fact stringlengths 8 1.54k | type stringclasses 19
values | library stringclasses 8
values | imports listlengths 1 10 | filename stringclasses 98
values | symbolic_name stringlengths 1 42 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
card_Zp: p > 0 -> #|Zp| = p.
Proof.
rewrite /Zp; case: p => [|[|p']] //= _; first by rewrite cards1.
by rewrite cardsT card_ord.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | card_Zp | |
mem_Zpx : p > 1 -> x \in Zp. Proof. by rewrite /Zp => ->. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | mem_Zp | |
units_Zp_group:= [group of units_Zp]. | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | units_Zp_group | |
card_units_Zp: p > 0 -> #|units_Zp| = totient p.
Proof.
move=> p_gt0; transitivity (totient p.-2.+2); last by case: p p_gt0 => [|[|p']].
rewrite cardsT card_sub -sum1_card big_mkcond /=.
by rewrite totient_count_coprime big_mkord.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | card_units_Zp | |
units_Zp_abelian: abelian units_Zp.
Proof. by apply/centsP=> u _ v _; apply: unit_Zp_mulgC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | units_Zp_abelian | |
char_Zp:= (pchar_Zp) (only parsing). | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | char_Zp | |
Fp_Zcast: Zp_trunc (pdiv p) = Zp_trunc p.
Proof. by rewrite /pdiv primes_prime. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Fp_Zcast | |
Fp_cast: (Zp_trunc (pdiv p)).+2 = p.
Proof. by rewrite Fp_Zcast ?Zp_cast ?prime_gt1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Fp_cast | |
card_Fp: #|'F_p| = p.
Proof. by rewrite card_ord Fp_cast. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | card_Fp | |
val_Fp_natn : (n%:R : 'F_p) = (n %% p)%N :> nat.
Proof. by rewrite Zp_nat /= Fp_cast. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | val_Fp_nat | |
Fp_nat_modm : (m %% p)%:R = m%:R :> 'F_p.
Proof. by apply: ord_inj; rewrite !val_Fp_nat // modn_mod. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Fp_nat_mod | |
pchar_Fp: p \in [pchar 'F_p].
Proof. by rewrite !inE -Fp_nat_mod p_pr ?modnn. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | pchar_Fp | |
pchar_Fp_0: p%:R = 0 :> 'F_p.
Proof. exact: GRing.pcharf0 pchar_Fp. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | pchar_Fp_0 | |
unitFpEx : ((x%:R : 'F_p) \is a GRing.unit) = coprime p x.
Proof. by rewrite pdiv_id // unitZpE // prime_gt1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | unitFpE | |
Fp_fieldMixin: GRing.ComUnitRing_isField 'F_p.
Proof.
constructor => x nzx.
rewrite qualifE /= prime_coprime ?gtnNdvd ?lt0n //.
case: (ltnP 1 p) => [lt1p | ]; last by case: p => [|[|p']].
by rewrite Zp_cast ?prime_gt1 ?pdiv_prime.
Qed.
HB.instance Definition _ := Fp_fieldMixin.
HB.instance Definition _ := FinRing.isFie... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Fp_fieldMixin | |
gen_tperm_stepn (k : 'I_n.+1) : coprime n.+1 k ->
<<[set tperm i (i + k) | i : 'I_n.+1]>>%g = [set: 'S_n.+1].
Proof.
case: n k => [|n] k.
move=> _; apply/eqP; rewrite eqEsubset subsetT/= -(gen_tperm 0)/= gen_subG.
apply/subsetP => s /imsetP[/= [][|//] lt01 _ ->].
have ->: (Ordinal lt01) = 0 by apply/val_inj.
... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | gen_tperm_step | |
perm_addr1Xn m (j k : 'I_n.+1) : (perm (addrI m%R) ^+ j)%g k = m *+ j + k.
Proof. by rewrite permX (eq_iter (permE _)) iter_addr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | perm_addr1X | |
gen_tpermn_circular_shiftn (i j : 'I_n.+2)
(c := perm (addrI 1)) : coprime n.+2 (j - i)%R ->
<<[set tperm i j ; c]>>%g = [set: 'S_n.+2].
Proof.
move=> jBi_coprime; apply/eqP; rewrite eqEsubset subsetT/=.
rewrite -(gen_tperm_step jBi_coprime) gen_subG.
apply/subsetP => s /imsetP[/= k _ ->].
suff -> : tperm k (k + ... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | gen_tpermn_circular_shift | |
char_Fp:= (pchar_Fp) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pchar_Fp_0 instead.")] | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | char_Fp | |
char_Fp_0:= (pchar_Fp_0) (only parsing). | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | char_Fp_0 | |
RecordisLaw T (op : T -> T -> T) := {
opA : associative op;
}.
#[export]
HB.structure Definition Law T := {op of isLaw T op}. | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | Record | |
law:= Law.type.
HB.mixin Record isCommutativeLaw T (op : T -> T -> T) := {
opC : commutative op;
}.
#[export]
HB.structure Definition ComLaw T := {op of Law T op & isCommutativeLaw T op}. | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | law | |
com_law:= ComLaw.type.
HB.factory Record isComLaw T (op : T -> T -> T) := {
opA : associative op;
opC : commutative op;
}.
HB.builders Context T op of isComLaw T op.
HB.instance Definition _ := isLaw.Build T op opA.
HB.instance Definition _ := isCommutativeLaw.Build T op opC.
HB.end. | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | com_law | |
mulmA: associative mul. Proof. exact: opA. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mulmA | |
mulmC: commutative mul. Proof. exact: opC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mulmC | |
mulmCA: left_commutative mul.
Proof. by move=> x y z; rewrite !mulmA [_ x _]mulmC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mulmCA | |
mulmAC: right_commutative mul.
Proof. by move=> x y z; rewrite -!mulmA [_ y _]mulmC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mulmAC | |
mulmACA: interchange mul mul.
Proof. by move=> x y z t; rewrite -!mulmA [_ y _]mulmCA. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mulmACA | |
RecordisMonoidLaw T (idm : T) (op : T -> T -> T) := {
op1m : left_id idm op;
opm1 : right_id idm op;
}.
#[export]
HB.structure Definition Law T idm :=
{op of SemiGroup.Law T op & isMonoidLaw T idm op}. | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | Record | |
law:= Law.type.
HB.factory Record isLaw T (idm : T) (op : T -> T -> T) := {
opA : associative op;
op1m : left_id idm op;
opm1 : right_id idm op;
}.
HB.builders Context T idm op of isLaw T idm op.
HB.instance Definition _ := SemiGroup.isLaw.Build T op opA.
HB.instance Definition _ := isMonoidLaw.Build T idm op op1... | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | law | |
com_law:= ComLaw.type.
HB.factory Record isComLaw T (idm : T) (op : T -> T -> T) := {
opA : associative op;
opC : commutative op;
op1m : left_id idm op;
}.
HB.builders Context T idm op of isComLaw T idm op. | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | com_law | |
opm1: right_id idm op. Proof. by move=> x; rewrite opC op1m. Qed.
HB.instance Definition _ := isLaw.Build T idm op opA op1m opm1.
HB.instance Definition _ := isCommutativeLaw.Build T op opC.
HB.end.
HB.mixin Record isMulLaw T (zero : T) (mul : T -> T -> T) := {
mul_zerol : left_zero zero mul;
mul_zeror : right_zero... | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | opm1 | |
mul_law:= MulLaw.type.
HB.mixin Record isAddLaw T (mul : T -> T -> T) (op : T -> T -> T) := {
mul_op_Dl : left_distributive mul op;
mul_op_Dr : right_distributive mul op;
}.
#[export]
HB.structure Definition AddLaw T zero mul :=
{add of ComLaw T zero add & isAddLaw T mul add}. | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mul_law | |
add_law:= AddLaw.type. | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | add_law | |
mulC_id: left_id one mul -> right_id one mul.
Proof. by move=> mul1x x; rewrite mulC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mulC_id | |
mulC_zero: left_zero zero mul -> right_zero zero mul.
Proof. by move=> mul0x x; rewrite mulC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mulC_zero | |
mulC_dist: left_distributive mul add -> right_distributive mul add.
Proof. by move=> mul_addl x y z; rewrite !(mulC x). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mulC_dist | |
mul1m: left_id idm mul. Proof. exact: op1m. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mul1m | |
mulm1: right_id idm mul. Proof. exact: opm1. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mulm1 | |
iteropEn x : iterop n mul x idm = iter n (mul x) idm.
Proof. by case: n => // n; rewrite iterSr mulm1 iteropS. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | iteropE | |
mul0m: left_zero idm mul. Proof. exact: mul_zerol. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mul0m | |
mulm0: right_zero idm mul. Proof. exact: mul_zeror. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mulm0 | |
addmA: associative add. Proof. exact: mulmA. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | addmA | |
addmC: commutative add. Proof. exact: mulmC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | addmC | |
addmCA: left_commutative add. Proof. exact: mulmCA. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | addmCA | |
addmAC: right_commutative add. Proof. exact: mulmAC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | addmAC | |
add0m: left_id idm add. Proof. exact: mul1m. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | add0m | |
addm0: right_id idm add. Proof. exact: mulm1. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | addm0 | |
mulmDl: left_distributive mul add. Proof. exact: mul_op_Dl. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mulmDl | |
mulmDr: right_distributive mul add. Proof. exact: mul_op_Dr. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mulmDr | |
simpm:= (mulm1, mulm0, mul1m, mul0m, mulmA). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | simpm | |
Definition_ := isComLaw.Build bool true andb andbA andbC andTb.
HB.instance Definition _ := isMulLaw.Build bool false andb andFb andbF.
HB.instance Definition _ := isComLaw.Build bool false orb orbA orbC orFb.
HB.instance Definition _ := isMulLaw.Build bool true orb orTb orbT.
HB.instance Definition _ := isComLaw.Build... | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | Definition | |
myp:= addn. Definition mym := muln. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | myp | |
myp_mon:= [law of myp]. | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | myp_mon | |
myp_cmon:= [com_law of myp]. | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | myp_cmon | |
mym_mul:= [mul_law of mym]. | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mym_mul | |
myp_add:= [add_law _ of myp].
Print myp_add.
Print Canonical Projections.
*)
Delimit Scope big_scope with BIG.
Open Scope big_scope. | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | myp_add | |
bigbodyR I := BigBody of I & (R -> R -> R) & bool & R. | Variant | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | bigbody | |
applybig{R I} (body : bigbody R I) x :=
let: BigBody _ op b v := body in if b then op v x else x. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | applybig | |
reducebigR I idx r (body : I -> bigbody R I) :=
foldr (applybig \o body) idx r.
HB.lock Definition bigop := reducebig. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | reducebig | |
bigop_unlock:= Unlockable bigop.unlock. | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | bigop_unlock | |
index_iotam n := iota m (n - m). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | index_iota | |
mem_index_iotam n i : i \in index_iota m n = (m <= i < n).
Proof.
rewrite mem_iota; case le_m_i: (m <= i) => //=.
by rewrite -leq_subLR subSn // -subn_gt0 -subnDA subnKC // subn_gt0.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mem_index_iota | |
index_enum(T : finType) :=
locked_with index_enum_key (Finite.enum T). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | index_enum | |
deprecated_filter_index_enumT P : filter P (index_enum T) = enum P.
Proof. by rewrite [index_enum T]unlock. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | deprecated_filter_index_enum | |
mem_index_enumT i : i \in index_enum T.
Proof. by rewrite [index_enum T]unlock -enumT mem_enum. Qed.
#[global] Hint Resolve mem_index_enum : core. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | mem_index_enum | |
index_enum_uniqT : uniq (index_enum T).
Proof. by rewrite [index_enum T]unlock -enumT enum_uniq. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | index_enum_uniq | |
BIG_F:= (F in \big[_/_]_(i <- _ | _) F i)%pattern. | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | BIG_F | |
BIG_P:= (P in \big[_/_]_(i <- _ | P i) _)%pattern.
Local Notation "+%N" := addn (only parsing). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | BIG_P | |
big_loadR (K K' : R -> Type) idx op I r (P : pred I) F :
K (\big[op/idx]_(i <- r | P i) F i) * K' (\big[op/idx]_(i <- r | P i) F i)
-> K' (\big[op/idx]_(i <- r | P i) F i).
Proof. by case. Qed.
Arguments big_load [R] K [K'] idx op [I]. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_load | |
big_rec3I r (P : pred I) F1 F2 F3
(K_F : forall i y1 y2 y3, P i -> K y1 y2 y3 ->
K (op1 (F1 i) y1) (op2 (F2 i) y2) (op3 (F3 i) y3)) :
K (\big[op1/id1]_(i <- r | P i) F1 i)
(\big[op2/id2]_(i <- r | P i) F2 i)
(\big[op3/id3]_(i <- r | P i) F3 i).
Proof. by rewrite unlock; elim: r => //= i r; case: if... | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_rec3 | |
big_ind3I r (P : pred I) F1 F2 F3
(K_F : forall i, P i -> K (F1 i) (F2 i) (F3 i)) :
K (\big[op1/id1]_(i <- r | P i) F1 i)
(\big[op2/id2]_(i <- r | P i) F2 i)
(\big[op3/id3]_(i <- r | P i) F3 i).
Proof. by apply: big_rec3 => i x1 x2 x3 /K_F; apply: Kop. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_ind3 | |
big_rec2I r (P : pred I) F1 F2
(K_F : forall i y1 y2, P i -> K y1 y2 ->
K (op1 (F1 i) y1) (op2 (F2 i) y2)) :
K (\big[op1/id1]_(i <- r | P i) F1 i) (\big[op2/id2]_(i <- r | P i) F2 i).
Proof. by rewrite unlock; elim: r => //= i r; case: ifP => //; apply: K_F. Qed.
Hypothesis Kop : forall x1 x2 y1 y2,
K x1... | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_rec2 | |
big_ind2I r (P : pred I) F1 F2 (K_F : forall i, P i -> K (F1 i) (F2 i)) :
K (\big[op1/id1]_(i <- r | P i) F1 i) (\big[op2/id2]_(i <- r | P i) F2 i).
Proof. by apply: big_rec2 => i x1 x2 /K_F; apply: Kop. Qed.
Hypotheses (f_op : {morph f : x y / op2 x y >-> op1 x y}) (f_id : f id2 = id1). | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_ind2 | |
big_morphI r (P : pred I) F :
f (\big[op2/id2]_(i <- r | P i) F i) = \big[op1/id1]_(i <- r | P i) f (F i).
Proof. by rewrite unlock; elim: r => //= i r <-; rewrite -f_op -fun_if. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_morph | |
big_recI r (P : pred I) F
(Kop : forall i x, P i -> K x -> K (op (F i) x)) :
K (\big[op/idx]_(i <- r | P i) F i).
Proof. by rewrite unlock; elim: r => //= i r; case: ifP => //; apply: Kop. Qed.
Hypothesis Kop : forall x y, K x -> K y -> K (op x y). | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_rec | |
big_indI r (P : pred I) F (K_F : forall i, P i -> K (F i)) :
K (\big[op/idx]_(i <- r | P i) F i).
Proof. by apply: big_rec => // i x /K_F /Kop; apply. Qed.
Hypothesis Kop' : forall x y, K x -> K y -> op x y = op' x y. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_ind | |
eq_big_opI r (P : pred I) F (K_F : forall i, P i -> K (F i)) :
\big[op/idx]_(i <- r | P i) F i = \big[op'/idx]_(i <- r | P i) F i.
Proof.
by elim/(big_load K): _; elim/big_rec2: _ => // i _ y Pi [Ky <-]; auto.
Qed.
Hypotheses (fM : {morph f : x y / op x y}) (f_id : f idx = idx). | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | eq_big_op | |
big_endoI r (P : pred I) F :
f (\big[op/idx]_(i <- r | P i) F i) = \big[op/idx]_(i <- r | P i) f (F i).
Proof. exact: big_morph. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_endo | |
big_morph_in(R1 R2 : Type) (Q : {pred R2}) (f : R2 -> R1)
(id1 : R1) (op1 : R1 -> R1 -> R1)
(id2 : R2) (op2 : R2 -> R2 -> R2) :
{in Q &, forall x y, op2 x y \in Q} ->
id2 \in Q ->
{in Q &, {morph f : x y / op2 x y >-> op1 x y}} ->
f id2 = id1 ->
forall [I : Type] (r : seq I) (P : pred I) (F ... | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_morph_in | |
oACof associative op & commutative op :=
fun x => oapp (fun y => Some (oapp (op^~ y) y x)) x.
Arguments oAC : simpl never.
Hypothesis (opA : associative op) (opC : commutative op).
Local Notation oop := (oAC opA opC). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | oAC | |
oACEx y : oop (Some x) (Some y) = some (op x y). Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | oACE | |
oopA_subdef: associative oop.
Proof. by move=> [x|] [y|] [z|]//; rewrite /oAC/= opA. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | oopA_subdef | |
oopx1_subdef: left_id None oop. Proof. by case. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | oopx1_subdef | |
oop1x_subdef: right_id None oop. Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | oop1x_subdef | |
oopC_subdef: commutative oop.
Proof. by move=> [x|] [y|]//; rewrite /oAC/= opC. Qed.
HB.instance Definition _ := Monoid.isComLaw.Build (option T) None oop
oopA_subdef oopC_subdef oopx1_subdef.
Context [x : T]. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | oopC_subdef | |
some_big_AC_mk_monoid[I : Type] r P (F : I -> T) :
Some (\big[op/x]_(i <- r | P i) F i) =
oop (\big[oop/None]_(i <- r | P i) Some (F i)) (Some x).
Proof. by elim/big_rec2 : _ => //= i [y|] _ Pi [] -> //=; rewrite opA. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | some_big_AC_mk_monoid | |
big_AC_mk_monoid[I : Type] r P (F : I -> T) :
\big[op/x]_(i <- r | P i) F i =
odflt x (oop (\big[oop/None]_(i <- r | P i) Some (F i)) (Some x)).
Proof. by apply: Some_inj; rewrite some_big_AC_mk_monoid. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_AC_mk_monoid | |
foldrEr : foldr op idx r = \big[op/idx]_(x <- r) x.
Proof. by rewrite unlock. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | foldrE | |
big_filterr (P : pred I) F :
\big[op/idx]_(i <- filter P r) F i = \big[op/idx]_(i <- r | P i) F i.
Proof. by rewrite unlock; elim: r => //= i r <-; case (P i). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_filter | |
big_filter_condr (P1 P2 : pred I) F :
\big[op/idx]_(i <- filter P1 r | P2 i) F i
= \big[op/idx]_(i <- r | P1 i && P2 i) F i.
Proof.
rewrite -big_filter -(big_filter r); congr bigop.
by rewrite -filter_predI; apply: eq_filter => i; apply: andbC.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_filter_cond | |
eq_biglr (P1 P2 : pred I) F :
P1 =1 P2 ->
\big[op/idx]_(i <- r | P1 i) F i = \big[op/idx]_(i <- r | P2 i) F i.
Proof. by move=> eqP12; rewrite -!(big_filter r) (eq_filter eqP12). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | eq_bigl | |
big_andbCr (P Q : pred I) F :
\big[op/idx]_(i <- r | P i && Q i) F i
= \big[op/idx]_(i <- r | Q i && P i) F i.
Proof. by apply: eq_bigl => i; apply: andbC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_andbC | |
eq_bigrr (P : pred I) F1 F2 : (forall i, P i -> F1 i = F2 i) ->
\big[op/idx]_(i <- r | P i) F1 i = \big[op/idx]_(i <- r | P i) F2 i.
Proof. by move=> eqF12; elim/big_rec2: _ => // i x _ /eqF12-> ->. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | eq_bigr | |
eq_bigr (P1 P2 : pred I) F1 F2 :
P1 =1 P2 -> (forall i, P1 i -> F1 i = F2 i) ->
\big[op/idx]_(i <- r | P1 i) F1 i = \big[op/idx]_(i <- r | P2 i) F2 i.
Proof. by move/eq_bigl <-; move/eq_bigr->. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | eq_big | |
congr_bigr1 r2 (P1 P2 : pred I) F1 F2 :
r1 = r2 -> P1 =1 P2 -> (forall i, P1 i -> F1 i = F2 i) ->
\big[op/idx]_(i <- r1 | P1 i) F1 i = \big[op/idx]_(i <- r2 | P2 i) F2 i.
Proof. by move=> <-{r2}; apply: eq_big. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | congr_big | |
big_nil(P : pred I) F : \big[op/idx]_(i <- [::] | P i) F i = idx.
Proof. by rewrite unlock. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_nil | |
big_consi r (P : pred I) F :
let x := \big[op/idx]_(j <- r | P j) F j in
\big[op/idx]_(j <- i :: r | P j) F j = if P i then op (F i) x else x.
Proof. by rewrite unlock. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_cons | |
big_rcons_opi r (P : pred I) F :
let idx' := if P i then op (F i) idx else idx in
\big[op/idx]_(j <- rcons r i | P j) F j = \big[op/idx']_(j <- r | P j) F j.
Proof.
by elim: r => /= [|j r]; rewrite !(big_nil, big_cons, unlock)// => ->.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_rcons_op | |
big_mapJ (h : J -> I) r (P : pred I) F :
\big[op/idx]_(i <- map h r | P i) F i
= \big[op/idx]_(j <- r | P (h j)) F (h j).
Proof. by rewrite unlock; elim: r => //= j r ->. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div fintype tuple finfun"
] | boot/bigop.v | big_map |
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