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_a____ *) | id_a____ : symb (* id_U11 *) | id_U11 : symb (* id_a__U11 *) | id_a__U11 : symb (* id_mark *) | id_mark : symb (* id_isNePal *) | id_isNePal : symb (* id_a__U12 *) | id_a__U12 : symb (* id___ *) | id___ : symb (* id_U12 *) | id_U12 : symb (* id_tt *) | id_tt : symb (* id_nil *) | id_nil : symb (* id_a__isNePal *) | id_a__isNePal : symb . Definition symb_eq_bool (f1 f2:symb) : bool := match f1,f2 with | id_a____,id_a____ => true | id_U11,id_U11 => true | id_a__U11,id_a__U11 => true | id_mark,id_mark => true | id_isNePal,id_isNePal => true | id_a__U12,id_a__U12 => true | id___,id___ => true | id_U12,id_U12 => true | id_tt,id_tt => true | id_nil,id_nil => true | id_a__isNePal,id_a__isNePal => 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_a____,id_a____ => refl_equal _ | id_U11,id_U11 => refl_equal _ | id_a__U11,id_a__U11 => refl_equal _ | id_mark,id_mark => refl_equal _ | id_isNePal,id_isNePal => refl_equal _ | id_a__U12,id_a__U12 => refl_equal _ | id___,id___ => refl_equal _ | id_U12,id_U12 => refl_equal _ | id_tt,id_tt => refl_equal _ | id_nil,id_nil => refl_equal _ | id_a__isNePal,id_a__isNePal => refl_equal _ | _,_ => _ end;intros abs;discriminate. Defined. Definition arity (f:symb) := match f with | id_a____ => term.Free 2 | id_U11 => term.Free 1 | id_a__U11 => term.Free 1 | id_mark => term.Free 1 | id_isNePal => term.Free 1 | id_a__U12 => term.Free 1 | id___ => term.Free 2 | id_U12 => term.Free 1 | id_tt => term.Free 0 | id_nil => term.Free 0 | id_a__isNePal => term.Free 1 end. Definition symb_order (f1 f2:symb) : bool := match f1,f2 with | id_a____,id_a____ => true | id_a____,id_U11 => false | id_a____,id_a__U11 => false | id_a____,id_mark => false | id_a____,id_isNePal => false | id_a____,id_a__U12 => false | id_a____,id___ => false | id_a____,id_U12 => false | id_a____,id_tt => false | id_a____,id_nil => false | id_a____,id_a__isNePal => false | id_U11,id_a____ => true | id_U11,id_U11 => true | id_U11,id_a__U11 => false | id_U11,id_mark => false | id_U11,id_isNePal => false | id_U11,id_a__U12 => false | id_U11,id___ => false | id_U11,id_U12 => false | id_U11,id_tt => false | id_U11,id_nil => false | id_U11,id_a__isNePal => false | id_a__U11,id_a____ => true | id_a__U11,id_U11 => true | id_a__U11,id_a__U11 => true | id_a__U11,id_mark => false | id_a__U11,id_isNePal => false | id_a__U11,id_a__U12 => false | id_a__U11,id___ => false | id_a__U11,id_U12 => false | id_a__U11,id_tt => false | id_a__U11,id_nil => false | id_a__U11,id_a__isNePal => false | id_mark,id_a____ => true | id_mark,id_U11 => true | id_mark,id_a__U11 => true | id_mark,id_mark => true | id_mark,id_isNePal => false | id_mark,id_a__U12 => false | id_mark,id___ => false | id_mark,id_U12 => false | id_mark,id_tt => false | id_mark,id_nil => false | id_mark,id_a__isNePal => false | id_isNePal,id_a____ => true | id_isNePal,id_U11 => true | id_isNePal,id_a__U11 => true | id_isNePal,id_mark => true | id_isNePal,id_isNePal => true | id_isNePal,id_a__U12 => false | id_isNePal,id___ => false | id_isNePal,id_U12 => false | id_isNePal,id_tt => false | id_isNePal,id_nil => false | id_isNePal,id_a__isNePal => false | id_a__U12,id_a____ => true | id_a__U12,id_U11 => true | id_a__U12,id_a__U11 => true | id_a__U12,id_mark => true | id_a__U12,id_isNePal => true | id_a__U12,id_a__U12 => true | id_a__U12,id___ => false | id_a__U12,id_U12 => false | id_a__U12,id_tt => false | id_a__U12,id_nil => false | id_a__U12,id_a__isNePal => false | id___,id_a____ => true | id___,id_U11 => true | id___,id_a__U11 => true | id___,id_mark => true | id___,id_isNePal => true | id___,id_a__U12 => true | id___,id___ => true | id___,id_U12 => false | id___,id_tt => false | id___,id_nil => false | id___,id_a__isNePal => false | id_U12,id_a____ => true | id_U12,id_U11 => true | id_U12,id_a__U11 => true | id_U12,id_mark => true | id_U12,id_isNePal => true | id_U12,id_a__U12 => true | id_U12,id___ => true | id_U12,id_U12 => true | id_U12,id_tt => false | id_U12,id_nil => false | id_U12,id_a__isNePal => false | id_tt,id_a____ => true | id_tt,id_U11 => true | id_tt,id_a__U11 => true | id_tt,id_mark => true | id_tt,id_isNePal => true | id_tt,id_a__U12 => true | id_tt,id___ => true | id_tt,id_U12 => true | id_tt,id_tt => true | id_tt,id_nil => false | id_tt,id_a__isNePal => false | id_nil,id_a____ => true | id_nil,id_U11 => true | id_nil,id_a__U11 => true | id_nil,id_mark => true | id_nil,id_isNePal => true | id_nil,id_a__U12 => true | id_nil,id___ => true | id_nil,id_U12 => true | id_nil,id_tt => true | id_nil,id_nil => true | id_nil,id_a__isNePal => false | id_a__isNePal,id_a____ => true | id_a__isNePal,id_U11 => true | id_a__isNePal,id_a__U11 => true | id_a__isNePal,id_mark => true | id_a__isNePal,id_isNePal => true | id_a__isNePal,id_a__U12 => true | id_a__isNePal,id___ => true | id_a__isNePal,id_U12 => true | id_a__isNePal,id_tt => true | id_a__isNePal,id_nil => true | id_a__isNePal,id_a__isNePal => 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 := (* a____(__(X_,Y_),Z_) -> a____(mark(X_),a____(mark(Y_),mark(Z_))) *) | R_xml_0_rule_0 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil)):: (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 2)::nil)):: (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 3)::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil)):: (algebra.Alg.Var 3)::nil)) (* a____(X_,nil) -> mark(X_) *) | R_xml_0_rule_1 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil)) (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Var 1):: (algebra.Alg.Term algebra.F.id_nil nil)::nil)) (* a____(nil,X_) -> mark(X_) *) | R_xml_0_rule_2 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil)) (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Var 1)::nil)) (* a__U11(tt) -> a__U12(tt) *) | R_xml_0_rule_3 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)) (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)) (* a__U12(tt) -> tt *) | R_xml_0_rule_4 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)) (* a__isNePal(__(I_,__(P_,I_))) -> a__U11(tt) *) | R_xml_0_rule_5 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)) (algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 4)::(algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 5):: (algebra.Alg.Var 4)::nil))::nil))::nil)) (* mark(__(X1_,X2_)) -> a____(mark(X1_),mark(X2_)) *) | R_xml_0_rule_6 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 6)::nil)):: (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil)) (* mark(U11(X_)) -> a__U11(mark(X_)) *) | R_xml_0_rule_7 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U11 ((algebra.Alg.Var 1)::nil))::nil)) (* mark(U12(X_)) -> a__U12(mark(X_)) *) | R_xml_0_rule_8 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U12 ((algebra.Alg.Var 1)::nil))::nil)) (* mark(isNePal(X_)) -> a__isNePal(mark(X_)) *) | R_xml_0_rule_9 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_isNePal ((algebra.Alg.Var 1)::nil))::nil)) (* mark(nil) -> nil *) | R_xml_0_rule_10 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_nil nil) (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_nil nil)::nil)) (* mark(tt) -> tt *) | R_xml_0_rule_11 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_tt nil)::nil)) (* a____(X1_,X2_) -> __(X1_,X2_) *) | R_xml_0_rule_12 : R_xml_0_rules (algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 6):: (algebra.Alg.Var 7)::nil)) (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Var 6):: (algebra.Alg.Var 7)::nil)) (* a__U11(X_) -> U11(X_) *) | R_xml_0_rule_13 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_U11 ((algebra.Alg.Var 1)::nil)) (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Var 1)::nil)) (* a__U12(X_) -> U12(X_) *) | R_xml_0_rule_14 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_U12 ((algebra.Alg.Var 1)::nil)) (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Var 1)::nil)) (* a__isNePal(X_) -> isNePal(X_) *) | R_xml_0_rule_15 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_isNePal ((algebra.Alg.Var 1)::nil)) (algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Var 1)::nil)) . Definition R_xml_0_rule_as_list_0 := ((algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil)):: (algebra.Alg.Var 3)::nil)), (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 2)::nil))::(algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 3)::nil))::nil))::nil)))::nil. Definition R_xml_0_rule_as_list_1 := ((algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Var 1):: (algebra.Alg.Term algebra.F.id_nil nil)::nil)), (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_0. Definition R_xml_0_rule_as_list_2 := ((algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_1. Definition R_xml_0_rule_as_list_3 := ((algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)), (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)))::R_xml_0_rule_as_list_2. Definition R_xml_0_rule_as_list_4 := ((algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_3. Definition R_xml_0_rule_as_list_5 := ((algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 4)::(algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 5)::(algebra.Alg.Var 4)::nil))::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)))::R_xml_0_rule_as_list_4. Definition R_xml_0_rule_as_list_6 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 7)::nil))::nil))):: R_xml_0_rule_as_list_5. Definition R_xml_0_rule_as_list_7 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U11 ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::nil))):: R_xml_0_rule_as_list_6. Definition R_xml_0_rule_as_list_8 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U12 ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_mark ((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_mark ((algebra.Alg.Term algebra.F.id_isNePal ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::nil))):: R_xml_0_rule_as_list_8. Definition R_xml_0_rule_as_list_10 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_nil nil)::nil)),(algebra.Alg.Term algebra.F.id_nil nil)):: R_xml_0_rule_as_list_9. Definition R_xml_0_rule_as_list_11 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_tt nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_10. Definition R_xml_0_rule_as_list_12 := ((algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Var 6):: (algebra.Alg.Var 7)::nil)), (algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 6):: (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_11. Definition R_xml_0_rule_as_list_13 := ((algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_U11 ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_12. Definition R_xml_0_rule_as_list_14 := ((algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_U12 ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_13. Definition R_xml_0_rule_as_list_15 := ((algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_isNePal ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_14. Definition R_xml_0_rule_as_list := R_xml_0_rule_as_list_15. 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.or17_equiv in H|-. case H;clear H;intros H. injection H;intros ;subst;constructor 16. injection H;intros ;subst;constructor 15. injection H;intros ;subst;constructor 14. 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 (forall x10, t = (algebra.Alg.Term algebra.F.id_U11 (x10::nil)) -> exists x9, t' = (algebra.Alg.Term algebra.F.id_U11 (x9::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10)) (forall x10, t = (algebra.Alg.Term algebra.F.id_isNePal (x10::nil)) -> exists x9, t' = (algebra.Alg.Term algebra.F.id_isNePal (x9::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10)) (forall x10 x12, t = (algebra.Alg.Term algebra.F.id___ (x10::x12::nil)) -> exists x9, exists x11, t' = (algebra.Alg.Term algebra.F.id___ (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)) (forall x10, t = (algebra.Alg.Term algebra.F.id_U12 (x10::nil)) -> exists x9, t' = (algebra.Alg.Term algebra.F.id_U12 (x9::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10)) (t = (algebra.Alg.Term algebra.F.id_tt nil) -> t' = (algebra.Alg.Term algebra.F.id_tt nil)) (t = (algebra.Alg.Term algebra.F.id_nil nil) -> t' = (algebra.Alg.Term algebra.F.id_nil 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 x10 H;exists x10;intuition;constructor 1. intros x10 H;exists x10;intuition;constructor 1. intros x10 x12 H;exists x10;exists x12;intuition;constructor 1. intros x10 H;exists x10;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_U11 H_id_isNePal H_id___ H_id_U12 H_id_tt H_id_nil]. constructor. clear H_id_isNePal H_id___ H_id_U12 H_id_tt H_id_nil;intros x10 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_U11 y (refl_equal _)) as [x9];intros ;intuition; exists x9;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_U11 H_id___ H_id_U12 H_id_tt H_id_nil;intros x10 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_isNePal y (refl_equal _)) as [x9];intros ;intuition; exists x9;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_U11 H_id_isNePal H_id_U12 H_id_tt H_id_nil;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___ 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___ 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_U11 H_id_isNePal H_id___ H_id_tt H_id_nil;intros x10 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_U12 y (refl_equal _)) as [x9];intros ;intuition; exists x9;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_U11 H_id_isNePal H_id___ H_id_U12 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 ). clear H_id_U11 H_id_isNePal H_id___ H_id_U12 H_id_tt;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_U11_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x10, t = (algebra.Alg.Term algebra.F.id_U11 (x10::nil)) -> exists x9, t' = (algebra.Alg.Term algebra.F.id_U11 (x9::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_isNePal_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x10, t = (algebra.Alg.Term algebra.F.id_isNePal (x10::nil)) -> exists x9, t' = (algebra.Alg.Term algebra.F.id_isNePal (x9::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id____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___ (x10::x12::nil)) -> exists x9, exists x11, t' = (algebra.Alg.Term algebra.F.id___ (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_U12_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x10, t = (algebra.Alg.Term algebra.F.id_U12 (x10::nil)) -> exists x9, t' = (algebra.Alg.Term algebra.F.id_U12 (x9::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_tt_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_tt nil) -> t' = (algebra.Alg.Term algebra.F.id_tt nil). 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. 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_U11 (?x9::nil)) |- _ => let x9 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U11_is_R_xml_0_constructor H (refl_equal _)) as [x9 [Heq 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_isNePal (?x9::nil)) |- _ => let x9 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_isNePal_is_R_xml_0_constructor H (refl_equal _)) as [x9 [Heq 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___ (?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____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_U12 (?x9::nil)) |- _ => let x9 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U12_is_R_xml_0_constructor H (refl_equal _)) as [x9 [Heq 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_tt nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_tt_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_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 )) 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_U11 (?x9::nil)) |- _ => let x9 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U11_is_R_xml_0_constructor H (refl_equal _)) as [x9 [Heq 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_isNePal (?x9::nil)) |- _ => let x9 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_isNePal_is_R_xml_0_constructor H (refl_equal _)) as [x9 [Heq 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___ (?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____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_U12 (?x9::nil)) |- _ => let x9 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U12_is_R_xml_0_constructor H (refl_equal _)) as [x9 [Heq 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_tt nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_tt_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_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 ))) 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_a____ : A ->A ->A. Hypothesis P_id_U11 : A ->A. Hypothesis P_id_a__U11 : A ->A. Hypothesis P_id_mark : A ->A. Hypothesis P_id_isNePal : A ->A. Hypothesis P_id_a__U12 : A ->A. Hypothesis P_id___ : A ->A ->A. Hypothesis P_id_U12 : A ->A. Hypothesis P_id_tt : A. Hypothesis P_id_nil : A. Hypothesis P_id_a__isNePal : A ->A. Hypothesis P_id_a_____monotonic : forall x12 x10 x9 x11, (A0 <= x12)/\ (x12 <= x11) -> (A0 <= x10)/\ (x10 <= x9) ->P_id_a____ x10 x12 <= P_id_a____ x9 x11. Hypothesis P_id_U11_monotonic : forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_U11 x10 <= P_id_U11 x9. Hypothesis P_id_a__U11_monotonic : forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_a__U11 x10 <= P_id_a__U11 x9. Hypothesis P_id_mark_monotonic : forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_mark x10 <= P_id_mark x9. Hypothesis P_id_isNePal_monotonic : forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_isNePal x10 <= P_id_isNePal x9. Hypothesis P_id_a__U12_monotonic : forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_a__U12 x10 <= P_id_a__U12 x9. Hypothesis P_id____monotonic : forall x12 x10 x9 x11, (A0 <= x12)/\ (x12 <= x11) -> (A0 <= x10)/\ (x10 <= x9) ->P_id___ x10 x12 <= P_id___ x9 x11. Hypothesis P_id_U12_monotonic : forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_U12 x10 <= P_id_U12 x9. Hypothesis P_id_a__isNePal_monotonic : forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_a__isNePal x10 <= P_id_a__isNePal x9. Hypothesis P_id_a_____bounded : forall x10 x9, (A0 <= x9) ->(A0 <= x10) ->A0 <= P_id_a____ x10 x9. Hypothesis P_id_U11_bounded : forall x9, (A0 <= x9) ->A0 <= P_id_U11 x9. Hypothesis P_id_a__U11_bounded : forall x9, (A0 <= x9) ->A0 <= P_id_a__U11 x9. Hypothesis P_id_mark_bounded : forall x9, (A0 <= x9) ->A0 <= P_id_mark x9. Hypothesis P_id_isNePal_bounded : forall x9, (A0 <= x9) ->A0 <= P_id_isNePal x9. Hypothesis P_id_a__U12_bounded : forall x9, (A0 <= x9) ->A0 <= P_id_a__U12 x9. Hypothesis P_id____bounded : forall x10 x9, (A0 <= x9) ->(A0 <= x10) ->A0 <= P_id___ x10 x9. Hypothesis P_id_U12_bounded : forall x9, (A0 <= x9) ->A0 <= P_id_U12 x9. Hypothesis P_id_tt_bounded : A0 <= P_id_tt . Hypothesis P_id_nil_bounded : A0 <= P_id_nil . Hypothesis P_id_a__isNePal_bounded : forall x9, (A0 <= x9) ->A0 <= P_id_a__isNePal x9. Fixpoint measure t { struct t } := match t with | (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) => P_id_a____ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_U11 (x9::nil)) => P_id_U11 (measure x9) | (algebra.Alg.Term algebra.F.id_a__U11 (x9::nil)) => P_id_a__U11 (measure x9) | (algebra.Alg.Term algebra.F.id_mark (x9::nil)) => P_id_mark (measure x9) | (algebra.Alg.Term algebra.F.id_isNePal (x9::nil)) => P_id_isNePal (measure x9) | (algebra.Alg.Term algebra.F.id_a__U12 (x9::nil)) => P_id_a__U12 (measure x9) | (algebra.Alg.Term algebra.F.id___ (x10::x9::nil)) => P_id___ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_U12 (x9::nil)) => P_id_U12 (measure x9) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_a__isNePal (x9::nil)) => P_id_a__isNePal (measure x9) | _ => A0 end. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) => P_id_a____ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_U11 (x9::nil)) => P_id_U11 (measure x9) | (algebra.Alg.Term algebra.F.id_a__U11 (x9::nil)) => P_id_a__U11 (measure x9) | (algebra.Alg.Term algebra.F.id_mark (x9::nil)) => P_id_mark (measure x9) | (algebra.Alg.Term algebra.F.id_isNePal (x9::nil)) => P_id_isNePal (measure x9) | (algebra.Alg.Term algebra.F.id_a__U12 (x9::nil)) => P_id_a__U12 (measure x9) | (algebra.Alg.Term algebra.F.id___ (x10::x9::nil)) => P_id___ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_U12 (x9::nil)) => P_id_U12 (measure x9) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_a__isNePal (x9::nil)) => P_id_a__isNePal (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_a____ => P_id_a____ | algebra.F.id_U11 => P_id_U11 | algebra.F.id_a__U11 => P_id_a__U11 | algebra.F.id_mark => P_id_mark | algebra.F.id_isNePal => P_id_isNePal | algebra.F.id_a__U12 => P_id_a__U12 | algebra.F.id___ => P_id___ | algebra.F.id_U12 => P_id_U12 | algebra.F.id_tt => P_id_tt | algebra.F.id_nil => P_id_nil | algebra.F.id_a__isNePal => P_id_a__isNePal 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_a____ => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U11 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U11 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_mark => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_isNePal => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U12 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id___ => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U12 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_tt => match l with | nil => _ | _::_ => _ end | algebra.F.id_nil => match l with | nil => _ | _::_ => _ end | algebra.F.id_a__isNePal => match l with | 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_a_____bounded;assumption. vm_compute in |-*;intros ;apply P_id_U11_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U11_bounded;assumption. vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isNePal_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U12_bounded;assumption. vm_compute in |-*;intros ;apply P_id____bounded;assumption. vm_compute in |-*;intros ;apply P_id_U12_bounded;assumption. vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption. vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isNePal_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_a_____monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U11_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U11_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_mark_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_isNePal_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U12_monotonic;assumption. vm_compute in |-*;intros ;apply P_id____monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U12_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_a__isNePal_monotonic;assumption. intros f. case f. vm_compute in |-*;intros ;apply P_id_a_____bounded;assumption. vm_compute in |-*;intros ;apply P_id_U11_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U11_bounded;assumption. vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isNePal_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U12_bounded;assumption. vm_compute in |-*;intros ;apply P_id____bounded;assumption. vm_compute in |-*;intros ;apply P_id_U12_bounded;assumption. vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption. vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isNePal_bounded;assumption. intros . do 2 (rewrite <- same_measure in |-*). apply rules_monotonic;assumption. assumption. Qed. Hypothesis P_id_MARK : A ->A. Hypothesis P_id_A__U12 : A ->A. Hypothesis P_id_A__U11 : A ->A. Hypothesis P_id_A____ : A ->A ->A. Hypothesis P_id_A__ISNEPAL : A ->A. Hypothesis P_id_MARK_monotonic : forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_MARK x10 <= P_id_MARK x9. Hypothesis P_id_A__U12_monotonic : forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_A__U12 x10 <= P_id_A__U12 x9. Hypothesis P_id_A__U11_monotonic : forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_A__U11 x10 <= P_id_A__U11 x9. Hypothesis P_id_A_____monotonic : forall x12 x10 x9 x11, (A0 <= x12)/\ (x12 <= x11) -> (A0 <= x10)/\ (x10 <= x9) ->P_id_A____ x10 x12 <= P_id_A____ x9 x11. Hypothesis P_id_A__ISNEPAL_monotonic : forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_A__ISNEPAL x10 <= P_id_A__ISNEPAL x9. Definition marked_measure t := match t with | (algebra.Alg.Term algebra.F.id_mark (x9::nil)) => P_id_MARK (measure x9) | (algebra.Alg.Term algebra.F.id_a__U12 (x9::nil)) => P_id_A__U12 (measure x9) | (algebra.Alg.Term algebra.F.id_a__U11 (x9::nil)) => P_id_A__U11 (measure x9) | (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) => P_id_A____ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_a__isNePal (x9::nil)) => P_id_A__ISNEPAL (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 x9;apply (P_id_A__ISNEPAL 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_A__U12 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_MARK 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_A__U11 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_A____ x10 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_a____ => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U11 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U11 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_mark => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_isNePal => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U12 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id___ => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U12 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_tt => match l with | nil => _ | _::_ => _ end | algebra.F.id_nil => match l with | nil => _ | _::_ => _ end | algebra.F.id_a__isNePal => match l with | nil => _ | _::nil => _ | _::_::_ => _ end end. 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 . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. 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 . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. 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 . 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 . 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 . Qed. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_mark (x9::nil)) => P_id_MARK (measure x9) | (algebra.Alg.Term algebra.F.id_a__U12 (x9::nil)) => P_id_A__U12 (measure x9) | (algebra.Alg.Term algebra.F.id_a__U11 (x9::nil)) => P_id_A__U11 (measure x9) | (algebra.Alg.Term algebra.F.id_a____ (x10:: x9::nil)) => P_id_A____ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_a__isNePal (x9::nil)) => P_id_A__ISNEPAL (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_a_____monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U11_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U11_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_mark_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_isNePal_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U12_monotonic;assumption. vm_compute in |-*;intros ;apply P_id____monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U12_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_a__isNePal_monotonic;assumption. clear f. intros f. case f. vm_compute in |-*;intros ;apply P_id_a_____bounded;assumption. vm_compute in |-*;intros ;apply P_id_U11_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U11_bounded;assumption. vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isNePal_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U12_bounded;assumption. vm_compute in |-*;intros ;apply P_id____bounded;assumption. vm_compute in |-*;intros ;apply P_id_U12_bounded;assumption. vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption. vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isNePal_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_A__ISNEPAL_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_A__U12_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_MARK_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_A__U11_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_A_____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_a____ : Z ->Z ->Z. Hypothesis P_id_U11 : Z ->Z. Hypothesis P_id_a__U11 : Z ->Z. Hypothesis P_id_mark : Z ->Z. Hypothesis P_id_isNePal : Z ->Z. Hypothesis P_id_a__U12 : Z ->Z. Hypothesis P_id___ : Z ->Z ->Z. Hypothesis P_id_U12 : Z ->Z. Hypothesis P_id_tt : Z. Hypothesis P_id_nil : Z. Hypothesis P_id_a__isNePal : Z ->Z. Hypothesis P_id_a_____monotonic : forall x12 x10 x9 x11, (min_value <= x12)/\ (x12 <= x11) -> (min_value <= x10)/\ (x10 <= x9) -> P_id_a____ x10 x12 <= P_id_a____ x9 x11. Hypothesis P_id_U11_monotonic : forall x10 x9, (min_value <= x10)/\ (x10 <= x9) ->P_id_U11 x10 <= P_id_U11 x9. Hypothesis P_id_a__U11_monotonic : forall x10 x9, (min_value <= x10)/\ (x10 <= x9) ->P_id_a__U11 x10 <= P_id_a__U11 x9. Hypothesis P_id_mark_monotonic : forall x10 x9, (min_value <= x10)/\ (x10 <= x9) ->P_id_mark x10 <= P_id_mark x9. Hypothesis P_id_isNePal_monotonic : forall x10 x9, (min_value <= x10)/\ (x10 <= x9) ->P_id_isNePal x10 <= P_id_isNePal x9. Hypothesis P_id_a__U12_monotonic : forall x10 x9, (min_value <= x10)/\ (x10 <= x9) ->P_id_a__U12 x10 <= P_id_a__U12 x9. Hypothesis P_id____monotonic : forall x12 x10 x9 x11, (min_value <= x12)/\ (x12 <= x11) -> (min_value <= x10)/\ (x10 <= x9) ->P_id___ x10 x12 <= P_id___ x9 x11. Hypothesis P_id_U12_monotonic : forall x10 x9, (min_value <= x10)/\ (x10 <= x9) ->P_id_U12 x10 <= P_id_U12 x9. Hypothesis P_id_a__isNePal_monotonic : forall x10 x9, (min_value <= x10)/\ (x10 <= x9) -> P_id_a__isNePal x10 <= P_id_a__isNePal x9. Hypothesis P_id_a_____bounded : forall x10 x9, (min_value <= x9) ->(min_value <= x10) ->min_value <= P_id_a____ x10 x9. Hypothesis P_id_U11_bounded : forall x9, (min_value <= x9) ->min_value <= P_id_U11 x9. Hypothesis P_id_a__U11_bounded : forall x9, (min_value <= x9) ->min_value <= P_id_a__U11 x9. Hypothesis P_id_mark_bounded : forall x9, (min_value <= x9) ->min_value <= P_id_mark x9. Hypothesis P_id_isNePal_bounded : forall x9, (min_value <= x9) ->min_value <= P_id_isNePal x9. Hypothesis P_id_a__U12_bounded : forall x9, (min_value <= x9) ->min_value <= P_id_a__U12 x9. Hypothesis P_id____bounded : forall x10 x9, (min_value <= x9) ->(min_value <= x10) ->min_value <= P_id___ x10 x9. Hypothesis P_id_U12_bounded : forall x9, (min_value <= x9) ->min_value <= P_id_U12 x9. Hypothesis P_id_tt_bounded : min_value <= P_id_tt . Hypothesis P_id_nil_bounded : min_value <= P_id_nil . Hypothesis P_id_a__isNePal_bounded : forall x9, (min_value <= x9) ->min_value <= P_id_a__isNePal x9. Definition measure := Interp.measure min_value P_id_a____ P_id_U11 P_id_a__U11 P_id_mark P_id_isNePal P_id_a__U12 P_id___ P_id_U12 P_id_tt P_id_nil P_id_a__isNePal. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) => P_id_a____ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_U11 (x9::nil)) => P_id_U11 (measure x9) | (algebra.Alg.Term algebra.F.id_a__U11 (x9::nil)) => P_id_a__U11 (measure x9) | (algebra.Alg.Term algebra.F.id_mark (x9::nil)) => P_id_mark (measure x9) | (algebra.Alg.Term algebra.F.id_isNePal (x9::nil)) => P_id_isNePal (measure x9) | (algebra.Alg.Term algebra.F.id_a__U12 (x9::nil)) => P_id_a__U12 (measure x9) | (algebra.Alg.Term algebra.F.id___ (x10::x9::nil)) => P_id___ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_U12 (x9::nil)) => P_id_U12 (measure x9) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_a__isNePal (x9::nil)) => P_id_a__isNePal (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_a_____monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. apply rules_monotonic. Qed. Hypothesis P_id_MARK : Z ->Z. Hypothesis P_id_A__U12 : Z ->Z. Hypothesis P_id_A__U11 : Z ->Z. Hypothesis P_id_A____ : Z ->Z ->Z. Hypothesis P_id_A__ISNEPAL : Z ->Z. Hypothesis P_id_MARK_monotonic : forall x10 x9, (min_value <= x10)/\ (x10 <= x9) ->P_id_MARK x10 <= P_id_MARK x9. Hypothesis P_id_A__U12_monotonic : forall x10 x9, (min_value <= x10)/\ (x10 <= x9) ->P_id_A__U12 x10 <= P_id_A__U12 x9. Hypothesis P_id_A__U11_monotonic : forall x10 x9, (min_value <= x10)/\ (x10 <= x9) ->P_id_A__U11 x10 <= P_id_A__U11 x9. Hypothesis P_id_A_____monotonic : forall x12 x10 x9 x11, (min_value <= x12)/\ (x12 <= x11) -> (min_value <= x10)/\ (x10 <= x9) -> P_id_A____ x10 x12 <= P_id_A____ x9 x11. Hypothesis P_id_A__ISNEPAL_monotonic : forall x10 x9, (min_value <= x10)/\ (x10 <= x9) -> P_id_A__ISNEPAL x10 <= P_id_A__ISNEPAL x9. Definition marked_measure := Interp.marked_measure min_value P_id_a____ P_id_U11 P_id_a__U11 P_id_mark P_id_isNePal P_id_a__U12 P_id___ P_id_U12 P_id_tt P_id_nil P_id_a__isNePal P_id_MARK P_id_A__U12 P_id_A__U11 P_id_A____ P_id_A__ISNEPAL. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_mark (x9::nil)) => P_id_MARK (measure x9) | (algebra.Alg.Term algebra.F.id_a__U12 (x9::nil)) => P_id_A__U12 (measure x9) | (algebra.Alg.Term algebra.F.id_a__U11 (x9::nil)) => P_id_A__U11 (measure x9) | (algebra.Alg.Term algebra.F.id_a____ (x10:: x9::nil)) => P_id_A____ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_a__isNePal (x9::nil)) => P_id_A__ISNEPAL (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_a_____monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. apply rules_monotonic. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__ISNEPAL_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) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::(algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x2::nil))::(algebra.Alg.Term algebra.F.id_mark (x3::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ (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) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_2 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x2::nil))::(algebra.Alg.Term algebra.F.id_mark (x3::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_3 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x2::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_4 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x3::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_5 : forall x10 x1 x9, (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_nil nil) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_6 : forall x10 x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_nil nil) 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_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_7 : forall x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)) (algebra.Alg.Term algebra.F.id_a__U11 (x9::nil)) (* *) | DP_R_xml_0_8 : forall x4 x9 x5, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x4::(algebra.Alg.Term algebra.F.id___ (x5::x4::nil))::nil)) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)) (algebra.Alg.Term algebra.F.id_a__isNePal (x9::nil)) (* *) | DP_R_xml_0_9 : forall x6 x9 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x6::x7::nil)) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x6::nil))::(algebra.Alg.Term algebra.F.id_mark (x7::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_10 : forall x6 x9 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x6::x7::nil)) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_11 : forall x6 x9 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x6::x7::nil)) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_12 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x1::nil)) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_13 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x1::nil)) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_14 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x1::nil)) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_15 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x1::nil)) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_16 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isNePal (x1::nil)) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_17 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isNePal (x1::nil)) x9) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (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 x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x1::nil)) x9) -> DP_R_xml_0_non_scc_1 (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (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_non_scc_2 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_2_0 : forall x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x9) -> DP_R_xml_0_non_scc_2 (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)) (algebra.Alg.Term algebra.F.id_a__U11 (x9::nil)) . Lemma acc_DP_R_xml_0_non_scc_2 : forall x y, (DP_R_xml_0_non_scc_2 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_non_scc_3 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_3_0 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x1::nil)) x9) -> DP_R_xml_0_non_scc_3 (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) . Lemma acc_DP_R_xml_0_non_scc_3 : forall x y, (DP_R_xml_0_non_scc_3 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; (eapply acc_DP_R_xml_0_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_non_scc_4 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_4_0 : forall x4 x9 x5, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x4::(algebra.Alg.Term algebra.F.id___ (x5::x4::nil))::nil)) x9) -> DP_R_xml_0_non_scc_4 (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)) (algebra.Alg.Term algebra.F.id_a__isNePal (x9::nil)) . Lemma acc_DP_R_xml_0_non_scc_4 : forall x y, (DP_R_xml_0_non_scc_4 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; (eapply acc_DP_R_xml_0_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_non_scc_5 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_5_0 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isNePal (x1::nil)) x9) -> DP_R_xml_0_non_scc_5 (algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) . Lemma acc_DP_R_xml_0_non_scc_5 : forall x y, (DP_R_xml_0_non_scc_5 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; (eapply acc_DP_R_xml_0_non_scc_4; 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_6 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_6_0 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0_scc_6 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_scc_6_1 : forall x6 x9 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x6::x7::nil)) x9) -> DP_R_xml_0_scc_6 (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x6::nil)):: (algebra.Alg.Term algebra.F.id_mark (x7::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_2 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0_scc_6 (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x1::nil)):: (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x2::nil)):: (algebra.Alg.Term algebra.F.id_mark (x3::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_scc_6_3 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0_scc_6 (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x2::nil)):: (algebra.Alg.Term algebra.F.id_mark (x3::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_scc_6_4 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0_scc_6 (algebra.Alg.Term algebra.F.id_mark (x2::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_scc_6_5 : forall x6 x9 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x6::x7::nil)) x9) -> DP_R_xml_0_scc_6 (algebra.Alg.Term algebra.F.id_mark (x6::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_6 : forall x6 x9 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x6::x7::nil)) x9) -> DP_R_xml_0_scc_6 (algebra.Alg.Term algebra.F.id_mark (x7::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_7 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x1::nil)) x9) -> DP_R_xml_0_scc_6 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_8 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x1::nil)) x9) -> DP_R_xml_0_scc_6 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_9 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isNePal (x1::nil)) x9) -> DP_R_xml_0_scc_6 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_10 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0_scc_6 (algebra.Alg.Term algebra.F.id_mark (x3::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_scc_6_11 : forall x10 x1 x9, (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_nil nil) x9) -> DP_R_xml_0_scc_6 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_scc_6_12 : forall x10 x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_nil nil) 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_6 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) . Module WF_DP_R_xml_0_scc_6. Inductive DP_R_xml_0_scc_6_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_6_large_0 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0_scc_6_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_scc_6_large_1 : forall x6 x9 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x6::x7::nil)) x9) -> DP_R_xml_0_scc_6_large (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x6::nil))::(algebra.Alg.Term algebra.F.id_mark (x7::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_large_2 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0_scc_6_large (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::(algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x2::nil)):: (algebra.Alg.Term algebra.F.id_mark (x3::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_scc_6_large_3 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0_scc_6_large (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x2::nil))::(algebra.Alg.Term algebra.F.id_mark (x3::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_scc_6_large_4 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0_scc_6_large (algebra.Alg.Term algebra.F.id_mark (x2::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_scc_6_large_5 : forall x6 x9 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x6::x7::nil)) x9) -> DP_R_xml_0_scc_6_large (algebra.Alg.Term algebra.F.id_mark (x6::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_large_6 : forall x6 x9 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x6::x7::nil)) x9) -> DP_R_xml_0_scc_6_large (algebra.Alg.Term algebra.F.id_mark (x7::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_large_7 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x1::nil)) x9) -> DP_R_xml_0_scc_6_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_large_8 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x1::nil)) x9) -> DP_R_xml_0_scc_6_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_large_9 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isNePal (x1::nil)) x9) -> DP_R_xml_0_scc_6_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_large_10 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0_scc_6_large (algebra.Alg.Term algebra.F.id_mark (x3::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) . Inductive DP_R_xml_0_scc_6_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_6_strict_0 : forall x10 x1 x9, (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_nil nil) x9) -> DP_R_xml_0_scc_6_strict (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_scc_6_strict_1 : forall x10 x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_nil nil) 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_6_strict (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) . Module WF_DP_R_xml_0_scc_6_large. Inductive DP_R_xml_0_scc_6_large_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_6_large_large_0 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0_scc_6_large_large (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::(algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x2::nil))::(algebra.Alg.Term algebra.F.id_mark (x3::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_scc_6_large_large_1 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isNePal (x1::nil)) x9) -> DP_R_xml_0_scc_6_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) . Inductive DP_R_xml_0_scc_6_large_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_6_large_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) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0_scc_6_large_strict (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_scc_6_large_strict_1 : forall x6 x9 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x6::x7::nil)) x9) -> DP_R_xml_0_scc_6_large_strict (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x6::nil))::(algebra.Alg.Term algebra.F.id_mark (x7::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_large_strict_2 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0_scc_6_large_strict (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x2::nil)):: (algebra.Alg.Term algebra.F.id_mark (x3::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_scc_6_large_strict_3 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0_scc_6_large_strict (algebra.Alg.Term algebra.F.id_mark (x2::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) (* *) | DP_R_xml_0_scc_6_large_strict_4 : forall x6 x9 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x6::x7::nil)) x9) -> DP_R_xml_0_scc_6_large_strict (algebra.Alg.Term algebra.F.id_mark (x6::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_large_strict_5 : forall x6 x9 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x6::x7::nil)) x9) -> DP_R_xml_0_scc_6_large_strict (algebra.Alg.Term algebra.F.id_mark (x7::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_large_strict_6 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x1::nil)) x9) -> DP_R_xml_0_scc_6_large_strict (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_large_strict_7 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x1::nil)) x9) -> DP_R_xml_0_scc_6_large_strict (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (* *) | DP_R_xml_0_scc_6_large_strict_8 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0_scc_6_large_strict (algebra.Alg.Term algebra.F.id_mark (x3::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) . Module WF_DP_R_xml_0_scc_6_large_large. Inductive DP_R_xml_0_scc_6_large_large_scc_1 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_6_large_large_scc_1_0 : forall x1 x9, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isNePal (x1::nil)) x9) -> DP_R_xml_0_scc_6_large_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x9::nil)) . Module WF_DP_R_xml_0_scc_6_large_large_scc_1. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x9:Z) (x10:Z) := 2* x9 + 1* x10. Definition P_id_U11 (x9:Z) := 0. Definition P_id_a__U11 (x9:Z) := 0. Definition P_id_mark (x9:Z) := 1* x9. Definition P_id_isNePal (x9:Z) := 2 + 2* x9. Definition P_id_a__U12 (x9:Z) := 0. Definition P_id___ (x9:Z) (x10:Z) := 2* x9 + 1* x10. Definition P_id_U12 (x9:Z) := 0. Definition P_id_tt := 0. Definition P_id_nil := 2. Definition P_id_a__isNePal (x9:Z) := 2 + 2* x9. Lemma P_id_a_____monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_a____ x10 x12 <= P_id_a____ 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_U11_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_U11 x10 <= P_id_U11 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_a__U11_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_a__U11 x10 <= P_id_a__U11 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_mark_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_mark x10 <= P_id_mark 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_isNePal_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_isNePal x10 <= P_id_isNePal 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_a__U12_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_a__U12 x10 <= P_id_a__U12 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____monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id___ x10 x12 <= P_id___ 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_U12_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_U12 x10 <= P_id_U12 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_a__isNePal_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_a__isNePal x10 <= P_id_a__isNePal 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_a_____bounded : forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id_a____ 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_U11_bounded : forall x9, (0 <= x9) ->0 <= P_id_U11 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__U11_bounded : forall x9, (0 <= x9) ->0 <= P_id_a__U11 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_mark_bounded : forall x9, (0 <= x9) ->0 <= P_id_mark 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_isNePal_bounded : forall x9, (0 <= x9) ->0 <= P_id_isNePal 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__U12_bounded : forall x9, (0 <= x9) ->0 <= P_id_a__U12 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____bounded : forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id___ 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_U12_bounded : forall x9, (0 <= x9) ->0 <= P_id_U12 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_tt_bounded : 0 <= P_id_tt . 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_a__isNePal_bounded : forall x9, (0 <= x9) ->0 <= P_id_a__isNePal 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_a____ P_id_U11 P_id_a__U11 P_id_mark P_id_isNePal P_id_a__U12 P_id___ P_id_U12 P_id_tt P_id_nil P_id_a__isNePal. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) => P_id_a____ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_U11 (x9::nil)) => P_id_U11 (measure x9) | (algebra.Alg.Term algebra.F.id_a__U11 (x9::nil)) => P_id_a__U11 (measure x9) | (algebra.Alg.Term algebra.F.id_mark (x9::nil)) => P_id_mark (measure x9) | (algebra.Alg.Term algebra.F.id_isNePal (x9::nil)) => P_id_isNePal (measure x9) | (algebra.Alg.Term algebra.F.id_a__U12 (x9::nil)) => P_id_a__U12 (measure x9) | (algebra.Alg.Term algebra.F.id___ (x10::x9::nil)) => P_id___ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_U12 (x9::nil)) => P_id_U12 (measure x9) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_a__isNePal (x9::nil)) => P_id_a__isNePal (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_a_____monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_MARK (x9:Z) := 3* x9. Definition P_id_A__U12 (x9:Z) := 0. Definition P_id_A__U11 (x9:Z) := 0. Definition P_id_A____ (x9:Z) (x10:Z) := 0. Definition P_id_A__ISNEPAL (x9:Z) := 0. Lemma P_id_MARK_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_MARK x10 <= P_id_MARK 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_A__U12_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_A__U12 x10 <= P_id_A__U12 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_A__U11_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_A__U11 x10 <= P_id_A__U11 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_A_____monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_A____ x10 x12 <= P_id_A____ 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_A__ISNEPAL_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_A__ISNEPAL x10 <= P_id_A__ISNEPAL 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_a____ P_id_U11 P_id_a__U11 P_id_mark P_id_isNePal P_id_a__U12 P_id___ P_id_U12 P_id_tt P_id_nil P_id_a__isNePal P_id_MARK P_id_A__U12 P_id_A__U11 P_id_A____ P_id_A__ISNEPAL. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_mark (x9::nil)) => P_id_MARK (measure x9) | (algebra.Alg.Term algebra.F.id_a__U12 (x9::nil)) => P_id_A__U12 (measure x9) | (algebra.Alg.Term algebra.F.id_a__U11 (x9::nil)) => P_id_A__U11 (measure x9) | (algebra.Alg.Term algebra.F.id_a____ (x10:: x9::nil)) => P_id_A____ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_a__isNePal (x9::nil)) => P_id_A__ISNEPAL (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_a_____monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. apply rules_monotonic. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__ISNEPAL_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 ). Lemma wf : well_founded WF_DP_R_xml_0_scc_6_large_large.DP_R_xml_0_scc_6_large_large_scc_1 . Proof. intros x. apply well_founded_ind with (R:=fun x y => (Zwf.Zwf 0) (marked_measure x) (marked_measure y)). apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. clear x. intros x IHx. repeat ( constructor;inversion 1;subst; full_prove_ineq algebra.Alg.Term ltac:(algebra.Alg_ext.find_replacement ) algebra.EQT_ext.one_step_list_refl_trans_clos marked_measure marked_measure_star_monotonic (Zwf.Zwf 0) (interp.o_Z 0) ltac:(fun _ => R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 ) ltac:(fun _ => rewrite_and_unfold ) ltac:(fun _ => generate_pos_hyp ) ltac:(fun _ => cbv beta iota zeta delta - [Zplus Zmult Zle Zlt] in * ; try (constructor)) IHx ). Qed. End WF_DP_R_xml_0_scc_6_large_large_scc_1. Definition wf_DP_R_xml_0_scc_6_large_large_scc_1 := WF_DP_R_xml_0_scc_6_large_large_scc_1.wf. Lemma acc_DP_R_xml_0_scc_6_large_large_scc_1 : forall x y, (DP_R_xml_0_scc_6_large_large_scc_1 x y) -> Acc WF_DP_R_xml_0_scc_6_large.DP_R_xml_0_scc_6_large_large x. Proof. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_6_large_large_scc_1). intros x' _ Hrec y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply Hrec;econstructor eassumption)|| ((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_6_large_large_scc_1. Qed. Inductive DP_R_xml_0_scc_6_large_large_scc_2 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_6_large_large_scc_2_0 : forall x2 x10 x1 x9 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x10) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x9) -> DP_R_xml_0_scc_6_large_large_scc_2 (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x1::nil)):: (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x2::nil)):: (algebra.Alg.Term algebra.F.id_mark (x3::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) . Module WF_DP_R_xml_0_scc_6_large_large_scc_2. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x9:Z) (x10:Z) := 1 + 2* x9 + 1* x10. Definition P_id_U11 (x9:Z) := 0. Definition P_id_a__U11 (x9:Z) := 0. Definition P_id_mark (x9:Z) := 1* x9. Definition P_id_isNePal (x9:Z) := 0. Definition P_id_a__U12 (x9:Z) := 0. Definition P_id___ (x9:Z) (x10:Z) := 1 + 2* x9 + 1* x10. Definition P_id_U12 (x9:Z) := 0. Definition P_id_tt := 0. Definition P_id_nil := 0. Definition P_id_a__isNePal (x9:Z) := 0. Lemma P_id_a_____monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_a____ x10 x12 <= P_id_a____ 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_U11_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_U11 x10 <= P_id_U11 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_a__U11_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_a__U11 x10 <= P_id_a__U11 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_mark_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_mark x10 <= P_id_mark 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_isNePal_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_isNePal x10 <= P_id_isNePal 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_a__U12_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_a__U12 x10 <= P_id_a__U12 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____monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id___ x10 x12 <= P_id___ 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_U12_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_U12 x10 <= P_id_U12 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_a__isNePal_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_a__isNePal x10 <= P_id_a__isNePal 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_a_____bounded : forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id_a____ 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_U11_bounded : forall x9, (0 <= x9) ->0 <= P_id_U11 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__U11_bounded : forall x9, (0 <= x9) ->0 <= P_id_a__U11 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_mark_bounded : forall x9, (0 <= x9) ->0 <= P_id_mark 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_isNePal_bounded : forall x9, (0 <= x9) ->0 <= P_id_isNePal 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__U12_bounded : forall x9, (0 <= x9) ->0 <= P_id_a__U12 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____bounded : forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id___ 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_U12_bounded : forall x9, (0 <= x9) ->0 <= P_id_U12 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_tt_bounded : 0 <= P_id_tt . 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_a__isNePal_bounded : forall x9, (0 <= x9) ->0 <= P_id_a__isNePal 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_a____ P_id_U11 P_id_a__U11 P_id_mark P_id_isNePal P_id_a__U12 P_id___ P_id_U12 P_id_tt P_id_nil P_id_a__isNePal. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) => P_id_a____ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_U11 (x9::nil)) => P_id_U11 (measure x9) | (algebra.Alg.Term algebra.F.id_a__U11 (x9::nil)) => P_id_a__U11 (measure x9) | (algebra.Alg.Term algebra.F.id_mark (x9::nil)) => P_id_mark (measure x9) | (algebra.Alg.Term algebra.F.id_isNePal (x9::nil)) => P_id_isNePal (measure x9) | (algebra.Alg.Term algebra.F.id_a__U12 (x9::nil)) => P_id_a__U12 (measure x9) | (algebra.Alg.Term algebra.F.id___ (x10::x9::nil)) => P_id___ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_U12 (x9::nil)) => P_id_U12 (measure x9) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_a__isNePal (x9::nil)) => P_id_a__isNePal (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_a_____monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_MARK (x9:Z) := 0. Definition P_id_A__U12 (x9:Z) := 0. Definition P_id_A__U11 (x9:Z) := 0. Definition P_id_A____ (x9:Z) (x10:Z) := 1* x9. Definition P_id_A__ISNEPAL (x9:Z) := 0. Lemma P_id_MARK_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_MARK x10 <= P_id_MARK 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_A__U12_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_A__U12 x10 <= P_id_A__U12 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_A__U11_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_A__U11 x10 <= P_id_A__U11 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_A_____monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_A____ x10 x12 <= P_id_A____ 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_A__ISNEPAL_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_A__ISNEPAL x10 <= P_id_A__ISNEPAL 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_a____ P_id_U11 P_id_a__U11 P_id_mark P_id_isNePal P_id_a__U12 P_id___ P_id_U12 P_id_tt P_id_nil P_id_a__isNePal P_id_MARK P_id_A__U12 P_id_A__U11 P_id_A____ P_id_A__ISNEPAL. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_mark (x9::nil)) => P_id_MARK (measure x9) | (algebra.Alg.Term algebra.F.id_a__U12 (x9::nil)) => P_id_A__U12 (measure x9) | (algebra.Alg.Term algebra.F.id_a__U11 (x9::nil)) => P_id_A__U11 (measure x9) | (algebra.Alg.Term algebra.F.id_a____ (x10:: x9::nil)) => P_id_A____ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_a__isNePal (x9::nil)) => P_id_A__ISNEPAL (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_a_____monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. apply rules_monotonic. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__ISNEPAL_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 ). Lemma wf : well_founded WF_DP_R_xml_0_scc_6_large_large.DP_R_xml_0_scc_6_large_large_scc_2 . Proof. intros x. apply well_founded_ind with (R:=fun x y => (Zwf.Zwf 0) (marked_measure x) (marked_measure y)). apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. clear x. intros x IHx. repeat ( constructor;inversion 1;subst; full_prove_ineq algebra.Alg.Term ltac:(algebra.Alg_ext.find_replacement ) algebra.EQT_ext.one_step_list_refl_trans_clos marked_measure marked_measure_star_monotonic (Zwf.Zwf 0) (interp.o_Z 0) ltac:(fun _ => R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 ) ltac:(fun _ => rewrite_and_unfold ) ltac:(fun _ => generate_pos_hyp ) ltac:(fun _ => cbv beta iota zeta delta - [Zplus Zmult Zle Zlt] in * ; try (constructor)) IHx ). Qed. End WF_DP_R_xml_0_scc_6_large_large_scc_2. Definition wf_DP_R_xml_0_scc_6_large_large_scc_2 := WF_DP_R_xml_0_scc_6_large_large_scc_2.wf. Lemma acc_DP_R_xml_0_scc_6_large_large_scc_2 : forall x y, (DP_R_xml_0_scc_6_large_large_scc_2 x y) -> Acc WF_DP_R_xml_0_scc_6_large.DP_R_xml_0_scc_6_large_large x. Proof. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_6_large_large_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)|| ((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_6_large_large_scc_2. Qed. Lemma wf : well_founded WF_DP_R_xml_0_scc_6_large.DP_R_xml_0_scc_6_large_large. Proof. constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_6_large_large_scc_2; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_6_large_large_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_6_large_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_6_large_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_a____ (x9:Z) (x10:Z) := 3 + 1* x9 + 1* x10. Definition P_id_U11 (x9:Z) := 1 + 2* x9. Definition P_id_a__U11 (x9:Z) := 1 + 2* x9. Definition P_id_mark (x9:Z) := 1* x9. Definition P_id_isNePal (x9:Z) := 2* x9. Definition P_id_a__U12 (x9:Z) := 1 + 2* x9. Definition P_id___ (x9:Z) (x10:Z) := 3 + 1* x9 + 1* x10. Definition P_id_U12 (x9:Z) := 1 + 2* x9. Definition P_id_tt := 3. Definition P_id_nil := 0. Definition P_id_a__isNePal (x9:Z) := 2* x9. Lemma P_id_a_____monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_a____ x10 x12 <= P_id_a____ 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_U11_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_U11 x10 <= P_id_U11 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_a__U11_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_a__U11 x10 <= P_id_a__U11 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_mark_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_mark x10 <= P_id_mark 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_isNePal_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_isNePal x10 <= P_id_isNePal 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_a__U12_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_a__U12 x10 <= P_id_a__U12 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____monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id___ x10 x12 <= P_id___ 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_U12_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_U12 x10 <= P_id_U12 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_a__isNePal_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_a__isNePal x10 <= P_id_a__isNePal 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_a_____bounded : forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id_a____ 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_U11_bounded : forall x9, (0 <= x9) ->0 <= P_id_U11 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__U11_bounded : forall x9, (0 <= x9) ->0 <= P_id_a__U11 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_mark_bounded : forall x9, (0 <= x9) ->0 <= P_id_mark 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_isNePal_bounded : forall x9, (0 <= x9) ->0 <= P_id_isNePal 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__U12_bounded : forall x9, (0 <= x9) ->0 <= P_id_a__U12 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____bounded : forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id___ 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_U12_bounded : forall x9, (0 <= x9) ->0 <= P_id_U12 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_tt_bounded : 0 <= P_id_tt . 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_a__isNePal_bounded : forall x9, (0 <= x9) ->0 <= P_id_a__isNePal 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_a____ P_id_U11 P_id_a__U11 P_id_mark P_id_isNePal P_id_a__U12 P_id___ P_id_U12 P_id_tt P_id_nil P_id_a__isNePal. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) => P_id_a____ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_U11 (x9::nil)) => P_id_U11 (measure x9) | (algebra.Alg.Term algebra.F.id_a__U11 (x9::nil)) => P_id_a__U11 (measure x9) | (algebra.Alg.Term algebra.F.id_mark (x9::nil)) => P_id_mark (measure x9) | (algebra.Alg.Term algebra.F.id_isNePal (x9::nil)) => P_id_isNePal (measure x9) | (algebra.Alg.Term algebra.F.id_a__U12 (x9::nil)) => P_id_a__U12 (measure x9) | (algebra.Alg.Term algebra.F.id___ (x10::x9::nil)) => P_id___ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_U12 (x9::nil)) => P_id_U12 (measure x9) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_a__isNePal (x9::nil)) => P_id_a__isNePal (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_a_____monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_MARK (x9:Z) := 2 + 1* x9. Definition P_id_A__U12 (x9:Z) := 0. Definition P_id_A__U11 (x9:Z) := 0. Definition P_id_A____ (x9:Z) (x10:Z) := 1 + 1* x9 + 1* x10. Definition P_id_A__ISNEPAL (x9:Z) := 0. Lemma P_id_MARK_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_MARK x10 <= P_id_MARK 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_A__U12_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_A__U12 x10 <= P_id_A__U12 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_A__U11_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_A__U11 x10 <= P_id_A__U11 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_A_____monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_A____ x10 x12 <= P_id_A____ 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_A__ISNEPAL_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_A__ISNEPAL x10 <= P_id_A__ISNEPAL 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_a____ P_id_U11 P_id_a__U11 P_id_mark P_id_isNePal P_id_a__U12 P_id___ P_id_U12 P_id_tt P_id_nil P_id_a__isNePal P_id_MARK P_id_A__U12 P_id_A__U11 P_id_A____ P_id_A__ISNEPAL. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_mark (x9::nil)) => P_id_MARK (measure x9) | (algebra.Alg.Term algebra.F.id_a__U12 (x9::nil)) => P_id_A__U12 (measure x9) | (algebra.Alg.Term algebra.F.id_a__U11 (x9::nil)) => P_id_A__U11 (measure x9) | (algebra.Alg.Term algebra.F.id_a____ (x10:: x9::nil)) => P_id_A____ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_a__isNePal (x9::nil)) => P_id_A__ISNEPAL (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_a_____monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. apply rules_monotonic. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__ISNEPAL_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_6_large_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_6_large_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_6_large_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_6_large_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_6_large_large := WF_DP_R_xml_0_scc_6_large_large.wf. Lemma wf : well_founded WF_DP_R_xml_0_scc_6.DP_R_xml_0_scc_6_large. Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_6_large_large). clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx';apply DP_R_xml_0_scc_6_large_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_6_large_large_in_le;econstructor eassumption])). apply wf_DP_R_xml_0_scc_6_large_large. Qed. End WF_DP_R_xml_0_scc_6_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_a____ (x9:Z) (x10:Z) := 1* x9 + 1* x10. Definition P_id_U11 (x9:Z) := 2* x9. Definition P_id_a__U11 (x9:Z) := 2* x9. Definition P_id_mark (x9:Z) := 1* x9. Definition P_id_isNePal (x9:Z) := 1* x9. Definition P_id_a__U12 (x9:Z) := 1* x9. Definition P_id___ (x9:Z) (x10:Z) := 1* x9 + 1* x10. Definition P_id_U12 (x9:Z) := 1* x9. Definition P_id_tt := 0. Definition P_id_nil := 2. Definition P_id_a__isNePal (x9:Z) := 1* x9. Lemma P_id_a_____monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_a____ x10 x12 <= P_id_a____ 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_U11_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_U11 x10 <= P_id_U11 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_a__U11_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_a__U11 x10 <= P_id_a__U11 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_mark_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_mark x10 <= P_id_mark 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_isNePal_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_isNePal x10 <= P_id_isNePal 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_a__U12_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_a__U12 x10 <= P_id_a__U12 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____monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id___ x10 x12 <= P_id___ 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_U12_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_U12 x10 <= P_id_U12 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_a__isNePal_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_a__isNePal x10 <= P_id_a__isNePal 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_a_____bounded : forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id_a____ 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_U11_bounded : forall x9, (0 <= x9) ->0 <= P_id_U11 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__U11_bounded : forall x9, (0 <= x9) ->0 <= P_id_a__U11 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_mark_bounded : forall x9, (0 <= x9) ->0 <= P_id_mark 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_isNePal_bounded : forall x9, (0 <= x9) ->0 <= P_id_isNePal 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__U12_bounded : forall x9, (0 <= x9) ->0 <= P_id_a__U12 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____bounded : forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id___ 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_U12_bounded : forall x9, (0 <= x9) ->0 <= P_id_U12 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_tt_bounded : 0 <= P_id_tt . 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_a__isNePal_bounded : forall x9, (0 <= x9) ->0 <= P_id_a__isNePal 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_a____ P_id_U11 P_id_a__U11 P_id_mark P_id_isNePal P_id_a__U12 P_id___ P_id_U12 P_id_tt P_id_nil P_id_a__isNePal. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x10::x9::nil)) => P_id_a____ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_U11 (x9::nil)) => P_id_U11 (measure x9) | (algebra.Alg.Term algebra.F.id_a__U11 (x9::nil)) => P_id_a__U11 (measure x9) | (algebra.Alg.Term algebra.F.id_mark (x9::nil)) => P_id_mark (measure x9) | (algebra.Alg.Term algebra.F.id_isNePal (x9::nil)) => P_id_isNePal (measure x9) | (algebra.Alg.Term algebra.F.id_a__U12 (x9::nil)) => P_id_a__U12 (measure x9) | (algebra.Alg.Term algebra.F.id___ (x10::x9::nil)) => P_id___ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_U12 (x9::nil)) => P_id_U12 (measure x9) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_a__isNePal (x9::nil)) => P_id_a__isNePal (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_a_____monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_MARK (x9:Z) := 1* x9. Definition P_id_A__U12 (x9:Z) := 0. Definition P_id_A__U11 (x9:Z) := 0. Definition P_id_A____ (x9:Z) (x10:Z) := 1* x9 + 1* x10. Definition P_id_A__ISNEPAL (x9:Z) := 0. Lemma P_id_MARK_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_MARK x10 <= P_id_MARK 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_A__U12_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_A__U12 x10 <= P_id_A__U12 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_A__U11_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_A__U11 x10 <= P_id_A__U11 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_A_____monotonic : forall x12 x10 x9 x11, (0 <= x12)/\ (x12 <= x11) -> (0 <= x10)/\ (x10 <= x9) ->P_id_A____ x10 x12 <= P_id_A____ 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_A__ISNEPAL_monotonic : forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_A__ISNEPAL x10 <= P_id_A__ISNEPAL 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_a____ P_id_U11 P_id_a__U11 P_id_mark P_id_isNePal P_id_a__U12 P_id___ P_id_U12 P_id_tt P_id_nil P_id_a__isNePal P_id_MARK P_id_A__U12 P_id_A__U11 P_id_A____ P_id_A__ISNEPAL. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_mark (x9::nil)) => P_id_MARK (measure x9) | (algebra.Alg.Term algebra.F.id_a__U12 (x9::nil)) => P_id_A__U12 (measure x9) | (algebra.Alg.Term algebra.F.id_a__U11 (x9::nil)) => P_id_A__U11 (measure x9) | (algebra.Alg.Term algebra.F.id_a____ (x10:: x9::nil)) => P_id_A____ (measure x10) (measure x9) | (algebra.Alg.Term algebra.F.id_a__isNePal (x9::nil)) => P_id_A__ISNEPAL (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_a_____monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. apply rules_monotonic. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__ISNEPAL_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_6_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_6_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_6_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_6_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_6_large := WF_DP_R_xml_0_scc_6_large.wf. Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_6. Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_6_large). clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx';apply DP_R_xml_0_scc_6_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_6_large_in_le;econstructor eassumption])). apply wf_DP_R_xml_0_scc_6_large. Qed. End WF_DP_R_xml_0_scc_6. Definition wf_DP_R_xml_0_scc_6 := WF_DP_R_xml_0_scc_6.wf. Lemma acc_DP_R_xml_0_scc_6 : forall x y, (DP_R_xml_0_scc_6 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_6). 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_5; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_3; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((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_6. 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_5; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_4; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_3; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_2; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((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_6; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_5; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_4; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_3; 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___TRCSR___PALINDROME_nosorts_noand_GM.trs/a3pat.v" *** *** End: *** *)