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 |
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Mathlib.RingTheory.Polynomial.Selmer | {
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"left_eq_add",
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Mathlib.RingTheory.Polynomial.Selmer | {
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{
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"NormedCommRing.toNormedRing",
"congrArg",
"NormedDivisionRing.toNormMulClass",
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Mathlib.RingTheory.Polynomial.Selmer | {
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{
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Mathlib.RingTheory.Polynomial.Selmer | {
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{
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Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
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} | {
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{
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Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
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{
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Mathlib.RingTheory.Polynomial.Selmer | {
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} | {
"line": 63,
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} | {
"line": 63,
"column": 35
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{
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Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
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} | {
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} | {
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} | [
{
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Mathlib.RingTheory.Polynomial.Selmer | {
"line": 67,
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} | {
"line": 67,
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} | {
"line": 67,
"column": 17
} | [
{
"pp": "n : ℕ\nhn1 : n ≠ 1\nhn0 : ¬n = 0\nhn : 1 < n\nhp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1\nz : ℂ\nh1 : z ^ n = z + 1\nh2 : 1 * z ^ 0 * z + -1 * (z + 1) + -1 * (z + 1) * z = 0\n⊢ z + 1 + z ^ 2 = 0",
"ppTerm": "?m.370",
"assigned": true,
"usedConstants": [
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Mathlib.RingTheory.Polynomial.Selmer | {
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} | {
"line": 49,
"column": 96
} | {
"line": 50,
"column": 2
} | [
{
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"usedConstants": [
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Mathlib.RingTheory.Polynomial.Selmer | {
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} | [
{
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Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
"line": 61,
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} | {
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} | {
"line": 62,
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} | [
{
"pp": "n : ℕ\n⊢ (⇑derivative)^[n] (∑ m ∈ range (n + 1), n.choose m • (-1) ^ m * X ^ (n + m)) =\n ∑ x ∈ range (n + 1), ↑((n + x)! / x !) * C (↑(n.choose x) * (-1) ^ x) * X ^ x",
"ppTerm": "?m.213",
"assigned": true,
"usedConstants": [
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"Polyno... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
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} | {
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} | {
"line": 24,
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{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]\nhp : p.degree ≤ 2\n⊢ p = C (p.coeff 2) * X ^ 2 + C (p.coeff 1) * X + C (p.coeff 0)",
"ppTerm": "?m.65",
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"usedConstants": [
"Iff.mpr",
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"Eq.mpr",
"Polynomial.C",
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Mathlib.RingTheory.Polynomial.Selmer | {
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} | {
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} | {
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} | [
{
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Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
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{
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"ppTerm": "?m.133",
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Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
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"ppTerm": "?m.244",
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... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
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} | {
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} | {
"line": 39,
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{
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"ppTerm": "?m.260",
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Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
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} | {
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} | {
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{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na b c x1 x2 : R\nhroots : (C a * X ^ 2 + C b * X + C c).roots = {x1, x2}\n⊢ b = -a * (x1 + x2)",
"ppTerm": "?m.71",
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"NegZeroClass.toNeg",
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Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
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} | {
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} | {
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} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na b c x1 x2 : R\nhroots : (C a * X ^ 2 + C b * X + C c).roots = {x1, x2}\np : R[X] := C a * X ^ 2 + C b * X + C c\n⊢ 2 ≤ p.natDegree",
"ppTerm": "?m.131",
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"usedConstants": [
"Eq.mpr",
"Polynomial.C",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
"line": 48,
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} | {
"line": 48,
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} | {
"line": 49,
"column": 4
} | [
{
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"ppTerm": "?m.242",
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... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
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} | {
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} | {
"line": 79,
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} | [
{
"pp": "n x : ℕ\na✝ : x ∈ range (n + 1)\n⊢ 0 < n !",
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"usedConstants": [
"Nat.factorial_pos"
],
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"n"
],
"usedGoals": []
}
] | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
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} | {
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{
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"assigned... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
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} | {
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{
"pp": "n : ℕ\n⊢ ∑ x ∈ range (n + 1), ↑((n + x)! / x !) * C (↑(n.choose x) * (-1) ^ x) * X ^ x =\n ∑ i ∈ range (n + 1), ↑n ! * C ((-1) ^ i * ↑(n.choose i) * ↑((n + i).choose n)) * X ^ i",
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Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
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{
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Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
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{
"pp": "n : ℕ\n⊢ ∑ i ∈ range (n + 1), ↑n ! * C ((-1) ^ i * ↑(n.choose i) * ↑((n + i).choose n)) * X ^ i = ↑n ! * shiftedLegendre n",
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Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
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{
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Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
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{
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"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
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} | {
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} | {
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{
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Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
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Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
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{
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Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
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Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
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{
"pp": "R : Type u_1\ninst✝ : Field R\na b c x1 x2 : R\nha : a ≠ 0\n⊢ (C a * X ^ 2 + C b * X + C c).roots = {x1, x2} ↔ x1 + x2 = -b / a ∧ x1 * x2 = c / a",
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Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
"line": 92,
"column": 83
} | {
"line": 92,
"column": 85
} | {
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"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (shiftedLegendre n).degree = ↑n",
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Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
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} | {
"line": 108,
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{
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Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
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} | {
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} | {
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{
"pp": "n : ℕ\n⊢ (-1) ^ n * (shiftedLegendre n).comp (1 - X) = shiftedLegendre n",
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Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
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} | {
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{
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Mathlib.RingTheory.PowerSeries.Catalan | {
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} | {
"line": 39,
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} | {
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{
"pp": "n : ℕ\n⊢ (coeff n) catalanSeries = catalan n",
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Catalan | {
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} | {
"line": 43,
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} | {
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{
"pp": "⊢ constantCoeff catalanSeries = 1",
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Mathlib.RingTheory.PowerSeries.Catalan | {
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} | {
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} | {
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{
"pp": "⊢ catalanSeries ^ 2 * X + 1 = catalanSeries",
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Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
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{
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Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
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} | {
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} | {
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{
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Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
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} | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
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} | {
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} | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
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} | {
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} | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
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} | {
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} | {
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{
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"Ideal.one_eq_top",
"Submodule.mem_top._simp_1",
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Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
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} | {
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} | {
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{
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"usedConstants": [
"Ideal.one_eq_top",
"Submodule.mem_top._simp_1",
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Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
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} | {
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} | {
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} | [
{
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"assigned": true,
"usedConstants": [
"Ideal.one_eq_top",
"Submodule.mem_top._simp_1",
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
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} | {
"line": 72,
"column": 62
} | {
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} | [
{
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Mathlib.RingTheory.Polynomial.Morse | {
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} | {
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} | {
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{
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Mathlib.RingTheory.Polynomial.Morse | {
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} | {
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} | {
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{
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Mathlib.RingTheory.Polynomial.Morse | {
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} | {
"line": 74,
"column": 39
} | {
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{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : IsDomain S\nG : Type u_3\ninst✝⁴ : Group G\ninst✝³ : MulSemiringAction G S\ninst✝² : SMulCommClass G R S\nf : R[X]\ninst✝¹ : DecidableEq ↑(f.rootSet S)\nhf : (map (algebraMap R S) f).Splits\np : Ideal S... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Polynomial.Morse | {
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} | {
"line": 74,
"column": 71
} | {
"line": 74,
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{
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Mathlib.RingTheory.Polynomial.Morse | {
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} | {
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"column": 100
} | {
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{
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Mathlib.RingTheory.Polynomial.Morse | {
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} | {
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} | {
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{
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Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
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} | {
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} | {
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{
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Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
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} | {
"line": 96,
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} | {
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{
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Mathlib.RingTheory.PowerSeries.Expand | {
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} | {
"line": 36,
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} | {
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{
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"CommSemiring.toSemiring",
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Expand | {
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} | {
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} | {
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{
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Mathlib.RingTheory.PowerSeries.Expand | {
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} | {
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{
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"usedConstants": [
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"congrArg... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Expand | {
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} | {
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} | {
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{
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"usedConstants": [
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"Semir... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Expand | {
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} | {
"line": 65,
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} | {
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{
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"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Expand | {
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} | {
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} | {
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{
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Mathlib.RingTheory.PowerSeries.Expand | {
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} | {
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{
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"usedConstants": [
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Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
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} | {
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} | {
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{
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Mathlib.RingTheory.PowerSeries.Expand | {
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} | {
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} | {
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{
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"usedConstants": [
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Expand | {
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} | {
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} | {
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} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : R⟦X⟧\nm : ℕ\n⊢ (coeff (p * m)) ((expand p hp) φ) = (coeff m) φ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MvPowerSeries.expand",
"Eq.mpr",
"Finsupp.smulZeroClass",
"Unit.unit",
"Nat.instMu... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Expand | {
"line": 93,
"column": 55
} | {
"line": 93,
"column": 57
} | {
"line": 94,
"column": 2
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : R⟦X⟧\n⊢ constantCoeff ((expand p hp) φ) = constantCoeff φ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PowerSeries.coeff_expand_mul",
"Nat.instMulZeroClass",
"Semiring.toModule",
"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
"line": 131,
"column": 44
} | {
"line": 131,
"column": 46
} | {
"line": 132,
"column": 4
} | [
{
"pp": "R : Type u\nM : Type u_1\nN : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\ninst✝ : DecidableEq { P // P.FG }\nP : { P // P.FG }\nu : ↥↑P ⊗[R] N\nthis :\n ↑(directLimit R M N) ∘ₗ\n Module.DirectLimit.of R { P // P.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Expand | {
"line": 98,
"column": 35
} | {
"line": 98,
"column": 37
} | {
"line": 99,
"column": 2
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : R⟦X⟧\nm : ℕ\nh : ¬p ∣ m\n⊢ (coeff m) ((expand p hp) φ) = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"MvPowerSeries.expand",
"Finsupp.instFunLike",
"Eq.mpr",
"Unit.unit",
"Nat.instMulZeroC... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Expand | {
"line": 103,
"column": 57
} | {
"line": 103,
"column": 59
} | {
"line": 104,
"column": 2
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : R⟦X⟧\n⊢ Function.support ((expand p hp) φ) ⊆ (fun x ↦ p • x) '' Function.support φ",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"MvPowerSeries.expand",
"Eq.mpr",
"MvPowerSeries.support_expand",
"le... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Expand | {
"line": 107,
"column": 57
} | {
"line": 107,
"column": 59
} | {
"line": 108,
"column": 2
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : R⟦X⟧\n⊢ Function.support ((expand p hp) φ) = (fun x ↦ p • x) '' Function.support φ",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"MvPowerSeries.expand",
"Eq.mpr",
"MvPowerSeries.support_expand",
"Na... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Expand | {
"line": 111,
"column": 70
} | {
"line": 111,
"column": 72
} | {
"line": 112,
"column": 2
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : R⟦X⟧\nn : ℕ\n⊢ (coeff n) ((expand p hp) φ) = if p ∣ n then (coeff (n / p)) φ else 0",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PowerSeries.coeff_expand_mul",
"Dvd.dvd",
"instHDiv",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Expand | {
"line": 118,
"column": 62
} | {
"line": 118,
"column": 64
} | {
"line": 119,
"column": 2
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : R⟦X⟧\n⊢ ((expand p hp) φ).order = p • φ.order",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"MvPowerSeries.expand",
"Eq.mpr",
"instHSMul",
"instAddMonoidWithOneENat",
"congrArg",
"CommSe... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
"line": 127,
"column": 47
} | {
"line": 127,
"column": 49
} | {
"line": 128,
"column": 2
} | [
{
"pp": "R : Type u\nM : Type u_1\nN : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\ninst✝ : DecidableEq { P // P.FG }\nP : { P // P.FG }\nu : ↥↑P ⊗[R] N\n⊢ (directLimit R M N)\n ((Module.DirectLimit.of R { P // P.FG } (fun ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 34,
"column": 68
} | {
"line": 34,
"column": 70
} | {
"line": 35,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nc : ℝ\nf : R⟦X⟧\n⊢ (fun t ↦ ‖(MvPowerSeries.coeff t) f‖ * t.prod fun x x_1 ↦ c ^ x_1) =\n (fun n ↦ ‖(coeff n) f‖ * c ^ n) ∘ ⇑(Finsupp.uniqueEquiv ())",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Norm.nor... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 6
} | {
"line": 44,
"column": 7
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nc : ℝ\nf : R⟦X⟧\nH : Tendsto ((fun n ↦ ‖(coeff n) f‖ * c ^ n) ∘ ⇑(Finsupp.uniqueEquiv ())) cofinite (𝓝 0)\nn : ℕ\n⊢ (((fun n ↦ ‖(coeff n) f‖ * c ^ n) ∘ ⇑(Finsupp.uniqueEquiv ())) ∘ ⇑(Finsupp.uniqueEquiv ()).symm) n =\n ‖(coeff n) f‖ * c ^ n",
"ppTerm": "?m.69... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 41,
"column": 66
} | {
"line": 41,
"column": 68
} | {
"line": 42,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nc : ℝ\nf : R⟦X⟧\n⊢ IsRestricted c f ↔ Tendsto (fun t ↦ ‖(coeff t) f‖ * c ^ t) cofinite (𝓝 0)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Norm.norm",
"Eq.mpr",
"Unit.unit",
"Nat.instMulZe... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 47,
"column": 63
} | {
"line": 47,
"column": 65
} | {
"line": 48,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nc : ℝ\nf : R⟦X⟧\n⊢ IsRestricted c f ↔ Tendsto (fun t ↦ ‖(coeff t) f‖ * c ^ t) atTop (𝓝 0)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"Semiring.toModule",
"NormedRing.toRing",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 47,
"column": 63
} | {
"line": 48,
"column": 51
} | {
"line": 50,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nc : ℝ\nf : R⟦X⟧\n⊢ IsRestricted c f ↔ Tendsto (fun t ↦ ‖(coeff t) f‖ * c ^ t) atTop (𝓝 0)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"Semiring.toModule",
"NormedRing.toRing",
... | [] | by
simp_rw [isRestricted_iff, Nat.cofinite_eq_atTop] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
"line": 142,
"column": 47
} | {
"line": 142,
"column": 49
} | {
"line": 143,
"column": 2
} | [
{
"pp": "R : Type u\nM : Type u_1\nN : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\ninst✝ : DecidableEq { P // P.FG }\nP : Submodule R M\nhP : P.FG\nu : ↥P ⊗[R] N\n⊢ (directLimit R M N)\n ((Module.DirectLimit.of R { P // P.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
"line": 150,
"column": 18
} | {
"line": 150,
"column": 20
} | {
"line": 151,
"column": 4
} | [
{
"pp": "R : Type u\nM : Type u_1\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nι : Type u_3\ninst✝² : Preorder ι\nF : ι → Type u_4\ninst✝¹ : (i : ι) → AddCommMonoid (F i)\ninst✝ : (i : ι) → Module R (F i)\nf : ⦃i j : ι⦄ → i ≤ j → F i →ₗ[R] F j\nD : DirectedSystem F fun x x_1 h ↦ ⇑(f ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 95,
"column": 94
} | {
"line": 95,
"column": 96
} | {
"line": 96,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommSemiring R\nM : Type u_1\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nN : Type u_2\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nf : M →ₚₗ[R] N\nS : Type u\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nS' : Type u\ninst✝¹ : CommSemiring S'\ninst✝ : Algebra R S'\nφ : S →ₐ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Schroder | {
"line": 42,
"column": 55
} | {
"line": 42,
"column": 57
} | {
"line": 43,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (coeff n) largeSchroderSeries = n.largeSchroder",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Nat.largeSchroder",
"Semiring.toModule",
"PowerSeries.largeSchroderSeries",
"congrArg",
"LinearMap.instFunLike",
"PowerSeries.coeff",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Schroder | {
"line": 47,
"column": 45
} | {
"line": 47,
"column": 47
} | {
"line": 48,
"column": 2
} | [
{
"pp": "⊢ constantCoeff largeSchroderSeries = 1",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Nat.largeSchroder",
"Semiring.toModule",
"PowerSeries.largeSchroderSeries",
"congrArg",
"LinearMap.instFunLike",
"RingHom",
"PowerSeries.coeff",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Schroder | {
"line": 52,
"column": 65
} | {
"line": 52,
"column": 67
} | {
"line": 53,
"column": 2
} | [
{
"pp": "n : ℕ\nhn : 0 < n\n⊢ (coeff n) (X * largeSchroderSeries) = (n - 1).largeSchroder",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Finset.Nat.sum_antidiagonal_eq_sum_range_succ",
"Eq.mpr",
"NonAssocSemir... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 150,
"column": 50
} | {
"line": 150,
"column": 52
} | {
"line": 151,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommSemiring R\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_2\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\na b : R\nf : M →ₚₗ[R] N\n⊢ (a + b) • f = a • f + b • f",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Pi.Function.m... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 153,
"column": 39
} | {
"line": 153,
"column": 41
} | {
"line": 154,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommSemiring R\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_2\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nf : M →ₚₗ[R] N\n⊢ 0 • f = 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"PolynomialLaw.instSMul",
"instHSMu... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
"line": 155,
"column": 28
} | {
"line": 155,
"column": 30
} | {
"line": 156,
"column": 4
} | [
{
"pp": "R : Type u\nM : Type u_1\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nι : Type u_3\ninst✝² : Preorder ι\nF : ι → Type u_4\ninst✝¹ : (i : ι) → AddCommMonoid (F i)\ninst✝ : (i : ι) → Module R (F i)\nf : ⦃i j : ι⦄ → i ≤ j → F i →ₗ[R] F j\nD : DirectedSystem F fun x x_1 h ↦ ⇑(f ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 156,
"column": 38
} | {
"line": 156,
"column": 40
} | {
"line": 157,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommSemiring R\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_2\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nf : M →ₚₗ[R] N\n⊢ 1 • f = f",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"TensorProduct.instDistribMulAction",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Schroder | {
"line": 78,
"column": 47
} | {
"line": 78,
"column": 49
} | {
"line": 78,
"column": 50
} | [
{
"pp": "n : ℕ\nhn : 0 < n\nx : ℕ\na : x < n\nh : 0 < x\n⊢ 0 < x",
"ppTerm": "?m.196",
"assigned": true,
"usedConstants": [
"_private.Mathlib.RingTheory.PowerSeries.Schroder.0.PowerSeries.coeff_X_mul_largeSchroderSeriesSeries_sq._proof_1_3"
],
"usedFVars": [
"x",
"h"
],... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
"line": 185,
"column": 44
} | {
"line": 185,
"column": 46
} | {
"line": 186,
"column": 4
} | [
{
"pp": "R : Type u\nM : Type u_1\nN : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\ninst✝ : DecidableEq { P // P.FG }\nQ : { Q // Q.FG }\nu : M ⊗[R] ↥↑Q\nthis :\n ↑(directLimit R M N) ∘ₗ\n Module.DirectLimit.of R { Q // Q.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 161,
"column": 20
} | {
"line": 161,
"column": 22
} | {
"line": 161,
"column": 23
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommSemiring R\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_2\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nr a✝ b✝ : R\nf✝ g : M →ₚₗ[R] N\na b : R\nf : M →ₚₗ[R] N\n⊢ (a * b) • f = a • b • f",
"ppTerm": "?m.37",
"assigned": true,
"usedConsta... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 164,
"column": 21
} | {
"line": 164,
"column": 23
} | {
"line": 164,
"column": 24
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommSemiring R\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_2\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nr a b : R\nf✝ g✝ f g h : M →ₚₗ[R] N\n⊢ f + g + h = f + (g + h)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Pi.ad... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Schroder | {
"line": 71,
"column": 91
} | {
"line": 71,
"column": 93
} | {
"line": 72,
"column": 4
} | [
{
"pp": "n : ℕ\nhn : 0 < n\n⊢ ∑ x ∈ range n, (coeff x) (X * largeSchroderSeries) * (n - x).largeSchroder =\n ∑ x ∈ range n, if 0 < x then (x - 1).largeSchroder * (n - x).largeSchroder else 0",
"ppTerm": "?m.157",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 165,
"column": 16
} | {
"line": 165,
"column": 18
} | {
"line": 165,
"column": 19
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommSemiring R\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_2\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nr a b : R\nf✝ g f : M →ₚₗ[R] N\n⊢ 0 + f = f",
"ppTerm": "?m.89",
"assigned": true,
"usedConstants": [
"congrArg",
"CommSe... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 166,
"column": 16
} | {
"line": 166,
"column": 18
} | {
"line": 166,
"column": 19
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommSemiring R\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_2\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nr a b : R\nf✝ g f : M →ₚₗ[R] N\n⊢ f + 0 = f",
"ppTerm": "?m.129",
"assigned": true,
"usedConstants": [
"congrArg",
"CommS... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
"line": 181,
"column": 51
} | {
"line": 181,
"column": 53
} | {
"line": 182,
"column": 2
} | [
{
"pp": "R : Type u\nM : Type u_1\nN : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\ninst✝ : DecidableEq { P // P.FG }\nQ : { Q // Q.FG }\nu : M ⊗[R] ↥↑Q\n⊢ (directLimit R M N)\n ((Module.DirectLimit.of R { Q // Q.FG } (fun ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 168,
"column": 18
} | {
"line": 168,
"column": 20
} | {
"line": 168,
"column": 21
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommSemiring R\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_2\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nr a b : R\nf✝ g f : M →ₚₗ[R] N\n⊢ 0 • f = 0",
"ppTerm": "?m.169",
"assigned": true,
"usedConstants": [
"PolynomialLaw.instSMul"... | [] | by | [anonymous] | by |
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