Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 |
|---|---|---|---|---|---|---|
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Measure.GiryMonad
#align_import probability.kernel.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open MeasureTheory
open scoped MeasureTheory ENNReal NNReal
namespace ProbabilityTheory
noncomputab... | Mathlib/Probability/Kernel/Basic.lean | 113 | 114 | theorem finset_sum_apply (I : Finset ι) (κ : ι → kernel α β) (a : α) :
(∑ i ∈ I, κ i) a = ∑ i ∈ I, κ i a := by | rw [coe_finset_sum, Finset.sum_apply]
| 1 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ι : Sort u} {α : Type v} {β : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder α] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 102 | 103 | theorem iUnion_Ico_left (b : α) : ⋃ a, Ico a b = Iio b := by |
simp only [← Ici_inter_Iio, ← iUnion_inter, iUnion_Ici, univ_inter]
| 1 |
import Mathlib.Order.Filter.Ultrafilter
import Mathlib.Order.Filter.Germ
#align_import order.filter.filter_product from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
universe u v
variable {α : Type u} {β : Type v} {φ : Ultrafilter α}
open scoped Classical
namespace Filter
local not... | Mathlib/Order/Filter/FilterProduct.lean | 65 | 66 | theorem coe_lt [Preorder β] {f g : α → β} : (f : β*) < g ↔ ∀* x, f x < g x := by |
simp only [lt_iff_le_not_le, eventually_and, coe_le, eventually_not, EventuallyLE]
| 1 |
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Finset Set
variable (𝕜 E : Type*) {ι : Type*} [OrderedRing 𝕜] [AddCommGroup E] [Mod... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 86 | 87 | theorem mem_space_iff : x ∈ K.space ↔ ∃ s ∈ K.faces, x ∈ convexHull 𝕜 (s : Set E) := by |
simp [space]
| 1 |
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.Group
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathli... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 152 | 154 | theorem mem_zpowers_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ} [hp : Fact p.Prime]
(h : Fintype.card G = p) {g g' : G} (hg : g ≠ 1) : g' ∈ zpowers g := by |
simp_rw [zpowers_eq_top_of_prime_card h hg, Subgroup.mem_top]
| 1 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 245 | 246 | theorem get_toList (n : ℕ) (hn : n < length (toList p x)) :
(toList p x).get ⟨n, hn⟩ = (p ^ n) x := by | simp [toList]
| 1 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Algebra.Star.Unitary
import Mathlib.Topology.Algebra.Module.Star
#align_import analysis.no... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 140 | 141 | theorem star_mul_self_ne_zero_iff (x : E) : x⋆ * x ≠ 0 ↔ x ≠ 0 := by |
simp only [Ne, star_mul_self_eq_zero_iff]
| 1 |
import Mathlib.Data.List.Range
import Mathlib.Algebra.Order.Ring.Nat
variable {α : Type*}
namespace List
@[simp]
theorem length_iterate (f : α → α) (a : α) (n : ℕ) : length (iterate f a n) = n := by
induction n generalizing a <;> simp [*]
@[simp]
theorem iterate_eq_nil {f : α → α} {a : α} {n : ℕ} : iterate f ... | Mathlib/Data/List/Iterate.lean | 44 | 46 | theorem range_map_iterate (n : ℕ) (f : α → α) (a : α) :
(List.range n).map (f^[·] a) = List.iterate f a n := by |
apply List.ext_get <;> simp
| 1 |
import Mathlib.Data.Fintype.Card
import Mathlib.Computability.Language
import Mathlib.Tactic.NormNum
#align_import computability.DFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Computability
universe u v
-- Porting note: Required as `DFA` is used in mathlib3
set_option li... | Mathlib/Computability/DFA.lean | 64 | 66 | theorem evalFrom_append_singleton (s : σ) (x : List α) (a : α) :
M.evalFrom s (x ++ [a]) = M.step (M.evalFrom s x) a := by |
simp only [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil]
| 1 |
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 | 84 | 85 | theorem tendsto_ceil_left_pure (n : ℤ) : Tendsto (ceil : α → ℤ) (𝓝[≤] n) (pure n) := by |
simpa only [ceil_intCast] using tendsto_ceil_left_pure_ceil (n : α)
| 1 |
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.ordinal from "leanprover-community/mathlib"@"b90e72c7eebbe8de7c8293a80208ea2ba135c834"
universe u
open SetTheory PGame
open scoped NaturalOps PGame
namespace Ordinal
noncomputable def toPGame : Ordin... | Mathlib/SetTheory/Game/Ordinal.lean | 130 | 131 | theorem toPGame_lf {a b : Ordinal} (h : a < b) : a.toPGame ⧏ b.toPGame := by |
convert moveLeft_lf (toLeftMovesToPGame ⟨a, h⟩); rw [toPGame_moveLeft]
| 1 |
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-community/mathlib"@"4e24c4bfcff371c71f7ba22050308aa17815626c"
variable {α β : Type*} [LinearOrder α]
open Function
namespace Set
def projIci (a x : α) : Ici a := ⟨max a x,... | Mathlib/Order/Interval/Set/ProjIcc.lean | 124 | 124 | theorem projIci_coe (x : Ici a) : projIci a x = x := by | cases x; apply projIci_of_mem
| 1 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 73 | 73 | theorem eval₂_X : X.eval₂ f x = x := by | simp [eval₂_eq_sum]
| 1 |
import Mathlib.FieldTheory.SeparableClosure
import Mathlib.Algebra.CharP.IntermediateField
open FiniteDimensional Polynomial IntermediateField Field
noncomputable section
universe u v w
variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
variable (K : Type w) [Field K] [Algebra F K]
section per... | Mathlib/FieldTheory/PurelyInseparable.lean | 287 | 289 | theorem mem_perfectClosure_iff_natSepDegree_eq_one {x : E} :
x ∈ perfectClosure F E ↔ (minpoly F x).natSepDegree = 1 := by |
rw [mem_perfectClosure_iff, minpoly.natSepDegree_eq_one_iff_pow_mem (ringExpChar F)]
| 1 |
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.reverse from "leanprover-community/mathlib"@"44de64f183393284a16016dfb2a48ac97382f2bd"
namespace Polynomial
open Polynomial Finsupp Finset
open... | Mathlib/Algebra/Polynomial/Reverse.lean | 166 | 167 | theorem reflect_monomial (N n : ℕ) : reflect N ((X : R[X]) ^ n) = X ^ revAt N n := by |
rw [← one_mul (X ^ n), ← one_mul (X ^ revAt N n), ← C_1, reflect_C_mul_X_pow]
| 1 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 87 | 87 | theorem transvection_zero : transvection i j (0 : R) = 1 := by | simp [transvection]
| 1 |
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.add_torsor from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
class AddTorsor (G : outParam Type*) (P : Type*) [AddGroup G] extends AddAction G P,
VSub G P where
[nonempty : Nonempty P]
vsub_vadd' : ∀ p₁ p₂ : P, (p₁ ... | Mathlib/Algebra/AddTorsor.lean | 124 | 125 | theorem vsub_self (p : P) : p -ᵥ p = (0 : G) := by |
rw [← zero_add (p -ᵥ p), ← vadd_vsub_assoc, vadd_vsub]
| 1 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 307 | 308 | theorem map_neg₂ (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (x : M) (y : F) : f (-x) y = -f x y := by |
rw [f.map_neg, neg_apply]
| 1 |
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 119 | 120 | theorem add_left_cancel (a) {b c : Ordinal} : a + b = a + c ↔ b = c := by |
simp only [le_antisymm_iff, add_le_add_iff_left]
| 1 |
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 | 127 | 127 | theorem c_of_eq (h : P.toPoly = Q.toPoly) : P.c = Q.c := by | rw [← coeff_eq_c, h, coeff_eq_c]
| 1 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : ℝ}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 76 | 77 | theorem logb_div (hx : x ≠ 0) (hy : y ≠ 0) : logb b (x / y) = logb b x - logb b y := by |
simp_rw [logb, log_div hx hy, sub_div]
| 1 |
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 84 | 84 | theorem add_one (n : PosNum) : n + 1 = succ n := by | cases n <;> rfl
| 1 |
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.GroupTheory.GroupAction.SubMulAction
open scoped BigOperators Pointwise
namespace MulAction
section SMul
variable (G : Type*) {X : Type*} [SMul G X]
-- Change termin... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 90 | 92 | theorem IsBlock.mk_notempty {B : Set X} :
IsBlock G B ↔ ∀ g g' : G, g • B ∩ g' • B ≠ ∅ → g • B = g' • B := by |
simp_rw [IsBlock.def, or_iff_not_imp_right, Set.disjoint_iff_inter_eq_empty]
| 1 |
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e228"
open scoped Classical Topology
open Finset Filter
namespace FormalMultilinearSeries
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} ... | Mathlib/Analysis/Analytic/Inverse.lean | 177 | 178 | theorem rightInv_coeff_zero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
p.rightInv i 0 = 0 := by | rw [rightInv]
| 1 |
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 62 | 64 | theorem equalizerSubobject_arrow_comp :
(equalizerSubobject f g).arrow ≫ f = (equalizerSubobject f g).arrow ≫ g := by |
rw [← equalizerSubobject_arrow, Category.assoc, Category.assoc, equalizer.condition]
| 1 |
import Mathlib.FieldTheory.SplittingField.IsSplittingField
import Mathlib.Algebra.CharP.Algebra
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Classical Polynomial
universe u v w
variable {... | Mathlib/FieldTheory/SplittingField/Construction.lean | 103 | 104 | theorem natDegree_removeFactor' {f : K[X]} {n : ℕ} (hfn : f.natDegree = n + 1) :
f.removeFactor.natDegree = n := by | rw [natDegree_removeFactor, hfn, n.add_sub_cancel]
| 1 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : ℝ}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 49 | 49 | theorem logb_zero : logb b 0 = 0 := by | simp [logb]
| 1 |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 161 | 161 | theorem refl : IntervalIntegrable f μ a a := by | constructor <;> simp
| 1 |
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
variable {α β : Type*}
namespace Set
section Einfsep
open ENNReal
open Function
noncomputable def einfsep [EDist α] (s : Set α) : ℝ≥0∞ :=
⨅ (x... | Mathlib/Topology/MetricSpace/Infsep.lean | 79 | 81 | theorem einfsep_lt_iff {d} :
s.einfsep < d ↔ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ edist x y < d := by |
simp_rw [einfsep, iInf_lt_iff, exists_prop]
| 1 |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Lebesgue
open scoped Classical ENNReal
open Set Function Equiv Finset
noncomputable section
namespace MeasureTheory
section LMarginal
variable {δ δ' : Type*} {π : δ → Type*} [∀ x, MeasurableSpace (π x)]
variable {μ : ∀ i, Measu... | Mathlib/MeasureTheory/Integral/Marginal.lean | 105 | 108 | theorem lmarginal_congr {x y : ∀ i, π i} (f : (∀ i, π i) → ℝ≥0∞)
(h : ∀ i ∉ s, x i = y i) :
(∫⋯∫⁻_s, f ∂μ) x = (∫⋯∫⁻_s, f ∂μ) y := by |
dsimp [lmarginal, updateFinset_def]; rcongr; exact h _ ‹_›
| 1 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section Inducing
variable [To... | Mathlib/Topology/Maps.lean | 132 | 134 | theorem continuous_iff (hg : Inducing g) :
Continuous f ↔ Continuous (g ∘ f) := by |
simp_rw [continuous_iff_continuousAt, hg.continuousAt_iff]
| 1 |
import Mathlib.CategoryTheory.Preadditive.Injective
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Algebra.Homology.QuasiIso
#align_import category_theory.preadditive.injective_resolution from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
noncomputable s... | Mathlib/CategoryTheory/Preadditive/InjectiveResolution.lean | 111 | 113 | theorem complex_d_comp (n : ℕ) :
I.cocomplex.d n (n + 1) ≫ I.cocomplex.d (n + 1) (n + 2) = 0 := by |
simp
| 1 |
import Mathlib.ModelTheory.Syntax
import Mathlib.ModelTheory.Semantics
import Mathlib.Algebra.Ring.Equiv
variable {α : Type*}
namespace FirstOrder
open FirstOrder
inductive ringFunc : ℕ → Type
| add : ringFunc 2
| mul : ringFunc 2
| neg : ringFunc 1
| zero : ringFunc 0
| one : ringFunc 0
deriving D... | Mathlib/ModelTheory/Algebra/Ring/Basic.lean | 185 | 187 | theorem realize_mul (x y : ring.Term α) (v : α → R) :
Term.realize v (x * y) = Term.realize v x * Term.realize v y := by |
simp [mul_def, funMap_mul]
| 1 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp Ad... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 133 | 135 | theorem degrees_sum {ι : Type*} [DecidableEq σ] (s : Finset ι) (f : ι → MvPolynomial σ R) :
(∑ i ∈ s, f i).degrees ≤ s.sup fun i => (f i).degrees := by |
simp_rw [degrees_def]; exact supDegree_sum_le
| 1 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.Topology.Sheaves.Stalks
#align_import algebraic_geometry.stalks from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
universe v u v' u'
open Opposite Cate... | Mathlib/Geometry/RingedSpace/Stalks.lean | 181 | 184 | theorem congr_hom {X Y : PresheafedSpace.{_, _, v} C} (α β : X ⟶ Y) (h : α = β) (x : X) :
stalkMap α x =
eqToHom (show Y.stalk (α.base x) = Y.stalk (β.base x) by rw [h]) ≫ stalkMap β x := by |
rw [← stalkMap.congr α β h x x rfl, eqToHom_refl, Category.comp_id]
| 1 |
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.ordinal.principal from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
universe u v w
noncomputable section
open Order
namespace Ordinal
-- Porting note: commented out, doesn't seem necessary
--local infixr:0 "^" => ... | Mathlib/SetTheory/Ordinal/Principal.lean | 52 | 54 | theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :
Principal op o ↔ Principal (Function.swap op) o := by |
constructor <;> exact fun h a b ha hb => h hb ha
| 1 |
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 | 116 | 117 | theorem iterateFrobeniusEquiv_one_apply (x : R) : iterateFrobeniusEquiv R p 1 x = x ^ p := by |
rw [iterateFrobeniusEquiv_def, pow_one]
| 1 |
import Batteries.Data.Array.Lemmas
namespace ByteArray
@[ext] theorem ext : {a b : ByteArray} → a.data = b.data → a = b
| ⟨_⟩, ⟨_⟩, rfl => rfl
theorem getElem_eq_data_getElem (a : ByteArray) (h : i < a.size) : a[i] = a.data[i] := rfl
@[simp] theorem uset_eq_set (a : ByteArray) {i : USize} (h : i.toNat < a.size... | .lake/packages/batteries/Batteries/Data/ByteArray.lean | 79 | 82 | theorem get_append_left {a b : ByteArray} (hlt : i < a.size)
(h : i < (a ++ b).size := size_append .. ▸ Nat.lt_of_lt_of_le hlt (Nat.le_add_right ..)) :
(a ++ b)[i] = a[i] := by |
simp [getElem_eq_data_getElem]; exact Array.get_append_left hlt
| 1 |
import Batteries.Tactic.Alias
import Batteries.Data.Nat.Basic
namespace Nat
@[simp] theorem recAux_zero {motive : Nat → Sort _} (zero : motive 0)
(succ : ∀ n, motive n → motive (n+1)) :
Nat.recAux zero succ 0 = zero := rfl
theorem recAux_succ {motive : Nat → Sort _} (zero : motive 0)
(succ : ∀ n, mo... | .lake/packages/batteries/Batteries/Data/Nat/Lemmas.lean | 44 | 46 | theorem strongRec_eq {motive : Nat → Sort _} (ind : ∀ n, (∀ m, m < n → motive m) → motive n)
(t : Nat) : Nat.strongRec ind t = ind t fun m _ => Nat.strongRec ind m := by |
conv => lhs; unfold Nat.strongRec
| 1 |
import Mathlib.Topology.ContinuousOn
#align_import topology.algebra.order.left_right from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Set Filter Topology
section TopologicalSpace
variable {α β : Type*} [TopologicalSpace α] [LinearOrder α] [TopologicalSpace β]
theorem nhds_lef... | Mathlib/Topology/Order/LeftRight.lean | 119 | 120 | theorem nhds_left_sup_nhds_right' (a : α) : 𝓝[≤] a ⊔ 𝓝[>] a = 𝓝 a := by |
rw [← nhdsWithin_union, Iic_union_Ioi, nhdsWithin_univ]
| 1 |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 524 | 526 | theorem map_one_of_isMulOneCocycle {f : G → M} (hf : IsMulOneCocycle f) :
f 1 = 1 := by |
simpa only [mul_one, one_smul, self_eq_mul_right] using hf 1 1
| 1 |
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [T... | Mathlib/Topology/NhdsSet.lean | 124 | 124 | theorem nhdsSet_singleton : 𝓝ˢ {x} = 𝓝 x := by | simp [nhdsSet]
| 1 |
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Analysis.InnerProductSpace.Projection
#align_import analysis.convex.cone.dual from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
open Set LinearMap
open scoped Classical
open Pointwise
variable {𝕜 E F G : Type*}
section Dua... | Mathlib/Analysis/Convex/Cone/InnerDual.lean | 105 | 107 | theorem innerDualCone_insert (x : H) (s : Set H) :
(insert x s).innerDualCone = Set.innerDualCone {x} ⊓ s.innerDualCone := by |
rw [insert_eq, innerDualCone_union]
| 1 |
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 | 93 | 93 | theorem to_real_im (x : ℤ[i]) : ((x.im : ℤ) : ℝ) = (x : ℂ).im := by | simp [toComplex_def]
| 1 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.Interval.Set.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open scoped Classical
open Set
variable {ι : ... | Mathlib/Order/CompleteLatticeIntervals.lean | 265 | 265 | theorem coe_biSup : (↑(⨆ i, ⨆ (_ : p i), f i) : α) = ⨆ i, ⨆ (_ : p i), (f i : α) := by | simp
| 1 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Ring.Regular
import Mathlib.Tactic.Common
#align_import algebra.gcd_monoid.basic from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
variable {α : Type*}
-- Porting note: mathlib3 had a `@[protect_proj]` here, but adding `protect... | Mathlib/Algebra/GCDMonoid/Basic.lean | 148 | 148 | theorem normalize_coe_units (u : αˣ) : normalize (u : α) = 1 := by | simp
| 1 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section BagInter
@[simp]
| Mathlib/Data/List/Lattice.lean | 195 | 195 | theorem nil_bagInter (l : List α) : [].bagInter l = [] := by | cases l <;> rfl
| 1 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Instances.Irrational
#align_import number_theory.liouville.liouville_with from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Filter Metric Real Set
open sc... | Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | 142 | 143 | theorem rat_mul_iff (hr : r ≠ 0) : LiouvilleWith p (r * x) ↔ LiouvilleWith p x := by |
rw [mul_comm, mul_rat_iff hr]
| 1 |
import Mathlib.RingTheory.Derivation.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.derivation.to_square_zero from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
section ToSquareZero
universe u v w
variable {R : Type u} {A : Type v} {B : Type w} [Co... | Mathlib/RingTheory/Derivation/ToSquareZero.lean | 114 | 116 | theorem liftOfDerivationToSquareZero_mk_apply' (d : Derivation R A I) (x : A) :
(Ideal.Quotient.mk I) (d x) + (algebraMap A (B ⧸ I)) x = algebraMap A (B ⧸ I) x := by |
simp only [Ideal.Quotient.eq_zero_iff_mem.mpr (d x).prop, zero_add]
| 1 |
import Batteries.Data.RBMap.Alter
import Batteries.Data.List.Lemmas
namespace Batteries
namespace RBNode
open RBColor
attribute [simp] fold foldl foldr Any forM foldlM Ordered
@[simp] theorem min?_reverse (t : RBNode α) : t.reverse.min? = t.max? := by
unfold RBNode.max?; split <;> simp [RBNode.min?]
unfold RB... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 42 | 43 | theorem mem_congr [@TransCmp α cmp] {t : RBNode α} (h : cmp x y = .eq) :
Mem cmp x t ↔ Mem cmp y t := by | simp [Mem, TransCmp.cmp_congr_left' h]
| 1 |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
#align_import number_theory.legendre_symbol.basic from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Nat
section Euler
section Legendre
open ZMod
variable (p : ℕ) [Fact p.Prime]
def legendreSym (a : ℤ) : ℤ :=
... | Mathlib/NumberTheory/LegendreSymbol/Basic.lean | 302 | 303 | theorem exists_sq_eq_neg_one_iff : IsSquare (-1 : ZMod p) ↔ p % 4 ≠ 3 := by |
rw [FiniteField.isSquare_neg_one_iff, card p]
| 1 |
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 412 | 415 | theorem imageSubobjectCompIso_hom_arrow (f : X ⟶ Y) [HasImage f] {Y' : C} (h : Y ⟶ Y') [IsIso h] :
(imageSubobjectCompIso f h).hom ≫ (imageSubobject f).arrow =
(imageSubobject (f ≫ h)).arrow ≫ inv h := by |
simp [imageSubobjectCompIso]
| 1 |
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 | 68 | 69 | theorem image_Iic (e : α ≃o β) (a : α) : e '' Iic a = Iic (e a) := by |
rw [e.image_eq_preimage, e.symm.preimage_Iic, e.symm_symm]
| 1 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 101 | 104 | theorem HasDerivWithinAt.scomp_of_eq (hg : HasDerivWithinAt g₁ g₁' t' y)
(hh : HasDerivWithinAt h h' s x) (hst : MapsTo h s t') (hy : y = h x) :
HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x := by |
rw [hy] at hg; exact hg.scomp x hh hst
| 1 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
#align_import analysis.convex.hull from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set
open Pointwise
variable {𝕜 E F : Type*}
section convexHull
section OrderedSemiring
variable [OrderedSemiring 𝕜]
secti... | Mathlib/Analysis/Convex/Hull.lean | 62 | 63 | theorem mem_convexHull_iff : x ∈ convexHull 𝕜 s ↔ ∀ t, s ⊆ t → Convex 𝕜 t → x ∈ t := by |
simp_rw [convexHull_eq_iInter, mem_iInter]
| 1 |
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 | 45 | 45 | theorem gcd_mul_right_add_right (m n k : ℕ) : gcd m (k * m + n) = gcd m n := by | simp [add_comm _ n]
| 1 |
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.PPWithUniv
#align_import logic.small.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
universe u w v v'
@[mk_iff, pp_with_univ]
class Small (α : Type v) : Prop where
equiv_sma... | Mathlib/Logic/Small/Defs.lean | 56 | 58 | theorem Shrink.ext {α : Type v} [Small.{w} α] {x y : Shrink α}
(w : (equivShrink _).symm x = (equivShrink _).symm y) : x = y := by |
simpa using w
| 1 |
import Mathlib.Algebra.Algebra.Basic
import Mathlib.Algebra.Periodic
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Algebra.UniformMulAction
import Mathlib.Topology.Algebra.Star
import Mathlib.Topology.Instances.Int
import Mathlib.Topology.Order.Bornology
#align_import topology.instances.real fro... | Mathlib/Topology/Instances/Real.lean | 92 | 94 | theorem Real.mem_closure_iff {s : Set ℝ} {x : ℝ} :
x ∈ closure s ↔ ∀ ε > 0, ∃ y ∈ s, |y - x| < ε := by |
simp [mem_closure_iff_nhds_basis nhds_basis_ball, Real.dist_eq]
| 1 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Finset
#align_import algebra.tropical.big_operators from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
variable {R S :... | Mathlib/Algebra/Tropical/BigOperators.lean | 141 | 143 | theorem Finset.untrop_sum [ConditionallyCompleteLinearOrder R] (s : Finset S)
(f : S → Tropical (WithTop R)) : untrop (∑ i ∈ s, f i) = ⨅ i : s, untrop (f i) := by |
simpa [← _root_.untrop_sum] using (sum_attach _ _).symm
| 1 |
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 145 | 146 | theorem add_right_cancel {a b : Ordinal} (n : ℕ) : a + n = b + n ↔ a = b := by |
simp only [le_antisymm_iff, add_le_add_iff_right]
| 1 |
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.ordinal from "leanprover-community/mathlib"@"b90e72c7eebbe8de7c8293a80208ea2ba135c834"
universe u
open SetTheory PGame
open scoped NaturalOps PGame
namespace Ordinal
noncomputable def toPGame : Ordin... | Mathlib/SetTheory/Game/Ordinal.lean | 121 | 121 | theorem one_toPGame_moveLeft (x) : (toPGame 1).moveLeft x = toPGame 0 := by | simp
| 1 |
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.Index
#align_import group_theory.commensurable from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
variable {G : Type*} [Group G]
def Commensurable (H K : Subgroup... | Mathlib/GroupTheory/Commensurable.lean | 81 | 82 | theorem commensurable_inv (H : Subgroup G) (g : ConjAct G) :
Commensurable (g • H) H ↔ Commensurable H (g⁻¹ • H) := by | rw [commensurable_conj, inv_smul_smul]
| 1 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 133 | 133 | theorem prop_x (a : Solution₁ d) : a.x ^ 2 = 1 + d * a.y ^ 2 := by | rw [← a.prop]; ring
| 1 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin_root from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open scoped Polynomial
open Polynomial
noncomputable sec... | Mathlib/RingTheory/IsAdjoinRoot.lean | 179 | 181 | theorem repr_add_sub_repr_add_repr_mem_span (h : IsAdjoinRoot S f) (x y : S) :
h.repr (x + y) - (h.repr x + h.repr y) ∈ Ideal.span ({f} : Set R[X]) := by |
rw [← h.ker_map, RingHom.mem_ker, map_sub, h.map_repr, map_add, h.map_repr, h.map_repr, sub_self]
| 1 |
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.StrongTopology
#align_import analysis.normed_space.compact_operator from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Function Set Filter Bornology Metric Pointwise Topology
def IsCompactOperat... | Mathlib/Analysis/NormedSpace/CompactOperator.lean | 228 | 230 | theorem IsCompactOperator.sub [TopologicalAddGroup M₄] {f g : M₁ → M₄} (hf : IsCompactOperator f)
(hg : IsCompactOperator g) : IsCompactOperator (f - g) := by |
rw [sub_eq_add_neg]; exact hf.add hg.neg
| 1 |
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.CategoryTheory.Monoidal.Linear
#align_import algebra.category.Module.monoidal.basic from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
-- Porting note: Module
set_option linte... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean | 158 | 162 | theorem pentagon (W X Y Z : ModuleCat R) :
whiskerRight (associator W X Y).hom Z ≫
(associator W (tensorObj X Y) Z).hom ≫ whiskerLeft W (associator X Y Z).hom =
(associator (tensorObj W X) Y Z).hom ≫ (associator W X (tensorObj Y Z)).hom := by |
convert pentagon_aux R W X Y Z using 1
| 1 |
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 | 101 | 102 | theorem val_natCast_of_lt {n a : ℕ} (h : a < n) : (a : ZMod n).val = a := by |
rwa [val_natCast, Nat.mod_eq_of_lt]
| 1 |
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 | 48 | 48 | theorem unop_mem_unop {s : Set αᵒᵖ} {a : αᵒᵖ} : unop a ∈ s.unop ↔ a ∈ s := by | rfl
| 1 |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics Fil... | Mathlib/Analysis/ODE/Gronwall.lean | 92 | 93 | theorem gronwallBound_ε0_δ0 (K x : ℝ) : gronwallBound 0 K 0 x = 0 := by |
simp only [gronwallBound_ε0, zero_mul]
| 1 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 285 | 286 | theorem map_add₂ (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (x x' : M) (y : F) :
f (x + x') y = f x y + f x' y := by | rw [f.map_add, add_apply]
| 1 |
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.FieldDivision
#align_import ring_theory.polynomial.content from "leanprover-community/mathlib"@"7a030ab8eb5d99f05a891dccc49c5b5b90c947d3"
namespace Po... | Mathlib/RingTheory/Polynomial/Content.lean | 142 | 142 | theorem content_X : content (X : R[X]) = 1 := by | rw [← mul_one X, content_X_mul, content_one]
| 1 |
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Order.Synonym
#align_import algebra.field.basic from "leanprover-community/mathlib"@"05101c3df9d9cfe9430edc205860c79b6d660102"
open Function ... | Mathlib/Algebra/Field/Basic.lean | 138 | 138 | theorem inv_neg_one : (-1 : K)⁻¹ = -1 := by | rw [← neg_inv, inv_one]
| 1 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 61 | 61 | theorem ascPochhammer_one : ascPochhammer S 1 = X := by | simp [ascPochhammer]
| 1 |
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 1,056 | 1,056 | theorem zneg_bit1 (n : ZNum) : -n.bit1 = (-n).bitm1 := by | cases n <;> rfl
| 1 |
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 | 226 | 228 | theorem vars_monomial_single (i : σ) {e : ℕ} {r : R} (he : e ≠ 0) (hr : r ≠ 0) :
(monomial (Finsupp.single i e) r).vars = {i} := by |
rw [vars_monomial hr, Finsupp.support_single_ne_zero _ he]
| 1 |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 128 | 129 | theorem Iio_mem_nhds (h : γ₁ < γ₂) : Iio γ₂ ∈ 𝓝 γ₁ := by |
rcases eq_or_ne γ₁ 0 with (rfl | h₀) <;> simp [*, h.ne', Iio_mem_nhds_zero]
| 1 |
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Linear.Basic
#align_import category_theory.linear.linear_functor from "leanprover-community/mathlib"@"829895f162a1f29d0133f4b3538f4cd1fb5bffd3"
namespace CategoryTheory
variable (R : Type*) [Semiring R]
class Functor.Linear... | Mathlib/CategoryTheory/Linear/LinearFunctor.lean | 53 | 54 | theorem map_units_smul {X Y : C} (r : Rˣ) (f : X ⟶ Y) : F.map (r • f) = r • F.map f := by |
apply map_smul
| 1 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 101 | 103 | theorem frontier_closedBall (x : E) {r : ℝ} (hr : r ≠ 0) :
frontier (closedBall x r) = sphere x r := by |
rw [frontier, closure_closedBall, interior_closedBall x hr, closedBall_diff_ball]
| 1 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b"
universe u v
namespace SimpleGraph
@[ext]
structure Subgraph {V : Type u} (G : SimpleGra... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 177 | 178 | theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by |
simp [Subgraph.spanningCoe]
| 1 |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.Probability.Kernel.Disintegration.CdfToKernel
#align_import probability.kernel.cond_cdf from "leanprover-community/mathlib"@"3b88f4005dc2e28d42f974cc1ce838f0dafb39b8"
open MeasureTheory Set Filter TopologicalSpace
open scoped NNReal ENNReal Me... | Mathlib/Probability/Kernel/Disintegration/CondCdf.lean | 87 | 89 | theorem iInf_IicSnd_gt (t : ℚ) {s : Set α} (hs : MeasurableSet s) [IsFiniteMeasure ρ] :
⨅ r : { r' : ℚ // t < r' }, ρ.IicSnd r s = ρ.IicSnd t s := by |
simp_rw [ρ.IicSnd_apply _ hs, Measure.iInf_rat_gt_prod_Iic hs]
| 1 |
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Lemma
import Mathlib.Tactic.TypeStar
#align_import data.option.defs from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Option
#align option.lift_or_get Option.liftOrGet
protected def traverse.{u, v}
{F : Type u → Type... | Mathlib/Data/Option/Defs.lean | 96 | 97 | theorem mem_toList {a : α} {o : Option α} : a ∈ toList o ↔ a ∈ o := by |
cases o <;> simp [toList, eq_comm]
| 1 |
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 | 212 | 212 | theorem top_mul' : ∞ * a = if a = 0 then 0 else ∞ := by | convert WithTop.top_mul' a
| 1 |
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.Dual
#align_import linear_algebra.clifford_algebra.contraction from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2... | Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 159 | 161 | theorem contractRight_mul_algebraMap (a : CliffordAlgebra Q) (r : R) :
a * algebraMap _ _ r⌊d = a⌊d * algebraMap _ _ r := by |
rw [← Algebra.commutes, contractRight_algebraMap_mul, Algebra.commutes]
| 1 |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.pullbacks from "leanprover-community/mathlib"@"f11e306adb9f2a393539d2bb4293bf1b42caa7ac"
noncomputable section
universe v₁ v₂ u₁ u₂
-- Porting note: ne... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | 239 | 241 | theorem PreservesPushout.inl_iso_inv :
G.map pushout.inl ≫ (PreservesPushout.iso G f g).inv = pushout.inl := by |
simp [PreservesPushout.iso, Iso.comp_inv_eq]
| 1 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 297 | 298 | theorem mem_support {x : α} : x ∈ f.support ↔ f x ≠ x := by |
rw [support, mem_filter, and_iff_right (mem_univ x)]
| 1 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 126 | 127 | theorem toLp_congr {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) (hfg : f =ᵐ[μ] g) :
hf.toLp f = hg.toLp g := by | simp [toLp, hfg]
| 1 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ι : Sort*} {𝕜 E : Type*}
section OrderedSemiring
variable (𝕜) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t s₁ s₂ t₁ t₂ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 98 | 100 | theorem convexJoin_iUnion_right (s : Set E) (t : ι → Set E) :
convexJoin 𝕜 s (⋃ i, t i) = ⋃ i, convexJoin 𝕜 s (t i) := by |
simp_rw [convexJoin_comm s, convexJoin_iUnion_left]
| 1 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import algebra.polynomial.group_ring_action from "leanprover-community/mathlib"@"afad8e438d03f... | Mathlib/Algebra/Polynomial/GroupRingAction.lean | 76 | 78 | theorem smul_eval [MulSemiringAction G S] (g : G) (f : S[X]) (x : S) :
(g • f).eval x = g • f.eval (g⁻¹ • x) := by |
rw [← smul_eval_smul, smul_inv_smul]
| 1 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 113 | 114 | theorem isSeparating_unop_iff (𝒢 : Set Cᵒᵖ) : IsSeparating 𝒢.unop ↔ IsCoseparating 𝒢 := by |
rw [← isCoseparating_op_iff, Set.unop_op]
| 1 |
import Batteries.Data.RBMap.Basic
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Mathport.Rename
import Mathlib.Tactic.TypeStar
import Mathlib.Util.CompileInductive
#align_import data.tree from "leanprover-community/mathlib"@"ed989ff568099019c6533a4d94b27d852a5710d8"
inductive Tree.{u} (α : Type u) : Type ... | Mathlib/Data/Tree/Basic.lean | 90 | 91 | theorem numLeaves_eq_numNodes_succ (x : Tree α) : x.numLeaves = x.numNodes + 1 := by |
induction x <;> simp [*, Nat.add_comm, Nat.add_assoc, Nat.add_left_comm]
| 1 |
import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
#align_import algebra.quaternion_basis from "leanprover-community/mathlib"@"3aa5b8a9ed7a7cabd36e6e1d022c9858ab8a8c2d"
open Quaternion
namespace QuaternionAlgebra
structure Basis {R : Type*} (A : Type*) [CommRing R] [Ring A] [Algebra R A] (c₁ c₂ : R) ... | Mathlib/Algebra/QuaternionBasis.lean | 138 | 139 | theorem lift_smul (r : R) (x : ℍ[R,c₁,c₂]) : q.lift (r • x) = r • q.lift x := by |
simp [lift, mul_smul, ← Algebra.smul_def]
| 1 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e600... | Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 72 | 74 | theorem Finsupp.toFreeAbelianGroup_toFinsupp {X} (x : FreeAbelianGroup X) :
Finsupp.toFreeAbelianGroup (FreeAbelianGroup.toFinsupp x) = x := by |
rw [← AddMonoidHom.comp_apply, Finsupp.toFreeAbelianGroup_comp_toFinsupp, AddMonoidHom.id_apply]
| 1 |
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 | 154 | 155 | theorem intCast_real_norm (x : ℤ[i]) : (x.norm : ℝ) = Complex.normSq (x : ℂ) := by |
rw [Zsqrtd.norm, normSq]; simp
| 1 |
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
universe u v w x
variable {α : Type u} {α' : Type w} {β : Type v} {β' : Type x} {γ δ : Type*}
namespace Sum
#align sum.foral... | Mathlib/Data/Sum/Basic.lean | 63 | 64 | theorem getLeft_eq_getLeft? (h₁ : x.isLeft) (h₂ : x.getLeft?.isSome) :
x.getLeft h₁ = x.getLeft?.get h₂ := by | simp [← getLeft?_eq_some_iff]
| 1 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 875 | 876 | theorem div_lt_div_iff_right (c : α) : a / c < b / c ↔ a < b := by |
simpa only [div_eq_mul_inv] using mul_lt_mul_iff_right _
| 1 |
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 88 | 89 | theorem eq_div_of_mul_eq_left {a b c : R} (hb : b ≠ 0) (h : a * b = c) : a = c / b := by |
rw [← h, mul_div_cancel_right₀ _ hb]
| 1 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.normed_space.add_torsor from "leanprover-community/mathlib"@... | Mathlib/Analysis/NormedSpace/AddTorsor.lean | 57 | 58 | theorem dist_homothety_center (p₁ p₂ : P) (c : 𝕜) :
dist (homothety p₁ c p₂) p₁ = ‖c‖ * dist p₁ p₂ := by | rw [dist_comm, dist_center_homothety]
| 1 |
import Mathlib.Tactic.Ring.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Tactic.Conv
import Mathlib.Util.Qq
set_option autoImplicit true
-- In this file we would like to be able to use multi-character auto-implicits.
set_option relaxedAutoImplicit true
namespace Mathlib.Tactic
open Lean hiding Rat
open Qq Me... | Mathlib/Tactic/Ring/RingNF.lean | 124 | 125 | theorem rat_rawCast_pos {R} [DivisionRing R] :
(Rat.rawCast (.ofNat n) d : R) = Nat.rawCast n / Nat.rawCast d := by | simp
| 1 |
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Fintype
#align_import group_theory.perm.fin from "leanprover-community/mathlib"@"7e1c1263b6a25eb90bf16e80d8f47a657e403c4c"
open Equiv
def Equiv.Perm.decomposeFin {n : ℕ} : ... | Mathlib/GroupTheory/Perm/Fin.lean | 29 | 31 | theorem Equiv.Perm.decomposeFin_symm_of_refl {n : ℕ} (p : Fin (n + 1)) :
Equiv.Perm.decomposeFin.symm (p, Equiv.refl _) = swap 0 p := by |
simp [Equiv.Perm.decomposeFin, Equiv.permCongr_def]
| 1 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Ring.Opposite
import Mathlib.GroupTheory.GroupAction.Opposite
#align_import ring_theory.non_zero_divisors from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f6721fe"
variable (M₀ : Type*) [... | Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean | 129 | 130 | theorem mul_left_mem_nonZeroDivisors_eq_zero_iff {r x : M₁} (hr : r ∈ M₁⁰) : r * x = 0 ↔ x = 0 := by |
rw [mul_comm, mul_right_mem_nonZeroDivisors_eq_zero_iff hr]
| 1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.