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) 2024 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen, Mitchell Lee
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.GroupTheory.PresentedGroup
import Mathlib.GroupTheory.Coxeter.Matrix
/-!
# Coxeter groups and Coxeter syst... | Mathlib/GroupTheory/Coxeter/Basic.lean | 237 | 240 | theorem submonoid_closure_range_simple : Submonoid.closure (range cs.simple) = ⊤ := by |
have : range cs.simple = range cs.simple ∪ (range cs.simple)⁻¹ := by
simp_rw [inv_range, inv_simple, union_self]
rw [this, ← Subgroup.closure_toSubmonoid, subgroup_closure_range_simple, Subgroup.top_toSubmonoid]
|
/-
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.Algebra.Group.Embedding
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Finset.Union
#align_imp... | Mathlib/Data/Finset/Image.lean | 175 | 180 | theorem subset_map_symm {t : Finset β} {f : α ≃ β} : s ⊆ t.map f.symm ↔ s.map f ⊆ t := by |
constructor <;> intro h x hx
· simp only [mem_map_equiv, Equiv.symm_symm] at hx
simpa using h hx
· simp only [mem_map_equiv]
exact h (by simp [hx])
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 780 | 786 | theorem setToL1S_smul_real (T : Set α → E →L[ℝ] F)
(h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) (c : ℝ)
(f : α →₁ₛ[μ] E) : setToL1S T (c • f) = c • setToL1S T f := by |
simp_rw [setToL1S]
rw [← SimpleFunc.setToSimpleFunc_smul_real T h_add c (SimpleFunc.integrable f)]
refine SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable _) ?_
exact smul_toSimpleFunc c f
|
/-
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.Algebra.Group.Embedding
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Finset.Union
#align_imp... | Mathlib/Data/Finset/Image.lean | 183 | 184 | theorem map_symm_subset {t : Finset β} {f : α ≃ β} : t.map f.symm ⊆ s ↔ t ⊆ s.map f := by |
simp only [← subset_map_symm, Equiv.symm_symm]
|
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.M... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 376 | 382 | theorem weightedHomogeneousComponent_eq_zero'
(h : ∀ d : σ →₀ ℕ, d ∈ φ.support → weightedDegree w d ≠ n) :
weightedHomogeneousComponent w n φ = 0 := by |
classical
rw [weightedHomogeneousComponent_apply, sum_eq_zero]
intro d hd; rw [mem_filter] at hd
exfalso; exact h _ hd.1 hd.2
|
/-
Copyright (c) 2019 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Paul Lezeau, Junyan Xu
-/
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 103 | 108 | theorem ker_eval {s : S} (hs : IsIntegral R s) :
RingHom.ker ((Polynomial.aeval s).toRingHom : R[X] →+* S) =
Ideal.span ({minpoly R s} : Set R[X]) := by |
ext p
simp_rw [RingHom.mem_ker, AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom,
isIntegrallyClosed_dvd_iff hs, ← Ideal.mem_span_singleton]
|
/-
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.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
#align_import order.hom.complete_lattice from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a11... | Mathlib/Order/Hom/CompleteLattice.lean | 134 | 135 | theorem map_iSup₂ [SupSet α] [SupSet β] [sSupHomClass F α β] (f : F) (g : ∀ i, κ i → α) :
f (⨆ (i) (j), g i j) = ⨆ (i) (j), f (g i j) := by | simp_rw [map_iSup]
|
/-
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 measure_theory.group.arithmetic from "leanprover-community/mathlib"@"a75898643b2d774cced9ae7c0b28c2... | Mathlib/MeasureTheory/Group/Arithmetic.lean | 419 | 424 | theorem ae_eq_trim_of_measurable {α E} {m m0 : MeasurableSpace α} {μ : Measure α}
[MeasurableSpace E] [AddGroup E] [MeasurableSingletonClass E] [MeasurableSub₂ E]
(hm : m ≤ m0) {f g : α → E} (hf : Measurable[m] f) (hg : Measurable[m] g) (hfg : f =ᵐ[μ] g) :
f =ᵐ[μ.trim hm] g := by |
rwa [Filter.EventuallyEq, ae_iff, trim_measurableSet_eq hm _]
exact @MeasurableSet.compl α _ m (@measurableSet_eq_fun α m E _ _ _ _ _ _ hf hg)
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.measure.probability_measure from "leanprove... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 380 | 385 | theorem self_eq_mass_mul_normalize (s : Set Ω) : μ s = μ.mass * μ.normalize s := by |
obtain rfl | h := eq_or_ne μ 0
· simp
have mass_nonzero : μ.mass ≠ 0 := by rwa [μ.mass_nonzero_iff]
simp only [normalize, dif_neg mass_nonzero]
simp [ProbabilityMeasure.coe_mk, toMeasure_smul, mul_inv_cancel_left₀ mass_nonzero, coeFn_def]
|
/-
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 | 142 | 146 | theorem contDiffOn_of_differentiableOn_deriv {n : ℕ∞}
(h : ∀ m : ℕ, (m : ℕ∞) ≤ n → DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s) :
ContDiffOn 𝕜 n f s := by |
apply contDiffOn_of_differentiableOn
simpa only [iteratedFDerivWithin_eq_equiv_comp, LinearIsometryEquiv.comp_differentiableOn_iff]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Grou... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 528 | 535 | theorem map_of_injective [MonoidHomClass F M N] (h : IsPrimitiveRoot ζ k) (hf : Injective f) :
IsPrimitiveRoot (f ζ) k where
pow_eq_one := by | rw [← map_pow, h.pow_eq_one, _root_.map_one]
dvd_of_pow_eq_one := by
rw [h.eq_orderOf]
intro l hl
rw [← map_pow, ← map_one f] at hl
exact orderOf_dvd_of_pow_eq_one (hf hl)
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathl... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 1,383 | 1,387 | theorem zero_le_toSignedMeasure : 0 ≤ μ.toSignedMeasure := by |
rw [← le_restrict_univ_iff_le]
refine restrict_le_restrict_of_subset_le _ _ fun j hj₁ _ => ?_
simp only [Measure.toSignedMeasure_apply_measurable hj₁, coe_zero, Pi.zero_apply,
ENNReal.toReal_nonneg, VectorMeasure.coe_zero]
|
/-
Copyright (c) 2021 Alena Gusakov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Jeremy Tan
-/
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
im... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 84 | 95 | theorem IsSRGWith.card_neighborFinset_union_eq {v w : V} (h : G.IsSRGWith n k ℓ μ) :
(G.neighborFinset v ∪ G.neighborFinset w).card =
2 * k - Fintype.card (G.commonNeighbors v w) := by |
apply Nat.add_right_cancel (m := Fintype.card (G.commonNeighbors v w))
rw [Nat.sub_add_cancel, ← Set.toFinset_card]
-- Porting note: Set.toFinset_inter needs workaround to use unification to solve for one of the
-- instance arguments:
· simp [commonNeighbors, @Set.toFinset_inter _ _ _ _ _ _ (_),
← neig... |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 1,205 | 1,208 | theorem IsPrefix.filter (p : α → Bool) ⦃l₁ l₂ : List α⦄ (h : l₁ <+: l₂) :
l₁.filter p <+: l₂.filter p := by |
obtain ⟨xs, rfl⟩ := h
rw [filter_append]; apply prefix_append
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
/-!
# Counting in lists
T... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 207 | 210 | theorem le_count_iff_replicate_sublist {l : List α} : n ≤ count a l ↔ replicate n a <+ l := by |
refine ⟨fun h => ?_, fun h => ?_⟩
· exact ((replicate_sublist_replicate a).2 h).trans <| filter_eq l a ▸ filter_sublist _
· simpa only [count_replicate_self] using h.count_le a
|
/-
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.Data.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order... | Mathlib/Order/CompleteLattice.lean | 1,848 | 1,849 | theorem swap_iInf [InfSet α] [InfSet β] (f : ι → α × β) : (iInf f).swap = ⨅ i, (f i).swap := by |
simp_rw [iInf, swap_sInf, ← range_comp, Function.comp] -- Porting note: need to unfold `∘`
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSe... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 187 | 189 | theorem coeff_sin_bit1 : coeff A (bit1 n) (sin A) = (-1) ^ n * coeff A (bit1 n) (exp A) := by |
rw [sin, coeff_mk, if_neg n.not_even_bit1, Nat.bit1_div_two, ← mul_one_div, map_mul, map_pow,
map_neg, map_one, coeff_exp]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | Mathlib/MeasureTheory/Function/Jacobian.lean | 924 | 1,026 | theorem lintegral_abs_det_fderiv_le_addHaar_image_aux1 (hs : MeasurableSet s)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) {ε : ℝ≥0} (εpos : 0 < ε) :
(∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) ≤ μ (f '' s) + 2 * ε * μ s := by |
/- To bound `∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ`, we cover `s` by sets where `f` is
well-approximated by linear maps `A n` (and where `f'` is almost everywhere close to `A n`),
and then use that `f` expands the measure of such a set by at least `(A n).det - ε`. -/
have :
∀ A : E →L[ℝ] E,
∃... |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTh... | Mathlib/GroupTheory/CoprodI.lean | 609 | 615 | theorem smul_induction {C : Word M → Prop} (h_empty : C empty)
(h_smul : ∀ (i) (m : M i) (w), C w → C (of m • w)) (w : Word M) : C w := by |
induction w using consRecOn with
| h_empty => exact h_empty
| h_cons _ _ _ _ _ ih =>
rw [cons_eq_smul]
exact h_smul _ _ _ ih
|
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot, Eric Wieser
-/
import Mathlib.Algebra.Group.Prod
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.prod from "leanprover-co... | Mathlib/GroupTheory/GroupAction/Prod.lean | 80 | 81 | theorem smul_mk_zero {β : Type*} [Monoid M] [AddMonoid β] [DistribMulAction M β] (a : M) (b : α) :
a • (b, (0 : β)) = (a • b, 0) := by | rw [Prod.smul_mk, smul_zero]
|
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... | Mathlib/Data/Num/Lemmas.lean | 958 | 980 | theorem castNum_shiftRight (m : Num) (n : Nat) : ↑(m >>> n) = (m : ℕ) >>> (n : ℕ) := by |
cases' m with m <;> dsimp only [← shiftr_eq_shiftRight, shiftr];
· symm
apply Nat.zero_shiftRight
induction' n with n IH generalizing m
· cases m <;> rfl
cases' m with m m <;> dsimp only [PosNum.shiftr, ← PosNum.shiftr_eq_shiftRight]
· rw [Nat.shiftRight_eq_div_pow]
symm
apply Nat.div_eq_of_lt
... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 845 | 846 | theorem Ioc_diff_Ioo_same (h : a < b) : Ioc a b \ Ioo a b = {b} := by |
rw [← Ioc_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 <| right_mem_Ioc.2 h)]
|
/-
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 | 683 | 684 | theorem mem_iSup₂_iff {f : ∀ i, κ i → UpperSet α} : (a ∈ ⨆ (i) (j), f i j) ↔ ∀ i j, a ∈ f i j := by |
simp_rw [mem_iSup_iff]
|
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Algebra.Module.LinearMap.Basic
/-!
# Endomorphisms ... | Mathlib/Algebra/Module/LinearMap/End.lean | 174 | 174 | theorem iterate_succ (n : ℕ) : f' ^ (n + 1) = comp (f' ^ n) f' := by | rw [pow_succ, mul_eq_comp]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.... | Mathlib/Order/Interval/Set/OrderIso.lean | 88 | 89 | theorem image_Ioo (e : α ≃o β) (a b : α) : e '' Ioo a b = Ioo (e a) (e b) := by |
rw [e.image_eq_preimage, e.symm.preimage_Ioo, e.symm_symm]
|
/-
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.Order.CompleteLattice
import Mathlib.Order.Synonym
import Mathlib.Order.Hom.Set
import Mathlib.Order.Bounds.Basic
#align_import order.galois_connectio... | Mathlib/Order/GaloisConnection.lean | 564 | 568 | theorem l_inf_u [SemilatticeInf α] [SemilatticeInf β] (gi : GaloisInsertion l u) (a b : β) :
l (u a ⊓ u b) = a ⊓ b :=
calc
l (u a ⊓ u b) = l (u (a ⊓ b)) := congr_arg l gi.gc.u_inf.symm
_ = a ⊓ b := by | simp only [gi.l_u_eq]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267... | Mathlib/Data/Finset/Prod.lean | 76 | 78 | theorem product_image_fst [DecidableEq α] (ht : t.Nonempty) : (s ×ˢ t).image Prod.fst = s := by |
ext i
simp [mem_image, ht.exists_mem]
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 2,043 | 2,053 | theorem reachable_iff_reflTransGen (u v : V) :
G.Reachable u v ↔ Relation.ReflTransGen G.Adj u v := by |
constructor
· rintro ⟨h⟩
induction h with
| nil => rfl
| cons h' _ ih => exact (Relation.ReflTransGen.single h').trans ih
· intro h
induction h with
| refl => rfl
| tail _ ha hr => exact Reachable.trans hr ⟨Walk.cons ha Walk.nil⟩
|
/-
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.NNReal
#align_import anal... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 259 | 266 | theorem IsBigOWith.rpow (h : IsBigOWith c l f g) (hc : 0 ≤ c) (hr : 0 ≤ r) (hg : 0 ≤ᶠ[l] g) :
IsBigOWith (c ^ r) l (fun x => f x ^ r) fun x => g x ^ r := by |
apply IsBigOWith.of_bound
filter_upwards [hg, h.bound] with x hgx hx
calc
|f x ^ r| ≤ |f x| ^ r := abs_rpow_le_abs_rpow _ _
_ ≤ (c * |g x|) ^ r := rpow_le_rpow (abs_nonneg _) hx hr
_ = c ^ r * |g x ^ r| := by rw [mul_rpow hc (abs_nonneg _), abs_rpow_of_nonneg hgx]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "l... | Mathlib/Algebra/Polynomial/Monic.lean | 388 | 389 | theorem monic_X_sub_C (x : R) : Monic (X - C x) := by |
simpa only [sub_eq_add_neg, C_neg] using monic_X_add_C (-x)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.Measur... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 74 | 106 | theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedSpace 𝕜 E]
{t : Set E} (ht : TopologicalSpace.IsSeparable t) {f : α → E}
(hf : ∀ c : Dual 𝕜 E, (fun x => ⟪f x, c⟫) =ᵐ[μ] 0) (h't : ∀ᵐ x ∂μ, f x ∈ t) : f =ᵐ[μ] 0 := by |
rcases ht with ⟨d, d_count, hd⟩
haveI : Encodable d := d_count.toEncodable
have : ∀ x : d, ∃ g : E →L[𝕜] 𝕜, ‖g‖ ≤ 1 ∧ g x = ‖(x : E)‖ :=
fun x => exists_dual_vector'' 𝕜 (x : E)
choose s hs using this
have A : ∀ a : E, a ∈ t → (∀ x, ⟪a, s x⟫ = (0 : 𝕜)) → a = 0 := by
intro a hat ha
contrapose! ... |
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.shapes.kernels from "leanprover-community/mathlib"@"95... | Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 909 | 912 | theorem cokernelIsoOfEq_inv_comp_desc {Z} {f g : X ⟶ Y} [HasCokernel f] [HasCokernel g] (h : f = g)
(e : Y ⟶ Z) (he) :
(cokernelIsoOfEq h).inv ≫ cokernel.desc _ e he = cokernel.desc _ e (by simp [← h, he]) := by |
cases h; simp
|
/-
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 | 876 | 883 | theorem prod_of_right {f : α × β → γ} {μ : Measure α} {ν : Measure β} {τ : Measure γ}
(hf : Measurable f) [SFinite ν]
(h2f : ∀ᵐ x ∂μ, QuasiMeasurePreserving (fun y => f (x, y)) ν τ) :
QuasiMeasurePreserving f (μ.prod ν) τ := by |
refine ⟨hf, ?_⟩
refine AbsolutelyContinuous.mk fun s hs h2s => ?_
rw [map_apply hf hs, prod_apply (hf hs)]; simp_rw [preimage_preimage]
rw [lintegral_congr_ae (h2f.mono fun x hx => hx.preimage_null h2s), lintegral_zero]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprove... | Mathlib/Analysis/Convex/Strict.lean | 353 | 355 | theorem StrictConvex.smul_mem_of_zero_mem (hs : StrictConvex 𝕜 s) (zero_mem : (0 : E) ∈ s)
(hx : x ∈ s) (hx₀ : x ≠ 0) {t : 𝕜} (ht₀ : 0 < t) (ht₁ : t < 1) : t • x ∈ interior s := by |
simpa using hs.add_smul_mem zero_mem (by simpa using hx) hx₀ ht₀ ht₁
|
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johanes Hölzl, Patrick Massot, Yury Kudryashov, Kevin Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@... | Mathlib/Order/Filter/Prod.lean | 479 | 483 | theorem tendsto_prod_iff' {g' : Filter γ} {s : α → β × γ} :
Tendsto s f (g ×ˢ g') ↔ Tendsto (fun n => (s n).1) f g ∧ Tendsto (fun n => (s n).2) f g' := by |
dsimp only [SProd.sprod]
unfold Filter.prod
simp only [tendsto_inf, tendsto_comap_iff, (· ∘ ·)]
|
/-
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.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 242 | 246 | theorem comp_right_hasFDerivWithinAt_iff' {f : F → G} {s : Set F} {x : E} {f' : E →L[𝕜] G} :
HasFDerivWithinAt (f ∘ iso) f' (iso ⁻¹' s) x ↔
HasFDerivWithinAt f (f'.comp (iso.symm : F →L[𝕜] E)) s (iso x) := by |
rw [← iso.comp_right_hasFDerivWithinAt_iff, ContinuousLinearMap.comp_assoc,
iso.coe_symm_comp_coe, ContinuousLinearMap.comp_id]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Sum.Order
import Mathlib.Order.InitialSeg
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
#align_impor... | Mathlib/SetTheory/Ordinal/Basic.lean | 1,648 | 1,663 | theorem List.Sorted.lt_ord_of_lt [LinearOrder α] [IsWellOrder α (· < ·)] {l m : List α}
{o : Ordinal} (hl : l.Sorted (· > ·)) (hm : m.Sorted (· > ·)) (hmltl : m < l)
(hlt : ∀ i ∈ l, Ordinal.typein (· < ·) i < o) : ∀ i ∈ m, Ordinal.typein (· < ·) i < o := by |
replace hmltl : List.Lex (· < ·) m l := hmltl
cases l with
| nil => simp at hmltl
| cons a as =>
cases m with
| nil => intro i hi; simp at hi
| cons b bs =>
intro i hi
suffices h : i ≤ a by refine lt_of_le_of_lt ?_ (hlt a (mem_cons_self a as)); simpa
cases hi with
| head as ... |
/-
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 | 2,054 | 2,056 | theorem mk_sUnion_le {α : Type u} (A : Set (Set α)) : #(⋃₀ A) ≤ #A * ⨆ s : A, #s := by |
rw [sUnion_eq_iUnion]
apply mk_iUnion_le
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury G. Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
impor... | Mathlib/Analysis/SpecificLimits/Normed.lean | 525 | 534 | theorem NormedRing.tsum_geometric_of_norm_lt_one (x : R) (h : ‖x‖ < 1) :
‖∑' n : ℕ, x ^ n‖ ≤ ‖(1 : R)‖ - 1 + (1 - ‖x‖)⁻¹ := by |
rw [tsum_eq_zero_add (summable_geometric_of_norm_lt_one x h)]
simp only [_root_.pow_zero]
refine le_trans (norm_add_le _ _) ?_
have : ‖∑' b : ℕ, (fun n ↦ x ^ (n + 1)) b‖ ≤ (1 - ‖x‖)⁻¹ - 1 := by
refine tsum_of_norm_bounded ?_ fun b ↦ norm_pow_le' _ (Nat.succ_pos b)
convert (hasSum_nat_add_iff' 1).mpr (h... |
/-
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 | 387 | 400 | theorem ContinuousAt.comp_lowerSemicontinuousWithinAt {g : γ → δ} {f : α → γ}
(hg : ContinuousAt g (f x)) (hf : LowerSemicontinuousWithinAt f s x) (gmon : Monotone g) :
LowerSemicontinuousWithinAt (g ∘ f) s x := by |
intro y hy
by_cases h : ∃ l, l < f x
· obtain ⟨z, zlt, hz⟩ : ∃ z < f x, Ioc z (f x) ⊆ g ⁻¹' Ioi y :=
exists_Ioc_subset_of_mem_nhds (hg (Ioi_mem_nhds hy)) h
filter_upwards [hf z zlt] with a ha
calc
y < g (min (f x) (f a)) := hz (by simp [zlt, ha, le_refl])
_ ≤ g (f a) := gmon (min_le_rig... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.Disjoint
#a... | Mathlib/Data/Real/Archimedean.lean | 296 | 299 | theorem sInf_nonpos (S : Set ℝ) (hS : ∀ x ∈ S, x ≤ (0 : ℝ)) : sInf S ≤ 0 := by |
rcases S.eq_empty_or_nonempty with (rfl | ⟨y, hy⟩)
· exact sInf_empty.le
· apply dite _ (fun h => csInf_le_of_le h hy <| hS y hy) fun h => (sInf_of_not_bddBelow h).le
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import probability.process.stopping from "leanp... | Mathlib/Probability/Process/Stopping.lean | 202 | 207 | theorem IsStoppingTime.measurableSet_ge (hτ : IsStoppingTime f τ) (i : ι) :
MeasurableSet[f i] {ω | i ≤ τ ω} := by |
have : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ := by
ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt]
rw [this]
exact (hτ.measurableSet_lt i).compl
|
/-
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.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | Mathlib/Order/Lattice.lean | 249 | 250 | theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by |
rw [sup_assoc, sup_assoc, sup_comm b]
|
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.Se... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 263 | 269 | theorem IsSelfAdjoint.nnnorm_pow_two_pow [NormedRing E] [StarRing E] [CstarRing E] {x : E}
(hx : IsSelfAdjoint x) (n : ℕ) : ‖x ^ 2 ^ n‖₊ = ‖x‖₊ ^ 2 ^ n := by |
induction' n with k hk
· simp only [pow_zero, pow_one, Nat.zero_eq]
· rw [pow_succ', pow_mul', sq]
nth_rw 1 [← selfAdjoint.mem_iff.mp hx]
rw [← star_pow, CstarRing.nnnorm_star_mul_self, ← sq, hk, pow_mul']
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Sébastien Gouëzel, Alex J. Best
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.Group.Nat
import ... | Mathlib/Algebra/BigOperators/Group/List.lean | 558 | 559 | theorem headI_add_tail_sum (L : List ℕ) : L.headI + L.tail.sum = L.sum := by |
cases L <;> simp
|
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Damiano Testa, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.Polynomial.Degree.Definitions
import ... | Mathlib/Algebra/Polynomial/Inductions.lean | 116 | 117 | theorem divX_C_mul_X_pow : divX (C a * X ^ n) = if n = 0 then 0 else C a * X ^ (n - 1) := by |
simp only [divX_C_mul, divX_X_pow, mul_ite, mul_zero]
|
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Data.Real.Pi.Bounds
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
/-!
# Number field discriminant
This file defines the discrimi... | Mathlib/NumberTheory/NumberField/Discriminant.lean | 320 | 325 | theorem natDegree_le_rankOfDiscrBdd (a : 𝓞 K) (h : ℚ⟮(a : K)⟯ = ⊤) :
natDegree (minpoly ℤ (a : K)) ≤ rankOfDiscrBdd N := by |
rw [Field.primitive_element_iff_minpoly_natDegree_eq,
minpoly.isIntegrallyClosed_eq_field_fractions' ℚ a.isIntegral_coe,
(minpoly.monic a.isIntegral_coe).natDegree_map] at h
exact h.symm ▸ rank_le_rankOfDiscrBdd hK
|
/-
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 | 118 | 119 | theorem laverage_eq (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂μ = (∫⁻ x, f x ∂μ) / μ univ := by |
rw [laverage_eq', lintegral_smul_measure, ENNReal.div_eq_inv_mul]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7... | Mathlib/Data/Set/Prod.lean | 326 | 327 | theorem prod_eq_empty_iff : s ×ˢ t = ∅ ↔ s = ∅ ∨ t = ∅ := by |
simp only [not_nonempty_iff_eq_empty.symm, prod_nonempty_iff, not_and_or]
|
/-
Copyright (c) 2022 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
import Mathlib.Algebra.Order.Hom.Ring
import Mathlib.Algebra.Order.Pointwise
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import algebra.order.... | Mathlib/Algebra/Order/CompleteField.lean | 144 | 146 | theorem cutMap_bddAbove (a : α) : BddAbove (cutMap β a) := by |
obtain ⟨q, hq⟩ := exists_rat_gt a
exact ⟨q, forall_mem_image.2 fun r hr => mod_cast (hq.trans' hr).le⟩
|
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
#align_import data.finmap from "leanprover-community/mathlib"@"c... | Mathlib/Data/Finmap.lean | 273 | 274 | theorem dlookup_list_toFinmap (a : α) (s : List (Sigma β)) : lookup a s.toFinmap = s.dlookup a := by |
rw [List.toFinmap, lookup_toFinmap, lookup_to_alist]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import geometry.euclidean.angle.un... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | 183 | 185 | theorem cos_angle_mul_norm_mul_norm (x y : V) : Real.cos (angle x y) * (‖x‖ * ‖y‖) = ⟪x, y⟫ := by |
rw [cos_angle, div_mul_cancel_of_imp]
simp (config := { contextual := true }) [or_imp]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Sym... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 291 | 292 | theorem polar_zero_right (y : M) : polar Q y 0 = 0 := by |
simp only [add_zero, polar, QuadraticForm.map_zero, sub_self]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury G. Kudryashov
-/
import Batteries.Data.Sum.Basic
import Batteries.Logic
/-!
# Disjoint union of types
Theorems about the definitions introduced in `Batteries.Da... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 219 | 225 | theorem lex_acc_inl (aca : Acc r a) : Acc (Lex r s) (inl a) := by |
induction aca with
| intro _ _ IH =>
constructor
intro y h
cases h with
| inl h' => exact IH _ h'
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 134 | 152 | theorem scaleRoots_eval₂_mul_of_commute {p : S[X]} (f : S →+* A) (a : A) (s : S)
(hsa : Commute (f s) a) (hf : ∀ s₁ s₂, Commute (f s₁) (f s₂)) :
eval₂ f (f s * a) (scaleRoots p s) = f s ^ p.natDegree * eval₂ f a p := by |
calc
_ = (scaleRoots p s).support.sum fun i =>
f (coeff p i * s ^ (p.natDegree - i)) * (f s * a) ^ i := by
simp [eval₂_eq_sum, sum_def]
_ = p.support.sum fun i => f (coeff p i * s ^ (p.natDegree - i)) * (f s * a) ^ i :=
(Finset.sum_subset (support_scaleRoots_le p s) fun i _hi hi' => by... |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
import Mathlib.Order.WellFounded
#align_import order.succ_pred.basic from "l... | Mathlib/Order/SuccPred/Basic.lean | 563 | 563 | theorem lt_succ_bot_iff [NoMaxOrder α] : a < succ ⊥ ↔ a = ⊥ := by | rw [lt_succ_iff, le_bot_iff]
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
#align_import number_theory.modular_forms.jacob... | Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean | 45 | 54 | theorem jacobiTheta_S_smul (τ : ℍ) :
jacobiTheta ↑(ModularGroup.S • τ) = (-I * τ) ^ (1 / 2 : ℂ) * jacobiTheta τ := by |
have h0 : (τ : ℂ) ≠ 0 := ne_of_apply_ne im (zero_im.symm ▸ ne_of_gt τ.2)
have h1 : (-I * τ) ^ (1 / 2 : ℂ) ≠ 0 := by
rw [Ne, cpow_eq_zero_iff, not_and_or]
exact Or.inl <| mul_ne_zero (neg_ne_zero.mpr I_ne_zero) h0
simp_rw [UpperHalfPlane.modular_S_smul, jacobiTheta_eq_jacobiTheta₂]
conv_rhs => erw [← of... |
/-
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.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.engel from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3... | Mathlib/Algebra/Lie/Engel.lean | 173 | 183 | theorem Function.Surjective.isEngelian {f : L →ₗ⁅R⁆ L₂} (hf : Function.Surjective f)
(h : LieAlgebra.IsEngelian.{u₁, u₂, u₄} R L) : LieAlgebra.IsEngelian.{u₁, u₃, u₄} R L₂ := by |
intro M _i1 _i2 _i3 _i4 h'
letI : LieRingModule L M := LieRingModule.compLieHom M f
letI : LieModule R L M := compLieHom M f
have hnp : ∀ x, IsNilpotent (toEnd R L M x) := fun x => h' (f x)
have surj_id : Function.Surjective (LinearMap.id : M →ₗ[R] M) := Function.surjective_id
haveI : LieModule.IsNilpotent... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
imp... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 650 | 662 | theorem leftUnitor_monoidal (X₁ X₂ : C) :
(λ_ X₁).hom ⊗ (λ_ X₂).hom =
tensor_μ C (𝟙_ C, X₁) (𝟙_ C, X₂) ≫ ((λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂)) ≫ (λ_ (X₁ ⊗ X₂)).hom := by |
dsimp only [tensor_μ]
have :
(λ_ X₁).hom ⊗ (λ_ X₂).hom =
(α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫
(𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv) ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ ((ρ_ X₁).hom ▷ X₂) := by
coherence
rw [this]; clear this
rw [← braiding_leftUnitor]
dsimp only [tensor_obj, prodMonoidal_tensorOb... |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,862 | 1,879 | theorem integral_mul_le_Lp_mul_Lq_of_nonneg {p q : ℝ} (hpq : p.IsConjExponent q) {f g : α → ℝ}
(hf_nonneg : 0 ≤ᵐ[μ] f) (hg_nonneg : 0 ≤ᵐ[μ] g) (hf : Memℒp f (ENNReal.ofReal p) μ)
(hg : Memℒp g (ENNReal.ofReal q) μ) :
∫ a, f a * g a ∂μ ≤ (∫ a, f a ^ p ∂μ) ^ (1 / p) * (∫ a, g a ^ q ∂μ) ^ (1 / q) := by |
have h_left : ∫ a, f a * g a ∂μ = ∫ a, ‖f a‖ * ‖g a‖ ∂μ := by
refine integral_congr_ae ?_
filter_upwards [hf_nonneg, hg_nonneg] with x hxf hxg
rw [Real.norm_of_nonneg hxf, Real.norm_of_nonneg hxg]
have h_right_f : ∫ a, f a ^ p ∂μ = ∫ a, ‖f a‖ ^ p ∂μ := by
refine integral_congr_ae ?_
filter_upwa... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.Affine... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 796 | 797 | theorem sum_centroidWeights_eq_one_of_cast_card_ne_zero (h : (card s : k) ≠ 0) :
∑ i ∈ s, s.centroidWeights k i = 1 := by | simp [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.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 415 | 416 | theorem UniformCauchySeqOn.uniformCauchySeqOnFilter (hF : UniformCauchySeqOn F p s) :
UniformCauchySeqOnFilter F p (𝓟 s) := by | rwa [← uniformCauchySeqOn_iff_uniformCauchySeqOnFilter]
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Sub
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_t... | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | 610 | 628 | theorem rnDeriv_smul_right (ν μ : Measure α) [IsFiniteMeasure ν]
[ν.HaveLebesgueDecomposition μ] {r : ℝ≥0} (hr : r ≠ 0) :
ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by |
refine (absolutelyContinuous_smul <| ENNReal.coe_ne_zero.2 hr).ae_le
(?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
rw [← withDensity_eq_iff]
rotate_left
· exact (measurable_rnDeriv _ _).aemeasurable
· exact (measurable_rnDeriv _ _).aemeasurable.const_smul _
· exact (lintegral_rnDeriv_lt_top ν _)... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Michael Howes
-/
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Finiteness
#align_import group_theory.abelianization from "lean... | Mathlib/GroupTheory/Abelianization.lean | 71 | 79 | theorem commutator_centralizer_commutator_le_center :
⁅centralizer (commutator G : Set G), centralizer (commutator G)⁆ ≤ Subgroup.center G := by |
rw [← Subgroup.centralizer_univ, ← Subgroup.coe_top, ←
Subgroup.commutator_eq_bot_iff_le_centralizer]
suffices ⁅⁅⊤, centralizer (commutator G : Set G)⁆, centralizer (commutator G : Set G)⁆ = ⊥ by
refine Subgroup.commutator_commutator_eq_bot_of_rotate ?_ this
rwa [Subgroup.commutator_comm (centralizer (... |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableClosure
import Mathlib.Algebra.CharP.IntermediateField
/-!
# Purely inseparable extension and relative perfect closure
This file contains basic... | Mathlib/FieldTheory/PurelyInseparable.lean | 666 | 681 | theorem adjoin_eq_adjoin_pow_expChar_pow_of_isSeparable (S : Set E) [IsSeparable F (adjoin F S)]
(q : ℕ) [ExpChar F q] (n : ℕ) : adjoin F S = adjoin F ((· ^ q ^ n) '' S) := by |
set L := adjoin F S
set M := adjoin F ((· ^ q ^ n) '' S)
have hi : M ≤ L := by
rw [adjoin_le_iff]
rintro _ ⟨y, hy, rfl⟩
exact pow_mem (subset_adjoin F S hy) _
letI := (inclusion hi).toAlgebra
haveI : IsSeparable M (extendScalars hi) := isSeparable_tower_top_of_isSeparable F M L
haveI : IsPurely... |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableClosure
import Mathlib.Algebra.CharP.IntermediateField
/-!
# Purely inseparable extension and relative perfect closure
This file contains basic... | Mathlib/FieldTheory/PurelyInseparable.lean | 327 | 332 | theorem le_perfectClosure_iff (L : IntermediateField F E) :
L ≤ perfectClosure F E ↔ IsPurelyInseparable F L := by |
refine ⟨fun h ↦ (isPurelyInseparable_iff_pow_mem F (ringExpChar F)).2 fun x ↦ ?_,
fun _ ↦ le_perfectClosure F E L⟩
obtain ⟨n, y, hy⟩ := h x.2
exact ⟨n, y, (algebraMap L E).injective hy⟩
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.... | Mathlib/Data/Nat/GCD/Basic.lean | 176 | 177 | theorem coprime_add_self_left {m n : ℕ} : Coprime (m + n) n ↔ Coprime m n := by |
rw [Coprime, Coprime, gcd_add_self_left]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 1,058 | 1,063 | theorem iUnion_Ioc_add_zsmul : ⋃ n : ℤ, Ioc (a + n • p) (a + (n + 1) • p) = univ := by |
refine eq_univ_iff_forall.mpr fun b => mem_iUnion.mpr ?_
rcases sub_toIocDiv_zsmul_mem_Ioc hp a b with ⟨hl, hr⟩
refine ⟨toIocDiv hp a b, ⟨lt_sub_iff_add_lt.mp hl, ?_⟩⟩
rw [add_smul, one_smul, ← add_assoc]
convert sub_le_iff_le_add.mp hr using 1; abel
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520... | Mathlib/Data/ENNReal/Inv.lean | 230 | 231 | theorem le_inv_iff_le_inv : a ≤ b⁻¹ ↔ b ≤ a⁻¹ := by |
simpa only [inv_inv] using @ENNReal.inv_le_inv a⁻¹ b
|
/-
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 | 1,043 | 1,047 | theorem comp_open_symm_mem_uniformity_sets {s : Set (α × α)} (hs : s ∈ 𝓤 α) :
∃ t ∈ 𝓤 α, IsOpen t ∧ SymmetricRel t ∧ t ○ t ⊆ s := by |
obtain ⟨t, ht₁, ht₂⟩ := comp_mem_uniformity_sets hs
obtain ⟨u, ⟨hu₁, hu₂, hu₃⟩, hu₄ : u ⊆ t⟩ := uniformity_hasBasis_open_symmetric.mem_iff.mp ht₁
exact ⟨u, hu₁, hu₂, hu₃, (compRel_mono hu₄ hu₄).trans ht₂⟩
|
/-
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.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Topology.UniformSpace.Compact
import Mathl... | Mathlib/Analysis/BoxIntegral/Basic.lean | 386 | 396 | theorem norm_integral_le_of_norm_le {g : ℝⁿ → ℝ} (hle : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ g x)
(μ : Measure ℝⁿ) [IsLocallyFiniteMeasure μ] (hg : Integrable I l g μ.toBoxAdditive.toSMul) :
‖(integral I l f μ.toBoxAdditive.toSMul : E)‖ ≤ integral I l g μ.toBoxAdditive.toSMul := by |
by_cases hfi : Integrable.{u, v, v} I l f μ.toBoxAdditive.toSMul
· refine le_of_tendsto_of_tendsto' hfi.hasIntegral.norm hg.hasIntegral fun π => ?_
refine norm_sum_le_of_le _ fun J _ => ?_
simp only [BoxAdditiveMap.toSMul_apply, norm_smul, smul_eq_mul, Real.norm_eq_abs,
μ.toBoxAdditive_apply, abs_of_... |
/-
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.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | Mathlib/Order/Lattice.lean | 245 | 246 | theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by |
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a... | Mathlib/MeasureTheory/Function/L1Space.lean | 656 | 657 | theorem integrable_zero : Integrable (fun _ => (0 : β)) μ := by |
simp [Integrable, aestronglyMeasurable_const]
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.RingTheory.Ideal.Maps
#align_import ring_theory.ideal.prod from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
/-!
# Ideals ... | Mathlib/RingTheory/Ideal/Prod.lean | 72 | 78 | theorem map_snd_prod (I : Ideal R) (J : Ideal S) : map (RingHom.snd R S) (prod I J) = J := by |
ext x
rw [mem_map_iff_of_surjective (RingHom.snd R S) Prod.snd_surjective]
exact
⟨by
rintro ⟨x, ⟨h, rfl⟩⟩
exact h.2, fun h => ⟨⟨0, x⟩, ⟨⟨Ideal.zero_mem _, h⟩, rfl⟩⟩⟩
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | Mathlib/RingTheory/Polynomial/Content.lean | 409 | 417 | theorem primPart_mul {p q : R[X]} (h0 : p * q ≠ 0) :
(p * q).primPart = p.primPart * q.primPart := by |
rw [Ne, ← content_eq_zero_iff, ← C_eq_zero] at h0
apply mul_left_cancel₀ h0
conv_lhs =>
rw [← (p * q).eq_C_content_mul_primPart, p.eq_C_content_mul_primPart,
q.eq_C_content_mul_primPart]
rw [content_mul, RingHom.map_mul]
ring
|
/-
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 | 151 | 164 | theorem comp_right_triangle_aux (adj₁ : f₁ ⊣ g₁) (adj₂ : f₂ ⊣ g₂) :
rightZigzag (compUnit adj₁ adj₂) (compCounit adj₁ adj₂) = (ρ_ _).hom ≫ (λ_ _).inv := by |
calc
_ = 𝟙 _ ⊗≫
(g₂ ≫ g₁) ◁ adj₁.unit ⊗≫
g₂ ◁ ((g₁ ≫ f₁) ◁ adj₂.unit ≫ adj₁.counit ▷ (f₂ ≫ g₂)) ▷ g₁ ⊗≫
adj₂.counit ▷ (g₂ ≫ g₁) ⊗≫ 𝟙 _ := by
simp [bicategoricalComp]; coherence
_ = 𝟙 _ ⊗≫
g₂ ◁ (rightZigzag adj₁.unit adj₁.counit) ⊗≫
(rightZigz... |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Data.Countable.Basic
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.Interval.Finset.Defs
#align_import orde... | Mathlib/Order/SuccPred/LinearLocallyFinite.lean | 233 | 236 | theorem toZ_iterate_succ_le (n : ℕ) : toZ i0 (succ^[n] i0) ≤ n := by |
rw [toZ_of_ge (le_succ_iterate _ _)]
norm_cast
exact Nat.find_min' _ rfl
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Algebra.MulA... | Mathlib/Topology/Algebra/Module/Basic.lean | 212 | 215 | theorem Submodule.isClosed_or_dense_of_isCoatom (s : Submodule R M) (hs : IsCoatom s) :
IsClosed (s : Set M) ∨ Dense (s : Set M) := by |
refine (hs.le_iff.mp s.le_topologicalClosure).symm.imp ?_ dense_iff_topologicalClosure_eq_top.mpr
exact fun h ↦ h ▸ isClosed_closure
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathl... | Mathlib/Data/Real/GoldenRatio.lean | 64 | 66 | theorem goldConj_mul_gold : ψ * φ = -1 := by |
rw [mul_comm]
exact gold_mul_goldConj
|
/-
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.Divisibility.Basic
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib... | Mathlib/Algebra/Ring/Divisibility/Basic.lean | 129 | 129 | theorem dvd_add_right (h : a ∣ b) : a ∣ b + c ↔ a ∣ c := by | rw [add_comm]; exact dvd_add_left h
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Mathlib/FieldTheory/RatFunc/Basic.lean | 145 | 147 | theorem ofFractionRing_mul (p q : FractionRing K[X]) :
ofFractionRing (p * q) = ofFractionRing p * ofFractionRing q := by |
simp only [Mul.mul, HMul.hMul, RatFunc.mul]
|
/-
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 | 1,813 | 1,816 | theorem DenseRange.comp {g : Y → Z} {f : α → Y} (hg : DenseRange g) (hf : DenseRange f)
(cg : Continuous g) : DenseRange (g ∘ f) := by |
rw [DenseRange, range_comp]
exact hg.dense_image cg hf
|
/-
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
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.LinearAlgebra.Span
#align_import linear_algebra.quotient from... | Mathlib/LinearAlgebra/Quotient.lean | 452 | 462 | theorem mapQ_pow {f : M →ₗ[R] M} (h : p ≤ p.comap f) (k : ℕ)
(h' : p ≤ p.comap (f ^ k) := p.le_comap_pow_of_le_comap h k) :
p.mapQ p (f ^ k) h' = p.mapQ p f h ^ k := by |
induction' k with k ih
· simp [LinearMap.one_eq_id]
· simp only [LinearMap.iterate_succ]
-- Porting note: why does any of these `optParams` need to be applied? Why didn't `simp` handle
-- all of this for us?
convert mapQ_comp p p p f (f ^ k) h (p.le_comap_pow_of_le_comap h k)
(h.trans (comap_mo... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "lean... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 996 | 1,003 | theorem contDiffOn_zero : ContDiffOn 𝕜 0 f s ↔ ContinuousOn f s := by |
refine ⟨fun H => H.continuousOn, fun H => ?_⟩
intro x hx m hm
have : (m : ℕ∞) = 0 := le_antisymm hm bot_le
rw [this]
refine ⟨insert x s, self_mem_nhdsWithin, ftaylorSeriesWithin 𝕜 f s, ?_⟩
rw [hasFTaylorSeriesUpToOn_zero_iff]
exact ⟨by rwa [insert_eq_of_mem hx], fun x _ => by simp [ftaylorSeriesWithin]⟩... |
/-
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.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPower... | Mathlib/RingTheory/PowerSeries/Basic.lean | 478 | 487 | theorem coeff_X_pow_mul' (p : R⟦X⟧) (n d : ℕ) :
coeff R d (X ^ n * p) = ite (n ≤ d) (coeff R (d - n) p) 0 := by |
split_ifs with h
· rw [← tsub_add_cancel_of_le h, coeff_X_pow_mul]
simp
· refine (coeff_mul _ _ _).trans (Finset.sum_eq_zero fun x hx => ?_)
rw [coeff_X_pow, if_neg, zero_mul]
have := mem_antidiagonal.mp hx
rw [add_comm] at this
exact ((le_of_add_le_right this.le).trans_lt <| not_le.mp h).ne
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Whiskering
import Mathlib.CategoryTheory.Sites.Plus
#align_import category_theory.sites.compatible_plus from "leanprover-community/mathli... | Mathlib/CategoryTheory/Sites/CompatiblePlus.lean | 61 | 66 | theorem diagramCompIso_hom_ι (X : C) (W : (J.Cover X)ᵒᵖ) (i : W.unop.Arrow) :
(J.diagramCompIso F P X).hom.app W ≫ Multiequalizer.ι ((unop W).index (P ⋙ F)) i =
F.map (Multiequalizer.ι _ _) := by |
delta diagramCompIso
dsimp
simp
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Algebra.MvPolynomial.Polynomial
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.RingTheory.Pol... | Mathlib/Algebra/MvPolynomial/Funext.lean | 46 | 59 | theorem funext {σ : Type*} {p q : MvPolynomial σ R} (h : ∀ x : σ → R, eval x p = eval x q) :
p = q := by |
suffices ∀ p, (∀ x : σ → R, eval x p = 0) → p = 0 by
rw [← sub_eq_zero, this (p - q)]
simp only [h, RingHom.map_sub, forall_const, sub_self]
clear h p q
intro p h
obtain ⟨n, f, hf, p, rfl⟩ := exists_fin_rename p
suffices p = 0 by rw [this, AlgHom.map_zero]
apply funext_fin
intro x
classical
... |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 365 | 368 | theorem sin_eq_zero_iff {θ : Angle} : sin θ = 0 ↔ θ = 0 ∨ θ = π := by |
nth_rw 1 [← sin_zero]
rw [sin_eq_iff_eq_or_add_eq_pi]
simp
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,665 | 1,675 | theorem setToFun_congr_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞)
(hT : DominatedFinMeasAdditive μ T C) (hT_smul : DominatedFinMeasAdditive (c • μ) T C')
(f : α → E) : setToFun μ T hT f = setToFun (c • μ) T hT_smul f := by |
by_cases hc0 : c = 0
· simp [hc0] at hT_smul
have h : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0 := fun s hs _ => hT_smul.eq_zero hs
rw [setToFun_zero_left' _ h, setToFun_measure_zero]
simp [hc0]
refine setToFun_congr_measure c⁻¹ c ?_ hc_ne_top (le_of_eq ?_) le_rfl hT hT_smul f
· simp [hc0]
· rw [s... |
/-
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.Fin.Fin2
import Mathlib.Data.PFun
import Mathlib.Data.Vector3
import Mathlib.NumberTheory.PellMatiyasevic
#align_import number_theory.dioph from ... | Mathlib/NumberTheory/Dioph.lean | 302 | 302 | theorem ext (d : Dioph S) (H : ∀ v, v ∈ S ↔ v ∈ S') : Dioph S' := by | rwa [← Set.ext H]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathlib.GroupTheory.GroupAction.Ring
im... | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 267 | 271 | theorem det_smul (A : Matrix n n R) (c : R) : det (c • A) = c ^ Fintype.card n * det A :=
calc
det (c • A) = det ((diagonal fun _ => c) * A) := by | rw [smul_eq_diagonal_mul]
_ = det (diagonal fun _ => c) * det A := det_mul _ _
_ = c ^ Fintype.card n * det A := by simp [card_univ]
|
/-
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 | 2,129 | 2,130 | theorem iSup_finset_image {f : γ → α} {g : α → β} {s : Finset γ} :
⨆ x ∈ s.image f, g x = ⨆ y ∈ s, g (f y) := by | rw [← iSup_coe, coe_image, iSup_image, iSup_coe]
|
/-
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.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
import Mathlib.MeasureTheory.Decomposition.Jordan
import Mathlib.MeasureThe... | Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean | 426 | 433 | theorem rnDeriv_smul (s : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ] (r : ℝ) :
(r • s).rnDeriv μ =ᵐ[μ] r • s.rnDeriv μ := by |
refine
Integrable.ae_eq_of_withDensityᵥ_eq (integrable_rnDeriv _ _)
((integrable_rnDeriv _ _).smul r) ?_
rw [withDensityᵥ_smul (rnDeriv s μ) r, ← add_right_inj ((r • s).singularPart μ),
singularPart_add_withDensity_rnDeriv_eq, singularPart_smul, ← smul_add,
singularPart_add_withDensity_rnDeriv_eq... |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries
import Mathlib.Algebra.ContinuedFractions.Computation.Translations
import... | Mathlib/NumberTheory/DiophantineApproximation.lean | 364 | 368 | theorem convergent_of_int {ξ : ℤ} (n : ℕ) : convergent ξ n = ξ := by |
cases n
· simp only [Nat.zero_eq, convergent_zero, floor_intCast]
· simp only [convergent_succ, floor_intCast, fract_intCast, convergent_of_zero, add_zero,
inv_zero]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "... | Mathlib/Data/Set/Function.lean | 274 | 278 | theorem _root_.StrictMonoOn.congr (h₁ : StrictMonoOn f₁ s) (h : s.EqOn f₁ f₂) :
StrictMonoOn f₂ s := by |
intro a ha b hb hab
rw [← h ha, ← h hb]
exact h₁ ha hb hab
|
/-
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.Equiv
import Mathlib.Algebra.Module.Hom
import Mathl... | Mathlib/LinearAlgebra/Basic.lean | 169 | 173 | theorem ofSubmodule'_toLinearMap [Module R M] [Module R₂ M₂] (f : M ≃ₛₗ[σ₁₂] M₂)
(U : Submodule R₂ M₂) :
(f.ofSubmodule' U).toLinearMap = (f.toLinearMap.domRestrict _).codRestrict _ Subtype.prop := by |
ext
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.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Topology.UniformSpace.Basic
#align_import topology.uniform... | Mathlib/Topology/UniformSpace/Cauchy.lean | 366 | 371 | theorem isComplete_iff_ultrafilter {s : Set α} :
IsComplete s ↔ ∀ l : Ultrafilter α, Cauchy (l : Filter α) → ↑l ≤ 𝓟 s → ∃ x ∈ s, ↑l ≤ 𝓝 x := by |
refine ⟨fun h l => h l, fun H => isComplete_iff_clusterPt.2 fun l hl hls => ?_⟩
haveI := hl.1
rcases H (Ultrafilter.of l) hl.ultrafilter_of ((Ultrafilter.of_le l).trans hls) with ⟨x, hxs, hxl⟩
exact ⟨x, hxs, (ClusterPt.of_le_nhds hxl).mono (Ultrafilter.of_le l)⟩
|
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tactic.NoncommRing
#align_import algebra.algebra.spectrum from "leanprover-c... | Mathlib/Algebra/Algebra/Spectrum.lean | 143 | 146 | theorem algebraMap_mem_iff (S : Type*) {R A : Type*} [CommSemiring R] [CommSemiring S]
[Ring A] [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] {a : A} {r : R} :
algebraMap R S r ∈ spectrum S a ↔ r ∈ spectrum R a := by |
simp only [spectrum.mem_iff, Algebra.algebraMap_eq_smul_one, smul_assoc, one_smul]
|
/-
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, Sander Dahmen, Scott Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_im... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 369 | 370 | theorem rank_tensorProduct' :
Module.rank S (M ⊗[S] M₁) = Module.rank S M * Module.rank S M₁ := by | simp
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 256 | 256 | theorem descPochhammer_one : descPochhammer R 1 = X := by | simp [descPochhammer]
|
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