Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
open Finset Submodule FiniteDimensional
variable (𝕜 : Type*) {E : Type*} [RCLike �... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 321 | 323 | theorem span_gramSchmidtNormed_range (f : ι → E) :
span 𝕜 (range (gramSchmidtNormed 𝕜 f)) = span 𝕜 (range (gramSchmidt 𝕜 f)) := by |
simpa only [image_univ.symm] using span_gramSchmidtNormed f univ
|
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib.Order.PiLex
import Mathlib.Order.Interval.Set.Basic
#align_import data.fin.tuple.basic from "leanprover-community/mathlib"@"ef997baa41b5c428be3fb50089a7139bf4ee886b"
assert_not_exists MonoidWithZero
un... | Mathlib/Data/Fin/Tuple/Basic.lean | 128 | 136 | theorem update_cons_zero : update (cons x p) 0 z = cons z p := by |
ext j
by_cases h : j = 0
· rw [h]
simp
· simp only [h, update_noteq, Ne, not_false_iff]
let j' := pred j h
have : j'.succ = j := succ_pred j h
rw [← this, cons_succ, cons_succ]
|
import Mathlib.Combinatorics.SimpleGraph.Regularity.Bound
import Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise
import Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform
#align_import combinatorics.simple_graph.regularity.chunk from "leanprover-community/mathlib"@"bf7ef0e83e5b7e6c1169e97f055e58a2e4e9d52d... | Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean | 473 | 521 | theorem edgeDensity_chunk_not_uniform [Nonempty α] (hPα : P.parts.card * 16 ^ P.parts.card ≤ card α)
(hPε : ↑100 ≤ ↑4 ^ P.parts.card * ε ^ 5) (hε₁ : ε ≤ 1) {hU : U ∈ P.parts} {hV : V ∈ P.parts}
(hUVne : U ≠ V) (hUV : ¬G.IsUniform ε U V) :
(G.edgeDensity U V : ℝ) ^ 2 - ε ^ 5 / ↑25 + ε ^ 4 / ↑3 ≤
(∑ ab ∈ ... |
apply add_le_add_left
have Ul : 4 / 5 * ε ≤ (star hP G ε hU V).card / _ :=
eps_le_card_star_div hPα hPε hε₁ hU hV hUVne hUV
have Vl : 4 / 5 * ε ≤ (star hP G ε hV U).card / _ :=
eps_le_card_star_div hPα hPε hε₁ hV hU hUVne.symm fun h => hUV h.symm
rw [show (16 : ℝ) = ↑4 ^ 2 by no... |
import Mathlib.Order.Monotone.Odd
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
#align_import analysis.special_functions.trigonometric.deriv from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
open s... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | 134 | 138 | theorem hasStrictDerivAt_cosh (x : ℂ) : HasStrictDerivAt cosh (sinh x) x := by |
simp only [sinh, div_eq_mul_inv]
convert ((hasStrictDerivAt_exp x).add (hasStrictDerivAt_id x).neg.cexp).mul_const (2 : ℂ)⁻¹
using 1
rw [id, mul_neg_one, sub_eq_add_neg]
|
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 327 | 329 | theorem limitRecOn_succ {C} (o H₁ H₂ H₃) :
@limitRecOn C (succ o) H₁ H₂ H₃ = H₂ o (@limitRecOn C o H₁ H₂ H₃) := by |
simp_rw [limitRecOn, SuccOrder.limitRecOn_succ _ _ (not_isMax _)]
|
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
import Mathlib.CategoryTheory.Monoidal.Functorial
import Mathlib.CategoryTheory.Monoidal.Types.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.CategoryTheory.Linear.LinearFunctor
#align_import algebra.category.Module.adjunctions from "leanpr... | Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean | 152 | 179 | theorem associativity (X Y Z : Type u) :
((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).obj (α_ X Y Z).hom =
(α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
(𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom := by |
-- Porting note (#11041): broken ext
apply TensorProduct.ext
apply TensorProduct.ext
apply Finsupp.lhom_ext'
intro x
apply LinearMap.ext_ring
apply Finsupp.lhom_ext'
intro y
apply LinearMap.ext_ring
apply Finsupp.lhom_ext'
intro z
apply LinearMap.ext_ring
apply Finsupp.ext
intro a
-- Port... |
import Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace
#align_import geometry.manifold.bump_function from "leanprover-community/mathlib"@"b018406ad2f2a73223a3a9e198ccae61e6f05318"
universe uE uF uH uM
va... | Mathlib/Geometry/Manifold/BumpFunction.lean | 290 | 298 | theorem nhds_basis_tsupport :
(𝓝 c).HasBasis (fun _ : SmoothBumpFunction I c => True) fun f => tsupport f := by |
have :
(𝓝 c).HasBasis (fun _ : SmoothBumpFunction I c => True) fun f =>
(extChartAt I c).symm '' (closedBall (extChartAt I c c) f.rOut ∩ range I) := by
rw [← map_extChartAt_symm_nhdsWithin_range I c]
exact nhdsWithin_range_basis.map _
exact this.to_hasBasis' (fun f _ => ⟨f, trivial, f.tsupport_s... |
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 | 117 | 119 | theorem comp_mul_left (x y : α) : (x * ·) ∘ (y * ·) = (x * y * ·) := by |
ext z
simp [mul_assoc]
|
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Defs
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
variable {α : Type*}
open Opposite
namespace Set
protected def op (s : Set α) : Set αᵒᵖ :=
unop ⁻¹' s
#align set.op Set.op
protected def u... | Mathlib/Data/Set/Opposite.lean | 100 | 104 | theorem singleton_unop_op (x : αᵒᵖ) : ({unop x} : Set α).op = {x} := by |
ext
constructor
· apply unop_injective
· apply op_injective
|
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
protected theorem div_eq_inv_mul : a / b = b⁻¹ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 137 | 138 | theorem inv_lt_top {x : ℝ≥0∞} : x⁻¹ < ∞ ↔ 0 < x := by |
simp only [lt_top_iff_ne_top, inv_ne_top, pos_iff_ne_zero]
|
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 | 106 | 113 | theorem evariance_eq_lintegral_ofReal (X : Ω → ℝ) (μ : Measure Ω) :
evariance X μ = ∫⁻ ω, ENNReal.ofReal ((X ω - μ[X]) ^ 2) ∂μ := by |
rw [evariance]
congr
ext1 ω
rw [pow_two, ← ENNReal.coe_mul, ← nnnorm_mul, ← pow_two]
congr
exact (Real.toNNReal_eq_nnnorm_of_nonneg <| sq_nonneg _).symm
|
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.UpperLower.Basic
#align_import algebra.order.upper_lower from "leanprover-community/mathlib"@"c0c52abb75074ed8b73a948341f50521fbf43b4c"
open Function Set
open Pointw... | Mathlib/Algebra/Order/UpperLower.lean | 56 | 58 | theorem Set.OrdConnected.smul (hs : s.OrdConnected) : (a • s).OrdConnected := by |
rw [← hs.upperClosure_inter_lowerClosure, smul_set_inter]
exact (upperClosure _).upper.smul.ordConnected.inter (lowerClosure _).lower.smul.ordConnected
|
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import analysis.special_functions.pow.complex from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open scoped Classical
open Real Topology Filter ComplexConjugate Finset Set
namespace Complex
noncomputable def cpow (x y : ℂ) ... | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 55 | 55 | theorem zero_cpow {x : ℂ} (h : x ≠ 0) : (0 : ℂ) ^ x = 0 := by | simp [cpow_def, *]
|
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Tactic.Common
#align_import data.nat.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
namespace Nat
variable {α : Type*}
@[simp]
theorem cast_div [DivisionSemiring α] {m n : ℕ} (n_... | Mathlib/Data/Nat/Cast/Field.lean | 76 | 79 | theorem one_div_lt_one_div {n m : ℕ} (h : n < m) : 1 / ((m : α) + 1) < 1 / ((n : α) + 1) := by |
refine one_div_lt_one_div_of_lt ?_ ?_
· exact Nat.cast_add_one_pos _
· simpa
|
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Algebra.Lie.Quotient
#align_import algebra.lie.normalizer from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102"
variable {R L M M' : Type*}
variable [CommRing R] [LieRing L] [LieAlgebra R L]
varia... | Mathlib/Algebra/Lie/Normalizer.lean | 82 | 83 | theorem comap_normalizer (f : M' →ₗ⁅R,L⁆ M) : N.normalizer.comap f = (N.comap f).normalizer := by |
ext; simp
|
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
noncomputable section
open Set Filter
universe u v w x
def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : ∅ ∈... | Mathlib/Topology/Basic.lean | 502 | 504 | theorem Set.Finite.closure_biUnion {ι : Type*} {s : Set ι} (hs : s.Finite) (f : ι → Set X) :
closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) := by |
simp [closure_eq_compl_interior_compl, hs.interior_biInter]
|
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
#align_import analysis.special_functions.trigonometric.complex from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
noncomputable secti... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 215 | 218 | theorem sin_surjective : Function.Surjective sin := by |
intro x
rcases cos_surjective x with ⟨z, rfl⟩
exact ⟨z + π / 2, sin_add_pi_div_two z⟩
|
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.Tactic.ApplyFun
import Mathlib.Tactic.CategoryTheory.Elementwise
#align_import category_theory.subobject.basic from "leanprover-community/mathlib"@"70fd9563a21e7b... | Mathlib/CategoryTheory/Subobject/Basic.lean | 431 | 436 | theorem ofMkLEMk_comp_ofMkLE {B A₁ A₂ : C} (f : A₁ ⟶ B) [Mono f] (g : A₂ ⟶ B) [Mono g]
(X : Subobject B) (h₁ : mk f ≤ mk g) (h₂ : mk g ≤ X) :
ofMkLEMk f g h₁ ≫ ofMkLE g X h₂ = ofMkLE f X (h₁.trans h₂) := by |
simp only [ofMkLE, ofLEMk, ofLE, ofMkLEMk, ← Functor.map_comp underlying,
assoc, Iso.hom_inv_id_assoc]
congr 1
|
import Mathlib.Init.Algebra.Classes
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Order.BoundedOrder
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Lift
import Mathlib.Data.Option.Basic
#align_import order.with_bot from "leanprover-community/mathlib"@"0111834459f5d7400215223ea95ae38a1265a907"
variabl... | Mathlib/Order/WithBot.lean | 266 | 269 | theorem le_unbot_iff {a : α} {b : WithBot α} (h : b ≠ ⊥) :
a ≤ unbot b h ↔ (a : WithBot α) ≤ b := by |
match b, h with
| some _, _ => simp only [unbot_coe, coe_le_coe]
|
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 | 64 | 66 | theorem exp_log_of_neg (hx : x < 0) : exp (log x) = -x := by |
rw [exp_log_eq_abs (ne_of_lt hx)]
exact abs_of_neg hx
|
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import algebra.category.Ring.colimits from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v
open CategoryTheory
open CategoryTheory.Limits
-- [ROBOT VOICE]:
-- You should prete... | Mathlib/Algebra/Category/Ring/Colimits.lean | 253 | 255 | theorem cocone_naturality_components (j j' : J) (f : j ⟶ j') (x : F.obj j) :
(coconeMorphism F j') (F.map f x) = (coconeMorphism F j) x := by |
rw [← cocone_naturality F f, comp_apply]
|
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 98 | 101 | theorem goldConj_sq : ψ ^ 2 = ψ + 1 := by |
rw [goldenConj, ← sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
|
import Mathlib.Order.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
import Mathlib.Topology.Order.LeftRight
import Mathlib.Topology.Order.OrderClosed
#align_import topology.order.basic from "leanprover-community/mathlib"@"3efd324a3a31eaa40c9d5bfc669c4fafe... | Mathlib/Topology/Order/Basic.lean | 169 | 171 | theorem nhds_order_unbounded {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) :
𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by |
simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]; rfl
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin_root from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open scoped Polynomial
open Polynomial
noncomputable sec... | Mathlib/RingTheory/IsAdjoinRoot.lean | 186 | 188 | theorem ext_map (h h' : IsAdjoinRoot S f) (eq : ∀ x, h.map x = h'.map x) : h = h' := by |
cases h; cases h'; congr
exact RingHom.ext eq
|
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