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 | goals listlengths 0 224 | goals_before listlengths 0 220 |
|---|---|---|---|---|---|---|---|
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
#align_import algebra.lie.abelian from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
class LieModule.IsTrivial (L : Type v) (M : Type w) [Bracket L M] [Zero M] : Prop where
triv... | Mathlib/Algebra/Lie/Abelian.lean | 91 | 96 | theorem commutative_ring_iff_abelian_lie_ring {A : Type v} [Ring A] :
Std.Commutative (α := A) (· * ·) ↔ IsLieAbelian A := by |
have h₁ : Std.Commutative (α := A) (· * ·) ↔ ∀ a b : A, a * b = b * a :=
⟨fun h => h.1, fun h => ⟨h⟩⟩
have h₂ : IsLieAbelian A ↔ ∀ a b : A, ⁅a, b⁆ = 0 := ⟨fun h => h.1, fun h => ⟨h⟩⟩
simp only [h₁, h₂, LieRing.of_associative_ring_bracket, sub_eq_zero]
| [
" ⁅x, m⁆ = 0",
" ⁅x, y⁆ = 0",
" ⁅f u, y⁆ = 0",
" ⁅f u, f v⁆ = 0",
" (Std.Commutative fun x x_1 => x * x_1) ↔ IsLieAbelian A"
] | [
" ⁅x, m⁆ = 0",
" ⁅x, y⁆ = 0",
" ⁅f u, y⁆ = 0",
" ⁅f u, f v⁆ = 0"
] |
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheory.MeasurableSpace.Basic
open Set
namespace MeasureTheory
variable {ι : Type _} {α : ι → Type _}
section cylinder
def cylinder (s : Finset ι) (S : Set (∀ i : s, α... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 169 | 183 | theorem cylinder_eq_empty_iff [h_nonempty : Nonempty (∀ i, α i)] (s : Finset ι)
(S : Set (∀ i : s, α i)) :
cylinder s S = ∅ ↔ S = ∅ := by |
refine ⟨fun h ↦ ?_, fun h ↦ by (rw [h]; exact cylinder_empty _)⟩
by_contra hS
rw [← Ne, ← nonempty_iff_ne_empty] at hS
let f := hS.some
have hf : f ∈ S := hS.choose_spec
classical
let f' : ∀ i, α i := fun i ↦ if hi : i ∈ s then f ⟨i, hi⟩ else h_nonempty.some i
have hf' : f' ∈ cylinder s S := by
rw ... | [
" cylinder s ∅ = ∅",
" cylinder s univ = univ",
" cylinder s S = ∅ ↔ S = ∅",
" cylinder s S = ∅",
" S = ∅",
" False",
" f' ∈ cylinder s S",
" (fun i => f' ↑i) ∈ S"
] | [
" cylinder s ∅ = ∅",
" cylinder s univ = univ"
] |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localization.Basic
#align_import data.polynomial.laurent from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Polynomial Func... | Mathlib/Algebra/Polynomial/Laurent.lean | 203 | 204 | theorem mul_T_assoc (f : R[T;T⁻¹]) (m n : ℤ) : f * T m * T n = f * T (m + n) := by |
simp [← T_add, mul_assoc]
| [
" (C t) n = if n = 0 then t else 0",
" (if 0 = n then t else 0) = if n = 0 then t else 0",
" T (m + n) = T m * T n",
" T (m - n) = T m * T (-n)",
" T m ^ n = T (↑n * m)",
" f * T m * T n = f * T (m + n)"
] | [
" (C t) n = if n = 0 then t else 0",
" (if 0 = n then t else 0) = if n = 0 then t else 0",
" T (m + n) = T m * T n",
" T (m - n) = T m * T (-n)",
" T m ^ n = T (↑n * m)"
] |
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.BilinearForm.DualLattice
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Trace
#align_import ring_theory.dedekind_domain.... | Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean | 106 | 112 | theorem integralClosure_le_span_dualBasis [IsSeparable K L] {ι : Type*} [Fintype ι] [DecidableEq ι]
(b : Basis ι K L) (hb_int : ∀ i, IsIntegral A (b i)) [IsIntegrallyClosed A] :
Subalgebra.toSubmodule (integralClosure A L) ≤
Submodule.span A (Set.range <| (traceForm K L).dualBasis (traceForm_nondegenerate K... |
refine le_trans ?_ (IsIntegralClosure.range_le_span_dualBasis (integralClosure A L) b hb_int)
intro x hx
exact ⟨⟨x, hx⟩, rfl⟩
| [
" IsLocalization (algebraMapSubmonoid C A⁰) L",
" ∀ (y : ↥(algebraMapSubmonoid C A⁰)), IsUnit ((algebraMap C L) ↑y)",
" IsUnit ((algebraMap C L) ↑⟨(algebraMap A C) x, ⋯⟩)",
" x ≠ 0",
" ∃ x, z * (algebraMap C L) ↑x.2 = (algebraMap C L) x.1",
" z * (algebraMap C L) ↑(x, ⟨(algebraMap A C) ↑m, ⋯⟩).2 = (algebr... | [
" IsLocalization (algebraMapSubmonoid C A⁰) L",
" ∀ (y : ↥(algebraMapSubmonoid C A⁰)), IsUnit ((algebraMap C L) ↑y)",
" IsUnit ((algebraMap C L) ↑⟨(algebraMap A C) x, ⋯⟩)",
" x ≠ 0",
" ∃ x, z * (algebraMap C L) ↑x.2 = (algebraMap C L) x.1",
" z * (algebraMap C L) ↑(x, ⟨(algebraMap A C) ↑m, ⋯⟩).2 = (algebr... |
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 144 | 144 | theorem right_mem_Icc : b ∈ Icc a b ↔ a ≤ b := by | simp only [mem_Icc, and_true_iff, le_rfl]
| [
" (Icc a b).Nonempty ↔ a ≤ b",
" (Ico a b).Nonempty ↔ a < b",
" (Ioc a b).Nonempty ↔ a < b",
" (Ioo a b).Nonempty ↔ a < b",
" Icc a b = ∅ ↔ ¬a ≤ b",
" Ico a b = ∅ ↔ ¬a < b",
" Ioc a b = ∅ ↔ ¬a < b",
" Ioo a b = ∅ ↔ ¬a < b",
" a ∈ Icc a b ↔ a ≤ b",
" a ∈ Ico a b ↔ a < b",
" b ∈ Icc a b ↔ a ≤ b"
] | [
" (Icc a b).Nonempty ↔ a ≤ b",
" (Ico a b).Nonempty ↔ a < b",
" (Ioc a b).Nonempty ↔ a < b",
" (Ioo a b).Nonempty ↔ a < b",
" Icc a b = ∅ ↔ ¬a ≤ b",
" Ico a b = ∅ ↔ ¬a < b",
" Ioc a b = ∅ ↔ ¬a < b",
" Ioo a b = ∅ ↔ ¬a < b",
" a ∈ Icc a b ↔ a ≤ b",
" a ∈ Ico a b ↔ a < b"
] |
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
noncomputable section
universe u v v' v''
variable {K : Type u} {V V₁ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
open Cardinal Basis Submodule Function Set
namespace LinearMap
section Ring
... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 58 | 60 | theorem lift_rank_comp_le_right (g : V →ₗ[K] V') (f : V' →ₗ[K] V'') :
Cardinal.lift.{v'} (rank (f.comp g)) ≤ Cardinal.lift.{v''} (rank g) := by |
rw [rank, rank, LinearMap.range_comp]; exact lift_rank_map_le _ _
| [
" rank 0 = 0",
" (f ∘ₗ g).rank ≤ f.rank",
" range (f ∘ₗ g) ≤ range f",
" Submodule.map f (range g) ≤ range f",
" lift.{v', v''} (f ∘ₗ g).rank ≤ lift.{v'', v'} g.rank",
" lift.{v', v''} (Module.rank K ↥(Submodule.map f (range g))) ≤ lift.{v'', v'} (Module.rank K ↥(range g))"
] | [
" rank 0 = 0",
" (f ∘ₗ g).rank ≤ f.rank",
" range (f ∘ₗ g) ≤ range f",
" Submodule.map f (range g) ≤ range f"
] |
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 | 112 | 112 | theorem laverage_zero_measure (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂(0 : Measure α) = 0 := by | simp [laverage]
| [
" ⨍⁻ (_x : α), 0 ∂μ = 0",
" ⨍⁻ (x : α), f x ∂0 = 0"
] | [
" ⨍⁻ (_x : α), 0 ∂μ = 0"
] |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
#align_import analysis.special_functions.trigonometric.inverse from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter
open S... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 133 | 134 | theorem arcsin_of_le_neg_one {x : ℝ} (hx : x ≤ -1) : arcsin x = -(π / 2) := by |
rw [← arcsin_projIcc, projIcc_of_le_left _ hx, Subtype.coe_mk, arcsin_neg_one]
| [
" range arcsin = Icc (-(π / 2)) (π / 2)",
" Subtype.val '' range (IccExtend arcsin.proof_2 ⇑sinOrderIso.symm) = Icc (-(π / 2)) (π / 2)",
" (↑(projIcc (-1) 1 ⋯ x)).arcsin = x.arcsin",
" x.arcsin.sin = x",
" x.sin.arcsin.sin = x.sin",
" y.arcsin = x",
" x.sin.arcsin = x",
" x.arcsin = π / 2",
" (-(π /... | [
" range arcsin = Icc (-(π / 2)) (π / 2)",
" Subtype.val '' range (IccExtend arcsin.proof_2 ⇑sinOrderIso.symm) = Icc (-(π / 2)) (π / 2)",
" (↑(projIcc (-1) 1 ⋯ x)).arcsin = x.arcsin",
" x.arcsin.sin = x",
" x.sin.arcsin.sin = x.sin",
" y.arcsin = x",
" x.sin.arcsin = x",
" x.arcsin = π / 2",
" (-(π /... |
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 | 114 | 115 | theorem sinh_log {x : ℝ} (hx : 0 < x) : sinh (log x) = (x - x⁻¹) / 2 := by |
rw [sinh_eq, exp_neg, exp_log hx]
| [
" x.log = expOrderIso.symm ⟨x, hx⟩",
" expOrderIso.symm ⟨|x|, ⋯⟩ = expOrderIso.symm ⟨x, hx⟩",
" |x| = x",
" rexp x.log = |x|",
" rexp x.log = x",
" rexp x.log = -x",
" |x| = -x",
" x ≤ rexp x.log",
" 0 ≤ 1",
" x ≤ |x|",
" rexp (log 1) = rexp 0",
" |x|.log = x.log",
" (-x).log = x.log",
" x... | [
" x.log = expOrderIso.symm ⟨x, hx⟩",
" expOrderIso.symm ⟨|x|, ⋯⟩ = expOrderIso.symm ⟨x, hx⟩",
" |x| = x",
" rexp x.log = |x|",
" rexp x.log = x",
" rexp x.log = -x",
" |x| = -x",
" x ≤ rexp x.log",
" 0 ≤ 1",
" x ≤ |x|",
" rexp (log 1) = rexp 0",
" |x|.log = x.log",
" (-x).log = x.log"
] |
import Mathlib.Analysis.SpecialFunctions.Integrals
#align_import data.real.pi.wallis from "leanprover-community/mathlib"@"980755c33b9168bc82f774f665eaa27878140fac"
open scoped Real Topology Nat
open Filter Finset intervalIntegral
namespace Real
namespace Wallis
set_option linter.uppercaseLean3 false
noncomp... | Mathlib/Data/Real/Pi/Wallis.lean | 62 | 75 | theorem W_eq_factorial_ratio (n : ℕ) :
W n = 2 ^ (4 * n) * n ! ^ 4 / ((2 * n)! ^ 2 * (2 * n + 1)) := by |
induction' n with n IH
· simp only [W, prod_range_zero, Nat.factorial_zero, mul_zero, pow_zero,
algebraMap.coe_one, one_pow, mul_one, algebraMap.coe_zero, zero_add, div_self, Ne,
one_ne_zero, not_false_iff]
norm_num
· unfold W at IH ⊢
rw [prod_range_succ, IH, _root_.div_mul_div_comm, _root_.d... | [
" 0 < W k",
" 0 < W 0",
" 0 < ∏ i ∈ range 0, (2 * ↑i + 2) / (2 * ↑i + 1) * ((2 * ↑i + 2) / (2 * ↑i + 3))",
" 0 < W (k + 1)",
" 0 < W k * ((2 * ↑k + 2) / (2 * ↑k + 1) * ((2 * ↑k + 2) / (2 * ↑k + 3)))",
" 0 < 2 * ↑k + 2",
" 0 < 2 * ↑k + 1",
" 0 < 2 * ↑k + 3",
" W n = 2 ^ (4 * n) * ↑n ! ^ 4 / (↑(2 * n)... | [
" 0 < W k",
" 0 < W 0",
" 0 < ∏ i ∈ range 0, (2 * ↑i + 2) / (2 * ↑i + 1) * ((2 * ↑i + 2) / (2 * ↑i + 3))",
" 0 < W (k + 1)",
" 0 < W k * ((2 * ↑k + 2) / (2 * ↑k + 1) * ((2 * ↑k + 2) / (2 * ↑k + 3)))",
" 0 < 2 * ↑k + 2",
" 0 < 2 * ↑k + 1",
" 0 < 2 * ↑k + 3"
] |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toNNReal_iInf (hf : ∀ i, f ... | Mathlib/Data/ENNReal/Real.lean | 548 | 553 | theorem toNNReal_sInf (s : Set ℝ≥0∞) (hs : ∀ r ∈ s, r ≠ ∞) :
(sInf s).toNNReal = sInf (ENNReal.toNNReal '' s) := by |
have hf : ∀ i, ((↑) : s → ℝ≥0∞) i ≠ ∞ := fun ⟨r, rs⟩ => hs r rs
-- Porting note: `← sInf_image'` had to be replaced by `← image_eq_range` as the lemmas are used
-- in a different order.
simpa only [← sInf_range, ← image_eq_range, Subtype.range_coe_subtype] using (toNNReal_iInf hf)
| [
" (iInf f).toNNReal = ⨅ i, (f i).toNNReal",
" (⨅ i, ↑(f i)).toNNReal = ⨅ i, ((fun i => ↑(f i)) i).toNNReal",
" (sInf s).toNNReal = sInf (ENNReal.toNNReal '' s)"
] | [
" (iInf f).toNNReal = ⨅ i, (f i).toNNReal",
" (⨅ i, ↑(f i)).toNNReal = ⨅ i, ((fun i => ↑(f i)) i).toNNReal"
] |
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 | 75 | 79 | theorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by |
apply Subset.antisymm
· exact closure_minimal Ico_subset_Icc_self isClosed_Icc
· apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)
rw [closure_Ioo hab]
| [
" closure (Ioi a) = Ici a",
" closure (Ioi a) ⊆ Ici a",
" Ici a ⊆ closure (Ioi a)",
" a ∈ closure (Ioi a)",
" closure (Ioo a b) = Icc a b",
" closure (Ioo a b) ⊆ Icc a b",
" Icc a b ⊆ closure (Ioo a b)",
" {a, b} ⊆ closure (Ioo a b)",
" a ∈ closure (Ioo a b) ∧ b ∈ closure (Ioo a b)",
" ∅ ⊆ closure... | [
" closure (Ioi a) = Ici a",
" closure (Ioi a) ⊆ Ici a",
" Ici a ⊆ closure (Ioi a)",
" a ∈ closure (Ioi a)",
" closure (Ioo a b) = Icc a b",
" closure (Ioo a b) ⊆ Icc a b",
" Icc a b ⊆ closure (Ioo a b)",
" {a, b} ⊆ closure (Ioo a b)",
" a ∈ closure (Ioo a b) ∧ b ∈ closure (Ioo a b)",
" ∅ ⊆ closure... |
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic.Abel
open Lean Elab Meta Tactic Qq
initialize registerTraceClass `abel
initialize registerTraceClass `abel.detail
structure Context where
α : Expr
univ :... | Mathlib/Tactic/Abel.lean | 132 | 134 | theorem const_add_termg {α} [AddCommGroup α] (k n x a a') (h : k + a = a') :
k + @termg α _ n x a = termg n x a' := by |
simp [h.symm, termg, add_comm, add_assoc]
| [
" k + term n x a = term n x a'",
" k + termg n x a = termg n x a'"
] | [
" k + term n x a = term n x a'"
] |
import Mathlib.Topology.LocalAtTarget
import Mathlib.AlgebraicGeometry.Morphisms.Basic
#align_import algebraic_geometry.morphisms.open_immersion from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace... | Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean | 34 | 38 | theorem isOpenImmersion_iff_stalk {f : X ⟶ Y} : IsOpenImmersion f ↔
OpenEmbedding f.1.base ∧ ∀ x, IsIso (PresheafedSpace.stalkMap f.1 x) := by |
constructor
· intro h; exact ⟨h.1, inferInstance⟩
· rintro ⟨h₁, h₂⟩; exact IsOpenImmersion.of_stalk_iso f h₁
| [
" IsOpenImmersion f ↔\n OpenEmbedding ⇑f.val.base ∧ ∀ (x : ↑↑X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)",
" IsOpenImmersion f →\n OpenEmbedding ⇑f.val.base ∧ ∀ (x : ↑↑X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)",
" OpenEmbedding ⇑f.val.base ∧ ∀ (x : ↑↑X.toPresheafedS... | [] |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 192 | 194 | theorem Fintype.not_linearIndependent_iff [Fintype ι] :
¬LinearIndependent R v ↔ ∃ g : ι → R, ∑ i, g i • v i = 0 ∧ ∃ i, g i ≠ 0 := by |
simpa using not_iff_not.2 Fintype.linearIndependent_iff
| [
" LinearIndependent R v ↔ ∀ (l : ι →₀ R), (Finsupp.total ι M R v) l = 0 → l = 0",
" (Finsupp.total ι M R v) (∑ i ∈ s, Finsupp.single i (g i)) = 0",
" g i = (Finsupp.lapply i) (Finsupp.single i (g i))",
" (Finsupp.lapply i) (Finsupp.single j (g j)) = 0",
" LinearIndependent R v ↔ ∀ (s : Finset ι) (g : ι → R)... | [
" LinearIndependent R v ↔ ∀ (l : ι →₀ R), (Finsupp.total ι M R v) l = 0 → l = 0",
" (Finsupp.total ι M R v) (∑ i ∈ s, Finsupp.single i (g i)) = 0",
" g i = (Finsupp.lapply i) (Finsupp.single i (g i))",
" (Finsupp.lapply i) (Finsupp.single j (g j)) = 0",
" LinearIndependent R v ↔ ∀ (s : Finset ι) (g : ι → R)... |
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ ℵ₀
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 90 | 90 | theorem mk_set_nat : #(Set ℕ) = 𝔠 := by | simp
| [
" lift.{v, u_1} 𝔠 = 𝔠",
" 𝔠 ≤ lift.{v, u} c ↔ 𝔠 ≤ c",
" lift.{v, u} c ≤ 𝔠 ↔ c ≤ 𝔠",
" 𝔠 < lift.{v, u} c ↔ 𝔠 < c",
" lift.{v, u} c < 𝔠 ↔ c < 𝔠",
" beth 1 = 𝔠",
" #(Set ℕ) = 𝔠"
] | [
" lift.{v, u_1} 𝔠 = 𝔠",
" 𝔠 ≤ lift.{v, u} c ↔ 𝔠 ≤ c",
" lift.{v, u} c ≤ 𝔠 ↔ c ≤ 𝔠",
" 𝔠 < lift.{v, u} c ↔ 𝔠 < c",
" lift.{v, u} c < 𝔠 ↔ c < 𝔠",
" beth 1 = 𝔠"
] |
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNR... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 122 | 127 | theorem of_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞) (hT : FinMeasAdditive (c • μ) T) :
FinMeasAdditive μ T := by |
refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT
rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top] at hμs
simp only [hc_ne_top, or_false_iff, Ne, false_and_iff] at hμs
exact hμs.2
| [
" 0 (s ∪ t) = 0 s + 0 t",
" FinMeasAdditive μ (T + T')",
" (T + T') (s ∪ t) = (T + T') s + (T + T') t",
" T s + T t + (T' s + T' t) = T s + T' s + (T t + T' t)",
" (fun s => c • T s) (s ∪ t) = (fun s => c • T s) s + (fun s => c • T s) t",
" FinMeasAdditive μ T",
" μ s = ⊤"
] | [
" 0 (s ∪ t) = 0 s + 0 t",
" FinMeasAdditive μ (T + T')",
" (T + T') (s ∪ t) = (T + T') s + (T + T') t",
" T s + T t + (T' s + T' t) = T s + T' s + (T t + T' t)",
" (fun s => c • T s) (s ∪ t) = (fun s => c • T s) s + (fun s => c • T s) t"
] |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 65 | 75 | theorem exists_coprime (m n : Nat) :
∃ m' n', Coprime m' n' ∧ m = m' * gcd m n ∧ n = n' * gcd m n := by |
cases eq_zero_or_pos (gcd m n) with
| inl h0 =>
rw [gcd_eq_zero_iff] at h0
refine ⟨1, 1, gcd_one_left 1, ?_⟩
simp [h0]
| inr hpos =>
exact ⟨_, _, coprime_div_gcd_div_gcd hpos,
(Nat.div_mul_cancel (gcd_dvd_left m n)).symm,
(Nat.div_mul_cancel (gcd_dvd_right m n)).symm⟩
| [
" k ∣ m",
" k ∣ n * m",
" ((k * m).gcd n).Coprime k",
" (m * k).gcd n = m.gcd n",
" m.gcd (k * n) = m.gcd n",
" m.gcd (n * k) = m.gcd n",
" (m / m.gcd n).Coprime (n / m.gcd n)",
" d ∣ 1",
" d ∣ m.gcd n",
" ∃ m' n', m'.Coprime n' ∧ m = m' * m.gcd n ∧ n = n' * m.gcd n",
" m = 1 * m.gcd n ∧ n = 1 *... | [
" k ∣ m",
" k ∣ n * m",
" ((k * m).gcd n).Coprime k",
" (m * k).gcd n = m.gcd n",
" m.gcd (k * n) = m.gcd n",
" m.gcd (n * k) = m.gcd n",
" (m / m.gcd n).Coprime (n / m.gcd n)",
" d ∣ 1",
" d ∣ m.gcd n"
] |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 304 | 312 | theorem isBigO_rpow_top_log_smul [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} (hab : b < a)
(hf : f =O[atTop] (· ^ (-a))) :
(fun t : ℝ => log t • f t) =O[atTop] (· ^ (-b)) := by |
refine
((isLittleO_log_rpow_atTop (sub_pos.mpr hab)).isBigO.smul hf).congr'
(eventually_of_forall fun t => by rfl)
((eventually_gt_atTop 0).mp (eventually_of_forall fun t ht => ?_))
simp only
rw [smul_eq_mul, ← rpow_add ht, ← sub_eq_add_neg, sub_eq_add_neg a, add_sub_cancel_left]
| [
" (fun t => t.log • f t) =O[atTop] fun x => x ^ (-b)",
" (fun x => x.log • f x) t = (fun t => t.log • f t) t",
" (fun x => x ^ (a - b) • x ^ (-a)) t = (fun x => x ^ (-b)) t",
" t ^ (a - b) • t ^ (-a) = t ^ (-b)"
] | [] |
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.ProbabilityMassFunction.Constructions
open scoped Classical MeasureTheory NNReal ENNReal
-- TODO: We can't `open ProbabilityTheory` without opening the `ProbabilityThe... | Mathlib/Probability/Distributions/Uniform.lean | 114 | 121 | theorem pdf_eq_zero_of_measure_eq_zero_or_top {X : Ω → E} {s : Set E}
(hu : IsUniform X s ℙ μ) (hμs : μ s = 0 ∨ μ s = ∞) : pdf X ℙ μ =ᵐ[μ] 0 := by |
rcases hμs with H|H
· simp only [IsUniform, ProbabilityTheory.cond, H, ENNReal.inv_zero, restrict_eq_zero.mpr H,
smul_zero] at hu
simp [pdf, hu]
· simp only [IsUniform, ProbabilityTheory.cond, H, ENNReal.inv_top, zero_smul] at hu
simp [pdf, hu]
| [
" AEMeasurable X ℙ",
" False",
" 0 = 1",
" 0 Set.univ = 1",
" Measure.map X ℙ ≪ μ",
" ProbabilityTheory.cond μ s ≪ μ",
" ℙ (X ⁻¹' A) = μ (s ∩ A) / μ s",
" ℙ Set.univ = 1",
" IsUniform X (toMeasurable μ s) ℙ μ ↔ IsUniform X s ℙ μ",
" Measure.map X ℙ = ProbabilityTheory.cond μ (toMeasurable μ s) ↔ M... | [
" AEMeasurable X ℙ",
" False",
" 0 = 1",
" 0 Set.univ = 1",
" Measure.map X ℙ ≪ μ",
" ProbabilityTheory.cond μ s ≪ μ",
" ℙ (X ⁻¹' A) = μ (s ∩ A) / μ s",
" ℙ Set.univ = 1",
" IsUniform X (toMeasurable μ s) ℙ μ ↔ IsUniform X s ℙ μ",
" Measure.map X ℙ = ProbabilityTheory.cond μ (toMeasurable μ s) ↔ M... |
import Mathlib.MeasureTheory.Function.AEEqFun.DomAct
import Mathlib.MeasureTheory.Function.LpSpace
set_option autoImplicit true
open MeasureTheory Filter
open scoped ENNReal
namespace DomMulAct
variable {M N α E : Type*} [MeasurableSpace M] [MeasurableSpace N]
[MeasurableSpace α] [NormedAddCommGroup E] {μ : Me... | Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean | 103 | 104 | theorem edist_smul_Lp (c : Mᵈᵐᵃ) (f g : Lp E p μ) : edist (c • f) (c • g) = edist f g := by |
simp only [Lp.edist_dist, dist_smul_Lp]
| [
" ∀ (f g : ↥(Lp E p μ)), c • (f + g) = c • f + c • g",
" c • (⟨Quot.mk Setoid.r a✝¹, property✝¹⟩ + ⟨Quot.mk Setoid.r a✝, property✝⟩) =\n c • ⟨Quot.mk Setoid.r a✝¹, property✝¹⟩ + c • ⟨Quot.mk Setoid.r a✝, property✝⟩",
" c • -f = -(c • f)",
" c • -⟨Quot.mk Setoid.r a✝, property✝⟩ = -(c • ⟨Quot.mk Setoid.r a✝... | [
" ∀ (f g : ↥(Lp E p μ)), c • (f + g) = c • f + c • g",
" c • (⟨Quot.mk Setoid.r a✝¹, property✝¹⟩ + ⟨Quot.mk Setoid.r a✝, property✝⟩) =\n c • ⟨Quot.mk Setoid.r a✝¹, property✝¹⟩ + c • ⟨Quot.mk Setoid.r a✝, property✝⟩",
" c • -f = -(c • f)",
" c • -⟨Quot.mk Setoid.r a✝, property✝⟩ = -(c • ⟨Quot.mk Setoid.r a✝... |
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Fin.Tuple.Reflection
#align_import data.matrix.reflection from "leanprover-community/mathlib"@"820b22968a2bc4a47ce5cf1d2f36a9ebe52510aa"
open Matrix
namespace Matrix
variable {l m n : ℕ} {α β : Type*}
def Forall : ∀ {m n}... | Mathlib/Data/Matrix/Reflection.lean | 159 | 162 | theorem mulᵣ_eq [Mul α] [AddCommMonoid α] (A : Matrix (Fin l) (Fin m) α)
(B : Matrix (Fin m) (Fin n) α) : mulᵣ A B = A * B := by |
simp [mulᵣ, Function.comp, Matrix.transpose]
rfl
| [
" Forall P ↔ ∀ (x : Matrix (Fin (m + 1)) (Fin n) α), P x",
" (∀ (x : Fin n → α) (x_1 : Matrix (Fin m) (Fin n) α), P (of (vecCons x x_1))) ↔\n ∀ (x : Matrix (Fin (m + 1)) (Fin n) α), P x",
" Exists P ↔ ∃ x, P x",
" (∃ r A, P (of (vecCons r A))) ↔ ∃ x, P x",
" A.transposeᵣ i j = Aᵀ i j",
" of (vecCons (F... | [
" Forall P ↔ ∀ (x : Matrix (Fin (m + 1)) (Fin n) α), P x",
" (∀ (x : Fin n → α) (x_1 : Matrix (Fin m) (Fin n) α), P (of (vecCons x x_1))) ↔\n ∀ (x : Matrix (Fin (m + 1)) (Fin n) α), P x",
" Exists P ↔ ∃ x, P x",
" (∃ r A, P (of (vecCons r A))) ↔ ∃ x, P x",
" A.transposeᵣ i j = Aᵀ i j",
" of (vecCons (F... |
import Mathlib.Analysis.Convex.StrictConvexBetween
import Mathlib.Geometry.Euclidean.Basic
#align_import geometry.euclidean.sphere.basic from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RealInnerProductSpace
namespace EuclideanGeometry
variable {V : Type... | Mathlib/Geometry/Euclidean/Sphere/Basic.lean | 124 | 128 | theorem Sphere.center_eq_iff_eq_of_mem {s₁ s₂ : Sphere P} {p : P} (hs₁ : p ∈ s₁) (hs₂ : p ∈ s₂) :
s₁.center = s₂.center ↔ s₁ = s₂ := by |
refine ⟨fun h => Sphere.ext _ _ h ?_, fun h => h ▸ rfl⟩
rw [mem_sphere] at hs₁ hs₂
rw [← hs₁, ← hs₂, h]
| [
" { center := s.center, radius := s.radius } = s",
" { center := s.center, radius := s.radius }.center = s.center",
" { center := s.center, radius := s.radius }.radius = s.radius",
" s₁ ≠ s₂ ↔ s₁.center ≠ s₂.center ∨ s₁.radius ≠ s₂.radius",
" s₁.center = s₂.center ↔ s₁ = s₂",
" s₁.radius = s₂.radius"
] | [
" { center := s.center, radius := s.radius } = s",
" { center := s.center, radius := s.radius }.center = s.center",
" { center := s.center, radius := s.radius }.radius = s.radius",
" s₁ ≠ s₂ ↔ s₁.center ≠ s₂.center ∨ s₁.radius ≠ s₂.radius"
] |
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 59 | 61 | theorem NNReal.tendsto_const_div_atTop_nhds_zero_nat (C : ℝ≥0) :
Tendsto (fun n : ℕ ↦ C / n) atTop (𝓝 0) := by |
simpa using tendsto_const_nhds.mul NNReal.tendsto_inverse_atTop_nhds_zero_nat
| [
" Tendsto (fun n => C / ↑n) atTop (𝓝 0)",
" Tendsto (fun n => (↑n)⁻¹) atTop (𝓝 0)",
" Tendsto (fun a => ↑(↑a)⁻¹) atTop (𝓝 ↑0)"
] | [
" Tendsto (fun n => C / ↑n) atTop (𝓝 0)",
" Tendsto (fun n => (↑n)⁻¹) atTop (𝓝 0)",
" Tendsto (fun a => ↑(↑a)⁻¹) atTop (𝓝 ↑0)"
] |
import Mathlib.Order.Filter.Ultrafilter
import Mathlib.Order.Filter.Germ
#align_import order.filter.filter_product from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
universe u v
variable {α : Type u} {β : Type v} {φ : Ultrafilter α}
open scoped Classical
namespace Filter
local not... | Mathlib/Order/Filter/FilterProduct.lean | 82 | 84 | theorem lt_def [Preorder β] : ((· < ·) : β* → β* → Prop) = LiftRel (· < ·) := by |
ext ⟨f⟩ ⟨g⟩
exact coe_lt
| [
" ((fun x => Inv.inv ∘ x) fun x => 0) =ᶠ[↑φ] fun x => 0",
" ↑f < ↑g ↔ ∀* (x : α), f x < g x",
" (fun x x_1 => x < x_1) = LiftRel fun x x_1 => x < x_1",
" Quot.mk Setoid.r f < Quot.mk Setoid.r g ↔ LiftRel (fun x x_1 => x < x_1) (Quot.mk Setoid.r f) (Quot.mk Setoid.r g)"
] | [
" ((fun x => Inv.inv ∘ x) fun x => 0) =ᶠ[↑φ] fun x => 0",
" ↑f < ↑g ↔ ∀* (x : α), f x < g x"
] |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.ord_connected_component from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Interval Function OrderDual
namespace Set
variable {α : Type*} [LinearOrder α] {s t : Set α}... | Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 82 | 84 | theorem mem_ordConnectedComponent_comm :
y ∈ ordConnectedComponent s x ↔ x ∈ ordConnectedComponent s y := by |
rw [mem_ordConnectedComponent, mem_ordConnectedComponent, uIcc_comm]
| [
" toDual x ∈ (⇑ofDual ⁻¹' s).ordConnectedComponent (toDual x✝) ↔ toDual x ∈ ⇑ofDual ⁻¹' s.ordConnectedComponent x✝",
" ⇑ofDual ⁻¹' [[x✝, x]] ⊆ ⇑ofDual ⁻¹' s ↔ toDual x ∈ ⇑ofDual ⁻¹' s.ordConnectedComponent x✝",
" x ∈ s.ordConnectedComponent x ↔ x ∈ s",
" s.ordConnectedComponent x = ∅ ↔ x ∉ s",
" univ.ordCon... | [
" toDual x ∈ (⇑ofDual ⁻¹' s).ordConnectedComponent (toDual x✝) ↔ toDual x ∈ ⇑ofDual ⁻¹' s.ordConnectedComponent x✝",
" ⇑ofDual ⁻¹' [[x✝, x]] ⊆ ⇑ofDual ⁻¹' s ↔ toDual x ∈ ⇑ofDual ⁻¹' s.ordConnectedComponent x✝",
" x ∈ s.ordConnectedComponent x ↔ x ∈ s",
" s.ordConnectedComponent x = ∅ ↔ x ∉ s",
" univ.ordCon... |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}
def map₂ (f : α → β → γ) (a : Option α) ... | Mathlib/Data/Option/NAry.lean | 63 | 63 | theorem map₂_none_right (f : α → β → γ) (a : Option α) : map₂ f a none = none := by | cases a <;> rfl
| [
" map₂ f a b = Seq.seq (f <$> a) fun x => b",
" map₂ f none b = Seq.seq (f <$> none) fun x => b",
" map₂ f (some val✝) b = Seq.seq (f <$> some val✝) fun x => b",
" map₂ f a none = none",
" map₂ f none none = none",
" map₂ f (some val✝) none = none"
] | [
" map₂ f a b = Seq.seq (f <$> a) fun x => b",
" map₂ f none b = Seq.seq (f <$> none) fun x => b",
" map₂ f (some val✝) b = Seq.seq (f <$> some val✝) fun x => b"
] |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 116 | 119 | theorem withDensity_sum {ι : Type*} {m : MeasurableSpace α} (μ : ι → Measure α) (f : α → ℝ≥0∞) :
(sum μ).withDensity f = sum fun n => (μ n).withDensity f := by |
ext1 s hs
simp_rw [sum_apply _ hs, withDensity_apply f hs, restrict_sum μ hs, lintegral_sum_measure]
| [
" (fun s x => ∫⁻ (a : α) in s, f a ∂μ) ∅ ⋯ = 0",
" ∫⁻ (a : α) in s, f a ∂μ ≤ (μ.withDensity f) s",
" (μ.withDensity f) s = ∫⁻ (a : α) in s, f a ∂μ",
" (μ.withDensity f) s ≤ ∫⁻ (a : α) in s, f a ∂μ",
" ∫⁻ (a : α) in t, f a ∂μ = ∫⁻ (a : α) in s, f a ∂μ",
" μ.restrict t = μ.restrict s",
" withDensity 0 f =... | [
" (fun s x => ∫⁻ (a : α) in s, f a ∂μ) ∅ ⋯ = 0",
" ∫⁻ (a : α) in s, f a ∂μ ≤ (μ.withDensity f) s",
" (μ.withDensity f) s = ∫⁻ (a : α) in s, f a ∂μ",
" (μ.withDensity f) s ≤ ∫⁻ (a : α) in s, f a ∂μ",
" ∫⁻ (a : α) in t, f a ∂μ = ∫⁻ (a : α) in s, f a ∂μ",
" μ.restrict t = μ.restrict s",
" withDensity 0 f =... |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 60 | 66 | theorem summable_of_summable_hasDerivAt_of_isPreconnected (hu : Summable u) (ht : IsOpen t)
(h't : IsPreconnected t) (hg : ∀ n y, y ∈ t → HasDerivAt (g n) (g' n y) y)
(hg' : ∀ n y, y ∈ t → ‖g' n y‖ ≤ u n) (hy₀ : y₀ ∈ t) (hg0 : Summable (g · y₀))
(hy : y ∈ t) : Summable fun n => g n y := by |
simp_rw [hasDerivAt_iff_hasFDerivAt] at hg
refine summable_of_summable_hasFDerivAt_of_isPreconnected hu ht h't hg ?_ hy₀ hg0 hy
simpa? says simpa only [ContinuousLinearMap.norm_smulRight_apply, norm_one, one_mul]
| [
" Summable fun n => f n x",
" CauchySeq fun s => ∑ b ∈ s, f b x",
" HasFDerivAt ((fun t x => ∑ i ∈ t, f i x) t) (∑ i ∈ t, f' i y) y",
" Summable fun n => g n y",
" ∀ (n : α), ∀ x ∈ t, ‖ContinuousLinearMap.smulRight 1 (g' n x)‖ ≤ u n"
] | [
" Summable fun n => f n x",
" CauchySeq fun s => ∑ b ∈ s, f b x",
" HasFDerivAt ((fun t x => ∑ i ∈ t, f i x) t) (∑ i ∈ t, f' i y) y"
] |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {α β R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 92 | 95 | theorem IsDiag.add [AddZeroClass α] {A B : Matrix n n α} (ha : A.IsDiag) (hb : B.IsDiag) :
(A + B).IsDiag := by |
intro i j h
simp [ha h, hb h]
| [
" diagonal A.diag i j = A i j",
" diagonal A.diag i i = A i i",
" (A.map f).IsDiag",
" A.map f i j = 0",
" (-A).IsDiag",
" (-A) i j = 0",
" (A + B).IsDiag",
" (A + B) i j = 0"
] | [
" diagonal A.diag i j = A i j",
" diagonal A.diag i i = A i i",
" (A.map f).IsDiag",
" A.map f i j = 0",
" (-A).IsDiag",
" (-A) i j = 0"
] |
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Set.Lattice
#align_import data.nat.pairing from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
assert_not_exists MonoidWithZero
open Prod Decidable Function
namespace Nat
-- Porting note: no pp_nodot
--@[pp_nodot]
def pair (a b : ... | Mathlib/Data/Nat/Pairing.lean | 59 | 60 | theorem pair_unpair' {n a b} (H : unpair n = (a, b)) : pair a b = n := by |
simpa [H] using pair_unpair n
| [
" n.unpair.1.pair n.unpair.2 = n",
" (if n - n.sqrt * n.sqrt < n.sqrt then (n - n.sqrt * n.sqrt, n.sqrt)\n else (n.sqrt, n - n.sqrt * n.sqrt - n.sqrt)).1.pair\n (if n - n.sqrt * n.sqrt < n.sqrt then (n - n.sqrt * n.sqrt, n.sqrt)\n else (n.sqrt, n - n.sqrt * n.sqrt - n.sqrt)).2 =\n n",
" ... | [
" n.unpair.1.pair n.unpair.2 = n",
" (if n - n.sqrt * n.sqrt < n.sqrt then (n - n.sqrt * n.sqrt, n.sqrt)\n else (n.sqrt, n - n.sqrt * n.sqrt - n.sqrt)).1.pair\n (if n - n.sqrt * n.sqrt < n.sqrt then (n - n.sqrt * n.sqrt, n.sqrt)\n else (n.sqrt, n - n.sqrt * n.sqrt - n.sqrt)).2 =\n n",
" ... |
import Mathlib.Data.Set.Image
#align_import order.directed from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
open Function
universe u v w
variable {α : Type u} {β : Type v} {ι : Sort w} (r r' s : α → α → Prop)
local infixl:50 " ≼ " => r
def Directed (f : ι → α) :=
∀ x y, ∃ z, ... | Mathlib/Order/Directed.lean | 58 | 60 | theorem directedOn_iff_directed {s} : @DirectedOn α r s ↔ Directed r (Subtype.val : s → α) := by |
simp only [DirectedOn, Directed, Subtype.exists, exists_and_left, exists_prop, Subtype.forall]
exact forall₂_congr fun x _ => by simp [And.comm, and_assoc]
| [
" DirectedOn r s ↔ Directed r Subtype.val",
" (∀ x ∈ s, ∀ y ∈ s, ∃ z ∈ s, r x z ∧ r y z) ↔ ∀ a ∈ s, ∀ a_1 ∈ s, ∃ a_2, r a a_2 ∧ a_2 ∈ s ∧ r a_1 a_2",
" (∀ y ∈ s, ∃ z ∈ s, r x z ∧ r y z) ↔ ∀ a ∈ s, ∃ a_1, r x a_1 ∧ a_1 ∈ s ∧ r a a_1"
] | [] |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 77 | 79 | theorem AffineEquiv.pointReflection_midpoint_right (x y : P) :
pointReflection R (midpoint R x y) y = x := by |
rw [midpoint_comm, AffineEquiv.pointReflection_midpoint_left]
| [
" (pointReflection R (midpoint R x y)) x = y",
" (pointReflection (midpoint R x y)) x = y",
" midpoint R x y = midpoint R y x",
" (pointReflection R (midpoint R x y)) y = x"
] | [
" (pointReflection R (midpoint R x y)) x = y",
" (pointReflection (midpoint R x y)) x = y",
" midpoint R x y = midpoint R y x"
] |
import Mathlib.Tactic.NormNum.Inv
set_option autoImplicit true
open Lean Meta Qq
namespace Mathlib.Meta.NormNum
theorem isNat_eq_false [AddMonoidWithOne α] [CharZero α] : {a b : α} → {a' b' : ℕ} →
IsNat a a' → IsNat b b' → Nat.beq a' b' = false → ¬a = b
| _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, h => by simp; exact Nat.n... | Mathlib/Tactic/NormNum/Eq.lean | 26 | 29 | theorem Rat.invOf_denom_swap [Ring α] (n₁ n₂ : ℤ) (a₁ a₂ : α)
[Invertible a₁] [Invertible a₂] : n₁ * ⅟a₁ = n₂ * ⅟a₂ ↔ n₁ * a₂ = n₂ * a₁ := by |
rw [mul_invOf_eq_iff_eq_mul_right, ← Int.commute_cast, mul_assoc,
← mul_left_eq_iff_eq_invOf_mul, Int.commute_cast]
| [
" ¬↑n✝¹ = ↑n✝",
" ¬n✝¹ = n✝",
" ↑n₁ * ⅟a₁ = ↑n₂ * ⅟a₂ ↔ ↑n₁ * a₂ = ↑n₂ * a₁"
] | [
" ¬↑n✝¹ = ↑n✝",
" ¬n✝¹ = n✝"
] |
import Mathlib.RingTheory.EisensteinCriterion
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomial.eisenstein.basic from "leanprover-community/mathlib"@"2032a878972d5672e7c27c957e7a6e297b044973"
universe u v w z
variable {R : Type u}
open Ideal Algebra Finset
open Polynomial
na... | Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean | 66 | 69 | theorem map {A : Type v} [CommRing A] (φ : R →+* A) : (f.map φ).IsWeaklyEisensteinAt (𝓟.map φ) := by |
refine (isWeaklyEisensteinAt_iff _ _).2 fun hn => ?_
rw [coeff_map]
exact mem_map_of_mem _ (hf.mem (lt_of_lt_of_le hn (natDegree_map_le _ _)))
| [
" (Polynomial.map φ f).IsWeaklyEisensteinAt (Ideal.map φ 𝓟)",
" (Polynomial.map φ f).coeff n✝ ∈ Ideal.map φ 𝓟",
" φ (f.coeff n✝) ∈ Ideal.map φ 𝓟"
] | [] |
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 | 174 | 179 | theorem coe_bot : (⊥ : YoungDiagram).cells = (∅ : Set (ℕ × ℕ)) := by |
refine Set.eq_of_subset_of_subset ?_ ?_
· intros x h
simp? [mem_mk, Finset.coe_empty, Set.mem_empty_iff_false] at h says
simp only [cells_bot, Finset.coe_empty, Set.mem_empty_iff_false] at h
· simp only [cells_bot, Finset.coe_empty, Set.empty_subset]
| [
" μ = ν",
" IsLowerSet ↑(μ.cells ∪ ν.cells)",
" IsLowerSet (↑μ.cells ∪ ↑ν.cells)",
" IsLowerSet ↑(μ.cells ∩ ν.cells)",
" IsLowerSet (↑μ.cells ∩ ↑ν.cells)",
" IsLowerSet ↑∅",
" b ∈ ↑∅",
" False",
" x✝ ∈ ⊥ → x✝ ∈ x✝¹",
" x✝ ∈ x✝¹",
" ↑⊥.cells = ∅",
" ↑⊥.cells ⊆ ∅",
" x ∈ ∅",
" ∅ ⊆ ↑⊥.cells"
... | [
" μ = ν",
" IsLowerSet ↑(μ.cells ∪ ν.cells)",
" IsLowerSet (↑μ.cells ∪ ↑ν.cells)",
" IsLowerSet ↑(μ.cells ∩ ν.cells)",
" IsLowerSet (↑μ.cells ∩ ↑ν.cells)",
" IsLowerSet ↑∅",
" b ∈ ↑∅",
" False",
" x✝ ∈ ⊥ → x✝ ∈ x✝¹",
" x✝ ∈ x✝¹"
] |
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
import Mathlib.Analysis.Fourier.PoissonSummation
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform RealInnerProductSpace
open Complex hiding exp continuous_exp abs_of_nonneg sq_abs
noncomp... | Mathlib/Analysis/SpecialFunctions/Gaussian/PoissonSummation.lean | 88 | 122 | theorem Complex.tsum_exp_neg_quadratic {a : ℂ} (ha : 0 < a.re) (b : ℂ) :
(∑' n : ℤ, cexp (-π * a * n ^ 2 + 2 * π * b * n)) =
1 / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n + I * b) ^ 2) := by |
let f : ℝ → ℂ := fun x ↦ cexp (-π * a * x ^ 2 + 2 * π * b * x)
have hCf : Continuous f := by
refine Complex.continuous_exp.comp (Continuous.add ?_ ?_)
· exact continuous_const.mul (Complex.continuous_ofReal.pow 2)
· exact continuous_const.mul Complex.continuous_ofReal
have hFf : 𝓕 f = fun x : ℝ ↦ 1 ... | [
" (fun x => rexp (a * x ^ 2 + b * x)) =o[atTop] fun x => x ^ s",
" (fun x => rexp (-x)) =o[atTop] fun x => x ^ s",
" (fun x => rexp (a * x ^ 2 + b * x)) =o[atTop] fun x => rexp (-x)",
" Tendsto (fun x => -x - (a * x ^ 2 + b * x)) atTop atTop",
" (fun x => -x - (a * x ^ 2 + b * x)) = fun x => x * (-a * x - (... | [
" (fun x => rexp (a * x ^ 2 + b * x)) =o[atTop] fun x => x ^ s",
" (fun x => rexp (-x)) =o[atTop] fun x => x ^ s",
" (fun x => rexp (a * x ^ 2 + b * x)) =o[atTop] fun x => rexp (-x)",
" Tendsto (fun x => -x - (a * x ^ 2 + b * x)) atTop atTop",
" (fun x => -x - (a * x ^ 2 + b * x)) = fun x => x * (-a * x - (... |
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
#align_import data.complex.exponential_bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
namespace Real
open IsAbsoluteValue Finset CauSeq Complex
| Mathlib/Data/Complex/ExponentialBounds.lean | 20 | 25 | theorem exp_one_near_10 : |exp 1 - 2244083 / 825552| ≤ 1 / 10 ^ 10 := by |
apply exp_approx_start
iterate 13 refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_
norm_num1
refine exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) ?_
rw [_root_.abs_one, abs_of_pos] <;> norm_num1
| [
" |rexp 1 - 2244083 / 825552| ≤ 1 / 10 ^ 10",
" |rexp 1 - expNear 0 1 (2244083 / 825552)| ≤ |1| ^ 0 / ↑(Nat.factorial 0) * (1 / 10 ^ 10)",
" 0 + 1 = ?m.532",
" 1 = ?m.532",
" ↑1 = ?m.543",
" |rexp 1 - expNear 1 1 ((2244083 / 825552 - 1) * 1)| ≤ |1| ^ 1 / ↑(Nat.factorial 1) * (1 / 10 ^ 10 * 1)",
" 1 + 1 ... | [] |
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.GroupTheory.SpecificGroups.Cyclic
#align_import ring_theory.integral_domain from "leanprover-community/mathlib"@"6e70e0d419bf686784937d64ed4bfde866ff229e"
section
open Finset Polynomial Function Nat
section CancelMonoidWithZero... | Mathlib/RingTheory/IntegralDomain.lean | 61 | 69 | theorem exists_eq_pow_of_mul_eq_pow_of_coprime {R : Type*} [CommSemiring R] [IsDomain R]
[GCDMonoid R] [Unique Rˣ] {a b c : R} {n : ℕ} (cp : IsCoprime a b) (h : a * b = c ^ n) :
∃ d : R, a = d ^ n := by |
refine exists_eq_pow_of_mul_eq_pow (isUnit_of_dvd_one ?_) h
obtain ⟨x, y, hxy⟩ := cp
rw [← hxy]
exact -- Porting note: added `GCDMonoid.` twice
dvd_add (dvd_mul_of_dvd_right (GCDMonoid.gcd_dvd_left _ _) _)
(dvd_mul_of_dvd_right (GCDMonoid.gcd_dvd_right _ _) _)
| [
" 0⁻¹ = 0",
" a * a⁻¹ = 1",
" a * bijInv ⋯ 1 = 1",
" ∃ d, a = d ^ n",
" GCDMonoid.gcd a b ∣ 1",
" GCDMonoid.gcd a b ∣ x * a + y * b"
] | [
" 0⁻¹ = 0",
" a * a⁻¹ = 1",
" a * bijInv ⋯ 1 = 1"
] |
import Mathlib.Algebra.IsPrimePow
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.Tactic.WLOG
#align_import set_theory.cardinal.divisibility from "leanprover-community/mathlib"@"ea050b44c0f9aba9d16a948c7cc7d2e7c8493567"
namespace Cardinal
open Cardinal
universe u
variable {a b : Cardinal.{u}} {n m : ℕ... | Mathlib/SetTheory/Cardinal/Divisibility.lean | 43 | 58 | theorem isUnit_iff : IsUnit a ↔ a = 1 := by |
refine
⟨fun h => ?_, by
rintro rfl
exact isUnit_one⟩
rcases eq_or_ne a 0 with (rfl | ha)
· exact (not_isUnit_zero h).elim
rw [isUnit_iff_forall_dvd] at h
cases' h 1 with t ht
rw [eq_comm, mul_eq_one_iff'] at ht
· exact ht.1
· exact one_le_iff_ne_zero.mpr ha
· apply one_le_iff_ne_zero.... | [
" IsUnit a ↔ a = 1",
" a = 1 → IsUnit a",
" IsUnit 1",
" a = 1",
" 0 = 1",
" 1 ≤ a",
" 1 ≤ t",
" t ≠ 0",
" 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 | 81 | 87 | theorem finite_of_fin_dim_affineIndependent [FiniteDimensional k V] {p : ι → P}
(hi : AffineIndependent k p) : Finite ι := by |
nontriviality ι; inhabit ι
rw [affineIndependent_iff_linearIndependent_vsub k p default] at hi
letI : IsNoetherian k V := IsNoetherian.iff_fg.2 inferInstance
exact
(Set.finite_singleton default).finite_of_compl (Set.finite_coe_iff.1 hi.finite_of_isNoetherian)
| [
" _root_.Finite ι"
] | [] |
import Mathlib.RingTheory.EisensteinCriterion
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomial.eisenstein.basic from "leanprover-community/mathlib"@"2032a878972d5672e7c27c957e7a6e297b044973"
universe u v w z
variable {R : Type u}
open Ideal Algebra Finset
open Polynomial
na... | Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean | 161 | 166 | theorem scaleRoots.isWeaklyEisensteinAt (p : R[X]) {x : R} {P : Ideal R} (hP : x ∈ P) :
(scaleRoots p x).IsWeaklyEisensteinAt P := by |
refine ⟨fun i => ?_⟩
rw [coeff_scaleRoots]
rw [natDegree_scaleRoots, ← tsub_pos_iff_lt] at i
exact Ideal.mul_mem_left _ _ (Ideal.pow_mem_of_mem P hP _ i)
| [
" (p.scaleRoots x).IsWeaklyEisensteinAt P",
" (p.scaleRoots x).coeff n✝ ∈ P",
" p.coeff n✝ * x ^ (p.natDegree - n✝) ∈ P"
] | [] |
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import measure_theory.measure.sub from "leanprover-community/mathlib"@"562bbf524c595c153470e53d36c57b6f891cc480"
open Set
namespace MeasureTheory
namespace Measure
noncomputable instance instSub {α : Type*} [MeasurableSpace α] : Sub (Measure α) :=
⟨fun ... | Mathlib/MeasureTheory/Measure/Sub.lean | 105 | 134 | theorem restrict_sub_eq_restrict_sub_restrict (h_meas_s : MeasurableSet s) :
(μ - ν).restrict s = μ.restrict s - ν.restrict s := by |
repeat rw [sub_def]
have h_nonempty : { d | μ ≤ d + ν }.Nonempty := ⟨μ, Measure.le_add_right le_rfl⟩
rw [restrict_sInf_eq_sInf_restrict h_nonempty h_meas_s]
apply le_antisymm
· refine sInf_le_sInf_of_forall_exists_le ?_
intro ν' h_ν'_in
rw [mem_setOf_eq] at h_ν'_in
refine ⟨ν'.restrict s, ?_, rest... | [
" μ ≤ 0 + ν",
" (μ - ν) s = μ s - ν s",
" (fun t x => μ t - ν t) ∅ ⋯ = 0",
" (fun t x => μ t - ν t) (⋃ i, g i) ⋯ = ∑' (i : ℕ), (fun t x => μ t - ν t) (g i) ⋯",
" ∑' (i : ℕ), μ (g i) - ∑' (i : ℕ), ν (g i) = ∑' (i : ℕ), (μ (g i) - ν (g i))",
" ∑' (i : ℕ), ν (g i) ≠ ⊤",
" ν (⋃ i, g i) ≠ ⊤",
" ν + measure... | [
" μ ≤ 0 + ν",
" (μ - ν) s = μ s - ν s",
" (fun t x => μ t - ν t) ∅ ⋯ = 0",
" (fun t x => μ t - ν t) (⋃ i, g i) ⋯ = ∑' (i : ℕ), (fun t x => μ t - ν t) (g i) ⋯",
" ∑' (i : ℕ), μ (g i) - ∑' (i : ℕ), ν (g i) = ∑' (i : ℕ), (μ (g i) - ν (g i))",
" ∑' (i : ℕ), ν (g i) ≠ ⊤",
" ν (⋃ i, g i) ≠ ⊤",
" ν + measure... |
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import algebra.bounds from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29... | Mathlib/Algebra/Bounds.lean | 185 | 186 | theorem ciInf_div (hf : BddBelow (range f)) (a : G) : (⨅ i, f i) / a = ⨅ i, f i / a := by |
simp only [div_eq_mul_inv, ciInf_mul hf]
| [
" (⨆ i, f i) / a = ⨆ i, f i / a",
" (⨅ i, f i) / a = ⨅ i, f i / a"
] | [
" (⨆ i, f i) / a = ⨆ i, f i / a"
] |
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 | 580 | 582 | theorem map_id (x : Subobject X) : (map (𝟙 X)).obj x = x := by |
induction' x using Quotient.inductionOn' with f
exact Quotient.sound ⟨(MonoOver.mapId _).app f⟩
| [
" PartialOrder (Subobject X)",
" PartialOrder (ThinSkeleton (MonoOver X))",
" 𝟭 (Subobject A) ≅ lower e.functor ⋙ lower e.inverse",
" 𝟭 (Subobject A) = lower e.functor ⋙ lower e.inverse",
" 𝟭 (Subobject A) = ThinSkeleton.map (𝟭 (MonoOver A))",
" lower e.functor ⋙ lower e.inverse = ThinSkeleton.map (e.... | [
" PartialOrder (Subobject X)",
" PartialOrder (ThinSkeleton (MonoOver X))",
" 𝟭 (Subobject A) ≅ lower e.functor ⋙ lower e.inverse",
" 𝟭 (Subobject A) = lower e.functor ⋙ lower e.inverse",
" 𝟭 (Subobject A) = ThinSkeleton.map (𝟭 (MonoOver A))",
" lower e.functor ⋙ lower e.inverse = ThinSkeleton.map (e.... |
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.weighted_homogeneous from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Fins... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 89 | 91 | theorem weightedTotalDegree'_zero (w : σ → M) :
weightedTotalDegree' w (0 : MvPolynomial σ R) = ⊥ := by |
simp only [weightedTotalDegree', support_zero, Finset.sup_empty]
| [
" (weightedDegree w) f = f.sum fun i c => c • w i",
" weightedTotalDegree' w p = ⊥ ↔ p = 0",
" (∀ (s : σ →₀ ℕ), coeff s p ≠ 0 → False) ↔ ∀ (d : σ →₀ ℕ), coeff d p = 0",
" weightedTotalDegree' w 0 = ⊥"
] | [
" (weightedDegree w) f = f.sum fun i c => c • w i",
" weightedTotalDegree' w p = ⊥ ↔ p = 0",
" (∀ (s : σ →₀ ℕ), coeff s p ≠ 0 → False) ↔ ∀ (d : σ →₀ ℕ), coeff d p = 0"
] |
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheory.MeasurableSpace.Basic
open Set
namespace MeasureTheory
variable {ι : Type _} {α : ι → Type _}
section cylinder
def cylinder (s : Finset ι) (S : Set (∀ i : s, α... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 205 | 207 | theorem union_cylinder_same (s : Finset ι) (S₁ : Set (∀ i : s, α i)) (S₂ : Set (∀ i : s, α i)) :
cylinder s S₁ ∪ cylinder s S₂ = cylinder s (S₁ ∪ S₂) := by |
classical rw [union_cylinder]; rfl
| [
" cylinder s ∅ = ∅",
" cylinder s univ = univ",
" cylinder s S = ∅ ↔ S = ∅",
" cylinder s S = ∅",
" S = ∅",
" False",
" f' ∈ cylinder s S",
" (fun i => f' ↑i) ∈ S",
" cylinder s₁ S₁ ∩ cylinder s₂ S₂ = cylinder (s₁ ∪ s₂) ((fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₁ ∩ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₂)",
" f ∈ cylind... | [
" cylinder s ∅ = ∅",
" cylinder s univ = univ",
" cylinder s S = ∅ ↔ S = ∅",
" cylinder s S = ∅",
" S = ∅",
" False",
" f' ∈ cylinder s S",
" (fun i => f' ↑i) ∈ S",
" cylinder s₁ S₁ ∩ cylinder s₂ S₂ = cylinder (s₁ ∪ s₂) ((fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₁ ∩ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₂)",
" f ∈ cylind... |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
open Set Filter MeasureTheory MeasurableSpace
open scoped Classical Topology NNReal ENNReal MeasureTheory
univers... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 76 | 82 | theorem borel_eq_generateFrom_Ici_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Ici (a : ℝ)}) := by |
rw [borel_eq_generateFrom_Iio_rat, iUnion_singleton_eq_range, iUnion_singleton_eq_range]
refine le_antisymm (generateFrom_le ?_) (generateFrom_le ?_) <;>
rintro _ ⟨q, rfl⟩ <;>
dsimp only <;>
[rw [← compl_Ici]; rw [← compl_Iio]] <;>
exact MeasurableSet.compl (GenerateMeasurable.basic _ (mem_range_self q))
| [
" borel ℝ = generateFrom (⋃ a, {Iio ↑a})",
" generateFrom (range Iio) = generateFrom (⋃ a, {Iio ↑a})",
" ∀ t ∈ range Iio, MeasurableSet t",
" MeasurableSet (Iio a)",
" IsLUB (range Rat.cast ∩ Iio a) a",
" MeasurableSet (⋃ y ∈ Rat.cast ⁻¹' Iio a, Iio ↑y)",
" Iio ↑b ∈ ⋃ a, {Iio ↑a}",
" borel ℝ = generat... | [
" borel ℝ = generateFrom (⋃ a, {Iio ↑a})",
" generateFrom (range Iio) = generateFrom (⋃ a, {Iio ↑a})",
" ∀ t ∈ range Iio, MeasurableSet t",
" MeasurableSet (Iio a)",
" IsLUB (range Rat.cast ∩ Iio a) a",
" MeasurableSet (⋃ y ∈ Rat.cast ⁻¹' Iio a, Iio ↑y)",
" Iio ↑b ∈ ⋃ a, {Iio ↑a}",
" borel ℝ = generat... |
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
#align_import algebraic_geometry.open_immersion.basic from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
-- Porting note: due to `PresheafedSpace`, `SheafedSpace` and `Locally... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 145 | 146 | theorem isoRestrict_inv_ofRestrict : H.isoRestrict.inv ≫ f = Y.ofRestrict _ := by |
rw [Iso.inv_comp_eq, isoRestrict_hom_ofRestrict]
| [] | [] |
import Mathlib.Data.Matrix.Basis
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Pi
#align_import linear_algebra.std_basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
namespace LinearMap
variable (R : Type*) {ι : Type*} [Semiring R] ... | Mathlib/LinearAlgebra/StdBasis.lean | 123 | 129 | theorem iSup_range_stdBasis_eq_iInf_ker_proj {I J : Set ι} (hd : Disjoint I J)
(hu : Set.univ ⊆ I ∪ J) (hI : Set.Finite I) :
⨆ i ∈ I, range (stdBasis R φ i) = ⨅ i ∈ J, ker (proj i : (∀ i, φ i) →ₗ[R] φ i) := by |
refine le_antisymm (iSup_range_stdBasis_le_iInf_ker_proj _ _ _ _ hd) ?_
have : Set.univ ⊆ ↑hI.toFinset ∪ J := by rwa [hI.coe_toFinset]
refine le_trans (iInf_ker_proj_le_iSup_range_stdBasis R φ this) (iSup_mono fun i => ?_)
rw [Set.Finite.mem_toFinset]
| [
" (stdBasis R (fun _x => R) i) 1 i' = if i = i' then 1 else 0",
" (if i' = i then 1 else 0) = if i = i' then 1 else 0",
" (i' = i) = (i = i')",
" stdBasis R φ i = pi (diag i)",
" (stdBasis R φ i) x j = (pi (diag i)) x j",
" x = id x",
" proj i ∘ₗ stdBasis R φ j = diag j i",
" ⨆ i ∈ I, range (stdBasis ... | [
" (stdBasis R (fun _x => R) i) 1 i' = if i = i' then 1 else 0",
" (if i' = i then 1 else 0) = if i = i' then 1 else 0",
" (i' = i) = (i = i')",
" stdBasis R φ i = pi (diag i)",
" (stdBasis R φ i) x j = (pi (diag i)) x j",
" x = id x",
" proj i ∘ₗ stdBasis R φ j = diag j i",
" ⨆ i ∈ I, range (stdBasis ... |
import Batteries.Data.List.Lemmas
import Batteries.Data.Array.Basic
import Batteries.Tactic.SeqFocus
import Batteries.Util.ProofWanted
namespace Array
theorem forIn_eq_data_forIn [Monad m]
(as : Array α) (b : β) (f : α → β → m (ForInStep β)) :
forIn as b f = forIn as.data b f := by
let rec loop : ∀ {i h b ... | .lake/packages/batteries/Batteries/Data/Array/Lemmas.lean | 121 | 125 | theorem size_shrink_loop (a : Array α) (n) : (shrink.loop n a).size = a.size - n := by |
induction n generalizing a with simp[shrink.loop]
| succ n ih =>
simp[ih]
omega
| [
" forIn as b f = forIn as.data b f",
" forIn.loop as f 0 x✝¹ x✝ = forIn (List.drop as.data.length as.data) x✝ f",
" forIn.loop as f 0 x✝¹ x✝ = forIn [] x✝ f",
" forIn.loop as f (i + 1) x✝¹ x✝ = forIn (List.drop j as.data) x✝ f",
" (do\n let __do_lift ← f as[as.size - 1 - i] x✝\n match __do_lift wi... | [
" forIn as b f = forIn as.data b f",
" forIn.loop as f 0 x✝¹ x✝ = forIn (List.drop as.data.length as.data) x✝ f",
" forIn.loop as f 0 x✝¹ x✝ = forIn [] x✝ f",
" forIn.loop as f (i + 1) x✝¹ x✝ = forIn (List.drop j as.data) x✝ f",
" (do\n let __do_lift ← f as[as.size - 1 - i] x✝\n match __do_lift wi... |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 124 | 134 | theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) := by |
suffices h : ascPochhammer ℕ (n + 1) = ascPochhammer ℕ n * (X + (n : ℕ[X])) by
apply_fun Polynomial.map (algebraMap ℕ S) at h
simpa only [ascPochhammer_map, Polynomial.map_mul, Polynomial.map_add, map_X,
Polynomial.map_natCast] using h
induction' n with n ih
· simp
· conv_lhs =>
rw [ascPoch... | [
" ascPochhammer S 1 = X",
" ascPochhammer S (n + 1) = X * (ascPochhammer S n).comp (X + 1)",
" (ascPochhammer S n).Monic",
" (ascPochhammer S 0).Monic",
" (ascPochhammer S (n + 1)).Monic",
" map f (ascPochhammer S n) = ascPochhammer T n",
" map f (ascPochhammer S 0) = ascPochhammer T 0",
" map f (ascP... | [
" ascPochhammer S 1 = X",
" ascPochhammer S (n + 1) = X * (ascPochhammer S n).comp (X + 1)",
" (ascPochhammer S n).Monic",
" (ascPochhammer S 0).Monic",
" (ascPochhammer S (n + 1)).Monic",
" map f (ascPochhammer S n) = ascPochhammer T n",
" map f (ascPochhammer S 0) = ascPochhammer T 0",
" map f (ascP... |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F ... | Mathlib/MeasureTheory/Group/Integral.lean | 79 | 83 | theorem integral_div_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
(∫ x, f (x / g) ∂μ) = ∫ x, f x ∂μ := by |
simp_rw [div_eq_mul_inv]
-- Porting note: was `simp_rw`
rw [integral_mul_right_eq_self f g⁻¹]
| [
" ∫ (x : G), f (g * x) ∂μ = ∫ (x : G), f x ∂μ",
" ∫ (x : G), f (x * g) ∂μ = ∫ (x : G), f x ∂μ",
" ∫ (x : G), f (x / g) ∂μ = ∫ (x : G), f x ∂μ",
" ∫ (x : G), f (x * g⁻¹) ∂μ = ∫ (x : G), f x ∂μ"
] | [
" ∫ (x : G), f (g * x) ∂μ = ∫ (x : G), f x ∂μ",
" ∫ (x : G), f (x * g) ∂μ = ∫ (x : G), f x ∂μ"
] |
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
open Nat Function
namespace List
theorem rel_of_pairwise_cons (p : (a :: l).Pairwise R) : ∀ {a'}, a' ∈ l → R a a' :=
(pairwise_cons.1 p).1 _
theorem Pairwise.of_cons (p : (a :: l).Pairwise R) : Pairwise R l :=
(pairwise_cons.1 p).2
theorem... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 114 | 118 | theorem pairwise_middle {R : α → α → Prop} (s : ∀ {x y}, R x y → R y x) {a : α} {l₁ l₂ : List α} :
Pairwise R (l₁ ++ a :: l₂) ↔ Pairwise R (a :: (l₁ ++ l₂)) := by |
show Pairwise R (l₁ ++ ([a] ++ l₂)) ↔ Pairwise R ([a] ++ l₁ ++ l₂)
rw [← append_assoc, pairwise_append, @pairwise_append _ _ ([a] ++ l₁), pairwise_append_comm s]
simp only [mem_append, or_comm]
| [
" Pairwise S l",
" Pairwise S []",
" Pairwise S (a :: l)",
" ∀ (a' : α), a' ∈ l → S a a'",
" Pairwise (fun a b => R a b ∧ S a b) l",
" Pairwise (fun a b => R a b ∧ S a b) []",
" Pairwise (fun a b => R a b ∧ S a b) (a✝¹ :: l✝)",
" (∀ (a' : α✝), a' ∈ l✝ → R a✝¹ a' ∧ S a✝¹ a') ∧ Pairwise (fun a b => R a ... | [
" Pairwise S l",
" Pairwise S []",
" Pairwise S (a :: l)",
" ∀ (a' : α), a' ∈ l → S a a'",
" Pairwise (fun a b => R a b ∧ S a b) l",
" Pairwise (fun a b => R a b ∧ S a b) []",
" Pairwise (fun a b => R a b ∧ S a b) (a✝¹ :: l✝)",
" (∀ (a' : α✝), a' ∈ l✝ → R a✝¹ a' ∧ S a✝¹ a') ∧ Pairwise (fun a b => R a ... |
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 | 109 | 114 | theorem moveLeft_card {b : Board} {m : ℤ × ℤ} (h : m ∈ left b) :
Finset.card (moveLeft b m) + 2 = Finset.card b := by |
dsimp [moveLeft]
rw [Finset.card_erase_of_mem (snd_pred_mem_erase_of_mem_left h)]
rw [Finset.card_erase_of_mem (Finset.mem_of_mem_inter_left h)]
exact tsub_add_cancel_of_le (card_of_mem_left h)
| [
" (m.1 - 1, m.2) ∈ Finset.erase b m",
" (m.1 - 1, m.2) ≠ m",
" (m.1, m.2 - 1) ∈ Finset.erase b m",
" (m.1, m.2 - 1) ≠ m",
" 2 ≤ Finset.card b",
" Finset.card (moveLeft b m) + 2 = Finset.card b",
" ((Finset.erase b m).erase (m.1, m.2 - 1)).card + 2 = Finset.card b",
" (Finset.erase b m).card - 1 + 2 = ... | [
" (m.1 - 1, m.2) ∈ Finset.erase b m",
" (m.1 - 1, m.2) ≠ m",
" (m.1, m.2 - 1) ∈ Finset.erase b m",
" (m.1, m.2 - 1) ≠ m",
" 2 ≤ Finset.card b"
] |
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheory.MeasurableSpace.Basic
open Set
namespace MeasureTheory
variable {ι : Type _} {α : ι → Type _}
section squareCylinders
def squareCylinders (C : ∀ i, Set (Set (α... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 63 | 105 | theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
(hC_univ : ∀ i, univ ∈ C i) :
IsPiSystem (squareCylinders C) := by |
rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
classical
let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
have h1 : ∀ i ∈ (s₁ : Set ι), t₁ i = t₁' i :=
fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
have h1' : ∀ i ∉ (s₁ : Set ι), t₁' i = univ :=
... | [
" squareCylinders C = ⋃ s, (fun t => (↑s).pi t) '' univ.pi C",
" f ∈ squareCylinders C ↔ f ∈ ⋃ s, (fun t => (↑s).pi t) '' univ.pi C",
" IsPiSystem (squareCylinders C)",
" (↑s₁).pi t₁ ∩ (↑s₂).pi t₂ ∈ squareCylinders C",
" ((↑s₁ ∪ ↑s₂).pi fun i => t₁' i ∩ t₂' i) ∈ squareCylinders C",
" (fun i => t₁' i ∩ t₂'... | [
" squareCylinders C = ⋃ s, (fun t => (↑s).pi t) '' univ.pi C",
" f ∈ squareCylinders C ↔ f ∈ ⋃ s, (fun t => (↑s).pi t) '' univ.pi C"
] |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 120 | 130 | theorem condDistrib_ae_eq_of_measure_eq_compProd (hX : Measurable X) (hY : Measurable Y)
(κ : kernel β Ω) [IsFiniteKernel κ] (hκ : μ.map (fun x => (X x, Y x)) = μ.map X ⊗ₘ κ) :
∀ᵐ x ∂μ.map X, κ x = condDistrib Y X μ x := by |
have heq : μ.map X = (μ.map (fun x ↦ (X x, Y x))).fst := by
ext s hs
rw [Measure.map_apply hX hs, Measure.fst_apply hs, Measure.map_apply]
exacts [rfl, Measurable.prod hX hY, measurable_fst hs]
rw [heq, condDistrib]
refine eq_condKernel_of_measure_eq_compProd _ ?_
convert hκ
exact heq.symm
| [
" IsMarkovKernel (condDistrib Y X μ)",
" IsMarkovKernel (Measure.map (fun a => (X a, Y a)) μ).condKernel",
" ((condDistrib Y X μ) x) s = ((Measure.map X μ) {x})⁻¹ * (Measure.map (fun a => (X a, Y a)) μ) ({x} ×ˢ s)",
" ((Measure.map (fun a => (X a, Y a)) μ).fst {x})⁻¹ * (Measure.map (fun a => (X a, Y a)) μ) ({... | [
" IsMarkovKernel (condDistrib Y X μ)",
" IsMarkovKernel (Measure.map (fun a => (X a, Y a)) μ).condKernel",
" ((condDistrib Y X μ) x) s = ((Measure.map X μ) {x})⁻¹ * (Measure.map (fun a => (X a, Y a)) μ) ({x} ×ˢ s)",
" ((Measure.map (fun a => (X a, Y a)) μ).fst {x})⁻¹ * (Measure.map (fun a => (X a, Y a)) μ) ({... |
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.Spectrum
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Algebra.Star.StarAlgHom
#align_import analysis.normed_space.star.spectrum from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
l... | Mathlib/Analysis/NormedSpace/Star/Spectrum.lean | 60 | 69 | theorem IsSelfAdjoint.spectralRadius_eq_nnnorm {a : A} (ha : IsSelfAdjoint a) :
spectralRadius ℂ a = ‖a‖₊ := by |
have hconst : Tendsto (fun _n : ℕ => (‖a‖₊ : ℝ≥0∞)) atTop _ := tendsto_const_nhds
refine tendsto_nhds_unique ?_ hconst
convert
(spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius (a : A)).comp
(Nat.tendsto_pow_atTop_atTop_of_one_lt one_lt_two) using 1
refine funext fun n => ?_
rw [Function... | [
" spectralRadius ℂ a = ↑‖a‖₊",
" Tendsto (fun _n => ↑‖a‖₊) atTop (𝓝 (spectralRadius ℂ a))",
" (fun _n => ↑‖a‖₊) = (fun n => ↑‖a ^ n‖₊ ^ (1 / ↑n)) ∘ fun n => 2 ^ n",
" ↑‖a‖₊ = ((fun n => ↑‖a ^ n‖₊ ^ (1 / ↑n)) ∘ fun n => 2 ^ n) n",
" ↑‖a‖₊ = ↑‖a‖₊ ^ (↑(2 ^ n) * (1 / ↑(2 ^ n)))"
] | [] |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.LinearAlgebra.AffineSpace.Slope
#align_import analysis.calculus.deriv.slope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open Topology Filter TopologicalSpace
open Filter Set
secti... | Mathlib/Analysis/Calculus/Deriv/Slope.lean | 51 | 63 | theorem hasDerivAtFilter_iff_tendsto_slope {x : 𝕜} {L : Filter 𝕜} :
HasDerivAtFilter f f' x L ↔ Tendsto (slope f x) (L ⊓ 𝓟 {x}ᶜ) (𝓝 f') :=
calc HasDerivAtFilter f f' x L
↔ Tendsto (fun y ↦ slope f x y - (y - x)⁻¹ • (y - x) • f') L (𝓝 0) := by |
simp only [hasDerivAtFilter_iff_tendsto, ← norm_inv, ← norm_smul,
← tendsto_zero_iff_norm_tendsto_zero, slope_def_module, smul_sub]
_ ↔ Tendsto (fun y ↦ slope f x y - (y - x)⁻¹ • (y - x) • f') (L ⊓ 𝓟 {x}ᶜ) (𝓝 0) :=
.symm <| tendsto_inf_principal_nhds_iff_of_forall_eq <| by simp
_ ↔ Tend... | [
" HasDerivAtFilter f f' x L ↔ Tendsto (fun y => slope f x y - (y - x)⁻¹ • (y - x) • f') L (𝓝 0)",
" ∀ a ∉ {x}ᶜ, slope f x a - (a - x)⁻¹ • (a - x) • f' = 0",
" (fun y => slope f x y - (y - x)⁻¹ • (y - x) • f') =ᶠ[L ⊓ 𝓟 {x}ᶜ] fun y => slope f x y - f'",
" slope f x y - (y - x)⁻¹ • (y - x) • f' = slope f x y -... | [] |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 804 | 809 | theorem sum_centroidWeights_eq_one_of_card_ne_zero [CharZero k] (h : card s ≠ 0) :
∑ i ∈ s, s.centroidWeights k i = 1 := by |
-- Porting note: `simp` cannot find `mul_inv_cancel` and does not use `norm_cast`
simp only [centroidWeights_apply, sum_const, nsmul_eq_mul, ne_eq, Nat.cast_eq_zero, card_eq_zero]
refine mul_inv_cancel ?_
norm_cast
| [
" ∑ i ∈ s, centroidWeights k s i = 1",
" ↑s.card * (↑s.card)⁻¹ = 1",
" ↑s.card ≠ 0"
] | [
" ∑ i ∈ s, centroidWeights k s i = 1"
] |
import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion
import Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
universe ... | Mathlib/AlgebraicGeometry/Morphisms/Separated.lean | 57 | 60 | theorem respectsIso : MorphismProperty.RespectsIso @IsSeparated := by |
rw [isSeparated_eq_diagonal_isClosedImmersion]
apply MorphismProperty.RespectsIso.diagonal
exact IsClosedImmersion.respectsIso
| [
" @IsSeparated = MorphismProperty.diagonal @IsClosedImmersion",
" IsSeparated x✝ ↔ MorphismProperty.diagonal (@IsClosedImmersion) x✝",
" MorphismProperty.RespectsIso @IsSeparated",
" (MorphismProperty.diagonal @IsClosedImmersion).RespectsIso",
" MorphismProperty.RespectsIso @IsClosedImmersion"
] | [
" @IsSeparated = MorphismProperty.diagonal @IsClosedImmersion",
" IsSeparated x✝ ↔ MorphismProperty.diagonal (@IsClosedImmersion) x✝"
] |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Join
#align_import analysis.convex.stone_separation from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
variable {𝕜 E ι : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E}
| Mathlib/Analysis/Convex/StoneSeparation.lean | 30 | 77 | theorem not_disjoint_segment_convexHull_triple {p q u v x y z : E} (hz : z ∈ segment 𝕜 x y)
(hu : u ∈ segment 𝕜 x p) (hv : v ∈ segment 𝕜 y q) :
¬Disjoint (segment 𝕜 u v) (convexHull 𝕜 {p, q, z}) := by |
rw [not_disjoint_iff]
obtain ⟨az, bz, haz, hbz, habz, rfl⟩ := hz
obtain rfl | haz' := haz.eq_or_lt
· rw [zero_add] at habz
rw [zero_smul, zero_add, habz, one_smul]
refine ⟨v, by apply right_mem_segment, segment_subset_convexHull ?_ ?_ hv⟩ <;> simp
obtain ⟨av, bv, hav, hbv, habv, rfl⟩ := hv
obtain r... | [
" ¬Disjoint (segment 𝕜 u v) ((convexHull 𝕜) {p, q, z})",
" ∃ x ∈ segment 𝕜 u v, x ∈ (convexHull 𝕜) {p, q, z}",
" ∃ x_1 ∈ segment 𝕜 u v, x_1 ∈ (convexHull 𝕜) {p, q, az • x + bz • y}",
" ∃ x_1 ∈ segment 𝕜 u v, x_1 ∈ (convexHull 𝕜) {p, q, 0 • x + bz • y}",
" ∃ x ∈ segment 𝕜 u v, x ∈ (convexHull 𝕜) {p... | [] |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics Fil... | Mathlib/Analysis/ODE/Gronwall.lean | 79 | 83 | theorem gronwallBound_x0 (δ K ε : ℝ) : gronwallBound δ K ε 0 = δ := by |
by_cases hK : K = 0
· simp only [gronwallBound, if_pos hK, mul_zero, add_zero]
· simp only [gronwallBound, if_neg hK, mul_zero, exp_zero, sub_self, mul_one,
add_zero]
| [
" HasDerivAt (gronwallBound δ K ε) (K * gronwallBound δ K ε x + ε) x",
" HasDerivAt (gronwallBound δ 0 ε) (0 * gronwallBound δ 0 ε x + ε) x",
" HasDerivAt (fun x => δ + ε * x) ε x",
" ε = ε * 1",
" HasDerivAt (fun x => δ * rexp (K * x) + ε / K * (rexp (K * x) - 1))\n (K * (δ * rexp (K * x) + ε / K * (rex... | [
" HasDerivAt (gronwallBound δ K ε) (K * gronwallBound δ K ε x + ε) x",
" HasDerivAt (gronwallBound δ 0 ε) (0 * gronwallBound δ 0 ε x + ε) x",
" HasDerivAt (fun x => δ + ε * x) ε x",
" ε = ε * 1",
" HasDerivAt (fun x => δ * rexp (K * x) + ε / K * (rexp (K * x) - 1))\n (K * (δ * rexp (K * x) + ε / K * (rex... |
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Multilinear.Basic
#align_import linear_algebra.multilinear.basis from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
open MultilinearMap
variable {R : Type*} {ι : Type*} {n : ℕ} {M : Fin n → Type*} {M₂ : Type*} {M₃ : Type*... | Mathlib/LinearAlgebra/Multilinear/Basis.lean | 56 | 61 | theorem Basis.ext_multilinear [Finite ι] {f g : MultilinearMap R (fun _ : ι => M₂) M₃} {ι₁ : Type*}
(e : Basis ι₁ R M₂) (h : ∀ v : ι → ι₁, (f fun i => e (v i)) = g fun i => e (v i)) : f = g := by |
cases nonempty_fintype ι
exact
(domDomCongr_eq_iff (Fintype.equivFin ι) f g).mp
(Basis.ext_multilinear_fin (fun _ => e) fun i => h (i ∘ _))
| [
" f = g",
" f x = g x",
" (fun x => x.curryLeft) f = (fun x => x.curryLeft) g",
" ∀ (i : ι₁ 0), ((fun x => x.curryLeft) f) ((e 0) i) = ((fun x => x.curryLeft) g) ((e 0) i)",
" ((fun x => x.curryLeft) f) ((e 0) i) = ((fun x => x.curryLeft) g) ((e 0) i)",
" ∀ (v : (i : Fin m) → ι₁ i.succ),\n ((((fun x =>... | [
" f = g",
" f x = g x",
" (fun x => x.curryLeft) f = (fun x => x.curryLeft) g",
" ∀ (i : ι₁ 0), ((fun x => x.curryLeft) f) ((e 0) i) = ((fun x => x.curryLeft) g) ((e 0) i)",
" ((fun x => x.curryLeft) f) ((e 0) i) = ((fun x => x.curryLeft) g) ((e 0) i)",
" ∀ (v : (i : Fin m) → ι₁ i.succ),\n ((((fun x =>... |
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mathlib"@"2f588be38bb5bec02f218ba14f82fc82eb663f87"
noncomputable section
open scoped Classical nonZeroDivisors
open Set Function UniqueFactorizationMonoid IsDedekindDomain IsDedekind... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 81 | 90 | theorem Associates.finite_factors {I : Ideal R} (hI : I ≠ 0) :
∀ᶠ v : HeightOneSpectrum R in Filter.cofinite,
((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ) = 0 := by |
have h_supp : {v : HeightOneSpectrum R | ¬((Associates.mk v.asIdeal).count
(Associates.mk I).factors : ℤ) = 0} = {v : HeightOneSpectrum R | v.asIdeal ∣ I} := by
ext v
simp_rw [Int.natCast_eq_zero]
exact Associates.count_ne_zero_iff_dvd hI v.irreducible
rw [Filter.eventually_cofinite, h_supp]
ex... | [
" {v | v.asIdeal ∣ I}.Finite",
" Finite { x // x.asIdeal ∣ I }",
" Injective fun v => ⟨(↑v).asIdeal, ⋯⟩",
" v = w",
" ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, ↑((Associates.mk v.asIdeal).count (Associates.mk I).factors) = 0",
" {v | ¬↑((Associates.mk v.asIdeal).count (Associates.mk I).factors) = 0... | [
" {v | v.asIdeal ∣ I}.Finite",
" Finite { x // x.asIdeal ∣ I }",
" Injective fun v => ⟨(↑v).asIdeal, ⋯⟩",
" v = w"
] |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 320 | 323 | theorem finsuppTensorFinsuppRid_single_tmul_single (a : ι) (b : κ) (m : M) (r : R) :
finsuppTensorFinsuppRid R M ι κ (Finsupp.single a m ⊗ₜ[R] Finsupp.single b r) =
Finsupp.single (a, b) (r • m) := by |
simp [finsuppTensorFinsuppRid]
| [
" (finsuppTensorFinsupp R S M N ι κ) (Finsupp.single i m ⊗ₜ[R] Finsupp.single k n) = Finsupp.single (i, k) (m ⊗ₜ[R] n)",
" ((finsuppTensorFinsupp R S M N ι κ) (f ⊗ₜ[R] g)) (i, k) = f i ⊗ₜ[R] g k",
" ((finsuppTensorFinsupp R S M N ι κ) (0 ⊗ₜ[R] g)) (i, k) = 0 i ⊗ₜ[R] g k",
" ∀ (f g_1 : ι →₀ M),\n ((finsuppT... | [
" (finsuppTensorFinsupp R S M N ι κ) (Finsupp.single i m ⊗ₜ[R] Finsupp.single k n) = Finsupp.single (i, k) (m ⊗ₜ[R] n)",
" ((finsuppTensorFinsupp R S M N ι κ) (f ⊗ₜ[R] g)) (i, k) = f i ⊗ₜ[R] g k",
" ((finsuppTensorFinsupp R S M N ι κ) (0 ⊗ₜ[R] g)) (i, k) = 0 i ⊗ₜ[R] g k",
" ∀ (f g_1 : ι →₀ M),\n ((finsuppT... |
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Set
open Filter
namespace Filter
variable {α β γ δ : Type*} {ι : Sort*}
section Prod
variable {s : Set α} {t : Set β} {f : Filter α} {g : Filter β}
protected ... | Mathlib/Order/Filter/Prod.lean | 121 | 123 | theorem sup_prod (f₁ f₂ : Filter α) (g : Filter β) : (f₁ ⊔ f₂) ×ˢ g = (f₁ ×ˢ g) ⊔ (f₂ ×ˢ g) := by |
dsimp only [SProd.sprod]
rw [Filter.prod, comap_sup, inf_sup_right, ← Filter.prod, ← Filter.prod]
| [
" s ∈ f ×ˢ g ↔ ∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁ ×ˢ t₂ ⊆ s",
" s ∈ comap Prod.fst f ⊓ comap Prod.snd g ↔ ∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁.prod t₂ ⊆ s",
" s ∈ comap Prod.fst f ⊓ comap Prod.snd g → ∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁.prod t₂ ⊆ s",
" ∃ t₁_1 ∈ f, ∃ t₂_1 ∈ g, t₁_1.prod t₂_1 ⊆ t₁ ∩ t₂",
" (∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁.prod t₂ ⊆ s) → s ... | [
" s ∈ f ×ˢ g ↔ ∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁ ×ˢ t₂ ⊆ s",
" s ∈ comap Prod.fst f ⊓ comap Prod.snd g ↔ ∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁.prod t₂ ⊆ s",
" s ∈ comap Prod.fst f ⊓ comap Prod.snd g → ∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁.prod t₂ ⊆ s",
" ∃ t₁_1 ∈ f, ∃ t₂_1 ∈ g, t₁_1.prod t₂_1 ⊆ t₁ ∩ t₂",
" (∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁.prod t₂ ⊆ s) → s ... |
import Mathlib.Topology.MetricSpace.PseudoMetric
#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
open Set Filter Bornology
open scoped NNReal Uniformity
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
variable [PseudoMetricS... | Mathlib/Topology/MetricSpace/Basic.lean | 140 | 144 | theorem uniformEmbedding_iff' [MetricSpace β] {f : γ → β} :
UniformEmbedding f ↔
(∀ ε > 0, ∃ δ > 0, ∀ {a b : γ}, dist a b < δ → dist (f a) (f b) < ε) ∧
∀ δ > 0, ∃ ε > 0, ∀ {a b : γ}, dist (f a) (f b) < ε → dist a b < δ := by |
rw [uniformEmbedding_iff_uniformInducing, uniformInducing_iff, uniformContinuous_iff]
| [
" m = m'",
" mk eq_of_dist_eq_zero✝ = m'",
" mk eq_of_dist_eq_zero✝¹ = mk eq_of_dist_eq_zero✝",
" toPseudoMetricSpace✝¹ = toPseudoMetricSpace✝",
" PseudoMetricSpace.toDist = PseudoMetricSpace.toDist",
" 0 = dist x y ↔ x = y",
" dist x y ≠ 0 ↔ x ≠ y",
" dist x y ≤ 0 ↔ x = y",
" 0 < dist x y ↔ x ≠ y",... | [
" m = m'",
" mk eq_of_dist_eq_zero✝ = m'",
" mk eq_of_dist_eq_zero✝¹ = mk eq_of_dist_eq_zero✝",
" toPseudoMetricSpace✝¹ = toPseudoMetricSpace✝",
" PseudoMetricSpace.toDist = PseudoMetricSpace.toDist",
" 0 = dist x y ↔ x = y",
" dist x y ≠ 0 ↔ x ≠ y",
" dist x y ≤ 0 ↔ x = y",
" 0 < dist x y ↔ x ≠ y",... |
import Mathlib.LinearAlgebra.Dual
open Function Module
variable (R M N : Type*) [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
structure PerfectPairing :=
toLin : M →ₗ[R] N →ₗ[R] R
bijectiveLeft : Bijective toLin
bijectiveRight : Bijective toLin.flip
attribute [nolint docBlame] P... | Mathlib/LinearAlgebra/PerfectPairing.lean | 112 | 115 | theorem reflexive_left : IsReflexive R M where
bijective_dual_eval' := by |
rw [← p.toDualRight_symm_comp_toDualLeft]
exact p.bijective_toDualRight_symm_toDualLeft
| [
" x = y",
" { toLin := toLin✝, bijectiveLeft := bijectiveLeft✝, bijectiveRight := bijectiveRight✝ } = y",
" { toLin := toLin✝¹, bijectiveLeft := bijectiveLeft✝¹, bijectiveRight := bijectiveRight✝¹ } =\n { toLin := toLin✝, bijectiveLeft := bijectiveLeft✝, bijectiveRight := bijectiveRight✝ }",
" (p (p.toDual... | [
" x = y",
" { toLin := toLin✝, bijectiveLeft := bijectiveLeft✝, bijectiveRight := bijectiveRight✝ } = y",
" { toLin := toLin✝¹, bijectiveLeft := bijectiveLeft✝¹, bijectiveRight := bijectiveRight✝¹ } =\n { toLin := toLin✝, bijectiveLeft := bijectiveLeft✝, bijectiveRight := bijectiveRight✝ }",
" (p (p.toDual... |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 459 | 465 | theorem HasFDerivWithinAt.eventually_ne (h : HasFDerivWithinAt f f' s x)
(hf' : ∃ C, ∀ z, ‖z‖ ≤ C * ‖f' z‖) : ∀ᶠ z in 𝓝[s \ {x}] x, f z ≠ f x := by |
rw [nhdsWithin, diff_eq, ← inf_principal, ← inf_assoc, eventually_inf_principal]
have A : (fun z => z - x) =O[𝓝[s] x] fun z => f' (z - x) :=
isBigO_iff.2 <| hf'.imp fun C hC => eventually_of_forall fun z => hC _
have : (fun z => f z - f x) ~[𝓝[s] x] fun z => f' (z - x) := h.isLittleO.trans_isBigO A
simpa... | [
" HasStrictFDerivAt g (↑f'.symm) a",
" (fun p => g p.1 - g p.2 - f'.symm (p.1 - p.2)) =O[𝓝 (a, a)] fun p => f' (g p.1 - g p.2) - (p.1 - p.2)",
" ↑f'.symm (f' (g x.1 - g x.2) - (x.1 - x.2)) = g x.1 - g x.2 - f'.symm (x.1 - x.2)",
" (fun p => f' (g p.1 - g p.2) - (p.1 - p.2)) =o[𝓝 (a, a)] fun p => p.1 - p.2",... | [
" HasStrictFDerivAt g (↑f'.symm) a",
" (fun p => g p.1 - g p.2 - f'.symm (p.1 - p.2)) =O[𝓝 (a, a)] fun p => f' (g p.1 - g p.2) - (p.1 - p.2)",
" ↑f'.symm (f' (g x.1 - g x.2) - (x.1 - x.2)) = g x.1 - g x.2 - f'.symm (x.1 - x.2)",
" (fun p => f' (g p.1 - g p.2) - (p.1 - p.2)) =o[𝓝 (a, a)] fun p => p.1 - p.2",... |
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Tactic.IntervalCases
#align_import number_theory.padics.padic_val from "leanprover-community/mathlib"@"60fa54e778c9e85d930efae172435f42fb0d71f7"
universe u
ope... | Mathlib/NumberTheory/Padics/PadicVal.lean | 162 | 169 | theorem of_ne_one_ne_zero {z : ℤ} (hp : p ≠ 1) (hz : z ≠ 0) :
padicValInt p z =
(multiplicity (p : ℤ) z).get
(by
apply multiplicity.finite_int_iff.2
simp [hp, hz]) := by |
rw [padicValInt, padicValNat, dif_pos (And.intro hp (Int.natAbs_pos.mpr hz))]
simp only [multiplicity.Int.natAbs p z]
| [
" (multiplicity (↑p) z).Dom",
" (↑p).natAbs ≠ 1 ∧ z ≠ 0",
" padicValInt p z = (multiplicity (↑p) z).get ⋯",
" (multiplicity p z.natAbs).get ⋯ = (multiplicity (↑p) z).get ⋯"
] | [] |
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 | 69 | 72 | theorem tan_eq_zero_iff {θ : ℂ} : tan θ = 0 ↔ ∃ k : ℤ, k * π / 2 = θ := by |
rw [tan, div_eq_zero_iff, ← mul_eq_zero, ← mul_right_inj' two_ne_zero, mul_zero,
← mul_assoc, ← sin_two_mul, sin_eq_zero_iff]
field_simp [mul_comm, eq_comm]
| [
" θ.cos = 0 ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2",
" (cexp (θ * I) + cexp (-θ * I)) / 2 = 0 ↔ cexp (2 * θ * I) = -1",
" cexp (θ * I - -θ * I) = -1 ↔ cexp (2 * θ * I) = -1",
" (∃ n, 2 * I * θ = ↑π * I + ↑n * (2 * ↑π * I)) ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2",
" 2 * I * θ = ↑π * I + ↑x * (2 * ↑π * I) ↔ θ = (2 * ↑x +... | [
" θ.cos = 0 ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2",
" (cexp (θ * I) + cexp (-θ * I)) / 2 = 0 ↔ cexp (2 * θ * I) = -1",
" cexp (θ * I - -θ * I) = -1 ↔ cexp (2 * θ * I) = -1",
" (∃ n, 2 * I * θ = ↑π * I + ↑n * (2 * ↑π * I)) ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2",
" 2 * I * θ = ↑π * I + ↑x * (2 * ↑π * I) ↔ θ = (2 * ↑x +... |
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Rat.Cast.CharZero
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algebra.Order.Field.Basic
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {F ι α β : Type*}
namespace Rat
variable {p... | Mathlib/Data/Rat/Cast/Order.lean | 31 | 33 | theorem cast_pos_of_pos (hq : 0 < q) : (0 : K) < q := by |
rw [Rat.cast_def]
exact div_pos (Int.cast_pos.2 <| num_pos.2 hq) (Nat.cast_pos.2 q.pos)
| [
" 0 < ↑q",
" 0 < ↑q.num / ↑q.den"
] | [] |
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 | 64 | 66 | theorem reduceOption_length_eq_iff {l : List (Option α)} :
l.reduceOption.length = l.length ↔ ∀ x ∈ l, Option.isSome x := by |
rw [reduceOption_length_eq, List.filter_length_eq_length]
| [
" (some x :: l).reduceOption = x :: l.reduceOption",
" (none :: l).reduceOption = l.reduceOption",
" (map (Option.map f) l).reduceOption = map f l.reduceOption",
" (map (Option.map f) []).reduceOption = map f [].reduceOption",
" (map (Option.map f) (hd :: tl)).reduceOption = map f (hd :: tl).reduceOption",
... | [
" (some x :: l).reduceOption = x :: l.reduceOption",
" (none :: l).reduceOption = l.reduceOption",
" (map (Option.map f) l).reduceOption = map f l.reduceOption",
" (map (Option.map f) []).reduceOption = map f [].reduceOption",
" (map (Option.map f) (hd :: tl)).reduceOption = map f (hd :: tl).reduceOption",
... |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 78 | 87 | theorem MellinConvergent.comp_rpow {f : ℝ → E} {s : ℂ} {a : ℝ} (ha : a ≠ 0) :
MellinConvergent (fun t => f (t ^ a)) s ↔ MellinConvergent f (s / a) := by |
refine Iff.trans ?_ (integrableOn_Ioi_comp_rpow_iff' _ ha)
rw [MellinConvergent]
refine integrableOn_congr_fun (fun t ht => ?_) measurableSet_Ioi
dsimp only [Pi.smul_apply]
rw [← Complex.coe_smul (t ^ (a - 1)), ← mul_smul, ← cpow_mul_ofReal_nonneg (le_of_lt ht),
ofReal_cpow (le_of_lt ht), ← cpow_add _ _ ... | [
" MellinConvergent (fun t => c • f t) s",
" MellinConvergent (fun t => ↑t ^ a • f t) s ↔ MellinConvergent f (s + a)",
" ↑t ^ (s - 1) • (fun t => ↑t ^ a • f t) t = ↑t ^ (s + a - 1) • f t",
" MellinConvergent (fun t => f t / a) s",
" MellinConvergent (fun t => f (a * t)) s ↔ MellinConvergent f s",
" (fun t ... | [
" MellinConvergent (fun t => c • f t) s",
" MellinConvergent (fun t => ↑t ^ a • f t) s ↔ MellinConvergent f (s + a)",
" ↑t ^ (s - 1) • (fun t => ↑t ^ a • f t) t = ↑t ^ (s + a - 1) • f t",
" MellinConvergent (fun t => f t / a) s",
" MellinConvergent (fun t => f (a * t)) s ↔ MellinConvergent f s",
" (fun t ... |
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
#align list.length_enum_from List.enumFrom_length
#align list.length_enum List.enum_length
@[simp]
theorem get?_enumFrom :
∀ n (l : List α) m, get? (enumFrom n l) m = (get? l m).map fun a => (n + m, a)
| n, [], m => rfl
| n, a :: l, 0 =... | Mathlib/Data/List/Enum.lean | 54 | 56 | theorem get_enum (l : List α) (i : Fin l.enum.length) :
l.enum.get i = (i.1, l.get (i.cast enum_length)) := by |
simp [enum]
| [
" Option.map (fun a => (n + 1 + m, a)) (l.get? m) = Option.map (fun a => (n + (m + 1), a)) ((a :: l).get? (m + 1))",
" Option.map (fun a => (n + m + 1, a)) (l.get? m) = Option.map (fun a => (n + (m + 1), a)) ((a :: l).get? (m + 1))",
" l.enum.get? n = Option.map (fun a => (n, a)) (l.get? n)",
" (enumFrom n l)... | [
" Option.map (fun a => (n + 1 + m, a)) (l.get? m) = Option.map (fun a => (n + (m + 1), a)) ((a :: l).get? (m + 1))",
" Option.map (fun a => (n + m + 1, a)) (l.get? m) = Option.map (fun a => (n + (m + 1), a)) ((a :: l).get? (m + 1))",
" l.enum.get? n = Option.map (fun a => (n, a)) (l.get? n)",
" (enumFrom n l)... |
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.Algebra.CharP.Reduced
open Function Polynomial
class PerfectRing (R : Type*) (p : ℕ) [CommSemiring R] [ExpChar R p] : Prop where
bijective_frobenius : Bijective <| frobenius R p
section PerfectRing
va... | Mathlib/FieldTheory/Perfect.lean | 146 | 148 | theorem frobenius_comp_frobeniusEquiv_symm :
(frobenius R p).comp (frobeniusEquiv R p).symm = RingHom.id R := by |
ext; simp
| [
" (iterateFrobeniusEquiv R p (m + n)) ((iterateFrobeniusEquiv R p (m + n)).symm x) =\n (iterateFrobeniusEquiv R p (m + n)) ((iterateFrobeniusEquiv R p m).symm ((iterateFrobeniusEquiv R p n).symm x))",
" (iterateFrobeniusEquiv R p 0) x = x",
" (iterateFrobeniusEquiv R p 1) x = x ^ p",
" ⇑(iterateFrobeniusEq... | [
" (iterateFrobeniusEquiv R p (m + n)) ((iterateFrobeniusEquiv R p (m + n)).symm x) =\n (iterateFrobeniusEquiv R p (m + n)) ((iterateFrobeniusEquiv R p m).symm ((iterateFrobeniusEquiv R p n).symm x))",
" (iterateFrobeniusEquiv R p 0) x = x",
" (iterateFrobeniusEquiv R p 1) x = x ^ p",
" ⇑(iterateFrobeniusEq... |
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monotonicity
#align_import data.nat.totient from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
open Finset
namespace Nat
... | Mathlib/Data/Nat/Totient.lean | 51 | 57 | theorem totient_eq_card_lt_and_coprime (n : ℕ) : φ n = Nat.card { m | m < n ∧ n.Coprime m } := by |
let e : { m | m < n ∧ n.Coprime m } ≃ Finset.filter n.Coprime (Finset.range n) :=
{ toFun := fun m => ⟨m, by simpa only [Finset.mem_filter, Finset.mem_range] using m.property⟩
invFun := fun m => ⟨m, by simpa only [Finset.mem_filter, Finset.mem_range] using m.property⟩
left_inv := fun m => by simp onl... | [
" φ n = Nat.card ↑{m | m < n ∧ n.Coprime m}",
" ↑m ∈ filter n.Coprime (range n)",
" ↑m ∈ {m | m < n ∧ n.Coprime m}",
" (fun m => ⟨↑m, ⋯⟩) ((fun m => ⟨↑m, ⋯⟩) m) = m"
] | [] |
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
universe u v w
namespace MvPolynomial
variable {σ τ : Type*} {R : Type u} {S : Type v} {r : R} {e : ℕ} {n m : σ}
section CommSemiring
variable... | Mathlib/Algebra/MvPolynomial/Supported.lean | 123 | 127 | theorem supported_le_supported_iff [Nontrivial R] : supported R s ≤ supported R t ↔ s ⊆ t := by |
constructor
· intro h i
simpa using @h (X i)
· exact supported_mono
| [
" supported R s = (rename Subtype.val).range",
" (aeval fun x => X ↑x).range = (aeval (X ∘ Subtype.val)).range",
" (supportedEquivMvPolynomial s).symm (C x) = (algebraMap R ↥(supported R s)) x",
" ↑((supportedEquivMvPolynomial s).symm (C x)) = ↑((algebraMap R ↥(supported R s)) x)",
" ↑((supportedEquivMvPoly... | [
" supported R s = (rename Subtype.val).range",
" (aeval fun x => X ↑x).range = (aeval (X ∘ Subtype.val)).range",
" (supportedEquivMvPolynomial s).symm (C x) = (algebraMap R ↥(supported R s)) x",
" ↑((supportedEquivMvPolynomial s).symm (C x)) = ↑((algebraMap R ↥(supported R s)) x)",
" ↑((supportedEquivMvPoly... |
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Data.Rat.Cast.CharZero
import Mathlib.Algebra.Field.Basic
set_option autoImplicit true
namespace Mathlib.Meta.NormNum
open Lean.Meta Qq
def inferCharZeroOfRing {α : Q(Type u)} (_i : Q(Ring $α) := by with_reducible assumption) :
MetaM Q(CharZero $α) :=
ret... | Mathlib/Tactic/NormNum/Inv.lean | 124 | 131 | theorem isRat_inv_neg {α} [DivisionRing α] [CharZero α] {a : α} {n d : ℕ} :
IsRat a (.negOfNat (Nat.succ n)) d → IsRat a⁻¹ (.negOfNat d) (Nat.succ n) := by |
rintro ⟨_, rfl⟩
simp only [Int.negOfNat_eq]
have := invertibleOfNonzero (α := α) (Nat.cast_ne_zero.2 (Nat.succ_ne_zero n))
generalize Nat.succ n = n at *
use this; simp only [Int.ofNat_eq_coe, Int.cast_neg,
Int.cast_natCast, invOf_eq_inv, inv_neg, neg_mul, mul_inv_rev, inv_inv]
| [
" IsRat (mkRat ↑n✝¹ ↑n✝) num✝ denom✝",
" IsRat (↑↑n✝¹ / ↑↑n✝) num✝ denom✝",
" ↑↑n✝ = ↑n✝",
" ↑qi * ↑denom✝ = 1",
" ↑denom✝ * ↑qi = 1",
" ↑(↑num✝ * ⅟↑denom✝) = ↑num✝ * ⅟↑denom✝",
" ↑(↑num✝ * qi) = ↑num✝ * ↑qi",
" IsRat a (Int.ofNat n.succ) d → IsRat a⁻¹ (Int.ofNat d) n.succ",
" IsRat (↑(Int.ofNat n.s... | [
" IsRat (mkRat ↑n✝¹ ↑n✝) num✝ denom✝",
" IsRat (↑↑n✝¹ / ↑↑n✝) num✝ denom✝",
" ↑↑n✝ = ↑n✝",
" ↑qi * ↑denom✝ = 1",
" ↑denom✝ * ↑qi = 1",
" ↑(↑num✝ * ⅟↑denom✝) = ↑num✝ * ⅟↑denom✝",
" ↑(↑num✝ * qi) = ↑num✝ * ↑qi",
" IsRat a (Int.ofNat n.succ) d → IsRat a⁻¹ (Int.ofNat d) n.succ",
" IsRat (↑(Int.ofNat n.s... |
import Mathlib.Analysis.BoxIntegral.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.Tactic.Generalize
#align_import analysis.box_integral.integrability from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open scoped Classical NNReal ENNReal Topology
universe u v
... | Mathlib/Analysis/BoxIntegral/Integrability.lean | 104 | 155 | theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ι} {f : (ι → ℝ) → E}
{μ : Measure (ι → ℝ)} [IsLocallyFiniteMeasure μ] (hf : f =ᵐ[μ.restrict I] 0)
(hl : l.bRiemann = false) : HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul 0 := by |
/- Each set `{x | n < ‖f x‖ ≤ n + 1}`, `n : ℕ`, has measure zero. We cover it by an open set of
measure less than `ε / 2 ^ n / (n + 1)`. Then the norm of the integral sum is less than `ε`. -/
refine hasIntegral_iff.2 fun ε ε0 => ?_
lift ε to ℝ≥0 using ε0.lt.le; rw [gt_iff_lt, NNReal.coe_pos] at ε0
rcases N... | [
" HasIntegral I l (s.indicator fun x => y) μ.toBoxAdditive.toSMul ((μ (s ∩ ↑I)).toReal • y)",
" ∃ r,\n (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n ∀ (c : ℝ≥0) (π : TaggedPrepartition I),\n l.MemBaseSet I c (r c) π →\n π.IsPartition →\n dist (integralSum (s.indicator fun x => y) μ.toBoxAddi... | [
" HasIntegral I l (s.indicator fun x => y) μ.toBoxAdditive.toSMul ((μ (s ∩ ↑I)).toReal • y)",
" ∃ r,\n (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n ∀ (c : ℝ≥0) (π : TaggedPrepartition I),\n l.MemBaseSet I c (r c) π →\n π.IsPartition →\n dist (integralSum (s.indicator fun x => y) μ.toBoxAddi... |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 140 | 152 | theorem exists_finite_submodule_of_finite' (s : Set (M₁ ⊗[R] N₁)) (hs : s.Finite) :
∃ (M' : Submodule R M) (N' : Submodule R N) (hM : M' ≤ M₁) (hN : N' ≤ N₁),
Module.Finite R M' ∧ Module.Finite R N' ∧
s ⊆ LinearMap.range (TensorProduct.map (inclusion hM) (inclusion hN)) := by |
obtain ⟨M', N', _, _, h⟩ := exists_finite_submodule_of_finite s hs
have hM := map_subtype_le M₁ M'
have hN := map_subtype_le N₁ N'
refine ⟨_, _, hM, hN, .map _ _, .map _ _, ?_⟩
rw [mapIncl,
show M'.subtype = inclusion hM ∘ₗ M₁.subtype.submoduleMap M' by ext; simp,
show N'.subtype = inclusion hN ∘ₗ N₁... | [
" ∃ S, x = (Multiset.map (fun i => i.1 ⊗ₜ[R] i.2) S).sum",
" ∃ S, 0 = (Multiset.map (fun i => i.1 ⊗ₜ[R] i.2) S).sum",
" 0 = (Multiset.map (fun i => i.1 ⊗ₜ[R] i.2) 0).sum",
" ∃ S, x ⊗ₜ[R] y = (Multiset.map (fun i => i.1 ⊗ₜ[R] i.2) S).sum",
" x ⊗ₜ[R] y = (Multiset.map (fun i => i.1 ⊗ₜ[R] i.2) {(x, y)}).sum",
... | [
" ∃ S, x = (Multiset.map (fun i => i.1 ⊗ₜ[R] i.2) S).sum",
" ∃ S, 0 = (Multiset.map (fun i => i.1 ⊗ₜ[R] i.2) S).sum",
" 0 = (Multiset.map (fun i => i.1 ⊗ₜ[R] i.2) 0).sum",
" ∃ S, x ⊗ₜ[R] y = (Multiset.map (fun i => i.1 ⊗ₜ[R] i.2) S).sum",
" x ⊗ₜ[R] y = (Multiset.map (fun i => i.1 ⊗ₜ[R] i.2) {(x, y)}).sum",
... |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
noncomputable section
namespace Polynomial
universe u v w
section Semiring
variable {R : Type... | Mathlib/Algebra/Polynomial/Lifts.lean | 87 | 91 | theorem C'_mem_lifts {f : R →+* S} {s : S} (h : s ∈ Set.range f) : C s ∈ lifts f := by |
obtain ⟨r, rfl⟩ := Set.mem_range.1 h
use C r
simp only [coe_mapRingHom, map_C, Set.mem_univ, Subsemiring.coe_top, eq_self_iff_true,
and_self_iff]
| [
" p ∈ lifts f ↔ ∃ q, map f q = p",
" p ∈ lifts f ↔ p ∈ Set.range (map f)",
" p ∈ lifts f ↔ p ∈ (mapRingHom f).rangeS",
" p ∈ lifts f ↔ ∀ (n : ℕ), p.coeff n ∈ Set.range ⇑f",
" (∀ (n : ℕ), p.coeff n ∈ f.rangeS) ↔ ∀ (n : ℕ), p.coeff n ∈ Set.range ⇑f",
" (mapRingHom f) (C r) = C (f r)",
" C s ∈ lifts f",
... | [
" p ∈ lifts f ↔ ∃ q, map f q = p",
" p ∈ lifts f ↔ p ∈ Set.range (map f)",
" p ∈ lifts f ↔ p ∈ (mapRingHom f).rangeS",
" p ∈ lifts f ↔ ∀ (n : ℕ), p.coeff n ∈ Set.range ⇑f",
" (∀ (n : ℕ), p.coeff n ∈ f.rangeS) ↔ ∀ (n : ℕ), p.coeff n ∈ Set.range ⇑f",
" (mapRingHom f) (C r) = C (f r)"
] |
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
set_option linter.uppercaseLean3 false
open Polynomial
... | Mathlib/RingTheory/Polynomial/Quotient.lean | 205 | 209 | theorem quotient_map_C_eq_zero {I : Ideal R} {i : R} (hi : i ∈ I) :
(Ideal.Quotient.mk (Ideal.map (C : R →+* MvPolynomial σ R) I :
Ideal (MvPolynomial σ R))).comp C i = 0 := by |
simp only [Function.comp_apply, RingHom.coe_comp, Ideal.Quotient.eq_zero_iff_mem]
exact Ideal.mem_map_of_mem _ hi
| [
" ((Ideal.Quotient.mk (Ideal.map C I)).comp C) i = 0",
" C i ∈ Ideal.map C I"
] | [] |
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 | 120 | 122 | theorem dedup_map_dedup_eq [DecidableEq β] (f : α → β) (s : Multiset α) :
dedup (map f (dedup s)) = dedup (map f s) := by |
simp [dedup_ext]
| [
" count a (dedup (Quot.mk Setoid.r x✝)) = if a ∈ Quot.mk Setoid.r x✝ then 1 else 0",
" List.count a x✝.dedup = if a ∈ x✝ then 1 else 0",
" s ≤ s.dedup ↔ s.Nodup",
" s.dedup = t.dedup ↔ ∀ (a : α), a ∈ s ↔ a ∈ t",
" (map f s.dedup).dedup = (map f s).dedup"
] | [
" count a (dedup (Quot.mk Setoid.r x✝)) = if a ∈ Quot.mk Setoid.r x✝ then 1 else 0",
" List.count a x✝.dedup = if a ∈ x✝ then 1 else 0",
" s ≤ s.dedup ↔ s.Nodup",
" s.dedup = t.dedup ↔ ∀ (a : α), a ∈ s ↔ a ∈ t"
] |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 409 | 411 | theorem map_one_snd_of_isTwoCocycle {f : G × G → A} (hf : IsTwoCocycle f) (g : G) :
f (g, 1) = g • f (1, 1) := by |
simpa only [mul_one, add_left_inj] using hf g 1 1
| [
" f 1 = 0",
" f (1, g) = f (1, 1)",
" f (g, 1) = g • f (1, 1)"
] | [
" f 1 = 0",
" f (1, g) = f (1, 1)"
] |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 322 | 325 | theorem differentiableOn_euclidean :
DifferentiableOn 𝕜 f t ↔ ∀ i, DifferentiableOn 𝕜 (fun x => f x i) t := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiableOn_iff, differentiableOn_pi]
rfl
| [
" DifferentiableWithinAt 𝕜 f t y ↔ ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x => f x i) t y",
" (∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x => (⇑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i) t y) ↔\n ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x => f x i) t y",
" DifferentiableAt 𝕜 f y ↔ ∀ (i : ι), DifferentiableA... | [
" DifferentiableWithinAt 𝕜 f t y ↔ ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x => f x i) t y",
" (∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x => (⇑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i) t y) ↔\n ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x => f x i) t y",
" DifferentiableAt 𝕜 f y ↔ ∀ (i : ι), DifferentiableA... |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 157 | 160 | theorem _root_.MeasureTheory.Integrable.integral_norm_condDistrib_map
(hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂condDistrib Y X μ x) (μ.map X) := by |
rw [condDistrib, ← Measure.fst_map_prod_mk₀ (X := X) hY]; exact hf_int.integral_norm_condKernel
| [
" IsMarkovKernel (condDistrib Y X μ)",
" IsMarkovKernel (Measure.map (fun a => (X a, Y a)) μ).condKernel",
" ((condDistrib Y X μ) x) s = ((Measure.map X μ) {x})⁻¹ * (Measure.map (fun a => (X a, Y a)) μ) ({x} ×ˢ s)",
" ((Measure.map (fun a => (X a, Y a)) μ).fst {x})⁻¹ * (Measure.map (fun a => (X a, Y a)) μ) ({... | [
" IsMarkovKernel (condDistrib Y X μ)",
" IsMarkovKernel (Measure.map (fun a => (X a, Y a)) μ).condKernel",
" ((condDistrib Y X μ) x) s = ((Measure.map X μ) {x})⁻¹ * (Measure.map (fun a => (X a, Y a)) μ) ({x} ×ˢ s)",
" ((Measure.map (fun a => (X a, Y a)) μ).fst {x})⁻¹ * (Measure.map (fun a => (X a, Y a)) μ) ({... |
import Mathlib.Probability.ProbabilityMassFunction.Monad
#align_import probability.probability_mass_function.constructions from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
universe u
namespace PMF
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENN... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 125 | 128 | theorem seq_apply : (seq q p) b = ∑' (f : α → β) (a : α), if b = f a then q f * p a else 0 := by |
simp only [seq, mul_boole, bind_apply, pure_apply]
refine tsum_congr fun f => ENNReal.tsum_mul_left.symm.trans (tsum_congr fun a => ?_)
simpa only [mul_zero] using mul_ite (b = f a) (q f) (p a) 0
| [
" (map f p) b = ∑' (a : α), if b = f a then p a else 0",
" b ∈ (map f p).support ↔ b ∈ f '' p.support",
" b ∈ (map f p).support ↔ ∃ a ∈ p.support, f a = b",
" map g (map f p) = map (g ∘ f) p",
" map (Function.const α b) p = pure b",
" (q.seq p) b = ∑' (f : α → β) (a : α), if b = f a then q f * p a else 0"... | [
" (map f p) b = ∑' (a : α), if b = f a then p a else 0",
" b ∈ (map f p).support ↔ b ∈ f '' p.support",
" b ∈ (map f p).support ↔ ∃ a ∈ p.support, f a = b",
" map g (map f p) = map (g ∘ f) p",
" map (Function.const α b) p = pure b"
] |
import Mathlib.RingTheory.Ideal.Maps
#align_import ring_theory.ideal.prod from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
universe u v
variable {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I I' : Ideal R) (J J' : Ideal S)
namespace Ideal
def prod : Ideal (R × S) where
... | Mathlib/RingTheory/Ideal/Prod.lean | 103 | 105 | theorem prod.ext_iff {I I' : Ideal R} {J J' : Ideal S} :
prod I J = prod I' J' ↔ I = I' ∧ J = J' := by |
simp only [← idealProdEquiv_symm_apply, idealProdEquiv.symm.injective.eq_iff, Prod.mk.inj_iff]
| [
" ∀ {a b : R × S}, a ∈ {x | x.1 ∈ I ∧ x.2 ∈ J} → b ∈ {x | x.1 ∈ I ∧ x.2 ∈ J} → a + b ∈ {x | x.1 ∈ I ∧ x.2 ∈ J}",
" (a₁, a₂) + (b₁, b₂) ∈ {x | x.1 ∈ I ∧ x.2 ∈ J}",
" 0 ∈ { carrier := {x | x.1 ∈ I ∧ x.2 ∈ J}, add_mem' := ⋯ }.carrier",
" ∀ (c : R × S) {x : R × S},\n x ∈ { carrier := {x | x.1 ∈ I ∧ x.2 ∈ J}, a... | [
" ∀ {a b : R × S}, a ∈ {x | x.1 ∈ I ∧ x.2 ∈ J} → b ∈ {x | x.1 ∈ I ∧ x.2 ∈ J} → a + b ∈ {x | x.1 ∈ I ∧ x.2 ∈ J}",
" (a₁, a₂) + (b₁, b₂) ∈ {x | x.1 ∈ I ∧ x.2 ∈ J}",
" 0 ∈ { carrier := {x | x.1 ∈ I ∧ x.2 ∈ J}, add_mem' := ⋯ }.carrier",
" ∀ (c : R × S) {x : R × S},\n x ∈ { carrier := {x | x.1 ∈ I ∧ x.2 ∈ J}, a... |
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_anti... | Mathlib/RingTheory/PowerSeries/Order.lean | 121 | 129 | theorem le_order (φ : R⟦X⟧) (n : PartENat) (h : ∀ i : ℕ, ↑i < n → coeff R i φ = 0) :
n ≤ order φ := by |
induction n using PartENat.casesOn
· show _ ≤ _
rw [top_le_iff, order_eq_top]
ext i
exact h _ (PartENat.natCast_lt_top i)
· apply nat_le_order
simpa only [PartENat.coe_lt_coe] using h
| [
" (∃ n, (coeff R n) φ ≠ 0) ↔ φ ≠ 0",
" (¬∃ n, (coeff R n) φ ≠ 0) ↔ ¬φ ≠ 0",
" (∀ (n : ℕ), (coeff R n) φ = 0) ↔ φ = 0",
" φ.order.Dom ↔ φ ≠ 0",
" (if h : φ = 0 then ⊤ else ↑(Nat.find ⋯)).Dom ↔ φ ≠ 0",
" (if h : φ = 0 then ⊤ else ↑(Nat.find ⋯)).Dom → φ ≠ 0",
" ⊤.Dom → φ ≠ 0",
" (↑(Nat.find ⋯)).Dom → φ ≠... | [
" (∃ n, (coeff R n) φ ≠ 0) ↔ φ ≠ 0",
" (¬∃ n, (coeff R n) φ ≠ 0) ↔ ¬φ ≠ 0",
" (∀ (n : ℕ), (coeff R n) φ = 0) ↔ φ = 0",
" φ.order.Dom ↔ φ ≠ 0",
" (if h : φ = 0 then ⊤ else ↑(Nat.find ⋯)).Dom ↔ φ ≠ 0",
" (if h : φ = 0 then ⊤ else ↑(Nat.find ⋯)).Dom → φ ≠ 0",
" ⊤.Dom → φ ≠ 0",
" (↑(Nat.find ⋯)).Dom → φ ≠... |
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 | 150 | 152 | theorem card_pair_eq_one_or_two : ({a,b} : Finset α).card = 1 ∨ ({a,b} : Finset α).card = 2 := by |
simp [card_insert_eq_ite]
tauto
| [
" Monotone card",
" ({a} ∩ s).card ≤ 1",
" (insert a s).card = s.card + 1",
" (insert a s).card = s.card",
" (insert a s).card ≤ s.card + 1",
" s.card ≤ s.card + 1",
" (insert a s).card = if a ∈ s then s.card else s.card + 1",
" {a, b}.card = 1 ∨ {a, b}.card = 2",
" a = b ∨ ¬a = b"
] | [
" Monotone card",
" ({a} ∩ s).card ≤ 1",
" (insert a s).card = s.card + 1",
" (insert a s).card = s.card",
" (insert a s).card ≤ s.card + 1",
" s.card ≤ s.card + 1",
" (insert a s).card = if a ∈ s then s.card else s.card + 1"
] |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 315 | 324 | theorem descPochhammer_natDegree (n : ℕ) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n := by |
induction' n with n hn
· simp
· have : natDegree (X - (n : R[X])) = 1 := natDegree_X_sub_C (n : R)
rw [descPochhammer_succ_right,
natDegree_mul _ (ne_zero_of_natDegree_gt <| this.symm ▸ Nat.zero_lt_one), hn, this]
cases n
· simp
· refine ne_zero_of_natDegree_gt <| hn.symm ▸ Nat.add_one_po... | [
" descPochhammer R 1 = X",
" descPochhammer R (n + 1) = X * (descPochhammer R n).comp (X - 1)",
" (descPochhammer R n).Monic",
" (descPochhammer R 0).Monic",
" (descPochhammer R (n + 1)).Monic",
" map f (descPochhammer R n) = descPochhammer T n",
" map f (descPochhammer R 0) = descPochhammer T 0",
" m... | [
" descPochhammer R 1 = X",
" descPochhammer R (n + 1) = X * (descPochhammer R n).comp (X - 1)",
" (descPochhammer R n).Monic",
" (descPochhammer R 0).Monic",
" (descPochhammer R (n + 1)).Monic",
" map f (descPochhammer R n) = descPochhammer T n",
" map f (descPochhammer R 0) = descPochhammer T 0",
" m... |
import Mathlib.MeasureTheory.OuterMeasure.Basic
open Filter Set
open scoped ENNReal
namespace MeasureTheory
variable {α β F : Type*} [FunLike F (Set α) ℝ≥0∞] [OuterMeasureClass F α] {μ : F} {s t : Set α}
def ae (μ : F) : Filter α :=
.ofCountableUnion (μ · = 0) (fun _S hSc ↦ (measure_sUnion_null_iff hSc).2) fu... | Mathlib/MeasureTheory/OuterMeasure/AE.lean | 107 | 109 | theorem all_ae_of {ι : Sort*} {p : α → ι → Prop} (hp : ∀ᵐ a ∂μ, ∀ i, p a i) (i : ι) :
∀ᵐ a ∂μ, p a i := by |
filter_upwards [hp] with a ha using ha i
| [
" sᶜ ∈ ae μ ↔ μ s = 0",
" CountableInterFilter (ae μ)",
" CountableInterFilter (ofCountableUnion (fun x => μ x = 0) ⋯ ⋯)",
" ∀ᵐ (a : α) ∂μ, p a i"
] | [
" sᶜ ∈ ae μ ↔ μ s = 0",
" CountableInterFilter (ae μ)",
" CountableInterFilter (ofCountableUnion (fun x => μ x = 0) ⋯ ⋯)"
] |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.Interval.Set.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open scoped Classical
open Set
variable {ι : ... | Mathlib/Order/CompleteLatticeIntervals.lean | 62 | 64 | theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by |
simp [sSup]
| [
" sSup (Subtype.val '' t) = ↑(sSup t)",
" sSup ∅ = default"
] | [
" sSup (Subtype.val '' t) = ↑(sSup t)"
] |
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.ideal.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
variable {R : Type*} [CommRing R]
namespace Ideal
open Submodule
variable (R) in
def isPrincipalSubmonoid : Submonoid (Ideal R) where
carrier := ... | Mathlib/RingTheory/Ideal/IsPrincipal.lean | 75 | 78 | theorem associatesEquivIsPrincipal_map_zero :
(associatesEquivIsPrincipal R 0 : Ideal R) = 0 := by |
rw [← Associates.mk_zero, ← Associates.quotient_mk_eq_mk, associatesEquivIsPrincipal_apply,
Set.singleton_zero, span_zero, zero_eq_bot]
| [
" ∀ {a b : Ideal R}, a ∈ {I | IsPrincipal I} → b ∈ {I | IsPrincipal I} → a * b ∈ {I | IsPrincipal I}",
" Submodule.span R {x} * Submodule.span R {y} ∈ {I | IsPrincipal I}",
" (fun x => ⟨span {x}, ⋯⟩) x✝² = (fun x => ⟨span {x}, ⋯⟩) x✝¹",
" (fun I => Associates.mk (IsPrincipal.generator ↑I)) (Quotient.lift (fun... | [
" ∀ {a b : Ideal R}, a ∈ {I | IsPrincipal I} → b ∈ {I | IsPrincipal I} → a * b ∈ {I | IsPrincipal I}",
" Submodule.span R {x} * Submodule.span R {y} ∈ {I | IsPrincipal I}",
" (fun x => ⟨span {x}, ⋯⟩) x✝² = (fun x => ⟨span {x}, ⋯⟩) x✝¹",
" (fun I => Associates.mk (IsPrincipal.generator ↑I)) (Quotient.lift (fun... |
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped NNReal Matrix
namespace Matrix
variable {R l m n α β : Type*} [Fintype l] [Fintyp... | Mathlib/Analysis/Matrix.lean | 90 | 91 | theorem norm_le_iff {r : ℝ} (hr : 0 ≤ r) {A : Matrix m n α} : ‖A‖ ≤ r ↔ ∀ i j, ‖A i j‖ ≤ r := by |
simp_rw [norm_def, pi_norm_le_iff_of_nonneg hr]
| [
" ‖A‖ = ↑(Finset.univ.sup fun i => Finset.univ.sup fun j => ‖A i j‖₊)",
" ‖A‖ ≤ r ↔ ∀ (i : m) (j : n), ‖A i j‖ ≤ r"
] | [
" ‖A‖ = ↑(Finset.univ.sup fun i => Finset.univ.sup fun j => ‖A i j‖₊)"
] |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.Algebra.Category.ModuleCat.Basic
#align_import algebra.category.Module.epi_mono from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe v u
open CategoryTheory
namespace ModuleCat
variable {R : Type u} [Ring R] {X Y : ModuleCa... | Mathlib/Algebra/Category/ModuleCat/EpiMono.lean | 44 | 45 | theorem mono_iff_injective : Mono f ↔ Function.Injective f := by |
rw [mono_iff_ker_eq_bot, LinearMap.ker_eq_bot]
| [
" Function.Injective ⇑f",
" Mono f ↔ Function.Injective ⇑f"
] | [
" Function.Injective ⇑f"
] |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : ℕ} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 498 | 500 | theorem nthHom_zero : nthHom f 0 = 0 := by |
simp (config := { unfoldPartialApp := true }) [nthHom]
rfl
| [
" nthHom f 0 = 0",
" (fun n => 0) = 0"
] | [] |
import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.IsSplittingField
import Mathlib.RingTheory.TensorProduct.Basic
#align_import field_theory.adjoin from "leanprover-community/mathlib"@"df76f4335... | Mathlib/FieldTheory/Adjoin.lean | 62 | 67 | theorem mem_adjoin_simple_iff {α : E} (x : E) :
x ∈ adjoin F {α} ↔ ∃ r s : F[X], x = aeval α r / aeval α s := by |
simp only [adjoin, mem_mk, Subring.mem_toSubsemiring, Subfield.mem_toSubring,
Subfield.mem_closure_iff, ← Algebra.adjoin_eq_ring_closure, Subalgebra.mem_toSubring,
Algebra.adjoin_singleton_eq_range_aeval, AlgHom.mem_range, exists_exists_eq_and]
tauto
| [
" x ∈ adjoin F S ↔ ∃ r s, x = (MvPolynomial.aeval Subtype.val) r / (MvPolynomial.aeval Subtype.val) s",
" (∃ a a_1, (MvPolynomial.aeval Subtype.val) a / (MvPolynomial.aeval Subtype.val) a_1 = x) ↔\n ∃ r s, x = (MvPolynomial.aeval Subtype.val) r / (MvPolynomial.aeval Subtype.val) s",
" x ∈ adjoin F {α} ↔ ∃ r ... | [
" x ∈ adjoin F S ↔ ∃ r s, x = (MvPolynomial.aeval Subtype.val) r / (MvPolynomial.aeval Subtype.val) s",
" (∃ a a_1, (MvPolynomial.aeval Subtype.val) a / (MvPolynomial.aeval Subtype.val) a_1 = x) ↔\n ∃ r s, x = (MvPolynomial.aeval Subtype.val) r / (MvPolynomial.aeval Subtype.val) s"
] |
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