Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
|---|---|---|---|---|---|
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sebastien Gouezel, Heather Macbeth, Patrick Massot, Floris van Doorn
-/
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.Topology.FiberBundle.Ba... | Mathlib/Topology/VectorBundle/Basic.lean | 1,029 | 1,037 | theorem inCoordinates_eq (x₀ x : B) (y₀ y : B') (ϕ : E x →SL[σ] E' y)
(hx : x ∈ (trivializationAt F E x₀).baseSet) (hy : y ∈ (trivializationAt F' E' y₀).baseSet) :
inCoordinates F E F' E' x₀ x y₀ y ϕ =
((trivializationAt F' E' y₀).continuousLinearEquivAt 𝕜₂ y hy : E' y →L[𝕜₂] F').comp
(ϕ.comp <|... |
ext
simp_rw [inCoordinates, ContinuousLinearMap.coe_comp', ContinuousLinearEquiv.coe_coe,
Trivialization.coe_continuousLinearEquivAt_eq, Trivialization.symm_continuousLinearEquivAt_eq]
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Init.Align
import Mathlib.Data.Fintype.Order
import Mathlib.Algebra.DirectLimit
import Mathlib.ModelTheory.Quotients
import Mathlib.ModelT... | Mathlib/ModelTheory/DirectLimit.lean | 279 | 291 | theorem exists_quotient_mk'_sigma_mk'_eq {α : Type*} [Finite α] (x : α → DirectLimit G f) :
∃ (i : ι) (y : α → G i), x = fun a => ⟦.mk f i (y a)⟧ := by |
obtain ⟨i, hi⟩ := Finite.bddAbove_range fun a => (x a).out.1
refine ⟨i, unify f (Quotient.out ∘ x) i hi, ?_⟩
ext a
rw [Quotient.eq_mk_iff_out, unify]
generalize_proofs r
change _ ≈ .mk f i (f (Quotient.out (x a)).fst i r (Quotient.out (x a)).snd)
have : (.mk f i (f (Quotient.out (x a)).fst i r (Quotient.... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,010 | 1,014 | theorem lintegral_strict_mono {f g : α → ℝ≥0∞} (hμ : μ ≠ 0) (hg : AEMeasurable g μ)
(hfi : ∫⁻ x, f x ∂μ ≠ ∞) (h : ∀ᵐ x ∂μ, f x < g x) : ∫⁻ x, f x ∂μ < ∫⁻ x, g x ∂μ := by |
rw [Ne, ← Measure.measure_univ_eq_zero] at hμ
refine lintegral_strict_mono_of_ae_le_of_ae_lt_on hg hfi (ae_le_of_ae_lt h) hμ ?_
simpa using h
|
/-
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.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 122 | 123 | theorem contDiffOn_of_subsingleton [Subsingleton F] : ContDiffOn 𝕜 n f s := by |
rw [Subsingleton.elim f fun _ => 0]; exact contDiffOn_const
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin... | Mathlib/RingTheory/IsAdjoinRoot.lean | 243 | 244 | theorem lift_root (h : IsAdjoinRoot S f) : h.lift i x hx h.root = x := by |
rw [← h.map_X, lift_map, eval₂_X]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
impor... | Mathlib/Dynamics/PeriodicPts.lean | 321 | 322 | theorem minimalPeriod_eq_zero_iff_nmem_periodicPts : minimalPeriod f x = 0 ↔ x ∉ periodicPts f := by |
rw [← minimalPeriod_pos_iff_mem_periodicPts, not_lt, nonpos_iff_eq_zero]
|
/-
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.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.special_... | Mathlib/Analysis/SpecialFunctions/Exp.lean | 45 | 61 | theorem locally_lipschitz_exp {r : ℝ} (hr_nonneg : 0 ≤ r) (hr_le : r ≤ 1) (x y : ℂ)
(hyx : ‖y - x‖ < r) : ‖exp y - exp x‖ ≤ (1 + r) * ‖exp x‖ * ‖y - x‖ := by |
have hy_eq : y = x + (y - x) := by abel
have hyx_sq_le : ‖y - x‖ ^ 2 ≤ r * ‖y - x‖ := by
rw [pow_two]
exact mul_le_mul hyx.le le_rfl (norm_nonneg _) hr_nonneg
have h_sq : ∀ z, ‖z‖ ≤ 1 → ‖exp (x + z) - exp x‖ ≤ ‖z‖ * ‖exp x‖ + ‖exp x‖ * ‖z‖ ^ 2 := by
intro z hz
have : ‖exp (x + z) - exp x - z • ex... |
/-
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.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import ... | Mathlib/Data/Set/Basic.lean | 2,097 | 2,102 | theorem subset_pair_iff_eq {x y : α} : s ⊆ {x, y} ↔ s = ∅ ∨ s = {x} ∨ s = {y} ∨ s = {x, y} := by |
refine ⟨?_, by rintro (rfl | rfl | rfl | rfl) <;> simp [pair_subset_iff]⟩
rw [subset_insert_iff, subset_singleton_iff_eq, subset_singleton_iff_eq,
← subset_empty_iff (s := s \ {x}), diff_subset_iff, union_empty, subset_singleton_iff_eq]
have h : x ∈ s → {y} = s \ {x} → s = {x,y} := fun h₁ h₂ ↦ by simp [h₁, h... |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.Algebraic
#align_import field_the... | Mathlib/FieldTheory/Minpoly/Field.lean | 238 | 243 | theorem prime (hx : IsIntegral A x) : Prime (minpoly A x) := by |
refine ⟨minpoly.ne_zero hx, not_isUnit A x, ?_⟩
rintro p q ⟨d, h⟩
have : Polynomial.aeval x (p * q) = 0 := by simp [h, aeval A x]
replace : Polynomial.aeval x p = 0 ∨ Polynomial.aeval x q = 0 := by simpa
exact Or.imp (dvd A x) (dvd A x) this
|
/-
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, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 946 | 948 | theorem HasCompactMulSupport.exists_bound_of_continuous [TopologicalSpace α]
{f : α → E} (hf : HasCompactMulSupport f) (h'f : Continuous f) : ∃ C, ∀ x, ‖f x‖ ≤ C := by |
simpa using (hf.isCompact_range h'f).isBounded.exists_norm_le'
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebr... | Mathlib/LinearAlgebra/Finsupp.lean | 133 | 136 | theorem LinearEquiv.finsuppUnique_symm_apply [Unique α] (m : M) :
(LinearEquiv.finsuppUnique R M α).symm m = Finsupp.single default m := by |
ext; simp [LinearEquiv.finsuppUnique, Equiv.funUnique, single, Pi.single,
equivFunOnFinite, Function.update]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Eric Wieser, Yaël Dillies
-/
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_spa... | Mathlib/Analysis/NormedSpace/Real.lean | 50 | 59 | theorem dist_smul_add_one_sub_smul_le {r : ℝ} {x y : E} (h : r ∈ Icc 0 1) :
dist (r • x + (1 - r) • y) x ≤ dist y x :=
calc
dist (r • x + (1 - r) • y) x = ‖1 - r‖ * ‖x - y‖ := by |
simp_rw [dist_eq_norm', ← norm_smul, sub_smul, one_smul, smul_sub, ← sub_sub, ← sub_add,
sub_right_comm]
_ = (1 - r) * dist y x := by
rw [Real.norm_eq_abs, abs_eq_self.mpr (sub_nonneg.mpr h.2), dist_eq_norm']
_ ≤ (1 - 0) * dist y x := by gcongr; exact h.1
_ = dist y x := by rw [sub_zero... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Sets.Opens
#align_import topology.local_homeomorph from "leanprover-community/mathlib"@"431589b... | Mathlib/Topology/PartialHomeomorph.lean | 381 | 382 | theorem tendsto_symm {x} (hx : x ∈ e.source) : Tendsto e.symm (𝓝 (e x)) (𝓝 x) := by |
simpa only [ContinuousAt, e.left_inv hx] using e.continuousAt_symm (e.map_source hx)
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padic... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 918 | 937 | theorem norm_int_lt_one_iff_dvd (k : ℤ) : ‖(k : ℚ_[p])‖ < 1 ↔ ↑p ∣ k := by |
constructor
· intro h
contrapose! h
apply le_of_eq
rw [eq_comm]
calc
‖(k : ℚ_[p])‖ = ‖((k : ℚ) : ℚ_[p])‖ := by norm_cast
_ = padicNorm p k := padicNormE.eq_padicNorm _
_ = 1 := mod_cast (int_eq_one_iff k).mpr h
· rintro ⟨x, rfl⟩
push_cast
rw [padicNormE.mul]
calc
... |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Order.WellFounded
import Mathlib.Tactic.Common
#align_import data.pi.lex from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
/... | Mathlib/Order/PiLex.lean | 65 | 68 | theorem lex_lt_of_lt [∀ i, PartialOrder (β i)] {r} (hwf : WellFounded r) {x y : ∀ i, β i}
(hlt : x < y) : Pi.Lex r (@fun i => (· < ·)) x y := by |
simp_rw [Pi.Lex, le_antisymm_iff]
exact lex_lt_of_lt_of_preorder hwf hlt
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import ... | Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean | 279 | 283 | theorem FarFromTriangleFree.nonpos (h₀ : G.FarFromTriangleFree ε) (h₁ : G.CliqueFree 3) :
ε ≤ 0 := by |
have := h₀ (empty_subset _)
rw [coe_empty, Finset.card_empty, cast_zero, deleteEdges_empty] at this
exact nonpos_of_mul_nonpos_left (this h₁) (cast_pos.2 <| sq_pos_of_pos Fintype.card_pos)
|
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathl... | Mathlib/NumberTheory/ZetaValues.lean | 59 | 64 | theorem bernoulliFun_eval_one (k : ℕ) : bernoulliFun k 1 = bernoulliFun k 0 + ite (k = 1) 1 0 := by |
rw [bernoulliFun, bernoulliFun_eval_zero, Polynomial.eval_one_map, Polynomial.bernoulli_eval_one]
split_ifs with h
· rw [h, bernoulli_one, bernoulli'_one, eq_ratCast]
push_cast; ring
· rw [bernoulli_eq_bernoulli'_of_ne_one h, add_zero, eq_ratCast]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Orde... | Mathlib/Order/CompactlyGenerated/Basic.lean | 498 | 502 | theorem isCompactlyGenerated_of_wellFounded (h : WellFounded ((· > ·) : α → α → Prop)) :
IsCompactlyGenerated α := by |
rw [wellFounded_iff_isSupFiniteCompact, isSupFiniteCompact_iff_all_elements_compact] at h
-- x is the join of the set of compact elements {x}
exact ⟨fun x => ⟨{x}, ⟨fun x _ => h x, sSup_singleton⟩⟩⟩
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
/-!
# Restricting a measure to a subset or a subtype
Given a measure `μ` on a type `α` and a subse... | Mathlib/MeasureTheory/Measure/Restrict.lean | 287 | 290 | theorem restrict_add_restrict_compl (hs : MeasurableSet s) :
μ.restrict s + μ.restrict sᶜ = μ := by |
rw [← restrict_union (@disjoint_compl_right (Set α) _ _) hs.compl, union_compl_self,
restrict_univ]
|
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
/-!
# Operator norm as an `NNNorm`
Operator norm as an `NNNorm`, i.e. takin... | Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean | 158 | 172 | theorem exists_lt_apply_of_lt_opNNNorm {𝕜 𝕜₂ E F : Type*} [NormedAddCommGroup E]
[SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂}
[NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) {r : ℝ≥0}
(hr : r < ‖f‖₊) : ∃ x : E, ‖x‖₊ < ... |
obtain ⟨y, hy⟩ := f.exists_mul_lt_apply_of_lt_opNNNorm hr
have hy' : ‖y‖₊ ≠ 0 :=
nnnorm_ne_zero_iff.2 fun heq => by
simp [heq, nnnorm_zero, map_zero, not_lt_zero'] at hy
have hfy : ‖f y‖₊ ≠ 0 := (zero_le'.trans_lt hy).ne'
rw [← inv_inv ‖f y‖₊, NNReal.lt_inv_iff_mul_lt (inv_ne_zero hfy), mul_assoc, mu... |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.circle_transform from "leanprover-commun... | Mathlib/MeasureTheory/Integral/CircleTransform.lean | 58 | 65 | theorem circleTransformDeriv_eq (f : ℂ → E) : circleTransformDeriv R z w f =
fun θ => (circleMap z R θ - w)⁻¹ • circleTransform R z w f θ := by |
ext
simp_rw [circleTransformDeriv, circleTransform, ← mul_smul, ← mul_assoc]
ring_nf
rw [inv_pow]
congr
ring
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
imp... | Mathlib/FieldTheory/Adjoin.lean | 484 | 486 | theorem restrictScalars_adjoin (K : IntermediateField F E) (S : Set E) :
restrictScalars F (adjoin K S) = adjoin F (K ∪ S) := by |
rw [← adjoin_self _ K, adjoin_adjoin_left, adjoin_self _ K]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#alig... | Mathlib/Algebra/Polynomial/Roots.lean | 272 | 276 | theorem roots_prod_X_sub_C (s : Finset R) : (s.prod fun a => X - C a).roots = s.val := by |
apply (roots_prod (fun a => X - C a) s ?_).trans
· simp_rw [roots_X_sub_C]
rw [Multiset.bind_singleton, Multiset.map_id']
· refine prod_ne_zero_iff.mpr (fun a _ => X_sub_C_ne_zero a)
|
/-
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 | 682 | 684 | theorem full_empty : full ∅ = (⊥ : Subgroupoid C) := by |
ext
simp only [Bot.bot, mem_full_iff, mem_empty_iff_false, and_self_iff]
|
/-
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.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... | Mathlib/Data/Num/Lemmas.lean | 485 | 486 | theorem of_natCast {α} [AddMonoidWithOne α] (n : ℕ) : ((n : Num) : α) = n := by |
rw [← cast_to_nat, to_of_nat]
|
/-
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
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 578 | 581 | theorem equicontinuousOn_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'}
{S : Set X} :
EquicontinuousOn (uα := ⨅ k, u k) F S ↔ ∀ k, EquicontinuousOn (uα := u k) F S := by |
simp_rw [EquicontinuousOn, equicontinuousWithinAt_iInf_rng, @forall_swap _ κ]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemm... | Mathlib/Algebra/Order/Floor.lean | 1,546 | 1,556 | theorem round_le (x : α) (z : ℤ) : |x - round x| ≤ |x - z| := by |
rw [abs_sub_round_eq_min, min_le_iff]
rcases le_or_lt (z : α) x with (hx | hx) <;> [left; right]
· conv_rhs => rw [abs_eq_self.mpr (sub_nonneg.mpr hx), ← fract_add_floor x, add_sub_assoc]
simpa only [le_add_iff_nonneg_right, sub_nonneg, cast_le] using le_floor.mpr hx
· rw [abs_eq_neg_self.mpr (sub_neg.mpr ... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Antoine Chambert-Loir
-/
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Group.Hom.CompType... | Mathlib/GroupTheory/GroupAction/Hom.lean | 473 | 476 | theorem toMulActionHom_injective {f g : A →ₑ+[φ] B} (h : (f : A →ₑ[φ] B) = (g : A →ₑ[φ] B)) :
f = g := by |
ext a
exact MulActionHom.congr_fun h a
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
#align_import analysis.box_integr... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 657 | 659 | theorem iUnion_disjUnion (h : Disjoint π₁.iUnion π₂.iUnion) :
(π₁.disjUnion π₂ h).iUnion = π₁.iUnion ∪ π₂.iUnion := by |
simp [disjUnion, Prepartition.iUnion, iUnion_or, iUnion_union_distrib]
|
/-
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.Algebra.MonoidAlgebra.Basic
#align_import algebra.monoid_algebra.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
/-... | Mathlib/Algebra/MonoidAlgebra/Division.lean | 193 | 200 | theorem of'_dvd_iff_modOf_eq_zero {x : k[G]} {g : G} :
of' k G g ∣ x ↔ x %ᵒᶠ g = 0 := by |
constructor
· rintro ⟨x, rfl⟩
rw [of'_mul_modOf]
· intro h
rw [← divOf_add_modOf x g, h, add_zero]
exact dvd_mul_right _ _
|
/-
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.Topology.Algebra.Nonarchimedean.Basic
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Algebra.Module.Submodule.Pointwise
#align_import topo... | Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean | 194 | 199 | theorem nonarchimedean : @NonarchimedeanRing A _ hB.topology := by |
letI := hB.topology
constructor
intro U hU
obtain ⟨i, -, hi : (B i : Set A) ⊆ U⟩ := hB.hasBasis_nhds_zero.mem_iff.mp hU
exact ⟨hB.openAddSubgroup i, hi⟩
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 398 | 400 | theorem disjoint_iff_disjoint_support : Disjoint f g ↔ _root_.Disjoint f.support g.support := by |
simp [disjoint_iff_eq_or_eq, disjoint_iff, disjoint_iff, Finset.ext_iff, not_and_or,
imp_iff_not_or]
|
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Sum
#align_import combinatoric... | Mathlib/Combinatorics/HalesJewett.lean | 190 | 193 | theorem vertical_apply {α ι ι'} (v : ι → α) (l : Line α ι') (x : α) :
l.vertical v x = Sum.elim v (l x) := by |
funext i
cases i <;> rfl
|
/-
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.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysi... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 194 | 200 | theorem decay_add_le_aux (k n : ℕ) (f g : 𝓢(E, F)) (x : E) :
‖x‖ ^ k * ‖iteratedFDeriv ℝ n ((f : E → F) + (g : E → F)) x‖ ≤
‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ + ‖x‖ ^ k * ‖iteratedFDeriv ℝ n g x‖ := by |
rw [← mul_add]
refine mul_le_mul_of_nonneg_left ?_ (by positivity)
rw [iteratedFDeriv_add_apply (f.smooth _) (g.smooth _)]
exact norm_add_le _ _
|
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216... | Mathlib/Data/Bool/Basic.lean | 221 | 221 | theorem and_le_left : ∀ x y : Bool, (x && y) ≤ x := by | decide
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Thin
#align_import category_theory.limits.shapes.wide_pullbacks from "lean... | Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean | 368 | 376 | theorem eq_lift_of_comp_eq (g : X ⟶ widePullback _ _ arrows) :
(∀ j : J, g ≫ π arrows j = fs j) → g ≫ base arrows = f → g = lift f fs w := by |
intro h1 h2
apply
(limit.isLimit (WidePullbackShape.wideCospan B objs arrows)).uniq
(WidePullbackShape.mkCone f fs <| w)
rintro (_ | _)
· apply h2
· apply h1
|
/-
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 | 599 | 664 | theorem det_blockDiagonal {o : Type*} [Fintype o] [DecidableEq o] (M : o → Matrix n n R) :
(blockDiagonal M).det = ∏ k, (M k).det := by |
-- Rewrite the determinants as a sum over permutations.
simp_rw [det_apply']
-- The right hand side is a product of sums, rewrite it as a sum of products.
rw [Finset.prod_sum]
simp_rw [Finset.prod_attach_univ, Finset.univ_pi_univ]
-- We claim that the only permutations contributing to the sum are those tha... |
/-
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.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,142 | 1,154 | theorem coeff_mul_add_eq_of_natDegree_le {df dg : ℕ} {f g : R[X]}
(hdf : natDegree f ≤ df) (hdg : natDegree g ≤ dg) :
(f * g).coeff (df + dg) = f.coeff df * g.coeff dg := by |
rw [coeff_mul, Finset.sum_eq_single_of_mem (df, dg)]
· rw [mem_antidiagonal]
rintro ⟨df', dg'⟩ hmem hne
obtain h | hdf' := lt_or_le df df'
· rw [coeff_eq_zero_of_natDegree_lt (hdf.trans_lt h), zero_mul]
obtain h | hdg' := lt_or_le dg dg'
· rw [coeff_eq_zero_of_natDegree_lt (hdg.trans_lt h), mul_zero]
o... |
/-
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.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 860 | 862 | theorem Icc_diff_Ioo_same (h : a ≤ b) : Icc a b \ Ioo a b = {a, b} := by |
rw [← Icc_diff_both, diff_diff_cancel_left]
simp [insert_subset_iff, h]
|
/-
Copyright (c) 2022 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson, Devon Tuma, Eric Rodriguez, Oliver Nash
-/
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.... | Mathlib/Topology/Algebra/Order/Field.lean | 189 | 191 | theorem tendsto_const_mul_pow_nhds_iff {n : ℕ} {c d : 𝕜} (hc : c ≠ 0) :
Tendsto (fun x : 𝕜 => c * x ^ n) atTop (𝓝 d) ↔ n = 0 ∧ c = d := by |
simp [tendsto_const_mul_pow_nhds_iff', hc]
|
/-
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.CompleteLattice
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
import Mathlib.Order.WellFounded
#align_import order.succ_pred.basic from "l... | Mathlib/Order/SuccPred/Basic.lean | 331 | 332 | theorem Ico_succ_right_of_not_isMax (hb : ¬IsMax b) : Ico a (succ b) = Icc a b := by |
rw [← Ici_inter_Iio, Iio_succ_of_not_isMax hb, Ici_inter_Iic]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes, Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monoto... | Mathlib/Data/Nat/Factorial/Basic.lean | 182 | 185 | theorem add_factorial_le_factorial_add (i : ℕ) {n : ℕ} (n1 : 1 ≤ n) : i + n ! ≤ (i + n)! := by |
cases' n1 with h
· exact self_le_factorial _
exact add_factorial_succ_le_factorial_add_succ i h
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.Bin... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 221 | 222 | theorem inl_pushoutZeroZeroIso_inv (X Y : C) [HasBinaryCoproduct X Y] :
coprod.inl ≫ (pushoutZeroZeroIso X Y).inv = pushout.inl := by | simp [Iso.comp_inv_eq]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mat... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 729 | 745 | theorem image_comap_zeroLocus_eq_zeroLocus_comap (hf : Surjective f) (I : Ideal S) :
comap f '' zeroLocus I = zeroLocus (I.comap f) := by |
simp only [Set.ext_iff, Set.mem_image, mem_zeroLocus, SetLike.coe_subset_coe]
refine fun p => ⟨?_, fun h_I_p => ?_⟩
· rintro ⟨p, hp, rfl⟩ a ha
exact hp ha
· have hp : ker f ≤ p.asIdeal := (Ideal.comap_mono bot_le).trans h_I_p
refine ⟨⟨p.asIdeal.map f, Ideal.map_isPrime_of_surjective hf hp⟩, fun x hx =>... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-com... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 293 | 297 | theorem iteratedDeriv_eq_iterate : iteratedDeriv n f = deriv^[n] f := by |
ext x
rw [← iteratedDerivWithin_univ]
convert iteratedDerivWithin_eq_iterate uniqueDiffOn_univ (F := F) (mem_univ x)
simp [derivWithin_univ]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.rig... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 709 | 713 | theorem sin_oangle_right_mul_dist_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
(h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : Real.Angle.sin (∡ p₂ p₃ p₁) * dist p₁ p₃ = dist p₁ p₂ := by |
have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe,
sin_angle_mul_dist_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Pi
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.CharP.Subring
import Mathlib.Algebra.Ring.Pi
import Mathlib.Analysis.SpecialFunctio... | Mathlib/RingTheory/Perfection.lean | 406 | 413 | theorem preVal_mk {x : O} (hx : (Ideal.Quotient.mk _ x : ModP K v O hv p) ≠ 0) :
preVal K v O hv p (Ideal.Quotient.mk _ x) = v (algebraMap O K x) := by |
obtain ⟨r, hr⟩ : ∃ (a : O), a * (p : O) = (Quotient.mk'' x).out' - x :=
Ideal.mem_span_singleton'.1 <| Ideal.Quotient.eq.1 <| Quotient.sound' <| Quotient.mk_out' _
refine (if_neg hx).trans (v.map_eq_of_sub_lt <| lt_of_not_le ?_)
erw [← RingHom.map_sub, ← hr, hv.le_iff_dvd]
exact fun hprx =>
hx (Ideal.Q... |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
... | Mathlib/Probability/Martingale/Upcrossing.lean | 778 | 801 | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i ∈ Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by |
by_cases hN : N = 0
· simp [hN]
rw [← Finset.sum_Ico_consecutive _ (Nat.succ_le_succ zero_le')
(Nat.succ_le_succ (upcrossingsBefore_le f ω hab))]
have h₁ : ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1),
{n : ℕ | upperCrossingTime a b f N n ω < N}.indicator 1 k = 1 := by
rintro k hk
rw ... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.DeriveFintype
#align_import geometry.eucl... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 728 | 735 | theorem cospherical_iff_exists_mem_of_complete {s : AffineSubspace ℝ P} {ps : Set P} (h : ps ⊆ s)
[Nonempty s] [HasOrthogonalProjection s.direction] :
Cospherical ps ↔ ∃ center ∈ s, ∃ radius : ℝ, ∀ p ∈ ps, dist p center = radius := by |
constructor
· rintro ⟨c, hcr⟩
rw [exists_dist_eq_iff_exists_dist_orthogonalProjection_eq h c] at hcr
exact ⟨orthogonalProjection s c, orthogonalProjection_mem _, hcr⟩
· exact fun ⟨c, _, hd⟩ => ⟨c, hd⟩
|
/-
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.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib... | Mathlib/Data/List/Sublists.lean | 82 | 93 | 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]
constructor <;> intro h
· rcases h with (h | ⟨s, h, rfl⟩)
· exact sublist_cons_of_sublist _ h
· exact h.cons_con... |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENa... | Mathlib/RingTheory/Multiplicity.lean | 137 | 139 | theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
k = get (multiplicity a b) ⟨k, hsucc⟩ := by |
rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate f... | Mathlib/Data/List/Rotate.lean | 577 | 578 | theorem length_cyclicPermutations_of_ne_nil (l : List α) (h : l ≠ []) :
length (cyclicPermutations l) = length l := by | simp [cyclicPermutations_of_ne_nil _ h]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | Mathlib/Data/Prod/Lex.lean | 115 | 119 | theorem toLex_mono : Monotone (toLex : α × β → α ×ₗ β) := by |
rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ ⟨ha, hb⟩
obtain rfl | ha : a₁ = a₂ ∨ _ := ha.eq_or_lt
· exact right _ hb
· exact left _ _ ha
|
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.... | Mathlib/Algebra/GeomSum.lean | 375 | 384 | theorem geom_sum_Ico_le_of_lt_one [LinearOrderedField α] {x : α} (hx : 0 ≤ x) (h'x : x < 1)
{m n : ℕ} : ∑ i ∈ Ico m n, x ^ i ≤ x ^ m / (1 - x) := by |
rcases le_or_lt m n with (hmn | hmn)
· rw [geom_sum_Ico' h'x.ne hmn]
apply div_le_div (pow_nonneg hx _) _ (sub_pos.2 h'x) le_rfl
simpa using pow_nonneg hx _
· rw [Ico_eq_empty, sum_empty]
· apply div_nonneg (pow_nonneg hx _)
simpa using h'x.le
· simpa using hmn.le
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 310 | 312 | theorem support_eq_empty_iff {σ : Perm α} : σ.support = ∅ ↔ σ = 1 := by |
simp_rw [Finset.ext_iff, mem_support, Finset.not_mem_empty, iff_false_iff, not_not,
Equiv.Perm.ext_iff, one_apply]
|
/-
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 | 105 | 106 | theorem eval₂_bit1 : (bit1 p).eval₂ f x = bit1 (p.eval₂ f x) := by |
rw [bit1, eval₂_add, eval₂_bit0, eval₂_one, bit1]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
/-!
# Infinite sums and products in topological groups
Lemmas on topo... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 137 | 142 | theorem hasProd_ite_div_hasProd [DecidableEq β] (hf : HasProd f a) (b : β) :
HasProd (fun n ↦ ite (n = b) 1 (f n)) (a / f b) := by |
convert hf.update b 1 using 1
· ext n
rw [Function.update_apply]
· rw [div_mul_eq_mul_div, one_mul]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "lea... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 182 | 185 | theorem leftDistributor_inv_comp_biproduct_π {J : Type} [Fintype J] (X : C) (f : J → C) (j : J) :
(leftDistributor X f).inv ≫ (X ◁ biproduct.π _ j) = biproduct.π _ j := by |
simp [leftDistributor_inv, Preadditive.sum_comp, ← MonoidalCategory.whiskerLeft_comp,
biproduct.ι_π, whiskerLeft_dite, comp_dite]
|
/-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.RingTheory.AdicCompletion.Algebra
import Mathlib.Algebra.DirectSum.Bas... | Mathlib/RingTheory/AdicCompletion/Functoriality.lean | 344 | 346 | theorem piEquivFin_apply (n : ℕ) (x : AdicCompletion I (Fin n → R)) :
piEquivFin I n x = pi I (fun _ : Fin n ↦ R) x := by |
simp only [piEquivFin, piEquivOfFintype_apply]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
/-!
# Infinite sums and products over `ℕ` and `ℤ`
This file contains lemma... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 174 | 179 | theorem rel_sup_mul [CompleteLattice α] (m : α → M) (m0 : m ⊥ = 1) (R : M → M → Prop)
(m_iSup : ∀ s : ℕ → α, R (m (⨆ i, s i)) (∏' i, m (s i))) (s₁ s₂ : α) :
R (m (s₁ ⊔ s₂)) (m s₁ * m s₂) := by |
convert rel_iSup_tprod m m0 R m_iSup fun b ↦ cond b s₁ s₂
· simp only [iSup_bool_eq, cond]
· rw [tprod_fintype, Fintype.prod_bool, cond, cond]
|
/-
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.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | Mathlib/Analysis/Convex/Gauge.lean | 257 | 279 | theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set E)] {s : Set E} {a : α}
(ha : 0 ≤ a) (x : E) : gauge s (a • x) = a • gauge s x := by |
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_smul, gauge_zero, zero_smul]
rw [gauge_def', gauge_def', ← Real.sInf_smul_of_nonneg ha]
congr 1
ext r
simp_rw [Set.mem_smul_set, Set.mem_sep_iff]
constructor
· rintro ⟨hr, hx⟩
simp_rw [mem_Ioi] at hr ⊢
rw [← mem_smul_set_iff_inv_smul_mem₀ hr.ne'] at hx... |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 776 | 779 | theorem Orthonormal.inner_right_finsupp {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι →₀ 𝕜) (i : ι) :
⟪v i, Finsupp.total ι E 𝕜 v l⟫ = l i := by |
classical
simpa [Finsupp.total_apply, Finsupp.inner_sum, orthonormal_iff_ite.mp hv] using Eq.symm
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.RingTheory.Ideal.Quotient
#align_import topology.algebra.ring.ideal from "leanprover-community/mathlib"@"9a... | Mathlib/Topology/Algebra/Ring/Ideal.lean | 61 | 65 | theorem QuotientRing.isOpenMap_coe : IsOpenMap (mk N) := by |
intro s s_op
change IsOpen (mk N ⁻¹' (mk N '' s))
rw [quotient_ring_saturate]
exact isOpen_iUnion fun ⟨n, _⟩ => isOpenMap_add_left n s s_op
|
/-
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.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.Rat
#align_import data.rat.order from "leanprover-community/mathlib"@"a59dad53320b7... | Mathlib/Algebra/Order/Ring/Rat.lean | 240 | 246 | theorem div_lt_div_iff_mul_lt_mul {a b c d : ℤ} (b_pos : 0 < b) (d_pos : 0 < d) :
(a : ℚ) / b < c / d ↔ a * d < c * b := by |
simp only [lt_iff_le_not_le]
apply and_congr
· simp [div_def', Rat.divInt_le_divInt b_pos d_pos]
· apply not_congr
simp [div_def', Rat.divInt_le_divInt d_pos b_pos]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 2,079 | 2,084 | theorem isLittleO_pow_pow {m n : ℕ} (h : m < n) : (fun x : 𝕜 => x ^ n) =o[𝓝 0] fun x => x ^ m := by |
rcases lt_iff_exists_add.1 h with ⟨p, hp0 : 0 < p, rfl⟩
suffices (fun x : 𝕜 => x ^ m * x ^ p) =o[𝓝 0] fun x => x ^ m * 1 ^ p by
simpa only [pow_add, one_pow, mul_one]
exact IsBigO.mul_isLittleO (isBigO_refl _ _)
(IsLittleO.pow ((isLittleO_one_iff _).2 tendsto_id) hp0)
|
/-
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.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
#align_import set_theory.lists from "leanprover-community/mathlib"@"497d1e06409995dd8ec95301fa8... | Mathlib/SetTheory/Lists.lean | 99 | 99 | theorem to_ofList (l : List (Lists α)) : toList (ofList l) = l := by | induction l <;> simp [*]
|
/-
Copyright (c) 2022 Yuyang Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuyang Zhao
-/
import Mathlib.Algebra.Order.Floor
import Mathlib.Data.Nat.Prime
/-!
# Existence of a sufficiently large prime for which `a * c ^ p / (p - 1)! < 1`
This is a technical r... | Mathlib/Algebra/Order/Floor/Prime.lean | 22 | 34 | theorem exists_prime_mul_pow_lt_factorial [LinearOrderedRing K] [FloorRing K] (n : ℕ) (a c : K) :
∃ p > n, p.Prime ∧ a * c ^ p < (p - 1)! := by |
obtain ⟨p, pn, pp, h⟩ := n.exists_prime_mul_pow_lt_factorial ⌈|a|⌉.natAbs ⌈|c|⌉.natAbs
use p, pn, pp
calc a * c ^ p
_ ≤ |a * c ^ p| := le_abs_self _
_ ≤ ⌈|a|⌉ * (⌈|c|⌉ : K) ^ p := ?_
_ = ↑(Int.natAbs ⌈|a|⌉ * Int.natAbs ⌈|c|⌉ ^ p) := ?_
_ < ↑(p - 1)! := Nat.cast_lt.mpr h
· rw [abs_mul, abs_pow]
... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.GroupTheory.GroupAction.BigOperators
import Mathlib.LinearAlgebra.Prod
#align_import algebra.triv_sq_zero_ext f... | Mathlib/Algebra/TrivSqZeroExt.lean | 1,074 | 1,075 | theorem snd_map (f : M →ₗ[R'] N) (x : TrivSqZeroExt R' M) : snd (map f x) = f (snd x) := by |
simp [map, lift_def, Algebra.ofId_apply, algebraMap_eq_inl]
|
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 640 | 650 | theorem orderOf_root_cyclotomic_dvd {n : ℕ} (hpos : 0 < n) {p : ℕ} [Fact p.Prime] {a : ℕ}
(hroot : IsRoot (cyclotomic n (ZMod p)) (Nat.castRingHom (ZMod p) a)) :
orderOf (ZMod.unitOfCoprime a (coprime_of_root_cyclotomic hpos hroot)) ∣ n := by |
apply orderOf_dvd_of_pow_eq_one
suffices hpow : eval (Nat.castRingHom (ZMod p) a) (X ^ n - 1 : (ZMod p)[X]) = 0 by
simp only [eval_X, eval_one, eval_pow, eval_sub, eq_natCast] at hpow
apply Units.val_eq_one.1
simp only [sub_eq_zero.mp hpow, ZMod.coe_unitOfCoprime, Units.val_pow_eq_pow_val]
rw [IsRoot... |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mat... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 847 | 851 | theorem isBasis_basic_opens : TopologicalSpace.Opens.IsBasis (Set.range (@basicOpen R _)) := by |
unfold TopologicalSpace.Opens.IsBasis
convert isTopologicalBasis_basic_opens (R := R)
rw [← Set.range_comp]
rfl
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 158 | 159 | theorem toIocMod_add_toIocDiv_zsmul (a b : α) : toIocMod hp a b + toIocDiv hp a b • p = b := by |
rw [toIocMod, sub_add_cancel]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 231 | 238 | theorem of_measure_le {μ' : Measure α} (h : μ ≤ μ') (hT : DominatedFinMeasAdditive μ T C)
(hC : 0 ≤ C) : DominatedFinMeasAdditive μ' T C := by |
have h' : ∀ s, μ s = ∞ → μ' s = ∞ := fun s hs ↦ top_unique <| hs.symm.trans_le (h _)
refine ⟨hT.1.of_eq_top_imp_eq_top fun s _ ↦ h' s, fun s hs hμ's ↦ ?_⟩
have hμs : μ s < ∞ := (h s).trans_lt hμ's
calc
‖T s‖ ≤ C * (μ s).toReal := hT.2 s hs hμs
_ ≤ C * (μ' s).toReal := by gcongr; exacts [hμ's.ne, 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.Abelian.Basic
#align_import category_theory.idempotents.basic from "leanprover-community/mathlib"@"3a061790136d13594ec10c7c90d202335ac5d854"
/-!... | Mathlib/CategoryTheory/Idempotents/Basic.lean | 99 | 101 | theorem idem_of_id_sub_idem [Preadditive C] {X : C} (p : X ⟶ X) (hp : p ≫ p = p) :
(𝟙 _ - p) ≫ (𝟙 _ - p) = 𝟙 _ - p := by |
simp only [comp_sub, sub_comp, id_comp, comp_id, hp, sub_self, sub_zero]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 2,208 | 2,211 | theorem isBigO_pi {ι : Type*} [Fintype ι] {E' : ι → Type*} [∀ i, NormedAddCommGroup (E' i)]
{f : α → ∀ i, E' i} : f =O[l] g' ↔ ∀ i, (fun x => f x i) =O[l] g' := by |
simp only [isBigO_iff_eventually_isBigOWith, ← eventually_all]
exact eventually_congr (eventually_atTop.2 ⟨0, fun c => isBigOWith_pi⟩)
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.Extr
import Mathlib.Topology... | Mathlib/Analysis/Complex/AbsMax.lean | 369 | 382 | theorem exists_mem_frontier_isMaxOn_norm [FiniteDimensional ℂ E] {f : E → F} {U : Set E}
(hb : IsBounded U) (hne : U.Nonempty) (hd : DiffContOnCl ℂ f U) :
∃ z ∈ frontier U, IsMaxOn (norm ∘ f) (closure U) z := by |
have hc : IsCompact (closure U) := hb.isCompact_closure
obtain ⟨w, hwU, hle⟩ : ∃ w ∈ closure U, IsMaxOn (norm ∘ f) (closure U) w :=
hc.exists_isMaxOn hne.closure hd.continuousOn.norm
rw [closure_eq_interior_union_frontier, mem_union] at hwU
cases' hwU with hwU hwU; rotate_left; · exact ⟨w, hwU, hle⟩
have... |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Tactic.SeqFocus
/-! ## Ordering -/
namespace Ordering
@[simp] theorem swap_swap {o : Ordering} : o.swap.swap = o := by cases o <;> rfl
@[simp] th... | .lake/packages/batteries/Batteries/Classes/Order.lean | 163 | 166 | theorem compareOfLessAndEq_eq_lt {x y : α} [LT α] [Decidable (x < y)] [DecidableEq α] :
compareOfLessAndEq x y = .lt ↔ x < y := by |
simp [compareOfLessAndEq]
split <;> simp
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors ... | Mathlib/NumberTheory/Divisors.lean | 116 | 131 | theorem mem_divisorsAntidiagonal {x : ℕ × ℕ} :
x ∈ divisorsAntidiagonal n ↔ x.fst * x.snd = n ∧ n ≠ 0 := by |
simp only [divisorsAntidiagonal, Finset.mem_Ico, Ne, Finset.mem_filter, Finset.mem_product]
rw [and_comm]
apply and_congr_right
rintro rfl
constructor <;> intro h
· contrapose! h
simp [h]
· rw [Nat.lt_add_one_iff, Nat.lt_add_one_iff]
rw [mul_eq_zero, not_or] at h
simp only [succ_le_of_lt (Nat... |
/-
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.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "l... | Mathlib/Algebra/Polynomial/Monic.lean | 182 | 187 | theorem degree_mul_comm (hp : p.Monic) (q : R[X]) : (p * q).degree = (q * p).degree := by |
by_cases h : q = 0
· simp [h]
rw [degree_mul', hp.degree_mul]
· exact add_comm _ _
· rwa [hp.leadingCoeff, one_mul, leadingCoeff_ne_zero]
|
/-
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.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.... | Mathlib/LinearAlgebra/Determinant.lean | 424 | 427 | theorem LinearEquiv.isUnit_det (f : M ≃ₗ[R] M') (v : Basis ι R M) (v' : Basis ι R M') :
IsUnit (LinearMap.toMatrix v v' f).det := by |
apply isUnit_det_of_left_inverse
simpa using (LinearMap.toMatrix_comp v v' v f.symm f).symm
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate f... | Mathlib/Data/List/Rotate.lean | 49 | 49 | theorem rotate'_nil (n : ℕ) : ([] : List α).rotate' n = [] := by | cases n <;> rfl
|
/-
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
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 703 | 705 | theorem rpow_lt_one_of_one_lt_of_neg {x z : ℝ} (hx : 1 < x) (hz : z < 0) : x ^ z < 1 := by |
convert rpow_lt_rpow_of_exponent_lt hx hz
exact (rpow_zero x).symm
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Sylow
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schur_zassenhaus from "leanprover-community/mathlib"@"d57133e49cf06... | Mathlib/GroupTheory/SchurZassenhaus.lean | 287 | 304 | theorem exists_right_complement'_of_coprime {N : Subgroup G} [N.Normal]
(hN : Nat.Coprime (Nat.card N) N.index) : ∃ H : Subgroup G, IsComplement' N H := by |
by_cases hN1 : Nat.card N = 0
· rw [hN1, Nat.coprime_zero_left, index_eq_one] at hN
rw [hN]
exact ⟨⊥, isComplement'_top_bot⟩
by_cases hN2 : N.index = 0
· rw [hN2, Nat.coprime_zero_right] at hN
haveI := (Cardinal.toNat_eq_one_iff_unique.mp hN).1
rw [N.eq_bot_of_subsingleton]
exact ⟨⊤, isComp... |
/-
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.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Ri... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 636 | 640 | theorem LinearMap.toMatrix_reindexRange [DecidableEq M₁] (f : M₁ →ₗ[R] M₂) (k : m) (i : n) :
LinearMap.toMatrix v₁.reindexRange v₂.reindexRange f ⟨v₂ k, Set.mem_range_self k⟩
⟨v₁ i, Set.mem_range_self i⟩ =
LinearMap.toMatrix v₁ v₂ f k i := by |
simp_rw [LinearMap.toMatrix_apply, Basis.reindexRange_self, Basis.reindexRange_repr]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathl... | Mathlib/Algebra/MvPolynomial/Basic.lean | 539 | 542 | theorem support_monomial [h : Decidable (a = 0)] :
(monomial s a).support = if a = 0 then ∅ else {s} := by |
rw [← Subsingleton.elim (Classical.decEq R a 0) h]
rfl
|
/-
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, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 796 | 796 | theorem div_mul_eq_mul_div : a / b * c = a * c / b := by | simp
|
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semicon... | Mathlib/Data/Int/GCD.lean | 80 | 82 | theorem gcdB_zero_left {s : ℕ} : gcdB 0 s = 1 := by |
unfold gcdB
rw [xgcd, xgcd_zero_left]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_imp... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 297 | 312 | theorem sq_add_sq_of_nat_prime_of_not_irreducible (p : ℕ) [hp : Fact p.Prime]
(hpi : ¬Irreducible (p : ℤ[i])) : ∃ a b, a ^ 2 + b ^ 2 = p :=
have hpu : ¬IsUnit (p : ℤ[i]) :=
mt norm_eq_one_iff.2 <| by
rw [norm_natCast, Int.natAbs_mul, mul_eq_one]
exact fun h => (ne_of_lt hp.1.one_lt).symm h.1
hav... |
-- Porting note: was
-- simpa [irreducible_iff, hpu, not_forall, not_or] using hpi
simpa only [true_and, not_false_iff, exists_prop, irreducible_iff, hpu, not_forall, not_or]
using hpi
let ⟨a, b, hpab, hau, hbu⟩ := hab
have hnap : (norm a).natAbs = p :=
((hp.1.mul_eq_prime_sq_iff (mt norm_eq_... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | Mathlib/Data/Int/Cast/Basic.lean | 137 | 138 | theorem cast_bit1 (n : ℤ) : ((bit1 n : ℤ) : R) = bit1 (n : R) := by |
rw [bit1, Int.cast_add, Int.cast_one, cast_bit0]; rfl
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsymptoti... | Mathlib/Analysis/SpecialFunctions/Polynomials.lean | 218 | 223 | theorem abs_div_tendsto_atTop_of_degree_gt (hdeg : Q.degree < P.degree) (hQ : Q ≠ 0) :
Tendsto (fun x => |eval x P / eval x Q|) atTop atTop := by |
by_cases h : 0 ≤ P.leadingCoeff / Q.leadingCoeff
· exact tendsto_abs_atTop_atTop.comp (P.div_tendsto_atTop_of_degree_gt Q hdeg hQ h)
· push_neg at h
exact tendsto_abs_atBot_atTop.comp (P.div_tendsto_atBot_of_degree_gt Q hdeg hQ h.le)
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-... | Mathlib/Order/LiminfLimsup.lean | 953 | 962 | theorem CompleteLatticeHom.apply_limsup_iterate (f : CompleteLatticeHom α α) (a : α) :
f (limsup (fun n => f^[n] a) atTop) = limsup (fun n => f^[n] a) atTop := by |
rw [limsup_eq_iInf_iSup_of_nat', map_iInf]
simp_rw [_root_.map_iSup, ← Function.comp_apply (f := f), ← Function.iterate_succ' f,
← Nat.add_succ]
conv_rhs => rw [iInf_split _ (0 < ·)]
simp only [not_lt, Nat.le_zero, iInf_iInf_eq_left, add_zero, iInf_nat_gt_zero_eq, left_eq_inf]
refine (iInf_le (fun i => ⨆... |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Geometry.Euclidean.Inversion.Basic
import Mathlib.Geometry.Euclidean.PerpBisector
/-!
# Image of a hyperplane under inversion
In this file we prove... | Mathlib/Geometry/Euclidean/Inversion/ImageHyperplane.lean | 73 | 76 | theorem image_inversion_sphere_dist_center (hR : R ≠ 0) (hy : y ≠ c) :
inversion c R '' sphere y (dist y c) = insert c (perpBisector c (inversion c R y) : Set P) := by |
rw [image_eq_preimage_of_inverse (inversion_involutive _ hR) (inversion_involutive _ hR),
preimage_inversion_sphere_dist_center hR hy]
|
/-
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, Jeremy Avigad
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@... | Mathlib/Topology/Basic.lean | 164 | 167 | theorem TopologicalSpace.ext_iff_isClosed {t₁ t₂ : TopologicalSpace X} :
t₁ = t₂ ↔ ∀ s, IsClosed[t₁] s ↔ IsClosed[t₂] s := by |
rw [TopologicalSpace.ext_iff, compl_surjective.forall]
simp only [@isOpen_compl_iff _ _ t₁, @isOpen_compl_iff _ _ t₂]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.linear_pmap from "leanprover-community/mathlib"@"8b981918a93bc45a8600de608cde7944a80d92... | Mathlib/LinearAlgebra/LinearPMap.lean | 1,081 | 1,084 | theorem inverse_domain : (inverse f).domain = LinearMap.range f.toFun := by |
rw [inverse, Submodule.toLinearPMap_domain, ← graph_map_snd_eq_range,
← LinearEquiv.fst_comp_prodComm, Submodule.map_comp]
rfl
|
/-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
imp... | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 305 | 310 | theorem colorable_chromaticNumber {m : ℕ} (hc : G.Colorable m) :
G.Colorable (ENat.toNat G.chromaticNumber) := by |
classical
rw [hc.chromaticNumber_eq_sInf, Nat.sInf_def]
· apply Nat.find_spec
· exact colorable_set_nonempty_of_colorable hc
|
/-
Copyright (c) 2022 Rémy Degenne, Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.MeasureTheory.Function.Egorov
import Mathlib.MeasureTheory.Function.LpSpace
#a... | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | 178 | 182 | theorem seqTendstoAeSeq_strictMono (hfg : TendstoInMeasure μ f atTop g) :
StrictMono (seqTendstoAeSeq hfg) := by |
refine strictMono_nat_of_lt_succ fun n => ?_
rw [seqTendstoAeSeq_succ]
exact lt_of_lt_of_le (lt_add_one <| seqTendstoAeSeq hfg n) (le_max_right _ _)
|
/-
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.Init.Data.Ordering.Basic
import Mathlib.Order.Synonym
#align_import order.compare from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb... | Mathlib/Order/Compare.lean | 167 | 168 | theorem cmp_compares [LinearOrder α] (a b : α) : (cmp a b).Compares a b := by |
obtain h | h | h := lt_trichotomy a b <;> simp [cmp, cmpUsing, h, h.not_lt]
|
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.... | Mathlib/Algebra/GeomSum.lean | 175 | 179 | theorem Commute.mul_neg_geom_sum₂ [Ring α] {x y : α} (h : Commute x y) (n : ℕ) :
((y - x) * ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i)) = y ^ n - x ^ n := by |
apply op_injective
simp only [op_mul, op_sub, op_geom_sum₂, op_pow]
simp [(Commute.op h.symm).geom_sum₂_mul n]
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 116 | 117 | theorem encard_insert_of_not_mem {a : α} (has : a ∉ s) : (insert a s).encard = s.encard + 1 := by |
rw [← union_singleton, encard_union_eq (by simpa), encard_singleton]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 347 | 348 | theorem span_eq_iSup_of_singleton_spans (s : Set M) : span R s = ⨆ x ∈ s, R ∙ x := by |
simp only [← span_iUnion, Set.biUnion_of_singleton s]
|
/-
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.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.... | Mathlib/Algebra/Order/Interval/Set/Group.lean | 219 | 221 | theorem pairwise_disjoint_Ico_zpow :
Pairwise (Disjoint on fun n : ℤ => Ico (b ^ n) (b ^ (n + 1))) := by |
simpa only [one_mul] using pairwise_disjoint_Ico_mul_zpow 1 b
|
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