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phanerozoic
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Coq-Finmap

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msubUsetP A B C : reflect (A `<=` C /\ B `<=` C) (A `|` B `<=` C). Proof. by rewrite msubUset; exact: andP. Qed.
Lemma
root
[ "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop tuple.", "Require Import finmap." ]
multiset.v
msubUsetP
msetU_eq0 A B : (A `|` B == mset0) = (A == mset0) && (B == mset0). Proof. by rewrite -!msubset0 msubUset. Qed.
Lemma
root
[ "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop tuple.", "Require Import finmap." ]
multiset.v
msetU_eq0
setD_eq0 A B : (A `\` B == mset0) = (A `<=` B). Proof. by rewrite -msubset0 subset_msetBLR msetD0. Qed.
Lemma
root
[ "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop tuple.", "Require Import finmap." ]
multiset.v
setD_eq0
msub1set A a : ([mset a] `<=` A) = (a \in A). Proof. apply/msubsetP/idP; first by move/(_ a); rewrite msetnxx in_mset. by move=> ainA b; rewrite msetnE; case: eqP => // ->; rewrite -in_mset. Qed.
Lemma
root
[ "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop tuple.", "Require Import finmap." ]
multiset.v
msub1set
msetDBA A B C : C `<=` B -> A `+` B `\` C = (A `+` B) `\` C. Proof. by move=> /msubsetP CB; apply/msetP=> a; rewrite !msetE2 addnBA. Qed.
Lemma
root
[ "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop tuple.", "Require Import finmap." ]
multiset.v
msetDBA
mset_0Vmem A : (A = mset0) + {x : K | x \in A}. Proof. have [/fsetP Aisfset0 | [a ainA]] := fset_0Vmem (finsupp A); last first. by right; exists a; rewrite -msuppE. left; apply/msetP => a; rewrite mset0E; apply/mset_eq0P. by rewrite -msuppE Aisfset0 inE. Qed.
Lemma
root
[ "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop tuple.", "Require Import finmap." ]
multiset.v
mset_0Vmem
size_mset A : size A = \sum_(a <- finsupp A) A a. Proof. by rewrite -sum1_size sum_mset; apply: eq_bigr => i; rewrite muln1. Qed.
Definition
root
[ "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop tuple.", "Require Import finmap." ]
multiset.v
size_mset
size_mset0 : size (mset0 : {mset K}) = 0. Proof. by rewrite -sum1_size big_mset0. Qed.
Lemma
root
[ "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop tuple.", "Require Import finmap." ]
multiset.v
size_mset0
sum_nat_seq_eq0 (I : eqType) r (P : pred I) (E : I -> nat) : (\sum_(i <- r | P i) E i == 0) = all [pred i | P i ==> (E i == 0)] r. Proof. rewrite big_tnth sum_nat_eq0; apply/forallP/allP => /= HE x. by move=> /seq_tnthP[i ->]; apply: HE. by apply: HE; rewrite mem_tnth. Qed.
Lemma
root
[ "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop tuple.", "Require Import finmap." ]
multiset.v
sum_nat_seq_eq0
size_mset_eq0 A : (size A == 0) = (A == mset0). Proof. apply/idP/eqP => [|->]; last by rewrite size_mset0. rewrite size_mset sum_nat_seq_eq0 => /allP AP. apply/msetP => a /=; rewrite msetE. by have /= := AP a; case: finsuppP => // _ /(_ _)/eqP->. Qed.
Lemma
root
[ "From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq.", "From mathcomp Require Import choice finset finfun fintype bigop tuple.", "Require Import finmap." ]
multiset.v
size_mset_eq0