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Set Implicit Arguments.
Unset Strict Implicit.
(****)
Definition is_transitive (A : Type) (r : A -> A -> Prop) :=
forall x y z : A, r x y -> r y z -> r x z.
Definition is_reflexive (A : Type) (r : A -> A -> Prop) :=
forall x : A, r x x.
Definition is_symmetric (A : Type) (r : A -> A -> Prop) :=
forall x y : A, r x y -> r y x.
(****)
Record po : Type :=
{ po_carrier :> Type;
sub : po_carrier -> po_carrier -> Prop;
sub_transitive : is_transitive sub;
sub_reflexive : is_reflexive sub }.
Section equivalence.
Variable A : po.
Definition equiv (x y : A) := sub x y /\ sub y x.
(* Antisymmetry *)
Lemma equiv_intro : forall x y : A, sub x y -> sub y x -> equiv x y.
intros; split; trivial.
Qed.
Lemma equiv_elim_1 : forall x y : A, equiv x y -> sub x y.
intros x y (H, H'); trivial.
Qed.
Lemma equiv_elim_2 : forall x y : A, equiv x y -> sub y x.
intros x y (H, H'); trivial.
Qed.
Lemma equiv_transitive : is_transitive equiv.
intros x y z (H1, H2) (H3, H4); split;
[ exact (sub_transitive H1 H3) | exact (sub_transitive H4 H2) ].
Qed.
Lemma equiv_reflexive : is_reflexive equiv.
intros x; split; apply sub_reflexive.
Qed.
Lemma equiv_symmetric : is_symmetric equiv.
intros x y (H1, H2); split; trivial.
Qed.
End equivalence.
Opaque equiv.
(****)
Lemma opposite_transitive :
forall (A : Type) (r : A -> A -> Prop),
is_transitive r -> is_transitive (fun x y => r y x).
unfold is_transitive in |- *; eauto.
Qed.
Lemma opposite_reflexive :
forall (A : Type) (r : A -> A -> Prop),
is_reflexive r -> is_reflexive (fun x y => r y x).
trivial.
Qed.
Section opposite.
Variable A : po.
Definition opposite_po :=
Build_po (opposite_transitive (sub_transitive (p:=A)))
(opposite_reflexive (sub_reflexive (p:=A))).
Lemma equiv_opp_intro :
forall x y : A, equiv x y -> equiv (A:=opposite_po) x y.
intros x y H; apply equiv_intro;
[ apply equiv_elim_2; trivial | apply equiv_elim_1; trivial ].
Qed.
Lemma equiv_opp_elim :
forall x y : A, equiv (A:=opposite_po) x y -> equiv x y.
intros x y H; apply equiv_intro;
[ apply equiv_elim_2; trivial | apply equiv_elim_1; trivial ].
Qed.
End opposite.
(****)
Definition is_monotone (A B : po) (f : A -> B) :=
forall x y : A, sub x y -> sub (f x) (f y).
Record monotone_fun (A B : po) : Type :=
{ m_fun :> A -> B;
monotone : is_monotone m_fun }.
Definition monotone_sub_def (A B : po) (f g : monotone_fun A B) :=
forall x : A, sub (f x) (g x).
Lemma monotone_sub_refl :
forall A B : po, is_reflexive (monotone_sub_def (A:=A) (B:=B)).
intros A B f x; apply sub_reflexive.
Qed.
Lemma monotone_sub_trans :
forall A B : po, is_transitive (monotone_sub_def (A:=A) (B:=B)).
intros A B f g h H1 H2 x; exact (sub_transitive (H1 x) (H2 x)).
Qed.
Definition monotone_fun_po (A B : po) :=
Build_po (monotone_sub_trans (A:=A) (B:=B))
(monotone_sub_refl (A:=A) (B:=B)).
Lemma monotone_eq :
forall (A B : po) (f : monotone_fun A B) (x y : A),
equiv x y -> equiv (f x) (f y).
intros A B f x y H; apply equiv_intro;
[ apply monotone; exact (equiv_elim_1 H)
| apply monotone; exact (equiv_elim_2 H) ].
Qed.
Let compose_monotone :
forall (A B C : po) (f : monotone_fun A B)
(g : monotone_fun B C), is_monotone (fun x => g (f x)).
unfold is_monotone in |- *; intros; apply monotone; apply monotone; trivial.
Qed.
Definition compose (A B C : po) (f : monotone_fun A B)
(g : monotone_fun B C) := Build_monotone_fun (compose_monotone f g).
Let opp_monotone :
forall (A B : po) (f : A -> B),
is_monotone f -> is_monotone (A:=opposite_po A) (B:=opposite_po B) f.
unfold is_monotone in |- *; simpl in |- *; eauto.
Qed.
Definition opposite_monotone (A B : po) (f : monotone_fun A B) :=
Build_monotone_fun (opp_monotone (monotone f)).
Lemma constant_implies_monotone :
forall (A B : po) (y : B), is_monotone (fun x : A => y).
intros A B y x x' H; apply sub_reflexive.
Qed.
Definition constant_monotone (A B : po) (y : B) :=
Build_monotone_fun (@constant_implies_monotone A B y).
(****)
Definition is_fixpoint (A : po) (f : A -> A) (a : A) := equiv a (f a).
Definition is_upper_bound (A : po) (P : A -> Prop) (x : A) :=
forall y : A, P y -> sub y x.
Definition postfixpoint (A : po) (f : A -> A) (x : A) := sub x (f x).
Definition prefixpoint (A : po) (f : A -> A) (x : A) := sub (f x) x.
(****)
Definition sub_product (A B : po) (x y : prodT A B) :=
sub (fstT x) (fstT y) /\ sub (sndT x) (sndT y).
Lemma sub_product_transitive :
forall A B : po, is_transitive (sub_product (A:=A) (B:=B)).
intros A B x y z (H1, H1') (H2, H2'); split;
[ exact (sub_transitive H1 H2) | exact (sub_transitive H1' H2') ].
Qed.
Lemma sub_product_reflexive :
forall A B : po, is_reflexive (sub_product (A:=A) (B:=B)).
intros A B x; split; apply sub_reflexive.
Qed.
Definition product_po (A B : po) :=
Build_po (sub_product_transitive (A:=A) (B:=B))
(sub_product_reflexive (A:=A) (B:=B)).
(****)
Definition pwe_lift (A : Type) (B : po) (r : B -> B -> Prop) (f g : A -> B) :=
forall x : A, r (f x) (g x).
Definition sub_pwe (A : Type) (B : po) (f g : A -> B) :=
forall x : A, sub (f x) (g x).
Lemma sub_pwe_transitive :
forall (A : Type) (B : po), is_transitive (sub_pwe (A:=A) (B:=B)).
intros A B f g h H1 H2 x; apply sub_transitive with (1 := H1 x); apply H2.
Qed.
Lemma sub_pwe_reflexive :
forall (A : Type) (B : po), is_reflexive (sub_pwe (A:=A) (B:=B)).
intros A B f x; apply sub_reflexive.
Qed.
Definition pwe_po (A : Type) (B : po) :=
Build_po (sub_pwe_transitive (A:=A) (B:=B))
(sub_pwe_reflexive (A:=A) (B:=B)).
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