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.Calculus.Deriv.Mul | {
"line": 426,
"column": 2
} | {
"line": 426,
"column": 77
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\nι : Type u_2\ninst✝² : DecidableEq ι\n𝔸' : Type u_3\ninst✝¹ : NormedCommRing 𝔸'\ninst✝ : NormedAlgebra 𝕜 𝔸'\nu : Finset ι\nf : ι → 𝕜 → 𝔸'\nf' : ι → 𝔸'\nhf : ∀ i ∈ u, HasDerivWithinAt (f i) (f' i) s x\n⊢ HasDerivWithinAt (fun x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 441,
"column": 2
} | {
"line": 441,
"column": 77
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nx : 𝕜\nι : Type u_2\ninst✝² : DecidableEq ι\n𝔸' : Type u_3\ninst✝¹ : NormedCommRing 𝔸'\ninst✝ : NormedAlgebra 𝕜 𝔸'\nu : Finset ι\nf : ι → 𝕜 → 𝔸'\nf' : ι → 𝔸'\nhf : ∀ i ∈ u, HasStrictDerivAt (f i) (f' i) x\n⊢ HasStrictDerivAt (fun x ↦ ∏ i ∈ u, f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 563,
"column": 2
} | {
"line": 563,
"column": 35
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝕜' : Type u_2\ninst✝¹ : NormedDivisionRing 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nc : 𝕜 → 𝕜'\nc' : 𝕜'\nhc : HasDerivAt c c' x\nd : 𝕜'\n⊢ HasDerivAt (fun x ↦ c x / d) (c' / d) x",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 567,
"column": 2
} | {
"line": 567,
"column": 35
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\n𝕜' : Type u_2\ninst✝¹ : NormedDivisionRing 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nc : 𝕜 → 𝕜'\nc' : 𝕜'\nhc : HasDerivWithinAt c c' s x\nd : 𝕜'\n⊢ HasDerivWithinAt (fun x ↦ c x / d) (c' / d) s x",
"usedConstants": [
"HasDeri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 571,
"column": 2
} | {
"line": 571,
"column": 35
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝕜' : Type u_2\ninst✝¹ : NormedDivisionRing 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nc : 𝕜 → 𝕜'\nc' : 𝕜'\nhc : HasStrictDerivAt c c' x\nd : 𝕜'\n⊢ HasStrictDerivAt (fun x ↦ c x / d) (c' / d) x",
"usedConstants": [
"Eq.mpr",
"NormedC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 613,
"column": 2
} | {
"line": 613,
"column": 24
} | [
{
"pp": "𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nx : 𝕜\nG : Type u_2\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nc : 𝕜 → F →L[𝕜] G\nc' : F →L[𝕜]... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 24
} | [
{
"pp": "𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nx : 𝕜\ns : Set 𝕜\nG : Type u_2\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nc : 𝕜 → F →L[𝕜] G\nc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 639,
"column": 2
} | {
"line": 639,
"column": 24
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nx : 𝕜\nG : Type u_2\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nc : 𝕜 → F →L[𝕜] G\nc' : F →L[𝕜] G\nu : 𝕜 → F\nu' : F\nhc : HasStrictDerivAt c c' x\nhu : HasStrictDe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 644,
"column": 2
} | {
"line": 644,
"column": 24
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nx : 𝕜\ns : Set 𝕜\nG : Type u_2\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nc : 𝕜 → F →L[𝕜] G\nc' : F →L[𝕜] G\nu : 𝕜 → F\nu' : F\nhc : HasDerivWithinAt c c' s x\nhu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 648,
"column": 2
} | {
"line": 648,
"column": 24
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nx : 𝕜\nG : Type u_2\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nc : 𝕜 → F →L[𝕜] G\nc' : F →L[𝕜] G\nu : 𝕜 → F\nu' : F\nhc : HasDerivAt c c' x\nhu : HasDerivAt u u' x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Slope | {
"line": 199,
"column": 53
} | {
"line": 199,
"column": 64
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nx✝ : 𝕜\ns : Set 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\ninst✝ : OrderTopology 𝕜\ng : 𝕜 → 𝕜\ng' : 𝕜\nhx✝ : AccPt x✝ (𝓟 s)\nhd : HasDerivWithinAt g g' s x✝\nhg : AntitoneOn g s\nx : 𝕜\nhx : x ∈ s\ny : 𝕜\nhy : y ∈ s\nhxy : x ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Slope | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 13
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\ninst✝ : OrderTopology 𝕜\ng : 𝕜 → 𝕜\ng' : 𝕜\nhx : AccPt x (𝓟 s)\nhd : HasDerivWithinAt g g' s x\nhg : AntitoneOn g s\nthis : MonotoneOn (-g) s\n⊢ g' ≤ 0",
"usedConstan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Slope | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 35
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\ninst✝ : OrderTopology 𝕜\ng : 𝕜 → 𝕜\nhg : AntitoneOn g s\n⊢ derivWithin g s x ≤ 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.DSlope | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 41
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ng : 𝕜 → E\na b : 𝕜\nH : a = b → DifferentiableAt 𝕜 g a\nhne : b ≠ a\n⊢ dslope (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.DSlope | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 52
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\na b : 𝕜\ns : Set 𝕜\nh : ContinuousWithinAt (dslope f a) s b\nthis : ContinuousWithinAt (fun x ↦ (x - a) • dslope f a x + f a) s b\n⊢ ContinuousWithinAt f s b",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.DSlope | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 53
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\na b : 𝕜\ns : Set 𝕜\nh : b ≠ a\n⊢ ContinuousWithinAt (dslope f a) s b ↔ ContinuousWithinAt f s b",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
... | refine ⟨ContinuousWithinAt.of_dslope, fun hc => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.DSlope | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 60
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\na b : 𝕜\ns : Set 𝕜\nh : DifferentiableWithinAt 𝕜 (dslope f a) s b\n⊢ DifferentiableWithinAt 𝕜 f s b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 33
} | [
{
"pp": "a : ℝ\n⊢ interior {z | z.re ≤ a} = {z | z.re < a}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 33
} | [
{
"pp": "a : ℝ\n⊢ interior {z | z.im ≤ a} = {z | z.im < a}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 33
} | [
{
"pp": "a : ℝ\n⊢ interior {z | a ≤ z.re} = {z | a < z.re}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 33
} | [
{
"pp": "a : ℝ\n⊢ interior {z | a ≤ z.im} = {z | a < z.im}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 32
} | [
{
"pp": "a : ℝ\n⊢ closure {z | z.re < a} = {z | z.re ≤ a}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 32
} | [
{
"pp": "a : ℝ\n⊢ closure {z | z.im < a} = {z | z.im ≤ a}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 32
} | [
{
"pp": "a : ℝ\n⊢ closure {z | a < z.re} = {z | a ≤ z.re}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 32
} | [
{
"pp": "a : ℝ\n⊢ closure {z | a < z.im} = {z | a ≤ z.im}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 33
} | [
{
"pp": "a : ℝ\n⊢ frontier {z | z.re ≤ a} = {z | z.re = a}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 33
} | [
{
"pp": "a : ℝ\n⊢ frontier {z | z.im ≤ a} = {z | z.im = a}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 33
} | [
{
"pp": "a : ℝ\n⊢ frontier {z | a ≤ z.re} = {z | z.re = a}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 33
} | [
{
"pp": "a : ℝ\n⊢ frontier {z | a ≤ z.im} = {z | z.im = a}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 33
} | [
{
"pp": "a : ℝ\n⊢ frontier {z | z.re < a} = {z | z.re = a}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 33
} | [
{
"pp": "a : ℝ\n⊢ frontier {z | z.im < a} = {z | z.im = a}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 33
} | [
{
"pp": "a : ℝ\n⊢ frontier {z | a < z.re} = {z | z.re = a}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 33
} | [
{
"pp": "a : ℝ\n⊢ frontier {z | a < z.im} = {z | z.im = a}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 147,
"column": 2
} | {
"line": 148,
"column": 62
} | [
{
"pp": "s t : Set ℝ\n⊢ closure (s ×ℂ t) = closure s ×ℂ closure t",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"ContinuousLinearEquiv.symm",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"congrArg",
"Nor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 63
} | [
{
"pp": "s t : Set ℝ\n⊢ frontier (s ×ℂ t) = closure s ×ℂ frontier t ∪ frontier s ×ℂ closure t",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"frontier",
"ContinuousLinearEquiv.symm",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Se... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 46
} | [
{
"pp": "a b : ℝ\n⊢ frontier {z | a ≤ z.re ∧ b ≤ z.im} = {z | a ≤ z.re ∧ z.im = b ∨ z.re = a ∧ b ≤ z.im}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 73
} | [
{
"pp": "a b : ℝ\n⊢ frontier {z | a ≤ z.re ∧ z.im ≤ b} = {z | a ≤ z.re ∧ z.im = b ∨ z.re = a ∧ z.im ≤ b}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Mul | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nx : E\n𝕜' : Type u_5\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : NormedAlgebra 𝕜 𝕜'\ninst✝² : Module 𝕜' F\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Mul | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nx : E\ns : Set E\n𝕜' : Type u_5\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : NormedAlgebra 𝕜 𝕜'\ninst✝² : Module �... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Mul | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nx : E\n𝕜' : Type u_5\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : NormedAlgebra 𝕜 𝕜'\ninst✝² : Module 𝕜' F\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Mul | {
"line": 470,
"column": 27
} | {
"line": 470,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nι : Type u_5\n𝔸' : Type u_7\ninst✝³ : NormedCommRing 𝔸'\ninst✝² : NormedAlgebra 𝕜 𝔸'\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nu : Multiset ι\nx : ι → 𝔸'\nthis : Fintype ι\nl : List ι\n⊢ HasStrictFDerivAt (fun x ↦ (Multiset.map x ⟦l⟧).prod)\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.CircleMap | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 37
} | [
{
"pp": "c : ℂ\nR : ℝ\nhR : 0 ≤ R\nθ : ℝ\n⊢ circleMap c R θ ∈ sphere c R",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Mul | {
"line": 604,
"column": 2
} | {
"line": 604,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nι : Type u_5\n𝔸' : Type u_7\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸'\nu : Finset ι\ng : ι → E → 𝔸'\ng' : ι → E →L[𝕜] 𝔸'\ninst✝ : DecidableEq ι\nx : E\nhg : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Mul | {
"line": 614,
"column": 2
} | {
"line": 614,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nι : Type u_5\n𝔸' : Type u_7\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸'\nu : Finset ι\ng : ι → E → 𝔸'\ng' : ι → E →L[𝕜] 𝔸'\ninst✝ : DecidableEq ι\nx : E\nhg : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Mul | {
"line": 623,
"column": 2
} | {
"line": 623,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ns : Set E\nι : Type u_5\n𝔸' : Type u_7\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸'\nu : Finset ι\ng : ι → E → 𝔸'\ng' : ι → E →L[𝕜] 𝔸'\ninst✝ : DecidableEq ι\nx... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Mul | {
"line": 665,
"column": 2
} | {
"line": 665,
"column": 51
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nR : Type u_5\ninst✝² : NormedRing R\ninst✝¹ : HasSummableGeomSeries R\ninst✝ : NormedAlgebra 𝕜 R\nx : Rˣ\nthis : (fun t ↦ (↑x + t)⁻¹ʳ - ↑x⁻¹ + ↑x⁻¹ * t * ↑x⁻¹) =o[𝓝 0] id\n⊢ HasFDerivAt Ring.inverse (-((mulLeftRight 𝕜 R) ↑x⁻¹) ↑x⁻¹) ↑x",
"usedC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Mul | {
"line": 729,
"column": 2
} | {
"line": 729,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nR : Type u_5\ninst✝¹ : NormedDivisionRing R\ninst✝ : NormedAlgebra 𝕜 R\nx : R\nhx : x ≠ 0\n⊢ HasStrictFDerivAt Inv.inv (-((mulLeftRight 𝕜 R) x⁻¹) x⁻¹) x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Mul | {
"line": 737,
"column": 2
} | {
"line": 737,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nR : Type u_5\ninst✝¹ : NormedDivisionRing R\ninst✝ : NormedAlgebra 𝕜 R\nx : R\nhx : x ≠ 0\n⊢ HasFDerivAt Inv.inv (-((mulLeftRight 𝕜 R) x⁻¹) x⁻¹) x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Shift | {
"line": 29,
"column": 2
} | {
"line": 29,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : F\na x : 𝕜\nhf : HasDerivAt f f' (a + x)\n⊢ HasDerivAt (fun x ↦ f (a + x)) f' x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Shift | {
"line": 34,
"column": 2
} | {
"line": 34,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : F\nx a : 𝕜\nhf : HasDerivAt f f' (x + a)\n⊢ HasDerivAt (fun x ↦ f (x + a)) f' x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Shift | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : F\na x : 𝕜\nhf : HasDerivAt f f' (a - x)\n⊢ HasDerivAt (fun x ↦ f (a - x)) (-f') x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Shift | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : F\nx a : 𝕜\nhf : HasDerivAt f f' (x - a)\n⊢ HasDerivAt (fun x ↦ f (x - a)) f' x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Shift | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\n⊢ derivWithin (fun x ↦ f (-x)) s x = -derivWithin f (-s) (-x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Shift | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\n⊢ deriv (fun x ↦ f (-x)) x = -deriv f (-x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Shift | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 24
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\na x : 𝕜\n⊢ deriv (fun x ↦ f (x + a)) x = deriv f (x + a)",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrAr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.ZPow | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 30
} | [
{
"pp": "𝕜 : Type u\ninst✝ : NontriviallyNormedField 𝕜\nk : ℕ\nc d : 𝕜\n⊢ (deriv^[k] fun x ↦ (c * x - d)⁻¹) = fun x ↦ (-1) ^ k * ↑k ! * c ^ k * (c * x - d) ^ (-1 - ↑k)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 62,
"column": 34
} | {
"line": 62,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : E\nn : ℕ\nz : 𝕜\na : ℕ → E\nhs : HasSum (fun m ↦ z ^ m • a m) s\nha : ∀ k < n, a k = 0\nhn : n > 0\nh : z = 0\n⊢ HasSum (fun m ↦ z ^ m • a m) 0",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 63,
"column": 46
} | {
"line": 63,
"column": 61
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : E\nn : ℕ\nz : 𝕜\na : ℕ → E\nhs : HasSum (fun m ↦ z ^ m • a m) s\nha : ∀ k < n, a k = 0\nhn : n > 0\nh : z = 0\nthis : s = 0\n⊢ HasSum (fun m ↦ z ^ m • a (m + n)) (a n)",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : E\nn : ℕ\nz : 𝕜\na : ℕ → E\nhs : HasSum (fun m ↦ z ^ m • a m) s\nha : ∀ k < n, a k = 0\nhn : n > 0\nh : ¬z = 0\nh1 : ∑ i ∈ Finset.range n, z ^ i • a i = 0\n⊢ HasSum (fun m ↦ z... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Comp | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 13
} | [
{
"pp": "𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nL : Filter (𝕜 × 𝕜)\n𝕜' : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nh : 𝕜 → 𝕜'\nh'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 52
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : FormalMultilinearSeries 𝕜 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhpd : deriv f z₀ = p.coeff 1\nhp0 : p.coeff 0 = f z₀\nhp : ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n ↦ z ^ n • p.coeff n) (f (z₀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 21
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : FormalMultilinearSeries 𝕜 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhpd : deriv f z₀ = p.coeff 1\nhp0 : p.coeff 0 = f z₀\nhp : ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n ↦ z ^ n • p.coeff ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 95,
"column": 17
} | {
"line": 95,
"column": 28
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nz₀ : 𝕜\nn : ℕ\nih :\n ∀ {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E},\n HasFPowerSeriesAt f p z₀ → HasFPowerSeriesAt ((swap dslope z₀)^[n] f) (fslope^[n] p) z... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 29
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : FormalMultilinearSeries 𝕜 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhp : HasFPowerSeriesAt f p z₀\nh : p ≠ 0\n⊢ (fslope^[p.order] p 0) 1 ≠ 0",
"usedConstants": [
"NormedCommRing.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 58
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : FormalMultilinearSeries 𝕜 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhp : HasFPowerSeriesAt f p z₀\nh : p ≠ 0\nh2 : ContinuousAt ((swap dslope z₀)^[p.order] f) z₀\nh3 : ∀ᶠ (z : 𝕜) in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 127,
"column": 2
} | {
"line": 130,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f z₀\n⊢ (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
... | rcases hf with ⟨p, hp⟩
by_cases h : p = 0
· exact Or.inl (HasFPowerSeriesAt.eventually_eq_zero (by rwa [h] at hp))
· exact Or.inr (hp.locally_ne_zero h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 127,
"column": 2
} | {
"line": 130,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f z₀\n⊢ (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
... | rcases hf with ⟨p, hp⟩
by_cases h : p = 0
· exact Or.inl (HasFPowerSeriesAt.eventually_eq_zero (by rwa [h] at hp))
· exact Or.inr (hp.locally_ne_zero h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf g : 𝕜 → E\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f z₀\nhg : AnalyticAt 𝕜 g z₀\n⊢ (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = g z) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ g z",
"usedConstants": [
"No... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf g : 𝕜 → E\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f z₀\nhg : AnalyticAt 𝕜 g z₀\n⊢ (∃ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = g z) ↔ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = g z",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Comp | {
"line": 330,
"column": 17
} | {
"line": 330,
"column": 33
} | [
{
"pp": "𝕜 : Type u\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\nf : 𝕜 → 𝕜\nf' : 𝕜\nhf : HasDerivAt f f' x\nhx : f x = x\nn : ℕ\n⊢ Tendsto (Prod.map f f) (𝓝 x ×ˢ pure x) (𝓝 x ×ˢ pure x)",
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"SProd.sp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Comp | {
"line": 359,
"column": 2
} | {
"line": 359,
"column": 13
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → F\nf' : F\nx : 𝕜\ns : Set 𝕜\nl : F → E\nl' : F →L[𝕜] E\nt : Set F\nhl : HasFDerivWithinAt l l' t (f x)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Comp | {
"line": 371,
"column": 2
} | {
"line": 371,
"column": 13
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → F\nf' : F\nx : 𝕜\nl : F → E\nl' : F →L[𝕜] E\nt : Set F\nhl : HasFDerivWithinAt l l' t (f x)\nhf : HasDe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.AffineMap | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\na b : E\nx : 𝕜\n⊢ HasStrictDerivAt (⇑(lineMap a b)) (b - a) x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Comp | {
"line": 402,
"column": 2
} | {
"line": 402,
"column": 13
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → F\nf' : F\nx : 𝕜\nl : F → E\nl' : F →L[𝕜] E\nhl : HasStrictFDerivAt l l' (f x)\nhf : HasStrictDerivAt f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 192,
"column": 22
} | {
"line": 192,
"column": 89
} | [
{
"pp": "case mp.h\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f z₀\nn : ℕ\ng : 𝕜 → E\nhg_an : AnalyticAt 𝕜 g z₀\nhg_eq : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z\nhg_ne : ∀ᶠ (z : 𝕜) ... | ← AnalyticAt.frequently_eq_iff_eventually_eq hg_an analyticAt_const | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 189,
"column": 4
} | {
"line": 196,
"column": 44
} | [
{
"pp": "case mp\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f z₀\n⊢ (∃ n g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z) → ¬∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0",
... | rintro ⟨n, g, hg_an, hg_ne, hg_eq⟩
contrapose hg_ne
apply EventuallyEq.eq_of_nhds
rw [EventuallyEq, ← AnalyticAt.frequently_eq_iff_eventually_eq hg_an analyticAt_const]
refine (eventually_nhdsWithin_iff.mpr ?_).frequently
filter_upwards [hg_eq, hg_ne] with z hf_eq hf0 hz
rwa [hf0, eq_comm, smul_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 189,
"column": 4
} | {
"line": 196,
"column": 44
} | [
{
"pp": "case mp\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f z₀\n⊢ (∃ n g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z) → ¬∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0",
... | rintro ⟨n, g, hg_an, hg_ne, hg_eq⟩
contrapose hg_ne
apply EventuallyEq.eq_of_nhds
rw [EventuallyEq, ← AnalyticAt.frequently_eq_iff_eventually_eq hg_an analyticAt_const]
refine (eventually_nhdsWithin_iff.mpr ?_).frequently
filter_upwards [hg_eq, hg_ne] with z hf_eq hf0 hz
rwa [hf0, eq_comm, smul_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf g : 𝕜 → E\nz₀ : 𝕜\nU : Set 𝕜\nhf : AnalyticOnNhd 𝕜 f U\nhg : AnalyticOnNhd 𝕜 g U\nhU : IsPreconnected U\nh₀ : z₀ ∈ U\nhfg : ∃ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = g z\nhfg' : ∃ᶠ (z ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 286,
"column": 4
} | {
"line": 286,
"column": 22
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nU : Set 𝕜\nA : Type u_3\ninst✝⁶ : NormedRing A\ninst✝⁵ : IsDomain A\ninst✝⁴ : NormedAlgebra 𝕜 A\nB : Type u_4\ninst✝³ : NormedAddCommGroup B\ninst✝² : NormedSpace 𝕜 B\ninst✝¹ : Module A B\ninst✝ : IsTorsionFree A B\nf : 𝕜 → A\ng : 𝕜 → B... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LogDeriv | {
"line": 83,
"column": 6
} | {
"line": 83,
"column": 43
} | [
{
"pp": "case cons.hg\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NontriviallyNormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nι : Type u_3\nf : ι → 𝕜 → 𝕜'\nx : 𝕜\na : ι\ns : Finset ι\nha : a ∉ s\nih :\n (∀ i ∈ s, f i x ≠ 0) →\n (∀ i ∈ s, DifferentiableAt 𝕜 (f i) x) → log... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LogDeriv | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\n⊢ logDeriv (fun x ↦ x⁻¹) x = -1 / x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LogDeriv | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 15
} | [
{
"pp": "case inl\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NontriviallyNormedField 𝕜'\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : IsRCLikeNormedField 𝕜\nf g : 𝕜 → 𝕜'\nhf : DifferentiableOn 𝕜 f ∅\nhg : DifferentiableOn 𝕜 g ∅\nhs2 : IsOpen[PseudoMetricSpace.toUniformSpace.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 319,
"column": 4
} | {
"line": 319,
"column": 15
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nf' : ℝ → E\nC : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x\nbound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C\ng : ℝ → E := fun x ↦ f x - f a\nhg : ContinuousOn g (Icc a b)\nhg' : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 38
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → E\nC : ℝ\nhf : ∀ x ∈ Icc 0 1, HasDerivWithinAt f (f' x) (Icc 0 1) x\nbound : ∀ x ∈ Ico 0 1, ‖f' x‖ ≤ C\n⊢ ‖f 1 - f 0‖ ≤ C",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 38
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nC : ℝ\nhf : DifferentiableOn ℝ f (Icc 0 1)\nbound : ∀ x ∈ Ico 0 1, ‖derivWithin f (Icc 0 1) x‖ ≤ C\n⊢ ‖f 1 - f 0‖ ≤ C",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 364,
"column": 2
} | {
"line": 364,
"column": 60
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nhcont : ContinuousOn f (Icc a b)\nhderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x\nthis : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a)\n⊢ ∀ x ∈ Icc a b, f x = f a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 369,
"column": 4
} | {
"line": 369,
"column": 39
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nhdiff : DifferentiableOn ℝ f (Icc a b)\nhderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0\n⊢ ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 370,
"column": 2
} | {
"line": 370,
"column": 60
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nhdiff : DifferentiableOn ℝ f (Icc a b)\nhderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0\nH : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0\n⊢ ∀ x ∈ Icc a b, f x = f a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 31
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nf' g : ℝ → E\nderivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x\nderivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x\nfcont : ContinuousOn f (Icc a b)\ngcont : ContinuousOn g (Icc a b)\nhi : f a ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 436,
"column": 4
} | {
"line": 436,
"column": 15
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\n𝕜 : Type u_3\nG : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : E → G\nC : ℝ\ns : Set E\nx y : E\nf' : E → E ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 440,
"column": 2
} | {
"line": 440,
"column": 17
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\n𝕜 : Type u_3\nG : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : E → G\nC : ℝ\ns : Set E\nx y : E\nf' : E → E ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 562,
"column": 2
} | {
"line": 562,
"column": 77
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\n𝕜 : Type u_3\nG : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : E → G\ns : Set E\nx y : E\nhs : Convex ℝ s\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 595,
"column": 4
} | {
"line": 595,
"column": 67
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\n𝕜 : Type u_3\nG : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : E → G\ns : Set E\nhs : IsOpen[... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 599,
"column": 2
} | {
"line": 599,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\n𝕜 : Type u_3\nG : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : E → G\nh₁ : Differentiable 𝕜 f\nh₂ : ∀ (x : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 612,
"column": 49
} | {
"line": 612,
"column": 74
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\n𝕜 : Type u_3\nG : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : E → G\ns : Set E\nhs : IsOpen[PseudoMetricSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 639,
"column": 2
} | {
"line": 639,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\n𝕜 : Type u_3\nG : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → G\ns : Set E\nx : E\nhs : IsOpen[Pseudo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 649,
"column": 37
} | {
"line": 649,
"column": 48
} | [
{
"pp": "𝕜 : Type u_3\nG : Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : IsRCLikeNormedField 𝕜\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf g : E → G\nhf : Differentiable 𝕜 f\nhg : Differentiable 𝕜 g\nhf' : ∀ (x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 680,
"column": 4
} | {
"line": 680,
"column": 15
} | [
{
"pp": "case refine_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → E\nx₀ : ℝ\nn : ℕ\ns : Set ℝ\nhs : Convex ℝ s\nhx₀s : x₀ ∈ s\nhff' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf' : f' =o[𝓝[s] x₀] fun x ↦ (x - x₀) ^ n\nh : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℝ) in 𝓝[s] x₀, ‖f x - f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 107,
"column": 8
} | {
"line": 107,
"column": 19
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nn : ℕ∞ω\ns : Set E\nt : Set F\ng : F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 336,
"column": 4
} | {
"line": 336,
"column": 76
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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