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 |
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/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs fr... | Mathlib/Data/Finsupp/Defs.lean | 1,127 | 1,130 | theorem erase_add (a : α) (f f' : α →₀ M) : erase a (f + f') = erase a f + erase a f' := by |
ext s; by_cases hs : s = a
· rw [hs, add_apply, erase_same, erase_same, erase_same, add_zero]
rw [add_apply, erase_ne hs, erase_ne hs, erase_ne hs, add_apply]
|
/-
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 | 1,624 | 1,626 | theorem map_toDual_max (s : Finset α) : s.max.map toDual = (s.image toDual).min := by |
rw [min_eq_inf_withTop, inf_image]
exact congr_fun Option.map_id _
|
/-
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.Equiv.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Order.Hom.Basic
#align_import alg... | Mathlib/Algebra/Order/Sub/Basic.lean | 36 | 38 | theorem le_tsub_mul {R : Type*} [CommSemiring R] [Preorder R] [Sub R] [OrderedSub R]
[CovariantClass R R (· * ·) (· ≤ ·)] {a b c : R} : a * c - b * c ≤ (a - b) * c := by |
simpa only [mul_comm _ c] using le_mul_tsub
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearA... | Mathlib/Algebra/Quaternion.lean | 770 | 772 | theorem mul_star_eq_coe : a * star a = (a * star a).re := by |
rw [← star_comm_self']
exact a.star_mul_eq_coe
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
#align_import ring_theory.fractional_ideal from "lean... | Mathlib/RingTheory/FractionalIdeal/Basic.lean | 424 | 428 | theorem zero_le (I : FractionalIdeal S P) : 0 ≤ I := by |
intro x hx
-- Porting note: changed the proof from convert; simp into rw; exact
rw [(mem_zero_iff _).mp hx]
exact zero_mem (I : Submodule R P)
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Scott Morrison
-/
import Mathlib.Algebra.Homology.Exact
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.Adjunction.Limits
import Math... | Mathlib/CategoryTheory/Preadditive/Projective.lean | 255 | 261 | theorem enoughProjectives_iff (F : C ≌ D) : EnoughProjectives C ↔ EnoughProjectives D := by |
constructor
all_goals intro H; constructor; intro X; constructor
· exact F.symm.projectivePresentationOfMapProjectivePresentation _
(Nonempty.some (H.presentation (F.inverse.obj X)))
· exact F.projectivePresentationOfMapProjectivePresentation X
(Nonempty.some (H.presentation (F.functor.obj X)))
|
/-
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.Algebra.Pi
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
#align_im... | Mathlib/Algebra/Polynomial/AlgebraMap.lean | 185 | 189 | theorem adjoin_X : Algebra.adjoin R ({X} : Set R[X]) = ⊤ := by |
refine top_unique fun p _hp => ?_
set S := Algebra.adjoin R ({X} : Set R[X])
rw [← sum_monomial_eq p]; simp only [← smul_X_eq_monomial, Sum]
exact S.sum_mem fun n _hn => S.smul_mem (S.pow_mem (Algebra.subset_adjoin rfl) _) _
|
/-
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.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homo... | Mathlib/Algebra/Homology/Homotopy.lean | 109 | 112 | theorem prevD_comp_right (f : ∀ i j, C.X i ⟶ D.X j) (g : D ⟶ E) (j : ι) :
(prevD j fun i j => f i j ≫ g.f j) = prevD j f ≫ g.f j := by |
dsimp [prevD]
simp only [assoc, g.comm]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-communit... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 319 | 322 | theorem condexpInd_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
(hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) : (condexpInd G hm μ (s ∪ t) : G →L[ℝ] α →₁[μ] G) =
condexpInd G hm μ s + condexpInd G hm μ t := by |
ext1 x; push_cast; exact condexpInd_disjoint_union_apply hs ht hμs hμt hst 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, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 350 | 350 | theorem image_const_add_Iic : (fun x => a + x) '' Iic b = Iic (a + b) := by | simp [add_comm]
|
/-
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 | 70 | 80 | theorem _root_.Function.Injective.wSameSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).WSameSide (f x) (f y) ↔ s.WSameSide x y := by |
refine ⟨fun h => ?_, fun h => h.map _⟩
rcases h with ⟨fp₁, hfp₁, fp₂, hfp₂, h⟩
rw [mem_map] at hfp₁ hfp₂
rcases hfp₁ with ⟨p₁, hp₁, rfl⟩
rcases hfp₂ with ⟨p₂, hp₂, rfl⟩
refine ⟨p₁, hp₁, p₂, hp₂, ?_⟩
simp_rw [← linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h
exact h
|
/-
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 | 1,360 | 1,371 | theorem fract_eq_zero_or_add_one_sub_ceil (a : α) : fract a = 0 ∨ fract a = a + 1 - (⌈a⌉ : α) := by |
rcases eq_or_ne (fract a) 0 with ha | ha
· exact Or.inl ha
right
suffices (⌈a⌉ : α) = ⌊a⌋ + 1 by
rw [this, ← self_sub_fract]
abel
norm_cast
rw [ceil_eq_iff]
refine ⟨?_, _root_.le_of_lt <| by simp⟩
rw [cast_add, cast_one, add_tsub_cancel_right, ← self_sub_fract a, sub_lt_self_iff]
exact ha.sym... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 91 | 94 | theorem isPiSystem_Iio_rat : IsPiSystem (⋃ a : ℚ, {Iio (a : ℝ)}) := by |
convert isPiSystem_image_Iio (((↑) : ℚ → ℝ) '' univ)
ext x
simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and]
|
/-
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.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingle... | Mathlib/Combinatorics/Enumerative/Composition.lean | 416 | 420 | theorem index_embedding (i : Fin c.length) (j : Fin (c.blocksFun i)) :
c.index (c.embedding i j) = i := by |
symm
rw [← mem_range_embedding_iff']
apply Set.mem_range_self
|
/-
Copyright (c) 2021 Alex Kontorovich, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth
-/
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Homeomorph
import Mathlib.GroupTheory.GroupAction.Basic
im... | Mathlib/Topology/Algebra/ConstMulAction.lean | 591 | 594 | theorem set_smul_mem_nhds_zero_iff {s : Set α} {c : G₀} (hc : c ≠ 0) :
c • s ∈ 𝓝 (0 : α) ↔ s ∈ 𝓝 (0 : α) := by |
refine Iff.trans ?_ (set_smul_mem_nhds_smul_iff hc)
rw [smul_zero]
|
/-
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.Algebra.Module.Submodule.Ker
/-!
# Iterate maps and comaps of submodules
Some preliminary work for establishing the strong rank condition for noetherian rings.
Giv... | Mathlib/Algebra/Module/Submodule/IterateMapComap.lean | 88 | 92 | theorem ker_le_of_iterateMapComap_eq_succ (K : Submodule R N)
(m : ℕ) (heq : f.iterateMapComap i m K = f.iterateMapComap i (m + 1) K)
(hf : Surjective f) (hi : Injective i) : LinearMap.ker f ≤ K := by |
rw [show K = _ from f.iterateMapComap_eq_succ i K m heq hf hi 0]
exact f.ker_le_comap
|
/-
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.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 274 | 278 | theorem vadd_mem_iff_mem_of_mem_direction {s : AffineSubspace k P} {v : V} (hv : v ∈ s.direction)
{p : P} : v +ᵥ p ∈ s ↔ p ∈ s := by |
refine ⟨fun h => ?_, fun h => vadd_mem_of_mem_direction hv h⟩
convert vadd_mem_of_mem_direction (Submodule.neg_mem _ hv) h
simp
|
/-
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 | 480 | 481 | theorem mem_core_res (f : α → β) (s : Set α) (t : Set β) (x : α) :
x ∈ (res f s).core t ↔ x ∈ s → f x ∈ t := by | simp [mem_core, mem_res]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlg... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 238 | 239 | theorem trace_adjMatrix [AddCommMonoid α] [One α] : Matrix.trace (G.adjMatrix α) = 0 := by |
simp [Matrix.trace]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 2,544 | 2,626 | theorem tr_respects_aux₂ {k : K} {q : Stmt₂₁} {v : σ} {S : ∀ k, List (Γ k)}
{L : ListBlank (∀ k, Option (Γ k))}
(hL : ∀ k, L.map (proj k) = ListBlank.mk ((S k).map some).reverse) (o : StAct₂ k) :
let v' := stVar v (S k) o
let Sk' := stWrite v (S k) o
let S' := update S k Sk'
∃ L' : ListBlank (∀ ... |
dsimp only; simp; cases o with simp only [stWrite, stVar, trStAct, TM1.stepAux]
| push f =>
have := Tape.write_move_right_n fun a : Γ' ↦ (a.1, update a.2 k (some (f v)))
refine
⟨_, fun k' ↦ ?_, by
-- Porting note: `rw [...]` to `erw [...]; rfl`.
-- https://github.com/leanprover-commun... |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTh... | Mathlib/MeasureTheory/Function/LpSpace.lean | 320 | 322 | theorem edist_toLp_zero (f : α → E) (hf : Memℒp f p μ) : edist (hf.toLp f) 0 = snorm f p μ := by |
convert edist_toLp_toLp f 0 hf zero_memℒp
simp
|
/-
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.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "lean... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,490 | 1,492 | theorem contDiff_zero : ContDiff 𝕜 0 f ↔ Continuous f := by |
rw [← contDiffOn_univ, continuous_iff_continuousOn_univ]
exact contDiffOn_zero
|
/-
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.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | Mathlib/Topology/ContinuousFunction/Compact.lean | 141 | 143 | theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by |
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalculus
#align_import analysis.special_functions.non_integrable fr... | Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean | 98 | 121 | theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter
{f : ℝ → E} {g : ℝ → F}
{k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
(hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
(hfg : deriv f =O[l] g) : ¬IntegrableOn g k := by |
let a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
let f' := a ∘ f
have h'd : ∀ᶠ x in l, DifferentiableAt ℝ f' x := by
filter_upwards [hd] with x hx using a.toContinuousLinearMap.differentiableAt.comp x hx
have h'f : Tendsto (fun x => ‖f' x‖) l atTop := hf.congr (fun x ↦ by sim... |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... | Mathlib/Analysis/Convex/Segment.lean | 260 | 261 | theorem mem_segment_translate (a : E) {x b c} : a + x ∈ [a + b -[𝕜] a + c] ↔ x ∈ [b -[𝕜] c] := by |
simp_rw [← vadd_eq_add, ← vadd_segment, vadd_mem_vadd_set_iff]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | Mathlib/Topology/Order/DenselyOrdered.lean | 25 | 29 | theorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by |
apply Subset.antisymm
· exact closure_minimal Ioi_subset_Ici_self isClosed_Ici
· rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]
exact isGLB_Ioi.mem_closure h
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.M... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 549 | 551 | theorem norm_integral_min_max (f : ℝ → E) :
‖∫ x in min a b..max a b, f x ∂μ‖ = ‖∫ x in a..b, f x ∂μ‖ := by |
cases le_total a b <;> simp [*, integral_symm a b]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.DensityTheorem
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.covering.one_dim from "l... | Mathlib/MeasureTheory/Covering/OneDim.lean | 26 | 30 | theorem Icc_mem_vitaliFamily_at_right {x y : ℝ} (hxy : x < y) :
Icc x y ∈ (vitaliFamily (volume : Measure ℝ) 1).setsAt x := by |
rw [Icc_eq_closedBall]
refine closedBall_mem_vitaliFamily_of_dist_le_mul _ ?_ (by linarith)
rw [dist_comm, Real.dist_eq, abs_of_nonneg] <;> linarith
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux, Jon Bannon
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.Star.Unitization
import Mathlib.Analysis.SpecialFunctions.... | Mathlib/Analysis/NormedSpace/Star/Multiplier.lean | 648 | 649 | theorem norm_fst (a : 𝓜(𝕜, A)) : ‖a.fst‖ = ‖a‖ := by |
simp only [norm_def, toProdHom_apply, Prod.norm_def, norm_fst_eq_snd, max_eq_right le_rfl]
|
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Ring.Parity
import Mathlib.Combinatorics.SimpleGraph.Connectivity
#align_import combinatorics.simple_graph.trails from "leanprover-community/mathlib... | Mathlib/Combinatorics/SimpleGraph/Trails.lean | 134 | 142 | theorem IsEulerian.even_degree_iff {x u v : V} {p : G.Walk u v} (ht : p.IsEulerian) [Fintype V]
[DecidableRel G.Adj] : Even (G.degree x) ↔ u ≠ v → x ≠ u ∧ x ≠ v := by |
convert ht.isTrail.even_countP_edges_iff x
rw [← Multiset.coe_countP, Multiset.countP_eq_card_filter, ← card_incidenceFinset_eq_degree]
change Multiset.card _ = _
congr 1
convert_to _ = (ht.isTrail.edgesFinset.filter (Membership.mem x)).val
have : Fintype G.edgeSet := fintypeEdgeSet ht
rw [ht.edgesFinset... |
/-
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 | 335 | 336 | theorem Ioo_succ_right_of_not_isMax (hb : ¬IsMax b) : Ioo a (succ b) = Ioc a b := by |
rw [← Ioi_inter_Iio, Iio_succ_of_not_isMax hb, Ioi_inter_Iic]
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Order.Ring.Finset
import Mat... | Mathlib/Analysis/Normed/Field/Basic.lean | 233 | 235 | theorem nnnorm_mul_le (a b : α) : ‖a * b‖₊ ≤ ‖a‖₊ * ‖b‖₊ := by |
simpa only [← norm_toNNReal, ← Real.toNNReal_mul (norm_nonneg _)] using
Real.toNNReal_mono (norm_mul_le _ _)
|
/-
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.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive... | Mathlib/Data/Option/Basic.lean | 61 | 62 | theorem exists_mem_map {f : α → β} {o : Option α} {p : β → Prop} :
(∃ y ∈ o.map f, p y) ↔ ∃ x ∈ o, p (f x) := by | simp
|
/-
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.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathl... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 915 | 934 | theorem restrict_le_restrict_iUnion {f : ℕ → Set α} (hf₁ : ∀ n, MeasurableSet (f n))
(hf₂ : ∀ n, v ≤[f n] w) : v ≤[⋃ n, f n] w := by |
refine restrict_le_restrict_of_subset_le v w fun a ha₁ ha₂ => ?_
have ha₃ : ⋃ n, a ∩ disjointed f n = a := by
rwa [← Set.inter_iUnion, iUnion_disjointed, Set.inter_eq_left]
have ha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n) :=
(disjoint_disjointed _).mono fun i j => Disjoint.mono inf_le_right ... |
/-
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.Data.Num.Basic
import Mathlib.Data.Vector.Basic
#align_import data.num.bitwise from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e... | Mathlib/Data/Num/Bitwise.lean | 368 | 368 | theorem bit_one (b : Bool) : (b :: zero (Not b)) = msb b := by | cases b <;> rfl
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 487 | 491 | theorem eval_range {ι : Type*} [Finite ι] (b : Basis ι R M) :
LinearMap.range (Dual.eval R M) = ⊤ := by |
classical
cases nonempty_fintype ι
rw [← b.toDual_toDual, range_comp, b.toDual_range, Submodule.map_top, toDual_range _]
|
/-
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 | 107 | 109 | theorem ae_uIoc_iff [LinearOrder α] {a b : α} {P : α → Prop} :
(∀ᵐ x ∂μ, x ∈ Ι a b → P x) ↔ (∀ᵐ x ∂μ, x ∈ Ioc a b → P x) ∧ ∀ᵐ x ∂μ, x ∈ Ioc b a → P x := by |
simp only [uIoc_eq_union, mem_union, or_imp, eventually_and]
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Prod
#align_import data.set.n_ary from "leanprover-community/mathlib"@"5e526d18cea33550268dcbbddcb822d5cde40654"
/-!
# N-ary images of s... | Mathlib/Data/Set/NAry.lean | 255 | 259 | theorem image2_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
image2 f (image2 g s t) u = image2 g' (image2 f' s u) t := by |
rw [image2_swap g, image2_swap g']
exact image2_assoc fun _ _ _ => h_right_comm _ _ _
|
/-
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, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Da... | Mathlib/Algebra/Group/Subsemigroup/Basic.lean | 452 | 455 | theorem mem_iSup {ι : Sort*} (p : ι → Subsemigroup M) {m : M} :
(m ∈ ⨆ i, p i) ↔ ∀ N, (∀ i, p i ≤ N) → m ∈ N := by |
rw [← closure_singleton_le_iff_mem, le_iSup_iff]
simp only [closure_singleton_le_iff_mem]
|
/-
Copyright (c) 2023 Luke Mantle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Mantle
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.h... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 59 | 62 | theorem hermite_eq_iterate (n : ℕ) : hermite n = (fun p => X * p - derivative p)^[n] 1 := by |
induction' n with n ih
· rfl
· rw [Function.iterate_succ_apply', ← ih, hermite_succ]
|
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
/-!
# The length function, reduced words, and descents
Throughout this file, `B` is a type and `... | Mathlib/GroupTheory/Coxeter/Length.lean | 305 | 312 | theorem not_isLeftDescent_iff {w : W} {i : B} :
¬cs.IsLeftDescent w i ↔ ℓ (s i * w) = ℓ w + 1 := by |
unfold IsLeftDescent
constructor
· intro _
exact (cs.length_simple_mul w i).resolve_right (by linarith)
· intro _
linarith
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Eq... | Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | 370 | 371 | theorem sub_sub_sub {X Y : C} (a b c d : X ⟶ Y) : a - c - (b - d) = a - b - (c - d) := by |
rw [sub_def, ← lift_sub_lift, sub_def, Category.assoc, σ_comp, prod.lift_map_assoc]; rfl
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Linarith.Frontend
#align_import alge... | Mathlib/Algebra/QuadraticDiscriminant.lean | 148 | 155 | theorem discrim_lt_zero (ha : a ≠ 0) (h : ∀ x : K, 0 < a * x * x + b * x + c) :
discrim a b c < 0 := by |
have : ∀ x : K, 0 ≤ a * x * x + b * x + c := fun x => le_of_lt (h x)
refine lt_of_le_of_ne (discrim_le_zero this) fun h' ↦ ?_
have := h (-b / (2 * a))
have : a * (-b / (2 * a)) * (-b / (2 * a)) + b * (-b / (2 * a)) + c = 0 := by
rw [quadratic_eq_zero_iff_of_discrim_eq_zero ha h' (-b / (2 * a))]
linarith
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl
-/
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryPro... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 1,398 | 1,410 | theorem pullback.congrHom_inv {X Y Z : C} {f₁ f₂ : X ⟶ Z} {g₁ g₂ : Y ⟶ Z} (h₁ : f₁ = f₂)
(h₂ : g₁ = g₂) [HasPullback f₁ g₁] [HasPullback f₂ g₂] :
(pullback.congrHom h₁ h₂).inv =
pullback.map _ _ _ _ (𝟙 _) (𝟙 _) (𝟙 _) (by simp [h₁]) (by simp [h₂]) := by |
ext
· erw [pullback.lift_fst]
rw [Iso.inv_comp_eq]
erw [pullback.lift_fst_assoc]
rw [Category.comp_id, Category.comp_id]
· erw [pullback.lift_snd]
rw [Iso.inv_comp_eq]
erw [pullback.lift_snd_assoc]
rw [Category.comp_id, Category.comp_id]
|
/-
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.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.... | Mathlib/Data/Rat/Lemmas.lean | 165 | 168 | theorem substr_num_den' (q r : ℚ) :
(q - r).num * q.den * r.den = (q.num * r.den - r.num * q.den) * (q - r).den := by |
rw [sub_eq_add_neg, sub_eq_add_neg, ← neg_mul, ← num_neg_eq_neg_num, ← den_neg_eq_den r,
add_num_den' q (-r)]
|
/-
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 | 2,109 | 2,110 | theorem iSup_union {f : α → β} {s t : Finset α} :
⨆ x ∈ s ∪ t, f x = (⨆ x ∈ s, f x) ⊔ ⨆ x ∈ t, f x := by | simp [iSup_or, iSup_sup_eq]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 138 | 148 | theorem measure_iUnion_of_tendsto_zero {ι} (μ : F) {s : ι → Set α} (l : Filter ι) [NeBot l]
(h0 : Tendsto (fun k => μ ((⋃ n, s n) \ s k)) l (𝓝 0)) : μ (⋃ n, s n) = ⨆ n, μ (s n) := by |
refine le_antisymm ?_ <| iSup_le fun n ↦ measure_mono <| subset_iUnion _ _
set S := ⋃ n, s n
set M := ⨆ n, μ (s n)
have A : ∀ k, μ S ≤ M + μ (S \ s k) := fun k ↦ calc
μ S ≤ μ (S ∩ s k) + μ (S \ s k) := measure_le_inter_add_diff _ _ _
_ ≤ μ (s k) + μ (S \ s k) := by gcongr; apply inter_subset_right
... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.... | Mathlib/Data/Nat/GCD/Basic.lean | 238 | 239 | theorem coprime_self_sub_right {m n : ℕ} (h : m ≤ n) : Coprime n (n - m) ↔ Coprime n m := by |
rw [Coprime, Coprime, gcd_self_sub_right h]
|
/-
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 | 735 | 741 | theorem _root_.LinearIsometryEquiv.piLpCongrLeft_single [DecidableEq ι] [DecidableEq ι']
(e : ι ≃ ι') (i : ι) (v : E) :
LinearIsometryEquiv.piLpCongrLeft p 𝕜 E e ((WithLp.equiv p (_ → E)).symm <| Pi.single i v) =
(WithLp.equiv p (_ → E)).symm (Pi.single (e i) v) := by |
funext x
simp [LinearIsometryEquiv.piLpCongrLeft_apply, LinearEquiv.piCongrLeft', Equiv.piCongrLeft',
Pi.single, Function.update, Equiv.symm_apply_eq]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Sébastien Gouëzel, Zhouhang Zhou, Reid Barton
-/
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.DenseEmbedding
import Mathlib.Topology.Support
impo... | Mathlib/Topology/Homeomorph.lean | 370 | 371 | theorem isOpen_image (h : X ≃ₜ Y) {s : Set X} : IsOpen (h '' s) ↔ IsOpen s := by |
rw [← preimage_symm, isOpen_preimage]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 940 | 944 | theorem forIn_eq_bindList [Monad m] [LawfulMonad m]
(f : α → β → m (ForInStep β)) (l : List α) (init : β) :
forIn l init f = ForInStep.run <$> (ForInStep.yield init).bindList f l := by |
induction l generalizing init <;> simp [*, map_eq_pure_bind]
congr; ext (b | b) <;> simp
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.ca... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 438 | 440 | theorem hasFDerivWithinAt_diff_singleton (y : E) :
HasFDerivWithinAt f f' (s \ {y}) x ↔ HasFDerivWithinAt f f' s x := by |
rw [← hasFDerivWithinAt_insert, insert_diff_singleton, hasFDerivWithinAt_insert]
|
/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Logic.Function.Iterate
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Tactic.GCon... | Mathlib/Topology/EMetricSpace/Lipschitz.lean | 335 | 344 | theorem ediam_image2_le (f : α → β → γ) {K₁ K₂ : ℝ≥0} (s : Set α) (t : Set β)
(hf₁ : ∀ b ∈ t, LipschitzOnWith K₁ (f · b) s) (hf₂ : ∀ a ∈ s, LipschitzOnWith K₂ (f a) t) :
EMetric.diam (Set.image2 f s t) ≤ ↑K₁ * EMetric.diam s + ↑K₂ * EMetric.diam t := by |
simp only [EMetric.diam_le_iff, forall_image2_iff]
intro a₁ ha₁ b₁ hb₁ a₂ ha₂ b₂ hb₂
refine (edist_triangle _ (f a₂ b₁) _).trans ?_
exact
add_le_add
((hf₁ b₁ hb₁ ha₁ ha₂).trans <| ENNReal.mul_left_mono <| EMetric.edist_le_diam_of_mem ha₁ ha₂)
((hf₂ a₂ ha₂ hb₁ hb₂).trans <| ENNReal.mul_left_mono... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.NAry
import Mathlib.Order.Directed
#align_import order.bounds.basic from "leanprover... | Mathlib/Order/Bounds/Basic.lean | 1,504 | 1,506 | theorem BddAbove.image2_bddBelow : BddAbove s → BddAbove t → BddBelow (Set.image2 f s t) := by |
rintro ⟨a, ha⟩ ⟨b, hb⟩
exact ⟨f a b, mem_lowerBounds_image2_of_mem_upperBounds h₀ h₁ ha hb⟩
|
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subg... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 911 | 913 | theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by |
ext <;> simp [or_comm, and_comm]
|
/-
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.Analysis.NormedSpace.Multilinear.Curry
#align_import analysis.calculus.formal_multilinear_series from "leanprover-community/mathlib"@"f2ce608671... | Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean | 310 | 311 | theorem norm_apply_eq_norm_coef : ‖p n‖ = ‖coeff p n‖ := by |
rw [← mkPiRing_coeff_eq p, ContinuousMultilinearMap.norm_mkPiRing]
|
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Ma... | Mathlib/Data/Nat/Factorization/Basic.lean | 614 | 618 | theorem prod_primeFactors_dvd (n : ℕ) : ∏ p ∈ n.primeFactors, p ∣ n := by |
by_cases hn : n = 0
· subst hn
simp
simpa [prod_factors hn] using Multiset.toFinset_prod_dvd_prod (n.factors : Multiset ℕ)
|
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 1,240 | 1,241 | theorem sup_eq_closure (H H' : Subgroup G) : H ⊔ H' = closure ((H : Set G) ∪ (H' : Set G)) := by |
simp_rw [closure_union, closure_eq]
|
/-
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.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Basic
import Mathl... | Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean | 42 | 43 | theorem energy_nonneg : 0 ≤ P.energy G := by |
exact div_nonneg (Finset.sum_nonneg fun _ _ => sq_nonneg _) <| sq_nonneg _
|
/-
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 | 808 | 810 | theorem zero_prod (ν : Measure β) : (0 : Measure α).prod ν = 0 := by |
rw [Measure.prod]
exact bind_zero_left _
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Yaël Dillies
-/
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.Group.Units
import Ma... | Mathlib/Algebra/Order/Ring/Defs.lean | 971 | 972 | theorem one_le_bit1 : (1 : α) ≤ bit1 a ↔ 0 ≤ a := by |
rw [bit1, le_add_iff_nonneg_left, bit0, ← two_mul, mul_nonneg_iff_of_pos_left (zero_lt_two' α)]
|
/-
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 | 672 | 673 | theorem one_le_cast [LinearOrderedSemiring α] (n : PosNum) : (1 : α) ≤ n := by |
rw [← cast_to_nat, ← Nat.cast_one, Nat.cast_le (α := α)]; apply to_nat_pos
|
/-
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.Complex.Log
#align_import ana... | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 248 | 253 | theorem inv_cpow_eq_ite' (x : ℂ) (n : ℂ) :
(x ^ n)⁻¹ = if x.arg = π then conj (x⁻¹ ^ conj n) else x⁻¹ ^ n := by |
rw [inv_cpow_eq_ite, apply_ite conj, conj_conj, conj_conj]
split_ifs with h
· rfl
· rw [inv_cpow _ _ h]
|
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.GroupTheory.Subsemigroup.Center
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
/-!
# `NonUnitalSubring... | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | 491 | 492 | theorem map_range : f.range.map g = (g.comp f).range := by |
simpa only [range_eq_map] using (⊤ : NonUnitalSubring R).map_map g f
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Alex J. Best
-/
import Mathlib.MeasureTheory.Group.Arithmetic
#align_import measure_theory.group.pointwise from "leanprover-community/mathlib"@"66f7114a1d5cba4... | Mathlib/MeasureTheory/Group/Pointwise.lean | 39 | 44 | theorem MeasurableSet.const_smul₀ {G₀ α : Type*} [GroupWithZero G₀] [Zero α]
[MulActionWithZero G₀ α] [MeasurableSpace G₀] [MeasurableSpace α] [MeasurableSMul G₀ α]
[MeasurableSingletonClass α] {s : Set α} (hs : MeasurableSet s) (a : G₀) :
MeasurableSet (a • s) := by |
rcases eq_or_ne a 0 with (rfl | ha)
exacts [(subsingleton_zero_smul_set s).measurableSet, hs.const_smul_of_ne_zero ha]
|
/-
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.SmoothManifoldWithCorners
import Mathlib.Geometry.Manifold.LocalInvariantProperties
#align_import geometry.m... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 164 | 169 | theorem contDiffWithinAtProp_id (x : H) : ContDiffWithinAtProp I I n id univ x := by |
simp only [ContDiffWithinAtProp, id_comp, preimage_univ, univ_inter]
have : ContDiffWithinAt 𝕜 n id (range I) (I x) := contDiff_id.contDiffAt.contDiffWithinAt
refine this.congr (fun y hy => ?_) ?_
· simp only [ModelWithCorners.right_inv I hy, mfld_simps]
· simp only [mfld_simps]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Ma... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 230 | 251 | theorem contDiff_tsum (hf : ∀ i, ContDiff 𝕜 N (f i)) (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
(h'f : ∀ (k : ℕ) (i : α) (x : E), (k : ℕ∞) ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) :
ContDiff 𝕜 N fun x => ∑' i, f i x := by |
rw [contDiff_iff_continuous_differentiable]
constructor
· intro m hm
rw [iteratedFDeriv_tsum hf hv h'f hm]
refine continuous_tsum ?_ (hv m hm) ?_
· intro i
exact ContDiff.continuous_iteratedFDeriv hm (hf i)
· intro n x
exact h'f _ _ _ hm
· intro m hm
have h'm : ((m + 1 : ℕ) : ℕ∞... |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.M... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 252 | 255 | theorem coeff_eq_zero {w : σ → M} (hφ : IsWeightedHomogeneous w φ n) (d : σ →₀ ℕ)
(hd : weightedDegree w d ≠ n) : coeff d φ = 0 := by |
have aux := mt (@hφ d) hd
rwa [Classical.not_not] at aux
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 731 | 736 | theorem norm_integral_le (f : α →₁[μ] E) : ‖integral f‖ ≤ ‖f‖ :=
calc
‖integral f‖ = ‖integralCLM (E := E) f‖ := by | simp only [integral]
_ ≤ ‖integralCLM (α := α) (E := E) (μ := μ)‖ * ‖f‖ := le_opNorm _ _
_ ≤ 1 * ‖f‖ := mul_le_mul_of_nonneg_right norm_Integral_le_one <| norm_nonneg _
_ = ‖f‖ := one_mul _
|
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomia... | Mathlib/Algebra/Polynomial/Reverse.lean | 113 | 119 | theorem coeff_reflect (N : ℕ) (f : R[X]) (i : ℕ) : coeff (reflect N f) i = f.coeff (revAt N i) := by |
rcases f with ⟨f⟩
simp only [reflect, coeff]
calc
Finsupp.embDomain (revAt N) f i = Finsupp.embDomain (revAt N) f (revAt N (revAt N i)) := by
rw [revAt_invol]
_ = f (revAt N i) := Finsupp.embDomain_apply _ _ _
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
#align_import analysis.calculus.deriv.inv from "leanpro... | Mathlib/Analysis/Calculus/Deriv/Inv.lean | 48 | 61 | theorem hasStrictDerivAt_inv (hx : x ≠ 0) : HasStrictDerivAt Inv.inv (-(x ^ 2)⁻¹) x := by |
suffices
(fun p : 𝕜 × 𝕜 => (p.1 - p.2) * ((x * x)⁻¹ - (p.1 * p.2)⁻¹)) =o[𝓝 (x, x)] fun p =>
(p.1 - p.2) * 1 by
refine this.congr' ?_ (eventually_of_forall fun _ => mul_one _)
refine Eventually.mono ((isOpen_ne.prod isOpen_ne).mem_nhds ⟨hx, hx⟩) ?_
rintro ⟨y, z⟩ ⟨hy, hz⟩
simp only [mem_se... |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 52 | 56 | theorem hasDerivWithinAt_of_bilinear
(hu : HasDerivWithinAt u u' s x) (hv : HasDerivWithinAt v v' s x) :
HasDerivWithinAt (fun x ↦ B (u x) (v x)) (B (u x) v' + B u' (v x)) s x := by |
simpa using (B.hasFDerivWithinAt_of_bilinear
hu.hasFDerivWithinAt hv.hasFDerivWithinAt).hasDerivWithinAt
|
/-
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 | 99 | 104 | theorem Functor.initial_of_exists_of_isCofiltered [IsCofilteredOrEmpty C]
(h₁ : ∀ d, ∃ c, Nonempty (F.obj c ⟶ d)) (h₂ : ∀ {d : D} {c : C} (s s' : F.obj c ⟶ d),
∃ (c' : C) (t : c' ⟶ c), F.map t ≫ s = F.map t ≫ s') : Functor.Initial F := by |
suffices ∀ d, IsCofiltered (CostructuredArrow F d) from
initial_of_isCofiltered_costructuredArrow F
exact isCofiltered_costructuredArrow_of_isCofiltered_of_exists F h₁ h₂
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
/-!
# Ideal... | Mathlib/Algebra/Lie/IdealOperations.lean | 103 | 104 | theorem lie_coe_mem_lie (x : I) (m : N) : ⁅(x : L), (m : M)⁆ ∈ ⁅I, N⁆ := by |
rw [lieIdeal_oper_eq_span]; apply subset_lieSpan; use x, m
|
/-
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.Lattice
#align_import order.min_max from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# `max` and `min`
This ... | Mathlib/Order/MinMax.lean | 180 | 184 | theorem min_eq_iff : min a b = c ↔ a = c ∧ a ≤ b ∨ b = c ∧ b ≤ a := by |
constructor
· intro h
refine Or.imp (fun h' => ?_) (fun h' => ?_) (le_total a b) <;> exact ⟨by simpa [h'] using h, h'⟩
· rintro (⟨rfl, h⟩ | ⟨rfl, h⟩) <;> simp [h]
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 89 | 93 | theorem det_of_card_zero (h : Fintype.card n = 0) (M : Matrix n n R) : M.det = 1 := by |
rw [Fintype.card_eq_zero_iff] at h
suffices M = 1 by simp [this]
ext i
exact h.elim i
|
/-
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 | 72 | 74 | theorem Filter.Tendsto.mul_atTop {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l (𝓝 C))
(hg : Tendsto g l atTop) : Tendsto (fun x => f x * g x) l atTop := by |
simpa only [mul_comm] using hg.atTop_mul hC hf
|
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 320 | 322 | theorem inclusion_right (h : H ≤ K) (x : K) (hx : (x : G) ∈ H) : inclusion h ⟨x, hx⟩ = x := by |
cases x
rfl
|
/-
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 | 685 | 686 | theorem weightedVSubVSubWeights_apply_of_ne [DecidableEq ι] {i j t : ι} (hi : t ≠ i) (hj : t ≠ j) :
weightedVSubVSubWeights k i j t = 0 := by | simp [weightedVSubVSubWeights, hi, hj]
|
/-
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 | 189 | 196 | theorem tsum_meas_le_meas_iUnion_of_disjoint₀ {ι : Type*} [MeasurableSpace α] (μ : Measure α)
{As : ι → Set α} (As_mble : ∀ i : ι, NullMeasurableSet (As i) μ)
(As_disj : Pairwise (AEDisjoint μ on As)) : (∑' i, μ (As i)) ≤ μ (⋃ i, As i) := by |
rw [ENNReal.tsum_eq_iSup_sum, iSup_le_iff]
intro s
simp only [← measure_biUnion_finset₀ (fun _i _hi _j _hj hij => As_disj hij) fun i _ => As_mble i]
gcongr
exact iUnion_subset fun _ ↦ Subset.rfl
|
/-
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.Algebra.Order.Field.Pi
import Mathlib.Algebra.Order.UpperLower
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.Normed.Order.Basic
im... | Mathlib/Analysis/Normed/Order/UpperLower.lean | 94 | 109 | theorem IsUpperSet.mem_interior_of_forall_lt (hs : IsUpperSet s) (hx : x ∈ closure s)
(h : ∀ i, x i < y i) : y ∈ interior s := by |
cases nonempty_fintype ι
obtain ⟨ε, hε, hxy⟩ := Pi.exists_forall_pos_add_lt h
obtain ⟨z, hz, hxz⟩ := Metric.mem_closure_iff.1 hx _ hε
rw [dist_pi_lt_iff hε] at hxz
have hyz : ∀ i, z i < y i := by
refine fun i => (hxy _).trans_le' (sub_le_iff_le_add'.1 <| (le_abs_self _).trans ?_)
rw [← Real.norm_eq_a... |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
im... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 432 | 446 | theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) := by |
refine ⟨fun x y hxy => ?_, fun φ => ?_⟩
· contrapose! hxy
rcases exists_continuous_zero_one_of_isClosed (isClosed_singleton : _root_.IsClosed {x})
(isClosed_singleton : _root_.IsClosed {y}) (Set.disjoint_singleton.mpr hxy) with
⟨f, fx, fy, -⟩
rw [DFunLike.ne_iff]
use (⟨fun (x : ℝ) => (x :... |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 282 | 286 | theorem posPart_sub_negPart (f : α →ₛ ℝ) : f.posPart - f.negPart = f := by |
simp only [posPart, negPart]
ext a
rw [coe_sub]
exact max_zero_sub_eq_self (f a)
|
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mat... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 158 | 164 | theorem natTrailingDegree_le_of_ne_zero (h : coeff p n ≠ 0) : natTrailingDegree p ≤ n := by |
have : WithTop.some (natTrailingDegree p) = Nat.cast (natTrailingDegree p) := rfl
rw [← WithTop.coe_le_coe, this, ← trailingDegree_eq_natTrailingDegree]
· exact trailingDegree_le_of_ne_zero h
· intro h
subst h
exact h rfl
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7... | Mathlib/Data/Set/Prod.lean | 745 | 746 | theorem pi_nonempty_iff : (s.pi t).Nonempty ↔ ∀ i, ∃ x, i ∈ s → x ∈ t i := by |
simp [Classical.skolem, Set.Nonempty]
|
/-
Copyright (c) 2020 Zhangir Azerbayev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Zhangir Azerbayev
-/
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Data.Fintype.Perm
import Mathlib.LinearAlgebra.Multilinear.Basis
#align_import linear_algebra.... | Mathlib/LinearAlgebra/Alternating/Basic.lean | 571 | 576 | theorem compLinearMap_zero [Nonempty ι] (f : M [⋀^ι]→ₗ[R] N) :
f.compLinearMap (0 : M₂ →ₗ[R] M) = 0 := by |
ext
-- Porting note: Was `simp_rw [.., ← Pi.zero_def, map_zero, zero_apply]`.
simp_rw [compLinearMap_apply, LinearMap.zero_apply, ← Pi.zero_def, zero_apply]
exact map_zero f
|
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d... | Mathlib/Order/Interval/Set/Disjoint.lean | 245 | 248 | theorem iUnion_Ici_eq_Ioi_iInf {R : Type*} [CompleteLinearOrder R] {f : ι → R}
(no_least_elem : ⨅ i, f i ∉ range f) : ⋃ i : ι, Ici (f i) = Ioi (⨅ i, f i) := by |
simp only [← IsGLB.biUnion_Ici_eq_Ioi (@isGLB_iInf _ _ _ f) no_least_elem, mem_range,
iUnion_exists, iUnion_iUnion_eq']
|
/-
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, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde9... | Mathlib/MeasureTheory/Integral/Average.lean | 662 | 677 | theorem measure_le_setLaverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞)
(hf : AEMeasurable f (μ.restrict s)) : 0 < μ {x ∈ s | f x ≤ ⨍⁻ a in s, f a ∂μ} := by |
obtain h | h := eq_or_ne (∫⁻ a in s, f a ∂μ) ∞
· simpa [mul_top, hμ₁, laverage, h, top_div_of_ne_top hμ₁, pos_iff_ne_zero] using hμ
have := measure_le_setAverage_pos hμ hμ₁ (integrable_toReal_of_lintegral_ne_top hf h)
rw [← setOf_inter_eq_sep, ← Measure.restrict_apply₀
(hf.aestronglyMeasurable.nullMeasurab... |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Co... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 283 | 287 | theorem CPolynomialOn.iterated_deriv (h : CPolynomialOn 𝕜 f s) (n : ℕ) :
CPolynomialOn 𝕜 (deriv^[n] f) s := by |
induction' n with n IH
· exact h
· simpa only [Function.iterate_succ', Function.comp_apply] using IH.deriv
|
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johanes Hölzl, Patrick Massot, Yury Kudryashov, Kevin Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@... | Mathlib/Order/Filter/Prod.lean | 426 | 428 | theorem map_pure_prod (f : α → β → γ) (a : α) (B : Filter β) :
map (Function.uncurry f) (pure a ×ˢ B) = map (f a) B := by |
rw [Filter.pure_prod]; 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
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.Special... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 206 | 209 | theorem HasDerivWithinAt.cpow (hf : HasDerivWithinAt f f' s x) (hg : HasDerivWithinAt g g' s x)
(h0 : f x ∈ slitPlane) : HasDerivWithinAt (fun x => f x ^ g x)
(g x * f x ^ (g x - 1) * f' + f x ^ g x * Complex.log (f x) * g') s x := by |
simpa only [aux] using (hf.hasFDerivWithinAt.cpow hg h0).hasDerivWithinAt
|
/-
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 | 253 | 254 | theorem oangle_neg_self_left {x : V} (hx : x ≠ 0) : o.oangle (-x) x = π := by |
simp [oangle_neg_left, hx]
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
#align_import category_theory.abelian.images from "leanprover-community/mathlib"@"9e7c80f638149bfb3... | Mathlib/CategoryTheory/Abelian/Images.lean | 122 | 123 | theorem coimage_image_factorisation : coimage.π f ≫ coimageImageComparison f ≫ image.ι f = f := by |
simp [coimageImageComparison]
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 888 | 893 | theorem realize_alls {φ : L.BoundedFormula α n} {v : α → M} :
φ.alls.Realize v ↔ ∀ xs : Fin n → M, φ.Realize v xs := by |
induction' n with n ih
· exact Unique.forall_iff.symm
· simp only [alls, ih, Realize]
exact ⟨fun h xs => Fin.snoc_init_self xs ▸ h _ _, fun h xs x => h (Fin.snoc xs x)⟩
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Scott Morrison, Ainsley Pahljina
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathli... | Mathlib/NumberTheory/LucasLehmer.lean | 154 | 158 | theorem sZMod_eq_s (p' : ℕ) (i : ℕ) : sZMod (p' + 2) i = (s i : ZMod (2 ^ (p' + 2) - 1)) := by |
induction' i with i ih
· dsimp [s, sZMod]
norm_num
· push_cast [s, sZMod, ih]; rfl
|
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical.Basic
import Mathlib.Algebra.Or... | Mathlib/Data/Real/NNReal.lean | 919 | 920 | theorem inv_le {r p : ℝ≥0} (h : r ≠ 0) : r⁻¹ ≤ p ↔ 1 ≤ r * p := by |
rw [← mul_le_mul_left (pos_iff_ne_zero.2 h), mul_inv_cancel h]
|
/-
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 | 230 | 230 | theorem left_le_or : ∀ x y : Bool, x ≤ (x || y) := by | decide
|
/-
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.Analysis.Convex.StrictConvexBetween
import Mathlib.Geometry.Euclidean.Basic
#align_import geometry.euclidean.sphere.basic from "leanprover-community/mathl... | Mathlib/Geometry/Euclidean/Sphere/Basic.lean | 331 | 356 | theorem inner_pos_or_eq_of_dist_le_radius {s : Sphere P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s)
(hp₂ : dist p₂ s.center ≤ s.radius) : 0 < ⟪p₁ -ᵥ p₂, p₁ -ᵥ s.center⟫ ∨ p₁ = p₂ := by |
by_cases h : p₁ = p₂; · exact Or.inr h
refine Or.inl ?_
rw [mem_sphere] at hp₁
rw [← vsub_sub_vsub_cancel_right p₁ p₂ s.center, inner_sub_left,
real_inner_self_eq_norm_mul_norm, sub_pos]
refine lt_of_le_of_ne
((real_inner_le_norm _ _).trans (mul_le_mul_of_nonneg_right ?_ (norm_nonneg _))) ?_
· rwa ... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Set.Image
#align_import data.fintype.car... | Mathlib/Data/Fintype/Card.lean | 1,217 | 1,261 | theorem List.exists_pw_disjoint_with_card {α : Type*} [Fintype α]
{c : List ℕ} (hc : c.sum ≤ Fintype.card α) :
∃ o : List (List α),
o.map length = c ∧ (∀ s ∈ o, s.Nodup) ∧ Pairwise List.Disjoint o := by |
let klift (n : ℕ) (hn : n < Fintype.card α) : Fin (Fintype.card α) :=
(⟨n, hn⟩ : Fin (Fintype.card α))
let klift' (l : List ℕ) (hl : ∀ a ∈ l, a < Fintype.card α) :
List (Fin (Fintype.card α)) := List.pmap klift l hl
have hc'_lt : ∀ l ∈ c.ranges, ∀ n ∈ l, n < Fintype.card α := by
intro l hl n hn
a... |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
import Mathlib.Ord... | Mathlib/Order/Filter/Bases.lean | 419 | 423 | theorem hasBasis_self {l : Filter α} {P : Set α → Prop} :
HasBasis l (fun s => s ∈ l ∧ P s) id ↔ ∀ t ∈ l, ∃ r ∈ l, P r ∧ r ⊆ t := by |
simp only [hasBasis_iff, id, and_assoc]
exact forall_congr' fun s =>
⟨fun h => h.1, fun h => ⟨h, fun ⟨t, hl, _, hts⟩ => mem_of_superset hl hts⟩⟩
|
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