Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.57k | proof stringlengths 5 7.36k | hint bool 2
classes |
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import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.locally_convex.continuous_of_bounded from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open TopologicalSpace Bornology Filter Topology Pointwise
variable {𝕜 𝕜' E F : Type*}
var... | Mathlib/Analysis/LocallyConvex/ContinuousOfBounded.lean | 96 | 166 | theorem LinearMap.continuousAt_zero_of_locally_bounded (f : E →ₛₗ[σ] F)
(hf : ∀ s, IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (f '' s)) : ContinuousAt f 0 := by
-- Assume that f is not continuous at 0 |
-- Assume that f is not continuous at 0
by_contra h
-- We use a decreasing balanced basis for 0 : E and a balanced basis for 0 : F
-- and reformulate non-continuity in terms of these bases
rcases (nhds_basis_balanced 𝕜 E).exists_antitone_subbasis with ⟨b, bE1, bE⟩
simp only [_root_.id] at bE
have bE' : ... | true |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 85 | 89 | theorem incMatrix_apply_mul_incMatrix_apply_of_not_adj (hab : a ≠ b) (h : ¬G.Adj a b) :
G.incMatrix R a e * G.incMatrix R b e = 0 := by
rw [incMatrix_apply_mul_incMatrix_apply, Set.indicator_of_not_mem] |
rw [incMatrix_apply_mul_incMatrix_apply, Set.indicator_of_not_mem]
rw [G.incidenceSet_inter_incidenceSet_of_not_adj h hab]
exact Set.not_mem_empty e
| true |
import Mathlib.Order.Filter.AtTopBot
#align_import order.filter.indicator_function from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
variable {α β M E : Type*}
open Set Filter
@[to_additive]
theorem Monotone.mulIndicator_eventuallyEq_iUnion {ι} [Preorder ι] [One β] (s : ι → Set α)
... | Mathlib/Order/Filter/IndicatorFunction.lean | 89 | 94 | theorem mulIndicator_biUnion_finset_eventuallyEq {ι} [One β] (s : ι → Set α) (f : α → β) (a : α) :
(fun n : Finset ι => mulIndicator (⋃ i ∈ n, s i) f a) =ᶠ[atTop]
fun _ ↦ mulIndicator (iUnion s) f a := by
rw [iUnion_eq_iUnion_finset s] |
rw [iUnion_eq_iUnion_finset s]
apply Monotone.mulIndicator_eventuallyEq_iUnion
exact fun _ _ ↦ biUnion_subset_biUnion_left
| true |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 183 | 194 | theorem smul_Ioo : r • Ioo a b = Ioo (r • a) (r • b) := by
ext x |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Ioo]
constructor
· rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩
constructor
· exact (mul_lt_mul_left hr).mpr a_h_left_left
· exact (mul_lt_mul_left hr).mpr a_h_left_right
· rintro ⟨a_left, a_right⟩
use x / r
refine ⟨⟨(lt_div_iff' hr).mpr... | true |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Data.Fintype.Order
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.Analysis.NormedSpace.WithLp
#align_import analysis.normed_space.pi_Lp from "leanprover-community/mathlib"@"9d013ad8430ddddd350cff5c3db830278ded3c79"
set_option linter.uppercaseLean3 f... | Mathlib/Analysis/NormedSpace/PiLp.lean | 247 | 249 | theorem dist_eq_iSup (f g : PiLp ∞ α) : dist f g = ⨆ i, dist (f i) (g i) := by
dsimp [dist] |
dsimp [dist]
exact if_neg ENNReal.top_ne_zero
| true |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefi... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 88 | 93 | theorem length_mul_left_ne (w : W) : ℓ (w * t) ≠ ℓ w := by
suffices cs.lengthParity (w * t) ≠ cs.lengthParity w by |
suffices cs.lengthParity (w * t) ≠ cs.lengthParity w by
contrapose! this
simp only [lengthParity_eq_ofAdd_length, this]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
| true |
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import linear_algebra.exterior_algebra.grading from "leanprover-community/mathlib"@"34020e531ebc4e8aac6d449d9eecbcd1508ea8d0"
namespace ExteriorAlgebra
variable {R M : Type*} [CommRing R] [AddCommGroup M] [Modu... | Mathlib/LinearAlgebra/ExteriorAlgebra/Grading.lean | 64 | 80 | theorem GradedAlgebra.liftι_eq (i : ℕ) (x : ⋀[R]^i M) :
GradedAlgebra.liftι R M x = DirectSum.of (fun i => ⋀[R]^i M) i x := by
cases' x with x hx |
cases' x with x hx
dsimp only [Subtype.coe_mk, DirectSum.lof_eq_of]
-- Porting note: original statement was
-- refine Submodule.pow_induction_on_left' _ (fun r => ?_) (fun x y i hx hy ihx ihy => ?_)
-- (fun m hm i x hx ih => ?_) hx
-- but it created invalid goals
induction hx using Submodule.pow_indu... | true |
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.Geometry.RingedSpace.SheafedSpace
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Limits
#align_import algebraic_geometry.ringed_space from "leanprover-community/mathlib"@"5dc... | Mathlib/Geometry/RingedSpace/Basic.lean | 58 | 79 | theorem isUnit_res_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U)) (x : U)
(h : IsUnit (X.presheaf.germ x f)) :
∃ (V : Opens X) (i : V ⟶ U) (_ : x.1 ∈ V), IsUnit (X.presheaf.map i.op f) := by
obtain ⟨g', heq⟩ := h.exists_right_inv |
obtain ⟨g', heq⟩ := h.exists_right_inv
obtain ⟨V, hxV, g, rfl⟩ := X.presheaf.germ_exist x.1 g'
let W := U ⊓ V
have hxW : x.1 ∈ W := ⟨x.2, hxV⟩
-- Porting note: `erw` can't write into `HEq`, so this is replaced with another `HEq` in the
-- desired form
replace heq : (X.presheaf.germ ⟨x.val, hxW⟩) ((X.pres... | true |
import Mathlib.Algebra.Module.Submodule.Ker
open Function Submodule
namespace LinearMap
variable {R N M : Type*} [Semiring R] [AddCommMonoid N] [Module R N]
[AddCommMonoid M] [Module R M] (f i : N →ₗ[R] M)
def iterateMapComap (n : ℕ) := (fun K : Submodule R N ↦ (K.map i).comap f)^[n]
theorem iterateMapComap... | Mathlib/Algebra/Module/Submodule/IterateMapComap.lean | 65 | 79 | theorem iterateMapComap_eq_succ (K : Submodule R N)
(m : ℕ) (heq : f.iterateMapComap i m K = f.iterateMapComap i (m + 1) K)
(hf : Surjective f) (hi : Injective i) (n : ℕ) :
f.iterateMapComap i n K = f.iterateMapComap i (n + 1) K := by
induction n with |
induction n with
| zero =>
contrapose! heq
induction m with
| zero => exact heq
| succ m ih =>
rw [iterateMapComap, iterateMapComap, iterate_succ', iterate_succ']
exact fun H ↦ ih (map_injective_of_injective hi (comap_injective_of_surjective hf H))
| succ n ih =>
rw [iterateMapCom... | true |
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.Equiv
#align_import topology.uniform_space.abstract_completion from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
attribute [local instance] Classical.propDecidable
open F... | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | 158 | 161 | theorem extend_unique (hf : UniformContinuous f) {g : hatα → β} (hg : UniformContinuous g)
(h : ∀ a : α, f a = g (ι a)) : pkg.extend f = g := by
apply pkg.funext pkg.continuous_extend hg.continuous |
apply pkg.funext pkg.continuous_extend hg.continuous
simpa only [pkg.extend_coe hf] using h
| true |
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α : Type*} (p : α → Bool) (l : List α) (n : ℕ)
namespace List
def rdrop : List α :=
l.take (l.leng... | Mathlib/Data/List/DropRight.lean | 166 | 174 | theorem rdropWhile_eq_self_iff : rdropWhile p l = l ↔ ∀ hl : l ≠ [], ¬p (l.getLast hl) := by
simp only [rdropWhile, reverse_eq_iff, dropWhile_eq_self_iff, getLast_eq_get] |
simp only [rdropWhile, reverse_eq_iff, dropWhile_eq_self_iff, getLast_eq_get]
refine ⟨fun h hl => ?_, fun h hl => ?_⟩
· rw [← length_pos, ← length_reverse] at hl
have := h hl
rwa [get_reverse'] at this
· rw [length_reverse, length_pos] at hl
have := h hl
rwa [get_reverse']
| true |
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
@[simp]
theorem reduceOption_cons_of_some (x : α) (l : List (Option α)) :
reduceOption (some x :: l) = x :: l.reduceOption := by
simp only [reduceOption, filterMap, id, eq_self_iff_true, and_self_iff]
#align list.reduce_option_cons_of_some... | Mathlib/Data/List/ReduceOption.lean | 59 | 61 | theorem reduceOption_length_le (l : List (Option α)) : l.reduceOption.length ≤ l.length := by
rw [length_eq_reduceOption_length_add_filter_none] |
rw [length_eq_reduceOption_length_add_filter_none]
apply Nat.le_add_right
| true |
import Mathlib.Algebra.Lie.Matrix
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.Tactic.NoncommRing
#align_import algebra.lie.skew_adjoint from "leanprover-community/mathlib"@"075b3f7d19b9da85a0b54b3e33055a74fc388dec"
universe u v w w₁
section SkewAdjointMatrices
open scoped Matrix
variabl... | Mathlib/Algebra/Lie/SkewAdjoint.lean | 103 | 112 | theorem Matrix.isSkewAdjoint_bracket {A B : Matrix n n R} (hA : A ∈ skewAdjointMatricesSubmodule J)
(hB : B ∈ skewAdjointMatricesSubmodule J) : ⁅A, B⁆ ∈ skewAdjointMatricesSubmodule J := by
simp only [mem_skewAdjointMatricesSubmodule] at * |
simp only [mem_skewAdjointMatricesSubmodule] at *
change ⁅A, B⁆ᵀ * J = J * (-⁅A, B⁆)
change Aᵀ * J = J * (-A) at hA
change Bᵀ * J = J * (-B) at hB
rw [Matrix.lie_transpose, LieRing.of_associative_ring_bracket,
LieRing.of_associative_ring_bracket, sub_mul, mul_assoc, mul_assoc, hA, hB, ← mul_assoc,
← ... | true |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toNNReal_iInf (hf : ∀ i, f ... | Mathlib/Data/ENNReal/Real.lean | 581 | 582 | theorem toReal_iSup (hf : ∀ i, f i ≠ ∞) : (iSup f).toReal = ⨆ i, (f i).toReal := by |
simp only [ENNReal.toReal, toNNReal_iSup hf, NNReal.coe_iSup]
| true |
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.LinearAlgebra.TensorProduct.Graded.Internal
import Mathlib.LinearAlgebra.QuadraticForm.Prod
suppress_compilation
variable {R M₁ M₂ N : Type*}
variable [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup N]
variable [Module R M₁] [Module... | Mathlib/LinearAlgebra/CliffordAlgebra/Prod.lean | 101 | 104 | theorem map_mul_map_eq_neg_of_isOrtho_of_mem_evenOdd_one
(hm₁ : m₁ ∈ evenOdd Q₁ 1) (hm₂ : m₂ ∈ evenOdd Q₂ 1) :
map f₁ m₁ * map f₂ m₂ = - map f₂ m₂ * map f₁ m₁ := by |
simp [map_mul_map_of_isOrtho_of_mem_evenOdd _ _ hf _ _ hm₁ hm₂]
| true |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
import Mathlib.Tactic.FieldSimp
#align_import number_theory.bernoulli from "leanprover-community/mat... | Mathlib/NumberTheory/Bernoulli.lean | 78 | 80 | theorem bernoulli'_def (n : ℕ) :
bernoulli' n = 1 - ∑ k ∈ range n, n.choose k / (n - k + 1) * bernoulli' k := by |
rw [bernoulli'_def', ← Fin.sum_univ_eq_sum_range]
| true |
import Mathlib.Data.PFunctor.Multivariate.Basic
#align_import data.qpf.multivariate.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u
open MvFunctor
class MvQPF {n : ℕ} (F : TypeVec.{u} n → Type*) [MvFunctor F] where
P : MvPFunctor.{u} n
abs : ∀ {α}, P α → F α
... | Mathlib/Data/QPF/Multivariate/Basic.lean | 164 | 177 | theorem mem_supp {α : TypeVec n} (x : F α) (i) (u : α i) :
u ∈ supp x i ↔ ∀ a f, abs ⟨a, f⟩ = x → u ∈ f i '' univ := by
rw [supp]; dsimp; constructor |
rw [supp]; dsimp; constructor
· intro h a f haf
have : LiftP (fun i u => u ∈ f i '' univ) x := by
rw [liftP_iff]
refine ⟨a, f, haf.symm, ?_⟩
intro i u
exact mem_image_of_mem _ (mem_univ _)
exact h this
intro h p; rw [liftP_iff]
rintro ⟨a, f, xeq, h'⟩
rcases h a f xeq.symm with... | true |
import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import number_theory.number_field.units from "leanprover-community/mathlib"@"00f91228655eecdcd3ac... | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 100 | 106 | theorem mult_log_place_eq_zero {x : (𝓞 K)ˣ} {w : InfinitePlace K} :
mult w * Real.log (w x) = 0 ↔ w x = 1 := by
rw [mul_eq_zero, or_iff_right, Real.log_eq_zero, or_iff_right, or_iff_left] |
rw [mul_eq_zero, or_iff_right, Real.log_eq_zero, or_iff_right, or_iff_left]
· linarith [(apply_nonneg _ _ : 0 ≤ w x)]
· simp only [ne_eq, map_eq_zero, coe_ne_zero x, not_false_eq_true]
· refine (ne_of_gt ?_)
rw [mult]; split_ifs <;> norm_num
| true |
import Mathlib.Topology.Connected.Basic
open Set Function
universe u v
variable {α : Type u} {β : Type v} {ι : Type*} {π : ι → Type*} [TopologicalSpace α]
{s t u v : Set α}
section TotallyDisconnected
def IsTotallyDisconnected (s : Set α) : Prop :=
∀ t, t ⊆ s → IsPreconnected t → t.Subsingleton
#align is_t... | Mathlib/Topology/Connected/TotallyDisconnected.lean | 123 | 128 | theorem totallyDisconnectedSpace_iff_connectedComponent_singleton :
TotallyDisconnectedSpace α ↔ ∀ x : α, connectedComponent x = {x} := by
rw [totallyDisconnectedSpace_iff_connectedComponent_subsingleton] |
rw [totallyDisconnectedSpace_iff_connectedComponent_subsingleton]
refine forall_congr' fun x => ?_
rw [subsingleton_iff_singleton]
exact mem_connectedComponent
| true |
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68ab8cf7dd83d549eb78a68a1"
open Finset
variable {α : Type*}
theorem sup_sdiff_injOn [GeneralizedBooleanAlgebra α] (u v : α) :
{ x | Disjoint u x ∧ v ≤ x }.... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 156 | 158 | theorem mem_compression :
a ∈ 𝓒 u v s ↔ a ∈ s ∧ compress u v a ∈ s ∨ a ∉ s ∧ ∃ b ∈ s, compress u v b = a := by |
simp_rw [compression, mem_union, mem_filter, mem_image, and_comm]
| true |
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 104 | 107 | theorem diff_mem_nhdsWithin_diff {x : α} {s t : Set α} (hs : s ∈ 𝓝[t] x) (t' : Set α) :
s \ t' ∈ 𝓝[t \ t'] x := by
rw [nhdsWithin, diff_eq, diff_eq, ← inf_principal, ← inf_assoc] |
rw [nhdsWithin, diff_eq, diff_eq, ← inf_principal, ← inf_assoc]
exact inter_mem_inf hs (mem_principal_self _)
| true |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 129 | 131 | theorem neg_one_lt_goldConj : -1 < ψ := by
rw [neg_lt, ← inv_gold] |
rw [neg_lt, ← inv_gold]
exact inv_lt_one one_lt_gold
| true |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
var... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 108 | 109 | theorem rpow_inv_rpow_self {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0) : (x ^ y) ^ (1 / y) = x := by |
field_simp [← rpow_mul]
| true |
import Mathlib.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} {a : α}
open Finset
-- The namespace is here to distinguish fro... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 258 | 261 | theorem erase_mem_compression_of_mem_compression : s ∈ 𝓓 a 𝒜 → s.erase a ∈ 𝓓 a 𝒜 := by
simp_rw [mem_compression, erase_idem] |
simp_rw [mem_compression, erase_idem]
refine Or.imp (fun h => ⟨h.2, h.2⟩) fun h => ?_
rwa [erase_eq_of_not_mem (insert_ne_self.1 <| ne_of_mem_of_not_mem h.2 h.1)]
| true |
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
import Mathlib.Order.PartialSups
#align_import linear_algebra.prod from "leanprover-community/mathlib"@"cd391184c85986113f8c00844cfe6dda1d34be3d"
universe u v w x y z u' v' w' y'
variable {R : Type u} {K : Ty... | Mathlib/LinearAlgebra/Prod.lean | 148 | 155 | theorem range_inl : range (inl R M M₂) = ker (snd R M M₂) := by
ext x |
ext x
simp only [mem_ker, mem_range]
constructor
· rintro ⟨y, rfl⟩
rfl
· intro h
exact ⟨x.fst, Prod.ext rfl h.symm⟩
| true |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 167 | 169 | theorem csInf_div (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t) :
sInf (s / t) = sInf s / sSup t := by |
rw [div_eq_mul_inv, csInf_mul hs₀ hs₁ ht₀.inv ht₁.inv, csInf_inv ht₀ ht₁, div_eq_mul_inv]
| true |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Instances.Irrational
#align_import number_theory.liouville.liouville_with from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Filter Metric Real Set
open sc... | Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | 99 | 110 | theorem frequently_lt_rpow_neg (h : LiouvilleWith p x) (hlt : q < p) :
∃ᶠ n : ℕ in atTop, ∃ m : ℤ, x ≠ m / n ∧ |x - m / n| < n ^ (-q) := by
rcases h.exists_pos with ⟨C, _hC₀, hC⟩ |
rcases h.exists_pos with ⟨C, _hC₀, hC⟩
have : ∀ᶠ n : ℕ in atTop, C < n ^ (p - q) := by
simpa only [(· ∘ ·), neg_sub, one_div] using
((tendsto_rpow_atTop (sub_pos.2 hlt)).comp tendsto_natCast_atTop_atTop).eventually
(eventually_gt_atTop C)
refine (this.and_frequently hC).mono ?_
rintro n ⟨hnC,... | true |
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.BoundedOrder
import Mathlib.Mathport.Notation
import Mathlib.Data.Sigma.Basic
#align_import data.sigma.order from "leanprover-community/mathlib"@"1fc36cc9c8264e6e81253f88be7fb2cb6c92d76a"
namespace Sigma
variable {ι : Type*} {α : ι → Type*}
-- Porting note: I... | Mathlib/Data/Sigma/Order.lean | 89 | 96 | theorem lt_def [∀ i, LT (α i)] {a b : Σi, α i} : a < b ↔ ∃ h : a.1 = b.1, h.rec a.2 < b.2 := by
constructor |
constructor
· rintro ⟨i, a, b, h⟩
exact ⟨rfl, h⟩
· obtain ⟨i, a⟩ := a
obtain ⟨j, b⟩ := b
rintro ⟨rfl : i = j, h⟩
exact lt.fiber _ _ _ h
| true |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace List
variable [D... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 58 | 70 | theorem formPerm_disjoint_iff (hl : Nodup l) (hl' : Nodup l') (hn : 2 ≤ l.length)
(hn' : 2 ≤ l'.length) : Perm.Disjoint (formPerm l) (formPerm l') ↔ l.Disjoint l' := by
rw [disjoint_iff_eq_or_eq, List.Disjoint] |
rw [disjoint_iff_eq_or_eq, List.Disjoint]
constructor
· rintro h x hx hx'
specialize h x
rw [formPerm_apply_mem_eq_self_iff _ hl _ hx, formPerm_apply_mem_eq_self_iff _ hl' _ hx'] at h
omega
· intro h x
by_cases hx : x ∈ l
on_goal 1 => by_cases hx' : x ∈ l'
· exact (h hx hx').elim
al... | true |
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import number_theory.von_mangoldt from "leanprover-community/mathlib"@"c946d6097a6925ad16d7ec55677bbc977f9846de"
namespace ArithmeticFunction
open Finset Nat
open scoped Arit... | Mathlib/NumberTheory/VonMangoldt.lean | 79 | 79 | theorem vonMangoldt_apply_one : Λ 1 = 0 := by | simp [vonMangoldt_apply]
| true |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.between from "leanprover-community/mathlib"@"571e13cacbed7bf042fd3058c... | Mathlib/Analysis/Convex/Between.lean | 133 | 135 | theorem mem_vsub_const_affineSegment {x y z : P} (p : P) :
z -ᵥ p ∈ affineSegment R (x -ᵥ p) (y -ᵥ p) ↔ z ∈ affineSegment R x y := by |
rw [← affineSegment_vsub_const_image, (vsub_left_injective p).mem_set_image]
| true |
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 |
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_... | true |
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Complex.RemovableSingularity
#align_import analysis.complex.schwarz from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric Set Function Filter TopologicalSpace
open scoped Topology
namespace Complex
section Space... | Mathlib/Analysis/Complex/Schwarz.lean | 65 | 88 | theorem schwarz_aux {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball c R₁))
(h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
‖dslope f c z‖ ≤ R₂ / R₁ := by
have hR₁ : 0 < R₁ := nonempty_ball.1 ⟨z, hz⟩ |
have hR₁ : 0 < R₁ := nonempty_ball.1 ⟨z, hz⟩
suffices ∀ᶠ r in 𝓝[<] R₁, ‖dslope f c z‖ ≤ R₂ / r by
refine ge_of_tendsto ?_ this
exact (tendsto_const_nhds.div tendsto_id hR₁.ne').mono_left nhdsWithin_le_nhds
rw [mem_ball] at hz
filter_upwards [Ioo_mem_nhdsWithin_Iio ⟨hz, le_rfl⟩] with r hr
have hr₀ : ... | true |
import Mathlib.MeasureTheory.SetSemiring
open MeasurableSpace Set
namespace MeasureTheory
variable {α : Type*} {𝒜 : Set (Set α)} {s t : Set α}
structure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where
empty_mem : ∅ ∈ 𝒜
compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜
union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜
... | Mathlib/MeasureTheory/SetAlgebra.lean | 122 | 134 | theorem generateFrom_generateSetAlgebra_eq :
generateFrom (generateSetAlgebra 𝒜) = generateFrom 𝒜 := by
refine le_antisymm (fun s ms ↦ ?_) (generateFrom_mono self_subset_generateSetAlgebra) |
refine le_antisymm (fun s ms ↦ ?_) (generateFrom_mono self_subset_generateSetAlgebra)
refine @generateFrom_induction _ _ (generateSetAlgebra 𝒜) (fun t ht ↦ ?_)
(@MeasurableSet.empty _ (generateFrom 𝒜))
(fun t ↦ MeasurableSet.compl)
(fun f hf ↦ MeasurableSet.iUnion hf)
s ms
induction ht with
|... | true |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.RingTheory.Polynomial.Nilpotent
#align_import linear_algebra.matrix.charpoly.coeff from "leanprover-community/mathlib"@"9745b... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 260 | 276 | theorem matPolyEquiv_eq_X_pow_sub_C {K : Type*} (k : ℕ) [Field K] (M : Matrix n n K) :
matPolyEquiv ((expand K k : K[X] →+* K[X]).mapMatrix (charmatrix (M ^ k))) =
X ^ k - C (M ^ k) := by
-- Porting note: `i` and `j` are used later on, but were not mentioned in mathlib3 |
-- Porting note: `i` and `j` are used later on, but were not mentioned in mathlib3
ext m i j
rw [coeff_sub, coeff_C, matPolyEquiv_coeff_apply, RingHom.mapMatrix_apply, Matrix.map_apply,
AlgHom.coe_toRingHom, DMatrix.sub_apply, coeff_X_pow]
by_cases hij : i = j
· rw [hij, charmatrix_apply_eq, AlgHom.map_s... | true |
import Mathlib.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Function Set Filter Topology
universe u v w
namespace TopologicalSpace
variable {α : Type u}
inductive GenerateOpen (g : Set (Set ... | Mathlib/Topology/Order.lean | 129 | 138 | theorem nhds_mkOfNhds_single [DecidableEq α] {a₀ : α} {l : Filter α} (h : pure a₀ ≤ l) (b : α) :
@nhds α (TopologicalSpace.mkOfNhds (update pure a₀ l)) b =
(update pure a₀ l : α → Filter α) b := by
refine nhds_mkOfNhds _ _ (le_update_iff.mpr ⟨h, fun _ _ => le_rfl⟩) fun a s hs => ?_ |
refine nhds_mkOfNhds _ _ (le_update_iff.mpr ⟨h, fun _ _ => le_rfl⟩) fun a s hs => ?_
rcases eq_or_ne a a₀ with (rfl | ha)
· filter_upwards [hs] with b hb
rcases eq_or_ne b a with (rfl | hb)
· exact hs
· rwa [update_noteq hb]
· simpa only [update_noteq ha, mem_pure, eventually_pure] using hs
| true |
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.Polynomial.Eval
#align_import data.mv_polynomial.polynomial from "leanprover-community/mathlib"@"0b89934139d3be96f9dab477f10c20f9f93da580"
namespace MvPolynomial
variable {R S σ : Type*}
theorem polynomial_eval_eval₂ [CommSemiring R] [CommSemiring ... | Mathlib/Algebra/MvPolynomial/Polynomial.lean | 30 | 40 | theorem eval_polynomial_eval_finSuccEquiv {n : ℕ} {x : Fin n → R}
[CommSemiring R] (f : MvPolynomial (Fin (n + 1)) R) (q : MvPolynomial (Fin n) R) :
(eval x) (Polynomial.eval q (finSuccEquiv R n f)) = eval (Fin.cases (eval x q) x) f := by
simp only [finSuccEquiv_apply, coe_eval₂Hom, polynomial_eval_eval₂, eva... |
simp only [finSuccEquiv_apply, coe_eval₂Hom, polynomial_eval_eval₂, eval_eval₂]
conv in RingHom.comp _ _ =>
refine @RingHom.ext _ _ _ _ _ (RingHom.id _) fun r => ?_
simp
simp only [eval₂_id]
congr
funext i
refine Fin.cases (by simp) (by simp) i
| true |
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.FreeAlgebra
import Mathlib.RingTheory.Localization.FractionRing
#align_import algebra.char_p.algebra from "leanprover-community/mathlib"@"96782a2d6dcded92116d8ac9ae48efb41d46a27c"
theorem charP_of_injective_ringHom {R A : Type*} [NonAssocSemiring R] [NonAs... | Mathlib/Algebra/CharP/Algebra.lean | 64 | 67 | theorem RingHom.charP {R A : Type*} [NonAssocSemiring R] [NonAssocSemiring A] (f : R →+* A)
(H : Function.Injective f) (p : ℕ) [CharP A p] : CharP R p := by
obtain ⟨q, h⟩ := CharP.exists R |
obtain ⟨q, h⟩ := CharP.exists R
exact CharP.eq _ (charP_of_injective_ringHom H q) ‹CharP A p› ▸ h
| true |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Common
#align_import order.heyting.boundary from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
variable {α : Type*}
namespace Coheyting
variable [CoheytingAlgebra α] {a b : α}
def boundary (a : α) : α :=
a ⊓ ￢a
#align cohe... | Mathlib/Order/Heyting/Boundary.lean | 76 | 76 | theorem hnot_boundary (a : α) : ￢∂ a = ⊤ := by | rw [boundary, hnot_inf_distrib, sup_hnot_self]
| true |
import Mathlib.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
#align_import ring_theory.witt_vector.structure_polynomial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open MvPolynomial Set
open Finset (range)
o... | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 140 | 148 | theorem wittStructureRat_prop (Φ : MvPolynomial idx ℚ) (n : ℕ) :
bind₁ (wittStructureRat p Φ) (W_ ℚ n) = bind₁ (fun i => rename (Prod.mk i) (W_ ℚ n)) Φ :=
calc
bind₁ (wittStructureRat p Φ) (W_ ℚ n) =
bind₁ (fun k => bind₁ (fun i => (rename (Prod.mk i)) (W_ ℚ k)) Φ)
(bind₁ (xInTermsOfW p ℚ) (... |
rw [bind₁_bind₁]; exact eval₂Hom_congr (RingHom.ext_rat _ _) rfl rfl
_ = bind₁ (fun i => rename (Prod.mk i) (W_ ℚ n)) Φ := by
rw [bind₁_xInTermsOfW_wittPolynomial p _ n, bind₁_X_right]
| true |
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
namespace PowerSeries
open Polynomial Derivation Nat
section CommutativeSemiring
variable {R} [CommSemiring R]
noncomputable def derivativeFun (f : R⟦X⟧) : R⟦X⟧ := mk fun n ↦ coef... | Mathlib/RingTheory/PowerSeries/Derivative.lean | 90 | 92 | theorem derivativeFun_smul (r : R) (f : R⟦X⟧) : derivativeFun (r • f) = r • derivativeFun f := by
rw [smul_eq_C_mul, smul_eq_C_mul, derivativeFun_mul, derivativeFun_C, smul_zero, add_zero, |
rw [smul_eq_C_mul, smul_eq_C_mul, derivativeFun_mul, derivativeFun_C, smul_zero, add_zero,
smul_eq_mul]
| true |
import Mathlib.MeasureTheory.Group.Arithmetic
#align_import measure_theory.group.pointwise from "leanprover-community/mathlib"@"66f7114a1d5cba41c47d417a034bbb2e96cf564a"
open Pointwise
open Set
@[to_additive]
theorem MeasurableSet.const_smul {G α : Type*} [Group G] [MulAction G α] [MeasurableSpace G]
[Measu... | Mathlib/MeasureTheory/Group/Pointwise.lean | 32 | 36 | theorem MeasurableSet.const_smul_of_ne_zero {G₀ α : Type*} [GroupWithZero G₀] [MulAction G₀ α]
[MeasurableSpace G₀] [MeasurableSpace α] [MeasurableSMul G₀ α] {s : Set α}
(hs : MeasurableSet s) {a : G₀} (ha : a ≠ 0) : MeasurableSet (a • s) := by
rw [← preimage_smul_inv₀ ha] |
rw [← preimage_smul_inv₀ ha]
exact measurable_const_smul _ hs
| true |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RCLike
open scoped ComplexConjugate
variable {𝕜 E F G : Type... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 144 | 147 | theorem apply_norm_sq_eq_inner_adjoint_left (A : E →L[𝕜] F) (x : E) :
‖A x‖ ^ 2 = re ⟪(A† ∘L A) x, x⟫ := by
have h : ⟪(A† ∘L A) x, x⟫ = ⟪A x, A x⟫ := by rw [← adjoint_inner_left]; rfl |
have h : ⟪(A† ∘L A) x, x⟫ = ⟪A x, A x⟫ := by rw [← adjoint_inner_left]; rfl
rw [h, ← inner_self_eq_norm_sq (𝕜 := 𝕜) _]
| true |
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Analysis.Calculus.FDeriv.Extend
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 305 | 313 | theorem contDiffAt_rpow_of_ne (p : ℝ × ℝ) (hp : p.1 ≠ 0) {n : ℕ∞} :
ContDiffAt ℝ n (fun p : ℝ × ℝ => p.1 ^ p.2) p := by
cases' hp.lt_or_lt with hneg hpos |
cases' hp.lt_or_lt with hneg hpos
exacts
[(((contDiffAt_fst.log hneg.ne).mul contDiffAt_snd).exp.mul
(contDiffAt_snd.mul contDiffAt_const).cos).congr_of_eventuallyEq
((continuousAt_fst.eventually (gt_mem_nhds hneg)).mono fun p hp => rpow_def_of_neg hp _),
((contDiffAt_fst.log hpos.ne').mul ... | true |
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Asymptotics.SuperpolynomialDecay
#align_import analysis.complex.phragmen_lindelof from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter Asymptotics Metric Complex Bornology
open scoped Topology Filter R... | Mathlib/Analysis/Complex/PhragmenLindelof.lean | 115 | 221 | theorem horizontal_strip (hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo a b))
(hB : ∃ c < π / (b - a), ∃ B, f =O[comap (_root_.abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b)]
fun z ↦ expR (B * expR (c * |z.re|)))
(hle_a : ∀ z : ℂ, im z = a → ‖f z‖ ≤ C) (hle_b : ∀ z, im z = b → ‖f z‖ ≤ C) (hza : a ≤ im z)
(hzb : im z ≤ b)... |
-- If `im z = a` or `im z = b`, then we apply `hle_a` or `hle_b`, otherwise `im z ∈ Ioo a b`.
rw [le_iff_eq_or_lt] at hza hzb
cases' hza with hza hza; · exact hle_a _ hza.symm
cases' hzb with hzb hzb; · exact hle_b _ hzb
wlog hC₀ : 0 < C generalizing C
· refine le_of_forall_le_of_dense fun C' hC' => this (... | true |
import Mathlib.NumberTheory.Cyclotomic.Embeddings
import Mathlib.NumberTheory.Cyclotomic.Rat
import Mathlib.NumberTheory.NumberField.Units.DirichletTheorem
open NumberField Units InfinitePlace nonZeroDivisors Polynomial
namespace IsCyclotomicExtension.Rat.Three
variable {K : Type*} [Field K] [NumberField K] [IsC... | Mathlib/NumberTheory/Cyclotomic/Three.lean | 85 | 111 | theorem eq_one_or_neg_one_of_unit_of_congruent (hcong : ∃ n : ℤ, λ ^ 2 ∣ (u - n : 𝓞 K)) :
u = 1 ∨ u = -1 := by
replace hcong : ∃ n : ℤ, (3 : 𝓞 K) ∣ (↑u - n : 𝓞 K) := by |
replace hcong : ∃ n : ℤ, (3 : 𝓞 K) ∣ (↑u - n : 𝓞 K) := by
obtain ⟨n, x, hx⟩ := hcong
exact ⟨n, -η * x, by rw [← mul_assoc, mul_neg, ← neg_mul, ← lambda_sq, hx]⟩
have hζ := IsCyclotomicExtension.zeta_spec 3 ℚ K
have := Units.mem hζ u
fin_cases this
· left; rfl
· right; rfl
all_goals exfalso
· ... | true |
import Mathlib.Topology.Basic
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.OmegaCompletePartialOrder
#align_import topology.omega_complete_partial_order from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
open Set OmegaCompletePartialOrder
open scoped Classical
universe ... | Mathlib/Topology/OmegaCompletePartialOrder.lean | 41 | 43 | theorem isωSup_iff_isLUB {α : Type u} [Preorder α] {c : Chain α} {x : α} :
IsωSup c x ↔ IsLUB (range c) x := by |
simp [IsωSup, IsLUB, IsLeast, upperBounds, lowerBounds]
| true |
import Mathlib.Algebra.CharP.Defs
#align_import algebra.char_p.invertible from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable {K : Type*}
section Field
variable [Field K]
def invertibleOfRingCharNotDvd {t : ℕ} (not_dvd : ¬ringChar K ∣ t) : Invertible (t : K) :=
invertible... | Mathlib/Algebra/CharP/Invertible.lean | 32 | 34 | theorem not_ringChar_dvd_of_invertible {t : ℕ} [Invertible (t : K)] : ¬ringChar K ∣ t := by
rw [← ringChar.spec, ← Ne] |
rw [← ringChar.spec, ← Ne]
exact nonzero_of_invertible (t : K)
| true |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Instances.Irrational
#align_import number_theory.liouville.liouville_with from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Filter Metric Real Set
open sc... | Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | 89 | 94 | theorem mono (h : LiouvilleWith p x) (hle : q ≤ p) : LiouvilleWith q x := by
rcases h.exists_pos with ⟨C, hC₀, hC⟩ |
rcases h.exists_pos with ⟨C, hC₀, hC⟩
refine ⟨C, hC.mono ?_⟩; rintro n ⟨hn, m, hne, hlt⟩
refine ⟨m, hne, hlt.trans_le <| ?_⟩
gcongr
exact_mod_cast hn
| true |
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {α β γ : Type*}
class Fi... | Mathlib/Data/Fintype/Basic.lean | 178 | 178 | theorem not_mem_compl : a ∉ sᶜ ↔ a ∈ s := by | rw [mem_compl, not_not]
| true |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 384 | 384 | theorem graph_id : graph id = @Eq α := by | simp (config := { unfoldPartialApp := true }) [graph]
| true |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 123 | 126 | theorem primeSpectrumProd_symm_inr_asIdeal (x : PrimeSpectrum S) :
((primeSpectrumProd R S).symm <| Sum.inr x).asIdeal = Ideal.prod ⊤ x.asIdeal := by
cases x |
cases x
rfl
| true |
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.EulerProduct.Basic
import Mathlib.NumberTheory.LSeries.Basic
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex
variable {s : ℂ}
noncomputable
def riemannZetaSummandHom (hs : s ≠ 0) : ℕ →*₀ ℂ where
toFun n := (n : ℂ) ^ ... | Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | 104 | 108 | theorem riemannZeta_eulerProduct (hs : 1 < s.re) :
Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, (1 - (p : ℂ) ^ (-s))⁻¹) atTop
(𝓝 (riemannZeta s)) := by
rw [← tsum_riemannZetaSummand hs] |
rw [← tsum_riemannZetaSummand hs]
apply eulerProduct_completely_multiplicative <| summable_riemannZetaSummand hs
| true |
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.bezout from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
universe u v
variable {R : Type u} [CommRing R]
namespace IsBezout
| Mathlib/RingTheory/Bezout.lean | 30 | 39 | theorem iff_span_pair_isPrincipal :
IsBezout R ↔ ∀ x y : R, (Ideal.span {x, y} : Ideal R).IsPrincipal := by
classical |
classical
constructor
· intro H x y; infer_instance
· intro H
constructor
apply Submodule.fg_induction
· exact fun _ => ⟨⟨_, rfl⟩⟩
· rintro _ _ ⟨⟨x, rfl⟩⟩ ⟨⟨y, rfl⟩⟩; rw [← Submodule.span_insert]; exact H _ _
| true |
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.circle_transform from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
open Set MeasureTheory Metric Filter Function
open scoped Interval Real
noncomputable secti... | Mathlib/MeasureTheory/Integral/CircleTransform.lean | 98 | 106 | theorem continuousOn_prod_circle_transform_function {R r : ℝ} (hr : r < R) {z : ℂ} :
ContinuousOn (fun w : ℂ × ℝ => (circleMap z R w.snd - w.fst)⁻¹ ^ 2)
(closedBall z r ×ˢ univ) := by
simp_rw [← one_div] |
simp_rw [← one_div]
apply_rules [ContinuousOn.pow, ContinuousOn.div, continuousOn_const]
· exact ((continuous_circleMap z R).comp_continuousOn continuousOn_snd).sub continuousOn_fst
· rintro ⟨a, b⟩ ⟨ha, -⟩
have ha2 : a ∈ ball z R := closedBall_subset_ball hr ha
exact sub_ne_zero.2 (circleMap_ne_mem_bal... | true |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 165 | 167 | theorem hom.inj_on_objects : Function.Injective (hom S).obj := by
rintro ⟨c, hc⟩ ⟨d, hd⟩ hcd |
rintro ⟨c, hc⟩ ⟨d, hd⟩ hcd
simp only [Subtype.mk_eq_mk]; exact hcd
| true |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set TopologicalSpace MeasureTheory MeasureTheory.Measure FiniteDimensional
open sco... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 76 | 94 | theorem parallelepiped_comp_equiv (v : ι → E) (e : ι' ≃ ι) :
parallelepiped (v ∘ e) = parallelepiped v := by
simp only [parallelepiped] |
simp only [parallelepiped]
let K : (ι' → ℝ) ≃ (ι → ℝ) := Equiv.piCongrLeft' (fun _a : ι' => ℝ) e
have : Icc (0 : ι → ℝ) 1 = K '' Icc (0 : ι' → ℝ) 1 := by
rw [← Equiv.preimage_eq_iff_eq_image]
ext x
simp only [K, mem_preimage, mem_Icc, Pi.le_def, Pi.zero_apply, Equiv.piCongrLeft'_apply,
Pi.one_a... | true |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 36 | 38 | theorem leftUnitor_tensor' (X Y : C) :
(λ_ (X ⊗ Y)).hom = (α_ (𝟙_ C) X Y).inv ≫ ((λ_ X).hom ⊗ 𝟙 Y) := by |
coherence
| true |
import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Card
import Mathlib.Order.RelSeries
#align_import order.jordan_holder from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u
... | Mathlib/Order/JordanHolder.lean | 109 | 113 | theorem isMaximal_of_eq_inf (x b : X) {a y : X} (ha : x ⊓ y = a) (hxy : x ≠ y) (hxb : IsMaximal x b)
(hyb : IsMaximal y b) : IsMaximal a y := by
have hb : x ⊔ y = b := sup_eq_of_isMaximal hxb hyb hxy |
have hb : x ⊔ y = b := sup_eq_of_isMaximal hxb hyb hxy
substs a b
exact isMaximal_inf_right_of_isMaximal_sup hxb hyb
| true |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}
def map₂ (f : α → β → γ) (a : Option α) ... | Mathlib/Data/Option/NAry.lean | 124 | 127 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) := by |
cases a <;> cases b <;> cases c <;> simp [h_assoc]
| true |
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
noncomputable section
open Filter Set
open Topology
universe u v
section Ultrafilter
def ultrafilterBasis (α : Type u) : Set ... | Mathlib/Topology/StoneCech.lean | 58 | 62 | theorem ultrafilter_isClosed_basic (s : Set α) : IsClosed { u : Ultrafilter α | s ∈ u } := by
rw [← isOpen_compl_iff] |
rw [← isOpen_compl_iff]
convert ultrafilter_isOpen_basic sᶜ using 1
ext u
exact Ultrafilter.compl_mem_iff_not_mem.symm
| true |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Tree.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.GCongr
import Mathlib... | Mathlib/Combinatorics/Enumerative/Catalan.lean | 144 | 145 | theorem catalan_two : catalan 2 = 2 := by |
norm_num [catalan_eq_centralBinom_div, Nat.centralBinom, Nat.choose]
| true |
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 255 | 264 | theorem ker_mulVecLin_transpose_mul_self (A : Matrix m n R) :
LinearMap.ker (Aᵀ * A).mulVecLin = LinearMap.ker (mulVecLin A) := by
ext x |
ext x
simp only [LinearMap.mem_ker, mulVecLin_apply, ← mulVec_mulVec]
constructor
· intro h
replace h := congr_arg (dotProduct x) h
rwa [dotProduct_mulVec, dotProduct_zero, vecMul_transpose, dotProduct_self_eq_zero] at h
· intro h
rw [h, mulVec_zero]
| true |
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.FieldDivision
#align_import ring_theory.polynomial.content from "leanprover-community/mathlib"@"7a030ab8eb5d99f05a891dccc49c5b5b90c947d3"
namespace Po... | Mathlib/RingTheory/Polynomial/Content.lean | 203 | 212 | theorem content_eq_gcd_leadingCoeff_content_eraseLead (p : R[X]) :
p.content = GCDMonoid.gcd p.leadingCoeff (eraseLead p).content := by
by_cases h : p = 0 |
by_cases h : p = 0
· simp [h]
rw [← leadingCoeff_eq_zero, leadingCoeff, ← Ne, ← mem_support_iff] at h
rw [content, ← Finset.insert_erase h, Finset.gcd_insert, leadingCoeff, content,
eraseLead_support]
refine congr rfl (Finset.gcd_congr rfl fun i hi => ?_)
rw [Finset.mem_erase] at hi
rw [eraseLead_coe... | true |
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.int.gcd from "leanprover-community/mathlib"@"47a1a73351de8dd6c8d3d32b569c8e434b03ca47"
namespace Nat
... | Mathlib/Data/Int/GCD.lean | 108 | 109 | theorem xgcdAux_val (x y) : xgcdAux x 1 0 y 0 1 = (gcd x y, xgcd x y) := by |
rw [xgcd, ← xgcdAux_fst x y 1 0 0 1]
| true |
import Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence
import Mathlib.Algebra.ContinuedFractions.TerminatedStable
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
#align_import algebra.continued_fractions.convergents_equiv from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40a... | Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean | 121 | 129 | theorem squashSeq_nth_of_lt {m : ℕ} (m_lt_n : m < n) : (squashSeq s n).get? m = s.get? m := by
cases s_succ_nth_eq : s.get? (n + 1) with |
cases s_succ_nth_eq : s.get? (n + 1) with
| none => rw [squashSeq_eq_self_of_terminated s_succ_nth_eq]
| some =>
obtain ⟨gp_n, s_nth_eq⟩ : ∃ gp_n, s.get? n = some gp_n :=
s.ge_stable n.le_succ s_succ_nth_eq
obtain ⟨gp_m, s_mth_eq⟩ : ∃ gp_m, s.get? m = some gp_m :=
s.ge_stable (le_of_lt m_lt_n... | true |
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.Algebra.GCDMonoid.IntegrallyClosed
import Mathlib.FieldTheory.Finite.Basic
#align_import ring_theory.roots_of_unity.minpoly from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f"
... | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | 118 | 169 | theorem minpoly_eq_pow {p : ℕ} [hprime : Fact p.Prime] (hdiv : ¬p ∣ n) :
minpoly ℤ μ = minpoly ℤ (μ ^ p) := by
classical |
classical
by_cases hn : n = 0
· simp_all
have hpos := Nat.pos_of_ne_zero hn
by_contra hdiff
set P := minpoly ℤ μ
set Q := minpoly ℤ (μ ^ p)
have Pmonic : P.Monic := minpoly.monic (h.isIntegral hpos)
have Qmonic : Q.Monic := minpoly.monic ((h.pow_of_prime hprime.1 hdiv).isIntegral hpos)
have Pirr : ... | true |
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Filter.CountableInter
import Mathlib.Order.Filter.CardinalInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.Order.Filter.Bases
open Set Filter Cardinal
universe u
variable {ι : Type u} {α β : Type u}... | Mathlib/Order/Filter/Cocardinal.lean | 61 | 68 | theorem hasBasis_cocardinal : HasBasis (cocardinal α hreg) {s : Set α | #s < c} compl :=
⟨fun s =>
⟨fun h => ⟨sᶜ, h, (compl_compl s).subset⟩, fun ⟨_t, htf, hts⟩ => by
have : #↑sᶜ < c := by
apply lt_of_le_of_lt _ htf |
apply lt_of_le_of_lt _ htf
rw [compl_subset_comm] at hts
apply Cardinal.mk_le_mk_of_subset hts
simp_all only [mem_cocardinal] ⟩⟩
| true |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
inductive DihedralGroup (n : ℕ) : Type
| r : ZMod n → DihedralGroup n
| sr : ZMod n → DihedralGroup n
derivin... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 125 | 126 | theorem card [NeZero n] : Fintype.card (DihedralGroup n) = 2 * n := by |
rw [← Fintype.card_eq.mpr ⟨fintypeHelper⟩, Fintype.card_sum, ZMod.card, two_mul]
| true |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 93 | 96 | theorem summable_cantor_function (f : ℕ → Bool) (h1 : 0 ≤ c) (h2 : c < 1) :
Summable (cantorFunctionAux c f) := by
apply (summable_geometric_of_lt_one h1 h2).summable_of_eq_zero_or_self |
apply (summable_geometric_of_lt_one h1 h2).summable_of_eq_zero_or_self
intro n; cases h : f n <;> simp [h]
| true |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 312 | 323 | theorem nnnorm_eq_sup_normAtPlace (x : E K) :
‖x‖₊ = univ.sup fun w ↦ ⟨normAtPlace w x, normAtPlace_nonneg w x⟩ := by
rw [show (univ : Finset (InfinitePlace K)) = (univ.image |
rw [show (univ : Finset (InfinitePlace K)) = (univ.image
(fun w : {w : InfinitePlace K // IsReal w} ↦ w.1)) ∪
(univ.image (fun w : {w : InfinitePlace K // IsComplex w} ↦ w.1))
by ext; simp [isReal_or_isComplex], sup_union, univ.sup_image, univ.sup_image, sup_eq_max,
Prod.nnnorm_def', Pi.nnnorm_def, P... | true |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 137 | 142 | theorem eq_zero_of_hasseDeriv_eq_zero {R} [CommRing R] (f : R[X]) (r : R)
(h : ∀ k, (hasseDeriv k f).eval r = 0) : f = 0 := by
apply taylor_injective r |
apply taylor_injective r
rw [LinearMap.map_zero]
ext k
simp only [taylor_coeff, h, coeff_zero]
| true |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 46 | 50 | theorem oangle_add_left_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x + y) y = Real.arccos (‖y‖ / ‖x + y‖) := by
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two h
| true |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 448 | 451 | theorem X_sub_C_pow_dvd_iff {p : R[X]} {t : R} {n : ℕ} :
(X - C t) ^ n ∣ p ↔ X ^ n ∣ p.comp (X + C t) := by
convert (map_dvd_iff <| algEquivAevalXAddC t).symm using 2 |
convert (map_dvd_iff <| algEquivAevalXAddC t).symm using 2
simp [C_eq_algebraMap]
| true |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 210 | 212 | theorem mul_inv (t : TransvectionStruct n R) : t.toMatrix * t.inv.toMatrix = 1 := by
rcases t with ⟨_, _, t_hij⟩ |
rcases t with ⟨_, _, t_hij⟩
simp [toMatrix, transvection_mul_transvection_same, t_hij]
| true |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Finsupp
#align_import algebra.big_operators.associated from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α β γ δ : Type*}
-- the same local notation used in `Algebra.Associated`
local infixl:50 " ~ᵤ " => ... | Mathlib/Algebra/BigOperators/Associated.lean | 159 | 168 | theorem exists_mem_multiset_le_of_prime {s : Multiset (Associates α)} {p : Associates α}
(hp : Prime p) : p ≤ s.prod → ∃ a ∈ s, p ≤ a :=
Multiset.induction_on s (fun ⟨d, Eq⟩ => (hp.ne_one (mul_eq_one_iff.1 Eq.symm).1).elim)
fun a s ih h =>
have : p ≤ a * s.prod := by simpa using h | simpa using h
match Prime.le_or_le hp this with
| Or.inl h => ⟨a, Multiset.mem_cons_self a s, h⟩
| Or.inr h =>
let ⟨a, has, h⟩ := ih h
⟨a, Multiset.mem_cons_of_mem has, h⟩
| true |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 522 | 527 | theorem IsHermitian.fromBlocks₂₂ [Fintype n] [DecidableEq n] (A : Matrix m m 𝕜) (B : Matrix m n 𝕜)
{D : Matrix n n 𝕜} (hD : D.IsHermitian) :
(Matrix.fromBlocks A B Bᴴ D).IsHermitian ↔ (A - B * D⁻¹ * Bᴴ).IsHermitian := by
rw [← isHermitian_submatrix_equiv (Equiv.sumComm n m), Equiv.sumComm_apply, |
rw [← isHermitian_submatrix_equiv (Equiv.sumComm n m), Equiv.sumComm_apply,
fromBlocks_submatrix_sum_swap_sum_swap]
convert IsHermitian.fromBlocks₁₁ _ _ hD <;> simp
| true |
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Conj
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import category_theory.adjunction.reflective from "leanprover-community/mathlib"@"239d882c4fb58361ee8b3b39fb2091320edef10a"
universe v₁ v₂ v₃ u₁ u₂ u₃
noncomputable s... | Mathlib/CategoryTheory/Adjunction/Reflective.lean | 99 | 109 | theorem mem_essImage_of_unit_isSplitMono [Reflective i] {A : C}
[IsSplitMono ((reflectorAdjunction i).unit.app A)] : A ∈ i.essImage := by
let η : 𝟭 C ⟶ reflector i ⋙ i := (reflectorAdjunction i).unit |
let η : 𝟭 C ⟶ reflector i ⋙ i := (reflectorAdjunction i).unit
haveI : IsIso (η.app (i.obj ((reflector i).obj A))) :=
Functor.essImage.unit_isIso ((i.obj_mem_essImage _))
have : Epi (η.app A) := by
refine @epi_of_epi _ _ _ _ _ (retraction (η.app A)) (η.app A) ?_
rw [show retraction _ ≫ η.app A = _ fr... | true |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 189 | 192 | theorem lipschitzExtensionConstant_pos (E' : Type*) [NormedAddCommGroup E'] [NormedSpace ℝ E']
[FiniteDimensional ℝ E'] : 0 < lipschitzExtensionConstant E' := by
rw [lipschitzExtensionConstant] |
rw [lipschitzExtensionConstant]
exact zero_lt_one.trans_le (le_max_right _ _)
| true |
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.Data.Finsupp.PWO
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a484a7d0eade4e1268f4fb402859b6686037f965"
set_option linter.uppercaseLean3 false
open Finset Function
open sco... | Mathlib/RingTheory/HahnSeries/PowerSeries.lean | 117 | 128 | theorem ofPowerSeries_C (r : R) : ofPowerSeries Γ R (PowerSeries.C R r) = HahnSeries.C r := by
ext n |
ext n
simp only [ofPowerSeries_apply, C, RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk, ne_eq,
single_coeff]
split_ifs with hn
· subst hn
convert @embDomain_coeff ℕ R _ _ Γ _ _ _ 0 <;> simp
· rw [embDomain_notin_image_support]
simp only [not_exists, Set.mem_image, toPowerSeries_symm_apply_coeff... | true |
import Mathlib.Data.List.Range
import Mathlib.Algebra.Order.Ring.Nat
variable {α : Type*}
namespace List
@[simp]
| Mathlib/Data/List/Iterate.lean | 21 | 22 | theorem length_iterate (f : α → α) (a : α) (n : ℕ) : length (iterate f a n) = n := by |
induction n generalizing a <;> simp [*]
| true |
import Mathlib.Geometry.RingedSpace.PresheafedSpace
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.Topology.Sheaves.Limits
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
#align_import algebraic_geometry.presheafed_space.has_colimits from "leanprover-community/mathlib"@"178a32653e369dc... | Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean | 70 | 79 | theorem map_comp_c_app (F : J ⥤ PresheafedSpace.{_, _, v} C) {j₁ j₂ j₃}
(f : j₁ ⟶ j₂) (g : j₂ ⟶ j₃) (U) :
(F.map (f ≫ g)).c.app (op U) =
(F.map g).c.app (op U) ≫
(pushforwardMap (F.map g).base (F.map f).c).app (op U) ≫
(Pushforward.comp (F.obj j₁).presheaf (F.map f).base (F.map g).base).... |
cases U
simp [PresheafedSpace.congr_app (F.map_comp f g)]
| true |
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.cantor_normal_form from "leanprover-community/mathlib"@"991ff3b5269848f6dd942ae8e9dd3c946035dc8b"
noncomputable section
universe u
open List
namespace Ordinal
@[elab_as_elim]
noncomputabl... | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | 140 | 145 | theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2 := by
refine CNFRec b (by simp) (fun o ho IH ↦ ?_) o |
refine CNFRec b (by simp) (fun o ho IH ↦ ?_) o
rw [CNF_ne_zero ho]
rintro (h | ⟨_, h⟩)
· exact div_opow_log_pos b ho
· exact IH h
| true |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 71 | 81 | theorem intDegree_mul {x y : RatFunc K} (hx : x ≠ 0) (hy : y ≠ 0) :
intDegree (x * y) = intDegree x + intDegree y := by
simp only [intDegree, add_sub, sub_add, sub_sub_eq_add_sub, sub_sub, sub_eq_sub_iff_add_eq_add] |
simp only [intDegree, add_sub, sub_add, sub_sub_eq_add_sub, sub_sub, sub_eq_sub_iff_add_eq_add]
norm_cast
rw [← Polynomial.natDegree_mul x.denom_ne_zero y.denom_ne_zero, ←
Polynomial.natDegree_mul (RatFunc.num_ne_zero (mul_ne_zero hx hy))
(mul_ne_zero x.denom_ne_zero y.denom_ne_zero),
← Polynomial.... | true |
import Mathlib.Analysis.InnerProductSpace.Adjoint
#align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
open InnerProductSpace RCLike ContinuousLinearMap
open scoped InnerProduct ComplexConjugate
namespace ContinuousLinearMap
variable... | Mathlib/Analysis/InnerProductSpace/Positive.lean | 95 | 98 | theorem IsPositive.adjoint_conj {T : E →L[𝕜] E} (hT : T.IsPositive) (S : F →L[𝕜] E) :
(S† ∘L T ∘L S).IsPositive := by
convert hT.conj_adjoint (S†) |
convert hT.conj_adjoint (S†)
rw [adjoint_adjoint]
| true |
import Mathlib.Dynamics.BirkhoffSum.Basic
import Mathlib.Algebra.Module.Basic
open Finset
section birkhoffAverage
variable (R : Type*) {α M : Type*} [DivisionSemiring R] [AddCommMonoid M] [Module R M]
def birkhoffAverage (f : α → α) (g : α → M) (n : ℕ) (x : α) : M := (n : R)⁻¹ • birkhoffSum f g n x
theorem bir... | Mathlib/Dynamics/BirkhoffSum/Average.lean | 50 | 51 | theorem birkhoffAverage_one (f : α → α) (g : α → M) (x : α) :
birkhoffAverage R f g 1 x = g x := by | simp [birkhoffAverage]
| true |
import Mathlib.Data.Set.Function
import Mathlib.Logic.Function.Iterate
import Mathlib.GroupTheory.Perm.Basic
#align_import dynamics.fixed_points.basic from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Equiv
universe u v
variable {α : Type u} {β : Type v} {f fa g : α → α} {x y :... | Mathlib/Dynamics/FixedPoints/Basic.lean | 97 | 100 | theorem preimage_iterate {s : Set α} (h : IsFixedPt (Set.preimage f) s) (n : ℕ) :
IsFixedPt (Set.preimage f^[n]) s := by
rw [Set.preimage_iterate_eq] |
rw [Set.preimage_iterate_eq]
exact h.iterate n
| true |
import Mathlib.CategoryTheory.Comma.Basic
#align_import category_theory.arrow from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
namespace CategoryTheory
universe v u
-- morphism levels before object levels. See note [CategoryTheory universes].
variable {T : Type u} [Category.{v} T]
... | Mathlib/CategoryTheory/Comma/Arrow.lean | 171 | 174 | theorem iso_w {f g : Arrow T} (e : f ≅ g) : g.hom = e.inv.left ≫ f.hom ≫ e.hom.right := by
have eq := Arrow.hom.congr_right e.inv_hom_id |
have eq := Arrow.hom.congr_right e.inv_hom_id
rw [Arrow.comp_right, Arrow.id_right] at eq
erw [Arrow.w_assoc, eq, Category.comp_id]
| true |
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.Algebra.Exact
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.Derivation
#align_import ring_theory.kaehler from "leanprover-community/mathli... | Mathlib/RingTheory/Kaehler.lean | 131 | 141 | theorem KaehlerDifferential.span_range_eq_ideal :
Ideal.span (Set.range fun s : S => (1 : S) ⊗ₜ[R] s - s ⊗ₜ[R] (1 : S)) =
KaehlerDifferential.ideal R S := by
apply le_antisymm |
apply le_antisymm
· rw [Ideal.span_le]
rintro _ ⟨s, rfl⟩
exact KaehlerDifferential.one_smul_sub_smul_one_mem_ideal _ _
· change (KaehlerDifferential.ideal R S).restrictScalars S ≤ (Ideal.span _).restrictScalars S
rw [← KaehlerDifferential.submodule_span_range_eq_ideal, Ideal.span]
conv_rhs => rw ... | true |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 133 | 137 | theorem comp_hasStrictFDerivAt_iff {f : G → E} {x : G} {f' : G →L[𝕜] E} :
HasStrictFDerivAt (iso ∘ f) ((iso : E →L[𝕜] F).comp f') x ↔ HasStrictFDerivAt f f' x := by
refine ⟨fun H => ?_, fun H => iso.hasStrictFDerivAt.comp x H⟩ |
refine ⟨fun H => ?_, fun H => iso.hasStrictFDerivAt.comp x H⟩
convert iso.symm.hasStrictFDerivAt.comp x H using 1 <;>
ext z <;> apply (iso.symm_apply_apply _).symm
| true |
import Mathlib.RingTheory.Finiteness
import Mathlib.LinearAlgebra.FreeModule.Basic
#align_import linear_algebra.free_module.finite.basic from "leanprover-community/mathlib"@"59628387770d82eb6f6dd7b7107308aa2509ec95"
universe u v w
variable (R : Type u) (M : Type v) (N : Type w)
namespace Module.Free
section Co... | Mathlib/LinearAlgebra/FreeModule/Finite/Basic.lean | 53 | 58 | theorem _root_.Module.Finite.of_basis {R M ι : Type*} [Semiring R] [AddCommMonoid M] [Module R M]
[_root_.Finite ι] (b : Basis ι R M) : Module.Finite R M := by
cases nonempty_fintype ι |
cases nonempty_fintype ι
classical
refine ⟨⟨Finset.univ.image b, ?_⟩⟩
simp only [Set.image_univ, Finset.coe_univ, Finset.coe_image, Basis.span_eq]
| true |
import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
#align_import algebra.quaternion_basis from "leanprover-community/mathlib"@"3aa5b8a9ed7a7cabd36e6e1d022c9858ab8a8c2d"
open Quaternion
namespace QuaternionAlgebra
structure Basis {R : Type*} (A : Type*) [CommRing R] [Ring A] [Algebra R A] (c₁ c₂ : R) ... | Mathlib/Algebra/QuaternionBasis.lean | 99 | 100 | theorem j_mul_k : q.j * q.k = -c₂ • q.i := by |
rw [← i_mul_j, ← mul_assoc, j_mul_i, neg_mul, k_mul_j, neg_smul]
| true |
import Mathlib.Data.Nat.Bits
import Mathlib.Order.Lattice
#align_import data.nat.size from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
namespace Nat
section
set_option linter.deprecated false
theorem shiftLeft_eq_mul_pow (m) : ∀ n, m <<< n = m * 2 ^ n := shiftLeft_eq _
#align nat.... | Mathlib/Data/Nat/Size.lean | 85 | 97 | theorem size_shiftLeft' {b m n} (h : shiftLeft' b m n ≠ 0) :
size (shiftLeft' b m n) = size m + n := by
induction' n with n IH <;> simp [shiftLeft'] at h ⊢ |
induction' n with n IH <;> simp [shiftLeft'] at h ⊢
rw [size_bit h, Nat.add_succ]
by_cases s0 : shiftLeft' b m n = 0 <;> [skip; rw [IH s0]]
rw [s0] at h ⊢
cases b; · exact absurd rfl h
have : shiftLeft' true m n + 1 = 1 := congr_arg (· + 1) s0
rw [shiftLeft'_tt_eq_mul_pow] at this
obtain rfl := succ.in... | true |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Finset.Pointwise
import Mathlib.Tactic.GCongr
#align_import combinatorics.additive.pluennecke_ruzsa from "leanprover-community/mathlib"@"4aab2abced69a9e579b1e6dc... | Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 73 | 76 | theorem card_mul_mul_le_card_div_mul_card_mul (A B C : Finset α) :
(A * C).card * B.card ≤ (A / B).card * (B * C).card := by
rw [← div_inv_eq_mul, ← div_inv_eq_mul B] |
rw [← div_inv_eq_mul, ← div_inv_eq_mul B]
exact card_div_mul_le_card_div_mul_card_div _ _ _
| true |
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
#align_import topology.metric_space.hausdorff_dimension from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
open scoped MeasureTheory ENNReal NNReal Topology
open MeasureTheory MeasureTheory... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 110 | 111 | theorem dimH_def (s : Set X) : dimH s = ⨆ (d : ℝ≥0) (_ : μH[d] s = ∞), (d : ℝ≥0∞) := by |
borelize X; rw [dimH]
| true |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 137 | 139 | theorem length_mul_mod_two (w₁ w₂ : W) : ℓ (w₁ * w₂) % 2 = (ℓ w₁ + ℓ w₂) % 2 := by
rw [← ZMod.natCast_eq_natCast_iff', Nat.cast_add] |
rw [← ZMod.natCast_eq_natCast_iff', Nat.cast_add]
simpa only [lengthParity_eq_ofAdd_length, ofAdd_add] using map_mul cs.lengthParity w₁ w₂
| true |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 117 | 118 | theorem mellin_div_const (f : ℝ → ℂ) (s a : ℂ) : mellin (fun t => f t / a) s = mellin f s / a := by |
simp_rw [mellin, smul_eq_mul, ← mul_div_assoc, integral_div]
| true |
import Mathlib.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
#align_import category_theory.idempotents.karoubi from "leanprover-community/mathlib"@"200eda15d8ff5669854ff6bcc10aaf37cb70498f"
noncomputable section
open CategoryT... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 144 | 147 | theorem eqToHom_f {P Q : Karoubi C} (h : P = Q) :
Karoubi.Hom.f (eqToHom h) = P.p ≫ eqToHom (congr_arg Karoubi.X h) := by
subst h |
subst h
simp only [eqToHom_refl, Karoubi.id_eq, comp_id]
| true |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 120 | 121 | theorem transvection_mul_apply_same (b : n) (c : R) (M : Matrix n n R) :
(transvection i j c * M) i b = M i b + c * M j b := by | simp [transvection, Matrix.add_mul]
| true |
import Mathlib.GroupTheory.Solvable
import Mathlib.FieldTheory.PolynomialGaloisGroup
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import field_theory.abel_ruffini from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Classical Polynomial Intermedi... | Mathlib/FieldTheory/AbelRuffini.lean | 82 | 93 | theorem gal_isSolvable_tower (p q : F[X]) (hpq : p.Splits (algebraMap F q.SplittingField))
(hp : IsSolvable p.Gal) (hq : IsSolvable (q.map (algebraMap F p.SplittingField)).Gal) :
IsSolvable q.Gal := by
let K := p.SplittingField |
let K := p.SplittingField
let L := q.SplittingField
haveI : Fact (p.Splits (algebraMap F L)) := ⟨hpq⟩
let ϕ : (L ≃ₐ[K] L) ≃* (q.map (algebraMap F K)).Gal :=
(IsSplittingField.algEquiv L (q.map (algebraMap F K))).autCongr
have ϕ_inj : Function.Injective ϕ.toMonoidHom := ϕ.injective
haveI : IsSolvable (K... | true |
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