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.DerivativeTest | {
"line": 101,
"column": 83
} | {
"line": 101,
"column": 91
} | [
{
"pp": "case refine_2.hf'_nonpos\nf : ℝ → ℝ\na b c : ℝ\nh : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Ico b c), deriv f x ≤ 0",
"usedConstants": [
"Real.instIsOr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 110,
"column": 84
} | {
"line": 110,
"column": 92
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\na b c : ℝ\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Ioc a b))",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 110,
"column": 84
} | {
"line": 110,
"column": 92
} | [
{
"pp": "case refine_1.hf'_nonneg\nf : ℝ → ℝ\na b c : ℝ\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Ioc a b), 0 ≤ deriv f x",
"usedConstants"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 111,
"column": 87
} | {
"line": 111,
"column": 95
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\na b c : ℝ\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Icc b c))",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 111,
"column": 87
} | {
"line": 111,
"column": 95
} | [
{
"pp": "case refine_2.hf'_nonpos\nf : ℝ → ℝ\na b c : ℝ\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Icc b c), deriv f x ≤ 0",
"usedConstants"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 120,
"column": 87
} | {
"line": 120,
"column": 95
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Icc a b))",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 120,
"column": 87
} | {
"line": 120,
"column": 95
} | [
{
"pp": "case refine_1.hf'_nonneg\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Icc a b), 0 ≤ deriv f x",
"usedConstants"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 121,
"column": 84
} | {
"line": 121,
"column": 92
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Ico b c))",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 121,
"column": 84
} | {
"line": 121,
"column": 92
} | [
{
"pp": "case refine_2.hf'_nonpos\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Ico b c), deriv f x ≤ 0",
"usedConstants"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 130,
"column": 87
} | {
"line": 130,
"column": 95
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Icc a b))",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 130,
"column": 87
} | {
"line": 130,
"column": 95
} | [
{
"pp": "case refine_1.hf'_nonneg\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Icc a b), 0 ≤ deriv f ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 131,
"column": 87
} | {
"line": 131,
"column": 95
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Icc b c))",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 131,
"column": 87
} | {
"line": 131,
"column": 95
} | [
{
"pp": "case refine_2.hf'_nonpos\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Icc b c), deriv f x ≤ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 140,
"column": 84
} | {
"line": 140,
"column": 92
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\na b : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioi b, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Ioc a b))",
"usedConstants": [
"Real.instIsOrderedRing"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 140,
"column": 84
} | {
"line": 140,
"column": 92
} | [
{
"pp": "case refine_1.hf'_nonneg\nf : ℝ → ℝ\na b : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioi b, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Ioc a b), 0 ≤ deriv f x",
"usedConstants": [
"Real.instIsOrdered... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 141,
"column": 82
} | {
"line": 141,
"column": 90
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\na b : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioi b, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Ici b))",
"usedConstants": [
"Real.instIsOrderedRing",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 141,
"column": 82
} | {
"line": 141,
"column": 90
} | [
{
"pp": "case refine_2.hf'_nonpos\nf : ℝ → ℝ\na b : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioi b, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Ici b), deriv f x ≤ 0",
"usedConstants": [
"Real.instIsOrderedRi... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 150,
"column": 87
} | {
"line": 150,
"column": 95
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\na b : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioi b, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Icc a b))",
"usedConstants": [
"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 150,
"column": 87
} | {
"line": 150,
"column": 95
} | [
{
"pp": "case refine_1.hf'_nonneg\nf : ℝ → ℝ\na b : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioi b, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Icc a b), 0 ≤ deriv f x",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 151,
"column": 82
} | {
"line": 151,
"column": 90
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\na b : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioi b, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Ici b))",
"usedConstants": [
"Re... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 151,
"column": 82
} | {
"line": 151,
"column": 90
} | [
{
"pp": "case refine_2.hf'_nonpos\nf : ℝ → ℝ\na b : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioi b, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Ici b), deriv f x ≤ 0",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 160,
"column": 82
} | {
"line": 160,
"column": 90
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Iic b))",
"usedConstants": [
"Real.instIsOrderedRing",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 160,
"column": 82
} | {
"line": 160,
"column": 90
} | [
{
"pp": "case refine_1.hf'_nonneg\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Iic b), 0 ≤ deriv f x",
"usedConstants": [
"Real.instIsOrderedRi... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 161,
"column": 84
} | {
"line": 161,
"column": 92
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Ico b c))",
"usedConstants": [
"Real.instIsOrderedRing"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 161,
"column": 84
} | {
"line": 161,
"column": 92
} | [
{
"pp": "case refine_2.hf'_nonpos\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Ico b c), deriv f x ≤ 0",
"usedConstants": [
"Real.instIsOrdered... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 170,
"column": 82
} | {
"line": 170,
"column": 90
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Iic b))",
"usedConstants": [
"Re... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 170,
"column": 82
} | {
"line": 170,
"column": 90
} | [
{
"pp": "case refine_1.hf'_nonneg\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Iic b), 0 ≤ deriv f x",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 171,
"column": 87
} | {
"line": 171,
"column": 95
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Icc b c))",
"usedConstants": [
"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 171,
"column": 87
} | {
"line": 171,
"column": 95
} | [
{
"pp": "case refine_2.hf'_nonpos\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Icc b c), deriv f x ≤ 0",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 180,
"column": 82
} | {
"line": 180,
"column": 90
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\nb : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Iio b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioi b, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Iic b))",
"usedConstants": [
"Real.instIsOrderedRing",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 180,
"column": 82
} | {
"line": 180,
"column": 90
} | [
{
"pp": "case refine_1.hf'_nonneg\nf : ℝ → ℝ\nb : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Iio b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioi b, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Iic b), 0 ≤ deriv f x",
"usedConstants": [
"Real.instIsOrderedRing",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 181,
"column": 82
} | {
"line": 181,
"column": 90
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\nb : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Iio b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioi b, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Ici b))",
"usedConstants": [
"Real.instIsOrderedRing",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 181,
"column": 82
} | {
"line": 181,
"column": 90
} | [
{
"pp": "case refine_2.hf'_nonpos\nf : ℝ → ℝ\nb : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Iio b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioi b, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Ici b), deriv f x ≤ 0",
"usedConstants": [
"Real.instIsOrderedRing",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 199,
"column": 83
} | {
"line": 199,
"column": 91
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\na b c : ℝ\nh : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Ioc a b))",
"usedConstants": [
"Real.instIsOrdered... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 199,
"column": 83
} | {
"line": 199,
"column": 91
} | [
{
"pp": "case refine_1.hf'_nonpos\nf : ℝ → ℝ\na b c : ℝ\nh : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Ioc a b), deriv f x ≤ 0",
"usedConstants": [
"Real.instIsOr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 200,
"column": 83
} | {
"line": 200,
"column": 91
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\na b c : ℝ\nh : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Ico b c))",
"usedConstants": [
"Real.instIsOrdered... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 200,
"column": 83
} | {
"line": 200,
"column": 91
} | [
{
"pp": "case refine_2.hf'_nonneg\nf : ℝ → ℝ\na b c : ℝ\nh : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Ico b c), 0 ≤ deriv f x",
"usedConstants": [
"Real.instIsOr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 209,
"column": 84
} | {
"line": 209,
"column": 92
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\na b c : ℝ\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Ioc a b))",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 209,
"column": 84
} | {
"line": 209,
"column": 92
} | [
{
"pp": "case refine_1.hf'_nonpos\nf : ℝ → ℝ\na b c : ℝ\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Ioc a b), deriv f x ≤ 0",
"usedConstants"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 210,
"column": 87
} | {
"line": 210,
"column": 95
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\na b c : ℝ\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Icc b c))",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 210,
"column": 87
} | {
"line": 210,
"column": 95
} | [
{
"pp": "case refine_2.hf'_nonneg\nf : ℝ → ℝ\na b c : ℝ\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Icc b c), 0 ≤ deriv f x",
"usedConstants"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 219,
"column": 87
} | {
"line": 219,
"column": 95
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Icc a b))",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 219,
"column": 87
} | {
"line": 219,
"column": 95
} | [
{
"pp": "case refine_1.hf'_nonpos\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Icc a b), deriv f x ≤ 0",
"usedConstants"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 220,
"column": 84
} | {
"line": 220,
"column": 92
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Ico b c))",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 220,
"column": 84
} | {
"line": 220,
"column": 92
} | [
{
"pp": "case refine_2.hf'_nonneg\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Ico b c), 0 ≤ deriv f x",
"usedConstants"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 92
} | [
{
"pp": "case refine_2\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ MonotoneOn f (Ico b c)",
"usedConstants": [
"Real.instIsOrdere... | · apply monotoneOn_of_deriv_nonneg (convex_Ico b c) (continuousOn_Ico hb hd₁) <;> simp_all | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 229,
"column": 87
} | {
"line": 229,
"column": 95
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Icc a b))",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 229,
"column": 87
} | {
"line": 229,
"column": 95
} | [
{
"pp": "case refine_1.hf'_nonpos\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Icc a b), deriv f x ≤ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 230,
"column": 87
} | {
"line": 230,
"column": 95
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Icc b c))",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 230,
"column": 87
} | {
"line": 230,
"column": 95
} | [
{
"pp": "case refine_2.hf'_nonneg\nf : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Icc b c), 0 ≤ deriv f ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 239,
"column": 84
} | {
"line": 239,
"column": 92
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\na b : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Ioc a b))",
"usedConstants": [
"Real.instIsOrderedRing"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 239,
"column": 84
} | {
"line": 239,
"column": 92
} | [
{
"pp": "case refine_1.hf'_nonpos\nf : ℝ → ℝ\na b : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Ioc a b), deriv f x ≤ 0",
"usedConstants": [
"Real.instIsOrdered... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 240,
"column": 82
} | {
"line": 240,
"column": 90
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\na b : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Ici b))",
"usedConstants": [
"Real.instIsOrderedRing",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 240,
"column": 82
} | {
"line": 240,
"column": 90
} | [
{
"pp": "case refine_2.hf'_nonneg\nf : ℝ → ℝ\na b : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Ici b), 0 ≤ deriv f x",
"usedConstants": [
"Real.instIsOrderedRi... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 249,
"column": 87
} | {
"line": 249,
"column": 95
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\na b : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Icc a b))",
"usedConstants": [
"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 249,
"column": 87
} | {
"line": 249,
"column": 95
} | [
{
"pp": "case refine_1.hf'_nonpos\nf : ℝ → ℝ\na b : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Icc a b), deriv f x ≤ 0",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 250,
"column": 82
} | {
"line": 250,
"column": 90
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\na b : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Ici b))",
"usedConstants": [
"Re... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 250,
"column": 82
} | {
"line": 250,
"column": 90
} | [
{
"pp": "case refine_2.hf'_nonneg\nf : ℝ → ℝ\na b : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Ici b), 0 ≤ deriv f x",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 259,
"column": 82
} | {
"line": 259,
"column": 90
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Iic b))",
"usedConstants": [
"Real.instIsOrderedRing",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 259,
"column": 82
} | {
"line": 259,
"column": 90
} | [
{
"pp": "case refine_1.hf'_nonpos\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Iic b), deriv f x ≤ 0",
"usedConstants": [
"Real.instIsOrderedRi... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 260,
"column": 84
} | {
"line": 260,
"column": 92
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Ico b c))",
"usedConstants": [
"Real.instIsOrderedRing"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 260,
"column": 84
} | {
"line": 260,
"column": 92
} | [
{
"pp": "case refine_2.hf'_nonneg\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Ico b c), 0 ≤ deriv f x",
"usedConstants": [
"Real.instIsOrdered... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 92
} | [
{
"pp": "case refine_2\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ MonotoneOn f (Ico b c)",
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOr... | · apply monotoneOn_of_deriv_nonneg (convex_Ico b c) (continuousOn_Ico hb hd₁) <;> simp_all | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 269,
"column": 82
} | {
"line": 269,
"column": 90
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Iic b))",
"usedConstants": [
"Re... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 269,
"column": 82
} | {
"line": 269,
"column": 90
} | [
{
"pp": "case refine_1.hf'_nonpos\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Iic b), deriv f x ≤ 0",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 270,
"column": 87
} | {
"line": 270,
"column": 95
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Icc b c))",
"usedConstants": [
"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 270,
"column": 87
} | {
"line": 270,
"column": 95
} | [
{
"pp": "case refine_2.hf'_nonneg\nf : ℝ → ℝ\nb c : ℝ\nhb : ContinuousAt f b\nhc : ContinuousAt f c\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Iio b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Icc b c), 0 ≤ deriv f x",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 279,
"column": 82
} | {
"line": 279,
"column": 90
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\nb : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Iio b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Iic b))",
"usedConstants": [
"Real.instIsOrderedRing",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 279,
"column": 82
} | {
"line": 279,
"column": 90
} | [
{
"pp": "case refine_1.hf'_nonpos\nf : ℝ → ℝ\nb : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Iio b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Iic b), deriv f x ≤ 0",
"usedConstants": [
"Real.instIsOrderedRing",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 280,
"column": 82
} | {
"line": 280,
"column": 90
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\nb : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Iio b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x\n⊢ DifferentiableOn ℝ f (interior (Ici b))",
"usedConstants": [
"Real.instIsOrderedRing",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 280,
"column": 82
} | {
"line": 280,
"column": 90
} | [
{
"pp": "case refine_2.hf'_nonneg\nf : ℝ → ℝ\nb : ℝ\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Iio b)\nhd₁ : DifferentiableOn ℝ f (Ioi b)\nh₀ : ∀ x ∈ Iio b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x\n⊢ ∀ x ∈ interior (Ici b), 0 ≤ deriv f x",
"usedConstants": [
"Real.instIsOrderedRing",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DifferentialForm.Basic | {
"line": 214,
"column": 6
} | {
"line": 214,
"column": 81
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nr : WithTop ℕ∞\nω : E → E [⋀^Fin n]→L[𝕜] F\ns : Set E\nx : E\nhω : ContDiffWithinAt 𝕜 r ω s x\nhr :... | refine (hω.fderivWithin_right hs ?_ h'x).differentiableWithinAt one_ne_zero | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.FDeriv.Symmetric | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 30
} | [
{
"pp": "case h\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\n... | replace hpos : 0 < h := hpos | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Analysis.Calculus.LineDeriv.Basic | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nv : E\nf : E → F\nf' : F\nx₀ : E\nhf : HasLineDerivAt 𝕜 f f' x₀ v\nC : ℝ\nhC₀ : 0 ≤ C\nhlip : ∀ᶠ (x : E) in... | have A : Continuous (fun (t : 𝕜) ↦ x₀ + t • v) := by fun_prop | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.LineDeriv.Basic | {
"line": 433,
"column": 2
} | {
"line": 433,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nv : E\nf : E → F\nx₀ : E\nC : ℝ\nhC₀ : 0 ≤ C\nhlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖\n⊢ ∀ᶠ ... | have A : Continuous (fun (t : 𝕜) ↦ x₀ + t • v) := by fun_prop | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.FDeriv.Norm | {
"line": 164,
"column": 31
} | {
"line": 164,
"column": 49
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\nh : DifferentiableAt ℝ (fun x ↦ ‖x‖) x\nthis : ∀ (t : ℝ), ‖x + t • x‖ = |1 + t| * ‖x‖\n⊢ ↑(SignType.sign (1 + 0)) * deriv (fun t ↦ 1 + t) 0 * ‖x‖ = ‖x‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"SignT... | deriv_const_add_id | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Symmetric | {
"line": 596,
"column": 4
} | {
"line": 596,
"column": 12
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\nhf : ContDiffWithinAt 𝕜 n f s x\nhn : minSmoothness 𝕜 2 ≤ n\... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.Implicit | {
"line": 143,
"column": 6
} | {
"line": 143,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : CompleteSpace E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : CompleteSpace F\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : N... | φ.hasStrictFDerivAt.hasFDerivAt.fderiv | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.LHopital | {
"line": 305,
"column": 2
} | {
"line": 305,
"column": 51
} | [
{
"pp": "a : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\ns : Set ℝ\nhs : Convex ℝ s\nhff' : ∀ᶠ (x : ℝ) in 𝓝[s \\ {a}] a, HasDerivWithinAt f (f' x) (s \\ {a}) x\nhgg' : ∀ᶠ (x : ℝ) in 𝓝[s \\ {a}] a, HasDerivWithinAt g (g' x) (s \\ {a}) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝[s \\ {a}] a, g' x ≠ 0\nhfa : Tendsto f (𝓝[s \\ {a}] a) (�... | have h := hs.diff_singleton_eventually_mem_nhds a | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 40
} | [
{
"pp": "case refine_3\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x... | simpa [c] using disjoint_sdiff_right | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 40
} | [
{
"pp": "case refine_3\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x... | simpa [c] using disjoint_sdiff_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 40
} | [
{
"pp": "case refine_3\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x... | simpa [c] using disjoint_sdiff_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.LHopital | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 58
} | [
{
"pp": "l : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) atBot l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atBot\nhgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃... | have hs : s ∈ atBot := inter_mem (inter_mem hs₁ hs₂) hs₃ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 125,
"column": 10
} | {
"line": 125,
"column": 54
} | [
{
"pp": "s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ... | apply le_antisymm (hf hy.1 hx.1.1 hy.2.2.le) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 55
} | [
{
"pp": "s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ... | have : (𝓝[s ∩ Iio x] x).NeBot := neBot_iff.2 hx.1.2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 335,
"column": 90
} | {
"line": 339,
"column": 96
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝⁴ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\nβ : Type u_3\ninst✝³ : TopologicalSpace β\ninst✝² : PseudoMetrizableSpace β\ninst✝¹ : l.IsCountablyGenerated\ninst✝ : l.NeBot\nf : α → β\nφ : ι → Set α\nhφ : AECover μ l φ\nhfm : ∀ (i : ι), AEStronglyMeasurable f (μ.re... | by
obtain ⟨u, hu⟩ := l.exists_seq_tendsto
have := aestronglyMeasurable_iUnion_iff.mpr fun n : ℕ => hfm (u n)
rwa [Measure.restrict_eq_self_of_ae_mem] at this
filter_upwards [hφ.ae_eventually_mem] with x hx using mem_iUnion.mpr (hu.eventually hx).exists | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 634,
"column": 2
} | {
"line": 634,
"column": 21
} | [
{
"pp": "case neg\nι : Type u_1\nE : Type u_2\nμ : Measure ℝ\nl : Filter ι\ninst✝² : l.IsCountablyGenerated\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na : ι → ℝ\nf : ℝ → E\nha : Tendsto a l atBot\nh : ∃ b, IntegrableOn f (Iic b) μ\n⊢ Tendsto (fun i ↦ ∫ (x : ℝ) in Iic (a i), f x ∂μ) l (𝓝 0)",
... | obtain ⟨b, hb⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 639,
"column": 2
} | {
"line": 639,
"column": 40
} | [
{
"pp": "case neg\nι : Type u_1\nE : Type u_2\nμ : Measure ℝ\nl : Filter ι\ninst✝² : l.IsCountablyGenerated\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na : ι → ℝ\nf : ℝ → E\nha : Tendsto a l atBot\nb : ℝ\nhb : IntegrableOn f (Iic b) μ\nthis : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Iic b, f x ∂μ - ∫ (x : ℝ) ... | rw [← sub_self (∫ x in Iic b, f x ∂μ)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 652,
"column": 2
} | {
"line": 652,
"column": 21
} | [
{
"pp": "case neg\nι : Type u_1\nE : Type u_2\nμ : Measure ℝ\nl : Filter ι\ninst✝² : l.IsCountablyGenerated\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na : ι → ℝ\nf : ℝ → E\nha : Tendsto a l atBot\nh : ∃ b, IntegrableOn f (Iio b) μ\n⊢ Tendsto (fun i ↦ ∫ (x : ℝ) in Iio (a i), f x ∂μ) l (𝓝 0)",
... | obtain ⟨b, hb⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 692,
"column": 2
} | {
"line": 692,
"column": 21
} | [
{
"pp": "case neg\nι : Type u_1\nE : Type u_2\nμ : Measure ℝ\nl : Filter ι\ninst✝² : l.IsCountablyGenerated\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na : ι → ℝ\nf : ℝ → E\nha : Tendsto a l atTop\nh : ∃ b, IntegrableOn f (Ici b) μ\n⊢ Tendsto (fun i ↦ ∫ (x : ℝ) in Ici (a i), f x ∂μ) l (𝓝 0)",
... | obtain ⟨b, hb⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 776,
"column": 4
} | {
"line": 776,
"column": 25
} | [
{
"pp": "E : Type u_1\nf f' : ℝ → E\na : ℝ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nfint : IntegrableOn f (Ioi a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Ioi a, HasDerivAt (⇑F ∘ f... | exact measure_mono hb | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 806,
"column": 2
} | {
"line": 806,
"column": 44
} | [
{
"pp": "case h\nE : Type u_1\nf f' : ℝ → E\na : ℝ\nm : E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhcont✝ : ContinuousWithinAt f (Ici a) a\nhderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nhf : Tendsto f atTop (𝓝 m)\nhcont : Continuo... | exact f'int.mono (fun y hy => hy.1) le_rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 973,
"column": 4
} | {
"line": 973,
"column": 25
} | [
{
"pp": "E : Type u_1\nf f' : ℝ → E\na : ℝ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f... | exact measure_mono hb | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 409,
"column": 2
} | {
"line": 409,
"column": 50
} | [
{
"pp": "case a.inr\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ↑m < ENNReal.ofReal |A.det|\nmpos : 0 < m\nhA : A.det ≠ 0\n⊢ {x |... | let B := A.toContinuousLinearEquivOfDetNeZero hA | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1040,
"column": 4
} | {
"line": 1040,
"column": 38
} | [
{
"pp": "case h.e'_3\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nhf : ContDiff ℝ 1 f\nh'f : HasCompactSupport f\nx : ℝ\nI : F →L[ℝ] Completion F := Completion.toComplL\nf' : ℝ → Completion F := ⇑I ∘ f\nhf' : ContDiff ℝ 1 f'\nh'f' : HasCompactSupport f'\nthis : ‖f' x‖ₑ ≤ ∫⁻ ... | simp [f', I, Completion.enorm_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1040,
"column": 4
} | {
"line": 1040,
"column": 38
} | [
{
"pp": "case h.e'_3\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nhf : ContDiff ℝ 1 f\nh'f : HasCompactSupport f\nx : ℝ\nI : F →L[ℝ] Completion F := Completion.toComplL\nf' : ℝ → Completion F := ⇑I ∘ f\nhf' : ContDiff ℝ 1 f'\nh'f' : HasCompactSupport f'\nthis : ‖f' x‖ₑ ≤ ∫⁻ ... | simp [f', I, Completion.enorm_coe] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1040,
"column": 4
} | {
"line": 1040,
"column": 38
} | [
{
"pp": "case h.e'_3\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nhf : ContDiff ℝ 1 f\nh'f : HasCompactSupport f\nx : ℝ\nI : F →L[ℝ] Completion F := Completion.toComplL\nf' : ℝ → Completion F := ⇑I ∘ f\nhf' : ContDiff ℝ 1 f'\nh'f' : HasCompactSupport f'\nthis : ‖f' x‖ₑ ≤ ∫⁻ ... | simp [f', I, Completion.enorm_coe] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 483,
"column": 2
} | {
"line": 485,
"column": 33
} | [
{
"pp": "f : ℝ → ℝ\nhf : MeasurableEmbedding f\ns : Set ℝ\nhs : MeasurableSet s\ng : ℝ → ℝ\nhg : ∀ᵐ (x : ℝ), x ∈ f '' s → 0 ≤ g x\nhf_int : IntegrableOn g (f '' s) volume\nf' : ℝ → ℝ\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\n⊢ (Measure.comap f (volume.withDensity fun x ↦ ENNReal.ofReal (g x))) s =\n ENN... | rw [hf.withDensity_ofReal_comap_apply_eq_integral_abs_det_fderiv_mul volume hs
hg hf_int hf']
simp only [det_toSpanSingleton] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 483,
"column": 2
} | {
"line": 485,
"column": 33
} | [
{
"pp": "f : ℝ → ℝ\nhf : MeasurableEmbedding f\ns : Set ℝ\nhs : MeasurableSet s\ng : ℝ → ℝ\nhg : ∀ᵐ (x : ℝ), x ∈ f '' s → 0 ≤ g x\nhf_int : IntegrableOn g (f '' s) volume\nf' : ℝ → ℝ\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\n⊢ (Measure.comap f (volume.withDensity fun x ↦ ENNReal.ofReal (g x))) s =\n ENN... | rw [hf.withDensity_ofReal_comap_apply_eq_integral_abs_det_fderiv_mul volume hs
hg hf_int hf']
simp only [det_toSpanSingleton] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 557,
"column": 2
} | {
"line": 557,
"column": 83
} | [
{
"pp": "a b : ℝ\nf f' g : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x\n⊢ IntervalIntegrable (fun x ↦ (g ∘ f) x * f' x) volume a b ↔ IntervalIntegrable g volume (f a) (f b)",
"usedConstants": [
"NonUnit... | simpa [mul_comm] using integrable_deriv_smul_comp_iff_of_deriv_nonneg hf hff' hf' | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
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