module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 162,
"column": 2
} | {
"line": 169,
"column": 68
} | {
"line": 171,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\n⊢ log⁺ (∏ t ∈ s, f t) ≤ ∑ t ∈ s, log⁺ (f t)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"Real.instLE",
"Real",
"Trans.trans",
"Finset.prod_insert",
"HMul.hMul",
"Real.po... | [] | induction s using Finset.induction with
| empty => simp [posLog]
| insert a s ha hs =>
calc log⁺ (∏ t ∈ insert a s, f t)
_ = log⁺ (f a * ∏ t ∈ s, f t) := by rw [Finset.prod_insert ha]
_ ≤ log⁺ (f a) + log⁺ (∏ t ∈ s, f t) := posLog_mul
_ ≤ log⁺ (f a) + ∑ t ∈ s, log⁺ (f t) := add_le_add (by rfl) hs
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 27
} | {
"line": 196,
"column": 2
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhs : s.Nonempty\nt_max : α\nht_max : t_max ∈ s ∧ ∀ x' ∈ s, |f x'| ≤ |f t_max|\n⊢ log⁺ (∑ t ∈ s, |f t_max|) = log⁺ (↑s.card * |f t_max|)",
"ppTerm": "?m.115",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"R... | [] | simp [Finset.sum_const] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 27
} | {
"line": 196,
"column": 2
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhs : s.Nonempty\nt_max : α\nht_max : t_max ∈ s ∧ ∀ x' ∈ s, |f x'| ≤ |f t_max|\n⊢ log⁺ (∑ t ∈ s, |f t_max|) = log⁺ (↑s.card * |f t_max|)",
"ppTerm": "?m.115",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"R... | [] | simp [Finset.sum_const] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 27
} | {
"line": 196,
"column": 2
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhs : s.Nonempty\nt_max : α\nht_max : t_max ∈ s ∧ ∀ x' ∈ s, |f x'| ≤ |f t_max|\n⊢ log⁺ (∑ t ∈ s, |f t_max|) = log⁺ (↑s.card * |f t_max|)",
"ppTerm": "?m.115",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"R... | [] | simp [Finset.sum_const] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 164,
"column": 8
} | {
"line": 164,
"column": 61
} | {
"line": 164,
"column": 61
} | [
{
"pp": "case e'_3\na b r : ℝ\nh : -1 < r ∨ r ≠ -1 ∧ 0 ∉ [[a, b]]\nh' : -1 < (↑r).re ∨ ↑r ≠ -1 ∧ 0 ∉ [[a, b]]\nthis : (∫ (x : ℝ) in a..b, ↑x ^ ↑r).re = ((↑b ^ (↑r + 1) - ↑a ^ (↑r + 1)) / (↑r + 1)).re\n⊢ (b ^ (r + 1) - a ^ (r + 1)) / (r + 1) = ((↑b ^ (↑r + 1) - ↑a ^ (↑r + 1)) / (↑r + 1)).re",
"ppTerm": "?e'_... | [
"case e'_3\na b r : ℝ\nh : -1 < r ∨ r ≠ -1 ∧ 0 ∉ [[a, b]]\nh' : -1 < (↑r).re ∨ ↑r ≠ -1 ∧ 0 ∉ [[a, b]]\nthis : (∫ (x : ℝ) in a..b, ↑x ^ ↑r).re = ((↑b ^ (↑r + 1) - ↑a ^ (↑r + 1)) / (↑r + 1)).re\n⊢ (b ^ (r + 1) - a ^ (r + 1)) / (r + 1) = ((↑b ^ ↑(r + 1) - ↑a ^ ↑(r + 1)) / ↑(r + 1)).re"
] | (by push_cast; rfl : (r : ℂ) + 1 = ((r + 1 : ℝ) : ℂ)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 57,
"column": 4
} | {
"line": 59,
"column": 65
} | {
"line": 60,
"column": 4
} | [
{
"pp": "a : ℂ\nh : ‖a‖ < 1\n⊢ circleAverage (fun x ↦ log ‖1 - x⁻¹ * a‖) 0 1 = 0",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"MulOne.toOne",
"Real",
"NonUn... | [
"a : ℂ\nh : ‖a‖ < 1\n⊢ InnerProductSpace.HarmonicOnNhd (fun x ↦ log ‖1 - x * a‖) (closedBall 0 |1|)"
] | rw [circleAverage_zero_one_congr_inv (f := fun x ↦ log ‖1 - x * a‖),
HarmonicOnNhd.circleAverage_eq, zero_mul, sub_zero,
CStarRing.norm_of_mem_unitary (unitary ℂ).one_mem, log_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 123,
"column": 4
} | {
"line": 124,
"column": 48
} | {
"line": 125,
"column": 4
} | [
{
"pp": "a : ℂ\nh : ‖a‖ = 1\nζ : ℝ\nhζ : a⁻¹ = circleMap 0 1 ζ\n⊢ ∫ (x : ℝ) in 0..2 * π, log ‖circleMap 0 1 (ζ + x) - 1‖ = ∫ (x : ℝ) in 0..2 * π, log ‖circleMap 0 1 x - 1‖",
"ppTerm": "?m.222",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"Real.pi",
"HMul.hMul",
... | [
"a : ℂ\nh : ‖a‖ = 1\nζ : ℝ\nhζ : a⁻¹ = circleMap 0 1 ζ\nthis : Function.Periodic (fun x ↦ log ‖circleMap 0 1 x - 1‖) (2 * π)\n⊢ ∫ (x : ℝ) in 0..2 * π, log ‖circleMap 0 1 (ζ + x) - 1‖ = ∫ (x : ℝ) in 0..2 * π, log ‖circleMap 0 1 x - 1‖"
] | have : Function.Periodic (log ‖circleMap 0 1 · - 1‖) (2 * π) :=
fun x ↦ by simp [periodic_circleMap 0 1 x] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 738,
"column": 4
} | {
"line": 738,
"column": 12
} | {
"line": 739,
"column": 4
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nz✝ : ℂ\nhre : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ ‖f ↑x‖\nhim : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhz✝ : 0 ≤ z✝.re\nε : ℝ\nε₀ : ε < 0\ng : ℂ → E := fun z ↦ cexp (↑ε * z) • f z\nhd : DiffContO... | [
"case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nz✝ : ℂ\nhre : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ ‖f ↑x‖\nhim : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhz✝ : 0 ≤ z✝.re\nε : ℝ\nε₀ : ε < 0\ng : ℂ → E := fun z ↦ cexp (↑ε * z) • f z\nhd : DiffContOnCl ℂ g {z |... | rw [hgn] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 133,
"column": 4
} | {
"line": 149,
"column": 10
} | {
"line": 150,
"column": 2
} | [
{
"pp": "case e_a\na : ℂ\nh : ‖a‖ = 1\nζ : ℝ\nhζ : a⁻¹ = circleMap 0 1 ζ\nx : ℝ\nhx : x ∈ [[0, 2 * π]]\n⊢ Complex.normSq (circleMap 0 1 x - 1) = 4 * sin (x / 2) ^ 2",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Co... | [] | calc Complex.normSq (circleMap 0 1 x - 1)
_ = (cos x - 1) * (cos x - 1) + sin x * sin x := by
simp [circleMap, Complex.normSq_apply]
_ = sin x ^ 2 + cos x ^ 2 + 1 - 2 * cos x := by
ring
_ = 2 - 2 * cos x := by
rw [sin_sq_add_cos_sq]
norm_num
_ = 2 - 2 * cos (2 * (x / 2)) := by
... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 248,
"column": 6
} | {
"line": 248,
"column": 67
} | {
"line": 248,
"column": 68
} | [
{
"pp": "case neg\na c : ℂ\nR : ℝ\nhu : a ∈ closedBall c |R|\nhR : ¬R = 0\n⊢ circleAverage (fun x ↦ log ‖x - a‖) c R = log R",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"HMul.hMul",
"R... | [
"case neg\na c : ℂ\nR : ℝ\nhu : a ∈ closedBall c |R|\nhR : ¬R = 0\n⊢ log R + log⁺ (R⁻¹ * ‖c - a‖) = log R"
] | circleAverage_log_norm_sub_const_eq_log_radius_add_posLog hR, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 787,
"column": 4
} | {
"line": 787,
"column": 96
} | {
"line": 788,
"column": 4
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nhd : DiffContOnCl ℂ f {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x ↦ ‖f ↑x‖\nC : ℝ\nhC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\ng : ℕ → ℂ → E := fun n z ↦ cexp z ^ n • f z\nhg : ∀ (n : ℕ) (z : ℂ), ‖g n ... | [
"case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nhd : DiffContOnCl ℂ f {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x ↦ ‖f ↑x‖\nC : ℝ\nhC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\ng : ℕ → ℂ → E := fun n z ↦ cexp z ^ n • f z\nhg : ∀ (n : ℕ) (z : ℂ), ‖g n z‖ = expR z.... | refine ((isBigO_refl (fun z : ℂ => expR z.re ^ n) _).mul hO.norm_left).trans (.of_bound' ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 113,
"column": 8
} | {
"line": 113,
"column": 34
} | {
"line": 113,
"column": 34
} | [
{
"pp": "f : ℂ → ℂ\nz₀ : ℂ\nhf : AnalyticAt ℂ f z₀\nh : ¬∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀\nR : ℝ\nhR : 0 < R\nh1 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ f z₀\nh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z\nρ : ℝ\nhρ : ρ > 0\nh4 : ∀ z ∈ closedBall z₀ ρ, z ≠ z₀ → f z ≠ f z₀\nh3 : DiffContOnCl ℂ f (ball z₀ ρ)\nr : ℝ := min ρ ... | [] | gcongr; exact inf_le_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 113,
"column": 8
} | {
"line": 113,
"column": 34
} | {
"line": 113,
"column": 34
} | [
{
"pp": "f : ℂ → ℂ\nz₀ : ℂ\nhf : AnalyticAt ℂ f z₀\nh : ¬∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀\nR : ℝ\nhR : 0 < R\nh1 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ f z₀\nh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z\nρ : ℝ\nhρ : ρ > 0\nh4 : ∀ z ∈ closedBall z₀ ρ, z ≠ z₀ → f z ≠ f z₀\nh3 : DiffContOnCl ℂ f (ball z₀ ρ)\nr : ℝ := min ρ ... | [] | gcongr; exact inf_le_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 38
} | {
"line": 142,
"column": 4
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\ng : E → ℂ\nz₀ : E\nhg : AnalyticAt ℂ g z₀\nray : E → ℂ → E := fun z t ↦ z₀ + t • z\ngray : E → ℂ → ℂ := fun z ↦ g ∘ ray z\nr : ℝ\nhr : r > 0\nhgr : ball z₀ r ⊆ {x | AnalyticAt ℂ g x}\nh1 : ∀ z ∈ sphere 0 1, AnalyticOnNhd ℂ ... | [
"case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\ng : E → ℂ\nz₀ : E\nhg : AnalyticAt ℂ g z₀\nray : E → ℂ → E := fun z t ↦ z₀ + t • z\ngray : E → ℂ → ℂ := fun z ↦ g ∘ ray z\nr : ℝ\nhr : r > 0\nhgr : ball z₀ r ⊆ {x | AnalyticAt ℂ g x}\nh1 : ∀ z ∈ sphere 0 1, AnalyticOnNhd ℂ (gray z) (ba... | let w : E := ‖z - z₀‖⁻¹ • (z - z₀) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 582,
"column": 68
} | {
"line": 584,
"column": 30
} | {
"line": 586,
"column": 0
} | [
{
"pp": "a b : ℝ\n⊢ ∫ (x : ℝ) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"MulOne.toOne",
"Mathlib.Meta.NormNum.isNat_add",
"Real",
"MeasureTheory.Measure",
"... | [] | by
have := @integral_sin_pow_mul_cos_pow_odd a b 2 0
norm_num at this; exact this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 653,
"column": 10
} | {
"line": 653,
"column": 76
} | {
"line": 654,
"column": 10
} | [
{
"pp": "⊢ ∫ (x : ℝ) in -(π / 2)..π / 2, √(1 - sin x ^ 2) * cos x = ∫ (x : ℝ) in -(π / 2)..π / 2, cos x ^ 2",
"ppTerm": "?m.189",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real",
"MeasureTheory.Measure",
"instHDiv",
"Real.pi",
"HMul.... | [
"x✝ : ℝ\nh : x✝ ∈ Ι (-(π / 2)) (π / 2)\n⊢ √(1 - sin x✝ ^ 2) * cos x✝ = cos x✝ ^ 2"
] | refine integral_congr_ae (MeasureTheory.ae_of_all _ fun _ h => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.Interval.Set.IsoIoo | {
"line": 41,
"column": 2
} | {
"line": 43,
"column": 42
} | {
"line": 44,
"column": 0
} | [
{
"pp": "case refine_3\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : LinearOrder k\ninst✝ : IsStrictOrderedRing k\n⊢ Function.RightInverse (fun x ↦ ↑x / (1 - |↑x|)) (codRestrict (fun x ↦ x / (1 + |x|)) (Ioo (-1) 1) ⋯)",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.... | [] | · refine fun x ↦ Subtype.ext ?_
have : 0 < 1 - |(x : k)| := sub_pos.2 (abs_lt.2 x.2)
simp [field, abs_div, abs_of_pos this] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.RCLike.Sqrt | {
"line": 32,
"column": 93
} | {
"line": 37,
"column": 36
} | {
"line": 39,
"column": 0
} | [
{
"pp": "a : ℂ\n⊢ a.sqrt = ↑√((‖a‖ + a.re) / 2) + (if 0 ≤ a.im then 1 else -1) * ↑√((‖a‖ - a.re) / 2) * I",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"instHDiv",
"NonUnitalCommRing.... | [] | by
rw [← cpow_inv_two_re, sqrt]
by_cases! h : 0 ≤ a.im
· simp [← cpow_inv_two_im_eq_sqrt h, h]
simp only [re_add_im, ↓reduceIte, h.not_ge, neg_one_mul, ← ofReal_neg,
← cpow_inv_two_im_eq_neg_sqrt h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.RCLike.Sqrt | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 83
} | {
"line": 117,
"column": 0
} | [
{
"pp": "α : ℝ\nhα : α ≥ 0\nha : 0 ≤ ↑α\n⊢ ↑(-↑α).sqrt.re + ↑(-↑α).sqrt.im * I = I * ↑√α",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Norm.norm",
"SeminormedAddGroup.toN... | [] | simp [sqrt, cpow_inv_two_im_eq_sqrt, abs_of_nonneg hα, cpow_inv_two_re, mul_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 407,
"column": 4
} | {
"line": 407,
"column": 87
} | {
"line": 408,
"column": 4
} | [
{
"pp": "case hf\nc : ℂ\nr R M : ℝ\nf : ℂ → ℂ\nr_pos : 0 < |r|\nr_lt_R : |r| < |R|\nhM : 1 ≤ M\nh₁f : AnalyticOnNhd ℂ f (closedBall c |R|)\nh₂f : f c ≠ 0\nf_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M\nhrR : 1 < |R / r|\njensen :\n circleAverage (fun x ↦ Real.log ‖f x‖) c R =\n ∑ᶠ (u : ℂ), ↑((divisor f (closedBal... | [
"case h₂f\nc : ℂ\nr R M : ℝ\nf : ℂ → ℂ\nr_pos : 0 < |r|\nr_lt_R : |r| < |R|\nhM : 1 ≤ M\nh₁f : AnalyticOnNhd ℂ f (closedBall c |R|)\nh₂f : f c ≠ 0\nf_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M\nhrR : 1 < |R / r|\njensen :\n circleAverage (fun x ↦ Real.log ‖f x‖) c R =\n ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|)) u... | · exact (h₁f.mono sphere_subset_closedBall).meromorphicOn.circleIntegrable_log_norm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.Tietze | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 8
} | {
"line": 91,
"column": 4
} | [
{
"pp": "𝕜 : Type v\ninst✝³ : RCLike 𝕜\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\ny : E\nr : ℝ\nhr : 0 < r\n⊢ ↑(closedBall y r) ≃ₜ ↑(closedBall 0 1)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Real",
"PseudoMetr... | [
"𝕜 : Type v\ninst✝³ : RCLike 𝕜\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\ny : E\nr : ℝ\nhr : 0 < r\n⊢ ↑(closedBall 0 1) ≃ₜ ↑(closedBall y r)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Topology.TietzeExtension | {
"line": 207,
"column": 6
} | {
"line": 207,
"column": 54
} | {
"line": 208,
"column": 6
} | [
{
"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... | [
"case inr.refine_2.inr.inl\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))\nhc₂ ... | rcases le_total (f x) (‖f‖ / 3) with hle₂ | hle₂ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 81,
"column": 75
} | {
"line": 83,
"column": 71
} | {
"line": 85,
"column": 0
} | [
{
"pp": "g : GL (Fin 2) ℝ\nz : ℍ\nh : (↑g).det < 0\nhtrace : (↑g).trace = 0\nhc : ↑g 1 0 ≠ 0\n⊢ g • z = z ↔ dist (↑z) (-↑(↑g 1 1) / ↑(↑g 1 0)) = √(-(↑g).det) / |↑g 1 0|",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.glAction",
"Iff.mpr",
"Real.sqrt_eq_iff... | [] | by
rw [gl_smul_eq_self_iff_dist_sq_eq h htrace hc, eq_comm, ← Real.sqrt_eq_iff_eq_sq, eq_comm,
Real.sqrt_div', Real.sqrt_sq_eq_abs] <;> positivity [neg_pos.mpr h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 147,
"column": 62
} | {
"line": 150,
"column": 6
} | {
"line": 152,
"column": 0
} | [
{
"pp": "g : GL (Fin 2) ℝ\nhg : (-g).IsElliptic\n⊢ fixedPt (-g) hg = fixedPt g ⋯",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"one_pow",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Units.val",
"Eq.mpr",
"NegZeroCl... | [] | by
ext
simp [fixedPt, Matrix.discr_fin_two, Matrix.det_neg]
ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 245,
"column": 4
} | {
"line": 246,
"column": 15
} | {
"line": 248,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝ : CommRing R\ng : GL (Fin 2) R\n⊢ ↑g ∈ Set.range ⇑(Matrix.scalar (Fin 2)) → C (↑g 1 0) * X ^ 2 + C (↑g 1 1 - ↑g 0 0) * X - C (↑g 0 1) = 0",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Units.val",
"Polynomial.C",
"RingHom.instRingHo... | [] | rintro ⟨a, ha⟩
simp [← ha] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 245,
"column": 4
} | {
"line": 246,
"column": 15
} | {
"line": 248,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝ : CommRing R\ng : GL (Fin 2) R\n⊢ ↑g ∈ Set.range ⇑(Matrix.scalar (Fin 2)) → C (↑g 1 0) * X ^ 2 + C (↑g 1 1 - ↑g 0 0) * X - C (↑g 0 1) = 0",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Units.val",
"Polynomial.C",
"RingHom.instRingHo... | [] | rintro ⟨a, ha⟩
simp [← ha] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 264,
"column": 4
} | {
"line": 272,
"column": 13
} | {
"line": 273,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u_2\ninst✝¹ : Field K\ng : GL (Fin 2) K\ninst✝ : CharZero K\nn : ℕ\nhn : 1 ≤ 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\n⊢ ((Matrix.scalar (Fin 2)) a + m) ^ n = (Matrix.scalar (Fin 2)) (a ^ n) + (↑n * a ^ (n - 1... | [] | induction n, hn using Nat.le_induction with
| base => simp
| succ n hn IH =>
simp only [pow_succ, IH, add_mul, Nat.add_sub_cancel, mul_add, ← map_mul, add_assoc]
simp only [scalar_apply, ← smul_eq_mul_diagonal, ← mul_smul,
← smul_eq_diagonal_mul, smul_mul, ← sq, hmsq, smul_zero, add_zero, ← ... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Geometry.Euclidean.Inversion.Basic | {
"line": 184,
"column": 2
} | {
"line": 185,
"column": 14
} | {
"line": 186,
"column": 2
} | [
{
"pp": "case inr.inl\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nb c d : P\nhb : b ≠ c\n⊢ dist c c * dist b d ≤ dist c b * dist c d + dist b c * dist c d",
"ppTerm": "?inr.inl",
"assigned": true,
"u... | [
"case inr.inr\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d : P\nhb : b ≠ a\nhc : c ≠ a\n⊢ dist a c * dist b d ≤ dist a b * dist c d + dist b c * dist a d"
] | · rw [dist_self, zero_mul]
positivity | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 51,
"column": 63
} | {
"line": 51,
"column": 75
} | {
"line": 51,
"column": 76
} | [
{
"pp": "z w : ℍ\n⊢ 1 + dist ↑z ↑w ^ 2 / (2 * √(z.im * w.im)) ^ 2 = dist (↑z) ((starRingEnd ℂ) ↑w) ^ 2 / (2 * √(z.im * w.im)) ^ 2",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [
"z w : ℍ\n⊢ ((2 * √(z.im * w.im)) ^ 2 + dist ↑z ↑w ^ 2) / (2 * √(z.im * w.im)) ^ 2 =\n dist (↑z) ((starRingEnd ℂ) ↑w) ^ 2 / (2 * √(z.im * w.im)) ^ 2",
"z w : ℍ\n⊢ (2 * √(z.im * w.im)) ^ 2 ≠ 0",
"case ha\nz w : ℍ\n⊢ 0 ≤ cosh (dist z w / 2)",
"case hb\nz w : ℍ\n⊢ 0 ≤ dist (↑z) ((starRingEnd ℂ) ↑w) / (2 * √(z... | one_add_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 164,
"column": 2
} | {
"line": 165,
"column": 79
} | {
"line": 166,
"column": 2
} | [
{
"pp": "g : GL (Fin 2) ℝ\nk : ℤ\nτ : ℍ\n⊢ HasDerivAt (fun z ↦ denom g z ^ k) (↑k * ↑(↑g 1 0) * denom g ↑τ ^ (k - 1)) ↑τ",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"NormedCommRing.toNormedRing",
"Units.val",
"Eq.mpr",
... | [
"g : GL (Fin 2) ℝ\nk : ℤ\nτ : ℍ\nhd : HasDerivAt (fun x ↦ denom g x) ↑(↑g 1 0) ↑τ\n⊢ HasDerivAt (fun z ↦ denom g z ^ k) (↑k * ↑(↑g 1 0) * denom g ↑τ ^ (k - 1)) ↑τ"
] | have hd : HasDerivAt (denom g ·) (g 1 0) τ := by
simpa [denom] using hasDerivAt_id _ |>.const_mul _ |>.add_const (g 1 1 : ℂ) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Compactification.StoneCech | {
"line": 192,
"column": 2
} | {
"line": 196,
"column": 63
} | {
"line": 197,
"column": 2
} | [
{
"pp": "α : Type u\nγ : Type u_1\ninst✝² : TopologicalSpace γ\ninst✝¹ : T2Space γ\ninst✝ : CompactSpace γ\nf : α → γ\n⊢ Continuous[_, inst✝²] (Ultrafilter.extend f)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Filter.instMembership",
"Eq.mpr",
"Fil... | [
"α : Type u\nγ : Type u_1\ninst✝² : TopologicalSpace γ\ninst✝¹ : T2Space γ\ninst✝ : CompactSpace γ\nf : α → γ\nh : ∀ (b : Ultrafilter α), ∃ c, Tendsto f (comap pure (𝓝 b)) (𝓝 c)\n⊢ Continuous[_, inst✝²] (Ultrafilter.extend f)"
] | have h (b : Ultrafilter α) : ∃ c, Tendsto f (comap pure (𝓝 b)) (𝓝 c) :=
-- b.map f is an ultrafilter on γ, which is compact, so it converges to some c in γ.
let ⟨c, _, h'⟩ :=
isCompact_univ.ultrafilter_le_nhds (b.map f) (by rw [le_principal_iff]; exact univ_mem)
⟨c, le_trans (map_mono (ultrafilter_c... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 320,
"column": 80
} | {
"line": 325,
"column": 62
} | {
"line": 327,
"column": 0
} | [
{
"pp": "a : { x // 0 < x }\n⊢ Isometry fun x ↦ a • x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"mul_self_nonneg",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"GroupWith... | [] | by
refine Isometry.of_dist_eq fun y₁ y₂ => ?_
simp only [dist_eq, coe_pos_real_smul, pos_real_im]; congr 2
rw [dist_smul₀, mul_mul_mul_comm, Real.sqrt_mul (mul_self_nonneg _), Real.sqrt_mul_self_eq_abs,
Real.norm_eq_abs, mul_left_comm]
exact mul_div_mul_left _ _ (mt _root_.abs_eq_zero.1 a.2.ne') | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.ProperAction.Basic | {
"line": 216,
"column": 70
} | {
"line": 226,
"column": 93
} | {
"line": 228,
"column": 0
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝⁴ : Group G\ninst✝³ : MulAction G X\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalSpace X\ninst✝ : ProperSMul G X\nt : Set X\n⊢ IsProperMap fun gx ↦ (gx.1 • ↑gx.2, gx.2)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Homeomorph.Set.univ... | [] | by
let Φ : G × X → X × X := fun gx ↦ (gx.1 • gx.2, gx.2)
have Φ_proper : IsProperMap Φ := ProperSMul.isProperMap_smul_pair
let α : G × t ≃ₜ (Φ ⁻¹' snd ⁻¹' t) :=
have : univ ×ˢ t = Φ ⁻¹' snd ⁻¹' t := by ext; simp [Φ]
Homeomorph.Set.univ G |>.symm.prodCongr (.refl t) |>.trans
((Homeomorph.Set.prod _ t... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.IntegralPresentation | {
"line": 103,
"column": 4
} | {
"line": 104,
"column": 61
} | {
"line": 105,
"column": 4
} | [
{
"pp": "f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nμ : Measure ℝ := volume.restrict (Ioc 0 (2 * π))\nh_meas_K : Measurable fun a ↦ (cartanKernel f R a.1 a.2)⁺\nh_int_posLog : Integrable (fun β ↦ log⁺ ‖f (circleMap 0 R β)‖) μ\nh_int_Bound : Integrable (fun β ↦ log⁺ ‖f (circleMap 0 R β)‖ + log 2) μ\nβ : ℝ\nh_int_nonne... | [
"f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nμ : Measure ℝ := volume.restrict (Ioc 0 (2 * π))\nh_meas_K : Measurable fun a ↦ (cartanKernel f R a.1 a.2)⁺\nh_int_posLog : Integrable (fun β ↦ log⁺ ‖f (circleMap 0 R β)‖) μ\nh_int_Bound : Integrable (fun β ↦ log⁺ ‖f (circleMap 0 R β)‖ + log 2) μ\nβ : ℝ\nh_int_nonneg : 0 ≤ ∫ (α... | have h_bound_nonneg : 0 ≤ (2 * π) * (log⁺ ‖f (circleMap 0 R β)‖ + log 2) := by
positivity [posLog_nonneg (x := ‖f (circleMap 0 R β)‖)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.ConstantSpeed | {
"line": 121,
"column": 6
} | {
"line": 121,
"column": 40
} | {
"line": 122,
"column": 6
} | [
{
"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... | [
"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) = ENNReal.ofReal (↑l... | simp only [NNReal.val_eq_coe] at q | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.ConstantSpeed | {
"line": 157,
"column": 2
} | {
"line": 164,
"column": 63
} | {
"line": 165,
"column": 2
} | [
{
"pp": "case mp\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\n⊢ (∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = 0) → eVariationOn f s = 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"PseudoEMetri... | [
"case mpr\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\n⊢ eVariationOn f s = 0 → ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = 0"
] | · by_contra! ⟨h, hfs⟩
simp_rw [ne_eq, eVariationOn.eq_zero_iff] at hfs h
push Not at hfs
obtain ⟨x, xs, y, ys, hxy⟩ := hfs
rcases le_total x y with (xy | yx)
· exact hxy (h xs ys x ⟨xs, le_rfl, xy⟩ y ⟨ys, xy, le_rfl⟩)
· rw [edist_comm] at hxy
exact hxy (h ys xs y ⟨ys, le_rfl, yx⟩ x ⟨xs, yx... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.ConstantSpeed | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 6
} | {
"line": 176,
"column": 2
} | [
{
"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 :\n LocallyBoundedVariationOn (f ∘ φ) s ∧\n ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → variationOnFromTo (f ∘ φ) s x y = ↑l * (y - x)\nhf :\n LocallyBoundedVariationOn f (φ ''... | [
"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 :\n LocallyBoundedVariationOn (f ∘ φ) s ∧\n ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → variationOnFromTo (f ∘ φ) s x y = ↑l * (y - x)\nhf :\n LocallyBoundedVariationOn f (φ '' s) ∧\n ∀... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.Convex.BetweenList | {
"line": 112,
"column": 6
} | {
"line": 121,
"column": 52
} | {
"line": 122,
"column": 6
} | [
{
"pp": "case cons.refine_1.refine_1\nR : 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\nhead : P\ntail : List P\nih : List.Wbtw R tail ∧ Pairwise (fun x1 x2 ↦ x1 ≠ x2) tail ↔ Triplewi... | [
"case cons.refine_1.refine_2\nR : 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\nhead : P\ntail : List P\nih : List.Wbtw R tail ∧ Pairwise (fun x1 x2 ↦ x1 ≠ x2) tail ↔ Triplewise (Sbtw R) ... | · clear ih
induction tail with
| nil => simp
| cons head2 tail ih' =>
rw [pairwise_cons] at hp hpne hpne ⊢
refine ⟨fun a ha ↦ ⟨hp.1 a ha, ?_⟩, ?_⟩
· refine ⟨(hpne.1 head2 ?_).symm, hpne.2.1 a ha⟩
simp
· rw [wbtw_cons] at ht
grind [L... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Between | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 31
} | {
"line": 108,
"column": 32
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : IsOrderedRing R\nx y : V\n⊢ affineSegment R x y = segment R x y",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"affineSegment",
"Eq.mpr",
... | [
"R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : IsOrderedRing R\nx y : V\n⊢ affineSegment R x y = ⇑(lineMap x y) '' Set.Icc 0 1"
] | segment_eq_image_lineMap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Between | {
"line": 586,
"column": 2
} | {
"line": 586,
"column": 33
} | {
"line": 588,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁷ : Ring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : IsOrderedRing R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R V\nw x z : P\nh₁ : Wbtw R w x z\nh : w ≠ x\nh₂ : Wbtw R x w z\n⊢ False",
"... | [] | exact h (h₁.swap_left_iff.1 h₂) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Between | {
"line": 585,
"column": 2
} | {
"line": 586,
"column": 33
} | {
"line": 588,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁷ : Ring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : IsOrderedRing R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R V\nw x y z : P\nh₁ : Wbtw R w x z\nh₂ : Wbtw R x y z\nh : w ≠ x\n⊢ w ≠ y",
... | [] | rintro rfl
exact h (h₁.swap_left_iff.1 h₂) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Between | {
"line": 585,
"column": 2
} | {
"line": 586,
"column": 33
} | {
"line": 588,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁷ : Ring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : IsOrderedRing R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R V\nw x y z : P\nh₁ : Wbtw R w x z\nh₂ : Wbtw R x y z\nh : w ≠ x\n⊢ w ≠ y",
... | [] | rintro rfl
exact h (h₁.swap_left_iff.1 h₂) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.PEquiv | {
"line": 249,
"column": 23
} | {
"line": 249,
"column": 73
} | {
"line": 251,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α ≃. β\n⊢ (f.symm.trans f).symm = (ofSet {b | (f.symm b).isSome = true}).symm",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"PEquiv.instFunLikeOption",
"PEquiv.ofSet",
"congrArg",
"setOf",
"instDecidableEqBool",
"Se... | [] | simp [symm_trans_rev, self_trans_symm, -symm_symm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.PEquiv | {
"line": 249,
"column": 23
} | {
"line": 249,
"column": 73
} | {
"line": 251,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α ≃. β\n⊢ (f.symm.trans f).symm = (ofSet {b | (f.symm b).isSome = true}).symm",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"PEquiv.instFunLikeOption",
"PEquiv.ofSet",
"congrArg",
"setOf",
"instDecidableEqBool",
"Se... | [] | simp [symm_trans_rev, self_trans_symm, -symm_symm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.PEquiv | {
"line": 249,
"column": 23
} | {
"line": 249,
"column": 73
} | {
"line": 251,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α ≃. β\n⊢ (f.symm.trans f).symm = (ofSet {b | (f.symm b).isSome = true}).symm",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"PEquiv.instFunLikeOption",
"PEquiv.ofSet",
"congrArg",
"setOf",
"instDecidableEqBool",
"Se... | [] | simp [symm_trans_rev, self_trans_symm, -symm_symm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Matrix.PEquiv | {
"line": 116,
"column": 35
} | {
"line": 116,
"column": 57
} | {
"line": 116,
"column": 58
} | [
{
"pp": "l : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝² : Fintype m\ninst✝¹ : DecidableEq n\ninst✝ : NonAssocSemiring α\nM : Matrix l m α\nf : m ≃ n\ni : l\nj : n\n⊢ Option.casesOn (f.toPEquiv.symm j) 0 (M i) = M.submatrix id (⇑f.symm) i j",
"ppTerm": "?m.50",
"assigned": true,
"used... | [
"l : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝² : Fintype m\ninst✝¹ : DecidableEq n\ninst✝ : NonAssocSemiring α\nM : Matrix l m α\nf : m ≃ n\ni : l\nj : n\n⊢ Option.casesOn (f.symm.toPEquiv j) 0 (M i) = M.submatrix id (⇑f.symm) i j"
] | ← Equiv.toPEquiv_symm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Matrix.PEquiv | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 31
} | {
"line": 153,
"column": 4
} | [
{
"pp": "case h.none\nm : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝² : DecidableEq n\ninst✝¹ : MulZeroOneClass α\ninst✝ : Nontrivial α\nf g : m ≃. n\ni : m\nhi : ¬f i = g i\nhf : f i = none\n⊢ ∃ x, ¬(if x ∈ none then 1 else 0) = if x ∈ g i then 1 else 0",
"ppTerm": "?h.none",
"assigned": true,
"us... | [
"case h.none.none\nm : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝² : DecidableEq n\ninst✝¹ : MulZeroOneClass α\ninst✝ : Nontrivial α\nf g : m ≃. n\ni : m\nhi : ¬f i = g i\nhf : f i = none\nhg : g i = none\n⊢ ∃ x, ¬(if x ∈ none then 1 else 0) = if x ∈ none then 1 else 0",
"case h.none.some\nm : Type u_3\nn : Type... | rcases hg : g i with - | gi | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Data.Matrix.PEquiv | {
"line": 156,
"column": 2
} | {
"line": 157,
"column": 30
} | {
"line": 159,
"column": 0
} | [
{
"pp": "case h.some\nm : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝² : DecidableEq n\ninst✝¹ : MulZeroOneClass α\ninst✝ : Nontrivial α\nf g : m ≃. n\ni : m\nhi : ¬f i = g i\nfi : n\nhf : f i = some fi\n⊢ ∃ x, ¬(if x ∈ some fi then 1 else 0) = if x ∈ g i then 1 else 0",
"ppTerm": "?h.some",
"assigned":... | [] | · use fi
simp [hf.symm, Ne.symm hi] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Between | {
"line": 700,
"column": 39
} | {
"line": 700,
"column": 45
} | {
"line": 700,
"column": 46
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁷ : Ring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AffineSpace V P\ninst✝² : IsOrderedRing R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R V\ns : Simplex R P 1\np : P\n⊢ 0 ≠ 1",
"ppTerm": "?m.116",
"assigned... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Analysis.Convex.Between | {
"line": 700,
"column": 39
} | {
"line": 700,
"column": 45
} | {
"line": 700,
"column": 46
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁷ : Ring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AffineSpace V P\ninst✝² : IsOrderedRing R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R V\ns : Simplex R P 1\np : P\n⊢ 0 ≠ 1",
"ppTerm": "?m.116",
"assigned... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Between | {
"line": 700,
"column": 39
} | {
"line": 700,
"column": 45
} | {
"line": 700,
"column": 46
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁷ : Ring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AffineSpace V P\ninst✝² : IsOrderedRing R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R V\ns : Simplex R P 1\np : P\n⊢ 0 ≠ 1",
"ppTerm": "?m.116",
"assigned... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Permutation | {
"line": 52,
"column": 30
} | {
"line": 52,
"column": 52
} | {
"line": 52,
"column": 53
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝² : DecidableEq n\nσ : Perm n\ninst✝¹ : Zero R\ninst✝ : One R\n⊢ (toPEquiv σ).symm.toMatrix = Perm.permMatrix R σ⁻¹",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.instInv",
"Equiv.toPEquiv",
"congrArg"... | [
"n : Type u_1\nR : Type u_2\ninst✝² : DecidableEq n\nσ : Perm n\ninst✝¹ : Zero R\ninst✝ : One R\n⊢ (Equiv.symm σ).toPEquiv.toMatrix = Perm.permMatrix R σ⁻¹"
] | ← Equiv.toPEquiv_symm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Permutation | {
"line": 94,
"column": 45
} | {
"line": 94,
"column": 59
} | {
"line": 94,
"column": 59
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝² : DecidableEq n\nσ✝ τ✝ : Perm n\ninst✝¹ : Fintype n\ninst✝ : NonAssocSemiring R\nσ τ : Perm n\n⊢ Perm.permMatrix R (τ⁻¹ * σ⁻¹) = Perm.permMatrix R σ⁻¹ * Perm.permMatrix R τ⁻¹",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"n : Type u_1\nR : Type u_2\ninst✝² : DecidableEq n\nσ✝ τ✝ : Perm n\ninst✝¹ : Fintype n\ninst✝ : NonAssocSemiring R\nσ τ : Perm n\n⊢ Perm.permMatrix R σ⁻¹ * Perm.permMatrix R τ⁻¹ = Perm.permMatrix R σ⁻¹ * Perm.permMatrix R τ⁻¹"
] | permMatrix_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Caratheodory | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 24
} | {
"line": 84,
"column": 4
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : DecidableEq E\nt : Finset E\nf : E → 𝕜\nfpos : ∀ y ∈ t, 0 ≤ f y\nfsum : ∑ y ∈ t, f y = 1\ng : E → 𝕜\ngcombo : ∑ e ∈ t, g e • e ... | [
"case pos\n𝕜 : Type u_1\nE : Type u\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : DecidableEq E\nt : Finset E\nf : E → 𝕜\nfpos : ∀ y ∈ t, 0 ≤ f y\nfsum : ∑ y ∈ t, f y = 1\ng : E → 𝕜\ngcombo : ∑ e ∈ t, g e • e = 0\ngsum : ∑ e ∈... | by_cases hes : e ∈ s | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Convex.Body | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 72
} | {
"line": 218,
"column": 4
} | [
{
"pp": "case refine_1\nV : Type u_1\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : T2Space V\nu : ℕ → ℝ≥0\nK : ConvexBody V\nh_zero : 0 ∈ K\nhu : Tendsto u atTop (𝓝 0)\nx : V\nh : x ∈ ⋂ n, (1 + ↑(u n)) • ↑K\nC : ℝ\nhC_pos : C > 0\nhC_bdd : ∀ x ∈ ↑K, ‖x‖ ≤ C\n⊢ x ∈ ↑K",
"ppTerm": "?r... | [
"case refine_1\nV : Type u_1\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : T2Space V\nu : ℕ → ℝ≥0\nK : ConvexBody V\nh_zero : 0 ∈ K\nhu : Tendsto u atTop (𝓝 0)\nx : V\nh : x ∈ ⋂ n, (1 + ↑(u n)) • ↑K\nC : ℝ\nhC_pos : C > 0\nhC_bdd : ∀ x ∈ ↑K, ‖x‖ ≤ C\n⊢ ∀ (ε : ℝ), 0 < ε → ∃ b ∈ ↑K, ‖x - b‖ <... | rw [← K.isClosed.closure_eq, SeminormedAddCommGroup.mem_closure_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Between | {
"line": 927,
"column": 2
} | {
"line": 927,
"column": 47
} | {
"line": 928,
"column": 2
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\ns : AffineSubspace R P\nhx : x ∈ s\nε : R\nhy : (lineMap x z) ε ∈ s\nhxy : x ≠ (lineMap x z) ε\n⊢ z ∈ s",
"ppTerm": "?m.53",
"assigne... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\ns : AffineSubspace R P\nhx : x ∈ s\nε : R\nhy : (lineMap x z) ε ∈ s\nhxy : x ≠ (lineMap x z) ε\nhε : ε ≠ 0\n⊢ z ∈ s"
] | have hε : ε ≠ 0 := by rintro rfl; simp at hxy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Between | {
"line": 932,
"column": 2
} | {
"line": 932,
"column": 95
} | {
"line": 933,
"column": 2
} | [
{
"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 : P\nv : V\nr₁ r₂ : R\nhr₁ : 0 ≤ r₁\nhr₂ : r₁ ≤ r₂\n⊢ Wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x)",
"ppTerm": "?m.35",
... | [
"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 : P\nv : V\nr₁ r₂ : R\nhr₁ : 0 ≤ r₁\nhr₂ : r₁ ≤ r₂\n⊢ (lineMap x (r₂ • v +ᵥ x)) (r₁ / r₂) = r₁ • v +ᵥ x"
] | refine ⟨r₁ / r₂, ⟨div_nonneg hr₁ (hr₁.trans hr₂), div_le_one_of_le₀ hr₂ (hr₁.trans hr₂)⟩, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 180,
"column": 2
} | {
"line": 197,
"column": 7
} | {
"line": 199,
"column": 0
} | [
{
"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\n⊢ Set.extremePoints R ↑(doublyStochastic R n) = {x | ∃ σ, Equiv.Perm.permMatrix R σ = x}",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [... | [] | refine subset_antisymm ?_ ?_
· rw [doublyStochastic_eq_convexHull_permMatrix]
exact extremePoints_convexHull_subset
rintro _ ⟨σ, rfl⟩
refine ⟨permMatrix_mem_doublyStochastic, fun x₁ hx₁ x₂ hx₂ hσ ↦ ?_⟩
suffices ∀ i j : n, x₁ i j = x₂ i j by
obtain rfl : x₁ = x₂ := by simpa [← Matrix.ext_iff]
simp_al... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 180,
"column": 2
} | {
"line": 197,
"column": 7
} | {
"line": 199,
"column": 0
} | [
{
"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\n⊢ Set.extremePoints R ↑(doublyStochastic R n) = {x | ∃ σ, Equiv.Perm.permMatrix R σ = x}",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [... | [] | refine subset_antisymm ?_ ?_
· rw [doublyStochastic_eq_convexHull_permMatrix]
exact extremePoints_convexHull_subset
rintro _ ⟨σ, rfl⟩
refine ⟨permMatrix_mem_doublyStochastic, fun x₁ hx₁ x₂ hx₂ hσ ↦ ?_⟩
suffices ∀ i j : n, x₁ i j = x₂ i j by
obtain rfl : x₁ = x₂ := by simpa [← Matrix.ext_iff]
simp_al... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Between | {
"line": 1060,
"column": 4
} | {
"line": 1065,
"column": 71
} | {
"line": 1067,
"column": 0
} | [
{
"pp": "case inr.inr\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\nx : P\nv : V\nty tz : R\nh : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = tz • v +ᵥ x → ∃ r, p = r • v +ᵥ x\... | [] | rcases lt_trichotomy tz 0 with (hz0 | rfl | hz0)
· refine Or.inr (Or.inr (wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg _ _ hz0.le hy0.le))
· simp
· rw [wbtw_comm (z := x)]
rw [← or_assoc]
exact Or.inl (wbtw_or_wbtw_smul_vadd_of_nonneg _ _ hy0.le hz0.le) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Between | {
"line": 1060,
"column": 4
} | {
"line": 1065,
"column": 71
} | {
"line": 1067,
"column": 0
} | [
{
"pp": "case inr.inr\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\nx : P\nv : V\nty tz : R\nh : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = tz • v +ᵥ x → ∃ r, p = r • v +ᵥ x\... | [] | rcases lt_trichotomy tz 0 with (hz0 | rfl | hz0)
· refine Or.inr (Or.inr (wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg _ _ hz0.le hy0.le))
· simp
· rw [wbtw_comm (z := x)]
rw [← or_assoc]
exact Or.inl (wbtw_or_wbtw_smul_vadd_of_nonneg _ _ hy0.le hz0.le) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Extrema | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 64
} | {
"line": 42,
"column": 2
} | [
{
"pp": "case inr\nβ : Type u_2\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module ℝ β\ninst✝¹ : IsOrderedModule ℝ β\ninst✝ : PosSMulReflectLE ℝ β\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[≥] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\n... | [
"case inr\nβ : Type u_2\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module ℝ β\ninst✝¹ : IsOrderedModule ℝ β\ninst✝ : PosSMulReflectLE ℝ β\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsMinFilter f (𝓝[≥] a) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ Icc... | have H₂ : ∀ᶠ y in 𝓝[>] a, y ∈ Ioc a c := Ioc_mem_nhdsGT a_lt_c | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Continuous | {
"line": 40,
"column": 55
} | {
"line": 40,
"column": 88
} | {
"line": 41,
"column": 8
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nε r M : ℝ\nhf : ConvexOn ℝ (ball x₀ r) f\nhε : 0 < ε\nhM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M\nK : ℝ := 2 * M / ε\nhK : K = 2 * M / ε\nx y : E\nhx : x ∈ ball x₀ (r - ε)\nhy : y ∈ ball x₀ (r - ε)\nhx₀r : ball x₀ (... | [] | simp only [z, add_sub_right_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.Continuous | {
"line": 40,
"column": 55
} | {
"line": 40,
"column": 88
} | {
"line": 41,
"column": 8
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nε r M : ℝ\nhf : ConvexOn ℝ (ball x₀ r) f\nhε : 0 < ε\nhM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M\nK : ℝ := 2 * M / ε\nhK : K = 2 * M / ε\nx y : E\nhx : x ∈ ball x₀ (r - ε)\nhy : y ∈ ball x₀ (r - ε)\nhx₀r : ball x₀ (... | [] | simp only [z, add_sub_right_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Continuous | {
"line": 40,
"column": 55
} | {
"line": 40,
"column": 88
} | {
"line": 41,
"column": 8
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nε r M : ℝ\nhf : ConvexOn ℝ (ball x₀ r) f\nhε : 0 < ε\nhM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M\nK : ℝ := 2 * M / ε\nhK : K = 2 * M / ε\nx y : E\nhx : x ∈ ball x₀ (r - ε)\nhy : y ∈ ball x₀ (r - ε)\nhx₀r : ball x₀ (... | [] | simp only [z, add_sub_right_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Independent | {
"line": 139,
"column": 2
} | {
"line": 142,
"column": 13
} | {
"line": 143,
"column": 2
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\n⊢ ConvexIndependent 𝕜 Subtype.val → ∀ t ⊆ s, s ∩ (convexHull 𝕜) t ⊆ t",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case mpr\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\n⊢ (∀ t ⊆ s, s ∩ (convexHull 𝕜) t ⊆ t) → ConvexIndependent 𝕜 Subtype.val"
] | · rintro hc t h x ⟨hxs, hxt⟩
refine hc { x | ↑x ∈ t } ⟨x, hxs⟩ ?_
rw [Subtype.coe_image_of_subset h]
exact hxt | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.GaugeRescale | {
"line": 121,
"column": 19
} | {
"line": 121,
"column": 50
} | {
"line": 121,
"column": 50
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℝ E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : T1Space E\ns t : Set E\nhs : Convex ℝ s\nhs₀ : s ∈ 𝓝 0\nht : Convex ℝ t\nht₀ : t ∈ 𝓝 0\nhtb : IsVonNBounded ℝ t\nhta : Absorbent ℝ ... | [
"case inl\nE : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℝ E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : T1Space E\ns t : Set E\nhs : Convex ℝ s\nhs₀ : s ∈ 𝓝 0\nht : Convex ℝ t\nht₀ : t ∈ 𝓝 0\nhtb : IsVonNBounded ℝ t\nhta : Absorbent ℝ t\n⊢ Tendsto... | ← comap_gauge_nhds_zero htb ht₀ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Intrinsic | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 54
} | {
"line": 207,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_5\ninst✝⁴ : Ring 𝕜\ninst✝³ : AddCommGroup V\ninst✝² : Module 𝕜 V\ninst✝¹ : TopologicalSpace P\ninst✝ : AddTorsor V P\ns : Set P\n⊢ IsClosed[instTopologicalSpaceSubtype] (Subtype.val ⁻¹' intrinsicClosure 𝕜 s)",
"ppTerm": "?m.36",
"assigned": true,
"... | [
"𝕜 : Type u_1\nV : Type u_2\nP : Type u_5\ninst✝⁴ : Ring 𝕜\ninst✝³ : AddCommGroup V\ninst✝² : Module 𝕜 V\ninst✝¹ : TopologicalSpace P\ninst✝ : AddTorsor V P\ns : Set P\nt : AffineSubspace 𝕜 P := affineSpan 𝕜 (intrinsicClosure 𝕜 s)\nht : t = affineSpan 𝕜 (intrinsicClosure 𝕜 s)\n⊢ IsClosed[instTopologicalSpac... | set t := affineSpan 𝕜 (intrinsicClosure 𝕜 s) with ht | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Analysis.Convex.Intrinsic | {
"line": 224,
"column": 29
} | {
"line": 224,
"column": 54
} | {
"line": 224,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\nQ : Type u_4\nP : Type u_5\ninst✝⁸ : Ring 𝕜\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module 𝕜 V\ninst✝⁵ : TopologicalSpace P\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup W\ninst✝² : Module 𝕜 W\ninst✝¹ : TopologicalSpace Q\ninst✝ : AddTorsor W Q\ns : Set P\nt : ... | [] | simp [affineSpan_prod_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.Intrinsic | {
"line": 224,
"column": 29
} | {
"line": 224,
"column": 54
} | {
"line": 224,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\nQ : Type u_4\nP : Type u_5\ninst✝⁸ : Ring 𝕜\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module 𝕜 V\ninst✝⁵ : TopologicalSpace P\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup W\ninst✝² : Module 𝕜 W\ninst✝¹ : TopologicalSpace Q\ninst✝ : AddTorsor W Q\ns : Set P\nt : ... | [] | simp [affineSpan_prod_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Intrinsic | {
"line": 224,
"column": 29
} | {
"line": 224,
"column": 54
} | {
"line": 224,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\nQ : Type u_4\nP : Type u_5\ninst✝⁸ : Ring 𝕜\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module 𝕜 V\ninst✝⁵ : TopologicalSpace P\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup W\ninst✝² : Module 𝕜 W\ninst✝¹ : TopologicalSpace Q\ninst✝ : AddTorsor W Q\ns : Set P\nt : ... | [] | simp [affineSpan_prod_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Intrinsic | {
"line": 402,
"column": 4
} | {
"line": 404,
"column": 23
} | {
"line": 405,
"column": 2
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nV : Type u_2\nP : Type u_5\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : CompleteSpace 𝕜\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace 𝕜 V\ninst✝² : FiniteDimensional 𝕜 V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\nx : P\n⊢ (∃ y,\n (∀ (... | [] | rintro ⟨x, h, rfl⟩ t ht hx
obtain ⟨z, hz₁, hz₂⟩ := h _ (continuous_induced_dom.isOpen_preimage t ht) hx
exact ⟨z, hz₁, hz₂⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Intrinsic | {
"line": 402,
"column": 4
} | {
"line": 404,
"column": 23
} | {
"line": 405,
"column": 2
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nV : Type u_2\nP : Type u_5\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : CompleteSpace 𝕜\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace 𝕜 V\ninst✝² : FiniteDimensional 𝕜 V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\nx : P\n⊢ (∃ y,\n (∀ (... | [] | rintro ⟨x, h, rfl⟩ t ht hx
obtain ⟨z, hz₁, hz₂⟩ := h _ (continuous_induced_dom.isOpen_preimage t ht) hx
exact ⟨z, hz₁, hz₂⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Convex.ConvexSpace.AffineSpace | {
"line": 145,
"column": 4
} | {
"line": 147,
"column": 47
} | {
"line": 148,
"column": 4
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AffineSpace V P\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nx y : P\n⊢ ∑ i ∈ (Finsupp.single x s + Finsupp.single y t).support, ... | [
"R : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AffineSpace V P\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nx y : P\n⊢ ((Finsupp.single x s + Finsupp.single y t).sum fun x r ↦ r) = 1"
] | · apply Finset.sum_congr rfl
intro i _
simp only [Finsupp.coe_add, Pi.add_apply] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Convex.Star | {
"line": 62,
"column": 35
} | {
"line": 62,
"column": 84
} | {
"line": 64,
"column": 0
} | [
{
"pp": "R : Type u_1\nX : Type u_2\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R X\nx : X\ns t : Set X\nhs : IsStarConvexSet R x s\nht : IsStarConvexSet R x t\n⊢ IsStarConvexSet R x (s ∩ t)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants"... | [] | by simp +contextual [IsStarConvexSet, hs _, ht _] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Integral | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 61
} | {
"line": 345,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nt : Set α\nf : α → E\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nht : μ t ≠ ∞\nh_le : ∀ᵐ (x : α) ∂μ.restrict t, ‖f x‖ ≤ C\nthis : Fact (μ t < ∞)\n⊢ f =ᵐ[μ.res... | [] | exact ae_eq_const_or_norm_integral_lt_of_norm_le_const h_le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Radon | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 39
} | {
"line": 53,
"column": 2
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nf : ι → E\ns : Finset ι\nw : ι → 𝕜\nh_wsum : s.sum w = 0\nh_vsum : ∑ e ∈ s, w e • f e = 0\nnonzero_w_index : ι\nh1 : nonzero_w_index ∈ s... | [
"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nf : ι → E\ns : Finset ι\nw : ι → 𝕜\nh_wsum : s.sum w = 0\nh_vsum : ∑ e ∈ s, w e • f e = 0\nnonzero_w_index : ι\nh1 : nonzero_w_index ∈ s\nh2 : w non... | let J : Finset ι := {i ∈ s | w i < 0} | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 68
} | {
"line": 219,
"column": 2
} | [
{
"pp": "X : Type u_2\ninst✝³ : ConvexSpace ℝ X\ninst✝² : MetricSpace X\ninst✝¹ : IsConvexDist X\nT : Type u_3\ninst✝ : TopologicalSpace T\nf : T → ℝ\nhf : Continuous f\nhf0 : ∀ (t : T), 0 ≤ f t\nhf1 : ∀ (t : T), f t ≤ 1\nx y : T → X\nhx : ContinuousOn x (f ⁻¹' {0}ᶜ)\nhy : ContinuousOn y (f ⁻¹' {1}ᶜ)\nhx' : Bor... | [] | simpa [le_antisymm_iff, hf0, hf1, -not_and, not_and_or] using ht | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 68
} | {
"line": 219,
"column": 2
} | [
{
"pp": "X : Type u_2\ninst✝³ : ConvexSpace ℝ X\ninst✝² : MetricSpace X\ninst✝¹ : IsConvexDist X\nT : Type u_3\ninst✝ : TopologicalSpace T\nf : T → ℝ\nhf : Continuous f\nhf0 : ∀ (t : T), 0 ≤ f t\nhf1 : ∀ (t : T), f t ≤ 1\nx y : T → X\nhx : ContinuousOn x (f ⁻¹' {0}ᶜ)\nhy : ContinuousOn y (f ⁻¹' {1}ᶜ)\nhx' : Bor... | [] | simpa [le_antisymm_iff, hf0, hf1, -not_and, not_and_or] using ht | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 68
} | {
"line": 219,
"column": 2
} | [
{
"pp": "X : Type u_2\ninst✝³ : ConvexSpace ℝ X\ninst✝² : MetricSpace X\ninst✝¹ : IsConvexDist X\nT : Type u_3\ninst✝ : TopologicalSpace T\nf : T → ℝ\nhf : Continuous f\nhf0 : ∀ (t : T), 0 ≤ f t\nhf1 : ∀ (t : T), f t ≤ 1\nx y : T → X\nhx : ContinuousOn x (f ⁻¹' {0}ᶜ)\nhy : ContinuousOn y (f ⁻¹' {1}ᶜ)\nhx' : Bor... | [] | simpa [le_antisymm_iff, hf0, hf1, -not_and, not_and_or] using ht | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Side | {
"line": 433,
"column": 81
} | {
"line": 436,
"column": 22
} | {
"line": 438,
"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\ns : AffineSubspace R P\nx y p₁ : P\nh : p₁ ∈ s\n⊢ s.SSameSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (y -... | [] | by
rw [SSameSide, and_comm, wSameSide_iff_exists_left h, and_assoc, and_congr_right_iff]
intro hx
rw [or_iff_right hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.StoneSeparation | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 28
} | {
"line": 110,
"column": 0
} | [
{
"pp": "𝕜 : 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\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhsC : s ⊆ C\nhmax ... | [] | rwa [← hC.1.convexHull_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.Convex.StrictCombination | {
"line": 59,
"column": 8
} | {
"line": 59,
"column": 47
} | {
"line": 60,
"column": 6
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nι : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : TopologicalSpace V\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\ns : Set V\nw : ι → R\nz : ι → V\nhs : StrictConvex R s\ni : ι\nt : Finset ι\nhi✝ : i ∉ t\nht :\n (∀ i ∈ t, 0 ≤ w... | [] | grw [← hwi, ← sum_nonneg hs₀, add_zero] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.Analysis.Convex.Side | {
"line": 643,
"column": 4
} | {
"line": 643,
"column": 18
} | {
"line": 645,
"column": 0
} | [
{
"pp": "case refine_2\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 p : P\nhp : p ∈ s\nh : s.SOppSide x p\nhw : Wbtw R x p p\n⊢ False",
"ppTe... | [] | exact h.2.2 hp | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Visible | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 75
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case inr.refine_1.refine_3\n𝕜 : Type u_1\nV : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\ns : Set V\nx : V\nι : Type u_4\nt : Finset ι\na : ι → V\nw : ι → 𝕜\nhw₀ : ∀ i ∈ t, 0 ≤ w i\nhw₁ : ∑ i ∈ t, w i = 1\nha : ∀... | [] | exact fun j hj ↦ subset_convexHull _ _ <| ha _ <| erase_subset _ _ hj | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Visible | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 75
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case inr.refine_1.refine_3\n𝕜 : Type u_1\nV : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\ns : Set V\nx : V\nι : Type u_4\nt : Finset ι\na : ι → V\nw : ι → 𝕜\nhw₀ : ∀ i ∈ t, 0 ≤ w i\nhw₁ : ∑ i ∈ t, w i = 1\nha : ∀... | [] | exact fun j hj ↦ subset_convexHull _ _ <| ha _ <| erase_subset _ _ hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Visible | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 75
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case inr.refine_1.refine_3\n𝕜 : Type u_1\nV : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\ns : Set V\nx : V\nι : Type u_4\nt : Finset ι\na : ι → V\nw : ι → 𝕜\nhw₀ : ∀ i ∈ t, 0 ≤ w i\nhw₁ : ∑ i ∈ t, w i = 1\nha : ∀... | [] | exact fun j hj ↦ subset_convexHull _ _ <| ha _ <| erase_subset _ _ hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Side | {
"line": 796,
"column": 2
} | {
"line": 798,
"column": 30
} | {
"line": 799,
"column": 2
} | [
{
"pp": "case inl\nV : Type u_2\nP : Type u_4\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\nh : ↑s = ∅\n⊢ IsPreconnected {y | s.WSameSide x y}",
"ppTerm": "?inl",
"assigned": true,
"usedConstant... | [
"case inr\nV : Type u_2\nP : Type u_4\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\nh : (↑s).Nonempty\n⊢ IsPreconnected {y | s.WSameSide x y}"
] | · rw [coe_eq_bot_iff] at h
simp only [h, not_wSameSide_bot]
exact isPreconnected_empty | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Side | {
"line": 794,
"column": 46
} | {
"line": 799,
"column": 58
} | {
"line": 801,
"column": 0
} | [
{
"pp": "V : Type u_2\nP : Type u_4\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nx : P\n⊢ IsPreconnected {y | s.WSameSide x y}",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | by
rcases Set.eq_empty_or_nonempty (s : Set P) with (h | h)
· rw [coe_eq_bot_iff] at h
simp only [h, not_wSameSide_bot]
exact isPreconnected_empty
· exact (isConnected_setOf_wSameSide x h).isPreconnected | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Side | {
"line": 1077,
"column": 77
} | {
"line": 1078,
"column": 74
} | {
"line": 1080,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw : Fin (n + 1) → R\nhw : ∑ j, w j = 1\ni : Fin (n + 1)\n⊢ (affineSpan ... | [] | by
rw [wOppSide_comm, s.wOppSide_affineSpan_faceOpposite_point_left_iff hw] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 352,
"column": 8
} | {
"line": 352,
"column": 51
} | {
"line": 353,
"column": 8
} | [
{
"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\nH2 : ∀ s > 0, 0 <... | [
"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\nH2 : ∀ s > 0, 0 < ∫ (t : ℝ) i... | apply ae_mono (restrict_mono ?_ le_rfl) hgM | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 517,
"column": 16
} | {
"line": 517,
"column": 39
} | {
"line": 518,
"column": 4
} | [
{
"pp": "case coe.inl\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : MeasurableSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0\nhp : ↑p = 0\n⊢ ∃ k, eLpNorm (fun x ↦ (1 + ‖x‖) ^ (-↑k)) (↑p) μ < ∞",
"ppTerm": "?coe.inl",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero"... | [] | exact ⟨0, by simp [hp]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 517,
"column": 16
} | {
"line": 517,
"column": 39
} | {
"line": 518,
"column": 4
} | [
{
"pp": "case coe.inl\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : MeasurableSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0\nhp : ↑p = 0\n⊢ ∃ k, eLpNorm (fun x ↦ (1 + ‖x‖) ^ (-↑k)) (↑p) μ < ∞",
"ppTerm": "?coe.inl",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero"... | [] | exact ⟨0, by simp [hp]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 517,
"column": 16
} | {
"line": 517,
"column": 39
} | {
"line": 518,
"column": 4
} | [
{
"pp": "case coe.inl\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : MeasurableSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0\nhp : ↑p = 0\n⊢ ∃ k, eLpNorm (fun x ↦ (1 + ‖x‖) ^ (-↑k)) (↑p) μ < ∞",
"ppTerm": "?coe.inl",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero"... | [] | exact ⟨0, by simp [hp]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Matrix.Spectrum | {
"line": 156,
"column": 18
} | {
"line": 156,
"column": 38
} | {
"line": 156,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\n| A.charpoly",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Matrix.smul",
"Real",
"Algebra.to_smulCommClass",
"MonoidHom.ins... | [
"𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\n| (((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (RCLike.ofReal ∘ hA.eigenvalues))).charpoly"
] | hA.spectral_theorem, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Analysis.Matrix.Spectrum | {
"line": 156,
"column": 2
} | {
"line": 157,
"column": 26
} | {
"line": 159,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\n⊢ A.charpoly = ∏ i, (X - C ↑(hA.eigenvalues i))",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Matrix.sm... | [] | conv_lhs => rw [hA.spectral_theorem, conjStarAlgAut_apply, charpoly_mul_comm, ← mul_assoc]
simp [charpoly_diagonal] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Matrix.Spectrum | {
"line": 156,
"column": 2
} | {
"line": 157,
"column": 26
} | {
"line": 159,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\n⊢ A.charpoly = ∏ i, (X - C ↑(hA.eigenvalues i))",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Matrix.sm... | [] | conv_lhs => rw [hA.spectral_theorem, conjStarAlgAut_apply, charpoly_mul_comm, ← mul_assoc]
simp [charpoly_diagonal] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Matrix.Spectrum | {
"line": 167,
"column": 2
} | {
"line": 169,
"column": 6
} | {
"line": 171,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\n⊢ A.charpoly.roots = Multiset.map (RCLike.ofReal ∘ hA.eigenvalues₀) Finset.univ.val",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [hA.roots_charpoly_eq_eigenvalues]
simp only [← Multiset.map_map, eigenvalues, ← Function.comp_apply (f := hA.eigenvalues₀)]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Matrix.Spectrum | {
"line": 167,
"column": 2
} | {
"line": 169,
"column": 6
} | {
"line": 171,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\n⊢ A.charpoly.roots = Multiset.map (RCLike.ofReal ∘ hA.eigenvalues₀) Finset.univ.val",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [hA.roots_charpoly_eq_eigenvalues]
simp only [← Multiset.map_map, eigenvalues, ← Function.comp_apply (f := hA.eigenvalues₀)]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Matrix.Spectrum | {
"line": 195,
"column": 18
} | {
"line": 195,
"column": 38
} | {
"line": 195,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\n| A.rank",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Matrix.smul",
"Real",
"Algebra.to_smulCommClass",
"MonoidHom.instFun... | [
"𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\n| (((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (RCLike.ofReal ∘ hA.eigenvalues))).rank"
] | hA.spectral_theorem, | Lean.Elab.Tactic.Conv.evalRewrite | null |
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