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
enum1 x : enum (pred1 x) = [:: x].	Proof. rewrite [enum _](all_pred1P x _ _); last by rewrite size_filter enumP. by apply/allP=> y; rewrite mem_enum. Qed.	Lemma	enum1	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "allP", "all_pred1P", "apply", "enum", "enumP", "last", "mem_enum", "pred1", "size_filter" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pick_spec : option T -> Type	:= \| Pick x of P x : pick_spec (Some x) \| Nopick of P =1 xpred0 : pick_spec None.	Variant	pick_spec	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "Pick" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pickP : pick_spec (pick P).	Proof. rewrite /pick; case: (enum _) (mem_enum P) => [\|x s] Pxs /=. by right; apply: fsym. by left; rewrite -[P _]Pxs mem_head. Qed.	Lemma	pickP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "enum", "mem_enum", "mem_head", "pick", "pick_spec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_enum A B : A =i B -> enum A = enum B.	Proof. by move=> eqAB; apply: eq_filter. Qed.	Lemma	eq_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "enum", "eq_filter" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_pick P Q : P =1 Q -> pick P = pick Q.	Proof. by move=> eqPQ; rewrite /pick (eq_enum eqPQ). Qed.	Lemma	eq_pick	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "eq_enum", "pick" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardE A : #\|A\| = size (enum A).	Proof. by rewrite unlock. Qed.	Lemma	cardE	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "enum", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_card A B : A =i B -> #\|A\| = #\|B\|.	Proof. by move=> eqAB; rewrite !cardE (eq_enum eqAB). Qed.	Lemma	eq_card	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cardE", "eq_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_card_trans A B n : #\|A\| = n -> B =i A -> #\|B\| = n.	Proof. by move <-; apply: eq_card. Qed.	Lemma	eq_card_trans	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "eq_card" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card0 : #\|@pred0 T\| = 0.	Proof. by rewrite cardE enum0. Qed.	Lemma	card0	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cardE", "enum0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardT : #\|T\| = size (enum T).	Proof. by rewrite cardE. Qed.	Lemma	cardT	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cardE", "enum", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card1 x : #\|pred1 x\| = 1.	Proof. by rewrite cardE enum1. Qed.	Lemma	card1	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cardE", "enum1", "pred1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_card0 A : A =i pred0 -> #\|A\| = 0.	Proof. exact: eq_card_trans card0. Qed.	Lemma	eq_card0	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "card0", "eq_card_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_cardT A : A =i predT -> #\|A\| = size (enum T).	Proof. exact: eq_card_trans cardT. Qed.	Lemma	eq_cardT	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cardT", "enum", "eq_card_trans", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_card1 x A : A =i pred1 x -> #\|A\| = 1.	Proof. exact: eq_card_trans (card1 x). Qed.	Lemma	eq_card1	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "card1", "eq_card_trans", "pred1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardUI A B : #\|[predU A & B]\| + #\|[predI A & B]\| = #\|A\| + #\|B\|.	Proof. by rewrite !cardE !size_filter count_predUI. Qed.	Lemma	cardUI	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cardE", "count_predUI", "size_filter" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardID B A : #\|[predI A & B]\| + #\|[predD A & B]\| = #\|A\|.	Proof. rewrite -cardUI addnC [#\|predI _ _\|]eq_card0 => [x\|] /=. by rewrite !inE -!andbA andbC andbA andbN. by apply: eq_card => x; rewrite !inE andbC -andb_orl orbN. Qed.	Lemma	cardID	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "addnC", "apply", "cardUI", "eq_card", "eq_card0", "inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardC A : #\|A\| + #\|[predC A]\| = #\|T\|.	Proof. by rewrite !cardE !size_filter count_predC. Qed.	Lemma	cardC	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cardE", "count_predC", "size_filter" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardU1 x A : #\|[predU1 x & A]\| = (x \notin A) + #\|A\|.	Proof. case Ax: (x \in A). by apply: eq_card => y /[1!inE]/=; case: eqP => // ->. rewrite /= -(card1 x) -cardUI addnC. rewrite [#\|predI _ _\|]eq_card0 => [y /=\|]; last exact: eq_card. by rewrite !inE; case: eqP => // ->. Qed.	Lemma	cardU1	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "addnC", "apply", "card1", "cardUI", "eq_card", "eq_card0", "inE", "last", "predU1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card2 x y : #\|pred2 x y\| = (x != y).+1.	Proof. by rewrite cardU1 card1 addn1. Qed.	Lemma	card2	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "addn1", "card1", "cardU1", "pred2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardC1 x : #\|predC1 x\| = #\|T\|.-1.	Proof. by rewrite -(cardC (pred1 x)) card1. Qed.	Lemma	cardC1	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "card1", "cardC", "pred1", "predC1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardD1 x A : #\|A\| = (x \in A) + #\|[predD1 A & x]\|.	Proof. case Ax: (x \in A); last first. by apply: eq_card => y /[!inE]/=; case: eqP => // ->. rewrite /= -(card1 x) -cardUI addnC /=. rewrite [#\|predI _ _\|]eq_card0 => [y\|]; first by rewrite !inE; case: eqP. by apply: eq_card => y /[!inE]; case: eqP => // ->. Qed.	Lemma	cardD1	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "addnC", "apply", "card1", "cardUI", "eq_card", "eq_card0", "inE", "last", "predD1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_card A : #\|A\| <= #\|T\|.	Proof. by rewrite -(cardC A) leq_addr. Qed.	Lemma	max_card	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cardC", "leq_addr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_size s : #\|s\| <= size s.	Proof. elim: s => [\|x s IHs] /=; first by rewrite card0. by rewrite cardU1 /=; case: (~~ _) => //; apply: leqW. Qed.	Lemma	card_size	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "card0", "cardU1", "leqW", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_uniqP s : reflect (#\|s\| = size s) (uniq s).	Proof. elim: s => [\|x s IHs]; first by left; apply: card0. rewrite cardU1 /= /addn; case: {+}(x \in s) => /=. by right=> card_Ssz; have:= card_size s; rewrite card_Ssz ltnn. by apply: (iffP IHs) => [<-\| [<-]]. Qed.	Lemma	card_uniqP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "addn", "apply", "card0", "cardU1", "card_size", "ltnn", "size", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card0_eq A : #\|A\| = 0 -> A =i pred0.	Proof. by move=> A0 x; apply/idP => Ax; rewrite (cardD1 x) Ax in A0. Qed.	Lemma	card0_eq	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "cardD1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fintype0 : T -> #\|T\| <> 0.	Proof. by move=> x /card0_eq/(_ x). Qed.	Lemma	fintype0	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "card0_eq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pred0P P : reflect (P =1 pred0) (pred0b P).	Proof. by apply: (iffP eqP); [apply: card0_eq \| apply: eq_card0]. Qed.	Lemma	pred0P	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "card0_eq", "eq_card0", "pred0b" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pred0Pn P : reflect (exists x, P x) (~~ pred0b P).	Proof. case: (pickP P) => [x Px \| P0]. by rewrite (introN (pred0P P)) => [P0\|]; [rewrite P0 in Px \| left; exists x]. by rewrite -lt0n eq_card0 //; right=> [[x]]; rewrite P0. Qed.	Lemma	pred0Pn	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "P0", "Px", "eq_card0", "lt0n", "pickP", "pred0P", "pred0b" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_gt0P A : reflect (exists i, i \in A) (#\|A\| > 0).	Proof. by rewrite lt0n; apply: pred0Pn. Qed.	Lemma	card_gt0P	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "lt0n", "pred0Pn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_le1P {A} : reflect {in A, forall x, A =i pred1 x} (#\|A\| <= 1).	Proof. apply: (iffP idP) => [A1 x xA y\|]; last first. by have [/= x xA /(_ _ xA)/eq_card1->\|/eq_card0->//] := pickP [in A]. move: A1; rewrite (cardD1 x) xA ltnS leqn0 => /eqP/card0_eq/(_ y). by rewrite !inE; have [->\|]:= eqP. Qed.	Lemma	card_le1P	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "card0_eq", "cardD1", "eq_card0", "eq_card1", "inE", "last", "leqn0", "ltnS", "pickP", "pred1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_card1 A : #\|A\| = 1 -> {x \| A =i pred1 x}.	Proof. move=> A1; have /card_gt0P/sigW[x xA]: #\|A\| > 0 by rewrite A1. by exists x; apply/card_le1P; rewrite ?A1. Qed.	Lemma	mem_card1	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "card_gt0P", "card_le1P", "pred1", "sigW" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card1P A : reflect (exists x, A =i pred1 x) (#\|A\| == 1).	Proof. by apply: (iffP idP) => [/eqP/mem_card1[x inA]\|[x /eq_card1/eqP//]]; exists x. Qed.	Lemma	card1P	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "eq_card1", "inA", "mem_card1", "pred1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_le1_eqP A : reflect {in A &, forall x, all_equal_to x} (#\|A\| <= 1).	Proof. apply: (iffP card_le1P) => [Ale1 x y xA yA /=\|all_eq x xA y]. by apply/eqP; rewrite -[_ == _]/(y \in pred1 x) -Ale1. by rewrite inE; case: (altP (y =P x)) => [->//\|]; exact/contra_neqF/all_eq. Qed.	Lemma	card_le1_eqP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "card_le1P", "contra_neqF", "inE", "pred1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fintype_le1P : reflect (forall x : T, all_equal_to x) (#\|T\| <= 1).	Proof. apply: (iffP (card_le1_eqP {:T})); [exact: in2T \| exact: in2W]. Qed.	Lemma	fintype_le1P	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "card_le1_eqP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fintype1 : #\|T\| = 1 -> {x : T \| all_equal_to x}.	Proof. by move=> /mem_card1[x ex]; exists x => y; suff: y \in T by rewrite ex => /eqP. Qed.	Lemma	fintype1	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "ex", "mem_card1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fintype1P : reflect (exists x, all_equal_to x) (#\|T\| == 1).	Proof. apply: (iffP idP) => [/eqP/fintype1\|] [x eqx]; first by exists x. by apply/card1P; exists x => y; rewrite eqx !inE eqxx. Qed.	Lemma	fintype1P	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "card1P", "eqxx", "fintype1", "inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetE A B : (A \subset B) = pred0b [predD A & B].	Proof. by rewrite unlock. Qed.	Lemma	subsetE	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "pred0b" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetP A B : reflect {subset A <= B} (A \subset B).	Proof. rewrite unlock; apply: (iffP (pred0P _)) => [AB0 x \| sAB x /=]. by apply/implyP; apply/idPn; rewrite negb_imply andbC [_ && _]AB0. by rewrite andbC -negb_imply; apply/negbF/implyP; apply: sAB. Qed.	Lemma	subsetP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "pred0P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetPn A B : reflect (exists2 x, x \in A & x \notin B) (~~ (A \subset B)).	Proof. rewrite unlock; apply: (iffP (pred0Pn _)) => [[x] \| [x Ax nBx]]. by case/andP; exists x. by exists x; rewrite /= nBx. Qed.	Lemma	subsetPn	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "pred0Pn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_leq_card A B : A \subset B -> #\|A\| <= #\|B\|.	Proof. move=> sAB. rewrite -(cardID A B) [#\|predI _ _\|](@eq_card _ A) ?leq_addr //= => x. by rewrite !inE andbC; case Ax: (x \in A) => //; apply: subsetP Ax. Qed.	Lemma	subset_leq_card	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "cardID", "eq_card", "inE", "leq_addr", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subxx_hint (mA : mem_pred T) : subset mA mA.	Proof. by case: mA => A; have:= introT (subsetP A A); rewrite !unlock => ->. Qed.	Lemma	subxx_hint	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subxx (pT : predType T) (pA : pT) : pA \subset pA.	Proof. by []. Qed.	Lemma	subxx	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "pA" ]	The parametrization by predType makes it easier to apply subxx.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_subset A B : A =i B -> subset (mem A) =1 subset (mem B).	Proof. move=> eqAB [C]; rewrite !unlock; congr (_ == 0). by apply: eq_card => x; rewrite inE /= eqAB. Qed.	Lemma	eq_subset	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "eq_card", "inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_subset_r A B : A =i B -> (@subset T)^~ (mem A) =1 (@subset T)^~ (mem B).	Proof. move=> eqAB [C]; rewrite !unlock; congr (_ == 0). by apply: eq_card => x; rewrite !inE /= eqAB. Qed.	Lemma	eq_subset_r	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "eq_card", "inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_subxx A B : A =i B -> A \subset B.	Proof. by move/eq_subset->. Qed.	Lemma	eq_subxx	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "eq_subset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_predT A : A \subset T.	Proof. exact/subsetP. Qed.	Lemma	subset_predT	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
predT_subset A : T \subset A -> forall x, x \in A.	Proof. by move/subsetP=> allA x; apply: allA. Qed.	Lemma	predT_subset	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_pred1 A x : (pred1 x \subset A) = (x \in A).	Proof. by apply/subsetP/idP=> [-> // \| Ax y /eqP-> //]; apply: eqxx. Qed.	Lemma	subset_pred1	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "eqxx", "pred1", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_eqP A B : reflect (A =i B) ((A \subset B) && (B \subset A)).	Proof. apply: (iffP andP) => [[sAB sBA] x\| eqAB]; last by rewrite !eq_subxx. by apply/idP/idP; apply: subsetP. Qed.	Lemma	subset_eqP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "eq_subxx", "last", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_cardP A B : #\|A\| = #\|B\| -> reflect (A =i B) (A \subset B).	Proof. move=> eqcAB; case: (subsetP A B) (subset_eqP A B) => //= sAB. case: (subsetP B A) => [//\|[]] x Bx; apply/idPn => Ax. case/idP: (ltnn #\|A\|); rewrite {2}eqcAB (cardD1 x B) Bx /=. apply: subset_leq_card; apply/subsetP=> y Ay; rewrite inE /= andbC. by rewrite sAB //; apply/eqP => eqyx; rewrite -eqyx Ay in Ax. Qed.	Lemma	subset_cardP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "cardD1", "inE", "ltnn", "subsetP", "subset_eqP", "subset_leq_card" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_leqif_card A B : A \subset B -> #\|A\| <= #\|B\| ?= iff (B \subset A).	Proof. move=> sAB; split; [exact: subset_leq_card \| apply/eqP/idP]. by move/subset_cardP=> sABP; rewrite (eq_subset_r (sABP sAB)). by move=> sBA; apply: eq_card; apply/subset_eqP; rewrite sAB. Qed.	Lemma	subset_leqif_card	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "eq_card", "eq_subset_r", "split", "subset_cardP", "subset_eqP", "subset_leq_card" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_trans A B C : A \subset B -> B \subset C -> A \subset C.	Proof. by move/subsetP=> sAB /subsetP=> sBC; apply/subsetP=> x /sAB; apply: sBC. Qed.	Lemma	subset_trans	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_all s A : (s \subset A) = all [in A] s.	Proof. exact: (sameP (subsetP _ _) allP). Qed.	Lemma	subset_all	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "all", "allP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_cons s x : s \subset x :: s.	Proof. by apply/subsetP => y /[!inE] ->; rewrite orbT. Qed.	Lemma	subset_cons	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "inE", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_cons2 s1 s2 x : s1 \subset s2 -> x :: s1 \subset x :: s2.	Proof. by move=> ?; apply/subsetP => y /[!inE]; case: eqP => // _; apply: subsetP. Qed.	Lemma	subset_cons2	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "inE", "s1", "s2", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_catl s s' : s \subset s ++ s'.	Proof. by apply/subsetP=> x xins; rewrite mem_cat xins. Qed.	Lemma	subset_catl	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "mem_cat", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_catr s s' : s \subset s' ++ s.	Proof. by apply/subsetP => x xins; rewrite mem_cat xins orbT. Qed.	Lemma	subset_catr	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "mem_cat", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_cat2 s1 s2 s3 : s1 \subset s2 -> s3 ++ s1 \subset s3 ++ s2.	Proof. move=> /subsetP s12; apply/subsetP => x. by rewrite !mem_cat => /orP[->\|/s12->]; rewrite ?orbT. Qed.	Lemma	subset_cat2	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "mem_cat", "s1", "s12", "s2", "s3", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
filter_subset p s : [seq a <- s \| p a] \subset s.	Proof. by apply/subsetP=> x; rewrite mem_filter => /andP[]. Qed.	Lemma	filter_subset	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "mem_filter", "seq", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_filter p s1 s2 : s1 \subset s2 -> [seq a <- s1 \| p a] \subset [seq a <- s2 \| p a].	Proof. by move/subsetP=> s12; apply/subsetP=> x; rewrite !mem_filter=> /andP[-> /s12]. Qed.	Lemma	subset_filter	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "mem_filter", "s1", "s12", "s2", "seq", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
properE A B : (A \proper B) = (A \subset B) && ~~ (B \subset A).	Proof. by []. Qed.	Lemma	properE	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "proper" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
properP A B : reflect (A \subset B /\ (exists2 x, x \in B & x \notin A)) (A \proper B).	Proof. by rewrite properE; apply: (iffP andP) => [] [-> /subsetPn]. Qed.	Lemma	properP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "proper", "properE", "subsetPn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
proper_sub A B : A \proper B -> A \subset B.	Proof. by case/andP. Qed.	Lemma	proper_sub	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "proper" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
proper_subn A B : A \proper B -> ~~ (B \subset A).	Proof. by case/andP. Qed.	Lemma	proper_subn	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "proper" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
proper_trans A B C : A \proper B -> B \proper C -> A \proper C.	Proof. case/properP=> sAB [x Bx nAx] /properP[sBC [y Cy nBy]]. rewrite properE (subset_trans sAB) //=; apply/subsetPn; exists y => //. by apply: contra nBy; apply: subsetP. Qed.	Lemma	proper_trans	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "proper", "properE", "properP", "subsetP", "subsetPn", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
proper_sub_trans A B C : A \proper B -> B \subset C -> A \proper C.	Proof. case/properP=> sAB [x Bx nAx] sBC; rewrite properE (subset_trans sAB) //. by apply/subsetPn; exists x; rewrite ?(subsetP _ _ sBC). Qed.	Lemma	proper_sub_trans	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "proper", "properE", "properP", "subsetP", "subsetPn", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_proper_trans A B C : A \subset B -> B \proper C -> A \proper C.	Proof. move=> sAB /properP[sBC [x Cx nBx]]; rewrite properE (subset_trans sAB) //. by apply/subsetPn; exists x => //; apply: contra nBx; apply: subsetP. Qed.	Lemma	sub_proper_trans	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "proper", "properE", "properP", "subsetP", "subsetPn", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
proper_card A B : A \proper B -> #\|A\| < #\|B\|.	Proof. by case/andP=> sAB nsBA; rewrite ltn_neqAle !(subset_leqif_card sAB) andbT. Qed.	Lemma	proper_card	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "ltn_neqAle", "proper", "subset_leqif_card" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
proper_irrefl A : ~~ (A \proper A).	Proof. by rewrite properE subxx. Qed.	Lemma	proper_irrefl	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "proper", "properE", "subxx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
properxx A : (A \proper A) = false.	Proof. by rewrite properE subxx. Qed.	Lemma	properxx	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "proper", "properE", "subxx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_proper A B : A =i B -> proper (mem A) =1 proper (mem B).	Proof. move=> eAB [C]; congr (_ && _); first exact: (eq_subset eAB). by rewrite (eq_subset_r eAB). Qed.	Lemma	eq_proper	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "eq_subset", "eq_subset_r", "proper" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_proper_r A B : A =i B -> (@proper T)^~ (mem A) =1 (@proper T)^~ (mem B).	Proof. move=> eAB [C]; congr (_ && _); first exact: (eq_subset_r eAB). by rewrite (eq_subset eAB). Qed.	Lemma	eq_proper_r	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "eq_subset", "eq_subset_r", "proper" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_geqP {A n} : reflect (exists s, [/\ uniq s, size s = n & {subset s <= A}]) (n <= #\|A\|).	Proof. apply: (iffP idP) => [n_le_A\|[s] [uniq_s size_s /subsetP subA]]; last first. by rewrite -size_s -(card_uniqP _ uniq_s); exact: subset_leq_card. exists (take n (enum A)); rewrite take_uniq ?enum_uniq // size_take. split => //; last by move => x /mem_take; rewrite mem_enum. case: (ltnP n (size (enum A))) => // s...	Lemma	card_geqP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "cardE", "card_uniqP", "enum", "enum_uniq", "eqn_leq", "last", "ltnP", "mem_enum", "mem_take", "size", "size_take", "split", "subsetP", "subset_leq_card", "take", "take_uniq", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_gt1P A : reflect (exists x y, [/\ x \in A, y \in A & x != y]) (1 < #\|A\|).	Proof. apply: (iffP card_geqP) => [[s] []\|[x] [y] [xA yA xDy]]. case: s => [\|a [\|b []]]//= /[!(inE, andbT)] aDb _ subD. by exists a, b; rewrite aDb !subD ?inE ?eqxx ?orbT. by exists [:: x; y]; rewrite /= !inE xDy; split=> // z /[!inE] /pred2P[]->. Qed.	Lemma	card_gt1P	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "card_geqP", "eqxx", "inE", "pred2P", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_gt2P A : reflect (exists x y z, [/\ x \in A, y \in A & z \in A] /\ [/\ x != y, y != z & z != x]) (2 < #\|A\|).	Proof. apply: (iffP card_geqP) => [[s] []\|[x] [y] [z] [[xD yD zD] [xDy xDz yDz]]]. case: s => [\|x [\|y [\|z []]]]//=; rewrite !inE !andbT negb_or -andbA. case/and3P => xDy xDz yDz _ subA. by exists x, y, z; rewrite xDy yDz eq_sym xDz !subA ?inE ?eqxx ?orbT. exists [:: x; y; z]; rewrite /= !inE negb_or xDy xDz eq_sy...	Lemma	card_gt2P	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "card_geqP", "eq_sym", "eqxx", "inE", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disjoint_sym A B : [disjoint A & B] = [disjoint B & A].	Proof. by congr (_ == 0); apply: eq_card => x; apply: andbC. Qed.	Lemma	disjoint_sym	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "disjoint", "eq_card" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_disjoint A B : A =i B -> disjoint (mem A) =1 disjoint (mem B).	Proof. by move=> eqAB [C]; congr (_ == 0); apply: eq_card => x; rewrite !inE eqAB. Qed.	Lemma	eq_disjoint	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "disjoint", "eq_card", "inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_disjoint_r A B : A =i B -> (@disjoint T)^~ (mem A) =1 (@disjoint T)^~ (mem B).	Proof. by move=> eqAB [C]; congr (_ == 0); apply: eq_card => x; rewrite !inE eqAB. Qed.	Lemma	eq_disjoint_r	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "disjoint", "eq_card", "inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_disjoint A B : (A \subset B) = [disjoint A & [predC B]].	Proof. by rewrite disjoint_sym unlock. Qed.	Lemma	subset_disjoint	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "disjoint", "disjoint_sym" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disjoint_subset A B : [disjoint A & B] = (A \subset [predC B]).	Proof. by rewrite subset_disjoint; apply: eq_disjoint_r => x; rewrite !inE /= negbK. Qed.	Lemma	disjoint_subset	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "disjoint", "eq_disjoint_r", "inE", "subset_disjoint" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disjointFr A B x : [disjoint A & B] -> x \in A -> (x \in B) = false.	Proof. by move/pred0P/(_ x) => /=; case: (x \in A). Qed.	Lemma	disjointFr	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "disjoint", "pred0P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disjointFl A B x : [disjoint A & B] -> x \in B -> (x \in A) = false.	Proof. rewrite disjoint_sym; exact: disjointFr. Qed.	Lemma	disjointFl	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "disjoint", "disjointFr", "disjoint_sym" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disjointWl A B C : A \subset B -> [disjoint B & C] -> [disjoint A & C].	Proof. by rewrite 2!disjoint_subset; apply: subset_trans. Qed.	Lemma	disjointWl	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "disjoint", "disjoint_subset", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disjointWr A B C : A \subset B -> [disjoint C & B] -> [disjoint C & A].	Proof. rewrite ![[disjoint C & _]]disjoint_sym. exact:disjointWl. Qed.	Lemma	disjointWr	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "disjoint", "disjointWl", "disjoint_sym" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disjointW A B C D : A \subset B -> C \subset D -> [disjoint B & D] -> [disjoint A & C].	Proof. by move=> subAB subCD BD; apply/(disjointWl subAB)/(disjointWr subCD). Qed.	Lemma	disjointW	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "disjoint", "disjointWl", "disjointWr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disjoint0 A : [disjoint pred0 & A].	Proof. exact/pred0P. Qed.	Lemma	disjoint0	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "disjoint", "pred0P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_disjoint0 A B : A =i pred0 -> [disjoint A & B].	Proof. by move/eq_disjoint->; apply: disjoint0. Qed.	Lemma	eq_disjoint0	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "disjoint", "disjoint0", "eq_disjoint" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disjoint1 x A : [disjoint pred1 x & A] = (x \notin A).	Proof. apply/negbRL/(sameP (pred0Pn _))=> /=. apply: introP => [Ax \| notAx [_ /andP[/eqP->]]]; last exact: negP. by exists x; rewrite inE eqxx. Qed.	Lemma	disjoint1	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "disjoint", "eqxx", "inE", "last", "pred0Pn", "pred1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_disjoint1 x A B : A =i pred1 x -> [disjoint A & B] = (x \notin B).	Proof. by move/eq_disjoint->; apply: disjoint1. Qed.	Lemma	eq_disjoint1	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "disjoint", "disjoint1", "eq_disjoint", "pred1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disjointU A B C : [disjoint predU A B & C] = [disjoint A & C] && [disjoint B & C].	Proof. case: [disjoint A & C] / (pred0P (xpredI A C)) => [A0 \| nA0] /=. by congr (_ == 0); apply: eq_card => x; rewrite [x \in _]andb_orl A0. apply/pred0P=> nABC; case: nA0 => x; apply/idPn=> /=; move/(_ x): nABC. by rewrite [_ x]andb_orl; case/norP. Qed.	Lemma	disjointU	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "disjoint", "eq_card", "pred0P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disjointU1 x A B : [disjoint predU1 x A & B] = (x \notin B) && [disjoint A & B].	Proof. by rewrite disjointU disjoint1. Qed.	Lemma	disjointU1	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "disjoint", "disjoint1", "disjointU", "predU1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disjoint_cons x s B : [disjoint x :: s & B] = (x \notin B) && [disjoint s & B].	Proof. exact: disjointU1. Qed.	Lemma	disjoint_cons	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "disjoint", "disjointU1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disjoint_has s A : [disjoint s & A] = ~~ has [in A] s.	Proof. apply/negbRL; apply/pred0Pn/hasP => [[x /andP[]]\|[x]]; exists x => //. exact/andP. Qed.	Lemma	disjoint_has	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "disjoint", "has", "hasP", "pred0Pn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disjoint_cat s1 s2 A : [disjoint s1 ++ s2 & A] = [disjoint s1 & A] && [disjoint s2 & A].	Proof. by rewrite !disjoint_has has_cat negb_or. Qed.	Lemma	disjoint_cat	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "disjoint", "disjoint_has", "has_cat", "s1", "s2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_subset {T T' : finType} (s1 s2 : seq T) (f : T -> T') : s1 \subset s2 -> [seq f x \| x <- s1 ] \subset [seq f x \| x <- s2].	Proof. move=> s1s2; apply/subsetP => _ /mapP[y] /[swap] -> ys1. by apply/mapP; exists y => //; move/subsetP : s1s2; exact. Qed.	Lemma	map_subset	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "T'", "apply", "mapP", "s1", "s2", "seq", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
viewP : forall x, reflect (PP x) (P x).		Hypothesis	viewP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
existsPP : reflect (exists x, PP x) [exists x, P x].	Proof. by apply: (iffP pred0Pn) => -[x /viewP]; exists x. Qed.	Lemma	existsPP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "pred0Pn", "viewP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
forallPP : reflect (forall x, PP x) [forall x, P x].	Proof. by apply: (iffP pred0P) => /= allP x; have /viewP//=-> := allP x. Qed.	Lemma	forallPP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "allP", "apply", "pred0P", "viewP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'exists_ view"	:= (existsPP (fun _ => view)) (at level 4, right associativity, format "''exists_' view").	Notation	'exists_ view	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "existsPP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'forall_ view"	:= (forallPP (fun _ => view)) (at level 4, right associativity, format "''forall_' view").	Notation	'forall_ view	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "forallPP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum1 x : enum (pred1 x) = [:: x].

Proof. rewrite [enum _](all_pred1P x _ _); last by rewrite size_filter enumP. by apply/allP=> y; rewrite mem_enum. Qed.

Lemma

enum1

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "allP", "all_pred1P", "apply", "enum", "enumP", "last", "mem_enum", "pred1", "size_filter" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pick_spec : option T -> Type

:= | Pick x of P x : pick_spec (Some x) | Nopick of P =1 xpred0 : pick_spec None.

Variant

pick_spec

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "Pick" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pickP : pick_spec (pick P).

Proof. rewrite /pick; case: (enum _) (mem_enum P) => [|x s] Pxs /=. by right; apply: fsym. by left; rewrite -[P _]Pxs mem_head. Qed.

Lemma

pickP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "enum", "mem_enum", "mem_head", "pick", "pick_spec" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_enum A B : A =i B -> enum A = enum B.

Proof. by move=> eqAB; apply: eq_filter. Qed.

Lemma

eq_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "enum", "eq_filter" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_pick P Q : P =1 Q -> pick P = pick Q.

Proof. by move=> eqPQ; rewrite /pick (eq_enum eqPQ). Qed.

Lemma

eq_pick

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "eq_enum", "pick" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardE A : #|A| = size (enum A).

Proof. by rewrite unlock. Qed.

Lemma

cardE

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "enum", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_card A B : A =i B -> #|A| = #|B|.

Proof. by move=> eqAB; rewrite !cardE (eq_enum eqAB). Qed.

Lemma

eq_card

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cardE", "eq_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_card_trans A B n : #|A| = n -> B =i A -> #|B| = n.

Proof. by move <-; apply: eq_card. Qed.

Lemma

eq_card_trans

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "eq_card" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card0 : #|@pred0 T| = 0.

Proof. by rewrite cardE enum0. Qed.

Lemma

card0

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cardE", "enum0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardT : #|T| = size (enum T).

Proof. by rewrite cardE. Qed.

Lemma

cardT

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cardE", "enum", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card1 x : #|pred1 x| = 1.

Proof. by rewrite cardE enum1. Qed.

Lemma

card1

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cardE", "enum1", "pred1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_card0 A : A =i pred0 -> #|A| = 0.

Proof. exact: eq_card_trans card0. Qed.

Lemma

eq_card0

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "card0", "eq_card_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_cardT A : A =i predT -> #|A| = size (enum T).

Proof. exact: eq_card_trans cardT. Qed.

Lemma

eq_cardT

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cardT", "enum", "eq_card_trans", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_card1 x A : A =i pred1 x -> #|A| = 1.

Proof. exact: eq_card_trans (card1 x). Qed.

Lemma

eq_card1

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "card1", "eq_card_trans", "pred1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardUI A B : #|[predU A & B]| + #|[predI A & B]| = #|A| + #|B|.

Proof. by rewrite !cardE !size_filter count_predUI. Qed.

Lemma

cardUI

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cardE", "count_predUI", "size_filter" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardID B A : #|[predI A & B]| + #|[predD A & B]| = #|A|.

Proof. rewrite -cardUI addnC [#|predI _ _|]eq_card0 => [x|] /=. by rewrite !inE -!andbA andbC andbA andbN. by apply: eq_card => x; rewrite !inE andbC -andb_orl orbN. Qed.

Lemma

cardID

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "addnC", "apply", "cardUI", "eq_card", "eq_card0", "inE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardC A : #|A| + #|[predC A]| = #|T|.

Proof. by rewrite !cardE !size_filter count_predC. Qed.

Lemma

cardC

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cardE", "count_predC", "size_filter" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardU1 x A : #|[predU1 x & A]| = (x \notin A) + #|A|.

Proof. case Ax: (x \in A). by apply: eq_card => y /[1!inE]/=; case: eqP => // ->. rewrite /= -(card1 x) -cardUI addnC. rewrite [#|predI _ _|]eq_card0 => [y /=|]; last exact: eq_card. by rewrite !inE; case: eqP => // ->. Qed.

Lemma

cardU1

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "addnC", "apply", "card1", "cardUI", "eq_card", "eq_card0", "inE", "last", "predU1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card2 x y : #|pred2 x y| = (x != y).+1.

Proof. by rewrite cardU1 card1 addn1. Qed.

Lemma

card2

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "addn1", "card1", "cardU1", "pred2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardC1 x : #|predC1 x| = #|T|.-1.

Proof. by rewrite -(cardC (pred1 x)) card1. Qed.

Lemma

cardC1

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "card1", "cardC", "pred1", "predC1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardD1 x A : #|A| = (x \in A) + #|[predD1 A & x]|.

Proof. case Ax: (x \in A); last first. by apply: eq_card => y /[!inE]/=; case: eqP => // ->. rewrite /= -(card1 x) -cardUI addnC /=. rewrite [#|predI _ _|]eq_card0 => [y|]; first by rewrite !inE; case: eqP. by apply: eq_card => y /[!inE]; case: eqP => // ->. Qed.

Lemma

cardD1

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "addnC", "apply", "card1", "cardUI", "eq_card", "eq_card0", "inE", "last", "predD1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

max_card A : #|A| <= #|T|.

Proof. by rewrite -(cardC A) leq_addr. Qed.

Lemma

max_card

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cardC", "leq_addr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_size s : #|s| <= size s.

Proof. elim: s => [|x s IHs] /=; first by rewrite card0. by rewrite cardU1 /=; case: (~~ _) => //; apply: leqW. Qed.

Lemma

card_size

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "card0", "cardU1", "leqW", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_uniqP s : reflect (#|s| = size s) (uniq s).

Proof. elim: s => [|x s IHs]; first by left; apply: card0. rewrite cardU1 /= /addn; case: {+}(x \in s) => /=. by right=> card_Ssz; have:= card_size s; rewrite card_Ssz ltnn. by apply: (iffP IHs) => [<-| [<-]]. Qed.

Lemma

card_uniqP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "addn", "apply", "card0", "cardU1", "card_size", "ltnn", "size", "uniq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card0_eq A : #|A| = 0 -> A =i pred0.

Proof. by move=> A0 x; apply/idP => Ax; rewrite (cardD1 x) Ax in A0. Qed.

Lemma

card0_eq

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "cardD1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fintype0 : T -> #|T| <> 0.

Proof. by move=> x /card0_eq/(_ x). Qed.

Lemma

fintype0

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "card0_eq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pred0P P : reflect (P =1 pred0) (pred0b P).

Proof. by apply: (iffP eqP); [apply: card0_eq | apply: eq_card0]. Qed.

Lemma

pred0P

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "card0_eq", "eq_card0", "pred0b" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pred0Pn P : reflect (exists x, P x) (~~ pred0b P).

Proof. case: (pickP P) => [x Px | P0]. by rewrite (introN (pred0P P)) => [P0|]; [rewrite P0 in Px | left; exists x]. by rewrite -lt0n eq_card0 //; right=> [[x]]; rewrite P0. Qed.

Lemma

pred0Pn

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "P0", "Px", "eq_card0", "lt0n", "pickP", "pred0P", "pred0b" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_gt0P A : reflect (exists i, i \in A) (#|A| > 0).

Proof. by rewrite lt0n; apply: pred0Pn. Qed.

Lemma

card_gt0P

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "lt0n", "pred0Pn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_le1P {A} : reflect {in A, forall x, A =i pred1 x} (#|A| <= 1).

Proof. apply: (iffP idP) => [A1 x xA y|]; last first. by have [/= x xA /(_ _ xA)/eq_card1->|/eq_card0->//] := pickP [in A]. move: A1; rewrite (cardD1 x) xA ltnS leqn0 => /eqP/card0_eq/(_ y). by rewrite !inE; have [->|]:= eqP. Qed.

Lemma

card_le1P

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "card0_eq", "cardD1", "eq_card0", "eq_card1", "inE", "last", "leqn0", "ltnS", "pickP", "pred1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_card1 A : #|A| = 1 -> {x | A =i pred1 x}.

Proof. move=> A1; have /card_gt0P/sigW[x xA]: #|A| > 0 by rewrite A1. by exists x; apply/card_le1P; rewrite ?A1. Qed.

Lemma

mem_card1

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "card_gt0P", "card_le1P", "pred1", "sigW" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card1P A : reflect (exists x, A =i pred1 x) (#|A| == 1).

Proof. by apply: (iffP idP) => [/eqP/mem_card1[x inA]|[x /eq_card1/eqP//]]; exists x. Qed.

Lemma

card1P

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "eq_card1", "inA", "mem_card1", "pred1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_le1_eqP A : reflect {in A &, forall x, all_equal_to x} (#|A| <= 1).

Proof. apply: (iffP card_le1P) => [Ale1 x y xA yA /=|all_eq x xA y]. by apply/eqP; rewrite -[_ == _]/(y \in pred1 x) -Ale1. by rewrite inE; case: (altP (y =P x)) => [->//|]; exact/contra_neqF/all_eq. Qed.

Lemma

card_le1_eqP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "card_le1P", "contra_neqF", "inE", "pred1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fintype_le1P : reflect (forall x : T, all_equal_to x) (#|T| <= 1).

Proof. apply: (iffP (card_le1_eqP {:T})); [exact: in2T | exact: in2W]. Qed.

Lemma

fintype_le1P

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "card_le1_eqP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fintype1 : #|T| = 1 -> {x : T | all_equal_to x}.

Proof. by move=> /mem_card1[x ex]; exists x => y; suff: y \in T by rewrite ex => /eqP. Qed.

Lemma

fintype1

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "ex", "mem_card1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fintype1P : reflect (exists x, all_equal_to x) (#|T| == 1).

Proof. apply: (iffP idP) => [/eqP/fintype1|] [x eqx]; first by exists x. by apply/card1P; exists x => y; rewrite eqx !inE eqxx. Qed.

Lemma

fintype1P

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "card1P", "eqxx", "fintype1", "inE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetE A B : (A \subset B) = pred0b [predD A & B].

Proof. by rewrite unlock. Qed.

Lemma

subsetE

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "pred0b" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetP A B : reflect {subset A <= B} (A \subset B).

Proof. rewrite unlock; apply: (iffP (pred0P _)) => [AB0 x | sAB x /=]. by apply/implyP; apply/idPn; rewrite negb_imply andbC [_ && _]AB0. by rewrite andbC -negb_imply; apply/negbF/implyP; apply: sAB. Qed.

Lemma

subsetP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "pred0P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetPn A B : reflect (exists2 x, x \in A & x \notin B) (~~ (A \subset B)).

Proof. rewrite unlock; apply: (iffP (pred0Pn _)) => [[x] | [x Ax nBx]]. by case/andP; exists x. by exists x; rewrite /= nBx. Qed.

Lemma

subsetPn

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "pred0Pn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_leq_card A B : A \subset B -> #|A| <= #|B|.

Proof. move=> sAB. rewrite -(cardID A B) [#|predI _ _|](@eq_card _ A) ?leq_addr //= => x. by rewrite !inE andbC; case Ax: (x \in A) => //; apply: subsetP Ax. Qed.

Lemma

subset_leq_card

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "cardID", "eq_card", "inE", "leq_addr", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subxx_hint (mA : mem_pred T) : subset mA mA.

Proof. by case: mA => A; have:= introT (subsetP A A); rewrite !unlock => ->. Qed.

Lemma

subxx_hint

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subxx (pT : predType T) (pA : pT) : pA \subset pA.

Proof. by []. Qed.

Lemma

subxx

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "pA" ]

The parametrization by predType makes it easier to apply subxx.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_subset A B : A =i B -> subset (mem A) =1 subset (mem B).

Proof. move=> eqAB [C]; rewrite !unlock; congr (_ == 0). by apply: eq_card => x; rewrite inE /= eqAB. Qed.

Lemma

eq_subset

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "eq_card", "inE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_subset_r A B : A =i B -> (@subset T)^~ (mem A) =1 (@subset T)^~ (mem B).

Proof. move=> eqAB [C]; rewrite !unlock; congr (_ == 0). by apply: eq_card => x; rewrite !inE /= eqAB. Qed.

Lemma

eq_subset_r

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "eq_card", "inE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_subxx A B : A =i B -> A \subset B.

Proof. by move/eq_subset->. Qed.

Lemma

eq_subxx

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "eq_subset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_predT A : A \subset T.

Proof. exact/subsetP. Qed.

Lemma

subset_predT

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

predT_subset A : T \subset A -> forall x, x \in A.

Proof. by move/subsetP=> allA x; apply: allA. Qed.

Lemma

predT_subset

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_pred1 A x : (pred1 x \subset A) = (x \in A).

Proof. by apply/subsetP/idP=> [-> // | Ax y /eqP-> //]; apply: eqxx. Qed.

Lemma

subset_pred1

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "eqxx", "pred1", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_eqP A B : reflect (A =i B) ((A \subset B) && (B \subset A)).

Proof. apply: (iffP andP) => [[sAB sBA] x| eqAB]; last by rewrite !eq_subxx. by apply/idP/idP; apply: subsetP. Qed.

Lemma

subset_eqP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "eq_subxx", "last", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_cardP A B : #|A| = #|B| -> reflect (A =i B) (A \subset B).

Proof. move=> eqcAB; case: (subsetP A B) (subset_eqP A B) => //= sAB. case: (subsetP B A) => [//|[]] x Bx; apply/idPn => Ax. case/idP: (ltnn #|A|); rewrite {2}eqcAB (cardD1 x B) Bx /=. apply: subset_leq_card; apply/subsetP=> y Ay; rewrite inE /= andbC. by rewrite sAB //; apply/eqP => eqyx; rewrite -eqyx Ay in Ax. Qed.

Lemma

subset_cardP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "cardD1", "inE", "ltnn", "subsetP", "subset_eqP", "subset_leq_card" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_leqif_card A B : A \subset B -> #|A| <= #|B| ?= iff (B \subset A).

Proof. move=> sAB; split; [exact: subset_leq_card | apply/eqP/idP]. by move/subset_cardP=> sABP; rewrite (eq_subset_r (sABP sAB)). by move=> sBA; apply: eq_card; apply/subset_eqP; rewrite sAB. Qed.

Lemma

subset_leqif_card

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "eq_card", "eq_subset_r", "split", "subset_cardP", "subset_eqP", "subset_leq_card" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_trans A B C : A \subset B -> B \subset C -> A \subset C.

Proof. by move/subsetP=> sAB /subsetP=> sBC; apply/subsetP=> x /sAB; apply: sBC. Qed.

Lemma

subset_trans

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_all s A : (s \subset A) = all [in A] s.

Proof. exact: (sameP (subsetP _ _) allP). Qed.

Lemma

subset_all

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "all", "allP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_cons s x : s \subset x :: s.

Proof. by apply/subsetP => y /[!inE] ->; rewrite orbT. Qed.

Lemma

subset_cons

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "inE", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_cons2 s1 s2 x : s1 \subset s2 -> x :: s1 \subset x :: s2.

Proof. by move=> ?; apply/subsetP => y /[!inE]; case: eqP => // _; apply: subsetP. Qed.

Lemma

subset_cons2

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "inE", "s1", "s2", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_catl s s' : s \subset s ++ s'.

Proof. by apply/subsetP=> x xins; rewrite mem_cat xins. Qed.

Lemma

subset_catl

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "mem_cat", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_catr s s' : s \subset s' ++ s.

Proof. by apply/subsetP => x xins; rewrite mem_cat xins orbT. Qed.

Lemma

subset_catr

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "mem_cat", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_cat2 s1 s2 s3 : s1 \subset s2 -> s3 ++ s1 \subset s3 ++ s2.

Proof. move=> /subsetP s12; apply/subsetP => x. by rewrite !mem_cat => /orP[->|/s12->]; rewrite ?orbT. Qed.

Lemma

subset_cat2

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "mem_cat", "s1", "s12", "s2", "s3", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

filter_subset p s : [seq a <- s | p a] \subset s.

Proof. by apply/subsetP=> x; rewrite mem_filter => /andP[]. Qed.

Lemma

filter_subset

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "mem_filter", "seq", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_filter p s1 s2 : s1 \subset s2 -> [seq a <- s1 | p a] \subset [seq a <- s2 | p a].

Proof. by move/subsetP=> s12; apply/subsetP=> x; rewrite !mem_filter=> /andP[-> /s12]. Qed.

Lemma

subset_filter

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "mem_filter", "s1", "s12", "s2", "seq", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

properE A B : (A \proper B) = (A \subset B) && ~~ (B \subset A).

Proof. by []. Qed.

Lemma

properE

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "proper" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

properP A B : reflect (A \subset B /\ (exists2 x, x \in B & x \notin A)) (A \proper B).

Proof. by rewrite properE; apply: (iffP andP) => [] [-> /subsetPn]. Qed.

Lemma

properP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "proper", "properE", "subsetPn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

proper_sub A B : A \proper B -> A \subset B.

Proof. by case/andP. Qed.

Lemma

proper_sub

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "proper" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

proper_subn A B : A \proper B -> ~~ (B \subset A).

Proof. by case/andP. Qed.

Lemma

proper_subn

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "proper" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

proper_trans A B C : A \proper B -> B \proper C -> A \proper C.

Proof. case/properP=> sAB [x Bx nAx] /properP[sBC [y Cy nBy]]. rewrite properE (subset_trans sAB) //=; apply/subsetPn; exists y => //. by apply: contra nBy; apply: subsetP. Qed.

Lemma

proper_trans

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "proper", "properE", "properP", "subsetP", "subsetPn", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

proper_sub_trans A B C : A \proper B -> B \subset C -> A \proper C.

Proof. case/properP=> sAB [x Bx nAx] sBC; rewrite properE (subset_trans sAB) //. by apply/subsetPn; exists x; rewrite ?(subsetP _ _ sBC). Qed.

Lemma

proper_sub_trans

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "proper", "properE", "properP", "subsetP", "subsetPn", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_proper_trans A B C : A \subset B -> B \proper C -> A \proper C.

Proof. move=> sAB /properP[sBC [x Cx nBx]]; rewrite properE (subset_trans sAB) //. by apply/subsetPn; exists x => //; apply: contra nBx; apply: subsetP. Qed.

Lemma

sub_proper_trans

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "proper", "properE", "properP", "subsetP", "subsetPn", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

proper_card A B : A \proper B -> #|A| < #|B|.

Proof. by case/andP=> sAB nsBA; rewrite ltn_neqAle !(subset_leqif_card sAB) andbT. Qed.

Lemma

proper_card

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "ltn_neqAle", "proper", "subset_leqif_card" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

proper_irrefl A : ~~ (A \proper A).

Proof. by rewrite properE subxx. Qed.

Lemma

proper_irrefl

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "proper", "properE", "subxx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

properxx A : (A \proper A) = false.

Proof. by rewrite properE subxx. Qed.

Lemma

properxx

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "proper", "properE", "subxx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_proper A B : A =i B -> proper (mem A) =1 proper (mem B).

Proof. move=> eAB [C]; congr (_ && _); first exact: (eq_subset eAB). by rewrite (eq_subset_r eAB). Qed.

Lemma

eq_proper

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "eq_subset", "eq_subset_r", "proper" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_proper_r A B : A =i B -> (@proper T)^~ (mem A) =1 (@proper T)^~ (mem B).

Proof. move=> eAB [C]; congr (_ && _); first exact: (eq_subset_r eAB). by rewrite (eq_subset eAB). Qed.

Lemma

eq_proper_r

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "eq_subset", "eq_subset_r", "proper" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_geqP {A n} : reflect (exists s, [/\ uniq s, size s = n & {subset s <= A}]) (n <= #|A|).

Proof. apply: (iffP idP) => [n_le_A|[s] [uniq_s size_s /subsetP subA]]; last first. by rewrite -size_s -(card_uniqP _ uniq_s); exact: subset_leq_card. exists (take n (enum A)); rewrite take_uniq ?enum_uniq // size_take. split => //; last by move => x /mem_take; rewrite mem_enum. case: (ltnP n (size (enum A))) => // s...

Lemma

card_geqP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "cardE", "card_uniqP", "enum", "enum_uniq", "eqn_leq", "last", "ltnP", "mem_enum", "mem_take", "size", "size_take", "split", "subsetP", "subset_leq_card", "take", "take_uniq", "uniq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_gt1P A : reflect (exists x y, [/\ x \in A, y \in A & x != y]) (1 < #|A|).

Proof. apply: (iffP card_geqP) => [[s] []|[x] [y] [xA yA xDy]]. case: s => [|a [|b []]]//= /[!(inE, andbT)] aDb _ subD. by exists a, b; rewrite aDb !subD ?inE ?eqxx ?orbT. by exists [:: x; y]; rewrite /= !inE xDy; split=> // z /[!inE] /pred2P[]->. Qed.

Lemma

card_gt1P

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "card_geqP", "eqxx", "inE", "pred2P", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_gt2P A : reflect (exists x y z, [/\ x \in A, y \in A & z \in A] /\ [/\ x != y, y != z & z != x]) (2 < #|A|).

Proof. apply: (iffP card_geqP) => [[s] []|[x] [y] [z] [[xD yD zD] [xDy xDz yDz]]]. case: s => [|x [|y [|z []]]]//=; rewrite !inE !andbT negb_or -andbA. case/and3P => xDy xDz yDz _ subA. by exists x, y, z; rewrite xDy yDz eq_sym xDz !subA ?inE ?eqxx ?orbT. exists [:: x; y; z]; rewrite /= !inE negb_or xDy xDz eq_sy...

Lemma

card_gt2P

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "card_geqP", "eq_sym", "eqxx", "inE", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disjoint_sym A B : [disjoint A & B] = [disjoint B & A].

Proof. by congr (_ == 0); apply: eq_card => x; apply: andbC. Qed.

Lemma

disjoint_sym

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "disjoint", "eq_card" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_disjoint A B : A =i B -> disjoint (mem A) =1 disjoint (mem B).

Proof. by move=> eqAB [C]; congr (_ == 0); apply: eq_card => x; rewrite !inE eqAB. Qed.

Lemma

eq_disjoint

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "disjoint", "eq_card", "inE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_disjoint_r A B : A =i B -> (@disjoint T)^~ (mem A) =1 (@disjoint T)^~ (mem B).

Proof. by move=> eqAB [C]; congr (_ == 0); apply: eq_card => x; rewrite !inE eqAB. Qed.

Lemma

eq_disjoint_r

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "disjoint", "eq_card", "inE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_disjoint A B : (A \subset B) = [disjoint A & [predC B]].

Proof. by rewrite disjoint_sym unlock. Qed.

Lemma

subset_disjoint

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "disjoint", "disjoint_sym" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disjoint_subset A B : [disjoint A & B] = (A \subset [predC B]).

Proof. by rewrite subset_disjoint; apply: eq_disjoint_r => x; rewrite !inE /= negbK. Qed.

Lemma

disjoint_subset

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "disjoint", "eq_disjoint_r", "inE", "subset_disjoint" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disjointFr A B x : [disjoint A & B] -> x \in A -> (x \in B) = false.

Proof. by move/pred0P/(_ x) => /=; case: (x \in A). Qed.

Lemma

disjointFr

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "disjoint", "pred0P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disjointFl A B x : [disjoint A & B] -> x \in B -> (x \in A) = false.

Proof. rewrite disjoint_sym; exact: disjointFr. Qed.

Lemma

disjointFl

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "disjoint", "disjointFr", "disjoint_sym" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disjointWl A B C : A \subset B -> [disjoint B & C] -> [disjoint A & C].

Proof. by rewrite 2!disjoint_subset; apply: subset_trans. Qed.

Lemma

disjointWl

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "disjoint", "disjoint_subset", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disjointWr A B C : A \subset B -> [disjoint C & B] -> [disjoint C & A].

Proof. rewrite ![[disjoint C & _]]disjoint_sym. exact:disjointWl. Qed.

Lemma

disjointWr

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "disjoint", "disjointWl", "disjoint_sym" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disjointW A B C D : A \subset B -> C \subset D -> [disjoint B & D] -> [disjoint A & C].

Proof. by move=> subAB subCD BD; apply/(disjointWl subAB)/(disjointWr subCD). Qed.

Lemma

disjointW

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "disjoint", "disjointWl", "disjointWr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disjoint0 A : [disjoint pred0 & A].

Proof. exact/pred0P. Qed.

Lemma

disjoint0

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "disjoint", "pred0P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_disjoint0 A B : A =i pred0 -> [disjoint A & B].

Proof. by move/eq_disjoint->; apply: disjoint0. Qed.

Lemma

eq_disjoint0

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "disjoint", "disjoint0", "eq_disjoint" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disjoint1 x A : [disjoint pred1 x & A] = (x \notin A).

Proof. apply/negbRL/(sameP (pred0Pn _))=> /=. apply: introP => [Ax | notAx [_ /andP[/eqP->]]]; last exact: negP. by exists x; rewrite inE eqxx. Qed.

Lemma

disjoint1

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "disjoint", "eqxx", "inE", "last", "pred0Pn", "pred1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_disjoint1 x A B : A =i pred1 x -> [disjoint A & B] = (x \notin B).

Proof. by move/eq_disjoint->; apply: disjoint1. Qed.

Lemma

eq_disjoint1

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "disjoint", "disjoint1", "eq_disjoint", "pred1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disjointU A B C : [disjoint predU A B & C] = [disjoint A & C] && [disjoint B & C].

Proof. case: [disjoint A & C] / (pred0P (xpredI A C)) => [A0 | nA0] /=. by congr (_ == 0); apply: eq_card => x; rewrite [x \in _]andb_orl A0. apply/pred0P=> nABC; case: nA0 => x; apply/idPn=> /=; move/(_ x): nABC. by rewrite [_ x]andb_orl; case/norP. Qed.

Lemma

disjointU

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "disjoint", "eq_card", "pred0P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disjointU1 x A B : [disjoint predU1 x A & B] = (x \notin B) && [disjoint A & B].

Proof. by rewrite disjointU disjoint1. Qed.

Lemma

disjointU1

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "disjoint", "disjoint1", "disjointU", "predU1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disjoint_cons x s B : [disjoint x :: s & B] = (x \notin B) && [disjoint s & B].

Proof. exact: disjointU1. Qed.

Lemma

disjoint_cons

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "disjoint", "disjointU1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disjoint_has s A : [disjoint s & A] = ~~ has [in A] s.

Proof. apply/negbRL; apply/pred0Pn/hasP => [[x /andP[]]|[x]]; exists x => //. exact/andP. Qed.

Lemma

disjoint_has

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "disjoint", "has", "hasP", "pred0Pn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disjoint_cat s1 s2 A : [disjoint s1 ++ s2 & A] = [disjoint s1 & A] && [disjoint s2 & A].

Proof. by rewrite !disjoint_has has_cat negb_or. Qed.

Lemma

disjoint_cat

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "disjoint", "disjoint_has", "has_cat", "s1", "s2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_subset {T T' : finType} (s1 s2 : seq T) (f : T -> T') : s1 \subset s2 -> [seq f x | x <- s1 ] \subset [seq f x | x <- s2].

Proof. move=> s1s2; apply/subsetP => _ /mapP[y] /[swap] -> ys1. by apply/mapP; exists y => //; move/subsetP : s1s2; exact. Qed.

Lemma

map_subset

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "T'", "apply", "mapP", "s1", "s2", "seq", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

viewP : forall x, reflect (PP x) (P x).

Hypothesis

viewP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

existsPP : reflect (exists x, PP x) [exists x, P x].

Proof. by apply: (iffP pred0Pn) => -[x /viewP]; exists x. Qed.

Lemma

existsPP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "pred0Pn", "viewP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

forallPP : reflect (forall x, PP x) [forall x, P x].

Proof. by apply: (iffP pred0P) => /= allP x; have /viewP//=-> := allP x. Qed.

Lemma

forallPP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "allP", "apply", "pred0P", "viewP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'exists_ view"

:= (existsPP (fun _ => view)) (at level 4, right associativity, format "''exists_' view").

Notation

'exists_ view

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "existsPP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'forall_ view"

:= (forallPP (fun _ => view)) (at level 4, right associativity, format "''forall_' view").

Notation

'forall_ view

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "forallPP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d