Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.57k | proof stringlengths 5 7.36k | hint bool 2
classes |
|---|---|---|---|---|---|---|
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 84 | 89 | theorem trace_eq_matrix_trace_of_finset {s : Finset M} (b : Basis s R M) (f : M ββ[R] M) :
trace R M f = Matrix.trace (LinearMap.toMatrix b b f) := by
have : β s : Finset M, Nonempty (Basis s R M) := β¨s, β¨bβ©β© |
have : β s : Finset M, Nonempty (Basis s R M) := β¨s, β¨bβ©β©
rw [trace, dif_pos this, β traceAux_def]
congr 1
apply traceAux_eq
| true |
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
#align_import group_theory.subsemigroup.center from "leanprover-community/mathlib"@"1ac8d4304efba9d03fa720d06516fac845aa5353"
variable {M : Type*}
namespace Set
variable (M)
@[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : β) :... | Mathlib/Algebra/Ring/Center.lean | 72 | 77 | theorem add_mem_center [Distrib M] {a b : M} (ha : a β Set.center M) (hb : b β Set.center M) :
a + b β Set.center M where
comm _ := by rw [add_mul, mul_add, ha.comm, hb.comm] | rw [add_mul, mul_add, ha.comm, hb.comm]
left_assoc _ _ := by rw [add_mul, ha.left_assoc, hb.left_assoc, β add_mul, β add_mul]
mid_assoc _ _ := by rw [mul_add, add_mul, ha.mid_assoc, hb.mid_assoc, β mul_add, β add_mul]
right_assoc _ _ := by rw [mul_add, ha.right_assoc, hb.right_assoc, β mul_add, β mul_add]
| true |
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community/mathlib"@"2f3994e1b117b1e1da49bcfb67334f33460c3ce4"
open Nat
namespace Nat
def choose : β β β β β
| _, 0 => 1
| 0, _ + 1 => 0
| n + 1, k + 1 => choose n k + choose n ... | Mathlib/Data/Nat/Choose/Basic.lean | 54 | 54 | theorem choose_zero_right (n : β) : choose n 0 = 1 := by | cases n <;> rfl
| true |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {Ξ±... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 56 | 57 | theorem nhds_zero : π (0 : Ξβ) = β¨
Ξ³ β 0, π (Iio Ξ³) := by |
rw [nhds_eq_update, update_same]
| true |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingleton
#align_import combinatorics.composition from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open List
variable {n : β}
... | Mathlib/Combinatorics/Enumerative/Composition.lean | 187 | 189 | theorem length_le : c.length β€ n := by
conv_rhs => rw [β c.blocks_sum] |
conv_rhs => rw [β c.blocks_sum]
exact length_le_sum_of_one_le _ fun i hi => c.one_le_blocks hi
| true |
import Mathlib.Analysis.SpecialFunctions.Bernstein
import Mathlib.Topology.Algebra.Algebra
#align_import topology.continuous_function.weierstrass from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
open ContinuousMap Filter
open scoped unitInterval
theorem polynomialFunctions_closure... | Mathlib/Topology/ContinuousFunction/Weierstrass.lean | 114 | 122 | theorem exists_polynomial_near_of_continuousOn (a b : β) (f : β β β)
(c : ContinuousOn f (Set.Icc a b)) (Ξ΅ : β) (pos : 0 < Ξ΅) :
β p : β[X], β x β Set.Icc a b, |p.eval x - f x| < Ξ΅ := by
let f' : C(Set.Icc a b, β) := β¨fun x => f x, continuousOn_iff_continuous_restrict.mp cβ© |
let f' : C(Set.Icc a b, β) := β¨fun x => f x, continuousOn_iff_continuous_restrict.mp cβ©
obtain β¨p, bβ© := exists_polynomial_near_continuousMap a b f' Ξ΅ pos
use p
rw [norm_lt_iff _ pos] at b
intro x m
exact b β¨x, mβ©
| true |
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.Topology.VectorBundle.Constructions
#align_import geometry.manifold.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
assert_not_exists mfde... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 178 | 196 | theorem contMDiffWithinAt_totalSpace (f : M β TotalSpace F E) {s : Set M} {xβ : M} :
ContMDiffWithinAt IM (IB.prod π(π, F)) n f s xβ β
ContMDiffWithinAt IM IB n (fun x => (f x).proj) s xβ β§
ContMDiffWithinAt IM π(π, F) n (fun x β¦ (trivializationAt F E (f xβ).proj (f x)).2) s xβ := by
simp (config ... |
simp (config := { singlePass := true }) only [contMDiffWithinAt_iff_target]
rw [and_and_and_comm, β FiberBundle.continuousWithinAt_totalSpace, and_congr_right_iff]
intro hf
simp_rw [modelWithCornersSelf_prod, FiberBundle.extChartAt, Function.comp,
PartialEquiv.trans_apply, PartialEquiv.prod_coe, PartialEqu... | true |
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Group.Pointwise
#align_import measure_theory.group.fundamental_domain from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
open scoped ENNReal Pointwise Topology NNRea... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 595 | 597 | theorem mem_fundamentalInterior :
x β fundamentalInterior G s β x β s β§ β g : G, g β 1 β x β g β’ s := by |
simp [fundamentalInterior]
| true |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.between from "leanprover-community/mathlib"@"571e13cacbed7bf042fd3058c... | Mathlib/Analysis/Convex/Between.lean | 49 | 55 | theorem affineSegment_comm (x y : P) : affineSegment R x y = affineSegment R y x := by
refine Set.ext fun z => ?_ |
refine Set.ext fun z => ?_
constructor <;>
Β· rintro β¨t, ht, hxyβ©
refine β¨1 - t, ?_, ?_β©
Β· rwa [Set.sub_mem_Icc_iff_right, sub_self, sub_zero]
Β· rwa [lineMap_apply_one_sub]
| true |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Constructions.Prod.Integral
open Fintype MeasureTheory MeasureTheory.Measure
variable {π : Type*} [RCLike π]
namespace MeasureTheory
theorem Integrable.fin_nat_prod {n : β} {E : Fin n β Type*}
[β i, MeasureSpace (E i)] [β i, SigmaF... | Mathlib/MeasureTheory/Integral/Pi.lean | 65 | 84 | theorem integral_fin_nat_prod_eq_prod {n : β} {E : Fin n β Type*}
[β i, MeasureSpace (E i)] [β i, SigmaFinite (volume : Measure (E i))]
(f : (i : Fin n) β E i β π) :
β« x : (i : Fin n) β E i, β i, f i (x i) = β i, β« x, f i x := by
induction n with |
induction n with
| zero =>
simp only [Nat.zero_eq, volume_pi, Finset.univ_eq_empty, Finset.prod_empty, integral_const,
pi_empty_univ, ENNReal.one_toReal, smul_eq_mul, mul_one, pow_zero, one_smul]
| succ n n_ih =>
calc
_ = β« x : E 0 Γ ((i : Fin n) β E (Fin.succ i)),
f 0 x.1... | true |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 75 | 80 | theorem lintegral_edist_triangle {f g h : Ξ± β Ξ²} (hf : AEStronglyMeasurable f ΞΌ)
(hh : AEStronglyMeasurable h ΞΌ) :
(β«β» a, edist (f a) (g a) βΞΌ) β€ (β«β» a, edist (f a) (h a) βΞΌ) + β«β» a, edist (g a) (h a) βΞΌ := by
rw [β lintegral_add_left' (hf.edist hh)] |
rw [β lintegral_add_left' (hf.edist hh)]
refine lintegral_mono fun a => ?_
apply edist_triangle_right
| true |
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 | 86 | 89 | theorem coeff_hermite_succ_succ (n k : β) : coeff (hermite (n + 1)) (k + 1) =
coeff (hermite n) k - (k + 2) * coeff (hermite n) (k + 2) := by
rw [hermite_succ, coeff_sub, coeff_X_mul, coeff_derivative, mul_comm] |
rw [hermite_succ, coeff_sub, coeff_X_mul, coeff_derivative, mul_comm]
norm_cast
| true |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 93 | 94 | theorem T_sub_two (n : β€) : T R (n - 2) = 2 * X * T R (n - 1) - T R n := by |
linear_combination (norm := ring_nf) T_add_two R (n - 2)
| true |
import Mathlib.Analysis.Calculus.FDeriv.Measurable
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Integral.DominatedConve... | Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 1,024 | 1,114 | theorem sub_le_integral_of_hasDeriv_right_of_le_Ico (hab : a β€ b)
(hcont : ContinuousOn g (Icc a b)) (hderiv : β x β Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x)
(Οint : IntegrableOn Ο (Icc a b)) (hΟg : β x β Ico a b, g' x β€ Ο x) :
g b - g a β€ β« y in a..b, Ο y := by
refine le_of_forall_pos_le_add fun Ξ΅ Ξ΅... |
refine le_of_forall_pos_le_add fun Ξ΅ Ξ΅pos => ?_
-- Bound from above `g'` by a lower-semicontinuous function `G'`.
rcases exists_lt_lowerSemicontinuous_integral_lt Ο Οint Ξ΅pos with
β¨G', f_lt_G', G'cont, G'int, G'lt_top, hG'β©
-- we will show by "induction" that `g t - g a β€ β« u in a..t, G' u` for all `t β [a... | true |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 130 | 138 | theorem zero_memβp : Memβp (0 : β i, E i) p := by
rcases p.trichotomy with (rfl | rfl | hp) |
rcases p.trichotomy with (rfl | rfl | hp)
Β· apply memβp_zero
simp
Β· apply memβp_infty
simp only [norm_zero, Pi.zero_apply]
exact bddAbove_singleton.mono Set.range_const_subset
Β· apply memβp_gen
simp [Real.zero_rpow hp.ne', summable_zero]
| true |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Contraction
import Mathlib.RingTheory.TensorProduct.Basic
#align_import representation_... | Mathlib/RepresentationTheory/Basic.lean | 106 | 107 | theorem asAlgebraHom_single (g : G) (r : k) : asAlgebraHom Ο (Finsupp.single g r) = r β’ Ο g := by |
simp only [asAlgebraHom_def, MonoidAlgebra.lift_single]
| true |
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
#align_import number_theory.number_field.norm from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
open scoped NumberField
open Finset NumberField Algebra FiniteDimensional
namespace RingOfIn... | Mathlib/NumberTheory/NumberField/Norm.lean | 65 | 69 | theorem norm_algebraMap [IsSeparable K L] (x : π K) :
norm K (algebraMap (π K) (π L) x) = x ^ finrank K L := by
rw [RingOfIntegers.ext_iff, RingOfIntegers.coe_eq_algebraMap, |
rw [RingOfIntegers.ext_iff, RingOfIntegers.coe_eq_algebraMap,
RingOfIntegers.algebraMap_norm_algebraMap, Algebra.norm_algebraMap,
RingOfIntegers.coe_eq_algebraMap, map_pow]
| true |
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ΞΉ : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 78 | 83 | theorem IsLindelof.inter_right (hs : IsLindelof s) (ht : IsClosed t) : IsLindelof (s β© t) := by
intro f hnf _ hstf |
intro f hnf _ hstf
rw [β inf_principal, le_inf_iff] at hstf
obtain β¨x, hsx, hxβ© : β x β s, ClusterPt x f := hs hstf.1
have hxt : x β t := ht.mem_of_nhdsWithin_neBot <| hx.mono hstf.2
exact β¨x, β¨hsx, hxtβ©, hxβ©
| true |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
open Topology InnerProductSpace Set
noncomputable section
variable {π F : Type*} [RCLike π]
variable [NormedAddCommGroup F] [InnerProductSpace π F] [CompleteSpace F]
variabl... | Mathlib/Analysis/Calculus/Gradient/Basic.lean | 118 | 121 | theorem DifferentiableAt.hasGradientAt (h : DifferentiableAt π f x) :
HasGradientAt f (β f x) x := by
rw [hasGradientAt_iff_hasFDerivAt, gradient, (toDual π F).apply_symm_apply (fderiv π f x)] |
rw [hasGradientAt_iff_hasFDerivAt, gradient, (toDual π F).apply_symm_apply (fderiv π f x)]
exact h.hasFDerivAt
| true |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {Ξ± I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 281 | 289 | theorem pairwise_isRelPrime_iff_isRelPrime_prod [DecidableEq I] :
Pairwise (IsRelPrime on fun i : t β¦ s i) β β i β t, IsRelPrime (s i) (β j β t \ {i}, s j) := by
refine β¨fun hp i hi β¦ IsRelPrime.prod_right_iff.mpr fun j hj β¦ ?_, fun hp β¦ ?_β© |
refine β¨fun hp i hi β¦ IsRelPrime.prod_right_iff.mpr fun j hj β¦ ?_, fun hp β¦ ?_β©
Β· rw [Finset.mem_sdiff, Finset.mem_singleton] at hj
obtain β¨hj, jiβ© := hj
exact @hp β¨i, hiβ© β¨j, hjβ© fun h β¦ ji (congrArg Subtype.val h).symm
Β· rintro β¨i, hiβ© β¨j, hjβ© h
apply IsRelPrime.prod_right_iff.mp (hp i hi)
exac... | true |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 66 | 72 | theorem mirror_natDegree : p.mirror.natDegree = p.natDegree := by
by_cases hp : p = 0 |
by_cases hp : p = 0
Β· rw [hp, mirror_zero]
nontriviality R
rw [mirror, natDegree_mul', reverse_natDegree, natDegree_X_pow,
tsub_add_cancel_of_le p.natTrailingDegree_le_natDegree]
rwa [leadingCoeff_X_pow, mul_one, reverse_leadingCoeff, Ne, trailingCoeff_eq_zero]
| true |
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.ApplyFun
import Mathlib.Control.Fix
import Mathlib.Order.OmegaCompletePartialOrder
#align_import control.lawful_fix from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
universe u v
open scoped Classical
variable {Ξ± : Type*} {Ξ² : Ξ± β... | Mathlib/Control/LawfulFix.lean | 71 | 91 | theorem mem_iff (a : Ξ±) (b : Ξ² a) : b β Part.fix f a β β i, b β approx f i a := by
by_cases hβ : β i : β, (approx f i a).Dom |
by_cases hβ : β i : β, (approx f i a).Dom
Β· simp only [Part.fix_def f hβ]
constructor <;> intro hh
Β· exact β¨_, hhβ©
have hβ := Nat.find_spec hβ
rw [dom_iff_mem] at hβ
cases' hβ with y hβ
replace hβ := approx_mono' f _ _ hβ
suffices y = b by
subst this
exact hβ
cases' hh w... | true |
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.GroupTheory.SpecificGroups.Cyclic
#align_import ring_theory.integral_domain from "leanprover-community/mathlib"@"6e70e0d419bf686784937d64ed4bfde866ff229e"
section
open Finset Polynomial Function Nat
section CancelMonoidWithZero... | Mathlib/RingTheory/IntegralDomain.lean | 73 | 84 | theorem Finset.exists_eq_pow_of_mul_eq_pow_of_coprime {ΞΉ R : Type*} [CommSemiring R] [IsDomain R]
[GCDMonoid R] [Unique RΛ£] {n : β} {c : R} {s : Finset ΞΉ} {f : ΞΉ β R}
(h : β i β s, β j β s, i β j β IsCoprime (f i) (f j))
(hprod : β i β s, f i = c ^ n) : β i β s, β d : R, f i = d ^ n := by
classical |
classical
intro i hi
rw [β insert_erase hi, prod_insert (not_mem_erase i s)] at hprod
refine
exists_eq_pow_of_mul_eq_pow_of_coprime
(IsCoprime.prod_right fun j hj => h i hi j (erase_subset i s hj) fun hij => ?_) hprod
rw [hij] at hj
exact (s.not_mem_erase _) hj
| true |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {Ξ± : Type*} {m0 : MeasurableSpace Ξ±} {ΞΌ : Measure Ξ±}
noncomputable
def Measure.withDensity {m : MeasurableSpace Ξ±} (ΞΌ : Measure Ξ±) (f : Ξ± β ββ₯... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 83 | 87 | theorem withDensity_congr_ae {f g : Ξ± β ββ₯0β} (h : f =α΅[ΞΌ] g) :
ΞΌ.withDensity f = ΞΌ.withDensity g := by
refine Measure.ext fun s hs => ?_ |
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, withDensity_apply _ hs]
exact lintegral_congr_ae (ae_restrict_of_ae h)
| true |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 134 | 134 | theorem T_neg_two : T R (-2) = 2 * X ^ 2 - 1 := by | simp [T_two]
| true |
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposites
#align_import linear_algebra.clifford_algebra.conjugation from "leanprover-community/mathlib"@"34020e531ebc4e8aac6d449d9eecbcd1508ea8d0"
variable {R : Type*} [CommRing R]
variable {M : Type*} [AddCommGroup M] [Module R M]... | Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 111 | 111 | theorem reverse_ΞΉ (m : M) : reverse (ΞΉ Q m) = ΞΉ Q m := by | simp [reverse]
| true |
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.Order.Minimal
#align_import ring_theory.ideal.minimal_prime from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
section
variable {R S : Type*} [CommSemiring R] [CommSemiring S] (I J ... | Mathlib/RingTheory/Ideal/MinimalPrime.lean | 56 | 74 | theorem Ideal.exists_minimalPrimes_le [J.IsPrime] (e : I β€ J) : β p β I.minimalPrimes, p β€ J := by
suffices |
suffices
β m β { p : (Ideal R)α΅α΅ | Ideal.IsPrime p β§ I β€ OrderDual.ofDual p },
OrderDual.toDual J β€ m β§ β z β { p : (Ideal R)α΅α΅ | Ideal.IsPrime p β§ I β€ p }, m β€ z β z = m by
obtain β¨p, hβ, hβ, hββ© := this
simp_rw [β @eq_comm _ p] at hβ
exact β¨p, β¨hβ, fun a b c => le_of_eq (hβ a b c)β©, hββ©
app... | true |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {Ξ± : Type*} {Ξ² : Type v} {Ξ³ Ξ΄ : Ty... | Mathlib/Data/Multiset/Bind.lean | 163 | 163 | theorem card_bind : card (s.bind f) = (s.map (card β f)).sum := by | simp [bind]
| true |
import Mathlib.Analysis.Complex.RealDeriv
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas
#align_import analysis.special_functions.exp_deriv from "leanprover-community/mathlib"@"6a5c85000ab93fe5dcfdf620676f614ba8e18c26"
noncomputable section
open Filter Asym... | Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean | 36 | 42 | theorem hasDerivAt_exp (x : β) : HasDerivAt exp (exp x) x := by
rw [hasDerivAt_iff_isLittleO_nhds_zero] |
rw [hasDerivAt_iff_isLittleO_nhds_zero]
have : (1 : β) < 2 := by norm_num
refine (IsBigO.of_bound βexp xβ ?_).trans_isLittleO (isLittleO_pow_id this)
filter_upwards [Metric.ball_mem_nhds (0 : β) zero_lt_one]
simp only [Metric.mem_ball, dist_zero_right, norm_pow]
exact fun z hz => exp_bound_sq x z hz.le
| true |
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import group_theory.index from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace Subgroup
open Ca... | Mathlib/GroupTheory/Index.lean | 146 | 148 | theorem relindex_inf_mul_relindex : H.relindex (K β L) * K.relindex L = (H β K).relindex L := by
rw [β inf_relindex_right H (K β L), β inf_relindex_right K L, β inf_relindex_right (H β K) L, |
rw [β inf_relindex_right H (K β L), β inf_relindex_right K L, β inf_relindex_right (H β K) L,
inf_assoc, relindex_mul_relindex (H β (K β L)) (K β L) L inf_le_right inf_le_right]
| true |
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.int.gcd from "leanprover-community/mathlib"@"47a1a73351de8dd6c8d3d32b569c8e434b03ca47"
namespace Nat
... | Mathlib/Data/Int/GCD.lean | 74 | 76 | theorem gcdA_zero_left {s : β} : gcdA 0 s = 0 := by
unfold gcdA |
unfold gcdA
rw [xgcd, xgcd_zero_left]
| true |
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.nakayama from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
open Ideal
namespace Submodule
theorem eq_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {... | Mathlib/RingTheory/Nakayama.lean | 109 | 111 | theorem eq_bot_of_le_smul_of_le_jacobson_bot (I : Ideal R) (N : Submodule R M) (hN : N.FG)
(hIN : N β€ I β’ N) (hIjac : I β€ jacobson β₯) : N = β₯ := by |
rw [eq_smul_of_le_smul_of_le_jacobson hN hIN hIjac, Submodule.bot_smul]
| true |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section S... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 48 | 58 | theorem snorm'_le_snormEssSup_mul_rpow_measure_univ {q : β} (hq_pos : 0 < q) :
snorm' f q ΞΌ β€ snormEssSup f ΞΌ * ΞΌ Set.univ ^ (1 / q) := by
have h_le : (β«β» a : Ξ±, (βf aββ : ββ₯0β) ^ q βΞΌ) β€ β«β» _ : Ξ±, snormEssSup f ΞΌ ^ q βΞΌ := by |
have h_le : (β«β» a : Ξ±, (βf aββ : ββ₯0β) ^ q βΞΌ) β€ β«β» _ : Ξ±, snormEssSup f ΞΌ ^ q βΞΌ := by
refine lintegral_mono_ae ?_
have h_nnnorm_le_snorm_ess_sup := coe_nnnorm_ae_le_snormEssSup f ΞΌ
exact h_nnnorm_le_snorm_ess_sup.mono fun x hx => by gcongr
rw [snorm', β ENNReal.rpow_one (snormEssSup f ΞΌ)]
nth_rw 2 ... | true |
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.lhopital from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Filter Set
open scoped Filter Topology Pointwise
variable {a b : β} (hab : a < b) {l : Filter β} {f f... | Mathlib/Analysis/Calculus/LHopital.lean | 107 | 129 | theorem lhopital_zero_left_on_Ioo (hff' : β x β Ioo a b, HasDerivAt f (f' x) x)
(hgg' : β x β Ioo a b, HasDerivAt g (g' x) x) (hg' : β x β Ioo a b, g' x β 0)
(hfb : Tendsto f (π[<] b) (π 0)) (hgb : Tendsto g (π[<] b) (π 0))
(hdiv : Tendsto (fun x => f' x / g' x) (π[<] b) l) :
Tendsto (fun x => f x / ... |
-- Here, we essentially compose by `Neg.neg`. The following is mostly technical details.
have hdnf : β x β -Ioo a b, HasDerivAt (f β Neg.neg) (f' (-x) * -1) x := fun x hx =>
comp x (hff' (-x) hx) (hasDerivAt_neg x)
have hdng : β x β -Ioo a b, HasDerivAt (g β Neg.neg) (g' (-x) * -1) x := fun x hx =>
comp ... | true |
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.CategoryTheory.Bicategory.LocallyDiscrete
#align_import category_theory.bicategory.coherence from "leanprover-community/mathlib"@"f187f1074fa1857c94589cc653c786ca... | Mathlib/CategoryTheory/Bicategory/Coherence.lean | 188 | 193 | theorem normalizeAux_nil_comp {a b c : B} (f : Hom a b) (g : Hom b c) :
normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g) := by
induction g generalizing a with |
induction g generalizing a with
| id => rfl
| of => rfl
| comp g _ ihf ihg => erw [ihg (f.comp g), ihf f, ihg g, comp_assoc]
| true |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : β) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 99 | 110 | theorem remainder_vars (n : β) : (remainder p n).vars β univ ΓΛ’ range (n + 1) := by
rw [remainder] |
rw [remainder]
apply Subset.trans (vars_mul _ _)
refine union_subset ?_ ?_ <;>
Β· refine Subset.trans (vars_sum_subset _ _) ?_
rw [biUnion_subset]
intro x hx
rw [rename_monomial, vars_monomial, Finsupp.mapDomain_single]
Β· apply Subset.trans Finsupp.support_single_subset
simpa using mem_ran... | true |
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Complex.RemovableSingularity
#align_import analysis.complex.schwarz from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric Set Function Filter TopologicalSpace
open scoped Topology
namespace Complex
section Space... | Mathlib/Analysis/Complex/Schwarz.lean | 113 | 130 | theorem affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div [CompleteSpace E] [StrictConvexSpace β E]
(hd : DifferentiableOn β f (ball c Rβ)) (h_maps : Set.MapsTo f (ball c Rβ) (ball (f c) Rβ))
(h_zβ : zβ β ball c Rβ) (h_eq : βdslope f c zββ = Rβ / Rβ) :
Set.EqOn f (fun z => f c + (z - c) β’ dslope f c zβ) (b... |
set g := dslope f c
rintro z hz
by_cases h : z = c; Β· simp [h]
have h_Rβ : 0 < Rβ := nonempty_ball.mp β¨_, h_zββ©
have g_le_div : β z β ball c Rβ, βg zβ β€ Rβ / Rβ := fun z hz =>
norm_dslope_le_div_of_mapsTo_ball hd h_maps hz
have g_max : IsMaxOn (norm β g) (ball c Rβ) zβ :=
isMaxOn_iff.mpr fun z hz =... | true |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Order.SupIndep
import Mathlib.Order.Atoms
#align_import order.partition.finpartition from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Finset Function
variable {Ξ± : Type*}
@[ext]
structure Finpartition [Lattice Ξ±]... | Mathlib/Order/Partition/Finpartition.lean | 191 | 196 | theorem parts_eq_empty_iff : P.parts = β
β a = β₯ := by
simp_rw [β P.sup_parts] |
simp_rw [β P.sup_parts]
refine β¨fun h β¦ ?_, fun h β¦ eq_empty_iff_forall_not_mem.2 fun b hb β¦ P.not_bot_mem ?_β©
Β· rw [h]
exact Finset.sup_empty
Β· rwa [β le_bot_iff.1 ((le_sup hb).trans h.le)]
| true |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Finsupp.Multiset
#align_import data.nat.choose.multinomial from "leanprover-community/mathlib"@"2738d2ca56cbc... | Mathlib/Data/Nat/Choose/Multinomial.lean | 123 | 131 | theorem binomial_succ_succ [DecidableEq Ξ±] (h : a β b) :
multinomial {a, b} (Function.update (Function.update f a (f a).succ) b (f b).succ) =
multinomial {a, b} (Function.update f a (f a).succ) +
multinomial {a, b} (Function.update f b (f b).succ) := by
simp only [binomial_eq_choose, Function.update_a... |
simp only [binomial_eq_choose, Function.update_apply,
h, Ne, ite_true, ite_false, not_false_eq_true]
rw [if_neg h.symm]
rw [add_succ, choose_succ_succ, succ_add_eq_add_succ]
ring
| true |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ββ₯0β} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 206 | 206 | theorem mul_top' : a * β = if a = 0 then 0 else β := by | convert WithTop.mul_top' a
| true |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Order.Partition.Equipartition
import Mathlib.SetTheory.Ordinal.Basic
#align_import combinatorics.simple_graph.regularity.uniform from "leanprover-community/mathlib"@"bf7ef0e83e5b... | Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | 142 | 145 | theorem left_nonuniformWitnesses_card (h : Β¬G.IsUniform Ξ΅ s t) :
(s.card : π) * Ξ΅ β€ (G.nonuniformWitnesses Ξ΅ s t).1.card := by
rw [nonuniformWitnesses, dif_pos h] |
rw [nonuniformWitnesses, dif_pos h]
exact (not_isUniform_iff.1 h).choose_spec.2.choose_spec.2.1
| true |
import Mathlib.Topology.FiberBundle.Trivialization
import Mathlib.Topology.Order.LeftRightNhds
#align_import topology.fiber_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
variable {ΞΉ B F X : Type*} [TopologicalSpace X]
open TopologicalSpace Filter Set Bundle Topology
... | Mathlib/Topology/FiberBundle/Basic.lean | 474 | 477 | theorem mem_trivChange_source (i j : ΞΉ) (p : B Γ F) :
p β (Z.trivChange i j).source β p.1 β Z.baseSet i β© Z.baseSet j := by
erw [mem_prod] |
erw [mem_prod]
simp
| true |
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 | 601 | 606 | theorem oangle_right_eq_arcsin_of_oangle_eq_pi_div_two {pβ pβ pβ : P} (h : β‘ pβ pβ pβ = β(Ο / 2)) :
β‘ pβ pβ pβ = Real.arcsin (dist pβ pβ / dist pβ pβ) := by
have hs : (β‘ pβ pβ pβ).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two] |
have hs : (β‘ pβ pβ pβ).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs,
angle_eq_arcsin_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(Or.inl (left_ne_of_oangle_eq_pi_div_two h))]
| true |
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vandermonde
import Mathlib.Tactic.FieldSimp
#align_import data.polynomial.hasse_deriv from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 111 | 124 | theorem hasseDeriv_monomial (n : β) (r : R) :
hasseDeriv k (monomial n r) = monomial (n - k) (β(n.choose k) * r) := by
ext i |
ext i
simp only [hasseDeriv_coeff, coeff_monomial]
by_cases hnik : n = i + k
Β· rw [if_pos hnik, if_pos, β hnik]
apply tsub_eq_of_eq_add_rev
rwa [add_comm]
Β· rw [if_neg hnik, mul_zero]
by_cases hkn : k β€ n
Β· rw [β tsub_eq_iff_eq_add_of_le hkn] at hnik
rw [if_neg hnik]
Β· push_neg at h... | true |
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Util.AssertExists
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
namespace List
universe u v
variable {Ξ± : Type u} {Ξ² : Type v} (l :... | Mathlib/Data/List/GetD.lean | 73 | 73 | theorem getD_singleton_default_eq (n : β) : [d].getD n d = d := by | cases n <;> simp
| true |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 303 | 304 | theorem map_subβ (f : M βSL[Οββ] F βSL[Οββ] G') (x x' : M) (y : F) :
f (x - x') y = f x y - f x' y := by | rw [f.map_sub, sub_apply]
| true |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.nat.prime from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82... | Mathlib/Data/Nat/Prime.lean | 89 | 96 | theorem Prime.eq_one_or_self_of_dvd {p : β} (pp : p.Prime) (m : β) (hm : m β£ p) :
m = 1 β¨ m = p := by
obtain β¨n, hnβ© := hm |
obtain β¨n, hnβ© := hm
have := pp.isUnit_or_isUnit hn
rw [Nat.isUnit_iff, Nat.isUnit_iff] at this
apply Or.imp_right _ this
rintro rfl
rw [hn, mul_one]
| true |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {Ξ± Ξ² Ξ³ : Type*}
section ConditionallyCompleteLinearOrder
variable [ConditionallyCompleteLinearOrder Ξ±] [TopologicalSpace Ξ±] [OrderTopology Ξ±]
[ConditionallyCompleteLinearOrder Ξ²] [Top... | Mathlib/Topology/Order/Monotone.lean | 221 | 225 | theorem Monotone.map_csSup_of_continuousAt {f : Ξ± β Ξ²} {s : Set Ξ±} (Cf : ContinuousAt f (sSup s))
(Mf : Monotone f) (ne : s.Nonempty) (H : BddAbove s) : f (sSup s) = sSup (f '' s) := by
refine ((isLUB_csSup (ne.image f) (Mf.map_bddAbove H)).unique ?_).symm |
refine ((isLUB_csSup (ne.image f) (Mf.map_bddAbove H)).unique ?_).symm
refine (isLUB_csSup ne H).isLUB_of_tendsto (fun x _ y _ xy => Mf xy) ne ?_
exact Cf.mono_left inf_le_left
| true |
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 51 | 54 | theorem NNReal.tendsto_inverse_atTop_nhds_zero_nat :
Tendsto (fun n : β β¦ (n : ββ₯0)β»ΒΉ) atTop (π 0) := by
rw [β NNReal.tendsto_coe] |
rw [β NNReal.tendsto_coe]
exact _root_.tendsto_inverse_atTop_nhds_zero_nat
| true |
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.pid from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"
variable {R : Type*} [CommRing R]
open Ideal
open UniqueFactorizationMonoid
open scoped nonZer... | Mathlib/RingTheory/DedekindDomain/PID.lean | 78 | 102 | theorem FractionalIdeal.isPrincipal_of_unit_of_comap_mul_span_singleton_eq_top {R A : Type*}
[CommRing R] [CommRing A] [Algebra R A] {S : Submonoid R} [IsLocalization S A]
(I : (FractionalIdeal S A)Λ£) {v : A} (hv : v β (βIβ»ΒΉ : FractionalIdeal S A))
(h : Submodule.comap (Algebra.linearMap R A) ((I : Submodul... |
have hinv := I.mul_inv
set J := Submodule.comap (Algebra.linearMap R A) ((I : Submodule R A) * Submodule.span R {v})
have hJ : IsLocalization.coeSubmodule A J = βI * Submodule.span R {v} := by
-- Porting note: had to insert `val_eq_coe` into this rewrite.
-- Arguably this is because `Subtype.ext_iff` is ... | true |
import Mathlib.MeasureTheory.Measure.Typeclasses
open scoped ENNReal
namespace MeasureTheory
variable {Ξ± : Type*}
noncomputable
def Measure.trim {m m0 : MeasurableSpace Ξ±} (ΞΌ : @Measure Ξ± m0) (hm : m β€ m0) : @Measure Ξ± m :=
@OuterMeasure.toMeasure Ξ± m ΞΌ.toOuterMeasure (hm.trans (le_toOuterMeasure_caratheodory... | Mathlib/MeasureTheory/Measure/Trim.lean | 37 | 38 | theorem trim_eq_self [MeasurableSpace Ξ±] {ΞΌ : Measure Ξ±} : ΞΌ.trim le_rfl = ΞΌ := by |
simp [Measure.trim]
| true |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.arsinh from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Function Filter Set
open scoped Topology
name... | Mathlib/Analysis/SpecialFunctions/Arsinh.lean | 83 | 84 | theorem cosh_arsinh (x : β) : cosh (arsinh x) = β(1 + x ^ 2) := by |
rw [β sqrt_sq (cosh_pos _).le, cosh_sq', sinh_arsinh]
| true |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 134 | 138 | theorem EuclideanSpace.ball_zero_eq {n : Type*} [Fintype n] (r : β) (hr : 0 β€ r) :
Metric.ball (0 : EuclideanSpace β n) r = {x | β i, x i ^ 2 < r ^ 2} := by
ext x |
ext x
have : (0 : β) β€ β i, x i ^ 2 := Finset.sum_nonneg fun _ _ => sq_nonneg _
simp_rw [mem_setOf, mem_ball_zero_iff, norm_eq, norm_eq_abs, sq_abs, sqrt_lt this hr]
| true |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9a... | Mathlib/Logic/Relation.lean | 192 | 196 | theorem _root_.Acc.of_fibration (fib : Fibration rΞ± rΞ² f) {a} (ha : Acc rΞ± a) : Acc rΞ² (f a) := by
induction' ha with a _ ih |
induction' ha with a _ ih
refine Acc.intro (f a) fun b hr β¦ ?_
obtain β¨a', hr', rflβ© := fib hr
exact ih a' hr'
| true |
import Mathlib.Topology.Order.LeftRightNhds
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {Ξ± Ξ² Ξ³ : Type*}
section OrderTopology
variable [TopologicalSpace Ξ±] [TopologicalSpace Ξ²] [LinearOrder Ξ±] [LinearOrder Ξ²] [OrderTopology Ξ±]
[OrderTopology Ξ²]
| Mathlib/Topology/Order/IsLUB.lean | 24 | 32 | theorem IsLUB.frequently_mem {a : Ξ±} {s : Set Ξ±} (ha : IsLUB s a) (hs : s.Nonempty) :
βαΆ x in π[β€] a, x β s := by
rcases hs with β¨a', ha'β© |
rcases hs with β¨a', ha'β©
intro h
rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)
Β· exact h.self_of_nhdsWithin le_rfl ha'
Β· rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with β¨b, hba, hbβ©
rcases ha.exists_between hba with β¨b', hb's, hb'β©
exact hb hb' hb's
| true |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.pullbacks from "leanprover-community/mathlib"@"f11e306adb9f2a393539d2bb4293bf1b42caa7ac"
noncomputable section
universe vβ vβ uβ uβ
-- Porting note: ne... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | 245 | 247 | theorem PreservesPushout.inr_iso_inv :
G.map pushout.inr β« (PreservesPushout.iso G f g).inv = pushout.inr := by |
simp [PreservesPushout.iso, Iso.comp_inv_eq]
| true |
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Finset.Option
#align_import data.fintype.option from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {Ξ± Ξ² Ξ³ : Type*}
open Fin... | Mathlib/Data/Fintype/Option.lean | 94 | 106 | theorem induction_empty_option {P : β (Ξ± : Type u) [Fintype Ξ±], Prop}
(of_equiv : β (Ξ± Ξ²) [Fintype Ξ²] (e : Ξ± β Ξ²), @P Ξ± (@Fintype.ofEquiv Ξ± Ξ² βΉ_βΊ e.symm) β @P Ξ² βΉ_βΊ)
(h_empty : P PEmpty) (h_option : β (Ξ±) [Fintype Ξ±], P Ξ± β P (Option Ξ±)) (Ξ± : Type u)
[h_fintype : Fintype Ξ±] : P Ξ± := by
obtain β¨pβ© := |
obtain β¨pβ© :=
let f_empty := fun i => by convert h_empty
let h_option : β {Ξ± : Type u} [Fintype Ξ±] [DecidableEq Ξ±],
(β (h : Fintype Ξ±), P Ξ±) β β (h : Fintype (Option Ξ±)), P (Option Ξ±) := by
rintro Ξ± hΞ± - PΞ± hΞ±'
convert h_option Ξ± (PΞ± _)
@truncRecEmptyOption (fun Ξ± => β h, @P Ξ± h) (... | true |
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.quotient_group from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
open scope... | Mathlib/GroupTheory/QuotientGroup.lean | 108 | 113 | theorem sound (U : Set (G β§Έ N)) (g : N.op) :
g β’ (mk' N) β»ΒΉ' U = (mk' N) β»ΒΉ' U := by
ext x |
ext x
simp only [Set.mem_preimage, Set.mem_smul_set_iff_inv_smul_mem]
congr! 1
exact Quotient.sound β¨gβ»ΒΉ, rflβ©
| true |
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
open Monoid Coprod Multiplicative Subgroup Function
def HNNExtension.con (G : Type*) [Group G] (A B : Subgroup G) (Ο : A β* B) :
Con (G β Multiplicative β€) :=
conGen (fun x y => β (a : A),
x = inr (ofAdd 1) * inl (a : G) β§
... | Mathlib/GroupTheory/HNNExtension.lean | 164 | 170 | theorem toSubgroupEquiv_neg_apply (u : β€Λ£) (a : toSubgroup A B u) :
(toSubgroupEquiv Ο (-u) (toSubgroupEquiv Ο u a) : G) = a := by
rcases Int.units_eq_one_or u with rfl | rfl |
rcases Int.units_eq_one_or u with rfl | rfl
Β· -- This used to be `simp` before leanprover/lean4#2644
simp; erw [MulEquiv.symm_apply_apply]
Β· simp only [toSubgroup_neg_one, toSubgroupEquiv_neg_one, SetLike.coe_eq_coe]
exact Ο.apply_symm_apply a
| true |
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-community/mathlib"@"4e24c4bfcff371c71f7ba22050308aa17815626c"
variable {Ξ± Ξ² : Type*} [LinearOrder Ξ±]
open Function
namespace Set
def projIci (a x : Ξ±) : Ici a := β¨max a x,... | Mathlib/Order/Interval/Set/ProjIcc.lean | 116 | 116 | theorem projIic_of_mem (hx : x β Iic b) : projIic b x = β¨x, hxβ© := by | simpa [projIic]
| true |
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
#align_import measure_theory.function.simple_func_dense from "leanprover-community/mathlib"@"7317149f12f55affbc900fc873d0d422485122b9"
open Set Function Filter TopologicalSpace ENNReal EMetric Finset
... | Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | 102 | 113 | theorem edist_nearestPt_le (e : β β Ξ±) (x : Ξ±) {k N : β} (hk : k β€ N) :
edist (nearestPt e N x) x β€ edist (e k) x := by
induction' N with N ihN generalizing k |
induction' N with N ihN generalizing k
Β· simp [nonpos_iff_eq_zero.1 hk, le_refl]
Β· simp only [nearestPt, nearestPtInd_succ, map_apply]
split_ifs with h
Β· rcases hk.eq_or_lt with (rfl | hk)
exacts [le_rfl, (h k (Nat.lt_succ_iff.1 hk)).le]
Β· push_neg at h
rcases h with β¨l, hlN, hxlβ©
r... | true |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : β} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 124 | 134 | theorem norm_sub_modPart (h : β(r : β_[p])β β€ 1) : β(β¨r, hβ© - modPart p r : β€_[p])β < 1 := by
let n := modPart p r |
let n := modPart p r
rw [norm_lt_one_iff_dvd, β (isUnit_den r h).dvd_mul_right]
suffices βp β£ r.num - n * r.den by
convert (Int.castRingHom β€_[p]).map_dvd this
simp only [sub_mul, Int.cast_natCast, eq_intCast, Int.cast_mul, sub_left_inj, Int.cast_sub]
apply Subtype.coe_injective
simp only [coe_mu... | true |
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.calculus.fderiv.basic from "leanprover-community/mathlib"@"41bef4ae1254365bc190aee63b947674d2977f01"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open To... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 225 | 228 | theorem fderivWithin_zero_of_not_differentiableWithinAt (h : Β¬DifferentiableWithinAt π f s x) :
fderivWithin π f s x = 0 := by
have : Β¬β f', HasFDerivWithinAt f f' s x := h |
have : Β¬β f', HasFDerivWithinAt f f' s x := h
simp [fderivWithin, this]
| true |
import Mathlib.Algebra.Module.Submodule.Lattice
import Mathlib.Data.ZMod.Basic
import Mathlib.Order.OmegaCompletePartialOrder
variable {n : β} {M Mβ : Type*}
abbrev AddCommMonoid.zmodModule [NeZero n] [AddCommMonoid M] (h : β (x : M), n β’ x = 0) :
Module (ZMod n) M := by
have h_mod (c : β) (x : M) : (c % n)... | Mathlib/Data/ZMod/Module.lean | 54 | 56 | theorem smul_mem (hx : x β K) (c : ZMod n) : c β’ x β K := by
rw [β ZMod.intCast_zmod_cast c, β zsmul_eq_smul_cast] |
rw [β ZMod.intCast_zmod_cast c, β zsmul_eq_smul_cast]
exact zsmul_mem hx (cast c)
| true |
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
#align_import measure_theory.measure.open_pos from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal MeasureTheory
open Set Function Filter
namespace Measur... | Mathlib/MeasureTheory/Measure/OpenPos.lean | 57 | 59 | theorem _root_.IsOpen.measure_eq_zero_iff (hU : IsOpen U) : ΞΌ U = 0 β U = β
:= by
simpa only [not_lt, nonpos_iff_eq_zero, not_nonempty_iff_eq_empty] using |
simpa only [not_lt, nonpos_iff_eq_zero, not_nonempty_iff_eq_empty] using
not_congr (hU.measure_pos_iff ΞΌ)
| true |
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.deriv.zpow from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter
open Filter Asymptotics Set
variable {π : Typ... | Mathlib/Analysis/Calculus/Deriv/ZPow.lean | 86 | 92 | theorem deriv_zpow (m : β€) (x : π) : deriv (fun x => x ^ m) x = m * x ^ (m - 1) := by
by_cases H : x β 0 β¨ 0 β€ m |
by_cases H : x β 0 β¨ 0 β€ m
Β· exact (hasDerivAt_zpow m x H).deriv
Β· rw [deriv_zero_of_not_differentiableAt (mt differentiableAt_zpow.1 H)]
push_neg at H
rcases H with β¨rfl, hmβ©
rw [zero_zpow _ ((sub_one_lt _).trans hm).ne, mul_zero]
| true |
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
inductive Heap (Ξ± : Type u) where
| nil : Heap Ξ±
| node (a : Ξ±) (child sibling : Heap Ξ±) : Heap Ξ±
deriving Repr
def Heap.size : Heap Ξ± β Nat
| .nil => 0
| .node _ c s => c.size + 1 + s.size
def Heap.singleton (a : Ξ±) : Heap Ξ± := .... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 90 | 93 | theorem Heap.noSibling_merge (le) (sβ sβ : Heap Ξ±) :
(sβ.merge le sβ).NoSibling := by
unfold merge |
unfold merge
(split <;> try split) <;> constructor
| true |
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {Ξ± : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 57 | 60 | theorem lt_of_tsub_lt_tsub_right_of_le (h : c β€ b) (h2 : a - c < b - c) : a < b := by
refine ((tsub_le_tsub_iff_right h).mp h2.le).lt_of_ne ?_ |
refine ((tsub_le_tsub_iff_right h).mp h2.le).lt_of_ne ?_
rintro rfl
exact h2.false
| true |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.NumberTheory.Bernoulli
#align_import number_theory.bernoulli_polynomials from "leanprover-community/mathlib"@"ca3d21f7f4fd613c2a3c54ac7871163e1e5ecb3a"
noncomputable section... | Mathlib/NumberTheory/BernoulliPolynomials.lean | 76 | 82 | theorem bernoulli_eval_zero (n : β) : (bernoulli n).eval 0 = _root_.bernoulli n := by
rw [bernoulli, eval_finset_sum, sum_range_succ] |
rw [bernoulli, eval_finset_sum, sum_range_succ]
have : β x β range n, _root_.bernoulli x * n.choose x * 0 ^ (n - x) = 0 := by
apply sum_eq_zero fun x hx => _
intros x hx
simp [tsub_eq_zero_iff_le, mem_range.1 hx]
simp [this]
| true |
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 {Ξ± Ξ² : Type*} [Finite Ξ±]
noncomputable def toCompl {p q : Ξ± β Prop} (e ... | Mathlib/Logic/Equiv/Fintype.lean | 132 | 135 | theorem extendSubtype_mem (e : { x // p x } β { x // q x }) (x) (hx : p x) :
q (e.extendSubtype x) := by
convert (e β¨x, hxβ©).2 |
convert (e β¨x, hxβ©).2
rw [e.extendSubtype_apply_of_mem _ hx]
| true |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.List.Chain
#align_import data.bool.count from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
namespace List
@[simp]
theorem count_not_add_count (l : List Bool) (b : Bool) : count (!b) l + count b l = length l := by
-- Porting ... | Mathlib/Data/Bool/Count.lean | 79 | 87 | theorem count_not_le_count_add_one (hl : Chain' (Β· β Β·) l) (b : Bool) :
count (!b) l β€ count b l + 1 := by
cases' l with x l |
cases' l with x l
Β· exact zero_le _
obtain rfl | rfl : b = x β¨ b = !x := by simp only [Bool.eq_not_iff, em]
Β· rw [count_cons_of_ne b.not_ne_self, count_cons_self, hl.count_not, add_assoc]
exact add_le_add_left (Nat.mod_lt _ two_pos).le _
Β· rw [Bool.not_not, count_cons_self, count_cons_of_ne x.not_ne_self... | true |
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 {Ξ± : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 304 | 306 | theorem coe_support_eq_set_support (f : Perm Ξ±) : (f.support : Set Ξ±) = { x | f x β x } := by
ext |
ext
simp
| true |
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Finset Set
variable (π E : Type*) {ΞΉ : Type*} [OrderedRing π] [AddCommGroup E] [Mod... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 158 | 162 | theorem vertices_eq : K.vertices = β k β K.faces, (k : Set E) := by
ext x |
ext x
refine β¨fun h => mem_biUnion h <| mem_coe.2 <| mem_singleton_self x, fun h => ?_β©
obtain β¨s, hs, hxβ© := mem_iUnionβ.1 h
exact K.down_closed hs (Finset.singleton_subset_iff.2 <| mem_coe.1 hx) (singleton_ne_empty _)
| true |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Set.Card
import Mathlib.GroupTheory.Subgroup.Center
open MulAction ConjClasses
variable (G : Type*) [Group G]
theorem sum_conjClasses_card_eq_card [Fintype <| Conj... | Mathlib/GroupTheory/ClassEquation.lean | 72 | 81 | theorem Group.card_center_add_sum_card_noncenter_eq_card (G) [Group G]
[β x : ConjClasses G, Fintype x.carrier] [Fintype G] [Fintype <| Subgroup.center G]
[Fintype <| noncenter G] : Fintype.card (Subgroup.center G) +
β x β (noncenter G).toFinset, x.carrier.toFinset.card = Fintype.card G := by
convert Group.... |
convert Group.nat_card_center_add_sum_card_noncenter_eq_card G using 2
Β· simp
Β· rw [β finsum_set_coe_eq_finsum_mem (noncenter G), finsum_eq_sum_of_fintype,
β Finset.sum_set_coe]
simp
Β· simp
| true |
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped NNReal Matrix
namespace Matrix
variable {R l m n Ξ± Ξ² : Type*} [Fintype l] [Fintyp... | Mathlib/Analysis/Matrix.lean | 290 | 292 | theorem linfty_opNNNorm_col (v : m β Ξ±) : βcol vββ = βvββ := by
rw [linfty_opNNNorm_def, Pi.nnnorm_def] |
rw [linfty_opNNNorm_def, Pi.nnnorm_def]
simp
| true |
import Mathlib.Analysis.NormedSpace.Multilinear.Curry
#align_import analysis.calculus.formal_multilinear_series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Set Fin Topology
-- Porting note: added explicit universes to fix compile
universe u u' v w x
... | Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean | 111 | 114 | theorem removeZero_of_pos (p : FormalMultilinearSeries π E F) {n : β} (h : 0 < n) :
p.removeZero n = p n := by
rw [β Nat.succ_pred_eq_of_pos h] |
rw [β Nat.succ_pred_eq_of_pos h]
rfl
| true |
import Batteries.Data.UInt
@[ext] theorem Char.ext : {a b : Char} β a.val = b.val β a = b
| β¨_,_β©, β¨_,_β©, rfl => rfl
theorem Char.ext_iff {x y : Char} : x = y β x.val = y.val := β¨congrArg _, Char.extβ©
theorem Char.le_antisymm_iff {x y : Char} : x = y β x β€ y β§ y β€ x :=
Char.ext_iff.trans UInt32.le_antisymm_iff
... | .lake/packages/batteries/Batteries/Data/Char.lean | 33 | 34 | theorem csize_le_4 (c) : csize c β€ 4 := by |
rcases csize_eq c with _|_|_|_ <;> simp_all (config := {decide := true})
| true |
import Mathlib.MeasureTheory.Measure.Restrict
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter Function MeasurableSpace ENNReal
variable {Ξ± Ξ² Ξ΄ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] {ΞΌ Ξ½ Ξ½β Ξ½β: Measure Ξ±}
{s t : Set Ξ±}
theorem ite_ae_... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 501 | 508 | theorem ite_ae_eq_of_measure_compl_zero {Ξ³} (f : Ξ± β Ξ³) (g : Ξ± β Ξ³)
(s : Set Ξ±) [DecidablePred (Β· β s)] (hs_zero : ΞΌ sαΆ = 0) :
(fun x => ite (x β s) (f x) (g x)) =α΅[ΞΌ] f := by
rw [β mem_ae_iff] at hs_zero |
rw [β mem_ae_iff] at hs_zero
filter_upwards [hs_zero]
intros
split_ifs
rfl
| true |
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
variable {Ξ± Ξ² Ξ³ : Type*}
namespace Prod.Lex
@[inherit_doc] notation:35 Ξ± " Γβ " Ξ²:34 => Lex (Prod Ξ± Ξ²)
instance decidableEq (Ξ± Ξ² : Type*) [DecidableEq Ξ±] [DecidableEq Ξ²] ... | Mathlib/Data/Prod/Lex.lean | 122 | 126 | theorem toLex_strictMono : StrictMono (toLex : Ξ± Γ Ξ² β Ξ± Γβ Ξ²) := by
rintro β¨aβ, bββ© β¨aβ, bββ© h |
rintro β¨aβ, bββ© β¨aβ, bββ© h
obtain rfl | ha : aβ = aβ β¨ _ := h.le.1.eq_or_lt
Β· exact right _ (Prod.mk_lt_mk_iff_right.1 h)
Β· exact left _ _ ha
| true |
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.IsLocalHomeomorph
#align_import topology.instances.add_circle from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/AddCircle.lean | 231 | 237 | theorem liftIoc_coe_apply {f : π β B} {x : π} (hx : x β Ioc a (a + p)) :
liftIoc p a f βx = f x := by
have : (equivIoc p a) x = β¨x, hxβ© := by |
have : (equivIoc p a) x = β¨x, hxβ© := by
rw [Equiv.apply_eq_iff_eq_symm_apply]
rfl
rw [liftIoc, comp_apply, this]
rfl
| true |
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.ideal from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
namespace IsLocalization
section CommSemiring
variable {R : Type*} [CommSemiring R] (M : Sub... | Mathlib/RingTheory/Localization/Ideal.lean | 108 | 132 | theorem isPrime_iff_isPrime_disjoint (J : Ideal S) :
J.IsPrime β
(Ideal.comap (algebraMap R S) J).IsPrime β§
Disjoint (M : Set R) β(Ideal.comap (algebraMap R S) J) := by
constructor |
constructor
Β· refine fun h =>
β¨β¨?_, ?_β©,
Set.disjoint_left.mpr fun m hm1 hm2 =>
h.ne_top (Ideal.eq_top_of_isUnit_mem _ hm2 (map_units S β¨m, hm1β©))β©
Β· refine fun hJ => h.ne_top ?_
rw [eq_top_iff, β (orderEmbedding M S).le_iff_le]
exact le_of_eq hJ.symm
Β· intro x y hxy
... | true |
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Bornology.Hom
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.Bounded
#align_import topology.metric_space.lipschitz from "leanprove... | Mathlib/Topology/MetricSpace/Lipschitz.lean | 51 | 55 | theorem lipschitzOnWith_iff_dist_le_mul [PseudoMetricSpace Ξ±] [PseudoMetricSpace Ξ²] {K : ββ₯0}
{s : Set Ξ±} {f : Ξ± β Ξ²} :
LipschitzOnWith K f s β β x β s, β y β s, dist (f x) (f y) β€ K * dist x y := by
simp only [LipschitzOnWith, edist_nndist, dist_nndist] |
simp only [LipschitzOnWith, edist_nndist, dist_nndist]
norm_cast
| true |
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mathlib"@"2f588be38bb5bec02f218ba14f82fc82eb663f87"
noncomputable section
open scoped Classical nonZeroDivisors
open Set Function UniqueFactorizationMonoid IsDedekindDomain IsDedekind... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 122 | 127 | theorem finite_mulSupport_inv {I : Ideal R} (hI : I β 0) :
(mulSupport fun v : HeightOneSpectrum R => (v.asIdeal : FractionalIdeal Rβ° K) ^
(-((Associates.mk v.asIdeal).count (Associates.mk I).factors : β€))).Finite := by
rw [mulSupport] |
rw [mulSupport]
simp_rw [zpow_neg, Ne, inv_eq_one]
exact finite_mulSupport_coe hI
| true |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-community/mathlib"@"fa256f00ce018e7b40e1dc756e403c86680bf448"
noncomputable section
open Polynomial
open Polynomial Finset
namespace Polynomial
variable {R : Type*}... | Mathlib/Algebra/Polynomial/EraseLead.lean | 52 | 52 | theorem eraseLead_coeff_natDegree : f.eraseLead.coeff f.natDegree = 0 := by | simp [eraseLead_coeff]
| true |
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 | 100 | 102 | theorem degrees_sub [DecidableEq Ο] (p q : MvPolynomial Ο R) :
(p - q).degrees β€ p.degrees β q.degrees := by |
simpa only [sub_eq_add_neg] using le_trans (degrees_add p (-q)) (by rw [degrees_neg])
| true |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 107 | 109 | theorem map_nsmul_const [AddMonoid G] [AddMonoid H] [AddConstMapClass F G H a b]
(f : F) (n : β) : f (n β’ a) = f 0 + n β’ b := by |
simpa using map_add_nsmul f 0 n
| true |
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 β]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 57 | 58 | theorem smeval_C : (C r).smeval x = r β’ x ^ 0 := by |
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_C_index]
| true |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 116 | 116 | theorem ascPochhammer_zero_eval_zero : (ascPochhammer S 0).eval 0 = 1 := by | simp
| true |
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
open Monoid Coprod Multiplicative Subgroup Function
def HNNExtension.con (G : Type*) [Group G] (A B : Subgroup G) (Ο : A β* B) :
Con (G β Multiplicative β€) :=
conGen (fun x y => β (a : A),
x = inr (ofAdd 1) * inl (a : G) β§
... | Mathlib/GroupTheory/HNNExtension.lean | 69 | 71 | theorem of_mul_t (b : B) :
(of (b : G) : HNNExtension G A B Ο) * t = t * of (Ο.symm b : G) := by |
rw [t_mul_of]; simp
| true |
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.Group
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathli... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 145 | 149 | theorem zpowers_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : β}
[hp : Fact p.Prime] (h : Fintype.card G = p) {g : G} (hg : g β 1) : zpowers g = β€ := by
subst h |
subst h
have := (zpowers g).eq_bot_or_eq_top_of_prime_card
rwa [zpowers_eq_bot, or_iff_right hg] at this
| true |
import Mathlib.Algebra.DualNumber
import Mathlib.Analysis.NormedSpace.TrivSqZeroExt
#align_import analysis.normed_space.dual_number from "leanprover-community/mathlib"@"806c0bb86f6128cfa2f702285727518eb5244390"
open NormedSpace -- For `NormedSpace.exp`.
namespace DualNumber
open TrivSqZeroExt
variable (π : Typ... | Mathlib/Analysis/NormedSpace/DualNumber.lean | 38 | 39 | theorem exp_smul_eps (r : R) : exp π (r β’ eps : DualNumber R) = 1 + r β’ eps := by |
rw [eps, β inr_smul, exp_inr]
| true |
import Mathlib.ModelTheory.Satisfiability
#align_import model_theory.types from "leanprover-community/mathlib"@"98bd247d933fb581ff37244a5998bd33d81dd46d"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal Set
open scoped Classical
open Cardinal FirstOrder
namespace FirstOrder
namespace La... | Mathlib/ModelTheory/Types.lean | 129 | 132 | theorem setOf_mem_eq_univ_iff (Ο : L[[Ξ±]].Sentence) :
{ p : T.CompleteType Ξ± | Ο β p } = Set.univ β (L.lhomWithConstants Ξ±).onTheory T β¨α΅ Ο := by
rw [models_iff_not_satisfiable, β compl_empty_iff, compl_setOf_mem, β setOf_subset_eq_empty_iff] |
rw [models_iff_not_satisfiable, β compl_empty_iff, compl_setOf_mem, β setOf_subset_eq_empty_iff]
simp
| true |
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Data.Nat.Cast.Order
#align_import algebra.order.invertible from "leanprover-community/mathlib"@"ee0c179cd3c8a45aa5bffbf1b41d8dbede452865"
variable {Ξ± : Type*} [LinearOrderedSemiring Ξ±] {a : Ξ±}
@[simp]
theorem invOf_pos [I... | Mathlib/Algebra/Order/Invertible.lean | 35 | 35 | theorem invOf_lt_zero [Invertible a] : β
a < 0 β a < 0 := by | simp only [β not_le, invOf_nonneg]
| true |
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.GroupAction.Units
#align_import data.int.absolute_value from "leanprover-community/mathlib"@"9aba7801eeecebb61f58a5763c2b6dd1b47dc6ef"
variable {R S : Type*} [Ring R] [Linea... | Mathlib/Data/Int/AbsoluteValue.lean | 28 | 29 | theorem AbsoluteValue.map_units_int (abv : AbsoluteValue β€ S) (x : β€Λ£) : abv x = 1 := by |
rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp
| true |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {Ξ± I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 245 | 248 | theorem IsRelPrime.prod_left_iff : IsRelPrime (β i β t, s i) x β β i β t, IsRelPrime (s i) x := by
classical |
classical
refine Finset.induction_on t (iff_of_true isRelPrime_one_left fun _ β¦ by simp) fun b t hbt ih β¦ ?_
rw [Finset.prod_insert hbt, IsRelPrime.mul_left_iff, ih, Finset.forall_mem_insert]
| true |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 92 | 95 | theorem HasDerivAt.smul (hc : HasDerivAt c c' x) (hf : HasDerivAt f f' x) :
HasDerivAt (fun y => c y β’ f y) (c x β’ f' + c' β’ f x) x := by
rw [β hasDerivWithinAt_univ] at * |
rw [β hasDerivWithinAt_univ] at *
exact hc.smul hf
| true |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
noncomputable section
open Set Filter
universe u v w x
def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : β
β... | Mathlib/Topology/Basic.lean | 115 | 116 | theorem IsOpen.union (hβ : IsOpen sβ) (hβ : IsOpen sβ) : IsOpen (sβ βͺ sβ) := by |
rw [union_eq_iUnion]; exact isOpen_iUnion (Bool.forall_bool.2 β¨hβ, hββ©)
| true |
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.ae_disjoint from "leanprover-community/mathlib"@"bc7d81beddb3d6c66f71449c5bc76c38cb77cf9e"
open Set Function
namespace MeasureTheory
variable {ΞΉ Ξ± : Type*} {m : MeasurableSpace Ξ±} (ΞΌ : Measure Ξ±)
def AEDisjoint (s t : Se... | Mathlib/MeasureTheory/Measure/AEDisjoint.lean | 94 | 96 | theorem iUnion_left_iff [Countable ΞΉ] {s : ΞΉ β Set Ξ±} :
AEDisjoint ΞΌ (β i, s i) t β β i, AEDisjoint ΞΌ (s i) t := by |
simp only [AEDisjoint, iUnion_inter, measure_iUnion_null_iff]
| true |
import Mathlib.Analysis.NormedSpace.PiTensorProduct.ProjectiveSeminorm
import Mathlib.LinearAlgebra.Isomorphisms
universe uΞΉ uπ uE uF
variable {ΞΉ : Type uΞΉ} [Fintype ΞΉ]
variable {π : Type uπ} [NontriviallyNormedField π]
variable {E : ΞΉ β Type uE} [β i, SeminormedAddCommGroup (E i)] [β i, NormedSpace π (E i)]
... | Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean | 116 | 119 | theorem toDualContinuousMultilinearMap_le_projectiveSeminorm (x : β¨[π] i, E i) :
βtoDualContinuousMultilinearMap F xβ β€ projectiveSeminorm x := by
simp only [toDualContinuousMultilinearMap, LinearMap.coe_mk, AddHom.coe_mk] |
simp only [toDualContinuousMultilinearMap, LinearMap.coe_mk, AddHom.coe_mk]
apply LinearMap.mkContinuous_norm_le _ (apply_nonneg _ _)
| true |
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Multilinear.Basic
#align_import linear_algebra.multilinear.basis from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
open MultilinearMap
variable {R : Type*} {ΞΉ : Type*} {n : β} {M : Fin n β Type*} {Mβ : Type*} {Mβ : Type*... | Mathlib/LinearAlgebra/Multilinear/Basis.lean | 32 | 49 | theorem Basis.ext_multilinear_fin {f g : MultilinearMap R M Mβ} {ΞΉβ : Fin n β Type*}
(e : β i, Basis (ΞΉβ i) R (M i))
(h : β v : β i, ΞΉβ i, (f fun i => e i (v i)) = g fun i => e i (v i)) : f = g := by
induction' n with m hm |
induction' n with m hm
Β· ext x
convert h finZeroElim
Β· apply Function.LeftInverse.injective uncurry_curryLeft
refine Basis.ext (e 0) ?_
intro i
apply hm (Fin.tail e)
intro j
convert h (Fin.cons i j)
iterate 2
rw [curryLeft_apply]
congr 1 with x
refine Fin.cases rfl (... | true |
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