fact stringlengths 7 4.15k | 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 8 | deps listlengths 0 51 | docstring stringclasses 188
values | line_start int64 8 4.84k | line_end int64 9 4.84k | has_proof bool 2
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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 }. | 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",
"Arith",
"Lia"
] | [] | ** Ordered type | 11 | 15 | true | https://github.com/coq-community/alea | 0cd68687229aaa26623c7092d4b40e9a812f17bc |
Oeq (O:ord) (x y : O) := x <= y /\ y <= x. | Oeq (O:ord) (x y : O) | := x <= y /\ y <= x. | Definition | Oeq | Root | Ccpo.v | [
"Coq",
"Arith",
"Lia"
] | [
"ord"
] | *** Associated equality | 27 | 27 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 32 | 34 | true | https://github.com/coq-community/alea | 0cd68687229aaa26623c7092d4b40e9a812f17bc |
Ole_antisym : forall (O:ord) (x y:O), x<=y -> y <=x -> x==y.
red; auto.
Qed. | Ole_antisym : forall (O:ord) (x y:O), x<=y -> y <=x -> x==y. | red; auto.
Qed. | Lemma | Ole_antisym | Root | Ccpo.v | [
"Coq",
"Arith",
"Lia"
] | [
"ord"
] | null | 38 | 40 | true | https://github.com/coq-community/alea | 0cd68687229aaa26623c7092d4b40e9a812f17bc |
Oeq_refl : forall (O:ord) (x:O), x == x.
red; auto.
Qed. | Oeq_refl : forall (O:ord) (x:O), x == x. | red; auto.
Qed. | Lemma | Oeq_refl | Root | Ccpo.v | [
"Coq",
"Arith",
"Lia"
] | [
"ord"
] | null | 43 | 45 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 48 | 50 | true | https://github.com/coq-community/alea | 0cd68687229aaa26623c7092d4b40e9a812f17bc |
Oeq_sym : forall (O:ord) (x y:O), x == y -> y == x.
unfold Oeq; intuition.
Qed. | Oeq_sym : forall (O:ord) (x y:O), x == y -> y == x. | unfold Oeq; intuition.
Qed. | Lemma | Oeq_sym | Root | Ccpo.v | [
"Coq",
"Arith",
"Lia"
] | [
"Oeq",
"ord"
] | null | 53 | 55 | true | https://github.com/coq-community/alea | 0cd68687229aaa26623c7092d4b40e9a812f17bc |
Oeq_le : forall (O:ord) (x y:O), x == y -> x <= y.
unfold Oeq; intuition.
Qed. | Oeq_le : forall (O:ord) (x y:O), x == y -> x <= y. | unfold Oeq; intuition.
Qed. | Lemma | Oeq_le | Root | Ccpo.v | [
"Coq",
"Arith",
"Lia"
] | [
"Oeq",
"ord"
] | null | 57 | 59 | true | https://github.com/coq-community/alea | 0cd68687229aaa26623c7092d4b40e9a812f17bc |
Oeq_le_sym : forall (O:ord) (x y:O), x == y -> y <= x.
unfold Oeq; intuition.
Qed. | 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",
"Arith",
"Lia"
] | [
"Oeq",
"ord"
] | null | 61 | 63 | true | 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. | 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",
"Arith",
"Lia"
] | [
"Oeq",
"ord"
] | null | 68 | 70 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 96 | 101 | true | 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. | 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",
"Arith",
"Lia"
] | [
"Ole_eq_compat",
"ord"
] | null | 103 | 106 | true | 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. | 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",
"Arith",
"Lia"
] | [
"Ole_eq_compat",
"ord"
] | null | 108 | 111 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | - Iord x y := y <= x | 116 | 119 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | - ford f g := forall x, f x <= g x | 124 | 127 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 132 | 134 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 137 | 139 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 142 | 144 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 147 | 149 | true | https://github.com/coq-community/alea | 0cd68687229aaa26623c7092d4b40e9a812f17bc |
monotonic (O1 O2:ord) (f : O1 -> O2) := forall x y, x <= y -> f x <= f y. | monotonic (O1 O2:ord) (f : O1 -> O2) | := forall x y, x <= y -> f x <= f y. | Definition | monotonic | Root | Ccpo.v | [
"Coq",
"Arith",
"Lia"
] | [
"ord"
] | *** Definition and properties | 158 | 158 | true | https://github.com/coq-community/alea | 0cd68687229aaa26623c7092d4b40e9a812f17bc |
stable (O1 O2:ord) (f : O1 -> O2) := forall x y, x == y -> f x == f y. | stable (O1 O2:ord) (f : O1 -> O2) | := forall x y, x == y -> f x == f y. | Definition | stable | Root | Ccpo.v | [
"Coq",
"Arith",
"Lia"
] | [
"ord"
] | null | 161 | 161 | true | 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. | 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",
"Arith",
"Lia"
] | [
"monotonic",
"ord",
"stable"
] | null | 164 | 167 | true | https://github.com/coq-community/alea | 0cd68687229aaa26623c7092d4b40e9a812f17bc |
fmono (O1 O2:ord) : Type := mk_fmono
{fmonot :> O1 -> O2; fmonotonic: monotonic fmonot}. | fmono (O1 O2:ord) : Type | := mk_fmono
{fmonot :> O1 -> O2; fmonotonic: monotonic fmonot}. | Record | fmono | Root | Ccpo.v | [
"Coq",
"Arith",
"Lia"
] | [
"monotonic",
"ord"
] | *** Type of monotonic functions | 172 | 173 | true | 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. | 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",
"Arith",
"Lia"
] | [
"fmono",
"ord"
] | - fmon O1 O2 (f g : fmono O1 O2) := forall x, f x <= g x | 177 | 180 | true | 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. | 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",
"Arith",
"Lia"
] | [
"monotonic_stable",
"ord",
"stable"
] | null | 185 | 187 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 190 | 192 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 195 | 197 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 200 | 202 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 205 | 207 | true | 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. | 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",
"Arith",
"Lia"
] | [
"Iord"
] | - [lmon f] uses f as monotonic function over the dual order. | 215 | 218 | true | 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. | 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",
"Arith",
"Lia"
] | [
"Imon",
"Iord"
] | null | 220 | 223 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | *** Monotonic functions with 2 arguments | 226 | 230 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | - natO n m = n <= m | 236 | 239 | true | 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. | 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",
"Arith",
"Lia"
] | [
"natO",
"ord"
] | null | 241 | 245 | true | 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. | 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",
"Arith",
"Lia"
] | [
"natO",
"ord"
] | null | 247 | 249 | true | 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. | 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",
"Arith",
"Lia"
] | [
"natO",
"ord"
] | - (mseq_lift_left f n) k = f (n+k) | 254 | 257 | true | 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. | 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",
"Arith",
"Lia"
] | [
"mseq_lift_left",
"natO",
"ord"
] | null | 259 | 262 | true | 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. | 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",
"Arith",
"Lia"
] | [
"mseq_lift_left",
"natO",
"ord"
] | null | 264 | 267 | true | 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. | 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",
"Arith",
"Lia"
] | [
"natO",
"ord"
] | - (mseq_lift_right f n) k = f (k+n) | 278 | 281 | true | 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. | 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",
"Arith",
"Lia"
] | [
"mseq_lift_right",
"natO",
"ord"
] | null | 283 | 286 | true | 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. | 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",
"Arith",
"Lia"
] | [
"mseq_lift_right",
"natO",
"ord"
] | null | 289 | 292 | true | 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. | 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",
"Arith",
"Lia"
] | [
"fmon_eq_intro",
"mseq_lift_left",
"mseq_lift_right",
"natO",
"ord"
] | null | 301 | 305 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | - (ford_app f x) n = f n x | 309 | 314 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 319 | 322 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 324 | 328 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | - ford_shift f x y == f y x | 338 | 342 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ford_shift",
"ord"
] | null | 344 | 347 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | - (fmon_app f x) n = f n x | 359 | 364 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 369 | 372 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 374 | 379 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | - fmon_id c = c | 389 | 392 | true | https://github.com/coq-community/alea | 0cd68687229aaa26623c7092d4b40e9a812f17bc |
fmon_id_simpl : forall (O:ord) (x:O), fmon_id O x = x.
trivial.
Qed. | fmon_id_simpl : forall (O:ord) (x:O), fmon_id O x = x. | trivial.
Qed. | Lemma | fmon_id_simpl | Root | Ccpo.v | [
"Coq",
"Arith",
"Lia"
] | [
"fmon_id",
"ord"
] | null | 394 | 396 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | - (fmon_cte c) n = c | 399 | 402 | true | 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. | 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",
"Arith",
"Lia"
] | [
"fmon_cte",
"ord"
] | null | 404 | 406 | true | https://github.com/coq-community/alea | 0cd68687229aaa26623c7092d4b40e9a812f17bc |
mseq_cte : forall O:ord, O -> natO -m> O := fmon_cte natO. | mseq_cte : forall O:ord, O -> natO -m> O | := fmon_cte natO. | Definition | mseq_cte | Root | Ccpo.v | [
"Coq",
"Arith",
"Lia"
] | [
"fmon_cte",
"natO",
"ord"
] | null | 408 | 408 | true | 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. | 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",
"Arith",
"Lia"
] | [
"fmon_cte",
"ord"
] | null | 410 | 413 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | - (fmon_diag h) n = h n n | 422 | 429 | true | 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. | 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",
"Arith",
"Lia"
] | [
"fmon_diag",
"ord"
] | null | 431 | 434 | true | 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. | 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",
"Arith",
"Lia"
] | [
"fmon_diag",
"ord"
] | null | 437 | 440 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | - (fmon_shift h) n m = h m n | 449 | 453 | true | 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. | 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",
"Arith",
"Lia"
] | [
"fmon_shift",
"ord"
] | null | 455 | 458 | true | 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. | 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",
"Arith",
"Lia"
] | [
"fmon_shift",
"ord"
] | null | 460 | 464 | true | 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. | 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",
"Arith",
"Lia"
] | [
"fmon_eq_intro",
"fmon_shift",
"ord"
] | null | 473 | 476 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | - (f@g) x = f (g x) | 479 | 483 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 488 | 491 | true | 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 (fmonoto... | 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",
"Arith",
"Lia"
] | [
"ord"
] | - (f@2 g) h x = f (g x) (h x) | 494 | 502 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 507 | 511 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 521 | 525 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 536 | 540 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 543 | 547 | true | 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. | 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",
"Arith",
"Lia"
] | [
"fmon_comp",
"fmon_comp_monotonic2",
"ord"
] | null | 550 | 556 | true | 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. | 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",
"Arith",
"Lia"
] | [
"fcomp",
"ord"
] | null | 558 | 561 | true | 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. | 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",
"Arith",
"Lia"
] | [
"fcomp",
"ord"
] | null | 563 | 569 | true | 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. | 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",
"Arith",
"Lia"
] | [
"fcomp2",
"ord"
] | null | 571 | 574 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 576 | 578 | true | 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. | 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",
"Arith",
"Lia"
] | [
"ord"
] | null | 581 | 586 | true | 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. | 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",
"Arith",
"Lia"
] | [
"fmon_cte",
"fmon_eq_intro",
"ord"
] | null | 589 | 592 | true | 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
}. | 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",
"Arith",
"Lia"
] | [
"natO",
"ord"
] | *** Definition of cpos | 602 | 610 | true | https://github.com/coq-community/alea | 0cd68687229aaa26623c7092d4b40e9a812f17bc |
"0" := D0 : O_scope. | "0" | := D0 : O_scope. | Notation | 0 | Root | Ccpo.v | [
"Coq",
"Arith",
"Lia"
] | [] | null | 613 | 613 | true | 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. | 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",
"Arith",
"Lia"
] | [
"cpo",
"natO"
] | null | 627 | 630 | true | 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. | 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",
"Arith",
"Lia"
] | [
"cpo",
"natO"
] | null | 633 | 635 | true | 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. | 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",
"Arith",
"Lia"
] | [
"Ole_antisym",
"cpo",
"fmon_cte",
"natO"
] | null | 644 | 647 | true | 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. | 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",
"Arith",
"Lia"
] | [
"Ole_antisym",
"cpo",
"lub_le_compat",
"mseq_lift_right",
"natO"
] | null | 651 | 656 | true | 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. | 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",
"Arith",
"Lia"
] | [
"Ole_antisym",
"cpo",
"lub_le_compat",
"mseq_lift_left",
"natO"
] | null | 659 | 664 | true | 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. | 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",
"Arith",
"Lia"
] | [
"cpo",
"natO"
] | null | 667 | 674 | true | 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. | 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",
"Arith",
"Lia"
] | [
"Oeq_le_sym",
"Ole_antisym",
"cpo",
"lub_le_lift",
"natO"
] | null | 676 | 680 | true | 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. | 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",
"Arith",
"Lia"
] | [
"cpo",
"lub_le_compat",
"natO",
"ord"
] | - (lub_fun h) x = lub_n (h n x) | 683 | 688 | true | 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. | 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",
"Arith",
"Lia"
] | [
"cpo",
"lub_fun",
"natO",
"ord"
] | null | 690 | 693 | true | 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. | 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",
"Arith",
"Lia"
] | [
"Lub",
"cpo",
"fmon_eq_intro",
"fmon_shift",
"lub_fun",
"natO"
] | null | 695 | 698 | true | 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 (fmonoton... | 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",
"Arith",
"Lia"
] | [
"Lub",
"Ole_antisym",
"cpo",
"fmon_diag",
"lub_le_compat",
"natO"
] | null | 700 | 711 | true | 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. | 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",
"Arith",
"Lia"
] | [
"Lub",
"cpo",
"lub_fun",
"lub_le_compat",
"natO"
] | null | 713 | 721 | true | 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. | 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",
"Arith",
"Lia"
] | [
"Lub",
"Ole_antisym",
"cpo",
"lub_fun",
"natO"
] | null | 724 | 727 | true | 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. | 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",
"Arith",
"Lia"
] | [
"cpo",
"ford"
] | *** Functional cpos | 733 | 736 | true | 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. | 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",
"Arith",
"Lia"
] | [
"cpo",
"natO"
] | null | 741 | 744 | true | 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. | 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",
"Arith",
"Lia"
] | [
"cpo",
"natO"
] | ** Continuity | 748 | 752 | true | 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. | 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",
"Arith",
"Lia"
] | [
"cpo",
"natO"
] | null | 755 | 758 | true | 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). | 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",
"Arith",
"Lia"
] | [
"cpo",
"natO"
] | null | 761 | 762 | true | 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. | 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",
"Arith",
"Lia"
] | [
"continuous",
"cpo"
] | null | 764 | 770 | true | 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. | 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",
"Arith",
"Lia"
] | [
"Ole_antisym",
"continuous",
"cpo",
"natO"
] | null | 780 | 783 | true | https://github.com/coq-community/alea | 0cd68687229aaa26623c7092d4b40e9a812f17bc |
mon0 (O1:ord) (D2 : cpo) : O1 -m> D2 := fmon_cte O1 (0:D2). | mon0 (O1:ord) (D2 : cpo) : O1 -m> D2 | := fmon_cte O1 (0:D2). | Definition | mon0 | Root | Ccpo.v | [
"Coq",
"Arith",
"Lia"
] | [
"cpo",
"fmon_cte",
"ord"
] | - mon0 x == 0 | 788 | 788 | true | https://github.com/coq-community/alea | 0cd68687229aaa26623c7092d4b40e9a812f17bc |
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