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 |
|---|---|---|---|---|---|
/-
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, Yury Kudryashov, Yuyang Zhao
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calcu... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 80 | 83 | theorem HasDerivAtFilter.scomp_of_eq (hg : HasDerivAtFilter g₁ g₁' y L')
(hh : HasDerivAtFilter h h' x L) (hy : y = h x) (hL : Tendsto h L L') :
HasDerivAtFilter (g₁ ∘ h) (h' • g₁') x L := by |
rw [hy] at hg; exact hg.scomp x hh hL
|
/-
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.RingTheory.TensorProduct.Basic
import Mathlib.Algebra.Module.ULift
#align_import ring_theory.is_tensor_product from "leanprover-community/mathlib"@"c4926d76... | Mathlib/RingTheory/IsTensorProduct.lean | 83 | 87 | theorem IsTensorProduct.equiv_symm_apply (h : IsTensorProduct f) (x₁ : M₁) (x₂ : M₂) :
h.equiv.symm (f x₁ x₂) = x₁ ⊗ₜ x₂ := by |
apply h.equiv.injective
refine (h.equiv.apply_symm_apply _).trans ?_
simp
|
/-
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.FieldTheory.Tower
import Mathlib.RingTheory.Algebraic
import Mathlib.FieldTheory.Minpoly.Basic
#align_import field_theory.intermediate_field from "leanprove... | Mathlib/FieldTheory/IntermediateField.lean | 316 | 319 | theorem toSubalgebra_toIntermediateField' (S : Subalgebra K L) (hS : IsField S) :
(S.toIntermediateField' hS).toSubalgebra = S := by |
ext
rfl
|
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint f... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 85 | 87 | theorem adjointAux_inner_right (A : E →L[𝕜] F) (x : E) (y : F) :
⟪x, adjointAux A y⟫ = ⟪A x, y⟫ := by |
rw [← inner_conj_symm, adjointAux_inner_left, inner_conj_symm]
|
/-
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.Rat.Basic
import Batteries.Tactic.SeqFocus
/-! # Additional lemmas about the Rational Numbers -/
namespace Rat
theorem ext : {p q : Rat} → p.... | .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | 167 | 168 | theorem divInt_mul_right {a : Int} (a0 : a ≠ 0) : (n * a) /. (d * a) = n /. d := by |
simp [← divInt_mul_left (d := d) a0, Int.mul_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.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Adjunction.Over
#align_import categ... | Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 87 | 96 | theorem pullback_diagonal_map_snd_fst_fst :
(pullback.snd :
pullback (diagonal f)
(map (i₁ ≫ snd) (i₂ ≫ snd) f f (i₁ ≫ fst) (i₂ ≫ fst) i (by simp [condition])
(by simp [condition])) ⟶
_) ≫
fst ≫ i₁ ≫ fst =
pullback.fst := by |
conv_rhs => rw [← Category.comp_id pullback.fst]
rw [← diagonal_fst f, pullback.condition_assoc, pullback.lift_fst]
|
/-
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, Floris van Doorn
-/
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,635 | 1,636 | theorem mk_le_aleph0_iff : #α ≤ ℵ₀ ↔ Countable α := by |
rw [countable_iff_nonempty_embedding, aleph0, ← lift_uzero #α, lift_mk_le']
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mat... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 359 | 362 | theorem union_zeroLocus (s s' : Set R) :
zeroLocus s ∪ zeroLocus s' = zeroLocus (Ideal.span s ⊓ Ideal.span s' : Ideal R) := by |
rw [zeroLocus_inf]
simp
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Algebra.Grou... | Mathlib/Algebra/Group/Submonoid/Operations.lean | 985 | 986 | theorem mrange_id : mrange (MonoidHom.id M) = ⊤ := by |
simp [mrange_eq_map]
|
/-
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 | 474 | 475 | theorem isLittleO_map {k : β → α} {l : Filter β} : f =o[map k l] g ↔ (f ∘ k) =o[l] (g ∘ k) := by |
simp only [IsLittleO_def, isBigOWith_map]
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.prod from "leanprover-communi... | Mathlib/MeasureTheory/Group/Prod.lean | 440 | 445 | theorem quasiMeasurePreserving_inv_of_right_invariant [IsMulRightInvariant μ] :
QuasiMeasurePreserving (Inv.inv : G → G) μ μ := by |
rw [← μ.inv_inv]
exact
(quasiMeasurePreserving_inv μ.inv).mono (inv_absolutelyContinuous μ.inv)
(absolutelyContinuous_inv μ.inv)
|
/-
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 | 906 | 907 | theorem mk'_mul_cancel_left (x) (y : S) : f.mk' ((y : M) * x) y = f.toMap x := by |
rw [mul_comm, mk'_mul_cancel_right]
|
/-
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.QuasiCompact
import Mathlib.Topology.QuasiSeparated
#align_import algebraic_geometry.morphisms.quasi_separated from "leanprover-... | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | 117 | 118 | theorem quasiSeparated_eq_diagonal_is_quasiCompact :
@QuasiSeparated = MorphismProperty.diagonal @QuasiCompact := by | ext; exact quasiSeparated_iff _
|
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Init.Order.Defs
set_option autoImplicit true
structure UFModel (n) where
parent : Fin n → Fin n
rank : Nat ... | Mathlib/Data/UnionFind.lean | 163 | 164 | theorem model' (self : UnionFind α) : ∃ (m : UFModel self.arr.size), m.Models self.arr := by |
let ⟨n, m, hm⟩ := self.model; cases hm.size_eq; exact ⟨m, hm⟩
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanpro... | Mathlib/RingTheory/Ideal/Operations.lean | 74 | 75 | theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := by |
simp_rw [annihilator, Module.mem_annihilator, Subtype.forall, Subtype.ext_iff]; rfl
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Measure.GiryMonad
#align_import probability.kernel.basic from "leanprover-community/mathlib"@"... | Mathlib/Probability/Kernel/Basic.lean | 421 | 424 | theorem set_lintegral_deterministic {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
[MeasurableSingletonClass β] (s : Set β) [Decidable (g a ∈ s)] :
∫⁻ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by |
rw [kernel.deterministic_apply, set_lintegral_dirac f s]
|
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "... | Mathlib/NumberTheory/Padics/RingHoms.lean | 104 | 121 | theorem norm_sub_modPart_aux (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) :
↑p ∣ r.num - r.num * r.den.gcdA p % p * ↑r.den := by |
rw [← ZMod.intCast_zmod_eq_zero_iff_dvd]
simp only [Int.cast_natCast, ZMod.natCast_mod, Int.cast_mul, Int.cast_sub]
have := congr_arg (fun x => x % p : ℤ → ZMod p) (gcd_eq_gcd_ab r.den p)
simp only [Int.cast_natCast, CharP.cast_eq_zero, EuclideanDomain.mod_zero, Int.cast_add,
Int.cast_mul, zero_mul, add_ze... |
/-
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.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 1,711 | 1,713 | theorem supportsStmt_write {S : Finset Λ'₁} {l : List Bool} {q : Stmt'₁} :
SupportsStmt S (write l q) = SupportsStmt S q := by |
induction' l with _ l IH <;> simp only [write, SupportsStmt, *]
|
/-
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 | 270 | 274 | theorem norm_sub_eq_add_norm_of_angle_eq_pi {x y : V} (h : angle x y = π) :
‖x - y‖ = ‖x‖ + ‖y‖ := by |
rw [← sq_eq_sq (norm_nonneg (x - y)) (add_nonneg (norm_nonneg x) (norm_nonneg y)),
norm_sub_pow_two_real, inner_eq_neg_mul_norm_of_angle_eq_pi h]
ring
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7... | Mathlib/Data/Seq/WSeq.lean | 1,384 | 1,385 | theorem get?_ofSeq (s : Seq α) (n) : get? (ofSeq s) n = Computation.pure (Seq.get? s n) := by |
dsimp [get?]; rw [dropn_ofSeq, head_ofSeq, Seq.head_dropn]
|
/-
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.Fintype.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Set.Image
#align_import data.fintype.car... | Mathlib/Data/Fintype/Card.lean | 933 | 946 | theorem wellFounded_of_trans_of_irrefl (r : α → α → Prop) [IsTrans α r] [IsIrrefl α r] :
WellFounded r := by |
classical
cases nonempty_fintype α
have :
∀ x y, r x y → (univ.filter fun z => r z x).card < (univ.filter fun z => r z y).card :=
fun x y hxy =>
Finset.card_lt_card <| by
simp only [Finset.lt_iff_ssubset.symm, lt_iff_le_not_le, Finset.le_iff_subset,
Finset.subset_iff, mem_filter, true... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
/-!
# Map... | Mathlib/RingTheory/Ideal/Maps.lean | 645 | 647 | theorem injective_iff_ker_eq_bot : Function.Injective f ↔ ker f = ⊥ := by |
rw [SetLike.ext'_iff, ker_eq, Set.ext_iff]
exact injective_iff_map_eq_zero' f
|
/-
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.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_imp... | Mathlib/Topology/Separation.lean | 261 | 264 | theorem t0Space_iff_exists_isOpen_xor'_mem (X : Type u) [TopologicalSpace X] :
T0Space X ↔ Pairwise fun x y => ∃ U : Set X, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U) := by |
simp only [t0Space_iff_not_inseparable, xor_iff_not_iff, not_forall, exists_prop,
inseparable_iff_forall_open, Pairwise]
|
/-
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 | 848 | 849 | theorem preimage_image_preimage {f : α → β} {s : Set β} : f ⁻¹' (f '' (f ⁻¹' s)) = f ⁻¹' s := by |
rw [image_preimage_eq_range_inter, preimage_range_inter]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Topology.Instances.RealVectorSpace
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import analysis.normed_space.mazur_ulam from "leanprove... | Mathlib/Analysis/NormedSpace/MazurUlam.lean | 45 | 83 | theorem midpoint_fixed {x y : PE} :
∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y := by |
set z := midpoint ℝ x y
-- Consider the set of `e : E ≃ᵢ E` such that `e x = x` and `e y = y`
set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y }
haveI : Nonempty s := ⟨⟨IsometryEquiv.refl PE, rfl, rfl⟩⟩
-- On the one hand, `e` cannot send the midpoint `z` of `[x, y]` too far
have h_bdd : BddAbove (range fun e :... |
/-
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 | 1,174 | 1,176 | theorem toSpanNonzeroSingleton_one :
LinearEquiv.toSpanNonzeroSingleton R M x h 1 =
(⟨x, Submodule.mem_span_singleton_self x⟩ : R ∙ x) := by | simp
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 2,427 | 2,428 | theorem erase_eq_empty_iff (s : Finset α) (a : α) : s.erase a = ∅ ↔ s = ∅ ∨ s = {a} := by |
rw [← sdiff_singleton_eq_erase, sdiff_eq_empty_iff_subset, subset_singleton_iff]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 441 | 442 | theorem Ioc_subset_Iic_self : Ioc a b ⊆ Iic b := by |
simpa [← coe_subset] using Set.Ioc_subset_Iic_self
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Sym... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 1,142 | 1,145 | theorem toMatrix'_comp (Q : QuadraticForm R (m → R)) (f : (n → R) →ₗ[R] m → R) :
(Q.comp f).toMatrix' = (LinearMap.toMatrix' f)ᵀ * Q.toMatrix' * (LinearMap.toMatrix' f) := by |
ext
simp only [QuadraticForm.associated_comp, LinearMap.toMatrix₂'_compl₁₂, toMatrix']
|
/-
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.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/ma... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 294 | 299 | theorem maximal_linearIndependent_eq_infinite_basis {ι : Type w} (b : Basis ι R M) [Infinite ι]
{κ : Type w} (v : κ → M) (i : LinearIndependent R v) (m : i.Maximal) : #κ = #ι := by |
apply le_antisymm
· exact linearIndependent_le_basis b v i
· haveI : Nontrivial R := nontrivial_of_invariantBasisNumber R
exact infinite_basis_le_maximal_linearIndependent b v i m
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib... | Mathlib/Order/SymmDiff.lean | 655 | 655 | theorem bihimp_bihimp_self : a ⇔ b ⇔ a = b := by | rw [bihimp_comm, bihimp_bihimp_cancel_left]
|
/-
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.Rat.Basic
import Batteries.Tactic.SeqFocus
/-! # Additional lemmas about the Rational Numbers -/
namespace Rat
theorem ext : {p q : Rat} → p.... | .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | 238 | 251 | theorem sub_def (a b : Rat) :
a - b = normalize (a.num * b.den - b.num * a.den) (a.den * b.den)
(Nat.mul_ne_zero a.den_nz b.den_nz) := by |
show Rat.sub .. = _; delta Rat.sub; dsimp only; split
· exact (normalize_self _).symm
· have : a.den.gcd b.den ≠ 0 := Nat.gcd_ne_zero_left a.den_nz
rw [maybeNormalize_eq_normalize _ _
(Int.ofNat_dvd_left.2 <| Nat.gcd_dvd_left ..)
(Nat.dvd_trans (Nat.gcd_dvd_right ..) <|
Nat.dvd_trans... |
/-
Copyright (c) 2023 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.AdicValuation
import Mathlib.RingTheory.DedekindDomain.Factorization
import Mathlib.Algebra.O... | Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean | 203 | 217 | theorem add {x y : K_hat R K} (hx : x.IsFiniteAdele) (hy : y.IsFiniteAdele) :
(x + y).IsFiniteAdele := by |
rw [IsFiniteAdele, Filter.eventually_cofinite] at hx hy ⊢
have h_subset :
{v : HeightOneSpectrum R | ¬(x + y) v ∈ v.adicCompletionIntegers K} ⊆
{v : HeightOneSpectrum R | ¬x v ∈ v.adicCompletionIntegers K} ∪
{v : HeightOneSpectrum R | ¬y v ∈ v.adicCompletionIntegers K} := by
intro v hv
rw... |
/-
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.Option.NAry
import Mathlib.Data.Seq.Computation
#align_import data.seq.seq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b34... | Mathlib/Data/Seq/Seq.lean | 937 | 939 | theorem join_cons (a b : α) (s S) :
join ((a, Seq.cons b s), S) = (a, Seq.join (Seq.cons (b, s) S)) := by |
dsimp [join]; rw [destruct_cons]
|
/-
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.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel ... | Mathlib/Data/Rel.lean | 86 | 88 | theorem codom_inv : r.inv.codom = r.dom := by |
ext x
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.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTh... | Mathlib/RingTheory/Localization/Basic.lean | 725 | 728 | theorem ringEquivOfRingEquiv_mk' {j : R ≃+* P} (H : M.map j.toMonoidHom = T) (x : R) (y : M) :
ringEquivOfRingEquiv S Q j H (mk' S x y) =
mk' Q (j x) ⟨j y, show j y ∈ T from H ▸ Set.mem_image_of_mem j y.2⟩ := by |
simp [map_mk']
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
imp... | Mathlib/FieldTheory/Adjoin.lean | 638 | 642 | theorem adjoin_simple_toSubalgebra_of_integral (hα : IsIntegral F α) :
F⟮α⟯.toSubalgebra = Algebra.adjoin F {α} := by |
apply adjoin_algebraic_toSubalgebra
rintro x (rfl : x = α)
rwa [isAlgebraic_iff_isIntegral]
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Additive.Corner.Defs
import Mathlib.Combinatorics.SimpleGraph... | Mathlib/Combinatorics/Additive/Corner/Roth.lean | 163 | 189 | theorem roth_3ap_theorem_nat (ε : ℝ) (hε : 0 < ε) (hG : cornersTheoremBound (ε / 3) ≤ n)
(A : Finset ℕ) (hAn : A ⊆ range n) (hAε : ε * n ≤ A.card) : ¬ ThreeAPFree (A : Set ℕ) := by |
rintro hA
rw [← coe_subset, coe_range] at hAn
have : A = Fin.val '' (Nat.cast '' A : Set (Fin (2 * n).succ)) := by
rw [Set.image_image, Set.image_congr, Set.image_id]
simp only [mem_coe, Nat.succ_eq_add_one, Fin.val_natCast, id_eq, Nat.mod_succ_eq_iff_lt]
rintro a ha
have := hAn ha
simp at th... |
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
imp... | Mathlib/FieldTheory/Adjoin.lean | 54 | 60 | theorem mem_adjoin_iff (x : E) :
x ∈ adjoin F S ↔ ∃ r s : MvPolynomial S F,
x = MvPolynomial.aeval Subtype.val r / MvPolynomial.aeval Subtype.val s := by |
simp only [adjoin, mem_mk, Subring.mem_toSubsemiring, Subfield.mem_toSubring,
Subfield.mem_closure_iff, ← Algebra.adjoin_eq_ring_closure, Subalgebra.mem_toSubring,
Algebra.adjoin_eq_range, AlgHom.mem_range, exists_exists_eq_and]
tauto
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate f... | Mathlib/Data/List/Rotate.lean | 167 | 168 | theorem rotate_length (l : List α) : rotate l l.length = l := by |
rw [rotate_eq_rotate', rotate'_length]
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.Sieves
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e622... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 760 | 762 | theorem familyOfElements_compatible : hx.familyOfElements.Compatible := by |
rintro Y₁ Y₂ Z g₁ g₂ f₁ f₂ ⟨i⟩ ⟨j⟩ hgf
simp [hx i j Z g₁ g₂ hgf]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.PUnitInstances
import Mathlib.GroupTheory.Congr... | Mathlib/GroupTheory/Coprod/Basic.lean | 573 | 580 | theorem con_mul_left_inv (x : FreeMonoid (G ⊕ H)) :
coprodCon G H (ofList (x.toList.map (Sum.map Inv.inv Inv.inv)).reverse * x) 1 := by |
rw [← mk_eq_mk, map_mul, map_one]
induction x using FreeMonoid.recOn with
| h0 => simp [map_one mk] -- TODO: fails without `[map_one mk]`
| ih x xs ihx =>
simp only [toList_of_mul, map_cons, reverse_cons, ofList_append, map_mul, ihx, ofList_singleton]
rwa [mul_assoc, ← mul_assoc (mk (of _)), mk_of_inv_... |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.Equiv
#align_import topology... | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 1,123 | 1,140 | theorem uniformSpace_eq_iInf_precomp_of_cover {δ : ι → Type*} (φ : Π i, δ i → α)
(𝔗 : ∀ i, Set (Set (δ i))) (h_image : ∀ i, MapsTo (φ i '' ·) (𝔗 i) 𝔖)
(h_preimage : ∀ i, MapsTo (φ i ⁻¹' ·) 𝔖 (𝔗 i))
(h_cover : ∀ S ∈ 𝔖, ∃ I : Set ι, I.Finite ∧ S ⊆ ⋃ i ∈ I, range (φ i)) :
𝒱(α, β, 𝔖, _) = ⨅ i, .coma... |
set ψ : Π S : Set α, Π i : ι, (φ i) ⁻¹' S → S := fun S i ↦ S.restrictPreimage (φ i)
have : ∀ S ∈ 𝔖, 𝒰(S, β, _) = ⨅ i, .comap (· ∘ ψ S i) 𝒰(_, β, _) := fun S hS ↦ by
rcases h_cover S hS with ⟨I, I_finite, I_cover⟩
refine UniformFun.uniformSpace_eq_iInf_precomp_of_cover β _ ⟨I, I_finite, ?_⟩
simpa onl... |
/-
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 | 462 | 465 | theorem add_cons (v : Fin n.succ → α) (y : α) (w : Fin n → α) :
v + vecCons y w = vecCons (vecHead v + y) (vecTail v + w) := by |
ext i
refine Fin.cases ?_ ?_ i <;> simp [vecHead, vecTail]
|
/-
Copyright (c) 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Basic
#align_import topology.fiber_bundle.constructions from "leanprove... | Mathlib/Topology/FiberBundle/Constructions.lean | 310 | 315 | theorem inducing_pullbackTotalSpaceEmbedding (f : B' → B) :
Inducing (@pullbackTotalSpaceEmbedding B F E B' f) := by |
constructor
simp_rw [instTopologicalSpaceProd, induced_inf, induced_compose,
Pullback.TotalSpace.topologicalSpace, pullbackTopology_def]
rfl
|
/-
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.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import M... | Mathlib/Order/WellFoundedSet.lean | 76 | 88 | theorem wellFoundedOn_iff :
s.WellFoundedOn r ↔ WellFounded fun a b : α => r a b ∧ a ∈ s ∧ b ∈ s := by |
have f : RelEmbedding (fun (a : s) (b : s) => r a b) fun a b : α => r a b ∧ a ∈ s ∧ b ∈ s :=
⟨⟨(↑), Subtype.coe_injective⟩, by simp⟩
refine ⟨fun h => ?_, f.wellFounded⟩
rw [WellFounded.wellFounded_iff_has_min]
intro t ht
by_cases hst : (s ∩ t).Nonempty
· rw [← Subtype.preimage_coe_nonempty] at hst
... |
/-
Copyright (c) 2019 Johannes Hölzl, Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Order.Filter.Germ
import Mathlib.Topology.ContinuousFunction.Algebra
impor... | Mathlib/MeasureTheory/Function/AEEqFun.lean | 346 | 349 | theorem pair_eq_mk (f : α →ₘ[μ] β) (g : α →ₘ[μ] γ) :
f.pair g =
mk (fun x => (f x, g x)) (f.aestronglyMeasurable.prod_mk g.aestronglyMeasurable) := by |
simp only [← pair_mk_mk, mk_coeFn, f.aestronglyMeasurable, g.aestronglyMeasurable]
|
/-
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 | 601 | 602 | theorem toFinset_congr {s t : Set α} [Fintype s] [Fintype t] (h : s = t) :
toFinset s = toFinset t := by | subst h; congr; exact Subsingleton.elim _ _
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb8... | Mathlib/Data/Finset/Card.lean | 69 | 69 | theorem card_mono : Monotone (@card α) := by | apply card_le_card
|
/-
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.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.FieldTheory.Finite.Trace
import Mathlib.Algebra.Group.AddChar
import Mathlib.Data.ZMod.Units
import... | Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean | 177 | 185 | theorem zmod_char_isNontrivial_iff (n : ℕ+) (ψ : AddChar (ZMod n) C) :
IsNontrivial ψ ↔ ψ 1 ≠ 1 := by |
refine ⟨?_, fun h => ⟨1, h⟩⟩
contrapose!
rintro h₁ ⟨a, ha⟩
have ha₁ : a = a.val • (1 : ZMod ↑n) := by
rw [nsmul_eq_mul, mul_one]; exact (ZMod.natCast_zmod_val a).symm
rw [ha₁, map_nsmul_eq_pow, h₁, one_pow] at ha
exact ha rfl
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
... | Mathlib/Combinatorics/SimpleGraph/Clique.lean | 235 | 248 | theorem not_cliqueFree_of_top_embedding {n : ℕ} (f : (⊤ : SimpleGraph (Fin n)) ↪g G) :
¬G.CliqueFree n := by |
simp only [CliqueFree, isNClique_iff, isClique_iff_induce_eq, not_forall, Classical.not_not]
use Finset.univ.map f.toEmbedding
simp only [card_map, Finset.card_fin, eq_self_iff_true, and_true_iff]
ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [coe_map, Set.mem_image, coe_univ, Set.mem_univ, true_and_iff] at hv hw
obtain ⟨... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.NAry
import Mathlib.Order.Directed
#align_import order.bounds.basic from "leanprover... | Mathlib/Order/Bounds/Basic.lean | 861 | 861 | theorem not_bddAbove_univ [NoMaxOrder α] : ¬BddAbove (univ : Set α) := by | simp [BddAbove]
|
/-
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 | 794 | 800 | theorem image_mul_const_uIcc (a b c : α) : (· * a) '' [[b, c]] = [[b * a, c * a]] :=
if ha : a = 0 then by simp [ha]
else calc
(fun x => x * a) '' [[b, c]] = (· * a⁻¹) ⁻¹' [[b, c]] :=
(Units.mk0 a ha).mulRight.image_eq_preimage _
_ = (fun x => x / a) ⁻¹' [[b, c]] := by | simp only [div_eq_mul_inv]
_ = [[b * a, c * a]] := preimage_div_const_uIcc ha _ _
|
/-
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 | 312 | 313 | theorem dual_node' (l : Ordnode α) (x : α) (r : Ordnode α) :
dual (node' l x r) = node' (dual r) x (dual l) := by | simp [node', add_comm]
|
/-
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 | 346 | 352 | theorem summable_geometric_iff_norm_lt_one : (Summable fun n : ℕ ↦ ξ ^ n) ↔ ‖ξ‖ < 1 := by |
refine ⟨fun h ↦ ?_, summable_geometric_of_norm_lt_one⟩
obtain ⟨k : ℕ, hk : dist (ξ ^ k) 0 < 1⟩ :=
(h.tendsto_cofinite_zero.eventually (ball_mem_nhds _ zero_lt_one)).exists
simp only [norm_pow, dist_zero_right] at hk
rw [← one_pow k] at hk
exact lt_of_pow_lt_pow_left _ zero_le_one hk
|
/-
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
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 2,409 | 2,410 | theorem map_top (f : α → β) : map f ⊤ = 𝓟 (range f) := by |
rw [← principal_univ, map_principal, image_univ]
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 140 | 143 | theorem edgeDensity_le_one (s : Finset α) (t : Finset β) : edgeDensity r s t ≤ 1 := by |
apply div_le_one_of_le
· exact mod_cast card_interedges_le_mul r s t
· exact mod_cast Nat.zero_le _
|
/-
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
-/
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.Antichain
import Mathlib.Order.SetNotation
#align_import data.set.intervals.ord_connect... | Mathlib/Order/Interval/Set/OrdConnected.lean | 257 | 260 | theorem ordConnected_image {E : Type*} [EquivLike E α β] [OrderIsoClass E α β] (e : E) {s : Set α}
[hs : OrdConnected s] : OrdConnected (e '' s) := by |
erw [(e : α ≃o β).image_eq_preimage]
apply ordConnected_preimage (e : α ≃o β).symm
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimag... | Mathlib/Data/Finsupp/Basic.lean | 1,458 | 1,460 | theorem comapSMul_apply (g : G) (f : α →₀ M) (a : α) : (g • f) a = f (g⁻¹ • a) := by |
conv_lhs => rw [← smul_inv_smul g a]
exact mapDomain_apply (MulAction.injective g) _ (g⁻¹ • a)
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupActio... | Mathlib/GroupTheory/Index.lean | 333 | 336 | theorem index_map_dvd {G' : Type*} [Group G'] {f : G →* G'} (hf : Function.Surjective f) :
(H.map f).index ∣ H.index := by |
rw [index_map, f.range_top_of_surjective hf, index_top, mul_one]
exact index_dvd_of_le le_sup_left
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | Mathlib/Order/Lattice.lean | 242 | 242 | theorem sup_right_idem (a b : α) : a ⊔ b ⊔ b = a ⊔ b := by | simp
|
/-
Copyright (c) 2019 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
#align_import analysis.con... | Mathlib/Analysis/Convex/Combination.lean | 502 | 510 | theorem convexHull_basis_eq_stdSimplex :
convexHull R (range fun i j : ι => if i = j then (1 : R) else 0) = stdSimplex R ι := by |
refine Subset.antisymm (convexHull_min ?_ (convex_stdSimplex R ι)) ?_
· rintro _ ⟨i, rfl⟩
exact ite_eq_mem_stdSimplex R i
· rintro w ⟨hw₀, hw₁⟩
rw [pi_eq_sum_univ w, ← Finset.univ.centerMass_eq_of_sum_1 _ hw₁]
exact Finset.univ.centerMass_mem_convexHull (fun i _ => hw₀ i) (hw₁.symm ▸ zero_lt_one)
... |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.Topology.Constructions
#align_import measure_theo... | Mathlib/MeasureTheory/Constructions/Pi.lean | 363 | 376 | theorem pi_eq_generateFrom {C : ∀ i, Set (Set (α i))}
(hC : ∀ i, generateFrom (C i) = by apply_assumption) (h2C : ∀ i, IsPiSystem (C i))
(h3C : ∀ i, (μ i).FiniteSpanningSetsIn (C i)) {μν : Measure (∀ i, α i)}
(h₁ : ∀ s : ∀ i, Set (α i), (∀ i, s i ∈ C i) → μν (pi univ s) = ∏ i, μ i (s i)) :
Measure.pi μ ... |
have h4C : ∀ (i) (s : Set (α i)), s ∈ C i → MeasurableSet s := by
intro i s hs; rw [← hC]; exact measurableSet_generateFrom hs
refine
(FiniteSpanningSetsIn.pi h3C).ext
(generateFrom_eq_pi hC fun i => (h3C i).isCountablySpanning).symm (IsPiSystem.pi h2C) ?_
rintro _ ⟨s, hs, rfl⟩
rw [mem_univ_pi] a... |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Scott Morrison
-/
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Multiset.Functor
#align_import data.finset.functor from "leanprove... | Mathlib/Data/Finset/Functor.lean | 92 | 95 | theorem image₂_def {α β γ : Type u} (f : α → β → γ) (s : Finset α) (t : Finset β) :
image₂ f s t = f <$> s <*> t := by |
ext
simp [mem_sup]
|
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Cont... | Mathlib/Data/Real/Cardinality.lean | 64 | 65 | theorem cantorFunctionAux_true (h : f n = true) : cantorFunctionAux c f n = c ^ n := by |
simp [cantorFunctionAux, h]
|
/-
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, Floris van Doorn
-/
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,241 | 2,242 | theorem two_le_iff' (x : α) : (2 : Cardinal) ≤ #α ↔ ∃ y : α, y ≠ x := by |
rw [two_le_iff, ← nontrivial_iff, nontrivial_iff_exists_ne x]
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Order.Closure
import Mathlib.ModelTheory.Semantics
import Mathlib.ModelTheory.Encoding
#align_import model_theory.substructures from "lea... | Mathlib/ModelTheory/Substructures.lean | 319 | 322 | theorem lift_card_closure_le_card_term : Cardinal.lift.{max u w} #(closure L s) ≤ #(L.Term s) := by |
rw [← SetLike.coe_sort_coe, coe_closure_eq_range_term_realize]
rw [← Cardinal.lift_id'.{w, max u w} #(L.Term s)]
exact Cardinal.mk_range_le_lift
|
/-
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.Group.Even
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Gr... | Mathlib/Algebra/Associated.lean | 814 | 817 | theorem eq_of_prime_pow_eq' (hp₁ : Prime p₁) (hp₂ : Prime p₂) (hk₁ : 0 < k₂)
(h : p₁ ^ k₁ = p₂ ^ k₂) : p₁ = p₂ := by |
rw [← associated_iff_eq] at h ⊢
apply h.of_pow_associated_of_prime' hp₁ hp₂ hk₁
|
/-
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.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Sets.Opens
#align_import topology.local_homeomorph from "leanprover-community/mathlib"@"431589b... | Mathlib/Topology/PartialHomeomorph.lean | 1,011 | 1,018 | theorem Set.EqOn.restr_eqOn_source {e e' : PartialHomeomorph X Y}
(h : EqOn e e' (e.source ∩ e'.source)) : e.restr e'.source ≈ e'.restr e.source := by |
constructor
· rw [e'.restr_source' _ e.open_source]
rw [e.restr_source' _ e'.open_source]
exact Set.inter_comm _ _
· rw [e.restr_source' _ e'.open_source]
refine (EqOn.trans ?_ h).trans ?_ <;> simp only [mfld_simps, eqOn_refl]
|
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical.Basic
import Mathlib.Algebra.Or... | Mathlib/Data/Real/NNReal.lean | 1,252 | 1,256 | theorem Real.exists_lt_of_strictMono [h : Nontrivial Γ₀ˣ] {f : Γ₀ →*₀ ℝ≥0} (hf : StrictMono f)
{r : ℝ} (hr : 0 < r) : ∃ d : Γ₀ˣ, (f d : ℝ) < r := by |
set s : NNReal := ⟨r, le_of_lt hr⟩
have hs : 0 < s := hr
exact NNReal.exists_lt_of_strictMono hf hs
|
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.monad fro... | Mathlib/Algebra/MvPolynomial/Monad.lean | 285 | 288 | theorem map_bind₁ (f : R →+* S) (g : σ → MvPolynomial τ R) (φ : MvPolynomial σ R) :
map f (bind₁ g φ) = bind₁ (fun i : σ => (map f) (g i)) (map f φ) := by |
rw [hom_bind₁, map_comp_C, ← eval₂Hom_map_hom]
rfl
|
/-
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.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction... | Mathlib/Algebra/Polynomial/Derivative.lean | 359 | 373 | theorem degree_derivative_eq [NoZeroSMulDivisors ℕ R] (p : R[X]) (hp : 0 < natDegree p) :
degree (derivative p) = (natDegree p - 1 : ℕ) := by |
apply le_antisymm
· rw [derivative_apply]
apply le_trans (degree_sum_le _ _) (Finset.sup_le _)
intro n hn
apply le_trans (degree_C_mul_X_pow_le _ _) (WithBot.coe_le_coe.2 (tsub_le_tsub_right _ _))
apply le_natDegree_of_mem_supp _ hn
· refine le_sup ?_
rw [mem_support_derivative, tsub_add_canc... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
#align_import logic.equiv.set from "leanprover-... | Mathlib/Logic/Equiv/Set.lean | 353 | 355 | theorem sumCompl_symm_apply {α : Type*} {s : Set α} [DecidablePred (· ∈ s)] {x : s} :
(Equiv.Set.sumCompl s).symm x = Sum.inl x := by |
cases' x with x hx; exact Set.sumCompl_symm_apply_of_mem hx
|
/-
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 | 418 | 420 | theorem inv_tensor {W X Y Z : C} (f : W ⟶ X) [IsIso f] (g : Y ⟶ Z) [IsIso g] :
inv (f ⊗ g) = inv f ⊗ inv g := by |
simp [tensorHom_def ,whisker_exchange]
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-... | Mathlib/Order/LiminfLimsup.lean | 1,064 | 1,070 | theorem OrderIso.apply_blimsup [CompleteLattice γ] (e : α ≃o γ) :
DFunLike.coe e (blimsup u f p) = blimsup ((DFunLike.coe e) ∘ u) f p := by |
simp only [blimsup_eq, map_sInf, Function.comp_apply]
congr
ext c
obtain ⟨a, rfl⟩ := e.surjective c
simp
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
#align_import data.matri... | Mathlib/Data/Matrix/Notation.lean | 527 | 528 | theorem vec2_dotProduct' {a₀ a₁ b₀ b₁ : α} : ![a₀, a₁] ⬝ᵥ ![b₀, b₁] = a₀ * b₀ + a₁ * b₁ := by |
rw [cons_dotProduct_cons, cons_dotProduct_cons, dotProduct_empty, add_zero]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Da... | Mathlib/GroupTheory/OrderOfElement.lean | 1,096 | 1,098 | theorem zpow_mod_card (a : G) (n : ℤ) : a ^ (n % Fintype.card G : ℤ) = a ^ n := by |
rw [eq_comm, ← zpow_mod_orderOf, ← Int.emod_emod_of_dvd n
(Int.natCast_dvd_natCast.2 orderOf_dvd_card), zpow_mod_orderOf]
|
/-
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, Yury Kudryashov, Yuyang Zhao
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calcu... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 90 | 93 | theorem HasDerivWithinAt.scomp_hasDerivAt_of_eq (hg : HasDerivWithinAt g₁ g₁' s' y)
(hh : HasDerivAt h h' x) (hs : ∀ x, h x ∈ s') (hy : y = h x) :
HasDerivAt (g₁ ∘ h) (h' • g₁') x := by |
rw [hy] at hg; exact hg.scomp_hasDerivAt x hh hs
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,866 | 1,870 | theorem bsup_eq_blsub_of_lt_succ_limit {o : Ordinal.{u}} (ho : IsLimit o)
{f : ∀ a < o, Ordinal.{max u v}} (hf : ∀ a ha, f a ha < f (succ a) (ho.2 a ha)) :
bsup.{_, v} o f = blsub.{_, v} o f := by |
rw [bsup_eq_blsub_iff_lt_bsup]
exact fun i hi => (hf i hi).trans_le (le_bsup f _ _)
|
/-
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 | 225 | 232 | theorem f_add_nat_le (hf_conv : ConvexOn ℝ (Ioi 0) f)
(hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hn : n ≠ 0) (hx : 0 < x) (hx' : x ≤ 1) :
f (n + x) ≤ f n + x * log n := by |
have hn' : 0 < (n : ℝ) := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn)
have : f n + x * log n = (1 - x) * f n + x * f (n + 1) := by rw [hf_feq hn']; ring
rw [this, (by ring : (n : ℝ) + x = (1 - x) * n + x * (n + 1))]
simpa only [smul_eq_mul] using
hf_conv.2 hn' (by linarith : 0 < (n + 1 : ℝ)) (by linarith : 0 ... |
/-
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, Sébastien Gouëzel
-/
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.M... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 554 | 556 | theorem norm_integral_eq_norm_integral_Ioc (f : ℝ → E) :
‖∫ x in a..b, f x ∂μ‖ = ‖∫ x in Ι a b, f x ∂μ‖ := by |
rw [← norm_integral_min_max, integral_of_le min_le_max, uIoc]
|
/-
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 | 91 | 102 | theorem lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors'
{p : R[X]} {t : R} {n : ℕ} (h : p ≠ 0)
(hroot : ∀ m ≤ n, (derivative^[m] p).IsRoot t)
(hnzd : ∀ m ≤ n, m ≠ 0 → (m : R) ∈ nonZeroDivisors R) :
n < p.rootMultiplicity t := by |
apply lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors h hroot
clear hroot
induction' n with n ih
· simp only [Nat.zero_eq, Nat.factorial_zero, Nat.cast_one]
exact Submonoid.one_mem _
· rw [Nat.factorial_succ, Nat.cast_mul, mul_mem_nonZeroDivisors]
exact ⟨hnzd _ le_rfl n.succ_n... |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic... | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 108 | 120 | theorem logEmbedding_eq_zero_iff {x : (𝓞 K)ˣ} :
logEmbedding K x = 0 ↔ x ∈ torsion K := by |
rw [mem_torsion]
refine ⟨fun h w => ?_, fun h => ?_⟩
· by_cases hw : w = w₀
· suffices -mult w₀ * Real.log (w₀ (x : K)) = 0 by
rw [neg_mul, neg_eq_zero, ← hw] at this
exact mult_log_place_eq_zero.mp this
rw [← sum_logEmbedding_component, sum_eq_zero]
exact fun w _ => congrFun h w
... |
/-
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 | 752 | 756 | theorem nhds_basis_uniformity {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s)
{x : α} : (𝓝 x).HasBasis p fun i => { y | (y, x) ∈ s i } := by |
replace h := h.comap Prod.swap
rw [comap_swap_uniformity] at h
exact nhds_basis_uniformity' h
|
/-
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.Group.Equiv.TypeTags
import Mathlib.Data.ZMod.Quotient
import Mathlib.RingTheory.DedekindDomain.AdicValuation
#align_impor... | Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean | 209 | 234 | theorem fromUnit_ker [hn : Fact <| 0 < n] :
(@fromUnit R _ _ K _ _ _ n).ker = (powMonoidHom n : Rˣ →* Rˣ).range := by |
ext ⟨_, _, _, _⟩
constructor
· intro hx
rcases (QuotientGroup.eq_one_iff _).mp (Subtype.mk.inj hx) with ⟨⟨v, i, vi, iv⟩, hx⟩
have hv : ↑(_ ^ n : Kˣ) = algebraMap R K _ := congr_arg Units.val hx
have hi : ↑(_ ^ n : Kˣ)⁻¹ = algebraMap R K _ := congr_arg Units.inv hx
rw [Units.val_pow_eq_pow_val] at... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebr... | Mathlib/LinearAlgebra/Lagrange.lean | 227 | 229 | theorem basis_pair_left (hij : i ≠ j) : Lagrange.basis {i, j} v i = basisDivisor (v i) (v j) := by |
simp only [Lagrange.basis, hij, erase_insert_eq_erase, erase_eq_of_not_mem, mem_singleton,
not_false_iff, prod_singleton]
|
/-
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.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space
#align_import measure_theory.function.uniform_integrable from "lea... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 231 | 254 | theorem Memℒp.snormEssSup_indicator_norm_ge_eq_zero (hf : Memℒp f ∞ μ)
(hmeas : StronglyMeasurable f) :
∃ M : ℝ, snormEssSup ({ x | M ≤ ‖f x‖₊ }.indicator f) μ = 0 := by |
have hbdd : snormEssSup f μ < ∞ := hf.snorm_lt_top
refine ⟨(snorm f ∞ μ + 1).toReal, ?_⟩
rw [snormEssSup_indicator_eq_snormEssSup_restrict]
· have : μ.restrict { x : α | (snorm f ⊤ μ + 1).toReal ≤ ‖f x‖₊ } = 0 := by
simp only [coe_nnnorm, snorm_exponent_top, Measure.restrict_eq_zero]
have : { x : α... |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
[`data.finset.sym`@`98e83c3d541c77cdb7da20d79611a780ff8e7d90`..`02ba8949f486ebecf93fe7460f1ed0564b5e442c`](https://leanprover-community.github.io/mathlib-port-status/file/d... | Mathlib/Data/Finset/Sym.lean | 96 | 97 | theorem sym2_eq_empty : s.sym2 = ∅ ↔ s = ∅ := by |
rw [← val_eq_zero, sym2_val, Multiset.sym2_eq_zero_iff, val_eq_zero]
|
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrabl... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 1,341 | 1,348 | theorem integral_comp_comm' (L : E →L[𝕜] F) {K} (hL : AntilipschitzWith K L) (φ : X → E) :
∫ x, L (φ x) ∂μ = L (∫ x, φ x ∂μ) := by |
by_cases h : Integrable φ μ
· exact integral_comp_comm L h
have : ¬Integrable (fun x => L (φ x)) μ := by
rwa [← Function.comp_def,
LipschitzWith.integrable_comp_iff_of_antilipschitz L.lipschitz hL L.map_zero]
simp [integral_undef, h, this]
|
/-
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 | 247 | 250 | theorem append_nil {u v : V} (p : G.Walk u v) : p.append nil = p := by |
induction p with
| nil => rfl
| cons _ _ ih => rw [cons_append, ih]
|
/-
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.Group.Aut
import Mathlib.Algebra.Group.Invertible.Basic
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.GroupTheory.GroupAction.Uni... | Mathlib/GroupTheory/GroupAction/Group.lean | 119 | 122 | theorem smul_zpow [Group β] [SMulCommClass α β β] [IsScalarTower α β β] (c : α) (x : β) (p : ℤ) :
(c • x) ^ p = c ^ p • x ^ p := by |
cases p <;>
simp [smul_pow, smul_inv]
|
/-
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.WF
/-!
# Path operations; `modify` and `alter`
This develops the necessary theorems to construct the `modify` and `alter` functions on `... | .lake/packages/batteries/Batteries/Data/RBMap/Alter.lean | 278 | 279 | theorem modify_eq_alter (t : RBNode α) : t.modify cut f = t.alter cut (.map f) := by |
simp [modify, alter]
|
/-
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.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0... | Mathlib/Geometry/Euclidean/MongePoint.lean | 471 | 483 | theorem altitude_eq_mongePlane (t : Triangle ℝ P) {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃)
(h₂₃ : i₂ ≠ i₃) : t.altitude i₁ = t.mongePlane i₂ i₃ := by |
have hs : ({i₂, i₃}ᶜ : Finset (Fin 3)) = {i₁} := by
-- Porting note (#11043): was `decide!`
fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
<;> simp (config := {decide := true}) at h₁₂ h₁₃ h₂₃ ⊢
have he : univ.erase i₁ = {i₂, i₃} := by
-- Porting note (#11043): was `decide!`
fin_cases i₁ <;> f... |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanp... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 104 | 106 | theorem smul_closedBall' {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ) :
c • closedBall x r = closedBall (c • x) (‖c‖ * r) := by |
simp only [← ball_union_sphere, Set.smul_set_union, _root_.smul_ball hc, smul_sphere' hc]
|
/-
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.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Orde... | Mathlib/Order/CompactlyGenerated/Basic.lean | 83 | 105 | theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
CompleteLattice.IsCompactElement k ↔
∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t : Finset ι, k ≤ t.sup s := by |
classical
constructor
· intro H ι s hs
obtain ⟨t, ht, ht'⟩ := H (Set.range s) hs
have : ∀ x : t, ∃ i, s i = x := fun x => ht x.prop
choose f hf using this
refine ⟨Finset.univ.image f, ht'.trans ?_⟩
rw [Finset.sup_le_iff]
intro b hb
rw [← show s (f ⟨b, hb⟩) = id b fro... |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#alig... | Mathlib/LinearAlgebra/Dimension/Free.lean | 88 | 90 | theorem _root_.FiniteDimensional.finrank_eq_card_chooseBasisIndex [Module.Finite R M] :
finrank R M = Fintype.card (ChooseBasisIndex R M) := by |
simp [finrank, rank_eq_card_chooseBasisIndex]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 488 | 491 | theorem UniformCauchySeqOn.mono {s' : Set α} (hf : UniformCauchySeqOn F p s) (hss' : s' ⊆ s) :
UniformCauchySeqOn F p s' := by |
rw [uniformCauchySeqOn_iff_uniformCauchySeqOnFilter] at hf ⊢
exact hf.mono_right (le_principal_iff.mpr <| mem_principal.mpr hss')
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Scott Morrison
-/
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.List.InsertNth
import Mathlib.Logic.Relation
import Mathlib... | Mathlib/SetTheory/Game/PGame.lean | 1,445 | 1,445 | theorem neg_lf_zero_iff {x : PGame} : -x ⧏ 0 ↔ 0 ⧏ x := by | rw [neg_lf_iff, neg_zero]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.RingTheory.MatrixAlgebra
#align_impor... | Mathlib/RingTheory/PolynomialAlgebra.lean | 286 | 290 | theorem matPolyEquiv_symm_apply_coeff (p : (Matrix n n R)[X]) (i j : n) (k : ℕ) :
coeff (matPolyEquiv.symm p i j) k = coeff p k i j := by |
have t : p = matPolyEquiv (matPolyEquiv.symm p) := by simp
conv_rhs => rw [t]
simp only [matPolyEquiv_coeff_apply]
|
/-
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
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
#align_import measure_theory.function.locally_integrable from "leanprover-community/mathlib"@"08a4542bec7242a5... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 179 | 182 | theorem locallyIntegrable_comap (hs : MeasurableSet s) :
LocallyIntegrable (fun x : s ↦ f x) (μ.comap Subtype.val) ↔ LocallyIntegrableOn f s μ := by |
simp_rw [LocallyIntegrableOn, Subtype.forall', ← map_nhds_subtype_val]
exact forall_congr' fun _ ↦ (MeasurableEmbedding.subtype_coe hs).integrableAtFilter_iff_comap.symm
|
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