module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.Convex.Set | {
"line": 200,
"column": 45
} | {
"line": 200,
"column": 61
} | {
"line": 200,
"column": 62
} | [
{
"pp": "K : Type u_4\nX : Type u_5\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : ConvexSpace K X\ns : Set X\nhs : ∀ (a b : K) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1), ∀ x ∈ s, ∀ y ∈ s, convexCombPair a b ha hb hab x y ∈ s\nt✝ : Finset X\nx : X\nt : Finset X\nhx : x ∉ t\... | [
"K : Type u_4\nX : Type u_5\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : ConvexSpace K X\ns : Set X\nhs : ∀ (a b : K) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1), ∀ x ∈ s, ∀ y ∈ s, convexCombPair a b ha hb hab x y ∈ s\nt✝ : Finset X\nx : X\nt : Finset X\nhx : x ∉ t\nih : ∀ ⦃w :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Quasiconvex | {
"line": 217,
"column": 47
} | {
"line": 217,
"column": 73
} | {
"line": 217,
"column": 74
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_3\ninst✝⁴ : Semiring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : LinearOrder β\ninst✝ : SMul 𝕜 E\ns : Set E\nf : E → β\n⊢ (Convex 𝕜 s ∧\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_3\ninst✝⁴ : Semiring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : LinearOrder β\ninst✝ : SMul 𝕜 E\ns : Set E\nf : E → β\n⊢ ((Convex 𝕜 s ∧\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • ... | quasiconcaveOn_iff_min_le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Radon | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 37
} | {
"line": 55,
"column": 38
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nf : ι → E\ns : Finset ι\nw : ι → 𝕜\nh_wsum : s.sum w = 0\nh_vsum : ∑ e ∈ s, w e • f e = 0\nnonzero_w_index : ι\nh1 : nonzero_w_index ∈ s... | [
"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nf : ι → E\ns : Finset ι\nw : ι → 𝕜\nh_wsum : s.sum w = 0\nh_vsum : ∑ e ∈ s, w e • f e = 0\nnonzero_w_index : ι\nh1 : nonzero_w_index ∈ s\nh2 : w non... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Radon | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 39
} | {
"line": 62,
"column": 40
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nf : ι → E\ns : Finset ι\nw : ι → 𝕜\nh_wsum : s.sum w = 0\nh_vsum : ∑ e ∈ s, w e • f e = 0\nnonzero_w_index : ι\nh1 : nonzero_w_index ∈ s... | [
"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nf : ι → E\ns : Finset ι\nw : ι → 𝕜\nh_wsum : s.sum w = 0\nh_vsum : ∑ e ∈ s, w e • f e = 0\nnonzero_w_index : ι\nh1 : nonzero_w_index ∈ s\nh2 : w non... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 72,
"column": 2
} | {
"line": 73,
"column": 9
} | {
"line": 73,
"column": 10
} | [
{
"pp": "X : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\nι : Type u_3\nf : StdSimplex ℝ ι\nx y : ι → X\n⊢ dist (iConvexComb f x) (iConvexComb f y) ≤ iConvexComb f fun i ↦ dist (x i) (y i)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"X : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\nι : Type u_3\nf : StdSimplex ℝ ι\nx y : ι → X\n⊢ dist (iConvexComb f x) (iConvexComb f y) ≤ f.weights.sum fun i r ↦ r * dist (x i) (y i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 13
} | {
"line": 79,
"column": 14
} | [
{
"pp": "I : Type u_1\nX : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\nf : StdSimplex ℝ I\ng : I → X\nx : X\n⊢ dist (iConvexComb f g) x ≤ iConvexComb f fun i ↦ dist (g i) x",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAss... | [
"I : Type u_1\nX : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\nf : StdSimplex ℝ I\ng : I → X\nx : X\n⊢ dist (iConvexComb f g) x ≤ f.weights.sum fun i r ↦ r * dist (g i) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 13
} | {
"line": 83,
"column": 14
} | [
{
"pp": "I : Type u_1\nX : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\nx : X\nf : StdSimplex ℝ I\ng : I → X\n⊢ dist x (iConvexComb f g) ≤ iConvexComb f fun i ↦ dist x (g i)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAss... | [
"I : Type u_1\nX : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\nx : X\nf : StdSimplex ℝ I\ng : I → X\n⊢ dist x (iConvexComb f g) ≤ f.weights.sum fun i r ↦ r * dist x (g i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 13
} | {
"line": 87,
"column": 14
} | [
{
"pp": "X : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\nf : StdSimplex ℝ X\nx : X\n⊢ dist (sConvexComb f) x ≤ iConvexComb f fun x_1 ↦ dist x_1 x",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidW... | [
"X : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\nf : StdSimplex ℝ X\nx : X\n⊢ dist (sConvexComb f) x ≤ f.weights.sum fun i r ↦ r * dist i x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 13
} | {
"line": 91,
"column": 14
} | [
{
"pp": "X : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\nx : X\nf : StdSimplex ℝ X\n⊢ dist x (sConvexComb f) ≤ iConvexComb f (dist x)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [
"X : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\nx : X\nf : StdSimplex ℝ X\n⊢ dist x (sConvexComb f) ≤ f.weights.sum fun i r ↦ r * dist x i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 124,
"column": 17
} | {
"line": 124,
"column": 42
} | {
"line": 124,
"column": 42
} | [
{
"pp": "X : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\ns t : ℝ\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nx y : X\n⊢ dist (convexCombPair s t hs ht h x y) y = s * dist x y",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.par... | [
"X : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\ns t : ℝ\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nx y : X\n⊢ s * dist x y = s * dist x y"
] | dist_convexCombPair_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 179,
"column": 40
} | {
"line": 179,
"column": 51
} | {
"line": 179,
"column": 52
} | [
{
"pp": "I : Type u_1\nX : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\nx : ↑(Set.Icc 0 1) × X × X\n⊢ 0 ≤ 1 - ↑x.1",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"Real.instZero",
"R... | [
"I : Type u_1\nX : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\nx : ↑(Set.Icc 0 1) × X × X\n⊢ ↑x.1 ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 187,
"column": 12
} | {
"line": 187,
"column": 23
} | {
"line": 187,
"column": 24
} | [
{
"pp": "case ha\nX : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\ni : ↑(Set.Icc 0 1)\nx y : X × X\n⊢ 0 ≤ ↑i",
"ppTerm": "?ha✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case ha\nX : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\ni : ↑(Set.Icc 0 1)\nx y : X × X\n⊢ 0 ≤ ↑i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 189,
"column": 12
} | {
"line": 189,
"column": 23
} | {
"line": 189,
"column": 24
} | [
{
"pp": "case ha\nX : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\ni : ↑(Set.Icc 0 1)\nx y : X × X\n⊢ 0 ≤ 1 - ↑i",
"ppTerm": "?ha✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"Real.instSub",
"covariant... | [
"case ha\nX : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\ni : ↑(Set.Icc 0 1)\nx y : X × X\n⊢ ↑i ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 204,
"column": 66
} | {
"line": 204,
"column": 77
} | {
"line": 204,
"column": 78
} | [
{
"pp": "I : Type u_1\nX : Type u_2\ninst✝³ : ConvexSpace ℝ X\ninst✝² : MetricSpace X\ninst✝¹ : IsConvexDist X\nT : Type u_3\ninst✝ : TopologicalSpace T\nf : T → ℝ\nhf : Continuous f\nhf0 : ∀ (t : T), 0 ≤ f t\nhf1 : ∀ (t : T), f t ≤ 1\nx y : T → X\nhx : ContinuousOn x (f ⁻¹' {0}ᶜ)\nhy : ContinuousOn y (f ⁻¹' {1... | [
"I : Type u_1\nX : Type u_2\ninst✝³ : ConvexSpace ℝ X\ninst✝² : MetricSpace X\ninst✝¹ : IsConvexDist X\nT : Type u_3\ninst✝ : TopologicalSpace T\nf : T → ℝ\nhf : Continuous f\nhf0 : ∀ (t : T), 0 ≤ f t\nhf1 : ∀ (t : T), f t ≤ 1\nx y : T → X\nhx : ContinuousOn x (f ⁻¹' {0}ᶜ)\nhy : ContinuousOn y (f ⁻¹' {1}ᶜ)\nhx' : B... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 65
} | {
"line": 218,
"column": 66
} | [
{
"pp": "X : Type u_2\ninst✝³ : ConvexSpace ℝ X\ninst✝² : MetricSpace X\ninst✝¹ : IsConvexDist X\nT : Type u_3\ninst✝ : TopologicalSpace T\nf : T → ℝ\nhf : Continuous f\nhf0 : ∀ (t : T), 0 ≤ f t\nhf1 : ∀ (t : T), f t ≤ 1\nx y : T → X\nhx : ContinuousOn x (f ⁻¹' {0}ᶜ)\nhy : ContinuousOn y (f ⁻¹' {1}ᶜ)\nhx' : Bor... | [
"X : Type u_2\ninst✝³ : ConvexSpace ℝ X\ninst✝² : MetricSpace X\ninst✝¹ : IsConvexDist X\nT : Type u_3\ninst✝ : TopologicalSpace T\nf : T → ℝ\nhf : Continuous f\nhf0 : ∀ (t : T), 0 ≤ f t\nhf1 : ∀ (t : T), f t ≤ 1\nx y : T → X\nhx : ContinuousOn x (f ⁻¹' {0}ᶜ)\nhy : ContinuousOn y (f ⁻¹' {1}ᶜ)\nhx' : Bornology.IsBou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 226,
"column": 69
} | {
"line": 226,
"column": 80
} | {
"line": 226,
"column": 81
} | [
{
"pp": "X : Type u_2\ninst✝³ : ConvexSpace ℝ X\ninst✝² : MetricSpace X\ninst✝¹ : IsConvexDist X\nT : Type u_3\ninst✝ : TopologicalSpace T\nf : T → ℝ\nhf : Continuous f\nhf0 : ∀ (t : T), 0 ≤ f t\nhf1 : ∀ (t : T), f t ≤ 1\nx y : T → X\nhx : ContinuousOn x (f ⁻¹' {0}ᶜ)\nhy : ContinuousOn y (f ⁻¹' {1}ᶜ)\nhx' : Bor... | [
"X : Type u_2\ninst✝³ : ConvexSpace ℝ X\ninst✝² : MetricSpace X\ninst✝¹ : IsConvexDist X\nT : Type u_3\ninst✝ : TopologicalSpace T\nf : T → ℝ\nhf : Continuous f\nhf0 : ∀ (t : T), 0 ≤ f t\nhf1 : ∀ (t : T), f t ≤ 1\nx y : T → X\nhx : ContinuousOn x (f ⁻¹' {0}ᶜ)\nhy : ContinuousOn y (f ⁻¹' {1}ᶜ)\nhx' : Bornology.IsBou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.ConvexSpace.Module | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 13
} | {
"line": 87,
"column": 14
} | [
{
"pp": "R : Type u_2\nM : Type u_3\nN : Type u_4\nI : Type u_5\ninst✝¹⁰ : Semiring R\ninst✝⁹ : PartialOrder R\ninst✝⁸ : IsStrictOrderedRing R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nf : M → N\ninst✝³ : ConvexSpace R M\ninst✝² : IsModuleConvexSpace R M\nins... | [
"R : Type u_2\nM : Type u_3\nN : Type u_4\nI : Type u_5\ninst✝¹⁰ : Semiring R\ninst✝⁹ : PartialOrder R\ninst✝⁸ : IsStrictOrderedRing R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nf : M → N\ninst✝³ : ConvexSpace R M\ninst✝² : IsModuleConvexSpace R M\ninst✝¹ : Convex... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.ConvexSpace.Module | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 13
} | {
"line": 91,
"column": 14
} | [
{
"pp": "R : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Semiring R\ninst✝⁹ : PartialOrder R\ninst✝⁸ : IsStrictOrderedRing R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nf : M → N\ninst✝³ : ConvexSpace R M\ninst✝² : IsModuleConvexSpace R M\ninst✝¹ : ConvexSp... | [
"R : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Semiring R\ninst✝⁹ : PartialOrder R\ninst✝⁸ : IsStrictOrderedRing R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nf : M → N\ninst✝³ : ConvexSpace R M\ninst✝² : IsModuleConvexSpace R M\ninst✝¹ : ConvexSpace R N\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 239,
"column": 69
} | {
"line": 239,
"column": 80
} | {
"line": 239,
"column": 81
} | [
{
"pp": "X : Type u_2\ninst✝³ : ConvexSpace ℝ X\ninst✝² : MetricSpace X\ninst✝¹ : IsConvexDist X\nT : Type u_3\ninst✝ : TopologicalSpace T\nf : T → ℝ\nhf : Continuous f\nhf0 : ∀ (t : T), 0 ≤ f t\nhf1 : ∀ (t : T), f t ≤ 1\nx y : T → X\nhx : ContinuousOn x (f ⁻¹' {0}ᶜ)\nhy : ContinuousOn y (f ⁻¹' {1}ᶜ)\nhx' : Bor... | [
"X : Type u_2\ninst✝³ : ConvexSpace ℝ X\ninst✝² : MetricSpace X\ninst✝¹ : IsConvexDist X\nT : Type u_3\ninst✝ : TopologicalSpace T\nf : T → ℝ\nhf : Continuous f\nhf0 : ∀ (t : T), 0 ≤ f t\nhf1 : ∀ (t : T), f t ≤ 1\nx y : T → X\nhx : ContinuousOn x (f ⁻¹' {0}ᶜ)\nhy : ContinuousOn y (f ⁻¹' {1}ᶜ)\nhx' : Bornology.IsBou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.ConvexSpace.Module | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 30
} | {
"line": 152,
"column": 31
} | [
{
"pp": "R : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Semiring R\ninst✝⁹ : PartialOrder R\ninst✝⁸ : IsStrictOrderedRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : ConvexSpace R M\ninst✝² : IsModuleConvexSpace R M\ninst✝¹ : ConvexSpace R N\ninst... | [
"R : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Semiring R\ninst✝⁹ : PartialOrder R\ninst✝⁸ : IsStrictOrderedRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : ConvexSpace R M\ninst✝² : IsModuleConvexSpace R M\ninst✝¹ : ConvexSpace R N\ninst✝ : IsModule... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.SimplicialComplex.Basic | {
"line": 75,
"column": 14
} | {
"line": 75,
"column": 25
} | {
"line": 75,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\nh : ∅ ∈ K.faces\n⊢ False",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\nh : ∅ ∈ K.faces\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.SimplicialComplex.Basic | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 29
} | {
"line": 110,
"column": 30
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t : Finset E\nhs : s ∈ K.faces\nht : t ∈ K.faces\nh :\n ((convexHull 𝕜) ↑s ∩ (convexHull 𝕜) ↑t).Nonempty ∧\n ∀ u ∈ K.faces, (convexHu... | [
"case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t : Finset E\nhs : s ∈ K.faces\nht : t ∈ K.faces\nh :\n ((convexHull 𝕜) ↑s ∩ (convexHull 𝕜) ↑t).Nonempty ∧\n ∀ u ∈ K.faces, (convexHull 𝕜) ↑s ∩ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.SimplicialComplex.Basic | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 27
} | {
"line": 124,
"column": 28
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t : Finset E\nx : E\nfaces : Set (Finset E)\nindep : ∀ s ∈ faces, AffineIndependent 𝕜 Subtype.val\ndown_closed : IsLowerSet faces\ninter_subset_convexHul... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t : Finset E\nx : E\nfaces : Set (Finset E)\nindep : ∀ s ∈ faces, AffineIndependent 𝕜 Subtype.val\ndown_closed : IsLowerSet faces\ninter_subset_convexHull : ∀ s ∈ fa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 254,
"column": 66
} | {
"line": 254,
"column": 77
} | {
"line": 254,
"column": 78
} | [
{
"pp": "I : Type u_1\nX : Type u_2\ninst✝⁴ : ConvexSpace ℝ X\ninst✝³ : MetricSpace X\ninst✝² : IsConvexDist X\ninst✝¹ : BoundedSpace X\nT : Type u_3\ninst✝ : TopologicalSpace T\nf : T → ℝ\nhf : Continuous f\nhf0 : ∀ (t : T), 0 ≤ f t\nhf1 : ∀ (t : T), f t ≤ 1\nx y : T → X\nhx : ContinuousOn x (f ⁻¹' {0}ᶜ)\nhy :... | [
"I : Type u_1\nX : Type u_2\ninst✝⁴ : ConvexSpace ℝ X\ninst✝³ : MetricSpace X\ninst✝² : IsConvexDist X\ninst✝¹ : BoundedSpace X\nT : Type u_3\ninst✝ : TopologicalSpace T\nf : T → ℝ\nhf : Continuous f\nhf0 : ∀ (t : T), 0 ≤ f t\nhf1 : ∀ (t : T), f t ≤ 1\nx y : T → X\nhx : ContinuousOn x (f ⁻¹' {0}ᶜ)\nhy : ContinuousO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Radon | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 22
} | {
"line": 208,
"column": 2
} | [
{
"pp": "case inr.hs\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : T2Space E\nF : ι → Set E\nh_convex : ∀ (i : ι), Conve... | [] | exact h_compact i0 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Radon | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 22
} | {
"line": 208,
"column": 2
} | [
{
"pp": "case inr.hs\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : T2Space E\nF : ι → Set E\nh_convex : ∀ (i : ι), Conve... | [] | exact h_compact i0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Radon | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 22
} | {
"line": 208,
"column": 2
} | [
{
"pp": "case inr.hs\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : T2Space E\nF : ι → Set E\nh_convex : ∀ (i : ι), Conve... | [] | exact h_compact i0 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 34
} | {
"line": 114,
"column": 35
} | [
{
"pp": "V : Type u\nadj : V → V → Bool\nproperty✝¹ : (∀ (x y : V), adj x y = adj y x) ∧ ∀ (x : V), ¬adj x x = true\nadj' : V → V → Bool\nproperty✝ : (∀ (x y : V), adj' x y = adj' y x) ∧ ∀ (x : V), ¬adj' x x = true\nh : (fun v w ↦ adj v w = true) = fun v w ↦ adj' v w = true\nv w : V\n⊢ adj v w = adj' v w",
... | [
"V : Type u\nadj : V → V → Bool\nproperty✝¹ : (∀ (x y : V), adj x y = adj y x) ∧ ∀ (x : V), ¬adj x x = true\nadj' : V → V → Bool\nproperty✝ : (∀ (x y : V), adj' x y = adj' y x) ∧ ∀ (x : V), ¬adj' x x = true\nh : (fun v w ↦ adj v w = true) = fun v w ↦ adj' v w = true\nv w : V\n⊢ adj v w = adj' v w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Radon | {
"line": 211,
"column": 4
} | {
"line": 211,
"column": 15
} | {
"line": 211,
"column": 16
} | [
{
"pp": "case inr.hst\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : T2Space E\nF : ι → Set E\nh_convex : ∀ (i : ι), Conv... | [
"case inr.hst\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : T2Space E\nF : ι → Set E\nh_convex : ∀ (i : ι), Convex 𝕜 (F i)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 369,
"column": 41
} | {
"line": 370,
"column": 57
} | {
"line": 370,
"column": 58
} | [
{
"pp": "ι : Sort u_1\nV : Type u\nG H : SimpleGraph V\na b c u v w : V\ne : Sym2 V\ninst✝ : Nontrivial V\nh : ⊥ = ⊤\n⊢ ∀ (a b : V), a = b",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Sort u_1\nV : Type u\nG H : SimpleGraph V\na b c u v w : V\ne : Sym2 V\ninst✝ : Nontrivial V\nh : ⊥ = ⊤\n⊢ ∀ (a b : V), a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 515,
"column": 2
} | {
"line": 515,
"column": 77
} | {
"line": 515,
"column": 78
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\n⊢ G.edgeSet ⊆ Sym2.diagSetᶜ",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Std.Irrefl",
"Eq.mpr",
"Compl.compl",
"SimpleGraph.Adj",
"Disjoint",
"SemilatticeInf.toPartialOrder",
"id",
"BiheytingAlgebra.... | [
"V : Type u\nG : SimpleGraph V\n⊢ Std.Irrefl G.Adj"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 535,
"column": 2
} | {
"line": 535,
"column": 13
} | {
"line": 535,
"column": 14
} | [
{
"pp": "case mk\nV : Type u\ns : Set (SimpleGraph V)\nh : s.Nonempty\nx✝ : Sym2 V\nx y : V\nG : SimpleGraph V\nhG : G ∈ s\n⊢ Quot.mk (Sym2.Rel V) (x, y) ∈ (sInf s).edgeSet ↔ Quot.mk (Sym2.Rel V) (x, y) ∈ ⋂₀ (edgeSet '' s)",
"ppTerm": "?mk",
"assigned": true,
"usedConstants": [
"SimpleGraph.sI... | [
"case mk\nV : Type u\ns : Set (SimpleGraph V)\nh : s.Nonempty\nx✝ : Sym2 V\nx y : V\nG : SimpleGraph V\nhG : G ∈ s\n⊢ (∀ G ∈ s, G.Adj x y) → ¬x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 546,
"column": 2
} | {
"line": 546,
"column": 13
} | {
"line": 546,
"column": 14
} | [
{
"pp": "case mk\nV : Type u\nι : Sort u_2\ninst✝ : Nonempty ι\nf : ι → SimpleGraph V\nx✝ : Sym2 V\nx y : V\ni : ι\n⊢ Quot.mk (Sym2.Rel V) (x, y) ∈ (⨅ i, f i).edgeSet ↔ Quot.mk (Sym2.Rel V) (x, y) ∈ ⋂ i, (f i).edgeSet",
"ppTerm": "?mk",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Sym... | [
"case mk\nV : Type u\nι : Sort u_2\ninst✝ : Nonempty ι\nf : ι → SimpleGraph V\nx✝ : Sym2 V\nx y : V\ni : ι\n⊢ (∀ (i : ι), (f i).Adj x y) → ¬x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.SpecificFunctions.Pow | {
"line": 59,
"column": 2
} | {
"line": 60,
"column": 47
} | {
"line": 61,
"column": 2
} | [
{
"pp": "⊢ StrictConcaveOn ℝ≥0 univ ⇑sqrt",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"NNReal.sqrt_eq_rpow",
"congrArg",
"Real.instDivInvMonoid",
"PartialOrder.toPreorder",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithO... | [
"this : ⇑sqrt = fun x ↦ x ^ (1 / 2)\n⊢ StrictConcaveOn ℝ≥0 univ ⇑sqrt"
] | have : NNReal.sqrt = fun x : ℝ≥0 ↦ x ^ (1 / (2 : ℝ)) := by
ext x; exact mod_cast NNReal.sqrt_eq_rpow x | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.SpecificFunctions.Pow | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 31
} | {
"line": 92,
"column": 2
} | [
{
"pp": "⊢ StrictConcaveOn ℝ (Ici 0) fun x ↦ √x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"StrictConcaveOn",
"Real.partialOrder",
"Real",
"instHDiv",
"instSMulOfMul",
"Set.Ici",
"Real.instZero",
"co... | [
"⊢ StrictConcaveOn ℝ (Ici 0) fun x ↦ (fun x ↦ x ^ (1 / 2)) x"
] | rw [funext Real.sqrt_eq_rpow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Strict.Extreme | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 77
} | {
"line": 47,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℝ E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : Nontrivial E\nS : Set E\nx : E\nx✝ : x ∈ interior S ∩ extremePoints ℝ S\nx_int : S ∈ 𝓝 x\nx_ext : x ∈ extremePoints ℝ S\nh₁ : ∀ᶠ (v : E) in 𝓝... | [
"E : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℝ E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : Nontrivial E\nS : Set E\nx : E\nx✝ : x ∈ interior S ∩ extremePoints ℝ S\nx_int : S ∈ 𝓝 x\nx_ext : x ∈ extremePoints ℝ S\nh₁ : ∀ᶠ (v : E) in 𝓝[≠] 0, x - v... | have key : x ∈ openSegment ℝ (x - v) (x + v) := mem_openSegment_sub_add _ _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 1116,
"column": 4
} | {
"line": 1116,
"column": 20
} | {
"line": 1116,
"column": 21
} | [
{
"pp": "case refine_1\nV : Type u\nG : SimpleGraph V\nv : V\nh : Gᶜ.IsUniversal v\nx : V\nhx : G.Adj v x\n⊢ False",
"ppTerm": "?refine_1",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_1\nV : Type u\nG : SimpleGraph V\nv : V\nh : Gᶜ.IsUniversal v\nx : V\nhx : G.Adj v x\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 1117,
"column": 4
} | {
"line": 1117,
"column": 20
} | {
"line": 1117,
"column": 21
} | [
{
"pp": "case refine_2\nV : Type u\nG : SimpleGraph V\nv : V\nh : G.IsIsolated v\nx : V\nhx : v ≠ x\n⊢ Gᶜ.Adj v x",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"eq_false",
"congrArg",
"Compl.compl",
"SimpleGraph.Adj",
"id",... | [
"case refine_2\nV : Type u\nG : SimpleGraph V\nv : V\nh : G.IsIsolated v\nx : V\nhx : v ≠ x\n⊢ ¬G.Adj v x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 1124,
"column": 4
} | {
"line": 1124,
"column": 15
} | {
"line": 1124,
"column": 16
} | [
{
"pp": "case refine_1\nV : Type u\nG : SimpleGraph V\nv : V\nh : Gᶜ.IsIsolated v\n⊢ G.IsUniversal v",
"ppTerm": "?refine_1",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_1\nV : Type u\nG : SimpleGraph V\nv : V\nh : Gᶜ.IsIsolated v\n⊢ G.IsUniversal v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.StrictConvexBetween | {
"line": 34,
"column": 39
} | {
"line": 34,
"column": 50
} | {
"line": 34,
"column": 51
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : StrictConvexSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\np p₁ p₂ p₃ : P\nh : Sbtw ℝ p₁ p₂ p₃\n⊢ p₁ -ᵥ p ≠ p₃ -ᵥ p",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [
"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : StrictConvexSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\np p₁ p₂ p₃ : P\nh : Sbtw ℝ p₁ p₂ p₃\n⊢ ¬p₁ = p₃"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.StrictConvexBetween | {
"line": 136,
"column": 23
} | {
"line": 136,
"column": 33
} | {
"line": 136,
"column": 34
} | [
{
"pp": "case hyz\nE : Type u_3\nPE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : StrictConvexSpace ℝ E\ninst✝¹ : MetricSpace PE\ninst✝ : NormedAddTorsor E PE\nx y z : PE\nhx : dist x y = dist x z / 2\nhy : dist y z = dist x z / 2\n⊢ dist y z = (1 - 2⁻¹) * dist x z",
"ppTerm"... | [
"case hyz\nE : Type u_3\nPE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : StrictConvexSpace ℝ E\ninst✝¹ : MetricSpace PE\ninst✝ : NormedAddTorsor E PE\nx y z : PE\nhx : dist x y = dist x z / 2\nhy : dist y z = dist x z / 2\n⊢ dist y z = (1 - 1 / 2) * dist x z"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.StrictCombination | {
"line": 123,
"column": 57
} | {
"line": 123,
"column": 68
} | {
"line": 123,
"column": 69
} | [
{
"pp": "V : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : StrictConvexSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\nt : Finset ι\nw : ι → ℝ\np₀ : P\nr : ℝ\np : ι → P\nh0 : ∀ i ∈ t, 0 ≤ w i\nh1 : ∑ i ∈ t, w i = 1\ni j : ι\nhi : i ... | [
"V : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : StrictConvexSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\nt : Finset ι\nw : ι → ℝ\np₀ : P\nr : ℝ\np : ι → P\nh0 : ∀ i ∈ t, 0 ≤ w i\nh1 : ∑ i ∈ t, w i = 1\ni j : ι\nhi : i ∈ t\nhj : j ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Strong | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 13
} | {
"line": 69,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nφ : ℝ → ℝ\ns : Set E\nf : E → ℝ\nhf : UniformConvexOn s φ f\nhφ : 0 ≤ φ\n⊢ ConvexOn ℝ s f",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nφ : ℝ → ℝ\ns : Set E\nf : E → ℝ\nhf : UniformConvexOn s φ f\nhφ : 0 ≤ φ\n⊢ ConvexOn ℝ s f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Strong | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 13
} | {
"line": 72,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nφ : ℝ → ℝ\ns : Set E\nf : E → ℝ\nhf : UniformConcaveOn s φ f\nhφ : 0 ≤ φ\n⊢ ConcaveOn ℝ s f",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nφ : ℝ → ℝ\ns : Set E\nf : E → ℝ\nhf : UniformConcaveOn s φ f\nhφ : 0 ≤ φ\n⊢ ConcaveOn ℝ s f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Strong | {
"line": 91,
"column": 2
} | {
"line": 92,
"column": 9
} | {
"line": 92,
"column": 10
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nφ ψ : ℝ → ℝ\ns : Set E\nf g : E → ℝ\nhf : UniformConvexOn s φ f\nhg : UniformConvexOn s ψ g\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ (f + g) (a • x + b • y) ≤ a • (f + g) x + b • (f +... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nφ ψ : ℝ → ℝ\ns : Set E\nf g : E → ℝ\nhf : UniformConvexOn s φ f\nhg : UniformConvexOn s ψ g\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ f (a • x + b • y) + g (a • x + b • y) ≤\n a * f x + a * g x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Strong | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 41
} | {
"line": 97,
"column": 42
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nφ ψ : ℝ → ℝ\ns : Set E\nf g : E → ℝ\nhf : UniformConcaveOn s φ f\nhg : UniformConcaveOn s ψ g\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • (f + g) x + b • (f + g) y + a * b * (φ + ψ) ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nφ ψ : ℝ → ℝ\ns : Set E\nf g : E → ℝ\nhf : UniformConcaveOn s φ f\nhg : UniformConcaveOn s ψ g\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a * f x + a * g x + (b * f y + b * g y) + (a * b * φ ‖x - y‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Strong | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 61
} | {
"line": 101,
"column": 62
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nφ : ℝ → ℝ\ns : Set E\nf : E → ℝ\nhf : UniformConvexOn s φ f\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ -(-f) (a • x + b • y) ≤ -(a • (-f) x + b • (-f) y + a * b * φ ‖x - y‖)",
"ppTe... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nφ : ℝ → ℝ\ns : Set E\nf : E → ℝ\nhf : UniformConvexOn s φ f\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ f (a • x + b • y) + a * b * φ ‖x - y‖ ≤ a * f x + b * f y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Strong | {
"line": 105,
"column": 2
} | {
"line": 106,
"column": 9
} | {
"line": 106,
"column": 10
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nφ : ℝ → ℝ\ns : Set E\nf : E → ℝ\nhf : UniformConcaveOn s φ f\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ -(a • (-f) x + b • (-f) y - a * b * φ ‖x - y‖) ≤ -(-f) (a • x + b • y)",
"ppT... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nφ : ℝ → ℝ\ns : Set E\nf : E → ℝ\nhf : UniformConcaveOn s φ f\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a * b * φ ‖x + -y‖ ≤ f (a • x + b • y) + (-(a * f x) + -(b * f y))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Visible | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 33
} | {
"line": 93,
"column": 2
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nV : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\ns : Set V\nx : V\nι : Type u_4\nt : Finset ι\na : ι → V\nw : ι → 𝕜\nhw₀ : ∀ i ∈ t, 0 ≤ w i\nhw₁ : ∑ i ∈ t, w i = 1\nha : ∀ i ∈ t, a i ∈ s\nh... | [
"case inr\n𝕜 : Type u_1\nV : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\ns : Set V\nx : V\nι : Type u_4\nt : Finset ι\na : ι → V\nw : ι → 𝕜\nhw₀ : ∀ i ∈ t, 0 ≤ w i\nhw₁ : ∑ i ∈ t, w i = 1\nha : ∀ i ∈ t, a i ∈ s\nhx : x ∉ (con... | have : 0 < 1 - ε := by linarith | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Side | {
"line": 931,
"column": 4
} | {
"line": 931,
"column": 15
} | {
"line": 931,
"column": 16
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw₁ w₂ : Fin (n + 1) → R\nhw₁ : ∑ j, w₁ j = 1\nhw₂ : ∑ j, w₂ j = 1\ni : ... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw₁ w₂ : Fin (n + 1) → R\nhw₁ : ∑ j, w₁ j = 1\nhw₂ : ∑ j, w₂ j = 1\ni : Fin (n + 1)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Visible | {
"line": 132,
"column": 10
} | {
"line": 132,
"column": 21
} | {
"line": 132,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\ninst✝⁹ : Field 𝕜\ninst✝⁸ : LinearOrder 𝕜\ninst✝⁷ : IsStrictOrderedRing 𝕜\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module 𝕜 V\ns : Set V\nx y : V\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : OrderTopology 𝕜\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : Co... | [
"𝕜 : Type u_1\nV : Type u_2\ninst✝⁹ : Field 𝕜\ninst✝⁸ : LinearOrder 𝕜\ninst✝⁷ : IsStrictOrderedRing 𝕜\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module 𝕜 V\ns : Set V\nx y : V\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : OrderTopology 𝕜\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Holder | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 69
} | {
"line": 148,
"column": 4
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nm : MeasurableSpace α\nμ : Measure α\np q : ℝ≥0∞\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedAddCommGroup G\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace �... | [
"α : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nm : MeasurableSpace α\nμ : Measure α\np q : ℝ≥0∞\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedAddCommGroup G\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 F\ninst✝⁶ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Side | {
"line": 1097,
"column": 21
} | {
"line": 1097,
"column": 73
} | {
"line": 1097,
"column": 73
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw : Fin (n + 1) → R\nhw : ∑ j, w j = 1\ni : Fin (n + 1)\n⊢ (affineSpan ... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw : Fin (n + 1) → R\nhw : ∑ j, w j = 1\ni : Fin (n + 1)\n⊢ w i < 0 ↔ w i < 0"
] | s.sOppSide_affineSpan_faceOpposite_point_left_iff hw | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Side | {
"line": 1096,
"column": 77
} | {
"line": 1097,
"column": 74
} | {
"line": 1099,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw : Fin (n + 1) → R\nhw : ∑ j, w j = 1\ni : Fin (n + 1)\n⊢ (affineSpan ... | [] | by
rw [sOppSide_comm, s.sOppSide_affineSpan_faceOpposite_point_left_iff hw] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.JapaneseBracket | {
"line": 62,
"column": 51
} | {
"line": 62,
"column": 76
} | {
"line": 64,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nr : ℝ\nx : E\nhr : 0 < r\n⊢ 2 ^ (r / 2) * ((1 + ‖x‖) ^ r)⁻¹ = 2 ^ (r / 2) * (1 + ‖x‖) ^ (-r)",
"ppTerm": "?m.217",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"Real.instPow",
... | [] | rw [rpow_neg]; positivity | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.JapaneseBracket | {
"line": 62,
"column": 51
} | {
"line": 62,
"column": 76
} | {
"line": 64,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nr : ℝ\nx : E\nhr : 0 < r\n⊢ 2 ^ (r / 2) * ((1 + ‖x‖) ^ r)⁻¹ = 2 ^ (r / 2) * (1 + ‖x‖) ^ (-r)",
"ppTerm": "?m.217",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"Real.instPow",
... | [] | rw [rpow_neg]; positivity | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.JapaneseBracket | {
"line": 87,
"column": 74
} | {
"line": 87,
"column": 85
} | {
"line": 87,
"column": 86
} | [
{
"pp": "r : ℝ\nn : ℕ\nhnr : ↑n < r\nhr : 0 < r\nx : ℝ\nhx : x ∈ Ioc 0 1\n⊢ -r⁻¹ ≤ 0",
"ppTerm": "?m.170",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"Real.partialOrder",
"... | [
"r : ℝ\nn : ℕ\nhnr : ↑n < r\nhr : 0 < r\nx : ℝ\nhx : x ∈ Ioc 0 1\n⊢ 0 ≤ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 56
} | {
"line": 183,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\ninst✝ : SFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nint... | [
"α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\ninst✝ : SFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 233,
"column": 14
} | {
"line": 233,
"column": 78
} | {
"line": 233,
"column": 79
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\nf_nonneg : ∀ (ω : α), 0 ≤ f ω\nH1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0\ns : ℝ\ns_pos : s ... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\nf_nonneg : ∀ (ω : α), 0 ≤ f ω\nH1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0\ns : ℝ\ns_pos : s > 0\nhs : 0 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 254,
"column": 10
} | {
"line": 254,
"column": 64
} | {
"line": 254,
"column": 65
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\nf_nonneg : ∀ (ω : α), 0 ≤ f ω\nH1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0\ns : ℝ\ns_pos : s ... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\nf_nonneg : ∀ (ω : α), 0 ≤ f ω\nH1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0\ns : ℝ\ns_pos : s > 0\nhs : 0 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 316,
"column": 49
} | {
"line": 316,
"column": 60
} | {
"line": 316,
"column": 61
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\nf_nonneg : ∀ (ω : α), 0 ≤ f ω\nH1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0\nH2 : ∀ s > 0, 0 <... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\nf_nonneg : ∀ (ω : α), 0 ≤ f ω\nH1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0\nH2 : ∀ s > 0, 0 < ∫ (t : ℝ) i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 83,
"column": 28
} | {
"line": 83,
"column": 39
} | {
"line": 83,
"column": 40
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nhf_temperate : HasTemperateGrowth f\nN : ℕ\nk : ℕ → ℕ\nhk : ∀ (n : ℕ), iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k n\nn : ℕ\nhn : n ≤ N\nx : E\n⊢ ... | [
"E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nhf_temperate : HasTemperateGrowth f\nN : ℕ\nk : ℕ → ℕ\nhk : ∀ (n : ℕ), iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k n\nn : ℕ\nhn : n ≤ N\nx : E\n⊢ n ≤ N"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 59
} | {
"line": 92,
"column": 4
} | [
{
"pp": "E : Type u_5\nF✝ : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nf : E → F✝\nhf_temperate : HasTemperateGrowth f\nn k : ℕ\nhk : ∀ n_1 ≤ n, iteratedFDeriv ℝ n_1 f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k\nF : E → (N : Fin (n + 1)) → E [×↑... | [
"E : Type u_5\nF✝ : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nf : E → F✝\nhf_temperate : HasTemperateGrowth f\nn k : ℕ\nhk : ∀ n_1 ≤ n, iteratedFDeriv ℝ n_1 f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k\nF : E → (N : Fin (n + 1)) → E [×↑N]→L[ℝ] F✝ :... | simp_rw [F, isBigO_pi, Fin.forall_iff, Nat.lt_succ_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 104,
"column": 28
} | {
"line": 104,
"column": 39
} | {
"line": 104,
"column": 40
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nh'f : HasTemperateGrowth (fderiv ℝ f)\nhf : Differentiable ℝ f\nk : ℕ\nC : ℝ\nh : ∀ (x : E), ‖f x‖ ≤ C * (1 + ‖x‖) ^ k\nx : E\n⊢ ‖iteratedFDeriv ℝ 0 f ... | [
"E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nh'f : HasTemperateGrowth (fderiv ℝ f)\nhf : Differentiable ℝ f\nk : ℕ\nC : ℝ\nh : ∀ (x : E), ‖f x‖ ≤ C * (1 + ‖x‖) ^ k\nx : E\n⊢ ‖f x‖ ≤ C * (1 + ‖x‖) ^ k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 51
} | {
"line": 107,
"column": 52
} | [
{
"pp": "case succ\nE : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nh'f : HasTemperateGrowth (fderiv ℝ f)\nhf : Differentiable ℝ f\nk : ℕ\nC : ℝ\nh : ∀ (x : E), ‖f x‖ ≤ C * (1 + ‖x‖) ^ k\nm k' : ℕ\nC' : ℝ\nh'... | [
"case succ\nE : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nh'f : HasTemperateGrowth (fderiv ℝ f)\nhf : Differentiable ℝ f\nk : ℕ\nC : ℝ\nh : ∀ (x : E), ‖f x‖ ≤ C * (1 + ‖x‖) ^ k\nm k' : ℕ\nC' : ℝ\nh' : ∀ (x : E)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 118,
"column": 17
} | {
"line": 118,
"column": 28
} | {
"line": 118,
"column": 29
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nc : F\n⊢ HasTemperateGrowth (fderiv ℝ fun x ↦ c)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSem... | [
"E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nc : F\n⊢ HasTemperateGrowth 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 13
} | {
"line": 128,
"column": 14
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nh₁ : HasCompactSupport f\nh₂ : ContDiff ℝ ∞ f\nn : ℕ\ng : E → ℝ := fun x ↦ ‖iteratedFDeriv ℝ n f x‖\nhg : Continuous g\nx₀ : E\nhx₀ : ∀ (y : E), g y ≤ ... | [
"E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nh₁ : HasCompactSupport f\nh₂ : ContDiff ℝ ∞ f\nn : ℕ\ng : E → ℝ := fun x ↦ ‖iteratedFDeriv ℝ n f x‖\nhg : Continuous g\nx₀ : E\nhx₀ : ∀ (y : E), g y ≤ g x₀\nx : E\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 355,
"column": 8
} | {
"line": 355,
"column": 19
} | {
"line": 355,
"column": 20
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\nf_nonneg : ∀ (ω : α), 0 ≤ f ω\nH1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0\nH2 : ∀ s > 0, 0 <... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\nf_nonneg : ∀ (ω : α), 0 ≤ f ω\nH1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0\nH2 : ∀ s > 0, 0 < ∫ (t : ℝ) i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 142,
"column": 52
} | {
"line": 142,
"column": 63
} | {
"line": 142,
"column": 64
} | [
{
"pp": "D : Type u_4\nE : Type u_5\nF : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup D\ninst✝ : NormedSpace ℝ D\ng : E → F\nf : D → E\nt : Set E\nht : Set.range f ⊆ t\nht' : UniqueDiffOn ℝ t\nhg₁ : ContDi... | [
"D : Type u_4\nE : Type u_5\nF : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup D\ninst✝ : NormedSpace ℝ D\ng : E → F\nf : D → E\nt : Set E\nht : Set.range f ⊆ t\nht' : UniqueDiffOn ℝ t\nhg₁ : ContDiffOn ℝ ∞ g t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 153,
"column": 8
} | {
"line": 153,
"column": 31
} | {
"line": 154,
"column": 8
} | [
{
"pp": "D : Type u_4\nE : Type u_5\nF : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup D\ninst✝ : NormedSpace ℝ D\ng : E → F\nf : D → E\nt : Set E\nht : Set.range f ⊆ t\nht' : UniqueDiffOn ℝ t\nhg₁ : ContDi... | [
"D : Type u_4\nE : Type u_5\nF : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup D\ninst✝ : NormedSpace ℝ D\ng : E → F\nf : D → E\nt : Set E\nht : Set.range f ⊆ t\nht' : UniqueDiffOn ℝ t\nhg₁ : ContDiffOn ℝ ∞ g t... | apply Finset.sum_le_sum | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 169,
"column": 7
} | {
"line": 171,
"column": 54
} | {
"line": 173,
"column": 0
} | [] | [] | _ ≤ n ! * (C₃ * (1 + ‖x‖) ^ (k₁ * k₂)) * ((1 + C₁) * (1 + ‖x‖) ^ k₁) ^ n :=
norm_iteratedFDeriv_comp_le' ht ht' hg₁ hf.1 (mod_cast le_top) x hg' hf'
_ = _ := by rw [mul_pow, ← pow_mul, pow_add]; ring | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 32
} | {
"line": 178,
"column": 2
} | [
{
"pp": "D : Type u_4\nE : Type u_5\nF : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup D\ninst✝ : NormedSpace ℝ D\ng : E → F\nf : D → E\nhg : HasTemperateGrowth g\nhf : HasTemperateGrowth f\n⊢ HasTemperateG... | [
"case ht\nD : Type u_4\nE : Type u_5\nF : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup D\ninst✝ : NormedSpace ℝ D\ng : E → F\nf : D → E\nhg : HasTemperateGrowth g\nhf : HasTemperateGrowth f\n⊢ Set.range f ⊆ Se... | apply hf.comp' (t := Set.univ) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 43
} | {
"line": 182,
"column": 44
} | [
{
"pp": "case hg₂\nD : Type u_4\nE : Type u_5\nF : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup D\ninst✝ : NormedSpace ℝ D\ng : E → F\nf : D → E\nhg : HasTemperateGrowth g\nhf : HasTemperateGrowth f\n⊢ ∀ (... | [
"case hg₂\nD : Type u_4\nE : Type u_5\nF : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup D\ninst✝ : NormedSpace ℝ D\ng : E → F\nf : D → E\nhg : HasTemperateGrowth g\nhf : HasTemperateGrowth f\n⊢ ∀ (N : ℕ), ∃ k ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 192,
"column": 26
} | {
"line": 192,
"column": 64
} | {
"line": 192,
"column": 65
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nhf : HasTemperateGrowth f\nn k : ℕ\nC : ℝ\nh : ∀ (x : E), ‖iteratedFDeriv ℝ n f x‖ ≤ C * (1 + ‖x‖) ^ k\nx : E\n⊢ ‖iteratedFDeriv ℝ n (-f) x‖ ≤ C * (1 +... | [
"E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nhf : HasTemperateGrowth f\nn k : ℕ\nC : ℝ\nh : ∀ (x : E), ‖iteratedFDeriv ℝ n f x‖ ≤ C * (1 + ‖x‖) ^ k\nx : E\n⊢ ‖iteratedFDeriv ℝ n f x‖ ≤ C * (1 + ‖x‖) ^ k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 204,
"column": 4
} | {
"line": 204,
"column": 79
} | {
"line": 205,
"column": 2
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf g : E → F\nhf : ContDiff ℝ ∞ f ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k\nhg : ContDiff ℝ ∞ g ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n g =O... | [] | filter_upwards with _ using (le_add_iff_nonneg_right _).mpr (by positivity) | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 376,
"column": 29
} | {
"line": 376,
"column": 44
} | {
"line": 376,
"column": 45
} | [
{
"pp": "H : Type u_8\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\nr : ℝ\nt : Set ℝ := {y | 1 / 2 < y}\nht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t\nhdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t\nhunique : UniqueDiffOn ℝ t\nN k : ℕ\nhk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k\nhk₁ : r ≤... | [
"H : Type u_8\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\nr : ℝ\nt : Set ℝ := {y | 1 / 2 < y}\nht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t\nhdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t\nhunique : UniqueDiffOn ℝ t\nN k : ℕ\nhk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k\nhk₁ : r ≤ ↑k\nhk₂ : R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 382,
"column": 27
} | {
"line": 382,
"column": 38
} | {
"line": 382,
"column": 39
} | [
{
"pp": "H : Type u_8\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\nr : ℝ\nt : Set ℝ := {y | 1 / 2 < y}\nht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t\nhdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t\nhunique : UniqueDiffOn ℝ t\nN k : ℕ\nhk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k\nhk₁ : r ≤... | [
"H : Type u_8\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\nr : ℝ\nt : Set ℝ := {y | 1 / 2 < y}\nht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t\nhdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t\nhunique : UniqueDiffOn ℝ t\nN k : ℕ\nhk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k\nhk₁ : r ≤ ↑k\nhk₂ : R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.Basic | {
"line": 224,
"column": 14
} | {
"line": 224,
"column": 25
} | {
"line": 224,
"column": 26
} | [
{
"pp": "M₁ : Type u_3\nM₂ : Type u_4\ninst✝⁶ : TopologicalSpace M₁\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : TopologicalSpace M₂\ninst✝³ : AddCommMonoid M₂\nS : Type u_6\ninst✝² : Monoid S\ninst✝¹ : DistribMulAction S M₂\ninst✝ : ContinuousConstSMul S M₂\nf : M₁ → M₂\nc : Sˣ\nh : IsCompactOperator (c • f)\n⊢ IsComp... | [
"M₁ : Type u_3\nM₂ : Type u_4\ninst✝⁶ : TopologicalSpace M₁\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : TopologicalSpace M₂\ninst✝³ : AddCommMonoid M₂\nS : Type u_6\ninst✝² : Monoid S\ninst✝¹ : DistribMulAction S M₂\ninst✝ : ContinuousConstSMul S M₂\nf : M₁ → M₂\nc : Sˣ\nh : IsCompactOperator (c • f)\n⊢ IsCompactOperator ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 393,
"column": 6
} | {
"line": 393,
"column": 17
} | {
"line": 393,
"column": 18
} | [
{
"pp": "case e_a\nH : Type u_8\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\nr : ℝ\nt : Set ℝ := {y | 1 / 2 < y}\nht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t\nhdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t\nhunique : UniqueDiffOn ℝ t\nN k : ℕ\nhk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k\... | [
"case e_a\nH : Type u_8\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\nr : ℝ\nt : Set ℝ := {y | 1 / 2 < y}\nht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t\nhdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t\nhunique : UniqueDiffOn ℝ t\nN k : ℕ\nhk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k\nhk₁ : r ≤ ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 436,
"column": 2
} | {
"line": 436,
"column": 42
} | {
"line": 436,
"column": 43
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : MeasurableSpace E\nμ : Measure E\nh : μ.HasTemperateGrowth\n⊢ Integrable (fun x ↦ (1 + ‖x‖) ^ (-↑μ.integrablePower)) μ",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"dite_cond_eq_true",
"Norm.... | [
"E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : MeasurableSpace E\nμ : Measure E\nh : μ.HasTemperateGrowth\n⊢ Integrable (fun x ↦ ((1 + ‖x‖) ^ ⋯.choose)⁻¹) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 206,
"column": 50
} | {
"line": 206,
"column": 71
} | {
"line": 207,
"column": 4
} | [
{
"pp": "case e'_12\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nT : F →L[ℝ] F\nhT : (↑T).IsSymmetric\nx₀ : F\ne_8✝ : Real.instAddCommGroup = Real.instRCLike.toDivisionRing.toAddCommGroup\ne_9✝ : Semiring.toModule ≍ toInnerProductSpaceReal.toModule\ny : F\n⊢ 2 • ((innerSL ℝ) (T x... | [
"case e'_12\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nT : F →L[ℝ] F\nhT : (↑T).IsSymmetric\nx₀ : F\ne_8✝ : Real.instAddCommGroup = Real.instRCLike.toDivisionRing.toAddCommGroup\ne_9✝ : Semiring.toModule ≍ toInnerProductSpaceReal.toModule\ny : F\n⊢ 2 • ((innerSL ℝ) (T x₀)) y =\n ... | fderivInnerCLM_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 59
} | {
"line": 231,
"column": 60
} | [
{
"pp": "F : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : CompleteSpace F\nT : F →L[ℝ] F\nhT : IsSelfAdjoint T\nx₀ : F\nhextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\nH : IsLocalExtrOn T.reApplyInnerSelf {x | ‖x‖ ^ 2 = ‖x₀‖ ^ 2} x₀\na b : ℝ\nh₁ : (a, b) ≠ 0\nh₂ ... | [
"F : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : CompleteSpace F\nT : F →L[ℝ] F\nhT : IsSelfAdjoint T\nx₀ : F\nhextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\nH : IsLocalExtrOn T.reApplyInnerSelf {x | ‖x‖ ^ 2 = ‖x₀‖ ^ 2} x₀\na b : ℝ\nh₁ : (a, b) ≠ 0\nh₂ : a • 2 • (i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 286,
"column": 26
} | {
"line": 286,
"column": 37
} | {
"line": 286,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsSelfAdjoint T\nx₀ : E\nhx₀ : x₀ ≠ 0\nhextr : IsMaxOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\nhx₀' : 0 < ‖x₀‖\nhx₀'' : x₀ ∈ sphere 0 ‖x₀‖\nx : E... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsSelfAdjoint T\nx₀ : E\nhx₀ : x₀ ≠ 0\nhextr : IsMaxOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\nhx₀' : 0 < ‖x₀‖\nhx₀'' : x₀ ∈ sphere 0 ‖x₀‖\nx : E\nhx : x ∈ s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 305,
"column": 26
} | {
"line": 305,
"column": 37
} | {
"line": 305,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsSelfAdjoint T\nx₀ : E\nhx₀ : x₀ ≠ 0\nhextr : IsMinOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\nhx₀' : 0 < ‖x₀‖\nhx₀'' : x₀ ∈ sphere 0 ‖x₀‖\nx : E... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsSelfAdjoint T\nx₀ : E\nhx₀ : x₀ ≠ 0\nhextr : IsMinOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\nhx₀' : 0 < ‖x₀‖\nhx₀'' : x₀ ∈ sphere 0 ‖x₀‖\nx : E\nhx : x ∈ s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 335,
"column": 30
} | {
"line": 335,
"column": 41
} | {
"line": 335,
"column": 42
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : Nontrivial E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ :... | [
"𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : Nontrivial E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 336,
"column": 66
} | {
"line": 336,
"column": 90
} | {
"line": 336,
"column": 91
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : Nontrivial E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ :... | [
"𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : Nontrivial E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 520,
"column": 38
} | {
"line": 520,
"column": 63
} | {
"line": 520,
"column": 64
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : MeasurableSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0\nhp : ↑p ≠ 0\nh_one_add : ∀ (x : E), 0 < 1 + ‖x‖\n⊢ 0 < ↑p",
"ppTerm": "?m.147",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LinearOrderedCommGroupWithZe... | [
"E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : MeasurableSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0\nhp : ↑p ≠ 0\nh_one_add : ∀ (x : E), 0 < 1 + ‖x‖\n⊢ ¬p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 72,
"column": 29
} | {
"line": 72,
"column": 40
} | {
"line": 72,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nhK : ∀ K > 0, ∃ x, ‖(T - μ • 1) x‖ < K * ‖x‖\nC : 𝕜\nhC : 1 < ‖C‖\nε : ... | [
"𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nhK : ∀ K > 0, ∃ x, ‖(T - μ • 1) x‖ < K * ‖x‖\nC : 𝕜\nhC : 1 < ‖C‖\nε : ℝ\nhε : ε > ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 523,
"column": 39
} | {
"line": 523,
"column": 71
} | {
"line": 523,
"column": 72
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : MeasurableSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0\nhp : ↑p ≠ 0\nh_one_add : ∀ (x : E), 0 < 1 + ‖x‖\nhp_pos : 0 < ↑p\nl : ℕ\nhl : Integrable (fun x ↦ (1 + ‖x‖) ^ (-↑l)) μ\nk : ℕ := ⌈↑l / ↑p⌉₊\n⊢ ↑l ≤ ↑k * ↑p",
"ppTerm": "?m.191"... | [
"E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : MeasurableSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0\nhp : ↑p ≠ 0\nh_one_add : ∀ (x : E), 0 < 1 + ‖x‖\nhp_pos : 0 < ↑p\nl : ℕ\nhl : Integrable (fun x ↦ (1 + ‖x‖) ^ (-↑l)) μ\nk : ℕ := ⌈↑l / ↑p⌉₊\n⊢ ↑l ≤ ↑k * ↑p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 354,
"column": 30
} | {
"line": 354,
"column": 41
} | {
"line": 354,
"column": 42
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : Nontrivial E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ :... | [
"𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : Nontrivial E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 355,
"column": 66
} | {
"line": 355,
"column": 90
} | {
"line": 355,
"column": 91
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : Nontrivial E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ :... | [
"𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : Nontrivial E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Charpoly | {
"line": 52,
"column": 52
} | {
"line": 52,
"column": 63
} | {
"line": 52,
"column": 64
} | [
{
"pp": "K : Type u_3\nV : Type u_4\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nf : End K V\nh : (charpoly f).Splits\n⊢ ((toMatrix (Free.chooseBasis K V) (Free.chooseBasis K V)) f).charpoly.Splits",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants":... | [
"K : Type u_3\nV : Type u_4\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nf : End K V\nh : (charpoly f).Splits\n⊢ (charpoly f).Splits"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Charpoly | {
"line": 58,
"column": 53
} | {
"line": 58,
"column": 64
} | {
"line": 58,
"column": 65
} | [
{
"pp": "K : Type u_3\nV : Type u_4\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nf : End K V\nh : (charpoly f).Splits\nb : Basis (Free.ChooseBasisIndex K V) K V := Free.chooseBasis K V\n⊢ ((toMatrix (Free.chooseBasis K V) (Free.chooseBasis K V)) f).charpoly.Splits"... | [
"K : Type u_3\nV : Type u_4\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nf : End K V\nh : (charpoly f).Splits\nb : Basis (Free.ChooseBasisIndex K V) K V := Free.chooseBasis K V\n⊢ (charpoly f).Splits"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 88,
"column": 46
} | {
"line": 88,
"column": 57
} | {
"line": 88,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nc : ℝ\nhc₀ : c > 0\nhc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) ... | [
"𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nc : ℝ\nhc₀ : c > 0\nhc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) x‖ < ε\nφ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 29
} | {
"line": 90,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nc : ℝ\nhc₀ : c > 0\nhc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) ... | [
"𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nc : ℝ\nhc₀ : c > 0\nhc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) x‖ < ε\nφ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 35
} | {
"line": 99,
"column": 36
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nc : ℝ\nhc₀ : c > 0\nhc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) ... | [
"𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nc : ℝ\nhc₀ : c > 0\nhc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) x‖ < ε\nφ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.ContinuousLinearMap | {
"line": 29,
"column": 2
} | {
"line": 29,
"column": 33
} | {
"line": 29,
"column": 34
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : TopologicalSpace M\ninst✝² : T0Space M\ninst✝¹ : ContinuousConstSMul R M\ninst✝ : IsTopologicalAddGroup M\nf : M →L[R] M\nμ : R\nn : ℕ\n⊢ IsClosed ↑((genEigenspace (↑f) μ) ↑n)",
"ppTerm": "?m.30"... | [
"R : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : TopologicalSpace M\ninst✝² : T0Space M\ninst✝¹ : ContinuousConstSMul R M\ninst✝ : IsTopologicalAddGroup M\nf : M →L[R] M\nμ : R\nn : ℕ\n⊢ IsClosed ↑((↑f - μ • 1) ^ n).ker"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Fin.Tuple.Sort | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 40
} | {
"line": 67,
"column": 41
} | [
{
"pp": "n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nx✝ : ↥(graph f)\nx : α\ni : Fin n\nh : (x, i) ∈ graph f\n⊢ (fun i ↦ ⟨(f i, i), ⋯⟩) ((fun p ↦ (↑p).2) ⟨(x, i), h⟩) = ⟨(x, i), h⟩",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Tuple.graphEquiv₁._proo... | [
"n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nx✝ : ↥(graph f)\nx : α\ni : Fin n\nh : (x, i) ∈ graph f\n⊢ x = f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Fin.Tuple.Sort | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 15
} | {
"line": 201,
"column": 16
} | [
{
"pp": "n : ℕ\nσ : Equiv.Perm (Fin n)\n⊢ Monotone (⇑σ ∘ ⇑σ⁻¹)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"Equiv.Perm.instInv",
"congrArg",
"PartialOrder.toPreorder",
"Monotone",
"Function.comp",
"Semilat... | [
"n : ℕ\nσ : Equiv.Perm (Fin n)\n⊢ Monotone id"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 79
} | {
"line": 114,
"column": 80
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nc : ℝ\nhc₀ : c > 0\nhc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) ... | [
"𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nc : ℝ\nhc₀ : c > 0\nhc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) x‖ < ε\nφ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.