module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 512,
"column": 2
} | {
"line": 512,
"column": 29
} | [
{
"pp": "𝕜 : Type u_3\nA : Type u_4\nSA : Type u_5\ninst✝⁵ : NormedRing A\ninst✝⁴ : CompleteSpace A\ninst✝³ : SetLike SA A\ninst✝² : SubringClass SA A\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedAlgebra 𝕜 A\ninstSMulMem : SMulMemClass SA 𝕜 A\nS : SA\nhS : IsClosed ↑S\nl : Filter ↥S\na : ↥S\nha : IsUnit ↑a\nhla :... | apply hl.mono fun x hx ↦ ?_ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 565,
"column": 4
} | {
"line": 567,
"column": 39
} | [
{
"pp": "𝕜 : Type u_3\nA : Type u_4\nSA : Type u_5\ninst✝⁵ : NormedRing A\ninst✝⁴ : CompleteSpace A\ninst✝³ : SetLike SA A\ninst✝² : SubringClass SA A\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedAlgebra 𝕜 A\ninstSMulMem : SMulMemClass SA 𝕜 A\nS : SA\nhS : IsClosed ↑S\nx : ↥S\nthis : CompleteSpace ↥S\n⊢ IsOpen (σ... | rw [← (spectrum.isClosed (𝕜 := 𝕜) x).closure_eq, closure_eq_interior_union_frontier,
union_diff_distrib, diff_eq_empty.mpr (frontier_spectrum S x),
diff_eq_compl_inter, union_empty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Module.WeakDual | {
"line": 307,
"column": 2
} | {
"line": 309,
"column": 88
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set (WeakDual 𝕜 E)\nhb : Bornology.IsBounded s\n⊢ Bornology.IsBounded (closure s)",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
... | obtain ⟨R, hR⟩ := (Metric.isBounded_iff_subset_closedBall (0 : StrongDual 𝕜 E)).mp hb
exact (isBounded_closedBall 0 R).subset
(closure_minimal (fun y hy ↦ hR (a := toStrongDual y) hy) (isClosed_closedBall 0 R)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.WeakDual | {
"line": 307,
"column": 2
} | {
"line": 309,
"column": 88
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set (WeakDual 𝕜 E)\nhb : Bornology.IsBounded s\n⊢ Bornology.IsBounded (closure s)",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
... | obtain ⟨R, hR⟩ := (Metric.isBounded_iff_subset_closedBall (0 : StrongDual 𝕜 E)).mp hb
exact (isBounded_closedBall 0 R).subset
(closure_minimal (fun y hy ↦ hR (a := toStrongDual y) hy) (isClosed_closedBall 0 R)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Ordered | {
"line": 85,
"column": 29
} | {
"line": 85,
"column": 71
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝⁸ : Ring k\ninst✝⁷ : PartialOrder k\ninst✝⁶ : IsOrderedRing k\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : PartialOrder E\ninst✝³ : IsOrderedAddMonoid E\ninst✝² : Module k E\ninst✝¹ : IsStrictOrderedModule k E\na b : E\nr r' : k\ninst✝ : PosSMulReflectLT k E\nh : r < r'\n⊢ (r' - ... | smul_lt_smul_iff_of_pos_left (sub_pos.2 h) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.ContinuousSqrt | {
"line": 35,
"column": 22
} | {
"line": 35,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nx : 𝕜 × 𝕜\nhx : 0 ≤ x.2 - x.1\nhx' : ↑(re (x.2 - x.1)) = x.2 - x.1\n⊢ x.2 = x.1 + ↑(√(re (x.2 - x.1)) * √(re (x.2 - x.1)))",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"Real.instAddMonoid",
"c... | Real.mul_self_sqrt, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 166,
"column": 8
} | {
"line": 166,
"column": 14
} | [
{
"pp": "case h\nA : Type u_1\ninst✝ : CommCStarAlgebra A\na : A\n⊢ (gelfandTransform ℂ A).toRingHom (star a) = star ((gelfandTransform ℂ A).toRingHom a)",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"NormedCommRing.toCommRing",
"NonUnitalCommCStarAlgebra.toNonUnitalCStarAlge... | ext1 φ | Lean.Elab.Tactic.Ext._aux_Init_Ext___macroRules_Lean_Elab_Tactic_Ext_tacticExt1____1 | Lean.Elab.Tactic.Ext.tacticExt1___ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi | {
"line": 135,
"column": 4
} | {
"line": 136,
"column": 78
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝²³ : CommSemiring R\ninst✝²² : StarRing R\ninst✝²¹ : MetricSpace R\ninst✝²⁰ : IsTopologicalSemiring R\ninst✝¹⁹ : ContinuousStar R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra R S\ninst✝¹⁶ : Ring A\ninst✝¹⁵ : Ring B\ninst✝¹⁴ : Algebra S A\ninst✝¹³... | let φ := StarAlgHom.snd S A B
exact φ.map_cfc f (a, b) (by rwa [Prod.spectrum_eq]) continuous_snd hab hb | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi | {
"line": 135,
"column": 4
} | {
"line": 136,
"column": 78
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝²³ : CommSemiring R\ninst✝²² : StarRing R\ninst✝²¹ : MetricSpace R\ninst✝²⁰ : IsTopologicalSemiring R\ninst✝¹⁹ : ContinuousStar R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra R S\ninst✝¹⁶ : Ring A\ninst✝¹⁵ : Ring B\ninst✝¹⁴ : Algebra S A\ninst✝¹³... | let φ := StarAlgHom.snd S A B
exact φ.map_cfc f (a, b) (by rwa [Prod.spectrum_eq]) continuous_snd hab hb | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric | {
"line": 72,
"column": 27
} | {
"line": 75,
"column": 17
} | [
{
"pp": "A : Type u_1\ninst✝⁶ : NormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedAlgebra ℝ A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonnegSpectrumClass ℝ A\ninst✝ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\na : A\nr : ℝ\nhr : 0 < r\nha : 0 ≤ a\n⊢ ‖a ^ r‖₊ = ‖a‖₊ ^ r",
... | by
lift r to ℝ≥0 using hr.le
rw [← nnrpow_eq_rpow, ← nnnorm_nnrpow a]
all_goals simpa | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric | {
"line": 112,
"column": 42
} | {
"line": 113,
"column": 64
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : CStarRing A\ninst✝⁵ : NormedSpace ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : N... | by
simpa [hb.star_eq] using norm_mul_mul_star_self_of_nonneg b ha | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 217,
"column": 21
} | {
"line": 217,
"column": 42
} | [
{
"pp": "case add\nX : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : ContinuousStar A\ninst✝ : TopologicalSpace X\ns : Set 𝕜\nhs : IsCompact s\na :... | simpa using hf.add hg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 217,
"column": 21
} | {
"line": 217,
"column": 42
} | [
{
"pp": "case add\nX : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : ContinuousStar A\ninst✝ : TopologicalSpace X\ns : Set 𝕜\nhs : IsCompact s\na :... | simpa using hf.add hg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 217,
"column": 21
} | {
"line": 217,
"column": 42
} | [
{
"pp": "case add\nX : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : ContinuousStar A\ninst✝ : TopologicalSpace X\ns : Set 𝕜\nhs : IsCompact s\na :... | simpa using hf.add hg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 298,
"column": 23
} | {
"line": 298,
"column": 32
} | [
{
"pp": "A : Type u_1\ninst✝¹¹ : PartialOrder A\ninst✝¹⁰ : NonUnitalRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : StarRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : No... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 990,
"column": 2
} | {
"line": 990,
"column": 77
} | [
{
"pp": "X : Type u_1\nA : Type u_2\ninst✝¹² : NonUnitalNormedRing A\ninst✝¹¹ : StarRing A\ninst✝¹⁰ : NormedSpace ℝ A\ninst✝⁹ : IsScalarTower ℝ A A\ninst✝⁸ : SMulCommClass ℝ A A\ninst✝⁷ : ContinuousStar A\ninst✝⁶ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁵ : PartialOrder A\ninst✝⁴... | refine fun x hx ↦ (ha_cont x hx).cfcₙ_nnreal (hs x hx) f hx ?_ ?_ (hf x hx) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Matrix.Normed | {
"line": 296,
"column": 2
} | {
"line": 297,
"column": 6
} | [
{
"pp": "m : Type u_3\nα : Type u_5\nι : Type u_7\ninst✝² : Fintype m\ninst✝¹ : Unique ι\ninst✝ : SeminormedAddCommGroup α\nv : m → α\n⊢ ‖replicateCol ι v‖₊ = ‖v‖₊",
"usedConstants": [
"Eq.mpr",
"Inhabited.default",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"Finse... | rw [linfty_opNNNorm_def, Pi.nnnorm_def]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Matrix.Normed | {
"line": 296,
"column": 2
} | {
"line": 297,
"column": 6
} | [
{
"pp": "m : Type u_3\nα : Type u_5\nι : Type u_7\ninst✝² : Fintype m\ninst✝¹ : Unique ι\ninst✝ : SeminormedAddCommGroup α\nv : m → α\n⊢ ‖replicateCol ι v‖₊ = ‖v‖₊",
"usedConstants": [
"Eq.mpr",
"Inhabited.default",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"Finse... | rw [linfty_opNNNorm_def, Pi.nnnorm_def]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Matrix.Normed | {
"line": 450,
"column": 2
} | {
"line": 451,
"column": 30
} | [
{
"pp": "case inr\nm : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : NontriviallyNormedField α\ninst✝ : NormedAlgebra ℝ α\nA : Matrix m n α\nN : ℝ≥0\ni : m\nx✝ : i ∈ Finset.univ\nh✝ : Nonempty n\nx : n → α := fun j ↦ unitOf (A i j)\nhxn : ‖x‖₊ = 1\nhN : ‖∑ i_1, A i i_1 *... | simp_rw [x, mul_unitOf, ← map_sum, nnnorm_algebraMap, ← NNReal.coe_sum, NNReal.nnnorm_eq,
nnnorm_one, mul_one] at hN | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Matrix.Normed | {
"line": 614,
"column": 23
} | {
"line": 614,
"column": 98
} | [
{
"pp": "n : Type u_4\nα : Type u_5\ninst✝² : Fintype n\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : DecidableEq n\nv : n → α\ns : Finset (n × n) := Finset.map { toFun := fun i ↦ (i, i), inj' := ⋯ } Finset.univ\ni : n × n\n_hi : i ∈ Finset.univ\nhis : i ∉ s\nthis : i.1 ≠ i.2\n⊢ ‖diagonal v i.1 i.2‖₊ ^ 2 = 0",
... | by rw [diagonal_apply_ne _ this, nnnorm_zero, NNReal.zero_rpow two_ne_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.PositiveLinearMap | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 33
} | [
{
"pp": "A₁ : Type u_1\nA₂ : Type u_2\ninst✝⁵ : NonUnitalCStarAlgebra A₁\ninst✝⁴ : NonUnitalCStarAlgebra A₂\ninst✝³ : PartialOrder A₁\ninst✝² : StarOrderedRing A₁\ninst✝¹ : PartialOrder A₂\ninst✝ : StarOrderedRing A₂\nf : A₁ →ₚ[ℂ] A₂\nC : ℝ≥0\nhmain : ∀ (a : A₁), 0 ≤ a → ‖f a‖ ≤ ↑C * ‖a‖\nx : A₁\ny : Fin 4 → A₁... | simp only [map_sum, map_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 93,
"column": 2
} | {
"line": 95,
"column": 26
} | [
{
"pp": "⊢ InvOn (fun x ↦ 1 - (1 + x)⁻¹) (fun x ↦ x * (1 - x)⁻¹) {x | x < 1} {x | x < 1}",
"usedConstants": [
"zero_le",
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Mat... | have : (fun x : ℝ≥0 ↦ x * (1 + x)⁻¹) = fun x ↦ 1 - (1 + x)⁻¹ := by
ext x : 1
simp [field, mul_tsub] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 459,
"column": 43
} | {
"line": 467,
"column": 70
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na b : A\nhb : IsSelfAdjoint b\n⊢ star a * b * a ≤ ‖b‖ • (star a * a)",
"usedConstants": [
"Unitization.instSMul",
"Norm.norm",
"Eq.mpr",
"NonUnitalCStarAlgebra.toStarModule",
... | by
suffices ∀ a b : A⁺¹, IsSelfAdjoint b → star a * b * a ≤ ‖b‖ • (star a * a) by
rw [← Unitization.inr_le_iff _ _ (by aesop) ((IsSelfAdjoint.all _).smul (.star_mul_self a))]
simpa [Unitization.norm_inr] using this a b <| hb.inr ℂ
intro a b hb
calc
star a * b * a ≤ star a * (algebraMap ℝ A⁺¹ ‖b‖) * a ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.CompletelyPositiveMap | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 52
} | [
{
"pp": "F : Type u_1\nA₁ : Type u_2\nA₂ : Type u_3\ninst✝⁷ : NonUnitalCStarAlgebra A₁\ninst✝⁶ : NonUnitalCStarAlgebra A₂\ninst✝⁵ : PartialOrder A₁\ninst✝⁴ : PartialOrder A₂\ninst✝³ : StarOrderedRing A₁\ninst✝² : StarOrderedRing A₂\ninst✝¹ : FunLike F A₁ A₂\ninst✝ : LinearMapClass F ℂ A₁ A₂\nh : ∀ (φ : F) (k : ... | exact map_nonneg (toOneByOne (Fin 1) ℂ A₂).symm h₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.Dual | {
"line": 206,
"column": 75
} | {
"line": 207,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\n⊢ 0♯ = 0",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"InnerProductSpace.toDual",
"NormedCommRing.toSeminormedCommRing",
"Cont... | by
simp [continuousLinearMapOfBilin] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range | {
"line": 246,
"column": 94
} | {
"line": 249,
"column": 42
} | [
{
"pp": "A : Type u_1\ninst✝¹⁴ : NonUnitalRing A\ninst✝¹³ : StarRing A\ninst✝¹² : Module ℝ A\ninst✝¹¹ : IsScalarTower ℝ A A\ninst✝¹⁰ : SMulCommClass ℝ A A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : IsTopologicalRing A\ninst✝⁷ : T2Space A\ninst✝⁶ : PartialOrder A\ninst✝⁵ : NonnegSpectrumClass ℝ A\ninst✝⁴ : StarOrder... | by
grw [range_cfcₙ_nnreal_eq_image_cfcₙ_real a ha, Set.setOf_and, SetLike.setOf_mem_eq,
← range_cfcₙ_subset _ ha.isSelfAdjoint, Set.inter_comm, ← Set.image_preimage_eq_inter_range]
exact Set.image_mono fun _ ↦ cfcₙ_nonneg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Completion | {
"line": 74,
"column": 19
} | {
"line": 74,
"column": 52
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\n⊢ ((innerₛₗ 𝕜) 0).toAddMonoidHom = 0",
"usedConstants": [
"LinearMap.toAddMonoidHom",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
... | by ext x; exact inner_zero_left _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal | {
"line": 181,
"column": 81
} | {
"line": 185,
"column": 41
} | [
{
"pp": "A : Type u_2\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nf : A →ₚ[ℂ] ℂ\n⊢ f.gnsNonUnitalStarAlgHom 1 = 1",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"CStarAlgebra.toNonUnitalCStarAlgebra",
"... | by
ext b
induction b using Completion.induction_on with
| hp => apply isClosed_eq <;> fun_prop
| ih b => simp [gnsNonUnitalStarAlgHom] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Circle | {
"line": 206,
"column": 29
} | {
"line": 206,
"column": 38
} | [
{
"pp": "⊢ Circle.exp (2 * π * 0) = 1",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"MulZeroClass.toMul",
"Real.instAddMonoid",
"Monoid.toMulOneClass",
"congrArg",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Circle | {
"line": 207,
"column": 32
} | {
"line": 207,
"column": 40
} | [
{
"pp": "x y : ℝ\n⊢ Circle.exp (2 * π * (x + y)) = Circle.exp (2 * π * x) * Circle.exp (2 * π * y)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"AddMonoid.toAddSemigroup",... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 49
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Opens X\nhs : Nonempty ↥s\nf f' : OpenPartialHomeomorph X Y\nopenness₁ : IsOpen (f.target ∩ ↑f.symm ⁻¹' ↑s)\nset_identity : f.symm.source ∩ (f.target ∩ ↑f.symm ⁻¹' ↑s) = f.symm.source ∩ ↑f.symm ⁻¹' ↑s\nopenness₂ : ... | refine EqOnSource.trans' (eqOnSource_refl _) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 604,
"column": 4
} | {
"line": 605,
"column": 13
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace F\nproj : Z → B\ninst✝² : TopologicalSpace Z\ninst✝¹ : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\nZ' : Type u_5\ninst✝ : TopologicalSpace Z'\nh : Z' ≃ₜ Z\np : Z'\... | have hp : h p ∈ e.source := by simpa using hp
simp [hp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 604,
"column": 4
} | {
"line": 605,
"column": 13
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace F\nproj : Z → B\ninst✝² : TopologicalSpace Z\ninst✝¹ : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\nZ' : Type u_5\ninst✝ : TopologicalSpace Z'\nh : Z' ≃ₜ Z\np : Z'\... | have hp : h p ∈ e.source := by simpa using hp
simp [hp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.IsLocalHomeomorph | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 30
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nh : ∀ x ∈ s, ∃ e, x ∈ e.source ∧ Set.EqOn f (↑e) e.source\nx : X\nhx : x ∈ s\n⊢ ∃ e, x ∈ e.source ∧ f = ↑e",
"usedConstants": []
}
] | obtain ⟨e, hx, he⟩ := h x hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.FiberBundle.Basic | {
"line": 617,
"column": 4
} | {
"line": 621,
"column": 42
} | [
{
"pp": "case refine_2\nι : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\nv : F\nh : ∀ (i j : ι), ∀ x ∈ Z.baseSet i ∩ Z.baseSet j, Z.coordChange i j x v = v\nx : B\nA : Z.baseSet (Z.indexAt x) ∈ 𝓝 x\n⊢ ContinuousAt\n (↑(Z.localTrivA... | apply continuousAt_id.prodMk
simp only [mfld_simps]
have : ContinuousOn (fun _ : B => v) (Z.baseSet (Z.indexAt x)) := continuousOn_const
refine (this.congr fun y hy ↦ ?_).continuousAt A
exact h _ _ _ ⟨mem_baseSet_at _ _, hy⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.FiberBundle.Basic | {
"line": 617,
"column": 4
} | {
"line": 621,
"column": 42
} | [
{
"pp": "case refine_2\nι : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\nv : F\nh : ∀ (i j : ι), ∀ x ∈ Z.baseSet i ∩ Z.baseSet j, Z.coordChange i j x v = v\nx : B\nA : Z.baseSet (Z.indexAt x) ∈ 𝓝 x\n⊢ ContinuousAt\n (↑(Z.localTrivA... | apply continuousAt_id.prodMk
simp only [mfld_simps]
have : ContinuousOn (fun _ : B => v) (Z.baseSet (Z.indexAt x)) := continuousOn_const
refine (this.congr fun y hy ↦ ?_).continuousAt A
exact h _ _ _ ⟨mem_baseSet_at _ _, hy⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.IsLocalHomeomorph | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 30
} | [
{
"pp": "case refine_2\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\ng : Y → Z\ns : Set X\nx : X\nhx : x ∈ s\nf : OpenPartialHomeomorph X Y\nhxf : x ∈ f.source\nhgf✝ : IsLocalHomeomorphOn (g ∘ ↑f) s\nhf : IsLocalHomeomorphOn (↑f)... | change g y = gf (f.symm y) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Topology.FiberBundle.Basic | {
"line": 872,
"column": 4
} | {
"line": 872,
"column": 81
} | [
{
"pp": "case refine_2\nB : Type u_2\nF : Type u_3\nE : B → Type u_5\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace F\ninst✝¹ : (x : B) → TopologicalSpace (E x)\na : FiberPrebundle F E\nX : Type u_6\ninst✝ : TopologicalSpace X\nf : TotalSpace F E → X\ns : Set B\nhs : IsOpen s\nhf : ∀ b ∈ s, ContinuousO... | exact (hs.inter (a.pretrivializationAt z.proj).open_baseSet).prod isOpen_univ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.FiberBundle.Basic | {
"line": 872,
"column": 4
} | {
"line": 872,
"column": 81
} | [
{
"pp": "case refine_2\nB : Type u_2\nF : Type u_3\nE : B → Type u_5\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace F\ninst✝¹ : (x : B) → TopologicalSpace (E x)\na : FiberPrebundle F E\nX : Type u_6\ninst✝ : TopologicalSpace X\nf : TotalSpace F E → X\ns : Set B\nhs : IsOpen s\nhf : ∀ b ∈ s, ContinuousO... | exact (hs.inter (a.pretrivializationAt z.proj).open_baseSet).prod isOpen_univ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.FiberBundle.Basic | {
"line": 872,
"column": 4
} | {
"line": 872,
"column": 81
} | [
{
"pp": "case refine_2\nB : Type u_2\nF : Type u_3\nE : B → Type u_5\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace F\ninst✝¹ : (x : B) → TopologicalSpace (E x)\na : FiberPrebundle F E\nX : Type u_6\ninst✝ : TopologicalSpace X\nf : TotalSpace F E → X\ns : Set B\nhs : IsOpen s\nhf : ∀ b ∈ s, ContinuousO... | exact (hs.inter (a.pretrivializationAt z.proj).open_baseSet).prod isOpen_univ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 186,
"column": 10
} | {
"line": 186,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NonUnitalNormedRing A\ninst✝² : NormedSpace 𝕜 A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : IsScalarTower 𝕜 A A\nn : ℕ\nx y : A\n⊢ (↑n).2 x * y = x * (↑n).1 y",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq... | Prod.snd_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 605,
"column": 8
} | {
"line": 611,
"column": 71
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁸ : DenselyNormedField 𝕜\ninst✝⁷ : StarRing 𝕜\ninst✝⁶ : NonUnitalNormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : CStarRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : SMulCommClass 𝕜 A A\ninst✝¹ : IsScalarTower 𝕜 A A\ninst✝ : StarModule 𝕜 A\na : 𝓜(𝕜, A)\nhball : (Metric.... | calc
‖star (a.fst (star x)) * a.fst y‖₊ ≤ ‖a.fst (star x)‖₊ * ‖a.fst y‖₊ :=
nnnorm_star (a.fst (star x)) ▸ nnnorm_mul_le _ _
_ ≤ ‖a.fst‖₊ * 1 * (‖a.fst‖₊ * 1) :=
(mul_le_mul' (a.fst.le_opNorm_of_le ((nnnorm_star x).trans_le hx))
(a.fst.le_opNorm_of_le hy))
... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 78
} | [
{
"pp": "case inr\nT : ℝ\nhT : T ≠ 0\n⊢ Periodic (fun x ↦ Circle.exp (2 * π / T * x)) T",
"usedConstants": [
"Circle.periodic_exp",
"Distrib.leftDistribClass",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"div_mul_cancel₀",
"Real",
"instHDiv",
"NonUnitalCommRin... | · intro x; simp_rw [mul_add]; rw [div_mul_cancel₀ _ hT, Circle.periodic_exp] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 149,
"column": 41
} | {
"line": 149,
"column": 49
} | [
{
"pp": "case H.H\nT z✝¹ z✝ : ℝ\n⊢ Circle.exp (2 * π / T * (z✝¹ + z✝)) = Circle.exp (2 * π / T * z✝¹) * Circle.exp (2 * π / T * z✝)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 152,
"column": 53
} | {
"line": 152,
"column": 62
} | [
{
"pp": "T : ℝ\n⊢ Circle.exp (2 * π / T * 0) = 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"InvOneClass.toOne",
"HMul.hMul",
"DivisionCommMonoid.toDivisi... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Basic | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 50
} | [
{
"pp": "A : Type u_1\ninst✝⁶ : NormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedAlgebra ℝ A\ninst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : NonnegSpectrumClass ℝ A\na : A\nha : IsStrictlyPositive a\n⊢ exp (log a) = a",
"usedConstants":... | have ha₂ : ∀ x ∈ spectrum ℝ a, x ≠ 0 := by grind | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 574,
"column": 4
} | {
"line": 574,
"column": 47
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\nhmem1 : s.medial.points 0 ∈ affineSpan k (Set.range s.medial.points)\nhmem2 : s.medial.points 0 ∈ a... | rw [this, Submodule.span_smul_eq_of_isUnit] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Affine.AddTorsorBases | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 53
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\ninst✝² : Finite ι\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : AffineBasis ι ℝ E\nh✝ : Nontrivial ι\nthis : FiniteDimensional ℝ E\n⊢ (convexHull ℝ) (range ⇑b) = ⋂ i, ⇑(b.coord i) ⁻¹' Ici 0",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
... | rw [b.convexHull_eq_nonneg_coord, setOf_forall] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Affine.AddTorsorBases | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 45
} | [
{
"pp": "case refine_1\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt... | · intro p hp; use ⟨p, ht₁ hp⟩; simp [w, hp] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Affine.AddTorsorBases | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 21
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\nι : Type u_3\ninst✝ : Fintype ι\nb : AffineBasis ι ℝ V\n⊢ Finset.centroid ℝ Finset.univ ⇑b ∈ interior ((convexHull ℝ) (range ⇑b))",
"usedConstants": [
"Real",
"NormedSpace.toModule",
"Real.instRing",
"Nor... | haveI := b.nonempty | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 457,
"column": 66
} | {
"line": 480,
"column": 45
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\n⊢ Collinear k s ↔ ∃ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀",
"usedConstants": [
"Eq.mpr",
"IsNoetherianRing.strongRankCondit... | by
simp_rw [collinear_iff_rank_le_one, rank_submodule_le_one_iff', Submodule.le_span_singleton_iff]
constructor
· rintro ⟨v₀, hv⟩
use v₀
intro p hp
obtain ⟨r, hr⟩ := hv (p -ᵥ p₀) (vsub_mem_vectorSpan k hp h)
use r
rw [eq_vadd_iff_vsub_eq]
exact hr.symm
· rintro ⟨v, hp₀v⟩
use v
in... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.BumpFunction.Basic | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 57
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : HasContDiffBump E\nc : E\nf : ContDiffBump c\nx : E\nhx : x ∈ closedBall c f.rIn\n⊢ ↑f x = 1",
"usedConstants": [
"Real",
"instHSMul",
"instHDiv",
"someContDiffBumpBase",
"Real.instInv",
... | apply ContDiffBumpBase.eq_one _ _ f.one_lt_rOut_div_rIn | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Group.Integral | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 16
} | [
{
"pp": "G : Type u_4\nF : Type u_6\ninst✝⁴ : MeasurableSpace G\ninst✝³ : NormedAddCommGroup F\nμ : Measure G\ninst✝² : Group G\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsInvInvariant\nf : G → F\ns : Set G\nhf : IntegrableOn f s μ\nthis : s⁻¹ = ⇑(MeasurableEquiv.inv G) ⁻¹' s\n⊢ Integrable f ((map (⇑(MeasurableEquiv... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Group.Integral | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 26
} | [
{
"pp": "G : Type u_4\nF : Type u_6\ninst✝⁴ : MeasurableSpace G\ninst✝³ : NormedAddCommGroup F\nμ : Measure G\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\nf : G → F\ninst✝ : μ.IsMulRightInvariant\nhf : Integrable f μ\ng : G\n⊢ Integrable (fun t ↦ f (t / g)) μ",
"usedConstants": [
"Eq.mpr",
"DivI... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 118,
"column": 2
} | {
"line": 167,
"column": 92
} | [
{
"pp": "ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\nf' : ι → E → E →L[𝕜] G\nx : E\n... | letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
letI : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _
rw [SeminormedAddGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_zero] at hf' ⊢
suffices
TendstoUniformlyOnFilter (fun (n : ι × ι) (z : E) => f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) 0
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 118,
"column": 2
} | {
"line": 167,
"column": 92
} | [
{
"pp": "ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\nf' : ι → E → E →L[𝕜] G\nx : E\n... | letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
letI : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _
rw [SeminormedAddGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_zero] at hf' ⊢
suffices
TendstoUniformlyOnFilter (fun (n : ι × ι) (z : E) => f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) 0
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 105,
"column": 49
} | {
"line": 105,
"column": 63
} | [
{
"pp": "α : Type u_3\nE : α → Type u_4\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhf : Summable fun i ↦ ‖f i‖ ^ ENNReal.toReal 0\n⊢ Summable fun x ↦ 1",
"usedConstants": [
"Norm.norm",
"Real.instPow",
"Real",
"Real.instZero",
"congrArg",
"SummationFi... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 108,
"column": 49
} | {
"line": 108,
"column": 63
} | [
{
"pp": "α : Type u_3\nE : α → Type u_4\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhf : Summable fun i ↦ ‖f i‖ ^ ∞.toReal\n⊢ Summable fun x ↦ 1",
"usedConstants": [
"Norm.norm",
"Real.instPow",
"Real",
"Real.instZero",
"congrArg",
"SummationFilter",
... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 93,
"column": 48
} | {
"line": 93,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace E\nA : E →L[𝕜] F\nx : E\ny : F\n⊢ (starRingEnd 𝕜) ⟪y, A x⟫_𝕜 = ⟪A x, y⟫_𝕜",
"usedC... | inner_conj_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 396,
"column": 2
} | {
"line": 400,
"column": 78
} | [
{
"pp": "case inr.inl\nα : Type u_3\nE : α → Type u_4\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : ↥(lp E ∞)\n⊢ 0 ≤ ‖f‖",
"usedConstants": [
"lp.isLUB_norm",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Inhabited.default",
"Real.instLE",
"Real",
"R... | · rcases isEmpty_or_nonempty α with _i | _i
· rw [lp.norm_eq_ciSup]
simp [Real.iSup_of_isEmpty]
inhabit α
exact (norm_nonneg (f default)).trans ((lp.isLUB_norm f).1 ⟨default, rfl⟩) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 205,
"column": 41
} | {
"line": 208,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nT U : E →L[𝕜] E\nhT : (↑T).IsSymmetric\nhU : (↑U).IsSymmetric\n⊢ (↑U).ker ≤ (↑T).ker ↔ (↑T).range ≤ (↑U).range",
"usedConstants": [
"LinearMap.IsSy... | by
refine ⟨fun h ↦ ?_, LinearMap.ker_le_ker_of_range hT hU⟩
have := FiniteDimensional.complete 𝕜 E
simpa [orthogonal_ker, hT, hU] using Submodule.orthogonal_le h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 613,
"column": 2
} | {
"line": 627,
"column": 24
} | [
{
"pp": "case inr.inr\n𝕜 : Type u_1\nα : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝³ : (i : α) → NormedAddCommGroup (E i)\ninst✝² : NormedRing 𝕜\ninst✝¹ : (i : α) → Module 𝕜 (E i)\ninst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : ↥(lp E p)\nhp : 0 < p.toReal\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"... | · letI inst : NNNorm (lp E p) := ⟨fun f => ⟨‖f‖, norm_nonneg' _⟩⟩
have coe_nnnorm : ∀ f : lp E p, ↑‖f‖₊ = ‖f‖ := fun _ => rfl
suffices ‖c • f‖₊ ^ p.toReal ≤ (‖c‖₊ * ‖f‖₊) ^ p.toReal by
rwa [NNReal.rpow_le_rpow_iff hp] at this
clear_value inst
rw [NNReal.mul_rpow]
have hLHS := lp.hasSum_norm hp... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Sqrt | {
"line": 50,
"column": 38
} | {
"line": 50,
"column": 47
} | [
{
"pp": "case inl\nx : ℝ\nhx✝ : x ≠ 0\nhx : x < 0\n⊢ HasStrictDerivAt (fun x ↦ √x) (1 / (2 * 0)) x ∧ ∀ (n : WithTop ℕ∞), ContDiffAt ℝ n (fun x ↦ √x) x",
"usedConstants": [
"ContDiffAt",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"NonUnitalCommRin... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Ball.Homeomorph | {
"line": 131,
"column": 19
} | {
"line": 131,
"column": 36
} | [
{
"pp": "E : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nP : Type u_2\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nr : ℝ\n⊢ (if h : 0 < r then univUnitBall.trans' (unitBallBall c r h) ⋯\n else (IsometryEquiv.vaddConst c).toHomeomorph.toOpenPartialHomeomorph)... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 1219,
"column": 4
} | {
"line": 1219,
"column": 33
} | [
{
"pp": "case mpr\nα : Type u_3\nι : Type u_5\ninst✝ : PseudoMetricSpace α\nf : α → ↥(lp (fun i ↦ ℝ) ∞)\ns : Set α\nK : ℝ≥0\nhgl : ∀ (i : ι), ∀ x ∈ s, ∀ y ∈ s, dist (↑(f x) i) (↑(f y) i) ≤ ↑K * dist x y\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\n⊢ ‖f x - f y‖ ≤ ↑K * dist x y",
"usedConstants": [
"NormedCo... | apply lp.norm_le_of_forall_le | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.SpecialFunctions.SmoothTransition | {
"line": 108,
"column": 8
} | {
"line": 108,
"column": 27
} | [
{
"pp": "case inr.inl\np : ℝ[X]\n⊢ HasDerivAt (fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x)\n (Polynomial.eval 0⁻¹ (X ^ 2 * (p - derivative p)) * expNegInvGlue 0) 0",
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Polynomial.eval",
"NormedCommRing.toSeminormedCommRing"... | expNegInvGlue.zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.SmoothTransition | {
"line": 108,
"column": 28
} | {
"line": 108,
"column": 37
} | [
{
"pp": "case inr.inl\np : ℝ[X]\n⊢ HasDerivAt (fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x) (Polynomial.eval 0⁻¹ (X ^ 2 * (p - derivative p)) * 0) 0",
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Polynomial.eval",
"NormedCommRing.toSeminormedCommRing",
"Real",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Calculus | {
"line": 359,
"column": 68
} | {
"line": 367,
"column": 60
} | [
{
"pp": "n : ℕ∞\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\ny : E\nhy : y ∈ ball 0 1\n⊢ ContDiffWithinAt ℝ (↑n) (↑univUnitBall.symm) (ball 0 1) y",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Iff.mpr",
"sub_pos",
"Normed... | by
apply ContDiffAt.contDiffWithinAt
suffices ContDiffAt ℝ n (fun y : E => (√(1 - ‖y‖ ^ 2 : ℝ))⁻¹) y from this.smul contDiffAt_id
have h : (0 : ℝ) < (1 : ℝ) - ‖(y : E)‖ ^ 2 := by
rwa [mem_ball_zero_iff, ← _root_.abs_one, ← abs_norm, ← sq_lt_sq, one_pow, ← sub_pos] at hy
refine ContDiffAt.inv ?_ (Real.sqrt_n... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.EuclideanDist | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 39
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : IsTopologicalAddGroup E\ninst✝³ : T2Space E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : FiniteDimensional ℝ E\nR : ℝ\ns : Set E\nx : E\nhR : 0 < R\nhs : IsClosed s\nh : ⇑toEuclidean '' s ⊆ Metric.ball (toEuclide... | exact ⟨r, hr, image_subset_iff.1 hsr⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convolution | {
"line": 570,
"column": 4
} | {
"line": 570,
"column": 35
} | [
{
"pp": "𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\nP : Type uP\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace... | refine hg.comp (by fun_prop) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.ParametricIntegral | {
"line": 223,
"column": 2
} | {
"line": 229,
"column": 54
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\ns : ... | have : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (F · a) (ball x₀ ε) := by
apply (h_diff.and h_bound).mono
rintro a ⟨ha_deriv, ha_bound⟩
refine (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le
(fun x x_in ↦ (ha_deriv x (hε x_in)).hasFDerivWithinAt) fun x x_in ↦ ?_
rw [← NNReal.... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convolution | {
"line": 748,
"column": 2
} | {
"line": 749,
"column": 39
} | [
{
"pp": "𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ng : G → E'\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace ... | refine (norm_integral_le_of_norm_le ((L.integrable_comp hif).norm.mul_const ε)
(Eventually.of_forall h2)).trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convolution | {
"line": 770,
"column": 2
} | {
"line": 770,
"column": 69
} | [
{
"pp": "G : Type uG\nE' : Type uE'\ninst✝⁸ : NormedAddCommGroup E'\ng : G → E'\ninst✝⁷ : MeasurableSpace G\nμ : Measure G\ninst✝⁶ : SeminormedAddCommGroup G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : SecondCountableTopology G\ninst✝³ : μ.IsAddLeftInvariant\ninst✝² : SFinite μ\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : Complete... | convert (dist_convolution_le' (lsmul ℝ ℝ) hε hif hf hmg hg).trans _ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.Analysis.Convolution | {
"line": 772,
"column": 13
} | {
"line": 772,
"column": 41
} | [
{
"pp": "case convert_2\nG : Type uG\nE' : Type uE'\ninst✝⁸ : NormedAddCommGroup E'\ng : G → E'\ninst✝⁷ : MeasurableSpace G\nμ : Measure G\ninst✝⁶ : SeminormedAddCommGroup G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : SecondCountableTopology G\ninst✝³ : μ.IsAddLeftInvariant\ninst✝² : SFinite μ\ninst✝¹ : NormedSpace ℝ E'\n... | Real.norm_of_nonneg (hnf _), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Calculus.ContDiff.Convolution | {
"line": 181,
"column": 8
} | {
"line": 181,
"column": 67
} | [
{
"pp": "case refine_1\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\nP : Type uP\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace ℝ... | exact Subset.trans (thickening_mono (min_le_left _ _) _) hε | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.ContDiff.Convolution | {
"line": 181,
"column": 8
} | {
"line": 181,
"column": 67
} | [
{
"pp": "case refine_1\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\nP : Type uP\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace ℝ... | exact Subset.trans (thickening_mono (min_le_left _ _) _) hε | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Convolution | {
"line": 181,
"column": 8
} | {
"line": 181,
"column": 67
} | [
{
"pp": "case refine_1\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\nP : Type uP\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace ℝ... | exact Subset.trans (thickening_mono (min_le_left _ _) _) hε | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 65,
"column": 4
} | {
"line": 68,
"column": 55
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nx : E\nn : ℕ∞\nhs : s ∈ 𝓝 x\nd : ℝ\nd_pos : 0 < d\nhd : Euclidean.closedBall x d ⊆ s\nc : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut... | refine isCompact_of_isClosed_isBounded isClosed_closure ?_
have : IsBounded (Euclidean.closedBall x d) := Euclidean.isCompact_closedBall.isBounded
refine this.subset (Euclidean.isClosed_closedBall.closure_subset_iff.2 ?_)
exact f_supp.trans Euclidean.ball_subset_closedBall | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 65,
"column": 4
} | {
"line": 68,
"column": 55
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nx : E\nn : ℕ∞\nhs : s ∈ 𝓝 x\nd : ℝ\nd_pos : 0 < d\nhd : Euclidean.closedBall x d ⊆ s\nc : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut... | refine isCompact_of_isClosed_isBounded isClosed_closure ?_
have : IsBounded (Euclidean.closedBall x d) := Euclidean.isCompact_closedBall.isBounded
refine this.subset (Euclidean.isClosed_closedBall.closure_subset_iff.2 ?_)
exact f_supp.trans Euclidean.ball_subset_closedBall | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 291,
"column": 48
} | {
"line": 296,
"column": 33
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nD : ℝ\nx : E\n⊢ 0 ≤ w D x",
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZ... | by
apply mul_nonneg _ (u_nonneg _)
apply inv_nonneg.2
apply mul_nonneg (u_int_pos E).le
norm_cast
apply pow_nonneg (abs_nonneg D) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 452,
"column": 20
} | {
"line": 452,
"column": 41
} | [
{
"pp": "case refine_3.hf.hf.h0\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\nx : ℝ × E\nhx : x ∈ Ioo 0 1 ×ˢ univ\n⊢ x.1 ≠ 0",
"usedConstants": ... | exact ne_of_gt hx.1.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 515,
"column": 29
} | {
"line": 521,
"column": 13
} | [
{
"pp": "E✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E✝\ninst✝⁴ : NormedSpace ℝ E✝\ninst✝³ : FiniteDimensional ℝ E✝\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nthis✝¹ : MeasurableSpace E := borel E\nthis✝ : BorelSpace E\nIR : ∀ (R : ℝ), 1 < R → 0 < (R - ... | by
have A : 0 < (R + 1) / 2 := by linarith
have C : (R - 1) / (R + 1) < 1 := by apply (div_lt_one _).2 <;> linarith
simp only [hR, if_true, support_comp_inv_smul₀ A.ne', y_support _ (IR R hR) C,
_root_.smul_ball A.ne', Real.norm_of_nonneg A.le, smul_zero]
refine congr (congr_ar... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 876,
"column": 38
} | {
"line": 881,
"column": 28
} | [
{
"pp": "G : Type u_1\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : Group G\ninst✝⁸ : IsTopologicalGroup G\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : BorelSpace G\ninst✝⁵ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝⁴ : μ.IsHaarMeasure\ninst✝³ : IsFiniteMeasureOnCompacts μ'\ninst✝² : μ'.IsMulLeftInvariant\ninst✝¹ : μ.I... | by
ext s hs
rw [hs.measure_eq_iSup_isCompact, hs.measure_eq_iSup_isCompact]
congr! 4 with K _Ks K_comp
exact measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ K_comp.measure_lt_top.ne
K_comp.measure_lt_top.ne | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 916,
"column": 72
} | {
"line": 921,
"column": 33
} | [
{
"pp": "G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : IsTopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : LocallyCompactSpace G\ninst✝³ : SecondCountableTopology G\nμ ν : Measure G\ninst✝² : SigmaFinite μ\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsHaarMeas... | by
have K : PositiveCompacts G := Classical.arbitrary _
have h : haarMeasure K = (haarScalarFactor (haarMeasure K) ν : ℝ≥0∞) • ν :=
isMulLeftInvariant_eq_smul (haarMeasure K) ν
rw [haarMeasure_unique μ K, h, smul_smul]
exact smul_absolutelyContinuous | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 1047,
"column": 2
} | {
"line": 1048,
"column": 43
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝¹⁰ : Group G\ninst✝⁹ : AddCommGroup A\ninst✝⁸ : DistribMulAction G A\ninst✝⁷ : MeasurableSpace A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : BorelSpace A\ninst✝⁴ : IsTopologicalAddGroup A\ninst✝³ : LocallyCompactSpace A\ninst✝² : ContinuousConstSMul G A\nμ ν : Measure A\nins... | rw [addHaarScalarFactor_eq_mul _ (g • ν), addHaarScalarFactor_domSMul,
mul_comm, ← addHaarScalarFactor_eq_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 1047,
"column": 2
} | {
"line": 1048,
"column": 43
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝¹⁰ : Group G\ninst✝⁹ : AddCommGroup A\ninst✝⁸ : DistribMulAction G A\ninst✝⁷ : MeasurableSpace A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : BorelSpace A\ninst✝⁴ : IsTopologicalAddGroup A\ninst✝³ : LocallyCompactSpace A\ninst✝² : ContinuousConstSMul G A\nμ ν : Measure A\nins... | rw [addHaarScalarFactor_eq_mul _ (g • ν), addHaarScalarFactor_domSMul,
mul_comm, ← addHaarScalarFactor_eq_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 1047,
"column": 2
} | {
"line": 1048,
"column": 43
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝¹⁰ : Group G\ninst✝⁹ : AddCommGroup A\ninst✝⁸ : DistribMulAction G A\ninst✝⁷ : MeasurableSpace A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : BorelSpace A\ninst✝⁴ : IsTopologicalAddGroup A\ninst✝³ : LocallyCompactSpace A\ninst✝² : ContinuousConstSMul G A\nμ ν : Measure A\nins... | rw [addHaarScalarFactor_eq_mul _ (g • ν), addHaarScalarFactor_domSMul,
mul_comm, ← addHaarScalarFactor_eq_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 1053,
"column": 2
} | {
"line": 1054,
"column": 43
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝¹⁰ : Group G\ninst✝⁹ : AddCommGroup A\ninst✝⁸ : DistribMulAction G A\ninst✝⁷ : MeasurableSpace A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : BorelSpace A\ninst✝⁴ : IsTopologicalAddGroup A\ninst✝³ : LocallyCompactSpace A\ninst✝² : ContinuousConstSMul G A\nμ ν : Measure A\nins... | rw [addHaarScalarFactor_eq_mul _ (g • ν), addHaarScalarFactor_domSMul,
mul_comm, ← addHaarScalarFactor_eq_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 1053,
"column": 2
} | {
"line": 1054,
"column": 43
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝¹⁰ : Group G\ninst✝⁹ : AddCommGroup A\ninst✝⁸ : DistribMulAction G A\ninst✝⁷ : MeasurableSpace A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : BorelSpace A\ninst✝⁴ : IsTopologicalAddGroup A\ninst✝³ : LocallyCompactSpace A\ninst✝² : ContinuousConstSMul G A\nμ ν : Measure A\nins... | rw [addHaarScalarFactor_eq_mul _ (g • ν), addHaarScalarFactor_domSMul,
mul_comm, ← addHaarScalarFactor_eq_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 1053,
"column": 2
} | {
"line": 1054,
"column": 43
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝¹⁰ : Group G\ninst✝⁹ : AddCommGroup A\ninst✝⁸ : DistribMulAction G A\ninst✝⁷ : MeasurableSpace A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : BorelSpace A\ninst✝⁴ : IsTopologicalAddGroup A\ninst✝³ : LocallyCompactSpace A\ninst✝² : ContinuousConstSMul G A\nμ ν : Measure A\nins... | rw [addHaarScalarFactor_eq_mul _ (g • ν), addHaarScalarFactor_domSMul,
mul_comm, ← addHaarScalarFactor_eq_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Holder | {
"line": 198,
"column": 31
} | {
"line": 198,
"column": 80
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nr : ℝ≥0\nf : X → Y\nC D s : ℝ≥0\nA : Set X\nhA : ∀ x ∈ A, ∀ y ∈ A, edist x y ≤ ↑D\nhf : HolderOnWith C r f A\nhsr : ↑s ≤ ↑r\nht : 0 < s\nhr : 0 < ↑r\nθ₁ : ℝ≥0 := ⟨↑s / ↑r, ⋯⟩\n⊢ 0 ≤ 1 - ↑s / ↑r",
"usedConstants... | simpa using div_le_one_of_le₀ hsr (by positivity) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.MetricSpace.Holder | {
"line": 198,
"column": 31
} | {
"line": 198,
"column": 80
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nr : ℝ≥0\nf : X → Y\nC D s : ℝ≥0\nA : Set X\nhA : ∀ x ∈ A, ∀ y ∈ A, edist x y ≤ ↑D\nhf : HolderOnWith C r f A\nhsr : ↑s ≤ ↑r\nht : 0 < s\nhr : 0 < ↑r\nθ₁ : ℝ≥0 := ⟨↑s / ↑r, ⋯⟩\n⊢ 0 ≤ 1 - ↑s / ↑r",
"usedConstants... | simpa using div_le_one_of_le₀ hsr (by positivity) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Holder | {
"line": 198,
"column": 31
} | {
"line": 198,
"column": 80
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nr : ℝ≥0\nf : X → Y\nC D s : ℝ≥0\nA : Set X\nhA : ∀ x ∈ A, ∀ y ∈ A, edist x y ≤ ↑D\nhf : HolderOnWith C r f A\nhsr : ↑s ≤ ↑r\nht : 0 < s\nhr : 0 < ↑r\nθ₁ : ℝ≥0 := ⟨↑s / ↑r, ⋯⟩\n⊢ 0 ≤ 1 - ↑s / ↑r",
"usedConstants... | simpa using div_le_one_of_le₀ hsr (by positivity) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 84
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nDu : Type u\ninst✝⁷ : NormedAddCommGroup Du\ninst✝⁶ : NormedSpace 𝕜 Du\ns : Set Du\nx : Du\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\nn : ℕ\nIH :\n ∀ {Eu Fu Gu : Type u} [inst : NormedAddCommGroup Eu] [inst_1 : NormedSpace 𝕜 Eu] [inst_2 : Norm... | have In : (n : WithTop ℕ∞) + 1 ≤ n.succ := by simp only [Nat.cast_succ, le_refl] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.Deriv.Abs | {
"line": 95,
"column": 57
} | {
"line": 98,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nf' : StrongDual ℝ E\nx : E\nhf : HasFDerivAt f f' x\nh₀ : 0 < f x\n⊢ HasFDerivAt (fun x ↦ |f x|) f' x",
"usedConstants": [
"ContinuousLinearMap.comp",
"IsModuleTopology.toContinuousSMul",
"HasFDerivAt... | by
convert (hasDerivAt_abs_pos h₀).hasFDerivAt.comp x hf using 1
ext y
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 169,
"column": 32
} | {
"line": 169,
"column": 84
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type uF\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type uG\ninst✝¹ : NormedAddCommGro... | simp only [xu, LinearIsometryEquiv.apply_symm_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 785,
"column": 6
} | {
"line": 785,
"column": 72
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : HasBesicovitchCovering α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → Set ℝ\ns : Set α\nhf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty\nN : ℕ\nτ : ℝ\nhτ... | apply ENNReal.Tendsto.mul_const _ (Or.inr (measure_lt_top μ s).ne) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 789,
"column": 8
} | {
"line": 789,
"column": 76
} | [
{
"pp": "case hr\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : HasBesicovitchCovering α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → Set ℝ\ns : Set α\nhf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty\nN : ℕ\n... | exact ENNReal.add_lt_add_left (ENNReal.natCast_ne_top N) zero_lt_one | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.FDeriv.Star | {
"line": 125,
"column": 8
} | {
"line": 125,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : StarRing 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : StarAddMonoid F\ninst✝⁶ : NormedSpace 𝕜 F\ninst✝⁵ : StarModule 𝕜 F\ninst✝⁴ : ContinuousStar F\nins... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.FDeriv.Star | {
"line": 125,
"column": 8
} | {
"line": 125,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : StarRing 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : StarAddMonoid F\ninst✝⁶ : NormedSpace 𝕜 F\ninst✝⁵ : StarModule 𝕜 F\ninst✝⁴ : ContinuousStar F\nins... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.