module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 97,
"column": 56
} | {
"line": 97,
"column": 67
} | {
"line": 97,
"column": 68
} | [
{
"pp": "x : ℂ\n⊢ x ^ (-1) = x⁻¹",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℂ\n⊢ x ^ (-1) = x⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 124,
"column": 65
} | {
"line": 124,
"column": 76
} | {
"line": 124,
"column": 77
} | [
{
"pp": "x : ℂ\nn : ℕ\n⊢ x ^ ↑n = x ^ n",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℂ\nn : ℕ\n⊢ x ^ ↑n = x ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 134,
"column": 65
} | {
"line": 134,
"column": 76
} | {
"line": 134,
"column": 77
} | [
{
"pp": "x : ℂ\nn : ℤ\n⊢ x ^ ↑n = x ^ n",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℂ\nn : ℤ\n⊢ x ^ ↑n = x ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 28
} | {
"line": 317,
"column": 2
} | [
{
"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... | [
"case inl.inr.inr\nx : ℂ\nhr : x.re < 0\nhi : 0 < x.im\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 if 0 ≤ x.im... | · simp [hr, hr.not_ge, hi] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 211,
"column": 4
} | {
"line": 211,
"column": 15
} | {
"line": 211,
"column": 16
} | [
{
"pp": "case convert_1\nz : ℝ\nhre : (↑z).re < 0\n⊢ ‖↑z‖ ≠ 0",
"ppTerm": "?convert_1",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"R... | [
"case convert_1\nz : ℝ\nhre : (↑z).re < 0\n⊢ ¬z = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 13
} | {
"line": 222,
"column": 14
} | [
{
"pp": "case convert_1\nz : ℝ\nhre : (↑z).re < 0\n⊢ ‖↑z‖ ≠ 0",
"ppTerm": "?convert_1",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"R... | [
"case convert_1\nz : ℝ\nhre : (↑z).re < 0\n⊢ ¬z = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 52
} | {
"line": 227,
"column": 4
} | [
{
"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))",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 304,
"column": 53
} | {
"line": 304,
"column": 74
} | {
"line": 304,
"column": 75
} | [
{
"pp": "x y : ℝ\nh₁ : -π < y\nh₂ : y < π\n⊢ y ≤ -π + 2 * π",
"ppTerm": "?m.96",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.pi",
"HMul.hMul",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"PartialOrder.toPreorder",
"Nat.instAtLeastTwoHA... | [
"x y : ℝ\nh₁ : -π < y\nh₂ : y < π\n⊢ y ≤ π"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 347,
"column": 7
} | {
"line": 347,
"column": 18
} | {
"line": 347,
"column": 19
} | [
{
"pp": "x✝ : ℂ\n⊢ x✝ ∈ (fun θ ↦ cexp (↑θ * I)) '' Ioc (-π) π ↔ x✝ ∈ sphere 0 1",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Set.Ioc",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Preorder.toLT",
"Real.pi",
"HMul... | [
"x✝ : ℂ\n⊢ (∃ x, (-π < x ∧ x ≤ π) ∧ cexp (↑x * I) = x✝) ↔ ‖x✝‖ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 448,
"column": 2
} | {
"line": 449,
"column": 47
} | {
"line": 449,
"column": 48
} | [
{
"pp": "case h.h\nx✝¹ x✝ : ℝ\nh : (↑x✝¹).toReal = (↑x✝).toReal\n⊢ ↑x✝¹ = ↑x✝",
"ppTerm": "?h.h",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case h.h\nx✝¹ x✝ : ℝ\nh : (↑x✝¹).toReal = (↑x✝).toReal\n⊢ ↑x✝¹ = ↑x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 62,
"column": 65
} | {
"line": 62,
"column": 76
} | {
"line": 62,
"column": 77
} | [
{
"pp": "x : ℝ\nn : ℕ\n⊢ x ^ ↑n = x ^ n",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nn : ℕ\n⊢ x ^ ↑n = x ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 167,
"column": 80
} | {
"line": 169,
"column": 63
} | {
"line": 171,
"column": 0
} | [
{
"pp": "x y : ℝ\nhx_nonneg : 0 ≤ x\n⊢ |x ^ y| = |x| ^ y",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"Real.lattice",
"Real.instZero",
"abs",
"congrArg",
"PartialOrder... | [] | 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.Trigonometric.Angle | {
"line": 801,
"column": 42
} | {
"line": 801,
"column": 57
} | {
"line": 801,
"column": 58
} | [
{
"pp": "θ ψ : Angle\nh : θ.sign = ψ.sign\n⊢ θ = ψ ↔ |θ.toReal| = |ψ.toReal|",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"θ ψ : Angle\nh : θ.sign = ψ.sign\n⊢ θ = ψ ↔ |θ.toReal| = |ψ.toReal|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 251,
"column": 8
} | {
"line": 251,
"column": 23
} | {
"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",
"ppTerm": "?cons",
"assigned": true,
"usedConst... | [
"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 : 0 ≤ f i ∧ ∀ x ∈ s, 0 ≤ f x\n⊢ a ^ ∑ x ∈ cons i s hi, f x = ∏ x ∈ cons i s hi, a ^ f x"
] | forall_mem_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 523,
"column": 31
} | {
"line": 523,
"column": 58
} | {
"line": 523,
"column": 58
} | [
{
"pp": "z : ℂ\nθ : Real.Angle\n⊢ (↑z.arg).toReal = θ.toReal ↔ z.arg = θ.toReal",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.Angle.coe",
"congrArg",
"Complex.arg",
"id",
"Iff",
"Complex.arg_coe_angle_toReal_eq_arg",
... | [
"z : ℂ\nθ : Real.Angle\n⊢ z.arg = θ.toReal ↔ z.arg = θ.toReal"
] | arg_coe_angle_toReal_eq_arg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 527,
"column": 36
} | {
"line": 527,
"column": 63
} | {
"line": 527,
"column": 63
} | [
{
"pp": "x y : ℂ\n⊢ (↑x.arg).toReal = (↑y.arg).toReal ↔ x.arg = y.arg",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Real",
"Real.Angle.coe",
"congrArg",
"Complex.arg",
"iff_self",
"Iff",
"congr",
"True",
"Complex.arg_coe_angle_toReal_... | [] | arg_coe_angle_toReal_eq_arg | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 255,
"column": 63
} | {
"line": 256,
"column": 71
} | {
"line": 258,
"column": 0
} | [
{
"pp": "x y : ℝ\n⊢ x ^ (-y) = x⁻¹ ^ y",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"RingHom.instRingHomClass",
"Real.instPow",
"Real",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"Real.pi",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
... | [] | by
simp [rpow_def, Complex.cpow_neg, Complex.inv_cpow_eq_ite, apply_ite] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 835,
"column": 2
} | {
"line": 836,
"column": 9
} | {
"line": 836,
"column": 10
} | [
{
"pp": "θ : Angle\n⊢ (2 • θ).sign = -θ.sign ↔ θ = 0 ∨ π / 2 < |θ.toReal|",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle.0.Real.Angle.sign_two_nsmul_eq_neg_sign_iff._simp_1_2",
"Real",
"instH... | [
"θ : Angle\n⊢ (2 • θ).sign = -θ.sign ↔ θ = 0 ∨ θ.cos < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 614,
"column": 4
} | {
"line": 614,
"column": 15
} | {
"line": 614,
"column": 16
} | [
{
"pp": "case convert_3\nz : ℝ\nhre : (↑z).re < 0\n⊢ ‖↑z‖ ≠ 0",
"ppTerm": "?convert_3",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NormedCommRing.toSem... | [
"case convert_3\nz : ℝ\nhre : (↑z).re < 0\n⊢ ¬z = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 338,
"column": 2
} | {
"line": 339,
"column": 67
} | {
"line": 341,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : 0 < x\ny : ℂ\n⊢ ‖↑x ^ y‖ = x ^ y.re",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"InvOneClass.toOne",
"HMul.hMul",
"DivisionCommMonoid.toDivis... | [] | 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
} | {
"line": 341,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : 0 < x\ny : ℂ\n⊢ ‖↑x ^ y‖ = x ^ y.re",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"InvOneClass.toOne",
"HMul.hMul",
"DivisionCommMonoid.toDivis... | [] | 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
} | {
"line": 341,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : 0 < x\ny : ℂ\n⊢ ‖↑x ^ y‖ = x ^ y.re",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"InvOneClass.toOne",
"HMul.hMul",
"DivisionCommMonoid.toDivis... | [] | 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.Trigonometric.Angle | {
"line": 898,
"column": 79
} | {
"line": 898,
"column": 90
} | {
"line": 898,
"column": 91
} | [
{
"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",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 630,
"column": 4
} | {
"line": 630,
"column": 15
} | {
"line": 630,
"column": 16
} | [
{
"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",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 635,
"column": 2
} | {
"line": 635,
"column": 47
} | {
"line": 636,
"column": 4
} | [
{
"pp": "z : ℂ\nhre : z.re < 0\nhim : z.im = 0\n⊢ Tendsto arg (𝓝[{z | 0 ≤ z.im}] z) (𝓝 π)",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z : ℂ\nhre : z.re < 0\nhim : z.im = 0\n⊢ Tendsto arg (𝓝[{z | 0 ≤ z.im}] z) (𝓝 π)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 432,
"column": 2
} | {
"line": 432,
"column": 13
} | {
"line": 432,
"column": 14
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\ny : ℝ\nn : ℕ\n⊢ x ^ (y + ↑n) = x ^ y * x ^ n",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nhx : x ≠ 0\ny : ℝ\nn : ℕ\n⊢ x ^ (y + ↑n) = x ^ y * x ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 438,
"column": 2
} | {
"line": 438,
"column": 13
} | {
"line": 438,
"column": 14
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\ny : ℝ\nn : ℕ\n⊢ x ^ (y - ↑n) = x ^ y / x ^ n",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nhx : x ≠ 0\ny : ℝ\nn : ℕ\n⊢ x ^ (y - ↑n) = x ^ y / x ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 40,
"column": 2
} | {
"line": 40,
"column": 65
} | {
"line": 40,
"column": 66
} | [
{
"pp": "y : 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",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"Set.Ici",
"Preorder.toLE",
"Membership.mem",
"id",
... | [
"y : ℝ\nhy : 0 < y\nb : ℝ\nhb : 0 ≤ b\nx : ℝ\nhx₀ : 0 ≤ x\nhx : b ^ (1 / y) ≤ x\n⊢ b ≤ x ^ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 89
} | {
"line": 63,
"column": 6
} | [
{
"pp": "b : ℝ\nhb₀ : -1 < b\nhb₁ : b < 1\nhb : b < 0\n⊢ Tendsto (fun x ↦ rexp (log b * x)) atTop (𝓝 0)",
"ppTerm": "?m.112",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real",
"HMul.hMul",
"Real.instZero",
"Filter.tendsto_id",
"PartialOrder.toPreorder",
... | [
"b : ℝ\nhb₀ : -1 < b\nhb₁ : b < 1\nhb : b < 0\n⊢ log b < 0"
] | 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.Real | {
"line": 453,
"column": 2
} | {
"line": 453,
"column": 13
} | {
"line": 453,
"column": 14
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\ny : ℝ\n⊢ x ^ (y + 1) = x ^ y * x",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nhx : x ≠ 0\ny : ℝ\n⊢ x ^ (y + 1) = x ^ y * x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 13
} | {
"line": 456,
"column": 14
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\ny : ℝ\n⊢ x ^ (y - 1) = x ^ y / x",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nhx : x ≠ 0\ny : ℝ\n⊢ x ^ (y - 1) = x ^ y / x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 87
} | {
"line": 76,
"column": 4
} | [
{
"pp": "b : ℝ\nhb₀ : -1 < b\nhb₁ : b < 1\nhb : 0 < b\n⊢ Tendsto (fun x ↦ rexp (log b * x)) atTop (𝓝 0)",
"ppTerm": "?m.217",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real",
"HMul.hMul",
"Real.instZero",
"Filter.tendsto_id",
"PartialOrder.toPreorder",
... | [
"b : ℝ\nhb₀ : -1 < b\nhb₁ : b < 1\nhb : 0 < b\n⊢ log b < 0"
] | 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": 104,
"column": 14
} | {
"line": 104,
"column": 50
} | {
"line": 105,
"column": 16
} | [
{
"pp": "a b c : ℝ\nhb : 0 ≠ b\n⊢ Tendsto ?m.40 atTop (𝓝 0)",
"ppTerm": "?m.45",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b c : ℝ\nhb : 0 ≠ b\n⊢ Tendsto ?m.40 atTop (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 13
} | {
"line": 130,
"column": 14
} | [
{
"pp": "case hdb\ns : ℝ\nn : ℕ\nhn : s < ↑n\nx : ℝ\nhx₀ : 0 < x\nhx₁ : 1 ≤ x\n⊢ x ^ s ≤ x ^ n",
"ppTerm": "?hdb",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case hdb\ns : ℝ\nn : ℕ\nhn : s < ↑n\nx : ℝ\nhx₀ : 0 < x\nhx₁ : 1 ≤ x\n⊢ x ^ s ≤ x ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 539,
"column": 27
} | {
"line": 539,
"column": 47
} | {
"line": 539,
"column": 47
} | [
{
"pp": "x y z : ℝ\nhxy : x < y\nhz : 0 < z\nhx : 0 = x\n⊢ 0 < y",
"ppTerm": "?m.53",
"assigned": true,
"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.Asymptotics | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 90
} | {
"line": 220,
"column": 91
} | [
{
"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",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"Filter.Eventually",
"Comp... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 238,
"column": 27
} | {
"line": 238,
"column": 48
} | {
"line": 239,
"column": 4
} | [
{
"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",
"ppTerm": "?m.116",
"assigned": true,
"usedConstants": [
"Real",
"HMul.hMul",
"... | [] | by gcongr; assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 33
} | {
"line": 263,
"column": 34
} | [
{
"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)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
... | [
"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 ^ 2⁻¹) =O[l] fun x ↦ g x ^ 2⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 267,
"column": 2
} | {
"line": 267,
"column": 33
} | {
"line": 267,
"column": 34
} | [
{
"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)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
... | [
"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 ^ 2⁻¹) =o[l] fun x ↦ g x ^ 2⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 308,
"column": 4
} | {
"line": 309,
"column": 10
} | {
"line": 309,
"column": 11
} | [
{
"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}",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"Lattice.toSemilattice... | [
"a b : ℝ\nh : a ≤ b\nhimp : b = 0 → a = 0\nx : ℝ\nhx : x ∈ Set.Icc 0 1\n⊢ x ^ b ≤ x ^ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 329,
"column": 4
} | {
"line": 329,
"column": 55
} | {
"line": 330,
"column": 6
} | [
{
"pp": "s b : ℝ\nhb : 0 < b\n⊢ Tendsto (fun x ↦ x ^ s / rexp (b * x)) atTop (𝓝 0)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"cong... | [
"s b : ℝ\nhb : 0 < b\n⊢ Tendsto (fun x ↦ x ^ s * (rexp (b * x))⁻¹) atTop (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 335,
"column": 2
} | {
"line": 335,
"column": 38
} | {
"line": 335,
"column": 39
} | [
{
"pp": "k : ℤ\nb : ℝ\nhb : 0 < b\n⊢ (fun x ↦ x ^ k) =o[atTop] fun x ↦ rexp (b * x)",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k : ℤ\nb : ℝ\nhb : 0 < b\n⊢ (fun x ↦ x ^ k) =o[atTop] fun x ↦ rexp (b * x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 13
} | {
"line": 340,
"column": 14
} | [
{
"pp": "k : ℕ\nb : ℝ\nhb : 0 < b\n⊢ (fun x ↦ x ^ k) =o[atTop] fun x ↦ rexp (b * x)",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k : ℕ\nb : ℝ\nhb : 0 < b\n⊢ (fun x ↦ x ^ k) =o[atTop] fun x ↦ rexp (b * x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 28
} | {
"line": 344,
"column": 29
} | [
{
"pp": "s : ℝ\n⊢ (fun x ↦ x ^ s) =o[atTop] rexp",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"s : ℝ\n⊢ (fun x ↦ x ^ s) =o[atTop] rexp"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 44
} | {
"line": 78,
"column": 2
} | [
{
"pp": "p : ℂ × ℂ\nhp_fst : p.1 ∈ slitPlane\n⊢ ContinuousAt (fun x ↦ cexp (Complex.log x.1 * x.2)) p",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Complex.log",
"HMul.hMul",
"Complex.instNormedField",
"PseudoMetricSpa... | [
"p : ℂ × ℂ\nhp_fst : p.1 ∈ slitPlane\n⊢ ContinuousAt (fun x ↦ Complex.log x.1 * x.2) p"
] | refine continuous_exp.continuousAt.comp ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 736,
"column": 2
} | {
"line": 737,
"column": 9
} | {
"line": 737,
"column": 10
} | [
{
"pp": "x y : ℝ\nh₁ : 0 ≤ x\nh₂ : x ≤ 1\nh₃ : y ≤ 1\n⊢ x ≤ x ^ y",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℝ\nh₁ : 0 ≤ x\nh₂ : x ≤ 1\nh₃ : y ≤ 1\n⊢ x ≤ x ^ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 740,
"column": 2
} | {
"line": 740,
"column": 29
} | {
"line": 740,
"column": 30
} | [
{
"pp": "x y : ℝ\nh₁ : 1 ≤ x\nh₂ : 1 ≤ y\n⊢ x ≤ x ^ y",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℝ\nh₁ : 1 ≤ x\nh₂ : 1 ≤ y\n⊢ x ≤ x ^ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 743,
"column": 2
} | {
"line": 744,
"column": 9
} | {
"line": 744,
"column": 10
} | [
{
"pp": "x y : ℝ\nh₁ : 0 ≤ x\nh₂ : x ≤ 1\nh₃ : 1 ≤ y\n⊢ x ^ y ≤ x",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℝ\nh₁ : 0 ≤ x\nh₂ : x ≤ 1\nh₃ : 1 ≤ y\n⊢ x ^ y ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 747,
"column": 2
} | {
"line": 747,
"column": 29
} | {
"line": 747,
"column": 30
} | [
{
"pp": "x y : ℝ\nh₁ : 1 ≤ x\nh₂ : y ≤ 1\n⊢ x ^ y ≤ x",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℝ\nh₁ : 1 ≤ x\nh₂ : y ≤ 1\n⊢ x ^ y ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 750,
"column": 2
} | {
"line": 750,
"column": 29
} | {
"line": 750,
"column": 30
} | [
{
"pp": "x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : y < 1\n⊢ x < x ^ y",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : y < 1\n⊢ x < x ^ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 750,
"column": 2
} | {
"line": 750,
"column": 66
} | {
"line": 752,
"column": 0
} | [
{
"pp": "x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : y < 1\n⊢ x < x ^ y",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real",
"congrArg",
"Real.instLT",
"Eq.mp",
"Real.rpow_one",
"Real.instOne",
"HPow.hPow",
"LT.lt",
"... | [] | 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": 750,
"column": 2
} | {
"line": 750,
"column": 66
} | {
"line": 752,
"column": 0
} | [
{
"pp": "x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : y < 1\n⊢ x < x ^ y",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real",
"congrArg",
"Real.instLT",
"Eq.mp",
"Real.rpow_one",
"Real.instOne",
"HPow.hPow",
"LT.lt",
"... | [] | 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.NNReal | {
"line": 79,
"column": 18
} | {
"line": 79,
"column": 54
} | {
"line": 79,
"column": 55
} | [
{
"pp": "x : ℝ≥0\nn : ℕ\n⊢ ↑(x ^ ↑n) = ↑(x ^ n)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Real",
"id",
"NNReal",
"Nat.cast",
"NPow.toPow",
"HPow.hPow",
"NNReal.instPowReal",
"Nat",
"Semiring.toMonoid",
"NNReal.instSemiring"... | [
"x : ℝ≥0\nn : ℕ\n⊢ ↑x ^ ↑n = ↑x ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 750,
"column": 2
} | {
"line": 750,
"column": 66
} | {
"line": 752,
"column": 0
} | [
{
"pp": "x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : y < 1\n⊢ x < x ^ y",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real",
"congrArg",
"Real.instLT",
"Eq.mp",
"Real.rpow_one",
"Real.instOne",
"HPow.hPow",
"LT.lt",
"... | [] | 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": 753,
"column": 2
} | {
"line": 753,
"column": 29
} | {
"line": 753,
"column": 30
} | [
{
"pp": "x y : ℝ\nh₁ : 1 < x\nh₂ : 1 < y\n⊢ x < x ^ y",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℝ\nh₁ : 1 < x\nh₂ : 1 < y\n⊢ x < x ^ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 756,
"column": 2
} | {
"line": 756,
"column": 29
} | {
"line": 756,
"column": 30
} | [
{
"pp": "x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : 1 < y\n⊢ x ^ y < x",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : 1 < y\n⊢ x ^ y < x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 756,
"column": 2
} | {
"line": 756,
"column": 66
} | {
"line": 758,
"column": 0
} | [
{
"pp": "x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : 1 < y\n⊢ x ^ y < x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real",
"congrArg",
"Real.instLT",
"Eq.mp",
"Real.rpow_one",
"Real.instOne",
"HPow.hPow",
"LT.lt",
"... | [] | 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": 756,
"column": 2
} | {
"line": 756,
"column": 66
} | {
"line": 758,
"column": 0
} | [
{
"pp": "x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : 1 < y\n⊢ x ^ y < x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real",
"congrArg",
"Real.instLT",
"Eq.mp",
"Real.rpow_one",
"Real.instOne",
"HPow.hPow",
"LT.lt",
"... | [] | 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": 756,
"column": 2
} | {
"line": 756,
"column": 66
} | {
"line": 758,
"column": 0
} | [
{
"pp": "x y : ℝ\nh₁ : 0 < x\nh₂ : x < 1\nh₃ : 1 < y\n⊢ x ^ y < x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real",
"congrArg",
"Real.instLT",
"Eq.mp",
"Real.rpow_one",
"Real.instOne",
"HPow.hPow",
"LT.lt",
"... | [] | 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": 759,
"column": 2
} | {
"line": 759,
"column": 29
} | {
"line": 759,
"column": 30
} | [
{
"pp": "x y : ℝ\nh₁ : 1 < x\nh₂ : y < 1\n⊢ x ^ y < x",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℝ\nh₁ : 1 < x\nh₂ : y < 1\n⊢ x ^ y < x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 13
} | {
"line": 121,
"column": 14
} | [
{
"pp": "x : ℝ≥0\nhx : x ≠ 0\ny : ℝ\n⊢ x ^ (y + 1) = x ^ y * x",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ≥0\nhx : x ≠ 0\ny : ℝ\n⊢ x ^ (y + 1) = x ^ y * x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 13
} | {
"line": 124,
"column": 14
} | [
{
"pp": "x : ℝ≥0\nhx : x ≠ 0\ny : ℝ\n⊢ x ^ (y - 1) = x ^ y / x",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ≥0\nhx : x ≠ 0\ny : ℝ\n⊢ x ^ (y - 1) = x ^ y / x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 208,
"column": 2
} | {
"line": 209,
"column": 41
} | {
"line": 209,
"column": 42
} | [
{
"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)",
"ppTerm": "?inr",
"assigned": tru... | [
"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⊢ Tendsto (fun p ↦ p.1 ^ p.2) (𝓝 0 ×ˢ 𝓝 y) (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 13
} | {
"line": 246,
"column": 14
} | [
{
"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",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"List.map",
"Membership.mem",
"Exists",
... | [
"ι : Type u_1\nl : List ι\nf : ι → ℝ\nhl : ∀ i ∈ l, 0 ≤ f i\nr : ℝ\n⊢ ∀ a ∈ l, 0 ≤ f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 13
} | {
"line": 253,
"column": 14
} | [
{
"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",
"ppTerm": "?mk",
"assigned": true... | [
"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⊢ (List.map (fun x ↦ f x ^ r) l).prod = (List.map f l).prod ^ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 278,
"column": 29
} | {
"line": 278,
"column": 39
} | {
"line": 278,
"column": 40
} | [
{
"pp": "x y : ℝ≥0\nz : ℝ\nhz : 0 < z\n⊢ x ^ z ≤ (y ^ z⁻¹) ^ z ↔ x ^ z ≤ y",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"Real.instInv",... | [
"x y : ℝ≥0\nz : ℝ\nhz : 0 < z\n⊢ x ^ z ≤ (y ^ (1 / z)) ^ z ↔ x ^ z ≤ y"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 281,
"column": 29
} | {
"line": 281,
"column": 39
} | {
"line": 281,
"column": 40
} | [
{
"pp": "x y : ℝ≥0\nz : ℝ\nhz : 0 < z\n⊢ (x ^ z⁻¹) ^ z ≤ y ^ z ↔ x ≤ y ^ z",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"Real.instInv",... | [
"x y : ℝ≥0\nz : ℝ\nhz : 0 < z\n⊢ (x ^ (1 / z)) ^ z ≤ y ^ z ↔ x ≤ y ^ z"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 391,
"column": 20
} | {
"line": 391,
"column": 60
} | {
"line": 391,
"column": 61
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\ny z : ℝ≥0\nhyz : (fun y ↦ y ^ x) y = (fun y ↦ y ^ x) z\n⊢ y = z",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nhx : x ≠ 0\ny z : ℝ≥0\nhyz : (fun y ↦ y ^ x) y = (fun y ↦ y ^ x) z\n⊢ y = z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 990,
"column": 37
} | {
"line": 990,
"column": 55
} | {
"line": 990,
"column": 56
} | [
{
"pp": "case inr\nx : ℝ\nh : x < 0\nthis : 1 / 2 * π = π / 2\n⊢ 0 = x ^ (1 / 2)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.instZero",
"Real.cos",
"congrArg... | [
"case inr\nx : ℝ\nh : x < 0\nthis : 1 / 2 * π = π / 2\n⊢ 0 = rexp (log x * (1 / 2)) * cos (1 / 2 * π)"
] | rpow_def_of_neg h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 415,
"column": 2
} | {
"line": 415,
"column": 24
} | {
"line": 415,
"column": 25
} | [
{
"pp": "case inr.inr\nx : ℝ≥0\ny z : ℝ\nhx₀ : x ≠ 0\nhx₁ : x ≠ 1\n⊢ x ^ y = x ^ z ↔ y = z ∨ x = 1 ∨ x = 0 ∧ (y = 0 ↔ z = 0)",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Real",
"eq_false",
"Real.instZero",
"congrArg",
"fal... | [
"case inr.inr\nx : ℝ≥0\ny z : ℝ\nhx₀ : x ≠ 0\nhx₁ : x ≠ 1\n⊢ x ^ y = x ^ z ↔ y = z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 419,
"column": 2
} | {
"line": 419,
"column": 28
} | {
"line": 419,
"column": 29
} | [
{
"pp": "x : ℝ≥0\ny : ℝ\n⊢ x ^ y = x ↔ x = 1 ∨ y = 1 ∨ x = 0 ∧ y ≠ 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"id",
"NNReal",
"Ne",
"NNReal.instZero",
"Real.instOne",
"And",
"Iff",
"HPow.hPow",
... | [
"x : ℝ≥0\ny : ℝ\n⊢ x ^ y = x ↔ x = 1 ∨ y = 1 ∨ x = 0 ∧ ¬y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 1004,
"column": 70
} | {
"line": 1004,
"column": 80
} | {
"line": 1005,
"column": 4
} | [
{
"pp": "x : ℂ\n⊢ ‖x‖ ^ 2⁻¹ * Real.cos (x.arg / 2) = √((‖x‖ + x.re) / 2)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"MulOne.toOne",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Real... | [
"x : ℂ\n⊢ ‖x‖ ^ (1 / 2) * Real.cos (x.arg / 2) = √((‖x‖ + x.re) / 2)"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 422,
"column": 29
} | {
"line": 422,
"column": 39
} | {
"line": 422,
"column": 40
} | [
{
"pp": "x y : ℝ≥0\nz : ℝ\nhz : z ≠ 0\n⊢ x ^ z = (y ^ z⁻¹) ^ z ↔ x ^ z = y",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"Real.instInv",... | [
"x y : ℝ≥0\nz : ℝ\nhz : z ≠ 0\n⊢ x ^ z = (y ^ (1 / z)) ^ z ↔ x ^ z = y"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 425,
"column": 29
} | {
"line": 425,
"column": 39
} | {
"line": 425,
"column": 40
} | [
{
"pp": "x y : ℝ≥0\nz : ℝ\nhz : z ≠ 0\n⊢ (x ^ z⁻¹) ^ z = y ^ z ↔ x = y ^ z",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"Real.instInv",... | [
"x y : ℝ≥0\nz : ℝ\nhz : z ≠ 0\n⊢ (x ^ (1 / z)) ^ z = y ^ z ↔ x = y ^ z"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 1010,
"column": 70
} | {
"line": 1010,
"column": 80
} | {
"line": 1011,
"column": 4
} | [
{
"pp": "x : ℂ\nhx : 0 ≤ x.im\n⊢ ‖x‖ ^ 2⁻¹ * Real.sin (x.arg / 2) = √((‖x‖ - x.re) / 2)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"MulOne.toOne",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMu... | [
"x : ℂ\nhx : 0 ≤ x.im\n⊢ ‖x‖ ^ (1 / 2) * Real.sin (x.arg / 2) = √((‖x‖ - x.re) / 2)"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 393,
"column": 4
} | {
"line": 393,
"column": 15
} | {
"line": 393,
"column": 16
} | [
{
"pp": "case refine_1\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",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",... | [
"case refine_1\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 = 0 ∨ 0 < y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 402,
"column": 2
} | {
"line": 403,
"column": 30
} | {
"line": 403,
"column": 31
} | [
{
"pp": "x y : ℝ≥0\nhy : 1 < y\nm : ℕ\nhm : x < y ^ m\nn : ℕ\nhn : m + 1 ≤ n\n⊢ x ^ (↑n)⁻¹ ≤ y",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
... | [
"x y : ℝ≥0\nhy : 1 < y\nm : ℕ\nhm : x < y ^ m\nn : ℕ\nhn : m + 1 ≤ n\n⊢ x ≤ y ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 1018,
"column": 70
} | {
"line": 1018,
"column": 80
} | {
"line": 1019,
"column": 4
} | [
{
"pp": "x : ℂ\nhx : x.im < 0\n⊢ ‖x‖ ^ 2⁻¹ * Real.sin (x.arg / 2) = -√((‖x‖ - x.re) / 2)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"MulOne.toOne",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hM... | [
"x : ℂ\nhx : x.im < 0\n⊢ ‖x‖ ^ (1 / 2) * Real.sin (x.arg / 2) = -√((‖x‖ - x.re) / 2)"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 1025,
"column": 70
} | {
"line": 1025,
"column": 80
} | {
"line": 1026,
"column": 4
} | [
{
"pp": "x : ℂ\n⊢ |‖x‖ ^ 2⁻¹ * Real.sin (x.arg / 2)| = √((‖x‖ - x.re) / 2)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"MulOne.toOne",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Re... | [
"x : ℂ\n⊢ |‖x‖ ^ (1 / 2) * Real.sin (x.arg / 2)| = √((‖x‖ - x.re) / 2)"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 459,
"column": 4
} | {
"line": 461,
"column": 12
} | {
"line": 462,
"column": 2
} | [
{
"pp": "case pos\nx : ℝ≥0∞\ny : ℝ\nh : 0 < y\nhx : x = ∞\n⊢ ContinuousAt (fun a ↦ a ^ y) x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"ContinuousAt",
"ENNReal.instPowReal",
"HEq.refl",
"nhds",
"ENNReal.tend... | [] | 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
} | {
"line": 462,
"column": 2
} | [
{
"pp": "case pos\nx : ℝ≥0∞\ny : ℝ\nh : 0 < y\nhx : x = ∞\n⊢ ContinuousAt (fun a ↦ a ^ y) x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"ContinuousAt",
"ENNReal.instPowReal",
"HEq.refl",
"nhds",
"ENNReal.tend... | [] | 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
} | {
"line": 476,
"column": 46
} | [
{
"pp": "y : ℝ\nx : ℝ≥0∞\nhy : y < 0\nz : ℝ\nhz : y = -z\n⊢ 0 < z",
"ppTerm": "?m.94",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"y : ℝ\nx : ℝ≥0∞\nhy : y < 0\nz : ℝ\nhz : y = -z\n⊢ 0 < z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Midpoint | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 40
} | {
"line": 131,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁴ : Ring R\ninst✝³ : Invertible 2\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\np₁ p₂ : P\n⊢ p₂ -ᵥ midpoint R p₁ p₂ = ⅟2 • (p₂ -ᵥ p₁)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instH... | [] | rw [midpoint_comm, left_vsub_midpoint] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.Midpoint | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 40
} | {
"line": 131,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁴ : Ring R\ninst✝³ : Invertible 2\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\np₁ p₂ : P\n⊢ p₂ -ᵥ midpoint R p₁ p₂ = ⅟2 • (p₂ -ᵥ p₁)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instH... | [] | rw [midpoint_comm, left_vsub_midpoint] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Midpoint | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 40
} | {
"line": 131,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁴ : Ring R\ninst✝³ : Invertible 2\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\np₁ p₂ : P\n⊢ p₂ -ᵥ midpoint R p₁ p₂ = ⅟2 • (p₂ -ᵥ p₁)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instH... | [] | rw [midpoint_comm, left_vsub_midpoint] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Midpoint | {
"line": 211,
"column": 64
} | {
"line": 211,
"column": 75
} | {
"line": 211,
"column": 76
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : Invertible 2\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nx : V\n⊢ midpoint R (-x) x = 0",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : Invertible 2\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nx : V\n⊢ midpoint R (-x) x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Algebra | {
"line": 44,
"column": 24
} | {
"line": 44,
"column": 54
} | {
"line": 44,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁶ : CommSemiring α\ninst✝⁵ : PartialOrder α\ninst✝⁴ : Semiring β\ninst✝³ : PartialOrder β\ninst✝² : Algebra α β\ninst✝¹ : PosMulMono β\ninst✝ : MulPosMono β\nh : Monotone ⇑(algebraMap α β)\n⊢ Monotone fun x ↦ x • 1",
"ppTerm": "?m.33",
"assigned": true,
"use... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁶ : CommSemiring α\ninst✝⁵ : PartialOrder α\ninst✝⁴ : Semiring β\ninst✝³ : PartialOrder β\ninst✝² : Algebra α β\ninst✝¹ : PosMulMono β\ninst✝ : MulPosMono β\nh : Monotone ⇑(algebraMap α β)\n⊢ Monotone fun x ↦ (algebraMap α β) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Algebra | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 32
} | {
"line": 54,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁶ : CommSemiring α\ninst✝⁵ : PartialOrder α\ninst✝⁴ : Semiring β\ninst✝³ : PartialOrder β\ninst✝² : Algebra α β\ninst✝¹ : ZeroLEOneClass β\ninst✝ : SMulPosMono α β\n⊢ Monotone ⇑(algebraMap α β)",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁶ : CommSemiring α\ninst✝⁵ : PartialOrder α\ninst✝⁴ : Semiring β\ninst✝³ : PartialOrder β\ninst✝² : Algebra α β\ninst✝¹ : ZeroLEOneClass β\ninst✝ : SMulPosMono α β\n⊢ Monotone ⇑(algebraMap α β)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Algebra | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 13
} | {
"line": 57,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁶ : CommSemiring α\ninst✝⁵ : PartialOrder α\ninst✝⁴ : Semiring β\ninst✝³ : PartialOrder β\ninst✝² : Algebra α β\ninst✝¹ : ZeroLEOneClass β\ninst✝ : SMulPosMono α β\na : α\nha : 0 ≤ a\n⊢ 0 ≤ (algebraMap α β) a",
"ppTerm": "?m.22",
"assigned": false,
"usedCons... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁶ : CommSemiring α\ninst✝⁵ : PartialOrder α\ninst✝⁴ : Semiring β\ninst✝³ : PartialOrder β\ninst✝² : Algebra α β\ninst✝¹ : ZeroLEOneClass β\ninst✝ : SMulPosMono α β\na : α\nha : 0 ≤ a\n⊢ 0 ≤ (algebraMap α β) a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Algebra | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 32
} | {
"line": 78,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁷ : CommSemiring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : Semiring β\ninst✝⁴ : PartialOrder β\ninst✝³ : Algebra α β\ninst✝² : ZeroLEOneClass β\ninst✝¹ : Nontrivial β\ninst✝ : SMulPosStrictMono α β\n⊢ StrictMono ⇑(algebraMap α β)",
"ppTerm": "?m.18",
"assigned": fals... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁷ : CommSemiring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : Semiring β\ninst✝⁴ : PartialOrder β\ninst✝³ : Algebra α β\ninst✝² : ZeroLEOneClass β\ninst✝¹ : Nontrivial β\ninst✝ : SMulPosStrictMono α β\n⊢ StrictMono ⇑(algebraMap α β)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Algebra | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 13
} | {
"line": 81,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁷ : CommSemiring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : Semiring β\ninst✝⁴ : PartialOrder β\ninst✝³ : Algebra α β\ninst✝² : ZeroLEOneClass β\ninst✝¹ : Nontrivial β\ninst✝ : SMulPosStrictMono α β\na : α\nha : 0 < a\n⊢ 0 < (algebraMap α β) a",
"ppTerm": "?m.22",
"as... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁷ : CommSemiring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : Semiring β\ninst✝⁴ : PartialOrder β\ninst✝³ : Algebra α β\ninst✝² : ZeroLEOneClass β\ninst✝¹ : Nontrivial β\ninst✝ : SMulPosStrictMono α β\na : α\nha : 0 < a\n⊢ 0 < (algebraMap α β) a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 741,
"column": 2
} | {
"line": 741,
"column": 35
} | {
"line": 743,
"column": 0
} | [
{
"pp": "x y : ℝ≥0∞\nz : ℝ\nhz : 0 ≤ z\n⊢ (x * y) ^ z = x ^ z * y ^ z",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"False",
"Real",
"Preorder.toLT",
"HMul.hMul",
"eq_false",
"LinearOrder.toDecidableEq",
"Real.instZero",
"congrArg",
"... | [] | simp [hz.not_gt, mul_rpow_eq_ite] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 741,
"column": 2
} | {
"line": 741,
"column": 35
} | {
"line": 743,
"column": 0
} | [
{
"pp": "x y : ℝ≥0∞\nz : ℝ\nhz : 0 ≤ z\n⊢ (x * y) ^ z = x ^ z * y ^ z",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"False",
"Real",
"Preorder.toLT",
"HMul.hMul",
"eq_false",
"LinearOrder.toDecidableEq",
"Real.instZero",
"congrArg",
"... | [] | simp [hz.not_gt, mul_rpow_eq_ite] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 741,
"column": 2
} | {
"line": 741,
"column": 35
} | {
"line": 743,
"column": 0
} | [
{
"pp": "x y : ℝ≥0∞\nz : ℝ\nhz : 0 ≤ z\n⊢ (x * y) ^ z = x ^ z * y ^ z",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"False",
"Real",
"Preorder.toLT",
"HMul.hMul",
"eq_false",
"LinearOrder.toDecidableEq",
"Real.instZero",
"congrArg",
"... | [] | simp [hz.not_gt, mul_rpow_eq_ite] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Ray | {
"line": 371,
"column": 4
} | {
"line": 371,
"column": 23
} | {
"line": 371,
"column": 24
} | [
{
"pp": "case inr.inl\nR : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsStrictOrderedRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nx : M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nr : R\nhr : r < 0\nh₀ : r • x = 0\n⊢ x = 0",
"ppTerm": "?inr.inl",
"assign... | [
"case inr.inl\nR : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsStrictOrderedRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nx : M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nr : R\nhr : r < 0\nh₀ : r • x = 0\n⊢ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Segment | {
"line": 122,
"column": 6
} | {
"line": 122,
"column": 89
} | {
"line": 122,
"column": 90
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Semiring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : ZeroLEOneClass 𝕜\ninst✝ : Module 𝕜 E\nx z : E\nx✝ : z ∈ [x -[𝕜] x]\na b : 𝕜\nleft✝¹ : 0 ≤ a\nleft✝ : 0 ≤ b\nhab : a + b = 1\nhz : a • x + b • x = z\n⊢ z ∈ {x}",
"ppTerm": "?m.40",
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Semiring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : ZeroLEOneClass 𝕜\ninst✝ : Module 𝕜 E\nx z : E\nx✝ : z ∈ [x -[𝕜] x]\na b : 𝕜\nleft✝¹ : 0 ≤ a\nleft✝ : 0 ≤ b\nhab : a + b = 1\nhz : a • x + b • x = z\n⊢ x = z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Segment | {
"line": 155,
"column": 27
} | {
"line": 155,
"column": 49
} | {
"line": 155,
"column": 50
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝¹⁰ : Semiring 𝕜\ninst✝⁹ : PartialOrder 𝕜\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : ZeroLEOneClass 𝕜\ninst✝⁶ : Module 𝕜 E\nR : Type u_7\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\ninst✝³ : Module R E\ninst✝² : Module R 𝕜\ninst✝¹ : IsScalarTower R 𝕜 E\ninst✝ : SMulPos... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝¹⁰ : Semiring 𝕜\ninst✝⁹ : PartialOrder 𝕜\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : ZeroLEOneClass 𝕜\ninst✝⁶ : Module 𝕜 E\nR : Type u_7\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\ninst✝³ : Module R E\ninst✝² : Module R 𝕜\ninst✝¹ : IsScalarTower R 𝕜 E\ninst✝ : SMulPosMono R 𝕜\nx... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Segment | {
"line": 161,
"column": 27
} | {
"line": 161,
"column": 49
} | {
"line": 161,
"column": 50
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝¹¹ : Semiring 𝕜\ninst✝¹⁰ : PartialOrder 𝕜\ninst✝⁹ : AddCommMonoid E\ninst✝⁸ : ZeroLEOneClass 𝕜\ninst✝⁷ : Module 𝕜 E\nR : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : Module R E\ninst✝³ : Module R 𝕜\ninst✝² : IsScalarTower R 𝕜 E\ninst✝¹ : Nontr... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝¹¹ : Semiring 𝕜\ninst✝¹⁰ : PartialOrder 𝕜\ninst✝⁹ : AddCommMonoid E\ninst✝⁸ : ZeroLEOneClass 𝕜\ninst✝⁷ : Module 𝕜 E\nR : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : Module R E\ninst✝³ : Module R 𝕜\ninst✝² : IsScalarTower R 𝕜 E\ninst✝¹ : Nontrivial 𝕜\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Segment | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 78
} | {
"line": 185,
"column": 79
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : Ring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : AddRightMono 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : ZeroLEOneClass 𝕜\ninst✝¹ : Nontrivial 𝕜\ninst✝ : DenselyOrdered 𝕜\nx z : E\nx✝ : z ∈ openSegment 𝕜 x x\na b : 𝕜\nleft✝¹ : 0 < a\nleft✝ : 0 < b\... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : Ring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : AddRightMono 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : ZeroLEOneClass 𝕜\ninst✝¹ : Nontrivial 𝕜\ninst✝ : DenselyOrdered 𝕜\nx z : E\nx✝ : z ∈ openSegment 𝕜 x x\na b : 𝕜\nleft✝¹ : 0 < a\nleft✝ : 0 < b\nhab : a + b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Ray | {
"line": 535,
"column": 25
} | {
"line": 535,
"column": 41
} | {
"line": 535,
"column": 42
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsTorsionFree R M\nx y : M\nhx : x = 0\n⊢ SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y]",
"ppTerm": "?pos✝",
... | [
"case pos\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsTorsionFree R M\nx y : M\nhx : x = 0\n⊢ ¬LinearIndependent R ![0, y]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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