Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 |
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import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import number_theory.number_field.units from "leanprover-community/mathlib"@"00f91228655eecdcd3ac... | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 122 | 126 | theorem logEmbedding_component_le {r : ℝ} {x : (𝓞 K)ˣ} (hr : 0 ≤ r) (h : ‖logEmbedding K x‖ ≤ r)
(w : {w : InfinitePlace K // w ≠ w₀}) : |logEmbedding K x w| ≤ r := by |
lift r to NNReal using hr
simp_rw [Pi.norm_def, NNReal.coe_le_coe, Finset.sup_le_iff, ← NNReal.coe_le_coe] at h
exact h w (mem_univ _)
| 3 |
import Mathlib.Combinatorics.Enumerative.Composition
import Mathlib.Tactic.ApplyFun
#align_import combinatorics.partition from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Multiset
namespace Nat
@[ext]
structure Partition (n : ℕ) where
parts : Multiset ℕ
parts_pos : ∀... | Mathlib/Combinatorics/Enumerative/Partition.lean | 77 | 80 | theorem ofComposition_surj {n : ℕ} : Function.Surjective (ofComposition n) := by |
rintro ⟨b, hb₁, hb₂⟩
induction b using Quotient.inductionOn with | _ b => ?_
exact ⟨⟨b, hb₁, by simpa using hb₂⟩, Partition.ext _ _ rfl⟩
| 3 |
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.Separation
#align_import topology.uniform_space.separation from "leanprover-community/mathlib"@"0c1f285a9f6e608ae2bdffa3f993eafb01eba829"
open Filter Set Function Topology Uniformity UniformSpace
open scoped Classical... | Mathlib/Topology/UniformSpace/Separation.lean | 142 | 146 | theorem inseparable_iff_clusterPt_uniformity {x y : α} :
Inseparable x y ↔ ClusterPt (x, y) (𝓤 α) := by |
refine ⟨fun h ↦ .of_nhds_le h.nhds_le_uniformity, fun h ↦ ?_⟩
simp_rw [uniformity_hasBasis_closed.inseparable_iff_uniformity, isClosed_iff_clusterPt]
exact fun U ⟨hU, hUc⟩ ↦ hUc _ <| h.mono <| le_principal_iff.2 hU
| 3 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Data.Complex.ExponentialBounds
#align_import combinatorics.additive.behrend from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open N... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 125 | 129 | theorem norm_of_mem_sphere {x : Fin n → ℕ} (hx : x ∈ sphere n d k) :
‖(WithLp.equiv 2 _).symm ((↑) ∘ x : Fin n → ℝ)‖ = √↑k := by |
rw [EuclideanSpace.norm_eq]
dsimp
simp_rw [abs_cast, ← cast_pow, ← cast_sum, (mem_filter.1 hx).2]
| 3 |
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Data.Rat.Floor
#align_import algebra.continued_fractions.computation.terminates_iff_rat from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b3... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 112 | 115 | theorem exists_rat_eq_nth_denominator : ∃ q : ℚ, (of v).denominators n = (q : K) := by |
rcases exists_gcf_pair_rat_eq_nth_conts v n with ⟨⟨_, b⟩, nth_cont_eq⟩
use b
simp [denom_eq_conts_b, nth_cont_eq]
| 3 |
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.cartan_subalgebra from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102"
universe u v w w₁ w₂
variable {R : Type u} {L : Type v}
variable [CommRing R] [LieRing L] [LieAlgebra R L] (H : Lie... | Mathlib/Algebra/Lie/CartanSubalgebra.lean | 65 | 69 | theorem ucs_eq_self_of_isCartanSubalgebra (H : LieSubalgebra R L) [H.IsCartanSubalgebra] (k : ℕ) :
H.toLieSubmodule.ucs k = H.toLieSubmodule := by |
induction' k with k ih
· simp
· simp [ih]
| 3 |
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padics.padic_numbers from "leanprover-community/mathlib"@"b9b2114f7711fec1c1e055d507f082f8ceb2c3b7"
noncomputable section
open scoped Classical
open Nat m... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 234 | 238 | theorem val_eq_iff_norm_eq {f g : PadicSeq p} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) :
f.valuation = g.valuation ↔ f.norm = g.norm := by |
rw [norm_eq_pow_val hf, norm_eq_pow_val hg, ← neg_inj, zpow_inj]
· exact mod_cast (Fact.out : p.Prime).pos
· exact mod_cast (Fact.out : p.Prime).ne_one
| 3 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 82 | 86 | theorem edgeSet_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeSet =
(G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s)) \ {s(t, t)} := by |
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
| 3 |
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Asymptotics.Theta
import Mathlib.Analysis.Normed.Order.Basic
#align_import analysis.asymptotics.asymptotic_equivalent from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
namespace Asymptotics
open Filter Function
... | Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 151 | 154 | theorem IsEquivalent.tendsto_const {c : β} (hu : u ~[l] const _ c) : Tendsto u l (𝓝 c) := by |
rcases em <| c = 0 with rfl | h
· exact (tendsto_congr' <| isEquivalent_zero_iff_eventually_zero.mp hu).mpr tendsto_const_nhds
· exact (isEquivalent_const_iff_tendsto h).mp hu
| 3 |
import Mathlib.AlgebraicTopology.DoldKan.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComplex
#align_import algebraic_topology.dold_kan.gamma_comp_n from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
no... | Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 86 | 91 | theorem N₁Γ₀_hom_app (K : ChainComplex C ℕ) :
N₁Γ₀.hom.app K = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv ≫
(toKaroubi _).map (Γ₀NondegComplexIso K).hom := by |
change (N₁Γ₀.app K).hom = _
simp only [N₁Γ₀_app]
rfl
| 3 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheor... | Mathlib/Probability/Martingale/Basic.lean | 109 | 113 | theorem setIntegral_eq [SigmaFiniteFiltration μ ℱ] (hf : Martingale f ℱ μ) {i j : ι} (hij : i ≤ j)
{s : Set Ω} (hs : MeasurableSet[ℱ i] s) : ∫ ω in s, f i ω ∂μ = ∫ ω in s, f j ω ∂μ := by |
rw [← @setIntegral_condexp _ _ _ _ _ (ℱ i) m0 _ _ _ (ℱ.le i) _ (hf.integrable j) hs]
refine setIntegral_congr_ae (ℱ.le i s hs) ?_
filter_upwards [hf.2 i j hij] with _ heq _ using heq.symm
| 3 |
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι... | Mathlib/Data/Set/Lattice.lean | 207 | 211 | theorem exists_set_mem_of_union_eq_top {ι : Type*} (t : Set ι) (s : ι → Set β)
(w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by |
have p : x ∈ ⊤ := Set.mem_univ x
rw [← w, Set.mem_iUnion] at p
simpa using p
| 3 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section AddEdge
def edge : SimpleGraph V := fromEdgeSet {s(s, t)}
lemma edge_adj (v w : V) : (edge s t).Adj v w ↔ ... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 171 | 175 | theorem edgeFinset_sup_edge [Fintype (edgeSet (G ⊔ edge s t))] (hn : ¬G.Adj s t) (h : s ≠ t) :
(G ⊔ edge s t).edgeFinset = G.edgeFinset.cons s(s, t) (by simp_all) := by |
letI := Classical.decEq V
rw [edgeFinset_sup, cons_eq_insert, insert_eq, union_comm]
simp_rw [edgeFinset, edge_edgeSet_of_ne h]; rfl
| 3 |
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.PartENat
#align_import set_theory.cardinal.finite from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
set_option autoImplicit true
open Cardinal Function
noncomputable section
variable {α β : Typ... | Mathlib/SetTheory/Cardinal/Finite.lean | 167 | 170 | theorem card_sum [Finite α] [Finite β] : Nat.card (α ⊕ β) = Nat.card α + Nat.card β := by |
have := Fintype.ofFinite α
have := Fintype.ofFinite β
simp_rw [Nat.card_eq_fintype_card, Fintype.card_sum]
| 3 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
namespace MulAction
open Pointwise
variable {α : Type*}
variable {G : Type*} [Group G] [MulAction G α]
variable {M : Type*} [Monoid M] [MulAction M α]
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 195 | 199 | theorem fixedBy_mem_fixedBy_of_commute {g h : G} (comm: Commute g h) :
(fixedBy α g) ∈ fixedBy (Set α) h := by |
ext x
rw [Set.mem_smul_set_iff_inv_smul_mem, mem_fixedBy, ← mul_smul, comm.inv_right, mul_smul,
smul_left_cancel_iff, mem_fixedBy]
| 3 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.unoriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 69 | 73 | theorem angle_add_eq_arccos_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
angle x (x + y) = Real.arccos (‖x‖ / ‖x + y‖) := by |
rw [angle, inner_add_right, h, add_zero, real_inner_self_eq_norm_mul_norm]
by_cases hx : ‖x‖ = 0; · simp [hx]
rw [div_mul_eq_div_div, mul_self_div_self]
| 3 |
import Mathlib.Logic.Pairwise
import Mathlib.Logic.Relation
import Mathlib.Data.List.Basic
#align_import data.list.pairwise from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
open Nat Function
namespace List
variable {α β : Type*} {R S T : α → α → Prop} {a : α} {l : List α}
mk_iff_o... | Mathlib/Data/List/Pairwise.lean | 152 | 156 | theorem pairwise_of_forall_mem_list {l : List α} {r : α → α → Prop} (h : ∀ a ∈ l, ∀ b ∈ l, r a b) :
l.Pairwise r := by |
rw [pairwise_iff_forall_sublist]
intro a b hab
apply h <;> (apply hab.subset; simp)
| 3 |
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.Order.Copy
import Mathlib.Data.Set.Subsingleton
#align_import category_theory.sites.grothendieck fr... | Mathlib/CategoryTheory/Sites/Grothendieck.lean | 105 | 109 | theorem ext {J₁ J₂ : GrothendieckTopology C} (h : (J₁ : ∀ X : C, Set (Sieve X)) = J₂) :
J₁ = J₂ := by |
cases J₁
cases J₂
congr
| 3 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import analysis.special_functions.pow.continuity from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter ComplexConjugate
open Filter Finset... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 105 | 109 | theorem Filter.Tendsto.const_cpow {l : Filter α} {f : α → ℂ} {a b : ℂ} (hf : Tendsto f l (𝓝 b))
(h : a ≠ 0 ∨ b ≠ 0) : Tendsto (fun x => a ^ f x) l (𝓝 (a ^ b)) := by |
cases h with
| inl h => exact (continuousAt_const_cpow h).tendsto.comp hf
| inr h => exact (continuousAt_const_cpow' h).tendsto.comp hf
| 3 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 91 | 94 | theorem gold_sq : φ ^ 2 = φ + 1 := by |
rw [goldenRatio, ← sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
| 3 |
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
open Function Set
variable ... | Mathlib/RingTheory/Nilpotent/Lemmas.lean | 123 | 126 | theorem IsNilpotent.mapQ (hnp : IsNilpotent f) : IsNilpotent (p.mapQ p f hp) := by |
obtain ⟨k, hk⟩ := hnp
use k
simp [← p.mapQ_pow, hk]
| 3 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
#align_import measure_theory.function.conditional_expectation.condexp_L2 from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 113 | 117 | theorem norm_condexpL2_coe_le (hm : m ≤ m0) (f : α →₂[μ] E) :
‖(condexpL2 E 𝕜 hm f : α →₂[μ] E)‖ ≤ ‖f‖ := by |
rw [Lp.norm_def, Lp.norm_def, ← lpMeas_coe]
refine (ENNReal.toReal_le_toReal ?_ (Lp.snorm_ne_top _)).mpr (snorm_condexpL2_le hm f)
exact Lp.snorm_ne_top _
| 3 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R α β δ γ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ]
variable {μ μ₁ μ₂ μ₃ ν ν' ν... | Mathlib/MeasureTheory/Measure/Restrict.lean | 110 | 113 | theorem restrict_apply₀' (hs : NullMeasurableSet s μ) : μ.restrict s t = μ (t ∩ s) := by |
rw [← restrict_congr_set hs.toMeasurable_ae_eq,
restrict_apply' (measurableSet_toMeasurable _ _),
measure_congr ((ae_eq_refl t).inter hs.toMeasurable_ae_eq)]
| 3 |
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.comm_ring from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/CommRing.lean | 207 | 212 | theorem totalDegree_sub (a b : MvPolynomial σ R) :
(a - b).totalDegree ≤ max a.totalDegree b.totalDegree :=
calc
(a - b).totalDegree = (a + -b).totalDegree := by | rw [sub_eq_add_neg]
_ ≤ max a.totalDegree (-b).totalDegree := totalDegree_add a (-b)
_ = max a.totalDegree b.totalDegree := by rw [totalDegree_neg]
| 3 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 92 | 95 | theorem Iic_mul_Iio_subset' (a b : α) : Iic a * Iio b ⊆ Iio (a * b) := by |
haveI := covariantClass_le_of_lt
rintro x ⟨y, hya, z, hzb, rfl⟩
exact mul_lt_mul_of_le_of_lt hya hzb
| 3 |
import Mathlib.CategoryTheory.Generator
import Mathlib.CategoryTheory.Preadditive.Yoneda.Basic
#align_import category_theory.preadditive.generator from "leanprover-community/mathlib"@"09f981f72d43749f1fa072deade828d9c1e185bb"
universe v u
open CategoryTheory Opposite
namespace CategoryTheory
variable {C : Type... | Mathlib/CategoryTheory/Preadditive/Generator.lean | 77 | 81 | theorem isCoseparator_iff_faithful_preadditiveYonedaObj (G : C) :
IsCoseparator G ↔ (preadditiveYonedaObj G).Faithful := by |
rw [isCoseparator_iff_faithful_preadditiveYoneda, preadditiveYoneda_obj]
exact ⟨fun h => Functor.Faithful.of_comp _ (forget₂ _ AddCommGroupCat.{v}),
fun h => Functor.Faithful.comp _ _⟩
| 3 |
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Bornology
universe u v w
variable {ι : Sort*} {α : Type u} {β :... | Mathlib/Topology/MetricSpace/Thickening.lean | 271 | 275 | theorem closedBall_subset_cthickening_singleton {α : Type*} [PseudoMetricSpace α] (x : α) (δ : ℝ) :
closedBall x δ ⊆ cthickening δ ({x} : Set α) := by |
rcases lt_or_le δ 0 with (hδ | hδ)
· simp only [closedBall_eq_empty.mpr hδ, empty_subset]
· simp only [cthickening_singleton x hδ, Subset.rfl]
| 3 |
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Topology.MetricSpace.Holder
import Mathlib.Topology.MetricSpace.MetricSeparated
#align_import measure_theory.measure.hausdorff from "leanprover-communit... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 286 | 290 | theorem tendsto_pre (m : Set X → ℝ≥0∞) (s : Set X) :
Tendsto (fun r => pre m r s) (𝓝[>] 0) (𝓝 <| mkMetric' m s) := by |
rw [← map_coe_Ioi_atBot, tendsto_map'_iff]
simp only [mkMetric', OuterMeasure.iSup_apply, iSup_subtype']
exact tendsto_atBot_iSup fun r r' hr => mono_pre _ hr _
| 3 |
import Mathlib.Order.RelClasses
#align_import data.sigma.lex from "leanprover-community/mathlib"@"41cf0cc2f528dd40a8f2db167ea4fb37b8fde7f3"
namespace Sigma
variable {ι : Type*} {α : ι → Type*} {r r₁ r₂ : ι → ι → Prop} {s s₁ s₂ : ∀ i, α i → α i → Prop}
{a b : Σ i, α i}
inductive Lex (r : ι → ι → Prop) (s : ∀ ... | Mathlib/Data/Sigma/Lex.lean | 63 | 67 | theorem Lex.mono (hr : ∀ a b, r₁ a b → r₂ a b) (hs : ∀ i a b, s₁ i a b → s₂ i a b) {a b : Σ i, α i}
(h : Lex r₁ s₁ a b) : Lex r₂ s₂ a b := by |
obtain ⟨a, b, hij⟩ | ⟨a, b, hab⟩ := h
· exact Lex.left _ _ (hr _ _ hij)
· exact Lex.right _ _ (hs _ _ _ hab)
| 3 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 82 | 86 | theorem SimpleFunc.lintegral_eq_lintegral {m : MeasurableSpace α} (f : α →ₛ ℝ≥0∞) (μ : Measure α) :
∫⁻ a, f a ∂μ = f.lintegral μ := by |
rw [MeasureTheory.lintegral]
exact le_antisymm (iSup₂_le fun g hg => lintegral_mono hg <| le_rfl)
(le_iSup₂_of_le f le_rfl le_rfl)
| 3 |
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "leanprover-community/mathlib"@"9b2660e1b25419042c8da10bf411aa3c67f14383"
open Nat
def ack : ℕ → ℕ → ℕ
| 0, n => n + 1
| m + 1, 0 ... | Mathlib/Computability/Ackermann.lean | 89 | 92 | theorem ack_two (n : ℕ) : ack 2 n = 2 * n + 3 := by |
induction' n with n IH
· rfl
· simpa [mul_succ]
| 3 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 83 | 87 | theorem oangle_add_left_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x + y) y = Real.arctan (‖x‖ / ‖y‖) := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two h
| 3 |
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.ProdSigma
#align_import model_theory.semantics from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable {L : Language.{u, v}} {... | Mathlib/ModelTheory/Semantics.lean | 130 | 134 | theorem realize_subst {t : L.Term α} {tf : α → L.Term β} {v : β → M} :
(t.subst tf).realize v = t.realize fun a => (tf a).realize v := by |
induction' t with _ _ _ _ ih
· rfl
· simp [ih]
| 3 |
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Asymptotics.Theta
import Mathlib.Analysis.Normed.Order.Basic
#align_import analysis.asymptotics.asymptotic_equivalent from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
namespace Asymptotics
open Filter Function
... | Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 133 | 136 | theorem isEquivalent_zero_iff_isBigO_zero : u ~[l] 0 ↔ u =O[l] (0 : α → β) := by |
refine ⟨IsEquivalent.isBigO, fun h ↦ ?_⟩
rw [isEquivalent_zero_iff_eventually_zero, eventuallyEq_iff_exists_mem]
exact ⟨{ x : α | u x = 0 }, isBigO_zero_right_iff.mp h, fun x hx ↦ hx⟩
| 3 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Set.Finite
#align_import combinatorics.pigeonhole from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
un... | Mathlib/Combinatorics/Pigeonhole.lean | 134 | 141 | theorem exists_lt_sum_fiber_of_sum_fiber_nonpos_of_nsmul_lt_sum
(ht : ∀ y ∉ t, ∑ x ∈ s.filter fun x => f x = y, w x ≤ 0)
(hb : t.card • b < ∑ x ∈ s, w x) : ∃ y ∈ t, b < ∑ x ∈ s.filter fun x => f x = y, w x :=
exists_lt_of_sum_lt <|
calc
∑ _y ∈ t, b < ∑ x ∈ s, w x := by | simpa
_ ≤ ∑ y ∈ t, ∑ x ∈ s.filter fun x => f x = y, w x :=
sum_le_sum_fiberwise_of_sum_fiber_nonpos ht
| 3 |
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 76 | 79 | theorem conj_eigenvalue_eq_self {μ : 𝕜} (hμ : HasEigenvalue T μ) : conj μ = μ := by |
obtain ⟨v, hv₁, hv₂⟩ := hμ.exists_hasEigenvector
rw [mem_eigenspace_iff] at hv₁
simpa [hv₂, inner_smul_left, inner_smul_right, hv₁] using hT v v
| 3 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Lie.Basic
#align_import algebra.lie.direct_sum from "leanprover-community/mathlib"@"c0cc689babd41c0e9d5f02429211ffbe2403472a"
universe u v w w₁
namespace DirectSum
open DF... | Mathlib/Algebra/Lie/DirectSum.lean | 140 | 144 | theorem lie_of [DecidableEq ι] {i j : ι} (x : L i) (y : L j) :
⁅of L i x, of L j y⁆ = if hij : i = j then of L i ⁅x, hij.symm.recOn y⁆ else 0 := by |
obtain rfl | hij := Decidable.eq_or_ne i j
· simp only [lie_of_same L x y, dif_pos]
· simp only [lie_of_of_ne L hij x y, hij, dif_neg, dite_false]
| 3 |
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.measure.regular from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Set Filter ENNReal Topology NNReal TopologicalSpace
namespace MeasureTh... | Mathlib/MeasureTheory/Measure/Regular.lean | 260 | 264 | theorem trans {q' : Set α → Prop} (H : InnerRegularWRT μ p q) (H' : InnerRegularWRT μ q q') :
InnerRegularWRT μ p q' := by |
intro U hU r hr
rcases H' hU r hr with ⟨F, hFU, hqF, hF⟩; rcases H hqF _ hF with ⟨K, hKF, hpK, hrK⟩
exact ⟨K, hKF.trans hFU, hpK, hrK⟩
| 3 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
open Set Filter MeasureTheory MeasurableSpace
open scoped Classical Topology NNReal ENNReal MeasureTheory
univers... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 91 | 94 | theorem isPiSystem_Iio_rat : IsPiSystem (⋃ a : ℚ, {Iio (a : ℝ)}) := by |
convert isPiSystem_image_Iio (((↑) : ℚ → ℝ) '' univ)
ext x
simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and]
| 3 |
import Mathlib.MeasureTheory.Covering.DensityTheorem
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.covering.one_dim from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set MeasureTheory IsUnifLocDoublingMeasure Filter
open scoped Topology
names... | Mathlib/MeasureTheory/Covering/OneDim.lean | 26 | 30 | theorem Icc_mem_vitaliFamily_at_right {x y : ℝ} (hxy : x < y) :
Icc x y ∈ (vitaliFamily (volume : Measure ℝ) 1).setsAt x := by |
rw [Icc_eq_closedBall]
refine closedBall_mem_vitaliFamily_of_dist_le_mul _ ?_ (by linarith)
rw [dist_comm, Real.dist_eq, abs_of_nonneg] <;> linarith
| 3 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 59 | 62 | theorem hermite_eq_iterate (n : ℕ) : hermite n = (fun p => X * p - derivative p)^[n] 1 := by |
induction' n with n ih
· rfl
· rw [Function.iterate_succ_apply', ← ih, hermite_succ]
| 3 |
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open CategoryTheory.Limits CategoryTheory.Functor Opposite
section ArbitraryUniverses
variable {C : Type u₁} [Category.{v₁}... | Mathlib/CategoryTheory/Filtered/Final.lean | 99 | 104 | theorem Functor.initial_of_exists_of_isCofiltered [IsCofilteredOrEmpty C]
(h₁ : ∀ d, ∃ c, Nonempty (F.obj c ⟶ d)) (h₂ : ∀ {d : D} {c : C} (s s' : F.obj c ⟶ d),
∃ (c' : C) (t : c' ⟶ c), F.map t ≫ s = F.map t ≫ s') : Functor.Initial F := by |
suffices ∀ d, IsCofiltered (CostructuredArrow F d) from
initial_of_isCofiltered_costructuredArrow F
exact isCofiltered_costructuredArrow_of_isCofiltered_of_exists F h₁ h₂
| 3 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section ConditionallyCompleteLinearOrder
variable [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [Top... | Mathlib/Topology/Order/Monotone.lean | 58 | 62 | theorem Monotone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Mf : Monotone f)
(bdd : BddBelow (range g) := by | bddDefault) : f (⨅ i, g i) = ⨅ i, f (g i) := by
rw [iInf, Monotone.map_sInf_of_continuousAt' Cf Mf (range_nonempty g) bdd, ← range_comp, iInf]
rfl
| 3 |
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Baire.LocallyCompactRegular
import Mathlib.Topology.Instances.Irrational
#align_import number_theory.liouville.residual from "leanprover-community/mathlib"@"32b08ef840dd25ca2e47e035c5da03ce16d2dc3c"
open scope... | Mathlib/NumberTheory/Liouville/Residual.lean | 25 | 31 | theorem setOf_liouville_eq_iInter_iUnion :
{ x | Liouville x } =
⋂ n : ℕ, ⋃ (a : ℤ) (b : ℤ) (_ : 1 < b),
ball ((a : ℝ) / b) (1 / (b : ℝ) ^ n) \ {(a : ℝ) / b} := by |
ext x
simp only [mem_iInter, mem_iUnion, Liouville, mem_setOf_eq, exists_prop, mem_diff,
mem_singleton_iff, mem_ball, Real.dist_eq, and_comm]
| 3 |
import Mathlib.Topology.Bornology.Basic
#align_import topology.bornology.constructions from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
open Set Filter Bornology Function
open Filter
variable {α β ι : Type*} {π : ι → Type*} [Bornology α] [Bornology β]
[∀ i, Bornology (π i)]
inst... | Mathlib/Topology/Bornology/Constructions.lean | 88 | 91 | theorem isBounded_prod : IsBounded (s ×ˢ t) ↔ s = ∅ ∨ t = ∅ ∨ IsBounded s ∧ IsBounded t := by |
rcases s.eq_empty_or_nonempty with (rfl | hs); · simp
rcases t.eq_empty_or_nonempty with (rfl | ht); · simp
simp only [hs.ne_empty, ht.ne_empty, isBounded_prod_of_nonempty (hs.prod ht), false_or_iff]
| 3 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerS... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 64 | 68 | theorem map_invUnitsSub (f : R →+* S) (u : Rˣ) :
map f (invUnitsSub u) = invUnitsSub (Units.map (f : R →* S) u) := by |
ext
simp only [← map_pow, coeff_map, coeff_invUnitsSub, one_divp]
rfl
| 3 |
import Mathlib.CategoryTheory.Preadditive.Injective
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Logic.Equiv.TransferInstance
#align_import algebra.module.injective from "leanprover-community/mathlib"@"f8d8465c3c392a93b9... | Mathlib/Algebra/Module/Injective.lean | 70 | 76 | theorem Module.injective_module_of_injective_object
[inj : CategoryTheory.Injective <| ModuleCat.of R Q] :
Module.Injective R Q where
out X Y _ _ _ _ f hf g := by |
have : CategoryTheory.Mono (ModuleCat.ofHom f) := (ModuleCat.mono_iff_injective _).mpr hf
obtain ⟨l, rfl⟩ := inj.factors (ModuleCat.ofHom g) (ModuleCat.ofHom f)
exact ⟨l, fun _ ↦ rfl⟩
| 3 |
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
-- Porting note: universe metavariables behave oddly
universe w u v₁ v₂ v₃ v₄
variable {ι : Type... | Mathlib/LinearAlgebra/Contraction.lean | 96 | 101 | theorem dualTensorHom_prodMap_zero (f : Module.Dual R M) (p : P) :
((dualTensorHom R M P) (f ⊗ₜ[R] p)).prodMap (0 : N →ₗ[R] Q) =
dualTensorHom R (M × N) (P × Q) ((f ∘ₗ fst R M N) ⊗ₜ inl R P Q p) := by |
ext <;>
simp only [coe_comp, coe_inl, Function.comp_apply, prodMap_apply, dualTensorHom_apply,
fst_apply, Prod.smul_mk, LinearMap.zero_apply, smul_zero]
| 3 |
import Mathlib.Data.Option.Basic
import Mathlib.Data.Set.Basic
#align_import data.pequiv from "leanprover-community/mathlib"@"7c3269ca3fa4c0c19e4d127cd7151edbdbf99ed4"
universe u v w x
structure PEquiv (α : Type u) (β : Type v) where
toFun : α → Option β
invFun : β → Option α
inv : ∀ (a : α) (b :... | Mathlib/Data/PEquiv.lean | 161 | 165 | theorem trans_eq_none (f : α ≃. β) (g : β ≃. γ) (a : α) :
f.trans g a = none ↔ ∀ b c, b ∉ f a ∨ c ∉ g b := by |
simp only [eq_none_iff_forall_not_mem, mem_trans, imp_iff_not_or.symm]
push_neg
exact forall_swap
| 3 |
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community/mathlib"@"9407b03373c8cd201df99d6bc5514fc2db44054f"
namespace Equiv
variable {α β : Type*} [Finite α]
noncomputable def toCompl {p q : α → Prop} (e ... | Mathlib/Logic/Equiv/Fintype.lean | 125 | 129 | theorem extendSubtype_apply_of_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : p x) :
e.extendSubtype x = e ⟨x, hx⟩ := by |
dsimp only [extendSubtype]
simp only [subtypeCongr, Equiv.trans_apply, Equiv.sumCongr_apply]
rw [sumCompl_apply_symm_of_pos _ _ hx, Sum.map_inl, sumCompl_apply_inl]
| 3 |
import Mathlib.Init.Control.Combinators
import Mathlib.Init.Function
import Mathlib.Tactic.CasesM
import Mathlib.Tactic.Attr.Core
#align_import control.basic from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
universe u v w
variable {α β γ : Type u}
section Applicative
variable {F : ... | Mathlib/Control/Basic.lean | 60 | 64 | theorem seq_map_assoc (x : F (α → β)) (f : γ → α) (y : F γ) :
x <*> f <$> y = (· ∘ f) <$> x <*> y := by |
simp only [← pure_seq]
simp only [seq_assoc, Function.comp, seq_pure, ← comp_map]
simp [pure_seq]
| 3 |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
#align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 168 | 171 | theorem apply_mono (μ : FiniteMeasure Ω) {s₁ s₂ : Set Ω} (h : s₁ ⊆ s₂) : μ s₁ ≤ μ s₂ := by |
change ((μ : Measure Ω) s₁).toNNReal ≤ ((μ : Measure Ω) s₂).toNNReal
have key : (μ : Measure Ω) s₁ ≤ (μ : Measure Ω) s₂ := (μ : Measure Ω).mono h
apply (ENNReal.toNNReal_le_toNNReal (measure_ne_top _ s₁) (measure_ne_top _ s₂)).mpr key
| 3 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.Core
import Mathlib.Util.SynthesizeUsing
import Mathlib.Util.Qq
open Lean Parser Tactic Mathlib Meta NormNum Qq
initialize registerTraceClass `CancelDen... | Mathlib/Tactic/CancelDenoms/Core.lean | 73 | 78 | theorem cancel_factors_lt {α} [LinearOrderedField α] {a b ad bd a' b' gcd : α}
(ha : ad * a = a') (hb : bd * b = b') (had : 0 < ad) (hbd : 0 < bd) (hgcd : 0 < gcd) :
(a < b) = (1 / gcd * (bd * a') < 1 / gcd * (ad * b')) := by |
rw [mul_lt_mul_left, ← ha, ← hb, ← mul_assoc, ← mul_assoc, mul_comm bd, mul_lt_mul_left]
· exact mul_pos had hbd
· exact one_div_pos.2 hgcd
| 3 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 65 | 69 | theorem oangle_add_left_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x + y) y = Real.arcsin (‖x‖ / ‖x + y‖) := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two h
| 3 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.GroupTheory.Torsion
import Mathlib.RingTheory.Coprime.Ideal
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Set.Lattice
#align_import algebra.module.torsion from "leanpr... | Mathlib/Algebra/Module/Torsion.lean | 92 | 95 | theorem torsionOf_eq_top_iff (m : M) : torsionOf R M m = ⊤ ↔ m = 0 := by |
refine ⟨fun h => ?_, fun h => by simp [h]⟩
rw [← one_smul R m, ← mem_torsionOf_iff m (1 : R), h]
exact Submodule.mem_top
| 3 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 89 | 92 | theorem IsCoprime.ne_zero_or_ne_zero [Nontrivial R] (h : IsCoprime x y) : x ≠ 0 ∨ y ≠ 0 := by |
apply not_or_of_imp
rintro rfl rfl
exact not_isCoprime_zero_zero h
| 3 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 241 | 246 | theorem finrank_directSum {ι : Type v} [Fintype ι] (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] [∀ i : ι, Module.Finite R (M i)] :
finrank R (⨁ i, M i) = ∑ i, finrank R (M i) := by |
letI := nontrivial_of_invariantBasisNumber R
simp only [finrank, fun i => rank_eq_card_chooseBasisIndex R (M i), rank_directSum, ← mk_sigma,
mk_toNat_eq_card, card_sigma]
| 3 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 57 | 60 | theorem gold_mul_goldConj : φ * ψ = -1 := by |
field_simp
rw [← sq_sub_sq]
norm_num
| 3 |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.restrict from "leanprover-community/mathlib"@"09258fb7f75d741b7eda9fa18d5c869e2135d9f1"
variable {k V₁ P₁ V₂ P₂ : Type*} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁]
[Module k V₂] [AddTorsor V₁ P₁] [A... | Mathlib/LinearAlgebra/AffineSpace/Restrict.lean | 73 | 78 | theorem AffineMap.restrict.injective {φ : P₁ →ᵃ[k] P₂} (hφ : Function.Injective φ)
{E : AffineSubspace k P₁} {F : AffineSubspace k P₂} [Nonempty E] [Nonempty F]
(hEF : E.map φ ≤ F) : Function.Injective (AffineMap.restrict φ hEF) := by |
intro x y h
simp only [Subtype.ext_iff, Subtype.coe_mk, AffineMap.restrict.coe_apply] at h ⊢
exact hφ h
| 3 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 87 | 90 | theorem disjoint_refl_iff : Disjoint f f ↔ f = 1 := by |
refine ⟨fun h => ?_, fun h => h.symm ▸ disjoint_one_left 1⟩
ext x
cases' h x with hx hx <;> simp [hx]
| 3 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 86 | 89 | theorem Ico_mul_Ioc_subset' (a b c d : α) : Ico a b * Ioc c d ⊆ Ioo (a * c) (b * d) := by |
haveI := covariantClass_le_of_lt
rintro x ⟨y, ⟨hya, hyb⟩, z, ⟨hzc, hzd⟩, rfl⟩
exact ⟨mul_lt_mul_of_le_of_lt hya hzc, mul_lt_mul_of_lt_of_le hyb hzd⟩
| 3 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {α : Type v} {β : Type w}
namespace Matrix
def col (w : m → α) : Matrix m Unit α :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 154 | 158 | theorem vecMulVec_eq [Mul α] [AddCommMonoid α] (w : m → α) (v : n → α) :
vecMulVec w v = col w * row v := by |
ext
simp only [vecMulVec, mul_apply, Fintype.univ_punit, Finset.sum_singleton]
rfl
| 3 |
import Mathlib.Data.Finset.Order
import Mathlib.Algebra.DirectSum.Module
import Mathlib.RingTheory.FreeCommRing
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.SuppressCompilation
#align_import algebra.direct_limit from "leanprover-community/mathlib"@"f0c8bf9245297a... | Mathlib/Algebra/DirectLimit.lean | 164 | 170 | theorem lift_unique [IsDirected ι (· ≤ ·)] (F : DirectLimit G f →ₗ[R] P) (x) :
F x =
lift R ι G f (fun i => F.comp <| of R ι G f i)
(fun i j hij x => by rw [LinearMap.comp_apply, of_f]; rfl) x := by |
cases isEmpty_or_nonempty ι
· simp_rw [Subsingleton.elim x 0, _root_.map_zero]
· exact DirectLimit.induction_on x fun i x => by rw [lift_of]; rfl
| 3 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.normed_space.add_torsor from "leanprover-community/mathlib"@... | Mathlib/Analysis/NormedSpace/AddTorsor.lean | 68 | 72 | theorem dist_lineMap_lineMap (p₁ p₂ : P) (c₁ c₂ : 𝕜) :
dist (lineMap p₁ p₂ c₁) (lineMap p₁ p₂ c₂) = dist c₁ c₂ * dist p₁ p₂ := by |
rw [dist_comm p₁ p₂]
simp only [lineMap_apply, dist_eq_norm_vsub, vadd_vsub_vadd_cancel_right,
← sub_smul, norm_smul, vsub_eq_sub]
| 3 |
import Mathlib.Order.RelClasses
#align_import data.sigma.lex from "leanprover-community/mathlib"@"41cf0cc2f528dd40a8f2db167ea4fb37b8fde7f3"
namespace Sigma
variable {ι : Type*} {α : ι → Type*} {r r₁ r₂ : ι → ι → Prop} {s s₁ s₂ : ∀ i, α i → α i → Prop}
{a b : Σ i, α i}
inductive Lex (r : ι → ι → Prop) (s : ∀ ... | Mathlib/Data/Sigma/Lex.lean | 80 | 83 | theorem lex_swap : Lex (Function.swap r) s a b ↔ Lex r (fun i => Function.swap (s i)) b a := by |
constructor <;>
· rintro (⟨a, b, h⟩ | ⟨a, b, h⟩)
exacts [Lex.left _ _ h, Lex.right _ _ h]
| 3 |
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 146 | 153 | theorem yang_baxter (X Y Z : C) :
(α_ X Y Z).inv ≫ (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫
Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv ≫ (β_ Y Z).hom ▷ X ≫ (α_ Z Y X).hom =
X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫ (β_ X Z).hom ▷ Y ≫
(α_ Z X Y).hom ≫ Z ◁ (β_ X Y).hom := by |
rw [← braiding_tensor_right_assoc X Y Z, ← cancel_mono (α_ Z Y X).inv]
repeat rw [assoc]
rw [Iso.hom_inv_id, comp_id, ← braiding_naturality_right, braiding_tensor_right]
| 3 |
import Mathlib.Data.Fintype.Basic
import Mathlib.ModelTheory.Substructures
#align_import model_theory.elementary_maps from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
open FirstOrder
namespace FirstOrder
namespace Language
open Structure
variable (L : Language) (M : Type*) (N : T... | Mathlib/ModelTheory/ElementaryMaps.lean | 132 | 136 | theorem map_fun (φ : M ↪ₑ[L] N) {n : ℕ} (f : L.Functions n) (x : Fin n → M) :
φ (funMap f x) = funMap f (φ ∘ x) := by |
have h := φ.map_formula (Formula.graph f) (Fin.cons (funMap f x) x)
rw [Formula.realize_graph, Fin.comp_cons, Formula.realize_graph] at h
rw [eq_comm, h]
| 3 |
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 63 | 67 | theorem eventually_nhdsWithin_nhdsWithin {a : α} {s : Set α} {p : α → Prop} :
(∀ᶠ y in 𝓝[s] a, ∀ᶠ x in 𝓝[s] y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x := by |
refine ⟨fun h => ?_, fun h => (eventually_nhds_nhdsWithin.2 h).filter_mono inf_le_left⟩
simp only [eventually_nhdsWithin_iff] at h ⊢
exact h.mono fun x hx hxs => (hx hxs).self_of_nhds hxs
| 3 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 162 | 166 | theorem DifferentiableAt.conformalAt (h : DifferentiableAt ℂ f z) (hf' : deriv f z ≠ 0) :
ConformalAt f z := by |
rw [conformalAt_iff_isConformalMap_fderiv, (h.hasFDerivAt.restrictScalars ℝ).fderiv]
apply isConformalMap_complex_linear
simpa only [Ne, ext_ring_iff]
| 3 |
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Finset.Sum
import Mathlib.Logic.Embedding.Set
#align_import data.fintype.sum from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
universe u v
variable {α β : Type*}
open Finset
instance (α : Type u) (β : Type v) [Fintype α] [Fintyp... | Mathlib/Data/Fintype/Sum.lean | 118 | 123 | theorem Fintype.card_subtype_or (p q : α → Prop) [Fintype { x // p x }] [Fintype { x // q x }]
[Fintype { x // p x ∨ q x }] :
Fintype.card { x // p x ∨ q x } ≤ Fintype.card { x // p x } + Fintype.card { x // q x } := by |
classical
convert Fintype.card_le_of_embedding (subtypeOrLeftEmbedding p q)
rw [Fintype.card_sum]
| 3 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.OpenImmersion
#align_import algebraic_geometry.gluing from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
set_option linter.uppercaseLean3 false
noncomputable section
universe u
open Topologica... | Mathlib/AlgebraicGeometry/Gluing.lean | 331 | 335 | theorem glued_cover_cocycle (x y z : 𝒰.J) :
gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y = 𝟙 _ := by |
apply pullback.hom_ext <;> simp_rw [Category.id_comp, Category.assoc]
· apply glued_cover_cocycle_fst
· apply glued_cover_cocycle_snd
| 3 |
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 68 | 72 | theorem invariant_orthogonalComplement_eigenspace (μ : 𝕜) (v : E) (hv : v ∈ (eigenspace T μ)ᗮ) :
T v ∈ (eigenspace T μ)ᗮ := by |
intro w hw
have : T w = (μ : 𝕜) • w := by rwa [mem_eigenspace_iff] at hw
simp [← hT w, this, inner_smul_left, hv w hw]
| 3 |
import Mathlib.MeasureTheory.Measure.Restrict
#align_import measure_theory.measure.mutually_singular from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
open Set
open MeasureTheory NNReal ENNReal
namespace MeasureTheory
namespace Measure
variable {α : Type*} {m0 : MeasurableSpace α}... | Mathlib/MeasureTheory/Measure/MutuallySingular.lean | 48 | 52 | theorem mk {s t : Set α} (hs : μ s = 0) (ht : ν t = 0) (hst : univ ⊆ s ∪ t) :
MutuallySingular μ ν := by |
use toMeasurable μ s, measurableSet_toMeasurable _ _, (measure_toMeasurable _).trans hs
refine measure_mono_null (fun x hx => (hst trivial).resolve_left fun hxs => hx ?_) ht
exact subset_toMeasurable _ _ hxs
| 3 |
import Mathlib.Data.List.Range
import Mathlib.Algebra.Order.Ring.Nat
variable {α : Type*}
namespace List
@[simp]
theorem length_iterate (f : α → α) (a : α) (n : ℕ) : length (iterate f a n) = n := by
induction n generalizing a <;> simp [*]
@[simp]
theorem iterate_eq_nil {f : α → α} {a : α} {n : ℕ} : iterate f ... | Mathlib/Data/List/Iterate.lean | 48 | 52 | theorem iterate_add (f : α → α) (a : α) (m n : ℕ) :
iterate f a (m + n) = iterate f a m ++ iterate f (f^[m] a) n := by |
induction m generalizing a with
| zero => simp
| succ n ih => rw [iterate, add_right_comm, iterate, ih, Nat.iterate, cons_append]
| 3 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Finset
#align_import algebra.tropical.big_operators from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
variable {R S :... | Mathlib/Algebra/Tropical/BigOperators.lean | 70 | 74 | theorem untrop_prod [AddCommMonoid R] (s : Finset S) (f : S → Tropical R) :
untrop (∏ i ∈ s, f i) = ∑ i ∈ s, untrop (f i) := by |
convert Multiset.untrop_prod (s.val.map f)
simp only [Multiset.map_map, Function.comp_apply]
rfl
| 3 |
import Mathlib.Geometry.Euclidean.Inversion.Basic
import Mathlib.Geometry.Euclidean.PerpBisector
open Metric Function AffineMap Set AffineSubspace
open scoped Topology
variable {V P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
[NormedAddTorsor V P] {c x y : P} {R : ℝ}
namespace Euclid... | Mathlib/Geometry/Euclidean/Inversion/ImageHyperplane.lean | 66 | 71 | theorem preimage_inversion_sphere_dist_center (hR : R ≠ 0) (hy : y ≠ c) :
inversion c R ⁻¹' sphere y (dist y c) =
insert c (perpBisector c (inversion c R y) : Set P) := by |
ext x
rcases eq_or_ne x c with rfl | hx; · simp [dist_comm]
rw [mem_preimage, mem_sphere, ← inversion_mem_perpBisector_inversion_iff hR] <;> simp [*]
| 3 |
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 87 | 90 | theorem take_range (m n : ℕ) : take m (range n) = range (min m n) := by |
apply List.ext_get
· simp
· simp (config := { contextual := true }) [← get_take, Nat.lt_min]
| 3 |
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
-- Porting note: universe metavariables behave oddly
universe w u v₁ v₂ v₃ v₄
variable {ι : Type... | Mathlib/LinearAlgebra/Contraction.lean | 113 | 118 | theorem map_dualTensorHom (f : Module.Dual R M) (p : P) (g : Module.Dual R N) (q : Q) :
TensorProduct.map (dualTensorHom R M P (f ⊗ₜ[R] p)) (dualTensorHom R N Q (g ⊗ₜ[R] q)) =
dualTensorHom R (M ⊗[R] N) (P ⊗[R] Q) (dualDistrib R M N (f ⊗ₜ g) ⊗ₜ[R] p ⊗ₜ[R] q) := by |
ext m n
simp only [compr₂_apply, mk_apply, map_tmul, dualTensorHom_apply, dualDistrib_apply, ←
smul_tmul_smul]
| 3 |
import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Card
import Mathlib.Order.RelSeries
#align_import order.jordan_holder from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u
... | Mathlib/Order/JordanHolder.lean | 102 | 106 | theorem isMaximal_inf_right_of_isMaximal_sup {x y : X} (hxz : IsMaximal x (x ⊔ y))
(hyz : IsMaximal y (x ⊔ y)) : IsMaximal (x ⊓ y) y := by |
rw [inf_comm]
rw [sup_comm] at hxz hyz
exact isMaximal_inf_left_of_isMaximal_sup hyz hxz
| 3 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Finset
#align_import algebra.tropical.big_operators from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
variable {R S :... | Mathlib/Algebra/Tropical/BigOperators.lean | 58 | 62 | theorem List.untrop_prod [AddMonoid R] (l : List (Tropical R)) :
untrop l.prod = List.sum (l.map untrop) := by |
induction' l with hd tl IH
· simp
· simp [← IH]
| 3 |
import Mathlib.Data.Int.Bitwise
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.matrix.zpow from "leanprover-community/mathlib"@"03fda9112aa6708947da13944a19310684bfdfcb"
open Matrix
namespace Matrix
variable {n' : Type*} [Decidab... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 50 | 54 | theorem pow_sub' (A : M) {m n : ℕ} (ha : IsUnit A.det) (h : n ≤ m) :
A ^ (m - n) = A ^ m * (A ^ n)⁻¹ := by |
rw [← tsub_add_cancel_of_le h, pow_add, Matrix.mul_assoc, mul_nonsing_inv,
tsub_add_cancel_of_le h, Matrix.mul_one]
simpa using ha.pow n
| 3 |
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Conj
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import category_theory.adjunction.reflective from "leanprover-community/mathlib"@"239d882c4fb58361ee8b3b39fb2091320edef10a"
universe v₁ v₂ v₃ u₁ u₂ u₃
noncomputable s... | Mathlib/CategoryTheory/Adjunction/Reflective.lean | 62 | 67 | theorem unit_obj_eq_map_unit [Reflective i] (X : C) :
(reflectorAdjunction i).unit.app (i.obj ((reflector i).obj X)) =
i.map ((reflector i).map ((reflectorAdjunction i).unit.app X)) := by |
rw [← cancel_mono (i.map ((reflectorAdjunction i).counit.app ((reflector i).obj X))),
← i.map_comp]
simp
| 3 |
import Mathlib.RingTheory.Flat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Vanishing
import Mathlib.Algebra.Module.FinitePresentation
universe u
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M]
open Classical DirectSum LinearMap TensorProduct Finsupp
open scoped BigOperators
namespace Modu... | Mathlib/RingTheory/Flat/EquationalCriterion.lean | 88 | 92 | theorem sum_smul_eq_zero_of_isTrivialRelation (h : IsTrivialRelation f x) :
∑ i, f i • x i = 0 := by |
simpa using
congr_arg (TensorProduct.lid R M) <|
sum_tmul_eq_zero_of_vanishesTrivially R (isTrivialRelation_iff_vanishesTrivially.mp h)
| 3 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S ℕ]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 69 | 74 | theorem eval₂_eq_smeval (R : Type*) [Semiring R] {S : Type*} [Semiring S] (f : R →+* S) (p : R[X])
(x: S) : letI : Module R S := RingHom.toModule f
p.eval₂ f x = p.smeval x := by |
letI : Module R S := RingHom.toModule f
rw [smeval_eq_sum, eval₂_eq_sum]
rfl
| 3 |
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Invertible.Defs
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Logic.Equiv.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
assert_not_exists MonoidWithZero
assert_not_ex... | Mathlib/Algebra/Group/Invertible/Basic.lean | 77 | 82 | theorem Commute.invOf_left [Monoid α] {a b : α} [Invertible b] (h : Commute b a) :
Commute (⅟ b) a :=
calc
⅟ b * a = ⅟ b * (a * b * ⅟ b) := by | simp [mul_assoc]
_ = ⅟ b * (b * a * ⅟ b) := by rw [h.eq]
_ = a * ⅟ b := by simp [mul_assoc]
| 3 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.log.monotone from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {x y : ℝ}
theorem log_mul_self_monotoneOn... | Mathlib/Analysis/SpecialFunctions/Log/Monotone.lean | 85 | 88 | theorem log_div_sqrt_antitoneOn : AntitoneOn (fun x : ℝ => log x / √x) { x | exp 2 ≤ x } := by |
simp_rw [sqrt_eq_rpow]
convert @log_div_self_rpow_antitoneOn (1 / 2) (by norm_num)
norm_num
| 3 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 113 | 116 | theorem charZero_of_expChar_one' [hq : ExpChar R 1] : CharZero R := by |
cases hq
· assumption
· exact False.elim (CharP.char_ne_one R 1 rfl)
| 3 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 95 | 99 | theorem coeff_opRingEquiv (p : R[X]ᵐᵒᵖ) (n : ℕ) :
(opRingEquiv R p).coeff n = op ((unop p).coeff n) := by |
induction' p using MulOpposite.rec' with p
cases p
rfl
| 3 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.PNat.Basic
import Mathlib.GroupTheory.GroupAction.Prod
variable {M : Type*}
class PNatPowAssoc (M : Type*) [Mul M] [Pow M ℕ+] : Prop where
protected ppow_add : ∀ (k n : ℕ+) (x : M), x ^ (k + n) = x ^ k * x ^ n
prote... | Mathlib/Algebra/Group/PNatPowAssoc.lean | 67 | 70 | theorem ppow_mul (x : M) (m n : ℕ+) : x ^ (m * n) = (x ^ m) ^ n := by |
refine PNat.recOn n ?_ fun k hk ↦ ?_
· rw [ppow_one, mul_one]
· rw [ppow_add, ppow_one, mul_add, ppow_add, mul_one, hk]
| 3 |
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Data.ZMod.Algebra
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
import Mathlib.FieldTheory.Perfect
#align_import ring_theory.witt_vector.frobenius from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb3304bef77472"... | Mathlib/RingTheory/WittVector/Frobenius.lean | 123 | 127 | theorem map_frobeniusPoly.key₁ (n j : ℕ) (hj : j < p ^ n) :
p ^ (n - v p ⟨j + 1, j.succ_pos⟩) ∣ (p ^ n).choose (j + 1) := by |
apply multiplicity.pow_dvd_of_le_multiplicity
rw [hp.out.multiplicity_choose_prime_pow hj j.succ_ne_zero]
rfl
| 3 |
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.CategoryTheory.Monoidal.Linear
#align_import algebra.category.Module.monoidal.basic from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
-- Porting note: Module
set_option linte... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean | 81 | 85 | theorem tensor_comp {X₁ Y₁ Z₁ X₂ Y₂ Z₂ : ModuleCat R} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : Y₁ ⟶ Z₁)
(g₂ : Y₂ ⟶ Z₂) : tensorHom (f₁ ≫ g₁) (f₂ ≫ g₂) = tensorHom f₁ f₂ ≫ tensorHom g₁ g₂ := by |
-- Porting note: even with high priority ext fails to find this
apply TensorProduct.ext
rfl
| 3 |
import Mathlib.Algebra.AddTorsor
import Mathlib.Topology.Algebra.Constructions
import Mathlib.GroupTheory.GroupAction.SubMulAction
import Mathlib.Topology.Algebra.ConstMulAction
#align_import topology.algebra.mul_action from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
open Topology Po... | Mathlib/Topology/Algebra/MulAction.lean | 276 | 280 | theorem continuousSMul_inf {t₁ t₂ : TopologicalSpace X} [@ContinuousSMul M X _ _ t₁]
[@ContinuousSMul M X _ _ t₂] : @ContinuousSMul M X _ _ (t₁ ⊓ t₂) := by |
rw [inf_eq_iInf]
refine continuousSMul_iInf fun b => ?_
cases b <;> assumption
| 3 |
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Directed
import Mathlib.Order.Hom.Set
#align_import order.antichain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open Function Set
section General
variable {α β : Type*} {r r₁ r₂ : α →... | Mathlib/Order/Antichain.lean | 313 | 317 | theorem image (hs : IsStrongAntichain r s) {f : α → β} (hf : Surjective f)
(h : ∀ a b, r' (f a) (f b) → r a b) : IsStrongAntichain r' (f '' s) := by |
rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ hab c
obtain ⟨c, rfl⟩ := hf c
exact (hs ha hb (ne_of_apply_ne _ hab) _).imp (mt <| h _ _) (mt <| h _ _)
| 3 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
open Set Filter MeasureTheory MeasurableSpace
open scoped Classical Topology NNReal ENNReal MeasureTheory
univers... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 96 | 99 | theorem isPiSystem_Ioi_rat : IsPiSystem (⋃ a : ℚ, {Ioi (a : ℝ)}) := by |
convert isPiSystem_image_Ioi (((↑) : ℚ → ℝ) '' univ)
ext x
simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and]
| 3 |
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padics.padic_numbers from "leanprover-community/mathlib"@"b9b2114f7711fec1c1e055d507f082f8ceb2c3b7"
noncomputable section
open scoped Classical
open Nat m... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 156 | 160 | theorem norm_eq_norm_app_of_nonzero {f : PadicSeq p} (hf : ¬f ≈ 0) :
∃ k, f.norm = padicNorm p k ∧ k ≠ 0 :=
have heq : f.norm = padicNorm p (f <| stationaryPoint hf) := by | simp [norm, hf]
⟨f <| stationaryPoint hf, heq, fun h ↦
norm_nonzero_of_not_equiv_zero hf (by simpa [h] using heq)⟩
| 3 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.ZMod.Basic
#align_import data.zmod.parity from "leanprover-community/mathlib"@"048240e809f04e2bde02482ab44bc230744cc6c9"
namespace ZMod
theorem eq_zero_iff_even {n : ℕ} : (n : ZMod 2) = 0 ↔ Even n :=
(CharP.cast_eq_zero_iff (ZMod 2) 2 n).trans even_iff_... | Mathlib/Data/ZMod/Parity.lean | 32 | 35 | theorem ne_zero_iff_odd {n : ℕ} : (n : ZMod 2) ≠ 0 ↔ Odd n := by |
constructor <;>
· contrapose
simp [eq_zero_iff_even]
| 3 |
import Mathlib.Topology.Order.Basic
open Set Filter OrderDual
open scoped Topology
section OrderClosedTopology
variable {α : Type*} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c d : α}
@[simp] theorem nhdsSet_Ioi : 𝓝ˢ (Ioi a) = 𝓟 (Ioi a) := isOpen_Ioi.nhdsSet_eq
@[simp] theorem nhdsSet... | Mathlib/Topology/Order/NhdsSet.lean | 47 | 50 | theorem nhdsSet_Icc (h : a ≤ b) : 𝓝ˢ (Icc a b) = 𝓝 a ⊔ 𝓝 b ⊔ 𝓟 (Ioo a b) := by |
rcases h.eq_or_lt with rfl | hlt
· simp
· rw [← Ioc_insert_left h, nhdsSet_insert, nhdsSet_Ioc hlt, sup_assoc]
| 3 |
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Ring.Int
#align_import algebra.order.field.power from "leanprover-community/mathlib"@"acb3d204d4ee883eb686f45d486a2a6811a01329"
... | Mathlib/Algebra/Order/Field/Power.lean | 97 | 100 | theorem zpow_injective (h₀ : 0 < a) (h₁ : a ≠ 1) : Injective (a ^ · : ℤ → α) := by |
rcases h₁.lt_or_lt with (H | H)
· exact (zpow_strictAnti h₀ H).injective
· exact (zpow_strictMono H).injective
| 3 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpace
namespace MeasureTheory
namespace Measure
variable {M : Type*} [Monoid M] [MeasurableSpace M]
@[to_additive conv "Additive convolution of measures."]
noncomputable def mconv (μ : Measure M) (ν : Measure M) :
... | Mathlib/MeasureTheory/Group/Convolution.lean | 77 | 81 | theorem add_mconv [MeasurableMul₂ M] (μ : Measure M) (ν : Measure M) (ρ : Measure M) [SFinite μ]
[SFinite ν] [SFinite ρ] : (μ + ν) ∗ ρ = μ ∗ ρ + ν ∗ ρ := by |
unfold mconv
rw [add_prod, map_add]
measurability
| 3 |
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
namespace Con... | Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean | 148 | 151 | theorem decomp (f : V →ᴬ[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by |
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
| 3 |
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