module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 115,
"column": 12
} | {
"line": 115,
"column": 23
} | {
"line": 115,
"column": 24
} | [
{
"pp": "case inr.zero\np : ℕ\nhp : Nat.Prime p\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nm : ℕ\nhm : m > 0\nhmn : m ≤ m + 0\nh : Irreducible (cyclotomic (p ^ (m + 0)) R)\n⊢ Irreducible (cyclotomic (p ^ m) R)",
"ppTerm": "?inr.zero",
"assigned": false,
"usedConstants": [],
"usedFVa... | [
"case inr.zero\np : ℕ\nhp : Nat.Prime p\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nm : ℕ\nhm : m > 0\nhmn : m ≤ m + 0\nh : Irreducible (cyclotomic (p ^ (m + 0)) R)\n⊢ Irreducible (cyclotomic (p ^ m) R)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 91,
"column": 8
} | {
"line": 91,
"column": 69
} | {
"line": 91,
"column": 70
} | [
{
"pp": "case h.inl.hb.inr.inl\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nn : ℕ\nih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)\nhn : 2 < n\nhn' : 0 < n\nhn'' : 1 < n\nthis : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x ... | [
"case h.inl.hb.inr.inl\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nn : ℕ\nih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)\nhn : 2 < n\nhn' : 0 < n\nhn'' : 1 < n\nthis : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ r... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 111,
"column": 6
} | {
"line": 111,
"column": 67
} | {
"line": 111,
"column": 68
} | [
{
"pp": "case hb\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nn : ℕ\nih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)\nhn : 2 < n\nhn' : 0 < n\nhn'' : 1 < n\nthis : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ ran... | [
"case hb\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nn : ℕ\nih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)\nhn : 2 < n\nhn' : 0 < n\nhn'' : 1 < n\nthis : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 66
} | {
"line": 144,
"column": 67
} | [
{
"pp": "case inr.inl\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\nh✝ : ∀ {p : ℕ}, Nat.Prime p → ∀ (k : ℕ), p ^ k ≠ n\nhn' : n > 0\nhn : 1 < n\nh : eval 1 (cyclotomic n ℤ) = 1\nthis : eval (↑1) (map (Int.castRingHom R) (cyclotomic n ℤ)) = (Int.castRingHom R) (eval (↑1) (cyclotomic n ℤ))\n⊢ eval 1 (cyclotomic n R) = 1"... | [
"case inr.inl\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\nh✝ : ∀ {p : ℕ}, Nat.Prime p → ∀ (k : ℕ), p ^ k ≠ n\nhn' : n > 0\nhn : 1 < n\nh : eval 1 (cyclotomic n ℤ) = 1\nthis : eval (↑1) (map (Int.castRingHom R) (cyclotomic n ℤ)) = (Int.castRingHom R) (eval (↑1) (cyclotomic n ℤ))\n⊢ eval 1 (cyclotomic n R) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 195,
"column": 47
} | {
"line": 195,
"column": 58
} | {
"line": 195,
"column": 59
} | [
{
"pp": "n : ℕ\nq : ℝ\nhn' : 2 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nhex : ∃ ζ' ∈ primitiveRoots n ℂ, q - 1 < ‖↑q - ζ'‖\n⊢ ¬eval (↑q) (cyclotomic n ℂ) = 0",
"ppTerm": "?m.... | [
"n : ℕ\nq : ℝ\nhn' : 2 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nhex : ∃ ζ' ∈ primitiveRoots n ℂ, q - 1 < ‖↑q - ζ'‖\n⊢ ¬eval q (cyclotomic n ℝ) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 108,
"column": 35
} | {
"line": 108,
"column": 94
} | {
"line": 108,
"column": 95
} | [
{
"pp": "n : ℕ\ninst✝⁶ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nh : IsCyclotomicExtension ∅ A B\n⊢ ⊥ = ⊤",
"ppTerm": "?m.18",
"assigned": true,
"us... | [
"n : ℕ\ninst✝⁶ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nh : IsCyclotomicExtension ∅ A B\n⊢ ∀ (x : B), x ∈ ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 38
} | {
"line": 117,
"column": 39
} | [
{
"pp": "A : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nh : IsCyclotomicExtension {1} A B\nx : B\n⊢ x ∈ ⊥",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nh : IsCyclotomicExtension {1} A B\nx : B\n⊢ x ∈ ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 240,
"column": 4
} | {
"line": 240,
"column": 22
} | {
"line": 240,
"column": 23
} | [
{
"pp": "case inr\nn : ℕ\ninst✝³ : NeZero n\nK : Type u\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nζ : K\nl : ℕ\nhζ : IsPrimitiveRoot ζ l\nhl✝ : l ≠ 0\nhl : NeZero l\nhroot : IsPrimitiveRoot (zeta n ℚ K) n\nr : ℕ\nhr : GCDMonoid.lcm l n = 2 * n\nineq : φ n * φ r ≤ φ (GCDMo... | [
"case inr\nn : ℕ\ninst✝³ : NeZero n\nK : Type u\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nζ : K\nl : ℕ\nhζ : IsPrimitiveRoot ζ l\nhl✝ : l ≠ 0\nhl : NeZero l\nhroot : IsPrimitiveRoot (zeta n ℚ K) n\nr : ℕ\nhr : GCDMonoid.lcm l n = 2 * n\nineq : φ n * φ r ≤ φ (GCDMonoid.lcm l n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 50
} | {
"line": 212,
"column": 51
} | [
{
"pp": "case convert_5\nn : ℕ\nq : ℝ\nhn' : 2 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nhex : ∃ ζ' ∈ primitiveRoots n ℂ, q - 1 < ‖↑q - ζ'‖\nthis : ¬eval (↑q) (cyclotomic n ℂ) = 0... | [
"case convert_5\nn : ℕ\nq : ℝ\nhn' : 2 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nhex : ∃ ζ' ∈ primitiveRoots n ℂ, q - 1 < ‖↑q - ζ'‖\nthis : ¬eval (↑q) (cyclotomic n ℂ) = 0\nx : ℂ\nhx ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 215,
"column": 4
} | {
"line": 215,
"column": 69
} | {
"line": 215,
"column": 70
} | [
{
"pp": "case convert_6\nn : ℕ\nq : ℝ\nhn' : 2 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nhex : ∃ ζ' ∈ primitiveRoots n ℂ, q - 1 < ‖↑q - ζ'‖\nthis : ¬eval (↑q) (cyclotomic n ℂ) = 0... | [
"case convert_6\nn : ℕ\nq : ℝ\nhn' : 2 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nhex : ∃ ζ' ∈ primitiveRoots n ℂ, q - 1 < ‖↑q - ζ'‖\nthis : ¬eval (↑q) (cyclotomic n ℂ) = 0\n⊢ ∃ a ∈ pr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 111,
"column": 6
} | {
"line": 111,
"column": 49
} | {
"line": 112,
"column": 4
} | [
{
"pp": "case succ.refine_1.inl\nn : ℕ\nh : n ≠ 0\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nthis : Fintype (GaloisField (n✝ + 1) n)\ng_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X\nhp : 1 < n✝ + 1\naux : X ^ (n✝ + 1) ^ n - X ≠ 0\nkey : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ... | [] | rw [← map_pow, ZMod.pow_card_pow, sub_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 259,
"column": 47
} | {
"line": 259,
"column": 58
} | {
"line": 259,
"column": 59
} | [
{
"pp": "n : ℕ\nq : ℝ\nhn' : 3 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, ‖↑q - ζ'‖ ≤ q + 1\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nhex : ∃ ζ' ∈ primitiveRoots n ℂ, ‖↑q - ζ'‖ < q + 1\n⊢ ¬eval (↑q) (cyclotomic n ℂ) = 0",
"ppTerm": "?m.... | [
"n : ℕ\nq : ℝ\nhn' : 3 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, ‖↑q - ζ'‖ ≤ q + 1\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nhex : ∃ ζ' ∈ primitiveRoots n ℂ, ‖↑q - ζ'‖ < q + 1\n⊢ ¬eval q (cyclotomic n ℝ) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 269,
"column": 6
} | {
"line": 269,
"column": 40
} | {
"line": 269,
"column": 41
} | [
{
"pp": "case neg.refine_1\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nhS : ¬∃ s ∈ S, s ≠ 0\nH : IsCyclotomicExtension ∅ A B\n⊢ adjoin A {b | ∃ n ∈ {1}, n ≠ 0 ∧ b ^ n = 1} = ⊤",
"ppTerm": "?neg.refine_1✝",
"assigned": true,
"usedConstants": [
... | [
"case neg.refine_1\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nhS : ¬∃ s ∈ S, s ≠ 0\nH : IsCyclotomicExtension ∅ A B\n⊢ ⊥ = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 345,
"column": 4
} | {
"line": 345,
"column": 32
} | {
"line": 346,
"column": 4
} | [
{
"pp": "case mem\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsCyclotomicExtension S A B\ninst✝ : IsDomain B\nf g : B ≃ₐ[A] B\nH : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n ∧ f r = g r\nx y : B\nhy : y ∈ {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}\n⊢ f y = ... | [
"case mem\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsCyclotomicExtension S A B\ninst✝ : IsDomain B\nf g : B ≃ₐ[A] B\nH : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n ∧ f r = g r\nx y : B\nn : ℕ\nhn : n ∈ S\nh1 : n ≠ 0\nh2 : y ^ n = 1\n⊢ f y = g y"
] | obtain ⟨n, hn, h1, h2⟩ := hy | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 341,
"column": 2
} | {
"line": 352,
"column": 44
} | {
"line": 354,
"column": 0
} | [
{
"pp": "S : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsCyclotomicExtension S A B\ninst✝ : IsDomain B\nf g : B ≃ₐ[A] B\nH : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n ∧ f r = g r\n⊢ f = g",
"ppTerm": "?m.35",
"assigned": true,
"usedConsta... | [] | ext x
have hx := ‹IsCyclotomicExtension S A B›.adjoin_roots x
induction hx using Algebra.adjoin_induction with
| mem y hy =>
obtain ⟨n, hn, h1, h2⟩ := hy
obtain ⟨r, hr1, hr2⟩ := H n hn h1
have := NeZero.mk h1
obtain ⟨m, -, rfl⟩ := hr1.eq_pow_of_pow_eq_one h2
simp [hr2]
| algebraMap y => simp... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 341,
"column": 2
} | {
"line": 352,
"column": 44
} | {
"line": 354,
"column": 0
} | [
{
"pp": "S : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsCyclotomicExtension S A B\ninst✝ : IsDomain B\nf g : B ≃ₐ[A] B\nH : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n ∧ f r = g r\n⊢ f = g",
"ppTerm": "?m.35",
"assigned": true,
"usedConsta... | [] | ext x
have hx := ‹IsCyclotomicExtension S A B›.adjoin_roots x
induction hx using Algebra.adjoin_induction with
| mem y hy =>
obtain ⟨n, hn, h1, h2⟩ := hy
obtain ⟨r, hr1, hr2⟩ := H n hn h1
have := NeZero.mk h1
obtain ⟨m, -, rfl⟩ := hr1.eq_pow_of_pow_eq_one h2
simp [hr2]
| algebraMap y => simp... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 377,
"column": 34
} | {
"line": 377,
"column": 63
} | {
"line": 377,
"column": 64
} | [
{
"pp": "n : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nh✝ : IsCyclotomicExtension {n} A B\nx : B\nh : x ∈ {b | n ≠ 0 ∧ b ^ n = 1}\n⊢ x ∈ ↑(nthRoots n 1).toFinset",
"ppTerm": "?m.94",
"assigned": true,
"usedConsta... | [
"n : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nh✝ : IsCyclotomicExtension {n} A B\nx : B\nh : x ∈ {b | n ≠ 0 ∧ b ^ n = 1}\n⊢ x ^ n = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 378,
"column": 19
} | {
"line": 378,
"column": 61
} | {
"line": 378,
"column": 62
} | [
{
"pp": "n : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nh✝ : IsCyclotomicExtension {n} A B\nx : B\nh : x ∈ ↑(nthRoots n 1).toFinset\n⊢ x ∈ {b | n ≠ 0 ∧ b ^ n = 1}",
"ppTerm": "?m.97",
"assigned": true,
"usedConsta... | [
"n : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nh✝ : IsCyclotomicExtension {n} A B\nx : B\nh : x ∈ ↑(nthRoots n 1).toFinset\n⊢ x ^ n = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 416,
"column": 63
} | {
"line": 416,
"column": 74
} | {
"line": 416,
"column": 75
} | [
{
"pp": "p : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ... | [
"p : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 383,
"column": 52
} | {
"line": 383,
"column": 77
} | {
"line": 383,
"column": 78
} | [
{
"pp": "n✝ : ℕ\ninst✝⁴ : NeZero n✝\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nh : IsCyclotomicExtension {n✝} A B\nb : B\nx✝ : b ∈ {b | ∃ n ∈ {n✝}, n ≠ 0 ∧ b ^ n = 1}\nn : ℕ\nhb : n = n✝ ∧ n ≠ 0 ∧ b ^ n = 1\n⊢ eval₂ (algebraMap A B) b (X ^ n - 1)... | [
"n✝ : ℕ\ninst✝⁴ : NeZero n✝\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nh : IsCyclotomicExtension {n✝} A B\nb : B\nx✝ : b ∈ {b | ∃ n ∈ {n✝}, n ≠ 0 ∧ b ^ n = 1}\nn : ℕ\nhb : n = n✝ ∧ n ≠ 0 ∧ b ^ n = 1\n⊢ b ^ n = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 66
} | {
"line": 193,
"column": 67
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝ : Field K\nval✝ : Fintype K\nthis : (Polynomial.map (algebraMap K K) (X ^ Fintype.card K - X)).Splits\n⊢ (X ^ Nat.card K - X).Splits",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"HSub.hSub",
"Field.toDiv... | [
"case inl\nK : Type u_1\ninst✝ : Field K\nval✝ : Fintype K\nthis : (Polynomial.map (algebraMap K K) (X ^ Fintype.card K - X)).Splits\n⊢ (X ^ Fintype.card K - X).Splits"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 65
} | {
"line": 195,
"column": 66
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝ : Field K\nval✝ : Infinite K\n⊢ (-(X ^ Nat.card K - X)).Splits",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"neg_sub",
"Polynomial.instNeg",
"Monoid.toMulOneClass",
"congrArg",
"Nat... | [
"case inr\nK : Type u_1\ninst✝ : Field K\nval✝ : Infinite K\n⊢ (X - 1).Splits"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.AddCharacter | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 32
} | {
"line": 68,
"column": 33
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommMonoid R'\nR'' : Type u_1\ninst✝ : CommMonoid R''\nφ : AddChar R R'\nf : R' →* R''\nhφ : φ.IsPrimitive\nhf : Function.Injective ⇑f\na : R\nha : a ≠ 0\n⊢ (f.compAddChar φ).mulShift a ≠ 1",
"ppTerm": "?m.24",
"assigned": true,
"usedCo... | [
"R : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommMonoid R'\nR'' : Type u_1\ninst✝ : CommMonoid R''\nφ : AddChar R R'\nf : R' →* R''\nhφ : φ.IsPrimitive\nhf : Function.Injective ⇑f\na : R\nha : a ≠ 0\n⊢ ∃ x, ¬f (φ (a * x)) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.AddCharacter | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 45
} | {
"line": 77,
"column": 46
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nR' : Type v\ninst✝ : CommMonoid R'\nψ : AddChar R R'\nhψ : ψ.IsPrimitive\na b : R\nh : ψ.mulShift (a + -b) = 1\n⊢ a = b",
"ppTerm": "?m.95",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝¹ : CommRing R\nR' : Type v\ninst✝ : CommMonoid R'\nψ : AddChar R R'\nhψ : ψ.IsPrimitive\na b : R\nh : ψ.mulShift (a + -b) = 1\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 73
} | {
"line": 273,
"column": 2
} | [
{
"pp": "p✝ : ℕ\ninst✝⁶ : Fact (Nat.Prime p✝)\nn✝ : ℕ\nK : Type u_1\nK' : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : Field K'\ninst✝³ : Fintype K\ninst✝² : Fintype K'\np : ℕ\nh_prime : Fact (Nat.Prime p)\ninst✝¹ : Algebra (ZMod p) K\ninst✝ : Algebra (ZMod p) K'\nthis✝ : CharP K p\nthis : CharP K' p\nn : ℕ+\na : Nat.P... | [
"p✝ : ℕ\ninst✝⁶ : Fact (Nat.Prime p✝)\nn✝ : ℕ\nK : Type u_1\nK' : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : Field K'\ninst✝³ : Fintype K\ninst✝² : Fintype K'\np : ℕ\nh_prime : Fact (Nat.Prime p)\ninst✝¹ : Algebra (ZMod p) K\ninst✝ : Algebra (ZMod p) K'\nthis✝ : CharP K p\nthis : CharP K' p\nn : ℕ+\na : Nat.Prime p\nhK :... | have hK'Gal := (GaloisField.algEquivGaloisFieldOfFintype p n' hK').symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 285,
"column": 4
} | {
"line": 285,
"column": 45
} | {
"line": 285,
"column": 46
} | [
{
"pp": "p✝ : ℕ\ninst✝⁴ : Fact (Nat.Prime p✝)\nn✝ : ℕ\nK : Type u_1\nK' : Type u_2\ninst✝³ : Field K\ninst✝² : Field K'\ninst✝¹ : Fintype K\ninst✝ : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : ℕ\n_char_p_K : CharP K p\np' : ℕ\n_char_p'_K' : CharP K' p'\nn : ℕ+\nhp : Nat.Prime p\nhK : Fintype.card K... | [
"p✝ : ℕ\ninst✝⁴ : Fact (Nat.Prime p✝)\nn✝ : ℕ\nK : Type u_1\nK' : Type u_2\ninst✝³ : Field K\ninst✝² : Field K'\ninst✝¹ : Fintype K\ninst✝ : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : ℕ\n_char_p_K : CharP K p\np' : ℕ\n_char_p'_K' : CharP K' p'\nn : ℕ+\nhp : Nat.Prime p\nhK : Fintype.card K = p ^ ↑n\nn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 26
} | {
"line": 323,
"column": 2
} | [
{
"pp": "F : Type u_3\nK : Type u_4\nL : Type u_5\ninst✝⁵ : Field F\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Field L\ninst✝¹ : Algebra F L\ninst✝ : Finite L\nh : Module.finrank F K ∣ Module.finrank F L\nf : K →ₐ[F] L\n⊢ Nat.card (K →ₐ[F] L) = Module.finrank F K",
"ppTerm": "?m.40",
"assigned": ... | [
"F : Type u_3\nK : Type u_4\nL : Type u_5\ninst✝⁵ : Field F\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Field L\ninst✝¹ : Algebra F L\ninst✝ : Finite L\nh : Module.finrank F K ∣ Module.finrank F L\nf : K →ₐ[F] L\nalgInst✝ : Algebra K L := f.toAlgebra\n⊢ Nat.card (K →ₐ[F] L) = Module.finrank F K"
] | algebraize [f.toRingHom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 334,
"column": 2
} | {
"line": 334,
"column": 26
} | {
"line": 335,
"column": 2
} | [
{
"pp": "F : Type u_3\nK : Type u_4\nL : Type u_5\ninst✝⁵ : Field F\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Field L\ninst✝¹ : Algebra F L\ninst✝ : Finite L\nx✝ : Nonempty (K →ₐ[F] L)\nf : K →ₐ[F] L\n⊢ Module.finrank F K ∣ Module.finrank F L",
"ppTerm": "?m.24",
"assigned": true,
"usedConst... | [
"F : Type u_3\nK : Type u_4\nL : Type u_5\ninst✝⁵ : Field F\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Field L\ninst✝¹ : Algebra F L\ninst✝ : Finite L\nx✝ : Nonempty (K →ₐ[F] L)\nf : K →ₐ[F] L\nalgInst✝ : Algebra K L := f.toAlgebra\n⊢ Module.finrank F K ∣ Module.finrank F L"
] | algebraize [f.toRingHom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 64
} | {
"line": 63,
"column": 65
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoid M\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : HasEnoughRootsOfUnity M n\nthis : IsCyclic ↥(rootsOfUnity n M)\ng : ↥(rootsOfUnity n M)\nhg : ∀ (x : ↥(rootsOfUnity n M)), x ∈ Subgroup.zpowers g\nhg' : g ^ n = 1\nf : ZMod n → ↥(rootsOfUnity n M) := fun j ↦ g ^ ↑j.val\nx : ↥(rootsO... | [
"M : Type u_1\ninst✝² : CommMonoid M\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : HasEnoughRootsOfUnity M n\nthis : IsCyclic ↥(rootsOfUnity n M)\ng : ↥(rootsOfUnity n M)\nhg : ∀ (x : ↥(rootsOfUnity n M)), x ∈ Subgroup.zpowers g\nhg' : g ^ n = 1\nf : ZMod n → ↥(rootsOfUnity n M) := fun j ↦ g ^ ↑j.val\nx : ↥(rootsOfUnity n M)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 457,
"column": 4
} | {
"line": 457,
"column": 86
} | {
"line": 458,
"column": 6
} | [
{
"pp": "case refine_2\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : IsDomain B\nζ : B\nhζ : IsPrimitiveRoot ζ n\nx : B\nhx : x = ζ\n⊢ x ∈ (cyclotomic n A).rootSet B",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants... | [
"case refine_2\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : IsDomain B\nζ : B\nhζ : IsPrimitiveRoot ζ n\nx : B\nhx : x = ζ\n⊢ cyclotomic n B ≠ 0 ∧ eval ζ (cyclotomic n B) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 461,
"column": 2
} | {
"line": 461,
"column": 13
} | {
"line": 461,
"column": 14
} | [
{
"pp": "p : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nh : p ≠ 2\n⊢ (Algebra.norm K) (ζ - 1)... | [
"p : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nh : p ≠ 2\n⊢ (Algebra.norm K) (ζ - 1) = ↑p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 13
} | {
"line": 471,
"column": 14
} | [
{
"pp": "p : ℕ\nK : Type u\nL : Type v\ninst✝² : Field L\nζ : L\ninst✝¹ : Field K\ninst✝ : Algebra K L\nhpri : Fact (Nat.Prime p)\nhcyc : IsCyclotomicExtension {p} K L\nh : p ≠ 2\nhirr : Irreducible (cyclotomic (p ^ (0 + 1)) K)\nhζ : IsPrimitiveRoot ζ (p ^ (0 + 1))\nthis : IsCyclotomicExtension {p ^ (0 + 1)} K ... | [
"p : ℕ\nK : Type u\nL : Type v\ninst✝² : Field L\nζ : L\ninst✝¹ : Field K\ninst✝ : Algebra K L\nhpri : Fact (Nat.Prime p)\nhcyc : IsCyclotomicExtension {p} K L\nh : p ≠ 2\nhirr : Irreducible (cyclotomic (p ^ (0 + 1)) K)\nhζ : IsPrimitiveRoot ζ (p ^ (0 + 1))\nthis : IsCyclotomicExtension {p ^ (0 + 1)} K L\n⊢ (Algebr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 502,
"column": 2
} | {
"line": 502,
"column": 19
} | {
"line": 502,
"column": 20
} | [
{
"pp": "K : Type u\nL : Type v\ninst✝² : Field L\nζ : L\ninst✝¹ : Field K\ninst✝ : Algebra K L\nk : ℕ\nhk : 2 ≤ k\nH : IsCyclotomicExtension {2 ^ k} K L\nthis : 2 < 2 ^ k\nhirr : Irreducible (cyclotomic (2 ^ k) K)\nhζ : IsPrimitiveRoot ζ (2 ^ k)\nk₁ : ℕ\nhk₁ : k = k₁.succ\n⊢ (Algebra.norm K) (ζ - 1) = 2",
... | [
"K : Type u\nL : Type v\ninst✝² : Field L\nζ : L\ninst✝¹ : Field K\ninst✝ : Algebra K L\nk : ℕ\nhk : 2 ≤ k\nH : IsCyclotomicExtension {2 ^ k} K L\nthis : 2 < 2 ^ k\nhirr : Irreducible (cyclotomic (2 ^ k) K)\nhζ : IsPrimitiveRoot ζ (2 ^ k)\nk₁ : ℕ\nhk₁ : k = k₁.succ\n⊢ (Algebra.norm K) (ζ - 1) = 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 558,
"column": 61
} | {
"line": 558,
"column": 72
} | {
"line": 558,
"column": 73
} | [
{
"pp": "S : Set ℕ\nK : Type w\nL : Type z\ninst✝⁶ : Field K\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\ninst✝³ : IsCyclotomicExtension S K L\nM : Type u_1\ninst✝² : Field M\ninst✝¹ : Algebra K M\ninst✝ : IsSepClosed M\nthis : Algebra.IsSeparable K L\ni : L →ₐ[K] M := IsSepClosed.lift\nhtop : IntermediateField.adj... | [
"S : Set ℕ\nK : Type w\nL : Type z\ninst✝⁶ : Field K\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\ninst✝³ : IsCyclotomicExtension S K L\nM : Type u_1\ninst✝² : Field M\ninst✝¹ : Algebra K M\ninst✝ : IsSepClosed M\nthis : Algebra.IsSeparable K L\ni : L →ₐ[K] M := IsSepClosed.lift\nhtop : IntermediateField.adjoin K {x | ∃... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 509,
"column": 48
} | {
"line": 519,
"column": 62
} | {
"line": 521,
"column": 0
} | [
{
"pp": "p : ℕ\nK : Type u\nL : Type v\ninst✝² : Field L\nζ : L\ninst✝¹ : Field K\ninst✝ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhk : k ≠ 0\n⊢ (Algebra.n... | [] | by
by_cases htwo : p ^ (k - s + 1) = 2
· obtain ⟨hp, hks⟩ := (Nat.prime_two.pow_eq_iff).1 htwo
simp only [add_eq_right] at hks
replace hs : s = k := le_antisymm hs (Nat.sub_eq_zero_iff_le.mp hks)
simp only [hp, hs] at hζ hirr hcycl ⊢
obtain ⟨k₁, hk₁⟩ := Nat.exists_eq_succ_of_ne_zero hk
rw [hζ.no... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 13
} | {
"line": 62,
"column": 14
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ℕ\n⊢ ↑(toCircle ↑j) = cexp (2 * ↑π * I * ↑j / ↑N)",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"N : ℕ\ninst✝ : NeZero N\nj : ℕ\n⊢ ↑(toCircle ↑j) = cexp (2 * ↑π * I * ↑j / ↑N)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 41
} | {
"line": 64,
"column": 0
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ℕ\n⊢ ↑(toCircle ↑j) = cexp (2 * ↑π * I * ↑j / ↑N)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Int.cast",
"NormedCommRing.toSeminormedCommRing",
"Int.cast_natCast",
"instHDiv",
"Real.pi",
"HMul.hMul",
"ZMod... | [] | simpa using toCircle_intCast (N := N) j | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 41
} | {
"line": 64,
"column": 0
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ℕ\n⊢ ↑(toCircle ↑j) = cexp (2 * ↑π * I * ↑j / ↑N)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Int.cast",
"NormedCommRing.toSeminormedCommRing",
"Int.cast_natCast",
"instHDiv",
"Real.pi",
"HMul.hMul",
"ZMod... | [] | simpa using toCircle_intCast (N := N) j | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 41
} | {
"line": 64,
"column": 0
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ℕ\n⊢ ↑(toCircle ↑j) = cexp (2 * ↑π * I * ↑j / ↑N)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Int.cast",
"NormedCommRing.toSeminormedCommRing",
"Int.cast_natCast",
"instHDiv",
"Real.pi",
"HMul.hMul",
"ZMod... | [] | simpa using toCircle_intCast (N := N) j | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 147,
"column": 6
} | {
"line": 147,
"column": 54
} | {
"line": 147,
"column": 55
} | [
{
"pp": "n : ℤ\nhelp : ∀ (m : ℤ), 0 ≤ m → m < 8 → χ₈ ↑m = if m % 2 = 0 then 0 else if m = 1 ∨ m = 7 then 1 else -1\n⊢ χ₈ ↑n = if n % 2 = 0 then 0 else if n % 8 = 1 ∨ n % 8 = 7 then 1 else -1",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.LegendreSymbol... | [
"n : ℤ\nhelp : ∀ (m : ℤ), 0 ≤ m → m < 8 → χ₈ ↑m = if m % 2 = 0 then 0 else if m = 1 ∨ m = 7 then 1 else -1\n⊢ χ₈ ↑n = if n % 8 % 2 = 0 then 0 else if n % 8 = 1 ∨ n % 8 = 7 then 1 else -1"
] | ← Int.emod_emod_of_dvd n (by lia : (2 : ℤ) ∣ 8), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 178,
"column": 6
} | {
"line": 178,
"column": 54
} | {
"line": 178,
"column": 55
} | [
{
"pp": "n : ℤ\nhelp : ∀ (m : ℤ), 0 ≤ m → m < 8 → χ₈' ↑m = if m % 2 = 0 then 0 else if m = 1 ∨ m = 3 then 1 else -1\n⊢ χ₈' ↑n = if n % 2 = 0 then 0 else if n % 8 = 1 ∨ n % 8 = 3 then 1 else -1",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"ZMod.com... | [
"n : ℤ\nhelp : ∀ (m : ℤ), 0 ≤ m → m < 8 → χ₈' ↑m = if m % 2 = 0 then 0 else if m = 1 ∨ m = 3 then 1 else -1\n⊢ χ₈' ↑n = if n % 8 % 2 = 0 then 0 else if n % 8 = 1 ∨ n % 8 = 3 then 1 else -1"
] | ← Int.emod_emod_of_dvd n (by lia : (2 : ℤ) ∣ 8), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 166,
"column": 8
} | {
"line": 166,
"column": 57
} | {
"line": 167,
"column": 8
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\nf : Rˣ →* R'ˣ\nx y : R\nhx : IsUnit x\n⊢ (if hx : IsUnit (x * y) then ↑(f hx.unit) else 0) =\n (if hx : IsUnit x then ↑(f hx.unit) else 0) * if hx : IsUnit y then ↑(f hx.unit) else 0",
"ppTerm": "?pos✝",... | [
"case pos\nR : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\nf : Rˣ →* R'ˣ\nx y : R\nhx : IsUnit x\n⊢ (if h : IsUnit y then ↑(f ⋯.unit) else 0) = ↑(f ⋯.unit) * if hx : IsUnit y then ↑(f hx.unit) else 0"
] | simp only [hx, IsUnit.mul_iff, true_and, dif_pos] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 31
} | {
"line": 86,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn m : ℕ\ninst✝ : NeZero m\nhm : n ∣ m\na₁✝ a₂✝ : DirichletCharacter R n\nh : ∀ (a : (ZMod m)ˣ), ((changeLevel hm) a₁✝) ↑a = ((changeLevel hm) a₂✝) ↑a\nz : (ZMod m)ˣ\n⊢ a₁✝ ↑((ZMod.unitsMap hm) z) = a₂✝ ↑((ZMod.unitsMap hm) z)",
"ppTerm": "?m.59",
"as... | [
"R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn m : ℕ\ninst✝ : NeZero m\nhm : n ∣ m\na₁✝ a₂✝ : DirichletCharacter R n\nh : ∀ (a : (ZMod m)ˣ), ((changeLevel hm) a₁✝) ↑a = ((changeLevel hm) a₂✝) ↑a\nz : (ZMod m)ˣ\n⊢ a₁✝ ↑((ZMod.unitsMap hm) z) = a₂✝ ↑((ZMod.unitsMap hm) z)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 215,
"column": 6
} | {
"line": 215,
"column": 17
} | {
"line": 215,
"column": 18
} | [
{
"pp": "R : Type u_1\ninst✝² : CommMonoid R\nR' : Type u_2\ninst✝¹ : CommMonoidWithZero R'\ninst✝ : Nontrivial R'\nχ : MulChar R R'\na : R\n⊢ χ a ≠ 0 → IsUnit a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"IsUnit",
"id",
"MulChar",
"MulChar.instFunLike",
... | [
"R : Type u_1\ninst✝² : CommMonoid R\nR' : Type u_2\ninst✝¹ : CommMonoidWithZero R'\ninst✝ : Nontrivial R'\nχ : MulChar R R'\na : R\n⊢ ¬χ a = 0 → IsUnit a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 219,
"column": 2
} | {
"line": 219,
"column": 13
} | {
"line": 219,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝² : CommMonoid R\nR' : Type u_2\ninst✝¹ : CommMonoidWithZero R'\ninst✝ : Nontrivial R'\nχ : MulChar R R'\na : R\n⊢ χ a = 0 ↔ ¬IsUnit a",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝² : CommMonoid R\nR' : Type u_2\ninst✝¹ : CommMonoidWithZero R'\ninst✝ : Nontrivial R'\nχ : MulChar R R'\na : R\n⊢ χ a = 0 ↔ ¬IsUnit a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 183,
"column": 8
} | {
"line": 183,
"column": 51
} | {
"line": 183,
"column": 52
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nm : ℕ\ninst✝ : NeZero n\nψ : DirichletCharacter R m\nh : (changeLevel ⋯) χ = (changeLevel ⋯) ψ\nx : (ZMod n)ˣ\nhx : x ∈ (ZMod.unitsMap ⋯).ker\nthis : (↑x).val ≡ 1 [MOD n.gcd m]\nz : ℕ\nhz₁ : z ≡ (↑x).val [MOD... | [
"case refine_1\nR : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nm : ℕ\ninst✝ : NeZero n\nψ : DirichletCharacter R m\nh : (changeLevel ⋯) χ = (changeLevel ⋯) ψ\nx : (ZMod n)ˣ\nhx : x ∈ (ZMod.unitsMap ⋯).ker\nthis : (↑x).val ≡ 1 [MOD n.gcd m]\nz : ℕ\nhz₁ : z ≡ (↑x).val [MOD n]\nhz₂ : z... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 602,
"column": 2
} | {
"line": 602,
"column": 46
} | {
"line": 602,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommMonoid R\ninst✝² : Fintype R\nR' : Type u_2\ninst✝¹ : CommRing R'\ninst✝ : IsDomain R'\nχ : MulChar R R'\nhχ : χ ≠ 1\nb : Rˣ\nhb : χ ↑b ≠ 1\n⊢ χ ↑b * ∑ a, χ a = ∑ a, χ a",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
... | [
"R : Type u_1\ninst✝³ : CommMonoid R\ninst✝² : Fintype R\nR' : Type u_2\ninst✝¹ : CommRing R'\ninst✝ : IsDomain R'\nχ : MulChar R R'\nhχ : χ ≠ 1\nb : Rˣ\nhb : χ ↑b ≠ 1\n⊢ ∑ x, χ (↑b * x) = ∑ a, χ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 367,
"column": 2
} | {
"line": 370,
"column": 82
} | {
"line": 371,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nm : ℕ\ninst✝ : NeZero m\nhm : n ∣ m\nthis : NeZero n\n⊢ ((changeLevel hm) χ).conductor = χ.conductor",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"DirichletCharacter.conductor",
"Eq.mpr"... | [
"R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nm : ℕ\ninst✝ : NeZero m\nhm : n ∣ m\nthis : NeZero n\nh : ((changeLevel hm) χ).conductor ∣ χ.conductor\n⊢ ((changeLevel hm) χ).conductor = χ.conductor"
] | have h : (changeLevel hm χ).conductor ∣ χ.conductor := by
refine conductor_dvd_of_mem_conductorSet _
⟨χ.conductor_dvd_level.trans hm, χ.primitiveCharacter, ?_⟩
rw [changeLevel_trans _ χ.conductor_dvd_level, changeLevel_primitiveCharacter] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 474,
"column": 17
} | {
"line": 474,
"column": 71
} | {
"line": 474,
"column": 72
} | [
{
"pp": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nH : Set (ZMod n)ˣ\nχ : DirichletCharacter R n\nhχ : ∀ (x : (↥(Submonoid.map (Units.coeHom (ZMod n)) (Submonoid.closure H)))ˣ), χ ↑↑x = 1\nx : (ZMod n)ˣ\nhx : x ∈ Submonoid.closure H\n⊢ x⁻¹ ∈ ↑(Submonoid.closure H)",
"ppTerm": "?m.54",
"assigned... | [
"R : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nH : Set (ZMod n)ˣ\nχ : DirichletCharacter R n\nhχ : ∀ (x : (↥(Submonoid.map (Units.coeHom (ZMod n)) (Submonoid.closure H)))ˣ), χ ↑↑x = 1\nx : (ZMod n)ˣ\nhx : x ∈ Submonoid.closure H\n⊢ x ∈ Subgroup.closure H"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DirichletCharacter.GaussSum | {
"line": 30,
"column": 4
} | {
"line": 31,
"column": 11
} | {
"line": 31,
"column": 12
} | [
{
"pp": "N : ℕ\ninst✝¹ : NeZero N\nR : Type u_1\ninst✝ : CommRing R\ne : AddChar (ZMod N) R\nχ : DirichletCharacter R N\nd : ℕ\nhd : d ∣ N\nhe : e.mulShift ↑d = 1\nu : (ZMod N)ˣ\nhu : (ZMod.unitsMap hd) u = 1\na : ℤ\nha : ↑(↑u).val - 1 = ↑d * a\nthis : ↑u - 1 = ↑(↑(↑u).val - 1)\ny : ZMod N\n⊢ (e.mulShift ↑(↑d *... | [
"N : ℕ\ninst✝¹ : NeZero N\nR : Type u_1\ninst✝ : CommRing R\ne : AddChar (ZMod N) R\nχ : DirichletCharacter R N\nd : ℕ\nhd : d ∣ N\nhe : e.mulShift ↑d = 1\nu : (ZMod N)ˣ\nhu : (ZMod.unitsMap hd) u = 1\na : ℤ\nha : ↑(↑u).val - 1 = ↑d * a\nthis : ↑u - 1 = ↑(↑(↑u).val - 1)\ny : ZMod N\n⊢ e (↑d * (↑a * y)) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DirichletCharacter.GaussSum | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 37
} | {
"line": 44,
"column": 38
} | [
{
"pp": "N : ℕ\ninst✝² : NeZero N\nR : Type u_1\ninst✝¹ : CommRing R\ne : AddChar (ZMod N) R\ninst✝ : IsDomain R\nχ : DirichletCharacter R N\nd : ℕ\nhd : d ∣ N\nhe : e.mulShift ↑d = 1\nh_ne : gaussSum χ e ≠ 0\nx✝ : (ZMod N)ˣ\nhu : x✝ ∈ (ZMod.unitsMap hd).ker\n⊢ x✝ ∈ (MulChar.toUnitHom χ).ker",
"ppTerm": "?m... | [
"N : ℕ\ninst✝² : NeZero N\nR : Type u_1\ninst✝¹ : CommRing R\ne : AddChar (ZMod N) R\ninst✝ : IsDomain R\nχ : DirichletCharacter R N\nd : ℕ\nhd : d ∣ N\nhe : e.mulShift ↑d = 1\nh_ne : gaussSum χ e ≠ 0\nx✝ : (ZMod N)ˣ\nhu : x✝ ∈ (ZMod.unitsMap hd).ker\n⊢ χ ↑x✝ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DirichletCharacter.GaussSum | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 30
} | {
"line": 61,
"column": 31
} | [
{
"pp": "case pos\nN : ℕ\ninst✝² : NeZero N\nR : Type u_1\ninst✝¹ : CommRing R\ne : AddChar (ZMod N) R\ninst✝ : IsDomain R\nχ : DirichletCharacter R N\nhχ : χ.IsPrimitive\na : ZMod N\nha : IsUnit a\n⊢ gaussSum χ (e.mulShift a) = χ⁻¹ a * gaussSum χ e",
"ppTerm": "?pos✝",
"assigned": false,
"usedConst... | [
"case pos\nN : ℕ\ninst✝² : NeZero N\nR : Type u_1\ninst✝¹ : CommRing R\ne : AddChar (ZMod N) R\ninst✝ : IsDomain R\nχ : DirichletCharacter R N\nhχ : χ.IsPrimitive\na : ZMod N\nha : IsUnit a\n⊢ gaussSum χ (e.mulShift a) = χ⁻¹ a * gaussSum χ e"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.GaussSum | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 24
} | {
"line": 116,
"column": 25
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\ninst✝³ : CommRing R\ninst✝² : Fintype R\ninst✝¹ : CommRing R'\ninst✝ : IsDomain R'\nχ : MulChar R R'\nhχ : χ ≠ 1\n⊢ gaussSum χ 1 = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Finset.univ",
"congrArg... | [
"R : Type u_1\nR' : Type u_2\ninst✝³ : CommRing R\ninst✝² : Fintype R\ninst✝¹ : CommRing R'\ninst✝ : IsDomain R'\nχ : MulChar R R'\nhχ : χ ≠ 1\n⊢ ∑ x, χ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 13
} | {
"line": 65,
"column": 14
} | [
{
"pp": "x p : ℝ\nhp : 1 < p\n⊢ HasDerivAt (fun x ↦ |x| ^ p) (p * |x| ^ (p - 2) * x) x",
"ppTerm": "?m.44",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x p : ℝ\nhp : 1 < p\n⊢ HasDerivAt (fun x ↦ |x| ^ p) (p * |x| ^ (p - 2) * x) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 900,
"column": 4
} | {
"line": 901,
"column": 25
} | {
"line": 901,
"column": 26
} | [
{
"pp": "case hS\nn : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nC : Subalgebra A B\nζ : B\nhζ : IsPrimitiveRoot ζ n\n⊢ ∀ n_1 ∈ {n}, n_1 ≠ 0 → ∃ r, IsPrimitiveRoot r n_1",
"ppTerm": "?hS",
"assigned": true,
"usedC... | [
"case hS\nn : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nC : Subalgebra A B\nζ : B\nhζ : IsPrimitiveRoot ζ n\n⊢ ∃ r, IsPrimitiveRoot r n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.ZMod | {
"line": 55,
"column": 10
} | {
"line": 55,
"column": 35
} | {
"line": 56,
"column": 4
} | [
{
"pp": "N : ℕ\ninst✝² : NeZero N\nE : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℂ E\nΦ : ZMod N → E\nk : ZMod N\n⊢ auxDFT (fun j ↦ Φ (-j)) k = auxDFT Φ (-k)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"instHSMul",
"HMul.h... | [
"N : ℕ\ninst✝² : NeZero N\nE : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℂ E\nΦ : ZMod N → E\nk : ZMod N\n⊢ ∑ x, stdAddChar (-(x * k)) • Φ (-x) = ∑ j, stdAddChar (-(j * -k)) • Φ j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 923,
"column": 2
} | {
"line": 924,
"column": 74
} | {
"line": 925,
"column": 4
} | [
{
"pp": "A : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsDomain B\nn₁ n₂ : ℕ\nC₁ C₂ : Subalgebra A B\nh₁ : IsCyclotomicExtension {n₁} A ↥C₁\nh₂ : IsCyclotomicExtension {n₂} A ↥C₂\ninst✝ : NeZero n₂\nthis : NeZero n₁\nζ₂ : ↥C₂\nhζ₂ : IsPrimitiveRoot ((algebraMap... | [
"A : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsDomain B\nn₁ n₂ : ℕ\nC₁ C₂ : Subalgebra A B\nh₁ : IsCyclotomicExtension {n₁} A ↥C₁\nh₂ : IsCyclotomicExtension {n₂} A ↥C₂\ninst✝ : NeZero n₂\nthis : NeZero n₁\nζ₂ : ↥C₂\nhζ₂ : IsPrimitiveRoot ((algebraMap (↥C₂) B) ζ₂... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 105,
"column": 2
} | {
"line": 106,
"column": 28
} | {
"line": 106,
"column": 29
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : F → E\nhf : Differentiable ℝ f\nx : F\np : ℝ≥0\nhp : 1 < p\n⊢ ‖fderiv ℝ (fun x ↦ ‖f x‖ ^ ↑p) x‖ₑ ≤ ↑p * ‖f x‖ₑ ^ (↑p - 1) * ‖fderiv ℝ f x‖ₑ",
"ppTer... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : F → E\nhf : Differentiable ℝ f\nx : F\np : ℝ≥0\nhp : 1 < p\n⊢ ‖fderiv ℝ (fun x ↦ ‖f x‖ ^ ↑p) x‖₊ ≤ p * ‖f x‖₊ ^ (↑p - 1) * ‖fderiv ℝ f x‖₊"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.ZMod | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 73
} | {
"line": 194,
"column": 74
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nh : ∀ {f : ZMod N → ℂ}, Function.Even f → Function.Even (𝓕 f)\nhΦ : Function.Even (𝓕 Φ)\nx : ZMod N\n⊢ Φ (-x) = Φ x",
"ppTerm": "?m.38",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nh : ∀ {f : ZMod N → ℂ}, Function.Even f → Function.Even (𝓕 f)\nhΦ : Function.Even (𝓕 Φ)\nx : ZMod N\n⊢ Φ (-x) = Φ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 58
} | {
"line": 119,
"column": 59
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\np : ℝ\nhp : 1 < p\nx : E\nhx : x = 0\nthis : ContinuousAt (fun x ↦ p * ‖x‖ ^ (p - 1)) 0\n⊢ Filter.Tendsto (fun x ↦ p * ‖x‖ ^ (p - 1)) (𝓝 0) (𝓝 0)",
"ppTerm": "?m.136",
"assigned": false,
"usedConstants": [],
"... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\np : ℝ\nhp : 1 < p\nx : E\nhx : x = 0\nthis : ContinuousAt (fun x ↦ p * ‖x‖ ^ (p - 1)) 0\n⊢ Filter.Tendsto (fun x ↦ p * ‖x‖ ^ (p - 1)) (𝓝 0) (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.ZMod | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 85
} | {
"line": 202,
"column": 86
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nh : ∀ {f : ZMod N → ℂ}, Function.Odd f → Function.Odd (𝓕 f)\nhΦ : Function.Odd (𝓕 Φ)\nx : ZMod N\n⊢ Φ (-x) = -Φ x",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nh : ∀ {f : ZMod N → ℂ}, Function.Odd f → Function.Odd (𝓕 f)\nhΦ : Function.Odd (𝓕 Φ)\nx : ZMod N\n⊢ Φ (-x) = -Φ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.ZMod | {
"line": 216,
"column": 43
} | {
"line": 216,
"column": 60
} | {
"line": 216,
"column": 61
} | [
{
"pp": "case e_f\nN : ℕ\ninst✝ : NeZero N\nχ : DirichletCharacter ℂ N\nk j : ZMod N\n⊢ stdAddChar (-(k * j)) * χ j = χ j * stdAddChar (-(k * j))",
"ppTerm": "?e_f",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"NegZero... | [
"case e_f\nN : ℕ\ninst✝ : NeZero N\nχ : DirichletCharacter ℂ N\nk j : ZMod N\n⊢ ↑(toCircle (-(k * j))) * χ j = χ j * ↑(toCircle (-(k * j)))"
] | stdAddChar_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finset.Grade | {
"line": 36,
"column": 4
} | {
"line": 36,
"column": 15
} | {
"line": 36,
"column": 16
} | [
{
"pp": "α : Type u_1\ns : Multiset α\na : α\nt : Multiset α\nhst : s < t\nhts : t < a ::ₘ s\n⊢ t.card < succ s.card",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"Order.succ_eq_add_one",
"Nat.instOne",
"congrAr... | [
"α : Type u_1\ns : Multiset α\na : α\nt : Multiset α\nhst : s < t\nhts : t < a ::ₘ s\n⊢ t.card ≤ s.card"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Grade | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 13
} | {
"line": 116,
"column": 14
} | [
{
"pp": "α : Type u_1\ns t : Finset α\ninst✝ : DecidableEq α\nh : s ⋖ t\n⊢ ∃ a ∉ s, insert a s = t",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ns t : Finset α\ninst✝ : DecidableEq α\nh : s ⋖ t\n⊢ ∃ a ∉ s, insert a s = t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.GaussSum | {
"line": 246,
"column": 51
} | {
"line": 249,
"column": 5
} | {
"line": 251,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : Fintype R\nR' : Type v\ninst✝ : CommRing R'\np : ℕ\nfp : Fact (Nat.Prime p)\nhch : CharP R' p\nχ : MulChar R R'\nψ : AddChar R R'\n⊢ gaussSum χ ψ ^ p = gaussSum (χ ^ p) (ψ ^ p)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [] | by
rw [← frobenius_def, gaussSum, gaussSum, map_sum]
simp_rw [pow_apply' χ fp.1.ne_zero, map_mul, frobenius_def]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.GaussSum | {
"line": 257,
"column": 2
} | {
"line": 259,
"column": 27
} | {
"line": 261,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : Fintype R\nR' : Type v\ninst✝ : CommRing R'\np : ℕ\nfp : Fact (Nat.Prime p)\nhch : CharP R' p\nhp : IsUnit ↑p\nχ : MulChar R R'\nhχ : χ.IsQuadratic\nψ : AddChar R R'\n⊢ gaussSum χ ψ ^ p = χ ↑p * gaussSum χ ψ",
"ppTerm": "?m.32",
"assigned": true,
"u... | [] | rw [_root_.gaussSum_frob, pow_mulShift, hχ.pow_char p, ← gaussSum_mulShift χ ψ hp.unit,
← mul_assoc, hp.unit_spec, ← pow_two, ← pow_apply' _ two_ne_zero, hχ.sq_eq_one, ← hp.unit_spec,
one_apply_coe, one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.GaussSum | {
"line": 257,
"column": 2
} | {
"line": 259,
"column": 27
} | {
"line": 261,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : Fintype R\nR' : Type v\ninst✝ : CommRing R'\np : ℕ\nfp : Fact (Nat.Prime p)\nhch : CharP R' p\nhp : IsUnit ↑p\nχ : MulChar R R'\nhχ : χ.IsQuadratic\nψ : AddChar R R'\n⊢ gaussSum χ ψ ^ p = χ ↑p * gaussSum χ ψ",
"ppTerm": "?m.32",
"assigned": true,
"u... | [] | rw [_root_.gaussSum_frob, pow_mulShift, hχ.pow_char p, ← gaussSum_mulShift χ ψ hp.unit,
← mul_assoc, hp.unit_spec, ← pow_two, ← pow_apply' _ two_ne_zero, hχ.sq_eq_one, ← hp.unit_spec,
one_apply_coe, one_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.GaussSum | {
"line": 257,
"column": 2
} | {
"line": 259,
"column": 27
} | {
"line": 261,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : Fintype R\nR' : Type v\ninst✝ : CommRing R'\np : ℕ\nfp : Fact (Nat.Prime p)\nhch : CharP R' p\nhp : IsUnit ↑p\nχ : MulChar R R'\nhχ : χ.IsQuadratic\nψ : AddChar R R'\n⊢ gaussSum χ ψ ^ p = χ ↑p * gaussSum χ ψ",
"ppTerm": "?m.32",
"assigned": true,
"u... | [] | rw [_root_.gaussSum_frob, pow_mulShift, hχ.pow_char p, ← gaussSum_mulShift χ ψ hp.unit,
← mul_assoc, hp.unit_spec, ← pow_two, ← pow_apply' _ two_ne_zero, hχ.sq_eq_one, ← hp.unit_spec,
one_apply_coe, one_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.Interval | {
"line": 50,
"column": 10
} | {
"line": 50,
"column": 64
} | {
"line": 50,
"column": 65
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq α\ns t : Finset α\nu₁ u₂ : ↥(t \\ s).powerset\nh : (fun u ↦ (↑u).disjUnion s ⋯) u₁ = (fun u ↦ (↑u).disjUnion s ⋯) u₂\n⊢ u₁ = u₂",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset",
"Finset.instSDiff",... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq α\ns t : Finset α\nu₁ u₂ : ↥(t \\ s).powerset\nh : (fun u ↦ (↑u).disjUnion s ⋯) u₁ = (fun u ↦ (↑u).disjUnion s ⋯) u₂\n⊢ ↑u₁ = ↑u₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.GaussSum | {
"line": 367,
"column": 4
} | {
"line": 368,
"column": 12
} | {
"line": 369,
"column": 4
} | [
{
"pp": "case refine_1\nF : Type u_1\ninst✝¹ : Fintype F\ninst✝ : Field F\nhF : ringChar F ≠ 2\nhp2 : ∀ (n : ℕ), 2 ^ n ≠ 0\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ ↑n\nFF : Type u_1 := CyclotomicField 8 F\nhchar : ringChar F = ringChar FF\nFFp : Nat.Prime (ringChar FF)\nthis : Fa... | [
"case refine_2\nF : Type u_1\ninst✝¹ : Fintype F\ninst✝ : Field F\nhF : ringChar F ≠ 2\nhp2 : ∀ (n : ℕ), 2 ^ n ≠ 0\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ ↑n\nFF : Type u_1 := CyclotomicField 8 F\nhchar : ringChar F = ringChar FF\nFFp : Nat.Prime (ringChar FF)\nthis : Fact (Nat.Prim... | · rw [← pow_mul, ← map_nsmul_eq_pow ψ₈.char, ψ₈.prim.zmod_char_eq_one_iff]
decide | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.InnerProductSpace.Affine | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 13
} | {
"line": 92,
"column": 14
} | [
{
"pp": "V : Type u_2\nP : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b p : P\nh_inner : ⟪p -ᵥ a, b -ᵥ a⟫ = 0\n⊢ dist p b ^ 2 = dist p a ^ 2 + dist a b ^ 2",
"ppTerm": "?m.60",
"assigned": false,
"usedConstants": []... | [
"V : Type u_2\nP : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b p : P\nh_inner : ⟪p -ᵥ a, b -ᵥ a⟫ = 0\n⊢ dist p b ^ 2 = dist p a ^ 2 + dist a b ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Hofer | {
"line": 72,
"column": 6
} | {
"line": 74,
"column": 52
} | {
"line": 75,
"column": 6
} | [
{
"pp": "case hi\nX : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : CompleteSpace X\nx : X\nε : ℝ\nε_pos : 0 < ε\nϕ : X → ℝ\ncont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ\nnonneg : ∀ (y : X), 0 ≤ ϕ y\nreformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'\n... | [
"case hi\nX : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : CompleteSpace X\nx : X\nε : ℝ\nε_pos : 0 < ε\nϕ : X → ℝ\ncont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ\nnonneg : ∀ (y : X), 0 ≤ ϕ y\nreformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'\nthis : Nonem... | have B : 2 ^ (n + 1) * ϕ x ≤ ϕ (u (n + 1)) := by
refine @geom_le (ϕ ∘ u) _ zero_le_two (n + 1) fun m hm => ?_
exact (IH _ <| Nat.lt_add_one_iff.1 hm).2.le | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Coalgebra | {
"line": 76,
"column": 8
} | {
"line": 76,
"column": 26
} | {
"line": 76,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : FiniteDimensional 𝕜 E\nA : Type u_3\ninst✝³ : Ring A\ninst✝² : Module 𝕜 A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : IsScalarTower 𝕜 A A\ne : E ≃ₗ[𝕜] A\n⊢ ↑(TensorProduct.assoc 𝕜 ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : FiniteDimensional 𝕜 E\nA : Type u_3\ninst✝³ : Ring A\ninst✝² : Module 𝕜 A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : IsScalarTower 𝕜 A A\ne : E ≃ₗ[𝕜] A\n⊢ ↑(TensorProduct.assoc 𝕜 E E E) ∘ₗ\n ... | ← adjoint_lTensor, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 25
} | {
"line": 133,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nT : ι → E →ₗ[𝕜] E\ns : Finset ι\nhT : ∀ i ∈ s, (T i).IsPositive\nx✝ : E\n⊢ 0 ≤ re ⟪(∑ i ∈ s, T i) x✝, x✝⟫",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nT : ι → E →ₗ[𝕜] E\ns : Finset ι\nhT : ∀ i ∈ s, (T i).IsPositive\nx✝ : E\n⊢ 0 ≤ ∑ x ∈ s, re ⟪(T x) x✝, x✝⟫"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Coalgebra | {
"line": 85,
"column": 35
} | {
"line": 85,
"column": 53
} | {
"line": 85,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : FiniteDimensional 𝕜 E\nA : Type u_3\ninst✝³ : Ring A\ninst✝² : Module 𝕜 A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : IsScalarTower 𝕜 A A\ne : E ≃ₗ[𝕜] A\n⊢ LinearMap.lTensor E (adjo... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : FiniteDimensional 𝕜 E\nA : Type u_3\ninst✝³ : Ring A\ninst✝² : Module 𝕜 A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : IsScalarTower 𝕜 A A\ne : E ≃ₗ[𝕜] A\n⊢ adjoint (LinearMap.lTensor E (toSpanS... | ← adjoint_lTensor, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 151,
"column": 23
} | {
"line": 151,
"column": 48
} | {
"line": 151,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsPositive\nhT' : T ≠ 0\nα : 𝕜\nh : (α • T).IsPositive\n⊢ ?m.48",
"ppTerm": "?m.53",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsPositive\nhT' : T ≠ 0\nα : 𝕜\nh : (α • T).IsPositive\n⊢ ?m.48"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 152,
"column": 13
} | {
"line": 152,
"column": 43
} | {
"line": 152,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsPositive\nhT' : T ≠ 0\nα : 𝕜\nh : (α • T).IsPositive\nx : E\nhx : 0 < ⟪x, T x⟫\n⊢ ?m.78",
"ppTerm": "?m.79",
"assigned": false,
"usedConstants": [],
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsPositive\nhT' : T ≠ 0\nα : 𝕜\nh : (α • T).IsPositive\nx : E\nhx : 0 < ⟪x, T x⟫\n⊢ ?m.78"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 158,
"column": 2
} | {
"line": 160,
"column": 11
} | {
"line": 160,
"column": 12
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\nn : ℕ\nhT : T.IsPositive\nhn : Module.finrank 𝕜 E = n\ni : Fin n\n⊢ 0 ≤ ⋯.eigenvalues hn i",
"ppTerm": "?m.44",
"assigned": false,
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\nn : ℕ\nhT : T.IsPositive\nhn : Module.finrank 𝕜 E = n\ni : Fin n\n⊢ 0 ≤ ⋯.eigenvalues hn i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 169,
"column": 29
} | {
"line": 169,
"column": 40
} | {
"line": 169,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace 𝕜 F\nx✝² x✝¹ x✝ : E →ₗ[𝕜] E\nh₁ : (x✝¹ - x✝²).IsPositive\nh₂ : (x✝ - x✝¹).IsPositive\n⊢ (x✝ - x✝²).IsPositive",
"... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace 𝕜 F\nx✝² x✝¹ x✝ : E →ₗ[𝕜] E\nh₁ : (x✝¹ - x✝²).IsPositive\nh₂ : (x✝ - x✝¹).IsPositive\n⊢ (x✝ - x✝²).IsPositive"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 13
} | {
"line": 181,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf : E →ₗ[𝕜] E\n⊢ 0 ≤ f ↔ f.IsPositive",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf : E →ₗ[𝕜] E\n⊢ 0 ≤ f ↔ f.IsPositive"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 30
} | {
"line": 268,
"column": 31
} | [
{
"pp": "case inr\nι : Type u_1\nA : ι → Type u_2\ninst✝³ : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\np : ℝ\nhp₀ : 0 ≤ p\nhp : (↑#ι - 1) * p ≤ 1\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\nh✝ : Nonempty ((i... | [
"case inr\nι : Type u_1\nA : ι → Type u_2\ninst✝³ : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\np : ℝ\nhp₀ : 0 ≤ p\nhp : (↑#ι - 1) * p ≤ 1\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\nh✝ : Nonempty ((i : ι) → A i)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 200,
"column": 32
} | {
"line": 200,
"column": 43
} | {
"line": 200,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace 𝕜 F\nT : E →ₗ[𝕜] E\nf : E ≃ₗᵢ[𝕜] F\nx✝ : T.IsSymmetric\nh : ∀ (x : F), 0 ≤ re ⟪T (f.symm x), f.symm x⟫\nx : E\n⊢ 0 ≤... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace 𝕜 F\nT : E →ₗ[𝕜] E\nf : E ≃ₗᵢ[𝕜] F\nx✝ : T.IsSymmetric\nh : ∀ (x : F), 0 ≤ re ⟪T (f.symm x), f.symm x⟫\nx : E\n⊢ 0 ≤ re ⟪T x, x⟫... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Coalgebra | {
"line": 135,
"column": 38
} | {
"line": 135,
"column": 56
} | {
"line": 135,
"column": 57
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : Coalgebra 𝕜 E\nx : E\n⊢ (adjoint comul ∘ₗ LinearMap.lTensor E (adjoint counit)) (x ⊗ₜ[𝕜] One.one) = x",
"ppTerm": "?m.241",
"assigned": tru... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : Coalgebra 𝕜 E\nx : E\n⊢ (adjoint comul ∘ₗ adjoint (LinearMap.lTensor E counit)) (x ⊗ₜ[𝕜] One.one) = x"
] | ← adjoint_lTensor, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Coalgebra | {
"line": 163,
"column": 45
} | {
"line": 163,
"column": 63
} | {
"line": 163,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : Coalgebra 𝕜 E\nr : 𝕜\nx : E\n⊢ (adjoint comul) ((LinearMap.rTensor E (adjoint counit)) (r ⊗ₜ[𝕜] x)) =\n (adjoint comul) ((LinearMap.lTensor E (... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : Coalgebra 𝕜 E\nr : 𝕜\nx : E\n⊢ (adjoint comul) ((LinearMap.rTensor E (adjoint counit)) (r ⊗ₜ[𝕜] x)) =\n (adjoint comul) ((adjoint (LinearMap.lTensor E coun... | ← adjoint_lTensor, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 73
} | {
"line": 239,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\np : E →ₗ[𝕜] E\nhp : p.IsSymmetricProjection\na : E\nha : a ∈ p.range\nhh : ∀ {T : E →ₗ[𝕜] E}, T.IsSymmetricProjection → re ⟪T a, a⟫ = ‖T a‖ ^ 2\nU : Submodule 𝕜 E\nw✝ : U.HasOrthogonalProj... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\np : E →ₗ[𝕜] E\nhp : p.IsSymmetricProjection\na : E\nha : a ∈ p.range\nhh : ∀ {T : E →ₗ[𝕜] E}, T.IsSymmetricProjection → re ⟪T a, a⟫ = ‖T a‖ ^ 2\nU : Submodule 𝕜 E\nw✝ : U.HasOrthogonalProjection\nhq :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.JointEigenspace | {
"line": 95,
"column": 2
} | {
"line": 96,
"column": 9
} | {
"line": 96,
"column": 10
} | [
{
"pp": "case e_p\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nα : 𝕜\nA B : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nhB : B.IsSymmetric\nhAB : Commute A B\n⊢ ⨆ i, (genEigenspace (B.restrict ⋯) i) 1 = ⊤",
"ppTerm": "?e_p",
"assigned... | [
"case e_p\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nα : 𝕜\nA B : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nhB : B.IsSymmetric\nhAB : Commute A B\n⊢ ⨆ i, (genEigenspace (B.restrict ⋯) i) 1 = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.JointEigenspace | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 64
} | {
"line": 105,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nA B : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nhA : A.IsSymmetric\nhB : B.IsSymmetric\nhAB : Commute A B\n⊢ ⨆ α, ⨆ γ, eigenspace A α ⊓ eigenspace B γ = ⊤",
"ppTerm": "?m.69",
"ass... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nA B : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nhA : A.IsSymmetric\nhB : B.IsSymmetric\nhAB : Commute A B\n⊢ ⨆ α, eigenspace A α = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.LaxMilgram | {
"line": 63,
"column": 23
} | {
"line": 63,
"column": 34
} | {
"line": 63,
"column": 35
} | [
{
"pp": "V : Type u\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : CompleteSpace V\nB : V →L[ℝ] V →L[ℝ] ℝ\nC : ℝ\nC_ge_0 : 0 < C\ncoercivity : ∀ (u : V), C * ‖u‖ * ‖u‖ ≤ (B u) u\nv : V\nh : ¬0 < ‖v‖\n⊢ v = 0",
"ppTerm": "?m.174",
"assigned": false,
"usedConstants": [],
"... | [
"V : Type u\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : CompleteSpace V\nB : V →L[ℝ] V →L[ℝ] ℝ\nC : ℝ\nC_ge_0 : 0 < C\ncoercivity : ∀ (u : V), C * ‖u‖ * ‖u‖ ≤ (B u) u\nv : V\nh : ¬0 < ‖v‖\n⊢ v = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 443,
"column": 44
} | {
"line": 443,
"column": 91
} | {
"line": 443,
"column": 92
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : E →L[𝕜] E\nc : ℝ≥0\nhc : 0 < c\nh : ∀ (x : E), ‖x‖ ^ 2 * ↑c ≤ ‖⟪f x, x⟫‖\nh_anti : AntilipschitzWith c⁻¹ ⇑f\nx : E\nhx : x ∈ (↑f).rangeᗮ\n⊢ ‖x‖ ^ 2 * ↑c ≤ 0",
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : E →L[𝕜] E\nc : ℝ≥0\nhc : 0 < c\nh : ∀ (x : E), ‖x‖ ^ 2 * ↑c ≤ ‖⟪f x, x⟫‖\nh_anti : AntilipschitzWith c⁻¹ ⇑f\nx : E\nhx : x ∈ (↑f).rangeᗮ\n⊢ ‖x‖ ^ 2 * ↑c ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.LaxMilgram | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 13
} | {
"line": 72,
"column": 14
} | [
{
"pp": "V : Type u\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : CompleteSpace V\nB : V →L[ℝ] V →L[ℝ] ℝ\ncoercive : IsCoercive B\nC : ℝ\nC_pos : 0 < C\nbelow_bound : ∀ (v : V), C * ‖v‖ ≤ ‖(continuousLinearMapOfBilin B) v‖\n⊢ ∀ (x : V), C⁻¹⁻¹ * ‖x‖ ≤ ‖(continuousLinearMapOfBilin B) x‖"... | [
"V : Type u\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : CompleteSpace V\nB : V →L[ℝ] V →L[ℝ] ℝ\ncoercive : IsCoercive B\nC : ℝ\nC_pos : 0 < C\nbelow_bound : ∀ (v : V), C * ‖v‖ ≤ ‖(continuousLinearMapOfBilin B) v‖\n⊢ ∀ (x : V), C * ‖x‖ ≤ ‖(continuousLinearMapOfBilin B) x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 465,
"column": 29
} | {
"line": 465,
"column": 40
} | {
"line": 465,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace 𝕜 F\nx✝² x✝¹ x✝ : E →L[𝕜] E\nh₁ : (x✝¹ - x✝²).IsPositive\nh₂ : (x✝ - x✝¹).IsPositive\n⊢ (x✝ - x✝²).IsPositive",
"... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace 𝕜 F\nx✝² x✝¹ x✝ : E →L[𝕜] E\nh₁ : (x✝¹ - x✝²).IsPositive\nh₂ : (x✝ - x✝¹).IsPositive\n⊢ (x✝ - x✝²).IsPositive"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 475,
"column": 2
} | {
"line": 475,
"column": 13
} | {
"line": 475,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf : E →L[𝕜] E\n⊢ 0 ≤ f ↔ f.IsPositive",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf : E →L[𝕜] E\n⊢ 0 ≤ f ↔ f.IsPositive"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 530,
"column": 13
} | {
"line": 530,
"column": 24
} | {
"line": 530,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E ≃ₗ[𝕜] E\nhT : (↑T).IsPositive\nx : E\n⊢ ?m.59",
"ppTerm": "?m.60",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E ≃ₗ[𝕜] E\nhT : (↑T).IsPositive\nx : E\n⊢ ?m.59"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 546,
"column": 2
} | {
"line": 555,
"column": 76
} | {
"line": 557,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nT : E →L[𝕜] E\n⊢ T.IsPositive ↔ ∃ m u, T = ∑ i, ((rankOne 𝕜) (u i)) (u i)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Linea... | [] | refine ⟨fun hT ↦ ?_, fun ⟨m, u, hT⟩ ↦ hT ▸ isPositive_sum _ fun _ _ ↦ isPositive_rankOne_self _⟩
let a (i : Fin (Module.finrank 𝕜 E)) : E :=
((hT.isSymmetric.eigenvalues rfl i).sqrt : 𝕜) • hT.isSymmetric.eigenvectorBasis rfl i
refine ⟨Module.finrank 𝕜 E, a, ext fun _ ↦ ?_⟩
simp_rw [_root_.sum_apply, rankOn... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 546,
"column": 2
} | {
"line": 555,
"column": 76
} | {
"line": 557,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nT : E →L[𝕜] E\n⊢ T.IsPositive ↔ ∃ m u, T = ∑ i, ((rankOne 𝕜) (u i)) (u i)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Linea... | [] | refine ⟨fun hT ↦ ?_, fun ⟨m, u, hT⟩ ↦ hT ▸ isPositive_sum _ fun _ _ ↦ isPositive_rankOne_self _⟩
let a (i : Fin (Module.finrank 𝕜 E)) : E :=
((hT.isSymmetric.eigenvalues rfl i).sqrt : 𝕜) • hT.isSymmetric.eigenvectorBasis rfl i
refine ⟨Module.finrank 𝕜 E, a, ext fun _ ↦ ?_⟩
simp_rw [_root_.sum_apply, rankOn... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Dynamics.BirkhoffSum.Basic | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 27
} | {
"line": 61,
"column": 28
} | [
{
"pp": "α : Type u_1\nM : Type u_2\ninst✝ : AddCommMonoid M\nf : α → α\ng g' : α → M\nn : ℕ\nx : α\n⊢ birkhoffSum f (g + g') n x = birkhoffSum f g n x + birkhoffSum f g' n x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Membership... | [
"α : Type u_1\nM : Type u_2\ninst✝ : AddCommMonoid M\nf : α → α\ng g' : α → M\nn : ℕ\nx : α\n⊢ ∑ x_1 ∈ range n, (g (f^[x_1] x) + g' (f^[x_1] x)) = ∑ k ∈ range n, g (f^[k] x) + ∑ k ∈ range n, g' (f^[k] x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.BirkhoffSum.NormedSpace | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 17
} | {
"line": 123,
"column": 18
} | [
{
"pp": "case h\n𝕜 : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝³ : PseudoEMetricSpace X\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : X → X\ng : X → E\nhf : LipschitzWith 1 f\nhg : UniformContinuous g\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0∞\nhδ₀ : 0 < δ\nhδε : ∀ (x y : X), (x, y) ∈ {p... | [
"case h\n𝕜 : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝³ : PseudoEMetricSpace X\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : X → X\ng : X → E\nhf : LipschitzWith 1 f\nhg : UniformContinuous g\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0∞\nhδ₀ : 0 < δ\nhδε : ∀ (x y : X), (x, y) ∈ {p | edist p.1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 496,
"column": 32
} | {
"line": 496,
"column": 43
} | {
"line": 496,
"column": 44
} | [
{
"pp": "E : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : InnerProductSpace ℝ F'\nu : E → F'\nhu : ContDiff ... | [
"E : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : InnerProductSpace ℝ F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 59,
"column": 4
} | {
"line": 60,
"column": 11
} | {
"line": 60,
"column": 12
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(f.eqLocus 1)\nhg_proj : ∀ (x : ↥(f.eqLocus 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(f.eqLocus 1)\nhg_proj : ∀ (x : ↥(f.eqLocus 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑(f - 1).rang... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 21
} | {
"line": 72,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(f.eqLocus 1)\nhg_proj : ∀ (x : ↥(f.eqLocus 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(f.eqLocus 1)\nhg_proj : ∀ (x : ↥(f.eqLocus 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑(f - 1).rang... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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