module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 127,
"column": 25
} | {
"line": 127,
"column": 36
} | {
"line": 127,
"column": 37
} | [
{
"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))\nh : (span {hζ.toInteger - 1}).IsPrime\n⊢ 𝒑 ≠ ⊥",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
... | [
"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))\nh : (span {hζ.toInteger - 1}).IsPrime\n⊢ ¬p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 81
} | {
"line": 205,
"column": 0
} | [
{
"pp": "p : ℕ\nK : Type u_1\ninst✝¹ : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ p\ninst✝ : NeZero p\nη : K\nhη : IsPrimitiveRoot η p\ni : ℕ\nhi : i.Coprime p\nhζη : ζ ^ i = η\n⊢ Associated (hζ.toInteger - 1) (hζ.toInteger ^ i - 1)",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"IsPri... | [] | exact hζ.toInteger_isPrimitiveRoot.associated_sub_one_pow_sub_one_of_coprime hi | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 219,
"column": 2
} | {
"line": 219,
"column": 13
} | {
"line": 219,
"column": 14
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝ : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ p\nη : 𝓞 K\nhη : η ∈ primitiveRoots p (𝓞 K)\nhη' : IsPrimitiveRoot (↑η) p\n⊢ Associated (hζ.toInteger - 1) (1 - η)",
"ppTerm": "?m.121",
"assigned": false,
"usedConstants": [],
"usedFVars": []... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝ : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ p\nη : 𝓞 K\nhη : η ∈ primitiveRoots p (𝓞 K)\nhη' : IsPrimitiveRoot (↑η) p\n⊢ Associated (hζ.toInteger - 1) (1 - η)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 15
} | {
"line": 307,
"column": 16
} | [
{
"pp": "m p : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\nP : Ideal (𝓞 K)\nhP₁ : P.IsPrime\nhP₂ : P.LiesOver 𝒑\ninst✝ : NeZero m\nhK : IsCyclotomicExtension {m} ℚ K\nhm : p.Coprime m\nζ : 𝓞 K := ⋯.toInteger\nh₁ : ¬p ∣ exponent ζ\nh₂ :\n Irreducible ↑((primesOverSpanE... | [
"m p : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\nP : Ideal (𝓞 K)\nhP₁ : P.IsPrime\nhP₂ : P.LiesOver 𝒑\ninst✝ : NeZero m\nhK : IsCyclotomicExtension {m} ℚ K\nhm : p.Coprime m\nζ : 𝓞 K := ⋯.toInteger\nh₁ : ¬p ∣ exponent ζ\nh₂ :\n Irreducible ↑((primesOverSpanEquivMonicFac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.House | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 31
} | {
"line": 56,
"column": 32
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ns : Finset K\n⊢ house (∏ x ∈ s, x) ≤ ∏ x ∈ s, house x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real.instLE... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ns : Finset K\n⊢ ‖∏ x ∈ s, (canonicalEmbedding K) x‖ ≤ ∏ x ∈ s, ‖(canonicalEmbedding K) x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.House | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 35
} | {
"line": 62,
"column": 36
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nα : K\ni : ℕ\n⊢ house (α ^ i) ≤ house α ^ i",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real.instLE",
"... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nα : K\ni : ℕ\n⊢ ‖(canonicalEmbedding K) α ^ i‖ ≤ ‖(canonicalEmbedding K) α‖ ^ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.House | {
"line": 132,
"column": 38
} | {
"line": 134,
"column": 12
} | {
"line": 136,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : DecidableEq (K →+* ℂ)\n⊢ 0 ≤ c K",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"mul_nonneg",
"NormedCommRing.toSeminormedCommRing",
... | [] | by
rw [c]
positivity | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 327,
"column": 2
} | {
"line": 327,
"column": 9
} | {
"line": 328,
"column": 2
} | [
{
"pp": "m p : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\nP : Ideal (𝓞 K)\nhP₁ : P.IsPrime\nhP₂ : P.LiesOver 𝒑\ninst✝ : NeZero m\nhK : IsCyclotomicExtension {m} ℚ K\nhm : ¬p ∣ m\nζ : 𝓞 K := ⋯.toInteger\nh₁ : ¬p ∣ exponent ζ\nh₂ :\n Irreducible ↑((primesOverSpanEquivM... | [
"m p : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\nP : Ideal (𝓞 K)\nhP₁ : P.IsPrime\nhP₂ : P.LiesOver 𝒑\ninst✝ : NeZero m\nhK : IsCyclotomicExtension {m} ℚ K\nhm : ¬p ∣ m\nζ : 𝓞 K := ⋯.toInteger\nh₁ : ¬p ∣ exponent ζ\nh₂ :\n Irreducible ↑((primesOverSpanEquivMonicFactorsM... | rw [h₃] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Ostrowski | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 81
} | {
"line": 98,
"column": 2
} | [
{
"pp": "f g : AbsoluteValue ℚ ℝ\n⊢ (∃ c, 0 < c ∧ ∀ (n : ℕ), f ↑n ^ c = g ↑n) ↔ ∃ c, 0 < c ∧ (fun x ↦ f x ^ c) = ⇑g",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real.partialOrder",
"Real",
"Real.instZero",
"Rat",
"Real.instLT",
... | [
"f g : AbsoluteValue ℚ ℝ\nx✝ : ∃ c, 0 < c ∧ ∀ (n : ℕ), f ↑n ^ c = g ↑n\nc : ℝ\nhc : 0 < c\nh : ∀ (n : ℕ), f ↑n ^ c = g ↑n\n⊢ (fun x ↦ f x ^ c) = ⇑g"
] | refine ⟨fun ⟨c, hc, h⟩ ↦ ⟨c, hc, ?_⟩, fun ⟨c, hc, h⟩ ↦ ⟨c, hc, (congrFun h ·)⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Padics.ProperSpace | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 55
} | {
"line": 51,
"column": 56
} | [
{
"pp": "case refine_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nε : ℝ\nhε : ε > 0\nk : ℕ\nhk : ↑p ^ (-↑k) < ε\nz : ℤ_[p]\nx✝ : z ∈ Set.univ\n⊢ z.appr k ∈ ↑(Finset.range (p ^ k))",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset",
"Nat.instMonoid",
"_p... | [
"case refine_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nε : ℝ\nhε : ε > 0\nk : ℕ\nhk : ↑p ^ (-↑k) < ε\nz : ℤ_[p]\nx✝ : z ∈ Set.univ\n⊢ z.appr k < p ^ k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Nonarchimedean | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 13
} | {
"line": 60,
"column": 14
} | [
{
"pp": "case h\nα : Type u_1\nG : Type u_2\ninst✝³ : CommGroup G\ninst✝² : UniformSpace G\ninst✝¹ : IsUniformGroup G\ninst✝ : NonarchimedeanGroup G\nf : α → G\nhf : Tendsto f cofinite (𝓝 1)\nU : Set G\nhU : U ∈ 𝓝 1\nV : OpenSubgroup G\nhV : ↑V ⊆ U\nt : Finset α\nht : Disjoint t (Set.Finite.toFinset ⋯)\ni : α... | [
"case h\nα : Type u_1\nG : Type u_2\ninst✝³ : CommGroup G\ninst✝² : UniformSpace G\ninst✝¹ : IsUniformGroup G\ninst✝ : NonarchimedeanGroup G\nf : α → G\nhf : Tendsto f cofinite (𝓝 1)\nU : Set G\nhU : U ∈ 𝓝 1\nV : OpenSubgroup G\nhV : ↑V ⊆ U\nt : Finset α\nht : Disjoint t (Set.Finite.toFinset ⋯)\ni : α\nhi : i ∈ t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Ostrowski | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 60
} | {
"line": 256,
"column": 61
} | [
{
"pp": "case refine_2\nf : AbsoluteValue ℚ ℝ\nhf_nontriv : f.IsNontrivial\nbdd : ∀ (n : ℕ), f ↑n ≤ 1\np : ℕ\nhfp : 0 < f ↑p ∧ f ↑p < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\nhprime : Nat.Prime p\nhprime_fact : Fact (Nat.Prime p)\nt : ℝ\nh : 0 < t ∧ f ↑p = ↑p ^ (-t)\nq : ℕ\nx✝ : (fun p ↦ ∃ (h : Fact (N... | [
"case refine_2\nf : AbsoluteValue ℚ ℝ\nhf_nontriv : f.IsNontrivial\nbdd : ∀ (n : ℕ), f ↑n ≤ 1\np : ℕ\nhfp : 0 < f ↑p ∧ f ↑p < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\nhprime : Nat.Prime p\nhprime_fact : Fact (Nat.Prime p)\nt : ℝ\nh : 0 < t ∧ f ↑p = ↑p ^ (-t)\nq : ℕ\nx✝ : (fun p ↦ ∃ (h : Fact (Nat.Prime p))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Ostrowski | {
"line": 275,
"column": 20
} | {
"line": 275,
"column": 31
} | {
"line": 275,
"column": 32
} | [
{
"pp": "f g : AbsoluteValue ℚ ℝ\nx y : ℚ\n⊢ |↑(x + y)| ≤ |↑x| + |↑y|",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"Real.lattice",
"DivisionRing.toRatCast",
"FloorRing.toFloorSemiring",
"abs",
"congrA... | [
"f g : AbsoluteValue ℚ ℝ\nx y : ℚ\n⊢ |↑x + ↑y| ≤ |↑x| + |↑y|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 42
} | {
"line": 65,
"column": 43
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : ℕ\nx : ℤ_[p]\nf : ℤ_[p] → ℤ_[p] := fun x ↦ Polynomial.eval x (ascPochhammer ℤ_[p] k)\nhC : ↑k.factorial ≠ 0\nhf : ContinuousAt f x\nδ : ℝ\nhδp : δ > 0\nhδ : ∀ ⦃x_1 : ℤ_[p]⦄, dist x_1 x < δ → dist (f x_1) (f x) < ‖↑k.factorial‖\nn : ℕ\nhn' : dist x ↑n < δ\n⊢ ∃ n, ‖f x... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nk : ℕ\nx : ℤ_[p]\nf : ℤ_[p] → ℤ_[p] := fun x ↦ Polynomial.eval x (ascPochhammer ℤ_[p] k)\nhC : ↑k.factorial ≠ 0\nhf : ContinuousAt f x\nδ : ℝ\nhδp : δ > 0\nhδ : ∀ ⦃x_1 : ℤ_[p]⦄, dist x_1 x < δ → dist (f x_1) (f x) < ‖↑k.factorial‖\nn : ℕ\nhn' : dist x ↑n < δ\n⊢ ∃ n, dist (f ↑n) (f x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.AddChar | {
"line": 106,
"column": 31
} | {
"line": 106,
"column": 42
} | {
"line": 106,
"column": 43
} | [
{
"pp": "p : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nR : Type u_1\ninst✝⁴ : NormedRing R\ninst✝³ : Algebra ℤ_[p] R\ninst✝² : IsBoundedSMul ℤ_[p] R\ninst✝¹ : IsUltrametricDist R\ninst✝ : CompleteSpace R\nx✝ : { κ // Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑κ }\nκ : AddChar ℤ_[p] R\nhκ : Contin... | [
"p : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nR : Type u_1\ninst✝⁴ : NormedRing R\ninst✝³ : Algebra ℤ_[p] R\ninst✝² : IsBoundedSMul ℤ_[p] R\ninst✝¹ : IsUltrametricDist R\ninst✝ : CompleteSpace R\nx✝ : { κ // Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑κ }\nκ : AddChar ℤ_[p] R\nhκ : Continuous[_, Pseu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 81,
"column": 33
} | {
"line": 81,
"column": 44
} | {
"line": 81,
"column": 45
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nk : ℕ\n⊢ 0 < ‖↑k.factorial‖",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"Real.instZero",
... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nk : ℕ\n⊢ ¬k.factorial = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 82
} | {
"line": 143,
"column": 6
} | [
{
"pp": "M : Type u_1\nG : Type u_2\ninst✝¹ : AddCommMonoidWithOne M\ninst✝ : AddCommGroup G\nf : M → G\nn R : ℕ\nhR : 1 ≤ R\naux : Δ_[1]^[n + R] f 0 = R.choose (R - 1 + 1) • Δ_[1]^[n + R] f 0\n⊢ ∑ k ∈ range (R + 1), R.choose k • Δ_[1]^[n + k] f 0 = Δ_[1]^[n] f ↑R",
"ppTerm": "?m.287",
"assigned": false... | [
"M : Type u_1\nG : Type u_2\ninst✝¹ : AddCommMonoidWithOne M\ninst✝ : AddCommGroup G\nf : M → G\nn R : ℕ\nhR : 1 ≤ R\naux : Δ_[1]^[n + R] f 0 = R.choose (R - 1 + 1) • Δ_[1]^[n + R] f 0\n⊢ ∑ k ∈ range (R + 1), R.choose k • Δ_[1]^[n + k] f 0 = Δ_[1]^[n] f ↑R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 193,
"column": 6
} | {
"line": 193,
"column": 38
} | {
"line": 193,
"column": 39
} | [
{
"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 i : ℕ\nx✝ : i ∈ range (n + 1)\n⊢ ‖↑(-1... | [
"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 i : ℕ\nx✝ : i ∈ range (n + 1)\n⊢ ‖↑(-1) ^ (n - i) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Ostrowski | {
"line": 467,
"column": 2
} | {
"line": 469,
"column": 45
} | {
"line": 471,
"column": 0
} | [
{
"pp": "f : AbsoluteValue ℚ ℝ\nhf_nontriv : f.IsNontrivial\n⊢ f ≈ real ∨ ∃! p, ∃ (x : Fact (Nat.Prime p)), f ≈ padic p",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real.instLE",
"Real",
"Nat.Prime",
"Rat",
"Rat.AbsoluteValue.equ... | [] | by_cases bdd : ∀ n : ℕ, f n ≤ 1
· exact .inr <| equiv_padic_of_bounded hf_nontriv bdd
· exact .inl <| equiv_real_of_unbounded bdd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Ostrowski | {
"line": 467,
"column": 2
} | {
"line": 469,
"column": 45
} | {
"line": 471,
"column": 0
} | [
{
"pp": "f : AbsoluteValue ℚ ℝ\nhf_nontriv : f.IsNontrivial\n⊢ f ≈ real ∨ ∃! p, ∃ (x : Fact (Nat.Prime p)), f ≈ padic p",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real.instLE",
"Real",
"Nat.Prime",
"Rat",
"Rat.AbsoluteValue.equ... | [] | by_cases bdd : ∀ n : ℕ, f n ≤ 1
· exact .inr <| equiv_padic_of_bounded hf_nontriv bdd
· exact .inl <| equiv_real_of_unbounded bdd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 210,
"column": 4
} | {
"line": 210,
"column": 60
} | {
"line": 210,
"column": 61
} | [
{
"pp": "case zero\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 : ℕ\nhk : 0 ≤ s\n⊢ ‖Δ_[1]^[... | [
"case zero\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 : ℕ\nhk : 0 ≤ s\n⊢ ‖Δ_[1]^[n] (⇑f) 0‖ ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 239,
"column": 32
} | {
"line": 239,
"column": 67
} | {
"line": 239,
"column": 68
} | [
{
"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)\nε : ℝ\nhε : ε > 0\nthis✝ : Tendsto (fun s ↦ ‖f‖ / ↑p ^ s) atTop (𝓝 0)\ns : ℕ\nhs : ‖f‖ / ↑p ^ s < ε\nhf : f ≠ 0\nthis : 0 ... | [
"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)\nε : ℝ\nhε : ε > 0\nthis✝ : Tendsto (fun s ↦ ‖f‖ / ↑p ^ s) atTop (𝓝 0)\ns : ℕ\nhs : ‖f‖ / ↑p ^ s < ε\nhf : f ≠ 0\nthis : 0 < ‖f‖ / ↑p ^... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 42
} | {
"line": 241,
"column": 43
} | [
{
"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)\nε : ℝ\nhε : ε > 0\nthis : Tendsto (fun s ↦ ‖f‖ / ↑p ^ s) atTop (𝓝 0)\ns : ℕ\nhs : ‖f‖ / ↑p ^ s < ε\nt : ℕ\nht : ∀ (x y : ℤ... | [
"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)\nε : ℝ\nhε : ε > 0\nthis : Tendsto (fun s ↦ ‖f‖ / ↑p ^ s) atTop (𝓝 0)\ns : ℕ\nhs : ‖f‖ / ↑p ^ s < ε\nt : ℕ\nht : ∀ (x y : ℤ_[p]), ‖x - ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 59
} | {
"line": 289,
"column": 60
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : Module ℤ_[p] E\ninst✝² : IsBoundedSMul ℤ_[p] E\ninst✝¹ : IsUltrametricDist E\ninst✝ : CompleteSpace E\na : ℕ → E\nha : Tendsto (fun x ↦ ‖a x‖) atTop (𝓝 0)\n⊢ Tendsto (fun x ↦ ‖mahlerTerm (a x) x‖) cofinite (𝓝 0)",
... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : Module ℤ_[p] E\ninst✝² : IsBoundedSMul ℤ_[p] E\ninst✝¹ : IsUltrametricDist E\ninst✝ : CompleteSpace E\na : ℕ → E\nha : Tendsto (fun x ↦ ‖a x‖) atTop (𝓝 0)\n⊢ Tendsto (fun x ↦ ‖a x‖) atTop (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 305,
"column": 4
} | {
"line": 305,
"column": 39
} | {
"line": 305,
"column": 40
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : Module ℤ_[p] E\ninst✝² : IsBoundedSMul ℤ_[p] E\ninst✝¹ : IsUltrametricDist E\ninst✝ : CompleteSpace E\na : ℕ → E\nha : Tendsto a atTop (𝓝 0)\nm n : ℕ\nhmn : m ≤ n\nh_van : ∀ (i : ℕ), m.choose (i + (n + 1)) = 0\n⊢ Sum... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : Module ℤ_[p] E\ninst✝² : IsBoundedSMul ℤ_[p] E\ninst✝¹ : IsUltrametricDist E\ninst✝ : CompleteSpace E\na : ℕ → E\nha : Tendsto a atTop (𝓝 0)\nm n : ℕ\nhmn : m ≤ n\nh_van : ∀ (i : ℕ), m.choose (i + (n + 1)) = 0\n⊢ Summable fun i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 348,
"column": 2
} | {
"line": 349,
"column": 9
} | {
"line": 349,
"column": 10
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : Module ℤ_[p] E\ninst✝² : IsBoundedSMul ℤ_[p] E\ninst✝¹ : IsUltrametricDist E\ninst✝ : CompleteSpace E\nf : C(ℤ_[p], E)\nthis : HasSum (fun n ↦ mahlerTerm (Δ_[1]^[n] (⇑f) 0) n) (mahlerSeries fun x ↦ Δ_[1]^[x] (⇑f) 0)\n... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : Module ℤ_[p] E\ninst✝² : IsBoundedSMul ℤ_[p] E\ninst✝¹ : IsUltrametricDist E\ninst✝ : CompleteSpace E\nf : C(ℤ_[p], E)\nthis : HasSum (fun n ↦ mahlerTerm (Δ_[1]^[n] (⇑f) 0) n) (mahlerSeries fun x ↦ Δ_[1]^[x] (⇑f) 0)\nn : ℕ\n⊢ f ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.DisjointCover | {
"line": 68,
"column": 25
} | {
"line": 68,
"column": 88
} | {
"line": 68,
"column": 89
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TotallyDisconnectedSpace X\ninst✝¹ : T2Space X\ninst✝ : CompactSpace X\nU : ι → Opens X\nhU : IsOpenCover U\nn : ℕ\nV : Fin n → Clopens X\nhVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)\nhVun : univ ⊆ ⋃ j, ↑(V j)\nW : Fin n → Clopens X\nhWle... | [
"ι : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TotallyDisconnectedSpace X\ninst✝¹ : T2Space X\ninst✝ : CompactSpace X\nU : ι → Opens X\nhU : IsOpenCover U\nn : ℕ\nV : Fin n → Clopens X\nhVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)\nhVun : univ ⊆ ⋃ j, ↑(V j)\nW : Fin n → Clopens X\nhWle : W ≤ V\nhW... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.DisjointCover | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 49
} | {
"line": 98,
"column": 50
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nS : Set (X × X)\ninst✝ : CompactSpace X\nhS : S ∈ 𝓝ˢ (diagonal X)\nU : X → Set X\nhUo : ∀ (x : X), IsOpen[inst✝¹] (U x)\nhUx : ∀ (x : X), x ∈ U x\nhUp : ∀ (x : X), U x ×ˢ U x ⊆ S\nt : Finset X\nht : univ ⊆ ⋃ i ∈ t, U i\n⊢ ⋃ i, U ↑i = univ",
"ppTerm": "?m.... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\nS : Set (X × X)\ninst✝ : CompactSpace X\nhS : S ∈ 𝓝ˢ (diagonal X)\nU : X → Set X\nhUo : ∀ (x : X), IsOpen[inst✝¹] (U x)\nhUx : ∀ (x : X), x ∈ U x\nhUp : ∀ (x : X), U x ×ˢ U x ⊆ S\nt : Finset X\nht : univ ⊆ ⋃ i ∈ t, U i\n⊢ univ ⊆ ⋃ x ∈ t, U x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.WithVal | {
"line": 70,
"column": 8
} | {
"line": 70,
"column": 58
} | {
"line": 71,
"column": 10
} | [
{
"pp": "case neg\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp0' : 0 < ↑p\nhp0 : 0 < (↑p)⁻¹\nhp1' : 1 < ↑p\nhp1 : (↑p)⁻¹ < 1\nn : ℕ\nhn : Valued.v (↑p ^ n) = exp (-↑n)\nx y : WithVal (Rat.padicValuation p)\nx' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x\nhx : x' = (WithVal.equiv (Rat.padicValuation p)) x\ny' : ℚ ... | [
"case neg\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp0' : 0 < ↑p\nhp0 : 0 < (↑p)⁻¹\nhp1' : 1 < ↑p\nhp1 : (↑p)⁻¹ < 1\nn : ℕ\nhn : Valued.v (↑p ^ n) = exp (-↑n)\nx y : WithVal (Rat.padicValuation p)\nx' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x\nhx : x' = (WithVal.equiv (Rat.padicValuation p)) x\ny' : ℚ := (WithVal.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.ProdApproximation | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 13
} | {
"line": 110,
"column": 14
} | [
{
"pp": "X : Type u_5\nY : Type u_6\nR : Type u_7\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : TopologicalSpace Y\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : IsTopologicalRing R\ninst✝³ : CompactSpace X\ninst✝² : T2Space X\ninst✝¹ : CompactSpace Y\ninst✝ : TotallyDisconnectedSpace X\nthis✝¹ : UniformS... | [
"X : Type u_5\nY : Type u_6\nR : Type u_7\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : TopologicalSpace Y\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : IsTopologicalRing R\ninst✝³ : CompactSpace X\ninst✝² : T2Space X\ninst✝¹ : CompactSpace Y\ninst✝ : TotallyDisconnectedSpace X\nthis✝¹ : UniformSpace R := Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.ProdApproximation | {
"line": 110,
"column": 27
} | {
"line": 110,
"column": 50
} | {
"line": 110,
"column": 51
} | [
{
"pp": "X : Type u_5\nY : Type u_6\nR : Type u_7\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : TopologicalSpace Y\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : IsTopologicalRing R\ninst✝³ : CompactSpace X\ninst✝² : T2Space X\ninst✝¹ : CompactSpace Y\ninst✝ : TotallyDisconnectedSpace X\nthis✝¹ : UniformS... | [
"X : Type u_5\nY : Type u_6\nR : Type u_7\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : TopologicalSpace Y\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : IsTopologicalRing R\ninst✝³ : CompactSpace X\ninst✝² : T2Space X\ninst✝¹ : CompactSpace Y\ninst✝ : TotallyDisconnectedSpace X\nthis✝¹ : UniformSpace R := Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.WithVal | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 30
} | {
"line": 97,
"column": 31
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx✝ : ℚ_[p]\n⊢ x✝ ∈ closure (Set.range ⇑((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom))",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"DivisionRing.t... | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx✝ : ℚ_[p]\n⊢ x✝ ∈ closure (Set.range Rat.cast)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.WithVal | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 54
} | {
"line": 172,
"column": 55
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : ℚ\nh : ‖↑x‖ ≤ 1\n⊢ ¬p ∣ x.den",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : ℚ\nh : ‖↑x‖ ≤ 1\n⊢ ¬p ∣ x.den"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.WithVal | {
"line": 179,
"column": 8
} | {
"line": 179,
"column": 32
} | {
"line": 179,
"column": 32
} | [
{
"pp": "case hp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ IsClosed {a | ‖withValUniformEquiv a‖ ≤ 1 ↔ Valued.v a ≤ 1}",
"ppTerm": "?hp",
"assigned": true,
"usedConstants": [
"Set.ext",
"Norm.norm",
"Eq.mpr",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
... | [
"case hp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ IsClosed {x | Valued.v x ≤ 1 ↔ ‖withValUniformEquiv x‖ ≤ 1}"
] | Set.ext fun _ ↦ Iff.comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.WithVal | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 35
} | {
"line": 183,
"column": 36
} | [
{
"pp": "case hp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ IsClopen {y | ‖y‖ ≤ 1}",
"ppTerm": "?hp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case hp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ IsClopen {y | ‖y‖ ≤ 1}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 94,
"column": 42
} | {
"line": 94,
"column": 53
} | {
"line": 94,
"column": 54
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nncs : CauSeq ℤ_[p] norm\nF : Polynomial R\nhnorm : Tendsto (fun i ↦ ‖(Polynomial.aeval (↑ncs i)) F‖) atTop (𝓝 0)\n⊢ Tendsto (fun e ↦ ‖(Polynomial.aeval (↑ncs e)) F - 0‖) atTop (𝓝 0)",
"ppTerm": "?m... | [
"p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nncs : CauSeq ℤ_[p] norm\nF : Polynomial R\nhnorm : Tendsto (fun i ↦ ‖(Polynomial.aeval (↑ncs i)) F‖) atTop (𝓝 0)\n⊢ Tendsto (fun e ↦ ‖(Polynomial.aeval (↑ncs e)) F‖) atTop (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 116,
"column": 47
} | {
"line": 116,
"column": 58
} | {
"line": 116,
"column": 59
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nha : (Polynomial.aeval a) F = 0\nz' : ℤ_[p]\nhz' : (Polynomial.aeval z') F = 0\nhnormz' : ‖z' - a‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq ... | [
"p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nha : (Polynomial.aeval a) F = 0\nz' : ℤ_[p]\nhz' : (Polynomial.aeval z') F = 0\nhnormz' : ‖z' - a‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq : ℤ_[p]\nhq ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 138,
"column": 72
} | {
"line": 139,
"column": 14
} | {
"line": 141,
"column": 0
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\n⊢ T_gen p F a = ‖(Polynomial.aeval a) F‖ / ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"P... | [] | by
simp [T_gen] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 15
} | {
"line": 219,
"column": 16
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nhnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2\nz z' z1 : ℤ_[p]\nhz' : z' = z - z1\nn : ℕ\nhz : ih_gen n z\nh1 : ‖↑((Polynomial.aeval... | [
"p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nhnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2\nz z' z1 : ℤ_[p]\nhz' : z' = z - z1\nn : ℕ\nhz : ih_gen n z\nh1 : ‖↑((Polynomial.aeval z) F) / ↑((... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 226,
"column": 24
} | {
"line": 226,
"column": 73
} | {
"line": 226,
"column": 73
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nhnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2\nz z' z1 : ℤ_[p]\nhz' : z' = z - z1\nn : ℕ\nhz : ih_gen n z\nh1 : ‖↑((Polynomial.aeval... | [] | by simp only [PadicInt.coe_neg, PadicInt.coe_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 228,
"column": 15
} | {
"line": 228,
"column": 74
} | {
"line": 228,
"column": 75
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nhnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2\nz z' z1 : ℤ_[p]\nhz' : z' = z - z1\nn : ℕ\nhz : ih_gen n z\nh1 : ‖↑((Polynomial.aeval... | [
"p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nhnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2\nz z' z1 : ℤ_[p]\nhz' : z' = z - z1\nn : ℕ\nhz : ih_gen n z\nh1 : ‖↑((Polynomial.aeval z) F) / ↑((... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Pell | {
"line": 393,
"column": 2
} | {
"line": 393,
"column": 71
} | {
"line": 393,
"column": 72
} | [
{
"pp": "x y : ℤ\nhy : y ≠ 0\na : ℤ\nh₀ : 0 < a * a\nhxy : (x + a * y) * (x - a * y) = 1\n⊢ False",
"ppTerm": "?m.146",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℤ\nhy : y ≠ 0\na : ℤ\nh₀ : 0 < a * a\nhxy : (x + a * y) * (x - a * y) = 1\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 387,
"column": 4
} | {
"line": 387,
"column": 15
} | {
"line": 387,
"column": 16
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nhnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2\nhnsol : (Polynomial.aeval a) F ≠ 0\nn : ℕ\nthis : (fun x ↦ 2 ^ x) 1 ≤ (fun x ↦ 2 ^ x)... | [
"p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nhnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2\nhnsol : (Polynomial.aeval a) F ≠ 0\nn : ℕ\nthis : (fun x ↦ 2 ^ x) 1 ≤ (fun x ↦ 2 ^ x) (n + 1)\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Pell | {
"line": 565,
"column": 35
} | {
"line": 565,
"column": 50
} | {
"line": 565,
"column": 50
} | [
{
"pp": "d : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\n⊢ d * a.y * (a.y * a₁.x) = (a.x ^ 2 - 1) * a₁.x",
"ppTerm": "?m.130",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Pell.Solution₁.x",
"HSub... | [
"d : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\n⊢ d * a.y * (a.y * a₁.x) = d * a.y ^ 2 * a₁.x"
] | rw [← a.prop_y] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 50,
"column": 36
} | {
"line": 57,
"column": 34
} | {
"line": 59,
"column": 0
} | [
{
"pp": "K : Type u_1\nΓ : Type u_2\ninst✝³ : Field K\ninst✝² : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\ninst✝¹ : DecidableEq K⟮X⟯\ninst✝ : IsTrivialOn K v\nhlt : 1 < v X\n⊢ v.IsEquiv (inftyValuation K)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"WithZero.instNont... | [] | by
refine isEquiv_iff_val_lt_one.mpr fun {f} ↦ ?_
rcases eq_or_ne f 0 with rfl | hf
· simp
· have hlt' : 1 < inftyValuation K X := by simp [← exp_zero]
rw [valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt hf,
valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt' hf]
g... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Rayleigh | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 25
} | {
"line": 70,
"column": 2
} | [
{
"pp": "r s : ℝ\nhrs : r.HolderConjugate s\n⊢ ∀ ⦃a : ℤ⦄, a ∈ {x | ∃ k, beattySeq r k = x} → a ∉ {x | ∃ k, beattySeq' s k = x}",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.Rayleigh.0.Beatty.no_collision.match_1_3",
"False",
"setOf",
... | [
"r s : ℝ\nhrs : r.HolderConjugate s\nj k : ℤ\nh₁ : beattySeq r k = j\nm : ℤ\nh₂ : beattySeq' s m = j\n⊢ False"
] | intro j ⟨k, h₁⟩ ⟨m, h₂⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 15
} | {
"line": 84,
"column": 16
} | [
{
"pp": "K : Type u_1\nΓ : Type u_2\ninst✝¹ : Field K\ninst✝ : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\nhne : {p | v ↑p < 1 ∧ p ≠ 0}.Nonempty\na b : K[X]\nhab : v ↑(a * b) < 1 ∧ a ≠ 0 ∧ b ≠ 0\nhb : ¬IsUnit b\nπᵥ : K[X] := ⋯.min {p | v ↑p < 1 ∧ p ≠ 0} hne\nhπᵥ : πᵥ = a * b\nhbpos : 0 < ↑b.natDegre... | [
"K : Type u_1\nΓ : Type u_2\ninst✝¹ : Field K\ninst✝ : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\nhne : {p | v ↑p < 1 ∧ p ≠ 0}.Nonempty\na b : K[X]\nhab : v ↑(a * b) < 1 ∧ a ≠ 0 ∧ b ≠ 0\nhb : ¬IsUnit b\nπᵥ : K[X] := ⋯.min {p | v ↑p < 1 ∧ p ≠ 0} hne\nhπᵥ : πᵥ = a * b\nhbpos : 0 < ↑b.natDegree\n⊢ 0 < b.n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 132,
"column": 15
} | {
"line": 132,
"column": 26
} | {
"line": 132,
"column": 27
} | [
{
"pp": "K : 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\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)",
"ppTerm": "?m.62",
"... | [
"K : 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\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Rayleigh | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 46
} | {
"line": 135,
"column": 0
} | [
{
"pp": "r s : ℝ\nhrs : r.HolderConjugate s\n⊢ {x | ∃ k, beattySeq' r k = x}ᶜ = {x | ∃ k, beattySeq s k = x}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"compl_compl",
"congrArg",
"Compl.compl",
"setOf",
"Real.HolderConjugate.symm",
"... | [] | rw [← compl_beattySeq hrs.symm, compl_compl] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Rayleigh | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 46
} | {
"line": 135,
"column": 0
} | [
{
"pp": "r s : ℝ\nhrs : r.HolderConjugate s\n⊢ {x | ∃ k, beattySeq' r k = x}ᶜ = {x | ∃ k, beattySeq s k = x}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"compl_compl",
"congrArg",
"Compl.compl",
"setOf",
"Real.HolderConjugate.symm",
"... | [] | rw [← compl_beattySeq hrs.symm, compl_compl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Rayleigh | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 46
} | {
"line": 135,
"column": 0
} | [
{
"pp": "r s : ℝ\nhrs : r.HolderConjugate s\n⊢ {x | ∃ k, beattySeq' r k = x}ᶜ = {x | ∃ k, beattySeq s k = x}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"compl_compl",
"congrArg",
"Compl.compl",
"setOf",
"Real.HolderConjugate.symm",
"... | [] | rw [← compl_beattySeq hrs.symm, compl_compl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.SelbergSieve | {
"line": 156,
"column": 8
} | {
"line": 156,
"column": 60
} | {
"line": 157,
"column": 4
} | [
{
"pp": "case h_ne\ns : BoundingSieve\nd : ℕ\nhdP : d ∣ s.prodPrimes\nhd_ne_one : d ≠ 1\nhd_sq : Squarefree d\nthis : d ≠ 0\n⊢ d.primeFactors.Nonempty",
"ppTerm": "?h_ne",
"assigned": true,
"usedConstants": [
"instOfNatNat",
"Nat",
"LT.lt",
"True",
"Finset.Nonempty",
... | [] | simp only [nonempty_primeFactors, show 1 < d by lia] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.SelbergSieve | {
"line": 156,
"column": 8
} | {
"line": 156,
"column": 60
} | {
"line": 157,
"column": 4
} | [
{
"pp": "case h_ne\ns : BoundingSieve\nd : ℕ\nhdP : d ∣ s.prodPrimes\nhd_ne_one : d ≠ 1\nhd_sq : Squarefree d\nthis : d ≠ 0\n⊢ d.primeFactors.Nonempty",
"ppTerm": "?h_ne",
"assigned": true,
"usedConstants": [
"instOfNatNat",
"Nat",
"LT.lt",
"True",
"Finset.Nonempty",
... | [] | simp only [nonempty_primeFactors, show 1 < d by lia] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.SelbergSieve | {
"line": 156,
"column": 8
} | {
"line": 156,
"column": 60
} | {
"line": 157,
"column": 4
} | [
{
"pp": "case h_ne\ns : BoundingSieve\nd : ℕ\nhdP : d ∣ s.prodPrimes\nhd_ne_one : d ≠ 1\nhd_sq : Squarefree d\nthis : d ≠ 0\n⊢ d.primeFactors.Nonempty",
"ppTerm": "?h_ne",
"assigned": true,
"usedConstants": [
"instOfNatNat",
"Nat",
"LT.lt",
"True",
"Finset.Nonempty",
... | [] | simp only [nonempty_primeFactors, show 1 < d by lia] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 169,
"column": 4
} | {
"line": 170,
"column": 11
} | {
"line": 170,
"column": 12
} | [
{
"pp": "case right\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\np : K[X]\nhp : p ≠ 0\nπ : K[X] := ⋯\nhne : {p | v ↑p < 1 ∧ p ≠ 0}.Nonempty\nhπirr : Irreducible π\nk : ℕ\nq : K[X]\... | [
"case right\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\np : K[X]\nhp : p ≠ 0\nπ : K[X] := πᵥ\nhne : {p | v ↑p < 1 ∧ p ≠ 0}.Nonempty\nhπirr : Irreducible π\nk : ℕ\nq : K[X]\nhnq : ¬q %... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 174,
"column": 27
} | {
"line": 174,
"column": 38
} | {
"line": 174,
"column": 39
} | [
{
"pp": "K : 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\nf : K⟮X⟯\nhf : f ≠ 0\n⊢ v ↑πᵥ ≠ 0",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"K : 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\nf : K⟮X⟯\nhf : f ≠ 0\n⊢ ¬πᵥ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 186,
"column": 27
} | {
"line": 186,
"column": 38
} | {
"line": 186,
"column": 39
} | [
{
"pp": "K : 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\n⊢ v ↑πᵥ ≠ 0",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"K : 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\n⊢ ¬πᵥ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.SelbergSieve | {
"line": 307,
"column": 40
} | {
"line": 307,
"column": 51
} | {
"line": 307,
"column": 52
} | [
{
"pp": "s : BoundingSieve\nl : ℕ\nhl : Squarefree l\nhnu_nonzero : s.nu l ≠ 0\nd e : ℕ\nhd : (d, e) ∈ l.divisorsAntidiagonal\n⊢ d * e = l ∧ ?m.92",
"ppTerm": "?m.94",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"s : BoundingSieve\nl : ℕ\nhl : Squarefree l\nhnu_nonzero : s.nu l ≠ 0\nd e : ℕ\nhd : (d, e) ∈ l.divisorsAntidiagonal\n⊢ d * e = l ∧ ?m.92"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.SelbergSieve | {
"line": 308,
"column": 42
} | {
"line": 308,
"column": 79
} | {
"line": 308,
"column": 80
} | [
{
"pp": "s : BoundingSieve\nd e : ℕ\nhl : Squarefree (d * e)\nhnu_nonzero : s.nu (d * e) ≠ 0\nhd : (d, e) ∈ (d * e).divisorsAntidiagonal\n⊢ d.Coprime e ∧ ?m.123",
"ppTerm": "?m.125",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"s : BoundingSieve\nd e : ℕ\nhl : Squarefree (d * e)\nhnu_nonzero : s.nu (d * e) ≠ 0\nhd : (d, e) ∈ (d * e).divisorsAntidiagonal\n⊢ d.Coprime e ∧ ?m.123"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 194,
"column": 56
} | {
"line": 194,
"column": 67
} | {
"line": 194,
"column": 68
} | [
{
"pp": "x y : ℤ[i]\nthis : |2⁻¹| = 2⁻¹\n⊢ |(toComplex x / toComplex y).re - ↑(round (toComplex x / toComplex y).re)| ≤ 2⁻¹",
"ppTerm": "?m.149",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℤ[i]\nthis : |2⁻¹| = 2⁻¹\n⊢ |(toComplex x / toComplex y).re - ↑(round (toComplex x / toComplex y).re)| ≤ 2⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 195,
"column": 56
} | {
"line": 195,
"column": 67
} | {
"line": 195,
"column": 68
} | [
{
"pp": "x y : ℤ[i]\nthis : |2⁻¹| = 2⁻¹\n⊢ |(toComplex x / toComplex y).im - ↑(round (toComplex x / toComplex y).im)| ≤ 2⁻¹",
"ppTerm": "?m.171",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℤ[i]\nthis : |2⁻¹| = 2⁻¹\n⊢ |(toComplex x / toComplex y).im - ↑(round (toComplex x / toComplex y).im)| ≤ 2⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 58
} | {
"line": 250,
"column": 59
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nhpi : ¬Irreducible ↑p\nhpu : ¬IsUnit ↑p\n⊢ ∃ a b, ↑p = a * b ∧ ¬IsUnit a ∧ ¬IsUnit b",
"ppTerm": "?m.84",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nhpi : ¬Irreducible ↑p\nhpu : ¬IsUnit ↑p\n⊢ ∃ a b, ↑p = a * b ∧ ¬IsUnit a ∧ ¬IsUnit b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 253,
"column": 91
} | {
"line": 254,
"column": 85
} | {
"line": 254,
"column": 85
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nhpi : ¬Irreducible ↑p\nhpu : ¬IsUnit ↑p\nhab : ∃ a b, ↑p = a * b ∧ ¬IsUnit a ∧ ¬IsUnit b\na b : ℤ[i]\nhpab : ↑p = a * b\nhau : ¬IsUnit a\nhbu : ¬IsUnit b\n⊢ (norm a).natAbs * (norm b).natAbs = p ^ 2",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
... | [] | by
rw [← Int.natCast_inj, Int.natCast_pow, sq, ← @norm_natCast (-1), hpab]; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 255,
"column": 32
} | {
"line": 255,
"column": 64
} | {
"line": 255,
"column": 65
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nhpi : ¬Irreducible ↑p\nhpu : ¬IsUnit ↑p\nhab : ∃ a b, ↑p = a * b ∧ ¬IsUnit a ∧ ¬IsUnit b\na b : ℤ[i]\nhpab : ↑p = a * b\nhau : ¬IsUnit a\nhbu : ¬IsUnit b\nhnap : (norm a).natAbs = p\n⊢ a.re.natAbs ^ 2 + a.im.natAbs ^ 2 = p",
"ppTerm": "?m.127",
"assigned": true,
... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nhpi : ¬Irreducible ↑p\nhpu : ¬IsUnit ↑p\nhab : ∃ a b, ↑p = a * b ∧ ¬IsUnit a ∧ ¬IsUnit b\na b : ℤ[i]\nhpab : ↑p = a * b\nhau : ¬IsUnit a\nhbu : ¬IsUnit b\nhnap : (norm a).natAbs = p\n⊢ a.re.natAbs * a.re.natAbs + a.im.natAbs * a.im.natAbs = p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.SumFourSquares | {
"line": 130,
"column": 20
} | {
"line": 130,
"column": 31
} | {
"line": 130,
"column": 32
} | [
{
"pp": "case h\np : ℕ\nhp : Prime p\nthis : Fact (Prime p)\nnatAbs_iff :\n ∀ {a b c d : ℤ} {k : ℕ},\n a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k\nhm : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p\na b c d : ℕ\nhmin : ∀ m < 1, ¬(m ... | [
"case h\np : ℕ\nhp : Prime p\nthis : Fact (Prime p)\nnatAbs_iff :\n ∀ {a b c d : ℤ} {k : ℕ},\n a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k\nhm : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p\na b c d : ℕ\nhmin : ∀ m < 1, ¬(m < p ∧ 0 < m ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.SumTwoSquares | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 42
} | {
"line": 92,
"column": 43
} | [
{
"pp": "m n : ℕ\nhc : m.Coprime n\nhm : IsSquare (-1)\nhn : IsSquare (-1)\nthis : IsSquare (-1)\n⊢ IsSquare (-1)",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"m n : ℕ\nhc : m.Coprime n\nhm : IsSquare (-1)\nhn : IsSquare (-1)\nthis : IsSquare (-1)\n⊢ IsSquare (-1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.SumFourSquares | {
"line": 161,
"column": 63
} | {
"line": 161,
"column": 86
} | {
"line": 161,
"column": 87
} | [
{
"pp": "p : ℕ\nhp : Prime p\nthis✝ : Fact (Prime p)\nnatAbs_iff :\n ∀ {a b c d : ℤ} {k : ℕ},\n a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k\nhm✝ : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p\nm : ℕ\nhmin : ∀ m_1 < m, ¬(m_1 < p ∧ 0 ... | [
"p : ℕ\nhp : Prime p\nthis✝ : Fact (Prime p)\nnatAbs_iff :\n ∀ {a b c d : ℤ} {k : ℕ},\n a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k\nhm✝ : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p\nm : ℕ\nhmin : ∀ m_1 < m, ¬(m_1 < p ∧ 0 < m_1 ∧ ∃ a ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 33
} | {
"line": 203,
"column": 2
} | [
{
"pp": "case h\nR : Type u\ninst✝¹⁵ : CommRing R\nS : Type v\ninst✝¹⁴ : CommRing S\ninst✝¹³ : Algebra R S\nK : Type u_1\ninst✝¹² : Field K\ninst✝¹¹ : Algebra R K\nV : Type u_3\nV' : Type u_4\nV'' : Type u_5\ninst✝¹⁰ : AddCommGroup V\ninst✝⁹ : Module R V\ninst✝⁸ : Module K V\ninst✝⁷ : IsScalarTower R K V\ninst✝... | [
"case h\nR : Type u\ninst✝¹⁵ : CommRing R\nS : Type v\ninst✝¹⁴ : CommRing S\ninst✝¹³ : Algebra R S\nK : Type u_1\ninst✝¹² : Field K\ninst✝¹¹ : Algebra R K\nV : Type u_3\nV' : Type u_4\nV'' : Type u_5\ninst✝¹⁰ : AddCommGroup V\ninst✝⁹ : Module R V\ninst✝⁸ : Module K V\ninst✝⁷ : IsScalarTower R K V\ninst✝⁶ : AddCommG... | letI := Classical.propDecidable | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.NumberTheory.SumTwoSquares | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 90
} | {
"line": 230,
"column": 0
} | [
{
"pp": "case inr.refine_2\nn : ℕ\nhn₀ : n > 0\nH : ∀ q ∈ n.primeFactors, q % 4 = 3 → Even (padicValNat q n)\nb a : ℕ\nhb₀ : 0 < b\nha₀ : 0 < a\nhab : a ^ 2 * b = n\nhb : Squarefree b\nq : ℕ\nhq : q ∈ b.primeFactors\nhq4 : q % 4 = 3\nthis✝¹ : Fact (Prime q)\nthis✝ : n ≠ 0 → b.primeFactors ⊆ n.primeFactors\nthis... | [] | grind [factorization_def, prime_of_mem_primeFactors, padicValNat.mul, padicValNat.pow] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.NumberTheory.SumFourSquares | {
"line": 191,
"column": 8
} | {
"line": 192,
"column": 30
} | {
"line": 192,
"column": 31
} | [
{
"pp": "p : ℕ\nhp : Prime p\nthis✝² : Fact (Prime p)\nnatAbs_iff :\n ∀ {a b c d : ℤ} {k : ℕ},\n a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k\nhm✝ : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p\nm : ℕ\nhmin : ∀ m_1 < m, ¬(m_1 < p ∧ 0... | [
"p : ℕ\nhp : Prime p\nthis✝² : Fact (Prime p)\nnatAbs_iff :\n ∀ {a b c d : ℤ} {k : ℕ},\n a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k\nhm✝ : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p\nm : ℕ\nhmin : ∀ m_1 < m, ¬(m_1 < p ∧ 0 < m_1 ∧ ∃ a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Irrational | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 35
} | {
"line": 95,
"column": 36
} | [
{
"pp": "x : ℝ\nhx : Irrational x\nn : ℕ\nε : ℝ\nH : ∀ k ≤ n, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑k)\nr : ℚ\nhr : r.den ≤ n\n⊢ ε ≤ dist x ↑r",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.instLE",
"Real",
"Rat.num",
"instHDiv",
... | [
"x : ℝ\nhx : Irrational x\nn : ℕ\nε : ℝ\nH : ∀ k ≤ n, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑k)\nr : ℚ\nhr : r.den ≤ n\n⊢ ε ≤ dist x (↑r.num / ↑r.den)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart | {
"line": 179,
"column": 6
} | {
"line": 179,
"column": 26
} | {
"line": 179,
"column": 26
} | [
{
"pp": "case h.right.refine_1\nf : ℤ[X]\nhf : eval 0 f ≠ 0\nc' : ℂ → ℝ\nc'0 : ∀ (s : ℂ), c' s ≥ 0\nPp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p\np : ℕ\np_gt : p > (eval 0 f).natAbs\nprime_p : Nat.Prime p\ngp' : ℤ[X]\nh' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^... | [
"case h.right.refine_1\nf : ℤ[X]\nhf : eval 0 f ≠ 0\nc' : ℂ → ℝ\nc'0 : ∀ (s : ℂ), c' s ≥ 0\nPp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p\np : ℕ\np_gt : p > (eval 0 f).natAbs\nprime_p : Nat.Prime p\ngp' : ℤ[X]\nh' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - ... | add_tsub_cancel_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 39
} | {
"line": 84,
"column": 2
} | [
{
"pp": "p x C : ℝ\nhC : ∃ᶠ (n : ℕ) in atTop, ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p\nn : ℕ\nhle : 1 ≤ n\nm : ℤ\nhne : x ≠ ↑m / ↑n\nhlt : |x - ↑m / ↑n| < C / ↑n ^ p\n⊢ 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < max C 1 / ↑n ^ p",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
... | [
"p x C : ℝ\nhC : ∃ᶠ (n : ℕ) in atTop, ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p\nn : ℕ\nhle : 1 ≤ n\nm : ℤ\nhne : x ≠ ↑m / ↑n\nhlt : |x - ↑m / ↑n| < C / ↑n ^ p\n⊢ C / ↑n ^ p ≤ max C 1 / ↑n ^ p"
] | refine ⟨hle, m, hne, hlt.trans_le ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleNumber | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 41
} | {
"line": 189,
"column": 42
} | [
{
"pp": "m : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑m ^ n !\nhpos : 0 < remainder (↑m) n\n⊢ partialSum (↑m) n + remainder (↑m) n ≠ partialSum (↑m) n ∧\n |partialSum (↑m) n + remainder (↑m) n - partialSum (↑m) n| < 1 / (↑m ^ n !) ^ n",
"ppTerm": "?m.78",
"ass... | [
"m : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑m ^ n !\nhpos : 0 < remainder (↑m) n\n⊢ remainder (↑m) n < ((↑m ^ n !) ^ n)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 48
} | {
"line": 102,
"column": 6
} | [
{
"pp": "p q x : ℝ\nh : LiouvilleWith p x\nhlt : q < p\nC : ℝ\n_hC₀ : 0 < C\nhC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p\n⊢ ∀ᶠ (n : ℕ) in atTop, C < ↑n ^ (p - q)",
"ppTerm": "?m.64",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"p q x : ℝ\nh : LiouvilleWith p x\nhlt : q < p\nC : ℝ\n_hC₀ : 0 < C\nhC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p\n⊢ ∀ᶠ (n : ℕ) in atTop, C < ↑n ^ (p - q)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 91
} | {
"line": 132,
"column": 6
} | [
{
"pp": "p x : ℝ\nr : ℚ\nhr : r ≠ 0\nh : LiouvilleWith p (x * ↑r)\n⊢ LiouvilleWith p x",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p x : ℝ\nr : ℚ\nhr : r ≠ 0\nh : LiouvilleWith p (x * ↑r)\n⊢ LiouvilleWith p x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 183,
"column": 15
} | {
"line": 183,
"column": 26
} | {
"line": 183,
"column": 27
} | [
{
"pp": "p x : ℝ\nr : ℚ\nh : LiouvilleWith p (x + ↑r)\n⊢ LiouvilleWith p x",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p x : ℝ\nr : ℚ\nh : LiouvilleWith p (x + ↑r)\n⊢ LiouvilleWith p x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Liouville.Measure | {
"line": 54,
"column": 30
} | {
"line": 54,
"column": 40
} | {
"line": 54,
"column": 41
} | [
{
"pp": "p : ℝ\nhp : p > 2\nn : ℕ\nhn : 2 + 1 / (↑n + 1) < p\nx : ℝ\nhxp : LiouvilleWith p x\nhx01 : x ∈ Ico 0 1\nb : ℕ\nhb : 1 ≤ b\na : ℤ\nhlt : |x - ↑a / ↑b| < (↑b ^ (2 + 1 / (↑n + 1)))⁻¹\n⊢ ∃ a ∈ Finset.Icc 0 ↑b, |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑(n + 1))",
"ppTerm": "?m.308",
"assigned": true,
... | [
"p : ℝ\nhp : p > 2\nn : ℕ\nhn : 2 + 1 / (↑n + 1) < p\nx : ℝ\nhxp : LiouvilleWith p x\nhx01 : x ∈ Ico 0 1\nb : ℕ\nhb : 1 ≤ b\na : ℤ\nhlt : |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / (↑n + 1))\n⊢ ∃ a ∈ Finset.Icc 0 ↑b, |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑(n + 1))"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Transcendental.Liouville.Measure | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 15
} | {
"line": 64,
"column": 16
} | [
{
"pp": "p : ℝ\nhp : p > 2\nn : ℕ\nhn : 2 + 1 / (↑n + 1) < p\nx : ℝ\nhxp : LiouvilleWith p x\nhx01 : x ∈ Ico 0 1\nb : ℕ\na : ℤ\nhlt : |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑n.succ)\nhb : 1 ≤ ↑b\nhb0 : 0 < ↑b\n⊢ 2 ≤ 2 + 1 / ↑(n + 1)",
"ppTerm": "?m.445",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"p : ℝ\nhp : p > 2\nn : ℕ\nhn : 2 + 1 / (↑n + 1) < p\nx : ℝ\nhxp : LiouvilleWith p x\nhx01 : x ∈ Ico 0 1\nb : ℕ\na : ℤ\nhlt : |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑n.succ)\nhb : 1 ≤ ↑b\nhb0 : 0 < ↑b\n⊢ 0 ≤ ↑n + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 275,
"column": 20
} | {
"line": 275,
"column": 30
} | {
"line": 275,
"column": 31
} | [
{
"pp": "p : ℝ\nhp : 1 < p\nM : ℤ\nh : LiouvilleWith p ↑M\nn : ℕ\nhn : 0 < n\nm : ℤ\nhne : ↑M ≠ ↑m / ↑n\nhlt : |↑M - ↑m / ↑n| < ↑n ^ (-1)\nhn' : 0 < ↑n\n⊢ (↑n)⁻¹ ≤ |↑M - ↑m / ↑n|",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"MulOne.toOne",
... | [
"p : ℝ\nhp : 1 < p\nM : ℤ\nh : LiouvilleWith p ↑M\nn : ℕ\nhn : 0 < n\nm : ℤ\nhne : ↑M ≠ ↑m / ↑n\nhlt : |↑M - ↑m / ↑n| < ↑n ^ (-1)\nhn' : 0 < ↑n\n⊢ 1 / ↑n ≤ |↑M - ↑m / ↑n|"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Transcendental.Liouville.Measure | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 72
} | {
"line": 111,
"column": 4
} | [
{
"pp": "⊢ ∀ᵐ (x : ℝ), ∀ p > 2, ¬LiouvilleWith p x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"Iff.of_eq",
"congrArg",
"_private.Mathlib.NumberTheory.Transcendental.Liouville.Measu... | [
"⊢ volume (⋃ i, ⋃ (_ : i > 2), {x | LiouvilleWith i x}) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Liouville.Measure | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 46
} | {
"line": 119,
"column": 47
} | [
{
"pp": "⊢ volume {x | Liouville x} = 0",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ volume {x | Liouville x} = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart | {
"line": 204,
"column": 4
} | {
"line": 204,
"column": 44
} | {
"line": 204,
"column": 45
} | [
{
"pp": "f : ℤ[X]\nhf : eval 0 f ≠ 0\nc' : ℂ → ℝ\nc'0 : ∀ (s : ℂ), c' s ≥ 0\nPp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p\np : ℕ\np_gt : p > (eval 0 f).natAbs\nprime_p : Nat.Prime p\ngp' : ℤ[X]\nh' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - 1)! • eval ... | [
"f : ℤ[X]\nhf : eval 0 f ≠ 0\nc' : ℂ → ℝ\nc'0 : ∀ (s : ℂ), c' s ≥ 0\nPp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p\np : ℕ\np_gt : p > (eval 0 f).natAbs\nprime_p : Nat.Prime p\ngp' : ℤ[X]\nh' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - 1)! • eval 0 (f ^ p) + ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 35
} | {
"line": 206,
"column": 36
} | [
{
"pp": "case h.right.refine_2\nf : ℤ[X]\nhf : eval 0 f ≠ 0\nc' : ℂ → ℝ\nc'0 : ∀ (s : ℂ), c' s ≥ 0\nPp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p\np : ℕ\np_gt : p > (eval 0 f).natAbs\nprime_p : Nat.Prime p\ngp' : ℤ[X]\nh' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^... | [
"case h.right.refine_2\nf : ℤ[X]\nhf : eval 0 f ≠ 0\nc' : ℂ → ℝ\nc'0 : ∀ (s : ℂ), c' s ≥ 0\nPp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p\np : ℕ\np_gt : p > (eval 0 f).natAbs\nprime_p : Nat.Prime p\ngp' : ℤ[X]\nh' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Wilson | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 58
} | {
"line": 66,
"column": 59
} | [
{
"pp": "case refine_3.refine_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp : 0 < p\nb : ℕ\nhb : b ≠ 0 ∧ b < p\nh : (↑b).val = val 0\n⊢ b = 0",
"ppTerm": "?refine_3.refine_1",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_3.refine_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp : 0 < p\nb : ℕ\nhb : b ≠ 0 ∧ b < p\nh : (↑b).val = val 0\n⊢ b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Wilson | {
"line": 44,
"column": 2
} | {
"line": 68,
"column": 48
} | {
"line": 70,
"column": 0
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ ↑(p - 1)! = -1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Finset.mem_univ",
"Units.val",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | refine
calc
((p - 1)! : ZMod p) = ∏ x ∈ Ico 1 (succ (p - 1)), (x : ZMod p) := by
rw [← Finset.prod_Ico_id_eq_factorial, prod_natCast]
_ = ∏ x : (ZMod p)ˣ, (x : ZMod p) := ?_
_ = -1 := by
simp_rw [← Units.coeHom_apply, ← map_prod (Units.coeHom (ZMod p)),
prod_univ_units_id... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Wilson | {
"line": 44,
"column": 2
} | {
"line": 68,
"column": 48
} | {
"line": 70,
"column": 0
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ ↑(p - 1)! = -1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Finset.mem_univ",
"Units.val",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | refine
calc
((p - 1)! : ZMod p) = ∏ x ∈ Ico 1 (succ (p - 1)), (x : ZMod p) := by
rw [← Finset.prod_Ico_id_eq_factorial, prod_natCast]
_ = ∏ x : (ZMod p)ˣ, (x : ZMod p) := ?_
_ = -1 := by
simp_rw [← Units.coeHom_apply, ← map_prod (Units.coeHom (ZMod p)),
prod_univ_units_id... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 340,
"column": 36
} | {
"line": 340,
"column": 58
} | {
"line": 340,
"column": 59
} | [
{
"pp": "x : ℝ\nH : ∀ (p : ℝ), LiouvilleWith p x\nn b : ℕ\nhb : 1 < b\na : ℤ\nhne : x ≠ ↑a / ↑b\nhlt : |x - ↑a / ↑b| < ↑b ^ (-↑n)\n⊢ |x - ↑a / ↑↑b| < 1 / ↑↑b ^ n",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"Real",
... | [
"x : ℝ\nH : ∀ (p : ℝ), LiouvilleWith p x\nn b : ℕ\nhb : 1 < b\na : ℤ\nhne : x ≠ ↑a / ↑b\nhlt : |x - ↑a / ↑b| < ↑b ^ (-↑n)\n⊢ |x - ↑a / ↑b| < (↑b ^ n)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Bounds.Lattice | {
"line": 31,
"column": 2
} | {
"line": 32,
"column": 9
} | {
"line": 32,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\n⊢ GaloisConnection (⇑OrderDual.toDual ∘ upperBounds) (lowerBounds ∘ ⇑OrderDual.ofDual)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"OrderDual.instLE",
"OrderDual.toDual",
"Eq.mpr",
"Equiv.instEquivLike",
"OrderDua... | [
"α : Type u_1\ninst✝ : Preorder α\n⊢ ∀ (a a_1 : Set α), (∀ x ∈ a_1, ∀ x_1 ∈ a, x_1 ≤ x) ↔ ∀ x ∈ a, ∀ x_1 ∈ a_1, x ≤ x_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.WellApproximable | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 65
} | {
"line": 154,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝ : SeminormedCommGroup A\na : A\nn : ℕ\nδ : ℝ\nhn : 0 < n\nhan : ∀ {b : A}, orderOf b = n → orderOf (a * b) = n\nf : ↑{b | orderOf b = n} → ↑{b | orderOf b = n} := fun b ↦ ⟨a * ↑b, ⋯⟩\nhf : Surjective f\n⊢ ⋃ i, ⋃ (_ : orderOf i = n), ball (a * i) δ = ⋃ x, ⋃ (_ : orderOf x = n), ball ... | [
"A : Type u_1\ninst✝ : SeminormedCommGroup A\na : A\nn : ℕ\nδ : ℝ\nhn : 0 < n\nhan : ∀ {b : A}, orderOf b = n → orderOf (a * b) = n\nf : ↑{b | orderOf b = n} → ↑{b | orderOf b = n} := fun b ↦ ⟨a * ↑b, ⋯⟩\nhf : Surjective f\n⊢ ⋃ i, ⋃ (_ : orderOf i = n), ball (a * i) δ = ⋃ x, ⋃ (_ : orderOf x = n), ball x δ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Comparable | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 13
} | {
"line": 347,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : PartialOrder α\na b : α\n⊢ a < b ∨ a = b ∨ b < a ∨ IncompRel (fun x1 x2 ↦ x1 ≤ x2) a b",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : PartialOrder α\na b : α\n⊢ a < b ∨ a = b ∨ b < a ∨ IncompRel (fun x1 x2 ↦ x1 ≤ x2) a b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.CompletePartialOrder | {
"line": 87,
"column": 2
} | {
"line": 89,
"column": 16
} | {
"line": 91,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : CompletePartialOrder α\ninst✝ : Preorder β\nf : α → β\n⊢ ScottContinuous f ↔ ∀ ⦃d : Set α⦄, d.Nonempty → DirectedOn (fun x1 x2 ↦ x1 ≤ x2) d → IsLUB (f '' d) (f (sSup d))",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"con... | [] | refine ⟨fun h d hd₁ hd₂ ↦ h hd₁ hd₂ hd₂.isLUB_sSup, fun h d hne hd a hda ↦ ?_⟩
rw [hda.unique hd.isLUB_sSup]
exact h hne hd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.CompletePartialOrder | {
"line": 87,
"column": 2
} | {
"line": 89,
"column": 16
} | {
"line": 91,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : CompletePartialOrder α\ninst✝ : Preorder β\nf : α → β\n⊢ ScottContinuous f ↔ ∀ ⦃d : Set α⦄, d.Nonempty → DirectedOn (fun x1 x2 ↦ x1 ≤ x2) d → IsLUB (f '' d) (f (sSup d))",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"con... | [] | refine ⟨fun h d hd₁ hd₂ ↦ h hd₁ hd₂ hd₂.isLUB_sSup, fun h d hne hd a hda ↦ ?_⟩
rw [hda.unique hd.isLUB_sSup]
exact h hne hd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Completion | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 13
} | {
"line": 106,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\na b : α\n⊢ principal a ≤ principal b ↔ a ≤ b",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : Preorder α\na b : α\n⊢ principal a ≤ principal b ↔ a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Completion | {
"line": 180,
"column": 2
} | {
"line": 184,
"column": 6
} | {
"line": 186,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : PartialOrder β\nf : β ↪o α\nx : β\n⊢ (factorEmbedding f) (principal x) = f x",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"sSup_le_iff._simp_2",
"Eq.mpr",
"instReflLe",
"congrArg",
"Se... | [] | rw [factorEmbedding_apply]
apply le_antisymm (by simp)
rw [le_sSup_iff]
refine fun y hy ↦ hy ?_
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Completion | {
"line": 180,
"column": 2
} | {
"line": 184,
"column": 6
} | {
"line": 186,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : PartialOrder β\nf : β ↪o α\nx : β\n⊢ (factorEmbedding f) (principal x) = f x",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"sSup_le_iff._simp_2",
"Eq.mpr",
"instReflLe",
"congrArg",
"Se... | [] | rw [factorEmbedding_apply]
apply le_antisymm (by simp)
rw [le_sSup_iff]
refine fun y hy ↦ hy ?_
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Completion | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 52
} | {
"line": 227,
"column": 53
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b : DedekindCut α\nh : a.extent ⊆ b.extent ∧ ∃ x ∈ b.extent, x ∉ a.extent\n⊢ ∃ c, a < principal c ∧ principal c ≤ b",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.t... | [
"α : Type u_1\ninst✝ : LinearOrder α\na b : DedekindCut α\nh : a.extent ⊆ b.extent ∧ ∃ x ∈ b.extent, x ∉ a.extent\n⊢ ∃ c ∉ a.left, c ∈ b.left"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Completion | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 42
} | {
"line": 234,
"column": 43
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b : DedekindCut α\nh : b.intent ⊆ a.intent ∧ ∃ x ∈ a.intent, x ∉ b.intent\n⊢ ∃ c, a ≤ principal c ∧ principal c < b",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.t... | [
"α : Type u_1\ninst✝ : LinearOrder α\na b : DedekindCut α\nh : b.intent ⊆ a.intent ∧ ∃ x ∈ a.intent, x ∉ b.intent\n⊢ ∃ c ∈ a.right, c ∉ b.right"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Completion | {
"line": 243,
"column": 4
} | {
"line": 247,
"column": 74
} | {
"line": 249,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : DenselyOrdered α\na b : DedekindCut α\nh : a < b\n⊢ ∃ a_1, a < a_1 ∧ a_1 < b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"DedekindCut.principal_lt_principal._simp_1",
"Eq.mpr",
"Preorder.toLT",
... | [] | obtain ⟨c, hac, hcb⟩ := lt_iff_exists.mp h
obtain ⟨d, had, hdc⟩ := lt_iff_exists'.mp hac
simp only [principal_lt_principal] at hdc
obtain ⟨u, _, _⟩ := DenselyOrdered.dense d c hdc
exact ⟨principal u, had.trans_lt (by simpa), hcb.trans_lt' (by simpa)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Completion | {
"line": 243,
"column": 4
} | {
"line": 247,
"column": 74
} | {
"line": 249,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : DenselyOrdered α\na b : DedekindCut α\nh : a < b\n⊢ ∃ a_1, a < a_1 ∧ a_1 < b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"DedekindCut.principal_lt_principal._simp_1",
"Eq.mpr",
"Preorder.toLT",
... | [] | obtain ⟨c, hac, hcb⟩ := lt_iff_exists.mp h
obtain ⟨d, had, hdc⟩ := lt_iff_exists'.mp hac
simp only [principal_lt_principal] at hdc
obtain ⟨u, _, _⟩ := DenselyOrdered.dense d c hdc
exact ⟨principal u, had.trans_lt (by simpa), hcb.trans_lt' (by simpa)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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