Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 |
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import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 81 | 83 | theorem Integrable.const_inner (c : E) (hf : Integrable f μ) :
Integrable (fun x => ⟪c, f x⟫) μ := by |
rw [← memℒp_one_iff_integrable] at hf ⊢; exact hf.const_inner c
| 1 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 315 | 315 | theorem average_zero : ⨍ _, (0 : E) ∂μ = 0 := by | rw [average, integral_zero]
| 1 |
import Mathlib.Control.Functor
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6eb4348fb1749c302d8abc303d34a"
universe u₀ u₁ u₂ v₀ v₁ v₂
open Function
class Bifunctor (F : Type u₀ → Type u₁ → Type u₂) where
bimap : ∀ {α α' β β'}, (α → α') → (β → β'... | Mathlib/Control/Bifunctor.lean | 92 | 93 | theorem fst_snd {α₀ α₁ β₀ β₁} (f : α₀ → α₁) (f' : β₀ → β₁) (x : F α₀ β₀) :
fst f (snd f' x) = bimap f f' x := by | simp [fst, bimap_bimap]
| 1 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 97 | 98 | theorem prod_univ_get [CommMonoid α] (l : List α) : ∏ i, l.get i = l.prod := by |
simp [Finset.prod_eq_multiset_prod]
| 1 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}
def map₂ (f : α → β → γ) (a : Option α) ... | Mathlib/Data/Option/NAry.lean | 99 | 100 | theorem map₂_map_right (f : α → γ → δ) (g : β → γ) :
map₂ f a (b.map g) = map₂ (fun a b => f a (g b)) a b := by | cases b <;> rfl
| 1 |
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Multiset.Basic
#align_import algebra.big_operators.multiset.basic from "leanprover-community/mathlib"@"6c5f73fd6f6cc83122788a80a27cdd54663609f4"
assert_not_exists MonoidWithZero
variable {F ι α β γ : Type*}
names... | Mathlib/Algebra/BigOperators/Group/Multiset.lean | 130 | 131 | theorem prod_replicate (n : ℕ) (a : α) : (replicate n a).prod = a ^ n := by |
simp [replicate, List.prod_replicate]
| 1 |
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
assert_not_exists Basis -- See `RingTheory.Ideal.Basis`
assert_not_exists ... | Mathlib/RingTheory/Ideal/Operations.lean | 99 | 100 | theorem mem_annihilator_span_singleton (g : M) (r : R) :
r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by | simp [mem_annihilator_span]
| 1 |
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {α β γ : Type*}
class Fi... | Mathlib/Data/Fintype/Basic.lean | 84 | 84 | theorem eq_univ_iff_forall : s = univ ↔ ∀ x, x ∈ s := by | simp [ext_iff]
| 1 |
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 80 | 80 | theorem exp_one_rpow (x : ℝ) : exp 1 ^ x = exp x := by | rw [← exp_mul, one_mul]
| 1 |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 60 | 61 | theorem natAbs_inj_of_nonneg_of_nonneg {a b : ℤ} (ha : 0 ≤ a) (hb : 0 ≤ b) :
natAbs a = natAbs b ↔ a = b := by | rw [← sq_eq_sq ha hb, ← natAbs_eq_iff_sq_eq]
| 1 |
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 210 | 210 | theorem add_zero (n : Num) : n + 0 = n := by | cases n <;> rfl
| 1 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.OpenImmersion
#align_import algebraic_geometry.gluing from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
set_option linter.uppercaseLean3 false
noncomputable section
universe u
open Topologica... | Mathlib/AlgebraicGeometry/Gluing.lean | 308 | 310 | theorem gluedCoverT'_snd_fst (x y z : 𝒰.J) :
gluedCoverT' 𝒰 x y z ≫ pullback.snd ≫ pullback.fst = pullback.fst ≫ pullback.snd := by |
delta gluedCoverT'; simp
| 1 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 125 | 127 | theorem mul_transvection_apply_same (a : n) (c : R) (M : Matrix n n R) :
(M * transvection i j c) a j = M a j + c * M a i := by |
simp [transvection, Matrix.mul_add, mul_comm]
| 1 |
import Mathlib.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
#align_import linear_algebra.free_module.finite.matrix from "leanprover-community/mathlib"@"b1c23399f01266afe392a0d8f71f599a0dad4f7b"
universe u u' v w
variable (R : Type u) (S : Type u') (M : Type v) (N ... | Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean | 59 | 61 | theorem FiniteDimensional.finrank_linearMap :
finrank S (M →ₗ[R] N) = finrank R M * finrank S N := by |
simp_rw [finrank, rank_linearMap, toNat_mul, toNat_lift]
| 1 |
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α : Type*} {N : α → Type*}
namespace DFinsupp
variable [DecidableEq α]
section NHasZero
variable [∀ a, DecidableEq (N a)] [∀ a, Zero (N a)] (f g : Π₀... | Mathlib/Data/DFinsupp/NeLocus.lean | 67 | 68 | theorem neLocus_comm : f.neLocus g = g.neLocus f := by |
simp_rw [neLocus, Finset.union_comm, ne_comm]
| 1 |
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
import Mathlib.Logic.Basic
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort*} {f : α → β}
@[reducible, simp] de... | Mathlib/Logic/Function/Basic.lean | 109 | 110 | theorem not_injective_iff : ¬ Injective f ↔ ∃ a b, f a = f b ∧ a ≠ b := by |
simp only [Injective, not_forall, exists_prop]
| 1 |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ι α β : Type*} {π : ι → Type*}
def symmDiff [Sup α] [SDiff α] (a b : α) : α :=
a \ b ⊔ b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 165 | 166 | theorem symmDiff_sdiff : a ∆ b \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) := by |
rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left]
| 1 |
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.RingTheory.Valuation.RankOne
import Mathlib.Topology.Algebra.Valuation
noncomputable section
open Filter Set Valuation
open scoped NNReal
variable {K : Type*} [hK : NormedField K] (h : IsNonarchimedean (norm : K → ℝ))
namespace Valued
variable {L : Typ... | Mathlib/Topology/Algebra/NormedValued.lean | 68 | 68 | theorem norm_nonneg (x : L) : 0 ≤ norm x := by | simp only [norm, NNReal.zero_le_coe]
| 1 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin_root from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open scoped Polynomial
open Polynomial
noncomputable sec... | Mathlib/RingTheory/IsAdjoinRoot.lean | 136 | 137 | theorem map_eq_zero_iff (h : IsAdjoinRoot S f) {p} : h.map p = 0 ↔ f ∣ p := by |
rw [← h.mem_ker_map, RingHom.mem_ker]
| 1 |
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α : Type*} (p : α → Bool) (l : List α) (n : ℕ)
namespace List
def rdrop : List α :=
l.take (l.leng... | Mathlib/Data/List/DropRight.lean | 74 | 74 | theorem rtake_nil : rtake ([] : List α) n = [] := by | simp [rtake]
| 1 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 668 | 670 | theorem preimage_mul_const_Ioc_of_neg (a b : α) {c : α} (h : c < 0) :
(fun x => x * c) ⁻¹' Ioc a b = Ico (b / c) (a / c) := by |
simp [← Ioi_inter_Iic, ← Ici_inter_Iio, h, inter_comm]
| 1 |
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Finset
namespace Nat
variable (p : ℕ → Prop)
section Count
variable [DecidablePred p]
def count (n : ℕ) : ℕ :=
(List.range n).... | Mathlib/Data/Nat/Count.lean | 94 | 96 | theorem count_succ' (n : ℕ) :
count p (n + 1) = count (fun k ↦ p (k + 1)) n + if p 0 then 1 else 0 := by |
rw [count_add', count_one]
| 1 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section Inter
@[simp]
theorem inter_nil (l : L... | Mathlib/Data/List/Lattice.lean | 134 | 135 | theorem inter_cons_of_mem (l₁ : List α) (h : a ∈ l₂) : (a :: l₁) ∩ l₂ = a :: l₁ ∩ l₂ := by |
simp [Inter.inter, List.inter, h]
| 1 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 104 | 106 | theorem ofFractionRing_sub (p q : FractionRing K[X]) :
ofFractionRing (p - q) = ofFractionRing p - ofFractionRing q := by |
simp only [Sub.sub, HSub.hSub, RatFunc.sub]
| 1 |
import Mathlib.Algebra.Group.Support
import Mathlib.Data.Int.Cast.Field
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.int.char_zero from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Nat Set
variable {α β : Type*}
namespace Int
@[simp, norm_cast]
theorem cast_div_charZe... | Mathlib/Data/Int/CharZero.lean | 33 | 35 | theorem cast_div_ofNat_charZero {k : Type*} [DivisionRing k] [CharZero k] {m n : ℕ}
(n_dvd : n ∣ m) : (((m : ℤ) / (n : ℤ) : ℤ) : k) = m / n := by |
rw [cast_div_charZero (Int.ofNat_dvd.mpr n_dvd), cast_natCast, cast_natCast]
| 1 |
import Mathlib.Deprecated.Group
#align_import deprecated.ring from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
universe u v w
variable {α : Type u}
structure IsSemiringHom {α : Type u} {β : Type v} [Semiring α] [Semiring β] (f : α → β) : Prop where
map_zero : f 0 = 0
map... | Mathlib/Deprecated/Ring.lean | 67 | 68 | theorem to_isAddMonoidHom (hf : IsSemiringHom f) : IsAddMonoidHom f :=
{ ‹IsSemiringHom f› with map_add := by | apply @‹IsSemiringHom f›.map_add }
| 1 |
import Mathlib.Order.Filter.Prod
#align_import order.filter.n_ary from "leanprover-community/mathlib"@"78f647f8517f021d839a7553d5dc97e79b508dea"
open Function Set
open Filter
namespace Filter
variable {α α' β β' γ γ' δ δ' ε ε' : Type*} {m : α → β → γ} {f f₁ f₂ : Filter α}
{g g₁ g₂ : Filter β} {h h₁ h₂ : Filt... | Mathlib/Order/Filter/NAry.lean | 120 | 121 | theorem map₂_sup_left : map₂ m (f₁ ⊔ f₂) g = map₂ m f₁ g ⊔ map₂ m f₂ g := by |
simp_rw [← map_prod_eq_map₂, sup_prod, map_sup]
| 1 |
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 | 107 | 108 | theorem SameCycle.conj : SameCycle f x y → SameCycle (g * f * g⁻¹) (g x) (g y) := by |
simp [sameCycle_conj]
| 1 |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section General
variable {α : Type*} {g : Gen... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 35 | 35 | theorem terminatedAt_iff_s_terminatedAt : g.TerminatedAt n ↔ g.s.TerminatedAt n := by | rfl
| 1 |
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
protected theorem div_eq_inv_mul : a / b = b⁻¹ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 72 | 73 | theorem coe_div (hr : r ≠ 0) : (↑(p / r) : ℝ≥0∞) = p / r := by |
rw [div_eq_mul_inv, div_eq_mul_inv, coe_mul, coe_inv hr]
| 1 |
import Mathlib.Algebra.CharP.Pi
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.CharP.Subring
import Mathlib.Algebra.Ring.Pi
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Ring.Subring.Basic
... | Mathlib/RingTheory/Perfection.lean | 137 | 138 | theorem coeff_frobenius (f : Ring.Perfection R p) (n : ℕ) :
coeff R p (n + 1) (frobenius _ p f) = coeff R p n f := by | apply coeff_pow_p f n
| 1 |
import Mathlib.Algebra.Order.Field.Power
import Mathlib.NumberTheory.Padics.PadicVal
#align_import number_theory.padics.padic_norm from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
def padicNorm (p : ℕ) (q : ℚ) : ℚ :=
if q = 0 then 0 else (p : ℚ) ^ (-padicValRat p q)
#align padic_n... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 81 | 82 | theorem padicNorm_p (hp : 1 < p) : padicNorm p p = (p : ℚ)⁻¹ := by |
simp [padicNorm, (pos_of_gt hp).ne', padicValNat.self hp]
| 1 |
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "leanprover-community/mathlib"@"9b2660e1b25419042c8da10bf411aa3c67f14383"
open Nat
def ack : ℕ → ℕ → ℕ
| 0, n => n + 1
| m + 1, 0 ... | Mathlib/Computability/Ackermann.lean | 70 | 70 | theorem ack_zero (n : ℕ) : ack 0 n = n + 1 := by | rw [ack]
| 1 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S ℕ]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 79 | 80 | theorem smeval_zero : (0 : R[X]).smeval x = 0 := by |
simp only [smeval_eq_sum, smul_pow, sum_zero_index]
| 1 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 178 | 179 | theorem vanishingIdeal_singleton (x : PrimeSpectrum R) :
vanishingIdeal ({x} : Set (PrimeSpectrum R)) = x.asIdeal := by | simp [vanishingIdeal]
| 1 |
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Data.Rat.Floor
#align_import algebra.continued_fractions.computation.terminates_iff_rat from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b3... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 197 | 200 | theorem coe_stream'_rat_eq :
((IntFractPair.stream q).map (Option.map (mapFr (↑))) : Stream' <| Option <| IntFractPair K) =
IntFractPair.stream v := by |
funext n; exact IntFractPair.coe_stream_nth_rat_eq v_eq_q n
| 1 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 106 | 108 | theorem HasDerivAt.complexToReal_fderiv' {f : ℂ → E} {x : ℂ} {f' : E} (h : HasDerivAt f f' x) :
HasFDerivAt f (reCLM.smulRight f' + I • imCLM.smulRight f') x := by |
simpa only [Complex.restrictScalars_one_smulRight'] using h.hasFDerivAt.restrictScalars ℝ
| 1 |
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 112 | 112 | theorem and_intro : ∀ {a b : Bool}, a → b → a && b := by | decide
| 1 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 119 | 120 | theorem angle_const_sub (v : V) (v₁ v₂ v₃ : V) : ∠ (v - v₁) (v - v₂) (v - v₃) = ∠ v₁ v₂ v₃ := by |
simpa only [vsub_eq_sub] using angle_const_vsub v v₁ v₂ v₃
| 1 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 87 | 88 | theorem fib_add_two {n : ℕ} : fib (n + 2) = fib n + fib (n + 1) := by |
simp [fib, Function.iterate_succ_apply']
| 1 |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 138 | 139 | theorem card_fintypeIic : Fintype.card (Set.Iic b) = b + 1 := by |
rw [Fintype.card_ofFinset, card_Iic]
| 1 |
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 | 128 | 130 | theorem hasFiniteIntegral_iff_ofNNReal {f : α → ℝ≥0} :
HasFiniteIntegral (fun x => (f x : ℝ)) μ ↔ (∫⁻ a, f a ∂μ) < ∞ := by |
simp [hasFiniteIntegral_iff_norm]
| 1 |
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.DifferentialObject
#align_import algebra.homology.differential_object from "leanprover-community/mathlib"@"b535c2d5d996acd9b0554b76395d9c920e186f4f"
open CategoryTheory CategoryTheory.Limits
open scoped Classical
noncomputable secti... | Mathlib/Algebra/Homology/DifferentialObject.lean | 53 | 54 | theorem objEqToHom_d {x y : β} (h : x = y) :
X.objEqToHom h ≫ X.d y = X.d x ≫ X.objEqToHom (by cases h; rfl) := by | cases h; dsimp; simp
| 1 |
import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Init.Data.Int.Order
set_option autoImplicit true
namespace Linarith
theorem lt_irrefl {α : Type u} ... | Mathlib/Tactic/Linarith/Lemmas.lean | 36 | 37 | theorem le_of_le_of_eq {α} [OrderedSemiring α] {a b : α} (ha : a ≤ 0) (hb : b = 0) : a + b ≤ 0 := by |
simp [*]
| 1 |
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Multiset.Basic
#align_import algebra.big_operators.multiset.basic from "leanprover-community/mathlib"@"6c5f73fd6f6cc83122788a80a27cdd54663609f4"
assert_not_exists MonoidWithZero
variable {F ι α β γ : Type*}
names... | Mathlib/Algebra/BigOperators/Group/Multiset.lean | 105 | 106 | theorem prod_pair (a b : α) : ({a, b} : Multiset α).prod = a * b := by |
rw [insert_eq_cons, prod_cons, prod_singleton]
| 1 |
import Mathlib.CategoryTheory.Adjunction.Unique
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Sites.Sheaf
import Mathlib.CategoryTheory.Limits.Preserves.Finite
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open Limits
variable {C : Type u₁} [Category.{v₁} C] (J : Grothendiec... | Mathlib/CategoryTheory/Sites/Sheafification.lean | 100 | 102 | theorem sheafifyMap_comp {P Q R : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (γ : Q ⟶ R) :
sheafifyMap J (η ≫ γ) = sheafifyMap J η ≫ sheafifyMap J γ := by |
simp [sheafifyMap, sheafify]
| 1 |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.Analytic.Basic
#align_import measure_theory.integral.circle_integral from "leanprover-communit... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 105 | 106 | theorem circleMap_sub_center (c : ℂ) (R : ℝ) (θ : ℝ) : circleMap c R θ - c = circleMap 0 R θ := by |
simp [circleMap]
| 1 |
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 | 110 | 110 | theorem asAlgebraHom_single_one (g : G) : asAlgebraHom ρ (Finsupp.single g 1) = ρ g := by | simp
| 1 |
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 uₘ uₙ uₒ
variable {α : Type u} {o n m... | Mathlib/Data/Matrix/Notation.lean | 174 | 175 | theorem cons_dotProduct_cons (x : α) (v : Fin n → α) (y : α) (w : Fin n → α) :
dotProduct (vecCons x v) (vecCons y w) = x * y + dotProduct v w := by | simp
| 1 |
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {α β γ : Type*}
class Fi... | Mathlib/Data/Fintype/Basic.lean | 113 | 114 | theorem univ_eq_empty_iff : (univ : Finset α) = ∅ ↔ IsEmpty α := by |
rw [← not_nonempty_iff, ← univ_nonempty_iff, not_nonempty_iff_eq_empty]
| 1 |
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 | 114 | 116 | theorem EuclideanSpace.norm_eq {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n]
(x : EuclideanSpace 𝕜 n) : ‖x‖ = √(∑ i, ‖x i‖ ^ 2) := by |
simpa only [Real.coe_sqrt, NNReal.coe_sum] using congr_arg ((↑) : ℝ≥0 → ℝ) x.nnnorm_eq
| 1 |
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
universe u v w
namespace MvPolynomial
variable {σ τ : Type*} {R : Type u} {S : Type v} {r : R} {e : ℕ} {n m : σ}
section CommSemiring
variable... | Mathlib/Algebra/MvPolynomial/Supported.lean | 91 | 92 | theorem mem_supported_vars (p : MvPolynomial σ R) : p ∈ supported R (↑p.vars : Set σ) := by |
rw [mem_supported]
| 1 |
import Mathlib.Algebra.Group.WithOne.Defs
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.GroupWithZero.WithZero
import Mathlib.Algebra.Order.Group.Units
import Mathlib.Algebra.Order.GroupWithZero.Synonym
import Mathlib.Algebra.Order.Monoid.Basic
imp... | Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | 128 | 129 | theorem le_of_le_mul_right (h : c ≠ 0) (hab : a * c ≤ b * c) : a ≤ b := by |
simpa only [mul_inv_cancel_right₀ h] using mul_le_mul_right' hab c⁻¹
| 1 |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open Filter Topology
variable {ι : Sort*} {α β X Y : Type*}... | Mathlib/Topology/Filter.lean | 125 | 125 | theorem nhds_top : 𝓝 (⊤ : Filter α) = ⊤ := by | simp [nhds_eq]
| 1 |
import Mathlib.Tactic.Ring.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Tactic.Conv
import Mathlib.Util.Qq
set_option autoImplicit true
-- In this file we would like to be able to use multi-character auto-implicits.
set_option relaxedAutoImplicit true
namespace Mathlib.Tactic
open Lean hiding Rat
open Qq Me... | Mathlib/Tactic/Ring/RingNF.lean | 123 | 123 | theorem int_rawCast_neg {R} [Ring R] : (Int.rawCast (.negOfNat n) : R) = -Nat.rawCast n := by | simp
| 1 |
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
universe u v w
open scoped Classical Polynomial
open Polynomial
vari... | Mathlib/FieldTheory/IsAlgClosed/Basic.lean | 68 | 69 | theorem IsAlgClosed.splits_codomain {k K : Type*} [Field k] [IsAlgClosed k] [Field K] {f : K →+* k}
(p : K[X]) : p.Splits f := by | convert IsAlgClosed.splits (p.map f); simp [splits_map_iff]
| 1 |
import Mathlib.Algebra.Category.ModuleCat.Abelian
import Mathlib.CategoryTheory.Limits.Shapes.Images
#align_import algebra.category.Module.images from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open CategoryTheory
open CategoryTheory.Limits
universe u v
namespace ModuleCat
set_op... | Mathlib/Algebra/Category/ModuleCat/Images.lean | 117 | 119 | theorem imageIsoRange_hom_subtype {G H : ModuleCat.{v} R} (f : G ⟶ H) :
(imageIsoRange f).hom ≫ ModuleCat.ofHom f.range.subtype = Limits.image.ι f := by |
erw [← imageIsoRange_inv_image_ι f, Iso.hom_inv_id_assoc]
| 1 |
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Cardinality
#align_import data.complex.cardinality from "leanprover-community/mathlib"@"1c4e18434eeb5546b212e830b2b39de6a83c473c"
-- Porting note: the lemmas `mk_complex` and `mk_univ_complex` should be in the namespace `Cardinal`
-- like their real counter... | Mathlib/Data/Complex/Cardinality.lean | 31 | 31 | theorem mk_univ_complex : #(Set.univ : Set ℂ) = 𝔠 := by | rw [mk_univ, mk_complex]
| 1 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
... | Mathlib/RingTheory/PowerSeries/Basic.lean | 150 | 151 | theorem coeff_def {s : Unit →₀ ℕ} {n : ℕ} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by |
erw [coeff, ← h, ← Finsupp.unique_single s]
| 1 |
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
#align_import combinatorics.quiver.single_obj from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
namespace Quiver
-- Porting note: Removed `deriving Unique`.
@[nolint unusedArguments]
def SingleObj ... | Mathlib/Combinatorics/Quiver/SingleObj.lean | 110 | 112 | theorem toPrefunctor_symm_comp (f : SingleObj α ⥤q SingleObj β) (g : SingleObj β ⥤q SingleObj γ) :
toPrefunctor.symm (f ⋙q g) = toPrefunctor.symm g ∘ toPrefunctor.symm f := by |
simp only [Equiv.symm_apply_eq, toPrefunctor_comp, Equiv.apply_symm_apply]
| 1 |
import Mathlib.Data.List.Join
#align_import data.list.permutation from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we don't import algebra
assert_not_exists Monoid
open Nat
variable {α β : Type*}
namespace List
theorem permutationsAux2_fst (t : α) (ts : List α) (r : L... | Mathlib/Data/List/Permutation.lean | 77 | 79 | theorem permutationsAux2_append (t : α) (ts : List α) (r : List β) (ys : List α) (f : List α → β) :
(permutationsAux2 t ts nil ys f).2 ++ r = (permutationsAux2 t ts r ys f).2 := by |
induction ys generalizing f <;> simp [*]
| 1 |
import Mathlib.Geometry.Manifold.ContMDiff.Defs
open Set Filter Function
open scoped Topology Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
(I : Mode... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 262 | 262 | theorem smooth_one [One M'] : Smooth I I' (1 : M → M') := by | simp only [Pi.one_def, smooth_const]
| 1 |
import Mathlib.Analysis.InnerProductSpace.Adjoint
#align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
open InnerProductSpace RCLike ContinuousLinearMap
open scoped InnerProduct ComplexConjugate
namespace ContinuousLinearMap
variable... | Mathlib/Analysis/InnerProductSpace/Positive.lean | 67 | 68 | theorem IsPositive.inner_nonneg_right {T : E →L[𝕜] E} (hT : IsPositive T) (x : E) :
0 ≤ re ⟪x, T x⟫ := by | rw [inner_re_symm]; exact hT.inner_nonneg_left x
| 1 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Analysis.Complex.Arg
#align_import ring_theory.polynomial.cyclotomic.eval from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf16... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 36 | 37 | theorem eval₂_one_cyclotomic_prime {R S : Type*} [CommRing R] [Semiring S] (f : R →+* S) {p : ℕ}
[Fact p.Prime] : eval₂ f 1 (cyclotomic p R) = p := by | simp
| 1 |
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 125 | 125 | theorem rpow_zero (x : ℝ) : x ^ (0 : ℝ) = 1 := by | simp [rpow_def]
| 1 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Data.Rat.Cast.Defs
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {F ι α β : Type*}
namespace Rat
open Rat
section WithDivRing
variable [DivisionRing α]
@[simp, norm_cast]
th... | Mathlib/Data/Rat/Cast/CharZero.lean | 46 | 46 | theorem cast_eq_zero [CharZero α] {n : ℚ} : (n : α) = 0 ↔ n = 0 := by | rw [← cast_zero, cast_inj]
| 1 |
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 230 | 233 | theorem fderiv_continuousMultilinear_apply_const_apply (hc : DifferentiableAt 𝕜 c x)
(u : ∀ i, M i) (m : E) :
(fderiv 𝕜 (fun y ↦ (c y) u) x) m = (fderiv 𝕜 c x) m u := by |
simp [fderiv_continuousMultilinear_apply_const hc]
| 1 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
#align_import linear_algebra.clifford_algebra.star from "leanprover-community/mathlib"@"4d66277cfec381260ba05c68f9ae6ce2a118031d"
variable {R : Type*} [CommRing R]
variable {M : Type*} [AddCommGroup M] [Module R M]
variable {Q : QuadraticForm R M}
namespac... | Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean | 57 | 58 | theorem star_smul (r : R) (x : CliffordAlgebra Q) : star (r • x) = r • star x := by |
rw [star_def, star_def, map_smul, map_smul]
| 1 |
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set F... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 84 | 86 | theorem measure_iUnion_fintype_le [Fintype ι] (μ : F) (s : ι → Set α) :
μ (⋃ i, s i) ≤ ∑ i, μ (s i) := by |
simpa using measure_biUnion_finset_le Finset.univ s
| 1 |
import Mathlib.Algebra.Order.Pointwise
import Mathlib.Analysis.NormedSpace.SphereNormEquiv
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
open Set Function Metric MeasurableSpace intervalIntegral
open s... | Mathlib/MeasureTheory/Constructions/HaarToSphere.lean | 62 | 63 | theorem toSphere_apply_univ' : μ.toSphere univ = dim E * μ (ball 0 1 \ {0}) := by |
rw [μ.toSphere_apply' .univ, image_univ, Subtype.range_coe, Ioo_smul_sphere_zero] <;> simp
| 1 |
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 58 | 59 | theorem norm_exp_ofReal_mul_I (t : ℝ) : ‖exp (t * I)‖ = 1 := by |
simp only [norm_eq_abs, abs_exp_ofReal_mul_I]
| 1 |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 83 | 84 | theorem intervalIntegrable_iff : IntervalIntegrable f μ a b ↔ IntegrableOn f (Ι a b) μ := by |
rw [uIoc_eq_union, integrableOn_union, IntervalIntegrable]
| 1 |
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import ring_theory.algebraic_independent from "leanprove... | Mathlib/RingTheory/AlgebraicIndependent.lean | 134 | 135 | theorem coe_range : AlgebraicIndependent R ((↑) : range x → A) := by |
simpa using hx.comp _ (rangeSplitting_injective x)
| 1 |
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 112 | 113 | theorem contDiffAt_of_subsingleton [Subsingleton F] : ContDiffAt 𝕜 n f x := by |
rw [Subsingleton.elim f fun _ => 0]; exact contDiffAt_const
| 1 |
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) (φ : A ≃* B) :
Con (G ∗ Multiplicative ℤ) :=
conGen (fun x y => ∃ (a : A),
x = inr (ofAdd 1) * inl (a : G) ∧
... | Mathlib/GroupTheory/HNNExtension.lean | 81 | 83 | theorem inv_t_mul_of (b : B) :
t⁻¹ * (of (b : G) : HNNExtension G A B φ) = of (φ.symm b : G) * t⁻¹ := by |
rw [equiv_symm_eq_conj]; simp
| 1 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 116 | 117 | theorem Ico_mem_nhds_iff [NoMinOrder α] {a b x : α} : Ico a b ∈ 𝓝 x ↔ x ∈ Ioo a b := by |
rw [← interior_Ico, mem_interior_iff_mem_nhds]
| 1 |
import Mathlib.Data.List.Nodup
#align_import data.prod.tprod from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open List Function
universe u v
variable {ι : Type u} {α : ι → Type v} {i j : ι} {l : List ι} {f : ∀ i, α i}
namespace List
variable (α)
abbrev TProd (l : List ι) : Type v... | Mathlib/Data/Prod/TProd.lean | 90 | 90 | theorem elim_self (v : TProd α (i :: l)) : v.elim (l.mem_cons_self i) = v.1 := by | simp [TProd.elim]
| 1 |
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_topology.fundamental_groupoid.basic from "leanprover-community/mathlib"@"3d7987cda72abc473... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 148 | 149 | theorem transReflReparamAux_one : transReflReparamAux 1 = 1 := by |
set_option tactic.skipAssignedInstances false in norm_num [transReflReparamAux]
| 1 |
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
open Nat Function
namespace List
theorem rel_of_pairwise_cons (p : (a :: l).Pairwise R) : ∀ {a'}, a' ∈ l → R a a' :=
(pairwise_cons.1 p).1 _
theorem Pairwise.of_cons (p : (a :: l).Pairwise R) : Pairwise R l :=
(pairwise_cons.1 p).2
theorem... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 106 | 106 | theorem pairwise_pair {a b : α} : Pairwise R [a, b] ↔ R a b := by | simp
| 1 |
import Mathlib.Algebra.Polynomial.Smeval
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.RingTheory.Polynomial.Pochhammer
section Multichoose
open Function Polynomial
class BinomialRing (R : Type*) [AddCommMonoid R] [Pow R ℕ] where
nsmul_right_injective (n : ℕ) (h : n ≠ 0) : Injective (n • · : R →... | Mathlib/RingTheory/Binomial.lean | 101 | 103 | theorem ascPochhammer_smeval_eq_eval [Semiring R] (r : R) (n : ℕ) :
(ascPochhammer ℕ n).smeval r = (ascPochhammer R n).eval r := by |
rw [eval_eq_smeval, ascPochhammer_smeval_cast R]
| 1 |
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith
#align_import data.ordmap.ordset from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
variable... | Mathlib/Data/Ordmap/Ordset.lean | 157 | 157 | theorem size_dual (t : Ordnode α) : size (dual t) = size t := by | cases t <;> rfl
| 1 |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Star.Unitary
import Mathlib.Data.Nat.ModEq
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic.Monotonicity
#align_import number_theory.pell_matiyasevic from "leanprover-community/mathlib"@"795b501869b9f... | Mathlib/NumberTheory/PellMatiyasevic.lean | 151 | 151 | theorem xn_one : xn a1 1 = a := by | simp
| 1 |
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.torus_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
variable {n : ℕ}
variable {E : Type*} [NormedAddCommGroup E]
noncomputa... | Mathlib/MeasureTheory/Integral/TorusIntegral.lean | 88 | 89 | theorem torusMap_eq_center_iff {c : ℂⁿ} {R : ℝⁿ} {θ : ℝⁿ} : torusMap c R θ = c ↔ R = 0 := by |
simp [funext_iff, torusMap, exp_ne_zero]
| 1 |
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 | 59 | 64 | theorem nim_def (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) := inferInstance
nim o =
PGame.mk o.out.α o.out.α (fun o₂ => nim (Ordinal.typein (· < ·) o₂)) fun o₂ =>
nim (Ordinal.typein (· < ·) o₂) := by |
rw [nim]; rfl
| 1 |
import Mathlib.Algebra.Lie.Matrix
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.Tactic.NoncommRing
#align_import algebra.lie.skew_adjoint from "leanprover-community/mathlib"@"075b3f7d19b9da85a0b54b3e33055a74fc388dec"
universe u v w w₁
section SkewAdjointMatrices
open scoped Matrix
variabl... | Mathlib/Algebra/Lie/SkewAdjoint.lean | 142 | 145 | theorem skewAdjointMatricesLieSubalgebraEquiv_apply (P : Matrix n n R) (h : Invertible P)
(A : skewAdjointMatricesLieSubalgebra J) :
↑(skewAdjointMatricesLieSubalgebraEquiv J P h A) = P⁻¹ * (A : Matrix n n R) * P := by |
simp [skewAdjointMatricesLieSubalgebraEquiv]
| 1 |
import Mathlib.Deprecated.Group
#align_import deprecated.ring from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
universe u v w
variable {α : Type u}
structure IsSemiringHom {α : Type u} {β : Type v} [Semiring α] [Semiring β] (f : α → β) : Prop where
map_zero : f 0 = 0
map... | Mathlib/Deprecated/Ring.lean | 114 | 115 | theorem map_sub (hf : IsRingHom f) : f (x - y) = f x - f y := by |
simp [sub_eq_add_neg, hf.map_add, hf.map_neg]
| 1 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.Solvable
import Mathlib.LinearAlgebra.Dual
#align_import algebra.lie.character from "leanprover-community/mathlib"@"132328c4dd48da87adca5d408ca54f315282b719"
universe u v w w₁
namespace LieAlgebra
variable (R : Type u) (L : Type v) [CommRing R] [LieR... | Mathlib/Algebra/Lie/Character.lean | 44 | 45 | theorem lieCharacter_apply_lie (χ : LieCharacter R L) (x y : L) : χ ⁅x, y⁆ = 0 := by |
rw [LieHom.map_lie, LieRing.of_associative_ring_bracket, mul_comm, sub_self]
| 1 |
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f : α → β} [DecidablePred p] : DecidablePred (p ∘ f) :=
inferInstanceAs <| DecidablePred fun x => p (f x)
@[deprecated] alias proofIrrel := proof_irrel
theorem Function.id_def : @id α = fun x => x := rfl
al... | .lake/packages/batteries/Batteries/Logic.lean | 106 | 109 | theorem heq_eqRec_iff_heq {α : Sort _} {a : α} {motive : (a' : α) → a = a' → Sort _}
(x : motive a (rfl : a = a)) {a' : α} (e : a = a') {β : Sort _} (y : β) :
HEq y (@Eq.rec α a motive x a' e) ↔ HEq y x := by |
subst e; rfl
| 1 |
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 | 588 | 590 | theorem mem_fundamentalFrontier :
x ∈ fundamentalFrontier G s ↔ x ∈ s ∧ ∃ g : G, g ≠ 1 ∧ x ∈ g • s := by |
simp [fundamentalFrontier]
| 1 |
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 213 | 213 | theorem zero_add (n : Num) : 0 + n = n := by | cases n <;> rfl
| 1 |
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Flip
variable (xs : Vector α n) (ys : Vector β n)
| Mathlib/Data/Vector/MapLemmas.lean | 385 | 387 | theorem map₂_flip (f : α → β → γ) :
map₂ f xs ys = map₂ (flip f) ys xs := by |
induction xs, ys using Vector.inductionOn₂ <;> simp_all[flip]
| 1 |
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 | 112 | 113 | theorem xgcd_val (x y) : xgcd x y = (gcdA x y, gcdB x y) := by |
unfold gcdA gcdB; cases xgcd x y; rfl
| 1 |
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Unary
variable (xs : Vector α n) (f₁ : β → σ₁ → σ₁ × γ) (f₂ : α → σ₂ → σ₂ × β)
@[simp]
theorem mapAccumr_mapAccumr :
mapAccumr f₁ (mapAccumr f₂ xs s₂).snd s₁
= let m := (... | Mathlib/Data/Vector/MapLemmas.lean | 43 | 47 | theorem map_mapAccumr (f₁ : β → γ) :
(map f₁ (mapAccumr f₂ xs s).snd) = (mapAccumr (fun x s =>
let r := (f₂ x s); (r.fst, f₁ r.snd)
) xs s).snd := by |
induction xs using Vector.revInductionOn generalizing s <;> simp_all
| 1 |
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
open Complex Set
open scoped Topology
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E]
variable {f g : E →... | Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean | 24 | 25 | theorem analyticOn_cexp : AnalyticOn ℂ exp univ := by |
rw [analyticOn_univ_iff_differentiable]; exact differentiable_exp
| 1 |
import Mathlib.Order.BoundedOrder
import Mathlib.Order.MinMax
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Order.Monoid.Defs
#align_import algebra.order.monoid.canonical.defs from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
universe u
variable {α : Type u}
class ExistsMulOf... | Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean | 199 | 200 | theorem le_iff_exists_mul' : a ≤ b ↔ ∃ c, b = c * a := by |
simp only [mul_comm _ a, le_iff_exists_mul]
| 1 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 109 | 110 | theorem interior_setOf_le_im (a : ℝ) : interior { z : ℂ | a ≤ z.im } = { z | a < z.im } := by |
simpa only [interior_Ici] using interior_preimage_im (Ici a)
| 1 |
import Mathlib.Data.Set.Lattice
import Mathlib.Init.Set
import Mathlib.Control.Basic
import Mathlib.Lean.Expr.ExtraRecognizers
#align_import data.set.functor from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u
open Function
namespace Set
variable {α β : Type u} {s : Set α} ... | Mathlib/Data/Set/Functor.lean | 149 | 150 | theorem mem_of_mem_image_val (ha : a ∈ (γ : Set α)) : ⟨a, image_val_subset ha⟩ ∈ γ := by |
rcases ha with ⟨_, ha, rfl⟩; exact ha
| 1 |
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 |
import Mathlib.CategoryTheory.Iso
import Mathlib.CategoryTheory.EssentialImage
import Mathlib.CategoryTheory.Types
import Mathlib.CategoryTheory.Opposites
import Mathlib.Data.Rel
#align_import category_theory.category.Rel from "leanprover-community/mathlib"@"afad8e438d03f9d89da2914aa06cb4964ba87a18"
namespace Cate... | Mathlib/CategoryTheory/Category/RelCat.lean | 65 | 66 | theorem rel_comp_apply₂ {X Y Z : RelCat} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) (z : Z) :
(f ≫ g) x z ↔ ∃ y, f x y ∧ g y z := by | rfl
| 1 |
import Mathlib.Dynamics.BirkhoffSum.Basic
import Mathlib.Algebra.Module.Basic
open Finset
section birkhoffAverage
variable (R : Type*) {α M : Type*} [DivisionSemiring R] [AddCommMonoid M] [Module R M]
def birkhoffAverage (f : α → α) (g : α → M) (n : ℕ) (x : α) : M := (n : R)⁻¹ • birkhoffSum f g n x
| Mathlib/Dynamics/BirkhoffSum/Average.lean | 44 | 45 | theorem birkhoffAverage_zero (f : α → α) (g : α → M) (x : α) :
birkhoffAverage R f g 0 x = 0 := by | simp [birkhoffAverage]
| 1 |
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