Context stringlengths 57 92.3k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
import Mathlib.Topology.LocallyConstant.Algebra
import Mathlib.Init.Data.Bool.Lemmas
universe u
namespace Profinite
namespace NobelingProof
variable {I : Ty... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,513 | 1,540 | theorem Products.max_eq_eval [Inhabited I] (l : Products I) (hl : l.val ≠ [])
(hlh : l.val.head! = term I ho) :
Linear_CC' C hsC ho (l.eval C) = l.Tail.eval (C' C ho) := by |
have hlc : ((term I ho) :: l.Tail.val).Chain' (·>·) := by
rw [← max_eq_o_cons_tail ho l hl hlh]; exact l.prop
rw [max_eq_o_cons_tail' ho l hl hlh hlc, Products.evalCons]
ext x
simp only [Linear_CC', Linear_CC'₁, LocallyConstant.comapₗ, Linear_CC'₀, Subtype.coe_eta,
LinearMap.sub_apply, LinearMap.coe_mk... |
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
@[ext]
structure YoungDiagram where
cells : Finset (ℕ × ℕ)
isLowerSet : IsLowerSet (cel... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 506 | 509 | theorem ofRowLens_to_rowLens_eq_self {μ : YoungDiagram} : ofRowLens _ (rowLens_sorted μ) = μ := by |
ext ⟨i, j⟩
simp only [mem_cells, mem_ofRowLens, length_rowLens, get_rowLens]
simpa [← mem_iff_lt_colLen, mem_iff_lt_rowLen] using j.zero_le.trans_lt
|
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 369 | 371 | theorem apply_pow_apply_eq_iff (f : Perm α) (n : ℕ) {x : α} :
f ((f ^ n) x) = (f ^ n) x ↔ f x = x := by |
rw [← mul_apply, Commute.self_pow f, mul_apply, apply_eq_iff_eq]
|
import Mathlib.Probability.Kernel.Basic
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
#align_import probability.kernel.measurable_integral from "leanprover-community/mathlib"@"28b2a92f2996d28e580450863c130955de0ed398"
open MeasureTheory Probabilit... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 257 | 302 | theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂κ x := by |
classical
by_cases hE : CompleteSpace E; swap
· simp [integral, hE, stronglyMeasurable_const]
borelize E
haveI : TopologicalSpace.SeparableSpace (range (uncurry f) ∪ {0} : Set E) :=
hf.separableSpace_range_union_singleton
let s : ℕ → SimpleFunc (α × β) E :=
SimpleFunc.approxOn _ hf.measurable (rang... |
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.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.NumberTheory.Bernoulli
#align_import number_theory.bernoulli_polynomials from "leanprover-community/mathlib"@"ca3d21f7f4fd613c2a3c54ac7871163e1e5ecb3a"
noncomputable section... | Mathlib/NumberTheory/BernoulliPolynomials.lean | 188 | 211 | theorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :
(bernoulli n).eval (1 + x) = (bernoulli n).eval x + n * x ^ (n - 1) := by |
refine Nat.strong_induction_on n fun d hd => ?_
have nz : ((d.succ : ℕ) : ℚ) ≠ 0 := by
norm_cast
apply (mul_right_inj' nz).1
rw [← smul_eq_mul, ← eval_smul, bernoulli_eq_sub_sum, mul_add, ← smul_eq_mul, ← eval_smul,
bernoulli_eq_sub_sum, eval_sub, eval_finset_sum]
conv_lhs =>
congr
· skip
... |
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.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
variable {α β : Type*}
namespace Set
section Einfsep
open ENNReal
open Function
noncomputable def einfsep [EDist α] (s : Set α) : ℝ≥0∞ :=
⨅ (x... | Mathlib/Topology/MetricSpace/Infsep.lean | 257 | 260 | theorem Nontrivial.einfsep_ne_top (hs : s.Nontrivial) : s.einfsep ≠ ∞ := by |
contrapose! hs
rw [not_nontrivial_iff]
exact subsingleton_of_einfsep_eq_top hs
|
import Mathlib.Topology.FiberBundle.Trivialization
import Mathlib.Topology.Order.LeftRightNhds
#align_import topology.fiber_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
variable {ι B F X : Type*} [TopologicalSpace X]
open TopologicalSpace Filter Set Bundle Topology
... | Mathlib/Topology/FiberBundle/Basic.lean | 318 | 381 | theorem FiberBundle.exists_trivialization_Icc_subset [ConditionallyCompleteLinearOrder B]
[OrderTopology B] [FiberBundle F E] (a b : B) :
∃ e : Trivialization F (π F E), Icc a b ⊆ e.baseSet := by |
obtain ⟨ea, hea⟩ : ∃ ea : Trivialization F (π F E), a ∈ ea.baseSet :=
⟨trivializationAt F E a, mem_baseSet_trivializationAt F E a⟩
-- If `a < b`, then `[a, b] = ∅`, and the statement is trivial
cases' lt_or_le b a with hab hab
· exact ⟨ea, by simp [*]⟩
/- Let `s` be the set of points `x ∈ [a, b]` such th... |
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.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
theorem isCobounde... | Mathlib/Order/LiminfLimsup.lean | 1,078 | 1,082 | theorem SupHom.apply_blimsup_le [CompleteLattice γ] (g : sSupHom α γ) :
g (blimsup u f p) ≤ blimsup (g ∘ u) f p := by |
simp only [blimsup_eq_iInf_biSup, Function.comp]
refine ((OrderHomClass.mono g).map_iInf₂_le _).trans ?_
simp only [_root_.map_iSup, le_refl]
|
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.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
#align_import logic.equiv.local_equiv from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Lean Meta Elab Tactic
def mfld_cfg : Simps.Config where
attrs :=... | Mathlib/Logic/Equiv/PartialEquiv.lean | 639 | 641 | theorem refl_restr_target (s : Set α) : ((PartialEquiv.refl α).restr s).target = s := by |
change univ ∩ id ⁻¹' s = s
simp
|
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.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace EN... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,629 | 1,648 | theorem hasSum_integral_measure {ι} {m : MeasurableSpace α} {f : α → G} {μ : ι → Measure α}
(hf : Integrable f (Measure.sum μ)) :
HasSum (fun i => ∫ a, f a ∂μ i) (∫ a, f a ∂Measure.sum μ) := by |
have hfi : ∀ i, Integrable f (μ i) := fun i => hf.mono_measure (Measure.le_sum _ _)
simp only [HasSum, ← integral_finset_sum_measure fun i _ => hfi i]
refine Metric.nhds_basis_ball.tendsto_right_iff.mpr fun ε ε0 => ?_
lift ε to ℝ≥0 using ε0.le
have hf_lt : (∫⁻ x, ‖f x‖₊ ∂Measure.sum μ) < ∞ := hf.2
have hme... |
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.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 582 | 586 | theorem norm_le_of_forall_le {f : lp E ∞} {C : ℝ} (hC : 0 ≤ C) (hCf : ∀ i, ‖f i‖ ≤ C) :
‖f‖ ≤ C := by |
cases isEmpty_or_nonempty α
· simpa [eq_zero' f] using hC
· exact norm_le_of_forall_le' C hCf
|
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.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Symmetric
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.Algebra.DirectSum.Decomposition
#align_import analysis.inner_product_space.proje... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 922 | 945 | theorem orthogonalProjection_tendsto_closure_iSup [CompleteSpace E] {ι : Type*} [SemilatticeSup ι]
(U : ι → Submodule 𝕜 E) [∀ i, CompleteSpace (U i)] (hU : Monotone U) (x : E) :
Filter.Tendsto (fun i => (orthogonalProjection (U i) x : E)) atTop
(𝓝 (orthogonalProjection (⨆ i, U i).topologicalClosure x : ... |
cases isEmpty_or_nonempty ι
· exact tendsto_of_isEmpty
let y := (orthogonalProjection (⨆ i, U i).topologicalClosure x : E)
have proj_x : ∀ i, orthogonalProjection (U i) x = orthogonalProjection (U i) y := fun i =>
(orthogonalProjection_orthogonalProjection_of_le
((le_iSup U i).trans (iSup U).le_top... |
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.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,079 | 1,082 | theorem norm_inner_le_norm (x y : E) : ‖⟪x, y⟫‖ ≤ ‖x‖ * ‖y‖ := by |
rw [norm_eq_sqrt_inner (𝕜 := 𝕜) x, norm_eq_sqrt_inner (𝕜 := 𝕜) y]
letI : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore
exact InnerProductSpace.Core.norm_inner_le_norm x y
|
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 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 | 1,036 | 1,036 | theorem eq_mul_of_div_eq (h : a / c = b) : a = b * c := by | simp [← h]
|
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
#align_import category_theory.limits.shapes.pullbacks from "leanprover-community/mathlib"@"7316286ff2942aa14e540add9058c6b0aa1c8070"
noncomputable section
open CategoryTheory
universe w v₁ v₂ v u... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 2,219 | 2,222 | theorem pullbackRightPullbackFstIso_inv_snd_fst :
(pullbackRightPullbackFstIso f g f').inv ≫ pullback.snd ≫ pullback.fst = pullback.fst ≫ f' := by |
rw [← pullback.condition]
exact pullbackRightPullbackFstIso_inv_fst_assoc _ _ _ _
|
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.DiscreteSubset
import Mathlib.Tactic.Abel... | Mathlib/Topology/Algebra/UniformGroup.lean | 378 | 386 | theorem uniformContinuous_of_tendsto_one {hom : Type*} [UniformSpace β] [Group β] [UniformGroup β]
[FunLike hom α β] [MonoidHomClass hom α β] {f : hom} (h : Tendsto f (𝓝 1) (𝓝 1)) :
UniformContinuous f := by |
have :
((fun x : β × β => x.2 / x.1) ∘ fun x : α × α => (f x.1, f x.2)) = fun x : α × α =>
f (x.2 / x.1) := by ext; simp only [Function.comp_apply, map_div]
rw [UniformContinuous, uniformity_eq_comap_nhds_one α, uniformity_eq_comap_nhds_one β,
tendsto_comap_iff, this]
exact Tendsto.comp h tendsto_c... |
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.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemmas
import Mathlib.Data.Set.Subsingleton
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Order.GaloisConnection
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith... | Mathlib/Algebra/Order/Floor.lean | 1,143 | 1,154 | theorem fract_div_natCast_eq_div_natCast_mod {m n : ℕ} : fract ((m : k) / n) = ↑(m % n) / n := by |
rcases n.eq_zero_or_pos with (rfl | hn)
· simp
have hn' : 0 < (n : k) := by
norm_cast
refine fract_eq_iff.mpr ⟨?_, ?_, m / n, ?_⟩
· positivity
· simpa only [div_lt_one hn', Nat.cast_lt] using m.mod_lt hn
· rw [sub_eq_iff_eq_add', ← mul_right_inj' hn'.ne', mul_div_cancel₀ _ hn'.ne', mul_add,
mul... |
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subring.Units
#align_import linear_algebra.ray from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
noncomputable section
... | Mathlib/LinearAlgebra/Ray.lean | 701 | 705 | theorem exists_pos_left_iff_sameRay (hx : x ≠ 0) (hy : y ≠ 0) :
(∃ r : R, 0 < r ∧ r • x = y) ↔ SameRay R x y := by |
refine ⟨fun h => ?_, fun h => h.exists_pos_left hx hy⟩
rcases h with ⟨r, hr, rfl⟩
exact SameRay.sameRay_pos_smul_right x hr
|
import Mathlib.MeasureTheory.Integral.IntegrableOn
#align_import measure_theory.function.locally_integrable from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49de8c0d677b1b"
open MeasureTheory MeasureTheory.Measure Set Function TopologicalSpace Bornology
open scoped Topology Interval ENNReal
variabl... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 341 | 343 | theorem locallyIntegrable_finset_sum {ι} (s : Finset ι) {f : ι → X → E}
(hf : ∀ i ∈ s, LocallyIntegrable (f i) μ) : LocallyIntegrable (fun a ↦ ∑ i ∈ s, f i a) μ := by |
simpa only [← Finset.sum_apply] using locallyIntegrable_finset_sum' s hf
|
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.Data.Sum.Order
import Mathlib.Order.InitialSeg
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
#align_import set_theory.ordinal.basic from "leanprover-community/mathlib"@"8ea5598db6caeddde6cb734aa179cc2408dbd345"
assert_not_exists Module
assert_not_exists Field
noncomputabl... | Mathlib/SetTheory/Ordinal/Basic.lean | 1,641 | 1,641 | theorem type_fin (n : ℕ) : @type (Fin n) (· < ·) _ = n := by | simp
|
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.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Set.Image
import Mathlib.Order.Atoms
import Mathlib.Tactic.ApplyFun
#align_import g... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 931 | 933 | theorem bot_or_exists_ne_one (H : Subgroup G) : H = ⊥ ∨ ∃ x ∈ H, x ≠ (1 : G) := by |
convert H.bot_or_nontrivial
rw [nontrivial_iff_exists_ne_one]
|
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists AddCommMonoidWithOne
assert_not_exists MonoidWithZero... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 1,071 | 1,074 | theorem prod_subtype_of_mem (f : α → β) {p : α → Prop} [DecidablePred p] (h : ∀ x ∈ s, p x) :
∏ x ∈ s.subtype p, f x = ∏ x ∈ s, f x := by |
rw [prod_subtype_eq_prod_filter, filter_true_of_mem]
simpa using h
|
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
|
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.ByContra
import Mathlib.Util.Delaborators
#align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
... | Mathlib/Data/Set/Basic.lean | 2,334 | 2,335 | theorem ite_inter (t s₁ s₂ s : Set α) : t.ite (s₁ ∩ s) (s₂ ∩ s) = t.ite s₁ s₂ ∩ s := by |
rw [ite_inter_inter, ite_same]
|
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Geometry.Manifold.Instances.Real
import Mathlib.Geometry.Manifold.PartitionOfUnity
#align_import geometry.manifold.whitney_embedding from "leanprover-community/mathlib"@"86c29aefdba50b3f33e86e52e3b2f51a0d8f0282"
universe uι uE uH uM
variable {ι : Type u... | Mathlib/Geometry/Manifold/WhitneyEmbedding.lean | 68 | 75 | theorem embeddingPiTangent_injOn : InjOn f.embeddingPiTangent s := by |
intro x hx y _ h
simp only [embeddingPiTangent_coe, funext_iff] at h
obtain ⟨h₁, h₂⟩ := Prod.mk.inj_iff.1 (h (f.ind x hx))
rw [f.apply_ind x hx] at h₂
rw [← h₂, f.apply_ind x hx, one_smul, one_smul] at h₁
have := f.mem_extChartAt_source_of_eq_one h₂.symm
exact (extChartAt I (f.c _)).injOn (f.mem_extChart... |
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 | 1,491 | 1,495 | theorem sigmaCurry_smul [Monoid γ] [∀ i j, AddMonoid (δ i j)] [∀ i j, DistribMulAction γ (δ i j)]
(r : γ) (f : Π₀ (i : Σ _, _), δ i.1 i.2) :
sigmaCurry (r • f) = r • sigmaCurry f := by |
ext (i j)
rfl
|
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
#align_import data.typevec from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
universe u v w
@[pp_with_univ]
def TypeVec (n : ℕ) :=
Fin2 n → Type*
#align typevec TypeVec
instance {n} : Inh... | Mathlib/Data/TypeVec.lean | 530 | 534 | theorem prod_fst_mk {α β : TypeVec n} (i : Fin2 n) (a : α i) (b : β i) :
TypeVec.prod.fst i (prod.mk i a b) = a := by |
induction' i with _ _ _ i_ih
· simp_all only [prod.fst, prod.mk]
apply i_ih
|
import Mathlib.Algebra.Group.Embedding
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Finset.Union
#align_import data.finset.image from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists OrderedCommMonoid
assert_not_exists MonoidWithZero
assert_not_exists MulA... | Mathlib/Data/Finset/Image.lean | 327 | 329 | theorem range_add_one' (n : ℕ) :
range (n + 1) = insert 0 ((range n).map ⟨fun i => i + 1, fun i j => by simp⟩) := by |
ext (⟨⟩ | ⟨n⟩) <;> simp [Nat.succ_eq_add_one, Nat.zero_lt_succ n]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 233 | 233 | theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by | simp [arg, hx]
|
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 902 | 918 | theorem Pi.isPWO {α : ι → Type*} [∀ i, LinearOrder (α i)] [∀ i, IsWellOrder (α i) (· < ·)]
[Finite ι] (s : Set (∀ i, α i)) : s.IsPWO := by |
cases nonempty_fintype ι
suffices ∀ (s : Finset ι) (f : ℕ → ∀ s, α s),
∃ g : ℕ ↪o ℕ, ∀ ⦃a b : ℕ⦄, a ≤ b → ∀ x, x ∈ s → (f ∘ g) a x ≤ (f ∘ g) b x by
refine isPWO_iff_exists_monotone_subseq.2 fun f _ => ?_
simpa only [Finset.mem_univ, true_imp_iff] using this Finset.univ f
refine Finset.cons_induction ... |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 296 | 300 | theorem normAtPlace_apply (w : InfinitePlace K) (x : K) :
normAtPlace w (mixedEmbedding K x) = w x := by |
simp_rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, mixedEmbedding,
RingHom.prod_apply, Pi.ringHom_apply, norm_embedding_of_isReal, norm_embedding_eq, dite_eq_ite,
ite_id]
|
import Mathlib.Algebra.Group.Hom.End
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.module.basic from "leanprover-community/mathlib"@"30413fc89f202a090a54d78e540963ed3de0056e"
assert_n... | Mathlib/Algebra/Module/Defs.lean | 434 | 437 | theorem map_natCast_smul [AddCommMonoid M] [AddCommMonoid M₂] {F : Type*} [FunLike F M M₂]
[AddMonoidHomClass F M M₂] (f : F) (R S : Type*) [Semiring R] [Semiring S] [Module R M]
[Module S M₂] (x : ℕ) (a : M) : f ((x : R) • a) = (x : S) • f a := by |
simp only [← nsmul_eq_smul_cast, AddMonoidHom.map_nsmul, map_nsmul]
|
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
import Mathlib.LinearAlgebra.Dimension.DivisionRing
#align_import linear_algebra.finite_dimensional from "leanprover-community/mathlib"@"e95e4f92c8f8da3c7f693c3ec948bcf9b6683f51"
universe u v v' w
open Cardina... | Mathlib/LinearAlgebra/FiniteDimensional.lean | 781 | 790 | theorem isUnit_iff_ker_eq_bot [FiniteDimensional K V] (f : V →ₗ[K] V) :
IsUnit f ↔ (LinearMap.ker f) = ⊥ := by |
constructor
· rintro ⟨u, rfl⟩
exact LinearMap.ker_eq_bot_of_inverse u.inv_mul
· intro h_inj
rw [ker_eq_bot] at h_inj
exact ⟨⟨f, (LinearEquiv.ofInjectiveEndo f h_inj).symm.toLinearMap,
LinearEquiv.ofInjectiveEndo_right_inv f h_inj, LinearEquiv.ofInjectiveEndo_left_inv f h_inj⟩,
rfl⟩
|
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.calculus.deriv.basic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical Topology Filter ENNReal ... | Mathlib/Analysis/Calculus/Deriv/Basic.lean | 792 | 795 | theorem norm_deriv_le_of_lip' {f : 𝕜 → F} {x₀ : 𝕜}
{C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ x in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖) :
‖deriv f x₀‖ ≤ C := by |
simpa [norm_deriv_eq_norm_fderiv] using norm_fderiv_le_of_lip' 𝕜 hC₀ hlip
|
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algebra.Ring.NegOnePow
#align_import analysis.special_functions.trigonometric.basic from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
open scoped Classical
open Top... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 885 | 895 | theorem cos_pi_div_three : cos (π / 3) = 1 / 2 := by |
have h₁ : (2 * cos (π / 3) - 1) ^ 2 * (2 * cos (π / 3) + 2) = 0 := by
have : cos (3 * (π / 3)) = cos π := by
congr 1
ring
linarith [cos_pi, cos_three_mul (π / 3)]
cases' mul_eq_zero.mp h₁ with h h
· linarith [pow_eq_zero h]
· have : cos π < cos (π / 3) := by
refine cos_lt_cos_of_nonne... |
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.calculus.fderiv.basic from "leanprover-community/mathlib"@"41bef4ae1254365bc190aee63b947674d2977f01"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open To... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 403 | 405 | theorem hasFDerivWithinAt_univ : HasFDerivWithinAt f f' univ x ↔ HasFDerivAt f f' x := by |
simp only [HasFDerivWithinAt, nhdsWithin_univ]
rfl
|
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 | 183 | 186 | theorem measure_mul_setLaverage (f : α → ℝ≥0∞) (h : μ s ≠ ∞) :
μ s * ⨍⁻ x in s, f x ∂μ = ∫⁻ x in s, f x ∂μ := by |
have := Fact.mk h.lt_top
rw [← measure_mul_laverage, restrict_apply_univ]
|
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Set.Image
#align_import data.fintype.card from "leanprover-community/mathlib"@"bf2428c9486c407ca38b5b3fb10b87dad0bc99fa"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Fu... | Mathlib/Data/Fintype/Card.lean | 315 | 322 | theorem Fintype.card_fin_lt_of_le {m n : ℕ} (h : m ≤ n) :
Fintype.card {i : Fin n // i < m} = m := by |
conv_rhs => rw [← Fintype.card_fin m]
apply Fintype.card_congr
exact { toFun := fun ⟨⟨i, _⟩, hi⟩ ↦ ⟨i, hi⟩
invFun := fun ⟨i, hi⟩ ↦ ⟨⟨i, lt_of_lt_of_le hi h⟩, hi⟩
left_inv := fun i ↦ rfl
right_inv := fun i ↦ rfl }
|
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 | 65 | 66 | theorem expand_monomial (r : R) : expand R p (monomial q r) = monomial (q * p) r := by |
simp_rw [← smul_X_eq_monomial, AlgHom.map_smul, AlgHom.map_pow, expand_X, mul_comm, pow_mul]
|
import Mathlib.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
noncomputable section
open scoped Classical
open scoped RealInnerProductSpace
namespace Affine
namespace Simplex
open Finset AffineSubspac... | Mathlib/Geometry/Euclidean/MongePoint.lean | 210 | 240 | theorem inner_mongePoint_vsub_face_centroid_vsub {n : ℕ} (s : Simplex ℝ P (n + 2))
{i₁ i₂ : Fin (n + 3)} :
⟪s.mongePoint -ᵥ ({i₁, i₂}ᶜ : Finset (Fin (n + 3))).centroid ℝ s.points,
s.points i₁ -ᵥ s.points i₂⟫ =
0 := by |
by_cases h : i₁ = i₂
· simp [h]
simp_rw [mongePoint_vsub_face_centroid_eq_weightedVSub_of_pointsWithCircumcenter s h,
point_eq_affineCombination_of_pointsWithCircumcenter, affineCombination_vsub]
have hs : ∑ i, (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0 := by
simp
rw [i... |
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Data.Nat.Factorization.PrimePow
#align_import data.nat.squarefree from "leanprover-community/mathlib"@"3c1368cac4abd5a5cbe44317ba7e87379d51ed88"
open Finset
theorem Squarefree.nodup_factors {n : ℕ} (hn : Squarefree n) : n.factors.Nodup :=
(Nat.squarefree_if... | Mathlib/Data/Nat/Squarefree.lean | 229 | 232 | theorem minSqFac_dvd {n d : ℕ} (h : n.minSqFac = some d) : d * d ∣ n := by |
have := minSqFac_has_prop n
rw [h] at this
exact this.2.1
|
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section S... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 105 | 118 | theorem snorm'_lt_top_of_snorm'_lt_top_of_exponent_le {p q : ℝ} [IsFiniteMeasure μ]
(hf : AEStronglyMeasurable f μ) (hfq_lt_top : snorm' f q μ < ∞) (hp_nonneg : 0 ≤ p)
(hpq : p ≤ q) : snorm' f p μ < ∞ := by |
rcases le_or_lt p 0 with hp_nonpos | hp_pos
· rw [le_antisymm hp_nonpos hp_nonneg]
simp
have hq_pos : 0 < q := lt_of_lt_of_le hp_pos hpq
calc
snorm' f p μ ≤ snorm' f q μ * μ Set.univ ^ (1 / p - 1 / q) :=
snorm'_le_snorm'_mul_rpow_measure_univ hp_pos hpq hf
_ < ∞ := by
rw [ENNReal.mul_lt... |
import Mathlib.Data.PNat.Defs
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Set.Basic
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Positive.Ring
import Mathlib.Order.Hom.Basic
#align_import data.pnat.basic from "leanprover-community/mathlib"@"172bf2812857f5e56938cc148b7a5... | Mathlib/Data/PNat/Basic.lean | 308 | 313 | theorem sub_coe (a b : ℕ+) : ((a - b : ℕ+) : ℕ) = ite (b < a) (a - b : ℕ) 1 := by |
change (toPNat' _ : ℕ) = ite _ _ _
split_ifs with h
· exact toPNat'_coe (tsub_pos_of_lt h)
· rw [tsub_eq_zero_iff_le.mpr (le_of_not_gt h : (a : ℕ) ≤ b)]
rfl
|
import Mathlib.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} {a : α}
namespace Finset
def nonMemberSubfamily (a : α) (𝒜 : ... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 56 | 57 | theorem mem_nonMemberSubfamily : s ∈ 𝒜.nonMemberSubfamily a ↔ s ∈ 𝒜 ∧ a ∉ s := by |
simp [nonMemberSubfamily]
|
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
noncomputable section
open scoped Classical
open ... | Mathlib/Geometry/Euclidean/Basic.lean | 622 | 627 | theorem reflection_mem_of_le_of_mem {s₁ s₂ : AffineSubspace ℝ P} [Nonempty s₁]
[HasOrthogonalProjection s₁.direction] (hle : s₁ ≤ s₂) {p : P} (hp : p ∈ s₂) :
reflection s₁ p ∈ s₂ := by |
rw [reflection_apply]
have ho : ↑(orthogonalProjection s₁ p) ∈ s₂ := hle (orthogonalProjection_mem p)
exact vadd_mem_of_mem_direction (vsub_mem_direction ho hp) ho
|
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.GroupWithZero.Canonical
import Mathlib.Order.Hom.Basic
#align_import algebra.order.hom.monoid from "leanprover-community/mathlib"@"3342d1b2178381196f818146ff79bc0e7ccd9e2d"
open Function
variable {F... | Mathlib/Algebra/Order/Hom/Monoid.lean | 216 | 221 | theorem strictMono_iff_map_pos : StrictMono (f : α → β) ↔ ∀ a, 0 < a → 0 < f a := by |
refine ⟨fun h a => ?_, fun h a b hl => ?_⟩
· rw [← map_zero f]
apply h
· rw [← sub_add_cancel b a, map_add f]
exact lt_add_of_pos_left _ (h _ <| sub_pos.2 hl)
|
import Mathlib.FieldTheory.Galois
#align_import field_theory.polynomial_galois_group from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Polynomial
open FiniteDimensional
namespace Polynomial
variable {F : Type*} [Field F] (p q : F[X]) (E : Type*) [... | Mathlib/FieldTheory/PolynomialGaloisGroup.lean | 322 | 336 | theorem mul_splits_in_splittingField_of_mul {p₁ q₁ p₂ q₂ : F[X]} (hq₁ : q₁ ≠ 0) (hq₂ : q₂ ≠ 0)
(h₁ : p₁.Splits (algebraMap F q₁.SplittingField))
(h₂ : p₂.Splits (algebraMap F q₂.SplittingField)) :
(p₁ * p₂).Splits (algebraMap F (q₁ * q₂).SplittingField) := by |
apply splits_mul
· rw [←
(SplittingField.lift q₁
(splits_of_splits_of_dvd (algebraMap F (q₁ * q₂).SplittingField) (mul_ne_zero hq₁ hq₂)
(SplittingField.splits _) (dvd_mul_right q₁ q₂))).comp_algebraMap]
exact splits_comp_of_splits _ _ h₁
· rw [←
(SplittingField.lift q₂
... |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 54 | 57 | theorem memℒp_two_iff_integrable_sq {f : α → ℝ} (hf : AEStronglyMeasurable f μ) :
Memℒp f 2 μ ↔ Integrable (fun x => f x ^ 2) μ := by |
convert memℒp_two_iff_integrable_sq_norm hf using 3
simp
|
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Submodule.RestrictScalars
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.LinearAlgebra.Basic
import Mathlib.Order.CompactlyGenerated.Basic
import Mathlib.Order.OmegaCompletePartialOrder
#align_... | Mathlib/LinearAlgebra/Span.lean | 458 | 461 | theorem sup_toAddSubmonoid : (p ⊔ p').toAddSubmonoid = p.toAddSubmonoid ⊔ p'.toAddSubmonoid := by |
ext x
rw [mem_toAddSubmonoid, mem_sup, AddSubmonoid.mem_sup]
rfl
|
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Order.Filter.ModEq
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.List.TFAE
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.specific_lim... | Mathlib/Analysis/SpecificLimits/Normed.lean | 566 | 568 | theorem geom_series_mul_one_add (x : R) (h : ‖x‖ < 1) :
(1 + x) * ∑' i : ℕ, x ^ i = 2 * ∑' i : ℕ, x ^ i - 1 := by |
rw [add_mul, one_mul, geom_series_mul_shift x h, geom_series_succ x h, two_mul, add_sub_assoc]
|
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.NumberTheory.Divisors
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Tactic.Zify
#align_import rin... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 688 | 691 | theorem eq_neg_one_of_two_right [NoZeroDivisors R] {ζ : R} (h : IsPrimitiveRoot ζ 2) : ζ = -1 := by |
apply (eq_or_eq_neg_of_sq_eq_sq ζ 1 _).resolve_left
· rw [← pow_one ζ]; apply h.pow_ne_one_of_pos_of_lt <;> decide
· simp only [h.pow_eq_one, one_pow]
|
import Mathlib.SetTheory.Game.State
#align_import set_theory.game.domineering from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
namespace SetTheory
namespace PGame
namespace Domineering
open Function
@[simps!]
def shiftUp : ℤ × ℤ ≃ ℤ × ℤ :=
(Equiv.refl ℤ).prodCongr (Equiv.addRig... | Mathlib/SetTheory/Game/Domineering.lean | 125 | 126 | theorem moveLeft_smaller {b : Board} {m : ℤ × ℤ} (h : m ∈ left b) :
Finset.card (moveLeft b m) / 2 < Finset.card b / 2 := by | simp [← moveLeft_card h, lt_add_one]
|
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 258 | 263 | theorem two_zsmul_oangle_of_vectorSpan_eq {p₁ p₂ p₃ p₄ p₅ p₆ : P}
(h₁₂₄₅ : vectorSpan ℝ ({p₁, p₂} : Set P) = vectorSpan ℝ ({p₄, p₅} : Set P))
(h₃₂₆₅ : vectorSpan ℝ ({p₃, p₂} : Set P) = vectorSpan ℝ ({p₆, p₅} : Set P)) :
(2 : ℤ) • ∡ p₁ p₂ p₃ = (2 : ℤ) • ∡ p₄ p₅ p₆ := by |
simp_rw [vectorSpan_pair] at h₁₂₄₅ h₃₂₆₅
exact o.two_zsmul_oangle_of_span_eq_of_span_eq h₁₂₄₅ h₃₂₆₅
|
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Order.Filter.Germ
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import measure_theory.function.ae_eq_fun from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a21598... | Mathlib/MeasureTheory/Function/AEEqFun.lean | 228 | 231 | theorem compQuasiMeasurePreserving_eq_mk (g : β →ₘ[ν] γ) (hf : QuasiMeasurePreserving f μ ν) :
g.compQuasiMeasurePreserving f hf =
mk (g ∘ f) (g.aestronglyMeasurable.comp_quasiMeasurePreserving hf) := by |
rw [← compQuasiMeasurePreserving_mk g.aestronglyMeasurable hf, mk_coeFn]
|
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.pi from "leanprover-community/mathlib"@"e4bc74cbaf429d706cb9140902f7ca6c431e75a4"
-- Porting note: Added, since dot nota... | Mathlib/Order/Interval/Set/Pi.lean | 148 | 151 | theorem image_update_Ioc (f : ∀ i, α i) (i : ι) (a b : α i) :
update f i '' Ioc a b = Ioc (update f i a) (update f i b) := by |
rw [← Icc_diff_left, ← Icc_diff_left, image_diff (update_injective _ _), image_singleton,
image_update_Icc]
|
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Instances.Irrational
#align_import number_theory.liouville.liouville_with from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Filter Metric Real Set
open sc... | Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | 166 | 167 | theorem mul_nat_iff (hn : n ≠ 0) : LiouvilleWith p (x * n) ↔ LiouvilleWith p x := by |
rw [← Rat.cast_natCast, mul_rat_iff (Nat.cast_ne_zero.2 hn)]
|
import Mathlib.Data.Setoid.Partition
import Mathlib.Topology.Separation
import Mathlib.Topology.LocallyConstant.Basic
#align_import topology.discrete_quotient from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function TopologicalSpace
variable {α X Y Z : Type*} [TopologicalS... | Mathlib/Topology/DiscreteQuotient.lean | 198 | 198 | theorem comap_mono {A B : DiscreteQuotient Y} (h : A ≤ B) : A.comap f ≤ B.comap f := by | tauto
|
import Mathlib.Probability.Kernel.Basic
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
#align_import probability.kernel.measurable_integral from "leanprover-community/mathlib"@"28b2a92f2996d28e580450863c130955de0ed398"
open MeasureTheory Probabilit... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 231 | 235 | theorem _root_.Measurable.set_lintegral_kernel {f : β → ℝ≥0∞} (hf : Measurable f) {s : Set β}
(hs : MeasurableSet s) : Measurable fun a => ∫⁻ b in s, f b ∂κ a := by |
-- Porting note: was term mode proof (`Function.comp` reducibility)
refine Measurable.set_lintegral_kernel_prod_right ?_ hs
convert hf.comp measurable_snd
|
import Mathlib.ModelTheory.Syntax
import Mathlib.ModelTheory.Semantics
import Mathlib.Algebra.Ring.Equiv
variable {α : Type*}
namespace FirstOrder
open FirstOrder
inductive ringFunc : ℕ → Type
| add : ringFunc 2
| mul : ringFunc 2
| neg : ringFunc 1
| zero : ringFunc 0
| one : ringFunc 0
deriving D... | Mathlib/ModelTheory/Algebra/Ring/Basic.lean | 180 | 182 | theorem realize_add (x y : ring.Term α) (v : α → R) :
Term.realize v (x + y) = Term.realize v x + Term.realize v y := by |
simp [add_def, funMap_add]
|
import Mathlib.Probability.Kernel.Disintegration.Basic
open MeasureTheory ProbabilityTheory MeasurableSpace
open scoped ENNReal
namespace ProbabilityTheory
variable {α β Ω : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
[MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω]
namespace MeasureTheory... | Mathlib/Probability/Kernel/Disintegration/Integral.lean | 277 | 281 | theorem Integrable.integral_norm_condKernel {f : α × Ω → F} (hf_int : Integrable f ρ) :
Integrable (fun x ↦ ∫ y, ‖f (x, y)‖ ∂ρ.condKernel x) ρ.fst := by |
have hf_ae : AEStronglyMeasurable f ρ := hf_int.1
rw [← hf_ae.ae_integrable_condKernel_iff] at hf_int
exact hf_int.2
|
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.ArithMult
#align_import number_theory.arithmetic_functi... | Mathlib/NumberTheory/ArithmeticFunction.lean | 898 | 905 | theorem zeta_mul_pow_eq_sigma {k : ℕ} : ζ * pow k = σ k := by |
ext
rw [sigma, zeta_mul_apply]
apply sum_congr rfl
intro x hx
rw [pow_apply, if_neg (not_and_of_not_right _ _)]
contrapose! hx
simp [hx]
|
import Mathlib.Data.Finset.Finsupp
import Mathlib.Data.Finsupp.Order
import Mathlib.Order.Interval.Finset.Basic
#align_import data.finsupp.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
noncomputable section
open Finset Finsupp Function
open scoped Classical
open Pointwis... | Mathlib/Data/Finsupp/Interval.lean | 150 | 151 | theorem card_Iio : (Iio f).card = (∏ i ∈ f.support, (Iic (f i)).card) - 1 := by |
rw [card_Iio_eq_card_Iic_sub_one, card_Iic]
|
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot
import Mathlib.Order.Hom.Basic
#align_import data.multiset.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
universe v
... | Mathlib/Data/Multiset/Basic.lean | 1,192 | 1,196 | theorem map_congr {f g : α → β} {s t : Multiset α} :
s = t → (∀ x ∈ t, f x = g x) → map f s = map g t := by |
rintro rfl h
induction s using Quot.inductionOn
exact congr_arg _ (List.map_congr h)
|
import Mathlib.Data.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Hom.Set
import Mathlib.Order.SetNotation
#align_import order.complete_lattice from "leanprover-community/mathlib"@"5709b0d8725255e76f47debca6400c07b5c2... | Mathlib/Order/CompleteLattice.lean | 1,564 | 1,565 | theorem iSup_option_elim (a : α) (f : β → α) : ⨆ o : Option β, o.elim a f = a ⊔ ⨆ b, f b := by |
simp [iSup_option]
|
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Measure.GiryMonad
#align_import probability.kernel.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open MeasureTheory
open scoped MeasureTheory ENNReal NNReal
namespace ProbabilityTheory
noncomputab... | Mathlib/Probability/Kernel/Basic.lean | 574 | 576 | theorem set_lintegral_restrict (κ : kernel α β) (hs : MeasurableSet s) (a : α) (f : β → ℝ≥0∞)
(t : Set β) : ∫⁻ b in t, f b ∂kernel.restrict κ hs a = ∫⁻ b in t ∩ s, f b ∂κ a := by |
rw [restrict_apply, Measure.restrict_restrict' hs]
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Measure... | Mathlib/Analysis/Convolution.lean | 859 | 884 | theorem convolution_tendsto_right {ι} {g : ι → G → E'} {l : Filter ι} {x₀ : G} {z₀ : E'}
{φ : ι → G → ℝ} {k : ι → G} (hnφ : ∀ᶠ i in l, ∀ x, 0 ≤ φ i x)
(hiφ : ∀ᶠ i in l, ∫ x, φ i x ∂μ = 1)
-- todo: we could weaken this to "the integral tends to 1"
(hφ : Tendsto (fun n => support (φ n)) l (𝓝 0).smallSets... |
simp_rw [tendsto_smallSets_iff] at hφ
rw [Metric.tendsto_nhds] at hcg ⊢
simp_rw [Metric.eventually_prod_nhds_iff] at hcg
intro ε hε
have h2ε : 0 < ε / 3 := div_pos hε (by norm_num)
obtain ⟨p, hp, δ, hδ, hgδ⟩ := hcg _ h2ε
dsimp only [uncurry] at hgδ
have h2k := hk.eventually (ball_mem_nhds x₀ <| half_po... |
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.MetricSpace.IsometricSMul
import Mathlib.Topology.Sequences
#align_import analysis.normed.group.basic from "leanprover-community/mat... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,979 | 1,980 | theorem _root_.Int.norm_cast_rat (m : ℤ) : ‖(m : ℚ)‖ = ‖m‖ := by |
rw [← Rat.norm_cast_real, ← Int.norm_cast_real]; congr 1
|
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Tactic.AdaptationNote
#align_import linear_algebra.pi_tensor_product from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
suppress_compilati... | Mathlib/LinearAlgebra/PiTensorProduct.lean | 535 | 540 | theorem map_range_eq_span_tprod :
LinearMap.range (map f) =
Submodule.span R {t | ∃ (m : Π i, s i), tprod R (fun i ↦ f i (m i)) = t} := by |
rw [← Submodule.map_top, ← span_tprod_eq_top, Submodule.map_span, ← Set.range_comp]
apply congrArg; ext x
simp only [Set.mem_range, comp_apply, map_tprod, Set.mem_setOf_eq]
|
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#align_import linear_algebra.sesquilinear_form from "leanprover-community/mathlib"@"87c54600fe3cdc7d32ff5b50873ac724d86aef8d"
variable {R R₁ R₂ R₃ M M₁ M₂ M₃... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 776 | 789 | theorem IsOrthoᵢ.not_isOrtho_basis_self_of_separatingLeft [Nontrivial R]
{B : M →ₛₗ[I] M →ₛₗ[I'] M₁} {v : Basis n R M} (h : B.IsOrthoᵢ v) (hB : B.SeparatingLeft)
(i : n) : ¬B.IsOrtho (v i) (v i) := by |
intro ho
refine v.ne_zero i (hB (v i) fun m ↦ ?_)
obtain ⟨vi, rfl⟩ := v.repr.symm.surjective m
rw [Basis.repr_symm_apply, Finsupp.total_apply, Finsupp.sum, map_sum]
apply Finset.sum_eq_zero
rintro j -
rw [map_smulₛₗ]
suffices B (v i) (v j) = 0 by rw [this, smul_zero]
obtain rfl | hij := eq_or_ne i j
... |
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 131 | 136 | theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b) := by |
rcases zero_or_succ_or_limit b with (e | ⟨b, rfl⟩ | l')
· exact absurd e hb
· rw [opow_succ]
exact mul_isLimit (opow_pos _ l.pos) l
· exact opow_isLimit l.one_lt l'
|
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
variable (p q : ℤ[X])
def IsUnitTrinomial :=
... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 146 | 148 | theorem card_support_eq_three (hp : p.IsUnitTrinomial) : p.support.card = 3 := by |
obtain ⟨k, m, n, hkm, hmn, u, v, w, rfl⟩ := hp
exact card_support_trinomial hkm hmn u.ne_zero v.ne_zero w.ne_zero
|
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.Group.Units
import Mathlib.Data.List.Perm
import Mathlib.Data.List.ProdSigma
import Mathlib.Data.List.Range
import Mathlib.Data.List.Rotate
#align_im... | Mathlib/Algebra/BigOperators/Group/List.lean | 128 | 129 | theorem prod_join {l : List (List M)} : l.join.prod = (l.map List.prod).prod := by |
induction l <;> [rfl; simp only [*, List.join, map, prod_append, prod_cons]]
|
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 196 | 197 | theorem lt_inv_mul_iff_lt : 1 < b⁻¹ * a ↔ b < a := by |
rw [← mul_lt_mul_iff_left b, mul_one, mul_inv_cancel_left]
|
import Mathlib.Order.Filter.EventuallyConst
import Mathlib.Order.PartialSups
import Mathlib.Algebra.Module.Submodule.IterateMapComap
import Mathlib.RingTheory.OrzechProperty
import Mathlib.RingTheory.Nilpotent.Lemmas
#align_import ring_theory.noetherian from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3... | Mathlib/RingTheory/Noetherian.lean | 642 | 645 | theorem IsNoetherianRing.isNilpotent_nilradical (R : Type*) [CommRing R] [IsNoetherianRing R] :
IsNilpotent (nilradical R) := by |
obtain ⟨n, hn⟩ := Ideal.exists_radical_pow_le_of_fg (⊥ : Ideal R) (IsNoetherian.noetherian _)
exact ⟨n, eq_bot_iff.mpr hn⟩
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Ring.Subsemiring.Basic
#align_import ring_theory.subring.basic from "leanprover-community/mathlib"@"b915e9392ecb2a861e1e766f0e1df6ac481188ca"
universe u v w
variable {R : Type u} {S : Type v} {T : Type w} [Ring R]
namespace Subring
instance ... | Mathlib/Algebra/Ring/Subring/Basic.lean | 685 | 686 | theorem mem_iInf {ι : Sort*} {S : ι → Subring R} {x : R} : (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by |
simp only [iInf, mem_sInf, Set.forall_mem_range]
|
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.ProdSigma
#align_import model_theory.semantics from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable {L : Language.{u, v}} {... | Mathlib/ModelTheory/Semantics.lean | 109 | 113 | theorem realize_functions_apply₁ {f : L.Functions 1} {t : L.Term α} {v : α → M} :
(f.apply₁ t).realize v = funMap f ![t.realize v] := by |
rw [Functions.apply₁, Term.realize]
refine congr rfl (funext fun i => ?_)
simp only [Matrix.cons_val_fin_one]
|
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Init.Data.Fin.Basic
#align_import data.list.nodup from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
universe u v
open Nat Function
variable {α : Type u} {β : Type v} {l l₁ l₂ : List α} {r : α → α ... | Mathlib/Data/List/Nodup.lean | 430 | 438 | theorem Nodup.pairwise_of_forall_ne {l : List α} {r : α → α → Prop} (hl : l.Nodup)
(h : ∀ a ∈ l, ∀ b ∈ l, a ≠ b → r a b) : l.Pairwise r := by |
rw [pairwise_iff_forall_sublist]
intro a b hab
if heq : a = b then
cases heq; have := nodup_iff_sublist.mp hl _ hab; contradiction
else
apply h <;> try (apply hab.subset; simp)
exact heq
|
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