phanerozoic/Coq-ALEA · Datasets at Hugging Face

fact stringlengths 14 3.66k	type stringclasses 11 values	library stringclasses 1 value	imports listlengths 1 4	filename stringclasses 10 values	symbolic_name stringlengths 1 26
ord : Type := mk_ord { tord:>Type; Ole : tord->tord->Prop; Ole_refl : forall x :tord, Ole x x; Ole_trans : forall x y z:tord, Ole x y -> Ole y z -> Ole x z }. Hint Resolve Ole_refl Ole_trans: core. Hint Extern 2 (@Ole ?X1 ?X2 ?X3 ) => simpl Ole: core. Declare Scope O_scope. Infix "<=" := Ole : O_scope. Open Scope O_scope. ( * Associated equality *)	Record	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	ord
Oeq (O:ord) (x y : O) := x <= y /\ y <= x. (** printing == %\ensuremath{\equiv}% #≡# *) Infix "==" := Oeq (at level 70) : O_scope.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Oeq
Ole_refl_eq : forall (O:ord) (x y:O), x=y -> x <= y. intros O x y H; rewrite H; auto. Qed. Hint Resolve Ole_refl_eq: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Ole_refl_eq
Ole_antisym : forall (O:ord) (x y:O), x<=y -> y <=x -> x==y. red; auto. Qed. Hint Immediate Ole_antisym: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Ole_antisym
Oeq_refl : forall (O:ord) (x:O), x == x. red; auto. Qed. Hint Resolve Oeq_refl: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Oeq_refl
Oeq_refl_eq : forall (O:ord) (x y:O), x=y -> x == y. intros O x y H; rewrite H; auto. Qed. Hint Resolve Oeq_refl_eq: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Oeq_refl_eq
Oeq_sym : forall (O:ord) (x y:O), x == y -> y == x. unfold Oeq; intuition. Qed.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Oeq_sym
Oeq_le : forall (O:ord) (x y:O), x == y -> x <= y. unfold Oeq; intuition. Qed.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Oeq_le
Oeq_le_sym : forall (O:ord) (x y:O), x == y -> y <= x. unfold Oeq; intuition. Qed. Hint Resolve Oeq_le: core. Hint Immediate Oeq_sym Oeq_le_sym: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Oeq_le_sym
Oeq_trans : forall (O:ord) (x y z:O), x == y -> y == z -> x == z. unfold Oeq; split; apply Ole_trans with y; auto. Qed. Hint Resolve Oeq_trans: core. ( * Setoid relations *)	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Oeq_trans
Parametric Relation (o:ord) : (tord o) (Oeq (O:=o)) reflexivity proved by (Oeq_refl (O:=o)) symmetry proved by (Oeq_sym (O:=o)) transitivity proved by (Oeq_trans (O:=o)) as Oeq_Relation.	Add	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Parametric
Parametric Relation (o:ord) : (tord o) (Ole (o:=o)) reflexivity proved by (Ole_refl (o:=o)) transitivity proved by (Ole_trans (o:=o)) as Ole_Relation. (** printing ==> %\ensuremath\Longrightarrow% #⇾# *)	Add	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Parametric
Parametric Morphism (o:ord) : (Ole (o:=o)) with signature (Oeq (O:=o)) ==> (Oeq (O:=o)) ==> iff as Ole_eq_compat_iff. Proof. split; firstorder. apply Ole_trans with x; trivial. apply Ole_trans with x0; trivial. apply Ole_trans with y; trivial. apply Ole_trans with y0; trivial. Qed.	Add	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Parametric
Ole_eq_compat : forall (O : ord) (x1 x2 : O), x1 == x2 -> forall x3 x4 : O, x3 == x4 -> x1 <= x3 -> x2 <= x4. firstorder; apply Ole_trans with x1; trivial. apply Ole_trans with x3; trivial. Qed.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Ole_eq_compat
Ole_eq_right : forall (O : ord) (x y z: O), x <= y -> y == z -> x <= z. intros; apply Ole_eq_compat with x y; auto. Qed.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Ole_eq_right
Ole_eq_left : forall (O : ord) (x y z: O), x == y -> y <= z -> x <= z. intros; apply Ole_eq_compat with y z; auto. Qed. ( * Dual order ) (* - Iord x y := y <= x *)	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Ole_eq_left
Iord : ord -> ord. intros O; exists O (fun x y : O => y <= x); intros; auto. apply Ole_trans with y; auto. Defined. ( * Order on functions ) (* - ford f g := forall x, f x <= g x *)	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Iord
ford : Type -> ord -> ord. intros A O; exists (A->O) (fun f g:A->O => forall x, f x <= g x); intros; auto. apply Ole_trans with (y x0); auto. Defined. (** printing -o> %\ensuremath{\stackrel{o}{\rightarrow}}% *) Infix "-o>" := ford (right associativity, at level 30) : O_scope .	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	ford
ford_le_elim : forall A (O:ord) (f g:A -o> O), f <= g ->forall n, f n <= g n. auto. Qed. Hint Immediate ford_le_elim: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	ford_le_elim
ford_le_intro : forall A (O:ord) (f g:A -o> O), (forall n, f n <= g n) -> f <= g. auto. Qed. Hint Resolve ford_le_intro: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	ford_le_intro
ford_eq_elim : forall A (O:ord) (f g:A -o> O), f == g ->forall n, f n == g n. firstorder. Qed. Hint Immediate ford_eq_elim: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	ford_eq_elim
ford_eq_intro : forall A (O:ord) (f g:A -o> O), (forall n, f n == g n) -> f == g. red; auto. Qed. Hint Resolve ford_eq_intro: core. Hint Extern 2 (Ole (o:=ford ?X1 ?X2) ?X3 ?X4) => intro: core. ( Monotonicity ) (* *** Definition and properties *)	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	ford_eq_intro
monotonic (O1 O2:ord) (f : O1 -> O2) := forall x y, x <= y -> f x <= f y. Hint Unfold monotonic: core.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	monotonic
stable (O1 O2:ord) (f : O1 -> O2) := forall x y, x == y -> f x == f y. Hint Unfold stable: core.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	stable
monotonic_stable : forall (O1 O2 : ord) (f:O1 -> O2), monotonic f -> stable f. unfold monotonic, stable; firstorder. Qed. Hint Resolve monotonic_stable: core. ( * Type of monotonic functions *)	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	monotonic_stable
fmono (O1 O2:ord) : Type := mk_fmono {fmonot :> O1 -> O2; fmonotonic: monotonic fmonot}. Hint Resolve fmonotonic: core. (** - fmon O1 O2 (f g : fmono O1 O2) := forall x, f x <= g x *)	Record	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmono
fmon : ord -> ord -> ord. intros O1 O2; exists (fmono O1 O2) (fun f g:fmono O1 O2 => forall x, f x <= g x); intros; auto. apply Ole_trans with (y x0); auto. Defined. (** printing -m> %\ensuremath{\stackrel{m}{\rightarrow}}%*) Infix "-m>" := fmon (at level 30, right associativity) : O_scope.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon
fmon_stable : forall (O1 O2:ord) (f:O1 -m> O2), stable f. intros; apply monotonic_stable; auto. Qed. Hint Resolve fmon_stable: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_stable
fmon_le_elim : forall (O1 O2:ord) (f g:O1 -m> O2), f <= g -> forall n, f n <= g n. auto. Qed. Hint Immediate fmon_le_elim: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_le_elim
fmon_le_intro : forall (O1 O2:ord) (f g:O1 -m> O2), (forall n, f n <= g n) -> f <= g. auto. Qed. Hint Resolve fmon_le_intro: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_le_intro
fmon_eq_elim : forall (O1 O2:ord) (f g:O1 -m> O2), f == g ->forall n, f n == g n. firstorder. Qed. Hint Immediate fmon_eq_elim: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_eq_elim
fmon_eq_intro : forall (O1 O2:ord) (f g:O1 -m> O2), (forall n, f n == g n) -> f == g. red; auto. Qed. Hint Resolve fmon_eq_intro: core. Hint Extern 2 (Ole (o:=fmon ?X1 ?X2) ?X3 ?X4) => intro: core. ( * Monotonicity and dual order ) (* - [lmon f] uses f as monotonic function over the dual order. *)	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_eq_intro
Imon : forall O1 O2, (O1 -m> O2) -> Iord O1 -m> Iord O2. intros O1 O2 f; exists (f: Iord O1 -> Iord O2); red; simpl; intros. apply (fmonotonic f); auto. Defined.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Imon
Imon2 : forall O1 O2 O3, (O1 -m> O2 -m> O3) -> Iord O1 -m> Iord O2 -m> Iord O3. intros O1 O2 O3 f; exists (fun (x:Iord O1) => Imon (f x)); red; simpl; intros. apply (fmonotonic f); auto. Defined. ( * Monotonic functions with 2 arguments *)	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Imon2
le_compat2_mon : forall (O1 O2 O3:ord)(f:O1 -> O2 -> O3), (forall (x y:O1) (z t:O2), x<=y -> z <= t -> f x z <= f y t) -> (O1 -m> O2 -m> O3). intros O1 O2 O3 f Hle; exists (fun (x:O1) => mk_fmono (fun z t => Hle x x z t (Ole_refl x))). red; intros; intro a; simpl; auto. Defined. ( Sequences ) (* *** Order on natural numbers ) (* - natO n m = n <= m *)	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	le_compat2_mon
natO : ord. exists nat (fun n m : nat => (n <= m)%nat); intros; auto with arith. apply le_trans with y; auto. Defined.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	natO
fnatO_intro : forall (O:ord) (f:nat -> O), (forall n, f n <= f (S n)) -> natO -m> O. intros; exists f; red; simpl; intros. elim H0; intros; auto. apply Ole_trans with (f m); trivial. Defined.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fnatO_intro
fnatO_elim : forall (O:ord) (f:natO -m> O) (n:nat), f n <= f (S n). intros; apply (fmonotonic f); auto. Qed. Hint Resolve fnatO_elim: core. (** - (mseq_lift_left f n) k = f (n+k) *)	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fnatO_elim
mseq_lift_left : forall (O:ord) (f:natO -m> O) (n:nat), natO -m> O. intros; exists (fun k => f (n+k)%nat); red; intros. apply (fmonotonic f); auto with arith. Defined.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	mseq_lift_left
mseq_lift_left_simpl : forall (O:ord) (f:natO -m> O) (n k:nat), mseq_lift_left f n k = f (n+k)%nat. trivial. Qed.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	mseq_lift_left_simpl
mseq_lift_left_le_compat : forall (O:ord) (f g:natO -m> O) (n:nat), f <= g -> mseq_lift_left f n <= mseq_lift_left g n. intros; intro; simpl; auto. Qed. Hint Resolve mseq_lift_left_le_compat: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	mseq_lift_left_le_compat
Parametric Morphism (o:ord) : (mseq_lift_left (O:=o)) with signature (Oeq (O:=natO -m> o)) ==> eq (A:=nat) ==> (Oeq (O:=natO -m> o)) as mseq_lift_left_eq_compat. intros; apply Ole_antisym; auto. Qed. Hint Resolve mseq_lift_left_eq_compat: core. (** - (mseq_lift_right f n) k = f (k+n) *)	Add	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Parametric
mseq_lift_right : forall (O:ord) (f:natO -m> O) (n:nat), natO -m> O. intros; exists (fun k => f (k+n)%nat); red; intros. apply (fmonotonic f); auto with arith. Defined.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	mseq_lift_right
mseq_lift_right_simpl : forall (O:ord) (f:natO -m> O) (n k:nat), mseq_lift_right f n k = f (k+n)%nat. trivial. Qed.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	mseq_lift_right_simpl
mseq_lift_right_le_compat : forall (O:ord) (f g:natO -m> O) (n:nat), f <= g -> mseq_lift_right f n <= mseq_lift_right g n. intros; intro; simpl; auto. Qed. Hint Resolve mseq_lift_right_le_compat: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	mseq_lift_right_le_compat
Parametric Morphism (o:ord) : (mseq_lift_right (O:=o)) with signature Oeq (O:=natO -m> o) ==> eq (A:=nat) ==> Oeq (O:=natO -m> o) as mseq_lift_right_eq_compat. intros; apply Ole_antisym; auto. Qed.	Add	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Parametric
mseq_lift_right_left : forall (O:ord) (f:natO -m> O) n, mseq_lift_left f n == mseq_lift_right f n. intros; apply fmon_eq_intro; unfold mseq_lift_left,mseq_lift_right; simpl; intros. replace (n0+n)%nat with (n+n0)%nat; auto with arith. Qed. ( * Monotonicity and functions ) (* - (ford_app f x) n = f n x *)	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	mseq_lift_right_left
ford_app : forall (A:Type)(O1 O2:ord)(f:O1 -m> (A -o> O2))(x:A), O1 -m> O2. intros; exists (fun n => f n x); intros. intro n; intros. assert (f n <= f y); auto. apply (fmonotonic f); trivial. Defined. (** printing <o> %\ensuremath{\stackrel{o}{\diamond}}% *) Infix "<o>" := ford_app (at level 30, no associativity) : O_scope.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	ford_app
ford_app_simpl : forall (A:Type)(O1 O2:ord) (f : O1 -m> A -o> O2) (x:A)(y:O1), (f <o> x) y = f y x. trivial. Qed.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	ford_app_simpl
ford_app_le_compat : forall (A:Type)(O1 O2:ord) (f g:O1 -m> A -o> O2) (x:A), f <= g -> f <o> x <= g <o> x. intros; intro; simpl. apply (H x0). Qed. Hint Resolve ford_app_le_compat: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	ford_app_le_compat
Parametric Morphism (A:Type)(O1 O2:ord) : (ford_app (A:=A) (O1:=O1) (O2:=O2)) with signature Oeq (O:=O1 -m> (A -o> O2)) ==> eq (A:=A) ==> Oeq (O:=O1 -m> O2) as ford_app_eq_compat. intros; apply Ole_antisym; auto. Qed. (** - ford_shift f x y == f y x *)	Add	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Parametric
ford_shift : forall (A:Type)(O1 O2:ord)(f:A -o> (O1 -m> O2)), O1 -m> (A -o> O2). intros; exists (fun x y => f y x); intros. intros n x H y. apply (fmonotonic (f y)); trivial. Defined.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	ford_shift
ford_shift_le_compat : forall (A:Type)(O1 O2:ord) (f g: A -o> (O1 -m> O2)), f <= g -> ford_shift f <= ford_shift g. intros; intro; simpl; auto. Qed. Hint Resolve ford_shift_le_compat: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	ford_shift_le_compat
Parametric Morphism (A:Type)(O1 O2:ord) : (ford_shift (A:=A) (O1:=O1) (O2:=O2)) with signature Oeq (O:=A -o> (O1 -m> O2)) ==> Oeq (O:=O1 -m> (A -o> O2)) as ford_shift_eq_compat. intros; apply Ole_antisym; auto. Qed. (** - (fmon_app f x) n = f n x *)	Add	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Parametric
fmon_app : forall (O1 O2 O3:ord)(f:O1 -m> O2 -m> O3)(x:O2), O1 -m> O3. intros; exists (fun n => f n x); intros. intro n; intros. assert (f n <= f y); auto. apply (fmonotonic f); trivial. Defined. (** printing <_> %\ensuremath{\leftrightarroweq}%*) Infix "<_>" := fmon_app (at level 35, no associativity) : O_scope.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_app
fmon_app_simpl : forall (O1 O2 O3:ord)(f:O1 -m> O2 -m> O3)(x:O2)(y:O1), (f <_> x) y = f y x. trivial. Qed.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_app_simpl
fmon_app_le_compat : forall (O1 O2 O3:ord) (f g:O1 -m> (O2 -m> O3)) (x y:O2), f <= g -> x <= y -> f <_> x <= g <_> y. red; intros; simpl; intros; auto. apply Ole_trans with (f x0 y); auto. apply (fmonotonic (f x0)); auto. Qed. Hint Resolve fmon_app_le_compat: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_app_le_compat
Parametric Morphism (O1 O2 O3:ord) : (fmon_app (O1:=O1) (O2:=O2) (O3:=O3)) with signature Oeq (O:=O1 -m> O2 -m> O3) ==> Oeq (O:=O2) ==> Oeq (O:=O1-m>O3) as fmon_app_eq_compat. intros; apply Ole_antisym; intros; auto. Qed. (** - fmon_id c = c *)	Add	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Parametric
fmon_id : forall (O:ord), O -m> O. intros; exists (fun (x:O)=>x). intro n; auto. Defined.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_id
fmon_id_simpl : forall (O:ord) (x:O), fmon_id O x = x. trivial. Qed. (** - (fmon_cte c) n = c *)	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_id_simpl
fmon_cte : forall (O1 O2:ord)(c:O2), O1 -m> O2. intros; exists (fun (x:O1)=>c). intro n; auto. Defined.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_cte
fmon_cte_simpl : forall (O1 O2:ord)(c:O2)(c:O2) (x:O1), fmon_cte O1 c x = c. trivial. Qed.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_cte_simpl
mseq_cte : forall O:ord, O -> natO -m> O := fmon_cte natO.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	mseq_cte
fmon_cte_le_compat : forall (O1 O2:ord) (c1 c2:O2), c1 <= c2 -> fmon_cte O1 c1 <= fmon_cte O1 c2. intros; intro; auto. Qed.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_cte_le_compat
Parametric Morphism (O1 O2:ord) : (fmon_cte O1 (O2:=O2)) with signature Oeq (O:=O2) ==> Oeq (O:=O1 -m> O2) as fmon_cte_eq_compat. intros; apply Ole_antisym; auto. Qed. (** - (fmon_diag h) n = h n n *)	Add	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Parametric
fmon_diag : forall (O1 O2:ord)(h:O1 -m> (O1 -m> O2)), O1 -m> O2. intros; exists (fun n => h n n). red; intros. apply Ole_trans with (h x y); auto. apply (fmonotonic (h x)); auto. assert (h x <= h y); auto. apply (fmonotonic h); trivial. Defined.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_diag
fmon_diag_le_compat : forall (O1 O2:ord) (f g:O1 -m> (O1 -m> O2)), f <= g -> fmon_diag f <= fmon_diag g. intros; intro; simpl; auto. Qed. Hint Resolve fmon_diag_le_compat: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_diag_le_compat
fmon_diag_simpl : forall (O1 O2:ord) (f:O1 -m> (O1 -m> O2)) (x:O1), fmon_diag f x = f x x. trivial. Qed.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_diag_simpl
Parametric Morphism (O1 O2:ord) : (fmon_diag (O1:=O1) (O2:=O2)) with signature Oeq (O:=O1 -m> (O1 -m> O2)) ==> Oeq (O:=O1 -m> O2) as fmon_diag_eq_compat. intros; apply Ole_antisym; auto. Qed. (** - (fmon_shift h) n m = h m n *)	Add	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Parametric
fmon_shift : forall (O1 O2 O3:ord)(h:O1 -m> O2 -m> O3), O2 -m> O1 -m> O3. intros; exists (fun m => h <_> m). intro n; simpl; intros. apply (fmonotonic (h x)); trivial. Defined.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_shift
fmon_shift_simpl : forall (O1 O2 O3:ord)(h:O1 -m> O2 -m> O3) (x : O2) (y:O1), fmon_shift h x y = h y x. trivial. Qed.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_shift_simpl
fmon_shift_le_compat : forall (O1 O2 O3:ord) (f g:O1 -m> O2 -m> O3), f <= g -> fmon_shift f <= fmon_shift g. intros; intro; simpl; intros. assert (f x0 <= g x0); auto. Qed. Hint Resolve fmon_shift_le_compat: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_shift_le_compat
Parametric Morphism (O1 O2 O3:ord) : (fmon_shift (O1:=O1) (O2:=O2) (O3:=O3)) with signature Oeq (O:=O1 -m> O2 -m> O3) ==> Oeq (O:=O2 -m> O1 -m> O3) as fmon_shift_eq_compat. intros; apply Ole_antisym; auto. Qed.	Add	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Parametric
fmon_shift_shift_eq : forall (O1 O2 O3:ord) (h : O1 -m> O2 -m> O3), fmon_shift (fmon_shift h) == h. intros; apply fmon_eq_intro; unfold fmon_shift; simpl; auto. Qed. (** - (f@g) x = f (g x) *)	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_shift_shift_eq
fmon_comp : forall O1 O2 O3:ord, (O2 -m> O3) -> (O1 -m> O2) -> O1 -m> O3. intros O1 O2 O3 f g; exists (fun n => f (g n)); red; intros. apply (fmonotonic f). apply (fmonotonic g); auto. Defined. (** printing @ %\ensuremath{\stackrel{m}{\circ}}% *) Infix "@" := fmon_comp (at level 35) : O_scope.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_comp
fmon_comp_simpl : forall (O1 O2 O3:ord) (f :O2 -m> O3) (g:O1 -m> O2) (x:O1), (f @ g) x = f (g x). trivial. Qed. (** - (f@2 g) h x = f (g x) (h x) *)	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_comp_simpl
fmon_comp2 : forall O1 O2 O3 O4:ord, (O2 -m> O3 -m> O4) -> (O1 -m> O2) -> (O1 -m> O3) -> O1-m>O4. intros O1 O2 O3 O4 f g h; exists (fun n => f (g n) (h n)); red; intros. apply Ole_trans with (f (g x) (h y)); auto. apply (fmonotonic (f (g x))). apply (fmonotonic h); auto. apply (fmonotonic f); auto. apply (fmonotonic g); auto. Defined. (** printing @2 %\ensuremath{\stackrel{m}{\circ_2}}% *) Infix "@2" := fmon_comp2 (at level 70) : O_scope.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_comp2
fmon_comp2_simpl : forall (O1 O2 O3 O4:ord) (f:O2 -m> O3 -m> O4) (g:O1 -m> O2) (h:O1 -m> O3) (x:O1), (f @2 g) h x = f (g x) (h x). trivial. Qed.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_comp2_simpl
Parametric Morphism (O1 O2 O3:ord) : (fmon_comp (O1:=O1) (O2:=O2) (O3:=O3)) with signature Ole (o:=O2 -m> O3) ++> Ole (o:=O1 -m> O2) ++> Ole (o:=O1 -m> O3) as fmon_comp_le_compat_morph. red; intros f1 f2 H g1 g2 H1 x; simpl. apply Ole_trans with (f2 (g1 x)); auto. apply (fmonotonic f2); auto. Qed.	Add	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Parametric
fmon_comp_le_compat : forall (O1 O2 O3:ord) (f1 f2: O2 -m> O3) (g1 g2:O1 -m> O2), f1 <= f2 -> g1<= g2 -> f1 @ g1 <= f2 @ g2. intros; exact (fmon_comp_le_compat_morph H H0). Qed. Hint Immediate fmon_comp_le_compat: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_comp_le_compat
Parametric Morphism (O1 O2 O3:ord) : (fmon_comp (O1:=O1) (O2:=O2) (O3:=O3)) with signature Oeq (O:=O2 -m> O3) ==> Oeq (O:=O1 -m> O2) ==> Oeq (O:=O1 -m> O3) as fmon_comp_eq_compat. intros; apply Ole_antisym; apply fmon_comp_le_compat; auto. Qed. Hint Immediate fmon_comp_eq_compat: core.	Add	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Parametric
fmon_comp_monotonic2 : forall (O1 O2 O3:ord) (f: O2 -m> O3) (g1 g2:O1 -m> O2), g1<= g2 -> f @ g1 <= f @ g2. auto. Qed. Hint Resolve fmon_comp_monotonic2: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_comp_monotonic2
fmon_comp_monotonic1 : forall (O1 O2 O3:ord) (f1 f2: O2 -m> O3) (g:O1 -m> O2), f1<= f2 -> f1 @ g <= f2 @ g. auto. Qed. Hint Resolve fmon_comp_monotonic1: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_comp_monotonic1
fcomp : forall O1 O2 O3:ord, (O2 -m> O3) -m> (O1 -m> O2) -m> (O1 -m> O3). intros; exists (fun f : O2 -m> O3 => mk_fmono (fmonot:=fun g : O1 -m> O2 => fmon_comp f g) (fmon_comp_monotonic2 f)). red; intros; simpl; intros. apply H. Defined.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fcomp
fcomp_simpl : forall (O1 O2 O3:ord) (f:O2 -m> O3) (g:O1 -m> O2), fcomp O1 O2 O3 f g = f @ g. trivial. Qed.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fcomp_simpl
fcomp2 : forall O1 O2 O3 O4:ord, (O3 -m> O4) -m> (O1 -m> O2-m>O3) -m> (O1 -m> O2 -m> O4). intros; exists (fun f : O3 -m> O4 => fcomp O1 (O2-m> O3) (O2-m>O4) (fcomp O2 O3 O4 f)). red; intros; simpl; intros. apply H. Defined.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fcomp2
fcomp2_simpl : forall (O1 O2 O3 O4:ord) (f:O3 -m> O4) (g:O1 -m> O2-m>O3) (x:O1)(y:O2), fcomp2 O1 O2 O3 O4 f g x y = f (g x y). trivial. Qed.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fcomp2_simpl
fmon_le_compat : forall (O1 O2:ord) (f: O1 -m> O2) (x y:O1), x<=y -> f x <= f y. intros; apply (fmonotonic f); auto. Qed. Hint Resolve fmon_le_compat: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_le_compat
fmon_le_compat2 : forall (O1 O2 O3:ord) (f: O1 -m> O2 -m> O3) (x y:O1) (z t:O2), x<=y -> z <=t -> f x z <= f y t. intros; apply Ole_trans with (f x t). apply (fmonotonic (f x)); auto. apply (fmonotonic f); auto. Qed. Hint Resolve fmon_le_compat2: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_le_compat2
fmon_cte_comp : forall (O1 O2 O3:ord)(c:O3)(f:O1-m>O2), fmon_cte O2 c @ f == fmon_cte O1 c. intros; apply fmon_eq_intro; intro x; auto. Qed. ( Basic operators of omega-cpos ) (* - Constant : $0$ - lub : limit of monotonic sequences ) (* *** Definition of cpos *)	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	fmon_cte_comp
cpo : Type := mk_cpo { tcpo :> ord; D0 : tcpo; lub: (natO -m> tcpo) -> tcpo; Dbot : forall x : tcpo, D0 <= x; le_lub : forall (f : natO -m> tcpo) (n : nat), f n <= lub f; lub_le : forall (f : natO -m> tcpo) (x : tcpo), (forall n, f n <= x) -> lub f <= x }. Arguments D0 {c}.	Record	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	cpo
Parametric Morphism (c:cpo) : (lub (c:=c)) with signature Ole (o:=natO -m> c) ++> Ole (o:=c) as lub_le_compat_morph. intros f g H; apply lub_le; intros. apply Ole_trans with (g n); auto. Qed. Hint Resolve lub_le_compat_morph: core.	Add	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Parametric
lub_le_compat : forall (D:cpo) (f g:natO -m> D), f <= g -> lub f <= lub g. intros; apply lub_le; intros. apply Ole_trans with (g n); auto. Qed. Hint Resolve lub_le_compat: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	lub_le_compat
Lub : forall (D:cpo), (natO -m> D) -m> D. intro D; exists (fun (f :natO-m>D) => lub f); red; auto. Defined.	Definition	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Lub
Parametric Morphism (c:cpo) : (lub (c:=c)) with signature Oeq (O:=natO -m> c) ==> Oeq (O:=c) as lub_eq_compat. intros; apply Ole_antisym; auto. Qed. Hint Resolve lub_eq_compat: core.	Add	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	Parametric
lub_cte : forall (D:cpo) (c:D), lub (fmon_cte natO c) == c. intros; apply Ole_antisym; auto. apply le_lub with (f:=fmon_cte natO c) (n:=O); auto. Qed. Hint Resolve lub_cte: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	lub_cte
lub_lift_right : forall (D:cpo) (f:natO -m> D) n, lub f == lub (mseq_lift_right f n). intros; apply Ole_antisym; auto. apply lub_le_compat; intro. unfold mseq_lift_right; simpl. apply (fmonotonic f); auto with arith. Qed. Hint Resolve lub_lift_right: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	lub_lift_right
lub_lift_left : forall (D:cpo) (f:natO -m> D) n, lub f == lub (mseq_lift_left f n). intros; apply Ole_antisym; auto. apply lub_le_compat; intro. unfold mseq_lift_left; simpl. apply (fmonotonic f); auto with arith. Qed. Hint Resolve lub_lift_left: core.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	lub_lift_left
lub_le_lift : forall (D:cpo) (f g:natO -m> D) (n:natO), (forall k, n <= k -> f k <= g k) -> lub f <= lub g. intros; apply lub_le; intros. apply Ole_trans with (f (n+n0)). apply (fmonotonic f); simpl; auto with arith. apply Ole_trans with (g (n+n0)); auto. apply H; simpl; auto with arith. Qed.	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	lub_le_lift
lub_eq_lift : forall (D:cpo) (f g:natO -m> D) (n:natO), (forall k, n <= k -> f k == g k) -> lub f == lub g. intros; apply Ole_antisym; apply lub_le_lift with n; intros; auto. apply Oeq_le_sym; auto. Qed. (** - (lub_fun h) x = lub_n (h n x) *)	Lemma	root	[ "From Coq Require Export Setoid.", "From Coq Require Export Arith.", "From Coq Require Export Lia." ]	Ccpo.v	lub_eq_lift

ord : Type := mk_ord { tord:>Type; Ole : tord->tord->Prop; Ole_refl : forall x :tord, Ole x x; Ole_trans : forall x y z:tord, Ole x y -> Ole y z -> Ole x z }. Hint Resolve Ole_refl Ole_trans: core. Hint Extern 2 (@Ole ?X1 ?X2 ?X3 ) => simpl Ole: core. Declare Scope O_scope. Infix "<=" := Ole : O_scope. Open Scope O_scope. (** *** Associated equality *)

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