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phanerozoic
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Coq-ALEA

statement stringlengths 2 276	proof stringlengths 0 4.14k	type stringclasses 12 values	symbolic_name stringlengths 1 26	library stringclasses 1 value	filename stringclasses 10 values	imports listlengths 2 10	deps listlengths 0 51	docstring stringclasses 189 values	source_url stringclasses 1 value	commit stringclasses 1 value
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 }.	Record	ord	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[]	** Ordered type	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
Oeq (O:ord) (x y : O)	:= x <= y /\ y <= x.	Definition	Oeq	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]	*** Associated equality	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
Ole_refl_eq : forall (O:ord) (x y:O), x=y -> x <= y.	intros O x y H; rewrite H; auto. Qed.	Lemma	Ole_refl_eq	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
Ole_antisym : forall (O:ord) (x y:O), x<=y -> y <=x -> x==y.	red; auto. Qed.	Lemma	Ole_antisym	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
Oeq_refl : forall (O:ord) (x:O), x == x.	red; auto. Qed.	Lemma	Oeq_refl	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
Oeq_refl_eq : forall (O:ord) (x y:O), x=y -> x == y.	intros O x y H; rewrite H; auto. Qed.	Lemma	Oeq_refl_eq	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
Oeq_sym : forall (O:ord) (x y:O), x == y -> y == x.	unfold Oeq; intuition. Qed.	Lemma	Oeq_sym	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Oeq", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
Oeq_le : forall (O:ord) (x y:O), x == y -> x <= y.	unfold Oeq; intuition. Qed.	Lemma	Oeq_le	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Oeq", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
Oeq_le_sym : forall (O:ord) (x y:O), x == y -> y <= x.	unfold Oeq; intuition. Qed.	Lemma	Oeq_le_sym	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Oeq", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	Oeq_trans	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Oeq", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	Ole_eq_compat	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	Ole_eq_right	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Ole_eq_compat", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	Ole_eq_left	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Ole_eq_compat", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
Iord: ord -> ord.	intros O; exists O (fun x y : O => y <= x); intros; auto. apply Ole_trans with y; auto. Defined.	Definition	Iord	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]	- Iord x y := y <= x	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Definition	ford	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]	- ford f g := forall x, f x <= g x	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
ford_le_elim : forall A (O:ord) (f g:A -o> O), f <= g ->forall n, f n <= g n.	auto. Qed.	Lemma	ford_le_elim	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
ford_le_intro : forall A (O:ord) (f g:A -o> O), (forall n, f n <= g n) -> f <= g.	auto. Qed.	Lemma	ford_le_intro	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
ford_eq_elim : forall A (O:ord) (f g:A -o> O), f == g ->forall n, f n == g n.	firstorder. Qed.	Lemma	ford_eq_elim	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
ford_eq_intro : forall A (O:ord) (f g:A -o> O), (forall n, f n == g n) -> f == g.	red; auto. Qed.	Lemma	ford_eq_intro	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
monotonic (O1 O2:ord) (f : O1 -> O2)	:= forall x y, x <= y -> f x <= f y.	Definition	monotonic	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]	*** Definition and properties	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
stable (O1 O2:ord) (f : O1 -> O2)	:= forall x y, x == y -> f x == f y.	Definition	stable	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
monotonic_stable : forall (O1 O2 : ord) (f:O1 -> O2), monotonic f -> stable f.	unfold monotonic, stable; firstorder. Qed.	Lemma	monotonic_stable	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "monotonic", "ord", "stable" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
fmono (O1 O2:ord) : Type	:= mk_fmono {fmonot :> O1 -> O2; fmonotonic: monotonic fmonot}.	Record	fmono	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "monotonic", "ord" ]	*** Type of monotonic functions	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Definition	fmon	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "fmono", "ord" ]	- fmon O1 O2 (f g : fmono O1 O2) := forall x, f x <= g x	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
fmon_stable : forall (O1 O2:ord) (f:O1 -m> O2), stable f.	intros; apply monotonic_stable; auto. Qed.	Lemma	fmon_stable	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "monotonic_stable", "ord", "stable" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
fmon_le_elim : forall (O1 O2:ord) (f g:O1 -m> O2), f <= g -> forall n, f n <= g n.	auto. Qed.	Lemma	fmon_le_elim	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
fmon_le_intro : forall (O1 O2:ord) (f g:O1 -m> O2), (forall n, f n <= g n) -> f <= g.	auto. Qed.	Lemma	fmon_le_intro	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
fmon_eq_elim : forall (O1 O2:ord) (f g:O1 -m> O2), f == g ->forall n, f n == g n.	firstorder. Qed.	Lemma	fmon_eq_elim	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
fmon_eq_intro : forall (O1 O2:ord) (f g:O1 -m> O2), (forall n, f n == g n) -> f == g.	red; auto. Qed.	Lemma	fmon_eq_intro	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	Imon	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Iord" ]	- [lmon f] uses f as monotonic function over the dual order.	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Definition	Imon2	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Imon", "Iord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Definition	le_compat2_mon	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]	*** Monotonic functions with 2 arguments	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
natO : ord.	exists nat (fun n m : nat => (n <= m)%nat); intros; auto with arith. apply le_trans with y; auto. Defined.	Definition	natO	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]	- natO n m = n <= m	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	fnatO_intro	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "natO", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
fnatO_elim : forall (O:ord) (f:natO -m> O) (n:nat), f n <= f (S n).	intros; apply (fmonotonic f); auto. Qed.	Lemma	fnatO_elim	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "natO", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	mseq_lift_left	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "natO", "ord" ]	- (mseq_lift_left f n) k = f (n+k)	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	mseq_lift_left_simpl	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "mseq_lift_left", "natO", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	mseq_lift_left_le_compat	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "mseq_lift_left", "natO", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	mseq_lift_right	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "natO", "ord" ]	- (mseq_lift_right f n) k = f (k+n)	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	mseq_lift_right_simpl	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "mseq_lift_right", "natO", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	mseq_lift_right_le_compat	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "mseq_lift_right", "natO", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	mseq_lift_right_left	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "fmon_eq_intro", "mseq_lift_left", "mseq_lift_right", "natO", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Definition	ford_app	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]	- (ford_app f x) n = f n x	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	ford_app_simpl	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	ford_app_le_compat	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	ford_shift	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]	- ford_shift f x y == f y x	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	ford_shift_le_compat	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ford_shift", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Definition	fmon_app	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]	- (fmon_app f x) n = f n x	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	fmon_app_simpl	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	fmon_app_le_compat	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
fmon_id : forall (O:ord), O -m> O.	intros; exists (fun (x:O)=>x). intro n; auto. Defined.	Definition	fmon_id	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]	- fmon_id c = c	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
fmon_id_simpl : forall (O:ord) (x:O), fmon_id O x = x.	trivial. Qed.	Lemma	fmon_id_simpl	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "fmon_id", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
fmon_cte : forall (O1 O2:ord)(c:O2), O1 -m> O2.	intros; exists (fun (x:O1)=>c). intro n; auto. Defined.	Definition	fmon_cte	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]	- (fmon_cte c) n = c	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
fmon_cte_simpl : forall (O1 O2:ord)(c:O2)(c:O2) (x:O1), fmon_cte O1 c x = c.	trivial. Qed.	Lemma	fmon_cte_simpl	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "fmon_cte", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
mseq_cte : forall O:ord, O -> natO -m> O	:= fmon_cte natO.	Definition	mseq_cte	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "fmon_cte", "natO", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	fmon_cte_le_compat	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "fmon_cte", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	fmon_diag	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]	- (fmon_diag h) n = h n n	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	fmon_diag_le_compat	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "fmon_diag", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	fmon_diag_simpl	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "fmon_diag", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	fmon_shift	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]	- (fmon_shift h) n m = h m n	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	fmon_shift_simpl	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "fmon_shift", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	fmon_shift_le_compat	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "fmon_shift", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	fmon_shift_shift_eq	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "fmon_eq_intro", "fmon_shift", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Definition	fmon_comp	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]	- (f@g) x = f (g x)	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	fmon_comp_simpl	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Definition	fmon_comp2	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]	- (f@2 g) h x = f (g x) (h x)	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	fmon_comp2_simpl	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	fmon_comp_le_compat	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	fmon_comp_monotonic2	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	fmon_comp_monotonic1	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	fcomp	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "fmon_comp", "fmon_comp_monotonic2", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	fcomp_simpl	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "fcomp", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	fcomp2	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "fcomp", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	fcomp2_simpl	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "fcomp2", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	fmon_le_compat	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	fmon_le_compat2	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	fmon_cte_comp	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "fmon_cte", "fmon_eq_intro", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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 }.	Record	cpo	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "natO", "ord" ]	*** Definition of cpos	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
"0"	:= D0 : O_scope.	Notation	0	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	lub_le_compat	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "cpo", "natO" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
Lub : forall (D:cpo), (natO -m> D) -m> D.	intro D; exists (fun (f :natO-m>D) => lub f); red; auto. Defined.	Definition	Lub	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "cpo", "natO" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	lub_cte	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Ole_antisym", "cpo", "fmon_cte", "natO" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	lub_lift_right	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Ole_antisym", "cpo", "lub_le_compat", "mseq_lift_right", "natO" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	lub_lift_left	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Ole_antisym", "cpo", "lub_le_compat", "mseq_lift_left", "natO" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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	lub_le_lift	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "cpo", "natO" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
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.	Lemma	lub_eq_lift	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Oeq_le_sym", "Ole_antisym", "cpo", "lub_le_lift", "natO" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
lub_fun : forall (O:ord) (D:cpo) (h : natO -m> O -m> D), O -m> D.	intros; exists (fun m => lub (h <_> m)). red; intros. apply lub_le_compat; simpl; intros. apply (fmonotonic (h x0)); auto. Defined.	Definition	lub_fun	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "cpo", "lub_le_compat", "natO", "ord" ]	- (lub_fun h) x = lub_n (h n x)	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
lub_fun_eq : forall (O:ord) (D:cpo) (h : natO -m> O -m> D) (x:O), lub_fun h x == lub (h <_> x).	auto. Qed.	Lemma	lub_fun_eq	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "cpo", "lub_fun", "natO", "ord" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
lub_fun_shift : forall (D:cpo) (h : natO -m> (natO -m> D)), lub_fun h == Lub D @ (fmon_shift h).	intros; apply fmon_eq_intro; unfold lub_fun; simpl; auto. Qed.	Lemma	lub_fun_shift	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Lub", "cpo", "fmon_eq_intro", "fmon_shift", "lub_fun", "natO" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
double_lub_simpl : forall (D:cpo) (h : natO -m> natO -m> D), lub (Lub D @ h) == lub (fmon_diag h).	intros; apply Ole_antisym. apply lub_le; intros; simpl; apply lub_le; intros. apply Ole_trans with (h n (n+n0)). apply (fmonotonic (h n)); auto with arith. apply Ole_trans with (h (n+n0) (n+n0)). apply (fmonotonic h); auto with arith. apply le_lub with (f:=fmon_diag h) (n:=(n + n0)%nat). apply lub_le_compat. unfold fmo...	Lemma	double_lub_simpl	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Lub", "Ole_antisym", "cpo", "fmon_diag", "lub_le_compat", "natO" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
lub_exch_le : forall (D:cpo) (h : natO -m> (natO -m> D)), lub (Lub D @ h) <= lub (lub_fun h).	intros; apply lub_le; intros; simpl. apply lub_le; intros. apply Ole_trans with (lub (h n)); auto. apply lub_le_compat; intro. unfold lub_fun; simpl. apply le_lub with (f:=h <_> x) (n:=n). Qed.	Lemma	lub_exch_le	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Lub", "cpo", "lub_fun", "lub_le_compat", "natO" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
lub_exch_eq : forall (D:cpo) (h : natO -m> (natO -m> D)), lub (Lub D @ h) == lub (lub_fun h).	intros; apply Ole_antisym; auto. Qed.	Lemma	lub_exch_eq	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Lub", "Ole_antisym", "cpo", "lub_fun", "natO" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
fcpo : Type -> cpo -> cpo.	intros A D; exists (ford A D) (fun x:A => (0:D)) (fun f (x:A) => lub(c:=D) (f <o> x)); intros; auto. intro x; apply Ole_trans with ((f <o> x) n); auto. Defined.	Definition	fcpo	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "cpo", "ford" ]	*** Functional cpos	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
fcpo_lub_simpl : forall A (D:cpo) (h:natO-m> A-O->D)(x:A), (lub h) x = lub(c:=D) (h <o> x).	trivial. Qed.	Lemma	fcpo_lub_simpl	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "cpo", "natO" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
lub_comp_le : forall (D1 D2 : cpo) (f:D1 -m> D2) (h : natO -m> D1), lub (f @ h) <= f (lub h).	intros; apply lub_le; simpl; intros. apply (fmonotonic f); auto. Qed.	Lemma	lub_comp_le	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "cpo", "natO" ]	** Continuity	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
lub_comp2_le : forall (D1 D2 D3: cpo) (F:D1 -m> D2-m>D3) (f : natO -m> D1) (g: natO -m> D2), lub ((F @2 f) g) <= F (lub f) (lub g).	intros; apply lub_le; simpl; auto. Qed.	Lemma	lub_comp2_le	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "cpo", "natO" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
continuous (D1 D2 : cpo) (f:D1 -m> D2)	:= forall h : natO -m> D1, f (lub h) <= lub (f @ h).	Definition	continuous	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "cpo", "natO" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
continuous_eq_compat : forall (D1 D2 : cpo) (f g:D1 -m> D2), f==g -> continuous f -> continuous g.	red; intros. apply Ole_trans with (f (lub h)). assert (g <= f); auto. rewrite <- H; auto. Qed.	Lemma	continuous_eq_compat	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "continuous", "cpo" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
lub_comp_eq : forall (D1 D2 : cpo) (f:D1 -m> D2) (h : natO -m> D1), continuous f -> f (lub h) == lub (f @ h).	intros; apply Ole_antisym; auto. Qed.	Lemma	lub_comp_eq	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "Ole_antisym", "continuous", "cpo", "natO" ]		https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc
mon0 (O1:ord) (D2 : cpo) : O1 -m> D2	:= fmon_cte O1 (0:D2).	Definition	mon0	Root	Ccpo.v	[ "Coq", "Setoid", "Arith", "Lia" ]	[ "cpo", "fmon_cte", "ord" ]	- mon0 x == 0	https://github.com/coq-community/alea	0cd68687229aaa26623c7092d4b40e9a812f17bc

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Coq-ALEA

Structured dataset from ALEA — Reasoning on randomized algorithms.

Source

Repository: https://github.com/coq-community/alea
Commit: 0cd68687229aaa26623c7092d4b40e9a812f17bc
Files: 10
License: lgpl-2.1

Schema

Column	Type	Description
statement	string	Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof
proof	string	Verbatim proof/body, empty if the declaration has none
type	string	Declaration keyword
symbolic_name	string	Declaration identifier
library	string	Sub-library
filename	string	Repository-relative source path
imports	list[string]	File-level `Require`/`Import` modules
deps	list[string]	Intra-corpus identifiers referenced
docstring	string	Preceding documentation comment, empty if absent
source_url	string	Upstream repository
commit	string	Upstream commit extracted

Statistics

Entries: 1,577
With proof: 1,529 (97.0%)
With docstring: 190 (12.0%)
Libraries: 1

By type

Type	Count
Lemma	1,178
Definition	318
Parameter	31
Hypothesis	15
Fixpoint	11
Record	9
Notation	7
Ltac	3
Theorem	2
Coercion	1
Inductive	1
Parameters	1

Example

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  }.

type: Record | symbolic_name: ord | Ccpo.v

Use

Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint and together form the complete declaration, for proof modeling, autoformalization, retrieval, and dependency analysis via deps.

Citation

@misc{coq_alea_dataset,
  title  = {Coq-ALEA},
  author = {Norton, Charles},
  year   = {2026},
  note   = {Extracted from https://github.com/coq-community/alea, commit 0cd68687229a},
  url    = {https://huggingface.co/datasets/phanerozoic/Coq-ALEA}
}

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