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.Valuation.Extension | {
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Mathlib.RingTheory.Valuation.Extension | {
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"line": 198,
"column": 70
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{
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Mathlib.RingTheory.Valuation.Extension | {
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Mathlib.RingTheory.Smooth.Quotient | {
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Mathlib.RingTheory.WittVector.Compare | {
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{
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"usedConstants": [
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"NonAssocSemiring.toAddCommMonoidWithOne",
"Ring... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.Compare | {
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} | {
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{
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"usedConstants": [
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Mathlib.RingTheory.WittVector.Compare | {
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} | {
"line": 61,
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} | {
"line": 62,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ Fintype.card (TruncatedWittVector p n (ZMod p)) = p ^ n",
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"usedConstants": [
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Mathlib.RingTheory.WittVector.Compare | {
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} | {
"line": 79,
"column": 35
} | {
"line": 79,
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} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nR : Type u_1\ninst✝ : CommRing R\nx : ZMod (p ^ n)\n⊢ p ^ n ∣ p ^ n",
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"Monoid.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Compare | {
"line": 105,
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} | {
"line": 105,
"column": 75
} | {
"line": 106,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn m : ℕ\nhm : n ≤ m\nx : TruncatedWittVector p m (ZMod p)\n⊢ (zmodEquivTrunc p n).symm ((truncate hm) x) = (ZMod.castHom ⋯ (ZMod (p ^ n))) ((zmodEquivTrunc p m).symm x)",
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Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
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} | {
"line": 71,
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} | {
"line": 71,
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{
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Mathlib.RingTheory.WittVector.Compare | {
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} | {
"line": 124,
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"line": 125,
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{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn m : ℕ\nhm : n ≤ m\n⊢ (zmodEquivTrunc p n).symm.toRingHom.comp (truncate hm) =\n (ZMod.castHom ⋯ (ZMod (p ^ n))).comp (zmodEquivTrunc p m).symm.toRingHom",
"ppTerm": "?m.53",
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Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
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} | {
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} | {
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{
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Mathlib.RingTheory.WittVector.Compare | {
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} | {
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} | {
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} | [
{
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"ppTerm": "?m.101",
"assigned": true,
"us... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Compare | {
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} | {
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} | {
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} | [
{
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"ppTerm": "?m.127",
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"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Smooth.Quotient | {
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} | {
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} | {
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{
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Mathlib.RingTheory.WittVector.Compare | {
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} | {
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} | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
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} | {
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{
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Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
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} | {
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} | {
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{
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"usedConstants": [
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"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
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} | {
"line": 100,
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} | {
"line": 101,
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Compare | {
"line": 172,
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} | {
"line": 172,
"column": 100
} | {
"line": 173,
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} | [
{
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Mathlib.RingTheory.WittVector.Compare | {
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} | {
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} | {
"line": 182,
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{
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"PadicInt",
"congrArg",
"WittVector.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Compare | {
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} | {
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} | {
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{
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Mathlib.RingTheory.WittVector.Compare | {
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} | {
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} | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
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} | {
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} | {
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{
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Mathlib.RingTheory.Smooth.Quotient | {
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Mathlib.RingTheory.Smooth.Quotient | {
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Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
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} | {
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{
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Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
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Mathlib.RingTheory.Smooth.Quotient | {
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Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
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Mathlib.RingTheory.WittVector.MulCoeff | {
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{
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Mathlib.RingTheory.WittVector.MulCoeff | {
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{
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Mathlib.RingTheory.WittVector.MulCoeff | {
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"assigned": true,
"usedConstants": [
"Finset.union_subset",
"Finsupp.instAddZeroClass",
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"RingHom.ins... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 33
} | {
"line": 104,
"column": 4
} | [
{
"pp": "case refine_1\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range (n + 1)\n⊢ {(0, x)} ⊆ univ ×ˢ range (n + 1)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Finset.singleton_subset_iff._simp_1",
"Eq.mpr",
"Finset.mem_range._simp_1",
"Finset.univ"... | [] | simpa using mem_range.mp hx | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 33
} | {
"line": 104,
"column": 4
} | [
{
"pp": "case refine_2\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range (n + 1)\n⊢ {(1, x)} ⊆ univ ×ˢ range (n + 1)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Finset.singleton_subset_iff._simp_1",
"Eq.mpr",
"Finset.mem_range._simp_1",
"Finset.univ"... | [] | simpa using mem_range.mp hx | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 94,
"column": 81
} | {
"line": 94,
"column": 83
} | {
"line": 95,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ (remainder p n).vars ⊆ univ ×ˢ range (n + 1)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Finset.union_subset",
"Finset.singleton_subset_iff._simp_1",
"Eq.mpr",
"Finset.mem_range._simp_1",
"Int.instIsStrict... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 121,
"column": 67
} | {
"line": 121,
"column": 69
} | {
"line": 122,
"column": 6
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn i : ℕ\nm✝ : Fin 2 × ℕ →₀ ℕ\n⊢ (Finsupp.single i (p ^ (n - i))).support = {i}",
"ppTerm": "?m.227",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Nat.instMulZeroClass",
"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 154,
"column": 2
} | {
"line": 155,
"column": 50
} | {
"line": 156,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommRing R\nS : Type u_2\ninst✝¹⁰ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R R'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S'\ninst✝² : IsScalarTower R S S'\ninst✝¹ : Is... | [
"R : Type u_1\ninst✝¹¹ : CommRing R\nS : Type u_2\ninst✝¹⁰ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R R'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S'\ninst✝² : IsScalarTower R S S'\ninst✝¹ : IsScalarTower ... | have cotker : LinearMap.ker mapcot = (Submodule.comap J.subtype (_ ⊓ J)).map J.toCotangent :=
Ideal.mapCotangent_ker_of_surjective surjPP' h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 114,
"column": 87
} | {
"line": 114,
"column": 89
} | {
"line": 115,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ ∑ i ∈ range (n + 1), ↑p ^ i * wittMul p i ^ p ^ (n - i) = wittPolyProd p n",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"Int.cast",
"Eq.mpr",
"Nat.instCanon... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 130,
"column": 80
} | {
"line": 130,
"column": 82
} | {
"line": 131,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ ↑p ^ n * wittMul p n + wittPolyProdRemainder p n = wittPolyProd p n",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"HMul.hMul",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 142,
"column": 72
} | {
"line": 142,
"column": 74
} | {
"line": 142,
"column": 75
} | [
{
"pp": "p n : ℕ\n⊢ ↑p ^ (n + 1) = C (↑p ^ (n + 1))",
"ppTerm": "?m.227",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"congrArg",
"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 83,
"column": 76
} | {
"line": 83,
"column": 78
} | {
"line": 84,
"column": 4
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : IsDomain k\nn : ℕ\na₁ a₂ : 𝕎 k\nbs : Fin (n + 1) → k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\n⊢ (X ^ p * C (a₁.coeff 0 ^ p ^ (n + 1))).degree = ↑p",
"ppTerm": "?m.69",
"assigned": true,
"used... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 88,
"column": 25
} | {
"line": 88,
"column": 27
} | {
"line": 89,
"column": 4
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : IsDomain k\nn : ℕ\na₁ a₂ : 𝕎 k\nbs : Fin (n + 1) → k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\nthis : (X ^ p * C (a₁.coeff 0 ^ p ^ (n + 1))).degree = ↑p\n⊢ (X ^ p * C (a₁.coeff 0 ^ p ^ (n + 1)) - X * C (a... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 82,
"column": 53
} | {
"line": 82,
"column": 55
} | {
"line": 83,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : IsDomain k\nn : ℕ\na₁ a₂ : 𝕎 k\nbs : Fin (n + 1) → k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\n⊢ (succNthDefiningPoly p n a₁ a₂ bs).degree = ↑p",
"ppTerm": "?m.30",
"assigned": true,
"usedCons... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 106,
"column": 31
} | {
"line": 106,
"column": 33
} | {
"line": 107,
"column": 4
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\nn : ℕ\na₁ a₂ : 𝕎 k\nbs : Fin (n + 1) → k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\n⊢ (succNthDefiningPoly p n a₁ a₂ bs).degree ≠ 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConst... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 127,
"column": 56
} | {
"line": 127,
"column": 58
} | {
"line": 128,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\nn : ℕ\na₁ a₂ : 𝕎 k\nbs : Fin (n + 1) → k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\n⊢ succNthVal p n a₁ a₂ bs ha₁ ha₂ ^ p * a₁.coeff 0 ^ p ^ (n + 1) + a₁.coeff (n + 1) * (bs 0 ^ p) ^ p ^ (n + 1... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 156,
"column": 28
} | {
"line": 156,
"column": 30
} | {
"line": 157,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\na₁ a₂ : 𝕎 k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\n⊢ solution p a₁ a₂ ≠ 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"False",
"Nat... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 164,
"column": 73
} | {
"line": 164,
"column": 75
} | {
"line": 165,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\na₁ : 𝕎 k\nha₁ : a₁.coeff 0 ≠ 0\na₂ : 𝕎 k\n⊢ solution p a₁ a₂ ^ p * a₁.coeff 0 = solution p a₁ a₂ * a₂.coeff 0",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"instHDiv... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 192,
"column": 39
} | {
"line": 192,
"column": 41
} | {
"line": 193,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\na₁ a₂ : 𝕎 k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\n⊢ frobeniusRotation p ha₁ ha₂ ≠ 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"WittVector.instZero",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 199,
"column": 87
} | {
"line": 199,
"column": 89
} | {
"line": 200,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\na₁ a₂ : 𝕎 k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\n⊢ frobenius (frobeniusRotation p ha₁ ha₂) * a₁ = frobeniusRotation p ha₁ ha₂ * a₂",
"ppTerm": "?m.45",
"assigned": true,
"used... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 232,
"column": 53
} | {
"line": 232,
"column": 55
} | {
"line": 233,
"column": 4
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\nm n : ℕ\nr' q' : 𝕎 k\nhr' : r'.coeff 0 ≠ 0\nhq' : q'.coeff 0 ≠ 0\nhq : ↑p ^ n * q' ∈ nonZeroDivisors (𝕎 k)\nb : 𝕎 k := frobeniusRotation p hr' hq'\n⊢ frobenius b * r' = q' * b",
"ppTerm": "... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 235,
"column": 41
} | {
"line": 235,
"column": 43
} | {
"line": 235,
"column": 44
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\nm n : ℕ\nr' q' : 𝕎 k\nhr' : r'.coeff 0 ≠ 0\nhq' : q'.coeff 0 ≠ 0\nhq : ↑p ^ n * q' ∈ nonZeroDivisors (𝕎 k)\nb : 𝕎 k := frobeniusRotation p hr' hq'\nkey : frobenius b * r' = q' * b\nh : q' = 0\n... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 234,
"column": 62
} | {
"line": 234,
"column": 64
} | {
"line": 235,
"column": 4
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\nm n : ℕ\nr' q' : 𝕎 k\nhr' : r'.coeff 0 ≠ 0\nhq' : q'.coeff 0 ≠ 0\nhq : ↑p ^ n * q' ∈ nonZeroDivisors (𝕎 k)\nb : 𝕎 k := frobeniusRotation p hr' hq'\nkey : frobenius b * r' = q' * b\n⊢ (algebraMa... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 169,
"column": 65
} | {
"line": 169,
"column": 67
} | {
"line": 170,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommRing R\nS : Type u_2\ninst✝¹⁰ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R R'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S'\ninst✝² : IsScalarTower R S S'\ninst✝¹ : Is... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 230,
"column": 51
} | {
"line": 230,
"column": 53
} | {
"line": 231,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\nm n : ℕ\nr' q' : 𝕎 k\nhr' : r'.coeff 0 ≠ 0\nhq' : q'.coeff 0 ≠ 0\nhq : ↑p ^ n * q' ∈ nonZeroDivisors (𝕎 k)\n⊢ let b := frobeniusRotation p hr' hq';\n φ ((algebraMap (𝕎 k) (FractionRing (𝕎 k))... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 164,
"column": 87
} | {
"line": 164,
"column": 89
} | {
"line": 165,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommRing R\nS : Type u_2\ninst✝¹⁰ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R R'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S'\ninst✝² : IsScalarTower R S S'\ninst✝¹ : Is... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 250,
"column": 36
} | {
"line": 250,
"column": 38
} | {
"line": 250,
"column": 39
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\na : FractionRing (𝕎 k)\nr q : 𝕎 k\nhq : q ∈ nonZeroDivisors (𝕎 k)\nhrq : Localization.mk (r, ⟨q, hq⟩).1 (r, ⟨q, hq⟩).2 ≠ 0\nhq0 : q ≠ 0\nh : r = 0\n⊢ Localization.mk (r, ⟨q, hq⟩).1 (r, ⟨q, hq⟩)... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 245,
"column": 70
} | {
"line": 245,
"column": 72
} | {
"line": 246,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\na : FractionRing (𝕎 k)\nha : a ≠ 0\n⊢ ∃ b, b ≠ 0 ∧ ∃ m, φ b * a = ↑p ^ m * b",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"WittVector.instZero",
"GroupWithZer... | [] | by | [anonymous] | by |
Mathlib.RingTheory.ZMod.Torsion | {
"line": 23,
"column": 43
} | {
"line": 23,
"column": 45
} | {
"line": 24,
"column": 2
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ rootsOfUnity (p - 1) (ZMod p) = ⊤",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"MulOne.toOne",
"ZMod.commRing",
"Monoid.toMulOneClass",
"congrArg",
"HSub.hSub",
"iff_true... | [] | by | [anonymous] | by |
Mathlib.RingTheory.ZMod.Torsion | {
"line": 28,
"column": 28
} | {
"line": 28,
"column": 30
} | {
"line": 28,
"column": 31
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ p - 1 ≠ 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"instOfNatNat",
"LE.le",
"instLENat",
"_private.Mathlib.RingTheory.ZMod.Torsion.0.ZMod.instHasEnoughRootsOfUnityHSubNatOfNat._proof_1",
"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.ZMod.Torsion | {
"line": 27,
"column": 76
} | {
"line": 27,
"column": 78
} | {
"line": 28,
"column": 2
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ HasEnoughRootsOfUnity (ZMod p) (p - 1)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"ZMod.rootsOfUnity_eq_top",
"Nat.instMulZeroClass",
"HasEnoughRootsOfUnity.of_card_l... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 140,
"column": 21
} | {
"line": 140,
"column": 23
} | {
"line": 142,
"column": 2
} | [
{
"pp": "p n : ℕ\n⊢ wittPolyProd p (n + 1) =\n -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +\n ↑p ^ (n + 1) * X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) +\n ↑p ^ (n + 1) * X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)) +\n remain... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 171,
"column": 57
} | {
"line": 171,
"column": 59
} | {
"line": 172,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ ↑p ^ (n + 1) * wittMul p (n + 1) =\n -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +\n ↑p ^ (n + 1) * X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) +\n ↑p ^ (n + 1) * X (1, n + 1) * (rename (Prod.mk 0)) (wit... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 176,
"column": 96
} | {
"line": 176,
"column": 98
} | {
"line": 177,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ ↑p ^ (n + 1) * polyOfInterest p n = remainder p n - wittPolyProdRemainder p (n + 1)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Distrib.leftDistribClass",
"Mathlib.Tactic.Ring... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 182,
"column": 77
} | {
"line": 182,
"column": 79
} | {
"line": 183,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ (↑p ^ (n + 1) * polyOfInterest p n).vars ⊆ univ ×ˢ range (n + 1)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"WittVector.remainder_vars",
"Finset.union_subset",
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidA... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 193,
"column": 54
} | {
"line": 193,
"column": 56
} | {
"line": 193,
"column": 57
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ ↑p ^ (n + 1) = C (↑p ^ (n + 1))",
"ppTerm": "?m.201",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiri... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 192,
"column": 89
} | {
"line": 192,
"column": 91
} | {
"line": 193,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ (polyOfInterest p n).vars =\n (↑p ^ (n + 1) *\n (wittMul p (n + 1) + ↑p ^ (n + 1) * X (0, n + 1) * X (1, n + 1) -\n X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) -\n X (1, n + 1) * (rename (Prod.mk 0)) (wittPolyno... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 198,
"column": 91
} | {
"line": 198,
"column": 93
} | {
"line": 199,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ (polyOfInterest p n).vars ⊆ univ ×ˢ range (n + 1)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"wittPolynomial",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"HMul.hMul",
"Finset.univ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 205,
"column": 88
} | {
"line": 205,
"column": 90
} | {
"line": 206,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝ : CommRing k\nn : ℕ\nx y : 𝕎 k\n⊢ peval (polyOfInterest p n) ![fun i ↦ x.coeff i, fun i ↦ y.coeff i] =\n (x * y).coeff (n + 1) + ↑p ^ (n + 1) * x.coeff (n + 1) * y.coeff (n + 1) -\n y.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * x.coef... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 225,
"column": 77
} | {
"line": 225,
"column": 79
} | {
"line": 226,
"column": 4
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : CommRing k\ninst✝ : CharP k p\nn : ℕ\nx y✝ : 𝕎 k\nthis : ↑p = 0\ny : 𝕎 k\n⊢ ∑ x ∈ range (n + 1 + 1), 0 ^ x * y.coeff x ^ p ^ (n + 1 - x) = y.coeff 0 ^ p ^ (n + 1)",
"ppTerm": "?m.181",
"assigned": true,
"usedConstants": [
"IsRig... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.Isocrystal | {
"line": 188,
"column": 26
} | {
"line": 188,
"column": 28
} | {
"line": 188,
"column": 29
} | [
{
"pp": "p : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nk : Type u_2\ninst✝⁴ : Field k\ninst✝³ : IsAlgClosed k\ninst✝² : CharP k p\nV : Type u_3\ninst✝¹ : AddCommGroup V\ninst✝ : Isocrystal p k V\nh_dim : finrank K(p, k) V = 1\nthis : Nontrivial V\nx : V\nhx : x ≠ 0\n⊢ Φ(p, k) x ≠ 0",
"ppTerm": "?m.67",
"assigned"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.Isocrystal | {
"line": 189,
"column": 68
} | {
"line": 189,
"column": 70
} | {
"line": 190,
"column": 4
} | [
{
"pp": "p : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nk : Type u_2\ninst✝⁴ : Field k\ninst✝³ : IsAlgClosed k\ninst✝² : CharP k p\nV : Type u_3\ninst✝¹ : AddCommGroup V\ninst✝ : Isocrystal p k V\nh_dim : finrank K(p, k) V = 1\nthis✝ : Nontrivial V\nx : V\nhx : x ≠ 0\nthis : Φ(p, k) x ≠ 0\n⊢ ∃ a, a ≠ 0 ∧ Φ(p, k) x = a • x... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 219,
"column": 53
} | {
"line": 219,
"column": 55
} | {
"line": 220,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : CommRing k\ninst✝ : CharP k p\nn : ℕ\nx y : 𝕎 k\n⊢ peval (polyOfInterest p n) ![fun i ↦ x.coeff i, fun i ↦ y.coeff i] =\n (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1)",
"ppTer... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 190,
"column": 30
} | {
"line": 190,
"column": 32
} | {
"line": 190,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommRing R\nS : Type u_2\ninst✝¹⁰ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R R'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S'\ninst✝² : IsScalarTower R S S'\ninst✝¹ : Is... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.Isocrystal | {
"line": 197,
"column": 57
} | {
"line": 197,
"column": 59
} | {
"line": 197,
"column": 60
} | [
{
"pp": "p : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nk : Type u_2\ninst✝⁴ : Field k\ninst✝³ : IsAlgClosed k\ninst✝² : CharP k p\nV : Type u_3\ninst✝¹ : AddCommGroup V\ninst✝ : Isocrystal p k V\nh_dim : finrank K(p, k) V = 1\nthis✝ : Nontrivial V\nx : V\nhx : x ≠ 0\nthis : Φ(p, k) x ≠ 0\na : K(p, k)\nha : a ≠ 0\nhax : Φ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 245,
"column": 24
} | {
"line": 245,
"column": 26
} | {
"line": 245,
"column": 27
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : CommRing k\ninst✝ : CharP k p\nn : ℕ\nf₀ : (↑↑(univ ×ˢ range (n + 1)) → k) → k\nhf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)\nx y : TruncatedWittVector p (n + 1) k\na : Fin 2 × ℕ\nha : a ∈ ↑(univ ×ˢ range (n + ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 185,
"column": 53
} | {
"line": 185,
"column": 55
} | {
"line": 186,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommRing R\nS : Type u_2\ninst✝¹⁰ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R R'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S'\ninst✝² : IsScalarTower R S S'\ninst✝¹ : Is... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 239,
"column": 83
} | {
"line": 239,
"column": 85
} | {
"line": 240,
"column": 4
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : CommRing k\ninst✝ : CharP k p\nn : ℕ\nf₀ : (↑↑(univ ×ˢ range (n + 1)) → k) → k\nhf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)\n⊢ TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k",
"ppTer... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 71,
"column": 21
} | {
"line": 71,
"column": 23
} | {
"line": 72,
"column": 2
} | [
{
"pp": "α : Type v\ninst✝¹ : LinearOrder α\ns : Set (Set α)\ninst✝ : OrderTop α\nhs : ∀ x ∈ s, IsClub x\n⊢ IsClub (⋂₀ s)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsClub.dirSupClosed",
"IsCofinal.top_mem",
"IsClub.mk",
"congrArg",
"Part... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 79,
"column": 29
} | {
"line": 79,
"column": 31
} | {
"line": 79,
"column": 32
} | [
{
"pp": "α : Type v\ninst✝¹ : LinearOrder α\nι : Type u_1\nf : ι → Set α\ninst✝ : OrderTop α\nhs : ∀ (i : ι), IsClub (f i)\n⊢ ∀ x ∈ range f, IsClub x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"Exists",
"id",
"Set.mem_range._si... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 77,
"column": 25
} | {
"line": 77,
"column": 27
} | {
"line": 78,
"column": 2
} | [
{
"pp": "α : Type v\ninst✝¹ : LinearOrder α\nι : Type u_1\nf : ι → Set α\ninst✝ : OrderTop α\nhs : ∀ (i : ι), IsClub (f i)\n⊢ IsClub (⋂ i, f i)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.iInter",
"Membership.mem",
"Exists",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Isocrystal | {
"line": 200,
"column": 58
} | {
"line": 200,
"column": 60
} | {
"line": 201,
"column": 4
} | [
{
"pp": "p : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nk : Type u_2\ninst✝⁴ : Field k\ninst✝³ : IsAlgClosed k\ninst✝² : CharP k p\nV : Type u_3\ninst✝¹ : AddCommGroup V\ninst✝ : Isocrystal p k V\nh_dim : finrank K(p, k) V = 1\nthis✝ : Nontrivial V\nx : V\nhx : x ≠ 0\nthis : Φ(p, k) x ≠ 0\na : K(p, k)\nha : a ≠ 0\nhax : Φ... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 82,
"column": 21
} | {
"line": 82,
"column": 23
} | {
"line": 83,
"column": 2
} | [
{
"pp": "α : Type v\ninst✝ : LinearOrder α\ns : Set (Set α)\nhα : cof α ≤ 1\nhs : ∀ x ∈ s, IsClub x\n⊢ IsClub (⋂₀ s)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"PSum.casesOn",
"False",
"Cardinal.instOne",
"topOrderOrNoTopOrder",
"Cardinal",
"Partial... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 91,
"column": 34
} | {
"line": 91,
"column": 36
} | {
"line": 91,
"column": 37
} | [
{
"pp": "α : Type v\ninst✝ : LinearOrder α\nι : Type u_1\nf : ι → Set α\nhα : cof α ≤ 1\nhs : ∀ (i : ι), IsClub (f i)\n⊢ ∀ x ∈ range f, IsClub x",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"Exists",
"id",
"Set.mem_range._simp_1"... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 89,
"column": 25
} | {
"line": 89,
"column": 27
} | {
"line": 90,
"column": 2
} | [
{
"pp": "α : Type v\ninst✝ : LinearOrder α\nι : Type u_1\nf : ι → Set α\nhα : cof α ≤ 1\nhs : ∀ (i : ι), IsClub (f i)\n⊢ IsClub (⋂ i, f i)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.iInter",
"Membership.mem",
"Exists",
"id... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Isocrystal | {
"line": 208,
"column": 56
} | {
"line": 208,
"column": 87
} | {
"line": 209,
"column": 4
} | [
{
"pp": "case h\np : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nk : Type u_2\ninst✝⁴ : Field k\ninst✝³ : IsAlgClosed k\ninst✝² : CharP k p\nV : Type u_3\ninst✝¹ : AddCommGroup V\ninst✝ : Isocrystal p k V\nh_dim : finrank K(p, k) V = 1\nthis✝ : Nontrivial V\nx : V\nhx : x ≠ 0\nthis : Φ(p, k) x ≠ 0\na : K(p, k)\nha : a ≠ 0\... | [
"case h\np : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nk : Type u_2\ninst✝⁴ : Field k\ninst✝³ : IsAlgClosed k\ninst✝² : CharP k p\nV : Type u_3\ninst✝¹ : AddCommGroup V\ninst✝ : Isocrystal p k V\nh_dim : finrank K(p, k) V = 1\nthis✝ : Nontrivial V\nx : V\nhx : x ≠ 0\nthis : Φ(p, k) x ≠ 0\na : K(p, k)\nha : a ≠ 0\nhax : Φ(p, ... | LinearEquiv.smulOfNeZero_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 107,
"column": 35
} | {
"line": 107,
"column": 37
} | {
"line": 108,
"column": 4
} | [
{
"pp": "α : Type v\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\ns : Set (Set α)\nhα✝ : cof α ≠ ℵ₀\nhsα : #↑s < cof α\nhs : ∀ x ∈ s, IsClub x\nh✝ : Nonempty α\nhα : ℵ₀ < cof α\na : α\nf : ↑s → α → α\nhf : ∀ (x : ↑s) (x_1 : α), f x x_1 ∈ ↑x ∧ x_1 ≤ f x x_1\ng : ℕ → α := fun t ↦ Nat.rec a (fun x IH ↦ sSup (r... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.Isocrystal | {
"line": 209,
"column": 4
} | {
"line": 209,
"column": 35
} | {
"line": 209,
"column": 36
} | [
{
"pp": "case h\np : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nk : Type u_2\ninst✝⁴ : Field k\ninst✝³ : IsAlgClosed k\ninst✝² : CharP k p\nV : Type u_3\ninst✝¹ : AddCommGroup V\ninst✝ : Isocrystal p k V\nh_dim : finrank K(p, k) V = 1\nthis✝ : Nontrivial V\nx : V\nhx : x ≠ 0\nthis : Φ(p, k) x ≠ 0\na : K(p, k)\nha : a ≠ 0\... | [
"case h\np : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nk : Type u_2\ninst✝⁴ : Field k\ninst✝³ : IsAlgClosed k\ninst✝² : CharP k p\nV : Type u_3\ninst✝¹ : AddCommGroup V\ninst✝ : Isocrystal p k V\nh_dim : finrank K(p, k) V = 1\nthis✝ : Nontrivial V\nx : V\nhx : x ≠ 0\nthis : Φ(p, k) x ≠ 0\na : K(p, k)\nha : a ≠ 0\nhax : Φ(p, ... | LinearEquiv.smulOfNeZero_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 193,
"column": 77
} | {
"line": 193,
"column": 79
} | {
"line": 194,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommRing R\nS : Type u_2\ninst✝¹⁰ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R R'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S'\ninst✝² : IsScalarTower R S S'\ninst✝¹ : Is... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 235,
"column": 53
} | {
"line": 235,
"column": 55
} | {
"line": 236,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : CommRing k\ninst✝ : CharP k p\nn : ℕ\n⊢ ∃ f,\n ∀ (x y : 𝕎 k),\n f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =\n (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1)",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 100,
"column": 48
} | {
"line": 100,
"column": 50
} | {
"line": 101,
"column": 2
} | [
{
"pp": "α : Type v\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\ns : Set (Set α)\nhα : cof α ≠ ℵ₀\nhsα : #↑s < cof α\nhs : ∀ x ∈ s, IsClub x\n⊢ IsClub (⋂₀ s)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsClub.dirSupClosed",
"csSup_le'",
"Preorder... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 123,
"column": 30
} | {
"line": 123,
"column": 32
} | {
"line": 123,
"column": 33
} | [
{
"pp": "α : Type v\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\nι : Type u\nf : ι → Set α\nhα : cof α ≠ ℵ₀\nhι : lift.{v, u} #ι < lift.{u, v} (cof α)\nhf : ∀ (i : ι), IsClub (f i)\n⊢ ∀ x ∈ range f, IsClub x",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Member... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 262,
"column": 59
} | {
"line": 262,
"column": 61
} | {
"line": 263,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : CommRing k\ninst✝ : CharP k p\nn : ℕ\n⊢ ∃ f,\n ∀ (x y : 𝕎 k),\n (x * y).coeff (n + 1) =\n x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1) + y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) +\n f (truncateFun (n + 1) x) (truncateFun (n +... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 121,
"column": 25
} | {
"line": 121,
"column": 27
} | {
"line": 122,
"column": 2
} | [
{
"pp": "α : Type v\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\nι : Type u\nf : ι → Set α\nhα : cof α ≠ ℵ₀\nhι : lift.{v, u} #ι < lift.{u, v} (cof α)\nhf : ∀ (i : ι), IsClub (f i)\n⊢ IsClub (⋂ i, f i)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toL... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.EventuallyConst | {
"line": 31,
"column": 80
} | {
"line": 31,
"column": 82
} | {
"line": 32,
"column": 2
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : LinearOrder α\ninst✝¹ : PartialOrder β\nf : α → β\ninst✝ : Nonempty α\nhf : Monotone f\nhf' : ¬IsCofinal (range (rangeSplitting f))\n⊢ EventuallyConst f atTop",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"not_isCofinal_iff",
"Eq.mpr"... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.EventuallyConst | {
"line": 39,
"column": 23
} | {
"line": 39,
"column": 25
} | {
"line": 39,
"column": 26
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nhf : Monotone f\nhα : lift.{u, v} #β < lift.{v, u} (cof α)\n⊢ Nonempty α",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"not_lt_zero._simp_1",
"NonAssocSemiring.toAddCommMonoidWithOne",... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.SetTheory.Cardinal.EventuallyConst | {
"line": 38,
"column": 31
} | {
"line": 38,
"column": 33
} | {
"line": 39,
"column": 2
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nhf : Monotone f\nhα : lift.{u, v} #β < lift.{v, u} (cof α)\n⊢ EventuallyConst f atTop",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"not_lt_zero._simp_1",
"Eq.mpr",
"NonAssocSemi... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.EventuallyConst | {
"line": 55,
"column": 83
} | {
"line": 55,
"column": 85
} | {
"line": 56,
"column": 2
} | [
{
"pp": "β : Type v\ninst✝¹ : PartialOrder β\nf : Cardinal.{v} → β\ninst✝ : Small.{v, v} β\nhf : Monotone f\n⊢ EventuallyConst f atTop",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal.lift_univ",
"Preorder.toLT",
"Cardinal",
"congrArg",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Isocrystal | {
"line": 185,
"column": 70
} | {
"line": 185,
"column": 72
} | {
"line": 186,
"column": 2
} | [
{
"pp": "p : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nk : Type u_2\ninst✝⁴ : Field k\ninst✝³ : IsAlgClosed k\ninst✝² : CharP k p\nV : Type u_3\ninst✝¹ : AddCommGroup V\ninst✝ : Isocrystal p k V\nh_dim : finrank K(p, k) V = 1\n⊢ ∃ m, Nonempty (StandardOneDimIsocrystal p k m ≃ᶠⁱ[p, k] V)",
"ppTerm": "?m.26",
"assi... | [] | by | [anonymous] | by |
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