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 | meta_tactic_error bool 2
classes |
|---|---|---|---|---|---|---|
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4972407cd4f1222"
noncomputable section
@[ext]
structure Cubic (R : Type*) where
(a b c d : R)
#align cubic Cubic
namespace Cubic
open Cubic Polynomial
open Polynom... | Mathlib/Algebra/CubicDiscriminant.lean | 124 | 124 | theorem b_of_eq (h : P.toPoly = Q.toPoly) : P.b = Q.b := by | rw [← coeff_eq_b, h, coeff_eq_b]
| false |
import Mathlib.NumberTheory.Zsqrtd.GaussianInt
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.zsqrtd.quadratic_reciprocity from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open scoped Comple... | Mathlib/NumberTheory/Zsqrtd/QuadraticReciprocity.lean | 33 | 83 | theorem mod_four_eq_three_of_nat_prime_of_prime (p : ℕ) [hp : Fact p.Prime]
(hpi : Prime (p : ℤ[i])) : p % 4 = 3 :=
hp.1.eq_two_or_odd.elim
(fun hp2 =>
absurd hpi
(mt irreducible_iff_prime.2 fun ⟨_, h⟩ => by
have := h ⟨1, 1⟩ ⟨1, -1⟩ (hp2.symm ▸ rfl)
rw [← norm_eq_one_iff, ← n... |
rw [← Nat.mod_mul_left_mod p 2 2, show 2 * 2 = 4 from rfl] at hp1
have := Nat.mod_lt p (show 0 < 4 by decide)
revert this hp3 hp1
generalize p % 4 = m
intros; interval_cases m <;> simp_all -- Porting note (#11043): was `decide!`
let ⟨k, hk⟩ := (ZMod.exists_sq_eq_neg_one_if... | false |
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.Convex.Deriv
#align_import analysis.convex.specific_functions.deriv from "leanprover-communi... | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 115 | 119 | theorem hasDerivAt_sqrt_mul_log {x : ℝ} (hx : x ≠ 0) :
HasDerivAt (fun x => √x * log x) ((2 + log x) / (2 * √x)) x := by |
convert (hasDerivAt_sqrt hx).mul (hasDerivAt_log hx) using 1
rw [add_div, div_mul_cancel_left₀ two_ne_zero, ← div_eq_mul_inv, sqrt_div_self', add_comm,
one_div, one_div, ← div_eq_inv_mul]
| false |
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.RingTheory.AlgebraTower
import Mathlib.Algebra.Algebra.Subalgebra.Tower
#align_import linear_algebra.matrix.to_lin from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e8491d6"
... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 102 | 110 | theorem range_vecMulLinear (M : Matrix m n R) :
LinearMap.range M.vecMulLinear = span R (range M) := by |
letI := Classical.decEq m
simp_rw [range_eq_map, ← iSup_range_stdBasis, Submodule.map_iSup, range_eq_map, ←
Ideal.span_singleton_one, Ideal.span, Submodule.map_span, image_image, image_singleton,
Matrix.vecMulLinear_apply, iSup_span, range_eq_iUnion, iUnion_singleton_eq_range,
LinearMap.stdBasis, coe_s... | true |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [Decida... | Mathlib/Data/Matrix/Basis.lean | 139 | 140 | theorem apply_of_row_ne {i i' : m} (hi : i ≠ i') (j j' : n) (a : α) :
stdBasisMatrix i j a i' j' = 0 := by | simp [hi]
| true |
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.GCDMonoid.Nat
#align_import ring_theory.int.basic from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
theorem Int.Prime.dvd_mul {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p ... | Mathlib/RingTheory/Int/Basic.lean | 139 | 141 | theorem span_natAbs (a : ℤ) : Ideal.span ({(a.natAbs : ℤ)} : Set ℤ) = Ideal.span {a} := by |
rw [Ideal.span_singleton_eq_span_singleton]
exact (associated_natAbs _).symm
| false |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 358 | 360 | theorem toQuaternion_comp_ofQuaternion :
toQuaternion.comp ofQuaternion = AlgHom.id R ℍ[R,c₁,c₂] := by |
ext : 1 <;> simp
| false |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp Ad... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 128 | 130 | theorem degrees_add [DecidableEq σ] (p q : MvPolynomial σ R) :
(p + q).degrees ≤ p.degrees ⊔ q.degrees := by |
simp_rw [degrees_def]; exact supDegree_add_le
| true |
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib"@"ccad6d5093bd2f5c6ca621fc74674cce51355af6"
universe u v w
variable {α : Type u} {β : Type v} {γ : Type w}
open Nat
namespace List
@[simp]
theo... | Mathlib/Data/List/Sublists.lean | 120 | 129 | theorem sublistsAux_eq_array_foldl :
sublistsAux = fun (a : α) (r : List (List α)) =>
(r.toArray.foldl (init := #[])
fun r l => (r.push l).push (a :: l)).toList := by |
funext a r
simp only [sublistsAux, Array.foldl_eq_foldl_data, Array.mkEmpty]
have := foldl_hom Array.toList (fun r l => (r.push l).push (a :: l))
(fun (r : List (List α)) l => r ++ [l, a :: l]) r #[]
(by simp)
simpa using this
| false |
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.complex.locally_uniform_limit from "leanprover-community/mathlib"@"fe44cd36149e675eb5dec87acc7e8f1d6568e081"
open Set Metric MeasureThe... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 79 | 86 | theorem norm_cderiv_lt (hr : 0 < r) (hfM : ∀ w ∈ sphere z r, ‖f w‖ < M)
(hf : ContinuousOn f (sphere z r)) : ‖cderiv r f z‖ < M / r := by |
obtain ⟨L, hL1, hL2⟩ : ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L := by
have e1 : (sphere z r).Nonempty := NormedSpace.sphere_nonempty.mpr hr.le
have e2 : ContinuousOn (fun w => ‖f w‖) (sphere z r) := continuous_norm.comp_continuousOn hf
obtain ⟨x, hx, hx'⟩ := (isCompact_sphere z r).exists_isMaxOn e1 e2
ex... | false |
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.linear_pmap from "leanprover-community/mathlib"@"8b981918a93bc45a8600de608cde7944a80d92b9"
universe u v w
structure LinearPMap (R : Type u) [Ring R] (E : Type v) [AddCommGroup E] [Module R E] (F : Type w)
[AddCommGroup F] [Module R F] where
domai... | Mathlib/LinearAlgebra/LinearPMap.lean | 64 | 70 | theorem ext {f g : E →ₗ.[R] F} (h : f.domain = g.domain)
(h' : ∀ ⦃x : f.domain⦄ ⦃y : g.domain⦄ (_h : (x : E) = y), f x = g y) : f = g := by |
rcases f with ⟨f_dom, f⟩
rcases g with ⟨g_dom, g⟩
obtain rfl : f_dom = g_dom := h
obtain rfl : f = g := LinearMap.ext fun x => h' rfl
rfl
| false |
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 | 68 | 68 | theorem coe_inv_two : ((2⁻¹ : ℝ≥0) : ℝ≥0∞) = 2⁻¹ := by | rw [coe_inv _root_.two_ne_zero, coe_two]
| false |
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 73 | 73 | theorem rpow_natCast (x : ℝ) (n : ℕ) : x ^ (n : ℝ) = x ^ n := by | simpa using rpow_intCast x n
| false |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Equiv.Fin
#align_import data.fin.tuple.nat_antidiagonal from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
namespace List.Nat
def antidiagona... | Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean | 131 | 139 | theorem antidiagonalTuple_one (n : ℕ) : antidiagonalTuple 1 n = [![n]] := by |
simp_rw [antidiagonalTuple, antidiagonal, List.range_succ, List.map_append, List.map_singleton,
tsub_self, List.append_bind, List.bind_singleton, List.map_bind]
conv_rhs => rw [← List.nil_append [![n]]]
congr 1
simp_rw [List.bind_eq_nil, List.mem_range, List.map_eq_nil]
intro x hx
obtain ⟨m, rfl⟩ := Na... | false |
import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.Algebra.Group.Ext
#align_import topology.homotopy.homotopy_group from "leanprover-community/mathlib"@"4c3e1721c58ef9087bbc2c8c38b540f70eda2e53"
... | Mathlib/Topology/Homotopy/HomotopyGroup.lean | 74 | 78 | theorem insertAt_boundary (i : N) {t₀ : I} {t}
(H : (t₀ = 0 ∨ t₀ = 1) ∨ t ∈ boundary { j // j ≠ i }) : insertAt i ⟨t₀, t⟩ ∈ boundary N := by |
obtain H | ⟨j, H⟩ := H
· use i; rwa [funSplitAt_symm_apply, dif_pos rfl]
· use j; rwa [funSplitAt_symm_apply, dif_neg j.prop, Subtype.coe_eta]
| true |
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Quotient
import Mathlib.Combinatorics.Quiver.Path
#align_import category_theory.path_category from "leanprover-community/mathlib"@"c6dd521ebdce53bb372c527569dd7c25de53a08b"
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
section
def Paths (V : ... | Mathlib/CategoryTheory/PathCategory.lean | 93 | 100 | theorem lift_spec {C} [Category C] (φ : V ⥤q C) : of ⋙q (lift φ).toPrefunctor = φ := by |
fapply Prefunctor.ext
· rintro X
rfl
· rintro X Y f
rcases φ with ⟨φo, φm⟩
dsimp [lift, Quiver.Hom.toPath]
simp only [Category.id_comp]
| true |
import Mathlib.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import probability.strong_law from "leanprover-community/mathlib"@"f2ce60867... | Mathlib/Probability/StrongLaw.lean | 151 | 175 | theorem moment_truncation_eq_intervalIntegral_of_nonneg (hf : AEStronglyMeasurable f μ) {A : ℝ}
{n : ℕ} (hn : n ≠ 0) (h'f : 0 ≤ f) :
∫ x, truncation f A x ^ n ∂μ = ∫ y in (0)..A, y ^ n ∂Measure.map f μ := by |
have M : MeasurableSet (Set.Ioc 0 A) := measurableSet_Ioc
have M' : MeasurableSet (Set.Ioc A 0) := measurableSet_Ioc
rw [truncation_eq_of_nonneg h'f]
change ∫ x, (fun z => indicator (Set.Ioc 0 A) id z ^ n) (f x) ∂μ = _
rcases le_or_lt 0 A with (hA | hA)
· rw [← integral_map (f := fun z => _ ^ n) hf.aemeasu... | false |
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
open Filter ENNReal
namespace ENNReal
variable {α : Type*} {f : Filter α}
theorem eventually_le_limsup [CountableInterFilter f] (u : α → ℝ≥0∞) :
∀ᶠ y i... | Mathlib/Order/Filter/ENNReal.lean | 50 | 68 | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by |
by_cases ha_top : a ≠ ⊤
· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =ᶠ[f] 0
· have hau : (a * u ·) =ᶠ[f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, ← ENNReal.bot_eq_zero,
limsup_const_bot]
simp
· have hu_... | false |
import Mathlib.Topology.Order.LeftRight
import Mathlib.Topology.Order.Monotone
#align_import topology.algebra.order.left_right_lim from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
open Set Filter
open Topology
section
variable {α β : Type*} [LinearOrder α] [TopologicalSpace β]
n... | Mathlib/Topology/Order/LeftRightLim.lean | 125 | 136 | theorem le_leftLim (h : x < y) : f x ≤ leftLim f y := by |
letI : TopologicalSpace α := Preorder.topology α
haveI : OrderTopology α := ⟨rfl⟩
rcases eq_or_ne (𝓝[<] y) ⊥ with (h' | h')
· rw [leftLim_eq_of_eq_bot _ h']
exact hf h.le
rw [leftLim_eq_sSup hf h']
refine le_csSup ⟨f y, ?_⟩ (mem_image_of_mem _ h)
simp only [upperBounds, mem_image, mem_Iio, forall_ex... | true |
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.ContinuousFunction.CocompactMap
open Filter Metric
variable {𝕜 E F 𝓕 : Type*}
variable [NormedAddCommGroup E] [NormedAddCommGroup F] [ProperSpace E] [ProperSpace F]
variable {f : 𝓕}
| Mathlib/Analysis/Normed/Group/CocompactMap.lean | 29 | 39 | theorem CocompactMapClass.norm_le [FunLike 𝓕 E F] [CocompactMapClass 𝓕 E F] (ε : ℝ) :
∃ r : ℝ, ∀ x : E, r < ‖x‖ → ε < ‖f x‖ := by |
have h := cocompact_tendsto f
rw [tendsto_def] at h
specialize h (Metric.closedBall 0 ε)ᶜ (mem_cocompact_of_closedBall_compl_subset 0 ⟨ε, rfl.subset⟩)
rcases closedBall_compl_subset_of_mem_cocompact h 0 with ⟨r, hr⟩
use r
intro x hx
suffices x ∈ f⁻¹' (Metric.closedBall 0 ε)ᶜ by aesop
apply hr
simp [h... | false |
import Mathlib.Data.Fintype.Basic
import Mathlib.ModelTheory.Substructures
#align_import model_theory.elementary_maps from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
open FirstOrder
namespace FirstOrder
namespace Language
open Structure
variable (L : Language) (M : Type*) (N : T... | Mathlib/ModelTheory/ElementaryMaps.lean | 132 | 136 | theorem map_fun (φ : M ↪ₑ[L] N) {n : ℕ} (f : L.Functions n) (x : Fin n → M) :
φ (funMap f x) = funMap f (φ ∘ x) := by |
have h := φ.map_formula (Formula.graph f) (Fin.cons (funMap f x) x)
rw [Formula.realize_graph, Fin.comp_cons, Formula.realize_graph] at h
rw [eq_comm, h]
| true |
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4972407cd4f1222"
noncomputable section
@[ext]
structure Cubic (R : Type*) where
(a b c d : R)
#align cubic Cubic
namespace Cubic
open Cubic Polynomial
open Polynom... | Mathlib/Algebra/CubicDiscriminant.lean | 67 | 71 | theorem C_mul_prod_X_sub_C_eq [CommRing S] {w x y z : S} :
C w * (X - C x) * (X - C y) * (X - C z) =
toPoly ⟨w, w * -(x + y + z), w * (x * y + x * z + y * z), w * -(x * y * z)⟩ := by |
simp only [toPoly, C_neg, C_add, C_mul]
ring1
| true |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.Core
import Mathlib.Util.SynthesizeUsing
import Mathlib.Util.Qq
open Lean Parser Tactic Mathlib Meta NormNum Qq
initialize registerTraceClass `CancelDen... | Mathlib/Tactic/CancelDenoms/Core.lean | 66 | 68 | theorem pow_subst {α} [CommRing α] {n e1 t1 k l : α} {e2 : ℕ}
(h1 : n * e1 = t1) (h2 : l * n ^ e2 = k) : k * (e1 ^ e2) = l * t1 ^ e2 := by |
rw [← h2, ← h1, mul_pow, mul_assoc]
| false |
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
#align_import measure_theory.function.simple_func_dense from "leanprover-community/mathlib"@"7317149f12f55affbc900fc873d0d422485122b9"
open Set Function Filter TopologicalSpace ENNReal EMetric Finset
... | Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | 102 | 113 | theorem edist_nearestPt_le (e : ℕ → α) (x : α) {k N : ℕ} (hk : k ≤ N) :
edist (nearestPt e N x) x ≤ edist (e k) x := by |
induction' N with N ihN generalizing k
· simp [nonpos_iff_eq_zero.1 hk, le_refl]
· simp only [nearestPt, nearestPtInd_succ, map_apply]
split_ifs with h
· rcases hk.eq_or_lt with (rfl | hk)
exacts [le_rfl, (h k (Nat.lt_succ_iff.1 hk)).le]
· push_neg at h
rcases h with ⟨l, hlN, hxl⟩
r... | true |
import Mathlib.Init.Data.Ordering.Basic
import Mathlib.Order.Synonym
#align_import order.compare from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {α β : Type*}
def cmpLE {α} [LE α] [@DecidableRel α (· ≤ ·)] (x y : α) : Ordering :=
if x ≤ y then if y ≤ x then Ordering.eq ... | Mathlib/Order/Compare.lean | 34 | 37 | theorem cmpLE_swap {α} [LE α] [IsTotal α (· ≤ ·)] [@DecidableRel α (· ≤ ·)] (x y : α) :
(cmpLE x y).swap = cmpLE y x := by |
by_cases xy:x ≤ y <;> by_cases yx:y ≤ x <;> simp [cmpLE, *, Ordering.swap]
cases not_or_of_not xy yx (total_of _ _ _)
| false |
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.basis from "leanprover-community/mathlib"@"6c263e4bfc2e6714de30f22178b4d0ca4d149a76"
noncomputable section
open LinearMap Matrix Set Submodule
open Matrix
section BasisToMatrix
variable {ι... | Mathlib/LinearAlgebra/Matrix/Basis.lean | 117 | 122 | theorem toMatrix_smul {R₁ S : Type*} [CommRing R₁] [Ring S] [Algebra R₁ S] [Fintype ι]
[DecidableEq ι] (x : S) (b : Basis ι R₁ S) (w : ι → S) :
(b.toMatrix (x • w)) = (Algebra.leftMulMatrix b x) * (b.toMatrix w) := by |
ext
rw [Basis.toMatrix_apply, Pi.smul_apply, smul_eq_mul, ← Algebra.leftMulMatrix_mulVec_repr]
rfl
| false |
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.LocallyFinite
open Set Filter Topology TopologicalSpace Classical Function
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [Topolog... | Mathlib/Topology/Compactness/Compact.lean | 104 | 116 | theorem IsCompact.image_of_continuousOn {f : X → Y} (hs : IsCompact s) (hf : ContinuousOn f s) :
IsCompact (f '' s) := by |
intro l lne ls
have : NeBot (l.comap f ⊓ 𝓟 s) :=
comap_inf_principal_neBot_of_image_mem lne (le_principal_iff.1 ls)
obtain ⟨x, hxs, hx⟩ : ∃ x ∈ s, ClusterPt x (l.comap f ⊓ 𝓟 s) := @hs _ this inf_le_right
haveI := hx.neBot
use f x, mem_image_of_mem f hxs
have : Tendsto f (𝓝 x ⊓ (comap f l ⊓ 𝓟 s)) (�... | false |
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Pointwise
import Mathlib.Data.Real.Archimedean
#align_import data.real.pointwise from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
open Set
open Pointwise
variable {ι : Sort*} {α : Type*} [LinearOrde... | Mathlib/Data/Real/Pointwise.lean | 91 | 100 | theorem Real.sSup_smul_of_nonpos (ha : a ≤ 0) (s : Set ℝ) : sSup (a • s) = a • sInf s := by |
obtain rfl | hs := s.eq_empty_or_nonempty
· rw [smul_set_empty, Real.sSup_empty, Real.sInf_empty, smul_zero]
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_smul_set hs, zero_smul]
exact csSup_singleton 0
by_cases h : BddBelow s
· exact ((OrderIso.smulRightDual ℝ ha').map_csInf' hs h).symm
· rw [Real.sSup... | false |
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 | 103 | 104 | theorem integrableOn_univ : IntegrableOn f univ μ ↔ Integrable f μ := by |
rw [IntegrableOn, Measure.restrict_univ]
| false |
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
namespace Con... | Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean | 148 | 151 | theorem decomp (f : V →ᴬ[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by |
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
| false |
import Mathlib.Order.Cover
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.GaloisConnection
#align_import order.modular_lattice from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Set
variable {α : Type*}
class IsWeakUpperModularLattice (α : Type*) [Lattice α] : Prop ... | Mathlib/Order/ModularLattice.lean | 233 | 237 | theorem eq_of_le_of_inf_le_of_le_sup (hxy : x ≤ y) (hinf : y ⊓ z ≤ x) (hsup : y ≤ x ⊔ z) :
x = y := by |
refine hxy.antisymm ?_
rw [← inf_eq_right, sup_inf_assoc_of_le _ hxy] at hsup
rwa [← hsup, sup_le_iff, and_iff_right rfl.le, inf_comm]
| false |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.FieldTheory.IsAlgClosed.Basic
#align_import field_theory.is_alg_closed.spectrum from "leanprover-community/mathlib"@"58a272265b5e05f258161260dd2c5d247213cbd3"
namespace spectrum
open Set Polynomial
open scoped Pointwise Polynomial
universe u v
section Scal... | Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean | 81 | 91 | theorem subset_polynomial_aeval (a : A) (p : 𝕜[X]) : (eval · p) '' σ a ⊆ σ (aeval a p) := by |
rintro _ ⟨k, hk, rfl⟩
let q := C (eval k p) - p
have hroot : IsRoot q k := by simp only [q, eval_C, eval_sub, sub_self, IsRoot.def]
rw [← mul_div_eq_iff_isRoot, ← neg_mul_neg, neg_sub] at hroot
have aeval_q_eq : ↑ₐ (eval k p) - aeval a p = aeval a q := by
simp only [q, aeval_C, AlgHom.map_sub, sub_left_i... | false |
import Mathlib.CategoryTheory.Sites.IsSheafFor
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Tactic.ApplyFun
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w v u
namespace CategoryTheory
open Opposite ... | Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | 216 | 223 | theorem w : forkMap P R ≫ firstMap P R = forkMap P R ≫ secondMap P R := by |
dsimp
ext fg
simp only [firstMap, secondMap, forkMap]
simp only [limit.lift_π, limit.lift_π_assoc, assoc, Fan.mk_π_app]
haveI := Presieve.hasPullbacks.has_pullbacks fg.1.2.2 fg.2.2.2
rw [← P.map_comp, ← op_comp, pullback.condition]
simp
| false |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Antisymmetrization
#align_import order.cover from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Set OrderDual
variable {α β : Type*}
section WeaklyCovers
section LT
variable [LT α] {a b : α}
def CovBy (a b :... | Mathlib/Order/Cover.lean | 233 | 234 | theorem not_covBy_iff (h : a < b) : ¬a ⋖ b ↔ ∃ c, a < c ∧ c < b := by |
simp_rw [CovBy, h, true_and_iff, not_forall, exists_prop, not_not]
| false |
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.DeriveFintype
#align_import geometry.euclidean.circumcenter from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
noncomputable section
open scoped Classical
o... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 48 | 56 | theorem dist_eq_iff_dist_orthogonalProjection_eq {s : AffineSubspace ℝ P} [Nonempty s]
[HasOrthogonalProjection s.direction] {p1 p2 : P} (p3 : P) (hp1 : p1 ∈ s) (hp2 : p2 ∈ s) :
dist p1 p3 = dist p2 p3 ↔
dist p1 (orthogonalProjection s p3) = dist p2 (orthogonalProjection s p3) := by |
rw [← mul_self_inj_of_nonneg dist_nonneg dist_nonneg, ←
mul_self_inj_of_nonneg dist_nonneg dist_nonneg,
dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq p3 hp1,
dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq p3 hp2]
simp
| false |
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs from "leanprover-community/mathlib"@"842328d9df7e96fd90fc424e115679c15fb23a71"
noncomputable section
open Finset Function
variable {α β γ ι M M' N P G H R S : Type*}... | Mathlib/Data/Finsupp/Defs.lean | 209 | 209 | theorem card_support_eq_zero {f : α →₀ M} : card f.support = 0 ↔ f = 0 := by | simp
| false |
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
-- Porting note: added to make the syntax work below.
open scoped TensorProd... | Mathlib/RingTheory/Unramified/Basic.lean | 139 | 152 | theorem comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by |
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F... | false |
import Mathlib.Combinatorics.SimpleGraph.Basic
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
structure Dart extends V × V where
adj : G.Adj fst snd
deriving DecidableEq
#align simple_graph.dart SimpleGraph.Dart
initialize_simps_projections Dart (+toProd, -fst, -snd)
attribute [simp] Dart.a... | Mathlib/Combinatorics/SimpleGraph/Dart.lean | 118 | 123 | theorem dart_edge_eq_mk'_iff' :
∀ {d : G.Dart} {u v : V},
d.edge = s(u, v) ↔ d.fst = u ∧ d.snd = v ∨ d.fst = v ∧ d.snd = u := by |
rintro ⟨⟨a, b⟩, h⟩ u v
rw [dart_edge_eq_mk'_iff]
simp
| false |
import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
assert_not_exists MonoidWithZero
open Set
namespace Function
variable {α β A B M N P G : Type*}
section One
variable [One M] [One N] [One P]
... | Mathlib/Algebra/Group/Support.lean | 93 | 95 | theorem mulSupport_update_one [DecidableEq α] (f : α → M) (x : α) :
mulSupport (update f x 1) = mulSupport f \ {x} := by |
ext a; rcases eq_or_ne a x with rfl | hne <;> simp [*]
| false |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.prod from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set hiding prod_eq
open Function MeasureTheory
open Filter hiding ma... | Mathlib/MeasureTheory/Group/Prod.lean | 424 | 429 | theorem measurePreserving_mul_prod_inv_right [IsMulRightInvariant μ] [IsMulRightInvariant ν] :
MeasurePreserving (fun z : G × G => (z.1 * z.2, z.1⁻¹)) (μ.prod ν) (μ.prod ν) := by |
convert (measurePreserving_prod_div_swap ν μ).comp (measurePreserving_prod_mul_swap_right μ ν)
using 1
ext1 ⟨x, y⟩
simp_rw [Function.comp_apply, div_mul_eq_div_div_swap, div_self', one_div]
| false |
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
#align_import linear_algebra.clifford_algebra.even from "leanprover-community/mathlib"@"9264b15ee696b7ca83f13c8ad67c83d6eb70b730"
namespace CliffordAlgebra
-- Porting note: explicit universes
universe uR uM uA ... | Mathlib/LinearAlgebra/CliffordAlgebra/Even.lean | 116 | 128 | theorem even.algHom_ext ⦃f g : even Q →ₐ[R] A⦄ (h : (even.ι Q).compr₂ f = (even.ι Q).compr₂ g) :
f = g := by |
rw [EvenHom.ext_iff] at h
ext ⟨x, hx⟩
induction x, hx using even_induction with
| algebraMap r =>
exact (f.commutes r).trans (g.commutes r).symm
| add x y hx hy ihx ihy =>
have := congr_arg₂ (· + ·) ihx ihy
exact (f.map_add _ _).trans (this.trans <| (g.map_add _ _).symm)
| ι_mul_ι_mul m₁ m₂ x h... | true |
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 | 113 | 119 | theorem measurable_kernel_prod_mk_left' [IsSFiniteKernel η] {s : Set (β × γ)} (hs : MeasurableSet s)
(a : α) : Measurable fun b => η (a, b) (Prod.mk b ⁻¹' s) := by |
have : ∀ b, Prod.mk b ⁻¹' s = {c | ((a, b), c) ∈ {p : (α × β) × γ | (p.1.2, p.2) ∈ s}} := by
intro b; rfl
simp_rw [this]
refine (measurable_kernel_prod_mk_left ?_).comp measurable_prod_mk_left
exact (measurable_fst.snd.prod_mk measurable_snd) hs
| true |
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Init.Order.Defs
set_option autoImplicit true
structure UFModel (n) where
parent : Fin n → Fin n
rank : Nat → Nat
rank_lt : ∀ i, (parent i).1 ≠ i → rank i < rank (parent i)
structure UFNode (α : Type*) where
parent : Nat
value : α
rank : Nat
inductive... | Mathlib/Data/UnionFind.lean | 79 | 80 | theorem size_eq {arr : Array α} {m : Fin n → β} (H : Agrees arr f m) : n = arr.size := by |
cases H; rfl
| false |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Tactic.Linarith
#align_import combinatorics.simple_graph.acyclic from "leanprover-community/mathlib"@"b07688016d62f81d14508ff339ea3415558d6353"
universe u v
namespace SimpleGraph
open Walk
variable {V : Type u} (G : SimpleGraph V)
def IsAcy... | Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | 134 | 154 | theorem isTree_iff_existsUnique_path :
G.IsTree ↔ Nonempty V ∧ ∀ v w : V, ∃! p : G.Walk v w, p.IsPath := by |
classical
rw [isTree_iff, isAcyclic_iff_path_unique]
constructor
· rintro ⟨hc, hu⟩
refine ⟨hc.nonempty, ?_⟩
intro v w
let q := (hc v w).some.toPath
use q
simp only [true_and_iff, Path.isPath]
intro p hp
specialize hu ⟨p, hp⟩ q
exact Subtype.ext_iff.mp hu
· rintro ⟨hV, h⟩
r... | false |
import Mathlib.Data.List.Sublists
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α : Type*}
-- Porting note (#11215): TODO: Write a more efficient version
def powerset... | Mathlib/Data/Multiset/Powerset.lean | 55 | 57 | theorem powersetAux'_cons (a : α) (l : List α) :
powersetAux' (a :: l) = powersetAux' l ++ List.map (cons a) (powersetAux' l) := by |
simp only [powersetAux', sublists'_cons, map_append, List.map_map, append_cancel_left_eq]; rfl
| true |
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.Order.Filter.AtTopBot
import Mathlib.RingTheory.Artinian
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mat... | Mathlib/Algebra/Lie/Nilpotent.lean | 504 | 508 | theorem ucs_le_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) :
(⊥ : LieSubmodule R L M).ucs k ≤ N₁ := by |
rw [← ucs_eq_self_of_normalizer_eq_self h k]
mono
simp
| false |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Algebra.Ring.NegOnePow
namespace Matrix
variable {R : Type*} [CommRing R]
theorem submatrix_succAbove_det_eq_negOnePow_submatrix_succAbove_det {n : ℕ}
(M : Matrix (Fin (n + 1)) (Fin n) R) (hv : ∑ j, M j = 0) (j₁ j₂ : Fin (n + 1)) :
(M.s... | Mathlib/LinearAlgebra/Matrix/Determinant/Misc.lean | 51 | 59 | theorem submatrix_succAbove_det_eq_negOnePow_submatrix_succAbove_det' {n : ℕ}
(M : Matrix (Fin n) (Fin (n + 1)) R) (hv : ∀ i, ∑ j, M i j = 0) (j₁ j₂ : Fin (n + 1)) :
(M.submatrix id (Fin.succAbove j₁)).det =
Int.negOnePow (j₁ - j₂) • (M.submatrix id (Fin.succAbove j₂)).det := by |
rw [← det_transpose, transpose_submatrix,
submatrix_succAbove_det_eq_negOnePow_submatrix_succAbove_det M.transpose ?_ j₁ j₂,
← det_transpose, transpose_submatrix, transpose_transpose]
ext
simp_rw [Finset.sum_apply, transpose_apply, hv, Pi.zero_apply]
| false |
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem «forall» {p : α ⊕ β → Prop} :
(∀ x, p x) ↔ (∀ a, p (inl a)) ∧ ∀ b, p (inr b) :=
⟨fun h => ⟨fun _ => h _, fun _ => h _⟩, fun ⟨h₁, h₂⟩ => Sum.rec h₁ h₂⟩
@[simp] protected theorem «exists» {p : α ⊕ β ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 83 | 83 | theorem isLeft_iff : x.isLeft ↔ ∃ y, x = Sum.inl y := by | cases x <;> simp
| false |
import Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries
import Mathlib.Algebra.ContinuedFractions.Computation.Translations
import Mathlib.Data.Real.Irrational
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.Basic
#align_import number_theory.diophantine_approximation from "leanpro... | Mathlib/NumberTheory/DiophantineApproximation.lean | 139 | 144 | theorem exists_nat_abs_mul_sub_round_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :
∃ k : ℕ, 0 < k ∧ k ≤ n ∧ |↑k * ξ - round (↑k * ξ)| ≤ 1 / (n + 1) := by |
obtain ⟨j, k, hk₀, hk₁, h⟩ := exists_int_int_abs_mul_sub_le ξ n_pos
have hk := toNat_of_nonneg hk₀.le
rw [← hk] at hk₀ hk₁ h
exact ⟨k.toNat, natCast_pos.mp hk₀, Nat.cast_le.mp hk₁, (round_le (↑k.toNat * ξ) j).trans h⟩
| false |
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Data.Set.MulAntidiagonal
#align_import data.finset.mul_antidiagonal from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
namespace Set
open Pointwise
variable {α : Type*} {s t : Set α}
@[to_additive]
| Mathlib/Data/Finset/MulAntidiagonal.lean | 25 | 27 | theorem IsPWO.mul [OrderedCancelCommMonoid α] (hs : s.IsPWO) (ht : t.IsPWO) : IsPWO (s * t) := by |
rw [← image_mul_prod]
exact (hs.prod ht).image_of_monotone (monotone_fst.mul' monotone_snd)
| false |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
#align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163"
noncomputable section
variable {E : Type*} [NormedAddCommGroup E] [InnerProduct... | Mathlib/Analysis/InnerProductSpace/Orientation.lean | 91 | 96 | theorem det_eq_neg_det_of_opposite_orientation (h : e.toBasis.orientation ≠ f.toBasis.orientation) :
e.toBasis.det = -f.toBasis.det := by |
rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
-- Porting note: added `neg_one_smul` with explicit type
simp [e.det_to_matrix_orthonormalBasis_of_opposite_orientation f h,
neg_one_smul ℝ (M := E [⋀^ι]→ₗ[ℝ] ℝ)]
| false |
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
@[simp]
theorem reduceOption_cons_of_some (x : α) (l : List (Option α)) :
reduceOption (some x :: l) = x :: l.reduceOption := by
simp only [reduceOption, filterMap, id, eq_self_iff_true, and_self_iff]
#align list.reduce_option_cons_of_some... | Mathlib/Data/List/ReduceOption.lean | 25 | 26 | theorem reduceOption_cons_of_none (l : List (Option α)) :
reduceOption (none :: l) = l.reduceOption := by | simp only [reduceOption, filterMap, id]
| false |
import Mathlib.Algebra.Group.Commutator
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Bracket
import Mathlib.GroupTheory.Subgroup.Centralizer
import Mathlib.Tactic.Group
#align_import group_theory.commutator from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
variable... | Mathlib/GroupTheory/Commutator.lean | 61 | 62 | theorem commutatorElement_inv : ⁅g₁, g₂⁆⁻¹ = ⁅g₂, g₁⁆ := by |
simp_rw [commutatorElement_def, mul_inv_rev, inv_inv, mul_assoc]
| false |
import Mathlib.Analysis.NormedSpace.Star.Spectrum
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Analysis.NormedSpace.Algebra
import Mathlib.Topology.ContinuousFunction.Units
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunct... | Mathlib/Analysis/NormedSpace/Star/GelfandDuality.lean | 108 | 115 | theorem WeakDual.CharacterSpace.mem_spectrum_iff_exists {a : A} {z : ℂ} :
z ∈ spectrum ℂ a ↔ ∃ f : characterSpace ℂ A, f a = z := by |
refine ⟨fun hz => ?_, ?_⟩
· obtain ⟨f, hf⟩ := WeakDual.CharacterSpace.exists_apply_eq_zero hz
simp only [map_sub, sub_eq_zero, AlgHomClass.commutes] at hf
exact ⟨_, hf.symm⟩
· rintro ⟨f, rfl⟩
exact AlgHom.apply_mem_spectrum f a
| false |
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 | 26 | 27 | theorem then_eq_gt {o₁ o₂ : Ordering} : o₁.then o₂ = gt ↔ o₁ = gt ∨ o₁ = eq ∧ o₂ = gt := by |
cases o₁ <;> cases o₂ <;> decide
| false |
import Mathlib.AlgebraicTopology.DoldKan.PInfty
#align_import algebraic_topology.dold_kan.decomposition from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Preadditive
Opposite Simplicial
noncomputable section
namespace Alge... | Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean | 52 | 81 | theorem decomposition_Q (n q : ℕ) :
((Q q).f (n + 1) : X _[n + 1] ⟶ X _[n + 1]) =
∑ i ∈ Finset.filter (fun i : Fin (n + 1) => (i : ℕ) < q) Finset.univ,
(P i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ (Fin.rev i) := by |
induction' q with q hq
· simp only [Nat.zero_eq, Q_zero, HomologicalComplex.zero_f_apply, Nat.not_lt_zero,
Finset.filter_False, Finset.sum_empty]
· by_cases hqn : q + 1 ≤ n + 1
swap
· rw [Q_is_eventually_constant (show n + 1 ≤ q by omega), hq]
congr 1
ext ⟨x, hx⟩
simp only [Nat.su... | true |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 147 | 148 | theorem preimage_const_add_Icc : (fun x => a + x) ⁻¹' Icc b c = Icc (b - a) (c - a) := by |
simp [← Ici_inter_Iic]
| false |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.asymptotics.asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open ... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 113 | 114 | theorem isBigO_iff : f =O[l] g ↔ ∃ c : ℝ, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := by |
simp only [IsBigO_def, IsBigOWith_def]
| false |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 58 | 83 | theorem mk_eq_mk_of_basis (v : Basis ι R M) (v' : Basis ι' R M) :
Cardinal.lift.{w'} #ι = Cardinal.lift.{w} #ι' := by |
classical
haveI := nontrivial_of_invariantBasisNumber R
cases fintypeOrInfinite ι
· -- `v` is a finite basis, so by `basis_finite_of_finite_spans` so is `v'`.
-- haveI : Finite (range v) := Set.finite_range v
haveI := basis_finite_of_finite_spans _ (Set.finite_range v) v.span_eq v'
cases nonempty_f... | false |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 107 | 108 | theorem card_insert_of_not_mem (h : a ∉ s) : (insert a s).card = s.card + 1 := by |
rw [← cons_eq_insert _ _ h, card_cons]
| false |
import Mathlib.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_dimensional from "leanprover-community/mathlib"@"67e606eaea14c7854bdc556bd53d98aefdf76ec0"
noncomputable section
open Affine
section AffineSpace... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 120 | 125 | theorem AffineIndependent.finrank_vectorSpan [Fintype ι] {p : ι → P} (hi : AffineIndependent k p)
{n : ℕ} (hc : Fintype.card ι = n + 1) : finrank k (vectorSpan k (Set.range p)) = n := by |
classical
rw [← Finset.card_univ] at hc
rw [← Set.image_univ, ← Finset.coe_univ, ← Finset.coe_image]
exact hi.finrank_vectorSpan_image_finset hc
| false |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Prod
import Mathlib.Tactic.Common
variable {ι G₁ G₂ : Type*} {G : ι → Type*} [Semigroup G₁] [Semigroup G₂] [∀ i, Semigroup (G i)]
| Mathlib/Algebra/Divisibility/Prod.lean | 16 | 20 | theorem prod_dvd_iff {x y : G₁ × G₂} :
x ∣ y ↔ x.1 ∣ y.1 ∧ x.2 ∣ y.2 := by |
cases x; cases y
simp only [dvd_def, Prod.exists, Prod.mk_mul_mk, Prod.mk.injEq,
exists_and_left, exists_and_right, and_self, true_and]
| false |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 69 | 70 | theorem IsCoprime.prod_right : (∀ i ∈ t, IsCoprime x (s i)) → IsCoprime x (∏ i ∈ t, s i) := by |
simpa only [isCoprime_comm] using IsCoprime.prod_left (R := R)
| false |
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Bisim
variable {xs : Vector α n}
theorem mapAccumr_bisim {f₁ : α → σ₁ → σ₁ × β} {f₂ : α → σ₂ → σ₂ × β} {s₁ : σ₁} {s₂ : σ₂}
(R : σ₁ → σ₂ → Prop) (h₀ : R s₁ s₂)
(hR : ∀ {... | Mathlib/Data/Vector/MapLemmas.lean | 192 | 203 | theorem mapAccumr₂_bisim {ys : Vector β n} {f₁ : α → β → σ₁ → σ₁ × γ}
{f₂ : α → β → σ₂ → σ₂ × γ} {s₁ : σ₁} {s₂ : σ₂}
(R : σ₁ → σ₂ → Prop) (h₀ : R s₁ s₂)
(hR : ∀ {s q} a b, R s q → R (f₁ a b s).1 (f₂ a b q).1 ∧ (f₁ a b s).2 = (f₂ a b q).2) :
R (mapAccumr₂ f₁ xs ys s₁).1 (mapAccumr₂ f₂ xs ys s₂).1
∧ ... |
induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂
next => exact ⟨h₀, rfl⟩
next xs ys x y ih =>
rcases (hR x y h₀) with ⟨hR, _⟩
simp only [mapAccumr₂_snoc, ih hR, true_and]
congr 1
| true |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Set.Finite
#align_import order.conditionally_complete_lattice.finset from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
open Set
variable {ι α β γ : Type*}
section ConditionallyCompleteLinearOrder
variable [... | Mathlib/Order/ConditionallyCompleteLattice/Finset.lean | 42 | 44 | theorem Set.Nonempty.csSup_mem (h : s.Nonempty) (hs : s.Finite) : sSup s ∈ s := by |
lift s to Finset α using hs
exact Finset.Nonempty.csSup_mem h
| false |
import Mathlib.GroupTheory.Solvable
import Mathlib.FieldTheory.PolynomialGaloisGroup
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import field_theory.abel_ruffini from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Classical Polynomial Intermedi... | Mathlib/FieldTheory/AbelRuffini.lean | 45 | 45 | theorem gal_C_isSolvable (x : F) : IsSolvable (C x).Gal := by | infer_instance
| false |
import Mathlib.Data.Set.Lattice
import Mathlib.Init.Set
import Mathlib.Control.Basic
import Mathlib.Lean.Expr.ExtraRecognizers
#align_import data.set.functor from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u
open Function
namespace Set
variable {α β : Type u} {s : Set α} ... | Mathlib/Data/Set/Functor.lean | 149 | 150 | theorem mem_of_mem_image_val (ha : a ∈ (γ : Set α)) : ⟨a, image_val_subset ha⟩ ∈ γ := by |
rcases ha with ⟨_, ha, rfl⟩; exact ha
| false |
import Mathlib.AlgebraicTopology.SimplicialObject
import Mathlib.CategoryTheory.Limits.Shapes.Products
#align_import algebraic_topology.split_simplicial_object from "leanprover-community/mathlib"@"dd1f8496baa505636a82748e6b652165ea888733"
noncomputable section
open CategoryTheory CategoryTheory.Category Category... | Mathlib/AlgebraicTopology/SplitSimplicialObject.lean | 162 | 171 | theorem eqId_iff_mono : A.EqId ↔ Mono A.e := by |
constructor
· intro h
dsimp at h
subst h
dsimp only [id, e]
infer_instance
· intro h
rw [eqId_iff_len_le]
exact len_le_of_mono h
| false |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 144 | 152 | theorem nodup_of_pairwise_disjoint {l : List (Perm α)} (h1 : (1 : Perm α) ∉ l)
(h2 : l.Pairwise Disjoint) : l.Nodup := by |
refine List.Pairwise.imp_of_mem ?_ h2
intro τ σ h_mem _ h_disjoint _
subst τ
suffices (σ : Perm α) = 1 by
rw [this] at h_mem
exact h1 h_mem
exact ext fun a => or_self_iff.mp (h_disjoint a)
| false |
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheor... | Mathlib/Probability/Martingale/Basic.lean | 109 | 113 | theorem setIntegral_eq [SigmaFiniteFiltration μ ℱ] (hf : Martingale f ℱ μ) {i j : ι} (hij : i ≤ j)
{s : Set Ω} (hs : MeasurableSet[ℱ i] s) : ∫ ω in s, f i ω ∂μ = ∫ ω in s, f j ω ∂μ := by |
rw [← @setIntegral_condexp _ _ _ _ _ (ℱ i) m0 _ _ _ (ℱ.le i) _ (hf.integrable j) hs]
refine setIntegral_congr_ae (ℱ.le i s hs) ?_
filter_upwards [hf.2 i j hij] with _ heq _ using heq.symm
| false |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
inductive DihedralGroup (n : ℕ) : Type
| r : ZMod n → DihedralGroup n
| sr : ZMod n → DihedralGroup n
derivin... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 153 | 153 | theorem sr_mul_self (i : ZMod n) : sr i * sr i = 1 := by | rw [sr_mul_sr, sub_self, one_def]
| false |
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.ApplyFun
#align_import field_theory.finite.basic from "leanprover-community/mathlib"@"12a85fac627bea918960da036049d611b1a3ee43"
variable {K : Type*} {R : Type*}
local notation ... | Mathlib/FieldTheory/Finite/Basic.lean | 168 | 176 | theorem sum_subgroup_units [Ring K] [NoZeroDivisors K]
{G : Subgroup Kˣ} [Fintype G] [Decidable (G = ⊥)] :
∑ x : G, (x.val : K) = if G = ⊥ then 1 else 0 := by |
by_cases G_bot : G = ⊥
· subst G_bot
simp only [ite_true, Subgroup.mem_bot, Fintype.card_ofSubsingleton, Nat.cast_ite, Nat.cast_one,
Nat.cast_zero, univ_unique, Set.default_coe_singleton, sum_singleton, Units.val_one]
· simp only [G_bot, ite_false]
exact sum_subgroup_units_eq_zero G_bot
| true |
import Mathlib.Algebra.CharP.Pi
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.CharP.Subring
import Mathlib.Algebra.Ring.Pi
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Ring.Subring.Basic
... | Mathlib/RingTheory/Perfection.lean | 406 | 413 | theorem preVal_mk {x : O} (hx : (Ideal.Quotient.mk _ x : ModP K v O hv p) ≠ 0) :
preVal K v O hv p (Ideal.Quotient.mk _ x) = v (algebraMap O K x) := by |
obtain ⟨r, hr⟩ : ∃ (a : O), a * (p : O) = (Quotient.mk'' x).out' - x :=
Ideal.mem_span_singleton'.1 <| Ideal.Quotient.eq.1 <| Quotient.sound' <| Quotient.mk_out' _
refine (if_neg hx).trans (v.map_eq_of_sub_lt <| lt_of_not_le ?_)
erw [← RingHom.map_sub, ← hr, hv.le_iff_dvd]
exact fun hprx =>
hx (Ideal.Q... | false |
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 151 | 153 | theorem map_comm {f₁ : α → β} {f₂ : α → γ} {g₁ : β → δ} {g₂ : γ → δ} (h : g₁ ∘ f₁ = g₂ ∘ f₂)
(a : α) :
(Option.map f₁ a).map g₁ = (Option.map f₂ a).map g₂ := by | rw [map_map, h, ← map_map]
| false |
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 | 162 | 165 | theorem consCases_cons {P : (∀ i : Fin n.succ, α i) → Sort v} (h : ∀ x₀ x, P (Fin.cons x₀ x))
(x₀ : α 0) (x : ∀ i : Fin n, α i.succ) : @consCases _ _ _ h (cons x₀ x) = h x₀ x := by |
rw [consCases, cast_eq]
congr
| false |
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}
open Finset
-- The namespace is here to distinguish fro... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 264 | 268 | theorem mem_compression_of_insert_mem_compression (h : insert a s ∈ 𝓓 a 𝒜) : s ∈ 𝓓 a 𝒜 := by |
by_cases ha : a ∈ s
· rwa [insert_eq_of_mem ha] at h
· rw [← erase_insert ha]
exact erase_mem_compression_of_mem_compression h
| false |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open scoped ENNReal
namespace MeasureTheory
variable {α E : Type*} {m0 : MeasurableSpace α} [NormedAddCommGroup E]
{p : ℝ≥0∞} (μ... | Mathlib/MeasureTheory/Function/LpSeminorm/ChebyshevMarkov.lean | 31 | 40 | theorem mul_meas_ge_le_pow_snorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
(hf : AEStronglyMeasurable f μ) (ε : ℝ≥0∞) :
ε * μ { x | ε ≤ (‖f x‖₊ : ℝ≥0∞) ^ p.toReal } ≤ snorm f p μ ^ p.toReal := by |
have : 1 / p.toReal * p.toReal = 1 := by
refine one_div_mul_cancel ?_
rw [Ne, ENNReal.toReal_eq_zero_iff]
exact not_or_of_not hp_ne_zero hp_ne_top
rw [← ENNReal.rpow_one (ε * μ { x | ε ≤ (‖f x‖₊ : ℝ≥0∞) ^ p.toReal }), ← this, ENNReal.rpow_mul]
gcongr
exact pow_mul_meas_ge_le_snorm μ hp_ne_zero hp_n... | false |
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.Convex.Deriv
#align_import analysis.convex.specific_functions.deriv from "leanprover-communi... | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 168 | 171 | theorem strictConcaveOn_sin_Icc : StrictConcaveOn ℝ (Icc 0 π) sin := by |
apply strictConcaveOn_of_deriv2_neg (convex_Icc _ _) continuousOn_sin fun x hx => ?_
rw [interior_Icc] at hx
simp [sin_pos_of_mem_Ioo hx]
| true |
import Mathlib.Algebra.MvPolynomial.Rename
#align_import data.mv_polynomial.comap from "leanprover-community/mathlib"@"aba31c938d3243cc671be7091b28a1e0814647ee"
namespace MvPolynomial
variable {σ : Type*} {τ : Type*} {υ : Type*} {R : Type*} [CommSemiring R]
noncomputable def comap (f : MvPolynomial σ R →ₐ[R] M... | Mathlib/Algebra/MvPolynomial/Comap.lean | 62 | 74 | theorem comap_comp_apply (f : MvPolynomial σ R →ₐ[R] MvPolynomial τ R)
(g : MvPolynomial τ R →ₐ[R] MvPolynomial υ R) (x : υ → R) :
comap (g.comp f) x = comap f (comap g x) := by |
funext i
trans aeval x (aeval (fun i => g (X i)) (f (X i)))
· apply eval₂Hom_congr rfl rfl
rw [AlgHom.comp_apply]
suffices g = aeval fun i => g (X i) by rw [← this]
exact aeval_unique g
· simp only [comap, aeval_eq_eval₂Hom, map_eval₂Hom, AlgHom.comp_apply]
refine eval₂Hom_congr ?_ rfl rfl
... | false |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section Sigma
variable {α : ι → Type*} {β : Type*} (s s₁ s₂ : Finset ι) (... | Mathlib/Data/Finset/Sigma.lean | 91 | 94 | theorem sigma_eq_biUnion [DecidableEq (Σi, α i)] (s : Finset ι) (t : ∀ i, Finset (α i)) :
s.sigma t = s.biUnion fun i => (t i).map <| Embedding.sigmaMk i := by |
ext ⟨x, y⟩
simp [and_left_comm]
| false |
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 | 91 | 96 | theorem card_memberSubfamily_add_card_nonMemberSubfamily (a : α) (𝒜 : Finset (Finset α)) :
(𝒜.memberSubfamily a).card + (𝒜.nonMemberSubfamily a).card = 𝒜.card := by |
rw [memberSubfamily, nonMemberSubfamily, card_image_of_injOn]
· conv_rhs => rw [← filter_card_add_filter_neg_card_eq_card (fun s => (a ∈ s))]
· apply (erase_injOn' _).mono
simp
| true |
import Mathlib.Geometry.Manifold.MFDeriv.Atlas
noncomputable section
open scoped Manifold
open Set
section UniqueMDiff
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type*}
[Topolog... | Mathlib/Geometry/Manifold/MFDeriv/UniqueDifferential.lean | 120 | 131 | theorem UniqueMDiffWithinAt.smooth_bundle_preimage {p : TotalSpace F Z}
(hs : UniqueMDiffWithinAt I s p.proj) :
UniqueMDiffWithinAt (I.prod 𝓘(𝕜, F)) (π F Z ⁻¹' s) p := by |
set e := trivializationAt F Z p.proj
have hp : p ∈ e.source := FiberBundle.mem_trivializationAt_proj_source
have : UniqueMDiffWithinAt (I.prod 𝓘(𝕜, F)) (s ×ˢ univ) (e p) := by
rw [← Prod.mk.eta (p := e p), FiberBundle.trivializationAt_proj_fst]
exact hs.prod (uniqueMDiffWithinAt_univ _)
rw [← e.left_... | false |
import Mathlib.Algebra.Module.Submodule.Ker
open Function Submodule
namespace LinearMap
variable {R N M : Type*} [Semiring R] [AddCommMonoid N] [Module R N]
[AddCommMonoid M] [Module R M] (f i : N →ₗ[R] M)
def iterateMapComap (n : ℕ) := (fun K : Submodule R N ↦ (K.map i).comap f)^[n]
| Mathlib/Algebra/Module/Submodule/IterateMapComap.lean | 45 | 56 | theorem iterateMapComap_le_succ (K : Submodule R N) (h : K.map f ≤ K.map i) (n : ℕ) :
f.iterateMapComap i n K ≤ f.iterateMapComap i (n + 1) K := by |
nth_rw 2 [iterateMapComap]
rw [iterate_succ', Function.comp_apply, ← iterateMapComap, ← map_le_iff_le_comap]
induction n with
| zero => exact h
| succ n ih =>
simp_rw [iterateMapComap, iterate_succ', Function.comp_apply]
calc
_ ≤ (f.iterateMapComap i n K).map i := map_comap_le _ _
_ ≤ (((... | true |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 136 | 143 | theorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by |
rcases eq_or_ne a b with (rfl | h)
· simp
· calc
Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self
_ = closure (Ioo a b) := (closure_Ioo h).symm
_ ⊆ closure (interior (Ioc a b)) :=
closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)
| false |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.NormedSpace.WithLp
open Real Set Filter RCLike Bornology Uniformity Topology NNReal ENNReal
noncomputable section
variable (p : ℝ≥0∞) (𝕜 α β : Type*)
namespace WithLp
section DistNorm
section EDist
variable [EDist α] [EDist β]
open scope... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 171 | 174 | theorem prod_edist_eq_sup (f g : WithLp ∞ (α × β)) :
edist f g = edist f.fst g.fst ⊔ edist f.snd g.snd := by |
dsimp [edist]
exact if_neg ENNReal.top_ne_zero
| true |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 64 | 83 | theorem integral_comp_smul (f : E → F) (R : ℝ) :
∫ x, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ x, f x ∂μ := by |
by_cases hF : CompleteSpace F; swap
· simp [integral, hF]
rcases eq_or_ne R 0 with (rfl | hR)
· simp only [zero_smul, integral_const]
rcases Nat.eq_zero_or_pos (finrank ℝ E) with (hE | hE)
· have : Subsingleton E := finrank_zero_iff.1 hE
have : f = fun _ => f 0 := by ext x; rw [Subsingleton.elim ... | true |
import Mathlib.LinearAlgebra.Alternating.Basic
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import linear_algebra.alternating from "leanprover-community/mathlib"@"0c1d80f5a86b36c1db32e021e8d19ae7809d5b79"
suppress_compilation
open TensorProduct
vari... | Mathlib/LinearAlgebra/Alternating/DomCoprod.lean | 212 | 222 | theorem MultilinearMap.domCoprod_alternization_coe [DecidableEq ιa] [DecidableEq ιb]
(a : MultilinearMap R' (fun _ : ιa => Mᵢ) N₁) (b : MultilinearMap R' (fun _ : ιb => Mᵢ) N₂) :
MultilinearMap.domCoprod (MultilinearMap.alternatization a)
(MultilinearMap.alternatization b) =
∑ σa : Perm ιa, ∑ σb : P... |
simp_rw [← MultilinearMap.domCoprod'_apply, MultilinearMap.alternatization_coe]
simp_rw [TensorProduct.sum_tmul, TensorProduct.tmul_sum, _root_.map_sum,
← TensorProduct.smul_tmul', TensorProduct.tmul_smul]
rfl
| true |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.continuous_affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901fe9326d83"
structure ContinuousAffineMap (R : T... | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | 127 | 129 | theorem mk_coe (f : P →ᴬ[R] Q) (h) : (⟨(f : P →ᵃ[R] Q), h⟩ : P →ᴬ[R] Q) = f := by |
ext
rfl
| false |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section SigmaLift
variable {α β γ : ι → Type*} [DecidableEq ι]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 156 | 173 | theorem mem_sigmaLift (f : ∀ ⦃i⦄, α i → β i → Finset (γ i)) (a : Sigma α) (b : Sigma β)
(x : Sigma γ) :
x ∈ sigmaLift f a b ↔ ∃ (ha : a.1 = x.1) (hb : b.1 = x.1), x.2 ∈ f (ha ▸ a.2) (hb ▸ b.2) := by |
obtain ⟨⟨i, a⟩, j, b⟩ := a, b
obtain rfl | h := Decidable.eq_or_ne i j
· constructor
· simp_rw [sigmaLift]
simp only [dite_eq_ite, ite_true, mem_map, Embedding.sigmaMk_apply, forall_exists_index,
and_imp]
rintro x hx rfl
exact ⟨rfl, rfl, hx⟩
· rintro ⟨⟨⟩, ⟨⟩, hx⟩
rw [sigma... | true |
import Mathlib.RingTheory.FiniteType
#align_import ring_theory.rees_algebra from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open Polynomial
def reesAlgebra : Subalgebra... | Mathlib/RingTheory/ReesAlgebra.lean | 82 | 95 | theorem monomial_mem_adjoin_monomial {I : Ideal R} {n : ℕ} {r : R} (hr : r ∈ I ^ n) :
monomial n r ∈ Algebra.adjoin R (Submodule.map (monomial 1 : R →ₗ[R] R[X]) I : Set R[X]) := by |
induction' n with n hn generalizing r
· exact Subalgebra.algebraMap_mem _ _
· rw [pow_succ'] at hr
apply Submodule.smul_induction_on
-- Porting note: did not need help with motive previously
(p := fun r => (monomial (Nat.succ n)) r ∈ Algebra.adjoin R (Submodule.map (monomial 1) I)) hr
· intro... | false |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 114 | 116 | theorem EuclideanSpace.norm_eq {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n]
(x : EuclideanSpace 𝕜 n) : ‖x‖ = √(∑ i, ‖x i‖ ^ 2) := by |
simpa only [Real.coe_sqrt, NNReal.coe_sum] using congr_arg ((↑) : ℝ≥0 → ℝ) x.nnnorm_eq
| false |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
#align_import linear_algebra.free_module.pid from "leanprover-community/mathlib"@"d87199d51218d36a0a42c66c82d147b5a7ff87b3"
universe u v
section Ring
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/FreeModule/PID.lean | 72 | 81 | theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M} (b : Basis ι R O)
(hNO : N ≤ O) {ϕ : O →ₗ[R] R} (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
[(ϕ.submoduleImage N).IsPrincipal] (hgen : generator (ϕ.submoduleImage N) = 0) : N = ⊥ := by |
rw [Submodule.eq_bot_iff]
intro x hx
refine (mk_eq_zero _ _).mp (show (⟨x, hNO hx⟩ : O) = 0 from b.ext_elem fun i ↦ ?_)
rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ
rw [LinearEquiv.map_zero, Finsupp.zero_apply]
refine (Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)... | false |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.ShrinkingLemma
import Mathlib.Topology.UrysohnsLemma
#align_import topology.partition_of_unity from "leanprover-... | Mathlib/Topology/PartitionOfUnity.lean | 214 | 220 | theorem sum_finsupport_smul_eq_finsum {M : Type*} [AddCommGroup M] [Module ℝ M] (φ : ι → X → M) :
∑ i ∈ ρ.finsupport x₀, ρ i x₀ • φ i x₀ = ∑ᶠ i, ρ i x₀ • φ i x₀ := by |
apply (finsum_eq_sum_of_support_subset _ _).symm
have : (fun i ↦ (ρ i) x₀ • φ i x₀) = (fun i ↦ (ρ i) x₀) • (fun i ↦ φ i x₀) :=
funext fun _ => (Pi.smul_apply' _ _ _).symm
rw [ρ.coe_finsupport x₀, this, support_smul]
exact inter_subset_left
| false |
import Mathlib.CategoryTheory.Linear.LinearFunctor
import Mathlib.CategoryTheory.Monoidal.Preadditive
#align_import category_theory.monoidal.linear from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
namespace CategoryTheory
open CategoryTheory.Limits
open CategoryTheory.MonoidalCateg... | Mathlib/CategoryTheory/Monoidal/Linear.lean | 58 | 70 | theorem monoidalLinearOfFaithful {D : Type*} [Category D] [Preadditive D] [Linear R D]
[MonoidalCategory D] [MonoidalPreadditive D] (F : MonoidalFunctor D C) [F.Faithful]
[F.toFunctor.Additive] [F.toFunctor.Linear R] : MonoidalLinear R D :=
{ whiskerLeft_smul := by |
intros X Y Z r f
apply F.toFunctor.map_injective
rw [F.map_whiskerLeft]
simp
smul_whiskerRight := by
intros r X Y f Z
apply F.toFunctor.map_injective
rw [F.map_whiskerRight]
simp }
| false |
import Mathlib.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open scoped Topology
open Filter Set TopologicalSpace
variable {X α : Type*} {ι : Sort*}
section BaireTheorem
variable [TopologicalSpace... | Mathlib/Topology/Baire/Lemmas.lean | 50 | 55 | theorem dense_sInter_of_isOpen {S : Set (Set X)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) := by |
rcases S.eq_empty_or_nonempty with h | h
· simp [h]
· rcases hS.exists_eq_range h with ⟨f, rfl⟩
exact dense_iInter_of_isOpen_nat (forall_mem_range.1 ho) (forall_mem_range.1 hd)
| false |
import Mathlib.Data.Multiset.Nodup
#align_import data.multiset.dedup from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α β : Type*} [DecidableEq α]
def dedup (s : Multiset α) : Multiset α :=
Quot.liftOn s (fun l => (l.dedup : Multiset α)... | Mathlib/Data/Multiset/Dedup.lean | 112 | 113 | theorem le_dedup_self {s : Multiset α} : s ≤ dedup s ↔ Nodup s := by |
rw [le_dedup, and_iff_right le_rfl]
| false |
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Tactic.IntervalCases
namespace Polynomial
section IsDomain
variable {R : Type*} [CommRing R] [IsDomain R]
| Mathlib/Algebra/Polynomial/SpecificDegree.lean | 22 | 34 | theorem Monic.irreducible_iff_roots_eq_zero_of_degree_le_three {p : R[X]} (hp : p.Monic)
(hp2 : 2 ≤ p.natDegree) (hp3 : p.natDegree ≤ 3) : Irreducible p ↔ p.roots = 0 := by |
have hp0 : p ≠ 0 := hp.ne_zero
have hp1 : p ≠ 1 := by rintro rfl; rw [natDegree_one] at hp2; cases hp2
rw [hp.irreducible_iff_lt_natDegree_lt hp1]
simp_rw [show p.natDegree / 2 = 1 from
(Nat.div_le_div_right hp3).antisymm
(by apply Nat.div_le_div_right (c := 2) hp2),
show Finset.Ioc 0 1 = {1}... | true |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.Probability.Kernel.Disintegration.CdfToKernel
#align_import probability.kernel.cond_cdf from "leanprover-community/mathlib"@"3b88f4005dc2e28d42f974cc1ce838f0dafb39b8"
open MeasureTheory Set Filter TopologicalSpace
open scoped NNReal ENNReal Me... | Mathlib/Probability/Kernel/Disintegration/CondCdf.lean | 87 | 89 | theorem iInf_IicSnd_gt (t : ℚ) {s : Set α} (hs : MeasurableSet s) [IsFiniteMeasure ρ] :
⨅ r : { r' : ℚ // t < r' }, ρ.IicSnd r s = ρ.IicSnd t s := by |
simp_rw [ρ.IicSnd_apply _ hs, Measure.iInf_rat_gt_prod_Iic hs]
| true |
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
open MeasureTheory Measure FiniteDimensional
variable {E F G W : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F]
[NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedAddCo... | Mathlib/Analysis/Calculus/LineDeriv/IntegrationByParts.lean | 101 | 151 | theorem integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable
{f f' : E → F} {g g' : E → G} {v : E} {B : F →L[ℝ] G →L[ℝ] W}
(hf'g : Integrable (fun x ↦ B (f' x) (g x)) μ) (hfg' : Integrable (fun x ↦ B (f x) (g' x)) μ)
(hfg : Integrable (fun x ↦ B (f x) (g x)) μ)
(hf : ∀ x, HasLineDerivAt ℝ f ... |
by_cases hW : CompleteSpace W; swap
· simp [integral, hW]
rcases eq_or_ne v 0 with rfl|hv
· have Hf' x : f' x = 0 := by
simpa [(hasLineDerivAt_zero (f := f) (x := x)).lineDeriv] using (hf x).lineDeriv.symm
have Hg' x : g' x = 0 := by
simpa [(hasLineDerivAt_zero (f := g) (x := x)).lineDeriv] usi... | false |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 289 | 290 | theorem map_zero₂ (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (y : F) : f 0 y = 0 := by |
rw [f.map_zero, zero_apply]
| false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.