module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 173,
"column": 2
} | {
"line": 197,
"column": 48
} | {
"line": 198,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC✝ : ℝ\nf : ℂ → E\nz : ℂ\nC : ℝ\nhC₀ : 0 < C\na b : ℝ\nhza : a - b < z.im\nhle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C\nhzb : z.im < a + b\nhle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C\nhfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC✝ : ℝ\nf : ℂ → E\nz : ℂ\nC : ℝ\nhC₀ : 0 < C\na b : ℝ\nhza : a - b < z.im\nhle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C\nhzb : z.im < a + b\nhle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C\nhfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))\nhab : a... | obtain ⟨R, hzR, hR⟩ :
∃ R : ℝ, |z.re| < R ∧ ∀ w, |re w| = R → im w ∈ Ioo (a - b) (a + b) → ‖g ε w • f w‖ ≤ C := by
refine ((eventually_gt_atTop _).and ?_).exists
rcases hO.exists_pos with ⟨A, hA₀, hA⟩
simp only [isBigOWith_iff, eventually_inf_principal, eventually_comap, mem_Ioo,
mem_preimage, (· ... | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.InnerProductSpace.Laplacian | {
"line": 306,
"column": 6
} | {
"line": 306,
"column": 42
} | {
"line": 306,
"column": 42
} | [
{
"pp": "E : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nx : E\ns : Set E\nhs : UniqueDiffOn ℝ s\nhx : x ∈ s\n⊢ ∑ i, (iteratedFDerivWithin ℝ 2 (-f) s x) ![(stdOrthonormal... | [
"E : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nx : E\ns : Set E\nhs : UniqueDiffOn ℝ s\nhx : x ∈ s\n⊢ ∑ i, (-iteratedFDerivWithin ℝ 2 f s x) ![(stdOrthonormalBasis ℝ E) i, ... | iteratedFDerivWithin_neg_apply hs hx | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 9
} | {
"line": 44,
"column": 2
} | [
{
"pp": "f : ℂ → ℝ\nx : ℂ\nhf : HarmonicAt f x\nthis :\n (fun z ↦ ↑((fderiv ℝ f z) 1) - I * ↑((fderiv ℝ f z) I)) =\n (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nh₁f : ContDiffAt ℝ 2 f x\n⊢ ↑((fderiv ℝ (fun x ↦ (fderiv ℝ f x) 1) x) I) - I * ↑((fderiv ℝ (fun x ↦ (fderi... | [
"f : ℂ → ℝ\nx : ℂ\nhf : HarmonicAt f x\nthis :\n (fun z ↦ ↑((fderiv ℝ f z) 1) - I * ↑((fderiv ℝ f z) I)) =\n (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nh₁f : ContDiffAt ℝ 2 f x\n⊢ ↑((fderiv ℝ (fun x ↦ (fderiv ℝ f x) 1) x) I) - I * ↑((fderiv ℝ (fun x ↦ (fderiv ℝ f x) I) ... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.MeasureTheory.Integral.CircleAverage | {
"line": 155,
"column": 52
} | {
"line": 155,
"column": 91
} | {
"line": 155,
"column": 91
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nc : ℂ\nR : ℝ\nf₁ f₂ : ℂ → E\nhf : EqOn f₁ f₂ (sphere c |R|)\nx : ℝ\n⊢ x ∈ uIcc 0 (2 * π) → f₁ (circleMap c R x) = f₂ (circleMap c R x)",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Real",
"Real.pi... | [] | simp [hf (circleMap_mem_sphere' c R x)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.CircleAverage | {
"line": 155,
"column": 52
} | {
"line": 155,
"column": 91
} | {
"line": 155,
"column": 91
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nc : ℂ\nR : ℝ\nf₁ f₂ : ℂ → E\nhf : EqOn f₁ f₂ (sphere c |R|)\nx : ℝ\n⊢ x ∈ uIcc 0 (2 * π) → f₁ (circleMap c R x) = f₂ (circleMap c R x)",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Real",
"Real.pi... | [] | simp [hf (circleMap_mem_sphere' c R x)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CircleAverage | {
"line": 155,
"column": 52
} | {
"line": 155,
"column": 91
} | {
"line": 155,
"column": 91
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nc : ℂ\nR : ℝ\nf₁ f₂ : ℂ → E\nhf : EqOn f₁ f₂ (sphere c |R|)\nx : ℝ\n⊢ x ∈ uIcc 0 (2 * π) → f₁ (circleMap c R x) = f₂ (circleMap c R x)",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Real",
"Real.pi... | [] | simp [hf (circleMap_mem_sphere' c R x)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.CircleAverage | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 9
} | {
"line": 165,
"column": 2
} | [
{
"pp": "case e_a.e_f\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℂ → E\nc : ℂ\nR θ : ℝ\n⊢ f (c + ↑R * cexp (↑θ * I)) = (fun z ↦ f (↑R * z + c)) (0 + ↑1 * cexp (↑θ * I))",
"ppTerm": "?e_a.e_f✝",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul... | [
"case e_a.e_f\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℂ → E\nc : ℂ\nR θ : ℝ\n⊢ f (c + ↑R * cexp (↑θ * I)) = f (c + ↑R * cexp (↑θ * I) * ↑1)"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Complex.Poisson | {
"line": 62,
"column": 14
} | {
"line": 62,
"column": 40
} | {
"line": 62,
"column": 41
} | [
{
"pp": "a b : ℂ\n⊢ (a + b).re * (a - b).re / normSq (a - b) + (a + b).im * (a - b).im / normSq (a - b) =\n (‖a‖ ^ 2 - ‖b‖ ^ 2) / ‖a - b‖ ^ 2",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"cong... | [
"a b : ℂ\n⊢ (a + b).re * (a - b).re / ‖a - b‖ ^ 2 + (a + b).im * (a - b).im / ‖a - b‖ ^ 2 = (‖a‖ ^ 2 - ‖b‖ ^ 2) / ‖a - b‖ ^ 2"
] | normSq_eq_norm_sq (a - b), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.CircleAverage | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 9
} | {
"line": 247,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na : E\nc : ℂ\nR : ℝ\n⊢ ((2 * π)⁻¹ * (2 * π)) • a = a",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemir... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na : E\nc : ℂ\nR : ℝ\n⊢ (π * π⁻¹) • a = a"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Complex.IntegerCompl | {
"line": 59,
"column": 2
} | {
"line": 61,
"column": 47
} | {
"line": 63,
"column": 0
} | [
{
"pp": "x : ℂ\nhx : x ∈ ℂ_ℤ\nn : ℤ\n⊢ x ^ 2 ≠ ↑n ^ 2",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"_private.Mathlib.Analysis.Complex.IntegerCompl.0.Complex.integerComplement_pow_two_n... | [] | have := not_exists.mp hx n
have := not_exists.mp hx (-n)
simp_all [sq_eq_sq_iff_eq_or_eq_neg, eq_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.IntegerCompl | {
"line": 59,
"column": 2
} | {
"line": 61,
"column": 47
} | {
"line": 63,
"column": 0
} | [
{
"pp": "x : ℂ\nhx : x ∈ ℂ_ℤ\nn : ℤ\n⊢ x ^ 2 ≠ ↑n ^ 2",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"_private.Mathlib.Analysis.Complex.IntegerCompl.0.Complex.integerComplement_pow_two_n... | [] | have := not_exists.mp hx n
have := not_exists.mp hx (-n)
simp_all [sq_eq_sq_iff_eq_or_eq_neg, eq_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions | {
"line": 85,
"column": 6
} | {
"line": 85,
"column": 83
} | {
"line": 86,
"column": 6
} | [
{
"pp": "z : ℂ\ng : ℂ → ℂ\nh₁g : AnalyticAt ℂ g z\nh₂g : g z ≠ 0\nh₃g : g z ∈ slitPlane\nt₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z\nx : ℂ\n⊢ (⇑reCLM ∘ ⇑ofRealCLM ∘ Real.log ∘ ⇑normSq ∘ g) x = (⇑reCLM ∘ log ∘ (⇑conjCLE ∘ g * g)) x",
"ppTerm": "?m.303",
"assigned": true,
"usedConstants": [
"No... | [
"z : ℂ\ng : ℂ → ℂ\nh₁g : AnalyticAt ℂ g z\nh₂g : g z ≠ 0\nh₃g : g z ∈ slitPlane\nt₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z\nx : ℂ\n⊢ reCLM ↑(Real.log (normSq (g x))) = reCLM (log ((starRingEnd ℂ) (g x) * g x))"
] | simp only [Function.comp_apply, ofRealCLM_apply, Pi.mul_apply, conjCLE_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions | {
"line": 93,
"column": 8
} | {
"line": 93,
"column": 76
} | {
"line": 94,
"column": 6
} | [
{
"pp": "case e_6\nz : ℂ\ng : ℂ → ℂ\nh₁g : AnalyticAt ℂ g z\nh₂g : g z ≠ 0\nh₃g : g z ∈ slitPlane\nt₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z\nx : ℂ\nhx : x ∈ g ⁻¹' (slitPlane ∩ {y | y ≠ 0})\n⊢ (if (g x).arg = Real.pi then Real.pi else -(g x).arg) + (g x).arg ∈ Set.Ioc (-Real.pi) Real.pi",
"ppTerm": "?e_6"... | [] | simp [Complex.slitPlane_arg_ne_pi hx.1, Real.pi_pos, Real.pi_nonneg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions | {
"line": 93,
"column": 8
} | {
"line": 93,
"column": 76
} | {
"line": 94,
"column": 6
} | [
{
"pp": "case e_6\nz : ℂ\ng : ℂ → ℂ\nh₁g : AnalyticAt ℂ g z\nh₂g : g z ≠ 0\nh₃g : g z ∈ slitPlane\nt₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z\nx : ℂ\nhx : x ∈ g ⁻¹' (slitPlane ∩ {y | y ≠ 0})\n⊢ (if (g x).arg = Real.pi then Real.pi else -(g x).arg) + (g x).arg ∈ Set.Ioc (-Real.pi) Real.pi",
"ppTerm": "?e_6"... | [] | simp [Complex.slitPlane_arg_ne_pi hx.1, Real.pi_pos, Real.pi_nonneg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions | {
"line": 93,
"column": 8
} | {
"line": 93,
"column": 76
} | {
"line": 94,
"column": 6
} | [
{
"pp": "case e_6\nz : ℂ\ng : ℂ → ℂ\nh₁g : AnalyticAt ℂ g z\nh₂g : g z ≠ 0\nh₃g : g z ∈ slitPlane\nt₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z\nx : ℂ\nhx : x ∈ g ⁻¹' (slitPlane ∩ {y | y ≠ 0})\n⊢ (if (g x).arg = Real.pi then Real.pi else -(g x).arg) + (g x).arg ∈ Set.Ioc (-Real.pi) Real.pi",
"ppTerm": "?e_6"... | [] | simp [Complex.slitPlane_arg_ne_pi hx.1, Real.pi_pos, Real.pi_nonneg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 13
} | {
"line": 73,
"column": 6
} | [
{
"pp": "case inl\nx : ℝ\nhx : x < 0\n⊢ sin x * x⁻¹ * -x ≤ 1",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"GroupWithZero.toMonoidWithZe... | [
"case inl\nx : ℝ\nhx : x < 0\n⊢ -(sin x * x * x⁻¹) ≤ 1"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 20
} | {
"line": 88,
"column": 20
} | [
{
"pp": "x : ℝ\n⊢ ContinuousAt sinc x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.sinc_eq_dslope",
"Real.denselyNormedField",
"Real.instZero",
"congrArg",
"ContinuousAt",
"PseudoMetricSpace.toUniformSpace",
"... | [
"x : ℝ\n⊢ ContinuousAt (dslope sin 0) x"
] | sinc_eq_dslope | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Integrability.Basic | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 43
} | {
"line": 261,
"column": 0
} | [
{
"pp": "a b : ℝ\nμ : Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\n⊢ Continuous fun x ↦ 1 / (1 + x ^ 2)",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Real.instIsOrderedRing",
"Not.intro",
"Mathlib.Tactic.Ring.Common.neg_z... | [] | fun_prop (discharger := intro; nlinarith) | Mathlib.Meta.FunProp.funPropTac | Mathlib.Meta.FunProp.funPropTacStx |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 363,
"column": 51
} | {
"line": 363,
"column": 58
} | {
"line": 364,
"column": 2
} | [
{
"pp": "a b c : ℝ\n⊢ ∫ (x : ℝ) in a..b, c / (c ^ 2 + x ^ 2) = ∫ (x : ℝ) in a..b, c * (c ^ 2 + x ^ 2)⁻¹",
"ppTerm": "?m.109",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NonAssocSemiring.toAddCom... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 363,
"column": 51
} | {
"line": 363,
"column": 58
} | {
"line": 364,
"column": 2
} | [
{
"pp": "a b c : ℝ\n⊢ ∫ (x : ℝ) in a..b, c / (c ^ 2 + x ^ 2) = ∫ (x : ℝ) in a..b, c * (c ^ 2 + x ^ 2)⁻¹",
"ppTerm": "?m.109",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NonAssocSemiring.toAddCom... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 363,
"column": 51
} | {
"line": 363,
"column": 58
} | {
"line": 364,
"column": 2
} | [
{
"pp": "a b c : ℝ\n⊢ ∫ (x : ℝ) in a..b, c / (c ^ 2 + x ^ 2) = ∫ (x : ℝ) in a..b, c * (c ^ 2 + x ^ 2)⁻¹",
"ppTerm": "?m.109",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NonAssocSemiring.toAddCom... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 367,
"column": 10
} | {
"line": 367,
"column": 29
} | {
"line": 367,
"column": 30
} | [
{
"pp": "case neg\na b c : ℝ\nhc : ¬c = 0\n⊢ ∫ (x : ℝ) in a..b, c * (c ^ 2 + x ^ 2)⁻¹ = arctan (b / c) - arctan (a / c)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"instHDiv",
"RCLike.toNormedAlgebra",
... | [
"case neg\na b c : ℝ\nhc : ¬c = 0\n⊢ c * ∫ (x : ℝ) in a..b, (c ^ 2 + x ^ 2)⁻¹ = arctan (b / c) - arctan (a / c)"
] | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 204,
"column": 7
} | {
"line": 236,
"column": 23
} | {
"line": 238,
"column": 0
} | [] | [] | circleAverage (log ‖· - a‖) c R
_ = circleAverage (fun z ↦ log ‖R * (z + R⁻¹ * (c - a))‖) 0 1 := by
rw [circleAverage_eq_circleAverage_zero_one]
congr
ext z
congr
rw [Complex.ofReal_inv R]
field [Complex.ofReal_ne_zero.mpr hR]
_ = circleAverage (fun z ↦ log ‖R‖ + log ‖z + R⁻¹ * (c - a)‖) 0 1... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 27
} | {
"line": 107,
"column": 4
} | [
{
"pp": "case h₂\nw ρ : ℂ\nR r₀ r : ℝ\nhR : 0 < R\nhρ : ‖ρ‖ = R\nhr₀ : 0 < r₀\nhw : ‖w‖ < r₀\nhr₀r : r₀ ≤ r\nhrR : r ≤ R\nθ : ℝ\nhdR : 0 < ‖circleMap 0 R θ - ρ‖\nhrw : 0 < r₀ - ‖w‖\nhr : 0 < r\nh_norm_sub₁ : √(r₀ / R) * ‖circleMap 0 R θ - ρ‖ ≤ ‖circleMap 0 r θ - ρ‖\nh_norm_sub₂ : 0 < ‖circleMap 0 r θ - ρ‖\n⊢ |l... | [
"w ρ : ℂ\nR r₀ r : ℝ\nhR : 0 < R\nhρ : ‖ρ‖ = R\nhr₀ : 0 < r₀\nhw : ‖w‖ < r₀\nhr₀r : r₀ ≤ r\nhrR : r ≤ R\nθ : ℝ\nhdR : 0 < ‖circleMap 0 R θ - ρ‖\nhrw : 0 < r₀ - ‖w‖\nhr : 0 < r\nh_norm_sub₁ : √(r₀ / R) * ‖circleMap 0 R θ - ρ‖ ≤ ‖circleMap 0 r θ - ρ‖\nh_norm_sub₂ : 0 < ‖circleMap 0 r θ - ρ‖\n⊢ -(|log (2 * R)| + |log ... | apply abs_le.mpr ⟨_, _⟩ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 600,
"column": 6
} | {
"line": 603,
"column": 29
} | {
"line": 604,
"column": 4
} | [
{
"pp": "a b : ℝ\nm n : ℕ\nhc : Continuous fun u ↦ u ^ n * (1 - u ^ 2) ^ m\n⊢ -∫ (x : ℝ) in b..a, sin x ^ (2 * m + 1) * cos x ^ n = ∫ (x : ℝ) in b..a, (1 - cos x ^ 2) ^ m * -sin x * cos x ^ n",
"ppTerm": "?m.250",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormed... | [] | simp only [_root_.pow_succ, pow_mul, _root_.pow_zero, one_mul, mul_neg, neg_mul,
integral_neg, neg_inj]
congr! 5
rw [← sq, ← sq, sin_sq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 600,
"column": 6
} | {
"line": 603,
"column": 29
} | {
"line": 604,
"column": 4
} | [
{
"pp": "a b : ℝ\nm n : ℕ\nhc : Continuous fun u ↦ u ^ n * (1 - u ^ 2) ^ m\n⊢ -∫ (x : ℝ) in b..a, sin x ^ (2 * m + 1) * cos x ^ n = ∫ (x : ℝ) in b..a, (1 - cos x ^ 2) ^ m * -sin x * cos x ^ n",
"ppTerm": "?m.250",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormed... | [] | simp only [_root_.pow_succ, pow_mul, _root_.pow_zero, one_mul, mul_neg, neg_mul,
integral_neg, neg_inj]
congr! 5
rw [← sq, ← sq, sin_sq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 192,
"column": 6
} | {
"line": 194,
"column": 34
} | {
"line": 195,
"column": 2
} | [
{
"pp": "case hw\nw ρ : ℂ\nR : ℝ\nhρ : ‖ρ‖ = R\nhw : ‖w‖ < R\nhR : 0 < R\nr : ℕ → ℝ := fun n ↦ R - (R - ‖w‖) / (↑n + 2)\nhr_lt : ∀ (n : ℕ), r n < R\nhr_pos : ∀ (n : ℕ), 0 < r n\nhr_tendsto : Tendsto r atTop (𝓝 R)\nDCT :\n Tendsto (fun n ↦ circleAverage (herglotzLogIntegrand w ρ) 0 (r n)) atTop\n (𝓝 (circl... | [] | calc ‖w‖ * (n + 2) + (R - ‖w‖) = ‖w‖ * (n + 1) + R := by ring
_ < R * (n + 1) + R := by gcongr
_ = R * (n + 2) := by ring | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 40
} | {
"line": 239,
"column": 4
} | [
{
"pp": "R : ℝ\nc : ℂ\nD : Function.locallyFinsuppWithin (closedBall c |R|) ℤ\nh : D.support.Finite\n⊢ circleAverage (∑ᶠ (u : ℂ), fun x ↦ ↑(D u) * log ‖x - u‖) c R =\n circleAverage (∑ u ∈ h.toFinset, fun x ↦ ↑(D u) * log ‖x - u‖) c R",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
... | [
"case h\nR : ℝ\nc : ℂ\nD : Function.locallyFinsuppWithin (closedBall c |R|) ℤ\nh : D.support.Finite\n⊢ (Function.support fun u x ↦ ↑(D u) * log ‖x - u‖) ⊆ ↑h.toFinset"
] | rw [finsum_eq_sum_of_support_subset] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 40
} | {
"line": 257,
"column": 4
} | [
{
"pp": "R : ℝ\nc : ℂ\nD : Function.locallyFinsuppWithin (closedBall c |R|) ℤ\nh : D.support.Finite\n⊢ ∑ u ∈ h.toFinset, ↑(D u) * log R = ∑ᶠ (u : ℂ), ↑(D u) * log R",
"ppTerm": "?m.162",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing... | [
"case h\nR : ℝ\nc : ℂ\nD : Function.locallyFinsuppWithin (closedBall c |R|) ℤ\nh : D.support.Finite\n⊢ (Function.support fun u ↦ ↑(D u) * log R) ⊆ ↑h.toFinset"
] | rw [finsum_eq_sum_of_support_subset] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.RCLike.Sqrt | {
"line": 132,
"column": 35
} | {
"line": 132,
"column": 69
} | {
"line": 132,
"column": 70
} | [
{
"pp": "⊢ ↑(I ^ 2⁻¹).re + ↑(I ^ 2⁻¹).im * I = ↑√2⁻¹ * (1 + I)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"Real.instZero",
"Real.instZeroLEOneClass",
"congrArg"... | [
"⊢ ↑(I ^ 2⁻¹).re + ↑√((‖I‖ - I.re) / 2) * I = ↑√2⁻¹ * (1 + I)"
] | cpow_inv_two_im_eq_sqrt (by simp), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Exp | {
"line": 34,
"column": 45
} | {
"line": 34,
"column": 62
} | {
"line": 34,
"column": 63
} | [
{
"pp": "ξ : ℂ\nhξ : 1 / 2 ≤ ξ.im\n⊢ ‖cexp (2 * ↑π * Complex.I * ξ)‖ ≤ rexp (-π)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"Real.pi",
"HMul.hMul",
"congrArg",
"Complex.instMul",
"Compl... | [
"ξ : ℂ\nhξ : 1 / 2 ≤ ξ.im\n⊢ rexp (2 * ↑π * Complex.I * ξ).re ≤ rexp (-π)"
] | Complex.norm_exp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.TietzeExtension | {
"line": 207,
"column": 4
} | {
"line": 217,
"column": 40
} | {
"line": 219,
"column": 0
} | [
{
"pp": "case inr.refine_2.inr\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\ne : C(X, Y)\nhe : IsClosedEmbedding ⇑e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < ‖f‖\nhf3 : -‖f‖ / 3 < ‖f‖ / 3\nhc₁ : IsClosed[inst✝¹] (⇑e '' ⇑f ⁻¹' Iic (-‖f‖ / 3... | [] | · rcases le_total (f x) (‖f‖ / 3) with hle₂ | hle₂
· simp only [neg_div] at *
calc
dist (g (e x)) (f x) ≤ |g (e x)| + |f x| := dist_le_norm_add_norm _ _
_ ≤ ‖f‖ / 3 + ‖f‖ / 3 := (add_le_add (abs_le.2 <| hgf _) (abs_le.2 ⟨hle₁, hle₂⟩))
_ = 2 / 3 * ‖f‖ := by linarith
· ca... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 92,
"column": 17
} | {
"line": 92,
"column": 90
} | {
"line": 92,
"column": 91
} | [
{
"pp": "case a\ng : GL (Fin 2) ℝ\nh : (↑g).det < 0\nz : ℍ\nhz : g • z = z\n⊢ (↑g).trace + ((σ g) ↑z * denom g ↑z).im / z.im - (0 + (num g ↑z).im / z.im) = 0",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"Complex.mul_im",
"AddGroup.toSubtractionMonoid",
... | [] | simp [σ, h.not_gt, num, denom, z.im_ne_zero, Matrix.trace_fin_two, field] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 334,
"column": 73
} | {
"line": 335,
"column": 56
} | {
"line": 336,
"column": 2
} | [
{
"pp": "a b c d : ℝ\nh : a * d - b * c = 1\nh_denom : ∀ (z : ℍ), denom (toGL ⟨!![a, b; c, d], ⋯⟩) ↑z ≠ 0\nhc : c ≠ 0\nz : ℂ\nhz : 0 < z.im\nthis : (↑a * z + ↑b) / (↑c * z + ↑d) = ↑a / ↑c - (↑c * ↑d + ↑c * ↑c * z)⁻¹\n⊢ ↑(⟨!![a, b; c, d], ⋯⟩ • { coe := z, coe_im_pos := hz }) =\n ↑(((fun x ↦ a / c +ᵥ x) ∘ (fun... | [] | by
simpa [modular_S_smul, coe_specialLinearGroup_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 69
} | {
"line": 338,
"column": 2
} | [
{
"pp": "case h\na b c d : ℝ\nh : a * d - b * c = 1\nh_denom : ∀ (z : ℍ), denom (toGL ⟨!![a, b; c, d], ⋯⟩) ↑z ≠ 0\nz : ℂ\nhz : 0 < z.im\nhc : ↑c ≠ 0\n⊢ (↑a * z + ↑b) / (↑c * z + ↑d) = ↑a / ↑c - (↑c * ↑d + ↑c * ↑c * z)⁻¹",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"HMul.hMul",
... | [
"case h\na b c d : ℝ\nh : a * d - b * c = 1\nz : ℂ\nhz : 0 < z.im\nhc : ↑c ≠ 0\nh_denom : ↑c * z + ↑d ≠ 0\n⊢ (↑a * z + ↑b) / (↑c * z + ↑d) = ↑a / ↑c - (↑c * ↑d + ↑c * ↑c * z)⁻¹"
] | replace h_denom : ↑c * z + d ≠ 0 := by simpa using! h_denom ⟨z, hz⟩ | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 362,
"column": 6
} | {
"line": 362,
"column": 28
} | {
"line": 362,
"column": 29
} | [
{
"pp": "z : ℍ\nthis : ↑√z.im ≠ 0\n⊢ ↑z.re / ↑√z.im + ↑√z.im * Complex.I = ↑z * (↑√z.im)⁻¹",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"GroupWithZero.toDivInvMonoid",
"UpperHalfPlane.coe",
"congrArg",
"div_add... | [
"z : ℍ\nthis : ↑√z.im ≠ 0\n⊢ (↑z.re + ↑√z.im * Complex.I * ↑√z.im) / ↑√z.im = ↑z * (↑√z.im)⁻¹"
] | div_add' (hc := this), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 388,
"column": 44
} | {
"line": 388,
"column": 52
} | {
"line": 390,
"column": 0
} | [
{
"pp": "⊢ ↑(GeneralLinearGroup.det J) = ↑(-1)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Units.val",
"Matrix.GeneralLinearGroup.val_mkOfDetNeZero",
"MulOne.toOne",
"Real",
"MonoidHom.instFunLike",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | [] | simp [J] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 272,
"column": 6
} | {
"line": 272,
"column": 13
} | {
"line": 273,
"column": 2
} | [
{
"pp": "case refine_1.succ\nK : Type u_2\ninst✝¹ : Field K\ng : GL (Fin 2) K\ninst✝ : CharZero K\nn✝ : ℕ\na : K\nm : Matrix (Fin 2) (Fin 2) K\nhg : ↑g = (Matrix.scalar (Fin 2)) a + m\nhm0 : m ≠ 0\nhmsq : m ^ 2 = 0\nn : ℕ\nhn : 1 ≤ n\nIH : ((Matrix.scalar (Fin 2)) a + m) ^ n = (Matrix.scalar (Fin 2)) (a ^ n) + ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Geometry.Manifold.MFDeriv.FDeriv | {
"line": 81,
"column": 2
} | {
"line": 82,
"column": 52
} | {
"line": 84,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nE' : Type u_3\ninst✝¹ : NormedAddCommGroup E'\ninst✝ : NormedSpace 𝕜 E'\nf : E → E'\nx : E\n⊢ MDiffAt f x ↔ DifferentiableAt 𝕜 f x",
"ppTerm": "?m.51",
"assigned": true,... | [] | simp only [mdifferentiableAt_iff, differentiableWithinAt_univ, mfld_simps]
exact ⟨fun H => H.2, fun H => ⟨H.continuousAt, H⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.MFDeriv.FDeriv | {
"line": 81,
"column": 2
} | {
"line": 82,
"column": 52
} | {
"line": 84,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nE' : Type u_3\ninst✝¹ : NormedAddCommGroup E'\ninst✝ : NormedSpace 𝕜 E'\nf : E → E'\nx : E\n⊢ MDiffAt f x ↔ DifferentiableAt 𝕜 f x",
"ppTerm": "?m.51",
"assigned": true,... | [] | simp only [mdifferentiableAt_iff, differentiableWithinAt_univ, mfld_simps]
exact ⟨fun H => H.2, fun H => ⟨H.continuousAt, H⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.TietzeExtension | {
"line": 391,
"column": 2
} | {
"line": 393,
"column": 50
} | {
"line": 394,
"column": 2
} | [
{
"pp": "case inr.inr\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormalSpace Y\ninst✝ : Nonempty X\nf : X →ᵇ ℝ\ne : X → Y\nhe : IsClosedEmbedding e\ninhabited_h : Inhabited X\na : ℝ\nha : IsGLB (range ⇑f) a\nb : ℝ\nhb : IsLUB (range ⇑f) b\nhmem : ∀ (x : X), f... | [
"case inr.inr\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormalSpace Y\ninst✝ : Nonempty X\nf : X →ᵇ ℝ\ne : X → Y\nhe : IsClosedEmbedding e\ninhabited_h : Inhabited X\na : ℝ\nha : IsGLB (range ⇑f) a\nb : ℝ\nhb : IsLUB (range ⇑f) b\nhmem : ∀ (x : X), f x ∈ Icc a b... | replace hgf : ∀ x, (g - dg) (e x) = f x := by
intro x
simp [dg0 (Or.inl <| mem_range_self _), ← hgf] | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 24
} | {
"line": 309,
"column": 2
} | [
{
"pp": "z w : ℍ\nr : ℝ\n⊢ ProperSpace ℍ",
"ppTerm": "?m.2",
"assigned": true,
"usedConstants": [
"Real",
"UpperHalfPlane",
"UpperHalfPlane.instMetricSpace",
"ProperSpace.mk",
"MetricSpace.toPseudoMetricSpace"
],
"usedFVars": [],
"usedGoals": [
{
... | [
"z✝ w : ℍ\nr✝ : ℝ\nz : ℍ\nr : ℝ\n⊢ IsCompact (closedBall z r)"
] | refine ⟨fun z r => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.ProperAction.CompactlyGenerated | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 46
} | {
"line": 83,
"column": 2
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : MulAction G X\ninst✝² : CompactlyGeneratedSpace (X × X)\ninst✝¹ : T2Space X\ninst✝ : ContinuousSMul G X\nh : ∀ {U V : Set X}, IsCompact U → IsCompact V → IsCompact {g | (g • U ∩ V).Nonempty}... | [
"case ht\nG : Type u_1\nX : Type u_2\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : MulAction G X\ninst✝² : CompactlyGeneratedSpace (X × X)\ninst✝¹ : T2Space X\ninst✝ : ContinuousSMul G X\nh : ∀ {U V : Set X}, IsCompact U → IsCompact V → IsCompact {g | (g • U ∩ V).Nonempty}\nK... | apply ((h hV hU).prod hV).of_isClosed_subset | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 210,
"column": 4
} | {
"line": 214,
"column": 78
} | {
"line": 215,
"column": 4
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\nS : Set (Set (OnePoint X))\nho : ∀ t ∈ S, (∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' t)\n⊢ (∞ ∈ ⋃₀ S → IsCompact (some ⁻¹' ⋃₀ S)ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' ⋃₀ S)",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants":... | [
"X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\nS : Set (Set (OnePoint X))\nho : ∀ t ∈ S, (∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' t)\n⊢ IsOpen[inst✝] (some ⁻¹' ⋃₀ S)"
] | suffices IsOpen ((↑) ⁻¹' ⋃₀ S : Set X) by
refine ⟨?_, this⟩
rintro ⟨s, hsS : s ∈ S, hs : ∞ ∈ s⟩
refine IsCompact.of_isClosed_subset ((ho s hsS).1 hs) this.isClosed_compl ?_
exact compl_subset_compl.mpr (preimage_mono <| subset_sUnion_of_mem hsS) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 509,
"column": 47
} | {
"line": 512,
"column": 35
} | {
"line": 514,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝ : T0Space X\n⊢ T0Space (OnePoint X)",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"OnePoint.infty",
"OnePoint.some",
"Exists",
"Inseparable.eq",
"T0Space.mk",
... | [] | by
refine ⟨fun x y hxy => ?_⟩
rcases inseparable_iff.1 hxy with (⟨rfl, rfl⟩ | ⟨x, rfl, y, rfl, h⟩)
exacts [rfl, congr_arg some h.eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 600,
"column": 8
} | {
"line": 603,
"column": 42
} | {
"line": 603,
"column": 43
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nthis : Tendsto f (coclosedCompact X) (𝓝 y)\nq : Y\n⊢ (fun p ↦ p.elim y f) (... | [] | rcases eq_or_ne q y with rfl | hq
· simp
· have hq' : q ∈ range f := by simpa [hy]
simpa [hq] using hq'.choose_spec | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 600,
"column": 8
} | {
"line": 603,
"column": 42
} | {
"line": 603,
"column": 43
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nthis : Tendsto f (coclosedCompact X) (𝓝 y)\nq : Y\n⊢ (fun p ↦ p.elim y f) (... | [] | rcases eq_or_ne q y with rfl | hq
· simp
· have hq' : q ∈ range f := by simpa [hy]
simpa [hq] using hq'.choose_spec | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.ValueDistribution.Cartan | {
"line": 125,
"column": 2
} | {
"line": 126,
"column": 24
} | {
"line": 127,
"column": 2
} | [
{
"pp": "f : ℂ → ℂ\nh : meromorphicOrderAt f 0 < 0\n⊢ circleAverage (fun a ↦ log ‖meromorphicTrailingCoeffAt (fun x ↦ f x - a) 0‖) 0 1 =\n log ‖meromorphicTrailingCoeffAt f 0‖",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toN... | [
"f : ℂ → ℂ\nh : meromorphicOrderAt f 0 < 0\n⊢ EqOn (fun a ↦ log ‖meromorphicTrailingCoeffAt (fun x ↦ f x - a) 0‖) (fun x ↦ log ‖meromorphicTrailingCoeffAt f 0‖)\n (sphere 0 |1|)"
] | rw [circleAverage_congr_sphere (f₂ := fun _ ↦ log ‖meromorphicTrailingCoeffAt f 0‖),
circleAverage_const] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.ConstantSpeed | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 13
} | {
"line": 124,
"column": 4
} | [
{
"pp": "case inl.inr\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\nt : Set ℝ\nhfs : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))\nhft : ∀ ⦃x : ℝ⦄, x ∈ t → ∀ ⦃y : ℝ⦄, y ∈ t → x ≤ y → eVariationOn f (t ∩ Icc x y) = ENNRea... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.ConstantSpeed | {
"line": 171,
"column": 2
} | {
"line": 182,
"column": 44
} | {
"line": 184,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl l' : ℝ≥0\nhl' : l' ≠ 0\nφ : ℝ → ℝ\nφm : MonotoneOn φ s\nhfφ : HasConstantSpeedOnWith (f ∘ φ) s l\nhf : HasConstantSpeedOnWith f (φ '' s) l'\nx : ℝ\nxs : x ∈ s\n⊢ EqOn φ (fun y ↦ ↑l / ↑l' * (y - x) + φ x) s",
"ppTerm": "?m.39",
... | [] | rintro y ys
rw [← sub_eq_iff_eq_add, mul_comm, ← mul_div_assoc, eq_div_iff (NNReal.coe_ne_zero.mpr hl')]
rw [hasConstantSpeedOnWith_iff_variationOnFromTo_eq] at hf
rw [hasConstantSpeedOnWith_iff_variationOnFromTo_eq] at hfφ
symm
calc
(y - x) * l = l * (y - x) := by rw [mul_comm]
_ = variationOnFromTo ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ConstantSpeed | {
"line": 171,
"column": 2
} | {
"line": 182,
"column": 44
} | {
"line": 184,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl l' : ℝ≥0\nhl' : l' ≠ 0\nφ : ℝ → ℝ\nφm : MonotoneOn φ s\nhfφ : HasConstantSpeedOnWith (f ∘ φ) s l\nhf : HasConstantSpeedOnWith f (φ '' s) l'\nx : ℝ\nxs : x ∈ s\n⊢ EqOn φ (fun y ↦ ↑l / ↑l' * (y - x) + φ x) s",
"ppTerm": "?m.39",
... | [] | rintro y ys
rw [← sub_eq_iff_eq_add, mul_comm, ← mul_div_assoc, eq_div_iff (NNReal.coe_ne_zero.mpr hl')]
rw [hasConstantSpeedOnWith_iff_variationOnFromTo_eq] at hf
rw [hasConstantSpeedOnWith_iff_variationOnFromTo_eq] at hfφ
symm
calc
(y - x) * l = l * (y - x) := by rw [mul_comm]
_ = variationOnFromTo ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.BetweenList | {
"line": 236,
"column": 14
} | {
"line": 236,
"column": 21
} | {
"line": 237,
"column": 14
} | [
{
"pp": "case neg.refine_3\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nhead : P\nl'' : List R\nhl''0 : ∀ a ∈ l'', 0 ≤ a\nhl''s : l''.SortedLE\ntail : List P\nht2 : ¬tail ... | [
"case neg.refine_3\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nhead : P\nl'' : List R\nhl''0 : ∀ a ∈ l'', 0 ≤ a\nhl''s : l''.SortedLE\ntail : List P\nht2 : ¬tail = []\nr : R\... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Convex.Between | {
"line": 289,
"column": 18
} | {
"line": 289,
"column": 38
} | {
"line": 289,
"column": 39
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : V\np : P\n⊢ Wbtw R (x +ᵥ p) (y +ᵥ p) (z +ᵥ p) ∧ y +ᵥ p ≠ x +ᵥ p ∧ y +ᵥ p ≠ z +ᵥ p ↔ Wbtw R x y z ∧ y ≠ x ∧ y ≠ z",
"ppTerm": "?m.57",
... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : V\np : P\n⊢ Wbtw R x y z ∧ y +ᵥ p ≠ x +ᵥ p ∧ y +ᵥ p ≠ z +ᵥ p ↔ Wbtw R x y z ∧ y ≠ x ∧ y ≠ z"
] | wbtw_vadd_const_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Between | {
"line": 448,
"column": 2
} | {
"line": 448,
"column": 41
} | {
"line": 449,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : PartialOrder R\nx : R\n⊢ Wbtw R 0 x 1 ↔ x ∈ Set.Icc 0 1",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"affineSegment",
"Eq.mpr",
"Semiring.toModule",
"AffineMap.instFunLike",
"AddGroupWithOne.toAddGroup",
... | [
"R : Type u_1\ninst✝¹ : Ring R\ninst✝ : PartialOrder R\nx : R\n⊢ (∃ x_1 ∈ Set.Icc 0 1, (lineMap 0 1) x_1 = x) ↔ x ∈ Set.Icc 0 1"
] | rw [Wbtw, affineSegment, Set.mem_image] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Between | {
"line": 476,
"column": 2
} | {
"line": 476,
"column": 24
} | {
"line": 478,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nw x y z : P\nh₁ : Wbtw R z x w\nh₂ : Wbtw R z y x\n⊢ Wbtw R z y w",
"ppTerm": "?m.78",
"assigned": true,
"usedCo... | [] | exact h₁.trans_left h₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.CofilteredSystem | {
"line": 220,
"column": 6
} | {
"line": 220,
"column": 78
} | {
"line": 221,
"column": 6
} | [
{
"pp": "case refine_1\nJ : Type u\ninst✝¹ : Category.{v_1, u} J\nF : J ⥤ Type v\ni : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : F.IsMittagLeffler\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : F.eventualRange j₁ = range ⇑(ConcreteCategory.hom (F.map f₂))\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx ... | [
"case refine_1\nJ : Type u\ninst✝¹ : Category.{v_1, u} J\nF : J ⥤ Type v\ni : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nh : F.IsMittagLeffler\nj j₁ : J\ng₁ : j₁ ⟶ i\nf₁ : j₁ ⟶ j\nj₂ : J\nf₂ : j₂ ⟶ j₁\nh₂ : F.eventualRange j₁ = range ⇑(ConcreteCategory.hom (F.map f₂))\nj₃ : J\nf₃ : j₃ ⟶ j₂\nx : F.obj j₂\n... | obtain ⟨j₄, f₄, h₄⟩ := IsCofilteredOrEmpty.cone_maps g₂ ((f₃ ≫ f₂) ≫ g₁) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Convex.Approximation | {
"line": 223,
"column": 2
} | {
"line": 224,
"column": 42
} | {
"line": 225,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nφ : E → ℝ\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : LocallyConvexSpace ℝ E\ninst✝ : Hereditari... | [
"𝕜 : Type u_1\nE : Type u_2\nφ : E → ℝ\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : LocallyConvexSpace ℝ E\ninst✝ : HereditarilyLindelofSp... | obtain ⟨l, c, hle, hsup⟩ := hφcv.sSup_of_nat_affine_eq (𝕜 := 𝕜) (s := univ) isClosed_univ
(lowerSemicontinuousOn_univ_iff.2 hφc) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Data.PEquiv | {
"line": 304,
"column": 93
} | {
"line": 305,
"column": 49
} | {
"line": 307,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\na₁ a₂ : α\nb₁ b₂ : β\n⊢ b₁ ∈ (single a₂ b₂) a₁ ↔ a₁ = a₂ ∧ b₁ = b₂",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"PEquiv.instFunLikeOption",
"Eq.mpr",
"False",
"eq_false",
"PEqui... | [] | by
dsimp [single]; split_ifs <;> simp [*, eq_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.CofilteredSystem | {
"line": 329,
"column": 2
} | {
"line": 331,
"column": 78
} | {
"line": 332,
"column": 2
} | [
{
"pp": "J : Type u\ninst✝⁵ : Category.{v_1, u} J\nF : J ⥤ Type v\ninst✝⁴ : IsCofilteredOrEmpty J\ninst✝³ : ∀ (j : J), Nonempty (F.obj j)\ninst✝² : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective ⇑(ConcreteCategory.hom (F.map f))\ninst✝¹ : Nonempty J\ninst✝ : Finite ↑F.sections\... | [
"J : Type u\ninst✝⁵ : Category.{v_1, u} J\nF : J ⥤ Type v\ninst✝⁴ : IsCofilteredOrEmpty J\ninst✝³ : ∀ (j : J), Nonempty (F.obj j)\ninst✝² : ∀ (j : J), Finite (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective ⇑(ConcreteCategory.hom (F.map f))\ninst✝¹ : Nonempty J\ninst✝ : Finite ↑F.sections\nthis✝ : (j ... | refine ⟨fn.argmin,
fun i f => ((Fintype.bijective_iff_surjective_and_card _).2
⟨Fsur f, le_antisymm ?_ (Fintype.card_le_of_surjective _ <| Fsur f)⟩).1⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Matrix.Stochastic | {
"line": 51,
"column": 25
} | {
"line": 51,
"column": 30
} | {
"line": 51,
"column": 31
} | [
{
"pp": "R✝ : Type u_1\nn✝ : Type u_2\ninst✝⁹ : Fintype n✝\ninst✝⁸ : DecidableEq n✝\ninst✝⁷ : Semiring R✝\ninst✝⁶ : PartialOrder R✝\ninst✝⁵ : IsOrderedRing R✝\nM✝ : Matrix n✝ n✝ R✝\nx : n✝ → R✝\nR : Type u_3\nn : Type u_4\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\... | [
"R✝ : Type u_1\nn✝ : Type u_2\ninst✝⁹ : Fintype n✝\ninst✝⁸ : DecidableEq n✝\ninst✝⁷ : Semiring R✝\ninst✝⁶ : PartialOrder R✝\ninst✝⁵ : IsOrderedRing R✝\nM✝ : Matrix n✝ n✝ R✝\nx : n✝ → R✝\nR : Type u_3\nn : Type u_4\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsO... | hN.2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Stochastic | {
"line": 183,
"column": 57
} | {
"line": 189,
"column": 38
} | {
"line": 191,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nM : Matrix n n R\nx : n → R\nhM : M ∈ colStochastic R n\nhx : ∀ (i : n), 0 ≤ x i\n⊢ ∀ (j : n), 0 ≤ (x ᵥ* M) j",
"ppTerm": "?m.24",
"assigned": true,
... | [] | by
intro j
simp only [Matrix.vecMul, dotProduct]
apply Finset.sum_nonneg
intro k _
refine Left.mul_nonneg (hx k) ?_
exact nonneg_of_mem_colStochastic hM | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 156,
"column": 8
} | {
"line": 156,
"column": 42
} | {
"line": 156,
"column": 42
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nM : Matrix n n R\nhM : M ∈ doublyStochastic R n\nh✝ : Nonempty n\nw : Equiv.Perm n → R\nhw1 : ∀ (σ : Equiv.Perm n), 0 ≤ w σ\nhw3 : ∑ σ, w σ • Equiv.Perm.permM... | [
"R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nM : Matrix n n R\nhM : M ∈ doublyStochastic R n\nh✝ : Nonempty n\nw : Equiv.Perm n → R\nhw1 : ∀ (σ : Equiv.Perm n), 0 ≤ w σ\nhw3 : ∑ σ, w σ • Equiv.Perm.permMatrix R σ = ... | sum_row_of_mem_doublyStochastic hM | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Between | {
"line": 991,
"column": 4
} | {
"line": 991,
"column": 11
} | {
"line": 993,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nw z : P\nt₁ : R\nht₁ : t₁ ∈ Set.Icc 0 1\nt₂ : R\nht₂ : t₂ ∈ Set.Icc 0 1\nh : ¬1 - t₂ * t₁ = 0\n⊢ ((t₁ - t₂ *... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Convex.Between | {
"line": 1067,
"column": 90
} | {
"line": 1068,
"column": 81
} | {
"line": 1070,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\n⊢ Wbtw R x y z ↔ SameRay R (y -ᵥ x) (z -ᵥ y)",
"ppTerm": "?m.25",
"assigned": true,
"usedConsta... | [] | by
simp [← wbtw_vsub_const_iff x, ← mem_segment_iff_wbtw, mem_segment_iff_sameRay] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Between | {
"line": 1083,
"column": 2
} | {
"line": 1083,
"column": 9
} | {
"line": 1085,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nr₁ r₂ : R\nhr₁ : 0 < r₁\nhr₂ : 0 < r₂\nh : r₁ • (y -ᵥ x) = r₂ • (z -ᵥ x)\nhr : r₂ ≤ r₁\nh' : y = r₁⁻¹ • r₂ ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Geometry.Convex.Cone.TensorProduct | {
"line": 122,
"column": 2
} | {
"line": 123,
"column": 86
} | {
"line": 124,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\nG : Type u_2\ninst✝³ : AddCommGroup G\ninst✝² : Module R G\nH : Type u_3\ninst✝¹ : AddCommGroup H\ninst✝ : Module R H\nC₁ : PointedCone R G\nC₂ : PointedCone R H\nz : H ⊗[R] G\n⊢ (∃ x,\n (∀ φ ∈ dual (Dual.ev... | [
"case refine_1\nR : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\nG : Type u_2\ninst✝³ : AddCommGroup G\ninst✝² : Module R G\nH : Type u_3\ninst✝¹ : AddCommGroup H\ninst✝ : Module R H\nC₁ : PointedCone R G\nC₂ : PointedCone R H\nz : H ⊗[R] G\n⊢ (∃ x,\n (∀ φ ∈ dual (Dual... | refine ⟨?_, fun hz ↦
⟨(TensorProduct.comm R H G) z, ?_, (TensorProduct.comm R H G).symm_apply_apply z⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.Exposed | {
"line": 165,
"column": 2
} | {
"line": 168,
"column": 67
} | {
"line": 170,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : TopologicalSpace 𝕜\ninst✝⁵ : Ring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : TopologicalSpace E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderClosedTopology 𝕜\nA B : Set E\nhAB : IsExposed 𝕜 A B\nhA : IsClosed A\n⊢ IsClosed B",
"ppTerm": "?m.21",... | [] | obtain rfl | hB := B.eq_empty_or_nonempty
· simp
obtain ⟨l, a, rfl⟩ := hAB.eq_inter_halfSpace' hB
exact hA.isClosed_le continuousOn_const l.continuous.continuousOn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Exposed | {
"line": 165,
"column": 2
} | {
"line": 168,
"column": 67
} | {
"line": 170,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : TopologicalSpace 𝕜\ninst✝⁵ : Ring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : TopologicalSpace E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderClosedTopology 𝕜\nA B : Set E\nhAB : IsExposed 𝕜 A B\nhA : IsClosed A\n⊢ IsClosed B",
"ppTerm": "?m.21",... | [] | obtain rfl | hB := B.eq_empty_or_nonempty
· simp
obtain ⟨l, a, rfl⟩ := hAB.eq_inter_halfSpace' hB
exact hA.isClosed_le continuousOn_const l.continuous.continuousOn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.GaugeRescale | {
"line": 46,
"column": 57
} | {
"line": 48,
"column": 58
} | {
"line": 50,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\nc : ℝ\nhc : 0 ≤ c\nx : E\n⊢ gaugeRescale s t (c • x) = c • gaugeRescale s t x",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"MonoidWithZero.toMul... | [] | by
simp only [gaugeRescale, gauge_smul_of_nonneg hc, smul_smul, smul_eq_mul]
rw [mul_div_mul_comm, mul_right_comm, div_self_mul_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Join | {
"line": 38,
"column": 87
} | {
"line": 39,
"column": 19
} | {
"line": 41,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_3\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns t : Set E\nx : E\n⊢ x ∈ convexJoin 𝕜 s t ↔ ∃ a ∈ s, ∃ b ∈ t, x ∈ segment 𝕜 a b",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Iff.of_eq",
"... | [] | by
simp [convexJoin] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Convex.Set | {
"line": 115,
"column": 18
} | {
"line": 115,
"column": 41
} | {
"line": 116,
"column": 6
} | [
{
"pp": "R : Type u_3\nX : Type u_5\nY : Type u_6\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : ConvexSpace R X\ninst✝ : ConvexSpace R Y\nf : X → Y\ns : Set X\nhf : IsAffineMap R f\nhs : IsConvexSet R s\nw : StdSimplex R Y\nhw : ↑w.weights.support ⊆ f '' s\nu : Finset X... | [] | by simp; split <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 190,
"column": 12
} | {
"line": 192,
"column": 98
} | {
"line": 193,
"column": 2
} | [
{
"pp": "R : Type u_1\nX : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nI : Type u_6\nJ : Type u_7\nK : Type u_8\ninst✝² : Semifield K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\nw : StdSimplex K X\ns : Set X\nhs : ∃ x ∈ s, w.weights x ≠ 0\n⊢ 0 ≤ ((filter (fun x ↦ x ∈ s) w.weights).sum fun x ... | [] | by
classical
exact smul_nonneg (inv_nonneg.2 restrict_nonneg_aux) fun _ ↦ by simp [filter_apply, apply_ite] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 48
} | {
"line": 318,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝³ : PartialOrder R\ninst✝² : Semiring R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R M\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nhst : s + t = 1\nw w' : StdSimplex R M\n⊢ sConvexComb (convexCombPair s t hs ht hst w w') = convexCombPair s t hs ht hst (sConvexComb w) (s... | [] | simp [convexCombPair, sConvexComb_sConvexComb] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 48
} | {
"line": 318,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝³ : PartialOrder R\ninst✝² : Semiring R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R M\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nhst : s + t = 1\nw w' : StdSimplex R M\n⊢ sConvexComb (convexCombPair s t hs ht hst w w') = convexCombPair s t hs ht hst (sConvexComb w) (s... | [] | simp [convexCombPair, sConvexComb_sConvexComb] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 48
} | {
"line": 318,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝³ : PartialOrder R\ninst✝² : Semiring R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R M\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nhst : s + t = 1\nw w' : StdSimplex R M\n⊢ sConvexComb (convexCombPair s t hs ht hst w w') = convexCombPair s t hs ht hst (sConvexComb w) (s... | [] | simp [convexCombPair, sConvexComb_sConvexComb] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Piecewise | {
"line": 68,
"column": 8
} | {
"line": 68,
"column": 43
} | {
"line": 69,
"column": 2
} | [
{
"pp": "case hb\n𝕜 : Type u_1\nE : Type u_2\nβ : Type u_3\ninst✝¹¹ : Semiring 𝕜\ninst✝¹⁰ : PartialOrder 𝕜\ninst✝⁹ : AddCommMonoid E\ninst✝⁸ : LinearOrder E\ninst✝⁷ : IsOrderedAddMonoid E\ninst✝⁶ : Module 𝕜 E\ninst✝⁵ : PosSMulMono 𝕜 E\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAdd... | [] | exact h_anti hx Set.self_mem_Iic hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Radon | {
"line": 228,
"column": 4
} | {
"line": 232,
"column": 65
} | {
"line": 233,
"column": 2
} | [
{
"pp": "case neg\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : T2Space E\nF : ι → Set E\nh_card : ↑(finrank 𝕜 E) + 1 ≤... | [] | · have : Finite ι := Finite.of_not_infinite h
have : Fintype ι := Fintype.ofFinite ι
apply exists_superset_card_eq hI_card
simp only [ENat.card_eq_coe_fintype_card] at h_card
rwa [← Nat.cast_one, ← Nat.cast_add, Nat.cast_le] at h_card | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Radon | {
"line": 279,
"column": 9
} | {
"line": 279,
"column": 39
} | {
"line": 279,
"column": 40
} | [
{
"pp": "case a\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : T2Space E\nF : Set (Set E)\nh_card : ↑(finrank 𝕜 E) + 1 ≤ F.encard\nh_c... | [
"case a\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : T2Space E\nF : Set (Set E)\nh_card : ↑(finrank 𝕜 E) + 1 ≤ F.encard\nh_convex : ∀ X ... | encard_eq_coe_toFinset_card J, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 669,
"column": 2
} | {
"line": 669,
"column": 74
} | {
"line": 671,
"column": 0
} | [
{
"pp": "V : Type u\nv w : V\n⊢ (fromEdgeSet ∅).Adj v w ↔ ⊥.Adj v w",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"False",
"Sym2.mk",
"congrArg",
"SimpleGraph.fromEdgeSet",
"SimpleGraph.Adj",
"_private.Mathlib.Combinatorics.SimpleGraph.Basic.0.SimpleGr... | [] | simp only [fromEdgeSet_adj, Set.mem_empty_iff_false, false_and, bot_adj] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 832,
"column": 43
} | {
"line": 832,
"column": 53
} | {
"line": 832,
"column": 54
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\nh : ∀ {a b : V}, G.Adj a b → a ≠ b\n⊢ G.Adj v w ∨ Gᶜ.Adj v w ↔ w ∈ {v}ᶜ",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"SimpleGraph.Adj",
"Membership.mem",
"Set.instSi... | [
"V : Type u\nG : SimpleGraph V\nv w : V\nh : ∀ {a b : V}, G.Adj a b → a ≠ b\n⊢ G.Adj v w ∨ v ≠ w ∧ ¬G.Adj v w ↔ w ∈ {v}ᶜ"
] | compl_adj, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convex.StoneSeparation | {
"line": 41,
"column": 4
} | {
"line": 41,
"column": 78
} | {
"line": 41,
"column": 79
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np q u v x y : E\nhu : u ∈ segment 𝕜 x p\nhv : v ∈ segment 𝕜 y q\nbz : 𝕜\nhbz : 0 ≤ bz\nhaz : 0 ≤ 0\nhabz : bz = 1\n⊢ ∃ x ∈ segment 𝕜 u v,... | [
"case inl.refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np q u v x y : E\nhu : u ∈ segment 𝕜 x p\nhv : v ∈ segment 𝕜 y q\nbz : 𝕜\nhbz : 0 ≤ bz\nhaz : 0 ≤ 0\nhabz : bz = 1\n⊢ y ∈ {p, q, y}",
"case ... | refine ⟨v, by apply right_mem_segment, segment_subset_convexHull ?_ ?_ hv⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.StrictConvexBetween | {
"line": 42,
"column": 8
} | {
"line": 42,
"column": 29
} | {
"line": 42,
"column": 30
} | [
{
"pp": "case inr.inr\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : StrictConvexSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nh : p₂ -ᵥ p ∈ openSegment ℝ (p₁ -ᵥ p) (p₃ -ᵥ p)\nhp₂p₁ : p₂ ≠ p₁\nhp₂p... | [
"case inr.inr\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : StrictConvexSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nh : p₂ -ᵥ p ∈ (fun θ ↦ (1 - θ) • (p₁ -ᵥ p) + θ • (p₃ -ᵥ p)) '' Set.Ioo 0 1\nhp₂p₁ ... | openSegment_eq_image, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Side | {
"line": 625,
"column": 6
} | {
"line": 628,
"column": 76
} | {
"line": 629,
"column": 6
} | [
{
"pp": "case inr.inr.refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\nx y p₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nr₁ r₂ : R\nhr₁ : 0 < r₁... | [
"case inr.inr.refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\nx y p₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nr₁ r₂ : R\nhr₁ : 0 < r₁\nhr₂ : 0 < ... | have : (r₂ / (r₁ + r₂)) • (y -ᵥ p₂ + (p₂ -ᵥ p₁) - (x -ᵥ p₁)) + (x -ᵥ p₁) =
(r₂ / (r₁ + r₂)) • (p₂ -ᵥ p₁) := by
rw [← neg_vsub_eq_vsub_rev p₂ y]
linear_combination (norm := match_scalars <;> field) (r₁ + r₂)⁻¹ • h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Strong | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 9
} | {
"line": 162,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nb m : ℝ\nx y : E\nf : E → ℝ\nhb : 0 ≤ b\nha : 0 ≤ 1 - b\nhab : 1 - b + b = 1\n⊢ (1 - b) * (f x - m / 2 * ‖x‖ ^ 2) + b * (f y - m / 2 * ‖y‖ ^ 2) +\n m / 2 * ((1 - b) ^ 2 * ‖x‖ ^ 2 + 2 * ((1 - b) * (b * inner ℝ x y)) + b ^ 2... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Convex.Side | {
"line": 641,
"column": 4
} | {
"line": 641,
"column": 18
} | {
"line": 642,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\ny p : P\nhp : p ∈ s\nh : s.SOppSide p y\nhw : Wbtw R p p y\n⊢ False",
"ppTe... | [] | exact h.2.1 hp | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Visible | {
"line": 148,
"column": 67
} | {
"line": 155,
"column": 78
} | {
"line": 157,
"column": 0
} | [
{
"pp": "V : Type u_2\ninst✝¹ : AddCommGroup V\ninst✝ : Module ℝ V\ns : Set V\nx y : V\nhx : x ∉ (convexHull ℝ) s\nhy : y ∈ (convexHull ℝ) s\nhxy : IsVisible ℝ ((convexHull ℝ) s) x y\n⊢ y ∈ (convexHull ℝ) {z | z ∈ s ∧ IsVisible ℝ ((convexHull ℝ) s) x z}",
"ppTerm": "?m.61",
"assigned": true,
"usedCo... | [] | by
classical
obtain ⟨ι, _, w, a, hw₀, hw₁, ha, rfl⟩ := mem_convexHull_iff_exists_fintype.1 hy
rw [← Fintype.sum_subset (s := {i | w i ≠ 0})
fun i hi ↦ mem_filter.2 ⟨mem_univ _, left_ne_zero_of_smul hi⟩]
exact (convex_convexHull ..).sum_mem (fun i _ ↦ hw₀ _) (by rwa [sum_filter_ne_zero])
fun i hi ↦ subse... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.Holder | {
"line": 203,
"column": 2
} | {
"line": 206,
"column": 21
} | {
"line": 208,
"column": 0
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\np q r : ℝ≥0∞\nhpqr : p.HolderTriple q r\ninst✝³ : NormedRing 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nf₁ f₂ : ↥(Lp 𝕜 p μ)\ng : ↥(Lp E q μ)\n⊢ (f₁ + f₂) • g = f₁ • g + f₂ • g",
... | [] | simp only [smul_def, ← MemLp.toLp_add]
apply MemLp.toLp_congr
filter_upwards [AEEqFun.coeFn_add f₁.val f₂.val] with x hx
simp [hx, add_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.Holder | {
"line": 203,
"column": 2
} | {
"line": 206,
"column": 21
} | {
"line": 208,
"column": 0
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\np q r : ℝ≥0∞\nhpqr : p.HolderTriple q r\ninst✝³ : NormedRing 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nf₁ f₂ : ↥(Lp 𝕜 p μ)\ng : ↥(Lp E q μ)\n⊢ (f₁ + f₂) • g = f₁ • g + f₂ • g",
... | [] | simp only [smul_def, ← MemLp.toLp_add]
apply MemLp.toLp_congr
filter_upwards [AEEqFun.coeFn_add f₁.val f₂.val] with x hx
simp [hx, add_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 46
} | {
"line": 130,
"column": 47
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\ninst✝ : SFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nint... | [
"α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\ninst✝ : SFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ... | ← lintegral_indicator measurableSet_Ioi, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 238,
"column": 10
} | {
"line": 238,
"column": 20
} | {
"line": 239,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\nf_nonneg : ∀ (ω : α), 0 ≤ f ω\nH1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0\ns : ℝ\ns_pos : s ... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\nf_nonneg : ∀ (ω : α), 0 ≤ f ω\nH1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0\ns : ℝ\ns_pos : s > 0\nhs : 0 ... | rw [← h's] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 488,
"column": 2
} | {
"line": 488,
"column": 81
} | {
"line": 489,
"column": 2
} | [
{
"pp": "case hf\nE : Type u_5\nF : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : μ.HasTemperateGrowth\nf : E → F\nC₁ C₂ : ℝ\nk : ℕ\nhf : ∀ (x : E), ‖f x‖ ≤ C₁\nh'f : ∀ (x : E)... | [
"case h\nE : Type u_5\nF : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : μ.HasTemperateGrowth\nf : E → F\nC₁ C₂ : ℝ\nk : ℕ\nhf : ∀ (x : E), ‖f x‖ ≤ C₁\nh'f : ∀ (x : E), ‖x‖ ^ (k + ... | · exact AEStronglyMeasurable.mul (aestronglyMeasurable_id.norm.pow _) h''f.norm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 58,
"column": 2
} | {
"line": 116,
"column": 44
} | {
"line": 118,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ¬HasEigenvalue (↑T) μ\n⊢ ∃ K, AntilipschitzWith K ⇑(T - μ • 1)",
"ppTerm": "?m.60",
"assigned": true,
... | [] | rw [antilipschitzWith_iff_exists_mul_le_norm]
contrapose! h
-- then for every `K > 0`, there is some `x` such that `‖(T - μ • 1) x‖ < K * ‖x‖`.
replace hK : ∀ K > 0, ∃ x, ‖(T - μ • 1) x‖ < K * ‖x‖ := h
-- In fact, there is a lower bound `c` such that for every `ε > 0`, there is an `x` with norm
-- in the inte... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 58,
"column": 2
} | {
"line": 116,
"column": 44
} | {
"line": 118,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ¬HasEigenvalue (↑T) μ\n⊢ ∃ K, AntilipschitzWith K ⇑(T - μ • 1)",
"ppTerm": "?m.60",
"assigned": true,
... | [] | rw [antilipschitzWith_iff_exists_mul_le_norm]
contrapose! h
-- then for every `K > 0`, there is some `x` such that `‖(T - μ • 1) x‖ < K * ‖x‖`.
replace hK : ∀ K > 0, ∃ x, ‖(T - μ • 1) x‖ < K * ‖x‖ := h
-- In fact, there is a lower bound `c` such that for every `ε > 0`, there is an `x` with norm
-- in the inte... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 35
} | {
"line": 211,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\ninst✝ : CompleteSpace X\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh₁ : ¬HasEigenvalue (↑T) μ\nS : X →L[𝕜] X := T - μ • 1\nK✝ : NNReal\nhK✝ : AntilipschitzWi... | [
"𝕜 : Type u_1\nX : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\ninst✝ : CompleteSpace X\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh₁ : ¬HasEigenvalue (↑T) μ\nS : X →L[𝕜] X := T - μ • 1\nK✝ : NNReal\nhK✝ : AntilipschitzWith K✝ ⇑S\nh₂... | rw [Metric.cauchySeq_iff'] at hψy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.SmoothApprox | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 86
} | {
"line": 88,
"column": 2
} | [
{
"pp": "E : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : Chart... | [
"E : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\... | have dist_f_f : ∀ x, dist (f x) (f x) < ε x := by simpa only [dist_self] using ε_pos | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 179,
"column": 22
} | {
"line": 179,
"column": 93
} | {
"line": 180,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝² : RCLike 𝕜\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf f' g : ↥(Lp E 2 μ)\n⊢ ∫ (a : α), ⟪↑↑(f + f') a, ↑↑g a⟫ ∂μ = ∫ (a : α), ⟪↑↑f a, ↑↑g a⟫ ∂μ + ∫ (a : α), ⟪↑↑f' a, ↑↑g a⟫ ∂μ",
"ppTerm": ... | [
"α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝² : RCLike 𝕜\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf f' g : ↥(Lp E 2 μ)\n⊢ ∫ (a : α), ⟪↑↑(f + f') a, ↑↑g a⟫ ∂μ = ∫ (a : α), ⟪↑↑f a, ↑↑g a⟫ + ⟪↑↑f' a, ↑↑g a⟫ ∂μ"
] | ← integral_add (integrable_inner (𝕜 := 𝕜) f g) (integrable_inner f' g), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Lp.SmoothApprox | {
"line": 87,
"column": 2
} | {
"line": 95,
"column": 75
} | {
"line": 97,
"column": 0
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nhp : p ≠ ∞\nhp₂ ... | [] | have hε₂ : 0 < ε / 2 := by positivity
have hε₂' : 0 < ENNReal.ofReal (ε / 2) := by positivity
obtain ⟨g, hg₁, hg₂, hg₃, hg₄⟩ := hf.exists_hasCompactSupport_eLpNorm_sub_le hp hε₂'.ne'
obtain ⟨g', hg'₁, hg'₂, hg'₃⟩ := hg₁.exist_eLpNorm_sub_le_of_continuous μ hε₂ hg₃
refine ⟨g', hg'₁, hg'₂, ?_⟩
have : f - g' = (... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.SmoothApprox | {
"line": 87,
"column": 2
} | {
"line": 95,
"column": 75
} | {
"line": 97,
"column": 0
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nhp : p ≠ ∞\nhp₂ ... | [] | have hε₂ : 0 < ε / 2 := by positivity
have hε₂' : 0 < ENNReal.ofReal (ε / 2) := by positivity
obtain ⟨g, hg₁, hg₂, hg₃, hg₄⟩ := hf.exists_hasCompactSupport_eLpNorm_sub_le hp hε₂'.ne'
obtain ⟨g', hg'₁, hg'₂, hg'₃⟩ := hg₁.exist_eLpNorm_sub_le_of_continuous μ hε₂ hg₃
refine ⟨g', hg'₁, hg'₂, ?_⟩
have : f - g' = (... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Distribution.SchwartzSpace.Deriv | {
"line": 201,
"column": 2
} | {
"line": 205,
"column": 72
} | {
"line": 207,
"column": 0
} | [
{
"pp": "E : Type u_5\nF : Type u_8\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ E\nn : ℕ\nm : Fin n → E\nf : 𝓢(E, F)\n⊢ tsupport ⇑(∂^{m} f) ⊆ tsupport ⇑f",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | induction n with
| zero => simp
| succ n IH =>
rw [iteratedLineDerivOp_succ_left]
exact (tsupport_lineDerivOp_subset (m 0) _).trans (IH <| Fin.tail m) | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Distribution.SchwartzSpace.Deriv | {
"line": 201,
"column": 2
} | {
"line": 205,
"column": 72
} | {
"line": 207,
"column": 0
} | [
{
"pp": "E : Type u_5\nF : Type u_8\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ E\nn : ℕ\nm : Fin n → E\nf : 𝓢(E, F)\n⊢ tsupport ⇑(∂^{m} f) ⊆ tsupport ⇑f",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | induction n with
| zero => simp
| succ n IH =>
rw [iteratedLineDerivOp_succ_left]
exact (tsupport_lineDerivOp_subset (m 0) _).trans (IH <| Fin.tail m) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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