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) 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 | 1,098 | 1,100 | theorem restrict_const_lintegral (c : ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) :
((const α c).restrict s).lintegral μ = c * μ s := by |
rw [restrict_lintegral_eq_lintegral_restrict _ hs, const_lintegral_restrict]
|
/-
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 | 1,086 | 1,090 | theorem const_lintegral (c : ℝ≥0∞) : (const α c).lintegral μ = c * μ univ := by |
rw [lintegral]
cases isEmpty_or_nonempty α
· simp [μ.eq_zero_of_isEmpty]
· simp; unfold Function.const; rw [preimage_const_of_mem (mem_singleton c)]
|
/-
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 | 1,023 | 1,026 | theorem sup_dualAnnihilator_le_inf (U V : Submodule R M) :
U.dualAnnihilator ⊔ V.dualAnnihilator ≤ (U ⊓ V).dualAnnihilator := by |
rw [le_dualAnnihilator_iff_le_dualCoannihilator, dualCoannihilator_sup_eq]
apply inf_le_inf <;> exact le_dualAnnihilator_dualCoannihilator _
|
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.... | Mathlib/Order/UpperLower/Basic.lean | 1,541 | 1,543 | theorem upperClosure_singleton (a : α) : upperClosure ({a} : Set α) = UpperSet.Ici a := by |
ext
simp
|
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.Algebraic... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 537 | 538 | theorem mul_left_le_iff [IsDedekindDomain A] {J : FractionalIdeal A⁰ K} (hJ : J ≠ 0) {I I'} :
J * I ≤ J * I' ↔ I ≤ I' := by | convert mul_right_le_iff hJ using 1; simp only [mul_comm]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Basic
import Mathlib.Tactic.Coe
import Mathlib.Tactic.Lift
imp... | Mathlib/Data/PNat/Defs.lean | 254 | 263 | theorem mod_coe (m k : ℕ+) :
(mod m k : ℕ) = ite ((m : ℕ) % (k : ℕ) = 0) (k : ℕ) ((m : ℕ) % (k : ℕ)) := by |
dsimp [mod, modDiv]
cases (m : ℕ) % (k : ℕ) with
| zero =>
rw [if_pos rfl]
rfl
| succ n =>
rw [if_neg n.succ_ne_zero]
rfl
|
/-
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 | 1,038 | 1,043 | theorem mem_map_indicator_ae_iff_of_zero_nmem [Zero β] {t : Set β} (ht : (0 : β) ∉ t) :
t ∈ Filter.map (s.indicator f) (ae μ) ↔ μ ((f ⁻¹' t)ᶜ ∪ sᶜ) = 0 := by |
classical
rw [mem_map, mem_ae_iff, Set.indicator_preimage, Set.ite, Set.compl_union, Set.compl_inter]
change μ (((f ⁻¹' t)ᶜ ∪ sᶜ) ∩ ((fun _ => (0 : β)) ⁻¹' t \ s)ᶜ) = 0 ↔ μ ((f ⁻¹' t)ᶜ ∪ sᶜ) = 0
simp only [ht, if_false, Set.compl_empty, Set.empty_diff, Set.inter_univ, Set.preimage_const]
|
/-
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 | 328 | 332 | theorem Ico_filter_lt_of_le_right [DecidablePred (· < c)] (hcb : c ≤ b) :
(Ico a b).filter (· < c) = Ico a c := by |
ext x
rw [mem_filter, mem_Ico, mem_Ico, and_right_comm]
exact and_iff_left_of_imp fun h => h.2.trans_le hcb
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 581 | 593 | theorem stepRet_then {k k' : Cont} {v} : stepRet (k.then k') v = (stepRet k v).then k' := by |
induction k generalizing v with simp only [Cont.then, stepRet, *]
| cons₁ =>
rw [← stepNormal_then]
rfl
| comp =>
rw [← stepNormal_then]
| fix _ _ k_ih =>
split_ifs
· rw [← k_ih]
· rw [← stepNormal_then]
rfl
| _ => simp only [Cfg.then]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
#align_import number_theory.legendre_symbol.basic from "leanprover-community/mathlib"@"5b2f... | Mathlib/NumberTheory/LegendreSymbol/Basic.lean | 179 | 182 | theorem sq_one' {a : ℤ} (ha : (a : ZMod p) ≠ 0) : legendreSym p (a ^ 2) = 1 := by |
dsimp only [legendreSym]
rw [Int.cast_pow]
exact quadraticChar_sq_one' ha
|
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Adjoin.Ba... | Mathlib/RingTheory/AlgebraicIndependent.lean | 138 | 149 | theorem map {f : A →ₐ[R] A'} (hf_inj : Set.InjOn f (adjoin R (range x))) :
AlgebraicIndependent R (f ∘ x) := by |
have : aeval (f ∘ x) = f.comp (aeval x) := by ext; simp
have h : ∀ p : MvPolynomial ι R, aeval x p ∈ (@aeval R _ _ _ _ _ ((↑) : range x → A)).range := by
intro p
rw [AlgHom.mem_range]
refine ⟨MvPolynomial.rename (codRestrict x (range x) mem_range_self) p, ?_⟩
simp [Function.comp, aeval_rename]
in... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.BigOperators.Fin
im... | Mathlib/Algebra/BigOperators/Finsupp.lean | 69 | 75 | theorem prod_single_index {a : α} {b : M} {h : α → M → N} (h_zero : h a 0 = 1) :
(single a b).prod h = h a b :=
calc
(single a b).prod h = ∏ x ∈ {a}, h x (single a b x) :=
prod_of_support_subset _ support_single_subset h fun x hx =>
(mem_singleton.1 hx).symm ▸ h_zero
_ = h a b := by | simp
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
#align_import number_theory.well_approximable from "leanprover-community/mathl... | Mathlib/NumberTheory/WellApproximable.lean | 121 | 129 | theorem image_pow_subset (n : ℕ) (hm : 0 < m) :
(fun (y : A) => y ^ m) '' approxOrderOf A (n * m) δ ⊆ approxOrderOf A n (m * δ) := by |
rintro - ⟨a, ha, rfl⟩
obtain ⟨b, hb : orderOf b = n * m, hab : a ∈ ball b δ⟩ := mem_approxOrderOf_iff.mp ha
replace hb : b ^ m ∈ {y : A | orderOf y = n} := by
rw [mem_setOf_eq, orderOf_pow' b hm.ne', hb, Nat.gcd_mul_left_left, n.mul_div_cancel hm]
apply ball_subset_thickening hb (m * δ)
convert pow_mem_b... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.InsertNth
#align_import group_theory.f... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 787 | 790 | theorem closure_range_of (α) :
Subgroup.closure (Set.range (FreeGroup.of : α → FreeGroup α)) = ⊤ := by |
rw [← lift.range_eq_closure, lift_of_eq_id]
exact MonoidHom.range_top_of_surjective _ Function.surjective_id
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@... | Mathlib/Topology/Basic.lean | 1,192 | 1,194 | theorem isOpen_iff_ultrafilter :
IsOpen s ↔ ∀ x ∈ s, ∀ (l : Ultrafilter X), ↑l ≤ 𝓝 x → s ∈ l := by |
simp_rw [isOpen_iff_mem_nhds, ← mem_iff_ultrafilter]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Topology.Instan... | Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | 386 | 413 | theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f : (∫⁻ x, f x ∂μ) ≠ ∞)
{ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
∃ g : α → ℝ≥0, (∀ x, g x ≤ f x) ∧ UpperSemicontinuous g ∧
(∫⁻ x, f x ∂μ) ≤ (∫⁻ x, g x ∂μ) + ε := by |
obtain ⟨fs, fs_le_f, int_fs⟩ :
∃ fs : α →ₛ ℝ≥0, (∀ x, fs x ≤ f x) ∧ (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, fs x ∂μ) + ε / 2 := by
-- Porting note: need to name identifier (not `this`), because `conv_rhs at this` errors
have aux := ENNReal.lt_add_right int_f (ENNReal.half_pos ε0).ne'
conv_rhs at aux => rw [lintegral... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 338 | 340 | theorem abs_cpow_eq_rpow_re_of_pos {x : ℝ} (hx : 0 < x) (y : ℂ) : abs (x ^ y) = x ^ y.re := by |
rw [abs_cpow_of_ne_zero (ofReal_ne_zero.mpr hx.ne'), arg_ofReal_of_nonneg hx.le,
zero_mul, Real.exp_zero, div_one, abs_of_nonneg hx.le]
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Thomas Zhu, Mario Carneiro
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
#align_import number_theory.legendre_symbol.jacobi_symbol from "leanprover-... | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 159 | 163 | theorem mul_left (a₁ a₂ : ℤ) (b : ℕ) : J(a₁ * a₂ | b) = J(a₁ | b) * J(a₂ | b) := by |
simp_rw [jacobiSym, List.pmap_eq_map_attach, legendreSym.mul _ _ _];
exact List.prod_map_mul (α := ℤ) (l := (factors b).attach)
(f := fun x ↦ @legendreSym x {out := prime_of_mem_factors x.2} a₁)
(g := fun x ↦ @legendreSym x {out := prime_of_mem_factors x.2} a₂)
|
/-
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.Group.Measure
import Mathlib.Topology.UrysohnsLemma
/-!
# Everywhere positive sets in measure spaces
A set `s` in a topological s... | Mathlib/MeasureTheory/Measure/EverywherePos.lean | 271 | 296 | theorem innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_group :
InnerRegularWRT μ (fun s ↦ ∃ (f : G → ℝ), Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1})
(fun s ↦ MeasurableSet s ∧ μ s ≠ ∞) := by |
/- First, approximate a measurable set from inside by a compact closed set `K`. Then notice that
the everywhere positive subset of `K` is a Gδ,
by Lemma `IsEverywherePos.IsGdelta_of_isMulLeftInvariant`, and therefore the level set of a
continuous compactly supported function. Moreover, it has the same measure ... |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Data.Finsupp.Fintype
import Mathlib.Data.Int.Absolut... | Mathlib/RingTheory/Ideal/Norm.lean | 177 | 183 | theorem Ideal.mul_add_mem_pow_succ_unique [IsDedekindDomain S] {i : ℕ} (a d d' e e' : S)
(a_not_mem : a ∉ P ^ (i + 1)) (e_mem : e ∈ P ^ (i + 1)) (e'_mem : e' ∈ P ^ (i + 1))
(h : a * d + e - (a * d' + e') ∈ P ^ (i + 1)) : d - d' ∈ P := by |
have h' : a * (d - d') ∈ P ^ (i + 1) := by
convert Ideal.add_mem _ h (Ideal.sub_mem _ e'_mem e_mem) using 1
ring
exact Ideal.mem_prime_of_mul_mem_pow hP a_not_mem h'
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Heather Macbeth
-/
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Math... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 355 | 377 | theorem Subalgebra.SeparatesPoints.rclike_to_real {A : StarSubalgebra 𝕜 C(X, 𝕜)}
(hA : A.SeparatesPoints) :
((A.restrictScalars ℝ).comap
(ofRealAm.compLeftContinuous ℝ continuous_ofReal)).SeparatesPoints := by |
intro x₁ x₂ hx
-- Let `f` in the subalgebra `A` separate the points `x₁`, `x₂`
obtain ⟨_, ⟨f, hfA, rfl⟩, hf⟩ := hA hx
let F : C(X, 𝕜) := f - const _ (f x₂)
-- Subtract the constant `f x₂` from `f`; this is still an element of the subalgebra
have hFA : F ∈ A := by
refine A.sub_mem hfA (@Eq.subst _ (· ∈... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analys... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 529 | 533 | theorem rpow_add {x : ℝ≥0∞} (y z : ℝ) (hx : x ≠ 0) (h'x : x ≠ ⊤) : x ^ (y + z) = x ^ y * x ^ z := by |
cases' x with x
· exact (h'x rfl).elim
have : x ≠ 0 := fun h => by simp [h] at hx
simp [coe_rpow_of_ne_zero this, NNReal.rpow_add this]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 1,926 | 1,929 | theorem prod_update_of_mem [DecidableEq α] {s : Finset α} {i : α} (h : i ∈ s) (f : α → β) (b : β) :
∏ x ∈ s, Function.update f i b x = b * ∏ x ∈ s \ singleton i, f x := by |
rw [update_eq_piecewise, prod_piecewise]
simp [h]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Measure.AEMeasurable
#align_import dynamics.ergodic.measure_preserving from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409... | Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | 163 | 175 | theorem exists_mem_iterate_mem_of_volume_lt_mul_volume (hf : MeasurePreserving f μ μ)
(hs : MeasurableSet s) {n : ℕ} (hvol : μ (Set.univ : Set α) < n * μ s) :
∃ x ∈ s, ∃ m ∈ Set.Ioo 0 n, f^[m] x ∈ s := by |
have A : ∀ m, MeasurableSet (f^[m] ⁻¹' s) := fun m ↦ (hf.iterate m).measurable hs
have B : ∀ m, μ (f^[m] ⁻¹' s) = μ s := fun m ↦ (hf.iterate m).measure_preimage hs
have : μ (univ : Set α) < ∑ m ∈ Finset.range n, μ (f^[m] ⁻¹' s) := by simpa [B]
obtain ⟨i, hi, j, hj, hij, x, hxi : f^[i] x ∈ s, hxj : f^[j] x ∈ s⟩... |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
/-!
# Dropping or taki... | Mathlib/Data/List/DropRight.lean | 195 | 195 | theorem rtakeWhile_nil : rtakeWhile p ([] : List α) = [] := by | simp [rtakeWhile, takeWhile]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 138 | 141 | theorem degrees_mul (p q : MvPolynomial σ R) : (p * q).degrees ≤ p.degrees + q.degrees := by |
classical
simp_rw [degrees_def]
exact supDegree_mul_le (map_add _)
|
/-
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.Finset.Update
import Mathlib.Data.Prod.TProd
import Mathlib.GroupTheory.Coset
import Mathlib.Logic.Equiv.Fin
import Mathlib.Measur... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 1,763 | 1,764 | theorem coe_piCongrLeft (f : δ ≃ δ') :
⇑(MeasurableEquiv.piCongrLeft π f) = f.piCongrLeft π := 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.Algebra.BigOperators.Group.List
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.I... | Mathlib/Data/Vector/Basic.lean | 295 | 302 | theorem reverse_get_zero {v : Vector α (n + 1)} : v.reverse.head = v.last := by |
rw [← get_zero, last_def, get_eq_get, get_eq_get]
simp_rw [toList_reverse]
rw [← Option.some_inj, Fin.cast, Fin.cast, ← List.get?_eq_get, ← List.get?_eq_get,
List.get?_reverse]
· congr
simp
· simp
|
/-
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.Topology.Instances.Real
import Mathlib.Order.Filter.Archimedean
#align_import analysis.subadditive from "leanprover-community/mathlib"@"f2ce6086... | Mathlib/Analysis/Subadditive.lean | 51 | 59 | theorem apply_mul_add_le (k n r) : u (k * n + r) ≤ k * u n + u r := by |
induction k with
| zero => simp only [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_add]; rfl
| succ k IH =>
calc
u ((k + 1) * n + r) = u (n + (k * n + r)) := by congr 1; ring
_ ≤ u n + u (k * n + r) := h _ _
_ ≤ u n + (k * u n + u r) := add_le_add_left IH _
_ = (k + 1 : ℕ) * u n + u r :... |
/-
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.ContMDiff.Basic
/-!
## Smoothness of standard maps associated to the product of manifolds
This file contain... | Mathlib/Geometry/Manifold/ContMDiff/Product.lean | 341 | 345 | theorem ContMDiffAt.prod_map (hf : ContMDiffAt I I' n f x) (hg : ContMDiffAt J J' n g y) :
ContMDiffAt (I.prod J) (I'.prod J') n (Prod.map f g) (x, y) := by |
rw [← contMDiffWithinAt_univ] at *
convert hf.prod_map hg
exact univ_prod_univ.symm
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 936 | 936 | theorem star_dotProduct_star : star v ⬝ᵥ star w = star (w ⬝ᵥ v) := by | simp [dotProduct]
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanp... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 358 | 364 | theorem fst_openEmbedding_of_right_openEmbedding {X Y S : TopCat} (f : X ⟶ S) {g : Y ⟶ S}
(H : OpenEmbedding g) : OpenEmbedding <| ⇑(pullback.fst : pullback f g ⟶ X) := by |
convert (homeoOfIso (asIso (pullback.fst : pullback f (𝟙 S) ⟶ _))).openEmbedding.comp
(pullback_map_openEmbedding_of_open_embeddings (i₁ := 𝟙 X) f g f (𝟙 _)
(homeoOfIso (Iso.refl _)).openEmbedding H (𝟙 _) rfl (by simp))
erw [← coe_comp]
simp
|
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68a... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 205 | 209 | theorem compression_idem (u v : α) (s : Finset α) : 𝓒 u v (𝓒 u v s) = 𝓒 u v s := by |
have h : filter (compress u v · ∉ 𝓒 u v s) (𝓒 u v s) = ∅ :=
filter_false_of_mem fun a ha h ↦ h <| compress_mem_compression_of_mem_compression ha
rw [compression, filter_image, h, image_empty, ← h]
exact filter_union_filter_neg_eq _ (compression u v s)
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Bits
import Mathlib.Algebra.Ring.Nat
import Mathlib.Order.Basic
import Mathlib.Tactic.AdaptationNote
... | Mathlib/Data/Nat/Bitwise.lean | 205 | 209 | theorem zero_of_testBit_eq_false {n : ℕ} (h : ∀ i, testBit n i = false) : n = 0 := by |
induction' n using Nat.binaryRec with b n hn
· rfl
· have : b = false := by simpa using h 0
rw [this, bit_false, bit0_val, hn fun i => by rw [← h (i + 1), testBit_bit_succ], mul_zero]
|
/-
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 | 2,093 | 2,093 | theorem iSup_singleton (a : α) (s : α → β) : ⨆ x ∈ ({a} : Finset α), s x = s a := by | simp
|
/-
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 | 1,158 | 1,166 | theorem sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i ∈ range n, x ^ i / i ! ≤ exp x :=
calc
∑ i ∈ range n, x ^ i / i ! ≤ lim (⟨_, isCauSeq_re (exp' x)⟩ : CauSeq ℝ abs) := by |
refine le_lim (CauSeq.le_of_exists ⟨n, fun j hj => ?_⟩)
simp only [exp', const_apply, re_sum]
norm_cast
refine sum_le_sum_of_subset_of_nonneg (range_mono hj) fun _ _ _ ↦ ?_
positivity
_ = exp x := by rw [exp, Complex.exp, ← cauSeqRe, lim_re]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 2,626 | 2,626 | theorem filter_True {h} (s : Finset α) : @filter _ (fun _ => True) h s = s := by | ext; simp
|
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrabl... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 1,060 | 1,071 | theorem Lp_toLp_restrict_add (f g : Lp E p μ) (s : Set X) :
((Lp.memℒp (f + g)).restrict s).toLp (⇑(f + g)) =
((Lp.memℒp f).restrict s).toLp f + ((Lp.memℒp g).restrict s).toLp g := by |
ext1
refine (ae_restrict_of_ae (Lp.coeFn_add f g)).mp ?_
refine
(Lp.coeFn_add (Memℒp.toLp f ((Lp.memℒp f).restrict s))
(Memℒp.toLp g ((Lp.memℒp g).restrict s))).mp ?_
refine (Memℒp.coeFn_toLp ((Lp.memℒp f).restrict s)).mp ?_
refine (Memℒp.coeFn_toLp ((Lp.memℒp g).restrict s)).mp ?_
refine (Me... |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.Probability.Kernel.Disintegration.CdfToKernel
#align_import probability.kernel.cond_cdf from "lean... | Mathlib/Probability/Kernel/Disintegration/CondCdf.lean | 92 | 99 | theorem tendsto_IicSnd_atTop {s : Set α} (hs : MeasurableSet s) :
Tendsto (fun r : ℚ ↦ ρ.IicSnd r s) atTop (𝓝 (ρ.fst s)) := by |
simp_rw [ρ.IicSnd_apply _ hs, fst_apply hs, ← prod_univ]
rw [← Real.iUnion_Iic_rat, prod_iUnion]
refine tendsto_measure_iUnion fun r q hr_le_q x ↦ ?_
simp only [mem_prod, mem_Iic, and_imp]
refine fun hxs hxr ↦ ⟨hxs, hxr.trans ?_⟩
exact mod_cast hr_le_q
|
/-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra... | Mathlib/Algebra/Ring/Units.lean | 61 | 62 | theorem divp_add_divp_same (a b : α) (u : αˣ) : a /ₚ u + b /ₚ u = (a + b) /ₚ u := by |
simp only [divp, add_mul]
|
/-
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.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 756 | 757 | theorem snormEssSup_eq_zero_iff {f : α → F} : snormEssSup f μ = 0 ↔ f =ᵐ[μ] 0 := by |
simp [EventuallyEq, snormEssSup]
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.DFinsupp.Interval
import Mathlib.Data.DFinsupp.Multiset
import Mathlib.Order.Interval.Finset.Nat
#align_import data.multiset.interval from "leanprover-... | Mathlib/Data/Multiset/Interval.lean | 67 | 69 | theorem card_Ioc :
(Finset.Ioc s t).card = ∏ i ∈ s.toFinset ∪ t.toFinset, (t.count i + 1 - s.count i) - 1 := by |
rw [Finset.card_Ioc_eq_card_Icc_sub_one, card_Icc]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Patrick Massot
-/
import Mathlib.Algebra.Algebra.Subalgebra.Operations
import Mathlib.Algebra.Ring.Fin
import Mathlib.RingTheory.Ideal.Quotient
#align_import ring_theory.ideal.q... | Mathlib/RingTheory/Ideal/QuotientOperations.lean | 595 | 605 | theorem comp_quotientMap_eq_of_comp_eq {R' S' : Type*} [CommRing R'] [CommRing S'] {f : R →+* S}
{f' : R' →+* S'} {g : R →+* R'} {g' : S →+* S'} (hfg : f'.comp g = g'.comp f) (I : Ideal S') :
-- Porting note: was losing track of I
let leq := le_of_eq (_root_.trans (comap_comap (I := I) f g') (hfg ▸ comap_co... |
refine RingHom.ext fun a => ?_
obtain ⟨r, rfl⟩ := Quotient.mk_surjective a
simp only [RingHom.comp_apply, quotientMap_mk]
exact (Ideal.Quotient.mk I).congr_arg (_root_.trans (g'.comp_apply f r).symm
(hfg ▸ f'.comp_apply g 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.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 | 411 | 414 | theorem ListBlank.map_cons {Γ Γ'} [Inhabited Γ] [Inhabited Γ'] (f : PointedMap Γ Γ')
(l : ListBlank Γ) (a : Γ) : (l.cons a).map f = (l.map f).cons (f a) := by |
refine (ListBlank.cons_head_tail _).symm.trans ?_
simp only [ListBlank.head_map, ListBlank.head_cons, ListBlank.tail_map, ListBlank.tail_cons]
|
/-
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 | 485 | 488 | theorem preimage_mul_right_ball [IsometricSMul Gᵐᵒᵖ G] (a b : G) (r : ℝ) :
(fun x => x * a) ⁻¹' ball b r = ball (b / a) r := by |
rw [div_eq_mul_inv]
exact preimage_smul_ball (MulOpposite.op a) b r
|
/-
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.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanpro... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 148 | 150 | theorem real_smul_posPart_nonneg (r : ℝ) (hr : 0 ≤ r) :
(r • j).posPart = r.toNNReal • j.posPart := by |
rw [real_smul_def, ← smul_posPart, if_pos hr]
|
/-
Copyright (c) 2022 Jiale Miao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_s... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 63 | 65 | theorem gramSchmidt_def' (f : ι → E) (n : ι) :
f n = gramSchmidt 𝕜 f n + ∑ i ∈ Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) := by |
rw [gramSchmidt_def, sub_add_cancel]
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.ElementarySubstructures
#align_import model_theory.skolem from "leanprover-community/mathlib"@"3d7987cda72abc473c7cdbbb075170e9ac620042"
... | Mathlib/ModelTheory/Skolem.lean | 65 | 73 | theorem card_functions_sum_skolem₁_le : #(Σ n, (L.sum L.skolem₁).Functions n) ≤ max ℵ₀ L.card := by |
rw [card_functions_sum_skolem₁]
trans #(Σ n, L.BoundedFormula Empty n)
· exact
⟨⟨Sigma.map Nat.succ fun _ => id,
Nat.succ_injective.sigma_map fun _ => Function.injective_id⟩⟩
· refine _root_.trans BoundedFormula.card_le (lift_le.{max u v}.1 ?_)
simp only [mk_empty, lift_zero, lift_uzero, ze... |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.NormedSpace.ProdLp
import Mathlib.Topology.Instances.TrivSqZeroExt
#align_import analysis.normed_sp... | Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean | 104 | 111 | theorem hasSum_expSeries_of_smul_comm
(x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd)
{e : R} (h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
HasSum (fun n => expSeries 𝕜 (tsze R M) n fun _ => x) (inl e + inr (e • x.snd)) := by |
have : HasSum (fun n => fst (expSeries 𝕜 (tsze R M) n fun _ => x)) e := by
simpa [fst_expSeries] using h
simpa only [inl_fst_add_inr_snd_eq] using
(hasSum_inl _ <| this).add (hasSum_inr _ <| hasSum_snd_expSeries_of_smul_comm 𝕜 x hx h)
|
/-
Copyright (c) 2020 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.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
i... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 764 | 782 | theorem stronglyMeasurable_derivWithin_Ici :
StronglyMeasurable (fun x ↦ derivWithin f (Ici x) x) := by |
borelize F
apply stronglyMeasurable_iff_measurable_separable.2 ⟨measurable_derivWithin_Ici f, ?_⟩
obtain ⟨t, t_count, ht⟩ : ∃ t : Set ℝ, t.Countable ∧ Dense t := exists_countable_dense ℝ
suffices H : range (fun x ↦ derivWithin f (Ici x) x) ⊆ closure (Submodule.span ℝ (f '' t)) from
IsSeparable.mono (t_coun... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a7... | Mathlib/Data/List/Sigma.lean | 724 | 729 | theorem NodupKeys.kunion (nd₁ : l₁.NodupKeys) (nd₂ : l₂.NodupKeys) : (kunion l₁ l₂).NodupKeys := by |
induction l₁ generalizing l₂ with
| nil => simp only [nil_kunion, nd₂]
| cons s l₁ ih =>
simp? at nd₁ says simp only [nodupKeys_cons] at nd₁
simp [not_or, nd₁.1, nd₂, ih nd₁.2 (nd₂.kerase s.1)]
|
/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Mathport.Rename
import Mathlib.Init.Algebra.Classes
import Mathlib.Init.Data.Ordering.Basic
import Mathlib.Tactic.SplitIfs
import Mathlib.Tac... | Mathlib/Init/Order/Defs.lean | 425 | 430 | theorem compare_gt_iff_gt {a b : α} : (compare a b = .gt) ↔ a > b := by |
rw [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq]
split_ifs <;> simp only [*, lt_irrefl, not_lt_of_gt]
case _ h₁ h₂ =>
have h : b < a := lt_trichotomy a b |>.resolve_left h₁ |>.resolve_left h₂
exact true_iff_iff.2 h
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.... | Mathlib/Data/Matrix/Rank.lean | 192 | 195 | theorem rank_diagonal [Fintype m] [DecidableEq m] [DecidableEq R] (w : m → R) :
(diagonal w).rank = Fintype.card {i // (w i) ≠ 0} := by |
rw [Matrix.rank, ← Matrix.toLin'_apply', FiniteDimensional.finrank, ← LinearMap.rank,
LinearMap.rank_diagonal, Cardinal.toNat_natCast]
|
/-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.RepresentationTheory.Basic
import Mathlib.RepresentationTheory.FdRep
#align_import representation_theory.invariants from "leanprover-community/mathl... | Mathlib/RepresentationTheory/Invariants.lean | 133 | 139 | theorem mem_invariants_iff_comm {X Y : Rep k G} (f : X.V →ₗ[k] Y.V) (g : G) :
(linHom X.ρ Y.ρ) g f = f ↔ f.comp (X.ρ g) = (Y.ρ g).comp f := by |
dsimp
erw [← ρAut_apply_inv]
rw [← LinearMap.comp_assoc, ← ModuleCat.comp_def, ← ModuleCat.comp_def, Iso.inv_comp_eq,
ρAut_apply_hom]
exact comm
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/LinearIndependent.lean | 973 | 987 | theorem LinearIndependent.independent_span_singleton (hv : LinearIndependent R v) :
CompleteLattice.Independent fun i => R ∙ v i := by |
refine CompleteLattice.independent_def.mp fun i => ?_
rw [disjoint_iff_inf_le]
intro m hm
simp only [mem_inf, mem_span_singleton, iSup_subtype'] at hm
rw [← span_range_eq_iSup] at hm
obtain ⟨⟨r, rfl⟩, hm⟩ := hm
suffices r = 0 by simp [this]
apply linearIndependent_iff_not_smul_mem_span.mp hv i
-- Por... |
/-
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 | 1,125 | 1,126 | theorem insert_subset_iff : insert a s ⊆ t ↔ a ∈ t ∧ s ⊆ t := by |
simp only [subset_def, mem_insert_iff, or_imp, forall_and, forall_eq]
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Thomas Zhu, Mario Carneiro
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
#align_import number_theory.legendre_symbol.jacobi_symbol from "leanprover-... | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 493 | 499 | theorem quadratic_reciprocity_if {a b : ℕ} (ha2 : a % 2 = 1) (hb2 : b % 2 = 1) :
(if a % 4 = 3 ∧ b % 4 = 3 then -J(b | a) else J(b | a)) = J(a | b) := by |
rcases Nat.odd_mod_four_iff.mp ha2 with ha1 | ha3
· simpa [ha1] using jacobiSym.quadratic_reciprocity_one_mod_four' (Nat.odd_iff.mpr hb2) ha1
rcases Nat.odd_mod_four_iff.mp hb2 with hb1 | hb3
· simpa [hb1] using jacobiSym.quadratic_reciprocity_one_mod_four hb1 (Nat.odd_iff.mpr ha2)
simpa [ha3, hb3] using (ja... |
/-
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.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 671 | 680 | theorem _root_.PartialHomeomorph.isLocalStructomorphWithinAt_iff' {G : StructureGroupoid H}
[ClosedUnderRestriction G] (f : PartialHomeomorph H H) {s : Set H} {x : H} (hs : f.source ⊆ s)
(hx : x ∈ f.source ∪ sᶜ) :
G.IsLocalStructomorphWithinAt (⇑f) s x ↔
x ∈ s → ∃ e : PartialHomeomorph H H,
e ∈ ... |
rw [f.isLocalStructomorphWithinAt_iff hx]
refine imp_congr_right fun _ ↦ exists_congr fun e ↦ and_congr_right fun _ ↦ ?_
refine and_congr_right fun h2e ↦ ?_
rw [inter_eq_right.mpr (h2e.trans hs)]
|
/-
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.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
#align_import measure_theory... | Mathlib/MeasureTheory/Measure/Content.lean | 203 | 207 | theorem innerContent_iUnion_nat [R1Space G] ⦃U : ℕ → Set G⦄
(hU : ∀ i : ℕ, IsOpen (U i)) :
μ.innerContent ⟨⋃ i : ℕ, U i, isOpen_iUnion hU⟩ ≤ ∑' i : ℕ, μ.innerContent ⟨U i, hU i⟩ := by |
have := μ.innerContent_iSup_nat fun i => ⟨U i, hU i⟩
rwa [Opens.iSup_def] at this
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 863 | 863 | theorem div_eq_inv_self : a / b = b⁻¹ ↔ a = 1 := by | rw [div_eq_mul_inv, mul_left_eq_self]
|
/-
Copyright (c) 2020 Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jalex Stark, Scott Morrison, Eric Wieser, Oliver Nash, Wen Yang
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanpr... | Mathlib/Data/Matrix/Basis.lean | 190 | 191 | theorem mul_left_apply_of_ne (a b : n) (h : a ≠ i) (M : Matrix n n α) :
(stdBasisMatrix i j c * M) a b = 0 := by | simp [mul_apply, h.symm]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d4510... | Mathlib/Data/Fintype/Basic.lean | 976 | 979 | theorem Fintype.coe_image_univ [Fintype α] [DecidableEq β] {f : α → β} :
↑(Finset.image f Finset.univ) = Set.range f := by |
ext x
simp
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.C... | Mathlib/Topology/Instances/AddCircle.lean | 222 | 228 | theorem liftIco_coe_apply {f : 𝕜 → B} {x : 𝕜} (hx : x ∈ Ico a (a + p)) :
liftIco p a f ↑x = f x := by |
have : (equivIco p a) x = ⟨x, hx⟩ := by
rw [Equiv.apply_eq_iff_eq_symm_apply]
rfl
rw [liftIco, comp_apply, this]
rfl
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 429 | 441 | theorem lintegral_tendsto_of_tendsto_of_monotone {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞}
(hf : ∀ n, AEMeasurable (f n) μ) (h_mono : ∀ᵐ x ∂μ, Monotone fun n => f n x)
(h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 <| F x)) :
Tendsto (fun n => ∫⁻ x, f n x ∂μ) atTop (𝓝 <| ∫⁻ x, F x ∂μ) := by |
have : Monotone fun n => ∫⁻ x, f n x ∂μ := fun i j hij =>
lintegral_mono_ae (h_mono.mono fun x hx => hx hij)
suffices key : ∫⁻ x, F x ∂μ = ⨆ n, ∫⁻ x, f n x ∂μ by
rw [key]
exact tendsto_atTop_iSup this
rw [← lintegral_iSup' hf h_mono]
refine lintegral_congr_ae ?_
filter_upwards [h_mono, h_tendsto]... |
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_e... | Mathlib/FieldTheory/PrimitiveElement.lean | 404 | 413 | theorem primitive_element_iff_algHom_eq_of_eval (α : E)
(φ : E →ₐ[F] A) : F⟮α⟯ = ⊤ ↔ ∀ ψ : E →ₐ[F] A, φ α = ψ α → φ = ψ := by |
refine ⟨fun h ψ hψ ↦ (Field.primitive_element_iff_algHom_eq_of_eval' F A hA α).mp h hψ,
fun h ↦ eq_of_le_of_finrank_eq' le_top ?_⟩
letI : Algebra F⟮α⟯ A := (φ.comp F⟮α⟯.val).toAlgebra
haveI := isSeparable_tower_top_of_isSeparable F F⟮α⟯ E
rw [IntermediateField.finrank_top, ← AlgHom.card_of_splits _ _ A, Fi... |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.... | Mathlib/Data/Sign.lean | 347 | 347 | theorem sign_zero : sign (0 : α) = 0 := by | simp [sign_apply]
|
/-
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 | 167 | 169 | theorem matPolyEquiv_eval_eq_map (M : Matrix n n R[X]) (r : R) :
(matPolyEquiv M).eval (scalar n r) = M.map (eval r) := by |
simpa only [AlgEquiv.symm_apply_apply] using (matPolyEquiv_symm_map_eval (matPolyEquiv M) r).symm
|
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-communi... | Mathlib/Algebra/Polynomial/EraseLead.lean | 233 | 236 | theorem eraseLead_natDegree_le (f : R[X]) : (eraseLead f).natDegree ≤ f.natDegree - 1 := by |
rcases f.eraseLead_natDegree_lt_or_eraseLead_eq_zero with (h | h)
· exact Nat.le_sub_one_of_lt h
· simp only [h, natDegree_zero, zero_le]
|
/-
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 | 114 | 117 | theorem HasStrictDerivAt.smul_const (hc : HasStrictDerivAt c c' x) (f : F) :
HasStrictDerivAt (fun y => c y • f) (c' • f) x := by |
have := hc.smul (hasStrictDerivAt_const x f)
rwa [smul_zero, zero_add] at this
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 431 | 432 | theorem rpow_sub_nat {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℕ) : x ^ (y - n) = x ^ y / x ^ n := by |
simpa using rpow_sub_int hx y n
|
/-
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.Restrict
/-!
# Classes of measures
We introduce the following typeclasses for measures:
* `IsProbabilityMeasur... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 1,158 | 1,168 | theorem ae_of_forall_measure_lt_top_ae_restrict' {μ : Measure α} (ν : Measure α) [SigmaFinite μ]
[SigmaFinite ν] (P : α → Prop)
(h : ∀ s, MeasurableSet s → μ s < ∞ → ν s < ∞ → ∀ᵐ x ∂μ.restrict s, P x) : ∀ᵐ x ∂μ, P x := by |
have : ∀ n, ∀ᵐ x ∂μ, x ∈ spanningSets (μ + ν) n → P x := by
intro n
have := h
(spanningSets (μ + ν) n) (measurable_spanningSets _ _)
((self_le_add_right _ _).trans_lt (measure_spanningSets_lt_top (μ + ν) _))
((self_le_add_left _ _).trans_lt (measure_spanningSets_lt_top (μ + ν) _))
exact... |
/-
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.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.... | Mathlib/RingTheory/Trace.lean | 134 | 146 | theorem trace_trace_of_basis [Algebra S T] [IsScalarTower R S T] {ι κ : Type*} [Finite ι]
[Finite κ] (b : Basis ι R S) (c : Basis κ S T) (x : T) :
trace R S (trace S T x) = trace R T x := by |
haveI := Classical.decEq ι
haveI := Classical.decEq κ
cases nonempty_fintype ι
cases nonempty_fintype κ
rw [trace_eq_matrix_trace (b.smul c), trace_eq_matrix_trace b, trace_eq_matrix_trace c,
Matrix.trace, Matrix.trace, Matrix.trace, ← Finset.univ_product_univ, Finset.sum_product]
refine Finset.sum_con... |
/-
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.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "lea... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 620 | 622 | theorem _root_.IsClosed.not_mem_iff_infDist_pos (h : IsClosed s) (hs : s.Nonempty) :
x ∉ s ↔ 0 < infDist x s := by |
simp [h.mem_iff_infDist_zero hs, infDist_nonneg.gt_iff_ne]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
#align_import data.pnat.prime from "leanprover-community/mathlib"@"09597669f0242... | Mathlib/Data/PNat/Prime.lean | 279 | 285 | theorem Coprime.factor_eq_gcd_left {a b m n : ℕ+} (cop : m.Coprime n) (am : a ∣ m) (bn : b ∣ n) :
a = (a * b).gcd m := by |
rw [gcd_eq_left_iff_dvd] at am
conv_lhs => rw [← am]
rw [eq_comm]
apply Coprime.gcd_mul_right_cancel a
apply Coprime.coprime_dvd_left bn cop.symm
|
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-communi... | Mathlib/Algebra/Polynomial/EraseLead.lean | 127 | 130 | theorem card_support_eraseLead : f.eraseLead.support.card = f.support.card - 1 := by |
by_cases hf : f = 0
· rw [hf, eraseLead_zero, support_zero, card_empty]
· rw [← card_support_eraseLead_add_one hf, add_tsub_cancel_right]
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Multivariate.M
import Mathlib.Data... | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean | 432 | 445 | theorem Cofix.abs_repr {α} (x : Cofix F α) : Quot.mk _ (Cofix.repr x) = x := by |
let R := fun x y : Cofix F α => abs (repr y) = x
refine Cofix.bisim₂ R ?_ _ _ rfl
clear x
rintro x y h
subst h
dsimp [Cofix.dest, Cofix.abs]
induction y using Quot.ind
simp only [Cofix.repr, M.dest_corec, abs_map, MvQPF.abs_repr, Function.comp]
conv => congr; rfl; rw [Cofix.dest]
rw [MvFunctor.map_... |
/-
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 | 329 | 330 | theorem cosh_sub : cosh (x - y) = cosh x * cosh y - sinh x * sinh y := by |
simp [sub_eq_add_neg, cosh_add, sinh_neg, cosh_neg]
|
/-
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 | 650 | 652 | theorem integral_div {𝕜 : Type*} [RCLike 𝕜] (r : 𝕜) (f : ℝ → 𝕜) :
∫ x in a..b, f x / r ∂μ = (∫ x in a..b, f x ∂μ) / r := by |
simpa only [div_eq_mul_inv] using integral_mul_const r⁻¹ f
|
/-
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.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"7... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 194 | 196 | theorem coeff_add_eq_right_of_lt (pn : p.natDegree < n) : (p + q).coeff n = q.coeff n := by |
rw [add_comm]
exact coeff_add_eq_left_of_lt pn
|
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic
import Mathlib.RingTheory.RootsOfUnity.Minpoly
#align_import ring_theory.polynomial.cyclotomic.roots from "le... | Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean | 213 | 223 | theorem cyclotomic.isCoprime_rat {n m : ℕ} (h : n ≠ m) :
IsCoprime (cyclotomic n ℚ) (cyclotomic m ℚ) := by |
rcases n.eq_zero_or_pos with (rfl | hnzero)
· exact isCoprime_one_left
rcases m.eq_zero_or_pos with (rfl | hmzero)
· exact isCoprime_one_right
rw [Irreducible.coprime_iff_not_dvd <| cyclotomic.irreducible_rat <| hnzero]
exact fun hdiv => h <| cyclotomic_injective <|
eq_of_monic_of_associated (cyclotomi... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,910 | 1,925 | theorem eq_pow_of_mul_eq_pow {a b c : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
(hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) {k : ℕ} (h : a * b = c ^ k) :
∃ d : Associates α, a = d ^ k := by |
classical
nontriviality α
by_cases hk0 : k = 0
· use 1
rw [hk0, pow_zero] at h ⊢
apply (mul_eq_one_iff.1 h).1
· refine is_pow_of_dvd_count ha fun p hp ↦ ?_
apply dvd_count_of_dvd_count_mul hb hp hab
rw [h]
apply dvd_count_pow _ hp
rintro rfl
rw [zero_pow hk0] at h
cases mul_eq... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 150 | 154 | theorem degree_eq_iff_natDegree_eq_of_pos {p : R[X]} {n : ℕ} (hn : 0 < n) :
p.degree = n ↔ p.natDegree = n := by |
obtain rfl|h := eq_or_ne p 0
· simp [hn.ne]
· exact degree_eq_iff_natDegree_eq h
|
/-
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, Scott Morrison
-/
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathli... | Mathlib/Data/List/Range.lean | 125 | 126 | theorem pairwise_gt_iota (n : ℕ) : Pairwise (· > ·) (iota n) := by |
simpa only [iota_eq_reverse_range', pairwise_reverse] using pairwise_lt_range' 1 n
|
/-
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 | 140 | 146 | theorem IsCofiltered.of_exists_of_isCofiltered_of_fullyFaithful [IsCofiltered D] [F.Full]
[F.Faithful] (h : ∀ d, ∃ c, Nonempty (F.obj c ⟶ d)) : IsCofiltered C :=
{ IsCofilteredOrEmpty.of_exists_of_isCofiltered_of_fullyFaithful F h with
nonempty := by |
have : Nonempty D := IsCofiltered.nonempty
obtain ⟨c, -⟩ := h (Classical.arbitrary D)
exact ⟨c⟩ }
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
Coinductive formalization of unbounded computations.
-/
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
#align_import data.seq.computation from "le... | Mathlib/Data/Seq/Computation.lean | 599 | 610 | theorem eq_thinkN {s : Computation α} {a n} (h : Results s a n) : s = thinkN (pure a) n := by |
revert s
induction' n with n IH <;> intro s <;> apply recOn s (fun a' => _) fun s => _ <;> intro a h
· rw [← eq_of_pure_mem h.mem]
rfl
· cases' of_results_think h with n h
cases h
contradiction
· have := h.len_unique (results_pure _)
contradiction
· rw [IH (results_think_iff.1 h)]
rfl
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Da... | Mathlib/GroupTheory/OrderOfElement.lean | 1,242 | 1,248 | theorem orderOf_abs_ne_one (h : |x| ≠ 1) : orderOf x = 0 := by |
rw [orderOf_eq_zero_iff']
intro n hn hx
replace hx : |x| ^ n = 1 := by simpa only [abs_one, abs_pow] using congr_arg abs hx
cases' h.lt_or_lt with h h
· exact ((pow_lt_one (abs_nonneg x) h hn.ne').ne hx).elim
· exact ((one_lt_pow h hn.ne').ne' hx).elim
|
/-
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 | 440 | 440 | theorem arccos_lt_pi_div_two {x : ℝ} : arccos x < π / 2 ↔ 0 < x := by | simp [arccos]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 301 | 302 | theorem degree_C_mul_X (ha : a ≠ 0) : degree (C a * X) = 1 := by |
simpa only [pow_one] using degree_C_mul_X_pow 1 ha
|
/-
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.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of options
... | Mathlib/Data/Option/NAry.lean | 91 | 92 | theorem map_map₂ (f : α → β → γ) (g : γ → δ) :
(map₂ f a b).map g = map₂ (fun a b => g (f a b)) a b := by | cases a <;> cases b <;> rfl
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/LinearIndependent.lean | 569 | 578 | theorem LinearIndependent.injective [Nontrivial R] (hv : LinearIndependent R v) : Injective v := by |
intro i j hij
let l : ι →₀ R := Finsupp.single i (1 : R) - Finsupp.single j 1
have h_total : Finsupp.total ι M R v l = 0 := by
simp_rw [l, LinearMap.map_sub, Finsupp.total_apply]
simp [hij]
have h_single_eq : Finsupp.single i (1 : R) = Finsupp.single j 1 := by
rw [linearIndependent_iff] at hv
s... |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... | Mathlib/Analysis/Fourier/AddCircle.lean | 167 | 168 | theorem fourier_add {m n : ℤ} {x : AddCircle T} : fourier (m+n) x = fourier m x * fourier n x := by |
simp_rw [fourier_apply, add_zsmul, toCircle_add, coe_mul_unitSphere]
|
/-
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.Computability.Partrec
import Mathlib.Data.Option.Basic
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad... | Mathlib/Computability/PartrecCode.lean | 650 | 687 | theorem evaln_mono : ∀ {k₁ k₂ c n x}, k₁ ≤ k₂ → x ∈ evaln k₁ c n → x ∈ evaln k₂ c n
| 0, k₂, c, n, x, _, h => by simp [evaln] at h
| k + 1, k₂ + 1, c, n, x, hl, h => by
have hl' := Nat.le_of_succ_le_succ hl
have :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ → (x ∈ o₁ → x ∈ o₂) →
x ∈ ... |
simp only [Option.mem_def, bind, Option.bind_eq_some, Option.guard_eq_some', exists_and_left,
exists_const, and_imp]
introv h h₁ h₂ h₃
exact ⟨le_trans h₂ h, h₁ h₃⟩
simp? at h ⊢ says simp only [Option.mem_def] at h ⊢
induction' c with cf cg hf hg cf cg hf hg cf cg hf hg cf hf generaliz... |
/-
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.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 316 | 319 | theorem differentiableAt_euclidean :
DifferentiableAt 𝕜 f y ↔ ∀ i, DifferentiableAt 𝕜 (fun x => f x i) y := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiableAt_iff, differentiableAt_pi]
rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Math... | Mathlib/Analysis/SpecificLimits/Basic.lean | 485 | 491 | theorem dist_le_of_le_geometric_of_tendsto {a : α} (ha : Tendsto f atTop (𝓝 a)) (n : ℕ) :
dist (f n) a ≤ C * r ^ n / (1 - r) := by |
have := aux_hasSum_of_le_geometric hr hu
convert dist_le_tsum_of_dist_le_of_tendsto _ hu ⟨_, this⟩ ha n
simp only [pow_add, mul_left_comm C, mul_div_right_comm]
rw [mul_comm]
exact (this.mul_left _).tsum_eq.symm
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.Tactic.SuppressCo... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,374 | 1,376 | theorem rTensor_comp_map (f' : P →ₗ[R] S) (f : M →ₗ[R] P) (g : N →ₗ[R] Q) :
(f'.rTensor _).comp (map f g) = map (f'.comp f) g := by |
simp only [lTensor, rTensor, ← map_comp, id_comp, comp_id]
|
/-
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.BigOperators.Group.List
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.List.FinRange
import Mathlib.Algebra.Group.Action.Defs
import Mathli... | Mathlib/Algebra/GradedMonoid.lean | 446 | 451 | theorem GradedMonoid.mk_list_dProd (l : List α) (fι : α → ι) (fA : ∀ a, A (fι a)) :
GradedMonoid.mk _ (l.dProd fι fA) = (l.map fun a => GradedMonoid.mk (fι a) (fA a)).prod := by |
match l with
| [] => simp only [List.dProdIndex_nil, List.dProd_nil, List.map_nil, List.prod_nil]; rfl
| head::tail =>
simp [← GradedMonoid.mk_list_dProd tail _ _, GradedMonoid.mk_mul_mk, List.prod_cons]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tacti... | Mathlib/RingTheory/Ideal/Quotient.lean | 169 | 175 | theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by |
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Ta... | Mathlib/RingTheory/Valuation/Basic.lean | 348 | 349 | theorem one_lt_val_iff (v : Valuation K Γ₀) {x : K} (h : x ≠ 0) : 1 < v x ↔ v x⁻¹ < 1 := by |
simpa using (inv_lt_inv₀ (v.ne_zero_iff.2 h) one_ne_zero).symm
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,542 | 1,548 | theorem norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul {x : E} {r : 𝕜} (hx : x ≠ 0)
(hr : r ≠ 0) : ‖⟪x, r • x⟫‖ / (‖x‖ * ‖r • x‖) = 1 := by |
have hx' : ‖x‖ ≠ 0 := by simp [hx]
have hr' : ‖r‖ ≠ 0 := by simp [hr]
rw [inner_smul_right, norm_mul, ← inner_self_re_eq_norm, inner_self_eq_norm_mul_norm, norm_smul]
rw [← mul_assoc, ← div_div, mul_div_cancel_right₀ _ hx', ← div_div, mul_comm,
mul_div_cancel_right₀ _ hr', div_self hx']
|
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Order.Sublattice
import Mathlib.Order.Hom.CompleteLattice
/-!
# Complete Sublattices
This file defines complete sublattices. These are subsets of complete ... | Mathlib/Order/CompleteSublattice.lean | 84 | 85 | theorem coe_sSup' (S : Set L) : (↑(sSup S) : α) = ⨆ N ∈ S, (N : α) := by |
rw [coe_sSup, ← Set.image, sSup_image]
|
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