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.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 98 | 102 | theorem edgeFinset_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeFinset =
(G.edgeFinset \ G.incidenceFinset t ∪ (G.neighborFinset s).image (s(·, t))) \ {s(t, t)} := by |
simp only [incidenceFinset, neighborFinset, ← Set.toFinset_diff, ← Set.toFinset_image,
← Set.toFinset_union, ← Set.toFinset_singleton]
exact Set.toFinset_congr (G.edgeSet_replaceVertex_of_adj ha)
| 1,083 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 115 | 124 | theorem card_edgeFinset_replaceVertex_of_not_adj (hn : ¬G.Adj s t) :
(G.replaceVertex s t).edgeFinset.card = G.edgeFinset.card + G.degree s - G.degree t := by |
have inc : G.incidenceFinset t ⊆ G.edgeFinset := by simp [incidenceFinset, incidenceSet_subset]
rw [G.edgeFinset_replaceVertex_of_not_adj hn,
card_union_of_disjoint G.disjoint_sdiff_neighborFinset_image, card_sdiff inc,
← Nat.sub_add_comm <| card_le_card inc, card_incidenceFinset_eq_degree]
congr 2
rw ... | 1,083 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 126 | 135 | theorem card_edgeFinset_replaceVertex_of_adj (ha : G.Adj s t) :
(G.replaceVertex s t).edgeFinset.card = G.edgeFinset.card + G.degree s - G.degree t - 1 := by |
have inc : G.incidenceFinset t ⊆ G.edgeFinset := by simp [incidenceFinset, incidenceSet_subset]
rw [G.edgeFinset_replaceVertex_of_adj ha, card_sdiff (by simp [ha]),
card_union_of_disjoint G.disjoint_sdiff_neighborFinset_image, card_sdiff inc,
← Nat.sub_add_comm <| card_le_card inc, card_incidenceFinset_eq_... | 1,083 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section AddEdge
def edge : SimpleGraph V := fromEdgeSet {s(s, t)}
lemma edge_adj (v w : V) : (edge s t).Adj v w ↔ ... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 171 | 175 | theorem edgeFinset_sup_edge [Fintype (edgeSet (G ⊔ edge s t))] (hn : ¬G.Adj s t) (h : s ≠ t) :
(G ⊔ edge s t).edgeFinset = G.edgeFinset.cons s(s, t) (by simp_all) := by |
letI := Classical.decEq V
rw [edgeFinset_sup, cons_eq_insert, insert_eq, union_comm]
simp_rw [edgeFinset, edge_edgeSet_of_ne h]; rfl
| 1,083 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section AddEdge
def edge : SimpleGraph V := fromEdgeSet {s(s, t)}
lemma edge_adj (v w : V) : (edge s t).Adj v w ↔ ... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 177 | 179 | theorem card_edgeFinset_sup_edge [Fintype (edgeSet (G ⊔ edge s t))] (hn : ¬G.Adj s t) (h : s ≠ t) :
(G ⊔ edge s t).edgeFinset.card = G.edgeFinset.card + 1 := by |
rw [G.edgeFinset_sup_edge hn h, card_cons]
| 1,083 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
#align_import combinatorics.simple_graph.clique from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Finset Fintype Function SimpleGraph.Walk... | Mathlib/Combinatorics/SimpleGraph/Clique.lean | 55 | 66 | theorem isClique_iff_induce_eq : G.IsClique s ↔ G.induce s = ⊤ := by |
rw [isClique_iff]
constructor
· intro h
ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [comap_adj, Subtype.coe_mk, top_adj, Ne, Subtype.mk_eq_mk]
exact ⟨Adj.ne, h hv hw⟩
· intro h v hv w hw hne
have h2 : (G.induce s).Adj ⟨v, hv⟩ ⟨w, hw⟩ = _ := rfl
conv_lhs at h2 => rw [h]
simp only [top_adj, ne_eq, Subt... | 1,084 |
import Mathlib.Combinatorics.SimpleGraph.Clique
open Finset
namespace SimpleGraph
variable {V : Type*} [Fintype V] [DecidableEq V] (G H : SimpleGraph V) [DecidableRel G.Adj]
{n r : ℕ}
def IsTuranMaximal (r : ℕ) : Prop :=
G.CliqueFree (r + 1) ∧ ∀ (H : SimpleGraph V) [DecidableRel H.Adj],
H.CliqueFree (r +... | Mathlib/Combinatorics/SimpleGraph/Turan.lean | 54 | 62 | theorem turanGraph_eq_top : turanGraph n r = ⊤ ↔ r = 0 ∨ n ≤ r := by |
simp_rw [SimpleGraph.ext_iff, Function.funext_iff, turanGraph, top_adj, eq_iff_iff, not_iff_not]
refine ⟨fun h ↦ ?_, ?_⟩
· contrapose! h
use ⟨0, (Nat.pos_of_ne_zero h.1).trans h.2⟩, ⟨r, h.2⟩
simp [h.1.symm]
· rintro (rfl | h) a b
· simp [Fin.val_inj]
· rw [Nat.mod_eq_of_lt (a.2.trans_le h), Nat... | 1,085 |
import Mathlib.Combinatorics.SimpleGraph.Clique
open Finset
namespace SimpleGraph
variable {V : Type*} [Fintype V] [DecidableEq V] (G H : SimpleGraph V) [DecidableRel G.Adj]
{n r : ℕ}
def IsTuranMaximal (r : ℕ) : Prop :=
G.CliqueFree (r + 1) ∧ ∀ (H : SimpleGraph V) [DecidableRel H.Adj],
H.CliqueFree (r +... | Mathlib/Combinatorics/SimpleGraph/Turan.lean | 66 | 75 | theorem turanGraph_cliqueFree : (turanGraph n r).CliqueFree (r + 1) := by |
rw [cliqueFree_iff]
by_contra h
rw [not_isEmpty_iff] at h
obtain ⟨f, ha⟩ := h
simp only [turanGraph, top_adj] at ha
obtain ⟨x, y, d, c⟩ := Fintype.exists_ne_map_eq_of_card_lt (fun x ↦
(⟨(f x).1 % r, Nat.mod_lt _ hr⟩ : Fin r)) (by simp)
simp only [Fin.mk.injEq] at c
exact absurd c ((@ha x y).mpr d)
| 1,085 |
import Mathlib.Combinatorics.SimpleGraph.Clique
open Finset
namespace SimpleGraph
variable {V : Type*} [Fintype V] [DecidableEq V] (G H : SimpleGraph V) [DecidableRel G.Adj]
{n r : ℕ}
def IsTuranMaximal (r : ℕ) : Prop :=
G.CliqueFree (r + 1) ∧ ∀ (H : SimpleGraph V) [DecidableRel H.Adj],
H.CliqueFree (r +... | Mathlib/Combinatorics/SimpleGraph/Turan.lean | 84 | 92 | theorem not_cliqueFree_of_isTuranMaximal (hn : r ≤ Fintype.card V) (hG : G.IsTuranMaximal r) :
¬G.CliqueFree r := by |
rintro h
obtain ⟨K, _, rfl⟩ := exists_smaller_set (univ : Finset V) r hn
obtain ⟨a, -, b, -, hab, hGab⟩ : ∃ a ∈ K, ∃ b ∈ K, a ≠ b ∧ ¬ G.Adj a b := by
simpa only [isNClique_iff, IsClique, Set.Pairwise, mem_coe, ne_eq, and_true, not_forall,
exists_prop, exists_and_right] using h K
exact hGab <| le_sup_... | 1,085 |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 84 | 95 | theorem IsSRGWith.card_neighborFinset_union_eq {v w : V} (h : G.IsSRGWith n k ℓ μ) :
(G.neighborFinset v ∪ G.neighborFinset w).card =
2 * k - Fintype.card (G.commonNeighbors v w) := by |
apply Nat.add_right_cancel (m := Fintype.card (G.commonNeighbors v w))
rw [Nat.sub_add_cancel, ← Set.toFinset_card]
-- Porting note: Set.toFinset_inter needs workaround to use unification to solve for one of the
-- instance arguments:
· simp [commonNeighbors, @Set.toFinset_inter _ _ _ _ _ _ (_),
← neig... | 1,086 |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 102 | 106 | theorem IsSRGWith.card_neighborFinset_union_of_not_adj {v w : V} (h : G.IsSRGWith n k ℓ μ)
(hne : v ≠ w) (ha : ¬G.Adj v w) :
(G.neighborFinset v ∪ G.neighborFinset w).card = 2 * k - μ := by |
rw [← h.of_not_adj hne ha]
apply h.card_neighborFinset_union_eq
| 1,086 |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 110 | 113 | theorem IsSRGWith.card_neighborFinset_union_of_adj {v w : V} (h : G.IsSRGWith n k ℓ μ)
(ha : G.Adj v w) : (G.neighborFinset v ∪ G.neighborFinset w).card = 2 * k - ℓ := by |
rw [← h.of_adj v w ha]
apply h.card_neighborFinset_union_eq
| 1,086 |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 117 | 122 | theorem compl_neighborFinset_sdiff_inter_eq {v w : V} :
(G.neighborFinset v)ᶜ \ {v} ∩ ((G.neighborFinset w)ᶜ \ {w}) =
((G.neighborFinset v)ᶜ ∩ (G.neighborFinset w)ᶜ) \ ({w} ∪ {v}) := by |
ext
rw [← not_iff_not]
simp [imp_iff_not_or, or_assoc, or_comm, or_left_comm]
| 1,086 |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 125 | 134 | theorem sdiff_compl_neighborFinset_inter_eq {v w : V} (h : G.Adj v w) :
((G.neighborFinset v)ᶜ ∩ (G.neighborFinset w)ᶜ) \ ({w} ∪ {v}) =
(G.neighborFinset v)ᶜ ∩ (G.neighborFinset w)ᶜ := by |
ext
simp only [and_imp, mem_union, mem_sdiff, mem_compl, and_iff_left_iff_imp, mem_neighborFinset,
mem_inter, mem_singleton]
rintro hnv hnw (rfl | rfl)
· exact hnv h
· apply hnw
rwa [adj_comm]
| 1,086 |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 137 | 140 | theorem IsSRGWith.compl_is_regular (h : G.IsSRGWith n k ℓ μ) :
Gᶜ.IsRegularOfDegree (n - k - 1) := by |
rw [← h.card, Nat.sub_sub, add_comm, ← Nat.sub_sub]
exact h.regular.compl
| 1,086 |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 144 | 156 | theorem IsSRGWith.card_commonNeighbors_eq_of_adj_compl (h : G.IsSRGWith n k ℓ μ) {v w : V}
(ha : Gᶜ.Adj v w) : Fintype.card (Gᶜ.commonNeighbors v w) = n - (2 * k - μ) - 2 := by |
simp only [← Set.toFinset_card, commonNeighbors, Set.toFinset_inter, neighborSet_compl,
Set.toFinset_diff, Set.toFinset_singleton, Set.toFinset_compl, ← neighborFinset_def]
simp_rw [compl_neighborFinset_sdiff_inter_eq]
have hne : v ≠ w := ne_of_adj _ ha
rw [compl_adj] at ha
rw [card_sdiff, ← insert_eq, c... | 1,086 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.nat.prime from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82... | Mathlib/Data/Nat/Prime.lean | 89 | 96 | theorem Prime.eq_one_or_self_of_dvd {p : ℕ} (pp : p.Prime) (m : ℕ) (hm : m ∣ p) :
m = 1 ∨ m = p := by |
obtain ⟨n, hn⟩ := hm
have := pp.isUnit_or_isUnit hn
rw [Nat.isUnit_iff, Nat.isUnit_iff] at this
apply Or.imp_right _ this
rintro rfl
rw [hn, mul_one]
| 1,087 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.nat.prime from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82... | Mathlib/Data/Nat/Prime.lean | 99 | 109 | theorem prime_def_lt'' {p : ℕ} : Prime p ↔ 2 ≤ p ∧ ∀ m, m ∣ p → m = 1 ∨ m = p := by |
refine ⟨fun h => ⟨h.two_le, h.eq_one_or_self_of_dvd⟩, fun h => ?_⟩
-- Porting note: needed to make ℕ explicit
have h1 := (@one_lt_two ℕ ..).trans_le h.1
refine ⟨mt Nat.isUnit_iff.mp h1.ne', fun a b hab => ?_⟩
simp only [Nat.isUnit_iff]
apply Or.imp_right _ (h.2 a _)
· rintro rfl
rw [← mul_right_inj' ... | 1,087 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.nat.prime from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82... | Mathlib/Data/Nat/Prime.lean | 147 | 153 | theorem prime_of_coprime (n : ℕ) (h1 : 1 < n) (h : ∀ m < n, m ≠ 0 → n.Coprime m) : Prime n := by |
refine prime_def_lt.mpr ⟨h1, fun m mlt mdvd => ?_⟩
have hm : m ≠ 0 := by
rintro rfl
rw [zero_dvd_iff] at mdvd
exact mlt.ne' mdvd
exact (h m mlt hm).symm.eq_one_of_dvd mdvd
| 1,087 |
import Mathlib.Data.Nat.Prime
#align_import data.int.nat_prime from "leanprover-community/mathlib"@"422e70f7ce183d2900c586a8cda8381e788a0c62"
open Nat
namespace Int
theorem not_prime_of_int_mul {a b : ℤ} {c : ℕ} (ha : a.natAbs ≠ 1) (hb : b.natAbs ≠ 1)
(hc : a * b = (c : ℤ)) : ¬Nat.Prime c :=
not_prime_mul... | Mathlib/Data/Int/NatPrime.lean | 24 | 33 | theorem succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul {p : ℕ} (p_prime : Nat.Prime p) {m n : ℤ}
{k l : ℕ} (hpm : ↑(p ^ k) ∣ m) (hpn : ↑(p ^ l) ∣ n) (hpmn : ↑(p ^ (k + l + 1)) ∣ m * n) :
↑(p ^ (k + 1)) ∣ m ∨ ↑(p ^ (l + 1)) ∣ n :=
have hpm' : p ^ k ∣ m.natAbs := Int.natCast_dvd_natCast.1 <| Int.dvd_natAbs.2 hpm
ha... |
rw [← Int.natAbs_mul]; apply Int.natCast_dvd_natCast.1 <| Int.dvd_natAbs.2 hpmn
let hsd := Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn'
hsd.elim (fun hsd1 => Or.inl (by apply Int.dvd_natAbs.1; apply Int.natCast_dvd_natCast.2 hsd1))
fun hsd2 => Or.inr (by apply Int.dvd_natAbs.1; appl... | 1,088 |
import Mathlib.Data.Nat.Prime
#align_import data.int.nat_prime from "leanprover-community/mathlib"@"422e70f7ce183d2900c586a8cda8381e788a0c62"
open Nat
namespace Int
theorem not_prime_of_int_mul {a b : ℤ} {c : ℕ} (ha : a.natAbs ≠ 1) (hb : b.natAbs ≠ 1)
(hc : a * b = (c : ℤ)) : ¬Nat.Prime c :=
not_prime_mul... | Mathlib/Data/Int/NatPrime.lean | 36 | 39 | theorem Prime.dvd_natAbs_of_coe_dvd_sq {p : ℕ} (hp : p.Prime) (k : ℤ) (h : (p : ℤ) ∣ k ^ 2) :
p ∣ k.natAbs := by |
apply @Nat.Prime.dvd_of_dvd_pow _ _ 2 hp
rwa [sq, ← natAbs_mul, ← natCast_dvd, ← sq]
| 1,088 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.Data.Nat.Prime
import Mathlib.ModelTheory.Algebra.Ring.FreeCommRing
import Mathlib.ModelTheory.Algebra.Field.Basic
variable {p : ℕ} {K : Type*}
namespace FirstOrder
namespace Field
open Language Ring
noncomputable def eqZero (n : ℕ) : Language.ring.Sentence :=
... | Mathlib/ModelTheory/Algebra/Field/CharP.lean | 63 | 78 | theorem charP_iff_model_fieldOfChar [Field K] [CompatibleRing K] :
(Theory.fieldOfChar p).Model K ↔ CharP K p := by |
simp only [Theory.fieldOfChar, Theory.model_union_iff,
(show (Theory.field.Model K) by infer_instance), true_and]
split_ifs with hp0 hp
· subst hp0
simp only [Theory.model_iff, Set.mem_image, Set.mem_setOf_eq, Sentence.Realize,
forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, Formula.realize... | 1,089 |
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Prime
#align_import data.nat.choose.dvd from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
open Nat
namespace Prime
variable {p a b k : ℕ}
| Mathlib/Data/Nat/Choose/Dvd.lean | 24 | 29 | theorem dvd_choose_add (hp : Prime p) (hap : a < p) (hbp : b < p) (h : p ≤ a + b) :
p ∣ choose (a + b) a := by |
have h₁ : p ∣ (a + b)! := hp.dvd_factorial.2 h
rw [← add_choose_mul_factorial_mul_factorial, ← choose_symm_add, hp.dvd_mul, hp.dvd_mul,
hp.dvd_factorial, hp.dvd_factorial] at h₁
exact (h₁.resolve_right hbp.not_le).resolve_right hap.not_le
| 1,090 |
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Choose.Dvd
import Mathlib.Data.Nat.Prime
#align_import number_theory.primorial from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
open Finset
... | Mathlib/NumberTheory/Primorial.lean | 45 | 48 | theorem primorial_succ {n : ℕ} (hn1 : n ≠ 1) (hn : Odd n) : (n + 1)# = n# := by |
refine prod_congr ?_ fun _ _ ↦ rfl
rw [range_succ, filter_insert, if_neg fun h ↦ odd_iff_not_even.mp hn _]
exact fun h ↦ h.even_sub_one <| mt succ.inj hn1
| 1,091 |
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Choose.Dvd
import Mathlib.Data.Nat.Prime
#align_import number_theory.primorial from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
open Finset
... | Mathlib/NumberTheory/Primorial.lean | 51 | 55 | theorem primorial_add (m n : ℕ) :
(m + n)# = m# * ∏ p ∈ filter Nat.Prime (Ico (m + 1) (m + n + 1)), p := by |
rw [primorial, primorial, ← Ico_zero_eq_range, ← prod_union, ← filter_union, Ico_union_Ico_eq_Ico]
exacts [Nat.zero_le _, add_le_add_right (Nat.le_add_right _ _) _,
disjoint_filter_filter <| Ico_disjoint_Ico_consecutive _ _ _]
| 1,091 |
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Choose.Dvd
import Mathlib.Data.Nat.Prime
#align_import number_theory.primorial from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
open Finset
... | Mathlib/NumberTheory/Primorial.lean | 73 | 91 | theorem primorial_le_4_pow (n : ℕ) : n# ≤ 4 ^ n := by |
induction' n using Nat.strong_induction_on with n ihn
cases' n with n; · rfl
rcases n.even_or_odd with (⟨m, rfl⟩ | ho)
· rcases m.eq_zero_or_pos with (rfl | hm)
· decide
calc
(m + m + 1)# = (m + 1 + m)# := by rw [add_right_comm]
_ ≤ (m + 1)# * choose (m + 1 + m) (m + 1) := primorial_add_le ... | 1,091 |
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.prime_norm_num from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
open Nat Qq Lean Meta
namespace Mathlib.Meta.NormNum
theorem not_prime_mul_of_ble (a b n : ℕ) (h : a * b = n) (h₁ : a.ble 1 = fals... | Mathlib/Tactic/NormNum/Prime.lean | 50 | 56 | theorem MinFacHelper.one_lt {n k : ℕ} (h : MinFacHelper n k) : 1 < n := by |
have : 2 < minFac n := h.1.trans_le h.2.2
obtain rfl | h := n.eq_zero_or_pos
· contradiction
rcases (succ_le_of_lt h).eq_or_lt with rfl|h
· simp_all
exact h
| 1,092 |
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.prime_norm_num from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
open Nat Qq Lean Meta
namespace Mathlib.Meta.NormNum
theorem not_prime_mul_of_ble (a b n : ℕ) (h : a * b = n) (h₁ : a.ble 1 = fals... | Mathlib/Tactic/NormNum/Prime.lean | 58 | 65 | theorem minFacHelper_0 (n : ℕ)
(h1 : Nat.ble (nat_lit 2) n = true) (h2 : nat_lit 1 = n % (nat_lit 2)) :
MinFacHelper n (nat_lit 3) := by |
refine ⟨by norm_num, by norm_num, ?_⟩
refine (le_minFac'.mpr λ p hp hpn ↦ ?_).resolve_left (Nat.ne_of_gt (Nat.le_of_ble_eq_true h1))
rcases hp.eq_or_lt with rfl|h
· simp [(Nat.dvd_iff_mod_eq_zero ..).1 hpn] at h2
· exact h
| 1,092 |
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.prime_norm_num from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
open Nat Qq Lean Meta
namespace Mathlib.Meta.NormNum
theorem not_prime_mul_of_ble (a b n : ℕ) (h : a * b = n) (h₁ : a.ble 1 = fals... | Mathlib/Tactic/NormNum/Prime.lean | 67 | 82 | theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
(np : minFac n ≠ k) : MinFacHelper n k' := by |
rw [← e]
refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
· rw [add_mod, mod_self, add_zero, mod_mod]
exact h.2.1
rcases h.2.2.eq_or_lt with rfl|h2
· exact (np rfl).elim
rcases (succ_le_of_lt h2).eq_or_lt with h2|h2
· refine ((h.1.trans_le h.2.2).ne ?_).elim
have h3 : 2 ∣ minFac n := by
rw [Nat.dvd_... | 1,092 |
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.prime_norm_num from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
open Nat Qq Lean Meta
namespace Mathlib.Meta.NormNum
theorem not_prime_mul_of_ble (a b n : ℕ) (h : a * b = n) (h₁ : a.ble 1 = fals... | Mathlib/Tactic/NormNum/Prime.lean | 84 | 88 | theorem minFacHelper_2 {n k k' : ℕ} (e : k + 2 = k') (nk : ¬ Nat.Prime k)
(h : MinFacHelper n k) : MinFacHelper n k' := by |
refine minFacHelper_1 e h λ h2 ↦ ?_
rw [← h2] at nk
exact nk <| minFac_prime h.one_lt.ne'
| 1,092 |
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.prime_norm_num from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
open Nat Qq Lean Meta
namespace Mathlib.Meta.NormNum
theorem not_prime_mul_of_ble (a b n : ℕ) (h : a * b = n) (h₁ : a.ble 1 = fals... | Mathlib/Tactic/NormNum/Prime.lean | 90 | 95 | theorem minFacHelper_3 {n k k' : ℕ} (e : k + 2 = k') (nk : (n % k).beq 0 = false)
(h : MinFacHelper n k) : MinFacHelper n k' := by |
refine minFacHelper_1 e h λ h2 ↦ ?_
have nk := Nat.ne_of_beq_eq_false nk
rw [← Nat.dvd_iff_mod_eq_zero, ← h2] at nk
exact nk <| minFac_dvd n
| 1,092 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Prime
import Mathlib.Data.Nat.Digits
import Mathlib.RingTheory.Multiplicity
#align_import data.nat.multiplicity from "l... | Mathlib/Data/Nat/Multiplicity.lean | 61 | 77 | theorem multiplicity_eq_card_pow_dvd {m n b : ℕ} (hm : m ≠ 1) (hn : 0 < n) (hb : log m n < b) :
multiplicity m n = ↑((Finset.Ico 1 b).filter fun i => m ^ i ∣ n).card :=
calc
multiplicity m n = ↑(Ico 1 <| (multiplicity m n).get (finite_nat_iff.2 ⟨hm, hn⟩) + 1).card := by |
simp
_ = ↑((Finset.Ico 1 b).filter fun i => m ^ i ∣ n).card :=
congr_arg _ <|
congr_arg card <|
Finset.ext fun i => by
rw [mem_filter, mem_Ico, mem_Ico, Nat.lt_succ_iff, ← @PartENat.coe_le_coe i,
PartENat.natCast_get, ← pow_dvd_iff_le_multiplicity, and_right_... | 1,093 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Prime
import Mathlib.Data.Nat.Digits
import Mathlib.RingTheory.Multiplicity
#align_import data.nat.multiplicity from "l... | Mathlib/Data/Nat/Multiplicity.lean | 108 | 123 | theorem multiplicity_factorial {p : ℕ} (hp : p.Prime) :
∀ {n b : ℕ}, log p n < b → multiplicity p n ! = (∑ i ∈ Ico 1 b, n / p ^ i : ℕ)
| 0, b, _ => by simp [Ico, hp.multiplicity_one]
| n + 1, b, hb =>
calc
multiplicity p (n + 1)! = multiplicity p n ! + multiplicity p (n + 1) := by |
rw [factorial_succ, hp.multiplicity_mul, add_comm]
_ = (∑ i ∈ Ico 1 b, n / p ^ i : ℕ) +
((Finset.Ico 1 b).filter fun i => p ^ i ∣ n + 1).card := by
rw [multiplicity_factorial hp ((log_mono_right <| le_succ _).trans_lt hb), ←
multiplicity_eq_card_pow_dvd hp.ne_one (succ_pos _... | 1,093 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Prime
import Mathlib.Data.Nat.Digits
import Mathlib.RingTheory.Multiplicity
#align_import data.nat.multiplicity from "l... | Mathlib/Data/Nat/Multiplicity.lean | 138 | 158 | theorem multiplicity_factorial_mul_succ {n p : ℕ} (hp : p.Prime) :
multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1 := by |
have hp' := hp.prime
have h0 : 2 ≤ p := hp.two_le
have h1 : 1 ≤ p * n + 1 := Nat.le_add_left _ _
have h2 : p * n + 1 ≤ p * (n + 1) := by linarith
have h3 : p * n + 1 ≤ p * (n + 1) + 1 := by omega
have hm : multiplicity p (p * n)! ≠ ⊤ := by
rw [Ne, eq_top_iff_not_finite, Classical.not_not, finite_nat_if... | 1,093 |
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Tactic.IntervalCases
#align_import number_theory.padics.padic_val from "leanprover-community/mathlib"@"60fa54e778c9e85d930efae172435f42fb0d71f7"
universe u
ope... | Mathlib/NumberTheory/Padics/PadicVal.lean | 101 | 104 | theorem self (hp : 1 < p) : padicValNat p p = 1 := by |
have neq_one : ¬p = 1 ↔ True := iff_of_true hp.ne' trivial
have eq_zero_false : p = 0 ↔ False := iff_false_intro (zero_lt_one.trans hp).ne'
simp [padicValNat, neq_one, eq_zero_false]
| 1,094 |
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Tactic.IntervalCases
#align_import number_theory.padics.padic_val from "leanprover-community/mathlib"@"60fa54e778c9e85d930efae172435f42fb0d71f7"
universe u
ope... | Mathlib/NumberTheory/Padics/PadicVal.lean | 108 | 110 | theorem eq_zero_iff {n : ℕ} : padicValNat p n = 0 ↔ p = 1 ∨ n = 0 ∨ ¬p ∣ n := by |
simp only [padicValNat, dite_eq_right_iff, PartENat.get_eq_iff_eq_coe, Nat.cast_zero,
multiplicity_eq_zero, and_imp, pos_iff_ne_zero, Ne, ← or_iff_not_imp_left]
| 1,094 |
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Tactic.IntervalCases
#align_import number_theory.padics.padic_val from "leanprover-community/mathlib"@"60fa54e778c9e85d930efae172435f42fb0d71f7"
universe u
ope... | Mathlib/NumberTheory/Padics/PadicVal.lean | 119 | 124 | theorem maxPowDiv_eq_multiplicity {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv n = multiplicity p n := by |
apply multiplicity.unique <| pow_dvd p n
intro h
apply Nat.not_lt.mpr <| le_of_dvd hp hn h
simp
| 1,094 |
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Tactic.IntervalCases
#align_import number_theory.padics.padic_val from "leanprover-community/mathlib"@"60fa54e778c9e85d930efae172435f42fb0d71f7"
universe u
ope... | Mathlib/NumberTheory/Padics/PadicVal.lean | 126 | 129 | theorem maxPowDiv_eq_multiplicity_get {p n : ℕ} (hp : 1 < p) (hn : 0 < n) (h : Finite p n) :
p.maxPowDiv n = (multiplicity p n).get h := by |
rw [PartENat.get_eq_iff_eq_coe.mpr]
apply maxPowDiv_eq_multiplicity hp hn|>.symm
| 1,094 |
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Tactic.IntervalCases
#align_import number_theory.padics.padic_val from "leanprover-community/mathlib"@"60fa54e778c9e85d930efae172435f42fb0d71f7"
universe u
ope... | Mathlib/NumberTheory/Padics/PadicVal.lean | 133 | 146 | theorem padicValNat_eq_maxPowDiv : @padicValNat = @maxPowDiv := by |
ext p n
by_cases h : 1 < p ∧ 0 < n
· dsimp [padicValNat]
rw [dif_pos ⟨Nat.ne_of_gt h.1,h.2⟩, maxPowDiv_eq_multiplicity_get h.1 h.2]
· simp only [not_and_or,not_gt_eq,Nat.le_zero] at h
apply h.elim
· intro h
interval_cases p
· simp [Classical.em]
· dsimp [padicValNat, maxPowDiv]
... | 1,094 |
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Tactic.IntervalCases
#align_import number_theory.padics.padic_val from "leanprover-community/mathlib"@"60fa54e778c9e85d930efae172435f42fb0d71f7"
universe u
ope... | Mathlib/NumberTheory/Padics/PadicVal.lean | 162 | 169 | theorem of_ne_one_ne_zero {z : ℤ} (hp : p ≠ 1) (hz : z ≠ 0) :
padicValInt p z =
(multiplicity (p : ℤ) z).get
(by
apply multiplicity.finite_int_iff.2
simp [hp, hz]) := by |
rw [padicValInt, padicValNat, dif_pos (And.intro hp (Int.natAbs_pos.mpr hz))]
simp only [multiplicity.Int.natAbs p z]
| 1,094 |
import Mathlib.Algebra.Order.Field.Power
import Mathlib.NumberTheory.Padics.PadicVal
#align_import number_theory.padics.padic_norm from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
def padicNorm (p : ℕ) (q : ℚ) : ℚ :=
if q = 0 then 0 else (p : ℚ) ^ (-padicValRat p q)
#align padic_n... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 81 | 82 | theorem padicNorm_p (hp : 1 < p) : padicNorm p p = (p : ℚ)⁻¹ := by |
simp [padicNorm, (pos_of_gt hp).ne', padicValNat.self hp]
| 1,095 |
import Mathlib.Algebra.Order.Field.Power
import Mathlib.NumberTheory.Padics.PadicVal
#align_import number_theory.padics.padic_norm from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
def padicNorm (p : ℕ) (q : ℚ) : ℚ :=
if q = 0 then 0 else (p : ℚ) ^ (-padicValRat p q)
#align padic_n... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 94 | 98 | theorem padicNorm_of_prime_of_ne {q : ℕ} [p_prime : Fact p.Prime] [q_prime : Fact q.Prime]
(neq : p ≠ q) : padicNorm p q = 1 := by |
have p : padicValRat p q = 0 := mod_cast padicValNat_primes neq
rw [padicNorm, p]
simp [q_prime.1.ne_zero]
| 1,095 |
import Mathlib.Algebra.Order.Field.Power
import Mathlib.NumberTheory.Padics.PadicVal
#align_import number_theory.padics.padic_norm from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
def padicNorm (p : ℕ) (q : ℚ) : ℚ :=
if q = 0 then 0 else (p : ℚ) ^ (-padicValRat p q)
#align padic_n... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 104 | 106 | theorem padicNorm_p_lt_one (hp : 1 < p) : padicNorm p p < 1 := by |
rw [padicNorm_p hp, inv_lt_one_iff]
exact mod_cast Or.inr hp
| 1,095 |
import Mathlib.Algebra.Order.Field.Power
import Mathlib.NumberTheory.Padics.PadicVal
#align_import number_theory.padics.padic_norm from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
def padicNorm (p : ℕ) (q : ℚ) : ℚ :=
if q = 0 then 0 else (p : ℚ) ^ (-padicValRat p q)
#align padic_n... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 137 | 142 | theorem zero_of_padicNorm_eq_zero {q : ℚ} (h : padicNorm p q = 0) : q = 0 := by |
apply by_contradiction; intro hq
unfold padicNorm at h; rw [if_neg hq] at h
apply absurd h
apply zpow_ne_zero
exact mod_cast hp.1.ne_zero
| 1,095 |
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padics.padic_numbers from "leanprover-community/mathlib"@"b9b2114f7711fec1c1e055d507f082f8ceb2c3b7"
noncomputable section
open scoped Classical
open Nat m... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 121 | 135 | theorem norm_zero_iff (f : PadicSeq p) : f.norm = 0 ↔ f ≈ 0 := by |
constructor
· intro h
by_contra hf
unfold norm at h
split_ifs at h
· contradiction
apply hf
intro ε hε
exists stationaryPoint hf
intro j hj
have heq := stationaryPoint_spec hf le_rfl hj
simpa [h, heq]
· intro h
simp [norm, h]
| 1,096 |
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padics.padic_numbers from "leanprover-community/mathlib"@"b9b2114f7711fec1c1e055d507f082f8ceb2c3b7"
noncomputable section
open scoped Classical
open Nat m... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 156 | 160 | theorem norm_eq_norm_app_of_nonzero {f : PadicSeq p} (hf : ¬f ≈ 0) :
∃ k, f.norm = padicNorm p k ∧ k ≠ 0 :=
have heq : f.norm = padicNorm p (f <| stationaryPoint hf) := by | simp [norm, hf]
⟨f <| stationaryPoint hf, heq, fun h ↦
norm_nonzero_of_not_equiv_zero hf (by simpa [h] using heq)⟩
| 1,096 |
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padics.padic_numbers from "leanprover-community/mathlib"@"b9b2114f7711fec1c1e055d507f082f8ceb2c3b7"
noncomputable section
open scoped Classical
open Nat m... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 176 | 181 | theorem lift_index_left_left {f : PadicSeq p} (hf : ¬f ≈ 0) (v2 v3 : ℕ) :
padicNorm p (f (stationaryPoint hf)) =
padicNorm p (f (max (stationaryPoint hf) (max v2 v3))) := by |
apply stationaryPoint_spec hf
· apply le_max_left
· exact le_rfl
| 1,096 |
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padics.padic_numbers from "leanprover-community/mathlib"@"b9b2114f7711fec1c1e055d507f082f8ceb2c3b7"
noncomputable section
open scoped Classical
open Nat m... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 223 | 231 | theorem norm_eq_pow_val {f : PadicSeq p} (hf : ¬f ≈ 0) : f.norm = (p : ℚ) ^ (-f.valuation : ℤ) := by |
rw [norm, valuation, dif_neg hf, dif_neg hf, padicNorm, if_neg]
intro H
apply CauSeq.not_limZero_of_not_congr_zero hf
intro ε hε
use stationaryPoint hf
intro n hn
rw [stationaryPoint_spec hf le_rfl hn]
simpa [H] using hε
| 1,096 |
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padics.padic_numbers from "leanprover-community/mathlib"@"b9b2114f7711fec1c1e055d507f082f8ceb2c3b7"
noncomputable section
open scoped Classical
open Nat m... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 234 | 238 | theorem val_eq_iff_norm_eq {f g : PadicSeq p} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) :
f.valuation = g.valuation ↔ f.norm = g.norm := by |
rw [norm_eq_pow_val hf, norm_eq_pow_val hg, ← neg_inj, zpow_inj]
· exact mod_cast (Fact.out : p.Prime).pos
· exact mod_cast (Fact.out : p.Prime).ne_one
| 1,096 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 61 | 67 | theorem ExpChar.eq {p q : ℕ} (hp : ExpChar R p) (hq : ExpChar R q) : p = q := by |
cases' hp with hp _ hp' hp
· cases' hq with hq _ hq' hq
exacts [rfl, False.elim (Nat.not_prime_zero (CharP.eq R hq (CharP.ofCharZero R) ▸ hq'))]
· cases' hq with hq _ hq' hq
exacts [False.elim (Nat.not_prime_zero (CharP.eq R hp (CharP.ofCharZero R) ▸ hp')),
CharP.eq R hp hq]
| 1,097 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 74 | 79 | theorem ringExpChar.eq (q : ℕ) [h : ExpChar R q] : ringExpChar R = q := by |
cases' h with _ _ h _
· haveI := CharP.ofCharZero R
rw [ringExpChar, ringChar.eq R 0]; rfl
rw [ringExpChar, ringChar.eq R q]
exact Nat.max_eq_left h.one_lt.le
| 1,097 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 82 | 83 | theorem ringExpChar.eq_one (R : Type*) [NonAssocSemiring R] [CharZero R] : ringExpChar R = 1 := by |
rw [ringExpChar, ringChar.eq_zero, max_eq_right zero_le_one]
| 1,097 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 86 | 89 | theorem expChar_one_of_char_zero (q : ℕ) [hp : CharP R 0] [hq : ExpChar R q] : q = 1 := by |
cases' hq with q hq_one hq_prime hq_hchar
· rfl
· exact False.elim <| hq_prime.ne_zero <| hq_hchar.eq R hp
| 1,097 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 93 | 97 | theorem char_eq_expChar_iff (p q : ℕ) [hp : CharP R p] [hq : ExpChar R q] : p = q ↔ p.Prime := by |
cases' hq with q hq_one hq_prime hq_hchar
· rw [(CharP.eq R hp inferInstance : p = 0)]
decide
· exact ⟨fun hpq => hpq.symm ▸ hq_prime, fun _ => CharP.eq R hp hq_hchar⟩
| 1,097 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 105 | 108 | theorem char_zero_of_expChar_one (p : ℕ) [hp : CharP R p] [hq : ExpChar R 1] : p = 0 := by |
cases hq
· exact CharP.eq R hp inferInstance
· exact False.elim (CharP.char_ne_one R 1 rfl)
| 1,097 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 113 | 116 | theorem charZero_of_expChar_one' [hq : ExpChar R 1] : CharZero R := by |
cases hq
· assumption
· exact False.elim (CharP.char_ne_one R 1 rfl)
| 1,097 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 120 | 125 | theorem expChar_one_iff_char_zero (p q : ℕ) [CharP R p] [ExpChar R q] : q = 1 ↔ p = 0 := by |
constructor
· rintro rfl
exact char_zero_of_expChar_one R p
· rintro rfl
exact expChar_one_of_char_zero R q
| 1,097 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 133 | 136 | theorem char_prime_of_ne_zero {p : ℕ} [hp : CharP R p] (p_ne_zero : p ≠ 0) : Nat.Prime p := by |
cases' CharP.char_is_prime_or_zero R p with h h
· exact h
· contradiction
| 1,097 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 141 | 144 | theorem expChar_is_prime_or_one (q : ℕ) [hq : ExpChar R q] : Nat.Prime q ∨ q = 1 := by |
cases hq with
| zero => exact .inr rfl
| prime hp => exact .inl hp
| 1,097 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 148 | 150 | theorem expChar_pos (q : ℕ) [ExpChar R q] : 0 < q := by |
rcases expChar_is_prime_or_one R q with h | rfl
exacts [Nat.Prime.pos h, Nat.one_pos]
| 1,097 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ι : Type*}
namespace CharTwo
section Semiring
variable [Semiring R] [CharP R 2]
| Mathlib/Algebra/CharP/Two.lean | 29 | 29 | theorem two_eq_zero : (2 : R) = 0 := by | rw [← Nat.cast_two, CharP.cast_eq_zero]
| 1,098 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ι : Type*}
namespace CharTwo
section Semiring
variable [Semiring R] [CharP R 2]
theorem two_eq_zero : (2 : ... | Mathlib/Algebra/CharP/Two.lean | 33 | 33 | theorem add_self_eq_zero (x : R) : x + x = 0 := by | rw [← two_smul R x, two_eq_zero, zero_smul]
| 1,098 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ι : Type*}
namespace CharTwo
section Semiring
variable [Semiring R] [CharP R 2]
theorem two_eq_zero : (2 : ... | Mathlib/Algebra/CharP/Two.lean | 38 | 40 | theorem bit0_eq_zero : (bit0 : R → R) = 0 := by |
funext
exact add_self_eq_zero _
| 1,098 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ι : Type*}
namespace CharTwo
section Semiring
variable [Semiring R] [CharP R 2]
theorem two_eq_zero : (2 : ... | Mathlib/Algebra/CharP/Two.lean | 44 | 44 | theorem bit0_apply_eq_zero (x : R) : (bit0 x : R) = 0 := by | simp
| 1,098 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ι : Type*}
namespace CharTwo
section Semiring
variable [Semiring R] [CharP R 2]
theorem two_eq_zero : (2 : ... | Mathlib/Algebra/CharP/Two.lean | 49 | 51 | theorem bit1_eq_one : (bit1 : R → R) = 1 := by |
funext
simp [bit1]
| 1,098 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ι : Type*}
namespace CharTwo
section Semiring
variable [Semiring R] [CharP R 2]
theorem two_eq_zero : (2 : ... | Mathlib/Algebra/CharP/Two.lean | 55 | 55 | theorem bit1_apply_eq_one (x : R) : (bit1 x : R) = 1 := by | simp
| 1,098 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ι : Type*}
namespace CharTwo
section CommSemiring
variable [CommSemiring R] [CharP R 2]
theorem add_sq (x y... | Mathlib/Algebra/CharP/Two.lean | 91 | 92 | theorem add_mul_self (x y : R) : (x + y) * (x + y) = x * x + y * y := by |
rw [← pow_two, ← pow_two, ← pow_two, add_sq]
| 1,098 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ι : Type*}
namespace CharTwo
section CommSemiring
variable [CommSemiring R] [CharP R 2]
theorem add_sq (x y... | Mathlib/Algebra/CharP/Two.lean | 99 | 100 | theorem list_sum_mul_self (l : List R) : l.sum * l.sum = (List.map (fun x => x * x) l).sum := by |
simp_rw [← pow_two, list_sum_sq]
| 1,098 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ι : Type*}
namespace CharTwo
section CommSemiring
variable [CommSemiring R] [CharP R 2]
theorem add_sq (x y... | Mathlib/Algebra/CharP/Two.lean | 107 | 108 | theorem multiset_sum_mul_self (l : Multiset R) :
l.sum * l.sum = (Multiset.map (fun x => x * x) l).sum := by | simp_rw [← pow_two, multiset_sum_sq]
| 1,098 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ι : Type*}
namespace CharTwo
section CommSemiring
variable [CommSemiring R] [CharP R 2]
theorem add_sq (x y... | Mathlib/Algebra/CharP/Two.lean | 115 | 116 | theorem sum_mul_self (s : Finset ι) (f : ι → R) :
((∑ i ∈ s, f i) * ∑ i ∈ s, f i) = ∑ i ∈ s, f i * f i := by | simp_rw [← pow_two, sum_sq]
| 1,098 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.GroupTheory.GroupAction.Group
#align_import dynamics.periodic_pts from "leanp... | Mathlib/Dynamics/PeriodicPts.lean | 106 | 109 | theorem left_of_add (hn : IsPeriodicPt f (n + m) x) (hm : IsPeriodicPt f m x) :
IsPeriodicPt f n x := by |
rw [IsPeriodicPt, iterate_add] at hn
exact hn.left_of_comp hm
| 1,099 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.GroupTheory.GroupAction.Group
#align_import dynamics.periodic_pts from "leanp... | Mathlib/Dynamics/PeriodicPts.lean | 112 | 115 | theorem right_of_add (hn : IsPeriodicPt f (n + m) x) (hm : IsPeriodicPt f n x) :
IsPeriodicPt f m x := by |
rw [add_comm] at hn
exact hn.left_of_add hm
| 1,099 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.GroupTheory.GroupAction.Group
#align_import dynamics.periodic_pts from "leanp... | Mathlib/Dynamics/PeriodicPts.lean | 145 | 148 | theorem comp {g : α → α} (hco : Commute f g) (hf : IsPeriodicPt f n x) (hg : IsPeriodicPt g n x) :
IsPeriodicPt (f ∘ g) n x := by |
rw [IsPeriodicPt, hco.comp_iterate]
exact IsFixedPt.comp hf hg
| 1,099 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.GroupTheory.GroupAction.Group
#align_import dynamics.periodic_pts from "leanp... | Mathlib/Dynamics/PeriodicPts.lean | 156 | 159 | theorem left_of_comp {g : α → α} (hco : Commute f g) (hfg : IsPeriodicPt (f ∘ g) n x)
(hg : IsPeriodicPt g n x) : IsPeriodicPt f n x := by |
rw [IsPeriodicPt, hco.comp_iterate] at hfg
exact hfg.left_of_comp hg
| 1,099 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.GroupTheory.GroupAction.Group
#align_import dynamics.periodic_pts from "leanp... | Mathlib/Dynamics/PeriodicPts.lean | 162 | 163 | theorem iterate_mod_apply (h : IsPeriodicPt f n x) (m : ℕ) : f^[m % n] x = f^[m] x := by |
conv_rhs => rw [← Nat.mod_add_div m n, iterate_add_apply, (h.mul_const _).eq]
| 1,099 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.GroupTheory.GroupAction.Group
#align_import dynamics.periodic_pts from "leanp... | Mathlib/Dynamics/PeriodicPts.lean | 595 | 596 | theorem iterate_prod_map (f : α → α) (g : β → β) (n : ℕ) :
(Prod.map f g)^[n] = Prod.map (f^[n]) (g^[n]) := by | induction n <;> simp [*, Prod.map_comp_map]
| 1,099 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
namespace MulAction
open Pointwise
variable {α : Type*}
variable {G : Type*} [Group G] [MulAction G α]
variable {M : Type*} [Monoid M] [MulAction M α]
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 60 | 62 | theorem fixedBy_inv (g : G) : fixedBy α g⁻¹ = fixedBy α g := by |
ext
rw [mem_fixedBy, mem_fixedBy, inv_smul_eq_iff, eq_comm]
| 1,100 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
namespace MulAction
open Pointwise
variable {α : Type*}
variable {G : Type*} [Group G] [MulAction G α]
variable {M : Type*} [Monoid M] [MulAction M α]
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 65 | 68 | theorem smul_mem_fixedBy_iff_mem_fixedBy {a : α} {g : G} :
g • a ∈ fixedBy α g ↔ a ∈ fixedBy α g := by |
rw [mem_fixedBy, smul_left_cancel_iff]
rfl
| 1,100 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
namespace MulAction
open Pointwise
variable {α : Type*}
variable {G : Type*} [Group G] [MulAction G α]
variable {M : Type*} [Monoid M] [MulAction M α]
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 71 | 73 | theorem smul_inv_mem_fixedBy_iff_mem_fixedBy {a : α} {g : G} :
g⁻¹ • a ∈ fixedBy α g ↔ a ∈ fixedBy α g := by |
rw [← fixedBy_inv, smul_mem_fixedBy_iff_mem_fixedBy, fixedBy_inv]
| 1,100 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
namespace MulAction
open Pointwise
variable {α : Type*}
variable {G : Type*} [Group G] [MulAction G α]
variable {M : Type*} [Monoid M] [MulAction M α]
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 82 | 87 | theorem fixedBy_subset_fixedBy_zpow (g : G) (j : ℤ) :
fixedBy α g ⊆ fixedBy α (g ^ j) := by |
intro a a_in_fixedBy
rw [mem_fixedBy, zpow_smul_eq_iff_minimalPeriod_dvd,
minimalPeriod_eq_one_iff_fixedBy.mpr a_in_fixedBy, Nat.cast_one]
exact one_dvd j
| 1,100 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
namespace MulAction
open Pointwise
variable {α : Type*}
variable {G : Type*} [Group G] [MulAction G α]
variable {M : Type*} [Monoid M] [MulAction M α]
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 96 | 98 | theorem fixedBy_mul (m₁ m₂ : M) : fixedBy α m₁ ∩ fixedBy α m₂ ⊆ fixedBy α (m₁ * m₂) := by |
intro a ⟨h₁, h₂⟩
rw [mem_fixedBy, mul_smul, h₂, h₁]
| 1,100 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
namespace MulAction
open Pointwise
variable {α : Type*}
variable {G : Type*} [Group G] [MulAction G α]
variable {M : Type*} [Monoid M] [MulAction M α]
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 102 | 105 | theorem smul_fixedBy (g h: G) :
h • fixedBy α g = fixedBy α (h * g * h⁻¹) := by |
ext a
simp_rw [Set.mem_smul_set_iff_inv_smul_mem, mem_fixedBy, mul_smul, smul_eq_iff_eq_inv_smul h]
| 1,100 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
namespace MulAction
open Pointwise
variable {α : Type*}
variable {G : Type*} [Group G] [MulAction G α]
variable {M : Type*} [Monoid M] [MulAction M α]
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 124 | 126 | theorem set_mem_fixedBy_iff (s : Set α) (g : G) :
s ∈ fixedBy (Set α) g ↔ ∀ x, g • x ∈ s ↔ x ∈ s := by |
simp_rw [mem_fixedBy, ← eq_inv_smul_iff, Set.ext_iff, Set.mem_inv_smul_set_iff, Iff.comm]
| 1,100 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
namespace MulAction
open Pointwise
variable {α : Type*}
variable {G : Type*} [Group G] [MulAction G α]
variable {M : Type*} [Monoid M] [MulAction M α]
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 195 | 199 | theorem fixedBy_mem_fixedBy_of_commute {g h : G} (comm: Commute g h) :
(fixedBy α g) ∈ fixedBy (Set α) h := by |
ext x
rw [Set.mem_smul_set_iff_inv_smul_mem, mem_fixedBy, ← mul_smul, comm.inv_right, mul_smul,
smul_left_cancel_iff, mem_fixedBy]
| 1,100 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
namespace MulAction
open Pointwise
variable {α : Type*}
variable {G : Type*} [Group G] [MulAction G α]
variable {M : Type*} [Monoid M] [MulAction M α]
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 238 | 240 | theorem fixedBy_eq_univ_iff_eq_one {m : M} : fixedBy α m = Set.univ ↔ m = 1 := by |
rw [← (smul_left_injective' (M := M) (α := α)).eq_iff, Set.eq_univ_iff_forall]
simp_rw [Function.funext_iff, one_smul, mem_fixedBy]
| 1,100 |
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.FixedPoints
import Mathlib.GroupTheory.Perm.Support
open Equiv List MulAction Pointwise Set Subgroup
variable {G α : Type*} [Group G] [MulAction G α] [DecidableEq α]
theorem finite_compl_fixedBy_closure_iff {S : Set G} :
(∀ g ∈ closure S, ... | Mathlib/GroupTheory/Perm/ClosureSwap.lean | 41 | 44 | theorem Equiv.Perm.IsSwap.finite_compl_fixedBy {σ : Perm α} (h : σ.IsSwap) :
(fixedBy α σ)ᶜ.Finite := by |
obtain ⟨x, y, -, rfl⟩ := h
exact finite_compl_fixedBy_swap
| 1,101 |
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.FixedPoints
import Mathlib.GroupTheory.Perm.Support
open Equiv List MulAction Pointwise Set Subgroup
variable {G α : Type*} [Group G] [MulAction G α] [DecidableEq α]
theorem finite_compl_fixedBy_closure_iff {S : Set G} :
(∀ g ∈ closure S, ... | Mathlib/GroupTheory/Perm/ClosureSwap.lean | 47 | 55 | theorem SubmonoidClass.swap_mem_trans {a b c : α} {C} [SetLike C (Perm α)]
[SubmonoidClass C (Perm α)] (M : C) (hab : swap a b ∈ M) (hbc : swap b c ∈ M) :
swap a c ∈ M := by |
obtain rfl | hab' := eq_or_ne a b
· exact hbc
obtain rfl | hac := eq_or_ne a c
· exact swap_self a ▸ one_mem M
rw [swap_comm, ← swap_mul_swap_mul_swap hab' hac]
exact mul_mem (mul_mem hbc hab) hbc
| 1,101 |
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.FixedPoints
import Mathlib.GroupTheory.Perm.Support
open Equiv List MulAction Pointwise Set Subgroup
variable {G α : Type*} [Group G] [MulAction G α] [DecidableEq α]
theorem finite_compl_fixedBy_closure_iff {S : Set G} :
(∀ g ∈ closure S, ... | Mathlib/GroupTheory/Perm/ClosureSwap.lean | 59 | 70 | theorem exists_smul_not_mem_of_subset_orbit_closure (S : Set G) (T : Set α) {a : α}
(hS : ∀ g ∈ S, g⁻¹ ∈ S) (subset : T ⊆ orbit (closure S) a) (not_mem : a ∉ T)
(nonempty : T.Nonempty) : ∃ σ ∈ S, ∃ a ∈ T, σ • a ∉ T := by |
have key0 : ¬ closure S ≤ stabilizer G T := by
have ⟨b, hb⟩ := nonempty
obtain ⟨σ, rfl⟩ := subset hb
contrapose! not_mem with h
exact smul_mem_smul_set_iff.mp ((h σ.2).symm ▸ hb)
contrapose! key0
refine (closure_le _).mpr fun σ hσ ↦ ?_
simp_rw [SetLike.mem_coe, mem_stabilizer_iff, Set.ext_iff, ... | 1,101 |
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.FixedPoints
import Mathlib.GroupTheory.Perm.Support
open Equiv List MulAction Pointwise Set Subgroup
variable {G α : Type*} [Group G] [MulAction G α] [DecidableEq α]
theorem finite_compl_fixedBy_closure_iff {S : Set G} :
(∀ g ∈ closure S, ... | Mathlib/GroupTheory/Perm/ClosureSwap.lean | 74 | 88 | theorem swap_mem_closure_isSwap {S : Set (Perm α)} (hS : ∀ f ∈ S, f.IsSwap) {x y : α} :
swap x y ∈ closure S ↔ x ∈ orbit (closure S) y := by |
refine ⟨fun h ↦ ⟨⟨swap x y, h⟩, swap_apply_right x y⟩, fun hf ↦ ?_⟩
by_contra h
have := exists_smul_not_mem_of_subset_orbit_closure S {x | swap x y ∈ closure S}
(fun f hf ↦ ?_) (fun z hz ↦ ?_) h ⟨y, ?_⟩
· obtain ⟨σ, hσ, a, ha, hσa⟩ := this
obtain ⟨z, w, hzw, rfl⟩ := hS σ hσ
have := ne_of_mem_of_not... | 1,101 |
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.FixedPoints
import Mathlib.GroupTheory.Perm.Support
open Equiv List MulAction Pointwise Set Subgroup
variable {G α : Type*} [Group G] [MulAction G α] [DecidableEq α]
theorem finite_compl_fixedBy_closure_iff {S : Set G} :
(∀ g ∈ closure S, ... | Mathlib/GroupTheory/Perm/ClosureSwap.lean | 92 | 114 | theorem mem_closure_isSwap {S : Set (Perm α)} (hS : ∀ f ∈ S, f.IsSwap) {f : Perm α} :
f ∈ closure S ↔ (fixedBy α f)ᶜ.Finite ∧ ∀ x, f x ∈ orbit (closure S) x := by |
refine ⟨fun hf ↦ ⟨?_, fun x ↦ mem_orbit_iff.mpr ⟨⟨f, hf⟩, rfl⟩⟩, ?_⟩
· exact finite_compl_fixedBy_closure_iff.mpr (fun f hf ↦ (hS f hf).finite_compl_fixedBy) _ hf
rintro ⟨fin, hf⟩
set supp := (fixedBy α f)ᶜ with supp_eq
suffices h : (fixedBy α f)ᶜ ⊆ supp → f ∈ closure S from h supp_eq.symm.subset
clear_val... | 1,101 |
import Mathlib.Data.Num.Lemmas
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.Ring
#align_import data.num.prime from "leanprover-community/mathlib"@"58581d0fe523063f5651df0619be2bf65012a94a"
namespace PosNum
def minFacAux (n : PosNum) : ℕ → PosNum → PosNum
| 0, _ => n
| fuel + 1, k =>
if n < k.bit1... | Mathlib/Data/Num/Prime.lean | 44 | 54 | theorem minFacAux_to_nat {fuel : ℕ} {n k : PosNum} (h : Nat.sqrt n < fuel + k.bit1) :
(minFacAux n fuel k : ℕ) = Nat.minFacAux n k.bit1 := by |
induction' fuel with fuel ih generalizing k <;> rw [minFacAux, Nat.minFacAux]
· rw [Nat.zero_add, Nat.sqrt_lt] at h
simp only [h, ite_true]
simp_rw [← mul_to_nat]
simp only [cast_lt, dvd_to_nat]
split_ifs <;> try rfl
rw [ih] <;> [congr; convert Nat.lt_succ_of_lt h using 1] <;>
simp only [_root_.bit... | 1,102 |
import Mathlib.Data.Num.Lemmas
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.Ring
#align_import data.num.prime from "leanprover-community/mathlib"@"58581d0fe523063f5651df0619be2bf65012a94a"
namespace PosNum
def minFacAux (n : PosNum) : ℕ → PosNum → PosNum
| 0, _ => n
| fuel + 1, k =>
if n < k.bit1... | Mathlib/Data/Num/Prime.lean | 65 | 83 | theorem minFac_to_nat (n : PosNum) : (minFac n : ℕ) = Nat.minFac n := by |
cases' n with n
· rfl
· rw [minFac, Nat.minFac_eq, if_neg]
swap
· simp
rw [minFacAux_to_nat]
· rfl
simp only [cast_one, cast_bit1]
unfold _root_.bit1 _root_.bit0
rw [Nat.sqrt_lt]
calc
(n : ℕ) + (n : ℕ) + 1 ≤ (n : ℕ) + (n : ℕ) + (n : ℕ) := by simp
_ = (n : ℕ) * (1 + 1 +... | 1,102 |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
#align_import order.conditionally_complete_lattice.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Func... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 91 | 92 | theorem WithTop.iInf_empty [IsEmpty ι] [InfSet α] (f : ι → WithTop α) :
⨅ i, f i = ⊤ := by | rw [iInf, range_eq_empty, WithTop.sInf_empty]
| 1,103 |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
#align_import order.conditionally_complete_lattice.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Func... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 95 | 104 | theorem WithTop.coe_sInf' [InfSet α] {s : Set α} (hs : s.Nonempty) (h's : BddBelow s) :
↑(sInf s) = (sInf ((fun (a : α) ↦ ↑a) '' s) : WithTop α) := by |
obtain ⟨x, hx⟩ := hs
change _ = ite _ _ _
split_ifs with h
· rcases h with h1 | h2
· cases h1 (mem_image_of_mem _ hx)
· exact (h2 (Monotone.map_bddBelow coe_mono h's)).elim
· rw [preimage_image_eq]
exact Option.some_injective _
| 1,103 |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
#align_import order.conditionally_complete_lattice.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Func... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 110 | 113 | theorem WithTop.coe_iInf [Nonempty ι] [InfSet α] {f : ι → α} (hf : BddBelow (range f)) :
↑(⨅ i, f i) = (⨅ i, f i : WithTop α) := by |
rw [iInf, iInf, WithTop.coe_sInf' (range_nonempty f) hf, ← range_comp]
rfl
| 1,103 |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
#align_import order.conditionally_complete_lattice.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Func... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 116 | 121 | theorem WithTop.coe_sSup' [SupSet α] {s : Set α} (hs : BddAbove s) :
↑(sSup s) = (sSup ((fun (a : α) ↦ ↑a) '' s) : WithTop α) := by |
change _ = ite _ _ _
rw [if_neg, preimage_image_eq, if_pos hs]
· exact Option.some_injective _
· rintro ⟨x, _, ⟨⟩⟩
| 1,103 |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
#align_import order.conditionally_complete_lattice.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Func... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 127 | 129 | theorem WithTop.coe_iSup [SupSet α] (f : ι → α) (h : BddAbove (Set.range f)) :
↑(⨆ i, f i) = (⨆ i, f i : WithTop α) := by |
rw [iSup, iSup, WithTop.coe_sSup' h, ← range_comp]; rfl
| 1,103 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Int.LeastGreatest
#align_import data.int.conditionally_complete_order from "leanprover-community/mathlib"@"1e05171a5e8cf18d98d9cf7b207540acb044acae"
open Int
noncomputable section
open scoped Classical
instance instConditionallyComplet... | Mathlib/Data/Int/ConditionallyCompleteOrder.lean | 61 | 65 | theorem csSup_eq_greatest_of_bdd {s : Set ℤ} [DecidablePred (· ∈ s)] (b : ℤ) (Hb : ∀ z ∈ s, z ≤ b)
(Hinh : ∃ z : ℤ, z ∈ s) : sSup s = greatestOfBdd b Hb Hinh := by |
have : s.Nonempty ∧ BddAbove s := ⟨Hinh, b, Hb⟩
simp only [sSup, this, and_self, dite_true]
convert (coe_greatestOfBdd_eq Hb (Classical.choose_spec (⟨b, Hb⟩ : BddAbove s)) Hinh).symm
| 1,104 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Int.LeastGreatest
#align_import data.int.conditionally_complete_order from "leanprover-community/mathlib"@"1e05171a5e8cf18d98d9cf7b207540acb044acae"
open Int
noncomputable section
open scoped Classical
instance instConditionallyComplet... | Mathlib/Data/Int/ConditionallyCompleteOrder.lean | 78 | 82 | theorem csInf_eq_least_of_bdd {s : Set ℤ} [DecidablePred (· ∈ s)] (b : ℤ) (Hb : ∀ z ∈ s, b ≤ z)
(Hinh : ∃ z : ℤ, z ∈ s) : sInf s = leastOfBdd b Hb Hinh := by |
have : s.Nonempty ∧ BddBelow s := ⟨Hinh, b, Hb⟩
simp only [sInf, this, and_self, dite_true]
convert (coe_leastOfBdd_eq Hb (Classical.choose_spec (⟨b, Hb⟩ : BddBelow s)) Hinh).symm
| 1,104 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Int.LeastGreatest
#align_import data.int.conditionally_complete_order from "leanprover-community/mathlib"@"1e05171a5e8cf18d98d9cf7b207540acb044acae"
open Int
noncomputable section
open scoped Classical
instance instConditionallyComplet... | Mathlib/Data/Int/ConditionallyCompleteOrder.lean | 94 | 96 | theorem csSup_mem {s : Set ℤ} (h1 : s.Nonempty) (h2 : BddAbove s) : sSup s ∈ s := by |
convert (greatestOfBdd _ (Classical.choose_spec h2) h1).2.1
exact dif_pos ⟨h1, h2⟩
| 1,104 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.