statement stringlengths 1 374 | proof stringlengths 0 1.98k | type stringclasses 14
values | symbolic_name stringlengths 1 50 | library stringclasses 1
value | filename stringclasses 3
values | imports listlengths 14 19 | deps listlengths 0 21 | docstring stringclasses 27
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
msetIn a A n : A `&` msetn n a = msetn (minn (A a) n) a. | Proof.
by apply/msetP => x; rewrite !msetE; have [->|] := altP eqP; rewrite ?minn0.
Qed. | Lemma | msetIn | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP",
"msetn"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubIset A B C : (B `<=` A) || (C `<=` A) -> (B `&` C `<=` A). | Proof. by case/orP; apply: msubset_trans; rewrite (msubsetIl, msubsetIr). Qed. | Lemma | msubIset | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msubsetIl",
"msubsetIr",
"msubset_trans"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubsetI A B C : (A `<=` B `&` C) = (A `<=` B) && (A `<=` C). | Proof.
rewrite !(sameP msetIidPl eqP) msetIA; have [-> //| ] := altP (A `&` B =P A).
by apply: contraNF => /eqP <-; rewrite -msetIA -msetIIl msetIAC.
Qed. | Lemma | msubsetI | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetIA",
"msetIAC",
"msetIIl",
"msetIidPl"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubsetIP A B C : reflect (A `<=` B /\ A `<=` C) (A `<=` B `&` C). | Proof. by rewrite msubsetI; exact: andP. Qed. | Lemma | msubsetIP | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msubsetI"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubUset A B C : (B `|` C `<=` A) = (B `<=` A) && (C `<=` A). | Proof.
apply/idP/idP => [subA|/andP [AB CA]]; last by rewrite -[A]msetUid msetUSS.
by rewrite !(msubset_trans _ subA).
Qed. | Lemma | msubUset | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetUSS",
"msetUid",
"msubset_trans"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubUsetP A B C : reflect (A `<=` C /\ B `<=` C) (A `|` B `<=` C). | Proof. by rewrite msubUset; exact: andP. Qed. | Lemma | msubUsetP | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msubUset"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetU_eq0 A B : (A `|` B == mset0) = (A == mset0) && (B == mset0). | Proof. by rewrite -!msubset0 msubUset. Qed. | Lemma | msetU_eq0 | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"mset0",
"msubUset",
"msubset0"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
setD_eq0 A B : (A `\` B == mset0) = (A `<=` B). | Proof. by rewrite -msubset0 subset_msetBLR msetD0. Qed. | Lemma | setD_eq0 | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"mset0",
"msetD0",
"msubset0",
"subset_msetBLR"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
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 | msub1set | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"in_mset",
"msetnE",
"msetnxx",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
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 | msetDBA | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE2",
"msetP",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
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 | mset_0Vmem | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"fsetP",
"fset_0Vmem",
"inE",
"mset0",
"mset0E",
"msetP",
"mset_eq0P",
"msuppE"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
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 | size_mset | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"sum_mset"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
size_mset0 : size (mset0 : {mset K}) = 0. | Proof. by rewrite -sum1_size big_mset0. Qed. | Lemma | size_mset0 | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"big_mset0",
"mset0"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
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 | sum_nat_seq_eq0 | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
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 | size_mset_eq0 | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"finsuppP",
"mset0",
"msetE",
"msetP",
"size_mset",
"size_mset0",
"sum_nat_seq_eq0"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 |
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