Context stringlengths 295 65.3k | file_name stringlengths 21 74 | start int64 14 1.41k | end int64 20 1.41k | theorem stringlengths 27 1.42k | proof stringlengths 0 4.57k |
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/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov, Winston Yin
-/
import Mathlib.Algebra.Group.End
import Mathlib.Topology.EMetricSpace.Diam
/-!
# Lipschitz co... | Mathlib/Topology/EMetricSpace/Lipschitz.lean | 329 | 333 | theorem ediam_image2_le (f : α → β → γ) {K₁ K₂ : ℝ≥0} (s : Set α) (t : Set β)
(hf₁ : ∀ b ∈ t, LipschitzOnWith K₁ (f · b) s) (hf₂ : ∀ a ∈ s, LipschitzOnWith K₂ (f a) t) :
EMetric.diam (Set.image2 f s t) ≤ ↑K₁ * EMetric.diam s + ↑K₂ * EMetric.diam t := by | simp only [EMetric.diam_le_iff, forall_mem_image2]
intro a₁ ha₁ b₁ hb₁ a₂ ha₂ b₂ hb₂ |
/-
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.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 109 | 118 | theorem weightedVSubOfPoint_eq_of_sum_eq_zero (w : ι → k) (p : ι → P) (h : ∑ i ∈ s, w i = 0)
(b₁ b₂ : P) : s.weightedVSubOfPoint p b₁ w = s.weightedVSubOfPoint p b₂ w := by | apply eq_of_sub_eq_zero
rw [weightedVSubOfPoint_apply, weightedVSubOfPoint_apply, ← sum_sub_distrib]
conv_lhs =>
congr
· skip
· ext
rw [← smul_sub, vsub_sub_vsub_cancel_left]
rw [← sum_smul, h, zero_smul] |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 134 | 146 | theorem zero_rpow_nonneg (x : ℝ) : 0 ≤ (0 : ℝ) ^ x := by | by_cases h : x = 0 <;> simp [h, zero_le_one]
@[bound]
theorem rpow_nonneg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : 0 ≤ x ^ y := by
rw [rpow_def_of_nonneg hx]; split_ifs <;>
simp only [zero_le_one, le_refl, le_of_lt (exp_pos _)]
theorem abs_rpow_of_nonneg {x y : ℝ} (hx_nonneg : 0 ≤ x) : |x ^ y| = |x| ^ y := by
have h_rp... |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | Mathlib/Data/Num/Lemmas.lean | 733 | 734 | theorem cmp_eq (m n) : cmp m n = Ordering.eq ↔ m = n := by | have := cmp_to_nat m n |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro,
Kim Morrison
-/
import Mathlib.Data.List.Basic
/-!
# Lattice structure of lists
This files pro... | Mathlib/Data/List/Lattice.lean | 147 | 147 | theorem inter_reverse {xs ys : List α} : xs.inter ys.reverse = xs.inter ys := by | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... | Mathlib/Order/Heyting/Basic.lean | 388 | 389 | theorem sdiff_le_comm : a \ b ≤ c ↔ a \ c ≤ b := by | rw [sdiff_le_iff, sdiff_le_iff'] |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.Special... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 396 | 403 | theorem hasDerivAt_rpow_const {x p : ℝ} (h : x ≠ 0 ∨ 1 ≤ p) :
HasDerivAt (fun x => x ^ p) (p * x ^ (p - 1)) x := by | rcases ne_or_eq x 0 with (hx | rfl)
· exact (hasStrictDerivAt_rpow_const_of_ne hx _).hasDerivAt
replace h : 1 ≤ p := h.neg_resolve_left rfl
apply hasDerivAt_of_hasDerivAt_of_ne fun x hx =>
(hasStrictDerivAt_rpow_const_of_ne hx p).hasDerivAt
exacts [continuousAt_id.rpow_const (Or.inr (zero_le_one.trans h)), |
/-
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.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | Mathlib/Data/ENNReal/Real.lean | 37 | 40 | theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal + b.toReal := by | lift a to ℝ≥0 using ha
lift b to ℝ≥0 using hb
rfl |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... | Mathlib/Data/List/Sublists.lean | 68 | 72 | theorem mem_sublists' {s t : List α} : s ∈ sublists' t ↔ s <+ t := by | induction' t with a t IH generalizing s
· simp only [sublists'_nil, mem_singleton]
exact ⟨fun h => by rw [h], eq_nil_of_sublist_nil⟩
simp only [sublists'_cons, mem_append, IH, mem_map] |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
/-!
# Simplicial complexes
In this file, we define ... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 86 | 87 | theorem convexHull_subset_space (hs : s ∈ K.faces) : convexHull 𝕜 ↑s ⊆ K.space := by | convert subset_biUnion_of_mem hs |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Basic Translation Lemmas Between Functions Defined for Continued... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 49 | 50 | theorem partNum_eq_s_a {gp : Pair α} (s_nth_eq : g.s.get? n = some gp) :
g.partNums.get? n = some gp.a := by | simp [partNums, s_nth_eq] |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Trim
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
/-!
# Almost everywhere measurable functions
A funct... | Mathlib/MeasureTheory/Measure/AEMeasurable.lean | 244 | 249 | theorem MeasurableEmbedding.aemeasurable_comp_iff {g : β → γ} (hg : MeasurableEmbedding g)
{μ : Measure α} : AEMeasurable (g ∘ f) μ ↔ AEMeasurable f μ := by | refine ⟨fun H => ?_, hg.measurable.comp_aemeasurable⟩
suffices AEMeasurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f) μ by
rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this
exact hg.measurable_rangeSplitting.comp_aemeasurable H.subtype_mk |
/-
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.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | Mathlib/Order/SuccPred/Basic.lean | 431 | 434 | theorem not_isMin_succ [Nontrivial α] (a : α) : ¬ IsMin (succ a) := by | obtain ha | ha := (le_succ a).eq_or_lt
· exact (ha ▸ succ_eq_iff_isMax.1 ha.symm).not_isMin
· exact not_isMin_of_lt ha |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Patrick Massot
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Measure.Real
import Mathlib.Order.Filter.Indi... | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | 66 | 75 | theorem tendsto_integral_filter_of_dominated_convergence {ι} {l : Filter ι} [l.IsCountablyGenerated]
{F : ι → α → G} {f : α → G} (bound : α → ℝ) (hF_meas : ∀ᶠ n in l, AEStronglyMeasurable (F n) μ)
(h_bound : ∀ᶠ n in l, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
(h_lim : ∀ᵐ a ∂μ, Ten... | by_cases hG : CompleteSpace G
· simp only [integral, hG, L1.integral]
exact tendsto_setToFun_filter_of_dominated_convergence (dominatedFinMeasAdditive_weightedSMul μ)
bound hF_meas h_bound bound_integrable h_lim
· simp [integral, hG, tendsto_const_nhds] |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
import Mathlib.Algebra.Order.Floor.Ring
/-!
# ... | Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 163 | 165 | theorem of_h_eq_floor : (of v).h = ⌊v⌋ := by | simp [of_h_eq_intFractPair_seq1_fst_b, IntFractPair.of] |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 636 | 641 | theorem rpow_le_rpow_of_exponent_ge (hx0 : 0 < x) (hx1 : x ≤ 1) (hyz : z ≤ y) : x ^ y ≤ x ^ z := by | repeat' rw [rpow_def_of_pos hx0]
rw [exp_le_exp]; exact mul_le_mul_of_nonpos_left hyz (log_nonpos (le_of_lt hx0) hx1)
@[simp]
theorem rpow_le_rpow_left_iff_of_base_lt_one (hx0 : 0 < x) (hx1 : x < 1) : |
/-
Copyright (c) 2023 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Prod
import Mathlib.Tactic.Common
/-!
# Lemmas about the divisibility relation in product (sem... | Mathlib/Algebra/Divisibility/Prod.lean | 16 | 20 | theorem prod_dvd_iff {x y : G₁ × G₂} :
x ∣ y ↔ x.1 ∣ y.1 ∧ x.2 ∣ y.2 := by | cases x; cases y
simp only [dvd_def, Prod.exists, Prod.mk_mul_mk, Prod.mk.injEq,
exists_and_left, exists_and_right, and_self, true_and] |
/-
Copyright (c) 2024 Ira Fesefeldt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ira Fesefeldt
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
/-!
# Ordinal Approximants for the Fixed points on complete lattices
This file sets up the ordinal-indexed approximation t... | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | 209 | 211 | theorem lfpApprox_le_of_mem_fixedPoints {a : α}
(h_a : a ∈ fixedPoints f) (h_le_init : x ≤ a) (i : Ordinal) : lfpApprox f x i ≤ a := by | induction i using Ordinal.induction with |
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.UnorderedInterva... | Mathlib/Data/Complex/Basic.lean | 261 | 261 | theorem I_mul_re (z : ℂ) : (I * z).re = -z.im := by | simp |
/-
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.UniformConvergenceTopology
/-!
# Equicontinuity of a family of functions
Let `X` be a topological space and `α` a `UniformS... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 519 | 523 | theorem uniformEquicontinuous_iff_uniformContinuous {F : ι → β → α} :
UniformEquicontinuous F ↔ UniformContinuous (ofFun ∘ Function.swap F : β → ι →ᵤ α) := by | rw [UniformContinuous, (UniformFun.hasBasis_uniformity ι α).tendsto_right_iff]
rfl |
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
/-!
# Bicategories
In this file we define typeclass for bicategories.
A bicategory `B` consists of
* objects `a : B`,
* 1-morphisms ... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 399 | 400 | theorem leftUnitor_comp_inv (f : a ⟶ b) (g : b ⟶ c) :
(λ_ (f ≫ g)).inv = (λ_ f).inv ▷ g ≫ (α_ (𝟙 a) f g).hom := by | simp |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.... | Mathlib/RingTheory/Kaehler/Basic.lean | 351 | 354 | theorem KaehlerDifferential.End_equiv_aux (f : S →ₐ[R] S ⊗ S ⧸ KaehlerDifferential.ideal R S ^ 2) :
(Ideal.Quotient.mkₐ R (KaehlerDifferential.ideal R S).cotangentIdeal).comp f =
IsScalarTower.toAlgHom R S _ ↔
(TensorProduct.lmul' R : S ⊗[R] S →ₐ[R] S).kerSquareLift.comp f = AlgHom.id R S := by | |
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Topology.Homotopy.Path
import Mathlib.Topology.Homotopy.Equiv
/-!
# Contractible spaces
In this file, we define `ContractibleSpace`, a space ... | Mathlib/Topology/Homotopy/Contractible.lean | 28 | 29 | theorem Nullhomotopic.comp_right {f : C(X, Y)} (hf : f.Nullhomotopic) (g : C(Y, Z)) :
(g.comp f).Nullhomotopic := by | |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | Mathlib/Data/Set/Card.lean | 164 | 167 | theorem encard_le_encard (h : s ⊆ t) : s.encard ≤ t.encard := by | rw [← union_diff_cancel h, encard_union_eq disjoint_sdiff_right]; exact le_self_add
@[deprecated (since := "2025-01-05")] alias encard_le_card := encard_le_encard |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
/-!
# Finite maps over `Multiset`
-/
universe u v w
open List
... | Mathlib/Data/Finmap.lean | 502 | 504 | theorem union_toFinmap (s₁ s₂ : AList β) : (toFinmap s₁) ∪ (toFinmap s₂) = toFinmap (s₁ ∪ s₂) := by | simp [(· ∪ ·), union] |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over.Basic
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 765 | 767 | theorem coprod.map_desc {S T U V W : C} [HasBinaryCoproduct U W] [HasBinaryCoproduct T V]
(f : U ⟶ S) (g : W ⟶ S) (h : T ⟶ U) (k : V ⟶ W) :
coprod.map h k ≫ coprod.desc f g = coprod.desc (h ≫ f) (k ≫ g) := by | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Matrix.Block
import Mathl... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 701 | 703 | theorem LinearMap.toMatrixAlgEquiv_apply (f : M₁ →ₗ[R] M₁) (i j : n) :
LinearMap.toMatrixAlgEquiv v₁ f i j = v₁.repr (f (v₁ j)) i := by | simp [LinearMap.toMatrixAlgEquiv, LinearMap.toMatrix_apply] |
/-
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.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card α` as the numb... | Mathlib/Data/Fintype/Card.lean | 139 | 140 | theorem Finset.card_compl_add_card [DecidableEq α] [Fintype α] (s : Finset α) :
#sᶜ + #s = Fintype.card α := by | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... | Mathlib/Algebra/Polynomial/Coeff.lean | 50 | 54 | theorem support_smul [SMulZeroClass S R] (r : S) (p : R[X]) :
support (r • p) ⊆ support p := by | intro i hi
simp? [mem_support_iff] at hi ⊢ says simp only [mem_support_iff, coeff_smul, ne_eq] at hi ⊢
contrapose! hi |
/-
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.Countable
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.MeasureTheory.Me... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 83 | 87 | theorem withDensity_congr_ae {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) :
μ.withDensity f = μ.withDensity g := by | refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, withDensity_apply _ hs]
exact lintegral_congr_ae (ae_restrict_of_ae 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, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | Mathlib/Topology/Order/DenselyOrdered.lean | 223 | 225 | theorem comap_coe_nhdsLT_of_Ioo_subset (hb : s ⊆ Iio b) (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) :
comap ((↑) : s → α) (𝓝[<] b) = atTop := by | nontriviality |
/-
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 | 60 | 68 | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) := by | ext d
rw [coeff_trunc]
split_ifs with h
· have : d + 1 < n + 1 := succ_lt_succ_iff.2 h
rw [coeff_derivativeFun, coeff_derivative, coeff_trunc, if_pos this]
· have : ¬d + 1 < n + 1 := by rwa [succ_lt_succ_iff]
rw [coeff_derivative, coeff_trunc, if_neg this, zero_mul] |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Opposites
/-!
# Morphisms from equations between objects.
When working categorically, sometimes one encounters an equation `h ... | Mathlib/CategoryTheory/EqToHom.lean | 324 | 327 | theorem eqToHom_map_comp (F : C ⥤ D) {X Y Z : C} (p : X = Y) (q : Y = Z) :
F.map (eqToHom p) ≫ F.map (eqToHom q) = F.map (eqToHom <| p.trans q) := by | aesop_cat
/-- See the note on `eqToHom_map` regarding using this as a `simp` lemma. |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann, Kyle Miller, Mario Carneiro
-/
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.BinaryRec
import Mathlib.Logic.F... | Mathlib/Data/Nat/Fib/Basic.lean | 211 | 212 | theorem gcd_fib_add_self (m n : ℕ) : gcd (fib m) (fib (n + m)) = gcd (fib m) (fib n) := by | rcases Nat.eq_zero_or_pos n with rfl | h |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Finset.Option
import Mathlib.Data.PFun
import Mathlib.Data.Part
/-!
# Image of a `Finset α` under a partially defined function
In this file we... | Mathlib/Data/Finset/PImage.lean | 66 | 67 | theorem pimage_some (s : Finset α) (f : α → β) [∀ x, Decidable (Part.some <| f x).Dom] :
(s.pimage fun x => Part.some (f x)) = s.image f := by | |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Mathlib/FieldTheory/RatFunc/Basic.lean | 421 | 429 | theorem liftMonoidWithZeroHom_apply_ofFractionRing_mk (φ : R[X] →*₀ G₀) (hφ : R[X]⁰ ≤ G₀⁰.comap φ)
(n : R[X]) (d : R[X]⁰) :
liftMonoidWithZeroHom φ hφ (ofFractionRing (Localization.mk n d)) = φ n / φ d :=
liftOn_ofFractionRing_mk _ _ _ _
theorem liftMonoidWithZeroHom_injective [Nontrivial R] (φ : R[X] →*₀ G₀... | rintro ⟨x⟩ ⟨y⟩ |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | Mathlib/CategoryTheory/Monoidal/Category.lean | 621 | 623 | theorem associator_conjugation {X X' Y Y' Z Z' : C} (f : X ⟶ X') (g : Y ⟶ Y') (h : Z ⟶ Z') :
(f ⊗ g) ⊗ h = (α_ X Y Z).hom ≫ (f ⊗ g ⊗ h) ≫ (α_ X' Y' Z').inv := by | rw [associator_inv_naturality, hom_inv_id_assoc] |
/-
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.ClosedImmersion
import Mathlib.AlgebraicGeometry.PullbackCarrier
import Mathlib.Topology.LocalAtTarget
/-!
# Universally closed ... | Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean | 45 | 46 | theorem universallyClosed_eq : @UniversallyClosed = universally (topologically @IsClosedMap) := by | ext X Y f; rw [universallyClosed_iff] |
/-
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.Countable
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.MeasureTheory.Me... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 158 | 160 | theorem withDensity_const (c : ℝ≥0∞) : μ.withDensity (fun _ ↦ c) = c • μ := by | ext1 s hs
simp [withDensity_apply _ hs] |
/-
Copyright (c) 2024 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Topology.ContinuousMap.ZeroAtInfty
/-!
# ZeroAtInftyContinuousMapClass in normed additive groups
In this file we give a characterization of the predicate ... | Mathlib/Analysis/Normed/Group/ZeroAtInfty.lean | 24 | 34 | theorem ZeroAtInftyContinuousMapClass.norm_le (f : 𝓕) (ε : ℝ) (hε : 0 < ε) :
∃ (r : ℝ), ∀ (x : E) (_hx : r < ‖x‖), ‖f x‖ < ε := by | have h := zero_at_infty f
rw [tendsto_zero_iff_norm_tendsto_zero, tendsto_def] at h
specialize h (Metric.ball 0 ε) (Metric.ball_mem_nhds 0 hε)
rcases Metric.closedBall_compl_subset_of_mem_cocompact h 0 with ⟨r, hr⟩
use r
intro x hr'
suffices x ∈ (fun x ↦ ‖f x‖) ⁻¹' Metric.ball 0 ε by aesop
apply hr
aeso... |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence
/-!
# Lemmas which are consequences of monoidal coher... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 57 | 60 | theorem unitors_inv_equal : (λ_ (𝟙_ C)).inv = (ρ_ (𝟙_ C)).inv := by | monoidal_coherence
@[reassoc] |
/-
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.Data.Fintype.Order
import Mathlib.Order.Closure
import Mathlib.ModelTheory.Semantics
import Mathlib.ModelTheory.Encoding
/-!
# First-Orde... | Mathlib/ModelTheory/Substructures.lean | 335 | 339 | theorem dense_induction {p : M → Prop} (x : M) {s : Set M} (hs : closure L s = ⊤)
(Hs : ∀ x ∈ s, p x) (Hfun : ∀ {n : ℕ} (f : L.Functions n), ClosedUnder f (setOf p)) : p x := by | have : ∀ x ∈ closure L s, p x := fun x hx => closure_induction hx Hs fun {n} => Hfun
simpa [hs] using this x |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | Mathlib/Algebra/BigOperators/Finprod.lean | 994 | 997 | theorem single_le_finprod {M : Type*} [CommMonoid M] [PartialOrder M] [IsOrderedMonoid M]
(i : α) {f : α → M}
(hf : (mulSupport f).Finite) (h : ∀ j, 1 ≤ f j) : f i ≤ ∏ᶠ j, f j := by | classical calc |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.... | Mathlib/Algebra/Lie/Nilpotent.lean | 593 | 601 | theorem ucs_eq_top_iff (k : ℕ) : N.ucs k = ⊤ ↔ LieModule.lowerCentralSeries R L M k ≤ N := by | rw [eq_top_iff, ← lcs_le_iff]; rfl
variable (R) in
theorem _root_.LieModule.isNilpotent_iff_exists_ucs_eq_top :
LieModule.IsNilpotent L M ↔ ∃ k, (⊥ : LieSubmodule R L M).ucs k = ⊤ := by
rw [LieModule.isNilpotent_iff R]; exact exists_congr fun k => by simp [ucs_eq_top_iff]
theorem ucs_comap_incl (k : ℕ) : |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Ring.Basic
/-!
# Infini... | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | 93 | 94 | theorem tsum_mul_left [T2Space α] : ∑' x, a * f x = a * ∑' x, f x := by | classical |
/-
Copyright (c) 2019 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.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Normed.Module.FiniteDimension
/-!
# Higher differentiability in ... | Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean | 71 | 75 | theorem contDiffOn_succ_iff_fderiv_apply [FiniteDimensional 𝕜 D] (hs : UniqueDiffOn 𝕜 s) :
ContDiffOn 𝕜 (n + 1) f s ↔
DifferentiableOn 𝕜 f s ∧ (n = ω → AnalyticOn 𝕜 f s) ∧
∀ y, ContDiffOn 𝕜 n (fun x => fderivWithin 𝕜 f s x y) s := by | rw [contDiffOn_succ_iff_fderivWithin hs, contDiffOn_clm_apply] |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Canonical
import Mathlib.CategoryTheory.Sites.Coherent.Basic
import Mathlib.CategoryTheory.Sites.EffectiveEpimorphic
/-!
# Sheaves for th... | Mathlib/CategoryTheory/Sites/Coherent/CoherentSheaves.lean | 44 | 58 | theorem isSheaf_yoneda_obj (W : C) : Presieve.IsSheaf (coherentTopology C) (yoneda.obj W) := by | rw [isSheaf_coherent]
intro X α _ Y π H
have h_colim := isColimitOfEffectiveEpiFamilyStruct Y π H.effectiveEpiFamily.some
rw [← Sieve.generateFamily_eq] at h_colim
intro x hx
let x_ext := Presieve.FamilyOfElements.sieveExtend x
have hx_ext := Presieve.FamilyOfElements.Compatible.sieveExtend hx
let S := Si... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... | Mathlib/Data/Part.lean | 525 | 526 | theorem bind_le {α} (x : Part α) (f : α → Part β) (y : Part β) :
x >>= f ≤ y ↔ ∀ a, a ∈ x → f a ≤ y := by | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 304 | 307 | theorem one_le_iff_pos {c : Cardinal} : 1 ≤ c ↔ 0 < c := by | rw [← succ_zero, succ_le_iff]
theorem one_le_iff_ne_zero {c : Cardinal} : 1 ≤ c ↔ c ≠ 0 := by |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
/-!
# Variance of random ... | Mathlib/Probability/Variance.lean | 202 | 212 | theorem variance_le_expectation_sq [IsProbabilityMeasure μ] {X : Ω → ℝ}
(hm : AEStronglyMeasurable X μ) : variance X μ ≤ μ[X ^ 2] := by | by_cases hX : MemLp X 2 μ
· rw [variance_def' hX]
simp only [sq_nonneg, sub_le_self_iff]
rw [variance, evariance_eq_lintegral_ofReal, ← integral_eq_lintegral_of_nonneg_ae]
· by_cases hint : Integrable X μ; swap
· simp only [integral_undef hint, Pi.pow_apply, Pi.sub_apply, sub_zero]
exact le_rfl
... |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localizati... | Mathlib/Algebra/Polynomial/Laurent.lean | 559 | 560 | theorem eval₂_T (n : ℤ) : eval₂ f x (T n) = (x ^ n).val := by | by_cases hn : 0 ≤ n |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro,
Kim Morrison
-/
import Mathlib.Data.List.Basic
/-!
# Lattice structure of lists
This files pro... | Mathlib/Data/List/Lattice.lean | 167 | 169 | theorem cons_bagInter_of_pos (l₁ : List α) (h : a ∈ l₂) :
(a :: l₁).bagInter l₂ = a :: l₁.bagInter (l₂.erase a) := by | cases l₂ |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
impo... | Mathlib/Algebra/Polynomial/AlgebraMap.lean | 123 | 127 | theorem eval₂_algebraMap_X {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A] (p : R[X])
(f : R[X] →ₐ[R] A) : eval₂ (algebraMap R A) (f X) p = f p := by | conv_rhs => rw [← Polynomial.sum_C_mul_X_pow_eq p]
simp only [eval₂_eq_sum, sum_def]
simp only [map_sum, map_mul, map_pow, eq_intCast, map_intCast] |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... | Mathlib/Algebra/Polynomial/Derivative.lean | 640 | 644 | theorem dvd_derivative_iff {P : R[X]} : P ∣ derivative P ↔ derivative P = 0 where
mp h := by | by_cases hP : P = 0
· simp only [hP, derivative_zero]
exact eq_zero_of_dvd_of_degree_lt h (degree_derivative_lt hP) |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | Mathlib/MeasureTheory/Integral/Average.lean | 127 | 131 | theorem setLAverage_eq' (f : α → ℝ≥0∞) (s : Set α) :
⨍⁻ x in s, f x ∂μ = ∫⁻ x, f x ∂(μ s)⁻¹ • μ.restrict s := by | simp only [laverage_eq', restrict_apply_univ]
@[deprecated (since := "2025-04-22")] alias setLaverage_eq' := setLAverage_eq' |
/-
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.Data.Set.Prod
/-!
# N-ary images of sets
This file defines `Set.image2`, the binary image of sets.
This is mostly useful to define pointwise oper... | Mathlib/Data/Set/NAry.lean | 262 | 270 | theorem image2_distrib_subset_right {f : δ → γ → ε} {g : α → β → δ} {f₁ : α → γ → α'}
{f₂ : β → γ → β'} {g' : α' → β' → ε} (h_distrib : ∀ a b c, f (g a b) c = g' (f₁ a c) (f₂ b c)) :
image2 f (image2 g s t) u ⊆ image2 g' (image2 f₁ s u) (image2 f₂ t u) := by | rintro _ ⟨_, ⟨a, ha, b, hb, rfl⟩, c, hc, rfl⟩
rw [h_distrib]
exact mem_image2_of_mem (mem_image2_of_mem ha hc) (mem_image2_of_mem hb hc)
theorem image_image2_antidistrib {g : γ → δ} {f' : β' → α' → δ} {g₁ : β → β'} {g₂ : α → α'}
(h_antidistrib : ∀ a b, g (f a b) = f' (g₁ b) (g₂ a)) : |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
/-!
# Partial equivalences
This f... | Mathlib/Logic/Equiv/PartialEquiv.lean | 724 | 725 | theorem EqOnSource.source_inter_preimage_eq {e e' : PartialEquiv α β} (he : e ≈ e') (s : Set β) :
e.source ∩ e ⁻¹' s = e'.source ∩ e' ⁻¹' s := by | rw [he.eqOn.inter_preimage_eq, source_eq he] |
/-
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.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
/-!
# Lattice operations on multisets
-/
namespace Multiset
variable {α : Type*}
/-! ###... | Mathlib/Data/Multiset/Lattice.lean | 79 | 80 | theorem nodup_sup_iff {α : Type*} [DecidableEq α] {m : Multiset (Multiset α)} :
m.sup.Nodup ↔ ∀ a : Multiset α, a ∈ m → a.Nodup := by | |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 169 | 169 | theorem two_zsmul_ne_zero_iff {θ : Angle} : (2 : ℤ) • θ ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ π := by | |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Basic Translation Lemmas Between Functions Defined for Continued... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 146 | 147 | theorem convs'Aux_succ_none {s : Stream'.Seq (Pair K)} (h : s.head = none) (n : ℕ) :
convs'Aux s (n + 1) = 0 := by | simp [convs'Aux, h] |
/-
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
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | Mathlib/Data/Vector/MapLemmas.lean | 38 | 40 | theorem mapAccumr_map {s : σ₁} (f₂ : α → β) :
(mapAccumr f₁ (map f₂ xs) s) = (mapAccumr (fun x s => f₁ (f₂ x) s) xs s) := by | induction xs using Vector.revInductionOn generalizing s <;> simp_all |
/-
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.Algebra.Group.Embedding
import Mathlib.Order.Interval.Multiset
/-!
# Finite intervals of naturals
This file proves that `ℕ` is a `LocallyFiniteOrder` and... | Mathlib/Order/Interval/Finset/Nat.lean | 100 | 101 | theorem card_fintypeIic : Fintype.card (Set.Iic b) = b + 1 := by | simp |
/-
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.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
/-!
# Witt vecto... | Mathlib/RingTheory/WittVector/Basic.lean | 102 | 102 | theorem add : mapFun f (x + y) = mapFun f x + mapFun f y := by | map_fun_tac |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 336 | 351 | theorem mkMetric_top : (mkMetric (fun _ => ∞ : ℝ≥0∞ → ℝ≥0∞) : OuterMeasure X) = ⊤ := by | simp_rw [mkMetric, mkMetric', mkMetric'.pre, extend_top, boundedBy_top, eq_top_iff]
rw [le_iSup_iff]
intro b hb
simpa using hb ⊤
/-- If `m₁ d ≤ m₂ d` for `d < ε` for some `ε > 0` (we use `≤ᶠ[𝓝[≥] 0]` to state this), then
`mkMetric m₁ hm₁ ≤ mkMetric m₂ hm₂`. -/
theorem mkMetric_mono {m₁ m₂ : ℝ≥0∞ → ℝ≥0∞} (hle : ... |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.Da... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 281 | 284 | theorem rightAngleRotation_neg_orientation (x : E) :
(-o).rightAngleRotation x = -o.rightAngleRotation x := by | apply ext_inner_right ℝ
intro y |
/-
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.Data.Complex.FiniteDimensional
import Mathlib.MeasureTheory.Constructions.HaarToSphere
import Mathlib.MeasureTheory.Integral.Gamma
import Mathlib.Measure... | Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | 273 | 295 | theorem Complex.volume_sum_rpow_lt [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ) :
volume {x : ι → ℂ | (∑ i, ‖x i‖ ^ p) ^ (1 / p) < r} = (.ofReal r) ^ (2 * card ι) *
.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * card ι / p + 1)) := by | have h₁ (x : ι → ℂ) : 0 ≤ ∑ i, ‖x i‖ ^ p := by positivity
have h₂ : ∀ x : ι → ℂ, 0 ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p) := fun x => rpow_nonneg (h₁ x) _
obtain hr | hr := le_or_lt r 0
· have : {x : ι → ℂ | (∑ i, ‖x i‖ ^ p) ^ (1 / p) < r} = ∅ := by
ext x
refine ⟨fun hx => ?_, fun hx => hx.elim⟩
exact not... |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 212 | 215 | theorem val_eq_zero_iff [NeZero n] {a : Fin n} : a.val = 0 ↔ a = 0 := by | rw [Fin.ext_iff, val_zero]
theorem val_ne_zero_iff [NeZero n] {a : Fin n} : a.val ≠ 0 ↔ a ≠ 0 := |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 76 | 77 | theorem logb_div (hx : x ≠ 0) (hy : y ≠ 0) : logb b (x / y) = logb b x - logb b y := by | simp_rw [logb, log_div hx hy, sub_div] |
/-
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
/-!
# Adjoint of operators on Hilbert spaces
Given ... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 92 | 95 | theorem adjointAux_norm (A : E →L[𝕜] F) : ‖adjointAux A‖ = ‖A‖ := by | refine le_antisymm ?_ ?_
· refine ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg _) fun x => ?_
rw [adjointAux_apply, LinearIsometryEquiv.norm_map] |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber, Yaël Dillies, Kin Yau James Wong
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Topo... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 335 | 339 | theorem univ_mem_measurableCylinders (α : ι → Type*) [∀ i, MeasurableSpace (α i)] :
Set.univ ∈ measurableCylinders α := by | rw [← compl_empty]; exact compl_mem_measurableCylinders (empty_mem_measurableCylinders α)
theorem union_mem_measurableCylinders (hs : s ∈ measurableCylinders α) |
/-
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.Algebra.Rat
import Mathlib.Data.Nat.Cast.Field
import Mathlib.RingTheory.PowerSeries.Basic
/-!
# Definition of well-known power series
In t... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 260 | 261 | theorem exp_pow_eq_rescale_exp [Algebra ℚ A] (k : ℕ) : exp A ^ k = rescale (k : A) (exp A) := by | induction' k with k h |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | Mathlib/Topology/ContinuousOn.lean | 423 | 426 | theorem eventuallyEq_nhds_of_eventuallyEq_nhdsNE {f g : α → β} {a : α} (h₁ : f =ᶠ[𝓝[≠] a] g)
(h₂ : f a = g a) :
f =ᶠ[𝓝 a] g := by | filter_upwards [eventually_nhdsWithin_iff.1 h₁] |
/-
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.Regular
import Mathlib.SetTheory.Cardinal.Contin... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 156 | 164 | theorem cardinal_generateMeasurableRec_le (s : Set (Set α)) (i : Ordinal.{v}) :
#(generateMeasurableRec s i) ≤ max #s 2 ^ ℵ₀ := by | suffices ∀ i ≤ ω₁, #(generateMeasurableRec s i) ≤ max #s 2 ^ ℵ₀ by
obtain hi | hi := le_or_lt i ω₁
· exact this i hi
· rw [generateMeasurableRec_of_omega1_le s hi.le]
exact this _ le_rfl
intro i
apply WellFoundedLT.induction i |
/-
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.W
import Mathlib.Data.QPF.Multivariate.Basic
/-!
# The initial algebra of a multivariate qpf is again a qpf.
Fo... | Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean | 125 | 129 | theorem wrepr_equiv {α : TypeVec n} (x : q.P.W α) : WEquiv (wrepr x) x := by | apply q.P.w_ind _ x; intro a f' f ih
apply WEquiv.trans _ (q.P.wMk' (appendFun id wrepr <$$> ⟨a, q.P.appendContents f' f⟩))
· apply wEquiv.abs'
rw [wrepr_wMk, q.P.wDest'_wMk', q.P.wDest'_wMk', abs_repr] |
/-
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.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
/-!
# Maximal length of chains
This file contains lemmas to work with the ma... | Mathlib/Order/Height.lean | 142 | 144 | theorem le_chainHeight_add_nat_iff {n m : ℕ} :
↑n ≤ s.chainHeight + m ↔ ∃ l ∈ s.subchain, n ≤ length l + m := by | simp_rw [← tsub_le_iff_right, ← ENat.coe_sub, (le_chainHeight_TFAE s (n - m)).out 0 2] |
/-
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.Algebra.Group.Embedding
import Mathlib.Order.Interval.Multiset
/-!
# Finite intervals of naturals
This file proves that `ℕ` is a `LocallyFiniteOrder` and... | Mathlib/Order/Interval/Finset/Nat.lean | 114 | 115 | theorem Ico_succ_left : Ico a.succ b = Ioo a b := by | ext x |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | Mathlib/Topology/ContinuousOn.lean | 1,315 | 1,318 | theorem Topology.IsOpenEmbedding.map_nhdsWithin_preimage_eq {f : α → β} (hf : IsOpenEmbedding f)
(s : Set β) (x : α) : map f (𝓝[f ⁻¹' s] x) = 𝓝[s] f x := by | rw [hf.isEmbedding.map_nhdsWithin_eq, image_preimage_eq_inter_range]
apply nhdsWithin_eq_nhdsWithin (mem_range_self _) hf.isOpen_range |
/-
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.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | Mathlib/Data/ENNReal/Real.lean | 332 | 335 | theorem toReal_top_mul (a : ℝ≥0∞) : ENNReal.toReal (∞ * a) = 0 := by | rw [mul_comm]
exact toReal_mul_top _ |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Patrick Massot
-/
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
... | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 180 | 183 | theorem IsUniformEmbedding.of_comp_iff {g : β → γ} (hg : IsUniformEmbedding g) {f : α → β} :
IsUniformEmbedding (g ∘ f) ↔ IsUniformEmbedding f := by | simp_rw [isUniformEmbedding_iff, hg.isUniformInducing.of_comp_iff, hg.injective.of_comp_iff f] |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
/-... | Mathlib/Data/Sym/Sym2.lean | 354 | 355 | theorem other_spec {a : α} {z : Sym2 α} (h : a ∈ z) : s(a, Mem.other h) = z := by | erw [← Classical.choose_spec h] |
/-
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.Ring.Associated
import Mathlib.Algebra.Ring.Regular
/-!
# Monoids with normalization functions, `gcd`, and `lcm`
This file de... | Mathlib/Algebra/GCDMonoid/Basic.lean | 499 | 500 | theorem gcd_pow_right_dvd_pow_gcd [GCDMonoid α] {a b : α} {k : ℕ} :
gcd a (b ^ k) ∣ gcd a b ^ k := by | |
/-
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.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | Mathlib/GroupTheory/OrderOfElement.lean | 536 | 538 | theorem pow_inj_iff_of_orderOf_eq_zero (h : orderOf x = 0) {n m : ℕ} : x ^ n = x ^ m ↔ n = m := by | rw [pow_eq_pow_iff_modEq, h, modEq_zero_iff] |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 907 | 908 | theorem Iic_inter_Iic {a b : α} : Iic a ∩ Iic b = Iic (a ⊓ b) := by | ext x |
/-
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.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | Mathlib/Data/List/Rotate.lean | 433 | 434 | theorem isRotated_reverse_comm_iff : l.reverse ~r l' ↔ l ~r l'.reverse := by | constructor <;> |
/-
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.Data.Finset.Sym
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Linea... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 103 | 104 | theorem polar_add (f g : M → N) (x y : M) : polar (f + g) x y = polar f x y + polar g x y := by | simp only [polar, Pi.add_apply] |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Analysis.NormedSpac... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 180 | 183 | theorem log_neg (h0 : 0 < x) (h1 : x < 1) : log x < 0 :=
(log_neg_iff h0).2 h1
theorem log_neg_of_lt_zero (h0 : x < 0) (h1 : -1 < x) : log x < 0 := by | |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 474 | 476 | theorem addHaar_real_closedBall' (x : E) {r : ℝ} (hr : 0 ≤ r) :
μ.real (closedBall x r) = r ^ finrank ℝ E * μ.real (closedBall 0 1) := by | simp only [measureReal_def, addHaar_closedBall' μ x hr, ENNReal.toReal_mul, mul_eq_mul_right_iff, |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 160 | 163 | theorem lineDifferentiableWithinAt_univ :
LineDifferentiableWithinAt 𝕜 f univ x v ↔ LineDifferentiableAt 𝕜 f x v := by | simp only [LineDifferentiableWithinAt, LineDifferentiableAt, preimage_univ,
differentiableWithinAt_univ] |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Basic
/-!
# Intervals as multisets
This file defines intervals as multisets.
## Main declarations
In a `LocallyFiniteOrder`,
* `M... | Mathlib/Order/Interval/Multiset.lean | 287 | 290 | theorem Ico_filter_le (a b c : α) : ((Ico a b).filter fun x => c ≤ x) = Ico (max a c) b := by | rw [Ico, Ico, ← Finset.filter_val, Finset.Ico_filter_le]
@[simp] |
/-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.RepresentationTheory.FDRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
/-!
# Characters of representations
Th... | Mathlib/RepresentationTheory/Character.lean | 59 | 60 | theorem char_tensor (V W : FDRep k G) : (V ⊗ W).character = V.character * W.character := by | ext g; convert trace_tensorProduct' (V.ρ g) (W.ρ g) |
/-
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.Algebra.Order.Module.Defs
import Mathlib.Data.DFinsupp.Module
/-!
# Pointwise order on finitely supported dependent functions
This file lifts order struc... | Mathlib/Data/DFinsupp/Order.lean | 265 | 267 | theorem support_tsub : (f - g).support ⊆ f.support := by | simp +contextual only [subset_iff, tsub_eq_zero_iff_le, mem_support_iff,
Ne, coe_tsub, Pi.sub_apply, not_imp_not, zero_le, imp_true_iff] |
/-
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.Lemmas
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.Basic
/-!
# Period of a group action
This modul... | Mathlib/GroupTheory/GroupAction/Period.lean | 87 | 88 | theorem period_dvd_orderOf (m : M) (a : α) : period m a ∣ orderOf m := by | rw [← pow_smul_eq_iff_period_dvd, pow_orderOf_eq_one, one_smul] |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 991 | 992 | theorem iUnion_singleton_eq_range (f : α → β) : ⋃ x : α, {f x} = range f := by | ext x |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm.AbsNorm
import M... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 46 | 49 | theorem discr_odd_prime' [IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ p) (hodd : p ≠ 2) :
discr ℚ (hζ.subOnePowerBasis ℚ).basis = (-1) ^ (((p : ℕ) - 1) / 2) * p ^ ((p : ℕ) - 2) := by | rw [← discr_odd_prime hζ (cyclotomic.irreducible_rat hp.out.pos) hodd]
exact hζ.discr_zeta_eq_discr_zeta_sub_one.symm |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Multiset.Bind
/-!
# The cartesian product of multisets
## Main definitions
* `Multiset.pi`: Cartesian product of multisets indexed by a multise... | Mathlib/Data/Multiset/Pi.lean | 49 | 58 | theorem cons_swap {a a' : α} {b : δ a} {b' : δ a'} {m : Multiset α} {f : ∀ a ∈ m, δ a}
(h : a ≠ a') : HEq (Pi.cons (a' ::ₘ m) a b (Pi.cons m a' b' f))
(Pi.cons (a ::ₘ m) a' b' (Pi.cons m a b f)) := by | apply hfunext rfl
simp only [heq_iff_eq]
rintro a'' _ rfl
refine hfunext (by rw [Multiset.cons_swap]) fun ha₁ ha₂ _ => ?_
rcases Decidable.ne_or_eq a'' a with (h₁ | rfl)
on_goal 1 => rcases Decidable.eq_or_ne a'' a' with (rfl | h₂)
all_goals simp [*, Pi.cons_same, Pi.cons_ne] |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, Oliver Nash
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Identities
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.R... | Mathlib/Dynamics/Newton.lean | 106 | 126 | theorem existsUnique_nilpotent_sub_and_aeval_eq_zero
(h : IsNilpotent (aeval x P)) (h' : IsUnit (aeval x <| derivative P)) :
∃! r, IsNilpotent (x - r) ∧ aeval r P = 0 := by | simp_rw [(neg_sub _ x).symm, isNilpotent_neg_iff]
refine existsUnique_of_exists_of_unique ?_ fun r₁ r₂ ⟨hr₁, hr₁'⟩ ⟨hr₂, hr₂'⟩ ↦ ?_
· -- Existence
obtain ⟨n, hn⟩ := id h
refine ⟨P.newtonMap^[n] x, isNilpotent_iterate_newtonMap_sub_of_isNilpotent h n, ?_⟩
rw [← zero_dvd_iff, ← pow_eq_zero_of_le (n.lt_two... |
/-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Sites.Coherent.Comparison
import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves
import Mathlib.CategoryTheory.Sites.Coherent... | Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean | 55 | 76 | theorem exists_effectiveEpiFamily_iff_mem_induced (X : C) (S : Sieve X) :
(∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → (Y a ⟶ X)),
EffectiveEpiFamily Y π ∧ (∀ a : α, (S.arrows) (π a)) ) ↔
(S ∈ F.inducedTopology (coherentTopology _) X) := by | refine ⟨fun ⟨α, _, Y, π, ⟨H₁, H₂⟩⟩ ↦ ?_, fun hS ↦ ?_⟩
· apply (mem_sieves_iff_hasEffectiveEpiFamily (Sieve.functorPushforward _ S)).mpr
refine ⟨α, inferInstance, fun i => F.obj (Y i),
fun i => F.map (π i), ⟨?_,
fun a => Sieve.image_mem_functorPushforward F S (H₂ a)⟩⟩
exact F.map_finite_effectiveEp... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... | Mathlib/Data/List/Sigma.lean | 201 | 209 | theorem mem_dlookup_iff {a : α} {b : β a} {l : List (Sigma β)} (nd : l.NodupKeys) :
b ∈ dlookup a l ↔ Sigma.mk a b ∈ l :=
⟨of_mem_dlookup, mem_dlookup nd⟩
theorem perm_dlookup (a : α) {l₁ l₂ : List (Sigma β)} (nd₁ : l₁.NodupKeys) (nd₂ : l₂.NodupKeys)
(p : l₁ ~ l₂) : dlookup a l₁ = dlookup a l₂ := by | ext b; simp only [mem_dlookup_iff nd₁, mem_dlookup_iff nd₂]; exact p.mem_iff
theorem lookup_ext {l₀ l₁ : List (Sigma β)} (nd₀ : l₀.NodupKeys) (nd₁ : l₁.NodupKeys) |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.... | Mathlib/CategoryTheory/Subobject/Basic.lean | 349 | 350 | theorem ofLE_comp_ofLE {B : C} (X Y Z : Subobject B) (h₁ : X ≤ Y) (h₂ : Y ≤ Z) :
ofLE X Y h₁ ≫ ofLE Y Z h₂ = ofLE X Z (h₁.trans h₂) := by | |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | Mathlib/Data/Matroid/Basic.lean | 948 | 951 | theorem exists_isBasis (M : Matroid α) (X : Set α) (hX : X ⊆ M.E := by | aesop_mat) :
∃ I, M.IsBasis I X :=
let ⟨_, hI, _⟩ := M.empty_indep.subset_isBasis_of_subset (empty_subset X)
⟨_, hI⟩ |
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