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.WittVector.Truncated | {
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{
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Mathlib.RingTheory.WittVector.Truncated | {
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} | {
"line": 371,
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} | {
"line": 372,
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} | [
{
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"ppTerm": "?m.19",
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Mathlib.RingTheory.WittVector.Truncated | {
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} | {
"line": 385,
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} | {
"line": 386,
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} | [
{
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"Fintype.card_fin",
"congrArg",
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Mathlib.RingTheory.Perfection | {
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} | {
"line": 746,
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} | {
"line": 747,
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} | [
{
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Mathlib.RingTheory.WittVector.Teichmuller | {
"line": 114,
"column": 56
} | {
"line": 114,
"column": 58
} | {
"line": 115,
"column": 2
} | [
{
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Mathlib.RingTheory.WittVector.Truncated | {
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"column": 98
} | {
"line": 392,
"column": 100
} | {
"line": 393,
"column": 2
} | [
{
"pp": "p : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Fact (Nat.Prime p)\n⊢ ⨅ i, RingHom.ker (WittVector.truncate i) = ⊥",
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Mathlib.RingTheory.WittVector.InitTail | {
"line": 113,
"column": 30
} | {
"line": 113,
"column": 32
} | {
"line": 114,
"column": 4
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{
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"ppTerm": "?m.58",
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Mathlib.RingTheory.WittVector.Truncated | {
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"column": 82
} | {
"line": 425,
"column": 84
} | {
"line": 426,
"column": 2
} | [
{
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"p... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.InitTail | {
"line": 118,
"column": 44
} | {
"line": 118,
"column": 46
} | {
"line": 119,
"column": 4
} | [
{
"pp": "p n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nx y : 𝕎 R\nh : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0\nP : ℕ → Prop := fun n ↦ y.coeff n = 0\nthis : DecidablePred P\nz : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n\nhx : select P z = x\n⊢ select (fun i ↦ ¬P i) z = y"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.InitTail | {
"line": 124,
"column": 35
} | {
"line": 124,
"column": 37
} | {
"line": 124,
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} | [
{
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Mathlib.RingTheory.WittVector.Truncated | {
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} | {
"line": 437,
"column": 26
} | {
"line": 438,
"column": 2
} | [
{
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Mathlib.RingTheory.WittVector.InitTail | {
"line": 125,
"column": 33
} | {
"line": 125,
"column": 35
} | {
"line": 126,
"column": 6
} | [
{
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Mathlib.RingTheory.WittVector.InitTail | {
"line": 109,
"column": 47
} | {
"line": 109,
"column": 49
} | {
"line": 110,
"column": 2
} | [
{
"pp": "p n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nx y : 𝕎 R\nh : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0\n⊢ (x + y).coeff n = x.coeff n + y.coeff n",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Truncated | {
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} | {
"line": 454,
"column": 87
} | {
"line": 455,
"column": 2
} | [
{
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"ppT... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.InitTail | {
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} | {
"line": 149,
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} | {
"line": 150,
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{
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"usedConstants": [
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Mathlib.RingTheory.WittVector.InitTail | {
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} | {
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} | {
"line": 187,
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.InitTail | {
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} | {
"line": 192,
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} | {
"line": 193,
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.InitTail | {
"line": 195,
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} | {
"line": 195,
"column": 90
} | {
"line": 196,
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} | [
{
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"in... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.InitTail | {
"line": 198,
"column": 73
} | {
"line": 198,
"column": 75
} | {
"line": 199,
"column": 2
} | [
{
"pp": "p : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Fact (Nat.Prime p)\nx : 𝕎 R\nn : ℕ\n⊢ init n (-x) = init n (-init n x)",
"ppTerm": "?m.15",
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"instNeZeroNatHAdd_1",
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"instDeci... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.InitTail | {
"line": 201,
"column": 88
} | {
"line": 201,
"column": 90
} | {
"line": 202,
"column": 2
} | [
{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.InitTail | {
"line": 201,
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} | {
"line": 202,
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} | {
"line": 204,
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{
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... | [] | by
init_ring using wittSub_vars | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.WittVector.Truncated | {
"line": 459,
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} | {
"line": 459,
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} | {
"line": 460,
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{
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Mathlib.RingTheory.WittVector.InitTail | {
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"column": 89
} | {
"line": 204,
"column": 91
} | {
"line": 205,
"column": 2
} | [
{
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"instNeZeroNatHAdd_1",
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"Fintype.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.InitTail | {
"line": 207,
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} | {
"line": 207,
"column": 91
} | {
"line": 208,
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} | [
{
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Mathlib.RingTheory.WittVector.Truncated | {
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} | {
"line": 471,
"column": 37
} | {
"line": 472,
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} | [
{
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Mathlib.RingTheory.WittVector.Truncated | {
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} | {
"line": 474,
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} | {
"line": 474,
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} | [
{
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Mathlib.RingTheory.WittVector.InitTail | {
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} | {
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} | {
"line": 211,
"column": 2
} | [
{
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Mathlib.RingTheory.WittVector.Complete | {
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} | {
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} | {
"line": 39,
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} | [
{
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Mathlib.RingTheory.WittVector.Complete | {
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} | {
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} | {
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{
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"usedConstants": [
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"map_... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Complete | {
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} | {
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} | {
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{
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Mathlib.RingTheory.Perfection | {
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} | {
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} | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Complete | {
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} | {
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} | {
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{
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Mathlib.RingTheory.WittVector.Complete | {
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} | {
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} | {
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{
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Mathlib.RingTheory.WittVector.Complete | {
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} | {
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} | {
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} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx : 𝕎 k\nh : x.coeff 0 = 0\n⊢ ∀ i < 1, x.coeff i = 0",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"LinearOrderedCommMonoidWithZero.toIs... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Complete | {
"line": 70,
"column": 36
} | {
"line": 70,
"column": 38
} | {
"line": 71,
"column": 6
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx : 𝕎 k\nh : x.coeff 0 = 0\n⊢ x = verschiebung (x.shift 1)",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"WittVector.eq_iterate_verschiebung",
"Preorder.t... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Complete | {
"line": 72,
"column": 13
} | {
"line": 72,
"column": 15
} | {
"line": 73,
"column": 6
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx : 𝕎 k\nh : x.coeff 0 = 0\n⊢ verschiebung (x.shift 1) = (frobeniusEquiv p k).symm (x.shift 1) * ↑p",
"ppTerm": "?m.91",
"assigned": true,
"usedConstants": [
"RingEquiv.app... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Complete | {
"line": 64,
"column": 52
} | {
"line": 64,
"column": 54
} | {
"line": 65,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx : 𝕎 k\n⊢ x ∈ Ideal.span {↑p} ↔ x.coeff 0 = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"RingEquiv.apply_symm_apply",
"NonUnitalNonAssocCommRing.toNo... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Perfection | {
"line": 800,
"column": 72
} | {
"line": 800,
"column": 74
} | {
"line": 801,
"column": 2
} | [
{
"pp": "K : Type u₁\ninst✝⁴ : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝³ : CommRing O\ninst✝² : Algebra O K\nhv : v.Integers O\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fact ¬IsUnit ↑p\nf : PreTilt O p\n⊢ (val K v O hv p) f = 0 ↔ f = 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Complete | {
"line": 88,
"column": 40
} | {
"line": 88,
"column": 42
} | {
"line": 89,
"column": 6
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx : 𝕎 k\nn : ℕ\nh : ∀ m < n, x.coeff m = 0\n⊢ x = (⇑verschiebung)^[n] (x.shift n)",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": [
"WittVector.eq_iterate_verschie... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Perfection | {
"line": 817,
"column": 16
} | {
"line": 817,
"column": 18
} | {
"line": 818,
"column": 6
} | [
{
"pp": "K : Type u₁\ninst✝⁴ : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝³ : CommRing O\ninst✝² : Algebra O K\nhv : v.Integers O\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fact ¬IsUnit ↑p\nhp : Nat.Prime p\nthis : Nontrivial (PreTilt O p)\na✝ b✝ : PreTilt O p\nhfg : a✝ * b✝ = 0\n⊢ a✝ = 0 ∨ b✝ = 0",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Perfection | {
"line": 813,
"column": 45
} | {
"line": 813,
"column": 47
} | {
"line": 814,
"column": 2
} | [
{
"pp": "K : Type u₁\ninst✝⁴ : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝³ : CommRing O\ninst✝² : Algebra O K\nhv : v.Integers O\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fact ¬IsUnit ↑p\n⊢ IsDomain (PreTilt O p)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Nontrivial",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Complete | {
"line": 90,
"column": 13
} | {
"line": 90,
"column": 15
} | {
"line": 91,
"column": 6
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx : 𝕎 k\nn : ℕ\nh : ∀ m < n, x.coeff m = 0\n⊢ (⇑verschiebung)^[n] (x.shift n) =\n (⇑verschiebung)^[n] ((⇑frobenius)^[n] ((⇑(frobeniusEquiv p k).symm)^[n] (x.shift n)))",
"ppTerm": "?m... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Complete | {
"line": 82,
"column": 2
} | {
"line": 96,
"column": 88
} | {
"line": 98,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx : 𝕎 k\nn : ℕ\n⊢ x ∈ Ideal.span {↑p ^ n} ↔ ∀ m < n, x.coeff m = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Function.iterate_id",
"RingEquiv.apply_s... | [] | simp_rw [Ideal.mem_span_singleton, dvd_def, mul_comm]
refine ⟨fun ⟨u, hu⟩ m hm ↦ ?_, fun h ↦ ?_⟩
· rw [hu, mul_pow_charP_coeff_zero _ hm]
· use (frobeniusEquiv p k).symm^[n] (x.shift n)
rw [← iterate_verschiebung_iterate_frobenius]
calc
_ = verschiebung^[n] (x.shift n) := by
simpa using eq_itera... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.Complete | {
"line": 82,
"column": 2
} | {
"line": 96,
"column": 88
} | {
"line": 98,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx : 𝕎 k\nn : ℕ\n⊢ x ∈ Ideal.span {↑p ^ n} ↔ ∀ m < n, x.coeff m = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Function.iterate_id",
"RingEquiv.apply_s... | [] | simp_rw [Ideal.mem_span_singleton, dvd_def, mul_comm]
refine ⟨fun ⟨u, hu⟩ m hm ↦ ?_, fun h ↦ ?_⟩
· rw [hu, mul_pow_charP_coeff_zero _ hm]
· use (frobeniusEquiv p k).symm^[n] (x.shift n)
rw [← iterate_verschiebung_iterate_frobenius]
calc
_ = verschiebung^[n] (x.shift n) := by
simpa using eq_itera... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.Complete | {
"line": 81,
"column": 69
} | {
"line": 81,
"column": 71
} | {
"line": 82,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx : 𝕎 k\nn : ℕ\n⊢ x ∈ Ideal.span {↑p ^ n} ↔ ∀ m < n, x.coeff m = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Function.iterate_id",
"RingEquiv.apply_s... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Complete | {
"line": 98,
"column": 80
} | {
"line": 98,
"column": 82
} | {
"line": 99,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\n⊢ RingHom.ker constantCoeff = Ideal.span {↑p}",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"RingHom.instRingHomClass",
"Semiring.toModule",
"RingHom... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Complete | {
"line": 117,
"column": 11
} | {
"line": 117,
"column": 13
} | {
"line": 118,
"column": 4
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\n⊢ ∀ (x : 𝕎 k), (∀ (n : ℕ), x ≡ 0 [SMOD Ideal.span {↑p} ^ n • ⊤]) → x = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"instH... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.Complete | {
"line": 125,
"column": 11
} | {
"line": 125,
"column": 13
} | {
"line": 126,
"column": 4
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\n⊢ ∀ (f : ℕ → 𝕎 k),\n (∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD Ideal.span {↑p} ^ m • ⊤]) →\n ∃ L, ∀ (n : ℕ), f n ≡ L [SMOD Ideal.span {↑p} ^ n • ⊤]",
"ppTerm": "?m.19",
"assigned... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.TeichmullerSeries | {
"line": 51,
"column": 73
} | {
"line": 51,
"column": 75
} | {
"line": 52,
"column": 6
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : CommRing R\nα : Type u_2\nx : α → 𝕎 R\na : α\nS' : Finset α\nha : a ∉ S'\nhind :\n (∀ (n : ℕ), Subsingleton ↑{r | r ∈ S' ∧ (x r).coeff n ≠ 0}) →\n ∀ (n : ℕ), (∑ s ∈ S', x s).coeff n = ∑ s ∈ S', (x s).coeff n\nh : ∀ (n : ℕ), Subsingleton ↑{r | r... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.WittVector.TeichmullerSeries | {
"line": 58,
"column": 73
} | {
"line": 58,
"column": 75
} | {
"line": 59,
"column": 6
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : CommRing R\nα : Type u_2\nx : α → 𝕎 R\na : α\nS' : Finset α\nha : a ∉ S'\nh : ∀ (n : ℕ), Subsingleton ↑{r | r ∈ insert a S' ∧ (x r).coeff n ≠ 0}\nn : ℕ\nthis : ∀ (n : ℕ), Subsingleton ↑{r | r ∈ S' ∧ (x r).coeff n ≠ 0}\nhind : ∀ (n : ℕ), (∑ s ∈ S', ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Perfectoid.FontaineTheta | {
"line": 86,
"column": 47
} | {
"line": 86,
"column": 49
} | {
"line": 87,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nn : ℕ\n⊢ RingHom.ker (map (Ideal.Quotient.mk 𝔭)) ≤ RingHom.ker ((Ideal.Quotient.mk (𝔭 ^ (n + 1))).comp (ghostComponent n))",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Ideal.quotEquivOfEq",
"Eq.m... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Perfectoid.FontaineTheta | {
"line": 114,
"column": 72
} | {
"line": 114,
"column": 74
} | {
"line": 115,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : 𝕎 (R ⧸ 𝔭)\nh : 𝔭 ^ (0 + 1) = 𝔭\n⊢ (quotEquivOfEq h) ((ghostComponentModPPow 0) x) = (ghostComponent 0) x",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Ideal.quotEquivOfEq",
"wittPolynomial",... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Perfectoid.FontaineTheta | {
"line": 124,
"column": 48
} | {
"line": 124,
"column": 50
} | {
"line": 125,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete 𝔭 R\nn : ℕ\nx : R♭\n⊢ (ghostComponentModPPow n) ((teichmuller p) ((PreTilt.coeff n) x)) = (Ideal.Quotient.mk (𝔭 ^ (n + 1))) (untilt x)",
"ppTerm": "?m.55",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Perfectoid.FontaineTheta | {
"line": 140,
"column": 83
} | {
"line": 140,
"column": 85
} | {
"line": 141,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete 𝔭 R\nn : ℕ\nx : R♭\n⊢ (fontaineThetaModPPow R p n) ((teichmuller p) x) = (Ideal.Quotient.mk (𝔭 ^ (n + 1))) (untilt x)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Perfectoid.FontaineTheta | {
"line": 148,
"column": 52
} | {
"line": 148,
"column": 54
} | {
"line": 149,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete 𝔭 R\nn : ℕ\n⊢ ↑p = (Ideal.Quotient.mk (𝔭 ^ (n + 1))) ↑p",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Perfectoid.FontaineTheta | {
"line": 148,
"column": 4
} | {
"line": 149,
"column": 24
} | {
"line": 150,
"column": 4
} | [
{
"pp": "case h\nR : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete 𝔭 R\nn : ℕ\n⊢ ↑p ^ (n + 1) = 0",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomCla... | [
"case h\nR : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete 𝔭 R\nn : ℕ\nthis : ↑p = (Ideal.Quotient.mk (𝔭 ^ (n + 1))) ↑p\n⊢ ↑p ^ (n + 1) = 0"
] | have : p = Ideal.Quotient.mk (𝔭 ^ (n + 1)) p := by
simp [map_natCast] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.WittVector.TeichmullerSeries | {
"line": 45,
"column": 57
} | {
"line": 45,
"column": 59
} | {
"line": 46,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : CommRing R\nα : Type u_2\nS : Finset α\nx : α → 𝕎 R\nh : ∀ (n : ℕ), Subsingleton ↑{r | r ∈ S ∧ (x r).coeff n ≠ 0}\nn : ℕ\n⊢ (∑ s ∈ S, x s).coeff n = ∑ s ∈ S, (x s).coeff n",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Perfectoid.FontaineTheta | {
"line": 144,
"column": 96
} | {
"line": 144,
"column": 98
} | {
"line": 145,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete 𝔭 R\nn : ℕ\n⊢ (factorPowSucc 𝔭 (n + 1)).comp (fontaineThetaModPPow R p (n + 1)) = fontaineThetaModPPow R p n",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.TeichmullerSeries | {
"line": 74,
"column": 53
} | {
"line": 74,
"column": 55
} | {
"line": 75,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nn : ℕ\nx : R\n⊢ ((teichmuller p) x * ↑p ^ n).coeff n = x ^ p ^ n",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"MonoidHom.instFunLike",
"HMul.hMul",
"MonoidHom",
"WittVe... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Perfectoid.FontaineTheta | {
"line": 155,
"column": 95
} | {
"line": 155,
"column": 97
} | {
"line": 156,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete 𝔭 R\nn : ℕ\nx : 𝕎 R♭\n⊢ (factorPowSucc 𝔭 (n + 1)) ((fontaineThetaModPPow R p (n + 1)) x) = (fontaineThetaModPPow R p n) x",
"ppTerm": "?m.34",
"assigned": true,
"usedCons... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.TeichmullerSeries | {
"line": 78,
"column": 69
} | {
"line": 78,
"column": 71
} | {
"line": 79,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nx : R\nm n : ℕ\nh : m ≠ n\n⊢ ((teichmuller p) x * ↑p ^ n).coeff m = 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"False",
"Nat.Prime",
"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Perfectoid.FontaineTheta | {
"line": 173,
"column": 79
} | {
"line": 173,
"column": 81
} | {
"line": 174,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete 𝔭 R\nx : 𝕎 R♭\n⊢ (Ideal.Quotient.mk 𝔭) ((fontaineTheta R p) x) = (PreTilt.coeff 0) (x.coeff 0)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"WittVector... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Perfectoid.FontaineTheta | {
"line": 182,
"column": 95
} | {
"line": 182,
"column": 97
} | {
"line": 183,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete 𝔭 R\nx : R♭\n⊢ (fontaineTheta R p) ((teichmuller p) x) = untilt x",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"MulOne... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Perfectoid.FontaineTheta | {
"line": 197,
"column": 66
} | {
"line": 197,
"column": 68
} | {
"line": 198,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete 𝔭 R\nhF : Function.Surjective ⇑(_root_.frobenius (ModP R p) p)\n⊢ Ideal.map (fontaineTheta R p) (span {↑p}) = 𝔭",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
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{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\n⊢ dickson 2 1 (n + 2) = Chebyshev.S R ↑(n + 2)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"Polynomial.C",
"NegZero... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 54,
"column": 91
} | {
"line": 54,
"column": 93
} | {
"line": 55,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ hermite (n + 1) = X * hermite n - derivative (hermite n)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"LinearMap.instFunLike",
"HSub.hSub",
... | [] | by | [anonymous] | by |
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