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.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 464,
"column": 2
} | {
"line": 464,
"column": 13
} | [
{
"pp": "⊢ arccos '' Icc (-1) 1 = Icc 0 π",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 966,
"column": 4
} | {
"line": 966,
"column": 15
} | [
{
"pp": "R : Type u_4\nK : Type u_5\ninst✝¹⁰ : NormedRing K\ninst✝⁹ : IsDomain K\ninst✝⁸ : NormedAddCommGroup R\ninst✝⁷ : Module K R\ninst✝⁶ : IsTorsionFree K R\ninst✝⁵ : NormSMulClass K R\ninst✝⁴ : NormSMulClass ℤ K\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : FloorSemiring K\ninst✝ : HasS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 975,
"column": 51
} | {
"line": 975,
"column": 82
} | [
{
"pp": "R : Type u_4\nK : Type u_5\ninst✝⁹ : NormedRing K\ninst✝⁸ : NormedRing R\ninst✝⁷ : Module K R\ninst✝⁶ : IsTorsionFree K R\ninst✝⁵ : NormSMulClass K R\ninst✝⁴ : NormSMulClass ℤ K\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : FloorSemiring K\ninst✝ : HasSolidNorm K\ng : ℕ → R\nt : R\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 28
} | [
{
"pp": "case inl.inr.inl\nx : ℂ\nhr : x.re < 0\nhi : x.im = 0\n⊢ (if 0 ≤ x.re then -Real.arcsin (x.im / ‖x‖)\n else if 0 ≤ -x.im then Real.arcsin (x.im / ‖x‖) + π else Real.arcsin (x.im / ‖x‖) - π) =\n if x.re < 0 ∧ x.im = 0 then π\n else\n -if 0 ≤ x.re then Real.arcsin (x.im / ‖x‖)\n else... | · simp [hr, hr.not_ge, hi] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 27
} | [
{
"pp": "x : ℝ\n⊢ log x ≠ 0 ↔ x ≠ 0 ∧ x ≠ 1 ∧ x ≠ -1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 347,
"column": 7
} | {
"line": 347,
"column": 18
} | [
{
"pp": "case h\nx✝ : ℂ\n⊢ x✝ ∈ (fun θ ↦ cexp (↑θ * I)) '' Ioc (-π) π ↔ x✝ ∈ sphere 0 1",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Set.Ioc",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Preorder.toLT",
"Real.pi",
"HMul.hMul",
"congrArg",
"sub_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 24
} | [
{
"pp": "x : ℝ\nhx : 0 < x\n⊢ 1 - x⁻¹ ≤ log x",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"Real.instInv",
"Real.instSub",
"covariant_swap_add_of_covariant_add",
"AddGroup.toOrderedSub",
"HSub.hSub",
"id",
"Real.instAddGr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 326,
"column": 25
} | {
"line": 326,
"column": 60
} | [
{
"pp": "x : ℝ\nh1 : 0 < x\nh2 : x ≤ 1\n⊢ 0 < 1 / x",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real.partialOrder",
"Real",
"DivInvMonoid.toInv",
"Preorder.toLT",
"instHDiv",
"MulZeroClass.toMul",
"Real.instZero",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 147,
"column": 59
} | {
"line": 147,
"column": 89
} | [
{
"pp": "x : ℂ\nh : cexp x = 1\nn : ℤ\nhn : x.im + n • (2 * π) ∈ Ioc (-π) (-π + 2 * π)\n⊢ (x + ↑n * (2 * ↑π * I)).im ∈ Ioc (-π) π",
"usedConstants": [
"Complex.mul_im",
"Distrib.leftDistribClass",
"Int.cast",
"Eq.mpr",
"Set.Ioc",
"Real",
"Preorder.toLT",
"Comp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 341,
"column": 33
} | {
"line": 341,
"column": 59
} | [
{
"pp": "⊢ Tendsto (fun x ↦ log (rexp x)) atTop atTop",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.log_exp",
"congrArg",
"id",
"Real.exp",
"Real.log",
"Filter.atTop",
"funext",
"Filter.Tendsto",
"congrFun'",
"Eq",
"Filter",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 40
} | [
{
"pp": "⊢ Tendsto log (𝓝[>] 0) atBot",
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"Real.log_exp",
"Real.instZero",
"congrArg",
"nhdsWithin",
"PseudoMetricSpace.toUniformSpace",
"id",
"Real.exp",
"Real.log",
"funext",
"Filte... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 343,
"column": 62
} | {
"line": 344,
"column": 51
} | [
{
"pp": "⊢ Tendsto log (𝓝[>] 0) atBot",
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"Real.log_exp",
"Real.instZero",
"congrArg",
"Filter.tendsto_id",
"nhdsWithin",
"PseudoMetricSpace.toUniformSpace",
"id",
"Real.exp",
"Real.log",
... | by
simpa [← tendsto_comp_exp_atBot] using tendsto_id | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 24
} | [
{
"pp": "⊢ Tendsto log (𝓝[≠] 0) atBot",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 156,
"column": 31
} | {
"line": 156,
"column": 47
} | [
{
"pp": "x : ℂ\nhx : 0 ≤ x.im\nx✝ : ∃ n, x = ↑n * (2 * ↑π * I)\nn : ℤ\nhn : x = ↑n * (2 * ↑π * I)\n⊢ 0 ≤ ↑n * (2 * π)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 211,
"column": 4
} | {
"line": 211,
"column": 15
} | [
{
"pp": "case convert_1\nz : ℝ\nhre : (↑z).re < 0\n⊢ ‖↑z‖ ≠ 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.lattice",
"Real.instZero",
"abs",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 13
} | [
{
"pp": "case convert_1\nz : ℝ\nhre : (↑z).re < 0\n⊢ ‖↑z‖ ≠ 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.lattice",
"Real.instZero",
"abs",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 52
} | [
{
"pp": "z : ℂ\nhre : z.re < 0\nhim : z.im = 0\n⊢ Tendsto log (𝓝[{z | 0 ≤ z.im}] z) (𝓝 (↑(Real.log ‖z‖) + ↑π * I))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 436,
"column": 4
} | {
"line": 436,
"column": 15
} | [
{
"pp": "n : ℕ\n⊢ Tendsto (fun x ↦ log x ^ n / id x) atTop (𝓝 0)",
"usedConstants": [
"Real",
"instHDiv",
"NormedDivisionRing.toNormedRing",
"PseudoMetricSpace.toUniformSpace",
"NormedDivisionRing.toDivisionRing",
"nhds",
"DivisionRing.toDivisionSemiring",
"D... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 304,
"column": 53
} | {
"line": 304,
"column": 74
} | [
{
"pp": "x y : ℝ\nh₁ : -π < y\nh₂ : y < π\n⊢ y ≤ -π + 2 * π",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.pi",
"HMul.hMul",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"PartialOrder.toPreorder",
"Nat.instAtLeastTwoHAddOfNat",
"Preorder.toLE",
"two_m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 540,
"column": 2
} | {
"line": 540,
"column": 13
} | [
{
"pp": "case hx\ne : ℝ\nn : ℕ\nh : NormNum.IsNat e n\nw : Nat.blt 1 n = true\n⊢ 1 < ↑n",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"Real.instRCLike",
"Real.instZeroLEOneClass",
"AddGroupWithOne.toAddMonoidWithOne",
"Real.instLT",
"id",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 552,
"column": 2
} | {
"line": 552,
"column": 13
} | [
{
"pp": "case hx\ne : ℝ\nn : ℕ\nh : NormNum.IsInt e (Int.negOfNat n)\nw : Nat.blt 1 n = true\n⊢ 1 < ↑n",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"Real.instRCLike",
"Real.instZeroLEOneClass",
"AddGroupWithOne.toAddMonoidWithOne",
"Real.instLT",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 559,
"column": 6
} | {
"line": 559,
"column": 17
} | [
{
"pp": "e : ℝ\nd n : ℕ\ninv : Invertible ↑d\neq : e = ↑n * ⅟↑d\nh : decide (1 < ↑n / ↑d) = true\n⊢ 1 < ↑n / ↑d",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 574,
"column": 6
} | {
"line": 574,
"column": 17
} | [
{
"pp": "e : ℝ\nd n : ℕ\ninv : Invertible ↑d\neq : e = ↑n * ⅟↑d\nh₁ : decide (0 < ↑n / ↑d) = true\nh₂ : decide (↑n / ↑d < 1) = true\n⊢ 0 < ↑n / ↑d",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 576,
"column": 6
} | {
"line": 576,
"column": 17
} | [
{
"pp": "e : ℝ\nd n : ℕ\ninv : Invertible ↑d\neq : e = ↑n * ⅟↑d\nh₁ : decide (0 < ↑n / ↑d) = true\nh₂ : decide (↑n / ↑d < 1) = true\nh₁' : 0 < ↑n / ↑d\n⊢ ↑n / ↑d < 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 97,
"column": 56
} | {
"line": 97,
"column": 67
} | [
{
"pp": "x : ℂ\n⊢ x ^ (-1) = x⁻¹",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 124,
"column": 65
} | {
"line": 124,
"column": 76
} | [
{
"pp": "x : ℂ\nn : ℕ\n⊢ x ^ ↑n = x ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 134,
"column": 65
} | {
"line": 134,
"column": 76
} | [
{
"pp": "x : ℂ\nn : ℤ\n⊢ x ^ ↑n = x ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 523,
"column": 31
} | {
"line": 523,
"column": 58
} | [
{
"pp": "z : ℂ\nθ : Real.Angle\n⊢ (↑z.arg).toReal = θ.toReal ↔ z.arg = θ.toReal",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.Angle.coe",
"congrArg",
"Complex.arg",
"id",
"Iff",
"Complex.arg_coe_angle_toReal_eq_arg",
"Real.Angle.toReal",
"Eq"
]
... | arg_coe_angle_toReal_eq_arg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 527,
"column": 36
} | {
"line": 527,
"column": 63
} | [
{
"pp": "x y : ℂ\n⊢ (↑x.arg).toReal = (↑y.arg).toReal ↔ x.arg = y.arg",
"usedConstants": [
"Real",
"Real.Angle.coe",
"congrArg",
"Complex.arg",
"iff_self",
"Iff",
"congr",
"True",
"Complex.arg_coe_angle_toReal_eq_arg",
"of_eq_true",
"congrFun... | arg_coe_angle_toReal_eq_arg | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 33
} | [
{
"pp": "x : ℝ\nn : ℕ\n⊢ ↑(↑n * x) = n • ↑x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 33
} | [
{
"pp": "x : ℝ\nn : ℤ\n⊢ ↑(↑n * x) = n • ↑x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 44
} | [
{
"pp": "ψ θ : Angle\nthis : Int.natAbs 2 = 2\n⊢ ψ = θ ∨ ψ = θ + ↑(2 * π / 2) ↔ ψ = θ ∨ ψ = θ + ↑π",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.Angle",
"Real.Angle.coe",
"CharZero.NeZero.two",
"MulZeroClass.toMul",
... | mul_div_cancel_left₀ (_ : ℝ) two_ne_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 614,
"column": 4
} | {
"line": 614,
"column": 15
} | [
{
"pp": "case convert_3\nz : ℝ\nhre : (↑z).re < 0\n⊢ ‖↑z‖ ≠ 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.lattic... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 169,
"column": 64
} | {
"line": 170,
"column": 67
} | [
{
"pp": "θ : Angle\n⊢ θ = -θ ↔ θ = 0 ∨ θ = ↑π",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"instHSMul",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"Iff.rfl",
"AddMonoid.toAddZeroClass",
"Rea... | by
rw [← add_eq_zero_iff_eq_neg, ← two_nsmul, two_nsmul_eq_zero_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 630,
"column": 4
} | {
"line": 630,
"column": 15
} | [
{
"pp": "case refine_1\nz : ℝ\nhre : (↑z).re < 0\nthis : arg =ᶠ[𝓝[{z | 0 ≤ z.im}] ↑z] fun x ↦ Real.arcsin ((-x).im / ‖x‖) + π\n⊢ ‖↑z‖ ≠ 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 635,
"column": 2
} | {
"line": 635,
"column": 47
} | [
{
"pp": "z : ℂ\nhre : z.re < 0\nhim : z.im = 0\n⊢ Tendsto arg (𝓝[{z | 0 ≤ z.im}] z) (𝓝 π)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 448,
"column": 2
} | {
"line": 449,
"column": 47
} | [
{
"pp": "case h.h\nx✝¹ x✝ : ℝ\nh : (↑x✝¹).toReal = (↑x✝).toReal\n⊢ ↑x✝¹ = ↑x✝",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 801,
"column": 42
} | {
"line": 801,
"column": 57
} | [
{
"pp": "θ ψ : Angle\nh : θ.sign = ψ.sign\n⊢ θ = ψ ↔ |θ.toReal| = |ψ.toReal|",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 835,
"column": 2
} | {
"line": 836,
"column": 9
} | [
{
"pp": "θ : Angle\n⊢ (2 • θ).sign = -θ.sign ↔ θ = 0 ∨ π / 2 < |θ.toReal|",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle.0.Real.Angle.sign_two_nsmul_eq_neg_sign_iff._simp_1_2",
"Real",
"instHSMul",
"instHDiv",
"Real.pi",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 898,
"column": 79
} | {
"line": 898,
"column": 90
} | [
{
"pp": "θ ψ : Angle\nhθ : θ.sign = 1\nhψ : ψ.sign = -1\nthis : (↑(θ.toReal + ψ.toReal)).toReal = θ.toReal + ψ.toReal\n⊢ (θ + ψ).toReal = θ.toReal + ψ.toReal",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 40,
"column": 2
} | {
"line": 40,
"column": 65
} | [
{
"pp": "case h\ny : ℝ\nhy : 0 < y\nb : ℝ\nhb : 0 ≤ b\nx : ℝ\nhx₀ : 0 ≤ x\nhx : b ^ (1 / y) ≤ x\n⊢ x ^ y ∈ Set.Ici b",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"Set.Ici",
"Preorder.toLE",
"Membership.mem",
"id",
"LE.le",
"HPow.hPow",
"i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 89
} | [
{
"pp": "b : ℝ\nhb₀ : -1 < b\nhb₁ : b < 1\nhb : b < 0\n⊢ Tendsto (fun x ↦ rexp (log b * x)) atTop (𝓝 0)",
"usedConstants": [
"Iff.mpr",
"Real",
"HMul.hMul",
"Real.instZero",
"Filter.tendsto_id",
"PartialOrder.toPreorder",
"PseudoMetricSpace.toUniformSpace",
"... | refine tendsto_exp_atBot.comp <| (tendsto_const_mul_atBot_of_neg ?_).mpr tendsto_id | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 87
} | [
{
"pp": "b : ℝ\nhb₀ : -1 < b\nhb₁ : b < 1\nhb : 0 < b\n⊢ Tendsto (fun x ↦ rexp (log b * x)) atTop (𝓝 0)",
"usedConstants": [
"Iff.mpr",
"Real",
"HMul.hMul",
"Real.instZero",
"Filter.tendsto_id",
"PartialOrder.toPreorder",
"PseudoMetricSpace.toUniformSpace",
"... | refine tendsto_exp_atBot.comp <| (tendsto_const_mul_atBot_of_neg ?_).mpr tendsto_id | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 98,
"column": 14
} | {
"line": 98,
"column": 50
} | [
{
"pp": "a b c : ℝ\nhb : 0 ≠ b\n⊢ Tendsto ?m.40 atTop (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 13
} | [
{
"pp": "case h.hdb\ns : ℝ\nn : ℕ\nhn : s < ↑n\nx : ℝ\nhx₀ : 0 < x\nhx₁ : 1 ≤ x\n⊢ x ^ s ≤ x ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 62,
"column": 65
} | {
"line": 62,
"column": 76
} | [
{
"pp": "x : ℝ\nn : ℕ\n⊢ x ^ ↑n = x ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 90
} | [
{
"pp": "α : Type u_1\nl : Filter α\nf : α → ℂ\nb : ℂ\nhl : b.re = 0 → b ≠ 0 → ∀ᶠ (x : α) in l, f x ≠ 0\n⊢ ∀ᶠ (x : α) in l, f x = 0 → b.re = 0 → b = 0",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"Filter.Eventually",
"Complex.instZero",
"id",
"Filter.Freq... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 232,
"column": 27
} | {
"line": 232,
"column": 48
} | [
{
"pp": "α : Type u_1\nr c : ℝ\nl : Filter α\nf g : α → ℝ\nh : IsBigOWith c l f g\nhc : 0 ≤ c\nhr : 0 ≤ r\nhg : 0 ≤ᶠ[l] g\nx : α\nhgx : 0 x ≤ g x\nhx : ‖f x‖ ≤ c * ‖g x‖\n⊢ |f x| ^ r ≤ (c * |g x|) ^ r",
"usedConstants": [
"Real",
"HMul.hMul",
"Real.lattice",
"abs",
"covariant_s... | by gcongr; assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 257,
"column": 2
} | {
"line": 257,
"column": 33
} | [
{
"pp": "α : Type u_1\nl : Filter α\nf g : α → ℝ\nhfg : f =O[l] g\nhg : 0 ≤ᶠ[l] g\n⊢ (fun x ↦ √(f x)) =O[l] fun x ↦ √(g x)",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"Nat.instAt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 33
} | [
{
"pp": "α : Type u_1\nl : Filter α\nf g : α → ℝ\nhfg : f =o[l] g\nhg : 0 ≤ᶠ[l] g\n⊢ (fun x ↦ √(f x)) =o[l] fun x ↦ √(g x)",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"Nat.instAt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 167,
"column": 80
} | {
"line": 169,
"column": 63
} | [
{
"pp": "x y : ℝ\nhx_nonneg : 0 ≤ x\n⊢ |x ^ y| = |x| ^ y",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"Real.lattice",
"Real.instZero",
"abs",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"A... | by
have h_rpow_nonneg : 0 ≤ x ^ y := Real.rpow_nonneg hx_nonneg _
rw [abs_eq_self.mpr hx_nonneg, abs_eq_self.mpr h_rpow_nonneg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 302,
"column": 4
} | {
"line": 303,
"column": 10
} | [
{
"pp": "a b : ℝ\nh : a ≤ b\nhimp : b = 0 → a = 0\nx : ℝ\nhx : x ∈ Set.Icc 0 1\n⊢ x ∈ {x | (fun x ↦ ‖x ^ b‖ ≤ ‖x ^ a‖) x}",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"Lattice.toSemilatticeSup",
"Real.lattice",
"Real.instZ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 313,
"column": 2
} | {
"line": 313,
"column": 13
} | [
{
"pp": "a : ℝ\nha : a ≤ 1\n⊢ id =O[𝓝[≥] 0] fun x ↦ x ^ a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 55
} | [
{
"pp": "s b : ℝ\nhb : 0 < b\n⊢ Tendsto (fun x ↦ x ^ s / rexp (b * x)) atTop (𝓝 0)",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Real.instDivInvMonoid",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 329,
"column": 2
} | {
"line": 329,
"column": 38
} | [
{
"pp": "k : ℤ\nb : ℝ\nhb : 0 < b\n⊢ (fun x ↦ x ^ k) =o[atTop] fun x ↦ rexp (b * x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 334,
"column": 2
} | {
"line": 334,
"column": 13
} | [
{
"pp": "k : ℕ\nb : ℝ\nhb : 0 < b\n⊢ (fun x ↦ x ^ k) =o[atTop] fun x ↦ rexp (b * x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 338,
"column": 2
} | {
"line": 338,
"column": 28
} | [
{
"pp": "s : ℝ\n⊢ (fun x ↦ x ^ s) =o[atTop] rexp",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 251,
"column": 8
} | {
"line": 251,
"column": 23
} | [
{
"pp": "case cons\nι : Type u_1\na : ℝ\nha : 0 ≤ a\nf : ι → ℝ\ni : ι\ns : Finset ι\nhi : i ∉ s\nihs : (∀ x ∈ s, 0 ≤ f x) → a ^ ∑ x ∈ s, f x = ∏ x ∈ s, a ^ f x\nh : ∀ x ∈ cons i s hi, 0 ≤ f x\n⊢ a ^ ∑ x ∈ cons i s hi, f x = ∏ x ∈ cons i s hi, a ^ f x",
"usedConstants": [
"Real.instLE",
"Real",
... | forall_mem_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 255,
"column": 63
} | {
"line": 256,
"column": 71
} | [
{
"pp": "x y : ℝ\n⊢ x ^ (-y) = x⁻¹ ^ y",
"usedConstants": [
"RingHom.instRingHomClass",
"Real.instPow",
"Real",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"Real.pi",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"CommSemiring.toSemiring",
"Real.i... | by
simp [rpow_def, Complex.cpow_neg, Complex.inv_cpow_eq_ite, apply_ite] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 338,
"column": 2
} | {
"line": 339,
"column": 67
} | [
{
"pp": "x : ℝ\nhx : 0 < x\ny : ℂ\n⊢ ‖↑x ^ y‖ = x ^ y.re",
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"InvOneClass.toOne",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneCl... | rw [norm_cpow_of_ne_zero (ofReal_ne_zero.mpr hx.ne'), arg_ofReal_of_nonneg hx.le,
zero_mul, Real.exp_zero, div_one, Complex.norm_of_nonneg hx.le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 338,
"column": 2
} | {
"line": 339,
"column": 67
} | [
{
"pp": "x : ℝ\nhx : 0 < x\ny : ℂ\n⊢ ‖↑x ^ y‖ = x ^ y.re",
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"InvOneClass.toOne",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneCl... | rw [norm_cpow_of_ne_zero (ofReal_ne_zero.mpr hx.ne'), arg_ofReal_of_nonneg hx.le,
zero_mul, Real.exp_zero, div_one, Complex.norm_of_nonneg hx.le] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 338,
"column": 2
} | {
"line": 339,
"column": 67
} | [
{
"pp": "x : ℝ\nhx : 0 < x\ny : ℂ\n⊢ ‖↑x ^ y‖ = x ^ y.re",
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"InvOneClass.toOne",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneCl... | rw [norm_cpow_of_ne_zero (ofReal_ne_zero.mpr hx.ne'), arg_ofReal_of_nonneg hx.le,
zero_mul, Real.exp_zero, div_one, Complex.norm_of_nonneg hx.le] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 430,
"column": 2
} | {
"line": 430,
"column": 13
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\ny : ℝ\nn : ℕ\n⊢ x ^ (y + ↑n) = x ^ y * x ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 433,
"column": 2
} | {
"line": 433,
"column": 13
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\ny : ℝ\nn : ℤ\n⊢ x ^ (y - ↑n) = x ^ y / x ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 436,
"column": 2
} | {
"line": 436,
"column": 13
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\ny : ℝ\nn : ℕ\n⊢ x ^ (y - ↑n) = x ^ y / x ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 13
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\ny : ℝ\n⊢ x ^ (y + 1) = x ^ y * x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 454,
"column": 2
} | {
"line": 454,
"column": 13
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\ny : ℝ\n⊢ x ^ (y - 1) = x ^ y / x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 44
} | [
{
"pp": "p : ℂ × ℂ\nhp_fst : p.1 ∈ slitPlane\n⊢ ContinuousAt (fun x ↦ cexp (Complex.log x.1 * x.2)) p",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Complex.log",
"HMul.hMul",
"Complex.instNormedField",
"PseudoMetricSpace.toUniformSpace",
"instTopologicalSpa... | refine continuous_exp.continuousAt.comp ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 537,
"column": 27
} | {
"line": 537,
"column": 47
} | [
{
"pp": "x y z : ℝ\nhxy : x < y\nhz : 0 < z\nhx : 0 = x\n⊢ 0 < y",
"usedConstants": [
"Real",
"Real.instZero",
"congrArg",
"Real.instLT",
"Eq.mp",
"LT.lt",
"Zero.toOfNat0",
"OfNat.ofNat",
"Eq.symm"
]
}
] | by rwa [← hx] at hxy | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 208,
"column": 2
} | {
"line": 209,
"column": 41
} | [
{
"pp": "case inr\ny : ℝ\nhp : 0 < y\nA : Tendsto (fun p ↦ rexp (log p.1 * p.2)) (𝓝[≠] 0 ×ˢ 𝓝 y) (𝓝 0)\nB : Tendsto (fun p ↦ p.1 ^ p.2) (𝓝[≠] 0 ×ˢ 𝓝 y) (𝓝 0)\nC : Tendsto (fun p ↦ p.1 ^ p.2) (𝓝[{0}] 0 ×ˢ 𝓝 y) (pure 0)\n⊢ ContinuousAt (fun p ↦ p.1 ^ p.2) (0, y)",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 393,
"column": 4
} | {
"line": 393,
"column": 15
} | [
{
"pp": "case refine_1.h\nx : ℝ≥0\ny : ℝ\nh : x ≠ 0 ∨ 0 < y\nthis : (fun p ↦ p.1 ^ p.2) = toNNReal ∘ (fun p ↦ p.1 ^ p.2) ∘ fun p ↦ (↑p.1, p.2)\n⊢ (↑(x, y).1, (x, y).2).1 ≠ 0 ∨ 0 < (↑(x, y).1, (x, y).2).2",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 402,
"column": 2
} | {
"line": 403,
"column": 30
} | [
{
"pp": "x y : ℝ≥0\nhy : 1 < y\nm : ℕ\nhm : x < y ^ m\nn : ℕ\nhn : n ≥ m + 1\n⊢ x ^ (↑n)⁻¹ ≤ y",
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"DivInvMonoid.toInv",
"Pr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 459,
"column": 4
} | {
"line": 461,
"column": 12
} | [
{
"pp": "case pos\nx : ℝ≥0∞\ny : ℝ\nh : 0 < y\nhx : x = ∞\n⊢ ContinuousAt (fun a ↦ a ^ y) x",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"ContinuousAt",
"ENNReal.instPowReal",
"HEq.refl",
"nhds",
"ENNReal.tendsto_rpow_at_top",
"Eq.casesOn",
"... | rw [hx, ContinuousAt]
convert! ENNReal.tendsto_rpow_at_top h
simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 459,
"column": 4
} | {
"line": 461,
"column": 12
} | [
{
"pp": "case pos\nx : ℝ≥0∞\ny : ℝ\nh : 0 < y\nhx : x = ∞\n⊢ ContinuousAt (fun a ↦ a ^ y) x",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"ContinuousAt",
"ENNReal.instPowReal",
"HEq.refl",
"nhds",
"ENNReal.tendsto_rpow_at_top",
"Eq.casesOn",
"... | rw [hx, ContinuousAt]
convert! ENNReal.tendsto_rpow_at_top h
simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 476,
"column": 29
} | {
"line": 476,
"column": 45
} | [
{
"pp": "y : ℝ\nx : ℝ≥0∞\nhy : y < 0\nz : ℝ\nhz : y = -z\n⊢ 0 < z",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 734,
"column": 2
} | {
"line": 735,
"column": 9
} | [
{
"pp": "x y : ℝ\nh₁ : 0 ≤ x\nh₂ : x ≤ 1\nh₃ : y ≤ 1\n⊢ x ≤ x ^ y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 738,
"column": 2
} | {
"line": 738,
"column": 29
} | [
{
"pp": "x y : ℝ\nh₁ : 1 ≤ x\nh₂ : 1 ≤ y\n⊢ x ≤ x ^ y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 741,
"column": 2
} | {
"line": 742,
"column": 9
} | [
{
"pp": "x y : ℝ\nh₁ : 0 ≤ x\nh₂ : x ≤ 1\nh₃ : 1 ≤ y\n⊢ x ^ y ≤ x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 745,
"column": 2
} | {
"line": 745,
"column": 29
} | [
{
"pp": "x y : ℝ\nh₁ : 1 ≤ x\nh₂ : y ≤ 1\n⊢ x ^ y ≤ x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 748,
"column": 2
} | {
"line": 748,
"column": 29
} | [
{
"pp": "x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : y < 1\n⊢ x < x ^ y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 748,
"column": 2
} | {
"line": 748,
"column": 66
} | [
{
"pp": "x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : y < 1\n⊢ x < x ^ y",
"usedConstants": [
"Real.instPow",
"Real",
"congrArg",
"Real.instLT",
"Eq.mp",
"Real.rpow_one",
"Real.instOne",
"HPow.hPow",
"LT.lt",
"One.toOfNat1",
"congrFun'",
"inst... | simpa only [rpow_one] using rpow_lt_rpow_of_exponent_gt h₁ h₂ h₃ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 748,
"column": 2
} | {
"line": 748,
"column": 66
} | [
{
"pp": "x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : y < 1\n⊢ x < x ^ y",
"usedConstants": [
"Real.instPow",
"Real",
"congrArg",
"Real.instLT",
"Eq.mp",
"Real.rpow_one",
"Real.instOne",
"HPow.hPow",
"LT.lt",
"One.toOfNat1",
"congrFun'",
"inst... | simpa only [rpow_one] using rpow_lt_rpow_of_exponent_gt h₁ h₂ h₃ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 748,
"column": 2
} | {
"line": 748,
"column": 66
} | [
{
"pp": "x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : y < 1\n⊢ x < x ^ y",
"usedConstants": [
"Real.instPow",
"Real",
"congrArg",
"Real.instLT",
"Eq.mp",
"Real.rpow_one",
"Real.instOne",
"HPow.hPow",
"LT.lt",
"One.toOfNat1",
"congrFun'",
"inst... | simpa only [rpow_one] using rpow_lt_rpow_of_exponent_gt h₁ h₂ h₃ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 751,
"column": 2
} | {
"line": 751,
"column": 29
} | [
{
"pp": "x y : ℝ\nh₁ : 1 < x\nh₂ : 1 < y\n⊢ x < x ^ y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 754,
"column": 2
} | {
"line": 754,
"column": 29
} | [
{
"pp": "x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : 1 < y\n⊢ x ^ y < x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 754,
"column": 2
} | {
"line": 754,
"column": 66
} | [
{
"pp": "x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : 1 < y\n⊢ x ^ y < x",
"usedConstants": [
"Real.instPow",
"Real",
"congrArg",
"Real.instLT",
"Eq.mp",
"Real.rpow_one",
"Real.instOne",
"HPow.hPow",
"LT.lt",
"One.toOfNat1",
"instHPow",
"OfNat... | simpa only [rpow_one] using rpow_lt_rpow_of_exponent_gt h₁ h₂ h₃ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 754,
"column": 2
} | {
"line": 754,
"column": 66
} | [
{
"pp": "x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : 1 < y\n⊢ x ^ y < x",
"usedConstants": [
"Real.instPow",
"Real",
"congrArg",
"Real.instLT",
"Eq.mp",
"Real.rpow_one",
"Real.instOne",
"HPow.hPow",
"LT.lt",
"One.toOfNat1",
"instHPow",
"OfNat... | simpa only [rpow_one] using rpow_lt_rpow_of_exponent_gt h₁ h₂ h₃ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 754,
"column": 2
} | {
"line": 754,
"column": 66
} | [
{
"pp": "x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : 1 < y\n⊢ x ^ y < x",
"usedConstants": [
"Real.instPow",
"Real",
"congrArg",
"Real.instLT",
"Eq.mp",
"Real.rpow_one",
"Real.instOne",
"HPow.hPow",
"LT.lt",
"One.toOfNat1",
"instHPow",
"OfNat... | simpa only [rpow_one] using rpow_lt_rpow_of_exponent_gt h₁ h₂ h₃ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 757,
"column": 2
} | {
"line": 757,
"column": 29
} | [
{
"pp": "x y : ℝ\nh₁ : 1 < x\nh₂ : y < 1\n⊢ x ^ y < x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 79,
"column": 18
} | {
"line": 79,
"column": 54
} | [
{
"pp": "x : ℝ≥0\nn : ℕ\n⊢ ↑(x ^ ↑n) = ↑(x ^ n)",
"usedConstants": [
"Real",
"id",
"NNReal",
"Nat.cast",
"Monoid.toPow",
"HPow.hPow",
"NNReal.instPowReal",
"Nat",
"Semiring.toMonoid",
"NNReal.instSemiring",
"instHPow",
"Eq",
"NNRe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 13
} | [
{
"pp": "x : ℝ≥0\nhx : x ≠ 0\ny : ℝ\n⊢ x ^ (y + 1) = x ^ y * x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 13
} | [
{
"pp": "x : ℝ≥0\nhx : x ≠ 0\ny : ℝ\n⊢ x ^ (y - 1) = x ^ y / x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 988,
"column": 37
} | {
"line": 988,
"column": 55
} | [
{
"pp": "case inr\nx : ℝ\nh : x < 0\nthis : 1 / 2 * π = π / 2\n⊢ 0 = x ^ (1 / 2)",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.instZero",
"Real.cos",
"congrArg",
"Real.instDivInvMonoid",
"Nat... | rpow_def_of_neg h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 1002,
"column": 70
} | {
"line": 1002,
"column": 80
} | [
{
"pp": "x : ℂ\n⊢ ‖x‖ ^ 2⁻¹ * Real.cos (x.arg / 2) = √((‖x‖ + x.re) / 2)",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"MulOne.toOne",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Real.cos",
"Monoid.toMulOneClass",
"c... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 1008,
"column": 70
} | {
"line": 1008,
"column": 80
} | [
{
"pp": "x : ℂ\nhx : 0 ≤ x.im\n⊢ ‖x‖ ^ 2⁻¹ * Real.sin (x.arg / 2) = √((‖x‖ - x.re) / 2)",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"MulOne.toOne",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"cong... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 1016,
"column": 70
} | {
"line": 1016,
"column": 80
} | [
{
"pp": "x : ℂ\nhx : x.im < 0\n⊢ ‖x‖ ^ 2⁻¹ * Real.sin (x.arg / 2) = -√((‖x‖ - x.re) / 2)",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"MulOne.toOne",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"con... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 13
} | [
{
"pp": "ι : Type u_1\nl : List ι\nf : ι → ℝ\nhl : ∀ i ∈ l, 0 ≤ f i\nr : ℝ\n⊢ ∀ x ∈ List.map f l, 0 ≤ x",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"List.map",
"Membership.mem",
"Exists",
"id",
"LE.le",
"forall_exists_ind... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 13
} | [
{
"pp": "case mk\nι : Type u_1\ns : Multiset ι\nf : ι → ℝ\nr : ℝ\nl : List ι\nhs : ∀ i ∈ Quot.mk (⇑(List.isSetoid ι)) l, 0 ≤ f i\n⊢ (Multiset.map (fun x ↦ f x ^ r) (Quot.mk (⇑(List.isSetoid ι)) l)).prod =\n (Multiset.map f (Quot.mk (⇑(List.isSetoid ι)) l)).prod ^ r",
"usedConstants": [
"Real.instPo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 278,
"column": 29
} | {
"line": 278,
"column": 39
} | [
{
"pp": "x y : ℝ≥0\nz : ℝ\nhz : 0 < z\n⊢ x ^ z ≤ (y ^ z⁻¹) ^ z ↔ x ^ z ≤ y",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"Real.instInv",
"Real.instDivInvMonoid",
"Partia... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 281,
"column": 29
} | {
"line": 281,
"column": 39
} | [
{
"pp": "x y : ℝ≥0\nz : ℝ\nhz : 0 < z\n⊢ (x ^ z⁻¹) ^ z ≤ y ^ z ↔ x ≤ y ^ z",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"Real.instInv",
"Real.instDivInvMonoid",
"Partia... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
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