Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {Ξ± Ξ² ΞΉ : Type*}
namespace Finsupp
def toMultiset : (Ξ± ββ β) β+ Multiset Ξ± where
toFun f := Finsupp.sum f... | Mathlib/Data/Finsupp/Multiset.lean | 94 | 101 | theorem toFinset_toMultiset [DecidableEq Ξ±] (f : Ξ± ββ β) : f.toMultiset.toFinset = f.support := by |
refine f.induction ?_ ?_
Β· rw [toMultiset_zero, Multiset.toFinset_zero, support_zero]
Β· intro a n f ha hn ih
rw [toMultiset_add, Multiset.toFinset_add, ih, toMultiset_single, support_add_eq,
support_single_ne_zero _ hn, Multiset.toFinset_nsmul _ _ hn, Multiset.toFinset_singleton]
refine Disjoint.mo... | 1,536 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {Ξ± Ξ² ΞΉ : Type*}
namespace Finsupp
def toMultiset : (Ξ± ββ β) β+ Multiset Ξ± where
toFun f := Finsupp.sum f... | Mathlib/Data/Finsupp/Multiset.lean | 105 | 114 | theorem count_toMultiset [DecidableEq Ξ±] (f : Ξ± ββ β) (a : Ξ±) : (toMultiset f).count a = f a :=
calc
(toMultiset f).count a = Finsupp.sum f (fun x n => (n β’ {x} : Multiset Ξ±).count a) := by |
rw [toMultiset_apply]; exact map_sum (Multiset.countAddMonoidHom a) _ f.support
_ = f.sum fun x n => n * ({x} : Multiset Ξ±).count a := by simp only [Multiset.count_nsmul]
_ = f a * ({a} : Multiset Ξ±).count a :=
sum_eq_single _
(fun a' _ H => by simp only [Multiset.count_singleton, if_false,... | 1,536 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {Ξ± Ξ² ΞΉ : Type*}
namespace Finsupp
def toMultiset : (Ξ± ββ β) β+ Multiset Ξ± where
toFun f := Finsupp.sum f... | Mathlib/Data/Finsupp/Multiset.lean | 117 | 120 | theorem toMultiset_sup [DecidableEq Ξ±] (f g : Ξ± ββ β) :
toMultiset (f β g) = toMultiset f βͺ toMultiset g := by |
ext
simp_rw [Multiset.count_union, Finsupp.count_toMultiset, Finsupp.sup_apply, sup_eq_max]
| 1,536 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {Ξ± Ξ² ΞΉ : Type*}
namespace Finsupp
def toMultiset : (Ξ± ββ β) β+ Multiset Ξ± where
toFun f := Finsupp.sum f... | Mathlib/Data/Finsupp/Multiset.lean | 122 | 125 | theorem toMultiset_inf [DecidableEq Ξ±] (f g : Ξ± ββ β) :
toMultiset (f β g) = toMultiset f β© toMultiset g := by |
ext
simp_rw [Multiset.count_inter, Finsupp.count_toMultiset, Finsupp.inf_apply, inf_eq_min]
| 1,536 |
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Multiset.Antidiagonal
#align_import data.finsupp.antidiagonal from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
namespace Finsupp
open Finset
universe u
variable {Ξ± : Type u} [Decidab... | Mathlib/Data/Finsupp/Antidiagonal.lean | 61 | 79 | theorem antidiagonal_single (a : Ξ±) (n : β) :
antidiagonal (single a n) = (antidiagonal n).map
(Function.Embedding.prodMap β¨_, single_injective aβ© β¨_, single_injective aβ©) := by |
ext β¨x, yβ©
simp only [mem_antidiagonal, mem_map, mem_antidiagonal, Function.Embedding.coe_prodMap,
Function.Embedding.coeFn_mk, Prod.map_apply, Prod.mk.injEq, Prod.exists]
constructor
Β· intro h
refine β¨x a, y a, DFunLike.congr_fun h a |>.trans single_eq_same, ?_β©
simp_rw [DFunLike.ext_iff, β forall... | 1,537 |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 60 | 61 | theorem factorization_def (n : β) {p : β} (pp : p.Prime) : n.factorization p = padicValNat p n := by |
simpa [factorization] using absurd pp
| 1,538 |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 67 | 81 | theorem factors_count_eq {n p : β} : n.factors.count p = n.factorization p := by |
rcases n.eq_zero_or_pos with (rfl | hn0)
Β· simp [factorization, count]
if pp : p.Prime then ?_ else
rw [count_eq_zero_of_not_mem (mt prime_of_mem_factors pp)]
simp [factorization, pp]
simp only [factorization_def _ pp]
apply _root_.le_antisymm
Β· rw [le_padicValNat_iff_replicate_subperm_factors pp h... | 1,538 |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 84 | 87 | theorem factorization_eq_factors_multiset (n : β) :
n.factorization = Multiset.toFinsupp (n.factors : Multiset β) := by |
ext p
simp
| 1,538 |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 90 | 92 | theorem multiplicity_eq_factorization {n p : β} (pp : p.Prime) (hn : n β 0) :
multiplicity p n = n.factorization p := by |
simp [factorization, pp, padicValNat_def' pp.ne_one hn.bot_lt]
| 1,538 |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 99 | 102 | theorem factorization_prod_pow_eq_self {n : β} (hn : n β 0) : n.factorization.prod (Β· ^ Β·) = n := by |
rw [factorization_eq_factors_multiset n]
simp only [β prod_toMultiset, factorization, Multiset.prod_coe, Multiset.toFinsupp_toMultiset]
exact prod_factors hn
| 1,538 |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 116 | 116 | theorem factorization_zero : factorization 0 = 0 := by | ext; simp [factorization]
| 1,538 |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 120 | 120 | theorem factorization_one : factorization 1 = 0 := by | ext; simp [factorization]
| 1,538 |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 133 | 135 | theorem factorization_eq_zero_iff (n p : β) :
n.factorization p = 0 β Β¬p.Prime β¨ Β¬p β£ n β¨ n = 0 := by |
simp_rw [β not_mem_support_iff, support_factorization, mem_primeFactors, not_and_or, not_ne_iff]
| 1,538 |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 139 | 140 | theorem factorization_eq_zero_of_non_prime (n : β) {p : β} (hp : Β¬p.Prime) :
n.factorization p = 0 := by | simp [factorization_eq_zero_iff, hp]
| 1,538 |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 143 | 144 | theorem factorization_eq_zero_of_not_dvd {n p : β} (h : Β¬p β£ n) : n.factorization p = 0 := by |
simp [factorization_eq_zero_iff, h]
| 1,538 |
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monotonicity
#align_import data.nat.totient from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
open Finset
namespace Nat
... | Mathlib/Data/Nat/Totient.lean | 51 | 57 | theorem totient_eq_card_lt_and_coprime (n : β) : Ο n = Nat.card { m | m < n β§ n.Coprime m } := by |
let e : { m | m < n β§ n.Coprime m } β Finset.filter n.Coprime (Finset.range n) :=
{ toFun := fun m => β¨m, by simpa only [Finset.mem_filter, Finset.mem_range] using m.propertyβ©
invFun := fun m => β¨m, by simpa only [Finset.mem_filter, Finset.mem_range] using m.propertyβ©
left_inv := fun m => by simp onl... | 1,539 |
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monotonicity
#align_import data.nat.totient from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
open Finset
namespace Nat
... | Mathlib/Data/Nat/Totient.lean | 78 | 81 | theorem filter_coprime_Ico_eq_totient (a n : β) :
((Ico n (n + a)).filter (Coprime a)).card = totient a := by |
rw [totient, filter_Ico_card_eq_of_periodic, count_eq_card_filter_range]
exact periodic_coprime a
| 1,539 |
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monotonicity
#align_import data.nat.totient from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
open Finset
namespace Nat
... | Mathlib/Data/Nat/Totient.lean | 84 | 109 | theorem Ico_filter_coprime_le {a : β} (k n : β) (a_pos : 0 < a) :
((Ico k (k + n)).filter (Coprime a)).card β€ totient a * (n / a + 1) := by |
conv_lhs => rw [β Nat.mod_add_div n a]
induction' n / a with i ih
Β· rw [β filter_coprime_Ico_eq_totient a k]
simp only [add_zero, mul_one, mul_zero, le_of_lt (mod_lt n a_pos),
Nat.zero_eq, zero_add]
-- Porting note: below line was `mono`
refine Finset.card_mono ?_
refine monotone_filter_lef... | 1,539 |
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monotonicity
#align_import data.nat.totient from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
open Finset
namespace Nat
... | Mathlib/Data/Nat/Totient.lean | 117 | 126 | theorem _root_.ZMod.card_units_eq_totient (n : β) [NeZero n] [Fintype (ZMod n)Λ£] :
Fintype.card (ZMod n)Λ£ = Ο n :=
calc
Fintype.card (ZMod n)Λ£ = Fintype.card { x : ZMod n // x.val.Coprime n } :=
Fintype.card_congr ZMod.unitsEquivCoprime
_ = Ο n := by |
obtain β¨m, rflβ© : β m, n = m + 1 := exists_eq_succ_of_ne_zero NeZero.out
simp only [totient, Finset.card_eq_sum_ones, Fintype.card_subtype, Finset.sum_filter, β
Fin.sum_univ_eq_sum_range, @Nat.coprime_comm (m + 1)]
rfl
| 1,539 |
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monotonicity
#align_import data.nat.totient from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
open Finset
namespace Nat
... | Mathlib/Data/Nat/Totient.lean | 129 | 135 | theorem totient_even {n : β} (hn : 2 < n) : Even n.totient := by |
haveI : Fact (1 < n) := β¨one_lt_two.trans hnβ©
haveI : NeZero n := NeZero.of_gt hn
suffices 2 = orderOf (-1 : (ZMod n)Λ£) by
rw [β ZMod.card_units_eq_totient, even_iff_two_dvd, this]
exact orderOf_dvd_card
rw [β orderOf_units, Units.coe_neg_one, orderOf_neg_one, ringChar.eq (ZMod n) n, if_neg hn.ne']
| 1,539 |
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.Zify
import Mathlib.Data.Nat.Totient
#align_import number_theory.lucas_primality from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
| Mathlib/NumberTheory/LucasPrimality.lean | 42 | 63 | theorem lucas_primality (p : β) (a : ZMod p) (ha : a ^ (p - 1) = 1)
(hd : β q : β, q.Prime β q β£ p - 1 β a ^ ((p - 1) / q) β 1) : p.Prime := by |
have h0 : p β 0 := by
rintro β¨β©
exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _)
have h1 : p β 1 := by
rintro β¨β©
exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _)
have hp1 : 1 < p := lt_of_le_of_ne h0.bot_lt h1.symm
have order_of_a : orderOf a = p - 1 := by
apply orderOf_eq_of_pow_and_po... | 1,540 |
import Mathlib.Data.Nat.Totient
import Mathlib.Data.Nat.Nth
import Mathlib.NumberTheory.SmoothNumbers
#align_import number_theory.prime_counting from "leanprover-community/mathlib"@"7fdd4f3746cb059edfdb5d52cba98f66fce418c0"
namespace Nat
open Finset
def primeCounting' : β β β :=
Nat.count Prime
#align nat.pr... | Mathlib/NumberTheory/PrimeCounting.lean | 83 | 102 | theorem primeCounting'_add_le {a k : β} (h0 : 0 < a) (h1 : a < k) (n : β) :
Ο' (k + n) β€ Ο' k + Nat.totient a * (n / a + 1) :=
calc
Ο' (k + n) β€ ((range k).filter Prime).card + ((Ico k (k + n)).filter Prime).card := by |
rw [primeCounting', count_eq_card_filter_range, range_eq_Ico, β
Ico_union_Ico_eq_Ico (zero_le k) le_self_add, filter_union]
apply card_union_le
_ β€ Ο' k + ((Ico k (k + n)).filter Prime).card := by
rw [primeCounting', count_eq_card_filter_range]
_ β€ Ο' k + ((Ico k (k + n)).filter (Copr... | 1,541 |
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.IsLocalHomeomorph
#align_import topology.instances.add_circle from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/AddCircle.lean | 64 | 79 | theorem continuous_right_toIcoMod : ContinuousWithinAt (toIcoMod hp a) (Ici x) x := by |
intro s h
rw [Filter.mem_map, mem_nhdsWithin_iff_exists_mem_nhds_inter]
haveI : Nontrivial π := β¨β¨0, p, hp.neβ©β©
simp_rw [mem_nhds_iff_exists_Ioo_subset] at h β’
obtain β¨l, u, hxI, hIsβ© := h
let d := toIcoDiv hp a x β’ p
have hd := toIcoMod_mem_Ico hp a x
simp_rw [subset_def, mem_inter_iff]
refine β¨_, ... | 1,542 |
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.IsLocalHomeomorph
#align_import topology.instances.add_circle from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/AddCircle.lean | 82 | 89 | theorem continuous_left_toIocMod : ContinuousWithinAt (toIocMod hp a) (Iic x) x := by |
rw [(funext fun y => Eq.trans (by rw [neg_neg]) <| toIocMod_neg _ _ _ :
toIocMod hp a = (fun x => p - x) β toIcoMod hp (-a) β Neg.neg)]
-- Porting note: added
have : ContinuousNeg π := TopologicalAddGroup.toContinuousNeg
exact
(continuous_sub_left _).continuousAt.comp_continuousWithinAt <|
(co... | 1,542 |
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.IsLocalHomeomorph
#align_import topology.instances.add_circle from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/AddCircle.lean | 152 | 153 | theorem coe_eq_zero_iff {x : π} : (x : AddCircle p) = 0 β β n : β€, n β’ p = x := by |
simp [AddSubgroup.mem_zmultiples_iff]
| 1,542 |
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.IsLocalHomeomorph
#align_import topology.instances.add_circle from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/AddCircle.lean | 156 | 164 | theorem coe_eq_zero_of_pos_iff (hp : 0 < p) {x : π} (hx : 0 < x) :
(x : AddCircle p) = 0 β β n : β, n β’ p = x := by |
rw [coe_eq_zero_iff]
constructor <;> rintro β¨n, rflβ©
Β· replace hx : 0 < n := by
contrapose! hx
simpa only [β neg_nonneg, β zsmul_neg, zsmul_neg'] using zsmul_nonneg hp.le (neg_nonneg.2 hx)
exact β¨n.toNat, by rw [β natCast_zsmul, Int.toNat_of_nonneg hx.le]β©
Β· exact β¨(n : β€), by simpβ©
| 1,542 |
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.IsLocalHomeomorph
#align_import topology.instances.add_circle from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/AddCircle.lean | 175 | 176 | theorem coe_add_period (x : π) : ((x + p : π) : AddCircle p) = x := by |
rw [coe_add, β eq_sub_iff_add_eq', sub_self, coe_period]
| 1,542 |
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.IsLocalHomeomorph
#align_import topology.instances.add_circle from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/AddCircle.lean | 213 | 219 | theorem coe_eq_coe_iff_of_mem_Ico {x y : π} (hx : x β Ico a (a + p)) (hy : y β Ico a (a + p)) :
(x : AddCircle p) = y β x = y := by |
refine β¨fun h => ?_, by tautoβ©
suffices (β¨x, hxβ© : Ico a (a + p)) = β¨y, hyβ© by exact Subtype.mk.inj this
apply_fun equivIco p a at h
rw [β (equivIco p a).right_inv β¨x, hxβ©, β (equivIco p a).right_inv β¨y, hyβ©]
exact h
| 1,542 |
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.IsLocalHomeomorph
#align_import topology.instances.add_circle from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/AddCircle.lean | 222 | 228 | theorem liftIco_coe_apply {f : π β B} {x : π} (hx : x β Ico a (a + p)) :
liftIco p a f βx = f x := by |
have : (equivIco p a) x = β¨x, hxβ© := by
rw [Equiv.apply_eq_iff_eq_symm_apply]
rfl
rw [liftIco, comp_apply, this]
rfl
| 1,542 |
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.IsLocalHomeomorph
#align_import topology.instances.add_circle from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/AddCircle.lean | 231 | 237 | theorem liftIoc_coe_apply {f : π β B} {x : π} (hx : x β Ioc a (a + p)) :
liftIoc p a f βx = f x := by |
have : (equivIoc p a) x = β¨x, hxβ© := by
rw [Equiv.apply_eq_iff_eq_symm_apply]
rfl
rw [liftIoc, comp_apply, this]
rfl
| 1,542 |
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c"
noncomputable section
open Set
open Int hiding mem_zmultiples_iff
open AddSubgroup
namespace A... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 44 | 68 | theorem norm_coe_mul (x : β) (t : β) :
β(β(t * x) : AddCircle (t * p))β = |t| * β(x : AddCircle p)β := by |
have aux : β {a b c : β}, a β zmultiples b β c * a β zmultiples (c * b) := fun {a b c} h => by
simp only [mem_zmultiples_iff] at h β’
obtain β¨n, rflβ© := h
exact β¨n, (mul_smul_comm n c b).symmβ©
rcases eq_or_ne t 0 with (rfl | ht); Β· simp
have ht' : |t| β 0 := (not_congr abs_eq_zero).mpr ht
simp only ... | 1,543 |
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c"
noncomputable section
open Set
open Int hiding mem_zmultiples_iff
open AddSubgroup
namespace A... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 71 | 75 | theorem norm_neg_period (x : β) : β(x : AddCircle (-p))β = β(x : AddCircle p)β := by |
suffices β(β(-1 * x) : AddCircle (-1 * p))β = β(x : AddCircle p)β by
rw [β this, neg_one_mul]
simp
simp only [norm_coe_mul, abs_neg, abs_one, one_mul]
| 1,543 |
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c"
noncomputable section
open Set
open Int hiding mem_zmultiples_iff
open AddSubgroup
namespace A... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 79 | 83 | theorem norm_eq_of_zero {x : β} : β(x : AddCircle (0 : β))β = |x| := by |
suffices { y : β | (y : AddCircle (0 : β)) = (x : AddCircle (0 : β)) } = {x} by
rw [quotient_norm_eq, this, image_singleton, Real.norm_eq_abs, csInf_singleton]
ext y
simp [QuotientAddGroup.eq_iff_sub_mem, mem_zmultiples_iff, sub_eq_zero]
| 1,543 |
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c"
noncomputable section
open Set
open Int hiding mem_zmultiples_iff
open AddSubgroup
namespace A... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 86 | 117 | theorem norm_eq {x : β} : β(x : AddCircle p)β = |x - round (pβ»ΒΉ * x) * p| := by |
suffices β x : β, β(x : AddCircle (1 : β))β = |x - round x| by
rcases eq_or_ne p 0 with (rfl | hp)
Β· simp
have hx := norm_coe_mul p x pβ»ΒΉ
rw [abs_inv, eq_inv_mul_iff_mul_eqβ ((not_congr abs_eq_zero).mpr hp)] at hx
rw [β hx, inv_mul_cancel hp, this, β abs_mul, mul_sub, mul_inv_cancel_leftβ hp, mul... | 1,543 |
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c"
noncomputable section
open Set
open Int hiding mem_zmultiples_iff
open AddSubgroup
namespace A... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 120 | 124 | theorem norm_eq' (hp : 0 < p) {x : β} : β(x : AddCircle p)β = p * |pβ»ΒΉ * x - round (pβ»ΒΉ * x)| := by |
conv_rhs =>
congr
rw [β abs_eq_self.mpr hp.le]
rw [β abs_mul, mul_sub, mul_inv_cancel_leftβ hp.ne.symm, norm_eq, mul_comm p]
| 1,543 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 107 | 108 | theorem natCast_mul_eq_nsmul (x : β) (n : β) : β((n : β) * x) = n β’ (βx : Angle) := by |
simpa only [nsmul_eq_mul] using coeHom.map_nsmul x n
| 1,544 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 112 | 113 | theorem intCast_mul_eq_zsmul (x : β) (n : β€) : β((n : β) * x : β) = n β’ (βx : Angle) := by |
simpa only [zsmul_eq_mul] using coeHom.map_zsmul x n
| 1,544 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 119 | 125 | theorem angle_eq_iff_two_pi_dvd_sub {Ο ΞΈ : β} : (ΞΈ : Angle) = Ο β β k : β€, ΞΈ - Ο = 2 * Ο * k := by |
simp only [QuotientAddGroup.eq, AddSubgroup.zmultiples_eq_closure,
AddSubgroup.mem_closure_singleton, zsmul_eq_mul', (sub_eq_neg_add _ _).symm, eq_comm]
-- Porting note: added `rw`, `simp [Angle.coe, QuotientAddGroup.eq]` doesn't fire otherwise
rw [Angle.coe, Angle.coe, QuotientAddGroup.eq]
simp only [AddS... | 1,544 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 134 | 137 | theorem neg_coe_pi : -(Ο : Angle) = Ο := by |
rw [β coe_neg, angle_eq_iff_two_pi_dvd_sub]
use -1
simp [two_mul, sub_eq_add_neg]
| 1,544 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 141 | 142 | theorem two_nsmul_coe_div_two (ΞΈ : β) : (2 : β) β’ (β(ΞΈ / 2) : Angle) = ΞΈ := by |
rw [β coe_nsmul, two_nsmul, add_halves]
| 1,544 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 146 | 147 | theorem two_zsmul_coe_div_two (ΞΈ : β) : (2 : β€) β’ (β(ΞΈ / 2) : Angle) = ΞΈ := by |
rw [β coe_zsmul, two_zsmul, add_halves]
| 1,544 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 33 | 37 | theorem sin_arg (x : β) : Real.sin (arg x) = x.im / abs x := by |
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
| 1,545 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 40 | 50 | theorem cos_arg {x : β} (hx : x β 0) : Real.cos (arg x) = x.re / abs x := by |
rw [arg]
split_ifs with hβ hβ
Β· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
Β· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 hβ), *]
Β· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Re... | 1,545 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 54 | 58 | theorem abs_mul_exp_arg_mul_I (x : β) : β(abs x) * exp (arg x * I) = x := by |
rcases eq_or_ne x 0 with (rfl | hx)
Β· simp
Β· have : abs x β 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
| 1,545 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 63 | 64 | theorem abs_mul_cos_add_sin_mul_I (x : β) : (abs x * (cos (arg x) + sin (arg x) * I) : β) = x := by |
rw [β exp_mul_I, abs_mul_exp_arg_mul_I]
| 1,545 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 76 | 83 | theorem abs_eq_one_iff (z : β) : abs z = 1 β β ΞΈ : β, exp (ΞΈ * I) = z := by |
refine β¨fun hz => β¨arg z, ?_β©, ?_β©
Β· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
Β· rintro β¨ΞΈ, rflβ©
exact Complex.abs_exp_ofReal_mul_I ΞΈ
| 1,545 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 87 | 89 | theorem range_exp_mul_I : (Set.range fun x : β => exp (x * I)) = Metric.sphere 0 1 := by |
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
| 1,545 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 93 | 114 | theorem arg_mul_cos_add_sin_mul_I {r : β} (hr : 0 < r) {ΞΈ : β} (hΞΈ : ΞΈ β Set.Ioc (-Ο) Ο) :
arg (r * (cos ΞΈ + sin ΞΈ * I)) = ΞΈ := by |
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [re_ofReal_mul, im_ofReal_mul, neg_im, β ofReal_cos, β ofReal_sin, β
mk_eq_add_mul_I, neg_div, mul_div_cancel_leftβ _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases hβ : ΞΈ β Set.Icc (-(Ο / 2)) (Ο / 2)
Β· rw [if_... | 1,545 |
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Multiplicity
#align_import data.nat.choose.factorization from "leanprover-community/mathlib"@"dc9db541168768af03fe228703e758e649afdbfc"
namespace Nat
variable {p n k : β}
| Mathlib/Data/Nat/Choose/Factorization.lean | 36 | 45 | theorem factorization_choose_le_log : (choose n k).factorization p β€ log p n := by |
by_cases h : (choose n k).factorization p = 0
Β· simp [h]
have hp : p.Prime := Not.imp_symm (choose n k).factorization_eq_zero_of_non_prime h
have hkn : k β€ n := by
refine le_of_not_lt fun hnk => h ?_
simp [choose_eq_zero_of_lt hnk]
rw [factorization_def _ hp, @padicValNat_def _ β¨hpβ© _ (choose_pos hkn... | 1,546 |
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Multiplicity
#align_import data.nat.choose.factorization from "leanprover-community/mathlib"@"dc9db541168768af03fe228703e758e649afdbfc"
namespace Nat
variable {p n k : β}
theorem factorization_choose_le_l... | Mathlib/Data/Nat/Choose/Factorization.lean | 55 | 58 | theorem factorization_choose_le_one (p_large : n < p ^ 2) : (choose n k).factorization p β€ 1 := by |
apply factorization_choose_le_log.trans
rcases eq_or_ne n 0 with (rfl | hn0); Β· simp
exact Nat.lt_succ_iff.1 (log_lt_of_lt_pow hn0 p_large)
| 1,546 |
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Multiplicity
#align_import data.nat.choose.factorization from "leanprover-community/mathlib"@"dc9db541168768af03fe228703e758e649afdbfc"
namespace Nat
variable {p n k : β}
theorem factorization_choose_le_l... | Mathlib/Data/Nat/Choose/Factorization.lean | 61 | 88 | theorem factorization_choose_of_lt_three_mul (hp' : p β 2) (hk : p β€ k) (hk' : p β€ n - k)
(hn : n < 3 * p) : (choose n k).factorization p = 0 := by |
cases' em' p.Prime with hp hp
Β· exact factorization_eq_zero_of_non_prime (choose n k) hp
cases' lt_or_le n k with hnk hkn
Β· simp [choose_eq_zero_of_lt hnk]
rw [factorization_def _ hp, @padicValNat_def _ β¨hpβ© _ (choose_pos hkn)]
simp only [hp.multiplicity_choose hkn (lt_add_one _), PartENat.get_natCast, Fin... | 1,546 |
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Multiplicity
#align_import data.nat.choose.factorization from "leanprover-community/mathlib"@"dc9db541168768af03fe228703e758e649afdbfc"
namespace Nat
variable {p n k : β}
theorem factorization_choose_le_l... | Mathlib/Data/Nat/Choose/Factorization.lean | 93 | 97 | theorem factorization_centralBinom_of_two_mul_self_lt_three_mul (n_big : 2 < n) (p_le_n : p β€ n)
(big : 2 * n < 3 * p) : (centralBinom n).factorization p = 0 := by |
refine factorization_choose_of_lt_three_mul ?_ p_le_n (p_le_n.trans ?_) big
Β· omega
Β· rw [two_mul, add_tsub_cancel_left]
| 1,546 |
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Multiplicity
#align_import data.nat.choose.factorization from "leanprover-community/mathlib"@"dc9db541168768af03fe228703e758e649afdbfc"
namespace Nat
variable {p n k : β}
theorem factorization_choose_le_l... | Mathlib/Data/Nat/Choose/Factorization.lean | 100 | 103 | theorem factorization_factorial_eq_zero_of_lt (h : n < p) : (factorial n).factorization p = 0 := by |
induction' n with n hn; Β· simp
rw [factorial_succ, factorization_mul n.succ_ne_zero n.factorial_ne_zero, Finsupp.coe_add,
Pi.add_apply, hn (lt_of_succ_lt h), add_zero, factorization_eq_zero_of_lt h]
| 1,546 |
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Multiplicity
#align_import data.nat.choose.factorization from "leanprover-community/mathlib"@"dc9db541168768af03fe228703e758e649afdbfc"
namespace Nat
variable {p n k : β}
theorem factorization_choose_le_l... | Mathlib/Data/Nat/Choose/Factorization.lean | 106 | 110 | theorem factorization_choose_eq_zero_of_lt (h : n < p) : (choose n k).factorization p = 0 := by |
by_cases hnk : n < k; Β· simp [choose_eq_zero_of_lt hnk]
rw [choose_eq_factorial_div_factorial (le_of_not_lt hnk),
factorization_div (factorial_mul_factorial_dvd_factorial (le_of_not_lt hnk)), Finsupp.coe_tsub,
Pi.sub_apply, factorization_factorial_eq_zero_of_lt h, zero_tsub]
| 1,546 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : β)
| Mathlib/Data/Nat/Factorization/PrimePow.lean | 20 | 24 | theorem IsPrimePow.minFac_pow_factorization_eq {n : β} (hn : IsPrimePow n) :
n.minFac ^ n.factorization n.minFac = n := by |
obtain β¨p, k, hp, hk, rflβ© := hn
rw [β Nat.prime_iff] at hp
rw [hp.pow_minFac hk.ne', hp.factorization_pow, Finsupp.single_eq_same]
| 1,547 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : β)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 27 | 33 | theorem isPrimePow_of_minFac_pow_factorization_eq {n : β}
(h : n.minFac ^ n.factorization n.minFac = n) (hn : n β 1) : IsPrimePow n := by |
rcases eq_or_ne n 0 with (rfl | hn')
Β· simp_all
refine β¨_, _, (Nat.minFac_prime hn).prime, ?_, hβ©
simp [pos_iff_ne_zero, β Finsupp.mem_support_iff, Nat.support_factorization, hn',
Nat.minFac_prime hn, Nat.minFac_dvd]
| 1,547 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : β)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 41 | 54 | theorem isPrimePow_iff_factorization_eq_single {n : β} :
IsPrimePow n β β p k : β, 0 < k β§ n.factorization = Finsupp.single p k := by |
rw [isPrimePow_nat_iff]
refine existsβ_congr fun p k => ?_
constructor
Β· rintro β¨hp, hk, hnβ©
exact β¨hk, by rw [β hn, Nat.Prime.factorization_pow hp]β©
Β· rintro β¨hk, hnβ©
have hn0 : n β 0 := by
rintro rfl
simp_all only [Finsupp.single_eq_zero, eq_comm, Nat.factorization_zero, hk.ne']
rw ... | 1,547 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : β)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 57 | 60 | theorem isPrimePow_iff_card_primeFactors_eq_one {n : β} :
IsPrimePow n β n.primeFactors.card = 1 := by |
simp_rw [isPrimePow_iff_factorization_eq_single, β Nat.support_factorization,
Finsupp.card_support_eq_one', pos_iff_ne_zero]
| 1,547 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : β)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 63 | 73 | theorem IsPrimePow.exists_ord_compl_eq_one {n : β} (h : IsPrimePow n) :
β p : β, p.Prime β§ ord_compl[p] n = 1 := by |
rcases eq_or_ne n 0 with (rfl | hn0); Β· cases not_isPrimePow_zero h
rcases isPrimePow_iff_factorization_eq_single.mp h with β¨p, k, hk0, h1β©
rcases em' p.Prime with (pp | pp)
Β· refine absurd ?_ hk0.ne'
simp [β Nat.factorization_eq_zero_of_non_prime n pp, h1]
refine β¨p, pp, ?_β©
refine Nat.eq_of_factoriza... | 1,547 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : β)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 76 | 84 | theorem exists_ord_compl_eq_one_iff_isPrimePow {n : β} (hn : n β 1) :
IsPrimePow n β β p : β, p.Prime β§ ord_compl[p] n = 1 := by |
refine β¨fun h => IsPrimePow.exists_ord_compl_eq_one h, fun h => ?_β©
rcases h with β¨p, pp, hβ©
rw [isPrimePow_nat_iff]
rw [β Nat.eq_of_dvd_of_div_eq_one (Nat.ord_proj_dvd n p) h] at hn β’
refine β¨p, n.factorization p, pp, ?_, by simpβ©
contrapose! hn
simp [Nat.le_zero.1 hn]
| 1,547 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : β)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 89 | 108 | theorem isPrimePow_iff_unique_prime_dvd {n : β} : IsPrimePow n β β! p : β, p.Prime β§ p β£ n := by |
rw [isPrimePow_nat_iff]
constructor
Β· rintro β¨p, k, hp, hk, rflβ©
refine β¨p, β¨hp, dvd_pow_self _ hk.ne'β©, ?_β©
rintro q β¨hq, hq'β©
exact (Nat.prime_dvd_prime_iff_eq hq hp).1 (hq.dvd_of_dvd_pow hq')
rintro β¨p, β¨hp, hnβ©, hqβ©
rcases eq_or_ne n 0 with (rfl | hnβ)
Β· cases (hq 2 β¨Nat.prime_two, dvd_zero... | 1,547 |
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Data.Nat.Factorization.PrimePow
#align_import data.nat.squarefree from "leanprover-community/mathlib"@"3c1368cac4abd5a5cbe44317ba7e87379d51ed88"
open Finset
namespace Nat
| Mathlib/Data/Nat/Squarefree.lean | 28 | 30 | theorem squarefree_iff_nodup_factors {n : β} (h0 : n β 0) : Squarefree n β n.factors.Nodup := by |
rw [UniqueFactorizationMonoid.squarefree_iff_nodup_normalizedFactors h0, Nat.factors_eq]
simp
| 1,548 |
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section Fermat
open GaussianInt
| Mathlib/NumberTheory/SumTwoSquares.lean | 33 | 36 | theorem Nat.Prime.sq_add_sq {p : β} [Fact p.Prime] (hp : p % 4 β 3) :
β a b : β, a ^ 2 + b ^ 2 = p := by |
apply sq_add_sq_of_nat_prime_of_not_irreducible p
rwa [_root_.irreducible_iff_prime, prime_iff_mod_four_eq_three_of_nat_prime p]
| 1,549 |
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section General
theorem sq_add_sq_mul {R} [CommRing R] ... | Mathlib/NumberTheory/SumTwoSquares.lean | 56 | 61 | theorem Nat.sq_add_sq_mul {a b x y u v : β} (ha : a = x ^ 2 + y ^ 2) (hb : b = u ^ 2 + v ^ 2) :
β r s : β, a * b = r ^ 2 + s ^ 2 := by |
zify at ha hb β’
obtain β¨r, s, hβ© := _root_.sq_add_sq_mul ha hb
refine β¨r.natAbs, s.natAbs, ?_β©
simpa only [Int.natCast_natAbs, sq_abs]
| 1,549 |
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section NegOneSquare
-- This could be formulated for ... | Mathlib/NumberTheory/SumTwoSquares.lean | 77 | 81 | theorem ZMod.isSquare_neg_one_of_dvd {m n : β} (hd : m β£ n) (hs : IsSquare (-1 : ZMod n)) :
IsSquare (-1 : ZMod m) := by |
let f : ZMod n β+* ZMod m := ZMod.castHom hd _
rw [β RingHom.map_one f, β RingHom.map_neg]
exact hs.map f
| 1,549 |
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section NegOneSquare
-- This could be formulated for ... | Mathlib/NumberTheory/SumTwoSquares.lean | 86 | 94 | theorem ZMod.isSquare_neg_one_mul {m n : β} (hc : m.Coprime n) (hm : IsSquare (-1 : ZMod m))
(hn : IsSquare (-1 : ZMod n)) : IsSquare (-1 : ZMod (m * n)) := by |
have : IsSquare (-1 : ZMod m Γ ZMod n) := by
rw [show (-1 : ZMod m Γ ZMod n) = ((-1 : ZMod m), (-1 : ZMod n)) from rfl]
obtain β¨x, hxβ© := hm
obtain β¨y, hyβ© := hn
rw [hx, hy]
exact β¨(x, y), rflβ©
simpa only [RingEquiv.map_neg_one] using this.map (ZMod.chineseRemainder hc).symm
| 1,549 |
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section NegOneSquare
-- This could be formulated for ... | Mathlib/NumberTheory/SumTwoSquares.lean | 98 | 103 | theorem Nat.Prime.mod_four_ne_three_of_dvd_isSquare_neg_one {p n : β} (hpp : p.Prime) (hp : p β£ n)
(hs : IsSquare (-1 : ZMod n)) : p % 4 β 3 := by |
obtain β¨y, hβ© := ZMod.isSquare_neg_one_of_dvd hp hs
rw [β sq, eq_comm, show (-1 : ZMod p) = -1 ^ 2 by ring] at h
haveI : Fact p.Prime := β¨hppβ©
exact ZMod.mod_four_ne_three_of_sq_eq_neg_sq' one_ne_zero h
| 1,549 |
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section NegOneSquare
-- This could be formulated for ... | Mathlib/NumberTheory/SumTwoSquares.lean | 108 | 120 | theorem ZMod.isSquare_neg_one_iff {n : β} (hn : Squarefree n) :
IsSquare (-1 : ZMod n) β β {q : β}, q.Prime β q β£ n β q % 4 β 3 := by |
refine β¨fun H q hqp hqd => hqp.mod_four_ne_three_of_dvd_isSquare_neg_one hqd H, fun H => ?_β©
induction' n using induction_on_primes with p n hpp ih
Β· exact False.elim (hn.ne_zero rfl)
Β· exact β¨0, by simp only [mul_zero, eq_iff_true_of_subsingleton]β©
Β· haveI : Fact p.Prime := β¨hppβ©
have hcp : p.Coprime n ... | 1,549 |
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section NegOneSquare
-- This could be formulated for ... | Mathlib/NumberTheory/SumTwoSquares.lean | 125 | 138 | theorem ZMod.isSquare_neg_one_iff' {n : β} (hn : Squarefree n) :
IsSquare (-1 : ZMod n) β β {q : β}, q β£ n β q % 4 β 3 := by |
have help : β a b : ZMod 4, a β 3 β b β 3 β a * b β 3 := by decide
rw [ZMod.isSquare_neg_one_iff hn]
refine β¨?_, fun H q _ => Hβ©
intro H
refine @induction_on_primes _ ?_ ?_ (fun p q hp hq hpq => ?_)
Β· exact fun _ => by norm_num
Β· exact fun _ => by norm_num
Β· replace hp := H hp (dvd_of_mul_right_dvd hpq... | 1,549 |
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.Peel
#align_import group_... | Mathlib/GroupTheory/Exponent.lean | 94 | 97 | theorem _root_.MulOpposite.exponent : exponent (MulOpposite G) = exponent G := by |
simp only [Monoid.exponent, ExponentExists]
congr!
all_goals exact β¨(op_injective <| Β· <| op Β·), (unop_injective <| Β· <| unop Β·)β©
| 1,550 |
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.Peel
#align_import group_... | Mathlib/GroupTheory/Exponent.lean | 108 | 113 | theorem exponent_ne_zero : exponent G β 0 β ExponentExists G := by |
rw [exponent]
split_ifs with h
Β· simp [h, @not_lt_zero' β]
--if this isn't done this way, `to_additive` freaks
Β· tauto
| 1,550 |
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.Peel
#align_import group_... | Mathlib/GroupTheory/Exponent.lean | 145 | 148 | theorem exponent_eq_zero_iff_forall : exponent G = 0 β β n > 0, β g : G, g ^ n β 1 := by |
rw [exponent_eq_zero_iff, ExponentExists]
push_neg
rfl
| 1,550 |
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.Peel
#align_import group_... | Mathlib/GroupTheory/Exponent.lean | 151 | 155 | theorem pow_exponent_eq_one (g : G) : g ^ exponent G = 1 := by |
by_cases h : ExponentExists G
Β· simp_rw [exponent, dif_pos h]
exact (Nat.find_spec h).2 g
Β· simp_rw [exponent, dif_neg h, pow_zero]
| 1,550 |
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.Peel
#align_import group_... | Mathlib/GroupTheory/Exponent.lean | 160 | 163 | theorem pow_eq_mod_exponent {n : β} (g : G) : g ^ n = g ^ (n % exponent G) :=
calc
g ^ n = g ^ (n % exponent G + exponent G * (n / exponent G)) := by | rw [Nat.mod_add_div]
_ = g ^ (n % exponent G) := by simp [pow_add, pow_mul, pow_exponent_eq_one]
| 1,550 |
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.Peel
#align_import group_... | Mathlib/GroupTheory/Exponent.lean | 176 | 180 | theorem exponent_min' (n : β) (hpos : 0 < n) (hG : β g : G, g ^ n = 1) : exponent G β€ n := by |
rw [exponent, dif_pos]
Β· apply Nat.find_min'
exact β¨hpos, hGβ©
Β· exact β¨n, hpos, hGβ©
| 1,550 |
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.Basic
namespace MulAction
universe u v
variable {Ξ± : Type v}
variable {G : Type u} [Group G] [MulAction G Ξ±]
variable {M : Type u} [Monoid M] [MulAction M Ξ±]
@[to_additive "If the action is periodic, t... | Mathlib/GroupTheory/GroupAction/Period.lean | 71 | 75 | theorem period_inv (g : G) (a : Ξ±) : period gβ»ΒΉ a = period g a := by |
simp only [period_eq_minimalPeriod, Function.minimalPeriod_eq_minimalPeriod_iff,
isPeriodicPt_smul_iff]
intro n
rw [smul_eq_iff_eq_inv_smul, eq_comm, β zpow_natCast, inv_zpow, inv_inv, zpow_natCast]
| 1,551 |
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.Basic
namespace MulAction
universe u v
variable {Ξ± : Type v}
variable {G : Type u} [Group G] [MulAction G Ξ±]
variable {M : Type u} [Monoid M] [MulAction M Ξ±]
@[to_additive "If the action is periodic, t... | Mathlib/GroupTheory/GroupAction/Period.lean | 87 | 88 | theorem period_dvd_orderOf (m : M) (a : Ξ±) : period m a β£ orderOf m := by |
rw [β pow_smul_eq_iff_period_dvd, pow_orderOf_eq_one, one_smul]
| 1,551 |
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.Basic
namespace MulAction
universe u v
variable {Ξ± : Type v}
variable {G : Type u} [Group G] [MulAction G Ξ±]
variable {M : Type u} [Monoid M] [MulAction M Ξ±]
@[to_additive "If the action is periodic, t... | Mathlib/GroupTheory/GroupAction/Period.lean | 101 | 102 | theorem period_dvd_exponent (m : M) (a : Ξ±) : period m a β£ Monoid.exponent M := by |
rw [β pow_smul_eq_iff_period_dvd, Monoid.pow_exponent_eq_one, one_smul]
| 1,551 |
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.Basic
namespace MulAction
universe u v
variable {Ξ± : Type v}
variable {G : Type u} [Group G] [MulAction G Ξ±]
variable {M : Type u} [Monoid M] [MulAction M Ξ±]
@[to_additive "If the action is periodic, t... | Mathlib/GroupTheory/GroupAction/Period.lean | 117 | 120 | theorem period_bounded_of_exponent_pos (exp_pos : 0 < Monoid.exponent M) (m : M) :
BddAbove (Set.range (fun a : Ξ± => period m a)) := by |
use Monoid.exponent M
simpa [upperBounds] using period_le_exponent exp_pos _
| 1,551 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
inductive DihedralGroup (n : β) : Type
| r : ZMod n β DihedralGroup n
| sr : ZMod n β DihedralGroup n
derivin... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 125 | 126 | theorem card [NeZero n] : Fintype.card (DihedralGroup n) = 2 * n := by |
rw [β Fintype.card_eq.mpr β¨fintypeHelperβ©, Fintype.card_sum, ZMod.card, two_mul]
| 1,552 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
inductive DihedralGroup (n : β) : Type
| r : ZMod n β DihedralGroup n
| sr : ZMod n β DihedralGroup n
derivin... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 129 | 132 | theorem nat_card : Nat.card (DihedralGroup n) = 2 * n := by |
cases n
Β· rw [Nat.card_eq_zero_of_infinite]
Β· rw [Nat.card_eq_fintype_card, card]
| 1,552 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
inductive DihedralGroup (n : β) : Type
| r : ZMod n β DihedralGroup n
| sr : ZMod n β DihedralGroup n
derivin... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 135 | 142 | theorem r_one_pow (k : β) : (r 1 : DihedralGroup n) ^ k = r k := by |
induction' k with k IH
Β· rw [Nat.cast_zero]
rfl
Β· rw [pow_succ', IH, r_mul_r]
congr 1
norm_cast
rw [Nat.one_add]
| 1,552 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
inductive DihedralGroup (n : β) : Type
| r : ZMod n β DihedralGroup n
| sr : ZMod n β DihedralGroup n
derivin... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 146 | 149 | theorem r_one_pow_n : r (1 : ZMod n) ^ n = 1 := by |
rw [r_one_pow, one_def]
congr 1
exact ZMod.natCast_self _
| 1,552 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
inductive DihedralGroup (n : β) : Type
| r : ZMod n β DihedralGroup n
| sr : ZMod n β DihedralGroup n
derivin... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 153 | 153 | theorem sr_mul_self (i : ZMod n) : sr i * sr i = 1 := by | rw [sr_mul_sr, sub_self, one_def]
| 1,552 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
inductive DihedralGroup (n : β) : Type
| r : ZMod n β DihedralGroup n
| sr : ZMod n β DihedralGroup n
derivin... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 159 | 164 | theorem orderOf_sr (i : ZMod n) : orderOf (sr i) = 2 := by |
apply orderOf_eq_prime
Β· rw [sq, sr_mul_self]
Β· -- Porting note: Previous proof was `decide`
revert n
simp_rw [one_def, ne_eq, forall_const, not_false_eq_true]
| 1,552 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
inductive DihedralGroup (n : β) : Type
| r : ZMod n β DihedralGroup n
| sr : ZMod n β DihedralGroup n
derivin... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 170 | 184 | theorem orderOf_r_one : orderOf (r 1 : DihedralGroup n) = n := by |
rcases eq_zero_or_neZero n with (rfl | hn)
Β· rw [orderOf_eq_zero_iff']
intro n hn
rw [r_one_pow, one_def]
apply mt r.inj
simpa using hn.ne'
Β· apply (Nat.le_of_dvd (NeZero.pos n) <|
orderOf_dvd_of_pow_eq_one <| @r_one_pow_n n).lt_or_eq.resolve_left
intro h
have h1 : (r 1 : DihedralGr... | 1,552 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
inductive DihedralGroup (n : β) : Type
| r : ZMod n β DihedralGroup n
| sr : ZMod n β DihedralGroup n
derivin... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 189 | 191 | theorem orderOf_r [NeZero n] (i : ZMod n) : orderOf (r i) = n / Nat.gcd n i.val := by |
conv_lhs => rw [β ZMod.natCast_zmod_val i]
rw [β r_one_pow, orderOf_pow, orderOf_r_one]
| 1,552 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Line... | Mathlib/GroupTheory/CommutingProbability.lean | 47 | 52 | theorem commProb_prod (M' : Type*) [Mul M'] : commProb (M Γ M') = commProb M * commProb M' := by |
simp_rw [commProb_def, div_mul_div_comm, Nat.card_prod, Nat.cast_mul, mul_pow, β Nat.cast_mul,
β Nat.card_prod, Commute, SemiconjBy, Prod.ext_iff]
congr 2
exact Nat.card_congr β¨fun x => β¨β¨β¨x.1.1.1, x.1.2.1β©, x.2.1β©, β¨β¨x.1.1.2, x.1.2.2β©, x.2.2β©β©,
fun x => β¨β¨β¨x.1.1.1, x.2.1.1β©, β¨x.1.1.2, x.2.1.2β©β©, β¨x.1.2,... | 1,553 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Line... | Mathlib/GroupTheory/CommutingProbability.lean | 54 | 60 | theorem commProb_pi {Ξ± : Type*} (i : Ξ± β Type*) [Fintype Ξ±] [β a, Mul (i a)] :
commProb (β a, i a) = β a, commProb (i a) := by |
simp_rw [commProb_def, Finset.prod_div_distrib, Finset.prod_pow, β Nat.cast_prod,
β Nat.card_pi, Commute, SemiconjBy, Function.funext_iff]
congr 2
exact Nat.card_congr β¨fun x a => β¨β¨x.1.1 a, x.1.2 aβ©, x.2 aβ©, fun x => β¨β¨fun a => (x a).1.1,
fun a => (x a).1.2β©, fun a => (x a).2β©, fun x => rfl, fun x => rf... | 1,553 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Line... | Mathlib/GroupTheory/CommutingProbability.lean | 62 | 64 | theorem commProb_function {Ξ± Ξ² : Type*} [Fintype Ξ±] [Mul Ξ²] :
commProb (Ξ± β Ξ²) = (commProb Ξ²) ^ Fintype.card Ξ± := by |
rw [commProb_pi, Finset.prod_const, Finset.card_univ]
| 1,553 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Line... | Mathlib/GroupTheory/CommutingProbability.lean | 78 | 81 | theorem commProb_le_one : commProb M β€ 1 := by |
refine div_le_one_of_le ?_ (sq_nonneg (Nat.card M : β))
rw [β Nat.cast_pow, Nat.cast_le, sq, β Nat.card_prod]
apply Finite.card_subtype_le
| 1,553 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Line... | Mathlib/GroupTheory/CommutingProbability.lean | 86 | 93 | theorem commProb_eq_one_iff [h : Nonempty M] :
commProb M = 1 β Commutative ((Β· * Β·) : M β M β M) := by |
haveI := Fintype.ofFinite M
rw [commProb, β Set.coe_setOf, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card]
rw [div_eq_one_iff_eq, β Nat.cast_pow, Nat.cast_inj, sq, β card_prod,
set_fintype_card_eq_univ_iff, Set.eq_univ_iff_forall]
Β· exact β¨fun h x y β¦ h (x, y), fun h x β¦ h x.1 x.2β©
Β· exact pow_ne_zer... | 1,553 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Line... | Mathlib/GroupTheory/CommutingProbability.lean | 98 | 102 | theorem commProb_def' : commProb G = Nat.card (ConjClasses G) / Nat.card G := by |
rw [commProb, card_comm_eq_card_conjClasses_mul_card, Nat.cast_mul, sq]
by_cases h : (Nat.card G : β) = 0
Β· rw [h, zero_mul, div_zero, div_zero]
Β· exact mul_div_mul_right _ _ h
| 1,553 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Line... | Mathlib/GroupTheory/CommutingProbability.lean | 108 | 116 | theorem Subgroup.commProb_subgroup_le : commProb H β€ commProb G * (H.index : β) ^ 2 := by |
/- After rewriting with `commProb_def`, we reduce to showing that `G` has at least as many
commuting pairs as `H`. -/
rw [commProb_def, commProb_def, div_le_iff, mul_assoc, β mul_pow, β Nat.cast_mul,
mul_comm H.index, H.card_mul_index, div_mul_cancelβ, Nat.cast_le]
Β· refine Finite.card_le_of_injective ... | 1,553 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Line... | Mathlib/GroupTheory/CommutingProbability.lean | 119 | 128 | theorem Subgroup.commProb_quotient_le [H.Normal] : commProb (G β§Έ H) β€ commProb G * Nat.card H := by |
/- After rewriting with `commProb_def'`, we reduce to showing that `G` has at least as many
conjugacy classes as `G β§Έ H`. -/
rw [commProb_def', commProb_def', div_le_iff, mul_assoc, β Nat.cast_mul, β Subgroup.index,
H.card_mul_index, div_mul_cancelβ, Nat.cast_le]
Β· apply Finite.card_le_of_surjective
... | 1,553 |
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
open scoped Pointwise
namespace Subgroup
open MemRightTransversals
variable {G : T... | Mathlib/GroupTheory/Schreier.lean | 37 | 58 | theorem closure_mul_image_mul_eq_top
(hR : R β rightTransversals (H : Set G)) (hR1 : (1 : G) β R) (hS : closure S = β€) :
(closure ((R * S).image fun g => g * (toFun hR g : G)β»ΒΉ)) * R = β€ := by |
let f : G β R := fun g => toFun hR g
let U : Set G := (R * S).image fun g => g * (f g : G)β»ΒΉ
change (closure U : Set G) * R = β€
refine top_le_iff.mp fun g _ => ?_
refine closure_induction_right ?_ ?_ ?_ (eq_top_iff.mp hS (mem_top g))
Β· exact β¨1, (closure U).one_mem, 1, hR1, one_mul 1β©
Β· rintro - - s hs β¨... | 1,554 |
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
open scoped Pointwise
namespace Subgroup
open MemRightTransversals
variable {G : T... | Mathlib/GroupTheory/Schreier.lean | 64 | 79 | theorem closure_mul_image_eq (hR : R β rightTransversals (H : Set G)) (hR1 : (1 : G) β R)
(hS : closure S = β€) : closure ((R * S).image fun g => g * (toFun hR g : G)β»ΒΉ) = H := by |
have hU : closure ((R * S).image fun g => g * (toFun hR g : G)β»ΒΉ) β€ H := by
rw [closure_le]
rintro - β¨g, -, rflβ©
exact mul_inv_toFun_mem hR g
refine le_antisymm hU fun h hh => ?_
obtain β¨g, hg, r, hr, rflβ© :=
show h β _ from eq_top_iff.mp (closure_mul_image_mul_eq_top hR hR1 hS) (mem_top h)
suf... | 1,554 |
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
open scoped Pointwise
namespace Subgroup
open MemRightTransversals
variable {G : T... | Mathlib/GroupTheory/Schreier.lean | 85 | 89 | theorem closure_mul_image_eq_top (hR : R β rightTransversals (H : Set G)) (hR1 : (1 : G) β R)
(hS : closure S = β€) : closure ((R * S).image fun g =>
β¨g * (toFun hR g : G)β»ΒΉ, mul_inv_toFun_mem hR gβ© : Set H) = β€ := by |
rw [eq_top_iff, β map_subtype_le_map_subtype, MonoidHom.map_closure, Set.image_image]
exact (map_subtype_le β€).trans (ge_of_eq (closure_mul_image_eq hR hR1 hS))
| 1,554 |
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
open scoped Pointwise
namespace Subgroup
open MemRightTransversals
variable {G : T... | Mathlib/GroupTheory/Schreier.lean | 95 | 100 | theorem closure_mul_image_eq_top' [DecidableEq G] {R S : Finset G}
(hR : (R : Set G) β rightTransversals (H : Set G)) (hR1 : (1 : G) β R)
(hS : closure (S : Set G) = β€) :
closure (((R * S).image fun g => β¨_, mul_inv_toFun_mem hR gβ© : Finset H) : Set H) = β€ := by |
rw [Finset.coe_image, Finset.coe_mul]
exact closure_mul_image_eq_top hR hR1 hS
| 1,554 |
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
open scoped Pointwise
namespace Subgroup
open MemRightTransversals
variable {G : T... | Mathlib/GroupTheory/Schreier.lean | 105 | 124 | theorem exists_finset_card_le_mul [FiniteIndex H] {S : Finset G} (hS : closure (S : Set G) = β€) :
β T : Finset H, T.card β€ H.index * S.card β§ closure (T : Set H) = β€ := by |
letI := H.fintypeQuotientOfFiniteIndex
haveI : DecidableEq G := Classical.decEq G
obtain β¨Rβ, hR, hR1β© := H.exists_right_transversal 1
haveI : Fintype Rβ := Fintype.ofEquiv _ (toEquiv hR)
let R : Finset G := Set.toFinset Rβ
replace hR : (R : Set G) β rightTransversals (H : Set G) := by rwa [Set.coe_toFinse... | 1,554 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.