Author: Bruno Barras |
Require
Relation_Definitions.
Require
Relation_Operators.
Section
Wf_Transitive_Closure.
Variable
A: Set.
Variable
R: (relation A).
Syntactic
Definition
trans_clos := (clos_trans A R).
Lemma
incl_clos_trans: (inclusion A R trans_clos).
Red;Auto with sets.
Qed
.
Lemma
Acc_clos_trans: (x:A)(Acc A R x)->(Acc A trans_clos x).
Induction 1.
Intros x0 H0 H1.
Apply Acc_intro.
Intros y H2.
Generalize H1 .
Elim H2;Auto with sets.
Intros x1 y0 z H3 H4 H5 H6 H7.
Apply Acc_inv with y0 ;Auto with sets.
Qed
.
Hints
Resolve Acc_clos_trans.
Lemma
Acc_inv_trans: (x,y:A)(trans_clos y x)->(Acc A R x)->(Acc A R y).
Proof
.
Induction 1;Auto with sets.
Intros x0 y0 H0 H1.
Apply Acc_inv with y0 ;Auto with sets.
Qed
.
Theorem
wf_clos_trans: (well_founded A R) ->(well_founded A trans_clos).
Proof
.
Unfold well_founded;Auto with sets.
Qed
.
End
Wf_Transitive_Closure.