module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.FieldTheory.KrullTopology | {
"line": 326,
"column": 25
} | {
"line": 326,
"column": 58
} | {
"line": 326,
"column": 59
} | [
{
"pp": "k : Type u_1\nK : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nL : IntermediateField k K\nhnfd : FiniteDimensional k ↥L\nE : IntermediateField k K := normalClosure k (↥L) K\n⊢ L ≤ E",
"ppTerm": "?m.203",
"assigned": false,
"usedConstants": [],
... | [
"k : Type u_1\nK : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nL : IntermediateField k K\nhnfd : FiniteDimensional k ↥L\nE : IntermediateField k K := normalClosure k (↥L) K\n⊢ L ≤ E"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.BilinearForm.DualLattice | {
"line": 39,
"column": 34
} | {
"line": 39,
"column": 45
} | {
"line": 39,
"column": 46
} | [
{
"pp": "R : Type ?u.5\nS : Type ?u.7\nM : Type ?u.9\ninst✝⁶ : CommRing R\ninst✝⁵ : Field S\ninst✝⁴ : AddCommGroup M\ninst✝³ : Algebra R S\ninst✝² : Module R M\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\nB : BilinForm S M\nN : Submodule R M\na b : M\nha : a ∈ {x | ∀ y ∈ N, (B x) y ∈ 1}\nhb : b ∈ {x | ∀ y... | [
"R : Type ?u.5\nS : Type ?u.7\nM : Type ?u.9\ninst✝⁶ : CommRing R\ninst✝⁵ : Field S\ninst✝⁴ : AddCommGroup M\ninst✝³ : Algebra R S\ninst✝² : Module R M\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\nB : BilinForm S M\nN : Submodule R M\na b : M\nha : a ∈ {x | ∀ y ∈ N, (B x) y ∈ 1}\nhb : b ∈ {x | ∀ y ∈ N, (B x) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.BilinearForm.DualLattice | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 13
} | {
"line": 90,
"column": 14
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Field S\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : IsScalarTower R S M\nB : BilinForm S M\ninst✝¹ : IsDomain R\nhB : B.Nondegenerate\ninst✝ : IsTorsionFree R S\nN : Submodule ... | [
"R : Type u_1\nS : Type u_2\nM : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Field S\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : IsScalarTower R S M\nB : BilinForm S M\ninst✝¹ : IsDomain R\nhB : B.Nondegenerate\ninst✝ : IsTorsionFree R S\nN : Submodule R M\nhN : Su... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 40
} | {
"line": 67,
"column": 41
} | [
{
"pp": "case neg\nR : Type u_3\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : DecidableEq T\nx : S\ny : T\nσ : S →+* T\nhy : eval₂ σ y (minpolyDiv R x * (X - C x)) = 0\nh : ¬σ x = y\n⊢ eval₂ σ y (minpolyDiv R x) = 0... | [
"case neg\nR : Type u_3\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : DecidableEq T\nx : S\ny : T\nσ : S →+* T\nhy : eval₂ σ y (minpolyDiv R x * (X - C x)) = 0\nh : ¬σ x = y\n⊢ eval₂ σ y (minpolyDiv R x) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 91,
"column": 4
} | {
"line": 92,
"column": 53
} | {
"line": 93,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\ns : S\nhs : IsIntegral R s\np : R[X]\nhp : minpoly R s ∣ p\n⊢ (Polynomial.aeval s) p = 0",
"ppTerm": "?m.44... | [
"R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\ns : S\nhs : IsIntegral R s\np : R[X]\nhp : minpoly R s ∣ p\n⊢ (Polynomial.aeval s) p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 25
} | {
"line": 81,
"column": 26
} | [
{
"pp": "case hy\nR : Type u_2\nS : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nx : S\ninst✝¹ : IsDomain S\ninst✝ : DecidableEq S\ny : S\nhy : (aeval y) (minpoly R x) = 0\n⊢ eval₂ ((RingHom.id S).comp (algebraMap R S)) y (minpoly R x) = 0",
"ppTerm": "?hy",
"assigned": true... | [
"case hy\nR : Type u_2\nS : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nx : S\ninst✝¹ : IsDomain S\ninst✝ : DecidableEq S\ny : S\nhy : (aeval y) (minpoly R x) = 0\n⊢ eval₂ (algebraMap R S) y (minpoly R x) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.GaussLemma | {
"line": 205,
"column": 64
} | {
"line": 205,
"column": 82
} | {
"line": 205,
"column": 83
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsIntegrallyClosed R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\nr : K[X]\nhr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) q * r\nd' : R[X]\nhr' : Polynomi... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsIntegrallyClosed R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\nr : K[X]\nhr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) q * r\nd' : R[X]\nhr' : Polynomial.map (alge... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 27
} | {
"line": 123,
"column": 28
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\nx : S\np : R[X]\nhirr : Irreducible p\nhp : (Polynomial.aeval x) p = 0\nisUnit : IsUnit p.leadingCoeff\n⊢ IsInt... | [
"R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\nx : S\np : R[X]\nhirr : Irreducible p\nhp : (Polynomial.aeval x) p = 0\nisUnit : IsUnit p.leadingCoeff\n⊢ IsIntegral R x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 86,
"column": 2
} | {
"line": 98,
"column": 7
} | {
"line": 100,
"column": 0
} | [
{
"pp": "R : Type u_2\nS : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\ni : ℕ\n⊢ (minpolyDiv R x).coeff i ∈ R[x]",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne... | [] | by_contra H
have : ∀ j, coeff (minpolyDiv R x) (i + j) ∉ R[x] := by
intro j; induction j with
| zero => exact H
| succ j IH =>
intro H; apply IH
rw [coeff_minpolyDiv]
refine add_mem ?_ (mul_mem H (self_mem_adjoin_singleton R x))
exact Subalgebra.algebraMap_mem _ _
apply this (nat... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 86,
"column": 2
} | {
"line": 98,
"column": 7
} | {
"line": 100,
"column": 0
} | [
{
"pp": "R : Type u_2\nS : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\ni : ℕ\n⊢ (minpolyDiv R x).coeff i ∈ R[x]",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne... | [] | by_contra H
have : ∀ j, coeff (minpolyDiv R x) (i + j) ∉ R[x] := by
intro j; induction j with
| zero => exact H
| succ j IH =>
intro H; apply IH
rw [coeff_minpolyDiv]
refine add_mem ?_ (mul_mem H (self_mem_adjoin_singleton R x))
exact Subalgebra.algebraMap_mem _ _
apply this (nat... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 15
} | {
"line": 121,
"column": 16
} | [
{
"pp": "R : Type u_2\nS : Type u_1\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nx : S\nhx : IsIntegral R x\ninst✝ : Nontrivial S\n⊢ (minpolyDiv R x).leadingCoeff * (X - C x).leadingCoeff ≠ 0",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynom... | [
"R : Type u_2\nS : Type u_1\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nx : S\nhx : IsIntegral R x\ninst✝ : Nontrivial S\n⊢ ¬minpolyDiv R x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 38
} | {
"line": 191,
"column": 39
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\nr : R\nhr : r ≠ 0\ns : S\nhs : IsIntegral R s\nK : Type u_1 := FractionRing R\nL : Type u_2 := F... | [
"case refine_2\nR : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\nr : R\nhr : r ≠ 0\ns : S\nhs : IsIntegral R s\nK : Type u_1 := FractionRing R\nL : Type u_2 := FractionRing ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 195,
"column": 52
} | {
"line": 195,
"column": 66
} | {
"line": 195,
"column": 66
} | [
{
"pp": "case refine_4\nR : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\nr : R\nhr : r ≠ 0\ns : S\nhs : IsIntegral R s\nK : Type u_1 := FractionRing R\nL : Type u_2 := F... | [
"case refine_4\nR : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\nr : R\nhr : r ≠ 0\ns : S\nhs : IsIntegral R s\nK : Type u_1 := FractionRing R\nL : Type u_2 := FractionRing ... | scaleRoots_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 197,
"column": 14
} | {
"line": 198,
"column": 58
} | {
"line": 198,
"column": 59
} | [
{
"pp": "case refine_4\nR : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\nr : R\nhr : r ≠ 0\ns : S\nhs : IsIntegral R s\nK : Type u_1 := FractionRing R\nL : Type u_2 := F... | [
"case refine_4\nR : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\nr : R\nhr : r ≠ 0\ns : S\nhs : IsIntegral R s\nK : Type u_1 := FractionRing R\nL : Type u_2 := FractionRing ... | ← inv_mul_cancel_left₀ (b := algebraMap S L s)
(a := algebraMap K L (algebraMap R K r)) (by simpa), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 84
} | {
"line": 210,
"column": 85
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\nx : S\nhx : IsIntegral R x\nP : R[X]\nhP₁ : (Minpoly.toAdjoin R x) ((AdjoinRoot.mk (minpoly R x)) P) = 0\n⊢ (Ad... | [
"R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\nx : S\nhx : IsIntegral R x\nP : R[X]\nhP₁ : (Minpoly.toAdjoin R x) ((AdjoinRoot.mk (minpoly R x)) P) = 0\n⊢ minpoly R x ∣ P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsSepClosed | {
"line": 116,
"column": 15
} | {
"line": 116,
"column": 30
} | {
"line": 116,
"column": 31
} | [
{
"pp": "k : Type u\ninst✝¹ : Field k\ninst✝ : IsSepClosed k\nn : ℕ\na b c : k\nhn : ↑n = 0\nhn' : 2 ≤ n\nhb : b ≠ 0\nf : k[X] := C a * X ^ n + C b * X + C c\nhdeg : f.degree ≠ 0\nhsep : f.Separable\nx : k\nhx : f.IsRoot x\n⊢ a * x ^ n + b * x + c = 0",
"ppTerm": "?m.200",
"assigned": false,
"usedCo... | [
"k : Type u\ninst✝¹ : Field k\ninst✝ : IsSepClosed k\nn : ℕ\na b c : k\nhn : ↑n = 0\nhn' : 2 ≤ n\nhb : b ≠ 0\nf : k[X] := C a * X ^ n + C b * X + C c\nhdeg : f.degree ≠ 0\nhsep : f.Separable\nx : k\nhx : f.IsRoot x\n⊢ a * x ^ n + b * x + c = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsSepClosed | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 77
} | {
"line": 144,
"column": 78
} | [
{
"pp": "case h\nk : Type u\ninst✝¹ : Field k\ninst✝ : IsSepClosed k\nx : k\nn : ℕ\nhn : NeZero ↑n\nhn' : 0 < n\nthis : (X ^ n - C x).degree ≠ 0\nhx : ¬x = 0\nz : k\nhz : (X ^ n - C x).IsRoot z\n⊢ z ^ n = x",
"ppTerm": "?h",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals... | [
"case h\nk : Type u\ninst✝¹ : Field k\ninst✝ : IsSepClosed k\nx : k\nn : ℕ\nhn : NeZero ↑n\nhn' : 0 < n\nthis : (X ^ n - C x).degree ≠ 0\nhx : ¬x = 0\nz : k\nhz : (X ^ n - C x).IsRoot z\n⊢ z ^ n = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 275,
"column": 8
} | {
"line": 276,
"column": 44
} | {
"line": 276,
"column": 45
} | [
{
"pp": "case a\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nh : L.fixingSubgroup.Normal\ng : (x : K) → Subgroup Gal(↥(adjoin k {x}).toIntermediateField/k) :=\n fun x ↦ Subgroup.map (restrictNormalHom ↥(adjoin k {x}).toI... | [
"case a\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nh : L.fixingSubgroup.Normal\ng : (x : K) → Subgroup Gal(↥(adjoin k {x}).toIntermediateField/k) :=\n fun x ↦ Subgroup.map (restrictNormalHom ↥(adjoin k {x}).toIntermediateF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsSepClosed | {
"line": 184,
"column": 17
} | {
"line": 184,
"column": 41
} | {
"line": 184,
"column": 42
} | [
{
"pp": "k : Type u\ninst✝ : Field k\nH : ∀ (p : k[X]), p.Monic → Irreducible p → p.Separable → ∃ x, eval x p = 0\np : k[X]\nhp : Irreducible p\nhs : p.Separable\nx : k\nhx : eval x (p * C p.leadingCoeff⁻¹) = 0\n⊢ eval x p = 0",
"ppTerm": "?m.72",
"assigned": false,
"usedConstants": [],
"usedFVa... | [
"k : Type u\ninst✝ : Field k\nH : ∀ (p : k[X]), p.Monic → Irreducible p → p.Separable → ∃ x, eval x p = 0\np : k[X]\nhp : Irreducible p\nhs : p.Separable\nx : k\nhx : eval x (p * C p.leadingCoeff⁻¹) = 0\n⊢ eval x p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 279,
"column": 8
} | {
"line": 281,
"column": 15
} | {
"line": 281,
"column": 16
} | [
{
"pp": "case a\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nh : L.fixingSubgroup.Normal\ng : (x : K) → Subgroup Gal(↥(adjoin k {x}).toIntermediateField/k) := ⋯\nf : ↥L → IntermediateField k K := ⋯\nthis✝ : ∀ (x : K), (g ... | [
"case a\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nh : L.fixingSubgroup.Normal\ng : (x : K) → Subgroup Gal(↥(adjoin k {x}).toIntermediateField/k) :=\n fun x ↦ Subgroup.map (restrictNormalHom ↥(adjoin k {x}).toIntermediateF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsSepClosed | {
"line": 337,
"column": 15
} | {
"line": 337,
"column": 47
} | {
"line": 337,
"column": 48
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : IsSepClosed E\nh : separableClosure F E = ⊥\np : F[X]\nx✝ : p.Monic\nhirr : Irreducible p\nhsep : p.Separable\nx : F\nhx : (aeval ((Algebra.ofId F E).toRingHom x)) p = 0\n⊢ eval x p = 0",
"ppTerm": "?m.96"... | [
"F : Type u_1\nE : Type u_2\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : IsSepClosed E\nh : separableClosure F E = ⊥\np : F[X]\nx✝ : p.Monic\nhirr : Irreducible p\nhsep : p.Separable\nx : F\nhx : (aeval ((Algebra.ofId F E).toRingHom x)) p = 0\n⊢ eval x p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 284,
"column": 4
} | {
"line": 285,
"column": 38
} | {
"line": 285,
"column": 39
} | [
{
"pp": "case refine_2\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nh : IsGalois k ↥L\n⊢ L.fixingSubgroup.Normal",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsGalois.to... | [
"case refine_2\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nh : IsGalois k ↥L\n⊢ (restrictNormalHom ↥L).ker.Normal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 59
} | {
"line": 140,
"column": 60
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝⁵ : CommRing F\ninst✝⁴ : Ring E\ninst✝³ : Algebra F E\nK : Type u_3\ninst✝² : Ring K\ninst✝¹ : Algebra F K\ne : K ≃ₐ[F] E\ninst✝ : IsPurelyInseparable F K\nx : E\nh : (minpoly F (e.symm x)).Separable\n⊢ x ∈ (algebraMap F E).range",
"ppTerm": "?m.88",
"assigned":... | [
"F : Type u_1\nE : Type u_2\ninst✝⁵ : CommRing F\ninst✝⁴ : Ring E\ninst✝³ : Algebra F E\nK : Type u_3\ninst✝² : Ring K\ninst✝¹ : Algebra F K\ne : K ≃ₐ[F] E\ninst✝ : IsPurelyInseparable F K\nx : E\nh : (minpoly F (e.symm x)).Separable\n⊢ ∃ x_1, (algebraMap F E) x_1 = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 213,
"column": 6
} | {
"line": 213,
"column": 29
} | {
"line": 213,
"column": 29
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Ring E\ninst✝² : IsDomain E\ninst✝¹ : Algebra F E\nq : ℕ\ninst✝ : ExpChar F q\n⊢ IsPurelyInseparable F E ↔ ∀ (x : E), ∃ n, x ^ q ^ n ∈ (algebraMap F E).range",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"S... | [
"F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Ring E\ninst✝² : IsDomain E\ninst✝¹ : Algebra F E\nq : ℕ\ninst✝ : ExpChar F q\n⊢ (∀ (x : E), IsIntegral F x ∧ (IsSeparable F x → x ∈ (algebraMap F E).range)) ↔\n ∀ (x : E), ∃ n, x ^ q ^ n ∈ (algebraMap F E).range"
] | isPurelyInseparable_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 217,
"column": 6
} | {
"line": 217,
"column": 52
} | {
"line": 217,
"column": 53
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Ring E\ninst✝² : IsDomain E\ninst✝¹ : Algebra F E\nq : ℕ\ninst✝ : ExpChar F q\nh : ∀ (x : E), IsIntegral F x ∧ (IsSeparable F x → x ∈ (algebraMap F E).range)\nx : E\ng : F[X]\nh1 : g.Separable\nn : ℕ\nh2 : (expand F (q ^ n)) g = minpoly F x\n⊢ (aeval (... | [
"F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Ring E\ninst✝² : IsDomain E\ninst✝¹ : Algebra F E\nq : ℕ\ninst✝ : ExpChar F q\nh : ∀ (x : E), IsIntegral F x ∧ (IsSeparable F x → x ∈ (algebraMap F E).range)\nx : E\ng : F[X]\nh1 : g.Separable\nn : ℕ\nh2 : (expand F (q ^ n)) g = minpoly F x\n⊢ (aeval (x ^ q ^ n)) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Trace.Defs | {
"line": 102,
"column": 7
} | {
"line": 102,
"column": 18
} | {
"line": 102,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ (trace R R) 1 = LinearMap.id 1",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"CommSemiring.toSemiring",
"LinearMap.instFunLike",
... | [
"R : Type u_1\ninst✝ : CommRing R\n⊢ (trace R R) 1 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 23
} | {
"line": 269,
"column": 2
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\ninst✝¹ : Algebra E K\ninst✝ : IsScalarTower F E K\nq : ℕ\nh2✝ : ∀ (x : K), ∃ n, x ^ q ^ n ∈ (algebraMap E K).range\nh1✝ : ∀ (x : E), ∃ n, x ^ q ^ n ∈ (algebraMap F E).ra... | [
"F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\ninst✝¹ : Algebra E K\ninst✝ : IsScalarTower F E K\nq : ℕ\nh2✝ : ∀ (x : K), ∃ n, x ^ q ^ n ∈ (algebraMap E K).range\nh1✝ : ∀ (x : E), ∃ n, x ^ q ^ n ∈ (algebraMap F E).range\nh✝ : Ex... | refine ⟨n + m, z, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Polynomial.GaussLemma | {
"line": 305,
"column": 2
} | {
"line": 306,
"column": 9
} | {
"line": 306,
"column": 10
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁵ : CommRing R\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsFractionRing R K\ninst✝¹ : IsDomain R\ninst✝ : IsGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nr : R[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * (mapRingHom (algebraMap R K)) r\n⊢ q... | [
"case h\nR : Type u_1\ninst✝⁵ : CommRing R\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsFractionRing R K\ninst✝¹ : IsDomain R\ninst✝ : IsGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nr : R[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * (mapRingHom (algebraMap R K)) r\n⊢ q = p * r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 327,
"column": 4
} | {
"line": 327,
"column": 15
} | {
"line": 327,
"column": 16
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nq : ℕ\ninst✝² : ExpChar F q\ninst✝¹ : IsPurelyInseparable F E\ninst✝ : FiniteDimensional F E\nthis :\n ∀ (F E : Type v) [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] (q : ℕ) [ExpChar F q]\n [IsPurelyInsep... | [
"F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nq : ℕ\ninst✝² : ExpChar F q\ninst✝¹ : IsPurelyInseparable F E\ninst✝ : FiniteDimensional F E\nthis :\n ∀ (F E : Type v) [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] (q : ℕ) [ExpChar F q]\n [IsPurelyInseparable F E] ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 353,
"column": 8
} | {
"line": 353,
"column": 31
} | {
"line": 353,
"column": 31
} | [
{
"pp": "case pos\nF : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nhdeg : finSepDegree F E = 1\nH : Algebra.IsAlgebraic F E\n⊢ IsPurelyInseparable F E",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subring.instSetLike",
"Algebr... | [
"case pos\nF : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nhdeg : finSepDegree F E = 1\nH : Algebra.IsAlgebraic F E\n⊢ ∀ (x : E), IsIntegral F x ∧ (IsSeparable F x → x ∈ (algebraMap F E).range)"
] | isPurelyInseparable_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.IntegralClosure | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 72
} | {
"line": 134,
"column": 2
} | [
{
"pp": "A : Type u_1\nK : Type u_2\ninst✝⁹ : CommRing A\ninst✝⁸ : Field K\ninst✝⁷ : Algebra A K\ninst✝⁶ : IsFractionRing A K\nL : Type u_3\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra A L\ninst✝² : IsScalarTower A K L\ninst✝¹ : FiniteDimensional K L\ninst✝ : IsDomain A\nthis : DecidableEq L := Cla... | [] | exact (algebraMap K L).injective.comp (IsFractionRing.injective A K) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.IsGaloisGroup.Defs | {
"line": 62,
"column": 30
} | {
"line": 62,
"column": 46
} | {
"line": 62,
"column": 47
} | [
{
"pp": "G : Type u_1\nA : Type u_2\nB : Type u_4\ninst✝⁶ : Group G\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Semiring B\ninst✝³ : Algebra A B\ninst✝² : MulSemiringAction G B\nhG : IsGaloisGroup G A B\nH : Type u_5\ninst✝¹ : Group H\ninst✝ : MulSemiringAction H B\ne : H ≃* G\nhe : ∀ (h : H) (x : B), e h • x = h • x\nx... | [
"G : Type u_1\nA : Type u_2\nB : Type u_4\ninst✝⁶ : Group G\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Semiring B\ninst✝³ : Algebra A B\ninst✝² : MulSemiringAction G B\nhG : IsGaloisGroup G A B\nH : Type u_5\ninst✝¹ : Group H\ninst✝ : MulSemiringAction H B\ne : H ≃* G\nhe : ∀ (h : H) (x : B), e h • x = h • x\nx : H\na : A\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IsGaloisGroup.Defs | {
"line": 65,
"column": 45
} | {
"line": 65,
"column": 62
} | {
"line": 65,
"column": 63
} | [
{
"pp": "G : Type u_1\nA : Type u_2\nB : Type u_4\ninst✝⁶ : Group G\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Semiring B\ninst✝³ : Algebra A B\ninst✝² : MulSemiringAction G B\nhG : IsGaloisGroup G A B\nH : Type u_5\ninst✝¹ : Group H\ninst✝ : MulSemiringAction H B\ne : H ≃* G\nhe : ∀ (h : H) (x : B), e h • x = h • x\nb... | [
"G : Type u_1\nA : Type u_2\nB : Type u_4\ninst✝⁶ : Group G\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Semiring B\ninst✝³ : Algebra A B\ninst✝² : MulSemiringAction G B\nhG : IsGaloisGroup G A B\nH : Type u_5\ninst✝¹ : Group H\ninst✝ : MulSemiringAction H B\ne : H ≃* G\nhe : ∀ (h : H) (x : B), e h • x = h • x\nb : B\nh : ∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IsGaloisGroup.Defs | {
"line": 86,
"column": 65
} | {
"line": 86,
"column": 83
} | {
"line": 86,
"column": 84
} | [
{
"pp": "G : Type u_1\nA : Type u_2\nB : Type u_4\ninst✝⁷ : Group G\ninst✝⁶ : CommSemiring A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra A B\ninst✝³ : MulSemiringAction G B\nhG : IsGaloisGroup G A B\nB' : Type u_5\ninst✝² : Semiring B'\ninst✝¹ : Algebra A B'\ninst✝ : MulSemiringAction G B'\ne : B ≃ₐ[A] B'\nhe : ∀ (g... | [
"G : Type u_1\nA : Type u_2\nB : Type u_4\ninst✝⁷ : Group G\ninst✝⁶ : CommSemiring A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra A B\ninst✝³ : MulSemiringAction G B\nhG : IsGaloisGroup G A B\nB' : Type u_5\ninst✝² : Semiring B'\ninst✝¹ : Algebra A B'\ninst✝ : MulSemiringAction G B'\ne : B ≃ₐ[A] B'\nhe : ∀ (g : G) (x : B... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IsGaloisGroup.Defs | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 30
} | {
"line": 92,
"column": 31
} | [
{
"pp": "G : Type u_1\nA : Type u_2\nB : Type u_4\ninst✝⁷ : Group G\ninst✝⁶ : CommSemiring A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra A B\ninst✝³ : MulSemiringAction G B\nhG : IsGaloisGroup G A B\nB' : Type u_5\ninst✝² : Semiring B'\ninst✝¹ : Algebra A B'\ninst✝ : MulSemiringAction G B'\ne : B ≃ₐ[A] B'\nhe : ∀ (g... | [
"G : Type u_1\nA : Type u_2\nB : Type u_4\ninst✝⁷ : Group G\ninst✝⁶ : CommSemiring A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra A B\ninst✝³ : MulSemiringAction G B\nhG : IsGaloisGroup G A B\nB' : Type u_5\ninst✝² : Semiring B'\ninst✝¹ : Algebra A B'\ninst✝ : MulSemiringAction G B'\ne : B ≃ₐ[A] B'\nhe : ∀ (g : G) (x : B... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IsGaloisGroup.Defs | {
"line": 132,
"column": 20
} | {
"line": 132,
"column": 49
} | {
"line": 132,
"column": 50
} | [
{
"pp": "G : Type u_1\nA : Type u_2\nA' : Type u_3\nB : Type u_4\ninst✝⁵ : Group G\ninst✝⁴ : CommSemiring A\ninst✝³ : Semiring B\ninst✝² : Algebra A B\ninst✝¹ : MulSemiringAction G B\nhA : IsGaloisGroup G A B\ninst✝ : FaithfulSMul A B\nx : ↥(FixedPoints.subsemiring B G)\n⊢ (fun x ↦ ⟨(algebraMap A B) x, ⋯⟩) ((fu... | [
"G : Type u_1\nA : Type u_2\nA' : Type u_3\nB : Type u_4\ninst✝⁵ : Group G\ninst✝⁴ : CommSemiring A\ninst✝³ : Semiring B\ninst✝² : Algebra A B\ninst✝¹ : MulSemiringAction G B\nhA : IsGaloisGroup G A B\ninst✝ : FaithfulSMul A B\nx : ↥(FixedPoints.subsemiring B G)\n⊢ (algebraMap A B) ⋯.choose = ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 40
} | {
"line": 103,
"column": 41
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nhx : ¬IsIntegral K x\n⊢ (Algebra.trace K ↥K⟮x⟯) (gen K x) = 0",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"instSMulOfMul",
"congrA... | [
"K : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nhx : ¬IsIntegral K x\n⊢ 0 (gen K x) = 0",
"case h\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nhx : ¬IsIntegral K x\n⊢ ¬∃ s, Nonempty (Basis (↥s) K ↥K⟮x⟯)"
] | trace_eq_zero_of_not_exists_basis, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 31
} | {
"line": 138,
"column": 32
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nhx : IsIntegral K x\n⊢ (Algebra.trace K ↥K⟮x⟯) (AdjoinSimple.gen K x) = -(minpoly K x).nextCoeff",
"ppTerm": "?m.39",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"K : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nhx : IsIntegral K x\n⊢ (Algebra.trace K ↥K⟮x⟯) (AdjoinSimple.gen K x) = -(minpoly K x).nextCoeff"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 195,
"column": 45
} | {
"line": 195,
"column": 79
} | {
"line": 196,
"column": 4
} | [
{
"pp": "case neg\nA : Type u_7\nB : Type u_8\nC : Type u_9\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing C\ninst✝¹ : Algebra A C\ninst✝ : Algebra B C\ne : A ≃+* B\nhe : (algebraMap B C).comp ↑e = algebraMap A C\nx : C\nh : ¬∃ s, Nonempty (Basis (↥s) B C)\n⊢ e ((trace A C) x) = 0 x",
"ppTerm"... | [
"case neg\nA : Type u_7\nB : Type u_8\nC : Type u_9\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing C\ninst✝¹ : Algebra A C\ninst✝ : Algebra B C\ne : A ≃+* B\nhe : (algebraMap B C).comp ↑e = algebraMap A C\nx : C\nh : ¬∃ s, Nonempty (Basis (↥s) B C)\n⊢ e (0 x) = 0 x",
"case neg.h\nA : Type u_7\nB : T... | trace_eq_zero_of_not_exists_basis, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Prime.Int | {
"line": 33,
"column": 8
} | {
"line": 33,
"column": 78
} | {
"line": 33,
"column": 79
} | [
{
"pp": "p : ℕ\nhp : _root_.Prime ↑p\na b : ℕ\n⊢ p ∣ a * b → p ∣ a ∨ p ∣ b",
"ppTerm": "?m.49",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nhp : _root_.Prime ↑p\na b : ℕ\n⊢ p ∣ a * b → p ∣ a ∨ p ∣ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Prime.Int | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 13
} | {
"line": 49,
"column": 14
} | [
{
"pp": "p q : ℕ\nhp : Prime p\nhq : Prime q\nm : ℕ\nhm : m + 1 ≠ 0\nn : ℕ\nhn : n + 1 ≠ 0\nh : p ^ (m + 1) = q ^ (n + 1)\n⊢ p = q ∧ m + 1 = n + 1",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"instOfNatNat",
"instHAdd",
"An... | [
"p q : ℕ\nhp : Prime p\nhq : Prime q\nm : ℕ\nhm : m + 1 ≠ 0\nn : ℕ\nhn : n + 1 ≠ 0\nh : p ^ (m + 1) = q ^ (n + 1)\n⊢ p = q ∧ m = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 309,
"column": 41
} | {
"line": 309,
"column": 52
} | {
"line": 309,
"column": 53
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nH : ¬Algebra.IsSeparable K L\np : ℕ\nhp : ExpChar K p\nthis : p ≠ 0\nx : L\nh₀ : FiniteDimensional K L\nhx : ¬IsSeparable K x\ng : K[X]\nhg₁ : g.Separable\nhg₂ : (expand K (p ^ 0)) g = minpoly K x\n⊢ g = minpoly K x",
... | [
"K : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nH : ¬Algebra.IsSeparable K L\np : ℕ\nhp : ExpChar K p\nthis : p ≠ 0\nx : L\nh₀ : FiniteDimensional K L\nhx : ¬IsSeparable K x\ng : K[X]\nhg₁ : g.Separable\nhg₂ : (expand K (p ^ 0)) g = minpoly K x\n⊢ g = minpoly K x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 318,
"column": 45
} | {
"line": 318,
"column": 63
} | {
"line": 318,
"column": 64
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nH : ¬Algebra.IsSeparable K L\np : ℕ\nhp : ExpChar K p\nthis✝ : p ≠ 0\nx : L\nh₀ : FiniteDimensional K L\nhx : ¬IsSeparable K x\ng : K[X]\nhg₁ : g.Separable\nn : ℕ\nhg₂ : (expand K (p ^ (n + 1))) g = minpoly K x\nh : p ... | [
"K : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nH : ¬Algebra.IsSeparable K L\np : ℕ\nhp : ExpChar K p\nthis✝ : p ≠ 0\nx : L\nh₀ : FiniteDimensional K L\nhx : ¬IsSeparable K x\ng : K[X]\nhg₁ : g.Separable\nn : ℕ\nhg₂ : (expand K (p ^ (n + 1))) g = minpoly K x\nh : p ^ (n + 1) ∣ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ZMod.ValMinAbs | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 72
} | {
"line": 102,
"column": 0
} | [
{
"pp": "n : ℕ\na b : ZMod n\nh : a.valMinAbs = -b.valMinAbs\n⊢ a = -b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"neg_lt_neg_iff._simp_1",
"AddGroup.toSubtractionMonoid",
"Int.... | [] | rcases eq_zero_or_neZero n with rfl | hn <;> simp_all [valMinAbs_spec] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.ZMod.ValMinAbs | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 72
} | {
"line": 102,
"column": 0
} | [
{
"pp": "n : ℕ\na b : ZMod n\nh : a.valMinAbs = -b.valMinAbs\n⊢ a = -b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"neg_lt_neg_iff._simp_1",
"AddGroup.toSubtractionMonoid",
"Int.... | [] | rcases eq_zero_or_neZero n with rfl | hn <;> simp_all [valMinAbs_spec] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ZMod.ValMinAbs | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 72
} | {
"line": 102,
"column": 0
} | [
{
"pp": "n : ℕ\na b : ZMod n\nh : a.valMinAbs = -b.valMinAbs\n⊢ a = -b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"neg_lt_neg_iff._simp_1",
"AddGroup.toSubtractionMonoid",
"Int.... | [] | rcases eq_zero_or_neZero n with rfl | hn <;> simp_all [valMinAbs_spec] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ZMod.ValMinAbs | {
"line": 159,
"column": 52
} | {
"line": 159,
"column": 75
} | {
"line": 160,
"column": 4
} | [
{
"pp": "p q : ℕ\nhp : Fact (Nat.Prime p)\nhq : Fact (Nat.Prime q)\nhpq : p ≠ q\n⊢ ¬q ≡ 0 [MOD p]",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"id",
"instOfNatNat",
"Nat.ModEq",
"Nat.instDvd",
"Nat",
... | [
"p q : ℕ\nhp : Fact (Nat.Prime p)\nhq : Fact (Nat.Prime q)\nhpq : p ≠ q\n⊢ ¬p ∣ q"
] | Nat.modEq_zero_iff_dvd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 497,
"column": 2
} | {
"line": 497,
"column": 41
} | {
"line": 497,
"column": 42
} | [
{
"pp": "case refine_2\nK : Type u_4\nL : Type u_5\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\nι : Type w\ninst✝² : Algebra.IsSeparable K L\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι K L\nthis : FiniteDimensional K L\npb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L\n⊢ ... | [
"case refine_2\nK : Type u_4\nL : Type u_5\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\nι : Type w\ninst✝² : Algebra.IsSeparable K L\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι K L\nthis : FiniteDimensional K L\npb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L\n⊢ ((LinearMap.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Int.Associated | {
"line": 30,
"column": 4
} | {
"line": 30,
"column": 45
} | {
"line": 31,
"column": 4
} | [
{
"pp": "case mpr\na : ℤ\nu : ℤˣ\n⊢ a = a * ↑u ∨ a = -(a * ↑u)",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Units.val",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Monoid.toMulOneClass",
"Units.instNe... | [
"case mpr.inl\na : ℤ\n⊢ a = a * ↑1 ∨ a = -(a * ↑1)",
"case mpr.inr\na : ℤ\n⊢ a = a * ↑(-1) ∨ a = -(a * ↑(-1))"
] | obtain rfl | rfl := Int.units_eq_one_or u | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Invariant.Basic | {
"line": 292,
"column": 6
} | {
"line": 292,
"column": 39
} | {
"line": 293,
"column": 2
} | [
{
"pp": "case neg\nA : Type u_1\nB : Type u_2\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : CommRing B\ninst✝¹⁶ : Algebra A B\nG : Type u_3\ninst✝¹⁵ : Group G\ninst✝¹⁴ : Finite G\ninst✝¹³ : MulSemiringAction G B\ninst✝¹² : SMulCommClass G A B\nP : Ideal A\nQ : Ideal B\ninst✝¹¹ : Q.IsPrime\ninst✝¹⁰ : Q.LiesOver P\nK : Type u... | [] | rw [map_zero, map_zero, sub_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Quotient.Pi | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 17
} | {
"line": 68,
"column": 18
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝⁶ : CommRing R\nMs : ι → Type u_3\ninst✝⁵ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁴ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nNs : ι → Type u_5\ninst✝¹ : (i : ι) → AddCommGroup (Ns i)\ninst✝ : (i : ι) → Module R (Ns i)\np : (... | [
"ι : Type u_1\nR : Type u_2\ninst✝⁶ : CommRing R\nMs : ι → Type u_3\ninst✝⁵ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁴ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nNs : ι → Type u_5\ninst✝¹ : (i : ι) → AddCommGroup (Ns i)\ninst✝ : (i : ι) → Module R (Ns i)\np : (i : ι) → Sub... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Invariant.Basic | {
"line": 292,
"column": 6
} | {
"line": 292,
"column": 39
} | {
"line": 293,
"column": 2
} | [
{
"pp": "case neg\nA : Type u_1\nB : Type u_2\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : CommRing B\ninst✝¹⁶ : Algebra A B\nG : Type u_3\ninst✝¹⁵ : Group G\ninst✝¹⁴ : Finite G\ninst✝¹³ : MulSemiringAction G B\ninst✝¹² : SMulCommClass G A B\nP : Ideal A\nQ : Ideal B\ninst✝¹¹ : Q.IsPrime\ninst✝¹⁰ : Q.LiesOver P\nK : Type u... | [] | rw [map_zero, map_zero, sub_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Invariant.Basic | {
"line": 292,
"column": 6
} | {
"line": 292,
"column": 39
} | {
"line": 293,
"column": 2
} | [
{
"pp": "case neg\nA : Type u_1\nB : Type u_2\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : CommRing B\ninst✝¹⁶ : Algebra A B\nG : Type u_3\ninst✝¹⁵ : Group G\ninst✝¹⁴ : Finite G\ninst✝¹³ : MulSemiringAction G B\ninst✝¹² : SMulCommClass G A B\nP : Ideal A\nQ : Ideal B\ninst✝¹¹ : Q.IsPrime\ninst✝¹⁰ : Q.LiesOver P\nK : Type u... | [] | rw [map_zero, map_zero, sub_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Galois.IsGaloisGroup | {
"line": 259,
"column": 7
} | {
"line": 259,
"column": 56
} | {
"line": 259,
"column": 57
} | [
{
"pp": "G : Type u_1\nK : Type u_3\nL : Type u_4\ninst✝⁴ : Group G\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : MulSemiringAction G L\nhGKL : IsGaloisGroup G K L\nthis : FaithfulSMul G L\nx✝ : G\n⊢ x✝ ∈ fixingSubgroup G ↑⊤ ↔ x✝ ∈ ⊥",
"ppTerm": "?m.48",
"assigned": true,
"usedC... | [
"G : Type u_1\nK : Type u_3\nL : Type u_4\ninst✝⁴ : Group G\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : MulSemiringAction G L\nhGKL : IsGaloisGroup G K L\nthis : FaithfulSMul G L\nx✝ : G\n⊢ (∀ (y : L), x✝ • y = y) ↔ x✝ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 391,
"column": 6
} | {
"line": 391,
"column": 58
} | {
"line": 391,
"column": 59
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : Fintype K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Finite L\nthis✝ : Fintype L\nm : ℕ\nlt : m < Module.finrank K L\npos : 0 < m\neq : frobeniusAlgHom K L ^ m = 1\nx : L\nx✝ : x ∈ univ.val\nthis : (fun x ↦ x ^ q ^ m) x - x = 0\nh : X ^ q ^ m... | [
"K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : Fintype K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Finite L\nthis✝ : Fintype L\nm : ℕ\nlt : m < Module.finrank K L\npos : 0 < m\neq : frobeniusAlgHom K L ^ m = 1\nx : L\nx✝ : x ∈ univ.val\nthis : (fun x ↦ x ^ q ^ m) x - x = 0\nh : X ^ q ^ m - X = 0\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Finite.CardQuotient | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 70
} | {
"line": 53,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : AddCommGroup M\ninst✝³ : Free ℤ M\ninst✝² : Module.Finite ℤ M\nN : Submodule ℤ M\nE : Type u_2\ninst✝¹ : EquivLike E M ↥N\ninst✝ : AddEquivClass E M ↥N\ne : E\nb : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := Free.chooseBasis ℤ M\nh : finrank ℤ ↥N = finrank ℤ M\na : Free.ChooseBasisI... | [
"M : Type u_1\ninst✝⁴ : AddCommGroup M\ninst✝³ : Free ℤ M\ninst✝² : Module.Finite ℤ M\nN : Submodule ℤ M\nE : Type u_2\ninst✝¹ : EquivLike E M ↥N\ninst✝ : AddEquivClass E M ↥N\ne : E\nb : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := Free.chooseBasis ℤ M\nh : finrank ℤ ↥N = finrank ℤ M\na : Free.ChooseBasisIndex ℤ M → ℤ... | let f_apply : ∀ x, f x = b'.equiv ab (Equiv.refl _) x := fun x ↦ rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.FieldTheory.Finite.Basic | {
"line": 515,
"column": 2
} | {
"line": 515,
"column": 78
} | {
"line": 515,
"column": 79
} | [
{
"pp": "case inr\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ZMod p\nhp_odd : p % 2 = 1\nf : (ZMod p)[X] := X ^ 2\ng : (ZMod p)[X] := X ^ 2 - C x\na b : ZMod p\nhab : eval a f + eval b g = 0\n⊢ a ^ 2 + b ^ 2 - x = 0",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"case inr\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ZMod p\nhp_odd : p % 2 = 1\nf : (ZMod p)[X] := X ^ 2\ng : (ZMod p)[X] := X ^ 2 - C x\na b : ZMod p\nhab : eval a f + eval b g = 0\n⊢ a ^ 2 + b ^ 2 - x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 546,
"column": 2
} | {
"line": 546,
"column": 13
} | {
"line": 546,
"column": 14
} | [
{
"pp": "R : Type u_3\ninst✝³ : Ring R\ninst✝² : IsDomain R\np : ℕ\ninst✝¹ : NeZero p\ninst✝ : CharP R p\nx : ℤ\nthis : Fact (Nat.Prime p)\na b : ZMod p\nhab : a ^ 2 + b ^ 2 = ↑x\n⊢ ↑a.val ^ 2 + ↑b.val ^ 2 = ↑x",
"ppTerm": "?m.95",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr... | [
"R : Type u_3\ninst✝³ : Ring R\ninst✝² : IsDomain R\np : ℕ\ninst✝¹ : NeZero p\ninst✝ : CharP R p\nx : ℤ\nthis : Fact (Nat.Prime p)\na b : ZMod p\nhab : a ^ 2 + b ^ 2 = ↑x\n⊢ a.cast ^ 2 + b.cast ^ 2 = ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 645,
"column": 2
} | {
"line": 645,
"column": 45
} | {
"line": 645,
"column": 46
} | [
{
"pp": "p : ℕ\nhp : Nat.Prime p\nn : ℤ\nhpn : IsCoprime n ↑p\nthis✝ : Fact (Nat.Prime p)\nthis : ¬↑n = 0\n⊢ n ^ (p - 1) ≡ 1 [ZMOD ↑p]",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"ZMod.commRing",
"congrArg",
"AddGroupWithOne.toAddMono... | [
"p : ℕ\nhp : Nat.Prime p\nn : ℤ\nhpn : IsCoprime n ↑p\nthis✝ : Fact (Nat.Prime p)\nthis : ¬↑n = 0\n⊢ ↑n ^ (p - 1) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.IsGaloisGroup | {
"line": 475,
"column": 2
} | {
"line": 475,
"column": 27
} | {
"line": 475,
"column": 28
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\ninst✝¹⁷ : Group G\ninst✝¹⁶ : Group G'\nA : Type u_5\nB : Type u_6\nC : Type u_7\ninst✝¹⁵ : CommRing A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : CommRing C\ninst✝¹² : IsDomain C\ninst✝¹¹ : Algebra A B\ninst✝¹⁰ : Algebra A C\ninst✝⁹ : Algebra B C\ninst✝⁸ : FaithfulSMul A B\ninst✝⁷ : F... | [
"G : Type u_1\nG' : Type u_2\ninst✝¹⁷ : Group G\ninst✝¹⁶ : Group G'\nA : Type u_5\nB : Type u_6\nC : Type u_7\ninst✝¹⁵ : CommRing A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : CommRing C\ninst✝¹² : IsDomain C\ninst✝¹¹ : Algebra A B\ninst✝¹⁰ : Algebra A C\ninst✝⁹ : Algebra B C\ninst✝⁸ : FaithfulSMul A B\ninst✝⁷ : FaithfulSMul ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.IsGaloisGroup | {
"line": 514,
"column": 6
} | {
"line": 515,
"column": 13
} | {
"line": 515,
"column": 14
} | [
{
"pp": "G : Type u_1\nK : Type u_3\nL : Type u_4\ninst✝⁸ : Group G\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Algebra K L\ninst✝⁴ : MulSemiringAction G L\nH : Subgroup G\nF : IntermediateField K L\nN : Subgroup G\ninst✝³ : N.Normal\ninst✝² : IsGaloisGroup (↥N) (↥F) L\nE : IntermediateField K L\nhE : IsGaloi... | [
"G : Type u_1\nK : Type u_3\nL : Type u_4\ninst✝⁸ : Group G\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Algebra K L\ninst✝⁴ : MulSemiringAction G L\nH : Subgroup G\nF : IntermediateField K L\nN : Subgroup G\ninst✝³ : N.Normal\ninst✝² : IsGaloisGroup (↥N) (↥F) L\nE : IntermediateField K L\nhE : IsGaloisGroup (↥H) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative | {
"line": 63,
"column": 13
} | {
"line": 63,
"column": 24
} | {
"line": 63,
"column": 25
} | [
{
"pp": "case empty\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\nP : α → Prop\ni : α → ℕ\nh1 : ∀ {x : α}, IsUnit x → P x\nhpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)\nhcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)\nthis : DecidableEq α := Classical.decEq α\nis_p... | [
"case empty\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\nP : α → Prop\ni : α → ℕ\nh1 : ∀ {x : α}, IsUnit x → P x\nhpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)\nhcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)\nthis : DecidableEq α := Classical.decEq α\nis_prime : ∀ p ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 777,
"column": 2
} | {
"line": 777,
"column": 87
} | {
"line": 777,
"column": 88
} | [
{
"pp": "F : Type u_3\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nh : ¬Function.Injective fun x ↦ x * x\n⊢ ∃ a, ¬IsSquare a",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"HMul.hMul",
"congrArg",
"Exists",
"... | [
"F : Type u_3\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nh : ¬Function.Injective fun x ↦ x * x\n⊢ ∃ a, ∀ (x : F), ¬a = x * x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 810,
"column": 4
} | {
"line": 810,
"column": 81
} | {
"line": 811,
"column": 4
} | [
{
"pp": "F : Type u_3\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na g : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : (fun x ↦ g ^ x) n = a\n⊢ IsSquare a ↔ a ^ (Fintype.card F / 2) = 1",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.h... | [
"F : Type u_3\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na g : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : (fun x ↦ g ^ x) n = a\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\n⊢ IsSquare a ↔ a ^ (Fintype.card F / 2) = 1"
] | have hodd := Nat.two_mul_odd_div_two (FiniteField.odd_card_of_char_ne_two hF) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative | {
"line": 103,
"column": 13
} | {
"line": 103,
"column": 24
} | {
"line": 103,
"column": 25
} | [
{
"pp": "case empty\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_3\ninst✝ : CommMonoidWithZero β\nf : α → β\ni j : α → ℕ\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, IsRelPrime x ... | [
"case empty\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_3\ninst✝ : CommMonoidWithZero β\nf : α → β\ni j : α → ℕ\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, IsRelPrime x y → f (x * y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 833,
"column": 21
} | {
"line": 833,
"column": 41
} | {
"line": 833,
"column": 41
} | [
{
"pp": "F : Type u_3\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ Units.mk0 a ha ^ (Fintype.card F / 2) = 1 ↔ a ^ (Fintype.card F / 2) = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"instHDiv",
"Monoid.toMulOneClass... | [] | simp [Units.ext_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.Finite.Basic | {
"line": 833,
"column": 21
} | {
"line": 833,
"column": 41
} | {
"line": 833,
"column": 41
} | [
{
"pp": "F : Type u_3\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ Units.mk0 a ha ^ (Fintype.card F / 2) = 1 ↔ a ^ (Fintype.card F / 2) = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"instHDiv",
"Monoid.toMulOneClass... | [] | simp [Units.ext_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Finite.Basic | {
"line": 833,
"column": 21
} | {
"line": 833,
"column": 41
} | {
"line": 833,
"column": 41
} | [
{
"pp": "F : Type u_3\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ Units.mk0 a ha ^ (Fintype.card F / 2) = 1 ↔ a ^ (Fintype.card F / 2) = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"instHDiv",
"Monoid.toMulOneClass... | [] | simp [Units.ext_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Finite.Basic | {
"line": 837,
"column": 2
} | {
"line": 839,
"column": 37
} | {
"line": 841,
"column": 0
} | [
{
"pp": "case mpr\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ (∃ r, a = r * r) → ∃ r, a = ↑r * ↑r",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Units.val",
"GroupWithZero.toMonoidWithZero",
"False",
"HMul.hMul",
... | [] | · rintro ⟨y, rfl⟩
have hy : y ≠ 0 := by rintro rfl; simp at ha
refine ⟨Units.mk0 y hy, ?_⟩; simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Invariant.Basic | {
"line": 493,
"column": 40
} | {
"line": 493,
"column": 51
} | {
"line": 493,
"column": 52
} | [
{
"pp": "G : Type u_1\nA : Type u_2\nB : Type u_3\nK : Type u_4\nL : Type u_5\ninst✝¹⁸ : Group G\ninst✝¹⁷ : CommRing A\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra K L\ninst✝¹¹ : Algebra A K\ninst✝¹⁰ : Algebra B L\ninst✝⁹ : Algebra A L\ninst✝⁸ : IsFractio... | [
"G : Type u_1\nA : Type u_2\nB : Type u_3\nK : Type u_4\nL : Type u_5\ninst✝¹⁸ : Group G\ninst✝¹⁷ : CommRing A\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra K L\ninst✝¹¹ : Algebra A K\ninst✝¹⁰ : Algebra B L\ninst✝⁹ : Algebra A L\ninst✝⁸ : IsFractionRing A K\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Invariant.Basic | {
"line": 501,
"column": 8
} | {
"line": 501,
"column": 65
} | {
"line": 501,
"column": 66
} | [
{
"pp": "G : Type u_1\nA : Type u_2\nB : Type u_3\nK : Type u_4\nL : Type u_5\ninst✝¹⁸ : Group G\ninst✝¹⁷ : CommRing A\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra K L\ninst✝¹¹ : Algebra A K\ninst✝¹⁰ : Algebra B L\ninst✝⁹ : Algebra A L\ninst✝⁸ : IsFractio... | [
"G : Type u_1\nA : Type u_2\nB : Type u_3\nK : Type u_4\nL : Type u_5\ninst✝¹⁸ : Group G\ninst✝¹⁷ : CommRing A\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra K L\ninst✝¹¹ : Algebra A K\ninst✝¹⁰ : Algebra B L\ninst✝⁹ : Algebra A L\ninst✝⁸ : IsFractionRing A K\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RamificationInertia.Inertia | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 55
} | {
"line": 163,
"column": 56
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommRing R\nS : Type u_1\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDedekindDomain S\ninst✝⁵ : Free ℤ S\ninst✝⁴ : IsDedekindDomain R\ninst✝³ : Free ℤ R\ninst✝² : Algebra S R\ninst✝¹ : Module.Finite S R\nP : Ideal R\np : Ideal S\ninst✝ : P.LiesOver p\nhp : p.IsPrime\nhp_ne_bot : p ≠ ⊥\nthis :... | [
"R : Type u\ninst✝⁸ : CommRing R\nS : Type u_1\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDedekindDomain S\ninst✝⁵ : Free ℤ S\ninst✝⁴ : IsDedekindDomain R\ninst✝³ : Free ℤ R\ninst✝² : Algebra S R\ninst✝¹ : Module.Finite S R\nP : Ideal R\np : Ideal S\ninst✝ : P.LiesOver p\nhp : p.IsPrime\nhp_ne_bot : p ≠ ⊥\nthis : p.IsMaximal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RamificationInertia.Inertia | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 13
} | {
"line": 174,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Free ℤ R\ninst✝¹ : Module.Finite ℤ R\np : ℤ\nP : Ideal R\ninst✝ : P.LiesOver (span {p})\nhp : Prime p\n⊢ absNorm P = p.natAbs ^ (span {p}).inertiaDeg' P",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Id... | [
"R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Free ℤ R\ninst✝¹ : Module.Finite ℤ R\np : ℤ\nP : Ideal R\ninst✝ : P.LiesOver (span {p})\nhp : Prime p\n⊢ absNorm P = p.natAbs ^ finrank (ℤ ⧸ span {p}) (R ⧸ P)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RamificationInertia.Inertia | {
"line": 175,
"column": 51
} | {
"line": 175,
"column": 62
} | {
"line": 175,
"column": 63
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Free ℤ R\ninst✝¹ : Module.Finite ℤ R\np : ℤ\nP : Ideal R\ninst✝ : P.LiesOver (span {p})\nhp : Prime p\n⊢ span {p} ≠ ⊥",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Submodule.span_eq_bot._simp_1",
... | [
"R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Free ℤ R\ninst✝¹ : Module.Finite ℤ R\np : ℤ\nP : Ideal R\ninst✝ : P.LiesOver (span {p})\nhp : Prime p\n⊢ ¬p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 172,
"column": 4
} | {
"line": 173,
"column": 34
} | {
"line": 173,
"column": 35
} | [
{
"pp": "case succ.refine_1\nS : Type u_1\ninst✝¹ : CommRing S\nP : Ideal S\nP_prime : P.IsPrime\ninst✝ : IsDedekindDomain S\nhP : P ≠ ⊥\ni : ℕ\nih : cardQuot (P ^ i) = cardQuot P ^ i\nthis : P ^ (i + 1) < P ^ i\na : S\na_mem : a ∈ P ^ i\na_notMem : a ∉ P ^ (i + 1)\nf g : (c : S) → c ∈ P ^ i → S\nhg : ∀ (c : S)... | [
"case succ.refine_1\nS : Type u_1\ninst✝¹ : CommRing S\nP : Ideal S\nP_prime : P.IsPrime\ninst✝ : IsDedekindDomain S\nhP : P ≠ ⊥\ni : ℕ\nih : cardQuot (P ^ i) = cardQuot P ^ i\nthis : P ^ (i + 1) < P ^ i\na : S\na_mem : a ∈ P ^ i\na_notMem : a ∉ P ^ (i + 1)\nf g : (c : S) → c ∈ P ^ i → S\nhg : ∀ (c : S) (hc : c ∈ P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FiniteStability | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 15
} | {
"line": 72,
"column": 16
} | [
{
"pp": "case refine_3\nR : Type w₁\ninst✝⁵ : CommRing R\nA : Type w₂\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nB : Type w₃\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : FinitePresentation R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhsurj : Function.Surjective ⇑f\nhfg : (RingHom.ker f.toRingHom).FG\... | [
"case refine_3\nR : Type w₁\ninst✝⁵ : CommRing R\nA : Type w₂\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nB : Type w₃\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : FinitePresentation R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhsurj : Function.Surjective ⇑f\nhfg : (RingHom.ker f.toRingHom).FG\ng : B ⊗[R] ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 239,
"column": 24
} | {
"line": 239,
"column": 95
} | {
"line": 239,
"column": 96
} | [
{
"pp": "S : Type u_1\ninst✝² : CommRing S\ninst✝¹ : IsDedekindDomain S\ninst✝ : Free ℤ S\nI : Ideal S\nhI : Irreducible (absNorm I)\nh : IsUnit I\n⊢ IsUnit (absNorm I)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroOneClass",
"Semiring.toModule... | [
"S : Type u_1\ninst✝² : CommRing S\ninst✝¹ : IsDedekindDomain S\ninst✝ : Free ℤ S\nI : Ideal S\nhI : Irreducible (absNorm I)\nh : IsUnit I\n⊢ I = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 242,
"column": 6
} | {
"line": 242,
"column": 77
} | {
"line": 243,
"column": 8
} | [
{
"pp": "S : Type u_1\ninst✝² : CommRing S\ninst✝¹ : IsDedekindDomain S\ninst✝ : Free ℤ S\na b : Ideal S\nhI : Irreducible (absNorm (a * b))\n⊢ IsUnit a ∨ IsUnit b",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.RingTheory.Ideal.Norm.AbsNorm.0.Ideal.... | [
"S : Type u_1\ninst✝² : CommRing S\ninst✝¹ : IsDedekindDomain S\ninst✝ : Free ℤ S\na b : Ideal S\nhI : Irreducible (absNorm (a * b))\n⊢ a = ⊤ ∨ b = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Noetherian | {
"line": 61,
"column": 4
} | {
"line": 62,
"column": 48
} | {
"line": 62,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsArtinianRing R\np : Ideal R\ninst✝ : p.IsPrime\nr : R\nhr : (toPiLocalization R) r = Pi.single { asIdeal := p, isPrime := ⋯ } 1\n⊢ (algebraMap R (Localization p.primeCompl)) r = 1",
"ppTerm": "?m.58",
"assigned": false,
"usedConstants": [],
... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsArtinianRing R\np : Ideal R\ninst✝ : p.IsPrime\nr : R\nhr : (toPiLocalization R) r = Pi.single { asIdeal := p, isPrime := ⋯ } 1\n⊢ (algebraMap R (Localization p.primeCompl)) r = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Noetherian | {
"line": 71,
"column": 6
} | {
"line": 72,
"column": 51
} | {
"line": 72,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsArtinianRing R\np : Ideal R\ninst✝ : p.IsPrime\nr : R\nhr : (toPiLocalization R) r = Pi.single { asIdeal := p, isPrime := ⋯ } 1\nthis✝ : (algebraMap R (Localization p.primeCompl)) r = 1\nq : Ideal R\nhq : q.IsPrime\ne : q ≠ p\nthis : { asIdeal := q, isPrime... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsArtinianRing R\np : Ideal R\ninst✝ : p.IsPrime\nr : R\nhr : (toPiLocalization R) r = Pi.single { asIdeal := p, isPrime := ⋯ } 1\nthis✝ : (algebraMap R (Localization p.primeCompl)) r = 1\nq : Ideal R\nhq : q.IsPrime\ne : q ≠ p\nthis : { asIdeal := q, isPrime := ⋯ } ≠ { ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 388,
"column": 67
} | {
"line": 388,
"column": 78
} | {
"line": 388,
"column": 79
} | [
{
"pp": "S : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp : Prime p\nI : Ideal S\nhI : p ∣ ↑(absNorm I)\nthis✝ : IsAddTorsionFree S\nthis : CharZero S\n⊢ span {p} ≠ ⊥",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
... | [
"S : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp : Prime p\nI : Ideal S\nhI : p ∣ ↑(absNorm I)\nthis✝ : IsAddTorsionFree S\nthis : CharZero S\n⊢ ¬p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 395,
"column": 29
} | {
"line": 395,
"column": 40
} | {
"line": 395,
"column": 41
} | [
{
"pp": "S : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp : Prime p\nthis✝ : IsAddTorsionFree S\nthis : CharZero S\nhpMax : (span {p}).IsMaximal\nI : Ideal S\nhI' : IsUnit I\nhI : p ∣ ↑(absNorm I)\n⊢ I = ⊤",
"ppTerm": "?m.154",
"assi... | [
"S : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp : Prime p\nthis✝ : IsAddTorsionFree S\nthis : CharZero S\nhpMax : (span {p}).IsMaximal\nI : Ideal S\nhI' : IsUnit I\nhI : p ∣ ↑(absNorm I)\n⊢ I = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 396,
"column": 29
} | {
"line": 396,
"column": 40
} | {
"line": 396,
"column": 41
} | [
{
"pp": "S : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp : Prime p\nthis✝ : IsAddTorsionFree S\nthis : CharZero S\nhpMax : (span {p}).IsMaximal\nhI' : IsUnit ⊤\nhI : p ∣ ↑(absNorm ⊤)\n⊢ p ∣ 1",
"ppTerm": "?m.178",
"assigned": false,... | [
"S : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp : Prime p\nthis✝ : IsAddTorsionFree S\nthis : CharZero S\nhpMax : (span {p}).IsMaximal\nhI' : IsUnit ⊤\nhI : p ∣ ↑(absNorm ⊤)\n⊢ p ∣ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 405,
"column": 43
} | {
"line": 405,
"column": 54
} | {
"line": 405,
"column": 55
} | [
{
"pp": "S : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp : Prime p\nthis✝¹ : IsAddTorsionFree S\nthis✝ : CharZero S\nhpMax : (span {p}).IsMaximal\nI P : Ideal S\nhI' : I ≠ 0\nhP : Prime P\nIH : p ∣ ↑(absNorm I) → ∃ P, P.IsMaximal ∧ under ℤ ... | [
"S : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp : Prime p\nthis✝¹ : IsAddTorsionFree S\nthis✝ : CharZero S\nhpMax : (span {p}).IsMaximal\nI P : Ideal S\nhI' : I ≠ 0\nhP : Prime P\nIH : p ∣ ↑(absNorm I) → ∃ P, P.IsMaximal ∧ under ℤ P = span {p}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Artinian | {
"line": 52,
"column": 6
} | {
"line": 52,
"column": 32
} | {
"line": 52,
"column": 33
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Algebra.FiniteType R A\ninst✝ : IsArtinianRing R\nthis : IsNoetherianRing A\n⊢ Module.Finite R A ↔ Ring.KrullDimLE 0 A",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Algebra.FiniteType R A\ninst✝ : IsArtinianRing R\nthis : IsNoetherianRing A\n⊢ IsArtinianRing A ↔ Ring.KrullDimLE 0 A"
] | finite_iff_isArtinianRing, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Jacobson.Artinian | {
"line": 52,
"column": 33
} | {
"line": 52,
"column": 85
} | {
"line": 53,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Algebra.FiniteType R A\ninst✝ : IsArtinianRing R\nthis : IsNoetherianRing A\n⊢ IsArtinianRing A ↔ Ring.KrullDimLE 0 A",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Algebra.FiniteType R A\ninst✝ : IsArtinianRing R\nthis : IsNoetherianRing A\n⊢ IsNoetherianRing A ∧ Ring.KrullDimLE 0 A ↔ Ring.KrullDimLE 0 A"
] | isArtinianRing_iff_isNoetherianRing_krullDimLE_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 410,
"column": 80
} | {
"line": 410,
"column": 95
} | {
"line": 410,
"column": 96
} | [
{
"pp": "S : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp✝ : Prime p\nthis✝¹ : IsAddTorsionFree S\nthis✝ : CharZero S\nhpMax : (span {p}).IsMaximal\nI P : Ideal S\nhI' : I ≠ 0\nhP : Prime P\nIH : p ∣ ↑(absNorm I) → ∃ P, P.IsMaximal ∧ under ℤ... | [
"S : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp✝ : Prime p\nthis✝¹ : IsAddTorsionFree S\nthis✝ : CharZero S\nhpMax : (span {p}).IsMaximal\nI P : Ideal S\nhI' : I ≠ 0\nhP : Prime P\nIH : p ∣ ↑(absNorm I) → ∃ P, P.IsMaximal ∧ under ℤ P = span {p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Submonoid.Inverses | {
"line": 88,
"column": 2
} | {
"line": 94,
"column": 62
} | {
"line": 96,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\nS : Submonoid M\nhS : S ≤ IsUnit.submonoid M\n⊢ S.leftInv.leftInv = S",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"Monoid.toMulOneClass",
"congrArg",
"InvolutiveInv.toInv",
"Group.toDivi... | [] | refine le_antisymm S.leftInv_leftInv_le ?_
intro x hx
have : x = ((hS hx).unit⁻¹⁻¹ : Mˣ) := by
rw [inv_inv (hS hx).unit]
rfl
rw [this]
exact S.leftInv.unit_mem_leftInv _ (S.unit_mem_leftInv _ hx) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Submonoid.Inverses | {
"line": 88,
"column": 2
} | {
"line": 94,
"column": 62
} | {
"line": 96,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\nS : Submonoid M\nhS : S ≤ IsUnit.submonoid M\n⊢ S.leftInv.leftInv = S",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"Monoid.toMulOneClass",
"congrArg",
"InvolutiveInv.toInv",
"Group.toDivi... | [] | refine le_antisymm S.leftInv_leftInv_le ?_
intro x hx
have : x = ((hS hx).unit⁻¹⁻¹ : Mˣ) := by
rw [inv_inv (hS hx).unit]
rfl
rw [this]
exact S.leftInv.unit_mem_leftInv _ (S.unit_mem_leftInv _ hx) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 431,
"column": 6
} | {
"line": 431,
"column": 91
} | {
"line": 432,
"column": 8
} | [
{
"pp": "case inr.refine_1\nS : Type u_1\ninst✝⁴ : CommRing S\ninst✝³ : IsDedekindDomain S\ninst✝² : Free ℤ S\ninst✝¹ : Module.Finite ℤ S\ninst✝ : CharZero S\nn : ℕ\nhn : n > 0\nf : Ideal S → Ideal (S ⧸ span {↑n}) := fun I ↦ map (Quotient.mk (span {↑n})) I\n⊢ ((Algebra.norm ℤ) ↑n).natAbs ≠ 0",
"ppTerm": "?i... | [
"case inr.refine_1\nS : Type u_1\ninst✝⁴ : CommRing S\ninst✝³ : IsDedekindDomain S\ninst✝² : Free ℤ S\ninst✝¹ : Module.Finite ℤ S\ninst✝ : CharZero S\nn : ℕ\nhn : n > 0\nf : Ideal S → Ideal (S ⧸ span {↑n}) := fun I ↦ map (Quotient.mk (span {↑n})) I\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.NilpotentKer | {
"line": 47,
"column": 6
} | {
"line": 47,
"column": 93
} | {
"line": 48,
"column": 8
} | [
{
"pp": "case succ.refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Module.Finite R T\nf : S →ₐ[R] T\nhf₁ : Function.Surjective ⇑f\nI : Ideal S\nhI : RingHom.ker f = I\nhf₂ : I ≤ nilradical S\nh... | [
"case succ.refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Module.Finite R T\nf : S →ₐ[R] T\nhf₁ : Function.Surjective ⇑f\nI : Ideal S\nhI : RingHom.ker f = I\nhf₂ : I ≤ nilradical S\nhf₃ : I.FG\nt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Submonoid.Inverses | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 56
} | {
"line": 169,
"column": 57
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nS : Submonoid M\nhS : S ≤ IsUnit.submonoid M\nx : ↥S.leftInv\n⊢ ↑((S.leftInvEquiv hS) x) * ↑x = 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Submonoid.fromLeftInv",
"Eq.mpr",
"MulOne.toOne",
"MulEquiv.instEquivLike"... | [
"M : Type u_1\ninst✝ : CommMonoid M\nS : Submonoid M\nhS : S ≤ IsUnit.submonoid M\nx : ↥S.leftInv\n⊢ ↑(S.fromLeftInv x) * ↑x = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Submonoid.Inverses | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 61
} | {
"line": 183,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nS : Submonoid M\nhS : S ≤ IsUnit.submonoid M\nx : ↥S\n⊢ ↑x * ↑((S.leftInvEquiv hS).symm x) = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"MulEquiv.instEquivLike",
"Submonoid.mul",
"HM... | [
"M : Type u_1\ninst✝ : CommMonoid M\nS : Submonoid M\nhS : S ≤ IsUnit.submonoid M\nx : ↥S\n⊢ x = (S.leftInvEquiv hS) ((S.leftInvEquiv hS).symm x)"
] | convert! S.leftInvEquiv_mul hS ((S.leftInvEquiv hS).symm x) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 440,
"column": 6
} | {
"line": 441,
"column": 73
} | {
"line": 441,
"column": 73
} | [
{
"pp": "S : Type u_1\ninst✝⁴ : CommRing S\ninst✝³ : IsDedekindDomain S\ninst✝² : Free ℤ S\ninst✝¹ : Module.Finite ℤ S\ninst✝ : CharZero S\nn : ℕ\n⊢ {I | absNorm I ≤ n}.Finite",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Nat.instMulZeroOneClass",
... | [
"S : Type u_1\ninst✝⁴ : CommRing S\ninst✝³ : IsDedekindDomain S\ninst✝² : Free ℤ S\ninst✝¹ : Module.Finite ℤ S\ninst✝ : CharZero S\nn : ℕ\n⊢ (⋃ i ∈ Set.Icc 0 n, {I | absNorm I = i}).Finite"
] | show {I : Ideal S | Ideal.absNorm I ≤ n} =
(⋃ i ∈ Set.Icc 0 n, {I : Ideal S | Ideal.absNorm I = i}) by ext; simp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Finiteness.NilpotentKer | {
"line": 57,
"column": 8
} | {
"line": 57,
"column": 31
} | {
"line": 57,
"column": 32
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Module.Finite R T\nf : S →ₐ[R] T\nhf₁ : Function.Surjective ⇑f\nI : Ideal S\nhI : RingHom.ker f = I\nhf₂ : I ≤ nilradical S\nhf₃ : I.FG\nthis✝ : M... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Module.Finite R T\nf : S →ₐ[R] T\nhf₁ : Function.Surjective ⇑f\nI : Ideal S\nhI : RingHom.ker f = I\nhf₂ : I ≤ nilradical S\nhf₃ : I.FG\nthis✝ : Module.Finite... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.JacobsonSpace | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 13
} | {
"line": 80,
"column": 14
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : JacobsonSpace X\n⊢ closure[inst✝¹] (closedPoints X) = Set.univ",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : JacobsonSpace X\n⊢ closure[inst✝¹] (closedPoints X) = Set.univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.NilpotentKer | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 45
} | {
"line": 62,
"column": 46
} | [
{
"pp": "case e'_6\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Module.Finite R T\nf : S →ₐ[R] T\nhf₁ : Function.Surjective ⇑f\nI : Ideal S\nhI : RingHom.ker f = I\nhf₂ : I ≤ nilradical S\nhf₃ : I.FG... | [
"case e'_6\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Module.Finite R T\nf : S →ₐ[R] T\nhf₁ : Function.Surjective ⇑f\nI : Ideal S\nhI : RingHom.ker f = I\nhf₂ : I ≤ nilradical S\nhf₃ : I.FG\nthis✝ : Mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.JacobsonSpace | {
"line": 207,
"column": 27
} | {
"line": 207,
"column": 38
} | {
"line": 207,
"column": 39
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ninst✝ : JacobsonSpace X\nS : Set X\nhS : IsLocallyClosed S\nhf₁ : Continuous[inst✝², inst✝¹] f\nhf₂ : IsClosedMap f\nhfS : (f '' S).Finite\nhS'' : S.Nonempty\nhS' : IsIrreducible S\nH₁ : IsIrreducible (S ∩ ... | [
"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ninst✝ : JacobsonSpace X\nS : Set X\nhS : IsLocallyClosed S\nhf₁ : Continuous[inst✝², inst✝¹] f\nhf₂ : IsClosedMap f\nhfS : (f '' S).Finite\nhS'' : S.Nonempty\nhS' : IsIrreducible S\nH₁ : IsIrreducible (S ∩ closedPoints... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HopkinsLevitzki | {
"line": 182,
"column": 7
} | {
"line": 182,
"column": 59
} | {
"line": 182,
"column": 60
} | [
{
"pp": "R : Type u_3\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\n⊢ IsArtinianRing R ↔ Ring.KrullDimLE 0 R",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsArtinianRing",
"congrArg",
"CommSemiring.toSemiring",
"isArtinianRing_iff_isNoetherian... | [
"R : Type u_3\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\n⊢ IsNoetherianRing R ∧ Ring.KrullDimLE 0 R ↔ Ring.KrullDimLE 0 R"
] | isArtinianRing_iff_isNoetherianRing_krullDimLE_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LocalRing.Length | {
"line": 65,
"column": 19
} | {
"line": 65,
"column": 48
} | {
"line": 65,
"column": 49
} | [
{
"pp": "case pos.succ\nA : Type u_1\nB : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : IsLocalRing A\ninst✝⁶ : IsLocalRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsLocalHom (algebraMap A B)\ninst✝³ : AddCommGroup M\ninst✝² : Module A M\ninst✝¹ : Module B M\ninst✝ : IsScalarTower A B M\n... | [
"case pos.succ\nA : Type u_1\nB : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : IsLocalRing A\ninst✝⁶ : IsLocalRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsLocalHom (algebraMap A B)\ninst✝³ : AddCommGroup M\ninst✝² : Module A M\ninst✝¹ : Module B M\ninst✝ : IsScalarTower A B M\nh : IsFinite... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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