Require
Export
Relations_1.
Definition
Complement : (U: Type) (Relation U) -> (Relation U) :=
[U: Type] [R: (Relation U)] [x,y: U] ~ (R x y).
Theorem
Rsym_imp_notRsym: (U: Type) (R: (Relation U)) (Symmetric U R) ->
(Symmetric U (Complement U R)).
Proof
.
Unfold Symmetric Complement.
Intros U R H' x y H'0; Red; Intro H'1; Apply H'0; Auto with sets.
Qed
.
Theorem
Equiv_from_preorder :
(U: Type) (R: (Relation U)) (Preorder U R) ->
(Equivalence U [x,y: U] (R x y) /\ (R y x)).
Proof
.
Intros U R H'; Elim H'; Intros H'0 H'1.
Apply Definition_of_equivalence.
Red in H'0; Auto 10 with sets.
2:Red; Intros x y h; Elim h; Intros H'3 H'4; Auto 10 with sets.
Red in H'1; Red; Auto 10 with sets.
Intros x y z h; Elim h; Intros H'3 H'4; Clear h.
Intro h; Elim h; Intros H'5 H'6; Clear h.
Split; Apply H'1 with y; Auto 10 with sets.
Qed
.
Hints
Resolve Equiv_from_preorder.
Theorem
Equiv_from_order :
(U: Type) (R: (Relation U)) (Order U R) ->
(Equivalence U [x,y: U] (R x y) /\ (R y x)).
Proof
.
Intros U R H'; Elim H'; Auto 10 with sets.
Qed
.
Hints
Resolve Equiv_from_order.
Theorem
contains_is_preorder :
(U: Type) (Preorder (Relation U) (contains U)).
Proof
.
Auto 10 with sets.
Qed
.
Hints
Resolve contains_is_preorder.
Theorem
same_relation_is_equivalence :
(U: Type) (Equivalence (Relation U) (same_relation U)).
Proof
.
Unfold 1 same_relation; Auto 10 with sets.
Qed
.
Hints
Resolve same_relation_is_equivalence.
Theorem
cong_reflexive_same_relation:
(U:Type) (R, R':(Relation U)) (same_relation U R R') -> (Reflexive U R) ->
(Reflexive U R').
Proof
.
Unfold same_relation; Intuition.
Qed
.
Theorem
cong_symmetric_same_relation:
(U:Type) (R, R':(Relation U)) (same_relation U R R') -> (Symmetric U R) ->
(Symmetric U R').
Proof
.
Compute;Intros;Elim H;Intros;Clear H;Apply (H3 y x (H0 x y (H2 x y H1))).
Qed
.
Theorem
cong_antisymmetric_same_relation:
(U:Type) (R, R':(Relation U)) (same_relation U R R') ->
(Antisymmetric U R) -> (Antisymmetric U R').
Proof
.
Compute;Intros;Elim H;Intros;Clear H;Apply (H0 x y (H3 x y H1) (H3 y x H2)).
Qed
.
Theorem
cong_transitive_same_relation:
(U:Type) (R, R':(Relation U)) (same_relation U R R') -> (Transitive U R) ->
(Transitive U R').
Proof
.
Intros U R R' H' H'0; Red.
Elim H'.
Intros H'1 H'2 x y z H'3 H'4; Apply H'2.
Apply H'0 with y; Auto with sets.
Qed
.