module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 34
} | {
"line": 276,
"column": 35
} | [
{
"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 ... | [
"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 : x.PosDef\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 282,
"column": 7
} | {
"line": 282,
"column": 18
} | {
"line": 282,
"column": 19
} | [
{
"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",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 285,
"column": 12
} | {
"line": 285,
"column": 31
} | {
"line": 285,
"column": 32
} | [
{
"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",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 317,
"column": 2
} | {
"line": 317,
"column": 66
} | {
"line": 318,
"column": 6
} | [
{
"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",
"ppTerm": "?m.47",
"assigned": t... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 48
} | {
"line": 323,
"column": 49
} | [
{
"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",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"u... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 330,
"column": 12
} | {
"line": 330,
"column": 23
} | {
"line": 330,
"column": 24
} | [
{
"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",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
... | [
"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.PosSemidef"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 333,
"column": 4
} | {
"line": 333,
"column": 40
} | {
"line": 333,
"column": 41
} | [
{
"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",
"ppTerm": "?m.76",
"assigned": false,
"usedConstant... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 346,
"column": 4
} | {
"line": 346,
"column": 15
} | {
"line": 346,
"column": 16
} | [
{
"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",
"ppTerm": "?inl",
"assigned": true,
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 347,
"column": 4
} | {
"line": 347,
"column": 15
} | {
"line": 347,
"column": 16
} | [
{
"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",
"ppTerm": "?inr",
"assigned": true,
... | [
"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)⁻¹.PosSemidef"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 367,
"column": 2
} | {
"line": 367,
"column": 48
} | {
"line": 367,
"column": 49
} | [
{
"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",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": []... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 381,
"column": 2
} | {
"line": 381,
"column": 13
} | {
"line": 381,
"column": 14
} | [
{
"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",
"ppTerm": "?m.27",
"assigned": false,
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 13
} | {
"line": 406,
"column": 14
} | [
{
"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",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 450,
"column": 2
} | {
"line": 450,
"column": 66
} | {
"line": 450,
"column": 67
} | [
{
"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 ... | [
"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 x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 457,
"column": 2
} | {
"line": 457,
"column": 13
} | {
"line": 457,
"column": 14
} | [
{
"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",
"ppTerm": "?m.59",
"assigned": fals... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 463,
"column": 2
} | {
"line": 463,
"column": 13
} | {
"line": 463,
"column": 14
} | [
{
"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",
"ppTerm": "?m.28",
"assi... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 469,
"column": 2
} | {
"line": 469,
"column": 13
} | {
"line": 469,
"column": 14
} | [
{
"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",
"ppTerm": "?m.28",
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 475,
"column": 2
} | {
"line": 475,
"column": 54
} | {
"line": 475,
"column": 55
} | [
{
"pp": "n : Type u_2\ninst✝⁵ : Fintype n\nR : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : TrivialStar R\ninst✝ : DecidableEq n\nM : Matrix n n R\nhM : M.IsSymm\nhMq : QuadraticMap.PosDef M.toQuadraticForm'\nx : n → R\nhx : x ≠ 0\n⊢ 0 < star x ⬝ᵥ M *ᵥ x",
"ppTerm": ... | [
"n : Type u_2\ninst✝⁵ : Fintype n\nR : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : TrivialStar R\ninst✝ : DecidableEq n\nM : Matrix n n R\nhM : M.IsSymm\nhMq : QuadraticMap.PosDef M.toQuadraticForm'\nx : n → R\nhx : x ≠ 0\n⊢ 0 < x ⬝ᵥ M *ᵥ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 481,
"column": 2
} | {
"line": 481,
"column": 61
} | {
"line": 481,
"column": 62
} | [
{
"pp": "n : Type u_2\ninst✝⁵ : Fintype n\nR : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : TrivialStar R\ninst✝ : DecidableEq n\nM : Matrix n n R\nhM : M.PosDef\nx : n → R\nhx : x ≠ 0\n⊢ 0 < M.toQuadraticForm' x",
"ppTerm": "?m.27",
"assigned": true,
"usedCo... | [
"n : Type u_2\ninst✝⁵ : Fintype n\nR : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : TrivialStar R\ninst✝ : DecidableEq n\nM : Matrix n n R\nhM : M.PosDef\nx : n → R\nhx : x ≠ 0\n⊢ 0 < x ⬝ᵥ M *ᵥ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 493,
"column": 4
} | {
"line": 493,
"column": 89
} | {
"line": 494,
"column": 6
} | [
{
"pp": "case refine_2\nn : Type u_2\ninst✝⁷ : Fintype n\nR : Type u_5\nM : Type u_6\ninst✝⁶ : CommRing R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : TrivialStar R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : DecidableEq n\nb : Module.Basis n R M\nB : LinearMap.BilinForm R M\nhB_symm : B.I... | [
"case refine_2\nn : Type u_2\ninst✝⁷ : Fintype n\nR : Type u_5\nM : Type u_6\ninst✝⁶ : CommRing R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : TrivialStar R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : DecidableEq n\nb : Module.Basis n R M\nB : LinearMap.BilinForm R M\nhB_symm : B.IsSymm\naux :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 496,
"column": 4
} | {
"line": 497,
"column": 11
} | {
"line": 497,
"column": 12
} | [
{
"pp": "case refine_3\nn : Type u_2\ninst✝⁷ : Fintype n\nR : Type u_5\nM : Type u_6\ninst✝⁶ : CommRing R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : TrivialStar R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : DecidableEq n\nb : Module.Basis n R M\nB : LinearMap.BilinForm R M\nhB_symm : B.I... | [
"case refine_3\nn : Type u_2\ninst✝⁷ : Fintype n\nR : Type u_5\nM : Type u_6\ninst✝⁶ : CommRing R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : TrivialStar R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : DecidableEq n\nb : Module.Basis n R M\nB : LinearMap.BilinForm R M\nhB_symm : B.IsSymm\naux :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ZMatrix | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 13
} | {
"line": 37,
"column": 14
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝⁶ : Fintype ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : StarRing R\ninst✝ : TrivialStar R\nA : Matrix ι ι R\nd : ι → R\nhA : (diagonal d * A).PosDef\nhD : ∀ (i : ι), 0 < d i\ni : ι\n⊢ 0 < d i * A i i",... | [
"ι : Type u_1\nR : Type u_2\ninst✝⁶ : Fintype ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : StarRing R\ninst✝ : TrivialStar R\nA : Matrix ι ι R\nd : ι → R\nhA : (diagonal d * A).PosDef\nhD : ∀ (i : ι), 0 < d i\ni : ι\n⊢ 0 < d i * A i i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ZMatrix | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 54
} | {
"line": 52,
"column": 55
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝⁶ : Fintype ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : StarRing R\ninst✝ : TrivialStar R\nA : Matrix ι ι R\nd : ι → R\nhA : (diagonal d * A).PosDef\nhD : ∀ (i : ι), 0 < d i\nμ ρ : R\nhμ : ∀ (i j : ι),... | [
"ι : Type u_1\nR : Type u_2\ninst✝⁶ : Fintype ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : StarRing R\ninst✝ : TrivialStar R\nA : Matrix ι ι R\nd : ι → R\nhA : (diagonal d * A).PosDef\nhD : ∀ (i : ι), 0 < d i\nμ ρ : R\nhμ : ∀ (i j : ι), A i j ≤ if ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 512,
"column": 2
} | {
"line": 512,
"column": 19
} | {
"line": 512,
"column": 20
} | [
{
"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",
"ppTerm": "?m.87",
"assigned": false,
"usedConstants": []... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 517,
"column": 4
} | {
"line": 517,
"column": 15
} | {
"line": 517,
"column": 16
} | [
{
"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",
"ppTerm": "?refine_2",
"assign... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.ZMatrix | {
"line": 58,
"column": 6
} | {
"line": 58,
"column": 41
} | {
"line": 58,
"column": 42
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝⁶ : Fintype ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : StarRing R\ninst✝ : TrivialStar R\nA : Matrix ι ι R\nd : ι → R\nhA : (diagonal d * A).PosDef\nhD : ∀ (i : ι), 0 < d i\nμ ρ : R\nhμ : ∀ (i j : ι),... | [
"ι : Type u_1\nR : Type u_2\ninst✝⁶ : Fintype ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : StarRing R\ninst✝ : TrivialStar R\nA : Matrix ι ι R\nd : ι → R\nhA : (diagonal d * A).PosDef\nhD : ∀ (i : ι), 0 < d i\nμ ρ : R\nhμ : ∀ (i j : ι), A i j ≤ if ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 554,
"column": 2
} | {
"line": 554,
"column": 13
} | {
"line": 554,
"column": 14
} | [
{
"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",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Minpoly | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 42
} | {
"line": 42,
"column": 43
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : End R M\nc : R[X]ˣ\nm : M\nhm : ((aeval f) ↑c) m = 0\n⊢ m = 0",
"ppTerm": "?m.45",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type v\nM : Type w\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : End R M\nc : R[X]ˣ\nm : M\nhm : ((aeval f) ↑c) m = 0\n⊢ m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Minpoly | {
"line": 56,
"column": 2
} | {
"line": 59,
"column": 45
} | {
"line": 61,
"column": 0
} | [
{
"pp": "case refine_3\nR : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\nx : M\np : R[X]\nh : f.HasEigenvector μ x\n⊢ ∀ (n : ℕ) (a : R),\n ((aeval f) (C a * X ^ n)) x = eval μ (C a * X ^ n) • x →\n ((aeval f) (C a * X ^ (n + 1))) x = eval μ (C... | [] | · intro n a hna
rw [mul_comm, pow_succ', mul_assoc, map_mul, Module.End.mul_apply, mul_comm, hna]
simp only [mem_eigenspace_iff.1 h.1, smul_smul, aeval_X, eval_mul, eval_C, eval_pow, eval_X,
map_smulₛₗ, RingHom.id_apply, mul_comm] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Eigenspace.Minpoly | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 31
} | {
"line": 91,
"column": 32
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nf : End R M\nμ : R\ninst✝¹ : IsDomain R\ninst✝ : Module.Finite R M\nh : (minpoly R f).IsRoot μ\nq : R[X]\nhq : minpoly R f = (X - C μ) * q\nv : M\nhv : ((aeval f) q) v ≠ 0\n⊢ ((aeval f) q) v ∈ f.eigenspace μ",
... | [
"R : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nf : End R M\nμ : R\ninst✝¹ : IsDomain R\ninst✝ : Module.Finite R M\nh : (minpoly R f).IsRoot μ\nq : R[X]\nhq : minpoly R f = (X - C μ) * q\nv : M\nhv : ((aeval f) q) v ≠ 0\n⊢ f (((aeval f) q) v) = μ • ((aeval f) q) v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 64
} | {
"line": 149,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝³ : P.IsCrystallographic\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : Finite ι\ni j : ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝³ : P.IsCrystallographic\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : Finite ι\ni j : ↥b.support\n... | refine (not_linearIndependent_iff.mpr ?_) b.linearIndepOn_root | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 121,
"column": 4
} | {
"line": 122,
"column": 35
} | {
"line": 122,
"column": 36
} | [
{
"pp": "S : Type u\ninst✝ : Semiring S\nn : ℕ\nh : map (algebraMap ℕ S) (ascPochhammer ℕ (n + 1)) = map (algebraMap ℕ S) (ascPochhammer ℕ n * (X + ↑n))\n⊢ ascPochhammer S (n + 1) = ascPochhammer S n * (X + ↑n)",
"ppTerm": "?m.115",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"u... | [
"S : Type u\ninst✝ : Semiring S\nn : ℕ\nh : map (algebraMap ℕ S) (ascPochhammer ℕ (n + 1)) = map (algebraMap ℕ S) (ascPochhammer ℕ n * (X + ↑n))\n⊢ ascPochhammer S (n + 1) = ascPochhammer S n * (X + ↑n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 19
} | {
"line": 167,
"column": 20
} | [
{
"pp": "case inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁴ : P.IsCrystallographic\ninst✝³ : CharZero R\ninst✝² : IsDomain R\ninst✝¹ : Finit... | [
"case inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁴ : P.IsCrystallographic\ninst✝³ : CharZero R\ninst✝² : IsDomain R\ninst✝¹ : Finite ι\ninst✝ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 140,
"column": 7
} | {
"line": 140,
"column": 29
} | {
"line": 140,
"column": 30
} | [
{
"pp": "S : Type u\ninst✝ : Semiring S\nn : ℕ\nthis : (ascPochhammer ℕ (n + 1)).comp (X + 1) = ascPochhammer ℕ (n + 1) + (↑n + 1) * (ascPochhammer ℕ n).comp (X + 1)\n⊢ (ascPochhammer S (n + 1)).comp (X + 1) = ascPochhammer S (n + 1) + (n + 1) • (ascPochhammer S n).comp (X + 1)",
"ppTerm": "?m.111",
"as... | [
"S : Type u\ninst✝ : Semiring S\nn : ℕ\nthis : (ascPochhammer ℕ (n + 1)).comp (X + 1) = ascPochhammer ℕ (n + 1) + (↑n + 1) * (ascPochhammer ℕ n).comp (X + 1)\n⊢ (ascPochhammer S (n + 1)).comp (X + 1) = ascPochhammer S (n + 1) + (↑n + 1) * (ascPochhammer S n).comp (X + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 46
} | {
"line": 190,
"column": 4
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁴ : P.IsCrystallographic\ninst✝³ : CharZero R\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ni... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁴ : P.IsCrystallographic\ninst✝³ : CharZero R\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : P.IsR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 40
} | {
"line": 201,
"column": 4
} | [
{
"pp": "case refine_2\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁵ : P.IsCrystallographic\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² :... | [
"case refine_2\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁵ : P.IsCrystallographic\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\ni... | refine Matrix.PosDef.smul ?_ two_pos | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 202,
"column": 6
} | {
"line": 202,
"column": 71
} | {
"line": 202,
"column": 72
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁵ : P.IsCrystallographic\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\ninst... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁵ : P.IsCrystallographic\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\ninst✝¹ : Decidab... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 325,
"column": 4
} | {
"line": 326,
"column": 35
} | {
"line": 326,
"column": 36
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nn : ℕ\nh : map (algebraMap ℤ R) (descPochhammer ℤ (n + 1)) = map (algebraMap ℤ R) (descPochhammer ℤ n * (X - ↑n))\n⊢ descPochhammer R (n + 1) = descPochhammer R n * (X - ↑n)",
"ppTerm": "?m.115",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"u... | [
"R : Type u\ninst✝ : Ring R\nn : ℕ\nh : map (algebraMap ℤ R) (descPochhammer ℤ (n + 1)) = map (algebraMap ℤ R) (descPochhammer ℤ n * (X - ↑n))\n⊢ descPochhammer R (n + 1) = descPochhammer R n * (X - ↑n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 204,
"column": 4
} | {
"line": 204,
"column": 15
} | {
"line": 204,
"column": 16
} | [
{
"pp": "case refine_2\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁵ : P.IsCrystallographic\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² :... | [
"case refine_2\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁵ : P.IsCrystallographic\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 358,
"column": 7
} | {
"line": 358,
"column": 29
} | {
"line": 358,
"column": 30
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nn : ℕ\nthis : (descPochhammer ℤ (n + 1)).comp (X - 1) = descPochhammer ℤ (n + 1) - (↑n + 1) * (descPochhammer ℤ n).comp (X - 1)\n⊢ (descPochhammer R (n + 1)).comp (X - 1) = descPochhammer R (n + 1) - (↑n + 1) • (descPochhammer R n).comp (X - 1)",
"ppTerm": "?m.112",
... | [
"R : Type u\ninst✝ : Ring R\nn : ℕ\nthis : (descPochhammer ℤ (n + 1)).comp (X - 1) = descPochhammer ℤ (n + 1) - (↑n + 1) * (descPochhammer ℤ n).comp (X - 1)\n⊢ (descPochhammer R (n + 1)).comp (X - 1) = descPochhammer R (n + 1) - (↑n + 1) * (descPochhammer R n).comp (X - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 224,
"column": 6
} | {
"line": 224,
"column": 23
} | {
"line": 224,
"column": 24
} | [
{
"pp": "case inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁵ : P.IsCrystallographic\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : Fini... | [
"case inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁵ : P.IsCrystallographic\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\ninst✝¹... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 30
} | {
"line": 251,
"column": 6
} | [
{
"pp": "case add\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝⁵ : P.IsCrystallographic\ninst✝⁴ : CharZero R\ninst✝³ : IsDomain R\ninst✝² : Fini... | [] | | add x y hx hy hx' hy' => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 533,
"column": 55
} | {
"line": 533,
"column": 91
} | {
"line": 533,
"column": 91
} | [
{
"pp": "K : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : CharZero K\na b : ℕ\n⊢ ↑(a.descFactorial b) = eval (↑a) (descPochhammer K b)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
... | [
"K : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : CharZero K\na b : ℕ\n⊢ ↑(a.descFactorial b) = ↑(a.descFactorial b)"
] | descPochhammer_eval_eq_descFactorial | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Vandermonde | {
"line": 187,
"column": 15
} | {
"line": 187,
"column": 52
} | {
"line": 187,
"column": 52
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\nn✝ n : ℕ\nih : ∀ (v w : Fin n → K), (projVandermonde v w).det = ∏ i, ∏ j ∈ Ioi i, (v j * w i - v i * w j)\nv w : Fin (n + 1) → K\nh0 : w 0 ≠ 0\nr : K := v 0 / w 0\nhr : r = v 0 / w 0\nW : Matrix (Fin (n + 1)) (Fin (n + 1)) K :=\n of fun i ↦\n Fin.cons (projVandermonde... | [] | by simp [W, r, projVandermonde_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Basis | {
"line": 139,
"column": 27
} | {
"line": 139,
"column": 38
} | {
"line": 139,
"column": 39
} | [
{
"pp": "case mem.add\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\nx y : L\ni : ι\nv w : L\nhx✝ : v ∈ Submodule.span R (range b.h)\nhy✝ : w ∈ Submodule.span R (range b.h)\nhv : ⁅v, b.e i⁆ ∈ lieSpan R L (range b.e)... | [
"case mem.add\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\nx y : L\ni : ι\nv w : L\nhx✝ : v ∈ Submodule.span R (range b.h)\nhy✝ : w ∈ Submodule.span R (range b.h)\nhv : ⁅v, b.e i⁆ ∈ lieSpan R L (range b.e)\nhw : ⁅w, b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Basis | {
"line": 140,
"column": 23
} | {
"line": 140,
"column": 34
} | {
"line": 140,
"column": 35
} | [
{
"pp": "case mem.smul\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\nx y : L\ni : ι\nt : R\nv : L\nhx✝ : v ∈ Submodule.span R (range b.h)\nhv : ⁅v, b.e i⁆ ∈ lieSpan R L (range b.e)\n⊢ ⁅t • v, b.e i⁆ ∈ lieSpan R L (... | [
"case mem.smul\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\nx y : L\ni : ι\nt : R\nv : L\nhx✝ : v ∈ Submodule.span R (range b.h)\nhv : ⁅v, b.e i⁆ ∈ lieSpan R L (range b.e)\n⊢ t • ⁅v, b.e i⁆ ∈ lieSpan R L (range b.e)"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Basis | {
"line": 142,
"column": 25
} | {
"line": 142,
"column": 36
} | {
"line": 142,
"column": 37
} | [
{
"pp": "case add\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\nx y : L\nhx : x ∈ b.cartan\nu v : L\nhx✝ : u ∈ lieSpan R L (range b.e)\nhy✝ : v ∈ lieSpan R L (range b.e)\nhu : ⁅x, u⁆ ∈ lieSpan R L (range b.e)\nhv :... | [
"case add\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\nx y : L\nhx : x ∈ b.cartan\nu v : L\nhx✝ : u ∈ lieSpan R L (range b.e)\nhy✝ : v ∈ lieSpan R L (range b.e)\nhu : ⁅x, u⁆ ∈ lieSpan R L (range b.e)\nhv : ⁅x, v⁆ ∈ li... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Basis | {
"line": 143,
"column": 21
} | {
"line": 143,
"column": 32
} | {
"line": 143,
"column": 33
} | [
{
"pp": "case smul\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\nx y : L\nhx : x ∈ b.cartan\nt : R\nu : L\nhx✝ : u ∈ lieSpan R L (range b.e)\nhu : ⁅x, u⁆ ∈ lieSpan R L (range b.e)\n⊢ ⁅x, t • u⁆ ∈ lieSpan R L (range... | [
"case smul\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\nx y : L\nhx : x ∈ b.cartan\nt : R\nu : L\nhx✝ : u ∈ lieSpan R L (range b.e)\nhu : ⁅x, u⁆ ∈ lieSpan R L (range b.e)\n⊢ t • ⁅x, u⁆ ∈ lieSpan R L (range b.e)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 312,
"column": 4
} | {
"line": 312,
"column": 32
} | {
"line": 312,
"column": 33
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Finite ι\nK : Type u_6\ninst✝⁵ : Field K\ninst✝⁴ : CharZero K\ninst✝³ : Module K M\ninst✝² : Module K N\nP : RootPairing ι K M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallographic\nb : P.Base\np : S... | [
"ι : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Finite ι\nK : Type u_6\ninst✝⁵ : Field K\ninst✝⁴ : CharZero K\ninst✝³ : Module K M\ninst✝² : Module K N\nP : RootPairing ι K M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallographic\nb : P.Base\np : Submodule K (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Basis | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 15
} | {
"line": 153,
"column": 16
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\ny : L\nhy : y ∈ (lieSpan R L (range b.e)).carrier\nx : L\nhx : x ∈ b.cartan\n⊢ ⁅⟨x, hx⟩, y⁆ ∈ (lieSpan R L (range b.e)).carrier",
"ppTerm": "?m.52",
"as... | [
"ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\ny : L\nhy : y ∈ (lieSpan R L (range b.e)).carrier\nx : L\nhx : x ∈ b.cartan\n⊢ ⁅x, y⁆ ∈ lieSpan R L (range b.e)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Lagrange | {
"line": 276,
"column": 58
} | {
"line": 276,
"column": 78
} | {
"line": 277,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\nx y : F\nhxy : x ≠ y\n⊢ basisDivisor x y + basisDivisor y x = Lagrange.basis {x, y} id x + Lagrange.basis {x, y} id y",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"AddMonoid.toAddZeroClass",
... | [
"F : Type u_1\ninst✝ : Field F\nx y : F\nhxy : x ≠ y\n⊢ basisDivisor x y + basisDivisor y x = basisDivisor (id x) (id y) + Lagrange.basis {x, y} id y"
] | basis_pair_left hxy, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Lagrange | {
"line": 282,
"column": 4
} | {
"line": 282,
"column": 20
} | {
"line": 282,
"column": 21
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\ni : ι\nhvs : Set.InjOn v ↑s\nhi : i ∈ s\n⊢ (∏ j ∈ s.erase i, (X - C (v j))).coeff (#s - 1) = 1",
"ppTerm": "?m.64",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\ni : ι\nhvs : Set.InjOn v ↑s\nhi : i ∈ s\n⊢ (∏ j ∈ s.erase i, (X - C (v j))).coeff (#s - 1) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 32
} | {
"line": 323,
"column": 33
} | [
{
"pp": "case inr\nι : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Finite ι\nK : Type u_6\ninst✝⁵ : Field K\ninst✝⁴ : CharZero K\ninst✝³ : Module K M\ninst✝² : Module K N\nP : RootPairing ι K M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallographic\nb : P.B... | [
"case inr\nι : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Finite ι\nK : Type u_6\ninst✝⁵ : Field K\ninst✝⁴ : CharZero K\ninst✝³ : Module K M\ninst✝² : Module K N\nP : RootPairing ι K M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallographic\nb : P.Base\np : Sub... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 341,
"column": 43
} | {
"line": 341,
"column": 54
} | {
"line": 341,
"column": 55
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁵ : CommRing R\ninst✝¹⁴ : AddCommGroup M\ninst✝¹³ : Module R M\ninst✝¹² : AddCommGroup N\ninst✝¹¹ : Module R N\ninst✝¹⁰ : CharZero R\ninst✝⁹ : IsDomain R\ninst✝⁸ : Finite ι\nι₂ : Type u_6\nM₂ : Type u_7\nN₂ : Type u_8\ninst✝⁷ : AddCommGroup ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁵ : CommRing R\ninst✝¹⁴ : AddCommGroup M\ninst✝¹³ : Module R M\ninst✝¹² : AddCommGroup N\ninst✝¹¹ : Module R N\ninst✝¹⁰ : CharZero R\ninst✝⁹ : IsDomain R\ninst✝⁸ : Finite ι\nι₂ : Type u_6\nM₂ : Type u_7\nN₂ : Type u_8\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Lagrange | {
"line": 333,
"column": 4
} | {
"line": 333,
"column": 65
} | {
"line": 333,
"column": 66
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv r : ι → F\nhvs : Set.InjOn v ↑s\ni : ι\nhi : i ∈ s\nhr : r i = 0\n⊢ (C (r i)).degree + ↑(#s - 1) ≤ ↑(#s - 1)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polyno... | [
"case pos\nF : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv r : ι → F\nhvs : Set.InjOn v ↑s\ni : ι\nhi : i ∈ s\nhr : r i = 0\n⊢ ⊥ ≤ ↑(#s - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 344,
"column": 4
} | {
"line": 344,
"column": 63
} | {
"line": 344,
"column": 64
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁵ : CommRing R\ninst✝¹⁴ : AddCommGroup M\ninst✝¹³ : Module R M\ninst✝¹² : AddCommGroup N\ninst✝¹¹ : Module R N\ninst✝¹⁰ : CharZero R\ninst✝⁹ : IsDomain R\ninst✝⁸ : Finite ι\nι₂ : Type u_6\nM₂ : Type u_7\nN₂ : Type u_8\ninst✝⁷ : AddCommGroup ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁵ : CommRing R\ninst✝¹⁴ : AddCommGroup M\ninst✝¹³ : Module R M\ninst✝¹² : AddCommGroup N\ninst✝¹¹ : Module R N\ninst✝¹⁰ : CharZero R\ninst✝⁹ : IsDomain R\ninst✝⁸ : Finite ι\nι₂ : Type u_6\nM₂ : Type u_7\nN₂ : Type u_8\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 351,
"column": 6
} | {
"line": 351,
"column": 17
} | {
"line": 351,
"column": 18
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁵ : CommRing R\ninst✝¹⁴ : AddCommGroup M\ninst✝¹³ : Module R M\ninst✝¹² : AddCommGroup N\ninst✝¹¹ : Module R N\ninst✝¹⁰ : CharZero R\ninst✝⁹ : IsDomain R\ninst✝⁸ : Finite ι\nι₂ : Type u_6\nM₂ : Type u_7\nN₂ : Type u_8\ninst✝⁷ : AddCommGroup ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁵ : CommRing R\ninst✝¹⁴ : AddCommGroup M\ninst✝¹³ : Module R M\ninst✝¹² : AddCommGroup N\ninst✝¹¹ : Module R N\ninst✝¹⁰ : CharZero R\ninst✝⁹ : IsDomain R\ninst✝⁸ : Finite ι\nι₂ : Type u_6\nM₂ : Type u_7\nN₂ : Type u_8\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 362,
"column": 6
} | {
"line": 362,
"column": 17
} | {
"line": 362,
"column": 18
} | [
{
"pp": "case refine_1\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝²² : CommRing R\ninst✝²¹ : AddCommGroup M\ninst✝²⁰ : Module R M\ninst✝¹⁹ : AddCommGroup N\ninst✝¹⁸ : Module R N\nS : Type u_5\ninst✝¹⁷ : CommRing S\ninst✝¹⁶ : Algebra S R\nP✝ : RootPairing ι R M N\ninst✝¹⁵ : P✝.IsValuedIn S\nb✝... | [
"case refine_1\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝²² : CommRing R\ninst✝²¹ : AddCommGroup M\ninst✝²⁰ : Module R M\ninst✝¹⁹ : AddCommGroup N\ninst✝¹⁸ : Module R N\nS : Type u_5\ninst✝¹⁷ : CommRing S\ninst✝¹⁶ : Algebra S R\nP✝ : RootPairing ι R M N\ninst✝¹⁵ : P✝.IsValuedIn S\nb✝ : P✝.Base\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Lagrange | {
"line": 624,
"column": 29
} | {
"line": 624,
"column": 40
} | {
"line": 624,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nG : Subgroup Rˣ\ninst✝ : Fintype ↥G\nh : degree 1 < (X ^ Fintype.card ↥G).degree\ni : Rˣ\nhi : i ∈ (↑G).toFinset\n⊢ i ∈ G",
"ppTerm": "?m.142",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nG : Subgroup Rˣ\ninst✝ : Fintype ↥G\nh : degree 1 < (X ^ Fintype.card ↥G).degree\ni : Rˣ\nhi : i ∈ (↑G).toFinset\n⊢ i ∈ G"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Monad | {
"line": 329,
"column": 2
} | {
"line": 329,
"column": 73
} | {
"line": 329,
"column": 74
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\ninst✝ : CommSemiring R\nf : σ → MvPolynomial τ R\nφ : MvPolynomial σ R\nj : τ\nh : j ∈ ((bind₁ f) φ).vars\n⊢ ∃ i ∈ φ.vars, j ∈ (f i).vars",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"σ : Type u_1\nτ : Type u_2\nR : Type u_3\ninst✝ : CommSemiring R\nf : σ → MvPolynomial τ R\nφ : MvPolynomial σ R\nj : τ\nh : j ∈ ((bind₁ f) φ).vars\n⊢ ∃ i ∈ φ.vars, j ∈ (f i).vars"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Lagrange | {
"line": 695,
"column": 4
} | {
"line": 695,
"column": 78
} | {
"line": 696,
"column": 6
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\nx : F\nhvs : Set.InjOn v ↑s\nhx : ∀ i ∈ s, x ≠ v i\nhs : s.Nonempty\n⊢ eval x (nodal s v) * ∑ i ∈ s, nodalWeight s v i * (x - v i)⁻¹ = 1",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],... | [
"F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\nx : F\nhvs : Set.InjOn v ↑s\nhx : ∀ i ∈ s, x ≠ v i\nhs : s.Nonempty\n⊢ eval x (nodal s v) * ∑ i ∈ s, nodalWeight s v i * (x - v i)⁻¹ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Charpoly.Basic | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 56
} | {
"line": 66,
"column": 57
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.Free R M\ninst✝ : Module.Finite R M\nf : Module.End R M\nμ : R\n⊢ charpoly (f - μ • 1) = (charpoly f).comp (X + C μ)",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Mo... | [
"R : Type u\nM : Type v\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.Free R M\ninst✝ : Module.Finite R M\nf : Module.End R M\nμ : R\n⊢ ((toMatrix (chooseBasis R M) (chooseBasis R M)) f - diagonal fun x ↦ μ).charpoly =\n ((toMatrix (chooseBasis R M) (chooseBasis R M)) f).cha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Charpoly.Basic | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 38
} | {
"line": 143,
"column": 39
} | [
{
"pp": "R : Type u_2\nM : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : Ring M\ninst✝² : Algebra R M\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Free R M\nα : M\n⊢ (lmul R M) ((aeval α) (LinearMap.charpoly ((lmul R M) α))) = (lmul R M) 0",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq... | [
"R : Type u_2\nM : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : Ring M\ninst✝² : Algebra R M\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Free R M\nα : M\n⊢ (aeval ((LinearMap.mul R M) α)) ((LinearMap.mul R M) α).charpoly = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Submodule | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 15
} | {
"line": 80,
"column": 16
} | [
{
"pp": "ιA : Type u_1\nιM : Type u_2\nσA : Type u_3\nσM : Type u_4\nA : Type u_5\nM : Type u_6\ninst✝¹³ : Semiring A\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module A M\n𝒜 : ιA → σA\nℳ : ιM → σM\ninst✝¹⁰ : DecidableEq ιA\ninst✝⁹ : AddMonoid ιA\ninst✝⁸ : SetLike σA A\ninst✝⁷ : AddSubmonoidClass σA A\ninst✝⁶ : Gra... | [
"ιA : Type u_1\nιM : Type u_2\nσA : Type u_3\nσM : Type u_4\nA : Type u_5\nM : Type u_6\ninst✝¹³ : Semiring A\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module A M\n𝒜 : ιA → σA\nℳ : ιM → σM\ninst✝¹⁰ : DecidableEq ιA\ninst✝⁹ : AddMonoid ιA\ninst✝⁸ : SetLike σA A\ninst✝⁷ : AddSubmonoidClass σA A\ninst✝⁶ : GradedRing 𝒜\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Charpoly.ToMatrix | {
"line": 64,
"column": 60
} | {
"line": 69,
"column": 68
} | {
"line": 71,
"column": 0
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_3\nM₂ : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : Module.Finite R M₁\ninst✝⁴ : Free R M₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : Module.Finite R M₂\ninst✝ : Free R M₂\nf₁ : M₁ →ₗ[R] M₁\nf₂ : M₂ →ₗ[R] M₂\n⊢ (f₁.pro... | [] | by
let b₁ := chooseBasis R M₁
let b₂ := chooseBasis R M₂
let b := b₁.prod b₂
rw [← charpoly_toMatrix f₁ b₁, ← charpoly_toMatrix f₂ b₂, ← charpoly_toMatrix (f₁.prodMap f₂) b,
toMatrix_prodMap b₁ b₂ f₁ f₂, Matrix.charpoly_fromBlocks_zero₁₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 38
} | {
"line": 111,
"column": 39
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\nσ : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\nhm' : weightedTotalDegree' w p ≤ ↑m\n⊢ ↑(p.support.sup fun s ↦ (weight w) s) ≤ ... | [
"case a\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\nσ : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\nhm' : weightedTotalDegree' w p ≤ ↑m\n⊢ ∀ (b : σ →₀ ℕ), ¬coeff b p = 0 → (weight w) b ≤ m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 41
} | {
"line": 249,
"column": 2
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CharZero R\ninst✝⁴ : IsDomain R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : IsNoetherian R L\ninst✝ : Module.Free R L\n⊢ LieIdeal.killingCompl R L ⊤ ≤ radical R L",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\nL : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CharZero R\ninst✝⁴ : IsDomain R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : IsNoetherian R L\ninst✝ : Module.Free R L\n⊢ IsSolvable ↥(LieIdeal.killingCompl R L ⊤)"
] | rw [← LieIdeal.solvable_iff_le_radical] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal | {
"line": 604,
"column": 65
} | {
"line": 604,
"column": 76
} | {
"line": 604,
"column": 77
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁷ : Semiring A\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : PartialOrder ι\ninst✝³ : CanonicallyOrderedAdd ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nx : A\nhx : x ∈ 𝒜₊.toAddSubmonoid\nj : ι\... | [
"ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁷ : Semiring A\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : PartialOrder ι\ninst✝³ : CanonicallyOrderedAdd ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nx : A\nhx : x ∈ 𝒜₊.toAddSubmonoid\nj : ι\nhj : j ∈ DF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 259,
"column": 32
} | {
"line": 259,
"column": 43
} | {
"line": 259,
"column": 44
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : IsNoetherian R L\ninst✝¹ : Module.Free R L\ninst✝ : HasTrivialRadical R L\n⊢ LieIdeal.killingCompl R L ⊤ = ⊥",
"ppTerm": "?m.22",
"assig... | [
"R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : IsNoetherian R L\ninst✝¹ : Module.Free R L\ninst✝ : HasTrivialRadical R L\n⊢ LieIdeal.killingCompl R L ⊤ = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 323,
"column": 54
} | {
"line": 323,
"column": 65
} | {
"line": 323,
"column": 66
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nm : M\nmotive : (p : MvPolynomial σ R) → IsWeightedHomogeneous w p m → Prop\nzero : motive 0 ⋯\nadd :\n ∀ (p q : MvPolynomial σ R) (hp : IsWeightedHomogeneous w p m) (hq : IsWeightedHomogeneous w q m... | [
"R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nm : M\nmotive : (p : MvPolynomial σ R) → IsWeightedHomogeneous w p m → Prop\nzero : motive 0 ⋯\nadd :\n ∀ (p q : MvPolynomial σ R) (hp : IsWeightedHomogeneous w p m) (hq : IsWeightedHomogeneous w q m),\n moti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 70
} | {
"line": 340,
"column": 71
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nm : M\nmotive : (p : MvPolynomial σ R) → IsWeightedHomogeneous w p m → Prop\nzero : motive 0 ⋯\nadd :\n ∀ (p q : MvPolynomial σ R) (hp : IsWeightedHomogeneous w p m) (hq : IsWeightedHomogeneous w q m... | [
"R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nm : M\nmotive : (p : MvPolynomial σ R) → IsWeightedHomogeneous w p m → Prop\nzero : motive 0 ⋯\nadd :\n ∀ (p q : MvPolynomial σ R) (hp : IsWeightedHomogeneous w p m) (hq : IsWeightedHomogeneous w q m),\n moti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 29
} | {
"line": 250,
"column": 30
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\nφ : MvPolynomial σ R\nm : ℕ\nhφ : φ.IsHomogeneous m\nr : R\n⊢ (C r * φ).IsHomogeneous m",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"σ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\nφ : MvPolynomial σ R\nm : ℕ\nhφ : φ.IsHomogeneous m\nr : R\n⊢ (C r * φ).IsHomogeneous m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 28
} | {
"line": 263,
"column": 29
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\ni : σ\nn : ℕ\n⊢ (X i ^ n).IsHomogeneous n",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"σ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\ni : σ\nn : ℕ\n⊢ (X i ^ n).IsHomogeneous n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 23
} | {
"line": 285,
"column": 24
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\nS : Type u_4\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ : MvPolynomial σ R\nn : ℕ\nhφ : φ.IsHomogeneous n\nf : R →+* S\n⊢ ((eval₂Hom (C.comp f) X) φ).IsHomogeneous n",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClas... | [
"σ : Type u_1\nR : Type u_3\nS : Type u_4\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ : MvPolynomial σ R\nn : ℕ\nhφ : φ.IsHomogeneous n\nf : R →+* S\n⊢ (MvPolynomial.eval₂ (C.comp f) X φ).IsHomogeneous n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 355,
"column": 4
} | {
"line": 356,
"column": 68
} | {
"line": 356,
"column": 69
} | [
{
"pp": "R : Type u_3\ninst✝ : CommSemiring R\nN : ℕ\nφ : MvPolynomial (Fin (N + 1)) R\nn : ℕ\nhφ : φ.IsHomogeneous n\ni j : ℕ\nh : i + j = n\nd : Fin N →₀ ℕ\nhd : coeff (cons i d) φ ≠ 0\n⊢ (weight 1) (cons i d) = i + j",
"ppTerm": "?m.59",
"assigned": false,
"usedConstants": [],
"usedFVars": []... | [
"R : Type u_3\ninst✝ : CommSemiring R\nN : ℕ\nφ : MvPolynomial (Fin (N + 1)) R\nn : ℕ\nhφ : φ.IsHomogeneous n\ni j : ℕ\nh : i + j = n\nd : Fin N →₀ ℕ\nhd : coeff (cons i d) φ ≠ 0\n⊢ (weight 1) (cons i d) = i + j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Zero | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 42
} | {
"line": 70,
"column": 43
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Module.Finite R M\ninst✝¹ : Free R M\ninst✝ : IsNoetherian R M\nφ : End R M\ntfae_1_to_2 : IsNilpotent φ → charpoly φ = X ^ finrank R M\nh : charpoly φ = X ^ finrank R M\... | [
"case h\nR : Type u_1\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Module.Finite R M\ninst✝¹ : Free R M\ninst✝ : IsNoetherian R M\nφ : End R M\ntfae_1_to_2 : IsNilpotent φ → charpoly φ = X ^ finrank R M\nh : charpoly φ = X ^ finrank R M\nm : M\n⊢ φ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 412,
"column": 6
} | {
"line": 412,
"column": 17
} | {
"line": 412,
"column": 18
} | [
{
"pp": "case neg\nR : Type u_3\ninst✝ : CommSemiring R\nN : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\nhFn : constantCoeff (((finSuccEquiv R N) F... | [
"case neg\nR : Type u_3\ninst✝ : CommSemiring R\nN : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\nhFn : constantCoeff (((finSuccEquiv R N) F).coeff n) =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Zero | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 58
} | {
"line": 117,
"column": 59
} | [
{
"pp": "K : Type u_2\nM : Type u_3\ninst✝³ : Field K\ninst✝² : AddCommGroup M\ninst✝¹ : Module K M\ninst✝ : Module.Finite K M\nφ : End K M\ntfae_1_iff_2 : φ.HasEigenvalue 0 ↔ (minpoly K φ).IsRoot 0\ntfae_2_to_3 : (minpoly K φ).IsRoot 0 → constantCoeff (charpoly φ) = 0\ntfae_3_to_4 : constantCoeff (charpoly φ) ... | [
"K : Type u_2\nM : Type u_3\ninst✝³ : Field K\ninst✝² : AddCommGroup M\ninst✝¹ : Module K M\ninst✝ : Module.Finite K M\nφ : End K M\ntfae_1_iff_2 : φ.HasEigenvalue 0 ↔ (minpoly K φ).IsRoot 0\ntfae_2_to_3 : (minpoly K φ).IsRoot 0 → constantCoeff (charpoly φ) = 0\ntfae_3_to_4 : constantCoeff (charpoly φ) = 0 → Linear... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Zero | {
"line": 114,
"column": 2
} | {
"line": 117,
"column": 61
} | {
"line": 118,
"column": 2
} | [
{
"pp": "K : Type u_2\nM : Type u_3\ninst✝³ : Field K\ninst✝² : AddCommGroup M\ninst✝¹ : Module K M\ninst✝ : Module.Finite K M\nφ : End K M\ntfae_1_iff_2 : φ.HasEigenvalue 0 ↔ (minpoly K φ).IsRoot 0\ntfae_2_to_3 : (minpoly K φ).IsRoot 0 → constantCoeff (charpoly φ) = 0\ntfae_3_to_4 : constantCoeff (charpoly φ) ... | [
"K : Type u_2\nM : Type u_3\ninst✝³ : Field K\ninst✝² : AddCommGroup M\ninst✝¹ : Module K M\ninst✝ : Module.Finite K M\nφ : End K M\ntfae_1_iff_2 : φ.HasEigenvalue 0 ↔ (minpoly K φ).IsRoot 0\ntfae_2_to_3 : (minpoly K φ).IsRoot 0 → constantCoeff (charpoly φ) = 0\ntfae_3_to_4 : constantCoeff (charpoly φ) = 0 → Linear... | tfae_have 6 → 1
| ⟨x, h1, h2⟩ => by
apply Module.End.hasEigenvalue_of_hasEigenvector ⟨_, h1⟩
simpa only [Module.End.eigenspace_zero, mem_ker] using h2 | Mathlib.Tactic.TFAE._aux_Mathlib_Tactic_TFAE___macroRules_Mathlib_Tactic_TFAE_tfaeHave_1 | Mathlib.Tactic.TFAE.tfaeHave |
Mathlib.LinearAlgebra.Eigenspace.Zero | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 31
} | {
"line": 138,
"column": 32
} | [
{
"pp": "K : Type u_2\nM : Type u_3\ninst✝³ : Field K\ninst✝² : AddCommGroup M\ninst✝¹ : Module K M\ninst✝ : Module.Finite K M\nφ : End K M\nthis :\n [¬φ.HasEigenvalue 0, ¬(minpoly K φ).IsRoot 0, constantCoeff (charpoly φ) ≠ 0, LinearMap.det φ ≠ 0, ¬⊥ < ker φ,\n ∀ (m : M), m ≠ 0 → φ m ≠ 0].TFAE\naux₁ : ∀ ... | [
"K : Type u_2\nM : Type u_3\ninst✝³ : Field K\ninst✝² : AddCommGroup M\ninst✝¹ : Module K M\ninst✝ : Module.Finite K M\nφ : End K M\nthis :\n [¬φ.HasEigenvalue 0, ¬(minpoly K φ).IsRoot 0, constantCoeff (charpoly φ) ≠ 0, LinearMap.det φ ≠ 0, ¬⊥ < ker φ,\n ∀ (m : M), m ≠ 0 → φ m ≠ 0].TFAE\naux₁ : ∀ (m : M), m ≠... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 484,
"column": 2
} | {
"line": 484,
"column": 27
} | {
"line": 484,
"column": 28
} | [
{
"pp": "R : Type u_5\nσ : Type u_6\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nF G : MvPolynomial σ R\nn : ℕ\nhF : F.IsHomogeneous n\nhG : G.IsHomogeneous n\nh : ∀ (r : σ → R), (eval r) F = (eval r) G\nhnR : ↑n ≤ #R\n⊢ ∀ (r : σ → R), (eval r) (F - G) = 0",
"ppTerm": "?m.42",
"assigned": true,
"usedCo... | [
"R : Type u_5\nσ : Type u_6\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nF G : MvPolynomial σ R\nn : ℕ\nhF : F.IsHomogeneous n\nhG : G.IsHomogeneous n\nh : ∀ (r : σ → R), (eval r) F = (eval r) G\nhnR : ↑n ≤ #R\n⊢ ∀ (r : σ → R), (eval r) F = (eval r) G"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.LinearMap.Polynomial | {
"line": 489,
"column": 2
} | {
"line": 489,
"column": 57
} | {
"line": 489,
"column": 58
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup L\ninst✝⁷ : Module R L\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁴ : Free R M\ninst✝³ : Module.Finite R M\ninst✝² : Module.Finite R L\ninst✝¹ : Free R L\ninst✝ : Nontrivial R\n⊢ φ.nilRank... | [
"R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup L\ninst✝⁷ : Module R L\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁴ : Free R M\ninst✝³ : Module.Finite R M\ninst✝² : Module.Finite R L\ninst✝¹ : Free R L\ninst✝ : Nontrivial R\n⊢ φ.nilRank ≤ Fintype.c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Matrix | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 15
} | {
"line": 126,
"column": 16
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\n⊢ Function.Injective ⇑(toEnd R (Matrix n n R) (n → R))",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"LieHom",
"Module.End.instRing",
"Eq.mpr",
"Pi.Function.module",
... | [
"R : Type u\ninst✝² : CommRing R\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\n⊢ Function.Injective ⇑lieEquivMatrix'.symm"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.LinearMap | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 37
} | {
"line": 71,
"column": 38
} | [
{
"pp": "K : Type u\nV : Type v\nV' V'₁ : Type v'\ninst✝⁶ : Semiring K\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : Module K V\ninst✝³ : AddCommMonoid V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V'₁\ninst✝ : Module K V'₁\ng : V →ₗ[K] V'\nf : V' →ₗ[K] V'₁\n⊢ (f ∘ₗ g).rank ≤ g.rank",
"ppTerm": "?m.66",
"assign... | [
"K : Type u\nV : Type v\nV' V'₁ : Type v'\ninst✝⁶ : Semiring K\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : Module K V\ninst✝³ : AddCommMonoid V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V'₁\ninst✝ : Module K V'₁\ng : V →ₗ[K] V'\nf : V' →ₗ[K] V'₁\n⊢ (f ∘ₗ g).rank ≤ g.rank"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.LinearMap | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 37
} | {
"line": 78,
"column": 38
} | [
{
"pp": "K : Type u\nV : Type v\nV' V'₁ : Type v'\ninst✝⁶ : Semiring K\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : Module K V\ninst✝³ : AddCommMonoid V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V'₁\ninst✝ : Module K V'₁\ng : V →ₗ[K] V'\nf : V' →ₗ[K] V'₁\n⊢ (f ∘ₗ g).rank ≤ min f.rank g.rank",
"ppTerm": "?m.76",
... | [
"K : Type u\nV : Type v\nV' V'₁ : Type v'\ninst✝⁶ : Semiring K\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : Module K V\ninst✝³ : AddCommMonoid V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V'₁\ninst✝ : Module K V'₁\ng : V →ₗ[K] V'\nf : V' →ₗ[K] V'₁\n⊢ (f ∘ₗ g).rank ≤ min f.rank g.rank"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.SkewAdjoint | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 15
} | {
"line": 164,
"column": 16
} | [
{
"pp": "case mp\nR : Type u\nn : Type w\ninst✝² : CommRing R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nu : Rˣ\nJ A : Matrix n n R\nh : Aᵀ * u • J = u • J * -A\n⊢ Aᵀ * J = J * -A",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"NonUnitalCom... | [
"case mp\nR : Type u\nn : Type w\ninst✝² : CommRing R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nu : Rˣ\nJ A : Matrix n n R\nh : Aᵀ * u • J = u • J * -A\n⊢ Aᵀ * J = -(J * A)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.CartanExists | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 24
} | {
"line": 156,
"column": 25
} | [
{
"pp": "case inl\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\ny : L\nhyU : y ∈ U\nEy : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩\nhxU : 0 ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x}... | [
"case inl\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\ny : L\nhyU : y ∈ U\nEy : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩\nhxU : 0 ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SymplecticGroup | {
"line": 183,
"column": 16
} | {
"line": 183,
"column": 27
} | {
"line": 183,
"column": 28
} | [
{
"pp": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : Aᵀ ∈ symplecticGroup l R\n⊢ A ∈ symplecticGroup l R",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : Aᵀ ∈ symplecticGroup l R\n⊢ A ∈ symplecticGroup l R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SymplecticGroup | {
"line": 232,
"column": 6
} | {
"line": 233,
"column": 29
} | {
"line": 233,
"column": 30
} | [
{
"pp": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA B C D : Matrix l l R\nh : fromBlocks A B C D ∈ symplecticGroup l R\n⊢ fromBlocks (Cᵀ * A - Aᵀ * C) (Cᵀ * B - Aᵀ * D) (Dᵀ * A - Bᵀ * C) (Dᵀ * B - Bᵀ * D) = J l R",
"ppTerm": "?m.160",
"assigned": true,
... | [
"l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA B C D : Matrix l l R\nh : fromBlocks A B C D ∈ symplecticGroup l R\n⊢ Cᵀ * A + -(Aᵀ * C) = 0 ∧ Cᵀ * B + -(Aᵀ * D) = -1 ∧ Dᵀ * A + -(Bᵀ * C) = 1 ∧ Dᵀ * B + -(Bᵀ * D) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SymplecticGroup | {
"line": 238,
"column": 34
} | {
"line": 238,
"column": 45
} | {
"line": 238,
"column": 46
} | [
{
"pp": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA B C D : Matrix l l R\nh : Aᵀ * C = Cᵀ * A ∧ Bᵀ * D = Dᵀ * B ∧ Aᵀ * D - Cᵀ * B = 1\n⊢ Dᵀ * A - Bᵀ * C = 1",
"ppTerm": "?m.255",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA B C D : Matrix l l R\nh : Aᵀ * C = Cᵀ * A ∧ Bᵀ * D = Dᵀ * B ∧ Aᵀ * D - Cᵀ * B = 1\n⊢ Dᵀ * A - Bᵀ * C = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.CartanExists | {
"line": 247,
"column": 6
} | {
"line": 248,
"column": 13
} | {
"line": 248,
"column": 14
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} :=... | [
"K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SymplecticGroup | {
"line": 276,
"column": 52
} | {
"line": 276,
"column": 63
} | {
"line": 276,
"column": 64
} | [
{
"pp": "l : Type u_1\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\nR : Type u_3\ninst✝ : Field R\nA C : Matrix l l R\nhker : ∀ (x : l → R), A • x = 0 → C • x = 0 → x = 0\nhsymm : Aᵀ * C = Cᵀ * A\nV U : Matrix l l R\ns : l ≃ Fin C.rank ⊕ Fin (Fintype.card l - C.rank)\nhV : IsUnit V\nhU : IsUnit U\nP : Matrix l l... | [
"l : Type u_1\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\nR : Type u_3\ninst✝ : Field R\nA C : Matrix l l R\nhker : ∀ (x : l → R), A • x = 0 → C • x = 0 → x = 0\nhsymm : Aᵀ * C = Cᵀ * A\nV U : Matrix l l R\ns : l ≃ Fin C.rank ⊕ Fin (Fintype.card l - C.rank)\nhV : IsUnit V\nhU : IsUnit U\nP : Matrix l l R := V * C ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 76,
"column": 2
} | {
"line": 78,
"column": 9
} | {
"line": 79,
"column": 2
} | [
{
"pp": "m : Type um\nm₀ : Type um₀\nn : Type un\nn₀ : Type un₀\nR : Type uR\ninst✝ : Semiring R\nA : Matrix m n R\nr : m₀ → m\nc : n₀ → n\nh : ((A.submatrix r id).submatrix id c).cRank ≤ (A.submatrix r id).cRank\nf : (m → R) →ₗ[R] m₀ → R := LinearMap.funLeft R R r\n⊢ lift.{um, max uR um₀} (A.submatrix r id).cR... | [
"m : Type um\nm₀ : Type um₀\nn : Type un\nn₀ : Type un₀\nR : Type uR\ninst✝ : Semiring R\nA : Matrix m n R\nr : m₀ → m\nc : n₀ → n\nh : ((A.submatrix r id).submatrix id c).cRank ≤ (A.submatrix r id).cRank\nf : (m → R) →ₗ[R] m₀ → R := LinearMap.funLeft R R r\nh_eq : Submodule.map f (span R (range A.col)) = span R (r... | have h_eq : Submodule.map f (span R (range A.col)) = span R (range (A.submatrix r id).col) := by
simp_rw [LinearMap.map_span, ← image_univ, image_image, col_eq_transpose, transpose_submatrix]
aesop | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 13
} | {
"line": 87,
"column": 14
} | [
{
"pp": "n : Type un\nn₀ : Type un₀\nR : Type uR\ninst✝ : Semiring R\nm m₀ : Type um\nA : Matrix m n R\nr : m₀ → m\nc : n₀ → n\n⊢ (A.submatrix r c).cRank ≤ A.cRank",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : Type un\nn₀ : Type un₀\nR : Type uR\ninst✝ : Semiring R\nm m₀ : Type um\nA : Matrix m n R\nr : m₀ → m\nc : n₀ → n\n⊢ (A.submatrix r c).cRank ≤ A.cRank"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 96,
"column": 7
} | {
"line": 96,
"column": 37
} | {
"line": 96,
"column": 38
} | [
{
"pp": "m : Type um\nn : Type un\nR : Type uR\ninst✝² : Semiring R\ninst✝¹ : StrongRankCondition R\ninst✝ : Fintype n\nA : Matrix m n R\n⊢ #↑(range A.col) ≤ ↑(Fintype.card n)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"Cardinal",
... | [
"m : Type um\nn : Type un\nR : Type uR\ninst✝² : Semiring R\ninst✝¹ : StrongRankCondition R\ninst✝ : Fintype n\nA : Matrix m n R\n⊢ #↑(range (of.symm Aᵀ)) ≤ ↑(Fintype.card n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 115,
"column": 33
} | {
"line": 115,
"column": 44
} | {
"line": 115,
"column": 45
} | [
{
"pp": "m : Type um\nn : Type un\nR : Type uR\ninst✝¹ : Semiring R\ninst✝ : StrongRankCondition R\nA : Matrix m n R\nthis :\n ∀ {m : Type um} {n : Type un} {R : Type uR} [inst : Semiring R] [StrongRankCondition R] (A : Matrix m n R),\n Finite n → A.eRank ≤ ENat.card n\nhfin : ¬Finite n\n⊢ Infinite ?m.33",
... | [
"m : Type um\nn : Type un\nR : Type uR\ninst✝¹ : Semiring R\ninst✝ : StrongRankCondition R\nA : Matrix m n R\nthis :\n ∀ {m : Type um} {n : Type un} {R : Type uR} [inst : Semiring R] [StrongRankCondition R] (A : Matrix m n R),\n Finite n → A.eRank ≤ ENat.card n\nhfin : ¬Finite n\n⊢ Infinite ?m.33"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 123,
"column": 33
} | {
"line": 123,
"column": 44
} | {
"line": 123,
"column": 45
} | [
{
"pp": "m : Type um\nn : Type un\nR : Type uR\ninst✝¹ : Semiring R\ninst✝ : StrongRankCondition R\nA : Matrix m n R\nthis :\n ∀ {m : Type um} {n : Type un} {R : Type uR} [inst : Semiring R] [StrongRankCondition R] (A : Matrix m n R),\n Finite m → A.eRank ≤ ENat.card m\nhfin : ¬Finite m\n⊢ Infinite ?m.34",
... | [
"m : Type um\nn : Type un\nR : Type uR\ninst✝¹ : Semiring R\ninst✝ : StrongRankCondition R\nA : Matrix m n R\nthis :\n ∀ {m : Type um} {n : Type un} {R : Type uR} [inst : Semiring R] [StrongRankCondition R] (A : Matrix m n R),\n Finite m → A.eRank ≤ ENat.card m\nhfin : ¬Finite m\n⊢ Infinite ?m.34"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 269,
"column": 10
} | {
"line": 269,
"column": 21
} | {
"line": 269,
"column": 22
} | [
{
"pp": "m : Type um\nm₀ : Type um₀\nn₀ : Type un₀\nR : Type uR\nn : Type un\ninst✝ : Semiring R\nA : Matrix m n R\nem : m₀ ≃ m\nen : n₀ ≃ n\n⊢ lift.{um₀, max uR um} A.cRank ≤ lift.{um, max uR um₀} (A.submatrix ⇑em ⇑en).cRank",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"m : Type um\nm₀ : Type um₀\nn₀ : Type un₀\nR : Type uR\nn : Type un\ninst✝ : Semiring R\nA : Matrix m n R\nem : m₀ ≃ m\nen : n₀ ≃ n\n⊢ lift.{um₀, max uR um} A.cRank ≤ lift.{um, max uR um₀} (A.submatrix ⇑em ⇑en).cRank"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 37
} | {
"line": 275,
"column": 38
} | [
{
"pp": "m : Type um\nn₀ : Type un₀\nR : Type uR\nm₀ : Type um\nn : Type un\ninst✝ : Semiring R\nA : Matrix m n R\nem : m₀ ≃ m\nen : n₀ ≃ n\n⊢ (A.submatrix ⇑em ⇑en).cRank = A.cRank",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"m : Type um\nn₀ : Type un₀\nR : Type uR\nm₀ : Type um\nn : Type un\ninst✝ : Semiring R\nA : Matrix m n R\nem : m₀ ≃ m\nen : n₀ ≃ n\n⊢ (A.submatrix ⇑em ⇑en).cRank = A.cRank"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.CartanExists | {
"line": 323,
"column": 2
} | {
"line": 333,
"column": 75
} | {
"line": 334,
"column": 2
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} :=... | [
"K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y,... | have hz' : ∃ n : ℕ, (toEnd K U Q v ^ n) z' = 0 := by
rw [mem_engel_iff] at hz
obtain ⟨n, hn⟩ := hz
use n
apply_fun LieSubmodule.Quotient.mk' E at hn
rw [map_zero] at hn
rw [← hn]
clear hn
induction n with
| zero => simp only [z', pow_zero, Module.End.one_apply]
| succ n ih => rw ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.CartanExists | {
"line": 340,
"column": 4
} | {
"line": 340,
"column": 58
} | {
"line": 340,
"column": 59
} | [
{
"pp": "case inl\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K... | [
"case inl\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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