phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
le_bigmin_ord_cond [x0] n m (P P' : pred nat) (F : nat -> T) : (m <= n)%N -> (forall i : 'I_m, P' i -> P i) -> \big[min/x0]_(i < n \| P i) F i <= \big[min/x0]_(i < m \| P' i) F i.	Proof. exact: (le_big_ord_cond ge_refl). Qed.	Lemma	le_bigmin_ord_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_ord_cond", "min", "nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_bigmax_ord_cond [x0] n m (P P' : {pred nat}) (F : nat -> T) : (n <= m)%N -> (forall i : 'I_n, P i -> P' i) -> \big[max/x0]_(i < n \| P i) F i <= \big[max/x0]_(i < m \| P' i) F i.	Proof. exact: le_big_ord_cond. Qed.	Lemma	le_bigmax_ord_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_ord_cond", "max", "nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_bigmin [x0] [I : finType] [A A' P : {pred I}] (F : I -> T) : A' \subset A -> \big[min/x0]_(i in A \| P i) F i <= \big[min/x0]_(i in A' \| P i) F i.	Proof. exact: (subset_le_big ge_refl). Qed.	Lemma	subset_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",...	[ "A'", "ge_refl", "min", "subset_le_big" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_bigmax [x0] [I : finType] (A A' P : {pred I}) (F : I -> T) : A \subset A' -> \big[max/x0]_(i in A \| P i) F i <= \big[max/x0]_(i in A' \| P i) F i.	Proof. exact: subset_le_big. Qed.	Lemma	subset_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",...	[ "A'", "max", "subset_le_big" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_bigmin_cond [x0] (I : finType) (A A' P P' : {pred I}) (F : I -> T) : [set i in A' \| P' i] \subset [set i in A \| P i] -> \big[min/x0]_(i in A \| P i) F i <= \big[min/x0]_(i in A' \| P' i) F i.	Proof. exact: (subset_le_big_cond ge_refl). Qed.	Lemma	subset_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",...	[ "A'", "ge_refl", "min", "subset_le_big_cond" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_bigmax_cond [x0] (I : finType) (A A' P P' : {pred I}) (F : I -> T) : [set i in A \| P i] \subset [set i in A' \| P' i] -> \big[max/x0]_(i in A \| P i) F i <= \big[max/x0]_(i in A' \| P' i) F i.	Proof. exact: subset_le_big_cond. Qed.	Lemma	subset_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",...	[ "A'", "max", "subset_le_big_cond" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_le_id P F : \big[min/x]_(i <- r \| P i) F i <= x.	Proof. by rewrite bigmin_idl. Qed.	Lemma	bigmin_le_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",...	[ "bigmin_idl", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_ge_id P F : \big[max/x]_(i <- r \| P i) F i >= x.	Proof. by rewrite bigmax_idl. Qed.	Lemma	bigmax_ge_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",...	[ "bigmax_idl", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_eq_id P F : (forall i, P i -> x <= F i) -> \big[min/x]_(i <- r \| P i) F i = x.	Proof. by move=> x_le; apply: le_anti; rewrite bigmin_le_id le_bigmin. Qed.	Lemma	bigmin_eq_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",...	[ "apply", "bigmin_le_id", "le_anti", "le_bigmin", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_eq_id P F : (forall i, P i -> x >= F i) -> \big[max/x]_(i <- r \| P i) F i = x.	Proof. by move=> x_ge; apply: le_anti; rewrite bigmax_ge_id bigmax_le. Qed.	Lemma	bigmax_eq_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",...	[ "apply", "bigmax_ge_id", "bigmax_le", "le_anti", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ge_bigmin_seq i0 P F : i0 \in r -> P i0 -> \big[min/x]_(i <- r \| P i) F i <= F i0.	Proof. move=> + Pi0; elim: r => // h t ih; rewrite inE big_cons. move=> /predU1P[<-\|i0t]; first by rewrite Pi0 ge_min// lexx. by case: ifPn => Ph; [rewrite ge_min ih// orbT\|rewrite ih]. Qed.	Lemma	ge_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",...	[ "Pi0", "big_cons", "ge_min", "i0", "inE", "lexx", "min", "predU1P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_bigmax_seq i0 P F : i0 \in r -> P i0 -> F i0 <= \big[max/x]_(i <- r \| P i) F i.	Proof. move=> + Pi0; elim: r => // h t ih; rewrite inE big_cons. move=> /predU1P[<-\|i0t]; first by rewrite Pi0 le_max// lexx. by case: ifPn => Ph; [rewrite le_max ih// orbT\|rewrite ih]. Qed.	Lemma	le_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",...	[ "Pi0", "big_cons", "i0", "inE", "le_max", "lexx", "max", "predU1P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_inf_seq i0 P t F : i0 \in r -> P i0 -> F i0 <= t -> \big[min/x]_(i <- r \| P i) F i <= t.	Proof. by move=> ? ? ?; exact: le_trans (@ge_bigmin_seq i0 _ _ _ _) _. Qed.	Lemma	bigmin_inf_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_bigmin_seq", "i0", "le_trans", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_sup_seq i0 P t F : i0 \in r -> P i0 -> t <= F i0 -> t <= \big[max/x]_(i <- r \| P i) F i.	Proof. by move=> ? ? ?; exact: le_trans (@le_bigmax_seq i0 _ _ _ _). Qed.	Lemma	bigmax_sup_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",...	[ "i0", "le_bigmax_seq", "le_trans", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigminD1 j P F : P j -> \big[min/x]_(i \| P i) F i = min (F j) (\big[min/x]_(i \| P i && (i != j)) F i).	Proof. by move/(bigD1 _) ->. Qed.	Lemma	bigminD1	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",...	[ "bigD1", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmaxD1 j P F : P j -> \big[max/x]_(i \| P i) F i = max (F j) (\big[max/x]_(i \| P i && (i != j)) F i).	Proof. by move/(bigD1 _) ->. Qed.	Lemma	bigmaxD1	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",...	[ "bigD1", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_le_cond j P F : P j -> \big[min/x]_(i \| P i) F i <= F j.	Proof. have := mem_index_enum j; rewrite unlock; elim: (index_enum I) => //= i l ih. rewrite inE => /orP [/eqP-> ->\|/ih leminlfi Pi]; first by rewrite ge_min lexx. by case: ifPn => Pj; [rewrite ge_min leminlfi// orbC\|exact: leminlfi]. Qed.	Lemma	bigmin_le_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_min", "inE", "index_enum", "lexx", "mem_index_enum", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_bigmax_cond j P F : P j -> F j <= \big[max/x]_(i \| P i) F i.	Proof. by move=> Pj; rewrite (bigmaxD1 _ Pj) le_max lexx. Qed.	Lemma	le_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",...	[ "bigmaxD1", "le_max", "lexx", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_le j F : \big[min/x]_i F i <= F j.	Proof. exact: bigmin_le_cond. Qed.	Lemma	bigmin_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",...	[ "bigmin_le_cond", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_bigmax F j : F j <= \big[max/x]_i F i.	Proof. exact: le_bigmax_cond. Qed.	Lemma	le_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",...	[ "le_bigmax_cond", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_inf j P m F : P j -> F j <= m -> \big[min/x]_(i \| P i) F i <= m.	Proof. by move=> Pj ?; apply: le_trans (bigmin_le_cond _ Pj) _. Qed.	Lemma	bigmin_inf	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_le_cond", "le_trans", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_sup j P m F : P j -> m <= F j -> m <= \big[max/x]_(i \| P i) F i.	Proof. by move=> Pj ?; apply: le_trans (le_bigmax_cond _ Pj). Qed.	Lemma	bigmax_sup	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_bigmax_cond", "le_trans", "max" ]	NB: as of [2022-08-02], bigop.bigmax_sup already exists for nat	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_geP m P F : reflect (m <= x /\ forall i, P i -> m <= F i) (m <= \big[min/x]_(i \| P i) F i).	Proof. apply: (iffP idP) => [lemFi\|[lemx lemPi]]; [split\|exact: le_bigmin]. - by rewrite (le_trans lemFi)// bigmin_idl ge_min lexx. - by move=> i Pi; rewrite (le_trans lemFi)// (bigminD1 _ Pi)// le_minl lexx. Qed.	Lemma	bigmin_geP	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", "bigminD1", "bigmin_idl", "ge_min", "le_bigmin", "le_trans", "lexx", "min", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_leP m P F : reflect (x <= m /\ forall i, P i -> F i <= m) (\big[max/x]_(i \| P i) F i <= m).	Proof. apply: (iffP idP) => [\|[? ?]]; last exact: bigmax_le. rewrite bigmax_idl ge_max => /andP[-> leFm]; split=> // i Pi. by apply: le_trans leFm; exact: le_bigmax_cond. Qed.	Lemma	bigmax_leP	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_idl", "bigmax_le", "ge_max", "last", "le_bigmax_cond", "le_trans", "max", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_gtP m P F : reflect (m < x /\ forall i, P i -> m < F i) (m < \big[min/x]_(i \| P i) F i).	Proof. apply: (iffP idP) => [lemFi\|[lemx lemPi]]; [split\|exact: lt_bigmin]. - by rewrite (lt_le_trans lemFi)// bigmin_idl ge_min lexx. - by move=> i Pi; rewrite (lt_le_trans lemFi)// (bigminD1 _ Pi)// le_minl lexx. Qed.	Lemma	bigmin_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", "bigminD1", "bigmin_idl", "ge_min", "lexx", "lt_bigmin", "lt_le_trans", "min", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_ltP m P F : reflect (x < m /\ forall i, P i -> F i < m) (\big[max/x]_(i \| P i) F i < m).	Proof. apply: (iffP idP) => [\|[? ?]]; last exact: bigmax_lt. rewrite bigmax_idl gt_max => /andP[-> ltFm]; split=> // i Pi. by apply: le_lt_trans ltFm; exact: le_bigmax_cond. Qed.	Lemma	bigmax_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", "bigmax_idl", "bigmax_lt", "gt_max", "last", "le_bigmax_cond", "le_lt_trans", "max", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_eq_arg j P F : P j -> (forall i, P i -> F i <= x) -> \big[min/x]_(i \| P i) F i = F [arg min_(i < j \| P i) F i].	Proof. move=> Pi0; case: arg_minP => //= i Pi PF PFx. apply/eqP; rewrite eq_le bigmin_le_cond //=. by apply/bigmin_geP; split => //; exact: PFx. Qed.	Lemma	bigmin_eq_arg	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",...	[ "Pi0", "apply", "arg_minP", "bigmin_geP", "bigmin_le_cond", "eq_le", "min", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_eq_arg j P F : P j -> (forall i, P i -> x <= F i) -> \big[max/x]_(i \| P i) F i = F [arg max_(i > j \| P i) F i].	Proof. move=> Pi0; case: arg_maxP => //= i Pi PF PxF. apply/eqP; rewrite eq_le le_bigmax_cond // andbT. by apply/bigmax_leP; split => //; exact: PxF. Qed.	Lemma	bigmax_eq_arg	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",...	[ "Pi0", "apply", "arg_maxP", "bigmax_leP", "eq_le", "le_bigmax_cond", "max", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_bigmin j P F : P j -> (forall i, P i -> F i <= x) -> {i0 \| i0 \in P & \big[min/x]_(i \| P i) F i = F i0}.	Proof. by move=> Pi0 Hx; rewrite (bigmin_eq_arg Pi0) //; eexists=> //; case: arg_minP. Qed.	Lemma	eq_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",...	[ "Pi0", "arg_minP", "bigmin_eq_arg", "i0", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_bigmax j P F : P j -> (forall i, P i -> x <= F i) -> {i0 \| i0 \in P & \big[max/x]_(i \| P i) F i = F i0}.	Proof. by move=> Pi0 Hx; rewrite (bigmax_eq_arg Pi0) //; eexists=> //; case: arg_maxP. Qed.	Lemma	eq_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",...	[ "Pi0", "arg_maxP", "bigmax_eq_arg", "i0", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_bigmin2 P F1 F2 : (forall i, P i -> F1 i <= F2 i) -> \big[min/x]_(i \| P i) F1 i <= \big[min/x]_(i \| P i) F2 i.	Proof. move=> FG; elim/big_ind2 : _ => // a b e f ba fe. rewrite ge_min 2!le_min ba fe /= andbT. move: (le_total a e) => /orP[/(le_trans ba)-> // \| /(le_trans fe)->]. by rewrite orbT. Qed.	Lemma	le_bigmin2	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_ind2", "ge_min", "le_min", "le_total", "le_trans", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_bigmax2 P F1 F2 : (forall i, P i -> F1 i <= F2 i) -> \big[max/x]_(i \| P i) F1 i <= \big[max/x]_(i \| P i) F2 i.	Proof. move=> FG; elim/big_ind2 : _ => // a b e f ba fe. rewrite le_max 2!ge_max ba fe /= andbT; have [//\|/= af] := leP f a. by rewrite (le_trans ba) // (le_trans _ fe) // ltW. Qed.	Lemma	le_bigmax2	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_ind2", "ge_max", "leP", "le_max", "le_trans", "ltW", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmaxUl (A B : {set I}) F : \big[max/x]_(i in A) F i <= \big[max/x]_(i in A :\|: B) F i.	Proof. by apply: sub_bigmax => t; rewrite in_setU => ->. Qed.	Lemma	bigmaxUl	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", "in_setU", "max", "sub_bigmax" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmaxUr (A B : {set I}) F : \big[max/x]_(i in B) F i <= \big[max/x]_(i in A :\|: B) F i.	Proof. by under [leRHS]eq_bigl do rewrite setUC; apply: bigmaxUl. Qed.	Lemma	bigmaxUr	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", "bigmaxUl", "eq_bigl", "leRHS", "max", "setUC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigminUl (A B : {set I}) F : \big[min/x]_(i in A) F i >= \big[min/x]_(i in A :\|: B) F i.	Proof. by apply: sub_bigmin => t; rewrite in_setU => ->. Qed.	Lemma	bigminUl	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", "in_setU", "min", "sub_bigmin" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigminUr (A B : {set I}) F : \big[min/x]_(i in B) F i >= \big[min/x]_(i in A :\|: B) F i.	Proof. by under [leLHS]eq_bigl do rewrite setUC; apply: bigminUl. Qed.	Lemma	bigminUr	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", "bigminUl", "eq_bigl", "leLHS", "min", "setUC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmaxIl (A B : {set I}) F : \big[max/x]_(i in A) F i >= \big[max/x]_(i in A :&: B) F i.	Proof. by apply: sub_bigmax => t; rewrite in_setI => /andP[-> _]. Qed.	Lemma	bigmaxIl	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", "in_setI", "max", "sub_bigmax" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmaxIr (A B : {set I}) F : \big[max/x]_(i in B) F i >= \big[max/x]_(i in A :&: B) F i.	Proof. by under eq_bigl do rewrite setIC; apply: bigmaxIl. Qed.	Lemma	bigmaxIr	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", "bigmaxIl", "eq_bigl", "max", "setIC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigminIl (A B : {set I}) F : \big[min/x]_(i in A) F i <= \big[min/x]_(i in A :&: B) F i.	Proof. by apply: sub_bigmin => t; rewrite in_setI => /andP[->_]. Qed.	Lemma	bigminIl	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", "in_setI", "min", "sub_bigmin" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigminIr (A B : {set I}) F : \big[min/x]_(i in B) F i <= \big[min/x]_(i in A :&: B) F i.	Proof. by under [leRHS]eq_bigl do rewrite setIC; apply: bigminIl. Qed.	Lemma	bigminIr	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", "bigminIl", "eq_bigl", "leRHS", "min", "setIC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmaxD (A B : {set I}) F : \big[max/x]_(i in B) F i >= \big[max/x]_(i in B :\: A) F i.	Proof. by apply: sub_bigmax => t; rewrite in_setD => /andP[_->]. Qed.	Lemma	bigmaxD	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", "in_setD", "max", "sub_bigmax" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigminD (A B : {set I}) F : \big[min/x]_(i in B) F i <= \big[min/x]_(i in B :\: A) F i.	Proof. by apply: sub_bigmin => t; rewrite in_setD => /andP[_->]. Qed.	Lemma	bigminD	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", "in_setD", "min", "sub_bigmin" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmaxU (A B : {set I}) F : \big[max/x]_(i in A :\|: B) F i = max (\big[max/x]_(i in A) F i) (\big[max/x]_(i in B) F i).	Proof. apply: le_anti; rewrite ge_max bigmaxUl bigmaxUr !andbT; apply/bigmax_leP. split=> [\|i /[!in_setU]/orP[iA\|iB]]; first by rewrite le_max bigmax_ge_id. - by rewrite le_max le_bigmax_cond. - by rewrite le_max orbC le_bigmax_cond. Qed.	Lemma	bigmaxU	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", "bigmaxUl", "bigmaxUr", "bigmax_ge_id", "bigmax_leP", "ge_max", "in_setU", "le_anti", "le_bigmax_cond", "le_max", "max", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigminU (A B : {set I}) F : \big[min/x]_(i in A :\|: B) F i = min (\big[min/x]_(i in A) F i) (\big[min/x]_(i in B) F i).	Proof. apply: le_anti; rewrite le_min bigminUl bigminUr !andbT; apply/bigmin_geP. split=> [\|i /[!in_setU]/orP[iA\|iB]]; first by rewrite ge_min bigmin_le_id. - by rewrite ge_min bigmin_le_cond. - by rewrite ge_min orbC bigmin_le_cond. Qed.	Lemma	bigminU	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", "bigminUl", "bigminUr", "bigmin_geP", "bigmin_le_cond", "bigmin_le_id", "ge_min", "in_setU", "le_anti", "le_min", "min", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_set1 j F : \big[min/x]_(i in [set j]) F i = min (F j) x.	Proof. exact: big_set1E. Qed.	Lemma	bigmin_set1	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_set1E", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_set1 j F : \big[max/x]_(i in [set j]) F i = max (F j) x.	Proof. exact: big_set1E. Qed.	Lemma	bigmax_set1	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_set1E", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmin_imset [I J : finType] x [h : I -> J] [A : {set I}] (F : J -> T) : \big[min/x]_(j in [set h x \| x in A]) F j = \big[min/x]_(i in A) F (h i).	Proof. by apply: big_imset_idem; apply: minxx. Qed.	Lemma	bigmin_imset	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", "big_imset_idem", "min", "minxx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_imset [I J : finType] x [h : I -> J] [A : {set I}] (F : J -> T) : \big[max/x]_(j in [set h x \| x in A]) F j = \big[max/x]_(i in A) F (h i).	Proof. by apply: big_imset_idem; apply: maxxx. Qed.	Lemma	bigmax_imset	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", "big_imset_idem", "max", "maxxx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sorted_filter_gt x s : sorted <=%O s -> [seq y <- s \| x < y] = drop (count (<= x) s) s.	Proof. move=> s_sorted; rewrite count_le_gt -[LHS]revK -filter_rev. rewrite (@sorted_filter_lt _ T^d); last by rewrite take_rev revK count_rev. by rewrite rev_sorted. Qed.	Lemma	sorted_filter_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",...	[ "count", "count_le_gt", "count_rev", "drop", "filter_rev", "last", "revK", "rev_sorted", "seq", "sorted", "sorted_filter_lt", "take_rev" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sorted_filter_ge x s : sorted <=%O s -> [seq y <- s \| x <= y] = drop (count (< x) s) s.	Proof. move=> s_sorted; rewrite count_lt_ge -[LHS]revK -filter_rev. rewrite (@sorted_filter_le _ T^d); last by rewrite take_rev revK count_rev. by rewrite rev_sorted. Qed.	Lemma	sorted_filter_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",...	[ "count", "count_lt_ge", "count_rev", "drop", "filter_rev", "last", "revK", "rev_sorted", "seq", "sorted", "sorted_filter_le", "take_rev" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nth_count_ge x x0 s i : sorted <=%O s -> (count (< x) s <= i < size s)%N -> x <= nth x0 s i.	Proof. move=> ss /andP[ige ilt]; rewrite -(subnKC ige) -nth_drop -sorted_filter_ge //. apply/(all_nthP _ (filter_all _ _)). by rewrite size_filter ltn_subLR // count_lt_ge subnK // count_size. Qed.	Lemma	nth_count_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",...	[ "all_nthP", "apply", "count", "count_lt_ge", "count_size", "filter_all", "ltn_subLR", "nth", "nth_drop", "size", "size_filter", "sorted", "sorted_filter_ge", "subnK", "subnKC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nth_count_gt x x0 s i : sorted <=%O s -> (count (<= x) s <= i < size s)%N -> x < nth x0 s i.	Proof. move=> ss /andP[ige ilt]; rewrite -(subnKC ige) -nth_drop -sorted_filter_gt //. apply/(all_nthP _ (filter_all _ _)). by rewrite size_filter ltn_subLR // count_le_gt subnK // count_size. Qed.	Lemma	nth_count_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",...	[ "all_nthP", "apply", "count", "count_le_gt", "count_size", "filter_all", "ltn_subLR", "nth", "nth_drop", "size", "size_filter", "sorted", "sorted_filter_gt", "subnK", "subnKC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nth_count_eq x x0 s i : sorted <=%O s -> (count (< x) s <= i < count (<= x) s)%N -> nth x0 s i = x.	Proof. move=> ss /andP[ige ilt]; apply/le_anti. by rewrite nth_count_le// nth_count_ge// ige (leq_trans ilt (count_size _ _)). Qed.	Lemma	nth_count_eq	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", "count", "count_size", "le_anti", "leq_trans", "nth", "nth_count_ge", "nth_count_le", "sorted" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contraTle b z t : (t < z -> ~~ b) -> (b -> z <= t).	Proof. exact: comparable_contraTle. Qed.	Lemma	contraTle	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_contraTle" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contraTlt b z t : (t <= z -> ~~ b) -> (b -> z < t).	Proof. exact: comparable_contraTlt. Qed.	Lemma	contraTlt	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_contraTlt" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contraPle P z t : (t < z -> ~ P) -> (P -> z <= t).	Proof. exact: comparable_contraPle. Qed.	Lemma	contraPle	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_contraPle" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contraPlt P z t : (t <= z -> ~ P) -> (P -> z < t).	Proof. exact: comparable_contraPlt. Qed.	Lemma	contraPlt	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_contraPlt" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contraNle b z t : (t < z -> b) -> (~~ b -> z <= t).	Proof. exact: comparable_contraNle. Qed.	Lemma	contraNle	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_contraNle" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contraNlt b z t : (t <= z -> b) -> (~~ b -> z < t).	Proof. exact: comparable_contraNlt. Qed.	Lemma	contraNlt	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_contraNlt" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_not_le P z t : (t < z -> P) -> (~ P -> z <= t).	Proof. exact: comparable_contra_not_le. Qed.	Lemma	contra_not_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",...	[ "comparable_contra_not_le" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_not_lt P z t : (t <= z -> P) -> (~ P -> z < t).	Proof. exact: comparable_contra_not_lt. Qed.	Lemma	contra_not_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",...	[ "comparable_contra_not_lt" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contraFle b z t : (t < z -> b) -> (b = false -> z <= t).	Proof. exact: comparable_contraFle. Qed.	Lemma	contraFle	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_contraFle" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contraFlt b z t : (t <= z -> b) -> (b = false -> z < t).	Proof. exact: comparable_contraFlt. Qed.	Lemma	contraFlt	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_contraFlt" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_leq_le m n z t : (t < z -> (n < m)%N) -> ((m <= n)%N -> z <= t).	Proof. exact: comparable_contra_leq_le. Qed.	Lemma	contra_leq_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",...	[ "comparable_contra_leq_le" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_leq_lt m n z t : (t <= z -> (n < m)%N) -> ((m <= n)%N -> z < t).	Proof. exact: comparable_contra_leq_lt. Qed.	Lemma	contra_leq_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",...	[ "comparable_contra_leq_lt" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_ltn_le m n z t : (t < z -> (n <= m)%N) -> ((m < n)%N -> z <= t).	Proof. exact: comparable_contra_ltn_le. Qed.	Lemma	contra_ltn_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",...	[ "comparable_contra_ltn_le" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_ltn_lt m n z t : (t <= z -> (n <= m)%N) -> ((m < n)%N -> z < t).	Proof. exact: comparable_contra_ltn_lt. Qed.	Lemma	contra_ltn_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",...	[ "comparable_contra_ltn_lt" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_le x y z t : (t < z -> y < x) -> (x <= y -> z <= t).	Proof. exact: comparable_contra_le. Qed.	Lemma	contra_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",...	[ "comparable_contra_le" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_le_lt x y z t : (t <= z -> y < x) -> (x <= y -> z < t).	Proof. exact: comparable_contra_le_lt. Qed.	Lemma	contra_le_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",...	[ "comparable_contra_le_lt" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_lt_le x y z t : (t < z -> y <= x) -> (x < y -> z <= t).	Proof. exact: comparable_contra_lt_le. Qed.	Lemma	contra_lt_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",...	[ "comparable_contra_lt_le" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_lt x y z t : (t <= z -> y <= x) -> (x < y -> z < t).	Proof. exact: comparable_contra_lt. Qed.	Lemma	contra_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",...	[ "comparable_contra_lt" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leT_anti	:= @le_anti _ T.	Let	leT_anti	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_anti" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leT'_anti	:= @le_anti _ T'.	Let	leT'_anti	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",...	[ "T'", "le_anti" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltT_neqAle	:= @lt_neqAle _ T.	Let	ltT_neqAle	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",...	[ "lt_neqAle" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltT'_neqAle	:= @lt_neqAle _ T'.	Let	ltT'_neqAle	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",...	[ "T'", "lt_neqAle" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltT_def	:= @lt_def _ T.	Let	ltT_def	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",...	[ "lt_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leT_total	:= @le_total _ T.	Let	leT_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",...	[ "le_total" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_mono : {homo f : x y / x < y} -> {mono f : x y / x <= y}.	Proof. exact: total_homo_mono. Qed.	Lemma	le_mono	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",...	[ "total_homo_mono" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_nmono : {homo f : x y /~ x < y} -> {mono f : x y /~ x <= y}.	Proof. by apply: total_homo_mono => // x y; rewrite eq_sym. Qed.	Lemma	le_nmono	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_sym", "total_homo_mono" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_mono_in : {in D &, {homo f : x y / x < y}} -> {in D &, {mono f : x y / x <= y}}.	Proof. exact: total_homo_mono_in. Qed.	Lemma	le_mono_in	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",...	[ "total_homo_mono_in" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_nmono_in : {in D &, {homo f : x y /~ x < y}} -> {in D &, {mono f : x y /~ x <= y}}.	Proof. by apply: total_homo_mono_in => // x y; rewrite eq_sym. Qed.	Lemma	le_nmono_in	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_sym", "total_homo_mono_in" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcomplPmeet x y z : ((x `&` y) `\|` z) `&` rcompl x y z = x `&` y.	Proof. exact: rcomplPmeet. Qed.	Lemma	rcomplPmeet	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",...	[ "rcompl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcomplPjoin x y z : ((y `\|` x) `&` z) `\|` rcompl x y z = y `\|` x.	Proof. exact: rcomplPjoin. Qed.	Lemma	rcomplPjoin	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",...	[ "rcompl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcomplKI x y z : x <= y -> (x `\|` z) `&` rcompl x y z = x.	Proof. by move=> lexy; have := rcomplPmeet x y z; rewrite (meet_l lexy). Qed.	Lemma	rcomplKI	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",...	[ "meet_l", "rcompl", "rcomplPmeet" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcomplKU x y z : x <= y -> (y `&` z) `\|` rcompl x y z = y.	Proof. by move=> lexy; have := rcomplPjoin x y z; rewrite (join_l lexy). Qed.	Lemma	rcomplKU	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_l", "rcompl", "rcomplPjoin" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffErcompl x y : x `\` y = rcompl \bot x y.	Proof. exact: diffErcompl. Qed.	Lemma	diffErcompl	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",...	[ "bot", "rcompl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffKI x y : y `&` (x `\` y) = \bot.	Proof. by have := rcomplKI y (le0x x); rewrite join0x diffErcompl. Qed.	Lemma	diffKI	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",...	[ "bot", "diffErcompl", "join0x", "le0x", "rcomplKI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffIK x y : (x `\` y) `&` y = \bot.	Proof. by rewrite meetC diffKI. Qed.	Lemma	diffIK	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",...	[ "bot", "diffKI", "meetC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetIB z x y : (z `&` y) `&` (x `\` y) = \bot.	Proof. by rewrite -meetA diffKI meetx0. Qed.	Lemma	meetIB	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",...	[ "bot", "diffKI", "meetA", "meetx0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetBI z x y : (x `\` y) `&` (z `&` y) = \bot.	Proof. by rewrite meetC meetIB. Qed.	Lemma	meetBI	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",...	[ "bot", "meetC", "meetIB" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinIB y x : (x `&` y) `\|` (x `\` y) = x.	Proof. by rewrite diffErcompl rcomplKU. Qed.	Lemma	joinIB	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",...	[ "diffErcompl", "rcomplKU" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinBI y x : (x `\` y) `\|` (x `&` y) = x.	Proof. by rewrite joinC joinIB. Qed.	Lemma	joinBI	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",...	[ "joinC", "joinIB" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinIBC y x : (y `&` x) `\|` (x `\` y) = x.	Proof. by rewrite meetC joinIB. Qed.	Lemma	joinIBC	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",...	[ "joinIB", "meetC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinBIC y x : (x `\` y) `\|` (y `&` x) = x.	Proof. by rewrite meetC joinBI. Qed.	Lemma	joinBIC	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",...	[ "joinBI", "meetC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leBx x y : x `\` y <= x.	Proof. by rewrite -[leRHS](joinIB y) leUr. Qed.	Lemma	leBx	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",...	[ "joinIB", "leRHS", "leUr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffxx x : x `\` x = \bot.	Proof. by have := diffKI x x; rewrite meet_r. Qed.	Lemma	diffxx	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",...	[ "bot", "diffKI", "meet_r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leBl z x y : x <= y -> x `\` z <= y `\` z.	Proof. rewrite -[leLHS](joinIB z) -[leRHS](joinIB z). by rewrite leU2E ?meetIB ?meetBI // => /andP []. Qed.	Lemma	leBl	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",...	[ "joinIB", "leLHS", "leRHS", "leU2E", "meetBI", "meetIB" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffKU y x : y `\|` (x `\` y) = y `\|` x.	Proof. apply/eqP; rewrite eq_le leU2 //= leUx leUl. by apply/meet_idPl; have := joinIB y x; rewrite joinIl join_l. Qed.	Lemma	diffKU	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_le", "joinIB", "joinIl", "join_l", "leU2", "leUl", "leUx", "meet_idPl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffUK y x : (x `\` y) `\|` y = x `\|` y.	Proof. by rewrite joinC diffKU joinC. Qed.	Lemma	diffUK	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",...	[ "diffKU", "joinC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leBKU y x : y <= x -> y `\|` (x `\` y) = x.	Proof. by move=> /join_r {2}<-; rewrite diffKU. Qed.	Lemma	leBKU	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",...	[ "diffKU", "join_r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_bigmin_ord_cond [x0] n m (P P' : pred nat) (F : nat -> T) : (m <= n)%N -> (forall i : 'I_m, P' i -> P i) -> \big[min/x0]_(i < n | P i) F i <= \big[min/x0]_(i < m | P' i) F i.

Proof. exact: (le_big_ord_cond ge_refl). Qed.

Lemma

le_bigmin_ord_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_ord_cond", "min", "nat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_bigmax_ord_cond [x0] n m (P P' : {pred nat}) (F : nat -> T) : (n <= m)%N -> (forall i : 'I_n, P i -> P' i) -> \big[max/x0]_(i < n | P i) F i <= \big[max/x0]_(i < m | P' i) F i.

Proof. exact: le_big_ord_cond. Qed.

Lemma

le_bigmax_ord_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_ord_cond", "max", "nat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_bigmin [x0] [I : finType] [A A' P : {pred I}] (F : I -> T) : A' \subset A -> \big[min/x0]_(i in A | P i) F i <= \big[min/x0]_(i in A' | P i) F i.

Proof. exact: (subset_le_big ge_refl). Qed.

Lemma

subset_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",...

[ "A'", "ge_refl", "min", "subset_le_big" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_bigmax [x0] [I : finType] (A A' P : {pred I}) (F : I -> T) : A \subset A' -> \big[max/x0]_(i in A | P i) F i <= \big[max/x0]_(i in A' | P i) F i.

Proof. exact: subset_le_big. Qed.

Lemma

subset_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",...

[ "A'", "max", "subset_le_big" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_bigmin_cond [x0] (I : finType) (A A' P P' : {pred I}) (F : I -> T) : [set i in A' | P' i] \subset [set i in A | P i] -> \big[min/x0]_(i in A | P i) F i <= \big[min/x0]_(i in A' | P' i) F i.

Proof. exact: (subset_le_big_cond ge_refl). Qed.

Lemma

subset_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",...

[ "A'", "ge_refl", "min", "subset_le_big_cond" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_bigmax_cond [x0] (I : finType) (A A' P P' : {pred I}) (F : I -> T) : [set i in A | P i] \subset [set i in A' | P' i] -> \big[max/x0]_(i in A | P i) F i <= \big[max/x0]_(i in A' | P' i) F i.

Proof. exact: subset_le_big_cond. Qed.

Lemma

subset_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",...

[ "A'", "max", "subset_le_big_cond" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmin_le_id P F : \big[min/x]_(i <- r | P i) F i <= x.

Proof. by rewrite bigmin_idl. Qed.

Lemma

bigmin_le_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",...

[ "bigmin_idl", "min" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmax_ge_id P F : \big[max/x]_(i <- r | P i) F i >= x.

Proof. by rewrite bigmax_idl. Qed.

Lemma

bigmax_ge_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",...

[ "bigmax_idl", "max" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmin_eq_id P F : (forall i, P i -> x <= F i) -> \big[min/x]_(i <- r | P i) F i = x.

Proof. by move=> x_le; apply: le_anti; rewrite bigmin_le_id le_bigmin. Qed.

Lemma

bigmin_eq_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",...

[ "apply", "bigmin_le_id", "le_anti", "le_bigmin", "min" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmax_eq_id P F : (forall i, P i -> x >= F i) -> \big[max/x]_(i <- r | P i) F i = x.

Proof. by move=> x_ge; apply: le_anti; rewrite bigmax_ge_id bigmax_le. Qed.

Lemma

bigmax_eq_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",...

[ "apply", "bigmax_ge_id", "bigmax_le", "le_anti", "max" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ge_bigmin_seq i0 P F : i0 \in r -> P i0 -> \big[min/x]_(i <- r | P i) F i <= F i0.

Proof. move=> + Pi0; elim: r => // h t ih; rewrite inE big_cons. move=> /predU1P[<-|i0t]; first by rewrite Pi0 ge_min// lexx. by case: ifPn => Ph; [rewrite ge_min ih// orbT|rewrite ih]. Qed.

Lemma

ge_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",...

[ "Pi0", "big_cons", "ge_min", "i0", "inE", "lexx", "min", "predU1P" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_bigmax_seq i0 P F : i0 \in r -> P i0 -> F i0 <= \big[max/x]_(i <- r | P i) F i.

Proof. move=> + Pi0; elim: r => // h t ih; rewrite inE big_cons. move=> /predU1P[<-|i0t]; first by rewrite Pi0 le_max// lexx. by case: ifPn => Ph; [rewrite le_max ih// orbT|rewrite ih]. Qed.

Lemma

le_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",...

[ "Pi0", "big_cons", "i0", "inE", "le_max", "lexx", "max", "predU1P" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmin_inf_seq i0 P t F : i0 \in r -> P i0 -> F i0 <= t -> \big[min/x]_(i <- r | P i) F i <= t.

Proof. by move=> ? ? ?; exact: le_trans (@ge_bigmin_seq i0 _ _ _ _) _. Qed.

Lemma

bigmin_inf_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_bigmin_seq", "i0", "le_trans", "min" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmax_sup_seq i0 P t F : i0 \in r -> P i0 -> t <= F i0 -> t <= \big[max/x]_(i <- r | P i) F i.

Proof. by move=> ? ? ?; exact: le_trans (@le_bigmax_seq i0 _ _ _ _). Qed.

Lemma

bigmax_sup_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",...

[ "i0", "le_bigmax_seq", "le_trans", "max" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigminD1 j P F : P j -> \big[min/x]_(i | P i) F i = min (F j) (\big[min/x]_(i | P i && (i != j)) F i).

Proof. by move/(bigD1 _) ->. Qed.

Lemma

bigminD1

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",...

[ "bigD1", "min" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmaxD1 j P F : P j -> \big[max/x]_(i | P i) F i = max (F j) (\big[max/x]_(i | P i && (i != j)) F i).

Proof. by move/(bigD1 _) ->. Qed.

Lemma

bigmaxD1

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",...

[ "bigD1", "max" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmin_le_cond j P F : P j -> \big[min/x]_(i | P i) F i <= F j.

Proof. have := mem_index_enum j; rewrite unlock; elim: (index_enum I) => //= i l ih. rewrite inE => /orP [/eqP-> ->|/ih leminlfi Pi]; first by rewrite ge_min lexx. by case: ifPn => Pj; [rewrite ge_min leminlfi// orbC|exact: leminlfi]. Qed.

Lemma

bigmin_le_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_min", "inE", "index_enum", "lexx", "mem_index_enum", "min" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_bigmax_cond j P F : P j -> F j <= \big[max/x]_(i | P i) F i.

Proof. by move=> Pj; rewrite (bigmaxD1 _ Pj) le_max lexx. Qed.

Lemma

le_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",...

[ "bigmaxD1", "le_max", "lexx", "max" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmin_le j F : \big[min/x]_i F i <= F j.

Proof. exact: bigmin_le_cond. Qed.

Lemma

bigmin_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",...

[ "bigmin_le_cond", "min" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_bigmax F j : F j <= \big[max/x]_i F i.

Proof. exact: le_bigmax_cond. Qed.

Lemma

le_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",...

[ "le_bigmax_cond", "max" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmin_inf j P m F : P j -> F j <= m -> \big[min/x]_(i | P i) F i <= m.

Proof. by move=> Pj ?; apply: le_trans (bigmin_le_cond _ Pj) _. Qed.

Lemma

bigmin_inf

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_le_cond", "le_trans", "min" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmax_sup j P m F : P j -> m <= F j -> m <= \big[max/x]_(i | P i) F i.

Proof. by move=> Pj ?; apply: le_trans (le_bigmax_cond _ Pj). Qed.

Lemma

bigmax_sup

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_bigmax_cond", "le_trans", "max" ]

NB: as of [2022-08-02], bigop.bigmax_sup already exists for nat

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmin_geP m P F : reflect (m <= x /\ forall i, P i -> m <= F i) (m <= \big[min/x]_(i | P i) F i).

Proof. apply: (iffP idP) => [lemFi|[lemx lemPi]]; [split|exact: le_bigmin]. - by rewrite (le_trans lemFi)// bigmin_idl ge_min lexx. - by move=> i Pi; rewrite (le_trans lemFi)// (bigminD1 _ Pi)// le_minl lexx. Qed.

Lemma

bigmin_geP

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", "bigminD1", "bigmin_idl", "ge_min", "le_bigmin", "le_trans", "lexx", "min", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmax_leP m P F : reflect (x <= m /\ forall i, P i -> F i <= m) (\big[max/x]_(i | P i) F i <= m).

Proof. apply: (iffP idP) => [|[? ?]]; last exact: bigmax_le. rewrite bigmax_idl ge_max => /andP[-> leFm]; split=> // i Pi. by apply: le_trans leFm; exact: le_bigmax_cond. Qed.

Lemma

bigmax_leP

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_idl", "bigmax_le", "ge_max", "last", "le_bigmax_cond", "le_trans", "max", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmin_gtP m P F : reflect (m < x /\ forall i, P i -> m < F i) (m < \big[min/x]_(i | P i) F i).

Proof. apply: (iffP idP) => [lemFi|[lemx lemPi]]; [split|exact: lt_bigmin]. - by rewrite (lt_le_trans lemFi)// bigmin_idl ge_min lexx. - by move=> i Pi; rewrite (lt_le_trans lemFi)// (bigminD1 _ Pi)// le_minl lexx. Qed.

Lemma

bigmin_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", "bigminD1", "bigmin_idl", "ge_min", "lexx", "lt_bigmin", "lt_le_trans", "min", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmax_ltP m P F : reflect (x < m /\ forall i, P i -> F i < m) (\big[max/x]_(i | P i) F i < m).

Proof. apply: (iffP idP) => [|[? ?]]; last exact: bigmax_lt. rewrite bigmax_idl gt_max => /andP[-> ltFm]; split=> // i Pi. by apply: le_lt_trans ltFm; exact: le_bigmax_cond. Qed.

Lemma

bigmax_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", "bigmax_idl", "bigmax_lt", "gt_max", "last", "le_bigmax_cond", "le_lt_trans", "max", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmin_eq_arg j P F : P j -> (forall i, P i -> F i <= x) -> \big[min/x]_(i | P i) F i = F [arg min_(i < j | P i) F i].

Proof. move=> Pi0; case: arg_minP => //= i Pi PF PFx. apply/eqP; rewrite eq_le bigmin_le_cond //=. by apply/bigmin_geP; split => //; exact: PFx. Qed.

Lemma

bigmin_eq_arg

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",...

[ "Pi0", "apply", "arg_minP", "bigmin_geP", "bigmin_le_cond", "eq_le", "min", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmax_eq_arg j P F : P j -> (forall i, P i -> x <= F i) -> \big[max/x]_(i | P i) F i = F [arg max_(i > j | P i) F i].

Proof. move=> Pi0; case: arg_maxP => //= i Pi PF PxF. apply/eqP; rewrite eq_le le_bigmax_cond // andbT. by apply/bigmax_leP; split => //; exact: PxF. Qed.

Lemma

bigmax_eq_arg

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",...

[ "Pi0", "apply", "arg_maxP", "bigmax_leP", "eq_le", "le_bigmax_cond", "max", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_bigmin j P F : P j -> (forall i, P i -> F i <= x) -> {i0 | i0 \in P & \big[min/x]_(i | P i) F i = F i0}.

Proof. by move=> Pi0 Hx; rewrite (bigmin_eq_arg Pi0) //; eexists=> //; case: arg_minP. Qed.

Lemma

eq_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",...

[ "Pi0", "arg_minP", "bigmin_eq_arg", "i0", "min" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_bigmax j P F : P j -> (forall i, P i -> x <= F i) -> {i0 | i0 \in P & \big[max/x]_(i | P i) F i = F i0}.

Proof. by move=> Pi0 Hx; rewrite (bigmax_eq_arg Pi0) //; eexists=> //; case: arg_maxP. Qed.

Lemma

eq_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",...

[ "Pi0", "arg_maxP", "bigmax_eq_arg", "i0", "max" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_bigmin2 P F1 F2 : (forall i, P i -> F1 i <= F2 i) -> \big[min/x]_(i | P i) F1 i <= \big[min/x]_(i | P i) F2 i.

Proof. move=> FG; elim/big_ind2 : _ => // a b e f ba fe. rewrite ge_min 2!le_min ba fe /= andbT. move: (le_total a e) => /orP[/(le_trans ba)-> // | /(le_trans fe)->]. by rewrite orbT. Qed.

Lemma

le_bigmin2

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_ind2", "ge_min", "le_min", "le_total", "le_trans", "min" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_bigmax2 P F1 F2 : (forall i, P i -> F1 i <= F2 i) -> \big[max/x]_(i | P i) F1 i <= \big[max/x]_(i | P i) F2 i.

Proof. move=> FG; elim/big_ind2 : _ => // a b e f ba fe. rewrite le_max 2!ge_max ba fe /= andbT; have [//|/= af] := leP f a. by rewrite (le_trans ba) // (le_trans _ fe) // ltW. Qed.

Lemma

le_bigmax2

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_ind2", "ge_max", "leP", "le_max", "le_trans", "ltW", "max" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmaxUl (A B : {set I}) F : \big[max/x]_(i in A) F i <= \big[max/x]_(i in A :|: B) F i.

Proof. by apply: sub_bigmax => t; rewrite in_setU => ->. Qed.

Lemma

bigmaxUl

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", "in_setU", "max", "sub_bigmax" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmaxUr (A B : {set I}) F : \big[max/x]_(i in B) F i <= \big[max/x]_(i in A :|: B) F i.

Proof. by under [leRHS]eq_bigl do rewrite setUC; apply: bigmaxUl. Qed.

Lemma

bigmaxUr

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", "bigmaxUl", "eq_bigl", "leRHS", "max", "setUC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigminUl (A B : {set I}) F : \big[min/x]_(i in A) F i >= \big[min/x]_(i in A :|: B) F i.

Proof. by apply: sub_bigmin => t; rewrite in_setU => ->. Qed.

Lemma

bigminUl

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", "in_setU", "min", "sub_bigmin" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigminUr (A B : {set I}) F : \big[min/x]_(i in B) F i >= \big[min/x]_(i in A :|: B) F i.

Proof. by under [leLHS]eq_bigl do rewrite setUC; apply: bigminUl. Qed.

Lemma

bigminUr

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", "bigminUl", "eq_bigl", "leLHS", "min", "setUC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmaxIl (A B : {set I}) F : \big[max/x]_(i in A) F i >= \big[max/x]_(i in A :&: B) F i.

Proof. by apply: sub_bigmax => t; rewrite in_setI => /andP[-> _]. Qed.

Lemma

bigmaxIl

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", "in_setI", "max", "sub_bigmax" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmaxIr (A B : {set I}) F : \big[max/x]_(i in B) F i >= \big[max/x]_(i in A :&: B) F i.

Proof. by under eq_bigl do rewrite setIC; apply: bigmaxIl. Qed.

Lemma

bigmaxIr

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", "bigmaxIl", "eq_bigl", "max", "setIC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigminIl (A B : {set I}) F : \big[min/x]_(i in A) F i <= \big[min/x]_(i in A :&: B) F i.

Proof. by apply: sub_bigmin => t; rewrite in_setI => /andP[->_]. Qed.

Lemma

bigminIl

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", "in_setI", "min", "sub_bigmin" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigminIr (A B : {set I}) F : \big[min/x]_(i in B) F i <= \big[min/x]_(i in A :&: B) F i.

Proof. by under [leRHS]eq_bigl do rewrite setIC; apply: bigminIl. Qed.

Lemma

bigminIr

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", "bigminIl", "eq_bigl", "leRHS", "min", "setIC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmaxD (A B : {set I}) F : \big[max/x]_(i in B) F i >= \big[max/x]_(i in B :\: A) F i.

Proof. by apply: sub_bigmax => t; rewrite in_setD => /andP[_->]. Qed.

Lemma

bigmaxD

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", "in_setD", "max", "sub_bigmax" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigminD (A B : {set I}) F : \big[min/x]_(i in B) F i <= \big[min/x]_(i in B :\: A) F i.

Proof. by apply: sub_bigmin => t; rewrite in_setD => /andP[_->]. Qed.

Lemma

bigminD

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", "in_setD", "min", "sub_bigmin" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmaxU (A B : {set I}) F : \big[max/x]_(i in A :|: B) F i = max (\big[max/x]_(i in A) F i) (\big[max/x]_(i in B) F i).

Proof. apply: le_anti; rewrite ge_max bigmaxUl bigmaxUr !andbT; apply/bigmax_leP. split=> [|i /[!in_setU]/orP[iA|iB]]; first by rewrite le_max bigmax_ge_id. - by rewrite le_max le_bigmax_cond. - by rewrite le_max orbC le_bigmax_cond. Qed.

Lemma

bigmaxU

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", "bigmaxUl", "bigmaxUr", "bigmax_ge_id", "bigmax_leP", "ge_max", "in_setU", "le_anti", "le_bigmax_cond", "le_max", "max", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigminU (A B : {set I}) F : \big[min/x]_(i in A :|: B) F i = min (\big[min/x]_(i in A) F i) (\big[min/x]_(i in B) F i).

Proof. apply: le_anti; rewrite le_min bigminUl bigminUr !andbT; apply/bigmin_geP. split=> [|i /[!in_setU]/orP[iA|iB]]; first by rewrite ge_min bigmin_le_id. - by rewrite ge_min bigmin_le_cond. - by rewrite ge_min orbC bigmin_le_cond. Qed.

Lemma

bigminU

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", "bigminUl", "bigminUr", "bigmin_geP", "bigmin_le_cond", "bigmin_le_id", "ge_min", "in_setU", "le_anti", "le_min", "min", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmin_set1 j F : \big[min/x]_(i in [set j]) F i = min (F j) x.

Proof. exact: big_set1E. Qed.

Lemma

bigmin_set1

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_set1E", "min" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmax_set1 j F : \big[max/x]_(i in [set j]) F i = max (F j) x.

Proof. exact: big_set1E. Qed.

Lemma

bigmax_set1

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_set1E", "max" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmin_imset [I J : finType] x [h : I -> J] [A : {set I}] (F : J -> T) : \big[min/x]_(j in [set h x | x in A]) F j = \big[min/x]_(i in A) F (h i).

Proof. by apply: big_imset_idem; apply: minxx. Qed.

Lemma

bigmin_imset

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", "big_imset_idem", "min", "minxx" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmax_imset [I J : finType] x [h : I -> J] [A : {set I}] (F : J -> T) : \big[max/x]_(j in [set h x | x in A]) F j = \big[max/x]_(i in A) F (h i).

Proof. by apply: big_imset_idem; apply: maxxx. Qed.

Lemma

bigmax_imset

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", "big_imset_idem", "max", "maxxx" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sorted_filter_gt x s : sorted <=%O s -> [seq y <- s | x < y] = drop (count (<= x) s) s.

Proof. move=> s_sorted; rewrite count_le_gt -[LHS]revK -filter_rev. rewrite (@sorted_filter_lt _ T^d); last by rewrite take_rev revK count_rev. by rewrite rev_sorted. Qed.

Lemma

sorted_filter_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",...

[ "count", "count_le_gt", "count_rev", "drop", "filter_rev", "last", "revK", "rev_sorted", "seq", "sorted", "sorted_filter_lt", "take_rev" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sorted_filter_ge x s : sorted <=%O s -> [seq y <- s | x <= y] = drop (count (< x) s) s.

Proof. move=> s_sorted; rewrite count_lt_ge -[LHS]revK -filter_rev. rewrite (@sorted_filter_le _ T^d); last by rewrite take_rev revK count_rev. by rewrite rev_sorted. Qed.

Lemma

sorted_filter_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",...

[ "count", "count_lt_ge", "count_rev", "drop", "filter_rev", "last", "revK", "rev_sorted", "seq", "sorted", "sorted_filter_le", "take_rev" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nth_count_ge x x0 s i : sorted <=%O s -> (count (< x) s <= i < size s)%N -> x <= nth x0 s i.

Proof. move=> ss /andP[ige ilt]; rewrite -(subnKC ige) -nth_drop -sorted_filter_ge //. apply/(all_nthP _ (filter_all _ _)). by rewrite size_filter ltn_subLR // count_lt_ge subnK // count_size. Qed.

Lemma

nth_count_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",...

[ "all_nthP", "apply", "count", "count_lt_ge", "count_size", "filter_all", "ltn_subLR", "nth", "nth_drop", "size", "size_filter", "sorted", "sorted_filter_ge", "subnK", "subnKC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nth_count_gt x x0 s i : sorted <=%O s -> (count (<= x) s <= i < size s)%N -> x < nth x0 s i.

Proof. move=> ss /andP[ige ilt]; rewrite -(subnKC ige) -nth_drop -sorted_filter_gt //. apply/(all_nthP _ (filter_all _ _)). by rewrite size_filter ltn_subLR // count_le_gt subnK // count_size. Qed.

Lemma

nth_count_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",...

[ "all_nthP", "apply", "count", "count_le_gt", "count_size", "filter_all", "ltn_subLR", "nth", "nth_drop", "size", "size_filter", "sorted", "sorted_filter_gt", "subnK", "subnKC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nth_count_eq x x0 s i : sorted <=%O s -> (count (< x) s <= i < count (<= x) s)%N -> nth x0 s i = x.

Proof. move=> ss /andP[ige ilt]; apply/le_anti. by rewrite nth_count_le// nth_count_ge// ige (leq_trans ilt (count_size _ _)). Qed.

Lemma

nth_count_eq

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", "count", "count_size", "le_anti", "leq_trans", "nth", "nth_count_ge", "nth_count_le", "sorted" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contraTle b z t : (t < z -> ~~ b) -> (b -> z <= t).

Proof. exact: comparable_contraTle. Qed.

Lemma

contraTle

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_contraTle" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contraTlt b z t : (t <= z -> ~~ b) -> (b -> z < t).

Proof. exact: comparable_contraTlt. Qed.

Lemma

contraTlt

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_contraTlt" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contraPle P z t : (t < z -> ~ P) -> (P -> z <= t).

Proof. exact: comparable_contraPle. Qed.

Lemma

contraPle

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_contraPle" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contraPlt P z t : (t <= z -> ~ P) -> (P -> z < t).

Proof. exact: comparable_contraPlt. Qed.

Lemma

contraPlt

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_contraPlt" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contraNle b z t : (t < z -> b) -> (~~ b -> z <= t).

Proof. exact: comparable_contraNle. Qed.

Lemma

contraNle

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_contraNle" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contraNlt b z t : (t <= z -> b) -> (~~ b -> z < t).

Proof. exact: comparable_contraNlt. Qed.

Lemma

contraNlt

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_contraNlt" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_not_le P z t : (t < z -> P) -> (~ P -> z <= t).

Proof. exact: comparable_contra_not_le. Qed.

Lemma

contra_not_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",...

[ "comparable_contra_not_le" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_not_lt P z t : (t <= z -> P) -> (~ P -> z < t).

Proof. exact: comparable_contra_not_lt. Qed.

Lemma

contra_not_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",...

[ "comparable_contra_not_lt" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contraFle b z t : (t < z -> b) -> (b = false -> z <= t).

Proof. exact: comparable_contraFle. Qed.

Lemma

contraFle

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_contraFle" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contraFlt b z t : (t <= z -> b) -> (b = false -> z < t).

Proof. exact: comparable_contraFlt. Qed.

Lemma

contraFlt

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_contraFlt" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_leq_le m n z t : (t < z -> (n < m)%N) -> ((m <= n)%N -> z <= t).

Proof. exact: comparable_contra_leq_le. Qed.

Lemma

contra_leq_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",...

[ "comparable_contra_leq_le" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_leq_lt m n z t : (t <= z -> (n < m)%N) -> ((m <= n)%N -> z < t).

Proof. exact: comparable_contra_leq_lt. Qed.

Lemma

contra_leq_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",...

[ "comparable_contra_leq_lt" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_ltn_le m n z t : (t < z -> (n <= m)%N) -> ((m < n)%N -> z <= t).

Proof. exact: comparable_contra_ltn_le. Qed.

Lemma

contra_ltn_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",...

[ "comparable_contra_ltn_le" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_ltn_lt m n z t : (t <= z -> (n <= m)%N) -> ((m < n)%N -> z < t).

Proof. exact: comparable_contra_ltn_lt. Qed.

Lemma

contra_ltn_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",...

[ "comparable_contra_ltn_lt" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_le x y z t : (t < z -> y < x) -> (x <= y -> z <= t).

Proof. exact: comparable_contra_le. Qed.

Lemma

contra_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",...

[ "comparable_contra_le" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_le_lt x y z t : (t <= z -> y < x) -> (x <= y -> z < t).

Proof. exact: comparable_contra_le_lt. Qed.

Lemma

contra_le_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",...

[ "comparable_contra_le_lt" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_lt_le x y z t : (t < z -> y <= x) -> (x < y -> z <= t).

Proof. exact: comparable_contra_lt_le. Qed.

Lemma

contra_lt_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",...

[ "comparable_contra_lt_le" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_lt x y z t : (t <= z -> y <= x) -> (x < y -> z < t).

Proof. exact: comparable_contra_lt. Qed.

Lemma

contra_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",...

[ "comparable_contra_lt" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leT_anti

:= @le_anti _ T.

Let

leT_anti

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_anti" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leT'_anti

:= @le_anti _ T'.

Let

leT'_anti

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",...

[ "T'", "le_anti" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltT_neqAle

:= @lt_neqAle _ T.

Let

ltT_neqAle

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",...

[ "lt_neqAle" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltT'_neqAle

:= @lt_neqAle _ T'.

Let

ltT'_neqAle

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",...

[ "T'", "lt_neqAle" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltT_def

:= @lt_def _ T.

Let

ltT_def

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",...

[ "lt_def" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leT_total

:= @le_total _ T.

Let

leT_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",...

[ "le_total" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_mono : {homo f : x y / x < y} -> {mono f : x y / x <= y}.

Proof. exact: total_homo_mono. Qed.

Lemma

le_mono

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",...

[ "total_homo_mono" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_nmono : {homo f : x y /~ x < y} -> {mono f : x y /~ x <= y}.

Proof. by apply: total_homo_mono => // x y; rewrite eq_sym. Qed.

Lemma

le_nmono

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_sym", "total_homo_mono" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_mono_in : {in D &, {homo f : x y / x < y}} -> {in D &, {mono f : x y / x <= y}}.

Proof. exact: total_homo_mono_in. Qed.

Lemma

le_mono_in

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",...

[ "total_homo_mono_in" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_nmono_in : {in D &, {homo f : x y /~ x < y}} -> {in D &, {mono f : x y /~ x <= y}}.

Proof. by apply: total_homo_mono_in => // x y; rewrite eq_sym. Qed.

Lemma

le_nmono_in

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_sym", "total_homo_mono_in" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcomplPmeet x y z : ((x `&` y) `|` z) `&` rcompl x y z = x `&` y.

Proof. exact: rcomplPmeet. Qed.

Lemma

rcomplPmeet

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",...

[ "rcompl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcomplPjoin x y z : ((y `|` x) `&` z) `|` rcompl x y z = y `|` x.

Proof. exact: rcomplPjoin. Qed.

Lemma

rcomplPjoin

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",...

[ "rcompl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcomplKI x y z : x <= y -> (x `|` z) `&` rcompl x y z = x.

Proof. by move=> lexy; have := rcomplPmeet x y z; rewrite (meet_l lexy). Qed.

Lemma

rcomplKI

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",...

[ "meet_l", "rcompl", "rcomplPmeet" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcomplKU x y z : x <= y -> (y `&` z) `|` rcompl x y z = y.

Proof. by move=> lexy; have := rcomplPjoin x y z; rewrite (join_l lexy). Qed.

Lemma

rcomplKU

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_l", "rcompl", "rcomplPjoin" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffErcompl x y : x `\` y = rcompl \bot x y.

Proof. exact: diffErcompl. Qed.

Lemma

diffErcompl

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",...

[ "bot", "rcompl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffKI x y : y `&` (x `\` y) = \bot.

Proof. by have := rcomplKI y (le0x x); rewrite join0x diffErcompl. Qed.

Lemma

diffKI

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",...

[ "bot", "diffErcompl", "join0x", "le0x", "rcomplKI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffIK x y : (x `\` y) `&` y = \bot.

Proof. by rewrite meetC diffKI. Qed.

Lemma

diffIK

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",...

[ "bot", "diffKI", "meetC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetIB z x y : (z `&` y) `&` (x `\` y) = \bot.

Proof. by rewrite -meetA diffKI meetx0. Qed.

Lemma

meetIB

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",...

[ "bot", "diffKI", "meetA", "meetx0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetBI z x y : (x `\` y) `&` (z `&` y) = \bot.

Proof. by rewrite meetC meetIB. Qed.

Lemma

meetBI

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",...

[ "bot", "meetC", "meetIB" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinIB y x : (x `&` y) `|` (x `\` y) = x.

Proof. by rewrite diffErcompl rcomplKU. Qed.

Lemma

joinIB

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",...

[ "diffErcompl", "rcomplKU" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinBI y x : (x `\` y) `|` (x `&` y) = x.

Proof. by rewrite joinC joinIB. Qed.

Lemma

joinBI

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",...

[ "joinC", "joinIB" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinIBC y x : (y `&` x) `|` (x `\` y) = x.

Proof. by rewrite meetC joinIB. Qed.

Lemma

joinIBC

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",...

[ "joinIB", "meetC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinBIC y x : (x `\` y) `|` (y `&` x) = x.

Proof. by rewrite meetC joinBI. Qed.

Lemma

joinBIC

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",...

[ "joinBI", "meetC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leBx x y : x `\` y <= x.

Proof. by rewrite -[leRHS](joinIB y) leUr. Qed.

Lemma

leBx

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",...

[ "joinIB", "leRHS", "leUr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffxx x : x `\` x = \bot.

Proof. by have := diffKI x x; rewrite meet_r. Qed.

Lemma

diffxx

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",...

[ "bot", "diffKI", "meet_r" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leBl z x y : x <= y -> x `\` z <= y `\` z.

Proof. rewrite -[leLHS](joinIB z) -[leRHS](joinIB z). by rewrite leU2E ?meetIB ?meetBI // => /andP []. Qed.

Lemma

leBl

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",...

[ "joinIB", "leLHS", "leRHS", "leU2E", "meetBI", "meetIB" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffKU y x : y `|` (x `\` y) = y `|` x.

Proof. apply/eqP; rewrite eq_le leU2 //= leUx leUl. by apply/meet_idPl; have := joinIB y x; rewrite joinIl join_l. Qed.

Lemma

diffKU

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_le", "joinIB", "joinIl", "join_l", "leU2", "leUl", "leUx", "meet_idPl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffUK y x : (x `\` y) `|` y = x `|` y.

Proof. by rewrite joinC diffKU joinC. Qed.

Lemma

diffUK

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",...

[ "diffKU", "joinC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leBKU y x : y <= x -> y `|` (x `\` y) = x.

Proof. by move=> /join_r {2}<-; rewrite diffKU. Qed.

Lemma

leBKU

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",...

[ "diffKU", "join_r" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d