module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Artinian.Module | {
"line": 168,
"column": 21
} | {
"line": 168,
"column": 44
} | {
"line": 168,
"column": 45
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\ns : Injective ⇑f\nh : ¬Surjective ⇑f\nn : ℕ\n⊢ (f ^ n * f).range < (f ^ n).range",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"Submodule"... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\ns : Injective ⇑f\nh : ¬Surjective ⇑f\nn : ℕ\n⊢ ((f ^ n) ∘ₗ f).range < (f ^ n).range"
] | Module.End.mul_eq_comp, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 296,
"column": 10
} | {
"line": 296,
"column": 58
} | {
"line": 296,
"column": 59
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq' pq'' : κ\nhpq' : (c r).next pq' = pq''\ni₀' i₀... | [
"C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq' pq'' : κ\nhpq' : (c r).next pq' = pq''\ni₀' i₀ i₁ i₂ i₃ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Basic | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 38
} | {
"line": 181,
"column": 39
} | [
{
"pp": "L : Type v\ninst✝ : LieRing L\nx y : L\nh : ⁅x + y, x⁆ + ⁅x + y, y⁆ = 0\n⊢ -⁅y, x⁆ = ⁅x, y⁆",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"LieRing.toAddCommGroup",
"AddMonoid.toAddZeroC... | [
"L : Type v\ninst✝ : LieRing L\nx y : L\nh : ⁅x + y, x⁆ + ⁅x + y, y⁆ = 0\n⊢ ⁅y, x⁆ + ⁅x, y⁆ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 338,
"column": 19
} | {
"line": 338,
"column": 67
} | {
"line": 338,
"column": 68
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' : κ\ni₀ i₁ i₂ i₃ i₃' : ι\nhi₀ : i₀ = data.i₀... | [
"C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' : κ\ni₀ i₁ i₂ i₃ i₃' : ι\nhi₀ : i₀ = data.i₀ r pq' ⋯\nhi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Artinian.Module | {
"line": 278,
"column": 22
} | {
"line": 278,
"column": 38
} | {
"line": 278,
"column": 38
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type u_4\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup P\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R M\ninst✝⁴ : Module R P\ninst✝³ : Module R N\nι : Type u_5\ninst✝² : Finite ι\nα✝ : Type u_5\ninst✝¹ : Fintype α✝\nih : ∀ {M : α✝ → Submod... | [
"R : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type u_4\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup P\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R M\ninst✝⁴ : Module R P\ninst✝³ : Module R N\nι : Type u_5\ninst✝² : Finite ι\nα✝ : Type u_5\ninst✝¹ : Fintype α✝\nih : ∀ {M : α✝ → Submodule R P} [∀ ... | rw [iSup_option] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 97,
"column": 4
} | {
"line": 99,
"column": 21
} | {
"line": 101,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nL' : LieSubalgebra R L\n⊢ ∀ (x y z : ↥L'), ⟨⁅↑x, ↑⟨⁅↑y, ↑z⁆, ⋯⟩⁆, ⋯⟩ = ⟨⁅↑⟨⁅↑x, ↑y⁆, ⋯⟩, ↑z⁆, ⋯⟩ + ⟨⁅↑y, ↑⟨⁅↑x, ↑z⁆, ⋯⟩⁆, ⋯⟩",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"instIsLieTow... | [] | intros
apply SetCoe.ext
apply leibniz_lie | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 97,
"column": 4
} | {
"line": 99,
"column": 21
} | {
"line": 101,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nL' : LieSubalgebra R L\n⊢ ∀ (x y z : ↥L'), ⟨⁅↑x, ↑⟨⁅↑y, ↑z⁆, ⋯⟩⁆, ⋯⟩ = ⟨⁅↑⟨⁅↑x, ↑y⁆, ⋯⟩, ↑z⁆, ⋯⟩ + ⟨⁅↑y, ↑⟨⁅↑x, ↑z⁆, ⋯⟩⁆, ⋯⟩",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"instIsLieTow... | [] | intros
apply SetCoe.ext
apply leibniz_lie | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Artinian.Module | {
"line": 359,
"column": 15
} | {
"line": 359,
"column": 26
} | {
"line": 359,
"column": 27
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nr : R\nx : M\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → ∀ (x : M), x ∈ (r ^ n • LinearMap.id).range ↔ x ∈ (r ^ m • LinearMap.id).range\n⊢ ∃ y, r ^ n.succ • y = r ^ n • x",
"ppTerm": "?m.... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nr : R\nx : M\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → ∀ (x : M), x ∈ (r ^ n • LinearMap.id).range ↔ x ∈ (r ^ m • LinearMap.id).range\n⊢ ∃ y, r ^ (n + 1) • y = r ^ n • x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Artinian.Module | {
"line": 446,
"column": 2
} | {
"line": 446,
"column": 70
} | {
"line": 448,
"column": 0
} | [
{
"pp": "ι : Type u_2\ninst✝ : Finite ι\n⊢ ∀ {α : Type u_2} [Fintype α],\n (∀ {R : α → Type u_1} [inst : (i : α) → Semiring (R i)] [∀ (i : α), IsArtinianRing (R i)],\n IsArtinianRing ((i : α) → R i)) →\n ∀ {R : Option α → Type u_1} [inst : (i : Option α) → Semiring (R i)] [∀ (i : Option α), IsArt... | [] | · exact fun ih ↦ RingEquiv.isArtinianRing (.symm .piOptionEquivProd) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 358,
"column": 6
} | {
"line": 358,
"column": 17
} | {
"line": 358,
"column": 18
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nL₂ : Type w\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\nK K' : LieSubalgebra R L\nK₂ : LieSubalgebra R L₂\nx' : L\nhx' : x' ∈ ↑K.toSubmodule\ny' : L\nhy' : y' ∈ ↑K.toSubmodule\n⊢ ⁅↑f x', ↑f y'⁆ ... | [
"R : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nL₂ : Type w\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\nK K' : LieSubalgebra R L\nK₂ : LieSubalgebra R L₂\nx' : L\nhx' : x' ∈ ↑K.toSubmodule\ny' : L\nhy' : y' ∈ ↑K.toSubmodule\n⊢ ∃ x ∈ K, f x = ⁅f x', f y'⁆... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 394,
"column": 8
} | {
"line": 394,
"column": 56
} | {
"line": 394,
"column": 57
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).prev pq' = pq\ni₀ i₁ i₂ i₃... | [
"C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).prev pq' = pq\ni₀ i₁ i₂ i₃ i₃' : ι\nhi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Artinian.Module | {
"line": 589,
"column": 2
} | {
"line": 589,
"column": 13
} | {
"line": 589,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsArtinianRing R\n⊢ nilradical R = iInf MaximalSpectrum.asIdeal",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsArtinianRing R\n⊢ nilradical R = iInf MaximalSpectrum.asIdeal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Artinian.Module | {
"line": 592,
"column": 2
} | {
"line": 592,
"column": 42
} | {
"line": 592,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsArtinianRing R\n⊢ {I | I.IsPrime}.Finite",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"setOf",
"Set.Finite",
"id",
"Ideal",
"funext",
... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsArtinianRing R\n⊢ {I | I.IsMaximal}.Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 576,
"column": 4
} | {
"line": 576,
"column": 20
} | {
"line": 577,
"column": 2
} | [
{
"pp": "case mp\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nK K' : LieSubalgebra R L\nh : K ≤ K'\ny : ↥K\n⊢ ↑⟨↑y, ⋯⟩ ∈ K",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"LieSubalgebra.instSetLike",
"Membership.mem",
"LieSubalge... | [] | exact y.property | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Artinian.Module | {
"line": 653,
"column": 40
} | {
"line": 653,
"column": 68
} | {
"line": 653,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsArtinianRing R\nJac : Ideal R := Ring.jacobson R\nn : ℕ\nhn✝ : ∀ (m : ℕ), n ≤ m → { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } n = { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } m\nhn : Jac * Jac ^ n = Jac ^ n\nne✝ : ¬Ring.jacobson R ^ n = 0\nN : Ideal R\neq : Jac... | [] | rw [Jac.pow_zero, N.one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Artinian.Module | {
"line": 653,
"column": 40
} | {
"line": 653,
"column": 68
} | {
"line": 653,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsArtinianRing R\nJac : Ideal R := Ring.jacobson R\nn : ℕ\nhn✝ : ∀ (m : ℕ), n ≤ m → { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } n = { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } m\nhn : Jac * Jac ^ n = Jac ^ n\nne✝ : ¬Ring.jacobson R ^ n = 0\nN : Ideal R\neq : Jac... | [] | rw [Jac.pow_zero, N.one_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Artinian.Module | {
"line": 653,
"column": 40
} | {
"line": 653,
"column": 68
} | {
"line": 653,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsArtinianRing R\nJac : Ideal R := Ring.jacobson R\nn : ℕ\nhn✝ : ∀ (m : ℕ), n ≤ m → { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } n = { toFun := fun x ↦ Jac ^ x, monotone' := ⋯ } m\nhn : Jac * Jac ^ n = Jac ^ n\nne✝ : ¬Ring.jacobson R ^ n = 0\nN : Ideal R\neq : Jac... | [] | rw [Jac.pow_zero, N.one_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 624,
"column": 23
} | {
"line": 624,
"column": 34
} | {
"line": 624,
"column": 34
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nm : L\nhm : m ∈ s\n⊢ m ∈ lieSpan R L s",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LieSubalgebra.instSetLike",
"congrArg",
"Membership.mem",
... | [
"R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nm : L\nhm : m ∈ s\n⊢ ∀ (K : LieSubalgebra R L), s ⊆ ↑K → m ∈ K"
] | mem_lieSpan | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 636,
"column": 8
} | {
"line": 636,
"column": 19
} | {
"line": 636,
"column": 19
} | [
{
"pp": "case mpr\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nK : LieSubalgebra R L\nhs : s ⊆ ↑K\nm : L\nhm : m ∈ lieSpan R L s\n⊢ m ∈ K",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"LieSubalgebra.instSetLike",
"congrAr... | [
"case mpr\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nK : LieSubalgebra R L\nhs : s ⊆ ↑K\nm : L\nhm : ∀ (K : LieSubalgebra R L), s ⊆ ↑K → m ∈ K\n⊢ m ∈ K"
] | mem_lieSpan | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 719,
"column": 31
} | {
"line": 719,
"column": 42
} | {
"line": 719,
"column": 43
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nthis : ∀ (s : Set L), lieSpan R L (-s) ≤ lieSpan R L s\n⊢ lieSpan R L s ≤ lieSpan R L (-s)",
"ppTerm": "?m.44",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": [... | [
"R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nthis : ∀ (s : Set L), lieSpan R L (-s) ≤ lieSpan R L s\n⊢ lieSpan R L s ≤ lieSpan R L (-s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 248,
"column": 48
} | {
"line": 248,
"column": 59
} | {
"line": 248,
"column": 60
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nh : IsFaithful R L M\nx : L\nhx : ∀ (m : M), ⁅x, m⁆ = 0\nm : M\n⊢ ((toEnd R L M) x) m = 0 m",
"ppT... | [
"R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nh : IsFaithful R L M\nx : L\nhx : ∀ (m : M), ⁅x, m⁆ = 0\nm : M\n⊢ ⁅x, m⁆ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 249,
"column": 4
} | {
"line": 249,
"column": 15
} | {
"line": 249,
"column": 16
} | [
{
"pp": "case refine_1\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nh : IsFaithful R L M\nx : L\nhx : (toEnd R L M) x = 0\n⊢ x = 0",
"ppTerm": "?refine... | [
"case refine_1\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nh : IsFaithful R L M\nx : L\nhx : (toEnd R L M) x = 0\n⊢ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 250,
"column": 4
} | {
"line": 252,
"column": 80
} | {
"line": 254,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nh : ∀ (x : L), (∀ (m : M), ⁅x, m⁆ = 0) → x = 0\nx y : L\nhxy : (toEnd R L M) x = (toEnd... | [] | rw [← sub_eq_zero]
refine h _ fun m ↦ ?_
rw [sub_lie, sub_eq_zero, ← toEnd_apply_apply R, ← toEnd_apply_apply R, hxy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 250,
"column": 4
} | {
"line": 252,
"column": 80
} | {
"line": 254,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nh : ∀ (x : L), (∀ (m : M), ⁅x, m⁆ = 0) → x = 0\nx y : L\nhxy : (toEnd R L M) x = (toEnd... | [] | rw [← sub_eq_zero]
refine h _ fun m ↦ ?_
rw [sub_lie, sub_eq_zero, ← toEnd_apply_apply R, ← toEnd_apply_apply R, hxy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 260,
"column": 24
} | {
"line": 260,
"column": 35
} | {
"line": 260,
"column": 36
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\n⊢ Injective ⇑(toEnd R (Module.End R M) M)",
"ppTerm": "?m.33",
"assigned": true,
"usedCons... | [
"R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\n⊢ Injective id"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 515,
"column": 52
} | {
"line": 515,
"column": 86
} | {
"line": 515,
"column": 86
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr : ℤ\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).Rel pq pq'\n... | [] | by rw [h₃, data.hc₁₃ r pq pq' hpq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 516,
"column": 18
} | {
"line": 516,
"column": 48
} | {
"line": 516,
"column": 49
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr : ℤ\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).Rel pq pq'\n... | [
"C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr : ℤ\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).Rel pq pq'\ni₀ i₁ i₂ i₃ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 535,
"column": 24
} | {
"line": 535,
"column": 49
} | {
"line": 535,
"column": 50
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr : ℤ\nhr : r₀ ≤ r\npq : κ\nn : ℤ\nhn : n = data.deg pq\... | [
"C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr : ℤ\nhr : r₀ ≤ r\npq : κ\nn : ℤ\nhn : n = data.deg pq\ni₁ i₂ : ι\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 562,
"column": 21
} | {
"line": 563,
"column": 15
} | {
"line": 563,
"column": 16
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr : ℤ\nhr : r₀ ≤ r\npq pq' pq'' : κ\nhpq : (c r).Rel pq ... | [
"C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr : ℤ\nhr : r₀ ≤ r\npq pq' pq'' : κ\nhpq : (c r).Rel pq pq'\nhpq' : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 13
} | {
"line": 357,
"column": 14
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nx : L\nm : M\nhm : m ∈ ↑N\n⊢ ((toEnd R L M) x ∘ₗ (↑N).subtype) ⟨m, hm⟩ ∈ ↑N",
... | [
"R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nx : L\nm : M\nhm : m ∈ ↑N\n⊢ ⁅x, m⁆ ∈ N"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.OfAssociative | {
"line": 390,
"column": 47
} | {
"line": 390,
"column": 66
} | {
"line": 391,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nA : Type v\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nA' : Subalgebra R A\nx y : A\nhy : y ∈ (Subalgebra.toSubmodule A').carrier\nhx : x ∈ A'\n⊢ ⁅x, y⁆ ∈ A'",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Submodul... | [
"R : Type u\ninst✝² : CommRing R\nA : Type v\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nA' : Subalgebra R A\nx y : A\nhx : x ∈ A'\nhy : y ∈ A'\n⊢ ⁅x, y⁆ ∈ A'"
] | change y ∈ A' at hy | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Algebra.Lie.Ideal | {
"line": 197,
"column": 6
} | {
"line": 197,
"column": 12
} | {
"line": 197,
"column": 12
} | [
{
"pp": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\ninst✝ : LieAlgebra R L\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n⊢ map f I ≤ J ↔ I ≤ comap f J",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
... | [
"R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\ninst✝ : LieAlgebra R L\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n⊢ ⇑f '' ↑I ⊆ ↑J ↔ I ≤ comap f J"
] | map_le | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Ideal | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 56
} | {
"line": 234,
"column": 4
} | [
{
"pp": "case a\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\ninst✝ : LieAlgebra R L\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : ⇑f '' ↑I = ↑J\n⊢ map f I ≤ J",
"ppTerm": "?a✝",
"assigned": true,
"usedConst... | [
"case a\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\ninst✝ : LieAlgebra R L\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : ⇑f '' ↑I = ↑J\n⊢ ⇑↑f '' ↑(toLieSubalgebra R L I).toSubmodule ⊆ ↑J"
] | rw [map, LieSubmodule.lieSpan_le, Submodule.map_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Submodule | {
"line": 389,
"column": 36
} | {
"line": 389,
"column": 47
} | {
"line": 389,
"column": 48
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN N' : LieSubmodule R L M\nS : Set (LieSubmodule R L M)\nx : L\nm : M\nhs : ↑∅ ⊆ {x | ∃ p ∈ S, ↑p = x}\nhsm : m ∈ ⨆ i ∈ ∅, i\n⊢ m = 0",
"ppTerm": "?m... | [
"R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN N' : LieSubmodule R L M\nS : Set (LieSubmodule R L M)\nx : L\nm : M\nhs : ↑∅ ⊆ {x | ∃ p ∈ S, ↑p = x}\nhsm : m ∈ ⨆ i ∈ ∅, i\n⊢ m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Ideal | {
"line": 419,
"column": 2
} | {
"line": 419,
"column": 13
} | {
"line": 419,
"column": 14
} | [
{
"pp": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\ninst✝ : LieAlgebra R L\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\n⊢ map f I ⊔ map f f.ker = map f I",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"LieAlgeb... | [
"R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\ninst✝ : LieAlgebra R L\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\n⊢ map f f.ker ≤ map f I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Submodule | {
"line": 554,
"column": 4
} | {
"line": 554,
"column": 15
} | {
"line": 554,
"column": 16
} | [
{
"pp": "case mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN : LieSubmodule R L M\nh : ∀ (a b : ↥N), a = b\nm : M\nhm : m ∈ N\n⊢ m = 0",
"ppTerm": "?mpr",
"assigned": false,
"usedConstants":... | [
"case mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN : LieSubmodule R L M\nh : ∀ (a b : ↥N), a = b\nm : M\nhm : m ∈ N\n⊢ m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Ideal | {
"line": 482,
"column": 2
} | {
"line": 484,
"column": 43
} | {
"line": 485,
"column": 2
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI I₂ : LieIdeal R L\n⊢ comap I.incl I₂ = ⊥ ↔ Disjoint I I₂",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
"LieSubmodule.instSetLike",
"Eq.mpr",
... | [
"R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI I₂ : LieIdeal R L\n⊢ Submodule.comap (toLieSubalgebra R L I).subtype ↑I₂ = ⊥ ↔ ↑I ⊓ ↑I₂ = ⊥"
] | rw [disjoint_iff, ← LieSubmodule.toSubmodule_inj, LieIdeal.comap_toSubmodule,
LieSubmodule.bot_toSubmodule, ← LieSubmodule.toSubmodule_inj, LieSubmodule.inf_toSubmodule,
LieSubmodule.bot_toSubmodule, incl_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Submodule | {
"line": 614,
"column": 23
} | {
"line": 614,
"column": 34
} | {
"line": 614,
"column": 34
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\ns : Set M\nm : M\nhm : m ∈ s\n⊢ m ∈ lieSpan R L s",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"LieSubmodule.instSetLike",... | [
"R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\ns : Set M\nm : M\nhm : m ∈ s\n⊢ ∀ (N : LieSubmodule R L M), s ⊆ ↑N → m ∈ N"
] | mem_lieSpan | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Submodule | {
"line": 626,
"column": 23
} | {
"line": 626,
"column": 34
} | {
"line": 626,
"column": 34
} | [
{
"pp": "case mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\ns : Set M\nN : LieSubmodule R L M\nhs : s ⊆ ↑N\nm : M\nhm : m ∈ lieSpan R L s\n⊢ m ∈ N",
"ppTerm": "?mpr",
"assigned": true,
"usedC... | [
"case mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\ns : Set M\nN : LieSubmodule R L M\nhs : s ⊆ ↑N\nm : M\nhm : ∀ (N : LieSubmodule R L M), s ⊆ ↑N → m ∈ N\n⊢ m ∈ N"
] | mem_lieSpan | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 163,
"column": 29
} | {
"line": 163,
"column": 40
} | {
"line": 163,
"column": 40
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN N' : LieSubmodule R L M\ninst✝ : LieAlgebra R L\nI J : LieIdeal R L\nh₁ : I ≤ J\nh₂ : N ≤ N'\nm : M\nh : m ∈ lieSpan R L {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = x}... | [
"R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN N' : LieSubmodule R L M\ninst✝ : LieAlgebra R L\nI J : LieIdeal R L\nh₁ : I ≤ J\nh₂ : N ≤ N'\nm : M\nh : ∀ (N_1 : LieSubmodule R L M), {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = ... | mem_lieSpan | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 163,
"column": 75
} | {
"line": 163,
"column": 86
} | {
"line": 163,
"column": 86
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN N' : LieSubmodule R L M\ninst✝ : LieAlgebra R L\nI J : LieIdeal R L\nh₁ : I ≤ J\nh₂ : N ≤ N'\nm : M\nh : ∀ (N_1 : LieSubmodule R L M), {x | ∃ x_1 n, ⁅... | [
"R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN N' : LieSubmodule R L M\ninst✝ : LieAlgebra R L\nI J : LieIdeal R L\nh₁ : I ≤ J\nh₂ : N ≤ N'\nm : M\nh : ∀ (N_1 : LieSubmodule R L M), {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = ... | mem_lieSpan | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Submodule | {
"line": 705,
"column": 52
} | {
"line": 705,
"column": 63
} | {
"line": 705,
"column": 64
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN : LieSubmodule R L M\nm : M\nhm : m ∈ ↑N\nN' : Submodule R M\nhN' : ∀ p ∈ {x | ∃ s, (∃ x ∈ N, lieSpan R L {x} = s) ∧ ↑s = x}, p ≤ N'\n⊢ ∀ m ∈ N, ↑(lieS... | [
"R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN : LieSubmodule R L M\nm : M\nhm : m ∈ ↑N\nN' : Submodule R M\nhN' : ∀ p ∈ {x | ∃ s, (∃ x ∈ N, lieSpan R L {x} = s) ∧ ↑s = x}, p ≤ N'\n⊢ ∀ m ∈ N, ↑(lieSpan R L {m})... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 13
} | {
"line": 104,
"column": 14
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nh : minpoly A x = 1\n⊢ 1 = 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"NeZero.one",
"AddGroupWithOne.toAddMonoidWi... | [
"A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nh : minpoly A x = 1\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Abelian | {
"line": 99,
"column": 29
} | {
"line": 99,
"column": 66
} | {
"line": 99,
"column": 67
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nh : IsLieAbelian ↥(lieSpan R L s)\nx : L\nhx : x ∈ s\ny : L\nhy : y ∈ s\nx' : ↥(lieSpan R L s) := ⟨x, ⋯⟩\ny' : ↥(lieSpan R L s) := ⟨y, ⋯⟩\nthis : ⁅x', y'⁆ = 0\n⊢ ⁅x, y⁆ = 0",
"ppTerm": "?m.107",
... | [
"R : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nh : IsLieAbelian ↥(lieSpan R L s)\nx : L\nhx : x ∈ s\ny : L\nhy : y ∈ s\nx' : ↥(lieSpan R L s) := ⟨x, ⋯⟩\ny' : ↥(lieSpan R L s) := ⟨y, ⋯⟩\nthis : ⁅x', y'⁆ = 0\n⊢ ⁅x, y⁆ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Abelian | {
"line": 104,
"column": 20
} | {
"line": 104,
"column": 49
} | {
"line": 104,
"column": 50
} | [
{
"pp": "case refine_2.mem.mem\nR : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nh : ∀ x ∈ s, ∀ y ∈ s, ⁅x, y⁆ = 0\nx✝¹ x✝ : ↥(lieSpan R L s)\nx y w : L\nhw : w ∈ s\nu : L\nhu : u ∈ s\n⊢ ⁅⟨w, ⋯⟩, ⟨u, ⋯⟩⁆ = 0",
"ppTerm": "?refine_2.mem.mem",
"assigned... | [
"case refine_2.mem.mem\nR : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nh : ∀ x ∈ s, ∀ y ∈ s, ⁅x, y⁆ = 0\nx✝¹ x✝ : ↥(lieSpan R L s)\nx y w : L\nhw : w ∈ s\nu : L\nhu : u ∈ s\n⊢ ⁅w, u⁆ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Abelian | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 15
} | {
"line": 154,
"column": 16
} | [
{
"pp": "case refine_2\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\nh : I ≤ LieModule.ker R L ↥I\nx✝¹ x✝ : ↥I\nx : L\nhx : x ∈ I\ny : L\nhy : y ∈ I\n⊢ ⁅⟨x, hx⟩, ⟨y, hy⟩⁆ = 0",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
... | [
"case refine_2\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\nh : I ≤ LieModule.ker R L ↥I\nx✝¹ x✝ : ↥I\nx : L\nhx : x ∈ I\ny : L\nhy : y ∈ I\n⊢ ⁅x, ⟨y, hy⟩⁆ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 175,
"column": 6
} | {
"line": 175,
"column": 17
} | {
"line": 175,
"column": 18
} | [
{
"pp": "case inl\nA : Type u_1\nB : Type u_2\ninst✝² : CommRing A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nmonic : p.Monic\nhp0 : (Polynomial.aeval x) p = 0\nn : ℕ\nhpn : p.degree = ↑n\nind : LinearIndependent A fun i ↦ x ^ ↑i\nq : A[X]\nlt✝ : q.degree < p.degree\nne : ¬q = 0\nhq : (Polynomial.a... | [
"case inl\nA : Type u_1\nB : Type u_2\ninst✝² : CommRing A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nmonic : p.Monic\nhp0 : (Polynomial.aeval x) p = 0\nn : ℕ\nhpn : p.degree = ↑n\nind : LinearIndependent A fun i ↦ x ^ ↑i\nq : A[X]\nlt✝ : q.degree < p.degree\nne : ¬q = 0\nhq : (Polynomial.aeval x) (∑ i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 176,
"column": 12
} | {
"line": 176,
"column": 60
} | {
"line": 176,
"column": 61
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝² : CommRing A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nmonic : p.Monic\nhp0 : (Polynomial.aeval x) p = 0\nn : ℕ\nhpn : p.degree = ↑n\nind : LinearIndependent A fun i ↦ x ^ ↑i\nq : A[X]\nlt✝ : q.degree < p.degree\nne : ¬q = 0\nhq : (Polynomial.aeval x) (∑... | [
"A : Type u_1\nB : Type u_2\ninst✝² : CommRing A\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\np : A[X]\nmonic : p.Monic\nhp0 : (Polynomial.aeval x) p = 0\nn : ℕ\nhpn : p.degree = ↑n\nind : LinearIndependent A fun i ↦ x ^ ↑i\nq : A[X]\nlt✝ : q.degree < p.degree\nne : ¬q = 0\nhq : (Polynomial.aeval x) (∑ i ∈ Finset.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Abelian | {
"line": 188,
"column": 2
} | {
"line": 193,
"column": 52
} | {
"line": 195,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\n⊢ N ≤ maxTrivSubmodule R L M ↔ ⁅⊤, N⁆ = ⊥",
"ppTerm": "?m.48",
"assign... | [] | refine ⟨fun h => ?_, fun h m hm => ?_⟩
· rw [← le_bot_iff, ← ideal_oper_maxTrivSubmodule_eq_bot R L M ⊤]
exact LieSubmodule.mono_lie_right ⊤ h
· rw [mem_maxTrivSubmodule]
rw [LieSubmodule.lie_eq_bot_iff] at h
exact fun x => h x (LieSubmodule.mem_top x) m hm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Abelian | {
"line": 188,
"column": 2
} | {
"line": 193,
"column": 52
} | {
"line": 195,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\n⊢ N ≤ maxTrivSubmodule R L M ↔ ⁅⊤, N⁆ = ⊥",
"ppTerm": "?m.48",
"assign... | [] | refine ⟨fun h => ?_, fun h m hm => ?_⟩
· rw [← le_bot_iff, ← ideal_oper_maxTrivSubmodule_eq_bot R L M ⊤]
exact LieSubmodule.mono_lie_right ⊤ h
· rw [mem_maxTrivSubmodule]
rw [LieSubmodule.lie_eq_bot_iff] at h
exact fun x => h x (LieSubmodule.mem_top x) m hm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Abelian | {
"line": 252,
"column": 12
} | {
"line": 252,
"column": 29
} | {
"line": 252,
"column": 30
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieMo... | [
"R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\n... | LieHom.lie_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 77
} | {
"line": 164,
"column": 0
} | [
{
"pp": "case hp3\nA : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\np : A[X]\nhp1 : Irreducible p\nhp2 : (Polynomial.aeval x) p = 0\nthis : p.leadingCoeff ≠ 0\n⊢ (p * C p.leadingCoeff⁻¹).Monic",
"ppTerm": "?hp3",
"assigned": true,
"use... | [] | rwa [Polynomial.Monic, leadingCoeff_mul, leadingCoeff_C, mul_inv_cancel₀] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 77
} | {
"line": 164,
"column": 0
} | [
{
"pp": "case hp3\nA : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\np : A[X]\nhp1 : Irreducible p\nhp2 : (Polynomial.aeval x) p = 0\nthis : p.leadingCoeff ≠ 0\n⊢ (p * C p.leadingCoeff⁻¹).Monic",
"ppTerm": "?hp3",
"assigned": true,
"use... | [] | rwa [Polynomial.Monic, leadingCoeff_mul, leadingCoeff_C, mul_inv_cancel₀] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 77
} | {
"line": 164,
"column": 0
} | [
{
"pp": "case hp3\nA : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\np : A[X]\nhp1 : Irreducible p\nhp2 : (Polynomial.aeval x) p = 0\nthis : p.leadingCoeff ≠ 0\n⊢ (p * C p.leadingCoeff⁻¹).Monic",
"ppTerm": "?hp3",
"assigned": true,
"use... | [] | rwa [Polynomial.Monic, leadingCoeff_mul, leadingCoeff_C, mul_inv_cancel₀] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.SModEq.Basic | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 51
} | {
"line": 184,
"column": 52
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Submodule R M\nx y : M\nh : ↑U ⊆ (y - x) +ᵥ ↑U\n⊢ x - y ∈ U",
"ppTerm": "?refine_1",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_1\nR : Type u_1\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Submodule R M\nx y : M\nh : ↑U ⊆ (y - x) +ᵥ ↑U\n⊢ x - y ∈ U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SModEq.Basic | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 51
} | {
"line": 187,
"column": 52
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Submodule R M\nx y : M\nh : x - y ∈ U\nz : M\nhz : z ∈ ↑U\n⊢ z ∈ (y - x) +ᵥ ↑U",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
... | [
"case refine_2\nR : Type u_1\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Submodule R M\nx y : M\nh : x - y ∈ U\nz : M\nhz : z ∈ ↑U\n⊢ x - y + z ∈ U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.SModEq.Basic | {
"line": 185,
"column": 4
} | {
"line": 187,
"column": 66
} | {
"line": 189,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Submodule R M\nx y : M\nh : x - y ∈ U\n⊢ x +ᵥ ↑U ⊆ y +ᵥ ↑U",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"... | [] | rw [Set.vadd_set_subset_iff_subset_neg_vadd_set, vadd_vadd, neg_add_eq_sub]
intro z hz
simpa [Set.mem_vadd_set_iff_neg_vadd_mem] using U.add_mem h hz | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SModEq.Basic | {
"line": 185,
"column": 4
} | {
"line": 187,
"column": 66
} | {
"line": 189,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Submodule R M\nx y : M\nh : x - y ∈ U\n⊢ x +ᵥ ↑U ⊆ y +ᵥ ↑U",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"... | [] | rw [Set.vadd_set_subset_iff_subset_neg_vadd_set, vadd_vadd, neg_add_eq_sub]
intro z hz
simpa [Set.mem_vadd_set_iff_neg_vadd_mem] using U.add_mem h hz | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 174,
"column": 63
} | {
"line": 174,
"column": 87
} | {
"line": 174,
"column": 88
} | [
{
"pp": "A : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nx : B\na : A\nhx : IsIntegral A x\nq : A[X]\nqmo : q.Monic\nhq : (Polynomial.aeval (x + (algebraMap A B) a)) q = 0\n⊢ (Polynomial.aeval x) (q.comp (X + C a)) = 0",
"ppTerm": "?m.97",
"assigned": true,
"u... | [
"A : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nx : B\na : A\nhx : IsIntegral A x\nq : A[X]\nqmo : q.Monic\nhq : (Polynomial.aeval (x + (algebraMap A B) a)) q = 0\n⊢ (Polynomial.aeval (x + (algebraMap A B) a)) q = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 43
} | {
"line": 184,
"column": 44
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nx : B\na : A\nhx : ¬IsIntegral A x\nh : IsIntegral A (x + (algebraMap A B) a)\n⊢ IsIntegral A x",
"ppTerm": "?neg✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals"... | [
"case neg\nA : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nx : B\na : A\nhx : ¬IsIntegral A x\nh : IsIntegral A (x + (algebraMap A B) a)\n⊢ IsIntegral A x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 30
} | {
"line": 188,
"column": 31
} | [
{
"pp": "A : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nx : B\na : A\n⊢ minpoly A (x - (algebraMap A B) a) = (minpoly A x).comp (X + C a)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Algebra.algebraMap... | [
"A : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nx : B\na : A\n⊢ minpoly A (x + -(algebraMap A B) a) = (minpoly A x).comp (X + C a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 197,
"column": 8
} | {
"line": 197,
"column": 32
} | {
"line": 197,
"column": 33
} | [
{
"pp": "A : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\nq : A[X]\nqmo : q.Monic\nhq : (Polynomial.aeval (-x)) q = 0\n⊢ (Polynomial.aeval x) ((-1) ^ q.natDegree * q.comp (-X)) = 0",
"ppTerm": "?m.113",
"assigned": true,
"usedConstants":... | [
"A : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\nq : A[X]\nqmo : q.Monic\nhq : (Polynomial.aeval (-x)) q = 0\n⊢ (Polynomial.aeval (-x)) q = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 15
} | {
"line": 212,
"column": 16
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝⁹ : Field A\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : Ring S\ninst✝⁵ : Ring T\ninst✝⁴ : IsDomain S\ninst✝³ : IsDomain T\ninst✝² : Algebra R S\ninst✝¹ : Algebra A T\ninst✝ : Algebra.IsIntegral R S\nf : R ≃+* A\ng : S ≃... | [
"case refine_2\nA : Type u_1\ninst✝⁹ : Field A\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : Ring S\ninst✝⁵ : Ring T\ninst✝⁴ : IsDomain S\ninst✝³ : IsDomain T\ninst✝² : Algebra R S\ninst✝¹ : Algebra A T\ninst✝ : Algebra.IsIntegral R S\nf : R ≃+* A\ng : S ≃+* T\nhcomp ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 67
} | {
"line": 213,
"column": 2
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝⁹ : Field A\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : Ring S\ninst✝⁵ : Ring T\ninst✝⁴ : IsDomain S\ninst✝³ : IsDomain T\ninst✝² : Algebra R S\ninst✝¹ : Algebra A T\ninst✝ : Algebra.IsIntegral R S\nf : R ≃+* A\ng : S ≃... | [] | simpa using (map_aeval_eq_aeval_map hcomp (minpoly R x) x).symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 67
} | {
"line": 213,
"column": 2
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝⁹ : Field A\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : Ring S\ninst✝⁵ : Ring T\ninst✝⁴ : IsDomain S\ninst✝³ : IsDomain T\ninst✝² : Algebra R S\ninst✝¹ : Algebra A T\ninst✝ : Algebra.IsIntegral R S\nf : R ≃+* A\ng : S ≃... | [] | simpa using (map_aeval_eq_aeval_map hcomp (minpoly R x) x).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 67
} | {
"line": 213,
"column": 2
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝⁹ : Field A\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : Ring S\ninst✝⁵ : Ring T\ninst✝⁴ : IsDomain S\ninst✝³ : IsDomain T\ninst✝² : Algebra R S\ninst✝¹ : Algebra A T\ninst✝ : Algebra.IsIntegral R S\nf : R ≃+* A\ng : S ≃... | [] | simpa using (map_aeval_eq_aeval_map hcomp (minpoly R x) x).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 70
} | {
"line": 268,
"column": 71
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\ninst✝ : Nontrivial B\n⊢ minpoly A 0 = X",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\ninst✝ : Nontrivial B\n⊢ minpoly A 0 = X"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 49
} | {
"line": 273,
"column": 50
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\ninst✝ : Nontrivial B\n⊢ minpoly A 1 = X - 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.instOne",
"AddGroupWithOne.toAddGroup",
"congrArg",
... | [
"A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\ninst✝ : Nontrivial B\n⊢ minpoly A 1 = X + -1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 43
} | {
"line": 314,
"column": 44
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\nh : (minpoly A x).coeff 0 = 0\n⊢ x = 0",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\nh : (minpoly A x).coeff 0 = 0\n⊢ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 335,
"column": 4
} | {
"line": 335,
"column": 15
} | {
"line": 335,
"column": 16
} | [
{
"pp": "case refine_2\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : CommRing L\ninst✝¹ : IsDomain L\ninst✝ : Algebra K L\nσ : L ≃ₐ[K] L\nhσ : IsOfFinOrder σ\nq : K[X]\nhq : q.Monic\nH : q.natDegree < orderOf σ\nhs : ∑ x, q.coeff ↑x • (σ ^ ↑x).toLinearMap = 0\n⊢ q = 0",
"ppTerm": "?refine_2",
... | [
"case refine_2\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : CommRing L\ninst✝¹ : IsDomain L\ninst✝ : Algebra K L\nσ : L ≃ₐ[K] L\nhσ : IsOfFinOrder σ\nq : K[X]\nhq : q.Monic\nH : q.natDegree < orderOf σ\nhs : ∑ x, q.coeff ↑x • (σ ^ ↑x).toLinearMap = 0\n⊢ q = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Ideal | {
"line": 75,
"column": 31
} | {
"line": 75,
"column": 42
} | {
"line": 75,
"column": 43
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\ny a✝ b✝ : ↥R[x]\nha✝ : a✝ ∈ Submodule.span (↥R[x]) (⇑(algebraMap R ↥R[x]) '' ↑I)\nhb✝ : b✝ ∈ Submodule.span (↥R[x]) (⇑(algebraMap R ↥R[x]) '' ↑I)\na : R[X]\nha : ∀ (i : ℕ), a.coeff i ∈ I\n... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\ny a✝ b✝ : ↥R[x]\nha✝ : a✝ ∈ Submodule.span (↥R[x]) (⇑(algebraMap R ↥R[x]) '' ↑I)\nhb✝ : b✝ ∈ Submodule.span (↥R[x]) (⇑(algebraMap R ↥R[x]) '' ↑I)\na : R[X]\nha : ∀ (i : ℕ), a.coeff i ∈ I\nha' : (aeval... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerBasis | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 40
} | {
"line": 116,
"column": 41
} | [
{
"pp": "case neg\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nx y : S\nd : ℕ\nhd : d ≠ 0\nf : R[X]\nh : f.degree < ↑d\nhy : y = (aeval x) f\nhf : ¬f = 0\n⊢ f.natDegree < d",
"ppTerm": "?neg✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"case neg\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nx y : S\nd : ℕ\nhd : d ≠ 0\nf : R[X]\nh : f.degree < ↑d\nhy : y = (aeval x) f\nhf : ¬f = 0\n⊢ f.natDegree < d"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerBasis | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 40
} | {
"line": 116,
"column": 41
} | [
{
"pp": "case neg\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nx y : S\nd : ℕ\nhd : d ≠ 0\nf : R[X]\nh : f.natDegree < d\nhy : y = (aeval x) f\nhf : ¬f = 0\n⊢ f.degree < ↑d",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWi... | [
"case neg\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nx y : S\nd : ℕ\nhd : d ≠ 0\nf : R[X]\nh : f.natDegree < d\nhy : y = (aeval x) f\nhf : ¬f = 0\n⊢ f.natDegree < d"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerBasis | {
"line": 126,
"column": 39
} | {
"line": 126,
"column": 50
} | {
"line": 126,
"column": 51
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring S\ninst✝¹ : Algebra R S\ninst✝ : Nontrivial S\npb : PowerBasis R S\ny : S\n⊢ y ∈ Submodule.span R (Set.range fun i ↦ pb.gen ^ ↑i)",
"ppTerm": "?m.47",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring S\ninst✝¹ : Algebra R S\ninst✝ : Nontrivial S\npb : PowerBasis R S\ny : S\n⊢ y ∈ Submodule.span R (Set.range fun i ↦ pb.gen ^ ↑i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Ideal | {
"line": 102,
"column": 47
} | {
"line": 102,
"column": 58
} | {
"line": 102,
"column": 59
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nx : S\nI : Ideal R\nhI : I ≠ ⊤\ninst✝ : Invertible x\nh : ∃ i ∈ Ideal.map (algebraMap R ↥R[x]) I, ∃ j ∈ Ideal.span {⟨x, ⋯⟩}, i + j = 1\ny : ↥R[x]\nhy : y ∈ Ideal.map (algebraMap R ↥R[x]) I\nz : ↥R[x]\nhz : z ∈ I... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nx : S\nI : Ideal R\nhI : I ≠ ⊤\ninst✝ : Invertible x\nh : ∃ i ∈ Ideal.map (algebraMap R ↥R[x]) I, ∃ j ∈ Ideal.span {⟨x, ⋯⟩}, i + j = 1\ny : ↥R[x]\nhy : y ∈ Ideal.map (algebraMap R ↥R[x]) I\nz : ↥R[x]\nhz : z ∈ Ideal.span {⟨... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 41
} | {
"line": 94,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R[X]\nha : a ∈ map C I\n⊢ (a.sum fun n a ↦ eval₂ (C.comp (Quotient.mk I)) X ((monomial n) a)) = 0",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Ideal.Quotient.commSemiring",
"Polynomial.C",
"Semiring.toModule"... | [
"R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R[X]\nha : a ∈ map C I\nn : ℕ\nx✝ : n ∈ a.support\n⊢ (fun n a ↦ eval₂ (C.comp (Quotient.mk I)) X ((monomial n) a)) n (a.coeff n) = 0"
] | refine Finset.sum_eq_zero fun n _ => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 19
} | {
"line": 99,
"column": 20
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R[X]\nha : a ∈ map C I\nn : ℕ\nx✝ : n ∈ a.support\nm : ℕ\nh : m = 0\n⊢ (if m = 0 then (Quotient.mk I) (a.coeff n) else 0) = coeff 0 m",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"case pos\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R[X]\nha : a ∈ map C I\nn : ℕ\nx✝ : n ∈ a.support\nm : ℕ\nh : m = 0\n⊢ (Quotient.mk I) (a.coeff n) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 41
} | {
"line": 210,
"column": 2
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nI : Ideal R\na : MvPolynomial σ R\nha : a ∈ Ideal.map C I\n⊢ ∑ x ∈ a.support, eval₂ (C.comp (Ideal.Quotient.mk I)) X ((monomial x) (coeff x a)) = 0",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
... | [
"R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nI : Ideal R\na : MvPolynomial σ R\nha : a ∈ Ideal.map C I\nn : σ →₀ ℕ\nx✝ : n ∈ a.support\n⊢ eval₂ (C.comp (Ideal.Quotient.mk I)) X ((monomial n) (coeff n a)) = 0"
] | refine Finset.sum_eq_zero fun n _ => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 67
} | {
"line": 211,
"column": 2
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nI : Ideal R\na : MvPolynomial σ R\nha : a ∈ Ideal.map C I\nn : σ →₀ ℕ\nx✝ : n ∈ a.support\n⊢ eval₂ (C.comp (Ideal.Quotient.mk I)) X ((monomial n) (coeff n a)) = 0",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Finsupp.instAdd... | [
"R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nI : Ideal R\na : MvPolynomial σ R\nha : a ∈ Ideal.map C I\nn : σ →₀ ℕ\nx✝ : n ∈ a.support\n⊢ (C ((Ideal.Quotient.mk I) (coeff n a)) * n.prod fun n e ↦ X n ^ e) = 0"
] | simp only [eval₂_monomial, Function.comp_apply, RingHom.coe_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 213,
"column": 8
} | {
"line": 213,
"column": 30
} | {
"line": 213,
"column": 31
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nI : Ideal R\na : MvPolynomial σ R\nha : a ∈ Ideal.map C I\nn : σ →₀ ℕ\nx✝ : n ∈ a.support\nthis : coeff n a ∈ I\n⊢ C ((Ideal.Quotient.mk I) (coeff n a)) = 0",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"RingHom.instRingHomCl... | [
"R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nI : Ideal R\na : MvPolynomial σ R\nha : a ∈ Ideal.map C I\nn : σ →₀ ℕ\nx✝ : n ∈ a.support\nthis : coeff n a ∈ RingHom.ker (Ideal.Quotient.mk I)\n⊢ C ((Ideal.Quotient.mk I) (coeff n a)) = 0"
] | ← @Ideal.mk_ker R _ I, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.KummerPolynomial | {
"line": 36,
"column": 17
} | {
"line": 36,
"column": 60
} | {
"line": 38,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nn : ℕ\nhn : 1 < n\na : K\n⊢ X.natDegree < (X ^ n - C a).natDegree",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.natDegree_X",
"Polynomial.natDegree_X_pow_sub_C",
"congrArg",
"C... | [] | by rwa [natDegree_X_pow_sub_C, natDegree_X] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.KummerPolynomial | {
"line": 80,
"column": 16
} | {
"line": 80,
"column": 70
} | {
"line": 80,
"column": 71
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nn : ℕ\na : K\nH : Irreducible (X ^ n - C a)\nm : ℕ\nhm : m ∣ n\nhm' : m ≠ 1\nb : K\ne : n = 0\n⊢ Irreducible (C (1 - a))",
"ppTerm": "?m.41",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u\ninst✝ : Field K\nn : ℕ\na : K\nH : Irreducible (X ^ n - C a)\nm : ℕ\nhm : m ∣ n\nhm' : m ≠ 1\nb : K\ne : n = 0\n⊢ Irreducible (C (1 - a))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KummerPolynomial | {
"line": 86,
"column": 4
} | {
"line": 87,
"column": 59
} | {
"line": 87,
"column": 60
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nm : ℕ\nhm' : m ≠ 1\nb : K\nk : ℕ\nhn : m * k ≠ 0\nq : K[X]\nH : Irreducible ((X ^ k - C b) * q)\nhq : (X ^ k) ^ m - C b ^ m = (X ^ k - C b) * q\n⊢ q.degree = 0",
"ppTerm": "?m.154",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []... | [
"K : Type u\ninst✝ : Field K\nm : ℕ\nhm' : m ≠ 1\nb : K\nk : ℕ\nhn : m * k ≠ 0\nq : K[X]\nH : Irreducible ((X ^ k - C b) * q)\nhq : (X ^ k) ^ m - C b ^ m = (X ^ k - C b) * q\n⊢ q.degree = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Squarefree.Basic | {
"line": 140,
"column": 21
} | {
"line": 140,
"column": 32
} | {
"line": 140,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\ninst✝ : WfDvdMonoid R\nh : ∀ (x : R), Irreducible x → ¬x * x ∣ 0\n⊢ 0 = 0 ∧ ∀ (x : R), ¬Irreducible x",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Irreducible",
"id",
"CommMonoidWi... | [
"R : Type u_1\ninst✝¹ : CommMonoidWithZero R\ninst✝ : WfDvdMonoid R\nh : ∀ (x : R), Irreducible x → ¬x * x ∣ 0\n⊢ ∀ (x : R), ¬Irreducible x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Squarefree.Basic | {
"line": 143,
"column": 6
} | {
"line": 143,
"column": 17
} | {
"line": 143,
"column": 18
} | [
{
"pp": "case refine_2.inl\nR : Type u_1\ninst✝¹ : CommMonoidWithZero R\ninst✝ : WfDvdMonoid R\nh : ∀ (x : R), ¬Irreducible x\n⊢ ∀ (x : R), Irreducible x → ¬x * x ∣ 0",
"ppTerm": "?refine_2.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Dvd.dvd",
"HMul.hMul",
... | [
"case refine_2.inl\nR : Type u_1\ninst✝¹ : CommMonoidWithZero R\ninst✝ : WfDvdMonoid R\nh : ∀ (x : R), ¬Irreducible x\n⊢ ∀ (x : R), ¬Irreducible x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Squarefree.Basic | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 18
} | {
"line": 149,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\ninst✝ : WfDvdMonoid R\nr : R\nhr : r ≠ 0\n⊢ Squarefree r ↔ ∀ (x : R), Irreducible x → ¬x * x ∣ r",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : CommMonoidWithZero R\ninst✝ : WfDvdMonoid R\nr : R\nhr : r ≠ 0\n⊢ Squarefree r ↔ ∀ (x : R), Irreducible x → ¬x * x ∣ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Squarefree.Basic | {
"line": 285,
"column": 4
} | {
"line": 285,
"column": 12
} | {
"line": 286,
"column": 4
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝² : CommMonoidWithZero R\ninst✝¹ : UniqueFactorizationMonoid R\ninst✝ : NormalizationMonoid R\nx : R\nx0 : x ≠ 0\nthis : Nontrivial R\nh : ∀ (a : R), Multiset.count a (normalizedFactors x) ≤ 1\na : R\n⊢ ¬IsUnit a → emultiplicity a x ≤ 1",
"ppTerm": "?mpr",
"assigned... | [
"case mpr\nR : Type u_1\ninst✝² : CommMonoidWithZero R\ninst✝¹ : UniqueFactorizationMonoid R\ninst✝ : NormalizationMonoid R\nx : R\nx0 : x ≠ 0\nthis : Nontrivial R\nh : ∀ (a : R), Multiset.count a (normalizedFactors x) ≤ 1\na : R\nhu : ¬IsUnit a\n⊢ emultiplicity a x ≤ 1"
] | intro hu | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Algebra.Squarefree.Basic | {
"line": 285,
"column": 4
} | {
"line": 285,
"column": 12
} | {
"line": 286,
"column": 4
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝² : CommMonoidWithZero R\ninst✝¹ : UniqueFactorizationMonoid R\ninst✝ : NormalizationMonoid R\nx : R\nx0 : x ≠ 0\nthis : Nontrivial R\nh : ∀ (a : R), Multiset.count a (normalizedFactors x) ≤ 1\na : R\n⊢ ¬IsUnit a → emultiplicity a x ≤ 1",
"ppTerm": "?mpr",
"assigned... | [
"case mpr\nR : Type u_1\ninst✝² : CommMonoidWithZero R\ninst✝¹ : UniqueFactorizationMonoid R\ninst✝ : NormalizationMonoid R\nx : R\nx0 : x ≠ 0\nthis : Nontrivial R\nh : ∀ (a : R), Multiset.count a (normalizedFactors x) ≤ 1\na : R\nhu : ¬IsUnit a\n⊢ emultiplicity a x ≤ 1"
] | intro hu | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.FieldTheory.Separable | {
"line": 171,
"column": 2
} | {
"line": 175,
"column": 33
} | {
"line": 177,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\np q : R[X]\nhq : ¬IsUnit q\nhsep : p.Separable\n⊢ emultiplicity q p ≤ 1",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Mathlib.Tactic.Contrapose.contrapose₂",
"Preorder.toLT",
"Dvd.dvd",
"instAddMonoidWi... | [] | contrapose! hq
apply isUnit_of_self_mul_dvd_separable hsep
rw [← sq]
apply pow_dvd_of_le_emultiplicity
exact Order.add_one_le_of_lt hq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Separable | {
"line": 171,
"column": 2
} | {
"line": 175,
"column": 33
} | {
"line": 177,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\np q : R[X]\nhq : ¬IsUnit q\nhsep : p.Separable\n⊢ emultiplicity q p ≤ 1",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Mathlib.Tactic.Contrapose.contrapose₂",
"Preorder.toLT",
"Dvd.dvd",
"instAddMonoidWi... | [] | contrapose! hq
apply isUnit_of_self_mul_dvd_separable hsep
rw [← sq]
apply pow_dvd_of_le_emultiplicity
exact Order.add_one_le_of_lt hq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Separable | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 42
} | {
"line": 193,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nx : R\n⊢ (X - C x).Separable",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"AddGroupWithOne.toAddGroup",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"sub_eq... | [
"R : Type u\ninst✝ : CommRing R\nx : R\n⊢ (X + -C x).Separable"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Separable | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 58
} | {
"line": 235,
"column": 59
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\na : R\nt : Multiset R\nhs : (Multiset.map (fun a ↦ X - C a) (a ::ₘ a ::ₘ t)).prod.Separable\n⊢ (X - C a) * (X - C a) ∣ (Multiset.map (fun a ↦ X - C a) (a ::ₘ a ::ₘ t)).prod",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
... | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\na : R\nt : Multiset R\nhs : (Multiset.map (fun a ↦ X - C a) (a ::ₘ a ::ₘ t)).prod.Separable\n⊢ (X - C a) * (X - C a) ∣ (X - C a) * ((X - C a) * (Multiset.map (fun a ↦ X - C a) t).prod)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Separable | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 13
} | {
"line": 265,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nn : ℕ\na b c : R\nhn : ↑n = 0\nhb✝ : IsUnit b\nf : R[X] := C a * X ^ n + C b * X + C c\ne : R\nhb : e * b = 1\nhderiv : derivative f = C b\n⊢ -derivative f * f + (f + C e) * derivative f = 1",
"ppTerm": "?m.131",
"assigned": true,
"usedConstants": [
"Po... | [
"R : Type u\ninst✝ : CommRing R\nn : ℕ\na b c : R\nhn : ↑n = 0\nhb✝ : IsUnit b\nf : R[X] := C a * X ^ n + C b * X + C c\ne : R\nhb : e * b = 1\nhderiv : derivative f = C b\n⊢ -C b * f + (f + C e) * C b = 1"
] | hderiv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Separable | {
"line": 383,
"column": 15
} | {
"line": 383,
"column": 29
} | {
"line": 383,
"column": 30
} | [
{
"pp": "case h.inr.succ\nF : Type u\ninst✝ : Field F\np : ℕ\nHF : CharP F p\nhp : Nat.Prime p\nf : F[X]\nhf : Irreducible f\nh1 : ¬f.Separable\nN : ℕ\nih : ∀ m < N + 1, ∀ {f : F[X]}, Irreducible f → f.natDegree = m → ∃ n g, g.Separable ∧ (expand F (p ^ n)) g = f\nhn : f.natDegree = N + 1\nn : ℕ\ng : F[X]\nhg4 ... | [
"case h.inr.succ\nF : Type u\ninst✝ : Field F\np : ℕ\nHF : CharP F p\nhp : Nat.Prime p\nf : F[X]\nhf : Irreducible f\nh1 : ¬f.Separable\nN : ℕ\nih : ∀ m < N + 1, ∀ {f : F[X]}, Irreducible f → f.natDegree = m → ∃ n g, g.Separable ∧ (expand F (p ^ n)) g = f\nhn : f.natDegree = N + 1\nn : ℕ\ng : F[X]\nhg4 : g.Separabl... | expand_expand, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Separable | {
"line": 402,
"column": 4
} | {
"line": 402,
"column": 30
} | {
"line": 402,
"column": 31
} | [
{
"pp": "case inr\nF : Type u\ninst✝ : Field F\np : ℕ\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn₁ n₂ : ℕ\nthis :\n ∀ {F : Type u} [inst : Field F] (p : ℕ) [HF : CharP F p] {f : F[X]},\n Irreducible f →\n 0 < p →\n ∀ (n₁ n₂ : ℕ),\n n₁ ≤ n₂ →\n ∀ (g₁ : F[X]),\... | [
"case inr\nF : Type u\ninst✝ : Field F\np : ℕ\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn₁ n₂ : ℕ\nthis :\n ∀ {F : Type u} [inst : Field F] (p : ℕ) [HF : CharP F p] {f : F[X]},\n Irreducible f →\n 0 < p →\n ∀ (n₁ n₂ : ℕ),\n n₁ ≤ n₂ →\n ∀ (g₁ : F[X]),\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AnnihilatingPolynomial | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 57
} | {
"line": 149,
"column": 58
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : annIdealGenerator 𝕜 a = 0\np : 𝕜[X]\np_monic : p.Monic\nhp : (aeval a) p = 0\n⊢ p ∈ ⊥",
"ppTerm": "?pos✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoal... | [
"case pos\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : annIdealGenerator 𝕜 a = 0\np : 𝕜[X]\np_monic : p.Monic\nhp : (aeval a) p = 0\n⊢ p ∈ ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Separable | {
"line": 436,
"column": 4
} | {
"line": 436,
"column": 76
} | {
"line": 436,
"column": 77
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nn : ℕ\nx : F\nhn : 0 < n\nhx : x ≠ 0\nh : (X ^ n - C x).Separable\nhn' : ↑n = 0\n⊢ IsUnit (X ^ n - C x)",
"ppTerm": "?m.57",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"F : Type u\ninst✝ : Field F\nn : ℕ\nx : F\nhn : 0 < n\nhx : x ≠ 0\nh : (X ^ n - C x).Separable\nhn' : ↑n = 0\n⊢ IsUnit (X ^ n - C x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Separable | {
"line": 451,
"column": 45
} | {
"line": 451,
"column": 61
} | {
"line": 451,
"column": 61
} | [
{
"pp": "F : Type u\ninst✝² : Field F\nK : Type v\ninst✝¹ : Field K\ninst✝ : Algebra F K\np : F[X]\nhsep : p.Separable\nhsplit : (map (algebraMap F K) p).Splits\n⊢ Fintype.card ↥(p.aroots K).toFinset = p.natDegree",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
... | [
"F : Type u\ninst✝² : Field F\nK : Type v\ninst✝¹ : Field K\ninst✝ : Algebra F K\np : F[X]\nhsep : p.Separable\nhsplit : (map (algebraMap F K) p).Splits\n⊢ #(p.aroots K).toFinset = p.natDegree"
] | Fintype.card_coe | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.Separable | {
"line": 490,
"column": 36
} | {
"line": 490,
"column": 47
} | {
"line": 490,
"column": 48
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni : F →+* K\nx : F\nh : F[X]\nh_sep : h.Separable\nh_root : eval x h = 0\nh_splits : (map i h).Splits\nh_roots : ∀ y ∈ (map i h).roots, y = i x\nh_ne_zero : h ≠ 0\nthis : (map i h).roots = {i x}\n⊢ map i h = map i (C h.leadingCoeff * (X - C x))... | [
"F : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni : F →+* K\nx : F\nh : F[X]\nh_sep : h.Separable\nh_root : eval x h = 0\nh_splits : (map i h).Splits\nh_roots : ∀ y ∈ (map i h).roots, y = i x\nh_ne_zero : h ≠ 0\nthis : (map i h).roots = {i x}\n⊢ map i h = C (i h.leadingCoeff) * (X - C (i x))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 407,
"column": 77
} | {
"line": 407,
"column": 97
} | {
"line": 408,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nι : Type u_4\ninst✝² : Fintype ι\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\nf : ι → M →ₗ[R] M\nb : Basis (Free.ChooseBasisIndex R M) R M := Free.chooseBasis R M\nB : Basis (Free.ChooseBasisIndex R M × ι) ... | [
"R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nι : Type u_4\ninst✝² : Fintype ι\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\nf : ι → M →ₗ[R] M\nb : Basis (Free.ChooseBasisIndex R M) R M := Free.chooseBasis R M\nB : Basis (Free.ChooseBasisIndex R M × ι) R (ι → M) :=... | Equiv.prodComm_symm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
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