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 | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
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import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Manifold.Algebra.LieGroup
import Mathlib.Geometry.Manifol... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 98 | 104 | theorem contDiffOn_stereoToFun :
ContDiffOn โ โค (stereoToFun v) {x : E | innerSL _ v x โ (1 : โ)} := by |
refine ContDiffOn.smul ?_ (orthogonalProjection (โ โ v)แฎ).contDiff.contDiffOn
refine contDiff_const.contDiffOn.div ?_ ?_
ยท exact (contDiff_const.sub (innerSL โ v).contDiff).contDiffOn
ยท intro x h h'
exact h (sub_eq_zero.mp h').symm
| 5 | 148.413159 | 2 | 1.5 | 6 | 1,585 |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Manifold.Algebra.LieGroup
import Mathlib.Geometry.Manifol... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 131 | 142 | theorem stereoInvFunAux_mem (hv : โvโ = 1) {w : E} (hw : w โ (โ โ v)แฎ) :
stereoInvFunAux v w โ sphere (0 : E) 1 := by |
have hโ : (0 : โ) < โwโ ^ 2 + 4 := by positivity
suffices โ(4 : โ) โข w + (โwโ ^ 2 - 4) โข vโ = โwโ ^ 2 + 4 by
simp only [mem_sphere_zero_iff_norm, norm_smul, Real.norm_eq_abs, abs_inv, this,
abs_of_pos hโ, stereoInvFunAux_apply, inv_mul_cancel hโ.ne']
suffices โ(4 : โ) โข w + (โwโ ^ 2 - 4) โข vโ ^ 2 = (โw... | 10 | 22,026.465795 | 2 | 1.5 | 6 | 1,585 |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Manifold.Algebra.LieGroup
import Mathlib.Geometry.Manifol... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 145 | 160 | theorem hasFDerivAt_stereoInvFunAux (v : E) :
HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id โ E) 0 := by |
have hโ : HasFDerivAt (fun w : E => โwโ ^ 2) (0 : E โL[โ] โ) 0 := by
convert (hasStrictFDerivAt_norm_sq (0 : E)).hasFDerivAt
simp
have hโ : HasFDerivAt (fun w : E => (โwโ ^ 2 + 4)โปยน) (0 : E โL[โ] โ) 0 := by
convert (hasFDerivAt_inv _).comp _ (hโ.add (hasFDerivAt_const 4 0)) <;> simp
have hโ : HasFDer... | 14 | 1,202,604.284165 | 2 | 1.5 | 6 | 1,585 |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Manifold.Algebra.LieGroup
import Mathlib.Geometry.Manifol... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 163 | 167 | theorem hasFDerivAt_stereoInvFunAux_comp_coe (v : E) :
HasFDerivAt (stereoInvFunAux v โ ((โ) : (โ โ v)แฎ โ E)) (โ โ v)แฎ.subtypeL 0 := by |
have : HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id โ E) ((โ โ v)แฎ.subtypeL 0) :=
hasFDerivAt_stereoInvFunAux v
convert this.comp (0 : (โ โ v)แฎ) (by apply ContinuousLinearMap.hasFDerivAt)
| 3 | 20.085537 | 1 | 1.5 | 6 | 1,585 |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Manifold.Algebra.LieGroup
import Mathlib.Geometry.Manifol... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 170 | 179 | theorem contDiff_stereoInvFunAux : ContDiff โ โค (stereoInvFunAux v) := by |
have hโ : ContDiff โ โค fun w : E => โwโ ^ 2 := contDiff_norm_sq โ
have hโ : ContDiff โ โค fun w : E => (โwโ ^ 2 + 4)โปยน := by
refine (hโ.add contDiff_const).inv ?_
intro x
nlinarith
have hโ : ContDiff โ โค fun w => (4 : โ) โข w + (โwโ ^ 2 - 4) โข v := by
refine (contDiff_const.smul contDiff_id).add ?_... | 9 | 8,103.083928 | 2 | 1.5 | 6 | 1,585 |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Manifold.Algebra.LieGroup
import Mathlib.Geometry.Manifol... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 385 | 386 | theorem sphere_ext_iff (u v : sphere (0 : E) 1) : u = v โ โช(u : E), vโซ_โ = 1 := by |
simp [Subtype.ext_iff, inner_eq_one_iff_of_norm_one]
| 1 | 2.718282 | 0 | 1.5 | 6 | 1,585 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.jensen from "leanprover-community/mathlib"@"bfad3f455b388fbcc14c49d0cac884f774f14d20"
open Finset LinearMap Set
open scoped Classical
open Convex Pointwise
variable {๏ฟฝ... | Mathlib/Analysis/Convex/Jensen.lean | 52 | 58 | theorem ConvexOn.map_centerMass_le (hf : ConvexOn ๐ s f) (hโ : โ i โ t, 0 โค w i)
(hโ : 0 < โ i โ t, w i) (hmem : โ i โ t, p i โ s) :
f (t.centerMass w p) โค t.centerMass w (f โ p) := by |
have hmem' : โ i โ t, (p i, (f โ p) i) โ { p : E ร ฮฒ | p.1 โ s โง f p.1 โค p.2 } := fun i hi =>
โจhmem i hi, le_rflโฉ
convert (hf.convex_epigraph.centerMass_mem hโ hโ hmem').2 <;>
simp only [centerMass, Function.comp, Prod.smul_fst, Prod.fst_sum, Prod.smul_snd, Prod.snd_sum]
| 4 | 54.59815 | 2 | 1.5 | 2 | 1,586 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.jensen from "leanprover-community/mathlib"@"bfad3f455b388fbcc14c49d0cac884f774f14d20"
open Finset LinearMap Set
open scoped Classical
open Convex Pointwise
variable {๏ฟฝ... | Mathlib/Analysis/Convex/Jensen.lean | 69 | 72 | theorem ConvexOn.map_sum_le (hf : ConvexOn ๐ s f) (hโ : โ i โ t, 0 โค w i) (hโ : โ i โ t, w i = 1)
(hmem : โ i โ t, p i โ s) : f (โ i โ t, w i โข p i) โค โ i โ t, w i โข f (p i) := by |
simpa only [centerMass, hโ, inv_one, one_smul] using
hf.map_centerMass_le hโ (hโ.symm โธ zero_lt_one) hmem
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,586 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
suppress_compilation
set_option linter.uppercaseLean3 false
open Metric
open scoped Classical NNReal Topology Uniformity
variable {๐ E : Type*} [NontriviallyNormedField ๐]
section SemiNormed
variable [SeminormedAddCommGroup E] [NormedSpace ๐ E]
... | Mathlib/Analysis/NormedSpace/OperatorNorm/Mul.lean | 226 | 231 | theorem norm_toSpanSingleton (x : E) : โtoSpanSingleton ๐ xโ = โxโ := by |
refine opNorm_eq_of_bounds (norm_nonneg _) (fun x => ?_) fun N _ h => ?_
ยท rw [toSpanSingleton_apply, norm_smul, mul_comm]
ยท specialize h 1
rw [toSpanSingleton_apply, norm_smul, mul_comm] at h
exact (mul_le_mul_right (by simp)).mp h
| 5 | 148.413159 | 2 | 1.5 | 2 | 1,587 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
suppress_compilation
set_option linter.uppercaseLean3 false
open Metric
open scoped Classical NNReal Topology Uniformity
variable {๐ E : Type*} [NontriviallyNormedField ๐]
section SemiNormed
variable [SeminormedAddCommGroup E] [NormedSpace ๐ E]
... | Mathlib/Analysis/NormedSpace/OperatorNorm/Mul.lean | 243 | 246 | theorem opNorm_lsmul_le : โ(lsmul ๐ ๐' : ๐' โL[๐] E โL[๐] E)โ โค 1 := by |
refine ContinuousLinearMap.opNorm_le_bound _ zero_le_one fun x => ?_
simp_rw [one_mul]
exact opNorm_lsmul_apply_le _
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,587 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Function
#align_import data.set.intervals.surj_on from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
variable {ฮฑ : Type*} {ฮฒ : Type*} [LinearOrder ฮฑ] [PartialOrder ฮฒ] {f : ฮฑ โ ฮฒ}
open Set Function
open OrderDual (toDual)... | Mathlib/Order/Interval/Set/SurjOn.lean | 26 | 32 | theorem surjOn_Ioo_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
(a b : ฮฑ) : SurjOn f (Ioo a b) (Ioo (f a) (f b)) := by |
intro p hp
rcases h_surj p with โจx, rflโฉ
refine โจx, mem_Ioo.2 ?_, rflโฉ
contrapose! hp
exact fun h => h.2.not_le (h_mono <| hp <| h_mono.reflect_lt h.1)
| 5 | 148.413159 | 2 | 1.5 | 6 | 1,588 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Function
#align_import data.set.intervals.surj_on from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
variable {ฮฑ : Type*} {ฮฒ : Type*} [LinearOrder ฮฑ] [PartialOrder ฮฒ] {f : ฮฑ โ ฮฒ}
open Set Function
open OrderDual (toDual)... | Mathlib/Order/Interval/Set/SurjOn.lean | 35 | 44 | theorem surjOn_Ico_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
(a b : ฮฑ) : SurjOn f (Ico a b) (Ico (f a) (f b)) := by |
obtain hab | hab := lt_or_le a b
ยท intro p hp
rcases eq_left_or_mem_Ioo_of_mem_Ico hp with (rfl | hp')
ยท exact mem_image_of_mem f (left_mem_Ico.mpr hab)
ยท have := surjOn_Ioo_of_monotone_surjective h_mono h_surj a b hp'
exact image_subset f Ioo_subset_Ico_self this
ยท rw [Ico_eq_empty (h_mono hab... | 8 | 2,980.957987 | 2 | 1.5 | 6 | 1,588 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Function
#align_import data.set.intervals.surj_on from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
variable {ฮฑ : Type*} {ฮฒ : Type*} [LinearOrder ฮฑ] [PartialOrder ฮฒ] {f : ฮฑ โ ฮฒ}
open Set Function
open OrderDual (toDual)... | Mathlib/Order/Interval/Set/SurjOn.lean | 47 | 49 | theorem surjOn_Ioc_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
(a b : ฮฑ) : SurjOn f (Ioc a b) (Ioc (f a) (f b)) := by |
simpa using surjOn_Ico_of_monotone_surjective h_mono.dual h_surj (toDual b) (toDual a)
| 1 | 2.718282 | 0 | 1.5 | 6 | 1,588 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Function
#align_import data.set.intervals.surj_on from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
variable {ฮฑ : Type*} {ฮฒ : Type*} [LinearOrder ฮฑ] [PartialOrder ฮฒ] {f : ฮฑ โ ฮฒ}
open Set Function
open OrderDual (toDual)... | Mathlib/Order/Interval/Set/SurjOn.lean | 53 | 60 | theorem surjOn_Icc_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
{a b : ฮฑ} (hab : a โค b) : SurjOn f (Icc a b) (Icc (f a) (f b)) := by |
intro p hp
rcases eq_endpoints_or_mem_Ioo_of_mem_Icc hp with (rfl | rfl | hp')
ยท exact โจa, left_mem_Icc.mpr hab, rflโฉ
ยท exact โจb, right_mem_Icc.mpr hab, rflโฉ
ยท have := surjOn_Ioo_of_monotone_surjective h_mono h_surj a b hp'
exact image_subset f Ioo_subset_Icc_self this
| 6 | 403.428793 | 2 | 1.5 | 6 | 1,588 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Function
#align_import data.set.intervals.surj_on from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
variable {ฮฑ : Type*} {ฮฒ : Type*} [LinearOrder ฮฑ] [PartialOrder ฮฒ] {f : ฮฑ โ ฮฒ}
open Set Function
open OrderDual (toDual)... | Mathlib/Order/Interval/Set/SurjOn.lean | 63 | 67 | theorem surjOn_Ioi_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
(a : ฮฑ) : SurjOn f (Ioi a) (Ioi (f a)) := by |
rw [โ compl_Iic, โ compl_compl (Ioi (f a))]
refine MapsTo.surjOn_compl ?_ h_surj
exact fun x hx => (h_mono hx).not_lt
| 3 | 20.085537 | 1 | 1.5 | 6 | 1,588 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Function
#align_import data.set.intervals.surj_on from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
variable {ฮฑ : Type*} {ฮฒ : Type*} [LinearOrder ฮฑ] [PartialOrder ฮฒ] {f : ฮฑ โ ฮฒ}
open Set Function
open OrderDual (toDual)... | Mathlib/Order/Interval/Set/SurjOn.lean | 75 | 80 | theorem surjOn_Ici_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
(a : ฮฑ) : SurjOn f (Ici a) (Ici (f a)) := by |
rw [โ Ioi_union_left, โ Ioi_union_left]
exact
(surjOn_Ioi_of_monotone_surjective h_mono h_surj a).union_union
(@image_singleton _ _ f a โธ surjOn_image _ _)
| 4 | 54.59815 | 2 | 1.5 | 6 | 1,588 |
import Mathlib.Data.Real.NNReal
import Mathlib.RingTheory.Valuation.Basic
noncomputable section
open Function Multiplicative
open scoped NNReal
variable {R : Type*} [Ring R] {ฮโ : Type*} [LinearOrderedCommGroupWithZero ฮโ]
namespace Valuation
class RankOne (v : Valuation R ฮโ) where
hom : ฮโ โ*โ โโฅ0
st... | Mathlib/RingTheory/Valuation/RankOne.lean | 51 | 55 | theorem zero_of_hom_zero {x : ฮโ} (hx : hom v x = 0) : x = 0 := by |
refine (eq_of_le_of_not_lt (zero_le' (a := x)) fun h_lt โฆ ?_).symm
have hs := strictMono v h_lt
rw [_root_.map_zero, hx] at hs
exact hs.false
| 4 | 54.59815 | 2 | 1.5 | 2 | 1,589 |
import Mathlib.Data.Real.NNReal
import Mathlib.RingTheory.Valuation.Basic
noncomputable section
open Function Multiplicative
open scoped NNReal
variable {R : Type*} [Ring R] {ฮโ : Type*} [LinearOrderedCommGroupWithZero ฮโ]
namespace Valuation
class RankOne (v : Valuation R ฮโ) where
hom : ฮโ โ*โ โโฅ0
st... | Mathlib/RingTheory/Valuation/RankOne.lean | 67 | 69 | theorem unit_ne_one : unit v โ 1 := by |
rw [Ne, โ Units.eq_iff, Units.val_one]
exact ((nontrivial v).choose_spec ).2
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,589 |
import Mathlib.MeasureTheory.Covering.DensityTheorem
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.covering.one_dim from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set MeasureTheory IsUnifLocDoublingMeasure Filter
open scoped Topology
names... | Mathlib/MeasureTheory/Covering/OneDim.lean | 26 | 30 | theorem Icc_mem_vitaliFamily_at_right {x y : โ} (hxy : x < y) :
Icc x y โ (vitaliFamily (volume : Measure โ) 1).setsAt x := by |
rw [Icc_eq_closedBall]
refine closedBall_mem_vitaliFamily_of_dist_le_mul _ ?_ (by linarith)
rw [dist_comm, Real.dist_eq, abs_of_nonneg] <;> linarith
| 3 | 20.085537 | 1 | 1.5 | 4 | 1,590 |
import Mathlib.MeasureTheory.Covering.DensityTheorem
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.covering.one_dim from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set MeasureTheory IsUnifLocDoublingMeasure Filter
open scoped Topology
names... | Mathlib/MeasureTheory/Covering/OneDim.lean | 33 | 41 | theorem tendsto_Icc_vitaliFamily_right (x : โ) :
Tendsto (fun y => Icc x y) (๐[>] x) ((vitaliFamily (volume : Measure โ) 1).filterAt x) := by |
refine (VitaliFamily.tendsto_filterAt_iff _).2 โจ?_, ?_โฉ
ยท filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_right hy
ยท intro ฮต ฮตpos
have : x โ Ico x (x + ฮต) := โจle_refl _, by linarithโฉ
filter_upwards [Icc_mem_nhdsWithin_Ioi this] with y hy
rw [closedBall_eq_Icc]
exact I... | 7 | 1,096.633158 | 2 | 1.5 | 4 | 1,590 |
import Mathlib.MeasureTheory.Covering.DensityTheorem
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.covering.one_dim from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set MeasureTheory IsUnifLocDoublingMeasure Filter
open scoped Topology
names... | Mathlib/MeasureTheory/Covering/OneDim.lean | 44 | 48 | theorem Icc_mem_vitaliFamily_at_left {x y : โ} (hxy : x < y) :
Icc x y โ (vitaliFamily (volume : Measure โ) 1).setsAt y := by |
rw [Icc_eq_closedBall]
refine closedBall_mem_vitaliFamily_of_dist_le_mul _ ?_ (by linarith)
rw [Real.dist_eq, abs_of_nonneg] <;> linarith
| 3 | 20.085537 | 1 | 1.5 | 4 | 1,590 |
import Mathlib.MeasureTheory.Covering.DensityTheorem
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.covering.one_dim from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set MeasureTheory IsUnifLocDoublingMeasure Filter
open scoped Topology
names... | Mathlib/MeasureTheory/Covering/OneDim.lean | 51 | 59 | theorem tendsto_Icc_vitaliFamily_left (x : โ) :
Tendsto (fun y => Icc y x) (๐[<] x) ((vitaliFamily (volume : Measure โ) 1).filterAt x) := by |
refine (VitaliFamily.tendsto_filterAt_iff _).2 โจ?_, ?_โฉ
ยท filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_left hy
ยท intro ฮต ฮตpos
have : x โ Ioc (x - ฮต) x := โจby linarith, le_refl _โฉ
filter_upwards [Icc_mem_nhdsWithin_Iio this] with y hy
rw [closedBall_eq_Icc]
exact Ic... | 7 | 1,096.633158 | 2 | 1.5 | 4 | 1,590 |
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import category_theory.glue_data from "l... | Mathlib/CategoryTheory/GlueData.lean | 77 | 85 | theorem t'_iij (i j : D.J) : D.t' i i j = (pullbackSymmetry _ _).hom := by |
have eqโ := D.t_fac i i j
have eqโ := (IsIso.eq_comp_inv (D.f i i)).mpr (@pullback.condition _ _ _ _ _ _ (D.f i j) _)
rw [D.t_id, Category.comp_id, eqโ] at eqโ
have eqโ := (IsIso.eq_comp_inv (D.f i i)).mp eqโ
rw [Category.assoc, โ pullback.condition, โ Category.assoc] at eqโ
exact
Mono.right_cancellati... | 8 | 2,980.957987 | 2 | 1.5 | 6 | 1,591 |
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import category_theory.glue_data from "l... | Mathlib/CategoryTheory/GlueData.lean | 88 | 90 | theorem t'_jii (i j : D.J) : D.t' j i i = pullback.fst โซ D.t j i โซ inv pullback.snd := by |
rw [โ Category.assoc, โ D.t_fac]
simp
| 2 | 7.389056 | 1 | 1.5 | 6 | 1,591 |
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import category_theory.glue_data from "l... | Mathlib/CategoryTheory/GlueData.lean | 93 | 95 | theorem t'_iji (i j : D.J) : D.t' i j i = pullback.fst โซ D.t i j โซ inv pullback.snd := by |
rw [โ Category.assoc, โ D.t_fac]
simp
| 2 | 7.389056 | 1 | 1.5 | 6 | 1,591 |
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import category_theory.glue_data from "l... | Mathlib/CategoryTheory/GlueData.lean | 99 | 105 | theorem t_inv (i j : D.J) : D.t i j โซ D.t j i = ๐ _ := by |
have eq : (pullbackSymmetry (D.f i i) (D.f i j)).hom = pullback.snd โซ inv pullback.fst := by simp
have := D.cocycle i j i
rw [D.t'_iij, D.t'_jii, D.t'_iji, fst_eq_snd_of_mono_eq, eq] at this
simp only [Category.assoc, IsIso.inv_hom_id_assoc] at this
rw [โ IsIso.eq_inv_comp, โ Category.assoc, IsIso.comp_inv_e... | 6 | 403.428793 | 2 | 1.5 | 6 | 1,591 |
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import category_theory.glue_data from "l... | Mathlib/CategoryTheory/GlueData.lean | 108 | 111 | theorem t'_inv (i j k : D.J) :
D.t' i j k โซ (pullbackSymmetry _ _).hom โซ D.t' j i k โซ (pullbackSymmetry _ _).hom = ๐ _ := by |
rw [โ cancel_mono (pullback.fst : pullback (D.f i j) (D.f i k) โถ _)]
simp [t_fac, t_fac_assoc]
| 2 | 7.389056 | 1 | 1.5 | 6 | 1,591 |
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import category_theory.glue_data from "l... | Mathlib/CategoryTheory/GlueData.lean | 123 | 129 | theorem t'_comp_eq_pullbackSymmetry (i j k : D.J) :
D.t' j k i โซ D.t' k i j =
(pullbackSymmetry _ _).hom โซ D.t' j i k โซ (pullbackSymmetry _ _).hom := by |
trans inv (D.t' i j k)
ยท exact IsIso.eq_inv_of_hom_inv_id (D.cocycle _ _ _)
ยท rw [โ cancel_mono (pullback.fst : pullback (D.f i j) (D.f i k) โถ _)]
simp [t_fac, t_fac_assoc]
| 4 | 54.59815 | 2 | 1.5 | 6 | 1,591 |
import Mathlib.AlgebraicTopology.DoldKan.EquivalenceAdditive
import Mathlib.AlgebraicTopology.DoldKan.Compatibility
import Mathlib.CategoryTheory.Idempotents.SimplicialObject
#align_import algebraic_topology.dold_kan.equivalence_pseudoabelian from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b5... | Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean | 108 | 114 | theorem hฮท :
Compatibility.ฯโ =
Compatibility.ฯโ isoNโ isoฮโ
(Nโฮโ : ฮ โ Nโ โ
(toKaroubiEquivalence (ChainComplex C โ)).functor) := by |
ext K : 3
simp only [Compatibility.ฯโ_hom_app, Compatibility.ฯโ_hom_app]
exact (Nโฮโ_compatible_with_Nโฮโ K).trans (by simp )
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,592 |
import Mathlib.AlgebraicTopology.DoldKan.EquivalenceAdditive
import Mathlib.AlgebraicTopology.DoldKan.Compatibility
import Mathlib.CategoryTheory.Idempotents.SimplicialObject
#align_import algebraic_topology.dold_kan.equivalence_pseudoabelian from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b5... | Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean | 129 | 144 | theorem hฮต :
Compatibility.ฯ
(isoNโ) =
(ฮโNโ : (toKaroubiEquivalence _).functor โ
(Nโ : SimplicialObject C โฅค _) โ Preadditive.DoldKan.equivalence.inverse) := by |
dsimp only [isoNโ]
ext1
rw [โ cancel_epi ฮโNโ.inv, Iso.inv_hom_id]
ext X : 2
rw [NatTrans.comp_app]
erw [compatibility_ฮโNโ_ฮโNโ_natTrans X]
rw [Compatibility.ฯ
_hom_app, Preadditive.DoldKan.equivalence_unitIso, Iso.app_inv, assoc]
erw [โ NatTrans.comp_app_assoc, IsIso.hom_inv_id]
rw [NatTrans.id_app,... | 12 | 162,754.791419 | 2 | 1.5 | 2 | 1,592 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
#align_import number_theory.primes_congruent_one from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
namespace Nat
open Polynomial Nat Filter
open scoped Nat
| Mathlib/NumberTheory/PrimesCongruentOne.lean | 26 | 57 | theorem exists_prime_gt_modEq_one {k : โ} (n : โ) (hk0 : k โ 0) :
โ p : โ, Nat.Prime p โง n < p โง p โก 1 [MOD k] := by |
rcases (one_le_iff_ne_zero.2 hk0).eq_or_lt with (rfl | hk1)
ยท rcases exists_infinite_primes (n + 1) with โจp, hnp, hpโฉ
exact โจp, hp, hnp, modEq_oneโฉ
let b := k * (n !)
have hgt : 1 < (eval (โb) (cyclotomic k โค)).natAbs := by
rcases le_iff_exists_add'.1 hk1.le with โจk, rflโฉ
have hb : 2 โค b := le_mul_... | 30 | 10,686,474,581,524.463 | 2 | 1.5 | 2 | 1,593 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
#align_import number_theory.primes_congruent_one from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
namespace Nat
open Polynomial Nat Filter
open scoped Nat
theorem exists_prime_gt_modEq_one {k : โ} (n : โ) (hk0 : k โ 0) :
โ ... | Mathlib/NumberTheory/PrimesCongruentOne.lean | 60 | 64 | theorem frequently_atTop_modEq_one {k : โ} (hk0 : k โ 0) :
โแถ p in atTop, Nat.Prime p โง p โก 1 [MOD k] := by |
refine frequently_atTop.2 fun n => ?_
obtain โจp, hpโฉ := exists_prime_gt_modEq_one n hk0
exact โจp, โจhp.2.1.le, hp.1, hp.2.2โฉโฉ
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,593 |
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.Order.Atoms
#align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011... | Mathlib/CategoryTheory/Simple.lean | 61 | 77 | theorem Simple.of_iso {X Y : C} [Simple Y] (i : X โ
Y) : Simple X :=
{ mono_isIso_iff_nonzero := fun f m => by
haveI : Mono (f โซ i.hom) := mono_comp _ _
constructor
ยท intro h w
have j : IsIso (f โซ i.hom) := by | infer_instance
rw [Simple.mono_isIso_iff_nonzero] at j
subst w
simp at j
ยท intro h
have j : IsIso (f โซ i.hom) := by
apply isIso_of_mono_of_nonzero
intro w
apply h
simpa using (cancel_mono i.inv).2 w
rw [โ Category.comp_id f, โ i.hom_... | 12 | 162,754.791419 | 2 | 1.5 | 8 | 1,594 |
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.Order.Atoms
#align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011... | Mathlib/CategoryTheory/Simple.lean | 84 | 89 | theorem kernel_zero_of_nonzero_from_simple {X Y : C} [Simple X] {f : X โถ Y} [HasKernel f]
(w : f โ 0) : kernel.ฮน f = 0 := by |
classical
by_contra h
haveI := isIso_of_mono_of_nonzero h
exact w (eq_zero_of_epi_kernel f)
| 4 | 54.59815 | 2 | 1.5 | 8 | 1,594 |
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.Order.Atoms
#align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011... | Mathlib/CategoryTheory/Simple.lean | 96 | 100 | theorem epi_of_nonzero_to_simple [HasEqualizers C] {X Y : C} [Simple Y] {f : X โถ Y} [HasImage f]
(w : f โ 0) : Epi f := by |
rw [โ image.fac f]
haveI : IsIso (image.ฮน f) := isIso_of_mono_of_nonzero fun h => w (eq_zero_of_image_eq_zero h)
apply epi_comp
| 3 | 20.085537 | 1 | 1.5 | 8 | 1,594 |
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.Order.Atoms
#align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011... | Mathlib/CategoryTheory/Simple.lean | 103 | 107 | theorem mono_to_simple_zero_of_not_iso {X Y : C} [Simple Y] {f : X โถ Y} [Mono f]
(w : IsIso f โ False) : f = 0 := by |
classical
by_contra h
exact w (isIso_of_mono_of_nonzero h)
| 3 | 20.085537 | 1 | 1.5 | 8 | 1,594 |
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.Order.Atoms
#align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011... | Mathlib/CategoryTheory/Simple.lean | 119 | 120 | theorem Simple.not_isZero (X : C) [Simple X] : ยฌIsZero X := by |
simpa [Limits.IsZero.iff_id_eq_zero] using id_nonzero X
| 1 | 2.718282 | 0 | 1.5 | 8 | 1,594 |
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.Order.Atoms
#align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011... | Mathlib/CategoryTheory/Simple.lean | 170 | 175 | theorem cokernel_zero_of_nonzero_to_simple {X Y : C} [Simple Y] {f : X โถ Y} (w : f โ 0) :
cokernel.ฯ f = 0 := by |
classical
by_contra h
haveI := isIso_of_epi_of_nonzero h
exact w (eq_zero_of_mono_cokernel f)
| 4 | 54.59815 | 2 | 1.5 | 8 | 1,594 |
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.Order.Atoms
#align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011... | Mathlib/CategoryTheory/Simple.lean | 193 | 201 | theorem Biprod.isIso_inl_iff_isZero (X Y : C) : IsIso (biprod.inl : X โถ X โ Y) โ IsZero Y := by |
rw [biprod.isIso_inl_iff_id_eq_fst_comp_inl, โ biprod.total, add_right_eq_self]
constructor
ยท intro h
replace h := h =โซ biprod.snd
simpa [โ IsZero.iff_isSplitEpi_eq_zero (biprod.snd : X โ Y โถ Y)] using h
ยท intro h
rw [IsZero.iff_isSplitEpi_eq_zero (biprod.snd : X โ Y โถ Y)] at h
rw [h, zero_comp... | 8 | 2,980.957987 | 2 | 1.5 | 8 | 1,594 |
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.Order.Atoms
#align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011... | Mathlib/CategoryTheory/Simple.lean | 237 | 248 | theorem simple_of_isSimpleOrder_subobject (X : C) [IsSimpleOrder (Subobject X)] : Simple X := by |
constructor; intros Y f hf; constructor
ยท intro i
rw [Subobject.isIso_iff_mk_eq_top] at i
intro w
rw [โ Subobject.mk_eq_bot_iff_zero] at w
exact IsSimpleOrder.bot_ne_top (w.symm.trans i)
ยท intro i
rcases IsSimpleOrder.eq_bot_or_eq_top (Subobject.mk f) with (h | h)
ยท rw [Subobject.mk_eq_bo... | 11 | 59,874.141715 | 2 | 1.5 | 8 | 1,594 |
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
#align_import topology.vector_bundle.constructions from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Cl... | Mathlib/Topology/VectorBundle/Constructions.lean | 50 | 55 | theorem trivialization.coordChangeL (b : B) :
(trivialization B F).coordChangeL ๐ (trivialization B F) b =
ContinuousLinearEquiv.refl ๐ F := by |
ext v
rw [Trivialization.coordChangeL_apply']
exacts [rfl, โจmem_univ _, mem_univ _โฉ]
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,595 |
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
#align_import topology.vector_bundle.constructions from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Cl... | Mathlib/Topology/VectorBundle/Constructions.lean | 96 | 106 | theorem coordChangeL_prod [eโ.IsLinear ๐] [eโ'.IsLinear ๐] [eโ.IsLinear ๐] [eโ'.IsLinear ๐] โฆbโฆ
(hb : b โ (eโ.prod eโ).baseSet โฉ (eโ'.prod eโ').baseSet) :
((eโ.prod eโ).coordChangeL ๐ (eโ'.prod eโ') b : Fโ ร Fโ โL[๐] Fโ ร Fโ) =
(eโ.coordChangeL ๐ eโ' b : Fโ โL[๐] Fโ).prodMap (eโ.coordChangeL ๐ eโ... |
rw [ContinuousLinearMap.ext_iff, ContinuousLinearMap.coe_prodMap']
rintro โจvโ, vโโฉ
show
(eโ.prod eโ).coordChangeL ๐ (eโ'.prod eโ') b (vโ, vโ) =
(eโ.coordChangeL ๐ eโ' b vโ, eโ.coordChangeL ๐ eโ' b vโ)
rw [eโ.coordChangeL_apply eโ', eโ.coordChangeL_apply eโ', (eโ.prod eโ).coordChangeL_apply']
exa... | 7 | 1,096.633158 | 2 | 1.5 | 2 | 1,595 |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Haar.Unique
open MeasureTheory Measure Set
open scoped ENNReal
variable {๐ E F : Type*}
[NontriviallyNormedField ๐] [CompleteSpace ๐]
[NormedAddCommGroup E] [MeasurableSp... | Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean | 42 | 102 | theorem LinearMap.exists_map_addHaar_eq_smul_addHaar' (h : Function.Surjective L) :
โ (c : โโฅ0โ), 0 < c โง c < โ โง ฮผ.map L = (c * addHaar (univ : Set (LinearMap.ker L))) โข ฮฝ := by |
/- This is true for the second projection in product spaces, as the projection of the Haar
measure `ฮผS.prod ฮผT` is equal to the Haar measure `ฮผT` multiplied by the total mass of `ฮผS`. This
is also true for linear equivalences, as they map Haar measure to Haar measure. The general case
follows from these two an... | 59 | 42,012,104,037,905,144,000,000,000 | 2 | 1.5 | 2 | 1,596 |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Haar.Unique
open MeasureTheory Measure Set
open scoped ENNReal
variable {๐ E F : Type*}
[NontriviallyNormedField ๐] [CompleteSpace ๐]
[NormedAddCommGroup E] [MeasurableSp... | Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean | 106 | 109 | theorem LinearMap.exists_map_addHaar_eq_smul_addHaar (h : Function.Surjective L) :
โ (c : โโฅ0โ), 0 < c โง ฮผ.map L = c โข ฮฝ := by |
rcases L.exists_map_addHaar_eq_smul_addHaar' ฮผ ฮฝ h with โจc, c_pos, -, hcโฉ
exact โจ_, by simp [c_pos, NeZero.ne addHaar], hcโฉ
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,596 |
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.FieldTheory.IsAlgClosed.Basic
#align_import linear_algebra.matrix.charpoly.eigs from "leanprover-community/mathlib"@"48dc6abe71248bd6f4bffc9703dc87bdd4e37d0b"
variable {n : Type*} [Fintype n] [DecidableEq n]
variable {R : Type*} [Field R]
variable {A : Matrix... | Mathlib/LinearAlgebra/Matrix/Charpoly/Eigs.lean | 60 | 64 | theorem det_eq_prod_roots_charpoly_of_splits (hAps : A.charpoly.Splits (RingHom.id R)) :
A.det = (Matrix.charpoly A).roots.prod := by |
rw [det_eq_sign_charpoly_coeff, โ charpoly_natDegree_eq_dim A,
Polynomial.prod_roots_eq_coeff_zero_of_monic_of_split A.charpoly_monic hAps, โ mul_assoc,
โ pow_two, pow_right_comm, neg_one_sq, one_pow, one_mul]
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,597 |
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.FieldTheory.IsAlgClosed.Basic
#align_import linear_algebra.matrix.charpoly.eigs from "leanprover-community/mathlib"@"48dc6abe71248bd6f4bffc9703dc87bdd4e37d0b"
variable {n : Type*} [Fintype n] [DecidableEq n]
variable {R : Type*} [Field R]
variable {A : Matrix... | Mathlib/LinearAlgebra/Matrix/Charpoly/Eigs.lean | 67 | 75 | theorem trace_eq_sum_roots_charpoly_of_splits (hAps : A.charpoly.Splits (RingHom.id R)) :
A.trace = (Matrix.charpoly A).roots.sum := by |
cases' isEmpty_or_nonempty n with h
ยท rw [Matrix.trace, Fintype.sum_empty, Matrix.charpoly,
det_eq_one_of_card_eq_zero (Fintype.card_eq_zero_iff.2 h), Polynomial.roots_one,
Multiset.empty_eq_zero, Multiset.sum_zero]
ยท rw [trace_eq_neg_charpoly_coeff, neg_eq_iff_eq_neg,
โ Polynomial.sum_roots_eq... | 7 | 1,096.633158 | 2 | 1.5 | 2 | 1,597 |
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 34 | 40 | theorem rank_le {n : โ}
(H : โ s : Finset M, (LinearIndependent R fun i : s => (i : M)) โ s.card โค n) :
Module.rank R M โค n := by |
rw [Module.rank_def]
apply ciSup_le'
rintro โจs, liโฉ
exact linearIndependent_bounded_of_finset_linearIndependent_bounded H _ li
| 4 | 54.59815 | 2 | 1.5 | 4 | 1,598 |
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 70 | 73 | theorem rank_zero_iff_forall_zero :
Module.rank R M = 0 โ โ x : M, x = 0 := by |
simp_rw [rank_eq_zero_iff, smul_eq_zero, and_or_left, not_and_self_iff, false_or,
exists_and_right, and_iff_right (exists_ne (0 : R))]
| 2 | 7.389056 | 1 | 1.5 | 4 | 1,598 |
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 82 | 84 | theorem rank_pos_iff_exists_ne_zero : 0 < Module.rank R M โ โ x : M, x โ 0 := by |
rw [โ not_iff_not]
simpa using rank_zero_iff_forall_zero
| 2 | 7.389056 | 1 | 1.5 | 4 | 1,598 |
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 125 | 131 | theorem Module.finite_of_rank_eq_nat [Module.Free R M] {n : โ} (h : Module.rank R M = n) :
Module.Finite R M := by |
nontriviality R
obtain โจโจฮน, bโฉโฉ := Module.Free.exists_basis (R := R) (M := M)
have := mk_lt_aleph0_iff.mp <|
b.linearIndependent.cardinal_le_rank |>.trans_eq h |>.trans_lt <| nat_lt_aleph0 n
exact Module.Finite.of_basis b
| 5 | 148.413159 | 2 | 1.5 | 4 | 1,598 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.Algebra.Module.LocalizedModule
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
#align_import ring_theory.localization.module from "leanprover-community/mathlib"@"2e59a6de168f95d16b16d217b808a36290398c0a"
open nonZ... | Mathlib/RingTheory/Localization/Module.lean | 56 | 71 | theorem LinearIndependent.of_isLocalizedModule {ฮน : Type*} {v : ฮน โ M}
(hv : LinearIndependent R v) : LinearIndependent Rโ (f โ v) := by |
rw [linearIndependent_iff'] at hv โข
intro t g hg i hi
choose! a g' hg' using IsLocalization.exist_integer_multiples S t g
have h0 : f (โ i โ t, g' i โข v i) = 0 := by
apply_fun ((a : R) โข ยท) at hg
rw [smul_zero, Finset.smul_sum] at hg
rw [map_sum, โ hg]
refine Finset.sum_congr rfl fun i hi => ?_... | 14 | 1,202,604.284165 | 2 | 1.5 | 2 | 1,599 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.Algebra.Module.LocalizedModule
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
#align_import ring_theory.localization.module from "leanprover-community/mathlib"@"2e59a6de168f95d16b16d217b808a36290398c0a"
open nonZ... | Mathlib/RingTheory/Localization/Module.lean | 73 | 76 | theorem LinearIndependent.localization {ฮน : Type*} {b : ฮน โ M} (hli : LinearIndependent R b) :
LinearIndependent Rโ b := by |
have := isLocalizedModule_id S M Rโ
exact hli.of_isLocalizedModule Rโ S .id
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,599 |
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 ๐ E F) (n : โ) :
Continuous (p.partialSum n) := by |
unfold partialSum -- Porting note: added
continuity
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,600 |
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 : โr < p.radius) :
โ a โ Ioo (0 : โ) 1, (fun n => โp nโ * (r : โ) ^ n) =o[atTop] (a ^ ยท) := by |
have := (TFAE_exists_lt_isLittleO_pow (fun n => โp nโ * (r : โ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => โp nโ * (r : โ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with โจt, C, hC, rtโฉ
rw [ENNReal.coe_lt_coe, โ NNReal.coe_lt_coe] at ... | 14 | 1,202,604.284165 | 2 | 1.5 | 2 | 1,600 |
import Mathlib.Algebra.Group.Submonoid.Pointwise
#align_import group_theory.submonoid.inverses from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {M : Type*}
namespace Submonoid
@[to_additive]
noncomputable instance [Monoid M] : Group (IsUnit.submonoid M) :=
{ inferInstanc... | Mathlib/GroupTheory/Submonoid/Inverses.lean | 73 | 76 | theorem leftInv_leftInv_le : S.leftInv.leftInv โค S := by |
rintro x โจโจy, z, hโโฉ, hโ : x * y = 1โฉ
convert z.prop
rw [โ mul_one x, โ hโ, โ mul_assoc, hโ, one_mul]
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,601 |
import Mathlib.Algebra.Group.Submonoid.Pointwise
#align_import group_theory.submonoid.inverses from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {M : Type*}
namespace Submonoid
@[to_additive]
noncomputable instance [Monoid M] : Group (IsUnit.submonoid M) :=
{ inferInstanc... | Mathlib/GroupTheory/Submonoid/Inverses.lean | 87 | 94 | theorem leftInv_leftInv_eq (hS : S โค IsUnit.submonoid M) : S.leftInv.leftInv = S := by |
refine le_antisymm S.leftInv_leftInv_le ?_
intro x hx
have : x = ((hS hx).unitโปยนโปยน : Mหฃ) := by
rw [inv_inv (hS hx).unit]
rfl
rw [this]
exact S.leftInv.unit_mem_leftInv _ (S.unit_mem_leftInv _ hx)
| 7 | 1,096.633158 | 2 | 1.5 | 2 | 1,601 |
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 | 28 | 46 | theorem mul_eq_mul_prime_prod {ฮฑ : Type*} [DecidableEq ฮฑ] {x y a : R} {s : Finset ฮฑ} {p : ฮฑ โ R}
(hp : โ i โ s, Prime (p i)) (hx : x * y = a * โ i โ s, p i) :
โ (t u : Finset ฮฑ) (b c : R),
t โช u = s โง Disjoint t u โง a = b * c โง (x = b * โ i โ t, p i) โง y = c * โ i โ u, p i := by |
induction' s using Finset.induction with i s his ih generalizing x y a
ยท exact โจโ
, โ
, x, y, by simp [hx]โฉ
ยท rw [prod_insert his, โ mul_assoc] at hx
have hpi : Prime (p i) := hp i (mem_insert_self _ _)
rcases ih (fun i hi โฆ hp i (mem_insert_of_mem hi)) hx with
โจt, u, b, c, htus, htu, hbc, rfl, rflโฉ
... | 15 | 3,269,017.372472 | 2 | 1.5 | 4 | 1,602 |
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 : โ} (hp : Prime p) (hx : x * y = a * p ^ n) :
โ (i j : โ) (b c : R), i + j = n โง a = b * c โง x = b * p ^ i โง y = c * p ^ j := by |
rcases mul_eq_mul_prime_prod (fun _ _ โฆ hp)
(show x * y = a * (range n).prod fun _ โฆ p by simpa) with
โจt, u, b, c, htus, htu, rfl, rfl, rflโฉ
exact โจt.card, u.card, b, c, by rw [โ card_union_of_disjoint htu, htus, card_range], by simpโฉ
| 4 | 54.59815 | 2 | 1.5 | 4 | 1,602 |
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 CommRing
variable {ฮฑ : Ty... | Mathlib/RingTheory/Prime.lean | 65 | 67 | theorem Prime.neg {p : ฮฑ} (hp : Prime p) : Prime (-p) := by |
obtain โจh1, h2, h3โฉ := hp
exact โจneg_ne_zero.mpr h1, by rwa [IsUnit.neg_iff], by simpa [neg_dvd] using h3โฉ
| 2 | 7.389056 | 1 | 1.5 | 4 | 1,602 |
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 CommRing
variable {ฮฑ : Ty... | Mathlib/RingTheory/Prime.lean | 70 | 73 | theorem Prime.abs [LinearOrder ฮฑ] {p : ฮฑ} (hp : Prime p) : Prime (abs p) := by |
obtain h | h := abs_choice p <;> rw [h]
ยท exact hp
ยท exact hp.neg
| 3 | 20.085537 | 1 | 1.5 | 4 | 1,602 |
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ฮน : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 52 | 56 | theorem IsLindelof.compl_mem_sets (hs : IsLindelof s) {f : Filter X} [CountableInterFilter f]
(hf : โ x โ s, sแถ โ ๐ x โ f) : sแถ โ f := by |
contrapose! hf
simp only [not_mem_iff_inf_principal_compl, compl_compl, inf_assoc] at hf โข
exact hs inf_le_right
| 3 | 20.085537 | 1 | 1.5 | 6 | 1,603 |
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ฮน : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 60 | 64 | theorem IsLindelof.compl_mem_sets_of_nhdsWithin (hs : IsLindelof s) {f : Filter X}
[CountableInterFilter f] (hf : โ x โ s, โ t โ ๐[s] x, tแถ โ f) : sแถ โ f := by |
refine hs.compl_mem_sets fun x hx โฆ ?_
rw [โ disjoint_principal_right, disjoint_right_comm, (basis_sets _).disjoint_iff_left]
exact hf x hx
| 3 | 20.085537 | 1 | 1.5 | 6 | 1,603 |
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ฮน : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 69 | 75 | theorem IsLindelof.induction_on (hs : IsLindelof s) {p : Set X โ Prop}
(hmono : โ โฆs tโฆ, s โ t โ p t โ p s)
(hcountable_union : โ (S : Set (Set X)), S.Countable โ (โ s โ S, p s) โ p (โโ S))
(hnhds : โ x โ s, โ t โ ๐[s] x, p t) : p s := by |
let f : Filter X := ofCountableUnion p hcountable_union (fun t ht _ hsub โฆ hmono hsub ht)
have : sแถ โ f := hs.compl_mem_sets_of_nhdsWithin (by simpa [f] using hnhds)
rwa [โ compl_compl s]
| 3 | 20.085537 | 1 | 1.5 | 6 | 1,603 |
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ฮน : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 78 | 83 | theorem IsLindelof.inter_right (hs : IsLindelof s) (ht : IsClosed t) : IsLindelof (s โฉ t) := by |
intro f hnf _ hstf
rw [โ inf_principal, le_inf_iff] at hstf
obtain โจx, hsx, hxโฉ : โ x โ s, ClusterPt x f := hs hstf.1
have hxt : x โ t := ht.mem_of_nhdsWithin_neBot <| hx.mono hstf.2
exact โจx, โจhsx, hxtโฉ, hxโฉ
| 5 | 148.413159 | 2 | 1.5 | 6 | 1,603 |
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ฮน : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 98 | 110 | theorem IsLindelof.image_of_continuousOn {f : X โ Y} (hs : IsLindelof s) (hf : ContinuousOn f s) :
IsLindelof (f '' s) := by |
intro l lne _ ls
have : NeBot (l.comap f โ ๐ s) :=
comap_inf_principal_neBot_of_image_mem lne (le_principal_iff.1 ls)
obtain โจx, hxs, hxโฉ : โ x โ s, ClusterPt x (l.comap f โ ๐ s) := @hs _ this _ inf_le_right
haveI := hx.neBot
use f x, mem_image_of_mem f hxs
have : Tendsto f (๐ x โ (comap f l โ ๐ s)... | 11 | 59,874.141715 | 2 | 1.5 | 6 | 1,603 |
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ฮน : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 129 | 151 | theorem IsLindelof.elim_countable_subcover {ฮน : Type v} (hs : IsLindelof s) (U : ฮน โ Set X)
(hUo : โ i, IsOpen (U i)) (hsU : s โ โ i, U i) :
โ r : Set ฮน, r.Countable โง (s โ โ i โ r, U i) := by |
have hmono : โ โฆs t : Set Xโฆ, s โ t โ (โ r : Set ฮน, r.Countable โง t โ โ i โ r, U i)
โ (โ r : Set ฮน, r.Countable โง s โ โ i โ r, U i) := by
intro _ _ hst โจr, โจhrcountable, hsubโฉโฉ
exact โจr, hrcountable, Subset.trans hst hsubโฉ
have hcountable_union : โ (S : Set (Set X)), S.Countable
โ (โ s โ S, โ r... | 20 | 485,165,195.40979 | 2 | 1.5 | 6 | 1,603 |
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
-- Porting note: universe metavariables behave oddly
universe w u vโ vโ vโ vโ
variable {ฮน : Type... | Mathlib/LinearAlgebra/Contraction.lean | 85 | 92 | theorem transpose_dualTensorHom (f : Module.Dual R M) (m : M) :
Dual.transpose (R := R) (dualTensorHom R M M (f โโ m)) =
dualTensorHom R _ _ (Dual.eval R M m โโ f) := by |
ext f' m'
simp only [Dual.transpose_apply, coe_comp, Function.comp_apply, dualTensorHom_apply,
LinearMap.map_smulโโ, RingHom.id_apply, Algebra.id.smul_eq_mul, Dual.eval_apply,
LinearMap.smul_apply]
exact mul_comm _ _
| 5 | 148.413159 | 2 | 1.5 | 6 | 1,604 |
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
-- Porting note: universe metavariables behave oddly
universe w u vโ vโ vโ vโ
variable {ฮน : Type... | Mathlib/LinearAlgebra/Contraction.lean | 96 | 101 | theorem dualTensorHom_prodMap_zero (f : Module.Dual R M) (p : P) :
((dualTensorHom R M P) (f โโ[R] p)).prodMap (0 : N โโ[R] Q) =
dualTensorHom R (M ร N) (P ร Q) ((f โโ fst R M N) โโ inl R P Q p) := by |
ext <;>
simp only [coe_comp, coe_inl, Function.comp_apply, prodMap_apply, dualTensorHom_apply,
fst_apply, Prod.smul_mk, LinearMap.zero_apply, smul_zero]
| 3 | 20.085537 | 1 | 1.5 | 6 | 1,604 |
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
-- Porting note: universe metavariables behave oddly
universe w u vโ vโ vโ vโ
variable {ฮน : Type... | Mathlib/LinearAlgebra/Contraction.lean | 105 | 110 | theorem zero_prodMap_dualTensorHom (g : Module.Dual R N) (q : Q) :
(0 : M โโ[R] P).prodMap ((dualTensorHom R N Q) (g โโ[R] q)) =
dualTensorHom R (M ร N) (P ร Q) ((g โโ snd R M N) โโ inr R P Q q) := by |
ext <;>
simp only [coe_comp, coe_inr, Function.comp_apply, prodMap_apply, dualTensorHom_apply,
snd_apply, Prod.smul_mk, LinearMap.zero_apply, smul_zero]
| 3 | 20.085537 | 1 | 1.5 | 6 | 1,604 |
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
-- Porting note: universe metavariables behave oddly
universe w u vโ vโ vโ vโ
variable {ฮน : Type... | Mathlib/LinearAlgebra/Contraction.lean | 113 | 118 | theorem map_dualTensorHom (f : Module.Dual R M) (p : P) (g : Module.Dual R N) (q : Q) :
TensorProduct.map (dualTensorHom R M P (f โโ[R] p)) (dualTensorHom R N Q (g โโ[R] q)) =
dualTensorHom R (M โ[R] N) (P โ[R] Q) (dualDistrib R M N (f โโ g) โโ[R] p โโ[R] q) := by |
ext m n
simp only [comprโ_apply, mk_apply, map_tmul, dualTensorHom_apply, dualDistrib_apply, โ
smul_tmul_smul]
| 3 | 20.085537 | 1 | 1.5 | 6 | 1,604 |
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
-- Porting note: universe metavariables behave oddly
universe w u vโ vโ vโ vโ
variable {ฮน : Type... | Mathlib/LinearAlgebra/Contraction.lean | 122 | 128 | theorem comp_dualTensorHom (f : Module.Dual R M) (n : N) (g : Module.Dual R N) (p : P) :
dualTensorHom R N P (g โโ[R] p) โโ dualTensorHom R M N (f โโ[R] n) =
g n โข dualTensorHom R M P (f โโ p) := by |
ext m
simp only [coe_comp, Function.comp_apply, dualTensorHom_apply, LinearMap.map_smul,
RingHom.id_apply, LinearMap.smul_apply]
rw [smul_comm]
| 4 | 54.59815 | 2 | 1.5 | 6 | 1,604 |
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
-- Porting note: universe metavariables behave oddly
universe w u vโ vโ vโ vโ
variable {ฮน : Type... | Mathlib/LinearAlgebra/Contraction.lean | 133 | 140 | theorem toMatrix_dualTensorHom {m : Type*} {n : Type*} [Fintype m] [Finite n] [DecidableEq m]
[DecidableEq n] (bM : Basis m R M) (bN : Basis n R N) (j : m) (i : n) :
toMatrix bM bN (dualTensorHom R M N (bM.coord j โโ bN i)) = stdBasisMatrix i j 1 := by |
ext i' j'
by_cases hij : i = i' โง j = j' <;>
simp [LinearMap.toMatrix_apply, Finsupp.single_eq_pi_single, hij]
rw [and_iff_not_or_not, Classical.not_not] at hij
cases' hij with hij hij <;> simp [hij]
| 5 | 148.413159 | 2 | 1.5 | 6 | 1,604 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 36 | 46 | theorem tendsto_rpow_atTop {y : โ} (hy : 0 < y) : Tendsto (fun x : โ => x ^ y) atTop atTop := by |
rw [tendsto_atTop_atTop]
intro b
use max b 0 ^ (1 / y)
intro x hx
exact
le_of_max_le_left
(by
convert rpow_le_rpow (rpow_nonneg (le_max_right b 0) (1 / y)) hx (le_of_lt hy)
using 1
rw [โ rpow_mul (le_max_right b 0), (eq_div_iff (ne_of_gt hy)).mp rfl, Real.rpow_one])
| 10 | 22,026.465795 | 2 | 1.5 | 10 | 1,605 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 102 | 116 | theorem tendsto_rpow_div_mul_add (a b c : โ) (hb : 0 โ b) :
Tendsto (fun x => x ^ (a / (b * x + c))) atTop (๐ 1) := by |
refine
Tendsto.congr' ?_
((tendsto_exp_nhds_zero_nhds_one.comp
(by
simpa only [mul_zero, pow_one] using
(tendsto_const_nhds (x := a)).mul
(tendsto_div_pow_mul_exp_add_atTop b c 1 hb))).comp
tendsto_log_atTop)
apply eventuallyEq_of_mem (I... | 13 | 442,413.392009 | 2 | 1.5 | 10 | 1,605 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 120 | 122 | theorem tendsto_rpow_div : Tendsto (fun x => x ^ ((1 : โ) / x)) atTop (๐ 1) := by |
convert tendsto_rpow_div_mul_add (1 : โ) _ (0 : โ) zero_ne_one
ring
| 2 | 7.389056 | 1 | 1.5 | 10 | 1,605 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 126 | 128 | theorem tendsto_rpow_neg_div : Tendsto (fun x => x ^ (-(1 : โ) / x)) atTop (๐ 1) := by |
convert tendsto_rpow_div_mul_add (-(1 : โ)) _ (0 : โ) zero_ne_one
ring
| 2 | 7.389056 | 1 | 1.5 | 10 | 1,605 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 132 | 137 | theorem tendsto_exp_div_rpow_atTop (s : โ) : Tendsto (fun x : โ => exp x / x ^ s) atTop atTop := by |
cases' archimedean_iff_nat_lt.1 Real.instArchimedean s with n hn
refine tendsto_atTop_mono' _ ?_ (tendsto_exp_div_pow_atTop n)
filter_upwards [eventually_gt_atTop (0 : โ), eventually_ge_atTop (1 : โ)] with x hxโ hxโ
rw [div_le_div_left (exp_pos _) (pow_pos hxโ _) (rpow_pos_of_pos hxโ _), โ Real.rpow_natCast]
... | 5 | 148.413159 | 2 | 1.5 | 10 | 1,605 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 200 | 207 | theorem isTheta_exp_arg_mul_im (hl : IsBoundedUnder (ยท โค ยท) l fun x => |(g x).im|) :
(fun x => Real.exp (arg (f x) * im (g x))) =ฮ[l] fun _ => (1 : โ) := by |
rcases hl with โจb, hbโฉ
refine Real.isTheta_exp_comp_one.2 โจฯ * b, ?_โฉ
rw [eventually_map] at hb โข
refine hb.mono fun x hx => ?_
erw [abs_mul]
exact mul_le_mul (abs_arg_le_pi _) hx (abs_nonneg _) Real.pi_pos.le
| 6 | 403.428793 | 2 | 1.5 | 10 | 1,605 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 210 | 220 | theorem isBigO_cpow_rpow (hl : IsBoundedUnder (ยท โค ยท) l fun x => |(g x).im|) :
(fun x => f x ^ g x) =O[l] fun x => abs (f x) ^ (g x).re :=
calc
(fun x => f x ^ g x) =O[l]
(show ฮฑ โ โ from fun x => abs (f x) ^ (g x).re / Real.exp (arg (f x) * im (g x))) :=
isBigO_of_le _ fun x => (abs_cpow_le _ _... |
simp only [ofReal_one, div_one]
rfl
| 2 | 7.389056 | 1 | 1.5 | 10 | 1,605 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 223 | 234 | theorem isTheta_cpow_rpow (hl_im : IsBoundedUnder (ยท โค ยท) l fun x => |(g x).im|)
(hl : โแถ x in l, f x = 0 โ re (g x) = 0 โ g x = 0) :
(fun x => f x ^ g x) =ฮ[l] fun x => abs (f x) ^ (g x).re :=
calc
(fun x => f x ^ g x) =ฮ[l]
(show ฮฑ โ โ from fun x => abs (f x) ^ (g x).re / Real.exp (arg (f x) * i... |
simp only [ofReal_one, div_one]
rfl
| 2 | 7.389056 | 1 | 1.5 | 10 | 1,605 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 259 | 266 | theorem IsBigOWith.rpow (h : IsBigOWith c l f g) (hc : 0 โค c) (hr : 0 โค r) (hg : 0 โคแถ [l] g) :
IsBigOWith (c ^ r) l (fun x => f x ^ r) fun x => g x ^ r := by |
apply IsBigOWith.of_bound
filter_upwards [hg, h.bound] with x hgx hx
calc
|f x ^ r| โค |f x| ^ r := abs_rpow_le_abs_rpow _ _
_ โค (c * |g x|) ^ r := rpow_le_rpow (abs_nonneg _) hx hr
_ = c ^ r * |g x ^ r| := by rw [mul_rpow hc (abs_nonneg _), abs_rpow_of_nonneg hgx]
| 6 | 403.428793 | 2 | 1.5 | 10 | 1,605 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 279 | 283 | theorem IsLittleO.rpow (hr : 0 < r) (hg : 0 โคแถ [l] g) (h : f =o[l] g) :
(fun x => f x ^ r) =o[l] fun x => g x ^ r := by |
refine .of_isBigOWith fun c hc โฆ ?_
rw [โ rpow_inv_rpow hc.le hr.ne']
refine (h.forall_isBigOWith ?_).rpow ?_ ?_ hg <;> positivity
| 3 | 20.085537 | 1 | 1.5 | 10 | 1,605 |
import Mathlib.CategoryTheory.Category.Cat
import Mathlib.CategoryTheory.Elements
#align_import category_theory.grothendieck from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
universe u
namespace CategoryTheory
variable {C D : Type*} [Category C] [Category D]
variable (F : C โฅค Cat)
... | Mathlib/CategoryTheory/Grothendieck.lean | 78 | 83 | theorem ext {X Y : Grothendieck F} (f g : Hom X Y) (w_base : f.base = g.base)
(w_fiber : eqToHom (by rw [w_base]) โซ f.fiber = g.fiber) : f = g := by |
cases f; cases g
congr
dsimp at w_base
aesop_cat
| 4 | 54.59815 | 2 | 1.5 | 2 | 1,606 |
import Mathlib.CategoryTheory.Category.Cat
import Mathlib.CategoryTheory.Elements
#align_import category_theory.grothendieck from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
universe u
namespace CategoryTheory
variable {C D : Type*} [Category C] [Category D]
variable (F : C โฅค Cat)
... | Mathlib/CategoryTheory/Grothendieck.lean | 132 | 136 | theorem congr {X Y : Grothendieck F} {f g : X โถ Y} (h : f = g) :
f.fiber = eqToHom (by subst h; rfl) โซ g.fiber := by |
subst h
dsimp
simp
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,606 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.caratheodory from "leanprover-community/mathlib"@"e6fab1dc073396d45da082c644642c4f8bff2264"
open Set Finset
universe u
variable {๐ : Type*} {E : Type u} ... | Mathlib/Analysis/Convex/Caratheodory.lean | 52 | 98 | theorem mem_convexHull_erase [DecidableEq E] {t : Finset E} (h : ยฌAffineIndependent ๐ ((โ) : t โ E))
{x : E} (m : x โ convexHull ๐ (โt : Set E)) :
โ y : (โt : Set E), x โ convexHull ๐ (โ(t.erase y) : Set E) := by |
simp only [Finset.convexHull_eq, mem_setOf_eq] at m โข
obtain โจf, fpos, fsum, rflโฉ := m
obtain โจg, gcombo, gsum, gposโฉ := exists_nontrivial_relation_sum_zero_of_not_affine_ind h
replace gpos := exists_pos_of_sum_zero_of_exists_nonzero g gsum gpos
clear h
let s := @Finset.filter _ (fun z => 0 < g z) (fun _ =... | 44 | 12,851,600,114,359,308,000 | 2 | 1.5 | 2 | 1,607 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.caratheodory from "leanprover-community/mathlib"@"e6fab1dc073396d45da082c644642c4f8bff2264"
open Set Finset
universe u
variable {๐ : Type*} {E : Type u} ... | Mathlib/Analysis/Convex/Caratheodory.lean | 119 | 121 | theorem minCardFinsetOfMemConvexHull_nonempty : (minCardFinsetOfMemConvexHull hx).Nonempty := by |
rw [โ Finset.coe_nonempty, โ @convexHull_nonempty_iff ๐]
exact โจx, mem_minCardFinsetOfMemConvexHull hxโฉ
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,607 |
import Mathlib.Topology.Instances.Irrational
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.Compactification.OnePoint
#align_import topology.instances.rat_lemmas from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Metric Filter TopologicalSpace
open Topology One... | Mathlib/Topology/Instances/RatLemmas.lean | 56 | 62 | theorem not_countably_generated_cocompact : ยฌIsCountablyGenerated (cocompact โ) := by |
intro H
rcases exists_seq_tendsto (cocompact โ โ ๐ 0) with โจx, hxโฉ
rw [tendsto_inf] at hx; rcases hx with โจhxc, hx0โฉ
obtain โจn, hnโฉ : โ n : โ, x n โ insert (0 : โ) (range x) :=
(hxc.eventually hx0.isCompact_insert_range.compl_mem_cocompact).exists
exact hn (Or.inr โจn, rflโฉ)
| 6 | 403.428793 | 2 | 1.5 | 4 | 1,608 |
import Mathlib.Topology.Instances.Irrational
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.Compactification.OnePoint
#align_import topology.instances.rat_lemmas from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Metric Filter TopologicalSpace
open Topology One... | Mathlib/Topology/Instances/RatLemmas.lean | 65 | 69 | theorem not_countably_generated_nhds_infty_opc : ยฌIsCountablyGenerated (๐ (โ : โโ)) := by |
intro
have : IsCountablyGenerated (comap (OnePoint.some : โ โ โโ) (๐ โ)) := by infer_instance
rw [OnePoint.comap_coe_nhds_infty, coclosedCompact_eq_cocompact] at this
exact not_countably_generated_cocompact this
| 4 | 54.59815 | 2 | 1.5 | 4 | 1,608 |
import Mathlib.Topology.Instances.Irrational
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.Compactification.OnePoint
#align_import topology.instances.rat_lemmas from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Metric Filter TopologicalSpace
open Topology One... | Mathlib/Topology/Instances/RatLemmas.lean | 72 | 74 | theorem not_firstCountableTopology_opc : ยฌFirstCountableTopology โโ := by |
intro
exact not_countably_generated_nhds_infty_opc inferInstance
| 2 | 7.389056 | 1 | 1.5 | 4 | 1,608 |
import Mathlib.Topology.Instances.Irrational
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.Compactification.OnePoint
#align_import topology.instances.rat_lemmas from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Metric Filter TopologicalSpace
open Topology One... | Mathlib/Topology/Instances/RatLemmas.lean | 77 | 79 | theorem not_secondCountableTopology_opc : ยฌSecondCountableTopology โโ := by |
intro
exact not_firstCountableTopology_opc inferInstance
| 2 | 7.389056 | 1 | 1.5 | 4 | 1,608 |
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.Algebra.Exact
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.Derivation
#align_import ring_theory.kaehler from "leanprover-community/mathli... | Mathlib/RingTheory/Kaehler.lean | 62 | 63 | theorem KaehlerDifferential.one_smul_sub_smul_one_mem_ideal (a : S) :
(1 : S) โโ[R] a - a โโ[R] (1 : S) โ KaehlerDifferential.ideal R S := by | simp [RingHom.mem_ker]
| 1 | 2.718282 | 0 | 1.5 | 4 | 1,609 |
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.Algebra.Exact
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.Derivation
#align_import ring_theory.kaehler from "leanprover-community/mathli... | Mathlib/RingTheory/Kaehler.lean | 78 | 99 | theorem Derivation.tensorProductTo_mul (D : Derivation R S M) (x y : S โ[R] S) :
D.tensorProductTo (x * y) =
TensorProduct.lmul' (S := S) R x โข D.tensorProductTo y +
TensorProduct.lmul' (S := S) R y โข D.tensorProductTo x := by |
refine TensorProduct.induction_on x ?_ ?_ ?_
ยท rw [zero_mul, map_zero, map_zero, zero_smul, smul_zero, add_zero]
swap
ยท intro xโ yโ hโ hโ
rw [add_mul, map_add, map_add, map_add, add_smul, smul_add, hโ, hโ, add_add_add_comm]
intro xโ xโ
refine TensorProduct.induction_on y ?_ ?_ ?_
ยท rw [mul_zero, map_... | 18 | 65,659,969.137331 | 2 | 1.5 | 4 | 1,609 |
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.Algebra.Exact
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.Derivation
#align_import ring_theory.kaehler from "leanprover-community/mathli... | Mathlib/RingTheory/Kaehler.lean | 105 | 128 | theorem KaehlerDifferential.submodule_span_range_eq_ideal :
Submodule.span S (Set.range fun s : S => (1 : S) โโ[R] s - s โโ[R] (1 : S)) =
(KaehlerDifferential.ideal R S).restrictScalars S := by |
apply le_antisymm
ยท rw [Submodule.span_le]
rintro _ โจs, rflโฉ
exact KaehlerDifferential.one_smul_sub_smul_one_mem_ideal _ _
ยท rintro x (hx : _ = _)
have : x - TensorProduct.lmul' (S := S) R x โโ[R] (1 : S) = x := by
rw [hx, TensorProduct.zero_tmul, sub_zero]
rw [โ this]
clear this hx
... | 21 | 1,318,815,734.483215 | 2 | 1.5 | 4 | 1,609 |
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.Algebra.Exact
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.Derivation
#align_import ring_theory.kaehler from "leanprover-community/mathli... | Mathlib/RingTheory/Kaehler.lean | 131 | 141 | theorem KaehlerDifferential.span_range_eq_ideal :
Ideal.span (Set.range fun s : S => (1 : S) โโ[R] s - s โโ[R] (1 : S)) =
KaehlerDifferential.ideal R S := by |
apply le_antisymm
ยท rw [Ideal.span_le]
rintro _ โจs, rflโฉ
exact KaehlerDifferential.one_smul_sub_smul_one_mem_ideal _ _
ยท change (KaehlerDifferential.ideal R S).restrictScalars S โค (Ideal.span _).restrictScalars S
rw [โ KaehlerDifferential.submodule_span_range_eq_ideal, Ideal.span]
conv_rhs => rw ... | 8 | 2,980.957987 | 2 | 1.5 | 4 | 1,609 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Card
#align_import data.multiset.fintype from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
variable {ฮฑ : Type*} [DecidableEq ฮฑ] {m : Multiset ฮฑ}
def Multiset.ToType (m : Multiset ฮฑ) : Type _ := (x : ฮฑ) ร Fi... | Mathlib/Data/Multiset/Fintype.lean | 122 | 126 | theorem Multiset.toEnumFinset_mono {mโ mโ : Multiset ฮฑ} (h : mโ โค mโ) :
mโ.toEnumFinset โ mโ.toEnumFinset := by |
intro p
simp only [Multiset.mem_toEnumFinset]
exact gt_of_ge_of_gt (Multiset.le_iff_count.mp h p.1)
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,610 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Card
#align_import data.multiset.fintype from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
variable {ฮฑ : Type*} [DecidableEq ฮฑ] {m : Multiset ฮฑ}
def Multiset.ToType (m : Multiset ฮฑ) : Type _ := (x : ฮฑ) ร Fi... | Mathlib/Data/Multiset/Fintype.lean | 130 | 141 | theorem Multiset.toEnumFinset_subset_iff {mโ mโ : Multiset ฮฑ} :
mโ.toEnumFinset โ mโ.toEnumFinset โ mโ โค mโ := by |
refine โจfun h โฆ ?_, Multiset.toEnumFinset_monoโฉ
rw [Multiset.le_iff_count]
intro x
by_cases hx : x โ mโ
ยท apply Nat.le_of_pred_lt
have : (x, mโ.count x - 1) โ mโ.toEnumFinset := by
rw [Multiset.mem_toEnumFinset]
exact Nat.pred_lt (ne_of_gt (Multiset.count_pos.mpr hx))
simpa only [Multiset... | 10 | 22,026.465795 | 2 | 1.5 | 2 | 1,610 |
import Mathlib.Analysis.NormedSpace.ConformalLinearMap
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.conformal.normed_space from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
noncomputable section
variable {X Y Z : Type*} [NormedAddCommGroup X] [NormedAdd... | Mathlib/Analysis/Calculus/Conformal/NormedSpace.lean | 73 | 82 | theorem conformalAt_iff_isConformalMap_fderiv {f : X โ Y} {x : X} :
ConformalAt f x โ IsConformalMap (fderiv โ f x) := by |
constructor
ยท rintro โจf', hf, hf'โฉ
rwa [hf.fderiv]
ยท intro H
by_cases h : DifferentiableAt โ f x
ยท exact โจfderiv โ f x, h.hasFDerivAt, Hโฉ
ยท nontriviality X
exact absurd (fderiv_zero_of_not_differentiableAt h) H.ne_zero
| 8 | 2,980.957987 | 2 | 1.5 | 2 | 1,611 |
import Mathlib.Analysis.NormedSpace.ConformalLinearMap
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.conformal.normed_space from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
noncomputable section
variable {X Y Z : Type*} [NormedAddCommGroup X] [NormedAdd... | Mathlib/Analysis/Calculus/Conformal/NormedSpace.lean | 98 | 102 | theorem comp {f : X โ Y} {g : Y โ Z} (x : X) (hg : ConformalAt g (f x)) (hf : ConformalAt f x) :
ConformalAt (g โ f) x := by |
rcases hf with โจf', hfโ, cfโฉ
rcases hg with โจg', hgโ, cgโฉ
exact โจg'.comp f', hgโ.comp x hfโ, cg.comp cfโฉ
| 3 | 20.085537 | 1 | 1.5 | 2 | 1,611 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.RingTheory.Polynomial.Content
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import ring_theory.polynomial.basic from "leanprover-commun... | Mathlib/RingTheory/Polynomial/Basic.lean | 67 | 68 | theorem mem_degreeLE {n : WithBot โ} {f : R[X]} : f โ degreeLE R n โ degree f โค n := by |
simp only [degreeLE, Submodule.mem_iInf, degree_le_iff_coeff_zero, LinearMap.mem_ker]; rfl
| 1 | 2.718282 | 0 | 1.5 | 4 | 1,612 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.RingTheory.Polynomial.Content
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import ring_theory.polynomial.basic from "leanprover-commun... | Mathlib/RingTheory/Polynomial/Basic.lean | 76 | 94 | theorem degreeLE_eq_span_X_pow [DecidableEq R] {n : โ} :
degreeLE R n = Submodule.span R โ((Finset.range (n + 1)).image fun n => (X : R[X]) ^ n) := by |
apply le_antisymm
ยท intro p hp
replace hp := mem_degreeLE.1 hp
rw [โ Polynomial.sum_monomial_eq p, Polynomial.sum]
refine Submodule.sum_mem _ fun k hk => ?_
have := WithBot.coe_le_coe.1 (Finset.sup_le_iff.1 hp k hk)
rw [โ C_mul_X_pow_eq_monomial, C_mul']
refine
Submodule.smul_mem _ _
... | 17 | 24,154,952.753575 | 2 | 1.5 | 4 | 1,612 |
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