# Library mathcomp.ssreflect.finset

(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.
Distributed under the terms of CeCILL-B. *)
From HB Require Import structures.
From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq.
From mathcomp Require Import choice fintype finfun bigop.
This file defines a type for sets over a finite Type, similar to the type of functions over a finite Type defined in finfun.v (indeed, based in it): {set T} where T must have a finType structure. We equip {set T} itself with a finType structure, hence Leibnitz and extensional equalities coincide on {set T}, and we can form {set {set T}}. If A, B : {set T} and P : {set {set T}}, we define: x \in A == x belongs to A (i.e., {set T} implements predType, by coercion to pred_sort) mem A == the predicate corresponding to A finset p == the set corresponding to a predicate p [set x | P] == the set containing the x such that P is true (x may appear in P) [set x | P & Q] := [set x | P && Q] [set x in A] == the set containing the x in a collective predicate A [set x in A | P] == the set containing the x in A such that P is true [set x in A | P & Q] := [set x in A | P && Q] All these have typed variants [set x : T | P], [set x : T in A], etc. set0 == the empty set [set: T] or setT == the full set (the A containing all x : T) [set x] == the singleton {x} [set~ x] == the complement of the singleton {x} [set:: s] == the set spanned by the sequence s [set a1; a2;...; an] := a1 |: [set a2] :|: ... :|: [set an] A :|: B == the union of A and B x |: A == A with the element x added (:= [set x] :|: A) A :&: B == the intersection of A and B ~: A == the complement of A A :\: B == the difference A minus B A :\ x == A with the element x removed (:= A :\: [set x]) \bigcup_<range> A == the union of all A, for i in <range> (i is bound in A, see bigop.v) \bigcap_<range> A == the intersection of all A, for i in <range> cover P == the union of the set of sets P trivIset P <=> the elements of P are pairwise disjoint partition P A <=> P is a partition of A pblock P x == a block of P containing x, or else set0 equivalence_partition R D == the partition induced on D by the relation R (provided R is an equivalence relation in D) preim_partition f D == the partition induced on D by the equivalence [rel x y | f x == f y] is_transversal X P D <=> X is a transversal of the partition P of D transversal P D == a transversal of P, provided P is a partition of D transversal_repr x0 X B == a representative of B \in P selected by the transversal X of P, or else x0 powerset A == the set of all subset of the set A P ::&: A == those sets in P that are subsets of the set A setX A1 A2 == cartesian product of A1 and A2 := [set u | u.1 \in A1 & u.2 \in A2] setXn I f A == indexed cartesian product of A : forall i : I, {set f i} f @^-1: A == the preimage of the collective predicate A under f f @: A == the image set of the collective predicate A by f f @2:(A, B) == the image set of A x B by the binary function f [set E | x in A] == the set of all the values of the expression E, for x drawn from the collective predicate A [set E | x in A & P] == the set of values of E for x drawn from A, such that P is true [set E | x in A, y in B] == the set of values of E for x drawn from A and and y drawn from B; B may depend on x [set E | x in A, y in B & P] == the set of values of E for x drawn from A y drawn from B, such that P is true [set E | x : T] == the set of all values of E, with x in type T [set E | x : T & P] == the set of values of E for x : T s.t. P is true [set E | x : T, y : U in B], [set E | x : T, y : U in B & P], [set E | x : T in A, y : U], [set E | x : T in A, y : U & P], [set E | x : T, y : U], [set E | x : T, y : U & P] == type-ranging versions of the binary comprehensions [set E | x : T in A], [set E | x in A, y], [set E | x, y & P], etc. == typed and untyped variants of the comprehensions above The types may be required as type inference processes E before considering A or B. Note that type casts in the binary comprehension must either be both present or absent and that there are no untyped variants for single-type comprehension as Coq parsing confuses [x | P] and [E | x]. minset p A == A is a minimal set satisfying p maxset p A == A is a maximal set satisfying p Provided a monotonous function F : {set T} -> {set T}, we get fixpoints fixset F := iter #|T| F set0 == the least fixpoint of F == the minimal set such that F X == X fix_order F x == the minimum number of iterations so that x is in iter (fix_order F x) F set0 funsetC F := fun X => ~: F (~: X) cofixset F == the greatest fixpoint of F == the maximal set such that F X == X := ~: fixset (funsetC F) We also provide notations A :=: B, A :<>: B, A :==: B, A :!=: B, A :=P: B that specialize A = B, A <> B, A == B, etc., to {set _}. This is useful for subtypes of {set T}, such as {group T}, that coerce to {set T}. We give many lemmas on these operations, on card, and on set inclusion. In addition to the standard suffixes described in ssrbool.v, we associate the following suffixes to set operations: 0 -- the empty set, as in in_set0 : (x \in set0) = false T -- the full set, as in in_setT : x \in [set: T] 1 -- a singleton set, as in in_set1 : (x \in [set a]) = (x == a) 2 -- an unordered pair, as in in_set2 : (x \in [set a; b]) = (x == a) || (x == b) C -- complement, as in setCK : ~: ~: A = A I -- intersection, as in setIid : A :&: A = A U -- union, as in setUid : A :|: A = A D -- difference, as in setDv : A :\: A = set0 S -- a subset argument, as in setIS: B \subset C -> A :&: B \subset A :&: C These suffixes are sometimes preceded with an `s' to distinguish them from their basic ssrbool interpretation, e.g., card1 : #|pred1 x| = 1 and cards1 : #| [set x]| = 1 We also use a trailing `r' to distinguish a right-hand complement from commutativity, e.g., setIC : A :&: B = B :&: A and setICr : A :&: ~: A = set0.
Set Implicit Arguments.
Declare Scope set_scope.
Section SetType.
Variable T : finType.
Inductive set_type : predArgType := FinSet of {ffun pred T}.
Definition finfun_of_set A := let: FinSet f := A in f.
Definition set_of := set_type.
Identity Coercion type_of_set_of : set_of >-> set_type.
Definition set_isSub := Eval hnf in [isNew for finfun_of_set].
End SetType.
Delimit Scope set_scope with SET.
Bind Scope set_scope with set_type.
Bind Scope set_scope with set_of.
Open Scope set_scope.
Arguments set_of T%type.
Arguments finfun_of_set {T} A%SET.
Notation "{ 'set' T }" := (set_of T)
(at level 0, format "{ 'set' T }") : type_scope.
We later define several subtypes that coerce to set; for these it is preferable to state equalities at the {set _} level, even when comparing subtype values, because the primitive "injection" tactic tends to diverge on complex types (e.g., quotient groups). We provide some parse-only notation to make this technicality less obstructive.
Notation "A :=: B" := (A = B :> {set _})
(at level 70, no associativity, only parsing) : set_scope.
Notation "A :<>: B" := (A ≠ B :> {set _})
(at level 70, no associativity, only parsing) : set_scope.
Notation "A :==: B" := (A == B :> {set _})
(at level 70, no associativity, only parsing) : set_scope.
Notation "A :!=: B" := (A != B :> {set _})
(at level 70, no associativity, only parsing) : set_scope.
Notation "A :=P: B" := (A =P B :> {set _})
(at level 70, no associativity, only parsing) : set_scope.
Canonical finset_unlock := Unlockable finset.unlock.
The weird type of pred_of_set is imposed by the syntactic restrictions on coercion declarations; it is unfortunately not possible to use a functor to retype the declaration, because this triggers an ugly bug in the Coq coercion chaining code.
Canonical pred_of_set_unlock := Unlockable pred_of_set.unlock.
Notation "[ 'set' x : T | P ]" := (finset (fun x : T ⇒ P%B))
(at level 0, x at level 99, only parsing) : set_scope.
Notation "[ 'set' x | P ]" := [set x : _ | P]
(at level 0, x, P at level 99, format "[ 'set' x | P ]") : set_scope.
Notation "[ 'set' x 'in' A ]" := [set x | x \in A]
(at level 0, x at level 99, format "[ 'set' x 'in' A ]") : set_scope.
Notation "[ 'set' x : T 'in' A ]" := [set x : T | x \in A]
(at level 0, x at level 99, only parsing) : set_scope.
Notation "[ 'set' x : T | P & Q ]" := [set x : T | P && Q]
(at level 0, x at level 99, only parsing) : set_scope.
Notation "[ 'set' x | P & Q ]" := [set x | P && Q ]
(at level 0, x, P at level 99, format "[ 'set' x | P & Q ]") : set_scope.
Notation "[ 'set' x : T 'in' A | P ]" := [set x : T | x \in A & P]
(at level 0, x at level 99, only parsing) : set_scope.
Notation "[ 'set' x 'in' A | P ]" := [set x | x \in A & P]
(at level 0, x at level 99, format "[ 'set' x 'in' A | P ]") : set_scope.
Notation "[ 'set' x 'in' A | P & Q ]" := [set x in A | P && Q]
(at level 0, x at level 99,
format "[ 'set' x 'in' A | P & Q ]") : set_scope.
Notation "[ 'set' x : T 'in' A | P & Q ]" := [set x : T in A | P && Q]
(at level 0, x at level 99, only parsing) : set_scope.
Notation "[ 'set' :: s ]" := (finset [in pred_of_seq s])
(at level 0, format "[ 'set' :: s ]") : set_scope.
This lets us use set and subtypes of set, like group or coset_of, both as collective predicates and as arguments of the \pi(_) notation.
Coercion pred_of_set: set_type >-> fin_pred_sort.
Declare pred_of_set as a canonical instance of topred, but use the coercion to resolve mem A to @mem (predPredType T) (pred_of_set A).
Canonical set_predType T := @PredType _ (unkeyed (set_type T)) (@pred_of_set T).
Section BasicSetTheory.
Variable T : finType.
Implicit Types (x : T) (A B : {set T}) (pA : pred T).
Lemma in_set pA x : x \in finset pA = pA x.
Lemma setP A B : A =i B ↔ A = B.
Definition set0 := [set x : T | false].
Definition setTfor := [set x : T | true].
Lemma in_setT x : x \in setTfor.
Lemma eqsVneq A B : eq_xor_neq A B (B == A) (A == B).
Lemma eq_finset (pA pB : pred T) : pA =1 pB → finset pA = finset pB.
End BasicSetTheory.
Arguments eqsVneq {T} A B, {T A B}.
Arguments set0 {T}.
Arguments setTfor T%type.
Arguments eq_finset {T} [pA] pB eq_pAB.
#[global] Hint Resolve in_setT : core.
Notation "[ 'set' : T ]" := (setTfor T)
(at level 0, format "[ 'set' : T ]") : set_scope.
Notation setT := [set: _] (only parsing).
Section setOpsDefs.
Variable T : finType.
Implicit Types (a x : T) (A B D : {set T}) (P : {set {set T}}).
Definition setU A B := [set x | (x \in A) || (x \in B)].
Definition setI A B := [set x in A | x \in B].
Definition setC A := [set x | x \notin A].
Definition setD A B := [set x | x \notin B & x \in A].
Definition ssetI P D := [set A in P | A \subset D].
Definition powerset D := [set A : {set T} | A \subset D].
End setOpsDefs.
Notation "[ 'set' a ]" := (set1 a)
(at level 0, a at level 99, format "[ 'set' a ]") : set_scope.
Notation "[ 'set' a : T ]" := [set (a : T)]
(at level 0, a at level 99, format "[ 'set' a : T ]") : set_scope.
Notation "A :|: B" := (setU A B) : set_scope.
Notation "a |: A" := ([set a] :|: A) : set_scope.
This is left-associative due to historical limitations of the .. Notation.
Notation "[ 'set' a1 ; a2 ; .. ; an ]" := (setU .. (a1 |: [set a2]) .. [set an])
(at level 0, a1 at level 99,
format "[ 'set' a1 ; a2 ; .. ; an ]") : set_scope.
Notation "A :&: B" := (setI A B) : set_scope.
Notation "~: A" := (setC A) (at level 35, right associativity) : set_scope.
Notation "[ 'set' ~ a ]" := (~: [set a])
(at level 0, format "[ 'set' ~ a ]") : set_scope.
Notation "A :\: B" := (setD A B) : set_scope.
Notation "A :\ a" := (A :\: [set a]) : set_scope.
Notation "P ::&: D" := (ssetI P D) (at level 48) : set_scope.
Section setOps.
Variable T : finType.
Implicit Types (a x : T) (A B C D : {set T}) (pA pB pC : pred T).
Lemma eqEsubset A B : (A == B) = (A \subset B) && (B \subset A).
Lemma subEproper A B : A \subset B = (A == B) || (A \proper B).
Lemma eqVproper A B : A \subset B → A = B ∨ A \proper B.
Lemma properEneq A B : A \proper B = (A != B) && (A \subset B).
Lemma proper_neq A B : A \proper B → A != B.
Lemma eqEproper A B : (A == B) = (A \subset B) && ~~ (A \proper B).
Lemma eqEcard A B : (A == B) = (A \subset B) && (#|B| ≤ #|A|).
Lemma properEcard A B : (A \proper B) = (A \subset B) && (#|A| < #|B|).
Lemma subset_leqif_cards A B : A \subset B → (#|A| ≤ #|B| ?= iff (A == B)).
Lemma in_set0 x : x \in set0 = false.
Lemma sub0set A : set0 \subset A.
Lemma subset0 A : (A \subset set0) = (A == set0).
Lemma proper0 A : (set0 \proper A) = (A != set0).
Lemma subset_neq0 A B : A \subset B → A != set0 → B != set0.
Lemma set_0Vmem A : (A = set0) + {x : T | x \in A}.
Lemma set_enum A : [set x | x \in enum A] = A.
Lemma enum_set0 : enum set0 = [::] :> seq T.
Lemma subsetT A : A \subset setT.
Lemma subsetT_hint mA : subset mA (mem [set: T]).
Hint Resolve subsetT_hint : core.
Lemma subTset A : (setT \subset A) = (A == setT).
Lemma properT A : (A \proper setT) = (A != setT).
Lemma set1P x a : reflect (x = a) (x \in [set a]).
Lemma enum_setT : enum [set: T] = Finite.enum T.
Lemma in_set1 x a : (x \in [set a]) = (x == a).
Definition inE := (in_set, in_set1, inE).
Lemma set11 x : x \in [set x].
Lemma set1_inj : injective (@set1 T).
Lemma enum_set1 a : enum [set a] = [:: a].
Lemma setU1P x a B : reflect (x = a ∨ x \in B) (x \in a |: B).
Lemma in_setU1 x a B : (x \in a |: B) = (x == a) || (x \in B).
Lemma set_nil : [set:: nil] = @set0 T.
Lemma set_seq1 a : [set:: [:: a]] = [set a].
Lemma set_cons a s : [set:: a :: s] = a |: [set:: s].
Lemma setU11 x B : x \in x |: B.
Lemma setU1r x a B : x \in B → x \in a |: B.
We need separate lemmas for the explicit enumerations since they associate on the left.
Lemma set1Ul x A b : x \in A → x \in A :|: [set b].
Lemma set1Ur A b : b \in A :|: [set b].
Lemma in_setC1 x a : (x \in [set¬ a]) = (x != a).
Lemma setC11 x : (x \in [set¬ x]) = false.
Lemma setD1P x A b : reflect (x != b ∧ x \in A) (x \in A :\ b).
Lemma in_setD1 x A b : (x \in A :\ b) = (x != b) && (x \in A) .
Lemma setD11 b A : (b \in A :\ b) = false.
Lemma setD1K a A : a \in A → a |: (A :\ a) = A.
Lemma setU1K a B : a \notin B → (a |: B) :\ a = B.
Lemma set2P x a b : reflect (x = a ∨ x = b) (x \in [set a; b]).
Lemma in_set2 x a b : (x \in [set a; b]) = (x == a) || (x == b).
Lemma set21 a b : a \in [set a; b].
Lemma set22 a b : b \in [set a; b].
Lemma setUP x A B : reflect (x \in A ∨ x \in B) (x \in A :|: B).
Lemma in_setU x A B : (x \in A :|: B) = (x \in A) || (x \in B).
Lemma setUC A B : A :|: B = B :|: A.
Lemma setUS A B C : A \subset B → C :|: A \subset C :|: B.
Lemma setSU A B C : A \subset B → A :|: C \subset B :|: C.
Lemma setUSS A B C D : A \subset C → B \subset D → A :|: B \subset C :|: D.
Lemma set0U A : set0 :|: A = A.
Lemma setU0 A : A :|: set0 = A.
Lemma setUA A B C : A :|: (B :|: C) = A :|: B :|: C.
Lemma setUCA A B C : A :|: (B :|: C) = B :|: (A :|: C).
Lemma setUAC A B C : A :|: B :|: C = A :|: C :|: B.
Lemma setUACA A B C D : (A :|: B) :|: (C :|: D) = (A :|: C) :|: (B :|: D).
Lemma setTU A : setT :|: A = setT.
Lemma setUT A : A :|: setT = setT.
Lemma setUid A : A :|: A = A.
Lemma setUUl A B C : A :|: B :|: C = (A :|: C) :|: (B :|: C).
Lemma setUUr A B C : A :|: (B :|: C) = (A :|: B) :|: (A :|: C).
intersection setIdP is a generalisation of setIP that applies to comprehensions.
Lemma setIdP x pA pB : reflect (pA x ∧ pB x) (x \in [set y | pA y & pB y]).
Lemma setId2P x pA pB pC :
reflect [/\ pA x, pB x & pC x] (x \in [set y | pA y & pB y && pC y]).
Lemma setIdE A pB : [set x in A | pB x] = A :&: [set x | pB x].
Lemma setIP x A B : reflect (x \in A ∧ x \in B) (x \in A :&: B).
Lemma in_setI x A B : (x \in A :&: B) = (x \in A) && (x \in B).
Lemma setIC A B : A :&: B = B :&: A.
Lemma setIS A B C : A \subset B → C :&: A \subset C :&: B.
Lemma setSI A B C : A \subset B → A :&: C \subset B :&: C.
Lemma setISS A B C D : A \subset C → B \subset D → A :&: B \subset C :&: D.
Lemma setTI A : setT :&: A = A.
Lemma setIT A : A :&: setT = A.
Lemma set0I A : set0 :&: A = set0.
Lemma setI0 A : A :&: set0 = set0.
Lemma setIA A B C : A :&: (B :&: C) = A :&: B :&: C.
Lemma setICA A B C : A :&: (B :&: C) = B :&: (A :&: C).
Lemma setIAC A B C : A :&: B :&: C = A :&: C :&: B.
Lemma setIACA A B C D : (A :&: B) :&: (C :&: D) = (A :&: C) :&: (B :&: D).
Lemma setIid A : A :&: A = A.
Lemma setIIl A B C : A :&: B :&: C = (A :&: C) :&: (B :&: C).
Lemma setIIr A B C : A :&: (B :&: C) = (A :&: B) :&: (A :&: C).
distribute /cancel
Lemma setIUr A B C : A :&: (B :|: C) = (A :&: B) :|: (A :&: C).
Lemma setIUl A B C : (A :|: B) :&: C = (A :&: C) :|: (B :&: C).
Lemma setUIr A B C : A :|: (B :&: C) = (A :|: B) :&: (A :|: C).
Lemma setUIl A B C : (A :&: B) :|: C = (A :|: C) :&: (B :|: C).
Lemma setUK A B : (A :|: B) :&: A = A.
Lemma setKU A B : A :&: (B :|: A) = A.
Lemma setIK A B : (A :&: B) :|: A = A.
Lemma setKI A B : A :|: (B :&: A) = A.
complement
Lemma setCP x A : reflect (¬ x \in A) (x \in ~: A).
Lemma in_setC x A : (x \in ~: A) = (x \notin A).
Lemma setCK : involutive (@setC T).
Lemma setC_inj : injective (@setC T).
Lemma subsets_disjoint A B : (A \subset B) = [disjoint A & ~: B].
Lemma disjoints_subset A B : [disjoint A & B] = (A \subset ~: B).
Lemma powersetCE A B : (A \in powerset (~: B)) = [disjoint A & B].
Lemma setCS A B : (~: A \subset ~: B) = (B \subset A).
Lemma setCU A B : ~: (A :|: B) = ~: A :&: ~: B.
Lemma setCI A B : ~: (A :&: B) = ~: A :|: ~: B.
Lemma setUCr A : A :|: ~: A = setT.
Lemma setICr A : A :&: ~: A = set0.
Lemma setC0 : ~: set0 = [set: T].
Lemma setCT : ~: [set: T] = set0.
Lemma properC A B : (~: B \proper ~: A) = (A \proper B).
difference
Lemma setDP A B x : reflect (x \in A ∧ x \notin B) (x \in A :\: B).
Lemma in_setD A B x : (x \in A :\: B) = (x \notin B) && (x \in A).
Lemma setDE A B : A :\: B = A :&: ~: B.
Lemma setSD A B C : A \subset B → A :\: C \subset B :\: C.
Lemma setDS A B C : A \subset B → C :\: B \subset C :\: A.
Lemma setDSS A B C D : A \subset C → D \subset B → A :\: B \subset C :\: D.
Lemma setD0 A : A :\: set0 = A.
Lemma set0D A : set0 :\: A = set0.
Lemma setDT A : A :\: setT = set0.
Lemma setTD A : setT :\: A = ~: A.
Lemma setDv A : A :\: A = set0.
Lemma setCD A B : ~: (A :\: B) = ~: A :|: B.
Lemma setID A B : A :&: B :|: A :\: B = A.
Lemma setDUl A B C : (A :|: B) :\: C = (A :\: C) :|: (B :\: C).
Lemma setDUr A B C : A :\: (B :|: C) = (A :\: B) :&: (A :\: C).
Lemma setDIl A B C : (A :&: B) :\: C = (A :\: C) :&: (B :\: C).
Lemma setIDA A B C : A :&: (B :\: C) = (A :&: B) :\: C.
Lemma setIDAC A B C : (A :\: B) :&: C = (A :&: C) :\: B.
Lemma setDIr A B C : A :\: (B :&: C) = (A :\: B) :|: (A :\: C).
Lemma setDDl A B C : (A :\: B) :\: C = A :\: (B :|: C).
Lemma setDDr A B C : A :\: (B :\: C) = (A :\: B) :|: (A :&: C).
powerset
Lemma powersetE A B : (A \in powerset B) = (A \subset B).
Lemma powersetS A B : (powerset A \subset powerset B) = (A \subset B).
Lemma powerset0 : powerset set0 = [set set0] :> {set {set T}}.
Lemma powersetT : powerset [set: T] = [set: {set T}].
Lemma setI_powerset P A : P :&: powerset A = P ::&: A.
cardinal lemmas for sets
Lemma cardsE pA : #|[set x in pA]| = #|pA|.
Lemma sum1dep_card pA : \sum_(x | pA x) 1 = #|[set x | pA x]|.
Lemma sum_nat_cond_const pA n : \sum_(x | pA x) n = #|[set x | pA x]| × n.
Lemma cards0 : #|@set0 T| = 0.
Lemma cards_eq0 A : (#|A| == 0) = (A == set0).
Lemma set0Pn A : reflect (∃ x, x \in A) (A != set0).
Lemma card_gt0 A : (0 < #|A|) = (A != set0).
Lemma cards0_eq A : #|A| = 0 → A = set0.
Lemma cards1 x : #|[set x]| = 1.
Lemma cardsUI A B : #|A :|: B| + #|A :&: B| = #|A| + #|B|.
Lemma cardsU A B : #|A :|: B| = (#|A| + #|B| - #|A :&: B|)%N.
Lemma cardsI A B : #|A :&: B| = (#|A| + #|B| - #|A :|: B|)%N.
Lemma cardsT : #|[set: T]| = #|T|.
Lemma cardsID B A : #|A :&: B| + #|A :\: B| = #|A|.
Lemma cardsD A B : #|A :\: B| = (#|A| - #|A :&: B|)%N.
Lemma cardsC A : #|A| + #|~: A| = #|T|.
Lemma cardsCs A : #|A| = #|T| - #|~: A|.
Lemma cardsU1 a A : #|a |: A| = (a \notin A) + #|A|.
Lemma cards2 a b : #|[set a; b]| = (a != b).+1.
Lemma cardsC1 a : #|[set¬ a]| = #|T|.-1.
Lemma cardsD1 a A : #|A| = (a \in A) + #|A :\ a|.
other inclusions
Lemma subsetIl A B : A :&: B \subset A.
Lemma subsetIr A B : A :&: B \subset B.
Lemma subsetUl A B : A \subset A :|: B.
Lemma subsetUr A B : B \subset A :|: B.
Lemma subsetU1 x A : A \subset x |: A.
Lemma subsetDl A B : A :\: B \subset A.
Lemma subD1set A x : A :\ x \subset A.
Lemma subsetDr A B : A :\: B \subset ~: B.
Lemma sub1set A x : ([set x] \subset A) = (x \in A).
Variant cards_eq_spec A : seq T → {set T} → nat → Type :=
| CardEq (s : seq T) & uniq s : cards_eq_spec A s [set x | x \in s] (size s).
Lemma cards_eqP A : cards_eq_spec A (enum A) A #|A|.
Lemma cards1P A : reflect (∃ x, A = [set x]) (#|A| == 1).
Lemma cards2P A : reflect (∃ x y : T, x != y ∧ A = [set x; y])
(#|A| == 2).
Lemma subset1 A x : (A \subset [set x]) = (A == [set x]) || (A == set0).
Lemma powerset1 x : powerset [set x] = [set set0; [set x]].
Lemma setIidPl A B : reflect (A :&: B = A) (A \subset B).
Arguments setIidPl {A B}.
Lemma setIidPr A B : reflect (A :&: B = B) (B \subset A).
Lemma cardsDS A B : B \subset A → #|A :\: B| = (#|A| - #|B|)%N.
Lemma setUidPl A B : reflect (A :|: B = A) (B \subset A).
Lemma setUidPr A B : reflect (A :|: B = B) (A \subset B).
Lemma setDidPl A B : reflect (A :\: B = A) [disjoint A & B].
Lemma subIset A B C : (B \subset A) || (C \subset A) → (B :&: C \subset A).
Lemma subsetI A B C : (A \subset B :&: C) = (A \subset B) && (A \subset C).
Lemma subsetIP A B C : reflect (A \subset B ∧ A \subset C) (A \subset B :&: C).
Lemma subsetIidl A B : (A \subset A :&: B) = (A \subset B).
Lemma subsetIidr A B : (B \subset A :&: B) = (B \subset A).
Lemma powersetI A B : powerset (A :&: B) = powerset A :&: powerset B.
Lemma subUset A B C : (B :|: C \subset A) = (B \subset A) && (C \subset A).
Lemma subsetU A B C : (A \subset B) || (A \subset C) → A \subset B :|: C.
Lemma subUsetP A B C : reflect (A \subset C ∧ B \subset C) (A :|: B \subset C).
Lemma subsetC A B : (A \subset ~: B) = (B \subset ~: A).
Lemma subCset A B : (~: A \subset B) = (~: B \subset A).
Lemma subsetD A B C : (A \subset B :\: C) = (A \subset B) && [disjoint A & C].
Lemma subDset A B C : (A :\: B \subset C) = (A \subset B :|: C).
Lemma subsetDP A B C :
reflect (A \subset B ∧ [disjoint A & C]) (A \subset B :\: C).
Lemma setU_eq0 A B : (A :|: B == set0) = (A == set0) && (B == set0).
Lemma setD_eq0 A B : (A :\: B == set0) = (A \subset B).
Lemma setI_eq0 A B : (A :&: B == set0) = [disjoint A & B].
Lemma disjoint_setI0 A B : [disjoint A & B] → A :&: B = set0.
Lemma disjoints1 A x : [disjoint [set x] & A] = (x \notin A).
Lemma subsetD1 A B x : (A \subset B :\ x) = (A \subset B) && (x \notin A).
Lemma subsetD1P A B x : reflect (A \subset B ∧ x \notin A) (A \subset B :\ x).
Lemma properD1 A x : x \in A → A :\ x \proper A.
Lemma properIr A B : ~~ (B \subset A) → A :&: B \proper B.
Lemma properIl A B : ~~ (A \subset B) → A :&: B \proper A.
Lemma properUr A B : ~~ (A \subset B) → B \proper A :|: B.
Lemma properUl A B : ~~ (B \subset A) → A \proper A :|: B.
Lemma proper1set A x : ([set x] \proper A) → (x \in A).
Lemma properIset A B C : (B \proper A) || (C \proper A) → (B :&: C \proper A).
Lemma properI A B C : (A \proper B :&: C) → (A \proper B) && (A \proper C).
Lemma properU A B C : (B :|: C \proper A) → (B \proper A) && (C \proper A).
Lemma properD A B C : (A \proper B :\: C) → (A \proper B) && [disjoint A & C].
Lemma properCr A B : (A \proper ~: B) = (B \proper ~: A).
Lemma properCl A B : (~: A \proper B) = (~: B \proper A).
relationship with seq
Lemma enum_setU A B : perm_eq (enum (A :|: B)) (undup (enum A ++ enum B)).
Lemma enum_setI A B : perm_eq (enum (A :&: B)) (filter [in B] (enum A)).
Lemma has_set1 pA A a : has pA (enum [set a]) = pA a.
Lemma has_setU pA A B :
has pA (enum (A :|: B)) = (has pA (enum A)) || (has pA (enum B)).
Lemma all_set1 pA A a : all pA (enum [set a]) = pA a.
Lemma all_setU pA A B :
all pA (enum (A :|: B)) = (all pA (enum A)) && (all pA (enum B)).
End setOps.
Arguments set1P {T x a}.
Arguments set1_inj {T} [x1 x2].
Arguments set2P {T x a b}.
Arguments setIdP {T x pA pB}.
Arguments setIP {T x A B}.
Arguments setU1P {T x a B}.
Arguments setD1P {T x A b}.
Arguments setUP {T x A B}.
Arguments setDP {T A B x}.
Arguments cards1P {T A}.
Arguments setCP {T x A}.
Arguments setIidPl {T A B}.
Arguments setIidPr {T A B}.
Arguments setUidPl {T A B}.
Arguments setUidPr {T A B}.
Arguments setDidPl {T A B}.
Arguments subsetIP {T A B C}.
Arguments subUsetP {T A B C}.
Arguments subsetDP {T A B C}.
Arguments subsetD1P {T A B x}.
#[global] Hint Extern 0 (is_true (subset _ (mem (pred_of_set (setTfor _))))) ⇒
solve [apply: subsetT_hint] : core.
Section setOpsAlgebra.
Import Monoid.
Variable T : finType.
End setOpsAlgebra.
Section CartesianProd.
Variables fT1 fT2 : finType.
Variables (A1 : {set fT1}) (A2 : {set fT2}).
Definition setX := [set u | u.1 \in A1 & u.2 \in A2].
Lemma in_setX x1 x2 : ((x1, x2) \in setX) = (x1 \in A1) && (x2 \in A2).
Lemma setXP x1 x2 : reflect (x1 \in A1 ∧ x2 \in A2) ((x1, x2) \in setX).
Lemma cardsX : #|setX| = #|A1| × #|A2|.
End CartesianProd.
Arguments setXP {fT1 fT2 A1 A2 x1 x2}.
Section CartesianNProd.
Variables (I : finType) (fT : I → finType).
Variables (A : ∀ i, {set fT i}).
Implicit Types (x : {dffun ∀ i, fT i}).
Definition setXn := [set x : {dffun _} in family A].
Lemma in_setXn x : (x \in setXn) = [∀ i, x i \in A i].
Lemma setXnP x : reflect (∀ i, x i \in A i) (x \in setXn).
Lemma cardsXn : #|setXn| = \prod_i #|A i|.
End CartesianNProd.
Arguments setXnP {I fT A x}.
Canonical imset_unlock := Unlockable imset.unlock.
Canonical imset2_unlock := Unlockable imset2.unlock.
Definition preimset (aT : finType) rT f (R : mem_pred rT) :=
[set x : aT | in_mem (f x) R].
Notation "f @^-1: A" := (preimset f (mem A)) (at level 24) : set_scope.
Notation "f @: A" := (imset f (mem A)) (at level 24) : set_scope.
Notation "f @2: ( A , B )" := (imset2 f (mem A) (fun _ ⇒ mem B))
(at level 24, format "f @2: ( A , B )") : set_scope.
Comprehensions
Notation "[ 'set' E | x 'in' A ]" := ((fun x ⇒ E) @: A)
(at level 0, E, x at level 99,
format "[ '[hv' 'set' E '/ ' | x 'in' A ] ']'") : set_scope.
Notation "[ 'set' E | x 'in' A & P ]" := [set E | x in pred_of_set [set x in A | P]]
(at level 0, E, x at level 99,
format "[ '[hv' 'set' E '/ ' | x 'in' A '/ ' & P ] ']'") : set_scope.
Notation "[ 'set' E | x 'in' A , y 'in' B ]" :=
(imset2 (fun x y ⇒ E) (mem A) (fun x ⇒ mem B))
(at level 0, E, x, y at level 99, format
"[ '[hv' 'set' E '/ ' | x 'in' A , '/ ' y 'in' B ] ']'"
) : set_scope.
Notation "[ 'set' E | x 'in' A , y 'in' B & P ]" :=
[set E | x in A, y in pred_of_set [set y in B | P]]
(at level 0, E, x, y at level 99, format
"[ '[hv' 'set' E '/ ' | x 'in' A , '/ ' y 'in' B '/ ' & P ] ']'"
) : set_scope.
Typed variants
Notation "[ 'set' E | x : T 'in' A ]" := ((fun x : T ⇒ E) @: A)
(at level 0, E, x at level 99, only parsing) : set_scope.
Notation "[ 'set' E | x : T 'in' A & P ]" :=
[set E | x : T in [set x : T in A | P]]
(at level 0, E, x at level 99, only parsing) : set_scope.
Notation "[ 'set' E | x : T 'in' A , y : U 'in' B ]" :=
(imset2 (fun (x : T) (y : U) ⇒ E) (mem A) (fun (x : T) ⇒ mem B))
(at level 0, E, x, y at level 99, only parsing) : set_scope.
Notation "[ 'set' E | x : T 'in' A , y : U 'in' B & P ]" :=
[set E | x : T in A, y : U in [set y : U in B | P]]
(at level 0, E, x, y at level 99, only parsing) : set_scope.
Comprehensions over a type
Local Notation predOfType T := (pred_of_simpl (@pred_of_argType T)).
Notation "[ 'set' E | x : T ]" := [set E | x : T in predOfType T]
(at level 0, E, x at level 99,
format "[ '[hv' 'set' E '/ ' | x : T ] ']'") : set_scope.
Notation "[ 'set' E | x : T & P ]" :=
[set E | x : T in pred_of_set [set x : T | P]]
(at level 0, E, x at level 99,
format "[ '[hv' 'set' E '/ ' | x : T '/ ' & P ] ']'") : set_scope.
Notation "[ 'set' E | x : T , y : U 'in' B ]" :=
[set E | x : T in predOfType T, y : U in B]
(at level 0, E, x, y at level 99, format
"[ '[hv' 'set' E '/ ' | x : T , '/ ' y : U 'in' B ] ']'")
: set_scope.
Notation "[ 'set' E | x : T , y : U 'in' B & P ]" :=
[set E | x : T, y : U in pred_of_set [set y in B | P]]
(at level 0, E, x, y at level 99, format
"[ '[hv ' 'set' E '/' | x : T , '/ ' y : U 'in' B '/' & P ] ']'"
) : set_scope.
Notation "[ 'set' E | x : T 'in' A , y : U ]" :=
[set E | x : T in A, y : U in predOfType U]
(at level 0, E, x, y at level 99, format
"[ '[hv' 'set' E '/ ' | x : T 'in' A , '/ ' y : U ] ']'")
: set_scope.
Notation "[ 'set' E | x : T 'in' A , y : U & P ]" :=
[set E | x : T in A, y : U in pred_of_set [set y in P]]
(at level 0, E, x, y at level 99, format
"[ '[hv' 'set' E '/ ' | x : T 'in' A , '/ ' y : U & P ] ']'")
: set_scope.
Notation "[ 'set' E | x : T , y : U ]" :=
[set E | x : T, y : U in predOfType U]
(at level 0, E, x, y at level 99, format
"[ '[hv' 'set' E '/ ' | x : T , '/ ' y : U ] ']'")
: set_scope.
Notation "[ 'set' E | x : T , y : U & P ]" :=
[set E | x : T, y : U in pred_of_set [set y in P]]
(at level 0, E, x, y at level 99, format
"[ '[hv' 'set' E '/ ' | x : T , '/ ' y : U & P ] ']'")
: set_scope.
Untyped variants
Notation "[ 'set' E | x , y 'in' B ]" := [set E | x : _, y : _ in B]
(at level 0, E, x, y at level 99, only parsing) : set_scope.
Notation "[ 'set' E | x , y 'in' B & P ]" := [set E | x : _, y : _ in B & P]
(at level 0, E, x, y at level 99, only parsing) : set_scope.
Notation "[ 'set' E | x 'in' A , y ]" := [set E | x : _ in A, y : _]
(at level 0, E, x, y at level 99, only parsing) : set_scope.
Notation "[ 'set' E | x 'in' A , y & P ]" := [set E | x : _ in A, y : _ & P]
(at level 0, E, x, y at level 99, only parsing) : set_scope.
Notation "[ 'set' E | x , y ]" := [set E | x : _, y : _]
(at level 0, E, x, y at level 99, only parsing) : set_scope.
Notation "[ 'set' E | x , y & P ]" := [set E | x : _, y : _ & P ]
(at level 0, E, x, y at level 99, only parsing) : set_scope.
Section FunImage.
Variables aT aT2 : finType.
Section ImsetTheory.
Variable rT : finType.
Section ImsetProp.
Variables (f : aT → rT) (f2 : aT → aT2 → rT).
Lemma imsetP D y : reflect (exists2 x, in_mem x D & y = f x) (y \in imset f D).
Variant imset2_spec D1 D2 y : Prop :=
Imset2spec x1 x2 of in_mem x1 D1 & in_mem x2 (D2 x1) & y = f2 x1 x2.
Lemma imset2P D1 D2 y : reflect (imset2_spec D1 D2 y) (y \in imset2 f2 D1 D2).
Lemma imset_f (D : {pred aT}) x : x \in D → f x \in f @: D.
Lemma mem_imset (D : {pred aT}) x : injective f → f x \in f @: D = (x \in D).
Lemma imset0 : f @: set0 = set0.
Lemma imset_eq0 (A : {set aT}) : (f @: A == set0) = (A == set0).
Lemma imset_set1 x : f @: [set x] = [set f x].
Lemma imset_inj : injective f → injective (fun A : {set aT} ⇒ f @: A).
Lemma imset_disjoint (A B : {pred aT}) :
injective f → [disjoint f @: A & f @: B] = [disjoint A & B].
Lemma imset2_f (D : {pred aT}) (D2 : aT → {pred aT2}) x x2 :
x \in D → x2 \in D2 x →
f2 x x2 \in [set f2 y y2 | y in D, y2 in D2 y].
Lemma mem_imset2 (D : {pred aT}) (D2 : aT → {pred aT2}) x x2 :
injective2 f2 →
(f2 x x2 \in [set f2 y y2 | y in D, y2 in D2 y])
= (x \in D) && (x2 \in D2 x).
Lemma sub_imset_pre (A : {pred aT}) (B : {pred rT}) :
(f @: A \subset B) = (A \subset f @^-1: B).
Lemma preimsetS (A B : {pred rT}) :
A \subset B → (f @^-1: A) \subset (f @^-1: B).
Lemma preimset0 : f @^-1: set0 = set0.
Lemma preimsetT : f @^-1: setT = setT.
Lemma preimsetI (A B : {set rT}) :
f @^-1: (A :&: B) = (f @^-1: A) :&: (f @^-1: B).
Lemma preimsetU (A B : {set rT}) :
f @^-1: (A :|: B) = (f @^-1: A) :|: (f @^-1: B).
Lemma preimsetD (A B : {set rT}) :
f @^-1: (A :\: B) = (f @^-1: A) :\: (f @^-1: B).
Lemma preimsetC (A : {set rT}) : f @^-1: (~: A) = ~: f @^-1: A.
Lemma imsetS (A B : {pred aT}) : A \subset B → f @: A \subset f @: B.
Lemma imset_proper (A B : {set aT}) :
{in B &, injective f} → A \proper B → f @: A \proper f @: B.
Lemma preimset_proper (A B : {set rT}) :
B \subset codom f → A \proper B → (f @^-1: A) \proper (f @^-1: B).
Lemma imsetU (A B : {set aT}) : f @: (A :|: B) = (f @: A) :|: (f @: B).
Lemma imsetU1 a (A : {set aT}) : f @: (a |: A) = f a |: (f @: A).
Lemma imsetI (A B : {set aT}) :
{in A & B, injective f} → f @: (A :&: B) = f @: A :&: f @: B.
Lemma imset2Sl (A B : {pred aT}) (C : {pred aT2}) :
A \subset B → f2 @2: (A, C) \subset f2 @2: (B, C).
Lemma imset2Sr (A B : {pred aT2}) (C : {pred aT}) :
A \subset B → f2 @2: (C, A) \subset f2 @2: (C, B).
Lemma imset2S (A B : {pred aT}) (A2 B2 : {pred aT2}) :
A \subset B → A2 \subset B2 → f2 @2: (A, A2) \subset f2 @2: (B, B2).
End ImsetProp.
Implicit Types (f g : aT → rT) (D : {pred aT}) (R : {pred rT}).
Lemma eq_preimset f g R : f =1 g → f @^-1: R = g @^-1: R.
Lemma eq_imset f g D : f =1 g → f @: D = g @: D.
Lemma eq_in_imset f g D : {in D, f =1 g} → f @: D = g @: D.
Lemma eq_in_imset2 (f g : aT → aT2 → rT) (D : {pred aT}) (D2 : {pred aT2}) :
{in D & D2, f =2 g} → f @2: (D, D2) = g @2: (D, D2).
End ImsetTheory.
Lemma imset2_pair (A : {set aT}) (B : {set aT2}) :
[set (x, y) | x in A, y in B] = setX A B.
Lemma setXS (A1 B1 : {set aT}) (A2 B2 : {set aT2}) :
A1 \subset B1 → A2 \subset B2 → setX A1 A2 \subset setX B1 B2.
End FunImage.
Arguments imsetP {aT rT f D y}.
Arguments imset2P {aT aT2 rT f2 D1 D2 y}.
Arguments imset_disjoint {aT rT f A B}.
Lemma setXnS (I : finType) (T : I → finType) (A B : ∀ i, {set T i}) :
(∀ i, A i \subset B i) → setXn A \subset setXn B.
Lemma eq_setXn (I : finType) (T : I → finType) (A B : ∀ i, {set T i}) :
(∀ i, A i = B i) → setXn A = setXn B.
Section BigOpsAnyOp.
Variables (R : Type) (x : R) (op : R → R → R).
Variables I : finType.
Implicit Type F : I → R.
Lemma big_set0 F : \big[op/x]_(i in set0) F i = x.
Lemma big_set1E j F : \big[op/x]_(i in [set j]) F i = op (F j) x.
Lemma big_set (A : pred I) F :
\big[op/x]_(i in [set i | A i]) (F i) = \big[op/x]_(i in A) (F i).
End BigOpsAnyOp.
Section BigOpsSemiGroup.
Variables (R : Type) (op : SemiGroup.com_law R).
Variable (le : rel R).
Hypotheses (le_refl : reflexive le) (le_incr : ∀ x y, le x (op x y)).
Context [x : R].
Lemma subset_le_big_cond (I : finType) (A A' P P' : {pred I}) (F : I → R) :
[set i in A | P i] \subset [set i in A' | P' i] →
le (\big[op/x]_(i in A | P i) F i) (\big[op/x]_(i in A' | P' i) F i).
Lemma big_imset_idem [I J : finType] (h : I → J) (A : pred I) F :
idempotent_op op →
\big[op/x]_(j in h @: A) F j = \big[op/x]_(i in A) F (h i).
End BigOpsSemiGroup.
Section BigOps.
Variables (R : Type) (idx : R).
Variables (op : Monoid.law idx) (aop : Monoid.com_law idx).
Variables I J : finType.
Implicit Type A B : {set I}.
Implicit Type h : I → J.
Implicit Type P : pred I.
Implicit Type F : I → R.
Lemma big_set1 a F : \big[op/idx]_(i in [set a]) F i = F a.
Lemma big_setID A B F :
\big[aop/idx]_(i in A) F i =
aop (\big[aop/idx]_(i in A :&: B) F i)
(\big[aop/idx]_(i in A :\: B) F i).
Lemma big_setIDcond A B P F :
\big[aop/idx]_(i in A | P i) F i =
aop (\big[aop/idx]_(i in A :&: B | P i) F i)
(\big[aop/idx]_(i in A :\: B | P i) F i).
Lemma big_setD1 a A F : a \in A →
\big[aop/idx]_(i in A) F i = aop (F a) (\big[aop/idx]_(i in A :\ a) F i).
Lemma big_setU1 a A F : a \notin A →
\big[aop/idx]_(i in a |: A) F i = aop (F a) (\big[aop/idx]_(i in A) F i).
Lemma big_subset_idem_cond A B P F :
idempotent_op aop →
A \subset B →
aop (\big[aop/idx]_(i in A | P i) F i) (\big[aop/idx]_(i in B | P i) F i)
= \big[aop/idx]_(i in B | P i) F i.
Lemma big_subset_idem A B F :
idempotent_op aop →
A \subset B →
aop (\big[aop/idx]_(i in A) F i) (\big[aop/idx]_(i in B) F i)
= \big[aop/idx]_(i in B) F i.
Lemma big_setU_cond A B P F :
idempotent_op aop →
\big[aop/idx]_(i in A :|: B | P i) F i
= aop (\big[aop/idx]_(i in A | P i) F i) (\big[aop/idx]_(i in B | P i) F i).
Lemma big_setU A B F :
idempotent_op aop →
\big[aop/idx]_(i in A :|: B) F i
= aop (\big[aop/idx]_(i in A) F i) (\big[aop/idx]_(i in B) F i).
Lemma big_imset h (A : {pred I}) G : {in A &, injective h} →
\big[aop/idx]_(j in h @: A) G j = \big[aop/idx]_(i in A) G (h i).
Lemma big_imset_cond h (A : {pred I}) (P : pred J) G : {in A &, injective h} →
\big[aop/idx]_(j in h @: A | P j) G j =
\big[aop/idx]_(i in A | P (h i)) G (h i).
Lemma partition_big_imset h (A : {pred I}) F :
\big[aop/idx]_(i in A) F i =
\big[aop/idx]_(j in h @: A) \big[aop/idx]_(i in A | h i == j) F i.
End BigOps.
Lemma bigA_distr (R : Type) (zero one : R) (mul : Monoid.mul_law zero)
(add : Monoid.add_law zero mul) (I : finType) (F G : I → R) :
\big[mul/one]_i add (F i) (G i) =
\big[add/zero]_(J in {set I}) \big[mul/one]_i (if i \in J then F i else G i).
Arguments big_setID [R idx aop I A].
Arguments big_setD1 [R idx aop I] a [A F].
Arguments big_setU1 [R idx aop I] a [A F].
Arguments big_imset [R idx aop I J h A].
Arguments partition_big_imset [R idx aop I J].
Section Fun2Set1.
Variables aT1 aT2 rT : finType.
Variables (f : aT1 → aT2 → rT).
Lemma imset2_set1l x1 (D2 : {pred aT2}) : f @2: ([set x1], D2) = f x1 @: D2.
Lemma imset2_set1r x2 (D1 : {pred aT1}) : f @2: (D1, [set x2]) = f^~ x2 @: D1.
End Fun2Set1.
Section CardFunImage.
Variables aT aT2 rT : finType.
Variables (f : aT → rT) (g : rT → aT) (f2 : aT → aT2 → rT).
Variables (D : {pred aT}) (D2 : {pred aT}).
Lemma imset_card : #|f @: D| = #|image f D|.
Lemma leq_imset_card : #|f @: D| ≤ #|D|.
Lemma card_in_imset : {in D &, injective f} → #|f @: D| = #|D|.
Lemma card_imset : injective f → #|f @: D| = #|D|.
Lemma imset_injP : reflect {in D &, injective f} (#|f @: D| == #|D|).
Lemma can2_in_imset_pre :
{in D, cancel f g} → {on D, cancel g & f} → f @: D = g @^-1: D.
Lemma can2_imset_pre : cancel f g → cancel g f → f @: D = g @^-1: D.
End CardFunImage.
Arguments imset_injP {aT rT f D}.
Lemma on_card_preimset (aT rT : finType) (f : aT → rT) (R : {pred rT}) :
{on R, bijective f} → #|f @^-1: R| = #|R|.
Lemma can_imset_pre (T : finType) f g (A : {set T}) :
cancel f g → f @: A = g @^-1: A :> {set T}.
Lemma imset_id (T : finType) (A : {set T}) : [set x | x in A] = A.
Lemma card_preimset (T : finType) (f : T → T) (A : {set T}) :
injective f → #|f @^-1: A| = #|A|.
Lemma card_powerset (T : finType) (A : {set T}) : #|powerset A| = 2 ^ #|A|.
Section FunImageComp.
Variables T T' U : finType.
Lemma imset_comp (f : T' → U) (g : T → T') (H : {pred T}) :
(f \o g) @: H = f @: (g @: H).
End FunImageComp.
Notation "\bigcup_ ( i <- r | P ) F" :=
(\big[@setU _/set0]_(i <- r | P) F%SET) : set_scope.
Notation "\bigcup_ ( i <- r ) F" :=
(\big[@setU _/set0]_(i <- r) F%SET) : set_scope.
Notation "\bigcup_ ( m <= i < n | P ) F" :=
(\big[@setU _/set0]_(m ≤ i < n | P%B) F%SET) : set_scope.
Notation "\bigcup_ ( m <= i < n ) F" :=
(\big[@setU _/set0]_(m ≤ i < n) F%SET) : set_scope.
Notation "\bigcup_ ( i | P ) F" :=
(\big[@setU _/set0]_(i | P%B) F%SET) : set_scope.
Notation "\bigcup_ i F" :=
(\big[@setU _/set0]_i F%SET) : set_scope.
Notation "\bigcup_ ( i : t | P ) F" :=
(\big[@setU _/set0]_(i : t | P%B) F%SET) (only parsing): set_scope.
Notation "\bigcup_ ( i : t ) F" :=
(\big[@setU _/set0]_(i : t) F%SET) (only parsing) : set_scope.
Notation "\bigcup_ ( i < n | P ) F" :=
(\big[@setU _/set0]_(i < n | P%B) F%SET) : set_scope.
Notation "\bigcup_ ( i < n ) F" :=
(\big[@setU _/set0]_ (i < n) F%SET) : set_scope.
Notation "\bigcup_ ( i 'in' A | P ) F" :=
(\big[@setU _/set0]_(i in A | P%B) F%SET) : set_scope.
Notation "\bigcup_ ( i 'in' A ) F" :=
(\big[@setU _/set0]_(i in A) F%SET) : set_scope.
Notation "\bigcap_ ( i <- r | P ) F" :=
(\big[@setI _/setT]_(i <- r | P%B) F%SET) : set_scope.
Notation "\bigcap_ ( i <- r ) F" :=
(\big[@setI _/setT]_(i <- r) F%SET) : set_scope.
Notation "\bigcap_ ( m <= i < n | P ) F" :=
(\big[@setI _/setT]_(m ≤ i < n | P%B) F%SET) : set_scope.
Notation "\bigcap_ ( m <= i < n ) F" :=
(\big[@setI _/setT]_(m ≤ i < n) F%SET) : set_scope.
Notation "\bigcap_ ( i | P ) F" :=
(\big[@setI _/setT]_(i | P%B) F%SET) : set_scope.
Notation "\bigcap_ i F" :=
(\big[@setI _/setT]_i F%SET) : set_scope.
Notation "\bigcap_ ( i : t | P ) F" :=
(\big[@setI _/setT]_(i : t | P%B) F%SET) (only parsing): set_scope.
Notation "\bigcap_ ( i : t ) F" :=
(\big[@setI _/setT]_(i : t) F%SET) (only parsing) : set_scope.
Notation "\bigcap_ ( i < n | P ) F" :=
(\big[@setI _/setT]_(i < n | P%B) F%SET) : set_scope.
Notation "\bigcap_ ( i < n ) F" :=
(\big[@setI _/setT]_(i < n) F%SET) : set_scope.
Notation "\bigcap_ ( i 'in' A | P ) F" :=
(\big[@setI _/setT]_(i in A | P%B) F%SET) : set_scope.
Notation "\bigcap_ ( i 'in' A ) F" :=
(\big[@setI _/setT]_(i in A) F%SET) : set_scope.
Section BigSetOps.
Variables T I : finType.
Implicit Types (U : {pred T}) (P : pred I) (A B : {set I}) (F : I → {set T}).
It is very hard to use this lemma, because the unification fails to defer the F j pattern (even though it's a Miller pattern!).
Lemma bigcup_sup j P F : P j → F j \subset \bigcup_(i | P i) F i.
Lemma bigcup_max j U P F :
P j → U \subset F j → U \subset \bigcup_(i | P i) F i.
Lemma bigcupP x P F :
reflect (exists2 i, P i & x \in F i) (x \in \bigcup_(i | P i) F i).
Lemma bigcupsP U P F :
reflect (∀ i, P i → F i \subset U) (\bigcup_(i | P i) F i \subset U).
Lemma bigcup0P P F :
reflect (∀ i, P i → F i = set0) (\bigcup_(i | P i) F i == set0).
Lemma bigcup_disjointP U P F :
reflect (∀ i : I, P i → [disjoint U & F i])
[disjoint U & \bigcup_(i | P i) F i].
Lemma bigcup_disjoint U P F :
(∀ i, P i → [disjoint U & F i]) → [disjoint U & \bigcup_(i | P i) F i].
Lemma bigcup_setU A B F :
\bigcup_(i in A :|: B) F i =
(\bigcup_(i in A) F i) :|: (\bigcup_ (i in B) F i).
Lemma bigcup_seq r F : \bigcup_(i <- r) F i = \bigcup_(i in r) F i.
Unlike its setU counterpart, this lemma is useable.
Lemma bigcap_inf j P F : P j → \bigcap_(i | P i) F i \subset F j.
Lemma bigcap_min j U P F :
P j → F j \subset U → \bigcap_(i | P i) F i \subset U.
Lemma bigcapsP U P F :
reflect (∀ i, P i → U \subset F i) (U \subset \bigcap_(i | P i) F i).
Lemma bigcapP x P F :
reflect (∀ i, P i → x \in F i) (x \in \bigcap_(i | P i) F i).
Lemma setC_bigcup J r (P : pred J) (F : J → {set T}) :
~: (\bigcup_(j <- r | P j) F j) = \bigcap_(j <- r | P j) ~: F j.
Lemma setC_bigcap J r (P : pred J) (F : J → {set T}) :
~: (\bigcap_(j <- r | P j) F j) = \bigcup_(j <- r | P j) ~: F j.
Lemma bigcap_setU A B F :
(\bigcap_(i in A :|: B) F i) =
(\bigcap_(i in A) F i) :&: (\bigcap_(i in B) F i).
Lemma bigcap_seq r F : \bigcap_(i <- r) F i = \bigcap_(i in r) F i.
End BigSetOps.
Arguments bigcup_sup [T I] j [P F].
Arguments bigcup_max [T I] j [U P F].
Arguments bigcupP {T I x P F}.
Arguments bigcupsP {T I U P F}.
Arguments bigcup_disjointP {T I U P F}.
Arguments bigcap_inf [T I] j [P F].
Arguments bigcap_min [T I] j [U P F].
Arguments bigcapP {T I x P F}.
Arguments bigcapsP {T I U P F}.
Section ImsetCurry.
Variables (aT1 aT2 rT : finType) (f : aT1 → aT2 → rT).
Section Curry.
Variables (A1 : {set aT1}) (A2 : {set aT2}).
Variables (D1 : {pred aT1}) (D2 : {pred aT2}).
Lemma curry_imset2X : f @2: (A1, A2) = uncurry f @: (setX A1 A2).
Lemma curry_imset2l : f @2: (D1, D2) = \bigcup_(x1 in D1) f x1 @: D2.
Lemma curry_imset2r : f @2: (D1, D2) = \bigcup_(x2 in D2) f^~ x2 @: D1.
End Curry.
Lemma imset2Ul (A B : {set aT1}) (C : {set aT2}) :
f @2: (A :|: B, C) = f @2: (A, C) :|: f @2: (B, C).
Lemma imset2Ur (A : {set aT1}) (B C : {set aT2}) :
f @2: (A, B :|: C) = f @2: (A, B) :|: f @2: (A, C).
End ImsetCurry.
Section Partitions.
Variables T I : finType.
Implicit Types (x y z : T) (A B D X : {set T}) (P Q : {set {set T}}).
Implicit Types (J : pred I) (F : I → {set T}).
Definition cover P := \bigcup_(B in P) B.
Definition pblock P x := odflt set0 (pick [pred B in P | x \in B]).
Definition trivIset P := \sum_(B in P) #|B| == #|cover P|.
Definition partition P D := [&& cover P == D, trivIset P & set0 \notin P].
Definition is_transversal X P D :=
[&& partition P D, X \subset D & [∀ B in P, #|X :&: B| == 1]].
Definition transversal P D := [set odflt x [pick y in pblock P x] | x in D].
Definition transversal_repr x0 X B := odflt x0 [pick x in X :&: B].
Lemma leq_card_setU A B : #|A :|: B| ≤ #|A| + #|B| ?= iff [disjoint A & B].
Lemma leq_card_cover P : #|cover P| ≤ \sum_(A in P) #|A| ?= iff trivIset P.
Lemma imset_cover (T' : finType) P (f : T → T') :
[set f x | x in cover P] = \bigcup_(i in P) [set f x | x in i].
Lemma cover1 A : cover [set A] = A.
Lemma trivIset1 A : trivIset [set A].
Lemma trivIsetP P :
reflect {in P &, ∀ A B, A != B → [disjoint A & B]} (trivIset P).
Lemma trivIsetS P Q : P \subset Q → trivIset Q → trivIset P.
Lemma trivIsetD P Q : trivIset P → trivIset (P :\: Q).
Lemma trivIsetU P Q :
trivIset Q → trivIset P → [disjoint cover Q & cover P] → trivIset (Q :|: P).
Lemma coverD1 P B : trivIset P → B \in P → cover (P :\ B) = cover P :\: B.
Lemma trivIsetI P D : trivIset P → trivIset (P ::&: D).
Lemma cover_setI P D : cover (P ::&: D) \subset cover P :&: D.
Lemma mem_pblock P x : (x \in pblock P x) = (x \in cover P).
Lemma pblock_mem P x : x \in cover P → pblock P x \in P.
Lemma def_pblock P B x : trivIset P → B \in P → x \in B → pblock P x = B.
Lemma same_pblock P x y :
trivIset P → x \in pblock P y → pblock P x = pblock P y.
Lemma eq_pblock P x y :
trivIset P → x \in cover P →
(pblock P x == pblock P y) = (y \in pblock P x).
Lemma trivIsetU1 A P :
{in P, ∀ B, [disjoint A & B]} → trivIset P → set0 \notin P →
trivIset (A |: P) ∧ A \notin P.
Lemma cover_imset J F : cover (F @: J) = \bigcup_(i in J) F i.
Lemma trivIimset J F (P := F @: J) :
{in J &, ∀ i j, j != i → [disjoint F i & F j]} → set0 \notin P →
trivIset P ∧ {in J &, injective F}.
Lemma cover_partition P D : partition P D → cover P = D.
Lemma partition0 P D : partition P D → set0 \in P = false.
Lemma partition_neq0 P D B : partition P D → B \in P → B != set0.
Lemma partition_trivIset P D : partition P D → trivIset P.
Lemma partitionS P D B : partition P D → B \in P → B \subset D.
Lemma partitionD1 P D B :
partition P D → B \in P → partition (P :\ B) (D :\: B).
Lemma partitionU1 P D B :
partition P D → B != set0 → [disjoint B & D] → partition (B |: P) (B :|: D).
Lemma partition_set0 P : partition P set0 = (P == set0).
Lemma card_partition P D : partition P D → #|D| = \sum_(A in P) #|A|.
Lemma card_uniform_partition n P D :
{in P, ∀ A, #|A| = n} → partition P D → #|D| = #|P| × n.
Lemma partition_pigeonhole P D A :
partition P D → #|P| ≤ #|A| → A \subset D → {in P, ∀ B, #|A :&: B| ≤ 1} →
{in P, ∀ B, A :&: B != set0}.
Section BigOps.
Variables (R : Type) (idx : R) (op : Monoid.com_law idx).
Let rhs_cond P K E := \big[op/idx]_(A in P) \big[op/idx]_(x in A | K x) E x.
Let rhs P E := \big[op/idx]_(A in P) \big[op/idx]_(x in A) E x.
Lemma big_trivIset_cond P (K : pred T) (E : T → R) :
trivIset P → \big[op/idx]_(x in cover P | K x) E x = rhs_cond P K E.
Lemma big_trivIset P (E : T → R) :
trivIset P → \big[op/idx]_(x in cover P) E x = rhs P E.
Lemma set_partition_big_cond P D (K : pred T) (E : T → R) :
partition P D → \big[op/idx]_(x in D | K x) E x = rhs_cond P K E.
Lemma set_partition_big P D (E : T → R) :
partition P D → \big[op/idx]_(x in D) E x = rhs P E.
Lemma partition_disjoint_bigcup (F : I → {set T}) E :
(∀ i j, i != j → [disjoint F i & F j]) →
\big[op/idx]_(x in \bigcup_i F i) E x =
\big[op/idx]_i \big[op/idx]_(x in F i) E x.
End BigOps.
Section Equivalence.
Variables (R : rel T) (D : {set T}).
Let Px x := [set y in D | R x y].
Definition equivalence_partition := [set Px x | x in D].
Local Notation P := equivalence_partition.
Hypothesis eqiR : {in D & &, equivalence_rel R}.
Let Pxx x : x \in D → x \in Px x.
Let PPx x : x \in D → Px x \in P := fun Dx ⇒ imset_f _ Dx.
Lemma equivalence_partitionP : partition P D.
Lemma pblock_equivalence_partition :
{in D &, ∀ x y, (y \in pblock P x) = R x y}.
End Equivalence.
Lemma pblock_equivalence P D :
partition P D → {in D & &, equivalence_rel (fun x y ⇒ y \in pblock P x)}.
Lemma equivalence_partition_pblock P D :
partition P D → equivalence_partition (fun x y ⇒ y \in pblock P x) D = P.
Section Preim.
Variables (rT : eqType) (f : T → rT).
Definition preim_partition := equivalence_partition (fun x y ⇒ f x == f y).
Lemma preim_partitionP D : partition (preim_partition D) D.
End Preim.
Lemma preim_partition_pblock P D :
partition P D → preim_partition (pblock P) D = P.
Lemma transversalP P D : partition P D → is_transversal (transversal P D) P D.
Section Transversals.
Variables (X : {set T}) (P : {set {set T}}) (D : {set T}).
Hypothesis trPX : is_transversal X P D.
Lemma transversal_sub : X \subset D.
Let tiP : trivIset P.
Let sXP : {subset X ≤ cover P}.
Let trX : {in P, ∀ B, #|X :&: B| == 1}.
Lemma setI_transversal_pblock x0 B :
B \in P → X :&: B = [set transversal_repr x0 X B].
Lemma repr_mem_pblock x0 B : B \in P → transversal_repr x0 X B \in B.
Lemma repr_mem_transversal x0 B : B \in P → transversal_repr x0 X B \in X.
Lemma transversal_reprK x0 : {in P, cancel (transversal_repr x0 X) (pblock P)}.
Lemma pblockK x0 : {in X, cancel (pblock P) (transversal_repr x0 X)}.
Lemma pblock_inj : {in X &, injective (pblock P)}.
Lemma pblock_transversal : pblock P @: X = P.
Lemma card_transversal : #|X| = #|P|.
Lemma im_transversal_repr x0 : transversal_repr x0 X @: P = X.
End Transversals.
End Partitions.
Arguments trivIsetP {T P}.
Arguments big_trivIset_cond [T R idx op] P [K E].
Arguments set_partition_big_cond [T R idx op] P [D K E].
Arguments big_trivIset [T R idx op] P [E].
Arguments set_partition_big [T R idx op] P [D E].
Lemma partition_partition (T : finType) (D : {set T}) P Q :
partition P D → partition Q P →
partition (cover @: Q) D ∧ {in Q &, injective cover}.
Lemma indexed_partition (I T : finType) (J : {pred I}) (B : I → {set T}) :
let P := [set B i | i in J] in
{in J &, ∀ i j : I, j != i → [disjoint B i & B j]} →
(∀ i : I, J i → B i != set0) → partition P (cover P) ∧ {in J &, injective B}.
Section PartitionImage.
Variables (T : finType) (P : {set {set T}}) (D : {set T}).
Variables (T' : finType) (f : T → T') (inj_f : injective f).
Let fP := [set f @: (B : {set T}) | B in P].
Lemma imset_trivIset : trivIset fP = trivIset P.
Lemma imset0mem : (set0 \in fP) = (set0 \in P).
Lemma imset_partition : partition fP (f @: D) = partition P D.
End PartitionImage.
Maximum and minimum (sub)set with respect to a given pred
Section MaxSetMinSet.
Variable T : finType.
Notation sT := {set T}.
Implicit Types A B C : sT.
Implicit Type P : pred sT.
Definition minset P A := [∀ (B : sT | B \subset A), (B == A) == P B].
Lemma minset_eq P1 P2 A : P1 =1 P2 → minset P1 A = minset P2 A.
Lemma minsetP P A :
reflect ((P A) ∧ (∀ B, P B → B \subset A → B = A)) (minset P A).
Arguments minsetP {P A}.
Lemma minsetp P A : minset P A → P A.
Lemma minsetinf P A B : minset P A → P B → B \subset A → B = A.
Lemma ex_minset P : (∃ A, P A) → {A | minset P A}.
Lemma minset_exists P C : P C → {A | minset P A & A \subset C}.
The 'locked_with' allows Coq to find the value of P by unification.
Fact maxset_key : unit.
Definition maxset P A :=
minset (fun B ⇒ locked_with maxset_key P (~: B)) (~: A).
Lemma maxset_eq P1 P2 A : P1 =1 P2 → maxset P1 A = maxset P2 A.
Lemma maxminset P A : maxset P A = minset [pred B | P (~: B)] (~: A).
Lemma minmaxset P A : minset P A = maxset [pred B | P (~: B)] (~: A).
Lemma maxsetP P A :
reflect ((P A) ∧ (∀ B, P B → A \subset B → B = A)) (maxset P A).
Lemma maxsetp P A : maxset P A → P A.
Lemma maxsetsup P A B : maxset P A → P B → A \subset B → B = A.
Lemma ex_maxset P : (∃ A, P A) → {A | maxset P A}.
Lemma maxset_exists P C : P C → {A : sT | maxset P A & C \subset A}.
End MaxSetMinSet.
Arguments setCK {T}.
Arguments minsetP {T P A}.
Arguments maxsetP {T P A}.
Section SetFixpoint.
Section Least.
Variables (T : finType) (F : {set T} → {set T}).
Hypothesis (F_mono : {homo F : X Y / X \subset Y}).
Let n := #|T|.
Let iterF i := iter i F set0.
Lemma subset_iterS i : iterF i \subset iterF i.+1.
Lemma subset_iter : {homo iterF : i j / i ≤ j >-> i \subset j}.
Definition fixset := iterF n.
Lemma fixsetK : F fixset = fixset.
Hint Resolve fixsetK : core.
Lemma minset_fix : minset [pred X | F X == X] fixset.
Lemma fixsetKn k : iter k F fixset = fixset.
Lemma iter_sub_fix k : iterF k \subset fixset.
Lemma fix_order_proof x : x \in fixset → ∃ n, x \in iterF n.
Definition fix_order (x : T) :=
if (x \in fixset) =P true isn't ReflectT x_fix then 0
else (ex_minn (fix_order_proof x_fix)).
Lemma fix_order_le_max (x : T) : fix_order x ≤ n.
Lemma in_iter_fix_orderE (x : T) :
(x \in iterF (fix_order x)) = (x \in fixset).
Lemma fix_order_gt0 (x : T) : (fix_order x > 0) = (x \in fixset).
Lemma fix_order_eq0 (x : T) : (fix_order x == 0) = (x \notin fixset).
Lemma in_iter_fixE (x : T) k : (x \in iterF k) = (0 < fix_order x ≤ k).
Lemma in_iter (x : T) k : x \in fixset → fix_order x ≤ k → x \in iterF k.
Lemma notin_iter (x : T) k : k < fix_order x → x \notin iterF k.
Lemma fix_order_small x k : x \in iterF k → fix_order x ≤ k.
Lemma fix_order_big x k : x \in fixset → x \notin iterF k → fix_order x > k.
Lemma le_fix_order (x y : T) : y \in iterF (fix_order x) →
fix_order y ≤ fix_order x.
End Least.
Section Greatest.
Variables (T : finType) (F : {set T} → {set T}).
Hypothesis (F_mono : {homo F : X Y / X \subset Y}).
Definition funsetC X := ~: (F (~: X)).
Lemma funsetC_mono : {homo funsetC : X Y / X \subset Y}.
Hint Resolve funsetC_mono : core.
Definition cofixset := ~: fixset funsetC.
Lemma cofixsetK : F cofixset = cofixset.
Lemma maxset_cofix : maxset [pred X | F X == X] cofixset.
End Greatest.
End SetFixpoint.