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 |
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import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : ℕ) : List ℕ :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 136 | 139 | theorem pred_singleton {m : ℕ} (h : 0 < m) : Ico (m - 1) m = [m - 1] := by |
dsimp [Ico]
rw [Nat.sub_sub_self (succ_le_of_lt h)]
simp [← Nat.one_eq_succ_zero]
| 0.09375 |
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.IsomorphismClasses
import Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
#align_import category_theory.limits.shapes.zero_morphisms from "leanpr... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 150 | 151 | theorem eq_zero_of_image_eq_zero {X Y : C} {f : X ⟶ Y} [HasImage f] (w : image.ι f = 0) :
f = 0 := by | rw [← image.fac f, w, HasZeroMorphisms.comp_zero]
| 0.09375 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
#align_import measure_theory.function.conditional_expectation.condexp_L2 from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 113 | 117 | theorem norm_condexpL2_coe_le (hm : m ≤ m0) (f : α →₂[μ] E) :
‖(condexpL2 E 𝕜 hm f : α →₂[μ] E)‖ ≤ ‖f‖ := by |
rw [Lp.norm_def, Lp.norm_def, ← lpMeas_coe]
refine (ENNReal.toReal_le_toReal ?_ (Lp.snorm_ne_top _)).mpr (snorm_condexpL2_le hm f)
exact Lp.snorm_ne_top _
| 0.09375 |
import Mathlib.Data.W.Basic
#align_import data.pfunctor.univariate.basic from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
-- "W", "Idx"
set_option linter.uppercaseLean3 false
universe u v v₁ v₂ v₃
@[pp_with_univ]
structure PFunctor where
A : Type u
B : A → Type u
#align p... | Mathlib/Data/PFunctor/Univariate/Basic.lean | 154 | 154 | theorem fst_map (x : P α) (f : α → β) : (P.map f x).1 = x.1 := by | cases x; rfl
| 0.09375 |
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Tactic.Monotonicity
#align_import algebra.continued_fractions.computation.approximations from "leanprover-commu... | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 70 | 80 | theorem nth_stream_fr_nonneg_lt_one {ifp_n : IntFractPair K}
(nth_stream_eq : IntFractPair.stream v n = some ifp_n) : 0 ≤ ifp_n.fr ∧ ifp_n.fr < 1 := by |
cases n with
| zero =>
have : IntFractPair.of v = ifp_n := by injection nth_stream_eq
rw [← this, IntFractPair.of]
exact ⟨fract_nonneg _, fract_lt_one _⟩
| succ =>
rcases succ_nth_stream_eq_some_iff.1 nth_stream_eq with ⟨_, _, _, ifp_of_eq_ifp_n⟩
rw [← ifp_of_eq_ifp_n, IntFractPair.of]
ex... | 0.09375 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Set.Finite
#align_import combinatorics.pigeonhole from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
un... | Mathlib/Combinatorics/Pigeonhole.lean | 134 | 141 | theorem exists_lt_sum_fiber_of_sum_fiber_nonpos_of_nsmul_lt_sum
(ht : ∀ y ∉ t, ∑ x ∈ s.filter fun x => f x = y, w x ≤ 0)
(hb : t.card • b < ∑ x ∈ s, w x) : ∃ y ∈ t, b < ∑ x ∈ s.filter fun x => f x = y, w x :=
exists_lt_of_sum_lt <|
calc
∑ _y ∈ t, b < ∑ x ∈ s, w x := by | simpa
_ ≤ ∑ y ∈ t, ∑ x ∈ s.filter fun x => f x = y, w x :=
sum_le_sum_fiberwise_of_sum_fiber_nonpos ht
| 0.09375 |
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Defs
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
variable {α : Type*}
open Opposite
namespace Set
protected def op (s : Set α) : Set αᵒᵖ :=
unop ⁻¹' s
#align set.op Set.op
protected def u... | Mathlib/Data/Set/Opposite.lean | 76 | 80 | theorem singleton_op (x : α) : ({x} : Set α).op = {op x} := by |
ext
constructor
· apply unop_injective
· apply op_injective
| 0.09375 |
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Conj
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import category_theory.adjunction.reflective from "leanprover-community/mathlib"@"239d882c4fb58361ee8b3b39fb2091320edef10a"
universe v₁ v₂ v₃ u₁ u₂ u₃
noncomputable s... | Mathlib/CategoryTheory/Adjunction/Reflective.lean | 154 | 156 | theorem unitCompPartialBijective_symm_apply [Reflective i] (A : C) {B : C} (hB : B ∈ i.essImage)
(f) : (unitCompPartialBijective A hB).symm f = (reflectorAdjunction i).unit.app A ≫ f := by |
simp [unitCompPartialBijective, unitCompPartialBijectiveAux_symm_apply]
| 0.09375 |
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}
| Mathlib/Topology/Order/DenselyOrdered.lean | 25 | 29 | theorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by |
apply Subset.antisymm
· exact closure_minimal Ioi_subset_Ici_self isClosed_Ici
· rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]
exact isGLB_Ioi.mem_closure h
| 0.09375 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
noncomputable section
open Filter Finset Function Encodable
open scoped Topology
variable {M : Type*} [CommMonoid M] [TopologicalSpace M] {m m' : M}
variable {G : Type*} [CommGroup G] {g g' : G}
-- don't declare [Topologic... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 218 | 221 | theorem hasProd_nat_add_iff {f : ℕ → G} (k : ℕ) :
HasProd (fun n ↦ f (n + k)) g ↔ HasProd f (g * ∏ i ∈ range k, f i) := by |
refine Iff.trans ?_ (range k).hasProd_compl_iff
rw [← (notMemRangeEquiv k).symm.hasProd_iff, Function.comp_def, coe_notMemRangeEquiv_symm]
| 0.09375 |
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped NNReal Matrix
namespace Matrix
variable {R l m n α β : Type*} [Fintype l] [Fintyp... | Mathlib/Analysis/Matrix.lean | 318 | 323 | theorem linfty_opNNNorm_diagonal [DecidableEq m] (v : m → α) : ‖diagonal v‖₊ = ‖v‖₊ := by |
rw [linfty_opNNNorm_def, Pi.nnnorm_def]
congr 1 with i : 1
refine (Finset.sum_eq_single_of_mem _ (Finset.mem_univ i) fun j _hj hij => ?_).trans ?_
· rw [diagonal_apply_ne' _ hij, nnnorm_zero]
· rw [diagonal_apply_eq]
| 0.09375 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 84 | 87 | theorem eq_or_edist_eq_top_of_fixedPoints (hf : ContractingWith K f) {x y} (hx : IsFixedPt f x)
(hy : IsFixedPt f y) : x = y ∨ edist x y = ∞ := by |
refine or_iff_not_imp_right.2 fun h ↦ edist_le_zero.1 ?_
simpa only [hx.eq, edist_self, add_zero, ENNReal.zero_div] using hf.edist_le_of_fixedPoint h hy
| 0.09375 |
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
noncomputable section
open scoped Classical
open Topology Filter
open TopologicalSpace Set Metric Filter Function
attribute [local simp... | Mathlib/Topology/MetricSpace/PiNat.lean | 150 | 151 | theorem mem_cylinder_comm (x y : ∀ n, E n) (n : ℕ) : y ∈ cylinder x n ↔ x ∈ cylinder y n := by |
simp [mem_cylinder_iff_eq, eq_comm]
| 0.09375 |
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
#align_import data.complex.exponential_bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
namespace Real
open IsAbsoluteValue Finset CauSeq Complex
theorem exp_one_near_10 : |exp 1 - 224... | Mathlib/Data/Complex/ExponentialBounds.lean | 44 | 48 | theorem exp_neg_one_gt_d9 : 0.36787944116 < exp (-1) := by |
rw [exp_neg, lt_inv _ (exp_pos _)]
· refine lt_of_le_of_lt (sub_le_iff_le_add.1 (abs_sub_le_iff.1 exp_one_near_10).1) ?_
norm_num
· norm_num
| 0.09375 |
import Mathlib.Analysis.SpecialFunctions.Gamma.Beta
import Mathlib.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.PSeriesComplex
#align_import number_theory.zeta_function from "leanprover-community/mathlib"@"57f9349f2fe19d2de7207e99b0341808d977cdcf"
o... | Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 149 | 150 | theorem riemannZeta_zero : riemannZeta 0 = -1 / 2 := by |
simp_rw [riemannZeta, hurwitzZetaEven, Function.update_same, if_true]
| 0.09375 |
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 63 | 67 | theorem eventually_nhdsWithin_nhdsWithin {a : α} {s : Set α} {p : α → Prop} :
(∀ᶠ y in 𝓝[s] a, ∀ᶠ x in 𝓝[s] y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x := by |
refine ⟨fun h => ?_, fun h => (eventually_nhds_nhdsWithin.2 h).filter_mono inf_le_left⟩
simp only [eventually_nhdsWithin_iff] at h ⊢
exact h.mono fun x hx hxs => (hx hxs).self_of_nhds hxs
| 0.09375 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 111 | 113 | theorem length_mul_ge_length_sub_length' (w₁ w₂ : W) :
ℓ w₂ - ℓ w₁ ≤ ℓ (w₁ * w₂) := by |
simpa [Nat.sub_le_of_le_add, add_comm] using cs.length_mul_le w₁⁻¹ (w₁ * w₂)
| 0.09375 |
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 87 | 90 | theorem take_range (m n : ℕ) : take m (range n) = range (min m n) := by |
apply List.ext_get
· simp
· simp (config := { contextual := true }) [← get_take, Nat.lt_min]
| 0.09375 |
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
#align_import data.sum.interval from "leanprover-community/mathlib"@"48a058d7e39a80ed56858505719a0b2197900999"
open Function Sum
namespace Finset
variable {α₁ α₂ β₁ β₂ γ₁ γ₂ : Type*}
section SumLift₂
variabl... | Mathlib/Data/Sum/Interval.lean | 76 | 88 | theorem sumLift₂_eq_empty :
sumLift₂ f g a b = ∅ ↔
(∀ a₁ b₁, a = inl a₁ → b = inl b₁ → f a₁ b₁ = ∅) ∧
∀ a₂ b₂, a = inr a₂ → b = inr b₂ → g a₂ b₂ = ∅ := by |
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· constructor <;>
· rintro a b rfl rfl
exact map_eq_empty.1 h
cases a <;> cases b
· exact map_eq_empty.2 (h.1 _ _ rfl rfl)
· rfl
· rfl
· exact map_eq_empty.2 (h.2 _ _ rfl rfl)
| 0.09375 |
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.Layercake
#align_import analysis.special_functions.japanese_bracket from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
op... | Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean | 70 | 73 | theorem closedBall_rpow_sub_one_eq_empty_aux {r t : ℝ} (hr : 0 < r) (ht : 1 < t) :
Metric.closedBall (0 : E) (t ^ (-r⁻¹) - 1) = ∅ := by |
rw [Metric.closedBall_eq_empty, sub_neg]
exact Real.rpow_lt_one_of_one_lt_of_neg ht (by simp only [hr, Right.neg_neg_iff, inv_pos])
| 0.09375 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.ZMod.Basic
#align_import data.zmod.parity from "leanprover-community/mathlib"@"048240e809f04e2bde02482ab44bc230744cc6c9"
namespace ZMod
theorem eq_zero_iff_even {n : ℕ} : (n : ZMod 2) = 0 ↔ Even n :=
(CharP.cast_eq_zero_iff (ZMod 2) 2 n).trans even_iff_... | Mathlib/Data/ZMod/Parity.lean | 32 | 35 | theorem ne_zero_iff_odd {n : ℕ} : (n : ZMod 2) ≠ 0 ↔ Odd n := by |
constructor <;>
· contrapose
simp [eq_zero_iff_even]
| 0.09375 |
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subring.Units
#align_import linear_algebra.ray from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
noncomputable section
... | Mathlib/LinearAlgebra/Ray.lean | 102 | 111 | theorem trans (hxy : SameRay R x y) (hyz : SameRay R y z) (hy : y = 0 → x = 0 ∨ z = 0) :
SameRay R x z := by |
rcases eq_or_ne x 0 with (rfl | hx); · exact zero_left z
rcases eq_or_ne z 0 with (rfl | hz); · exact zero_right x
rcases eq_or_ne y 0 with (rfl | hy);
· exact (hy rfl).elim (fun h => (hx h).elim) fun h => (hz h).elim
rcases hxy.exists_pos hx hy with ⟨r₁, r₂, hr₁, hr₂, h₁⟩
rcases hyz.exists_pos hy hz with ... | 0.09375 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Topology.Algebra.Order.Group
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.floor from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Filter Function Int Set Topology
variable {α β γ : Type*} [LinearOrdere... | Mathlib/Topology/Algebra/Order/Floor.lean | 101 | 105 | theorem tendsto_ceil_right_pure_floor_add_one (x : α) :
Tendsto (ceil : α → ℤ) (𝓝[>] x) (pure (⌊x⌋ + 1)) :=
have : ↑(⌊x⌋ + 1) - 1 ≤ x := by | rw [cast_add, cast_one, add_sub_cancel_right]; exact floor_le _
tendsto_pure.2 <| mem_of_superset (Ioc_mem_nhdsWithin_Ioi' <| lt_succ_floor _) fun _y hy =>
ceil_eq_on_Ioc _ _ ⟨this.trans_lt hy.1, hy.2⟩
| 0.09375 |
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 | 1,059 | 1,059 | theorem zneg_bitm1 (n : ZNum) : -n.bitm1 = (-n).bit1 := by | cases n <;> rfl
| 0.09375 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.InsertNth
#align_import group_theory.free_group from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Relation
universe u v w
variable {α : Type u... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 151 | 155 | theorem not_step_nil : ¬Step [] L := by |
generalize h' : [] = L'
intro h
cases' h with L₁ L₂
simp [List.nil_eq_append] at h'
| 0.09375 |
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.measure.regular from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Set Filter ENNReal Topology NNReal TopologicalSpace
namespace MeasureTh... | Mathlib/MeasureTheory/Measure/Regular.lean | 349 | 353 | theorem _root_.Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular μ] :
μ A = ⨅ (U : Set α) (_ : A ⊆ U) (_ : IsOpen U), μ U := by |
refine le_antisymm (le_iInf₂ fun s hs => le_iInf fun _ => μ.mono hs) ?_
refine le_of_forall_lt' fun r hr => ?_
simpa only [iInf_lt_iff, exists_prop] using A.exists_isOpen_lt_of_lt r hr
| 0.09375 |
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.symplectic_group from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Matrix
variable {l R : Type*}
namespace Matrix
variable (l) [DecidableEq l] (R) [CommRing R]
section JMatrixLemmas
def J : ... | Mathlib/LinearAlgebra/SymplecticGroup.lean | 43 | 46 | theorem J_transpose : (J l R)ᵀ = -J l R := by |
rw [J, fromBlocks_transpose, ← neg_one_smul R (fromBlocks _ _ _ _ : Matrix (l ⊕ l) (l ⊕ l) R),
fromBlocks_smul, Matrix.transpose_zero, Matrix.transpose_one, transpose_neg]
simp [fromBlocks]
| 0.09375 |
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 | 170 | 171 | theorem Invertible.congr [Monoid α] (a b : α) [Invertible a] [Invertible b] (h : a = b) :
⅟a = ⅟b := by | subst h; congr; apply Subsingleton.allEq
| 0.09375 |
import Mathlib.Algebra.Order.Group.PiLex
import Mathlib.Data.DFinsupp.Order
import Mathlib.Data.DFinsupp.NeLocus
import Mathlib.Order.WellFoundedSet
#align_import data.dfinsupp.lex from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
variable {ι : Type*} {α : ι → Type*}
namespace DFinsu... | Mathlib/Data/DFinsupp/Lex.lean | 51 | 58 | theorem lex_lt_of_lt_of_preorder [∀ i, Preorder (α i)] (r) [IsStrictOrder ι r] {x y : Π₀ i, α i}
(hlt : x < y) : ∃ i, (∀ j, r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i := by |
obtain ⟨hle, j, hlt⟩ := Pi.lt_def.1 hlt
classical
have : (x.neLocus y : Set ι).WellFoundedOn r := (x.neLocus y).finite_toSet.wellFoundedOn
obtain ⟨i, hi, hl⟩ := this.has_min { i | x i < y i } ⟨⟨j, mem_neLocus.2 hlt.ne⟩, hlt⟩
refine ⟨i, fun k hk ↦ ⟨hle k, ?_⟩, hi⟩
exact of_not_not fun h ↦ hl ⟨k, mem_neLocus... | 0.09375 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 235 | 236 | theorem finrank_finsupp_self {ι : Type v} [Fintype ι] : finrank R (ι →₀ R) = card ι := by |
rw [finrank, rank_finsupp_self, ← mk_toNat_eq_card, toNat_lift]
| 0.09375 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 114 | 116 | theorem mul_charP_coeff_zero [CharP R p] (x : 𝕎 R) : (x * p).coeff 0 = 0 := by |
rw [← frobenius_verschiebung, coeff_frobenius_charP, verschiebung_coeff_zero,
zero_pow hp.out.ne_zero]
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import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
#align_import data.int.bitwise from "leanprover-community/mathlib"@"0743cc5d9d86bcd1bba10f480e948a257d65056f"
#align_import init.data.int.bitwise from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
... | Mathlib/Data/Int/Bitwise.lean | 153 | 155 | theorem bodd_negOfNat (n : ℕ) : bodd (negOfNat n) = n.bodd := by |
cases n <;> simp (config := {decide := true})
rfl
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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 | 265 | 275 | theorem smul_Ici : r • Ici a = Ici (r • a) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Ioi]
constructor
· rintro ⟨a_w, a_h_left, rfl⟩
exact (mul_le_mul_left hr).mpr a_h_left
· rintro h
use x / r
constructor
· exact (le_div_iff' hr).mpr h
· exact mul_div_cancel₀ _ (ne_of_gt hr)
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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 | 121 | 122 | theorem rdropWhile_singleton (x : α) : rdropWhile p [x] = if p x then [] else [x] := by |
rw [← nil_append [x], rdropWhile_concat, rdropWhile_nil]
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import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Int.GCD
instance : GCDMonoid ℕ where
gcd := Nat.gcd
lcm := Nat.lcm
gcd_dvd_left := Nat.gcd_dvd_left
gcd_dvd_right := Nat.gcd_dvd_right
dvd_gcd := Nat.dvd_gcd
gcd_mul_lcm a b := by rw [Nat.gcd_mul_lcm]; rfl
... | Mathlib/Algebra/GCDMonoid/Nat.lean | 139 | 145 | theorem exists_unit_of_abs (a : ℤ) : ∃ (u : ℤ) (_ : IsUnit u), (Int.natAbs a : ℤ) = u * a := by |
cases' natAbs_eq a with h h
· use 1, isUnit_one
rw [← h, one_mul]
· use -1, isUnit_one.neg
rw [← neg_eq_iff_eq_neg.mpr h]
simp only [neg_mul, one_mul]
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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 | 99 | 107 | theorem adjointAux_norm (A : E →L[𝕜] F) : ‖adjointAux A‖ = ‖A‖ := by |
refine le_antisymm ?_ ?_
· refine ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg _) fun x => ?_
rw [adjointAux_apply, LinearIsometryEquiv.norm_map]
exact toSesqForm_apply_norm_le
· nth_rw 1 [← adjointAux_adjointAux A]
refine ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg _) fun x => ?_
rw... | 0.09375 |
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.PSeries
#align_import number_theory.zeta_values from "leanprover-community/mathlib"@"f0c8bf9245297... | Mathlib/NumberTheory/ZetaValues.lean | 74 | 77 | theorem antideriv_bernoulliFun (k : ℕ) (x : ℝ) :
HasDerivAt (fun x => bernoulliFun (k + 1) x / (k + 1)) (bernoulliFun k x) x := by |
convert (hasDerivAt_bernoulliFun (k + 1) x).div_const _ using 1
field_simp [Nat.cast_add_one_ne_zero k]
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import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 178 | 180 | theorem not_dvd_of_degree_lt {p q : R[X]} (h0 : q ≠ 0) (hl : q.degree < p.degree) : ¬p ∣ q := by |
by_contra hcontra
exact h0 (eq_zero_of_dvd_of_degree_lt hcontra hl)
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import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.TensorProduct.Basis
#align_import linear_algebra.tensor_product.matrix from "leanprover-community/mathlib"@"f784cc6142443d9ee623a20788c282112c322081"
variable {R : Type*} {M N P M' N' : Type*} {ι κ τ ι' κ' ... | Mathlib/LinearAlgebra/TensorProduct/Matrix.lean | 68 | 77 | theorem TensorProduct.toMatrix_assoc :
toMatrix ((bM.tensorProduct bN).tensorProduct bP) (bM.tensorProduct (bN.tensorProduct bP))
(TensorProduct.assoc R M N P) =
(1 : Matrix (ι × κ × τ) (ι × κ × τ) R).submatrix _root_.id (Equiv.prodAssoc _ _ _) := by |
ext ⟨i, j, k⟩ ⟨⟨i', j'⟩, k'⟩
simp_rw [toMatrix_apply, Basis.tensorProduct_apply, LinearEquiv.coe_coe,
TensorProduct.assoc_tmul, Basis.tensorProduct_repr_tmul_apply, Matrix.submatrix_apply,
Equiv.prodAssoc_apply, _root_.id, Basis.repr_self_apply, Matrix.one_apply, Prod.ext_iff,
ite_and, @eq_comm _ i', @... | 0.09375 |
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.Algebra.Equicontinuity
import Mathlib.Topology.MetricSpace.Equicontinuity
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.Module.LocallyConvex
#align_import analysis.locally_convex.with_seminorms from "leanprover-community/mathlib"@"... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 115 | 118 | theorem basisSets_zero (U) (hU : U ∈ p.basisSets) : (0 : E) ∈ U := by |
rcases p.basisSets_iff.mp hU with ⟨ι', r, hr, hU⟩
rw [hU, mem_ball_zero, map_zero]
exact hr
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import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 151 | 151 | theorem rotation_pi_apply (x : V) : o.rotation π x = -x := by | simp
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import Mathlib.Topology.Category.TopCat.Adjunctions
#align_import topology.category.Top.epi_mono from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
open CategoryTheory
open TopCat
namespace TopCat
| Mathlib/Topology/Category/TopCat/EpiMono.lean | 27 | 34 | theorem epi_iff_surjective {X Y : TopCat.{u}} (f : X ⟶ Y) : Epi f ↔ Function.Surjective f := by |
suffices Epi f ↔ Epi ((forget TopCat).map f) by
rw [this, CategoryTheory.epi_iff_surjective]
rfl
constructor
· intro
infer_instance
· apply Functor.epi_of_epi_map
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import Mathlib.SetTheory.Ordinal.Arithmetic
namespace OrdinalApprox
universe u
variable {α : Type u}
variable [CompleteLattice α] (f : α →o α) (x : α)
open Function fixedPoints Cardinal Order OrderHom
set_option linter.unusedVariables false in
def lfpApprox (a : Ordinal.{u}) : α :=
sSup ({ f (lfpApprox b) | ... | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | 87 | 90 | theorem le_lfpApprox {a : Ordinal} : x ≤ lfpApprox f x a := by |
unfold lfpApprox
apply le_sSup
simp only [exists_prop, Set.union_singleton, Set.mem_insert_iff, Set.mem_setOf_eq, true_or]
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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 | 147 | 147 | theorem mem_of_mem_inter_right (h : a ∈ l₁ ∩ l₂) : a ∈ l₂ := by | simpa using of_mem_filter h
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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 ConditionallyCompleteLinearOrder
variable [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [Top... | Mathlib/Topology/Order/Monotone.lean | 230 | 232 | theorem Monotone.map_ciSup_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨆ i, g i))
(Mf : Monotone f) (H : BddAbove (range g)) : f (⨆ i, g i) = ⨆ i, f (g i) := by |
rw [iSup, Mf.map_csSup_of_continuousAt Cf (range_nonempty _) H, ← range_comp, iSup]; rfl
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import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def χ₄ : MulChar (ZMod 4) ℤ... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 56 | 56 | theorem χ₄_nat_mod_four (n : ℕ) : χ₄ n = χ₄ (n % 4 : ℕ) := by | rw [← ZMod.natCast_mod n 4]
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import Mathlib.Topology.EMetricSpace.Basic
#align_import topology.metric_space.metric_separated from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
open EMetric Set
noncomputable section
def IsMetricSeparated {X : Type*} [EMetricSpace X] (s t : Set X) :=
∃ r, r ≠ 0 ∧ ∀ x ∈ s, ∀ y ∈... | Mathlib/Topology/MetricSpace/MetricSeparated.lean | 78 | 85 | theorem union_left {s'} (h : IsMetricSeparated s t) (h' : IsMetricSeparated s' t) :
IsMetricSeparated (s ∪ s') t := by |
rcases h, h' with ⟨⟨r, r0, hr⟩, ⟨r', r0', hr'⟩⟩
refine ⟨min r r', ?_, fun x hx y hy => hx.elim ?_ ?_⟩
· rw [← pos_iff_ne_zero] at r0 r0' ⊢
exact lt_min r0 r0'
· exact fun hx => (min_le_left _ _).trans (hr _ hx _ hy)
· exact fun hx => (min_le_right _ _).trans (hr' _ hx _ hy)
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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 | 158 | 162 | theorem laverage_lt_top (hf : ∫⁻ x, f x ∂μ ≠ ∞) : ⨍⁻ x, f x ∂μ < ∞ := by |
obtain rfl | hμ := eq_or_ne μ 0
· simp
· rw [laverage_eq]
exact div_lt_top hf (measure_univ_ne_zero.2 hμ)
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import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
#align_import analysis.special_functions.trigonometric.inverse from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter
open S... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 133 | 134 | theorem arcsin_of_le_neg_one {x : ℝ} (hx : x ≤ -1) : arcsin x = -(π / 2) := by |
rw [← arcsin_projIcc, projIcc_of_le_left _ hx, Subtype.coe_mk, arcsin_neg_one]
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import Mathlib.Data.Set.Image
#align_import order.directed from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
open Function
universe u v w
variable {α : Type u} {β : Type v} {ι : Sort w} (r r' s : α → α → Prop)
local infixl:50 " ≼ " => r
def Directed (f : ι → α) :=
∀ x y, ∃ z, ... | Mathlib/Order/Directed.lean | 58 | 60 | theorem directedOn_iff_directed {s} : @DirectedOn α r s ↔ Directed r (Subtype.val : s → α) := by |
simp only [DirectedOn, Directed, Subtype.exists, exists_and_left, exists_prop, Subtype.forall]
exact forall₂_congr fun x _ => by simp [And.comm, and_assoc]
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import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.add 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/Add.lean | 131 | 133 | theorem derivWithin_const_add (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) :
derivWithin (fun y => c + f y) s x = derivWithin f s x := by |
simp only [derivWithin, fderivWithin_const_add hxs]
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import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheory.MeasurableSpace.Basic
open Set
namespace MeasureTheory
variable {ι : Type _} {α : ι → Type _}
section cylinder
def cylinder (s : Finset ι) (S : Set (∀ i : s, α... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 193 | 195 | theorem inter_cylinder_same (s : Finset ι) (S₁ : Set (∀ i : s, α i)) (S₂ : Set (∀ i : s, α i)) :
cylinder s S₁ ∩ cylinder s S₂ = cylinder s (S₁ ∩ S₂) := by |
classical rw [inter_cylinder]; rfl
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import Mathlib.Algebra.MonoidAlgebra.Basic
#align_import algebra.monoid_algebra.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
variable {k G : Type*} [Semiring k]
namespace AddMonoidAlgebra
section
variable [AddCancelCommMonoid G]
noncomputable def divOf (x : k[G]) (g... | Mathlib/Algebra/MonoidAlgebra/Division.lean | 139 | 141 | theorem modOf_apply_of_exists_add (x : k[G]) (g : G) (g' : G)
(h : ∃ d, g' = g + d) : (x %ᵒᶠ g) g' = 0 := by |
classical exact Finsupp.filter_apply_neg _ _ <| by rwa [Classical.not_not]
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import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e600... | Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 45 | 50 | theorem Finsupp.toFreeAbelianGroup_comp_singleAddHom (x : X) :
Finsupp.toFreeAbelianGroup.comp (Finsupp.singleAddHom x) =
(smulAddHom ℤ (FreeAbelianGroup X)).flip (of x) := by |
ext
simp only [AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply, one_smul,
toFreeAbelianGroup, Finsupp.liftAddHom_apply_single]
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import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.Init.Data.Prod
import Mathlib.RingTheory.OreLocalization.Basic
#align_import group_theory.monoid_localization from "leanprover-community/... | Mathlib/GroupTheory/MonoidLocalization.lean | 206 | 207 | theorem r_iff_exists {x y : M × S} : r S x y ↔ ∃ c : S, ↑c * (↑y.2 * x.1) = c * (x.2 * y.1) := by |
rw [r_eq_r' S]; rfl
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def SatisfiesM {m : Type u → Type v} [Functor m] (p : α → Prop) (x : m α) : Prop :=
∃ x' : m {a // p a}, Subtype.val <$> x' = x
@[simp] theorem SatisfiesM_Id_eq : SatisfiesM (m := Id) p x ↔ p x :=
⟨fun ⟨y, eq⟩ => eq ▸ y.2, fun h => ⟨⟨_, h⟩, rfl⟩⟩
@[simp] theorem SatisfiesM_Option_eq : SatisfiesM (m := Option... | .lake/packages/batteries/Batteries/Classes/SatisfiesM.lean | 165 | 166 | theorem SatisfiesM_StateRefT_eq [Monad m] :
SatisfiesM (m := StateRefT' ω σ m) p x ↔ ∀ s, SatisfiesM p (x s) := by | simp
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import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.IsLocalHomeomorph
#align_import topology.instances.add_circle from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/AddCircle.lean | 152 | 153 | theorem coe_eq_zero_iff {x : 𝕜} : (x : AddCircle p) = 0 ↔ ∃ n : ℤ, n • p = x := by |
simp [AddSubgroup.mem_zmultiples_iff]
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import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Constructions.Prod.Integral
open Fintype MeasureTheory MeasureTheory.Measure
variable {𝕜 : Type*} [RCLike 𝕜]
namespace MeasureTheory
theorem Integrable.fin_nat_prod {n : ℕ} {E : Fin n → Type*}
[∀ i, MeasureSpace (E i)] [∀ i, SigmaF... | Mathlib/MeasureTheory/Integral/Pi.lean | 95 | 98 | theorem integral_fintype_prod_eq_pow {E : Type*} (ι : Type*) [Fintype ι] (f : E → 𝕜)
[MeasureSpace E] [SigmaFinite (volume : Measure E)] :
∫ x : ι → E, ∏ i, f (x i) = (∫ x, f x) ^ (card ι) := by |
rw [integral_fintype_prod_eq_prod, Finset.prod_const, card]
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import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.ShrinkingLemma
import Mathlib.Topology.UrysohnsLemma
#align_import topology.partition_of_unity from "leanprover-... | Mathlib/Topology/PartitionOfUnity.lean | 188 | 190 | theorem mem_finsupport (x₀ : X) {i} :
i ∈ ρ.finsupport x₀ ↔ i ∈ support fun i ↦ ρ i x₀ := by |
simp only [finsupport, mem_support, Finite.mem_toFinset, mem_setOf_eq]
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import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 101 | 109 | theorem quotient_mk_maps_eq (P : Ideal R[X]) :
((Quotient.mk (map (mapRingHom (Quotient.mk (P.comap (C : R →+* R[X])))) P)).comp C).comp
(Quotient.mk (P.comap (C : R →+* R[X]))) =
(Ideal.quotientMap (map (mapRingHom (Quotient.mk (P.comap (C : R →+* R[X])))) P)
(mapRingHom (Quotient.mk (P.c... |
refine RingHom.ext fun x => ?_
repeat' rw [RingHom.coe_comp, Function.comp_apply]
rw [quotientMap_mk, coe_mapRingHom, map_C]
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import Mathlib.Data.Finset.Basic
variable {ι : Sort _} {π : ι → Sort _} {x : ∀ i, π i} [DecidableEq ι]
namespace Function
def updateFinset (x : ∀ i, π i) (s : Finset ι) (y : ∀ i : ↥s, π i) (i : ι) : π i :=
if hi : i ∈ s then y ⟨i, hi⟩ else x i
open Finset Equiv
theorem updateFinset_def {s : Finset ι} {y} :
... | Mathlib/Data/Finset/Update.lean | 35 | 41 | theorem updateFinset_singleton {i y} :
updateFinset x {i} y = Function.update x i (y ⟨i, mem_singleton_self i⟩) := by |
congr with j
by_cases hj : j = i
· cases hj
simp only [dif_pos, Finset.mem_singleton, update_same, updateFinset]
· simp [hj, updateFinset]
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import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Analysis.Normed.Group.ZeroAtInfty
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Ana... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 203 | 205 | theorem decay_neg_aux (k n : ℕ) (f : 𝓢(E, F)) (x : E) :
‖x‖ ^ k * ‖iteratedFDeriv ℝ n (-f : E → F) x‖ = ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ := by |
rw [iteratedFDeriv_neg_apply, norm_neg]
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import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory.AdicCompletion.Basic
#align_import ring_theory.discrete_valuation_ring.basic from "leanprover-community/mathlib"@"c163ec99dfc664628ca15d215fce0a5b9c2... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 107 | 109 | theorem exists_irreducible : ∃ ϖ : R, Irreducible ϖ := by |
simp_rw [irreducible_iff_uniformizer]
exact (IsPrincipalIdealRing.principal <| maximalIdeal R).principal
| 0.09375 |
import Mathlib.RingTheory.Ideal.Maps
#align_import ring_theory.ideal.prod from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
universe u v
variable {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I I' : Ideal R) (J J' : Ideal S)
namespace Ideal
def prod : Ideal (R × S) where
... | Mathlib/RingTheory/Ideal/Prod.lean | 50 | 58 | theorem ideal_prod_eq (I : Ideal (R × S)) :
I = Ideal.prod (map (RingHom.fst R S) I : Ideal R) (map (RingHom.snd R S) I) := by |
apply Ideal.ext
rintro ⟨r, s⟩
rw [mem_prod, mem_map_iff_of_surjective (RingHom.fst R S) Prod.fst_surjective,
mem_map_iff_of_surjective (RingHom.snd R S) Prod.snd_surjective]
refine ⟨fun h => ⟨⟨_, ⟨h, rfl⟩⟩, ⟨_, ⟨h, rfl⟩⟩⟩, ?_⟩
rintro ⟨⟨⟨r, s'⟩, ⟨h₁, rfl⟩⟩, ⟨⟨r', s⟩, ⟨h₂, rfl⟩⟩⟩
simpa using I.add_mem (I... | 0.09375 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.LiftingProperties.Basic
#align_import category_theory.limits.shapes.strong_epi from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
variable... | Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean | 172 | 175 | theorem StrongEpi.iff_of_arrow_iso {A B A' B' : C} {f : A ⟶ B} {g : A' ⟶ B'}
(e : Arrow.mk f ≅ Arrow.mk g) : StrongEpi f ↔ StrongEpi g := by |
constructor <;> intro
exacts [StrongEpi.of_arrow_iso e, StrongEpi.of_arrow_iso e.symm]
| 0.09375 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.FractionRing
import M... | Mathlib/RingTheory/Localization/Integral.lean | 112 | 115 | theorem integerNormalization_aeval_eq_zero [Algebra R R'] [Algebra S R'] [IsScalarTower R S R']
(p : S[X]) {x : R'} (hx : aeval x p = 0) : aeval x (integerNormalization M p) = 0 := by |
rw [aeval_def, IsScalarTower.algebraMap_eq R S R',
integerNormalization_eval₂_eq_zero _ (algebraMap _ _) _ hx]
| 0.09375 |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.pointwise from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
open Affine Pointwise
open Set
namespace AffineSubspace
variable {k : Type*} [Ring k]
variable {V P V₁ P₁ V₂ P₂ : Type*}
var... | Mathlib/LinearAlgebra/AffineSpace/Pointwise.lean | 74 | 79 | theorem map_pointwise_vadd (f : P₁ →ᵃ[k] P₂) (v : V₁) (s : AffineSubspace k P₁) :
(v +ᵥ s).map f = f.linear v +ᵥ s.map f := by |
erw [pointwise_vadd_eq_map, pointwise_vadd_eq_map, map_map, map_map]
congr 1
ext
exact f.map_vadd _ _
| 0.09375 |
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.circle_transform from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
open Set MeasureTheory Metric Filter Function
open scoped Interval Real
noncomputable secti... | Mathlib/MeasureTheory/Integral/CircleTransform.lean | 86 | 90 | theorem continuous_circleTransformDeriv {R : ℝ} (hR : 0 < R) {f : ℂ → E} {z w : ℂ}
(hf : ContinuousOn f (sphere z R)) (hw : w ∈ ball z R) :
Continuous (circleTransformDeriv R z w f) := by |
rw [circleTransformDeriv_eq]
exact (continuous_circleMap_inv hw).smul (continuous_circleTransform hR hf hw)
| 0.09375 |
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open Topology Uniformity Filter S... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 106 | 111 | theorem tendstoUniformlyOn_iff_tendstoUniformlyOnFilter :
TendstoUniformlyOn F f p s ↔ TendstoUniformlyOnFilter F f p (𝓟 s) := by |
simp only [TendstoUniformlyOn, TendstoUniformlyOnFilter]
apply forall₂_congr
simp_rw [eventually_prod_principal_iff]
simp
| 0.09375 |
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import probability.process.stopping from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open Filter Order TopologicalSpace
open scoped Classical MeasureTheory NNReal ENNReal Top... | Mathlib/Probability/Process/Stopping.lean | 72 | 82 | theorem IsStoppingTime.measurableSet_lt_of_pred [PredOrder ι] (hτ : IsStoppingTime f τ) (i : ι) :
MeasurableSet[f i] {ω | τ ω < i} := by |
by_cases hi_min : IsMin i
· suffices {ω : Ω | τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i)
ext1 ω
simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff]
rw [isMin_iff_forall_not_lt] at hi_min
exact hi_min (τ ω)
have : {ω : Ω | τ ω < i} = τ ⁻¹' Set.Iic (pred i) := by... | 0.09375 |
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 | 164 | 166 | theorem ofFractionRing_div (p q : FractionRing K[X]) :
ofFractionRing (p / q) = ofFractionRing p / ofFractionRing q := by |
simp only [Div.div, HDiv.hDiv, RatFunc.div]
| 0.09375 |
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathlib"@"74ad1c88c77e799d2fea62801d1dbbd698cff1b7"
assert_not_exists Submodule
open Function
namespace ZMod
instance charZero : CharZero (ZMod 0) :=... | Mathlib/Data/ZMod/Basic.lean | 183 | 186 | theorem cast_eq_val [NeZero n] (a : ZMod n) : (cast a : R) = a.val := by |
cases n
· cases NeZero.ne 0 rfl
rfl
| 0.09375 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e600... | Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 87 | 89 | theorem toFinsupp_toFreeAbelianGroup (f : X →₀ ℤ) :
FreeAbelianGroup.toFinsupp (Finsupp.toFreeAbelianGroup f) = f := by |
rw [← AddMonoidHom.comp_apply, toFinsupp_comp_toFreeAbelianGroup, AddMonoidHom.id_apply]
| 0.09375 |
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Directed
import Mathlib.Order.Hom.Set
#align_import order.antichain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open Function Set
section General
variable {α β : Type*} {r r₁ r₂ : α →... | Mathlib/Order/Antichain.lean | 120 | 124 | theorem image_relEmbedding (hs : IsAntichain r s) (φ : r ↪r r') : IsAntichain r' (φ '' s) := by |
intro b hb b' hb' h₁ h₂
rw [Set.mem_image] at hb hb'
obtain ⟨⟨a, has, rfl⟩, ⟨a', has', rfl⟩⟩ := hb, hb'
exact hs has has' (fun haa' => h₁ (by rw [haa'])) (φ.map_rel_iff.mp h₂)
| 0.09375 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Order.Basic
import Mathlib.Order.Monotone.Basic
#align_import algebra.covariant_and_contravariant from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"
-- TODO: convert `ExistsMulOfLE`, `ExistsAddOfLE`?
-- TODO: relationship with `Con/AddC... | Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean | 281 | 286 | theorem covariant_le_of_covariant_lt [PartialOrder N] :
Covariant M N μ (· < ·) → Covariant M N μ (· ≤ ·) := by |
intro h a b c bc
rcases bc.eq_or_lt with (rfl | bc)
· exact le_rfl
· exact (h _ bc).le
| 0.09375 |
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Nat
--@[pp_nodot] porting note: unknown attribute
def log (b : ℕ) : ℕ → ℕ
| n => i... | Mathlib/Data/Nat/Log.lean | 108 | 111 | theorem pow_le_of_le_log {b x y : ℕ} (hy : y ≠ 0) (h : x ≤ log b y) : b ^ x ≤ y := by |
refine (le_or_lt b 1).elim (fun hb => ?_) fun hb => (pow_le_iff_le_log hb hy).2 h
rw [log_of_left_le_one hb, Nat.le_zero] at h
rwa [h, Nat.pow_zero, one_le_iff_ne_zero]
| 0.09375 |
import Mathlib.Data.Set.Prod
#align_import data.set.n_ary from "leanprover-community/mathlib"@"5e526d18cea33550268dcbbddcb822d5cde40654"
open Function
namespace Set
variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*} {f f' : α → β → γ} {g g' : α → β → γ → δ}
variable {s s' : Set α} {t t' : Set β} {u u' : Set γ} {v... | Mathlib/Data/Set/NAry.lean | 72 | 73 | theorem image2_subset_iff_right : image2 f s t ⊆ u ↔ ∀ b ∈ t, (fun a => f a b) '' s ⊆ u := by |
simp_rw [image2_subset_iff, image_subset_iff, subset_def, mem_preimage, @forall₂_swap α]
| 0.09375 |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ι : Type*} {G : ι → Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [∀ i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 96 | 97 | theorem of_apply_eq_base (i : ι) (x : H) : of i (φ i x) = base φ x := by |
rw [← MonoidHom.comp_apply, of_comp_eq_base]
| 0.09375 |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 135 | 137 | theorem integral_comp_mul_left (g : ℝ → F) (a : ℝ) :
(∫ x : ℝ, g (a * x)) = |a⁻¹| • ∫ y : ℝ, g y := by |
simp_rw [← smul_eq_mul, Measure.integral_comp_smul, FiniteDimensional.finrank_self, pow_one]
| 0.09375 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSu... | Mathlib/Data/List/Chain.lean | 62 | 65 | theorem chain_split {a b : α} {l₁ l₂ : List α} :
Chain R a (l₁ ++ b :: l₂) ↔ Chain R a (l₁ ++ [b]) ∧ Chain R b l₂ := by |
induction' l₁ with x l₁ IH generalizing a <;>
simp only [*, nil_append, cons_append, Chain.nil, chain_cons, and_true_iff, and_assoc]
| 0.09375 |
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Basic
import Mathlib.Tactic.ApplyFun
#align_import computability.turing_machine from "leanprover-commu... | Mathlib/Computability/TuringMachine.lean | 91 | 95 | theorem BlankExtends.below_of_le {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} :
BlankExtends l l₁ → BlankExtends l l₂ → l₁.length ≤ l₂.length → BlankExtends l₁ l₂ := by |
rintro ⟨i, rfl⟩ ⟨j, rfl⟩ h; use j - i
simp only [List.length_append, Nat.add_le_add_iff_left, List.length_replicate] at h
simp only [← List.replicate_add, Nat.add_sub_cancel' h, List.append_assoc]
| 0.09375 |
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Set.Function
import Mathlib.Logic.Pairwise
#align_import algebra.group.pi from "leanprover-community/mathlib"@"e4bc74cbaf429d706cb9140902f7ca6c431e75a4"
assert_not_exists AddMonoidWithOne
assert_not_exists Mono... | Mathlib/Algebra/Group/Pi/Lemmas.lean | 552 | 555 | theorem uncurry_mulSingle_mulSingle [DecidableEq α] [∀ a, DecidableEq (β a)] [∀ a b, One (γ a b)]
(a : α) (b : β a) (x : γ a b) :
Sigma.uncurry (Pi.mulSingle a (Pi.mulSingle b x)) = Pi.mulSingle (Sigma.mk a b) x := by |
rw [← curry_mulSingle ⟨a, b⟩, uncurry_curry]
| 0.09375 |
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.modeq from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
namespace AddCommGroup
variable {α : Type*}
section AddCommGroup
variable [AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}
... | Mathlib/Algebra/ModEq.lean | 311 | 312 | theorem modEq_iff_int_modEq {a b z : ℤ} : a ≡ b [PMOD z] ↔ a ≡ b [ZMOD z] := by |
simp [ModEq, dvd_iff_exists_eq_mul_left, Int.modEq_iff_dvd]
| 0.09375 |
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 152 | 157 | theorem measure_biUnion₀ {s : Set β} {f : β → Set α} (hs : s.Countable)
(hd : s.Pairwise (AEDisjoint μ on f)) (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) :
μ (⋃ b ∈ s, f b) = ∑' p : s, μ (f p) := by |
haveI := hs.toEncodable
rw [biUnion_eq_iUnion]
exact measure_iUnion₀ (hd.on_injective Subtype.coe_injective fun x => x.2) fun x => h x x.2
| 0.09375 |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.RingTheory.Fintype
import Mathlib.Tactic.IntervalCases
#align_import number_the... | Mathlib/NumberTheory/LucasLehmer.lean | 148 | 151 | theorem sMod_lt (p : ℕ) (hp : p ≠ 0) (i : ℕ) : sMod p i < 2 ^ p - 1 := by |
rw [← sMod_mod]
refine (Int.emod_lt _ (mersenne_int_ne_zero p hp)).trans_eq ?_
exact abs_of_nonneg (mersenne_int_pos hp).le
| 0.09375 |
import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Analysis.Normed.Group.ZeroAtInfty
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Ana... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 173 | 175 | theorem isBigO_cocompact_zpow [ProperSpace E] (k : ℤ) :
f =O[cocompact E] fun x => ‖x‖ ^ k := by |
simpa only [Real.rpow_intCast] using isBigO_cocompact_rpow f k
| 0.09375 |
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.Data.Finsupp.PWO
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a484a7d0eade4e1268f4fb402859b6686037f965"
set_option linter.uppercaseLean3 false
open Finset Function
open sco... | Mathlib/RingTheory/HahnSeries/PowerSeries.lean | 112 | 113 | theorem ofPowerSeries_apply_coeff (x : PowerSeries R) (n : ℕ) :
(ofPowerSeries Γ R x).coeff n = PowerSeries.coeff R n x := by | simp [ofPowerSeries_apply]
| 0.09375 |
import Mathlib.Algebra.Quotient
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.SetTheory.Cardinal.Finite
#align_import group_theory.coset from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce4... | Mathlib/GroupTheory/Coset.lean | 302 | 308 | theorem leftRel_apply {x y : α} : @Setoid.r _ (leftRel s) x y ↔ x⁻¹ * y ∈ s :=
calc
(∃ a : s.op, y * MulOpposite.unop a = x) ↔ ∃ a : s, y * a = x :=
s.equivOp.symm.exists_congr_left
_ ↔ ∃ a : s, x⁻¹ * y = a⁻¹ := by |
simp only [inv_mul_eq_iff_eq_mul, Subgroup.coe_inv, eq_mul_inv_iff_mul_eq]
_ ↔ x⁻¹ * y ∈ s := by simp [exists_inv_mem_iff_exists_mem]
| 0.09375 |
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 | 137 | 139 | theorem log_inv (x : ℝ) : log x⁻¹ = -log x := by |
by_cases hx : x = 0; · simp [hx]
rw [← exp_eq_exp, exp_log_eq_abs (inv_ne_zero hx), exp_neg, exp_log_eq_abs hx, abs_inv]
| 0.09375 |
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
#align_import linear_algebra.quadratic_form.prod from "leanprover-community/mathlib"@"9b2755b951bc323c962bd072cd447b375cf58101"
universe u v w
variable {ι : Type*} {R : Type*} {M₁ M₂ N₁ N₂ : Type*} {Mᵢ Nᵢ : ι → Type*}
namespace QuadraticForm
section Pro... | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 313 | 325 | theorem anisotropic_of_pi [Fintype ι] {R} [OrderedCommRing R] [∀ i, Module R (Mᵢ i)]
{Q : ∀ i, QuadraticForm R (Mᵢ i)} (h : (pi Q).Anisotropic) : ∀ i, (Q i).Anisotropic := by |
simp_rw [Anisotropic, pi_apply, Function.funext_iff, Pi.zero_apply] at h
intro i x hx
classical
have := h (Pi.single i x) ?_ i
· rw [Pi.single_eq_same] at this
exact this
apply Finset.sum_eq_zero
intro j _
by_cases hji : j = i
· subst hji; rw [Pi.single_eq_same, hx]
· rw [Pi.single_eq_of_ne hji... | 0.09375 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 74 | 79 | theorem sign_intCast (z : ℤ) : sign (z : ℝ) = ↑(Int.sign z) := by |
obtain hn | rfl | hp := lt_trichotomy z (0 : ℤ)
· rw [sign_of_neg (Int.cast_lt_zero.mpr hn), Int.sign_eq_neg_one_of_neg hn, Int.cast_neg,
Int.cast_one]
· rw [Int.cast_zero, sign_zero, Int.sign_zero, Int.cast_zero]
· rw [sign_of_pos (Int.cast_pos.mpr hp), Int.sign_eq_one_of_pos hp, Int.cast_one]
| 0.09375 |
import Batteries.Data.List.Lemmas
import Batteries.Tactic.Classical
import Mathlib.Tactic.TypeStar
import Mathlib.Mathport.Rename
#align_import data.list.tfae from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
namespace List
def TFAE (l : List Prop) : Prop :=
∀ x ∈ l, ∀ y ∈ l, x ↔ ... | Mathlib/Data/List/TFAE.lean | 37 | 37 | theorem tfae_singleton (p) : TFAE [p] := by | simp [TFAE, -eq_iff_iff]
| 0.09375 |
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 133 | 134 | theorem toIcoMod_sub_self (a b : α) : toIcoMod hp a b - b = -toIcoDiv hp a b • p := by |
rw [toIcoMod, sub_sub_cancel_left, neg_smul]
| 0.09375 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.ProbabilityMassFunction.Constructions
open scoped Classical MeasureTheory NNReal ENNReal
-- TODO: We can't `open ProbabilityTheory` without opening the `ProbabilityThe... | Mathlib/Probability/Distributions/Uniform.lean | 95 | 98 | theorem toMeasurable_iff {X : Ω → E} {s : Set E} :
IsUniform X (toMeasurable μ s) ℙ μ ↔ IsUniform X s ℙ μ := by |
unfold IsUniform
rw [ProbabilityTheory.cond_toMeasurable_eq]
| 0.09375 |
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Finite.Set
#align_import combinatorics.simple_graph.ends.defs from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
universe u
variable {V : Type u} (G : SimpleGraph V... | Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean | 71 | 75 | theorem componentComplMk_eq_of_adj (G : SimpleGraph V) {v w : V} (vK : v ∉ K) (wK : w ∉ K)
(a : G.Adj v w) : G.componentComplMk vK = G.componentComplMk wK := by |
rw [ConnectedComponent.eq]
apply Adj.reachable
exact a
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import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.Antichain
import Mathlib.Order.SetNotation
#align_import data.set.intervals.ord_connected from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
open scoped Interval
open Set
open OrderDual (toDual ofDual)
namespace Set
... | Mathlib/Order/Interval/Set/OrdConnected.lean | 57 | 63 | theorem ordConnected_of_Ioo {α : Type*} [PartialOrder α] {s : Set α}
(hs : ∀ x ∈ s, ∀ y ∈ s, x < y → Ioo x y ⊆ s) : OrdConnected s := by |
rw [ordConnected_iff]
intro x hx y hy hxy
rcases eq_or_lt_of_le hxy with (rfl | hxy'); · simpa
rw [← Ioc_insert_left hxy, ← Ioo_insert_right hxy']
exact insert_subset_iff.2 ⟨hx, insert_subset_iff.2 ⟨hy, hs x hx y hy hxy'⟩⟩
| 0.09375 |
import Mathlib.Order.Cover
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.GaloisConnection
#align_import order.modular_lattice from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Set
variable {α : Type*}
class IsWeakUpperModularLattice (α : Type*) [Lattice α] : Prop ... | Mathlib/Order/ModularLattice.lean | 216 | 217 | theorem inf_sup_assoc_of_le {x : α} (y : α) {z : α} (h : z ≤ x) : x ⊓ y ⊔ z = x ⊓ (y ⊔ z) := by |
rw [inf_comm, sup_comm, ← sup_inf_assoc_of_le y h, inf_comm, sup_comm]
| 0.09375 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Ring.Int
#align_import data.int.associated from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
| Mathlib/Data/Int/Associated.lean | 21 | 30 | theorem Int.natAbs_eq_iff_associated {a b : ℤ} : a.natAbs = b.natAbs ↔ Associated a b := by |
refine Int.natAbs_eq_natAbs_iff.trans ?_
constructor
· rintro (rfl | rfl)
· rfl
· exact ⟨-1, by simp⟩
· rintro ⟨u, rfl⟩
obtain rfl | rfl := Int.units_eq_one_or u
· exact Or.inl (by simp)
· exact Or.inr (by simp)
| 0.09375 |
import Mathlib.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
#align_import probability.independence.zero_one from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
open MeasureTheory MeasurableSpace
open scoped MeasureTheory ENNReal
namespace Probabili... | Mathlib/Probability/Independence/ZeroOne.lean | 52 | 56 | theorem kernel.measure_eq_zero_or_one_of_indepSet_self [∀ a, IsFiniteMeasure (κ a)] {t : Set Ω}
(h_indep : IndepSet t t κ μα) :
∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 := by |
filter_upwards [measure_eq_zero_or_one_or_top_of_indepSet_self h_indep] with a h_0_1_top
simpa only [measure_ne_top (κ a), or_false] using h_0_1_top
| 0.09375 |
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.LocallyFinite
open Set Filter Topology TopologicalSpace Classical Function
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [Topolog... | Mathlib/Topology/Compactness/Compact.lean | 104 | 116 | theorem IsCompact.image_of_continuousOn {f : X → Y} (hs : IsCompact s) (hf : ContinuousOn f s) :
IsCompact (f '' s) := by |
intro l lne ls
have : NeBot (l.comap f ⊓ 𝓟 s) :=
comap_inf_principal_neBot_of_image_mem lne (le_principal_iff.1 ls)
obtain ⟨x, hxs, hx⟩ : ∃ x ∈ s, ClusterPt x (l.comap f ⊓ 𝓟 s) := @hs _ this inf_le_right
haveI := hx.neBot
use f x, mem_image_of_mem f hxs
have : Tendsto f (𝓝 x ⊓ (comap f l ⊓ 𝓟 s)) (�... | 0.09375 |
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