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.Category.MonCat.Basic
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
#align_import algebra.category.Mon.colimits from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe v
open CategoryTheory
open Ca... | Mathlib/Algebra/Category/MonCat/Colimits.lean | 188 | 191 | theorem cocone_naturality_components (j j' : J) (f : j βΆ j') (x : F.obj j) :
(coconeMorphism F j') (F.map f x) = (coconeMorphism F j) x := by |
rw [β cocone_naturality F f]
rfl
| [
" Inhabited (ColimitType F)",
" Inhabited (Quotient (colimitSetoid F))",
" Monoid (ColimitType F)",
" F.map f β« coconeMorphism F j' = coconeMorphism F j",
" (F.map f β« coconeMorphism F j') xβ = (coconeMorphism F j) xβ",
" Setoid.r (Prequotient.of j' ((F.map f) xβ)) (Prequotient.of j xβ)",
" (coconeMorph... | [
" Inhabited (ColimitType F)",
" Inhabited (Quotient (colimitSetoid F))",
" Monoid (ColimitType F)",
" F.map f β« coconeMorphism F j' = coconeMorphism F j",
" (F.map f β« coconeMorphism F j') xβ = (coconeMorphism F j) xβ",
" Setoid.r (Prequotient.of j' ((F.map f) xβ)) (Prequotient.of j xβ)"
] |
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import ring_theory.simple_module from "leanprover-community/mathlib"@"cce7f68a7eaadadf74c82bbac207... | Mathlib/RingTheory/SimpleModule.lean | 91 | 94 | theorem isSimpleModule_iff_isCoatom : IsSimpleModule R (M β§Έ m) β IsCoatom m := by |
rw [β Set.isSimpleOrder_Ici_iff_isCoatom]
apply OrderIso.isSimpleOrder_iff
exact Submodule.comapMkQRelIso m
| [
" β y, 0 β y",
" β₯ = β€",
" x β β₯ β x β β€",
" IsSimpleModule R β₯m β IsAtom m",
" IsSimpleModule R β₯m β IsSimpleOrder β(Set.Iic m)",
" IsSimpleModule R (M β§Έ m) β IsCoatom m",
" IsSimpleModule R (M β§Έ m) β IsSimpleOrder β(Set.Ici m)",
" Submodule R (M β§Έ m) βo β(Set.Ici m)"
] | [
" β y, 0 β y",
" β₯ = β€",
" x β β₯ β x β β€",
" IsSimpleModule R β₯m β IsAtom m",
" IsSimpleModule R β₯m β IsSimpleOrder β(Set.Iic m)"
] |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/mathlib"@"a50170a88a47570ed186b809ca754110590f9476"
universe u v w wβ wβ
variable (R : Type u) (L : Type v) (M : Type w) {L' : Type wβ}
variab... | Mathlib/Algebra/Lie/Solvable.lean | 116 | 124 | theorem derivedSeriesOfIdeal_add_le_add (J : LieIdeal R L) (k l : β) :
D (k + l) (I + J) β€ D k I + D l J := by |
let Dβ : LieIdeal R L βo LieIdeal R L :=
{ toFun := fun I => β
I, Iβ
monotone' := fun I J h => LieSubmodule.mono_lie I J I J h h }
have hβ : β I J : LieIdeal R L, Dβ (I β J) β€ Dβ I β J := by
simp [Dβ, LieSubmodule.lie_le_right, LieSubmodule.lie_le_left, le_sup_of_le_right]
rw [β Dβ.iterate_sup_le_su... | [
" D (k + l) I = D k (D l I)",
" D (0 + l) I = D 0 (D l I)",
" D (k + 1 + l) I = D (k + 1) (D l I)",
" D k I β€ D l J",
" β {l : β}, l β€ k β D k I β€ D l J",
" β {l : β}, l β€ 0 β D 0 I β€ D l J",
" β {l : β}, l β€ k + 1 β D (k + 1) I β€ D l J",
" D 0 I β€ D l J",
" I β€ D 0 J",
" D (k + 1) I β€ D l J",
"... | [
" D (k + l) I = D k (D l I)",
" D (0 + l) I = D 0 (D l I)",
" D (k + 1 + l) I = D (k + 1) (D l I)",
" D k I β€ D l J",
" β {l : β}, l β€ k β D k I β€ D l J",
" β {l : β}, l β€ 0 β D 0 I β€ D l J",
" β {l : β}, l β€ k + 1 β D (k + 1) I β€ D l J",
" D 0 I β€ D l J",
" I β€ D 0 J",
" D (k + 1) I β€ D l J",
"... |
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b37058d0"
open Function Set
variable {Ξ± Ξ±' Ξ² Ξ²' Ξ³ Ξ³' Ξ΄ Ξ΄' Ξ΅ Ξ΅' ΞΆ ΞΆ' Ξ½ : Type*}
namespace Finset
variable [DecidableEq Ξ±'] [DecidableEq Ξ²'] [Decidabl... | Mathlib/Data/Finset/NAry.lean | 73 | 74 | theorem mem_imageβ_iff (hf : Injective2 f) : f a b β imageβ f s t β a β s β§ b β t := by |
rw [β mem_coe, coe_imageβ, mem_image2_iff hf, mem_coe, mem_coe]
| [
" c β imageβ f s t β β a β s, β b β t, f a b = c",
" (imageβ f s t).card = s.card * t.card β InjOn (fun x => f x.1 x.2) (βs ΓΛ’ βt)",
" (imageβ f s t).card = (s ΓΛ’ t).card β InjOn (fun x => f x.1 x.2) β(s ΓΛ’ t)",
" f a b β imageβ f s t β a β s β§ b β t"
] | [
" c β imageβ f s t β β a β s, β b β t, f a b = c",
" (imageβ f s t).card = s.card * t.card β InjOn (fun x => f x.1 x.2) (βs ΓΛ’ βt)",
" (imageβ f s t).card = (s ΓΛ’ t).card β InjOn (fun x => f x.1 x.2) β(s ΓΛ’ t)"
] |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 144 | 151 | theorem add (hf : IsBoundedLinearMap π f) (hg : IsBoundedLinearMap π g) :
IsBoundedLinearMap π fun e => f e + g e :=
let β¨hlf, Mf, _, hMfβ© := hf
let β¨hlg, Mg, _, hMgβ© := hg
(hlf.mk' _ + hlg.mk' _).isLinear.with_bound (Mf + Mg) fun x =>
calc
βf x + g xβ β€ Mf * βxβ + Mg * βxβ := norm_add_le_of_le (... | rw [add_mul]
| [
" β (x : E), β0 xβ β€ 0 * βxβ",
" β (x : E), βLinearMap.id xβ β€ 1 * βxβ",
" IsBoundedLinearMap π fun x => x.1",
" β(LinearMap.fst π E F) xβ β€ 1 * βxβ",
" β(LinearMap.fst π E F) xβ β€ βxβ",
" IsBoundedLinearMap π fun x => x.2",
" β(LinearMap.snd π E F) xβ β€ 1 * βxβ",
" β(LinearMap.snd π E F) xβ β€ β... | [
" β (x : E), β0 xβ β€ 0 * βxβ",
" β (x : E), βLinearMap.id xβ β€ 1 * βxβ",
" IsBoundedLinearMap π fun x => x.1",
" β(LinearMap.fst π E F) xβ β€ 1 * βxβ",
" β(LinearMap.fst π E F) xβ β€ βxβ",
" IsBoundedLinearMap π fun x => x.2",
" β(LinearMap.snd π E F) xβ β€ 1 * βxβ",
" β(LinearMap.snd π E F) xβ β€ β... |
import Mathlib.Topology.MetricSpace.PseudoMetric
import Mathlib.Topology.UniformSpace.Equicontinuity
#align_import topology.metric_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Topology Uniformity
variable {Ξ± Ξ² ΞΉ : Type*} [PseudoMetricSpace Ξ±]
na... | Mathlib/Topology/MetricSpace/Equicontinuity.lean | 103 | 114 | theorem uniformEquicontinuous_of_continuity_modulus {ΞΉ : Type*} [PseudoMetricSpace Ξ²] (b : β β β)
(b_lim : Tendsto b (π 0) (π 0)) (F : ΞΉ β Ξ² β Ξ±)
(H : β (x y : Ξ²) (i), dist (F i x) (F i y) β€ b (dist x y)) : UniformEquicontinuous F := by |
rw [Metric.uniformEquicontinuous_iff]
intro Ξ΅ Ξ΅0
rcases tendsto_nhds_nhds.1 b_lim Ξ΅ Ξ΅0 with β¨Ξ΄, Ξ΄0, hΞ΄β©
refine β¨Ξ΄, Ξ΄0, fun x y hxy i => ?_β©
calc
dist (F i x) (F i y) β€ b (dist x y) := H x y i
_ β€ |b (dist x y)| := le_abs_self _
_ = dist (b (dist x y)) 0 := by simp [Real.dist_eq]
_ < Ξ΅ := hΞ΄ (... | [
" EquicontinuousAt F xβ β β Ξ΅ > 0, β U β π xβ, β x β U, β x' β U, β (i : ΞΉ), dist (F i x) (F i x') < Ξ΅",
" (β U β π€ Ξ±, β V β π xβ, β x β V, β y β V, β (i : ΞΉ), (F i x, F i y) β U) β\n β Ξ΅ > 0, β U β π xβ, β x β U, β x' β U, β (i : ΞΉ), dist (F i x) (F i x') < Ξ΅",
" (β U β π€ Ξ±, β V β π xβ, β x β V, β y β... | [
" EquicontinuousAt F xβ β β Ξ΅ > 0, β U β π xβ, β x β U, β x' β U, β (i : ΞΉ), dist (F i x) (F i x') < Ξ΅",
" (β U β π€ Ξ±, β V β π xβ, β x β V, β y β V, β (i : ΞΉ), (F i x, F i y) β U) β\n β Ξ΅ > 0, β U β π xβ, β x β U, β x' β U, β (i : ΞΉ), dist (F i x) (F i x') < Ξ΅",
" (β U β π€ Ξ±, β V β π xβ, β x β V, β y β... |
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68ab8cf7dd83d549eb78a68a1"
open Finset
variable {Ξ± : Type*}
theorem sup_sdiff_injOn [GeneralizedBooleanAlgebra Ξ±] (u v : Ξ±) :
{ x | Disjoint u x β§ v β€ x }.... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 98 | 102 | theorem compress_self (u a : Ξ±) : compress u u a = a := by |
unfold compress
split_ifs with h
Β· exact h.1.symm.sup_sdiff_cancel_right
Β· rfl
| [
" Set.InjOn (fun x => (x β u) \\ v) {x | Disjoint u x β§ v β€ x}",
" a = b",
" ((a β u) \\ v) \\ u β v = ((b β u) \\ v) \\ u β v",
" compress u v ((a β v) \\ u) = a",
" compress u u a = a",
" (if Disjoint u a β§ u β€ a then (a β u) \\ u else a) = a",
" (a β u) \\ u = a",
" a = a"
] | [
" Set.InjOn (fun x => (x β u) \\ v) {x | Disjoint u x β§ v β€ x}",
" a = b",
" ((a β u) \\ v) \\ u β v = ((b β u) \\ v) \\ u β v",
" compress u v ((a β v) \\ u) = a"
] |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
#align_import combinatorics.simple_graph.prod from "leanprover-community/mathlib"@"2985fa3c31a27274aed06c433510bc14b73d6488"
variable {Ξ± Ξ² Ξ³ : Type*}
namespace SimpleGraph
-- Porting note: pruned variables to keep things out of local contexts, which
-- can im... | Mathlib/Combinatorics/SimpleGraph/Prod.lean | 59 | 60 | theorem boxProd_adj_left : (G β‘ H).Adj (aβ, b) (aβ, b) β G.Adj aβ aβ := by |
simp only [boxProd_adj, and_true, SimpleGraph.irrefl, false_and, or_false]
| [
" (fun x y => G.Adj x.1 y.1 β§ x.2 = y.2 β¨ H.Adj x.2 y.2 β§ x.1 = y.1) x y β\n (fun x y => G.Adj x.1 y.1 β§ x.2 = y.2 β¨ H.Adj x.2 y.2 β§ x.1 = y.1) y x",
" Β¬(fun x y => G.Adj x.1 y.1 β§ x.2 = y.2 β¨ H.Adj x.2 y.2 β§ x.1 = y.1) x x",
" (G β‘ H).Adj (aβ, b) (aβ, b) β G.Adj aβ aβ"
] | [
" (fun x y => G.Adj x.1 y.1 β§ x.2 = y.2 β¨ H.Adj x.2 y.2 β§ x.1 = y.1) x y β\n (fun x y => G.Adj x.1 y.1 β§ x.2 = y.2 β¨ H.Adj x.2 y.2 β§ x.1 = y.1) y x",
" Β¬(fun x y => G.Adj x.1 y.1 β§ x.2 = y.2 β¨ H.Adj x.2 y.2 β§ x.1 = y.1) x x"
] |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 52 | 54 | theorem lineMap_mono_left (ha : a β€ a') (hr : r β€ 1) : lineMap a b r β€ lineMap a' b r := by |
simp only [lineMap_apply_module]
exact add_le_add_right (smul_le_smul_of_nonneg_left ha (sub_nonneg.2 hr)) _
| [
" (lineMap a b) r β€ (lineMap a' b) r",
" (1 - r) β’ a + r β’ b β€ (1 - r) β’ a' + r β’ b"
] | [] |
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 | 561 | 564 | theorem pullback_comp (f : X βΆ Y) (g : Y βΆ Z) (x : Subobject Z) :
(pullback (f β« g)).obj x = (pullback f).obj ((pullback g).obj x) := by |
induction' x using Quotient.inductionOn' with t
exact Quotient.sound β¨(MonoOver.pullbackComp _ _).app tβ©
| [
" PartialOrder (Subobject X)",
" PartialOrder (ThinSkeleton (MonoOver X))",
" π (Subobject A) β
lower e.functor β lower e.inverse",
" π (Subobject A) = lower e.functor β lower e.inverse",
" π (Subobject A) = ThinSkeleton.map (π (MonoOver A))",
" lower e.functor β lower e.inverse = ThinSkeleton.map (e.... | [
" PartialOrder (Subobject X)",
" PartialOrder (ThinSkeleton (MonoOver X))",
" π (Subobject A) β
lower e.functor β lower e.inverse",
" π (Subobject A) = lower e.functor β lower e.inverse",
" π (Subobject A) = ThinSkeleton.map (π (MonoOver A))",
" lower e.functor β lower e.inverse = ThinSkeleton.map (e.... |
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.int.gcd from "leanprover-community/mathlib"@"47a1a73351de8dd6c8d3d32b569c8e434b03ca47"
namespace Nat
... | Mathlib/Data/Int/GCD.lean | 48 | 48 | theorem xgcd_zero_left {s t r' s' t'} : xgcdAux 0 s t r' s' t' = (r', s', t') := by | simp [xgcdAux]
| [
" (invImage\n (fun x =>\n PSigma.casesOn x fun a a_1 =>\n PSigma.casesOn a_1 fun a_2 a_3 =>\n PSigma.casesOn a_3 fun a_4 a_5 => PSigma.casesOn a_5 fun a_6 a_7 => PSigma.casesOn a_7 fun a_8 a_9 => a)\n instWellFoundedRelationOfSizeOf).1\n β¨r' % k.succ, β¨s' - βq * s, ... | [
" (invImage\n (fun x =>\n PSigma.casesOn x fun a a_1 =>\n PSigma.casesOn a_1 fun a_2 a_3 =>\n PSigma.casesOn a_3 fun a_4 a_5 => PSigma.casesOn a_5 fun a_6 a_7 => PSigma.casesOn a_7 fun a_8 a_9 => a)\n instWellFoundedRelationOfSizeOf).1\n β¨r' % k.succ, β¨s' - βq * s, ... |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 97 | 99 | theorem integral_comp_inv_smul (f : E β F) (R : β) :
β« x, f (Rβ»ΒΉ β’ x) βΞΌ = |R ^ finrank β E| β’ β« x, f x βΞΌ := by |
rw [integral_comp_smul ΞΌ f Rβ»ΒΉ, inv_pow, inv_inv]
| [
" NoAtoms ΞΌ",
" β« (x : E), f (R β’ x) βΞΌ = |(R ^ finrank β E)β»ΒΉ| β’ β« (x : E), f x βΞΌ",
" β« (x : E), f (0 β’ x) βΞΌ = |(0 ^ finrank β E)β»ΒΉ| β’ β« (x : E), f x βΞΌ",
" (ΞΌ univ).toReal β’ f 0 = |(0 ^ finrank β E)β»ΒΉ| β’ β« (x : E), f x βΞΌ",
" f = fun x => f 0",
" f x = f 0",
"E : Type u_1\ninstββ· : NormedAddCommGrou... | [
" NoAtoms ΞΌ",
" β« (x : E), f (R β’ x) βΞΌ = |(R ^ finrank β E)β»ΒΉ| β’ β« (x : E), f x βΞΌ",
" β« (x : E), f (0 β’ x) βΞΌ = |(0 ^ finrank β E)β»ΒΉ| β’ β« (x : E), f x βΞΌ",
" (ΞΌ univ).toReal β’ f 0 = |(0 ^ finrank β E)β»ΒΉ| β’ β« (x : E), f x βΞΌ",
" f = fun x => f 0",
" f x = f 0",
"E : Type u_1\ninstββ· : NormedAddCommGrou... |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 530 | 534 | theorem one_div_Gamma_eq_self_mul_one_div_Gamma_add_one (s : β) :
(Gamma s)β»ΒΉ = s * (Gamma (s + 1))β»ΒΉ := by |
rcases ne_or_eq s 0 with (h | rfl)
Β· rw [Gamma_add_one s h, mul_inv, mul_inv_cancel_leftβ h]
Β· rw [zero_add, Gamma_zero, inv_zero, zero_mul]
| [
" s.Gammaβ»ΒΉ = s * (s + 1).Gammaβ»ΒΉ",
" (Gamma 0)β»ΒΉ = 0 * (0 + 1).Gammaβ»ΒΉ"
] | [] |
import Mathlib.Data.Set.Lattice
import Mathlib.Data.Set.Pairwise.Basic
#align_import data.set.pairwise.lattice from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Function Set Order
variable {Ξ± Ξ² Ξ³ ΞΉ ΞΉ' : Type*} {ΞΊ : Sort*} {r p q : Ξ± β Ξ± β Prop}
section Pairwise
variable {f g : ... | Mathlib/Data/Set/Pairwise/Lattice.lean | 147 | 153 | theorem Set.PairwiseDisjoint.subset_of_biUnion_subset_biUnion (hβ : (s βͺ t).PairwiseDisjoint f)
(hβ : β i β s, (f i).Nonempty) (h : β i β s, f i β β i β t, f i) : s β t := by |
rintro i hi
obtain β¨a, haiβ© := hβ i hi
obtain β¨j, hj, hajβ© := mem_iUnionβ.1 (h <| mem_iUnionβ_of_mem hi hai)
rwa [hβ.eq (subset_union_left hi) (subset_union_right hj)
(not_disjoint_iff.2 β¨a, hai, hajβ©)]
| [
" s β t",
" i β t"
] | [] |
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.category.Module.subobject from "leanprover-community/mathlib"@"6d584f1709bedbed9175bd9350df46599b... | Mathlib/Algebra/Category/ModuleCat/Subobject.lean | 111 | 120 | theorem cokernel_Ο_imageSubobject_ext {L M N : ModuleCat.{v} R} (f : L βΆ M) [HasImage f]
(g : (imageSubobject f : ModuleCat.{v} R) βΆ N) [HasCokernel g] {x y : N} (l : L)
(w : x = y + g (factorThruImageSubobject f l)) : cokernel.Ο g x = cokernel.Ο g y := by |
subst w
-- Porting note: The proof from here used to just be `simp`.
simp only [map_add, add_right_eq_self]
change ((cokernel.Ο g) β (g) β (factorThruImageSubobject f)) l = 0
rw [β coe_comp, β coe_comp, Category.assoc]
simp only [cokernel.condition, comp_zero]
rfl
| [
" (fun S => LinearMap.range S.arrow) ((fun N => Subobject.mk (βΎN.subtype)) N) = N",
" (fun S => LinearMap.range S.arrow) ((fun N => Subobject.mk (βΎN.subtype)) N) =\n LinearMap.range ((underlyingIso (βΎN.subtype)).inv β« (Subobject.mk (βΎN.subtype)).arrow)",
" (underlyingIso (βΎN.subtype)).inv = β(underlyingIso (... | [
" (fun S => LinearMap.range S.arrow) ((fun N => Subobject.mk (βΎN.subtype)) N) = N",
" (fun S => LinearMap.range S.arrow) ((fun N => Subobject.mk (βΎN.subtype)) N) =\n LinearMap.range ((underlyingIso (βΎN.subtype)).inv β« (Subobject.mk (βΎN.subtype)).arrow)",
" (underlyingIso (βΎN.subtype)).inv = β(underlyingIso (... |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section support
s... | Mathlib/GroupTheory/Perm/Support.lean | 270 | 271 | theorem set_support_apply_mem {p : Perm Ξ±} {a : Ξ±} :
p a β { x | p x β x } β a β { x | p x β x } := by | simp
| [
" {x | pβ»ΒΉ x β x} = {x | p x β x}",
" x β {x | pβ»ΒΉ x β x} β x β {x | p x β x}",
" Β¬pβ»ΒΉ x = x β Β¬p x = x",
" p a β {x | p x β x} β a β {x | p x β x}"
] | [
" {x | pβ»ΒΉ x β x} = {x | p x β x}",
" x β {x | pβ»ΒΉ x β x} β x β {x | p x β x}",
" Β¬pβ»ΒΉ x = x β Β¬p x = x"
] |
import Mathlib.Algebra.Module.Equiv
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Algebra.PUnitInstances
import Mathlib.Data.Set.Subsingleton
#align_import algebra.module.submodule.lattice from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
universe v
variable {R S M : Ty... | Mathlib/Algebra/Module/Submodule/Lattice.lean | 131 | 132 | theorem nontrivial_iff_ne_bot : Nontrivial p β p β β₯ := by |
rw [iff_not_comm, not_nontrivial_iff_subsingleton, subsingleton_iff_eq_bot]
| [
" β (c : R) {x : M},\n x β { carrier := {0}, add_mem' := β―, zero_mem' := β― }.carrier β\n c β’ x β { carrier := {0}, add_mem' := β―, zero_mem' := β― }.carrier",
" x β β₯ β x β p",
" x = y",
" β¨x, xmβ© = y",
" β¨x, xmβ© = β¨y, ymβ©",
" 0 = y",
" p β β₯ β β x β p, x β 0",
" Subsingleton β₯p β p = β₯",
" (β... | [
" β (c : R) {x : M},\n x β { carrier := {0}, add_mem' := β―, zero_mem' := β― }.carrier β\n c β’ x β { carrier := {0}, add_mem' := β―, zero_mem' := β― }.carrier",
" x β β₯ β x β p",
" x = y",
" β¨x, xmβ© = y",
" β¨x, xmβ© = β¨y, ymβ©",
" 0 = y",
" p β β₯ β β x β p, x β 0",
" Subsingleton β₯p β p = β₯",
" (β... |
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import algebra.squarefree from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
variable {R : Type*}
def Squarefree [Monoid R] (r : R) : Prop :=
β x : R, x * x β£ r β IsUnit x
#align sq... | Mathlib/Algebra/Squarefree/Basic.lean | 154 | 163 | theorem irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree (r : R) :
(β x : R, Irreducible x β Β¬x * x β£ r) β (r = 0 β§ β x : R, Β¬Irreducible x) β¨ Squarefree r := by |
refine β¨fun h β¦ ?_, ?_β©
Β· rcases eq_or_ne r 0 with (rfl | hr)
Β· exact .inl (by simpa using h)
Β· exact .inr ((squarefree_iff_no_irreducibles hr).mpr h)
Β· rintro (β¨rfl, hβ© | h)
Β· simpa using h
intro x hx t
exact hx.not_unit (h x t)
| [
" Β¬Squarefree 0",
" β x, Β¬(x * x β£ 0 β IsUnit x)",
" Β¬(0 * 0 β£ 0 β IsUnit 0)",
" m β 0",
" False",
" Squarefree x",
" IsUnit y",
" n = 0 β¨ n = 1",
" IsUnit x",
" 2 β€ n",
" x * x β£ x ^ n",
" x ^ 2 β£ x ^ n",
" Squarefree x β β (p : R), Irreducible p β Β¬p * p β£ x",
" (β (x : R), Irreducible x... | [
" Β¬Squarefree 0",
" β x, Β¬(x * x β£ 0 β IsUnit x)",
" Β¬(0 * 0 β£ 0 β IsUnit 0)",
" m β 0",
" False",
" Squarefree x",
" IsUnit y",
" n = 0 β¨ n = 1",
" IsUnit x",
" 2 β€ n",
" x * x β£ x ^ n",
" x ^ 2 β£ x ^ n",
" Squarefree x β β (p : R), Irreducible p β Β¬p * p β£ x"
] |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Function Set Filter
open scoped Topology Filter
variable... | Mathlib/Analysis/Calculus/BumpFunction/Basic.lean | 154 | 157 | theorem one_of_mem_closedBall (hx : x β closedBall c f.rIn) : f x = 1 := by |
apply ContDiffBumpBase.eq_one _ _ f.one_lt_rOut_div_rIn
simpa only [norm_smul, Real.norm_eq_abs, abs_inv, abs_of_nonneg f.rIn_pos.le, β div_eq_inv_mul,
div_le_one f.rIn_pos] using mem_closedBall_iff_norm.1 hx
| [
" 1 < f.rOut / f.rIn",
" f.rIn < f.rOut",
" βf (c - x) = βf (c + x)",
" βf (-x) = βf x",
" βf x = 1",
" β(fun x => f.rInβ»ΒΉ β’ (x - c)) xβ β€ 1"
] | [
" 1 < f.rOut / f.rIn",
" f.rIn < f.rOut",
" βf (c - x) = βf (c + x)",
" βf (-x) = βf x"
] |
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Data.Real.Sqrt
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Set Metric Pointwise
var... | Mathlib/Analysis/NormedSpace/HomeomorphBall.lean | 144 | 146 | theorem univBall_symm_apply_center (c : P) (r : β) : (univBall c r).symm c = 0 := by |
have : 0 β (univBall c r).source := by simp
simpa only [univBall_apply_zero] using (univBall c r).left_inv this
| [
" (fun x => (β(1 + βxβ ^ 2))β»ΒΉ β’ x) x β ball 0 1",
" 0 < 1 + βxβ ^ 2",
" βxβ ^ 2 < 1 + βxβ ^ 2",
" (fun y => (β(1 - βyβ ^ 2))β»ΒΉ β’ y) ((fun x => (β(1 + βxβ ^ 2))β»ΒΉ β’ x) x) = x",
" (fun x => (β(1 + βxβ ^ 2))β»ΒΉ β’ x) ((fun y => (β(1 - βyβ ^ 2))β»ΒΉ β’ y) y) = y",
" 0 < 1 - βyβ ^ 2",
" ContinuousOn\n β{ toFu... | [
" (fun x => (β(1 + βxβ ^ 2))β»ΒΉ β’ x) x β ball 0 1",
" 0 < 1 + βxβ ^ 2",
" βxβ ^ 2 < 1 + βxβ ^ 2",
" (fun y => (β(1 - βyβ ^ 2))β»ΒΉ β’ y) ((fun x => (β(1 + βxβ ^ 2))β»ΒΉ β’ x) x) = x",
" (fun x => (β(1 + βxβ ^ 2))β»ΒΉ β’ x) ((fun y => (β(1 - βyβ ^ 2))β»ΒΉ β’ y) y) = y",
" 0 < 1 - βyβ ^ 2",
" ContinuousOn\n β{ toFu... |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {Ξ± Ξ² Ξ³ : Type*}
section ConditionallyCompleteLinearOrder
variable [ConditionallyCompleteLinearOrder Ξ±] [TopologicalSpace Ξ±] [OrderTopology Ξ±]
[ConditionallyCompleteLinearOrder Ξ²] [Top... | Mathlib/Topology/Order/Monotone.lean | 282 | 292 | theorem Monotone.tendsto_nhdsWithin_Iio {Ξ± Ξ² : Type*} [LinearOrder Ξ±] [TopologicalSpace Ξ±]
[OrderTopology Ξ±] [ConditionallyCompleteLinearOrder Ξ²] [TopologicalSpace Ξ²] [OrderTopology Ξ²]
{f : Ξ± β Ξ²} (Mf : Monotone f) (x : Ξ±) : Tendsto f (π[<] x) (π (sSup (f '' Iio x))) := by |
rcases eq_empty_or_nonempty (Iio x) with (h | h); Β· simp [h]
refine tendsto_order.2 β¨fun l hl => ?_, fun m hm => ?_β©
Β· obtain β¨z, zx, lzβ© : β a : Ξ±, a < x β§ l < f a := by
simpa only [mem_image, exists_prop, exists_exists_and_eq_and] using
exists_lt_of_lt_csSup (h.image _) hl
exact mem_of_supers... | [
" f (sSup s) = sSup (f '' s)",
" IsLUB (f '' s) (f (sSup s))",
" Tendsto f (π[s] sSup s) (π (f (sSup s)))",
" f (β¨ i, g i) = β¨ i, f (g i)",
" sSup (range (f β fun i => g i)) = sSup (range fun i => f (g i))",
" Tendsto f (π[<] x) (π (sSup (f '' Iio x)))",
" βαΆ (b : Ξ±) in π[<] x, l < f b",
" β a < ... | [
" f (sSup s) = sSup (f '' s)",
" IsLUB (f '' s) (f (sSup s))",
" Tendsto f (π[s] sSup s) (π (f (sSup s)))",
" f (β¨ i, g i) = β¨ i, f (g i)",
" sSup (range (f β fun i => g i)) = sSup (range fun i => f (g i))"
] |
import Mathlib.RingTheory.RingHomProperties
import Mathlib.RingTheory.IntegralClosure
#align_import ring_theory.ring_hom.integral from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
namespace RingHom
open scoped TensorProduct
open TensorProduct Algebra.TensorProduct
theorem isIntegra... | Mathlib/RingTheory/RingHom/Integral.lean | 35 | 41 | theorem isIntegral_stableUnderBaseChange : StableUnderBaseChange fun f => f.IsIntegral := by |
refine StableUnderBaseChange.mk _ isIntegral_respectsIso ?_
introv h x
refine TensorProduct.induction_on x ?_ ?_ ?_
Β· apply isIntegral_zero
Β· intro x y; exact IsIntegral.tmul x (h y)
Β· intro x y hx hy; exact IsIntegral.add hx hy
| [
" StableUnderComposition fun {R S} [CommRing R] [CommRing S] f => f.IsIntegral",
" (g.comp f).IsIntegral",
" RespectsIso fun {R S} [CommRing R] [CommRing S] f => f.IsIntegral",
" β {R S : Type u_1} [inst : CommRing R] [inst_1 : CommRing S] (e : R β+* S), e.toRingHom.IsIntegral",
" e.toRingHom.IsIntegralElem... | [
" StableUnderComposition fun {R S} [CommRing R] [CommRing S] f => f.IsIntegral",
" (g.comp f).IsIntegral",
" RespectsIso fun {R S} [CommRing R] [CommRing S] f => f.IsIntegral",
" β {R S : Type u_1} [inst : CommRing R] [inst_1 : CommRing S] (e : R β+* S), e.toRingHom.IsIntegral",
" e.toRingHom.IsIntegralElem... |
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 69 | 74 | theorem card_support_mul_le : (p * q).support.card β€ p.support.card * q.support.card := by |
calc (p * q).support.card
_ = (p.toFinsupp * q.toFinsupp).support.card := by rw [β support_toFinsupp, toFinsupp_mul]
_ β€ (p.toFinsupp.support + q.toFinsupp.support).card :=
Finset.card_le_card (AddMonoidAlgebra.support_mul p.toFinsupp q.toFinsupp)
_ β€ p.support.card * q.support.card := Finset.card_image... | [
" (p + q).coeff n = p.coeff n + q.coeff n",
" ({ toFinsupp := toFinsuppβ } + q).coeff n = { toFinsupp := toFinsuppβ }.coeff n + q.coeff n",
" ({ toFinsupp := toFinsuppβΒΉ } + { toFinsupp := toFinsuppβ }).coeff n =\n { toFinsupp := toFinsuppβΒΉ }.coeff n + { toFinsupp := toFinsuppβ }.coeff n",
" (toFinsuppβΒΉ ... | [
" (p + q).coeff n = p.coeff n + q.coeff n",
" ({ toFinsupp := toFinsuppβ } + q).coeff n = { toFinsupp := toFinsuppβ }.coeff n + q.coeff n",
" ({ toFinsupp := toFinsuppβΒΉ } + { toFinsupp := toFinsuppβ }).coeff n =\n { toFinsupp := toFinsuppβΒΉ }.coeff n + { toFinsupp := toFinsuppβ }.coeff n",
" (toFinsuppβΒΉ ... |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {Ξ± I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 306 | 309 | theorem pow_left_iff (hm : 0 < m) : IsRelPrime (x ^ m) y β IsRelPrime x y := by |
refine β¨fun h β¦ ?_, IsRelPrime.pow_leftβ©
rw [β Finset.card_range m, β Finset.prod_const] at h
exact h.of_prod_left 0 (Finset.mem_range.mpr hm)
| [
" (β i β t, IsRelPrime (s i) x) β IsRelPrime (β i β t, s i) x",
" IsRelPrime (β i β insert b t, s i) x",
" IsRelPrime (s b * β x β t, s x) x",
" (β i β t, IsRelPrime x (s i)) β IsRelPrime x (β i β t, s i)",
" IsRelPrime (β i β t, s i) x β β i β t, IsRelPrime (s i) x",
" xβ β β
β IsRelPrime (s xβ) x",
" ... | [
" (β i β t, IsRelPrime (s i) x) β IsRelPrime (β i β t, s i) x",
" IsRelPrime (β i β insert b t, s i) x",
" IsRelPrime (s b * β x β t, s x) x",
" (β i β t, IsRelPrime x (s i)) β IsRelPrime x (β i β t, s i)",
" IsRelPrime (β i β t, s i) x β β i β t, IsRelPrime (s i) x",
" xβ β β
β IsRelPrime (s xβ) x",
" ... |
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Decomposition.Lebesgue
#align_import measure... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 97 | 113 | theorem ae_eventually_measure_pos [SecondCountableTopology Ξ±] :
βα΅ x βΞΌ, βαΆ a in v.filterAt x, 0 < ΞΌ a := by |
set s := {x | Β¬βαΆ a in v.filterAt x, 0 < ΞΌ a} with hs
simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
change ΞΌ s = 0
let f : Ξ± β Set (Set Ξ±) := fun _ => {a | ΞΌ a = 0}
have h : v.FineSubfamilyOn f s := by
intro x hx Ξ΅ Ξ΅pos
rw [hs] at hx
simp only [frequen... | [
" βα΅ (x : Ξ±) βΞΌ, βαΆ (a : Set Ξ±) in v.filterAt x, 0 < ΞΌ a",
" ΞΌ s = 0",
" v.FineSubfamilyOn f s",
" β a β v.setsAt x β© f x, a β closedBall x Ξ΅",
" ΞΌ s β€ 0",
" β' (x : βh.index), ΞΌ (h.covering βx) = β' (x : βh.index), 0",
" (fun x => ΞΌ (h.covering βx)) = fun x => 0",
" ΞΌ (h.covering βx) = 0",
" β' (x ... | [] |
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.fin from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
noncomputable section
namespace Finsupp
variable {n : β} (i : Fin n) {M : Type*} [Zero M] (y : M) (t : Fin (n + 1) ββ M) (s : Fin n ββ M)
def tail (s : Fin (n + 1) ββ ... | Mathlib/Data/Finsupp/Fin.lean | 68 | 73 | theorem cons_zero_zero : cons 0 (0 : Fin n ββ M) = 0 := by |
ext a
by_cases c : a = 0
Β· simp [c]
Β· rw [β Fin.succ_pred a c, cons_succ]
simp
| [
" (cons y s).tail k = s k",
" cons (t 0) t.tail = t",
" (cons (t 0) t.tail) a = t a",
" cons 0 0 = 0",
" (cons 0 0) a = 0 a",
" 0 (a.pred c) = 0 (a.pred c).succ"
] | [
" (cons y s).tail k = s k",
" cons (t 0) t.tail = t",
" (cons (t 0) t.tail) a = t a"
] |
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.Algebra.Module.Torsion
open Submodule
variable {R : Type*} [CommRing R] (I : Ideal R)
variable {M : Type*} [AddCommGroup M] [Module R M]
namespace AdicCompletion
attribute [-simp] smul_eq_mul Algebra.id.smul_eq_mul
@[local simp]
theorem transitionMap... | Mathlib/RingTheory/AdicCompletion/Algebra.lean | 133 | 139 | theorem smul_mk {m n : β} (hmn : m β€ n) (r : AdicCauchySequence I R)
(x : AdicCauchySequence I M) :
r.val n β’ Submodule.Quotient.mk (p := (I ^ m β’ β€ : Submodule R M)) (x.val n) =
r.val m β’ Submodule.Quotient.mk (p := (I ^ m β’ β€ : Submodule R M)) (x.val m) := by |
rw [β Submodule.Quotient.mk_smul, β Module.Quotient.mk_smul_mk,
AdicCauchySequence.mk_eq_mk hmn, Ideal.mk_eq_mk I hmn, Module.Quotient.mk_smul_mk,
Submodule.Quotient.mk_smul]
| [
" (transitionMap I R xβ) (1 nβ) = 1 mβ",
" (transitionMap I R hmn) ((x * y) n) = (x * y) m",
" I ^ n β’ β€ = I ^ n",
" x β I ^ n β’ β€ β x β I ^ n",
" (evalβ I n) ((mk I R) x) = (Ideal.Quotient.mk (I ^ n)) (βx n)",
" 1 m β‘ 1 n [SMOD I ^ m β’ β€]",
" 1 β‘ 1 [SMOD I ^ m]",
" (x * y) m β‘ (x * y) n [SMOD I ^ m β’... | [
" (transitionMap I R xβ) (1 nβ) = 1 mβ",
" (transitionMap I R hmn) ((x * y) n) = (x * y) m",
" I ^ n β’ β€ = I ^ n",
" x β I ^ n β’ β€ β x β I ^ n",
" (evalβ I n) ((mk I R) x) = (Ideal.Quotient.mk (I ^ n)) (βx n)",
" 1 m β‘ 1 n [SMOD I ^ m β’ β€]",
" 1 β‘ 1 [SMOD I ^ m]",
" (x * y) m β‘ (x * y) n [SMOD I ^ m β’... |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.RingTheory.Polynomial.Nilpotent
#align_import linear_algebra.matrix.charpoly.coeff from "leanprover-community/mathlib"@"9745b... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 81 | 86 | theorem charpoly_coeff_eq_prod_coeff_of_le {k : β} (h : Fintype.card n - 1 β€ k) :
M.charpoly.coeff k = (β i : n, (X - C (M i i))).coeff k := by |
apply eq_of_sub_eq_zero; rw [β coeff_sub]
apply Polynomial.coeff_eq_zero_of_degree_lt
apply lt_of_lt_of_le (charpoly_sub_diagonal_degree_lt M) ?_
rw [Nat.cast_le]; apply h
| [
" (M.charmatrix i j).natDegree = if i = j then 1 else 0",
" (M.charmatrix i j).natDegree β€ if i = j then 1 else 0",
" (M.charmatrix i j).natDegree β€ 1",
" (M.charmatrix i j).natDegree β€ 0",
" (M.charpoly - β i : n, (X - C (M i i))).degree < β(Fintype.card n - 1)",
" (β x β univ.erase (Equiv.refl n), ββ(Eq... | [
" (M.charmatrix i j).natDegree = if i = j then 1 else 0",
" (M.charmatrix i j).natDegree β€ if i = j then 1 else 0",
" (M.charmatrix i j).natDegree β€ 1",
" (M.charmatrix i j).natDegree β€ 0",
" (M.charpoly - β i : n, (X - C (M i i))).degree < β(Fintype.card n - 1)",
" (β x β univ.erase (Equiv.refl n), ββ(Eq... |
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
#align_import data.sum.interval from "leanprover-community/mathlib"@"48a058d7e39a80ed56858505719a0b2197900999"
open Function Sum
namespace Finset
variable {Ξ±β Ξ±β Ξ²β Ξ²β Ξ³β Ξ³β : Type*}
section SumLiftβ
variabl... | Mathlib/Data/Sum/Interval.lean | 91 | 95 | theorem sumLiftβ_nonempty :
(sumLiftβ f g a b).Nonempty β
(β aβ bβ, a = inl aβ β§ b = inl bβ β§ (f aβ bβ).Nonempty) β¨
β aβ bβ, a = inr aβ β§ b = inr bβ β§ (g aβ bβ).Nonempty := by |
simp only [nonempty_iff_ne_empty, Ne, sumLiftβ_eq_empty, not_and_or, not_forall, exists_prop]
| [
" c β sumLiftβ f g a b β\n (β aβ bβ cβ, a = inl aβ β§ b = inl bβ β§ c = inl cβ β§ cβ β f aβ bβ) β¨\n β aβ bβ cβ, a = inr aβ β§ b = inr bβ β§ c = inr cβ β§ cβ β g aβ bβ",
" c β sumLiftβ f g a b β\n (β aβ bβ cβ, a = inl aβ β§ b = inl bβ β§ c = inl cβ β§ cβ β f aβ bβ) β¨\n β aβ bβ cβ, a = inr aβ β§ b = inr bβ β§ ... | [
" c β sumLiftβ f g a b β\n (β aβ bβ cβ, a = inl aβ β§ b = inl bβ β§ c = inl cβ β§ cβ β f aβ bβ) β¨\n β aβ bβ cβ, a = inr aβ β§ b = inr bβ β§ c = inr cβ β§ cβ β g aβ bβ",
" c β sumLiftβ f g a b β\n (β aβ bβ cβ, a = inl aβ β§ b = inl bβ β§ c = inl cβ β§ cβ β f aβ bβ) β¨\n β aβ bβ cβ, a = inr aβ β§ b = inr bβ β§ ... |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section WithDivisionRing
variable {K : Type*}... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 162 | 163 | theorem first_numerator_eq {gp : Pair K} (zeroth_s_eq : g.s.get? 0 = some gp) :
g.numerators 1 = gp.b * g.h + gp.a := by | simp [num_eq_conts_a, first_continuant_eq zeroth_s_eq]
| [
" β conts, g.continuants n = conts β§ conts.a = A",
" β conts, g.continuants n = conts β§ conts.b = B",
" g.convergents 0 = g.h",
" g.continuantsAux 2 = { a := gp.b * g.h + gp.a, b := gp.b }",
" g.continuants 1 = { a := gp.b * g.h + gp.a, b := gp.b }",
" g.numerators 1 = gp.b * g.h + gp.a"
] | [
" β conts, g.continuants n = conts β§ conts.a = A",
" β conts, g.continuants n = conts β§ conts.b = B",
" g.convergents 0 = g.h",
" g.continuantsAux 2 = { a := gp.b * g.h + gp.a, b := gp.b }",
" g.continuants 1 = { a := gp.b * g.h + gp.a, b := gp.b }"
] |
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 104 | 112 | theorem chain'_range_succ (r : β β β β Prop) (n : β) :
Chain' r (range n.succ) β β m < n, r m m.succ := by |
rw [range_succ]
induction' n with n hn
Β· simp
Β· rw [range_succ]
simp only [append_assoc, singleton_append, chain'_append_cons_cons, chain'_singleton,
and_true_iff]
rw [hn, forall_lt_succ]
| [
" (range' n m).nthLe i H = n + i",
" Pairwise (fun x x_1 => x < x_1) (range n)",
" take m (range n) = range (min m n)",
" (take m (range n)).length = (range (min m n)).length",
" β (n_1 : β) (hβ : n_1 < (take m (range n)).length) (hβ : n_1 < (range (min m n)).length),\n (take m (range n)).get β¨n_1, hββ© =... | [
" (range' n m).nthLe i H = n + i",
" Pairwise (fun x x_1 => x < x_1) (range n)",
" take m (range n) = range (min m n)",
" (take m (range n)).length = (range (min m n)).length",
" β (n_1 : β) (hβ : n_1 < (take m (range n)).length) (hβ : n_1 < (range (min m n)).length),\n (take m (range n)).get β¨n_1, hββ© =... |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 58 | 83 | theorem mk_eq_mk_of_basis (v : Basis ΞΉ R M) (v' : Basis ΞΉ' R M) :
Cardinal.lift.{w'} #ΞΉ = Cardinal.lift.{w} #ΞΉ' := by |
classical
haveI := nontrivial_of_invariantBasisNumber R
cases fintypeOrInfinite ΞΉ
Β· -- `v` is a finite basis, so by `basis_finite_of_finite_spans` so is `v'`.
-- haveI : Finite (range v) := Set.finite_range v
haveI := basis_finite_of_finite_spans _ (Set.finite_range v) v.span_eq v'
cases nonempty_f... | [
" lift.{w', w} #ΞΉ = lift.{w, w'} #ΞΉ'",
" lift.{w', w} β(Fintype.card ΞΉ) = lift.{w, w'} β(Fintype.card ΞΉ')",
" Fintype.card ΞΉ = Fintype.card ΞΉ'",
" (ΞΉ β R) ββ[R] ΞΉ' β R"
] | [] |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Deprecated.Submonoid
#align_import deprecated.subgroup from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Set Function
variable {G : Type*} {H : Type*} {A : Type*} {a aβ aβ b c : G}
section Group
variable [Group G] [Add... | Mathlib/Deprecated/Subgroup.lean | 57 | 58 | theorem IsSubgroup.div_mem {s : Set G} (hs : IsSubgroup s) {x y : G} (hx : x β s) (hy : y β s) :
x / y β s := by | simpa only [div_eq_mul_inv] using hs.mul_mem hx (hs.inv_mem hy)
| [
" x / y β s"
] | [] |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 89 | 89 | theorem sSup_div : sSup (s / t) = sSup s / sInf t := by | simp_rw [div_eq_mul_inv, sSup_mul, sSup_inv]
| [
" sSup sβ»ΒΉ = (sInf s)β»ΒΉ",
" β¨ a β s, aβ»ΒΉ = (sInf s)β»ΒΉ",
" sInf sβ»ΒΉ = (sSup s)β»ΒΉ",
" β¨
a β s, aβ»ΒΉ = (sSup s)β»ΒΉ",
" sSup (s / t) = sSup s / sInf t"
] | [
" sSup sβ»ΒΉ = (sInf s)β»ΒΉ",
" β¨ a β s, aβ»ΒΉ = (sInf s)β»ΒΉ",
" sInf sβ»ΒΉ = (sSup s)β»ΒΉ",
" β¨
a β s, aβ»ΒΉ = (sSup s)β»ΒΉ"
] |
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {Ξ± : Type*} (p : Ξ± β Bool) (l : List Ξ±) (n : β)
namespace List
def rdrop : List Ξ± :=
l.take (l.leng... | Mathlib/Data/List/DropRight.lean | 47 | 47 | theorem rdrop_nil : rdrop ([] : List Ξ±) n = [] := by | simp [rdrop]
| [
" [].rdrop n = []"
] | [] |
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Instances.Real
def preCantorSet : β β Set β
| 0 => Set.Icc 0 1
| n + 1 => (Β· / 3) '' preCantorSet n βͺ (fun x β¦ (2 + x) / 3) '' preCantorSet n
@[simp] lemma preCantorSet_zero : preCantorSet 0 = Set.Ic... | Mathlib/Topology/Instances/CantorSet.lean | 75 | 75 | theorem zero_mem_cantorSet : 0 β cantorSet := by | simp [cantorSet, zero_mem_preCantorSet]
| [
" 1 / 4 β preCantorSet n β§ 3 / 4 β preCantorSet n",
" 1 / 4 β preCantorSet 0 β§ 3 / 4 β preCantorSet 0",
" 1 / 4 β Set.Icc 0 1 β§ 3 / 4 β Set.Icc 0 1",
" 0 β€ 1 / 4",
" 1 / 4 β€ 1",
" 0 β€ 3 / 4",
" 3 / 4 β€ 1",
" 1 / 4 β preCantorSet (n + 1) β§ 3 / 4 β preCantorSet (n + 1)",
" 1 / 4 β preCantorSet (n + 1)... | [
" 1 / 4 β preCantorSet n β§ 3 / 4 β preCantorSet n",
" 1 / 4 β preCantorSet 0 β§ 3 / 4 β preCantorSet 0",
" 1 / 4 β Set.Icc 0 1 β§ 3 / 4 β Set.Icc 0 1",
" 0 β€ 1 / 4",
" 1 / 4 β€ 1",
" 0 β€ 3 / 4",
" 3 / 4 β€ 1",
" 1 / 4 β preCantorSet (n + 1) β§ 3 / 4 β preCantorSet (n + 1)",
" 1 / 4 β preCantorSet (n + 1)... |
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 {Ξ± Ξ² R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 152 | 155 | theorem IsDiag.isSymm [Zero Ξ±] {A : Matrix n n Ξ±} (h : A.IsDiag) : A.IsSymm := by |
ext i j
by_cases g : i = j; Β· rw [g, transpose_apply]
simp [h g, h (Ne.symm g)]
| [
" 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",
" (A - B).IsDiag",
" (A - B) i j = 0",
" (k β’ A).IsDiag",
" (k β’ A) i j = 0",
" A.IsDiag",
" A = Aα΄΄α΄΄",
" (krone... | [
" 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",
" (A - B).IsDiag",
" (A - B) i j = 0",
" (k β’ A).IsDiag",
" (k β’ A) i j = 0",
" A.IsDiag",
" A = Aα΄΄α΄΄",
" (krone... |
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.indicator from "leanprover-community/mathlib"@"842328d9df7e96fd90fc424e115679c15fb23a71"
noncomputable section
open Finset Function
variable {ΞΉ Ξ± : Type*}
namespace Finsupp
variable [Zero Ξ±] {s : Finset ΞΉ} (f : β i β s, Ξ±) {i : ΞΉ}
def indicator (s ... | Mathlib/Data/Finsupp/Indicator.lean | 66 | 70 | theorem support_indicator_subset : ((indicator s f).support : Set ΞΉ) β s := by |
intro i hi
rw [mem_coe, mem_support_iff] at hi
by_contra h
exact hi (indicator_of_not_mem h _)
| [
" i β map (Embedding.subtype fun x => x β s) (filter (fun i => f βi β― β 0) s.attach) β\n (fun i => if H : i β s then f i H else 0) i β 0",
" (indicator s f) i = if hi : i β s then f i hi else 0",
" (if H : i β s then f i H else 0) = if hi : i β s then f i hi else 0",
" Injective fun f => indicator s f",
... | [
" i β map (Embedding.subtype fun x => x β s) (filter (fun i => f βi β― β 0) s.attach) β\n (fun i => if H : i β s then f i H else 0) i β 0",
" (indicator s f) i = if hi : i β s then f i hi else 0",
" (if H : i β s then f i H else 0) = if hi : i β s then f i hi else 0",
" Injective fun f => indicator s f",
... |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 86 | 90 | theorem cantorFunctionAux_succ (f : β β Bool) :
(fun n => cantorFunctionAux c f (n + 1)) = fun n =>
c * cantorFunctionAux c (fun n => f (n + 1)) n := by |
ext n
cases h : f (n + 1) <;> simp [h, _root_.pow_succ']
| [
" cantorFunctionAux c f n = c ^ n",
" cantorFunctionAux c f n = 0",
" 0 β€ cantorFunctionAux c f n",
" 0 β€ c ^ n",
" cantorFunctionAux c f n = cantorFunctionAux c g n",
" cantorFunctionAux c f 0 = bif f 0 then 1 else 0",
" cantorFunctionAux c f 0 = bif false then 1 else 0",
" cantorFunctionAux c f 0 = ... | [
" cantorFunctionAux c f n = c ^ n",
" cantorFunctionAux c f n = 0",
" 0 β€ cantorFunctionAux c f n",
" 0 β€ c ^ n",
" cantorFunctionAux c f n = cantorFunctionAux c g n",
" cantorFunctionAux c f 0 = bif f 0 then 1 else 0",
" cantorFunctionAux c f 0 = bif false then 1 else 0",
" cantorFunctionAux c f 0 = ... |
import Mathlib.Data.List.Basic
import Mathlib.Order.MinMax
import Mathlib.Order.WithBot
#align_import data.list.min_max from "leanprover-community/mathlib"@"6d0adfa76594f304b4650d098273d4366edeb61b"
namespace List
variable {Ξ± Ξ² : Type*}
section ArgAux
variable (r : Ξ± β Ξ± β Prop) [DecidableRel r] {l : List Ξ±} {o... | Mathlib/Data/List/MinMax.lean | 69 | 86 | theorem not_of_mem_foldl_argAux (hrβ : Irreflexive r) (hrβ : Transitive r) :
β {a m : Ξ±} {o : Option Ξ±}, a β l β m β foldl (argAux r) o l β Β¬r a m := by |
induction' l using List.reverseRecOn with tl a ih
Β· simp
intro b m o hb ho
rw [foldl_append, foldl_cons, foldl_nil, argAux] at ho
cases' hf : foldl (argAux r) o tl with c
Β· rw [hf] at ho
rw [foldl_argAux_eq_none] at hf
simp_all [hf.1, hf.2, hrβ _]
rw [hf, Option.mem_def] at ho
dsimp only at ho
... | [
" foldl (argAux r) o [] = none β [] = [] β§ o = none",
" (foldl (argAux r) o tl = none β tl = [] β§ o = none) β\n (foldl (argAux r) o (tl ++ [hd]) = none β tl ++ [hd] = [] β§ o = none)",
" (foldl (argAux r) o tl = none β tl = [] β§ o = none) β\n Β¬Option.rec (some hd) (fun val => if r hd val then some hd else ... | [
" foldl (argAux r) o [] = none β [] = [] β§ o = none",
" (foldl (argAux r) o tl = none β tl = [] β§ o = none) β\n (foldl (argAux r) o (tl ++ [hd]) = none β tl ++ [hd] = [] β§ o = none)",
" (foldl (argAux r) o tl = none β tl = [] β§ o = none) β\n Β¬Option.rec (some hd) (fun val => if r hd val then some hd else ... |
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.TensorPower
#align_import linear_algebra.tensor_algebra.to_tensor_power from "leanprover-community/mathlib"@"d97a0c9f7a7efe6d76d652c5a6b7c9c634b70e0a"
suppress_compilation
open scoped DirectSum TensorProduct
variable {R M : Type*} [Com... | Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean | 44 | 64 | theorem toTensorAlgebra_gMul {i j} (a : (β¨[R]^i) M) (b : (β¨[R]^j) M) :
TensorPower.toTensorAlgebra (@GradedMonoid.GMul.mul _ (fun n => β¨[R]^n M) _ _ _ _ a b) =
TensorPower.toTensorAlgebra a * TensorPower.toTensorAlgebra b := by |
-- change `a` and `b` to `tprod R a` and `tprod R b`
rw [TensorPower.gMul_eq_coe_linearMap, β LinearMap.comprβ_apply, β @LinearMap.mul_apply' R, β
LinearMap.complβ_apply, β LinearMap.comp_apply]
refine LinearMap.congr_fun (LinearMap.congr_fun ?_ a) b
clear! a b
ext (a b)
-- Porting note: pulled the nex... | [
" toTensorAlgebra (GradedMonoid.GMul.mul a b) = toTensorAlgebra a * toTensorAlgebra b",
" ((((TensorProduct.mk R (β¨[R]^i M) (β¨[R]^j M)).comprβ βmulEquiv).comprβ toTensorAlgebra) a) b =\n (((LinearMap.mul R (TensorAlgebra R M)).complβ toTensorAlgebra ββ toTensorAlgebra) a) b",
" ((TensorProduct.mk R (β¨[R]^i M... | [] |
import Mathlib.Order.Filter.Bases
#align_import order.filter.pi from "leanprover-community/mathlib"@"ce64cd319bb6b3e82f31c2d38e79080d377be451"
open Set Function
open scoped Classical
open Filter
namespace Filter
variable {ΞΉ : Type*} {Ξ± : ΞΉ β Type*} {f fβ fβ : (i : ΞΉ) β Filter (Ξ± i)} {s : (i : ΞΉ) β Set (Ξ± i)}
... | Mathlib/Order/Filter/Pi.lean | 74 | 77 | theorem pi_mem_pi {I : Set ΞΉ} (hI : I.Finite) (h : β i β I, s i β f i) : I.pi s β pi f := by |
rw [pi_def, biInter_eq_iInter]
refine mem_iInf_of_iInter hI (fun i => ?_) Subset.rfl
exact preimage_mem_comap (h i i.2)
| [
" Tendsto m l (pi f) β β (i : ΞΉ), Tendsto (fun x => m x i) l (f i)",
" (β (i : ΞΉ), Tendsto (eval i β m) l (f i)) β β (i : ΞΉ), Tendsto (fun x => m x i) l (f i)",
" I.pi s β pi f",
" β x, eval βx β»ΒΉ' s βx β pi f",
" eval βi β»ΒΉ' s βi β comap (eval βi) (f βi)"
] | [
" Tendsto m l (pi f) β β (i : ΞΉ), Tendsto (fun x => m x i) l (f i)",
" (β (i : ΞΉ), Tendsto (eval i β m) l (f i)) β β (i : ΞΉ), Tendsto (fun x => m x i) l (f i)"
] |
import Mathlib.GroupTheory.FreeGroup.Basic
import Mathlib.GroupTheory.QuotientGroup
#align_import group_theory.presented_group from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
variable {Ξ± : Type*}
def PresentedGroup (rels : Set (FreeGroup Ξ±)) :=
FreeGroup Ξ± β§Έ Subgroup.normalClosu... | Mathlib/GroupTheory/PresentedGroup.lean | 93 | 97 | theorem toGroup.unique (g : PresentedGroup rels β* G)
(hg : β x : Ξ±, g (PresentedGroup.of x) = f x) : β {x}, g x = toGroup h x := by |
intro x
refine QuotientGroup.induction_on x ?_
exact fun _ β¦ FreeGroup.lift.unique (g.comp (QuotientGroup.mk' _)) hg
| [
" Subgroup.closure (Set.range of) = β€",
" (QuotientGroup.mk' (Subgroup.normalClosure rels)).range = β€",
" β {x : PresentedGroup rels}, g x = (toGroup h) x",
" g x = (toGroup h) x",
" β (z : FreeGroup Ξ±), g βz = (toGroup h) βz"
] | [
" Subgroup.closure (Set.range of) = β€",
" (QuotientGroup.mk' (Subgroup.normalClosure rels)).range = β€"
] |
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Multiplicity
#align_import data.nat.choose.factorization from "leanprover-community/mathlib"@"dc9db541168768af03fe228703e758e649afdbfc"
namespace Nat
variable {p n k : β}
| Mathlib/Data/Nat/Choose/Factorization.lean | 36 | 45 | theorem factorization_choose_le_log : (choose n k).factorization p β€ log p n := by |
by_cases h : (choose n k).factorization p = 0
Β· simp [h]
have hp : p.Prime := Not.imp_symm (choose n k).factorization_eq_zero_of_non_prime h
have hkn : k β€ n := by
refine le_of_not_lt fun hnk => h ?_
simp [choose_eq_zero_of_lt hnk]
rw [factorization_def _ hp, @padicValNat_def _ β¨hpβ© _ (choose_pos hkn... | [
" (n.choose k).factorization p β€ p.log n",
" k β€ n",
" (n.choose k).factorization p = 0",
" (multiplicity p (n.choose k)).get β― β€ p.log n",
" (Finset.filter (fun i => p ^ i β€ k % p ^ i + (n - k) % p ^ i) (Finset.Ico 1 (p.log n + 1))).card β€ p.log n"
] | [] |
import Mathlib.NumberTheory.NumberField.Embeddings
#align_import number_theory.number_field.units from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
open scoped NumberField
noncomputable section
open NumberField Units
variable (K : Type*) [Field K]
namespace NumberField.Units
secti... | Mathlib/NumberTheory/NumberField/Units/Basic.lean | 78 | 79 | theorem coe_pow (x : (π K)Λ£) (n : β) : ((x ^ n : (π K)Λ£) : K) = (x : K) ^ n := by |
rw [β map_pow, β val_pow_eq_pow_val]
| [
" (algebraMap (π K) K) β(x ^ n) = (algebraMap (π K) K) βx ^ n"
] | [] |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : β} {x : β}
| Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 49 | 67 | theorem cosZeta_two_mul_nat (hk : k β 0) (hx : x β Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * Ο) ^ (2 * k) / 2 / (2 * k)! *
((Polynomial.bernoulli (2 * k)).map (algebraMap β β)).eval (x : β) := by |
rw [β (hasSum_nat_cosZeta x (?_ : 1 < re (2 * k))).tsum_eq]
refine Eq.trans ?_ <| (congr_arg ofReal' (hasSum_one_div_nat_pow_mul_cos hk hx).tsum_eq).trans ?_
Β· rw [ofReal_tsum]
refine tsum_congr fun n β¦ ?_
rw [mul_comm (1 / _), mul_one_div, ofReal_div, mul_assoc (2 * Ο), mul_comm x n, β mul_assoc,
... | [
" cosZeta (βx) (2 * βk) =\n (-1) ^ (k + 1) * (2 * βΟ) ^ (2 * k) / 2 / β(2 * k)! *\n Polynomial.eval (βx) (Polynomial.map (algebraMap β β) (Polynomial.bernoulli (2 * k)))",
" 1 < (2 * βk).re",
" β' (b : β), β(2 * Ο * x * βb).cos / βb ^ (2 * βk) = β(β' (b : β), 1 / βb ^ (2 * k) * (2 * Ο * βb * x).cos)",
... | [] |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.fin_range from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
universe u
namespace List
variable {Ξ± : Type u}
@[simp]
| Mathlib/Data/List/FinRange.lean | 25 | 27 | theorem map_coe_finRange (n : β) : ((finRange n) : List (Fin n)).map (Fin.val) = List.range n := by |
simp_rw [finRange, map_pmap, pmap_eq_map]
exact List.map_id _
| [
" map Fin.val (finRange n) = range n",
" map (fun a => a) (range n) = range n"
] | [] |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.projection from "leanprover-community/mathlib"@"6d584f1709bedbed9175bd9350df46599bdd7213"
noncomputable section Ring
variable {R : Type*} [Ring R] {E : Type*} [AddCommGroup E] [Module R E]
variable {F : Type*} [Ad... | Mathlib/LinearAlgebra/Projection.lean | 131 | 135 | theorem prodEquivOfIsCompl_symm_apply_fst_eq_zero (h : IsCompl p q) {x : E} :
((prodEquivOfIsCompl p q h).symm x).1 = 0 β x β q := by |
conv_rhs => rw [β (prodEquivOfIsCompl p q h).apply_symm_apply x]
rw [coe_prodEquivOfIsCompl', Submodule.add_mem_iff_left _ (Submodule.coe_mem _),
mem_right_iff_eq_zero_of_disjoint h.disjoint]
| [
" Function.Injective β(p.mkQ ββ q.subtype)",
" Function.Surjective β(p.mkQ ββ q.subtype)",
" (β₯p Γ β₯q) ββ[R] E",
" Function.Bijective β(p.subtype.coprod q.subtype)",
" Function.Injective β(p.subtype.coprod q.subtype)",
" Disjoint (range p.subtype) (range q.subtype)",
" Disjoint p q",
" Function.Surjec... | [
" Function.Injective β(p.mkQ ββ q.subtype)",
" Function.Surjective β(p.mkQ ββ q.subtype)",
" (β₯p Γ β₯q) ββ[R] E",
" Function.Bijective β(p.subtype.coprod q.subtype)",
" Function.Injective β(p.subtype.coprod q.subtype)",
" Disjoint (range p.subtype) (range q.subtype)",
" Disjoint p q",
" Function.Surjec... |
import Mathlib.Topology.MetricSpace.PseudoMetric
open Filter
open scoped Uniformity Topology
universe u v w
variable {Ξ± : Type u} {Ξ² : Type v} {X ΞΉ : Type*}
variable [PseudoMetricSpace Ξ±]
theorem Metric.complete_of_convergent_controlled_sequences (B : β β Real) (hB : β n, 0 < B n)
(H : β u : β β Ξ±, (β N n m... | Mathlib/Topology/MetricSpace/Cauchy.lean | 72 | 91 | theorem Metric.uniformCauchySeqOn_iff {Ξ³ : Type*} {F : Ξ² β Ξ³ β Ξ±} {s : Set Ξ³} :
UniformCauchySeqOn F atTop s β β Ξ΅ > (0 : β),
β N : Ξ², β m β₯ N, β n β₯ N, β x β s, dist (F m x) (F n x) < Ξ΅ := by |
constructor
Β· intro h Ξ΅ hΞ΅
let u := { a : Ξ± Γ Ξ± | dist a.fst a.snd < Ξ΅ }
have hu : u β π€ Ξ± := Metric.mem_uniformity_dist.mpr β¨Ξ΅, hΞ΅, by simp [u]β©
rw [β @Filter.eventually_atTop_prod_self' _ _ _ fun m =>
β x β s, dist (F m.fst x) (F m.snd x) < Ξ΅]
specialize h u hu
rw [prod_atTop_atTop_eq]... | [
" UniformCauchySeqOn F atTop s β β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (F m x) (F n x) < Ξ΅",
" UniformCauchySeqOn F atTop s β β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (F m x) (F n x) < Ξ΅",
" β N, β m β₯ N, β n β₯ N, β x β s, dist (F m x) (F n x) < Ξ΅",
" β {a b : Ξ±}, dist a b < Ξ΅ β (a, b) β u",
" ... | [] |
import Mathlib.Data.Multiset.Nodup
import Mathlib.Data.List.NatAntidiagonal
#align_import data.multiset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
namespace Nat
def antidiagonal (n : β) : Multiset (β Γ β) :=
List.Nat.antidiagonal n
#align... | Mathlib/Data/Multiset/NatAntidiagonal.lean | 64 | 67 | theorem antidiagonal_succ' {n : β} :
antidiagonal (n + 1) = (n + 1, 0) ::β (antidiagonal n).map (Prod.map id Nat.succ) := by |
rw [antidiagonal, List.Nat.antidiagonal_succ', β coe_add, add_comm, antidiagonal, map_coe,
coe_add, List.singleton_append, cons_coe]
| [
" x β antidiagonal n β x.1 + x.2 = n",
" card (antidiagonal n) = n + 1",
" antidiagonal (n + 1) = (0, n + 1) ::β map (Prod.map Nat.succ id) (antidiagonal n)",
" antidiagonal (n + 1) = (n + 1, 0) ::β map (Prod.map id Nat.succ) (antidiagonal n)"
] | [
" x β antidiagonal n β x.1 + x.2 = n",
" card (antidiagonal n) = n + 1",
" antidiagonal (n + 1) = (0, n + 1) ::β map (Prod.map Nat.succ id) (antidiagonal n)"
] |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/mathlib"@"a50170a88a47570ed186b809ca754110590f9476"
universe u v w wβ wβ
variable (R : Type u) (L : Type v) (M : Type w) {L' : Type wβ}
variab... | Mathlib/Algebra/Lie/Solvable.lean | 89 | 97 | theorem derivedSeriesOfIdeal_le {I J : LieIdeal R L} {k l : β} (hβ : I β€ J) (hβ : l β€ k) :
D k I β€ D l J := by |
revert l; induction' k with k ih <;> intro l hβ
Β· rw [le_zero_iff] at hβ; rw [hβ, derivedSeriesOfIdeal_zero]; exact hβ
Β· have h : l = k.succ β¨ l β€ k := by rwa [le_iff_eq_or_lt, Nat.lt_succ_iff] at hβ
cases' h with h h
Β· rw [h, derivedSeriesOfIdeal_succ, derivedSeriesOfIdeal_succ]
exact LieSubmodule... | [
" D (k + l) I = D k (D l I)",
" D (0 + l) I = D 0 (D l I)",
" D (k + 1 + l) I = D (k + 1) (D l I)",
" D k I β€ D l J",
" β {l : β}, l β€ k β D k I β€ D l J",
" β {l : β}, l β€ 0 β D 0 I β€ D l J",
" β {l : β}, l β€ k + 1 β D (k + 1) I β€ D l J",
" D 0 I β€ D l J",
" I β€ D 0 J",
" D (k + 1) I β€ D l J",
"... | [
" D (k + l) I = D k (D l I)",
" D (0 + l) I = D 0 (D l I)",
" D (k + 1 + l) I = D (k + 1) (D l I)"
] |
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Topology.Order.DenselyOrdered
import Mathlib.Topology.Order.Monotone
#align_import topology.algebra.order.intermediate_value from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filt... | Mathlib/Topology/Order/IntermediateValue.lean | 70 | 75 | theorem intermediate_value_univβ [PreconnectedSpace X] {a b : X} {f g : X β Ξ±} (hf : Continuous f)
(hg : Continuous g) (ha : f a β€ g a) (hb : g b β€ f b) : β x, f x = g x := by |
obtain β¨x, _, hfg, hgfβ© : (univ β© { x | f x β€ g x β§ g x β€ f x }).Nonempty :=
isPreconnected_closed_iff.1 PreconnectedSpace.isPreconnected_univ _ _ (isClosed_le hf hg)
(isClosed_le hg hf) (fun _ _ => le_total _ _) β¨a, trivial, haβ© β¨b, trivial, hbβ©
exact β¨x, le_antisymm hfg hgfβ©
| [
" β x, f x = g x"
] | [] |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ΞΉ : Type*}
namespace CharTwo
section CommSemiring
variable [CommSemiring R] [CharP R 2]
theorem add_sq (x y... | Mathlib/Algebra/CharP/Two.lean | 99 | 100 | theorem list_sum_mul_self (l : List R) : l.sum * l.sum = (List.map (fun x => x * x) l).sum := by |
simp_rw [β pow_two, list_sum_sq]
| [
" (x + y) * (x + y) = x * x + y * y",
" l.sum * l.sum = (List.map (fun x => x * x) l).sum"
] | [
" (x + y) * (x + y) = x * x + y * y"
] |
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import analysis.special_functions.pow.complex from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open scoped Classical
open Real Topology Filter ComplexConjugate Finset Set
namespace Complex
noncomputable def cpow (x y : β) ... | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 45 | 45 | theorem cpow_zero (x : β) : x ^ (0 : β) = 1 := by | simp [cpow_def]
| [
" x ^ 0 = 1"
] | [] |
import Mathlib.Algebra.Ring.Semiconj
import Mathlib.Algebra.Ring.Units
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Data.Bracket
#align_import algebra.ring.commute from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u v w x
variable {Ξ± : Type u} {Ξ² : Type v} {Ξ³ : T... | Mathlib/Algebra/Ring/Commute.lean | 72 | 74 | theorem mul_self_sub_mul_self_eq [NonUnitalNonAssocRing R] {a b : R} (h : Commute a b) :
a * a - b * b = (a + b) * (a - b) := by |
rw [add_mul, mul_sub, mul_sub, h.eq, sub_add_sub_cancel]
| [
" a * a - b * b = (a + b) * (a - b)"
] | [] |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
namespace MulAction
open Pointwise
variable {Ξ± : Type*}
variable {G : Type*} [Group G] [MulAction G Ξ±]
variable {M : Type*} [Monoid M] [MulAction M Ξ±]
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 124 | 126 | theorem set_mem_fixedBy_iff (s : Set Ξ±) (g : G) :
s β fixedBy (Set Ξ±) g β β x, g β’ x β s β x β s := by |
simp_rw [mem_fixedBy, β eq_inv_smul_iff, Set.ext_iff, Set.mem_inv_smul_set_iff, Iff.comm]
| [
" s β fixedBy (Set Ξ±) g β β (x : Ξ±), g β’ x β s β x β s"
] | [] |
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
variable {Ξ± Ξ² : Type*}
section Fold
variable (op : Ξ± β Ξ± β Ξ±) [hc : Std.Commutative op] [ha : Std.Associative op]
local notation a " * " b => ... | Mathlib/Data/Multiset/Fold.lean | 67 | 68 | theorem fold_cons'_right (b a : Ξ±) (s : Multiset Ξ±) : (a ::β s).fold op b = s.fold op (b * a) := by |
rw [fold_eq_foldl, foldl_cons, β fold_eq_foldl]
| [
" List.foldl (fun x y => op y x) b l = List.foldl op b l",
" fold op b (a ::β s) = op (fold op b s) a",
" fold op b (a ::β s) = fold op (op b a) s"
] | [
" List.foldl (fun x y => op y x) b l = List.foldl op b l",
" fold op b (a ::β s) = op (fold op b s) a"
] |
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Topology.Instances.EReal
#align_import measure_theory.integral.vitali_caratheodory from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
open sc... | Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | 164 | 195 | theorem exists_le_lowerSemicontinuous_lintegral_ge (f : Ξ± β ββ₯0β) (hf : Measurable f) {Ξ΅ : ββ₯0β}
(Ξ΅pos : Ξ΅ β 0) :
β g : Ξ± β ββ₯0β,
(β x, f x β€ g x) β§ LowerSemicontinuous g β§ (β«β» x, g x βΞΌ) β€ (β«β» x, f x βΞΌ) + Ξ΅ := by |
rcases ENNReal.exists_pos_sum_of_countable' Ξ΅pos β with β¨Ξ΄, Ξ΄pos, hΞ΄β©
have :
β n,
β g : Ξ± β ββ₯0,
(β x, SimpleFunc.eapproxDiff f n x β€ g x) β§
LowerSemicontinuous g β§
(β«β» x, g x βΞΌ) β€ (β«β» x, SimpleFunc.eapproxDiff f n x βΞΌ) + Ξ΄ n :=
fun n =>
SimpleFunc.exists_le_lowerS... | [
" β g, (β (x : Ξ±), βf x β€ g x) β§ LowerSemicontinuous g β§ β«β» (x : Ξ±), β(g x) βΞΌ β€ β«β» (x : Ξ±), β(βf x) βΞΌ + Ξ΅",
" β g,\n (β (x : Ξ±), β(piecewise s hs (const Ξ± c) (const Ξ± 0)) x β€ g x) β§\n LowerSemicontinuous g β§ β«β» (x : Ξ±), β(g x) βΞΌ β€ β«β» (x : Ξ±), β(β(piecewise s hs (const Ξ± c) (const Ξ± 0)) x) βΞΌ + Ξ΅",
" ... | [
" β g, (β (x : Ξ±), βf x β€ g x) β§ LowerSemicontinuous g β§ β«β» (x : Ξ±), β(g x) βΞΌ β€ β«β» (x : Ξ±), β(βf x) βΞΌ + Ξ΅",
" β g,\n (β (x : Ξ±), β(piecewise s hs (const Ξ± c) (const Ξ± 0)) x β€ g x) β§\n LowerSemicontinuous g β§ β«β» (x : Ξ±), β(g x) βΞΌ β€ β«β» (x : Ξ±), β(β(piecewise s hs (const Ξ± c) (const Ξ± 0)) x) βΞΌ + Ξ΅",
" ... |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 164 | 221 | theorem ae_le_of_forall_set_lintegral_le_of_sigmaFinite [SigmaFinite ΞΌ] {f g : Ξ± β ββ₯0β}
(hf : Measurable f) (hg : Measurable g)
(h : β s, MeasurableSet s β ΞΌ s < β β (β«β» x in s, f x βΞΌ) β€ β«β» x in s, g x βΞΌ) : f β€α΅[ΞΌ] g := by |
have A :
β (Ξ΅ N : ββ₯0) (p : β), 0 < Ξ΅ β ΞΌ ({x | g x + Ξ΅ β€ f x β§ g x β€ N} β© spanningSets ΞΌ p) = 0 := by
intro Ξ΅ N p Ξ΅pos
let s := {x | g x + Ξ΅ β€ f x β§ g x β€ N} β© spanningSets ΞΌ p
have s_meas : MeasurableSet s := by
have A : MeasurableSet {x | g x + Ξ΅ β€ f x} := measurableSet_le (hg.add measurable... | [
" (βα΅ (x : Ξ±) βΞΌ, c β€ f x) β β b < c, ΞΌ {x | f x β€ b} = 0",
" ΞΌ {a | Β¬c β€ f a} = 0 β β b < c, ΞΌ {x | f x β€ b} = 0",
" ΞΌ {a | f a < c} = 0 β β b < c, ΞΌ {x | f x β€ b} = 0",
" ΞΌ {a | f a < c} = 0 β β b < c, ΞΌ {x | f x β€ b} = 0",
" ΞΌ {x | f x β€ b} = 0",
" (β b < c, ΞΌ {x | f x β€ b} = 0) β ΞΌ {a | f a < c} = 0",... | [
" (βα΅ (x : Ξ±) βΞΌ, c β€ f x) β β b < c, ΞΌ {x | f x β€ b} = 0",
" ΞΌ {a | Β¬c β€ f a} = 0 β β b < c, ΞΌ {x | f x β€ b} = 0",
" ΞΌ {a | f a < c} = 0 β β b < c, ΞΌ {x | f x β€ b} = 0",
" ΞΌ {a | f a < c} = 0 β β b < c, ΞΌ {x | f x β€ b} = 0",
" ΞΌ {x | f x β€ b} = 0",
" (β b < c, ΞΌ {x | f x β€ b} = 0) β ΞΌ {a | f a < c} = 0",... |
import Mathlib.Combinatorics.SimpleGraph.Basic
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
structure Dart extends V Γ V where
adj : G.Adj fst snd
deriving DecidableEq
#align simple_graph.dart SimpleGraph.Dart
initialize_simps_projections Dart (+toProd, -fst, -snd)
attribute [simp] Dart.a... | Mathlib/Combinatorics/SimpleGraph/Dart.lean | 112 | 115 | theorem dart_edge_eq_mk'_iff :
β {d : G.Dart} {p : V Γ V}, d.edge = Sym2.mk p β d.toProd = p β¨ d.toProd = p.swap := by |
rintro β¨p, hβ©
apply Sym2.mk_eq_mk_iff
| [
" dβ = dβ β dβ.toProd = dβ.toProd",
" { toProd := toProdβ, adj := adjβ } = dβ β { toProd := toProdβ, adj := adjβ }.toProd = dβ.toProd",
" { toProd := toProdβΒΉ, adj := adjβΒΉ } = { toProd := toProdβ, adj := adjβ } β\n { toProd := toProdβΒΉ, adj := adjβΒΉ }.toProd = { toProd := toProdβ, adj := adjβ }.toProd",
"... | [
" dβ = dβ β dβ.toProd = dβ.toProd",
" { toProd := toProdβ, adj := adjβ } = dβ β { toProd := toProdβ, adj := adjβ }.toProd = dβ.toProd",
" { toProd := toProdβΒΉ, adj := adjβΒΉ } = { toProd := toProdβ, adj := adjβ } β\n { toProd := toProdβΒΉ, adj := adjβΒΉ }.toProd = { toProd := toProdβ, adj := adjβ }.toProd",
"... |
import Mathlib.Algebra.Quotient
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.SetTheory.Cardinal.Finite
#align_import group_theory.coset from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce4... | Mathlib/GroupTheory/Coset.lean | 111 | 112 | theorem rightCoset_assoc (s : Set Ξ±) (a b : Ξ±) : op b β’ op a β’ s = op (a * b) β’ s := by |
simp [β image_smul, (image_comp _ _ _).symm, Function.comp, mul_assoc]
| [
" a β’ b β’ s = (a * b) β’ s",
" op b β’ op a β’ s = op (a * b) β’ s"
] | [
" a β’ b β’ s = (a * b) β’ s"
] |
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.indicator from "leanprover-community/mathlib"@"842328d9df7e96fd90fc424e115679c15fb23a71"
noncomputable section
open Finset Function
variable {ΞΉ Ξ± : Type*}
namespace Finsupp
variable [Zero Ξ±] {s : Finset ΞΉ} (f : β i β s, Ξ±) {i : ΞΉ}
def indicator (s ... | Mathlib/Data/Finsupp/Indicator.lean | 54 | 56 | theorem indicator_apply [DecidableEq ΞΉ] : indicator s f i = if hi : i β s then f i hi else 0 := by |
simp only [indicator, ne_eq, coe_mk]
congr
| [
" i β map (Embedding.subtype fun x => x β s) (filter (fun i => f βi β― β 0) s.attach) β\n (fun i => if H : i β s then f i H else 0) i β 0",
" (indicator s f) i = if hi : i β s then f i hi else 0",
" (if H : i β s then f i H else 0) = if hi : i β s then f i hi else 0"
] | [
" i β map (Embedding.subtype fun x => x β s) (filter (fun i => f βi β― β 0) s.attach) β\n (fun i => if H : i β s then f i H else 0) i β 0"
] |
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 | 137 | 140 | theorem ascPochhammer_succ_eval {S : Type*} [Semiring S] (n : β) (k : S) :
(ascPochhammer S (n + 1)).eval k = (ascPochhammer S n).eval k * (k + n) := by |
rw [ascPochhammer_succ_right, mul_add, eval_add, eval_mul_X, β Nat.cast_comm, β C_eq_natCast,
eval_C_mul, Nat.cast_comm, β mul_add]
| [
" 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.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.LocallyConvex.Barrelled
import Mathlib.Topology.Baire.CompleteMetrizable
#align_import analysis.normed_space.banach_steinhaus from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set
variable {E F οΏ½... | Mathlib/Analysis/NormedSpace/BanachSteinhaus.lean | 34 | 38 | theorem banach_steinhaus {ΞΉ : Type*} [CompleteSpace E] {g : ΞΉ β E βSL[Οββ] F}
(h : β x, β C, β i, βg i xβ β€ C) : β C', β i, βg iβ β€ C' := by |
rw [show (β C, β i, βg iβ β€ C) β _ from (NormedSpace.equicontinuous_TFAE g).out 5 2]
refine (norm_withSeminorms πβ F).banach_steinhaus (fun _ x β¦ ?_)
simpa [bddAbove_def, forall_mem_range] using h x
| [
" β C', β (i : ΞΉ), βg iβ β€ C'",
" UniformEquicontinuous (DFunLike.coe β g)",
" BddAbove (range fun i => (normSeminorm πβ F) ((g i) x))"
] | [] |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {Ξ± Ξ² : Type*} {s t : Set Ξ±}
noncomputable def encard (s : Set Ξ±) : ββ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 73 | 76 | theorem Finite.encard_eq_coe_toFinset_card (h : s.Finite) : s.encard = h.toFinset.card := by |
have := h.fintype
rw [encard, PartENat.card_eq_coe_fintype_card,
PartENat.withTopEquiv_natCast, toFinite_toFinset, toFinset_card]
| [
" univ.encard = s.encard",
" univ.encard = PartENat.withTopEquiv (PartENat.card Ξ±)",
" s.encard = βh.toFinset.card"
] | [
" univ.encard = s.encard",
" univ.encard = PartENat.withTopEquiv (PartENat.card Ξ±)"
] |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.NormedSpace.ProdLp
import Mathlib.Topology.Instances.TrivSqZeroExt
#align_import analysis.normed_space.triv_sq_zero_ext from "leanprover-community/mathlib"@"88a563b158f59f2983cfad685664da95502e8cdd"
variable (π : Type*) {S R M : Type*}
loca... | Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean | 214 | 217 | theorem norm_def (x : tsze R M) : βxβ = βfst xβ + βsnd xβ := by |
rw [WithLp.prod_norm_eq_add (by norm_num)]
simp only [ENNReal.one_toReal, Real.rpow_one, div_one]
rfl
| [
" βxβ = βx.fstβ + βx.sndβ",
" 0 < ENNReal.toReal 1",
" (βx.1β ^ ENNReal.toReal 1 + βx.2β ^ ENNReal.toReal 1) ^ (1 / ENNReal.toReal 1) = βx.fstβ + βx.sndβ",
" βx.1β + βx.2β = βx.fstβ + βx.sndβ"
] | [] |
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Algebra.Constructions
#align_import topology.algebra.group.basic from "leanprover-community/mathlib"@"3b1890e71632be9e3... | Mathlib/Topology/Algebra/Group/Basic.lean | 114 | 117 | theorem Homeomorph.mulRight_symm (a : G) :
(Homeomorph.mulRight a).symm = Homeomorph.mulRight aβ»ΒΉ := by |
ext
rfl
| [
" (Homeomorph.mulLeft a).symm = Homeomorph.mulLeft aβ»ΒΉ",
" (Homeomorph.mulLeft a).symm xβ = (Homeomorph.mulLeft aβ»ΒΉ) xβ",
" (Homeomorph.mulRight a).symm = Homeomorph.mulRight aβ»ΒΉ",
" (Homeomorph.mulRight a).symm xβ = (Homeomorph.mulRight aβ»ΒΉ) xβ"
] | [
" (Homeomorph.mulLeft a).symm = Homeomorph.mulLeft aβ»ΒΉ",
" (Homeomorph.mulLeft a).symm xβ = (Homeomorph.mulLeft aβ»ΒΉ) xβ"
] |
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
#align_import analysis.normed.group.add_torsor from "leanprover-community/mathlib"@"837f72de63ad6cd96519cde5f1ffd5ed8d280ad0"
noncomputable section
open NNReal Topo... | Mathlib/Analysis/Normed/Group/AddTorsor.lean | 125 | 125 | theorem dist_vadd_right (v : V) (x : P) : dist x (v +α΅₯ x) = βvβ := by | rw [dist_comm, dist_vadd_left]
| [
" dist (c +α΅₯ x) (c +α΅₯ y) = dist x y",
" dist (vβ +α΅₯ x) (vβ +α΅₯ x) = dist vβ vβ",
" dist (v +α΅₯ x) x = βvβ",
" dist x (v +α΅₯ x) = βvβ"
] | [
" dist (c +α΅₯ x) (c +α΅₯ y) = dist x y",
" dist (vβ +α΅₯ x) (vβ +α΅₯ x) = dist vβ vβ",
" dist (v +α΅₯ x) x = βvβ"
] |
import Mathlib.Algebra.Category.MonCat.Basic
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
#align_import algebra.category.Mon.colimits from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe v
open CategoryTheory
open Ca... | Mathlib/Algebra/Category/MonCat/Colimits.lean | 179 | 183 | theorem cocone_naturality {j j' : J} (f : j βΆ j') :
F.map f β« coconeMorphism F j' = coconeMorphism F j := by |
ext
apply Quot.sound
apply Relation.map
| [
" Inhabited (ColimitType F)",
" Inhabited (Quotient (colimitSetoid F))",
" Monoid (ColimitType F)",
" F.map f β« coconeMorphism F j' = coconeMorphism F j",
" (F.map f β« coconeMorphism F j') xβ = (coconeMorphism F j) xβ",
" Setoid.r (Prequotient.of j' ((F.map f) xβ)) (Prequotient.of j xβ)"
] | [
" Inhabited (ColimitType F)",
" Inhabited (Quotient (colimitSetoid F))",
" Monoid (ColimitType F)"
] |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.Dynamics.Minimal
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.MeasureTheory.Group.MeasurableEquiv
import Mathlib.MeasureTheory.Measure.Regular
#align_import measure_theory.group.action from "leanprover-community/mathlib"@"f2ce6086713c78a7f8... | Mathlib/MeasureTheory/Group/Action.lean | 114 | 126 | theorem smulInvariantMeasure_map [SMul M Ξ±] [SMul M Ξ²]
[MeasurableSMul M Ξ²]
(ΞΌ : Measure Ξ±) [SMulInvariantMeasure M Ξ± ΞΌ] (f : Ξ± β Ξ²)
(hsmul : β (m : M) a, f (m β’ a) = m β’ f a) (hf : Measurable f) :
SMulInvariantMeasure M Ξ² (map f ΞΌ) where
measure_preimage_smul m S hS := calc
map f ΞΌ ((m β’ Β·) β»ΒΉ' S... | rw [preimage_preimage]
_ = ΞΌ ((f <| m β’ Β·) β»ΒΉ' S) := by simp_rw [hsmul]
_ = ΞΌ ((m β’ Β·) β»ΒΉ' (f β»ΒΉ' S)) := by rw [β preimage_preimage]
_ = ΞΌ (f β»ΒΉ' S) := by rw [SMulInvariantMeasure.measure_preimage_smul m (hS.preimage hf)]
_ = map f ΞΌ S := (map_apply hf hS).symm
| [
" ΞΌ (f β»ΒΉ' ((fun x => m β’ x) β»ΒΉ' S)) = ΞΌ ((fun x => m β’ f x) β»ΒΉ' S)",
" ΞΌ ((fun x => m β’ f x) β»ΒΉ' S) = ΞΌ ((fun x => f (m β’ x)) β»ΒΉ' S)",
" ΞΌ ((fun x => f (m β’ x)) β»ΒΉ' S) = ΞΌ ((fun x => m β’ x) β»ΒΉ' (f β»ΒΉ' S))",
" ΞΌ ((fun x => m β’ x) β»ΒΉ' (f β»ΒΉ' S)) = ΞΌ (f β»ΒΉ' S)"
] | [] |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
section IsLocalizedModule
universe u v
variable {R : Type*} [CommSemiring R] (S : Submonoid R)
variabl... | Mathlib/Algebra/Module/LocalizedModule.lean | 599 | 610 | theorem isLocalizedModule_iff_isLocalization {A Aβ} [CommSemiring A] [Algebra R A] [CommSemiring Aβ]
[Algebra A Aβ] [Algebra R Aβ] [IsScalarTower R A Aβ] :
IsLocalizedModule S (IsScalarTower.toAlgHom R A Aβ).toLinearMap β
IsLocalization (Algebra.algebraMapSubmonoid A S) Aβ := by |
rw [isLocalizedModule_iff, isLocalization_iff]
refine and_congr ?_ (and_congr (forall_congr' fun _ β¦ ?_) (forallβ_congr fun _ _ β¦ ?_))
Β· simp_rw [β (Algebra.lmul R Aβ).commutes, Algebra.lmul_isUnit_iff, Subtype.forall,
Algebra.algebraMapSubmonoid, β SetLike.mem_coe, Submonoid.coe_map,
Set.forall_mem_... | [
" f xβ = f xβ",
" IsUnit ((algebraMap R (Module.End R M'')) βs)",
" (algebraMap R (Module.End R M'')) βs = βe ββ (algebraMap R (Module.End R M')) βs ββ βe.symm",
" ((algebraMap R (Module.End R M'')) βs) xβ = (βe ββ (algebraMap R (Module.End R M')) βs ββ βe.symm) xβ",
" Function.Bijective β((algebraMap R (Mo... | [
" f xβ = f xβ",
" IsUnit ((algebraMap R (Module.End R M'')) βs)",
" (algebraMap R (Module.End R M'')) βs = βe ββ (algebraMap R (Module.End R M')) βs ββ βe.symm",
" ((algebraMap R (Module.End R M'')) βs) xβ = (βe ββ (algebraMap R (Module.End R M')) βs ββ βe.symm) xβ",
" Function.Bijective β((algebraMap R (Mo... |
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f : Ξ± β Ξ²} [DecidablePred p] : DecidablePred (p β f) :=
inferInstanceAs <| DecidablePred fun x => p (f x)
@[deprecated] alias proofIrrel := proof_irrel
theorem Function.id_def : @id Ξ± = fun x => x := rfl
al... | .lake/packages/batteries/Batteries/Logic.lean | 74 | 74 | theorem Eq.congr_right {x y z : Ξ±} (h : x = y) : z = x β z = y := by | rw [h]
| [
" h βΈ y = y",
" β― βΈ y = y",
" f x y = f x' y'",
" f x y = f x y",
" xβ = xβ β yβ = yβ",
" xβ = xβ β xβ = yβ",
" xβ = xβ β xβ = xβ",
" x = z β y = z",
" z = x β z = y"
] | [
" h βΈ y = y",
" β― βΈ y = y",
" f x y = f x' y'",
" f x y = f x y",
" xβ = xβ β yβ = yβ",
" xβ = xβ β xβ = yβ",
" xβ = xβ β xβ = xβ",
" x = z β y = z"
] |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 154 | 159 | theorem le_of_mem_A {r Ξ΅ : β} {L : E βL[π] F} {x : E} (hx : x β A f L r Ξ΅) {y z : E}
(hy : y β closedBall x (r / 2)) (hz : z β closedBall x (r / 2)) :
βf z - f y - L (z - y)β β€ Ξ΅ * r := by |
rcases hx with β¨r', r'mem, hr'β©
apply le_of_lt
exact hr' _ ((mem_closedBall.1 hy).trans_lt r'mem.1) _ ((mem_closedBall.1 hz).trans_lt r'mem.1)
| [
" IsOpen (A f L r Ξ΅)",
" β x β A f L r Ξ΅, β Ξ΅_1 > 0, ball x Ξ΅_1 β A f L r Ξ΅",
" β Ξ΅_1 > 0, ball x Ξ΅_1 β A f L r Ξ΅",
" r' - s > 0",
" β y β ball x' s, β z β ball x' s, βf z - f y - L (z - y)β < Ξ΅ * r",
" βf z - f y - L (z - y)β < Ξ΅ * r",
" IsOpen (B f K r s Ξ΅)",
" A f L r Ξ΅ β A f L r Ξ΄",
" x β A f L ... | [
" IsOpen (A f L r Ξ΅)",
" β x β A f L r Ξ΅, β Ξ΅_1 > 0, ball x Ξ΅_1 β A f L r Ξ΅",
" β Ξ΅_1 > 0, ball x Ξ΅_1 β A f L r Ξ΅",
" r' - s > 0",
" β y β ball x' s, β z β ball x' s, βf z - f y - L (z - y)β < Ξ΅ * r",
" βf z - f y - L (z - y)β < Ξ΅ * r",
" IsOpen (B f K r s Ξ΅)",
" A f L r Ξ΅ β A f L r Ξ΄",
" x β A f L ... |
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import linear_algebra.affine_space.independent from "leanprover-c... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 86 | 134 | theorem affineIndependent_iff_linearIndependent_vsub (p : ΞΉ β P) (i1 : ΞΉ) :
AffineIndependent k p β LinearIndependent k fun i : { x // x β i1 } => (p i -α΅₯ p i1 : V) := by |
classical
constructor
Β· intro h
rw [linearIndependent_iff']
intro s g hg i hi
set f : ΞΉ β k := fun x => if hx : x = i1 then -β y β s, g y else g β¨x, hxβ© with hfdef
let s2 : Finset ΞΉ := insert i1 (s.map (Embedding.subtype _))
have hfg : β x : { x // x β i1 }, g x = f x := by
... | [
" AffineIndependent k p β β (w : ΞΉ β k), β i : ΞΉ, w i = 0 β (univ.weightedVSub p) w = 0 β β (i : ΞΉ), w i = 0",
" AffineIndependent k p β β (w : ΞΉ β k), β i : ΞΉ, w i = 0 β (univ.weightedVSub p) w = 0 β β (i : ΞΉ), w i = 0",
" (β (w : ΞΉ β k), β i : ΞΉ, w i = 0 β (univ.weightedVSub p) w = 0 β β (i : ΞΉ), w i = 0) β A... | [
" AffineIndependent k p β β (w : ΞΉ β k), β i : ΞΉ, w i = 0 β (univ.weightedVSub p) w = 0 β β (i : ΞΉ), w i = 0",
" AffineIndependent k p β β (w : ΞΉ β k), β i : ΞΉ, w i = 0 β (univ.weightedVSub p) w = 0 β β (i : ΞΉ), w i = 0",
" (β (w : ΞΉ β k), β i : ΞΉ, w i = 0 β (univ.weightedVSub p) w = 0 β β (i : ΞΉ), w i = 0) β A... |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Tactic.Ring
#align_import data.nat.hyperoperation from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
def hyperoperation : β β β β β β β
| 0, _, k => k + 1
| 1, m, 0 => m
| 2, _, 0 => 0
| _ + 3, _, 0 => 1
| n + 1, m, k + 1 ... | Mathlib/Data/Nat/Hyperoperation.lean | 104 | 113 | theorem hyperoperation_ge_three_one (n : β) : β k : β, hyperoperation (n + 3) 1 k = 1 := by |
induction' n with nn nih
Β· intro k
rw [hyperoperation_three]
dsimp
rw [one_pow]
Β· intro k
cases k
Β· rw [hyperoperation_ge_three_eq_one]
Β· rw [hyperoperation_recursion, nih]
| [
" hyperoperation 0 m k = k.succ",
" hyperoperation (n + 3) m 0 = 1",
" hyperoperation (n + 1) m (k + 1) = hyperoperation n m (hyperoperation (n + 1) m k)",
" hyperoperation 1 = fun x x_1 => x + x_1",
" hyperoperation 1 m k = m + k",
" hyperoperation 1 m 0 = m + 0",
" hyperoperation 1 m (bn + 1) = m + (b... | [
" hyperoperation 0 m k = k.succ",
" hyperoperation (n + 3) m 0 = 1",
" hyperoperation (n + 1) m (k + 1) = hyperoperation n m (hyperoperation (n + 1) m k)",
" hyperoperation 1 = fun x x_1 => x + x_1",
" hyperoperation 1 m k = m + k",
" hyperoperation 1 m 0 = m + 0",
" hyperoperation 1 m (bn + 1) = m + (b... |
import Mathlib.Order.Filter.CountableInter
set_option autoImplicit true
open Function Set Filter
class HasCountableSeparatingOn (Ξ± : Type*) (p : Set Ξ± β Prop) (t : Set Ξ±) : Prop where
exists_countable_separating : β S : Set (Set Ξ±), S.Countable β§ (β s β S, p s) β§
β x β t, β y β t, (β s β S, x β s β y β s) ... | Mathlib/Order/Filter/CountableSeparatingOn.lean | 103 | 109 | theorem exists_seq_separating (Ξ± : Type*) {p : Set Ξ± β Prop} {sβ} (hp : p sβ) (t : Set Ξ±)
[HasCountableSeparatingOn Ξ± p t] :
β S : β β Set Ξ±, (β n, p (S n)) β§ β x β t, β y β t, (β n, x β S n β y β S n) β x = y := by |
rcases exists_nonempty_countable_separating Ξ± hp t with β¨S, hSne, hSc, hSβ©
rcases hSc.exists_eq_range hSne with β¨S, rflβ©
use S
simpa only [forall_mem_range] using hS
| [
" β S, (β (n : β), p (S n)) β§ β x β t, β y β t, (β (n : β), x β S n β y β S n) β x = y",
" (β (n : β), p (S n)) β§ β x β t, β y β t, (β (n : β), x β S n β y β S n) β x = y"
] | [] |
import Mathlib.RingTheory.FiniteType
#align_import ring_theory.rees_algebra from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open Polynomial
def reesAlgebra : Subalgebra... | Mathlib/RingTheory/ReesAlgebra.lean | 82 | 95 | theorem monomial_mem_adjoin_monomial {I : Ideal R} {n : β} {r : R} (hr : r β I ^ n) :
monomial n r β Algebra.adjoin R (Submodule.map (monomial 1 : R ββ[R] R[X]) I : Set R[X]) := by |
induction' n with n hn generalizing r
Β· exact Subalgebra.algebraMap_mem _ _
Β· rw [pow_succ'] at hr
apply Submodule.smul_induction_on
-- Porting note: did not need help with motive previously
(p := fun r => (monomial (Nat.succ n)) r β Algebra.adjoin R (Submodule.map (monomial 1) I)) hr
Β· intro... | [
" (aβ * bβ).coeff i β I ^ i",
" β x β Finset.antidiagonal i, aβ.coeff x.1 * bβ.coeff x.2 β I ^ i",
" β c β Finset.antidiagonal i, aβ.coeff c.1 * bβ.coeff c.2 β I ^ i",
" aβ.coeff (j, k).1 * bβ.coeff (j, k).2 β I ^ i",
" aβ.coeff (j, k).1 * bβ.coeff (j, k).2 β I ^ (j, k).1 * I ^ (j, k).2",
" coeff 1 i β I ... | [
" (aβ * bβ).coeff i β I ^ i",
" β x β Finset.antidiagonal i, aβ.coeff x.1 * bβ.coeff x.2 β I ^ i",
" β c β Finset.antidiagonal i, aβ.coeff c.1 * bβ.coeff c.2 β I ^ i",
" aβ.coeff (j, k).1 * bβ.coeff (j, k).2 β I ^ i",
" aβ.coeff (j, k).1 * bβ.coeff (j, k).2 β I ^ (j, k).1 * I ^ (j, k).2",
" coeff 1 i β I ... |
import Mathlib.Algebra.Group.Subsemigroup.Basic
#align_import group_theory.subsemigroup.membership from "leanprover-community/mathlib"@"6cb77a8eaff0ddd100e87b1591c6d3ad319514ff"
assert_not_exists MonoidWithZero
variable {ΞΉ : Sort*} {M A B : Type*}
section NonAssoc
variable [Mul M]
open Set
namespace Subsemigr... | Mathlib/Algebra/Group/Subsemigroup/Membership.lean | 123 | 128 | theorem iSup_induction (S : ΞΉ β Subsemigroup M) {C : M β Prop} {xβ : M} (hxβ : xβ β β¨ i, S i)
(mem : β i, β xβ β S i, C xβ) (mul : β x y, C x β C y β C (x * y)) : C xβ := by |
rw [iSup_eq_closure] at hxβ
refine closure_induction hxβ (fun xβ hxβ => ?_) mul
obtain β¨i, hiβ© := Set.mem_iUnion.mp hxβ
exact mem _ _ hi
| [
" x β β¨ i, S i β β i, x β S i",
" x β β¨ i, S i β β i, x β S i",
" x β closure (β i, β(S i)) β β i, x β S i",
" β (x y : M), (β i, x β S i) β (β i, y β S i) β β i, x * y β S i",
" β i, x * y β S i",
" x β β(β¨ i, S i) β x β β i, β(S i)",
" x β sSup S β β s β S, x β s",
" x β β(sSup S) β x β β s β S, βs"... | [
" x β β¨ i, S i β β i, x β S i",
" x β β¨ i, S i β β i, x β S i",
" x β closure (β i, β(S i)) β β i, x β S i",
" β (x y : M), (β i, x β S i) β (β i, y β S i) β β i, x * y β S i",
" β i, x * y β S i",
" x β β(β¨ i, S i) β x β β i, β(S i)",
" x β sSup S β β s β S, x β s",
" x β β(sSup S) β x β β s β S, βs"... |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.monotone.extension from "leanprover-community/mathlib"@"422e70f7ce183d2900c586a8cda8381e788a0c62"
open Set
variable {Ξ± Ξ² : Type*} [LinearOrder Ξ±] [ConditionallyCompleteLinearOrder Ξ²] {f : Ξ± β Ξ²} {s : Set Ξ±}
{a b : Ξ±}
| Mathlib/Order/Monotone/Extension.lean | 25 | 48 | theorem MonotoneOn.exists_monotone_extension (h : MonotoneOn f s) (hl : BddBelow (f '' s))
(hu : BddAbove (f '' s)) : β g : Ξ± β Ξ², Monotone g β§ EqOn f g s := by |
classical
/- The extension is defined by `f x = f a` for `x β€ a`, and `f x` is the supremum of the values
of `f` to the left of `x` for `x β₯ a`. -/
rcases hl with β¨a, haβ©
have hu' : β x, BddAbove (f '' (Iic x β© s)) := fun x =>
hu.mono (image_subset _ inter_subset_right)
let g : Ξ± β Ξ² := f... | [
" β g, Monotone g β§ EqOn f g s",
" EqOn f g s",
" f x = g x",
" f x = if Disjoint (Iic x) s then a else sSup (f '' (Iic x β© s))",
" g x β€ g y",
" a β€ sSup (f '' (Iic y β© s))",
" sSup (f '' (Iic x β© s)) β€ a",
" sSup (f '' (Iic x β© s)) β€ sSup (f '' (Iic y β© s))",
" Iic x β© s β Iic y β© s"
] | [] |
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Data.Rat.Floor
#align_import algebra.continued_fractions.computation.terminates_iff_rat from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b3... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 129 | 135 | theorem exists_rat_eq_of_terminates (terminates : (of v).Terminates) : β q : β, v = βq := by |
obtain β¨n, v_eq_convβ© : β n, v = (of v).convergents n :=
of_correctness_of_terminates terminates
obtain β¨q, conv_eq_qβ© : β q : β, (of v).convergents n = (βq : K) :=
exists_rat_eq_nth_convergent v n
have : v = (βq : K) := Eq.trans v_eq_conv conv_eq_q
use q, this
| [
" β (n : β),\n (β m < n, β conts, (of v).continuantsAux m = Pair.map Rat.cast conts) β\n β conts, (of v).continuantsAux n = Pair.map Rat.cast conts",
" β conts, (of v).continuantsAux n = Pair.map Rat.cast conts",
" β conts, (of v).continuantsAux 0 = Pair.map Rat.cast conts",
" β gp, { a := 1, b := 0 }... | [
" β (n : β),\n (β m < n, β conts, (of v).continuantsAux m = Pair.map Rat.cast conts) β\n β conts, (of v).continuantsAux n = Pair.map Rat.cast conts",
" β conts, (of v).continuantsAux n = Pair.map Rat.cast conts",
" β conts, (of v).continuantsAux 0 = Pair.map Rat.cast conts",
" β gp, { a := 1, b := 0 }... |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Topology.Algebra.InfiniteSum.Order
import Mathlib.Topology.Instances.Real
import Mathlib.Topology.Instances.ENNReal
#align_import topology.algebra.infinite_sum.real from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Filte... | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | 26 | 31 | theorem cauchySeq_of_dist_le_of_summable (d : β β β) (hf : β n, dist (f n) (f n.succ) β€ d n)
(hd : Summable d) : CauchySeq f := by |
lift d to β β ββ₯0 using fun n β¦ dist_nonneg.trans (hf n)
apply cauchySeq_of_edist_le_of_summable d (Ξ± := Ξ±) (f := f)
Β· exact_mod_cast hf
Β· exact_mod_cast hd
| [
" CauchySeq f",
" β (n : β), edist (f n) (f n.succ) β€ β(d n)",
" Summable d"
] | [] |
import Mathlib.Topology.Algebra.UniformConvergence
#align_import topology.algebra.module.strong_topology from "leanprover-community/mathlib"@"8905e5ed90859939681a725b00f6063e65096d95"
open scoped Topology UniformConvergence
section General
variable {πβ πβ : Type*} [NormedField πβ] [NormedField πβ] (Ο : π... | Mathlib/Topology/Algebra/Module/StrongTopology.lean | 113 | 115 | theorem uniformSpace_eq [UniformSpace F] [UniformAddGroup F] (π : Set (Set E)) :
instUniformSpace Ο F π = UniformSpace.comap DFunLike.coe (UniformOnFun.uniformSpace E F π) := by |
rw [instUniformSpace, UniformSpace.replaceTopology_eq]
| [
" instTopologicalSpace Ο F π = TopologicalSpace.induced DFunLike.coe (UniformOnFun.topologicalSpace E F π)",
" TopologicalSpace.induced DFunLike.coe (UniformOnFun.topologicalSpace E F π) =\n TopologicalSpace.induced DFunLike.coe (UniformOnFun.topologicalSpace E F π)",
" TopologicalAddGroup.toUniformSpace... | [
" instTopologicalSpace Ο F π = TopologicalSpace.induced DFunLike.coe (UniformOnFun.topologicalSpace E F π)",
" TopologicalSpace.induced DFunLike.coe (UniformOnFun.topologicalSpace E F π) =\n TopologicalSpace.induced DFunLike.coe (UniformOnFun.topologicalSpace E F π)",
" TopologicalAddGroup.toUniformSpace... |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 124 | 129 | theorem fold_insert_idem [DecidableEq Ξ±] [hi : Std.IdempotentOp op] :
(insert a s).fold op b f = f a * s.fold op b f := by |
by_cases h : a β s
Β· rw [β insert_erase h]
simp [β ha.assoc, hi.idempotent]
Β· apply fold_insert h
| [
" fold op b f (cons a s h) = op (f a) (fold op b f s)",
" Multiset.fold op b (Multiset.map f (cons a s h).val) = op (f a) (Multiset.fold op b (Multiset.map f s.val))",
" fold op b f (insert a s) = op (f a) (fold op b f s)",
" Multiset.fold op b (Multiset.map f (insert a s).val) = op (f a) (Multiset.fold op b ... | [
" fold op b f (cons a s h) = op (f a) (fold op b f s)",
" Multiset.fold op b (Multiset.map f (cons a s h).val) = op (f a) (Multiset.fold op b (Multiset.map f s.val))",
" fold op b f (insert a s) = op (f a) (fold op b f s)",
" Multiset.fold op b (Multiset.map f (insert a s).val) = op (f a) (Multiset.fold op b ... |
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 45 | 49 | theorem exp_log {x : β} (hx : x β 0) : exp (log x) = x := by |
rw [log, exp_add_mul_I, β ofReal_sin, sin_arg, β ofReal_cos, cos_arg hx, β ofReal_exp,
Real.exp_log (abs.pos hx), mul_add, ofReal_div, ofReal_div,
mul_div_cancelβ _ (ofReal_ne_zero.2 <| abs.ne_zero hx), β mul_assoc,
mul_div_cancelβ _ (ofReal_ne_zero.2 <| abs.ne_zero hx), re_add_im]
| [
" x.log.re = (abs x).log",
" x.log.im = x.arg",
" -Ο < x.log.im",
" x.log.im β€ Ο",
" cexp x.log = x"
] | [
" x.log.re = (abs x).log",
" x.log.im = x.arg",
" -Ο < x.log.im",
" x.log.im β€ Ο"
] |
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 {Mβ Mβ : Submodule R M} {Nβ Nβ : Submodule R N}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 131 | 136 | theorem exists_finite_submodule_right_of_finite (s : Set (M β[R] N)) (hs : s.Finite) :
β N' : Submodule R N, Module.Finite R N' β§ s β LinearMap.range (N'.subtype.lTensor M) := by |
obtain β¨_, N', _, hfin, hβ© := exists_finite_submodule_of_finite s hs
refine β¨N', hfin, ?_β©
rw [mapIncl, β LinearMap.lTensor_comp_rTensor] at h
exact h.trans (LinearMap.range_comp_le_range _ _)
| [
" β S, x = (Multiset.map (fun i => i.1 ββ[R] i.2) S).sum",
" β S, 0 = (Multiset.map (fun i => i.1 ββ[R] i.2) S).sum",
" 0 = (Multiset.map (fun i => i.1 ββ[R] i.2) 0).sum",
" β S, x ββ[R] y = (Multiset.map (fun i => i.1 ββ[R] i.2) S).sum",
" x ββ[R] y = (Multiset.map (fun i => i.1 ββ[R] i.2) {(x, y)}).sum",
... | [
" β S, x = (Multiset.map (fun i => i.1 ββ[R] i.2) S).sum",
" β S, 0 = (Multiset.map (fun i => i.1 ββ[R] i.2) S).sum",
" 0 = (Multiset.map (fun i => i.1 ββ[R] i.2) 0).sum",
" β S, x ββ[R] y = (Multiset.map (fun i => i.1 ββ[R] i.2) S).sum",
" x ββ[R] y = (Multiset.map (fun i => i.1 ββ[R] i.2) {(x, y)}).sum",
... |
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.GroupTheory.GroupAction.ConjAct
#align_import group_theory.subgroup.pointwise from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
open Set
open Pointwise
variable {Ξ± G A S... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 125 | 126 | theorem closure_inv (s : Set G) : closure sβ»ΒΉ = closure s := by |
simp only [β toSubmonoid_eq, closure_toSubmonoid, inv_inv, union_comm]
| [
" a β’ βs = βs",
" xβ β a β’ βs β xβ β βs",
" MulOpposite.op a β’ βs = βs",
" xβ β MulOpposite.op a β’ βs β xβ β βs",
" βH * βH = βH",
" βH / βH = βH",
" s β β(closure S)",
" sβ»ΒΉ β closure S",
" (closure S).toSubmonoid = Submonoid.closure (S βͺ Sβ»ΒΉ)",
" x β Submonoid.closure (S βͺ Sβ»ΒΉ)",
" xβ»ΒΉ β Submo... | [
" a β’ βs = βs",
" xβ β a β’ βs β xβ β βs",
" MulOpposite.op a β’ βs = βs",
" xβ β MulOpposite.op a β’ βs β xβ β βs",
" βH * βH = βH",
" βH / βH = βH",
" s β β(closure S)",
" sβ»ΒΉ β closure S",
" (closure S).toSubmonoid = Submonoid.closure (S βͺ Sβ»ΒΉ)",
" x β Submonoid.closure (S βͺ Sβ»ΒΉ)",
" xβ»ΒΉ β Submo... |
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.ae_disjoint from "leanprover-community/mathlib"@"bc7d81beddb3d6c66f71449c5bc76c38cb77cf9e"
open Set Function
namespace MeasureTheory
variable {ΞΉ Ξ± : Type*} {m : MeasurableSpace Ξ±} (ΞΌ : Measure Ξ±)
def AEDisjoint (s t : Se... | Mathlib/MeasureTheory/Measure/AEDisjoint.lean | 34 | 46 | theorem exists_null_pairwise_disjoint_diff [Countable ΞΉ] {s : ΞΉ β Set Ξ±}
(hd : Pairwise (AEDisjoint ΞΌ on s)) : β t : ΞΉ β Set Ξ±, (β i, MeasurableSet (t i)) β§
(β i, ΞΌ (t i) = 0) β§ Pairwise (Disjoint on fun i => s i \ t i) := by |
refine β¨fun i => toMeasurable ΞΌ (s i β© β j β ({i}αΆ : Set ΞΉ), s j), fun i =>
measurableSet_toMeasurable _ _, fun i => ?_, ?_β©
Β· simp only [measure_toMeasurable, inter_iUnion]
exact (measure_biUnion_null_iff <| to_countable _).2 fun j hj => hd (Ne.symm hj)
Β· simp only [Pairwise, disjoint_left, onFun, mem_d... | [
" β t, (β (i : ΞΉ), MeasurableSet (t i)) β§ (β (i : ΞΉ), ΞΌ (t i) = 0) β§ Pairwise (Disjoint on fun i => s i \\ t i)",
" ΞΌ ((fun i => toMeasurable ΞΌ (s i β© β j β {i}αΆ, s j)) i) = 0",
" ΞΌ (β i_1 β {i}αΆ, s i β© s i_1) = 0",
" Pairwise (Disjoint on fun i => s i \\ (fun i => toMeasurable ΞΌ (s i β© β j β {i}αΆ, s j)) i)",... | [] |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Ring.Divisibility.Basic
#align_import ring_theory.prime from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
section CancelCommMonoidWithZero
... | Mathlib/RingTheory/Prime.lean | 51 | 56 | theorem mul_eq_mul_prime_pow {x y a p : R} {n : β} (hp : Prime p) (hx : x * y = a * p ^ n) :
β (i j : β) (b c : R), i + j = n β§ a = b * c β§ x = b * p ^ i β§ y = c * p ^ j := by |
rcases mul_eq_mul_prime_prod (fun _ _ β¦ hp)
(show x * y = a * (range n).prod fun _ β¦ p by simpa) with
β¨t, u, b, c, htus, htu, rfl, rfl, rflβ©
exact β¨t.card, u.card, b, c, by rw [β card_union_of_disjoint htu, htus, card_range], by simpβ©
| [
" β t u b c, t βͺ u = s β§ Disjoint t u β§ a = b * c β§ x = b * β i β t, p i β§ y = c * β i β u, p i",
" β t u b c, t βͺ u = β
β§ Disjoint t u β§ a = b * c β§ x = b * β i β t, p i β§ y = c * β i β u, p i",
" β
βͺ β
= β
β§ Disjoint β
β
β§ a = x * y β§ x = x * β i β β
, p i β§ y = y * β i β β
, p i",
" β t u b c, t βͺ u = insert... | [
" β t u b c, t βͺ u = s β§ Disjoint t u β§ a = b * c β§ x = b * β i β t, p i β§ y = c * β i β u, p i",
" β t u b c, t βͺ u = β
β§ Disjoint t u β§ a = b * c β§ x = b * β i β t, p i β§ y = c * β i β u, p i",
" β
βͺ β
= β
β§ Disjoint β
β
β§ a = x * y β§ x = x * β i β β
, p i β§ y = y * β i β β
, p i",
" β t u b c, t βͺ u = insert... |
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Data.Rat.Floor
#align_import algebra.continued_fractions.computation.terminates_iff_rat from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b3... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 174 | 194 | theorem coe_stream_nth_rat_eq :
((IntFractPair.stream q n).map (mapFr (β)) : Option <| IntFractPair K) =
IntFractPair.stream v n := by |
induction n with
| zero =>
-- Porting note: was
-- simp [IntFractPair.stream, coe_of_rat_eq v_eq_q]
simp only [IntFractPair.stream, Option.map_some', coe_of_rat_eq v_eq_q]
| succ n IH =>
rw [v_eq_q] at IH
cases stream_q_nth_eq : IntFractPair.stream q n with
| none => simp [IntFractPair.st... | [
" mapFr Rat.cast (IntFractPair.of q) = IntFractPair.of v",
" Option.map (mapFr Rat.cast) (IntFractPair.stream q n) = IntFractPair.stream v n",
" Option.map (mapFr Rat.cast) (IntFractPair.stream q 0) = IntFractPair.stream v 0",
" Option.map (mapFr Rat.cast) (IntFractPair.stream q (n + 1)) = IntFractPair.stream... | [
" mapFr Rat.cast (IntFractPair.of q) = IntFractPair.of v"
] |
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 | 133 | 146 | theorem padicValNat_eq_maxPowDiv : @padicValNat = @maxPowDiv := by |
ext p n
by_cases h : 1 < p β§ 0 < n
Β· dsimp [padicValNat]
rw [dif_pos β¨Nat.ne_of_gt h.1,h.2β©, maxPowDiv_eq_multiplicity_get h.1 h.2]
Β· simp only [not_and_or,not_gt_eq,Nat.le_zero] at h
apply h.elim
Β· intro h
interval_cases p
Β· simp [Classical.em]
Β· dsimp [padicValNat, maxPowDiv]
... | [
" padicValNat p 0 = 0",
" padicValNat p 1 = 0",
" (if h : p β 1 β§ 0 < 1 then (multiplicity p 1).get β― else 0) = 0",
" (multiplicity p 1).get β― = 0",
" 0 = 0",
" padicValNat p p = 1",
" padicValNat p n = 0 β p = 1 β¨ n = 0 β¨ Β¬p β£ n",
" β(p.maxPowDiv n) = multiplicity p n",
" Β¬p ^ (p.maxPowDiv n + 1) β£... | [
" padicValNat p 0 = 0",
" padicValNat p 1 = 0",
" (if h : p β 1 β§ 0 < 1 then (multiplicity p 1).get β― else 0) = 0",
" (multiplicity p 1).get β― = 0",
" 0 = 0",
" padicValNat p p = 1",
" padicValNat p n = 0 β p = 1 β¨ n = 0 β¨ Β¬p β£ n",
" β(p.maxPowDiv n) = multiplicity p n",
" Β¬p ^ (p.maxPowDiv n + 1) β£... |
import Mathlib.CategoryTheory.Sites.Grothendieck
import Mathlib.CategoryTheory.Sites.Pretopology
import Mathlib.CategoryTheory.Limits.Lattice
import Mathlib.Topology.Sets.Opens
#align_import category_theory.sites.spaces from "leanprover-community/mathlib"@"b6fa3beb29f035598cf0434d919694c5e98091eb"
universe u
nam... | Mathlib/CategoryTheory/Sites/Spaces.lean | 78 | 86 | theorem pretopology_ofGrothendieck :
Pretopology.ofGrothendieck _ (Opens.grothendieckTopology T) = Opens.pretopology T := by |
apply le_antisymm
Β· intro X R hR x hx
rcases hR x hx with β¨U, f, β¨V, gβ, gβ, hgβ, _β©, hUβ©
exact β¨V, gβ, hgβ, gβ.le hUβ©
Β· intro X R hR x hx
rcases hR x hx with β¨U, f, hf, hUβ©
exact β¨U, f, Sieve.le_generate R U hf, hUβ©
| [
" β U f_1, (Sieve.pullback f S).arrows f_1 β§ y β U",
" (Sieve.pullback f S).arrows (homOfLE β―)",
" β U f, R.arrows f β§ x β U",
" β U f_1, Presieve.pullbackArrows f S f_1 β§ x β U",
" x β pullback g f",
" β U f, S.bind Ti f β§ x β U",
" Pretopology.ofGrothendieck (Opens T) (grothendieckTopology T) = pretop... | [
" β U f_1, (Sieve.pullback f S).arrows f_1 β§ y β U",
" (Sieve.pullback f S).arrows (homOfLE β―)",
" β U f, R.arrows f β§ x β U",
" β U f_1, Presieve.pullbackArrows f S f_1 β§ x β U",
" x β pullback g f",
" β U f, S.bind Ti f β§ x β U"
] |
import Mathlib.Algebra.Algebra.Quasispectrum
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
import Mathlib.Analysis.Complex.Liouville
import Mathlib.Analysis.Complex.Polynomial
import Mathlib.Analysis.Analytic.RadiusLiminf
import Mathlib.Topology.Algebra.Module.CharacterSpace
import Mathlib.Analysis.NormedSpace.Expon... | Mathlib/Analysis/NormedSpace/Spectrum.lean | 84 | 86 | theorem spectralRadius_zero : spectralRadius π (0 : A) = 0 := by |
nontriviality A
simp [spectralRadius]
| [
" spectralRadius π a = 0",
" spectralRadius π 0 = 0"
] | [
" spectralRadius π a = 0"
] |
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Data.FunLike.Fintype
open Function
namespace SimpleGraph
variable {V W X : Type*} (G : SimpleGraph V) (G' : SimpleGraph W) {u v : V}
protected def map (f : V βͺ W) (G : SimpleGraph V) : SimpleGraph W where
Adj := Relation.Map G.Adj f f
symm a b... | Mathlib/Combinatorics/SimpleGraph/Maps.lean | 154 | 155 | theorem map_comap_le (f : V βͺ W) (G : SimpleGraph W) : (G.comap f).map f β€ G := by |
rw [map_le_iff_le_comap]
| [
" Relation.Map G.Adj (βf) (βf) a b β Relation.Map G.Adj (βf) (βf) b a",
" Relation.Map G.Adj (βf) (βf) (f w) (f v)",
" Β¬Relation.Map G.Adj (βf) (βf) a a",
" False",
" (SimpleGraph.map f G).Adj (f a) (f b) β G.Adj a b",
" Monotone (SimpleGraph.map f)",
" (SimpleGraph.map f G').Adj (f u) (f v)",
" Simpl... | [
" Relation.Map G.Adj (βf) (βf) a b β Relation.Map G.Adj (βf) (βf) b a",
" Relation.Map G.Adj (βf) (βf) (f w) (f v)",
" Β¬Relation.Map G.Adj (βf) (βf) a a",
" False",
" (SimpleGraph.map f G).Adj (f a) (f b) β G.Adj a b",
" Monotone (SimpleGraph.map f)",
" (SimpleGraph.map f G').Adj (f u) (f v)",
" Simpl... |
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Data.FunLike.Fintype
open Function
namespace SimpleGraph
variable {V W X : Type*} (G : SimpleGraph V) (G' : SimpleGraph W) {u v : V}
protected def map (f : V βͺ W) (G : SimpleGraph V) : SimpleGraph W where
Adj := Relation.Map G.Adj f f
symm a b... | Mathlib/Combinatorics/SimpleGraph/Maps.lean | 76 | 78 | theorem map_monotone (f : V βͺ W) : Monotone (SimpleGraph.map f) := by |
rintro G G' h _ _ β¨u, v, ha, rfl, rflβ©
exact β¨_, _, h ha, rfl, rflβ©
| [
" Relation.Map G.Adj (βf) (βf) a b β Relation.Map G.Adj (βf) (βf) b a",
" Relation.Map G.Adj (βf) (βf) (f w) (f v)",
" Β¬Relation.Map G.Adj (βf) (βf) a a",
" False",
" (SimpleGraph.map f G).Adj (f a) (f b) β G.Adj a b",
" Monotone (SimpleGraph.map f)",
" (SimpleGraph.map f G').Adj (f u) (f v)"
] | [
" Relation.Map G.Adj (βf) (βf) a b β Relation.Map G.Adj (βf) (βf) b a",
" Relation.Map G.Adj (βf) (βf) (f w) (f v)",
" Β¬Relation.Map G.Adj (βf) (βf) a a",
" False",
" (SimpleGraph.map f G).Adj (f a) (f b) β G.Adj a b"
] |
import Mathlib.Dynamics.Ergodic.Ergodic
import Mathlib.MeasureTheory.Function.AEEqFun
open Function Set Filter MeasureTheory Topology TopologicalSpace
variable {Ξ± X : Type*} [MeasurableSpace Ξ±] {ΞΌ : MeasureTheory.Measure Ξ±}
theorem QuasiErgodic.ae_eq_const_of_ae_eq_comp_of_ae_rangeβ [Nonempty X] [MeasurableSpace... | Mathlib/Dynamics/Ergodic/Function.lean | 77 | 82 | theorem ae_eq_const_of_ae_eq_comp_ae {g : Ξ± β X} (h : QuasiErgodic f ΞΌ)
(hgm : AEStronglyMeasurable g ΞΌ) (hg_eq : g β f =α΅[ΞΌ] g) : β c, g =α΅[ΞΌ] const Ξ± c := by |
borelize X
rcases hgm.isSeparable_ae_range with β¨t, ht, hgtβ©
haveI := ht.secondCountableTopology
exact h.ae_eq_const_of_ae_eq_comp_of_ae_rangeβ hgt hgm.aemeasurable.nullMeasurable hg_eq
| [
" β c, g =αΆ [ae ΞΌ] const Ξ± c",
" β (U : Set X), MeasurableSet U β (βα΅ (x : Ξ±) βΞΌ, g x β U) β¨ βα΅ (x : Ξ±) βΞΌ, g x β U",
" f β»ΒΉ' (g β»ΒΉ' U) =αΆ [ae ΞΌ] g β»ΒΉ' U",
" x β f β»ΒΉ' (g β»ΒΉ' U) β x β g β»ΒΉ' U"
] | [
" β c, g =αΆ [ae ΞΌ] const Ξ± c",
" β (U : Set X), MeasurableSet U β (βα΅ (x : Ξ±) βΞΌ, g x β U) β¨ βα΅ (x : Ξ±) βΞΌ, g x β U",
" f β»ΒΉ' (g β»ΒΉ' U) =αΆ [ae ΞΌ] g β»ΒΉ' U",
" x β f β»ΒΉ' (g β»ΒΉ' U) β x β g β»ΒΉ' U"
] |
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b37058d0"
open Function Set
variable {Ξ± Ξ±' Ξ² Ξ²' Ξ³ Ξ³' Ξ΄ Ξ΄' Ξ΅ Ξ΅' ΞΆ ΞΆ' Ξ½ : Type*}
namespace Finset
variable [DecidableEq Ξ±'] [DecidableEq Ξ²'] [Decidabl... | Mathlib/Data/Finset/NAry.lean | 108 | 109 | theorem imageβ_subset_iff_left : imageβ f s t β u β β a β s, (t.image fun b => f a b) β u := by |
simp_rw [imageβ_subset_iff, image_subset_iff]
| [
" c β imageβ f s t β β a β s, β b β t, f a b = c",
" (imageβ f s t).card = s.card * t.card β InjOn (fun x => f x.1 x.2) (βs ΓΛ’ βt)",
" (imageβ f s t).card = (s ΓΛ’ t).card β InjOn (fun x => f x.1 x.2) β(s ΓΛ’ t)",
" f a b β imageβ f s t β a β s β§ b β t",
" imageβ f s t β imageβ f s' t'",
" image2 f βs βt β ... | [
" c β imageβ f s t β β a β s, β b β t, f a b = c",
" (imageβ f s t).card = s.card * t.card β InjOn (fun x => f x.1 x.2) (βs ΓΛ’ βt)",
" (imageβ f s t).card = (s ΓΛ’ t).card β InjOn (fun x => f x.1 x.2) β(s ΓΛ’ t)",
" f a b β imageβ f s t β a β s β§ b β t",
" imageβ f s t β imageβ f s' t'",
" image2 f βs βt β ... |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 87 | 88 | theorem mapβ_swap (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) (b : Option Ξ²) :
mapβ f a b = mapβ (fun a b => f b a) b a := by | cases a <;> cases b <;> rfl
| [
" mapβ f a b = Seq.seq (f <$> a) fun x => b",
" mapβ f none b = Seq.seq (f <$> none) fun x => b",
" mapβ f (some valβ) b = Seq.seq (f <$> some valβ) fun x => b",
" mapβ f a none = none",
" mapβ f none none = none",
" mapβ f (some valβ) none = none",
" mapβ f a (some b) = Option.map (fun a => f a b) a",
... | [
" mapβ f a b = Seq.seq (f <$> a) fun x => b",
" mapβ f none b = Seq.seq (f <$> none) fun x => b",
" mapβ f (some valβ) b = Seq.seq (f <$> some valβ) fun x => b",
" mapβ f a none = none",
" mapβ f none none = none",
" mapβ f (some valβ) none = none",
" mapβ f a (some b) = Option.map (fun a => f a b) a",
... |
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set Function Filter ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H ... | Mathlib/Geometry/Manifold/ContMDiff/Product.lean | 59 | 63 | theorem ContMDiffWithinAt.prod_mk {f : M β M'} {g : M β N'} (hf : ContMDiffWithinAt I I' n f s x)
(hg : ContMDiffWithinAt I J' n g s x) :
ContMDiffWithinAt I (I'.prod J') n (fun x => (f x, g x)) s x := by |
rw [contMDiffWithinAt_iff] at *
exact β¨hf.1.prod hg.1, hf.2.prod hg.2β©
| [
" ContMDiffWithinAt I (I'.prod J') n (fun x => (f x, g x)) s x",
" ContinuousWithinAt (fun x => (f x, g x)) s x β§\n ContDiffWithinAt π n (β(extChartAt (I'.prod J') (f x, g x)) β (fun x => (f x, g x)) β β(extChartAt I x).symm)\n (β(extChartAt I x).symm β»ΒΉ' s β© range βI) (β(extChartAt I x) x)"
] | [] |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 123 | 125 | theorem hasFiniteIntegral_iff_ofReal {f : Ξ± β β} (h : 0 β€α΅[ΞΌ] f) :
HasFiniteIntegral f ΞΌ β (β«β» a, ENNReal.ofReal (f a) βΞΌ) < β := by |
rw [HasFiniteIntegral, lintegral_nnnorm_eq_of_ae_nonneg h]
| [
" β«β» (a : Ξ±), ββf aββ βΞΌ = β«β» (a : Ξ±), edist (f a) 0 βΞΌ",
" β«β» (a : Ξ±), ENNReal.ofReal βf aβ βΞΌ = β«β» (a : Ξ±), edist (f a) 0 βΞΌ",
" β«β» (a : Ξ±), edist (f a) (g a) βΞΌ β€ β«β» (a : Ξ±), edist (f a) (h a) βΞΌ + β«β» (a : Ξ±), edist (g a) (h a) βΞΌ",
" β«β» (a : Ξ±), edist (f a) (g a) βΞΌ β€ β«β» (a : Ξ±), edist (f a) (h a) + edist... | [
" β«β» (a : Ξ±), ββf aββ βΞΌ = β«β» (a : Ξ±), edist (f a) 0 βΞΌ",
" β«β» (a : Ξ±), ENNReal.ofReal βf aβ βΞΌ = β«β» (a : Ξ±), edist (f a) 0 βΞΌ",
" β«β» (a : Ξ±), edist (f a) (g a) βΞΌ β€ β«β» (a : Ξ±), edist (f a) (h a) βΞΌ + β«β» (a : Ξ±), edist (g a) (h a) βΞΌ",
" β«β» (a : Ξ±), edist (f a) (g a) βΞΌ β€ β«β» (a : Ξ±), edist (f a) (h a) + edist... |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {Ξ± Ξ² : Type*} [Preorder Ξ±] [Preorder Ξ²]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 36 | 38 | theorem preimage_Iio (e : Ξ± βo Ξ²) (b : Ξ²) : e β»ΒΉ' Iio b = Iio (e.symm b) := by |
ext x
simp [β e.lt_iff_lt]
| [
" βe β»ΒΉ' Iic b = Iic (e.symm b)",
" x β βe β»ΒΉ' Iic b β x β Iic (e.symm b)",
" βe β»ΒΉ' Ici b = Ici (e.symm b)",
" x β βe β»ΒΉ' Ici b β x β Ici (e.symm b)",
" βe β»ΒΉ' Iio b = Iio (e.symm b)",
" x β βe β»ΒΉ' Iio b β x β Iio (e.symm b)"
] | [
" βe β»ΒΉ' Iic b = Iic (e.symm b)",
" x β βe β»ΒΉ' Iic b β x β Iic (e.symm b)",
" βe β»ΒΉ' Ici b = Ici (e.symm b)",
" x β βe β»ΒΉ' Ici b β x β Ici (e.symm b)"
] |
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