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 | eval_complexity float64 0 1 |
|---|---|---|---|---|---|---|
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 208 | 209 | theorem mul_rotate' (a b c : G) : a * (b * c) = b * (c * a) := by |
simp only [mul_left_comm, mul_comm]
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import Mathlib.Algebra.Group.Defs
import Mathlib.Control.Functor
#align_import control.applicative from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u v w
section Lemmas
open Function
variable {F : Type u → Type v}
variable [Applicative F] [LawfulApplicative F]
variable {α ... | Mathlib/Control/Applicative.lean | 31 | 33 | theorem Applicative.map_seq_map (f : α → β → γ) (g : σ → β) (x : F α) (y : F σ) :
f <$> x <*> g <$> y = ((· ∘ g) ∘ f) <$> x <*> y := by |
simp [flip, functor_norm]
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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 | 146 | 147 | theorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by |
simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)
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import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerS... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 206 | 208 | theorem map_exp : map (f : A →+* A') (exp A) = exp A' := by |
ext
simp
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import Mathlib.Logic.Equiv.Option
import Mathlib.Order.RelIso.Basic
import Mathlib.Order.Disjoint
import Mathlib.Order.WithBot
import Mathlib.Tactic.Monotonicity.Attr
import Mathlib.Util.AssertExists
#align_import order.hom.basic from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
open ... | Mathlib/Order/Hom/Basic.lean | 207 | 209 | theorem lt_map_inv_iff (f : F) {a : α} {b : β} : a < EquivLike.inv f b ↔ f a < b := by |
rw [← map_lt_map_iff f]
simp only [EquivLike.apply_inv_apply]
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import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 153 | 154 | theorem U_add_one (n : ℤ) : U R (n + 1) = 2 * X * U R n - U R (n - 1) := by |
linear_combination (norm := ring_nf) U_add_two R (n - 1)
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import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RCLike
open scoped ComplexConjugate
variable {𝕜 E F G : Type... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 92 | 95 | theorem adjointAux_adjointAux (A : E →L[𝕜] F) : adjointAux (adjointAux A) = A := by |
ext v
refine ext_inner_left 𝕜 fun w => ?_
rw [adjointAux_inner_right, adjointAux_inner_left]
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import Mathlib.Combinatorics.Young.YoungDiagram
#align_import combinatorics.young.semistandard_tableau from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
structure SemistandardYoungTableau (μ : YoungDiagram) where
entry : ℕ → ℕ → ℕ
row_weak' : ∀ {i j1 j2 : ℕ}, j1 < j2 → (i, ... | Mathlib/Combinatorics/Young/SemistandardTableau.lean | 136 | 140 | theorem col_weak {μ : YoungDiagram} (T : SemistandardYoungTableau μ) {i1 i2 j : ℕ} (hi : i1 ≤ i2)
(cell : (i2, j) ∈ μ) : T i1 j ≤ T i2 j := by |
cases' eq_or_lt_of_le hi with h h
· rw [h]
· exact le_of_lt (T.col_strict h cell)
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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).α (· < ·) := inferInstance
HEq (nim o).moveRight fun i : o.out.α => nim (typein (· < ·) i) := by | rw [nim_def]; rfl
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import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.lattice from "leanprover-community/mathlib"@"52fa514ec337dd970d71d8de8d0fd68b455a1e54"
assert_not_exists MonoidWithZero
open Set
namespace Nat
open scoped Classical
noncomputable instance : ... | Mathlib/Data/Nat/Lattice.lean | 91 | 98 | theorem nonempty_of_pos_sInf {s : Set ℕ} (h : 0 < sInf s) : s.Nonempty := by |
by_contra contra
rw [Set.not_nonempty_iff_eq_empty] at contra
have h' : sInf s ≠ 0 := ne_of_gt h
apply h'
rw [Nat.sInf_eq_zero]
right
assumption
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import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
import Mathlib.Data.Set.Lattice
#align_import group_theory.sylow from "leanprove... | Mathlib/GroupTheory/Sylow.lean | 493 | 495 | theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by |
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
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import Mathlib.Data.List.Basic
#align_import data.list.infix from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
open Nat
variable {α β : Type*}
namespace List
variable {l l₁ l₂ l₃ : List α} {a b : α} {m n : ℕ}
section Fix
#align list.prefix_append List.prefix_append
#align list.... | Mathlib/Data/List/Infix.lean | 70 | 70 | theorem prefix_concat (a : α) (l) : l <+: concat l a := by | simp
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import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.LinearAlgebra.AffineSpace.Restrict
import Mathlib.Tactic.FailIfNoProgress
#align_import analysis.normed_space.affine_isomet... | Mathlib/Analysis/NormedSpace/AffineIsometry.lean | 86 | 88 | theorem toAffineMap_injective : Injective (toAffineMap : (P →ᵃⁱ[𝕜] P₂) → P →ᵃ[𝕜] P₂) := by |
rintro ⟨f, _⟩ ⟨g, _⟩ rfl
rfl
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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 | 143 | 146 | theorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :
∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by |
rw [prod_univ_castSucc, prod_univ_four]
rfl
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import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb370b"
assert_not_exists MonoidWithZero
universe u
variable {m n : ℕ}
def finZeroEquiv : Fin 0 ≃ Empty :=
Equiv.equivEmpty _
#align fin_... | Mathlib/Logic/Equiv/Fin.lean | 56 | 60 | theorem Fin.preimage_apply_01_prod {α : Fin 2 → Type u} (s : Set (α 0)) (t : Set (α 1)) :
(fun f : ∀ i, α i => (f 0, f 1)) ⁻¹' s ×ˢ t =
Set.pi Set.univ (Fin.cons s <| Fin.cons t finZeroElim) := by |
ext f
simp [Fin.forall_fin_two]
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import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 69 | 74 | theorem le_exp_log (x : ℝ) : x ≤ exp (log x) := by |
by_cases h_zero : x = 0
· rw [h_zero, log, dif_pos rfl, exp_zero]
exact zero_le_one
· rw [exp_log_eq_abs h_zero]
exact le_abs_self _
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import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u w
open CategoryTheory CategoryTheory.Limits
variable {ι : Type*}
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
o... | Mathlib/Algebra/Homology/ImageToKernel.lean | 112 | 116 | theorem imageToKernel_comp_right {D : V} (h : C ⟶ D) (w : f ≫ g = 0) :
imageToKernel f (g ≫ h) (by simp [reassoc_of% w]) =
imageToKernel f g w ≫ Subobject.ofLE _ _ (kernelSubobject_comp_le g h) := by |
ext
simp
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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 | 123 | 125 | theorem HasDerivAt.complexToReal_fderiv {f : ℂ → ℂ} {f' x : ℂ} (h : HasDerivAt f f' x) :
HasFDerivAt f (f' • (1 : ℂ →L[ℝ] ℂ)) x := by |
simpa only [Complex.restrictScalars_one_smulRight] using h.hasFDerivAt.restrictScalars ℝ
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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 | 89 | 96 | theorem exists_pow_nat_eq [IsAlgClosed k] (x : k) {n : ℕ} (hn : 0 < n) : ∃ z, z ^ n = x := by |
have : degree (X ^ n - C x) ≠ 0 := by
rw [degree_X_pow_sub_C hn x]
exact ne_of_gt (WithBot.coe_lt_coe.2 hn)
obtain ⟨z, hz⟩ := exists_root (X ^ n - C x) this
use z
simp only [eval_C, eval_X, eval_pow, eval_sub, IsRoot.def] at hz
exact sub_eq_zero.1 hz
| 0.65625 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
#align_import algebra.group_with_zero.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
assert_not_exists DenselyOrdered
open scoped Classical
open Function
variable {α M... | Mathlib/Algebra/GroupWithZero/Basic.lean | 110 | 111 | theorem eq_zero_of_zero_eq_one (h : (0 : M₀) = 1) (a : M₀) : a = 0 := by |
rw [← mul_one a, ← h, mul_zero]
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import Mathlib.Data.Multiset.Nodup
#align_import data.multiset.sum from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Sum
namespace Multiset
variable {α β : Type*} (s : Multiset α) (t : Multiset β)
def disjSum : Multiset (Sum α β) :=
s.map inl + t.map inr
#align multiset.dis... | Mathlib/Data/Multiset/Sum.lean | 55 | 60 | theorem inl_mem_disjSum : inl a ∈ s.disjSum t ↔ a ∈ s := by |
rw [mem_disjSum, or_iff_left]
-- Porting note: Previous code for L62 was: simp only [exists_eq_right]
· simp only [inl.injEq, exists_eq_right]
rintro ⟨b, _, hb⟩
exact inr_ne_inl hb
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import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Nontriviality
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.group_with_zero.units.basic from "leanprover-community/... | Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 108 | 110 | theorem mul_inverse_cancel (x : M₀) (h : IsUnit x) : x * inverse x = 1 := by |
rcases h with ⟨u, rfl⟩
rw [inverse_unit, Units.mul_inv]
| 0.65625 |
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.calculus.fderiv.basic from "leanprover-community/mathlib"@"41bef4ae1254365bc190aee63b947674d2977f01"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open To... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 231 | 233 | theorem fderiv_zero_of_not_differentiableAt (h : ¬DifferentiableAt 𝕜 f x) : fderiv 𝕜 f x = 0 := by |
have : ¬∃ f', HasFDerivAt f f' x := h
simp [fderiv, this]
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import Mathlib.Analysis.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Init.Data.Bool.Lemmas
#align_import analysis.box_integral.basic from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open... | Mathlib/Analysis/BoxIntegral/Basic.lean | 143 | 145 | theorem integralSum_neg (f : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F) (π : TaggedPrepartition I) :
integralSum (-f) vol π = -integralSum f vol π := by |
simp only [integralSum, Pi.neg_apply, (vol _).map_neg, Finset.sum_neg_distrib]
| 0.65625 |
import Mathlib.Data.Multiset.Nodup
#align_import data.multiset.dedup from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α β : Type*} [DecidableEq α]
def dedup (s : Multiset α) : Multiset α :=
Quot.liftOn s (fun l => (l.dedup : Multiset α)... | Mathlib/Data/Multiset/Dedup.lean | 126 | 128 | theorem dedup_nsmul {s : Multiset α} {n : ℕ} (h0 : n ≠ 0) : (n • s).dedup = s.dedup := by |
ext a
by_cases h : a ∈ s <;> simp [h, h0]
| 0.65625 |
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Slope
noncomputable section
open scoped Topology Filter ENNReal NNReal
open Filter Asymptotics Set
variable {𝕜 : Type*} [NontriviallyNormedFiel... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 170 | 171 | theorem lineDerivWithin_univ : lineDerivWithin 𝕜 f univ x v = lineDeriv 𝕜 f x v := by |
simp [lineDerivWithin, lineDeriv]
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import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 94 | 94 | theorem sInf_div : sInf (s / t) = sInf s / sSup t := by | simp_rw [div_eq_mul_inv, sInf_mul, sInf_inv]
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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 | 65 | 66 | theorem map_commutatorElement : (f ⁅g₁, g₂⁆ : G') = ⁅f g₁, f g₂⁆ := by |
simp_rw [commutatorElement_def, map_mul f, map_inv f]
| 0.65625 |
import Mathlib.Analysis.Normed.Group.Basic
#align_import information_theory.hamming from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
section HammingDistNorm
open Finset Function
variable {α ι : Type*} {β : ι → Type*} [Fintype ι] [∀ i, DecidableEq (β i)]
variable {γ : ι → Type*} [∀ ... | Mathlib/InformationTheory/Hamming.lean | 56 | 57 | theorem hammingDist_comm (x y : ∀ i, β i) : hammingDist x y = hammingDist y x := by |
simp_rw [hammingDist, ne_comm]
| 0.65625 |
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Homeomorph
#align_import topology.algebra.group_with_zero from "leanprover-community/mathlib"@"c10e724be91096453ee3db13862b9fb9a992fef2"
open Topology Filter Function
variable {α β G₀ : Type*}
section DivConst... | Mathlib/Topology/Algebra/GroupWithZero.lean | 75 | 76 | theorem Continuous.div_const (hf : Continuous f) (y : G₀) : Continuous fun x => f x / y := by |
simpa only [div_eq_mul_inv] using hf.mul continuous_const
| 0.65625 |
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 | 313 | 315 | theorem HasFDerivWithinAt.smul_const (hc : HasFDerivWithinAt c c' s x) (f : F) :
HasFDerivWithinAt (fun y => c y • f) (c'.smulRight f) s x := by |
simpa only [smul_zero, zero_add] using hc.smul (hasFDerivWithinAt_const f x s)
| 0.65625 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 126 | 129 | theorem HasDerivAt.smul_const (hc : HasDerivAt c c' x) (f : F) :
HasDerivAt (fun y => c y • f) (c' • f) x := by |
rw [← hasDerivWithinAt_univ] at *
exact hc.smul_const f
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import Mathlib.Order.SuccPred.Basic
import Mathlib.Topology.Order.Basic
import Mathlib.Topology.Metrizable.Uniformity
#align_import topology.instances.discrete from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Order Set TopologicalSpace Filter
variable {α : Type*} [TopologicalSp... | Mathlib/Topology/Instances/Discrete.lean | 111 | 117 | theorem discreteTopology_iff_orderTopology_of_pred_succ [LinearOrder α] [PredOrder α]
[SuccOrder α] : DiscreteTopology α ↔ OrderTopology α := by |
refine ⟨fun h => ⟨?_⟩, fun h => ⟨?_⟩⟩
· rw [h.eq_bot]
exact LinearOrder.bot_topologicalSpace_eq_generateFrom
· rw [h.topology_eq_generate_intervals]
exact LinearOrder.bot_topologicalSpace_eq_generateFrom.symm
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namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 49 | 51 | theorem Coprime.gcd_mul_left_cancel_right (n : Nat)
(H : Coprime k m) : gcd m (k * n) = gcd m n := by |
rw [gcd_comm m n, gcd_comm m (k * n), H.gcd_mul_left_cancel n]
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import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
@[simp]
| Mathlib/Data/List/ReduceOption.lean | 19 | 21 | theorem reduceOption_cons_of_some (x : α) (l : List (Option α)) :
reduceOption (some x :: l) = x :: l.reduceOption := by |
simp only [reduceOption, filterMap, id, eq_self_iff_true, and_self_iff]
| 0.65625 |
import Mathlib.Algebra.FreeMonoid.Basic
#align_import algebra.free_monoid.count from "leanprover-community/mathlib"@"a2d2e18906e2b62627646b5d5be856e6a642062f"
variable {α : Type*} (p : α → Prop) [DecidablePred p]
namespace FreeAddMonoid
def countP : FreeAddMonoid α →+ ℕ where
toFun := List.countP p
map_zero... | Mathlib/Algebra/FreeMonoid/Count.lean | 43 | 45 | theorem count_of [DecidableEq α] (x y : α) : count x (of y) = (Pi.single x 1 : α → ℕ) y := by |
simp [Pi.single, Function.update, count, countP, List.countP, List.countP.go,
Bool.beq_eq_decide_eq]
| 0.65625 |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
#align_import linear_algebra.matrix.reindex from "leanprover-community/mathlib"@"1cfdf5f34e1044ecb65d10be753008baaf118edf"
namespace Matrix
open Equiv Matrix
variable {l m n o : Type*} {l' m' n' o' : Type*} {m'' n'' : Type*}
variable (R A : Type*)
section A... | Mathlib/LinearAlgebra/Matrix/Reindex.lean | 66 | 70 | theorem reindexLinearEquiv_trans (e₁ : m ≃ m') (e₂ : n ≃ n') (e₁' : m' ≃ m'') (e₂' : n' ≃ n'') :
(reindexLinearEquiv R A e₁ e₂).trans (reindexLinearEquiv R A e₁' e₂') =
(reindexLinearEquiv R A (e₁.trans e₁') (e₂.trans e₂') : _ ≃ₗ[R] _) := by |
ext
rfl
| 0.65625 |
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 | 125 | 127 | theorem prod_filter_mul_prod_filter_not (p) [DecidablePred p] :
(s.filter p).prod * (s.filter (fun a ↦ ¬ p a)).prod = s.prod := by |
rw [← prod_add, filter_add_not]
| 0.65625 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 211 | 222 | theorem smul_Ico : r • Ico a b = Ico (r • a) (r • b) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Ico]
constructor
· rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩
constructor
· exact (mul_le_mul_left hr).mpr a_h_left_left
· exact (mul_lt_mul_left hr).mpr a_h_left_right
· rintro ⟨a_left, a_right⟩
use x / r
refine ⟨⟨(le_div_iff' hr).mpr... | 0.65625 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Lattice
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_th... | Mathlib/RingTheory/Nilpotent/Basic.lean | 68 | 70 | theorem IsNilpotent.isUnit_add_one [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r + 1) := by |
rw [← IsUnit.neg_iff, neg_add']
exact isUnit_sub_one hnil.neg
| 0.65625 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {M : Type*} ... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 118 | 119 | theorem image_const_add_Ioi : (fun x => a + x) '' Ioi b = Ioi (a + b) := by |
simp only [add_comm a, image_add_const_Ioi]
| 0.65625 |
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.Algebra.CharP.Reduced
open Function Polynomial
class PerfectRing (R : Type*) (p : ℕ) [CommSemiring R] [ExpChar R p] : Prop where
bijective_frobenius : Bijective <| frobenius R p
section PerfectRing
va... | Mathlib/FieldTheory/Perfect.lean | 113 | 114 | theorem iterateFrobeniusEquiv_zero_apply (x : R) : iterateFrobeniusEquiv R p 0 x = x := by |
rw [iterateFrobeniusEquiv_def, pow_zero, pow_one]
| 0.65625 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 89 | 91 | theorem cons_self_properDivisors (h : n ≠ 0) :
cons n (properDivisors n) properDivisors.not_self_mem = divisors n := by |
rw [cons_eq_insert, insert_self_properDivisors h]
| 0.65625 |
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x
the... | Mathlib/Analysis/Convex/GaugeRescale.lean | 69 | 73 | theorem gauge_gaugeRescale_le (s t : Set E) (x : E) :
gauge t (gaugeRescale s t x) ≤ gauge s x := by |
by_cases hx : gauge t x = 0
· simp [gaugeRescale, hx, gauge_nonneg]
· exact (gauge_gaugeRescale' s hx).le
| 0.65625 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 87 | 89 | theorem HasDerivWithinAt.smul (hc : HasDerivWithinAt c c' s x) (hf : HasDerivWithinAt f f' s x) :
HasDerivWithinAt (fun y => c y • f y) (c x • f' + c' • f x) s x := by |
simpa using (HasFDerivWithinAt.smul hc hf).hasDerivWithinAt
| 0.65625 |
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.lattice from "leanprover-community/mathlib"@"52fa514ec337dd970d71d8de8d0fd68b455a1e54"
assert_not_exists MonoidWithZero
open Set
namespace Nat
open scoped Classical
noncomputable instance : ... | Mathlib/Data/Nat/Lattice.lean | 59 | 62 | theorem sInf_empty : sInf ∅ = 0 := by |
rw [sInf_eq_zero]
right
rfl
| 0.65625 |
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 | 153 | 154 | theorem of_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) : P.toPoly = C P.d := by |
rw [of_b_eq_zero ha hb, hc, C_0, zero_mul, zero_add]
| 0.65625 |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 54 | 55 | theorem intDegree_C (k : K) : intDegree (C k) = 0 := by |
rw [intDegree, num_C, natDegree_C, denom_C, natDegree_one, sub_self]
| 0.65625 |
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Logic.Lemmas
#align_import combinatorics.quiver.path from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
open Function
universe v v₁ v₂ u u₁ u₂
namespace Quiver
inductive Path {V : Type u} [Quiver.{v} V] (a : V) : V → Sort max ... | Mathlib/Combinatorics/Quiver/Path.lean | 81 | 84 | theorem eq_of_length_zero (p : Path a b) (hzero : p.length = 0) : a = b := by |
cases p
· rfl
· cases Nat.succ_ne_zero _ hzero
| 0.65625 |
import Mathlib.LinearAlgebra.Dual
open Function Module
variable (R M N : Type*) [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
structure PerfectPairing :=
toLin : M →ₗ[R] N →ₗ[R] R
bijectiveLeft : Bijective toLin
bijectiveRight : Bijective toLin.flip
attribute [nolint docBlame] P... | Mathlib/LinearAlgebra/PerfectPairing.lean | 85 | 89 | theorem apply_apply_toDualRight_symm (x : M) (f : Dual R M) :
(p x) (p.toDualRight.symm f) = f x := by |
have h := LinearEquiv.apply_symm_apply p.toDualRight f
rw [toDualRight_apply] at h
exact congrFun (congrArg DFunLike.coe h) x
| 0.65625 |
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 {α : Type u} {β : Type v} {γ : Type w} {R : Type x}
open Function
namespace SemiconjBy
@[simp... | Mathlib/Algebra/Ring/Semiconj.lean | 39 | 41 | theorem add_left [Distrib R] {a b x y : R} (ha : SemiconjBy a x y) (hb : SemiconjBy b x y) :
SemiconjBy (a + b) x y := by |
simp only [SemiconjBy, left_distrib, right_distrib, ha.eq, hb.eq]
| 0.65625 |
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Nontriviality
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.group_with_zero.units.basic from "leanprover-community/... | Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 126 | 127 | theorem mul_inverse_cancel_left (x y : M₀) (h : IsUnit x) : x * (inverse x * y) = y := by |
rw [← mul_assoc, mul_inverse_cancel x h, one_mul]
| 0.65625 |
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 | 149 | 150 | theorem toComplex_eq_zero {x : ℤ[i]} : (x : ℂ) = 0 ↔ x = 0 := by |
rw [← toComplex_zero, toComplex_inj]
| 0.65625 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 88 | 89 | theorem image_Ioo (e : α ≃o β) (a b : α) : e '' Ioo a b = Ioo (e a) (e b) := by |
rw [e.image_eq_preimage, e.symm.preimage_Ioo, e.symm_symm]
| 0.65625 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 134 | 140 | theorem vars_pow (φ : MvPolynomial σ R) (n : ℕ) : (φ ^ n).vars ⊆ φ.vars := by |
classical
induction' n with n ih
· simp
· rw [pow_succ']
apply Finset.Subset.trans (vars_mul _ _)
exact Finset.union_subset (Finset.Subset.refl _) ih
| 0.65625 |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 91 | 94 | theorem card_singleton_inter [DecidableEq α] : ({a} ∩ s).card ≤ 1 := by |
cases' Finset.decidableMem a s with h h
· simp [Finset.singleton_inter_of_not_mem h]
· simp [Finset.singleton_inter_of_mem h]
| 0.65625 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 223 | 234 | theorem isTheta_cpow_rpow (hl_im : IsBoundedUnder (· ≤ ·) l fun x => |(g x).im|)
(hl : ∀ᶠ x in l, f x = 0 → re (g x) = 0 → g x = 0) :
(fun x => f x ^ g x) =Θ[l] fun x => abs (f x) ^ (g x).re :=
calc
(fun x => f x ^ g x) =Θ[l]
(show α → ℝ from fun x => abs (f x) ^ (g x).re / Real.exp (arg (f x) * i... |
simp only [ofReal_one, div_one]
rfl
| 0.65625 |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 70 | 74 | theorem IsMatching.toEdge.surjective {M : Subgraph G} (h : M.IsMatching) :
Function.Surjective h.toEdge := by |
rintro ⟨e, he⟩
refine Sym2.ind (fun x y he => ?_) e he
exact ⟨⟨x, M.edge_vert he⟩, h.toEdge_eq_of_adj _ he⟩
| 0.65625 |
import Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
noncomputable section
open scoped Manifold
open Bundle Set Topology
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
(I : ModelWithCorners 𝕜 E H) {M : Type*} [To... | Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean | 200 | 210 | theorem symm_comp_deriv {x : M} (hx : x ∈ e.source) :
(mfderiv I' I e.symm (e x)).comp (mfderiv I I' e x) =
ContinuousLinearMap.id 𝕜 (TangentSpace I x) := by |
have : mfderiv I I (e.symm ∘ e) x = (mfderiv I' I e.symm (e x)).comp (mfderiv I I' e x) :=
mfderiv_comp x (he.mdifferentiableAt_symm (e.map_source hx)) (he.mdifferentiableAt hx)
rw [← this]
have : mfderiv I I (_root_.id : M → M) x = ContinuousLinearMap.id _ _ := mfderiv_id I
rw [← this]
apply Filter.Even... | 0.65625 |
import Mathlib.Algebra.Group.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
universe u
variable {α : Type u}
class Invertible [Mul α] [One α] (a : α) : Type u where
invOf... | Mathlib/Algebra/Group/Invertible/Defs.lean | 133 | 134 | theorem mul_invOf_mul_self_cancel' [Monoid α] (a b : α) {_ : Invertible b} : a * ⅟ b * b = a := by |
simp [mul_assoc]
| 0.65625 |
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.Basic
namespace MulAction
universe u v
variable {α : Type v}
variable {G : Type u} [Group G] [MulAction G α]
variable {M : Type u} [Monoid M] [MulAction M α]
@[to_additive "If the action is periodic, t... | Mathlib/GroupTheory/GroupAction/Period.lean | 101 | 102 | theorem period_dvd_exponent (m : M) (a : α) : period m a ∣ Monoid.exponent M := by |
rw [← pow_smul_eq_iff_period_dvd, Monoid.pow_exponent_eq_one, one_smul]
| 0.65625 |
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 : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
Orde... | Mathlib/Order/Interval/Finset/Fin.lean | 94 | 95 | theorem map_valEmbedding_Ioo : (Ioo a b).map Fin.valEmbedding = Ioo ↑a ↑b := by |
simp [Ioo_eq_finset_subtype, Finset.fin, Finset.map_map]
| 0.65625 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
noncomputable section
open Affine
open Set
section
variable (k : Type*) {V : Type*} {P : Type*} [Ring k] [AddCommGroup V]... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 78 | 79 | theorem vectorSpan_empty : vectorSpan k (∅ : Set P) = (⊥ : Submodule k V) := by |
rw [vectorSpan_def, vsub_empty, Submodule.span_empty]
| 0.65625 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 32 | 34 | theorem sq_ne_two_fin_zmod_four (z : ZMod 4) : z * z ≠ 2 := by |
change Fin 4 at z
fin_cases z <;> decide
| 0.65625 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Tree.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.GCongr
import Mathlib... | Mathlib/Combinatorics/Enumerative/Catalan.lean | 68 | 69 | theorem catalan_succ (n : ℕ) : catalan (n + 1) = ∑ i : Fin n.succ, catalan i * catalan (n - i) := by |
rw [catalan]
| 0.65625 |
import Mathlib.Data.Analysis.Filter
import Mathlib.Topology.Bases
import Mathlib.Topology.LocallyFinite
#align_import data.analysis.topology from "leanprover-community/mathlib"@"55d771df074d0dd020139ee1cd4b95521422df9f"
open Set
open Filter hiding Realizer
open Topology
structure Ctop (α σ : Type*) where
f ... | Mathlib/Data/Analysis/Topology.lean | 79 | 80 | theorem ofEquiv_val (E : σ ≃ τ) (F : Ctop α σ) (a : τ) : F.ofEquiv E a = F (E.symm a) := by |
cases F; rfl
| 0.65625 |
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 : ℕ)
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 [← Nat.card_Ioo, ← map_valEmbedding_Ioo, card_map]
| 0.65625 |
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 | 95 | 96 | theorem map₂_map_left (f : γ → β → δ) (g : α → γ) :
map₂ f (a.map g) b = map₂ (fun a b => f (g a) b) a b := by | cases a <;> rfl
| 0.65625 |
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
| 0.65625 |
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Util.AssertExists
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
namespace List
universe u v
variable {α : Type u} {β : Type v} (l :... | Mathlib/Data/List/GetD.lean | 47 | 53 | theorem getD_map {n : ℕ} (f : α → β) : (map f l).getD n (f d) = f (l.getD n d) := by |
induction l generalizing n with
| nil => rfl
| cons head tail ih =>
cases n
· rfl
· simp [ih]
| 0.65625 |
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace List
variable {α β : Type*}
section FormPerm
variable [DecidableEq α] (l :... | Mathlib/GroupTheory/Perm/List.lean | 100 | 103 | theorem support_formPerm_le [Fintype α] : support (formPerm l) ≤ l.toFinset := by |
intro x hx
have hx' : x ∈ { x | formPerm l x ≠ x } := by simpa using hx
simpa using support_formPerm_le' _ hx'
| 0.65625 |
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
#align_import topology.uniform_space.uniform_embedding from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Filter Function Set Uniformity Topology
sec... | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 116 | 118 | theorem UniformInducing.inducing {f : α → β} (h : UniformInducing f) : Inducing f := by |
obtain rfl := h.comap_uniformSpace
exact inducing_induced f
| 0.65625 |
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
| 0.65625 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Equiv.Fin
import Mathlib.ModelTheory.LanguageMap
#align_import model_theory.syntax from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable (L : Language.{u, v}) {L' : L... | Mathlib/ModelTheory/Syntax.lean | 274 | 279 | theorem id_onTerm : ((LHom.id L).onTerm : L.Term α → L.Term α) = id := by |
ext t
induction' t with _ _ _ _ ih
· rfl
· simp_rw [onTerm, ih]
rfl
| 0.65625 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section restrict
def restrict (... | Mathlib/Data/Set/Function.lean | 124 | 127 | theorem restrict_extend_compl_range (f : α → β) (g : α → γ) (g' : β → γ) :
(range f)ᶜ.restrict (extend f g g') = g' ∘ Subtype.val := by |
classical
exact restrict_dite_compl _ _
| 0.65625 |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 133 | 134 | theorem exists_duplicate_iff_not_nodup : (∃ x : α, x ∈+ l) ↔ ¬Nodup l := by |
simp [nodup_iff_forall_not_duplicate]
| 0.65625 |
import Mathlib.Data.DFinsupp.Interval
import Mathlib.Data.DFinsupp.Multiset
import Mathlib.Order.Interval.Finset.Nat
#align_import data.multiset.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
open Finset DFinsupp Function
open Pointwise
variable {α : Type*}
namespace Mu... | Mathlib/Data/Multiset/Interval.lean | 62 | 64 | theorem card_Ico :
(Finset.Ico s t).card = ∏ i ∈ s.toFinset ∪ t.toFinset, (t.count i + 1 - s.count i) - 1 := by |
rw [Finset.card_Ico_eq_card_Icc_sub_one, card_Icc]
| 0.65625 |
import Mathlib.Topology.MetricSpace.Isometry
#align_import topology.metric_space.gluing from "leanprover-community/mathlib"@"e1a7bdeb4fd826b7e71d130d34988f0a2d26a177"
noncomputable section
universe u v w
open Function Set Uniformity Topology
namespace Metric
namespace Sigma
variable {ι : Type*} {E : ι → Type... | Mathlib/Topology/MetricSpace/Gluing.lean | 352 | 355 | theorem one_le_dist_of_ne {i j : ι} (h : i ≠ j) (x : E i) (y : E j) :
1 ≤ dist (⟨i, x⟩ : Σk, E k) ⟨j, y⟩ := by |
rw [Sigma.dist_ne h x y]
linarith [@dist_nonneg _ _ x (Nonempty.some ⟨x⟩), @dist_nonneg _ _ (Nonempty.some ⟨y⟩) y]
| 0.65625 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 235 | 235 | theorem mul_ne_top : a ≠ ∞ → b ≠ ∞ → a * b ≠ ∞ := by | simpa only [lt_top_iff_ne_top] using mul_lt_top
| 0.65625 |
import Mathlib.Algebra.Module.Submodule.Lattice
import Mathlib.Order.Hom.CompleteLattice
namespace Submodule
variable (S : Type*) {R M : Type*} [Semiring R] [AddCommMonoid M] [Semiring S]
[Module S M] [Module R M] [SMul S R] [IsScalarTower S R M]
def restrictScalars (V : Submodule R M) : Submodule S M where
... | Mathlib/Algebra/Module/Submodule/RestrictScalars.lean | 116 | 117 | theorem restrictScalars_eq_top_iff {p : Submodule R M} : restrictScalars S p = ⊤ ↔ p = ⊤ := by |
simp [SetLike.ext_iff]
| 0.65625 |
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 | 343 | 343 | theorem symmDiff_top' : a ∆ ⊤ = ￢a := by | simp [symmDiff]
| 0.625 |
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Init.Data.List.Basic
import Mathlib.Data.List.Basic
#align_import data.stream.init from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
set_option autoImplicit true
open Nat Function Option
namespace Stre... | Mathlib/Data/Stream/Init.lean | 65 | 66 | theorem drop_drop (n m : Nat) (s : Stream' α) : drop n (drop m s) = drop (n + m) s := by |
ext; simp [Nat.add_assoc]
| 0.625 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
#align_import linear_algebra.exterior_algebra.of_alternating from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
variable {R M N N' : Type*}
variable [CommRing R] [AddCommGroup M] [AddCo... | Mathlib/LinearAlgebra/ExteriorAlgebra/OfAlternating.lean | 96 | 99 | theorem liftAlternating_algebraMap (f : ∀ i, M [⋀^Fin i]→ₗ[R] N) (r : R) :
liftAlternating (R := R) (M := M) (N := N) f (algebraMap _ (ExteriorAlgebra R M) r) =
r • f 0 0 := by |
rw [Algebra.algebraMap_eq_smul_one, map_smul, liftAlternating_one]
| 0.625 |
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
#align_import order.chain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open scoped Classical
open Set
variable {α β : Type*}
section Chain
variable (r : α → α → Prop)
... | Mathlib/Order/Chain.lean | 107 | 110 | theorem Monotone.isChain_range [LinearOrder α] [Preorder β] {f : α → β} (hf : Monotone f) :
IsChain (· ≤ ·) (range f) := by |
rw [← image_univ]
exact (isChain_of_trichotomous _).image (· ≤ ·) _ _ hf
| 0.625 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 68 | 69 | theorem gcd_mul_left_add_left (m n k : ℕ) : gcd (n * k + m) n = gcd m n := by |
rw [gcd_comm, gcd_mul_left_add_right, gcd_comm]
| 0.625 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 156 | 157 | theorem add_one_mul [RightDistribClass α] (a b : α) : (a + 1) * b = a * b + b := by |
rw [add_mul, one_mul]
| 0.625 |
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import analysis.special_functions.pow.complex from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open scoped Classical
open Real Topology Filter ComplexConjugate Finset Set
namespace Complex
noncomputable def cpow (x y : ℂ) ... | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 58 | 72 | theorem zero_cpow_eq_iff {x : ℂ} {a : ℂ} : (0 : ℂ) ^ x = a ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 := by |
constructor
· intro hyp
simp only [cpow_def, eq_self_iff_true, if_true] at hyp
by_cases h : x = 0
· subst h
simp only [if_true, eq_self_iff_true] at hyp
right
exact ⟨rfl, hyp.symm⟩
· rw [if_neg h] at hyp
left
exact ⟨h, hyp.symm⟩
· rintro (⟨h, rfl⟩ | ⟨rfl, rfl⟩)
·... | 0.625 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
noncomputable section
open Function Polynomial Finsupp Finset
open scoped Polynomial
namespace ... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 141 | 145 | theorem natTrailingDegree_le_trailingDegree : ↑(natTrailingDegree p) ≤ trailingDegree p := by |
by_cases hp : p = 0;
· rw [hp, trailingDegree_zero]
exact le_top
rw [trailingDegree_eq_natTrailingDegree hp]
| 0.625 |
import Mathlib.Logic.Function.Iterate
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Tactic.GCongr
#align_import topology.metric_space.lipschitz from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
universe u v w x
open Filter Function Set Topology NNReal ENNReal Bornology
va... | Mathlib/Topology/EMetricSpace/Lipschitz.lean | 86 | 88 | theorem lipschitzOnWith_iff_restrict [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0}
{f : α → β} {s : Set α} : LipschitzOnWith K f s ↔ LipschitzWith K (s.restrict f) := by |
simp only [LipschitzOnWith, LipschitzWith, SetCoe.forall', restrict, Subtype.edist_eq]
| 0.625 |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 118 | 119 | theorem cosh_log {x : ℝ} (hx : 0 < x) : cosh (log x) = (x + x⁻¹) / 2 := by |
rw [cosh_eq, exp_neg, exp_log hx]
| 0.625 |
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
#align_import analysis.calculus.deriv.pow from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter ENNReal
open Filter Asymptotics Set
variable {... | Mathlib/Analysis/Calculus/Deriv/Pow.lean | 99 | 102 | theorem HasDerivAt.pow (hc : HasDerivAt c c' x) :
HasDerivAt (fun y => c y ^ n) ((n : 𝕜) * c x ^ (n - 1) * c') x := by |
rw [← hasDerivWithinAt_univ] at *
exact hc.pow n
| 0.625 |
import Mathlib.RepresentationTheory.FdRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
#align_import representation_theory.character from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
noncomputable section
universe u
open CategoryTheory LinearMap ... | Mathlib/RepresentationTheory/Character.lean | 70 | 74 | theorem char_tensor' (V W : FdRep k G) :
character (Action.FunctorCategoryEquivalence.inverse.obj
(Action.FunctorCategoryEquivalence.functor.obj V ⊗
Action.FunctorCategoryEquivalence.functor.obj W)) = V.character * W.character := by |
simp [← char_tensor]
| 0.625 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.Topology.Sheaves.Limits
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
#align_import algebraic_geometry.presheafed_space.has_colimits from "leanprover-community/mathlib"@"178a32653e369dc... | Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean | 70 | 79 | theorem map_comp_c_app (F : J ⥤ PresheafedSpace.{_, _, v} C) {j₁ j₂ j₃}
(f : j₁ ⟶ j₂) (g : j₂ ⟶ j₃) (U) :
(F.map (f ≫ g)).c.app (op U) =
(F.map g).c.app (op U) ≫
(pushforwardMap (F.map g).base (F.map f).c).app (op U) ≫
(Pushforward.comp (F.obj j₁).presheaf (F.map f).base (F.map g).base).... |
cases U
simp [PresheafedSpace.congr_app (F.map_comp f g)]
| 0.625 |
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
open Finset Function Sym
universe u
variab... | Mathlib/Data/Sym/Sym2.lean | 91 | 92 | theorem rel_iff {x y z w : α} : Rel α (x, y) (z, w) ↔ x = z ∧ y = w ∨ x = w ∧ y = z := by |
simp
| 0.625 |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace α] [MeasurableSpace β] {s : Set α}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure α :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 84 | 88 | theorem count_apply_eq_top' (s_mble : MeasurableSet s) : count s = ∞ ↔ s.Infinite := by |
by_cases hs : s.Finite
· simp [Set.Infinite, hs, count_apply_finite' hs s_mble]
· change s.Infinite at hs
simp [hs, count_apply_infinite]
| 0.625 |
import Mathlib.Algebra.Category.MonCat.Basic
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
#align_import algebra.category.Mon.colimits from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe v
open CategoryTheory
open Ca... | Mathlib/Algebra/Category/MonCat/Colimits.lean | 188 | 191 | theorem cocone_naturality_components (j j' : J) (f : j ⟶ j') (x : F.obj j) :
(coconeMorphism F j') (F.map f x) = (coconeMorphism F j) x := by |
rw [← cocone_naturality F f]
rfl
| 0.625 |
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Instances.Real
def preCantorSet : ℕ → Set ℝ
| 0 => Set.Icc 0 1
| n + 1 => (· / 3) '' preCantorSet n ∪ (fun x ↦ (2 + x) / 3) '' preCantorSet n
@[simp] lemma preCantorSet_zero : preCantorSet 0 = Set.Ic... | Mathlib/Topology/Instances/CantorSet.lean | 75 | 75 | theorem zero_mem_cantorSet : 0 ∈ cantorSet := by | simp [cantorSet, zero_mem_preCantorSet]
| 0.625 |
import Mathlib.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open scoped Topology
open Filter Set TopologicalSpace
variable {X α : Type*} {ι : Sort*}
section BaireTheorem
variable [TopologicalSpace... | Mathlib/Topology/Baire/Lemmas.lean | 60 | 63 | theorem dense_biInter_of_isOpen {S : Set α} {f : α → Set X} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) := by |
rw [← sInter_image]
refine dense_sInter_of_isOpen ?_ (hS.image _) ?_ <;> rwa [forall_mem_image]
| 0.625 |
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.NumberTheory.Divisors
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Tactic.Zify
#align_import rin... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 119 | 122 | theorem rootsOfUnity_le_of_dvd (h : k ∣ l) : rootsOfUnity k M ≤ rootsOfUnity l M := by |
obtain ⟨d, rfl⟩ := h
intro ζ h
simp_all only [mem_rootsOfUnity, PNat.mul_coe, pow_mul, one_pow]
| 0.625 |
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 | 332 | 333 | theorem average_eq (f : α → E) : ⨍ x, f x ∂μ = (μ univ).toReal⁻¹ • ∫ x, f x ∂μ := by |
rw [average_eq', integral_smul_measure, ENNReal.toReal_inv]
| 0.625 |
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