fact stringlengths 8 1.54k | type stringclasses 19 values | library stringclasses 8 values | imports listlengths 1 10 | filename stringclasses 98 values | symbolic_name stringlengths 1 42 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
eq_fcard: fcard_mem f =1 fcard_mem f'.
Proof. exact: eq_n_comp eq_fconnect. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | eq_fcard | |
eq_finv: finv f =1 finv f'.
Proof.
by move=> x; rewrite /finv /order (eq_card (@eq_fconnect x)) (eq_iter eq_f).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | eq_finv | |
eq_froot: froot f =1 froot f'.
Proof. exact: eq_root eq_rf. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | eq_froot | |
eq_froots: froots f =1 froots f'.
Proof. exact: eq_roots eq_rf. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | eq_froots | |
finv_inv: finv (finv f) =1 f.
Proof. exact: (finv_eq_can (f_finv injf)). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | finv_inv | |
order_finv: order (finv f) =1 order f.
Proof. by move=> x; apply: eq_card (@same_fconnect_finv _ _ injf x). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | order_finv | |
order_set_finvn : order_set (finv f) n =i order_set f n.
Proof. by move=> x; rewrite !inE order_finv. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | order_set_finv | |
rel_adjunction_memm_a := RelAdjunction {
rel_unit x : in_mem x m_a -> {x' : T' | connect e x (h x')};
rel_functor x' y' :
in_mem (h x') m_a -> connect e' x' y' = connect e (h x') (h y')
}.
Variable a : {pred T}.
Hypothesis cl_a : closed e a.
Local Notation rel_adjunction := (rel_adjunction_mem (mem a)). | Record | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | rel_adjunction_mem | |
intro_adjunction(h' : forall x, x \in a -> T') :
(forall x a_x,
[/\ connect e x (h (h' x a_x))
& forall y a_y, e x y -> connect e' (h' x a_x) (h' y a_y)]) ->
(forall x' a_x,
[/\ connect e' x' (h' (h x') a_x)
& forall y', e' x' y' -> connect e (h x') (h y')]) ->
rel_adjunction.
Proof.
move=> Aee' Ae'e; split=> [y a_y | x' z' a_x].
by exists (h' y a_y); case/Aee': (a_y).
apply/idP/idP=> [/connectP[p e'p ->{z'}] | /connectP[p e_p p_z']].
elim: p x' a_x e'p => //= y' p IHp x' a_x.
case: (Ae'e x' a_x) => _ Ae'x /andP[/Ae'x e_xy /IHp e_yz] {Ae'x}.
by apply: connect_trans (e_yz _); rewrite // -(closed_connect cl_a e_xy).
case: (Ae'e x' a_x) => /connect_trans-> //.
elim: p {x'}(h x') p_z' a_x e_p => /= [|y p IHp] x p_z' a_x.
by rewrite -p_z' in a_x *; case: (Ae'e _ a_x); rewrite sym_e'.
case/andP=> e_xy /(IHp _ p_z') e'yz; have a_y: y \in a by rewrite -(cl_a e_xy).
by apply: connect_trans (e'yz a_y); case: (Aee' _ a_x) => _ ->.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | intro_adjunction | |
strict_adjunction:
injective h -> a \subset codom h -> rel_base h e e' [predC a] ->
rel_adjunction.
Proof.
move=> /= injh h_a a_ee'; pose h' x Hx := iinv (subsetP h_a x Hx).
apply: (@intro_adjunction h') => [x a_x | x' a_x].
rewrite f_iinv connect0; split=> // y a_y e_xy.
by rewrite connect1 // -a_ee' !f_iinv ?negbK.
rewrite [h' _ _]iinv_f //; split=> // y' e'xy.
by rewrite connect1 // a_ee' ?negbK.
Qed.
Let ccl_a := closed_connect cl_a. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | strict_adjunction | |
adjunction_closed: rel_adjunction -> closed e' [preim h of a].
Proof.
case=> _ Ae'e; apply: intro_closed => // x' y' /connect1 e'xy a_x.
by rewrite Ae'e // in e'xy; rewrite !inE -(ccl_a e'xy).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | adjunction_closed | |
adjunction_n_comp:
rel_adjunction -> n_comp e a = n_comp e' [preim h of a].
Proof.
case=> Aee' Ae'e.
have inj_h: {in predI (roots e') [preim h of a] &, injective (root e \o h)}.
move=> x' y' /andP[/eqP r_x' /= a_x'] /andP[/eqP r_y' _] /(rootP sym_e).
by rewrite -Ae'e // => /(rootP sym_e'); rewrite r_x' r_y'.
rewrite /n_comp_mem -(card_in_image inj_h); apply: eq_card => x.
apply/andP/imageP=> [[/eqP rx a_x] | [x' /andP[/eqP r_x' a_x'] ->]]; last first.
by rewrite /= -(ccl_a (connect_root _ _)) roots_root.
have [y' e_xy]:= Aee' x a_x; pose x' := root e' y'.
have ay': h y' \in a by rewrite -(ccl_a e_xy).
have e_yx: connect e (h y') (h x') by rewrite -Ae'e ?connect_root.
exists x'; first by rewrite inE /= -(ccl_a e_yx) ?roots_root.
by rewrite /= -(rootP sym_e e_yx) -(rootP sym_e e_xy).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | adjunction_n_comp | |
rel_adjunctionh e e' a := (rel_adjunction_mem h e e' (mem a)). | Notation | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | rel_adjunction | |
fun_adjunctionh f f' a := (rel_adjunction h (frel f) (frel f') a). | Notation | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fun_adjunction | |
set_type: predArgType := FinSet of {ffun pred T}. | Inductive | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set_type | |
finfun_of_setA := let: FinSet f := A in f. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | finfun_of_set | |
set_of:= set_type.
Identity Coercion type_of_set_of : set_of >-> set_type. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set_of | |
set_isSub:= Eval hnf in [isNew for finfun_of_set].
HB.instance Definition _ := set_isSub.
HB.instance Definition _ := [Finite of set_type by <:]. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set_isSub | |
finset(T : finType) (P : pred T) : {set T} := @FinSet T (finfun P). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | finset | |
finset_unlock:= Unlockable finset.unlock. | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | finset_unlock | |
pred_of_setT (A : set_type T) : fin_pred_sort (predPredType T)
:= val A. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | pred_of_set | |
pred_of_set_unlock:= Unlockable pred_of_set.unlock. | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | pred_of_set_unlock | |
pred_of_set: set_type >-> fin_pred_sort. | Coercion | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | pred_of_set | |
set_predTypeT := @PredType _ (unkeyed (set_type T)) (@pred_of_set T). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set_predType | |
Definition_ := Finite.on {set T}. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | Definition | |
in_setpA x : x \in finset pA = pA x.
Proof. by rewrite [@finset]unlock unlock [x \in _]ffunE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | in_set | |
setPA B : A =i B <-> A = B.
Proof.
by split=> [eqAB|-> //]; apply/val_inj/ffunP=> x; have:= eqAB x; rewrite unlock.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setP | |
set0:= [set x : T | false]. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set0 | |
setTfor:= [set x : T | true]. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setTfor | |
in_setTx : x \in setTfor.
Proof. by rewrite in_set. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | in_setT | |
eqsVneqA B : eq_xor_neq A B (B == A) (A == B).
Proof. exact: eqVneq. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | eqsVneq | |
eq_finset(pA pB : pred T) : pA =1 pB -> finset pA = finset pB.
Proof. by move=> eq_p; apply/setP => x; rewrite !(in_set, inE) eq_p. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | eq_finset | |
setT:= [set: _] (only parsing).
HB.lock | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setT | |
set1(T : finType) (a : T) := [set x | x == a]. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set1 | |
setUA B := [set x | (x \in A) || (x \in B)]. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setU | |
setIA B := [set x in A | x \in B]. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setI | |
setCA := [set x | x \notin A]. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setC | |
setDA B := [set x | x \notin B & x \in A]. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setD | |
ssetIP D := [set A in P | A \subset D]. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | ssetI | |
powersetD := [set A : {set T} | A \subset D]. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | powerset | |
eqEsubsetA B : (A == B) = (A \subset B) && (B \subset A).
Proof. by apply/eqP/subset_eqP=> /setP. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | eqEsubset | |
subEproperA B : A \subset B = (A == B) || (A \proper B).
Proof. by rewrite eqEsubset -andb_orr orbN andbT. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | subEproper | |
eqVproperA B : A \subset B -> A = B \/ A \proper B.
Proof. by rewrite subEproper => /predU1P. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | eqVproper | |
properEneqA B : A \proper B = (A != B) && (A \subset B).
Proof. by rewrite andbC eqEsubset negb_and andb_orr andbN. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | properEneq | |
proper_neqA B : A \proper B -> A != B.
Proof. by rewrite properEneq; case/andP. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | proper_neq | |
eqEproperA B : (A == B) = (A \subset B) && ~~ (A \proper B).
Proof. by rewrite negb_and negbK andb_orr andbN eqEsubset. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | eqEproper | |
eqEcardA B : (A == B) = (A \subset B) && (#|B| <= #|A|).
Proof.
rewrite eqEsubset; apply: andb_id2l => sAB.
by rewrite (geq_leqif (subset_leqif_card sAB)).
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | eqEcard | |
properEcardA B : (A \proper B) = (A \subset B) && (#|A| < #|B|).
Proof. by rewrite properEneq ltnNge andbC eqEcard; case: (A \subset B). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | properEcard | |
subset_leqif_cardsA B : A \subset B -> (#|A| <= #|B| ?= iff (A == B)).
Proof. by move=> sAB; rewrite eqEsubset sAB; apply: subset_leqif_card. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | subset_leqif_cards | |
in_set0x : x \in set0 = false.
Proof. by rewrite in_set. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | in_set0 | |
sub0setA : set0 \subset A.
Proof. by apply/subsetP=> x; rewrite in_set. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | sub0set | |
subset0A : (A \subset set0) = (A == set0).
Proof. by rewrite eqEsubset sub0set andbT. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | subset0 | |
proper0A : (set0 \proper A) = (A != set0).
Proof. by rewrite properE sub0set subset0. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | proper0 | |
subset_neq0A B : A \subset B -> A != set0 -> B != set0.
Proof. by rewrite -!proper0 => sAB /proper_sub_trans->. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | subset_neq0 | |
set_0VmemA : (A = set0) + {x : T | x \in A}.
Proof.
case: (pickP (mem A)) => [x Ax | A0]; [by right; exists x | left].
by apply/setP=> x; rewrite in_set; apply: A0.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set_0Vmem | |
set_enumA : [set x | x \in enum A] = A.
Proof. by apply/setP => x; rewrite in_set mem_enum. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set_enum | |
enum_set0: enum set0 = [::] :> seq T.
Proof. by rewrite (eq_enum (in_set _)) enum0. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | enum_set0 | |
subsetTA : A \subset setT.
Proof. by apply/subsetP=> x; rewrite in_set. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | subsetT | |
subsetT_hintmA : subset mA (mem [set: T]).
Proof. by rewrite unlock; apply/pred0P=> x; rewrite !inE in_set. Qed.
Hint Resolve subsetT_hint : core. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | subsetT_hint | |
subTsetA : (setT \subset A) = (A == setT).
Proof. by rewrite eqEsubset subsetT. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | subTset | |
properTA : (A \proper setT) = (A != setT).
Proof. by rewrite properEneq subsetT andbT. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | properT | |
set1Px a : reflect (x = a) (x \in [set a]).
Proof. by rewrite set1.unlock in_set; apply: eqP. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set1P | |
enum_setT: enum [set: T] = Finite.enum T.
Proof. by rewrite (eq_enum (in_set _)) enumT. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | enum_setT | |
in_set1x a : (x \in [set a]) = (x == a).
Proof. by rewrite set1.unlock in_set. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | in_set1 | |
inE:= (in_set, in_set1, inE). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | inE | |
set11x : x \in [set x].
Proof. by rewrite !inE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set11 | |
set1_inj: injective (@set1 T).
Proof. by move=> a b eqsab; apply/set1P; rewrite -eqsab set11. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set1_inj | |
enum_set1a : enum [set a] = [:: a].
Proof. by rewrite set1.unlock (eq_enum (in_set _)) enum1. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | enum_set1 | |
setU1Px a B : reflect (x = a \/ x \in B) (x \in a |: B).
Proof. by rewrite !inE; apply: predU1P. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setU1P | |
in_setU1x a B : (x \in a |: B) = (x == a) || (x \in B).
Proof. by rewrite !inE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | in_setU1 | |
set_nil: [set:: nil] = @set0 T. Proof. by rewrite -enum_set0 set_enum. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set_nil | |
set_seq1a : [set:: [:: a]] = [set a].
Proof. by rewrite -enum_set1 set_enum. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set_seq1 | |
set_consa s : [set:: a :: s] = a |: [set:: s].
Proof. by apply/setP=> x; rewrite !inE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set_cons | |
setU11x B : x \in x |: B.
Proof. by rewrite !inE eqxx. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setU11 | |
setU1rx a B : x \in B -> x \in a |: B.
Proof. by move=> Bx; rewrite !inE predU1r. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setU1r | |
set1Ulx A b : x \in A -> x \in A :|: [set b].
Proof. by move=> Ax; rewrite !inE Ax. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set1Ul | |
set1UrA b : b \in A :|: [set b].
Proof. by rewrite !inE eqxx orbT. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set1Ur | |
in_setC1x a : (x \in [set~ a]) = (x != a).
Proof. by rewrite !inE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | in_setC1 | |
setC11x : (x \in [set~ x]) = false.
Proof. by rewrite !inE eqxx. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setC11 | |
setD1Px A b : reflect (x != b /\ x \in A) (x \in A :\ b).
Proof. by rewrite !inE; apply: andP. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setD1P | |
in_setD1x A b : (x \in A :\ b) = (x != b) && (x \in A) .
Proof. by rewrite !inE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | in_setD1 | |
setD11b A : (b \in A :\ b) = false.
Proof. by rewrite !inE eqxx. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setD11 | |
setD1Ka A : a \in A -> a |: (A :\ a) = A.
Proof. by move=> Aa; apply/setP=> x /[!inE]; case: eqP => // ->. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setD1K | |
setU1Ka B : a \notin B -> (a |: B) :\ a = B.
Proof.
by move/negPf=> nBa; apply/setP=> x /[!inE]; case: eqP => // ->.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setU1K | |
set2Px a b : reflect (x = a \/ x = b) (x \in [set a; b]).
Proof. by rewrite !inE; apply: pred2P. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set2P | |
in_set2x a b : (x \in [set a; b]) = (x == a) || (x == b).
Proof. by rewrite !inE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | in_set2 | |
set21a b : a \in [set a; b].
Proof. by rewrite !inE eqxx. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set21 | |
set22a b : b \in [set a; b].
Proof. by rewrite !inE eqxx orbT. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set22 | |
setUPx A B : reflect (x \in A \/ x \in B) (x \in A :|: B).
Proof. by rewrite !inE; apply: orP. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setUP | |
in_setUx A B : (x \in A :|: B) = (x \in A) || (x \in B).
Proof. exact: in_set. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | in_setU | |
setUCA B : A :|: B = B :|: A.
Proof. by apply/setP => x; rewrite !inE orbC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setUC | |
setUSA B C : A \subset B -> C :|: A \subset C :|: B.
Proof.
move=> sAB; apply/subsetP=> x; rewrite !inE.
by case: (x \in C) => //; apply: (subsetP sAB).
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setUS | |
setSUA B C : A \subset B -> A :|: C \subset B :|: C.
Proof. by move=> sAB; rewrite -!(setUC C) setUS. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setSU | |
setUSSA B C D : A \subset C -> B \subset D -> A :|: B \subset C :|: D.
Proof. by move=> /(setSU B) /subset_trans sAC /(setUS C)/sAC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setUSS | |
set0UA : set0 :|: A = A.
Proof. by apply/setP => x; rewrite !inE orFb. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set0U | |
setU0A : A :|: set0 = A.
Proof. by rewrite setUC set0U. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setU0 | |
setUAA B C : A :|: (B :|: C) = A :|: B :|: C.
Proof. by apply/setP => x; rewrite !inE orbA. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setUA | |
setUCAA B C : A :|: (B :|: C) = B :|: (A :|: C).
Proof. by rewrite !setUA (setUC B). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setUCA | |
setUACA B C : A :|: B :|: C = A :|: C :|: B.
Proof. by rewrite -!setUA (setUC B). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setUAC | |
setUACAA B C D : (A :|: B) :|: (C :|: D) = (A :|: C) :|: (B :|: D).
Proof. by rewrite -!setUA (setUCA B). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setUACA |
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