module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 689,
"column": 10
} | {
"line": 689,
"column": 18
} | {
"line": 690,
"column": 8
} | [
{
"pp": "case zero\na : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\nij : 0 ≤ j\nh : xn a1 0 ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ 0 = 0 ∧ j = 2)\nij' : 0 < j\n⊢ 0 < xn a1 0 % xn a1 n",
"ppTerm": "?zero",
"assigned": true,
"usedConstant... | [] | exact x0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 689,
"column": 10
} | {
"line": 689,
"column": 18
} | {
"line": 690,
"column": 8
} | [
{
"pp": "case zero\na : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\nij : 0 ≤ j\nh : xn a1 0 ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ 0 = 0 ∧ j = 2)\nij' : 0 < j\n⊢ 0 < xn a1 0 % xn a1 n",
"ppTerm": "?zero",
"assigned": true,
"usedConstant... | [] | exact x0 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 695,
"column": 55
} | {
"line": 695,
"column": 61
} | {
"line": 695,
"column": 61
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : ℕ\nij : i + 1 ≤ j\nh : xn a1 (i + 1) ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i + 1 = 0 ∧ j = 2)\nij' : 2 < 1\nx✝ : a = 2 ∧ n = 1 ∧ 0 = 0 ∧ i + 1 = 2\nleft✝¹ : a = 2\nn1 : n = 1\nleft✝ : 0 ... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 695,
"column": 55
} | {
"line": 695,
"column": 61
} | {
"line": 695,
"column": 61
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : ℕ\nij : i + 1 ≤ j\nh : xn a1 (i + 1) ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i + 1 = 0 ∧ j = 2)\nij' : 2 < 1\nx✝ : a = 2 ∧ n = 1 ∧ 0 = 0 ∧ i + 1 = 2\nleft✝¹ : a = 2\nn1 : n = 1\nleft✝ : 0 ... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 695,
"column": 55
} | {
"line": 695,
"column": 61
} | {
"line": 695,
"column": 61
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nj n : ℕ\nj2n : j ≤ 2 * n\nnpos : ¬n = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : ℕ\nij : i + 1 ≤ j\nh : xn a1 (i + 1) ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i + 1 = 0 ∧ j = 2)\nij' : 2 < 1\nx✝ : a = 2 ∧ n = 1 ∧ 0 = 0 ∧ i + 1 = 2\nleft✝¹ : a = 2\nn1 : n = 1\nleft✝ : 0 ... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 712,
"column": 65
} | {
"line": 712,
"column": 71
} | {
"line": 712,
"column": 71
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : 2 ≤ 1\nj4n : j ≤ 4 * n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\ni2n : i ≤ 2 * n\nj2n : j ≤ 2 * n\nx✝¹ : a = 2\nn1 : n = 1\nx✝ : j = 0\ni2 : i = 2\n⊢ ¬2 ≤ 1",
"ppTerm": "?m.185",
"assigned": true,
"usedConstants": [
"instDecidableNot",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 712,
"column": 65
} | {
"line": 712,
"column": 71
} | {
"line": 712,
"column": 71
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : 2 ≤ 1\nj4n : j ≤ 4 * n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\ni2n : i ≤ 2 * n\nj2n : j ≤ 2 * n\nx✝¹ : a = 2\nn1 : n = 1\nx✝ : j = 0\ni2 : i = 2\n⊢ ¬2 ≤ 1",
"ppTerm": "?m.185",
"assigned": true,
"usedConstants": [
"instDecidableNot",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 712,
"column": 65
} | {
"line": 712,
"column": 71
} | {
"line": 712,
"column": 71
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : 2 ≤ 1\nj4n : j ≤ 4 * n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\ni2n : i ≤ 2 * n\nj2n : j ≤ 2 * n\nx✝¹ : a = 2\nn1 : n = 1\nx✝ : j = 0\ni2 : i = 2\n⊢ ¬2 ≤ 1",
"ppTerm": "?m.185",
"assigned": true,
"usedConstants": [
"instDecidableNot",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 727,
"column": 37
} | {
"line": 727,
"column": 43
} | {
"line": 727,
"column": 43
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j % (4 * n)\n⊢ 0 < 4",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Preorder.toLT",
"of_decide_eq_true",
"id",
"instOfNat... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 727,
"column": 37
} | {
"line": 727,
"column": 43
} | {
"line": 727,
"column": 43
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j % (4 * n)\n⊢ 0 < 4",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Preorder.toLT",
"of_decide_eq_true",
"id",
"instOfNat... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 727,
"column": 37
} | {
"line": 727,
"column": 43
} | {
"line": 727,
"column": 43
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j % (4 * n)\n⊢ 0 < 4",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Preorder.toLT",
"of_decide_eq_true",
"id",
"instOfNat... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.SmoothNumbers | {
"line": 181,
"column": 30
} | {
"line": 181,
"column": 68
} | {
"line": 181,
"column": 68
} | [
{
"pp": "s : Finset ℕ\np n : ℕ\nhp : Prime p\nhs : p ∉ s\nhn : n ∈ factoredNumbers s\n⊢ ¬p ∣ n",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Nat.mem_primeFactorsList_iff_dvd",
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"Finset",
"Membership.mem",
"id",
... | [
"s : Finset ℕ\np n : ℕ\nhp : Prime p\nhs : p ∉ s\nhn : n ∈ factoredNumbers s\n⊢ p ∉ n.primeFactorsList"
] | ← mem_primeFactorsList_iff_dvd hn.1 hp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Dioph | {
"line": 457,
"column": 8
} | {
"line": 473,
"column": 41
} | {
"line": 475,
"column": 0
} | [
{
"pp": "α : Type\nn : ℕ\nS : Set (α ⊕ Fin2 n.succ → ℕ)\nd : Dioph S\nf✝ : Vector3 ((α → ℕ) → ℕ) n.succ\nf : (α → ℕ) → ℕ\nfl : Vector3 ((α → ℕ) → ℕ) n\n⊢ VectorAllP DiophFn (f :: fl) → Dioph {v | (v ⊗ fun i ↦ (f :: fl) i v) ∈ S}",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [] | simp only [vectorAllP_cons, and_imp]
exact fun df dfl =>
have : Dioph {v | (v ∘ inl ⊗ f (v ∘ inl)::v ∘ inr) ∈ S} :=
ext (diophFn_comp1 (reindex_dioph _ (some ∘ inl ⊗ none :: some ∘ inr) d) <|
reindex_diophFn inl df)
fun v => by
dsimp
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Dioph | {
"line": 457,
"column": 8
} | {
"line": 473,
"column": 41
} | {
"line": 475,
"column": 0
} | [
{
"pp": "α : Type\nn : ℕ\nS : Set (α ⊕ Fin2 n.succ → ℕ)\nd : Dioph S\nf✝ : Vector3 ((α → ℕ) → ℕ) n.succ\nf : (α → ℕ) → ℕ\nfl : Vector3 ((α → ℕ) → ℕ) n\n⊢ VectorAllP DiophFn (f :: fl) → Dioph {v | (v ⊗ fun i ↦ (f :: fl) i v) ∈ S}",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [] | simp only [vectorAllP_cons, and_imp]
exact fun df dfl =>
have : Dioph {v | (v ∘ inl ⊗ f (v ∘ inl)::v ∘ inr) ∈ S} :=
ext (diophFn_comp1 (reindex_dioph _ (some ∘ inl ⊗ none :: some ∘ inr) d) <|
reindex_diophFn inl df)
fun v => by
dsimp
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.EulerProduct.ExpLog | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 63
} | {
"line": 46,
"column": 2
} | [
{
"pp": "f : ℕ →*₀ ℂ\nhsum : Summable fun x ↦ ‖f x‖\nhs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1\nhp : ∀ (p : Nat.Primes), 1 - f ↑p ≠ 0\nH :\n ∏' (b : Subtype Nat.Prime), (cexp ∘ (fun b ↦ -log (1 - f b)) ∘ Subtype.val) b =\n cexp (∑' (b : Subtype Nat.Prime), ((fun b ↦ -log (1 - f b)) ∘ Subtype.val) b)\n⊢ cexp (∑' (p ... | [
"f : ℕ →*₀ ℂ\nhsum : Summable fun x ↦ ‖f x‖\nhs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1\nhp : ∀ (p : Nat.Primes), 1 - f ↑p ≠ 0\nH : ∏' (b : Subtype Nat.Prime), (1 - f ↑b)⁻¹ = cexp (∑' (b : Subtype Nat.Prime), -log (1 - f ↑b))\n⊢ cexp (∑' (p : Nat.Primes), -log (1 - f ↑p)) = ∑' (n : ℕ), f n"
] | simp only [Function.comp_apply, exp_neg, exp_log (hp _)] at H | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 92
} | {
"line": 67,
"column": 2
} | [
{
"pp": "f : ℕ → ℂ\nx : ℝ\nh : ∀ (y : ℝ), x < y → LSeriesSummable f ↑y\ny : EReal\nhy : y ∈ lowerBounds (Real.toEReal '' {x | LSeriesSummable f ↑x})\na : EReal\n⊢ ↑x < a → y ≤ a",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Real",
"lowerBounds",
"CompletelyDistribLatti... | [
"f : ℕ → ℂ\nx : ℝ\nh : ∀ (y : ℝ), x < y → LSeriesSummable f ↑y\ny a : EReal\nhy : ∀ (a : ℝ), LSeriesSummable f ↑a → y ≤ ↑a\n⊢ ↑x < a → y ≤ a"
] | replace hy : ∀ (a : ℝ), LSeriesSummable f a → y ≤ a := by simpa [mem_lowerBounds] using hy | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 836,
"column": 54
} | {
"line": 836,
"column": 60
} | {
"line": 836,
"column": 60
} | [
{
"pp": "a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n 1 < a ∧\n k ≤ y... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 836,
"column": 54
} | {
"line": 836,
"column": 60
} | {
"line": 836,
"column": 60
} | [
{
"pp": "a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n 1 < a ∧\n k ≤ y... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 836,
"column": 54
} | {
"line": 836,
"column": 60
} | {
"line": 836,
"column": 60
} | [
{
"pp": "a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n 1 < a ∧\n k ≤ y... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.Convolution | {
"line": 82,
"column": 67
} | {
"line": 85,
"column": 84
} | {
"line": 87,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf g : ℕ → R\n⊢ f ⍟ g = fun n ↦ ∑ p ∈ n.divisorsAntidiagonal, f p.1 * g p.2",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"HMul.hMul",
"Nat.divisorsAntidiagonal",
"ArithmeticFunction... | [] | by
ext n
simpa [convolution, toArithmeticFunction] using
Finset.sum_congr rfl fun p hp ↦ by simp [ne_zero_of_mem_divisorsAntidiagonal hp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 496,
"column": 2
} | {
"line": 499,
"column": 63
} | {
"line": 500,
"column": 2
} | [
{
"pp": "case h.inl\nv : ℕ\nih : ∀ m < v, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = ξ.convergent n\nξ : ℝ\nu : ℤ\nh : ContfracLegendre.Ass ξ u ↑v\nht : v < 1\n⊢ ∃ n, ↑u / ↑v = ξ.convergent n",
"ppTerm": "?h.inl",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractio... | [
"case h.inr.inl\nξ : ℝ\nu : ℤ\nih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = ξ.convergent n\nh : ContfracLegendre.Ass ξ u ↑1\n⊢ ∃ n, ↑u / ↑1 = ξ.convergent n",
"case h.inr.inr\nv : ℕ\nih : ∀ m < v, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = ξ.convergent n\nξ : ... | · replace h := h.2.2
simp only [Nat.lt_one_iff.mp ht, Nat.cast_zero, div_zero, tsub_zero, zero_mul,
cast_zero, inv_zero] at h
exact False.elim (lt_irrefl _ <| (abs_nonneg ξ).trans_lt h) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 101,
"column": 54
} | {
"line": 101,
"column": 62
} | {
"line": 102,
"column": 4
} | [
{
"pp": "k : ℕ\na t : ℝ\nht : 0 < t\nthis : Summable fun n ↦ ↑n ^ k * rexp (-π * (↑n + a) ^ 2 * t)\n⊢ ∀ᶠ (i : ℕ) in atTop, |(↑i + a) ^ k| * |rexp (-π * (↑i + a) ^ 2 * t)| ≤ 2 ^ k * ↑i ^ k * rexp (-π * (↑i + a) ^ 2 * t)",
"ppTerm": "?m.334",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"k : ℕ\na t : ℝ\nht : 0 < t\nthis : Summable fun n ↦ ↑n ^ k * rexp (-π * (↑n + a) ^ 2 * t)\n⊢ ∀ᶠ (i : ℕ) in atTop, |(↑i + a) ^ k| * rexp (-π * (↑i + a) ^ 2 * t) ≤ 2 ^ k * ↑i ^ k * rexp (-π * (↑i + a) ^ 2 * t)"
] | abs_exp, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 51
} | {
"line": 95,
"column": 0
} | [
{
"pp": "S T : ℝ\nhT : 0 < T\nz τ : ℂ\nhz : |z.im| ≤ S\nhτ : T ≤ τ.im\nn : ℤ\n⊢ |↑n| * |z.im| ≤ |↑n| * S",
"ppTerm": "?m.189",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Int.cast",
"Real.partialOrder",
"Real",
"... | [] | exact mul_le_mul_of_nonneg_left hz (abs_nonneg _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 43
} | {
"line": 127,
"column": 2
} | [
{
"pp": "z τ : ℂ\n⊢ (Summable fun x ↦ jacobiTheta₂_term x z τ) ↔ 0 < τ.im",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"jacobiTheta₂_term",
"Real.instZero",
"Complex.im",
"Complex.instNormedField",
... | [
"case refine_1\nz τ : ℂ\nhτ : 0 < τ.im\n⊢ Summable fun x ↦ jacobiTheta₂_term x z τ",
"case refine_2\nz τ : ℂ\nh : Summable fun x ↦ jacobiTheta₂_term x z τ\n⊢ 0 < τ.im"
] | refine Iff.symm ⟨fun hτ ↦ ?_, fun h ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 144,
"column": 83
} | {
"line": 144,
"column": 91
} | {
"line": 144,
"column": 91
} | [
{
"pp": "case refine_2.inr\nz τ : ℂ\nhτ✝ : τ.im ≤ 0\nhτ : τ.im = 0\nh : Summable fun x ↦ rexp (0 - 2 * π * ↑x * z.im)\n⊢ False",
"ppTerm": "?refine_2.inr",
"assigned": true,
"usedConstants": [
"Int.cast",
"Real",
"Real.pi",
"HMul.hMul",
"congrArg",
"Complex.im",
... | [
"case refine_2.inr\nz τ : ℂ\nhτ✝ : τ.im ≤ 0\nhτ : τ.im = 0\nh : Summable fun x ↦ rexp (-(2 * π * ↑x * z.im))\n⊢ False"
] | zero_sub | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 273,
"column": 31
} | {
"line": 273,
"column": 76
} | {
"line": 274,
"column": 6
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nx : ℝ\nhx : 0 < x\nhx' : 1 < x\nthis : 1 / x < 1\n⊢ (Ioi 1).indicator (fun x ↦ P.f x - P.f₀) (1 / x) +\n (Ioo 0 1).indicator (fun x ↦ P.f x - (P.ε * ↑(x ^ (-P.k))) • P.g₀) (1 / x) =\n (P.ε * ↑(x ^ ... | [
"case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nx : ℝ\nhx : 0 < x\nhx' : 1 < x\nthis : 1 / x < 1\n⊢ 0 + (Ioo 0 1).indicator (fun x ↦ P.f x - (P.ε * ↑(x ^ (-P.k))) • P.g₀) (1 / x) = (P.ε * ↑(x ^ P.k)) • P.g_modif x"
] | indicator_of_notMem (notMem_Ioi.mpr this.le), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 328,
"column": 4
} | {
"line": 329,
"column": 62
} | {
"line": 330,
"column": 4
} | [
{
"pp": "case inr.inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nx : ℝ\nhx : 0 < x\nhx' : 1 < x\n⊢ (Ioi 1).indicator (fun x ↦ P.f x - P.f₀) x + (Ioo 0 1).indicator (fun x ↦ P.f x - (P.ε * ↑(x ^ (-P.k))) • P.g₀) x -\n P.f x +\n P.f₀ =\n (Ioo 0 1).indic... | [
"case inr.inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nx : ℝ\nhx : 0 < x\nhx' : 1 < x\n⊢ P.f x - P.f₀ + 0 - P.f x + P.f₀ = 0 + 0"
] | simp_rw [indicator_of_mem (mem_Ioi.mpr hx'), indicator_of_notMem (notMem_Ioo_of_ge hx'.le),
indicator_of_notMem (mem_singleton_iff.not.mpr hx'.ne')] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 345,
"column": 4
} | {
"line": 347,
"column": 41
} | {
"line": 348,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nP : WeakFEPair E\ninst✝ : CompleteSpace E\ns : ℂ\nhs : P.k < s.re\nh_re1 : -1 < (s - 1).re\nh_re2 : -1 < (s - ↑P.k - 1).re\n⊢ ∫ (x : ℝ) in Ioi 0,\n ↑x ^ (s - 1) •\n ((Ioo 0 1).indicator (fun t ↦ P.f₀ - (P.ε * ↑(t ^ (-P.k... | [] | refine setIntegral_congr_ae measurableSet_Ioi (eventually_of_mem (U := {1}ᶜ)
(compl_mem_ae_iff.mpr (subsingleton_singleton.measure_zero _)) (fun x hx _ ↦ ?_))
rw [indicator_of_notMem hx, add_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 343,
"column": 16
} | {
"line": 343,
"column": 54
} | {
"line": 344,
"column": 4
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\n⊢ ∀ (x y : ℂ × ℂ →L[ℂ] ℂ), (x + y) (1, 0) = x (1, 0) + y (1, 0)",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"congrArg",
... | [] | by simp only [add_apply, forall_const] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 345,
"column": 4
} | {
"line": 347,
"column": 41
} | {
"line": 348,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nP : WeakFEPair E\ninst✝ : CompleteSpace E\ns : ℂ\nhs : P.k < s.re\nh_re1 : -1 < (s - 1).re\nh_re2 : -1 < (s - ↑P.k - 1).re\n⊢ ∫ (x : ℝ) in Ioi 0,\n ↑x ^ (s - 1) •\n ((Ioo 0 1).indicator (fun t ↦ P.f₀ - (P.ε * ↑(t ^ (-P.k... | [] | refine setIntegral_congr_ae measurableSet_Ioi (eventually_of_mem (U := {1}ᶜ)
(compl_mem_ae_iff.mpr (subsingleton_singleton.measure_zero _)) (fun x hx _ ↦ ?_))
rw [indicator_of_notMem hx, add_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 415,
"column": 89
} | {
"line": 420,
"column": 31
} | {
"line": 422,
"column": 0
} | [
{
"pp": "z τ : ℂ\n⊢ jacobiTheta₂' z (τ + 2) = jacobiTheta₂' z τ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Distrib.leftDistribClass",
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSem... | [] | by
refine tsum_congr (fun n ↦ ?_)
simp_rw [jacobiTheta₂'_term, jacobiTheta₂_term, Complex.exp_add]
suffices cexp (π * I * n ^ 2 * 2 : ℂ) = 1 by rw [mul_add, Complex.exp_add, this, mul_one]
rw [(by push_cast; ring : (π * I * n ^ 2 * 2 : ℂ) = (n ^ 2 :) * (2 * π * I)), exp_int_mul,
exp_two_pi_mul_I, one_zpow] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.HurwitzZeta | {
"line": 89,
"column": 79
} | {
"line": 91,
"column": 96
} | {
"line": 93,
"column": 0
} | [
{
"pp": "a : UnitAddCircle\n⊢ DifferentiableAt ℂ (fun s ↦ hurwitzZeta a s - 1 / (s - 1) / s.Gammaℝ) 1",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"HurwitzZeta.differentiableAt_hurwitzZetaEven_sub_one_div",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedComm... | [] | by
simp only [hurwitzZeta, add_sub_right_comm]
exact (differentiableAt_hurwitzZetaEven_sub_one_div a).add (differentiable_hurwitzZetaOdd a 1) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 27
} | {
"line": 169,
"column": 2
} | [
{
"pp": "case inl\n⊢ riemannZeta 0 = (0 * completedRiemannZeta₀ 0 - 1 - 0 / (1 - 0)) / (2 * ↑π ^ (-0 / 2) * Gamma (0 / 2 + 1))",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"instHDiv",
"Real.pi",
"HMul.hMul",
"riemannZeta",
... | [] | simp [riemannZeta_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 27
} | {
"line": 169,
"column": 2
} | [
{
"pp": "case inl\n⊢ riemannZeta 0 = (0 * completedRiemannZeta₀ 0 - 1 - 0 / (1 - 0)) / (2 * ↑π ^ (-0 / 2) * Gamma (0 / 2 + 1))",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"instHDiv",
"Real.pi",
"HMul.hMul",
"riemannZeta",
... | [] | simp [riemannZeta_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 27
} | {
"line": 169,
"column": 2
} | [
{
"pp": "case inl\n⊢ riemannZeta 0 = (0 * completedRiemannZeta₀ 0 - 1 - 0 / (1 - 0)) / (2 * ↑π ^ (-0 / 2) * Gamma (0 / 2 + 1))",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"instHDiv",
"Real.pi",
"HMul.hMul",
"riemannZeta",
... | [] | simp [riemannZeta_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 635,
"column": 4
} | {
"line": 635,
"column": 61
} | {
"line": 636,
"column": 4
} | [
{
"pp": "a : UnitAddCircle\nthis : DifferentiableAt ℂ (fun s ↦ completedHurwitzZetaEven a s / s.Gammaℝ - 1 / (s - 1) / s.Gammaℝ) 1\n⊢ (fun s ↦ hurwitzZetaEven a s - 1 / (s - 1) / s.Gammaℝ) =ᶠ[𝓝 1] fun s ↦\n completedHurwitzZetaEven a s / s.Gammaℝ - 1 / (s - 1) / s.Gammaℝ",
"ppTerm": "?m.90",
"assign... | [
"a : UnitAddCircle\nthis : DifferentiableAt ℂ (fun s ↦ completedHurwitzZetaEven a s / s.Gammaℝ - 1 / (s - 1) / s.Gammaℝ) 1\nx : ℂ\nhx : x ≠ 0\n⊢ hurwitzZetaEven a x - 1 / (x - 1) / x.Gammaℝ = completedHurwitzZetaEven a x / x.Gammaℝ - 1 / (x - 1) / x.Gammaℝ"
] | filter_upwards [eventually_ne_nhds one_ne_zero] with x hx | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 279,
"column": 87
} | {
"line": 280,
"column": 53
} | {
"line": 281,
"column": 2
} | [
{
"pp": "s : ℂ\nhs : 1 < s.re\nthis : ∑' (n : ℕ), term (fun n ↦ if n = 0 then 0 else 1) s n = ∑' (n : ℕ), 1 / ↑n ^ s\n⊢ L (fun n ↦ ↑(ζ n)) s = riemannZeta s",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRin... | [] | by
simpa [LSeries, zeta_eq_tsum_one_div_nat_cpow hs] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 776,
"column": 2
} | {
"line": 778,
"column": 41
} | {
"line": 779,
"column": 2
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\n⊢ cosZeta a (1 - s) = s.Gammaℂ * Complex.cos (↑π * s / 2) * hurwitzZetaEven a s",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Eq.mpr",
"DivInvMonoid.toInv",
"HurwitzZeta.cosZeta.... | [
"a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\nthis : cosZeta a (1 - s) = completedCosZeta a (1 - s) * (1 - s).Gammaℝ⁻¹\n⊢ cosZeta a (1 - s) = s.Gammaℂ * Complex.cos (↑π * s / 2) * hurwitzZetaEven a s"
] | have : cosZeta a (1 - s) = completedCosZeta a (1 - s) * (Gammaℝ (1 - s))⁻¹ := by
rw [cosZeta, Function.update_of_ne, div_eq_mul_inv]
simpa [sub_eq_zero] using (hs 0).symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.FLT.Basic | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 38
} | {
"line": 133,
"column": 4
} | [
{
"pp": "n : ℕ\ntfae_1_to_2 : FermatLastTheoremWith ℕ n → FermatLastTheoremWith ℤ n\nh : FermatLastTheoremWith ℤ n\na b c : ℚ\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\nhabc : a ^ n + b ^ n = c ^ n\n⊢ False",
"ppTerm": "?m.327",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Rat.num",
... | [
"n : ℕ\ntfae_1_to_2 : FermatLastTheoremWith ℕ n → FermatLastTheoremWith ℤ n\nh : FermatLastTheoremWith ℤ n\na b c : ℚ\nha : a.num ≠ 0\nhb : b.num ≠ 0\nhc : c.num ≠ 0\nhabc : a ^ n + b ^ n = c ^ n\n⊢ False"
] | rw [← Rat.num_ne_zero] at ha hb hc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.FLT.Basic | {
"line": 212,
"column": 28
} | {
"line": 212,
"column": 34
} | {
"line": 213,
"column": 2
} | [
{
"pp": "case neg.inl.inl.inl\nn : ℕ\ntfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n\na b c : ℤ\nhn : ¬n = 0\nha : a ^ n = 1\nhb : b ^ n = 1\nhc : c ^ n = 1\n⊢ 1 + 1 ≠ 1",
"ppTerm": "?neg.inl.inl.inl✝",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_deci... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.FLT.Basic | {
"line": 212,
"column": 28
} | {
"line": 212,
"column": 34
} | {
"line": 213,
"column": 2
} | [
{
"pp": "case neg.inl.inl.inr\nn : ℕ\ntfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n\na b c : ℤ\nhn : ¬n = 0\nha : a ^ n = 1\nhb : b ^ n = 1\nhc : c ^ n = -1\n⊢ 1 + 1 ≠ -1",
"ppTerm": "?neg.inl.inl.inr✝",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_de... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.FLT.Basic | {
"line": 212,
"column": 28
} | {
"line": 212,
"column": 34
} | {
"line": 213,
"column": 2
} | [
{
"pp": "case neg.inl.inr.inl\nn : ℕ\ntfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n\na b c : ℤ\nhn : ¬n = 0\nha : a ^ n = 1\nhb : b ^ n = -1\nhc : c ^ n = 1\n⊢ 1 + -1 ≠ 1",
"ppTerm": "?neg.inl.inr.inl✝",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_de... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.FLT.Basic | {
"line": 212,
"column": 28
} | {
"line": 212,
"column": 34
} | {
"line": 213,
"column": 2
} | [
{
"pp": "case neg.inl.inr.inr\nn : ℕ\ntfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n\na b c : ℤ\nhn : ¬n = 0\nha : a ^ n = 1\nhb : b ^ n = -1\nhc : c ^ n = -1\n⊢ 1 + -1 ≠ -1",
"ppTerm": "?neg.inl.inr.inr✝",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.FLT.Basic | {
"line": 212,
"column": 28
} | {
"line": 212,
"column": 34
} | {
"line": 213,
"column": 2
} | [
{
"pp": "case neg.inr.inl.inl\nn : ℕ\ntfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n\na b c : ℤ\nhn : ¬n = 0\nha : a ^ n = -1\nhb : b ^ n = 1\nhc : c ^ n = 1\n⊢ -1 + 1 ≠ 1",
"ppTerm": "?neg.inr.inl.inl✝",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_de... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.FLT.Basic | {
"line": 212,
"column": 28
} | {
"line": 212,
"column": 34
} | {
"line": 213,
"column": 2
} | [
{
"pp": "case neg.inr.inl.inr\nn : ℕ\ntfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n\na b c : ℤ\nhn : ¬n = 0\nha : a ^ n = -1\nhb : b ^ n = 1\nhc : c ^ n = -1\n⊢ -1 + 1 ≠ -1",
"ppTerm": "?neg.inr.inl.inr✝",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.FLT.Basic | {
"line": 212,
"column": 28
} | {
"line": 212,
"column": 34
} | {
"line": 213,
"column": 2
} | [
{
"pp": "case neg.inr.inr.inl\nn : ℕ\ntfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n\na b c : ℤ\nhn : ¬n = 0\nha : a ^ n = -1\nhb : b ^ n = -1\nhc : c ^ n = 1\n⊢ -1 + -1 ≠ 1",
"ppTerm": "?neg.inr.inr.inl✝",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.FLT.Basic | {
"line": 212,
"column": 28
} | {
"line": 212,
"column": 34
} | {
"line": 213,
"column": 2
} | [
{
"pp": "case neg.inr.inr.inr\nn : ℕ\ntfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n\na b c : ℤ\nhn : ¬n = 0\nha : a ^ n = -1\nhb : b ^ n = -1\nhc : c ^ n = -1\n⊢ -1 + -1 ≠ -1",
"ppTerm": "?neg.inr.inr.inr✝",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"o... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 33,
"column": 18
} | {
"line": 33,
"column": 24
} | {
"line": 35,
"column": 0
} | [
{
"pp": "case «0»\n⊢ (fun i ↦ i) ⟨0, ⋯⟩ * (fun i ↦ i) ⟨0, ⋯⟩ ≠ 2",
"ppTerm": "?«0»",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"HMul.hMul",
"of_decide_eq_true",
"ZMod.commRing",
"Nat.le_refl",
"CommSemiring.toSemiring",
"Nat.instAtLeastTwoHAdd... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 33,
"column": 18
} | {
"line": 33,
"column": 24
} | {
"line": 35,
"column": 0
} | [
{
"pp": "case «1»\n⊢ (fun i ↦ i) ⟨1, ⋯⟩ * (fun i ↦ i) ⟨1, ⋯⟩ ≠ 2",
"ppTerm": "?«1»",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"HMul.hMul",
"of_decide_eq_true",
"ZMod.commRing",
"Nat.le_refl",
"CommSemiring.toSemiring",
"Nat.instAtLeastTwoHAdd... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 33,
"column": 18
} | {
"line": 33,
"column": 24
} | {
"line": 35,
"column": 0
} | [
{
"pp": "case «2»\n⊢ (fun i ↦ i) ⟨2, ⋯⟩ * (fun i ↦ i) ⟨2, ⋯⟩ ≠ 2",
"ppTerm": "?«2»",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"HMul.hMul",
"of_decide_eq_true",
"ZMod.commRing",
"Nat.le_refl",
"CommSemiring.toSemiring",
"Nat.instAtLeastTwoHAdd... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 33,
"column": 18
} | {
"line": 33,
"column": 24
} | {
"line": 35,
"column": 0
} | [
{
"pp": "case «3»\n⊢ (fun i ↦ i) ⟨3, ⋯⟩ * (fun i ↦ i) ⟨3, ⋯⟩ ≠ 2",
"ppTerm": "?«3»",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"HMul.hMul",
"of_decide_eq_true",
"ZMod.commRing",
"Nat.le_refl",
"CommSemiring.toSemiring",
"Nat.instAtLeastTwoHAdd... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 119,
"column": 44
} | {
"line": 119,
"column": 50
} | {
"line": 119,
"column": 51
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhx : x % 2 = 0\nhy : y % 2 = 0\n⊢ 1 < 2",
"ppTerm": "?m.95",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.de... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 119,
"column": 44
} | {
"line": 119,
"column": 50
} | {
"line": 119,
"column": 51
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhx : x % 2 = 0\nhy : y % 2 = 0\n⊢ 1 < 2",
"ppTerm": "?m.95",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.de... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 119,
"column": 44
} | {
"line": 119,
"column": 50
} | {
"line": 119,
"column": 51
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhx : x % 2 = 0\nhy : y % 2 = 0\n⊢ 1 < 2",
"ppTerm": "?m.95",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.de... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Wronskian | {
"line": 101,
"column": 8
} | {
"line": 101,
"column": 46
} | {
"line": 102,
"column": 8
} | [
{
"pp": "case hab\nR : Type u_1\ninst✝ : CommRing R\na b : R[X]\nha : a ≠ 0\nhb : b ≠ 0\n⊢ (derivative a * b).degree ≤ ?b",
"ppTerm": "?hab",
"assigned": true,
"usedConstants": [
"Polynomial.degree_mul_le",
"Polynomial.derivative",
"Semiring.toModule",
"CommSemiring.toSemirin... | [
"case hbc\nR : Type u_1\ninst✝ : CommRing R\na b : R[X]\nha : a ≠ 0\nhb : b ≠ 0\n⊢ (derivative a).degree + b.degree < a.degree + b.degree"
] | · exact degree_mul_le (derivative a) b | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 10
} | {
"line": 144,
"column": 0
} | [
{
"pp": "case inr.inr\nz x0 y0 : ℤ\nhx : (x0 * 2 + 1) % 2 = 1\nhy : (y0 * 2 + 1) % 2 = 1\nh : PythagoreanTriple (x0 * 2 + 1) (y0 * 2 + 1) z\nhc : (x0 * 2 + 1).gcd (y0 * 2 + 1) = 1\n⊢ 2 % 4 % 4 = 2",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"Int.inst... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.FLT.Four | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 12
} | {
"line": 124,
"column": 4
} | [
{
"pp": "case inl.inl\na b c : ℤ\nh : Fermat42 a b c\na0 b0 c0 : ℤ\nhf : Minimal a0 b0 c0\nhap : a0 % 2 = 0\nhbp : b0 % 2 = 0\n⊢ 2 ∣ ↑1 → False",
"ppTerm": "?inl.inl",
"assigned": true,
"usedConstants": [
"Int.decidableDvd",
"False",
"Dvd.dvd",
"of_decide_eq_true",
"id"... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.RingTheory.Polynomial.Wronskian | {
"line": 131,
"column": 4
} | {
"line": 133,
"column": 40
} | {
"line": 134,
"column": 4
} | [
{
"pp": "case left\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\na b : R[X]\nhc : IsCoprime a b\nhw : a * derivative b = derivative a * b\n⊢ derivative a = 0",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Dvd.dvd",
... | [
"case right\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\na b : R[X]\nhc : IsCoprime a b\nhw : a * derivative b = derivative a * b\n⊢ derivative b = 0"
] | · rw [← dvd_derivative_iff]
apply hc.dvd_of_dvd_mul_right
rw [← hw]; exact dvd_mul_right _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.FLT.Four | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 10
} | {
"line": 169,
"column": 2
} | [
{
"pp": "a b c : ℤ\nh : Minimal a b c\nha2 : a % 2 = 1\nhc : 0 < c\nht : PythagoreanTriple (a ^ 2) (b ^ 2) c\nh2 : (a ^ 2).gcd (b ^ 2) = 1\n⊢ 1 * 1 % 2 = 1",
"ppTerm": "?m.132",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"of_decide_eq_true",
"Int.instDecidableEq",
"id"... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 10
} | {
"line": 213,
"column": 2
} | [
{
"pp": "case h\ny z : ℤ\nh : PythagoreanTriple 0 y z\nhc : y.natAbs = 1\nhy : y = ↑y.natAbs\n⊢ (0 = 1 ^ 2 - 0 ^ 2 ∧ ↑1 = 2 * 1 * 0 ∨ 0 = 2 * 1 * 0 ∧ ↑1 = 1 ^ 2 - 0 ^ 2) ∧\n Int.natAbs 1 = 1 ∧ (1 % 2 = 0 ∧ 0 % 2 = 1 ∨ 1 % 2 = 1 ∧ 0 % 2 = 0)",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 215,
"column": 4
} | {
"line": 215,
"column": 10
} | {
"line": 217,
"column": 0
} | [
{
"pp": "case h\ny z : ℤ\nh : PythagoreanTriple 0 y z\nhc : y.natAbs = 1\nhy : y = -↑y.natAbs\n⊢ (0 = 0 ^ 2 - 1 ^ 2 ∧ -↑1 = 2 * 0 * 1 ∨ 0 = 2 * 0 * 1 ∧ -↑1 = 0 ^ 2 - 1 ^ 2) ∧\n Int.natAbs 1 = 1 ∧ (0 % 2 = 0 ∧ 1 % 2 = 1 ∨ 0 % 2 = 1 ∧ 1 % 2 = 0)",
"ppTerm": "?h",
"assigned": true,
"usedConstants": ... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 223,
"column": 10
} | {
"line": 223,
"column": 28
} | {
"line": 223,
"column": 28
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nH : ¬y.gcd z = 1\np : ℕ\nhp : Nat.Prime p\nhpy : p ∣ y.natAbs\nhpz : p ∣ z.natAbs\n⊢ ↑p ∣ x * x",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"Monoid.toMulOneClass... | [
"x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nH : ¬y.gcd z = 1\np : ℕ\nhp : Nat.Prime p\nhpy : p ∣ y.natAbs\nhpz : p ∣ z.natAbs\n⊢ ↑p ∣ z * z - y * y"
] | eq_sub_of_add_eq h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 305,
"column": 6
} | {
"line": 305,
"column": 12
} | {
"line": 306,
"column": 4
} | [
{
"pp": "m n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (m ^ 2 + n ^ 2) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ m ^ 2 + n ^ 2\nh2m : ↑p ∣ 2 * m ^ 2\nh2n : ↑p ∣ 2 * n ^ 2\nhmc : p = 2 ∨ p ∣ m.natAbs\nhnc : p = 2 ∨ p ∣ n.natAbs\nh2 : p = 2\n⊢ (0 % 2 %... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.RingTheory.Radical.Basic | {
"line": 404,
"column": 2
} | {
"line": 404,
"column": 6
} | {
"line": 404,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝² : EuclideanDomain E\ninst✝¹ : NormalizationMonoid E\ninst✝ : UniqueFactorizationMonoid E\na b : E\nhab : IsCoprime a b\n⊢ divRadical (a * b) = divRadical a * divRadical b",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"CommSemiring.to... | [
"E : Type u_1\ninst✝² : EuclideanDomain E\ninst✝¹ : NormalizationMonoid E\ninst✝ : UniqueFactorizationMonoid E\na b : E\nhab : IsCoprime a b\n⊢ divRadical a * divRadical b = divRadical (a * b)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 339,
"column": 6
} | {
"line": 339,
"column": 12
} | {
"line": 340,
"column": 4
} | [
{
"pp": "case inl.inl\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp2✝ : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(m.gcd n)\nhp2m : p ∣ Int.natAbs 2 * m.natAbs\nhp2 : p ∣ Int.natAbs 2\nhp2' : p = 2\n⊢ ¬-(1 % 2) % 2 = 0",
"ppTerm": "?inl.i... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 381,
"column": 4
} | {
"line": 381,
"column": 10
} | {
"line": 382,
"column": 2
} | [
{
"pp": "case H\nm0 n0 : ℤ\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : (m0 * 2 + 1).gcd (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\n⊢ 2 ≠ 0",
"ppTerm":... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 383,
"column": 31
} | {
"line": 383,
"column": 37
} | {
"line": 384,
"column": 2
} | [
{
"pp": "m0 n0 : ℤ\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : (m0 * 2 + 1).gcd (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * ... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 416,
"column": 69
} | {
"line": 416,
"column": 73
} | {
"line": 416,
"column": 74
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhzpos : 0 < z\nm n : ℤ\nhm2n2 : 0 < m ^ 2 + n ^ 2\nhv2 : ↑x / ↑z = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)\nhw2 : ↑y / ↑z = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)\nH : (m ^ 2 - n ^ 2).gcd (m ^ 2 + n ^ 2) = 1\nco : m.gcd n = 1\npp : m % 2 = 0 ∧ n % 2 = 1 ... | [
"x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhzpos : 0 < z\nm n : ℤ\nhm2n2 : 0 < m ^ 2 + n ^ 2\nhv2 : ↑x / ↑z = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)\nhw2 : ↑y / ↑z = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)\nH : (m ^ 2 - n ^ 2).gcd (m ^ 2 + n ^ 2) = 1\nco : m.gcd n = 1\npp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ... | hv2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 191,
"column": 8
} | {
"line": 191,
"column": 20
} | {
"line": 192,
"column": 8
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\np q r : ℕ\nu v w : k\nhp : p ≠ 0\nhq : q ≠ 0\nhr : r ≠ 0\nhineq : q * r + r * p + p * q ≤ p * q * r\nchp : ↑p ≠ 0\nchq : ↑q ≠ 0\nchr : ↑r ≠ 0\nhu : u ≠ 0\nhv : v ≠ 0\nhw : w ≠ 0\nd : ℕ\na b c : k[X]\nheq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0\nha : a ≠ 0\nhb : b ≠ ... | [
"k : Type u_1\ninst✝ : Field k\np q r : ℕ\nu v w : k\nhp : p ≠ 0\nhq : q ≠ 0\nhr : r ≠ 0\nhineq : q * r + r * p + p * q ≤ p * q * r\nchp : ↑p ≠ 0\nchq : ↑q ≠ 0\nchr : ↑r ≠ 0\nhu : u ≠ 0\nhv : v ≠ 0\nhw : w ≠ 0\nd : ℕ\na b c : k[X]\nheq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ ... | grw [← hch2] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 170,
"column": 4
} | {
"line": 172,
"column": 46
} | {
"line": 173,
"column": 4
} | [
{
"pp": "case inr\nk : Type u_1\ninst✝ : Field k\np q r : ℕ\na b c : k[X]\nu v w : k\nheq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0\nhp : p ≠ 0\nhq : q ≠ 0\nhr : r ≠ 0\nhineq : q * r + r * p + p * q ≤ p * q * r\nchp : ↑p ≠ 0\nchq : ↑q ≠ 0\nchr : ↑r ≠ 0\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\nhab : IsCoprime a ... | [] | induction d
using Nat.case_strong_induction_on
generalizing a b c ha hb hc hab heq with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 196,
"column": 6
} | {
"line": 196,
"column": 70
} | {
"line": 197,
"column": 6
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nk : ℕ+\nhc : (↑k).Coprime (torsionOrder K)\nζ : (𝓞 K)ˣ\nh : ζ ^ ↑k = 1\n⊢ ζ ∈ torsion K",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"NumberField.instCommRingRingOfIntegers",
"Monoid.toMulOneClass... | [
"K : Type u_1\ninst✝ : Field K\nk : ℕ+\nhc : (↑k).Coprime (torsionOrder K)\nζ : (𝓞 K)ˣ\nh : ζ ^ ↑k = 1\n⊢ ∃ n, 0 < n ∧ ζ ^ n = 1"
] | rw [torsion, CommGroup.mem_torsion, isOfFinOrder_iff_pow_eq_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 292,
"column": 2
} | {
"line": 292,
"column": 16
} | {
"line": 294,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nw : InfinitePlace K\n⊢ 1 ≤ w.mult",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"NumberField.InfinitePlace.mult_pos"
],
"usedFVars": [
"K",
"inst✝",
"w"
],
"usedGoals": []
}
] | [] | exact mult_pos | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 231,
"column": 33
} | {
"line": 231,
"column": 39
} | {
"line": 231,
"column": 39
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ ¬0 = 2",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"instDecidabl... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 231,
"column": 33
} | {
"line": 231,
"column": 39
} | {
"line": 231,
"column": 39
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ ¬0 = 2",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"instDecidabl... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 231,
"column": 33
} | {
"line": 231,
"column": 39
} | {
"line": 231,
"column": 39
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ ¬0 = 2",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"instDecidabl... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 247,
"column": 58
} | {
"line": 247,
"column": 64
} | {
"line": 247,
"column": 64
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh : Odd (Module.finrank ℚ K)\nx : ↥(torsion K)\nhi : orderOf ↑↑x = 2\nhc : 2 ≤ 2\n⊢ 0 ≠ 2",
"ppTerm": "?m.165",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"Ne",
"ins... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 247,
"column": 58
} | {
"line": 247,
"column": 64
} | {
"line": 247,
"column": 64
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh : Odd (Module.finrank ℚ K)\nx : ↥(torsion K)\nhi : orderOf ↑↑x = 2\nhc : 2 ≤ 2\n⊢ 0 ≠ 2",
"ppTerm": "?m.165",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"Ne",
"ins... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 247,
"column": 58
} | {
"line": 247,
"column": 64
} | {
"line": 247,
"column": 64
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh : Odd (Module.finrank ℚ K)\nx : ↥(torsion K)\nhi : orderOf ↑↑x = 2\nhc : 2 ≤ 2\n⊢ 0 ≠ 2",
"ppTerm": "?m.165",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"Ne",
"ins... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.EquivReindex | {
"line": 35,
"column": 8
} | {
"line": 35,
"column": 24
} | {
"line": 35,
"column": 25
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ Fintype.card (K →+* ℂ) = Fintype.card (ChooseBasisIndex ℤ (𝓞 K))",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NumberField.instCommRingRingOfIntegers",
"AddGroupWithOne.toAddGroup",
"con... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ finrank ℚ K = Fintype.card (ChooseBasisIndex ℤ (𝓞 K))"
] | Embeddings.card, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 475,
"column": 41
} | {
"line": 475,
"column": 57
} | {
"line": 475,
"column": 58
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ (-1) ^\n ((card (K →+* ℂ) - card ↑(Function.fixedPoints ⇑(Function.Involutive.toPerm ComplexEmbedding.conjugate ⋯))) / 2) =\n (-1) ^ nrComplexPlaces K",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Int.instCo... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ (-1) ^ ((finrank ℚ K - card ↑(Function.fixedPoints ⇑(Function.Involutive.toPerm ComplexEmbedding.conjugate ⋯))) / 2) =\n (-1) ^ nrComplexPlaces K",
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ Function.Involutive.toPerm ComplexEmbedding.con... | Embeddings.card, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 520,
"column": 47
} | {
"line": 520,
"column": 64
} | {
"line": 520,
"column": 64
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nζ : K\nk : ℕ\nhk : 2 < k\nhζ : IsPrimitiveRoot ζ k\nx✝ : { w // w.IsReal }\nw : InfinitePlace K\nhwreal : ComplexEmbedding.IsReal w.embedding\nf : K →+* ℂ := w.embedding\nhim : (f ζ).im = 0\nhnegone : (f ζ).re = -1\nhζ' : k = orderOf (f ζ)\n⊢ (f ζ)... | [] | by simp [hnegone] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 101,
"column": 78
} | {
"line": 101,
"column": 97
} | {
"line": 101,
"column": 97
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nf : InfinitePlace K → ℝ≥0\ninst✝ : NumberField K\n⊢ (∏ x, ENNReal.ofReal (2 * ↑(f ↑x))) * ∏ i, volume (ball 0 ↑(f ↑i)) =\n (∏ x, ENNReal.ofReal (2 * ↑(f ↑x))) * ∏ x, ENNReal.ofReal ↑(f ↑x) ^ 2 * ↑pi",
"ppTerm": "?m.264",
"assigned": true,
"usedConstants": ... | [] | Complex.volume_ball | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 164,
"column": 41
} | {
"line": 164,
"column": 58
} | {
"line": 165,
"column": 8
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\nx : K\nx✝ :\n (∀ (a : InfinitePlace K), a.IsReal → a x < ↑(f a)) ∧\n ∀ (a : InfinitePlace K) (b : a.IsComplex),\n a.embedding x ∈ if ⟨a, b⟩ = w₀ then {x | |x.re| < 1 ∧ |x.im| < ↑(f a) ^ 2} else ball ... | [
"case neg\nK : Type u_1\ninst✝ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\nx : K\nx✝ :\n (∀ (a : InfinitePlace K), a.IsReal → a x < ↑(f a)) ∧\n ∀ (a : InfinitePlace K) (b : a.IsComplex),\n a.embedding x ∈ if ⟨a, b⟩ = w₀ then {x | |x.re| < 1 ∧ |x.im| < ↑(f a) ^ 2} else ball 0 ↑(f a)\nh₁... | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 50
} | {
"line": 169,
"column": 2
} | [
{
"pp": "case refine_3\nK : Type u_1\ninst✝ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\nx : K\nx✝ :\n (∀ (w : InfinitePlace K), w ≠ ↑w₀ → w x < ↑(f w)) ∧\n |((↑w₀).embedding x).re| < 1 ∧ |((↑w₀).embedding x).im| < ↑(f ↑w₀) ^ 2\nh₁ : ∀ (w : InfinitePlace K), w ≠ ↑w₀ → w x < ↑(f w)\nh₂ : ... | [
"case refine_4\nK : Type u_1\ninst✝ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\nx : K\nx✝ :\n (∀ (w : InfinitePlace K), w ≠ ↑w₀ → w x < ↑(f w)) ∧\n |((↑w₀).embedding x).re| < 1 ∧ |((↑w₀).embedding x).im| < ↑(f ↑w₀) ^ 2\nh₁ : ∀ (w : InfinitePlace K), w ≠ ↑w₀ → w x < ↑(f w)\nh₂ : |((↑w₀).embe... | · exact h₁ w (ne_of_isReal_isComplex hw w₀.prop) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 213,
"column": 24
} | {
"line": 213,
"column": 41
} | {
"line": 213,
"column": 42
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\ninst✝ : NumberField K\nB : ℝ≥0\nx✝ : ℝ × ℝ\n⊢ x✝ ∈ {a | |a.1| < 1 ∧ |a.2| < ↑B ^ 2} ↔ x✝ ∈ Set.Ioo (-1) 1 ×ˢ Set.Ioo (-↑B ^ 2) (↑B ^ 2)",
"ppTerm": "?m.202",
"assigned": true,
"usedConstants": [
"Set... | [
"K : Type u_1\ninst✝¹ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\ninst✝ : NumberField K\nB : ℝ≥0\nx✝ : ℝ × ℝ\n⊢ |x✝.1| < 1 ∧ |x✝.2| < ↑B ^ 2 ↔ x✝ ∈ Set.Ioo (-1) 1 ×ˢ Set.Ioo (-↑B ^ 2) (↑B ^ 2)"
] | Set.mem_setOf_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 228,
"column": 49
} | {
"line": 228,
"column": 68
} | {
"line": 228,
"column": 68
} | [
{
"pp": "case e_a.e_a\nK : Type u_1\ninst✝¹ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\ninst✝ : NumberField K\nvol_box : ∀ (B : ℝ≥0), volume {x | |x.re| < 1 ∧ |x.im| < ↑B ^ 2} = 4 * ↑B ^ 2\nw' : { w // w.IsComplex }\nhw' : w' ∈ Finset.univ.erase w₀\n⊢ volume (ball 0 ↑(f ↑w')) = ENNReal.ofRe... | [
"case e_a.e_a\nK : Type u_1\ninst✝¹ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\ninst✝ : NumberField K\nvol_box : ∀ (B : ℝ≥0), volume {x | |x.re| < 1 ∧ |x.im| < ↑B ^ 2} = 4 * ↑B ^ 2\nw' : { w // w.IsComplex }\nhw' : w' ∈ Finset.univ.erase w₀\n⊢ ENNReal.ofReal ↑(f ↑w') ^ 2 * ↑pi = ENNReal.ofReal ... | Complex.volume_ball | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 102,
"column": 12
} | {
"line": 102,
"column": 18
} | {
"line": 104,
"column": 0
} | [
{
"pp": "case pos\nk : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlace K\nh₁ : (w.comap f).IsReal\nh₂ : w.IsReal\n⊢ 1 ≤ 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"LE.le",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 102,
"column": 12
} | {
"line": 102,
"column": 18
} | {
"line": 104,
"column": 0
} | [
{
"pp": "case pos\nk : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlace K\nh₁ : (w.comap f).IsReal\nh₂ : w.IsReal\n⊢ 1 ≤ 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"LE.le",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 102,
"column": 12
} | {
"line": 102,
"column": 18
} | {
"line": 104,
"column": 0
} | [
{
"pp": "case pos\nk : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlace K\nh₁ : (w.comap f).IsReal\nh₂ : w.IsReal\n⊢ 1 ≤ 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"LE.le",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 102,
"column": 12
} | {
"line": 102,
"column": 18
} | {
"line": 104,
"column": 0
} | [
{
"pp": "case neg\nk : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlace K\nh₁ : (w.comap f).IsReal\nh₂ : ¬w.IsReal\n⊢ 1 ≤ 2",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"LE.le",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 102,
"column": 12
} | {
"line": 102,
"column": 18
} | {
"line": 104,
"column": 0
} | [
{
"pp": "case neg\nk : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlace K\nh₁ : (w.comap f).IsReal\nh₂ : ¬w.IsReal\n⊢ 1 ≤ 2",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"LE.le",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 102,
"column": 12
} | {
"line": 102,
"column": 18
} | {
"line": 104,
"column": 0
} | [
{
"pp": "case neg\nk : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlace K\nh₁ : (w.comap f).IsReal\nh₂ : ¬w.IsReal\n⊢ 1 ≤ 2",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"LE.le",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 102,
"column": 12
} | {
"line": 102,
"column": 18
} | {
"line": 104,
"column": 0
} | [
{
"pp": "case neg\nk : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlace K\nh₁ : ¬(w.comap f).IsReal\nh₂ : ¬w.IsReal\n⊢ 2 ≤ 2",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"LE.le",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 102,
"column": 12
} | {
"line": 102,
"column": 18
} | {
"line": 104,
"column": 0
} | [
{
"pp": "case neg\nk : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlace K\nh₁ : ¬(w.comap f).IsReal\nh₂ : ¬w.IsReal\n⊢ 2 ≤ 2",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"LE.le",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 102,
"column": 12
} | {
"line": 102,
"column": 18
} | {
"line": 104,
"column": 0
} | [
{
"pp": "case neg\nk : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlace K\nh₁ : ¬(w.comap f).IsReal\nh₂ : ¬w.IsReal\n⊢ 2 ≤ 2",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"LE.le",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 348,
"column": 6
} | {
"line": 348,
"column": 37
} | {
"line": 348,
"column": 38
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nσ : Gal(K/k)\n⊢ φ.comp ↑σ.symm = φ ∨ conjugate (φ.comp ↑σ.symm) = φ ↔ σ = 1 ∨ ComplexEmbedding.IsConj φ σ",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgEqu... | [
"k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nσ : Gal(K/k)\n⊢ φ.comp ↑σ.symm = φ ∨ conjugate (φ.comp ↑σ.symm) = φ ↔ σ = 1 ∨ ComplexEmbedding.IsConj φ σ.symm"
] | ← ComplexEmbedding.isConj_symm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 327,
"column": 23
} | {
"line": 327,
"column": 40
} | {
"line": 327,
"column": 41
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nhB✝ : B ≤ 0\nhB : B = 0\nx✝ : mixedSpace K\n⊢ x✝ ∈ {x | convexBodySumFun x ≤ B} ↔ x✝ ∈ {0}",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NumberField.mixedEmbedding.convexBodySumFun",
... | [
"case inr\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nhB✝ : B ≤ 0\nhB : B = 0\nx✝ : mixedSpace K\n⊢ convexBodySumFun x✝ ≤ B ↔ x✝ ∈ {0}"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 333,
"column": 11
} | {
"line": 333,
"column": 28
} | {
"line": 333,
"column": 29
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nx : K\n⊢ (mixedEmbedding K) x ∈ convexBodySum K B ↔ ∑ w, ↑w.mult * ↑w x ≤ B",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real.instLE",
"Real",
"NumberField.mixedEmbedding... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nx : K\n⊢ convexBodySumFun ((mixedEmbedding K) x) ≤ B ↔ ∑ w, ↑w.mult * ↑w x ≤ B"
] | Set.mem_setOf_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 44
} | {
"line": 357,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\n⊢ IsCompact (convexBodySum K B)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Metric.isCompact_iff_isClosed_bounded",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Pi.t2Space"... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\n⊢ IsClosed (convexBodySum K B) ∧ Bornology.IsBounded (convexBodySum K B)"
] | rw [Metric.isCompact_iff_isClosed_bounded] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 396,
"column": 73
} | {
"line": 396,
"column": 79
} | {
"line": 398,
"column": 0
} | [
{
"pp": "case inl\nk : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra k K\nw : InfinitePlace K\ninst✝ : IsGalois k K\ne : Nat.card ↥(Stab w) = 1\n⊢ ¬1 = 1 ↔ 1 = 2",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
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