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) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Fin.Tuple.Reflection
#align_import data.matrix.reflection from "leanprover-communi... | Mathlib/Data/Matrix/Reflection.lean | 232 | 235 | theorem etaExpand_eq {m n} (A : Matrix (Fin m) (Fin n) α) : etaExpand A = A := by |
simp_rw [etaExpand, FinVec.etaExpand_eq, Matrix.of]
-- This to be in the above `simp_rw` before leanprover/lean4#2644
erw [Equiv.refl_apply]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Heather Macbeth
-/
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Math... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 88 | 91 | theorem polynomial_comp_attachBound_mem (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :
(g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attachBound ∈ A := by |
rw [polynomial_comp_attachBound]
apply SetLike.coe_mem
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Monad
#align_import probability.probability_mass_function.constructions from "leanprover-community/mat... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 296 | 297 | theorem filter_apply_ne_zero_iff (a : α) : (p.filter s h) a ≠ 0 ↔ a ∈ s ∧ a ∈ p.support := by |
rw [Ne, filter_apply_eq_zero_iff, not_or, Classical.not_not, Classical.not_not]
|
/-
Copyright (c) 2022 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Data.Vector.Basic
#align_import data.vector.mem from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
/-!
# Theorems about membershi... | Mathlib/Data/Vector/Mem.lean | 48 | 49 | theorem mem_cons_iff (v : Vector α n) : a' ∈ (a ::ᵥ v).toList ↔ a' = a ∨ a' ∈ v.toList := by |
rw [Vector.toList_cons, List.mem_cons]
|
/-
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.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingle... | Mathlib/Combinatorics/Enumerative/Composition.lean | 518 | 521 | theorem ne_ones_iff {c : Composition n} : c ≠ ones n ↔ ∃ i ∈ c.blocks, 1 < i := by |
refine (not_congr eq_ones_iff).trans ?_
have : ∀ j ∈ c.blocks, j = 1 ↔ j ≤ 1 := fun j hj => by simp [le_antisymm_iff, c.one_le_blocks hj]
simp (config := { contextual := true }) [this]
|
/-
Copyright (c) 2022 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 236 | 240 | theorem integrable_approxOn [BorelSpace E] {f : β → E} {μ : Measure β} (fmeas : Measurable f)
(hf : Integrable f μ) {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s]
(hi₀ : Integrable (fun _ => y₀) μ) (n : ℕ) : Integrable (approxOn f fmeas s y₀ h₀ n) μ := by |
rw [← memℒp_one_iff_integrable] at hf hi₀ ⊢
exact memℒp_approxOn fmeas hf h₀ hi₀ n
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
#align_import ring_theory.fractional_ideal from "lean... | Mathlib/RingTheory/FractionalIdeal/Basic.lean | 593 | 596 | theorem mul_left_mono (I : FractionalIdeal S P) : Monotone (I * ·) := by |
intro J J' h
simp only [mul_def]
exact mul_le.mpr fun x hx y hy => mul_mem_mul hx (h hy)
|
/-
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 | 580 | 582 | theorem triangle_assoc_comp_right (X Y : C) :
(α_ X (𝟙_ C) Y).inv ≫ ((ρ_ X).hom ▷ Y) = X ◁ (λ_ Y).hom := by |
rw [← triangle, Iso.inv_hom_id_assoc]
|
/-
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.MeasureSpace
/-!
# Restricting a measure to a subset or a subtype
Given a measure `μ` on a type `α` and a subse... | Mathlib/MeasureTheory/Measure/Restrict.lean | 1,025 | 1,035 | theorem mem_map_indicator_ae_iff_mem_map_restrict_ae_of_zero_mem [Zero β] {t : Set β}
(ht : (0 : β) ∈ t) (hs : MeasurableSet s) :
t ∈ Filter.map (s.indicator f) (ae μ) ↔ t ∈ Filter.map f (ae <| μ.restrict s) := by |
classical
simp_rw [mem_map, mem_ae_iff]
rw [Measure.restrict_apply' hs, Set.indicator_preimage, Set.ite]
simp_rw [Set.compl_union, Set.compl_inter]
change μ (((f ⁻¹' t)ᶜ ∪ sᶜ) ∩ ((fun _ => (0 : β)) ⁻¹' t \ s)ᶜ) = 0 ↔ μ ((f ⁻¹' t)ᶜ ∩ s) = 0
simp only [ht, ← Set.compl_eq_univ_diff, compl_compl, Set.compl_uni... |
/-
Copyright (c) 2019 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.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | Mathlib/Analysis/Calculus/MeanValue.lean | 455 | 471 | theorem norm_image_sub_le_of_norm_hasFDerivWithin_le
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s)
(xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by |
letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G
/- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined
on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`.
We just have to check the differentiability of the composition and ... |
/-
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.Projection
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
#align_... | Mathlib/Geometry/Euclidean/Basic.lean | 142 | 191 | theorem eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two {s : AffineSubspace ℝ P}
[FiniteDimensional ℝ s.direction] (hd : finrank ℝ s.direction = 2) {c₁ c₂ p₁ p₂ p : P}
(hc₁s : c₁ ∈ s) (hc₂s : c₂ ∈ s) (hp₁s : p₁ ∈ s) (hp₂s : p₂ ∈ s) (hps : p ∈ s) {r₁ r₂ : ℝ}
(hc : c₁ ≠ c₂) (hp : p₁ ≠ p₂) (hp₁c₁ : dist p₁ c... |
have ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0 :=
inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hp₂c₁.symm) (hp₁c₂.trans hp₂c₂.symm)
have hop : ⟪c₂ -ᵥ c₁, p -ᵥ p₁⟫ = 0 :=
inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hpc₁.symm) (hp₁c₂.trans hpc₂.symm)
let b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
have hb : ... |
/-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Ring.Defs
#align_import algebr... | Mathlib/Algebra/Ring/Semiconj.lean | 48 | 49 | theorem neg_right (h : SemiconjBy a x y) : SemiconjBy a (-x) (-y) := by |
simp only [SemiconjBy, h.eq, neg_mul, mul_neg]
|
/-
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.Matroid.Dual
/-!
# Matroid Restriction
Given `M : Matroid α` and `R : Set α`, the independent sets of `M` that are contained in `R`
are the independ... | Mathlib/Data/Matroid/Restrict.lean | 354 | 355 | theorem Indep.indep_restriction (hI : M.Indep I) (hNM : N ≤r M) (hIN : I ⊆ N.E) : N.Indep I := by |
obtain ⟨R, -, rfl⟩ := hNM; simpa [hI]
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.... | Mathlib/Data/Matrix/Rank.lean | 223 | 230 | theorem rank_conjTranspose_mul_self (A : Matrix m n R) : (Aᴴ * A).rank = A.rank := by |
dsimp only [rank]
refine add_left_injective (finrank R (LinearMap.ker (mulVecLin A))) ?_
dsimp only
trans finrank R { x // x ∈ LinearMap.range (mulVecLin (Aᴴ * A)) } +
finrank R { x // x ∈ LinearMap.ker (mulVecLin (Aᴴ * A)) }
· rw [ker_mulVecLin_conjTranspose_mul_self]
· simp only [LinearMap.finrank_ra... |
/-
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.ContMDiff.Basic
/-!
## Smoothness of charts and local structomorphisms
We show that the model with corners,... | Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean | 134 | 175 | theorem isLocalStructomorphOn_contDiffGroupoid_iff_aux {f : PartialHomeomorph M M'}
(hf : LiftPropOn (contDiffGroupoid ⊤ I).IsLocalStructomorphWithinAt f f.source) :
SmoothOn I I f f.source := by |
-- It suffices to show smoothness near each `x`
apply contMDiffOn_of_locally_contMDiffOn
intro x hx
let c := chartAt H x
let c' := chartAt H (f x)
obtain ⟨-, hxf⟩ := hf x hx
-- Since `f` is a local structomorph, it is locally equal to some transferred element `e` of
-- the `contDiffGroupoid`.
obtain
... |
/-
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, Scott Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_im... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 328 | 328 | theorem FiniteDimensional.finrank_fin_fun {n : ℕ} : finrank R (Fin n → R) = n := by | simp
|
/-
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.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.unoriented... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 316 | 320 | theorem tan_angle_sub_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y = 0) :
Real.tan (angle x (x - y)) * ‖x‖ = ‖y‖ := by |
rw [← neg_eq_zero, ← inner_neg_right] at h
rw [← neg_eq_zero] at h0
rw [sub_eq_add_neg, tan_angle_add_mul_norm_of_inner_eq_zero h h0, norm_neg]
|
/-
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.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/... | Mathlib/Algebra/Lie/Solvable.lean | 179 | 183 | theorem derivedSeries_map_le (k : ℕ) : (derivedSeries R L' k).map f ≤ derivedSeries R L k := by |
induction' k with k ih
· simp only [Nat.zero_eq, derivedSeries_def, derivedSeriesOfIdeal_zero, le_top]
· simp only [derivedSeries_def, derivedSeriesOfIdeal_succ] at ih ⊢
exact le_trans (map_bracket_le f) (LieSubmodule.mono_lie _ _ _ _ ih ih)
|
/-
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.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
/-!
# Infinite sums and products in topological groups
Lemmas on topo... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 249 | 252 | theorem multipliable_iff_vanishing :
Multipliable f ↔
∀ e ∈ 𝓝 (1 : α), ∃ s : Finset β, ∀ t, Disjoint t s → (∏ b ∈ t, f b) ∈ e := by |
rw [multipliable_iff_cauchySeq_finset, cauchySeq_finset_iff_prod_vanishing]
|
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.F... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 673 | 685 | theorem IsSwap.mul_mem_closure_three_cycles {σ τ : Perm α} (hσ : IsSwap σ) (hτ : IsSwap τ) :
σ * τ ∈ closure { σ : Perm α | IsThreeCycle σ } := by |
obtain ⟨a, b, ab, rfl⟩ := hσ
obtain ⟨c, d, cd, rfl⟩ := hτ
by_cases ac : a = c
· subst ac
exact swap_mul_swap_same_mem_closure_three_cycles ab cd
have h' : swap a b * swap c d = swap a b * swap a c * (swap c a * swap c d) := by
simp [swap_comm c a, mul_assoc]
rw [h']
exact
mul_mem (swap_mul_sw... |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johan Commelin
-/
import Mathlib.RingTheory.IntegralClosure
#align_import field_theory.minpoly.basic from "leanprover-community/mathlib"@"df0098f0db291900600f32070f6abb3e1... | Mathlib/FieldTheory/Minpoly/Basic.lean | 106 | 111 | theorem map_ne_one [Nontrivial B] {R : Type*} [Semiring R] [Nontrivial R] (f : A →+* R) :
(minpoly A x).map f ≠ 1 := by |
by_cases hx : IsIntegral A x
· exact mt ((monic hx).eq_one_of_map_eq_one f) (ne_one A x)
· rw [eq_zero hx, Polynomial.map_zero]
exact zero_ne_one
|
/-
Copyright (c) 2020 Zhangir Azerbayev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Zhangir Azerbayev
-/
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Data.Fintype.Perm
import Mathlib.LinearAlgebra.Multilinear.Basis
#align_import linear_algebra.... | Mathlib/LinearAlgebra/Alternating/Basic.lean | 165 | 168 | theorem coe_multilinearMap_mk (f : (ι → M) → N) (h₁ h₂ h₃) :
((⟨⟨f, h₁, h₂⟩, h₃⟩ : M [⋀^ι]→ₗ[R] N) : MultilinearMap R (fun _ : ι => M) N) =
⟨f, @h₁, @h₂⟩ := by |
simp
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin... | Mathlib/RingTheory/IsAdjoinRoot.lean | 651 | 652 | theorem aequiv_root (h : IsAdjoinRoot S f) (h' : IsAdjoinRoot T f) :
h.aequiv h' h.root = h'.root := by | rw [aequiv, AlgEquiv.coe_mk, liftHom_root]
|
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib... | Mathlib/Algebra/Polynomial/RingDivision.lean | 368 | 396 | theorem eq_zero_of_mul_eq_zero_of_smul (P : R[X]) (h : ∀ r : R, r • P = 0 → r = 0) :
∀ (Q : R[X]), P * Q = 0 → Q = 0 := by |
intro Q hQ
suffices ∀ i, P.coeff i • Q = 0 by
rw [← leadingCoeff_eq_zero]
apply h
simpa [ext_iff, mul_comm Q.leadingCoeff] using fun i ↦ congr_arg (·.coeff Q.natDegree) (this i)
apply Nat.strong_decreasing_induction
· use P.natDegree
intro i hi
rw [coeff_eq_zero_of_natDegree_lt hi, zero_smu... |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
#al... | Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 194 | 198 | theorem map_PInfty_f {D : Type*} [Category D] [Preadditive D] (G : C ⥤ D) [G.Additive]
(X : SimplicialObject C) (n : ℕ) :
(PInfty : K[((whiskering C D).obj G).obj X] ⟶ _).f n =
G.map ((PInfty : AlternatingFaceMapComplex.obj X ⟶ _).f n) := by |
simp only [PInfty_f, map_P]
|
/-
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.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
/-!
# Infinite sums and products in topological groups
Lemmas on topo... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 312 | 316 | theorem prod_mul_tprod_subtype_compl [T2Space α] {f : β → α} (hf : Multipliable f) (s : Finset β) :
(∏ x ∈ s, f x) * ∏' x : { x // x ∉ s }, f x = ∏' x, f x := by |
rw [← tprod_subtype_mul_tprod_subtype_compl hf s]
simp only [Finset.tprod_subtype', mul_right_inj]
rfl
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
import Mathli... | Mathlib/RingTheory/Localization/NumDen.lean | 97 | 105 | theorem isInteger_of_isUnit_den {x : K} (h : IsUnit (den A x : A)) : IsInteger A x := by |
cases' h with d hd
have d_ne_zero : algebraMap A K (den A x) ≠ 0 :=
IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors (den A x).2
use ↑d⁻¹ * num A x
refine _root_.trans ?_ (mk'_num_den A x)
rw [map_mul, map_units_inv, hd]
apply mul_left_cancel₀ d_ne_zero
rw [← mul_assoc, mul_inv_cancel d_ne_zero, ... |
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.shapes.kernels from "leanprover-community/mathlib"@"95... | Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 407 | 410 | 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 |
cases h; simp
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.Antilipschitz
#align_import topology.metric_space.isometry from "leanprover-community/mathlib"@"b1859b6d4636fdbb78c5d5cefd2... | Mathlib/Topology/MetricSpace/Isometry.lean | 40 | 42 | theorem isometry_iff_nndist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} :
Isometry f ↔ ∀ x y, nndist (f x) (f y) = nndist x y := by |
simp only [Isometry, edist_nndist, ENNReal.coe_inj]
|
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 894 | 895 | theorem sign_eq_zero_iff {θ : Angle} : θ.sign = 0 ↔ θ = 0 ∨ θ = π := by |
rw [sign, _root_.sign_eq_zero_iff, sin_eq_zero_iff]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.... | Mathlib/Order/Filter/AtTopBot.lean | 1,354 | 1,356 | theorem tendsto_const_mul_pow_atBot_iff {c : α} {n : ℕ} :
Tendsto (fun x => c * x ^ n) atTop atBot ↔ n ≠ 0 ∧ c < 0 := by |
simp only [← tendsto_neg_atTop_iff, ← neg_mul, tendsto_const_mul_pow_atTop_iff, neg_pos]
|
/-
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.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 945 | 954 | theorem direction_lt_of_nonempty {s1 s2 : AffineSubspace k P} (h : s1 < s2)
(hn : (s1 : Set P).Nonempty) : s1.direction < s2.direction := by |
cases' hn with p hp
rw [lt_iff_le_and_exists] at h
rcases h with ⟨hle, p2, hp2, hp2s1⟩
rw [SetLike.lt_iff_le_and_exists]
use direction_le hle, p2 -ᵥ p, vsub_mem_direction hp2 (hle hp)
intro hm
rw [vsub_right_mem_direction_iff_mem hp p2] at hm
exact hp2s1 hm
|
/-
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.Arithmetic
import Mathlib.Tactic.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-communit... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 293 | 298 | theorem nadd_assoc (a b c) : a ♯ b ♯ c = a ♯ (b ♯ c) := by |
rw [nadd_def a (b ♯ c), nadd_def, blsub_nadd_of_mono, blsub_nadd_of_mono, max_assoc]
· congr <;> ext <;> apply nadd_assoc
· exact fun _ _ h => nadd_le_nadd_left h a
· exact fun _ _ h => nadd_le_nadd_right h c
termination_by (a, b, 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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 529 | 530 | theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i := by |
rw [diff_eq, compl_iUnion, inter_iInter]; rfl
|
/-
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.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriente... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 265 | 266 | theorem two_zsmul_oangle_neg_self_left (x : V) : (2 : ℤ) • o.oangle (-x) x = 0 := by |
by_cases hx : x = 0 <;> simp [hx]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 1,018 | 1,027 | theorem lt_replicate_succ {m : Multiset α} {x : α} {n : ℕ} :
m < replicate (n + 1) x ↔ m ≤ replicate n x := by |
rw [lt_iff_cons_le]
constructor
· rintro ⟨x', hx'⟩
have := eq_of_mem_replicate (mem_of_le hx' (mem_cons_self _ _))
rwa [this, replicate_succ, cons_le_cons_iff] at hx'
· intro h
rw [replicate_succ]
exact ⟨x, cons_le_cons _ h⟩
|
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.NormalC... | Mathlib/FieldTheory/SeparableDegree.lean | 602 | 613 | theorem natSepDegree_eq_one_iff_pow_mem : (minpoly F x).natSepDegree = 1 ↔
∃ n : ℕ, x ^ q ^ n ∈ (algebraMap F E).range := by |
convert_to _ ↔ ∃ (n : ℕ) (y : F), Polynomial.aeval x (X ^ q ^ n - C y) = 0
· simp_rw [RingHom.mem_range, map_sub, map_pow, aeval_C, aeval_X, sub_eq_zero, eq_comm]
refine ⟨fun h ↦ ?_, fun ⟨n, y, h⟩ ↦ ?_⟩
· obtain ⟨n, y, hx⟩ := (minpoly.natSepDegree_eq_one_iff_eq_X_pow_sub_C q).1 h
exact ⟨n, y, hx ▸ aeval F ... |
/-
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.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
#align_import analysis.box_integr... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 184 | 199 | theorem injOn_setOf_mem_Icc_setOf_lower_eq (x : ι → ℝ) :
InjOn (fun J : Box ι => { i | J.lower i = x i }) { J | J ∈ π ∧ x ∈ Box.Icc J } := by |
rintro J₁ ⟨h₁, hx₁⟩ J₂ ⟨h₂, hx₂⟩ (H : { i | J₁.lower i = x i } = { i | J₂.lower i = x i })
suffices ∀ i, (Ioc (J₁.lower i) (J₁.upper i) ∩ Ioc (J₂.lower i) (J₂.upper i)).Nonempty by
choose y hy₁ hy₂ using this
exact π.eq_of_mem_of_mem h₁ h₂ hy₁ hy₂
intro i
simp only [Set.ext_iff, mem_setOf] at H
rcase... |
/-
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.Set.Lattice
import Mathlib.Logic.Small.Basic
import Mathlib.Logic.Function.OfArity
import Mathlib.Order.WellFounded
#align_import set_theory.zfc.... | Mathlib/SetTheory/ZFC/Basic.lean | 950 | 953 | theorem toSet_sep (a : ZFSet) (p : ZFSet → Prop) :
(ZFSet.sep p a).toSet = { x ∈ a.toSet | p x } := by |
ext
simp
|
/-
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
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,217 | 1,230 | theorem lintegral_tendsto_of_tendsto_of_antitone {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞}
(hf : ∀ n, AEMeasurable (f n) μ) (h_anti : ∀ᵐ x ∂μ, Antitone fun n ↦ f n x)
(h0 : ∫⁻ a, f 0 a ∂μ ≠ ∞)
(h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))) :
Tendsto (fun n ↦ ∫⁻ x, f n x ∂μ) atTop (𝓝 (∫⁻ x, F x... |
have : Antitone fun n ↦ ∫⁻ x, f n x ∂μ := fun i j hij ↦
lintegral_mono_ae (h_anti.mono fun x hx ↦ hx hij)
suffices key : ∫⁻ x, F x ∂μ = ⨅ n, ∫⁻ x, f n x ∂μ by
rw [key]
exact tendsto_atTop_iInf this
rw [← lintegral_iInf' hf h_anti h0]
refine lintegral_congr_ae ?_
filter_upwards [h_anti, h_tendsto]... |
/-
Copyright (c) 2022 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Order.Basic
import Mathlib.Tactic.NoncommRing
#align_import... | Mathlib/Analysis/NormedSpace/MStructure.lean | 268 | 276 | theorem distrib_lattice_lemma [FaithfulSMul M X] {P Q R : { P : M // IsLprojection X P }} :
((↑P : M) + ↑Pᶜ * R) * (↑P + ↑Q * ↑R * ↑Pᶜ) = ↑P + ↑Q * ↑R * ↑Pᶜ := by |
rw [add_mul, mul_add, mul_add, (mul_assoc _ (R : M) (↑Q * ↑R * ↑Pᶜ)),
← mul_assoc (R : M) (↑Q * ↑R) _, ← coe_inf Q, (Pᶜ.prop.commute R.prop).eq,
((Q ⊓ R).prop.commute Pᶜ.prop).eq, (R.prop.commute (Q ⊓ R).prop).eq, coe_inf Q,
mul_assoc (Q : M), ← mul_assoc, mul_assoc (R : M), (Pᶜ.prop.commute P.prop).eq, ... |
/-
Copyright (c) 2018 Mitchell Rowett. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Rowett, Scott Morrison
-/
import Mathlib.Algebra.Quotient
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Group... | Mathlib/GroupTheory/Coset.lean | 788 | 792 | theorem card_dvd_of_injective (f : α →* H) (hf : Function.Injective f) :
Nat.card α ∣ Nat.card H := by |
classical calc
Nat.card α = Nat.card (f.range : Subgroup H) := Nat.card_congr (Equiv.ofInjective f hf)
_ ∣ Nat.card H := card_subgroup_dvd_card _
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 558 | 558 | theorem image_neg_uIcc : Neg.neg '' [[a, b]] = [[-a, -b]] := by | simp
|
/-
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, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 534 | 537 | theorem span_singleton_smul_eq {r : R} (hr : IsUnit r) (x : M) : (R ∙ r • x) = R ∙ x := by |
lift r to Rˣ using hr
rw [← Units.smul_def]
exact span_singleton_group_smul_eq R r x
|
/-
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, Scott Morrison
-/
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dimension from "leanprover-community/mathl... | Mathlib/LinearAlgebra/Dimension/Basic.lean | 259 | 271 | theorem lift_rank_range_le (f : M →ₗ[R] M') : Cardinal.lift.{v}
(Module.rank R (LinearMap.range f)) ≤ Cardinal.lift.{v'} (Module.rank R M) := by |
simp only [Module.rank_def]
rw [Cardinal.lift_iSup (Cardinal.bddAbove_range.{v', v'} _)]
apply ciSup_le'
rintro ⟨s, li⟩
apply le_trans
swap
· apply Cardinal.lift_le.mpr
refine le_ciSup (Cardinal.bddAbove_range.{v, v} _) ⟨rangeSplitting f '' s, ?_⟩
apply LinearIndependent.of_comp f.rangeRestrict
... |
/-
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.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDo... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 210 | 211 | theorem roots_normalize {p : R[X]} : (normalize p).roots = p.roots := by |
rw [normalize_apply, mul_comm, coe_normUnit, roots_C_mul _ (normUnit (leadingCoeff p)).ne_zero]
|
/-
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.Order.PropInstances
#align_import order.heyting.basic from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# Heyting algebr... | Mathlib/Order/Heyting/Basic.lean | 1,049 | 1,050 | theorem codisjoint_hnot_hnot_right_iff : Codisjoint a (￢￢b) ↔ Codisjoint a b := by |
simp_rw [← hnot_le_iff_codisjoint_left, hnot_hnot_hnot]
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 88 | 92 | theorem realize_relabel {t : L.Term α} {g : α → β} {v : β → M} :
(t.relabel g).realize v = t.realize (v ∘ g) := by |
induction' t with _ n f ts ih
· rfl
· simp [ih]
|
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Analysis.Normed.Group.Basic
#align_import information_theory.hamming from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!... | Mathlib/InformationTheory/Hamming.lean | 211 | 213 | theorem hammingNorm_comp_le_hammingNorm (f : ∀ i, γ i → β i) {x : ∀ i, γ i} (hf : ∀ i, f i 0 = 0) :
(hammingNorm fun i => f i (x i)) ≤ hammingNorm x := by |
simpa only [← hammingDist_zero_right, hf] using hammingDist_comp_le_hammingDist f (y := fun _ ↦ 0)
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 1,024 | 1,025 | theorem lift_mul_right (z) : f.lift hg z * g (f.sec z).2 = g (f.sec z).1 := by |
erw [mul_assoc, IsUnit.liftRight_inv_mul, mul_one]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 512 | 517 | theorem IsLittleO.trans_isBigOWith (hfg : f =o[l] g) (hgk : IsBigOWith c l g k) (hc : 0 < c) :
f =o[l] k := by |
simp only [IsLittleO_def] at *
intro c' c'pos
have : 0 < c' / c := div_pos c'pos hc
exact ((hfg this).trans hgk this.le).congr_const (div_mul_cancel₀ _ hc.ne')
|
/-
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.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 843 | 847 | theorem sin_pi_div_eight : sin (π / 8) = √(2 - √2) / 2 := by |
trans sin (π / 2 ^ 3)
· congr
norm_num
· simp
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Oleksandr Manzyuk
-/
import Mathlib.CategoryTheory.Bicategory.Basic
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Eq... | Mathlib/CategoryTheory/Monoidal/Bimod.lean | 514 | 539 | theorem hom_left_act_hom' :
((P.tensorBimod Q).tensorBimod L).actLeft ≫ hom P Q L =
(R.X ◁ hom P Q L) ≫ (P.tensorBimod (Q.tensorBimod L)).actLeft := by |
dsimp; dsimp [hom, homAux]
refine (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 ?_
rw [tensorLeft_map]
slice_lhs 1 2 => rw [TensorBimod.whiskerLeft_π_actLeft]
slice_lhs 3 4 => rw [coequalizer.π_desc]
slice_rhs 1 2 => rw [← MonoidalCategory.whiskerLeft_comp, coequalizer.π_desc,
MonoidalCategor... |
/-
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 | 340 | 343 | theorem Affine.Simplex.span_eq_top [FiniteDimensional k V] {n : ℕ} (T : Affine.Simplex k V n)
(hrank : finrank k V = n) : affineSpan k (Set.range T.points) = ⊤ := by |
rw [AffineIndependent.affineSpan_eq_top_iff_card_eq_finrank_add_one T.independent,
Fintype.card_fin, hrank]
|
/-
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
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 152 | 154 | theorem comp_hasFDerivAt_iff' {f : G → E} {x : G} {f' : G →L[𝕜] F} :
HasFDerivAt (iso ∘ f) f' x ↔ HasFDerivAt f ((iso.symm : F →L[𝕜] E).comp f') x := by |
simp_rw [← hasFDerivWithinAt_univ, iso.comp_hasFDerivWithinAt_iff']
|
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 250 | 251 | theorem logb_nonneg_iff (hx : 0 < x) : 0 ≤ logb b x ↔ 1 ≤ x := by |
rw [← not_lt, logb_neg_iff hb hx, not_lt]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 640 | 641 | theorem inner_sub_right (x y z : E) : ⟪x, y - z⟫ = ⟪x, y⟫ - ⟪x, z⟫ := by |
simp [sub_eq_add_neg, inner_add_right]
|
/-
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.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
#align_import rin... | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 361 | 365 | theorem constantCoeff_wittStructureRat (Φ : MvPolynomial idx ℚ) (h : constantCoeff Φ = 0) (n : ℕ) :
constantCoeff (wittStructureRat p Φ n) = 0 := by |
simp only [wittStructureRat, eval₂Hom_zero'_apply, h, bind₁, map_aeval, constantCoeff_rename,
constantCoeff_wittPolynomial, constantCoeff_comp_algebraMap, RingHom.id_apply,
constantCoeff_xInTermsOfW]
|
/-
Copyright (c) 2021 François Sunatori. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: François Sunatori
-/
import Mathlib.Analysis.Complex.Circle
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
#align_import analysis.c... | Mathlib/Analysis/Complex/Isometry.lean | 125 | 139 | theorem linear_isometry_complex_aux {f : ℂ ≃ₗᵢ[ℝ] ℂ} (h : f 1 = 1) :
f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLIE := by |
have h0 : f I = I ∨ f I = -I := by
simp only [ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
constructor
· rw [← I_re]
exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
· apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
intro z
rw [... |
/-
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.LinearAlgebra.CliffordAlgebra.Conjugation
#align_import linear_algebra.clifford_algebra.star from "leanprover-community/mathlib"@"4d66277cfec381260ba05c68f9... | Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean | 62 | 64 | theorem star_algebraMap (r : R) :
star (algebraMap R (CliffordAlgebra Q) r) = algebraMap R (CliffordAlgebra Q) r := by |
rw [star_def, involute.commutes, reverse.commutes]
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.Derangements.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Tactic.Ring
#align_imp... | Mathlib/Combinatorics/Derangements/Finite.lean | 95 | 104 | theorem card_derangements_fin_eq_numDerangements {n : ℕ} :
card (derangements (Fin n)) = numDerangements n := by |
induction' n using Nat.strong_induction_on with n hyp
rcases n with _ | _ | n
-- knock out cases 0 and 1
· rfl
· rfl
-- now we have n ≥ 2. rewrite everything in terms of card_derangements, so that we can use
-- `card_derangements_fin_add_two`
rw [numDerangements_add_two, card_derangements_fin_add_two, ... |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.MeasureTheory.Integral.ExpDecay
import Mathlib.Analysis.MellinTransform
#align_import analysis.special_functions.gamma.basic from "leanprover-communit... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 551 | 553 | theorem Gamma_zero : Gamma 0 = 0 := by |
simpa only [← Complex.ofReal_zero, Complex.Gamma_ofReal, Complex.ofReal_inj] using
Complex.Gamma_zero
|
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.LinearAlgebra.BilinearForm.Hom
import Mathlib.LinearAlgebra.Dual
/-!
# Bilinear form
This file defines various properties of bilinea... | Mathlib/LinearAlgebra/BilinearForm/Properties.lean | 471 | 474 | theorem dualBasis_repr_apply (B : BilinForm K V) (hB : B.Nondegenerate) (b : Basis ι K V) (x i) :
(B.dualBasis hB b).repr x i = B x (b i) := by |
rw [dualBasis, Basis.map_repr, LinearEquiv.symm_symm, LinearEquiv.trans_apply,
Basis.dualBasis_repr, toDual_def]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.Differentiation
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Integral.Lebesgue
import M... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 1,131 | 1,139 | theorem ae_tendsto_measure_inter_div_of_measurableSet (μ : Measure β) [IsLocallyFiniteMeasure μ]
{s : Set β} (hs : MeasurableSet s) :
∀ᵐ x ∂μ,
Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0)
(𝓝 (s.indicator 1 x)) := by |
filter_upwards [VitaliFamily.ae_tendsto_measure_inter_div_of_measurableSet
(Besicovitch.vitaliFamily μ) hs]
intro x hx
exact hx.comp (tendsto_filterAt μ x)
|
/-
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.Ring
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Or... | Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | 182 | 187 | theorem le_card_nonuniformWitness (h : ¬G.IsUniform ε s t) :
(s.card : 𝕜) * ε ≤ (G.nonuniformWitness ε s t).card := by |
unfold nonuniformWitness
split_ifs
· exact G.left_nonuniformWitnesses_card h
· exact G.right_nonuniformWitnesses_card fun i => h i.symm
|
/-
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 | 568 | 570 | theorem forall_dual_apply_eq_zero_iff (v : V) : (∀ φ : Module.Dual K V, φ v = 0) ↔ v = 0 := by |
rw [← eval_apply_eq_zero_iff K v, LinearMap.ext_iff]
rfl
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Ta... | Mathlib/Logic/Relation.lean | 376 | 379 | theorem to_reflTransGen {a b} (h : TransGen r a b) : ReflTransGen r a b := by |
induction' h with b h b c _ bc ab
· exact ReflTransGen.single h
· exact ReflTransGen.tail ab bc
|
/-
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 | 474 | 487 | theorem collinear_iff_exists_forall_eq_smul_vadd (s : Set P) :
Collinear k s ↔ ∃ (p₀ : P) (v : V), ∀ p ∈ s, ∃ r : k, p = r • v +ᵥ p₀ := by |
rcases Set.eq_empty_or_nonempty s with (rfl | ⟨⟨p₁, hp₁⟩⟩)
· simp [collinear_empty]
· rw [collinear_iff_of_mem hp₁]
constructor
· exact fun h => ⟨p₁, h⟩
· rintro ⟨p, v, hv⟩
use v
intro p₂ hp₂
rcases hv p₂ hp₂ with ⟨r, rfl⟩
rcases hv p₁ hp₁ with ⟨r₁, rfl⟩
use r - r₁
... |
/-
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.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Interval.Finset
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tactic.Linarith
... | Mathlib/Algebra/BigOperators/Intervals.lean | 118 | 121 | theorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → M) :
∏ k ∈ Ico a b, f k = f a * ∏ k ∈ Ico (a + 1) b, f k := by |
have ha : a ∉ Ico (a + 1) b := by simp
rw [← prod_insert ha, Nat.Ico_insert_succ_left hab]
|
/-
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.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunction... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 417 | 437 | theorem integral_pow_abs_sub_uIoc : ∫ x in Ι a b, |x - a| ^ n = |b - a| ^ (n + 1) / (n + 1) := by |
rcases le_or_lt a b with hab | hab
· calc
∫ x in Ι a b, |x - a| ^ n = ∫ x in a..b, |x - a| ^ n := by
rw [uIoc_of_le hab, ← integral_of_le hab]
_ = ∫ x in (0)..(b - a), x ^ n := by
simp only [integral_comp_sub_right fun x => |x| ^ n, sub_self]
refine integral_congr fun x hx => co... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Ring.Regular
import Mathlib.Tactic.Common
#align_import algebra.gcd_monoid.basic from "leanp... | Mathlib/Algebra/GCDMonoid/Basic.lean | 695 | 703 | theorem extract_gcd {α : Type*} [CancelCommMonoidWithZero α] [GCDMonoid α] (x y : α) :
∃ x' y', x = gcd x y * x' ∧ y = gcd x y * y' ∧ IsUnit (gcd x' y') := by |
by_cases h : gcd x y = 0
· obtain ⟨rfl, rfl⟩ := (gcd_eq_zero_iff x y).1 h
simp_rw [← associated_one_iff_isUnit]
exact ⟨1, 1, by rw [h, zero_mul], by rw [h, zero_mul], gcd_one_left' 1⟩
obtain ⟨x', ex⟩ := gcd_dvd_left x y
obtain ⟨y', ey⟩ := gcd_dvd_right x y
exact ⟨x', y', ex, ey, isUnit_gcd_of_eq_mul_... |
/-
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 | 209 | 209 | theorem arg_one : arg 1 = 0 := by | simp [arg, zero_le_one]
|
/-
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.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "lea... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 501 | 501 | theorem infDist_empty : infDist x ∅ = 0 := by | simp [infDist]
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padic... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 1,175 | 1,177 | theorem norm_lt_pow_iff_norm_le_pow_sub_one (x : ℚ_[p]) (n : ℤ) :
‖x‖ < (p : ℝ) ^ n ↔ ‖x‖ ≤ (p : ℝ) ^ (n - 1) := by |
rw [norm_le_pow_iff_norm_lt_pow_add_one, sub_add_cancel]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.Function.SimpleFunc... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 1,720 | 1,734 | theorem _root_.aestronglyMeasurable_of_tendsto_ae {ι : Type*} [PseudoMetrizableSpace β]
(u : Filter ι) [NeBot u] [IsCountablyGenerated u] {f : ι → α → β} {g : α → β}
(hf : ∀ i, AEStronglyMeasurable (f i) μ) (lim : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))) :
AEStronglyMeasurable g μ := by |
borelize β
refine aestronglyMeasurable_iff_aemeasurable_separable.2 ⟨?_, ?_⟩
· exact aemeasurable_of_tendsto_metrizable_ae _ (fun n => (hf n).aemeasurable) lim
· rcases u.exists_seq_tendsto with ⟨v, hv⟩
have : ∀ n : ℕ, ∃ t : Set β, IsSeparable t ∧ f (v n) ⁻¹' t ∈ ae μ := fun n =>
(aestronglyMeasurabl... |
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Alistair Tucker, Wen Yang
-/
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Topology.Order.DenselyOrdered
... | Mathlib/Topology/Order/IntermediateValue.lean | 281 | 305 | theorem IsPreconnected.mem_intervals {s : Set α} (hs : IsPreconnected s) :
s ∈
({Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s),
Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅} : Set (Set α)) := by |
rcases s.eq_empty_or_nonempty with (rfl | hne)
· apply_rules [Or.inr, mem_singleton]
have hs' : IsConnected s := ⟨hne, hs⟩
by_cases hb : BddBelow s <;> by_cases ha : BddAbove s
· refine mem_of_subset_of_mem ?_ <| mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset
(hs'.Ioo_csInf_csSup_subset hb ha) (subset_Icc_csI... |
/-
Copyright (c) 2019 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Paul Lezeau, Junyan Xu
-/
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 138 | 145 | theorem prime_of_isIntegrallyClosed {x : S} (hx : IsIntegral R x) : Prime (minpoly R x) := by |
refine
⟨(minpoly.monic hx).ne_zero,
⟨fun h_contra => (ne_of_lt (minpoly.degree_pos hx)) (degree_eq_zero_of_isUnit h_contra).symm,
fun a b h => or_iff_not_imp_left.mpr fun h' => ?_⟩⟩
rw [← minpoly.isIntegrallyClosed_dvd_iff hx] at h' h ⊢
rw [aeval_mul] at h
exact eq_zero_of_ne_zero_of_mul_left... |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.Real.Basic
import Mathlib.Data.ENNReal.Real
import Mathlib.Data.Sign
#align_import data.real.ereal from "leanprover-community/mathlib"@"2196ab363eb... | Mathlib/Data/Real/EReal.lean | 347 | 349 | theorem range_coe_eq_Ioo : range Real.toEReal = Ioo ⊥ ⊤ := by |
ext x
induction x <;> simp
|
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Tactic.SplitIfs
#align_import data.nat.cast.defs from "leanprover-community/mathlib"@"a148d797a1094ab... | Mathlib/Data/Nat/Cast/Defs.lean | 154 | 155 | theorem cast_one [AddMonoidWithOne R] : ((1 : ℕ) : R) = 1 := by |
rw [cast_succ, Nat.cast_zero, zero_add]
|
/-
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 | 128 | 141 | theorem det_mul_aux {M N : Matrix n n R} {p : n → n} (H : ¬Bijective p) :
(∑ σ : Perm n, ε σ * ∏ x, M (σ x) (p x) * N (p x) x) = 0 := by |
obtain ⟨i, j, hpij, hij⟩ : ∃ i j, p i = p j ∧ i ≠ j := by
rw [← Finite.injective_iff_bijective, Injective] at H
push_neg at H
exact H
exact
sum_involution (fun σ _ => σ * Equiv.swap i j)
(fun σ _ => by
have : (∏ x, M (σ x) (p x)) = ∏ x, M ((σ * Equiv.swap i j) x) (p x) :=
Fi... |
/-
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.Analysis.Convex.Exposed
import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
import Mathlib.Topology.Algebra.ContinuousAffineMap
#align_import analys... | Mathlib/Analysis/Convex/KreinMilman.lean | 95 | 106 | theorem closure_convexHull_extremePoints (hscomp : IsCompact s) (hAconv : Convex ℝ s) :
closure (convexHull ℝ <| s.extremePoints ℝ) = s := by |
apply (closure_minimal (convexHull_min extremePoints_subset hAconv) hscomp.isClosed).antisymm
by_contra hs
obtain ⟨x, hxA, hxt⟩ := not_subset.1 hs
obtain ⟨l, r, hlr, hrx⟩ :=
geometric_hahn_banach_closed_point (convex_convexHull _ _).closure isClosed_closure hxt
have h : IsExposed ℝ s ({ y ∈ s | ∀ z ∈ s, ... |
/-
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 | 317 | 320 | theorem HasProd.mul_isCompl {s t : Set β} (hs : IsCompl s t) (ha : HasProd (f ∘ (↑) : s → α) a)
(hb : HasProd (f ∘ (↑) : t → α) b) : HasProd f (a * b) := by |
simpa [← hs.compl_eq] using
(hasProd_subtype_iff_mulIndicator.1 ha).mul (hasProd_subtype_iff_mulIndicator.1 hb)
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "lean... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 107 | 109 | theorem χ₄_int_one_mod_four {n : ℤ} (hn : n % 4 = 1) : χ₄ n = 1 := by |
rw [χ₄_int_mod_four, hn]
rfl
|
/-
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
-/
import Mathlib.Algebra.Polynomial.Monic
#align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594b... | Mathlib/Algebra/Polynomial/BigOperators.lean | 198 | 211 | theorem natDegree_multiset_prod_of_monic (h : ∀ f ∈ t, Monic f) :
t.prod.natDegree = (t.map natDegree).sum := by |
nontriviality R
apply natDegree_multiset_prod'
suffices (t.map fun f => leadingCoeff f).prod = 1 by
rw [this]
simp
convert prod_replicate (Multiset.card t) (1 : R)
· simp only [eq_replicate, Multiset.card_map, eq_self_iff_true, true_and_iff]
rintro i hi
obtain ⟨i, hi, rfl⟩ := Multiset.mem_map... |
/-
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.Topology.Spectral.Hom
import Mathlib.AlgebraicGeometry.Limits
#align_import algebraic_geometry.morphisms.qu... | Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean | 129 | 158 | theorem isCompact_basicOpen (X : Scheme) {U : Opens X.carrier} (hU : IsCompact (U : Set X.carrier))
(f : X.presheaf.obj (op U)) : IsCompact (X.basicOpen f : Set X.carrier) := by |
classical
refine ((isCompact_open_iff_eq_finset_affine_union _).mpr ?_).1
obtain ⟨s, hs, e⟩ := (isCompact_open_iff_eq_finset_affine_union _).mp ⟨hU, U.isOpen⟩
let g : s → X.affineOpens := by
intro V
use V.1 ⊓ X.basicOpen f
have : V.1.1 ⟶ U := by
apply homOfLE; change _ ⊆ (U : Set X.carrier); ... |
/-
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 | 201 | 210 | theorem isEquivalent_iff_exists_eq_mul :
u ~[l] v ↔ ∃ (φ : α → β) (_ : Tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v := by |
rw [IsEquivalent, isLittleO_iff_exists_eq_mul]
constructor <;> rintro ⟨φ, hφ, h⟩ <;> [refine ⟨φ + 1, ?_, ?_⟩; refine ⟨φ - 1, ?_, ?_⟩]
· conv in 𝓝 _ => rw [← zero_add (1 : β)]
exact hφ.add tendsto_const_nhds
· convert h.add (EventuallyEq.refl l v) <;> simp [add_mul]
· conv in 𝓝 _ => rw [← sub_self (1 : ... |
/-
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.Order.Interval.Set.Monotone
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Topology.Order.Monot... | Mathlib/Analysis/BoxIntegral/Box/Basic.lean | 312 | 313 | theorem withBotCoe_inj {I J : WithBot (Box ι)} : (I : Set (ι → ℝ)) = J ↔ I = J := by |
simp only [Subset.antisymm_iff, ← le_antisymm_iff, withBotCoe_subset_iff]
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Yury Kudryashov
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Na... | Mathlib/Combinatorics/Pigeonhole.lean | 248 | 253 | theorem exists_card_fiber_lt_of_card_lt_nsmul (ht : ↑s.card < t.card • b) :
∃ y ∈ t, ↑(s.filter fun x => f x = y).card < b := by |
simp_rw [cast_card] at ht ⊢
exact
exists_sum_fiber_lt_of_sum_fiber_nonneg_of_sum_lt_nsmul
(fun _ _ => sum_nonneg fun _ _ => zero_le_one) ht
|
/-
Copyright (c) 2022 Alex Kontorovich and Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Mea... | Mathlib/MeasureTheory/Measure/Haar/Quotient.lean | 160 | 184 | theorem MeasureTheory.Measure.IsMulLeftInvariant.quotientMeasureEqMeasurePreimage_of_set {s : Set G}
(fund_dom_s : IsFundamentalDomain Γ.op s ν) {V : Set (G ⧸ Γ)}
(meas_V : MeasurableSet V) (neZeroV : μ V ≠ 0) (hV : μ V = ν (π ⁻¹' V ∩ s))
(neTopV : μ V ≠ ⊤) : QuotientMeasureEqMeasurePreimage ν μ := by |
apply fund_dom_s.quotientMeasureEqMeasurePreimage
ext U _
have meas_π : Measurable (QuotientGroup.mk : G → G ⧸ Γ) := continuous_quotient_mk'.measurable
let μ' : Measure (G ⧸ Γ) := (ν.restrict s).map π
haveI has_fund : HasFundamentalDomain Γ.op G ν := ⟨⟨s, fund_dom_s⟩⟩
have i : QuotientMeasureEqMeasurePreim... |
/-
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.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 254 | 258 | theorem coeFn_add (μ ν : FiniteMeasure Ω) : (⇑(μ + ν) : Set Ω → ℝ≥0) = (⇑μ + ⇑ν : Set Ω → ℝ≥0) := by |
funext
simp only [Pi.add_apply, ← ENNReal.coe_inj, ne_eq, ennreal_coeFn_eq_coeFn_toMeasure,
ENNReal.coe_add]
norm_cast
|
/-
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.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import category_theory.monoidal.center from "leanprover... | Mathlib/CategoryTheory/Monoidal/Center.lean | 323 | 325 | theorem rightUnitor_inv_f (X : Center C) : Hom.f (ρ_ X).inv = (ρ_ X.1).inv := by |
apply Iso.inv_ext' -- Porting note: Originally `ext`
rw [← rightUnitor_hom_f, ← comp_f, Iso.hom_inv_id]; rfl
|
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathli... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 617 | 625 | theorem isTotallyDisconnected_iff :
S.IsTotallyDisconnected ↔ ∀ c d, (S.arrows c d).Nonempty → c = d := by |
constructor
· rintro h c d ⟨f, fS⟩
have := h ⟨c, mem_objs_of_src S fS⟩ ⟨d, mem_objs_of_tgt S fS⟩ ⟨f, fS⟩
exact congr_arg Subtype.val this
· rintro h ⟨c, hc⟩ ⟨d, hd⟩ ⟨f, fS⟩
simp only [Subtype.mk_eq_mk]
exact h c d ⟨f, fS⟩
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.RBMap.Alter
import Batteries.Data.List.Lemmas
/-!
# Additional lemmas for Red-black trees
-/
namespace Batteries
namespace RBNode
open RBColor... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 45 | 65 | theorem isOrdered_iff' [@TransCmp α cmp] {t : RBNode α} :
isOrdered cmp t L R ↔
(∀ a ∈ L, t.All (cmpLT cmp a ·)) ∧
(∀ a ∈ R, t.All (cmpLT cmp · a)) ∧
(∀ a ∈ L, ∀ b ∈ R, cmpLT cmp a b) ∧
Ordered cmp t := by |
induction t generalizing L R with
| nil =>
simp [isOrdered]; split <;> simp [cmpLT_iff]
next h => intro _ ha _ hb; cases h _ _ ha hb
| node _ l v r =>
simp [isOrdered, *]
exact ⟨
fun ⟨⟨Ll, lv, Lv, ol⟩, ⟨vr, rR, vR, or⟩⟩ => ⟨
fun _ h => ⟨Lv _ h, Ll _ h, (Lv _ h).trans_l vr⟩,
... |
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd0... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 347 | 348 | theorem mem_col_iff {μ : YoungDiagram} {j : ℕ} {c : ℕ × ℕ} : c ∈ μ.col j ↔ c ∈ μ ∧ c.snd = j := by |
simp [col]
|
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.finite.card from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
/-!
# Cardinality ... | Mathlib/Data/Finite/Card.lean | 166 | 169 | theorem card_sum [Finite α] [Finite β] : Nat.card (Sum α β) = Nat.card α + Nat.card β := by |
haveI := Fintype.ofFinite α
haveI := Fintype.ofFinite β
simp only [Nat.card_eq_fintype_card, Fintype.card_sum]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Wen Yang
-/
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.RingT... | Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean | 404 | 411 | theorem fin_two_exists_eq_mk_of_apply_zero_one_eq_zero {R : Type*} [Field R] (g : SL(2, R))
(hg : (g : Matrix (Fin 2) (Fin 2) R) 1 0 = 0) :
∃ (a b : R) (h : a ≠ 0), g = (⟨!![a, b; 0, a⁻¹], by simp [h]⟩ : SL(2, R)) := by |
induction' g using Matrix.SpecialLinearGroup.fin_two_induction with a b c d h_det
replace hg : c = 0 := by simpa using hg
have had : a * d = 1 := by rwa [hg, mul_zero, sub_zero] at h_det
refine ⟨a, b, left_ne_zero_of_mul_eq_one had, ?_⟩
simp_rw [eq_inv_of_mul_eq_one_right had, hg]
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Mario Carneiro, Johan Commelin
-/
import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.RingTheory.DiscreteValuationRing.Basic
#align_import number_theory.p... | Mathlib/NumberTheory/Padics/PadicIntegers.lean | 400 | 400 | theorem valuation_p : valuation (p : ℤ_[p]) = 1 := by | simp [valuation]
|
/-
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 | 527 | 528 | theorem all_balance' {P l x r} : @All α P (balance' l x r) ↔ All P l ∧ P x ∧ All P r := by |
rw [balance']; split_ifs <;> simp [all_node', all_rotateL, all_rotateR]
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Scott Morrison, Johannes Hölzl, Reid Barton
-/
import Mathlib.CategoryTheory.Category.Init
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Tactic.PPWithUniv
im... | Mathlib/CategoryTheory/Category/Basic.lean | 315 | 317 | theorem cancel_epi_id (f : X ⟶ Y) [Epi f] {h : Y ⟶ Y} : f ≫ h = f ↔ h = 𝟙 Y := by |
convert cancel_epi f
simp
|
/-
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.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Topology.UniformSpace.Basic
#align_import topology.uniform... | Mathlib/Topology/UniformSpace/Cauchy.lean | 387 | 405 | theorem isComplete_iUnion_separated {ι : Sort*} {s : ι → Set α} (hs : ∀ i, IsComplete (s i))
{U : Set (α × α)} (hU : U ∈ 𝓤 α) (hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j) :
IsComplete (⋃ i, s i) := by |
set S := ⋃ i, s i
intro l hl hls
rw [le_principal_iff] at hls
cases' cauchy_iff.1 hl with hl_ne hl'
obtain ⟨t, htS, htl, htU⟩ : ∃ t, t ⊆ S ∧ t ∈ l ∧ t ×ˢ t ⊆ U := by
rcases hl' U hU with ⟨t, htl, htU⟩
refine ⟨t ∩ S, inter_subset_right, inter_mem htl hls, Subset.trans ?_ htU⟩
gcongr <;> apply inte... |
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