module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Data.Finset.Grade | {
"line": 36,
"column": 4
} | {
"line": 36,
"column": 15
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"Order.succ_eq_add_one",
"Nat.instOne",
"congrArg",
"PartialOrder.toPreorder",
"P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Grade | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 13
} | [
{
"pp": "α : Type u_1\ns t : Finset α\ninst✝ : DecidableEq α\nh : s ⋖ t\n⊢ ∃ a ∉ s, insert a s = t",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Grade | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 41
} | [
{
"pp": "α : Type u_1\ns t : Finset α\ninst✝ : DecidableEq α\nh : s ⋖ t\n⊢ ∃ a ∈ t, t.erase a = s",
"usedConstants": [
"Eq.mpr",
"congrArg",
"_private.Mathlib.Data.Finset.Grade.0.CovBy.exists_finset_erase._simp_1_1",
"Finset",
"Finset.coe_erase",
"Membership.mem",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 13
} | [
{
"pp": "x p : ℝ\nhp : 1 < p\n⊢ HasDerivAt (fun x ↦ |x| ^ p) (p * |x| ^ (p - 2) * x) x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.GaussSum | {
"line": 203,
"column": 51
} | {
"line": 206,
"column": 5
} | [
{
"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\nχ : MulChar R R'\nψ : AddChar R R'\n⊢ gaussSum χ ψ ^ p = gaussSum (χ ^ p) (ψ ^ p)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWith... | by
rw [← frobenius_def, gaussSum, gaussSum, map_sum]
simp_rw [pow_apply' χ fp.1.ne_zero, map_mul, frobenius_def]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 105,
"column": 2
} | {
"line": 106,
"column": 28
} | [
{
"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",
"usedC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Interval | {
"line": 50,
"column": 10
} | {
"line": 50,
"column": 64
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Finset",
"Finset.instSDiff",
"Membership.mem",
"id",
"S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.GaussSum | {
"line": 214,
"column": 2
} | {
"line": 216,
"column": 27
} | [
{
"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",
"usedConstants": [
"Units.val",
"Eq... | 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.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.GaussSum | {
"line": 214,
"column": 2
} | {
"line": 216,
"column": 27
} | [
{
"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",
"usedConstants": [
"Units.val",
"Eq... | 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.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.GaussSum | {
"line": 214,
"column": 2
} | {
"line": 216,
"column": 27
} | [
{
"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",
"usedConstants": [
"Units.val",
"Eq... | 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.InnerProductSpace.NormPow | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 58
} | [
{
"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)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.GaussSum | {
"line": 324,
"column": 4
} | {
"line": 325,
"column": 12
} | [
{
"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... | · rw [← pow_mul, ← map_nsmul_eq_pow ψ₈.char, ψ₈.prim.zmod_char_eq_one_iff]
decide | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.GaussSum | {
"line": 358,
"column": 2
} | {
"line": 358,
"column": 68
} | [
{
"pp": "case convert_2.a\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 := ⋯\nhchar : ringChar F = ringChar FF\nFFp : Nat.Prime (ringChar FF)\nthis : Fact (Nat.Prime (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Affine | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 13
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Hofer | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 31
} | [
{
"pp": "X : 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\nH : ∀ ε' > 0, ∀ (x' : X), ε' ≤ ε → d x' x ≤ 2 * ε → ε * ϕ x ≤ ε' * ϕ x' → ∃ y, d x' y ≤ ε' ∧ 2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Hofer | {
"line": 58,
"column": 12
} | {
"line": 58,
"column": 37
} | [
{
"pp": "case hz\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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Hofer | {
"line": 72,
"column": 6
} | {
"line": 74,
"column": 52
} | [
{
"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... | 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.Hofer | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 39
} | [
{
"pp": "X : 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 : No... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 40,
"column": 74
} | {
"line": 42,
"column": 76
} | [
{
"pp": "n : Type u_2\nR : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Fintype n\nv w : n → R\nh : ∀ (u : n → R), v ⬝ᵥ u = w ⬝ᵥ u\n⊢ v = w",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"dotProduct",
"congrArg",
"Classical.propDecidable",
"NonUnitalN... | by
funext x
classical rw [← dotProduct_single_one v x, ← dotProduct_single_one w x, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 48
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\np : Type u_5\nA : Matrix m n R\nB : Matrix m p R\n⊢ A * Aᴴ * B = 0 ↔ Aᴴ * B = 0",
"usedCons... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 48
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\np : Type u_5\nA : Matrix m n R\nB : Matrix p n R\n⊢ B * (Aᴴ * A) = 0 ↔ B * Aᴴ = 0",
"usedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 71
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\nv : n → R\n⊢ (Aᴴ * A) *ᵥ v = 0 ↔ A *ᵥ v = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 48
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\nv : m → R\n⊢ (A * Aᴴ) *ᵥ v = 0 ↔ Aᴴ *ᵥ v = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 71
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\nv : n → R\n⊢ v ᵥ* (Aᴴ * A) = 0 ↔ v ᵥ* Aᴴ = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 48
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_4\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\nv : m → R\n⊢ v ᵥ* (A * Aᴴ) = 0 ↔ v ᵥ* A = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 13
} | [
{
"pp": "n : Type u_2\nR : Type u_4\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nv : n → R\n⊢ 0 < v ⬝ᵥ star v ↔ v ≠ 0",
"usedConstants": [
"Pi.instStarForall",
"Preorder.toLT",
"dotProduct... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Hadamard | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 13
} | [
{
"pp": "α : Type u_1\nn : Type u_3\ninst✝¹ : DecidableEq n\ninst✝ : MulZeroOneClass α\nA : Matrix n n α\nd : n → α\n⊢ 1 ⊙ A = diagonal d ↔ A.diag = d",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Hadamard | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 13
} | [
{
"pp": "α : Type u_1\nn : Type u_3\ninst✝¹ : DecidableEq n\ninst✝ : MulZeroOneClass α\nA : Matrix n n α\nd : n → α\n⊢ A ⊙ 1 = diagonal d ↔ A.diag = d",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Hadamard | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 13
} | [
{
"pp": "α : Type u_1\nn : Type u_3\ninst✝¹ : DecidableEq n\ninst✝ : MulZeroOneClass α\nA : Matrix n n α\n⊢ 1 ⊙ A = 0 ↔ A.diag = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Hadamard | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 13
} | [
{
"pp": "α : Type u_1\nn : Type u_3\ninst✝¹ : DecidableEq n\ninst✝ : MulZeroOneClass α\nA : Matrix n n α\n⊢ A ⊙ 1 = 0 ↔ A.diag = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 13
} | [
{
"pp": "case h\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm n : ℕ\nha : IsUnit A.det\nh : n ≤ m\n⊢ IsUnit (A ^ n).det",
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"Matrix",
"IsUnit",
"id... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 15
} | [
{
"pp": "case ofNat\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nh : IsUnit A.det\nn : ℕ\n⊢ IsUnit (A ^ ofNat n).det",
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"Matrix",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 15
} | [
{
"pp": "case negSucc\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nh : IsUnit A.det\nn : ℕ\n⊢ IsUnit (A ^ -[n+1]).det",
"usedConstants": [
"Eq.mpr",
"False",
"Nat.instMulZeroClass",
"DivInvMonoid.toInv",
"isUnit_pow_iff.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 198,
"column": 6
} | {
"line": 198,
"column": 17
} | [
{
"pp": "case inr.ofNat\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA B : M\nh : Commute A B\nhB : B⁻¹ = 0\na✝ : ℕ\n⊢ Commute A (B ^ ofNat a✝)",
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Hermitian | {
"line": 98,
"column": 15
} | {
"line": 98,
"column": 26
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nn : Type u_4\ninst✝ : Star α\nA : Matrix n n α\ne : m ≃ n\nh : (A.submatrix ⇑e ⇑e).IsHermitian\n⊢ A.IsHermitian",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Hermitian | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 13
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nn : Type u_4\ninst✝ : Star α\nA : Matrix n n α\nf : n ≃ m\nh : ((reindex f f) A).IsHermitian\n⊢ A.IsHermitian",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Hermitian | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 27
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nn : Type u_4\ninst✝² : AddMonoid α\ninst✝¹ : StarAddMonoid α\ninst✝ : DecidableEq n\nM : n → Matrix m m α\n⊢ (blockDiagonal M).IsHermitian ↔ ∀ (i : n), (M i).IsHermitian",
"usedConstants": [
"Eq.mpr",
"Matrix.blockDiagonal",
"congrArg",
"Matrix.bl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Hermitian | {
"line": 257,
"column": 2
} | {
"line": 257,
"column": 13
} | [
{
"pp": "α : Type u_1\nn : Type u_4\nR : Type u_5\ninst✝⁵ : Monoid R\ninst✝⁴ : Star R\ninst✝³ : Star α\ninst✝² : MulAction R α\ninst✝¹ : StarModule R α\nA : Matrix n n α\nk : R\ninst✝ : Invertible k\nh : k • Aᴴ = k • A\nhk : IsSelfAdjoint k\n⊢ A.IsHermitian",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Vec | {
"line": 89,
"column": 25
} | {
"line": 89,
"column": 37
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nR : Type u_1\ninst✝³ : AddCommMonoid R\ninst✝² : Mul R\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nA B : Matrix m n R\n⊢ A.vec ⬝ᵥ B.vec = ∑ i, (Aᵀ * B).diag i",
"usedConstants": [
"Eq.mpr",
"Matrix.diag",
"HMul.hMul",
"Finset.univ",
"dotProduct"... | Matrix.diag, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.Vec | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 47
} | [
{
"pp": "case hB\nl : Type u_2\nm : Type u_3\nn : Type u_1\nR : Type u_4\ninst✝³ : Semiring R\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix l m R\nB : Matrix m n R\nx : R\ni j : n\n⊢ Commute x (1 i j)",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"F... | obtain rfl | hij := eq_or_ne i j <;> simp [*] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.Matrix.Vec | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 47
} | [
{
"pp": "case hA\nm : Type u_1\nn : Type u_2\np : Type u_4\nR : Type u_3\ninst✝³ : Semiring R\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : DecidableEq m\nA : Matrix m n R\nB : Matrix n p R\nx : R\ni j : m\n⊢ Commute (1 i j) x",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"F... | obtain rfl | hij := eq_or_ne i j <;> simp [*] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 269,
"column": 2
} | {
"line": 269,
"column": 30
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 280,
"column": 2
} | {
"line": 290,
"column": 6
} | [
{
"pp": "ι : 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\nhp : (↑#ι).HolderConjugate p\nf : ((a : ι) → A a) → ℝ≥0∞\nhf : Measurable f\n⊢ ∫⁻ (x : (i : ι) → A i), ∏ i, (∫⁻... | have : Nontrivial ι :=
Fintype.one_lt_card_iff_nontrivial.mp (by exact_mod_cast hp.lt)
have h0 : (1 : ℝ) < #ι := by norm_cast; exact Fintype.one_lt_card
have h1 : (0 : ℝ) < #ι - 1 := by linarith
have h2 : 0 ≤ ((1 : ℝ) / (#ι - 1 : ℝ)) := by positivity
have h3 : (#ι - 1 : ℝ) * ((1 : ℝ) / (#ι - 1 : ℝ)) ≤ 1 := ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 280,
"column": 2
} | {
"line": 290,
"column": 6
} | [
{
"pp": "ι : 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\nhp : (↑#ι).HolderConjugate p\nf : ((a : ι) → A a) → ℝ≥0∞\nhf : Measurable f\n⊢ ∫⁻ (x : (i : ι) → A i), ∏ i, (∫⁻... | have : Nontrivial ι :=
Fintype.one_lt_card_iff_nontrivial.mp (by exact_mod_cast hp.lt)
have h0 : (1 : ℝ) < #ι := by norm_cast; exact Fintype.one_lt_card
have h1 : (0 : ℝ) < #ι - 1 := by linarith
have h2 : 0 ≤ ((1 : ℝ) / (#ι - 1 : ℝ)) := by positivity
have h3 : (#ι - 1 : ℝ) * ((1 : ℝ) / (#ι - 1 : ℝ)) ≤ 1 := ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 72,
"column": 22
} | {
"line": 72,
"column": 33
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : DecidableEq n\nd : n → R\nx✝ : (diagonal d).PosSemidef\ni : n\nleft✝ : (diagonal d).IsHermitian\nhP : ∀ (x : n →₀ R), 0 ≤ x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * diagonal d... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 61
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nM : Matrix n n R\nhM : M.PosSemidef\ne : m → n\nx : m →₀ R\n⊢ 0 ≤ x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * M.submatrix e e i j * xj",
"usedConstants": [
"Matrix.submatrix",
"HMul.h... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 48
} | [
{
"pp": "n : Type u_2\nR' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : PartialOrder R'\ninst✝ : StarRing R'\nM : Matrix n n R'\nhM : M.PosSemidef\nthis : ∀ (a b c : R'), a * b * c = c * b * a\nx : n →₀ R'\n⊢ 0 ≤ x.sum fun x' v' ↦ x.sum fun x v ↦ star v * Mᵀ x x' * v'",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 99,
"column": 7
} | {
"line": 99,
"column": 18
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nM : Matrix n n R\nx✝ : Mᴴ.PosSemidef\n⊢ M.PosSemidef",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 34
} | [
{
"pp": "m : Type u_1\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : AddLeftMono R\nA B : Matrix m m R\nhA : A.PosSemidef\nhB : B.PosSemidef\nx : m →₀ R\n⊢ 0 ≤ x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * (A + B) i j * xj",
"usedConstants": [
"add_mul",
"Dis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 65
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁹ : Ring R\ninst✝⁸ : PartialOrder R\ninst✝⁷ : StarRing R\nα : Type u_5\ninst✝⁶ : CommSemiring α\ninst✝⁵ : PartialOrder α\ninst✝⁴ : StarRing α\ninst✝³ : StarOrderedRing α\ninst✝² : Algebra α R\ninst✝¹ : StarModule α R\ninst✝ : PosSMulMono α R\nx : Matrix n n R\nhx : x.Po... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 49
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : DecidableEq n\nx : n →₀ R\ni : n\nx✝¹ : i ∈ x.support\nj : n\nx✝ : j ∈ x.support\n⊢ 0 ≤ star (x i) * 1 i j * x j",
"usedConstants": [
"Finsupp.instFunLike",
"Fa... | obtain rfl | hij := eq_or_ne i j <;> simp [*] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 49
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : DecidableEq n\nx : n →₀ R\ni : n\nx✝¹ : i ∈ x.support\nj : n\nx✝ : j ∈ x.support\n⊢ 0 ≤ star (x i) * 1 i j * x j",
"usedConstants": [
"Finsupp.instFunLike",
"Fa... | obtain rfl | hij := eq_or_ne i j <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 49
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : DecidableEq n\nx : n →₀ R\ni : n\nx✝¹ : i ∈ x.support\nj : n\nx✝ : j ∈ x.support\n⊢ 0 ≤ star (x i) * 1 i j * x j",
"usedConstants": [
"Finsupp.instFunLike",
"Fa... | obtain rfl | hij := eq_or_ne i j <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 13
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nA : Matrix n n R\nhA : A.PosSemidef\ni : n\n⊢ 0 ≤ A i i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 145,
"column": 15
} | {
"line": 145,
"column": 26
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nM : Matrix n n R\ne : m ≃ n\nh : (M.submatrix ⇑e ⇑e).PosSemidef\n⊢ M.PosSemidef",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 61
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nM : Matrix n n R\nhM : M.PosDef\ne : m → n\nhe : Function.Injective e\nx : m →₀ R\nhx : x ≠ 0\n⊢ 0 < x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * M.submatrix e e i j * xj",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 64
} | [
{
"pp": "n : Type u_2\nR' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : PartialOrder R'\ninst✝ : StarRing R'\nM : Matrix n n R'\nhM : M.PosDef\nthis : ∀ (a b c : R'), a * b * c = c * b * a\nx : n →₀ R'\n⊢ x ≠ 0 → 0 < x.sum fun x' v' ↦ x.sum fun x v ↦ star v * Mᵀ x x' * v'",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 186,
"column": 7
} | {
"line": 186,
"column": 18
} | [
{
"pp": "n : Type u_2\nR' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : PartialOrder R'\ninst✝ : StarRing R'\nM : Matrix n n R'\nx✝ : Mᵀ.PosDef\n⊢ M.PosDef",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 195,
"column": 26
} | {
"line": 195,
"column": 55
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : DecidableEq n\ninst✝ : NoZeroDivisors R\nd : n → R\nh : ∀ (i : n), 0 < d i\nx : n →₀ R\nhx : x ≠ 0\n⊢ ?m.73",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 204,
"column": 17
} | {
"line": 204,
"column": 28
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : DecidableEq n\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nd : n → R\nh : (diagonal d).PosDef\ni : n\n⊢ 0 < d i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 248,
"column": 4
} | {
"line": 248,
"column": 33
} | [
{
"pp": "m : Type u_1\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : AddLeftMono R\nA B : Matrix m m R\nhA : A.PosDef\nhB : B.PosSemidef\nx : m →₀ R\nhx : x ≠ 0\n⊢ 0 < x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * (A + B) i j * xj",
"usedConstants": [
"add_mul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 34
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁹ : Ring R\ninst✝⁸ : PartialOrder R\ninst✝⁷ : StarRing R\nα : Type u_5\ninst✝⁶ : CommSemiring α\ninst✝⁵ : PartialOrder α\ninst✝⁴ : StarRing α\ninst✝³ : StarOrderedRing α\ninst✝² : Algebra α R\ninst✝¹ : StarModule α R\ninst✝ : PosSMulStrictMono α R\nx : Matrix n n R\nhx ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 281,
"column": 7
} | {
"line": 281,
"column": 18
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nM : Matrix n n R\nx✝ : Mᴴ.PosDef\n⊢ M.PosDef",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 284,
"column": 12
} | {
"line": 284,
"column": 31
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : Nontrivial R\nA : Matrix n n R\nhA : A.PosDef\ni : n\n⊢ 0 < A i i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 66
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : Fintype n\ninst✝ : Finite m\nA : Matrix n n R\nhA : A.PosSemidef\nB : Matrix n m R\nthis : Fintype m\nx : m → R\n⊢ 0 ≤ star x ⬝ᵥ (Bᴴ * A * B) *ᵥ x",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 48
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : Fintype n\ninst✝ : Finite m\nA : Matrix n n R\nhA : A.PosSemidef\nB : Matrix m n R\n⊢ (B * A * Bᴴ).PosSemidef",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 329,
"column": 12
} | {
"line": 329,
"column": 23
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : Fintype n\ninst✝¹ : StarOrderedRing R\ninst✝ : DecidableEq n\nM : Matrix n n R\nhM : M.PosSemidef\nk : ℕ\n⊢ (M ^ 1).PosSemidef",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matrix",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 332,
"column": 4
} | {
"line": 332,
"column": 40
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : Fintype n\ninst✝¹ : StarOrderedRing R\ninst✝ : DecidableEq n\nM : Matrix n n R\nhM : M.PosSemidef\nk✝ k : ℕ\n⊢ (M * M ^ k * M).PosSemidef",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 345,
"column": 4
} | {
"line": 345,
"column": 15
} | [
{
"pp": "case inl\nn✝ : Type u_2\nR' : Type u_4\ninst✝⁵ : CommRing R'\ninst✝⁴ : PartialOrder R'\ninst✝³ : StarRing R'\ninst✝² : Fintype n✝\ninst✝¹ : StarOrderedRing R'\ninst✝ : DecidableEq n✝\nM : Matrix n✝ n✝ R'\nhM : M.PosSemidef\nn : ℕ\n⊢ (M ^ ↑n).PosSemidef",
"usedConstants": [
"zpow_natCast",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 346,
"column": 4
} | {
"line": 346,
"column": 15
} | [
{
"pp": "case inr\nn✝ : Type u_2\nR' : Type u_4\ninst✝⁵ : CommRing R'\ninst✝⁴ : PartialOrder R'\ninst✝³ : StarRing R'\ninst✝² : Fintype n✝\ninst✝¹ : StarOrderedRing R'\ninst✝ : DecidableEq n✝\nM : Matrix n✝ n✝ R'\nhM : M.PosSemidef\nn : ℕ\n⊢ (M ^ (-↑n)).PosSemidef",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 366,
"column": 2
} | {
"line": 366,
"column": 48
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : Fintype n\ninst✝¹ : Finite m\ninst✝ : StarOrderedRing R\nA : Matrix m n R\n⊢ (A * Aᴴ).PosSemidef",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 13
} | [
{
"pp": "m : Type u_1\nn : Type u_2\ninst✝⁶ : Fintype n\ninst✝⁵ : Fintype m\nR : Type u_5\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\n⊢ (A * Aᴴ).trace = 0 ↔ A = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 405,
"column": 2
} | {
"line": 405,
"column": 13
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nU x : Matrix n n R\nhU : IsUnit U\n⊢ (U * x * star U).PosSemidef ↔ x.PosSemidef",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 449,
"column": 2
} | {
"line": 449,
"column": 66
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : Fintype n\ninst✝ : Fintype m\nA : Matrix n n R\nB : Matrix n m R\nhA : A.PosDef\nhB : Function.Injective B.mulVec\nx : m → R\nhx : x ≠ 0\nthis : B *ᵥ x ≠ 0\n⊢ 0 < star x ⬝ᵥ (Bᴴ * A * B) *ᵥ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 13
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : Fintype n\ninst✝ : Fintype m\nA : Matrix n n R\nB : Matrix m n R\nhA : A.PosDef\nhB : Function.Injective fun x ↦ Bᴴ *ᵥ x\n⊢ (B * A * Bᴴ).PosDef",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 462,
"column": 2
} | {
"line": 462,
"column": 13
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : Fintype n\ninst✝² : Fintype m\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\nhA : Function.Injective A.mulVec\n⊢ (Aᴴ * A).PosDef",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 468,
"column": 2
} | {
"line": 468,
"column": 13
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : Fintype n\ninst✝² : Fintype m\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\nhA : Function.Injective fun v ↦ v ᵥ* A\n⊢ (A * Aᴴ).PosDef",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 496,
"column": 2
} | {
"line": 496,
"column": 19
} | [
{
"pp": "n : Type u_2\ninst✝⁴ : Fintype n\nK : Type u_5\ninst✝³ : Field K\ninst✝² : PartialOrder K\ninst✝¹ : StarRing K\ninst✝ : DecidableEq n\nM : Matrix n n K\nhM : M.PosDef\nh : ¬IsUnit M\na : n → K\nha : a ≠ 0\nha2 : M *ᵥ a = 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 501,
"column": 4
} | {
"line": 501,
"column": 15
} | [
{
"pp": "case refine_2\nn : Type u_2\ninst✝⁴ : Fintype n\nK : Type u_5\ninst✝³ : Field K\ninst✝² : PartialOrder K\ninst✝¹ : StarRing K\ninst✝ : DecidableEq n\nM : Matrix n n K\nhM : M.PosDef\nthis : (M⁻¹ * M * M⁻¹ᴴ).PosDef\nx✝ : Invertible M := ⋯.invertible\n⊢ M⁻¹.PosDef",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 538,
"column": 2
} | {
"line": 538,
"column": 13
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nx U : Matrix n n R\nhU : IsUnit U\n⊢ (U * x * star U).PosDef ↔ x.PosDef",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Coalgebra | {
"line": 76,
"column": 8
} | {
"line": 76,
"column": 26
} | [
{
"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 ... | ← adjoint_lTensor, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Coalgebra | {
"line": 85,
"column": 35
} | {
"line": 85,
"column": 53
} | [
{
"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... | ← adjoint_lTensor, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Coalgebra | {
"line": 135,
"column": 38
} | {
"line": 135,
"column": 56
} | [
{
"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",
"usedConstants": [
"NormedCommRin... | ← adjoint_lTensor, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 497,
"column": 32
} | {
"line": 497,
"column": 43
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Coalgebra | {
"line": 163,
"column": 45
} | {
"line": 163,
"column": 63
} | [
{
"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 (... | ← adjoint_lTensor, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 206,
"column": 70
} | {
"line": 206,
"column": 91
} | [
{
"pp": "case h.e'_12.h.h\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 = NormedField.toNormedCommRing.toAddCommGroup\ne_9✝ : Semiring.toModule ≍ toInnerProductSpaceReal.toModule\ny : F\n⊢ 2 • ((innerSL ℝ) ... | fderivInnerCLM_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 59
} | [
{
"pp": "case a.a\nF : 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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.Basic | {
"line": 222,
"column": 14
} | {
"line": 222,
"column": 25
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 241,
"column": 23
} | {
"line": 241,
"column": 39
} | [
{
"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\na b : ℝ\nh₁ : (a, b) ≠ 0\nh₂ : a • x₀ + b • T x₀ = 0\nhx₀ : ¬x₀ = 0\nhb : b = 0\n⊢ a ≠ 0",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 244,
"column": 4
} | {
"line": 244,
"column": 20
} | [
{
"pp": "case pos.a\nF : 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\na b : ℝ\nh₁ : (a, b) ≠ 0\nh₂ : a • x₀ + b • T x₀ = 0\nhx₀ : ¬x₀ = 0\nhb : b = 0\nth... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 50
} | [
{
"pp": "case h.e'_3.h.e'_5\nF : 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\na b : ℝ\nh₁ : (a, b) ≠ 0\nh₂ : a • x₀ + b • T x₀ = 0\nhx₀ : ¬x₀ = 0\nhb : ¬... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 286,
"column": 26
} | {
"line": 286,
"column": 37
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 92,
"column": 2
} | {
"line": 98,
"column": 78
} | [
{
"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": 92,
"column": 2
} | {
"line": 98,
"column": 78
} | [
{
"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": 92,
"column": 2
} | {
"line": 98,
"column": 78
} | [
{
"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.Rayleigh | {
"line": 305,
"column": 26
} | {
"line": 305,
"column": 37
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 15
} | [
{
"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] ↥... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 335,
"column": 30
} | {
"line": 335,
"column": 41
} | [
{
"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 :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 336,
"column": 66
} | {
"line": 336,
"column": 90
} | [
{
"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 :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 13
} | [
{
"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‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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