2_large_non_scc_2; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_2_large_non_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_2_large_non_scc_0; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_2_large_scc_4; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_2_large_scc_3; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_2_large_scc_2; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_2_large_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_2_large_scc_0; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail)))))))))). Qed. End WF_DP_R_xml_0_scc_2_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_g (x9:Z) := 0. Definition P_id_f' (x9:Z) (x10:Z) := 2 + 1* x9. Definition P_id_foldf (x9:Z) (x10:Z) := 1* x9 + 1* x10. Definition P_id_B := 0. Definition P_id_f'' (x9:Z) := 1 + 1* x9. Definition P_id_cons (x9:Z) (x10:Z) := 2 + 1* x9 + 1* x10. Definition P_id_A := 0. Definition P_id_triple (x9:Z) (x10:Z) (x11:Z) := 1* x10 + 1* x11. Definition P_id_nil := 0. Definition P_id_C := 0. Definition P_id_f (x9:Z) (x10:Z) := 2 + 1* x9 + 1* x10. Lemma P_id_g_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_g x10 <= P_id_g x9. Proof. intros x10 x9. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_f'_monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_f' x10 x12 <= P_id_f' x9 x11. Proof. intros x12 x11 x10 x9. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_foldf_monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_foldf x10 x12 <= P_id_foldf x9 x11. Proof. intros x12 x11 x10 x9. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_f''_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_f'' x10 <= P_id_f'' x9. Proof. intros x10 x9. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_cons_monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_cons x10 x12 <= P_id_cons x9 x11. Proof. intros x12 x11 x10 x9. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_triple_monotonic : forall x12 x10 x14 x9 x13 x11, (0 <= x14)/\ (x14 <= x13) -> (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) -> P_id_triple x10 x12 x14 <= P_id_triple x9 x11 x13. Proof. intros x14 x13 x12 x11 x10 x9. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_f_monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_f x10 x12 <= P_id_f x9 x11. Proof. intros x12 x11 x10 x9. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_g_bounded : forall x9, (0 <= x9) ->0 <= P_id_g x9. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_f'_bounded : forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id_f' x10 x9. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_foldf_bounded : forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id_foldf x10 x9. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_B_bounded : 0 <= P_id_B . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_f''_bounded : forall x9, (0 <= x9) ->0 <= P_id_f'' x9. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_cons_bounded : forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id_cons x10 x9. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_bounded : 0 <= P_id_A . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_triple_bounded : forall x10 x9 x11, (0 <= x9) ->(0 <= x10) ->(0 <= x11) ->0 <= P_id_triple x11 x10 x9. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_C_bounded : 0 <= P_id_C . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_f_bounded : forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id_f x10 x9. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_g P_id_f' P_id_foldf P_id_B P_id_f'' P_id_cons P_id_A P_id_triple P_id_nil P_id_C P_id_f. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_g (x9::nil)) => P_id_g (measure x9) | (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) => P_id_f' (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_foldf (x10::x9::nil)) => P_id_foldf (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_B nil) => P_id_B | (algebra.Alg.Term algebra.F.id_f'' (x9::nil)) => P_id_f'' (measure x9) | (algebra.Alg.Term algebra.F.id_cons (x10::x9::nil)) => P_id_cons (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_A nil) => P_id_A | (algebra.Alg.Term algebra.F.id_triple (x11::x10:: x9::nil)) => P_id_triple (measure x11) (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_C nil) => P_id_C | (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) => P_id_f (measure x10) (measure x9) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_g_monotonic;assumption. intros ;apply P_id_f'_monotonic;assumption. intros ;apply P_id_foldf_monotonic;assumption. intros ;apply P_id_f''_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_triple_monotonic;assumption. intros ;apply P_id_f_monotonic;assumption. intros ;apply P_id_g_bounded;assumption. intros ;apply P_id_f'_bounded;assumption. intros ;apply P_id_foldf_bounded;assumption. intros ;apply P_id_B_bounded;assumption. intros ;apply P_id_f''_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_A_bounded;assumption. intros ;apply P_id_triple_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_C_bounded;assumption. intros ;apply P_id_f_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_F (x9:Z) (x10:Z) := 2* x9. Definition P_id_G (x9:Z) := 0. Definition P_id_F' (x9:Z) (x10:Z) := 2* x9. Definition P_id_FOLDF (x9:Z) (x10:Z) := 2* x9 + 2* x10. Definition P_id_F'' (x9:Z) := 2* x9. Lemma P_id_F_monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_F x10 x12 <= P_id_F x9 x11. Proof. intros x12 x11 x10 x9. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_G_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_G x10 <= P_id_G x9. Proof. intros x10 x9. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_F'_monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_F' x10 x12 <= P_id_F' x9 x11. Proof. intros x12 x11 x10 x9. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_FOLDF_monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_FOLDF x10 x12 <= P_id_FOLDF x9 x11. Proof. intros x12 x11 x10 x9. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_F''_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_F'' x10 <= P_id_F'' x9. Proof. intros x10 x9. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_g P_id_f' P_id_foldf P_id_B P_id_f'' P_id_cons P_id_A P_id_triple P_id_nil P_id_C P_id_f P_id_F P_id_G P_id_F' P_id_FOLDF P_id_F''. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_f (x10::x9::nil)) => P_id_F (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_g (x9::nil)) => P_id_G (measure x9) | (algebra.Alg.Term algebra.F.id_f' (x10::x9::nil)) => P_id_F' (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_foldf (x10:: x9::nil)) => P_id_FOLDF (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_f'' (x9::nil)) => P_id_F'' (measure x9) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_g_monotonic;assumption. intros ;apply P_id_f'_monotonic;assumption. intros ;apply P_id_foldf_monotonic;assumption. intros ;apply P_id_f''_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_triple_monotonic;assumption. intros ;apply P_id_f_monotonic;assumption. intros ;apply P_id_g_bounded;assumption. intros ;apply P_id_f'_bounded;assumption. intros ;apply P_id_foldf_bounded;assumption. intros ;apply P_id_B_bounded;assumption. intros ;apply P_id_f''_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_A_bounded;assumption. intros ;apply P_id_triple_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_C_bounded;assumption. intros ;apply P_id_f_bounded;assumption. apply rules_monotonic. intros ;apply P_id_F_monotonic;assumption. intros ;apply P_id_G_monotonic;assumption. intros ;apply P_id_F'_monotonic;assumption. intros ;apply P_id_FOLDF_monotonic;assumption. intros ;apply P_id_F''_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_2_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_2_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_2_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_2_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_2_large := WF_DP_R_xml_0_scc_2_large.wf. Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_2. Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_2_large). clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx';apply DP_R_xml_0_scc_2_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_2_large_in_le;econstructor eassumption])). apply wf_DP_R_xml_0_scc_2_large. Qed. End WF_DP_R_xml_0_scc_2. Definition wf_DP_R_xml_0_scc_2 := WF_DP_R_xml_0_scc_2.wf. Lemma acc_DP_R_xml_0_scc_2 : forall x y, (DP_R_xml_0_scc_2 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_2). intros x' _ Hrec y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply Hrec;econstructor eassumption)|| ((eapply acc_DP_R_xml_0_non_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). apply wf_DP_R_xml_0_scc_2. Qed. Lemma wf : well_founded WF_R_xml_0_deep_rew.DP_R_xml_0. Proof. constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_non_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_0; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_2; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_0; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail)))))). Qed. End WF_DP_R_xml_0. Definition wf_H := WF_DP_R_xml_0.wf. Lemma wf : well_founded (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules). Proof. apply ddp.dp_criterion. apply R_xml_0_deep_rew.R_xml_0_non_var. apply R_xml_0_deep_rew.R_xml_0_reg. intros ; apply (ddp.constructor_defined_dec _ _ R_xml_0_deep_rew.R_xml_0_rules_included). refine (Inclusion.wf_incl _ _ _ _ wf_H). intros x y H. destruct (R_xml_0_dp_step_spec H) as [f [l1 [l2 [H1 [H2 H3]]]]]. destruct (ddp.dp_list_complete _ _ R_xml_0_deep_rew.R_xml_0_rules_included _ _ H3) as [x' [y' [sigma [h1 [h2 h3]]]]]. clear H3. subst. vm_compute in h3|-. let e := type of h3 in (dp_concl_tac h2 h3 ltac:(fun _ => idtac) e). Qed. End WF_R_xml_0_deep_rew. (* *** Local Variables: *** *** coq-prog-name: "coqtop" *** *** coq-prog-args: ("-emacs-U" "-I" "$COCCINELLE/examples" "-I" "$COCCINELLE/term_algebra" "-I" "$COCCINELLE/term_orderings" "-I" "$COCCINELLE/basis" "-I" "$COCCINELLE/list_extensions" "-I" "$COCCINELLE/examples/cime_trace/") *** *** compile-command: "coqc -I $COCCINELLE/term_algebra -I $COCCINELLE/term_orderings -I $COCCINELLE/basis -I $COCCINELLE/list_extensions -I $COCCINELLE/examples/cime_trace/ -I $COCCINELLE/examples/ c_output/strat/tpdb-5.0___TRS___Cime___filliatre.trs/a3pat.v" *** *** End: *** *)