module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.Modular | {
"line": 498,
"column": 4
} | {
"line": 498,
"column": 19
} | {
"line": 499,
"column": 4
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhc : ↑g 1 0 = 0\nhd' : ↑g 1 1 = 1 ∨ ↑g 1 1 = -1\nhd : ↑g 1 1 = 1\nha : ↑g 0 0 = 1\nb : ℤ := ↑g 0 1\ni j : Fin 2\n⊢ ↑g i j = ↑(T ^ b) i j",
"ppTerm": "?m.457",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhc : ↑g 1 0 = 0\nhd' : ↑g 1 1 = 1 ∨ ↑g 1 1 = -1\nhd : ↑g 1 1 = 1\nha : ↑g 0 0 = 1\nb : ℤ := ↑g 0 1\ni j : Fin 2\n⊢ ↑g i j = !![1, b; 0, 1] i j"
] | rw [coe_T_zpow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 30
} | {
"line": 242,
"column": 2
} | [
{
"pp": "Γ : Subgroup (GL (Fin 2) ℝ)\ninst✝² : Γ.IsArithmetic\nF : Type u_2\ninst✝¹ : FunLike F ℍ ℂ\nf : F\nC : ℝ\nτ : ℍ\nk : ℕ\ninst✝ : ModularFormClass F Γ ↑k\nhC : ‖f τ‖ ^ 2 ≤ C * (max τ.im (1 / τ.im) ^ ↑k / ‖↑τ.im ^ ↑k‖)\nhC' : 0 ≤ C\nh✝ : 0 < ‖↑τ.im ^ ↑k‖\nt : ℝ\nh : τ.im = t\nht : 0 < t\n⊢ max t (1 / t) ^... | [
"Γ : Subgroup (GL (Fin 2) ℝ)\ninst✝² : Γ.IsArithmetic\nF : Type u_2\ninst✝¹ : FunLike F ℍ ℂ\nf : F\nC : ℝ\nτ : ℍ\nk : ℕ\ninst✝ : ModularFormClass F Γ ↑k\nhC : ‖f τ‖ ^ 2 ≤ C * (max τ.im (1 / τ.im) ^ ↑k / ‖↑τ.im ^ ↑k‖)\nhC' : 0 ≤ C\nh✝ : 0 < ‖↑τ.im ^ ↑k‖\nt : NNReal\nh : τ.im = ↑t\nht : 0 < ↑t\n⊢ max (↑t) (1 / ↑t) ^ ... | lift t to NNReal using ht.le | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 137,
"column": 6
} | {
"line": 138,
"column": 43
} | {
"line": 139,
"column": 2
} | [
{
"pp": "z : ℍ\nt : ℂ := ∑' (m : ℤ) (n : ℤ), G2Term z ![m, n]\n⊢ Summable fun b ↦\n ∑' (n : ℤ), G2Term z ![b, n] + ∑'[symmetricIco ℤ] (n : ℤ), (1 / (↑b * ↑z + ↑n) - 1 / (↑b * ↑z + ↑n + 1))",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.ModularForms.... | [] | simpa only [tsum_symmetricIco_linear_sub_linear_add_one_eq_zero z, add_zero]
using (G2Term_prod_summable z).prod | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 137,
"column": 6
} | {
"line": 138,
"column": 43
} | {
"line": 139,
"column": 2
} | [
{
"pp": "z : ℍ\nt : ℂ := ∑' (m : ℤ) (n : ℤ), G2Term z ![m, n]\n⊢ Summable fun b ↦\n ∑' (n : ℤ), G2Term z ![b, n] + ∑'[symmetricIco ℤ] (n : ℤ), (1 / (↑b * ↑z + ↑n) - 1 / (↑b * ↑z + ↑n + 1))",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.ModularForms.... | [] | simpa only [tsum_symmetricIco_linear_sub_linear_add_one_eq_zero z, add_zero]
using (G2Term_prod_summable z).prod | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 137,
"column": 6
} | {
"line": 138,
"column": 43
} | {
"line": 139,
"column": 2
} | [
{
"pp": "z : ℍ\nt : ℂ := ∑' (m : ℤ) (n : ℤ), G2Term z ![m, n]\n⊢ Summable fun b ↦\n ∑' (n : ℤ), G2Term z ![b, n] + ∑'[symmetricIco ℤ] (n : ℤ), (1 / (↑b * ↑z + ↑n) - 1 / (↑b * ↑z + ↑n + 1))",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.ModularForms.... | [] | simpa only [tsum_symmetricIco_linear_sub_linear_add_one_eq_zero z, add_zero]
using (G2Term_prod_summable z).prod | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Modular | {
"line": 678,
"column": 58
} | {
"line": 678,
"column": 64
} | {
"line": 678,
"column": 64
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(T⁻¹ * S) 1 0\nhc : ↑(T⁻¹ * S) 1 0 = 1\nhd : ↑(T⁻¹ * S) 1 1 = 0\nhz : ρ ∈ 𝒟\nhg : (T⁻¹ * S) • ρ ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (T⁻¹ * S))) ↑ρ‖ ≤ 1\n⊢ T⁻¹ * S = ⟨!![-1, -1; 1, 0], ⋯⟩",
"ppTerm": "?m.625"... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.Modular | {
"line": 678,
"column": 58
} | {
"line": 678,
"column": 64
} | {
"line": 678,
"column": 64
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(T⁻¹ * S) 1 0\nhc : ↑(T⁻¹ * S) 1 0 = 1\nhd : ↑(T⁻¹ * S) 1 1 = 0\nhz : ρ ∈ 𝒟\nhg : (T⁻¹ * S) • ρ ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (T⁻¹ * S))) ↑ρ‖ ≤ 1\n⊢ T⁻¹ * S = ⟨!![-1, -1; 1, 0], ⋯⟩",
"ppTerm": "?m.625"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Modular | {
"line": 678,
"column": 58
} | {
"line": 678,
"column": 64
} | {
"line": 678,
"column": 64
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(T⁻¹ * S) 1 0\nhc : ↑(T⁻¹ * S) 1 0 = 1\nhd : ↑(T⁻¹ * S) 1 1 = 0\nhz : ρ ∈ 𝒟\nhg : (T⁻¹ * S) • ρ ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (T⁻¹ * S))) ↑ρ‖ ≤ 1\n⊢ T⁻¹ * S = ⟨!![-1, -1; 1, 0], ⋯⟩",
"ppTerm": "?m.625"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Modular | {
"line": 680,
"column": 55
} | {
"line": 680,
"column": 61
} | {
"line": 680,
"column": 61
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(T * S) 1 0\nhc : ↑(T * S) 1 0 = 1\nhd : ↑(T * S) 1 1 = 0\nhz : 1 +ᵥ ρ ∈ 𝒟\nhg : (T * S) • (1 +ᵥ ρ) ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (T * S))) ↑(1 +ᵥ ρ)‖ ≤ 1\n⊢ T * S = ⟨!![1, -1; 1, 0], ⋯⟩",
"ppTerm": "?... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.Modular | {
"line": 680,
"column": 55
} | {
"line": 680,
"column": 61
} | {
"line": 680,
"column": 61
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(T * S) 1 0\nhc : ↑(T * S) 1 0 = 1\nhd : ↑(T * S) 1 1 = 0\nhz : 1 +ᵥ ρ ∈ 𝒟\nhg : (T * S) • (1 +ᵥ ρ) ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (T * S))) ↑(1 +ᵥ ρ)‖ ≤ 1\n⊢ T * S = ⟨!![1, -1; 1, 0], ⋯⟩",
"ppTerm": "?... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Modular | {
"line": 680,
"column": 55
} | {
"line": 680,
"column": 61
} | {
"line": 680,
"column": 61
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(T * S) 1 0\nhc : ↑(T * S) 1 0 = 1\nhd : ↑(T * S) 1 1 = 0\nhz : 1 +ᵥ ρ ∈ 𝒟\nhg : (T * S) • (1 +ᵥ ρ) ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (T * S))) ↑(1 +ᵥ ρ)‖ ≤ 1\n⊢ T * S = ⟨!![1, -1; 1, 0], ⋯⟩",
"ppTerm": "?... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Modular | {
"line": 683,
"column": 55
} | {
"line": 683,
"column": 61
} | {
"line": 683,
"column": 61
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(S * T) 1 0\nhc : ↑(S * T) 1 0 = 1\nhd : ↑(S * T) 1 1 = 1\nhz : ρ ∈ 𝒟\nhg : (S * T) • ρ ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (S * T))) ↑ρ‖ ≤ 1\n⊢ S * T = ⟨!![0, -1; 1, 1], ⋯⟩",
"ppTerm": "?m.768",
"assign... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.Modular | {
"line": 683,
"column": 55
} | {
"line": 683,
"column": 61
} | {
"line": 683,
"column": 61
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(S * T) 1 0\nhc : ↑(S * T) 1 0 = 1\nhd : ↑(S * T) 1 1 = 1\nhz : ρ ∈ 𝒟\nhg : (S * T) • ρ ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (S * T))) ↑ρ‖ ≤ 1\n⊢ S * T = ⟨!![0, -1; 1, 1], ⋯⟩",
"ppTerm": "?m.768",
"assign... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Modular | {
"line": 683,
"column": 55
} | {
"line": 683,
"column": 61
} | {
"line": 683,
"column": 61
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(S * T) 1 0\nhc : ↑(S * T) 1 0 = 1\nhd : ↑(S * T) 1 1 = 1\nhz : ρ ∈ 𝒟\nhg : (S * T) • ρ ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (S * T))) ↑ρ‖ ≤ 1\n⊢ S * T = ⟨!![0, -1; 1, 1], ⋯⟩",
"ppTerm": "?m.768",
"assign... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Modular | {
"line": 683,
"column": 8
} | {
"line": 683,
"column": 62
} | {
"line": 684,
"column": 8
} | [
{
"pp": "case inr.inr.inl.inl\ng✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(S * T) 1 0\nhc : ↑(S * T) 1 0 = 1\nhd : ↑(S * T) 1 1 = 1\nhz : ρ ∈ 𝒟\nhg : (S * T) • ρ ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (S * T))) ↑ρ‖ ≤ 1\n⊢ normSq (denom (toGL ((SpecialLinearGroup.map... | [
"case inr.inr.inl.inl\ng✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(S * T) 1 0\nhc : ↑(S * T) 1 0 = 1\nhd : ↑(S * T) 1 1 = 1\nhz : ρ ∈ 𝒟\nhg : (S * T) • ρ ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (S * T))) ↑ρ‖ ≤ 1\n⊢ normSq (denom (toGL ((SpecialLinearGroup.map (Int.castRi... | rw [show S * T = ⟨!![0, -1; 1, 1], by simp⟩ by decide] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Modular | {
"line": 685,
"column": 58
} | {
"line": 685,
"column": 64
} | {
"line": 685,
"column": 64
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(T * S * T) 1 0\nhc : ↑(T * S * T) 1 0 = 1\nhd : ↑(T * S * T) 1 1 = 1\nhz : ρ ∈ 𝒟\nhg : (T * S * T) • ρ ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (T * S * T))) ↑ρ‖ ≤ 1\n⊢ T * S * T = ⟨!![1, 0; 1, 1], ⋯⟩",
"ppTerm"... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.Modular | {
"line": 685,
"column": 58
} | {
"line": 685,
"column": 64
} | {
"line": 685,
"column": 64
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(T * S * T) 1 0\nhc : ↑(T * S * T) 1 0 = 1\nhd : ↑(T * S * T) 1 1 = 1\nhz : ρ ∈ 𝒟\nhg : (T * S * T) • ρ ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (T * S * T))) ↑ρ‖ ≤ 1\n⊢ T * S * T = ⟨!![1, 0; 1, 1], ⋯⟩",
"ppTerm"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Modular | {
"line": 685,
"column": 58
} | {
"line": 685,
"column": 64
} | {
"line": 685,
"column": 64
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(T * S * T) 1 0\nhc : ↑(T * S * T) 1 0 = 1\nhd : ↑(T * S * T) 1 1 = 1\nhz : ρ ∈ 𝒟\nhg : (T * S * T) • ρ ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (T * S * T))) ↑ρ‖ ≤ 1\n⊢ T * S * T = ⟨!![1, 0; 1, 1], ⋯⟩",
"ppTerm"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Modular | {
"line": 688,
"column": 58
} | {
"line": 688,
"column": 64
} | {
"line": 688,
"column": 64
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(S * T⁻¹) 1 0\nhc : ↑(S * T⁻¹) 1 0 = 1\nhd : ↑(S * T⁻¹) 1 1 = -1\nhz : 1 +ᵥ ρ ∈ 𝒟\nhg : (S * T⁻¹) • (1 +ᵥ ρ) ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (S * T⁻¹))) ↑(1 +ᵥ ρ)‖ ≤ 1\n⊢ S * T⁻¹ = ⟨!![0, -1; 1, -1], ⋯⟩",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.Modular | {
"line": 688,
"column": 58
} | {
"line": 688,
"column": 64
} | {
"line": 688,
"column": 64
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(S * T⁻¹) 1 0\nhc : ↑(S * T⁻¹) 1 0 = 1\nhd : ↑(S * T⁻¹) 1 1 = -1\nhz : 1 +ᵥ ρ ∈ 𝒟\nhg : (S * T⁻¹) • (1 +ᵥ ρ) ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (S * T⁻¹))) ↑(1 +ᵥ ρ)‖ ≤ 1\n⊢ S * T⁻¹ = ⟨!![0, -1; 1, -1], ⋯⟩",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Modular | {
"line": 688,
"column": 58
} | {
"line": 688,
"column": 64
} | {
"line": 688,
"column": 64
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(S * T⁻¹) 1 0\nhc : ↑(S * T⁻¹) 1 0 = 1\nhd : ↑(S * T⁻¹) 1 1 = -1\nhz : 1 +ᵥ ρ ∈ 𝒟\nhg : (S * T⁻¹) • (1 +ᵥ ρ) ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (S * T⁻¹))) ↑(1 +ᵥ ρ)‖ ≤ 1\n⊢ S * T⁻¹ = ⟨!![0, -1; 1, -1], ⋯⟩",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Modular | {
"line": 690,
"column": 64
} | {
"line": 690,
"column": 70
} | {
"line": 690,
"column": 70
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(T⁻¹ * S * T⁻¹) 1 0\nhc : ↑(T⁻¹ * S * T⁻¹) 1 0 = 1\nhd : ↑(T⁻¹ * S * T⁻¹) 1 1 = -1\nhz : 1 +ᵥ ρ ∈ 𝒟\nhg : (T⁻¹ * S * T⁻¹) • (1 +ᵥ ρ) ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (T⁻¹ * S * T⁻¹))) ↑(1 +ᵥ ρ)‖ ≤ 1\n⊢ T⁻¹ * ... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.Modular | {
"line": 690,
"column": 64
} | {
"line": 690,
"column": 70
} | {
"line": 690,
"column": 70
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(T⁻¹ * S * T⁻¹) 1 0\nhc : ↑(T⁻¹ * S * T⁻¹) 1 0 = 1\nhd : ↑(T⁻¹ * S * T⁻¹) 1 1 = -1\nhz : 1 +ᵥ ρ ∈ 𝒟\nhg : (T⁻¹ * S * T⁻¹) • (1 +ᵥ ρ) ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (T⁻¹ * S * T⁻¹))) ↑(1 +ᵥ ρ)‖ ≤ 1\n⊢ T⁻¹ * ... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Modular | {
"line": 690,
"column": 64
} | {
"line": 690,
"column": 70
} | {
"line": 690,
"column": 70
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(T⁻¹ * S * T⁻¹) 1 0\nhc : ↑(T⁻¹ * S * T⁻¹) 1 0 = 1\nhd : ↑(T⁻¹ * S * T⁻¹) 1 1 = -1\nhz : 1 +ᵥ ρ ∈ 𝒟\nhg : (T⁻¹ * S * T⁻¹) • (1 +ᵥ ρ) ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (T⁻¹ * S * T⁻¹))) ↑(1 +ᵥ ρ)‖ ≤ 1\n⊢ T⁻¹ * ... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 277,
"column": 2
} | {
"line": 278,
"column": 63
} | {
"line": 279,
"column": 2
} | [
{
"pp": "case hf\nz : ℍ\nthis :\n ∀ (N : ℕ+),\n ∑ n ∈ Ico (-↑↑N) ↑↑N, ∑' (m : ℤ), (1 / (↑m * ↑z + ↑n) - 1 / (↑m * ↑z + ↑n + 1)) =\n -(2 / ↑↑N) +\n ∑' (m : ℕ+), (1 / (↑↑m * ↑z - ↑↑N) + 1 / (-↑↑m * ↑z + -↑↑N) - 1 / (↑↑m * ↑z + ↑↑N) - 1 / (-↑↑m * ↑z + ↑↑N))\n⊢ Tendsto (fun x ↦ -(2 / ↑↑x)) atTop (𝓝... | [
"case hg\nz : ℍ\nthis :\n ∀ (N : ℕ+),\n ∑ n ∈ Ico (-↑↑N) ↑↑N, ∑' (m : ℤ), (1 / (↑m * ↑z + ↑n) - 1 / (↑m * ↑z + ↑n + 1)) =\n -(2 / ↑↑N) +\n ∑' (m : ℕ+), (1 / (↑↑m * ↑z - ↑↑N) + 1 / (-↑↑m * ↑z + -↑↑N) - 1 / (↑↑m * ↑z + ↑↑N) - 1 / (-↑↑m * ↑z + ↑↑N))\n⊢ Tendsto\n (fun x ↦\n ∑' (m : ℕ+), (1 / (↑↑... | · simpa [← PNat.tendsto_comp_val_iff] using!
(tendsto_inv_atTop_nhds_zero_nat (𝕜 := ℂ)).const_mul (-2) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable | {
"line": 49,
"column": 71
} | {
"line": 49,
"column": 84
} | {
"line": 49,
"column": 84
} | [
{
"pp": "τ : ℍ\nh0 : ↑τ ≠ 0\nh1 : (-I * ↑τ) ^ (1 / 2) ≠ 0\n⊢ jacobiTheta₂ 0 (-↑τ)⁻¹ = (-I * ↑τ) ^ (1 / 2) * jacobiTheta₂ 0 ↑τ",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
"UpperHalfPlane.coe",
"Nat.instAtLeastTwoHAddOfNat",
"Complex... | [
"τ : ℍ\nh0 : ↑τ ≠ 0\nh1 : (-I * ↑τ) ^ (1 / 2) ≠ 0\n⊢ jacobiTheta₂ (↑0) (-↑τ)⁻¹ = (-I * ↑τ) ^ (1 / 2) * jacobiTheta₂ ↑0 ↑τ"
] | ← ofReal_zero | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Modular | {
"line": 781,
"column": 2
} | {
"line": 784,
"column": 15
} | {
"line": 785,
"column": 2
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟ᵒ\nhg : g • z ∈ 𝒟\nthis : ρ ∉ 𝒟ᵒ\n⊢ g = 1 ∨ g = -1",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"instHDiv",
"Mathl... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟ᵒ\nhg : g • z ∈ 𝒟\nthis✝ : ρ ∉ 𝒟ᵒ\nthis : 1 +ᵥ ρ ∉ 𝒟ᵒ\n⊢ g = 1 ∨ g = -1"
] | have : (1 : ℝ) +ᵥ ρ ∉ 𝒟ᵒ := by
intro h
have : ((1 : ℝ) +ᵥ ρ).re = 1 / 2 := by norm_num [← coe_re, coe_vadd, ρ]
grind [h.2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Modular | {
"line": 923,
"column": 27
} | {
"line": 923,
"column": 41
} | {
"line": 923,
"column": 42
} | [
{
"pp": "case left\nho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)\nho2 : interior 𝒟 ⊆ (fun τ ↦ ‖↑τ‖) ⁻¹' interior ((fun τ ↦ ‖↑τ‖) '' 𝒟)\nx : ℍ\nhx : x ∈ interior 𝒟\n⊢ 1 < ‖↑x‖",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr... | [
"case left\nho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)\nho2 : interior 𝒟 ⊆ (fun τ ↦ ‖↑τ‖) ⁻¹' interior ((fun τ ↦ ‖↑τ‖) '' 𝒟)\nx : ℍ\nhx : x ∈ interior 𝒟\n⊢ ‖↑x‖ ∈ Set.Ioi 1"
] | ← Set.mem_Ioi, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.LevelOne.GradedRing | {
"line": 32,
"column": 24
} | {
"line": 32,
"column": 30
} | {
"line": 32,
"column": 30
} | [
{
"pp": "⊢ ↑3 * 4 = 12",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"of_decide_eq_true",
"Int.instDecidableEq",
"id",
"instOfNatNat",
"Int",
"Nat.cast",
"Int.instMul",
"Bool.true",
"instOfNat",
"Nat",
"... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.ModularForms.LevelOne.GradedRing | {
"line": 32,
"column": 24
} | {
"line": 32,
"column": 30
} | {
"line": 32,
"column": 30
} | [
{
"pp": "⊢ ↑3 * 4 = 12",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"of_decide_eq_true",
"Int.instDecidableEq",
"id",
"instOfNatNat",
"Int",
"Nat.cast",
"Int.instMul",
"Bool.true",
"instOfNat",
"Nat",
"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.LevelOne.GradedRing | {
"line": 32,
"column": 24
} | {
"line": 32,
"column": 30
} | {
"line": 32,
"column": 30
} | [
{
"pp": "⊢ ↑3 * 4 = 12",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"of_decide_eq_true",
"Int.instDecidableEq",
"id",
"instOfNatNat",
"Int",
"Nat.cast",
"Int.instMul",
"Bool.true",
"instOfNat",
"Nat",
"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.LevelOne.GradedRing | {
"line": 32,
"column": 67
} | {
"line": 32,
"column": 73
} | {
"line": 32,
"column": 73
} | [
{
"pp": "⊢ ↑2 * 6 = 12",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"of_decide_eq_true",
"Int.instDecidableEq",
"id",
"instOfNatNat",
"Int",
"Nat.cast",
"Int.instMul",
"Bool.true",
"instOfNat",
"Nat",
"... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.ModularForms.LevelOne.GradedRing | {
"line": 32,
"column": 67
} | {
"line": 32,
"column": 73
} | {
"line": 32,
"column": 73
} | [
{
"pp": "⊢ ↑2 * 6 = 12",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"of_decide_eq_true",
"Int.instDecidableEq",
"id",
"instOfNatNat",
"Int",
"Nat.cast",
"Int.instMul",
"Bool.true",
"instOfNat",
"Nat",
"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.LevelOne.GradedRing | {
"line": 32,
"column": 67
} | {
"line": 32,
"column": 73
} | {
"line": 32,
"column": 73
} | [
{
"pp": "⊢ ↑2 * 6 = 12",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"of_decide_eq_true",
"Int.instDecidableEq",
"id",
"instOfNatNat",
"Int",
"Nat.cast",
"Int.instMul",
"Bool.true",
"instOfNat",
"Nat",
"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 57,
"column": 27
} | {
"line": 60,
"column": 55
} | {
"line": 62,
"column": 0
} | [
{
"pp": "𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝² : FunLike F ℍ ℂ\nk : ℤ\ninst✝¹ : SlashInvariantFormClass F 𝒢 k\ninst✝ : 𝒢.IsFiniteRelIndex ℋ\nh : GL (Fin 2) ℝ\nhh : h ∈ ℋ\n⊢ (let this := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ);\n ∑ q, quotientFunc f q) ∣[k]\n h =\n let this :... | [] | by
let := Fintype.ofFinite 𝒬
simpa [SlashAction.sum_slash, quotientFunc_smul f hh]
using Equiv.sum_comp (MulAction.toPerm (_ : ℋ)) _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Multiplicity | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 39
} | {
"line": 161,
"column": 2
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\nx y : R\np : ℕ\ninst✝ : IsDomain R\nhp : Prime ↑p\nhp1 : Odd p\nhxy : ↑p ∣ x - y\nhx : ¬↑p ∣ x\n⊢ ¬↑p ^ (1 + 1) ∣ ∑ i ∈ range p, x ^ i * y ^ (p - ↑1 - i)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"C... | [
"case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\nx y : R\np : ℕ\ninst✝ : IsDomain R\nhp : Prime ↑p\nhp1 : Odd p\nhxy : ↑p ∣ x - y\nhx : ¬↑p ∣ y\n⊢ ¬↑p ^ (1 + 1) ∣ ∑ i ∈ range p, x ^ i * y ^ (p - ↑1 - i)"
] | rw [dvd_iff_dvd_of_dvd_sub hxy] at hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace | {
"line": 265,
"column": 8
} | {
"line": 265,
"column": 25
} | {
"line": 266,
"column": 4
} | [
{
"pp": "case neg\nι : Type u_1\nR : ι → Type u_2\nA : (i : ι) → Set (R i)\ninst✝¹ : (i : ι) → TopologicalSpace (R i)\nS : Set ι\ninst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)\nhS : Sᶜ.Finite\nhAcompact : ∀ i ∈ S, IsCompact (A i)\nx : Πʳ (i : ι), [R i, A i]_[𝓟 S]\nK : (i : ι) → Set (R i)\nK_compact : ∀ (i... | [] | exact K_compact i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace | {
"line": 265,
"column": 8
} | {
"line": 265,
"column": 25
} | {
"line": 266,
"column": 4
} | [
{
"pp": "case neg\nι : Type u_1\nR : ι → Type u_2\nA : (i : ι) → Set (R i)\ninst✝¹ : (i : ι) → TopologicalSpace (R i)\nS : Set ι\ninst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)\nhS : Sᶜ.Finite\nhAcompact : ∀ i ∈ S, IsCompact (A i)\nx : Πʳ (i : ι), [R i, A i]_[𝓟 S]\nK : (i : ι) → Set (R i)\nK_compact : ∀ (i... | [] | exact K_compact i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace | {
"line": 265,
"column": 8
} | {
"line": 265,
"column": 25
} | {
"line": 266,
"column": 4
} | [
{
"pp": "case neg\nι : Type u_1\nR : ι → Type u_2\nA : (i : ι) → Set (R i)\ninst✝¹ : (i : ι) → TopologicalSpace (R i)\nS : Set ι\ninst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)\nhS : Sᶜ.Finite\nhAcompact : ∀ i ∈ S, IsCompact (A i)\nx : Πʳ (i : ι), [R i, A i]_[𝓟 S]\nK : (i : ι) → Set (R i)\nK_compact : ∀ (i... | [] | exact K_compact i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Multiplicity | {
"line": 245,
"column": 4
} | {
"line": 248,
"column": 8
} | {
"line": 250,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : CommRing R\nx y : R\nd : ℕ\nhd : x ^ 2 ^ d - y ^ 2 ^ d = (∏ i ∈ range d, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)\n⊢ x ^ 2 ^ (d + 1) - y ^ 2 ^ (d + 1) = (∏ i ∈ range (d + 1), (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants"... | [] | suffices x ^ 2 ^ d.succ - y ^ 2 ^ d.succ = (x ^ 2 ^ d + y ^ 2 ^ d) * (x ^ 2 ^ d - y ^ 2 ^ d) by
rw [this, hd, Finset.prod_range_succ, ← mul_assoc, mul_comm (x ^ 2 ^ d + y ^ 2 ^ d)]
rw [Nat.succ_eq_add_one]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Multiplicity | {
"line": 245,
"column": 4
} | {
"line": 248,
"column": 8
} | {
"line": 250,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : CommRing R\nx y : R\nd : ℕ\nhd : x ^ 2 ^ d - y ^ 2 ^ d = (∏ i ∈ range d, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)\n⊢ x ^ 2 ^ (d + 1) - y ^ 2 ^ (d + 1) = (∏ i ∈ range (d + 1), (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants"... | [] | suffices x ^ 2 ^ d.succ - y ^ 2 ^ d.succ = (x ^ 2 ^ d + y ^ 2 ^ d) * (x ^ 2 ^ d - y ^ 2 ^ d) by
rw [this, hd, Finset.prod_range_succ, ← mul_assoc, mul_comm (x ^ 2 ^ d + y ^ 2 ^ d)]
rw [Nat.succ_eq_add_one]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Multiplicity | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 8
} | {
"line": 258,
"column": 0
} | [
{
"pp": "w✝ : ℤ\n⊢ 1 % 4 = 1",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"of_decide_eq_true",
"CommSemiring.toSemiring",
"Nat.instAtLeastTwoHAddOfNat",
"Int.instDecidableEq",
"id",
"instHMod",
"A... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.Multiplicity | {
"line": 271,
"column": 70
} | {
"line": 271,
"column": 76
} | {
"line": 271,
"column": 76
} | [
{
"pp": "x y : ℤ\nhx : ¬2 ∣ x\nhxy : 4 ∣ x - y\ni : ℕ\nhx_odd : Odd x\n⊢ 2 ∣ 4",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"Int.decidableDvd",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dvd.dvd",
"of_decide_eq_true",
"Nat.instAtLeastTwoHAddOfNat",
"... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.Multiplicity | {
"line": 271,
"column": 70
} | {
"line": 271,
"column": 76
} | {
"line": 271,
"column": 76
} | [
{
"pp": "x y : ℤ\nhx : ¬2 ∣ x\nhxy : 4 ∣ x - y\ni : ℕ\nhx_odd : Odd x\n⊢ 2 ∣ 4",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"Int.decidableDvd",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dvd.dvd",
"of_decide_eq_true",
"Nat.instAtLeastTwoHAddOfNat",
"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Multiplicity | {
"line": 271,
"column": 70
} | {
"line": 271,
"column": 76
} | {
"line": 271,
"column": 76
} | [
{
"pp": "x y : ℤ\nhx : ¬2 ∣ x\nhxy : 4 ∣ x - y\ni : ℕ\nhx_odd : Odd x\n⊢ 2 ∣ 4",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"Int.decidableDvd",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dvd.dvd",
"of_decide_eq_true",
"Nat.instAtLeastTwoHAddOfNat",
"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Multiplicity | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 10
} | {
"line": 282,
"column": 2
} | [
{
"pp": "x y : ℤ\nhx : ¬2 ∣ x\nhxy : 4 ∣ x - y\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\ni : ℕ\nthis : ∀ (x : ℤ), Odd x → x ^ 2 ^ (i + 1) % 4 = 1\n⊢ ¬(1 + 1) % 4 = 0",
"ppTerm": "?m.243",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.Multiplicity | {
"line": 294,
"column": 70
} | {
"line": 294,
"column": 76
} | {
"line": 294,
"column": 76
} | [
{
"pp": "x y : ℤ\nn : ℕ\nhxy : 4 ∣ x - y\nhx : ¬2 ∣ x\nhx_odd : Odd x\n⊢ 2 ∣ 4",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"Int.decidableDvd",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dvd.dvd",
"of_decide_eq_true",
"Nat.instAtLeastTwoHAddOfNat",
"... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.Multiplicity | {
"line": 294,
"column": 70
} | {
"line": 294,
"column": 76
} | {
"line": 294,
"column": 76
} | [
{
"pp": "x y : ℤ\nn : ℕ\nhxy : 4 ∣ x - y\nhx : ¬2 ∣ x\nhx_odd : Odd x\n⊢ 2 ∣ 4",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"Int.decidableDvd",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dvd.dvd",
"of_decide_eq_true",
"Nat.instAtLeastTwoHAddOfNat",
"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Multiplicity | {
"line": 294,
"column": 70
} | {
"line": 294,
"column": 76
} | {
"line": 294,
"column": 76
} | [
{
"pp": "x y : ℤ\nn : ℕ\nhxy : 4 ∣ x - y\nhx : ¬2 ∣ x\nhx_odd : Odd x\n⊢ 2 ∣ 4",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"Int.decidableDvd",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dvd.dvd",
"of_decide_eq_true",
"Nat.instAtLeastTwoHAddOfNat",
"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.RestrictedProduct.Basic | {
"line": 529,
"column": 2
} | {
"line": 529,
"column": 30
} | {
"line": 531,
"column": 0
} | [
{
"pp": "ι : Type u_1\nS : ι → Type u_3\nG : ι → Type u_4\ninst✝⁴ : (i : ι) → SetLike (S i) (G i)\nA : (i : ι) → S i\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → MulOneClass (G i)\ninst✝¹ : ∀ (i : ι), OneMemClass (S i) (G i)\ninst✝ : ∀ (i : ι), MulMemClass (S i) (G i)\ni : ι\nr s : G i\n⊢ mulSingle A i (r * s) =... | [] | ext; simp [Pi.mulSingle_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.RestrictedProduct.Basic | {
"line": 529,
"column": 2
} | {
"line": 529,
"column": 30
} | {
"line": 531,
"column": 0
} | [
{
"pp": "ι : Type u_1\nS : ι → Type u_3\nG : ι → Type u_4\ninst✝⁴ : (i : ι) → SetLike (S i) (G i)\nA : (i : ι) → S i\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → MulOneClass (G i)\ninst✝¹ : ∀ (i : ι), OneMemClass (S i) (G i)\ninst✝ : ∀ (i : ι), MulMemClass (S i) (G i)\ni : ι\nr s : G i\n⊢ mulSingle A i (r * s) =... | [] | ext; simp [Pi.mulSingle_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.FiniteAdeleRing | {
"line": 60,
"column": 8
} | {
"line": 60,
"column": 25
} | {
"line": 60,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nk : K\nn d : R\nhd : d ∈ nonZeroDivisors R\nhd' : d ≠ 0\nthis : {v | (valuation K v) ((algebraMap R K) d) < 1}.Finite\nv : HeightOneSpectrum R\nhv : v ∈ Supp... | [
"R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nk : K\nn d : R\nhd : d ∈ nonZeroDivisors R\nhd' : d ≠ 0\nthis : {v | (valuation K v) ((algebraMap R K) d) < 1}.Finite\nv : HeightOneSpectrum R\nhv : v ∈ Support R k\nhk ... | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 336,
"column": 4
} | {
"line": 336,
"column": 10
} | {
"line": 337,
"column": 2
} | [
{
"pp": "x y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ 2 ≠ 1 ∧ 0 < 2",
"ppTerm": "?m.286",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"Ne",
"instOfNatNat",
"Bool.true",
"And",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.Multiplicity | {
"line": 337,
"column": 4
} | {
"line": 339,
"column": 75
} | {
"line": 341,
"column": 0
} | [
{
"pp": "x y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ ¬2 ∣ x ^ 2",
"ppTerm": "?m.228",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"Dvd.dvd",
"eq_false",
"congrArg",
... | [] | rw [← even_iff_two_dvd, Int.not_even_iff_odd]
apply Odd.pow
simp only [← Int.not_even_iff_odd, even_iff_two_dvd, hx, not_false_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Multiplicity | {
"line": 337,
"column": 4
} | {
"line": 339,
"column": 75
} | {
"line": 341,
"column": 0
} | [
{
"pp": "x y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ ¬2 ∣ x ^ 2",
"ppTerm": "?m.228",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"Dvd.dvd",
"eq_false",
"congrArg",
... | [] | rw [← even_iff_two_dvd, Int.not_even_iff_odd]
apply Odd.pow
simp only [← Int.not_even_iff_odd, even_iff_two_dvd, hx, not_false_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 219,
"column": 8
} | {
"line": 219,
"column": 25
} | {
"line": 219,
"column": 26
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\nc : ℝ\nhx : x ∈ fundamentalCone K\nhc : c ≠ 0\n⊢ c • x ∉ {x | mixedEmbedding.norm x = 0}",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCo... | [
"case refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\nc : ℝ\nhx : x ∈ fundamentalCone K\nhc : c ≠ 0\n⊢ ¬mixedEmbedding.norm (c • x) = 0"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 242,
"column": 8
} | {
"line": 242,
"column": 25
} | {
"line": 242,
"column": 26
} | [
{
"pp": "case right\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\nhx : x ∈ fundamentalCone K\nζ : (𝓞 K)ˣ\nhζ : ζ ∈ torsion K\n⊢ ζ • x ∉ {x | mixedEmbedding.norm x = 0}",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHSM... | [
"case right\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\nhx : x ∈ fundamentalCone K\nζ : (𝓞 K)ˣ\nhζ : ζ ∈ torsion K\n⊢ ¬mixedEmbedding.norm (ζ • x) = 0"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CMField | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 40
} | {
"line": 192,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : CharZero K\ninst✝¹ : IsCMField K\ninst✝ : Algebra.IsIntegral ℚ K\n⊢ Subgroup.zpowers (complexConj K) = ⊤",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Subgroup.instFiniteSubtypeMem",
"Subgroup.eq_top_of_card_eq",
"Subse... | [
"K : Type u_1\ninst✝³ : Field K\ninst✝² : CharZero K\ninst✝¹ : IsCMField K\ninst✝ : Algebra.IsIntegral ℚ K\n⊢ Nat.card ↥(Subgroup.zpowers (complexConj K)) = Nat.card Gal(K/↥K⁺)"
] | refine Subgroup.eq_top_of_card_eq _ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 426,
"column": 17
} | {
"line": 426,
"column": 49
} | {
"line": 426,
"column": 49
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx y : mixedSpace K\nc : ℝ\na✝ : Quotient (MulAction.orbitRel ↥(Units.torsion K) ↑(integerSet K))\n⊢ ↥(Units.torsion K) ⧸ MulAction.stabilizer (↥(Units.torsion K)) a✝.out ≃ ↥(Units.torsion K)",
"ppTerm": "?m.46",
"assigned": true,
"usedC... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx y : mixedSpace K\nc : ℝ\na✝ : Quotient (MulAction.orbitRel ↥(Units.torsion K) ↑(integerSet K))\n⊢ ↥(Units.torsion K) ⧸ ⊥ ≃ ↥(Units.torsion K)"
] | integerSetTorsionSMul_stabilizer | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 21
} | {
"line": 153,
"column": 22
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\n⊢ logMap x ∈ ZSpan.fundamentalDomain (Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K)) ∧\n mixedSpaceOfRealSpace (normAtAllPlaces x) ∉ {x | mixedEmbedding.norm x = 0} ↔\n logMap x ∈ ZSpan.fundamentalDomain (Ba... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\n⊢ logMap x ∈ ZSpan.fundamentalDomain (Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K)) ∧\n ¬mixedEmbedding.norm (mixedSpaceOfRealSpace (normAtAllPlaces x)) = 0 ↔\n logMap x ∈ ZSpan.fundamentalDomain (Basis.ofZLatticeBasis... | Set.mem_setOf_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 198,
"column": 11
} | {
"line": 198,
"column": 28
} | {
"line": 198,
"column": 29
} | [
{
"pp": "case refine_1.refine_3\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ny : mixedSpace K\nh₃ : y ∈ {x | mixedEmbedding.norm x ≤ 1}\nh₁ : y ∈ logMap ⁻¹' ZSpan.fundamentalDomain (Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K))\nh₂ : y ∉ {x | mixedEmbedding.norm x = 0}\n⊢ normAtAllPla... | [
"case refine_1.refine_3\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ny : mixedSpace K\nh₃ : y ∈ {x | mixedEmbedding.norm x ≤ 1}\nh₁ : y ∈ logMap ⁻¹' ZSpan.fundamentalDomain (Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K))\nh₂ : ¬mixedEmbedding.norm y = 0\n⊢ normAtAllPlaces y ∈ {x | mixedEmb... | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 199,
"column": 11
} | {
"line": 199,
"column": 28
} | {
"line": 199,
"column": 29
} | [
{
"pp": "case refine_1.refine_4\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ny : mixedSpace K\nh₃ : y ∈ {x | mixedEmbedding.norm x ≤ 1}\nh₁ : y ∈ logMap ⁻¹' ZSpan.fundamentalDomain (Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K))\nh₂ : y ∉ {x | mixedEmbedding.norm x = 0}\n⊢ normAtAllPla... | [
"case refine_1.refine_4\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ny : mixedSpace K\nh₃ : mixedEmbedding.norm y ≤ 1\nh₁ : y ∈ logMap ⁻¹' ZSpan.fundamentalDomain (Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K))\nh₂ : y ∉ {x | mixedEmbedding.norm x = 0}\n⊢ normAtAllPlaces y ∈ {x | mixedEmbe... | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CMField | {
"line": 413,
"column": 2
} | {
"line": 415,
"column": 37
} | {
"line": 416,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : CharZero K\ninst✝¹ : IsCMField K\ninst✝ : NumberField K\n⊢ Subgroup.closure (Set.range (realFundSystem K)) ⊔ torsion K = realUnits K ⊔ torsion K",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"RingHom.instRingHomClass",
"Monoid... | [
"K : Type u_1\ninst✝³ : Field K\ninst✝² : CharZero K\ninst✝¹ : IsCMField K\ninst✝ : NumberField K\nthis : Subgroup.map (Units.map ↑(algebraMap (𝓞 ↥K⁺) (𝓞 K))) (torsion ↥K⁺) ≤ torsion K\n⊢ Subgroup.closure (Set.range (realFundSystem K)) ⊔ torsion K = realUnits K ⊔ torsion K"
] | have : Subgroup.map (Units.map (algebraMap (𝓞 K⁺) (𝓞 K))) (torsion K⁺) ≤ torsion K := by
rintro _ ⟨x, hx, rfl⟩
exact MonoidHom.isOfFinOrder _ hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 247,
"column": 68
} | {
"line": 247,
"column": 80
} | {
"line": 247,
"column": 81
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\nw' : InfinitePlace K\ne : { w // w ≠ w' } ≃ Fin (rank K)\nf : Fin (rank K + 1) ≃ InfinitePlace K := (finSuccEquiv (rank K)).trans ↑((Equiv.optionSubtype w').symm e.symm)\ng : { w // w ≠ w' } ≃ Fin (rank K) := (f.symm.sub... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\nw' : InfinitePlace K\ne : { w // w ≠ w' } ≃ Fin (rank K)\nf : Fin (rank K + 1) ≃ InfinitePlace K := (finSuccEquiv (rank K)).trans ↑((Equiv.optionSubtype w').symm e.symm)\ng : { w // w ≠ w' } ≃ Fin (rank K) := (f.symm.subtypeEquiv ⋯)... | f.symm_symm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 381,
"column": 20
} | {
"line": 389,
"column": 41
} | {
"line": 391,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx : x ∈ Submodule.span ℝ (Set.range fun w ↦ completeFamily K ↑w)\n⊢ ∑ w, x w = 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Units.val",
"Eq.mpr",
"Pi.Function.mo... | [] | by
induction hx using Submodule.span_induction with
| mem _ h =>
obtain ⟨w, rfl⟩ := h
simp_rw [completeFamily, dif_neg w.prop, sum_expMap_symm_apply (coe_ne_zero _),
Units.norm, Rat.cast_one, Real.log_one]
| zero => simp
| add _ _ _ _ hx hy => simp [sum_add_distrib, hx, hy]
| smul _ _ _ hx... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CMField | {
"line": 442,
"column": 60
} | {
"line": 443,
"column": 23
} | {
"line": 443,
"column": 24
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : CharZero K\ninst✝¹ : IsCMField K\ninst✝ : NumberField K\n⊢ regulator K / regulator ↥K⁺ = 2 ^ rank K * (↑(Subgroup.closure (Set.range (realFundSystem K)) ⊔ torsion K).index)⁻¹",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"K : Type u_1\ninst✝³ : Field K\ninst✝² : CharZero K\ninst✝¹ : IsCMField K\ninst✝ : NumberField K\n⊢ regulator K / regulator ↥K⁺ = 2 ^ rank K * (regOfFamily (realFundSystem K) / regulator K)⁻¹"
] | ← regOfFamily_div_regulator
(realFundSystem K), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 255,
"column": 16
} | {
"line": 255,
"column": 28
} | {
"line": 255,
"column": 29
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\nw' : InfinitePlace K\ne : { w // w ≠ w' } ≃ Fin (rank K)\nf : Fin (rank K + 1) ≃ InfinitePlace K := (finSuccEquiv (rank K)).trans ↑((Equiv.optionSubtype w').symm e.symm)\ng : { w // w ≠ w' } ≃ Fin (rank K) := (f.symm.sub... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\nw' : InfinitePlace K\ne : { w // w ≠ w' } ≃ Fin (rank K)\nf : Fin (rank K + 1) ≃ InfinitePlace K := (finSuccEquiv (rank K)).trans ↑((Equiv.optionSubtype w').symm e.symm)\ng : { w // w ≠ w' } ≃ Fin (rank K) := (f.symm.subtypeEquiv ⋯)... | f.symm_symm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 555,
"column": 2
} | {
"line": 555,
"column": 24
} | {
"line": 556,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ns : Set (realSpace K)\n⊢ ∀ x ∈ ↑expMapBasis '' s, ∀ (w : InfinitePlace K), 0 ≤ x w",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"NumberField.mixedEmbedding.realSpace",
"Pi.topo... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ns : Set (realSpace K)\nx : realSpace K\nleft✝ : x ∈ s\nw : InfinitePlace K\n⊢ 0 ≤ ↑expMapBasis x w"
] | rintro _ ⟨x, _, rfl⟩ w | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.NumberTheory.NumberField.Ideal.KummerDedekind | {
"line": 246,
"column": 4
} | {
"line": 254,
"column": 75
} | {
"line": 256,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nθ : 𝓞 K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : NumberField K\nhp : ¬p ∣ exponent θ\nQ : ℤ[X]\nhQ : Polynomial.map (Int.castRingHom (ZMod p)) Q ∈ monicFactorsMod θ p\n⊢ multiplicity (↑((primesOverSpanEquivMonicFactorsMod hp).symm ⟨Polynomial.map (Int.castRingHom (Z... | [] | apply multiplicity_eq_of_emultiplicity_eq
rw [← emultiplicity_map_eq (mapEquiv (Int.quotientSpanNatEquivZMod p).symm),
emultiplicity_factors_map_eq_emultiplicity inferInstance (by simp [NeZero.ne p])
(not_dvd_exponent_iff.mp hp).eq_top θ.isIntegral]
· simp only [primesOverSpanEquivMonicFactorsMod_sy... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Ideal.KummerDedekind | {
"line": 246,
"column": 4
} | {
"line": 254,
"column": 75
} | {
"line": 256,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nθ : 𝓞 K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : NumberField K\nhp : ¬p ∣ exponent θ\nQ : ℤ[X]\nhQ : Polynomial.map (Int.castRingHom (ZMod p)) Q ∈ monicFactorsMod θ p\n⊢ multiplicity (↑((primesOverSpanEquivMonicFactorsMod hp).symm ⟨Polynomial.map (Int.castRingHom (Z... | [] | apply multiplicity_eq_of_emultiplicity_eq
rw [← emultiplicity_map_eq (mapEquiv (Int.quotientSpanNatEquivZMod p).symm),
emultiplicity_factors_map_eq_emultiplicity inferInstance (by simp [NeZero.ne p])
(not_dvd_exponent_iff.mp hp).eq_top θ.isIntegral]
· simp only [primesOverSpanEquivMonicFactorsMod_sy... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | {
"line": 66,
"column": 2
} | {
"line": 67,
"column": 58
} | {
"line": 69,
"column": 0
} | [
{
"pp": "n : ℕ\ninst✝² : NeZero n\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {n} ℚ K\nσ : Gal(K/ℚ)\na : ℕ\nhx : (zeta n ℚ K ^ a) ^ n = 1\n⊢ σ (zeta n ℚ K ^ a) = (zeta n ℚ K ^ a) ^ (↑((galEquivZMod n K) σ)).val",
"ppTerm": "?m.74",
"assigned": true,
"usedConsta... | [] | rw [map_pow, pow_right_comm, galEquivZMod, autEquivPow_apply, OneHom.toFun_eq_coe,
MonoidHom.toOneHom_coe, IsPrimitiveRoot.autToPow_spec] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 736,
"column": 2
} | {
"line": 737,
"column": 72
} | {
"line": 739,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ 0 ∈ compactSet K",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.image_nonempty",
"Eq.mpr",
"Pi.Function.module",
"Decidable.casesOn",
"Real",
"NumberField.mixedEmbedd... | [] | refine Set.zero_mem_smul_iff.mpr (Or.inl ⟨Set.left_mem_Icc.mpr zero_le_one, ?_⟩)
exact Set.image_nonempty.mpr (Set.univ_pi_nonempty_iff.mpr (by aesop)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 736,
"column": 2
} | {
"line": 737,
"column": 72
} | {
"line": 739,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ 0 ∈ compactSet K",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.image_nonempty",
"Eq.mpr",
"Pi.Function.module",
"Decidable.casesOn",
"Real",
"NumberField.mixedEmbedd... | [] | refine Set.zero_mem_smul_iff.mpr (Or.inl ⟨Set.left_mem_Icc.mpr zero_le_one, ?_⟩)
exact Set.image_nonempty.mpr (Set.univ_pi_nonempty_iff.mpr (by aesop)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 755,
"column": 4
} | {
"line": 755,
"column": 43
} | {
"line": 756,
"column": 2
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nc : ℝ\nhc : c ∈ Set.Icc 0 1\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then {0} else Set.Icc 0 1\nhx₀ : (fun x1 x2 ↦ x1 • x2) c (↑expMapBasis y) ≠ 0\nw : InfinitePlace K\nh : ¬w = w₀\n⊢ y w ∈ Set.Icc 0 1",
"ppTerm": "?ne... | [] | · simpa [h] using hy w (Set.mem_univ _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 778,
"column": 4
} | {
"line": 778,
"column": 43
} | {
"line": 780,
"column": 0
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then Set.Iic 0 else Set.Icc 0 1\nhx₀ : ↑expMapBasis y ≠ 0\nw : InfinitePlace K\nh : ¬w = w₀\n⊢ y w ∈ Set.Icc 0 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [... | [] | · simpa [h] using hy w (Set.mem_univ _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 102,
"column": 2
} | {
"line": 103,
"column": 30
} | {
"line": 105,
"column": 0
} | [
{
"pp": "p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nthis : (span {hζ.toInteger - 1}).IsMaximal\n⊢ (span {hζ.toInteger - 1}).inertiaDeg ℤ = 1",
"ppTerm": "?m.101",
"ass... | [] | rw [← Nat.pow_right_inj hp.out.one_lt, pow_one, pow_inertiaDeg,
absNorm_span_zeta_sub_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Ideal.Asymptotics | {
"line": 100,
"column": 2
} | {
"line": 105,
"column": 14
} | {
"line": 106,
"column": 2
} | [
{
"pp": "case e'_3\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nh₁ :\n ∀ (s : ℝ),\n {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =\n ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}\nh₂ : {x | x ∈ fundam... | [
"case e'_5\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nh₁ :\n ∀ (s : ℝ),\n {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =\n ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}\nh₂ : {x | x ∈ fundamentalCone K ... | · simp_rw [Ideal.tendsto_norm_le_and_mk_eq_div_atTop_aux₁ K hJ, id_eq,
Nat.card_congr (Ideal.tendsto_norm_le_and_mk_eq_div_atTop_aux₂ K),
← card_isPrincipal_dvd_norm_le, Function.comp_def, Nat.cast_mul, div_eq_mul_inv, mul_inv,
← mul_assoc, mul_comm _ (torsionOrder K : ℝ)⁻¹, mul_comm _ (torsionOrder K... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.House | {
"line": 232,
"column": 2
} | {
"line": 232,
"column": 10
} | {
"line": 232,
"column": 10
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\nα : Type u_2\nβ : Type u_3\na : Matrix α β (𝓞 K)\nx : β × (K →+* ℂ) → ℤ\nhxl : x ≠ 0\ninst✝ : Fintype β\nhmulvec0 : asiegel K a *ᵥ x = 0\n⊢ a *ᵥ ξ K x = 0",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"NonUnitalCommR... | [
"K : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\nα : Type u_2\nβ : Type u_3\na : Matrix α β (𝓞 K)\nx : β × (K →+* ℂ) → ℤ\nhxl : x ≠ 0\ninst✝ : Fintype β\nhmulvec0 : asiegel K a *ᵥ x = 0\nk : α\n⊢ (a *ᵥ ξ K x) k = 0 k"
] | funext k | _aux_Init_NotationExtra___macroRules_tacticFunext____1 | tacticFunext___ |
Mathlib.NumberTheory.NumberField.House | {
"line": 293,
"column": 4
} | {
"line": 296,
"column": 28
} | {
"line": 297,
"column": 4
} | [
{
"pp": "case calc_4\nK : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\nα : Type u_2\nβ : Type u_3\na : Matrix α β (𝓞 K)\ninst✝² : Fintype β\nA : ℝ\nhabs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A\ninst✝¹ : DecidableEq (K →+* ℂ)\ninst✝ : Fintype α\nApos : 0 ≤ A\nthis : 0 ≤ c K\nkr : α... | [
"case calc_5\nK : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\nα : Type u_2\nβ : Type u_3\na : Matrix α β (𝓞 K)\ninst✝² : Fintype β\nA : ℝ\nhabs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A\ninst✝¹ : DecidableEq (K →+* ℂ)\ninst✝ : Fintype α\nApos : 0 ≤ A\nthis : 0 ≤ c K\nkr : α × (K →+* ℂ)... | · rw [mul_assoc, mul_assoc]
gcongr _ * (?_ * _)
· apply house_nonneg
· exact habs kr.1 lu.1 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.House | {
"line": 331,
"column": 8
} | {
"line": 331,
"column": 24
} | {
"line": 331,
"column": 25
} | [
{
"pp": "case calc_5\nK : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\nα : Type u_2\nβ : Type u_3\na : Matrix α β (𝓞 K)\np q : ℕ\nhpq : p < q\nx : β × (K →+* ℂ) → ℤ\ninst✝² : Fintype β\nA : ℝ\nhabs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A\ninst✝¹ : DecidableEq (K →+* ℂ)\ninst✝ : Fi... | [
"case calc_5\nK : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\nα : Type u_2\nβ : Type u_3\na : Matrix α β (𝓞 K)\np q : ℕ\nhpq : p < q\nx : β × (K →+* ℂ) → ℤ\ninst✝² : Fintype β\nA : ℝ\nhabs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A\ninst✝¹ : DecidableEq (K →+* ℂ)\ninst✝ : Fintype α\nApo... | Embeddings.card, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Ostrowski | {
"line": 181,
"column": 2
} | {
"line": 215,
"column": 76
} | {
"line": 219,
"column": 0
} | [
{
"pp": "f : AbsoluteValue ℚ ℝ\nbdd : ∀ (n : ℕ), f ↑n ≤ 1\np : ℕ\nhp0 : 0 < f ↑p\nhp1 : f ↑p < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\nm : ℕ\nhpm : ¬p ∣ m\n⊢ f ↑m = 1",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
... | [] | apply le_antisymm (bdd m)
by_contra! hm
set M := f p ⊔ f m with hM
set k := Nat.ceil (M.logb (1 / 2)) + 1 with hk
obtain ⟨a, b, bezout⟩ : IsCoprime (p ^ k : ℤ) (m ^ k) :=
is_prime_of_minimal_nat_zero_lt_and_lt_one hp0 hp1 hmin
|>.coprime_iff_not_dvd |>.mpr hpm |>.isCoprime |>.pow
have le_half {x} (h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Ostrowski | {
"line": 181,
"column": 2
} | {
"line": 215,
"column": 76
} | {
"line": 219,
"column": 0
} | [
{
"pp": "f : AbsoluteValue ℚ ℝ\nbdd : ∀ (n : ℕ), f ↑n ≤ 1\np : ℕ\nhp0 : 0 < f ↑p\nhp1 : f ↑p < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\nm : ℕ\nhpm : ¬p ∣ m\n⊢ f ↑m = 1",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
... | [] | apply le_antisymm (bdd m)
by_contra! hm
set M := f p ⊔ f m with hM
set k := Nat.ceil (M.logb (1 / 2)) + 1 with hk
obtain ⟨a, b, bezout⟩ : IsCoprime (p ^ k : ℤ) (m ^ k) :=
is_prime_of_minimal_nat_zero_lt_and_lt_one hp0 hp1 hmin
|>.coprime_iff_not_dvd |>.mpr hpm |>.isCoprime |>.pow
have le_half {x} (h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Ostrowski | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 23
} | {
"line": 303,
"column": 2
} | [
{
"pp": "f : AbsoluteValue ℚ ℝ\nn m : ℕ\nhm : 1 < m\nL : List ℕ := m.digits n\nL' : List ℚ := List.map Nat.cast (List.mapIdx (fun i a ↦ a * m ^ i) L)\nhL' : L' = List.map Nat.cast (List.mapIdx (fun i a ↦ a * m ^ i) L)\nhcoef : ∀ {c : ℕ}, c ∈ m.digits n → f ↑c < ↑m\nx✝ : ℕ × ℕ\na i : ℕ\nhia✝ : (a, i) ∈ L.zipIdx\... | [
"f : AbsoluteValue ℚ ℝ\nn m : ℕ\nhm : 1 < m\nL : List ℕ := m.digits n\nL' : List ℚ := List.map Nat.cast (List.mapIdx (fun i a ↦ a * m ^ i) L)\nhL' : L' = List.map Nat.cast (List.mapIdx (fun i a ↦ a * m ^ i) L)\nhcoef : ∀ {c : ℕ}, c ∈ m.digits n → f ↑c < ↑m\nx✝ : ℕ × ℕ\na i : ℕ\nhia✝ : (a, i) ∈ L.zipIdx\nhia : 0 ≤ i... | rw [map_mul, map_pow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 181,
"column": 37
} | {
"line": 181,
"column": 56
} | {
"line": 182,
"column": 6
} | [
{
"pp": "case refine_1\np : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : Module ℤ_[p] E\ninst✝¹ : IsBoundedSMul ℤ_[p] E\ninst✝ : IsUltrametricDist E\nf : C(ℤ_[p], E)\ns t : ℕ\nhst : ∀ (x y : ℤ_[p]), ‖x - y‖ ≤ ↑p ^ (-↑t) → ‖f x - f y‖ ≤ ‖f‖ / ↑p ^ s\nn i : ℕ\nhi : i < p ^ t -... | [
"case refine_1\np : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : Module ℤ_[p] E\ninst✝¹ : IsBoundedSMul ℤ_[p] E\ninst✝ : IsUltrametricDist E\nf : C(ℤ_[p], E)\ns t : ℕ\nhst : ∀ (x y : ℤ_[p]), ‖x - y‖ ≤ ↑p ^ (-↑t) → ‖f x - f y‖ ≤ ‖f‖ / ↑p ^ s\nn i : ℕ\nhi : i < p ^ t - 1\nthis : 0... | NNReal.coe_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 204,
"column": 49
} | {
"line": 219,
"column": 85
} | {
"line": 221,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : Module ℤ_[p] E\ninst✝¹ : IsBoundedSMul ℤ_[p] E\ninst✝ : IsUltrametricDist E\nf : C(ℤ_[p], E)\ns t : ℕ\nhst : ∀ (x y : ℤ_[p]), ‖x - y‖ ≤ ↑p ^ (-↑t) → ‖f x - f y‖ ≤ ‖f‖ / ↑p ^ s\nn : ℕ\n⊢ ‖Δ_[1]^[n + s * p ^ t] (⇑f) 0‖ ... | [] | by
-- We show the following more general statement by induction on `k`:
suffices ∀ {k : ℕ}, k ≤ s → ‖Δ_[1]^[n + k * p ^ t] f 0‖ ≤ ‖f‖ / p ^ k from this le_rfl
intro k hk
induction k generalizing n with
| zero => -- base case just says that `‖Δ^[·] (⇑f) 0‖` is bounded by `‖f‖`
simpa only [zero_mul, pow_zer... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Separation.DisjointCover | {
"line": 131,
"column": 8
} | {
"line": 131,
"column": 30
} | {
"line": 131,
"column": 30
} | [
{
"pp": "X : Type u_1\nV : Type u_2\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace V\ninst✝² : TotallyDisconnectedSpace X\ninst✝¹ : T2Space X\ninst✝ : CompactSpace X\nS : Set (V × V)\nf : C(X, V)\nhS : S ∈ 𝓝ˢ (diagonal V)\n⊢ ⇑(f.prodMap f) ⁻¹' S ∈ 𝓝ˢ (diagonal X)",
"ppTerm": "?m.68",
"assigne... | [
"X : Type u_1\nV : Type u_2\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace V\ninst✝² : TotallyDisconnectedSpace X\ninst✝¹ : T2Space X\ninst✝ : CompactSpace X\nS : Set (V × V)\nf : C(X, V)\nhS : ∀ x ∈ diagonal V, S ∈ 𝓝 x\n⊢ ∀ x ∈ diagonal X, ⇑(f.prodMap f) ⁻¹' S ∈ 𝓝 x"
] | mem_nhdsSet_iff_forall | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.WithVal | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 85
} | {
"line": 174,
"column": 0
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : ℚ\nh : ‖↑x‖ ≤ 1\n⊢ ¬p ∣ x.den",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Nat.Coprime",
"Real",
"Nat.Prime",
"Dvd.dvd",
"PadicInt",
"CommSemiring.toSemiring",
"PadicInt.norm_na... | [] | simpa [Nat.Prime.coprime_iff_not_dvd Fact.out] using isUnit_iff.1 <| isUnit_den _ h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.Pell | {
"line": 197,
"column": 34
} | {
"line": 197,
"column": 42
} | {
"line": 197,
"column": 42
} | [
{
"pp": "d : ℤ\nh₀ : 0 ≤ d\na : Solution₁ d\nhx : a.x = 0\nh : 0 ≤ 0 - 1\n⊢ False",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"congrArg",
"AddMonoid.toAddZeroClass",
"HSub.hSub",
"AddZeroClass.toAddZero",
"Eq.mp",
"Int",
"LE.le",
"SubNegM... | [
"d : ℤ\nh₀ : 0 ≤ d\na : Solution₁ d\nhx : a.x = 0\nh : 0 ≤ -1\n⊢ False"
] | zero_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Pell | {
"line": 535,
"column": 54
} | {
"line": 535,
"column": 89
} | {
"line": 536,
"column": 4
} | [
{
"pp": "d : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\nH : d * (a₁.y ^ 2 - a.y ^ 2) = a₁.x ^ 2 - a.x ^ 2\n⊢ |a₁.x| ≤ |a.x|",
"ppTerm": "?m.147",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Int.instN... | [
"d : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\nH : d * (a₁.y ^ 2 - a.y ^ 2) = a₁.x ^ 2 - a.x ^ 2\n⊢ a₁.x ≤ |a.x|"
] | abs_of_pos (zero_lt_one.trans h.1), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Pell | {
"line": 560,
"column": 2
} | {
"line": 560,
"column": 75
} | {
"line": 561,
"column": 2
} | [
{
"pp": "d : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\n⊢ 0 < (a * a₁⁻¹).x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Eq.mpr",
"NonUni... | [
"d : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\n⊢ d * (a.y * a₁.y) < a.x * a₁.x"
] | simp only [x_mul, x_inv, y_inv, mul_neg, lt_add_neg_iff_add_lt, zero_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 104,
"column": 4
} | {
"line": 107,
"column": 84
} | {
"line": 109,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹⁶ : CommRing R\nS : Type v\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\np : Ideal R\nK : Type u_1\ninst✝¹³ : Field K\ninst✝¹² : Algebra R K\nL : Type u_2\ninst✝¹¹ : Field L\ninst✝¹⁰ : Algebra S L\ninst✝⁹ : IsFractionRing S L\ninst✝⁸ : IsDomain R\ninst✝⁷ : IsDomain S\ninst✝⁶ : Algebra... | [] | calc
p • ⊤ = Submodule.map M.mkQ (p • ⊤) := by
rw [Submodule.map_smul'', Submodule.map_top, M.range_mkQ]
_ = ⊤ := by rw [Ideal.smul_top_eq_map, (Submodule.map_mkQ_eq_top M _).mpr hb'] | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.NumberTheory.RamificationInertia.Valuation | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 33
} | {
"line": 92,
"column": 2
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝¹⁶ : CommRing A\ninst✝¹⁵ : IsDedekindDomain A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : IsDedekindDomain B\ninst✝¹² : Algebra A B\ninst✝¹¹ : Module.IsTorsionFree A B\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : Algebra A K\nins... | [
"A : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝¹⁶ : CommRing A\ninst✝¹⁵ : IsDedekindDomain A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : IsDedekindDomain B\ninst✝¹² : Algebra A B\ninst✝¹¹ : Module.IsTorsionFree A B\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : Algebra A K\ninst✝⁶ : IsFrac... | let m : ℤᵐ⁰ := σw (σL (σwV γL)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.Rayleigh | {
"line": 143,
"column": 10
} | {
"line": 143,
"column": 27
} | {
"line": 143,
"column": 28
} | [
{
"pp": "case a.inl\nr s : ℝ\nhrs : r.HolderConjugate s\nj k : ℤ\nhk : k > 0\nhjk : beattySeq r k = j\n⊢ j ∈ {n | 0 < n}",
"ppTerm": "?a.inl✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"setOf",
"Membership.mem",
"id",
"Int",
"Set.mem_setOf_... | [
"case a.inl\nr s : ℝ\nhrs : r.HolderConjugate s\nj k : ℤ\nhk : k > 0\nhjk : beattySeq r k = j\n⊢ 0 < j"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Rayleigh | {
"line": 145,
"column": 10
} | {
"line": 145,
"column": 27
} | {
"line": 145,
"column": 28
} | [
{
"pp": "case a.inr\nr s : ℝ\nhrs : r.HolderConjugate s\nj k : ℤ\nhk : k > 0\nhjk : beattySeq' s k = j\n⊢ j ∈ {n | 0 < n}",
"ppTerm": "?a.inr✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"setOf",
"Membership.mem",
"id",
"Int",
"Set.mem_setOf... | [
"case a.inr\nr s : ℝ\nhrs : r.HolderConjugate s\nj k : ℤ\nhk : k > 0\nhjk : beattySeq' s k = j\n⊢ 0 < j"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 53
} | {
"line": 191,
"column": 2
} | [
{
"pp": "case hg\nK : Type u_1\nΓ : Type u_2\ninst✝³ : Field K\ninst✝² : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\ninst✝¹ : v.IsNontrivial\ninst✝ : IsTrivialOn K v\nhle : v X ≤ 1\nhv : v.IsRankOneDiscrete\nh0 : v ↑πᵥ ≠ 0\n⊢ v ↑πᵥ < 1",
"ppTerm": "?hg",
"assigned": true,
"usedConstants"... | [] | exact valuation_uniformizingPolynomial_lt_one hle | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.SumFourSquares | {
"line": 48,
"column": 48
} | {
"line": 48,
"column": 54
} | {
"line": 48,
"column": 54
} | [
{
"pp": "m x y : ℤ\nh : 2 * m = x ^ 2 + y ^ 2\nthis : Even (x ^ 2 + y ^ 2)\n⊢ 2 * 2 ≠ 0",
"ppTerm": "?m.127",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"HMul.hMul",
"of_decide_eq_true",
"Int.instDecidableEq",
"id",
"Ne",
"Int",
"Int.inst... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.SumFourSquares | {
"line": 48,
"column": 48
} | {
"line": 48,
"column": 54
} | {
"line": 48,
"column": 54
} | [
{
"pp": "m x y : ℤ\nh : 2 * m = x ^ 2 + y ^ 2\nthis : Even (x ^ 2 + y ^ 2)\n⊢ 2 * 2 ≠ 0",
"ppTerm": "?m.127",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"HMul.hMul",
"of_decide_eq_true",
"Int.instDecidableEq",
"id",
"Ne",
"Int",
"Int.inst... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.SumFourSquares | {
"line": 48,
"column": 48
} | {
"line": 48,
"column": 54
} | {
"line": 48,
"column": 54
} | [
{
"pp": "m x y : ℤ\nh : 2 * m = x ^ 2 + y ^ 2\nthis : Even (x ^ 2 + y ^ 2)\n⊢ 2 * 2 ≠ 0",
"ppTerm": "?m.127",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"HMul.hMul",
"of_decide_eq_true",
"Int.instDecidableEq",
"id",
"Ne",
"Int",
"Int.inst... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 79,
"column": 72
} | {
"line": 79,
"column": 95
} | {
"line": 81,
"column": 0
} | [
{
"pp": "x y : ℤ\n⊢ toComplex { re := x, im := y } = ↑x + ↑y * I",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Int.cast",
"HMul.hMul",
"Complex.instMul",
"Complex.instIntCast",
"instHAdd",
"HAdd.hAdd",
"eq_self",
"of_eq_true",
"Compl... | [] | by simp [toComplex_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 85,
"column": 65
} | {
"line": 85,
"column": 88
} | {
"line": 87,
"column": 0
} | [
{
"pp": "x : ℤ[i]\n⊢ ↑x.re = (toComplex x).re",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Int.cast",
"GaussianInt",
"Real",
"Complex.mul_re",
"Zsqrtd.re",
"HMul.hMul",
"sub_self",
"Real.instZero",
"Real.instAddMonoid",
"congrA... | [] | by simp [toComplex_def] | [anonymous] | Lean.Parser.Term.byTactic |
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