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.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 | 111 | 113 | theorem singleton_prod : ({a} : Set α) ×ˢ t = Prod.mk a '' t := by |
ext ⟨x, y⟩
simp [and_left_comm, eq_comm]
| 0.71875 |
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 | 120 | 124 | theorem monomial_mem_lifts {s : S} (n : ℕ) (h : s ∈ Set.range f) : monomial n s ∈ lifts f := by |
obtain ⟨r, rfl⟩ := Set.mem_range.1 h
use monomial n r
simp only [coe_mapRingHom, Set.mem_univ, map_monomial, Subsemiring.coe_top, eq_self_iff_true,
and_self_iff]
| 0.71875 |
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 | 340 | 344 | theorem hasFDerivWithinAt_euclidean :
HasFDerivWithinAt f f' t y ↔
∀ i, HasFDerivWithinAt (fun x => f x i) (EuclideanSpace.proj i ∘L f') t y := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_hasFDerivWithinAt_iff, hasFDerivWithinAt_pi']
rfl
| 0.71875 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 127 | 129 | theorem nat_mul_inj {n : ℕ} {a b : R} (h : (n : R) * a = (n : R) * b) : n = 0 ∨ a = b := by |
rw [← sub_eq_zero, ← mul_sub, mul_eq_zero, sub_eq_zero] at h
exact mod_cast h
| 0.71875 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}
def map₂ (f : α → β → γ) (a : Option α) ... | Mathlib/Data/Option/NAry.lean | 124 | 127 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) := by |
cases a <;> cases b <;> cases c <;> simp [h_assoc]
| 0.71875 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Function Set Filter
open scoped Topology Filter
variable... | Mathlib/Analysis/Calculus/BumpFunction/Basic.lean | 118 | 120 | theorem one_lt_rOut_div_rIn {c : E} (f : ContDiffBump c) : 1 < f.rOut / f.rIn := by |
rw [one_lt_div f.rIn_pos]
exact f.rIn_lt_rOut
| 0.71875 |
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on from "leanprover-community/mathlib"@"8b8ba04e2f326f3f7cf24ad129beda58531ada61"
noncomputable section
open Set Filter TopologicalSpace MeasureTheory Function
ope... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 99 | 99 | theorem integrableOn_empty : IntegrableOn f ∅ μ := by | simp [IntegrableOn, integrable_zero_measure]
| 0.71875 |
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
set_option autoImplicit true
open Filter Set Function
variable {α β γ ι ι' : Type*}
namespace Filter
section Relation
... | Mathlib/Order/LiminfLimsup.lean | 83 | 84 | theorem isBounded_principal (s : Set α) : IsBounded r (𝓟 s) ↔ ∃ t, ∀ x ∈ s, r x t := by |
simp [IsBounded, subset_def]
| 0.71875 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {M : Type*} ... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 123 | 124 | theorem image_const_add_Icc : (fun x => a + x) '' Icc b c = Icc (a + b) (a + c) := by |
simp only [add_comm a, image_add_const_Icc]
| 0.71875 |
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
#align_import category_theory.limits.shapes.strict_initial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe v u
namespace CategoryTheory
namespace Limits
open C... | Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean | 74 | 77 | theorem IsInitial.strict_hom_ext (hI : IsInitial I) {A : C} (f g : A ⟶ I) : f = g := by |
haveI := hI.isIso_to f
haveI := hI.isIso_to g
exact eq_of_inv_eq_inv (hI.hom_ext (inv f) (inv g))
| 0.71875 |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {α β R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 82 | 84 | theorem IsDiag.neg [AddGroup α] {A : Matrix n n α} (ha : A.IsDiag) : (-A).IsDiag := by |
intro i j h
simp [ha h]
| 0.71875 |
import Mathlib.Order.SuccPred.Basic
#align_import order.succ_pred.relation from "leanprover-community/mathlib"@"9aba7801eeecebb61f58a5763c2b6dd1b47dc6ef"
open Function Order Relation Set
section PartialSucc
variable {α : Type*} [PartialOrder α] [SuccOrder α] [IsSuccArchimedean α]
theorem reflTransGen_of_succ_... | Mathlib/Order/SuccPred/Relation.lean | 40 | 43 | theorem reflTransGen_of_succ_of_ge (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Ico m n, r (succ i) i)
(hmn : m ≤ n) : ReflTransGen r n m := by |
rw [← reflTransGen_swap]
exact reflTransGen_of_succ_of_le (swap r) h hmn
| 0.71875 |
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 110 | 110 | theorem log_one : log 1 = 0 := by | simp [log]
| 0.71875 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Deprecated.Subring
#align_import deprecated.subfield from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
variable {F : Type*} [Field F] (S : Set F)
structure IsSubfield extends IsSubring S : Prop where
inv_mem : ∀ {x : F}, x ∈ S → x⁻... | Mathlib/Deprecated/Subfield.lean | 40 | 43 | theorem IsSubfield.div_mem {S : Set F} (hS : IsSubfield S) {x y : F} (hx : x ∈ S) (hy : y ∈ S) :
x / y ∈ S := by |
rw [div_eq_mul_inv]
exact hS.toIsSubring.toIsSubmonoid.mul_mem hx (hS.inv_mem hy)
| 0.71875 |
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.FieldDivision
#align_import ring_theory.polynomial.content from "leanprover-community/mathlib"@"7a030ab8eb5d99f05a891dccc49c5b5b90c947d3"
namespace Po... | Mathlib/RingTheory/Polynomial/Content.lean | 83 | 88 | theorem content_dvd_coeff {p : R[X]} (n : ℕ) : p.content ∣ p.coeff n := by |
by_cases h : n ∈ p.support
· apply Finset.gcd_dvd h
rw [mem_support_iff, Classical.not_not] at h
rw [h]
apply dvd_zero
| 0.71875 |
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
suppress_compilation
universe uR uM₁ uM₂ uM₃ uM₄
variable {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} {M₃ : Type uM₃} {M₄ : Type uM₄}
open scoped TensorProduct
namespace QuadraticForm
variable [Co... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean | 186 | 192 | theorem comp_tensorLId_eq (Q₂ : QuadraticForm R M₂) :
Q₂.comp (TensorProduct.lid R M₂) = (sq (R := R)).tmul Q₂ := by |
refine (QuadraticForm.associated_rightInverse R).injective ?_
ext m₂ m₂'
dsimp [-associated_apply]
simp only [associated_tmul, QuadraticForm.associated_comp]
simp [-associated_apply, mul_one]
| 0.71875 |
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 | 456 | 457 | theorem mul_div_assoc (a b c : G) : a * b / c = a * (b / c) := by |
rw [div_eq_mul_inv, div_eq_mul_inv, mul_assoc _ _ _]
| 0.71875 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 83 | 87 | theorem oangle_add_left_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x + y) y = Real.arctan (‖x‖ / ‖y‖) := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two h
| 0.71875 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 393 | 396 | theorem HasStrictFDerivAt.comp_hasStrictDerivAt_of_eq (hl : HasStrictFDerivAt l l' y)
(hf : HasStrictDerivAt f f' x) (hy : y = f x) :
HasStrictDerivAt (l ∘ f) (l' f') x := by |
rw [hy] at hl; exact hl.comp_hasStrictDerivAt x hf
| 0.71875 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 98 | 102 | theorem natDegree_taylor (p : R[X]) (r : R) : natDegree (taylor r p) = natDegree p := by |
refine map_natDegree_eq_natDegree _ ?_
nontriviality R
intro n c c0
simp [taylor_monomial, natDegree_C_mul_eq_of_mul_ne_zero, natDegree_pow_X_add_C, c0]
| 0.71875 |
import Batteries.Tactic.SeqFocus
namespace Ordering
@[simp] theorem swap_swap {o : Ordering} : o.swap.swap = o := by cases o <;> rfl
@[simp] theorem swap_inj {o₁ o₂ : Ordering} : o₁.swap = o₂.swap ↔ o₁ = o₂ :=
⟨fun h => by simpa using congrArg swap h, congrArg _⟩
theorem swap_then (o₁ o₂ : Ordering) : (o₁.then... | .lake/packages/batteries/Batteries/Classes/Order.lean | 23 | 24 | theorem then_eq_eq {o₁ o₂ : Ordering} : o₁.then o₂ = eq ↔ o₁ = eq ∧ o₂ = eq := by |
cases o₁ <;> simp [«then»]
| 0.71875 |
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Int.Order.Lemmas
#align_import group_theory.submonoid.membership fro... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 340 | 341 | theorem closure_singleton_one : closure ({1} : Set M) = ⊥ := by |
simp [eq_bot_iff_forall, mem_closure_singleton]
| 0.71875 |
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic.Abel
open Lean Elab Meta Tactic Qq
initialize registerTraceClass `abel
initialize registerTraceClass `abel.detail
structure Context where
α : Expr
univ :... | Mathlib/Tactic/Abel.lean | 157 | 158 | theorem zero_termg {α} [AddCommGroup α] (x a) : @termg α _ 0 x a = a := by |
simp [termg, zero_zsmul]
| 0.71875 |
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 | 103 | 104 | theorem image_Icc (e : α ≃o β) (a b : α) : e '' Icc a b = Icc (e a) (e b) := by |
rw [e.image_eq_preimage, e.symm.preimage_Icc, e.symm_symm]
| 0.71875 |
import Mathlib.Order.RelClasses
#align_import data.sigma.lex from "leanprover-community/mathlib"@"41cf0cc2f528dd40a8f2db167ea4fb37b8fde7f3"
namespace Sigma
variable {ι : Type*} {α : ι → Type*} {r r₁ r₂ : ι → ι → Prop} {s s₁ s₂ : ∀ i, α i → α i → Prop}
{a b : Σ i, α i}
inductive Lex (r : ι → ι → Prop) (s : ∀ ... | Mathlib/Data/Sigma/Lex.lean | 80 | 83 | theorem lex_swap : Lex (Function.swap r) s a b ↔ Lex r (fun i => Function.swap (s i)) b a := by |
constructor <;>
· rintro (⟨a, b, h⟩ | ⟨a, b, h⟩)
exacts [Lex.left _ _ h, Lex.right _ _ h]
| 0.71875 |
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
noncomputable section
universe u v v' v''
variable {K : Type u} {V V₁ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
open Cardinal Basis Submodule Function Set
namespace LinearMap
section Ring
... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 79 | 81 | theorem rank_comp_le (g : V →ₗ[K] V') (f : V' →ₗ[K] V'₁) :
rank (f.comp g) ≤ min (rank f) (rank g) := by |
simpa only [Cardinal.lift_id] using lift_rank_comp_le g f
| 0.71875 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
section Preimage
variable {f : α → β} {g : β → γ... | Mathlib/Data/Set/Image.lean | 133 | 136 | theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by |
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
| 0.71875 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Rat
import Mathlib.Data.PNat.Defs
#align_import data.rat.lemmas from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
namespace Rat
o... | Mathlib/Data/Rat/Lemmas.lean | 87 | 90 | theorem mul_den_dvd (q₁ q₂ : ℚ) : (q₁ * q₂).den ∣ q₁.den * q₂.den := by |
rw [mul_def, normalize_eq]
apply Nat.div_dvd_of_dvd
apply Nat.gcd_dvd_right
| 0.71875 |
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 | 53 | 54 | theorem preimage_Ico (e : α ≃o β) (a b : β) : e ⁻¹' Ico a b = Ico (e.symm a) (e.symm b) := by |
simp [← Ici_inter_Iio]
| 0.71875 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Int
theorem isUnit_iff_abs_eq {x : ℤ} : IsUnit x ↔ abs x = 1 := by
rw [isUnit_iff_natAbs_eq, abs_eq_natAbs, ← Int.ofNat_one, natCast_inj]
#align int.... | Mathlib/Data/Int/Order/Units.lean | 45 | 46 | theorem units_coe_mul_self (u : ℤˣ) : (u * u : ℤ) = 1 := by |
rw [← Units.val_mul, units_mul_self, Units.val_one]
| 0.71875 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 62 | 65 | theorem lcm_dvd_iff {a : α} : s.lcm f ∣ a ↔ ∀ b ∈ s, f b ∣ a := by |
apply Iff.trans Multiset.lcm_dvd
simp only [Multiset.mem_map, and_imp, exists_imp]
exact ⟨fun k b hb ↦ k _ _ hb rfl, fun k a' b hb h ↦ h ▸ k _ hb⟩
| 0.71875 |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import ring_theory.localization.num_denom from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
variable {R : Type*} [CommRing R] (... | Mathlib/RingTheory/Localization/NumDen.lean | 70 | 72 | theorem mk'_num_den' (x : K) : algebraMap A K (num A x) / algebraMap A K (den A x) = x := by |
rw [← mk'_eq_div]
apply mk'_num_den
| 0.71875 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 177 | 178 | theorem mem_Lp_iff_memℒp {f : α →ₘ[μ] E} : f ∈ Lp E p μ ↔ Memℒp f p μ := by |
simp [mem_Lp_iff_snorm_lt_top, Memℒp, f.stronglyMeasurable.aestronglyMeasurable]
| 0.71875 |
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 | 161 | 162 | theorem nnnorm_row (v : n → α) : ‖row v‖₊ = ‖v‖₊ := by |
simp [nnnorm_def, Pi.nnnorm_def]
| 0.71875 |
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.NormedSpace.BallAction
#align_import analysis.complex.unit_disc.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Set Function Metric
noncomputable section
local notation "conj'" => starRingEnd ℂ
namespace Co... | Mathlib/Analysis/Complex/UnitDisc/Basic.lean | 53 | 55 | theorem normSq_lt_one (z : 𝔻) : normSq z < 1 := by |
convert (Real.sqrt_lt' one_pos).1 z.abs_lt_one
exact (one_pow 2).symm
| 0.71875 |
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists MonoidWithZero
open Multiset
variable {α β γ : Type*}
namespace Finset
section Prod
variable {s s' : Finset α} {t t' : Finset β} {a : α} {b : β}
... | Mathlib/Data/Finset/Prod.lean | 81 | 83 | theorem product_image_snd [DecidableEq β] (ht : s.Nonempty) : (s ×ˢ t).image Prod.snd = t := by |
ext i
simp [mem_image, ht.exists_mem]
| 0.71875 |
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set F... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 72 | 76 | theorem measure_biUnion_le {I : Set ι} (μ : F) (hI : I.Countable) (s : ι → Set α) :
μ (⋃ i ∈ I, s i) ≤ ∑' i : I, μ (s i) := by |
have := hI.to_subtype
rw [biUnion_eq_iUnion]
apply measure_iUnion_le
| 0.71875 |
import Mathlib.Data.Vector.Basic
#align_import data.vector.mem from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
namespace Vector
variable {α β : Type*} {n : ℕ} (a a' : α)
@[simp]
theorem get_mem (i : Fin n) (v : Vector α n) : v.get i ∈ v.toList := by
rw [get_eq_get]
exact List.... | Mathlib/Data/Vector/Mem.lean | 48 | 49 | theorem mem_cons_iff (v : Vector α n) : a' ∈ (a ::ᵥ v).toList ↔ a' = a ∨ a' ∈ v.toList := by |
rw [Vector.toList_cons, List.mem_cons]
| 0.71875 |
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 | 100 | 100 | theorem eval₂_bit0 : (bit0 p).eval₂ f x = bit0 (p.eval₂ f x) := by | rw [bit0, eval₂_add, bit0]
| 0.71875 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F ... | Mathlib/MeasureTheory/Group/Integral.lean | 79 | 83 | theorem integral_div_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
(∫ x, f (x / g) ∂μ) = ∫ x, f x ∂μ := by |
simp_rw [div_eq_mul_inv]
-- Porting note: was `simp_rw`
rw [integral_mul_right_eq_self f g⁻¹]
| 0.71875 |
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 | 804 | 809 | theorem sum_centroidWeights_eq_one_of_card_ne_zero [CharZero k] (h : card s ≠ 0) :
∑ i ∈ s, s.centroidWeights k i = 1 := by |
-- Porting note: `simp` cannot find `mul_inv_cancel` and does not use `norm_cast`
simp only [centroidWeights_apply, sum_const, nsmul_eq_mul, ne_eq, Nat.cast_eq_zero, card_eq_zero]
refine mul_inv_cancel ?_
norm_cast
| 0.71875 |
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.Tactic.PPWithUniv
import Mathlib.Data.Set.Defs
#align_import category_theory.types from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
namespace CategoryTheory
-- morphism levels be... | Mathlib/CategoryTheory/Types.lean | 152 | 153 | theorem map_comp_apply (f : X ⟶ Y) (g : Y ⟶ Z) (a : F.obj X) :
(F.map (f ≫ g)) a = (F.map g) ((F.map f) a) := by | simp [types_comp]
| 0.71875 |
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dfinsupp from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
variable {ι : Type*} {R : Type*} {S : Type*} {M : ι → Type*} {N : Type*}
n... | Mathlib/LinearAlgebra/DFinsupp.lean | 170 | 172 | theorem lsum_single [Semiring S] [Module S N] [SMulCommClass R S N] (F : ∀ i, M i →ₗ[R] N) (i)
(x : M i) : lsum S (M := M) F (single i x) = F i x := by |
simp
| 0.71875 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 181 | 181 | theorem normalize_gcd : normalize (s.gcd f) = s.gcd f := by | simp [gcd_def]
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import Mathlib.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
#align_import category_theory.idempotents.karoubi from "leanprover-community/mathlib"@"200eda15d8ff5669854ff6bcc10aaf37cb70498f"
noncomputable section
open CategoryT... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 108 | 112 | theorem hom_ext_iff {P Q : Karoubi C} {f g : P ⟶ Q} : f = g ↔ f.f = g.f := by |
constructor
· intro h
rw [h]
· apply Hom.ext
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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 | 100 | 102 | theorem eqToIso_refl {n : ℕ} (h : n = n) : X.eqToIso h = Iso.refl _ := by |
ext
simp [eqToIso]
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import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Rat
import Mathlib.Data.PNat.Defs
#align_import data.rat.lemmas from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
namespace Rat
o... | Mathlib/Data/Rat/Lemmas.lean | 98 | 101 | theorem mul_den (q₁ q₂ : ℚ) :
(q₁ * q₂).den =
q₁.den * q₂.den / Nat.gcd (q₁.num * q₂.num).natAbs (q₁.den * q₂.den) := by |
rw [mul_def, normalize_eq]
| 0.71875 |
import Mathlib.Init.Function
import Mathlib.Logic.Function.Basic
#align_import data.sigma.basic from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
open Function
namespace PSigma
variable {α : Sort*} {β : α → Sort*}
def elim {γ} (f : ∀ a, β a → γ) (a : PSigma β) : γ :=
PSigma.cases... | Mathlib/Data/Sigma/Basic.lean | 273 | 274 | theorem ext_iff {x₀ x₁ : PSigma β} : x₀ = x₁ ↔ x₀.1 = x₁.1 ∧ HEq x₀.2 x₁.2 := by |
cases x₀; cases x₁; exact PSigma.mk.inj_iff
| 0.71875 |
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
#align_import measure_theory.measure.haar.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set Inv Function Topological... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 128 | 129 | theorem prehaar_nonneg (K₀ : PositiveCompacts G) {U : Set G} (K : Compacts G) :
0 ≤ prehaar (K₀ : Set G) U K := by | apply div_nonneg <;> norm_cast <;> apply zero_le
| 0.71875 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477... | Mathlib/Algebra/Order/Field/Basic.lean | 110 | 110 | theorem mul_inv_le_iff' (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ c * b := by | rw [mul_comm, inv_mul_le_iff' h]
| 0.71875 |
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 376 | 380 | theorem HasStrictFDerivAt.mul (hc : HasStrictFDerivAt c c' x) (hd : HasStrictFDerivAt d d' x) :
HasStrictFDerivAt (fun y => c y * d y) (c x • d' + d x • c') x := by |
convert hc.mul' hd
ext z
apply mul_comm
| 0.71875 |
import Mathlib.Init.Logic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Coe
set_option autoImplicit true
-- We align Lean 3 lemmas with lemmas in `Init.SimpLemmas` in Lean 4.
#align band_self Bool.and_self
#align band_tt Bool.and_true
#align band_ff Bool.and_false
#align tt_band Bool.true_and
#align f... | Mathlib/Init/Data/Bool/Lemmas.lean | 68 | 69 | theorem and_eq_true_eq_eq_true_and_eq_true (a b : Bool) :
((a && b) = true) = (a = true ∧ b = true) := by | simp
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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 | 125 | 126 | theorem derivative_of_natDegree_zero {p : R[X]} (hp : p.natDegree = 0) : derivative p = 0 := by |
rw [eq_C_of_natDegree_eq_zero hp, derivative_C]
| 0.71875 |
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Tactic.AdaptationNote
#align_import analysis.special_functions.log.deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
ope... | Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | 34 | 39 | theorem hasStrictDerivAt_log_of_pos (hx : 0 < x) : HasStrictDerivAt log x⁻¹ x := by |
have : HasStrictDerivAt log (exp <| log x)⁻¹ x :=
(hasStrictDerivAt_exp <| log x).of_local_left_inverse (continuousAt_log hx.ne')
(ne_of_gt <| exp_pos _) <|
Eventually.mono (lt_mem_nhds hx) @exp_log
rwa [exp_log hx] at this
| 0.71875 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 145 | 146 | theorem laverage_congr {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : ⨍⁻ x, f x ∂μ = ⨍⁻ x, g x ∂μ := by |
simp only [laverage_eq, lintegral_congr_ae h]
| 0.71875 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477... | Mathlib/Algebra/Order/Field/Basic.lean | 93 | 93 | theorem div_lt_iff' (hc : 0 < c) : b / c < a ↔ b < c * a := by | rw [mul_comm, div_lt_iff hc]
| 0.71875 |
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 | 31 | 34 | theorem sq_add_mul_sq_mul_sq_add_mul_sq :
(x₁ ^ 2 + n * x₂ ^ 2) * (y₁ ^ 2 + n * y₂ ^ 2) =
(x₁ * y₁ - n * x₂ * y₂) ^ 2 + n * (x₁ * y₂ + x₂ * y₁) ^ 2 := by |
ring
| 0.71875 |
import Mathlib.RingTheory.WittVector.InitTail
#align_import ring_theory.witt_vector.truncated from "leanprover-community/mathlib"@"acbe099ced8be9c9754d62860110295cde0d7181"
open Function (Injective Surjective)
noncomputable section
variable {p : ℕ} [hp : Fact p.Prime] (n : ℕ) (R : Type*)
local notation "𝕎" =>... | Mathlib/RingTheory/WittVector/Truncated.lean | 100 | 101 | theorem mk_coeff (x : TruncatedWittVector p n R) : (mk p fun i => x.coeff i) = x := by |
ext i; rw [coeff_mk]
| 0.71875 |
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 | 119 | 120 | theorem independent_pair_iff_neq (u v : ℙ K V) : Independent ![u, v] ↔ u ≠ v := by |
rw [independent_iff_not_dependent, dependent_pair_iff_eq u v]
| 0.71875 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 319 | 320 | theorem average_zero_measure (f : α → E) : ⨍ x, f x ∂(0 : Measure α) = 0 := by |
rw [average, smul_zero, integral_zero_measure]
| 0.71875 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Lattice
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_th... | Mathlib/RingTheory/Nilpotent/Basic.lean | 40 | 43 | theorem IsNilpotent.neg [Ring R] (h : IsNilpotent x) : IsNilpotent (-x) := by |
obtain ⟨n, hn⟩ := h
use n
rw [neg_pow, hn, mul_zero]
| 0.71875 |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 59 | 61 | theorem congr_heq {α β γ : Sort _} {f : α → γ} {g : β → γ} {x : α} {y : β}
(h₁ : HEq f g) (h₂ : HEq x y) : f x = g y := by |
cases h₂; cases h₁; rfl
| 0.71875 |
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.fintype.units from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {α : Type*}
instance UnitsInt.fintype : Fintype ℤˣ :=
⟨{1, -1}, fun x ↦ by cases Int... | Mathlib/Data/Fintype/Units.lean | 48 | 50 | theorem Fintype.card_units [GroupWithZero α] [Fintype α] [DecidableEq α] :
Fintype.card αˣ = Fintype.card α - 1 := by |
rw [@Fintype.card_eq_card_units_add_one α, Nat.add_sub_cancel]
| 0.71875 |
import Mathlib.Order.Filter.Bases
#align_import order.filter.pi from "leanprover-community/mathlib"@"ce64cd319bb6b3e82f31c2d38e79080d377be451"
open Set Function
open scoped Classical
open Filter
namespace Filter
variable {ι : Type*} {α : ι → Type*} {f f₁ f₂ : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)}
... | Mathlib/Order/Filter/Pi.lean | 244 | 245 | theorem coprodᵢ_neBot_iff [∀ i, Nonempty (α i)] : NeBot (Filter.coprodᵢ f) ↔ ∃ d, NeBot (f d) := by |
simp [coprodᵢ_neBot_iff', *]
| 0.71875 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Matrix
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.matrix_exponential from "l... | Mathlib/Analysis/NormedSpace/MatrixExponential.lean | 84 | 86 | theorem exp_blockDiagonal (v : m → Matrix n n 𝔸) :
exp 𝕂 (blockDiagonal v) = blockDiagonal (exp 𝕂 v) := by |
simp_rw [exp_eq_tsum, ← blockDiagonal_pow, ← blockDiagonal_smul, ← blockDiagonal_tsum]
| 0.71875 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.Option
#align_import algebra.big_operators.option from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
open Function
namespace Finset
variable {α M : Type*} [CommMonoid M]
@[to_additive (attr := simp)]
| Mathlib/Algebra/BigOperators/Option.lean | 25 | 26 | theorem prod_insertNone (f : Option α → M) (s : Finset α) :
∏ x ∈ insertNone s, f x = f none * ∏ x ∈ s, f (some x) := by | simp [insertNone]
| 0.71875 |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 158 | 159 | theorem mk_zero (p : K[X]) : RatFunc.mk p 0 = ofFractionRing (0 : FractionRing K[X]) := by |
rw [mk_eq_div', RingHom.map_zero, div_zero]
| 0.71875 |
import Mathlib.Data.Fintype.Quotient
import Mathlib.ModelTheory.Semantics
#align_import model_theory.quotients from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
namespace FirstOrder
namespace Language
variable (L : Language) {M : Type*}
open FirstOrder
open Structure
class Prest... | Mathlib/ModelTheory/Quotients.lean | 65 | 70 | theorem relMap_quotient_mk' {n : ℕ} (r : L.Relations n) (x : Fin n → M) :
(RelMap r fun i => (⟦x i⟧ : Quotient s)) ↔ @RelMap _ _ ps.toStructure _ r x := by |
change
Quotient.lift (@RelMap L M ps.toStructure n r) Prestructure.rel_equiv (Quotient.finChoice _) ↔
_
rw [Quotient.finChoice_eq, Quotient.lift_mk]
| 0.71875 |
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.euclidean_dist from "leanprover-community/mathlib"@"9425b6f8220e53b059f5a4904786c3c4b50fc057"
open scoped Topology
open Set
variable {E : Type*} [AddCommGroup E] [Topologi... | Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean | 82 | 84 | theorem closedBall_eq_image (x : E) (r : ℝ) :
closedBall x r = toEuclidean.symm '' Metric.closedBall (toEuclidean x) r := by |
rw [toEuclidean.image_symm_eq_preimage, closedBall_eq_preimage]
| 0.71875 |
import Mathlib.Data.Matrix.Block
#align_import linear_algebra.matrix.symmetric from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
variable {α β n m R : Type*}
namespace Matrix
open Matrix
def IsSymm (A : Matrix n n α) : Prop :=
Aᵀ = A
#align matrix.is_symm Matrix.IsSymm
instance... | Mathlib/LinearAlgebra/Matrix/Symmetric.lean | 86 | 89 | theorem IsSymm.pow [CommSemiring α] [Fintype n] [DecidableEq n] {A : Matrix n n α} (h : A.IsSymm)
(k : ℕ) :
(A ^ k).IsSymm := by |
rw [IsSymm, transpose_pow, h]
| 0.71875 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}
def map₂ (f : α → β → γ) (a : Option α) ... | Mathlib/Data/Option/NAry.lean | 83 | 84 | theorem map₂_eq_none_iff : map₂ f a b = none ↔ a = none ∨ b = none := by |
cases a <;> cases b <;> simp
| 0.71875 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 221 | 227 | theorem HasStrictDerivAt.mul (hc : HasStrictDerivAt c c' x) (hd : HasStrictDerivAt d d' x) :
HasStrictDerivAt (fun y => c y * d y) (c' * d x + c x * d') x := by |
have := (HasStrictFDerivAt.mul' hc hd).hasStrictDerivAt
rwa [ContinuousLinearMap.add_apply, ContinuousLinearMap.smul_apply,
ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.smulRight_apply,
ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply, one_smul, one_smul,
add_comm] at this... | 0.71875 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe v₁ v₂ v₃ u₁ u₂ u₃
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 138 | 141 | theorem congrArg_mpr_hom_right {X Y Z : C} (p : X ⟶ Y) (q : Z = Y) :
(congrArg (fun W : C => X ⟶ W) q).mpr p = p ≫ eqToHom q.symm := by |
cases q
simp
| 0.71875 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 218 | 221 | theorem ite_sub_ite {α} [Sub α] (P : Prop) [Decidable P] (a b c d : α) :
((if P then a else b) - if P then c else d) = if P then a - c else b - d := by |
split
repeat rfl
| 0.71875 |
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.prod from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal ... | Mathlib/Analysis/Calculus/FDeriv/Prod.lean | 400 | 403 | theorem hasStrictFDerivAt_pi' :
HasStrictFDerivAt Φ Φ' x ↔ ∀ i, HasStrictFDerivAt (fun x => Φ x i) ((proj i).comp Φ') x := by |
simp only [HasStrictFDerivAt, ContinuousLinearMap.coe_pi]
exact isLittleO_pi
| 0.71875 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Star
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section ProdDomain
variable [CommMonoid α] [TopologicalSpace α]
@[to_additive]
| Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | 33 | 35 | theorem hasProd_pi_single [DecidableEq β] (b : β) (a : α) : HasProd (Pi.mulSingle b a) a := by |
convert hasProd_ite_eq b a
simp [Pi.mulSingle_apply]
| 0.71875 |
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 50 | 52 | theorem equalizerSubobject_arrow :
(equalizerSubobjectIso f g).hom ≫ equalizer.ι f g = (equalizerSubobject f g).arrow := by |
simp [equalizerSubobjectIso]
| 0.71875 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {α : Type v} {β : Type w}
namespace Matrix
def col (w : m → α) : Matrix m Unit α :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 129 | 132 | theorem col_mulVec [Fintype n] [NonUnitalNonAssocSemiring α] (M : Matrix m n α) (v : n → α) :
Matrix.col (M *ᵥ v) = M * Matrix.col v := by |
ext
rfl
| 0.71875 |
import Mathlib.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
#align_import order.height from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe83530ae6949b"
open List hiding le_antisymm
open OrderDual
universe u v
variable {α β : Type*}
namespace Set
section LT
varia... | Mathlib/Order/Height.lean | 142 | 144 | theorem chainHeight_eq_zero_iff : s.chainHeight = 0 ↔ s = ∅ := by |
rw [← not_iff_not, ← Ne, ← ENat.one_le_iff_ne_zero, one_le_chainHeight_iff,
nonempty_iff_ne_empty]
| 0.71875 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F ... | Mathlib/MeasureTheory/Group/Integral.lean | 103 | 105 | theorem integral_eq_zero_of_mul_right_eq_neg [IsMulRightInvariant μ] (hf' : ∀ x, f (x * g) = -f x) :
∫ x, f x ∂μ = 0 := by |
simp_rw [← self_eq_neg ℝ E, ← integral_neg, ← hf', integral_mul_right_eq_self]
| 0.71875 |
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 | 119 | 121 | theorem mul_charP_coeff_succ [CharP R p] (x : 𝕎 R) (i : ℕ) :
(x * p).coeff (i + 1) = x.coeff i ^ p := by |
rw [← frobenius_verschiebung, coeff_frobenius_charP, verschiebung_coeff_succ]
| 0.71875 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
#align_import measure_theory.measure.open_pos from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal MeasureTheory
open Set Function Filter
namespace Measur... | Mathlib/MeasureTheory/Measure/OpenPos.lean | 102 | 105 | theorem _root_.IsClosed.measure_eq_univ_iff_eq [OpensMeasurableSpace X] [IsFiniteMeasure μ]
(hF : IsClosed F) :
μ F = μ univ ↔ F = univ := by |
rw [← ae_eq_univ_iff_measure_eq hF.measurableSet.nullMeasurableSet, hF.ae_eq_univ_iff_eq]
| 0.71875 |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {α β R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 76 | 79 | theorem IsDiag.map [Zero α] [Zero β] {A : Matrix n n α} (ha : A.IsDiag) {f : α → β} (hf : f 0 = 0) :
(A.map f).IsDiag := by |
intro i j h
simp [ha h, hf]
| 0.71875 |
import Mathlib.Data.Nat.Bits
import Mathlib.Data.Nat.Pairing
#align_import logic.equiv.nat from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Nat Function
namespace Equiv
variable {α : Type*}
@[simps]
def boolProdNatEquivNat : Bool × ℕ ≃ ℕ where
toFun := uncurry bit
invFun... | Mathlib/Logic/Equiv/Nat.lean | 48 | 49 | theorem natSumNatEquivNat_apply : ⇑natSumNatEquivNat = Sum.elim bit0 bit1 := by |
ext (x | x) <;> rfl
| 0.71875 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 80 | 80 | theorem expand_zero (f : R[X]) : expand R 0 f = C (eval 1 f) := by | simp [expand]
| 0.71875 |
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 | 219 | 221 | theorem gcd_ndinsert (a : α) (s : Multiset α) : (ndinsert a s).gcd = GCDMonoid.gcd a s.gcd := by |
rw [← gcd_dedup, dedup_ext.2, gcd_dedup, gcd_cons]
simp
| 0.71875 |
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 | 72 | 74 | theorem dslope_sub_smul_of_ne (f : 𝕜 → E) (h : b ≠ a) :
dslope (fun x => (x - a) • f x) a b = f b := by |
rw [dslope_of_ne _ h, slope_sub_smul _ h.symm]
| 0.71875 |
import Mathlib.Init.Function
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Inhabit
#align_import data.prod.basic from "leanprover-community/mathlib"@"d07245fd37786daa997af4f1a73a49fa3b748408"
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
@[simp]
theorem Prod.map_apply (f : α → γ) (g : β → δ... | Mathlib/Data/Prod/Basic.lean | 122 | 123 | theorem ext_iff {p q : α × β} : p = q ↔ p.1 = q.1 ∧ p.2 = q.2 := by |
rw [mk.inj_iff]
| 0.71875 |
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 | 84 | 85 | theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by |
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
| 0.71875 |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 52 | 55 | theorem Duplicate.mem_cons_self (h : x ∈+ x :: l) : x ∈ l := by |
cases' h with _ h _ _ h
· exact h
· exact h.mem
| 0.71875 |
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Order.Filter.Subsingleton
open Set
variable {α β γ δ : Type*} {l : Filter α} {f : α → β}
namespace Filter
def EventuallyConst (f : α → β) (l : Filter α) : Prop := (map f l).Subsingleton
theorem HasBasis.eventuallyConst_iff {ι : Sort*} {p : ι → Prop} {s : ι → S... | Mathlib/Order/Filter/EventuallyConst.lean | 145 | 147 | theorem mulIndicator_const_iff :
EventuallyConst (s.mulIndicator fun _ ↦ c) l ↔ c = 1 ∨ EventuallyConst s l := by |
rcases eq_or_ne c 1 with rfl | hc <;> simp [mulIndicator_const_iff_of_ne, *]
| 0.71875 |
import Mathlib.Topology.MetricSpace.PseudoMetric
#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
open Set Filter Bornology
open scoped NNReal Uniformity
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
variable [PseudoMetricS... | Mathlib/Topology/MetricSpace/Basic.lean | 87 | 88 | theorem dist_pos {x y : γ} : 0 < dist x y ↔ x ≠ y := by |
simpa only [not_le] using not_congr dist_le_zero
| 0.71875 |
import Mathlib.Geometry.Manifold.ContMDiff.Product
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
open Set ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[Norme... | Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.lean | 81 | 82 | theorem contMDiff_iff_contDiff {f : E → E'} : ContMDiff 𝓘(𝕜, E) 𝓘(𝕜, E') n f ↔ ContDiff 𝕜 n f := by |
rw [← contDiffOn_univ, ← contMDiffOn_univ, contMDiffOn_iff_contDiffOn]
| 0.71875 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 114 | 115 | theorem closure_setOf_re_lt (a : ℝ) : closure { z : ℂ | z.re < a } = { z | z.re ≤ a } := by |
simpa only [closure_Iio] using closure_preimage_re (Iio a)
| 0.71875 |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 63 | 64 | theorem unitors_equal : (λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom := by |
coherence
| 0.71875 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
section Image
variable {f : α → β} {s t : Set... | Mathlib/Data/Set/Image.lean | 263 | 263 | theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) := by | aesop
| 0.71875 |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 59 | 61 | theorem exp_log (hx : 0 < x) : exp (log x) = x := by |
rw [exp_log_eq_abs hx.ne']
exact abs_of_pos hx
| 0.71875 |
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 | 106 | 108 | theorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :
(∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by |
simp_rw [prod_univ_succ, cons_zero, cons_succ]
| 0.71875 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 64 | 65 | theorem apply_diag_ne [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i : V) :
¬A i i = 1 := by | simp [h.apply_diag i]
| 0.71875 |
import Mathlib.Data.Set.Finite
import Mathlib.Order.Partition.Finpartition
#align_import data.setoid.partition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
namespace Setoid
variable {α : Type*}
theorem eq_of_mem_eqv_class {c : Set (Set α)} (H : ∀ a, ∃! b ∈ c, a ∈ b) {x b b'}
... | Mathlib/Data/Setoid/Partition.lean | 67 | 71 | theorem classes_ker_subset_fiber_set {β : Type*} (f : α → β) :
(Setoid.ker f).classes ⊆ Set.range fun y => { x | f x = y } := by |
rintro s ⟨x, rfl⟩
rw [Set.mem_range]
exact ⟨f x, rfl⟩
| 0.71875 |
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