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.AlgebraicGeometry.Properties
#align_import algebraic_geometry.function_field from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #12737
set_option linter.uppercaseLean3 false
universe u v
open... | Mathlib/AlgebraicGeometry/FunctionField.lean | 67 | 75 | theorem germ_injective_of_isIntegral [IsIntegral X] {U : Opens X.carrier} (x : U) :
Function.Injective (X.presheaf.germ x) := by |
rw [injective_iff_map_eq_zero]
intro y hy
rw [← (X.presheaf.germ x).map_zero] at hy
obtain ⟨W, hW, iU, iV, e⟩ := X.presheaf.germ_eq _ x.prop x.prop _ _ hy
cases Subsingleton.elim iU iV
haveI : Nonempty W := ⟨⟨_, hW⟩⟩
exact map_injective_of_isIntegral X iU e
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import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
#align_import linear_algebra.clifford_algebra.star from "leanprover-community/mathlib"@"4d66277cfec381260ba05c68f9ae6ce2a118031d"
variable {R : Type*} [CommRing R]
variable {M : Type*} [AddCommGroup M] [Module R M]
variable {Q : QuadraticForm R M}
namespac... | Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean | 62 | 64 | theorem star_algebraMap (r : R) :
star (algebraMap R (CliffordAlgebra Q) r) = algebraMap R (CliffordAlgebra Q) r := by |
rw [star_def, involute.commutes, reverse.commutes]
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import Mathlib.Control.Bitraversable.Basic
#align_import control.bitraversable.lemmas from "leanprover-community/mathlib"@"58581d0fe523063f5651df0619be2bf65012a94a"
universe u
variable {t : Type u → Type u → Type u} [Bitraversable t]
variable {β : Type u}
namespace Bitraversable
open Functor LawfulApplicative
... | Mathlib/Control/Bitraversable/Lemmas.lean | 72 | 75 | theorem comp_tfst {α₀ α₁ α₂ β} (f : α₀ → F α₁) (f' : α₁ → G α₂) (x : t α₀ β) :
Comp.mk (tfst f' <$> tfst f x) = tfst (Comp.mk ∘ map f' ∘ f) x := by |
rw [← comp_bitraverse]
simp only [Function.comp, tfst, map_pure, Pure.pure]
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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 | 48 | 50 | theorem get_enumFrom (l : List α) (n) (i : Fin (l.enumFrom n).length) :
(l.enumFrom n).get i = (n + i, l.get (i.cast enumFrom_length)) := by |
simp [get_eq_get?]
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import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNR... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 122 | 127 | theorem of_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞) (hT : FinMeasAdditive (c • μ) T) :
FinMeasAdditive μ T := by |
refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT
rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top] at hμs
simp only [hc_ne_top, or_false_iff, Ne, false_and_iff] at hμs
exact hμs.2
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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 | 79 | 83 | theorem gronwallBound_x0 (δ K ε : ℝ) : gronwallBound δ K ε 0 = δ := by |
by_cases hK : K = 0
· simp only [gronwallBound, if_pos hK, mul_zero, add_zero]
· simp only [gronwallBound, if_neg hK, mul_zero, exp_zero, sub_self, mul_one,
add_zero]
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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 | 54 | 56 | theorem get_enum (l : List α) (i : Fin l.enum.length) :
l.enum.get i = (i.1, l.get (i.cast enum_length)) := by |
simp [enum]
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import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 157 | 160 | theorem _root_.MeasureTheory.Integrable.integral_norm_condDistrib_map
(hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂condDistrib Y X μ x) (μ.map X) := by |
rw [condDistrib, ← Measure.fst_map_prod_mk₀ (X := X) hY]; exact hf_int.integral_norm_condKernel
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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 | 123 | 124 | theorem degree_of_subsingleton [Subsingleton R] : degree p = ⊥ := by |
rw [Subsingleton.elim p 0, degree_zero]
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import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 186 | 189 | theorem Fintype.linearIndependent_iff' [Fintype ι] [DecidableEq ι] :
LinearIndependent R v ↔
LinearMap.ker (LinearMap.lsum R (fun _ ↦ R) ℕ fun i ↦ LinearMap.id.smulRight (v i)) = ⊥ := by |
simp [Fintype.linearIndependent_iff, LinearMap.ker_eq_bot', funext_iff]
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import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.PolynomialExp
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9... | Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean | 58 | 61 | theorem nonneg (x : ℝ) : 0 ≤ expNegInvGlue x := by |
cases le_or_gt x 0 with
| inl h => exact ge_of_eq (zero_of_nonpos h)
| inr h => exact le_of_lt (pos_of_pos h)
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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 | 206 | 211 | theorem prev_nat_zero : (ComplexShape.up ℕ).prev 0 = 0 := by |
classical
refine dif_neg ?_
push_neg
intro
apply Nat.noConfusion
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import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 57 | 59 | theorem opRingEquiv_op_C_mul_X_pow (r : R) (n : ℕ) :
opRingEquiv R (op (C r * X ^ n : R[X])) = C (op r) * X ^ n := by |
simp only [X_pow_mul, op_mul, op_pow, map_mul, map_pow, opRingEquiv_op_X, opRingEquiv_op_C]
| 0.28125 |
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 | 75 | 75 | theorem convexJoin_singletons (x : E) : convexJoin 𝕜 {x} {y} = segment 𝕜 x y := by | simp
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import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.multivariate.basic from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
universe u v
open MvFunctor
@[pp_with_univ]
structure MvPFunctor (n : ℕ) where
A : Type u
... | Mathlib/Data/PFunctor/Multivariate/Basic.lean | 148 | 149 | theorem comp.get_mk (x : P (fun i => Q i α)) : comp.get (comp.mk x) = x := by |
rfl
| 0.28125 |
import Mathlib.Algebra.Algebra.Subalgebra.Operations
import Mathlib.Algebra.Ring.Fin
import Mathlib.RingTheory.Ideal.Quotient
#align_import ring_theory.ideal.quotient_operations from "leanprover-community/mathlib"@"b88d81c84530450a8989e918608e5960f015e6c8"
universe u v w
namespace Ideal
open Function RingHom
var... | Mathlib/RingTheory/Ideal/QuotientOperations.lean | 189 | 191 | theorem mem_quotient_iff_mem {I J : Ideal R} (hIJ : I ≤ J) {x : R} :
Quotient.mk I x ∈ J.map (Quotient.mk I) ↔ x ∈ J := by |
rw [mem_quotient_iff_mem_sup, sup_eq_left.mpr hIJ]
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import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/mathlib"@"117e93f82b5f959f8193857370109935291f0cc4"
noncomputable section
open ProbabilityTheory
open MeasureTheory MeasurableSpace
namespace ProbabilityT... | Mathlib/Probability/CondCount.lean | 110 | 115 | theorem condCount_eq_one_of (hs : s.Finite) (hs' : s.Nonempty) (ht : s ⊆ t) :
condCount s t = 1 := by |
haveI := condCount_isProbabilityMeasure hs hs'
refine eq_of_le_of_not_lt prob_le_one ?_
rw [not_lt, ← condCount_self hs hs']
exact measure_mono ht
| 0.28125 |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.l2Space
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Function.L2Space
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.M... | Mathlib/Analysis/Fourier/AddCircle.lean | 144 | 146 | theorem fourier_eval_zero (n : ℤ) : fourier n (0 : AddCircle T) = 1 := by |
rw [← QuotientAddGroup.mk_zero, fourier_coe_apply, Complex.ofReal_zero, mul_zero,
zero_div, Complex.exp_zero]
| 0.28125 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Deprecated.Group
#align_import deprecated.submonoid from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {M : Type*} [Monoid M] {s : Set M}
variable {A : Type*} [AddMonoi... | Mathlib/Deprecated/Submonoid.lean | 246 | 250 | theorem multiset_prod_mem {M} [CommMonoid M] {s : Set M} (hs : IsSubmonoid s) (m : Multiset M) :
(∀ a ∈ m, a ∈ s) → m.prod ∈ s := by |
refine Quotient.inductionOn m fun l hl => ?_
rw [Multiset.quot_mk_to_coe, Multiset.prod_coe]
exact list_prod_mem hs hl
| 0.28125 |
import Mathlib.Data.Part
import Mathlib.Data.Rel
#align_import data.pfun from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Function
def PFun (α β : Type*) :=
α → Part β
#align pfun PFun
infixr:25 " →. " => PFun
namespace PFun
variable {α β γ δ ε ι : Type*}
instance inhab... | Mathlib/Data/PFun.lean | 80 | 80 | theorem mem_dom (f : α →. β) (x : α) : x ∈ Dom f ↔ ∃ y, y ∈ f x := by | simp [Dom, Part.dom_iff_mem]
| 0.28125 |
import Mathlib.Algebra.Group.Submonoid.Pointwise
#align_import group_theory.submonoid.inverses from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {M : Type*}
namespace Submonoid
@[to_additive]
noncomputable instance [Monoid M] : Group (IsUnit.submonoid M) :=
{ inferInstanc... | Mathlib/GroupTheory/Submonoid/Inverses.lean | 73 | 76 | theorem leftInv_leftInv_le : S.leftInv.leftInv ≤ S := by |
rintro x ⟨⟨y, z, h₁⟩, h₂ : x * y = 1⟩
convert z.prop
rw [← mul_one x, ← h₁, ← mul_assoc, h₂, one_mul]
| 0.28125 |
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
#align_import category_theory.functor.category from "leanprover-community/mathlib"@"63721b2c3eba6c325ecf8ae8cca27155a4f6306f"
namespace CategoryTheory
-- declare the `v`'s first; see note [CategoryTheory universes].
universe v₁ v₂ v₃ u₁ u₂ u... | Mathlib/CategoryTheory/Functor/Category.lean | 125 | 125 | theorem id_hcomp_app {H : E ⥤ C} (α : F ⟶ G) (X : E) : (𝟙 H ◫ α).app X = α.app _ := by | simp
| 0.28125 |
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 96 | 100 | theorem lie_le_iff : ⁅I, N⁆ ≤ N' ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N' := by |
rw [lieIdeal_oper_eq_span, LieSubmodule.lieSpan_le]
refine ⟨fun h x hx m hm => h ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩, ?_⟩
rintro h _ ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩
exact h x hx m hm
| 0.28125 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.normed_space.exponential from "leanprover-community/ma... | Mathlib/Analysis/NormedSpace/Exponential.lean | 119 | 120 | theorem expSeries_apply_eq (x : 𝔸) (n : ℕ) :
(expSeries 𝕂 𝔸 n fun _ => x) = (n !⁻¹ : 𝕂) • x ^ n := by | simp [expSeries]
| 0.28125 |
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.BoundedOrder
import Mathlib.Mathport.Notation
import Mathlib.Data.Sigma.Basic
#align_import data.sigma.order from "leanprover-community/mathlib"@"1fc36cc9c8264e6e81253f88be7fb2cb6c92d76a"
namespace Sigma
variable {ι : Type*} {α : ι → Type*}
-- Porting note: I... | Mathlib/Data/Sigma/Order.lean | 79 | 86 | theorem le_def [∀ i, LE (α i)] {a b : Σi, α i} : a ≤ b ↔ ∃ h : a.1 = b.1, h.rec a.2 ≤ b.2 := by |
constructor
· rintro ⟨i, a, b, h⟩
exact ⟨rfl, h⟩
· obtain ⟨i, a⟩ := a
obtain ⟨j, b⟩ := b
rintro ⟨rfl : i = j, h⟩
exact le.fiber _ _ _ h
| 0.28125 |
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Basic
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Yoneda
import Mathlib.Tactic.ApplyFun
#align_import category_theory.limits.concrete_category from "leanprover-community/math... | Mathlib/CategoryTheory/Limits/ConcreteCategory.lean | 76 | 83 | theorem Concrete.from_union_surjective_of_isColimit {D : Cocone F} (hD : IsColimit D) :
let ff : (Σj : J, F.obj j) → D.pt := fun a => D.ι.app a.1 a.2
Function.Surjective ff := by |
intro ff x
let E : Cocone (F ⋙ forget C) := (forget C).mapCocone D
let hE : IsColimit E := isColimitOfPreserves (forget C) hD
obtain ⟨j, y, hy⟩ := Types.jointly_surjective_of_isColimit hE x
exact ⟨⟨j, y⟩, hy⟩
| 0.28125 |
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
#align_import computability.NFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Set
open Computability
universe u v
-- Porting note: Required as `NFA` is used in mathlib3
set_option linter.uppercaseLean3 fa... | Mathlib/Computability/NFA.lean | 108 | 109 | theorem mem_accepts {x : List α} : x ∈ M.accepts ↔ ∃ S ∈ M.accept, S ∈ M.evalFrom M.start x := by |
rfl
| 0.28125 |
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 | 65 | 66 | theorem count_succ (n : ℕ) : count p (n + 1) = count p n + if p n then 1 else 0 := by |
split_ifs with h <;> simp [count, List.range_succ, h]
| 0.28125 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 51 | 52 | theorem verschiebung_zmod (x : 𝕎 (ZMod p)) : verschiebung x = x * p := by |
rw [← frobenius_verschiebung, frobenius_zmodp]
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import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace List
variable {α β : Type*}
section FormPerm
variable [DecidableEq α] (l :... | Mathlib/GroupTheory/Perm/List.lean | 142 | 146 | theorem formPerm_cons_concat_apply_last (x y : α) (xs : List α) :
formPerm (x :: (xs ++ [y])) y = x := by |
induction' xs with z xs IH generalizing x y
· simp
· simp [IH]
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import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
open Monoid Coprod Multiplicative Subgroup Function
def HNNExtension.con (G : Type*) [Group G] (A B : Subgroup G) (φ : A ≃* B) :
Con (G ∗ Multiplicative ℤ) :=
conGen (fun x y => ∃ (a : A),
x = inr (ofAdd 1) * inl (a : G) ∧
... | Mathlib/GroupTheory/HNNExtension.lean | 97 | 99 | theorem lift_t (f : G →* H) (x : H) (hx : ∀ a : A, x * f ↑a = f (φ a : G) * x) :
lift f x hx t = x := by |
delta HNNExtension; simp [lift, t]
| 0.28125 |
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Shapes.Diagonal
#align_import algebraic_geometry.pullbacks from "leanprover-community/mathlib"@"7316286ff2942aa14e540add9058c6b0aa1c8070"
set_opt... | Mathlib/AlgebraicGeometry/Pullbacks.lean | 64 | 67 | theorem t_fst_fst (i j : 𝒰.J) : t 𝒰 f g i j ≫ pullback.fst ≫ pullback.fst = pullback.snd := by |
simp only [t, Category.assoc, pullbackSymmetry_hom_comp_fst_assoc, pullbackAssoc_hom_snd_fst,
pullback.lift_fst_assoc, pullbackSymmetry_hom_comp_snd, pullbackAssoc_inv_fst_fst,
pullbackSymmetry_hom_comp_fst]
| 0.28125 |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Lang... | Mathlib/ModelTheory/Definability.lean | 141 | 144 | theorem definable_finset_biInter {ι : Type*} {f : ι → Set (α → M)}
(hf : ∀ i, A.Definable L (f i)) (s : Finset ι) : A.Definable L (⋂ i ∈ s, f i) := by |
rw [← Finset.inf_set_eq_iInter]
exact definable_finset_inf hf s
| 0.28125 |
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
namespace Nat
def dist (n m : ℕ) :=
n - m + (m - n)
#align nat.dist Nat.dist
-- Should be aligned to `Nat.dist.eq_def`, but that is generated on demand and isn't pr... | Mathlib/Data/Nat/Dist.lean | 85 | 89 | theorem dist_eq_intro {n m k l : ℕ} (h : n + m = k + l) : dist n k = dist l m :=
calc
dist n k = dist (n + m) (k + m) := by | rw [dist_add_add_right]
_ = dist (k + l) (k + m) := by rw [h]
_ = dist l m := by rw [dist_add_add_left]
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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 | 104 | 105 | theorem finrank_ulift : finrank R (ULift M) = finrank R M := by |
simp_rw [finrank, rank_ulift, toNat_lift]
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import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.extreme from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
open Function Set
open scoped Classical
open Affine
variable {𝕜 E F ι : Type*} {π : ι → Type*}
section SMul
variable (𝕜) [OrderedSemiring 𝕜] [AddCommMonoi... | Mathlib/Analysis/Convex/Extreme.lean | 120 | 123 | theorem isExtreme_biInter {F : Set (Set E)} (hF : F.Nonempty) (hA : ∀ B ∈ F, IsExtreme 𝕜 A B) :
IsExtreme 𝕜 A (⋂ B ∈ F, B) := by |
haveI := hF.to_subtype
simpa only [iInter_subtype] using isExtreme_iInter fun i : F ↦ hA _ i.2
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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 | 129 | 130 | theorem eq_of_vsub_eq_zero {p₁ p₂ : P} (h : p₁ -ᵥ p₂ = (0 : G)) : p₁ = p₂ := by |
rw [← vsub_vadd p₁ p₂, h, zero_vadd]
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import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 358 | 360 | theorem toQuaternion_comp_ofQuaternion :
toQuaternion.comp ofQuaternion = AlgHom.id R ℍ[R,c₁,c₂] := by |
ext : 1 <;> simp
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import Mathlib.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import probability.strong_law from "leanprover-community/mathlib"@"f2ce60867... | Mathlib/Probability/StrongLaw.lean | 99 | 103 | theorem abs_truncation_le_abs_self (f : α → ℝ) (A : ℝ) (x : α) : |truncation f A x| ≤ |f x| := by |
simp only [truncation, indicator, Set.mem_Icc, id, Function.comp_apply]
split_ifs
· exact le_rfl
· simp [abs_nonneg]
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import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
#align_import category_theory.limits.constructions.zero_objects from "leanprover-community/mathlib"@"52a270e2ea4e342c2587c106f8be904524214a4... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 115 | 117 | theorem inr_zeroCoprodIso_hom (X : C) : coprod.inr ≫ (zeroCoprodIso X).hom = 𝟙 X := by |
dsimp [zeroCoprodIso, binaryCofanZeroLeft]
simp
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import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
noncomputable section
namespace Polynomial
universe u v w
section Semiring
variable {R : Type... | Mathlib/Algebra/Polynomial/Lifts.lean | 61 | 62 | theorem mem_lifts (p : S[X]) : p ∈ lifts f ↔ ∃ q : R[X], map f q = p := by |
simp only [coe_mapRingHom, lifts, RingHom.mem_rangeS]
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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 | 132 | 134 | theorem update_inl_apply_inl [DecidableEq α] [DecidableEq (Sum α β)] {f : Sum α β → γ} {i j : α}
{x : γ} : update f (inl i) x (inl j) = update (f ∘ inl) i x j := by |
rw [← update_inl_comp_inl, Function.comp_apply]
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import Mathlib.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import probability.strong_law from "leanprover-community/mathlib"@"f2ce60867... | Mathlib/Probability/StrongLaw.lean | 135 | 137 | theorem _root_.MeasureTheory.AEStronglyMeasurable.integrable_truncation [IsFiniteMeasure μ]
(hf : AEStronglyMeasurable f μ) {A : ℝ} : Integrable (truncation f A) μ := by |
rw [← memℒp_one_iff_integrable]; exact hf.memℒp_truncation
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import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (antidiagonal mem_antidiagonal)
namespace MvPowerSeries
open Fi... | Mathlib/RingTheory/MvPowerSeries/Inverse.lean | 101 | 104 | theorem constantCoeff_invOfUnit (φ : MvPowerSeries σ R) (u : Rˣ) :
constantCoeff σ R (invOfUnit φ u) = ↑u⁻¹ := by |
classical
rw [← coeff_zero_eq_constantCoeff_apply, coeff_invOfUnit, if_pos rfl]
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import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.GroupTheory.Torsion
import Mathlib.RingTheory.Coprime.Ideal
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Set.Lattice
#align_import algebra.module.torsion from "leanpr... | Mathlib/Algebra/Module/Torsion.lean | 99 | 105 | theorem torsionOf_eq_bot_iff_of_noZeroSMulDivisors [Nontrivial R] [NoZeroSMulDivisors R M] (m : M) :
torsionOf R M m = ⊥ ↔ m ≠ 0 := by |
refine ⟨fun h contra => ?_, fun h => (Submodule.eq_bot_iff _).mpr fun r hr => ?_⟩
· rw [contra, torsionOf_zero] at h
exact bot_ne_top.symm h
· rw [mem_torsionOf_iff, smul_eq_zero] at hr
tauto
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import Mathlib.Data.Countable.Basic
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.Interval.Finset.Defs
#align_import order.succ_pred.linear_locally_finite from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
open Order
variable {ι : Type*}... | Mathlib/Order/SuccPred/LinearLocallyFinite.lean | 102 | 105 | theorem succFn_le_of_lt (i j : ι) (hij : i < j) : succFn i ≤ j := by |
have h := succFn_spec i
rw [IsGLB, IsGreatest, mem_lowerBounds] at h
exact h.1 j hij
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import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import data.dfinsupp.basic from "leanpr... | Mathlib/Data/DFinsupp/Basic.lean | 150 | 154 | theorem mapRange_comp (f : ∀ i, β₁ i → β₂ i) (f₂ : ∀ i, β i → β₁ i) (hf : ∀ i, f i 0 = 0)
(hf₂ : ∀ i, f₂ i 0 = 0) (h : ∀ i, (f i ∘ f₂ i) 0 = 0) (g : Π₀ i : ι, β i) :
mapRange (fun i => f i ∘ f₂ i) h g = mapRange f hf (mapRange f₂ hf₂ g) := by |
ext
simp only [mapRange_apply]; rfl
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import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped NNReal Matrix
namespace Matrix
variable {R l m n α β : Type*} [Fintype l] [Fintyp... | Mathlib/Analysis/Matrix.lean | 305 | 305 | theorem linfty_opNNNorm_row (v : n → α) : ‖row v‖₊ = ∑ i, ‖v i‖₊ := by | simp [linfty_opNNNorm_def]
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import Mathlib.Probability.ProbabilityMassFunction.Monad
#align_import probability.probability_mass_function.constructions from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
universe u
namespace PMF
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENN... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 136 | 136 | theorem mem_support_seq_iff : b ∈ (seq q p).support ↔ ∃ f ∈ q.support, b ∈ f '' p.support := by | simp
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import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Algebra.Star.Unitary
#align_import linear_algebra.unitary_group from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
universe u ... | Mathlib/LinearAlgebra/UnitaryGroup.lean | 76 | 80 | theorem det_of_mem_unitary {A : Matrix n n α} (hA : A ∈ Matrix.unitaryGroup n α) :
A.det ∈ unitary α := by |
constructor
· simpa [star, det_transpose] using congr_arg det hA.1
· simpa [star, det_transpose] using congr_arg det hA.2
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import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {α β γ : Type*} {ι : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 32 | 32 | theorem lift_top (g : Set α → Filter β) : (⊤ : Filter α).lift g = g univ := by | simp [Filter.lift]
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import Mathlib.Algebra.Order.Interval.Set.Instances
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Instances.Real
#align_import topology.unit_interval from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter
... | Mathlib/Topology/UnitInterval.lean | 154 | 155 | theorem half_le_symm_iff (t : I) : 1 / 2 ≤ (σ t : ℝ) ↔ (t : ℝ) ≤ 1 / 2 := by |
rw [coe_symm_eq, le_sub_iff_add_le, add_comm, ← le_sub_iff_add_le, sub_half]
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import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.lattice from "leanprover-community/mathlib"@"52fa514ec337dd970d71d8de8d0fd68b455a1e54"
assert_not_exists MonoidWithZero
open Set
namespace Nat
open scoped Classical
noncomputable instance : ... | Mathlib/Data/Nat/Lattice.lean | 66 | 67 | theorem iInf_of_empty {ι : Sort*} [IsEmpty ι] (f : ι → ℕ) : iInf f = 0 := by |
rw [iInf_of_isEmpty, sInf_empty]
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import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
noncomputable section
universe u
namespace SetTheory
open scoped PGame
namespace PGame... | Mathlib/SetTheory/Game/Nim.lean | 119 | 119 | theorem moveRight_nim {o : Ordinal} (i) : (nim o).moveRight (toRightMovesNim i) = nim i := by | simp
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import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.LinearAlgebra.Basis
#align_import analysis.normed_space.linear_isometry from "leanprover-community/mathlib"@"4601791ea62fea875b488dafc4e6dede19e8363f"
open Function Set
variable {R R₂ R₃ R₄ E E₂ E₃ E₄ F 𝓕 : Ty... | Mathlib/Analysis/NormedSpace/LinearIsometry.lean | 170 | 172 | theorem coe_injective : @Injective (E →ₛₗᵢ[σ₁₂] E₂) (E → E₂) (fun f => f) := by |
rintro ⟨_⟩ ⟨_⟩
simp
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import Mathlib.Deprecated.Group
#align_import deprecated.ring from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
universe u v w
variable {α : Type u}
structure IsSemiringHom {α : Type u} {β : Type v} [Semiring α] [Semiring β] (f : α → β) : Prop where
map_zero : f 0 = 0
map... | Mathlib/Deprecated/Ring.lean | 58 | 63 | theorem comp (hf : IsSemiringHom f) {γ} [Semiring γ] {g : β → γ} (hg : IsSemiringHom g) :
IsSemiringHom (g ∘ f) :=
{ map_zero := by | simpa [map_zero hf] using map_zero hg
map_one := by simpa [map_one hf] using map_one hg
map_add := fun {x y} => by simp [map_add hf, map_add hg]
map_mul := fun {x y} => by simp [map_mul hf, map_mul hg] }
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import Mathlib.Dynamics.Ergodic.MeasurePreserving
#align_import dynamics.ergodic.ergodic from "leanprover-community/mathlib"@"809e920edfa343283cea507aedff916ea0f1bd88"
open Set Function Filter MeasureTheory MeasureTheory.Measure
open ENNReal
variable {α : Type*} {m : MeasurableSpace α} (f : α → α) {s : Set α}
... | Mathlib/Dynamics/Ergodic/Ergodic.lean | 74 | 76 | theorem prob_eq_zero_or_one [IsProbabilityMeasure μ] (hf : PreErgodic f μ) (hs : MeasurableSet s)
(hs' : f ⁻¹' s = s) : μ s = 0 ∨ μ s = 1 := by |
simpa [hs] using hf.measure_self_or_compl_eq_zero hs hs'
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import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.independence from "leanprover-community/mathlib"@"1e82f5ec4645f6a92bb9e02fce51e44e3bc3e1fe"
open scoped LinearAlgebra.Projectivization
variable {ι K V : Type*} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ... | Mathlib/LinearAlgebra/Projectivization/Independence.lean | 98 | 99 | theorem dependent_iff_not_independent : Dependent f ↔ ¬Independent f := by |
rw [dependent_iff, independent_iff]
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import Mathlib.ModelTheory.ElementaryMaps
open FirstOrder
namespace FirstOrder
namespace Language
open Structure
variable {L : Language} {M : Type*} {N : Type*} {P : Type*} {Q : Type*}
variable [L.Structure M] [L.Structure N] [L.Structure P] [L.Structure Q]
def Substructure.IsElementary (S : L.Substructure M... | Mathlib/ModelTheory/ElementarySubstructures.lean | 111 | 112 | theorem theory_model_iff (S : L.ElementarySubstructure M) (T : L.Theory) : S ⊨ T ↔ M ⊨ T := by |
simp only [Theory.model_iff, realize_sentence]
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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 | 139 | 141 | theorem reflect_C_mul (f : R[X]) (r : R) (N : ℕ) : reflect N (C r * f) = C r * reflect N f := by |
ext
simp only [coeff_reflect, coeff_C_mul]
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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 | 61 | 62 | theorem eq_zero_of_lt {n ν : ℕ} (h : n < ν) : bernsteinPolynomial R n ν = 0 := by |
simp [bernsteinPolynomial, Nat.choose_eq_zero_of_lt h]
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import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Topological... | Mathlib/Probability/Martingale/Upcrossing.lean | 186 | 189 | theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by |
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
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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 | 62 | 65 | theorem hasBasis_nhds_zero : (𝓝 (0 : Γ₀)).HasBasis (fun γ : Γ₀ => γ ≠ 0) Iio := by |
rw [nhds_zero]
refine hasBasis_biInf_principal ?_ ⟨1, one_ne_zero⟩
exact directedOn_iff_directed.2 (Monotone.directed_ge fun a b hab => Iio_subset_Iio hab)
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import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 127 | 130 | theorem lineMap_le_lineMap_iff_of_lt (h : r < r') : lineMap a b r ≤ lineMap a b r' ↔ a ≤ b := by |
simp only [lineMap_apply_module]
rw [← le_sub_iff_add_le, add_sub_assoc, ← sub_le_iff_le_add', ← sub_smul, ← sub_smul,
sub_sub_sub_cancel_left, smul_le_smul_iff_of_pos_left (sub_pos.2 h)]
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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 | 116 | 118 | theorem to_leftMoves_one_toPGame_symm (i) :
(@toLeftMovesToPGame 1).symm i = ⟨0, Set.mem_Iio.mpr zero_lt_one⟩ := by |
simp [eq_iff_true_of_subsingleton]
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import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Sort
import Mathlib.Data.List.FinRange
import Mathlib.LinearAlgebra.Pi
import Mathlib.Logic.Equiv.Fintype
#align_import linear_algebra.multilinear.basic from ... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 183 | 185 | theorem map_zero [Nonempty ι] : f 0 = 0 := by |
obtain ⟨i, _⟩ : ∃ i : ι, i ∈ Set.univ := Set.exists_mem_of_nonempty ι
exact map_coord_zero f i rfl
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import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : ℕ) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 56 | 61 | theorem wittPolyProd_vars (n : ℕ) : (wittPolyProd p n).vars ⊆ univ ×ˢ range (n + 1) := by |
rw [wittPolyProd]
apply Subset.trans (vars_mul _ _)
refine union_subset ?_ ?_ <;>
· refine Subset.trans (vars_rename _ _) ?_
simp [wittPolynomial_vars, image_subset_iff]
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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 | 110 | 111 | theorem card_Ico_finset (h : s ⊆ t) : (Ico s t).card = 2 ^ (t.card - s.card) - 1 := by |
rw [card_Ico_eq_card_Icc_sub_one, card_Icc_finset h]
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import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Opposites
#align_import algebraic_topology.simplicial_object from "leanprover-community/mathlib"@"5ed51dc37c6b891b79314ee11a50adc2b1df6fd6"
o... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 154 | 156 | theorem δ_comp_σ_self {n} {i : Fin (n + 1)} : X.σ i ≫ X.δ (Fin.castSucc i) = 𝟙 _ := by |
dsimp [δ, σ]
simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_σ_self, op_id, X.map_id]
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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 | 85 | 89 | theorem sign_neg {r : ℝ} : sign (-r) = -sign r := by |
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)
· rw [sign_of_neg hn, sign_of_pos (neg_pos.mpr hn), neg_neg]
· rw [sign_zero, neg_zero, sign_zero]
· rw [sign_of_pos hp, sign_of_neg (neg_lt_zero.mpr hp)]
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import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.StdBasis
#align_import linear_algebra.finsupp_vector_space from "leanprover-community/mathlib"@"59628387770d82eb6f6dd7b7107308aa2509ec95"
noncomputable section
open Set LinearMap Submodule
open scoped Cardinal
universe u v w
namespace Finsupp
... | Mathlib/LinearAlgebra/FinsuppVectorSpace.lean | 167 | 170 | theorem equivFun_symm_stdBasis [Finite n] (b : Basis n R M) (i : n) :
b.equivFun.symm (LinearMap.stdBasis R (fun _ => R) i 1) = b i := by |
cases nonempty_fintype n
simp
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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}
| Mathlib/Topology/NhdsSet.lean | 35 | 38 | theorem nhdsSet_diagonal (X) [TopologicalSpace (X × X)] :
𝓝ˢ (diagonal X) = ⨆ (x : X), 𝓝 (x, x) := by |
rw [nhdsSet, ← range_diag, ← range_comp]
rfl
| 0.28125 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 100 | 103 | theorem _root_.MeasureTheory.Memℒp.ofReal_variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
ENNReal.ofReal (variance X μ) = evariance X μ := by |
rw [variance, ENNReal.ofReal_toReal]
exact hX.evariance_lt_top.ne
| 0.28125 |
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
universe u v w z
open scoped Matrix
open Matrix FiniteDimensional Fintype Polynomial Fin... | Mathlib/RingTheory/Discriminant.lean | 121 | 124 | theorem discr_of_matrix_mulVec (b : ι → B) (P : Matrix ι ι A) :
discr A (P.map (algebraMap A B) *ᵥ b) = P.det ^ 2 * discr A b := by |
rw [discr_def, traceMatrix_of_matrix_mulVec, det_mul, det_mul, det_transpose, mul_comm, ←
mul_assoc, discr_def, pow_two]
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import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.dslope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open scoped Classical Topology Filter
open Function Set Filter
variable {𝕜 E : Type*} [NontriviallyNormed... | Mathlib/Analysis/Calculus/Dslope.lean | 46 | 52 | theorem ContinuousLinearMap.dslope_comp {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
(f : E →L[𝕜] F) (g : 𝕜 → E) (a b : 𝕜) (H : a = b → DifferentiableAt 𝕜 g a) :
dslope (f ∘ g) a b = f (dslope g a b) := by |
rcases eq_or_ne b a with (rfl | hne)
· simp only [dslope_same]
exact (f.hasFDerivAt.comp_hasDerivAt b (H rfl).hasDerivAt).deriv
· simpa only [dslope_of_ne _ hne] using f.toLinearMap.slope_comp g a b
| 0.28125 |
import Mathlib.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
#align_import probability.ident_distrib from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Filter Finset
noncomputable section
open scoped Topology MeasureTheory ENNReal NNR... | Mathlib/Probability/IdentDistrib.lean | 132 | 135 | theorem measure_mem_eq (h : IdentDistrib f g μ ν) {s : Set γ} (hs : MeasurableSet s) :
μ (f ⁻¹' s) = ν (g ⁻¹' s) := by |
rw [← Measure.map_apply_of_aemeasurable h.aemeasurable_fst hs, ←
Measure.map_apply_of_aemeasurable h.aemeasurable_snd hs, h.map_eq]
| 0.28125 |
import Mathlib.Data.List.Basic
import Mathlib.Data.Sigma.Basic
#align_import data.list.prod_sigma from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
variable {α β : Type*}
namespace List
@[simp]
theorem nil_product (l : List β) : (@nil α) ×ˢ l = [] :=
rfl
#align list.nil_product... | Mathlib/Data/List/ProdSigma.lean | 45 | 48 | theorem mem_product {l₁ : List α} {l₂ : List β} {a : α} {b : β} :
(a, b) ∈ l₁ ×ˢ l₂ ↔ a ∈ l₁ ∧ b ∈ l₂ := by |
simp_all [SProd.sprod, product, mem_bind, mem_map, Prod.ext_iff, exists_prop, and_left_comm,
exists_and_left, exists_eq_left, exists_eq_right]
| 0.28125 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.Algebra.PUnitInstances
#align_import category_theory.monoidal.Mon_ from "leanprover-community/... | Mathlib/CategoryTheory/Monoidal/Mon_.lean | 75 | 76 | theorem one_mul_hom {Z : C} (f : Z ⟶ M.X) : (M.one ⊗ f) ≫ M.mul = (λ_ Z).hom ≫ f := by |
rw [tensorHom_def'_assoc, M.one_mul, leftUnitor_naturality]
| 0.28125 |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 209 | 213 | theorem set_lintegral_condDistrib_of_measurableSet (hX : Measurable X) (hY : AEMeasurable Y μ)
(hs : MeasurableSet s) {t : Set α} (ht : MeasurableSet[mβ.comap X] t) :
∫⁻ a in t, condDistrib Y X μ (X a) s ∂μ = μ (t ∩ Y ⁻¹' s) := by |
obtain ⟨t', ht', rfl⟩ := ht
rw [set_lintegral_preimage_condDistrib hX hY hs ht']
| 0.28125 |
import Mathlib.Algebra.MvPolynomial.Rename
#align_import data.mv_polynomial.comap from "leanprover-community/mathlib"@"aba31c938d3243cc671be7091b28a1e0814647ee"
namespace MvPolynomial
variable {σ : Type*} {τ : Type*} {υ : Type*} {R : Type*} [CommSemiring R]
noncomputable def comap (f : MvPolynomial σ R →ₐ[R] M... | Mathlib/Algebra/MvPolynomial/Comap.lean | 90 | 92 | theorem comap_rename (f : σ → τ) (x : τ → R) : comap (rename f) x = x ∘ f := by |
funext
simp [rename_X, comap_apply, aeval_X]
| 0.28125 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
namespace MeasureTheory
open Filter
open scoped ENNReal
variable {α E : Type*} {m m0 : MeasurableSpace α} {p : ℝ≥0∞} {q : ℝ} {μ :... | Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean | 59 | 65 | theorem snorm_trim (hm : m ≤ m0) {f : α → E} (hf : StronglyMeasurable[m] f) :
snorm f p (μ.trim hm) = snorm f p μ := by |
by_cases h0 : p = 0
· simp [h0]
by_cases h_top : p = ∞
· simpa only [h_top, snorm_exponent_top] using snormEssSup_trim hm hf
simpa only [snorm_eq_snorm' h0 h_top] using snorm'_trim hm hf
| 0.28125 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 69 | 72 | theorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :
∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by |
rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb]
rfl
| 0.28125 |
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.GCDMonoid.Nat
#align_import ring_theory.int.basic from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
namespace Int
theorem gcd_eq_one_iff_coprime {a b : ℤ} : Int.gcd a b ... | Mathlib/RingTheory/Int/Basic.lean | 49 | 50 | theorem coprime_iff_nat_coprime {a b : ℤ} : IsCoprime a b ↔ Nat.Coprime a.natAbs b.natAbs := by |
rw [← gcd_eq_one_iff_coprime, Nat.coprime_iff_gcd_eq_one, gcd_eq_natAbs]
| 0.28125 |
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 302 | 304 | theorem convexBodySumFun_neg (x : E K) :
convexBodySumFun (- x) = convexBodySumFun x := by |
simp_rw [convexBodySumFun, normAtPlace_neg]
| 0.28125 |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Topology.Instances.ENNReal
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Metric TopologicalSpace Function Filter
open scoped Topology NNReal
variable {α β F : Type*} [N... | Mathlib/Analysis/NormedSpace/FunctionSeries.lean | 81 | 84 | theorem continuous_tsum [TopologicalSpace β] {f : α → β → F} (hf : ∀ i, Continuous (f i))
(hu : Summable u) (hfu : ∀ n x, ‖f n x‖ ≤ u n) : Continuous fun x => ∑' n, f n x := by |
simp_rw [continuous_iff_continuousOn_univ] at hf ⊢
exact continuousOn_tsum hf hu fun n x _ => hfu n x
| 0.28125 |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 117 | 122 | theorem compl_neighborFinset_sdiff_inter_eq {v w : V} :
(G.neighborFinset v)ᶜ \ {v} ∩ ((G.neighborFinset w)ᶜ \ {w}) =
((G.neighborFinset v)ᶜ ∩ (G.neighborFinset w)ᶜ) \ ({w} ∪ {v}) := by |
ext
rw [← not_iff_not]
simp [imp_iff_not_or, or_assoc, or_comm, or_left_comm]
| 0.28125 |
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 | 232 | 235 | theorem PreservesPushout.inr_iso_hom :
pushout.inr ≫ (PreservesPushout.iso G f g).hom = G.map pushout.inr := by |
delta PreservesPushout.iso
simp
| 0.28125 |
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.GCongr
#align_import order.filter.archimedean from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
variable {α R : Type*}
open Filter Set Function
@[simp]
theorem Nat.comap_cast_atTop [S... | Mathlib/Order/Filter/Archimedean.lean | 100 | 102 | theorem Filter.Eventually.intCast_atBot [StrictOrderedRing R] [Archimedean R] {p : R → Prop}
(h : ∀ᶠ (x:R) in atBot, p x) : ∀ᶠ (n:ℤ) in atBot, p n := by |
rw [← Int.comap_cast_atBot (R := R)]; exact h.comap _
| 0.28125 |
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Monoidal.Braided.Opposite
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
universe v₁ v₂ u₁ u₂ u
open CategoryTheory MonoidalCategor... | Mathlib/CategoryTheory/Monoidal/Comon_.lean | 77 | 78 | theorem comul_counit_hom {Z : C} (f : M.X ⟶ Z) : M.comul ≫ (f ⊗ M.counit) = f ≫ (ρ_ Z).inv := by |
rw [rightUnitor_inv_naturality, tensorHom_def', comul_counit_assoc]
| 0.28125 |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section WithDivisionRing
variable {K : Type*}... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 166 | 167 | theorem first_denominator_eq {gp : Pair K} (zeroth_s_eq : g.s.get? 0 = some gp) :
g.denominators 1 = gp.b := by | simp [denom_eq_conts_b, first_continuant_eq zeroth_s_eq]
| 0.28125 |
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 | 82 | 85 | theorem eval₂_X_pow {n : ℕ} : (X ^ n).eval₂ f x = x ^ n := by |
rw [X_pow_eq_monomial]
convert eval₂_monomial f x (n := n) (r := 1)
simp
| 0.28125 |
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Opposites
#align_import algebraic_topology.simplicial_object from "leanprover-community/mathlib"@"5ed51dc37c6b891b79314ee11a50adc2b1df6fd6"
o... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 146 | 149 | theorem δ_comp_σ_of_le {n} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : i ≤ Fin.castSucc j) :
X.σ j.succ ≫ X.δ (Fin.castSucc i) = X.δ i ≫ X.σ j := by |
dsimp [δ, σ]
simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_σ_of_le H]
| 0.28125 |
import Mathlib.Data.Nat.Prime
#align_import data.int.nat_prime from "leanprover-community/mathlib"@"422e70f7ce183d2900c586a8cda8381e788a0c62"
open Nat
namespace Int
theorem not_prime_of_int_mul {a b : ℤ} {c : ℕ} (ha : a.natAbs ≠ 1) (hb : b.natAbs ≠ 1)
(hc : a * b = (c : ℤ)) : ¬Nat.Prime c :=
not_prime_mul... | Mathlib/Data/Int/NatPrime.lean | 24 | 33 | theorem succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul {p : ℕ} (p_prime : Nat.Prime p) {m n : ℤ}
{k l : ℕ} (hpm : ↑(p ^ k) ∣ m) (hpn : ↑(p ^ l) ∣ n) (hpmn : ↑(p ^ (k + l + 1)) ∣ m * n) :
↑(p ^ (k + 1)) ∣ m ∨ ↑(p ^ (l + 1)) ∣ n :=
have hpm' : p ^ k ∣ m.natAbs := Int.natCast_dvd_natCast.1 <| Int.dvd_natAbs.2 hpm
ha... |
rw [← Int.natAbs_mul]; apply Int.natCast_dvd_natCast.1 <| Int.dvd_natAbs.2 hpmn
let hsd := Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn'
hsd.elim (fun hsd1 => Or.inl (by apply Int.dvd_natAbs.1; apply Int.natCast_dvd_natCast.2 hsd1))
fun hsd2 => Or.inr (by apply Int.dvd_natAbs.1; appl... | 0.28125 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
#align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c"
namespace Finset
variable {α : Type*}
@[simps]
protected def sym2 (s : Finset α) : Finset (Sym2 α) :... | Mathlib/Data/Finset/Sym.lean | 85 | 89 | theorem sym2_toFinset [DecidableEq α] (m : Multiset α) :
m.toFinset.sym2 = m.sym2.toFinset := by |
ext z
refine z.ind fun x y ↦ ?_
simp only [mk_mem_sym2_iff, Multiset.mem_toFinset, Multiset.mk_mem_sym2_iff]
| 0.28125 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
#align_import analysis.normed_space.conformal_linear_map from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open Function LinearIsometry ContinuousLinearMap
def IsConf... | Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean | 97 | 100 | theorem ne_zero [Nontrivial M'] {f' : M' →L[R] N} (hf' : IsConformalMap f') : f' ≠ 0 := by |
rintro rfl
rcases exists_ne (0 : M') with ⟨a, ha⟩
exact ha (hf'.injective rfl)
| 0.28125 |
import Mathlib.LinearAlgebra.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.LinearAlgebra.Trace
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosu... | Mathlib/RingTheory/Trace.lean | 109 | 111 | theorem trace_eq_matrix_trace [DecidableEq ι] (b : Basis ι R S) (s : S) :
trace R S s = Matrix.trace (Algebra.leftMulMatrix b s) := by |
rw [trace_apply, LinearMap.trace_eq_matrix_trace _ b, ← toMatrix_lmul_eq]; rfl
| 0.28125 |
import Mathlib.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.Bounded
#align_import topology.metric_space.antilipschitz from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
... | Mathlib/Topology/MetricSpace/Antilipschitz.lean | 129 | 134 | theorem comp {Kg : ℝ≥0} {g : β → γ} (hg : AntilipschitzWith Kg g) {Kf : ℝ≥0} {f : α → β}
(hf : AntilipschitzWith Kf f) : AntilipschitzWith (Kf * Kg) (g ∘ f) := fun x y =>
calc
edist x y ≤ Kf * edist (f x) (f y) := hf x y
_ ≤ Kf * (Kg * edist (g (f x)) (g (f y))) := ENNReal.mul_left_mono (hg _ _)
_ = _... | rw [ENNReal.coe_mul, mul_assoc]; rfl
| 0.28125 |
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.FullSubcategory
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fintype.Card
#align_import category_theory.Fintype from "leanprover-community/mathlib"@"c3019c79074b0619edb4b27553a91b2e82242395"
open scoped Classical
ope... | Mathlib/CategoryTheory/FintypeCat.lean | 211 | 213 | theorem incl_mk_nat_card (n : ℕ) : Fintype.card (incl.obj (mk n)) = n := by |
convert Finset.card_fin n
apply Fintype.ofEquiv_card
| 0.28125 |
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 | 142 | 144 | theorem prod_inter : s ×ˢ (t₁ ∩ t₂) = s ×ˢ t₁ ∩ s ×ˢ t₂ := by |
ext ⟨x, y⟩
simp only [← and_and_left, mem_inter_iff, mem_prod]
| 0.28125 |
import Mathlib.Data.Bracket
import Mathlib.LinearAlgebra.Basic
#align_import algebra.lie.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v w w₁ w₂
open Function
class LieRing (L : Type v) extends AddCommGroup L, Bracket L L where
protected add_lie : ∀ x y z ... | Mathlib/Algebra/Lie/Basic.lean | 169 | 171 | theorem lie_neg : ⁅x, -m⁆ = -⁅x, m⁆ := by |
rw [← sub_eq_zero, sub_neg_eq_add, ← lie_add]
simp
| 0.28125 |
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