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) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.GCongr
#align_import order.filter.archimedean fr... | Mathlib/Order/Filter/Archimedean.lean | 242 | 245 | theorem Tendsto.atBot_mul_const' (hr : 0 < r) (hf : Tendsto f l atBot) :
Tendsto (fun x => f x * r) l atBot := by |
simp only [← tendsto_neg_atTop_iff, ← neg_mul] at hf ⊢
exact hf.atTop_mul_const' hr
|
/-
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 | 1,978 | 1,983 | theorem prod_erase [DecidableEq α] (s : Finset α) {f : α → β} {a : α} (h : f a = 1) :
∏ x ∈ s.erase a, f x = ∏ x ∈ s, f x := by |
rw [← sdiff_singleton_eq_erase]
refine prod_subset sdiff_subset fun x hx hnx => ?_
rw [sdiff_singleton_eq_erase] at hnx
rwa [eq_of_mem_of_not_mem_erase hx hnx]
|
/-
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 | 328 | 328 | theorem natDegree_sub : (p - q).natDegree = (q - p).natDegree := by | rw [← natDegree_neg, neg_sub]
|
/-
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 | 2,043 | 2,044 | theorem factorization_eq_count {n p : α} :
factorization n p = Multiset.count p (normalizedFactors n) := by | simp [factorization]
|
/-
Copyright (c) 2020 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo
-/
import Mathlib.Dynamics.Flow
import Mathlib.Tactic.Monotonicity
#align_import dynamics.omega_limit from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
... | Mathlib/Dynamics/OmegaLimit.lean | 360 | 364 | theorem omegaLimit_image_eq (hf : ∀ t, Tendsto (· + t) f f) (t : τ) : ω f ϕ (ϕ t '' s) = ω f ϕ s :=
Subset.antisymm (omegaLimit_image_subset _ _ _ _ (hf t)) <|
calc
ω f ϕ s = ω f ϕ (ϕ (-t) '' (ϕ t '' s)) := by | simp [image_image, ← map_add]
_ ⊆ ω f ϕ (ϕ t '' s) := omegaLimit_image_subset _ _ _ _ (hf _)
|
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
/-!
# B... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 196 | 197 | theorem whiskerLeft_hom_inv (f : a ⟶ b) {g h : b ⟶ c} (η : g ≅ h) :
f ◁ η.hom ≫ f ◁ η.inv = 𝟙 (f ≫ g) := by | rw [← whiskerLeft_comp, hom_inv_id, whiskerLeft_id]
|
/-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Sites.Coherent.Comparison
import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves
import Mathlib.CategoryTheory.Sites.Coherent... | Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean | 244 | 248 | theorem isSheaf_iff_preservesFiniteProducts_of_projective [Preregular C] [FinitaryExtensive C]
[∀ (X : C), Projective X] :
IsSheaf (coherentTopology C) F ↔ Nonempty (PreservesFiniteProducts F) := by |
rw [isSheaf_coherent_iff_regular_and_extensive, and_iff_left (isSheaf_of_projective F),
isSheaf_iff_preservesFiniteProducts]
|
/-
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.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5... | Mathlib/Topology/MetricSpace/PiNat.lean | 510 | 519 | theorem firstDiff_lt_shortestPrefixDiff {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
(hx : x ∉ s) (hy : y ∈ s) : firstDiff x y < shortestPrefixDiff x s := by |
have A := exists_disjoint_cylinder hs hx
rw [shortestPrefixDiff, dif_pos A]
have B := Nat.find_spec A
contrapose! B
rw [not_disjoint_iff_nonempty_inter]
refine ⟨y, hy, ?_⟩
rw [mem_cylinder_comm]
exact cylinder_anti y B (mem_cylinder_firstDiff x y)
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finset.Pointwise
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import algebra.group.unique_prods from "l... | Mathlib/Algebra/Group/UniqueProds.lean | 67 | 68 | theorem of_subsingleton [Subsingleton G] : UniqueMul A B a0 b0 := by |
simp [UniqueMul, eq_iff_true_of_subsingleton]
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Oleksandr Manzyuk
-/
import Mathlib.CategoryTheory.Bicategory.Basic
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Eq... | Mathlib/CategoryTheory/Monoidal/Bimod.lean | 702 | 710 | theorem hom_right_act_hom' :
((regular R).tensorBimod P).actRight ≫ hom P = (hom P ▷ S.X) ≫ P.actRight := by |
dsimp; dsimp [hom, TensorBimod.actRight, regular]
refine (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 ?_
dsimp
slice_lhs 1 4 => rw [π_tensor_id_preserves_coequalizer_inv_colimMap_desc]
slice_rhs 1 2 => rw [← comp_whiskerRight, coequalizer.π_desc]
slice_rhs 1 2 => rw [middle_assoc]
simp only [... |
/-
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 | 1,491 | 1,495 | theorem norm_toL1 (f : α → β) (hf : Integrable f μ) :
‖hf.toL1 f‖ = ENNReal.toReal (∫⁻ a, edist (f a) 0 ∂μ) := by |
simp only [toL1, Lp.norm_toLp, snorm, one_ne_zero, snorm', one_toReal, ENNReal.rpow_one, ne_eq,
not_false_eq_true, div_self, ite_false]
simp [edist_eq_coe_nnnorm]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
#align_import order.heyting.basic from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# Heyting algebr... | Mathlib/Order/Heyting/Basic.lean | 546 | 549 | theorem sdiff_sdiff_sdiff_le_sdiff : (a \ b) \ (a \ c) ≤ c \ b := by |
rw [sdiff_le_iff, sdiff_le_iff, sup_left_comm, sup_sdiff_self, sup_left_comm, sdiff_sup_self,
sup_left_comm]
exact le_sup_left
|
/-
Copyright (c) 2021 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.Pad... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 70 | 111 | theorem cyclotomic_pos {n : ℕ} (hn : 2 < n) {R} [LinearOrderedCommRing R] (x : R) :
0 < eval x (cyclotomic n R) := by |
induction' n using Nat.strong_induction_on with n ih
have hn' : 0 < n := pos_of_gt hn
have hn'' : 1 < n := one_lt_two.trans hn
have := prod_cyclotomic_eq_geom_sum hn' R
apply_fun eval x at this
rw [← cons_self_properDivisors hn'.ne', Finset.erase_cons_of_ne _ hn''.ne', Finset.prod_cons,
eval_mul, eval_... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f6856... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 711 | 718 | theorem MDifferentiableWithinAt.comp (hg : MDifferentiableWithinAt I' I'' g u (f x))
(hf : MDifferentiableWithinAt I I' f s x) (h : s ⊆ f ⁻¹' u) :
MDifferentiableWithinAt I I'' (g ∘ f) s x := by |
rcases hf.2 with ⟨f', hf'⟩
have F : HasMFDerivWithinAt I I' f s x f' := ⟨hf.1, hf'⟩
rcases hg.2 with ⟨g', hg'⟩
have G : HasMFDerivWithinAt I' I'' g u (f x) g' := ⟨hg.1, hg'⟩
exact (HasMFDerivWithinAt.comp x G F h).mdifferentiableWithinAt
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.MeasureTheory.Measure.Dirac
/-!
# Counting measure
In this file we define the counting measure `MeasurTheory.Measure.count`
as `MeasureTheory.Measure... | Mathlib/MeasureTheory/Measure/Count.lean | 150 | 153 | theorem count_singleton' {a : α} (ha : MeasurableSet ({a} : Set α)) : count ({a} : Set α) = 1 := by |
rw [count_apply_finite' (Set.finite_singleton a) ha, Set.Finite.toFinset]
simp [@toFinset_card _ _ (Set.finite_singleton a).fintype,
@Fintype.card_unique _ _ (Set.finite_singleton a).fintype]
|
/-
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.Limits.Shapes.Products
import Mathlib.CategoryTheory.Functor.EpiMono
#align_import category_theory.adjunction.evaluation from "leanprover-commu... | Mathlib/CategoryTheory/Adjunction/Evaluation.lean | 140 | 145 | theorem NatTrans.epi_iff_epi_app {F G : C ⥤ D} (η : F ⟶ G) : Epi η ↔ ∀ c, Epi (η.app c) := by |
constructor
· intro h c
exact (inferInstance : Epi (((evaluation _ _).obj c).map η))
· intros
apply NatTrans.epi_of_epi_app
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Constructions.Prod.Integral
impor... | Mathlib/Analysis/Convolution.lean | 753 | 758 | theorem _root_.BddAbove.continuous_convolution_left_of_integrable
[FirstCountableTopology G] [SecondCountableTopologyEither G E]
(hbf : BddAbove (range fun x => ‖f x‖)) (hf : Continuous f) (hg : Integrable g μ) :
Continuous (f ⋆[L, μ] g) := by |
rw [← convolution_flip]
exact hbf.continuous_convolution_right_of_integrable L.flip hg hf
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 573 | 578 | theorem factors_prod_eq_basis (x : π C (· ∈ s)) :
(factors C s x).prod = spanFinBasis C s x := by |
ext y
dsimp [spanFinBasis]
split_ifs with h <;> [exact factors_prod_eq_basis_of_eq _ _ h;
exact factors_prod_eq_basis_of_ne _ _ h]
|
/-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 286 | 303 | theorem of_ufd_of_unique_irreducible {R : Type u} [CommRing R] [IsDomain R]
[UniqueFactorizationMonoid R] (h₁ : ∃ p : R, Irreducible p)
(h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q) :
DiscreteValuationRing R := by |
rw [iff_pid_with_one_nonzero_prime]
haveI PID : IsPrincipalIdealRing R := aux_pid_of_ufd_of_unique_irreducible R h₁ h₂
obtain ⟨p, hp⟩ := h₁
refine ⟨PID, ⟨Ideal.span {p}, ⟨?_, ?_⟩, ?_⟩⟩
· rw [Submodule.ne_bot_iff]
exact ⟨p, Ideal.mem_span_singleton.mpr (dvd_refl p), hp.ne_zero⟩
· rwa [Ideal.span_singlet... |
/-
Copyright (c) 2018 . All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import... | Mathlib/GroupTheory/PGroup.lean | 232 | 241 | theorem exists_fixed_point_of_prime_dvd_card_of_fixed_point (hpα : p ∣ card α) {a : α}
(ha : a ∈ fixedPoints G α) : ∃ b, b ∈ fixedPoints G α ∧ a ≠ b := by |
cases nonempty_fintype (fixedPoints G α)
have hpf : p ∣ card (fixedPoints G α) :=
Nat.modEq_zero_iff_dvd.mp ((hG.card_modEq_card_fixedPoints α).symm.trans hpα.modEq_zero_nat)
have hα : 1 < card (fixedPoints G α) :=
(Fact.out (p := p.Prime)).one_lt.trans_le (Nat.le_of_dvd (card_pos_iff.2 ⟨⟨a, ha⟩⟩) hpf)
... |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib... | Mathlib/Order/SymmDiff.lean | 204 | 205 | theorem symmDiff_symmDiff_inf : a ∆ b ∆ (a ⊓ b) = a ⊔ b := by |
rw [← symmDiff_sdiff_inf a, sdiff_symmDiff_eq_sup, symmDiff_sup_inf]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.or... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 162 | 168 | theorem rotation_rotation (θ₁ θ₂ : Real.Angle) (x : V) :
o.rotation θ₁ (o.rotation θ₂ x) = o.rotation (θ₁ + θ₂) x := by |
simp only [o.rotation_apply, ← mul_smul, Real.Angle.cos_add, Real.Angle.sin_add, add_smul,
sub_smul, LinearIsometryEquiv.trans_apply, smul_add, LinearIsometryEquiv.map_add,
LinearIsometryEquiv.map_smul, rightAngleRotation_rightAngleRotation, smul_neg]
ring_nf
abel
|
/-
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 | 559 | 559 | theorem one_div_div : 1 / (a / b) = b / a := by | simp
|
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | Mathlib/LinearAlgebra/Matrix/PosDef.lean | 326 | 331 | theorem toQuadraticForm' [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) :
M.toQuadraticForm'.PosDef := by |
intro x hx
simp only [Matrix.toQuadraticForm', LinearMap.BilinForm.toQuadraticForm_apply,
toLinearMap₂'_apply']
apply hM.2 x hx
|
/-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_imp... | Mathlib/Analysis/Convex/Slope.lean | 29 | 48 | theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y - f x) / (y - x) ≤ (f z - f y) / (z - y) := by |
have hxz := hxy.trans hyz
rw [← sub_pos] at hxy hxz hyz
suffices f y / (y - x) + f y / (z - y) ≤ f x / (y - x) + f z / (z - y) by
ring_nf at this ⊢
linarith
set a := (z - y) / (z - x)
set b := (y - x) / (z - x)
have hy : a • x + b • z = y := by field_simp [a, b]; ring
have key :=
hf.2 hx hz (... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,562 | 1,564 | theorem ContDiffWithinAt.div_const {f : E → 𝕜'} {n} (hf : ContDiffWithinAt 𝕜 n f s x) (c : 𝕜') :
ContDiffWithinAt 𝕜 n (fun x => f x / c) s x := by |
simpa only [div_eq_mul_inv] using hf.mul contDiffWithinAt_const
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathli... | Mathlib/Data/ZMod/Basic.lean | 836 | 840 | theorem natCast_mod (a : ℕ) (n : ℕ) : ((a % n : ℕ) : ZMod n) = a := by |
conv =>
rhs
rw [← Nat.mod_add_div a n]
simp
|
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence
import Mathlib.Algebra.ContinuedFractions.TerminatedStable
import Mathlib.Tactic.FieldSimp
import ... | Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean | 155 | 181 | theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq :
convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by |
cases s_succ_nth_eq : s.get? <| n + 1 with
| none =>
rw [squashSeq_eq_self_of_terminated s_succ_nth_eq,
convergents'Aux_stable_step_of_terminated s_succ_nth_eq]
| some gp_succ_n =>
induction n generalizing s gp_succ_n with
| zero =>
obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some g... |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Michael Stoll
-/
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import number_theory.l_series from "leanprover-community/ma... | Mathlib/NumberTheory/LSeries/Basic.lean | 358 | 362 | theorem LSeriesSummable_of_bounded_of_one_lt_re {f : ℕ → ℂ} {m : ℝ}
(h : ∀ n ≠ 0, Complex.abs (f n) ≤ m) {s : ℂ} (hs : 1 < s.re) :
LSeriesSummable f s := by |
refine LSeriesSummable_of_le_const_mul_rpow hs ⟨m, fun n hn ↦ ?_⟩
simp only [norm_eq_abs, sub_self, Real.rpow_zero, mul_one, h n hn]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs fr... | Mathlib/Data/Finsupp/Defs.lean | 1,380 | 1,382 | theorem update_eq_sub_add_single [AddGroup G] (f : α →₀ G) (a : α) (b : G) :
f.update a b = f - single a (f a) + single a b := by |
rw [update_eq_erase_add_single, erase_eq_sub_single]
|
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomia... | Mathlib/Algebra/Polynomial/Reverse.lean | 50 | 52 | theorem revAtFun_inj {N : ℕ} : Function.Injective (revAtFun N) := by |
intro a b hab
rw [← @revAtFun_invol N a, hab, revAtFun_invol]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 2,164 | 2,165 | theorem pure_le_principal (a : α) : pure a ≤ 𝓟 s ↔ a ∈ s := by |
simp
|
/-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.Data.Option.Basic
import Mathlib.SetTheory.Cardinal.Basic
#al... | Mathlib/Computability/Encoding.lean | 43 | 45 | theorem Encoding.encode_injective {α : Type u} (e : Encoding α) : Function.Injective e.encode := by |
refine fun _ _ h => Option.some_injective _ ?_
rw [← e.decode_encode, ← e.decode_encode, h]
|
/-
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 | 1,522 | 1,524 | theorem SimpleFunc.integral_eq_sum (f : α →ₛ E) (hfi : Integrable f μ) :
∫ x, f x ∂μ = ∑ x ∈ f.range, ENNReal.toReal (μ (f ⁻¹' {x})) • x := by |
rw [← f.integral_eq_integral hfi, SimpleFunc.integral, ← SimpleFunc.integral_eq]; rfl
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 469 | 470 | theorem isBigO_map {k : β → α} {l : Filter β} : f =O[map k l] g ↔ (f ∘ k) =O[l] (g ∘ k) := by |
simp only [IsBigO_def, isBigOWith_map]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Sigma
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Vect... | Mathlib/Data/Fintype/BigOperators.lean | 185 | 186 | theorem card_vector [Fintype α] (n : ℕ) : Fintype.card (Vector α n) = Fintype.card α ^ n := by |
rw [Fintype.ofEquiv_card]; simp
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel ... | Mathlib/Data/Rel.lean | 416 | 417 | theorem image_eq (f : α → β) (s : Set α) : f '' s = (Function.graph f).image s := by |
rfl
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathli... | Mathlib/CategoryTheory/Extensive.lean | 422 | 429 | theorem finitaryExtensive_of_preserves_and_reflects_isomorphism (F : C ⥤ D) [FinitaryExtensive D]
[HasFiniteCoproducts C] [HasPullbacks C] [PreservesLimitsOfShape WalkingCospan F]
[PreservesColimitsOfShape (Discrete WalkingPair) F] [F.ReflectsIsomorphisms] :
FinitaryExtensive C := by |
haveI : ReflectsLimitsOfShape WalkingCospan F := reflectsLimitsOfShapeOfReflectsIsomorphisms
haveI : ReflectsColimitsOfShape (Discrete WalkingPair) F :=
reflectsColimitsOfShapeOfReflectsIsomorphisms
exact finitaryExtensive_of_preserves_and_reflects F
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 1,149 | 1,151 | theorem isLeast_csInf (hs : s.Nonempty) : IsLeast s (sInf s) := by |
rw [sInf_eq_argmin_on hs]
exact ⟨argminOn_mem _ _ _ _, fun a ha => argminOn_le id _ _ ha⟩
|
/-
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 | 433 | 443 | theorem card_support_eq_two :
f.support.card = 2 ↔
∃ (k m : ℕ) (hkm : k < m) (x y : R) (hx : x ≠ 0) (hy : y ≠ 0),
f = C x * X ^ k + C y * X ^ m := by |
refine ⟨fun h => ?_, ?_⟩
· obtain ⟨k, x, hk, hx, rfl⟩ := card_support_eq.mp h
refine ⟨k 0, k 1, hk Nat.zero_lt_one, x 0, x 1, hx 0, hx 1, ?_⟩
rw [Fin.sum_univ_castSucc, Fin.sum_univ_one]
rfl
· rintro ⟨k, m, hkm, x, y, hx, hy, rfl⟩
exact card_support_binomial hkm.ne hx hy
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
#... | Mathlib/Tactic/Ring/Basic.lean | 451 | 451 | theorem natCast_nat (n) : ((Nat.rawCast n : ℕ) : R) = Nat.rawCast n := by | simp
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Option.NAry
import Mathlib.Data.Seq.Computation
#align_import data.seq.seq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b34... | Mathlib/Data/Seq/Seq.lean | 418 | 426 | theorem coinduction2 (s) (f g : Seq α → Seq β)
(H :
∀ s,
BisimO (fun s1 s2 : Seq β => ∃ s : Seq α, s1 = f s ∧ s2 = g s) (destruct (f s))
(destruct (g s))) :
f s = g s := by |
refine eq_of_bisim (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) ?_ ⟨s, rfl, rfl⟩
intro s1 s2 h; rcases h with ⟨s, h1, h2⟩
rw [h1, h2]; apply H
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.GCongr
#align_import order.filter.archimedean fr... | Mathlib/Order/Filter/Archimedean.lean | 292 | 293 | theorem Tendsto.atTop_zsmul_neg_const {f : α → ℤ} (hr : r < 0) (hf : Tendsto f l atTop) :
Tendsto (fun x => f x • r) l atBot := by | simpa using hf.atTop_zsmul_const (neg_pos.2 hr)
|
/-
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 | 592 | 594 | theorem toReal_eq_zero_iff {θ : Angle} : θ.toReal = 0 ↔ θ = 0 := by |
nth_rw 1 [← toReal_zero]
exact toReal_inj
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.RingHomProperties
#align_import ring_theory.ring_hom.finite from "leanprover-community/mathlib"@"b5aecf07a179c60b6b37c1ac9da952f3b565c785"
/-!
... | Mathlib/RingTheory/RingHom/Finite.lean | 23 | 25 | theorem finite_stableUnderComposition : StableUnderComposition @Finite := by |
introv R hf hg
exact hg.comp hf
|
/-
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 | 143 | 148 | theorem StrictConvex.linear_preimage {s : Set F} (hs : StrictConvex 𝕜 s) (f : E →ₗ[𝕜] F)
(hf : Continuous f) (hfinj : Injective f) : StrictConvex 𝕜 (s.preimage f) := by |
intro x hx y hy hxy a b ha hb hab
refine preimage_interior_subset_interior_preimage hf ?_
rw [mem_preimage, f.map_add, f.map_smul, f.map_smul]
exact hs hx hy (hfinj.ne hxy) ha hb hab
|
/-
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 | 355 | 358 | theorem perfectClosure.comap_eq_of_algHom (i : E →ₐ[F] K) :
(perfectClosure F K).comap i = perfectClosure F E := by |
ext x
exact map_mem_perfectClosure_iff i
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
/-!
# Real sign fu... | Mathlib/Data/Real/Sign.lean | 43 | 43 | theorem sign_zero : sign 0 = 0 := by | rw [sign, if_neg (lt_irrefl _), if_neg (lt_irrefl _)]
|
/-
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 | 1,463 | 1,467 | theorem norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (x y : E) (h : ⟪x, y⟫ = 0) :
‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ := by |
rw [@norm_add_mul_self 𝕜, add_right_cancel_iff, add_right_eq_self, mul_eq_zero]
apply Or.inr
simp only [h, zero_re']
|
/-
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 | 780 | 781 | theorem coe_iInf (f : ι → LowerSet α) : (↑(⨅ i, f i) : Set α) = ⋂ i, f i := by |
simp_rw [iInf, coe_sInf, mem_range, iInter_exists, iInter_iInter_eq']
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import ... | Mathlib/Data/DFinsupp/Basic.lean | 1,919 | 1,922 | theorem sumAddHom_single [∀ i, AddZeroClass (β i)] [AddCommMonoid γ] (φ : ∀ i, β i →+ γ) (i)
(x : β i) : sumAddHom φ (single i x) = φ i x := by |
dsimp [sumAddHom, single, Trunc.lift_mk]
rw [Multiset.toFinset_singleton, Finset.sum_singleton, Pi.single_eq_same]
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Anatole Dedecker
-/
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.Algebra.Equicontinuity
import Mathlib.Topology.MetricSpace.Equicontinuity
import Mathlib.Topology... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 564 | 567 | theorem WithSeminorms.image_isVonNBounded_iff_seminorm_bounded (f : G → E) {s : Set G}
(hp : WithSeminorms p) :
Bornology.IsVonNBounded 𝕜 (f '' s) ↔ ∀ i : ι, ∃ r > 0, ∀ x ∈ s, p i (f x) < r := by |
simp_rw [hp.isVonNBounded_iff_seminorm_bounded, Set.forall_mem_image]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Alistair Tucker, Wen Yang
-/
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Topology.Order.DenselyOrdered
... | Mathlib/Topology/Order/IntermediateValue.lean | 612 | 615 | theorem ContinuousOn.surjOn_uIcc {s : Set α} [hs : OrdConnected s] {f : α → δ}
(hf : ContinuousOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) :
SurjOn f s (uIcc (f a) (f b)) := by |
rcases le_total (f a) (f b) with hab | hab <;> simp [hf.surjOn_Icc, *]
|
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheory.Products.Basic
#align_import cat... | Mathlib/CategoryTheory/Monoidal/Category.lean | 530 | 534 | theorem pentagon_inv_inv_hom_hom_inv :
(α_ W (X ⊗ Y) Z).inv ≫ (α_ W X Y).inv ▷ Z ≫ (α_ (W ⊗ X) Y Z).hom =
W ◁ (α_ X Y Z).hom ≫ (α_ W X (Y ⊗ Z)).inv := by |
rw [← cancel_epi (W ◁ (α_ X Y Z).inv), ← cancel_mono (α_ (W ⊗ X) Y Z).inv]
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 | 323 | 325 | theorem coe_image_Ioc_eq : ((↑) : 𝕜 → AddCircle p) '' Ioc a (a + p) = univ := by |
rw [image_eq_range]
exact (equivIoc p a).symm.range_eq_univ
|
/-
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 | 368 | 374 | theorem openEmbedding_of_pullback_open_embeddings {X Y S : TopCat} {f : X ⟶ S} {g : Y ⟶ S}
(H₁ : OpenEmbedding f) (H₂ : OpenEmbedding g) :
OpenEmbedding (limit.π (cospan f g) WalkingCospan.one) := by |
convert H₂.comp (snd_openEmbedding_of_left_openEmbedding H₁ g)
erw [← coe_comp]
rw [← limit.w _ WalkingCospan.Hom.inr]
rfl
|
/-
Copyright (c) 2023 Bulhwi Cha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bulhwi Cha, Mario Carneiro
-/
import Batteries.Data.Char
import Batteries.Data.List.Lemmas
import Batteries.Data.String.Basic
import Batteries.Tactic.Lint.Misc
import Batteries.Tactic.SeqF... | .lake/packages/batteries/Batteries/Data/String/Lemmas.lean | 148 | 157 | theorem utf8GetAux_of_valid (cs cs' : List Char) {i p : Nat} (hp : i + utf8Len cs = p) :
utf8GetAux (cs ++ cs') ⟨i⟩ ⟨p⟩ = cs'.headD default := by |
match cs, cs' with
| [], [] => rfl
| [], c::cs' => simp [← hp, utf8GetAux]
| c::cs, cs' =>
simp [utf8GetAux, -List.headD_eq_head?]; rw [if_neg]
case hnc => simp [← hp, Pos.ext_iff]; exact ne_self_add_add_csize
refine utf8GetAux_of_valid cs cs' ?_
simpa [Nat.add_assoc, Nat.add_comm] using hp
|
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4... | Mathlib/Algebra/CubicDiscriminant.lean | 531 | 536 | theorem eq_sum_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
map φ P =
⟨φ P.a, φ P.a * -(x + y + z), φ P.a * (x * y + x * z + y * z), φ P.a * -(x * y * z)⟩ := by |
apply_fun @toPoly _ _
· rw [eq_prod_three_roots ha h3, C_mul_prod_X_sub_C_eq]
· exact fun P Q ↦ (toPoly_injective P Q).mp
|
/-
Copyright (c) 2021 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.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#a... | Mathlib/Algebra/Order/Pointwise.lean | 61 | 63 | theorem sSup_inv (s : Set α) : sSup s⁻¹ = (sInf s)⁻¹ := by |
rw [← image_inv, sSup_image]
exact ((OrderIso.inv α).map_sInf _).symm
|
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subg... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,225 | 1,235 | theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :
G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by |
ext
· simp
· constructor
· intro h
simp only [subgraphOfAdj_adj, Sym2.eq, Sym2.rel_iff] at h
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm]
· intro h
simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h
obtain ⟨rfl | rfl, rfl | rfl, ha⟩ := ... |
/-
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 | 546 | 551 | theorem isWeightedHomogeneous_zero_iff_weightedTotalDegree_eq_zero {p : MvPolynomial σ R} :
IsWeightedHomogeneous w p 0 ↔ p.weightedTotalDegree w = 0 := by |
rw [weightedTotalDegree, ← bot_eq_zero, Finset.sup_eq_bot_iff, bot_eq_zero, IsWeightedHomogeneous]
apply forall_congr'
intro m
rw [mem_support_iff]
|
/-
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.LinearAlgebra.FreeModule.PID
import Mathlib.MeasureTheory.Group.FundamentalDomain
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Rin... | Mathlib/Algebra/Module/Zlattice/Basic.lean | 166 | 167 | theorem fract_add_zspan (m : E) {v : E} (h : v ∈ span ℤ (Set.range b)) :
fract b (m + v) = fract b m := by | rw [add_comm, fract_zspan_add b m h]
|
/-
Copyright (c) 2022 Alex Kontorovich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Eric Wieser
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.MulOpposite
import Mathlib.Logic.Encodable.Basic
#align_import gr... | Mathlib/Algebra/Group/Subgroup/MulOpposite.lean | 151 | 154 | theorem op_closure (s : Set G) : (closure s).op = closure (MulOpposite.unop ⁻¹' s) := by |
simp_rw [closure, op_sInf, Set.preimage_setOf_eq, Subgroup.unop_coe]
congr with a
exact MulOpposite.unop_surjective.forall
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "lean... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,495 | 1,496 | theorem contDiffAt_zero : ContDiffAt 𝕜 0 f x ↔ ∃ u ∈ 𝓝 x, ContinuousOn f u := by |
rw [← contDiffWithinAt_univ]; simp [contDiffWithinAt_zero, nhdsWithin_univ]
|
/-
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.BigOperators.Ring
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
import Mat... | Mathlib/NumberTheory/ArithmeticFunction.lean | 1,146 | 1,154 | theorem IsMultiplicative.prodPrimeFactors_one_add_of_squarefree [CommSemiring R]
{f : ArithmeticFunction R} (h_mult : f.IsMultiplicative) {n : ℕ} (hn : Squarefree n) :
∏ p ∈ n.primeFactors, (1 + f p) = ∑ d ∈ n.divisors, f d := by |
trans (∏ᵖ p ∣ n, ((ζ:ArithmeticFunction R) + f) p)
· simp_rw [prodPrimeFactors_apply hn.ne_zero, add_apply, natCoe_apply]
apply Finset.prod_congr rfl; intro p hp;
rw [zeta_apply_ne (prime_of_mem_factors <| List.mem_toFinset.mp hp).ne_zero, cast_one]
rw [isMultiplicative_zeta.natCast.prodPrimeFactors_add_... |
/-
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 | 945 | 963 | theorem eq_iff_sign_eq_and_abs_toReal_eq {θ ψ : Angle} :
θ = ψ ↔ θ.sign = ψ.sign ∧ |θ.toReal| = |ψ.toReal| := by |
refine ⟨?_, fun h => ?_⟩;
· rintro rfl
exact ⟨rfl, rfl⟩
rcases h with ⟨hs, hr⟩
rw [abs_eq_abs] at hr
rcases hr with (hr | hr)
· exact toReal_injective hr
· by_cases h : θ = π
· rw [h, toReal_pi, ← neg_eq_iff_eq_neg] at hr
exact False.elim ((neg_pi_lt_toReal ψ).ne hr)
· by_cases h' : ψ =... |
/-
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.Set.Lattice
import Mathlib.Logic.Small.Basic
import Mathlib.Logic.Function.OfArity
import Mathlib.Order.WellFounded
#align_import set_theory.zfc.... | Mathlib/SetTheory/ZFC/Basic.lean | 1,677 | 1,683 | theorem mem_sInter {x y : Class.{u}} (h : x.Nonempty) : y ∈ ⋂₀ x ↔ ∀ z, z ∈ x → y ∈ z := by |
refine ⟨fun hy z => mem_of_mem_sInter hy, fun H => ?_⟩
simp_rw [mem_def, sInter_apply]
obtain ⟨z, hz⟩ := h
obtain ⟨y, rfl, _⟩ := H z (coe_mem.2 hz)
refine ⟨y, rfl, fun w hxw => ?_⟩
simpa only [coe_mem, coe_apply] using H w (coe_mem.2 hxw)
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 1,308 | 1,315 | theorem Valid'.balance'_aux {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
(H₁ : 2 * @size α r ≤ 9 * size l + 5 ∨ size r ≤ 3)
(H₂ : 2 * @size α l ≤ 9 * size r + 5 ∨ size l ≤ 3) : Valid' o₁ (@balance' α l x r) o₂ := by |
rw [balance']; split_ifs with h h_1 h_2
· exact hl.node' hr (Or.inl h)
· exact hl.rotateL hr h h_1 H₁
· exact hl.rotateR hr h h_2 H₂
· exact hl.node' hr (Or.inr ⟨not_lt.1 h_2, not_lt.1 h_1⟩)
|
/-
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.Algebra.Quasispectrum
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
import Mathlib.Analysis.Complex.Liouville
import Mathlib.Analysis.Complex.P... | Mathlib/Analysis/NormedSpace/Spectrum.lean | 297 | 323 | theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
(fun n => ContinuousMultilinearMap.mkPiRing 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ :=
{ r_le := by |
refine le_of_forall_nnreal_lt fun r hr =>
le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) fun n => ?_
rw [← norm_toNNReal, norm_mkPiRing, norm_toNNReal]
cases' n with n
· simp only [Nat.zero_eq, le_refl, mul_one, or_true_iff, le_max_iff, pow_zero]
· refine
le_trans (le_tr... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.Po... | Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | 581 | 588 | theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ι → M) (h₁ : ∏ i ∈ s, f i = 1)
(h₂ : ∃ i ∈ s, f i ≠ 1) : ∃ i ∈ s, 1 < f i := by |
contrapose! h₁
obtain ⟨i, m, i_ne⟩ : ∃ i ∈ s, f i ≠ 1 := h₂
apply ne_of_lt
calc
∏ j ∈ s, f j < ∏ j ∈ s, 1 := prod_lt_prod' h₁ ⟨i, m, (h₁ i m).lt_of_ne i_ne⟩
_ = 1 := prod_const_one
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheor... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 265 | 268 | theorem empty_mem_measurableCylinders (α : ι → Type*) [∀ i, MeasurableSpace (α i)] :
∅ ∈ measurableCylinders α := by |
simp_rw [measurableCylinders, mem_iUnion, mem_singleton_iff]
exact ⟨∅, ∅, MeasurableSet.empty, (cylinder_empty _).symm⟩
|
/-
Copyright (c) 2020 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.C... | Mathlib/Tactic/CancelDenoms/Core.lean | 89 | 102 | theorem cancel_factors_eq {α} [Field α] {a b ad bd a' b' gcd : α} (ha : ad * a = a')
(hb : bd * b = b') (had : ad ≠ 0) (hbd : bd ≠ 0) (hgcd : gcd ≠ 0) :
(a = b) = (1 / gcd * (bd * a') = 1 / gcd * (ad * b')) := by |
rw [← ha, ← hb, ← mul_assoc bd, ← mul_assoc ad, mul_comm bd]
ext; constructor
· rintro rfl
rfl
· intro h
simp only [← mul_assoc] at h
refine mul_left_cancel₀ (mul_ne_zero ?_ ?_) h
on_goal 1 => apply mul_ne_zero
on_goal 1 => apply div_ne_zero
· exact one_ne_zero
all_goals assumption
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib... | Mathlib/Algebra/Polynomial/RingDivision.lean | 572 | 575 | theorem coeff_coe_units_zero_ne_zero (u : R[X]ˣ) : coeff (u : R[X]) 0 ≠ 0 := by |
conv in 0 => rw [← natDegree_coe_units u]
rw [← leadingCoeff, Ne, leadingCoeff_eq_zero]
exact Units.ne_zero _
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.AddCharacter
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.Algebra.CharP.CharAndCard
#align_import numb... | Mathlib/NumberTheory/GaussSum.lean | 74 | 78 | theorem gaussSum_mulShift (χ : MulChar R R') (ψ : AddChar R R') (a : Rˣ) :
χ a * gaussSum χ (mulShift ψ a) = gaussSum χ ψ := by |
simp only [gaussSum, mulShift_apply, Finset.mul_sum]
simp_rw [← mul_assoc, ← map_mul]
exact Fintype.sum_bijective _ a.mulLeft_bijective _ _ fun x => rfl
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Group.Support
import Mathlib.Order.WellFoundedSet
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a484a7d0eade4e126... | Mathlib/RingTheory/HahnSeries/Basic.lean | 395 | 397 | theorem embDomain_zero {f : Γ ↪o Γ'} : embDomain f (0 : HahnSeries Γ R) = 0 := by |
ext
simp [embDomain_notin_image_support]
|
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.NormalC... | Mathlib/FieldTheory/SeparableDegree.lean | 280 | 281 | theorem natSepDegree_X : (X : F[X]).natSepDegree = 1 := by |
simp only [natSepDegree, aroots_X, Multiset.toFinset_singleton, Finset.card_singleton]
|
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi
import Mathlib.CategoryTheory.LiftingProperties.Adjunction
#align_import ca... | Mathlib/CategoryTheory/Functor/EpiMono.lean | 264 | 269 | theorem epi_map_iff_epi [hF₁ : PreservesEpimorphisms F] [hF₂ : ReflectsEpimorphisms F] :
Epi (F.map f) ↔ Epi f := by |
constructor
· exact F.epi_of_epi_map
· intro h
exact F.map_epi f
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathl... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,348 | 1,357 | theorem map_eval₂ (f : R →+* S₁) (g : S₂ → MvPolynomial S₃ R) (p : MvPolynomial S₂ R) :
map f (eval₂ C g p) = eval₂ C (map f ∘ g) (map f p) := by |
apply MvPolynomial.induction_on p
· intro r
rw [eval₂_C, map_C, map_C, eval₂_C]
· intro p q hp hq
rw [eval₂_add, (map f).map_add, hp, hq, (map f).map_add, eval₂_add]
· intro p s hp
rw [eval₂_mul, (map f).map_mul, hp, (map f).map_mul, map_X, eval₂_mul, eval₂_X, eval₂_X]
rfl
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 233 | 234 | theorem sin_pi : sin π = 0 := by |
rw [← mul_div_cancel_left₀ π (two_ne_zero' ℝ), two_mul, add_div, sin_add, cos_pi_div_two]; simp
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Char... | Mathlib/RingTheory/Norm.lean | 245 | 257 | theorem norm_eq_prod_embeddings_gen [Algebra R F] (pb : PowerBasis R S)
(hE : (minpoly R pb.gen).Splits (algebraMap R F)) (hfx : (minpoly R pb.gen).Separable) :
algebraMap R F (norm R pb.gen) =
(@Finset.univ _ (PowerBasis.AlgHom.fintype pb)).prod fun σ => σ pb.gen := by |
letI := Classical.decEq F
rw [PowerBasis.norm_gen_eq_prod_roots pb hE]
rw [@Fintype.prod_equiv (S →ₐ[R] F) _ _ (PowerBasis.AlgHom.fintype pb) _ _ pb.liftEquiv'
(fun σ => σ pb.gen) (fun x => x) ?_]
· rw [Finset.prod_mem_multiset, Finset.prod_eq_multiset_prod, Multiset.toFinset_val,
Multiset.dedup_eq_s... |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topol... | Mathlib/Topology/EMetricSpace/Basic.lean | 583 | 583 | theorem mem_ball_comm : x ∈ ball y ε ↔ y ∈ ball x ε := by | rw [mem_ball', mem_ball]
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
i... | Mathlib/Analysis/Calculus/Rademacher.lean | 378 | 381 | theorem LipschitzWith.ae_differentiableAt {f : E → F} (h : LipschitzWith C f) :
∀ᵐ x ∂μ, DifferentiableAt ℝ f x := by |
rw [← lipschitzOn_univ] at h
simpa [differentiableWithinAt_univ] using h.ae_differentiableWithinAt_of_mem
|
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.univariate.M from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
/-!
... | Mathlib/Data/PFunctor/Univariate/M.lean | 404 | 430 | theorem agree_iff_agree' {n : ℕ} (x y : M F) :
Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by |
constructor <;> intro h
· induction' n with _ n_ih generalizing x y
· constructor
· induction x using PFunctor.M.casesOn'
induction y using PFunctor.M.casesOn'
simp only [approx_mk] at h
cases' h with _ _ _ _ _ _ hagree
constructor <;> try rfl
intro i
apply n_ih
ap... |
/-
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 | 416 | 418 | theorem prod_principal_principal {s : Set α} {t : Set β} : 𝓟 s ×ˢ 𝓟 t = 𝓟 (s ×ˢ t) := by |
simp only [SProd.sprod, Filter.prod, comap_principal, principal_eq_iff_eq, comap_principal,
inf_principal]; rfl
|
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.Action.Defs
import Mathlib.Algebra.Group.Units
#align_import algebra.free_mono... | Mathlib/Algebra/FreeMonoid/Basic.lean | 111 | 112 | theorem toList_prod (xs : List (FreeMonoid α)) : toList xs.prod = (xs.map toList).join := by |
induction xs <;> simp [*, List.join]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
impo... | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 137 | 151 | theorem deriv2_sqrt_mul_log (x : ℝ) :
deriv^[2] (fun x => √x * log x) x = -log x / (4 * √x ^ 3) := by |
simp only [Nat.iterate, deriv_sqrt_mul_log']
rcases le_or_lt x 0 with hx | hx
· rw [sqrt_eq_zero_of_nonpos hx, zero_pow three_ne_zero, mul_zero, div_zero]
refine HasDerivWithinAt.deriv_eq_zero ?_ (uniqueDiffOn_Iic 0 x hx)
refine (hasDerivWithinAt_const _ _ 0).congr_of_mem (fun x hx => ?_) hx
rw [sqrt... |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
impo... | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 98 | 110 | theorem strictConvexOn_zpow {m : ℤ} (hm₀ : m ≠ 0) (hm₁ : m ≠ 1) :
StrictConvexOn ℝ (Ioi 0) fun x : ℝ => x ^ m := by |
apply strictConvexOn_of_deriv2_pos' (convex_Ioi 0)
· exact (continuousOn_zpow₀ m).mono fun x hx => ne_of_gt hx
intro x hx
rw [mem_Ioi] at hx
rw [iter_deriv_zpow]
refine mul_pos ?_ (zpow_pos_of_pos hx _)
norm_cast
refine int_prod_range_pos (by decide) fun hm => ?_
rw [← Finset.coe_Ico] at hm
norm_ca... |
/-
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 Mathlib.Init.Data.Int.Basic
import Mathlib.Init.ZeroOne
import Mathlib.Tactic.Lemma
import Mathlib.Tactic.TypeSta... | Mathlib/Algebra/Group/Defs.lean | 1,256 | 1,257 | theorem mul_inv_cancel_left (a b : G) : a * (a⁻¹ * b) = b := by |
rw [← mul_assoc, mul_right_inv, one_mul]
|
/-
Copyright (c) 2021 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson, Yaël Dillies
-/
import Mathlib.Data.Finset.Order
import Mathlib.Order.Atoms.Finite
#align_import data.fintype.order from "leanprover-community/mathlib"@"1126441d6bccf98c81... | Mathlib/Data/Fintype/Order.lean | 216 | 220 | theorem Finite.bddBelow_range [IsDirected α (· ≥ ·)] (f : β → α) : BddBelow (Set.range f) := by |
obtain ⟨M, hM⟩ := Finite.exists_ge f
refine ⟨M, fun a ha => ?_⟩
obtain ⟨b, rfl⟩ := ha
exact hM 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
-/
import Mathlib.MeasureTheory.OuterMeasure.Induced
import Mathlib.MeasureTheory.OuterMeasure.AE
import Mathlib.Order.Filter.CountableInter
#align_impo... | Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean | 310 | 319 | theorem _root_.MeasurableSpace.ae_induction_on_inter {β} [MeasurableSpace β] {μ : Measure β}
{C : β → Set α → Prop} {s : Set (Set α)} [m : MeasurableSpace α]
(h_eq : m = MeasurableSpace.generateFrom s)
(h_inter : IsPiSystem s) (h_empty : ∀ᵐ x ∂μ, C x ∅) (h_basic : ∀ᵐ x ∂μ, ∀ t ∈ s, C x t)
(h_compl : ∀ᵐ ... |
filter_upwards [h_empty, h_basic, h_compl, h_union] with x hx_empty hx_basic hx_compl hx_union
using MeasurableSpace.induction_on_inter h_eq h_inter hx_empty hx_basic hx_compl hx_union
|
/-
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 | 656 | 660 | theorem snorm_smul_measure_of_ne_top {p : ℝ≥0∞} (hp_ne_top : p ≠ ∞) {f : α → F} (c : ℝ≥0∞) :
snorm f p (c • μ) = c ^ (1 / p).toReal • snorm f p μ := by |
by_cases hp0 : p = 0
· simp [hp0]
· exact snorm_smul_measure_of_ne_zero_of_ne_top hp0 hp_ne_top c
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 219 | 226 | theorem Hom.realize_term (g : M →[L] N) {t : L.Term α} {v : α → M} :
t.realize (g ∘ v) = g (t.realize v) := by |
induction t
· rfl
· rw [Term.realize, Term.realize, g.map_fun]
refine congr rfl ?_
ext x
simp [*]
|
/-
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 | 209 | 211 | theorem det_units_conj (M : (Matrix m m R)ˣ) (N : Matrix m m R) :
det ((M : Matrix _ _ _) * N * (↑M⁻¹ : Matrix _ _ _)) = det N := by |
rw [det_mul_right_comm, Units.mul_inv, one_mul]
|
/-
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
-/
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb8... | Mathlib/Data/Finset/Card.lean | 190 | 194 | theorem pred_card_le_card_erase : s.card - 1 ≤ (s.erase a).card := by |
by_cases h : a ∈ s
· exact (card_erase_of_mem h).ge
· rw [erase_eq_of_not_mem h]
exact Nat.sub_le _ _
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34... | Mathlib/Data/Finsupp/Multiset.lean | 230 | 231 | theorem toMultiset_strictMono : StrictMono (@toMultiset ι) := by |
classical exact (@orderIsoMultiset ι _).strictMono
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Matthew Robert Ballard
-/
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
i... | Mathlib/NumberTheory/Padics/PadicVal.lean | 393 | 407 | theorem padicValRat_le_padicValRat_iff {n₁ n₂ d₁ d₂ : ℤ} (hn₁ : n₁ ≠ 0) (hn₂ : n₂ ≠ 0)
(hd₁ : d₁ ≠ 0) (hd₂ : d₂ ≠ 0) :
padicValRat p (n₁ /. d₁) ≤ padicValRat p (n₂ /. d₂) ↔
∀ n : ℕ, (p : ℤ) ^ n ∣ n₁ * d₂ → (p : ℤ) ^ n ∣ n₂ * d₁ := by |
have hf1 : Finite (p : ℤ) (n₁ * d₂) := finite_int_prime_iff.2 (mul_ne_zero hn₁ hd₂)
have hf2 : Finite (p : ℤ) (n₂ * d₁) := finite_int_prime_iff.2 (mul_ne_zero hn₂ hd₁)
conv =>
lhs
rw [padicValRat.defn p (Rat.divInt_ne_zero_of_ne_zero hn₁ hd₁) rfl,
padicValRat.defn p (Rat.divInt_ne_zero_of_ne_zero h... |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudriashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Analysis.Convex.Star
import Mathlib.LinearAlgebra.AffineSpace.Af... | Mathlib/Analysis/Convex/Basic.lean | 560 | 569 | theorem Convex_subadditive_le [SMul 𝕜 E] {f : E → 𝕜} (hf1 : ∀ x y, f (x + y) ≤ (f x) + (f y))
(hf2 : ∀ ⦃c⦄ x, 0 ≤ c → f (c • x) ≤ c * f x) (B : 𝕜) :
Convex 𝕜 { x | f x ≤ B } := by |
rw [convex_iff_segment_subset]
rintro x hx y hy z ⟨a, b, ha, hb, hs, rfl⟩
calc
_ ≤ a • (f x) + b • (f y) := le_trans (hf1 _ _) (add_le_add (hf2 x ha) (hf2 y hb))
_ ≤ a • B + b • B :=
add_le_add (smul_le_smul_of_nonneg_left hx ha) (smul_le_smul_of_nonneg_left hy hb)
_ ≤ B := by rw [← add_smul,... |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.LinearAlgebra.FinsuppVectorS... | Mathlib/RingTheory/MvPolynomial/Basic.lean | 107 | 110 | theorem mem_restrictTotalDegree (p : MvPolynomial σ R) :
p ∈ restrictTotalDegree σ R m ↔ p.totalDegree ≤ m := by |
rw [totalDegree, Finset.sup_le_iff]
rfl
|
/-
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.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
/-!
# Linear equivalences of tensor products as isometrie... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean | 79 | 85 | theorem tmul_comp_tensorComm (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) :
(Q₂.tmul Q₁).comp (TensorProduct.comm R M₁ M₂) = Q₁.tmul Q₂ := by |
refine (QuadraticForm.associated_rightInverse R).injective ?_
ext m₁ m₂ m₁' m₂'
dsimp [-associated_apply]
simp only [associated_tmul, QuadraticForm.associated_comp]
exact mul_comm _ _
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Antichain
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.RelIso.Set
#align_import order.minimal ... | Mathlib/Order/Minimal.lean | 345 | 347 | theorem image_minimals_univ :
Subtype.val '' minimals (· ≤ ·) (univ : Set s) = minimals (· ≤ ·) s := by |
rw [image_minimals_of_rel_iff_rel, image_univ, Subtype.range_val]; intros; rfl
|
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