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.Normed.Group.Basic | {
"line": 546,
"column": 63
} | {
"line": 546,
"column": 74
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nx✝ : ∃ x, ‖x‖₊ ≠ 0\nx : E\nhx : ‖x‖₊ ≠ 0\n⊢ ¬‖x⁻¹ * 1‖₊ = 0",
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
"NNNorm.nnnorm",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 551,
"column": 2
} | {
"line": 551,
"column": 13
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\n⊢ IndiscreteTopology E ↔ ∀ (x : E), ‖x‖₊ = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 584,
"column": 2
} | {
"line": 584,
"column": 13
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\n⊢ IndiscreteTopology E ↔ ∀ (x : E), ‖x‖ = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Order.Lattice | {
"line": 65,
"column": 20
} | {
"line": 65,
"column": 70
} | [
{
"pp": "α : Type u_1\ninst✝² : NormedAddCommGroup α\ninst✝¹ : Lattice α\ninst✝ : HasSolidNorm α\nx y : ℤ\nh : |x| ≤ |y|\n⊢ ‖x‖ ≤ ‖y‖",
"usedConstants": [
"Norm.norm",
"Int.cast",
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"Real.instZeroLEOneClass",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 700,
"column": 2
} | {
"line": 700,
"column": 13
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nf : Filter E\n⊢ Disjoint (𝓝 1) f ↔ ∃ δ > 0, ∀ᶠ (y : E) in f, δ ≤ ‖y‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Real.instZero",
"c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 771,
"column": 2
} | {
"line": 771,
"column": 47
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedCommGroup E\na b : E\n⊢ ‖a⁻¹ * b‖ = ‖a / b‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
... | rw [← dist_eq_norm_inv_mul, dist_eq_norm_div] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 771,
"column": 2
} | {
"line": 771,
"column": 47
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedCommGroup E\na b : E\n⊢ ‖a⁻¹ * b‖ = ‖a / b‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
... | rw [← dist_eq_norm_inv_mul, dist_eq_norm_div] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 771,
"column": 2
} | {
"line": 771,
"column": 47
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedCommGroup E\na b : E\n⊢ ‖a⁻¹ * b‖ = ‖a / b‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
... | rw [← dist_eq_norm_inv_mul, dist_eq_norm_div] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Algebra | {
"line": 255,
"column": 32
} | {
"line": 255,
"column": 67
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : PseudoMetricSpace β\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMul α β\ninst✝ : IsBoundedSMul α β\nx y₁ y₂ : ℝ\n⊢ dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Order.Lattice | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝³ : NormedAddCommGroup α\ninst✝² : Lattice α\ninst✝¹ : HasSolidNorm α\ninst✝ : IsOrderedAddMonoid α\nx y : α\nh : ‖x ⊓ y - 0 ⊓ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖\n⊢ ‖x ⊓ y‖ ≤ ‖x‖ + ‖y‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Algebra | {
"line": 256,
"column": 32
} | {
"line": 256,
"column": 67
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : PseudoMetricSpace β\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMul α β\ninst✝ : IsBoundedSMul α β\nx₁ x₂ y : ℝ\n⊢ dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Order.Lattice | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝³ : NormedAddCommGroup α\ninst✝² : Lattice α\ninst✝¹ : HasSolidNorm α\ninst✝ : IsOrderedAddMonoid α\nx y : α\nh : ‖x ⊔ y - 0 ⊔ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖\n⊢ ‖x ⊔ y‖ ≤ ‖x‖ + ‖y‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Algebra | {
"line": 265,
"column": 39
} | {
"line": 265,
"column": 73
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁷ : PseudoMetricSpace α\ninst✝⁶ : PseudoMetricSpace β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMul α β\ninst✝² : IsBoundedSMul α β\ninst✝¹ : SMul αᵐᵒᵖ β\ninst✝ : IsCentralScalar α β\nx : α\ny₁ y₂ : β\n⊢ dist (MulOpposite.op x • y₁) (MulOpposite.op x • y₂) ≤ dist (Mu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Algebra | {
"line": 268,
"column": 38
} | {
"line": 268,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁷ : PseudoMetricSpace α\ninst✝⁶ : PseudoMetricSpace β\ninst✝⁵ : Zero α\ninst✝⁴ : Zero β\ninst✝³ : SMul α β\ninst✝² : IsBoundedSMul α β\ninst✝¹ : SMul αᵐᵒᵖ β\ninst✝ : IsCentralScalar α β\nx₁ x₂ : α\ny : β\n⊢ dist (MulOpposite.op x₁ • y) (MulOpposite.op x₂ • y) ≤ dist (Mu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Order.Lattice | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 29
} | [
{
"pp": "α : Type u_1\ninst✝³ : NormedAddCommGroup α\ninst✝² : Lattice α\ninst✝¹ : HasSolidNorm α\ninst✝ : IsOrderedAddMonoid α\n⊢ LipschitzWith 1 negPart",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 921,
"column": 2
} | {
"line": 921,
"column": 33
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedCommGroup E\na b : E\nr : ℝ\nn : ℕ\nh : ‖a⁻¹ * b‖ ≤ r\n⊢ ↑n * ‖a⁻¹ * b‖ ≤ n • r",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"instHSMul",
"HMul.hMul",
"DivisionCommMonoid.toDi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 972,
"column": 2
} | {
"line": 972,
"column": 28
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\nf : E → F\nx : E\ny : F\n⊢ Tendsto f (𝓝 x) (𝓝 y) ↔ ∀ ε > 0, ∃ δ > 0, ∀ (x' : E), ‖x' / x‖ < δ → ‖f x' / y‖ < ε",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 977,
"column": 2
} | {
"line": 977,
"column": 28
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedCommGroup E\nx : E\n⊢ (𝓝 x).HasBasis (fun ε ↦ 0 < ε) fun ε ↦ {y | ‖y / x‖ < ε}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 982,
"column": 2
} | {
"line": 982,
"column": 28
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedCommGroup E\n⊢ (𝓤 E).HasBasis (fun ε ↦ 0 < ε) fun ε ↦ {p | ‖p.1 / p.2‖ < ε}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 1020,
"column": 2
} | {
"line": 1020,
"column": 23
} | [
{
"pp": "E : Type u_5\ninst✝ : NormedGroup E\na : E\n⊢ a = 1 ∨ 0 < ‖a‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Real.instZero",
"congrArg",
"Real.instLT",
"Group.toDivisionMonoid",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 61
} | [
{
"pp": "𝓕 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : MonoidHomClass 𝓕 E F\nf : 𝓕\nC : ℝ\nh : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖\nx y : E\n⊢ dist (f x) (f y) ≤ C * dist x y",
"usedConstants": [
"Norm.norm",
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 13
} | [
{
"pp": "𝓕 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : MonoidHomClass 𝓕 E F\nf : 𝓕\nh : ∀ (x y : E), ‖f (x⁻¹ * y)‖ = ‖x⁻¹ * y‖\nx : E\n⊢ ‖f x‖ = ‖x‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 61
} | [
{
"pp": "𝓕 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : MonoidHomClass 𝓕 E F\nf : 𝓕\nK : ℝ≥0\nh : ∀ (x : E), ‖x‖ ≤ ↑K * ‖f x‖\nx y : E\n⊢ dist x y ≤ ↑K * dist (f x) (f y)",
"usedConstants": [
"Norm.norm",
"Eq.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 142,
"column": 32
} | {
"line": 142,
"column": 69
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝¹ : SeminormedGroup E\ninst✝ : SeminormedGroup F\nf : E → F\nK : ℝ≥0\nh : LipschitzWith K f\nhf : f 1 = 1\nx : E\n⊢ ‖f x‖ ≤ ↑K * ‖x‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 39
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝¹ : SeminormedGroup E\ninst✝ : SeminormedGroup F\nf : E → F\nK : ℝ≥0\nh : AntilipschitzWith K f\nhf : f 1 = 1\nx : E\n⊢ ‖x‖ ≤ ↑K * ‖f x‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 13
} | [
{
"pp": "E : Type u_2\ninst✝ : SeminormedGroup E\n⊢ LipschitzWith 1 norm",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 232,
"column": 2
} | {
"line": 232,
"column": 50
} | [
{
"pp": "E : Type u_2\ninst✝ : SeminormedCommGroup E\na₁ a₂ b₁ b₂ : E\n⊢ dist (a₁ * a₂) (b₁ * b₂) ≤ dist a₁ b₁ + dist a₂ b₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 49
} | [
{
"pp": "E : Type u_2\ninst✝ : SeminormedCommGroup E\na₁ a₂ b₁ b₂ : E\n⊢ dist (a₁ / a₂) (b₁ / b₂) ≤ dist a₁ b₁ + dist a₂ b₂",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 64
} | [
{
"pp": "E : Type u_2\ninst✝ : SeminormedCommGroup E\na₁ a₂ b₁ b₂ : E\n⊢ |dist a₁ b₁ - dist a₂ b₂| ≤ dist (a₁ * a₂) (b₁ * b₂)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 38
} | [
{
"pp": "α : Type u_4\nE : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : PseudoEMetricSpace α\nKf Kg : ℝ≥0\nf g : α → E\nhf : LipschitzWith Kf f\nhg : LipschitzWith Kg g\n⊢ LipschitzWith (Kf + Kg) fun x ↦ f x * g x",
"usedConstants": [
"Eq.mpr",
"LipschitzWith",
"HMul.hMul",
"Mon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 41
} | [
{
"pp": "α : Type u_4\nE : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : PseudoEMetricSpace α\nf g : α → E\nhf : LocallyLipschitz f\nhg : LocallyLipschitz g\n⊢ LocallyLipschitz fun x ↦ f x * g x",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"Set.univ",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 322,
"column": 42
} | {
"line": 323,
"column": 89
} | [
{
"pp": "α : Type u_4\nE : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : PseudoEMetricSpace α\nf g : α → E\nhf : LocallyLipschitz f\nhg : LocallyLipschitz g\n⊢ LocallyLipschitz fun x ↦ f x * g x",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"Set.univ",
... | by
simpa [← locallyLipschitzOn_univ] using hf.locallyLipschitzOn.mul hg.locallyLipschitzOn | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 35
} | [
{
"pp": "α : Type u_4\nE : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : PseudoEMetricSpace α\nKf Kg : ℝ≥0\nf g : α → E\ns : Set α\nhf : LipschitzOnWith Kf f s\nhg : LipschitzOnWith Kg g s\n⊢ LipschitzOnWith (Kf + Kg) (fun x ↦ f x / g x) s",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 35
} | [
{
"pp": "α : Type u_4\nE : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : PseudoEMetricSpace α\nKf Kg : ℝ≥0\nf g : α → E\nhf : LipschitzWith Kf f\nhg : LipschitzWith Kg g\n⊢ LipschitzWith (Kf + Kg) fun x ↦ f x / g x",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"LipschitzWith",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 338,
"column": 2
} | {
"line": 338,
"column": 35
} | [
{
"pp": "α : Type u_4\nE : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : PseudoEMetricSpace α\nf g : α → E\ns : Set α\nhf : LocallyLipschitzOn s f\nhg : LocallyLipschitzOn s g\n⊢ LocallyLipschitzOn s fun x ↦ f x / g x",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 35
} | [
{
"pp": "α : Type u_4\nE : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : PseudoEMetricSpace α\nf g : α → E\nhf : LocallyLipschitz f\nhg : LocallyLipschitz g\n⊢ LocallyLipschitz fun x ↦ f x / g x",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 49
} | [
{
"pp": "α : Type u_4\nE : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : PseudoEMetricSpace α\nKf Kg : ℝ≥0\nf g : α → E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg (g / f)\nhK : Kg < Kf⁻¹\n⊢ AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ g",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 443,
"column": 23
} | {
"line": 443,
"column": 34
} | [
{
"pp": "G : Type u_4\ninst✝ : SeminormedGroup G\nu : ℕ → G\nhu : CauchySeq u\nC : ℝ\nhC : ∀ (m n : ℕ), ‖(u m)⁻¹ * u n‖ < C\nthis : ∀ (n : ℕ), ‖u n‖ ≤ C + ‖u 0‖\n⊢ ∀ y ∈ Set.range fun n ↦ ‖u n‖, y ≤ C + ‖u 0‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"Preorder.toLE",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 448,
"column": 2
} | {
"line": 448,
"column": 30
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\nf : E → F\nC : ℝ≥0\ns : Set E\n⊢ LipschitzOnWith C f s ↔ ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ‖f x / f y‖ ≤ ↑C * ‖x / y‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Re... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Bounded | {
"line": 34,
"column": 2
} | {
"line": 34,
"column": 35
} | [
{
"pp": "E : Type u_2\ninst✝ : SeminormedGroup E\n⊢ comap norm atTop = cobounded E",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Bounded | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 59
} | [
{
"pp": "E : Type u_2\ninst✝ : SeminormedGroup E\ns : Set E\n⊢ Bornology.IsBounded s ↔ ∃ C, ∀ x ∈ s, ‖x‖ ≤ C",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Bounded | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 13
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝¹ : SeminormedGroup E\ninst✝ : TopologicalSpace α\nf : α → E\nhf : HasCompactMulSupport f\nh'f : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\n⊢ ∃ C, ∀ (x : α), ‖f x‖ ≤ C",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Dilation | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 23
} | [
{
"pp": "case pos.inr\nα : Type u_1\nβ : Type u_2\nF : Type u_4\ninst✝³ : PseudoEMetricSpace α\ninst✝² : PseudoEMetricSpace β\ninst✝¹ : FunLike F α β\ninst✝ : DilationClass F α β\nf : F\nx y : α\nkey : ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞\nr : ℝ≥0\nhne : r ≠ 0\nhr : edist (f x) (f y) = ↑r * edist x y\nh :... | · simp [hr, h, hne] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Field.Basic | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 13
} | [
{
"pp": "α : Type u_2\ninst✝ : NormedDivisionRing α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\n⊢ ‖a * b * a⁻¹ * b⁻¹ - 1‖ ≤ 2 * ‖a‖⁻¹ * ‖b‖⁻¹ * ‖a - 1‖ * ‖b - 1‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Basic | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 13
} | [
{
"pp": "α : Type u_2\ninst✝ : NormedDivisionRing α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\n⊢ ‖a * b * a⁻¹ * b⁻¹ - 1‖₊ ≤ 2 * ‖a‖₊⁻¹ * ‖b‖₊⁻¹ * ‖a - 1‖₊ * ‖b - 1‖₊",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Dilation | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 70
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nF : Type u_4\ninst✝³ : PseudoEMetricSpace α\ninst✝² : PseudoEMetricSpace β\ninst✝¹ : FunLike F α β\ninst✝ : DilationClass F α β\nf : F\nx y : α\nr : ℝ≥0\nh₀ : edist x y ≠ 0\nhtop : edist x y ≠ ∞\nhr : edist (f x) (f y) = ↑r * edist x y\n⊢ r = ratio f",
"usedConstants": [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Bounded | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 28
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddGroup E\ninst✝ : TopologicalSpace α\nf : α → E\nhf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nh : HasCompactSupport f\n⊢ ∃ C, ∀ (x : α), ‖f x‖ ≤ C",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Dilation | {
"line": 246,
"column": 35
} | {
"line": 246,
"column": 46
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nF : Type u_4\ninst✝⁴ : PseudoEMetricSpace α\ninst✝³ : PseudoEMetricSpace β\ninst✝² : PseudoEMetricSpace γ\ninst✝¹ : FunLike F α β\ninst✝ : DilationClass F α β\nf✝ : F\nf : α → β\nhf : Isometry f\n⊢ ∀ (x y : α), edist (f x) (f y) = ↑1 * edist x y",
"usedCons... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Basic | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 39
} | [
{
"pp": "α : Type u_2\ninst✝ : NontriviallyNormedField α\n⊢ (𝓝[{x | IsUnit x}] 0).NeBot",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NormedCommRing.toSeminormedCommRing",
"congrArg",
"Filter.NeBot",
"nhdsWithin",
"setOf",
"PseudoMetricSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.DilationEquiv | {
"line": 129,
"column": 91
} | {
"line": 130,
"column": 86
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : PseudoEMetricSpace X\ninst✝¹ : PseudoEMetricSpace Y\ninst✝ : PseudoEMetricSpace Z\ne : X ≃ᵈ Y\ne' : Y ≃ᵈ Z\nhX : ∀ (x y : X), edist x y = 0 ∨ edist x y = ∞\nx y : X\n⊢ edist (e x) (e y) = 0 ∨ edist (e x) (e y) = ∞",
"usedConstants": [
"False"... | by
refine (hX x y).imp (fun h ↦ ?_) fun h ↦ ?_ <;> simp [*, Dilation.ratio_ne_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Ring.Lemmas | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝² : SeminormedRing α\ninst✝¹ : NormOneClass α\ninst✝ : NormMulClass α\nm : ℕ\nhm : m ≠ 0\n⊢ Tendsto (fun x ↦ x ^ m) (cobounded α) (cobounded α)",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"SeminormedRing.toNorm",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.DilationEquiv | {
"line": 187,
"column": 35
} | {
"line": 187,
"column": 46
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : PseudoEMetricSpace X\ninst✝¹ : PseudoEMetricSpace Y\ninst✝ : PseudoEMetricSpace Z\ne : X ≃ᵢ Y\n⊢ ∀ (x y : X), edist (e.toFun x) (e.toFun y) = ↑1 * edist x y",
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 195,
"column": 6
} | {
"line": 195,
"column": 31
} | [
{
"pp": "G : Type u_1\nα✝ : Type u_2\nβ : Type u_3\nι✝ : Type u_4\nι : Type u_5\nα : ι → Type u_6\ninst✝⁴ : Nonempty ι\ninst✝³ : Fintype ι\ninst✝² : (i : ι) → SeminormedAddCommGroup (α i)\ninst✝¹ : (i : ι) → One (α i)\ninst✝ : ∀ (i : ι), NormOneClass (α i)\n⊢ ‖1‖ = 1",
"usedConstants": [
"Norm.norm",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 224,
"column": 8
} | {
"line": 224,
"column": 34
} | [
{
"pp": "β : Type u_5\ninst✝¹ : NormedRing β\ninst✝ : Nontrivial β\n⊢ ‖1‖ ≤ ‖1‖ * ‖1‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 364,
"column": 28
} | {
"line": 364,
"column": 39
} | [
{
"pp": "ι : Type u_4\nα : Type u_5\ninst✝ : NormedCommRing α\nf : ι → α\nval✝ : Multiset ι\nl : List ι\nhl : Multiset.Nodup (Quot.mk (⇑(List.isSetoid ι)) l)\nhs : { val := Quot.mk (⇑(List.isSetoid ι)) l, nodup := hl }.Nonempty\n⊢ List.map f l ≠ []",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 365,
"column": 2
} | {
"line": 365,
"column": 13
} | [
{
"pp": "case mk\nι : Type u_4\nα : Type u_5\ninst✝ : NormedCommRing α\nf : ι → α\nval✝ : Multiset ι\nl : List ι\nhl : Multiset.Nodup (Quot.mk (⇑(List.isSetoid ι)) l)\nhs : { val := Quot.mk (⇑(List.isSetoid ι)) l, nodup := hl }.Nonempty\nthis : List.map f l ≠ []\n⊢ ‖∏ i ∈ { val := Quot.mk (⇑(List.isSetoid ι)) l... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 13
} | [
{
"pp": "case mk\nι : Type u_4\nα : Type u_5\ninst✝¹ : NormedCommRing α\ninst✝ : NormOneClass α\nf : ι → α\nval✝ : Multiset ι\nl : List ι\nhl : Multiset.Nodup (Quot.mk (⇑(List.isSetoid ι)) l)\n⊢ ‖∏ i ∈ { val := Quot.mk (⇑(List.isSetoid ι)) l, nodup := hl }, f i‖ ≤\n ∏ i ∈ { val := Quot.mk (⇑(List.isSetoid ι)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 403,
"column": 4
} | {
"line": 403,
"column": 42
} | [
{
"pp": "α : Type u_2\ninst✝ : SeminormedRing α\na : α\nn : ℕ\nx✝ : 0 < n + 2\n⊢ ‖a ^ (n + 2)‖₊ ≤ ‖a‖₊ ^ (n + 2)",
"usedConstants": [
"Monoid",
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"SeminormedAddGroup.toNNNorm",
"NNNorm.nnnorm",
"PartialO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 414,
"column": 2
} | {
"line": 414,
"column": 47
} | [
{
"pp": "α : Type u_2\ninst✝ : SeminormedRing α\na : α\nn : ℕ\nh : 0 < n\n⊢ ‖a ^ n‖ ≤ ‖a‖ ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 25
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : SeminormedAddGroup α\ninst✝² : SeminormedAddGroup β\ninst✝¹ : SMulZeroClass α β\ninst✝ : IsBoundedSMul α β\nr : α\nx : β\n⊢ ‖r • x‖ ≤ ‖r‖ * ‖x‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 442,
"column": 8
} | {
"line": 442,
"column": 33
} | [
{
"pp": "α : Type u_2\ninst✝ : SeminormedRing α\na b c : α\nha : ‖a‖ ≤ 1\n⊢ ‖c - a * b‖ ≤ ‖c - a‖ + ‖a * (1 - b)‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"SeminormedRing.toNorm",
"Real.instLE",
"Real",
"HMul.hMul",
"Ring.toNonAssocRing",
"AddGroupWithOne.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 40
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : SeminormedAddGroup α\ninst✝² : SeminormedAddGroup β\ninst✝¹ : SMulZeroClass α β\ninst✝ : IsBoundedSMul α β\nr : α\nx : β\n⊢ ‖r • x‖ₑ ≤ ‖r‖ₑ * ‖x‖ₑ",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"instHSMul",
"SeminormedAddG... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 61
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : SeminormedAddGroup α\ninst✝² : SeminormedAddGroup β\ninst✝¹ : SMulZeroClass α β\ninst✝ : IsBoundedSMul α β\ns : α\nx y : β\n⊢ dist (s • x) (s • y) ≤ ‖s‖ * dist x y",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAdd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 62,
"column": 32
} | {
"line": 62,
"column": 67
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NonUnitalSeminormedRing α\nx y₁ y₂ : α\n⊢ dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHSMul",
"instSMulOfMul",
"dist_eq_norm",
"HMul.hMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 63,
"column": 32
} | {
"line": 63,
"column": 67
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NonUnitalSeminormedRing α\nx₁ x₂ y : α\n⊢ dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHSMul",
"instSMulOfMul",
"dist_eq_norm",
"HMul.hMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 49
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NonUnitalSeminormedRing α\nx : αᵐᵒᵖ\ny₁ y₂ : α\n⊢ dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 49
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NonUnitalSeminormedRing α\nx₁ x₂ : αᵐᵒᵖ\ny : α\n⊢ dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 79,
"column": 41
} | {
"line": 79,
"column": 77
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : SeminormedRing α\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : Module α β\nh : ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖\na : α\nb₁ b₂ : β\n⊢ dist (a • b₁) (a • b₂) ≤ dist a 0 * dist b₁ b₂",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 80,
"column": 41
} | {
"line": 80,
"column": 77
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : SeminormedRing α\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : Module α β\nh : ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖\na₁ a₂ : α\nb : β\n⊢ dist (a₁ • b) (a₂ • b) ≤ dist a₁ a₂ * dist b 0",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 84,
"column": 23
} | {
"line": 84,
"column": 91
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : SeminormedRing α\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : Module α β\nh : ∀ (r : α) (x : β), ‖r • x‖ₑ ≤ ‖r‖ₑ * ‖x‖ₑ\n⊢ ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 475,
"column": 2
} | {
"line": 475,
"column": 13
} | [
{
"pp": "α : Type u_2\ninst✝ : SeminormedRing α\na b : αˣ\n⊢ ‖↑(a * b * a⁻¹ * b⁻¹) - 1‖₊ ≤ 2 * ‖↑a⁻¹‖₊ * ‖↑b⁻¹‖₊ * ‖↑a - 1‖₊ * ‖↑b - 1‖₊",
"usedConstants": [
"Units.val",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"SeminormedAddGroup.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 630,
"column": 2
} | {
"line": 630,
"column": 13
} | [
{
"pp": "R : Type u_5\nS : Type u_6\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nA : Subalgebra R S\nf : S → ℝ\nhf_pm : IsPowMul f\nx : ↥A\nn : ℕ\nhn : 1 ≤ n\n⊢ (fun x ↦ f ↑x) (x ^ n) = (fun x ↦ f ↑x) x ^ n",
"usedConstants": [
"Subalgebra.instSetLike",
"Real",
"SubmonoidCla... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 28
} | [
{
"pp": "case h.mp\nα : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\ny : β\nh1 : dist y x < ε\n⊢ dist (s • y) (s • x) < ‖s‖ * ε",
"usedConstants": [
"Norm.norm",
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 31
} | [
{
"pp": "case h.mpr\nα : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\np : β\nh : dist p (s • x) < ‖s‖ * ε\n⊢ dist (s • s⁻¹ • p) (s • x) < ‖s‖ * ε",
"usedConstants": [
"Norm.nor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 28
} | [
{
"pp": "case h.mp\nα : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\ny : β\nh1 : dist y x ≤ ε\n⊢ dist (s • y) (s • x) ≤ ‖s‖ * ε",
"usedConstants": [
"Norm.norm",
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 31
} | [
{
"pp": "case h.mpr\nα : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\np : β\nh : dist p (s • x) ≤ ‖s‖ * ε\n⊢ dist (s • s⁻¹ • p) (s • x) ≤ ‖s‖ * ε",
"usedConstants": [
"Norm.nor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 774,
"column": 4
} | {
"line": 774,
"column": 56
} | [
{
"pp": "α : Type u_2\ninst✝³ : NormedAddCommGroup α\ninst✝² : MulOneClass α\ninst✝¹ : NormMulClass α\ninst✝ : Nontrivial α\nu : α\nhu : u ≠ 0\n⊢ ‖1‖ = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 789,
"column": 4
} | {
"line": 789,
"column": 69
} | [
{
"pp": "G : Type u_1\nα : Type u_2\nβ : Type u_3\nι : Type u_4\ninst✝¹ : NormedRing α\ninst✝ : NormMulClass α\na✝ b✝ : α\nh : a✝ * b✝ = 0\n⊢ a✝ = 0 ∨ b✝ = 0",
"usedConstants": [
"norm_eq_zero",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real",
"NormedRing.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 922,
"column": 28
} | {
"line": 922,
"column": 78
} | [
{
"pp": "G : Type u_1\nα : Type u_2\nβ : Type u_3\nι : Type u_4\nR : Type u_5\ninst✝ : Ring R\nv : AbsoluteValue R ℝ\nx y z : R\n⊢ v (-x + z) ≤ v (-x + y) + v (-y + z)",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.partialOrder",
"Real.instLE",
"Real",
"AddGro... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 31
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nE : Set α\nx : α\nx_in_E : x ∈ E\ny : α\nhy : ENNReal.ofReal δ ≤ infEDist y E\n⊢ ENNReal.ofReal δ ≤ edist x y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 32
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nE : Set α\n⊢ Eᶜᶜ ⊆ (thickening δ (thickening δ E)ᶜ)ᶜ",
"usedConstants": [
"Eq.mpr",
"compl_compl",
"congrArg",
"Compl.compl",
"id",
"HasSubset.Subset",
"Set.instCompl",
"Metric.thickening",
"Set.i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 33
} | [
{
"pp": "α : Type u_1\ninst✝ : NormedDivisionRing α\nm : ℕ\nhm : -↑m < 0\n⊢ Tendsto (fun x ↦ x ^ (-↑m)) (𝓝[≠] 0) (cobounded α)",
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"GroupWithZero.toDivisionMonoid",
"PseudoMetricSpace.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 13
} | [
{
"pp": "𝕜 : Type u_4\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\n⊢ ContinuousAt Inv.inv x ↔ x ≠ 0",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"ContinuousAt",
"PseudoMetricSpace.toUniformS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDense | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 17
} | [
{
"pp": "case succ\nα : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nx : α\nN : ℕ\nihN : (nearestPtInd e N) x ≤ N\n⊢ (nearestPtInd e (N + 1)) x ≤ N + 1",
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",... | | succ N ihN => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 27
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace β\ninst✝⁵ : PseudoMetrizableSpace β\ninst✝⁴ : MeasurableSpace β\ninst✝³ : BorelSpace β\nι : Type u_3\ninst✝² : Countable ι\ninst✝¹ : Nonempty ι\nμ : Measure α\nf : ι → α → β\nL : Filter ι\ninst✝ : L.IsCountablyGen... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 20
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace β\ninst✝⁵ : PseudoMetrizableSpace β\ninst✝⁴ : MeasurableSpace β\ninst✝³ : BorelSpace β\nι : Type u_3\ninst✝² : Countable ι\ninst✝¹ : Nonempty ι\nμ : Measure α\nf : ι → α → β\nL : Filter ι\ninst✝ : L.IsCountablyGen... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 60
} | [
{
"pp": "α : Type u_3\ninst✝² : MeasurableSpace α\nA : Set α\nι : Type u_4\nL : Filter ι\ninst✝¹ : L.IsCountablyGenerated\nAs : ι → Set α\ninst✝ : L.NeBot\nμ : Measure α\nh_lim : ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A\nAs_mble : ∀ (i : ι), AEMeasurable ((As i).indicator fun x ↦ 1) μ\n⊢ ∀ᵐ (x : α) ∂μ, ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDense | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 17
} | [
{
"pp": "case succ\nα : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nx : α\nN : ℕ\nihN : ∀ {k : ℕ}, k ≤ N → edist ((nearestPt e N) x) x ≤ edist (e k) x\nk : ℕ\nhk : k ≤ N + 1\n⊢ edist ((nearestPt e (N + 1)) x) x ≤ edist (e k) x",
"usedConsta... | | succ N ihN => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 474,
"column": 2
} | {
"line": 474,
"column": 23
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → (s ∩ Ioc δ ε).Nonempty\nE : Set α\n⊢ cthickening δ E = ⋂ ε ∈ s, cthickening ε E",
"usedConstants": [
"Set.Subset.antisymm",
"Real",
"Set.iInter",
"Membership.mem",
"Set... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 522,
"column": 2
} | {
"line": 522,
"column": 23
} | [
{
"pp": "case neg\nα : Type u\ninst✝ : PseudoEMetricSpace α\nE : Set α\ns : Set ℝ\nhs : ∀ (ε : ℝ), 0 < ε → (s ∩ Ioc 0 ε).Nonempty\nhs₀ : ¬s ⊆ Ioi 0\nδ : ℝ\nhδs : δ ∈ s\nδ_nonpos : δ ≤ 0\n⊢ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E = ⋂ δ ∈ s, cthickening δ E",
"usedConstants": [
"... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 557,
"column": 2
} | {
"line": 557,
"column": 13
} | [
{
"pp": "α : Type u_2\ninst✝ : PseudoMetricSpace α\nx : α\nE : Set α\nhx : x ∈ E\nδ : ℝ\n⊢ {x} ⊆ E",
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"Set.instSingletonSet",
"id",
"HasSubset.Subset",
"Singleton.singleton",
"Eq",
"Set.instMembership",
"Set.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 599,
"column": 49
} | {
"line": 600,
"column": 60
} | [
{
"pp": "δ : ℝ\nα : Type u_2\ninst✝¹ : PseudoMetricSpace α\ninst✝ : ProperSpace α\nE : Set α\nhE : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] E\nhδ : 0 ≤ δ\n⊢ cthickening δ E = ⋃ x ∈ E, closedBall x δ",
"usedConstants": [
"Eq.mpr",
"congrArg",
"PseudoMetricSpace.toUnifor... | by
rw [cthickening_eq_biUnion_closedBall E hδ, hE.closure_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.SimpleFuncDense | {
"line": 217,
"column": 8
} | {
"line": 217,
"column": 51
} | [
{
"pp": "case pos\nX : Type u_3\nY : Type u_4\nα : Type u_5\ninst✝⁷ : Zero α\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : MeasurableSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : OpensMeasurableSpace Y\ninst✝ : PseudoMetricSpace α\nf : X × Y → α\nhf : Conti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDense | {
"line": 220,
"column": 8
} | {
"line": 220,
"column": 23
} | [
{
"pp": "case neg.a\nX : Type u_3\nY : Type u_4\nα : Type u_5\ninst✝⁷ : Zero α\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : MeasurableSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : OpensMeasurableSpace Y\ninst✝ : PseudoMetricSpace α\nf : X × Y → α\nhf : Con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 13
} | [
{
"pp": "α : Type u_6\ninst✝¹ : SeminormedRing α\ninst✝ : NormSMulClass ℤ α\nn : ℕ\n⊢ ‖↑n‖ = ↑n * ‖1‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 13
} | [
{
"pp": "α : Type u_6\ninst✝² : SeminormedRing α\ninst✝¹ : NormOneClass α\ninst✝ : NormSMulClass ℤ α\na : ℕ\n⊢ ‖↑a‖ = ↑a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDense | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 47
} | [
{
"pp": "case pos\nX : Type u_3\nY : Type u_4\nα : Type u_5\ninst✝⁷ : Zero α\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : MeasurableSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : OpensMeasurableSpace Y\ninst✝ : PseudoMetricSpace α\nf : X × Y → α\nhf : Conti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 135,
"column": 16
} | {
"line": 136,
"column": 11
} | [
{
"pp": "E : Type u_6\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\nhr : 0 ≤ r\na : ℝ\nha : a ≥ 0\nha' : a < 2\nr' : ℝ\nhr' : r' ≥ 0\nhr'' : r' < r\n⊢ 2 * r' ≤ diam (ball x r)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 190,
"column": 27
} | {
"line": 190,
"column": 62
} | [
{
"pp": "𝕜✝ : Type u_1\n𝕜' : Type u_2\nE✝ : Type u_3\nF✝ : Type u_4\nα : Type u_5\nF : Type u_6\n𝕜 : Type u_7\nE : Type u_8\nG : Type u_9\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : FunLike F E G\ninst✝ : Line... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Markov | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 45
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f μ\nε : ℝ≥0∞\n⊢ ε * μ {x | ε ≤ f x} ≤ ∫⁻ (a : α), f a ∂μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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