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) 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 | 890 | 891 | theorem integral_comp_neg : (∫ x in a..b, f (-x)) = ∫ x in -b..-a, f x := by |
simpa only [zero_sub] using integral_comp_sub_left f 0
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Init.Data.Fin.Basic
#align_import data.list.nodup from "leanprover-... | Mathlib/Data/List/Nodup.lean | 219 | 220 | theorem nodup_append_comm {l₁ l₂ : List α} : Nodup (l₁ ++ l₂) ↔ Nodup (l₂ ++ l₁) := by |
simp only [nodup_append, and_left_comm, disjoint_comm]
|
/-
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 | 1,392 | 1,393 | theorem sep_eq_empty_iff_mem_false : { x ∈ s | p x } = ∅ ↔ ∀ x ∈ s, ¬p x := by |
simp_rw [ext_iff, mem_sep_iff, mem_empty_iff_false, iff_false_iff, not_and]
|
/-
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.Algebra.Subalgebra.Basic
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.RingTheory.Adjoin.Basic
#align_import topology.algebra.al... | Mathlib/Topology/Algebra/Algebra.lean | 130 | 137 | theorem Subalgebra.topologicalClosure_comap_homeomorph (s : Subalgebra R A) {B : Type*}
[TopologicalSpace B] [Ring B] [TopologicalRing B] [Algebra R B] (f : B →ₐ[R] A) (f' : B ≃ₜ A)
(w : (f : B → A) = f') : s.topologicalClosure.comap f = (s.comap f).topologicalClosure := by |
apply SetLike.ext'
simp only [Subalgebra.topologicalClosure_coe]
simp only [Subalgebra.coe_comap, Subsemiring.coe_comap, AlgHom.coe_toRingHom]
rw [w]
exact f'.preimage_closure _
|
/-
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.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.No... | Mathlib/Analysis/NormedSpace/AffineIsometry.lean | 733 | 737 | theorem symm_constVSub (p : P) :
(constVSub 𝕜 p).symm =
(LinearIsometryEquiv.neg 𝕜).toAffineIsometryEquiv.trans (vaddConst 𝕜 p) := by |
ext
rfl
|
/-
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 | 305 | 312 | theorem coeff_eq_zero_of_lt_natTrailingDegree {p : R[X]} {n : ℕ} (h : n < p.natTrailingDegree) :
p.coeff n = 0 := by |
apply coeff_eq_zero_of_lt_trailingDegree
by_cases hp : p = 0
· rw [hp, trailingDegree_zero]
exact WithTop.coe_lt_top n
· rw [trailingDegree_eq_natTrailingDegree hp]
exact WithTop.coe_lt_coe.2 h
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set... | Mathlib/SetTheory/Cardinal/Basic.lean | 632 | 632 | theorem lift_bit1 (a : Cardinal) : lift.{v} (bit1 a) = bit1 (lift.{v} a) := by | simp [bit1]
|
/-
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.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.ContinuousOn
import Mathlib.Topology.Instances.ENNReal
#align_import topology.semicontin... | Mathlib/Topology/Semicontinuous.lean | 784 | 785 | theorem upperSemicontinuousOn_univ_iff : UpperSemicontinuousOn f univ ↔ UpperSemicontinuous f := by |
simp [UpperSemicontinuousOn, UpperSemicontinuous, upperSemicontinuousWithinAt_univ_iff]
|
/-
Copyright (c) 2023 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Bicategory.Coherence
/-!
# Adjunctions in bicategories
For 1-morphisms `f : a ⟶ b` and `g : b... | Mathlib/CategoryTheory/Bicategory/Adjunction.lean | 220 | 226 | theorem right_triangle_of_left_triangle (h : leftZigzag η.hom ε.hom = (λ_ f).hom ≫ (ρ_ f).inv) :
rightZigzag η.hom ε.hom = (ρ_ g).hom ≫ (λ_ g).inv := by |
rw [← cancel_epi (rightZigzag η.hom ε.hom ≫ (λ_ g).hom ≫ (ρ_ g).inv)]
calc
_ = rightZigzag η.hom ε.hom ⊗≫ rightZigzag η.hom ε.hom := by coherence
_ = rightZigzag η.hom ε.hom := rightZigzag_idempotent_of_left_triangle _ _ h
_ = _ := by simp
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Set.Card
import Mathlib.Order.Minimal
import Mathlib.Data.Matroid.Init
/-!
# Matroids
A `Matroid` is a structure that combinatorially abstracts
the ... | Mathlib/Data/Matroid/Basic.lean | 912 | 916 | theorem Basis.basis_sUnion {Xs : Set (Set α)} (hne : Xs.Nonempty) (h : ∀ X ∈ Xs, M.Basis I X) :
M.Basis I (⋃₀ Xs) := by |
rw [sUnion_eq_iUnion]
have := Iff.mpr nonempty_coe_sort hne
exact Basis.basis_iUnion _ fun X ↦ (h X X.prop)
|
/-
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.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb3... | Mathlib/Logic/Equiv/Fin.lean | 248 | 249 | theorem finSuccEquivLast_last : finSuccEquivLast (Fin.last n) = none := by |
simp [finSuccEquivLast]
|
/-
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.RingTheory.WittVector.Truncated
import Mathlib.RingTheory.WittVector.Identities
import Mathlib.NumberTheory.Padics.RingHoms
#align_im... | Mathlib/RingTheory/WittVector/Compare.lean | 127 | 130 | theorem commutes_symm {m : ℕ} (hm : n ≤ m) :
(zmodEquivTrunc p n).symm.toRingHom.comp (truncate hm) =
(ZMod.castHom (pow_dvd_pow p hm) _).comp (zmodEquivTrunc p m).symm.toRingHom := by |
ext; apply commutes_symm'
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
#align_import order.heyting.basic from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# Heyting algebr... | Mathlib/Order/Heyting/Basic.lean | 670 | 672 | theorem sdiff_le_sdiff_of_sup_le_sup_left (h : c ⊔ a ≤ c ⊔ b) : a \ c ≤ b \ c := by |
rw [← sup_sdiff_left_self, ← @sup_sdiff_left_self _ _ _ b]
exact sdiff_le_sdiff_right h
|
/-
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.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlg... | Mathlib/Algebra/Lie/Nilpotent.lean | 313 | 316 | theorem iInf_lcs_le_of_isNilpotent_quot (h : IsNilpotent R L (M ⧸ N)) :
⨅ k, lowerCentralSeries R L M k ≤ N := by |
obtain ⟨k, hk⟩ := (isNilpotent_quotient_iff R L M N).mp h
exact iInf_le_of_le k hk
|
/-
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 | 1,188 | 1,193 | theorem dual_finrank_eq : finrank K (Module.Dual K V) = finrank K V := by |
by_cases h : FiniteDimensional K V
· classical exact LinearEquiv.finrank_eq (Basis.ofVectorSpace K V).toDualEquiv.symm
rw [finrank_eq_zero_of_basis_imp_false, finrank_eq_zero_of_basis_imp_false]
· exact fun _ b ↦ h (Module.Finite.of_basis b)
· exact fun _ b ↦ h ((Module.finite_dual_iff K).mp <| Module.Finite... |
/-
Copyright (c) 2019 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Batteries.Data.List.Basic
namespace Batteries
/--
`AssocList α β` is "the same as" `List (α × β)`, but flattening the structure
lea... | .lake/packages/batteries/Batteries/Data/AssocList.lean | 139 | 141 | theorem find?_eq_findEntry? [BEq α] (a : α) (l : AssocList α β) :
find? a l = (l.findEntry? a).map (·.2) := by |
induction l <;> simp [find?, List.find?_cons]; split <;> simp [*]
|
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_imp... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 52 | 53 | theorem leftUnitor_inv_tensor_id (X Y : C) : (λ_ X).inv ⊗ 𝟙 Y = (λ_ _).inv ≫ (α_ _ _ _).inv := by |
coherence
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Aesop
import Mathlib.Order.BoundedOrder
#align_import order.disjoint from "leanprover-community/mathlib"@"22c4d2ff43714b6ff724b2745ccfdc0f236a4a76"
/-!
# Dis... | Mathlib/Order/Disjoint.lean | 528 | 533 | theorem inf_left_le_of_le_sup_right (h : IsCompl x y) (hle : a ≤ b ⊔ y) : a ⊓ x ≤ b :=
calc
a ⊓ x ≤ (b ⊔ y) ⊓ x := inf_le_inf hle le_rfl
_ = b ⊓ x ⊔ y ⊓ x := inf_sup_right _ _ _
_ = b ⊓ x := by | rw [h.symm.inf_eq_bot, sup_bot_eq]
_ ≤ b := inf_le_left
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Module.Equiv
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finsupp.Basic
#align_import data.finsupp.to_dfinsupp from "leanprover-community/... | Mathlib/Data/Finsupp/ToDFinsupp.lean | 123 | 126 | theorem DFinsupp.toFinsupp_single (i : ι) (m : M) :
(DFinsupp.single i m : Π₀ _ : ι, M).toFinsupp = Finsupp.single i m := by |
ext
simp [Finsupp.single_apply, DFinsupp.single_apply]
|
/-
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 | 386 | 387 | theorem closure_eq_mrange (s : Set M) : closure s = mrange (FreeMonoid.lift ((↑) : s → M)) := by |
rw [FreeMonoid.mrange_lift, Subtype.range_coe]
|
/-
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.Basic
import Mathlib.Topology.Order.ProjIcc
#al... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 390 | 390 | theorem arccos_one : arccos 1 = 0 := by | simp [arccos]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 1,345 | 1,348 | theorem _root_.Summable.countable_support_ennreal {f : α → ℝ≥0∞} (h : ∑' (i : α), f i ≠ ∞) :
f.support.Countable := by |
lift f to α → ℝ≥0 using ENNReal.ne_top_of_tsum_ne_top h
simpa [support] using (ENNReal.tsum_coe_ne_top_iff_summable.1 h).countable_support_nnreal
|
/-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Junyan Xu, Jack McKoen
-/
import Mathlib.RingTheory.Valuation.ValuationRing
import Mathlib.RingTheory.Localization.AsSubring
import Mathlib.Algebra.Ring.Subring.Pointwise
impor... | Mathlib/RingTheory/Valuation/ValuationSubring.lean | 461 | 462 | theorem valuationSubring_valuation : A.valuation.valuationSubring = A := by |
ext; rw [← A.valuation_le_one_iff]; rfl
|
/-
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 | 789 | 795 | theorem prod_add (ν' : Measure β) [SFinite ν'] : μ.prod (ν + ν') = μ.prod ν + μ.prod ν' := by |
simp_rw [← sum_sFiniteSeq ν, ← sum_sFiniteSeq ν', sum_add_sum, ← sum_sFiniteSeq μ, prod_sum,
sum_add_sum]
congr
ext1 i
refine prod_eq fun s t _ _ => ?_
simp_rw [add_apply, prod_prod, left_distrib]
|
/-
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.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 963 | 965 | theorem tan_pi_div_six : tan (π / 6) = 1 / sqrt 3 := by |
rw [tan_eq_sin_div_cos, sin_pi_div_six, cos_pi_div_six]
ring
|
/-
Copyright (c) 2023 Shogo Saito. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shogo Saito. Adapted for mathlib by Hunter Monroe
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.GCD.BigOperators
/-!
# Chinese Re... | Mathlib/Data/Nat/ChineseRemainder.lean | 160 | 164 | theorem chineseRemainderOfFinset_lt_prod {t : Finset ι}
(hs : ∀ i ∈ t, s i ≠ 0) (pp : Set.Pairwise t (Coprime on s)) :
chineseRemainderOfFinset a s t hs pp < ∏ i ∈ t, s i := by |
simpa [chineseRemainderOfFinset] using
chineseRemainderOfMultiset_lt_prod a s t.nodup (by simpa using hs) (by simpa using pp)
|
/-
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 | 292 | 293 | theorem oangle_smul_left_of_pos (x y : V) {r : ℝ} (hr : 0 < r) :
o.oangle (r • x) y = o.oangle x y := by | simp [oangle, Complex.arg_real_mul _ hr]
|
/-
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.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 1,297 | 1,305 | theorem Valid'.rotateR {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
(H1 : ¬size l + size r ≤ 1) (H2 : delta * size r < size l)
(H3 : 2 * size l ≤ 9 * size r + 5 ∨ size l ≤ 3) : Valid' o₁ (@rotateR α l x r) o₂ := by |
refine Valid'.dual_iff.2 ?_
rw [dual_rotateR]
refine hr.dual.rotateL hl.dual ?_ ?_ ?_
· rwa [size_dual, size_dual, add_comm]
· rwa [size_dual, size_dual]
· rwa [size_dual, size_dual]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 836 | 843 | theorem continuousWithinAt_update_same [DecidableEq α] {f : α → β} {s : Set α} {x : α} {y : β} :
ContinuousWithinAt (update f x y) s x ↔ Tendsto f (𝓝[s \ {x}] x) (𝓝 y) :=
calc
ContinuousWithinAt (update f x y) s x ↔ Tendsto (update f x y) (𝓝[s \ {x}] x) (𝓝 y) := by |
{ rw [← continuousWithinAt_diff_self, ContinuousWithinAt, update_same] }
_ ↔ Tendsto f (𝓝[s \ {x}] x) (𝓝 y) :=
tendsto_congr' <| eventually_nhdsWithin_iff.2 <| eventually_of_forall
fun z hz => update_noteq hz.2 _ _
|
/-
Copyright (c) 2019 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Data.Bracket
import Mathlib.LinearAlgebra.Basic
#align_import algebra.lie.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd298... | Mathlib/Algebra/Lie/Basic.lean | 833 | 834 | theorem coe_linear_mk (f : M →ₗ[R] N) (h) : ((⟨f, h⟩ : M →ₗ⁅R,L⁆ N) : M →ₗ[R] N) = f := by |
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.Data.Fin.Tuple.Basic
import Mathlib.Data.List.Join
#align_import data.list.of_fn from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe8353... | Mathlib/Data/List/OfFn.lean | 125 | 131 | theorem ofFn_succ' {n} (f : Fin (succ n) → α) :
ofFn f = (ofFn fun i => f (Fin.castSucc i)).concat (f (Fin.last _)) := by |
induction' n with n IH
· rw [ofFn_zero, concat_nil, ofFn_succ, ofFn_zero]
rfl
· rw [ofFn_succ, IH, ofFn_succ, concat_cons, Fin.castSucc_zero]
congr
|
/-
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 | 796 | 813 | theorem mem_span_finite_of_mem_span {S : Set M} {x : M} (hx : x ∈ span R S) :
∃ T : Finset M, ↑T ⊆ S ∧ x ∈ span R (T : Set M) := by |
classical
refine span_induction hx (fun x hx => ?_) ?_ ?_ ?_
· refine ⟨{x}, ?_, ?_⟩
· rwa [Finset.coe_singleton, Set.singleton_subset_iff]
· rw [Finset.coe_singleton]
exact Submodule.mem_span_singleton_self x
· use ∅
simp
· rintro x y ⟨X, hX, hxX⟩ ⟨Y, hY, hyY⟩
refine ⟨X ∪ Y, ?_, ?_⟩
... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
/-!
# Restricting a measure to a subset or a subtype
Given a measure `μ` on a type `α` and a subse... | Mathlib/MeasureTheory/Measure/Restrict.lean | 271 | 273 | theorem restrict_union₀ (h : AEDisjoint μ s t) (ht : NullMeasurableSet t μ) :
μ.restrict (s ∪ t) = μ.restrict s + μ.restrict t := by |
simp [← restrict_union_add_inter₀ s ht, restrict_zero_set h]
|
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.Algebra.FreeAlgebra
import Mathlib.Algebra.RingQuot
import Mathlib.Algebra.TrivSqZeroExt
import Mathlib.Algebra.Algebra.Operations
import Mathlib.LinearAlgebra... | Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean | 159 | 162 | theorem lift_ι_apply {A : Type*} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) (x) :
lift R f (ι R x) = f x := by |
conv_rhs => rw [← ι_comp_lift f]
rfl
|
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subri... | Mathlib/LinearAlgebra/Ray.lean | 348 | 351 | theorem units_smul_of_pos (u : Rˣ) (hu : 0 < (u.1 : R)) (v : Module.Ray R M) : u • v = v := by |
induction v using Module.Ray.ind
rw [smul_rayOfNeZero, ray_eq_iff]
exact SameRay.sameRay_pos_smul_left _ hu
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.... | Mathlib/RingTheory/WittVector/Identities.lean | 95 | 96 | theorem FractionRing.p_nonzero [Nontrivial R] [CharP R p] : (p : FractionRing (𝕎 R)) ≠ 0 := by |
simpa using (IsFractionRing.injective (𝕎 R) (FractionRing (𝕎 R))).ne (WittVector.p_nonzero _ _)
|
/-
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.Data.Matrix.PEquiv
import Mathlib.Data.Set.Card
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
/-!
# Permu... | Mathlib/LinearAlgebra/Matrix/Permutation.lean | 47 | 50 | theorem trace_permutation [AddCommMonoidWithOne R] :
trace (σ.permMatrix R) = (Function.fixedPoints σ).ncard := by |
delta trace
simp [toPEquiv_apply, ← Set.ncard_coe_Finset, Function.fixedPoints, Function.IsFixedPt]
|
/-
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 | 513 | 519 | theorem restrict_boxes_of_le (π : Prepartition I) (h : I ≤ J) : (π.restrict J).boxes = π.boxes := by |
simp only [restrict, ofWithBot, eraseNone_eq_biUnion]
refine Finset.image_biUnion.trans ?_
refine (Finset.biUnion_congr rfl ?_).trans Finset.biUnion_singleton_eq_self
intro J' hJ'
rw [inf_of_le_right, ← WithBot.some_eq_coe, Option.toFinset_some]
exact WithBot.coe_le_coe.2 ((π.le_of_mem hJ').trans h)
|
/-
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.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 1,281 | 1,282 | theorem mkPiRing_eq_zero_iff [Fintype ι] (z : M₂) : MultilinearMap.mkPiRing R ι z = 0 ↔ z = 0 := by |
rw [← mkPiRing_zero, mkPiRing_eq_iff]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c066... | Mathlib/Algebra/GCDMonoid/Finset.lean | 212 | 223 | theorem gcd_eq_zero_iff : s.gcd f = 0 ↔ ∀ x : β, x ∈ s → f x = 0 := by |
rw [gcd_def, Multiset.gcd_eq_zero_iff]
constructor <;> intro h
· intro b bs
apply h (f b)
simp only [Multiset.mem_map, mem_def.1 bs]
use b
simp only [mem_def.1 bs, eq_self_iff_true, and_self]
· intro a as
rw [Multiset.mem_map] at as
rcases as with ⟨b, ⟨bs, rfl⟩⟩
apply h b (mem_def.1... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,293 | 2,296 | theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : (a^c) ≤ a ^< b := by |
refine le_ciSup (f := fun y : Iio b => a ^ (y : Cardinal)) ?_ ⟨c, h⟩
rw [← image_eq_range]
exact bddAbove_image.{u, u} _ bddAbove_Iio
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprove... | Mathlib/Analysis/Complex/ReImTopology.lean | 139 | 140 | theorem frontier_setOf_im_le (a : ℝ) : frontier { z : ℂ | z.im ≤ a } = { z | z.im = a } := by |
simpa only [frontier_Iic] using frontier_preimage_im (Iic a)
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 643 | 649 | theorem card_support_prod_list_of_pairwise_disjoint {l : List (Perm α)} (h : l.Pairwise Disjoint) :
l.prod.support.card = (l.map (Finset.card ∘ support)).sum := by |
induction' l with a t ih
· exact card_support_eq_zero.mpr rfl
· obtain ⟨ha, ht⟩ := List.pairwise_cons.1 h
rw [List.prod_cons, List.map_cons, List.sum_cons, ← ih ht]
exact (disjoint_prod_right _ ha).card_support_mul
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.Algebraic
#align_import field_the... | Mathlib/FieldTheory/Minpoly/Field.lean | 68 | 76 | theorem dvd {p : A[X]} (hp : Polynomial.aeval x p = 0) : minpoly A x ∣ p := by |
by_cases hp0 : p = 0
· simp only [hp0, dvd_zero]
have hx : IsIntegral A x := IsAlgebraic.isIntegral ⟨p, hp0, hp⟩
rw [← modByMonic_eq_zero_iff_dvd (monic hx)]
by_contra hnz
apply degree_le_of_ne_zero A x hnz
((aeval_modByMonic_eq_self_of_root (monic hx) (aeval _ _)).trans hp) |>.not_lt
exact degree_mo... |
/-
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.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.... | Mathlib/Topology/Compactness/Compact.lean | 512 | 516 | theorem IsCompact.finite_of_discrete [DiscreteTopology X] (hs : IsCompact s) : s.Finite := by |
have : ∀ x : X, ({x} : Set X) ∈ 𝓝 x := by simp [nhds_discrete]
rcases hs.elim_nhds_subcover (fun x => {x}) fun x _ => this x with ⟨t, _, hst⟩
simp only [← t.set_biUnion_coe, biUnion_of_singleton] at hst
exact t.finite_toSet.subset hst
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Monad
#align_import probability.probability_mass_function.constructions from "leanprover-community/mat... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 101 | 105 | theorem toMeasure_map_apply [MeasurableSpace α] [MeasurableSpace β] (hf : Measurable f)
(hs : MeasurableSet s) : (p.map f).toMeasure s = p.toMeasure (f ⁻¹' s) := by |
rw [toMeasure_apply_eq_toOuterMeasure_apply _ s hs,
toMeasure_apply_eq_toOuterMeasure_apply _ (f ⁻¹' s) (measurableSet_preimage hf hs)]
exact toOuterMeasure_map_apply f p s
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth, Johan Commelin
-/
import Mathlib.RingTheory.WittVector.Domain
import Mathlib.RingTheory.WittVector.MulCoeff
import Mathlib.RingTheory.DiscreteValuati... | Mathlib/RingTheory/WittVector/DiscreteValuationRing.lean | 121 | 135 | theorem exists_eq_pow_p_mul (a : 𝕎 k) (ha : a ≠ 0) :
∃ (m : ℕ) (b : 𝕎 k), b.coeff 0 ≠ 0 ∧ a = (p : 𝕎 k) ^ m * b := by |
obtain ⟨m, c, hc, hcm⟩ := WittVector.verschiebung_nonzero ha
obtain ⟨b, rfl⟩ := (frobenius_bijective p k).surjective.iterate m c
rw [WittVector.iterate_frobenius_coeff] at hc
have := congr_fun (WittVector.verschiebung_frobenius_comm.comp_iterate m) b
simp only [Function.comp_apply] at this
rw [← this] at h... |
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.Algebra.Lie.Abelian
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.Algebra.Lie.... | Mathlib/Algebra/Lie/Classical.lean | 222 | 227 | theorem soIndefiniteEquiv_apply {i : R} (hi : i * i = -1) (A : so' p q R) :
(soIndefiniteEquiv p q R hi A : Matrix (Sum p q) (Sum p q) R) =
(Pso p q R i)⁻¹ * (A : Matrix (Sum p q) (Sum p q) R) * Pso p q R i := by |
rw [soIndefiniteEquiv, LieEquiv.trans_apply, LieEquiv.ofEq_apply]
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [skewAdjointMatricesLieSubalgebraEquiv_apply]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-com... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 119 | 121 | theorem iteratedDerivWithin_one {x : 𝕜} (h : UniqueDiffWithinAt 𝕜 s x) :
iteratedDerivWithin 1 f s x = derivWithin f s x := by |
simp only [iteratedDerivWithin, iteratedFDerivWithin_one_apply h]; rfl
|
/-
Copyright (c) 2021 Manuel Candales. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Manuel Candales, Benjamin Davidson
-/
import Mathlib.Geometry.Euclidean.Sphere.Power
import Mathlib.Geometry.Euclidean.Triangle
#align_import geometry.euclidean.sphere.ptolemy from "... | Mathlib/Geometry/Euclidean/Sphere/Ptolemy.lean | 53 | 70 | theorem mul_dist_add_mul_dist_eq_mul_dist_of_cospherical {a b c d p : P}
(h : Cospherical ({a, b, c, d} : Set P)) (hapc : ∠ a p c = π) (hbpd : ∠ b p d = π) :
dist a b * dist c d + dist b c * dist d a = dist a c * dist b d := by |
have h' : Cospherical ({a, c, b, d} : Set P) := by rwa [Set.insert_comm c b {d}]
have hmul := mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi h' hapc hbpd
have hbp := left_dist_ne_zero_of_angle_eq_pi hbpd
have h₁ : dist c d = dist c p / dist b p * dist a b := by
rw [dist_mul_of_eq_angle_of_dist_mul b p ... |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import fi... | Mathlib/FieldTheory/MvPolynomial.lean | 54 | 55 | theorem rank_mvPolynomial : Module.rank K (MvPolynomial σ K) = Cardinal.mk (σ →₀ ℕ) := by |
rw [← Cardinal.lift_inj, ← (basisMonomials σ K).mk_eq_rank]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_seri... | Mathlib/RingTheory/PowerSeries/Order.lean | 362 | 365 | theorem divided_by_X_pow_order_of_X_eq_one : divided_by_X_pow_order X_ne_zero = (1 : R⟦X⟧) := by |
rw [← mul_eq_left₀ X_ne_zero]
simpa only [order_X, X_ne_zero, PartENat.get_one, pow_one, Ne, not_false_iff] using
self_eq_X_pow_order_mul_divided_by_X_pow_order (@X_ne_zero R _ _)
|
/-
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 | 650 | 651 | theorem erase_ne {a a' : α} {f : α →₀ M} (h : a' ≠ a) : (f.erase a) a' = f a' := by |
classical simp only [erase, coe_mk, h, ite_false]
|
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "... | Mathlib/NumberTheory/Padics/RingHoms.lean | 425 | 440 | theorem zmod_cast_comp_toZModPow (m n : ℕ) (h : m ≤ n) :
(ZMod.castHom (pow_dvd_pow p h) (ZMod (p ^ m))).comp (@toZModPow p _ n) = @toZModPow p _ m := by |
apply ZMod.ringHom_eq_of_ker_eq
ext x
rw [RingHom.mem_ker, RingHom.mem_ker]
simp only [Function.comp_apply, ZMod.castHom_apply, RingHom.coe_comp]
simp only [toZModPow, toZModHom, RingHom.coe_mk]
dsimp
rw [ZMod.cast_natCast (pow_dvd_pow p h),
zmod_congr_of_sub_mem_span m (x.appr n) (x.appr n) (x.appr ... |
/-
Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.PeakFunction
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
/-!
# Fourier inversion formula
In a fin... | Mathlib/Analysis/Fourier/Inversion.lean | 186 | 190 | theorem Continuous.fourier_inversion_inv (h : Continuous f)
(hf : Integrable f) (h'f : Integrable (𝓕 f)) :
𝓕 (𝓕⁻ f) = f := by |
ext v
exact hf.fourier_inversion_inv h'f h.continuousAt
|
/-
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.Constructions
import Mathlib.Topology.ContinuousOn
#align_import topology.bases from "leanprover-community/mathlib"@"bcfa7268... | Mathlib/Topology/Bases.lean | 495 | 515 | theorem IsSeparable.univ_pi {ι : Type*} [Countable ι] {X : ι → Type*} {s : ∀ i, Set (X i)}
[∀ i, TopologicalSpace (X i)] (h : ∀ i, IsSeparable (s i)) :
IsSeparable (univ.pi s) := by |
classical
rcases eq_empty_or_nonempty (univ.pi s) with he | ⟨f₀, -⟩
· rw [he]
exact countable_empty.isSeparable
· choose c c_count hc using h
haveI := fun i ↦ (c_count i).to_subtype
set g : (I : Finset ι) × ((i : I) → c i) → (i : ι) → X i := fun ⟨I, f⟩ i ↦
if hi : i ∈ I then f ⟨i, hi⟩ else f₀... |
/-
Copyright (c) 2021 Gabriel Moise. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Moise, Yaël Dillies, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 102 | 103 | theorem incMatrix_apply_eq_zero_iff : G.incMatrix R a e = 0 ↔ e ∉ G.incidenceSet a := by |
simp only [incMatrix_apply, Set.indicator_apply_eq_zero, Pi.one_apply, one_ne_zero]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.MetricSpace.IsometricSMul
#alig... | Mathlib/Analysis/Normed/Group/Pointwise.lean | 252 | 252 | theorem div_ball : s / ball x δ = x⁻¹ • thickening δ s := by | simp [div_eq_mul_inv]
|
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.Category.GroupCat.EquivalenceGroupAddGroup
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.category.Group.epi_mono from "leanprover... | Mathlib/Algebra/Category/GroupCat/EpiMono.lean | 256 | 260 | theorem h_apply_infinity (x : B) (hx : x ∈ f.range) : (h x) ∞ = ∞ := by |
change ((τ).symm.trans (g x)).trans τ _ = _
simp only [MonoidHom.coe_mk, Equiv.toFun_as_coe, Equiv.coe_trans, Function.comp_apply]
rw [τ_symm_apply_infinity, g_apply_fromCoset]
simpa only using τ_apply_fromCoset' f x hx
|
/-
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.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 865 | 871 | theorem nhds_eq_uniformity_prod {a b : α} :
𝓝 (a, b) =
(𝓤 α).lift' fun s : Set (α × α) => { y : α | (y, a) ∈ s } ×ˢ { y : α | (b, y) ∈ s } := by |
rw [nhds_prod_eq, nhds_nhds_eq_uniformity_uniformity_prod, lift_lift'_same_eq_lift']
· exact fun s => monotone_const.set_prod monotone_preimage
· refine fun t => Monotone.set_prod ?_ monotone_const
exact monotone_preimage (f := fun y => (y, a))
|
/-
Copyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Galois
import Mathlib.LinearAlgebra.Eigenspace.Mi... | Mathlib/FieldTheory/KummerExtension.lean | 88 | 93 | theorem X_pow_sub_C_eq_prod'
{n : ℕ} {ζ : K} (hζ : IsPrimitiveRoot ζ n) {α a : K} (hn : 0 < n) (e : α ^ n = a) :
(X ^ n - C a) = ∏ i ∈ Finset.range n, (X - C (ζ ^ i * α)) := by |
rw [eq_prod_roots_of_monic_of_splits_id (monic_X_pow_sub_C _ (Nat.pos_iff_ne_zero.mp hn))
(X_pow_sub_C_splits_of_isPrimitiveRoot hζ e), ← nthRoots, hζ.nthRoots_eq e, Multiset.map_map]
rfl
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padic... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 1,162 | 1,172 | theorem norm_le_pow_iff_norm_lt_pow_add_one (x : ℚ_[p]) (n : ℤ) :
‖x‖ ≤ (p : ℝ) ^ n ↔ ‖x‖ < (p : ℝ) ^ (n + 1) := by |
have aux : ∀ n : ℤ, 0 < ((p : ℝ) ^ n) := by
apply Nat.zpow_pos_of_pos
exact hp.1.pos
by_cases hx0 : x = 0
· simp [hx0, norm_zero, aux, le_of_lt (aux _)]
rw [norm_eq_pow_val hx0]
have h1p : 1 < (p : ℝ) := mod_cast hp.1.one_lt
have H := zpow_strictMono h1p
rw [H.le_iff_le, H.lt_iff_lt, Int.lt_add_o... |
/-
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.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDo... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 218 | 219 | theorem X_eq_normalize : (X : Polynomial R) = normalize X := by |
simp only [normalize_apply, normUnit_X, Units.val_one, mul_one]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 139 | 139 | theorem left_mem_Ico : a ∈ Ico a b ↔ a < b := by | simp only [mem_Ico, true_and_iff, le_refl]
|
/-
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 | 719 | 722 | theorem measure_laverage_le_pos (hμ : μ ≠ 0) (hint : ∫⁻ a, f a ∂μ ≠ ∞) :
0 < μ {x | ⨍⁻ a, f a ∂μ ≤ f x} := by |
simpa [hint] using
@measure_setLaverage_le_pos _ _ _ _ f (measure_univ_ne_zero.2 hμ) nullMeasurableSet_univ
|
/-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.Atoms
import Mathlib.Order.Chain
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Set.Subsingleton
#align_... | Mathlib/Order/Ideal.lean | 497 | 501 | theorem IsProper.not_mem_of_compl_mem (hI : IsProper I) (hxc : xᶜ ∈ I) : x ∉ I := by |
intro hx
apply hI.top_not_mem
have ht : x ⊔ xᶜ ∈ I := sup_mem ‹_› ‹_›
rwa [sup_compl_eq_top] at ht
|
/-
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 | 601 | 613 | theorem Ideal.exists_smith_normal_form (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) :
∃ (b' : Basis ι R S) (a : ι → R) (ab' : Basis ι R I), ∀ i, (ab' i : S) = a i • b' i := by |
cases nonempty_fintype ι
let ⟨bS, bI, f, a, snf⟩ := I.smithNormalForm b hI
let e : Fin (Fintype.card ι) ≃ ι :=
Equiv.ofBijective f
((Fintype.bijective_iff_injective_and_card f).mpr ⟨f.injective, Fintype.card_fin _⟩)
have fe : ∀ i, f (e.symm i) = i := e.apply_symm_apply
exact
⟨bS, a ∘ e.symm, (b... |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algeb... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 431 | 434 | theorem linearIndependent_rows_iff_isUnit {A : Matrix m m K} :
LinearIndependent K (fun i ↦ A i) ↔ IsUnit A := by |
rw [← transpose_transpose A, ← mulVec_injective_iff, ← coe_mulVecLin, mulVecLin_transpose,
transpose_transpose, ← vecMul_injective_iff_isUnit, coe_vecMulLinear]
|
/-
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.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimag... | Mathlib/Data/Finsupp/Basic.lean | 824 | 826 | theorem some_single_none [Zero M] (m : M) : (single none m : Option α →₀ M).some = 0 := by |
ext
simp
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Sh... | Mathlib/AlgebraicGeometry/Pullbacks.lean | 177 | 181 | theorem cocycle_snd_fst_fst (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j ≫ pullback.snd ≫ pullback.fst ≫ pullback.fst =
pullback.snd ≫ pullback.fst ≫ pullback.fst := by |
rw [← cancel_mono (𝒰.map i)]
simp only [pullback.condition_assoc, t'_snd_fst_fst, t'_fst_snd, t'_snd_snd]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpa... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 194 | 198 | theorem log_neg_of_lt_zero (h0 : x < 0) (h1 : -1 < x) : log x < 0 := by |
rw [← neg_neg x, log_neg_eq_log]
have h0' : 0 < -x := by linarith
have h1' : -x < 1 := by linarith
exact log_neg h0' h1'
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.lemmas from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
/-!
# Further... | Mathlib/Data/Int/Order/Lemmas.lean | 28 | 30 | theorem natAbs_eq_iff_mul_self_eq {a b : ℤ} : a.natAbs = b.natAbs ↔ a * a = b * b := by |
rw [← abs_eq_iff_mul_self_eq, abs_eq_natAbs, abs_eq_natAbs]
exact Int.natCast_inj.symm
|
/-
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.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
#align_import topology.metric_space.hausdorff_dimension from "leanp... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 535 | 537 | theorem dimH_univ_pi (ι : Type*) [Fintype ι] : dimH (univ : Set (ι → ℝ)) = Fintype.card ι := by |
simp only [← Metric.iUnion_ball_nat_succ (0 : ι → ℝ), dimH_iUnion,
dimH_ball_pi _ (Nat.cast_add_one_pos _), iSup_const]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.MetricSpace.IsometricSMul
#alig... | Mathlib/Analysis/Normed/Group/Pointwise.lean | 148 | 149 | theorem singleton_div_ball_one : {x} / ball 1 δ = ball x δ := by |
rw [singleton_div_ball, div_one]
|
/-
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.Extensive
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
#align_import category_theory.adhesive from "leanprover-community/mathlib"@"... | Mathlib/CategoryTheory/Adhesive.lean | 318 | 334 | theorem adhesive_of_reflective [HasPullbacks D] [Adhesive C] [HasPullbacks C] [HasPushouts C]
[H₂ : ∀ {X Y S : D} (f : S ⟶ X) (g : S ⟶ Y) [Mono f], HasPushout f g]
{Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Gr.Full] [Gr.Faithful]
[PreservesLimitsOfShape WalkingCospan Gl] :
Adhesive D := by |
have := adj.leftAdjointPreservesColimits
have := adj.rightAdjointPreservesLimits
apply Adhesive.mk (hasPushout_of_mono_left := H₂)
intro W X Y Z f g h i _ H
have := Adhesive.van_kampen (IsPushout.of_hasPushout (Gr.map f) (Gr.map g))
rw [IsPushout.isVanKampen_iff] at this ⊢
refine (IsVanKampenColimit.prec... |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 2,124 | 2,126 | theorem isLittleO_pow_sub_sub (x₀ : E') {m : ℕ} (h : 1 < m) :
(fun x => ‖x - x₀‖ ^ m) =o[𝓝 x₀] fun x => x - x₀ := by |
simpa only [isLittleO_norm_right, pow_one] using isLittleO_pow_sub_pow_sub x₀ h
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 408 | 409 | theorem Ico_subset_Ici_self : Ico a b ⊆ Ici a := by |
simpa [← coe_subset] using Set.Ico_subset_Ici_self
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Bhavik Mehta
-/
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/... | Mathlib/Probability/CondCount.lean | 70 | 76 | theorem condCount_univ [Fintype Ω] {s : Set Ω} :
condCount Set.univ s = Measure.count s / Fintype.card Ω := by |
rw [condCount, cond_apply _ MeasurableSet.univ, ← ENNReal.div_eq_inv_mul, Set.univ_inter]
congr
rw [← Finset.coe_univ, Measure.count_apply, Finset.univ.tsum_subtype' fun _ => (1 : ENNReal)]
· simp [Finset.card_univ]
· exact (@Finset.coe_univ Ω _).symm ▸ MeasurableSet.univ
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.PEmpty
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.... | Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean | 494 | 497 | theorem limitConstTerminal_inv_π {J : Type*} [Category J] {C : Type*} [Category C] [HasTerminal C]
{j : J} :
limitConstTerminal.inv ≫ limit.π ((CategoryTheory.Functor.const J).obj (⊤_ C)) j =
terminal.from _ := by | aesop_cat
|
/-
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.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d6... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 827 | 832 | theorem HasFDerivAt.finset_prod [DecidableEq ι] {x : E}
(hg : ∀ i ∈ u, HasFDerivAt (g i) (g' i) x) :
HasFDerivAt (∏ i ∈ u, g i ·) (∑ i ∈ u, (∏ j ∈ u.erase i, g j x) • g' i) x := by |
simpa [← Finset.prod_attach u] using .congr_fderiv
(hasFDerivAt_finset_prod.comp x <| hasFDerivAt_pi.mpr fun i ↦ hg i i.prop)
(by ext; simp [Finset.prod_erase_attach (g · x), ← u.sum_attach])
|
/-
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 | 1,343 | 1,344 | theorem im_sq : a.im ^ 2 = -normSq a.im := by |
simp_rw [sq, ← star_mul_self, im_star, neg_mul, neg_neg]
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 490 | 495 | theorem NNReal.lintegral_mul_le_Lp_mul_Lq {p q : ℝ} (hpq : p.IsConjExponent q) {f g : α → ℝ≥0}
(hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
(∫⁻ a, (f * g) a ∂μ) ≤
(∫⁻ a, (f a : ℝ≥0∞) ^ p ∂μ) ^ (1 / p) * (∫⁻ a, (g a : ℝ≥0∞) ^ q ∂μ) ^ (1 / q) := by |
simp_rw [Pi.mul_apply, ENNReal.coe_mul]
exact ENNReal.lintegral_mul_le_Lp_mul_Lq μ hpq hf.coe_nnreal_ennreal hg.coe_nnreal_ennreal
|
/-
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
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 824 | 824 | theorem exp_zero : exp 0 = 1 := by | simp [Real.exp]
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Order.RelClasses
import Mathlib.Order.Interval.Set.Basic
#align_import order.bounded from "leanprover-community/mathlib"@"aba5... | Mathlib/Order/Bounded.lean | 378 | 381 | theorem bounded_lt_inter_lt [LinearOrder α] [NoMaxOrder α] (a : α) :
Bounded (· < ·) (s ∩ { b | a < b }) ↔ Bounded (· < ·) s := by |
rw [← bounded_le_iff_bounded_lt, ← bounded_le_iff_bounded_lt]
exact bounded_le_inter_lt a
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Topology.PartialHomeomorph... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean | 148 | 280 | theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f f' s c)
(f'symm : f'.NonlinearRightInverse) {ε : ℝ} {b : E} (ε0 : 0 ≤ ε) (hε : closedBall b ε ⊆ s) :
SurjOn f (closedBall b ε) (closedBall (f b) (((f'symm.nnnorm : ℝ)⁻¹ - c) * ε)) := by |
intro y hy
rcases le_or_lt (f'symm.nnnorm : ℝ)⁻¹ c with hc | hc
· refine ⟨b, by simp [ε0], ?_⟩
have : dist y (f b) ≤ 0 :=
(mem_closedBall.1 hy).trans (mul_nonpos_of_nonpos_of_nonneg (by linarith) ε0)
simp only [dist_le_zero] at this
rw [this]
have If' : (0 : ℝ) < f'symm.nnnorm := by rw [← inv... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34... | Mathlib/Data/Finsupp/Multiset.lean | 128 | 130 | theorem mem_toMultiset (f : α →₀ ℕ) (i : α) : i ∈ toMultiset f ↔ i ∈ f.support := by |
classical
rw [← Multiset.count_ne_zero, Finsupp.count_toMultiset, Finsupp.mem_support_iff]
|
/-
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.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 478 | 480 | theorem integral_prod_symm (f : α × β → E) (hf : Integrable f (μ.prod ν)) :
∫ z, f z ∂μ.prod ν = ∫ y, ∫ x, f (x, y) ∂μ ∂ν := by |
rw [← integral_prod_swap f]; exact integral_prod _ hf.swap
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import ... | Mathlib/Data/DFinsupp/Basic.lean | 852 | 857 | theorem update_eq_erase_add_single {β : ι → Type*} [∀ i, AddZeroClass (β i)] (f : Π₀ i, β i)
(i : ι) (b : β i) : f.update i b = f.erase i + single i b := by |
ext j
rcases eq_or_ne i j with (rfl | h)
· simp
· simp [Function.update_noteq h.symm, h, erase_ne, h.symm]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,118 | 1,119 | theorem mk_perm_eq_two_power : #(Equiv.Perm α) = 2 ^ #α := by |
rw [mk_perm_eq_self_power, power_self_eq (aleph0_le_mk α)]
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 627 | 629 | theorem dist_le_norm_add_norm' (a b : E) : dist a b ≤ ‖a‖ + ‖b‖ := by |
rw [dist_eq_norm_div]
apply norm_div_le
|
/-
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.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimag... | Mathlib/Data/Finsupp/Basic.lean | 1,846 | 1,851 | theorem mem_splitSupport_iff_nonzero (i : ι) : i ∈ splitSupport l ↔ split l i ≠ 0 := by |
rw [splitSupport, @mem_image _ _ (Classical.decEq _), Ne, ← support_eq_empty, ← Ne, ←
Finset.nonempty_iff_ne_empty, split, comapDomain, Finset.Nonempty]
-- porting note (#10754): had to add the `Classical.decEq` instance manually
simp only [exists_prop, Finset.mem_preimage, exists_and_right, exists_eq_right,... |
/-
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.Algebra.NonUnitalSubalgebra
import Mathlib.Algebra.Star.StarAlgHom
import Mathlib.Algebra.Star.Center
/-!
# Non-unital Star Subalgebras
In this... | Mathlib/Algebra/Star/NonUnitalSubalgebra.lean | 1,013 | 1,017 | theorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :
iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by |
subst hT
dsimp [iSupLift]
apply Set.iUnionLift_mk
|
/-
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.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-commu... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 136 | 138 | theorem Matrix.Represents.zero : (0 : Matrix ι ι R).Represents b 0 := by |
delta Matrix.Represents
rw [map_zero, map_zero]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Felix Weilacher
-/
import Mathlib.Data.Real.Cardinality
import Mathlib.Topology.MetricSpace.Perfect
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
i... | Mathlib/MeasureTheory/Constructions/Polish.lean | 213 | 221 | theorem AnalyticSet.image_of_continuousOn {β : Type*} [TopologicalSpace β] {s : Set α}
(hs : AnalyticSet s) {f : α → β} (hf : ContinuousOn f s) : AnalyticSet (f '' s) := by |
rcases analyticSet_iff_exists_polishSpace_range.1 hs with ⟨γ, γtop, γpolish, g, g_cont, gs⟩
have : f '' s = range (f ∘ g) := by rw [range_comp, gs]
rw [this]
apply analyticSet_range_of_polishSpace
apply hf.comp_continuous g_cont fun x => _
rw [← gs]
exact mem_range_self
|
/-
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.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
#align_import rin... | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 377 | 382 | theorem constantCoeff_wittStructureInt (Φ : MvPolynomial idx ℤ) (h : constantCoeff Φ = 0) (n : ℕ) :
constantCoeff (wittStructureInt p Φ n) = 0 := by |
have inj : Function.Injective (Int.castRingHom ℚ) := by intro m n; exact Int.cast_inj.mp
apply inj
rw [← constantCoeff_map, map_wittStructureInt, constantCoeff_wittStructureRat, RingHom.map_zero]
rw [constantCoeff_map, h, RingHom.map_zero]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 752 | 754 | theorem coe_restrict (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) :
⇑(restrict f s) = indicator s f := by |
rw [restrict, dif_pos hs, coe_piecewise, coe_zero, piecewise_eq_indicator]
|
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
#align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49d... | Mathlib/MeasureTheory/Integral/Layercake.lean | 105 | 183 | theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
(μ : Measure α) [SigmaFinite μ]
(f_nn : 0 ≤ f) (f_mble : Measurable f)
(g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
(g_nn : ∀ t > 0, 0 ≤ g t) :
∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ... |
have g_intble' : ∀ t : ℝ, 0 ≤ t → IntervalIntegrable g volume 0 t := by
intro t ht
cases' eq_or_lt_of_le ht with h h
· simp [← h]
· exact g_intble t h
have integrand_eq : ∀ ω,
ENNReal.ofReal (∫ t in (0)..f ω, g t) = ∫⁻ t in Ioc 0 (f ω), ENNReal.ofReal (g t) := by
intro ω
have g_ae_nn ... |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_spa... | Mathlib/Topology/MetricSpace/IsometricSMul.lean | 470 | 471 | theorem preimage_smul_sphere (c : G) (x : X) (r : ℝ) :
(c • ·) ⁻¹' sphere x r = sphere (c⁻¹ • x) r := by | rw [preimage_smul, smul_sphere]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
[`data.finset.sym`@`98e83c3d541c77cdb7da20d79611a780ff8e7d90`..`02ba8949f486ebecf93fe7460f1ed0564b5e442c`](https://leanprover-community.github.io/mathlib-port-status/file/d... | Mathlib/Data/Finset/Sym.lean | 69 | 72 | theorem sym2_mono (h : s ⊆ t) : s.sym2 ⊆ t.sym2 := by |
rw [← val_le_iff, sym2_val, sym2_val]
apply Multiset.sym2_mono
rwa [val_le_iff]
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012... | Mathlib/Order/Interval/Set/WithBotTop.lean | 175 | 175 | theorem preimage_coe_Ioc : (some : α → WithBot α) ⁻¹' Ioc a b = Ioc a b := by | simp [← Ioi_inter_Iic]
|
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