Context stringlengths 295 65.3k | file_name stringlengths 21 74 | start int64 14 1.41k | end int64 20 1.41k | theorem stringlengths 27 1.42k | proof stringlengths 0 4.57k |
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/-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this fil... | Mathlib/RingTheory/PowerSeries/Derivative.lean | 41 | 43 | theorem coeff_derivativeFun (f : R⟦X⟧) (n : ℕ) :
coeff R n f.derivativeFun = coeff R (n + 1) f * (n + 1) := by | rw [derivativeFun, coeff_mk] |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam, Yury Kudryashov
-/
import Mathlib.Algebra.MvPolynomial.Derivation
import Mathlib.Algebra.MvPolynomial.Variables
/-!
# Partial derivatives of polynomials
This file defi... | Mathlib/Algebra/MvPolynomial/PDeriv.lean | 96 | 96 | theorem pderiv_X_of_ne {i j : σ} (h : j ≠ i) : pderiv i (X j : MvPolynomial σ R) = 0 := by | |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Algebra.Order.Chebyshev
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Math... | Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean | 130 | 135 | theorem a_add_one_le_four_pow_parts_card : a + 1 ≤ 4 ^ #P.parts := by | have h : 1 ≤ 4 ^ #P.parts := one_le_pow₀ (by norm_num)
rw [stepBound, ← Nat.div_div_eq_div_mul]
conv_rhs => rw [← Nat.sub_add_cancel h]
rw [add_le_add_iff_right, tsub_le_iff_left, ← Nat.add_sub_assoc h]
exact Nat.le_sub_one_of_lt (Nat.lt_div_mul_add h) |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 665 | 669 | theorem prod_coe_rpow {ι} (s : Finset ι) (f : ι → ℝ≥0) (r : ℝ) :
∏ i ∈ s, (f i : ℝ≥0∞) ^ r = ((∏ i ∈ s, f i : ℝ≥0) : ℝ≥0∞) ^ r := by | classical
induction s using Finset.induction with
| empty => simp |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... | Mathlib/Algebra/Polynomial/Coeff.lean | 256 | 266 | theorem coeff_X_mul (p : R[X]) (n : ℕ) : coeff (X * p) (n + 1) = coeff p n := by | rw [(commute_X p).eq, coeff_mul_X]
theorem coeff_mul_monomial (p : R[X]) (n d : ℕ) (r : R) :
coeff (p * monomial n r) (d + n) = coeff p d * r := by
rw [← C_mul_X_pow_eq_monomial, ← X_pow_mul, ← mul_assoc, coeff_mul_C, coeff_mul_X_pow]
theorem coeff_monomial_mul (p : R[X]) (n d : ℕ) (r : R) :
coeff (monomial... |
/-
Copyright (c) 2022 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
import Mathlib.Algebra.Order.Archimedean.Hom
import Mathlib.Algebra.Order.Group.Pointwise.CompleteLattice
/-!
# Conditionally complete linear ordered field... | Mathlib/Algebra/Order/CompleteField.lean | 244 | 245 | theorem exists_mem_cutMap_mul_self_of_lt_inducedMap_mul_self (ha : 0 < a) (b : β)
(hba : b < inducedMap α β a * inducedMap α β a) : ∃ c ∈ cutMap β (a * a), b < c := by | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 260 | 265 | theorem eq_rotation_self_iff_angle_eq_zero {x : V} (hx : x ≠ 0) (θ : Real.Angle) :
x = o.rotation θ x ↔ θ = 0 := by | rw [← o.rotation_eq_self_iff_angle_eq_zero hx, eq_comm]
/-- A rotation of a vector equals that vector if and only if the vector or the angle is zero. -/
theorem rotation_eq_self_iff (x : V) (θ : Real.Angle) : o.rotation θ x = x ↔ x = 0 ∨ θ = 0 := by
by_cases h : x = 0 <;> simp [h] |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.... | Mathlib/Tactic/NormNum/Basic.lean | 104 | 105 | theorem isNat_natCast {R} [AddMonoidWithOne R] (n m : ℕ) :
IsNat n m → IsNat (n : R) m := by | rintro ⟨⟨⟩⟩; exact ⟨rfl⟩ |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vanderm... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 93 | 97 | theorem hasseDeriv_one' : hasseDeriv 1 f = derivative f := by | simp only [hasseDeriv_apply, derivative_apply, ← C_mul_X_pow_eq_monomial, Nat.choose_one_right,
(Nat.cast_commute _ _).eq]
@[simp] |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.ConeCategory
/-!
... | Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean | 581 | 582 | theorem condition (a) : I.fst a ≫ K.π (J.fst a) = I.snd a ≫ K.π (J.snd a) := by | rw [← K.snd_app_right, ← K.fst_app_right] |
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.FreeModule.IdealQuotient
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.RingTheory.Norm.Defs
/-!
# N... | Mathlib/LinearAlgebra/FreeModule/Norm.lean | 30 | 50 | theorem associated_norm_prod_smith [Fintype ι] (b : Basis ι R S) {f : S} (hf : f ≠ 0) :
Associated (Algebra.norm R f) (∏ i, smithCoeffs b _ (span_singleton_eq_bot.not.2 hf) i) := by | have hI := span_singleton_eq_bot.not.2 hf
let b' := ringBasis b (span {f}) hI
classical
rw [← Matrix.det_diagonal, ← LinearMap.det_toLin b']
let e :=
(b'.equiv ((span {f}).selfBasis b hI) <| Equiv.refl _).trans
((LinearEquiv.coord S S f hf).restrictScalars R)
refine (LinearMap.associated_det_of_eq_c... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.LinearAlgebra.LinearIndependent.Basic
import Mathlib.Data.Set.Card
/... | Mathlib/LinearAlgebra/Dimension/Basic.lean | 163 | 167 | theorem rank_le_of_surjective_injective (i : R → R') (j : M →+ M₁)
(hi : Surjective i) (hj : Injective j)
(hc : ∀ (r : R) (m : M), j (r • m) = i r • j m) :
Module.rank R M ≤ Module.rank R' M₁ := by | simpa only [lift_id] using lift_rank_le_of_surjective_injective i j hi hj hc |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
# Basic properties of the manifold Fréchet ... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 211 | 215 | theorem mdifferentiableWithinAt_iff_target :
MDifferentiableWithinAt I I' f s x ↔
ContinuousWithinAt f s x ∧
MDifferentiableWithinAt I 𝓘(𝕜, E') (extChartAt I' (f x) ∘ f) s x := by | simp_rw [MDifferentiableWithinAt, liftPropWithinAt_iff', ← and_assoc] |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Monad
/-!
## Expand multivariate polynomials
Given a multivariate polynomial `φ`, one may replace every occurre... | Mathlib/Algebra/MvPolynomial/Expand.lean | 64 | 68 | theorem map_expand (f : R →+* S) (p : ℕ) (φ : MvPolynomial σ R) :
map f (expand p φ) = expand p (map f φ) := by | simp [expand, map_bind₁]
@[simp]
theorem rename_expand (f : σ → τ) (p : ℕ) (φ : MvPolynomial σ R) : |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Fi... | Mathlib/Data/Sign.lean | 439 | 441 | theorem Right.sign_neg [AddRightStrictMono α] (a : α) :
sign (-a) = -sign a := by | simp_rw [sign_apply, Right.neg_pos_iff, Right.neg_neg_iff] |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Or... | Mathlib/Analysis/Convex/Basic.lean | 521 | 524 | theorem Convex.mem_smul_of_zero_mem (h : Convex 𝕜 s) {x : E} (zero_mem : (0 : E) ∈ s) (hx : x ∈ s)
{t : 𝕜} (ht : 1 ≤ t) : x ∈ t • s := by | rw [mem_smul_set_iff_inv_smul_mem₀ (zero_lt_one.trans_le ht).ne']
exact h.smul_mem_of_zero_mem zero_mem hx |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 113 | 113 | theorem log_inv_eq_ite (x : ℂ) : log x⁻¹ = if x.arg = π then -conj (log x) else -log x := by | |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Principal ordinals
We define principal or indecomposable ordinals, and we prove the standa... | Mathlib/SetTheory/Ordinal/Principal.lean | 248 | 263 | theorem principal_mul_one : Principal (· * ·) 1 := by | rw [principal_one_iff]
exact zero_mul _
theorem principal_mul_two : Principal (· * ·) 2 := by
intro a b ha hb
rw [← succ_one, lt_succ_iff] at *
convert mul_le_mul' ha hb
exact (mul_one 1).symm
theorem principal_mul_of_le_two (ho : o ≤ 2) : Principal (· * ·) o := by
rcases lt_or_eq_of_le ho with (ho | rfl)... |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Martingale.Basic
/-!
# Centering lemma for stochastic processes
Any `ℕ`-indexed stochastic process which is adapted and integrable can be wri... | Mathlib/Probability/Martingale/Centering.lean | 156 | 162 | theorem martingalePart_bdd_difference {R : ℝ≥0} {f : ℕ → Ω → ℝ} (ℱ : Filtration ℕ m0)
(hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, ∀ i, |martingalePart f ℱ μ (i + 1) ω - martingalePart f ℱ μ i ω| ≤ ↑(2 * R) := by | filter_upwards [hbdd, predictablePart_bdd_difference ℱ hbdd] with ω hω₁ hω₂ i
simp only [two_mul, martingalePart, Pi.sub_apply]
have : |f (i + 1) ω - predictablePart f ℱ μ (i + 1) ω - (f i ω - predictablePart f ℱ μ i ω)| =
|f (i + 1) ω - f i ω - (predictablePart f ℱ μ (i + 1) ω - predictablePart f ℱ μ i ω)| :... |
/-
Copyright (c) 2021 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Adam Topaz, Johan Commelin
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.AlgebraicTopology.MooreComplex
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Catego... | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | 177 | 180 | theorem map_alternatingFaceMapComplex {D : Type*} [Category D] [Preadditive D] (F : C ⥤ D)
[F.Additive] :
alternatingFaceMapComplex C ⋙ F.mapHomologicalComplex _ =
(SimplicialObject.whiskering C D).obj F ⋙ alternatingFaceMapComplex D := by | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | Mathlib/Data/Finsupp/Basic.lean | 734 | 738 | theorem some_single_none [Zero M] (m : M) : (single none m : Option α →₀ M).some = 0 := by | ext
simp
@[simp] |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | Mathlib/NumberTheory/Divisors.lean | 79 | 81 | theorem filter_dvd_eq_properDivisors (h : n ≠ 0) : {d ∈ range n | d ∣ n} = n.properDivisors := by | ext
simp only [properDivisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self] |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Eric Wieser, Jeremy Avigad, Johan Commelin
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Linear... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 100 | 104 | theorem invOf_fromBlocks_zero₁₂_eq (A : Matrix m m α) (C : Matrix n m α) (D : Matrix n n α)
[Invertible A] [Invertible D] [Invertible (fromBlocks A 0 C D)] :
⅟ (fromBlocks A 0 C D) = fromBlocks (⅟ A) 0 (-(⅟ D * C * ⅟ A)) (⅟ D) := by | letI := fromBlocksZero₁₂Invertible A C D
convert (rfl : ⅟ (fromBlocks A 0 C D) = _) |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.MvPowerSer... | Mathlib/RingTheory/PowerSeries/Basic.lean | 239 | 241 | theorem coeff_C (n : ℕ) (a : R) : coeff R n (C R a : R⟦X⟧) = if n = 0 then a else 0 := by | rw [← monomial_zero_eq_C_apply, coeff_monomial] |
/-
Copyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Data.Complex.Cardinality
import Mathlib.Data.Complex.Modu... | Mathlib/Data/Complex/FiniteDimensional.lean | 27 | 28 | theorem finrank_real_complex : finrank ℝ ℂ = 2 := by | rw [finrank_eq_card_basis basisOneI, Fintype.card_fin] |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
/-!
# Bind operation for multisets
This file defines a few basic operations on `Multiset`, notably the mona... | Mathlib/Data/Multiset/Bind.lean | 115 | 117 | theorem add_bind : (s + t).bind f = s.bind f + t.bind f := by | simp [bind]
@[simp] |
/-
Copyright (c) 2022 Tian Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tian Chen, Mantas Bakšys
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.D... | Mathlib/NumberTheory/Multiplicity.lean | 181 | 189 | theorem emultiplicity_pow_prime_pow_sub_pow_prime_pow (a : ℕ) :
emultiplicity (↑p) (x ^ p ^ a - y ^ p ^ a) = emultiplicity (↑p) (x - y) + a := by | induction a with
| zero => rw [Nat.cast_zero, add_zero, pow_zero, pow_one, pow_one]
| succ a h_ind =>
rw [Nat.cast_add, Nat.cast_one, ← add_assoc, ← h_ind, pow_succ, pow_mul, pow_mul]
apply emultiplicity_pow_prime_sub_pow_prime hp hp1
· rw [← geom_sum₂_mul]
exact dvd_mul_of_dvd_right hxy _ |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Principal ordinals
We define principal or indecomposable ordinals, and we prove the standa... | Mathlib/SetTheory/Ordinal/Principal.lean | 282 | 290 | theorem principal_mul_iff_mul_left_eq : Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o := by | refine ⟨fun h a ha₀ hao => ?_, fun h a b hao hbo => ?_⟩
· rcases le_or_gt o 2 with ho | ho
· convert one_mul o
apply le_antisymm
· rw [← lt_succ_iff, succ_one]
exact hao.trans_le ho
· rwa [← succ_le_iff, succ_zero] at ha₀
· exact op_eq_self_of_principal hao (isNormal_mul_right ha₀) h... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.Tuple.Basic
/-!
# Lists from functions
Theorems and lemmas for dealing with `List.ofFn`, which converts a function on `Fin n` to a list
of l... | Mathlib/Data/List/OfFn.lean | 44 | 45 | theorem ofFn_congr {m n : ℕ} (h : m = n) (f : Fin m → α) :
ofFn f = ofFn fun i : Fin n => f (Fin.cast h.symm i) := by | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathli... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 299 | 303 | theorem braiding_rightUnitor_aux₂ (X : C) :
(𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (𝟙_ C ◁ (ρ_ X).hom) = 𝟙_ C ◁ (λ_ X).hom :=
calc
(𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (𝟙_ C ◁ (ρ_ X).hom) =
(𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (α_ _ _ _).inv ≫ (α_ _ _ _).hom ≫ (𝟙_ C ◁ (ρ_ X).hom) := by | |
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Bitraversable.Basic
/-!
# Bitraversable Lemmas
## Main definitions
* tfst - traverse on first functor argument
* tsnd - traverse on second func... | Mathlib/Control/Bitraversable/Lemmas.lean | 95 | 99 | theorem tfst_eq_fst_id {α α' β} (f : α → α') (x : t α β) :
tfst (F := Id) (pure ∘ f) x = pure (fst f x) := by | apply bitraverse_eq_bimap_id
@[higher_order] |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-... | Mathlib/RingTheory/Coprime/Lemmas.lean | 33 | 40 | theorem Int.isCoprime_iff_gcd_eq_one {m n : ℤ} : IsCoprime m n ↔ Int.gcd m n = 1 := by | constructor
· rintro ⟨a, b, h⟩
refine Nat.dvd_one.mp (Int.gcd_dvd_iff.mpr ⟨a, b, ?_⟩)
rwa [mul_comm m, mul_comm n, eq_comm]
· rw [← Int.ofNat_inj, IsCoprime, Int.gcd_eq_gcd_ab, mul_comm m, mul_comm n, Nat.cast_one]
intro h
exact ⟨_, _, h⟩ |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Integral.Layercake
/-!
# The integral of the real power of a nonnegative function
In thi... | Mathlib/Analysis/SpecialFunctions/Pow/Integral.lean | 50 | 72 | theorem lintegral_rpow_eq_lintegral_meas_le_mul :
∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) := by | have one_lt_p : -1 < p - 1 := by linarith
have obs : ∀ x : ℝ, ∫ t : ℝ in (0)..x, t ^ (p - 1) = x ^ p / p := by
intro x
rw [integral_rpow (Or.inl one_lt_p)]
simp [Real.zero_rpow p_pos.ne.symm]
set g := fun t : ℝ => t ^ (p - 1)
have g_nn : ∀ᵐ t ∂volume.restrict (Ioi (0 : ℝ)), 0 ≤ g t := by
filter_up... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 505 | 505 | theorem sin_eq_zero_iff_cos_eq {x : ℝ} : sin x = 0 ↔ cos x = 1 ∨ cos x = -1 := by | |
/-
Copyright (c) 2024 Ira Fesefeldt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ira Fesefeldt
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
/-!
# Ordinal Approximants for the Fixed points on complete lattices
This file sets up the ordinal-indexed approximation t... | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | 92 | 112 | theorem lfpApprox_add_one (h : x ≤ f x) (a : Ordinal) :
lfpApprox f x (a+1) = f (lfpApprox f x a) := by | apply le_antisymm
· conv => left; rw [lfpApprox]
apply sSup_le
simp only [Ordinal.add_one_eq_succ, lt_succ_iff, exists_prop, Set.union_singleton,
Set.mem_insert_iff, Set.mem_setOf_eq, forall_eq_or_imp, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂]
apply And.intro
· apply le_trans... |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.Algebra.Ring.Subring.Units
import Mathlib.LinearAlgebra.LinearIndepende... | Mathlib/LinearAlgebra/Ray.lean | 483 | 485 | theorem sameRay_neg_smul_right_iff_of_ne {v : M} {r : R} (hv : v ≠ 0) (hr : r ≠ 0) :
SameRay R (-v) (r • v) ↔ r < 0 := by | simp only [sameRay_neg_smul_right_iff, hv, or_false, hr.le_iff_lt] |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
/-!
# Operator norm as an `NNNorm`
Operator norm as an `NNNorm`, i.e. takin... | Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean | 49 | 53 | theorem nnnorm_def (f : E →SL[σ₁₂] F) : ‖f‖₊ = sInf { c | ∀ x, ‖f x‖₊ ≤ c * ‖x‖₊ } := by | ext
rw [NNReal.coe_sInf, coe_nnnorm, norm_def, NNReal.coe_image]
simp_rw [← NNReal.coe_le_coe, NNReal.coe_mul, coe_nnnorm, mem_setOf_eq, NNReal.coe_mk,
exists_prop] |
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Data.Nat.Factors
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
impo... | Mathlib/NumberTheory/FLT/Four.lean | 114 | 120 | theorem exists_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 := by | obtain ⟨a0, b0, c0, hf⟩ := exists_minimal h
rcases Int.emod_two_eq_zero_or_one a0 with hap | hap
· rcases Int.emod_two_eq_zero_or_one b0 with hbp | hbp
· exfalso
have h1 : 2 ∣ (Int.gcd a0 b0 : ℤ) := |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.RingTheory.LocalProperties.Reduced
/-!
# Basic properties of schemes
We provide some basic properties of sche... | Mathlib/AlgebraicGeometry/Properties.lean | 84 | 92 | theorem isReduced_of_isOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : IsOpenImmersion f]
[IsReduced Y] : IsReduced X := by | constructor
intro U
have : U = f ⁻¹ᵁ f ''ᵁ U := by
ext1; exact (Set.preimage_image_eq _ H.base_open.injective).symm
rw [this]
exact isReduced_of_injective (inv <| f.app (f ''ᵁ U)).hom
(asIso <| f.app (f ''ᵁ U) : Γ(Y, f ''ᵁ U) ≅ _).symm.commRingCatIsoToRingEquiv.injective |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
import Batteries.Data.Int
/-!
# Bitwise operations on integers
Possi... | Mathlib/Data/Int/Bitwise.lean | 304 | 309 | theorem ldiff_bit (a m b n) : ldiff (bit a m) (bit b n) = bit (a && not b) (ldiff m n) := by | rw [← bitwise_diff, bitwise_bit]
@[simp]
theorem lxor_bit (a m b n) : Int.xor (bit a m) (bit b n) = bit (xor a b) (Int.xor m n) := by
rw [← bitwise_xor, bitwise_bit] |
/-
Copyright (c) 2024 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Star.Basic
/-!
# Morphisms of star rings
This file defines a new type of morphism between (non-unita... | Mathlib/Algebra/Star/StarRingHom.lean | 141 | 144 | theorem mk_coe (f : A →⋆ₙ+* B) (h₁ h₂ h₃ h₄) :
(⟨⟨⟨f, h₁⟩, h₂, h₃⟩, h₄⟩ : A →⋆ₙ+* B) = f := by | ext
rfl |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Analysis.SpecialFunctions.Complex.LogBounds
/-!
# Complex arctangent
This file defines the complex arctangent `Complex.arctan` as
$$\arctan z = -\frac i2 \lo... | Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean | 80 | 86 | theorem ofReal_arctan (x : ℝ) : (Real.arctan x : ℂ) = arctan x := by | conv_rhs => rw [← Real.tan_arctan x]
rw [ofReal_tan, arctan_tan]
all_goals norm_cast
· rw [← ne_eq]; exact (Real.arctan_lt_pi_div_two _).ne
· exact Real.neg_pi_div_two_lt_arctan _
· exact (Real.arctan_lt_pi_div_two _).le |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Ralf Stephan, Neil Strickland, Ruben Van de Velde
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Positive.Ring
import Mathlib.... | Mathlib/Data/PNat/Basic.lean | 311 | 313 | theorem mod_le (m k : ℕ+) : mod m k ≤ m ∧ mod m k ≤ k := by | change (mod m k : ℕ) ≤ (m : ℕ) ∧ (mod m k : ℕ) ≤ (k : ℕ)
rw [mod_coe] |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
# Basic properties of the manifold Fréchet ... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 734 | 737 | theorem MDifferentiableAt.hasMFDerivAt (h : MDifferentiableAt I I' f x) :
HasMFDerivAt I I' f x (mfderiv I I' f x) := by | refine ⟨h.continuousAt, ?_⟩
simp only [mfderiv, h, if_pos, mfld_simps] |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.Topology.ContinuousMap.Bounded.Normed
/-!
# Integration of bounded continuous functions
In this file,... | Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean | 42 | 46 | theorem lintegral_lt_top_of_nnreal (f : X →ᵇ ℝ≥0) : ∫⁻ x, f x ∂μ < ∞ := by | apply IsFiniteMeasure.lintegral_lt_top_of_bounded_to_ennreal
refine ⟨nndist f 0, fun x ↦ ?_⟩
have key := BoundedContinuousFunction.NNReal.upper_bound f x
rwa [ENNReal.coe_le_coe] |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Kevin Buzzard, Jujian Zhang, Fangming Li
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.DirectSum.Algebra... | Mathlib/Algebra/DirectSum/Internal.lean | 171 | 177 | theorem coe_of_mul_apply_aux [AddMonoid ι] [SetLike.GradedMonoid A] {i : ι} (r : A i)
(r' : ⨁ i, A i) {j n : ι} (H : ∀ x : ι, i + x = n ↔ x = j) :
((of (fun i => A i) i r * r') n : R) = r * r' j := by | classical
rw [coe_mul_apply_eq_dfinsuppSum]
apply (DFinsupp.sum_single_index _).trans
swap |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.NNRat.Order
import Mathlib.Topology.Algebra.Order.Archimedean
import Mathlib.Topology.Algebra.Ring.Real
import Mathlib.Topology.In... | Mathlib/Topology/Instances/Rat.lean | 61 | 62 | theorem Int.dist_cast_rat (x y : ℤ) : dist (x : ℚ) y = dist x y := by | rw [← Int.dist_cast_real, ← Rat.dist_cast]; congr |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Moritz Doll
-/
import Mathlib.Algebra.GroupWithZero.Action.Opposite
import Mathlib.LinearAlgebra.Finsupp.VectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
im... | Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 298 | 301 | theorem LinearMap.toMatrix₂'_mul (B : (n → R) →ₗ[R] (m → R) →ₗ[R] R) (M : Matrix m m' R) :
toMatrix₂' R B * M = toMatrix₂' R (B.compl₂ <| toLin' M) := by | simp only [B.toMatrix₂'_compl₂, toMatrix'_toLin'] |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... | Mathlib/Algebra/Polynomial/Coeff.lean | 154 | 154 | theorem C_mul' (a : R) (f : R[X]) : C a * f = a • f := by | |
/-
Copyright (c) 2020 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Johan Commelin
-/
import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 307 | 314 | theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (ProjectiveSpectrum 𝒜)) :
zeroLocus 𝒜 (vanishingIdeal t : Set A) = closure t := by | apply Set.Subset.antisymm
· rintro x hx t' ⟨ht', ht⟩
obtain ⟨fs, rfl⟩ : ∃ s, t' = zeroLocus 𝒜 s := by rwa [isClosed_iff_zeroLocus] at ht'
rw [subset_zeroLocus_iff_subset_vanishingIdeal] at ht
exact Set.Subset.trans ht hx
· rw [(isClosed_zeroLocus _ _).closure_subset_iff] |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.MonoidAlgebra.Defs
/-!
# Division of `AddMonoidAlgebra` by monomials
This file is most important for when `G = ℕ` (polynomials) or `G = σ →₀ ℕ` (mu... | Mathlib/Algebra/MonoidAlgebra/Division.lean | 175 | 186 | theorem of'_dvd_iff_modOf_eq_zero [IsCancelAdd G] {x : k[G]} {g : G} :
of' k G g ∣ x ↔ x %ᵒᶠ g = 0 := by | constructor
· rintro ⟨x, rfl⟩
rw [of'_mul_modOf]
· intro h
rw [← divOf_add_modOf x g, h, add_zero]
exact dvd_mul_right _ _
end
end AddMonoidAlgebra |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
/-... | Mathlib/Data/Sym/Sym2.lean | 69 | 69 | theorem Rel.trans {x y z : α × α} (a : Rel α x y) (b : Rel α y z) : Rel α x z := by | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Kernels and cokernels
In a category with zero morphisms, the kernel of a morphism `f : X... | Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 388 | 390 | theorem lift_comp_kernelIsoOfEq_inv {Z} {f g : X ⟶ Y} [HasKernel f] [HasKernel g] (h : f = g)
(e : Z ⟶ X) (he) :
kernel.lift _ e he ≫ (kernelIsoOfEq h).inv = kernel.lift _ e (by simp [h, he]) := by | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | Mathlib/MeasureTheory/Measure/Restrict.lean | 861 | 866 | theorem MeasurableSet.map_coe_volume {s : Set α} (hs : MeasurableSet s) :
volume.map ((↑) : s → α) = restrict volume s := by | rw [volume_set_coe_def, (MeasurableEmbedding.subtype_coe hs).map_comap volume, Subtype.range_coe]
theorem volume_image_subtype_coe {s : Set α} (hs : MeasurableSet s) (t : Set s) :
volume ((↑) '' t : Set α) = volume t := |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 926 | 927 | theorem div_eq_of_eq_mul' {a b c : G} (h : a = b * c) : a / b = c := by | rw [h, div_eq_mul_inv, mul_comm, inv_mul_cancel_left] |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.SetTheory.Cardinal.ENat
/-!
# Finite Cardinality Funct... | Mathlib/SetTheory/Cardinal/Finite.lean | 272 | 275 | theorem card_eq_coe_fintype_card [Fintype α] : card α = Fintype.card α := by | simp [card]
@[simp high] |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.CoreAttrs
/-!
# (Semi-)lattices
Semilatti... | Mathlib/Order/Lattice.lean | 672 | 672 | theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 718 | 722 | theorem dist_div_tan_oangle_left_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
(h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : dist p₃ p₂ / Real.Angle.tan (∡ p₃ p₁ p₂) = dist p₁ p₂ := by | have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm, Real.Angle.tan_coe,
dist_div_tan_angle_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h) |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
/-!
# Finite maps over `Multiset`
-/
universe u v w
open List
... | Mathlib/Data/Finmap.lean | 591 | 595 | theorem disjoint_union_right (x y z : Finmap β) :
Disjoint x (y ∪ z) ↔ Disjoint x y ∧ Disjoint x z := by | rw [Disjoint.symm_iff, disjoint_union_left, Disjoint.symm_iff _ x, Disjoint.symm_iff _ x]
theorem union_comm_of_disjoint {s₁ s₂ : Finmap β} : Disjoint s₁ s₂ → s₁ ∪ s₂ = s₂ ∪ s₁ := |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
import Mathlib.MeasureTh... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 101 | 106 | theorem lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top {p q : ℝ} (hpq : p.HolderConjugate q)
{f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf_nontop : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤)
(hg_nontop : (∫⁻ a, g a ^ q ∂μ) ≠ ⊤) (hf_nonzero : (∫⁻ a, f a ^ p ∂μ) ≠ 0)
(hg_nonzero : (∫⁻ a, g a ^ q ∂μ) ≠ 0) :
(∫⁻ a, (f * g) a ... | let npf := (∫⁻ c : α, f c ^ p ∂μ) ^ (1 / p) |
/-
Copyright (c) 2021 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.GroupTheory.MonoidLocalization.Away
impo... | Mathlib/SetTheory/Surreal/Dyadic.lean | 187 | 191 | theorem dyadic_aux {m₁ m₂ : ℤ} {y₁ y₂ : ℕ} (h₂ : m₁ * 2 ^ y₁ = m₂ * 2 ^ y₂) :
m₁ * powHalf y₂ = m₂ * powHalf y₁ := by | revert m₁ m₂
wlog h : y₁ ≤ y₂
· intro m₁ m₂ aux; exact (this (le_of_not_le h) aux.symm).symm |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Finite.Sum
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutations on... | Mathlib/GroupTheory/Perm/Finite.lean | 241 | 246 | theorem support_pow_coprime {σ : Perm α} {n : ℕ} (h : Nat.Coprime n (orderOf σ)) :
(σ ^ n).support = σ.support := by | obtain ⟨m, hm⟩ := exists_pow_eq_self_of_coprime h
exact
le_antisymm (support_pow_le σ n)
(le_trans (ge_of_eq (congr_arg support hm)) (support_pow_le (σ ^ n) m)) |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | Mathlib/Algebra/Order/ToIntervalMod.lean | 419 | 421 | theorem toIocMod_add_left (a b : α) : toIocMod hp a (p + b) = toIocMod hp a b := by | rw [add_comm, toIocMod_add_right] |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | Mathlib/SetTheory/Ordinal/Basic.lean | 1,209 | 1,213 | theorem univ_umax : univ.{u, max (u + 1) v} = univ.{u, v} :=
congr_fun lift_umax _
theorem lift_lt_univ (c : Cardinal) : lift.{u + 1, u} c < univ.{u, u + 1} := by | simpa only [liftPrincipalSeg_coe, lift_ord, lift_succ, ord_le, succ_le_iff] using |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Minor.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most o... | Mathlib/Data/Matroid/Constructions.lean | 126 | 127 | theorem eq_loopyOn_or_rankPos (M : Matroid α) : M = loopyOn M.E ∨ RankPos M := by | rw [← empty_isBase_iff, rankPos_iff]; apply em |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 438 | 464 | theorem UniformCauchySeqOn.mono (hf : UniformCauchySeqOn F p s) (hss' : s' ⊆ s) :
UniformCauchySeqOn F p s' := by | rw [uniformCauchySeqOn_iff_uniformCauchySeqOnFilter] at hf ⊢
exact hf.mono_right (le_principal_iff.mpr <| mem_principal.mpr hss')
/-- Composing on the right by a function preserves uniform Cauchy sequences -/
theorem UniformCauchySeqOnFilter.comp {γ : Type*} (hf : UniformCauchySeqOnFilter F p p')
(g : γ → α) : U... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | Mathlib/RingTheory/Ideal/Operations.lean | 629 | 633 | theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by | rw [IsCoprime, codisjoint_iff]
constructor
· rintro ⟨x, y, hxy⟩
rw [eq_top_iff_one] |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Derivative of the ca... | Mathlib/Analysis/Calculus/FDeriv/Prod.lean | 500 | 503 | theorem differentiableOn_apply (i : ι) (s' : Set (∀ i, F' i)) :
DifferentiableOn (𝕜 := 𝕜) (fun f : ∀ i, F' i => f i) s' := by | have h := ((differentiableOn_pi (𝕜 := 𝕜)
(Φ := fun (f : ∀ i, F' i) (i' : ι) => f i') (s := s'))).1 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison
-/
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAl... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 417 | 420 | theorem Module.finrank_pos_iff [NoZeroSMulDivisors R M] :
0 < finrank R M ↔ Nontrivial M := by | rw [← rank_pos_iff_nontrivial (R := R), ← finrank_eq_rank]
norm_cast |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
/-!
# One-dimensional iterated derivatives
We define the `n`-th de... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 228 | 231 | theorem iteratedFDeriv_apply_eq_iteratedDeriv_mul_prod {m : Fin n → 𝕜} :
(iteratedFDeriv 𝕜 n f x : (Fin n → 𝕜) → F) m = (∏ i, m i) • iteratedDeriv n f x := by | rw [iteratedDeriv_eq_iteratedFDeriv, ← ContinuousMultilinearMap.map_smul_univ]; simp |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sets.OpenCover
/-!
# Sober spaces
A quasi-sober space is a topological space where every irreducible closed s... | Mathlib/Topology/Sober.lean | 182 | 190 | theorem Topology.IsOpenEmbedding.quasiSober {f : α → β} (hf : IsOpenEmbedding f) [QuasiSober β] :
QuasiSober α where
sober hS hS' := by | have hS'' := hS.image f hf.continuous.continuousOn
obtain ⟨x, hx⟩ := QuasiSober.sober hS''.closure isClosed_closure
obtain ⟨T, hT, rfl⟩ := hf.isInducing.isClosed_iff.mp hS'
rw [image_preimage_eq_inter_range] at hx hS''
have hxT : x ∈ T := by
rw [← hT.closure_eq] |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
/-! # Ordinal exponential
In this file we define the power function and the lo... | Mathlib/SetTheory/Ordinal/Exponential.lean | 68 | 69 | theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :
a < b ^ c ↔ ∃ c' < c, a < b ^ c' := by | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Mario Carneiro, Sean Leather
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finite sets in `Option α`
In this file we define
* `Option.t... | Mathlib/Data/Finset/Option.lean | 112 | 114 | theorem eraseNone_union [DecidableEq (Option α)] [DecidableEq α] (s t : Finset (Option α)) :
eraseNone (s ∪ t) = eraseNone s ∪ eraseNone t := by | ext |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Split
/-!
# Filters used in box-based integrals
First ... | Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | 464 | 466 | theorem tendsto_embedBox_toFilteriUnion_top (l : IntegrationParams) (h : I ≤ J) :
Tendsto (TaggedPrepartition.embedBox I J h) (l.toFilteriUnion I ⊤)
(l.toFilteriUnion J (Prepartition.single J I h)) := by | |
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Data.Fintype.Pigeonhole
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.SplittingField.Constructi... | Mathlib/FieldTheory/PrimitiveElement.lean | 365 | 370 | theorem primitive_element_iff_minpoly_natDegree_eq (α : E) :
F⟮α⟯ = ⊤ ↔ (minpoly F α).natDegree = finrank F E := by | rw [← adjoin.finrank (IsIntegral.of_finite F α), ← finrank_top F E]
refine ⟨fun h => ?_, fun h => eq_of_le_of_finrank_eq le_top h⟩
exact congr_arg (fun K : IntermediateField F E => finrank F K) h |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
/-!
# Composition of analytic functions
In this fi... | Mathlib/Analysis/Analytic/Composition.lean | 311 | 326 | theorem compAlongComposition_nnnorm {n : ℕ} (q : FormalMultilinearSeries 𝕜 F G)
(p : FormalMultilinearSeries 𝕜 E F) (c : Composition n) :
‖q.compAlongComposition p c‖₊ ≤ ‖q c.length‖₊ * ∏ i, ‖p (c.blocksFun i)‖₊ := by | rw [← NNReal.coe_le_coe]; push_cast; exact q.compAlongComposition_norm p c
/-!
### The identity formal power series
We will now define the identity power series, and show that it is a neutral element for left and
right composition.
-/
section
variable (𝕜 E) |
/-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.FractionalIdeal.Norm
import Mathlib.RingTheory.FractionalIdeal.Operations
/-!
# Fractional ide... | Mathlib/NumberTheory/NumberField/FractionalIdeal.lean | 93 | 96 | theorem fractionalIdeal_rank (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ) :
finrank ℤ I = finrank ℤ (𝓞 K) := by | rw [finrank_eq_card_chooseBasisIndex, RingOfIntegers.rank,
finrank_eq_card_basis (basisOfFractionalIdeal K I)] |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Topology.Order.MonotoneContinuity
import M... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 523 | 531 | theorem le_iterate_pos_iff {x : ℝ} {m : ℤ} {n : ℕ} (hn : 0 < n) :
x + n * m ≤ f^[n] x ↔ x + m ≤ f x := by | simpa only [not_lt] using not_congr (f.iterate_pos_lt_iff hn)
theorem lt_iterate_pos_iff {x : ℝ} {m : ℤ} {n : ℕ} (hn : 0 < n) :
x + n * m < f^[n] x ↔ x + m < f x := by
simpa only [not_le] using not_congr (f.iterate_pos_le_iff hn)
theorem mul_floor_map_zero_le_floor_iterate_zero (n : ℕ) : ↑n * ⌊f 0⌋ ≤ ⌊f^[n] 0⌋ ... |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
/-! # Positive Definite Matrices
This file defi... | Mathlib/LinearAlgebra/Matrix/PosDef.lean | 90 | 93 | theorem transpose {M : Matrix n n R} (hM : M.PosSemidef) : Mᵀ.PosSemidef := by | refine ⟨IsHermitian.transpose hM.1, fun x => ?_⟩
convert hM.2 (star x) using 1
rw [mulVec_transpose, dotProduct_mulVec, star_star, dotProduct_comm] |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 136 | 137 | theorem edgeDensity_empty_left (t : Finset β) : edgeDensity r ∅ t = 0 := by | rw [edgeDensity, Finset.card_empty, Nat.cast_zero, zero_mul, div_zero] |
/-
Copyright (c) 2022 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson, Devon Tuma, Eric Rodriguez, Oliver Nash
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Order.Filter.AtTopBot.Field
import Mathlib.Topolog... | Mathlib/Topology/Algebra/Order/Field.lean | 87 | 89 | theorem Filter.Tendsto.atTop_mul_neg {C : 𝕜} (hC : C < 0) (hf : Tendsto f l atTop)
(hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atBot := by | have := hf.atTop_mul_pos (neg_pos.2 hC) hg.neg |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Tactic.Ring
import Mathlib.Data.PNat.Prime
/-!
# Euclidean algorithm for ℕ
This file sets up a version of the Euclidean algorithm that only works w... | Mathlib/Data/PNat/Xgcd.lean | 241 | 246 | theorem finish_isSpecial (hs : u.IsSpecial) : u.finish.IsSpecial := by | dsimp [IsSpecial, finish] at hs ⊢
rw [add_mul _ _ u.y, add_comm _ (u.x * u.y), ← hs]
ring
theorem finish_v (hr : u.r = 0) : u.finish.v = u.v := by |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.OuterMeasure.OfFunction
import Mathlib.MeasureTheory.PiSystem
/-!
# The Caratheodory σ-algebra of an outer measure
Give... | Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean | 87 | 94 | theorem measure_inter_union (h : s₁ ∩ s₂ ⊆ ∅) (h₁ : IsCaratheodory m s₁) {t : Set α} :
m (t ∩ (s₁ ∪ s₂)) = m (t ∩ s₁) + m (t ∩ s₂) := by | rw [h₁, Set.inter_assoc, Set.union_inter_cancel_left, inter_diff_assoc, union_diff_cancel_left h]
theorem isCaratheodory_iUnion_lt {s : ℕ → Set α} :
∀ {n : ℕ}, (∀ i < n, IsCaratheodory m (s i)) → IsCaratheodory m (⋃ i < n, s i)
| 0, _ => by simp [Nat.not_lt_zero]
| n + 1, h => by |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | Mathlib/Data/Set/Image.lean | 742 | 745 | theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by | rw [image_preimage_eq_range_inter, preimage_range_inter]
@[simp, mfld_simps] |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Sums of collections of Finsupp, ... | Mathlib/Data/Finsupp/BigOperators.lean | 55 | 57 | theorem List.mem_foldr_sup_support_iff [Zero M] {l : List (ι →₀ M)} {x : ι} :
x ∈ l.foldr (Finsupp.support · ⊔ ·) ∅ ↔ ∃ f ∈ l, x ∈ f.support := by | simp only [Finset.sup_eq_union, List.foldr_map, Finsupp.mem_support_iff, exists_prop] |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algeb... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 555 | 563 | theorem isWeightedHomogeneous_zero_iff_weightedTotalDegree_eq_zero {p : MvPolynomial σ R} :
IsWeightedHomogeneous w p 0 ↔ p.weightedTotalDegree w = 0 := by | rw [weightedTotalDegree, ← bot_eq_zero, Finset.sup_eq_bot_iff, bot_eq_zero, IsWeightedHomogeneous]
apply forall_congr'
intro m
rw [mem_support_iff]
/-- If `w` is a nontorsion weight function, then a multivariate polynomial has weighted total
degree zero if and only if for every `m ∈ p.support` and `x : σ`, `m ... |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | Mathlib/Combinatorics/SimpleGraph/Clique.lean | 578 | 594 | theorem cliqueSet_one (G : SimpleGraph α) : G.cliqueSet 1 = Set.range singleton :=
Set.ext fun s => by simp [eq_comm]
@[simp]
theorem cliqueSet_bot (hn : 1 < n) : (⊥ : SimpleGraph α).cliqueSet n = ∅ :=
(cliqueFree_bot hn).cliqueSet
@[simp]
theorem cliqueSet_map (hn : n ≠ 1) (G : SimpleGraph α) (f : α ↪ β) :
(... | ext s
constructor
· rintro ⟨hs, rfl⟩
have hs' : (s.preimage f f.injective.injOn).map f = s := by
classical
rw [map_eq_image, image_preimage, filter_true_of_mem]
rintro a ha |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.OrderedMonoid
i... | Mathlib/Data/PNat/Factors.lean | 259 | 271 | theorem factorMultiset_pow (n : ℕ+) (m : ℕ) :
factorMultiset (n ^ m) = m • factorMultiset n := by | let u := factorMultiset n
have : n = u.prod := (prod_factorMultiset n).symm
rw [this, ← PrimeMultiset.prod_smul]
repeat' rw [PrimeMultiset.factorMultiset_prod]
/-- Factoring a prime gives the corresponding one-element multiset. -/
theorem factorMultiset_ofPrime (p : Nat.Primes) :
(p : ℕ+).factorMultiset = Pr... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-... | Mathlib/RingTheory/Coprime/Lemmas.lean | 42 | 43 | theorem Nat.isCoprime_iff_coprime {m n : ℕ} : IsCoprime (m : ℤ) n ↔ Nat.Coprime m n := by | rw [Int.isCoprime_iff_gcd_eq_one, Int.gcd_natCast_natCast] |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | Mathlib/Data/Matroid/Basic.lean | 953 | 957 | theorem exists_isBasis' (M : Matroid α) (X : Set α) : ∃ I, M.IsBasis' I X :=
let ⟨_, hI, _⟩ := M.empty_indep.subset_isBasis'_of_subset (empty_subset X)
⟨_, hI⟩
theorem exists_isBasis_subset_isBasis (M : Matroid α) (hXY : X ⊆ Y) (hY : Y ⊆ M.E := by | aesop_mat) : |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 750 | 786 | theorem mul_lt_mul_iff_left {a b c : Ordinal} (a0 : 0 < a) : a * b < a * c ↔ b < c :=
(isNormal_mul_right a0).lt_iff
theorem mul_le_mul_iff_left {a b c : Ordinal} (a0 : 0 < a) : a * b ≤ a * c ↔ b ≤ c :=
(isNormal_mul_right a0).le_iff
theorem mul_lt_mul_of_pos_left {a b c : Ordinal} (h : a < b) (c0 : 0 < c) : c * ... | simpa only [mul_zero] using mul_lt_mul_of_pos_left h₂ h₁
theorem mul_ne_zero {a b : Ordinal} : a ≠ 0 → b ≠ 0 → a * b ≠ 0 := by
simpa only [Ordinal.pos_iff_ne_zero] using mul_pos
theorem le_of_mul_le_mul_left {a b c : Ordinal} (h : c * a ≤ c * b) (h0 : 0 < c) : a ≤ b :=
le_imp_le_of_lt_imp_lt (fun h' => mul_lt_mul... |
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Dynamics.PeriodicPts.Defs
import Mathlib.GroupTheory.G... | 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] |
/-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Algebra.... | Mathlib/LinearAlgebra/UnitaryGroup.lean | 71 | 73 | theorem det_of_mem_unitary {A : Matrix n n α} (hA : A ∈ Matrix.unitaryGroup n α) :
A.det ∈ unitary α := by | constructor |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 269 | 271 | theorem intervalIntegrable_one_div_one_add_sq :
IntervalIntegrable (fun x : ℝ => 1 / (↑1 + x ^ 2)) μ a b := by | refine (continuous_const.div ?_ fun x => ?_).intervalIntegrable a b |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.LinearAlgebra.Determinant
/-!
# Gershgorin's circle theorem
Th... | Mathlib/LinearAlgebra/Matrix/Gershgorin.lean | 26 | 56 | theorem eigenvalue_mem_ball {μ : K} (hμ : Module.End.HasEigenvalue (Matrix.toLin' A) μ) :
∃ k, μ ∈ Metric.closedBall (A k k) (∑ j ∈ Finset.univ.erase k, ‖A k j‖) := by | cases isEmpty_or_nonempty n
· exfalso
exact hμ Submodule.eq_bot_of_subsingleton
· obtain ⟨v, h_eg, h_nz⟩ := hμ.exists_hasEigenvector
obtain ⟨i, -, h_i⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty (fun i => ‖v i‖)
have h_nz : v i ≠ 0 := by
contrapose! h_nz
ext j
rw [Pi.zero_apply,... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.Prime.Basic
import Ma... | Mathlib/Data/Nat/Factors.lean | 188 | 194 | theorem perm_primeFactorsList_mul {a b : ℕ} (ha : a ≠ 0) (hb : b ≠ 0) :
(a * b).primeFactorsList ~ a.primeFactorsList ++ b.primeFactorsList := by | refine (primeFactorsList_unique ?_ ?_).symm
· rw [List.prod_append, prod_primeFactorsList ha, prod_primeFactorsList hb]
· intro p hp
rw [List.mem_append] at hp
rcases hp with hp' | hp' <;> exact prime_of_mem_primeFactorsList hp' |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 461 | 463 | theorem addHaar_closedBall_mul (x : E) {r : ℝ} (hr : 0 ≤ r) {s : ℝ} (hs : 0 ≤ s) :
μ (closedBall x (r * s)) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (closedBall 0 s) := by | have : closedBall (0 : E) (r * s) = r • closedBall (0 : E) s := by |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
/-! # Formal power series (in one va... | Mathlib/RingTheory/PowerSeries/Order.lean | 162 | 164 | theorem order_add_of_order_eq (φ ψ : R⟦X⟧) (h : order φ ≠ order ψ) :
order (φ + ψ) = order φ ⊓ order ψ := by | refine le_antisymm ?_ (min_order_le_order_add _ _) |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Induction
import Mathlib.Data.List.TakeWhile
/-!
# Dropping or taking from lists on the right
Taking or removing element from the tail e... | Mathlib/Data/List/DropRight.lean | 121 | 122 | theorem rdropWhile_eq_nil_iff : rdropWhile p l = [] ↔ ∀ x ∈ l, p x := by | simp [rdropWhile] |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.Order.Compact
import Mathlib.Topology.MetricSpace.ProperSpace
import M... | Mathlib/Topology/MetricSpace/Bounded.lean | 190 | 194 | theorem isBounded_range_of_tendsto_cofinite_uniformity {f : β → α}
(hf : Tendsto (Prod.map f f) (.cofinite ×ˢ .cofinite) (𝓤 α)) : IsBounded (range f) := by | rcases (hasBasis_cofinite.prod_self.tendsto_iff uniformity_basis_dist).1 hf 1 zero_lt_one with
⟨s, hsf, hs1⟩
rw [← image_union_image_compl_eq_range] |
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