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.Polynomial.Dickson | {
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} | {
"line": 185,
"column": 46
} | {
"line": 185,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nm n : ℕ\n⊢ 1 = (Int.castRingHom R) 1",
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"eq_intCast"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
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} | {
"line": 57,
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} | {
"line": 58,
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} | [
{
"pp": "n : ℕ\n⊢ hermite n = (fun p ↦ X * p - derivative p)^[n] 1",
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Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 66,
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} | {
"line": 66,
"column": 41
} | {
"line": 67,
"column": 2
} | [
{
"pp": "⊢ hermite 1 = X",
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
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} | {
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} | {
"line": 76,
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} | [
{
"pp": "n : ℕ\n⊢ (hermite (n + 1)).coeff 0 = -(hermite n).coeff 1",
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
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} | {
"line": 79,
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} | {
"line": 80,
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} | [
{
"pp": "n k : ℕ\n⊢ (hermite (n + 1)).coeff (k + 1) = (hermite n).coeff k - (↑k + 2) * (hermite n).coeff (k + 2)",
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"usedConstants": [
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Mathlib.RingTheory.Polynomial.Hermite.Basic | {
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} | {
"line": 89,
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} | {
"line": 89,
"column": 49
} | [
{
"pp": "n : ℕ\nih : ∀ (k : ℕ), (hermite n).coeff (n + k + 1) = 0\nk : ℕ\n⊢ n + k + 1 + 2 = n + (k + 2) + 1",
"ppTerm": "?m.82",
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Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 83,
"column": 81
} | {
"line": 83,
"column": 83
} | {
"line": 84,
"column": 2
} | [
{
"pp": "n k : ℕ\nhnk : n < k\n⊢ (hermite n).coeff k = 0",
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"usedConstants": [
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Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 93,
"column": 64
} | {
"line": 93,
"column": 66
} | {
"line": 94,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (hermite n).coeff n = 1",
"ppTerm": "?m.6",
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"Preorder.toLT",
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"LinearOrderedCommMon... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.TeichmullerSeries | {
"line": 134,
"column": 33
} | {
"line": 134,
"column": 35
} | {
"line": 134,
"column": 36
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : CharP R p\ninst✝¹ : PerfectRing R p\nS : Type u_2\ninst✝ : CommRing S\nf g : 𝕎 R →+* S\nh : ∀ (x : R), f ((teichmuller p) x) = g ((teichmuller p) x)\nn : ℕ\nhn : ↑p ^ n = 0\nx c : 𝕎 R\nhc : x - ∑ i ≤ n, (teichmuller p) (((_ro... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 101,
"column": 59
} | {
"line": 101,
"column": 61
} | {
"line": 102,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (hermite n).degree = ↑n",
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Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 113,
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} | {
"line": 113,
"column": 73
} | {
"line": 114,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (hermite n).leadingCoeff = 1",
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"Int",
"Polynomial.coeff_hermit... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Hermite.Gaussian | {
"line": 53,
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} | {
"line": 53,
"column": 23
} | {
"line": 53,
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} | [
{
"pp": "n : ℕ\nx : ℝ\nih : (deriv^[n] fun y ↦ Real.exp (-(y ^ 2 / 2))) = fun x ↦ (-1) ^ n * ((aeval x) (hermite n) * Real.exp (-(x ^ 2 / 2)))\n⊢ DifferentiableAt ℝ (fun y ↦ -(y ^ 2 / 2)) x",
"ppTerm": "?m.221",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"NegZero... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.TeichmullerSeries | {
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} | {
"line": 135,
"column": 17
} | {
"line": 135,
"column": 18
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : CharP R p\ninst✝¹ : PerfectRing R p\nS : Type u_2\ninst✝ : CommRing S\nf g : 𝕎 R →+* S\nh : ∀ (x : R), f ((teichmuller p) x) = g ((teichmuller p) x)\nn : ℕ\nhn : ↑p ^ n = 0\nx c : 𝕎 R\nhc : x - ∑ i ≤ n, (teichmuller p) (((_ro... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 132,
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} | {
"line": 132,
"column": 15
} | {
"line": 132,
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} | [
{
"pp": "n : ℕ\nih : ∀ {k : ℕ}, Odd (n + k) → (hermite n).coeff k = 0\nk : ℕ\nhnk : Odd (n + 1 + (k + 1))\n⊢ n.succ + k.succ = n + k + 2",
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"usedConstants": [
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"congrArg",
"id",
"instOfNatNat",
"instHAdd",
"HAdd.hAdd",... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Hermite.Gaussian | {
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} | {
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} | {
"line": 53,
"column": 6
} | [
{
"pp": "n : ℕ\nx : ℝ\nih : (deriv^[n] fun y ↦ Real.exp (-(y ^ 2 / 2))) = fun x ↦ (-1) ^ n * ((aeval x) (hermite n) * Real.exp (-(x ^ 2 / 2)))\n⊢ deriv (fun y ↦ Real.exp (-(y ^ 2 / 2))) x = -x * Real.exp (-(x ^ 2 / 2))",
"ppTerm": "?m.216",
"assigned": true,
"usedConstants": [
"deriv_exp",
... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 119,
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} | {
"line": 119,
"column": 94
} | {
"line": 120,
"column": 2
} | [
{
"pp": "n k : ℕ\nhnk : Odd (n + k)\n⊢ (hermite n).coeff k = 0",
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"Pol... | [] | by | [anonymous] | by |
Mathlib.RingTheory.WittVector.TeichmullerSeries | {
"line": 120,
"column": 74
} | {
"line": 120,
"column": 76
} | {
"line": 121,
"column": 2
} | [
{
"pp": "p : ℕ\nhp✝ : Fact (Nat.Prime p)\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : CharP R p\ninst✝¹ : PerfectRing R p\nS : Type u_2\ninst✝ : CommRing S\nf g : 𝕎 R →+* S\nhp : IsNilpotent ↑p\nh : ∀ (x : R), f ((teichmuller p) x) = g ((teichmuller p) x)\n⊢ f = g",
"ppTerm": "?m.22",
"assigned": true,... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 144,
"column": 12
} | {
"line": 144,
"column": 14
} | {
"line": 144,
"column": 15
} | [
{
"pp": "x✝ : ℕ\n⊢ (hermite (2 * 0 + x✝)).coeff x✝ = (-1) ^ 0 * ↑(2 * 0 - 1)‼ * ↑((2 * 0 + x✝).choose x✝)",
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"Nat.choose... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 147,
"column": 37
} | {
"line": 147,
"column": 39
} | {
"line": 147,
"column": 40
} | [
{
"pp": "n : ℕ\n⊢ 2 * (n + 1) - 1 = 2 * n + 1",
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"assigned": true,
"usedConstants": [
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],
"usedFVars": [
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],
"usedGoals": []
}
] | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Dickson | {
"line": 184,
"column": 72
} | {
"line": 184,
"column": 74
} | {
"line": 185,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nm n : ℕ\n⊢ dickson 1 1 (m * n) = (dickson 1 1 m).comp (dickson 1 1 n)",
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"_pr... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Dickson | {
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} | {
"line": 199,
"column": 91
} | {
"line": 200,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nm n : ℕ\n⊢ (dickson 1 1 m).comp (dickson 1 1 n) = (dickson 1 1 n).comp (dickson 1 1 m)",
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Mathlib.RingTheory.Polynomial.Hermite.Basic | {
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"column": 16
} | {
"line": 145,
"column": 18
} | {
"line": 146,
"column": 4
} | [
{
"pp": "n : ℕ\n⊢ (hermite (2 * (n + 1) + 0)).coeff 0 = (-1) ^ (n + 1) * ↑(2 * (n + 1) - 1)‼ * ↑((2 * (n + 1) + 0).choose 0)",
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"N... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Dickson | {
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"column": 24
} | {
"line": 211,
"column": 26
} | {
"line": 212,
"column": 6
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nK : Type := FractionRing (ZMod p)[X]\nf : ZMod p →+* K := (algebraMap (ZMod p)[X] (FractionRing (ZMod p)[X])).comp C\n⊢ CharP K p",
"ppTerm": "?m.48",
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... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.Dickson | {
"line": 208,
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} | {
"line": 208,
"column": 85
} | {
"line": 209,
"column": 4
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ ∃ K x, ∃ (_ : CharP K p), Infinite K",
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Mathlib.RingTheory.Polynomial.HilbertPoly | {
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"column": 30
} | {
"line": 68,
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} | {
"line": 68,
"column": 33
} | [
{
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Mathlib.RingTheory.Polynomial.Hermite.Basic | {
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"column": 11
} | {
"line": 168,
"column": 13
} | {
"line": 168,
"column": 14
} | [
{
"pp": "n✝ k✝ : ℕ\nhermite_explicit : ℕ → ℕ → ℤ := fun n k ↦ (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)\nn k : ℕ\n⊢ 2 * (n + 1) - 1 = 2 * n + 1",
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"usedConstants": [
"_private.Mathlib.RingTheory.Polynomial.Hermite.Basic.0.Polynomial.coeff_hermite_e... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Dickson | {
"line": 244,
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} | {
"line": 244,
"column": 33
} | {
"line": 244,
"column": 34
} | [
{
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"ppTerm": "?m.320",
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"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
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} | {
"line": 67,
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} | {
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} | [
{
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"RingHom.instRingHomClass",
"MulOne... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Dickson | {
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} | {
"line": 242,
"column": 27
} | {
"line": 243,
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} | [
{
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"ppTerm": "?m.308",
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... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
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} | {
"line": 75,
"column": 32
} | {
"line": 75,
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} | [
{
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"AddGroupWithOne.toAddMo... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
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"column": 11
} | {
"line": 173,
"column": 13
} | {
"line": 173,
"column": 14
} | [
{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
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} | {
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} | {
"line": 81,
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{
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Mathlib.RingTheory.Polynomial.Hermite.Basic | {
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} | {
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} | {
"line": 174,
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{
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Mathlib.RingTheory.Polynomial.HilbertPoly | {
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} | {
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} | {
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{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\nd k : ℕ\nhne : ↑d ! ≠ 0\nheq : d = ((ascPochhammer F d).comp (X - C ↑k + 1)).natDegree\n⊢ (↑d !)⁻¹ • ((ascPochhammer F d).comp (X - C (↑k - 1))).leadingCoeff = (↑d !)⁻¹",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"Nont... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 73,
"column": 52
} | {
"line": 73,
"column": 54
} | {
"line": 74,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\nd k : ℕ\n⊢ (preHilbertPoly F d k).coeff d = (↑d !)⁻¹",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Nontrivial",
"sub_add",
"one_pow",
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMono... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 175,
"column": 9
} | {
"line": 175,
"column": 11
} | {
"line": 175,
"column": 12
} | [
{
"pp": "n✝ k✝ : ℕ\nhermite_explicit : ℕ → ℕ → ℤ := fun n k ↦ (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)\nn k : ℕ\n⊢ 2 * n + (k + 2) = 2 * n + 1 + (k + 1)",
"ppTerm": "?m.604",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathli... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 87,
"column": 57
} | {
"line": 87,
"column": 59
} | {
"line": 88,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\nd k : ℕ\n⊢ (preHilbertPoly F d k).leadingCoeff = (↑d !)⁻¹",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.leadingCoeff.eq_1",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.to... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 92,
"column": 26
} | {
"line": 92,
"column": 28
} | {
"line": 92,
"column": 29
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\nd k n : ℕ\nhkn : k ≤ n\n⊢ ↑d ! ≠ 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder",
"AddGroup... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 94,
"column": 60
} | {
"line": 94,
"column": 62
} | {
"line": 94,
"column": 63
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\nd k n : ℕ\nhkn : k ≤ n\nthis : ↑d ! ≠ 0\n⊢ eval (↑n) (preHilbertPoly F d k) = (↑d !)⁻¹ * eval (↑(n - k + 1)) (ascPochhammer F d)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Polynomial.eval",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 95,
"column": 30
} | {
"line": 95,
"column": 32
} | {
"line": 96,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\nd k n : ℕ\nhkn : k ≤ n\nthis : ↑d ! ≠ 0\n⊢ (↑d !)⁻¹ * eval (↑(n - k + 1)) (ascPochhammer F d) = ↑((n - k + d).choose d)",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Eq.mpr",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 91,
"column": 66
} | {
"line": 91,
"column": 68
} | {
"line": 92,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\nd k n : ℕ\nhkn : k ≤ n\n⊢ eval (↑n) (preHilbertPoly F d k) = ↑((n - k + d).choose d)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Eq.mpr",
"Polynomial.C",
"Polynom... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 116,
"column": 12
} | {
"line": 116,
"column": 21
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case zero\nF : Type u_1\ninst✝ : Field F\n⊢ (match 0 with\n | 0 => 0\n | d.succ => ∑ i ∈ support 0, coeff 0 i • preHilbertPoly F d i) =\n 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Field.toSemifield",
"Polynomial",
"Semifield.toDivisionSemiring",... | [] | simp only | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 116,
"column": 12
} | {
"line": 116,
"column": 21
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case zero\nF : Type u_1\ninst✝ : Field F\n⊢ (match 0 with\n | 0 => 0\n | d.succ => ∑ i ∈ support 0, coeff 0 i • preHilbertPoly F d i) =\n 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Field.toSemifield",
"Polynomial",
"Semifield.toDivisionSemiring",... | [] | simp only | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 116,
"column": 12
} | {
"line": 116,
"column": 21
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case zero\nF : Type u_1\ninst✝ : Field F\n⊢ (match 0 with\n | 0 => 0\n | d.succ => ∑ i ∈ support 0, coeff 0 i • preHilbertPoly F d i) =\n 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Field.toSemifield",
"Polynomial",
"Semifield.toDivisionSemiring",... | [] | simp only | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 158,
"column": 79
} | {
"line": 158,
"column": 81
} | {
"line": 159,
"column": 6
} | [
{
"pp": "n k : ℕ\nhermite_explicit : ℕ → ℕ → ℤ := fun n k ↦ (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)\n⊢ ∀ (n k : ℕ), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (↑k + 2) * hermite_explicit n (k + 2)",
"ppTerm": "?m.341",
"assigned": true,
"usedConstants": [
"Mathl... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 114,
"column": 70
} | {
"line": 114,
"column": 72
} | {
"line": 115,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\nd : ℕ\n⊢ hilbertPoly 0 d = 0",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Nat.recAux",
"instHSMul",
"Semiring.toModule",
"congrArg",
"SMulWithZero.toSMulZeroClass",
"AddMonoid.toAddZeroClass",
"Polynomial.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 125,
"column": 67
} | {
"line": 125,
"column": 69
} | {
"line": 126,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\nd k : ℕ\n⊢ (X ^ k).hilbertPoly (d + 1) = preHilbertPoly F d k",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Polynomial.distribMulAction",
"instHSMul",
"Semiring.toModule",
"F... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Hermite.Gaussian | {
"line": 44,
"column": 68
} | {
"line": 44,
"column": 70
} | {
"line": 45,
"column": 2
} | [
{
"pp": "n : ℕ\nx : ℝ\n⊢ deriv^[n] (fun y ↦ Real.exp (-(y ^ 2 / 2))) x = (-1) ^ n * (aeval x) (hermite n) * Real.exp (-(x ^ 2 / 2))",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"deriv_exp",
"NormedCommRing.toNormedRing",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 15
} | {
"line": 135,
"column": 6
} | [
{
"pp": "case succ\nF : Type u_1\ninst✝ : Field F\np q : F[X]\nd : ℕ\na✝ :\n (match d with\n | 0 => 0\n | d.succ => ∑ i ∈ (p + q).support, (p + q).coeff i • preHilbertPoly F d i) =\n (match d with\n | 0 => 0\n | d.succ => ∑ i ∈ p.support, p.coeff i • preHilbertPoly F d i) +\n match d wi... | [
"case succ\nF : Type u_1\ninst✝ : Field F\np q : F[X]\nd : ℕ\na✝ :\n (match d with\n | 0 => 0\n | d.succ => ∑ i ∈ (p + q).support, (p + q).coeff i • preHilbertPoly F d i) =\n (match d with\n | 0 => 0\n | d.succ => ∑ i ∈ p.support, p.coeff i • preHilbertPoly F d i) +\n match d with\n | ... | simp only | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Hermite.Gaussian | {
"line": 62,
"column": 95
} | {
"line": 62,
"column": 97
} | {
"line": 63,
"column": 2
} | [
{
"pp": "n : ℕ\nx : ℝ\n⊢ (aeval x) (hermite n) = (-1) ^ n * deriv^[n] (fun y ↦ Real.exp (-(y ^ 2 / 2))) x / Real.exp (-(x ^ 2 / 2))",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"one_pow",
"Distrib.leftDistribClass",
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Eq.mpr",... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Dickson | {
"line": 236,
"column": 85
} | {
"line": 236,
"column": 87
} | {
"line": 237,
"column": 6
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nK : Type\nw✝¹ : Field K\nw✝ : CharP K p\nH : Set.univ.Infinite\nh : {x | ∃ y, x = y + y⁻¹ ∧ y ≠ 0}.Finite\nthis : Set.univ = ⋃ x ∈ {x | ∃ y, x = y + y⁻¹ ∧ y ≠ 0}, {y | x = y + y⁻¹ ∨ y = 0}\n⊢ Set.univ.Finite",
"ppTerm": "?m.252",
"assigned": true,
"usedCon... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 129,
"column": 65
} | {
"line": 129,
"column": 67
} | {
"line": 130,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\np q : F[X]\nd : ℕ\n⊢ (p + q).hilbertPoly d = p.hilbertPoly d + q.hilbertPoly d",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.recAux",
"instHSMul",
"Semiring.toModule",
"congrArg",
"AddMonoid.toAddZ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Hermite.Gaussian | {
"line": 69,
"column": 92
} | {
"line": 69,
"column": 94
} | {
"line": 70,
"column": 2
} | [
{
"pp": "n : ℕ\nx : ℝ\n⊢ (aeval x) (hermite n) = (-1) ^ n * deriv^[n] (fun y ↦ Real.exp (-(y ^ 2 / 2))) x * Real.exp (x ^ 2 / 2)",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Eq.mpr",
"GroupWith... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 144,
"column": 6
} | {
"line": 144,
"column": 15
} | {
"line": 145,
"column": 6
} | [
{
"pp": "case succ\nF : Type u_1\ninst✝ : Field F\na : F\np : F[X]\nd : ℕ\na✝ :\n (match d with\n | 0 => 0\n | d.succ => ∑ i ∈ (a • p).support, (a • p).coeff i • preHilbertPoly F d i) =\n a •\n match d with\n | 0 => 0\n | d.succ => ∑ i ∈ p.support, p.coeff i • preHilbertPoly F d i\n⊢ (m... | [
"case succ\nF : Type u_1\ninst✝ : Field F\na : F\np : F[X]\nd : ℕ\na✝ :\n (match d with\n | 0 => 0\n | d.succ => ∑ i ∈ (a • p).support, (a • p).coeff i • preHilbertPoly F d i) =\n a •\n match d with\n | 0 => 0\n | d.succ => ∑ i ∈ p.support, p.coeff i • preHilbertPoly F d i\n⊢ ∑ i ∈ (a • p).... | simp only | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 182,
"column": 11
} | {
"line": 182,
"column": 13
} | {
"line": 182,
"column": 14
} | [
{
"pp": "n k : ℕ\nhermite_explicit : ℕ → ℕ → ℤ := fun n k ↦ (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)\nhermite_explicit_recur :\n ∀ (n k : ℕ), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (↑k + 2) * hermite_explicit n (k + 2)\n⊢ 2 * (n + 1) + k = 2 * n + (k + 2)",
"ppTerm": "?m.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.IrreducibleRing | {
"line": 51,
"column": 13
} | {
"line": 51,
"column": 15
} | {
"line": 51,
"column": 16
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : (nilradical R).IsPrime\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nφ : R →+* S\nf : R[X]\nhm : f.Monic\nR' : Type u_1 := R ⧸ nilradical R\nψ : R' →+* S := Ideal.Quotient.lift (nilradical R) φ ⋯\nι : R →+* R' := algebraMap R R'\nhi : Irreducible (P... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 139,
"column": 51
} | {
"line": 139,
"column": 53
} | {
"line": 140,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\na : F\np : F[X]\nd : ℕ\n⊢ (a • p).hilbertPoly d = a • p.hilbertPoly d",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.recAux",
"Polynomial.sum_smul_index'",
"inst... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 152,
"column": 20
} | {
"line": 152,
"column": 22
} | {
"line": 153,
"column": 4
} | [
{
"pp": "n k : ℕ\n⊢ (hermite (2 * (n + 1) + (k + 1))).coeff (k + 1) =\n (-1) ^ (n + 1) * ↑(2 * (n + 1) - 1)‼ * ↑((2 * (n + 1) + (k + 1)).choose (k + 1))",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Nat.cast_mul._simp_1",
"N... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.IrreducibleRing | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 50
} | {
"line": 54,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : (nilradical R).IsPrime\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nφ : R →+* S\nf : R[X]\nhm : f.Monic\nR' : Type u_1 := R ⧸ nilradical R\nψ : R' →+* S := Ideal.Quotient.lift (nilradical R) φ ⋯\nι : R →+* R' := algebraMap R R'\nhi✝ : Irreducible (... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : (nilradical R).IsPrime\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nφ : R →+* S\nf : R[X]\nhm : f.Monic\nR' : Type u_1 := R ⧸ nilradical R\nψ : R' →+* S := Ideal.Quotient.lift (nilradical R) φ ⋯\nι : R →+* R' := algebraMap R R'\nhi✝ : Irreducible (Polynomial.m... | obtain ⟨_, _, h⟩ := Polynomial.isUnit_iff.1 hb | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Polynomial.IrreducibleRing | {
"line": 52,
"column": 60
} | {
"line": 52,
"column": 62
} | {
"line": 53,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : (nilradical R).IsPrime\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nφ : R →+* S\nf : R[X]\nhm : f.Monic\nR' : Type u_1 := R ⧸ nilradical R\nψ : R' →+* S := Ideal.Quotient.lift (nilradical R) φ ⋯\nι : R →+* R' := algebraMap R R'\nhi✝ : Irreducible (... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.Dickson | {
"line": 202,
"column": 92
} | {
"line": 202,
"column": 94
} | {
"line": 208,
"column": 2
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ dickson 1 1 p = X ^ p",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Multiset.toFinset",
"Polynomial.map_dickson",
"Iff.mpr",
"add_mul",
"AddGroup.toSubtractionMono... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Dickson | {
"line": 270,
"column": 54
} | {
"line": 270,
"column": 56
} | {
"line": 271,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\n⊢ 1 = (ZMod.castHom ⋯ R) 1",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"ZMod.commRing",
"congrArg",
"CommSemiring.toSemiring",
"Nat.instMonoid",
"AddGroupWit... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 189,
"column": 9
} | {
"line": 189,
"column": 11
} | {
"line": 189,
"column": 12
} | [
{
"pp": "n k : ℕ\nh_le : k ≤ n\nm : ℕ\nhm : n - k = m + m\n⊢ n = 2 * m + k",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": [
"_private.Mathlib.RingTheory.Polynomial.Hermite.Basic.0.Polynomial.coeff_hermite_of_even_add._proof_1_1"
],
"usedFVars": [
"n",
"k",
"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Dickson | {
"line": 269,
"column": 98
} | {
"line": 269,
"column": 100
} | {
"line": 270,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\n⊢ dickson 1 1 p = X ^ p",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Polynomial.map_dickson",
"Eq.mpr",
"ZMod.commRing",
"congrArg",
"CommSemiring.toSemiring... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 177,
"column": 90
} | {
"line": 177,
"column": 92
} | {
"line": 178,
"column": 8
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nn : ℕ\nhn : p.natDegree < n\nd : ℕ\nhd :\n (PowerSeries.coeff n) (↑p * ↑(invOneSubPow F d)) =\n eval (↑n)\n (match d with\n | 0 => 0\n | d.succ => ∑ i ∈ p.support, p.coeff i • preHilbertPoly F d i)\nh_le : ∀ (i : ↥p.suppor... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 185,
"column": 82
} | {
"line": 185,
"column": 84
} | {
"line": 186,
"column": 2
} | [
{
"pp": "n k : ℕ\nhnk : Even (n + k)\n⊢ (hermite n).coeff k = (-1) ^ ((n - k) / 2) * ↑(n - k - 1)‼ * ↑(n.choose k)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Eq.mpr",
"Preorder.toLT",
"Nat.choose",
"instHDiv",
"HMul.hMul",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 195,
"column": 94
} | {
"line": 195,
"column": 96
} | {
"line": 196,
"column": 2
} | [
{
"pp": "n k : ℕ\n⊢ (hermite n).coeff k = if Even (n + k) then (-1) ^ ((n - k) / 2) * ↑(n - k - 1)‼ * ↑(n.choose k) else 0",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.choose",
"instHDiv",
"HMul.hMul",
"congrArg",
"Odd",
"HSub.hSu... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.IrreducibleRing | {
"line": 41,
"column": 92
} | {
"line": 41,
"column": 94
} | {
"line": 42,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : (nilradical R).IsPrime\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nφ : R →+* S\nf : R[X]\nhm : f.Monic\nhi : Irreducible (Polynomial.map φ f)\n⊢ Irreducible f",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Finset.Nat.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 44,
"column": 68
} | {
"line": 44,
"column": 70
} | {
"line": 45,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nn : ℕ\nr : R\n⊢ (opRingEquiv R) (op ((monomial n) r)) = (monomial n) (op r)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"RingEquiv.op",
"RingEquiv.op_apply_apply",
"AddMonoidAlgebra.semiring",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 56,
"column": 66
} | {
"line": 56,
"column": 68
} | {
"line": 57,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nr : R\nn : ℕ\n⊢ (opRingEquiv R) (op (C r * X ^ n)) = C (op r) * X ^ n",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"HMul.hMul",
"congrArg",
"RingEquivClass.toNonUnitalRingHomClass",
"RingEquiv.instE... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 66,
"column": 29
} | {
"line": 66,
"column": 31
} | {
"line": 66,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nn : ℕ\nr : Rᵐᵒᵖ\n⊢ (opRingEquiv R) ((opRingEquiv R).symm ((monomial n) r)) = (opRingEquiv R) (op ((monomial n) (unop r)))",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"RingEquiv.apply_symm_apply",
"Polynomial.opRingEquiv_op_monomial... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 77,
"column": 78
} | {
"line": 77,
"column": 80
} | {
"line": 78,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nr : Rᵐᵒᵖ\nn : ℕ\n⊢ (opRingEquiv R).symm (C r * X ^ n) = op (C (unop r) * X ^ n)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"RingEquiv.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 85,
"column": 57
} | {
"line": 85,
"column": 59
} | {
"line": 85,
"column": 60
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]ᵐᵒᵖ\nn : ℕ\n⊢ ((opRingEquiv R) p).coeff n = op ((unop p).coeff n)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"RingEquiv.op",
"RingEquiv.op_apply_apply",
"AddMonoidAlgebra.semiring",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 88,
"column": 92
} | {
"line": 88,
"column": 94
} | {
"line": 89,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]ᵐᵒᵖ\n⊢ ((opRingEquiv R) p).support = (unop p).support",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"congrArg",
"RingEquiv.instEquivLike",
"Finset",
"AddMonoid.toAddZeroClass",
"MulOpposite",
"Finset.e... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 92,
"column": 98
} | {
"line": 92,
"column": 100
} | {
"line": 93,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]ᵐᵒᵖ\n⊢ ((opRingEquiv R) p).natDegree = (unop p).natDegree",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"False",
"MulOpposite.instDecidableEq",
"eq_false",
"Finset.max'.congr_simp",
"congrAr... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 100,
"column": 65
} | {
"line": 100,
"column": 67
} | {
"line": 101,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]ᵐᵒᵖ\n⊢ ((opRingEquiv R) p).leadingCoeff = op (unop p).leadingCoeff",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.leadingCoeff.eq_1",
"congrArg",
"RingEquiv.instEquivLike",
"MulOpposite"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 168,
"column": 78
} | {
"line": 168,
"column": 80
} | {
"line": 169,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd n : ℕ\nhn : p.natDegree < n\n⊢ (PowerSeries.coeff n) (↑p * ↑(invOneSubPow F d)) = eval (↑n) (p.hilbertPoly d)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"_private.Mathlib.RingTheory.Polynomial.HilbertPoly.0... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 109,
"column": 95
} | {
"line": 109,
"column": 97
} | {
"line": 110,
"column": 8
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nh : IsLeftCancelMulZero R[X]\na b c : R\neq : (fun x ↦ a + x) b = (fun x ↦ a + x) c\na✝ : Nontrivial R\ntrinomial : R → R[X] := fun r ↦ a • X ^ 2 + r • X + C a\nr : R\n⊢ (X + C 1) * trinomial r = a • X ^ 3 + (a + r) • X ^ 2 + (a + r) • X + C a",
"ppTerm": "?m.240",... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 113,
"column": 75
} | {
"line": 113,
"column": 77
} | {
"line": 113,
"column": 78
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nh : IsLeftCancelMulZero R[X]\na b c : R\neq : (fun x ↦ a + x) b = (fun x ↦ a + x) c\na✝ : Nontrivial R\ntrinomial : R → R[X] := fun r ↦ a • X ^ 2 + r • X + C a\nht : ∀ (r : R), (X + C 1) * trinomial r = a • X ^ 3 + (a + r) • X ^ 2 + (a + r) • X + C a\n⊢ (fun x ↦ (X + C... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 106,
"column": 51
} | {
"line": 106,
"column": 53
} | {
"line": 107,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nh : IsLeftCancelMulZero R[X]\na b c : R\neq : (fun x ↦ a + x) b = (fun x ↦ a + x) c\n⊢ b = c",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Nontrivial",
"add_mul",
"Distrib.leftDistribClass",
"Eq.mpr",
"Polynomial.C... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 118,
"column": 74
} | {
"line": 118,
"column": 76
} | {
"line": 119,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\n⊢ IsRightCancelMulZero R[X] ↔ IsRightCancelMulZero R ∧ IsCancelAdd R",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Polynomial.isLeftCancelMulZero_iff",
"Eq.mpr",
"MulOpposite.instMulZeroClass",
"MulZeroClass.toMul",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 123,
"column": 64
} | {
"line": 123,
"column": 66
} | {
"line": 124,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\n⊢ IsCancelMulZero R[X] ↔ IsCancelMulZero R ∧ IsCancelAdd R",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"and_self",
"Iff.rfl",
"id",
"Distrib.toAdd",
"_private.Mathlib.RingTheory.Po... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 197,
"column": 64
} | {
"line": 197,
"column": 66
} | {
"line": 198,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\n⊢ ∃! h, ∃ N, ∀ n > N, (PowerSeries.coeff n) (↑p * ↑(invOneSubPow F d)) = eval (↑n) h",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.Infinite.mono",
"Units.val",
"Eq.mpr",
"Polynomial... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 127,
"column": 69
} | {
"line": 127,
"column": 71
} | {
"line": 128,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\n⊢ IsDomain R[X] ↔ IsDomain R ∧ IsCancelAdd R",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"IsDomain",
"congrArg",
"_private.Mathlib.RingTheory.Polynomial.Opposites.0.Polynomial.isDomain_iff._... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 224,
"column": 60
} | {
"line": 224,
"column": 62
} | {
"line": 224,
"column": 63
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd n : ℕ\nhn : n > (p * (1 - X)).natDegree\n⊢ 1 - PowerSeries.X = ↑(1 - X)",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Polynomial.instOne",
"Polynomial.coe_one",
"AddGroupWithOne.toAddGroup",
... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 221,
"column": 59
} | {
"line": 221,
"column": 61
} | {
"line": 222,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\n⊢ (p * (1 - X)).hilbertPoly (d + 1) = p.hilbertPoly d",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"Polynomial.eval",
"Polynomial.instOne",
"Semigroup.toMul... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 229,
"column": 63
} | {
"line": 229,
"column": 65
} | {
"line": 230,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd e : ℕ\n⊢ (p * (1 - X) ^ e).hilbertPoly (d + e) = p.hilbertPoly d",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Polynomial.instOne",
"Semigroup.toMul",
"Nat.re... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 240,
"column": 87
} | {
"line": 240,
"column": 89
} | {
"line": 241,
"column": 6
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\nhdp : d ≤ rootMultiplicity 1 p\nhp : ¬p = 0\nq : F[X]\nhq1 : p = (X - C 1) ^ rootMultiplicity 1 p * q\nhq2 : ¬X - C 1 ∣ q\n⊢ p = q * (-1) ^ rootMultiplicity 1 p * (1 - X) ^ rootMultiplicity 1 p",
"ppTerm": "?m.102",
"assigned"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 236,
"column": 27
} | {
"line": 236,
"column": 29
} | {
"line": 237,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\nhdp : d ≤ rootMultiplicity 1 p\n⊢ p.hilbertPoly d = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Polynomial.C",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 250,
"column": 85
} | {
"line": 250,
"column": 87
} | {
"line": 251,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\nhp : p ≠ 0\nhpd : rootMultiplicity 1 p < d\nq : F[X]\nhq1 : p = (X - C 1) ^ rootMultiplicity 1 p * q\nhq2 : ¬X - C 1 ∣ q\n⊢ p = q * (-1) ^ rootMultiplicity 1 p * (1 - X) ^ rootMultiplicity 1 p",
"ppTerm": "?m.102",
"assigned":... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 260,
"column": 64
} | {
"line": 260,
"column": 66
} | {
"line": 260,
"column": 67
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\nhp : p ≠ 0\nhpd : rootMultiplicity 1 p < d\nq : F[X]\nhq1 : p = (X - C 1) ^ rootMultiplicity 1 p * q\nhq2 : ¬X - C 1 ∣ q\nheq : p = q * (-1) ^ rootMultiplicity 1 p * (1 - X) ^ rootMultiplicity 1 p\n⊢ (fun x a ↦ a) = fun x a ↦ a * 1 ^ ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 248,
"column": 66
} | {
"line": 248,
"column": 68
} | {
"line": 249,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\nhp : p ≠ 0\nhpd : rootMultiplicity 1 p < d\n⊢ (p.hilbertPoly d).natDegree = d - rootMultiplicity 1 p - 1",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"not_iff_not",
"Iff.mpr",
"one_pow",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 270,
"column": 21
} | {
"line": 270,
"column": 23
} | {
"line": 271,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\nhh : p.hilbertPoly d ≠ 0\n⊢ p ≠ 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"congrArg",
"Eq.mp",
"Polynomial.hilbertPoly_zero_left",
"Ne",
"Field.toSemifield",
"Polynomia... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 274,
"column": 41
} | {
"line": 274,
"column": 43
} | {
"line": 275,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\nhh : p.hilbertPoly d ≠ 0\nhp : p ≠ 0\n⊢ rootMultiplicity 1 p < d",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Polynomial.hilbertPoly_eq_zero_of_le_rootMultiplicity_one",
"Pa... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 270,
"column": 2
} | {
"line": 277,
"column": 70
} | {
"line": 279,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\nhh : p.hilbertPoly d ≠ 0\n⊢ (p.hilbertPoly d).natDegree = d - rootMultiplicity 1 p - 1",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Polynomial.hilbertPoly_eq_zero_of_le_rootMultip... | [] | have hp : p ≠ 0 := by
intro h
rw [h] at hh
exact hh (hilbertPoly_zero_left d)
have hpd : p.rootMultiplicity 1 < d := by
by_contra h
exact hh (hilbertPoly_eq_zero_of_le_rootMultiplicity_one <| not_lt.1 h)
exact natDegree_hilbertPoly_of_ne_zero_of_rootMultiplicity_lt hp hpd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 270,
"column": 2
} | {
"line": 277,
"column": 70
} | {
"line": 279,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\nhh : p.hilbertPoly d ≠ 0\n⊢ (p.hilbertPoly d).natDegree = d - rootMultiplicity 1 p - 1",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Polynomial.hilbertPoly_eq_zero_of_le_rootMultip... | [] | have hp : p ≠ 0 := by
intro h
rw [h] at hh
exact hh (hilbertPoly_zero_left d)
have hpd : p.rootMultiplicity 1 < d := by
by_contra h
exact hh (hilbertPoly_eq_zero_of_le_rootMultiplicity_one <| not_lt.1 h)
exact natDegree_hilbertPoly_of_ne_zero_of_rootMultiplicity_lt hp hpd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 269,
"column": 66
} | {
"line": 269,
"column": 68
} | {
"line": 270,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\nhh : p.hilbertPoly d ≠ 0\n⊢ (p.hilbertPoly d).natDegree = d - rootMultiplicity 1 p - 1",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Polynomial.hilbertPoly_eq_zero_of_le_rootMultip... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Selmer | {
"line": 35,
"column": 28
} | {
"line": 35,
"column": 30
} | {
"line": 36,
"column": 4
} | [
{
"pp": "n : ℕ\nz : ℂ\nh1 : z ^ n = z + 1\nh2 : z ^ n + z ^ 2 = 0\n⊢ z ^ 3 = 1",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroCl... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.Selmer | {
"line": 37,
"column": 54
} | {
"line": 37,
"column": 56
} | {
"line": 38,
"column": 4
} | [
{
"pp": "n : ℕ\nz : ℂ\nh1 : z ^ n = z + 1\nh2 : z ^ n + z ^ 2 = 0\nh3 : z ^ 3 = 1\n⊢ z ^ n = 1 ∨ z ^ n = z ∨ z ^ n = z ^ 2",
"ppTerm": "?m.163",
"assigned": true,
"usedConstants": [
"one_pow",
"Eq.mpr",
"Nat.zero_le",
"MulOne.toOne",
"instHDiv",
"Nat.instIsOrdered... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.Selmer | {
"line": 45,
"column": 21
} | {
"line": 45,
"column": 23
} | {
"line": 45,
"column": 24
} | [
{
"pp": "n : ℕ\nz : ℂ\nh1 : z ^ n = z + 1\nh2 : z ^ n + z ^ 2 = 0\nh3 : z ^ 3 = 1\nz_ne_zero : z ≠ 0\nkey : z ^ n = 1\n⊢ z = 0",
"ppTerm": "?m.325",
"assigned": true,
"usedConstants": [
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toA... | [] | by | [anonymous] | by |
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