fact string | type string | library string | imports list | filename string | symbolic_name string | docstring string |
|---|---|---|---|---|---|---|
fsetU A B := [fset[fsetU_key] x in enum_fset A ++ enum_fset B]. | Definition | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetU | |
fsetI_key : unit. Proof. exact: tt. Qed. | Fact | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetI_key | |
fsetI A B := [fset[fsetI_key] x in A | x \in B]. | Definition | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetI | |
fsetD_key : unit. Proof. exact: tt. Qed. | Fact | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetD_key | |
fsetD A B := [fset[fsetD_key] x in A | x \notin B]. | Definition | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetD | |
fsetM_key : unit. Proof. exact: tt. Qed. | Fact | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetM_key | |
fsetM A E := [fset[fsetM_key] (x, y) | x : K in A, y : K' in E]. | Definition | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetM | |
fsubset A B := fsetI A B == A. | Definition | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsubset | |
fproper A B := fsubset A B && ~~ fsubset B A. | Definition | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fproper | |
fdisjoint A B := (fsetI A B == fset0). | Definition | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fdisjoint | |
imfsetP (T : choiceType) (f : T -> K) (p : finmempred T) (k : K) : reflect (exists2 x : T, in_mem x p & k = f x) (k \in imfset key f p). Proof. rewrite unlock seq_fsetE /=; apply: (iffP mapP) => [] [x xp eqkf]; by exists x => //=; move: xp; rewrite enum_finmemE. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | imfsetP | |
in_imfset (T : choiceType) (f : T -> K) (p : finmempred T) (x : T) : in_mem x p -> f x \in imfset key f p. Proof. by move=> px; apply/imfsetP; exists x. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_imfset | |
imfset_rec (T : choiceType) (f : T -> K) (p : finmempred T) (P : imfset key f p -> Prop) : (forall (x : T) (px : in_mem x p), P [` in_imfset f px ]) -> forall k, P k. Proof. move=> PP v; have /imfsetP [k pk vv_eq] := valP v. pose vP := in_imfset f pk; suff -> : P v = P [` vP] by apply: PP. by congr P; apply/val_inj => ... | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | imfset_rec | |
mem_imfset (T : choiceType) (f : T -> K) (p : finmempred T) : injective f -> forall (x : T), (f x \in imfset key f p) = (in_mem x p). Proof. by move=> f_inj x; rewrite unlock seq_fsetE mem_map// enum_finmemE. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | mem_imfset | |
imfset2P (T1 : choiceType) (T2 : T1 -> choiceType) (f : forall x, T2 x -> K) (p1 : finmempred T1) (p2 : forall x, finmempred (T2 x)) k : reflect (exists2 x : T1, in_mem x p1 & exists2 y : T2 x, in_mem y (p2 x) & k = f x y) (k \in imfset2 key f p1 p2). Proof. rewrite unlock !seq_fsetE; apply: (iffP allpairsPdep). move=>... | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | imfset2P | |
in_imfset2 (T1 : choiceType) (T2 : T1 -> choiceType) (f : forall x, T2 x -> K) (p1 : finmempred T1) (p2 : forall x, finmempred (T2 x)) (x : T1) (y : T2 x) : in_mem x p1 -> in_mem y (p2 x) -> f x y \in imfset2 key f p1 p2. Proof. by move=> xD1 yD2; apply/imfset2P; exists x => //; exists y. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_imfset2 | |
mem_imfset2 (T1 : choiceType) (T2 : T1 -> choiceType) (f : forall x, T2 x -> K) (g := fun t : {x : T1 & T2 x} => f (tag t) (tagged t)) (p1 : finmempred T1) (p2 : forall x, finmempred (T2 x)) (x : T1) (y : T2 x) : injective g -> f x y \in imfset2 key f p1 p2 = (in_mem x p1) && (in_mem y (p2 x)). Proof. move=> f_inj; rew... | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | mem_imfset2 | |
enum_imfset (T : choiceType) (f : T -> K) (p : finmempred T) : {in p &, injective f} -> perm_eq (imfset key f p) [seq f x | x <- enum_finmem p]. Proof. move=> f_inj; rewrite unlock -[X in perm_eq _ X]undup_id ?seq_fset_perm//. rewrite map_inj_in_uniq ?enum_finmem_uniq // => ??. by rewrite ?enum_finmemE; apply: f_inj. Q... | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | enum_imfset | |
enum_imfset2 (T1 : choiceType) (T2 : T1 -> choiceType) (f : forall x, T2 x -> K) (p1 : finmempred T1) (p2 : forall x, finmempred (T2 x)) : {in [pred t | p1 (tag t) & p2 _ (tagged t)] &, injective (fun t : sigT T2 => f (tag t) (tagged t))} -> perm_eq (imfset2 key f p1 p2) [seq f x y | x <- enum_finmem p1, y <- enum_finm... | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | enum_imfset2 | |
in_fset (p : finmempred K) (k : K) : (k \in imfset key id p) = in_mem k p. Proof. by rewrite mem_imfset; apply: inj_id. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fset | |
val_in_fset A (p : finmempred _) (k : A) : (val k \in imfset key val p) = (in_mem k p). Proof. by rewrite mem_imfset ?in_finmempred //; exact: val_inj. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | val_in_fset | |
in_fset_val A (p : finmempred A) (k : K) : (k \in imfset key val p) = if insub k is Some a then in_mem a p else false. Proof. have [a _ <- /=|kNA] := insubP; first by rewrite val_in_fset. by apply/imfsetP => [] [a _ k_def]; move: kNA; rewrite k_def [_ \in _]valP. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fset_val | |
in_fset_valT A (p : finmempred _) (k : K) (kA : k \in A) : (k \in imfset key val p) = in_mem [` kA] p. Proof. by rewrite in_fset_val insubT /=. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fset_valT | |
in_fset_valP A (p : finmempred _) (k : K) : reflect {kA : k \in A & in_mem [` kA] p} (k \in imfset key val p). Proof. apply: (iffP (imfsetP _ _ _ _)) => [|[kA xkA]]; last by exists [`kA]. by move=> /sig2_eqW[/= x Xx ->]; exists (valP x); rewrite fsetsubE. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fset_valP | |
in_fset_valF A (p : finmempred A) (k : K) : k \notin A -> (k \in imfset key val p) = false. Proof. by apply: contraNF => /imfsetP[/= a Xa->]. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fset_valF | |
in_fset_nil a : a \in [fset[key] x in [::]] = false. Proof. by rewrite !mem_imfset. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fset_nil | |
in_fset_cons x (xs : seq K) a : (a \in [fset[key] x in x :: xs]) = ((a == x) || (a \in [fset[key] x in xs])). Proof. by rewrite !mem_imfset. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fset_cons | |
in_fset_cat (xs ys : seq K) a : (a \in [fset[key] x in xs ++ ys]) = ((a \in [fset[key] x in xs]) || (a \in [fset[key] x in ys])). Proof. by rewrite !mem_imfset//= mem_cat. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fset_cat | |
in_fset_ (key : unit) := (in_fset_cons, in_fset_nil, in_fset_cat, in_fset). | Definition | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fset_ | |
card_in_imfset (T T' : choiceType) (f : T -> T') (p : finmempred T) : {in p &, injective f} -> #|` (imfset key f p)| = (size (enum_finmem p)). Proof. move=> f_inj; rewrite unlock /= size_seq_fset undup_id ?size_map//. rewrite map_inj_in_uniq ?enum_finmem_uniq// => ??. by rewrite !enum_finmemE; apply: f_inj. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | card_in_imfset | |
card_imfset (T T' : choiceType) (f : T -> T') (p : finmempred _) : injective f -> #|` (imfset key f p)| = size (enum_finmem p). Proof. by move=> f_inj; rewrite card_in_imfset //= => x y ? ?; apply: f_inj. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | card_imfset | |
leq_imfset_card (T T' : choiceType) (f : T -> T') (p : finmempred _) : (#|` imfset key f p| <= size (enum_finmem p))%N. Proof. by rewrite unlock size_seq_fset (leq_trans (size_undup _)) ?size_map. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | leq_imfset_card | |
fsetP {A B} : A =i B <-> A = B. Proof. by split=> [eqAB|-> //]; apply/val_inj/canonical_eq_keys => //= a. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetP | |
in_fset_spec (A : {fset K}) (x : K) : K -> bool -> Prop := | InFset (u : A) & x = val u : in_fset_spec A x (val u) true | OutFset of x \notin A : in_fset_spec A x x false. | Variant | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fset_spec | |
in_fsetP A x : in_fset_spec A x x (x \in A). Proof. have [xA|xNA] := boolP (x \in A); last by constructor. by have {2}-> : x = val [` xA] by []; constructor. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fsetP | |
fset_eqP {A B} : reflect (A =i B) (A == B). Proof. exact: (equivP eqP (iff_sym fsetP)). Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fset_eqP | |
in_fset0 x : x \in fset0 = false. Proof. by []. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fset0 | |
in_fset1 a' a : a \in [fset a'] = (a == a'). Proof. by rewrite !in_fset_ orbF. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fset1 | |
in_fsetU A B a : (a \in A `|` B) = (a \in A) || (a \in B). Proof. by rewrite !in_fset_. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fsetU | |
in_fset1U a' A a : (a \in a' |` A) = (a == a') || (a \in A). Proof. by rewrite in_fsetU in_fset1. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fset1U | |
in_fsetI A B a : (a \in A `&` B) = (a \in A) && (a \in B). Proof. by rewrite in_fset. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fsetI | |
in_fsetD A B a : (a \in A `\` B) = (a \notin B) && (a \in A). Proof. by rewrite in_fset andbC. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fsetD | |
in_fsetD1 A b a : (a \in A `\ b) = (a != b) && (a \in A). Proof. by rewrite in_fsetD in_fset1. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fsetD1 | |
in_fsetM A E (u : K * K') : (u \in A `*` E) = (u.1 \in A) && (u.2 \in E). Proof. by case: u => /= x y; rewrite mem_imfset2//= => -[??] [??] [-> ->]. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fsetM | |
in_fsetE := (@in_fset_ imfset_key, val_in_fset, in_fset0, in_fset1, in_fsetU, in_fsetI, in_fsetD, in_fsetM, in_fset1U, in_fsetD1). | Definition | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fsetE | |
inE := (inE, in_fsetE). | Let | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | inE | |
fsetIC (A B : {fset K}) : A `&` B = B `&` A. Proof. by apply/fsetP => a; rewrite !inE andbC. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetIC | |
fsetUC (A B : {fset K}) : A `|` B = B `|` A. Proof. by apply/fsetP => a; rewrite !inE orbC. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetUC | |
fset0I A : fset0 `&` A = fset0. Proof. by apply/fsetP => x; rewrite !inE andFb. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fset0I | |
fsetI0 A : A `&` fset0 = fset0. Proof. by rewrite fsetIC fset0I. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetI0 | |
fsetIA A B C : A `&` (B `&` C) = A `&` B `&` C. Proof. by apply/fsetP=> x; rewrite !inE andbA. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetIA | |
fsetICA A B C : A `&` (B `&` C) = B `&` (A `&` C). Proof. by rewrite !fsetIA (fsetIC A). Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetICA | |
fsetIAC A B C : A `&` B `&` C = A `&` C `&` B. Proof. by rewrite -!fsetIA (fsetIC B). Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetIAC | |
fsetIACA A B C D : (A `&` B) `&` (C `&` D) = (A `&` C) `&` (B `&` D). Proof. by rewrite -!fsetIA (fsetICA B). Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetIACA | |
fsetIid A : A `&` A = A. Proof. by apply/fsetP=> x; rewrite inE andbb. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetIid | |
fsetIIl A B C : A `&` B `&` C = (A `&` C) `&` (B `&` C). Proof. by rewrite fsetIA !(fsetIAC _ C) -(fsetIA _ C) fsetIid. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetIIl | |
fsetIIr A B C : A `&` (B `&` C) = (A `&` B) `&` (A `&` C). Proof. by rewrite !(fsetIC A) fsetIIl. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetIIr | |
fsetUA A B C : A `|` (B `|` C) = A `|` B `|` C. Proof. by apply/fsetP => x; rewrite !inE orbA. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetUA | |
fsetUCA A B C : A `|` (B `|` C) = B `|` (A `|` C). Proof. by rewrite !fsetUA (fsetUC B). Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetUCA | |
fsetUAC A B C : A `|` B `|` C = A `|` C `|` B. Proof. by rewrite -!fsetUA (fsetUC B). Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetUAC | |
fsetUACA A B C D : (A `|` B) `|` (C `|` D) = (A `|` C) `|` (B `|` D). Proof. by rewrite -!fsetUA (fsetUCA B). Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetUACA | |
fsetUid A : A `|` A = A. Proof. by apply/fsetP=> x; rewrite inE orbb. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetUid | |
fsetUUl A B C : A `|` B `|` C = (A `|` C) `|` (B `|` C). Proof. by rewrite fsetUA !(fsetUAC _ C) -(fsetUA _ C) fsetUid. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetUUl | |
fsetUUr A B C : A `|` (B `|` C) = (A `|` B) `|` (A `|` C). Proof. by rewrite !(fsetUC A) fsetUUl. Qed. (* distribute /cancel *) | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetUUr | |
fsetIUr A B C : A `&` (B `|` C) = (A `&` B) `|` (A `&` C). Proof. by apply/fsetP=> x; rewrite !inE andb_orr. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetIUr | |
fsetIUl A B C : (A `|` B) `&` C = (A `&` C) `|` (B `&` C). Proof. by apply/fsetP=> x; rewrite !inE andb_orl. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetIUl | |
fsetUIr A B C : A `|` (B `&` C) = (A `|` B) `&` (A `|` C). Proof. by apply/fsetP=> x; rewrite !inE orb_andr. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetUIr | |
fsetUIl A B C : (A `&` B) `|` C = (A `|` C) `&` (B `|` C). Proof. by apply/fsetP=> x; rewrite !inE orb_andl. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetUIl | |
fsetUKC A B : (A `|` B) `&` A = A. Proof. by apply/fsetP=> x; rewrite !inE orbK. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetUKC | |
fsetUK A B : (B `|` A) `&` A = A. Proof. by rewrite fsetUC fsetUKC. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetUK | |
fsetKUC A B : A `&` (B `|` A) = A. Proof. by rewrite fsetIC fsetUK. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetKUC | |
fsetKU A B : A `&` (A `|` B) = A. Proof. by rewrite fsetIC fsetUKC. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetKU | |
fsetIKC A B : (A `&` B) `|` A = A. Proof. by apply/fsetP=> x; rewrite !inE andbK. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetIKC | |
fsetIK A B : (B `&` A) `|` A = A. Proof. by rewrite fsetIC fsetIKC. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetIK | |
fsetKIC A B : A `|` (B `&` A) = A. Proof. by rewrite fsetUC fsetIK. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetKIC | |
fsetKI A B : A `|` (A `&` B) = A. Proof. by rewrite fsetIC fsetKIC. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetKI | |
fsetUKid A B : B `|` A `|` A = B `|` A. Proof. by rewrite -fsetUA fsetUid. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetUKid | |
fsetUKidC A B : A `|` B `|` A = A `|` B. Proof. by rewrite fsetUAC fsetUid. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetUKidC | |
fsetKUid A B : A `|` (A `|` B) = A `|` B. Proof. by rewrite fsetUA fsetUid. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetKUid | |
fsetKUidC A B : A `|` (B `|` A) = B `|` A. Proof. by rewrite fsetUCA fsetUid. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetKUidC | |
fsetIKid A B : B `&` A `&` A = B `&` A. Proof. by rewrite -fsetIA fsetIid. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetIKid | |
fsetIKidC A B : A `&` B `&` A = A `&` B. Proof. by rewrite fsetIAC fsetIid. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetIKidC | |
fsetKIid A B : A `&` (A `&` B) = A `&` B. Proof. by rewrite fsetIA fsetIid. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetKIid | |
fsetKIidC A B : A `&` (B `&` A) = B `&` A. Proof. by rewrite fsetICA fsetIid. Qed. (* subset *) | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetKIidC | |
fsubsetP {A B} : reflect {subset A <= B} (A `<=` B). Proof. apply: (iffP fset_eqP) => AsubB a; first by rewrite -AsubB inE => /andP[]. by rewrite inE; have [/AsubB|] := boolP (a \in A). Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsubsetP | |
fset_sub_val A (p : finmempred A) : (imfset key val p) `<=` A. Proof. by apply/fsubsetP => k /in_fset_valP []. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fset_sub_val | |
fset_sub A (P : pred K) : [fset x in A | P x] `<=` A. Proof. by apply/fsubsetP => k; rewrite in_fset inE /= => /andP []. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fset_sub | |
fsetD_eq0 (A B : {fset K}) : (A `\` B == fset0) = (A `<=` B). Proof. apply/fset_eqP/fsubsetP => sAB a. by move=> aA; have := sAB a; rewrite !inE aA andbT => /negPn. by rewrite !inE andbC; apply/negP => /andP [/sAB ->]. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetD_eq0 | |
fsubset_refl A : A `<=` A. Proof. exact/fsubsetP. Qed. Hint Resolve fsubset_refl : core. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsubset_refl | |
fincl A B (AsubB : A `<=` B) (a : A) : B := [` (fsubsetP AsubB) _ (valP a)]. | Definition | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fincl | |
fsub B A : {set B} := [set x : B | val x \in A]. | Definition | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsub | |
fsubE A B (AsubB : A `<=` B) : fsub B A = [set fincl AsubB x | x : A]. Proof. apply/setP => x; rewrite in_set; apply/idP/imsetP => [|[[a aA] aA' ->]] //. by move=> xA; exists [` xA]=> //; apply: val_inj. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsubE | |
fincl_fsub A B (AsubB : A `<=` B) (a : A) : fincl AsubB a \in fsub B A. Proof. by rewrite inE /= (valP a). Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fincl_fsub | |
in_fsub B A (b : B) : (b \in fsub B A) = (val b \in A). Proof. by rewrite inE. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | in_fsub | |
subset_fsubE C A B : A `<=` C -> B `<=` C -> (fsub C A \subset fsub C B) = (A `<=` B). Proof. move=> sAC sBC; apply/subsetP/fsubsetP => sAB a; last first. by rewrite !inE => /sAB. by move=> aA; have := sAB _ (fincl_fsub sAC [` aA]); rewrite inE. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | subset_fsubE | |
fsubset_trans : transitive (@fsubset K). Proof. by move=>??? s t ; apply/fsubsetP => a /(fsubsetP s) /(fsubsetP t). Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsubset_trans | |
subset_fsub A B C : A `<=` B -> B `<=` C -> fsub C A \subset fsub C B. Proof. by move=> sAB sBC; rewrite subset_fsubE // (fsubset_trans sAB). Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | subset_fsub | |
fsetIidPl {A B} : reflect (A `&` B = A) (A `<=` B). Proof. exact: eqP. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetIidPl | |
fsetIidPr {A B} : reflect (A `&` B = B) (B `<=` A). Proof. by rewrite fsetIC; apply: fsetIidPl. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsetIidPr | |
fsubsetIidl A B : (A `<=` A `&` B) = (A `<=` B). Proof. by apply/fsubsetP/fsubsetP=> sAB a aA; have := sAB _ aA; rewrite !inE ?aA. Qed. | Lemma | root | [
"From Corelib Require Import Setoid.",
"From HB Require Import structures.",
"From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.",
"From mathcomp Require Import choice finset finfun fintype bigop."
] | finmap.v | fsubsetIidl |
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