Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.calculus.deriv.bas... | Mathlib/Analysis/Calculus/Deriv/Basic.lean | 779 | 781 | theorem HasDerivAt.le_of_lipschitzOn {f : 𝕜 → F} {f' : F} {x₀ : 𝕜} (hf : HasDerivAt f f' x₀)
{s : Set 𝕜} (hs : s ∈ 𝓝 x₀) {C : ℝ≥0} (hlip : LipschitzOnWith C f s) : ‖f'‖ ≤ C := by |
simpa using HasFDerivAt.le_of_lipschitzOn hf.hasFDerivAt hs hlip
|
/-
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.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | Mathlib/RingTheory/Polynomial/Content.lean | 291 | 303 | theorem isUnit_primPart_C (r : R) : IsUnit (C r).primPart := by |
by_cases h0 : r = 0
· simp [h0]
unfold IsUnit
refine
⟨⟨C ↑(normUnit r)⁻¹, C ↑(normUnit r), by rw [← RingHom.map_mul, Units.inv_mul, C_1], by
rw [← RingHom.map_mul, Units.mul_inv, C_1]⟩,
?_⟩
rw [← normalize_eq_zero, ← C_eq_zero] at h0
apply mul_left_cancel₀ h0
conv_rhs => rw [← content_C... |
/-
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 Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.... | Mathlib/Data/Nat/GCD/Basic.lean | 234 | 235 | theorem coprime_self_sub_left {m n : ℕ} (h : m ≤ n) : Coprime (n - m) n ↔ Coprime m n := by |
rw [Coprime, Coprime, gcd_self_sub_left h]
|
/-
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.Algebra.Polynomial.Taylor
import Mathlib.FieldTheory.RatFunc.AsPolynomial
#align_import field_theory.laurent from "leanprover-community/mathlib"@"70... | Mathlib/FieldTheory/Laurent.lean | 39 | 45 | theorem taylor_mem_nonZeroDivisors (hp : p ∈ R[X]⁰) : taylor r p ∈ R[X]⁰ := by |
rw [mem_nonZeroDivisors_iff]
intro x hx
have : x = taylor (r - r) x := by simp
rwa [this, sub_eq_add_neg, ← taylor_taylor, ← taylor_mul,
LinearMap.map_eq_zero_iff _ (taylor_injective _), mul_right_mem_nonZeroDivisors_eq_zero_iff hp,
LinearMap.map_eq_zero_iff _ (taylor_injective _)] at hx
|
/-
Copyright (c) 2021 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Yaël Dillies
-/
import Mathlib.Computability.NFA
#align_import computability.epsilon_NFA from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
/-!
... | Mathlib/Computability/EpsilonNFA.lean | 82 | 83 | theorem mem_stepSet_iff : s ∈ M.stepSet S a ↔ ∃ t ∈ S, s ∈ M.εClosure (M.step t a) := by |
simp_rw [stepSet, mem_iUnion₂, exists_prop]
|
/-
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, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
... | Mathlib/LinearAlgebra/Prod.lean | 548 | 551 | theorem map_inl : p.map (inl R M M₂) = prod p ⊥ := by |
ext ⟨x, y⟩
simp only [and_left_comm, eq_comm, mem_map, Prod.mk.inj_iff, inl_apply, mem_bot, exists_eq_left',
mem_prod]
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Algebra.Order.Floor
import Mathlib.Topology.Algebra.Order.Group
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.floor fro... | Mathlib/Topology/Algebra/Order/Floor.lean | 96 | 98 | theorem tendsto_floor_left_pure_sub_one (n : ℤ) :
Tendsto (floor : α → ℤ) (𝓝[<] n) (pure (n - 1)) := by |
simpa only [ceil_intCast] using tendsto_floor_left_pure_ceil_sub_one (n : α)
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 3,563 | 3,568 | theorem mem_map_swap (x : α) (y : β) (xs : List (α × β)) :
(y, x) ∈ map Prod.swap xs ↔ (x, y) ∈ xs := by |
induction' xs with x xs xs_ih
· simp only [not_mem_nil, map_nil]
· cases' x with a b
simp only [mem_cons, Prod.mk.inj_iff, map, Prod.swap_prod_mk, Prod.exists, xs_ih, and_comm]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.morphisms.ring_hom_properties from "leanprover-... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 250 | 268 | theorem sourceAffineLocally_of_source_open_cover_aux (h₁ : RingHom.RespectsIso @P)
(h₃ : RingHom.OfLocalizationSpanTarget @P) {X Y : Scheme.{u}} (f : X ⟶ Y) (U : X.affineOpens)
(s : Set (X.presheaf.obj (op U.1))) (hs : Ideal.span s = ⊤)
(hs' : ∀ r : s, P (Scheme.Γ.map (Scheme.ιOpens (X.basicOpen r.1) ≫ f).o... |
apply_fun Ideal.map (X.presheaf.map (eqToHom U.1.openEmbedding_obj_top).op) at hs
rw [Ideal.map_span, Ideal.map_top] at hs
apply h₃.ofIsLocalization h₁ _ _ hs
rintro ⟨s, r, hr, hs⟩
refine ⟨_, _, _, @AlgebraicGeometry.Γ_restrict_isLocalization (X ∣_ᵤ U.1) U.2 s, ?_⟩
rw [RingHom.algebraMap_toAlgebra, ← CommR... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury G. Kudryashov, Scott Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.BigOperators.Finsupp
impor... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 174 | 176 | theorem mul_def {f g : MonoidAlgebra k G} :
f * g = f.sum fun a₁ b₁ => g.sum fun a₂ b₂ => single (a₁ * a₂) (b₁ * b₂) := by |
with_unfolding_all rfl
|
/-
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
#align_import set_theory.ordinal.principal from "leanprover-community/mathlib"@"31b269b6093548394... | Mathlib/SetTheory/Ordinal/Principal.lean | 244 | 245 | theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c := by |
rw [← Ordinal.add_sub_cancel_of_le h₂, ← add_assoc, add_omega_opow h₁]
|
/-
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, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.So... | Mathlib/Algebra/Polynomial/Basic.lean | 943 | 944 | theorem smul_X_eq_monomial {n} : a • X ^ n = monomial n (a : R) := by |
rw [X_pow_eq_monomial, smul_monomial, smul_eq_mul, mul_one]
|
/-
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, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 237 | 238 | theorem SymmetricRel.inter {U V : Set (α × α)} (hU : SymmetricRel U) (hV : SymmetricRel V) :
SymmetricRel (U ∩ V) := by | rw [SymmetricRel, preimage_inter, hU.eq, hV.eq]
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Ashvni Narayanan
-/
import Mathlib.Algebra.Order.Group.TypeTags
import Mathlib.FieldTheory.RatFunc.Degree
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Math... | Mathlib/NumberTheory/FunctionField.lean | 168 | 176 | theorem InftyValuation.map_mul' (x y : RatFunc Fq) :
inftyValuationDef Fq (x * y) = inftyValuationDef Fq x * inftyValuationDef Fq y := by |
rw [inftyValuationDef, inftyValuationDef, inftyValuationDef]
by_cases hx : x = 0
· rw [hx, zero_mul, if_pos (Eq.refl _), zero_mul]
· by_cases hy : y = 0
· rw [hy, mul_zero, if_pos (Eq.refl _), mul_zero]
· rw [if_neg hx, if_neg hy, if_neg (mul_ne_zero hx hy), ← WithZero.coe_mul, WithZero.coe_inj,
... |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.L... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 495 | 496 | theorem kahler_apply_self (x : E) : o.kahler x x = ‖x‖ ^ 2 := by |
simp [kahler_apply_apply, real_inner_self_eq_norm_sq]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlg... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 216 | 220 | theorem adjMatrix_vecMul_apply [NonAssocSemiring α] (v : V) (vec : V → α) :
(vec ᵥ* G.adjMatrix α) v = ∑ u ∈ G.neighborFinset v, vec u := by |
simp only [← dotProduct_adjMatrix, vecMul]
refine congr rfl ?_; ext x
rw [← transpose_apply (adjMatrix α G) x v, transpose_adjMatrix]
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,588 | 1,655 | theorem maxTail_isGood (l : MaxProducts C ho)
(h₁: ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C (ord I · < o))))) :
l.val.Tail.isGood (C' C ho) := by |
have : Inhabited I := ⟨term I ho⟩
-- Write `l.Tail` as a linear combination of smaller products:
intro h
rw [Finsupp.mem_span_image_iff_total, ← max_eq_eval C hsC ho] at h
obtain ⟨m, ⟨hmmem, hmsum⟩⟩ := h
rw [Finsupp.total_apply] at hmsum
-- Write the image of `l` under `Linear_CC'` as `Linear_CC'` appli... |
/-
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.Algebra.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monoton... | Mathlib/Data/Nat/Totient.lean | 117 | 126 | theorem _root_.ZMod.card_units_eq_totient (n : ℕ) [NeZero n] [Fintype (ZMod n)ˣ] :
Fintype.card (ZMod n)ˣ = φ n :=
calc
Fintype.card (ZMod n)ˣ = Fintype.card { x : ZMod n // x.val.Coprime n } :=
Fintype.card_congr ZMod.unitsEquivCoprime
_ = φ n := by |
obtain ⟨m, rfl⟩ : ∃ m, n = m + 1 := exists_eq_succ_of_ne_zero NeZero.out
simp only [totient, Finset.card_eq_sum_ones, Fintype.card_subtype, Finset.sum_filter, ←
Fin.sum_univ_eq_sum_range, @Nat.coprime_comm (m + 1)]
rfl
|
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4... | Mathlib/Algebra/CubicDiscriminant.lean | 137 | 138 | theorem of_a_eq_zero (ha : P.a = 0) : P.toPoly = C P.b * X ^ 2 + C P.c * X + C P.d := by |
rw [toPoly, ha, C_0, zero_mul, zero_add]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mat... | Mathlib/LinearAlgebra/Matrix/Block.lean | 260 | 263 | theorem det_of_upperTriangular [LinearOrder m] (h : M.BlockTriangular id) :
M.det = ∏ i : m, M i i := by |
haveI : DecidableEq R := Classical.decEq _
simp_rw [h.det, image_id, det_toSquareBlock_id]
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathl... | Mathlib/Data/Real/GoldenRatio.lean | 140 | 146 | theorem gold_irrational : Irrational φ := by |
have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
have := this.rat_add 1
have := this.rat_mul (show (0.5 : ℚ) ≠ 0 by norm_num)
convert this
norm_num
field_simp
|
/-
Copyright (c) 2023 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.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 360 | 366 | theorem LineDifferentiableAt.congr_of_eventuallyEq
(h : LineDifferentiableAt 𝕜 f x v) (hL : f₁ =ᶠ[𝓝 x] f) :
LineDifferentiableAt 𝕜 f₁ x v := by |
apply DifferentiableAt.congr_of_eventuallyEq h
let F := fun (t : 𝕜) ↦ x + t • v
rw [show x = F 0 by simp [F]] at hL
exact (Continuous.continuousAt (by continuity)).preimage_mem_nhds hL
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
#align_import analysis.special_functions.gamma.bohr_mollerup from "leanprover-community/mathlib"@"... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 509 | 515 | theorem Gamma_mul_Gamma_add_half_of_pos {s : ℝ} (hs : 0 < s) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * √π := by |
rw [← doublingGamma_eq_Gamma (mul_pos two_pos hs), doublingGamma,
mul_div_cancel_left₀ _ (two_ne_zero' ℝ), (by abel : 1 - 2 * s = -(2 * s - 1)),
rpow_neg zero_le_two]
field_simp [(sqrt_pos_of_pos pi_pos).ne', (rpow_pos_of_pos two_pos (2 * s - 1)).ne']
ring
|
/-
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.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | Mathlib/RingTheory/Polynomial/Content.lean | 333 | 343 | theorem gcd_content_eq_of_dvd_sub {a : R} {p q : R[X]} (h : C a ∣ p - q) :
GCDMonoid.gcd a p.content = GCDMonoid.gcd a q.content := by |
rw [content_eq_gcd_range_of_lt p (max p.natDegree q.natDegree).succ
(lt_of_le_of_lt (le_max_left _ _) (Nat.lt_succ_self _))]
rw [content_eq_gcd_range_of_lt q (max p.natDegree q.natDegree).succ
(lt_of_le_of_lt (le_max_right _ _) (Nat.lt_succ_self _))]
apply Finset.gcd_eq_of_dvd_sub
intro x _
cases... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Init.Order.LinearOrder
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Subtype
import Mathlib.Tactic.Spread
import Mathlib.Tactic.Conv... | Mathlib/Order/Basic.lean | 985 | 987 | theorem update_le_update_iff :
Function.update x i a ≤ Function.update y i b ↔ a ≤ b ∧ ∀ (j) (_ : j ≠ i), x j ≤ y j := by |
simp (config := { contextual := true }) [update_le_iff]
|
/-
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.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 257 | 267 | theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by |
by_cases h₀ : z = 0
· simp [h₀, lt_irrefl, Real.pi_ne_zero.symm]
constructor
· intro h
rw [← abs_mul_cos_add_sin_mul_I z, h]
simp [h₀]
· cases' z with x y
rintro ⟨h : x < 0, rfl : y = 0⟩
rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)]
simp [← ofReal_def]
|
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheory.Products.Basic
#align_import cat... | Mathlib/CategoryTheory/Monoidal/Category.lean | 235 | 237 | theorem whiskerLeft_comp (W : C) {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) :
W ◁ (f ≫ g) = W ◁ f ≫ W ◁ g := by |
simp only [← id_tensorHom, ← tensor_comp, comp_id]
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Algebra.Module.Basic
import Mathlib.Topology.Separation
#align_import topology.supp... | Mathlib/Topology/Support.lean | 241 | 243 | theorem _root_.hasCompactMulSupport_comp_left (hg : ∀ {x}, g x = 1 ↔ x = 1) :
HasCompactMulSupport (g ∘ f) ↔ HasCompactMulSupport f := by |
simp_rw [hasCompactMulSupport_def, mulSupport_comp_eq g (@hg) f]
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.qpf.univariate.basic from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
/-!
... | Mathlib/Data/QPF/Univariate/Basic.lean | 297 | 306 | theorem Fix.ind_aux (a : q.P.A) (f : q.P.B a → q.P.W) :
Fix.mk (abs ⟨a, fun x => ⟦f x⟧⟩) = ⟦⟨a, f⟩⟧ := by |
have : Fix.mk (abs ⟨a, fun x => ⟦f x⟧⟩) = ⟦Wrepr ⟨a, f⟩⟧ := by
apply Quot.sound; apply Wequiv.abs'
rw [PFunctor.W.dest_mk, abs_map, abs_repr, ← abs_map, PFunctor.map_eq]
simp only [Wrepr, recF_eq, PFunctor.W.dest_mk, abs_repr, Function.comp]
rfl
rw [this]
apply Quot.sound
apply Wrepr_equiv
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e... | Mathlib/Topology/MetricSpace/Thickening.lean | 377 | 379 | theorem thickening_iUnion (δ : ℝ) (f : ι → Set α) :
thickening δ (⋃ i, f i) = ⋃ i, thickening δ (f i) := by |
simp_rw [thickening, infEdist_iUnion, iInf_lt_iff, setOf_exists]
|
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.D... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 170 | 171 | theorem coe_of_rat_eq : ((IntFractPair.of q).mapFr (↑) : IntFractPair K) = IntFractPair.of v := by |
simp [IntFractPair.of, v_eq_q]
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.measure.probability_measure from "leanprove... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 518 | 527 | theorem tendsto_normalize_iff_tendsto {γ : Type*} {F : Filter γ} {μs : γ → FiniteMeasure Ω}
(nonzero : μ ≠ 0) :
Tendsto (fun i => (μs i).normalize) F (𝓝 μ.normalize) ∧
Tendsto (fun i => (μs i).mass) F (𝓝 μ.mass) ↔
Tendsto μs F (𝓝 μ) := by |
constructor
· rintro ⟨normalized_lim, mass_lim⟩
exact tendsto_of_tendsto_normalize_testAgainstNN_of_tendsto_mass normalized_lim mass_lim
· intro μs_lim
exact ⟨tendsto_normalize_of_tendsto μs_lim nonzero, μs_lim.mass⟩
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Algebra.Order.Floor
import Mathlib.Topology.Algebra.Order.Group
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.floor fro... | Mathlib/Topology/Algebra/Order/Floor.lean | 194 | 212 | theorem ContinuousOn.comp_fract' {f : β → α → γ} (h : ContinuousOn (uncurry f) <| univ ×ˢ I)
(hf : ∀ s, f s 0 = f s 1) : Continuous fun st : β × α => f st.1 (fract st.2) := by |
change Continuous (uncurry f ∘ Prod.map id fract)
rw [continuous_iff_continuousAt]
rintro ⟨s, t⟩
rcases em (∃ n : ℤ, t = n) with (⟨n, rfl⟩ | ht)
· rw [ContinuousAt, nhds_prod_eq, ← nhds_left'_sup_nhds_right (n : α), prod_sup, tendsto_sup]
constructor
· refine (((h (s, 1) ⟨trivial, zero_le_one, le_rfl... |
/-
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.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c066... | Mathlib/Algebra/GCDMonoid/Finset.lean | 270 | 281 | theorem extract_gcd (f : β → α) (hs : s.Nonempty) :
∃ g : β → α, (∀ b ∈ s, f b = s.gcd f * g b) ∧ s.gcd g = 1 := by |
classical
by_cases h : ∀ x ∈ s, f x = (0 : α)
· refine ⟨fun _ ↦ 1, fun b hb ↦ by rw [h b hb, gcd_eq_zero_iff.2 h, mul_one], ?_⟩
rw [gcd_eq_gcd_image, image_const hs, gcd_singleton, id, normalize_one]
· choose g' hg using @gcd_dvd _ _ _ _ s f
push_neg at h
refine ⟨fun b ↦ if hb : b ∈ s t... |
/-
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.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 1,780 | 1,783 | theorem pos_size_of_mem {x : α} {t : Ordset α} (h_mem : x ∈ t) : 0 < size t := by |
simp? [Membership.mem, mem] at h_mem says
simp only [Membership.mem, mem, Bool.decide_eq_true] at h_mem
apply Ordnode.pos_size_of_mem t.property.sz h_mem
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Joël Riou
-/
import Mathlib.CategoryTheory.CommSq
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.C... | Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean | 559 | 562 | theorem of_isBilimit {b : BinaryBicone X Y} (h : b.IsBilimit) :
IsPullback b.fst b.snd (0 : X ⟶ 0) (0 : Y ⟶ 0) := by |
convert IsPullback.of_is_product' h.isLimit HasZeroObject.zeroIsTerminal
<;> apply Subsingleton.elim
|
/-
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
#align_import geometry.euclidean.angle.oriented.rig... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 629 | 634 | theorem oangle_left_eq_arctan_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₃ p₁ p₂ = Real.arctan (dist p₃ 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,
angle_eq_arctan_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(left_ne_of_oangle_eq_pi_div_two h)]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemm... | Mathlib/Algebra/Order/Floor.lean | 1,671 | 1,672 | theorem map_fract (f : F) (hf : StrictMono f) (a : α) : fract (f a) = f (fract a) := by |
simp_rw [fract, map_sub, map_intCast, map_floor _ hf]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Quotient
import Mathlib.Combinatorics.Quiver.Path
#align_import category_theory.path_category fro... | Mathlib/CategoryTheory/PathCategory.lean | 176 | 180 | theorem composePath_comp {X Y Z : C} (f : Path X Y) (g : Path Y Z) :
composePath (f.comp g) = composePath f ≫ composePath g := by |
induction' g with Y' Z' g e ih
· simp
· simp [ih]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import geometry.euclidean.angle.un... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | 219 | 221 | theorem angle_eq_pi_iff {x y : V} : angle x y = π ↔ x ≠ 0 ∧ ∃ r : ℝ, r < 0 ∧ y = r • x := by |
rw [angle, ← real_inner_div_norm_mul_norm_eq_neg_one_iff, Real.arccos_eq_pi, LE.le.le_iff_eq]
exact (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x y)).1
|
/-
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.Complex.Basic
import Mathlib.Analysis.SpecificLimits.Normed
/-!
# Abel's limit theorem
If a real or complex power series for a function has radius o... | Mathlib/Analysis/Complex/AbelLimit.lean | 155 | 231 | theorem tendsto_tsum_powerSeries_nhdsWithin_stolzSet
(h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) {M : ℝ} :
Tendsto (fun z ↦ ∑' n, f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l) := by |
-- If `M ≤ 1` the Stolz set is empty and the statement is trivial
cases' le_or_lt M 1 with hM hM
· simp_rw [stolzSet_empty hM, nhdsWithin_empty, tendsto_bot]
-- Abbreviations
let s := fun n ↦ ∑ i ∈ range n, f i
let g := fun z ↦ ∑' n, f n * z ^ n
have hm := Metric.tendsto_atTop.mp h
rw [Metric.tendsto_n... |
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a14... | Mathlib/Algebra/Homology/ImageToKernel.lean | 330 | 335 | theorem homology'.comp_right_eq_comp_left {V : Type*} [Category V] {A₁ B₁ C₁ A₂ B₂ C₂ A₃ B₃ C₃ : V}
{f₁ : A₁ ⟶ B₁} {g₁ : B₁ ⟶ C₁} {f₂ : A₂ ⟶ B₂} {g₂ : B₂ ⟶ C₂} {f₃ : A₃ ⟶ B₃} {g₃ : B₃ ⟶ C₃}
{α₁ : Arrow.mk f₁ ⟶ Arrow.mk f₂} {β₁ : Arrow.mk g₁ ⟶ Arrow.mk g₂}
{α₂ : Arrow.mk f₂ ⟶ Arrow.mk f₃} {β₂ : Arrow.mk g₂ ⟶... |
simp only [Arrow.comp_left, Arrow.comp_right, p₁, p₂]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 407 | 412 | theorem Disjoint.support_mul (h : Disjoint f g) : (f * g).support = f.support ∪ g.support := by |
refine le_antisymm (support_mul_le _ _) fun a => ?_
rw [mem_union, mem_support, mem_support, mem_support, mul_apply, ← not_and_or, not_imp_not]
exact
(h a).elim (fun hf h => ⟨hf, f.apply_eq_iff_eq.mp (h.trans hf.symm)⟩) fun hg h =>
⟨(congr_arg f hg).symm.trans h, hg⟩
|
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instance... | Mathlib/Algebra/AddConstMap/Basic.lean | 137 | 139 | theorem map_nsmul_add [AddCommMonoid G] [AddMonoid H] [AddConstMapClass F G H a b]
(f : F) (n : ℕ) (x : G) : f (n • a + x) = f x + n • b := by |
rw [add_comm, map_add_nsmul]
|
/-
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.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d4510... | Mathlib/Data/Fintype/Basic.lean | 779 | 782 | theorem toFinset_insert [DecidableEq α] {a : α} {s : Set α} [Fintype (insert a s : Set α)]
[Fintype s] : (insert a s).toFinset = insert a s.toFinset := by |
ext
simp
|
/-
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.Trigonometric.Complex
#align_import analysis.special_function... | Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean | 37 | 44 | theorem tendsto_abs_tan_of_cos_eq_zero {x : ℂ} (hx : cos x = 0) :
Tendsto (fun x => abs (tan x)) (𝓝[≠] x) atTop := by |
simp only [tan_eq_sin_div_cos, ← norm_eq_abs, norm_div]
have A : sin x ≠ 0 := fun h => by simpa [*, sq] using sin_sq_add_cos_sq x
have B : Tendsto cos (𝓝[≠] x) (𝓝[≠] 0) :=
hx ▸ (hasDerivAt_cos x).tendsto_punctured_nhds (neg_ne_zero.2 A)
exact continuous_sin.continuousWithinAt.norm.mul_atTop (norm_pos_iff... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison, Adam Topaz
-/
import Mathlib.Tactic.Linarith
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fintype.Sort
import Mathlib.Order.Category.Nonem... | Mathlib/AlgebraicTopology/SimplexCategory.lean | 858 | 862 | theorem image_ι_eq {Δ Δ'' : SimplexCategory} {φ : Δ ⟶ Δ''} {e : Δ ⟶ image φ} [Epi e]
{i : image φ ⟶ Δ''} [Mono i] (fac : e ≫ i = φ) : image.ι φ = i := by |
haveI := strongEpi_of_epi e
rw [← image.isoStrongEpiMono_hom_comp_ι e i fac,
SimplexCategory.eq_id_of_isIso (image.isoStrongEpiMono e i fac).hom, Category.id_comp]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Basic
#align_import data.multiset.locally_finite from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
/-!... | Mathlib/Order/Interval/Multiset.lean | 369 | 370 | theorem Ico_sub_Ico_right (a b c : α) : Ico a b - Ico c b = Ico a (min b c) := by |
rw [Ico, Ico, Ico, ← Finset.sdiff_val, Finset.Ico_diff_Ico_right]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, James Gallicchio
-/
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
/-!
# Pairwise relations on a list
This file provides basic results about `List.... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 223 | 235 | theorem sublist_eq_map_get (h : l' <+ l) : ∃ is : List (Fin l.length),
l' = map (get l) is ∧ is.Pairwise (· < ·) := by |
induction h with
| slnil => exact ⟨[], by simp⟩
| cons _ _ IH =>
let ⟨is, IH⟩ := IH
refine ⟨is.map (·.succ), ?_⟩
simp [comp, pairwise_map]
exact IH
| cons₂ _ _ IH =>
rcases IH with ⟨is,IH⟩
refine ⟨⟨0, by simp [Nat.zero_lt_succ]⟩ :: is.map (·.succ), ?_⟩
simp [comp_def, pairwise_map, ... |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Heather Macbeth, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topolo... | Mathlib/Analysis/Normed/Group/InfiniteSum.lean | 119 | 121 | theorem HasSum.norm_le_of_bounded {f : ι → E} {g : ι → ℝ} {a : E} {b : ℝ} (hf : HasSum f a)
(hg : HasSum g b) (h : ∀ i, ‖f i‖ ≤ g i) : ‖a‖ ≤ b := by |
classical exact le_of_tendsto_of_tendsto' hf.norm hg fun _s ↦ norm_sum_le_of_le _ fun i _hi ↦ h i
|
/-
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 Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc429200506... | Mathlib/Data/Set/Image.lean | 249 | 251 | theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by |
ext x
exact exists_congr fun a ↦ and_congr_right fun ha ↦ by rw [h a ha]
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
... | Mathlib/Probability/Martingale/Upcrossing.lean | 727 | 735 | theorem mul_integral_upcrossingsBefore_le_integral_pos_part_aux [IsFiniteMeasure μ]
(hf : Submartingale f ℱ μ) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[fun ω => (f N ω - a)⁺] := by |
refine le_trans (le_of_eq ?_)
(integral_mul_upcrossingsBefore_le_integral (hf.sub_martingale (martingale_const _ _ _)).pos
(fun ω => posPart_nonneg _)
(fun ω => posPart_nonneg _) (sub_pos.2 hab))
simp_rw [sub_zero, ← upcrossingsBefore_pos_eq hab]
rfl
|
/-
Copyright (c) 2021 Filippo A. E. Nuccio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Filippo A. E. Nuccio, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Linear... | Mathlib/Data/Matrix/Kronecker.lean | 155 | 160 | theorem kroneckerMap_reindex (f : α → β → γ) (el : l ≃ l') (em : m ≃ m') (en : n ≃ n') (ep : p ≃ p')
(M : Matrix l m α) (N : Matrix n p β) :
kroneckerMap f (reindex el em M) (reindex en ep N) =
reindex (el.prodCongr en) (em.prodCongr ep) (kroneckerMap f M N) := by |
ext ⟨i, i'⟩ ⟨j, j'⟩
rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mitchell Lee
-/
import Mathlib.Topology.Algebra.InfiniteSum.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.Monoid
/-!
# Lemmas on infini... | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 35 | 35 | theorem hasProd_one : HasProd (fun _ ↦ 1 : β → α) 1 := by | simp [HasProd, tendsto_const_nhds]
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
i... | Mathlib/Analysis/Calculus/Rademacher.lean | 313 | 320 | theorem ae_differentiableAt_of_real (hf : LipschitzWith C f) :
∀ᵐ x ∂μ, DifferentiableAt ℝ f x := by |
obtain ⟨s, s_count, s_dense⟩ : ∃ (s : Set E), s.Countable ∧ Dense s :=
TopologicalSpace.exists_countable_dense E
have hs : sphere 0 1 ⊆ closure s := by rw [s_dense.closure_eq]; exact subset_univ _
filter_upwards [hf.ae_exists_fderiv_of_countable s_count]
rintro x ⟨L, hL⟩
exact (hf.hasFderivAt_of_hasLineD... |
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.DFinsupp.Lex
import Mathlib.Order.GameAdd
import Mathlib.Order.Antisymmetrization
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Tactic.AdaptationNot... | Mathlib/Data/DFinsupp/WellFounded.lean | 134 | 153 | theorem Lex.acc_single [DecidableEq ι] {i : ι} (hi : Acc (rᶜ ⊓ (· ≠ ·)) i) :
∀ a, Acc (DFinsupp.Lex r s) (single i a) := by |
induction' hi with i _ ih
refine fun a => WellFounded.induction (hs i)
(C := fun x ↦ Acc (DFinsupp.Lex r s) (single i x)) a fun a ha ↦ ?_
refine Acc.intro _ fun x ↦ ?_
rintro ⟨k, hr, hs⟩
rw [single_apply] at hs
split_ifs at hs with hik
swap
· exact (hbot hs).elim
subst hik
classical
refine ... |
/-
Copyright (c) 2023 Kim Liesinger. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Liesinger
-/
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.List.Infix
import Mathlib.Data.List.MinMax
import Mathlib.Data.List.EditDistance.Defs
/-!
# Lowe... | Mathlib/Data/List/EditDistance/Bounds.lean | 75 | 79 | theorem le_suffixLevenshtein_append_minimum (xs : List α) (ys₁ ys₂) :
(suffixLevenshtein C xs ys₂).1.minimum ≤ (suffixLevenshtein C xs (ys₁ ++ ys₂)).1.minimum := by |
induction ys₁ with
| nil => exact le_refl _
| cons y ys₁ ih => exact ih.trans (le_suffixLevenshtein_cons_minimum _ _ _)
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Asymptotics.Theta
import Mathlib.Analysis.Normed.Order.Basic
#align_import analysis.asymp... | Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 218 | 221 | theorem isEquivalent_of_tendsto_one (hz : ∀ᶠ x in l, v x = 0 → u x = 0)
(huv : Tendsto (u / v) l (𝓝 1)) : u ~[l] v := by |
rw [isEquivalent_iff_exists_eq_mul]
exact ⟨u / v, huv, hz.mono fun x hz' ↦ (div_mul_cancel_of_imp hz').symm⟩
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Floris van Doorn
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.Opposites
import Mathlib.Algebra.Order.GroupW... | Mathlib/Data/Set/Pointwise/Basic.lean | 288 | 289 | theorem inv_insert (a : α) (s : Set α) : (insert a s)⁻¹ = insert a⁻¹ s⁻¹ := by |
rw [insert_eq, union_inv, inv_singleton, insert_eq]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 329 | 330 | theorem toDual_apply_left (m : M) (i : ι) : b.toDual m (b i) = b.repr m i := by |
rw [← b.toDual_total_left, b.total_repr]
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Fintype
#align_import group_the... | Mathlib/GroupTheory/Perm/Fin.lean | 118 | 120 | theorem support_finRotate {n : ℕ} : support (finRotate (n + 2)) = Finset.univ := by |
ext
simp
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.LinearAlgebra.Ray
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.ray from "leanprover-community/mathlib"... | Mathlib/Analysis/NormedSpace/Ray.lean | 49 | 52 | theorem norm_smul_eq (h : SameRay ℝ x y) : ‖x‖ • y = ‖y‖ • x := by |
rcases h.exists_eq_smul with ⟨u, a, b, ha, hb, -, rfl, rfl⟩
simp only [norm_smul_of_nonneg, *, mul_smul]
rw [smul_comm, smul_comm b, smul_comm a b u]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Data.List.Range
#align_import data.fin.vec_notation from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b... | Mathlib/Data/Fin/VecNotation.lean | 374 | 375 | theorem vecHead_vecAlt1 (hm : m + 2 = n + 1 + (n + 1)) (v : Fin (m + 2) → α) :
vecHead (vecAlt1 hm v) = v 1 := by | simp [vecHead, vecAlt1]
|
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Data.Set.UnionLift
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
import Mathlib.RingTh... | Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean | 980 | 983 | theorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :
(iSupLift K dir f hf T hT).comp (inclusion h) = f i := by |
ext
simp only [NonUnitalAlgHom.comp_apply, iSupLift_inclusion]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathlib.GroupTheory.GroupAction.Ring
im... | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 478 | 481 | theorem det_updateColumn_add_self (A : Matrix n n R) {i j : n} (hij : i ≠ j) :
det (updateColumn A i fun k => A k i + A k j) = det A := by |
rw [← det_transpose, ← updateRow_transpose, ← det_transpose A]
exact det_updateRow_add_self Aᵀ hij
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
/-!
# Finite inter... | Mathlib/Order/Interval/Finset/Nat.lean | 126 | 127 | theorem card_fintypeIoc : Fintype.card (Set.Ioc a b) = b - a := by |
rw [Fintype.card_ofFinset, card_Ioc]
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 394 | 403 | theorem evalFacProp {l : Products I} (J : I → Prop)
(h : ∀ a, a ∈ l.val → J a) [∀ j, Decidable (J j)] :
l.eval (π C J) ∘ ProjRestrict C J = l.eval C := by |
ext x
dsimp [ProjRestrict]
rw [Products.eval_eq, Products.eval_eq]
congr
apply forall_congr; intro i
apply forall_congr; intro hi
simp [h i hi, Proj]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.DeriveFintype
#align_import geometry.eucl... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 875 | 881 | theorem exists_circumsphere_eq_of_cospherical {ps : Set P} {n : ℕ} [FiniteDimensional ℝ V]
(hd : finrank ℝ V = n) (hc : Cospherical ps) :
∃ c : Sphere P, ∀ sx : Simplex ℝ P n, Set.range sx.points ⊆ ps → sx.circumsphere = c := by |
haveI : Nonempty (⊤ : AffineSubspace ℝ P) := Set.univ.nonempty
rw [← finrank_top, ← direction_top ℝ V P] at hd
refine exists_circumsphere_eq_of_cospherical_subset ?_ hd hc
exact Set.subset_univ _
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_d... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 636 | 645 | theorem collinear_insert_insert_insert_of_mem_affineSpan_pair {p₁ p₂ p₃ p₄ p₅ : P}
(h₁ : p₁ ∈ line[k, p₄, p₅]) (h₂ : p₂ ∈ line[k, p₄, p₅]) (h₃ : p₃ ∈ line[k, p₄, p₅]) :
Collinear k ({p₁, p₂, p₃, p₄, p₅} : Set P) := by |
rw [collinear_insert_iff_of_mem_affineSpan
((AffineSubspace.le_def' _ _).1
(affineSpan_mono k ((Set.subset_insert _ _).trans (Set.subset_insert _ _))) _ h₁),
collinear_insert_iff_of_mem_affineSpan
((AffineSubspace.le_def' _ _).1 (affineSpan_mono k (Set.subset_insert _ _)) _ h₂),
collinear... |
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "... | Mathlib/Combinatorics/Quiver/Cast.lean | 112 | 115 | theorem Path.cast_heq {u v u' v' : U} (hu : u = u') (hv : v = v') (p : Path u v) :
HEq (p.cast hu hv) p := by |
rw [Path.cast_eq_cast]
exact _root_.cast_heq _ _
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
i... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 473 | 483 | theorem A_mem_nhdsWithin_Ioi {L : F} {r ε x : ℝ} (hx : x ∈ A f L r ε) : A f L r ε ∈ 𝓝[>] x := by |
rcases hx with ⟨r', rr', hr'⟩
rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]
obtain ⟨s, s_gt, s_lt⟩ : ∃ s : ℝ, r / 2 < s ∧ s < r' := exists_between rr'.1
have : s ∈ Ioc (r / 2) r := ⟨s_gt, le_of_lt (s_lt.trans_le rr'.2)⟩
refine ⟨x + r' - s, by simp only [mem_Ioi]; linarith, fun x' hx' => ⟨s, this, ?_⟩⟩
have... |
/-
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.Logic.Pairwise
import Mathlib.Logic.Relation
import Mathlib.Data.List.Basic
#align_import data.list.pairwise from "leanprover-community/mathlib"@"f694... | Mathlib/Data/List/Pairwise.lean | 124 | 133 | theorem pairwise_pmap {p : β → Prop} {f : ∀ b, p b → α} {l : List β} (h : ∀ x ∈ l, p x) :
Pairwise R (l.pmap f h) ↔
Pairwise (fun b₁ b₂ => ∀ (h₁ : p b₁) (h₂ : p b₂), R (f b₁ h₁) (f b₂ h₂)) l := by |
induction' l with a l ihl
· simp
obtain ⟨_, hl⟩ : p a ∧ ∀ b, b ∈ l → p b := by simpa using h
simp only [ihl hl, pairwise_cons, exists₂_imp, pmap, and_congr_left_iff, mem_pmap]
refine fun _ => ⟨fun H b hb _ hpb => H _ _ hb rfl, ?_⟩
rintro H _ b hb rfl
exact H b hb _ _
|
/-
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.GroupTheory.QuotientGroup
#align_import algebra.char_zero.quotient from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
/-!
# Lem... | Mathlib/Algebra/CharZero/Quotient.lean | 54 | 64 | theorem zmultiples_zsmul_eq_zsmul_iff {ψ θ : R ⧸ AddSubgroup.zmultiples p} {z : ℤ} (hz : z ≠ 0) :
z • ψ = z • θ ↔ ∃ k : Fin z.natAbs, ψ = θ + ((k : ℕ) • (p / z) : R) := by |
induction ψ using Quotient.inductionOn'
induction θ using Quotient.inductionOn'
-- Porting note: Introduced Zp notation to shorten lines
let Zp := AddSubgroup.zmultiples p
have : (Quotient.mk'' : R → R ⧸ Zp) = ((↑) : R → R ⧸ Zp) := rfl
simp only [this]
simp_rw [← QuotientAddGroup.mk_zsmul, ← QuotientAddG... |
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 424 | 429 | theorem exists_length_eq_zero_iff {u v : V} : (∃ p : G.Walk u v, p.length = 0) ↔ u = v := by |
constructor
· rintro ⟨p, hp⟩
exact eq_of_length_eq_zero hp
· rintro rfl
exact ⟨nil, rfl⟩
|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedS... | Mathlib/Analysis/RCLike/Basic.lean | 166 | 166 | theorem one_im : im (1 : K) = 0 := by | rw [← ofReal_one, ofReal_im]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 375 | 376 | theorem coeSubmodule_eq_bot_iff : (N : Submodule R M) = ⊥ ↔ N = ⊥ := by |
rw [← coe_toSubmodule_eq_iff, bot_coeSubmodule]
|
/-
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
#align_import probability.martingale.centering from "leanprover-community/mathlib"@"bea6c853b6edbd15e9d0941825abd04d77933ed0"... | Mathlib/Probability/Martingale/Centering.lean | 93 | 131 | theorem martingale_martingalePart (hf : Adapted ℱ f) (hf_int : ∀ n, Integrable (f n) μ)
[SigmaFiniteFiltration μ ℱ] : Martingale (martingalePart f ℱ μ) ℱ μ := by |
refine ⟨adapted_martingalePart hf, fun i j hij => ?_⟩
-- ⊢ μ[martingalePart f ℱ μ j | ℱ i] =ᵐ[μ] martingalePart f ℱ μ i
have h_eq_sum : μ[martingalePart f ℱ μ j|ℱ i] =ᵐ[μ]
f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|ℱ i] - μ[μ[f (k + 1) - f k|ℱ k]|ℱ i]) := by
rw [martingalePart_eq_sum]
refine (c... |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Batteries.Data.List.Lemmas
import Batteries.Data.Array.Basic
import Batteries.Tactic.SeqFocus
import Batteries.Util.ProofWanted
namespace Arra... | .lake/packages/batteries/Batteries/Data/Array/Lemmas.lean | 33 | 73 | theorem zipWith_eq_zipWith_data (f : α → β → γ) (as : Array α) (bs : Array β) :
(as.zipWith bs f).data = as.data.zipWith f bs.data := by |
let rec loop : ∀ (i : Nat) cs, i ≤ as.size → i ≤ bs.size →
(zipWithAux f as bs i cs).data = cs.data ++ (as.data.drop i).zipWith f (bs.data.drop i) := by
intro i cs hia hib
unfold zipWithAux
by_cases h : i = as.size ∨ i = bs.size
case pos =>
have : ¬(i < as.size) ∨ ¬(i < bs.size) := by
... |
/-
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.Algebra.Ring.Divisibility.Basic
import Mathlib.Init.Data.Ordering.Lemmas
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.NormNum
#ali... | Mathlib/SetTheory/Ordinal/Notation.lean | 252 | 253 | theorem NFBelow.lt {e n a b} (h : NFBelow (ONote.oadd e n a) b) : repr e < b := by |
cases' h with _ _ _ _ eb _ h₁ h₂ h₃; exact h₃
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.LinearAlgebra.AffineSpace.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import ... | Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 583 | 585 | theorem lineMap_eq_right_iff [NoZeroSMulDivisors k V1] {p₀ p₁ : P1} {c : k} :
lineMap p₀ p₁ c = p₁ ↔ p₀ = p₁ ∨ c = 1 := by |
rw [← @lineMap_eq_lineMap_iff k V1, lineMap_apply_one]
|
/-
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.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 1,543 | 1,546 | theorem ediam_Ioo (a b : ℝ) : EMetric.diam (Ioo a b) = ENNReal.ofReal (b - a) := by |
rcases le_or_lt b a with (h | h)
· simp [h]
· rw [Real.ediam_eq (isBounded_Ioo _ _), csSup_Ioo h, csInf_Ioo 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
#align_import analysis.analytic.composition from "l... | Mathlib/Analysis/Analytic/Composition.lean | 586 | 595 | theorem compChangeOfVariables_blocksFun (m M N : ℕ) {i : Σ n, Fin n → ℕ}
(hi : i ∈ compPartialSumSource m M N) (j : Fin i.1) :
(compChangeOfVariables m M N i hi).2.blocksFun
⟨j, (compChangeOfVariables_length m M N hi).symm ▸ j.2⟩ =
i.2 j := by |
rcases i with ⟨n, f⟩
dsimp [Composition.blocksFun, Composition.blocks, compChangeOfVariables]
simp only [map_ofFn, List.get_ofFn, Function.comp_apply]
-- Porting note: didn't used to need `rfl`
rfl
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b3705... | Mathlib/Data/Finset/NAry.lean | 530 | 538 | theorem subset_image₂ {s : Set α} {t : Set β} (hu : ↑u ⊆ image2 f s t) :
∃ (s' : Finset α) (t' : Finset β), ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ image₂ f s' t' := by |
rw [← Set.image_prod, subset_image_iff] at hu
rcases hu with ⟨u, hu, rfl⟩
classical
use u.image Prod.fst, u.image Prod.snd
simp only [coe_image, Set.image_subset_iff, image₂_image_left, image₂_image_right,
image_subset_iff]
exact ⟨fun _ h ↦ (hu h).1, fun _ h ↦ (hu h).2, fun x hx ↦ mem_image₂_of_mem hx ... |
/-
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.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 1,567 | 1,595 | theorem Valid'.map_aux {β} [Preorder β] {f : α → β} (f_strict_mono : StrictMono f) {t a₁ a₂}
(h : Valid' a₁ t a₂) :
Valid' (Option.map f a₁) (map f t) (Option.map f a₂) ∧ (map f t).size = t.size := by |
induction t generalizing a₁ a₂ with
| nil =>
simp [map]; apply valid'_nil
cases a₁; · trivial
cases a₂; · trivial
simp only [Bounded]
exact f_strict_mono h.ord
| node _ _ _ _ t_ih_l t_ih_r =>
have t_ih_l' := t_ih_l h.left
have t_ih_r' := t_ih_r h.right
clear t_ih_l t_ih_r
case... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury G. Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
impor... | Mathlib/Analysis/SpecificLimits/Normed.lean | 779 | 789 | theorem Antitone.alternating_series_le_tendsto
(hfl : Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i) atTop (𝓝 l))
(hfa : Antitone f) (k : ℕ) : ∑ i ∈ range (2 * k), (-1) ^ i * f i ≤ l := by |
have hm : Monotone (fun n ↦ ∑ i ∈ range (2 * n), (-1) ^ i * f i) := by
refine monotone_nat_of_le_succ (fun n ↦ ?_)
rw [show 2 * (n + 1) = 2 * n + 1 + 1 by ring, sum_range_succ, sum_range_succ]
simp_rw [_root_.pow_succ', show (-1 : E) ^ (2 * n) = 1 by simp, neg_one_mul, one_mul,
← sub_eq_add_neg, le... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.calculus.... | Mathlib/Analysis/Calculus/TangentCone.lean | 162 | 178 | theorem subset_tangentCone_prod_right {t : Set F} {y : F} (hs : x ∈ closure s) :
LinearMap.inr 𝕜 E F '' tangentConeAt 𝕜 t y ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) := by |
rintro _ ⟨w, ⟨c, d, hd, hc, hy⟩, rfl⟩
have : ∀ n, ∃ d', x + d' ∈ s ∧ ‖c n • d'‖ < ((1 : ℝ) / 2) ^ n := by
intro n
rcases mem_closure_iff_nhds.1 hs _
(eventually_nhds_norm_smul_sub_lt (c n) x (pow_pos one_half_pos n)) with
⟨z, hz, hzs⟩
exact ⟨z - x, by simpa using hzs, by simpa using hz⟩
... |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.No... | Mathlib/Analysis/NormedSpace/AffineIsometry.lean | 802 | 803 | theorem dist_pointReflection_fixed (x y : P) : dist (pointReflection 𝕜 x y) x = dist y x := by |
rw [← (pointReflection 𝕜 x).dist_map y x, pointReflection_self]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 385 | 386 | theorem get?_set_eq (a : α) (n) (l : List α) : (set l n a).get? n = (fun _ => a) <$> l.get? n := by |
simp only [set_eq_modifyNth, get?_modifyNth_eq]
|
/-
Copyright (c) 2022 Jesse Reimann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jesse Reimann, Kalle Kytölä
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.Sets.Compacts
#align_import measure_theory.integral.riesz_markov_kakutani from... | Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean | 94 | 112 | theorem rieszContentAux_sup_le (K1 K2 : Compacts X) :
rieszContentAux Λ (K1 ⊔ K2) ≤ rieszContentAux Λ K1 + rieszContentAux Λ K2 := by |
apply NNReal.le_of_forall_pos_le_add
intro ε εpos
--get test functions s.t. `λ(Ki) ≤ Λfi ≤ λ(Ki) + ε/2, i=1,2`
obtain ⟨f1, f_test_function_K1⟩ := exists_lt_rieszContentAux_add_pos Λ K1 (half_pos εpos)
obtain ⟨f2, f_test_function_K2⟩ := exists_lt_rieszContentAux_add_pos Λ K2 (half_pos εpos)
--let `f := f1 +... |
/-
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.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
#align_import model_theory.satisfiability from "leanpro... | Mathlib/ModelTheory/Satisfiability.lean | 174 | 186 | theorem exists_large_model_of_infinite_model (T : L.Theory) (κ : Cardinal.{w}) (M : Type w')
[L.Structure M] [M ⊨ T] [Infinite M] :
∃ N : ModelType.{_, _, max u v w} T, Cardinal.lift.{max u v w} κ ≤ #N := by |
obtain ⟨N⟩ :=
isSatisfiable_union_distinctConstantsTheory_of_infinite T (Set.univ : Set κ.out) M
refine ⟨(N.is_model.mono Set.subset_union_left).bundled.reduct _, ?_⟩
haveI : N ⊨ distinctConstantsTheory _ _ := N.is_model.mono Set.subset_union_right
rw [ModelType.reduct_Carrier, coe_of]
refine _root_.tran... |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Floris van Doorn
-/
import Mathlib.Order.WellFoundedSet
#align_import data.set.mul_antidiagonal from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447... | Mathlib/Data/Set/MulAntidiagonal.lean | 53 | 55 | theorem swap_mem_mulAntidiagonal [CommSemigroup α] {s t : Set α} {a : α} {x : α × α} :
x.swap ∈ Set.mulAntidiagonal s t a ↔ x ∈ Set.mulAntidiagonal t s a := by |
simp [mul_comm, and_left_comm]
|
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
/-!
This file establishes a `snoc : Vector α n → α → Vector α (n+1)` operation, that appends a single
element to the back of a vector.... | Mathlib/Data/Vector/Snoc.lean | 54 | 62 | theorem replicate_succ_to_snoc (val : α) :
replicate (n+1) val = (replicate n val).snoc val := by |
clear xs
induction n with
| zero => rfl
| succ n ih =>
rw [replicate_succ]
conv => rhs; rw [replicate_succ]
rw [snoc_cons, ih]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import... | Mathlib/RingTheory/Filtration.lean | 408 | 414 | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable := by |
rw [← submodule_fg_iff_stable] at hF ⊢
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
|
/-
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.RingTheory.WittVector.Truncated
import Mathlib.RingTheory.WittVector.Identities
import Mathlib.NumberTheory.Padics.RingHoms
#align_im... | Mathlib/RingTheory/WittVector/Compare.lean | 167 | 172 | theorem zmodEquivTrunc_compat (k₁ k₂ : ℕ) (hk : k₁ ≤ k₂) :
(TruncatedWittVector.truncate hk).comp
((zmodEquivTrunc p k₂).toRingHom.comp (PadicInt.toZModPow k₂)) =
(zmodEquivTrunc p k₁).toRingHom.comp (PadicInt.toZModPow k₁) := by |
rw [← RingHom.comp_assoc, commutes, RingHom.comp_assoc,
PadicInt.zmod_cast_comp_toZModPow _ _ hk]
|
/-
Copyright (c) 2021 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.Interval.Finset
import Mathlib.Combinatorics.Additive.FreimanHom
import Mathlib.Data.Set.Pointwise.SMul
import Ma... | Mathlib/Combinatorics/Additive/AP/Three/Defs.lean | 351 | 360 | theorem mulRothNumber_union_le (s t : Finset α) :
mulRothNumber (s ∪ t) ≤ mulRothNumber s + mulRothNumber t :=
let ⟨u, hus, hcard, hu⟩ := mulRothNumber_spec (s ∪ t)
calc
mulRothNumber (s ∪ t) = u.card := hcard.symm
_ = (u ∩ s ∪ u ∩ t).card := by | rw [← inter_union_distrib_left, inter_eq_left.2 hus]
_ ≤ (u ∩ s).card + (u ∩ t).card := card_union_le _ _
_ ≤ mulRothNumber s + mulRothNumber t := _root_.add_le_add
((hu.mono inter_subset_left).le_mulRothNumber inter_subset_right)
((hu.mono inter_subset_left).le_mulRothNumber inter_subset_right)
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Patrick Stevens
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.BigOperators.Ring
import Mathlib... | Mathlib/Data/Nat/Choose/Sum.lean | 134 | 139 | theorem sum_Icc_choose (n k : ℕ) : ∑ m ∈ Icc k n, m.choose k = (n + 1).choose (k + 1) := by |
cases' le_or_gt k n with h h
· induction' n, h using le_induction with n _ ih; · simp
rw [← Ico_insert_right (by omega), sum_insert (by simp),
show Ico k (n + 1) = Icc k n by rfl, ih, choose_succ_succ' (n + 1)]
· rw [choose_eq_zero_of_lt (by omega), Icc_eq_empty_of_lt h, sum_empty]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
#align_import logic.equiv.local_equ... | Mathlib/Logic/Equiv/PartialEquiv.lean | 711 | 712 | theorem trans_symm_eq_symm_trans_symm : (e.trans e').symm = e'.symm.trans e.symm := by |
cases e; cases e'; rfl
|
/-
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.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean | 185 | 211 | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by |
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note: Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
... |
/-
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.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 528 | 533 | theorem ncard_univ (α : Type*) : (univ : Set α).ncard = Nat.card α := by |
cases' finite_or_infinite α with h h
· have hft := Fintype.ofFinite α
rw [ncard_eq_toFinset_card, Finite.toFinset_univ, Finset.card_univ, Nat.card_eq_fintype_card]
rw [Nat.card_eq_zero_of_infinite, Infinite.ncard]
exact infinite_univ
|
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
#align_import ring_... | Mathlib/RingTheory/Derivation/Basic.lean | 153 | 154 | theorem map_natCast (n : ℕ) : D (n : A) = 0 := by |
rw [← nsmul_one, D.map_smul_of_tower n, map_one_eq_zero, smul_zero]
|
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