module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
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.Combinatorics.SimpleGraph.Basic | {
"line": 1063,
"column": 4
} | {
"line": 1063,
"column": 20
} | {
"line": 1063,
"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": 1064,
"column": 4
} | {
"line": 1064,
"column": 20
} | {
"line": 1064,
"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": 1071,
"column": 4
} | {
"line": 1071,
"column": 15
} | {
"line": 1071,
"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.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.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": 135,
"column": 23
} | {
"line": 135,
"column": 33
} | {
"line": 135,
"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.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": 90,
"column": 2
} | {
"line": 90,
"column": 33
} | {
"line": 91,
"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.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.Visible | {
"line": 129,
"column": 10
} | {
"line": 129,
"column": 21
} | {
"line": 129,
"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.Analysis.Convex.Side | {
"line": 884,
"column": 4
} | {
"line": 884,
"column": 15
} | {
"line": 884,
"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.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": 1049,
"column": 21
} | {
"line": 1049,
"column": 73
} | {
"line": 1049,
"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": 1048,
"column": 77
} | {
"line": 1049,
"column": 74
} | {
"line": 1051,
"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": 181,
"column": 4
} | {
"line": 181,
"column": 56
} | {
"line": 182,
"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": 232,
"column": 14
} | {
"line": 232,
"column": 78
} | {
"line": 232,
"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": 253,
"column": 10
} | {
"line": 253,
"column": 64
} | {
"line": 253,
"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": 315,
"column": 49
} | {
"line": 315,
"column": 60
} | {
"line": 315,
"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.MeasureTheory.Integral.Layercake | {
"line": 354,
"column": 8
} | {
"line": 354,
"column": 19
} | {
"line": 354,
"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": 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.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.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.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.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": 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.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.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": 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.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.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.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.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.Data.Fin.Tuple.Sort | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 40
} | {
"line": 66,
"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": 200,
"column": 4
} | {
"line": 200,
"column": 15
} | {
"line": 200,
"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.Data.Fin.Tuple.Sort | {
"line": 202,
"column": 22
} | {
"line": 202,
"column": 33
} | {
"line": 202,
"column": 34
} | [
{
"pp": "n : ℕ\nσ : Equiv.Perm (Fin n)\ni✝ j✝ : Fin n\nhij : i✝ < j✝\nh : σ (σ⁻¹ i✝) = σ (σ⁻¹ j✝)\n⊢ i✝ = j✝",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nσ : Equiv.Perm (Fin n)\ni✝ j✝ : Fin n\nhij : i✝ < j✝\nh : σ (σ⁻¹ i✝) = σ (σ⁻¹ j✝)\n⊢ i✝ = j✝"
] | 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 |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 44
} | {
"line": 188,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\ninst✝ : CompleteSpace X\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh₁ : ¬HasEigenvalue (↑T) μ\nS : X →L[𝕜] X := T - μ • 1\nK : NNReal\nhK : AntilipschitzWith... | [
"𝕜 : Type u_1\nX : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\ninst✝ : CompleteSpace X\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh₁ : ¬HasEigenvalue (↑T) μ\nS : X →L[𝕜] X := T - μ • 1\nK : NNReal\nhK : AntilipschitzWith K ⇑S\nh₂ : ... | rw [iterate_succ', LinearMap.range_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 212,
"column": 55
} | {
"line": 212,
"column": 85
} | {
"line": 212,
"column": 86
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\ninst✝ : CompleteSpace X\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh₁ : ¬HasEigenvalue (↑T) μ\nS : X →L[𝕜] X := T - μ • 1\nK✝ : NNReal\nhK✝ : AntilipschitzWi... | [
"𝕜 : Type u_1\nX : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\ninst✝ : CompleteSpace X\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh₁ : ¬HasEigenvalue (↑T) μ\nS : X →L[𝕜] X := T - μ • 1\nK✝ : NNReal\nhK✝ : AntilipschitzWith K✝ ⇑S\nh₂... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.HermitianFunctionalCalculus | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 29
} | {
"line": 82,
"column": 30
} | [
{
"pp": "n : Type u_1\n𝕜 : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.IsHermitian\nh0 : FiniteDimensional ℝ C(↑(spectrum ℝ A), ℝ)\nf : C(↑(spectrum ℝ A), ℝ)\nhf : ↑hA.eigenvectorUnitary * diagonal (RCLike.ofReal ∘ ⇑f ∘ fun i ↦ ⟨hA.eigenvalues i, ⋯⟩) = 0\n... | [
"n : Type u_1\n𝕜 : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.IsHermitian\nh0 : FiniteDimensional ℝ C(↑(spectrum ℝ A), ℝ)\nf : C(↑(spectrum ℝ A), ℝ)\nhf : ↑hA.eigenvectorUnitary * diagonal (RCLike.ofReal ∘ ⇑f ∘ fun i ↦ ⟨hA.eigenvalues i, ⋯⟩) = 0\n⊢ (RCLike.of... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 59
} | {
"line": 98,
"column": 60
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ : 𝕜\nhμ : HasEigenvalue T μ\nv : E\nhv₁ : T v = μ • v\nhv₂ : v ≠ 0\n⊢ (starRingEnd 𝕜) μ = μ",
"ppTerm": "?m.66",
"assigned": false,
"usedCo... | [
"𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ : 𝕜\nhμ : HasEigenvalue T μ\nv : E\nhv₁ : T v = μ • v\nhv₂ : v ≠ 0\n⊢ (starRingEnd 𝕜) μ = μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 14
} | {
"line": 106,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\nhv' : v = 0\n⊢ ⟪((fun μ ↦ (eigenspace T μ).subtypeₗᵢ) μ) ⟨... | [] | simp [hv'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 14
} | {
"line": 106,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\nhv' : v = 0\n⊢ ⟪((fun μ ↦ (eigenspace T μ).subtypeₗᵢ) μ) ⟨... | [] | simp [hv'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 14
} | {
"line": 106,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\nhv' : v = 0\n⊢ ⟪((fun μ ↦ (eigenspace T μ).subtypeₗᵢ) μ) ⟨... | [] | simp [hv'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 60
} | {
"line": 109,
"column": 61
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv✝ : v ∈ eigenspace T μ\nhv : T v = μ • v\nw : E\nhw✝ : w ∈ eigenspace T ν\nhw : T w = ν • w\nhv' : ¬v = 0\nH : (s... | [
"case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv✝ : v ∈ eigenspace T μ\nhv : T v = μ • v\nw : E\nhw✝ : w ∈ eigenspace T ν\nhw : T w = ν • w\nhv' : ¬v = 0\nH : (starRingEnd �... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 197,
"column": 4
} | {
"line": 197,
"column": 97
} | {
"line": 198,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nhT : T.IsSymmetric\nv : E\nμ : Eigenvalues T\nthis :\n ∀ (w : PiLp 2 fun μ ↦ ↥(eigenspace T (↑T 1 μ))),\n T (hT.diagonalization.symm w) = ... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nhT : T.IsSymmetric\nv : E\nμ : Eigenvalues T\nthis :\n ∀ (w : PiLp 2 fun μ ↦ ↥(eigenspace T (↑T 1 μ))),\n T (hT.diagonalization.symm w) = hT.diagonali... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.PosDef | {
"line": 50,
"column": 40
} | {
"line": 50,
"column": 51
} | {
"line": 50,
"column": 52
} | [
{
"pp": "n : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype n\ninst✝¹ : RCLike 𝕜\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.PosSemidef\ni : n\nx✝ : i ∈ Finset.univ\n⊢ 0 ≤ ↑(⋯.eigenvalues i)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real"... | [
"n : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype n\ninst✝¹ : RCLike 𝕜\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.PosSemidef\ni : n\nx✝ : i ∈ Finset.univ\n⊢ 0 ≤ ⋯.eigenvalues i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.PosDef | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 13
} | {
"line": 89,
"column": 14
} | [
{
"pp": "n : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype n\ninst✝¹ : RCLike 𝕜\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.PosDef\ni : n\na✝ : i ∈ Finset.univ\n⊢ 0 < ↑(⋯.eigenvalues i)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
... | [
"n : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype n\ninst✝¹ : RCLike 𝕜\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.PosDef\ni : n\na✝ : i ∈ Finset.univ\n⊢ 0 < ⋯.eigenvalues i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.PosDef | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 50
} | {
"line": 124,
"column": 51
} | [
{
"pp": "m : Type u_1\nn : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : RCLike 𝕜\nA M : Matrix n n 𝕜\nhM : M.PosDef\nx : n → 𝕜\nhx : M *ᵥ x ⬝ᵥ star x = 0\nh : x ≠ 0\n⊢ False",
"ppTerm": "?m.53",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoal... | [
"m : Type u_1\nn : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : RCLike 𝕜\nA M : Matrix n n 𝕜\nhM : M.PosDef\nx : n → 𝕜\nhx : M *ᵥ x ⬝ᵥ star x = 0\nh : x ≠ 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 268,
"column": 26
} | {
"line": 268,
"column": 37
} | {
"line": 268,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\ni : Fin n\nv : E := (hT.unsortedEigenvectorBasis hn) i\nμ : 𝕜 := ↑T (DirectSum.IsInternal.s... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\ni : Fin n\nv : E := (hT.unsortedEigenvectorBasis hn) i\nμ : 𝕜 := ↑T (DirectSum.IsInternal.subordinateOr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 20
} | {
"line": 273,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\ni : Fin n\nv : E := (hT.unsortedEigenvectorBasis hn) i\nμ : 𝕜 := ↑T (DirectSum.IsInternal.s... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\ni : Fin n\nv : E := (hT.unsortedEigenvectorBasis hn) i\nμ : 𝕜 := ↑T (DirectSum.IsInternal.subordinateOr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 340,
"column": 4
} | {
"line": 340,
"column": 48
} | {
"line": 341,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\nv : E\ni : Fin n\nthis :\n ∀ (w : EuclideanSpace 𝕜 (Fin n)),\n T ((hT.eigenvectorBasis ... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\nv : E\ni : Fin n\nthis :\n ∀ (w : EuclideanSpace 𝕜 (Fin n)),\n T ((hT.eigenvectorBasis hn).repr.sym... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 522,
"column": 2
} | {
"line": 523,
"column": 68
} | {
"line": 523,
"column": 69
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn : ℕ∞\nK : Compacts E\nf g : 𝓓^{n}_{K}(E, F)\nhfg ... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn : ℕ∞\nK : Compacts E\nf g : 𝓓^{n}_{K}(E, F)\nhfg : (structure... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 71,
"column": 20
} | {
"line": 71,
"column": 31
} | {
"line": 71,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nh₁ : A.PosSemidef\nh₂ : (-A).PosSemidef\ni✝ j i : n\n⊢ A i i ≤ 0",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nh₁ : A.PosSemidef\nh₂ : (-A).PosSemidef\ni✝ j i : n\n⊢ A i i ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 71,
"column": 52
} | {
"line": 71,
"column": 63
} | {
"line": 71,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nh₁ : A.PosSemidef\nh₂ : (-A).PosSemidef\ni✝ j i : n\n⊢ 0 ≤ A i i",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nh₁ : A.PosSemidef\nh₂ : (-A).PosSemidef\ni✝ j i : n\n⊢ 0 ≤ A i i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 13
} | {
"line": 76,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nh₁ : A.PosSemidef\nh₂ : (-A).PosSemidef\ni j : n\nhdiag : True\nh1 : 0 ≤ A i j * star (A i j) + A i j * star (A i j)\nh2 : A i j * star (A i j) + A i j * star (A i j) ≤ 0\n⊢ A i j = 0",
"ppTerm": "?m.118",
"assigned": false,
... | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nh₁ : A.PosSemidef\nh₂ : (-A).PosSemidef\ni j : n\nhdiag : True\nh1 : 0 ≤ A i j * star (A i j) + A i j * star (A i j)\nh2 : A i j * star (A i j) + A i j * star (A i j) ≤ 0\n⊢ A i j = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 412,
"column": 38
} | {
"line": 412,
"column": 85
} | {
"line": 412,
"column": 86
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nμ : ℝ\nT : E →ₗ[𝕜] E\nhμ : HasEigenvalue T ↑μ\nhnn : ∀ (x : E), 0 ≤ RCLike.re ⟪x, T x⟫\nv : E\nhv₁ : v ∈ (genEigenspace T ↑μ) 1\nhv₂ : v ≠ 0\n⊢ 0 < ‖v‖ ^ 2",
"ppTerm": "?m.73",
"assi... | [
"𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nμ : ℝ\nT : E →ₗ[𝕜] E\nhμ : HasEigenvalue T ↑μ\nhnn : ∀ (x : E), 0 ≤ RCLike.re ⟪x, T x⟫\nv : E\nhv₁ : v ∈ (genEigenspace T ↑μ) 1\nhv₂ : v ≠ 0\n⊢ v ≠ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 672,
"column": 24
} | {
"line": 672,
"column": 35
} | {
"line": 672,
"column": 36
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn : ℕ∞\nK : Compacts E\ni : ℕ\nhin : n < ↑i\n⊢ ¬↑i ≤... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn : ℕ∞\nK : Compacts E\ni : ℕ\nhin : n < ↑i\n⊢ n < ↑i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 38
} | {
"line": 81,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA B : Matrix n n 𝕜\nh₁ : A ≤ B\nh₂ : B ≤ A\n⊢ A = B",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA B : Matrix n n 𝕜\nh₁ : A ≤ B\nh₂ : B ≤ A\n⊢ A = B"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 82,
"column": 9
} | {
"line": 82,
"column": 47
} | {
"line": 82,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA B : Matrix n n 𝕜\nh₁ : A ≤ B\nh₂ : B ≤ A\n⊢ (-(B - A)).PosSemidef",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA B : Matrix n n 𝕜\nh₁ : A ≤ B\nh₂ : B ≤ A\n⊢ (-(B - A)).PosSemidef"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 421,
"column": 38
} | {
"line": 421,
"column": 85
} | {
"line": 421,
"column": 86
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nμ : ℝ\nT : E →ₗ[𝕜] E\nhμ : HasEigenvalue T ↑μ\nhnn : ∀ (x : E), 0 < RCLike.re ⟪x, T x⟫\nv : E\nhv₁ : v ∈ (genEigenspace T ↑μ) 1\nhv₂ : v ≠ 0\n⊢ 0 < ‖v‖ ^ 2",
"ppTerm": "?m.73",
"assi... | [
"𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nμ : ℝ\nT : E →ₗ[𝕜] E\nhμ : HasEigenvalue T ↑μ\nhnn : ∀ (x : E), 0 < RCLike.re ⟪x, T x⟫\nv : E\nhv₁ : v ∈ (genEigenspace T ↑μ) 1\nhv₂ : v ≠ 0\n⊢ v ≠ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 459,
"column": 2
} | {
"line": 459,
"column": 17
} | {
"line": 459,
"column": 18
} | [
{
"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 : IsCompactOperator ⇑T\nhT' : (↑T).IsSymmetric\nS : ↥(⨆ μ, eigenspace (↑T) μ)ᗮ →L[𝕜] ↥(⨆ μ, eigenspace (↑T) μ)ᗮ := T.restrict ⋯\nhS_compact : IsC... | [
"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 : IsCompactOperator ⇑T\nhT' : (↑T).IsSymmetric\nS : ↥(⨆ μ, eigenspace (↑T) μ)ᗮ →L[𝕜] ↥(⨆ μ, eigenspace (↑T) μ)ᗮ := T.restrict ⋯\nhS_compact : IsCompactOperat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 41
} | {
"line": 151,
"column": 42
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\nx : n → 𝕜\n⊢ (((toLinearMap₂' 𝕜) A) (star x)) x = 0 ↔ A *ᵥ x = 0",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.m... | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\nx : n → 𝕜\n⊢ star x ⬝ᵥ A *ᵥ x = 0 ↔ A *ᵥ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 44
} | {
"line": 177,
"column": 45
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nx✝ : A.IsHermitian ∧ spectrum 𝕜 A ⊆ {a | 0 ≤ a}\nh1 : A.IsHermitian\nh2 : spectrum 𝕜 A ⊆ {a | 0 ≤ a}\ni : n\n⊢ 0 i ≤ h1.eigenvalues i",
"ppTerm": "?refine_2",
"assigne... | [
"case refine_2\n𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nx✝ : A.IsHermitian ∧ spectrum 𝕜 A ⊆ {a | 0 ≤ a}\nh1 : A.IsHermitian\nh2 : spectrum 𝕜 A ⊆ {a | 0 ≤ a}\ni : n\n⊢ 0 ≤ h1.eigenvalues i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 219,
"column": 2
} | {
"line": 219,
"column": 83
} | {
"line": 220,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Finite n\nm : Type u_3\ninst✝ : Finite m\nthis✝ : Fintype n\nthis : Fintype m\na : Matrix n n 𝕜\nhx : (star a * a).PosSemidef\nb : Matrix m m 𝕜\nhy : (star b * b).PosSemidef\n⊢ (kroneckerMap (fun x1 x2 ↦ x1 * x2) (star a * a) (star b * b)).Pos... | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Finite n\nm : Type u_3\ninst✝ : Finite m\nthis✝ : Fintype n\nthis : Fintype m\na : Matrix n n 𝕜\nhx : (star a * a).PosSemidef\nb : Matrix m m 𝕜\nhy : (star b * b).PosSemidef\n⊢ ((kroneckerMap (fun x1 x2 ↦ x1 * x2) a b)ᴴ * kroneckerMap (fun x1 x2 ↦ x1 * x2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 250,
"column": 2
} | {
"line": 251,
"column": 73
} | {
"line": 251,
"column": 74
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nι : Type u_3\nA B : Matrix ι ι 𝕜\nhA : A.PosSemidef\nhB : B.PosSemidef\nx : ι →₀ 𝕜\nhAB : ((A ⊙ B).submatrix Subtype.val Subtype.val).PosSemidef\n⊢ 0 ≤ x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * (A ⊙ B) i j * xj",
"ppTerm": "?m.66",
"assigned": true,
"us... | [
"𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nι : Type u_3\nA B : Matrix ι ι 𝕜\nhA : A.PosSemidef\nhB : B.PosSemidef\nx : ι →₀ 𝕜\nhAB : ((A ⊙ B).submatrix Subtype.val Subtype.val).PosSemidef\n⊢ 0 ≤\n ∑ x_1 ∈ (Finsupp.subtypeDomain (fun x_1 ↦ x_1 ∈ x.support) x).support,\n ∑ x_2 ∈ (Finsupp.subtypeDomain (fun x_2 ↦ x_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.GramMatrix | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 49
} | {
"line": 75,
"column": 2
} | [
{
"pp": "E : Type u_1\nn : Type u_2\n𝕜 : Type u_4\ninst✝³ : RCLike 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : Fintype n\nv : n → E\nx y : n → 𝕜\n⊢ star x ⬝ᵥ gram 𝕜 v *ᵥ y = ⟪∑ i, x i • v i, ∑ i, y i • v i⟫_𝕜",
"ppTerm": "?m.41",
"assigned": true,
"usedConstan... | [
"E : Type u_1\nn : Type u_2\n𝕜 : Type u_4\ninst✝³ : RCLike 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : Fintype n\nv : n → E\nx y : n → 𝕜\n⊢ star x ⬝ᵥ gram 𝕜 v *ᵥ y = ∑ i, ∑ j, (starRingEnd 𝕜) (x i) * y j * ⟪v i, v j⟫_𝕜",
"E : Type u_1\nn : Type u_2\n𝕜 : Type u_4\ninst✝³ :... | trans ∑ i, ∑ j, conj (x i) * y j * ⟪v i, v j⟫_𝕜 | Batteries.Tactic._aux_Batteries_Tactic_Trans___elabRules_Batteries_Tactic_tacticTrans____1 | Batteries.Tactic.tacticTrans___ |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 774,
"column": 4
} | {
"line": 774,
"column": 15
} | {
"line": 774,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedSpace 𝕜 F\ninst✝⁵ : SMulCommClass ℝ 𝕜 F\ninst✝⁴ : NormedAddCommGroup F'\nin... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedSpace 𝕜 F\ninst✝⁵ : SMulCommClass ℝ 𝕜 F\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : Norme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 46
} | {
"line": 272,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nι : Type u_3\nA B : Matrix ι ι 𝕜\nhA : A.PosDef\nhB : B.PosDef\nx : ι →₀ 𝕜\nhx : x ≠ 0\nhAB : ((A ⊙ B).submatrix Subtype.val Subtype.val).PosDef\n⊢ Finsupp.subtypeDomain (fun x_1 ↦ x_1 ∈ x.support) x ≠ 0",
"ppTerm": "?m.210",
"assigned": true,
"usedConsta... | [
"𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nι : Type u_3\nA B : Matrix ι ι 𝕜\nhA : A.PosDef\nhB : B.PosDef\nx : ι →₀ 𝕜\nhx : x ≠ 0\nhAB : ((A ⊙ B).submatrix Subtype.val Subtype.val).PosDef\n⊢ x.support.Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 330,
"column": 6
} | {
"line": 330,
"column": 31
} | {
"line": 330,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝¹ : RCLike 𝕜\ninst✝ : Fintype n\nx : Matrix n n 𝕜\ny : (Matrix n n 𝕜)ˣ\nhM : (star ↑y * ↑y).PosDef\n__spread✝⁻⁰ : PreInnerProductSpace.Core 𝕜 (Matrix n n 𝕜) := ⋯.matrixPreInnerProductSpace\nhx : ↑y * xᴴ = 0\n⊢ x = 0",
"ppTerm": "?m.176",
"assigned": false,... | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝¹ : RCLike 𝕜\ninst✝ : Fintype n\nx : Matrix n n 𝕜\ny : (Matrix n n 𝕜)ˣ\nhM : (star ↑y * ↑y).PosDef\n__spread✝⁻⁰ : PreInnerProductSpace.Core 𝕜 (Matrix n n 𝕜) := ⋯.matrixPreInnerProductSpace\nhx : ↑y * xᴴ = 0\n⊢ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 813,
"column": 4
} | {
"line": 813,
"column": 15
} | {
"line": 813,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : SMulCommClass ℝ 𝕜 F\ninst✝³ : NormedAddCommGroup F'\nins... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : SMulCommClass ℝ 𝕜 F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : Normed... | 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.