Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | eval_complexity float64 0 1 |
|---|---|---|---|---|---|---|
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section TopologicalGroup
variable [CommGroup α] [TopologicalSpace α] [TopologicalGroup α]
variable {f g : β → α} {a a₁... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 50 | 53 | theorem HasProd.div (hf : HasProd f a₁) (hg : HasProd g a₂) :
HasProd (fun b ↦ f b / g b) (a₁ / a₂) := by |
simp only [div_eq_mul_inv]
exact hf.mul hg.inv
| 0.90625 |
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Function
namespace Set
section Prod
variable {α β γ δ : Type*} {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {a : α} {b : β}
theorem Subsingleton.pro... | Mathlib/Data/Set/Prod.lean | 96 | 98 | theorem univ_prod_univ : @univ α ×ˢ @univ β = univ := by |
ext
exact true_and_iff _
| 0.90625 |
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 | 58 | 62 | theorem List.untrop_prod [AddMonoid R] (l : List (Tropical R)) :
untrop l.prod = List.sum (l.map untrop) := by |
induction' l with hd tl IH
· simp
· simp [← IH]
| 0.90625 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 151 | 152 | theorem map_nat_add [AddCommMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1]
(f : F) (n : ℕ) (x : G) : f (↑n + x) = f x + n := by | simp
| 0.90625 |
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Algebra.Algebra.Pi
#align_import order.filter.zero_and_bounded_at_filter from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
namespace Filt... | Mathlib/Order/Filter/ZeroAndBoundedAtFilter.lean | 51 | 53 | theorem ZeroAtFilter.smul [TopologicalSpace β] [Zero 𝕜] [Zero β]
[SMulWithZero 𝕜 β] [ContinuousConstSMul 𝕜 β] {l : Filter α} {f : α → β} (c : 𝕜)
(hf : ZeroAtFilter l f) : ZeroAtFilter l (c • f) := by | simpa using hf.const_smul c
| 0.90625 |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.finite.card from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
noncomputable section
open scoped Classical
variable {α β γ : Type*}
def Finite.equivFin (α : Type*) [Finite α] : α ≃ Fin (Nat.card α) := by
have := (Finite.... | Mathlib/Data/Finite/Card.lean | 78 | 80 | theorem card_le_one_iff_subsingleton [Finite α] : Nat.card α ≤ 1 ↔ Subsingleton α := by |
haveI := Fintype.ofFinite α
simp only [Nat.card_eq_fintype_card, Fintype.card_le_one_iff_subsingleton]
| 0.90625 |
import Mathlib.LinearAlgebra.Ray
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.ray from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Real
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*}
[NormedAddCommGroup F] [NormedSp... | Mathlib/Analysis/NormedSpace/Ray.lean | 68 | 69 | theorem norm_injOn_ray_right (hy : y ≠ 0) : { x | SameRay ℝ x y }.InjOn norm := by |
simpa only [SameRay.sameRay_comm] using norm_injOn_ray_left hy
| 0.90625 |
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Finset
namespace Nat
variable (p : ℕ → Prop)
section Count
variable [DecidablePred p]
def count (n : ℕ) : ℕ :=
(List.range n).... | Mathlib/Data/Nat/Count.lean | 86 | 88 | theorem count_add' (a b : ℕ) : count p (a + b) = count (fun k ↦ p (k + b)) a + count p b := by |
rw [add_comm, count_add, add_comm]
simp_rw [add_comm b]
| 0.90625 |
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b37058d0"
open Function Set
variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*}
namespace Finset
variable [DecidableEq α'] [DecidableEq β'] [Decidabl... | Mathlib/Data/Finset/NAry.lean | 77 | 79 | theorem image₂_subset (hs : s ⊆ s') (ht : t ⊆ t') : image₂ f s t ⊆ image₂ f s' t' := by |
rw [← coe_subset, coe_image₂, coe_image₂]
exact image2_subset hs ht
| 0.90625 |
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 | 229 | 229 | theorem length_toList : length (toList p x) = (cycleOf p x).support.card := by | simp [toList]
| 0.90625 |
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
#align list.length_enum_from List.enumFrom_length
#align list.length_enum List.enum_length
@[simp]
theorem get?_enumFrom :
∀ n (l : List α) m, get? (enumFrom n l) m = (get? l m).map fun a => (n + m, a)
| n, [], m => rfl
| n, a :: l, 0 =... | Mathlib/Data/List/Enum.lean | 30 | 31 | theorem get?_enum (l : List α) (n) : get? (enum l) n = (get? l n).map fun a => (n, a) := by |
rw [enum, get?_enumFrom, Nat.zero_add]
| 0.90625 |
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 107 | 108 | theorem contDiff_of_subsingleton [Subsingleton F] : ContDiff 𝕜 n f := by |
rw [Subsingleton.elim f fun _ => 0]; exact contDiff_const
| 0.90625 |
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 73 | 73 | theorem rpow_natCast (x : ℝ) (n : ℕ) : x ^ (n : ℝ) = x ^ n := by | simpa using rpow_intCast x n
| 0.90625 |
import Mathlib.SetTheory.Game.Short
#align_import set_theory.game.state from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
universe u
namespace SetTheory
namespace PGame
class State (S : Type u) where
turnBound : S → ℕ
l : S → Finset S
r : S → Finset S
left_bound : ∀ {s t : S... | Mathlib/SetTheory/Game/State.lean | 50 | 54 | theorem turnBound_ne_zero_of_left_move {s t : S} (m : t ∈ l s) : turnBound s ≠ 0 := by |
intro h
have t := left_bound m
rw [h] at t
exact Nat.not_succ_le_zero _ t
| 0.90625 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Module.Pi
#align_import data.holor from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u
open List
def HolorIndex (ds : List ℕ) : Type :=
{ is : List ℕ // Forall₂ (· < ·) is ds }
#align holor_index Hol... | Mathlib/Data/Holor.lean | 58 | 59 | theorem cast_type (is : List ℕ) (eq : ds₁ = ds₂) (h : Forall₂ (· < ·) is ds₁) :
(cast (congr_arg HolorIndex eq) ⟨is, h⟩).val = is := by | subst eq; rfl
| 0.90625 |
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.enorm from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
noncomputable section
attribute [local instance] Classical.propDecidable
open ENNReal
structure ENorm (𝕜 : Type*) (V : Type*) [NormedField 𝕜] [Ad... | Mathlib/Analysis/NormedSpace/ENorm.lean | 113 | 113 | theorem map_sub_rev (x y : V) : e (x - y) = e (y - x) := by | rw [← neg_sub, e.map_neg]
| 0.90625 |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
variable {α : Type*} [CancelCommMonoidWithZero α] [NormalizedG... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 110 | 112 | theorem lcm_union (s₁ s₂ : Multiset α) : (s₁ ∪ s₂).lcm = GCDMonoid.lcm s₁.lcm s₂.lcm := by |
rw [← lcm_dedup, dedup_ext.2, lcm_dedup, lcm_add]
simp
| 0.90625 |
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.SetTheory.Game.Ordinal
#align_import set_theory.surreal.basic from "leanprover-community/mathlib"@"8900d545017cd21961daa2a1734bb658ef52c618"
universe u
namespace SetTheory
open scoped PGame
namespace PGame
def Numeric : PGame → Prop
| ⟨_, _, L, R⟩ => (... | Mathlib/SetTheory/Surreal/Basic.lean | 71 | 75 | theorem numeric_def {x : PGame} :
Numeric x ↔
(∀ i j, x.moveLeft i < x.moveRight j) ∧
(∀ i, Numeric (x.moveLeft i)) ∧ ∀ j, Numeric (x.moveRight j) := by |
cases x; rfl
| 0.90625 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
Orde... | Mathlib/Order/Interval/Finset/Fin.lean | 114 | 115 | theorem card_Ioc : (Ioc a b).card = b - a := by |
rw [← Nat.card_Ioc, ← map_valEmbedding_Ioc, card_map]
| 0.90625 |
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
#align_import analysis.special_functions.trigonometric.complex from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
noncomputable secti... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 43 | 44 | theorem cos_ne_zero_iff {θ : ℂ} : cos θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ (2 * k + 1) * π / 2 := by |
rw [← not_exists, not_iff_not, cos_eq_zero_iff]
| 0.90625 |
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.RingTheory.Localization.InvSubmonoid
#align_import algebraic_geometry.AffineScheme from "leanprover-community/mathlib"@"88474d1b5af6d37c2ab728b757771bced7f5194c"... | Mathlib/AlgebraicGeometry/AffineScheme.lean | 193 | 196 | theorem topIsAffineOpen (X : Scheme) [IsAffine X] : IsAffineOpen (⊤ : Opens X) := by |
convert rangeIsAffineOpenOfOpenImmersion (𝟙 X)
ext1
exact Set.range_id.symm
| 0.90625 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 353 | 356 | theorem contDiffAt_euclidean {n : ℕ∞} :
ContDiffAt 𝕜 n f y ↔ ∀ i, ContDiffAt 𝕜 n (fun x => f x i) y := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiffAt_iff, contDiffAt_pi]
rfl
| 0.90625 |
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Set
open Filter
namespace Filter
variable {α β γ δ : Type*} {ι : Sort*}
section Prod
variable {s : Set α} {t : Set β} {f : Filter α} {g : Filter β}
protected ... | Mathlib/Order/Filter/Prod.lean | 117 | 119 | theorem top_prod : (⊤ : Filter α) ×ˢ g = g.comap Prod.snd := by |
dsimp only [SProd.sprod]
rw [Filter.prod, comap_top, top_inf_eq]
| 0.90625 |
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 | 42 | 42 | theorem zero_bot (n : ℕ) : Ico 0 n = range n := by | rw [Ico, Nat.sub_zero, range_eq_range']
| 0.90625 |
import Mathlib.Data.Int.GCD
import Mathlib.Tactic.NormNum
namespace Tactic
namespace NormNum
theorem int_gcd_helper' {d : ℕ} {x y : ℤ} (a b : ℤ) (h₁ : (d : ℤ) ∣ x) (h₂ : (d : ℤ) ∣ y)
(h₃ : x * a + y * b = d) : Int.gcd x y = d := by
refine Nat.dvd_antisymm ?_ (Int.natCast_dvd_natCast.1 (Int.dvd_gcd h₁ h₂))
... | Mathlib/Tactic/NormNum/GCD.lean | 68 | 70 | theorem int_lcm_helper {x y : ℤ} {x' y' d : ℕ}
(hx : x.natAbs = x') (hy : y.natAbs = y') (h : Nat.lcm x' y' = d) :
Int.lcm x y = d := by | subst_vars; rw [Int.lcm_def]
| 0.90625 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.integral_normalization from "leanprover-community/mathlib"@"6f401acf4faec3ab9ab13a42789c4f68064a61cd"
open Polynomial
namespace Polynomial
universe u... | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 44 | 45 | theorem integralNormalization_zero : integralNormalization (0 : R[X]) = 0 := by |
simp [integralNormalization]
| 0.90625 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 537 | 538 | theorem coeff_mul_X_sub_C {p : R[X]} {r : R} {a : ℕ} :
coeff (p * (X - C r)) (a + 1) = coeff p a - coeff p (a + 1) * r := by | simp [mul_sub]
| 0.90625 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Contraction
variable {R M : Type*}
variable [CommRing R] [AddCommGroup M] [Module R M] {Q : QuadraticForm R M}
namespace CliffordAlgebra
variable (Q)
def invertibleιOfInvertible (m : M) [Invertible (Q m)] : Invertible (ι Q m) where
invOf := ι Q (⅟ (Q m) • m)
invO... | Mathlib/LinearAlgebra/CliffordAlgebra/Inversion.lean | 31 | 34 | theorem invOf_ι (m : M) [Invertible (Q m)] [Invertible (ι Q m)] :
⅟ (ι Q m) = ι Q (⅟ (Q m) • m) := by |
letI := invertibleιOfInvertible Q m
convert (rfl : ⅟ (ι Q m) = _)
| 0.90625 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 70 | 71 | theorem map (f : R →+* S) (n ν : ℕ) :
(bernsteinPolynomial R n ν).map f = bernsteinPolynomial S n ν := by | simp [bernsteinPolynomial]
| 0.90625 |
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.GaloisConnection
import Mathlib.Order.Hom.Basic
#align_import order.hom.order from "leanprover-community/mathlib"@"ba2245edf0c8bb155f1569fd9b9492a9b384cde6"
namespace OrderHom
variable {α β : Type*}
section Preorder
variable [Preorder α]
instance [Sem... | Mathlib/Order/Hom/Order.lean | 97 | 99 | theorem coe_iInf {ι : Sort*} [CompleteLattice β] (f : ι → α →o β) :
((⨅ i, f i : α →o β) : α → β) = ⨅ i, (f i : α → β) := by |
funext x; simp [iInf_apply]
| 0.90625 |
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.SetTheory.Game.Ordinal
#align_import set_theory.surreal.basic from "leanprover-community/mathlib"@"8900d545017cd21961daa2a1734bb658ef52c618"
universe u
namespace SetTheory
open scoped PGame
namespace PGame
def Numeric : PGame → Prop
| ⟨_, _, L, R⟩ => (... | Mathlib/SetTheory/Surreal/Basic.lean | 93 | 94 | theorem moveRight {x : PGame} (o : Numeric x) (j : x.RightMoves) : Numeric (x.moveRight j) := by |
cases x; exact o.2.2 j
| 0.90625 |
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
protected theorem div_eq_inv_mul : a / b = b⁻¹ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 79 | 79 | theorem div_zero (h : a ≠ 0) : a / 0 = ∞ := by | simp [div_eq_mul_inv, h]
| 0.90625 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 48 | 50 | theorem zero_scaleRoots (s : R) : scaleRoots 0 s = 0 := by |
ext
simp
| 0.90625 |
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.products.bifunctor from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open CategoryTheory
namespace CategoryTheory.Bifunctor
universe v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} {D : Type u₂} {E : Type u₃}
varia... | Mathlib/CategoryTheory/Products/Bifunctor.lean | 52 | 55 | theorem diagonal' (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :
F.map ((f, 𝟙 Y) : (X, Y) ⟶ (X', Y)) ≫ F.map ((𝟙 X', g) : (X', Y) ⟶ (X', Y')) =
F.map ((f, g) : (X, Y) ⟶ (X', Y')) := by |
rw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id]
| 0.90625 |
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
@[simp]
theorem reduceOption_cons_of_some (x : α) (l : List (Option α)) :
reduceOption (some x :: l) = x :: l.reduceOption := by
simp only [reduceOption, filterMap, id, eq_self_iff_true, and_self_iff]
#align list.reduce_option_cons_of_some... | Mathlib/Data/List/ReduceOption.lean | 77 | 77 | theorem reduceOption_singleton (x : Option α) : [x].reduceOption = x.toList := by | cases x <;> rfl
| 0.90625 |
import Mathlib.Tactic.Ring
#align_import algebra.group_power.identities from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {R : Type*} [CommRing R] {a b x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ n : R}
theorem sq_add_sq_mul_sq_add_sq :
(x₁ ^ 2 + x₂ ^ 2) * (y₁ ^ 2 +... | Mathlib/Algebra/Ring/Identities.lean | 46 | 48 | theorem pow_four_add_four_mul_pow_four' :
a ^ 4 + 4 * b ^ 4 = (a ^ 2 - 2 * a * b + 2 * b ^ 2) * (a ^ 2 + 2 * a * b + 2 * b ^ 2) := by |
ring
| 0.90625 |
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Binary
variable (xs : Vector α n) (ys : Vector β n)
@[simp]
theorem mapAccumr₂_mapAccumr_left (f₁ : γ → β → σ₁ → σ₁ × ζ) (f₂ : α → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ (mapAccumr f₂... | Mathlib/Data/Vector/MapLemmas.lean | 103 | 105 | theorem map_map₂ (f₁ : γ → ζ) (f₂ : α → β → γ) :
map f₁ (map₂ f₂ xs ys) = map₂ (fun x y => f₁ <| f₂ x y) xs ys := by |
induction xs, ys using Vector.revInductionOn₂ <;> simp_all
| 0.90625 |
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Algebra.Homology.ShortComplex.Basic
#align_import algebra.homology.homological_complex from "leanprover-community/mathlib"@"88bca0ce5d22ebfd9e73e682e51d60ea13b48347"
... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 71 | 76 | theorem d_comp_d (C : HomologicalComplex V c) (i j k : ι) : C.d i j ≫ C.d j k = 0 := by |
by_cases hij : c.Rel i j
· by_cases hjk : c.Rel j k
· exact C.d_comp_d' i j k hij hjk
· rw [C.shape j k hjk, comp_zero]
· rw [C.shape i j hij, zero_comp]
| 0.90625 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 149 | 152 | theorem cons_eval (f fs) : (cons f fs).eval = fun v => do {
let n ← Code.eval f v
let ns ← Code.eval fs v
pure (n.headI :: ns) } := by | simp [eval]
| 0.90625 |
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Data.Set.UnionLift
#align_import algebra.algebra.subalgebra.basic from "leanprover-community/mathlib"@"b915e9392ecb2a861e1e766f0e1df6ac481188ca"
namespace Subalgebra
open Algebra
variable {R A B : Type*} [CommSemiring R] [Semiring A] [Algebra R A] [... | Mathlib/Algebra/Algebra/Subalgebra/Directed.lean | 96 | 99 | theorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :
iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by |
dsimp [iSupLift, inclusion]
rw [Set.iUnionLift_of_mem]
| 0.90625 |
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Pointwise.Basic
#align_import algebra.star.pointwise from "leanprover-community/mathlib"@"30413fc89f202a090a54d78e540963ed3de0056e"
namespace Set
open Pointwise
local postfix:max "⋆" => star
variable {α : Type*} {s t : Set... | Mathlib/Algebra/Star/Pointwise.lean | 70 | 72 | theorem image_star [InvolutiveStar α] : Star.star '' s = s⋆ := by |
simp only [← star_preimage]
rw [image_eq_preimage_of_inverse] <;> intro <;> simp only [star_star]
| 0.90625 |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.finite.card from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
noncomputable section
open scoped Classical
variable {α β γ : Type*}
def Finite.equivFin (α : Type*) [Finite α] : α ≃ Fin (Nat.card α) := by
have := (Finite.... | Mathlib/Data/Finite/Card.lean | 93 | 95 | theorem card_option [Finite α] : Nat.card (Option α) = Nat.card α + 1 := by |
haveI := Fintype.ofFinite α
simp only [Nat.card_eq_fintype_card, Fintype.card_option]
| 0.90625 |
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.instances from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
section OrderedSemiring
variable [OrderedSe... | Mathlib/Algebra/Order/Interval/Set/Instances.lean | 201 | 203 | theorem coe_eq_zero [Nontrivial α] {x : Ico (0 : α) 1} : (x : α) = 0 ↔ x = 0 := by |
symm
exact Subtype.ext_iff
| 0.90625 |
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
open Finset Function Sym
universe u
variab... | Mathlib/Data/Sym/Sym2.lean | 73 | 74 | theorem Rel.trans {x y z : α × α} (a : Rel α x y) (b : Rel α y z) : Rel α x z := by |
aesop (rule_sets := [Sym2])
| 0.90625 |
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 | 155 | 158 | theorem apply_norm_sq_eq_inner_adjoint_right (A : E →L[𝕜] F) (x : E) :
‖A x‖ ^ 2 = re ⟪x, (A† ∘L A) x⟫ := by |
have h : ⟪x, (A† ∘L A) x⟫ = ⟪A x, A x⟫ := by rw [← adjoint_inner_right]; rfl
rw [h, ← inner_self_eq_norm_sq (𝕜 := 𝕜) _]
| 0.90625 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 104 | 107 | theorem Ioi_mul_Ici_subset' (a b : α) : Ioi a * Ici b ⊆ Ioi (a * b) := by |
haveI := covariantClass_le_of_lt
rintro x ⟨y, hya, z, hzb, rfl⟩
exact mul_lt_mul_of_lt_of_le hya hzb
| 0.90625 |
import Mathlib.MeasureTheory.Covering.DensityTheorem
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.covering.one_dim from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set MeasureTheory IsUnifLocDoublingMeasure Filter
open scoped Topology
names... | Mathlib/MeasureTheory/Covering/OneDim.lean | 51 | 59 | theorem tendsto_Icc_vitaliFamily_left (x : ℝ) :
Tendsto (fun y => Icc y x) (𝓝[<] x) ((vitaliFamily (volume : Measure ℝ) 1).filterAt x) := by |
refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩
· filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_left hy
· intro ε εpos
have : x ∈ Ioc (x - ε) x := ⟨by linarith, le_refl _⟩
filter_upwards [Icc_mem_nhdsWithin_Iio this] with y hy
rw [closedBall_eq_Icc]
exact Ic... | 0.90625 |
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 | 257 | 261 | theorem pi_apply_single [Fintype ι] [DecidableEq ι]
(Q : ∀ i, QuadraticForm R (Mᵢ i)) (i : ι) (m : Mᵢ i) :
pi Q (Pi.single i m) = Q i m := by |
rw [pi_apply, Fintype.sum_eq_single i fun j hj => ?_, Pi.single_eq_same]
rw [Pi.single_eq_of_ne hj, map_zero]
| 0.90625 |
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 87 | 88 | theorem snorm_eq_snorm' (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
snorm f p μ = snorm' f (ENNReal.toReal p) μ := by | simp [snorm, hp_ne_zero, hp_ne_top]
| 0.90625 |
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.PartENat
#align_import set_theory.cardinal.finite from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
set_option autoImplicit true
open Cardinal Function
noncomputable section
variable {α β : Typ... | Mathlib/SetTheory/Cardinal/Finite.lean | 144 | 146 | theorem card_of_subsingleton (a : α) [Subsingleton α] : Nat.card α = 1 := by |
letI := Fintype.ofSubsingleton a
rw [card_eq_fintype_card, Fintype.card_ofSubsingleton a]
| 0.90625 |
import Mathlib.GroupTheory.Solvable
import Mathlib.FieldTheory.PolynomialGaloisGroup
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import field_theory.abel_ruffini from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Classical Polynomial Intermedi... | Mathlib/FieldTheory/AbelRuffini.lean | 42 | 42 | theorem gal_one_isSolvable : IsSolvable (1 : F[X]).Gal := by | infer_instance
| 0.90625 |
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Group.Semiconj.Units
import Mathlib.Init.Classical
#align_import algebra.group_with_zero.semiconj from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
assert_not_exists DenselyOrdered
variable {α M₀ G₀ M₀' G₀' F F' :... | Mathlib/Algebra/GroupWithZero/Semiconj.lean | 24 | 25 | theorem zero_right [MulZeroClass G₀] (a : G₀) : SemiconjBy a 0 0 := by |
simp only [SemiconjBy, mul_zero, zero_mul]
| 0.90625 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Localization.Basic
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Surreal.Basic
#align_import set_theory.surreal.dyadic from "leanprover-community/mathlib"@"9... | Mathlib/SetTheory/Surreal/Dyadic.lean | 64 | 64 | theorem powHalf_moveLeft (n i) : (powHalf n).moveLeft i = 0 := by | cases n <;> cases i <;> rfl
| 0.90625 |
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 | 183 | 183 | theorem rank_finsupp_self' {ι : Type u} : Module.rank R (ι →₀ R) = #ι := by | simp
| 0.90625 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 115 | 116 | theorem derivative_X_sq : derivative (X ^ 2 : R[X]) = C 2 * X := by |
rw [derivative_X_pow, Nat.cast_two, pow_one]
| 0.90625 |
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Sqrt
#align_import data.complex.basic from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
open Set ComplexConjugate
namespace Complex
namespace AbsTheory
-- We develop enough theory to bundle `abs` into an `AbsoluteValue` be... | Mathlib/Data/Complex/Abs.lean | 34 | 34 | theorem abs_conj (z : ℂ) : (abs conj z) = abs z := by | simp
| 0.90625 |
import Mathlib.Analysis.Quaternion
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
#align_import analysis.normed_space.quaternion_exponential from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Quaternion Nat
open... | Mathlib/Analysis/NormedSpace/QuaternionExponential.lean | 117 | 118 | theorem im_exp (q : ℍ[ℝ]) : (exp ℝ q).im = (exp ℝ q.re * (Real.sin ‖q.im‖ / ‖q.im‖)) • q.im := by |
simp [exp_eq, smul_smul]
| 0.90625 |
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Init.Data.List.Basic
import Mathlib.Data.List.Basic
#align_import data.stream.init from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
set_option autoImplicit true
open Nat Function Option
namespace Stre... | Mathlib/Data/Stream/Init.lean | 76 | 76 | theorem tail_drop (n : Nat) (s : Stream' α) : tail (drop n s) = drop n (tail s) := by | simp
| 0.90625 |
import Mathlib.GroupTheory.GroupAction.BigOperators
import Mathlib.Logic.Equiv.Fin
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.pi from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
un... | Mathlib/LinearAlgebra/Pi.lean | 60 | 61 | theorem ker_pi (f : (i : ι) → M₂ →ₗ[R] φ i) : ker (pi f) = ⨅ i : ι, ker (f i) := by |
ext c; simp [funext_iff]
| 0.90625 |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 293 | 295 | theorem finsuppTensorFinsuppLid_apply_apply (f : ι →₀ R) (g : κ →₀ N) (a : ι) (b : κ) :
finsuppTensorFinsuppLid R N ι κ (f ⊗ₜ[R] g) (a, b) = f a • g b := by |
simp [finsuppTensorFinsuppLid]
| 0.90625 |
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
universe u v w
namespace MvPolynomial
variable {σ τ : Type*} {R : Type u} {S : Type v} {r : R} {e : ℕ} {n m : σ}
section CommSemiring
variable... | Mathlib/Algebra/MvPolynomial/Supported.lean | 102 | 103 | theorem supported_univ : supported R (Set.univ : Set σ) = ⊤ := by |
simp [Algebra.eq_top_iff, mem_supported]
| 0.90625 |
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 | 48 | 49 | theorem preimage_Icc (e : α ≃o β) (a b : β) : e ⁻¹' Icc a b = Icc (e.symm a) (e.symm b) := by |
simp [← Ici_inter_Iic]
| 0.90625 |
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {α β γ : Type*}
class Fi... | Mathlib/Data/Fintype/Basic.lean | 92 | 92 | theorem coe_univ : ↑(univ : Finset α) = (Set.univ : Set α) := by | ext; simp
| 0.90625 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
#align_import algebra.big_operators.nat_antidiagonal from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
variable {M N : Type*} [CommMonoid M] [AddCommMonoid N]
namespace Finset
namespace Nat
t... | Mathlib/Algebra/BigOperators/NatAntidiagonal.lean | 42 | 45 | theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p ∈ antidiagonal (n + 1), f p) =
f (n + 1, 0) * ∏ p ∈ antidiagonal n, f (p.1, p.2 + 1) := by |
rw [← prod_antidiagonal_swap, prod_antidiagonal_succ, ← prod_antidiagonal_swap]
rfl
| 0.90625 |
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 | 39 | 39 | theorem sign_of_pos {r : ℝ} (hr : 0 < r) : sign r = 1 := by | rw [sign, if_pos hr, if_neg hr.not_lt]
| 0.90625 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.Alternating.Basic
#align_import linear_algebra.exterior_algebra.basic from "leanprover-community/mathlib"@"b8d2eaa69d69ce8f03179a5cda774fc0cde984e4"
universe u1 u2 u3 u4 u5
variable (R : Type u1) [CommRing R]
variable (M : Type u2) [... | Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 97 | 98 | theorem comp_ι_sq_zero (g : ExteriorAlgebra R M →ₐ[R] A) (m : M) : g (ι R m) * g (ι R m) = 0 := by |
rw [← AlgHom.map_mul, ι_sq_zero, AlgHom.map_zero]
| 0.90625 |
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 | 389 | 389 | theorem inv_lt' : a⁻¹ < b ↔ b⁻¹ < a := by | rw [← inv_lt_inv_iff, inv_inv]
| 0.90625 |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section General
variable {α : Type*} {g : Gen... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 53 | 55 | theorem terminatedAt_iff_part_denom_none :
g.TerminatedAt n ↔ g.partialDenominators.get? n = none := by |
rw [terminatedAt_iff_s_none, part_denom_none_iff_s_none]
| 0.90625 |
import Mathlib.Tactic.Ring
#align_import algebra.group_power.identities from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {R : Type*} [CommRing R] {a b x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ n : R}
theorem sq_add_sq_mul_sq_add_sq :
(x₁ ^ 2 + x₂ ^ 2) * (y₁ ^ 2 +... | Mathlib/Algebra/Ring/Identities.lean | 55 | 60 | theorem sum_four_sq_mul_sum_four_sq :
(x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2) * (y₁ ^ 2 + y₂ ^ 2 + y₃ ^ 2 + y₄ ^ 2) =
(x₁ * y₁ - x₂ * y₂ - x₃ * y₃ - x₄ * y₄) ^ 2 + (x₁ * y₂ + x₂ * y₁ + x₃ * y₄ - x₄ * y₃) ^ 2 +
(x₁ * y₃ - x₂ * y₄ + x₃ * y₁ + x₄ * y₂) ^ 2 +
(x₁ * y₄ + x₂ * y₃ - x₃ * y₂ + x₄ * y₁) ... |
ring
| 0.90625 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 54 | 56 | theorem pythagoreanTriple_comm {x y z : ℤ} : PythagoreanTriple x y z ↔ PythagoreanTriple y x z := by |
delta PythagoreanTriple
rw [add_comm]
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import Mathlib.Data.Set.Prod
import Mathlib.Logic.Equiv.Fin
import Mathlib.ModelTheory.LanguageMap
#align_import model_theory.syntax from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable (L : Language.{u, v}) {L' : L... | Mathlib/ModelTheory/Syntax.lean | 284 | 290 | theorem comp_onTerm {L'' : Language} (φ : L' →ᴸ L'') (ψ : L →ᴸ L') :
((φ.comp ψ).onTerm : L.Term α → L''.Term α) = φ.onTerm ∘ ψ.onTerm := by |
ext t
induction' t with _ _ _ _ ih
· rfl
· simp_rw [onTerm, ih]
rfl
| 0.90625 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpace
namespace MeasureTheory
namespace Measure
variable {M : Type*} [Monoid M] [MeasurableSpace M]
@[to_additive conv "Additive convolution of measures."]
noncomputable def mconv (μ : Measure M) (ν : Measure M) :
... | Mathlib/MeasureTheory/Group/Convolution.lean | 65 | 67 | theorem zero_mconv (μ : Measure M) : μ ∗ (0 : Measure M) = (0 : Measure M) := by |
unfold mconv
simp
| 0.90625 |
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 | 58 | 59 | theorem toPGame_rightMoves (o : Ordinal) : o.toPGame.RightMoves = PEmpty := by |
rw [toPGame, RightMoves]
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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 Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 50 | 50 | theorem Disjoint.symm : Disjoint f g → Disjoint g f := by | simp only [Disjoint, or_comm, imp_self]
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import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.continuous_affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901fe9326d83"
structure ContinuousAffineMap (R : T... | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | 108 | 111 | theorem to_continuousMap_injective {f g : P →ᴬ[R] Q} (h : (f : C(P, Q)) = (g : C(P, Q))) :
f = g := by |
ext a
exact ContinuousMap.congr_fun h a
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import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Nat.Cast.Order
#align_import algebra.order.ring.abs from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
#align_import data.nat.parity from "leanpr... | Mathlib/Algebra/Order/Ring/Abs.lean | 201 | 202 | theorem dvd_abs (a b : α) : a ∣ |b| ↔ a ∣ b := by |
cases' abs_choice b with h h <;> simp only [h, dvd_neg]
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import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Tree.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.GCongr
import Mathlib... | Mathlib/Combinatorics/Enumerative/Catalan.lean | 65 | 65 | theorem catalan_zero : catalan 0 = 1 := by | rw [catalan]
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import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
#align_import algebra.lie.matrix from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
universe u v w w₁ w₂
section Matrices
open scoped Matrix
variabl... | Mathlib/Algebra/Lie/Matrix.lean | 76 | 79 | theorem Matrix.lieConj_symm_apply (P A : Matrix n n R) (h : Invertible P) :
(P.lieConj h).symm A = P⁻¹ * A * P := by |
simp [LinearEquiv.symm_conj_apply, Matrix.lieConj, LinearMap.toMatrix'_comp,
LinearMap.toMatrix'_toLin']
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import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.PartENat
#align_import set_theory.cardinal.finite from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
set_option autoImplicit true
open Cardinal Function
noncomputable section
variable {α β : Typ... | Mathlib/SetTheory/Cardinal/Finite.lean | 97 | 98 | theorem card_eq_of_equiv_fin {α : Type*} {n : ℕ} (f : α ≃ Fin n) : Nat.card α = n := by |
simpa only [card_eq_fintype_card, Fintype.card_fin] using card_congr f
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import Mathlib.Topology.Homotopy.Path
import Mathlib.Topology.Homotopy.Equiv
#align_import topology.homotopy.contractible from "leanprover-community/mathlib"@"16728b3064a1751103e1dc2815ed8d00560e0d87"
noncomputable section
namespace ContinuousMap
variable {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y]... | Mathlib/Topology/Homotopy/Contractible.lean | 39 | 43 | theorem Nullhomotopic.comp_left {f : C(Y, Z)} (hf : f.Nullhomotopic) (g : C(X, Y)) :
(f.comp g).Nullhomotopic := by |
cases' hf with y hy
use y
exact Homotopic.hcomp (Homotopic.refl g) hy
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import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.PNat.Basic
import Mathlib.GroupTheory.GroupAction.Prod
variable {M : Type*}
class PNatPowAssoc (M : Type*) [Mul M] [Pow M ℕ+] : Prop where
protected ppow_add : ∀ (k n : ℕ+) (x : M), x ^ (k + n) = x ^ k * x ^ n
prote... | Mathlib/Algebra/Group/PNatPowAssoc.lean | 72 | 74 | theorem ppow_mul' (x : M) (m n : ℕ+) : x ^ (m * n) = (x ^ n) ^ m := by |
rw [mul_comm]
exact ppow_mul x n m
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import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
(f : 𝕜 →ᵃ[𝕜] E) {a b : E} {L : Filter 𝕜} {s : Set 𝕜} {x : 𝕜}
n... | Mathlib/Analysis/Calculus/Deriv/AffineMap.lean | 32 | 34 | theorem hasStrictDerivAt : HasStrictDerivAt f (f.linear 1) x := by |
rw [f.decomp]
exact f.linear.hasStrictDerivAt.add_const (f 0)
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import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Functor
#align_import control.traversable.instances from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
universe u v
section Option
open Functor
variab... | Mathlib/Control/Traversable/Instances.lean | 47 | 51 | theorem Option.naturality {α β} (f : α → F β) (x : Option α) :
η (Option.traverse f x) = Option.traverse (@η _ ∘ f) x := by |
-- Porting note: added `ApplicativeTransformation` theorems
cases' x with x <;> simp! [*, functor_norm, ApplicativeTransformation.preserves_map,
ApplicativeTransformation.preserves_seq, ApplicativeTransformation.preserves_pure]
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import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section General
variable {α : Type*} {g : Gen... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 62 | 63 | theorem part_denom_eq_s_b {gp : Pair α} (s_nth_eq : g.s.get? n = some gp) :
g.partialDenominators.get? n = some gp.b := by | simp [partialDenominators, s_nth_eq]
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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 | 106 | 107 | theorem interior_sphere (x : E) {r : ℝ} (hr : r ≠ 0) : interior (sphere x r) = ∅ := by |
rw [← frontier_closedBall x hr, interior_frontier isClosed_ball]
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import Mathlib.Data.Finset.Grade
import Mathlib.Order.Interval.Finset.Basic
#align_import data.finset.interval from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
variable {α β : Type*}
namespace Finset
section Decidable
variable [DecidableEq α] (s t : Finset α)
instance instLocally... | Mathlib/Data/Finset/Interval.lean | 120 | 121 | theorem card_Ioo_finset (h : s ⊆ t) : (Ioo s t).card = 2 ^ (t.card - s.card) - 2 := by |
rw [card_Ioo_eq_card_Icc_sub_two, card_Icc_finset h]
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import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 796 | 797 | theorem sum_centroidWeights_eq_one_of_cast_card_ne_zero (h : (card s : k) ≠ 0) :
∑ i ∈ s, s.centroidWeights k i = 1 := by | simp [h]
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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 | 72 | 78 | theorem map_snd_prod (I : Ideal R) (J : Ideal S) : map (RingHom.snd R S) (prod I J) = J := by |
ext x
rw [mem_map_iff_of_surjective (RingHom.snd R S) Prod.snd_surjective]
exact
⟨by
rintro ⟨x, ⟨h, rfl⟩⟩
exact h.2, fun h => ⟨⟨0, x⟩, ⟨⟨Ideal.zero_mem _, h⟩, rfl⟩⟩⟩
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import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 450 | 451 | theorem mul_one_div (x y : G) : x * (1 / y) = x / y := by |
rw [div_eq_mul_inv, one_mul, div_eq_mul_inv]
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import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 73 | 74 | theorem nonempty_Ioo [DenselyOrdered α] : (Ioo a b).Nonempty ↔ a < b := by |
rw [← coe_nonempty, coe_Ioo, Set.nonempty_Ioo]
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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 | 52 | 54 | theorem eval₂_congr {R S : Type*} [Semiring R] [Semiring S] {f g : R →+* S} {s t : S}
{φ ψ : R[X]} : f = g → s = t → φ = ψ → eval₂ f s φ = eval₂ g t ψ := by |
rintro rfl rfl rfl; rfl
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import Mathlib.Data.List.Basic
#align_import data.list.forall2 from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
open Nat Function
namespace List
variable {α β γ δ : Type*} {R S : α → β → Prop} {P : γ → δ → Prop} {Rₐ : α → α → Prop}
open Relator
mk_iff_of_inductive_prop List.Foral... | Mathlib/Data/List/Forall2.lean | 34 | 35 | theorem Forall₂.imp (H : ∀ a b, R a b → S a b) {l₁ l₂} (h : Forall₂ R l₁ l₂) : Forall₂ S l₁ l₂ := by |
induction h <;> constructor <;> solve_by_elim
| 0.90625 |
import Mathlib.Topology.MetricSpace.Isometry
#align_import topology.metric_space.gluing from "leanprover-community/mathlib"@"e1a7bdeb4fd826b7e71d130d34988f0a2d26a177"
noncomputable section
universe u v w
open Function Set Uniformity Topology
namespace Metric
section ApproxGluing
variable {X : Type u} {Y : Typ... | Mathlib/Topology/MetricSpace/Gluing.lean | 106 | 108 | theorem le_glueDist_inr_inl (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x y) :
ε ≤ glueDist Φ Ψ ε (.inr x) (.inl y) := by |
rw [glueDist_comm]; apply le_glueDist_inl_inr
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import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Algebra.Field.Basic
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
namespace Rat
variable {α : Type*} [DivisionRing α]
-- Porting note: rewrote proof
@[simp]
theorem cast_inv_nat (n : ℕ) : ((n⁻¹ : ℚ) : α... | Mathlib/Data/Rat/Cast/Lemmas.lean | 37 | 40 | theorem cast_inv_int (n : ℤ) : ((n⁻¹ : ℚ) : α) = (n : α)⁻¹ := by |
cases' n with n n
· simp [ofInt_eq_cast, cast_inv_nat]
· simp only [ofInt_eq_cast, Int.cast_negSucc, ← Nat.cast_succ, cast_neg, inv_neg, cast_inv_nat]
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import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vandermonde
import Mathlib.Tactic.FieldSimp
#align_import data.polynomial.hasse_deriv from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 133 | 134 | theorem hasseDeriv_apply_one (hk : 0 < k) : hasseDeriv k (1 : R[X]) = 0 := by |
rw [← C_1, hasseDeriv_C k _ hk]
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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 | 248 | 249 | theorem toList_get_zero (h : x ∈ p.support) :
(toList p x).get ⟨0, (length_toList_pos_of_mem_support _ _ h)⟩ = x := by | simp [toList]
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import Mathlib.Data.Finset.Pointwise
#align_import combinatorics.additive.e_transform from "leanprover-community/mathlib"@"207c92594599a06e7c134f8d00a030a83e6c7259"
open MulOpposite
open Pointwise
variable {α : Type*} [DecidableEq α]
namespace Finset
section Group
variable [Group α] (e : α) (x : Finset... | Mathlib/Combinatorics/Additive/ETransform.lean | 137 | 137 | theorem mulETransformRight_one : mulETransformRight 1 x = x := by | simp [mulETransformRight]
| 0.90625 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ι : Type*}
theorem coprime_list_prod_left_iff {l : List ℕ} {k : ℕ} :
Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by
... | Mathlib/Data/Nat/GCD/BigOperators.lean | 52 | 54 | theorem coprime_fintype_prod_left_iff [Fintype ι] {s : ι → ℕ} {x : ℕ} :
Coprime (∏ i, s i) x ↔ ∀ i, Coprime (s i) x := by |
simp [coprime_prod_left_iff]
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import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
import Mathlib.Logic.Basic
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort*} {f : α → β}
@[reducible, simp] de... | Mathlib/Logic/Function/Basic.lean | 109 | 110 | theorem not_injective_iff : ¬ Injective f ↔ ∃ a b, f a = f b ∧ a ≠ b := by |
simp only [Injective, not_forall, exists_prop]
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import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 138 | 139 | theorem card_fintypeIic : Fintype.card (Set.Iic b) = b + 1 := by |
rw [Fintype.card_ofFinset, card_Iic]
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