module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope | {
"line": 66,
"column": 23
} | {
"line": 66,
"column": 38
} | {
"line": 66,
"column": 38
} | [
{
"pp": "f : ℝ → ℝ\na b c : ℝ\nhf : MonotoneOn f (uIcc a (b + c))\nhab : a ≤ b\nhc✝ : 0 ≤ c\nhc : 0 < c\nhf' : IntervalIntegrable f volume a (b + c)\n⊢ uIcc (a + c) a ⊆ uIcc a (b + c)",
"ppTerm": "?m.199",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.Integral.IntervalIn... | [] | by grind [uIcc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope | {
"line": 70,
"column": 25
} | {
"line": 70,
"column": 40
} | {
"line": 70,
"column": 40
} | [
{
"pp": "f : ℝ → ℝ\na b c : ℝ\nhf : MonotoneOn f (uIcc a (b + c))\nhab : a ≤ b\nhc✝ : 0 ≤ c\nhc : 0 < c\nhf' : IntervalIntegrable f volume a (b + c)\n⊢ uIcc b (b + c) ⊆ uIcc a (b + c)",
"ppTerm": "?m.246",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.Integral.IntervalIn... | [] | by grind [uIcc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope | {
"line": 78,
"column": 27
} | {
"line": 78,
"column": 42
} | {
"line": 78,
"column": 42
} | [
{
"pp": "f : ℝ → ℝ\na b c : ℝ\nhf : MonotoneOn f (uIcc a (b + c))\nhab : a ≤ b\nhc✝ : 0 ≤ c\nhc : 0 < c\nhf' : IntervalIntegrable f volume a (b + c)\nfU : ∫ (x : ℝ) in b..b + c, f x ≤ c * f (b + c)\n⊢ uIcc a (a + c) ⊆ uIcc a (b + c)",
"ppTerm": "?m.321",
"assigned": true,
"usedConstants": [
"_... | [] | by grind [uIcc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 31
} | {
"line": 122,
"column": 2
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nhf : MonotoneOn f (Icc a b)\nhab : a ≤ b\nG : ℕ → ℝ → ℝ\nhGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))\nhG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))\nhG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTop ≤ ENNReal.... | [] | exact hf.derivWithin_nonneg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.Taylor | {
"line": 434,
"column": 4
} | {
"line": 434,
"column": 30
} | {
"line": 435,
"column": 4
} | [
{
"pp": "case inl\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na : ℝ\nn : ℕ\nhab : a ≤ a\nhf : ContDiffOn ℝ (↑n + 1) f (Icc a a)\n⊢ ∃ C, ∀ x ∈ Icc a a, ‖f x - taylorWithinEval f n (Icc a a) a x‖ ≤ C * (x - a) ^ (n + 1)",
"ppTerm": "?inl",
"assigned": true,
"usedC... | [
"case inl\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na : ℝ\nn : ℕ\nhab : a ≤ a\nhf : ContDiffOn ℝ (↑n + 1) f (Icc a a)\nx : ℝ\nhx : x ∈ Icc a a\n⊢ ‖f x - taylorWithinEval f n (Icc a a) a x‖ ≤ 0 * (x - a) ^ (n + 1)"
] | refine ⟨0, fun x hx => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.Taylor | {
"line": 476,
"column": 4
} | {
"line": 476,
"column": 56
} | {
"line": 477,
"column": 4
} | [
{
"pp": "case inr.zero\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nx x₀ : ℝ\nthis : x₀ ≠ x\nhf : ∫ (t : ℝ) in x₀..x, deriv (fun t ↦ f t) t = f x - f x₀\nx✝¹ : ℝ\nx✝ : x✝¹ ∈ uIoo x₀ x\nh1 : min x₀ x < x✝¹\nh2 : x✝¹ < max x₀ x\n⊢ deriv (fun t ↦ f t) x✝¹ = derivWithin f [[x₀, ... | [
"case inr.zero\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nx x₀ : ℝ\nthis : x₀ ≠ x\nhf : ∫ (t : ℝ) in x₀..x, deriv (fun t ↦ f t) t = f x - f x₀\nx✝¹ : ℝ\nx✝ : x✝¹ ∈ uIoo x₀ x\nh1 : min x₀ x < x✝¹\nh2 : x✝¹ < max x₀ x\n⊢ derivWithin (fun t ↦ f t) (Icc (min x₀ x) (max x₀ x)) x✝¹ ... | rw [← derivWithin_of_mem_nhds <| Icc_mem_nhds h1 h2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.TaylorIntegral | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 60
} | {
"line": 106,
"column": 4
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\nf : E → F\nx y : E\nn✝ : ℕ\ninst✝ : CompleteSpace F\nn : ℕ\nih :\n f (x + y) =\n (∑ k ∈ Finset.range (n + 1), (↑k !)⁻¹ • (iteratedFDeriv ℝ k f x) fun x ↦ y) ... | [
"E : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\nf : E → F\nx y : E\nn✝ : ℕ\ninst✝ : CompleteSpace F\nn : ℕ\nih :\n f (x + y) =\n (∑ k ∈ Finset.range (n + 1), (↑k !)⁻¹ • (iteratedFDeriv ℝ k f x) fun x ↦ y) +\n (↑n... | set u := fun (k : ℕ) (t : ℝ) ↦ (k ! : ℝ)⁻¹ * (1 - t) ^ k | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Analysis.Calculus.TaylorIntegral | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 14
} | {
"line": 109,
"column": 6
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\nf : E → F\nx y : E\nn✝ : ℕ\ninst✝ : CompleteSpace F\nn : ℕ\nih :\n f (x + y) =\n (∑ k ∈ Finset.range (n + 1), (↑k !)⁻¹ • (iteratedFDeriv ℝ k f x) fun x ↦ y) ... | [
"E : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\nf : E → F\nx y : E\nn✝ : ℕ\ninst✝ : CompleteSpace F\nn : ℕ\nih :\n f (x + y) =\n (∑ k ∈ Finset.range (n + 1), (↑k !)⁻¹ • (iteratedFDeriv ℝ k f x) fun x ↦ y) +\n (↑n... | unfold u | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.Analysis.Calculus.Taylor | {
"line": 500,
"column": 8
} | {
"line": 500,
"column": 60
} | {
"line": 501,
"column": 8
} | [
{
"pp": "case e'_3.e_a\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nx x₀ : ℝ\nthis✝ : x₀ ≠ x\nn : ℕ\nih :\n f x - taylorWithinEval f n [[x₀, x]] x₀ x =\n ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t\nhf :\n ∀ k ≤ n + 1,\n let u... | [
"case e'_3.e_a\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nx x₀ : ℝ\nthis✝ : x₀ ≠ x\nn : ℕ\nih :\n f x - taylorWithinEval f n [[x₀, x]] x₀ x =\n ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t\nhf :\n ∀ k ≤ n + 1,\n let u := fun t ↦ ... | rw [← derivWithin_of_mem_nhds <| Icc_mem_nhds h1 h2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Uniform | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 19
} | {
"line": 78,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : UniformConvexSpace E\nε : ℝ\ninst✝ : NormedSpace ℝ E\nhε : 0 < ε\nhε' : 0 < ε / 3\nδ : ℝ\nhδ : 0 < δ\nh : ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y : E⦄, ‖y‖ = 1 → ε / 3 ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ\nδ' : ℝ := min (1 / 2) (min (ε / 3) (δ / 3))\nx : E\nhx : ‖x‖ ≤... | [
"E : Type u_1\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : UniformConvexSpace E\nε : ℝ\ninst✝ : NormedSpace ℝ E\nhε : 0 < ε\nhε' : 0 < ε / 3\nδ : ℝ\nhδ : 0 < δ\nh : ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y : E⦄, ‖y‖ = 1 → ε / 3 ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ\nδ' : ℝ := min (1 / 2) (min (ε / 3) (δ / 3))\nx : E\nhx : ‖x‖ ≤ 1\ny : E\nh... | rintro z hz hδz | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 36
} | {
"line": 180,
"column": 2
} | [
{
"pp": "case inr\nf : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nM : ℝ\nhM : 1 < M\ns : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i\ng : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n\nhm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε\nε : ℝ\nεpos : ε > 0\nB₁ : ℕ\nhB₁ : ∀ n ≥ B₁, ‖∑ i ∈ rang... | [
"case right\nf : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nM : ℝ\nhM : 1 < M\ns : ℕ → ℂ := ⋯\ng : ℂ → ℂ := ⋯\nhm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε\nε : ℝ\nεpos : ε > 0\nB₁ : ℕ\nhB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M\nF : ℝ := ⋯\n⊢ ∀ ⦃x : ℂ⦄, x ∈ stolzSet ... | use ε / 4 / (F + 1), by positivity | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.Order.ExtrClosure | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 41
} | {
"line": 45,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : Preorder Y\ninst✝ : OrderClosedTopology Y\nf : X → Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure[inst✝³] s)\nU : Set X\nUo : IsOpen[inst✝³] U\naU : a ∈ U\nhU : U ∩ s ⊆ {x | (fun x ... | [
"X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : Preorder Y\ninst✝ : OrderClosedTopology Y\nf : X → Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure[inst✝³] s)\nU : Set X\nUo : IsOpen[inst✝³] U\naU : a ∈ U\nhU : U ∩ s ⊆ {x | (fun x ↦ f x ≤ f a)... | refine mem_nhdsWithin.2 ⟨U, Uo, aU, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 221,
"column": 8
} | {
"line": 221,
"column": 66
} | {
"line": 222,
"column": 8
} | [
{
"pp": "case hbc\nf : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nM : ℝ\nhM : 1 < M\ns : ℕ → ℂ := ⋯\ng : ℂ → ℂ := ⋯\nε : ℝ\nεpos : ε > 0\nB₁ : ℕ\nhB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M\nF : ℝ := ⋯\nz : ℂ\nzn : ‖z‖ < 1\nzm : ‖1 - z‖ < M * (1 - ‖z‖)\nzd : ‖z - 1‖ < ε / 4 / ... | [
"case hbc\nf : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nM : ℝ\nhM : 1 < M\ns : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i\ng : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n\nε : ℝ\nεpos : ε > 0\nB₁ : ℕ\nhB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M\nF : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖\nz... | have := hB₁ (i + 1) (by linarith only [(mem_Ico.mp hi).1]) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.AbsMax | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 12
} | {
"line": 188,
"column": 2
} | [
{
"pp": "E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nz : E\nr : ℝ\nhd : DiffContOnCl ℂ f (ball z r)\nhz : IsMaxOn (norm ∘ f) (ball z r) z\n⊢ EqOn (norm ∘ f) (Function.const E ‖f z‖) (closedBall z r)",
"pp... | [
"E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nz : E\nr : ℝ\nhd : DiffContOnCl ℂ f (ball z r)\nhz : IsMaxOn (norm ∘ f) (ball z r) z\nw : E\nhw : w ∈ closedBall z r\n⊢ (norm ∘ f) w = Function.const E ‖f z‖ w"
] | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Complex.AbsMax | {
"line": 345,
"column": 9
} | {
"line": 345,
"column": 52
} | {
"line": 345,
"column": 52
} | [
{
"pp": "E : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : StrictConvexSpace ℝ F\ninst✝ : ProperSpace E\nf : E → F\nr b : ℝ\nh_an : DifferentiableOn ℂ f (ball 0 b)\nhr_nn : 0 ≤ r\nhr_lt : r < b\nhr : ∀ z ∈ ball 0 b,... | [
"E : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : StrictConvexSpace ℝ F\ninst✝ : ProperSpace E\nf : E → F\nr b : ℝ\nh_an : DifferentiableOn ℂ f (ball 0 b)\nhr_nn : 0 ≤ r\nhr_lt : r < b\nhr : ∀ z ∈ ball 0 b, ∃ w ∈ close... | this (mem_ball_self (hr_nn.trans_lt hr_lt)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex | {
"line": 204,
"column": 51
} | {
"line": 204,
"column": 62
} | {
"line": 204,
"column": 62
} | [
{
"pp": "x : ℂ\n| sin (2 * (x / 2))",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
"GroupWithZero.toDivInvMonoid",
"Complex.cos",
"congrArg",
"Complex.sin_two_mul",
"Complex.sin",
"Nat.instAtLeastTwoHAddOfNat",
... | [
"x : ℂ\n| 2 * sin (x / 2) * cos (x / 2)"
] | sin_two_mul | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex | {
"line": 273,
"column": 6
} | {
"line": 273,
"column": 23
} | {
"line": 273,
"column": 23
} | [
{
"pp": "n : ℕ\na : ℂ\nh : ∀ (k : ℤ), a ≠ ↑k * (2 * ↑π)\nb : ℂ\n⊢ ∑ i ∈ Finset.range n, sin (a * ↑i + b) = sin (↑n * a / 2) * sin ((↑n - 1) * a / 2 + b) / sin (a / 2)",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"congrArg",
... | [
"n : ℕ\na : ℂ\nh : ∀ (k : ℤ), a ≠ ↑k * (2 * ↑π)\nb : ℂ\n⊢ ∑ i ∈ Finset.range n, sin (a * ↑i + b) = (sin (a / 2) * ∑ i ∈ Finset.range n, sin (a * ↑i + b)) / sin (a / 2)"
] | ← sin_mul_sum_sin | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 19
} | {
"line": 87,
"column": 4
} | [
{
"pp": "case inl\nx : ℝ\nhx_gt : -(π / 2) < x\nhx_lt : ¬π / 2 ≤ x\nr : ℤ\nhxr_eq : x = (2 * ↑r + 1) * π / 2\nh : 0 ≤ r\n⊢ False",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hx_lt"
],
"usedGoals": [
{
"new": true,
"index": 0,
... | [
"case inl\nx : ℝ\nhx_gt : -(π / 2) < x\nhx_lt : ¬π / 2 ≤ x\nr : ℤ\nhxr_eq : x = (2 * ↑r + 1) * π / 2\nh : 0 ≤ r\n⊢ π / 2 ≤ x"
] | refine hx_lt ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 144,
"column": 66
} | {
"line": 145,
"column": 67
} | {
"line": 147,
"column": 0
} | [
{
"pp": "x : ℝ\n⊢ sin (arctan x) = x / √(1 + x ^ 2)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Real.arctan",
"congrArg",
"Real.instDivInvMonoid",
"Real.tan_div_sqrt_one_add_tan_sq",
"id",
"HDiv.hDiv",... | [] | by
rw [← tan_div_sqrt_one_add_tan_sq (cos_arctan_pos x), tan_arctan] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Deriv | {
"line": 401,
"column": 20
} | {
"line": 401,
"column": 25
} | {
"line": 401,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nx : 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nhfc : ConvexOn 𝕜 s f\nhxs : x ∈ interior s\ny : 𝕜\nhyx : y < x\nhys : y ∈ s\ny' : 𝕜\nx✝ : y' ∈ slope f x '' {y | y ∈ s ... | [
"𝕜 : Type u_1\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nx : 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nhfc : ConvexOn 𝕜 s f\nhxs : x ∈ interior s\ny : 𝕜\nhyx : y < x\nhys : y ∈ s\ny' : 𝕜\nx✝ : y' ∈ slope f x '' {y | y ∈ s ∧ x < y}\nz ... | ← hz' | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convex.Deriv | {
"line": 411,
"column": 20
} | {
"line": 411,
"column": 25
} | {
"line": 411,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nx : 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nhfc : ConvexOn 𝕜 s f\nhxs : x ∈ interior s\ny : 𝕜\nhyx : x < y\nhys : y ∈ s\ny' : 𝕜\nx✝ : y' ∈ slope f x '' {y | y ∈ s ... | [
"𝕜 : Type u_1\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nx : 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nhfc : ConvexOn 𝕜 s f\nhxs : x ∈ interior s\ny : 𝕜\nhyx : x < y\nhys : y ∈ s\ny' : 𝕜\nx✝ : y' ∈ slope f x '' {y | y ∈ s ∧ y < x}\nz ... | ← hz' | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 328,
"column": 87
} | {
"line": 329,
"column": 79
} | {
"line": 331,
"column": 0
} | [
{
"pp": "⊢ 2 * arctan 2⁻¹ - arctan 7⁻¹ = π / 4",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"instHDiv",
"Mathlib.Meta.NormNum.IsNat.to_isNNRa... | [] | by
rw [two_mul_arctan, ← arctan_one, sub_eq_iff_eq_add, arctan_add] <;> norm_num | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Angle | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 51
} | {
"line": 62,
"column": 0
} | [
{
"pp": "case inr\na x y : ℂ\nha : a ≠ 0\n⊢ angle (x / a) y = angle x (y * a)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"div_mul_cancel₀",
"Real",
"instHDiv",
"HMu... | [] | rw [← angle_mul_right ha, div_mul_cancel₀ _ ha] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Angle | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 51
} | {
"line": 62,
"column": 0
} | [
{
"pp": "case inr\na x y : ℂ\nha : a ≠ 0\n⊢ angle (x / a) y = angle x (y * a)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"div_mul_cancel₀",
"Real",
"instHDiv",
"HMu... | [] | rw [← angle_mul_right ha, div_mul_cancel₀ _ ha] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Angle | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 51
} | {
"line": 62,
"column": 0
} | [
{
"pp": "case inr\na x y : ℂ\nha : a ≠ 0\n⊢ angle (x / a) y = angle x (y * a)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"div_mul_cancel₀",
"Real",
"instHDiv",
"HMu... | [] | rw [← angle_mul_right ha, div_mul_cancel₀ _ ha] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Schwarz | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 14
} | {
"line": 121,
"column": 4
} | [
{
"pp": "case inr.refine_1\nf : ℂ → ℂ\nc z : ℂ\nR₂ : ℝ\nn : ℕ\nhn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n\nR₁ : ℝ\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\nhd : DifferentiableOn ℂ f (closedBall c R₁)\nh_maps : MapsTo f (closedBall c R₁) (closedBall (f c) R₂)\nhne : z ≠ c\ng : ℂ → ℂ := fun w ↦ ((w - c) ^ (n ... | [
"case inr.refine_1\nf : ℂ → ℂ\nc z : ℂ\nR₂ : ℝ\nn : ℕ\nhn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n\nR₁ : ℝ\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\nhd : DifferentiableOn ℂ f (closedBall c R₁)\nh_maps : MapsTo f (closedBall c R₁) (closedBall (f c) R₂)\nhne : z ≠ c\ng : ℂ → ℂ := fun w ↦ ((w - c) ^ (n + 1))⁻¹ * (f... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Complex.Schwarz | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 14
} | {
"line": 165,
"column": 4
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nc z : E\nR₁ R₂ : ℝ\nn : ℕ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)\nhn : (fun x ↦ f x - f c) =o[𝓝 c... | [
"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nc z : E\nR₁ R₂ : ℝ\nn : ℕ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)\nhn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ ‖w... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.Homotopy.Lifting | {
"line": 83,
"column": 26
} | {
"line": 83,
"column": 28
} | {
"line": 83,
"column": 28
} | [
{
"pp": "E : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace A\np : E → X\nf : C(↑I × A, X)\ng : ↑I × A → E\ng_lifts : p ∘ g = ⇑f\ncont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)\na : A\ncont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)\... | [
"E : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace A\np : E → X\nf : C(↑I × A, X)\ng : ↑I × A → E\ng_lifts : p ∘ g = ⇑f\ncont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)\na : A\ncont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)\nq : E → Ope... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Meromorphic.Divisor | {
"line": 132,
"column": 36
} | {
"line": 132,
"column": 38
} | {
"line": 133,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nU : Set 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf₁ f₂ : 𝕜 → E\nhf₁ : MeromorphicOn f₁ U\nh₂ : Set.EqOn f₁ f₂ Uᶜ\nx : 𝕜\nhx : x ∈ U\nh₁ : ∀ x ∈ U, (U \\ {x | f₁ x = f₂ x})ᶜ ∈ 𝓝[≠] x\na : 𝕜\n⊢ a ∈ (U \\ {x | f₁ x =... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nU : Set 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf₁ f₂ : 𝕜 → E\nhf₁ : MeromorphicOn f₁ U\nh₂ : Set.EqOn f₁ f₂ Uᶜ\nx : 𝕜\nhx : x ∈ U\nh₁ : ∀ x ∈ U, (U \\ {x | f₁ x = f₂ x})ᶜ ∈ 𝓝[≠] x\na : 𝕜\nha : a ∈ (U \\ {x | f₁ x = f₂ x})ᶜ\... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 111,
"column": 2
} | {
"line": 112,
"column": 12
} | {
"line": 114,
"column": 0
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nh : MeromorphicAt f x ∧ meromorphicOrderAt f x < 0 ∧ f x = 0\n⊢ MeromorphicAt f x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [... | [] | · obtain ⟨hf, _⟩ := h
exact hf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 137,
"column": 61
} | {
"line": 137,
"column": 63
} | {
"line": 138,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nhf : MeromorphicOn f U\na✝ : 𝕜\n⊢ AnalyticAt 𝕜 f a✝ → MeromorphicNFAt f a✝",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"No... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nhf : MeromorphicOn f U\na✝ : 𝕜\nha : AnalyticAt 𝕜 f a✝\n⊢ MeromorphicNFAt f a✝"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 205,
"column": 6
} | {
"line": 205,
"column": 35
} | {
"line": 206,
"column": 4
} | [
{
"pp": "case mp.inl\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\ng : 𝕜 → E\nhfg : f =ᶠ[𝓝 x] g\nh : f =ᶠ[𝓝 x] 0\n⊢ MeromorphicNFAt g x",
"ppTerm": "?mp.inl",
"assigned": true,
"usedConstants": [
... | [] | exact .inl (hfg.symm.trans h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 205,
"column": 6
} | {
"line": 205,
"column": 35
} | {
"line": 206,
"column": 4
} | [
{
"pp": "case mp.inl\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\ng : 𝕜 → E\nhfg : f =ᶠ[𝓝 x] g\nh : f =ᶠ[𝓝 x] 0\n⊢ MeromorphicNFAt g x",
"ppTerm": "?mp.inl",
"assigned": true,
"usedConstants": [
... | [] | exact .inl (hfg.symm.trans h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 205,
"column": 6
} | {
"line": 205,
"column": 35
} | {
"line": 206,
"column": 4
} | [
{
"pp": "case mp.inl\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\ng : 𝕜 → E\nhfg : f =ᶠ[𝓝 x] g\nh : f =ᶠ[𝓝 x] 0\n⊢ MeromorphicNFAt g x",
"ppTerm": "?mp.inl",
"assigned": true,
"usedConstants": [
... | [] | exact .inl (hfg.symm.trans h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 322,
"column": 33
} | {
"line": 338,
"column": 55
} | {
"line": 340,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nf : 𝕜 → 𝕜\nn : ℤ\nx : 𝕜\nhf : MeromorphicNFAt f x\n⊢ MeromorphicNFAt (f ^ n) x",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"zero_zpow",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NormedCommRing.toS... | [] | by
by_cases hn : n = 0
· simp_all only [zpow_zero]
apply AnalyticAt.meromorphicNFAt
apply analyticAt_const
rcases hf with hf | hf
· left
filter_upwards [hf] with z hz
simp_all only [Pi.zero_apply, Pi.pow_apply, zero_zpow n hn]
· obtain ⟨m, g, h₁g, h₂g, h₃g⟩ := hf
right
use n * m, g ^ n... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 214,
"column": 32
} | {
"line": 214,
"column": 34
} | {
"line": 215,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\nh₁ : MeromorphicAt f x\nh₂ : ¬meromorphicOrderAt f x = ⊤\ng : 𝕜 → E\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphic... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\nh₁ : MeromorphicAt f x\nh₂ : ¬meromorphicOrderAt f x = ⊤\ng : 𝕜 → E\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x)... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 832,
"column": 67
} | {
"line": 832,
"column": 69
} | {
"line": 833,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nU : Set 𝕜\nhf : MeromorphicOn f U\nhx : x ∈ U\na : 𝕜\n⊢ AnalyticAt 𝕜 f a ∨ a ∈ Uᶜ → toMeromorphicNFOn f U a = f a",
"ppTerm": "?m.46",
"assigned": tr... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nU : Set 𝕜\nhf : MeromorphicOn f U\nhx : x ∈ U\na : 𝕜\nha : AnalyticAt 𝕜 f a ∨ a ∈ Uᶜ\n⊢ toMeromorphicNFOn f U a = f a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 890,
"column": 2
} | {
"line": 891,
"column": 49
} | {
"line": 893,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nU : Set 𝕜\nhf : MeromorphicOn f U\nhx : x ∈ U\n⊢ meromorphicOrderAt (toMeromorphicNFOn f U) x = meromorphicOrderAt f x",
"ppTerm": "?m.27",
"assigned":... | [] | apply meromorphicOrderAt_congr
exact hf.toMeromorphicNFOn_eq_self_on_nhdsNE hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 890,
"column": 2
} | {
"line": 891,
"column": 49
} | {
"line": 893,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nU : Set 𝕜\nhf : MeromorphicOn f U\nhx : x ∈ U\n⊢ meromorphicOrderAt (toMeromorphicNFOn f U) x = meromorphicOrderAt f x",
"ppTerm": "?m.27",
"assigned":... | [] | apply meromorphicOrderAt_congr
exact hf.toMeromorphicNFOn_eq_self_on_nhdsNE hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.FactorizedRational | {
"line": 113,
"column": 4
} | {
"line": 114,
"column": 23
} | {
"line": 115,
"column": 4
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nd : 𝕜 → ℤ\nu₀ : 𝕜\nhd : HasFiniteSupport d\nh₁d : ¬d u₀ = 0\n⊢ ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u = (fun x ↦ x - u₀) ^ d u₀ * ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ update d u₀ 0 u",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants":... | [
"case neg\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nd : 𝕜 → ℤ\nu₀ : 𝕜\nhd : HasFiniteSupport d\nh₁d : ¬d u₀ = 0\nthis : (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) ⊆ ↑(Finite.toFinset hd)\n⊢ ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u = (fun x ↦ x - u₀) ^ d u₀ * ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ update d u₀... | have : (fun u ↦ (fun x ↦ x - u) ^ d u).mulSupport ⊆ hd.toFinset := by
simp [mulSupport] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 481,
"column": 34
} | {
"line": 481,
"column": 36
} | {
"line": 482,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\nn : ℤ\nf : 𝕜 → 𝕜\nh₁ : MeromorphicAt f x\nh₂ : ¬meromorphicOrderAt f x = ⊤\ng : 𝕜 → 𝕜\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x).untop₀ • g z\na : 𝕜\n⊢ f a = (a - x) ^ (meromorph... | [
"𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\nn : ℤ\nf : 𝕜 → 𝕜\nh₁ : MeromorphicAt f x\nh₂ : ¬meromorphicOrderAt f x = ⊤\ng : 𝕜 → 𝕜\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x).untop₀ • g z\na : 𝕜\nha : f a = (a - x) ^ (meromorphicOrderAt... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Meromorphic.Order | {
"line": 320,
"column": 6
} | {
"line": 320,
"column": 72
} | {
"line": 321,
"column": 6
} | [
{
"pp": "case refine_1.coe\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nhf : MeromorphicAt f x\nnneg : 0 ≤ meromorphicOrderAt f x\nn : ℤ\nh₀ : meromorphicOrderAt f x = ↑n\n⊢ ∃ g, AnalyticAt 𝕜 g x ∧ f =ᶠ[𝓝[≠] x] ... | [
"case refine_1.coe\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nhf : MeromorphicAt f x\nnneg : 0 ≤ meromorphicOrderAt f x\nn : ℤ\nh₀ : meromorphicOrderAt f x = ↑n\ng : 𝕜 → E\nhg : AnalyticAt 𝕜 g x\nhfg : ∀ᶠ (z : 𝕜)... | obtain ⟨g, hg, -, hfg⟩ := (meromorphicOrderAt_eq_int_iff hf).mp h₀ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Meromorphic.Order | {
"line": 455,
"column": 60
} | {
"line": 455,
"column": 62
} | {
"line": 455,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nR : Type u_3\ninst✝⁶ : NormedRing R\ninst✝⁵ : NoZeroDivisors R\ninst✝⁴ : Module R E\ninst✝³ : IsBoundedSMul R E\ninst✝² : Module.IsTorsionFree R E\nx : 𝕜\ninst✝¹ : NormedAlgebra ... | [
"𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nR : Type u_3\ninst✝⁶ : NormedRing R\ninst✝⁵ : NoZeroDivisors R\ninst✝⁴ : Module R E\ninst✝³ : IsBoundedSMul R E\ninst✝² : Module.IsTorsionFree R E\nx : 𝕜\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Meromorphic.FactorizedRational | {
"line": 358,
"column": 6
} | {
"line": 358,
"column": 28
} | {
"line": 360,
"column": 2
} | [
{
"pp": "case mpr\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf g : 𝕜 → E\nD : locallyFinsuppWithin U ℤ\nhg : ∀ (u : ↑U), g ↑u ≠ 0\nh : f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) • g\nu : 𝕜\nhx : D u... | [] | simp_all [Pi.zero_def] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Meromorphic.FactorizedRational | {
"line": 358,
"column": 6
} | {
"line": 358,
"column": 28
} | {
"line": 360,
"column": 2
} | [
{
"pp": "case mpr\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf g : 𝕜 → E\nD : locallyFinsuppWithin U ℤ\nhg : ∀ (u : ↑U), g ↑u ≠ 0\nh : f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) • g\nu : 𝕜\nhx : D u... | [] | simp_all [Pi.zero_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.FactorizedRational | {
"line": 358,
"column": 6
} | {
"line": 358,
"column": 28
} | {
"line": 360,
"column": 2
} | [
{
"pp": "case mpr\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf g : 𝕜 → E\nD : locallyFinsuppWithin U ℤ\nhg : ∀ (u : ↑U), g ↑u ≠ 0\nh : f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) • g\nu : 𝕜\nhx : D u... | [] | simp_all [Pi.zero_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.CanonicalDecomposition | {
"line": 324,
"column": 75
} | {
"line": 324,
"column": 77
} | {
"line": 325,
"column": 8
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nf : ℂ → E\nh₁f : MeromorphicOn f (closedBall 0 R)\nh₂f : ∀ (u : ↑(closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤\nhR : R ≤ 0\na : ℂ\n⊢ a ∈ closedBall 0 R → f a = ((∏ᶠ (u : ℂ), canonicalFactor R u ^ (-(divisor f (ball 0 R)) u)) ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nf : ℂ → E\nh₁f : MeromorphicOn f (closedBall 0 R)\nh₂f : ∀ (u : ↑(closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤\nhR : R ≤ 0\na : ℂ\nha : a ∈ closedBall 0 R\n⊢ f a = ((∏ᶠ (u : ℂ), canonicalFactor R u ^ (-(divisor f (ball 0 R)) u)) • fun x ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Meromorphic.Order | {
"line": 662,
"column": 4
} | {
"line": 663,
"column": 63
} | {
"line": 665,
"column": 0
} | [
{
"pp": "case neg.refine_3\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf₁ f₂ : 𝕜 → E\nx : 𝕜\nhf₂ : MeromorphicAt f₂ x\nhf₁ : MeromorphicAt f₁ x\nn₂ : ℤ\nhn₂ : ↑n₂ = meromorphicOrderAt f₂ x\nn₁ : ℤ\nhn₁ : ↑n₁ = meromorphicOrderAt f... | [] | filter_upwards [h₃g₁, h₃g₂, self_mem_nhdsWithin]
simp_all [smul_add, ← smul_assoc, ← zpow_add', sub_ne_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.Order | {
"line": 662,
"column": 4
} | {
"line": 663,
"column": 63
} | {
"line": 665,
"column": 0
} | [
{
"pp": "case neg.refine_3\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf₁ f₂ : 𝕜 → E\nx : 𝕜\nhf₂ : MeromorphicAt f₂ x\nhf₁ : MeromorphicAt f₁ x\nn₂ : ℤ\nhn₂ : ↑n₂ = meromorphicOrderAt f₂ x\nn₁ : ℤ\nhn₁ : ↑n₁ = meromorphicOrderAt f... | [] | filter_upwards [h₃g₁, h₃g₂, self_mem_nhdsWithin]
simp_all [smul_add, ← smul_assoc, ← zpow_add', sub_ne_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.CanonicalDecomposition | {
"line": 369,
"column": 63
} | {
"line": 369,
"column": 65
} | {
"line": 369,
"column": 66
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nf : ℂ → E\nh₁f : MeromorphicOn f (closedBall 0 R)\nh₂f : ∀ (u : ↑(closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤\nhR : 0 < R\nφ : ℂ → E := (∏ᶠ (c : ℂ), canonicalFactor R c ^ (divisor f (ball 0 R)) c) • f\nhφ : MeromorphicOn φ (... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nf : ℂ → E\nh₁f : MeromorphicOn f (closedBall 0 R)\nh₂f : ∀ (u : ↑(closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤\nhR : 0 < R\nφ : ℂ → E := (∏ᶠ (c : ℂ), canonicalFactor R c ^ (divisor f (ball 0 R)) c) • f\nhφ : MeromorphicOn φ (closedBall 0... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Meromorphic.Order | {
"line": 709,
"column": 8
} | {
"line": 709,
"column": 18
} | {
"line": 710,
"column": 8
} | [
{
"pp": "case h.left\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nhf : MeromorphicOn f U\nz : ↑U\nhz : z ∈ {u | meromorphicOrderAt f ↑u = ⊤}ᶜ\nh : ∀ᶠ (z : 𝕜) in 𝓝[≠] ↑z, f z ≠ 0\nt' : Set 𝕜\nh₁t' : ∀ y ∈ t'... | [
"case h.left\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nhf : MeromorphicOn f U\nz : ↑U\nhz : z ∈ {u | meromorphicOrderAt f ↑u = ⊤}ᶜ\nh : ∀ᶠ (z : 𝕜) in 𝓝[≠] ↑z, f z ≠ 0\nt' : Set 𝕜\nh₁t' : ∀ y ∈ t', y ∈ {↑z}ᶜ ... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Complex.Conformal | {
"line": 130,
"column": 10
} | {
"line": 130,
"column": 16
} | {
"line": 130,
"column": 16
} | [
{
"pp": "case mpr.inr\ng : ℂ →L[ℝ] ℂ\nh₂ : g ≠ 0\nmap : ℂ →L[ℂ] ℂ\nhmap : restrictScalars ℝ map = g ∘SL ↑conjCLE\nminor₁ : g = restrictScalars ℝ map ∘SL ↑conjCLE\n⊢ IsConformalMap g",
"ppTerm": "?mpr.inr",
"assigned": true,
"usedConstants": [
"ContinuousLinearMap.comp",
"Eq.mpr",
"... | [
"case mpr.inr\ng : ℂ →L[ℝ] ℂ\nmap : ℂ →L[ℂ] ℂ\nh₂ : restrictScalars ℝ map ∘SL ↑conjCLE ≠ 0\nhmap : restrictScalars ℝ map = g ∘SL ↑conjCLE\nminor₁ : g = restrictScalars ℝ map ∘SL ↑conjCLE\n⊢ IsConformalMap (restrictScalars ℝ map ∘SL ↑conjCLE)"
] | minor₁ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Conformal | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 54
} | {
"line": 163,
"column": 2
} | [
{
"pp": "case h₁\nf : ℂ → ℂ\nz : ℂ\nh : fderiv ℝ f z ≠ 0\nh_diff : DifferentiableAt ℝ f z\n⊢ (∃ map, restrictScalars ℝ map = fderiv ℝ f z) ↔ DifferentiableAt ℂ f z",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
... | [
"case h₂\nf : ℂ → ℂ\nz : ℂ\nh : fderiv ℝ f z ≠ 0\nh_diff : DifferentiableAt ℝ f z\n⊢ (∃ map, restrictScalars ℝ map = fderiv ℝ f z ∘SL ↑conjCLE) ↔\n DifferentiableAt ℂ (f ∘ ⇑(starRingEnd ℂ)) ((starRingEnd ℂ) z)"
] | · rw [differentiableAt_iff_restrictScalars ℝ h_diff] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.CanonicalDecomposition | {
"line": 480,
"column": 67
} | {
"line": 480,
"column": 69
} | {
"line": 481,
"column": 8
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nf : ℂ → E\nh₁f : MeromorphicOn f ∅\nh₂f : ∀ (u : ↑(closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤\nhR : R < 0\na : ℂ\n⊢ a ∈ ∅ →\n f a =\n (((∏ᶠ (u : ℂ), canonicalFactor R u ^ (-(divisor f (ball 0 R)) u)) *\n ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nf : ℂ → E\nh₁f : MeromorphicOn f ∅\nh₂f : ∀ (u : ↑(closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤\nhR : R < 0\na : ℂ\nha : a ∈ ∅\n⊢ f a =\n (((∏ᶠ (u : ℂ), canonicalFactor R u ^ (-(divisor f (ball 0 R)) u)) *\n ∏ᶠ (v : ℂ), (fun ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Meromorphic.Order | {
"line": 730,
"column": 4
} | {
"line": 730,
"column": 14
} | {
"line": 731,
"column": 4
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nhf : MeromorphicOn f U\nz : ↑U\nt' : Set 𝕜\nh₁t' : ∀ y ∈ t', y ∈ {↑z}ᶜ → f y = 0\nh₂t' : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace... | [
"case h\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nhf : MeromorphicOn f U\nz : ↑U\nt' : Set 𝕜\nh₁t' : ∀ y ∈ t', y ∈ {↑z}ᶜ → f y = 0\nh₂t' : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t'\nh₃t' :... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Complex.CanonicalDecomposition | {
"line": 533,
"column": 2
} | {
"line": 588,
"column": 13
} | {
"line": 590,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nw : ℂ\nf h : ℂ → E\nD : ECanonicalDecomp f h R\nhw : w ∈ closedBall 0 R\nhR : 0 < R\n⊢ h w =\n ((∏ᶠ (i : ℂ), meromorphicTrailingCoeffAt (canonicalFactor R i) w ^ (divisor f (ball 0 R)) i) *\n ∏ᶠ (i : ℂ), meromorphicT... | [] | let B₀R := ball (0 : ℂ) R
let S₀R := sphere (0 : ℂ) R
lift (divisor f S₀R).support to Finset ℂ using divisor_sphere_support_finite with t₁ ht₁
lift (divisor f B₀R).support to Finset ℂ using D.meromorphicOn.divisor_ball_support_finite
with t₂ ht₂
have := (D.analyticOnNhd w hw).meromorphicAt
rw [Eq.comm]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.CanonicalDecomposition | {
"line": 533,
"column": 2
} | {
"line": 588,
"column": 13
} | {
"line": 590,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nw : ℂ\nf h : ℂ → E\nD : ECanonicalDecomp f h R\nhw : w ∈ closedBall 0 R\nhR : 0 < R\n⊢ h w =\n ((∏ᶠ (i : ℂ), meromorphicTrailingCoeffAt (canonicalFactor R i) w ^ (divisor f (ball 0 R)) i) *\n ∏ᶠ (i : ℂ), meromorphicT... | [] | let B₀R := ball (0 : ℂ) R
let S₀R := sphere (0 : ℂ) R
lift (divisor f S₀R).support to Finset ℂ using divisor_sphere_support_finite with t₁ ht₁
lift (divisor f B₀R).support to Finset ℂ using D.meromorphicOn.divisor_ball_support_finite
with t₂ ht₂
have := (D.analyticOnNhd w hw).meromorphicAt
rw [Eq.comm]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Homotopy.Lifting | {
"line": 535,
"column": 30
} | {
"line": 535,
"column": 55
} | {
"line": 535,
"column": 55
} | [
{
"pp": "E : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace A\np : E → X\ninst✝¹ : SimplyConnectedSpace A\ninst✝ : LocallyPathConnectedSpace A\ns : Set X\ncov : IsCoveringMapOn p s\na₀ : A\ne₀ : E\nf : C(A, ↑s)\nhe : p e₀ = ({ toFun := S... | [
"E : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace A\np : E → X\ninst✝¹ : SimplyConnectedSpace A\ninst✝ : LocallyPathConnectedSpace A\ns : Set X\ncov : IsCoveringMapOn p s\na₀ : A\ne₀ : E\nf : C(A, ↑s)\nhe : p e₀ = ({ toFun := Subtype.val, ... | rw [Set.mem_preimage, he] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.CanonicalTensor | {
"line": 49,
"column": 2
} | {
"line": 65,
"column": 33
} | {
"line": 66,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nι : Type u_2\ninst✝ : Fintype ι\nv : OrthonormalBasis ι ℝ E\nw : OrthonormalBasis (Fin (Module.finrank ℝ E)) ℝ E := stdOrthonormalBasis ℝ E\n⊢ canonicalCovariantTensor E = ∑ i, v i ⊗ₜ[ℝ] v i",
... | [] | calc ∑ m, w m ⊗ₜ[ℝ] w m
_ = ∑ m, ∑ n, ⟪w m, w n⟫_ℝ • w m ⊗ₜ[ℝ] w n := by
congr 1 with m
rw [Fintype.sum_eq_single m _, orthonormal_iff_ite.1 w.orthonormal]
· simp only [↓reduceIte, one_smul]
simp only [orthonormal_iff_ite.1 w.orthonormal, ite_smul, one_smul, zero_smul,
ite_eq_right_iff]
taut... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.Complex.Hadamard | {
"line": 568,
"column": 2
} | {
"line": 570,
"column": 6
} | {
"line": 572,
"column": 0
} | [
{
"pp": "l u : ℝ\nhul : l < u\nz : ℂ\n⊢ ↑l + (z / (↑u - ↑l) - ↑l / (↑u - ↑l)) * (↑u - ↑l) = z",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Not.intro",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"GroupWithZero.toMonoidWithZer... | [] | rw [sub_mul, div_mul_comm, div_self (by norm_cast; linarith),
div_mul_comm, div_self (by norm_cast; linarith)]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Hadamard | {
"line": 568,
"column": 2
} | {
"line": 570,
"column": 6
} | {
"line": 572,
"column": 0
} | [
{
"pp": "l u : ℝ\nhul : l < u\nz : ℂ\n⊢ ↑l + (z / (↑u - ↑l) - ↑l / (↑u - ↑l)) * (↑u - ↑l) = z",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Not.intro",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"GroupWithZero.toMonoidWithZer... | [] | rw [sub_mul, div_mul_comm, div_self (by norm_cast; linarith),
div_mul_comm, div_self (by norm_cast; linarith)]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Laplacian | {
"line": 364,
"column": 49
} | {
"line": 364,
"column": 51
} | {
"line": 365,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAlgebra ℝ 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : IsScala... | [
"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAlgebra ℝ 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : IsScalarTower ℝ 𝕜 ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 264,
"column": 64
} | {
"line": 264,
"column": 85
} | {
"line": 265,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\ninst✝ : CompleteSpace E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nh : IsConservativeOn f (ball c r)\nw : ℂ\nI₁ : E := ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)\nI₂ : E := ∫ (y : ℝ) in ... | [] | congr; rw [re_add_im] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 264,
"column": 64
} | {
"line": 264,
"column": 85
} | {
"line": 265,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\ninst✝ : CompleteSpace E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nh : IsConservativeOn f (ball c r)\nw : ℂ\nI₁ : E := ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)\nI₂ : E := ∫ (y : ℝ) in ... | [] | congr; rw [re_add_im] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Laplacian | {
"line": 396,
"column": 49
} | {
"line": 396,
"column": 51
} | {
"line": 397,
"column": 2
} | [
{
"pp": "E : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : E → F\nx : E\nl : F →L[ℝ] G\nh : ContDiffAt ℝ 2 f... | [
"E : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : E → F\nx : E\nl : F →L[ℝ] G\nh : ContDiffAt ℝ 2 f x\na : E\nh... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.CircleAverage | {
"line": 161,
"column": 73
} | {
"line": 165,
"column": 6
} | {
"line": 167,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℂ → E\nc : ℂ\nR : ℝ\n⊢ circleAverage f c R = circleAverage (fun z ↦ f (↑R * z + c)) 0 1",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"... | [] | by
unfold circleAverage circleMap
congr with θ
ring_nf
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.CircleAverage | {
"line": 284,
"column": 6
} | {
"line": 284,
"column": 36
} | {
"line": 284,
"column": 37
} | [
{
"pp": "c : ℂ\nR : ℝ\nf : ℂ → ℝ\na : ℝ\nhf : CircleIntegrable f c R\nh₂f : ∀ x ∈ sphere c |R|, f x ≤ a\n⊢ circleAverage f c R ≤ a",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.instLE",
"Real",
"Real.lattice... | [
"c : ℂ\nR : ℝ\nf : ℂ → ℝ\na : ℝ\nhf : CircleIntegrable f c R\nh₂f : ∀ x ∈ sphere c |R|, f x ≤ a\n⊢ circleAverage f c R ≤ circleAverage (fun x ↦ a) c |R|"
] | ← circleAverage_const a c |R|, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 517,
"column": 4
} | {
"line": 517,
"column": 14
} | {
"line": 518,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)\nhB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C\nhim : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.re\nhz_... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)\nhB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C\nhim : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C\nz : ℂ\nhz_re : 0 ≤ z.re\nhz_im : 0 ≤ z.i... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions | {
"line": 38,
"column": 81
} | {
"line": 38,
"column": 83
} | {
"line": 39,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : ℂ → F\nx : ℂ\nh : ContDiffAt ℂ 2 f x\na : ℂ\n⊢ (ContinuousMultilinearMap.restrictScalars ℝ ∘ iteratedFDeriv ℂ 2 f) a = iteratedFDeriv ℝ 2 f a →\n Laplacian.laplacian f a = 0 a",
"ppTerm": "?m.69",
"assigned": true,
... | [
"F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : ℂ → F\nx : ℂ\nh : ContDiffAt ℂ 2 f x\na : ℂ\nha : (ContinuousMultilinearMap.restrictScalars ℝ ∘ iteratedFDeriv ℂ 2 f) a = iteratedFDeriv ℝ 2 f a\n⊢ Laplacian.laplacian f a = 0 a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.SpecialFunctions.Integrability.Basic | {
"line": 182,
"column": 12
} | {
"line": 182,
"column": 14
} | {
"line": 183,
"column": 4
} | [
{
"pp": "s : ℂ\nt : ℝ\nht : 0 < t\nh : IntegrableOn (fun x ↦ ↑x ^ s) (Ioo 0 t) volume\na : ℝ\n⊢ a ∈ Ioo 0 t → (fun a ↦ ‖↑a ^ s‖) a = (fun a ↦ a ^ s.re) a",
"ppTerm": "?m.109",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"PartialOrder.toPreorder",
"Membership... | [
"s : ℂ\nt : ℝ\nht : 0 < t\nh : IntegrableOn (fun x ↦ ↑x ^ s) (Ioo 0 t) volume\na : ℝ\nha : a ∈ Ioo 0 t\n⊢ (fun a ↦ ‖↑a ^ s‖) a = (fun a ↦ a ^ s.re) a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 92,
"column": 33
} | {
"line": 92,
"column": 35
} | {
"line": 93,
"column": 4
} | [
{
"pp": "this✝ : AnalyticOnNhd ℝ (fun x ↦ 4 * sin x ^ 2) Set.univ\nthis : ((fun x ↦ 4 * sin x ^ 2) ⁻¹' {0})ᶜ ∈ codiscrete ℝ\na : ℝ\n⊢ a ∈ ((fun x ↦ 4 * sin x ^ 2) ⁻¹' {0})ᶜ → log (4 * sin a ^ 2) = log 4 + 2 * log (sin a)",
"ppTerm": "?m.418",
"assigned": true,
"usedConstants": [
"Real",
... | [
"this✝ : AnalyticOnNhd ℝ (fun x ↦ 4 * sin x ^ 2) Set.univ\nthis : ((fun x ↦ 4 * sin x ^ 2) ⁻¹' {0})ᶜ ∈ codiscrete ℝ\na : ℝ\nha : a ∈ ((fun x ↦ 4 * sin x ^ 2) ⁻¹' {0})ᶜ\n⊢ log (4 * sin a ^ 2) = log 4 + 2 * log (sin a)"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 402,
"column": 2
} | {
"line": 402,
"column": 74
} | {
"line": 403,
"column": 2
} | [
{
"pp": "a b : ℝ\nt : ℂ\nht : t ≠ -1\n⊢ ∫ (x : ℝ) in a..b, ↑x * (1 + ↑x ^ 2) ^ t =\n (1 + ↑b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + ↑a ^ 2) ^ (t + 1) / (2 * (t + 1))",
"ppTerm": "?m.149",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"... | [
"a b : ℝ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ ∫ (x : ℝ) in a..b, ↑x * (1 + ↑x ^ 2) ^ t =\n (1 + ↑b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + ↑a ^ 2) ^ (t + 1) / (2 * (t + 1))"
] | have : t + 1 ≠ 0 := by contrapose ht; rwa [add_eq_zero_iff_eq_neg] at ht | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 821,
"column": 45
} | {
"line": 822,
"column": 74
} | {
"line": 823,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : ℂ → E\nhfd : DiffContOnCl ℂ f {z | 0 < z.re}\nhgd : DiffContOnCl ℂ g {z | 0 < z.re}\nhfexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhgexp : ∃ c < 2, ∃ B, g =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z... | [] | by
simpa only [EqOn, Pi.sub_apply, Pi.zero_apply, sub_eq_zero] using this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 85
} | {
"line": 133,
"column": 4
} | [
{
"pp": "case refine_1\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}\nz : E\nhz : z ∈ sphere 0 1\nt : ℂ\... | [
"case refine_2\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}\nz : E\nhz : z ∈ sphere 0 1\nt : ℂ\nht : t ∈ ba... | · exact hgr (by simpa [ray, norm_smul, mem_sphere_zero_iff_norm.mp hz] using! ht) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 486,
"column": 95
} | {
"line": 493,
"column": 8
} | {
"line": 495,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ ∫ (x : ℝ) in 0..π, sin x ^ (2 * n + 1) = 2 * ∏ i ∈ Finset.range n, (2 * ↑i + 2) / (2 * ↑i + 3)",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Nat.cast_mul._simp_1",
"Iff.mpr",
"zero_le",
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Mathlib.Tactic... | [] | by
induction n with
| zero => norm_num
| succ k ih =>
rw [prod_range_succ_comm, mul_left_comm, ← ih, mul_succ, integral_sin_pow]
norm_cast
field_simp
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 43
} | {
"line": 157,
"column": 4
} | [
{
"pp": "case neg\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 neg\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}\nz : E\nh1 : AnalyticAt ℂ (gray z) 0\nhz : z ∈ spher... | specialize h1 z hz 0 (mem_ball_self hr) | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.Analysis.Complex.Polynomial.GaussLucas | {
"line": 45,
"column": 6
} | {
"line": 45,
"column": 16
} | {
"line": 46,
"column": 6
} | [
{
"pp": "case neg.h\nP : ℂ[X]\nhP : 0 < P.degree\nz : ℂ\nhP₀ : P ≠ 0\nhPz : ¬eval z P = 0\n⊢ ∀ i ∈ P.roots.toFinset, 0 < ↑(rootMultiplicity i P) / ‖z - i‖ ^ 2",
"ppTerm": "?neg.h✝",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"Polynomial.roots",
"Complex.commRing",
... | [
"case neg.h\nP : ℂ[X]\nhP : 0 < P.degree\nz : ℂ\nhP₀ : P ≠ 0\nhPz : ¬eval z P = 0\nw : ℂ\nhw : w ∈ P.roots.toFinset\n⊢ 0 < ↑(rootMultiplicity w P) / ‖z - w‖ ^ 2"
] | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 639,
"column": 66
} | {
"line": 639,
"column": 77
} | {
"line": 639,
"column": 77
} | [
{
"pp": "a b : ℝ\nh1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4\nh2 : Continuous fun x ↦ cos (2 * x) ^ 2\nx✝ : ℝ\n⊢ cos x✝ * sin x✝ = sin (2 * x✝) / 2",
"ppTerm": "?m.362",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"... | [
"a b : ℝ\nh1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4\nh2 : Continuous fun x ↦ cos (2 * x) ^ 2\nx✝ : ℝ\n⊢ cos x✝ * sin x✝ = 2 * sin x✝ * cos x✝ / 2"
] | sin_two_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.RCLike.Sqrt | {
"line": 137,
"column": 2
} | {
"line": 138,
"column": 34
} | {
"line": 140,
"column": 0
} | [
{
"pp": "⊢ (-I).sqrt = ↑√2⁻¹ * (1 - I)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrd... | [] | rw [sqrt, ← re_add_im ((-I) ^ 2⁻¹), cpow_inv_two_im_eq_neg_sqrt (by simp), cpow_inv_two_re]
simp [mul_sub, ← sub_eq_add_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Sqrt | {
"line": 137,
"column": 2
} | {
"line": 138,
"column": 34
} | {
"line": 140,
"column": 0
} | [
{
"pp": "⊢ (-I).sqrt = ↑√2⁻¹ * (1 - I)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrd... | [] | rw [sqrt, ← re_add_im ((-I) ^ 2⁻¹), cpow_inv_two_im_eq_neg_sqrt (by simp), cpow_inv_two_re]
simp [mul_sub, ← sub_eq_add_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Sqrt | {
"line": 143,
"column": 2
} | {
"line": 144,
"column": 67
} | {
"line": 145,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nh : im I = 1\n⊢ (complexRingEquiv h).symm ((complexRingEquiv h) I).sqrt = ↑√2⁻¹ * (1 - I) * I",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"RCLike.one_re",
"add_mul",
"Distrib.leftDistribClass",
"Eq.mpr",
... | [
"case neg\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nh : ¬im I = 1\n⊢ ↑√(re I) = ↑√2⁻¹ * (1 - I) * I"
] | · simp_rw [RingEquiv.symm_apply_eq, map_mul]
simp [h, mul_assoc, mul_add, add_comm, Complex.sqrt_I, add_mul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.TietzeExtension | {
"line": 245,
"column": 12
} | {
"line": 245,
"column": 19
} | {
"line": 245,
"column": 20
} | [
{
"pp": "X : 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\nF : (X →ᵇ ℝ) → Y →ᵇ ℝ\nhF_norm : ∀ (f : X →ᵇ ℝ), ‖F f‖ ≤ ‖f‖ / 3\nhF_dist : ∀ (f : X →ᵇ ℝ), dist ((F f).compContinuous e) f ≤ 2 / 3 * ‖f‖\ng :... | [
"X : 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\nF : (X →ᵇ ℝ) → Y →ᵇ ℝ\nhF_norm : ∀ (f : X →ᵇ ℝ), ‖F f‖ ≤ ‖f‖ / 3\nhF_dist : ∀ (f : X →ᵇ ℝ), dist ((F f).compContinuous e) f ≤ 2 / 3 * ‖f‖\ng : ℕ → Y →ᵇ ℝ ... | g_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 191,
"column": 6
} | {
"line": 191,
"column": 66
} | {
"line": 192,
"column": 6
} | [
{
"pp": "case mp.inl\ng : GL (Fin 2) ℝ\nhg : ∀ (z : ℍ), g • z = z\nhgc : g ∉ Subgroup.center (GL (Fin 2) ℝ)\nhlt : (↑g).det < 0\n⊢ False",
"ppTerm": "?mp.inl",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.glAction",
"Units.val",
"False",
"Real",
"instHSMul",
... | [
"case mp.inl\ng : GL (Fin 2) ℝ\nhg : ∀ (z : ℍ), g • z = z\nhgc : g ∉ Subgroup.center (GL (Fin 2) ℝ)\nhlt : (↑g).det < 0\nha : ↑g 0 0 = -↑g 1 1\nhb : ↑g 0 1 = ↑g 1 0\n⊢ False"
] | obtain ⟨ha, hb⟩ := (gl_smul_I_eq_I_iff_of_neg hlt).mp (hg _) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 53
} | {
"line": 217,
"column": 2
} | [
{
"pp": "τ : ℍ\nU : Set ℝ\nhU : U ∈ map (fun τ ↦ ‖↑τ‖) (𝓝 τ)\ns : Set ℍ\nhs' : (fun τ ↦ ‖↑τ‖) '' s ⊆ U\nε : ℝ\nhεpos : ε > 0\nhεs : Metric.ball (↑τ) ε ⊆ UpperHalfPlane.coe '' s\n⊢ ∃ ε > 0, Metric.ball ‖↑τ‖ ε ⊆ U",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Rea... | [
"τ : ℍ\nU : Set ℝ\nhU : U ∈ map (fun τ ↦ ‖↑τ‖) (𝓝 τ)\ns : Set ℍ\nhs' : (fun τ ↦ ‖↑τ‖) '' s ⊆ U\nε : ℝ\nhεpos : ε > 0\nhεs : Metric.ball (↑τ) ε ⊆ UpperHalfPlane.coe '' s\nr : ℝ\nhr : r ∈ Metric.ball ‖↑τ‖ ε\n⊢ r ∈ (fun τ ↦ ‖↑τ‖) '' s"
] | refine ⟨ε, hεpos, subset_trans (fun r hr ↦ ?_) hs'⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 271,
"column": 10
} | {
"line": 271,
"column": 35
} | {
"line": 271,
"column": 36
} | [
{
"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) + ... | [
"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) + (↑n * a ^ (n... | (by lia : n = n - 1 + 1), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.TietzeExtension | {
"line": 266,
"column": 4
} | {
"line": 267,
"column": 36
} | {
"line": 269,
"column": 0
} | [
{
"pp": "case refine_2\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\nF : (X →ᵇ ℝ) → Y →ᵇ ℝ\nhF_norm : ∀ (f : X →ᵇ ℝ), ‖F f‖ ≤ ‖f‖ / 3\nhF_dist : ∀ (f : X →ᵇ ℝ), dist ((F f).compContinuous e) f ≤ 2... | [] | rw [← hge]
exact norm_compContinuous_le _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.TietzeExtension | {
"line": 266,
"column": 4
} | {
"line": 267,
"column": 36
} | {
"line": 269,
"column": 0
} | [
{
"pp": "case refine_2\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\nF : (X →ᵇ ℝ) → Y →ᵇ ℝ\nhF_norm : ∀ (f : X →ᵇ ℝ), ‖F f‖ ≤ ‖f‖ / 3\nhF_dist : ∀ (f : X →ᵇ ℝ), dist ((F f).compContinuous e) f ≤ 2... | [] | rw [← hge]
exact norm_compContinuous_le _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.TietzeExtension | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 97
} | {
"line": 276,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\ne : X → Y\nhe : IsClosedEmbedding e\n⊢ ∃ g, ‖g‖ = ‖f‖ ∧ ⇑g ∘ e = ⇑f",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NormedCommRing.toS... | [
"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\ne : X → Y\nhe : IsClosedEmbedding e\ng : Y →ᵇ ℝ\nhg : ‖g‖ = ‖g.compContinuous { toFun := e, continuous_toFun := ⋯ }‖\n⊢ ∃ g_1,\n ‖g_1‖ = ‖g.compContinuous { toFun := e, continuous_toFun := ⋯ }‖ ∧\n ... | rcases exists_extension_norm_eq_of_isClosedEmbedding' f ⟨e, he.continuous⟩ he with ⟨g, hg, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Complex.UpperHalfPlane.FunctionsBoundedAtInfty | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 37
} | {
"line": 97,
"column": 2
} | [
{
"pp": "g : GL (Fin 2) ℝ\nhg : ↑g 1 0 = 0\n⊢ Tendsto (fun τ ↦ |↑g 0 0 / ↑g 1 1| * τ.im) atImInfty atTop",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Units.val",
"Filter.Tendsto.const_mul_atTop",
"Real",
"Preorder.toLT",
"instHDiv",
"Real.lattice",
... | [
"g : GL (Fin 2) ℝ\nhg : ↑g 1 0 = 0\n⊢ 0 < |↑g 0 0 / ↑g 1 1|"
] | apply tendsto_comap.const_mul_atTop | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 352,
"column": 4
} | {
"line": 352,
"column": 60
} | {
"line": 352,
"column": 60
} | [
{
"pp": "z : ℍ\n⊢ !![√z.im, z.re / √z.im; 0, 1 / √z.im].det = 1",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"Real",
"DivInvMonoid.toInv",
"UpperHalfPlane.im_pos",
"instHDiv",
"E... | [] | simp [mul_inv_cancel₀ (Real.sqrt_ne_zero'.mpr z.im_pos)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 352,
"column": 4
} | {
"line": 352,
"column": 60
} | {
"line": 352,
"column": 60
} | [
{
"pp": "z : ℍ\n⊢ !![√z.im, z.re / √z.im; 0, 1 / √z.im].det = 1",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"Real",
"DivInvMonoid.toInv",
"UpperHalfPlane.im_pos",
"instHDiv",
"E... | [] | simp [mul_inv_cancel₀ (Real.sqrt_ne_zero'.mpr z.im_pos)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 352,
"column": 4
} | {
"line": 352,
"column": 60
} | {
"line": 352,
"column": 60
} | [
{
"pp": "z : ℍ\n⊢ !![√z.im, z.re / √z.im; 0, 1 / √z.im].det = 1",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"Real",
"DivInvMonoid.toInv",
"UpperHalfPlane.im_pos",
"instHDiv",
"E... | [] | simp [mul_inv_cancel₀ (Real.sqrt_ne_zero'.mpr z.im_pos)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 356,
"column": 4
} | {
"line": 356,
"column": 60
} | {
"line": 356,
"column": 60
} | [
{
"pp": "z : ℍ\n⊢ !![√z.im, z.re / √z.im; 0, 1 / √z.im].det = 1",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"Real",
"DivInvMonoid.toInv",
"UpperHalfPlane.im_pos",
"instHDiv",
"E... | [] | simp [mul_inv_cancel₀ (Real.sqrt_ne_zero'.mpr z.im_pos)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 356,
"column": 4
} | {
"line": 356,
"column": 60
} | {
"line": 356,
"column": 60
} | [
{
"pp": "z : ℍ\n⊢ !![√z.im, z.re / √z.im; 0, 1 / √z.im].det = 1",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"Real",
"DivInvMonoid.toInv",
"UpperHalfPlane.im_pos",
"instHDiv",
"E... | [] | simp [mul_inv_cancel₀ (Real.sqrt_ne_zero'.mpr z.im_pos)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 356,
"column": 4
} | {
"line": 356,
"column": 60
} | {
"line": 356,
"column": 60
} | [
{
"pp": "z : ℍ\n⊢ !![√z.im, z.re / √z.im; 0, 1 / √z.im].det = 1",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"Real",
"DivInvMonoid.toInv",
"UpperHalfPlane.im_pos",
"instHDiv",
"E... | [] | simp [mul_inv_cancel₀ (Real.sqrt_ne_zero'.mpr z.im_pos)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 133,
"column": 31
} | {
"line": 133,
"column": 33
} | {
"line": 134,
"column": 4
} | [
{
"pp": "f : ℍ → ℂ\nhf : DifferentiableOn ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}\nτ : ℍ\nthis : AnalyticOnNhd ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}\nw : ℍ\nhτ : ∀ᶠ (a : ℍ) in 𝓝 τ, ↑a ∈ {↑τ}ᶜ → (f ∘ ↑ofComplex) ↑a ≠ 0\na : ℍ\n⊢ (↑a ∈ {↑τ}ᶜ → (f ∘ ↑ofComplex) ↑a ≠ 0) → a ∈ {τ}ᶜ → f a ≠ 0",
"ppTerm": "?m.219",
"as... | [
"f : ℍ → ℂ\nhf : DifferentiableOn ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}\nτ : ℍ\nthis : AnalyticOnNhd ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}\nw : ℍ\nhτ : ∀ᶠ (a : ℍ) in 𝓝 τ, ↑a ∈ {↑τ}ᶜ → (f ∘ ↑ofComplex) ↑a ≠ 0\na : ℍ\nha : ↑a ∈ {↑τ}ᶜ → (f ∘ ↑ofComplex) ↑a ≠ 0\n⊢ a ∈ {τ}ᶜ → f a ≠ 0"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 33
} | {
"line": 134,
"column": 4
} | [
{
"pp": "case convert_2\nf : ℍ → ℂ\nhf : DifferentiableOn ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}\nτ : ℍ\nthis : AnalyticOnNhd ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}\nw : ℍ\nhτ : ∀ᶠ (a : ℍ) in 𝓝 τ, ↑a ∈ {↑τ}ᶜ → (f ∘ ↑ofComplex) ↑a ≠ 0\n⊢ ∀ᶠ (x : ℍ) in 𝓝 τ, x ∈ {τ}ᶜ → f x ≠ 0",
"ppTerm": "?convert_2",
"assigned":... | [
"f : ℍ → ℂ\nhf : DifferentiableOn ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}\nτ : ℍ\nthis : AnalyticOnNhd ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}\nw : ℍ\nhτ : ∀ᶠ (a : ℍ) in 𝓝 τ, ↑a ∈ {↑τ}ᶜ → (f ∘ ↑ofComplex) ↑a ≠ 0\na : ℍ\nha : ↑a ∈ {↑τ}ᶜ → (f ∘ ↑ofComplex) ↑a ≠ 0\n⊢ a ∈ {τ}ᶜ → f a ≠ 0"
] | filter_upwards [hτ] with a ha | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 59
} | {
"line": 169,
"column": 0
} | [
{
"pp": "g : GL (Fin 2) ℝ\nk : ℤ\nτ : ℍ\nhd : HasDerivAt (fun x ↦ denom g x) ↑(↑g 1 0) ↑τ\nthis : HasDerivAt ((fun x ↦ x ^ k) ∘ denom g) (↑k * denom g ↑τ ^ (k - 1) * ↑(↑g 1 0)) ↑τ\n⊢ HasDerivAt (fun z ↦ denom g z ^ k) (↑k * ↑(↑g 1 0) * denom g ↑τ ^ (k - 1)) ↑τ",
"ppTerm": "?m.109",
"assigned": true,
... | [] | simpa only [Function.comp_def, mul_right_comm] using this | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Category.CompHausLike.Basic | {
"line": 212,
"column": 13
} | {
"line": 212,
"column": 15
} | {
"line": 212,
"column": 16
} | [
{
"pp": "case mp\nP : TopCat → Prop\nX Y : CompHausLike P\nf : X ⟶ Y\nhf : Mono f\n⊢ Function.Injective ⇑(ConcreteCategory.hom f)",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"TopCat.carrier",
"CompHausLike.toTop"
],
"usedFVars": [
"P",
"X"
],
"used... | [
"case mp\nP : TopCat → Prop\nX Y : CompHausLike P\nf : X ⟶ Y\nhf : Mono f\nx₁ : ↑X.toTop\n⊢ ∀ ⦃a₂ : (fun X ↦ ↑X.toTop) X⦄, (ConcreteCategory.hom f) x₁ = (ConcreteCategory.hom f) a₂ → x₁ = a₂"
] | x₁ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Compactification.StoneCech | {
"line": 90,
"column": 14
} | {
"line": 90,
"column": 16
} | {
"line": 91,
"column": 4
} | [
{
"pp": "case mp\nα : Type u\nu : Ultrafilter (Ultrafilter α)\nx : Ultrafilter α\nh : ∀ (i : Set (Ultrafilter α)), (x ∈ i ∧ i ∈ range fun s ↦ {u | s ∈ u}) → i ∈ ↑u\na : Set α\n⊢ a ∈ x → {v | a ∈ v} ∈ u",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Ultrafilter... | [
"case mp\nα : Type u\nu : Ultrafilter (Ultrafilter α)\nx : Ultrafilter α\nh : ∀ (i : Set (Ultrafilter α)), (x ∈ i ∧ i ∈ range fun s ↦ {u | s ∈ u}) → i ∈ ↑u\na : Set α\nha : a ∈ x\n⊢ {v | a ∈ v} ∈ u"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
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