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 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
#align_import set_theory.ordinal.topology from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
noncomputable section
universe u v
open Cardinal Order Topology
namespace Ordina... | Mathlib/SetTheory/Ordinal/Topology.lean | 64 | 65 | theorem nhds_left_eq_nhds (a : Ordinal) : π[β€] a = π a := by |
rw [β nhds_left_sup_nhds_right', nhds_right', sup_bot_eq]
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,259 |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 143 | 144 | theorem factorization_eq_zero_of_not_dvd {n p : β} (h : Β¬p β£ n) : n.factorization p = 0 := by |
simp [factorization_eq_zero_iff, h]
| 1 | 2.718282 | 0 | 0.4 | 10 | 388 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib.Order.PiLex
import Mathlib.Order.Interval.Set.Basic
#align_import data.fin.tuple.basic from "leanprover-community/mathlib"@"ef997baa41b5c428be3fb50089a7139bf4ee886b"
assert_not_exists MonoidWithZero
un... | Mathlib/Data/Fin/Tuple/Basic.lean | 82 | 82 | theorem cons_zero : cons x p 0 = x := by | simp [cons]
| 1 | 2.718282 | 0 | 0.875 | 8 | 761 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Ring.Opposite
import Mathlib.GroupTheory.GroupAction.Opposite
#align_import ring_theory.non_zero_divisors from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f6721fe"
variable (Mβ : Type*) [... | Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean | 129 | 130 | theorem mul_left_mem_nonZeroDivisors_eq_zero_iff {r x : Mβ} (hr : r β Mββ°) : r * x = 0 β x = 0 := by |
rw [mul_comm, mul_right_mem_nonZeroDivisors_eq_zero_iff hr]
| 1 | 2.718282 | 0 | 0 | 1 | 114 |
import Mathlib.Geometry.Manifold.Algebra.Monoid
#align_import geometry.manifold.algebra.lie_group from "leanprover-community/mathlib"@"f9ec187127cc5b381dfcf5f4a22dacca4c20b63d"
noncomputable section
open scoped Manifold
-- See note [Design choices about smooth algebraic structures]
class LieAddGroup {π : Type*... | Mathlib/Geometry/Manifold/Algebra/LieGroup.lean | 171 | 174 | theorem ContMDiffWithinAt.div {f g : M β G} {s : Set M} {xβ : M}
(hf : ContMDiffWithinAt I' I n f s xβ) (hg : ContMDiffWithinAt I' I n g s xβ) :
ContMDiffWithinAt I' I n (fun x => f x / g x) s xβ := by |
simp_rw [div_eq_mul_inv]; exact hf.mul hg.inv
| 1 | 2.718282 | 0 | 0 | 1 | 45 |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ΞΉ Ξ± Ξ² : Type*} {Ο : ΞΉ β Type*}
def symmDiff [Sup Ξ±] [SDiff Ξ±] (a b : Ξ±) : Ξ± :=
a \ b β b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 165 | 166 | theorem symmDiff_sdiff : a β b \ c = a \ (b β c) β b \ (a β c) := by |
rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left]
| 1 | 2.718282 | 0 | 0.181818 | 22 | 266 |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 1,092 | 1,093 | theorem bex_eq_left {a : Ξ±} : (β (x : _) (_ : x = a), p x) β p a := by |
simp only [exists_prop, exists_eq_left]
| 1 | 2.718282 | 0 | 0 | 8 | 105 |
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.equipartition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
open Finset Fintype
namespace Finpartition
variable {Ξ± : Type*} [DecidableEq Ξ±] ... | Mathlib/Order/Partition/Equipartition.lean | 89 | 100 | theorem IsEquipartition.card_large_parts_eq_mod (hP : P.IsEquipartition) :
(P.parts.filter fun p β¦ p.card = s.card / P.parts.card + 1).card = s.card % P.parts.card := by |
have z := P.sum_card_parts
rw [β sum_filter_add_sum_filter_not (s := P.parts)
(p := fun x β¦ x.card = s.card / P.parts.card + 1),
hP.filter_ne_average_add_one_eq_average,
sum_const_nat (m := s.card / P.parts.card + 1) (by simp),
sum_const_nat (m := s.card / P.parts.card) (by simp),
β hP.filter... | 10 | 22,026.465795 | 2 | 1.375 | 8 | 1,469 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e600... | Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 72 | 74 | theorem Finsupp.toFreeAbelianGroup_toFinsupp {X} (x : FreeAbelianGroup X) :
Finsupp.toFreeAbelianGroup (FreeAbelianGroup.toFinsupp x) = x := by |
rw [β AddMonoidHom.comp_apply, Finsupp.toFreeAbelianGroup_comp_toFinsupp, AddMonoidHom.id_apply]
| 1 | 2.718282 | 0 | 0.857143 | 7 | 747 |
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 | 34 | 46 | theorem exists_null_pairwise_disjoint_diff [Countable ΞΉ] {s : ΞΉ β Set Ξ±}
(hd : Pairwise (AEDisjoint ΞΌ on s)) : β t : ΞΉ β Set Ξ±, (β i, MeasurableSet (t i)) β§
(β i, ΞΌ (t i) = 0) β§ Pairwise (Disjoint on fun i => s i \ t i) := by |
refine β¨fun i => toMeasurable ΞΌ (s i β© β j β ({i}αΆ : Set ΞΉ), s j), fun i =>
measurableSet_toMeasurable _ _, fun i => ?_, ?_β©
Β· simp only [measure_toMeasurable, inter_iUnion]
exact (measure_biUnion_null_iff <| to_countable _).2 fun j hj => hd (Ne.symm hj)
Β· simp only [Pairwise, disjoint_left, onFun, mem_d... | 10 | 22,026.465795 | 2 | 0.4 | 5 | 390 |
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
suppress_compilation
variable (π A : Type*) [NontriviallyNormedField π] [NonUnitalNormedRing A]
variable [NormedSpace π A] [IsScalarTower π A A] [SMulCommClass π A A]
open ContinuousLinearMap
namespace Unitizati... | Mathlib/Analysis/NormedSpace/Unitization.lean | 139 | 141 | theorem norm_eq_sup (x : Unitization π A) :
βxβ = βx.fstβ β βalgebraMap π (A βL[π] A) x.fst + mul π A x.sndβ := by |
rw [norm_def, splitMul_apply, Prod.norm_def, sup_eq_max]
| 1 | 2.718282 | 0 | 1.6 | 5 | 1,728 |
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
noncomputable section
universe u v v' v''
variable {K : Type u} {V Vβ : Type v} {V' V'β : Type v'} {V'' : Type v''}
open Cardinal Basis Submodule Function Set
namespace LinearMap
section Ring
... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 72 | 73 | theorem rank_comp_le_right (g : V ββ[K] V') (f : V' ββ[K] V'β) : rank (f.comp g) β€ rank g := by |
simpa only [Cardinal.lift_id] using lift_rank_comp_le_right g f
| 1 | 2.718282 | 0 | 0.2 | 5 | 277 |
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 328 | 329 | theorem imageSubobject_arrow_comp : factorThruImageSubobject f β« (imageSubobject f).arrow = f := by |
simp [factorThruImageSubobject, imageSubobject_arrow]
| 1 | 2.718282 | 0 | 0.263158 | 19 | 308 |
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.Int.ConditionallyCompleteOrder
import Mathlib.Topology.Instances.Discrete
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Order.Filter.Archimedean
#align_import topology.instances.int from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/Int.lean | 84 | 85 | theorem cofinite_eq : (cofinite : Filter β€) = atBot β atTop := by |
rw [β cocompact_eq_cofinite, cocompact_eq_atBot_atTop]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 327 |
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem Β«forallΒ» {p : Ξ± β Ξ² β Prop} :
(β x, p x) β (β a, p (inl a)) β§ β b, p (inr b) :=
β¨fun h => β¨fun _ => h _, fun _ => h _β©, fun β¨hβ, hββ© => Sum.rec hβ hββ©
@[simp] protected theorem Β«existsΒ» {p : Ξ± β Ξ² ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 75 | 75 | theorem not_isLeft {x : Ξ± β Ξ²} : Β¬x.isLeft β x.isRight := by | simp
| 1 | 2.718282 | 0 | 0.285714 | 7 | 316 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Data.Rat.Cast.Defs
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {F ΞΉ Ξ± Ξ² : Type*}
namespace Rat
open Rat
section WithDivRing
variable [DivisionRing Ξ±]
@[simp, norm_cast]
th... | Mathlib/Data/Rat/Cast/CharZero.lean | 119 | 120 | theorem cast_mk (a b : β€) : (a /. b : Ξ±) = a / b := by |
simp only [divInt_eq_div, cast_div, cast_intCast]
| 1 | 2.718282 | 0 | 0 | 3 | 196 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : β)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
Orde... | Mathlib/Order/Interval/Finset/Fin.lean | 119 | 120 | theorem card_Ioo : (Ioo a b).card = b - a - 1 := by |
rw [β Nat.card_Ioo, β map_valEmbedding_Ioo, card_map]
| 1 | 2.718282 | 0 | 0.125 | 16 | 251 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 142 | 143 | theorem sameCycle_inv_apply_right : SameCycle f x (fβ»ΒΉ y) β SameCycle f x y := by |
rw [β sameCycle_apply_right, apply_inv_self]
| 1 | 2.718282 | 0 | 0 | 7 | 112 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 112 | 113 | theorem intCast_mul_eq_zsmul (x : β) (n : β€) : β((n : β) * x : β) = n β’ (βx : Angle) := by |
simpa only [zsmul_eq_mul] using coeHom.map_zsmul x n
| 1 | 2.718282 | 0 | 0.5 | 6 | 425 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp Ad... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 133 | 135 | theorem degrees_sum {ΞΉ : Type*} [DecidableEq Ο] (s : Finset ΞΉ) (f : ΞΉ β MvPolynomial Ο R) :
(β i β s, f i).degrees β€ s.sup fun i => (f i).degrees := by |
simp_rw [degrees_def]; exact supDegree_sum_le
| 1 | 2.718282 | 0 | 0.846154 | 13 | 743 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {Ξ± : Type u}
class OrderedAddCommGroup (Ξ± : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 305 | 306 | theorem mul_inv_lt_iff_lt_mul : a * bβ»ΒΉ < c β a < c * b := by |
rw [β mul_lt_mul_iff_right b, inv_mul_cancel_right]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {Ξ± Ξ² : Type*} {s t : Set Ξ±}
noncomputable def encard (s : Set Ξ±) : ββ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 152 | 153 | theorem encard_le_card (h : s β t) : s.encard β€ t.encard := by |
rw [β union_diff_cancel h, encard_union_eq disjoint_sdiff_right]; exact le_self_add
| 1 | 2.718282 | 0 | 0.5 | 14 | 454 |
import Mathlib.LinearAlgebra.Ray
import Mathlib.LinearAlgebra.Determinant
#align_import linear_algebra.orientation from "leanprover-community/mathlib"@"0c1d80f5a86b36c1db32e021e8d19ae7809d5b79"
noncomputable section
section OrderedCommSemiring
variable (R : Type*) [StrictOrderedCommSemiring R]
variable (M : Typ... | Mathlib/LinearAlgebra/Orientation.lean | 74 | 75 | theorem Orientation.map_refl : (Orientation.map ΞΉ <| LinearEquiv.refl R M) = Equiv.refl _ := by |
rw [Orientation.map, AlternatingMap.domLCongr_refl, Module.Ray.map_refl]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 558 |
import Mathlib.Algebra.Group.Commutator
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Bracket
import Mathlib.GroupTheory.Subgroup.Centralizer
import Mathlib.Tactic.Group
#align_import group_theory.commutator from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
variable... | Mathlib/GroupTheory/Commutator.lean | 61 | 62 | theorem commutatorElement_inv : β
gβ, gβββ»ΒΉ = β
gβ, gββ := by |
simp_rw [commutatorElement_def, mul_inv_rev, inv_inv, mul_assoc]
| 1 | 2.718282 | 0 | 0.8 | 5 | 707 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Ring.Basic
#align_import topology.algebra.infinite_sum.ring from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Filter Finset Function
open... | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | 38 | 39 | theorem HasSum.mul_right (aβ) (hf : HasSum f aβ) : HasSum (fun i β¦ f i * aβ) (aβ * aβ) := by |
simpa only using hf.map (AddMonoidHom.mulRight aβ) (continuous_id.mul continuous_const)
| 1 | 2.718282 | 0 | 0.333333 | 3 | 348 |
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open Topology Uniformity Filter S... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 147 | 148 | theorem TendstoUniformly.tendstoUniformlyOnFilter (h : TendstoUniformly F f p) :
TendstoUniformlyOnFilter F f p β€ := by | rwa [β tendstoUniformly_iff_tendstoUniformlyOnFilter]
| 1 | 2.718282 | 0 | 0.571429 | 7 | 517 |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import ring_theory.localization.num_denom from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
variable {R : Type*} [CommRing R] (... | Mathlib/RingTheory/Localization/NumDen.lean | 37 | 47 | theorem exists_reduced_fraction (x : K) :
β (a : A) (b : nonZeroDivisors A), IsRelPrime a b β§ mk' K a b = x := by |
obtain β¨β¨b, b_nonzeroβ©, a, habβ© := exists_integer_multiple (nonZeroDivisors A) x
obtain β¨a', b', c', no_factor, rfl, rflβ© :=
UniqueFactorizationMonoid.exists_reduced_factors' a b
(mem_nonZeroDivisors_iff_ne_zero.mp b_nonzero)
obtain β¨_, b'_nonzeroβ© := mul_mem_nonZeroDivisors.mp b_nonzero
refine β¨a', ... | 9 | 8,103.083928 | 2 | 1.666667 | 3 | 1,768 |
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
set_option autoImplicit true
open Filter Set Function
variable {Ξ± Ξ² Ξ³ ΞΉ ΞΉ' : Type*}
namespace Filter
section Relation
... | Mathlib/Order/LiminfLimsup.lean | 80 | 80 | theorem isBounded_top : IsBounded r β€ β β t, β x, r x t := by | simp [IsBounded, eq_univ_iff_forall]
| 1 | 2.718282 | 0 | 0.25 | 4 | 306 |
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheory.MeasurableSpace.Basic
open Set
namespace MeasureTheory
variable {ΞΉ : Type _} {Ξ± : ΞΉ β Type _}
section cylinder
def cylinder (s : Finset ΞΉ) (S : Set (β i : s, Ξ±... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 193 | 195 | theorem inter_cylinder_same (s : Finset ΞΉ) (Sβ : Set (β i : s, Ξ± i)) (Sβ : Set (β i : s, Ξ± i)) :
cylinder s Sβ β© cylinder s Sβ = cylinder s (Sβ β© Sβ) := by |
classical rw [inter_cylinder]; rfl
| 1 | 2.718282 | 0 | 0.6875 | 16 | 636 |
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import data.dfinsupp.basic from "leanpr... | Mathlib/Data/DFinsupp/Basic.lean | 158 | 161 | theorem mapRange_zero (f : β i, Ξ²β i β Ξ²β i) (hf : β i, f i 0 = 0) :
mapRange f hf (0 : Ξ β i, Ξ²β i) = 0 := by |
ext
simp only [mapRange_apply, coe_zero, Pi.zero_apply, hf]
| 2 | 7.389056 | 1 | 1 | 3 | 915 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 48 | 49 | theorem evalβ_eq_sum {f : R β+* S} {x : S} : p.evalβ f x = p.sum fun e a => f a * x ^ e := by |
rw [evalβ_def]
| 1 | 2.718282 | 0 | 0.6 | 15 | 534 |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 111 | 112 | theorem polar_comm (f : M β R) (x y : M) : polar f x y = polar f y x := by |
rw [polar, polar, add_comm, sub_sub, sub_sub, add_comm (f x) (f y)]
| 1 | 2.718282 | 0 | 0.5 | 6 | 482 |
import Mathlib.RingTheory.Localization.LocalizationLocalization
#align_import ring_theory.localization.as_subring from "leanprover-community/mathlib"@"649ca66bf4d62796b5eefef966e622d91aa471f3"
namespace Localization
open nonZeroDivisors
variable {A : Type*} (K : Type*) [CommRing A] (S : Submonoid A) (hS : S β€ A... | Mathlib/RingTheory/Localization/AsSubring.lean | 31 | 32 | theorem map_isUnit_of_le (hS : S β€ Aβ°) (s : S) : IsUnit (algebraMap A K s) := by |
apply IsLocalization.map_units K (β¨s.1, hS s.2β© : Aβ°)
| 1 | 2.718282 | 0 | 0 | 1 | 117 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Matrix
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.matrix_exponential from "l... | Mathlib/Analysis/NormedSpace/MatrixExponential.lean | 89 | 91 | theorem exp_blockDiagonal' (v : β i, Matrix (n' i) (n' i) πΈ) :
exp π (blockDiagonal' v) = blockDiagonal' (exp π v) := by |
simp_rw [exp_eq_tsum, β blockDiagonal'_pow, β blockDiagonal'_smul, β blockDiagonal'_tsum]
| 1 | 2.718282 | 0 | 0.4 | 5 | 398 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 107 | 111 | theorem invOf_fromBlocks_zeroββ_eq (A : Matrix m m Ξ±) (C : Matrix n m Ξ±) (D : Matrix n n Ξ±)
[Invertible A] [Invertible D] [Invertible (fromBlocks A 0 C D)] :
β
(fromBlocks A 0 C D) = fromBlocks (β
A) 0 (-(β
D * C * β
A)) (β
D) := by |
letI := fromBlocksZeroββInvertible A C D
convert (rfl : β
(fromBlocks A 0 C D) = _)
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,248 |
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.birthday from "leanprover-community/mathlib"@"a347076985674932c0e91da09b9961ed0a79508c"
universe u
open Ordinal
namespace SetTheory
open scoped NaturalOps PGame
namespace PGame
noncomputable def b... | Mathlib/SetTheory/Game/Birthday.lean | 103 | 103 | theorem birthday_zero : birthday 0 = 0 := by | simp [inferInstanceAs (IsEmpty PEmpty)]
| 1 | 2.718282 | 0 | 0.4 | 10 | 387 |
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Order.Filter.Cofinite
#align_import number_theory.fermat_psp from "leanprover-community/mathlib"@"c0439b4877c24a117bfdd9e32faf62eee9b115eb"
namespace Nat
def ProbablePrime (n b : β) : Prop :=
n β£ b ^ (n - 1) - 1
#align fermat_psp.probable_prime Nat.Probabl... | Mathlib/NumberTheory/FermatPsp.lean | 127 | 130 | theorem fermatPsp_base_one {n : β} (hβ : 1 < n) (hβ : Β¬n.Prime) : FermatPsp n 1 := by |
refine β¨show n β£ 1 ^ (n - 1) - 1 from ?_, hβ, hββ©
exact show 0 = 1 ^ (n - 1) - 1 by
set_option tactic.skipAssignedInstances false in norm_num βΈ dvd_zero n
| 3 | 20.085537 | 1 | 1.5 | 4 | 1,688 |
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Function.LocallyIntegrable
open Asymptotics MeasureTheory Set Filter
variable {Ξ± E F : Type*} [MeasurableSpace Ξ±] [NormedAddCommGroup E] [NormedAddCommGroup F]
{f : Ξ± β E} {g : Ξ± β F} {a b :... | Mathlib/MeasureTheory/Integral/Asymptotics.lean | 70 | 77 | theorem LocallyIntegrable.integrable_of_isBigO_atBot_atTop
[IsMeasurablyGenerated (atBot (Ξ± := Ξ±))] [IsMeasurablyGenerated (atTop (Ξ± := Ξ±))]
(hf : LocallyIntegrable f ΞΌ)
(ho : f =O[atBot] g) (hg : IntegrableAtFilter g atBot ΞΌ)
(ho' : f =O[atTop] g') (hg' : IntegrableAtFilter g' atTop ΞΌ) : Integrable f ΞΌ... |
refine integrable_iff_integrableAtFilter_atBot_atTop.mpr
β¨β¨ho.integrableAtFilter ?_ hg, ho'.integrableAtFilter ?_ hg'β©, hfβ©
all_goals exact hf.aestronglyMeasurable.stronglyMeasurableAtFilter
| 3 | 20.085537 | 1 | 1.125 | 8 | 1,209 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 53 | 53 | theorem one_sub_K_pos' (hf : ContractingWith K f) : (0 : ββ₯0β) < 1 - K := by | simp [hf.1]
| 1 | 2.718282 | 0 | 0.6 | 5 | 531 |
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
noncomputable section
universe u
namespace SetTheory
open scoped PGame
namespace PGame... | Mathlib/SetTheory/Game/Nim.lean | 78 | 80 | theorem moveRight_nim_hEq (o : Ordinal) :
have : IsWellOrder (Quotient.out o).Ξ± (Β· < Β·) := inferInstance
HEq (nim o).moveRight fun i : o.out.Ξ± => nim (typein (Β· < Β·) i) := by | rw [nim_def]; rfl
| 1 | 2.718282 | 0 | 0 | 7 | 205 |
import Mathlib.Data.Set.Basic
#align_import data.set.bool_indicator from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
open Bool
namespace Set
variable {Ξ± : Type*} (s : Set Ξ±)
noncomputable def boolIndicator (x : Ξ±) :=
@ite _ (x β s) (Classical.propDecidable _) true false
#align s... | Mathlib/Data/Set/BoolIndicator.lean | 47 | 51 | theorem preimage_boolIndicator_eq_union (t : Set Bool) :
s.boolIndicator β»ΒΉ' t = (if true β t then s else β
) βͺ if false β t then sαΆ else β
:= by |
ext x
simp only [boolIndicator, mem_preimage]
split_ifs <;> simp [*]
| 3 | 20.085537 | 1 | 1 | 4 | 906 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp Ad... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 84 | 85 | theorem degrees_def [DecidableEq Ο] (p : MvPolynomial Ο R) :
p.degrees = p.support.sup fun s : Ο ββ β => Finsupp.toMultiset s := by | rw [degrees]; convert rfl
| 1 | 2.718282 | 0 | 0.846154 | 13 | 743 |
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {π E : Type*}
variable [NormedField π]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 104 | 106 | theorem smul_closedBall' {c : π} (hc : c β 0) (x : E) (r : β) :
c β’ closedBall x r = closedBall (c β’ x) (βcβ * r) := by |
simp only [β ball_union_sphere, Set.smul_set_union, _root_.smul_ball hc, smul_sphere' hc]
| 1 | 2.718282 | 0 | 1 | 8 | 994 |
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Quotient
#align_import linear_algebra.quotient_pi from "leanprover-community/mathlib"@"398f60f60b43ef42154bd2bdadf5133daf1577a4"
namespace Submodule
open LinearMap
variable {ΞΉ R : Type*} [CommRing R]
variable {Ms : ΞΉ β Type*} [β i, AddCommGroup (Ms i)... | Mathlib/LinearAlgebra/QuotientPi.lean | 50 | 60 | theorem piQuotientLift_single [Fintype ΞΉ] [DecidableEq ΞΉ] (p : β i, Submodule R (Ms i))
(q : Submodule R N) (f : β i, Ms i ββ[R] N) (hf : β i, p i β€ q.comap (f i)) (i)
(x : Ms i β§Έ p i) : piQuotientLift p q f hf (Pi.single i x) = mapQ _ _ (f i) (hf i) x := by |
simp_rw [piQuotientLift, lsum_apply, sum_apply, comp_apply, proj_apply]
rw [Finset.sum_eq_single i]
Β· rw [Pi.single_eq_same]
Β· rintro j - hj
rw [Pi.single_eq_of_ne hj, _root_.map_zero]
Β· intros
have := Finset.mem_univ i
contradiction
| 8 | 2,980.957987 | 2 | 1.666667 | 3 | 1,806 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Sort
import Mathlib.Data.List.FinRange
import Mathlib.LinearAlgebra.Pi
import Mathlib.Logic.Equiv.Fintype
#align_import linear_algebra.multilinear.basic from ... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 171 | 174 | theorem map_coord_zero {m : β i, Mβ i} (i : ΞΉ) (h : m i = 0) : f m = 0 := by |
classical
have : (0 : R) β’ (0 : Mβ i) = 0 := by simp
rw [β update_eq_self i m, h, β this, f.map_smul, zero_smul R (M := Mβ)]
| 3 | 20.085537 | 1 | 1 | 2 | 1,151 |
import Mathlib.Algebra.Module.Equiv
import Mathlib.Algebra.Module.Hom
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Module.Submodule.Range
import Mathlib.Data.Set.Finite
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.Abel
#align_import linear_algebra.basic from "leanprover-c... | Mathlib/LinearAlgebra/Basic.lean | 169 | 173 | theorem ofSubmodule'_toLinearMap [Module R M] [Module Rβ Mβ] (f : M βββ[Οββ] Mβ)
(U : Submodule Rβ Mβ) :
(f.ofSubmodule' U).toLinearMap = (f.toLinearMap.domRestrict _).codRestrict _ Subtype.prop := by |
ext
rfl
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,316 |
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathlib"@"74ad1c88c77e799d2fea62801d1dbbd698cff1b7"
assert_not_exists Submodule
open Function
namespace ZMod
instance charZero : CharZero (ZMod 0) :=... | Mathlib/Data/ZMod/Basic.lean | 192 | 195 | theorem _root_.Prod.fst_zmod_cast (a : ZMod n) : (cast a : R Γ S).fst = cast a := by |
cases n
Β· rfl
Β· simp [ZMod.cast]
| 3 | 20.085537 | 1 | 1 | 11 | 900 |
import Mathlib.Algebra.Module.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import algebra.module.big_operators from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {ΞΉ ΞΊ Ξ± Ξ² R M : Type*}
section AddCommMonoid
variable [... | Mathlib/Algebra/Module/BigOperators.lean | 30 | 34 | theorem Multiset.sum_smul_sum {s : Multiset R} {t : Multiset M} :
s.sum β’ t.sum = ((s ΓΛ’ t).map fun p : R Γ M β¦ p.fst β’ p.snd).sum := by |
induction' s using Multiset.induction with a s ih
Β· simp
Β· simp [add_smul, ih, β Multiset.smul_sum]
| 3 | 20.085537 | 1 | 0.666667 | 3 | 600 |
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTh... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 72 | 73 | theorem cyclotomic'_zero (R : Type*) [CommRing R] [IsDomain R] : cyclotomic' 0 R = 1 := by |
simp only [cyclotomic', Finset.prod_empty, primitiveRoots_zero]
| 1 | 2.718282 | 0 | 1 | 7 | 1,027 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 294 | 299 | theorem maximal_linearIndependent_eq_infinite_basis {ΞΉ : Type w} (b : Basis ΞΉ R M) [Infinite ΞΉ]
{ΞΊ : Type w} (v : ΞΊ β M) (i : LinearIndependent R v) (m : i.Maximal) : #ΞΊ = #ΞΉ := by |
apply le_antisymm
Β· exact linearIndependent_le_basis b v i
Β· haveI : Nontrivial R := nontrivial_of_invariantBasisNumber R
exact infinite_basis_le_maximal_linearIndependent b v i m
| 4 | 54.59815 | 2 | 1.727273 | 11 | 1,843 |
import Batteries.Tactic.SeqFocus
import Batteries.Data.List.Lemmas
import Batteries.Data.List.Init.Attach
namespace Std.Range
def numElems (r : Range) : Nat :=
if r.step = 0 then
-- This is a very weird choice, but it is chosen to coincide with the `forIn` impl
if r.stop β€ r.start then 0 else r.stop
els... | .lake/packages/batteries/Batteries/Data/Range/Lemmas.lean | 26 | 27 | theorem numElems_step_1 (start stop) : numElems β¨start, stop, 1β© = stop - start := by |
simp [numElems]
| 1 | 2.718282 | 0 | 1.5 | 4 | 1,541 |
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Nat.SuccPred
#align_import data.int.succ_pred from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Order
namespace Int
-- so that Lean reads `Int.succ` through `SuccOrder.succ`
@[instance] abbrev instSuccOrder : Su... | Mathlib/Data/Int/SuccPred.lean | 79 | 79 | theorem sub_one_covBy (z : β€) : z - 1 β z := by | rw [Int.covBy_iff_succ_eq, sub_add_cancel]
| 1 | 2.718282 | 0 | 0.5 | 2 | 474 |
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4972407cd4f1222"
noncomputable section
@[ext]
structure Cubic (R : Type*) where
(a b c d : R)
#align cubic Cubic
namespace Cubic
open Cubic Polynomial
open Polynom... | Mathlib/Algebra/CubicDiscriminant.lean | 137 | 138 | theorem of_a_eq_zero (ha : P.a = 0) : P.toPoly = C P.b * X ^ 2 + C P.c * X + C P.d := by |
rw [toPoly, ha, C_0, zero_mul, zero_add]
| 1 | 2.718282 | 0 | 0.1 | 10 | 246 |
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 65 | 67 | theorem exp_inj_of_neg_pi_lt_of_le_pi {x y : β} (hxβ : -Ο < x.im) (hxβ : x.im β€ Ο) (hyβ : -Ο < y.im)
(hyβ : y.im β€ Ο) (hxy : exp x = exp y) : x = y := by |
rw [β log_exp hxβ hxβ, β log_exp hyβ hyβ, hxy]
| 1 | 2.718282 | 0 | 0.375 | 16 | 378 |
import Mathlib.Data.Finite.Defs
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Subtype
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.countable.defs from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
open Function
universe u v
variable {Ξ± : Sort u} {Ξ² : Sort v}
... | Mathlib/Data/Countable/Defs.lean | 159 | 161 | theorem uncountable_iff_forall_not_surjective [Nonempty Ξ±] :
Uncountable Ξ± β β f : β β Ξ±, Β¬Surjective f := by |
rw [β not_countable_iff, countable_iff_exists_surjective, not_exists]
| 1 | 2.718282 | 0 | 0 | 1 | 51 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import ring_theory.algebraic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
universe u v w
open scoped Classical
open Polynomi... | Mathlib/RingTheory/Algebraic.lean | 83 | 85 | theorem isAlgebraic_iff_not_injective {x : A} :
IsAlgebraic R x β Β¬Function.Injective (Polynomial.aeval x : R[X] ββ[R] A) := by |
simp only [IsAlgebraic, injective_iff_map_eq_zero, not_forall, and_comm, exists_prop]
| 1 | 2.718282 | 0 | 1.125 | 8 | 1,205 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 86 | 91 | theorem weightedVSubOfPoint_congr {wβ wβ : ΞΉ β k} (hw : β i β s, wβ i = wβ i) {pβ pβ : ΞΉ β P}
(hp : β i β s, pβ i = pβ i) (b : P) :
s.weightedVSubOfPoint pβ b wβ = s.weightedVSubOfPoint pβ b wβ := by |
simp_rw [weightedVSubOfPoint_apply]
refine sum_congr rfl fun i hi => ?_
rw [hw i hi, hp i hi]
| 3 | 20.085537 | 1 | 1.083333 | 12 | 1,183 |
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f4b3538f4cd1fb5bffd3"
namespace CategoryTheory
open CategoryTheory.Limits Preadditive
variable {C D : Type*} [Category C] [Category D] [Preadditive D... | Mathlib/CategoryTheory/Preadditive/FunctorCategory.lean | 127 | 129 | theorem app_sum {ΞΉ : Type*} (s : Finset ΞΉ) (X : C) (Ξ± : ΞΉ β (F βΆ G)) :
(β i β s, Ξ± i).app X = β i β s, (Ξ± i).app X := by |
simp only [β appHom_apply, map_sum]
| 1 | 2.718282 | 0 | 0 | 2 | 69 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 286 | 288 | theorem normAtPlace_apply_isReal {w : InfinitePlace K} (hw : IsReal w) (x : E K):
normAtPlace w x = βx.1 β¨w, hwβ©β := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, dif_pos]
| 1 | 2.718282 | 0 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antid... | Mathlib/RingTheory/PowerSeries/Trunc.lean | 84 | 86 | theorem trunc_succ (f : Rβ¦Xβ§) (n : β) :
trunc n.succ f = trunc n f + Polynomial.monomial n (coeff R n f) := by |
rw [trunc, Ico_zero_eq_range, sum_range_succ, trunc, Ico_zero_eq_range]
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,272 |
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import analysis.special_functions.complex.circle from "leanprover-community/mathlib"@"f333194f5ecd1482191452c5ea60b37d4d6afa08"
open Complex Function Set
open Real
| Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean | 37 | 38 | theorem arg_expMapCircle {x : β} (hβ : -Ο < x) (hβ : x β€ Ο) : arg (expMapCircle x) = x := by |
rw [expMapCircle_apply, exp_mul_I, arg_cos_add_sin_mul_I β¨hβ, hββ©]
| 1 | 2.718282 | 0 | 0 | 1 | 155 |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "Ο" => cs.wordProd
local prefi... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 61 | 61 | theorem isReflection_simple (i : B) : cs.IsReflection (s i) := by | use 1, i; simp
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,198 |
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Ring.Defs
#align_import algebra.ring.semiconj from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u v w x
variable {Ξ± : Type u} {Ξ² : Type v} {Ξ³ : Type w} {R : Type x}
open Function
namespace SemiconjBy
@[simp... | Mathlib/Algebra/Ring/Semiconj.lean | 89 | 91 | theorem sub_right (h : SemiconjBy a x y) (h' : SemiconjBy a x' y') :
SemiconjBy a (x - x') (y - y') := by |
simpa only [sub_eq_add_neg] using h.add_right h'.neg_right
| 1 | 2.718282 | 0 | 0 | 6 | 216 |
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 133 | 137 | theorem copy_copy {u v u' v' u'' v''} (p : G.Walk u v)
(hu : u = u') (hv : v = v') (hu' : u' = u'') (hv' : v' = v'') :
(p.copy hu hv).copy hu' hv' = p.copy (hu.trans hu') (hv.trans hv') := by |
subst_vars
rfl
| 2 | 7.389056 | 1 | 1 | 6 | 918 |
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 | 87 | 88 | theorem gold_pow_sub_gold_pow (n : β) : Ο ^ (n + 2) - Ο ^ (n + 1) = Ο ^ n := by |
rw [goldenRatio]; ring_nf; norm_num; ring
| 1 | 2.718282 | 0 | 0.894737 | 19 | 776 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 62 | 63 | theorem convex_ball (a : E) (r : β) : Convex β (Metric.ball a r) := by |
simpa only [Metric.ball, sep_univ] using (convexOn_univ_dist a).convex_lt r
| 1 | 2.718282 | 0 | 0.818182 | 11 | 721 |
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
#align_import data.matrix.notation from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matrix
universe u uβ uβ uβ
variable {Ξ± : Type u} {o n m... | Mathlib/Data/Matrix/Notation.lean | 230 | 232 | theorem tail_transpose (A : Matrix m' (Fin n.succ) Ξ±) : vecTail (of.symm Aα΅) = (vecTail β A)α΅ := by |
ext i j
rfl
| 2 | 7.389056 | 1 | 0.75 | 12 | 672 |
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' uβ' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {Mβ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 375 | 376 | theorem FiniteDimensional.finrank_tensorProduct :
finrank S (M β[S] M') = finrank S M * finrank S M' := by | simp [finrank]
| 1 | 2.718282 | 0 | 0.75 | 24 | 667 |
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open sc... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 97 | 97 | theorem toComplex_re (x y : β€) : ((β¨x, yβ© : β€[i]) : β).re = x := by | simp [toComplex_def]
| 1 | 2.718282 | 0 | 0.090909 | 11 | 243 |
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.box_integral.box.subbox_induction from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Finset Function Filter Metric Classical Topology Filter ENNReal
noncomputable... | Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean | 95 | 97 | theorem iUnion_coe_splitCenterBox (I : Box ΞΉ) : β s, (I.splitCenterBox s : Set (ΞΉ β β)) = I := by |
ext x
simp
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,487 |
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e228"
open scoped Classical Topology
open Finset Filter
namespace FormalMultilinearSeries
variable {π : Type*} [NontriviallyNormedField π] {E : Type*} ... | Mathlib/Analysis/Analytic/Inverse.lean | 73 | 74 | theorem leftInv_coeff_one (p : FormalMultilinearSeries π E F) (i : E βL[π] F) :
p.leftInv i 1 = (continuousMultilinearCurryFin1 π F E).symm i.symm := by | rw [leftInv]
| 1 | 2.718282 | 0 | 0.8 | 5 | 701 |
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
variable {Ξ± Ξ² : Type*}
section Fold
variable (op : Ξ± β Ξ± β Ξ±) [hc : Std.Commutative op] [ha : Std.Associative op]
local notation a " * " b => ... | Mathlib/Data/Multiset/Fold.lean | 67 | 68 | theorem fold_cons'_right (b a : Ξ±) (s : Multiset Ξ±) : (a ::β s).fold op b = s.fold op (b * a) := by |
rw [fold_eq_foldl, foldl_cons, β fold_eq_foldl]
| 1 | 2.718282 | 0 | 0.2 | 5 | 270 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Localization.Basic
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Surreal.Basic
#align_import set_theory.surreal.dyadic from "leanprover-community/mathlib"@"9... | Mathlib/SetTheory/Surreal/Dyadic.lean | 85 | 86 | theorem birthday_half : birthday (powHalf 1) = 2 := by |
rw [birthday_def]; simp
| 1 | 2.718282 | 0 | 0.714286 | 7 | 646 |
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 | 72 | 72 | theorem bernoulli_zero : bernoulli 0 = 1 := by | simp [bernoulli]
| 1 | 2.718282 | 0 | 1.6 | 5 | 1,715 |
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ΞΉ : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 371 | 372 | theorem natDegree_C_mul (a0 : a β 0) : (C a * p).natDegree = p.natDegree := by |
simp only [natDegree, degree_C_mul a0]
| 1 | 2.718282 | 0 | 0.857143 | 14 | 751 |
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 | 77 | 78 | theorem projIcc_of_right_le (hx : b β€ x) : projIcc a b h x = β¨b, right_mem_Icc.2 hβ© := by |
simp [projIcc, hx, h]
| 1 | 2.718282 | 0 | 0.083333 | 12 | 241 |
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Lemma
import Mathlib.Tactic.TypeStar
#align_import data.option.defs from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Option
#align option.lift_or_get Option.liftOrGet
protected def traverse.{u, v}
{F : Type u β Type... | Mathlib/Data/Option/Defs.lean | 61 | 61 | theorem mem_some_iff {Ξ± : Type*} {a b : Ξ±} : a β some b β b = a := by | simp
| 1 | 2.718282 | 0 | 0 | 2 | 8 |
import Mathlib.Data.Finset.Card
#align_import data.finset.option from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
variable {Ξ± Ξ² : Type*}
open Function
namespace Option
def toFinset (o : Option Ξ±) : Finset Ξ± :=
o.elim β
singleton
#align option.to_finset Option.toFinset
@[simp]
... | Mathlib/Data/Finset/Option.lean | 55 | 55 | theorem card_toFinset (o : Option Ξ±) : o.toFinset.card = o.elim 0 1 := by | cases o <;> rfl
| 1 | 2.718282 | 0 | 0 | 5 | 120 |
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]
exact (not_isUniform_iff.1 h).choose_spec.2.choose_spec.2.1
| 2 | 7.389056 | 1 | 1 | 5 | 922 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import number_theory.von_mangoldt from "leanprover-community/mathlib"@"c946d6097a6925ad16d7ec55677bbc977f9846de"
namespace ArithmeticFunction
open Finset Nat
open scoped Arit... | Mathlib/NumberTheory/VonMangoldt.lean | 140 | 141 | theorem moebius_mul_log_eq_vonMangoldt : (ΞΌ : ArithmeticFunction β) * log = Ξ := by |
rw [mul_comm]; simp
| 1 | 2.718282 | 0 | 0.636364 | 11 | 552 |
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.measure.probability_measure from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open MeasureTheory
open Set
open Filter
open BoundedCon... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 193 | 196 | theorem ennreal_coeFn_eq_coeFn_toMeasure (Ξ½ : ProbabilityMeasure Ξ©) (s : Set Ξ©) :
(Ξ½ s : ββ₯0β) = (Ξ½ : Measure Ξ©) s := by |
rw [β coeFn_comp_toFiniteMeasure_eq_coeFn, FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure,
toMeasure_comp_toFiniteMeasure_eq_toMeasure]
| 2 | 7.389056 | 1 | 1 | 5 | 1,134 |
import Mathlib.Topology.MetricSpace.PseudoMetric
#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
open Set Filter Bornology
open scoped NNReal Uniformity
universe u v w
variable {Ξ± : Type u} {Ξ² : Type v} {X ΞΉ : Type*}
variable [PseudoMetricS... | Mathlib/Topology/MetricSpace/Basic.lean | 74 | 74 | theorem zero_eq_dist {x y : Ξ³} : 0 = dist x y β x = y := by | rw [eq_comm, dist_eq_zero]
| 1 | 2.718282 | 0 | 0.166667 | 12 | 258 |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Topology.Instances.ENNReal
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Metric TopologicalSpace Function Filter
open scoped Topology NNReal
variable {Ξ± Ξ² F : Type*} [N... | Mathlib/Analysis/NormedSpace/FunctionSeries.lean | 53 | 56 | theorem tendstoUniformly_tsum {f : Ξ± β Ξ² β F} (hu : Summable u) (hfu : β n x, βf n xβ β€ u n) :
TendstoUniformly (fun t : Finset Ξ± => fun x => β n β t, f n x)
(fun x => β' n, f n x) atTop := by |
rw [β tendstoUniformlyOn_univ]; exact tendstoUniformlyOn_tsum hu fun n x _ => hfu n x
| 1 | 2.718282 | 0 | 1.25 | 4 | 1,341 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder Ξ±] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 117 | 118 | theorem iUnion_Ico_right [NoMaxOrder Ξ±] (a : Ξ±) : β b, Ico a b = Ici a := by |
simp only [β Ici_inter_Iio, β inter_iUnion, iUnion_Iio, inter_univ]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Order.Filter.AtTopBot
#align_import order.filter.indicator_function from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
variable {Ξ± Ξ² M E : Type*}
open Set Filter
@[to_additive]
| Mathlib/Order/Filter/IndicatorFunction.lean | 63 | 66 | theorem Monotone.mulIndicator_eventuallyEq_iUnion {ΞΉ} [Preorder ΞΉ] [One Ξ²] (s : ΞΉ β Set Ξ±)
(hs : Monotone s) (f : Ξ± β Ξ²) (a : Ξ±) :
(fun i => mulIndicator (s i) f a) =αΆ [atTop] fun _ β¦ mulIndicator (β i, s i) f a := by |
classical exact hs.piecewise_eventually_eq_iUnion f 1 a
| 1 | 2.718282 | 0 | 0.333333 | 3 | 344 |
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 84 | 85 | theorem div_self {a : R} (a0 : a β 0) : a / a = 1 := by |
simpa only [one_mul] using mul_div_cancel_rightβ 1 a0
| 1 | 2.718282 | 0 | 0.888889 | 9 | 769 |
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 | 55 | 56 | theorem involute_comp_involute : involute.comp involute = AlgHom.id R (CliffordAlgebra Q) := by |
ext; simp
| 1 | 2.718282 | 0 | 0 | 2 | 177 |
import Mathlib.Algebra.Periodic
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.periodic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace Nat
open Nat Function
| Mathlib/Data/Nat/Periodic.lean | 25 | 26 | theorem periodic_gcd (a : β) : Periodic (gcd a) a := by |
simp only [forall_const, gcd_add_self_right, eq_self_iff_true, Periodic]
| 1 | 2.718282 | 0 | 0.5 | 4 | 451 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.normed_space.exponential from "leanprover-community/ma... | Mathlib/Analysis/NormedSpace/Exponential.lean | 155 | 157 | theorem exp_unop [T2Space πΈ] (x : πΈα΅α΅α΅) :
exp π (MulOpposite.unop x) = MulOpposite.unop (exp π x) := by |
simp_rw [exp, expSeries_sum_eq, β MulOpposite.unop_pow, β MulOpposite.unop_smul, tsum_unop]
| 1 | 2.718282 | 0 | 0.428571 | 7 | 408 |
import Mathlib.Geometry.Euclidean.Sphere.Power
import Mathlib.Geometry.Euclidean.Triangle
#align_import geometry.euclidean.sphere.ptolemy from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
open Real
open scoped EuclideanGeometry RealInnerProductSpace Real
namespace EuclideanGeometry
... | Mathlib/Geometry/Euclidean/Sphere/Ptolemy.lean | 53 | 70 | theorem mul_dist_add_mul_dist_eq_mul_dist_of_cospherical {a b c d p : P}
(h : Cospherical ({a, b, c, d} : Set P)) (hapc : β a p c = Ο) (hbpd : β b p d = Ο) :
dist a b * dist c d + dist b c * dist d a = dist a c * dist b d := by |
have h' : Cospherical ({a, c, b, d} : Set P) := by rwa [Set.insert_comm c b {d}]
have hmul := mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi h' hapc hbpd
have hbp := left_dist_ne_zero_of_angle_eq_pi hbpd
have hβ : dist c d = dist c p / dist b p * dist a b := by
rw [dist_mul_of_eq_angle_of_dist_mul b p ... | 15 | 3,269,017.372472 | 2 | 2 | 1 | 2,389 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 192 | 194 | theorem Fintype.not_linearIndependent_iff [Fintype ΞΉ] :
Β¬LinearIndependent R v β β g : ΞΉ β R, β i, g i β’ v i = 0 β§ β i, g i β 0 := by |
simpa using not_iff_not.2 Fintype.linearIndependent_iff
| 1 | 2.718282 | 0 | 1 | 7 | 908 |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {Ξ± Ξ² : Type*} {s t : Set Ξ±}
noncomputable def encard (s : Set Ξ±) : ββ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 140 | 141 | theorem finite_of_encard_le_coe {k : β} (h : s.encard β€ k) : s.Finite := by |
rw [β encard_lt_top_iff]; exact h.trans_lt (WithTop.coe_lt_top _)
| 1 | 2.718282 | 0 | 0.5 | 14 | 454 |
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 46 | 46 | theorem mem_map {f : Ξ± β Ξ²} {y : Ξ²} {o : Option Ξ±} : y β o.map f β β x β o, f x = y := by | simp
| 1 | 2.718282 | 0 | 0 | 8 | 110 |
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
namespace List
variable {Ξ± : Type*}
section Sym2
protected def sym2 : List Ξ± β List (Sym2 Ξ±)
| [] => []
| x :: xs => (x :: xs).map (fun y => s(x, y)) ++ xs.sym2
theorem mem_sym2_cons_iff {x : Ξ±} {xs : List Ξ±} {z : Sym2 Ξ±} :
z β (x :: xs)... | Mathlib/Data/List/Sym.lean | 49 | 61 | theorem left_mem_of_mk_mem_sym2 {xs : List Ξ±} {a b : Ξ±}
(h : s(a, b) β xs.sym2) : a β xs := by |
induction xs with
| nil => exact (not_mem_nil _ h).elim
| cons x xs ih =>
rw [mem_cons]
rw [mem_sym2_cons_iff] at h
obtain (h | β¨c, hc, hβ© | h) := h
Β· rw [Sym2.eq_iff, β and_or_left] at h
exact .inl h.1
Β· rw [Sym2.eq_iff] at h
obtain (β¨rfl, rflβ© | β¨rfl, rflβ©) := h <;> simp [hc]
... | 11 | 59,874.141715 | 2 | 1.444444 | 9 | 1,529 |
import Mathlib.Tactic.NormNum.Core
import Mathlib.Tactic.HaveI
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Algebra.Ring.Int
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Tactic.ClearExclamation
import Mathlib.Data.Nat.Cast.Basic
set_option autoImplicit true
namespace Mathlib
open Lean hidi... | Mathlib/Tactic/NormNum/Basic.lean | 125 | 126 | theorem isintCast {R} [Ring R] (n m : β€) :
IsInt n m β IsInt (n : R) m := by | rintro β¨β¨β©β©; exact β¨rflβ©
| 1 | 2.718282 | 0 | 0 | 3 | 207 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 444 | 445 | theorem pow_rootMultiplicity_not_dvd {p : R[X]} (p0 : p β 0) (a : R) :
Β¬(X - C a) ^ (rootMultiplicity a p + 1) β£ p := by | rw [β rootMultiplicity_le_iff p0]
| 1 | 2.718282 | 0 | 1.5 | 32 | 1,561 |
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 | 42 | 43 | theorem eraseLead_support (f : R[X]) : f.eraseLead.support = f.support.erase f.natDegree := by |
simp only [eraseLead, support_erase]
| 1 | 2.718282 | 0 | 0.5 | 14 | 465 |
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 75 | 76 | theorem left_ne_of_oangle_ne_zero {pβ pβ pβ : P} (h : β‘ pβ pβ pβ β 0) : pβ β pβ := by |
rw [β @vsub_ne_zero V]; exact o.left_ne_zero_of_oangle_ne_zero h
| 1 | 2.718282 | 0 | 0.333333 | 6 | 356 |
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 | 43 | 45 | theorem toOuterMeasure_trim_eq_trim_toOuterMeasure (ΞΌ : Measure Ξ±) (hm : m β€ m0) :
@Measure.toOuterMeasure _ m (ΞΌ.trim hm) = @OuterMeasure.trim _ m ΞΌ.toOuterMeasure := by |
rw [Measure.trim, toMeasure_toOuterMeasure (ms := m)]
| 1 | 2.718282 | 0 | 0.888889 | 9 | 766 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
#align_import measure_theory.covering.vitali_family from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Metric Set Filter TopologicalSpace MeasureTheory.Measure
open Filter MeasureTheory Topology
variable {Ξ± : Type*}... | Mathlib/MeasureTheory/Covering/VitaliFamily.lean | 226 | 228 | theorem _root_.Filter.HasBasis.vitaliFamily {ΞΉ : Sort*} {p : ΞΉ β Prop} {s : ΞΉ β Set Ξ±} {x : Ξ±}
(h : (π x).HasBasis p s) : (v.filterAt x).HasBasis p (fun i β¦ {t β v.setsAt x | t β s i}) := by |
simpa only [β Set.setOf_inter_eq_sep] using h.smallSets.inf_principal _
| 1 | 2.718282 | 0 | 0 | 2 | 57 |
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