module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.ModelTheory.Semantics | {
"line": 1046,
"column": 90
} | {
"line": 1048,
"column": 67
} | {
"line": 1050,
"column": 0
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nr : L.Relations 2\n⊢ M ⊨ r.symmetric ↔ Std.Symm fun x y ↦ RelMap r ![x, y]",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"FirstOrder.Language.BoundedFormula.imp",
"FirstOrder.Language.Sentence.Realize",
"Eq.mpr",... | [] | by
rw [symm_def]
exact forall₂_congr fun _ _ ↦ imp_congr realize_rel₂ realize_rel₂ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Semantics | {
"line": 1082,
"column": 22
} | {
"line": 1082,
"column": 33
} | {
"line": 1082,
"column": 34
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nn : ℕ\nxs : Fin n → M\nh :\n ∀ (φ : L.BoundedFormula Empty n) (x x_1 : Fin n),\n (decide ¬x = x_1) = true → ∼(var (Sum.inr x) =' var (Sum.inr x_1)) = φ → φ.Realize default xs\ni j : Fin n\nij : i ≠ j\n⊢ (decide ¬i = j) = true",
"ppTerm": "?m.70",... | [
"L : Language\nM : Type w\ninst✝ : L.Structure M\nn : ℕ\nxs : Fin n → M\nh :\n ∀ (φ : L.BoundedFormula Empty n) (x x_1 : Fin n),\n (decide ¬x = x_1) = true → ∼(var (Sum.inr x) =' var (Sum.inr x_1)) = φ → φ.Realize default xs\ni j : Fin n\nij : i ≠ j\n⊢ ¬i = j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Semantics | {
"line": 1084,
"column": 4
} | {
"line": 1084,
"column": 15
} | {
"line": 1084,
"column": 16
} | [
{
"pp": "case refine_2\nL : Language\nM : Type w\ninst✝ : L.Structure M\nn : ℕ\nxs : Nonempty (Fin n ↪ M)\ni j : Fin n\nij : (decide ¬i = j) = true\n⊢ (∼(var (Sum.inr i) =' var (Sum.inr j))).Realize default ⇑xs.some",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Function.instEm... | [
"case refine_2\nL : Language\nM : Type w\ninst✝ : L.Structure M\nn : ℕ\nxs : Nonempty (Fin n ↪ M)\ni j : Fin n\nij : (decide ¬i = j) = true\n⊢ ¬i = j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Skolem | {
"line": 153,
"column": 6
} | {
"line": 153,
"column": 47
} | {
"line": 154,
"column": 6
} | [
{
"pp": "case refine_2\nL : Language\nM : Type w\ninst✝ : L.Structure M\ns : Set M\nκ : Cardinal.{w'}\nh3 : lift.{w', max u v} L.card ≤ lift.{max u v, w'} κ\nh4 : lift.{w, w'} κ ≤ lift.{w', w} #M\ns' : Set (ULift.{w', w} M)\nh2 : lift.{w', w} #↑s ≤ #↑s'\nh1 : ℵ₀ ≤ #↑s'\nhs' : #↑s' = lift.{w, w'} κ\nthis : Nonem... | [
"case refine_2\nL : Language\nM : Type w\ninst✝ : L.Structure M\ns : Set M\nκ : Cardinal.{w'}\nh3 : lift.{w', max u v} L.card ≤ lift.{max u v, w'} κ\nh4 : lift.{w, w'} κ ≤ lift.{w', w} #M\ns' : Set (ULift.{w', w} M)\nh2 : lift.{w', w} #↑s ≤ #↑s'\nh1 : ℵ₀ ≤ #↑s'\nhs' : #↑s' = lift.{w, w'} κ\nthis : Nonempty M\nh : l... | refine _root_.trans (lift_le.{w}.2 h3) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.MvPolynomial.FreeCommRing | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 64
} | {
"line": 68,
"column": 65
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nR : Type u_3\ninst✝² : DecidableEq κ\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nmonoms : ι → Finset (κ →₀ ℕ)\np : { p // ∀ (i : ι), (p i).support ⊆ monoms i }\ni : ι\nm : κ →₀ ℕ\nhm : m ∉ monoms i\nthis : m ∉ (↑p i).support\n⊢ 0 = (↑p i).coeff m",
"ppTerm": "?m.117",
... | [
"ι : Type u_1\nκ : Type u_2\nR : Type u_3\ninst✝² : DecidableEq κ\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nmonoms : ι → Finset (κ →₀ ℕ)\np : { p // ∀ (i : ι), (p i).support ⊆ monoms i }\ni : ι\nm : κ →₀ ℕ\nhm : m ∉ monoms i\nthis : m ∉ (↑p i).support\n⊢ 0 = (↑p i).coeff m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Satisfiability | {
"line": 376,
"column": 4
} | {
"line": 376,
"column": 15
} | {
"line": 376,
"column": 16
} | [
{
"pp": "case mp\nL : Language\nT : L.Theory\nφ : L.Sentence\nthis : DecidableEq L.Sentence := Classical.decEq L.Sentence\nh : ∀ (T0 : Finset L.Sentence), ↑T0 ⊆ T ∪ {Formula.not φ} → IsSatisfiable ↑T0\nT0 : Finset L.Sentence\nhT0 : ↑T0 ⊆ T\n⊢ (↑T0 ∪ {Formula.not φ}).IsSatisfiable",
"ppTerm": "?mp",
"ass... | [
"case mp\nL : Language\nT : L.Theory\nφ : L.Sentence\nthis : DecidableEq L.Sentence := Classical.decEq L.Sentence\nh : ∀ (T0 : Finset L.Sentence), ↑T0 ⊆ T ∪ {Formula.not φ} → IsSatisfiable ↑T0\nT0 : Finset L.Sentence\nhT0 : ↑T0 ⊆ T\n⊢ IsSatisfiable (insert (Formula.not φ) ↑T0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Satisfiability | {
"line": 378,
"column": 8
} | {
"line": 379,
"column": 60
} | {
"line": 379,
"column": 60
} | [
{
"pp": "L : Language\nT : L.Theory\nφ : L.Sentence\nthis : DecidableEq L.Sentence := Classical.decEq L.Sentence\nh : ∀ (T0 : Finset L.Sentence), ↑T0 ⊆ T ∪ {Formula.not φ} → IsSatisfiable ↑T0\nT0 : Finset L.Sentence\nhT0 : ↑T0 ⊆ T\n⊢ ↑(T0 ∪ {Formula.not φ}) ⊆ T ∪ {Formula.not φ}",
"ppTerm": "?m.41",
"as... | [] | simp only [Finset.coe_union, Finset.coe_singleton]
exact Set.union_subset_union hT0 (Set.Subset.refl _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Satisfiability | {
"line": 378,
"column": 8
} | {
"line": 379,
"column": 60
} | {
"line": 379,
"column": 60
} | [
{
"pp": "L : Language\nT : L.Theory\nφ : L.Sentence\nthis : DecidableEq L.Sentence := Classical.decEq L.Sentence\nh : ∀ (T0 : Finset L.Sentence), ↑T0 ⊆ T ∪ {Formula.not φ} → IsSatisfiable ↑T0\nT0 : Finset L.Sentence\nhT0 : ↑T0 ⊆ T\n⊢ ↑(T0 ∪ {Formula.not φ}) ⊆ T ∪ {Formula.not φ}",
"ppTerm": "?m.41",
"as... | [] | simp only [Finset.coe_union, Finset.coe_singleton]
exact Set.union_subset_union hT0 (Set.Subset.refl _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Satisfiability | {
"line": 382,
"column": 10
} | {
"line": 382,
"column": 21
} | {
"line": 382,
"column": 22
} | [
{
"pp": "L : Language\nT : L.Theory\nφ : L.Sentence\nthis : DecidableEq L.Sentence := Classical.decEq L.Sentence\nh : ∀ (T0 : Finset L.Sentence), ↑T0 ⊆ T → (↑T0 ∪ {Formula.not φ}).IsSatisfiable\nT0 : Finset L.Sentence\nhT0 : ↑T0 ⊆ T ∪ {Formula.not φ}\n⊢ ↑(T0.erase (Formula.not φ)) ⊆ T",
"ppTerm": "?m.72",
... | [
"L : Language\nT : L.Theory\nφ : L.Sentence\nthis : DecidableEq L.Sentence := Classical.decEq L.Sentence\nh : ∀ (T0 : Finset L.Sentence), ↑T0 ⊆ T → (↑T0 ∪ {Formula.not φ}).IsSatisfiable\nT0 : Finset L.Sentence\nhT0 : ↑T0 ⊆ T ∪ {Formula.not φ}\n⊢ ↑T0 ⊆ insert (Formula.not φ) T"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Algebra.Field.IsAlgClosed | {
"line": 201,
"column": 29
} | {
"line": 201,
"column": 67
} | {
"line": 201,
"column": 68
} | [
{
"pp": "φ : Language.ring.Sentence\nT0 : Finset Language.ring.Sentence\nhT0 : ↑T0 ⊆ Theory.ACF 0\nh : ↑T0 ⊨ᵇ φ\np : ℕ\nhp : p ∈ {q | Nat.Prime q}\nq : ℕ\nproperty✝ : Nat.Prime q\nhq : ⟨q, property✝⟩ ∉ {⟨p, hp⟩}\nK : Type\nstruc✝ : Language.ring.Structure K\nis_model✝ : K ⊨ Theory.ACF ↑⟨q, property✝⟩\nnonempty'... | [
"φ : Language.ring.Sentence\nT0 : Finset Language.ring.Sentence\nhT0 : ↑T0 ⊆ Theory.ACF 0\nh : ↑T0 ⊨ᵇ φ\np : ℕ\nhp : p ∈ {q | Nat.Prime q}\nq : ℕ\nproperty✝ : Nat.Prime q\nhq : ⟨q, property✝⟩ ∉ {⟨p, hp⟩}\nK : Type\nstruc✝ : Language.ring.Structure K\nis_model✝ : K ⊨ Theory.ACF ↑⟨q, property✝⟩\nnonempty'✝ : Nonempty... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Algebra.Field.IsAlgClosed | {
"line": 229,
"column": 2
} | {
"line": 230,
"column": 79
} | {
"line": 231,
"column": 4
} | [
{
"pp": "φ : Language.ring.Sentence\n⊢ ¬Theory.ACF 0 ⊨ᵇ φ → ¬{p | Theory.ACF ↑p ⊨ᵇ φ}.Infinite",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FirstOrder.Language.ring",
"Nat.Prime",
"Set.ofPred",
"FirstOrder.Language.Theory.ACF",
"FirstOrder.... | [
"φ : Language.ring.Sentence\n⊢ ¬Theory.ACF 0 ⊨ᵇ φ → {p | Theory.ACF ↑p ⊨ᵇ φ}.Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.AxGrothendieck | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 43
} | {
"line": 84,
"column": 44
} | [
{
"pp": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nS : Set (ι → R)\nhm : Set.MapsTo (fun v i ↦ (eval v) (ps i)) S S\nhinj : Set.InjOn (fun v i ↦... | [
"case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nS : Set (ι → R)\nhm : Set.MapsTo (fun v i ↦ (eval v) (ps i)) S S\nhinj : Set.InjOn (fun v i ↦ (eval v) (p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.OreLocalization.Cardinality | {
"line": 92,
"column": 2
} | {
"line": 107,
"column": 7
} | {
"line": 109,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type v\ninst✝ : MulAction R X\nhc : ∀ (s s' : ↥S), Commute s s'\n⊢ #(OreLocalization S X) ≤ lift.{u, v} #X",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"OreLocalization.oreDiv_eq_iff",
... | [] | rcases finite_or_infinite X with _ | _
· have := lift_mk_le_lift_mk_of_surjective (oreDiv_one_surjective_of_finite_right S X)
rwa [lift_umax.{v, u}, lift_id'] at this
have key (x : X) (s s' : S) (h : s • x = s' • x) (hc : Commute s s') : x /ₒ s = x /ₒ s' := by
rw [oreDiv_eq_iff]
refine ⟨s, s'.1, h, ?_⟩
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.OreLocalization.Cardinality | {
"line": 92,
"column": 2
} | {
"line": 107,
"column": 7
} | {
"line": 109,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type v\ninst✝ : MulAction R X\nhc : ∀ (s s' : ↥S), Commute s s'\n⊢ #(OreLocalization S X) ≤ lift.{u, v} #X",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"OreLocalization.oreDiv_eq_iff",
... | [] | rcases finite_or_infinite X with _ | _
· have := lift_mk_le_lift_mk_of_surjective (oreDiv_one_surjective_of_finite_right S X)
rwa [lift_umax.{v, u}, lift_id'] at this
have key (x : X) (s s' : S) (h : s • x = s' • x) (hc : Commute s s') : x /ₒ s = x /ₒ s' := by
rw [oreDiv_eq_iff]
refine ⟨s, s'.1, h, ?_⟩
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Cardinality | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 13
} | {
"line": 49,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝³ : CommSemiring R\nL : Type u\ninst✝² : CommSemiring L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\n⊢ #L ≤ #R",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝³ : CommSemiring R\nL : Type u\ninst✝² : CommSemiring L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\n⊢ #L ≤ #R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Cardinality | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 13
} | {
"line": 79,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nL : Type u\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\nhS : S ≤ R⁰\n⊢ #L = #R",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝³ : CommRing R\nL : Type u\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\nhS : S ≤ R⁰\n⊢ #L = #R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Cardinality | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 36
} | {
"line": 81,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nL : Type u\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\nhS : S ≤ R⁰\n⊢ #L = #R",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"IsLocalization.lift_cardinalMk",
"Cardinal",
"congrArg",
... | [] | simpa using lift_cardinalMk L S hS | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Localization.Cardinality | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 36
} | {
"line": 81,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nL : Type u\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\nhS : S ≤ R⁰\n⊢ #L = #R",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"IsLocalization.lift_cardinalMk",
"Cardinal",
"congrArg",
... | [] | simpa using lift_cardinalMk L S hS | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Cardinality | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 36
} | {
"line": 81,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nL : Type u\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\nhS : S ≤ R⁰\n⊢ #L = #R",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"IsLocalization.lift_cardinalMk",
"Cardinal",
"congrArg",
... | [] | simpa using lift_cardinalMk L S hS | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.AxGrothendieck | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 49
} | {
"line": 163,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝³ : Finite α\nK : Type u_3\ninst✝² : Field K\ninst✝¹ : CompatibleRing K\ninst✝ : Finite ι\nφ : ring.Formula (α ⊕ ι)\nmons : ι → Finset (ι →₀ ℕ)\ninjOnAlt : ∀ {S : Set (ι → K)} (f : (ι → K) → ι → K), Set.InjOn f S ↔ ∀ (x y : ι → K), x ∈ S → y ∈ S → f x = f y → x = y\n⊢ (... | [
"ι : Type u_1\nα : Type u_2\ninst✝³ : Finite α\nK : Type u_3\ninst✝² : Field K\ninst✝¹ : CompatibleRing K\ninst✝ : Finite ι\nφ : ring.Formula (α ⊕ ι)\nmons : ι → Finset (ι →₀ ℕ)\ninjOnAlt : ∀ {S : Set (ι → K)} (f : (ι → K) → ι → K), Set.InjOn f S ↔ ∀ (x y : ι → K), x ∈ S → y ∈ S → f x = f y → x = y\n⊢ (∀ (fa : ⋯) (... | simp +singlePass only [← Sum.elim_comp_inl_inr] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.SetTheory.Cardinal.Divisibility | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 96
} | {
"line": 92,
"column": 4
} | [
{
"pp": "a : Cardinal.{u}\nha : ℵ₀ ≤ a\nh : ∀ ⦃a_1 b : Cardinal.{u}⦄, a = a_1 * b → IsUnit a_1 ∨ IsUnit b\n⊢ False",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : Cardinal.{u}\nha : ℵ₀ ≤ a\nh : ∀ ⦃a_1 b : Cardinal.{u}⦄, a = a_1 * b → IsUnit a_1 ∨ IsUnit b\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Cardinality | {
"line": 67,
"column": 4
} | {
"line": 68,
"column": 56
} | {
"line": 68,
"column": 57
} | [
{
"pp": "case inl\nα : Type u\nh : Fintype α\n⊢ Nonempty (Field α) ↔ ℵ₀ ≤ #α ∨ ∃ n, #α = ↑n ∧ IsPrimePow n",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"eq_false",
"Cardinal",
"congrArg",
... | [
"case inl\nα : Type u\nh : Fintype α\n⊢ Nonempty (Field α) ↔ IsPrimePow ‖α‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Cardinality | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 68
} | {
"line": 69,
"column": 69
} | [
{
"pp": "case inr\nα : Type u\nh : Infinite α\n⊢ Nonempty (Field α) ↔ ℵ₀ ≤ #α ∨ ∃ n, #α = ↑n ∧ IsPrimePow n",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.FieldTheory.Cardinality.0.Field.nonempty_iff._simp_1_3",
"Cardinal",
"congrArg",
... | [
"case inr\nα : Type u\nh : Infinite α\n⊢ Nonempty (Field α)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.AxGrothendieck | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 13
} | {
"line": 243,
"column": 14
} | [
{
"pp": "K : Type u_1\nι : Type u_2\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : Finite ι\np : ι → MvPolynomial ι K\n⊢ (Function.Injective fun v i ↦ (eval v) (p i)) → Function.Surjective fun v i ↦ (eval v) (p i)",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"K : Type u_1\nι : Type u_2\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : Finite ι\np : ι → MvPolynomial ι K\n⊢ (Function.Injective fun v i ↦ (eval v) (p i)) → Function.Surjective fun v i ↦ (eval v) (p i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Extension | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 13
} | {
"line": 150,
"column": 14
} | [
{
"pp": "k : Type u_1\ninst✝⁶ : Field k\ninst✝⁵ : Finite k\np : ℕ\ninst✝⁴ : Fact (Nat.Prime p)\ninst✝³ : CharP k p\nl : Type u_2\ninst✝² : Field l\ninst✝¹ : Algebra k l\ninst✝ : Finite l\ng : Gal(l/k)\nn : ℕ := Module.finrank k l\nthis : NeZero n\ni : ℕ\nleft✝ : i < n\nhi : Extension.frob k p n ^ i = (algEquivE... | [
"k : Type u_1\ninst✝⁶ : Field k\ninst✝⁵ : Finite k\np : ℕ\ninst✝⁴ : Fact (Nat.Prime p)\ninst✝³ : CharP k p\nl : Type u_2\ninst✝² : Field l\ninst✝¹ : Algebra k l\ninst✝ : Finite l\ng : Gal(l/k)\nn : ℕ := Module.finrank k l\nthis : NeZero n\ni : ℕ\nleft✝ : i < n\nhi : Extension.frob k p n ^ i = (algEquivExtension k p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Extension | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 33
} | {
"line": 179,
"column": 2
} | [
{
"pp": "case inr.inl.refine_1\nk : Type u_1\ninst✝ : Field k\nf : k[X]\nhi : Irreducible f\nn : ℕ\nh : f ∣ X ^ Nat.card k ^ n - X\nhn : n ≠ 0\nh✝ : Finite k\np : ℕ\nhp : CharP k p\nthis✝ : Fact (Nat.Prime p)\nthis : NeZero n\n⊢ (Polynomial.map (algebraMap k (Extension k p n)) (X ^ Nat.card k ^ n - X)).Splits",... | [] | apply IsSplittingField.splits | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.FieldTheory.Finite.Extension | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 33
} | {
"line": 179,
"column": 2
} | [
{
"pp": "case inr.inl.refine_1\nk : Type u_1\ninst✝ : Field k\nf : k[X]\nhi : Irreducible f\nn : ℕ\nh : f ∣ X ^ Nat.card k ^ n - X\nhn : n ≠ 0\nh✝ : Finite k\np : ℕ\nhp : CharP k p\nthis✝ : Fact (Nat.Prime p)\nthis : NeZero n\n⊢ (Polynomial.map (algebraMap k (Extension k p n)) (X ^ Nat.card k ^ n - X)).Splits",... | [] | apply IsSplittingField.splits | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Finite.Extension | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 33
} | {
"line": 179,
"column": 2
} | [
{
"pp": "case inr.inl.refine_1\nk : Type u_1\ninst✝ : Field k\nf : k[X]\nhi : Irreducible f\nn : ℕ\nh : f ∣ X ^ Nat.card k ^ n - X\nhn : n ≠ 0\nh✝ : Finite k\np : ℕ\nhp : CharP k p\nthis✝ : Fact (Nat.Prime p)\nthis : NeZero n\n⊢ (Polynomial.map (algebraMap k (Extension k p n)) (X ^ Nat.card k ^ n - X)).Splits",... | [] | apply IsSplittingField.splits | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Finite.Extension | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 13
} | {
"line": 198,
"column": 14
} | [
{
"pp": "k : Type u_1\ninst✝¹ : Field k\ninst✝ : Finite k\nf : k[X]\nhi : Irreducible f\nn : ℕ\nhdvd : f.natDegree ∣ n\na : AdjoinRoot f := AdjoinRoot.root f\nthis✝¹ : NeZero f.natDegree\nthis✝ : Fact (Irreducible f)\np : ℕ\nhp : CharP k p\nthis : Fact (Nat.Prime p)\ne : AdjoinRoot f ≃ₐ[k] Extension k p f.natDe... | [
"k : Type u_1\ninst✝¹ : Field k\ninst✝ : Finite k\nf : k[X]\nhi : Irreducible f\nn : ℕ\nhdvd : f.natDegree ∣ n\na : AdjoinRoot f := AdjoinRoot.root f\nthis✝¹ : NeZero f.natDegree\nthis✝ : Fact (Irreducible f)\np : ℕ\nhp : CharP k p\nthis : Fact (Nat.Prime p)\ne : AdjoinRoot f ≃ₐ[k] Extension k p f.natDegree := algE... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 137,
"column": 6
} | {
"line": 137,
"column": 91
} | {
"line": 139,
"column": 0
} | [
{
"pp": "case intro.refine_2\nK : Type u_1\nσ : Type u_2\ninst✝² : Field K\ninst✝¹ : Fintype K\ninst✝ : Finite σ\nval✝ : Fintype σ\ne : (σ → K) → K\nx✝ : e ∈ ⊤\nn : σ → K\n⊢ e n * (eval n) (indicator n) = e n",
"ppTerm": "?intro.refine_2",
"assigned": true,
"usedConstants": [
"MvPolynomial.eva... | [] | aesop (add simp [eval_indicator_apply_eq_zero, eval_indicator_apply_eq_one, eq_comm]) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 137,
"column": 6
} | {
"line": 137,
"column": 91
} | {
"line": 139,
"column": 0
} | [
{
"pp": "case intro.refine_2.h₀\nK : Type u_1\nσ : Type u_2\ninst✝² : Field K\ninst✝¹ : Fintype K\ninst✝ : Finite σ\nval✝ : Fintype σ\ne : (σ → K) → K\nx✝ : e ∈ ⊤\nn : σ → K\n⊢ ∀ b ∈ Finset.univ, b ≠ n → e b * (eval n) (indicator b) = 0",
"ppTerm": "?intro.refine_2.h₀",
"assigned": true,
"usedConsta... | [] | aesop (add simp [eval_indicator_apply_eq_zero, eval_indicator_apply_eq_one, eq_comm]) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 137,
"column": 6
} | {
"line": 137,
"column": 91
} | {
"line": 139,
"column": 0
} | [
{
"pp": "case intro.refine_2.h₁\nK : Type u_1\nσ : Type u_2\ninst✝² : Field K\ninst✝¹ : Fintype K\ninst✝ : Finite σ\nval✝ : Fintype σ\ne : (σ → K) → K\nx✝ : e ∈ ⊤\nn : σ → K\n⊢ n ∉ Finset.univ → e n * (eval n) (indicator n) = 0",
"ppTerm": "?intro.refine_2.h₁",
"assigned": true,
"usedConstants": [
... | [] | aesop (add simp [eval_indicator_apply_eq_zero, eval_indicator_apply_eq_one, eq_comm]) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.FieldTheory.Differential.Liouville | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 57
} | {
"line": 93,
"column": 4
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝⁸ : Field F\ninst✝⁷ : Field K\ninst✝⁶ : Differential F\ninst✝⁵ : Differential K\ninst✝⁴ : Algebra F K\ninst✝³ : DifferentialAlgebra F K\ninst✝² : CharZero F\nB : IntermediateField F K\ninst✝¹ : FiniteDimensional F ↥B\ninst : IsLiouville F K\na : F\nι : Type\ninst✝ : Fin... | [
"F : Type u_1\nK : Type u_2\ninst✝⁸ : Field F\ninst✝⁷ : Field K\ninst✝⁶ : Differential F\ninst✝⁵ : Differential K\ninst✝⁴ : Algebra F K\ninst✝³ : DifferentialAlgebra F K\ninst✝² : CharZero F\nB : IntermediateField F K\ninst✝¹ : FiniteDimensional F ↥B\ninst : IsLiouville F K\na : F\nι : Type\ninst✝ : Fintype ι\nc : ... | apply inst.isLiouville a ι c hc (B.val ∘ u) (B.val v) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 27
} | {
"line": 203,
"column": 28
} | [
{
"pp": "case intro\nσ K : Type u\ninst✝² : Fintype K\ninst✝¹ : Field K\ninst✝ : Finite σ\nval✝ : Fintype σ\n⊢ Module.rank K (R σ K) < ℵ₀",
"ppTerm": "?intro",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"congrArg",
"AddCommGroup.toAddCom... | [
"case intro\nσ K : Type u\ninst✝² : Fintype K\ninst✝¹ : Field K\ninst✝ : Finite σ\nval✝ : Fintype σ\n⊢ ↑(Fintype.card (σ → K)) < ℵ₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Connected | {
"line": 27,
"column": 6
} | {
"line": 27,
"column": 72
} | {
"line": 27,
"column": 72
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nh : TotallyDisconnectedSpace X\nx : X\n⊢ IsClosed[inst✝] {x}",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"totallyDisconnectedSpace_iff_connectedComponent_singleton",
"congrArg",
"Set.instSingletonSet",... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nh : TotallyDisconnectedSpace X\nx : X\n⊢ IsClosed[inst✝] (connectedComponent x)"
] | ← totallyDisconnectedSpace_iff_connectedComponent_singleton.mp h x | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Differential.Liouville | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 89
} | {
"line": 117,
"column": 90
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝¹² : Field F\ninst✝¹¹ : Field K\ninst✝¹⁰ : Differential F\ninst✝⁹ : Differential K\ninst✝⁸ : Algebra F K\ninst✝⁷ : DifferentialAlgebra F K\ninst✝⁶ : CharZero F\nK' : Type u_3\ninst✝⁵ : Field K'\ninst✝⁴ : Differential K'\ninst✝³ : Algebra F K'\ninst✝² : DifferentialAlgeb... | [
"F : Type u_1\nK : Type u_2\ninst✝¹² : Field F\ninst✝¹¹ : Field K\ninst✝¹⁰ : Differential F\ninst✝⁹ : Differential K\ninst✝⁸ : Algebra F K\ninst✝⁷ : DifferentialAlgebra F K\ninst✝⁶ : CharZero F\nK' : Type u_3\ninst✝⁵ : Field K'\ninst✝⁴ : Differential K'\ninst✝³ : Algebra F K'\ninst✝² : DifferentialAlgebra F K'\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.NormalBasis | {
"line": 53,
"column": 35
} | {
"line": 53,
"column": 64
} | {
"line": 53,
"column": 64
} | [
{
"pp": "case e'_4\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Finite L\nthis✝ : Finite K\nthis : Fintype K\nx : L\nhx :\n Ideal.span {X ^ finrank K L - 1} =\n (toSpanSingleton K[X] (AEval' (frobeniusAlgHom K L).toLinearMap)\n ((AEval'.of (frobeniusA... | [
"case e'_4\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Finite L\nthis✝ : Finite K\nthis : Fintype K\nx : L\nhx :\n Ideal.span {X ^ finrank K L - 1} =\n (toSpanSingleton K[X] (AEval' (frobeniusAlgHom K L).toLinearMap)\n ((AEval'.of (frobeniusAlgHom K L).t... | LinearMap.coe_restrictScalars | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.Differential.Liouville | {
"line": 187,
"column": 6
} | {
"line": 193,
"column": 30
} | {
"line": 195,
"column": 0
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝⁹ : Field F\ninst✝⁸ : Field K\ninst✝⁷ : Differential F\ninst✝⁶ : Differential K\ninst✝⁵ : Algebra F K\ninst✝⁴ : DifferentialAlgebra F K\ninst✝³ : CharZero F\ninst✝² : FiniteDimensional F K\ninst✝¹ : IsGalois F K\na : F\nι : Type\ninst✝ : Fintype ι\nc : ι → F\nhc : ∀ (x ... | [] | · rcongr e
apply_fun e at h
simp only [AlgEquiv.commutes, map_add, map_sum, map_mul] at h
convert! h using 2
· rcongr x
simp [logDeriv, algEquiv_deriv']
· rw [algEquiv_deriv'] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.CardinalEmb | {
"line": 131,
"column": 8
} | {
"line": 132,
"column": 83
} | {
"line": 133,
"column": 8
} | [
{
"pp": "case inl\nF : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nrank_inf : Fact (ℵ₀ ≤ Module.rank F E)\ninst✝ : Algebra.IsAlgebraic F E\ni : (Module.rank F E).ord.ToType\nih : (y : (Module.rank F E).ord.ToType) → y < i → (Module.rank F E).ord.ToType\ns : Set E := failed to p... | [
"case inl\nF : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nrank_inf : Fact (ℵ₀ ≤ Module.rank F E)\ninst✝ : Algebra.IsAlgebraic F E\ni : (Module.rank F E).ord.ToType\nih : (y : (Module.rank F E).ord.ToType) → y < i → (Module.rank F E).ord.ToType\ns : Set E := failed to pretty print ... | have : FiniteDimensional F (adjoin F s) :=
finiteDimensional_adjoin fun x _ ↦ (IsAlgebraic.isAlgebraic x).isIntegral | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.CardinalEmb | {
"line": 143,
"column": 70
} | {
"line": 145,
"column": 71
} | {
"line": 147,
"column": 0
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nrank_inf : Fact (ℵ₀ ≤ Module.rank F E)\ninst✝ : Algebra.IsAlgebraic F E\ni : (Module.rank F E).ord.ToType\n⊢ IsLeast {k | b k ∉ adjoin F (⇑b ∘ φ '' Iio i)} (φ i)",
"ppTerm": "?m.50",
"assigned": true,
"usedCon... | [] | by
rw [image_eq_range, leastExt, wellFounded_lt.fix_eq]
exact ⟨wellFounded_lt.min_mem _ _, fun _ ↦ (wellFounded_lt.min_le ·)⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.IsPerfectClosure | {
"line": 165,
"column": 15
} | {
"line": 165,
"column": 51
} | {
"line": 165,
"column": 52
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : CommSemiring K\ninst✝¹ : CommSemiring L\ni : K →+* L\np : ℕ\ninst✝ : IsPRadical i p\nx : K\nn : ℕ\nh : x ^ p ^ n = 0\n⊢ i x ^ p ^ n = 0",
"ppTerm": "?m.141",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\nL : Type u_2\ninst✝² : CommSemiring K\ninst✝¹ : CommSemiring L\ni : K →+* L\np : ℕ\ninst✝ : IsPRadical i p\nx : K\nn : ℕ\nh : x ^ p ^ n = 0\n⊢ i x ^ p ^ n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsPerfectClosure | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 59
} | {
"line": 181,
"column": 60
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring K\ninst✝³ : CommSemiring L\ninst✝² : CommSemiring M\ni : K →+* L\nf : L →+* M\np : ℕ\ninst✝¹ : IsPRadical i p\ninst✝ : IsPRadical f p\nx : K\nh : i x ∈ RingHom.ker f\n⊢ x ∈ pNilradical K p",
"ppTerm": "?m.164",
"assigned": false,
... | [
"K : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring K\ninst✝³ : CommSemiring L\ninst✝² : CommSemiring M\ni : K →+* L\nf : L →+* M\np : ℕ\ninst✝¹ : IsPRadical i p\ninst✝ : IsPRadical f p\nx : K\nh : i x ∈ RingHom.ker f\n⊢ x ∈ pNilradical K p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.Profinite | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 23
} | {
"line": 316,
"column": 24
} | [
{
"pp": "case h\nk : Type u_3\nK : Type u_4\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nH : Set Gal(K/k)\nL : FiniteGaloisIntermediateField k K\nle : ↑L.fixingSubgroup ⊆ H\n⊢ (fun a ↦ (mulEquivToLimit k K) a) '' ↑L.fixingSubgroup ⊆ (fun a ↦ (mulEquivToLimit k K).toEquiv a) '... | [
"case h\nk : Type u_3\nK : Type u_4\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nH : Set Gal(K/k)\nL : FiniteGaloisIntermediateField k K\nle : ↑L.fixingSubgroup ⊆ H\n⊢ ↑L.fixingSubgroup ⊆ (fun a ↦ (mulEquivToLimit k K) a) ⁻¹' (fun a ↦ (mulEquivToLimit k K) a) '' H"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PerfectClosure | {
"line": 351,
"column": 17
} | {
"line": 352,
"column": 38
} | {
"line": 352,
"column": 38
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ × K\nn : ℕ\nih : mk K p x ^ n = mk K p (x.1, x.2 ^ n)\n⊢ (⇑(frobenius K p))^[(x.1, x.2 ^ n * x.2).1 + 0]\n ((x.1, x.2 ^ n).1 + x.1,\n (⇑(frobenius K p))^[x.1] (x.1, x.2 ^ n).2 * (⇑(frobenius K p)... | [] | simp_rw [iterate_frobenius, add_zero, mul_pow, ← pow_mul,
← pow_add, mul_assoc, ← pow_add] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.FieldTheory.PerfectClosure | {
"line": 351,
"column": 17
} | {
"line": 352,
"column": 38
} | {
"line": 352,
"column": 38
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ × K\nn : ℕ\nih : mk K p x ^ n = mk K p (x.1, x.2 ^ n)\n⊢ (⇑(frobenius K p))^[(x.1, x.2 ^ n * x.2).1 + 0]\n ((x.1, x.2 ^ n).1 + x.1,\n (⇑(frobenius K p))^[x.1] (x.1, x.2 ^ n).2 * (⇑(frobenius K p)... | [] | simp_rw [iterate_frobenius, add_zero, mul_pow, ← pow_mul,
← pow_add, mul_assoc, ← pow_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.PerfectClosure | {
"line": 351,
"column": 17
} | {
"line": 352,
"column": 38
} | {
"line": 352,
"column": 38
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ × K\nn : ℕ\nih : mk K p x ^ n = mk K p (x.1, x.2 ^ n)\n⊢ (⇑(frobenius K p))^[(x.1, x.2 ^ n * x.2).1 + 0]\n ((x.1, x.2 ^ n).1 + x.1,\n (⇑(frobenius K p))^[x.1] (x.1, x.2 ^ n).2 * (⇑(frobenius K p)... | [] | simp_rw [iterate_frobenius, add_zero, mul_pow, ← pow_mul,
← pow_add, mul_assoc, ← pow_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IsRealClosed.Basic | {
"line": 80,
"column": 40
} | {
"line": 80,
"column": 51
} | {
"line": 80,
"column": 52
} | [
{
"pp": "R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nn : ℕ\nhn : Odd n\nr : R\nhr : (X ^ n - C x).IsRoot r\n⊢ r ^ n - x = 0",
"ppTerm": "?m.55",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nn : ℕ\nhn : Odd n\nr : R\nhr : (X ^ n - C x).IsRoot r\n⊢ r ^ n - x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsRealClosed.Basic | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 15
} | {
"line": 84,
"column": 16
} | [
{
"pp": "case inl\nR : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nn : ℕ\nhk : Odd ↑n\n⊢ ∃ r, x = r ^ ↑n",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"congrArg",
"DivInvMonoid.toZPow",
"Exists",
"Field.toDivisionR... | [
"case inl\nR : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nn : ℕ\nhk : Odd ↑n\n⊢ ∃ r, x = r ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsRealClosed.Basic | {
"line": 84,
"column": 43
} | {
"line": 84,
"column": 54
} | {
"line": 84,
"column": 55
} | [
{
"pp": "R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nn : ℕ\nhk : Odd ↑n\n⊢ Odd n",
"ppTerm": "?m.54",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nn : ℕ\nhk : Odd ↑n\n⊢ Odd n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsRealClosed.Basic | {
"line": 85,
"column": 38
} | {
"line": 85,
"column": 49
} | {
"line": 85,
"column": 50
} | [
{
"pp": "R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nn : ℕ\nhk : Odd (-↑n)\n⊢ Odd ?m.72",
"ppTerm": "?m.73",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nn : ℕ\nhk : Odd (-↑n)\n⊢ Odd ?m.72"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsRealClosed.Basic | {
"line": 86,
"column": 19
} | {
"line": 86,
"column": 30
} | {
"line": 86,
"column": 31
} | [
{
"pp": "R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nn : ℕ\nhk : Odd (-↑n)\nr : R\nhr : x = r ^ n\n⊢ x = r⁻¹ ^ (-↑n)",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.to... | [
"R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nn : ℕ\nhk : Odd (-↑n)\nr : R\nhr : x = r ^ n\n⊢ x = r ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PerfectClosure | {
"line": 373,
"column": 4
} | {
"line": 373,
"column": 64
} | {
"line": 373,
"column": 65
} | [
{
"pp": "case mp\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y z : ℕ\nH : (⇑(frobenius K p))^[(0, ↑y).1 + z] (0, ↑x).2 = (⇑(frobenius K p))^[(0, ↑x).1 + z] (0, ↑y).2\n⊢ ↑x = ↑y",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": ... | [
"case mp\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y z : ℕ\nH : (⇑(frobenius K p))^[(0, ↑y).1 + z] (0, ↑x).2 = (⇑(frobenius K p))^[(0, ↑x).1 + z] (0, ↑y).2\n⊢ ↑x = ↑y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PerfectClosure | {
"line": 404,
"column": 21
} | {
"line": 404,
"column": 78
} | {
"line": 404,
"column": 79
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝¹ : PerfectClosure K p\nx : ℕ × K\nx✝ : IsNilpotent (mk K p x)\nn m : ℕ\nh : (iterateFrobenius K p m) (x.2 ^ p ^ n) = 0\n⊢ (iterateFrobenius K p (n + m)) x.2 = 0",
"ppTerm": "?m.96",
"assigned": true,
... | [
"K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝¹ : PerfectClosure K p\nx : ℕ × K\nx✝ : IsNilpotent (mk K p x)\nn m : ℕ\nh : (iterateFrobenius K p m) (x.2 ^ p ^ n) = 0\n⊢ (x.2 ^ p ^ n) ^ p ^ m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsRealClosed.Basic | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 15
} | {
"line": 103,
"column": 16
} | [
{
"pp": "case inl\nR : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nhx : IsSquare x\nn : ℕ\nhk : ↑n ≠ 0\n⊢ ∃ r, x = r ^ ↑n",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"congrArg",
"DivInvMonoid.toZPow",
"Exists",
"... | [
"case inl\nR : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nhx : IsSquare x\nn : ℕ\nhk : ↑n ≠ 0\n⊢ ∃ r, x = r ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsRealClosed.Basic | {
"line": 103,
"column": 49
} | {
"line": 103,
"column": 60
} | {
"line": 103,
"column": 61
} | [
{
"pp": "R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nhx : IsSquare x\nn : ℕ\nhk : ↑n ≠ 0\n⊢ n ≠ 0",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"id",
"Ne",
"instOfNatNat",
"Nat",
"OfNat.ofNat"
],
"usedFVars": [
"n"
],
"... | [
"R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nhx : IsSquare x\nn : ℕ\nhk : ↑n ≠ 0\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsRealClosed.Basic | {
"line": 104,
"column": 44
} | {
"line": 104,
"column": 55
} | {
"line": 104,
"column": 56
} | [
{
"pp": "R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nhx : IsSquare x\nn : ℕ\nhk : -↑n ≠ 0\n⊢ ?m.78 ≠ 0",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"id",
"Ne",
"instOfNatNat",
"Nat",
"OfNat.ofNat"
],
"usedFVars": [],
"usedGoals"... | [
"R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nhx : IsSquare x\nn : ℕ\nhk : -↑n ≠ 0\n⊢ ¬?m.78 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsRealClosed.Basic | {
"line": 105,
"column": 19
} | {
"line": 105,
"column": 30
} | {
"line": 105,
"column": 31
} | [
{
"pp": "R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nhx : IsSquare x\nn : ℕ\nhk : -↑n ≠ 0\nr : R\nhr : x = r ^ n\n⊢ x = r⁻¹ ^ (-↑n)",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"DivisionCommMonoid.toDivisionMonoid",
"Di... | [
"R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nhx : IsSquare x\nn : ℕ\nhk : -↑n ≠ 0\nr : R\nhr : x = r ^ n\n⊢ x = r ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CosetCover | {
"line": 143,
"column": 6
} | {
"line": 143,
"column": 81
} | {
"line": 143,
"column": 82
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq (Subgroup G)\nn : ℕ\nih :\n ∀ m < n,\n ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},\n ⋃ i ∈ s, g i • ↑(H i) = Set.univ →\n ∀ j ∈ s,\n ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ →\n m = (... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq (Subgroup G)\nn : ℕ\nih :\n ∀ m < n,\n ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},\n ⋃ i ∈ s, g i • ↑(H i) = Set.univ →\n ∀ j ∈ s,\n ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ →\n m = (Finset.image... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CosetCover | {
"line": 149,
"column": 8
} | {
"line": 149,
"column": 19
} | {
"line": 149,
"column": 20
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq (Subgroup G)\nn : ℕ\nih :\n ∀ m < n,\n ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},\n ⋃ i ∈ s, g i • ↑(H i) = Set.univ →\n ∀ j ∈ s,\n ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ →\n m = (... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq (Subgroup G)\nn : ℕ\nih :\n ∀ m < n,\n ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},\n ⋃ i ∈ s, g i • ↑(H i) = Set.univ →\n ∀ j ∈ s,\n ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ →\n m = (Finset.image... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CosetCover | {
"line": 154,
"column": 8
} | {
"line": 154,
"column": 85
} | {
"line": 154,
"column": 86
} | [
{
"pp": "case refine_2\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq (Subgroup G)\nn : ℕ\nih :\n ∀ m < n,\n ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},\n ⋃ i ∈ s, g i • ↑(H i) = Set.univ →\n ∀ j ∈ s,\n ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ →\n ... | [
"case refine_2\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq (Subgroup G)\nn : ℕ\nih :\n ∀ m < n,\n ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},\n ⋃ i ∈ s, g i • ↑(H i) = Set.univ →\n ∀ j ∈ s,\n ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ →\n m ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CosetCover | {
"line": 172,
"column": 48
} | {
"line": 172,
"column": 59
} | {
"line": 172,
"column": 60
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq (Subgroup G)\nn : ℕ\nih :\n ∀ m < n,\n ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},\n ⋃ i ∈ s, g i • ↑(H i) = Set.univ →\n ∀ j ∈ s,\n ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ →\n m = (... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq (Subgroup G)\nn : ℕ\nih :\n ∀ m < n,\n ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},\n ⋃ i ∈ s, g i • ↑(H i) = Set.univ →\n ∀ j ∈ s,\n ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ →\n m = (Finset.image... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Laurent | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 31
} | {
"line": 106,
"column": 32
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nr : R\ninst✝ : IsDomain R\nx✝¹ x✝ : R⟮X⟯\nh : (laurent r) x✝¹ = (laurent r) x✝\n⊢ x✝¹ = x✝",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝¹ : CommRing R\nr : R\ninst✝ : IsDomain R\nx✝¹ x✝ : R⟮X⟯\nh : (laurent r) x✝¹ = (laurent r) x✝\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.JacobsonNoether | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 41
} | {
"line": 125,
"column": 42
} | [
{
"pp": "case h\nD : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Algebra.IsAlgebraic (↥k) D\nH : k ≠ ⊤\np : ℕ\nhp : ExpChar D p\ninsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k\na : D\nha : ∃ x, ¬x * a = a * x\nha₀ : a ≠ 0\n⊢ a * ha.choose - ha.choose * a ≠ 0",
"ppTerm": "?h",
"assigned": true,
"usedCo... | [
"case h\nD : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Algebra.IsAlgebraic (↥k) D\nH : k ≠ ⊤\np : ℕ\nhp : ExpChar D p\ninsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k\na : D\nha : ∃ x, ¬x * a = a * x\nha₀ : a ≠ 0\n⊢ ¬a * ha.choose = ha.choose * a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 51
} | {
"line": 50,
"column": 52
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁵ : CommRing F\ninst✝⁴ : Nontrivial F\ninst✝³ : CommRing E\ninst✝² : IsDomain E\ninst✝¹ : Algebra F E\nι : Type w\nx : ι → E\ninst✝ : Nonempty ι\nhx : IsTranscendenceBasis F x\nK : Subalgebra F E := adjoin F (range x)\nthis✝ : Algebra.IsAlgebraic (↥K) E\nthis : Infinite ↥K\... | [
"F : Type u\nE : Type v\ninst✝⁵ : CommRing F\ninst✝⁴ : Nontrivial F\ninst✝³ : CommRing E\ninst✝² : IsDomain E\ninst✝¹ : Algebra F E\nι : Type w\nx : ι → E\ninst✝ : Nonempty ι\nhx : IsTranscendenceBasis F x\nK : Subalgebra F E := adjoin F (range x)\nthis✝ : Algebra.IsAlgebraic (↥K) E\nthis : Infinite ↥K\n⊢ #E ≤ #↥K"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 48
} | {
"line": 69,
"column": 49
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.Transcendental F E\nι : Type v\nx : ι → E\nhx : IsTranscendenceBasis F x\nthis : Nonempty ι\n⊢ Module.rank F E = #E",
"ppTerm": "?m.45",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.Transcendental F E\nι : Type v\nx : ι → E\nhx : IsTranscendenceBasis F x\nthis : Nonempty ι\n⊢ Module.rank F E = #E"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KummerExtension | {
"line": 92,
"column": 2
} | {
"line": 94,
"column": 9
} | {
"line": 94,
"column": 10
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nζ : R\nhζ : IsPrimitiveRoot ζ n\nα a : R\nhn : 0 < n\ne : α ^ n = a\nK : Type u_1 := FractionRing R\ni : R →+* K := algebraMap R K\nh : Function.Injective ⇑(algebraMap R K)\n⊢ Polynomial.map i (X ^ n - C a) = Polynomial.map i (∏ i ∈ Finset.r... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nζ : R\nhζ : IsPrimitiveRoot ζ n\nα a : R\nhn : 0 < n\ne : α ^ n = a\nK : Type u_1 := FractionRing R\ni : R →+* K := algebraMap R K\nh : Function.Injective ⇑(algebraMap R K)\n⊢ X ^ n - C (i a) = ∏ x ∈ Finset.range n, (X - C (i ζ) ^ x * C (i α))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KummerExtension | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 23
} | {
"line": 109,
"column": 24
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nn m : ℕ\na : K\nhm : Irreducible (X ^ m - C a)\nhn :\n ∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n minpoly K x = X ^ m - C a → Irreducible (X ^ n - C (AdjoinSimple.gen K x))\nhm' : m ≠ 0\n⊢ Irreducible (X ^ (n * m) - C a)",
"ppTerm": "?m.99",
... | [
"K : Type u\ninst✝ : Field K\nn m : ℕ\na : K\nhm : Irreducible (X ^ m - C a)\nhn :\n ∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n minpoly K x = X ^ m - C a → Irreducible (X ^ n - C (AdjoinSimple.gen K x))\nhm' : m ≠ 0\n⊢ Irreducible ((X ^ n) ^ m - C a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KummerExtension | {
"line": 110,
"column": 8
} | {
"line": 110,
"column": 52
} | {
"line": 110,
"column": 53
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nn m : ℕ\na : K\nhm : Irreducible (X ^ m - C a)\nhn :\n ∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n minpoly K x = X ^ m - C a → Irreducible (X ^ n - C (AdjoinSimple.gen K x))\nhm' : m ≠ 0\n⊢ ∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x ... | [
"K : Type u\ninst✝ : Field K\nn m : ℕ\na : K\nhm : Irreducible (X ^ m - C a)\nhn :\n ∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n minpoly K x = X ^ m - C a → Irreducible (X ^ n - C (AdjoinSimple.gen K x))\nhm' : m ≠ 0\n⊢ ∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KummerExtension | {
"line": 118,
"column": 11
} | {
"line": 118,
"column": 22
} | {
"line": 118,
"column": 23
} | [
{
"pp": "case one\nK : Type u\ninst✝ : Field K\nhn : Odd 1\na : K\nha : ∀ (p : ℕ), Nat.Prime p → p ∣ 1 → ∀ (b : K), b ^ p ≠ a\n⊢ Irreducible (X ^ 1 - C a)",
"ppTerm": "?one",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"HSub.hSub",
"RingHo... | [
"case one\nK : Type u\ninst✝ : Field K\nhn : Odd 1\na : K\nha : ∀ (p : ℕ), Nat.Prime p → p ∣ 1 → ∀ (b : K), b ^ p ≠ a\n⊢ Irreducible (X - C a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KummerExtension | {
"line": 125,
"column": 6
} | {
"line": 126,
"column": 37
} | {
"line": 126,
"column": 38
} | [
{
"pp": "p n : ℕ\nhp : Nat.Prime p\nIH :\n ∀ {K : Type u} [inst : Field K],\n Odd n → ∀ {a : K}, (∀ (p : ℕ), Nat.Prime p → p ∣ n → ∀ (b : K), b ^ p ≠ a) → Irreducible (X ^ n - C a)\nK : Type u\ninst✝² : Field K\nhn : Odd (p * n)\na : K\nha : ∀ (p_1 : ℕ), Nat.Prime p_1 → p_1 ∣ p * n → ∀ (b : K), b ^ p_1 ≠ a\... | [
"p n : ℕ\nhp : Nat.Prime p\nIH :\n ∀ {K : Type u} [inst : Field K],\n Odd n → ∀ {a : K}, (∀ (p : ℕ), Nat.Prime p → p ∣ n → ∀ (b : K), b ^ p ≠ a) → Irreducible (X ^ n - C a)\nK : Type u\ninst✝² : Field K\nhn : Odd (p * n)\na : K\nha : ∀ (p_1 : ℕ), Nat.Prime p_1 → p_1 ∣ p * n → ∀ (b : K), b ^ p_1 ≠ a\nE : Type u\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KummerExtension | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 77
} | {
"line": 150,
"column": 78
} | [
{
"pp": "K : Type u\ninst✝ : Field K\np : ℕ\nhp : Nat.Prime p\nhp' : p ≠ 2\nn : ℕ\na : K\nha : ∀ (b : K), b ^ p ≠ a\nq : ℕ\nhq : Nat.Prime q\nhq' : q ∣ p ^ n\n⊢ ∀ (b : K), b ^ q ≠ a",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"Na... | [
"K : Type u\ninst✝ : Field K\np : ℕ\nhp : Nat.Prime p\nhp' : p ≠ 2\nn : ℕ\na : K\nha : ∀ (b : K), b ^ p ≠ a\nq : ℕ\nhq : Nat.Prime q\nhq' : q ∣ p ^ n\n⊢ ∀ (b : K), ¬b ^ p = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KummerExtension | {
"line": 375,
"column": 10
} | {
"line": 375,
"column": 45
} | {
"line": 375,
"column": 46
} | [
{
"pp": "K : Type u\ninst✝⁶ : Field K\nn : ℕ\nhζ : (primitiveRoots n K).Nonempty\na✝ : K\nH : Irreducible (X ^ n - C a✝)\nL✝ : Type u_1\ninst✝⁵ : Field L✝\ninst✝⁴ : Algebra K L✝\ninst✝³ : IsSplittingField K L✝ (X ^ n - C a✝)\nα : L✝\nhα : α ^ n = (algebraMap K L✝) a✝\nhn : 0 < n\na : K\nL : Type ?u.56\ninst✝² :... | [
"K : Type u\ninst✝⁶ : Field K\nn : ℕ\nhζ : (primitiveRoots n K).Nonempty\na✝ : K\nH : Irreducible (X ^ n - C a✝)\nL✝ : Type u_1\ninst✝⁵ : Field L✝\ninst✝⁴ : Algebra K L✝\ninst✝³ : IsSplittingField K L✝ (X ^ n - C a✝)\nα : L✝\nhα : α ^ n = (algebraMap K L✝) a✝\nhn : 0 < n\na : K\nL : Type ?u.56\ninst✝² : Field L\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KummerExtension | {
"line": 381,
"column": 2
} | {
"line": 381,
"column": 75
} | {
"line": 381,
"column": 76
} | [
{
"pp": "K : Type u\ninst✝⁴ : Field K\nn : ℕ\na : K\nL : Type u_1\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : IsSplittingField K L (X ^ n - C a)\ninst✝ : NeZero n\nthis : eval (rootOfSplits ⋯ ⋯) (Polynomial.map (algebraMap K L) (X ^ n - C a)) = 0\n⊢ rootOfSplitsXPowSubC ⋯ a L ^ n = (algebraMap K L) a",
... | [
"K : Type u\ninst✝⁴ : Field K\nn : ℕ\na : K\nL : Type u_1\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : IsSplittingField K L (X ^ n - C a)\ninst✝ : NeZero n\nthis : eval (rootOfSplits ⋯ ⋯) (Polynomial.map (algebraMap K L) (X ^ n - C a)) = 0\n⊢ rootOfSplitsXPowSubC ⋯ a L ^ n = (algebraMap K L) a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KummerExtension | {
"line": 413,
"column": 2
} | {
"line": 413,
"column": 56
} | {
"line": 414,
"column": 2
} | [
{
"pp": "K : Type u\ninst✝⁴ : Field K\nn : ℕ\nhζ : (primitiveRoots n K).Nonempty\na : K\nH : Irreducible (X ^ n - C a)\nL : Type u_1\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : IsSplittingField K L (X ^ n - C a)\nα : L\ninst✝ : NeZero n\nσ : Gal(L/K)\nζ : K\nhα : α ∈ Multiset.map (fun x ↦ (algebraMap K L)... | [
"K : Type u\ninst✝⁴ : Field K\nn : ℕ\nhζ : (primitiveRoots n K).Nonempty\na : K\nH : Irreducible (X ^ n - C a)\nL : Type u_1\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : IsSplittingField K L (X ^ n - C a)\nα : L\ninst✝ : NeZero n\nσ : Gal(L/K)\nζ : K\nhζ'✝ : ζ ∈ primitiveRoots n K\nhζ' : IsPrimitiveRoot ζ n\nh... | simp only [Multiset.mem_map, Multiset.mem_range] at hα | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.KummerExtension | {
"line": 417,
"column": 2
} | {
"line": 417,
"column": 23
} | {
"line": 419,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝⁴ : Field K\nn : ℕ\nhζ : (primitiveRoots n K).Nonempty\na : K\nH : Irreducible (X ^ n - C a)\nL : Type u_1\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : IsSplittingField K L (X ^ n - C a)\ninst✝ : NeZero n\nσ : Gal(L/K)\nζ : K\nhζ'✝ : ζ ∈ primitiveRoots n K\nhζ' : IsPrimitiveRoot ζ... | [] | exact smul_comm _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.KummerExtension | {
"line": 439,
"column": 2
} | {
"line": 439,
"column": 51
} | {
"line": 439,
"column": 52
} | [
{
"pp": "K : Type u\ninst✝⁴ : Field K\nn : ℕ\na : K\nH : Irreducible (X ^ n - C a)\nL : Type u_1\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : IsSplittingField K L (X ^ n - C a)\nα : L\nhα : α ^ n = (algebraMap K L) a\ninst✝ : NeZero n\nζ : K\nhζ : IsPrimitiveRoot ζ n\nm : ℕ\n⊢ ((autEquivZmod H L hζ).symm (... | [
"K : Type u\ninst✝⁴ : Field K\nn : ℕ\na : K\nH : Irreducible (X ^ n - C a)\nL : Type u_1\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : IsSplittingField K L (X ^ n - C a)\nα : L\nhα : α ^ n = (algebraMap K L) a\ninst✝ : NeZero n\nζ : K\nhζ : IsPrimitiveRoot ζ n\nm : ℕ\n⊢ ((autEquivZmod H L hζ).symm (Multiplicati... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KummerExtension | {
"line": 486,
"column": 4
} | {
"line": 487,
"column": 24
} | {
"line": 487,
"column": 25
} | [
{
"pp": "K : Type u\ninst✝⁵ : Field K\nL : Type u_1\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : FiniteDimensional K L\nhK : (primitiveRoots (finrank K L) K).Nonempty\ninst✝¹ : IsGalois K L\ninst✝ : IsCyclic Gal(L/K)\nζ : K\nhζ : IsPrimitiveRoot ζ (finrank K L)\nσ : Gal(L/K)\nhσ : Function.Surjective fun x... | [
"K : Type u\ninst✝⁵ : Field K\nL : Type u_1\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : FiniteDimensional K L\nhK : (primitiveRoots (finrank K L) K).Nonempty\ninst✝¹ : IsGalois K L\ninst✝ : IsCyclic Gal(L/K)\nζ : K\nhζ : IsPrimitiveRoot ζ (finrank K L)\nσ : Gal(L/K)\nhσ : Function.Surjective fun x ↦ σ ^ x\nhσ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LinearDisjoint | {
"line": 360,
"column": 32
} | {
"line": 360,
"column": 61
} | {
"line": 360,
"column": 61
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nA B : Subalgebra R S\nι : Type u_1\na : ι → ↥A\ni : (ι →₀ ↥B) →ₗ[R] S := Submodule.mulLeftMap (toSubmodule B) a\nj : (ι →₀ ↥B) →ₗ[R] S :=\n ↑(MulOpposite.opLinearEquiv R).symm ∘ₗ\n ↑R (Finsupp.linearCombination (↥B.o... | [] | LinearMap.coe_restrictScalars | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.LinearDisjoint | {
"line": 362,
"column": 2
} | {
"line": 366,
"column": 53
} | {
"line": 367,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nA B : Subalgebra R S\nι : Type u_1\na : ι → ↥A\ni : (ι →₀ ↥B) →ₗ[R] S := Submodule.mulLeftMap (toSubmodule B) a\nj : (ι →₀ ↥B) →ₗ[R] S :=\n ↑(MulOpposite.opLinearEquiv R).symm ∘ₗ\n ↑R (Finsupp.linearCombination (↥B.o... | [
"R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nA B : Subalgebra R S\nι : Type u_1\na : ι → ↥A\ni : (ι →₀ ↥B) →ₗ[R] S := Submodule.mulLeftMap (toSubmodule B) a\nj : (ι →₀ ↥B) →ₗ[R] S :=\n ↑(MulOpposite.opLinearEquiv R).symm ∘ₗ\n ↑R (Finsupp.linearCombination (↥B.op) (MulOppos... | simp only [LinearMap.coe_comp, Function.comp_apply, Finsupp.lsingle_apply, coe_val,
Finsupp.mapRange.linearEquiv_toLinearMap, LinearEquiv.coe_coe,
MulOpposite.coe_opLinearEquiv_symm, LinearMap.coe_restrictScalars,
Finsupp.mapRange.linearMap_apply, Finsupp.mapRange_single, Finsupp.linearCombination_single,
... | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.CosetCover | {
"line": 346,
"column": 2
} | {
"line": 346,
"column": 13
} | {
"line": 347,
"column": 2
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝ : Group G\nι : Type u_2\nH : ι → Subgroup G\ng : ι → G\ns : Finset ι\nhcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ\nh : ∀ i ∈ s, (H i).FiniteIndex → s.card < (H i).index\nhs : s.Nonempty\n⊢ ∑ i ∈ s, (↑(H i).index)⁻¹ < 1",
"ppTerm": "?inr",
"assigned": true,
"used... | [] | | inr hs => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.FieldTheory.Minpoly.ConjRootClass | {
"line": 71,
"column": 24
} | {
"line": 71,
"column": 39
} | {
"line": 71,
"column": 40
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx✝ : L\n⊢ mk K x✝ = 0 ↔ x✝ ∈ {0}",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"Set.instSingletonSet",
"id",
"ConjRoot... | [
"K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx✝ : L\n⊢ x✝ = 0 ↔ x✝ ∈ {0}"
] | mk_eq_zero_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Minpoly.ConjRootClass | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 30
} | {
"line": 99,
"column": 4
} | [
{
"pp": "case mp\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx y : ConjRootClass K L\n⊢ (∃ a, mk K a = x ∧ ∃ b, mk K b = y ∧ a + b = 0) → x = -y",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Exists",
"ConjRootClass.instNeg",
"Dis... | [
"case mp\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\na b : L\nh : a + b = 0\n⊢ mk K a = -mk K b"
] | rintro ⟨a, rfl, b, rfl, h⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.FieldTheory.Minpoly.ConjRootClass | {
"line": 153,
"column": 2
} | {
"line": 155,
"column": 61
} | {
"line": 157,
"column": 0
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsAlgebraic K L\nc : ConjRootClass K L\n⊢ Irreducible c.minpoly",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebra.IsIntegral.isIntegral",
"congr... | [] | induction c
rw [minpoly_mk]
exact minpoly.irreducible (Algebra.IsIntegral.isIntegral _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Minpoly.ConjRootClass | {
"line": 153,
"column": 2
} | {
"line": 155,
"column": 61
} | {
"line": 157,
"column": 0
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsAlgebraic K L\nc : ConjRootClass K L\n⊢ Irreducible c.minpoly",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebra.IsIntegral.isIntegral",
"congr... | [] | induction c
rw [minpoly_mk]
exact minpoly.irreducible (Algebra.IsIntegral.isIntegral _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Minpoly.ConjRootClass | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 80
} | {
"line": 160,
"column": 81
} | [
{
"pp": "case h\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsAlgebraic K L\nx x✝ : L\n⊢ (aeval x) (mk K x✝).minpoly = 0 ↔ mk K x = mk K x✝",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebra.IsIntegral.isIn... | [
"case h\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsAlgebraic K L\nx x✝ : L\n⊢ IsConjRoot K x✝ x ↔ IsConjRoot K x x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LinearDisjoint | {
"line": 590,
"column": 6
} | {
"line": 590,
"column": 42
} | {
"line": 590,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝⁷ : CommRing R\nA : Type v\ninst✝⁶ : CommRing A\nB : Type w\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Flat R A\ninst✝¹ : Flat R B\ninst✝ : IsDomain (A ⊗[R] B)\nha : Function.Injective ⇑(algebraMap R A)\nhb : Function.Injective ⇑(algebraMap R B)\nK : Typ... | [
"R : Type u\ninst✝⁷ : CommRing R\nA : Type v\ninst✝⁶ : CommRing A\nB : Type w\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Flat R A\ninst✝¹ : Flat R B\ninst✝ : IsDomain (A ⊗[R] B)\nha : Function.Injective ⇑(algebraMap R A)\nhb : Function.Injective ⇑(algebraMap R B)\nK : Type (max w v) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.LinearDisjoint | {
"line": 372,
"column": 2
} | {
"line": 372,
"column": 59
} | {
"line": 373,
"column": 2
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝¹³ : Field F\ninst✝¹² : Field E\ninst✝¹¹ : Algebra F E\nA : IntermediateField F E\nL : Type w\ninst✝¹⁰ : Field L\ninst✝⁹ : Algebra F L\ninst✝⁸ : Algebra L E\ninst✝⁷ : IsScalarTower F L E\nH : A.LinearDisjoint L\nL' : Type u_1\ninst✝⁶ : Field L'\ninst✝⁵ : Algebra F L'\ninst✝... | [
"F : Type u\nE : Type v\ninst✝¹³ : Field F\ninst✝¹² : Field E\ninst✝¹¹ : Algebra F E\nA : IntermediateField F E\nL : Type w\ninst✝¹⁰ : Field L\ninst✝⁹ : Algebra F L\ninst✝⁸ : Algebra L E\ninst✝⁷ : IsScalarTower F L E\nH : A.LinearDisjoint L\nL' : Type u_1\ninst✝⁶ : Field L'\ninst✝⁵ : Algebra F L'\ninst✝⁴ : Algebra ... | refine Subalgebra.LinearDisjoint.of_le_right_of_flat H ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.TensorProduct.Subalgebra | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 80
} | {
"line": 150,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring S\ninst✝² : Algebra R S\ninst✝¹ : Semiring T\ninst✝ : Algebra R T\nx : S\ny : T\n⊢ (includeLeft.toLinearMap.range.mulMap includeRight.toLinearMap.range ∘ₗ\n _root_.TensorProduct.map includeLeft.toLinearMap.range... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring S\ninst✝² : Algebra R S\ninst✝¹ : Semiring T\ninst✝ : Algebra R T\nx : S\ny : T\n⊢ (includeLeft.toLinearMap.range.mulMap includeRight.toLinearMap.range)\n (includeLeft.toLinearMap.rangeRestrict x ⊗ₜ[R] includeRight.toLinearMa... | rw [LinearMap.comp_apply, LinearMap.id_apply, _root_.TensorProduct.map_tmul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.PurelyInseparable.PerfectClosure | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 45
} | {
"line": 143,
"column": 46
} | [
{
"pp": "case refine_2\nF : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK : Type w\ninst✝¹ : Field K\ninst✝ : Algebra F K\ni : E →ₐ[F] K\nx : E\nx✝ : ∃ n, x ^ ringExpChar F ^ n ∈ (algebraMap F E).range\nn : ℕ\ny : F\nh : (algebraMap F E) y = x ^ ringExpChar F ^ n\n⊢ (algebraMap... | [
"case refine_2\nF : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK : Type w\ninst✝¹ : Field K\ninst✝ : Algebra F K\ni : E →ₐ[F] K\nx : E\nx✝ : ∃ n, x ^ ringExpChar F ^ n ∈ (algebraMap F E).range\nn : ℕ\ny : F\nh : (algebraMap F E) y = x ^ ringExpChar F ^ n\n⊢ (algebraMap F K) y = i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.LinearDisjoint | {
"line": 574,
"column": 2
} | {
"line": 574,
"column": 37
} | {
"line": 574,
"column": 38
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nA : IntermediateField F E\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra F L\ninst✝¹ : Algebra L E\ninst✝ : IsScalarTower F L E\nH : A.LinearDisjoint L\nhalg : Algebra.IsAlgebraic F ↥A ∨ Algebra.IsAlgebraic F L\n⊢ Module... | [
"F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nA : IntermediateField F E\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra F L\ninst✝¹ : Algebra L E\ninst✝ : IsScalarTower F L E\nH : A.LinearDisjoint L\nhalg : Algebra.IsAlgebraic F ↥A ∨ Algebra.IsAlgebraic F L\n⊢ Module.rank ↥A ↥(e... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.LinearDisjoint | {
"line": 593,
"column": 2
} | {
"line": 593,
"column": 37
} | {
"line": 593,
"column": 38
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nA : IntermediateField F E\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra F L\ninst✝¹ : Algebra L E\ninst✝ : IsScalarTower F L E\nH : A.LinearDisjoint L\nhalg : Algebra.IsAlgebraic F ↥A ∨ Algebra.IsAlgebraic F L\n⊢ Module... | [
"F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nA : IntermediateField F E\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra F L\ninst✝¹ : Algebra L E\ninst✝ : IsScalarTower F L E\nH : A.LinearDisjoint L\nhalg : Algebra.IsAlgebraic F ↥A ∨ Algebra.IsAlgebraic F L\n⊢ Module.rank F L * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PurelyInseparable.PerfectClosure | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 13
} | {
"line": 278,
"column": 14
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\na : E\nha : IsSeparable F a\nq : ℕ\ninst✝ : ExpChar F q\nn : ℕ\nthis : Algebra.IsSeparable F ↥F⟮a⟯\n⊢ F⟮a⟯ = F⟮a ^ q ^ n⟯",
"ppTerm": "?m.41",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"... | [
"F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\na : E\nha : IsSeparable F a\nq : ℕ\ninst✝ : ExpChar F q\nn : ℕ\nthis : Algebra.IsSeparable F ↥F⟮a⟯\n⊢ F⟮a⟯ = F⟮a ^ q ^ n⟯"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PurelyInseparable.PerfectClosure | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 13
} | {
"line": 284,
"column": 14
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Algebra.IsSeparable F E\na : E\nq : ℕ\ninst✝ : ExpChar F q\nn : ℕ\n⊢ F⟮a⟯ = F⟮a ^ q ^ n⟯",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Algebra.IsSeparable F E\na : E\nq : ℕ\ninst✝ : ExpChar F q\nn : ℕ\n⊢ F⟮a⟯ = F⟮a ^ q ^ n⟯"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PurelyInseparable.PerfectClosure | {
"line": 339,
"column": 50
} | {
"line": 339,
"column": 88
} | {
"line": 339,
"column": 89
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\nh' : LinearIndepOn F id (Set.range v)\nι' : Set E := h'.extend ⋯\nb : Basis (... | [
"F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nq n : ℕ\nhF : ExpChar F q\nι : Type u_1\nv : ι → E\ninst✝¹ : FiniteDimensional F E\ninst✝ : Algebra.IsSeparable F E\nh : LinearIndependent F v\nh' : LinearIndepOn F id (Set.range v)\nι' : Set E := h'.extend ⋯\nb : Basis (↑ι') F E := ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PurelyInseparable.Tower | {
"line": 78,
"column": 14
} | {
"line": 78,
"column": 78
} | {
"line": 78,
"column": 78
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁷ : Field F\ninst✝⁶ : Field E\ninst✝⁵ : Algebra F E\nK : Type w\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Algebra E K\ninst✝¹ : IsScalarTower F E K\ninst✝ : IsPurelyInseparable F E\nι : Type u_1\nv : ι → K\nhsep : ∀ (i : ι), IsSeparable F (v i)\nh : LinearIndependen... | [] | by rw [map_zero, Finsupp.notMem_support_iff.1 hs, zero_pow this] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.PurelyInseparable.PerfectClosure | {
"line": 405,
"column": 2
} | {
"line": 405,
"column": 13
} | {
"line": 405,
"column": 14
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nq : ℕ\nhF : ExpChar F q\ninst✝ : ExpChar E q\na : E\nhsep : IsSeparable F a\n⊢ minpoly F ((frobenius E q) a) = Polynomial.map (frobenius F q) (minpoly F a)",
"ppTerm": "?m.29",
"assigned": false,
"usedConstant... | [
"F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nq : ℕ\nhF : ExpChar F q\ninst✝ : ExpChar E q\na : E\nhsep : IsSeparable F a\n⊢ minpoly F ((frobenius E q) a) = Polynomial.map (frobenius F q) (minpoly F a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.LinearDisjoint | {
"line": 749,
"column": 2
} | {
"line": 749,
"column": 53
} | {
"line": 749,
"column": 54
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nA B : IntermediateField F E\ninst✝ : FiniteDimensional F E\nh₁ : A.LinearDisjoint B.toSubalgebra\nh₂ : A ⊔ B = ⊤\nx : ↥B\n⊢ A.toSubalgebra ⊔ B.toSubalgebra = ⊤",
"ppTerm": "?m.73",
"assigned": false,
"usedCons... | [
"F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nA B : IntermediateField F E\ninst✝ : FiniteDimensional F E\nh₁ : A.LinearDisjoint B.toSubalgebra\nh₂ : A ⊔ B = ⊤\nx : ↥B\n⊢ A.toSubalgebra ⊔ B.toSubalgebra = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.LinearDisjoint | {
"line": 759,
"column": 2
} | {
"line": 759,
"column": 53
} | {
"line": 759,
"column": 54
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nA B : IntermediateField F E\ninst✝ : FiniteDimensional F E\nh₁ : A.LinearDisjoint B.toSubalgebra\nh₂ : A ⊔ B = ⊤\nx : ↥B\n⊢ A.toSubalgebra ⊔ B.toSubalgebra = ⊤",
"ppTerm": "?m.75",
"assigned": false,
"usedCons... | [
"F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nA B : IntermediateField F E\ninst✝ : FiniteDimensional F E\nh₁ : A.LinearDisjoint B.toSubalgebra\nh₂ : A ⊔ B = ⊤\nx : ↥B\n⊢ A.toSubalgebra ⊔ B.toSubalgebra = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.IntermediateField | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 27
} | {
"line": 70,
"column": 28
} | [
{
"pp": "case h\nK : Type u_1\ninst✝ : Field K\nf : K⟮X⟯\nhf : ↑((f.minpolyX ↥K⟮f⟯).coeff f.denom.natDegree) = 0\n⊢ C (f.num.coeff f.denom.natDegree) = f * C f.denom.leadingCoeff",
"ppTerm": "?h",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case h\nK : Type u_1\ninst✝ : Field K\nf : K⟮X⟯\nhf : ↑((f.minpolyX ↥K⟮f⟯).coeff f.denom.natDegree) = 0\n⊢ C (f.num.coeff f.denom.natDegree) = f * C f.denom.leadingCoeff"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PurelyInseparable.Tower | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 37
} | {
"line": 134,
"column": 38
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁷ : Field F\ninst✝⁶ : Field E\ninst✝⁵ : Algebra F E\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Algebra E K\ninst✝¹ : IsScalarTower F E K\ninst✝ : Algebra.IsSeparable F E\n⊢ Module.rank F E * sepDegree E K = sepDegree F K",
"ppTerm": "?m.34",
"assi... | [
"F : Type u\nE : Type v\ninst✝⁷ : Field F\ninst✝⁶ : Field E\ninst✝⁵ : Algebra F E\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Algebra E K\ninst✝¹ : IsScalarTower F E K\ninst✝ : Algebra.IsSeparable F E\n⊢ Module.rank F E * sepDegree E K = sepDegree F K"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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