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.NumberTheory.GaussSum | {
"line": 257,
"column": 2
} | {
"line": 259,
"column": 27
} | {
"line": 261,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : Fintype R\nR' : Type v\ninst✝ : CommRing R'\np : ℕ\nfp : Fact (Nat.Prime p)\nhch : CharP R' p\nhp : IsUnit ↑p\nχ : MulChar R R'\nhχ : χ.IsQuadratic\nψ : AddChar R R'\n⊢ gaussSum χ ψ ^ p = χ ↑p * gaussSum χ ψ",
"ppTerm": "?m.32",
"assigned": true,
"u... | [] | rw [_root_.gaussSum_frob, pow_mulShift, hχ.pow_char p, ← gaussSum_mulShift χ ψ hp.unit,
← mul_assoc, hp.unit_spec, ← pow_two, ← pow_apply' _ two_ne_zero, hχ.sq_eq_one, ← hp.unit_spec,
one_apply_coe, one_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.ZMod | {
"line": 55,
"column": 10
} | {
"line": 55,
"column": 35
} | {
"line": 56,
"column": 4
} | [
{
"pp": "N : ℕ\ninst✝² : NeZero N\nE : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℂ E\nΦ : ZMod N → E\nk : ZMod N\n⊢ auxDFT (fun j ↦ Φ (-j)) k = auxDFT Φ (-k)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"instHSMul",
"HMul.h... | [
"N : ℕ\ninst✝² : NeZero N\nE : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℂ E\nΦ : ZMod N → E\nk : ZMod N\n⊢ ∑ x, stdAddChar (-(x * k)) • Φ (-x) = ∑ j, stdAddChar (-(j * -k)) • Φ j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 105,
"column": 2
} | {
"line": 106,
"column": 28
} | {
"line": 106,
"column": 29
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : F → E\nhf : Differentiable ℝ f\nx : F\np : ℝ≥0\nhp : 1 < p\n⊢ ‖fderiv ℝ (fun x ↦ ‖f x‖ ^ ↑p) x‖ₑ ≤ ↑p * ‖f x‖ₑ ^ (↑p - 1) * ‖fderiv ℝ f x‖ₑ",
"ppTer... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : F → E\nhf : Differentiable ℝ f\nx : F\np : ℝ≥0\nhp : 1 < p\n⊢ ‖fderiv ℝ (fun x ↦ ‖f x‖ ^ ↑p) x‖₊ ≤ p * ‖f x‖₊ ^ (↑p - 1) * ‖fderiv ℝ f x‖₊"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Grade | {
"line": 36,
"column": 4
} | {
"line": 36,
"column": 15
} | {
"line": 36,
"column": 16
} | [
{
"pp": "α : Type u_1\ns : Multiset α\na : α\nt : Multiset α\nhst : s < t\nhts : t < a ::ₘ s\n⊢ t.card < succ s.card",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"Order.succ_eq_add_one",
"Nat.instOne",
"congrAr... | [
"α : Type u_1\ns : Multiset α\na : α\nt : Multiset α\nhst : s < t\nhts : t < a ::ₘ s\n⊢ t.card ≤ s.card"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Grade | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 13
} | {
"line": 116,
"column": 14
} | [
{
"pp": "α : Type u_1\ns t : Finset α\ninst✝ : DecidableEq α\nh : s ⋖ t\n⊢ ∃ a ∉ s, insert a s = t",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ns t : Finset α\ninst✝ : DecidableEq α\nh : s ⋖ t\n⊢ ∃ a ∉ s, insert a s = t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 58
} | {
"line": 119,
"column": 59
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\np : ℝ\nhp : 1 < p\nx : E\nhx : x = 0\nthis : ContinuousAt (fun x ↦ p * ‖x‖ ^ (p - 1)) 0\n⊢ Filter.Tendsto (fun x ↦ p * ‖x‖ ^ (p - 1)) (𝓝 0) (𝓝 0)",
"ppTerm": "?m.136",
"assigned": false,
"usedConstants": [],
"... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\np : ℝ\nhp : 1 < p\nx : E\nhx : x = 0\nthis : ContinuousAt (fun x ↦ p * ‖x‖ ^ (p - 1)) 0\n⊢ Filter.Tendsto (fun x ↦ p * ‖x‖ ^ (p - 1)) (𝓝 0) (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.ZMod | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 73
} | {
"line": 194,
"column": 74
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nh : ∀ {f : ZMod N → ℂ}, Function.Even f → Function.Even (𝓕 f)\nhΦ : Function.Even (𝓕 Φ)\nx : ZMod N\n⊢ Φ (-x) = Φ x",
"ppTerm": "?m.38",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nh : ∀ {f : ZMod N → ℂ}, Function.Even f → Function.Even (𝓕 f)\nhΦ : Function.Even (𝓕 Φ)\nx : ZMod N\n⊢ Φ (-x) = Φ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.ZMod | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 85
} | {
"line": 202,
"column": 86
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nh : ∀ {f : ZMod N → ℂ}, Function.Odd f → Function.Odd (𝓕 f)\nhΦ : Function.Odd (𝓕 Φ)\nx : ZMod N\n⊢ Φ (-x) = -Φ x",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nh : ∀ {f : ZMod N → ℂ}, Function.Odd f → Function.Odd (𝓕 f)\nhΦ : Function.Odd (𝓕 Φ)\nx : ZMod N\n⊢ Φ (-x) = -Φ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.ZMod | {
"line": 216,
"column": 43
} | {
"line": 216,
"column": 60
} | {
"line": 216,
"column": 61
} | [
{
"pp": "case e_f\nN : ℕ\ninst✝ : NeZero N\nχ : DirichletCharacter ℂ N\nk j : ZMod N\n⊢ stdAddChar (-(k * j)) * χ j = χ j * stdAddChar (-(k * j))",
"ppTerm": "?e_f",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"NegZero... | [
"case e_f\nN : ℕ\ninst✝ : NeZero N\nχ : DirichletCharacter ℂ N\nk j : ZMod N\n⊢ ↑(toCircle (-(k * j))) * χ j = χ j * ↑(toCircle (-(k * j)))"
] | stdAddChar_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finset.Interval | {
"line": 50,
"column": 10
} | {
"line": 50,
"column": 64
} | {
"line": 50,
"column": 65
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq α\ns t : Finset α\nu₁ u₂ : ↥(t \\ s).powerset\nh : (fun u ↦ (↑u).disjUnion s ⋯) u₁ = (fun u ↦ (↑u).disjUnion s ⋯) u₂\n⊢ u₁ = u₂",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset",
"Finset.instSDiff",... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq α\ns t : Finset α\nu₁ u₂ : ↥(t \\ s).powerset\nh : (fun u ↦ (↑u).disjUnion s ⋯) u₁ = (fun u ↦ (↑u).disjUnion s ⋯) u₂\n⊢ ↑u₁ = ↑u₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.GaussSum | {
"line": 367,
"column": 4
} | {
"line": 368,
"column": 12
} | {
"line": 369,
"column": 4
} | [
{
"pp": "case refine_1\nF : Type u_1\ninst✝¹ : Fintype F\ninst✝ : Field F\nhF : ringChar F ≠ 2\nhp2 : ∀ (n : ℕ), 2 ^ n ≠ 0\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ ↑n\nFF : Type u_1 := CyclotomicField 8 F\nhchar : ringChar F = ringChar FF\nFFp : Nat.Prime (ringChar FF)\nthis : Fa... | [
"case refine_2\nF : Type u_1\ninst✝¹ : Fintype F\ninst✝ : Field F\nhF : ringChar F ≠ 2\nhp2 : ∀ (n : ℕ), 2 ^ n ≠ 0\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ ↑n\nFF : Type u_1 := CyclotomicField 8 F\nhchar : ringChar F = ringChar FF\nFFp : Nat.Prime (ringChar FF)\nthis : Fact (Nat.Prim... | · rw [← pow_mul, ← map_nsmul_eq_pow ψ₈.char, ψ₈.prim.zmod_char_eq_one_iff]
decide | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.InnerProductSpace.Affine | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 13
} | {
"line": 94,
"column": 14
} | [
{
"pp": "V : Type u_2\nP : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b p : P\nh_inner : ⟪p -ᵥ a, b -ᵥ a⟫ = 0\n⊢ dist p b ^ 2 = dist p a ^ 2 + dist a b ^ 2",
"ppTerm": "?m.60",
"assigned": false,
"usedConstants": []... | [
"V : Type u_2\nP : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b p : P\nh_inner : ⟪p -ᵥ a, b -ᵥ a⟫ = 0\n⊢ dist p b ^ 2 = dist p a ^ 2 + dist a b ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Hofer | {
"line": 72,
"column": 6
} | {
"line": 74,
"column": 52
} | {
"line": 75,
"column": 6
} | [
{
"pp": "case hi\nX : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : CompleteSpace X\nx : X\nε : ℝ\nε_pos : 0 < ε\nϕ : X → ℝ\ncont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ\nnonneg : ∀ (y : X), 0 ≤ ϕ y\nreformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'\n... | [
"case hi\nX : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : CompleteSpace X\nx : X\nε : ℝ\nε_pos : 0 < ε\nϕ : X → ℝ\ncont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ\nnonneg : ∀ (y : X), 0 ≤ ϕ y\nreformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'\nthis : Nonem... | have B : 2 ^ (n + 1) * ϕ x ≤ ϕ (u (n + 1)) := by
refine @geom_le (ϕ ∘ u) _ zero_le_two (n + 1) fun m hm => ?_
exact (IH _ <| Nat.lt_add_one_iff.1 hm).2.le | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Coalgebra | {
"line": 76,
"column": 8
} | {
"line": 76,
"column": 26
} | {
"line": 76,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : FiniteDimensional 𝕜 E\nA : Type u_3\ninst✝³ : Ring A\ninst✝² : Module 𝕜 A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : IsScalarTower 𝕜 A A\ne : E ≃ₗ[𝕜] A\n⊢ ↑(TensorProduct.assoc 𝕜 ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : FiniteDimensional 𝕜 E\nA : Type u_3\ninst✝³ : Ring A\ninst✝² : Module 𝕜 A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : IsScalarTower 𝕜 A A\ne : E ≃ₗ[𝕜] A\n⊢ ↑(TensorProduct.assoc 𝕜 E E E) ∘ₗ\n ... | ← adjoint_lTensor, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Coalgebra | {
"line": 85,
"column": 35
} | {
"line": 85,
"column": 53
} | {
"line": 85,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : FiniteDimensional 𝕜 E\nA : Type u_3\ninst✝³ : Ring A\ninst✝² : Module 𝕜 A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : IsScalarTower 𝕜 A A\ne : E ≃ₗ[𝕜] A\n⊢ LinearMap.lTensor E (adjo... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : FiniteDimensional 𝕜 E\nA : Type u_3\ninst✝³ : Ring A\ninst✝² : Module 𝕜 A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : IsScalarTower 𝕜 A A\ne : E ≃ₗ[𝕜] A\n⊢ adjoint (LinearMap.lTensor E (toSpanS... | ← adjoint_lTensor, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Coalgebra | {
"line": 135,
"column": 38
} | {
"line": 135,
"column": 56
} | {
"line": 135,
"column": 57
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : Coalgebra 𝕜 E\nx : E\n⊢ (adjoint comul ∘ₗ LinearMap.lTensor E (adjoint counit)) (x ⊗ₜ[𝕜] One.one) = x",
"ppTerm": "?m.241",
"assigned": tru... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : Coalgebra 𝕜 E\nx : E\n⊢ (adjoint comul ∘ₗ adjoint (LinearMap.lTensor E counit)) (x ⊗ₜ[𝕜] One.one) = x"
] | ← adjoint_lTensor, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 25
} | {
"line": 133,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nT : ι → E →ₗ[𝕜] E\ns : Finset ι\nhT : ∀ i ∈ s, (T i).IsPositive\nx✝ : E\n⊢ 0 ≤ re ⟪(∑ i ∈ s, T i) x✝, x✝⟫",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nT : ι → E →ₗ[𝕜] E\ns : Finset ι\nhT : ∀ i ∈ s, (T i).IsPositive\nx✝ : E\n⊢ 0 ≤ ∑ x ∈ s, re ⟪(T x) x✝, x✝⟫"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Coalgebra | {
"line": 163,
"column": 45
} | {
"line": 163,
"column": 63
} | {
"line": 163,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : Coalgebra 𝕜 E\nr : 𝕜\nx : E\n⊢ (adjoint comul) ((LinearMap.rTensor E (adjoint counit)) (r ⊗ₜ[𝕜] x)) =\n (adjoint comul) ((LinearMap.lTensor E (... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : Coalgebra 𝕜 E\nr : 𝕜\nx : E\n⊢ (adjoint comul) ((LinearMap.rTensor E (adjoint counit)) (r ⊗ₜ[𝕜] x)) =\n (adjoint comul) ((adjoint (LinearMap.lTensor E coun... | ← adjoint_lTensor, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 151,
"column": 23
} | {
"line": 151,
"column": 48
} | {
"line": 151,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsPositive\nhT' : T ≠ 0\nα : 𝕜\nh : (α • T).IsPositive\n⊢ ?m.48",
"ppTerm": "?m.53",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsPositive\nhT' : T ≠ 0\nα : 𝕜\nh : (α • T).IsPositive\n⊢ ?m.48"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 152,
"column": 13
} | {
"line": 152,
"column": 43
} | {
"line": 152,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsPositive\nhT' : T ≠ 0\nα : 𝕜\nh : (α • T).IsPositive\nx : E\nhx : 0 < ⟪x, T x⟫\n⊢ ?m.78",
"ppTerm": "?m.79",
"assigned": false,
"usedConstants": [],
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsPositive\nhT' : T ≠ 0\nα : 𝕜\nh : (α • T).IsPositive\nx : E\nhx : 0 < ⟪x, T x⟫\n⊢ ?m.78"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 158,
"column": 2
} | {
"line": 160,
"column": 11
} | {
"line": 160,
"column": 12
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\nn : ℕ\nhT : T.IsPositive\nhn : Module.finrank 𝕜 E = n\ni : Fin n\n⊢ 0 ≤ ⋯.eigenvalues hn i",
"ppTerm": "?m.44",
"assigned": false,
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\nn : ℕ\nhT : T.IsPositive\nhn : Module.finrank 𝕜 E = n\ni : Fin n\n⊢ 0 ≤ ⋯.eigenvalues hn i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 169,
"column": 29
} | {
"line": 169,
"column": 40
} | {
"line": 169,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace 𝕜 F\nx✝² x✝¹ x✝ : E →ₗ[𝕜] E\nh₁ : (x✝¹ - x✝²).IsPositive\nh₂ : (x✝ - x✝¹).IsPositive\n⊢ (x✝ - x✝²).IsPositive",
"... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace 𝕜 F\nx✝² x✝¹ x✝ : E →ₗ[𝕜] E\nh₁ : (x✝¹ - x✝²).IsPositive\nh₂ : (x✝ - x✝¹).IsPositive\n⊢ (x✝ - x✝²).IsPositive"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 13
} | {
"line": 181,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf : E →ₗ[𝕜] E\n⊢ 0 ≤ f ↔ f.IsPositive",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf : E →ₗ[𝕜] E\n⊢ 0 ≤ f ↔ f.IsPositive"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 30
} | {
"line": 268,
"column": 31
} | [
{
"pp": "case inr\nι : Type u_1\nA : ι → Type u_2\ninst✝³ : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\np : ℝ\nhp₀ : 0 ≤ p\nhp : (↑#ι - 1) * p ≤ 1\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\nh✝ : Nonempty ((i... | [
"case inr\nι : Type u_1\nA : ι → Type u_2\ninst✝³ : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\np : ℝ\nhp₀ : 0 ≤ p\nhp : (↑#ι - 1) * p ≤ 1\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\nh✝ : Nonempty ((i : ι) → A i)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 200,
"column": 32
} | {
"line": 200,
"column": 43
} | {
"line": 200,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace 𝕜 F\nT : E →ₗ[𝕜] E\nf : E ≃ₗᵢ[𝕜] F\nx✝ : T.IsSymmetric\nh : ∀ (x : F), 0 ≤ re ⟪T (f.symm x), f.symm x⟫\nx : E\n⊢ 0 ≤... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace 𝕜 F\nT : E →ₗ[𝕜] E\nf : E ≃ₗᵢ[𝕜] F\nx✝ : T.IsSymmetric\nh : ∀ (x : F), 0 ≤ re ⟪T (f.symm x), f.symm x⟫\nx : E\n⊢ 0 ≤ re ⟪T x, x⟫... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.ExteriorPower | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 51
} | {
"line": 113,
"column": 2
} | [
{
"pp": "n : ℕ\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nx : ↥(⋀[ℝ]^n E)\n⊢ 0 ≤ RCLike.re (∑ s, ((Module.Basis.exteriorPower n (stdOrthonormalBasis ℝ E).toBasis).repr x) s ^ 2)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": ... | [] | exact Finset.sum_nonneg (fun _ _ ↦ sq_nonneg _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.ExteriorPower | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 15
} | {
"line": 117,
"column": 16
} | [
{
"pp": "n : ℕ\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nx : ↥(⋀[ℝ]^n E)\nh : ∀ i ∈ Finset.univ, ((Module.Basis.exteriorPower n (stdOrthonormalBasis ℝ E).toBasis).repr x) i ^ 2 = 0\n⊢ ∀ (i : ↑(Set.powersetCard (Fin (Module.finrank ℝ E)) n)),\n ... | [
"n : ℕ\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nx : ↥(⋀[ℝ]^n E)\nh : ∀ i ∈ Finset.univ, ((Module.Basis.exteriorPower n (stdOrthonormalBasis ℝ E).toBasis).repr x) i ^ 2 = 0\n⊢ ∀ (a : Finset (Fin (Module.finrank ℝ E))) (b : a.card = n),\n ((Module... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 73
} | {
"line": 240,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\np : E →ₗ[𝕜] E\nhp : p.IsSymmetricProjection\na : E\nha : a ∈ p.range\nhh : ∀ {T : E →ₗ[𝕜] E}, T.IsSymmetricProjection → re ⟪T a, a⟫ = ‖T a‖ ^ 2\nU : Submodule 𝕜 E\nw✝ : U.HasOrthogonalProj... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\np : E →ₗ[𝕜] E\nhp : p.IsSymmetricProjection\na : E\nha : a ∈ p.range\nhh : ∀ {T : E →ₗ[𝕜] E}, T.IsSymmetricProjection → re ⟪T a, a⟫ = ‖T a‖ ^ 2\nU : Submodule 𝕜 E\nw✝ : U.HasOrthogonalProjection\nhq :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.JointEigenspace | {
"line": 95,
"column": 2
} | {
"line": 96,
"column": 9
} | {
"line": 96,
"column": 10
} | [
{
"pp": "case e_p\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nα : 𝕜\nA B : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nhB : B.IsSymmetric\nhAB : Commute A B\n⊢ ⨆ i, (genEigenspace (B.restrict ⋯) i) 1 = ⊤",
"ppTerm": "?e_p",
"assigned... | [
"case e_p\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nα : 𝕜\nA B : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nhB : B.IsSymmetric\nhAB : Commute A B\n⊢ ⨆ i, (genEigenspace (B.restrict ⋯) i) 1 = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.JointEigenspace | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 64
} | {
"line": 105,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nA B : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nhA : A.IsSymmetric\nhB : B.IsSymmetric\nhAB : Commute A B\n⊢ ⨆ α, ⨆ γ, eigenspace A α ⊓ eigenspace B γ = ⊤",
"ppTerm": "?m.69",
"ass... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nA B : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nhA : A.IsSymmetric\nhB : B.IsSymmetric\nhAB : Commute A B\n⊢ ⨆ α, eigenspace A α = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.LaxMilgram | {
"line": 63,
"column": 23
} | {
"line": 63,
"column": 34
} | {
"line": 63,
"column": 35
} | [
{
"pp": "V : Type u\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : CompleteSpace V\nB : V →L[ℝ] V →L[ℝ] ℝ\nC : ℝ\nC_ge_0 : 0 < C\ncoercivity : ∀ (u : V), C * ‖u‖ * ‖u‖ ≤ (B u) u\nv : V\nh : ¬0 < ‖v‖\n⊢ v = 0",
"ppTerm": "?m.174",
"assigned": false,
"usedConstants": [],
"... | [
"V : Type u\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : CompleteSpace V\nB : V →L[ℝ] V →L[ℝ] ℝ\nC : ℝ\nC_ge_0 : 0 < C\ncoercivity : ∀ (u : V), C * ‖u‖ * ‖u‖ ≤ (B u) u\nv : V\nh : ¬0 < ‖v‖\n⊢ v = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 443,
"column": 44
} | {
"line": 443,
"column": 91
} | {
"line": 443,
"column": 92
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : E →L[𝕜] E\nc : ℝ≥0\nhc : 0 < c\nh : ∀ (x : E), ‖x‖ ^ 2 * ↑c ≤ ‖⟪f x, x⟫‖\nh_anti : AntilipschitzWith c⁻¹ ⇑f\nx : E\nhx : x ∈ (↑f).rangeᗮ\n⊢ ‖x‖ ^ 2 * ↑c ≤ 0",
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : E →L[𝕜] E\nc : ℝ≥0\nhc : 0 < c\nh : ∀ (x : E), ‖x‖ ^ 2 * ↑c ≤ ‖⟪f x, x⟫‖\nh_anti : AntilipschitzWith c⁻¹ ⇑f\nx : E\nhx : x ∈ (↑f).rangeᗮ\n⊢ ‖x‖ ^ 2 * ↑c ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 465,
"column": 29
} | {
"line": 465,
"column": 40
} | {
"line": 465,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace 𝕜 F\nx✝² x✝¹ x✝ : E →L[𝕜] E\nh₁ : (x✝¹ - x✝²).IsPositive\nh₂ : (x✝ - x✝¹).IsPositive\n⊢ (x✝ - x✝²).IsPositive",
"... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace 𝕜 F\nx✝² x✝¹ x✝ : E →L[𝕜] E\nh₁ : (x✝¹ - x✝²).IsPositive\nh₂ : (x✝ - x✝¹).IsPositive\n⊢ (x✝ - x✝²).IsPositive"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 475,
"column": 2
} | {
"line": 475,
"column": 13
} | {
"line": 475,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf : E →L[𝕜] E\n⊢ 0 ≤ f ↔ f.IsPositive",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf : E →L[𝕜] E\n⊢ 0 ≤ f ↔ f.IsPositive"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.LaxMilgram | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 13
} | {
"line": 72,
"column": 14
} | [
{
"pp": "V : Type u\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : CompleteSpace V\nB : V →L[ℝ] V →L[ℝ] ℝ\ncoercive : IsCoercive B\nC : ℝ\nC_pos : 0 < C\nbelow_bound : ∀ (v : V), C * ‖v‖ ≤ ‖(continuousLinearMapOfBilin B) v‖\n⊢ ∀ (x : V), C⁻¹⁻¹ * ‖x‖ ≤ ‖(continuousLinearMapOfBilin B) x‖"... | [
"V : Type u\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : CompleteSpace V\nB : V →L[ℝ] V →L[ℝ] ℝ\ncoercive : IsCoercive B\nC : ℝ\nC_pos : 0 < C\nbelow_bound : ∀ (v : V), C * ‖v‖ ≤ ‖(continuousLinearMapOfBilin B) v‖\n⊢ ∀ (x : V), C * ‖x‖ ≤ ‖(continuousLinearMapOfBilin B) x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 530,
"column": 13
} | {
"line": 530,
"column": 24
} | {
"line": 530,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E ≃ₗ[𝕜] E\nhT : (↑T).IsPositive\nx : E\n⊢ ?m.59",
"ppTerm": "?m.60",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E ≃ₗ[𝕜] E\nhT : (↑T).IsPositive\nx : E\n⊢ ?m.59"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 546,
"column": 2
} | {
"line": 555,
"column": 76
} | {
"line": 557,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nT : E →L[𝕜] E\n⊢ T.IsPositive ↔ ∃ m u, T = ∑ i, ((rankOne 𝕜) (u i)) (u i)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Linea... | [] | refine ⟨fun hT ↦ ?_, fun ⟨m, u, hT⟩ ↦ hT ▸ isPositive_sum _ fun _ _ ↦ isPositive_rankOne_self _⟩
let a (i : Fin (Module.finrank 𝕜 E)) : E :=
((hT.isSymmetric.eigenvalues rfl i).sqrt : 𝕜) • hT.isSymmetric.eigenvectorBasis rfl i
refine ⟨Module.finrank 𝕜 E, a, ext fun _ ↦ ?_⟩
simp_rw [_root_.sum_apply, rankOn... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 546,
"column": 2
} | {
"line": 555,
"column": 76
} | {
"line": 557,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nT : E →L[𝕜] E\n⊢ T.IsPositive ↔ ∃ m u, T = ∑ i, ((rankOne 𝕜) (u i)) (u i)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Linea... | [] | refine ⟨fun hT ↦ ?_, fun ⟨m, u, hT⟩ ↦ hT ▸ isPositive_sum _ fun _ _ ↦ isPositive_rankOne_self _⟩
let a (i : Fin (Module.finrank 𝕜 E)) : E :=
((hT.isSymmetric.eigenvalues rfl i).sqrt : 𝕜) • hT.isSymmetric.eigenvectorBasis rfl i
refine ⟨Module.finrank 𝕜 E, a, ext fun _ ↦ ?_⟩
simp_rw [_root_.sum_apply, rankOn... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Dynamics.BirkhoffSum.Basic | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 27
} | {
"line": 61,
"column": 28
} | [
{
"pp": "α : Type u_1\nM : Type u_2\ninst✝ : AddCommMonoid M\nf : α → α\ng g' : α → M\nn : ℕ\nx : α\n⊢ birkhoffSum f (g + g') n x = birkhoffSum f g n x + birkhoffSum f g' n x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Membership... | [
"α : Type u_1\nM : Type u_2\ninst✝ : AddCommMonoid M\nf : α → α\ng g' : α → M\nn : ℕ\nx : α\n⊢ ∑ x_1 ∈ range n, (g (f^[x_1] x) + g' (f^[x_1] x)) = ∑ k ∈ range n, g (f^[k] x) + ∑ k ∈ range n, g' (f^[k] x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.BirkhoffSum.NormedSpace | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 17
} | {
"line": 123,
"column": 18
} | [
{
"pp": "case h\n𝕜 : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝³ : PseudoEMetricSpace X\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : X → X\ng : X → E\nhf : LipschitzWith 1 f\nhg : UniformContinuous g\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0∞\nhδ₀ : 0 < δ\nhδε : ∀ (x y : X), (x, y) ∈ {p... | [
"case h\n𝕜 : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝³ : PseudoEMetricSpace X\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : X → X\ng : X → E\nhf : LipschitzWith 1 f\nhg : UniformContinuous g\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0∞\nhδ₀ : 0 < δ\nhδε : ∀ (x y : X), (x, y) ∈ {p | edist p.1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 496,
"column": 32
} | {
"line": 496,
"column": 43
} | {
"line": 496,
"column": 44
} | [
{
"pp": "E : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : InnerProductSpace ℝ F'\nu : E → F'\nhu : ContDiff ... | [
"E : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : InnerProductSpace ℝ F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 59,
"column": 4
} | {
"line": 60,
"column": 11
} | {
"line": 60,
"column": 12
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(f.eqLocus 1)\nhg_proj : ∀ (x : ↥(f.eqLocus 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(f.eqLocus 1)\nhg_proj : ∀ (x : ↥(f.eqLocus 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑(f - 1).rang... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 21
} | {
"line": 72,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(f.eqLocus 1)\nhg_proj : ∀ (x : ↥(f.eqLocus 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(f.eqLocus 1)\nhg_proj : ∀ (x : ↥(f.eqLocus 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑(f - 1).rang... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 74,
"column": 2
} | {
"line": 75,
"column": 9
} | {
"line": 75,
"column": 10
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(f.eqLocus 1)\nhg_proj : ∀ (x : ↥(f.eqLocus 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(f.eqLocus 1)\nhg_proj : ∀ (x : ↥(f.eqLocus 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑(f - 1).rang... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 17
} | {
"line": 105,
"column": 18
} | [
{
"pp": "case hg_ker.refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : E →L[𝕜] E\nhf : ‖f‖ ≤ 1\nx : E\nhx : x ∈ (↑f - 1).rangeᗮ\n⊢ ‖↑f x‖ ≤ ‖↑1 x‖",
"ppTerm": "?hg_ker.refine_1",
"assigned": true,
... | [
"case hg_ker.refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : E →L[𝕜] E\nhf : ‖f‖ ≤ 1\nx : E\nhx : x ∈ (↑f - 1).rangeᗮ\n⊢ ‖f x‖ ≤ ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 107,
"column": 8
} | {
"line": 107,
"column": 75
} | {
"line": 107,
"column": 76
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : E →L[𝕜] E\nhf : ‖f‖ ≤ 1\nx : E\nhx : x ∈ (↑f - 1).rangeᗮ\n⊢ ∀ (y : E), ⟪f y, x⟫ = ⟪y, x⟫",
"ppTerm": "?m.219",
"assigned": false,
"usedConstants": [... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : E →L[𝕜] E\nhf : ‖f‖ ≤ 1\nx : E\nhx : x ∈ (↑f - 1).rangeᗮ\n⊢ ∀ (y : E), ⟪f y, x⟫ = ⟪y, x⟫"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.SingularValues | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 51
} | {
"line": 144,
"column": 52
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : FiniteDimensional 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] F\ni j : ℕ\nhij : i ≤ j\... | [
"case pos\n𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : FiniteDimensional 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] F\ni j : ℕ\nhij : i ≤ j\nhj : finran... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 548,
"column": 69
} | {
"line": 554,
"column": 14
} | {
"line": 555,
"column": 4
} | [
{
"pp": "E : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : InnerProductSpace ℝ F'\nu : E → F'\nhu : ContDiff ... | [] | by
rw [mul_assoc, ← lintegral_const_mul γ]
· gcongr
simp_rw [← mul_assoc]
exact enorm_fderiv_norm_rpow_le (hu.differentiable one_ne_zero) h1γ
dsimp [enorm]
fun_prop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 89,
"column": 2
} | {
"line": 95,
"column": 78
} | {
"line": 97,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\nι : Type u_6\nι' : Type u_7\ninst✝¹ : Fintype ι\ninst✝ : Fintype ι'\nx : E ⊗[𝕜] F\ne : OrthonormalBasis ι 𝕜 E\... | [] | classical
have : x = ∑ i : ι, ∑ j : ι', (e.toBasis.tensorProduct f.toBasis).repr x (i, j) • e i ⊗ₜ f j := by
conv_lhs => rw [← (e.toBasis.tensorProduct f.toBasis).sum_repr x]
simp [← Finset.sum_product', Basis.tensorProduct_apply']
conv_lhs => rw [this]
simp only [inner_def, map_sum, LinearMap.sum_apply]
... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 89,
"column": 2
} | {
"line": 95,
"column": 78
} | {
"line": 97,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\nι : Type u_6\nι' : Type u_7\ninst✝¹ : Fintype ι\ninst✝ : Fintype ι'\nx : E ⊗[𝕜] F\ne : OrthonormalBasis ι 𝕜 E\... | [] | classical
have : x = ∑ i : ι, ∑ j : ι', (e.toBasis.tensorProduct f.toBasis).repr x (i, j) • e i ⊗ₜ f j := by
conv_lhs => rw [← (e.toBasis.tensorProduct f.toBasis).sum_repr x]
simp [← Finset.sum_product', Basis.tensorProduct_apply']
conv_lhs => rw [this]
simp only [inner_def, map_sum, LinearMap.sum_apply]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 89,
"column": 2
} | {
"line": 95,
"column": 78
} | {
"line": 97,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\nι : Type u_6\nι' : Type u_7\ninst✝¹ : Fintype ι\ninst✝ : Fintype ι'\nx : E ⊗[𝕜] F\ne : OrthonormalBasis ι 𝕜 E\... | [] | classical
have : x = ∑ i : ι, ∑ j : ι', (e.toBasis.tensorProduct f.toBasis).repr x (i, j) • e i ⊗ₜ f j := by
conv_lhs => rw [← (e.toBasis.tensorProduct f.toBasis).sum_repr x]
simp [← Finset.sum_product', Basis.tensorProduct_apply']
conv_lhs => rw [this]
simp only [inner_def, map_sum, LinearMap.sum_apply]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 15
} | {
"line": 112,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nE' : Submodule 𝕜 E\nF' : Submodule 𝕜 F\niE' : Module.Finite 𝕜 ↥E'\niF' : Module.Finite 𝕜 ↥F'\ny : ↥E' ⊗[𝕜] ↥... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nE' : Submodule 𝕜 E\nF' : Submodule 𝕜 F\niE' : Module.Finite 𝕜 ↥E'\niF' : Module.Finite 𝕜 ↥F'\ny : ↥E' ⊗[𝕜] ↥F'\nhz : {(m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 13
} | {
"line": 148,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nx : E\ny : F\n⊢ ‖x ⊗ₜ[𝕜] y‖ = ‖x‖ * ‖y‖",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nx : E\ny : F\n⊢ ‖x ⊗ₜ[𝕜] y‖ = ‖x‖ * ‖y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 90
} | {
"line": 183,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nx y : E ⊗[𝕜] F\n⊢ x = y ↔ ∀ (a : E) (b : F), inner 𝕜 (a ⊗ₜ[𝕜] b) x = inner 𝕜 (a ⊗ₜ[𝕜] b) y",
"ppTerm": "... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nx y : E ⊗[𝕜] F\n⊢ x = y ↔ ∀ (a : E) (b : F), inner 𝕜 (a ⊗ₜ[𝕜] b) x = inner 𝕜 (a ⊗ₜ[𝕜] b) y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 197,
"column": 2
} | {
"line": 197,
"column": 90
} | {
"line": 198,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : InnerProductSpace 𝕜 G\nx y : E ⊗[𝕜] F ⊗[𝕜] G\n⊢ x = y ↔ ... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : InnerProductSpace 𝕜 G\nx y : E ⊗[𝕜] F ⊗[𝕜] G\n⊢ x = y ↔ ∀ (a : E) (b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 572,
"column": 80
} | {
"line": 582,
"column": 16
} | {
"line": 584,
"column": 0
} | [
{
"pp": "E : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : InnerProductSpace ℝ F'\nu : E → F'\nhu : ContDiff ... | [] | by
suffices (C : ℝ) * γ = eLpNormLESNormFDerivOfEqInnerConst μ p by
rw [eLpNorm_nnreal_eq_lintegral h0p]
congr
norm_cast at this ⊢
simp_rw [eLpNormLESNormFDerivOfEqInnerConst, γ]
refold_let n n' C
rw [NNReal.coe_mul, NNReal.coe_mk, Real.coe_toNNReal', mul_eq_mul_left_iff,... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 13
} | {
"line": 52,
"column": 14
} | [
{
"pp": "d : ℕ\n⊢ μH[↑d].IsAddHaarMeasure",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Real",
"fact_one_le_two_ennreal",
"MeasureTheory.Measure.hausdorffMeasure",
"PseudoMetricSpace.toUniformSpace",
"AddCommGroup.toAddGroup",
"WithLp.instAddCommGroup... | [
"d : ℕ\n⊢ μH[↑d].IsAddHaarMeasure"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 206,
"column": 8
} | {
"line": 206,
"column": 19
} | {
"line": 206,
"column": 20
} | [
{
"pp": "X : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n : ℕ), AreS... | [
"X : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n : ℕ), AreSeparated (S ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 689,
"column": 10
} | {
"line": 689,
"column": 21
} | {
"line": 689,
"column": 22
} | [
{
"pp": "case e'_4.h\nF : Type u_3\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\nμ : Measure E\ninst✝¹ : μ.IsAddHaarMeasure\ninst✝ : FiniteDimensi... | [
"case e'_4.h\nF : Type u_3\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\nμ : Measure E\ninst✝¹ : μ.IsAddHaarMeasure\ninst✝ : FiniteDimensional ℝ F\nu ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 206,
"column": 31
} | {
"line": 206,
"column": 42
} | {
"line": 206,
"column": 43
} | [
{
"pp": "X : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n : ℕ), AreS... | [
"X : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n : ℕ), AreSeparated (S ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 41
} | {
"line": 99,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nbasis : OrthonormalBasis (Fin 0) ℝ (EuclideanSpace ℝ (Fin 0)) := EuclideanSpace.basisFun (Fin 0) ℝ\nheq : {0} = parallelepiped ⇑basis\nh✝ : volume = volume.addHaarScalarFactor μH[↑0] • μH[↑0]\nh : volume.addHaarSca... | [] | simp [euclideanHausdorffMeasure_def, h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 733,
"column": 2
} | {
"line": 733,
"column": 90
} | {
"line": 734,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : InnerProductSpace 𝕜 G\nx y : E ⊗[𝕜] (F ⊗[𝕜] G)\n⊢ x = y ... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : InnerProductSpace 𝕜 G\nx y : E ⊗[𝕜] (F ⊗[𝕜] G)\n⊢ x = y ↔ ∀ (a : E) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 113,
"column": 8
} | {
"line": 113,
"column": 19
} | {
"line": 113,
"column": 20
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹⁰ : EMetricSpace X\ninst✝⁹ : MeasurableSpace X\ninst✝⁸ : BorelSpace X\ninst✝⁷ : EMetricSpace Y\ninst✝⁶ : MeasurableSpace Y\ninst✝⁵ : BorelSpace Y\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : Measurable... | [
"X : Type u_1\nY : Type u_2\ninst✝¹⁰ : EMetricSpace X\ninst✝⁹ : MeasurableSpace X\ninst✝⁸ : BorelSpace X\ninst✝⁷ : EMetricSpace Y\ninst✝⁶ : MeasurableSpace Y\ninst✝⁵ : BorelSpace Y\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 121,
"column": 70
} | {
"line": 121,
"column": 81
} | {
"line": 121,
"column": 82
} | [
{
"pp": "X : Type u_1\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nd₁ d₂ : ℕ\nh : d₁ < d₂\ns : Set X\n⊢ ↑d₁ < ↑d₂",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"FloorRing.toFloorSemiring",
... | [
"X : Type u_1\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nd₁ d₂ : ℕ\nh : d₁ < d₂\ns : Set X\n⊢ d₁ < d₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 24
} | {
"line": 157,
"column": 25
} | [
{
"pp": "case a\nX : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\ns : Set X\ni : ℝ≥0\nhi : μH[↑i] s = 0\nj : ℝ≥0\nhj : μH[↑j] s = ∞\nhij : i < j\n⊢ False",
"ppTerm": "?a✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case a\nX : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\ns : Set X\ni : ℝ≥0\nhi : μH[↑i] s = 0\nj : ℝ≥0\nhj : μH[↑j] s = ∞\nhij : i < j\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 92,
"column": 2
} | {
"line": 93,
"column": 41
} | {
"line": 94,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\nι : Type u_5\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : U →ₗ[𝕜] V\nbu : O... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\nι : Type u_5\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : U →ₗ[𝕜] V\nbu : OrthonormalBa... | rw [← basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix (stdOrthonormalBasis 𝕜 U).toBasis
bu.toBasis h.some.toBasis bv.toBasis] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 294,
"column": 12
} | {
"line": 294,
"column": 23
} | {
"line": 294,
"column": 24
} | [
{
"pp": "V : Type u_3\nP : Type u_4\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : InnerProductSpace ℝ V\ninst✝⁶ : MeasurableSpace V\ninst✝⁵ : BorelSpace V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MetricSpace P\ninst✝² : MeasurableSpace P\ninst✝¹ : BorelSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nh... | [
"V : Type u_3\nP : Type u_4\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : InnerProductSpace ℝ V\ninst✝⁶ : MeasurableSpace V\ninst✝⁵ : BorelSpace V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MetricSpace P\ninst✝² : MeasurableSpace P\ninst✝¹ : BorelSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nhs : Nonempty... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 219,
"column": 54
} | {
"line": 219,
"column": 87
} | {
"line": 219,
"column": 88
} | [
{
"pp": "X : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n : ℕ), AreS... | [
"X : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n : ℕ), AreSeparated (S ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 117,
"column": 10
} | {
"line": 117,
"column": 21
} | {
"line": 117,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker = ⊥\n⊢ Function.Injective ⇑f.rangeRestrict",
"ppTe... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker = ⊥\n⊢ Function.Injective ⇑f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 62
} | {
"line": 123,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker = ⊥\ng : U ≃ₗ[𝕜] ↥f.range := LinearEquiv.ofBijective ... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker = ⊥\ng : U ≃ₗ[𝕜] ↥f.range := LinearEquiv.ofBijective f.rangeRestr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 124,
"column": 4
} | {
"line": 125,
"column": 44
} | {
"line": 125,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker = ⊥\ng : U ≃ₗ[𝕜] ↥f.range := LinearEquiv.ofBijective ... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker = ⊥\ng : U ≃ₗ[𝕜] ↥f.range := LinearEquiv.ofBijective f.rangeRestr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 132,
"column": 6
} | {
"line": 132,
"column": 17
} | {
"line": 132,
"column": 18
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nb : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range\n⊢ finrank... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nb : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range\n⊢ finrank 𝕜 ↥f.range... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 23
} | {
"line": 133,
"column": 24
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nb : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range\nhrank : f... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nb : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range\nhrank : finrank 𝕜 ↥f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 15
} | {
"line": 153,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\ntfae_1_iff_2 : f.normDet ≠ 0 ↔ f.ker = ⊥\ntfae_1_iff_3 : f.normD... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\ntfae_1_iff_2 : f.normDet ≠ 0 ↔ f.ker = ⊥\ntfae_1_iff_3 : f.normDet ≠ 0 ↔ fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 330,
"column": 2
} | {
"line": 330,
"column": 49
} | {
"line": 330,
"column": 50
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nr : ℝ≥0\nf : X → Y\nhr : 0 < r\nhf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, HolderOnWith C r f s\nx : X\nx✝ : x ∈ univ\n⊢ ∃ C, ∃ t ∈ 𝓝[univ] x, HolderOnWith C r f t",
"ppTerm": "?m.49",
"assig... | [
"X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nr : ℝ≥0\nf : X → Y\nhr : 0 < r\nhf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, HolderOnWith C r f s\nx : X\nx✝ : x ∈ univ\n⊢ ∃ C, ∃ t ∈ 𝓝 x, HolderOnWith C r f t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 13
} | {
"line": 339,
"column": 14
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nK : ℝ≥0\nf : X → Y\ns : Set X\nh : LipschitzOnWith K f s\n⊢ dimH (f '' s) ≤ dimH s",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nK : ℝ≥0\nf : X → Y\ns : Set X\nh : LipschitzOnWith K f s\n⊢ dimH (f '' s) ≤ dimH s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 39
} | {
"line": 360,
"column": 40
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\ns : Set X\nhf : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t\n⊢ ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, HolderOnWith C 1 f t",
"ppTerm": "?m.45",
"assigned": true,
"usedCon... | [
"X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\ns : Set X\nhf : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t\n⊢ ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 361,
"column": 2
} | {
"line": 361,
"column": 45
} | {
"line": 361,
"column": 46
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\ns : Set X\nhf : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t\nthis : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, HolderOnWith C 1 f t\n⊢ dimH (f '' s) ≤ dimH s",
"ppTerm": "?m.46",
... | [
"X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\ns : Set X\nhf : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t\nthis : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, HolderOnWith C 1 f t\n⊢ dimH (f '' s) ≤ dimH s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 172,
"column": 24
} | {
"line": 172,
"column": 35
} | {
"line": 172,
"column": 36
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\ntfae_1_iff_2 : f.normDet = 0 ↔ f.ker ≠ ⊥\ntfae_1_iff_3 : f.normD... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\ntfae_1_iff_2 : f.normDet = 0 ↔ f.ker ≠ ⊥\ntfae_1_iff_3 : f.normDet = 0 ↔ fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 370,
"column": 2
} | {
"line": 370,
"column": 49
} | {
"line": 370,
"column": 50
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\nhf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, LipschitzOnWith C f s\nx : X\nx✝ : x ∈ univ\n⊢ ∃ C, ∃ t ∈ 𝓝[univ] x, LipschitzOnWith C f t",
"ppTerm": "?m.40",
"assigned": true,
"us... | [
"X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\nhf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, LipschitzOnWith C f s\nx : X\nx✝ : x ∈ univ\n⊢ ∃ C, ∃ t ∈ 𝓝 x, LipschitzOnWith C f t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 15
} | {
"line": 175,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\ntfae_1_iff_2 : f.normDet = 0 ↔ f.ker ≠ ⊥\ntfae_1_iff_3 : f.normD... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\ntfae_1_iff_2 : f.normDet = 0 ↔ f.ker ≠ ⊥\ntfae_1_iff_3 : f.normDet = 0 ↔ fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 420,
"column": 4
} | {
"line": 420,
"column": 41
} | {
"line": 420,
"column": 42
} | [
{
"pp": "𝕜 : Type u_4\nE : Type u_5\nF : Type u_6\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ne : E ≃L[𝕜] F\ns : Set E\n⊢ dimH s ≤ dimH (⇑e '' s)",
"ppTerm": "?m.52",
"assigned": false,
"us... | [
"𝕜 : Type u_4\nE : Type u_5\nF : Type u_6\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ne : E ≃L[𝕜] F\ns : Set E\n⊢ dimH s ≤ dimH (⇑e '' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 198,
"column": 6
} | {
"line": 198,
"column": 17
} | {
"line": 198,
"column": 18
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\nι : Type u_5\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : U →ₗ[𝕜] V\nbu : O... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\nι : Type u_5\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : U →ₗ[𝕜] V\nbu : OrthonormalBa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 15
} | {
"line": 202,
"column": 16
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\nι : Type u_5\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : U →ₗ[𝕜]... | [
"case neg\n𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\nι : Type u_5\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : U →ₗ[𝕜] V\nbu : Ort... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 13
} | {
"line": 343,
"column": 14
} | [
{
"pp": "X : Type u_2\ninst✝ : EMetricSpace X\nb : OuterMeasure X\nhb : ∀ (i : ℝ≥0∞), ⨆ (_ : i > 0), ⊤ ≤ b\n⊢ ⊤ ≤ b",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PartialOrder.toPreorder",
"Preorder.toLE",
"CompleteLattice.toBoundedOrder",
"Measure... | [
"X : Type u_2\ninst✝ : EMetricSpace X\nb : OuterMeasure X\nhb : ∀ (i : ℝ≥0∞), ⨆ (_ : i > 0), ⊤ ≤ b\n⊢ b = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 470,
"column": 4
} | {
"line": 470,
"column": 49
} | {
"line": 470,
"column": 50
} | [
{
"pp": "case refine_2\nE : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nx : E\ns : Set E\nh : s ∈ 𝓝 x\ne : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ\nthis : ⇑e '' s ∈ 𝓝 (e x)\nr : ℝ\nhr0 : 0 < r\nhr : Metric.ball (e x) r ⊆ ⇑e '' s\n⊢ ↑(finrank ℝ E) ≤ dimH (⇑e '' s)",
... | [
"case refine_2\nE : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nx : E\ns : Set E\nh : s ∈ 𝓝 x\ne : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ\nthis : ⇑e '' s ∈ 𝓝 (e x)\nr : ℝ\nhr0 : 0 < r\nhr : Metric.ball (e x) r ⊆ ⇑e '' s\n⊢ ↑(finrank ℝ E) ≤ dimH (⇑e '' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 258,
"column": 36
} | {
"line": 258,
"column": 65
} | {
"line": 258,
"column": 66
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nc : 𝕜\nhc : ¬c = 0\nh : ¬f.ker = ⊥\n⊢ (c • f).ker ≠ ⊥",
"pp... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nc : 𝕜\nhc : ¬c = 0\nh : ¬f.ker = ⊥\n⊢ ¬f.ker = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 243,
"column": 55
} | {
"line": 259,
"column": 86
} | {
"line": 261,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nc : 𝕜\n⊢ (c • f).normDet = ‖c‖ ^ finrank 𝕜 U * f.normDet",
... | [] | by
by_cases hc : c = 0
· nontriviality U
simp [hc, zero_pow finrank_pos.ne.symm]
by_cases h : f.ker = ⊥
· obtain ⟨bv⟩ := (f.normDet_ne_zero_tfae.out 1 3).mp h
let bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
let bv' : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 (c • f).ra... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 13
} | {
"line": 263,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\n⊢ (-f).normDet = f.normDet",
"ppTerm": "?m.40",
"assigne... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\n⊢ (-f).normDet = f.normDet"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 487,
"column": 2
} | {
"line": 487,
"column": 33
} | {
"line": 487,
"column": 34
} | [
{
"pp": "E : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nhcvx : Convex ℝ s\nhne : s.Nonempty\nthis : Nonempty ↑s\nφ : ↥(affineSpan ℝ s) ≃ᵃⁱ[ℝ] ↥(affineSpan ℝ s).direction := AffineIsometryEquiv.constVSub ℝ ⟨hne.some, ⋯⟩\nhs_eq : s = Subtype.val ''... | [
"E : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nhcvx : Convex ℝ s\nhne : s.Nonempty\nthis : Nonempty ↑s\nφ : ↥(affineSpan ℝ s) ≃ᵃⁱ[ℝ] ↥(affineSpan ℝ s).direction := AffineIsometryEquiv.constVSub ℝ ⟨hne.some, ⋯⟩\nhs_eq : s = Subtype.val '' Subtype.val... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 349,
"column": 48
} | {
"line": 349,
"column": 64
} | {
"line": 349,
"column": 65
} | [
{
"pp": "V : Type u_3\nP : Type u_4\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : InnerProductSpace ℝ V\ninst✝⁶ : MeasurableSpace V\ninst✝⁵ : BorelSpace V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MetricSpace P\ninst✝² : MeasurableSpace P\ninst✝¹ : BorelSpace P\ninst✝ : NormedAddTorsor V P\np : P\nv : V\nhv : v ≠ 0\... | [
"V : Type u_3\nP : Type u_4\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : InnerProductSpace ℝ V\ninst✝⁶ : MeasurableSpace V\ninst✝⁵ : BorelSpace V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MetricSpace P\ninst✝² : MeasurableSpace P\ninst✝¹ : BorelSpace P\ninst✝ : NormedAddTorsor V P\np : P\nv : V\nhv : v ≠ 0\nt : Set P\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 346,
"column": 6
} | {
"line": 346,
"column": 24
} | {
"line": 346,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : InnerProductSpace 𝕜 W\nf ... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : InnerProductSpace 𝕜 W\nf : U →ₗ[𝕜] V... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 527,
"column": 2
} | {
"line": 527,
"column": 33
} | {
"line": 527,
"column": 34
} | [
{
"pp": "X : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nβ : Type u_4\nι : β → Type u_5\nhι : (n : β) → Fintype (ι n)\ns : Set X\nl : Filter β\nr : β → ℝ≥0∞\nhr : Tendsto r l (𝓝 0)\nt : (n : β) → ι n → Set X\nht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n\nhst :... | [
"X : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nβ : Type u_4\nι : β → Type u_5\nhι : (n : β) → Fintype (ι n)\ns : Set X\nl : Filter β\nr : β → ℝ≥0∞\nhr : Tendsto r l (𝓝 0)\nt : (n : β) → ι n → Set X\nht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n\nhst : ∀ᶠ (n : β) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 359,
"column": 6
} | {
"line": 359,
"column": 71
} | {
"line": 359,
"column": 72
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup U\ninst✝⁶ : InnerProductSpace 𝕜 U\ninst✝⁵ : FiniteDimensional 𝕜 U\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace 𝕜 V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ni... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup U\ninst✝⁶ : InnerProductSpace 𝕜 U\ninst✝⁵ : FiniteDimensional 𝕜 U\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace 𝕜 V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : Finit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 362,
"column": 6
} | {
"line": 362,
"column": 24
} | {
"line": 362,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup U\ninst✝⁶ : InnerProductSpace 𝕜 U\ninst✝⁵ : FiniteDimensional 𝕜 U\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace 𝕜 V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ni... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup U\ninst✝⁶ : InnerProductSpace 𝕜 U\ninst✝⁵ : FiniteDimensional 𝕜 U\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace 𝕜 V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : Finit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 592,
"column": 32
} | {
"line": 592,
"column": 74
} | {
"line": 592,
"column": 75
} | [
{
"pp": "X : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nd₁ d₂ : ℝ\nh : d₁ < d₂\ns : Set X\nH : μH[d₂] s ≠ 0 ∧ μH[d₁] s ≠ ∞\nc : ℝ≥0\nhc : c ≠ 0\nthis : 0 < ↑c ^ (d₂ - d₁)⁻¹\nr : ℝ≥0\nhr₀✝ : 0 ≤ ↑r\nhrc : ↑r < ↑c ^ (d₂ - d₁)⁻¹\nhr₀ : r ≠ 0\n⊢ ↑r ≠ 0",
"ppTerm": "?m.2... | [
"X : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nd₁ d₂ : ℝ\nh : d₁ < d₂\ns : Set X\nH : μH[d₂] s ≠ 0 ∧ μH[d₁] s ≠ ∞\nc : ℝ≥0\nhc : c ≠ 0\nthis : 0 < ↑c ^ (d₂ - d₁)⁻¹\nr : ℝ≥0\nhr₀✝ : 0 ≤ ↑r\nhrc : ↑r < ↑c ^ (d₂ - d₁)⁻¹\nhr₀ : r ≠ 0\n⊢ ¬r = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 391,
"column": 2
} | {
"line": 391,
"column": 13
} | {
"line": 391,
"column": 14
} | [
{
"pp": "U : Type u_5\ninst✝² : NormedAddCommGroup U\ninst✝¹ : InnerProductSpace ℝ U\ninst✝ : FiniteDimensional ℝ U\nf : U →ₗ[ℝ] U\n⊢ f.normDet = |LinearMap.det f|",
"ppTerm": "?m.45",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"U : Type u_5\ninst✝² : NormedAddCommGroup U\ninst✝¹ : InnerProductSpace ℝ U\ninst✝ : FiniteDimensional ℝ U\nf : U →ₗ[ℝ] U\n⊢ f.normDet = |LinearMap.det f|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.OfNorm | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 28
} | {
"line": 166,
"column": 29
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nr : ℚ\nx y : E\na b : 𝕜\n⊢ inner_ 𝕜 ((a + b) • x) y = inner_ 𝕜 (a • x) y + inner_ 𝕜 (b • x) y",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nr : ℚ\nx y : E\na b : 𝕜\n⊢ inner_ 𝕜 (a • x + b • x) y = inner_ 𝕜 (a • x) y + inner_ 𝕜 (b • x) y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.OfNorm | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 33
} | {
"line": 167,
"column": 34
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nr : ℚ\nx y : E\nhom : 𝕜 →ₗ[ℚ] 𝕜 := (AddMonoidHom.mk' (fun r ↦ inner_ 𝕜 (r • x) y) ⋯).toRatLinearMap\n⊢ inner_ 𝕜 (↑r • x) y = (starRingEnd 𝕜) ↑r * inner_ 𝕜 x ... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nr : ℚ\nx y : E\nhom : 𝕜 →ₗ[ℚ] 𝕜 := (AddMonoidHom.mk' (fun r ↦ inner_ 𝕜 (r • x) y) ⋯).toRatLinearMap\n⊢ inner_ 𝕜 (↑r • x) y = ↑r * inner_ 𝕜 x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 623,
"column": 4
} | {
"line": 623,
"column": 33
} | {
"line": 623,
"column": 34
} | [
{
"pp": "case a\nX : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nx : X\nr : ℕ → ℝ≥0∞ := fun x ↦ 0\nt : ℕ → Unit → Set X := fun x_1 x_2 ↦ {x}\nht : ∀ᶠ (n : ℕ) in atTop, ∀ (i : Unit), ediam (t n i) ≤ r n\n⊢ μH[0] {x} ≤ 1",
"ppTerm": "?a✝",
"assigned": true,
"us... | [
"case a\nX : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nx : X\nr : ℕ → ℝ≥0∞ := fun x ↦ 0\nt : ℕ → Unit → Set X := fun x_1 x_2 ↦ {x}\nht : ∀ᶠ (n : ℕ) in atTop, ∀ (i : Unit), ediam (t n i) ≤ r n\n⊢ μH[0] {x} ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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