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wlog_le P : (forall x y, P y x -> P x y) -> (forall x y, x <= y -> P x y) -> forall x y, P x y.
Proof. by move=> sP hP x y; case: (leP x y) => [| /ltW] /hP // /sP. Qed.
Lemma
wlog_le
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "leP", "ltW" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
wlog_lt P : (forall x, P x x) -> (forall x y, (P y x -> P x y)) -> (forall x y, x < y -> P x y) -> forall x y, P x y.
Proof. by move=> rP sP hP x y; case: (ltgtP x y) => [||->] // /hP // /sP. Qed.
Lemma
wlog_lt
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "ltgtP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
neq_lt x y : (x != y) = (x < y) || (y < x).
Proof. by case: ltgtP. Qed.
Lemma
neq_lt
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "ltgtP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_total x y : x != y -> (x < y) || (y < x).
Proof. by case: ltgtP. Qed.
Lemma
lt_total
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "ltgtP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_leLR x y z t : (x <= y -> z <= t) -> (y < x -> t < z) -> (x <= y) = (z <= t).
Proof. by rewrite !ltNge => ? /contraTT ?; apply/idP/idP. Qed.
Lemma
eq_leLR
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "ltNge" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_leRL x y z t : (x <= y -> z <= t) -> (y < x -> t < z) -> (z <= t) = (x <= y).
Proof. by move=> *; apply/esym/eq_leLR. Qed.
Lemma
eq_leRL
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "eq_leLR" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_ltLR x y z t : (x < y -> z < t) -> (y <= x -> t <= z) -> (x < y) = (z < t).
Proof. by rewrite !leNgt => ? /contraTT ?; apply/idP/idP. Qed.
Lemma
eq_ltLR
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "leNgt" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_ltRL x y z t : (x < y -> z < t) -> (y <= x -> t <= z) -> (z < t) = (x < y).
Proof. by move=> *; apply/esym/eq_ltLR. Qed.
Lemma
eq_ltRL
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "eq_ltLR" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_gtP {x y} : reflect (forall z, z < x -> z < y) (x <= y).
Proof. by apply: (iffP idP) => [xy z /lt_le_trans|/(_ y)/[!ltNge]/contraTT]; apply. Qed.
Lemma
le_gtP
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "ltNge", "lt_le_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_ltP {x y} : reflect (forall z, y < z -> x < z) (x <= y).
Proof. by apply: (iffP idP) => [xy z /(le_lt_trans _)|/(_ x)/[!ltNge]/contraTT]; apply. Qed.
Lemma
le_ltP
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "le_lt_trans", "ltNge" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_gtP {x y} : reflect (forall z, (z < x) = (z < y)) (x == y).
Proof. by apply: (iffP eq_leP) => + k => /(_ k)/[!ltNge]/(congr1 negb); rewrite ?negbK. Qed.
Lemma
eq_gtP
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "eq_leP", "ltNge" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_ltP {x y} : reflect (forall z, (x < z) = (y < z)) (x == y).
Proof. by apply: (iffP eq_geP) => + k => /(_ k)/[!ltNge]/(congr1 negb); rewrite ?negbK. Qed.
Lemma
eq_ltP
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "eq_geP", "ltNge" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetEtotal x y : x `&` y = min x y.
Proof. by case: leP. Qed.
Lemma
meetEtotal
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "leP", "min" ]
max and min is join and meet
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinEtotal x y : x `|` y = max x y.
Proof. by case: leP. Qed.
Lemma
joinEtotal
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "leP", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
minEgt x y : min x y = if x > y then y else x.
Proof. by case: ltP. Qed.
Lemma
minEgt
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "ltP", "min" ]
max and min theory
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxEgt x y : max x y = if x > y then x else y.
Proof. by case: ltP. Qed.
Lemma
maxEgt
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "ltP", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
minEge x y : min x y = if x >= y then y else x.
Proof. by case: leP. Qed.
Lemma
minEge
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "leP", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxEge x y : max x y = if x >= y then x else y.
Proof. by case: leP. Qed.
Lemma
maxEge
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "leP", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
minC : commutative (min : T -> T -> T).
Proof. by move=> x y; apply: comparable_minC. Qed.
Lemma
minC
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "comparable_minC", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxC : commutative (max : T -> T -> T).
Proof. by move=> x y; apply: comparable_maxC. Qed.
Lemma
maxC
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "comparable_maxC", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
minA : associative (min : T -> T -> T).
Proof. by move=> x y z; apply: comparable_minA. Qed.
Lemma
minA
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "comparable_minA", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxA : associative (max : T -> T -> T).
Proof. by move=> x y z; apply: comparable_maxA. Qed.
Lemma
maxA
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "comparable_maxA", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
minAC : right_commutative (min : T -> T -> T).
Proof. by move=> x y z; apply: comparable_minAC. Qed.
Lemma
minAC
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "comparable_minAC", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxAC : right_commutative (max : T -> T -> T).
Proof. by move=> x y z; apply: comparable_maxAC. Qed.
Lemma
maxAC
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "comparable_maxAC", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
minCA : left_commutative (min : T -> T -> T).
Proof. by move=> x y z; apply: comparable_minCA. Qed.
Lemma
minCA
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "comparable_minCA", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxCA : left_commutative (max : T -> T -> T).
Proof. by move=> x y z; apply: comparable_maxCA. Qed.
Lemma
maxCA
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "comparable_maxCA", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
minACA : interchange (min : T -> T -> T) min.
Proof. by move=> x y z t; apply: comparable_minACA. Qed.
Lemma
minACA
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "comparable_minACA", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxACA : interchange (max : T -> T -> T) max.
Proof. by move=> x y z t; apply: comparable_maxACA. Qed.
Lemma
maxACA
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "comparable_maxACA", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_minr x y : (min x y == y) = (y <= x).
Proof. exact: comparable_eq_minr. Qed.
Lemma
eq_minr
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_eq_minr", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_maxl x y : (max x y == x) = (y <= x).
Proof. exact: comparable_eq_maxl. Qed.
Lemma
eq_maxl
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_eq_maxl", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
min_idPr x y : reflect (min x y = y) (y <= x).
Proof. exact: comparable_min_idPr. Qed.
Lemma
min_idPr
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_min_idPr", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_idPl x y : reflect (max x y = x) (y <= x).
Proof. exact: comparable_max_idPl. Qed.
Lemma
max_idPl
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_max_idPl", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_min z x y : (z <= min x y) = (z <= x) && (z <= y).
Proof. exact: comparable_le_min. Qed.
Lemma
le_min
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_le_min", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ge_min z x y : (min x y <= z) = (x <= z) || (y <= z).
Proof. exact: comparable_ge_min. Qed.
Lemma
ge_min
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_ge_min", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_min z x y : (z < min x y) = (z < x) && (z < y).
Proof. exact: comparable_lt_min. Qed.
Lemma
lt_min
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_lt_min", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gt_min z x y : (min x y < z) = (x < z) || (y < z).
Proof. exact: comparable_gt_min. Qed.
Lemma
gt_min
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_gt_min", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_max z x y : (z <= max x y) = (z <= x) || (z <= y).
Proof. exact: comparable_le_max. Qed.
Lemma
le_max
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_le_max", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ge_max z x y : (max x y <= z) = (x <= z) && (y <= z).
Proof. exact: comparable_ge_max. Qed.
Lemma
ge_max
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_ge_max", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_max z x y : (z < max x y) = (z < x) || (z < y).
Proof. exact: comparable_lt_max. Qed.
Lemma
lt_max
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_lt_max", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gt_max z x y : (max x y < z) = (x < z) && (y < z).
Proof. exact: comparable_gt_max. Qed.
Lemma
gt_max
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_gt_max", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
minxK x y : max (min x y) y = y.
Proof. exact: comparable_minxK. Qed.
Lemma
minxK
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_minxK", "max", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
minKx x y : max x (min x y) = x.
Proof. exact: comparable_minKx. Qed.
Lemma
minKx
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_minKx", "max", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxxK x y : min (max x y) y = y.
Proof. exact: comparable_maxxK. Qed.
Lemma
maxxK
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_maxxK", "max", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxKx x y : min x (max x y) = x.
Proof. exact: comparable_maxKx. Qed.
Lemma
maxKx
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_maxKx", "max", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_minl : left_distributive (max : T -> T -> T) min.
Proof. by move=> x y z; apply: comparable_max_minl. Qed.
Lemma
max_minl
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "comparable_max_minl", "max", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
min_maxl : left_distributive (min : T -> T -> T) max.
Proof. by move=> x y z; apply: comparable_min_maxl. Qed.
Lemma
min_maxl
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "comparable_min_maxl", "max", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_minr : right_distributive (max : T -> T -> T) min.
Proof. by move=> x y z; apply: comparable_max_minr. Qed.
Lemma
max_minr
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "comparable_max_minr", "max", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
min_maxr : right_distributive (min : T -> T -> T) max.
Proof. by move=> x y z; apply: comparable_min_maxr. Qed.
Lemma
min_maxr
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "comparable_min_maxr", "max", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
leIx x y z : (meet y z <= x) = (y <= x) || (z <= x).
Proof. by rewrite meetEtotal ge_min. Qed.
Lemma
leIx
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "ge_min", "meet", "meetEtotal" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lexU x y z : (x <= join y z) = (x <= y) || (x <= z).
Proof. by rewrite joinEtotal le_max. Qed.
Lemma
lexU
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "join", "joinEtotal", "le_max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltxI x y z : (x < meet y z) = (x < y) && (x < z).
Proof. by rewrite !ltNge leIx negb_or. Qed.
Lemma
ltxI
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "leIx", "ltNge", "meet" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltIx x y z : (meet y z < x) = (y < x) || (z < x).
Proof. by rewrite !ltNge lexI negb_and. Qed.
Lemma
ltIx
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "lexI", "ltNge", "meet" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltxU x y z : (x < join y z) = (x < y) || (x < z).
Proof. by rewrite !ltNge leUx negb_and. Qed.
Lemma
ltxU
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "join", "leUx", "ltNge" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltUx x y z : (join y z < x) = (y < x) && (z < x).
Proof. by rewrite !ltNge lexU negb_or. Qed.
Lemma
ltUx
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "join", "lexU", "ltNge" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltexI
:= (@lexI _ T, ltxI).
Definition
ltexI
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "lexI", "ltxI" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lteIx
:= (leIx, ltIx).
Definition
lteIx
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "leIx", "ltIx" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltexU
:= (lexU, ltxU).
Definition
ltexU
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "lexU", "ltxU" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lteUx
:= (@leUx _ T, ltUx).
Definition
lteUx
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "leUx", "ltUx" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_min2 x y z t : x <= z -> y <= t -> Order.min x y <= Order.min z t.
Proof. exact: comparable_le_min2. Qed.
Lemma
le_min2
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_le_min2", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_max2 x y z t : x <= z -> y <= t -> Order.max x y <= Order.max z t.
Proof. exact: comparable_le_max2. Qed.
Lemma
le_max2
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "comparable_le_max2", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lteifNE x y C : (x < y ?<= if ~~ C) = ~~ (y < x ?<= if C).
Proof. by case: C => /=; case: leP. Qed.
Lemma
lteifNE
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "leP" ]
lteif
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lteif_minr z x y C : (z < min x y ?<= if C) = (z < x ?<= if C) && (z < y ?<= if C).
Proof. by case: C; rewrite /= (le_min, lt_min). Qed.
Lemma
lteif_minr
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "le_min", "lt_min", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lteif_minl z x y C : (min x y < z ?<= if C) = (x < z ?<= if C) || (y < z ?<= if C).
Proof. by case: C; rewrite /= (ge_min, gt_min). Qed.
Lemma
lteif_minl
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "ge_min", "gt_min", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lteif_maxr z x y C : (z < max x y ?<= if C) = (z < x ?<= if C) || (z < y ?<= if C).
Proof. by case: C; rewrite /= (le_max, lt_max). Qed.
Lemma
lteif_maxr
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "le_max", "lt_max", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lteif_maxl z x y C : (max x y < z ?<= if C) = (x < z ?<= if C) && (y < z ?<= if C).
Proof. by case: C; rewrite /= (ge_max, gt_max). Qed.
Lemma
lteif_maxl
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "ge_max", "gt_max", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
arg_minP: extremum_spec <=%O P F (arg_min i0 P F).
Proof. by apply: extremumP => //; apply: le_trans. Qed.
Lemma
arg_minP
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "arg_min", "extremumP", "extremum_spec", "i0", "le_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
arg_maxP: extremum_spec >=%O P F (arg_max i0 P F).
Proof. by apply: extremumP => //; [apply: ge_refl | apply: ge_trans]. Qed.
Lemma
arg_maxP
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "arg_max", "extremumP", "extremum_spec", "ge_refl", "ge_trans", "i0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
count_le_gt x s : count (<= x) s = size s - count (> x) s.
Proof. by rewrite -(count_predC (> x)) addKn; apply: eq_count => y; rewrite /= leNgt. Qed.
Lemma
count_le_gt
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "addKn", "apply", "count", "count_predC", "eq_count", "leNgt", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
count_lt_ge x s : count (< x) s = size s - count (>= x) s.
Proof. by rewrite -(count_predC (>= x)) addKn; apply: eq_count => y; rewrite /= ltNge. Qed.
Lemma
count_lt_ge
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "addKn", "apply", "count", "count_predC", "eq_count", "ltNge", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_mkcond P F : \big[min/x]_(i <- r | P i) F i = \big[min/x]_(i <- r) (if P i then F i else x).
Proof. by rewrite big_mkcond_idem //= minxx. Qed.
Lemma
bigmin_mkcond
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "big_mkcond_idem", "min", "minxx" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_mkcond P F : \big[max/x]_(i <- r | P i) F i = \big[max/x]_(i <- r) if P i then F i else x.
Proof. by rewrite big_mkcond_idem //= maxxx. Qed.
Lemma
bigmax_mkcond
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "big_mkcond_idem", "max", "maxxx" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_mkcondl P Q F : \big[min/x]_(i <- r | P i && Q i) F i = \big[min/x]_(i <- r | Q i) if P i then F i else x.
Proof. rewrite bigmin_mkcond [RHS]bigmin_mkcond. by apply: eq_bigr => i _; case: P; case: Q. Qed.
Lemma
bigmin_mkcondl
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "bigmin_mkcond", "eq_bigr", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_mkcondr P Q F : \big[min/x]_(i <- r | P i && Q i) F i = \big[min/x]_(i <- r | P i) if Q i then F i else x.
Proof. by under eq_bigl do rewrite andbC; apply: bigmin_mkcondl. Qed.
Lemma
bigmin_mkcondr
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "bigmin_mkcondl", "eq_bigl", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_mkcondl P Q F : \big[max/x]_(i <- r | P i && Q i) F i = \big[max/x]_(i <- r | Q i) if P i then F i else x.
Proof. rewrite bigmax_mkcond [RHS]bigmax_mkcond. by apply: eq_bigr => i _; case: P; case: Q. Qed.
Lemma
bigmax_mkcondl
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "bigmax_mkcond", "eq_bigr", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_mkcondr P Q F : \big[max/x]_(i <- r | P i && Q i) F i = \big[max/x]_(i <- r | P i) if Q i then F i else x.
Proof. by under eq_bigl do rewrite andbC; apply: bigmax_mkcondl. Qed.
Lemma
bigmax_mkcondr
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "apply", "bigmax_mkcondl", "eq_bigl", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_split P F1 F2 : \big[min/x]_(i <- r | P i) (min (F1 i) (F2 i)) = min (\big[min/x]_(i <- r | P i) F1 i) (\big[min/x]_(i <- r | P i) F2 i).
Proof. by rewrite big_split_idem //= minxx. Qed.
Lemma
bigmin_split
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "F1", "F2", "big_split_idem", "min", "minxx" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_split P F1 F2 : \big[max/x]_(i <- r | P i) (max (F1 i) (F2 i)) = max (\big[max/x]_(i <- r | P i) F1 i) (\big[max/x]_(i <- r | P i) F2 i).
Proof. by rewrite big_split_idem //= maxxx. Qed.
Lemma
bigmax_split
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "F1", "F2", "big_split_idem", "max", "maxxx" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_idl P F : \big[min/x]_(i <- r | P i) F i = min x (\big[min/x]_(i <- r | P i) F i).
Proof. by rewrite minC big_id_idem //= minxx. Qed.
Lemma
bigmin_idl
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "big_id_idem", "min", "minC", "minxx" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_idl P F : \big[max/x]_(i <- r | P i) F i = max x (\big[max/x]_(i <- r | P i) F i).
Proof. by rewrite maxC big_id_idem //= maxxx. Qed.
Lemma
bigmax_idl
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "big_id_idem", "max", "maxC", "maxxx" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_idr P F : \big[min/x]_(i <- r | P i) F i = min (\big[min/x]_(i <- r | P i) F i) x.
Proof. by rewrite [LHS]bigmin_idl minC. Qed.
Lemma
bigmin_idr
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "bigmin_idl", "min", "minC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_idr P F : \big[max/x]_(i <- r | P i) F i = max (\big[max/x]_(i <- r | P i) F i) x.
Proof. by rewrite [LHS]bigmax_idl maxC. Qed.
Lemma
bigmax_idr
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "bigmax_idl", "max", "maxC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigminID a P F : \big[min/x]_(i <- r | P i) F i = min (\big[min/x]_(i <- r | P i && a i) F i) (\big[min/x]_(i <- r | P i && ~~ a i) F i).
Proof. by rewrite (bigID_idem _ _ a) //= minxx. Qed.
Lemma
bigminID
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "bigID_idem", "min", "minxx" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmaxID a P F : \big[max/x]_(i <- r | P i) F i = max (\big[max/x]_(i <- r | P i && a i) F i) (\big[max/x]_(i <- r | P i && ~~ a i) F i).
Proof. by rewrite (bigID_idem _ _ a) //= maxxx. Qed.
Lemma
bigmaxID
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "bigID_idem", "max", "maxxx" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ge_min_id (x y : T) : x >= min x y.
Proof. by rewrite ge_min lexx. Qed.
Let
ge_min_id
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "ge_min", "lexx", "min" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_max_id (x y : T) : x <= max x y.
Proof. by rewrite le_max lexx. Qed.
Let
le_max_id
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "le_max", "lexx", "max" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_bigmin [x0] I r (P P' : {pred I}) (F : I -> T) : (forall i, P' i -> P i) -> \big[min/x0]_(i <- r | P i) F i <= \big[min/x0]_(i <- r | P' i) F i.
Proof. exact: (sub_le_big ge_refl). Qed.
Lemma
sub_bigmin
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "ge_refl", "min", "sub_le_big" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_bigmax [x0] I r (P P' : {pred I}) (F : I -> T) : (forall i, P i -> P' i) -> \big[max/x0]_(i <- r | P i) F i <= \big[max/x0]_(i <- r | P' i) F i.
Proof. exact: sub_le_big. Qed.
Lemma
sub_bigmax
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "max", "sub_le_big" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'{subset' x '<=' y '}'"
:= (sub_mem (mem x) (mem y)) (at level 0, x, y at level 1).
Notation
'{subset' x '<=' y '}'
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[]
FIXME: Remove that.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_bigmin_seq [x0] (I : eqType) r r' P (F : I -> T) : {subset r' <= r} -> \big[min/x0]_(i <- r | P i) F i <= \big[min/x0]_(i <- r' | P i) F i.
Proof. exact: (idem_sub_le_big ge_refl _ minxx). Qed.
Lemma
sub_bigmin_seq
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "ge_refl", "idem_sub_le_big", "min", "minxx" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_bigmax_seq [x0] (I : eqType) r r' P (F : I -> T) : {subset r <= r'} -> \big[max/x0]_(i <- r | P i) F i <= \big[max/x0]_(i <- r' | P i) F i.
Proof. exact: (idem_sub_le_big _ _ maxxx). Qed.
Lemma
sub_bigmax_seq
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "idem_sub_le_big", "max", "maxxx" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_bigmin_cond [x0] (I : eqType) r r' P P' (F : I -> T) : {subset ([seq i <- r | P i]) <= ([seq i <- r' | P' i])} -> \big[min/x0]_(i <- r' | P' i) F i <= \big[min/x0]_(i <- r | P i) F i.
Proof. exact: (idem_sub_le_big_cond ge_refl _ minxx). Qed.
Lemma
sub_bigmin_cond
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "ge_refl", "idem_sub_le_big_cond", "min", "minxx", "seq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_bigmax_cond [x0] (I : eqType) r r' P P' (F : I -> T) : {subset ([seq i <- r | P i]) <= ([seq i <- r' | P' i])} -> \big[max/x0]_(i <- r | P i) F i <= \big[max/x0]_(i <- r' | P' i) F i.
Proof. exact: (idem_sub_le_big_cond _ _ maxxx). Qed.
Lemma
sub_bigmax_cond
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "idem_sub_le_big_cond", "max", "maxxx", "seq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_in_bigmin [x0] [I : eqType] (r : seq I) (P P' : {pred I}) F : {in r, forall i, P' i -> P i} -> \big[min/x0]_(i <- r | P i) F i <= \big[min/x0]_(i <- r | P' i) F i.
Proof. exact: (sub_in_le_big ge_refl). Qed.
Lemma
sub_in_bigmin
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "ge_refl", "min", "seq", "sub_in_le_big" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_in_bigmax [x0] [I : eqType] (r : seq I) (P P' : {pred I}) F : {in r, forall i, P i -> P' i} -> \big[max/x0]_(i <- r | P i) F i <= \big[max/x0]_(i <- r | P' i) F i.
Proof. exact: sub_in_le_big. Qed.
Lemma
sub_in_bigmax
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "max", "seq", "sub_in_le_big" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_bigmin_nat [x0] n m n' m' P (F : nat -> T) : (n <= n')%N -> (m' <= m)%N -> \big[min/x0]_(n <= i < m | P i) F i <= \big[min/x0]_(n' <= i < m' | P i) F i.
Proof. exact: (le_big_nat ge_refl). Qed.
Lemma
le_bigmin_nat
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "ge_refl", "le_big_nat", "min", "n'", "nat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_bigmax_nat [x0] n m n' m' P (F : nat -> T) : (n' <= n)%N -> (m <= m')%N -> \big[max/x0]_(n <= i < m | P i) F i <= \big[max/x0]_(n' <= i < m' | P i) F i.
Proof. exact: le_big_nat. Qed.
Lemma
le_bigmax_nat
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "le_big_nat", "max", "n'", "nat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_bigmin_nat_cond [x0] n m n' m' (P P' : pred nat) (F : nat -> T) : (n <= n')%N -> (m' <= m)%N -> (forall i, (n' <= i < m')%N -> P' i -> P i) -> \big[min/x0]_(n <= i < m | P i) F i <= \big[min/x0]_(n' <= i < m' | P' i) F i.
Proof. exact: (le_big_nat_cond ge_refl). Qed.
Lemma
le_bigmin_nat_cond
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "ge_refl", "le_big_nat_cond", "min", "n'", "nat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_bigmax_nat_cond [x0] n m n' m' (P P' : {pred nat}) (F : nat -> T) : (n' <= n)%N -> (m <= m')%N -> (forall i, (n <= i < m)%N -> P i -> P' i) -> \big[max/x0]_(n <= i < m | P i) F i <= \big[max/x0]_(n' <= i < m' | P' i) F i.
Proof. exact: le_big_nat_cond. Qed.
Lemma
le_bigmax_nat_cond
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "le_big_nat_cond", "max", "n'", "nat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_bigmin_ord [x0] n m (P : pred nat) (F : nat -> T) : (m <= n)%N -> \big[min/x0]_(i < n | P i) F i <= \big[min/x0]_(i < m | P i) F i.
Proof. exact: (le_big_ord ge_refl). Qed.
Lemma
le_bigmin_ord
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "ge_refl", "le_big_ord", "min", "nat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_bigmax_ord [x0] n m (P : {pred nat}) (F : nat -> T) : (n <= m)%N -> \big[max/x0]_(i < n | P i) F i <= \big[max/x0]_(i < m | P i) F i.
Proof. exact: le_big_ord. Qed.
Lemma
le_bigmax_ord
order
order/order.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...
[ "le_big_ord", "max", "nat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d