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 |
|---|---|---|---|---|---|---|---|---|---|---|
msetDP x A B : reflect (x \in A \/ x \in B) (x \in A `+` B). | Proof. by rewrite !inE; exact: orP. Qed. | Lemma | msetDP | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"inE"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetULVR x A B : x \in A `|` B -> (x \in A) + (x \in B). | Proof. by rewrite inE; case: (x \in A); [left|right]. Qed. | Lemma | msetULVR | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"inE"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetDLVR x A B : x \in A `+` B -> (x \in A) + (x \in B). | Proof. by rewrite inE; case: (x \in A); [left|right]. Qed. | Lemma | msetDLVR | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"inE"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetIUr A B C : A `&` (B `|` C) = (A `&` B) `|` (A `&` C). | Proof. by apply/msetP=> x; rewrite !msetE minn_maxr. Qed. | Lemma | msetIUr | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP"
] | distribute /cancel | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 |
msetIUl A B C : (A `|` B) `&` C = (A `&` C) `|` (B `&` C). | Proof. by apply/msetP=> x; rewrite !msetE minn_maxl. Qed. | Lemma | msetIUl | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetUIr A B C : A `|` (B `&` C) = (A `|` B) `&` (A `|` C). | Proof. by apply/msetP=> x; rewrite !msetE maxn_minr. Qed. | Lemma | msetUIr | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetUIl A B C : (A `&` B) `|` C = (A `|` C) `&` (B `|` C). | Proof. by apply/msetP=> x; rewrite !msetE maxn_minl. Qed. | Lemma | msetUIl | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetUKC A B : (A `|` B) `&` A = A. | Proof. by apply/msetP=> x; rewrite !msetE maxnK. Qed. | Lemma | msetUKC | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetUK A B : (B `|` A) `&` A = A. | Proof. by rewrite msetUC msetUKC. Qed. | Lemma | msetUK | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetUC",
"msetUKC"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetKUC A B : A `&` (B `|` A) = A. | Proof. by rewrite msetIC msetUK. Qed. | Lemma | msetKUC | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetIC",
"msetUK"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetKU A B : A `&` (A `|` B) = A. | Proof. by rewrite msetIC msetUKC. Qed. | Lemma | msetKU | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetIC",
"msetUKC"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetIKC A B : (A `&` B) `|` A = A. | Proof. by apply/msetP=> x; rewrite !msetE minnK. Qed. | Lemma | msetIKC | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetIK A B : (B `&` A) `|` A = A. | Proof. by rewrite msetIC msetIKC. Qed. | Lemma | msetIK | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetIC",
"msetIKC"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetKIC A B : A `|` (B `&` A) = A. | Proof. by rewrite msetUC msetIK. Qed. | Lemma | msetKIC | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetIK",
"msetUC"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetKI A B : A `|` (A `&` B) = A. | Proof. by rewrite msetIC msetKIC. Qed. | Lemma | msetKI | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetIC",
"msetKIC"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetUKid A B : B `|` A `|` A = B `|` A. | Proof. by rewrite -msetUA msetUid. Qed. | Lemma | msetUKid | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetUA",
"msetUid"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetUKidC A B : A `|` B `|` A = A `|` B. | Proof. by rewrite msetUAC msetUid. Qed. | Lemma | msetUKidC | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetUAC",
"msetUid"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetKUid A B : A `|` (A `|` B) = A `|` B. | Proof. by rewrite msetUA msetUid. Qed. | Lemma | msetKUid | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetUA",
"msetUid"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetKUidC A B : A `|` (B `|` A) = B `|` A. | Proof. by rewrite msetUCA msetUid. Qed. | Lemma | msetKUidC | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetUCA",
"msetUid"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetIKid A B : B `&` A `&` A = B `&` A. | Proof. by rewrite -msetIA msetIid. Qed. | Lemma | msetIKid | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetIA",
"msetIid"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetIKidC A B : A `&` B `&` A = A `&` B. | Proof. by rewrite msetIAC msetIid. Qed. | Lemma | msetIKidC | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetIAC",
"msetIid"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetKIid A B : A `&` (A `&` B) = A `&` B. | Proof. by rewrite msetIA msetIid. Qed. | Lemma | msetKIid | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetIA",
"msetIid"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetKIidC A B : A `&` (B `&` A) = B `&` A. | Proof. by rewrite msetICA msetIid. Qed. | Lemma | msetKIidC | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetICA",
"msetIid"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetDIr A B C : A `+` (B `&` C) = (A `+` B) `&` (A `+` C). | Proof. by apply/msetP=> x; rewrite !msetE addn_minr. Qed. | Lemma | msetDIr | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetDIl A B C : (A `&` B) `+` C = (A `+` C) `&` (B `+` C). | Proof. by apply/msetP=> x; rewrite !msetE addn_minl. Qed. | Lemma | msetDIl | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetDKIC A B : (A `+` B) `&` A = A. | Proof. by apply/msetP=> x; rewrite !msetE (minn_idPr _) // leq_addr. Qed. | Lemma | msetDKIC | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetDKI A B : (B `+` A) `&` A = A. | Proof. by rewrite msetDC msetDKIC. Qed. | Lemma | msetDKI | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetDC",
"msetDKIC"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetKDIC A B : A `&` (B `+` A) = A. | Proof. by rewrite msetIC msetDKI. Qed. | Lemma | msetKDIC | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetDKI",
"msetIC"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetKDI A B : A `&` (A `+` B) = A. | Proof. by rewrite msetDC msetKDIC. Qed. | Lemma | msetKDI | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetDC",
"msetKDIC"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetDKB A : cancel (msetD A) (msetB^~ A). | Proof. by move=> B; apply/msetP => a; rewrite !msetE addKn. Qed. | Lemma | msetDKB | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetB",
"msetD",
"msetE",
"msetP"
] | adjunction / subtraction | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 |
msetDKBC A : cancel (msetD^~ A) (msetB^~ A). | Proof. by move=> B; rewrite msetDC msetDKB. Qed. | Lemma | msetDKBC | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetB",
"msetD",
"msetDC",
"msetDKB"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetBSKl A B a : ((a +` A) `\` B) `\ a = A `\` B. | Proof.
apply/msetP=> b; rewrite !msetE; case: ifPn; rewrite ?add0n ?subn0 //.
by rewrite add1n subn1 subSKn.
Qed. | Lemma | msetBSKl | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetBDl C A B : (C `+` A) `\` (C `+` B) = A `\` B. | Proof. by apply/msetP=> a; rewrite !msetE subnDl. Qed. | Lemma | msetBDl | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetBDr C A B : (A `+` C) `\` (B `+` C) = A `\` B. | Proof. by apply/msetP=> a; rewrite !msetE subnDr. Qed. | Lemma | msetBDr | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetBDA A B C : B `\` (A `+` C) = B `\` A `\` C. | Proof. by apply/msetP=> a; rewrite !msetE subnDA. Qed. | Lemma | msetBDA | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetUE A B C : msetU A B = A `+` (B `\` A). | Proof. by apply/msetP=> a; rewrite !msetE maxnE. Qed. | Lemma | msetUE | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP",
"msetU"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubsetP {A B} : reflect (forall x, A x <= B x) (A `<=` B). | Proof.
apply: (iffP forallP)=> // ? x; case: (in_fsetP (finsupp A) x) => //.
by rewrite msuppE => /mset_eq0P->.
Qed. | Lemma | msubsetP | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"in_fsetP",
"mset_eq0P",
"msuppE"
] | subset | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 |
msubset_subset {A B} : A `<=` B -> {subset A <= B}. | Proof.
by move=> /msubsetP AB x; rewrite !in_mset => ?; exact: (leq_trans _ (AB _)).
Qed. | Lemma | msubset_subset | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"in_mset",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetB_eq0 (A B : {mset K}) : (A `\` B == mset0) = (A `<=` B). | Proof.
apply/mset_eqP/msubsetP => AB a;
by have := AB a; rewrite !msetE -subn_eq0 => /eqP.
Qed. | Lemma | msetB_eq0 | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"mset0",
"msetE",
"mset_eqP",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubset_refl A : A `<=` A. | Proof. exact/msubsetP. Qed. | Lemma | msubset_refl | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubset_trans : transitive (@msubset K). | Proof.
move=> y x z /msubsetP xy /msubsetP yz ; apply/msubsetP => a.
by apply: (leq_trans (xy _)).
Qed. | Lemma | msubset_trans | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msubset",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetUS C A B : A `<=` B -> C `|` A `<=` C `|` B. | Proof.
move=> sAB; apply/msubsetP=> x; rewrite !msetE.
by rewrite geq_max !leq_max leqnn (msubsetP sAB) orbT.
Qed. | Lemma | msetUS | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetDS C A B : A `<=` B -> C `+` A `<=` C `+` B. | Proof.
by move=> /msubsetP sAB; apply/msubsetP=> x; rewrite !msetE leq_add2l.
Qed. | Lemma | msetDS | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetSU C A B : A `<=` B -> A `|` C `<=` B `|` C. | Proof. by move=> sAB; rewrite -!(msetUC C) msetUS. Qed. | Lemma | msetSU | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetUC",
"msetUS"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetSD C A B : A `<=` B -> A `+` C `<=` B `+` C. | Proof. by move=> sAB; rewrite -!(msetDC C) msetDS. Qed. | Lemma | msetSD | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetDC",
"msetDS"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetUSS A B C D : A `<=` C -> B `<=` D -> A `|` B `<=` C `|` D. | Proof. by move=> /(msetSU B) /msubset_trans sAC /(msetUS C)/sAC. Qed. | Lemma | msetUSS | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetSU",
"msetUS",
"msubset_trans"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetDSS A B C D : A `<=` C -> B `<=` D -> A `+` B `<=` C `+` D. | Proof. by move=> /(msetSD B) /msubset_trans sAC /(msetDS C)/sAC. Qed. | Lemma | msetDSS | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetDS",
"msetSD",
"msubset_trans"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetIidPl {A B} : reflect (A `&` B = A) (A `<=` B). | Proof.
apply: (iffP msubsetP) => [?|<- a]; last by rewrite !msetE geq_min leqnn orbT.
by apply/msetP => a; rewrite !msetE (minn_idPl _).
Qed. | Lemma | msetIidPl | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetIidPr {A B} : reflect (A `&` B = B) (B `<=` A). | Proof. by rewrite msetIC; apply: msetIidPl. Qed. | Lemma | msetIidPr | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetIC",
"msetIidPl"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubsetIidl A B : (A `<=` A `&` B) = (A `<=` B). | Proof.
apply/msubsetP/msubsetP=> sAB a; have := sAB a; rewrite !msetE.
by rewrite leq_min leqnn.
by move/minn_idPl->.
Qed. | Lemma | msubsetIidl | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubsetIidr A B : (B `<=` A `&` B) = (B `<=` A). | Proof. by rewrite msetIC msubsetIidl. Qed. | Lemma | msubsetIidr | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetIC",
"msubsetIidl"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetUidPr A B : reflect (A `|` B = B) (A `<=` B). | Proof.
apply: (iffP msubsetP) => [AB|<- a]; last by rewrite !msetE leq_max leqnn.
by apply/msetP=> a; rewrite !msetE (maxn_idPr _).
Qed. | Lemma | msetUidPr | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetUidPl A B : reflect (A `|` B = A) (B `<=` A). | Proof. by rewrite msetUC; apply/msetUidPr. Qed. | Lemma | msetUidPl | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetUC",
"msetUidPr"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubsetUl A B : A `<=` A `|` B. | Proof. by apply/msubsetP=> a; rewrite !msetE leq_maxl. Qed. | Lemma | msubsetUl | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubsetUr A B : B `<=` (A `|` B). | Proof. by rewrite msetUC. Qed. | Lemma | msubsetUr | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetUC"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubsetU1 x A : A `<=` (x |` A). | Proof. by rewrite msubsetUr. Qed. | Lemma | msubsetU1 | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msubsetUr"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubsetU A B C : (A `<=` B) || (A `<=` C) -> A `<=` (B `|` C). | Proof. by move=> /orP [] /msubset_trans ->. Qed. | Lemma | msubsetU | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msubset_trans"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
eqEmsubset A B : (A == B) = (A `<=` B) && (B `<=` A). | Proof.
apply/eqP/andP => [<-|[/msubsetP AB /msubsetP BA]]; first by split.
by apply/msetP=> a; apply/eqP; rewrite eqn_leq AB BA.
Qed. | Lemma | eqEmsubset | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetP",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubEproper A B : A `<=` B = (A == B) || (A `<` B). | Proof. by rewrite eqEmsubset -andb_orr orbN andbT. Qed. | Lemma | msubEproper | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"eqEmsubset"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
mproper_sub A B : A `<` B -> A `<=` B. | Proof. by rewrite msubEproper orbC => ->. Qed. | Lemma | mproper_sub | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msubEproper"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
eqVmproper A B : A `<=` B -> A = B \/ A `<` B. | Proof. by rewrite msubEproper => /predU1P. Qed. | Lemma | eqVmproper | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msubEproper"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
mproperEneq A B : A `<` B = (A != B) && (A `<=` B). | Proof. by rewrite andbC eqEmsubset negb_and andb_orr andbN. Qed. | Lemma | mproperEneq | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"eqEmsubset"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
mproper_neq A B : A `<` B -> A != B. | Proof. by rewrite mproperEneq; case/andP. Qed. | Lemma | mproper_neq | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"mproperEneq"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
eqEmproper A B : (A == B) = (A `<=` B) && ~~ (A `<` B). | Proof. by rewrite negb_and negbK andb_orr andbN eqEmsubset. Qed. | Lemma | eqEmproper | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"eqEmsubset"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msub0set A : msubset mset0 A. | Proof. by apply/msubsetP=> x; rewrite msetE. Qed. | Lemma | msub0set | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"mset0",
"msetE",
"msubset",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubset0 A : (A `<=` mset0) = (A == mset0). | Proof. by rewrite eqEmsubset msub0set andbT. Qed. | Lemma | msubset0 | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"eqEmsubset",
"mset0",
"msub0set"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
mproper0 A : (mproper mset0 A) = (A != mset0). | Proof. by rewrite /mproper msub0set msubset0. Qed. | Lemma | mproper0 | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"mproper",
"mset0",
"msub0set",
"msubset0"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
mproperE A B : (A `<` B) = (A `<=` B) && ~~ (msubset B A). | Proof. by []. Qed. | Lemma | mproperE | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msubset"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
mproper_sub_trans B A C : A `<` B -> B `<=` C -> A `<` C. | Proof.
move=> /andP [AB NBA] BC; rewrite /mproper (msubset_trans AB) //=.
by apply: contra NBA=> /(msubset_trans _)->.
Qed. | Lemma | mproper_sub_trans | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"mproper",
"msubset_trans"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msub_proper_trans B A C :
A `<=` B -> B `<` C -> A `<` C. | Proof.
move=> AB /andP [CB NCB]; rewrite /mproper (msubset_trans AB) //=.
by apply: contra NCB=> /msubset_trans->.
Qed. | Lemma | msub_proper_trans | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"mproper",
"msubset_trans"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubset_neq0 A B : A `<=` B -> A != mset0 -> B != mset0. | Proof. by rewrite -!mproper0 => sAB /mproper_sub_trans->. Qed. | Lemma | msubset_neq0 | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"mproper0",
"mproper_sub_trans",
"mset0"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetBDKC A B : A `<=` B -> A `+` (B `\` A) = B. | Proof. by move=> /msubsetP AB; apply/msetP=> a; rewrite !msetE subnKC. Qed. | Lemma | msetBDKC | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP",
"msubsetP"
] | msub is a morphism | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 |
msetBDK A B : A `<=` B -> B `\` A `+` A = B. | Proof. by move=> /msubsetP AB; apply/msetP => a; rewrite !msetE subnK. Qed. | Lemma | msetBDK | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetBBK A B : A `<=` B -> B `\` (B `\` A) = A. | Proof. by move=> /msubsetP AB; apply/msetP => a; rewrite !msetE subKn. Qed. | Lemma | msetBBK | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetBD1K A B a : A `<=` B -> A a < B a -> a +` (B `\` (a +` A)) = B `\` A. | Proof.
move=> /msubsetP AB ABa; apply/msetP => b; rewrite !msetE.
by case: ifP => //= /eqP->; rewrite !add1n subnSK.
Qed. | Lemma | msetBD1K | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
subset_msetBLR A B C : (msubset (A `\` B) C) = (A `<=` B `+` C). | Proof.
apply/msubsetP/msubsetP => [] sABC a;
by have := sABC a; rewrite !msetE ?leq_subLR.
Qed. | Lemma | subset_msetBLR | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msubset",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetnP n x a : reflect (0 < n /\ x = a) (x \in msetn n a). | Proof. by do [apply: (iffP idP); rewrite !inE] => [/andP[]|[]] -> /eqP. Qed. | Lemma | msetnP | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"inE",
"msetn"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
gt0_msetnP n x a : 0 < n -> reflect (x = a) (x \in msetn n a). | Proof. by move=> n_gt0; rewrite inE n_gt0 /=; exact: eqP. Qed. | Lemma | gt0_msetnP | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"inE",
"msetn"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetn1 n a : a \in msetn n a = (n > 0). | Proof. by rewrite inE eqxx andbT. Qed. | Lemma | msetn1 | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"inE",
"msetn"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
mset1P x a : reflect (x = a) (x \in [mset a]). | Proof. by rewrite inE; exact: eqP. Qed. | Lemma | mset1P | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"inE"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
mset11 a : a \in [mset a]. | Proof. by rewrite inE /=. Qed. | Lemma | mset11 | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"inE"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetn_inj n : n > 0 -> injective (@msetn K n). | Proof.
move=> n_gt0 a b eqsab; apply/(gt0_msetnP _ _ n_gt0).
by rewrite -eqsab inE n_gt0 eqxx.
Qed. | Lemma | msetn_inj | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"gt0_msetnP",
"inE",
"msetn"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
mset1UP x a B : reflect (x = a \/ x \in B) (x \in a |` B). | Proof. by rewrite !inE; exact: predU1P. Qed. | Lemma | mset1UP | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"inE"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
mset_cons a s : seq_mset (a :: s) = a +` (seq_mset s). | Proof. by apply/msetP=> x; rewrite !msetE /= eq_sym. Qed. | Lemma | mset_cons | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP",
"seq_mset"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetIP x A B : reflect (x \in A /\ x \in B) (x \in A `&` B). | Proof. by rewrite inE; apply: andP. Qed. | Lemma | msetIP | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"inE"
] | intersection | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 |
msetIS C A B : A `<=` B -> C `&` A `<=` C `&` B. | Proof.
move=> sAB; apply/msubsetP=> x; rewrite !msetE.
by rewrite leq_min !geq_min leqnn (msubsetP sAB) orbT.
Qed. | Lemma | msetIS | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetSI C A B : A `<=` B -> A `&` C `<=` B `&` C. | Proof. by move=> sAB; rewrite -!(msetIC C) msetIS. Qed. | Lemma | msetSI | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetIC",
"msetIS"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetISS A B C D : A `<=` C -> B `<=` D -> A `&` B `<=` C `&` D. | Proof. by move=> /(msetSI B) /msubset_trans sAC /(msetIS C) /sAC. Qed. | Lemma | msetISS | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetIS",
"msetSI",
"msubset_trans"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetSB C A B : A `<=` B -> A `\` C `<=` B `\` C. | Proof.
by move=> /msubsetP sAB; apply/msubsetP=> x; rewrite !msetE leq_sub2r.
Qed. | Lemma | msetSB | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msubsetP"
] | difference | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 |
msetBS C A B : A `<=` B -> C `\` B `<=` C `\` A. | Proof.
by move=> /msubsetP sAB; apply/msubsetP=> x; rewrite !msetE leq_sub2l.
Qed. | Lemma | msetBS | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetBSS A B C D : A `<=` C -> D `<=` B -> A `\` B `<=` C `\` D. | Proof. by move=> /(msetSB B) /msubset_trans sAC /(msetBS C) /sAC. Qed. | Lemma | msetBSS | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetBS",
"msetSB",
"msubset_trans"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetB0 A : A `\` mset0 = A. | Proof. by apply/msetP=> x; rewrite !msetE subn0. Qed. | Lemma | msetB0 | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"mset0",
"msetE",
"msetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
mset0B A : mset0 `\` A = mset0. | Proof. by apply/msetP=> x; rewrite !msetE sub0n. Qed. | Lemma | mset0B | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"mset0",
"msetE",
"msetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msetBxx A : A `\` A = mset0. | Proof. by apply/msetP=> x; rewrite !msetE subnn. Qed. | Lemma | msetBxx | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"mset0",
"msetE",
"msetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubsetIl A B : A `&` B `<=` A. | Proof. by apply/msubsetP=> x; rewrite msetE geq_minl. Qed. | Lemma | msubsetIl | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msubsetP"
] | other inclusions | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 |
msubsetIr A B : A `&` B `<=` B. | Proof. by apply/msubsetP=> x; rewrite msetE geq_minr. Qed. | Lemma | msubsetIr | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubsetDl A B : A `\` B `<=` A. | Proof. by apply/msubsetP=> x; rewrite msetE leq_subLR leq_addl. Qed. | Lemma | msubsetDl | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msubsetP"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
msubD1set A x : A `\ x `<=` A. | Proof. by rewrite msubsetDl. Qed. | Lemma | msubD1set | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msubsetDl"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 | |
mem_mset1U a A : a \in A -> a |` A = A. | Proof.
rewrite in_mset => aA; apply/msetP => x; rewrite !msetE (maxn_idPr _) //.
by have [->|//] := altP eqP; rewrite (leq_trans _ aA).
Qed. | Lemma | mem_mset1U | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"in_mset",
"msetE",
"msetP"
] | cardinal lemmas for msets | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 |
mem_msetD1 a A : a \notin A -> A `\ a = A. | Proof.
move=> /mset_eq0P aA; apply/msetP => x; rewrite !msetE.
by have [->|] := altP eqP; rewrite ?aA ?subn0.
Qed. | Lemma | mem_msetD1 | Root | multiset.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrnat",
"eqtype",
"ssrfun",
"seq",
"choice",
"finset",
"finfun",
"fintype",
"bigop",
"tuple",
"finmap"
] | [
"msetE",
"msetP",
"mset_eq0P"
] | https://github.com/math-comp/finmap | 53239c7997b1143592d7814ec51a0e3404844b54 |
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