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) 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 | 495 | 496 | theorem mem_restrict' : J₁ ∈ π.restrict J ↔ ∃ J' ∈ π, (J₁ : Set (ι → ℝ)) = ↑J ∩ ↑J' := by |
simp only [mem_restrict, ← Box.withBotCoe_inj, Box.coe_inf, Box.coe_coe]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriente... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 58 | 63 | theorem continuousAt_oangle {x : V × V} (hx1 : x.1 ≠ 0) (hx2 : x.2 ≠ 0) :
ContinuousAt (fun y : V × V => o.oangle y.1 y.2) x := by |
refine (Complex.continuousAt_arg_coe_angle ?_).comp ?_
· exact o.kahler_ne_zero hx1 hx2
exact ((continuous_ofReal.comp continuous_inner).add
((continuous_ofReal.comp o.areaForm'.continuous₂).mul continuous_const)).continuousAt
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Scott Carnahan
-/
import Mathlib.RingTheory.HahnSeries.Addition
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Data.Finset.MulAntidiagonal
#align_impor... | Mathlib/RingTheory/HahnSeries/Multiplication.lean | 294 | 295 | theorem mul_single_zero_coeff [NonUnitalNonAssocSemiring R] {r : R} {x : HahnSeries Γ R} {a : Γ} :
(x * single 0 r).coeff a = x.coeff a * r := by | rw [← add_zero a, mul_single_coeff_add, add_zero]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanpro... | Mathlib/RingTheory/Ideal/Operations.lean | 847 | 848 | theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by |
rw [← add_eq_one_iff, isCoprime_iff_add]
|
/-
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 | 173 | 183 | theorem mapAccumr_bisim {f₁ : α → σ₁ → σ₁ × β} {f₂ : α → σ₂ → σ₂ × β} {s₁ : σ₁} {s₂ : σ₂}
(R : σ₁ → σ₂ → Prop) (h₀ : R s₁ s₂)
(hR : ∀ {s q} a, R s q → R (f₁ a s).1 (f₂ a q).1 ∧ (f₁ a s).2 = (f₂ a q).2) :
R (mapAccumr f₁ xs s₁).fst (mapAccumr f₂ xs s₂).fst
∧ (mapAccumr f₁ xs s₁).snd = (mapAccumr f₂ xs s₂... |
induction xs using Vector.revInductionOn generalizing s₁ s₂
next => exact ⟨h₀, rfl⟩
next xs x ih =>
rcases (hR x h₀) with ⟨hR, _⟩
simp only [mapAccumr_snoc, ih hR, true_and]
congr 1
|
/-
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.Filter.Prod
#align_import order.filter.n_ary from "leanprover-community/mathlib"@"78f647f8517f021d839a7553d5dc97e79b508dea"
/-!
# N-ary maps of fil... | Mathlib/Order/Filter/NAry.lean | 146 | 146 | theorem map₂_pure : map₂ m (pure a) (pure b) = pure (m a b) := by | rw [map₂_pure_right, map_pure]
|
/-
Copyright (c) 2018 . All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import... | Mathlib/GroupTheory/PGroup.lean | 368 | 373 | theorem card_center_eq_prime_pow (hn : 0 < n) [Fintype (center G)] :
∃ k > 0, card (center G) = p ^ k := by |
have hcG := to_subgroup (of_card hGpn) (center G)
rcases iff_card.1 hcG with _
haveI : Nontrivial G := (nontrivial_iff_card <| of_card hGpn).2 ⟨n, hn, hGpn⟩
exact (nontrivial_iff_card hcG).mp (center_nontrivial (of_card hGpn))
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Sub... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 722 | 730 | theorem mul_right_mem_add_closure (ha : a ∈ AddSubmonoid.closure (S : Set R)) (hb : b ∈ S) :
a * b ∈ AddSubmonoid.closure (S : Set R) := by |
revert b
apply @AddSubmonoid.closure_induction _ _ _
(fun z => ∀ (b : R), b ∈ S → z * b ∈ AddSubmonoid.closure S) _ ha <;> clear ha a
· exact fun r hr b hb => AddSubmonoid.mem_closure.mpr fun y hy => hy (mul_mem hr hb)
· exact fun b _ => by simp only [zero_mul, (AddSubmonoid.closure (S : Set R)).zero_mem]
... |
/-
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 | 496 | 497 | theorem inter_nonempty_iff_exists_right : (s ∩ t).Nonempty ↔ ∃ x ∈ t, x ∈ s := by |
simp_rw [inter_nonempty, and_comm]
|
/-
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 | 984 | 985 | theorem denom_algebraMap (p : K[X]) : denom (algebraMap _ (RatFunc K) p) = 1 := by |
convert denom_div p one_ne_zero <;> simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 1,902 | 1,904 | theorem EventuallyLE.sup_le [SemilatticeSup β] {l : Filter α} {f g h : α → β} (hf : f ≤ᶠ[l] h)
(hg : g ≤ᶠ[l] h) : f ⊔ g ≤ᶠ[l] h := by |
filter_upwards [hf, hg] with x hfx hgx using _root_.sup_le hfx hgx
|
/-
Copyright (c) 2024 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
/-!
# Final functors wit... | Mathlib/CategoryTheory/Filtered/Final.lean | 160 | 165 | theorem Functor.initial_of_exists_of_isCofiltered_of_fullyFaithful [IsCofilteredOrEmpty D] [F.Full]
[Faithful F] (h : ∀ d, ∃ c, Nonempty (F.obj c ⟶ d)) : Initial F := by |
suffices Final F.op from initial_of_final_op _
refine Functor.final_of_exists_of_isFiltered_of_fullyFaithful F.op (fun d => ?_)
obtain ⟨c, ⟨f⟩⟩ := h d.unop
exact ⟨op c, ⟨f.op⟩⟩
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib... | Mathlib/Algebra/Polynomial/RingDivision.lean | 504 | 508 | theorem mem_nonZeroDivisors_of_leadingCoeff {p : R[X]} (h : p.leadingCoeff ∈ R⁰) : p ∈ R[X]⁰ := by |
refine mem_nonZeroDivisors_iff.2 fun x hx ↦ leadingCoeff_eq_zero.1 ?_
by_contra hx'
rw [← mul_right_mem_nonZeroDivisors_eq_zero_iff h] at hx'
simp only [← leadingCoeff_mul' hx', hx, leadingCoeff_zero, not_true] at hx'
|
/-
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 | 392 | 398 | theorem ContDiffWithinAt.comp_continuousLinearMap {x : G} (g : G →L[𝕜] E)
(hf : ContDiffWithinAt 𝕜 n f s (g x)) : ContDiffWithinAt 𝕜 n (f ∘ g) (g ⁻¹' s) x := by |
intro m hm
rcases hf m hm with ⟨u, hu, p, hp⟩
refine ⟨g ⁻¹' u, ?_, _, hp.compContinuousLinearMap g⟩
refine g.continuous.continuousWithinAt.tendsto_nhdsWithin ?_ hu
exact (mapsTo_singleton.2 <| mem_singleton _).union_union (mapsTo_preimage _ _)
|
/-
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 | 967 | 971 | theorem smul_leftMulMatrix_algebraMap (x : S) :
leftMulMatrix (b.smul c) (algebraMap _ _ x) = blockDiagonal fun _ ↦ leftMulMatrix b x := by |
ext ⟨i, k⟩ ⟨j, k'⟩
rw [smul_leftMulMatrix, AlgHom.commutes, blockDiagonal_apply, algebraMap_matrix_apply]
split_ifs with h <;> simp only at h <;> simp [h]
|
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Junyan Xu
-/
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
/-!
# Lo... | Mathlib/Data/DFinsupp/NeLocus.lean | 145 | 146 | theorem neLocus_eq_support_sub : f.neLocus g = (f - g).support := by |
rw [← @neLocus_add_right α N _ _ _ _ _ (-g), add_right_neg, neLocus_zero_right, sub_eq_add_neg]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f6856... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 645 | 646 | theorem mfderiv_congr_point {x' : M} (h : x = x') :
@Eq (E →L[𝕜] E') (mfderiv I I' f x) (mfderiv I I' f x') := by | subst h; rfl
|
/-
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.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Ring.Pi
import Ma... | Mathlib/Algebra/Order/CauSeq/Basic.lean | 882 | 890 | theorem inf_equiv_inf {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊓ b₁ ≈ a₂ ⊓ b₂ := by |
intro ε ε0
obtain ⟨ai, hai⟩ := ha ε ε0
obtain ⟨bi, hbi⟩ := hb ε ε0
exact
⟨ai ⊔ bi, fun i hi =>
(abs_min_sub_min_le_max (a₁ i) (b₁ i) (a₂ i) (b₂ i)).trans_lt
(max_lt (hai i (sup_le_iff.mp hi).1) (hbi i (sup_le_iff.mp hi).2))⟩
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yakov Pechersky, Eric Wieser
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.enum`
-/
namespace List
variable {α β : Type*}
#align list.length_enum_from... | Mathlib/Data/List/Enum.lean | 72 | 73 | theorem mk_mem_enum_iff_get? {i : ℕ} {x : α} {l : List α} : (i, x) ∈ enum l ↔ l.get? i = x := by |
simp [enum, mk_mem_enumFrom_iff_le_and_get?_sub]
|
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer
-/
import Mathlib.Algebra.Category.MonCat.Limits
import Mathlib.CategoryTheory.Limits.Preserves.Filtered
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
imp... | Mathlib/Algebra/Category/MonCat/FilteredColimits.lean | 95 | 98 | theorem colimit_one_eq (j : J) : (1 : M.{v, u} F) = M.mk F ⟨j, 1⟩ := by |
apply M.mk_eq
refine ⟨max' _ j, IsFiltered.leftToMax _ j, IsFiltered.rightToMax _ j, ?_⟩
simp
|
/-
Copyright (c) 2022 Siddhartha Prasad, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Siddhartha Prasad, Yaël Dillies
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Ring.Prod
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Tactic.Monotonici... | Mathlib/Algebra/Order/Kleene.lean | 163 | 163 | theorem add_le_iff : a + b ≤ c ↔ a ≤ c ∧ b ≤ c := by | simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGrou... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,816 | 1,826 | theorem exists_disjoint_smul_of_isCompact [NoncompactSpace G] {K L : Set G} (hK : IsCompact K)
(hL : IsCompact L) : ∃ g : G, Disjoint K (g • L) := by |
have A : ¬K * L⁻¹ = univ := (hK.mul hL.inv).ne_univ
obtain ⟨g, hg⟩ : ∃ g, g ∉ K * L⁻¹ := by
contrapose! A
exact eq_univ_iff_forall.2 A
refine ⟨g, ?_⟩
refine disjoint_left.2 fun a ha h'a => hg ?_
rcases h'a with ⟨b, bL, rfl⟩
refine ⟨g * b, ha, b⁻¹, by simpa only [Set.mem_inv, inv_inv] using bL, ?_⟩
... |
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
/-!
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 60 | 62 | theorem fixedBy_inv (g : G) : fixedBy α g⁻¹ = fixedBy α g := by |
ext
rw [mem_fixedBy, mem_fixedBy, inv_smul_eq_iff, eq_comm]
|
/-
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 | 217 | 220 | theorem C_mul_monomial : C a * monomial s a' = monomial s (a * a') := by |
-- Porting note: this `show` feels like defeq abuse, but I can't find the appropriate lemmas
show AddMonoidAlgebra.single _ _ * AddMonoidAlgebra.single _ _ = AddMonoidAlgebra.single _ _
simp [C_apply, single_mul_single]
|
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel, Yury Kudryashov, Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.c... | Mathlib/Analysis/Calculus/Deriv/Add.lean | 346 | 348 | theorem derivWithin_sub_const (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) :
derivWithin (fun y => f y - c) s x = derivWithin f s x := by |
simp only [derivWithin, fderivWithin_sub_const hxs]
|
/-
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.Nat.Defs
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Co... | Mathlib/Data/Fin/Basic.lean | 1,219 | 1,221 | theorem castPred_inj {i j : Fin (n + 1)} {hi : i ≠ last n} {hj : j ≠ last n} :
castPred i hi = castPred j hj ↔ i = j := by |
simp_rw [ext_iff, le_antisymm_iff, ← le_def, castPred_le_castPred_iff]
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Eric Wieser
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat... | Mathlib/Analysis/NormedSpace/Exponential.lean | 414 | 424 | theorem expSeries_radius_eq_top : (expSeries 𝕂 𝔸).radius = ∞ := by |
refine (expSeries 𝕂 𝔸).radius_eq_top_of_summable_norm fun r => ?_
refine .of_norm_bounded_eventually _ (Real.summable_pow_div_factorial r) ?_
filter_upwards [eventually_cofinite_ne 0] with n hn
rw [norm_mul, norm_norm (expSeries 𝕂 𝔸 n), expSeries]
rw [norm_smul (n ! : 𝕂)⁻¹ (ContinuousMultilinearMap.mkPi... |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Mea... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 798 | 804 | theorem add_prod (μ' : Measure α) [SFinite μ'] : (μ + μ').prod ν = μ.prod ν + μ'.prod ν := by |
simp_rw [← sum_sFiniteSeq μ, ← sum_sFiniteSeq μ', sum_add_sum, ← sum_sFiniteSeq ν, prod_sum,
sum_add_sum]
congr
ext1 i
refine prod_eq fun s t _ _ => ?_
simp_rw [add_apply, prod_prod, right_distrib]
|
/-
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.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Alg... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 141 | 143 | theorem wittPolynomial_one : wittPolynomial p R 1 = C (p : R) * X 1 + X 0 ^ p := by |
simp only [wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ_comm, range_one, sum_singleton,
one_mul, pow_one, C_1, pow_zero, tsub_self, tsub_zero]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7... | Mathlib/Data/Seq/WSeq.lean | 1,064 | 1,065 | theorem exists_of_liftRel_right {R : α → β → Prop} {s t} (H : LiftRel R s t) {b} (h : b ∈ t) :
∃ a, a ∈ s ∧ R a b := by | rw [← LiftRel.swap] at H; exact exists_of_liftRel_left H h
|
/-
Copyright (c) 2023 Joachim Breitner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joachim Breitner
-/
import Mathlib.Probability.ProbabilityMassFunction.Constructions
import Mathlib.Tactic.FinCases
/-!
# The binomial distribution
This file defines the probabilit... | Mathlib/Probability/ProbabilityMassFunction/Binomial.lean | 49 | 50 | theorem binomial_apply_self (p : ℝ≥0∞) (h : p ≤ 1) (n : ℕ) :
binomial p h n n = p^n := by | simp
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
#align_import data.int.bitwise from "leanprover-community/mathlib"@"0743cc... | Mathlib/Data/Int/Bitwise.lean | 480 | 480 | theorem shiftRight_coe_nat (m n : ℕ) : (m : ℤ) >>> (n : ℤ) = m >>> n := by | cases n <;> rfl
|
/-
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
#align_import data.finset.pimage from "leanprover-community/mathlib"@"f7fc89d5d... | Mathlib/Data/Finset/PImage.lean | 76 | 79 | 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 |
ext
simp [eq_comm]
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.... | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | 158 | 184 | theorem isAdic_iff [top : TopologicalSpace R] [TopologicalRing R] {J : Ideal R} :
IsAdic J ↔
(∀ n : ℕ, IsOpen ((J ^ n : Ideal R) : Set R)) ∧
∀ s ∈ 𝓝 (0 : R), ∃ n : ℕ, ((J ^ n : Ideal R) : Set R) ⊆ s := by |
constructor
· intro H
change _ = _ at H
rw [H]
letI := J.adicTopology
constructor
· intro n
exact (J.openAddSubgroup n).isOpen'
· intro s hs
simpa using J.hasBasis_nhds_zero_adic.mem_iff.mp hs
· rintro ⟨H₁, H₂⟩
apply TopologicalAddGroup.ext
· apply @TopologicalRing.to_... |
/-
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.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "lean... | Mathlib/Analysis/Convex/Side.lean | 214 | 215 | theorem sOppSide_comm {s : AffineSubspace R P} {x y : P} : s.SOppSide x y ↔ s.SOppSide y x := by |
rw [SOppSide, SOppSide, wOppSide_comm, and_comm (b := x ∉ s)]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 292 | 296 | theorem TendstoUniformly.prod_map {ι' α' β' : Type*} [UniformSpace β'] {F' : ι' → α' → β'}
{f' : α' → β'} {p' : Filter ι'} (h : TendstoUniformly F f p) (h' : TendstoUniformly F' f' p') :
TendstoUniformly (fun i : ι × ι' => Prod.map (F i.1) (F' i.2)) (Prod.map f f') (p ×ˢ p') := by |
rw [← tendstoUniformlyOn_univ, ← univ_prod_univ] at *
exact h.prod_map h'
|
/-
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 | 731 | 745 | theorem Monic.irreducible_iff_degree_lt {p : R[X]} (p_monic : Monic p) (p_1 : p ≠ 1) :
Irreducible p ↔ ∀ q, degree q ≤ ↑(p.natDegree / 2) → q ∣ p → IsUnit q := by |
simp only [p_monic.irreducible_iff_lt_natDegree_lt p_1, Finset.mem_Ioc, and_imp,
natDegree_pos_iff_degree_pos, natDegree_le_iff_degree_le]
constructor
· rintro h q deg_le dvd
by_contra q_unit
have := degree_pos_of_not_isUnit_of_dvd_monic q_unit dvd p_monic
have hu := p_monic.isUnit_leadingCoeff_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
-/
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 2,007 | 2,009 | theorem mem_ae_map_iff {f : α → β} (hf : AEMeasurable f μ) {s : Set β} (hs : MeasurableSet s) :
s ∈ ae (μ.map f) ↔ f ⁻¹' s ∈ ae μ := by |
simp only [mem_ae_iff, map_apply_of_aemeasurable hf hs.compl, preimage_compl]
|
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.List.ProdSigma
#align_import data.fin_enum from "leanprover-community/mathlib"@"90... | Mathlib/Data/FinEnum.lean | 74 | 75 | theorem nodup_toList [FinEnum α] : List.Nodup (toList α) := by |
simp [toList]; apply List.Nodup.map <;> [apply Equiv.injective; apply List.nodup_finRange]
|
/-
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
import Batteries.Data.List.Lemmas
import Batteries.Data.List.Init.Attach
namespace Std.Range
/-- The number of elements contained i... | .lake/packages/batteries/Batteries/Data/Range/Lemmas.lean | 26 | 27 | theorem numElems_step_1 (start stop) : numElems ⟨start, stop, 1⟩ = stop - start := by |
simp [numElems]
|
/-
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 | 545 | 558 | theorem floor_div_nat (a : α) (n : ℕ) : ⌊a / n⌋₊ = ⌊a⌋₊ / n := by |
rcases le_total a 0 with ha | ha
· rw [floor_of_nonpos, floor_of_nonpos ha]
· simp
apply div_nonpos_of_nonpos_of_nonneg ha n.cast_nonneg
obtain rfl | hn := n.eq_zero_or_pos
· rw [cast_zero, div_zero, Nat.div_zero, floor_zero]
refine (floor_eq_iff ?_).2 ?_
· exact div_nonneg ha n.cast_nonneg
const... |
/-
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.Data.Part
import Mathlib.Data.Rel
#align_import data.pfun from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f70... | Mathlib/Data/PFun.lean | 450 | 450 | theorem coe_preimage (f : α → β) (s : Set β) : (f : α →. β).preimage s = f ⁻¹' s := by | ext; simp
|
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Hom.End
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Algebra.SMulWithZero
imp... | Mathlib/Algebra/Module/Defs.lean | 529 | 531 | theorem two_nsmul_eq_zero {v : M} : 2 • v = 0 ↔ v = 0 := by |
haveI := Nat.noZeroSMulDivisors R M
simp [smul_eq_zero]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Ma... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 420 | 423 | theorem mapAction_μ_inv_hom (X Y : Action V G) :
(inv ((F.mapAction G).μ X Y)).hom = inv (F.μ X.V Y.V) := by |
rw [← cancel_mono (F.μ X.V Y.V), IsIso.inv_hom_id, ← F.mapAction_toLaxMonoidalFunctor_μ_hom G,
← Action.comp_hom, IsIso.inv_hom_id, Action.id_hom]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Category.ModuleCat.Adjunctions
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.Algebra.Category.ModuleCat.Colimits
import Mathl... | Mathlib/RepresentationTheory/Rep.lean | 200 | 202 | theorem linearization_single (X : Action (Type u) (MonCat.of G)) (g : G) (x : X.V) (r : k) :
((linearization k G).obj X).ρ g (Finsupp.single x r) = Finsupp.single (X.ρ g x) r := by |
rw [linearization_obj_ρ, Finsupp.lmapDomain_apply, Finsupp.mapDomain_single]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Option.NAry
import Mathlib.Data.Seq.Computation
#align_import data.seq.seq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b34... | Mathlib/Data/Seq/Seq.lean | 1,006 | 1,027 | theorem join_join (SS : Seq (Seq1 (Seq1 α))) :
Seq.join (Seq.join SS) = Seq.join (Seq.map join SS) := by |
apply
Seq.eq_of_bisim fun s1 s2 =>
∃ s SS,
s1 = Seq.append s (Seq.join (Seq.join SS)) ∧ s2 = Seq.append s (Seq.join (Seq.map join SS))
· intro s1 s2 h
exact
match s1, s2, h with
| _, _, ⟨s, SS, rfl, rfl⟩ => by
apply recOn s <;> simp
· apply recOn SS <;> simp
... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Da... | Mathlib/GroupTheory/OrderOfElement.lean | 1,159 | 1,162 | theorem image_range_orderOf [DecidableEq G] :
Finset.image (fun i => x ^ i) (Finset.range (orderOf x)) = (zpowers x : Set G).toFinset := by |
ext x
rw [Set.mem_toFinset, SetLike.mem_coe, mem_zpowers_iff_mem_range_orderOf]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 1,070 | 1,073 | theorem toSpanSingleton_isIdempotentElem_iff {e : R} :
IsIdempotentElem (toSpanSingleton R R e) ↔ IsIdempotentElem e := by |
simp_rw [IsIdempotentElem, ext_iff, mul_apply, toSpanSingleton_apply, smul_eq_mul, mul_assoc]
exact ⟨fun h ↦ by conv_rhs => rw [← one_mul e, ← h, one_mul], fun h _ ↦ by rw [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.Group.Even
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Gr... | Mathlib/Algebra/Associated.lean | 1,206 | 1,211 | theorem one_or_eq_of_le_of_prime {p m : Associates α} (hp : Prime p) (hle : m ≤ p) :
m = 1 ∨ m = p := by |
rcases mk_surjective p with ⟨p, rfl⟩
rcases mk_surjective m with ⟨m, rfl⟩
simpa [mk_eq_mk_iff_associated, Associated.comm, -Quotient.eq]
using (prime_mk.1 hp).irreducible.dvd_iff.mp (mk_le_mk_iff_dvd.1 hle)
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Mea... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 155 | 172 | theorem measurable_measure_prod_mk_left_finite [IsFiniteMeasure ν] {s : Set (α × β)}
(hs : MeasurableSet s) : Measurable fun x => ν (Prod.mk x ⁻¹' s) := by |
refine induction_on_inter (C := fun s => Measurable fun x => ν (Prod.mk x ⁻¹' s))
generateFrom_prod.symm isPiSystem_prod ?_ ?_ ?_ ?_ hs
· simp
· rintro _ ⟨s, hs, t, _, rfl⟩
simp only [mk_preimage_prod_right_eq_if, measure_if]
exact measurable_const.indicator hs
· intro t ht h2t
simp_rw [preimag... |
/-
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.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 806 | 809 | theorem sup'_cons {b : β} {hb : b ∉ s} :
(cons b s hb).sup' (nonempty_cons hb) f = f b ⊔ s.sup' H f := by |
rw [← WithBot.coe_eq_coe]
simp [WithBot.coe_sup]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Interval.Finset
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tactic.Linarith
... | Mathlib/Algebra/BigOperators/Intervals.lean | 143 | 145 | theorem prod_Ioc_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → M) :
(∏ k ∈ Ioc a (b + 1), f k) = (∏ k ∈ Ioc a b, f k) * f (b + 1) := by |
rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b), Nat.Ioc_succ_singleton, prod_singleton]
|
/-
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.Order.RelIso.Set
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintyp... | Mathlib/Data/Finset/Sort.lean | 97 | 106 | theorem sorted_zero_eq_min'_aux (s : Finset α) (h : 0 < (s.sort (· ≤ ·)).length) (H : s.Nonempty) :
(s.sort (· ≤ ·)).get ⟨0, h⟩ = s.min' H := by |
let l := s.sort (· ≤ ·)
apply le_antisymm
· have : s.min' H ∈ l := (Finset.mem_sort (α := α) (· ≤ ·)).mpr (s.min'_mem H)
obtain ⟨i, hi⟩ : ∃ i, l.get i = s.min' H := List.mem_iff_get.1 this
rw [← hi]
exact (s.sort_sorted (· ≤ ·)).rel_get_of_le (Nat.zero_le i)
· have : l.get ⟨0, h⟩ ∈ s := (Finset.mem... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Jireh Loreaux
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Data.Fintype.Order
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.Analysis.Norm... | Mathlib/Analysis/NormedSpace/PiLp.lean | 345 | 351 | theorem iSup_edist_ne_top_aux {ι : Type*} [Finite ι] {α : ι → Type*}
[∀ i, PseudoMetricSpace (α i)] (f g : PiLp ∞ α) : (⨆ i, edist (f i) (g i)) ≠ ⊤ := by |
cases nonempty_fintype ι
obtain ⟨M, hM⟩ := Finite.exists_le fun i => (⟨dist (f i) (g i), dist_nonneg⟩ : ℝ≥0)
refine ne_of_lt ((iSup_le fun i => ?_).trans_lt (@ENNReal.coe_lt_top M))
simp only [edist, PseudoMetricSpace.edist_dist, ENNReal.ofReal_eq_coe_nnreal dist_nonneg]
exact mod_cast hM i
|
/-
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.Ker
#align_import linear_algebra.basic fr... | Mathlib/Algebra/Module/Submodule/Range.lean | 268 | 270 | theorem range_smul' (f : V →ₗ[K] V₂) (a : K) :
range (a • f) = ⨆ _ : a ≠ 0, range f := by |
simpa only [range_eq_map] using Submodule.map_smul' f _ a
|
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Sqrt
#al... | Mathlib/Analysis/Seminorm.lean | 850 | 852 | theorem closedBall_zero_eq_preimage_closedBall {r : ℝ} :
p.closedBall 0 r = p ⁻¹' Metric.closedBall 0 r := by |
rw [closedBall_zero_eq, preimage_metric_closedBall]
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
#align_import algebra.order.sub.defs from "le... | Mathlib/Algebra/Order/Sub/Defs.lean | 150 | 152 | theorem le_tsub_add_add : a + b ≤ a - c + (b + c) := by |
rw [add_comm a, add_comm (a - c)]
exact add_le_add_add_tsub
|
/-
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 | 390 | 401 | theorem biUnion_assoc (πi : ∀ J, Prepartition J) (πi' : Box ι → ∀ J : Box ι, Prepartition J) :
(π.biUnion fun J => (πi J).biUnion (πi' J)) =
(π.biUnion πi).biUnion fun J => πi' (π.biUnionIndex πi J) J := by |
ext J
simp only [mem_biUnion, exists_prop]
constructor
· rintro ⟨J₁, hJ₁, J₂, hJ₂, hJ⟩
refine ⟨J₂, ⟨J₁, hJ₁, hJ₂⟩, ?_⟩
rwa [π.biUnionIndex_of_mem hJ₁ hJ₂]
· rintro ⟨J₁, ⟨J₂, hJ₂, hJ₁⟩, hJ⟩
refine ⟨J₂, hJ₂, J₁, hJ₁, ?_⟩
rwa [π.biUnionIndex_of_mem hJ₂ hJ₁] at hJ
|
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.LinearAlgebra.BilinearForm.Hom
import Mathlib.LinearAlgebra.Dual
/-!
# Bilinear form
This file defines various properties of bilinea... | Mathlib/LinearAlgebra/BilinearForm/Properties.lean | 484 | 486 | theorem apply_dualBasis_right (B : BilinForm K V) (hB : B.Nondegenerate) (sym : B.IsSymm)
(b : Basis ι K V) (i j) : B (b i) (B.dualBasis hB b j) = if i = j then 1 else 0 := by |
rw [sym.eq, apply_dualBasis_left]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
#align_import linear_algebra.free_module.pid from "leanprover-comm... | Mathlib/LinearAlgebra/FreeModule/PID.lean | 312 | 317 | theorem Submodule.basisOfPid_bot {ι : Type*} [Finite ι] (b : Basis ι R M) :
Submodule.basisOfPid b ⊥ = ⟨0, Basis.empty _⟩ := by |
obtain ⟨n, b'⟩ := Submodule.basisOfPid b ⊥
let e : Fin n ≃ Fin 0 := b'.indexEquiv (Basis.empty _ : Basis (Fin 0) R (⊥ : Submodule R M))
obtain rfl : n = 0 := by simpa using Fintype.card_eq.mpr ⟨e⟩
exact Sigma.eq rfl (Basis.eq_of_apply_eq <| finZeroElim)
|
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
/-!
# Dropping or taki... | Mathlib/Data/List/DropRight.lean | 239 | 241 | theorem mem_rtakeWhile_imp {x : α} (hx : x ∈ rtakeWhile p l) : p x := by |
rw [rtakeWhile, mem_reverse] at hx
exact mem_takeWhile_imp hx
|
/-
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.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.So... | Mathlib/Algebra/Polynomial/Basic.lean | 422 | 424 | theorem support_eq_empty : p.support = ∅ ↔ p = 0 := by |
rcases p with ⟨⟩
simp [support]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yourong Zang
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mat... | Mathlib/Analysis/Complex/RealDeriv.lean | 128 | 130 | theorem HasDerivWithinAt.complexToReal_fderiv {f : ℂ → ℂ} {s : Set ℂ} {f' x : ℂ}
(h : HasDerivWithinAt f f' s x) : HasFDerivWithinAt f (f' • (1 : ℂ →L[ℝ] ℂ)) s x := by |
simpa only [Complex.restrictScalars_one_smulRight] using h.hasFDerivWithinAt.restrictScalars ℝ
|
/-
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.Real
#align_import analys... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 547 | 548 | theorem rpow_sub {x : ℝ≥0∞} (y z : ℝ) (hx : x ≠ 0) (h'x : x ≠ ⊤) : x ^ (y - z) = x ^ y / x ^ z := by |
rw [sub_eq_add_neg, rpow_add _ _ hx h'x, rpow_neg, div_eq_mul_inv]
|
/-
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.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathli... | Mathlib/FieldTheory/Separable.lean | 70 | 72 | theorem separable_X_add_C (a : R) : (X + C a).Separable := by |
rw [separable_def, derivative_add, derivative_X, derivative_C, add_zero]
exact isCoprime_one_right
|
/-
Copyright (c) 2020 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo
-/
import Mathlib.Dynamics.Flow
import Mathlib.Tactic.Monotonicity
#align_import dynamics.omega_limit from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
... | Mathlib/Dynamics/OmegaLimit.lean | 183 | 185 | theorem omegaLimit_iUnion (p : ι → Set α) : ⋃ i, ω f ϕ (p i) ⊆ ω f ϕ (⋃ i, p i) := by |
rw [iUnion_subset_iff]
exact fun i ↦ omegaLimit_mono_right _ _ (subset_iUnion _ _)
|
/-
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 | 475 | 476 | theorem toIcoMod_add_right (a b : α) : toIcoMod hp a (b + p) = toIcoMod hp a b := by |
simpa only [one_zsmul] using toIcoMod_add_zsmul hp a b 1
|
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableClosure
import Mathlib.Algebra.CharP.IntermediateField
/-!
# Purely inseparable extension and relative perfect closure
This file contains basic... | Mathlib/FieldTheory/PurelyInseparable.lean | 1,059 | 1,071 | theorem Polynomial.Separable.map_irreducible_of_isPurelyInseparable {f : F[X]} (hsep : f.Separable)
(hirr : Irreducible f) [IsPurelyInseparable F E] : Irreducible (f.map (algebraMap F E)) := by |
let K := AlgebraicClosure E
obtain ⟨x, hx⟩ := IsAlgClosed.exists_aeval_eq_zero K f
(natDegree_pos_iff_degree_pos.1 hirr.natDegree_pos).ne'
have ha : Associated f (minpoly F x) := by
have := isUnit_C.2 (leadingCoeff_ne_zero.2 hirr.ne_zero).isUnit.inv
exact ⟨this.unit, by rw [IsUnit.unit_spec, minpoly.... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 143 | 152 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by |
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz)
· simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN
rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_ad... |
/-
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 | 935 | 943 | theorem le_graph_of_le {f g : E →ₗ.[R] F} (h : f ≤ g) : f.graph ≤ g.graph := by |
intro x hx
rw [mem_graph_iff] at hx ⊢
cases' hx with y hx
use ⟨y, h.1 y.2⟩
simp only [hx, Submodule.coe_mk, eq_self_iff_true, true_and_iff]
convert hx.2 using 1
refine (h.2 ?_).symm
simp only [hx.1, Submodule.coe_mk]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Interval.Finset
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tactic.Linarith
... | Mathlib/Algebra/BigOperators/Intervals.lean | 201 | 210 | theorem sum_Ico_Ico_comm' {M : Type*} [AddCommMonoid M] (a b : ℕ) (f : ℕ → ℕ → M) :
(∑ i ∈ Finset.Ico a b, ∑ j ∈ Finset.Ico (i + 1) b, f i j) =
∑ j ∈ Finset.Ico a b, ∑ i ∈ Finset.Ico a j, f i j := by |
rw [Finset.sum_sigma', Finset.sum_sigma']
refine sum_nbij' (fun x ↦ ⟨x.2, x.1⟩) (fun x ↦ ⟨x.2, x.1⟩) ?_ ?_ (fun _ _ ↦ rfl) (fun _ _ ↦ rfl)
(fun _ _ ↦ rfl) <;>
simp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma] <;>
rintro a b ⟨⟨h₁, h₂⟩, ⟨h₃, h₄⟩⟩ <;>
refine ⟨⟨_, _⟩, ⟨_, _⟩⟩ <;>
omega
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Ma... | Mathlib/CategoryTheory/Preadditive/Biproducts.lean | 415 | 418 | theorem fst_of_isColimit {X Y : C} {t : BinaryBicone X Y} (ht : IsColimit t.toCocone) :
t.fst = ht.desc (BinaryCofan.mk (𝟙 X) 0) := by |
apply ht.uniq (BinaryCofan.mk (𝟙 X) 0)
rintro ⟨⟨⟩⟩ <;> dsimp <;> simp
|
/-
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.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Constructions.Prod.Integral
/-!
# Integration with respect to a finite product of measures
... | Mathlib/MeasureTheory/Integral/Pi.lean | 45 | 54 | theorem Integrable.fintype_prod_dep {ι : Type*} [Fintype ι] {E : ι → Type*}
{f : (i : ι) → E i → 𝕜} [∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
(hf : ∀ i, Integrable (f i)) :
Integrable (fun (x : (i : ι) → E i) ↦ ∏ i, f i (x i)) := by |
let e := (equivFin ι).symm
simp_rw [← (volume_measurePreserving_piCongrLeft _ e).integrable_comp_emb
(MeasurableEquiv.measurableEmbedding _),
← e.prod_comp, MeasurableEquiv.coe_piCongrLeft, Function.comp_def,
Equiv.piCongrLeft_apply_apply]
exact .fin_nat_prod (fun i ↦ hf _)
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
/-!
# Map... | Mathlib/RingTheory/Ideal/Maps.lean | 253 | 259 | theorem smul_restrictScalars {R S M} [CommSemiring R] [CommSemiring S]
[Algebra R S] [AddCommMonoid M] [Module R M] [Module S M]
[IsScalarTower R S M] (I : Ideal R) (N : Submodule S M) :
(I.map (algebraMap R S) • N).restrictScalars R = I • N.restrictScalars R := by |
simp_rw [map, Submodule.span_smul_eq, ← Submodule.coe_set_smul,
Submodule.set_smul_eq_iSup, ← element_smul_restrictScalars, iSup_image]
exact (_root_.map_iSup₂ (Submodule.restrictScalarsLatticeHom R S M) _)
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 495 | 534 | theorem range_mfderiv_coe_sphere {n : ℕ} [Fact (finrank ℝ E = n + 1)] (v : sphere (0 : E) 1) :
LinearMap.range (mfderiv (𝓡 n) 𝓘(ℝ, E) ((↑) : sphere (0 : E) 1 → E) v :
TangentSpace (𝓡 n) v →L[ℝ] E) = (ℝ ∙ (v : E))ᗮ := by |
rw [((contMDiff_coe_sphere v).mdifferentiableAt le_top).mfderiv]
dsimp [chartAt]
-- rw [LinearIsometryEquiv.toHomeomorph_symm]
-- rw [← LinearIsometryEquiv.coe_toHomeomorph]
simp only [chartAt, stereographic_neg_apply, fderivWithin_univ,
LinearIsometryEquiv.toHomeomorph_symm, LinearIsometryEquiv.coe_toHo... |
/-
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.GroupTheory.GroupAction.Pointwise
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.... | Mathlib/Analysis/LocallyConvex/Bounded.lean | 431 | 438 | theorem isBounded_iff_subset_smul_closedBall {s : Set E} :
Bornology.IsBounded s ↔ ∃ a : 𝕜, s ⊆ a • Metric.closedBall (0 : E) 1 := by |
constructor
· rw [isBounded_iff_subset_smul_ball 𝕜]
exact Exists.imp fun a ha => ha.trans <| Set.smul_set_mono <| Metric.ball_subset_closedBall
· rw [← isVonNBounded_iff 𝕜]
rintro ⟨a, ha⟩
exact ((isVonNBounded_closedBall 𝕜 E 1).image (a • (1 : E →L[𝕜] E))).subset ha
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.lpSpace
import Mathlib.Topology.Sets.Compacts
#align_import topology.metric_space.kuratowski from "leanprover-community/mat... | Mathlib/Topology/MetricSpace/Kuratowski.lean | 140 | 164 | theorem LipschitzOnWith.extend_lp_infty [PseudoMetricSpace α] {s : Set α} {ι : Type*}
{f : α → ℓ^∞(ι)} {K : ℝ≥0} (hfl : LipschitzOnWith K f s) :
∃ g : α → ℓ^∞(ι), LipschitzWith K g ∧ EqOn f g s := by |
-- Construct the coordinate-wise extensions
rw [LipschitzOnWith.coordinate] at hfl
have (i : ι) : ∃ g : α → ℝ, LipschitzWith K g ∧ EqOn (fun x => f x i) g s :=
LipschitzOnWith.extend_real (hfl i) -- use the nonlinear Hahn-Banach theorem here!
choose g hgl hgeq using this
rcases s.eq_empty_or_nonempty wit... |
/-
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.Group.Instances
import Mathlib.Analysis.Convex.Segment
import Mathlib.Tactic.GCongr
#align_import analysis.convex.star from "leanprover-comm... | Mathlib/Analysis/Convex/Star.lean | 344 | 348 | theorem StarConvex.add_smul_sub_mem (hs : StarConvex 𝕜 x s) (hy : y ∈ s) {t : 𝕜} (ht₀ : 0 ≤ t)
(ht₁ : t ≤ 1) : x + t • (y - x) ∈ s := by |
apply hs.segment_subset hy
rw [segment_eq_image']
exact mem_image_of_mem _ ⟨ht₀, ht₁⟩
|
/-
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 | 399 | 401 | theorem monic_X_pow_sub_C {R : Type u} [Ring R] (a : R) {n : ℕ} (h : n ≠ 0) :
(X ^ n - C a).Monic := by |
simpa only [map_neg, ← sub_eq_add_neg] using monic_X_pow_add_C (-a) h
|
/-
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.PrimitiveElement
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Char... | Mathlib/RingTheory/Norm.lean | 260 | 264 | theorem norm_eq_prod_roots [IsSeparable K L] [FiniteDimensional K L] {x : L}
(hF : (minpoly K x).Splits (algebraMap K F)) :
algebraMap K F (norm K x) =
((minpoly K x).aroots F).prod ^ finrank K⟮x⟯ L := by |
rw [norm_eq_norm_adjoin K x, map_pow, IntermediateField.AdjoinSimple.norm_gen_eq_prod_roots _ hF]
|
/-
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 | 729 | 732 | theorem one_le_rpow_of_pos_of_le_one_of_nonpos (hx1 : 0 < x) (hx2 : x ≤ 1) (hz : z ≤ 0) :
1 ≤ x ^ z := by |
convert rpow_le_rpow_of_exponent_ge hx1 hx2 hz
exact (rpow_zero x).symm
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries
import Mathlib.Algebra.ContinuedFractions.Computation.Translations
import... | Mathlib/NumberTheory/DiophantineApproximation.lean | 533 | 576 | theorem exists_rat_eq_convergent' {v : ℕ} (h' : ContfracLegendre.Ass ξ u v) :
∃ n, (u / v : ℚ) = ξ.convergent n := by |
-- Porting note: `induction` got in trouble because of the irrelevant hypothesis `h`
clear h; have h := h'; clear h'
induction v using Nat.strong_induction_on generalizing ξ u with | h v ih => ?_
rcases lt_trichotomy v 1 with (ht | rfl | ht)
· replace h := h.2.2
simp only [Nat.lt_one_iff.mp ht, Nat.cast_... |
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi
import Mathlib.CategoryTheory.LiftingProperties.Adjunction
#align_import ca... | Mathlib/CategoryTheory/Functor/EpiMono.lean | 273 | 278 | theorem mono_map_iff_mono [hF₁ : PreservesMonomorphisms F] [hF₂ : ReflectsMonomorphisms F] :
Mono (F.map f) ↔ Mono f := by |
constructor
· exact F.mono_of_mono_map
· intro h
exact F.map_mono f
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.Cat... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 486 | 497 | theorem isVanKampenColimit_of_isEmpty [HasStrictInitialObjects C] [IsEmpty J] {F : J ⥤ C}
(c : Cocone F) (hc : IsColimit c) : IsVanKampenColimit c := by |
have : IsInitial c.pt := by
have := (IsColimit.precomposeInvEquiv (Functor.uniqueFromEmpty _) _).symm
(hc.whiskerEquivalence (equivalenceOfIsEmpty (Discrete PEmpty.{1}) J))
exact IsColimit.ofIsoColimit this (Cocones.ext (Iso.refl c.pt) (fun {X} ↦ isEmptyElim X))
replace this := IsInitial.isVanKampenC... |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpac... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 170 | 173 | theorem norm_combo_lt_of_ne (hx : ‖x‖ ≤ r) (hy : ‖y‖ ≤ r) (hne : x ≠ y) (ha : 0 < a) (hb : 0 < b)
(hab : a + b = 1) : ‖a • x + b • y‖ < r := by |
simp only [← mem_ball_zero_iff, ← mem_closedBall_zero_iff] at hx hy ⊢
exact combo_mem_ball_of_ne hx hy hne ha hb hab
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure fr... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 84 | 92 | theorem withDensityᵥ_add (hf : Integrable f μ) (hg : Integrable g μ) :
μ.withDensityᵥ (f + g) = μ.withDensityᵥ f + μ.withDensityᵥ g := by |
ext1 i hi
rw [withDensityᵥ_apply (hf.add hg) hi, VectorMeasure.add_apply, withDensityᵥ_apply hf hi,
withDensityᵥ_apply hg hi]
simp_rw [Pi.add_apply]
rw [integral_add] <;> rw [← integrableOn_univ]
· exact hf.integrableOn.restrict MeasurableSet.univ
· exact hg.integrableOn.restrict MeasurableSet.univ
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.RBMap.Alter
import Batteries.Data.List.Lemmas
/-!
# Additional lemmas for Red-black trees
-/
namespace Batteries
namespace RBNode
open RBColor... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 186 | 189 | theorem forIn_eq_forIn_toList [Monad m] [LawfulMonad m] {t : RBNode α} :
forIn (m := m) t init f = forIn t.toList init f := by |
conv => lhs; simp only [forIn, RBNode.forIn]
rw [List.forIn_eq_bindList, forIn_visit_eq_bindList]
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import M... | Mathlib/Order/WellFoundedSet.lean | 623 | 625 | theorem partiallyWellOrderedOn_bUnion (s : Finset ι) {f : ι → Set α} :
(⋃ i ∈ s, f i).PartiallyWellOrderedOn r ↔ ∀ i ∈ s, (f i).PartiallyWellOrderedOn r := by |
simpa only [Finset.sup_eq_iSup] using s.partiallyWellOrderedOn_sup
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 838 | 840 | theorem mk'_d_1_0 : (mk' X₀ X₁ d₀ succ').d 1 0 = d₀ := by |
change ite (1 = 0 + 1) (𝟙 X₁ ≫ d₀) 0 = d₀
rw [if_pos rfl, Category.id_comp]
|
/-
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.Constructions.BorelSpace.Order
#align_import measure_theory.function.ae_measurable_order from "leanprover-community/mathlib"@"bf6a... | Mathlib/MeasureTheory/Function/AEMeasurableOrder.lean | 113 | 127 | theorem ENNReal.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*} {m : MeasurableSpace α}
(μ : Measure α) (f : α → ℝ≥0∞)
(h : ∀ (p : ℝ≥0) (q : ℝ≥0), p < q →
∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
{ x | f x < p } ⊆ u ∧ { x | (q : ℝ≥0∞) < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
AEMeasurable f... |
obtain ⟨s, s_count, s_dense, _, s_top⟩ :
∃ s : Set ℝ≥0∞, s.Countable ∧ Dense s ∧ 0 ∉ s ∧ ∞ ∉ s :=
ENNReal.exists_countable_dense_no_zero_top
have I : ∀ x ∈ s, x ≠ ∞ := fun x xs hx => s_top (hx ▸ xs)
apply MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets μ s s_count s_dense _
rintro p hp q ... |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import geometry.euclidean.basic from "lean... | Mathlib/Geometry/Euclidean/PerpBisector.lean | 86 | 90 | theorem mem_perpBisector_iff_inner_eq :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = (dist p₁ p₂) ^ 2 / 2 := by |
rw [mem_perpBisector_iff_inner_eq_zero, ← vsub_sub_vsub_cancel_right _ _ p₁, inner_sub_left,
sub_eq_zero, midpoint_vsub_left, invOf_eq_inv, real_inner_smul_left, real_inner_self_eq_norm_sq,
dist_eq_norm_vsub' V, div_eq_inv_mul]
|
/-
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 | 287 | 289 | theorem det_neg_eq_smul (A : Matrix n n R) :
det (-A) = (-1 : Units ℤ) ^ Fintype.card n • det A := by |
rw [← det_smul_of_tower, Units.neg_smul, one_smul]
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.Pretopology
import Mathlib.CategoryTheory.Sites.IsSheafFor
#align_import category_theory.sites.sheaf_of_types from "leanprover-commun... | Mathlib/CategoryTheory/Sites/SheafOfTypes.lean | 105 | 118 | theorem isSheaf_pretopology [HasPullbacks C] (K : Pretopology C) :
IsSheaf (K.toGrothendieck C) P ↔ ∀ {X : C} (R : Presieve X), R ∈ K X → IsSheafFor P R := by |
constructor
· intro PJ X R hR
rw [isSheafFor_iff_generate]
apply PJ (Sieve.generate R) ⟨_, hR, le_generate R⟩
· rintro PK X S ⟨R, hR, RS⟩
have gRS : ⇑(generate R) ≤ S := by
apply giGenerate.gc.monotone_u
rwa [sets_iff_generate]
apply isSheafFor_subsieve P gRS _
intro Y f
rw [←... |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Measure.GiryMonad
#align_import probability.kernel.basic from "leanprover-community/mathlib"@"... | Mathlib/Probability/Kernel/Basic.lean | 281 | 285 | theorem sum_add [Countable ι] (κ η : ι → kernel α β) :
(kernel.sum fun n => κ n + η n) = kernel.sum κ + kernel.sum η := by |
ext a s hs
simp only [coeFn_add, Pi.add_apply, sum_apply, Measure.sum_apply _ hs, Pi.add_apply,
Measure.coe_add, tsum_add ENNReal.summable ENNReal.summable]
|
/-
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.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.Affine... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 72 | 74 | theorem weightedVSubOfPoint_apply (w : ι → k) (p : ι → P) (b : P) :
s.weightedVSubOfPoint p b w = ∑ i ∈ s, w i • (p i -ᵥ b) := by |
simp [weightedVSubOfPoint, LinearMap.sum_apply]
|
/-
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.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.StdB... | Mathlib/RingTheory/Finiteness.lean | 611 | 619 | theorem of_restrictScalars_finite (R A M : Type*) [CommSemiring R] [Semiring A] [AddCommMonoid M]
[Module R M] [Module A M] [Algebra R A] [IsScalarTower R A M] [hM : Finite R M] :
Finite A M := by |
rw [finite_def, Submodule.fg_def] at hM ⊢
obtain ⟨S, hSfin, hSgen⟩ := hM
refine ⟨S, hSfin, eq_top_iff.2 ?_⟩
have := Submodule.span_le_restrictScalars R A S
rw [hSgen] at this
exact this
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Johan Commelin, Andrew Yang
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Monoidal.End
... | Mathlib/CategoryTheory/Shift/Basic.lean | 504 | 511 | theorem shift_shiftFunctorCompIsoId_inv_app (n m : A) (h : n + m = 0) (X : C) :
((shiftFunctorCompIsoId C n m h).inv.app X)⟦n⟧' =
((shiftFunctorCompIsoId C m n
(by rw [← neg_eq_of_add_eq_zero_left h, add_right_neg])).inv.app (X⟦n⟧)) := by |
rw [← cancel_mono (((shiftFunctorCompIsoId C n m h).hom.app X)⟦n⟧'),
← Functor.map_comp, Iso.inv_hom_id_app, Functor.map_id,
shift_shiftFunctorCompIsoId_hom_app, Iso.inv_hom_id_app]
rfl
|
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombinat... | Mathlib/NumberTheory/FLT/Four.lean | 169 | 293 | theorem not_minimal {a b c : ℤ} (h : Minimal a b c) (ha2 : a % 2 = 1) (hc : 0 < c) : False := by |
-- Use the fact that a ^ 2, b ^ 2, c form a pythagorean triple to obtain m and n such that
-- a ^ 2 = m ^ 2 - n ^ 2, b ^ 2 = 2 * m * n and c = m ^ 2 + n ^ 2
-- first the formula:
have ht : PythagoreanTriple (a ^ 2) (b ^ 2) c := by
delta PythagoreanTriple
linear_combination h.1.2.2
-- coprime requirem... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Topology.UniformSpace.Basic
#align_import topology.uniform... | Mathlib/Topology/UniformSpace/Cauchy.lean | 465 | 467 | theorem completeSpace_iff_ultrafilter :
CompleteSpace α ↔ ∀ l : Ultrafilter α, Cauchy (l : Filter α) → ∃ x : α, ↑l ≤ 𝓝 x := by |
simp [completeSpace_iff_isComplete_univ, isComplete_iff_ultrafilter]
|
/-
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.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 296 | 299 | theorem smul_boundedBy {c : ℝ≥0∞} (hc : c ≠ ∞) : c • boundedBy m = boundedBy (c • m) := by |
simp only [boundedBy , smul_ofFunction hc]
congr 1 with s : 1
rcases s.eq_empty_or_nonempty with (rfl | hs) <;> simp [*]
|
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