module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 181,
"column": 75
} | {
"line": 181,
"column": 86
} | [
{
"pp": "α : Type u_1\ninst✝³ : Field α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : Archimedean α\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ n ≥ m, |f n| ≤ a\nhnm : ∀ n ≥ m, f n ≤ f n.succ\n⊢ ∀ n ≥ m, (-f) n.succ ≤ (-f) n",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 35
} | [
{
"pp": "α : Type u_1\ninst✝³ : Field α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : Archimedean α\na x : α\nhx1 : |x| < 1\n⊢ IsCauSeq abs fun m ↦ ∑ n ∈ range m, a * x ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 213,
"column": 4
} | {
"line": 213,
"column": 52
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : Field α\ninst✝⁴ : LinearOrder α\ninst✝³ : IsStrictOrderedRing α\ninst✝² : Ring β\nabv : β → α\ninst✝¹ : IsAbsoluteValue abv\ninst✝ : Archimedean α\nf : ℕ → β\nn m : ℕ\nhmn : n.succ ≤ m\nhr0 : 0 ≤ 0\nhr1 : 0 < 1\nh : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ 0 * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.LinearIsometry | {
"line": 395,
"column": 6
} | {
"line": 395,
"column": 17
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nR₄ : Type u_4\nE : Type u_5\nE₂ : Type u_6\nE₃ : Type u_7\nE₄ : Type u_8\nF : Type u_9\n𝓕 : Type u_10\ninst✝³³ : Semiring R\ninst✝³² : Semiring R₂\ninst✝³¹ : Semiring R₃\ninst✝³⁰ : Semiring R₄\nσ₁₂ : R →+* R₂\nσ₂₁ : R₂ →+* R\nσ₁₃ : R →+* R₃\nσ₃₁ : R₃ →+* R\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.LinearIsometry | {
"line": 528,
"column": 40
} | {
"line": 528,
"column": 77
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nR₄ : Type u_4\nE : Type u_5\nE₂ : Type u_6\nE₃ : Type u_7\nE₄ : Type u_8\nF : Type u_9\n𝓕 : Type u_10\ninst✝³³ : Semiring R\ninst✝³² : Semiring R₂\ninst✝³¹ : Semiring R₃\ninst✝³⁰ : Semiring R₄\nσ₁₂ : R →+* R₂\nσ₂₁ : R₂ →+* R\nσ₁₃ : R →+* R₃\nσ₃₁ : R₃ →+* R\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 359,
"column": 2
} | {
"line": 359,
"column": 13
} | [
{
"pp": "x y : ℂ\n⊢ sin x + sin y = 2 * sin ((x + y) / 2) * cos ((x - y) / 2)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 551,
"column": 6
} | {
"line": 551,
"column": 31
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\n⊢ ‖(cexp (x * I) + cexp (-(x * I))) / 2 - (1 - x ^ 2 / 2)‖ =\n ‖(cexp (-(x * I)) - (1 + -(x * I) + (x * I) ^ 2 / 2 + (-(x * I)) ^ 3 / 6)) / 2 +\n (cexp (x * I) - (1 + x * I + (x * I) ^ 2 / 2 + (x * I) ^ 3 / 6)) / 2‖",
"usedConstants": [
"_private.Mathlib.Analysi... | grind [I_sq, two_ne_zero] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 567,
"column": 6
} | {
"line": 567,
"column": 31
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\n⊢ ‖(cexp (-(x * I)) - cexp (x * I)) * I / 2 - (x - x ^ 3 / 6)‖ =\n ‖(cexp (-(x * I)) - (1 + -(x * I) + (x * I) ^ 2 / 2 + (-(x * I)) ^ 3 / 6)) * I / 2 -\n (cexp (x * I) - (1 + x * I + (x * I) ^ 2 / 2 + (x * I) ^ 3 / 6)) * I / 2‖",
"usedConstants": [
"_private.Mat... | grind [I_sq, two_ne_zero] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 727,
"column": 21
} | {
"line": 727,
"column": 32
} | [
{
"pp": "x : ℝ\nhx : cos x ≠ 0\nthis : Complex.cos ↑x ≠ 0\n⊢ ↑(1 + tan x ^ 2)⁻¹ = ↑(cos x ^ 2)",
"usedConstants": [
"Eq.mpr",
"Real",
"Complex.cos",
"Real.cos",
"congrArg",
"Real.instInv",
"Complex.ofReal_add",
"id",
"instOfNatNat",
"Complex.ofReal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 874,
"column": 2
} | {
"line": 874,
"column": 56
} | [
{
"pp": "x : ℝ\nhx : |x| ≤ 1\n⊢ |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96)",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"Real.lattice",
"Real.cos",
"abs",
"congrArg",
"Real.instDivInvMonoid",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 877,
"column": 2
} | {
"line": 877,
"column": 56
} | [
{
"pp": "x : ℝ\nhx : |x| ≤ 1\n⊢ |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96)",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"Real.lattice",
"abs",
"congrArg",
"Real.instDivInvMonoid",
"Real.instS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 72,
"column": 8
} | {
"line": 72,
"column": 54
} | [
{
"pp": "x y : ℂ\n⊢ dist y x < 1 → dist (cexp y) (cexp x) ≤ 2 * ‖cexp x‖ * dist y x",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"dist_eq_norm",
"HMul.hMul",
"congrArg",
"Nat.instAtLeastTwoHAdd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 15
} | [
{
"pp": "case inl\n⊢ (fun x ↦ cexp x - ∑ i ∈ Finset.range 0, x ^ i / ↑i !) =O[𝓝 0] fun x ↦ x ^ 0",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Nat.instMulZeroClass",
"Real.instLE",
"Real",
"instHDiv",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 136,
"column": 35
} | {
"line": 136,
"column": 76
} | [
{
"pp": "a ε : ℝ\nhε : ε > 0\nthis : ∀ (a : ℂ) (ε : ℝ), 0 < ε → ∀ᶠ (x : ℂ) in 𝓝 a, dist (cexp x) (cexp a) < ε\nha : 0 < ε / (2 * Real.exp a)\nδ : ℝ\nhδ : δ > 0 ∧ ∀ ⦃y : ℂ⦄, dist y 0 < δ → dist (cexp y) (cexp 0) < ε / (2 * Real.exp a)\nx : ℂ\na✝ : x ∈ {x | x.re ≤ a}\ny : ℂ\nhy : y ∈ {x | x.re ≤ a}\nhxy : dist x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 216,
"column": 36
} | {
"line": 216,
"column": 47
} | [
{
"pp": "y : ℝ\n⊢ y ≤ rexp (y - 1)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 45
} | [
{
"pp": "y : ℝ\nh_le : y ≤ rexp (y - 1)\nh_mul_le : y * rexp (-y) ≤ rexp (y - 1) * rexp (-y)\n⊢ y * rexp (-y) ≤ rexp (-1)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"id",
"LE.le",
"Real.exp",
"Real.instOne",
"ge_iff_le._simp_1",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 278,
"column": 4
} | {
"line": 278,
"column": 33
} | [
{
"pp": "case inl\nb c : ℝ\nhb : 0 < b\n⊢ Tendsto (fun x ↦ (b * rexp x + c) / x ^ 0) atTop atTop",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real",
"instHDiv",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"Monoid.toMulOneClass",
"congrArg",
"R... | simp only [pow_zero, div_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 75
} | [
{
"pp": "n : ℕ\n⊢ (fun x ↦ x ^ n) =o[atTop] rexp",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Real.instZero",
"NormedDivisionRing.toNorm",
"False.elim",
"NormedDivisionRing.toNormedRing",
"PseudoMetricSpace.toUniformSpace",
"NormedDivisionRing.toD... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 426,
"column": 2
} | {
"line": 426,
"column": 32
} | [
{
"pp": "⊢ Summable fun n ↦ rexp (-↑n)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.RCLike.Real | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 42
} | [
{
"pp": "case inr.inr\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx y : E\nhr✝ : dist y x ≠ 0\nhr : 0 < dist y x\nhy : y ∈ interior (closedBall x (dist y x))\nf : ℝ → E := fun c ↦ c • (y - x) + x\nc : ℝ\nhc : c ∈ f ⁻¹' closedBall x (dist y x)\n⊢ ‖c‖ * dist y x ≤ 1 * dist y x",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.RCLike.Real | {
"line": 131,
"column": 19
} | {
"line": 131,
"column": 30
} | [
{
"pp": "E : Type u_1\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NontrivialTopology E\nx : E\nr : ℝ\nhr : 0 ≤ r\ny : E\nhy : ‖y‖ = r\n⊢ x + y ∈ sphere x r",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"add_sub_cancel_left",
"Ad... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 44
} | [
{
"pp": "α : Type u_1\ninst✝² : NormedRing α\ninst✝¹ : NormSMulClass ℤ α\ninst✝ : Nontrivial α\n⊢ Tendsto (fun x ↦ ‖↑x‖) atTop atTop",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"norm_natCast_eq_mul_norm_one",
"SeminormedRing.toNorm",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 48
} | [
{
"pp": "α : Type u_1\ninst✝² : NormedRing α\ninst✝¹ : NormSMulClass ℤ α\ninst✝ : Nontrivial α\n⊢ Tendsto (fun x ↦ ‖↑x‖) (atBot ⊔ atTop) atTop",
"usedConstants": [
"Norm.norm",
"Int.cast",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"SeminormedRing.toNorm",
"Int.cast_abs",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 71
} | [
{
"pp": "f : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 : (∃ a ∈ Set.Ioo (-R) R, f =o[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x ↦ a ^ x\ntfae_2_to_1 : (∃ a ∈ Set.Ioo 0 R, f =o[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo (-R) R, f =o[atTop] fun x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 57
} | [
{
"pp": "x : ℝ\nn : ℤ\n⊢ sin (↑n * π - x) = -((-1) ^ n * sin x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 37
} | [
{
"pp": "x : ℝ\nn : ℕ\n⊢ sin (↑n * π - x) = -((-1) ^ n * sin x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 13
} | [
{
"pp": "n : ℤ\n⊢ cos (↑n * π) = (-1) ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 13
} | [
{
"pp": "n : ℕ\n⊢ cos (↑n * π) = (-1) ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 395,
"column": 2
} | {
"line": 395,
"column": 29
} | [
{
"pp": "n : ℕ\n⊢ cos (↑n * (2 * π) + π) = -1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 29
} | [
{
"pp": "n : ℤ\n⊢ cos (↑n * (2 * π) + π) = -1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 401,
"column": 2
} | {
"line": 401,
"column": 29
} | [
{
"pp": "n : ℕ\n⊢ cos (↑n * (2 * π) - π) = -1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 404,
"column": 2
} | {
"line": 404,
"column": 29
} | [
{
"pp": "n : ℤ\n⊢ cos (↑n * (2 * π) - π) = -1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 128,
"column": 8
} | {
"line": 128,
"column": 19
} | [
{
"pp": "case h\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 : (∃ a ∈ Set.Ioo (-R) R, f =o[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x ↦ a ^ x\ntfae_2_to_1 : (∃ a ∈ Set.Ioo 0 R, f =o[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo (-R) R, f =o[atTo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 437,
"column": 4
} | {
"line": 437,
"column": 41
} | [
{
"pp": "⊢ sin (π / 2) = 1 ∨ sin (π / 2) = -1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Quotient | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 15
} | [
{
"pp": "M : Type u_1\ninst✝ : SeminormedCommGroup M\nS T : Subgroup M\nx : M ⧸ S\nm : M\nr ε : ℝ\na : M\nthis✝ : Nonempty ↑{m | ↑m = ↑a}\nb : M\nthis : Nonempty ↑{m | ↑m = ↑b}\n⊢ dist 1 ↑⟨a * b, ⋯⟩ ≤ dist 1 ↑⟨a, ⋯⟩ + dist 1 ↑⟨b, ⋯⟩",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Ball.Pointwise | {
"line": 125,
"column": 61
} | {
"line": 125,
"column": 87
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Bornology.IsBounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis : closedBall 0 (ε /... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Quotient | {
"line": 204,
"column": 21
} | {
"line": 204,
"column": 60
} | [
{
"pp": "M : Type u_1\ninst✝ : SeminormedCommGroup M\nS : Subgroup M\nm : M\nε : ℝ\nhε : 0 < ε\nn : M\nhn : ↑n = (mk' S) m\nhn' : ‖n‖ < ‖(mk' S) m‖ + ε\n⊢ m⁻¹ * n ∈ S",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Ball.Pointwise | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 76
} | [
{
"pp": "case h.a\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Bornology.IsBounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis✝ : closed... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Quotient | {
"line": 284,
"column": 28
} | {
"line": 284,
"column": 49
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nm : M\n⊢ ‖(↑(mk' S)).toFun m‖ ≤ 1 * ‖m‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"QuotientAddGroup.instSemin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Quotient | {
"line": 308,
"column": 4
} | {
"line": 308,
"column": 19
} | [
{
"pp": "case inl\nM : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : ∀ x ∈ S, f x = 0\nh : ‖f‖ = 0\nx : M\n⊢ ‖(lift S f.toAddMonoidHom hf) ↑x‖ ≤ ‖f‖ * ‖↑x‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 593,
"column": 2
} | {
"line": 593,
"column": 44
} | [
{
"pp": "⊢ SurjOn sin (Icc (-(π / 2)) (π / 2)) (Icc (-1) 1)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 597,
"column": 2
} | {
"line": 597,
"column": 37
} | [
{
"pp": "⊢ SurjOn cos (Icc 0 π) (Icc (-1) 1)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Ball.Pointwise | {
"line": 187,
"column": 54
} | {
"line": 187,
"column": 92
} | [
{
"pp": "E : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx z : E\nδ ε : ℝ\nhδ : 0 < δ\nhε : 0 ≤ ε\nh : dist x z < ε + δ\n⊢ dist z x < δ + ε",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Real.instLT",
"id",
"dist_comm",
"Real.instAdd",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Quotient | {
"line": 328,
"column": 2
} | {
"line": 330,
"column": 11
} | [
{
"pp": "M : Type u_1\ninst✝ : SeminormedAddCommGroup M\nS : AddSubgroup M\nh : ↑S.topologicalClosure = univ\nx : M\n⊢ ‖S.normedMk x‖ ≤ 0 * ‖x‖",
"usedConstants": [
"Eq.mpr",
"QuotientAddGroup.instSeminormedAddCommGroup",
"congrArg",
"Set.univ",
"AddSubgroup.normedMk",
"P... | have hker : x ∈ S.normedMk.ker.topologicalClosure := by
rw [S.ker_normedMk, ← SetLike.mem_coe, h]
trivial | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Group.Quotient | {
"line": 367,
"column": 38
} | {
"line": 367,
"column": 66
} | [
{
"pp": "M : Type u_1\ninst✝ : SeminormedAddCommGroup M\nS : AddSubgroup M\nm : M\n⊢ ‖S.normedMk m‖ = sInf ((fun m_1 ↦ ‖m + m_1‖) '' ↑S.normedMk.ker)",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"NormedAddGroupHom",
"QuotientAddGroup.instSeminormedAddCommGroup",
"c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Quotient | {
"line": 404,
"column": 2
} | {
"line": 404,
"column": 13
} | [
{
"pp": "M : Type u_1\ninst✝¹ : SeminormedAddCommGroup M\nN : Type u_3\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : ∀ s ∈ S, f s = 0\nfb : f.NormNoninc\nx : M ⧸ S\nfb' : ‖f‖ ≤ ↑1\n⊢ ‖(lift S f hf) x‖ ≤ ‖x‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 73
} | [
{
"pp": "R : Type u_2\ninst✝ : NormedRing R\nk : ℕ\nr : ℝ\nhr : 1 < r\nthis : Tendsto (fun x ↦ x ^ k) (𝓝[>] 1) (𝓝 1)\nr' : ℝ\nhr' : r' ^ k < r\nh1 : 1 < r'\nh0 : 0 ≤ r'\nn : ℕ\n⊢ ↑n ≤ (r' - 1)⁻¹ * ‖r' ^ n‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 28
} | [
{
"pp": "R : Type u_2\ninst✝ : NormedRing R\nr : ℝ\nhr : 1 < r\n⊢ Nat.cast =o[atTop] fun n ↦ r ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 179,
"column": 4
} | {
"line": 179,
"column": 63
} | [
{
"pp": "R : Type u_2\ninst✝ : NormedRing R\nk : ℕ\nr₁ : R\nr₂ : ℝ\nh : ‖r₁‖ < r₂\nh0 : 0 < ‖r₁‖\nA : (fun n ↦ ↑n ^ k) =o[atTop] fun n ↦ (r₂ / ‖r₁‖) ^ n\nthis : (fun n ↦ r₁ ^ n) =O[atTop] fun n ↦ ‖r₁‖ ^ n\n⊢ (fun n ↦ ↑n ^ k * r₁ ^ n) =o[atTop] fun n ↦ r₂ ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 30
} | [
{
"pp": "case neg\nk : ℕ\nr : ℝ\nhr : |r| < 1\nh0 : ¬r = 0\nhr' : 1 < |r|⁻¹\n⊢ Tendsto (fun x ↦ ‖↑x ^ k * r ^ x‖) atTop (𝓝 0)",
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Semino... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 13
} | [
{
"pp": "r : ℝ\nk : ℕ\nhk : k ≠ 0\n⊢ Tendsto (fun n ↦ r / ↑n ^ k) atTop (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 28
} | [
{
"pp": "r : ℝ\nhr : |r| < 1\n⊢ Tendsto (fun n ↦ ↑n * r ^ n) atTop (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.PathConnected | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 13
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nx y : X\nF : Set X\nγ : Path x y\nγ_in : ∀ (t : ↑I), γ t ∈ F\nthis : γ 0 ∈ F ∧ γ 1 ∈ F\n⊢ x ∈ F ∧ y ∈ F",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 28
} | [
{
"pp": "r : ℝ\nhr : 0 ≤ r\nh'r : r < 1\n⊢ Tendsto (fun n ↦ ↑n * r ^ n) atTop (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 254,
"column": 54
} | {
"line": 254,
"column": 65
} | [
{
"pp": "R : Type u_2\nS : Type u_3\ninst✝⁵ : Field R\ninst✝⁴ : Field S\ninst✝³ : LinearOrder S\ninst✝² : TopologicalSpace S\ninst✝¹ : IsStrictOrderedRing S\ninst✝ : Archimedean S\n_i : OrderTopology S\nv : AbsoluteValue R S\na : R\nha : v a < 1\nh_add : Tendsto (fun x ↦ 1 + v a ^ x) atTop (𝓝 (1 + 0))\nh_sub :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 254,
"column": 77
} | {
"line": 254,
"column": 88
} | [
{
"pp": "R : Type u_2\nS : Type u_3\ninst✝⁵ : Field R\ninst✝⁴ : Field S\ninst✝³ : LinearOrder S\ninst✝² : TopologicalSpace S\ninst✝¹ : IsStrictOrderedRing S\ninst✝ : Archimedean S\n_i : OrderTopology S\nv : AbsoluteValue R S\na : R\nha : v a < 1\nh_add : Tendsto (fun x ↦ 1 + v a ^ x) atTop (𝓝 (1 + 0))\nh_sub :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.PathConnected | {
"line": 258,
"column": 4
} | {
"line": 258,
"column": 55
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y : X\nF : Set X\nf : X → Y\nhf : IsInducing f\nhx : x ∈ F\nhy : y ∈ F\nγ : Path (f x) (f y)\nγ' : ↑I → X\nhγ'F : ∀ (t : ↑I), γ' t ∈ F\nhγ' : ∀ (t : ↑I), f (γ' t) = γ t\nh₀ : x ⤳ γ' 0\nh₁ : γ' 1 ⤳ y\n⊢ Continuous[_, ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.PathConnected | {
"line": 368,
"column": 33
} | {
"line": 368,
"column": 44
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace Y\nx y z : X\nι : Type u_3\nF : Set X\nM : Type u_4\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm₁ m₂ : M\nhm₁ : m₁ ∈ pathComponent 1\nhm₂ : m₂ ∈ pathComponent 1\n⊢ m₁ * m₂ ∈ pathComponent 1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.PathConnected | {
"line": 377,
"column": 24
} | {
"line": 377,
"column": 35
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace Y\nx y z : X\nι : Type u_3\nF : Set X\nG : Type u_4\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\ng : G\nhg : g ∈ (Submonoid.pathComponentOne G).carrier\n⊢ g⁻¹ ∈ (Submonoid.pathComponentOn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.PathConnected | {
"line": 439,
"column": 46
} | {
"line": 439,
"column": 57
} | [
{
"pp": "G : Type u_4\ninst✝² : InvolutiveInv G\ninst✝¹ : TopologicalSpace G\ninst✝ : ContinuousInv G\ns : Set G\nhs : IsPathConnected s\na : G\nha_mem : a ∈ s\nha : ∀ ⦃y : G⦄, y ∈ s → JoinedIn s a y\nx : G\nhx : x ∈ s⁻¹\n⊢ JoinedIn s⁻¹ a⁻¹ x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 264,
"column": 2
} | {
"line": 264,
"column": 13
} | [
{
"pp": "R : Type u_2\nS : Type u_3\ninst✝⁵ : Field R\ninst✝⁴ : Field S\ninst✝³ : LinearOrder S\ninst✝² : TopologicalSpace S\ninst✝¹ : IsStrictOrderedRing S\ninst✝ : Archimedean S\n_i : OrderTopology S\nv : AbsoluteValue R S\na : R\nha : 1 < v a\n⊢ Tendsto (fun n ↦ v (a ^ n) - v 1) atTop atTop",
"usedConsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 843,
"column": 2
} | {
"line": 843,
"column": 13
} | [
{
"pp": "⊢ (4 • X ^ 2 - 2 • X - C 1).IsRoot (cos (π / 5))",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.eval",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"Polynomial.instOne",
"Polynomial.instNSMul",
"Real",
"inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.PathConnected | {
"line": 499,
"column": 7
} | {
"line": 499,
"column": 49
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nU W : Set X\nhW : IsPathConnected W\nhWU : W ⊆ U\n⊢ IsPathConnected (Subtype.val ⁻¹' W)",
"usedConstants": [
"IsPathConnected",
"Eq.mpr",
"congrArg",
"Membership.mem",
"id",
"Subtype",
"Set.preimage",
"propext... | IsInducing.subtypeVal.isPathConnected_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Connected.PathConnected | {
"line": 577,
"column": 35
} | {
"line": 577,
"column": 77
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nF : Set X\n⊢ IsPathConnected F ↔ IsPathConnected univ",
"usedConstants": [
"IsPathConnected",
"Eq.mpr",
"congrArg",
"Set.univ",
"Membership.mem",
"Set.Elem",
"id",
"Subtype",
"Iff",
"propext",
... | IsInducing.subtypeVal.isPathConnected_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Connected.PathConnected | {
"line": 613,
"column": 12
} | {
"line": 613,
"column": 23
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X\nι : Type u_3\nF : Set X\ninst✝ : PathConnectedSpace X\nx : X\n_x_in : x ∈ univ\nhx : pathComponentIn univ x = univ\n⊢ pathComponent x = univ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 874,
"column": 26
} | {
"line": 874,
"column": 42
} | [
{
"pp": "⊢ sin (π / 4) / cos (π / 4) = 1",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Real.pi",
"Real.cos",
"congrArg",
"Real.instDivInvMonoid",
"Real.cos_pi_div_four",
"Nat.instAtLeastTwoHAddOfNat",
"id",
"HDiv.hDiv",
"instOfNat... | cos_pi_div_four, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 366,
"column": 4
} | {
"line": 366,
"column": 65
} | [
{
"pp": "K : Type u_4\ninst✝ : NormedDivisionRing K\nξ : K\nh : ‖ξ‖ < 1\nxi_ne_one : ξ ≠ 1\nA : Tendsto (fun n ↦ (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 ((0 - 1) * (ξ - 1)⁻¹))\n⊢ Tendsto (fun n ↦ ∑ i ∈ Finset.range n, ξ ^ i) atTop (𝓝 (1 - ξ)⁻¹)",
"usedConstants": [
"Eq.mpr",
"False",
"DivInvMo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.AddCircle.Real | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 13
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ℕ\n⊢ toAddCircle ↑j = ↑(↑j / ↑N)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 930,
"column": 2
} | {
"line": 930,
"column": 58
} | [
{
"pp": "⊢ Function.Periodic tan π",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Real.pi",
"Real.cos",
"congrArg",
"Real.instDivInvMonoid",
"id",
"HDiv.hDiv",
"Real.instAdd",
"instHAdd",
"HAdd.hAdd",
"Function.Periodic",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.AddCircle.Real | {
"line": 84,
"column": 27
} | {
"line": 84,
"column": 84
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nx y : ZMod N\nhxy : ↑(↑x.val / ↑N) = ↑(↑y.val / ↑N)\nthis : 0 < ↑N\n⊢ ↑x.val / ↑N < 0 + 1",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"instHDiv",
"GroupWith... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.AddCircle.Real | {
"line": 84,
"column": 27
} | {
"line": 84,
"column": 84
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nx y : ZMod N\nhxy : ↑(↑x.val / ↑N) = ↑(↑y.val / ↑N)\nthis : 0 < ↑N\n⊢ ↑y.val / ↑N < 0 + 1",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"instHDiv",
"GroupWith... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 431,
"column": 2
} | {
"line": 431,
"column": 13
} | [
{
"pp": "R : Type u_4\ninst✝ : NormedRing R\nr : R\nhr : ‖r‖ < 1\n⊢ Summable fun n ↦ ‖r ^ n‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 439,
"column": 12
} | {
"line": 439,
"column": 23
} | [
{
"pp": "case zero\nR : Type u_4\ninst✝¹ : NormedRing R\ninst✝ : HasSummableGeomSeries R\nr : R\nhr : ‖r‖ < 1\n⊢ HasSum (fun n ↦ ↑((n + 0).choose 0) * r ^ n) ((1 - r)⁻¹ʳ ^ (0 + 1))",
"usedConstants": [
"Eq.mpr",
"Nat.choose",
"NormedRing.toRing",
"HMul.hMul",
"Ring.toNonAssocRi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 1136,
"column": 2
} | {
"line": 1136,
"column": 29
} | [
{
"pp": "n : ℕ\n⊢ cos (↑n * (2 * ↑π) + ↑π) = -1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 1139,
"column": 2
} | {
"line": 1139,
"column": 29
} | [
{
"pp": "n : ℤ\n⊢ cos (↑n * (2 * ↑π) + ↑π) = -1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 1142,
"column": 2
} | {
"line": 1142,
"column": 29
} | [
{
"pp": "n : ℕ\n⊢ cos (↑n * (2 * ↑π) - ↑π) = -1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 1145,
"column": 2
} | {
"line": 1145,
"column": 29
} | [
{
"pp": "n : ℤ\n⊢ cos (↑n * (2 * ↑π) - ↑π) = -1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 1161,
"column": 2
} | {
"line": 1161,
"column": 39
} | [
{
"pp": "⊢ Function.Periodic tan ↑π",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 1205,
"column": 2
} | {
"line": 1205,
"column": 52
} | [
{
"pp": "⊢ Function.Antiperiodic (fun x ↦ cexp (x * I)) ↑π",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 559,
"column": 39
} | {
"line": 559,
"column": 65
} | [
{
"pp": "α : Type u_1\ninst✝ : SeminormedAddCommGroup α\nC r : ℝ\nhr : r < 1\nu : ℕ → α\nh : ∀ (n : ℕ), ‖u n - u (n + 1)‖ ≤ C * r ^ n\n⊢ ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ C * r ^ n",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"dist_eq_norm",
"HMul.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 625,
"column": 2
} | {
"line": 625,
"column": 38
} | [
{
"pp": "α : Type u_1\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nh : CauchySeq fun n ↦ ∑ k ∈ Finset.range n, f k\nb : ℕ → ℝ\nleft✝ : ∀ (n : ℕ), 0 ≤ b n\nkey : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (∑ k ∈ Finset.range n, f k) (∑ k ∈ Finset.range m, f k) ≤ b N\nright✝ : Tendsto b atTop (𝓝 0)\nn : ℕ\n⊢ ‖f n‖ ≤ b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Path | {
"line": 540,
"column": 39
} | {
"line": 540,
"column": 50
} | [
{
"pp": "X✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx y z : X✝\nι : Type u_3\nγ✝ : Path x y\nX : Type u_4\ninst✝ : TopologicalSpace X\na b : X\nγ : Path a b\nt₀ t₁ : ℝ\nh₁ : t₀ ≤ ↑0\nh₂ : ¬↑0 ≤ t₁\nh₄ : t₀ ≤ t₁\n⊢ t₁ < 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Interval | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nr : R\nhr : 0 < r\nk m : ℤ\nh : r * ↑k ∈ Set.Ioo (r * ↑(m - 1)) (r * ↑(m + 1))\n⊢ k = m",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Path | {
"line": 548,
"column": 39
} | {
"line": 548,
"column": 50
} | [
{
"pp": "X✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx y z : X✝\nι : Type u_3\nγ✝ : Path x y\nX : Type u_4\ninst✝ : TopologicalSpace X\na b : X\nγ : Path a b\nt₀ t₁ : ℝ\nh₁ : ¬t₀ ≤ ↑1\nh₃ : t₀ ≤ t₁\n⊢ 1 < t₀",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Path | {
"line": 562,
"column": 10
} | {
"line": 562,
"column": 13
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\na b : X\nγ : Path a b\nt₀ t₁ x : ℝ\n⊢ ∀ (h : x ∈ I), (γ.truncate t₀ t₁) ⟨x, h⟩ ∈ range ⇑γ.extend",
"usedConstants": [
"Real",
"Membership.mem",
"Set.instMembership",
"unitInterval",
"Set"
]
}
] | _hx | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 82,
"column": 6
} | {
"line": 82,
"column": 33
} | [
{
"pp": "case refine_1.left\nx r : ℝ\nhr : ∀ (m : ℝ), ↑m = ↑x → r ≤ ‖m‖\n⊢ r ≤ fract x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 83,
"column": 6
} | {
"line": 83,
"column": 42
} | [
{
"pp": "case refine_1.right\nx r : ℝ\nhr : ∀ (m : ℝ), ↑m = ↑x → r ≤ ‖m‖\n⊢ r ≤ 1 - fract x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 84,
"column": 4
} | {
"line": 86,
"column": 11
} | [
{
"pp": "case refine_2\nx : ℝ\n⊢ ∀ (m : ℝ), ↑m = ↑x → |x - ↑(round x)| ≤ |m|",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"zsmul_eq_mul",
"Real.instLE",
"Real",
"i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 51
} | [
{
"pp": "p : ℝ\nhp : p ≠ 0\nx✝ : AddCircle p\nε : ℝ\nhε : |p| / 2 ≤ ε\nx : AddCircle p\n⊢ x ∈ closedBall x✝ ε",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"dist_eq_norm",
"QuotientAddGroup.instSeminormedAd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 75
} | [
{
"pp": "x : ℝ\nhx : x ∈ Icc (-1) 1\n⊢ sin (arcsin x) = x",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instLE",
"Real",
"instHDiv",
"Real.pi",
"Real.arcsin",
"congrArg",
"Real.instDivInvMonoid",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 886,
"column": 6
} | {
"line": 886,
"column": 60
} | [
{
"pp": "case inr.right\nE : Type u_5\ninst✝⁶ : Ring E\ninst✝⁵ : LinearOrder E\ninst✝⁴ : IsOrderedRing E\ninst✝³ : UniformSpace E\ninst✝² : IsUniformAddGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : OrderClosedTopology E\nf : ℕ → E\nhfa : Antitone f\nhfs : Summable f\nh✝ : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 73
} | [
{
"pp": "p : ℝ\nhp : Fact (0 < p)\nu : AddCircle p\nn : ℕ\nhn : ‖u‖ = p * (↑n / ↑(addOrderOf u))\nhu : ↑(addOrderOf u) ≠ 0\nhu' : n = 0\n⊢ u = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 933,
"column": 4
} | {
"line": 933,
"column": 76
} | [
{
"pp": "case h.inl\nα : Type u_1\nR : Type u_4\nK : Type u_5\ninst✝⁵ : NormedRing K\ninst✝⁴ : IsDomain K\ninst✝³ : NormedAddCommGroup R\ninst✝² : Module K R\ninst✝¹ : IsTorsionFree K R\ninst✝ : NormSMulClass K R\nf : α → K\ng : α → R\nl : Filter α\nhmul : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖f x • g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 43
} | [
{
"pp": "x : ℂ\n⊢ ‖x‖ * Real.cos x.arg = x.re",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 43
} | [
{
"pp": "x : ℂ\n⊢ ‖x‖ * Real.sin x.arg = x.im",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 41
} | [
{
"pp": "z : ℂ\n⊢ toIocMod Real.two_pi_pos (-π) z.arg = z.arg",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Real",
"Preorder.toLT",
"Real.instArchimedean",
"Real.pi",
"HMul.hMul",
"congrArg",
"toIocMod.congr_simp",
"AddCommGroup.toAddCommMonoid",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 951,
"column": 4
} | {
"line": 951,
"column": 15
} | [
{
"pp": "case hf\nα : Type u_1\nR : Type u_4\nK : Type u_5\ninst✝⁵ : NormedRing K\ninst✝⁴ : IsDomain K\ninst✝³ : NormedAddCommGroup R\ninst✝² : Module K R\ninst✝¹ : IsTorsionFree K R\ninst✝ : NormSMulClass K R\nf₁ f₂ : α → K\ng : α → R\nt : R\nl : Filter α\nhmul : Tendsto (fun x ↦ f₁ x • g x) l (𝓝 t)\nhf₁ : Te... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 420,
"column": 25
} | {
"line": 420,
"column": 47
} | [
{
"pp": "x✝ y x : ℝ\nhx : x ∈ Icc (-(π / 2)) (π / 2)\n⊢ sin x ∈ Icc (-1) 1",
"usedConstants": [
"Eq.mpr",
"Real",
"abs",
"PartialOrder.toPreorder",
"_private.Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse.0.Real.sinPartialEquiv._simp_1",
"Preorder.toLE",
"A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 430,
"column": 2
} | {
"line": 430,
"column": 13
} | [
{
"pp": "⊢ arcsin '' Icc (-1) 1 = Icc (-(π / 2)) (π / 2)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 455,
"column": 25
} | {
"line": 455,
"column": 47
} | [
{
"pp": "x✝ y x : ℝ\nhx : x ∈ Icc 0 π\n⊢ cos x ∈ Icc (-1) 1",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.cos",
"abs",
"PartialOrder.toPreorder",
"_private.Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse.0.Real.sinPartialEquiv._simp_1",
"Preorder.toLE",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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