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.IteratedDeriv.Lemmas | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nx : 𝕜\nf g : 𝕜 → F\nhf : ContDiffAt 𝕜 (↑n) f x\nhg : ContDiffAt 𝕜 (↑n) g x\n⊢ iteratedDeriv n (f + g) x = iteratedDeriv n f x + iteratedDeriv n g x",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nx : 𝕜\nf : 𝕜 → F\nhn : 0 < n\nc : F\n⊢ iteratedDeriv n (fun z ↦ c + f z) x = iteratedDeriv n f x",
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddCommG... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 325,
"column": 2
} | {
"line": 325,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nx : 𝕜\nf : 𝕜 → F\nhn : 0 < n\nc : F\n⊢ iteratedDeriv n (fun z ↦ c - f z) x = iteratedDeriv n (-f) x",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 330,
"column": 2
} | {
"line": 330,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nf : 𝕜 → F\na : 𝕜\n⊢ iteratedDeriv n (-f) a = -iteratedDeriv n f a",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Pi.instNeg",
"congrArg"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nx : 𝕜\nf g : 𝕜 → F\nhf : ContDiffAt 𝕜 (↑n) f x\nhg : ContDiffAt 𝕜 (↑n) g x\n⊢ iteratedDeriv n (f - g) x = iteratedDeriv n f x - iteratedDeriv n g x",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nx : 𝕜\nR : Type u_3\ninst✝² : DistribSMul R F\ninst✝¹ : SMulCommClass 𝕜 R F\ninst✝ : ContinuousConstSMul R F\nn : ℕ\nf : 𝕜 → F\nh : ContDiffAt 𝕜 (↑n) f x\nc : R\n⊢ iteratedDer... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 351,
"column": 2
} | {
"line": 351,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nx : 𝕜\n𝕝 : Type u_4\ninst✝³ : DivisionSemiring 𝕝\ninst✝² : Module 𝕝 F\ninst✝¹ : SMulCommClass 𝕜 𝕝 F\ninst✝ : ContinuousConstSMul 𝕝 F\nn : ℕ\nc : 𝕝\nf : 𝕜 → F\n⊢ iteratedD... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 359,
"column": 2
} | {
"line": 359,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nx : 𝕜\n𝕝 : Type u_4\ninst✝³ : DivisionSemiring 𝕝\ninst✝² : Module 𝕝 F\ninst✝¹ : SMulCommClass 𝕜 𝕝 F\ninst✝ : ContinuousConstSMul 𝕝 F\nn : ℕ\nc : 𝕝\nf : 𝕜 → F\n⊢ iteratedD... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 368,
"column": 2
} | {
"line": 368,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝔸 : Type u_5\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nn : ℕ\nf : 𝕜 → 𝔸\nc : 𝔸\nhf : ContDiffAt 𝕜 (↑n) f x\n⊢ iteratedDeriv n (fun x ↦ c * f x) x = c * iteratedDeriv n f x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 376,
"column": 2
} | {
"line": 376,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝕜' : Type u_6\ninst✝¹ : NormedDivisionRing 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nn : ℕ\nc : 𝕜'\nf : 𝕜 → 𝕜'\n⊢ iteratedDeriv n (fun x ↦ c * f x) x = c * iteratedDeriv n f x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 384,
"column": 2
} | {
"line": 384,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝕜' : Type u_6\ninst✝¹ : NormedDivisionRing 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nn : ℕ\nf : 𝕜 → 𝕜'\nc : 𝕜'\n⊢ iteratedDeriv n (fun x ↦ f x * c) x = iteratedDeriv n f x * c",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 395,
"column": 2
} | {
"line": 395,
"column": 45
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nf : 𝕜 → F\nh : ContDiff 𝕜 (↑n) f\nc x : 𝕜\n⊢ iteratedDeriv n (fun x ↦ f (c * x)) x = c ^ n • iteratedDeriv n f (c * x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 401,
"column": 2
} | {
"line": 401,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nn : ℕ\nf : 𝕜 → 𝕜\nh : ContDiff 𝕜 (↑n) f\nc : 𝕜\n⊢ (iteratedDeriv n fun x ↦ f (c * x)) = fun x ↦ c ^ n * iteratedDeriv n f (c * x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 421,
"column": 2
} | {
"line": 421,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nn : ℕ\nx : 𝕜\n𝔸 : Type u_5\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nf g : 𝕜 → 𝔸\nhf : ContDiffAt 𝕜 (↑n) f x\nhg : ContDiffAt 𝕜 (↑n) g x\n⊢ iteratedDeriv n (f * g) x =\n ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * iteratedDeriv i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 427,
"column": 2
} | {
"line": 427,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\nm k : ℕ\n⊢ iteratedDeriv k (fun x ↦ x ^ m) x = ↑(m.descFactorial k) * x ^ (m - k)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 461,
"column": 4
} | {
"line": 461,
"column": 45
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn✝ : ℕ\nf : 𝕜 → F\ns : 𝕜\nn : ℕ\nIH : (iteratedDeriv n fun z ↦ f (s + z)) = fun t ↦ iteratedDeriv n f (s + t)\n⊢ (iteratedDeriv (n + 1) fun z ↦ f (s + z)) = fun t ↦ it... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 469,
"column": 4
} | {
"line": 469,
"column": 45
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn✝ : ℕ\nf : 𝕜 → F\ns : 𝕜\nn : ℕ\nIH : (iteratedDeriv n fun z ↦ f (z + s)) = fun t ↦ iteratedDeriv n f (t + s)\n⊢ (iteratedDeriv (n + 1) fun z ↦ f (z + s)) = fun t ↦ it... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 477,
"column": 2
} | {
"line": 477,
"column": 72
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nf : 𝕜 → F\ns : 𝕜\n⊢ (iteratedDeriv n fun z ↦ f (s - z)) = fun t ↦ (-1) ^ n • iteratedDeriv n f (s - t)",
"usedConstants": [
"NormedCommRing.toNormedRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 505,
"column": 2
} | {
"line": 505,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nι : Type u_7\nn : ℕ\nx : 𝕜\nf : ι → 𝕜 → F\nI : Finset ι\ns : Set 𝕜\nhx : x ∈ s\nhs : UniqueDiffOn 𝕜 s\nhf : ∀ i ∈ I, ContDiffWithinAt 𝕜 (↑n) (f i) s x\n⊢ iteratedDerivWithin n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 509,
"column": 2
} | {
"line": 509,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nι : Type u_7\nn : ℕ\nx : 𝕜\nf : ι → 𝕜 → F\nI : Finset ι\nhf : ∀ i ∈ I, ContDiffAt 𝕜 (↑n) (f i) x\n⊢ iteratedDeriv n (∑ i ∈ I, f i) x = ∑ i ∈ I, iteratedDeriv n (f i) x",
"us... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 513,
"column": 2
} | {
"line": 513,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nι : Type u_7\nn : ℕ\nx : 𝕜\nf : ι → 𝕜 → F\nI : Finset ι\nhf : ∀ i ∈ I, ContDiffAt 𝕜 (↑n) (f i) x\n⊢ iteratedDeriv n (fun z ↦ ∑ i ∈ I, f i z) x = ∑ i ∈ I, iteratedDeriv n (f i) x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Polar | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 15
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ns : Set E\nx : E\nhx : x ∈ s\ny : F\nhy : y ∈ B.polar s\n⊢ ‖(B x) y‖ ≤ 1",
"usedConstants": []
}
] | exact hy x hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.LocallyConvex.Polar | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 35
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ns : Set E\nx : F\nhx : ∀ (i : Set E), i.Finite → i ⊆ s → x ∈ B.polar i\na : E\nha : a ∈ s\n⊢ ‖(B a) x‖ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Polar | {
"line": 164,
"column": 8
} | {
"line": 164,
"column": 18
} | [
{
"pp": "case h.mp\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\nS : Type u_4\ninst✝¹ : SetLike S E\ninst✝ : SMulMemClass S 𝕜 E\nm : S\ny : F\nhy : y ∈ B.... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.Polar | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 15
} | [
{
"pp": "case h.mp\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\nS : Type u_4\ninst✝¹ : SetLike S E\ninst✝ : SMulMemClass S 𝕜 E\nm : S\ny : F\nhy : y ∈ B.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 47
} | [
{
"pp": "x : ℝ\nh : |x| < 1\nn : ℕ\nF : ℝ → ℝ := fun x ↦ ∑ i ∈ Finset.range n, x ^ (i + 1) / (↑i + 1) + log (1 - x)\nF' : ℝ → ℝ := fun x ↦ -x ^ n / (1 - x)\nA : ∀ y ∈ Set.Ioo (-1) 1, HasDerivAt F (F' y) y\nB : ∀ y ∈ Set.Icc (-|x|) |x|, |F' y| ≤ |x| ^ n / (1 - |x|)\nC : ‖F x - F 0‖ ≤ |x| ^ n / (1 - |x|) * ‖x - 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 306,
"column": 61
} | {
"line": 306,
"column": 81
} | [
{
"pp": "x : ℝ\nh : |x| < 1\nn : ℕ\nF : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)\nF' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)\nhI : Set.Icc (-|x|) |x| ⊆ Set.Ioo (-1) 1\nA : ∀ y ∈ Set.Ioo (-1) 1, HasDerivAt F (F' y) y\ny : ℝ\nhy : y ∈ Set.Icc (-|... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 307,
"column": 68
} | {
"line": 307,
"column": 88
} | [
{
"pp": "case hd\nx : ℝ\nh : |x| < 1\nn : ℕ\nF : ℝ → ℝ := ⋯\nF' : ℝ → ℝ := ⋯\nhI : Set.Icc (-|x|) |x| ⊆ Set.Ioo (-1) 1\nA : ∀ y ∈ Set.Ioo (-1) 1, HasDerivAt F (F' y) y\ny : ℝ\nhy : y ∈ Set.Icc (-|x|) |x|\nthis : y ^ 2 ≤ x ^ 2\n⊢ 0 < 1 - x ^ 2",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_o... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 47
} | [
{
"pp": "x : ℝ\nh : |x| < 1\nn : ℕ\nF : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)\nF' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)\nhI : Set.Icc (-|x|) |x| ⊆ Set.Ioo (-1) 1\nA : ∀ y ∈ Set.Ioo (-1) 1, HasDerivAt F (F' y) y\nB : ∀ y ∈ Set.Icc (-|x|) |x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 332,
"column": 37
} | {
"line": 332,
"column": 52
} | [
{
"pp": "x : ℝ\nh₀ : 0 ≤ x\nh : x < 1\nn : ℕ\nF : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)\nF' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)\nA : ∀ y ∈ Set.Icc 0 x, HasDerivAt F (F' y) y\nthis : MonotoneOn F (Set.Icc 0 x)\n⊢ ∑ i ∈ Finset.range n, x ^... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 374,
"column": 6
} | {
"line": 374,
"column": 28
} | [
{
"pp": "x : ℝ\nh : |x| < 1\ni : ℕ\n⊢ |x| ^ (i + 1) / (0 + 1) ≤ |x| ^ i",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"Real.lattice",
"DivisionCommMonoid.toDivisionMonoid",
"Real.instZero",
"Real.instAddMonoid",
"Mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Cone.Extension | {
"line": 75,
"column": 6
} | {
"line": 75,
"column": 89
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : ConvexCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\nhdom : f.domain ≠ ⊤\ny : E\nhy : y ∉ f.domain\nSp : Set ℝ := ↑f '' {x | ↑x + y ∈ s}\nSn : Set ℝ := ↑f '' {x | -↑x - y ∈ s}\nt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nx : E\nn : ℕ∞ω\ns : Set E\nf g : E → F\nhf : ContDiffWithinAt 𝕜 n f s x\nhg : ContDiffWithinAt 𝕜 n g s x\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 315,
"column": 48
} | {
"line": 315,
"column": 81
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nx : E\nn : ℕ∞ω\nf g : E → F\nhf : ContDiffAt 𝕜 n f x\nhg : ContDiffAt 𝕜 n g x\n⊢ ContDiffAt 𝕜 n (fun x ↦ f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ∞ω\ns : Set E\nf g : E → F\nhf : ContDiffOn 𝕜 n f s\nhg : ContDiffOn 𝕜 n g s\n⊢ ContDiffOn 𝕜 n (fun x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 326,
"column": 42
} | {
"line": 326,
"column": 75
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ∞ω\nf g : E → F\nhf : ContDiff 𝕜 n f\nhg : ContDiff 𝕜 n g\n⊢ ContDiff 𝕜 n fun x ↦ f x - g x",
"use... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 427,
"column": 20
} | {
"line": 427,
"column": 59
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nι : Type u_3\nf : ι → E → F\nu : Finset ι\ni : ℕ\nh : ∀ j ∈ u, ContDiff 𝕜 (↑i) (f j)\nx : E\n⊢ iteratedFDeriv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 476,
"column": 2
} | {
"line": 476,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ns : Set E\nx : E\nn : ℕ∞ω\n𝔸' : Type u_4\nι : Type u_5\ninst✝¹ : NormedCommRing 𝔸'\ninst✝ : NormedAlgebra 𝕜 𝔸'\nt : Finset ι\nf : ι → E → 𝔸'\nh : ∀ i ∈ t, ContDiffWithinAt 𝕜 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 509,
"column": 12
} | {
"line": 509,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nn : ℕ∞ω\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nf : E → 𝔸\nhf : ContDiff 𝕜 n f\n⊢ ContDiff 𝕜 n fun x ↦ f x ^ 0",
"usedConstants": [
"Eq.mp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 509,
"column": 12
} | {
"line": 509,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nn : ℕ∞ω\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nf : E → 𝔸\nhf : ContDiff 𝕜 n f\n⊢ ContDiff 𝕜 n fun x ↦ f x ^ 0",
"usedConstants": [
"Eq.mp... | simpa using contDiff_const | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 509,
"column": 12
} | {
"line": 509,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nn : ℕ∞ω\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nf : E → 𝔸\nhf : ContDiff 𝕜 n f\n⊢ ContDiff 𝕜 n fun x ↦ f x ^ 0",
"usedConstants": [
"Eq.mp... | simpa using contDiff_const | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 509,
"column": 12
} | {
"line": 509,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nn : ℕ∞ω\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nf : E → 𝔸\nhf : ContDiff 𝕜 n f\n⊢ ContDiff 𝕜 n fun x ↦ f x ^ 0",
"usedConstants": [
"Eq.mp... | simpa using contDiff_const | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 510,
"column": 16
} | {
"line": 510,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nn : ℕ∞ω\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nf : E → 𝔸\nhf : ContDiff 𝕜 n f\nm : ℕ\n⊢ ContDiff 𝕜 n fun x ↦ f x ^ (m + 1)",
"usedConstants": [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 529,
"column": 2
} | {
"line": 529,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ns : Set E\nx : E\n𝕜' : Type u_6\ninst✝¹ : NormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nf : E → 𝕜'\nn : ℕ∞ω\nhf : ContDiffWithinAt 𝕜 n f s x\nc : 𝕜'\n⊢ ContDiffWithinAt 𝕜 n (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 542,
"column": 40
} | {
"line": 542,
"column": 73
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\n𝕜' : Type u_6\ninst✝¹ : NormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nf : E → 𝕜'\nn : ℕ∞ω\nhf : ContDiff 𝕜 n f\nc : 𝕜'\n⊢ ContDiff 𝕜 n fun x ↦ f x / c",
"usedConstants": ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 740,
"column": 2
} | {
"line": 740,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nn : ℕ∞ω\nE' : Type u_3\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\nF' : Type u_4\ninst✝¹ : N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 790,
"column": 2
} | {
"line": 790,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\n𝕜' : Type u_4\ninst✝¹ : NormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nx : 𝕜'\nhx : x ≠ 0\nn : ℕ∞ω\n⊢ ContDiffAt 𝕜 n Inv.inv x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 822,
"column": 2
} | {
"line": 822,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\nx : E\nf g : E → 𝕜\nn : ℕ∞ω\nhf : ContDiffWithinAt 𝕜 n f s x\nhg : ContDiffWithinAt 𝕜 n g s x\nhx : g x ≠ 0\n⊢ ContDiffWithinAt 𝕜 n (fun x ↦ f x / g x) s x",
"use... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Cone.Extension | {
"line": 148,
"column": 71
} | {
"line": 153,
"column": 35
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : ConvexCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\n⊢ ∃ g, (∀ (x : ↥f.domain), g ↑x = ↑f x) ∧ ∀ x ∈ s, 0 ≤ g x",
"usedConstants": [
"LinearMap.id",
"Submodule",
... | by
rcases RieszExtension.exists_top s f nonneg dense
with ⟨⟨g_dom, g⟩, ⟨-, hfg⟩, rfl : g_dom = ⊤, hgs⟩
refine ⟨g.comp (LinearMap.id.codRestrict ⊤ fun _ ↦ trivial), ?_, ?_⟩
· exact fun x => (hfg rfl).symm
· exact fun x hx => hgs ⟨x, _⟩ hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Cone.Extension | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 20
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nf : E →ₗ.[ℝ] ℝ\nN : E → ℝ\nN_hom : ∀ (c : ℝ), 0 < c → ∀ (x : E), N (c • x) = c * N x\nN_add : ∀ (x y : E), N (x + y) ≤ N x + N y\nhf : ∀ (x : ↥f.domain), ↑f x ≤ N ↑x\ns : ConvexCone ℝ (E × ℝ) := { carrier := {p | N p.1 ≤ p.2}, smul_mem' := ⋯, a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Cone.Extension | {
"line": 184,
"column": 33
} | {
"line": 184,
"column": 44
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nf : E →ₗ.[ℝ] ℝ\nN : E → ℝ\nN_hom : ∀ (c : ℝ), 0 < c → ∀ (x : E), N (c • x) = c * N x\nN_add : ∀ (x y : E), N (x + y) ≤ N x + N y\nhf : ∀ (x : ↥f.domain), ↑f x ≤ N ↑x\ns : ConvexCone ℝ (E × ℝ) := { carrier := {p | N p.1 ≤ p.2}, smul_mem' := ⋯, a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Cone.Extension | {
"line": 185,
"column": 20
} | {
"line": 185,
"column": 31
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nf : E →ₗ.[ℝ] ℝ\nN : E → ℝ\nN_hom : ∀ (c : ℝ), 0 < c → ∀ (x : E), N (c • x) = c * N x\nN_add : ∀ (x y : E), N (x + y) ≤ N x + N y\nhf : ∀ (x : ↥f.domain), ↑f x ≤ N ↑x\ns : ConvexCone ℝ (E × ℝ) := { carrier := {p | N p.1 ≤ p.2}, smul_mem' := ⋯, a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 309,
"column": 48
} | {
"line": 309,
"column": 59
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nc : 𝕜\nhc : 1 < ‖c‖\nx ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.Cone.Basic | {
"line": 104,
"column": 18
} | {
"line": 104,
"column": 34
} | [
{
"pp": "𝕜 : Type u_1\nR : Type u_2\nG : Type u_3\nM : Type u_4\nN : Type u_5\nO : Type u_6\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : AddCommMonoid M\ninst✝ : SMul R M\nC✝ C₁ C₂ : ConvexCone R M\ns✝ : Set M\nc : R\nx : M\nC : ConvexCone R M\ns : Set M\nhs : s = ↑C\n⊢ ∀ ⦃c : R⦄, 0 < c → ∀ ⦃x : M⦄,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.Cone.Basic | {
"line": 323,
"column": 29
} | {
"line": 323,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\nR : Type u_2\nG : Type u_3\nM : Type u_4\nN : Type u_5\nO : Type u_6\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : AddCommGroup G\ninst✝ : SMul R G\nC✝ C₁ C₂ C : ConvexCone R G\nh₁ : C.Pointed\nx y z : G\nxy : y - x ∈ C\nzy : z - y ∈ C\n⊢ z - x ∈ C",
"usedConstants": []
}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.Cone.Basic | {
"line": 452,
"column": 2
} | {
"line": 452,
"column": 96
} | [
{
"pp": "R : Type u_2\nM : Type u_4\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ ⊥.IsGenerating ↔ Subsingleton M",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Submodule.subsingleton_iff",
"congrArg",
"DistribMulAction.toDistribSM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.Cone.Basic | {
"line": 491,
"column": 16
} | {
"line": 493,
"column": 89
} | [
{
"pp": "R : Type u_7\nM : Type u_8\ninst✝⁵ : Ring R\ninst✝⁴ : LinearOrder R\ninst✝³ : AddLeftStrictMono R\ninst✝² : AddCommGroup M\ninst✝¹ : Nontrivial M\ninst✝ : Module R M\nC : ConvexCone R M\nh : Submodule.span R ↑C = ⊤\nx : M\nhne : (↑C).Nonempty\n⊢ ∀ {a b : M}, a ∈ ↑C - ↑C → b ∈ ↑C - ↑C → a + b ∈ ↑C - ↑C"... | by
rintro _ _ ⟨y₁, hy₁, z₁, hz₁, rfl⟩ ⟨y₂, hy₂, z₂, hz₂, rfl⟩
exact ⟨y₁ + y₂, C.add_mem hy₁ hy₂, z₁ + z₂, C.add_mem hz₁ hz₂, add_sub_add_comm ..⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 330,
"column": 8
} | {
"line": 330,
"column": 99
} | [
{
"pp": "case hz\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nc : 𝕜\nhc : 1 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 335,
"column": 10
} | {
"line": 335,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nc : 𝕜\nhc : 1 < ‖c‖\nx ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.Cone.Basic | {
"line": 639,
"column": 4
} | {
"line": 639,
"column": 15
} | [
{
"pp": "R : Type u_2\ninst✝⁶ : Semiring R\ninst✝⁵ : PartialOrder R\nG : Type u_7\ninst✝⁴ : AddCommGroup G\ninst✝³ : PartialOrder G\ninst✝² : IsOrderedAddMonoid G\ninst✝¹ : Module R G\ninst✝ : PosSMulMono R G\nx : G\nhx_nonneg : x ∈ positive R G\nhx_ne_zero : x ≠ 0\nhx_nonpos : -x ∈ positive R G\n⊢ 0 < 0",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.AbsConvex | {
"line": 181,
"column": 22
} | {
"line": 181,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : SeminormedRing 𝕜\ninst✝³ : SMul 𝕜 E\ninst✝² : AddCommMonoid E\ninst✝¹ : PartialOrder 𝕜\ninst✝ : TopologicalSpace E\ns : Set E\n⊢ (closedAbsConvexHull 𝕜) (closure[inst✝] s) ⊆ (closedAbsConvexHull 𝕜) s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 61
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nf : E → F\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : CompleteSpace F\n⊢ Measurab... | have : IsComplete (univ : Set (E →L[𝕜] F)) := complete_univ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 402,
"column": 2
} | {
"line": 402,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : CompleteSpace F\ninst✝³ : MeasurableSpace 𝕜\ninst✝² : OpensMeasurableSpace 𝕜\ninst✝¹ : MeasurableSpace F\ninst✝ : BorelSpace F\nf : 𝕜 → F\n⊢ Measurable (deriv f)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 464,
"column": 4
} | {
"line": 464,
"column": 64
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nr s ε x : ℝ\nhx : x ∈ B f K r s ε\n⊢ ∃ L ∈ K, x ∈ A f L r ε ∧ x ∈ A f L s ε",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 68,
"column": 4
} | {
"line": 70,
"column": 48
} | [
{
"pp": "case refine_2\nE : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : Module ℝ E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhs₀ : 0 ∈ s\nhs₁ : Convex ℝ s\nhs₂ : IsOpen s\nx₀ : E\nhx₀ : x₀ ∉ s\nf : E →ₗ.[ℝ] ℝ := LinearPMap.mkSpanSingleton x₀ 1 ⋯\nφ : ... | refine
φ.continuous_of_nonzero_on_open _ (hs₂.vadd (-x₀)) (Nonempty.vadd_set ⟨0, hs₀⟩)
(vadd_set_subset_iff.mpr fun x hx => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 51
} | [
{
"pp": "case inr.inl\nE : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns : Set E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsOpen s\na₀ : E\nha₀ : a₀ ∈ s\nht : Convex ℝ ∅\ndisj : Disjoint s ∅\n⊢ ∃ f u, (∀ a ∈ s, f a < u) ∧ ∀ b... | exact ⟨0, 1, fun a _ha => zero_lt_one, by simp⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 51
} | [
{
"pp": "case inr.inl\nE : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns : Set E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsOpen s\na₀ : E\nha₀ : a₀ ∈ s\nht : Convex ℝ ∅\ndisj : Disjoint s ∅\n⊢ ∃ f u, (∀ a ∈ s, f a < u) ∧ ∀ b... | exact ⟨0, 1, fun a _ha => zero_lt_one, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 51
} | [
{
"pp": "case inr.inl\nE : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns : Set E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsOpen s\na₀ : E\nha₀ : a₀ ∈ s\nht : Convex ℝ ∅\ndisj : Disjoint s ∅\n⊢ ∃ f u, (∀ a ∈ s, f a < u) ∧ ∀ b... | exact ⟨0, 1, fun a _ha => zero_lt_one, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Gauge | {
"line": 122,
"column": 54
} | {
"line": 122,
"column": 65
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nsymmetric : ∀ x ∈ s, -x ∈ s\nx✝ x : E\nh : -x ∈ s\n⊢ x ∈ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Gauge | {
"line": 203,
"column": 2
} | {
"line": 207,
"column": 82
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nhs : Convex ℝ s\nh₀ : 0 ∈ s\nabsorbs : Absorbent ℝ s\na : ℝ\n⊢ Convex ℝ {x | gauge s x ≤ a}",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"instHSMul",
... | by_cases ha : 0 ≤ a
· rw [gauge_le_eq hs h₀ absorbs ha]
exact convex_iInter fun i => convex_iInter fun _ => hs.smul _
· convert! convex_empty (𝕜 := ℝ)
exact eq_empty_iff_forall_notMem.2 fun x hx => ha <| (gauge_nonneg _).trans hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Gauge | {
"line": 203,
"column": 2
} | {
"line": 207,
"column": 82
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nhs : Convex ℝ s\nh₀ : 0 ∈ s\nabsorbs : Absorbent ℝ s\na : ℝ\n⊢ Convex ℝ {x | gauge s x ≤ a}",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"instHSMul",
... | by_cases ha : 0 ≤ a
· rw [gauge_le_eq hs h₀ absorbs ha]
exact convex_iInter fun i => convex_iInter fun _ => hs.smul _
· convert! convex_empty (𝕜 := ℝ)
exact eq_empty_iff_forall_notMem.2 fun x hx => ha <| (gauge_nonneg _).trans hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 164,
"column": 33
} | {
"line": 164,
"column": 52
} | [
{
"pp": "E : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns t : Set E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\nhs : Convex ℝ s\nht : Convex ℝ t\nhst : Disjoint (interior s) t\nf : StrongDual ℝ E\nu : ℝ\nhfA : ∀ a ∈ interior s, f a < u\nhfB : ∀ b ∈ t,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 165,
"column": 33
} | {
"line": 165,
"column": 52
} | [
{
"pp": "E : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns t : Set E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\nhs : Convex ℝ s\nht : Convex ℝ t\nhst : Disjoint (interior s) t\nf : StrongDual ℝ E\nu : ℝ\nhfA : ∀ a ∈ interior s, f a < u\nhfB : ∀ b ∈ t,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 75
} | [
{
"pp": "case refine_2.a\nE : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns t : Set E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\nhs : Convex ℝ s\nht : Convex ℝ t\nhst : Disjoint (interior s) t\nhsint : (interior s).Nonempty\nhtne : t.Nonempty\nf : Str... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 221,
"column": 18
} | {
"line": 221,
"column": 29
} | [
{
"pp": "E : Type u_2\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ns✝ t✝ : Set E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s✝\nhs₂ : IsClosed s✝\nht₁ : Convex ℝ t✝\nht₂ : IsCompact t✝\ndisj : Disjoint s✝ t✝\nf : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 221,
"column": 37
} | {
"line": 221,
"column": 48
} | [
{
"pp": "E : Type u_2\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ns✝ t✝ : Set E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s✝\nhs₂ : IsClosed s✝\nht₁ : Convex ℝ t✝\nht₂ : IsCompact t✝\ndisj : Disjoint s✝ t✝\nf : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 221,
"column": 56
} | {
"line": 221,
"column": 67
} | [
{
"pp": "E : Type u_2\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ns✝ t✝ : Set E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s✝\nhs₂ : IsClosed s✝\nht₁ : Convex ℝ t✝\nht₂ : IsCompact t✝\ndisj : Disjoint s✝ t✝\nf : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 37
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ns t : Set E\ninst✝⁴ : RCLike 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nhs₁ : Convex ℝ s\nhs₂ : IsOpen s\nht : Con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 37
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ns t : Set E\ninst✝⁴ : RCLike 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nhs₁ : Convex ℝ s\nhs₂ : IsOpen s\nht₁ : Co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 305,
"column": 68
} | {
"line": 305,
"column": 79
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ns t : Set E\ninst✝⁴ : RCLike 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nhs : Convex ℝ s\nht : Convex ℝ t\nhst : Disjoint (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 306,
"column": 4
} | {
"line": 306,
"column": 39
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ns t : Set E\ninst✝⁴ : RCLike 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nhs : Convex ℝ s\nht : Convex ℝ t\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 39
} | [
{
"pp": "case refine_3\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ns t : Set E\ninst✝⁴ : RCLike 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nhs : Convex ℝ s\nht : Convex ℝ t\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Gauge | {
"line": 379,
"column": 2
} | {
"line": 386,
"column": 15
} | [
{
"pp": "E : Type u_2\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns : Set E\nx : E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\nhc : Convex ℝ s\nhs₀ : 0 ∈ s\nha : Absorbent ℝ s\nh : gauge s x ≤ 1\n⊢ x ∈ closure[inst✝¹] s",
"usedConstants": [
"Ico_mem_nhdsLT",
"Real.instIsOrderedR... | have : ∀ᶠ r : ℝ in 𝓝[<] 1, r • x ∈ s := by
filter_upwards [Ico_mem_nhdsLT one_pos] with r ⟨hr₀, hr₁⟩
apply gauge_lt_one_subset_self hc hs₀ ha
rw [mem_setOf_eq, gauge_smul_of_nonneg hr₀]
exact mul_lt_one_of_nonneg_of_lt_one_left hr₀ hr₁ h
refine mem_closure_of_tendsto ?_ this
exact Filter.Tendsto.mo... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Gauge | {
"line": 379,
"column": 2
} | {
"line": 386,
"column": 15
} | [
{
"pp": "E : Type u_2\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns : Set E\nx : E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\nhc : Convex ℝ s\nhs₀ : 0 ∈ s\nha : Absorbent ℝ s\nh : gauge s x ≤ 1\n⊢ x ∈ closure[inst✝¹] s",
"usedConstants": [
"Ico_mem_nhdsLT",
"Real.instIsOrderedR... | have : ∀ᶠ r : ℝ in 𝓝[<] 1, r • x ∈ s := by
filter_upwards [Ico_mem_nhdsLT one_pos] with r ⟨hr₀, hr₁⟩
apply gauge_lt_one_subset_self hc hs₀ ha
rw [mem_setOf_eq, gauge_smul_of_nonneg hr₀]
exact mul_lt_one_of_nonneg_of_lt_one_left hr₀ hr₁ h
refine mem_closure_of_tendsto ?_ this
exact Filter.Tendsto.mo... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 333,
"column": 68
} | {
"line": 333,
"column": 79
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\nx : E\ninst✝⁴ : RCLike 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nA : Set E\nhA : Convex ℝ A\nhxA : x ∉ interior A\nhAint ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Gauge | {
"line": 427,
"column": 6
} | {
"line": 427,
"column": 31
} | [
{
"pp": "E : Type u_2\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ns : Set E\nx : E\ninst✝² : TopologicalSpace E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\nhc : Convex ℝ s\nhs₀ : s ∈ 𝓝 0\nha : Absorbent ℝ s\nε : ℝ\nhε₀ : 0 < ε\n⊢ ∀ᶠ (x_1 : E) in 𝓝 x, gauge s x_1 ∈ Icc (gauge s x - ε) (ga... | ← map_add_left_nhds_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 334,
"column": 4
} | {
"line": 334,
"column": 39
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\nx : E\ninst✝⁴ : RCLike 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nA : Set E\nhA : Convex ℝ A\nhxA : x ∉ int... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 54
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module ℝ E\ns t : Set E\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 350,
"column": 18
} | {
"line": 350,
"column": 29
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module ℝ E\ns✝ t✝ : Set E\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 350,
"column": 37
} | {
"line": 350,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module ℝ E\ns✝ t✝ : Set E\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 350,
"column": 56
} | {
"line": 350,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module ℝ E\ns✝ t✝ : Set E\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.SeparatingDual | {
"line": 76,
"column": 15
} | {
"line": 76,
"column": 40
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup V\ninst✝³ : TopologicalSpace V\ninst✝² : TopologicalSpace R\ninst✝¹ : Module R V\ninst✝ : SeparatingDual R V\nx y : V\nh : x ≠ y\nf : StrongDual R V\nhf : f (x - y) ≠ 0\n⊢ f x ≠ f y",
"usedConstants": [
"Semiring.toModule",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 399,
"column": 4
} | {
"line": 399,
"column": 60
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module ℝ E\ns : Set E\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : LocallyConvexSpace ℝ E\ninst✝ : Hereditari... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Dual | {
"line": 62,
"column": 8
} | {
"line": 62,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\n⊢ IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace]\n (((topDualPairing 𝕜 E).flip.flip.polar ∘ ⇑OrderDual.ofDual) ((topDualPairing 𝕜 E).flip.polar s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.SeparatingDual | {
"line": 176,
"column": 23
} | {
"line": 176,
"column": 39
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝¹⁵ : Field R\ninst✝¹⁴ : AddCommGroup V\ninst✝¹³ : TopologicalSpace R\ninst✝¹² : TopologicalSpace V\ninst✝¹¹ : IsTopologicalRing R\ninst✝¹⁰ : Module R V\ninst✝⁹ : SeparatingDual R V\ninst✝⁸ : IsTopologicalAddGroup V\ninst✝⁷ : ContinuousSMul R V\nS : Type u_3\ninst✝⁶ : Co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Dual | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 28
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_4\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\nhr : 0 < r\nx' : StrongDual 𝕜 E\nh : x' ∈ polar 𝕜 (closedBall 0 r)\nz : E\nx✝ : z ∈ ball 0 r\n⊢ ‖x' z‖ ≤ r⁻¹ * ‖z‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.SeparatingDual | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 63
} | [
{
"pp": "R : Type u_4\nV : Type u_5\nW : Type u_6\ninst✝¹³ : NormedField R\ninst✝¹² : AddCommGroup V\ninst✝¹¹ : AddCommGroup W\ninst✝¹⁰ : TopologicalSpace R\ninst✝⁹ : TopologicalSpace V\ninst✝⁸ : TopologicalSpace W\ninst✝⁷ : IsTopologicalRing R\ninst✝⁶ : Module R V\ninst✝⁵ : Module R W\ninst✝⁴ : SeparatingDual ... | ← ContinuousLinearMap.comp_assoc _ f.toContinuousLinearMap, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.LocallyConvex.SeparatingDual | {
"line": 194,
"column": 58
} | {
"line": 194,
"column": 74
} | [
{
"pp": "R : Type u_4\nV : Type u_5\nW : Type u_6\ninst✝¹³ : NormedField R\ninst✝¹² : AddCommGroup V\ninst✝¹¹ : AddCommGroup W\ninst✝¹⁰ : TopologicalSpace R\ninst✝⁹ : TopologicalSpace V\ninst✝⁸ : TopologicalSpace W\ninst✝⁷ : IsTopologicalRing R\ninst✝⁶ : Module R V\ninst✝⁵ : Module R W\ninst✝⁴ : SeparatingDual ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 42
} | [
{
"pp": "case inl\nι : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace E\ninst✝² : Countable ι\nf : ι → X → E\nhE : Nonempty E\ninst✝¹ : IsCompletelyMetrizableSpace E\nhf : ∀ (i : ι), StronglyMeasurable (f i)\ninst✝ : ⊥.IsCountablyGenerated\n⊢ StronglyMeasurable fun x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.SeparatingDual | {
"line": 195,
"column": 52
} | {
"line": 195,
"column": 79
} | [
{
"pp": "R : Type u_4\nV : Type u_5\nW : Type u_6\ninst✝¹³ : NormedField R\ninst✝¹² : AddCommGroup V\ninst✝¹¹ : AddCommGroup W\ninst✝¹⁰ : TopologicalSpace R\ninst✝⁹ : TopologicalSpace V\ninst✝⁸ : TopologicalSpace W\ninst✝⁷ : IsTopologicalRing R\ninst✝⁶ : Module R V\ninst✝⁵ : Module R W\ninst✝⁴ : SeparatingDual ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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