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.RingTheory.HahnSeries.PowerSeries | {
"line": 67,
"column": 18
} | {
"line": 67,
"column": 20
} | {
"line": 68,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nf g : R⟦ℕ⟧\n⊢ PowerSeries.mk (f * g).coeff = PowerSeries.mk f.coeff * PowerSeries.mk g.coeff",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"Eq.mpr",
"_private.Mathlib.RingTheory.HahnSeries.Po... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.PowerSeries | {
"line": 64,
"column": 18
} | {
"line": 64,
"column": 20
} | {
"line": 65,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nf g : R⟦ℕ⟧\n⊢ PowerSeries.mk (f + g).coeff = PowerSeries.mk f.coeff + PowerSeries.mk g.coeff",
"ppTerm": "?m.139",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"AddMonoid.toAddSemigroup",
"congrArg",
"AddMo... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 504,
"column": 52
} | {
"line": 504,
"column": 54
} | {
"line": 505,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nα : Type u_5\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : VAdd Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\nR : Type u_7\nV : Type u_8\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\nx : R⟦Γ⟧\ns : SummableFamily Γ' V α\n⊢ (x • s).hsum = (of R... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 520,
"column": 33
} | {
"line": 520,
"column": 35
} | {
"line": 520,
"column": 36
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁸ : AddCommMonoid Γ\ninst✝⁷ : PartialOrder Γ\ninst✝⁶ : IsOrderedCancelAddMonoid Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : AddAction Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 517,
"column": 29
} | {
"line": 517,
"column": 31
} | {
"line": 517,
"column": 32
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁸ : AddCommMonoid Γ\ninst✝⁷ : PartialOrder Γ\ninst✝⁶ : IsOrderedCancelAddMonoid Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : AddAction Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.PowerSeries | {
"line": 106,
"column": 61
} | {
"line": 106,
"column": 63
} | {
"line": 107,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝³ : Semiring R\ninst✝² : Semiring Γ\ninst✝¹ : PartialOrder Γ\ninst✝ : IsStrictOrderedRing Γ\nx : PowerSeries R\nn : ℕ\n⊢ ((ofPowerSeries Γ R) x).coeff ↑n = (PowerSeries.coeff n) x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 515,
"column": 30
} | {
"line": 515,
"column": 32
} | {
"line": 515,
"column": 33
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁸ : AddCommMonoid Γ\ninst✝⁷ : PartialOrder Γ\ninst✝⁶ : IsOrderedCancelAddMonoid Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : AddAction Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 519,
"column": 33
} | {
"line": 519,
"column": 35
} | {
"line": 519,
"column": 36
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁸ : AddCommMonoid Γ\ninst✝⁷ : PartialOrder Γ\ninst✝⁶ : IsOrderedCancelAddMonoid Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : AddAction Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 518,
"column": 33
} | {
"line": 518,
"column": 35
} | {
"line": 518,
"column": 36
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁸ : AddCommMonoid Γ\ninst✝⁷ : PartialOrder Γ\ninst✝⁶ : IsOrderedCancelAddMonoid Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : AddAction Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 516,
"column": 30
} | {
"line": 516,
"column": 32
} | {
"line": 516,
"column": 33
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁸ : AddCommMonoid Γ\ninst✝⁷ : PartialOrder Γ\ninst✝⁶ : IsOrderedCancelAddMonoid Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : AddAction Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 523,
"column": 33
} | {
"line": 523,
"column": 35
} | {
"line": 524,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nα : Type u_5\ninst✝³ : AddCommMonoid Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\nx : R⟦Γ⟧\ns : SummableFamily Γ R α\n⊢ (x • s).hsum = x * s.hsum",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 535,
"column": 38
} | {
"line": 535,
"column": 40
} | {
"line": 536,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nα : Type u_5\ninst✝³ : AddCommMonoid Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\nR : Type u_7\ninst✝ : Ring R\ns t : SummableFamily Γ R α\n⊢ (s - t).hsum = s.hsum - t.hsum",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Hah... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 563,
"column": 77
} | {
"line": 563,
"column": 79
} | {
"line": 564,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommMonoid Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\ns : SummableFamily Γ R α\nt : SummableFamily Γ R β\ng : Γ\n⊢ (s.mul t).hsum.coeff g = ∑ gh ∈ antidiagonal ⋯ ⋯ g, s.hsum.coeff gh.1 * t.hsum... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 568,
"column": 40
} | {
"line": 568,
"column": 42
} | {
"line": 569,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommMonoid Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\ns : SummableFamily Γ R α\nt : SummableFamily Γ R β\n⊢ (s.mul t).hsum = s.hsum * t.hsum",
"ppTerm": "?m.39",
"assigned": true,
"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 584,
"column": 32
} | {
"line": 584,
"column": 34
} | {
"line": 585,
"column": 6
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\nf : α →₀ R⟦Γ⟧\na : α\ng : Γ\nhg : g ∈ (f a).support\n⊢ a ∈ f.support",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 581,
"column": 27
} | {
"line": 581,
"column": 29
} | {
"line": 582,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\nf : α →₀ R⟦Γ⟧\n⊢ (⋃ a, (f a).support).IsPWO",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"Iff.mpr",
"Se... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 588,
"column": 26
} | {
"line": 588,
"column": 28
} | {
"line": 589,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\nf : α →₀ R⟦Γ⟧\ng : Γ\n⊢ {a | (f a).coeff g ≠ 0}.Finite",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.PowerSeries | {
"line": 114,
"column": 90
} | {
"line": 114,
"column": 92
} | {
"line": 115,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝³ : Semiring R\ninst✝² : Semiring Γ\ninst✝¹ : PartialOrder Γ\ninst✝ : IsStrictOrderedRing Γ\nr : R\n⊢ (ofPowerSeries Γ R) (PowerSeries.C r) = C r",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"CharP.cast_eq_zero"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 599,
"column": 83
} | {
"line": 599,
"column": 85
} | {
"line": 600,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nα : Type u_5\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\nf : α →₀ R⟦Γ⟧\n⊢ (ofFinsupp f).hsum = f.sum fun x ↦ id",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"False",
"AddMonoidHom.instAd... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 619,
"column": 27
} | {
"line": 619,
"column": 29
} | {
"line": 620,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\ns : SummableFamily Γ R α\nf : α ↪ β\n⊢ (⋃ a, (if h : a ∈ Set.range ⇑f then s (Classical.choose h) else 0).support).IsPWO",
"ppTerm": "?m.43",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 631,
"column": 10
} | {
"line": 631,
"column": 66
} | {
"line": 632,
"column": 8
} | [
{
"pp": "case pos\nΓ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\ns : SummableFamily Γ R α\nf : α ↪ β\ng : Γ\nb : β\nhb : b ∈ Set.range ⇑f\nh : ¬(s (Classical.choose hb)).coeff g = 0\n⊢ b ∈ ⇑f '' Function.support fun a ↦ (s ... | [] | exact ⟨Classical.choose hb, h, Classical.choose_spec hb⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 627,
"column": 7
} | {
"line": 627,
"column": 9
} | {
"line": 628,
"column": 8
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\ns : SummableFamily Γ R α\nf : α ↪ β\ng : Γ\n⊢ {a | (if h : a ∈ Set.range ⇑f then s (Classical.choose h) else 0).coeff g ≠ 0} ⊆\n ⇑f '' Function.support fun a ↦ (s a)... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 642,
"column": 55
} | {
"line": 642,
"column": 57
} | {
"line": 643,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\ns : SummableFamily Γ R α\nf : α ↪ β\na : α\n⊢ (s.embDomain f) (f a) = s a",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Set.mem_range_self",
"HahnSeries.Summabl... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.PowerSeries | {
"line": 128,
"column": 74
} | {
"line": 128,
"column": 76
} | {
"line": 129,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝³ : Semiring R\ninst✝² : Semiring Γ\ninst✝¹ : PartialOrder Γ\ninst✝ : IsStrictOrderedRing Γ\n⊢ (ofPowerSeries Γ R) PowerSeries.X = (single 1) 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"Eq.mpr",
"NonAs... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.PowerSeries | {
"line": 141,
"column": 64
} | {
"line": 141,
"column": 66
} | {
"line": 142,
"column": 2
} | [
{
"pp": "Γ : Type u_1\ninst✝³ : Semiring Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsStrictOrderedRing Γ\nR : Type u_3\ninst✝ : Semiring R\nn : ℕ\n⊢ (ofPowerSeries Γ R) (PowerSeries.X ^ n) = (single ↑n) 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"one_pow",
"NonAssocSemiring.t... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 647,
"column": 81
} | {
"line": 647,
"column": 83
} | {
"line": 648,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\ns : SummableFamily Γ R α\nf : α ↪ β\nb : β\nh : b ∉ Set.range ⇑f\n⊢ (s.embDomain f) b = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"HahnSeries.SummableFamily.embD... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.PowerSeries | {
"line": 156,
"column": 16
} | {
"line": 156,
"column": 18
} | {
"line": 157,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : Semiring Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsStrictOrderedRing Γ\nσ : Type u_3\ninst✝ : Finite σ\nf : R⟦σ →₀ ℕ⟧\n⊢ (fun f ↦ { coeff := f, isPWO_support' := ⋯ }) ((fun f ↦ f.coeff) f) = f",
"ppTerm": "?m.45",
"assigned": true,
"use... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 653,
"column": 58
} | {
"line": 653,
"column": 60
} | {
"line": 654,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\ns : SummableFamily Γ R α\nf : α ↪ β\n⊢ (s.embDomain f).hsum = s.hsum",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"apply_dite",
"dite_congr",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.PowerSeries | {
"line": 159,
"column": 17
} | {
"line": 159,
"column": 19
} | {
"line": 160,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : Semiring Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsStrictOrderedRing Γ\nσ : Type u_3\ninst✝ : Finite σ\nf : MvPowerSeries σ R\n⊢ (fun f ↦ f.coeff) ((fun f ↦ { coeff := f, isPWO_support' := ⋯ }) f) = f",
"ppTerm": "?m.60",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 665,
"column": 68
} | {
"line": 665,
"column": 70
} | {
"line": 666,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\nx : R⟦Γ⟧\nn : ℕ\n⊢ (x ^ n).support ⊆ ↑(AddSubmonoid.closure x.support)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"HahnSeries.support"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.HahnEmbedding | {
"line": 62,
"column": 86
} | {
"line": 62,
"column": 88
} | {
"line": 76,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\n⊢ ∃ f,\n Function.Injective ⇑f ∧\n ∀ (a : M), ArchimedeanClass.mk a = (FiniteArchimedeanClass.withTopOrderIso M) (ofLex (f a)).orderTop",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 686,
"column": 49
} | {
"line": 686,
"column": 51
} | {
"line": 687,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\ng : Γ\n⊢ {a | ¬(0 ^ a).coeff g = 0} ⊆ {0}",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"HahnSeries.instSe... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Valuation | {
"line": 46,
"column": 13
} | {
"line": 46,
"column": 15
} | {
"line": 47,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝⁴ : AddCancelCommMonoid Γ\ninst✝³ : LinearOrder Γ\ninst✝² : IsOrderedCancelAddMonoid Γ\ninst✝¹ : Ring R\ninst✝ : IsDomain R\nx y : R⟦Γ⟧\n⊢ (x * y).orderTop = x.orderTop + y.orderTop",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.PowerSeries | {
"line": 165,
"column": 18
} | {
"line": 165,
"column": 20
} | {
"line": 166,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : Semiring Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsStrictOrderedRing Γ\nσ : Type u_3\ninst✝ : Finite σ\nf g : R⟦σ →₀ ℕ⟧\n⊢ (f * g).coeff = f.coeff * g.coeff",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"HahnSeries.sup... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Henselian | {
"line": 71,
"column": 60
} | {
"line": 71,
"column": 62
} | {
"line": 72,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ ⊥.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\n⊢ IsUnit ((Ideal.Quotient.mk ⊥.jacobson) a)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulO... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Henselian | {
"line": 83,
"column": 35
} | {
"line": 83,
"column": 37
} | {
"line": 83,
"column": 38
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh✝ : I ≤ ⊥.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R\nh1 : IsUnit ((a * y - 1) * 1 + 1)\nh2 : ∀ (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\n⊢ IsUnit a ∧ IsUnit y",
"ppTerm": "?m.323",
"assigned": true,
"usedConstants": [
... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.HahnSeries.PowerSeries | {
"line": 162,
"column": 18
} | {
"line": 162,
"column": 20
} | {
"line": 163,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : Semiring Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsStrictOrderedRing Γ\nσ : Type u_3\ninst✝ : Finite σ\nf g : R⟦σ →₀ ℕ⟧\n⊢ (f + g).coeff = f.coeff + g.coeff",
"ppTerm": "?m.152",
"assigned": true,
"usedConstants": [
"Nat.instMulZe... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Henselian | {
"line": 68,
"column": 69
} | {
"line": 68,
"column": 71
} | {
"line": 69,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh : I ≤ ⊥.jacobson\n⊢ IsLocalHom (Ideal.Quotient.mk I)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Semiring.toModule"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Henselian | {
"line": 115,
"column": 30
} | {
"line": 115,
"column": 32
} | {
"line": 116,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nf : K[X]\nx✝¹ : f.Monic\na₀ : K\nh₁ : Polynomial.eval a₀ f ∈ maximalIdeal K\nx✝ : IsUnit (Polynomial.eval a₀ (derivative f))\n⊢ ∃ a, f.IsRoot a ∧ a - a₀ ∈ maximalIdeal K",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.PowerSeries | {
"line": 204,
"column": 26
} | {
"line": 204,
"column": 28
} | {
"line": 205,
"column": 6
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nr : R\n⊢ toPowerSeries.toFun ((algebraMap R A⟦ℕ⟧) r) = (algebraMap R (PowerSeries A)) r",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"ZeroHom.funLike",
"Fals... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.GroupLike | {
"line": 27,
"column": 28
} | {
"line": 27,
"column": 30
} | {
"line": 28,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\nha : IsGroupLikeElem R a\n⊢ (antipode R) a * a = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"IsGroupLikeElem.comul_eq_tmul_self",
"NonAssocSemiring.toAddCom... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.GroupLike | {
"line": 31,
"column": 28
} | {
"line": 31,
"column": 30
} | {
"line": 32,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\nha : IsGroupLikeElem R a\n⊢ a * (antipode R) a = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"IsGroupLikeElem.comul_eq_tmul_self",
"NonAssocSemiring.toAddCom... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.GroupLike | {
"line": 44,
"column": 14
} | {
"line": 44,
"column": 16
} | {
"line": 44,
"column": 17
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na b : A\n⊢ { val := ↑1, inv := (antipode R) ↑1, val_inv := ⋯, inv_val := ⋯ } = 1",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Units.val",
"MulOne.toOne",
"HopfA... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.GroupLike | {
"line": 45,
"column": 18
} | {
"line": 45,
"column": 20
} | {
"line": 45,
"column": 21
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na✝ b✝ : A\na b : GroupLike R A\n⊢ { val := ↑(a * b), inv := (antipode R) ↑(a * b), val_inv := ⋯, inv_val := ⋯ } =\n { val := ↑a, inv := (antipode R) ↑a, val_inv := ⋯, inv_val := ⋯ } *\n { val := ↑... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.GroupLike | {
"line": 65,
"column": 22
} | {
"line": 65,
"column": 24
} | {
"line": 65,
"column": 25
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na✝ b : A\na : GroupLike R A\n⊢ a⁻¹ * a = 1",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"GroupLike.ext",
"Semigroup.to... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 43,
"column": 31
} | {
"line": 43,
"column": 33
} | {
"line": 44,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : Zero S\nx : R⟦Γ⟧\ny : S⟦Γ⟧\nh : x.support = y.support\n⊢ x.cardSupp = y.cardSupp",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"Cardinal",
"congrAr... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 51,
"column": 51
} | {
"line": 51,
"column": 53
} | {
"line": 52,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : Zero R\n⊢ cardSupp 0 = 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"Cardinal.mk_eq_zero",
"Cardinal",
"congrArg",
"Cardinal.mk",
"Set.Elem",
"HahnSeri... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 54,
"column": 89
} | {
"line": 54,
"column": 91
} | {
"line": 55,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : Zero R\na : Γ\nr : R\nh : r ≠ 0\n⊢ ((single a) r).cardSupp = 1",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"Eq.mpr",
"ZeroHom.funLike",
"Cardinal.instOne",
"Cardina... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 99,
"column": 46
} | {
"line": 99,
"column": 48
} | {
"line": 100,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝³ : PartialOrder Γ\ninst✝² : AddCommMonoid Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx : R⟦Γ⟧\na : Γ\nr : R\n⊢ ((single a) r * x).cardSupp ≤ x.cardSupp",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"ZeroHo... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 104,
"column": 46
} | {
"line": 104,
"column": 48
} | {
"line": 105,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝³ : PartialOrder Γ\ninst✝² : AddCommMonoid Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx : R⟦Γ⟧\na : Γ\nr : R\n⊢ (x * (single a) r).cardSupp ≤ x.cardSupp",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"ZeroHo... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 108,
"column": 62
} | {
"line": 108,
"column": 64
} | {
"line": 109,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝³ : PartialOrder Γ\ninst✝² : AddCommMonoid Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\nx : R⟦Γ⟧\nn : ℕ\n⊢ (x ^ n).cardSupp ≤ x.cardSupp ^ n",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAd... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 125,
"column": 80
} | {
"line": 125,
"column": 82
} | {
"line": 126,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : AddCommMonoid Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\n⊢ (SummableFamily.powers x).hsum.cardSupp ≤ max ℵ₀ x.cardSupp",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Wit... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 134,
"column": 40
} | {
"line": 134,
"column": 42
} | {
"line": 135,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : AddCommGroup Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\nx : R⟦Γ⟧\n⊢ x⁻¹.cardSupp ≤ max ℵ₀ x.cardSupp",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"AddGroup.toSubtractionMonoid... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 159,
"column": 15
} | {
"line": 159,
"column": 17
} | {
"line": 159,
"column": 18
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\nS : Type u_3\nα : Type u_4\nκ : Cardinal.{u_1}\ninst✝¹ : PartialOrder Γ\ninst✝ : AddMonoid R\nhκ : Fact (ℵ₀ ≤ κ)\n⊢ 0 ∈ {x | x.cardSupp < κ}",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 709,
"column": 4
} | {
"line": 714,
"column": 39
} | {
"line": 716,
"column": 0
} | [
{
"pp": "case pos.refine_2\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nhx : 0 < x.orderTop\ng : Γ\nhpwo : (⋃ n, (x ^ n).support).IsPWO\nh0 : ¬x = 0\nhg : g ∈ ⋃ n, {g | (x ^ n).coeff g ≠ 0}\ny : Γ\nys : y ∈ ⋃ n,... | [] | rintro (_ | n) hn
· exact Set.mem_union_right _ (Set.mem_singleton 0)
· obtain ⟨i, hi, j, hj, rfl⟩ := support_mul_subset hn
refine Set.mem_union_left _ ⟨n, Set.mem_iUnion.2 ⟨⟨j, i⟩, Set.mem_iUnion.2 ⟨?_, hi⟩⟩, rfl⟩
simp only [mem_coe, mem_antidiagonal, mem_support, ne_eq, Set.mem_iUnion]
exact... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 709,
"column": 4
} | {
"line": 714,
"column": 39
} | {
"line": 716,
"column": 0
} | [
{
"pp": "case pos.refine_2\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nhx : 0 < x.orderTop\ng : Γ\nhpwo : (⋃ n, (x ^ n).support).IsPWO\nh0 : ¬x = 0\nhg : g ∈ ⋃ n, {g | (x ^ n).coeff g ≠ 0}\ny : Γ\nys : y ∈ ⋃ n,... | [] | rintro (_ | n) hn
· exact Set.mem_union_right _ (Set.mem_singleton 0)
· obtain ⟨i, hi, j, hj, rfl⟩ := support_mul_subset hn
refine Set.mem_union_left _ ⟨n, Set.mem_iUnion.2 ⟨⟨j, i⟩, Set.mem_iUnion.2 ⟨?_, hi⟩⟩, rfl⟩
simp only [mem_coe, mem_antidiagonal, mem_support, ne_eq, Set.mem_iUnion]
exact... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 175,
"column": 14
} | {
"line": 175,
"column": 16
} | {
"line": 175,
"column": 17
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\nS : Type u_3\nα : Type u_4\nκ : Cardinal.{u_1}\ninst✝¹ : PartialOrder Γ\ninst✝ : AddGroup R\nhκ : Fact (ℵ₀ ≤ κ)\n⊢ ∀ {x : R⟦Γ⟧}, x ∈ (cardSuppLTAddSubmonoid Γ R κ).carrier → -x ∈ (cardSuppLTAddSubmonoid Γ R κ).carrier",
"ppTerm": "?m.13",
"assigned": true,
"usedC... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 695,
"column": 56
} | {
"line": 695,
"column": 58
} | {
"line": 696,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nhx : 0 < x.orderTop\ng : Γ\n⊢ {a | ((fun n ↦ x ^ n) a).coeff g ≠ 0}.Finite",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"HahnSe... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Cardinal | {
"line": 208,
"column": 50
} | {
"line": 208,
"column": 52
} | {
"line": 208,
"column": 53
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\nS : Type u_3\nα : Type u_4\nκ : Cardinal.{u_1}\ninst✝³ : LinearOrder Γ\ninst✝² : AddCommGroup Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\nhκ : Fact (ℵ₀ < κ)\nx✝¹ : R⟦Γ⟧\nx✝ :\n x✝¹ ∈\n (have this := ⋯;\n cardSuppLTSubring Γ R κ).carrier\n⊢ max ℵ₀ x✝¹.cardSupp... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 721,
"column": 27
} | {
"line": 721,
"column": 29
} | {
"line": 722,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\n⊢ (⋃ a, ((if 0 < x.orderTop then x else 0) ^ a).support).IsPWO",
"ppTerm": "?m.34",
"assigned... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 728,
"column": 26
} | {
"line": 728,
"column": 28
} | {
"line": 729,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\ng : Γ\n⊢ {a | ((if 0 < x.orderTop then x else 0) ^ a).coeff g ≠ 0}.Finite",
"ppTerm": "?m.90",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 736,
"column": 26
} | {
"line": 736,
"column": 28
} | {
"line": 737,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nhx : 0 < x.orderTop\nn : ℕ\n⊢ (powers x) n = x ^ n",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"WithTop.decidableLT",
"H... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Henselian | {
"line": 130,
"column": 9
} | {
"line": 130,
"column": 11
} | {
"line": 131,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\ntfae_3_to_2 :\n (∀ {K : Type u} [inst : Field K] (φ : R →+* K),\n Surjective ⇑φ →\n ∀ (f : R[X]),\n f.Monic → ∀ (a₀ : K), eval₂ φ a₀ f = 0 → eval₂ φ a₀ (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ φ a = a₀) →\n ∀ (f : R[X]),\n ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 740,
"column": 30
} | {
"line": 740,
"column": 32
} | {
"line": 741,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nhx : ¬0 < x.orderTop\n⊢ powers x = single 0 1",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"Set.IsP... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 747,
"column": 30
} | {
"line": 747,
"column": 32
} | {
"line": 747,
"column": 33
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nn : ℕ\nx✝ : Γ\n⊢ 0 < orderTop 0",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"WithTop.instPreorder",
"congrA... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 745,
"column": 57
} | {
"line": 745,
"column": 59
} | {
"line": 746,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\n⊢ powers 0 = single 0 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mu... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 754,
"column": 50
} | {
"line": 754,
"column": 52
} | {
"line": 755,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nhx : 0 < x.orderTop\n⊢ ⇑(powers x) = HPow.hPow x",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"WithTop.decidableLT",
"Hah... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 761,
"column": 51
} | {
"line": 761,
"column": 53
} | {
"line": 762,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nhx : 0 < x.orderTop\n⊢ (x • powers x).embDomain { toFun := Nat.succ, inj' := Nat.succ_injective } = powers x - ofFinsupp (Finsupp.single 0 1)",
"ppTerm": ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 770,
"column": 72
} | {
"line": 770,
"column": 74
} | {
"line": 771,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nhx : 0 < x.orderTop\n⊢ (1 - x) * (powers x).hsum = 1",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Henselian | {
"line": 146,
"column": 34
} | {
"line": 146,
"column": 36
} | {
"line": 147,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\ntfae_3_to_2 :\n (∀ {K : Type u} [inst : Field K] (φ : R →+* K),\n Surjective ⇑φ →\n ∀ (f : R[X]),\n f.Monic → ∀ (a₀ : K), eval₂ φ a₀ f = 0 → eval₂ φ a₀ (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ φ a = a₀) →\n ∀ (f : R[X]),\n ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 787,
"column": 52
} | {
"line": 787,
"column": 54
} | {
"line": 788,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx y : R⟦Γ⟧\nr : R\nhr : r * x.leadingCoeff = 1\nhxy : x = y + (single x.order) x.leadingCoeff\noinv : Γ\nhxo : oinv + x.order = 0\n⊢ 1 - (single oinv) r * x = -((single... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 797,
"column": 42
} | {
"line": 797,
"column": 44
} | {
"line": 798,
"column": 6
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nr : R\nhr : r * x.leadingCoeff = 1\noinv : Γ\nhxo : oinv + x.order = 0\ny : R⟦Γ⟧ := x - (single x.order) x.leadingCoeff\nhy : y = 0\n⊢ (single oinv) r * x = 1... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Henselian | {
"line": 142,
"column": 43
} | {
"line": 142,
"column": 45
} | {
"line": 143,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\ntfae_3_to_2 :\n (∀ {K : Type u} [inst : Field K] (φ : R →+* K),\n Surjective ⇑φ →\n ∀ (f : R[X]),\n f.Monic → ∀ (a₀ : K), eval₂ φ a₀ f = 0 → eval₂ φ a₀ (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ φ a = a₀) →\n ∀ (f : R[X]),\n ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 801,
"column": 48
} | {
"line": 801,
"column": 50
} | {
"line": 802,
"column": 6
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nr : R\nhr : r * x.leadingCoeff = 1\noinv : Γ\nhxo : oinv + x.order = 0\ny : R⟦Γ⟧ := x - (single x.order) x.leadingCoeff\nhy : ¬y = 0\nhr' : IsRegular r\n⊢ 0 <... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Henselian | {
"line": 126,
"column": 96
} | {
"line": 126,
"column": 98
} | {
"line": 127,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\n⊢ [HenselianLocalRing R,\n ∀ (f : R[X]),\n f.Monic →\n ∀ (a₀ : ResidueField R),\n (aeval a₀) f = 0 → (aeval a₀) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = a₀,\n ∀ {K : Type u} [inst : Field K] (φ : ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 794,
"column": 44
} | {
"line": 794,
"column": 46
} | {
"line": 795,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nr : R\nhr : r * x.leadingCoeff = 1\noinv : Γ\nhxo : oinv + x.order = 0\n⊢ 0 < (1 - (single oinv) r * x).orderTop",
"ppTerm": "?m.51",
"assigned": true... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Henselian | {
"line": 156,
"column": 9
} | {
"line": 156,
"column": 11
} | {
"line": 157,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nhR : HenselianLocalRing R\n⊢ maximalIdeal R ≤ ⊥.jacobson",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_sInf_iff",
"Eq.ge",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"Set.of... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 810,
"column": 44
} | {
"line": 810,
"column": 46
} | {
"line": 811,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nhx : IsUnit x.leadingCoeff\nhxo : IsAddUnit x.order\n⊢ IsUnit x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Units.val",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Henselian | {
"line": 160,
"column": 18
} | {
"line": 160,
"column": 20
} | {
"line": 161,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nhR : HenselianLocalRing R\n⊢ ∀ (f : R[X]),\n f.Monic →\n ∀ (a₀ : R),\n Polynomial.eval a₀ f ∈ maximalIdeal R →\n IsUnit ((Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a₀ (derivative f))) →\n ∃ a, f.IsRoot a ∧ a - a₀ ∈ maximalI... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 819,
"column": 16
} | {
"line": 819,
"column": 18
} | {
"line": 820,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nh : 0 < (x - 1).orderTop\n⊢ IsUnit x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Iff.mpr",
"Eq.mpr"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Henselian | {
"line": 181,
"column": 76
} | {
"line": 181,
"column": 78
} | {
"line": 182,
"column": 8
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nf : R[X]\nx✝ : f.Monic\na₀ : R\nh₁ : Polynomial.eval a₀ f ∈ I\nh₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))\nf' : R[X] := derivative f\nc : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eva... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Henselian | {
"line": 181,
"column": 6
} | {
"line": 183,
"column": 21
} | {
"line": 188,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nf : R[X]\nx✝ : f.Monic\na₀ : R\nh₁ : Polynomial.eval a₀ f ∈ I\nh₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))\nf' : R[X] := derivative f\nc : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eva... | [
"R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nf : R[X]\nx✝ : f.Monic\na₀ : R\nh₁ : Polynomial.eval a₀ f ∈ I\nh₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))\nf' : R[X] := derivative f\nc : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Pol... | have hc : ∀ n, c (n + 1) = c n - f.eval (c n) * (f'.eval (c n))⁻¹ʳ := by
intro n
simp only [c] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 852,
"column": 4
} | {
"line": 852,
"column": 100
} | {
"line": 853,
"column": 2
} | [
{
"pp": "case mp.hc\nΓ : Type u_1\nR : Type u_3\ninst✝⁴ : AddCommGroup Γ\ninst✝³ : LinearOrder Γ\ninst✝² : IsOrderedAddMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nu i : R⟦Γ⟧\nui : u * i = 1\niu : i * u = 1\n⊢ u.order + i.order = 0",
"ppTerm": "?mp.hc",
"assigned": true,
"usedConstants": [
... | [] | rw [← order_mul (left_ne_zero_of_mul_eq_one ui) (right_ne_zero_of_mul_eq_one ui), ui, order_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 845,
"column": 70
} | {
"line": 845,
"column": 72
} | {
"line": 846,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝⁴ : AddCommGroup Γ\ninst✝³ : LinearOrder Γ\ninst✝² : IsOrderedAddMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : R⟦Γ⟧\n⊢ IsUnit x ↔ IsUnit x.leadingCoeff",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 874,
"column": 73
} | {
"line": 874,
"column": 75
} | {
"line": 875,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\na : Γ\nr : R\n⊢ ((single a) r)⁻¹ = (single (-a)) r⁻¹",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"HahnSeries.order",
"GroupWithZero.toMonoid... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Henselian | {
"line": 188,
"column": 46
} | {
"line": 188,
"column": 48
} | {
"line": 189,
"column": 8
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nf : R[X]\nx✝ : f.Monic\na₀ : R\nh₁ : Polynomial.eval a₀ f ∈ I\nh₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))\nf' : R[X] := derivative f\nc : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eva... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 881,
"column": 56
} | {
"line": 881,
"column": 58
} | {
"line": 882,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\na b : Γ\nr s : R\n⊢ (single a) r / (single b) s = (single (a - b)) (r / s)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroCla... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Henselian | {
"line": 199,
"column": 49
} | {
"line": 199,
"column": 51
} | {
"line": 200,
"column": 8
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nf : R[X]\nx✝ : f.Monic\na₀ : R\nh₁ : Polynomial.eval a₀ f ∈ I\nh₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))\nf' : R[X] := derivative f\nc : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eva... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 886,
"column": 25
} | {
"line": 886,
"column": 27
} | {
"line": 887,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\nx : R⟦Γ⟧\nx0 : x ≠ 0\n⊢ x * x⁻¹ = 1",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Iff.mpr"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 885,
"column": 14
} | {
"line": 885,
"column": 16
} | {
"line": 885,
"column": 17
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\n⊢ 0⁻¹ = 0",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": [
"HahnSeries.order",
"GroupWi... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 896,
"column": 21
} | {
"line": 896,
"column": 23
} | {
"line": 897,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\nq : ℚ≥0\n⊢ ↑q = ↑q.num / ↑q.den",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"Semiring.to... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.Quotient | {
"line": 42,
"column": 94
} | {
"line": 42,
"column": 96
} | {
"line": 43,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : HopfAlgebra R A\ninst✝ : HopfAlgebraStruct R B\ng : A →ₐ[R] B\n⊢ g.toLinearMap ∘ₗ ofConv 1 = ofConv 1",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"AlgHom... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 894,
"column": 21
} | {
"line": 894,
"column": 23
} | {
"line": 894,
"column": 24
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\nq : ℚ≥0\nx : R⟦Γ⟧\n⊢ q • x = ↑q * x",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"NonAsso... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 898,
"column": 19
} | {
"line": 898,
"column": 21
} | {
"line": 899,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\nq : ℚ\n⊢ ↑q = ↑q.num / ↑q.den",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"Semiring.toNa... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 895,
"column": 19
} | {
"line": 895,
"column": 21
} | {
"line": 895,
"column": 22
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\nq : ℚ\nx : R⟦Γ⟧\n⊢ q • x = ↑q * x",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
"NonAssocS... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 905,
"column": 94
} | {
"line": 905,
"column": 96
} | {
"line": 906,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\nm : ℕ\ne : Bool\ns : ℕ\n⊢ (single 0) (OfScientific.ofScientific m e s) = OfScientific.ofScientific m e s",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra | {
"line": 46,
"column": 95
} | {
"line": 46,
"column": 97
} | {
"line": 47,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : HopfAlgebra R A\nG : Type u_3\ninst✝ : Group G\ng : G\na : A\n⊢ (antipode R) (single g a) = single g⁻¹ ((antipode R) a)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.Quotient | {
"line": 47,
"column": 94
} | {
"line": 47,
"column": 96
} | {
"line": 48,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : HopfAlgebra R A\ninst✝ : HopfAlgebraStruct R B\ng : A →ₗc[R] B\n⊢ ofConv 1 ∘ₗ g.toLinearMap = ofConv 1",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Coalg... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra | {
"line": 52,
"column": 32
} | {
"line": 52,
"column": 34
} | {
"line": 53,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : HopfAlgebra R A\nG : Type u_3\ninst✝ : Group G\n⊢ LinearMap.mul' R A[G] ∘ₗ LinearMap.rTensor A[G] (antipode R) ∘ₗ comul = Algebra.linearMap R A[G] ∘ₗ counit",
"ppTerm": "?m.12",
"assigned": true,
"usedConstan... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.Finsupp | {
"line": 23,
"column": 69
} | {
"line": 23,
"column": 71
} | {
"line": 24,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Type u_3\np : ι → Submodule R M\nI : Ideal R\n⊢ (submodule fun i ↦ I • p i) = I • submodule p",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWi... | [] | by | [anonymous] | by |
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