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 | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Algebra.DirectSum.Algebra
suppress_compilation
open scoped TensorProduct DirectSum
variable {R ΞΉ A B : Type*}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 85 | 90 | theorem gradedCommAux_lof_tmul (i j : ΞΉ) (a : π i) (b : β¬ j) :
gradedCommAux R π β¬ (lof R _ πβ¬ (i, j) (a ββ b)) =
(-1 : β€Λ£)^(j * i) β’ lof R _ β¬π (j, i) (b ββ a) := by |
rw [gradedCommAux]
dsimp
simp [mul_comm i j]
| 1,742 |
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Algebra.DirectSum.Algebra
suppress_compilation
open scoped TensorProduct DirectSum
variable {R ΞΉ A B : Type*}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 93 | 98 | theorem gradedCommAux_comp_gradedCommAux :
gradedCommAux R π β¬ ββ gradedCommAux R β¬ π = LinearMap.id := by |
ext i a b
dsimp
rw [gradedCommAux_lof_tmul, LinearMap.map_smul_of_tower, gradedCommAux_lof_tmul, smul_smul,
mul_comm i.2 i.1, Int.units_mul_self, one_smul]
| 1,742 |
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Algebra.DirectSum.Algebra
suppress_compilation
open scoped TensorProduct DirectSum
variable {R ΞΉ A B : Type*}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 111 | 114 | theorem gradedComm_symm : (gradedComm R π β¬).symm = gradedComm R β¬ π := by |
rw [gradedComm, gradedComm, LinearEquiv.trans_symm, LinearEquiv.symm_symm]
ext
rfl
| 1,742 |
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Algebra.DirectSum.Algebra
suppress_compilation
open scoped TensorProduct DirectSum
variable {R ΞΉ A B : Type*}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 116 | 124 | theorem gradedComm_of_tmul_of (i j : ΞΉ) (a : π i) (b : β¬ j) :
gradedComm R π β¬ (lof R _ π i a ββ lof R _ β¬ j b) =
(-1 : β€Λ£)^(j * i) β’ (lof R _ β¬ _ b ββ lof R _ π _ a) := by |
rw [gradedComm]
dsimp only [LinearEquiv.trans_apply, LinearEquiv.ofLinear_apply]
rw [TensorProduct.directSum_lof_tmul_lof, gradedCommAux_lof_tmul, Units.smul_def,
-- Note: #8386 specialized `map_smul` to `LinearEquiv.map_smul` to avoid timeouts.
zsmul_eq_smul_cast R, LinearEquiv.map_smul, TensorProduct.d... | 1,742 |
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Algebra.DirectSum.Algebra
suppress_compilation
open scoped TensorProduct DirectSum
variable {R ΞΉ A B : Type*}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 126 | 135 | theorem gradedComm_tmul_of_zero (a : β¨ i, π i) (b : β¬ 0) :
gradedComm R π β¬ (a ββ lof R _ β¬ 0 b) = lof R _ β¬ _ b ββ a := by |
suffices
(gradedComm R π β¬).toLinearMap ββ
(TensorProduct.mk R (β¨ i, π i) (β¨ i, β¬ i)).flip (lof R _ β¬ 0 b) =
TensorProduct.mk R _ _ (lof R _ β¬ 0 b) from
DFunLike.congr_fun this a
ext i a
dsimp
rw [gradedComm_of_tmul_of, zero_mul, uzpow_zero, one_smul]
| 1,742 |
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Algebra.DirectSum.Algebra
suppress_compilation
open scoped TensorProduct DirectSum
variable {R ΞΉ A B : Type*}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 137 | 145 | theorem gradedComm_of_zero_tmul (a : π 0) (b : β¨ i, β¬ i) :
gradedComm R π β¬ (lof R _ π 0 a ββ b) = b ββ lof R _ π _ a := by |
suffices
(gradedComm R π β¬).toLinearMap ββ (TensorProduct.mk R (β¨ i, π i) (β¨ i, β¬ i)) (lof R _ π 0 a) =
(TensorProduct.mk R _ _).flip (lof R _ π 0 a) from
DFunLike.congr_fun this b
ext i b
dsimp
rw [gradedComm_of_tmul_of, mul_zero, uzpow_zero, one_smul]
| 1,742 |
import Mathlib.LinearAlgebra.TensorProduct.Graded.External
import Mathlib.RingTheory.GradedAlgebra.Basic
import Mathlib.GroupTheory.GroupAction.Ring
suppress_compilation
open scoped TensorProduct
variable {R ΞΉ A B : Type*}
variable [CommSemiring ΞΉ] [Module ΞΉ (Additive β€Λ£)] [DecidableEq ΞΉ]
variable [CommRing R] [R... | Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean | 133 | 135 | theorem auxEquiv_one : auxEquiv R π β¬ 1 = 1 := by |
rw [β of_one, Algebra.TensorProduct.one_def, auxEquiv_tmul π β¬, DirectSum.decompose_one,
DirectSum.decompose_one, Algebra.TensorProduct.one_def]
| 1,743 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.LinearAlgebra.TensorProduct.Graded.Internal
import Mathlib.LinearAlgebra.QuadraticForm.Prod
suppress_compilation
variable {R Mβ Mβ N : Type*}
variable [CommRing R] [AddCommGroup Mβ] [AddCommGroup Mβ] [AddCommGroup N]
variable [Module R Mβ] [Module... | Mathlib/LinearAlgebra/CliffordAlgebra/Prod.lean | 101 | 104 | theorem map_mul_map_eq_neg_of_isOrtho_of_mem_evenOdd_one
(hmβ : mβ β evenOdd Qβ 1) (hmβ : mβ β evenOdd Qβ 1) :
map fβ mβ * map fβ mβ = - map fβ mβ * map fβ mβ := by |
simp [map_mul_map_of_isOrtho_of_mem_evenOdd _ _ hf _ _ hmβ hmβ]
| 1,744 |
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.RingTheory.Adjoin.Basic
#align_import topology.algebra.algebra from "leanprover-community/mathlib"@"43afc5ad87891456c57b5a183e3e617d67c2b1db"
open scoped Classical
open Set TopologicalSpace Algebra
open sc... | Mathlib/Topology/Algebra/Algebra.lean | 42 | 44 | theorem continuous_algebraMap [ContinuousSMul R A] : Continuous (algebraMap R A) := by |
rw [algebraMap_eq_smul_one']
exact continuous_id.smul continuous_const
| 1,745 |
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.RingTheory.Adjoin.Basic
#align_import topology.algebra.algebra from "leanprover-community/mathlib"@"43afc5ad87891456c57b5a183e3e617d67c2b1db"
open scoped Classical
open Set TopologicalSpace Algebra
open sc... | Mathlib/Topology/Algebra/Algebra.lean | 47 | 51 | theorem continuous_algebraMap_iff_smul [TopologicalSemiring A] :
Continuous (algebraMap R A) β Continuous fun p : R Γ A => p.1 β’ p.2 := by |
refine β¨fun h => ?_, fun h => have : ContinuousSMul R A := β¨hβ©; continuous_algebraMap _ _β©
simp only [Algebra.smul_def]
exact (h.comp continuous_fst).mul continuous_snd
| 1,745 |
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.RingTheory.Adjoin.Basic
#align_import topology.algebra.algebra from "leanprover-community/mathlib"@"43afc5ad87891456c57b5a183e3e617d67c2b1db"
open scoped Classical
open Set TopologicalSpace Algebra
open sc... | Mathlib/Topology/Algebra/Algebra.lean | 110 | 111 | theorem Subalgebra.isClosed_topologicalClosure (s : Subalgebra R A) :
IsClosed (s.topologicalClosure : Set A) := by | convert @isClosed_closure A s _
| 1,745 |
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.RingTheory.Adjoin.Basic
#align_import topology.algebra.algebra from "leanprover-community/mathlib"@"43afc5ad87891456c57b5a183e3e617d67c2b1db"
open scoped Classical
open Set TopologicalSpace Algebra
open sc... | Mathlib/Topology/Algebra/Algebra.lean | 130 | 137 | theorem Subalgebra.topologicalClosure_comap_homeomorph (s : Subalgebra R A) {B : Type*}
[TopologicalSpace B] [Ring B] [TopologicalRing B] [Algebra R B] (f : B ββ[R] A) (f' : B ββ A)
(w : (f : B β A) = f') : s.topologicalClosure.comap f = (s.comap f).topologicalClosure := by |
apply SetLike.ext'
simp only [Subalgebra.topologicalClosure_coe]
simp only [Subalgebra.coe_comap, Subsemiring.coe_comap, AlgHom.coe_toRingHom]
rw [w]
exact f'.preimage_closure _
| 1,745 |
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.Star
#align_import topology.algebra.star_subalgebra from "leanprover-community/mathlib"@"b7f5a77fa29ad9a3ccc484109b0d7534178e7ecd"
open scoped Classical
open Set TopologicalSpace
open scoped Classical
... | Mathlib/Topology/Algebra/StarSubalgebra.lean | 122 | 127 | theorem _root_.Subalgebra.topologicalClosure_star_comm (s : Subalgebra R A) :
(star s).topologicalClosure = star s.topologicalClosure := by |
suffices β t : Subalgebra R A, (star t).topologicalClosure β€ star t.topologicalClosure from
le_antisymm (this s) (by simpa only [star_star] using Subalgebra.star_mono (this (star s)))
exact fun t => (star t).topologicalClosure_minimal (Subalgebra.star_mono subset_closure)
(isClosed_closure.preimage continu... | 1,746 |
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.Star
#align_import topology.algebra.star_subalgebra from "leanprover-community/mathlib"@"b7f5a77fa29ad9a3ccc484109b0d7534178e7ecd"
open scoped Classical
open Set TopologicalSpace
open scoped Classical
... | Mathlib/Topology/Algebra/StarSubalgebra.lean | 146 | 163 | theorem _root_.StarAlgHom.ext_topologicalClosure [T2Space B] {S : StarSubalgebra R A}
{Ο Ο : S.topologicalClosure βββ[R] B} (hΟ : Continuous Ο) (hΟ : Continuous Ο)
(h :
Ο.comp (inclusion (le_topologicalClosure S)) = Ο.comp (inclusion (le_topologicalClosure S))) :
Ο = Ο := by |
rw [DFunLike.ext'_iff]
have : Dense (Set.range <| inclusion (le_topologicalClosure S)) := by
refine embedding_subtype_val.toInducing.dense_iff.2 fun x => ?_
convert show βx β closure (S : Set A) from x.prop
rw [β Set.range_comp]
exact
Set.ext fun y =>
β¨by
rintro β¨y, rflβ©
... | 1,746 |
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 39 | 41 | theorem tendsto_const_div_atTop_nhds_zero_nat (C : β) :
Tendsto (fun n : β β¦ C / n) atTop (π 0) := by |
simpa only [mul_zero] using tendsto_const_nhds.mul tendsto_inverse_atTop_nhds_zero_nat
| 1,747 |
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 51 | 54 | theorem NNReal.tendsto_inverse_atTop_nhds_zero_nat :
Tendsto (fun n : β β¦ (n : ββ₯0)β»ΒΉ) atTop (π 0) := by |
rw [β NNReal.tendsto_coe]
exact _root_.tendsto_inverse_atTop_nhds_zero_nat
| 1,747 |
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 59 | 61 | theorem NNReal.tendsto_const_div_atTop_nhds_zero_nat (C : ββ₯0) :
Tendsto (fun n : β β¦ C / n) atTop (π 0) := by |
simpa using tendsto_const_nhds.mul NNReal.tendsto_inverse_atTop_nhds_zero_nat
| 1,747 |
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 74 | 79 | theorem NNReal.tendsto_algebraMap_inverse_atTop_nhds_zero_nat (π : Type*) [Semiring π]
[Algebra ββ₯0 π] [TopologicalSpace π] [ContinuousSMul ββ₯0 π] :
Tendsto (algebraMap ββ₯0 π β fun n : β β¦ (n : ββ₯0)β»ΒΉ) atTop (π 0) := by |
convert (continuous_algebraMap ββ₯0 π).continuousAt.tendsto.comp
tendsto_inverse_atTop_nhds_zero_nat
rw [map_zero]
| 1,747 |
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 97 | 113 | theorem tendsto_natCast_div_add_atTop {π : Type*} [DivisionRing π] [TopologicalSpace π]
[CharZero π] [Algebra β π] [ContinuousSMul β π] [TopologicalDivisionRing π] (x : π) :
Tendsto (fun n : β β¦ (n : π) / (n + x)) atTop (π 1) := by |
convert Tendsto.congr' ((eventually_ne_atTop 0).mp (eventually_of_forall fun n hn β¦ _)) _
Β· exact fun n : β β¦ 1 / (1 + x / n)
Β· field_simp [Nat.cast_ne_zero.mpr hn]
Β· have : π (1 : π) = π (1 / (1 + x * (0 : π))) := by
rw [mul_zero, add_zero, div_one]
rw [this]
refine tendsto_const_nhds.div (t... | 1,747 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.specific_limits.floor_pow from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Filter Finset
open Topology
| Mathlib/Analysis/SpecificLimits/FloorPow.lean | 28 | 182 | theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : β β β) (l : β)
(hmono : Monotone u)
(hlim : β a : β, 1 < a β β c : β β β, (βαΆ n in atTop, (c (n + 1) : β) β€ a * c n) β§
Tendsto c atTop atTop β§ Tendsto (fun n => u (c n) / c n) atTop (π l)) :
Tendsto (fun n => u n / n) atTop (π l) := b... |
/- To check the result up to some `Ξ΅ > 0`, we use a sequence `c` for which the ratio
`c (N+1) / c N` is bounded by `1 + Ξ΅`. Sandwiching a given `n` between two consecutive values of
`c`, say `c N` and `c (N+1)`, one can then bound `u n / n` from above by `u (c N) / c (N - 1)`
and from below by `u (c (N -... | 1,748 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.RCLike.Basic
open Set Algebra Filter
open scoped Topology
variable (π : Type*) [RCLike π]
| Mathlib/Analysis/SpecificLimits/RCLike.lean | 19 | 22 | theorem RCLike.tendsto_inverse_atTop_nhds_zero_nat :
Tendsto (fun n : β => (n : π)β»ΒΉ) atTop (π 0) := by |
convert tendsto_algebraMap_inverse_atTop_nhds_zero_nat π
simp
| 1,749 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 167 | 167 | theorem map_one_add (x : β) : f (1 + x) = 1 + f x := by | rw [add_comm, map_add_one, add_comm 1]
| 1,750 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 213 | 214 | theorem units_inv_apply_apply (f : CircleDeg1LiftΛ£) (x : β) :
(fβ»ΒΉ : CircleDeg1LiftΛ£) (f x) = x := by | simp only [β mul_apply, f.inv_mul, coe_one, id]
| 1,750 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 218 | 219 | theorem units_apply_inv_apply (f : CircleDeg1LiftΛ£) (x : β) :
f ((fβ»ΒΉ : CircleDeg1LiftΛ£) x) = x := by | simp only [β mul_apply, f.mul_inv, coe_one, id]
| 1,750 |
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.hofer from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open Topology
open Filter Finset
local notation "d" => dist
#noalign pos_div_pow_pos
| Mathlib/Analysis/Hofer.lean | 33 | 104 | theorem hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (Ξ΅ : β) (Ξ΅_pos : 0 < Ξ΅)
{Ο : X β β} (cont : Continuous Ο) (nonneg : β y, 0 β€ Ο y) : β Ξ΅' > 0, β x' : X,
Ξ΅' β€ Ξ΅ β§ d x' x β€ 2 * Ξ΅ β§ Ξ΅ * Ο x β€ Ξ΅' * Ο x' β§ β y, d x' y β€ Ξ΅' β Ο y β€ 2 * Ο x' := by |
by_contra H
have reformulation : β (x') (k : β), Ξ΅ * Ο x β€ Ξ΅ / 2 ^ k * Ο x' β 2 ^ k * Ο x β€ Ο x' := by
intro x' k
rw [div_mul_eq_mul_div, le_div_iff, mul_assoc, mul_le_mul_left Ξ΅_pos, mul_comm]
positivity
-- Now let's specialize to `Ξ΅/2^k`
replace H : β k : β, β x', d x' x β€ 2 * Ξ΅ β§ 2 ^ k * Ο x β€ Ο... | 1,751 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 53 | 53 | theorem one_sub_K_pos' (hf : ContractingWith K f) : (0 : ββ₯0β) < 1 - K := by | simp [hf.1]
| 1,752 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 62 | 64 | theorem one_sub_K_ne_top : (1 : ββ₯0β) - K β β := by |
norm_cast
exact ENNReal.coe_ne_top
| 1,752 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 68 | 76 | theorem edist_inequality (hf : ContractingWith K f) {x y} (h : edist x y β β) :
edist x y β€ (edist x (f x) + edist y (f y)) / (1 - K) :=
suffices edist x y β€ edist x (f x) + edist y (f y) + K * edist x y by
rwa [ENNReal.le_div_iff_mul_le (Or.inl hf.one_sub_K_ne_zero) (Or.inl one_sub_K_ne_top),
mul_comm,... | rw [edist_comm y, add_right_comm]
_ β€ edist x (f x) + edist y (f y) + K * edist x y := add_le_add le_rfl (hf.2 _ _)
| 1,752 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 79 | 81 | theorem edist_le_of_fixedPoint (hf : ContractingWith K f) {x y} (h : edist x y β β)
(hy : IsFixedPt f y) : edist x y β€ edist x (f x) / (1 - K) := by |
simpa only [hy.eq, edist_self, add_zero] using hf.edist_inequality h
| 1,752 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 84 | 87 | theorem eq_or_edist_eq_top_of_fixedPoints (hf : ContractingWith K f) {x y} (hx : IsFixedPt f x)
(hy : IsFixedPt f y) : x = y β¨ edist x y = β := by |
refine or_iff_not_imp_right.2 fun h β¦ edist_le_zero.1 ?_
simpa only [hx.eq, edist_self, add_zero, ENNReal.zero_div] using hf.edist_le_of_fixedPoint h hy
| 1,752 |
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.Ordered
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.GDelta
import Mathlib.Analysis.NormedSpace.FunctionSeries
import Mathlib.Analysis.SpecificLimits.Basic
#align_import topology.urysohns_lemma from "lea... | Mathlib/Topology/UrysohnsLemma.lean | 161 | 166 | theorem approx_of_mem_C (c : CU P) (n : β) {x : X} (hx : x β c.C) : c.approx n x = 0 := by |
induction' n with n ihn generalizing c
Β· exact indicator_of_not_mem (fun (hU : x β c.UαΆ) => hU <| c.subset hx) _
Β· simp only [approx]
rw [ihn, ihn, midpoint_self]
exacts [c.subset_right_C hx, hx]
| 1,753 |
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.Ordered
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.GDelta
import Mathlib.Analysis.NormedSpace.FunctionSeries
import Mathlib.Analysis.SpecificLimits.Basic
#align_import topology.urysohns_lemma from "lea... | Mathlib/Topology/UrysohnsLemma.lean | 169 | 175 | theorem approx_of_nmem_U (c : CU P) (n : β) {x : X} (hx : x β c.U) : c.approx n x = 1 := by |
induction' n with n ihn generalizing c
Β· rw [β mem_compl_iff] at hx
exact indicator_of_mem hx _
Β· simp only [approx]
rw [ihn, ihn, midpoint_self]
exacts [hx, fun hU => hx <| c.left_U_subset hU]
| 1,753 |
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.Ordered
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.GDelta
import Mathlib.Analysis.NormedSpace.FunctionSeries
import Mathlib.Analysis.SpecificLimits.Basic
#align_import topology.urysohns_lemma from "lea... | Mathlib/Topology/UrysohnsLemma.lean | 178 | 182 | theorem approx_nonneg (c : CU P) (n : β) (x : X) : 0 β€ c.approx n x := by |
induction' n with n ihn generalizing c
Β· exact indicator_nonneg (fun _ _ => zero_le_one) _
Β· simp only [approx, midpoint_eq_smul_add, invOf_eq_inv]
refine mul_nonneg (inv_nonneg.2 zero_le_two) (add_nonneg ?_ ?_) <;> apply ihn
| 1,753 |
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.Ordered
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.GDelta
import Mathlib.Analysis.NormedSpace.FunctionSeries
import Mathlib.Analysis.SpecificLimits.Basic
#align_import topology.urysohns_lemma from "lea... | Mathlib/Topology/UrysohnsLemma.lean | 185 | 192 | theorem approx_le_one (c : CU P) (n : β) (x : X) : c.approx n x β€ 1 := by |
induction' n with n ihn generalizing c
Β· exact indicator_apply_le' (fun _ => le_rfl) fun _ => zero_le_one
Β· simp only [approx, midpoint_eq_smul_add, invOf_eq_inv, smul_eq_mul, β div_eq_inv_mul]
have := add_le_add (ihn (left c)) (ihn (right c))
set_option tactic.skipAssignedInstances false in
norm_num... | 1,753 |
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.Ordered
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.GDelta
import Mathlib.Analysis.NormedSpace.FunctionSeries
import Mathlib.Analysis.SpecificLimits.Basic
#align_import topology.urysohns_lemma from "lea... | Mathlib/Topology/UrysohnsLemma.lean | 199 | 207 | theorem approx_le_approx_of_U_sub_C {cβ cβ : CU P} (h : cβ.U β cβ.C) (nβ nβ : β) (x : X) :
cβ.approx nβ x β€ cβ.approx nβ x := by |
by_cases hx : x β cβ.U
Β· calc
approx nβ cβ x = 0 := approx_of_mem_C _ _ (h hx)
_ β€ approx nβ cβ x := approx_nonneg _ _ _
Β· calc
approx nβ cβ x β€ 1 := approx_le_one _ _ _
_ = approx nβ cβ x := (approx_of_nmem_U _ _ hx).symm
| 1,753 |
import Mathlib.CategoryTheory.Adjunction.Reflective
import Mathlib.Topology.StoneCech
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.CategoryTheory.Elementwise
#align_import topol... | Mathlib/Topology/Category/CompHaus/Basic.lean | 123 | 135 | theorem isIso_of_bijective {X Y : CompHaus.{u}} (f : X βΆ Y) (bij : Function.Bijective f) :
IsIso f := by |
let E := Equiv.ofBijective _ bij
have hE : Continuous E.symm := by
rw [continuous_iff_isClosed]
intro S hS
rw [β E.image_eq_preimage]
exact isClosedMap f S hS
refine β¨β¨β¨E.symm, hEβ©, ?_, ?_β©β©
Β· ext x
apply E.symm_apply_apply
Β· ext x
apply E.apply_symm_apply
| 1,754 |
import Mathlib.CategoryTheory.Monad.Types
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.CategoryTheory.Equivalence
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.Data.Set.Constructions
#align_import topology.category.Compactum from "leanprov... | Mathlib/Topology/Category/Compactum.lean | 143 | 146 | theorem str_incl (X : Compactum) (x : X) : X.str (X.incl x) = x := by |
change ((Ξ² ).Ξ·.app _ β« X.a) _ = _
rw [Monad.Algebra.unit]
rfl
| 1,755 |
import Mathlib.CategoryTheory.Monad.Types
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.CategoryTheory.Equivalence
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.Data.Set.Constructions
#align_import topology.category.Compactum from "leanprov... | Mathlib/Topology/Category/Compactum.lean | 150 | 154 | theorem str_hom_commute (X Y : Compactum) (f : X βΆ Y) (xs : Ultrafilter X) :
f (X.str xs) = Y.str (map f xs) := by |
change (X.a β« f.f) _ = _
rw [β f.h]
rfl
| 1,755 |
import Mathlib.CategoryTheory.Monad.Types
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.CategoryTheory.Equivalence
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.Data.Set.Constructions
#align_import topology.category.Compactum from "leanprov... | Mathlib/Topology/Category/Compactum.lean | 158 | 162 | theorem join_distrib (X : Compactum) (uux : Ultrafilter (Ultrafilter X)) :
X.str (X.join uux) = X.str (map X.str uux) := by |
change ((Ξ² ).ΞΌ.app _ β« X.a) _ = _
rw [Monad.Algebra.assoc]
rfl
| 1,755 |
import Mathlib.CategoryTheory.Monad.Types
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.CategoryTheory.Equivalence
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.Data.Set.Constructions
#align_import topology.category.Compactum from "leanprov... | Mathlib/Topology/Category/Compactum.lean | 173 | 185 | theorem isClosed_iff {X : Compactum} (S : Set X) :
IsClosed S β β F : Ultrafilter X, S β F β X.str F β S := by |
rw [β isOpen_compl_iff]
constructor
Β· intro cond F h
by_contra c
specialize cond F c
rw [compl_mem_iff_not_mem] at cond
contradiction
Β· intro h1 F h2
specialize h1 F
cases' F.mem_or_compl_mem S with h h
exacts [absurd (h1 h) h2, h]
| 1,755 |
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Preserves.Finite
namespace CompHaus
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
universe u w
open Categor... | Mathlib/Topology/Category/CompHaus/Limits.lean | 131 | 134 | theorem pullback_fst_eq :
CompHaus.pullback.fst f g = (pullbackIsoPullback f g).hom β« Limits.pullback.fst := by |
dsimp [pullbackIsoPullback]
simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_Ο]
| 1,756 |
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Preserves.Finite
namespace CompHaus
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
universe u w
open Categor... | Mathlib/Topology/Category/CompHaus/Limits.lean | 136 | 139 | theorem pullback_snd_eq :
CompHaus.pullback.snd f g = (pullbackIsoPullback f g).hom β« Limits.pullback.snd := by |
dsimp [pullbackIsoPullback]
simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_Ο]
| 1,756 |
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Preserves.Finite
namespace CompHaus
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
universe u w
open Categor... | Mathlib/Topology/Category/CompHaus/Limits.lean | 205 | 207 | theorem Sigma.ΞΉ_comp_toFiniteCoproduct (a : Ξ±) :
(Limits.Sigma.ΞΉ X a) β« (coproductIsoCoproduct X).inv = finiteCoproduct.ΞΉ X a := by |
simp [coproductIsoCoproduct]
| 1,756 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.Metrizable.Basic
#align_import topology.metric_space.metrizable from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter... | Mathlib/Topology/Metrizable/Urysohn.lean | 37 | 106 | theorem exists_inducing_l_infty : β f : X β β βα΅ β, Inducing f := by |
-- Choose a countable basis, and consider the set `s` of pairs of set `(U, V)` such that `U β B`,
-- `V β B`, and `closure U β V`.
rcases exists_countable_basis X with β¨B, hBc, -, hBβ©
let s : Set (Set X Γ Set X) := { UV β B ΓΛ’ B | closure UV.1 β UV.2 }
-- `s` is a countable set.
haveI : Encodable s := ((hB... | 1,757 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.ShrinkingLemma
import Mathlib.Topology.UrysohnsLemma
#align_import topology.partition_of_unity from "leanprover-... | Mathlib/Topology/PartitionOfUnity.lean | 161 | 164 | theorem exists_pos {x : X} (hx : x β s) : β i, 0 < f i x := by |
have H := f.sum_eq_one hx
contrapose! H
simpa only [fun i => (H i).antisymm (f.nonneg i x), finsum_zero] using zero_ne_one
| 1,758 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.ShrinkingLemma
import Mathlib.Topology.UrysohnsLemma
#align_import topology.partition_of_unity from "leanprover-... | Mathlib/Topology/PartitionOfUnity.lean | 188 | 190 | theorem mem_finsupport (xβ : X) {i} :
i β Ο.finsupport xβ β i β support fun i β¦ Ο i xβ := by |
simp only [finsupport, mem_support, Finite.mem_toFinset, mem_setOf_eq]
| 1,758 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.ShrinkingLemma
import Mathlib.Topology.UrysohnsLemma
#align_import topology.partition_of_unity from "leanprover-... | Mathlib/Topology/PartitionOfUnity.lean | 193 | 196 | theorem coe_finsupport (xβ : X) :
(Ο.finsupport xβ : Set ΞΉ) = support fun i β¦ Ο i xβ := by |
ext
rw [Finset.mem_coe, mem_finsupport]
| 1,758 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.ShrinkingLemma
import Mathlib.Topology.UrysohnsLemma
#align_import topology.partition_of_unity from "leanprover-... | Mathlib/Topology/PartitionOfUnity.lean | 200 | 201 | theorem sum_finsupport (hxβ : xβ β s) : β i β Ο.finsupport xβ, Ο i xβ = 1 := by |
rw [β Ο.sum_eq_one hxβ, finsum_eq_sum_of_support_subset _ (Ο.coe_finsupport xβ).superset]
| 1,758 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.ShrinkingLemma
import Mathlib.Topology.UrysohnsLemma
#align_import topology.partition_of_unity from "leanprover-... | Mathlib/Topology/PartitionOfUnity.lean | 203 | 212 | theorem sum_finsupport' (hxβ : xβ β s) {I : Finset ΞΉ} (hI : Ο.finsupport xβ β I) :
β i β I, Ο i xβ = 1 := by |
classical
rw [β Finset.sum_sdiff hI, Ο.sum_finsupport hxβ]
suffices β i β I \ Ο.finsupport xβ, (Ο i) xβ = β i β I \ Ο.finsupport xβ, 0 by
rw [this, add_left_eq_self, Finset.sum_const_zero]
apply Finset.sum_congr rfl
rintro x hx
simp only [Finset.mem_sdiff, Ο.mem_finsupport, mem_support, Classical.not_n... | 1,758 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.ShrinkingLemma
import Mathlib.Topology.UrysohnsLemma
#align_import topology.partition_of_unity from "leanprover-... | Mathlib/Topology/PartitionOfUnity.lean | 214 | 220 | theorem sum_finsupport_smul_eq_finsum {M : Type*} [AddCommGroup M] [Module β M] (Ο : ΞΉ β X β M) :
β i β Ο.finsupport xβ, Ο i xβ β’ Ο i xβ = βαΆ i, Ο i xβ β’ Ο i xβ := by |
apply (finsum_eq_sum_of_support_subset _ _).symm
have : (fun i β¦ (Ο i) xβ β’ Ο i xβ) = (fun i β¦ (Ο i) xβ) β’ (fun i β¦ Ο i xβ) :=
funext fun _ => (Pi.smul_apply' _ _ _).symm
rw [Ο.coe_finsupport xβ, this, support_smul]
exact inter_subset_left
| 1,758 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.ShrinkingLemma
import Mathlib.Topology.UrysohnsLemma
#align_import topology.partition_of_unity from "leanprover-... | Mathlib/Topology/PartitionOfUnity.lean | 229 | 234 | theorem finite_tsupport : {i | xβ β tsupport (Ο i)}.Finite := by |
rcases Ο.locallyFinite xβ with β¨t, t_in, htβ©
apply ht.subset
rintro i hi
simp only [inter_comm]
exact mem_closure_iff_nhds.mp hi t t_in
| 1,758 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.ShrinkingLemma
import Mathlib.Topology.UrysohnsLemma
#align_import topology.partition_of_unity from "leanprover-... | Mathlib/Topology/PartitionOfUnity.lean | 244 | 249 | theorem eventually_fintsupport_subset :
βαΆ y in π xβ, Ο.fintsupport y β Ο.fintsupport xβ := by |
apply (Ο.locallyFinite.closure.eventually_subset (fun _ β¦ isClosed_closure) xβ).mono
intro y hy z hz
rw [PartitionOfUnity.mem_fintsupport_iff] at *
exact hy hz
| 1,758 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.ShrinkingLemma
import Mathlib.Topology.UrysohnsLemma
#align_import topology.partition_of_unity from "leanprover-... | Mathlib/Topology/PartitionOfUnity.lean | 289 | 295 | theorem exists_finset_nhd' {s : Set X} (Ο : PartitionOfUnity ΞΉ X s) (xβ : X) :
β I : Finset ΞΉ, (βαΆ x in π[s] xβ, β i β I, Ο i x = 1) β§
βαΆ x in π xβ, support (Ο Β· x) β I := by |
rcases Ο.locallyFinite.exists_finset_support xβ with β¨I, hIβ©
refine β¨I, eventually_nhdsWithin_iff.mpr (hI.mono fun x hx x_in β¦ ?_), hIβ©
have : βαΆ i : ΞΉ, Ο i x = β i β I, Ο i x := finsum_eq_sum_of_support_subset _ hx
rwa [eq_comm, Ο.sum_eq_one x_in] at this
| 1,758 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.ShrinkingLemma
import Mathlib.Topology.UrysohnsLemma
#align_import topology.partition_of_unity from "leanprover-... | Mathlib/Topology/PartitionOfUnity.lean | 297 | 301 | theorem exists_finset_nhd (Ο : PartitionOfUnity ΞΉ X univ) (xβ : X) :
β I : Finset ΞΉ, βαΆ x in π xβ, β i β I, Ο i x = 1 β§ support (Ο Β· x) β I := by |
rcases Ο.exists_finset_nhd' xβ with β¨I, Hβ©
use I
rwa [nhdsWithin_univ, β eventually_and] at H
| 1,758 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.Topology.Sets.Compacts
#align_import topology.metric_space.closeds from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topo... | Mathlib/Topology/MetricSpace/Closeds.lean | 56 | 69 | theorem continuous_infEdist_hausdorffEdist :
Continuous fun p : Ξ± Γ Closeds Ξ± => infEdist p.1 p.2 := by |
refine continuous_of_le_add_edist 2 (by simp) ?_
rintro β¨x, sβ© β¨y, tβ©
calc
infEdist x s β€ infEdist x t + hausdorffEdist (t : Set Ξ±) s :=
infEdist_le_infEdist_add_hausdorffEdist
_ β€ infEdist y t + edist x y + hausdorffEdist (t : Set Ξ±) s :=
(add_le_add_right infEdist_le_infEdist_add_edist _)
... | 1,759 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.Topology.Sets.Compacts
#align_import topology.metric_space.closeds from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topo... | Mathlib/Topology/MetricSpace/Closeds.lean | 74 | 84 | theorem isClosed_subsets_of_isClosed (hs : IsClosed s) :
IsClosed { t : Closeds Ξ± | (t : Set Ξ±) β s } := by |
refine isClosed_of_closure_subset fun
(t : Closeds Ξ±) (ht : t β closure {t : Closeds Ξ± | (t : Set Ξ±) β s}) (x : Ξ±) (hx : x β t) => ?_
have : x β closure s := by
refine mem_closure_iff.2 fun Ξ΅ Ξ΅pos => ?_
obtain β¨u : Closeds Ξ±, hu : u β {t : Closeds Ξ± | (t : Set Ξ±) β s}, Dtu : edist t u < Ξ΅β© :=
mem... | 1,759 |
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Order.Filter.ModEq
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.List.TFAE
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.specific_lim... | Mathlib/Analysis/SpecificLimits/Normed.lean | 62 | 68 | theorem tendsto_norm_zpow_nhdsWithin_0_atTop {π : Type*} [NormedDivisionRing π] {m : β€}
(hm : m < 0) :
Tendsto (fun x : π β¦ βx ^ mβ) (π[β ] 0) atTop := by |
rcases neg_surjective m with β¨m, rflβ©
rw [neg_lt_zero] at hm; lift m to β using hm.le; rw [Int.natCast_pos] at hm
simp only [norm_pow, zpow_neg, zpow_natCast, β inv_pow]
exact (tendsto_pow_atTop hm.ne').comp NormedField.tendsto_norm_inverse_nhdsWithin_0_atTop
| 1,760 |
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Order.Filter.ModEq
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.List.TFAE
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.specific_lim... | Mathlib/Analysis/SpecificLimits/Normed.lean | 72 | 77 | theorem tendsto_zero_smul_of_tendsto_zero_of_bounded {ΞΉ π πΈ : Type*} [NormedDivisionRing π]
[NormedAddCommGroup πΈ] [Module π πΈ] [BoundedSMul π πΈ] {l : Filter ΞΉ} {Ξ΅ : ΞΉ β π} {f : ΞΉ β πΈ}
(hΞ΅ : Tendsto Ξ΅ l (π 0)) (hf : Filter.IsBoundedUnder (Β· β€ Β·) l (norm β f)) :
Tendsto (Ξ΅ β’ f) l (π 0) := by |
rw [β isLittleO_one_iff π] at hΞ΅ β’
simpa using IsLittleO.smul_isBigO hΞ΅ (hf.isBigO_const (one_ne_zero : (1 : π) β 0))
| 1,760 |
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Order.Filter.ModEq
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.List.TFAE
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.specific_lim... | Mathlib/Analysis/SpecificLimits/Normed.lean | 81 | 86 | theorem continuousAt_zpow {π : Type*} [NontriviallyNormedField π] {m : β€} {x : π} :
ContinuousAt (fun x β¦ x ^ m) x β x β 0 β¨ 0 β€ m := by |
refine β¨?_, continuousAt_zpowβ _ _β©
contrapose!; rintro β¨rfl, hmβ© hc
exact not_tendsto_atTop_of_tendsto_nhds (hc.tendsto.mono_left nhdsWithin_le_nhds).norm
(tendsto_norm_zpow_nhdsWithin_0_atTop hm)
| 1,760 |
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Order.Filter.ModEq
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.List.TFAE
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.specific_lim... | Mathlib/Analysis/SpecificLimits/Normed.lean | 90 | 92 | theorem continuousAt_inv {π : Type*} [NontriviallyNormedField π] {x : π} :
ContinuousAt Inv.inv x β x β 0 := by |
simpa [(zero_lt_one' β€).not_le] using @continuousAt_zpow _ _ (-1) x
| 1,760 |
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Order.Filter.ModEq
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.List.TFAE
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.specific_lim... | Mathlib/Analysis/SpecificLimits/Normed.lean | 111 | 114 | theorem isLittleO_pow_pow_of_abs_lt_left {rβ rβ : β} (h : |rβ| < |rβ|) :
(fun n : β β¦ rβ ^ n) =o[atTop] fun n β¦ rβ ^ n := by |
refine (IsLittleO.of_norm_left ?_).of_norm_right
exact (isLittleO_pow_pow_of_lt_left (abs_nonneg rβ) h).congr (pow_abs rβ) (pow_abs rβ)
| 1,760 |
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Order.Filter.ModEq
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.List.TFAE
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.specific_lim... | Mathlib/Analysis/SpecificLimits/Normed.lean | 132 | 189 | theorem TFAE_exists_lt_isLittleO_pow (f : β β β) (R : β) :
TFAE
[β a β Ioo (-R) R, f =o[atTop] (a ^ Β·), β a β Ioo 0 R, f =o[atTop] (a ^ Β·),
β a β Ioo (-R) R, f =O[atTop] (a ^ Β·), β a β Ioo 0 R, f =O[atTop] (a ^ Β·),
β a < R, β C : β, (0 < C β¨ 0 < R) β§ β n, |f n| β€ C * a ^ n,
β a β Ioo 0... |
have A : Ico 0 R β Ioo (-R) R :=
fun x hx β¦ β¨(neg_lt_zero.2 (hx.1.trans_lt hx.2)).trans_le hx.1, hx.2β©
have B : Ioo 0 R β Ioo (-R) R := Subset.trans Ioo_subset_Ico_self A
-- First we prove that 1-4 are equivalent using 2 β 3 β 4, 1 β 3, and 2 β 1
tfae_have 1 β 3
Β· exact fun β¨a, ha, Hβ© β¦ β¨a, ha, H.isBigOβ©... | 1,760 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.normed.group.controlled_closure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Finset
open Topology
variable {G : Type*} [NormedAddCommGroup G] [CompleteSpace... | Mathlib/Analysis/Normed/Group/ControlledClosure.lean | 32 | 106 | theorem controlled_closure_of_complete {f : NormedAddGroupHom G H} {K : AddSubgroup H} {C Ξ΅ : β}
(hC : 0 < C) (hΞ΅ : 0 < Ξ΅) (hyp : f.SurjectiveOnWith K C) :
f.SurjectiveOnWith K.topologicalClosure (C + Ξ΅) := by |
rintro (h : H) (h_in : h β K.topologicalClosure)
-- We first get rid of the easy case where `h = 0`.
by_cases hyp_h : h = 0
Β· rw [hyp_h]
use 0
simp
/- The desired preimage will be constructed as the sum of a series. Convergence of
the series will be guaranteed by completeness of `G`. We first wri... | 1,761 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.normed.group.controlled_closure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Finset
open Topology
variable {G : Type*} [NormedAddCommGroup G] [CompleteSpace... | Mathlib/Analysis/Normed/Group/ControlledClosure.lean | 116 | 125 | theorem controlled_closure_range_of_complete {f : NormedAddGroupHom G H} {K : Type*}
[SeminormedAddCommGroup K] {j : NormedAddGroupHom K H} (hj : β x, βj xβ = βxβ) {C Ξ΅ : β}
(hC : 0 < C) (hΞ΅ : 0 < Ξ΅) (hyp : β k, β g, f g = j k β§ βgβ β€ C * βkβ) :
f.SurjectiveOnWith j.range.topologicalClosure (C + Ξ΅) := by |
replace hyp : β h β j.range, β g, f g = h β§ βgβ β€ C * βhβ := by
intro h h_in
rcases (j.mem_range _).mp h_in with β¨k, rflβ©
rw [hj]
exact hyp k
exact controlled_closure_of_complete hC hΞ΅ hyp
| 1,761 |
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
noncomputable... | Mathlib/Analysis/Analytic/Basic.lean | 102 | 105 | theorem partialSum_continuous (p : FormalMultilinearSeries π E F) (n : β) :
Continuous (p.partialSum n) := by |
unfold partialSum -- Porting note: added
continuity
| 1,762 |
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
noncomputable... | Mathlib/Analysis/Analytic/Basic.lean | 187 | 202 | theorem isLittleO_of_lt_radius (h : βr < p.radius) :
β a β Ioo (0 : β) 1, (fun n => βp nβ * (r : β) ^ n) =o[atTop] (a ^ Β·) := by |
have := (TFAE_exists_lt_isLittleO_pow (fun n => βp nβ * (r : β) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => βp nβ * (r : β) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with β¨t, C, hC, rtβ©
rw [ENNReal.coe_lt_coe, β NNReal.coe_lt_coe] at ... | 1,762 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.analytic.radius_liminf from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
variable {π : Type*} [NontriviallyNormedField π] {E : Type*} [NormedAddCommGroup E]
[NormedSpa... | Mathlib/Analysis/Analytic/RadiusLiminf.lean | 35 | 61 | theorem radius_eq_liminf :
p.radius = liminf (fun n => (1 / (βp nββ ^ (1 / (n : β)) : ββ₯0) : ββ₯0β)) atTop := by |
-- Porting note: added type ascription to make elaborated statement match Lean 3 version
have :
β (r : ββ₯0) {n : β},
0 < n β ((r : ββ₯0β) β€ 1 / β(βp nββ ^ (1 / (n : β))) β βp nββ * r ^ n β€ 1) := by
intro r n hn
have : 0 < (n : β) := Nat.cast_pos.2 hn
conv_lhs =>
rw [one_div, ENNReal.le_i... | 1,763 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
noncomputable section
variable {π : Type*} {E F G H : Type*}
open Filter List
open scoped Topol... | Mathlib/Analysis/Analytic/Composition.lean | 106 | 114 | theorem applyComposition_ones (p : FormalMultilinearSeries π E F) (n : β) :
p.applyComposition (Composition.ones n) = fun v i =>
p 1 fun _ => v (Fin.castLE (Composition.length_le _) i) := by |
funext v i
apply p.congr (Composition.ones_blocksFun _ _)
intro j hjn hj1
obtain rfl : j = 0 := by omega
refine congr_arg v ?_
rw [Fin.ext_iff, Fin.coe_castLE, Composition.ones_embedding, Fin.val_mk]
| 1,764 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
noncomputable section
variable {π : Type*} {E F G H : Type*}
open Filter List
open scoped Topol... | Mathlib/Analysis/Analytic/Composition.lean | 117 | 127 | theorem applyComposition_single (p : FormalMultilinearSeries π E F) {n : β} (hn : 0 < n)
(v : Fin n β E) : p.applyComposition (Composition.single n hn) v = fun _j => p n v := by |
ext j
refine p.congr (by simp) fun i hi1 hi2 => ?_
dsimp
congr 1
convert Composition.single_embedding hn β¨i, hi2β© using 1
cases' j with j_val j_property
have : j_val = 0 := le_bot_iff.1 (Nat.lt_succ_iff.1 j_property)
congr!
simp
| 1,764 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
noncomputable section
variable {π : Type*} {E F G H : Type*}
open Filter List
open scoped Topol... | Mathlib/Analysis/Analytic/Composition.lean | 131 | 134 | theorem removeZero_applyComposition (p : FormalMultilinearSeries π E F) {n : β}
(c : Composition n) : p.removeZero.applyComposition c = p.applyComposition c := by |
ext v i
simp [applyComposition, zero_lt_one.trans_le (c.one_le_blocksFun i), removeZero_of_pos]
| 1,764 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
noncomputable section
variable {π : Type*} {E F G H : Type*}
open Filter List
open scoped Topol... | Mathlib/Analysis/Analytic/Composition.lean | 140 | 162 | theorem applyComposition_update (p : FormalMultilinearSeries π E F) {n : β} (c : Composition n)
(j : Fin n) (v : Fin n β E) (z : E) :
p.applyComposition c (Function.update v j z) =
Function.update (p.applyComposition c v) (c.index j)
(p (c.blocksFun (c.index j))
(Function.update (v β c.... |
ext k
by_cases h : k = c.index j
Β· rw [h]
let r : Fin (c.blocksFun (c.index j)) β Fin n := c.embedding (c.index j)
simp only [Function.update_same]
change p (c.blocksFun (c.index j)) (Function.update v j z β r) = _
let j' := c.invEmbedding j
suffices B : Function.update v j z β r = Function.u... | 1,764 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
noncomputable section
variable {π : Type*} {E F G H : Type*}
open Filter List
open scoped Topol... | Mathlib/Analysis/Analytic/Composition.lean | 166 | 169 | theorem compContinuousLinearMap_applyComposition {n : β} (p : FormalMultilinearSeries π F G)
(f : E βL[π] F) (c : Composition n) (v : Fin n β E) :
(p.compContinuousLinearMap f).applyComposition c v = p.applyComposition c (f β v) := by |
simp (config := {unfoldPartialApp := true}) [applyComposition]; rfl
| 1,764 |
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.NormedSpace.Completion
#align_import analysis.analytic.uniqueness from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
variable {π : Type*} [NontriviallyNormedField π] {E : Type... | Mathlib/Analysis/Analytic/Uniqueness.lean | 32 | 70 | theorem eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux [CompleteSpace F] {f : E β F} {U : Set E}
(hf : AnalyticOn π f U) (hU : IsPreconnected U) {zβ : E} (hβ : zβ β U) (hfzβ : f =αΆ [π zβ] 0) :
EqOn f 0 U := by |
/- Let `u` be the set of points around which `f` vanishes. It is clearly open. We have to show
that its limit points in `U` still belong to it, from which the inclusion `U β u` will follow
by connectedness. -/
let u := {x | f =αΆ [π x] 0}
suffices main : closure u β© U β u by
have Uu : U β u :=
h... | 1,765 |
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.NormedSpace.Completion
#align_import analysis.analytic.uniqueness from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
variable {π : Type*} [NontriviallyNormedField π] {E : Type... | Mathlib/Analysis/Analytic/Uniqueness.lean | 77 | 89 | theorem eqOn_zero_of_preconnected_of_eventuallyEq_zero {f : E β F} {U : Set E}
(hf : AnalyticOn π f U) (hU : IsPreconnected U) {zβ : E} (hβ : zβ β U) (hfzβ : f =αΆ [π zβ] 0) :
EqOn f 0 U := by |
let F' := UniformSpace.Completion F
set e : F βL[π] F' := UniformSpace.Completion.toComplL
have : AnalyticOn π (e β f) U := fun x hx => (e.analyticAt _).comp (hf x hx)
have A : EqOn (e β f) 0 U := by
apply eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux this hU hβ
filter_upwards [hfzβ] with x hx
... | 1,765 |
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.NormedSpace.Completion
#align_import analysis.analytic.uniqueness from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
variable {π : Type*} [NontriviallyNormedField π] {E : Type... | Mathlib/Analysis/Analytic/Uniqueness.lean | 96 | 101 | theorem eqOn_of_preconnected_of_eventuallyEq {f g : E β F} {U : Set E} (hf : AnalyticOn π f U)
(hg : AnalyticOn π g U) (hU : IsPreconnected U) {zβ : E} (hβ : zβ β U) (hfg : f =αΆ [π zβ] g) :
EqOn f g U := by |
have hfg' : f - g =αΆ [π zβ] 0 := hfg.mono fun z h => by simp [h]
simpa [sub_eq_zero] using fun z hz =>
(hf.sub hg).eqOn_zero_of_preconnected_of_eventuallyEq_zero hU hβ hfg' hz
| 1,765 |
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e228"
open scoped Classical Topology
open Finset Filter
namespace FormalMultilinearSeries
variable {π : Type*} [NontriviallyNormedField π] {E : Type*} ... | Mathlib/Analysis/Analytic/Inverse.lean | 68 | 69 | theorem leftInv_coeff_zero (p : FormalMultilinearSeries π E F) (i : E βL[π] F) :
p.leftInv i 0 = 0 := by | rw [leftInv]
| 1,766 |
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e228"
open scoped Classical Topology
open Finset Filter
namespace FormalMultilinearSeries
variable {π : Type*} [NontriviallyNormedField π] {E : Type*} ... | Mathlib/Analysis/Analytic/Inverse.lean | 73 | 74 | theorem leftInv_coeff_one (p : FormalMultilinearSeries π E F) (i : E βL[π] F) :
p.leftInv i 1 = (continuousMultilinearCurryFin1 π F E).symm i.symm := by | rw [leftInv]
| 1,766 |
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e228"
open scoped Classical Topology
open Finset Filter
namespace FormalMultilinearSeries
variable {π : Type*} [NontriviallyNormedField π] {E : Type*} ... | Mathlib/Analysis/Analytic/Inverse.lean | 79 | 92 | theorem leftInv_removeZero (p : FormalMultilinearSeries π E F) (i : E βL[π] F) :
p.removeZero.leftInv i = p.leftInv i := by |
ext1 n
induction' n using Nat.strongRec' with n IH
match n with
| 0 => simp -- if one replaces `simp` with `refl`, the proof times out in the kernel.
| 1 => simp -- TODO: why?
| n + 2 =>
simp only [leftInv, neg_inj]
refine Finset.sum_congr rfl fun c cuniv => ?_
rcases c with β¨c, hcβ©
ext v
... | 1,766 |
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e228"
open scoped Classical Topology
open Finset Filter
namespace FormalMultilinearSeries
variable {π : Type*} [NontriviallyNormedField π] {E : Type*} ... | Mathlib/Analysis/Analytic/Inverse.lean | 97 | 148 | theorem leftInv_comp (p : FormalMultilinearSeries π E F) (i : E βL[π] F)
(h : p 1 = (continuousMultilinearCurryFin1 π E F).symm i) : (leftInv p i).comp p = id π E := by |
ext (n v)
match n with
| 0 =>
simp only [leftInv_coeff_zero, ContinuousMultilinearMap.zero_apply, id_apply_ne_one, Ne,
not_false_iff, zero_ne_one, comp_coeff_zero']
| 1 =>
simp only [leftInv_coeff_one, comp_coeff_one, h, id_apply_one, ContinuousLinearEquiv.coe_apply,
ContinuousLinearEquiv.s... | 1,766 |
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e228"
open scoped Classical Topology
open Finset Filter
namespace FormalMultilinearSeries
variable {π : Type*} [NontriviallyNormedField π] {E : Type*} ... | Mathlib/Analysis/Analytic/Inverse.lean | 177 | 178 | theorem rightInv_coeff_zero (p : FormalMultilinearSeries π E F) (i : E βL[π] F) :
p.rightInv i 0 = 0 := by | rw [rightInv]
| 1,766 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 39 | 44 | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by |
refine h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right ?_)
refine isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, ?_, EventuallyEq.rflβ©
refine (continuous_id.sub continuous_const).norm.tendsto' _ _ ?_
rw [_root_.id, sub_self, norm_zero]
| 1,767 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 91 | 101 | theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f) p.derivSeries x r := by |
refine .congr (f := fun z β¦ continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1)) ?_
fun z hz β¦ ?_
Β· refine continuousMultilinearCurryFin1 π E F
|>.toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall ?_
simpa using ((p.hasFPowerSeriesOnBall_changeOrigin 1
(h.r_... | 1,767 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 105 | 109 | theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by |
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
| 1,767 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 113 | 122 | theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by |
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert ContinuousLin... | 1,767 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 314 | 346 | theorem changeOrigin_toFormalMultilinearSeries [DecidableEq ΞΉ] :
continuousMultilinearCurryFin1 π (β i, E i) F (f.toFormalMultilinearSeries.changeOrigin x 1) =
f.linearDeriv x := by |
ext y
rw [continuousMultilinearCurryFin1_apply, linearDeriv_apply,
changeOrigin, FormalMultilinearSeries.sum]
cases isEmpty_or_nonempty ΞΉ
Β· have (l) : 1 + l β Fintype.card ΞΉ := by
rw [add_comm, Fintype.card_eq_zero]; exact Nat.succ_ne_zero _
simp_rw [Fintype.sum_empty, changeOriginSeries_suppor... | 1,767 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 449 | 458 | theorem derivSeries_apply_diag (n : β) (x : E) :
derivSeries p n (fun _ β¦ x) x = (n + 1) β’ p (n + 1) fun _ β¦ x := by |
simp only [derivSeries, compFormalMultilinearSeries_apply, changeOriginSeries,
compContinuousMultilinearMap_coe, ContinuousLinearEquiv.coe_coe, LinearIsometryEquiv.coe_coe,
Function.comp_apply, ContinuousMultilinearMap.sum_apply, map_sum, coe_sum', Finset.sum_apply,
continuousMultilinearCurryFin1_apply, ... | 1,767 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 469 | 474 | theorem iteratedFDeriv_zero_apply_diag : iteratedFDeriv π 0 f x = p 0 := by |
ext
convert (h.hasSum <| EMetric.mem_ball_self h.r_pos).tsum_eq.symm
Β· rw [iteratedFDeriv_zero_apply, add_zero]
Β· rw [tsum_eq_single 0 fun n hn β¦ by haveI := NeZero.mk hn; exact (p n).map_zero]
exact congr(p 0 $(Subsingleton.elim _ _))
| 1,767 |
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.Dslope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Analytic.Uniqueness
#align_import analysis.analytic.isolated_zeros from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
open sco... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 44 | 45 | theorem hasSum_at_zero (a : β β E) : HasSum (fun n => (0 : π) ^ n β’ a n) (a 0) := by |
convert hasSum_single (Ξ± := E) 0 fun b h β¦ _ <;> simp [*]
| 1,768 |
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.Dslope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Analytic.Uniqueness
#align_import analysis.analytic.isolated_zeros from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
open sco... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 48 | 62 | theorem exists_hasSum_smul_of_apply_eq_zero (hs : HasSum (fun m => z ^ m β’ a m) s)
(ha : β k < n, a k = 0) : β t : E, z ^ n β’ t = s β§ HasSum (fun m => z ^ m β’ a (m + n)) t := by |
obtain rfl | hn := n.eq_zero_or_pos
Β· simpa
by_cases h : z = 0
Β· have : s = 0 := hs.unique (by simpa [ha 0 hn, h] using hasSum_at_zero a)
exact β¨a n, by simp [h, hn.ne', this], by simpa [h] using hasSum_at_zero fun m => a (m + n)β©
Β· refine β¨(z ^ n)β»ΒΉ β’ s, by field_simp [smul_smul], ?_β©
have h1 : β i ... | 1,768 |
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.Dslope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Analytic.Uniqueness
#align_import analysis.analytic.isolated_zeros from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
open sco... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 69 | 80 | theorem has_fpower_series_dslope_fslope (hp : HasFPowerSeriesAt f p zβ) :
HasFPowerSeriesAt (dslope f zβ) p.fslope zβ := by |
have hpd : deriv f zβ = p.coeff 1 := hp.deriv
have hp0 : p.coeff 0 = f zβ := hp.coeff_zero 1
simp only [hasFPowerSeriesAt_iff, apply_eq_pow_smul_coeff, coeff_fslope] at hp β’
refine hp.mono fun x hx => ?_
by_cases h : x = 0
Β· convert hasSum_single (Ξ± := E) 0 _ <;> intros <;> simp [*]
Β· have hxx : β n : β,... | 1,768 |
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.Dslope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Analytic.Uniqueness
#align_import analysis.analytic.isolated_zeros from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
open sco... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 83 | 87 | theorem has_fpower_series_iterate_dslope_fslope (n : β) (hp : HasFPowerSeriesAt f p zβ) :
HasFPowerSeriesAt ((swap dslope zβ)^[n] f) (fslope^[n] p) zβ := by |
induction' n with n ih generalizing f p
Β· exact hp
Β· simpa using ih (has_fpower_series_dslope_fslope hp)
| 1,768 |
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.Dslope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Analytic.Uniqueness
#align_import analysis.analytic.isolated_zeros from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
open sco... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 90 | 93 | theorem iterate_dslope_fslope_ne_zero (hp : HasFPowerSeriesAt f p zβ) (h : p β 0) :
(swap dslope zβ)^[p.order] f zβ β 0 := by |
rw [β coeff_zero (has_fpower_series_iterate_dslope_fslope p.order hp) 1]
simpa [coeff_eq_zero] using apply_order_ne_zero h
| 1,768 |
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.box_integral.box.subbox_induction from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Finset Function Filter Metric Classical Topology Filter ENNReal
noncomputable... | Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean | 53 | 62 | theorem mem_splitCenterBox {s : Set ΞΉ} {y : ΞΉ β β} :
y β I.splitCenterBox s β y β I β§ β i, (I.lower i + I.upper i) / 2 < y i β i β s := by |
simp only [splitCenterBox, mem_def, β forall_and]
refine forall_congr' fun i β¦ ?_
dsimp only [Set.piecewise]
split_ifs with hs <;> simp only [hs, iff_true_iff, iff_false_iff, not_lt]
exacts [β¨fun H β¦ β¨β¨(left_lt_add_div_two.2 (I.lower_lt_upper i)).trans H.1, H.2β©, H.1β©,
fun H β¦ β¨H.2, H.1.2β©β©,
β¨fun H... | 1,769 |
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.box_integral.box.subbox_induction from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Finset Function Filter Metric Classical Topology Filter ENNReal
noncomputable... | Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean | 69 | 75 | theorem disjoint_splitCenterBox (I : Box ΞΉ) {s t : Set ΞΉ} (h : s β t) :
Disjoint (I.splitCenterBox s : Set (ΞΉ β β)) (I.splitCenterBox t) := by |
rw [disjoint_iff_inf_le]
rintro y β¨hs, htβ©; apply h
ext i
rw [mem_coe, mem_splitCenterBox] at hs ht
rw [β hs.2, β ht.2]
| 1,769 |
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.box_integral.box.subbox_induction from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Finset Function Filter Metric Classical Topology Filter ENNReal
noncomputable... | Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean | 95 | 97 | theorem iUnion_coe_splitCenterBox (I : Box ΞΉ) : β s, (I.splitCenterBox s : Set (ΞΉ β β)) = I := by |
ext x
simp
| 1,769 |
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.box_integral.box.subbox_induction from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Finset Function Filter Metric Classical Topology Filter ENNReal
noncomputable... | Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean | 101 | 103 | theorem upper_sub_lower_splitCenterBox (I : Box ΞΉ) (s : Set ΞΉ) (i : ΞΉ) :
(I.splitCenterBox s).upper i - (I.splitCenterBox s).lower i = (I.upper i - I.lower i) / 2 := by |
by_cases i β s <;> field_simp [splitCenterBox] <;> field_simp [mul_two, two_mul]
| 1,769 |
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.box_integral.box.subbox_induction from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Finset Function Filter Metric Classical Topology Filter ENNReal
noncomputable... | Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean | 122 | 170 | theorem subbox_induction_on' {p : Box ΞΉ β Prop} (I : Box ΞΉ)
(H_ind : β J β€ I, (β s, p (splitCenterBox J s)) β p J)
(H_nhds : β z β Box.Icc I, β U β π[Box.Icc I] z, β J β€ I, β (m : β), z β Box.Icc J β
Box.Icc J β U β (β i, J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) β p J) :
p I := by |
by_contra hpI
-- First we use `H_ind` to construct a decreasing sequence of boxes such that `β m, Β¬p (J m)`.
replace H_ind := fun J hJ β¦ not_imp_not.2 (H_ind J hJ)
simp only [exists_imp, not_forall] at H_ind
choose! s hs using H_ind
set J : β β Box ΞΉ := fun m β¦ (fun J β¦ splitCenterBox J (s J))^[m] I
have... | 1,769 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.UrysohnsBounded
#align_import topology.tietze_extension from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section TietzeExten... | Mathlib/Topology/TietzeExtension.lean | 73 | 77 | theorem ContinuousMap.exists_extension (f : C(Xβ, Y)) :
β (g : C(X, Y)), g.comp β¨e, he.continuousβ© = f := by |
let e' : Xβ ββ Set.range e := Homeomorph.ofEmbedding _ he.toEmbedding
obtain β¨g, hgβ© := (f.comp e'.symm).exists_restrict_eq he.isClosed_range
exact β¨g, by ext x; simpa using congr($(hg) β¨e' x, x, rflβ©)β©
| 1,770 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.UrysohnsBounded
#align_import topology.tietze_extension from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section TietzeExten... | Mathlib/Topology/TietzeExtension.lean | 96 | 100 | theorem ContinuousMap.exists_forall_mem_restrict_eq {Y : Type v} [TopologicalSpace Y] (f : C(s, Y))
{t : Set Y} (hf : β x, f x β t) [ht : TietzeExtension.{u, v} t] :
β (g : C(X, Y)), (β x, g x β t) β§ g.restrict s = f := by |
obtain β¨g, hgβ© := mk _ (map_continuous f |>.codRestrict hf) |>.exists_restrict_eq hs
exact β¨comp β¨Subtype.val, by continuityβ© g, by simp, by ext x; congrm(($(hg) x : Y))β©
| 1,770 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.UrysohnsBounded
#align_import topology.tietze_extension from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section TietzeExten... | Mathlib/Topology/TietzeExtension.lean | 108 | 112 | theorem ContinuousMap.exists_extension_forall_mem {Y : Type v} [TopologicalSpace Y] (f : C(Xβ, Y))
{t : Set Y} (hf : β x, f x β t) [ht : TietzeExtension.{u, v} t] :
β (g : C(X, Y)), (β x, g x β t) β§ g.comp β¨e, he.continuousβ© = f := by |
obtain β¨g, hgβ© := mk _ (map_continuous f |>.codRestrict hf) |>.exists_extension he
exact β¨comp β¨Subtype.val, by continuityβ© g, by simp, by ext x; congrm(($(hg) x : Y))β©
| 1,770 |
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