Context stringlengths 285 6.98k | file_name stringlengths 21 79 | start int64 14 184 | end int64 18 184 | theorem stringlengths 25 1.34k | proof stringlengths 5 3.43k |
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/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Order.Sublattice
import Mathlib.Order.Hom.CompleteLattice
/-!
# Complete Sublattices
This file defines complete sublattices. These are subsets of complete ... | Mathlib/Order/CompleteSublattice.lean | 89 | 90 | theorem coe_sInf' (S : Set L) : (↑(sInf S) : α) = ⨅ N ∈ S, (N : α) := by |
rw [coe_sInf, ← Set.image, sInf_image]
|
/-
Copyright (c) 2021 Chris Hughes, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Junyan Xu
-/
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Data.Finsupp.Fintype
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import data.mv_polyn... | Mathlib/Algebra/MvPolynomial/Cardinal.lean | 58 | 59 | theorem cardinal_mk_eq_max [Nonempty σ] [Nontrivial R] :
#(MvPolynomial σ R) = max (max #R #σ) ℵ₀ := by | simp
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.Normed.Group.Basic
#align_import analysis.normed.group.hom from "leanprover-community/mathlib"@"3c4225288b55380a90df078ebae0991080b12393"
/-... | Mathlib/Analysis/Normed/Group/Hom.lean | 67 | 74 | theorem exists_pos_bound_of_bound {V W : Type*} [SeminormedAddCommGroup V]
[SeminormedAddCommGroup W] {f : V → W} (M : ℝ) (h : ∀ x, ‖f x‖ ≤ M * ‖x‖) :
∃ N, 0 < N ∧ ∀ x, ‖f x‖ ≤ N * ‖x‖ :=
⟨max M 1, lt_of_lt_of_le zero_lt_one (le_max_right _ _), fun x =>
calc
‖f x‖ ≤ M * ‖x‖ := h x
_ ≤ max M 1 ... | gcongr; apply le_max_left
⟩
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Order.BoundedOrder
import Mathlib.Order.MinMax
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean | 56 | 58 | theorem exists_one_lt_mul_of_lt' (h : a < b) : ∃ c, 1 < c ∧ a * c = b := by |
obtain ⟨c, rfl⟩ := exists_mul_of_le h.le
exact ⟨c, one_lt_of_lt_mul_right h, rfl⟩
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
#align_import data.pfunctor.multivariate.W from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e3... | Mathlib/Data/PFunctor/Multivariate/W.lean | 109 | 111 | theorem wPathCasesOn_eta {α : TypeVec n} {a : P.A} {f : P.last.B a → P.last.W}
(h : P.WPath ⟨a, f⟩ ⟹ α) : P.wPathCasesOn (P.wPathDestLeft h) (P.wPathDestRight h) = h := by |
ext i x; cases x <;> 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 | 73 | 73 | theorem oangle_zero_right (x : V) : o.oangle x 0 = 0 := by | simp [oangle]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.P... | Mathlib/NumberTheory/Bernoulli.lean | 128 | 131 | theorem bernoulli'_four : bernoulli' 4 = -1 / 30 := by |
have : Nat.choose 4 2 = 6 := by decide -- shrug
rw [bernoulli'_def]
norm_num [sum_range_succ, sum_range_succ, sum_range_zero, this]
|
/-
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.BigOperators.Ring.Finset
import Mathlib.Data.NNRat.Defs
/-! # Casting lemmas for non-negative rational numbers i... | Mathlib/Data/NNRat/BigOperators.lean | 41 | 44 | theorem toNNRat_sum_of_nonneg {s : Finset α} {f : α → ℚ} (hf : ∀ a, a ∈ s → 0 ≤ f a) :
(∑ a ∈ s, f a).toNNRat = ∑ a ∈ s, (f a).toNNRat := by |
rw [← coe_inj, coe_sum, Rat.coe_toNNRat _ (Finset.sum_nonneg hf)]
exact Finset.sum_congr rfl fun x hxs ↦ by rw [Rat.coe_toNNRat _ (hf x hxs)]
|
/-
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 | 85 | 89 | theorem sym2_toFinset [DecidableEq α] (m : Multiset α) :
m.toFinset.sym2 = m.sym2.toFinset := by |
ext z
refine z.ind fun x y ↦ ?_
simp only [mk_mem_sym2_iff, Multiset.mem_toFinset, Multiset.mk_mem_sym2_iff]
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Scott Morrison
-/
import Mathlib.CategoryTheory.Preadditive.Yoneda.Basic
import Mathlib.CategoryTheory.Preadditive.Injective
import Mathlib.Algebra.Category.GroupCat.EpiM... | Mathlib/CategoryTheory/Preadditive/Yoneda/Injective.lean | 32 | 40 | theorem injective_iff_preservesEpimorphisms_preadditiveYoneda_obj (J : C) :
Injective J ↔ (preadditiveYoneda.obj J).PreservesEpimorphisms := by |
rw [injective_iff_preservesEpimorphisms_yoneda_obj]
refine
⟨fun h : (preadditiveYoneda.obj J ⋙ (forget AddCommGroupCat)).PreservesEpimorphisms => ?_, ?_⟩
· exact
Functor.preservesEpimorphisms_of_preserves_of_reflects (preadditiveYoneda.obj J) (forget _)
· intro
exact (inferInstance : (preadditive... |
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
/-!
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 96 | 98 | theorem fixedBy_mul (m₁ m₂ : M) : fixedBy α m₁ ∩ fixedBy α m₂ ⊆ fixedBy α (m₁ * m₂) := by |
intro a ⟨h₁, h₂⟩
rw [mem_fixedBy, mul_smul, h₂, h₁]
|
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most one bas... | Mathlib/Data/Matroid/Constructions.lean | 126 | 127 | theorem not_rkPos_iff : ¬RkPos M ↔ M = loopyOn M.E := by |
rw [rkPos_iff_empty_not_base, not_iff_comm, empty_base_iff]
|
/-
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 | 113 | 124 | theorem numDerangements_sum (n : ℕ) :
(numDerangements n : ℤ) =
∑ k ∈ Finset.range (n + 1), (-1 : ℤ) ^ k * Nat.ascFactorial (k + 1) (n - k) := by |
induction' n with n hn; · rfl
rw [Finset.sum_range_succ, numDerangements_succ, hn, Finset.mul_sum, tsub_self,
Nat.ascFactorial_zero, Int.ofNat_one, mul_one, pow_succ', neg_one_mul, sub_eq_add_neg,
add_left_inj, Finset.sum_congr rfl]
-- show that (n + 1) * (-1)^x * asc_fac x (n - x) = (-1)^x * asc_fac x (... |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.t... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 79 | 81 | theorem thickenedIndicatorAux_closure_eq (δ : ℝ) (E : Set α) :
thickenedIndicatorAux δ (closure E) = thickenedIndicatorAux δ E := by |
simp (config := { unfoldPartialApp := true }) only [thickenedIndicatorAux, infEdist_closure]
|
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Batteries.Data.List.Lemmas
import Batteries.Data.Array.Basic
import Batteries.Tactic.SeqFocus
import Batteries.Util.ProofWanted
namespace Arra... | .lake/packages/batteries/Batteries/Data/Array/Lemmas.lean | 75 | 77 | theorem size_zipWith (as : Array α) (bs : Array β) (f : α → β → γ) :
(as.zipWith bs f).size = min as.size bs.size := by |
rw [size_eq_length_data, zipWith_eq_zipWith_data, List.length_zipWith]
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.MeasurableIntegral
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.with_density from "leanprover-com... | Mathlib/Probability/Kernel/WithDensity.lean | 56 | 57 | theorem withDensity_of_not_measurable (κ : kernel α β) [IsSFiniteKernel κ]
(hf : ¬Measurable (Function.uncurry f)) : withDensity κ f = 0 := by | classical exact dif_neg hf
|
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.Polynomial.Basic
#align_import algebraic_geometry.prime_spectrum.is_open_comap_C from "l... | Mathlib/AlgebraicGeometry/PrimeSpectrum/IsOpenComapC.lean | 54 | 66 | theorem imageOfDf_eq_comap_C_compl_zeroLocus :
imageOfDf f = PrimeSpectrum.comap (C : R →+* R[X]) '' (zeroLocus {f})ᶜ := by |
ext x
refine ⟨fun hx => ⟨⟨map C x.asIdeal, isPrime_map_C_of_isPrime x.IsPrime⟩, ⟨?_, ?_⟩⟩, ?_⟩
· rw [mem_compl_iff, mem_zeroLocus, singleton_subset_iff]
cases' hx with i hi
exact fun a => hi (mem_map_C_iff.mp a i)
· ext x
refine ⟨fun h => ?_, fun h => subset_span (mem_image_of_mem C.1 h)⟩
rw [←... |
/-
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.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
/-!
... | Mathlib/Data/ENNReal/Real.lean | 88 | 91 | theorem toReal_mono' (h : a ≤ b) (ht : b = ∞ → a = ∞) : a.toReal ≤ b.toReal := by |
rcases eq_or_ne a ∞ with rfl | ha
· exact toReal_nonneg
· exact toReal_mono (mt ht ha) h
|
/-
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.CategoryTheory.LiftingProperties.Basic
import Mathlib.CategoryTheory.Adjunction.Basic
#align_import category_theory.lifting_properties.adjunction from "leanprov... | Mathlib/CategoryTheory/LiftingProperties/Adjunction.lean | 66 | 68 | theorem right_adjoint_hasLift_iff : HasLift (sq.right_adjoint adj) ↔ HasLift sq := by |
simp only [HasLift.iff]
exact Equiv.nonempty_congr (sq.rightAdjointLiftStructEquiv adj).symm
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanpro... | Mathlib/Data/Sym/Sym2.lean | 69 | 69 | theorem Rel.symm {x y : α × α} : Rel α x y → Rel α y x := by | aesop (rule_sets := [Sym2])
|
/-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Group... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 38 | 42 | theorem orbit.eq_or_disjoint (a b : X) :
orbit G a = orbit G b ∨ Disjoint (orbit G a) (orbit G b) := by |
apply (em (Disjoint (orbit G a) (orbit G b))).symm.imp _ id
simp (config := { contextual := true })
only [Set.not_disjoint_iff, ← orbit_eq_iff, forall_exists_index, and_imp, eq_comm, implies_true]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
/-!
# Lemmas a... | Mathlib/Data/Int/Order/Units.lean | 40 | 41 | theorem units_div_eq_mul (u₁ u₂ : ℤˣ) : u₁ / u₂ = u₁ * u₂ := by |
rw [div_eq_mul_inv, units_inv_eq_self]
|
/-
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 | 91 | 91 | theorem det_one : det (1 : Matrix n n R) = 1 := by | rw [← diagonal_one]; simp [-diagonal_one]
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.Probability.Kernel.Disintegration.CdfToKernel
#align_import probability.kernel.cond_cdf from "lean... | Mathlib/Probability/Kernel/Disintegration/CondCdf.lean | 72 | 75 | theorem IicSnd_le_fst (r : ℝ) : ρ.IicSnd r ≤ ρ.fst := by |
refine Measure.le_iff.2 fun s hs ↦ ?_
simp_rw [fst_apply hs, IicSnd_apply ρ r hs]
exact measure_mono (prod_subset_preimage_fst _ _)
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Rodriguez
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.... | Mathlib/GroupTheory/ClassEquation.lean | 38 | 43 | theorem Group.sum_card_conj_classes_eq_card [Finite G] :
∑ᶠ x : ConjClasses G, x.carrier.ncard = Nat.card G := by |
classical
cases nonempty_fintype G
rw [Nat.card_eq_fintype_card, ← sum_conjClasses_card_eq_card, finsum_eq_sum_of_fintype]
simp [Set.ncard_eq_toFinset_card']
|
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Field.Rat
im... | Mathlib/Algebra/Module/Basic.lean | 28 | 43 | theorem map_inv_natCast_smul [AddCommMonoid M] [AddCommMonoid M₂] {F : Type*} [FunLike F M M₂]
[AddMonoidHomClass F M M₂] (f : F) (R S : Type*)
[DivisionSemiring R] [DivisionSemiring S] [Module R M]
[Module S M₂] (n : ℕ) (x : M) : f ((n⁻¹ : R) • x) = (n⁻¹ : S) • f x := by |
by_cases hR : (n : R) = 0 <;> by_cases hS : (n : S) = 0
· simp [hR, hS, map_zero f]
· suffices ∀ y, f y = 0 by rw [this, this, smul_zero]
clear x
intro x
rw [← inv_smul_smul₀ hS (f x), ← map_natCast_smul f R S]
simp [hR, map_zero f]
· suffices ∀ y, f y = 0 by simp [this]
clear x
intro x... |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Fintype.Prod
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Option
... | Mathlib/GroupTheory/Perm/Option.lean | 38 | 43 | theorem Equiv.optionCongr_sign {α : Type*} [DecidableEq α] [Fintype α] (e : Perm α) :
Perm.sign e.optionCongr = Perm.sign e := by |
refine Perm.swap_induction_on e ?_ ?_
· simp [Perm.one_def]
· intro f x y hne h
simp [h, hne, Perm.mul_def, ← Equiv.optionCongr_trans]
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Int.Bitwise
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.m... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 44 | 47 | theorem inv_pow' (A : M) (n : ℕ) : A⁻¹ ^ n = (A ^ n)⁻¹ := by |
induction' n with n ih
· simp
· rw [pow_succ A, mul_inv_rev, ← ih, ← pow_succ']
|
/-
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 | 74 | 83 | theorem eq_of_degree_le_of_eval_finset_eq
(h_deg_le : f.degree ≤ s.card)
(h_deg_eq : f.degree = g.degree)
(hlc : f.leadingCoeff = g.leadingCoeff)
(h_eval : ∀ x ∈ s, f.eval x = g.eval x) :
f = g := by |
rcases eq_or_ne f 0 with rfl | hf
· rwa [degree_zero, eq_comm, degree_eq_bot, eq_comm] at h_deg_eq
· exact eq_of_degree_sub_lt_of_eval_finset_eq s
(lt_of_lt_of_le (degree_sub_lt h_deg_eq hf hlc) h_deg_le) h_eval
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.Layercake
#align_import analy... | Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean | 36 | 38 | theorem sqrt_one_add_norm_sq_le (x : E) : √((1 : ℝ) + ‖x‖ ^ 2) ≤ 1 + ‖x‖ := by |
rw [sqrt_le_left (by positivity)]
simp [add_sq]
|
/-
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.UniqueFactorizationDomain
import Mathlib.RingTheory.Localization.Basic
#align_import ... | Mathlib/RingTheory/Localization/Away/Basic.lean | 58 | 61 | theorem mul_invSelf : algebraMap R S x * invSelf x = 1 := by |
convert IsLocalization.mk'_mul_mk'_eq_one (M := Submonoid.powers x) (S := S) _ 1
symm
apply IsLocalization.mk'_one
|
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Sites.Coherent.Comparison
import Mathlib.Topology.Category.CompHaus.Limits
/-!
# Effective epimorphisms and finite effective ... | Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean | 72 | 85 | theorem effectiveEpi_tfae
{B X : CompHaus.{u}} (π : X ⟶ B) :
TFAE
[ EffectiveEpi π
, Epi π
, Function.Surjective π
] := by |
tfae_have 1 → 2
· intro; infer_instance
tfae_have 2 ↔ 3
· exact epi_iff_surjective π
tfae_have 3 → 1
· exact fun hπ ↦ ⟨⟨struct π hπ⟩⟩
tfae_finish
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
#align_import dynamics.ergodic.ergodic from "leanprover-community/mathlib"@"809e920edfa343283cea507aedff916ea0f1bd88"
/-... | Mathlib/Dynamics/Ergodic/Ergodic.lean | 99 | 106 | theorem preErgodic_conjugate_iff {e : α ≃ᵐ β} (h : MeasurePreserving e μ μ') :
PreErgodic (e ∘ f ∘ e.symm) μ' ↔ PreErgodic f μ := by |
refine ⟨fun hf => preErgodic_of_preErgodic_conjugate (h.symm e) hf ?_,
fun hf => preErgodic_of_preErgodic_conjugate h hf ?_⟩
· change (e.symm ∘ e) ∘ f ∘ e.symm = f ∘ e.symm
rw [MeasurableEquiv.symm_comp_self, id_comp]
· change e ∘ f = e ∘ f ∘ e.symm ∘ e
rw [MeasurableEquiv.symm_comp_self, comp_id]
|
/-
Copyright (c) 2024 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn,
Mario Carneiro
-/
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data... | Mathlib/Data/List/GetD.lean | 57 | 63 | theorem getD_eq_default {n : ℕ} (hn : l.length ≤ n) : l.getD n d = d := by |
induction l generalizing n with
| nil => exact getD_nil
| cons head tail ih =>
cases n
· simp at hn
· exact ih (Nat.le_of_succ_le_succ hn)
|
/-
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.LegendreSymbol.QuadraticChar.Basic
import Mathlib.NumberTheory.GaussSum
#align_import number_theory.legendre_symbol.quadratic_char.gauss_su... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean | 42 | 61 | theorem FiniteField.isSquare_two_iff :
IsSquare (2 : F) ↔ Fintype.card F % 8 ≠ 3 ∧ Fintype.card F % 8 ≠ 5 := by |
classical
by_cases hF : ringChar F = 2
focus
have h := FiniteField.even_card_of_char_two hF
simp only [FiniteField.isSquare_of_char_two hF, true_iff_iff]
rotate_left
focus
have h := FiniteField.odd_card_of_char_ne_two hF
rw [← quadraticChar_one_iff_isSquare (Ring.two_ne_zero hF), quadraticCha... |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.Multilinear.Basic
/-!
# Images of (von Neumann) bounded sets under continuous ... | Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean | 44 | 83 | theorem image_multilinear' [Nonempty ι] {s : Set (∀ i, E i)} (hs : IsVonNBounded 𝕜 s)
(f : ContinuousMultilinearMap 𝕜 E F) : IsVonNBounded 𝕜 (f '' s) := fun V hV ↦ by
classical
if h₁ : ∀ c : 𝕜, ‖c‖ ≤ 1 then
exact absorbs_iff_norm.2 ⟨2, fun c hc ↦ by linarith [h₁ c]⟩
else
let _ : NontriviallyNormed... |
have hfV : f ⁻¹' V ∈ 𝓝 0 := (map_continuous f).tendsto' _ _ f.map_zero hV
rwa [nhds_pi, Filter.mem_pi, exists_finite_iff_finset] at hfV
have : ∀ i, ∃ c : 𝕜, c ≠ 0 ∧ ∀ c' : 𝕜, ‖c'‖ ≤ ‖c‖ → ∀ x ∈ s, c' • x i ∈ t i := fun i ↦ by
rw [isVonNBounded_pi_iff] at hs
have := (hs i).tendsto_smallSe... |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.RingHomProperties
#align_import ring_theory.ring_hom.finite from "leanprover-community/mathlib"@"b5aecf07a179c60b6b37c1ac9da952f3b565c785"
/-!
... | Mathlib/RingTheory/RingHom/Finite.lean | 28 | 31 | theorem finite_respectsIso : RespectsIso @Finite := by |
apply finite_stableUnderComposition.respectsIso
intros
exact Finite.of_surjective _ (RingEquiv.toEquiv _).surjective
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Finite
import Mathlib.Order.Atoms
#align_import order.atoms.finite from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b383... | Mathlib/Order/Atoms/Finite.lean | 45 | 49 | theorem univ : (Finset.univ : Finset α) = {⊤, ⊥} := by |
change Finset.map _ (Finset.univ : Finset Bool) = _
rw [Fintype.univ_bool]
simp only [Finset.map_insert, Function.Embedding.coeFn_mk, Finset.map_singleton]
rfl
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Scott Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
#align_import algebra.... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean | 65 | 71 | theorem hexagon_reverse (X Y Z : ModuleCat.{u} R) :
(α_ X Y Z).inv ≫ (braiding _ Z).hom ≫ (α_ Z X Y).inv =
X ◁ (Y.braiding Z).hom ≫ (α_ X Z Y).inv ≫ (X.braiding Z).hom ▷ Y := by |
apply (cancel_epi (α_ X Y Z).hom).1
apply TensorProduct.ext_threefold
intro x y z
rfl
|
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Logic.UnivLE
import Mathlib.SetTheory.Ordinal.Basic
/-!
# UnivLE and cardinals
-/
set_option autoImplicit true
noncomputable section
open Cardinal
theorem u... | Mathlib/SetTheory/Cardinal/UnivLE.lean | 30 | 31 | theorem univLE_total : UnivLE.{u, v} ∨ UnivLE.{v, u} := by |
simp_rw [univLE_iff_cardinal_le]; apply le_total
|
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpace
/-!
# The multiplicative and additive convolution of measures
In... | Mathlib/MeasureTheory/Group/Convolution.lean | 41 | 46 | theorem dirac_one_mconv [MeasurableMul₂ M] (μ : Measure M) [SFinite μ] :
(Measure.dirac 1) ∗ μ = μ := by |
unfold mconv
rw [MeasureTheory.Measure.dirac_prod, map_map]
· simp only [Function.comp_def, one_mul, map_id']
all_goals { measurability }
|
/-
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, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.... | Mathlib/Order/Interval/Set/OrderIso.lean | 24 | 26 | theorem preimage_Iic (e : α ≃o β) (b : β) : e ⁻¹' Iic b = Iic (e.symm b) := by |
ext x
simp [← e.le_iff_le]
|
/-
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 | 82 | 87 | theorem inv_mem_iff {c d : C} (f : c ⟶ d) :
Groupoid.inv f ∈ S.arrows d c ↔ f ∈ S.arrows c d := by |
constructor
· intro h
simpa only [inv_eq_inv, IsIso.inv_inv] using S.inv h
· apply S.inv
|
/-
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.Topology.MetricSpace.Isometry
#align_import topology.metric_space.gluing from "leanprover-community/mathlib"@"e1a7bdeb4fd826b7e71d130d34988f0a2d... | Mathlib/Topology/MetricSpace/Gluing.lean | 106 | 108 | theorem le_glueDist_inr_inl (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x y) :
ε ≤ glueDist Φ Ψ ε (.inr x) (.inl y) := by |
rw [glueDist_comm]; apply le_glueDist_inl_inr
|
/-
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.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
#align_import linear_algebra.clifford_algebra.even from "leanprover-... | Mathlib/LinearAlgebra/CliffordAlgebra/Even.lean | 116 | 128 | theorem even.algHom_ext ⦃f g : even Q →ₐ[R] A⦄ (h : (even.ι Q).compr₂ f = (even.ι Q).compr₂ g) :
f = g := by |
rw [EvenHom.ext_iff] at h
ext ⟨x, hx⟩
induction x, hx using even_induction with
| algebraMap r =>
exact (f.commutes r).trans (g.commutes r).symm
| add x y hx hy ihx ihy =>
have := congr_arg₂ (· + ·) ihx ihy
exact (f.map_add _ _).trans (this.trans <| (g.map_add _ _).symm)
| ι_mul_ι_mul m₁ m₂ x h... |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | Mathlib/Logic/Equiv/Fintype.lean | 85 | 87 | theorem Equiv.Perm.viaFintypeEmbedding_apply_not_mem_range {b : β} (h : b ∉ Set.range f) :
e.viaFintypeEmbedding f b = b := by |
rwa [Equiv.Perm.viaFintypeEmbedding, Equiv.Perm.extendDomain_apply_not_subtype]
|
/-
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 | 80 | 86 | theorem isCompact_open_iff_eq_finset_affine_union {X : Scheme} (U : Set X.carrier) :
IsCompact U ∧ IsOpen U ↔
∃ s : Set X.affineOpens, s.Finite ∧ U = ⋃ (i : X.affineOpens) (_ : i ∈ s), i := by |
apply Opens.IsBasis.isCompact_open_iff_eq_finite_iUnion
(fun (U : X.affineOpens) => (U : Opens X.carrier))
· rw [Subtype.range_coe]; exact isBasis_affine_open X
· exact fun i => i.2.isCompact
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
#al... | Mathlib/Analysis/Normed/Group/AddTorsor.lean | 104 | 105 | theorem dist_vadd_cancel_right (v₁ v₂ : V) (x : P) : dist (v₁ +ᵥ x) (v₂ +ᵥ x) = dist v₁ v₂ := by |
rw [dist_eq_norm_vsub V, dist_eq_norm, vadd_vsub_vadd_cancel_right]
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathl... | Mathlib/Data/Real/GoldenRatio.lean | 75 | 76 | theorem one_sub_goldConj : 1 - φ = ψ := by |
linarith [gold_add_goldConj]
|
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
/-!
This file establishes a `snoc : Vector α n → α → Vector α (n+1)` operation, that appends a single
element to the back of a vector.... | Mathlib/Data/Vector/Snoc.lean | 134 | 141 | theorem mapAccumr_snoc :
mapAccumr f (xs.snoc x) s
= let q := f x s
let r := mapAccumr f xs q.1
(r.1, r.2.snoc q.2) := by |
induction xs
· rfl
· simp [*]
|
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Ma... | Mathlib/Data/Nat/Factorization/Basic.lean | 84 | 87 | theorem factorization_eq_factors_multiset (n : ℕ) :
n.factorization = Multiset.toFinsupp (n.factors : Multiset ℕ) := by |
ext p
simp
|
/-
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.Basic
/-!
# Subsingleton
Defines the predicate `Subsingleton s : Prop`, saying that `s` has at most one element.
Also defi... | Mathlib/Data/Set/Subsingleton.lean | 68 | 71 | theorem Subsingleton.induction_on {p : Set α → Prop} (hs : s.Subsingleton) (he : p ∅)
(h₁ : ∀ x, p {x}) : p s := by |
rcases hs.eq_empty_or_singleton with (rfl | ⟨x, rfl⟩)
exacts [he, h₁ _]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Joseph Myers
-/
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
#align_import data.complex.exponential_bounds from "leanprover... | Mathlib/Data/Complex/ExponentialBounds.lean | 28 | 33 | theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by |
apply exp_approx_start
iterate 21 refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_
norm_num1
refine exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) ?_
rw [_root_.abs_one, abs_of_pos] <;> norm_num1
|
/-
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.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 69 | 69 | theorem eval₂_C : (C a).eval₂ f x = f a := by | simp [eval₂_eq_sum]
|
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.Basic
/-!
# Period of a group action
This module defin... | Mathlib/GroupTheory/GroupAction/Period.lean | 101 | 102 | theorem period_dvd_exponent (m : M) (a : α) : period m a ∣ Monoid.exponent M := by |
rw [← pow_smul_eq_iff_period_dvd, Monoid.pow_exponent_eq_one, one_smul]
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.Taylor
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.AdicCompletion.Basic
#align_import ring_theory.henselian... | Mathlib/RingTheory/Henselian.lean | 65 | 82 | theorem isLocalRingHom_of_le_jacobson_bot {R : Type*} [CommRing R] (I : Ideal R)
(h : I ≤ Ideal.jacobson ⊥) : IsLocalRingHom (Ideal.Quotient.mk I) := by |
constructor
intro a h
have : IsUnit (Ideal.Quotient.mk (Ideal.jacobson ⊥) a) := by
rw [isUnit_iff_exists_inv] at *
obtain ⟨b, hb⟩ := h
obtain ⟨b, rfl⟩ := Ideal.Quotient.mk_surjective b
use Ideal.Quotient.mk _ b
rw [← (Ideal.Quotient.mk _).map_one, ← (Ideal.Quotient.mk _).map_mul, Ideal.Quotie... |
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv
/-!... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/ContDiff.lean | 68 | 75 | theorem to_localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
(hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) :
ContDiffAt 𝕂 n (hf.localInverse hf' hn) (f a) := by |
have := hf.localInverse_apply_image hf' hn
apply (hf.toPartialHomeomorph f hf' hn).contDiffAt_symm
(image_mem_toPartialHomeomorph_target hf hf' hn)
· convert hf'
· convert hf
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Kyle Miller
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.Util.AddRelatedDecl
import Batteries.Tactic.Lint
/-!
# Tools to reformulate categor... | Mathlib/Tactic/CategoryTheory/Elementwise.lean | 52 | 53 | theorem hom_elementwise [Category C] [ConcreteCategory C]
{X Y : C} {f g : X ⟶ Y} (h : f = g) (x : X) : f x = g x := by | rw [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
-/
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.Int.ConditionallyCompleteOrder
import Mathlib.Topology.Instances... | Mathlib/Topology/Instances/Int.lean | 34 | 34 | theorem dist_eq' (m n : ℤ) : dist m n = |m - n| := by | rw [dist_eq]; norm_cast
|
/-
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.CategoryTheory.Category.Cat
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_the... | Mathlib/CategoryTheory/Category/Cat/Limit.lean | 127 | 132 | theorem limit_π_homDiagram_eqToHom {F : J ⥤ Cat.{v, v}} (X Y : limit (F ⋙ Cat.objects.{v, v}))
(j : J) (h : X = Y) :
limit.π (homDiagram X Y) j (eqToHom h) =
eqToHom (congr_arg (limit.π (F ⋙ Cat.objects.{v, v}) j) h) := by |
subst h
simp
|
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.MonoidAlgebra.Ideal
import Mathlib.Algebra.MvPolynomial.Division
#align_import ring_theory.mv_polynomial.ideal from "leanprover-community/mathlib"@"... | Mathlib/RingTheory/MvPolynomial/Ideal.lean | 32 | 36 | theorem mem_ideal_span_monomial_image {x : MvPolynomial σ R} {s : Set (σ →₀ ℕ)} :
x ∈ Ideal.span ((fun s => monomial s (1 : R)) '' s) ↔ ∀ xi ∈ x.support, ∃ si ∈ s, si ≤ xi := by |
refine AddMonoidAlgebra.mem_ideal_span_of'_image.trans ?_
simp_rw [le_iff_exists_add, add_comm]
rfl
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
/-!
# Finite int... | Mathlib/Order/Interval/Finset/Fin.lean | 94 | 95 | theorem map_valEmbedding_Ioo : (Ioo a b).map Fin.valEmbedding = Ioo ↑a ↑b := by |
simp [Ioo_eq_finset_subtype, Finset.fin, Finset.map_map]
|
/-
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.Algebra.Group.Int
import Mathlib.GroupTheory.GroupAction.Opposite
import Mathlib.Logic.Function.Iterate
#align_import algebra.hom.iterate from "lean... | Mathlib/Algebra/GroupPower/IterateHom.lean | 111 | 111 | theorem mul_right_iterate_apply_one : (· * a)^[n] 1 = a ^ n := by | simp [mul_right_iterate]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.... | Mathlib/Data/Nat/GCD/Basic.lean | 102 | 103 | theorem gcd_sub_self_right {m n : ℕ} (h : m ≤ n) : gcd m (n - m) = gcd m n := by |
rw [gcd_comm, gcd_sub_self_left h, gcd_comm]
|
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 99 | 101 | theorem inv_mul_le_iff (h : 0 < b) : b⁻¹ * a ≤ c ↔ a ≤ b * c := by |
rw [inv_eq_one_div, mul_comm, ← div_eq_mul_one_div]
exact div_le_iff' h
|
/-
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.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mat... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 50 | 57 | theorem continuousAt_angle {x : P × P × P} (hx12 : x.1 ≠ x.2.1) (hx32 : x.2.2 ≠ x.2.1) :
ContinuousAt (fun y : P × P × P => ∠ y.1 y.2.1 y.2.2) x := by |
let f : P × P × P → V × V := fun y => (y.1 -ᵥ y.2.1, y.2.2 -ᵥ y.2.1)
have hf1 : (f x).1 ≠ 0 := by simp [hx12]
have hf2 : (f x).2 ≠ 0 := by simp [hx32]
exact (InnerProductGeometry.continuousAt_angle hf1 hf2).comp
((continuous_fst.vsub continuous_snd.fst).prod_mk
(continuous_snd.snd.vsub continuous_snd... |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import rin... | Mathlib/RingTheory/Algebraic.lean | 78 | 80 | theorem Algebra.isAlgebraic_iff : Algebra.IsAlgebraic R A ↔ (⊤ : Subalgebra R A).IsAlgebraic := by |
delta Subalgebra.IsAlgebraic
simp only [Algebra.isAlgebraic_def, Algebra.mem_top, forall_prop_of_true, iff_self_iff]
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 143 | 143 | theorem succ_eval : succ.eval = fun v => pure [v.headI.succ] := by | simp [eval]
|
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most one bas... | Mathlib/Data/Matroid/Constructions.lean | 67 | 69 | theorem eq_emptyOn_or_nonempty (M : Matroid α) : M = emptyOn α ∨ Matroid.Nonempty M := by |
rw [← ground_eq_empty_iff]
exact M.E.eq_empty_or_nonempty.elim Or.inl (fun h ↦ Or.inr ⟨h⟩)
|
/-
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.List.Basic
#align_import data.bool.all_any from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
/-!
# Boolean quantifi... | Mathlib/Data/Bool/AllAny.lean | 48 | 48 | theorem any_iff_exists_prop : (any l fun a => p a) ↔ ∃ a ∈ l, p a := by | simp [any_iff_exists]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community... | Mathlib/Data/Nat/Choose/Basic.lean | 93 | 95 | theorem triangle_succ (n : ℕ) : (n + 1) * (n + 1 - 1) / 2 = n * (n - 1) / 2 + n := by |
rw [← add_mul_div_left, Nat.mul_comm 2 n, ← Nat.mul_add, Nat.add_sub_cancel, Nat.mul_comm]
cases n <;> rfl; apply zero_lt_succ
|
/-
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 | 89 | 93 | theorem rank_unit [StrongRankCondition R] [DecidableEq n] (A : (Matrix n n R)ˣ) :
(A : Matrix n n R).rank = Fintype.card n := by |
apply le_antisymm (rank_le_card_width (A : Matrix n n R)) _
have := rank_mul_le_left (A : Matrix n n R) (↑A⁻¹ : Matrix n n R)
rwa [← Units.val_mul, mul_inv_self, Units.val_one, rank_one] at this
|
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#ali... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 91 | 98 | theorem isOrthoᵢ_flip (B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M) {v : n → M₁} :
B.IsOrthoᵢ v ↔ B.flip.IsOrthoᵢ v := by |
simp_rw [isOrthoᵢ_def]
constructor <;> intro h i j hij
· rw [flip_apply]
exact h j i (Ne.symm hij)
simp_rw [flip_apply] at h
exact h j i (Ne.symm hij)
|
/-
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.Basic
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.G... | Mathlib/Algebra/Ring/Basic.lean | 112 | 113 | theorem inv_neg' (a : α) : (-a)⁻¹ = -a⁻¹ := by |
rw [eq_comm, eq_inv_iff_mul_eq_one, neg_mul, mul_neg, neg_neg, mul_left_inv]
|
/-
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.Basic
#align_import ring_theory.localization.integer from "leanprover-co... | Mathlib/RingTheory/Localization/Integer.lean | 85 | 87 | theorem exists_integer_multiple (a : S) : ∃ b : M, IsInteger R ((b : R) • a) := by |
simp_rw [Algebra.smul_def, mul_comm _ a]
apply exists_integer_multiple'
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-comm... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 108 | 112 | theorem lt_or_le_of_mul_le_mul [CovariantClass α α (· * ·) (· < ·)]
[CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ ≤ a₂ * b₂ → a₁ < a₂ ∨ b₁ ≤ b₂ := by |
contrapose!
exact fun h => mul_lt_mul_of_le_of_lt h.1 h.2
|
/-
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.MeasureTheory.Integral.Lebesgue
/-!
# Measure with a given density with respect to another measure
For a measure `μ` on `α` and a fun... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 97 | 102 | theorem withDensity_add_left {f : α → ℝ≥0∞} (hf : Measurable f) (g : α → ℝ≥0∞) :
μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g := by |
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, Measure.add_apply, withDensity_apply _ hs, withDensity_apply _ hs,
← lintegral_add_left hf]
simp only [Pi.add_apply]
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.... | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | 116 | 126 | theorem adic_module_basis :
I.ringFilterBasis.SubmodulesBasis fun n : ℕ => I ^ n • (⊤ : Submodule R M) :=
{ inter := fun i j =>
⟨max i j,
le_inf_iff.mpr
⟨smul_mono_left <| pow_le_pow_right (le_max_left i j),
smul_mono_left <| pow_le_pow_right (le_max_right i j)⟩⟩
smul := fu... | simpa [(I ^ i).mul_top] using a_in
exact smul_mem_smul a_in mem_top⟩ }
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27dd... | Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean | 68 | 71 | theorem snorm_trim_ae (hm : m ≤ m0) {f : α → E} (hf : AEStronglyMeasurable f (μ.trim hm)) :
snorm f p (μ.trim hm) = snorm f p μ := by |
rw [snorm_congr_ae hf.ae_eq_mk, snorm_congr_ae (ae_eq_of_ae_eq_trim hf.ae_eq_mk)]
exact snorm_trim hm hf.stronglyMeasurable_mk
|
/-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this fil... | Mathlib/RingTheory/PowerSeries/Derivative.lean | 87 | 88 | theorem derivativeFun_one : derivativeFun (1 : R⟦X⟧) = 0 := by |
rw [← map_one (C R), derivativeFun_C (1 : R)]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Logic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Coe
/-!
# Lemmas about booleans
These are the lemmas about booleans w... | Mathlib/Init/Data/Bool/Lemmas.lean | 76 | 76 | theorem not_eq_true_eq_eq_false (a : Bool) : (not a = true) = (a = false) := by | cases a <;> simp
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.Topology.Algebra.GroupCompletion
import Mathlib.Topolog... | Mathlib/Topology/Algebra/UniformRing.lean | 77 | 84 | theorem continuous_mul : Continuous fun p : Completion α × Completion α => p.1 * p.2 := by |
let m := (AddMonoidHom.mul : α →+ α →+ α).compr₂ toCompl
have : Continuous fun p : α × α => m p.1 p.2 := by
apply (continuous_coe α).comp _
simp only [AddMonoidHom.coe_mul, AddMonoidHom.coe_mulLeft]
exact _root_.continuous_mul
have di : DenseInducing (toCompl : α → Completion α) := denseInducing_coe
... |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathli... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 104 | 105 | theorem edgeFinset_sdiff [DecidableEq V] :
(G₁ \ G₂).edgeFinset = G₁.edgeFinset \ G₂.edgeFinset := by | simp [edgeFinset]
|
/-
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.Defs
/-!
## Basic properties of smooth functions between manifolds
In this file, we show that sta... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 52 | 77 | theorem ContMDiffWithinAt.comp {t : Set M'} {g : M' → M''} (x : M)
(hg : ContMDiffWithinAt I' I'' n g t (f x)) (hf : ContMDiffWithinAt I I' n f s x)
(st : MapsTo f s t) : ContMDiffWithinAt I I'' n (g ∘ f) s x := by |
rw [contMDiffWithinAt_iff] at hg hf ⊢
refine ⟨hg.1.comp hf.1 st, ?_⟩
set e := extChartAt I x
set e' := extChartAt I' (f x)
have : e' (f x) = (writtenInExtChartAt I I' x f) (e x) := by simp only [e, e', mfld_simps]
rw [this] at hg
have A : ∀ᶠ y in 𝓝[e.symm ⁻¹' s ∩ range I] e x, f (e.symm y) ∈ t ∧ f (e.sy... |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Violeta Hernández Palacios
-/
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Con... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 62 | 65 | theorem empty_mem_generateMeasurableRec (s : Set (Set α)) (i : ω₁) :
∅ ∈ generateMeasurableRec s i := by |
unfold generateMeasurableRec
exact mem_union_left _ (mem_union_left _ (mem_union_right _ (mem_singleton ∅)))
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Order.PartialSups
#align_import order.disjointed from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
/-!
# C... | Mathlib/Order/Disjointed.lean | 114 | 118 | theorem partialSups_disjointed (f : ℕ → α) : partialSups (disjointed f) = partialSups f := by |
ext n
induction' n with k ih
· rw [partialSups_zero, partialSups_zero, disjointed_zero]
· rw [partialSups_succ, partialSups_succ, disjointed_succ, ih, sup_sdiff_self_right]
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... | Mathlib/Analysis/Fourier/AddCircle.lean | 132 | 135 | theorem fourier_zero {x : AddCircle T} : fourier 0 x = 1 := by |
induction x using QuotientAddGroup.induction_on'
simp only [fourier_coe_apply]
norm_num
|
/-
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.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-communit... | Mathlib/MeasureTheory/Group/Integral.lean | 40 | 43 | theorem integral_inv_eq_self (f : G → E) (μ : Measure G) [IsInvInvariant μ] :
∫ x, f x⁻¹ ∂μ = ∫ x, f x ∂μ := by |
have h : MeasurableEmbedding fun x : G => x⁻¹ := (MeasurableEquiv.inv G).measurableEmbedding
rw [← h.integral_map, map_inv_eq_self]
|
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang
-/
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
#align_import analysis.normed_space.conformal_linear_map from "leanprover-co... | Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean | 97 | 100 | theorem ne_zero [Nontrivial M'] {f' : M' →L[R] N} (hf' : IsConformalMap f') : f' ≠ 0 := by |
rintro rfl
rcases exists_ne (0 : M') with ⟨a, ha⟩
exact ha (hf'.injective rfl)
|
/-
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.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.... | Mathlib/RingTheory/WittVector/Identities.lean | 87 | 87 | theorem coeff_p_one [CharP R p] : (p : 𝕎 R).coeff 1 = 1 := by | rw [coeff_p, if_pos rfl]
|
/-
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.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.... | Mathlib/RingTheory/WittVector/Identities.lean | 42 | 47 | theorem frobenius_verschiebung (x : 𝕎 R) : frobenius (verschiebung x) = x * p := by |
have : IsPoly p fun {R} [CommRing R] x ↦ frobenius (verschiebung x) :=
IsPoly.comp (hg := frobenius_isPoly p) (hf := verschiebung_isPoly)
have : IsPoly p fun {R} [CommRing R] x ↦ x * p := mulN_isPoly p p
ghost_calc x
ghost_simp [mul_comm]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Algebra.Field.Basic
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6e... | Mathlib/Data/Rat/Cast/Lemmas.lean | 37 | 40 | theorem cast_inv_int (n : ℤ) : ((n⁻¹ : ℚ) : α) = (n : α)⁻¹ := by |
cases' n with n n
· simp [ofInt_eq_cast, cast_inv_nat]
· simp only [ofInt_eq_cast, Int.cast_negSucc, ← Nat.cast_succ, cast_neg, inv_neg, cast_inv_nat]
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.ContinuousOn
#align_import topology.algebra.order.left_right from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e... | Mathlib/Topology/Order/LeftRight.lean | 111 | 112 | theorem nhds_left_sup_nhds_right (a : α) : 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a := by |
rw [← nhdsWithin_union, Iic_union_Ici, nhdsWithin_univ]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Monad
#align_import data.mv_polynomial.expand from "leanprover-community/mathlib"@"5da451b4c96b4c2e122c0325a7fce... | Mathlib/Algebra/MvPolynomial/Expand.lean | 88 | 92 | theorem rename_comp_expand (f : σ → τ) (p : ℕ) :
(rename f).comp (expand p) =
(expand p).comp (rename f : MvPolynomial σ R →ₐ[R] MvPolynomial τ R) := by |
ext1 φ
simp only [rename_expand, AlgHom.comp_apply]
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Eric Wieser
-/
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathl... | Mathlib/Analysis/Matrix.lean | 102 | 104 | theorem nnnorm_lt_iff {r : ℝ≥0} (hr : 0 < r) {A : Matrix m n α} :
‖A‖₊ < r ↔ ∀ i j, ‖A i j‖₊ < r := by |
simp_rw [nnnorm_def, pi_nnnorm_lt_iff hr]
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Subsemigroup.Basic
#align_import group_theory.subsemigroup.membership from "leanprover-community/mathlib"@"6cb77a8eaff0ddd100e87b1591c6d3a... | Mathlib/Algebra/Group/Subsemigroup/Membership.lean | 67 | 70 | theorem mem_sSup_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) {x : M} :
x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s := by |
simp only [sSup_eq_iSup', mem_iSup_of_directed hS.directed_val, SetCoe.exists, Subtype.coe_mk,
exists_prop]
|
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
... | Mathlib/Algebra/MvPolynomial/Supported.lean | 117 | 118 | theorem X_mem_supported [Nontrivial R] {i : σ} : X i ∈ supported R s ↔ i ∈ s := by |
simp [mem_supported]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.LocalExtr.Basic
#align_import analysis.calcu... | Mathlib/Analysis/Calculus/Darboux.lean | 76 | 90 | theorem Set.OrdConnected.image_hasDerivWithinAt {s : Set ℝ} (hs : OrdConnected s)
(hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) : OrdConnected (f' '' s) := by |
apply ordConnected_of_Ioo
rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ - m ⟨hma, hmb⟩
rcases le_total a b with hab | hab
· have : Icc a b ⊆ s := hs.out ha hb
rcases exists_hasDerivWithinAt_eq_of_gt_of_lt hab (fun x hx => (hf x <| this hx).mono this) hma
hmb with
⟨c, cmem, hc⟩
exact ⟨c, this <| Io... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.Prime
import Mathlib.Data.List.Prime
import Mathlib.Dat... | Mathlib/Data/Nat/Factors.lean | 53 | 53 | theorem factors_two : factors 2 = [2] := by | simp [factors]
|
/-
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.LinearAlgebra.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.... | Mathlib/RingTheory/Trace.lean | 109 | 111 | theorem trace_eq_matrix_trace [DecidableEq ι] (b : Basis ι R S) (s : S) :
trace R S s = Matrix.trace (Algebra.leftMulMatrix b s) := by |
rw [trace_apply, LinearMap.trace_eq_matrix_trace _ b, ← toMatrix_lmul_eq]; rfl
|
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