module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 12
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝² : CommRing K\nμ : K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nh : IsPrimitiveRoot μ 0\nhdiv : ¬p ∣ 0\n⊢ minpoly ℤ μ ∣ (expand ℤ p) (minpoly ℤ (μ ^ p))",
"usedConstants": [
"False",
"Dvd.dvd",
"CommRing.toNonUnitalCommRing",
"Nat.instSemi... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 399,
"column": 2
} | {
"line": 399,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nd n : ℕ\nhdvd : d ∣ n\nhn : n ≠ 0\n⊢ (X ^ d - 1) * ∏ x ∈ n.divisors \\ d.divisors, cyclotomic x R = X ^ n - 1",
"usedConstants": [
"zero_le",
"Nat.instCanonicallyOrderedAdd",
"Nat.instMulZeroClass",
"instIsBotZeroClass",
"AddMonoid.toA... | have h0d : 0 < d := Nat.pos_of_dvd_of_pos hdvd (by positivity) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 12
} | [
{
"pp": "case pos\nn : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nhn : n = 0\n⊢ minpoly ℤ μ = minpoly ℤ (μ ^ p)",
"usedConstants": [
"False",
"Dvd.dvd",
"Nat.instSemigroup... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 12
} | [
{
"pp": "case pos\nn : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nhn : n = 0\n⊢ minpoly ℤ μ = minpoly ℤ (μ ^ p)",
"usedConstants": [
"False",
"Dvd.dvd",
"Nat.instSemigroup... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 12
} | [
{
"pp": "case pos\nn : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nhn : n = 0\n⊢ minpoly ℤ μ = minpoly ℤ (μ ^ p)",
"usedConstants": [
"False",
"Dvd.dvd",
"Nat.instSemigroup... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 432,
"column": 49
} | {
"line": 432,
"column": 57
} | [
{
"pp": "n✝ : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhpos : n✝ > 0\nn : ℕ\nhn : 0 < n\n⊢ ∏ i ∈ n.divisors, (algebraMap R[X] (RatFunc R)) (cyclotomic i R) =\n (algebraMap R[X] (RatFunc R)) (∏ i ∈ n.divisors, cyclotomic i R)",
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHom... | map_prod | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 719,
"column": 49
} | {
"line": 719,
"column": 57
} | [
{
"pp": "case mk\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ : Rˣ\nh✝¹ : IsPrimitiveRoot ζ k\ninst✝ : IsDomain R\na b n : ℕ\nh✝ : a * b ≡ 1 [MOD n]\nh : n ≤ 0\nva... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 724,
"column": 49
} | {
"line": 724,
"column": 57
} | [
{
"pp": "case mk\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ : Rˣ\nh✝¹ : IsPrimitiveRoot ζ k\ninst✝ : IsDomain R\na b n : ℕ\nh✝ : a * b ≡ 1 [MOD n]\nh : n ≤ 0\nva... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 760,
"column": 59
} | {
"line": 760,
"column": 67
} | [
{
"pp": "R : Type u_4\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : DecidableEq R\nn : ℕ\ninst✝ : NeZero n\nx : R\na : ℕ\nha : x ∈ primitiveRoots a R\nd : ℕ\nhd : n = a * d\nha₀ : a = 0\n⊢ False",
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"HMul.hMul",
"congrArg",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 760,
"column": 59
} | {
"line": 760,
"column": 67
} | [
{
"pp": "R : Type u_4\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : DecidableEq R\nn : ℕ\ninst✝ : NeZero n\nx : R\na : ℕ\nha : x ∈ primitiveRoots a R\nd : ℕ\nhd : n = a * d\nha₀ : a = 0\n⊢ False",
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"HMul.hMul",
"congrArg",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 760,
"column": 59
} | {
"line": 760,
"column": 67
} | [
{
"pp": "R : Type u_4\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : DecidableEq R\nn : ℕ\ninst✝ : NeZero n\nx : R\na : ℕ\nha : x ∈ primitiveRoots a R\nd : ℕ\nhd : n = a * d\nha₀ : a = 0\n⊢ False",
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"HMul.hMul",
"congrArg",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 573,
"column": 4
} | {
"line": 574,
"column": 38
} | [
{
"pp": "case pos\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).coeff 0 = 0\nh : ↑a = 0\nhone : n = 1\n⊢ False",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"Polynomial.instOne",
"NeZero.one",
"ZMod.c... | simp only [hone, cyclotomic_one, zero_sub, coeff_one_zero, coeff_X_zero, neg_eq_zero,
one_ne_zero, coeff_sub] at hroot | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 573,
"column": 4
} | {
"line": 574,
"column": 38
} | [
{
"pp": "case pos\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).coeff 0 = 0\nh : ↑a = 0\nhone : n = 1\n⊢ False",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"Polynomial.instOne",
"NeZero.one",
"ZMod.c... | simp only [hone, cyclotomic_one, zero_sub, coeff_one_zero, coeff_X_zero, neg_eq_zero,
one_ne_zero, coeff_sub] at hroot | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 573,
"column": 4
} | {
"line": 574,
"column": 38
} | [
{
"pp": "case pos\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).coeff 0 = 0\nh : ↑a = 0\nhone : n = 1\n⊢ False",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"Polynomial.instOne",
"NeZero.one",
"ZMod.c... | simp only [hone, cyclotomic_one, zero_sub, coeff_one_zero, coeff_X_zero, neg_eq_zero,
one_ne_zero, coeff_sub] at hroot | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 195,
"column": 25
} | {
"line": 195,
"column": 33
} | [
{
"pp": "case pos\nn : ℕ\nK : Type u_1\ninst✝³ : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝² : IsDomain K\ninst✝¹ : CharZero K\ninst✝ : DecidableEq K\nhn : n = 0\n⊢ primitiveRoots n K ⊆ (map (Int.castRingHom K) (minpoly ℤ μ)).roots.toFinset",
"usedConstants": [
"Multiset.toFinset",
"Polyn... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 195,
"column": 25
} | {
"line": 195,
"column": 33
} | [
{
"pp": "case pos\nn : ℕ\nK : Type u_1\ninst✝³ : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝² : IsDomain K\ninst✝¹ : CharZero K\ninst✝ : DecidableEq K\nhn : n = 0\n⊢ primitiveRoots n K ⊆ (map (Int.castRingHom K) (minpoly ℤ μ)).roots.toFinset",
"usedConstants": [
"Multiset.toFinset",
"Polyn... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 195,
"column": 25
} | {
"line": 195,
"column": 33
} | [
{
"pp": "case pos\nn : ℕ\nK : Type u_1\ninst✝³ : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝² : IsDomain K\ninst✝¹ : CharZero K\ninst✝ : DecidableEq K\nhn : n = 0\n⊢ primitiveRoots n K ⊆ (map (Int.castRingHom K) (minpoly ℤ μ)).roots.toFinset",
"usedConstants": [
"Multiset.toFinset",
"Polyn... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 633,
"column": 33
} | {
"line": 633,
"column": 41
} | [
{
"pp": "case a\nR : Type u_1\ninst✝¹ : CommRing R\nζ : R\nn : ℕ\nx y : R\ninst✝ : IsDomain R\nhpos : 0 < n\nh : IsPrimitiveRoot ζ n\nK : Type u_1 := FractionRing R\n⊢ (algebraMap R K) x ^ n - (algebraMap R K) y ^ n = (algebraMap R K) (∏ ζ ∈ nthRootsFinset n 1, (x - ζ * y))",
"usedConstants": [
"Eq.mp... | map_prod | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 72,
"column": 2
} | {
"line": 96,
"column": 51
} | [
{
"pp": "n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\ninst✝ : NeZero ↑n\n⊢ (cyclotomic n K).IsRoot μ ↔ IsPrimitiveRoot μ n",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"Finset.singleton_subset_iff._simp_1",
"Nat.mem_divisors._simp_1",
"Mathlib.Tact... | have hnpos : 0 < n := (NeZero.of_neZero_natCast K).out.bot_lt
refine ⟨fun hμ => ?_, IsPrimitiveRoot.isRoot_cyclotomic hnpos⟩
have hμn : μ ^ n = 1 := by
rw [isRoot_of_unity_iff hnpos _]
exact ⟨n, n.mem_divisors_self hnpos.ne', hμ⟩
by_contra hnμ
have ho : 0 < orderOf μ := (isOfFinOrder_iff_pow_eq_one.2 <|... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 72,
"column": 2
} | {
"line": 96,
"column": 51
} | [
{
"pp": "n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\ninst✝ : NeZero ↑n\n⊢ (cyclotomic n K).IsRoot μ ↔ IsPrimitiveRoot μ n",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"Finset.singleton_subset_iff._simp_1",
"Nat.mem_divisors._simp_1",
"Mathlib.Tact... | have hnpos : 0 < n := (NeZero.of_neZero_natCast K).out.bot_lt
refine ⟨fun hμ => ?_, IsPrimitiveRoot.isRoot_cyclotomic hnpos⟩
have hμn : μ ^ n = 1 := by
rw [isRoot_of_unity_iff hnpos _]
exact ⟨n, n.mem_divisors_self hnpos.ne', hμ⟩
by_contra hnμ
have ho : 0 < orderOf μ := (isOfFinOrder_iff_pow_eq_one.2 <|... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.KrullTopology | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 45
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : FiniteDimensional K L\n⊢ IsOpen ↑⊥",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrArg",
"IntermediateField",
"Intermediate... | rw [← IntermediateField.fixingSubgroup_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.KrullTopology | {
"line": 331,
"column": 6
} | {
"line": 331,
"column": 41
} | [
{
"pp": "k : Type u_1\nK : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nL : IntermediateField k K\nhnfd : FiniteDimensional k ↥L\nE : IntermediateField k K := normalClosure k (↥L) K\nhle : L ≤ E\nL' : IntermediateField k ↥E := restrict hle\nh : Module.finrank k ↥L' *... | ← L'.fixingSubgroup.index_mul_card, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Galois.Abelian | {
"line": 37,
"column": 6
} | {
"line": 37,
"column": 66
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Algebra K L\ninst✝⁴ : Field M\ninst✝³ : Algebra K M\ninst✝² : Algebra L M\ninst✝¹ : IsScalarTower K L M\ninst✝ : IsAbelianGalois K M\nthis : IsGalois K L\nx y : Gal(L/K)\n⊢ x * y = y * x",
"usedConstants": [
... | obtain ⟨x, rfl⟩ := AlgEquiv.restrictNormalHom_surjective M x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 159,
"column": 12
} | {
"line": 160,
"column": 95
} | [
{
"pp": "R : Type u_1\np m : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Ring R\ninst✝ : CharP R p\nhm : ¬p ∣ m\nx✝ : 0 < 1\n⊢ cyclotomic (p ^ 1 * m) R = cyclotomic m R ^ (p ^ 1 - p ^ (1 - 1))",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"Com... | by
rw [pow_one, Nat.sub_self, pow_zero, mul_comm, cyclotomic_mul_prime_eq_pow_of_not_dvd R hm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 182,
"column": 82
} | {
"line": 182,
"column": 91
} | [
{
"pp": "case inr.refine_2\nm k p : ℕ\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nhp : Fact (Nat.Prime p)\nhchar : CharP R p\nμ : R\ninst✝ : NeZero ↑m\nhk : k > 0\nh : eval μ (cyclotomic m R) = 0\n⊢ eval μ (cyclotomic m R ^ (p ^ k - p ^ (k - 1))) = 0",
"usedConstants": [
"Eq.mpr",
"... | eval_pow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 203,
"column": 24
} | {
"line": 205,
"column": 7
} | [
{
"pp": "k : Type u_1\nK✝ : Type u_2\ninst✝³ : Field k\ninst✝² : Field K✝\ninst✝¹ : Algebra k K✝\ninst✝ : IsGalois k K✝\nK L : IntermediateField k K✝\n⊢ { toFun := fun L ↦ { toSubgroup := L.fixingSubgroup, isClosed' := ⋯ },\n invFun := fun H ↦ IntermediateField.fixedField ↑H, left_inv := ⋯, right_inv :... | by
rw [← fixedField_fixingSubgroup L, IntermediateField.le_iff_le, fixedField_fixingSubgroup L]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 52,
"column": 21
} | {
"line": 56,
"column": 90
} | [
{
"pp": "K : Type u_1\nF : Type u_2\ninst✝³ : Field K\ninst✝² : Fintype K\ninst✝¹ : Field F\ninst✝ : Algebra F K\n⊢ Algebra.adjoin F ((X ^ Fintype.card K - X).rootSet K) = ⊤",
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
"Polynomial.roots",
"Lattice.toSemilatticeSup",
"Fin... | by
classical
trans Algebra.adjoin F ((roots (X ^ Fintype.card K - X : K[X])).toFinset : Set K)
· simp only [rootSet, aroots, Polynomial.map_pow, map_X, Polynomial.map_sub]
· rw [FiniteField.roots_X_pow_card_sub_X, val_toFinset, coe_univ, Algebra.adjoin_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 196,
"column": 48
} | {
"line": 196,
"column": 56
} | [
{
"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'‖\nthis : ¬eval (↑q) (cyclotomic n ℂ) = 0\n⊢ ‖eval (↑q) (... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 196,
"column": 48
} | {
"line": 196,
"column": 56
} | [
{
"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'‖\nthis : ¬eval (↑q) (cyclotomic n ℂ) = 0\n⊢ ‖eval (↑q) (... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 196,
"column": 48
} | {
"line": 196,
"column": 56
} | [
{
"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'‖\nthis : ¬eval (↑q) (cyclotomic n ℂ) = 0\n⊢ ‖eval (↑q) (... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 259,
"column": 53
} | {
"line": 259,
"column": 61
} | [
{
"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\nthis : ¬eval (↑q) (cyclotomic n ℂ) = 0\n⊢ ‖eval (↑q) (... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 259,
"column": 53
} | {
"line": 259,
"column": 61
} | [
{
"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\nthis : ¬eval (↑q) (cyclotomic n ℂ) = 0\n⊢ ‖eval (↑q) (... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 259,
"column": 53
} | {
"line": 259,
"column": 61
} | [
{
"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\nthis : ¬eval (↑q) (cyclotomic n ℂ) = 0\n⊢ ‖eval (↑q) (... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 269,
"column": 2
} | {
"line": 269,
"column": 49
} | [
{
"pp": "n : ℕ\ninst✝³ : NeZero n\nK : Type u\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nhno : Odd n\nζ x : K\nhζ : IsPrimitiveRoot ζ n\nhx : IsOfFinOrder x\nr : ℕ\nhr : x = (-ζ) ^ r\n⊢ ∃ r < n, x = ζ ^ r ∨ x = -ζ ^ r",
"usedConstants": [
"NegZeroClass.toNeg",
... | refine ⟨r % n, Nat.mod_lt _ (NeZero.pos _), ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 267,
"column": 49
} | {
"line": 271,
"column": 73
} | [
{
"pp": "n : ℕ\ninst✝³ : NeZero n\nK : Type u\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nhno : Odd n\nζ x : K\nhζ : IsPrimitiveRoot ζ n\nhx : IsOfFinOrder x\n⊢ ∃ r < n, x = ζ ^ r ∨ x = -ζ ^ r",
"usedConstants": [
"Monoid",
"IsPrimitiveRoot.eq_orderOf",
... | by
obtain ⟨r, hr⟩ := hζ.exists_neg_pow_of_isOfFinOrder hno hx
refine ⟨r % n, Nat.mod_lt _ (NeZero.pos _), ?_⟩
rw [show ζ ^ (r % n) = ζ ^ r from (IsPrimitiveRoot.eq_orderOf hζ).symm ▸ pow_mod_orderOf .., hr]
rcases Nat.even_or_odd r with (h | h) <;> simp [neg_pow, h.neg_one_pow] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 274,
"column": 4
} | {
"line": 276,
"column": 17
} | [
{
"pp": "case convert_5\nn : 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\nthis : ¬eval (↑q) (cyclotomic n ℂ) = 0... | simp only [Finset.mem_attach, forall_true_left, Subtype.forall, ←
Units.val_le_val, ← NNReal.coe_le_coe, Units.val_mk0,
coe_nnnorm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.ZMod.Units | {
"line": 105,
"column": 4
} | {
"line": 106,
"column": 40
} | [
{
"pp": "N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\np q : ℤ\nhpq : ↑d = ↑a.val * p + ↑N * q\n⊢ IsCoprime ↑a₀ ↑N₀",
"usedConstants": [
"HMul.hMul",
"congrArg",
"Eq.mp",
"instMulNat",
"... | rw [ha₀, hN₀, Nat.cast_mul, Nat.cast_mul, mul_assoc, mul_assoc, ← mul_add, eq_comm,
mul_comm _ p, mul_comm _ q] at hpq | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 240,
"column": 13
} | {
"line": 240,
"column": 27
} | [
{
"pp": "K : Type u_1\nK' : Type u_2\ninst✝³ : Field K\ninst✝² : Field K'\ninst✝¹ : Algebra K K'\ninst✝ : Finite K'\nthis : Finite K\n⊢ Nat.card ↥(Units.map (Algebra.norm K)).ker ≤ Nat.card K'ˣ / Nat.card Kˣ",
"usedConstants": [
"Eq.mpr",
"Nat.card_units",
"instHDiv",
"Monoid.toMulOn... | Nat.card_units | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 252,
"column": 84
} | {
"line": 256,
"column": 31
} | [
{
"pp": "K : Type u_1\nK' : Type u_2\ninst✝³ : Field K\ninst✝² : Field K'\ninst✝¹ : Algebra K K'\ninst✝ : Finite K'\nk : K\n⊢ ∃ a, (Algebra.norm K) a = k",
"usedConstants": [
"Units.val",
"MonoidHom.instFunLike",
"PrincipalIdealRing.isNoetherianRing",
"MonoidHom",
"CommSemiring... | by
obtain rfl | ne := eq_or_ne k 0
· exact ⟨0, Algebra.norm_zero ..⟩
have ⟨x, eq⟩ := unitsMap_norm_surjective K K' (Units.mk0 k ne)
exact ⟨x, congr_arg (·.1) eq⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LegendreSymbol.AddCharacter | {
"line": 266,
"column": 4
} | {
"line": 266,
"column": 22
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : Fintype R\ninst✝² : DecidableEq R\nR' : Type u_2\ninst✝¹ : CommRing R'\ninst✝ : IsDomain R'\nψ : AddChar R R'\nb : R\nhψ : ψ.IsPrimitive\nh : ¬b = 0\n⊢ ∑ x, ψ (x * b) = ↑0",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssoc... | simp_rw [mul_comm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 46
} | [
{
"pp": "case neg.refine_1.inl\nn : ℕ\ninst✝³ : NeZero n\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nhB : IsCyclotomicExtension S A B\nr : B\nhr : IsPrimitiveRoot r n\nhn : ¬n = 0\nm : ℕ\nhm₂ : m ≠ 0\nhm₁ : m ∈ S\n⊢ ∃ r, IsPrimitiveRoot r m",
"usedConst... | · exact exists_isPrimitiveRoot A B hm₁ hm₂ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 86,
"column": 57
} | {
"line": 86,
"column": 95
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ℤ\n⊢ stdAddChar ↑j = cexp (2 * ↑π * I * ↑j / ↑N)",
"usedConstants": [
"Int.cast",
"instHDiv",
"Real.pi",
"HMul.hMul",
"ZMod.commRing",
"congrArg",
"ZMod.stdAddChar",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.to... | by simp [stdAddChar, toCircle_intCast] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 396,
"column": 46
} | {
"line": 396,
"column": 54
} | [
{
"pp": "S✝ : Set ℕ\nn : ℕ\nS : Set ℕ\na✝ : n ∉ S\nhs✝ : S.Finite\nH :\n ∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [IsDomain B]\n [h₂ : IsCyclotomicExtension S A B], Module.Finite A B\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 45
} | [
{
"pp": "n : ℤ\n⊢ χ₄ ↑n = χ₄ ↑(n % 4)",
"usedConstants": [
"Int.cast",
"Nat.cast_ofNat",
"Eq.mpr",
"ZMod.χ₄",
"ZMod.commRing",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"CommSemiring.toCommMonoidWithZero",
"id",
"instHMod",
"MulChar",
... | rw [← ZMod.intCast_mod n 4, Nat.cast_ofNat] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 45
} | [
{
"pp": "n : ℤ\n⊢ χ₄ ↑n = χ₄ ↑(n % 4)",
"usedConstants": [
"Int.cast",
"Nat.cast_ofNat",
"Eq.mpr",
"ZMod.χ₄",
"ZMod.commRing",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"CommSemiring.toCommMonoidWithZero",
"id",
"instHMod",
"MulChar",
... | rw [← ZMod.intCast_mod n 4, Nat.cast_ofNat] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 45
} | [
{
"pp": "n : ℤ\n⊢ χ₄ ↑n = χ₄ ↑(n % 4)",
"usedConstants": [
"Int.cast",
"Nat.cast_ofNat",
"Eq.mpr",
"ZMod.χ₄",
"ZMod.commRing",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"CommSemiring.toCommMonoidWithZero",
"id",
"instHMod",
"MulChar",
... | rw [← ZMod.intCast_mod n 4, Nat.cast_ofNat] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 479,
"column": 8
} | {
"line": 479,
"column": 11
} | [
{
"pp": "case h.inr.inr\nR : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommRing R'\nχ : MulChar R R'\nhχ : χ.IsQuadratic\nx : Rˣ\nh₂ : χ ↑x = -1\nthis : -1 = ↑(-1)\n⊢ (χ ↑x)⁻¹ʳ = χ ↑x",
"usedConstants": [
"Units.val",
"Eq.mpr",
"NegZeroClass.toNeg",
"congrArg",
"A... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.MulChar.Lemmas | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 87
} | [
{
"pp": "case h\nR : Type u_1\ninst✝³ : CommMonoid R\nS : Type u_2\ninst✝² : SetLike S R\ninst✝¹ : SubmonoidClass S R\nT : S\nR' : Type u_3\ninst✝ : CommMonoidWithZero R'\nx : (↥T)ˣ\n⊢ (restrict T 1) ↑x = 1 ↑x",
"usedConstants": [
"Units.val",
"Eq.mpr",
"MulOne.toOne",
"MulChar.one_a... | rw [restrict_apply, if_pos x.isUnit, MulChar.one_apply x.isUnit.coe, one_apply_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 693,
"column": 6
} | {
"line": 694,
"column": 39
} | [
{
"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\nx : CyclotomicField 0 K\n⊢ finrank K (CyclotomicField 0 K) = 1",
"usedConstants": [
"... | have : Polynomial.IsSplittingField K K (Polynomial.cyclotomic 0 K) :=
Polynomial.isSplittingField_C 1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 859,
"column": 2
} | {
"line": 860,
"column": 82
} | [
{
"pp": "n : ℕ\ninst✝⁸ : NeZero n\nS✝ T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : IsSepClosed K\ninst✝ : CharZero K\nS : Set ℕ\n⊢ IsCyclotomicExtension S K K",
"u... | rw [IsCyclotomicExtension.eq_self_sdiff_zero]
exact IsSepClosed.isCyclotomicExtension _ K fun _ _ h ↦ ⟨Nat.cast_ne_zero.mpr h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 859,
"column": 2
} | {
"line": 860,
"column": 82
} | [
{
"pp": "n : ℕ\ninst✝⁸ : NeZero n\nS✝ T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : IsSepClosed K\ninst✝ : CharZero K\nS : Set ℕ\n⊢ IsCyclotomicExtension S K K",
"u... | rw [IsCyclotomicExtension.eq_self_sdiff_zero]
exact IsSepClosed.isCyclotomicExtension _ K fun _ _ h ↦ ⟨Nat.cast_ne_zero.mpr h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 899,
"column": 6
} | {
"line": 899,
"column": 84
} | [
{
"pp": "case a\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⊢ A[ζ] ≤ adjoin A {b | n ≠ 0 ∧ b ^ n = 1}",
"usedConstants": [
"Iff.mpr",
"Set.singleton_sub... | exact adjoin_mono <| Set.singleton_subset_iff.mpr ⟨NeZero.ne n, hζ.pow_eq_one⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 899,
"column": 6
} | {
"line": 899,
"column": 84
} | [
{
"pp": "case a\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⊢ A[ζ] ≤ adjoin A {b | n ≠ 0 ∧ b ^ n = 1}",
"usedConstants": [
"Iff.mpr",
"Set.singleton_sub... | exact adjoin_mono <| Set.singleton_subset_iff.mpr ⟨NeZero.ne n, hζ.pow_eq_one⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 899,
"column": 6
} | {
"line": 899,
"column": 84
} | [
{
"pp": "case a\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⊢ A[ζ] ≤ adjoin A {b | n ≠ 0 ∧ b ^ n = 1}",
"usedConstants": [
"Iff.mpr",
"Set.singleton_sub... | exact adjoin_mono <| Set.singleton_subset_iff.mpr ⟨NeZero.ne n, hζ.pow_eq_one⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 105,
"column": 2
} | {
"line": 106,
"column": 61
} | [
{
"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",
"usedC... | simpa [enorm, ← ENNReal.coe_rpow_of_nonneg _ (sub_nonneg.2 <| NNReal.one_le_coe.2 hp.le),
← ENNReal.coe_mul] using nnnorm_fderiv_norm_rpow_le hf hp | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 105,
"column": 2
} | {
"line": 106,
"column": 61
} | [
{
"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",
"usedC... | simpa [enorm, ← ENNReal.coe_rpow_of_nonneg _ (sub_nonneg.2 <| NNReal.one_le_coe.2 hp.le),
← ENNReal.coe_mul] using nnnorm_fderiv_norm_rpow_le hf hp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 105,
"column": 2
} | {
"line": 106,
"column": 61
} | [
{
"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",
"usedC... | simpa [enorm, ← ENNReal.coe_rpow_of_nonneg _ (sub_nonneg.2 <| NNReal.one_le_coe.2 hp.le),
← ENNReal.coe_mul] using nnnorm_fderiv_norm_rpow_le hf hp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.GaussSum | {
"line": 267,
"column": 2
} | {
"line": 267,
"column": 35
} | [
{
"pp": "F : Type u_1\ninst✝³ : Field F\ninst✝² : Fintype F\nF' : Type u_2\ninst✝¹ : Field F'\ninst✝ : Fintype F'\nχ : MulChar F F'\nhχ₁ : χ ≠ 1\nhχ₂ : χ.IsQuadratic\nhch₁ : ringChar F' ≠ ringChar F\nhch₂ : ringChar F' ≠ 2\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ ↑n\nn' : ℕ+\nhp'... | let FF' := CyclotomicField ψ.n F' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.LinearAlgebra.Matrix.DotProduct | {
"line": 168,
"column": 39
} | {
"line": 168,
"column": 52
} | [
{
"pp": "n : Type u_2\nR : Type u_4\ninst✝⁵ : Fintype n\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nv : n → R\na✝ : Nontrivial R\n⊢ ((0 ≤ fun i ↦ star (v i) * v i) ∧ ∃ i, 0 i < star (v i) * v i) ↔ ∃ a, v a ≠ 0 a",
"usedConsta... | Pi.zero_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 53,
"column": 29
} | {
"line": 53,
"column": 43
} | [
{
"pp": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm n : ℕ\nha : IsUnit A.det\nh : n ≤ m\n⊢ A ^ (m - n) = A ^ (m - n) * 1",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"CommRing.toNon... | Matrix.mul_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 143,
"column": 64
} | {
"line": 143,
"column": 78
} | [
{
"pp": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nh : IsUnit A.det\nn : ℕ\n⊢ (A ^ n)⁻¹ = (A ^ n)⁻¹ * 1",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
... | Matrix.mul_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Hermitian | {
"line": 341,
"column": 26
} | {
"line": 341,
"column": 78
} | [
{
"pp": "α : Type u_1\nm : Type u_3\ninst✝³ : CommRing α\ninst✝² : StarRing α\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nA : Matrix m m α\nhA : A.IsHermitian\n⊢ A⁻¹.IsHermitian",
"usedConstants": [
"Matrix.IsHermitian.eq",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUn... | simp [IsHermitian, conjTranspose_nonsing_inv, hA.eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.Hermitian | {
"line": 341,
"column": 26
} | {
"line": 341,
"column": 78
} | [
{
"pp": "α : Type u_1\nm : Type u_3\ninst✝³ : CommRing α\ninst✝² : StarRing α\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nA : Matrix m m α\nhA : A.IsHermitian\n⊢ A⁻¹.IsHermitian",
"usedConstants": [
"Matrix.IsHermitian.eq",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUn... | simp [IsHermitian, conjTranspose_nonsing_inv, hA.eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Hermitian | {
"line": 341,
"column": 26
} | {
"line": 341,
"column": 78
} | [
{
"pp": "α : Type u_1\nm : Type u_3\ninst✝³ : CommRing α\ninst✝² : StarRing α\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nA : Matrix m m α\nhA : A.IsHermitian\n⊢ A⁻¹.IsHermitian",
"usedConstants": [
"Matrix.IsHermitian.eq",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUn... | simp [IsHermitian, conjTranspose_nonsing_inv, hA.eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Vec | {
"line": 165,
"column": 54
} | {
"line": 165,
"column": 68
} | [
{
"pp": "l : Type u_2\nm : Type u_3\nn : Type u_1\nR : Type u_4\ninst✝³ : Semiring R\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix l m R\nB : Matrix m n R\n⊢ (A * B).vec = (A * B * 1).vec",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | Matrix.mul_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 267,
"column": 8
} | {
"line": 267,
"column": 16
} | [
{
"pp": "case pos\nm : Type u_1\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\nι : Type u_5\ninst✝ : AddLeftMono R\nA : ι → Matrix m m R\ni : ι\nhi : Finset ι\nhins : i ∉ hi\nH : hi.Nonempty → (∀ i ∈ hi, (A i).PosDef) → (∑ i ∈ hi, A i).PosDef\nhs : (insert i hi).Nonempty\nhA : ∀ i... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 267,
"column": 8
} | {
"line": 267,
"column": 16
} | [
{
"pp": "case pos\nm : Type u_1\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\nι : Type u_5\ninst✝ : AddLeftMono R\nA : ι → Matrix m m R\ni : ι\nhi : Finset ι\nhins : i ∉ hi\nH : hi.Nonempty → (∀ i ∈ hi, (A i).PosDef) → (∑ i ∈ hi, A i).PosDef\nhs : (insert i hi).Nonempty\nhA : ∀ i... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 267,
"column": 8
} | {
"line": 267,
"column": 16
} | [
{
"pp": "case pos\nm : Type u_1\nR : Type u_3\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\nι : Type u_5\ninst✝ : AddLeftMono R\nA : ι → Matrix m m R\ni : ι\nhi : Finset ι\nhins : i ∉ hi\nH : hi.Nonempty → (∀ i ∈ hi, (A i).PosDef) → (∑ i ∈ hi, A i).PosDef\nhs : (insert i hi).Nonempty\nhA : ∀ i... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 128,
"column": 6
} | {
"line": 128,
"column": 14
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →L[𝕜] E\nhT : (↑T).IsSymmetric\nM : ℝ := ⨆ x, |T.rayleighQuotient x|\nnonneg : 0 ≤ M\nx : E\nhM : |T.rayleighQuotient x| ≤ M\nhx : ¬0 < ‖x‖ ^ 2\n⊢ |re ⟪T x, x⟫_𝕜| ≤ M * ‖x‖ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 128,
"column": 6
} | {
"line": 128,
"column": 14
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →L[𝕜] E\nhT : (↑T).IsSymmetric\nM : ℝ := ⨆ x, |T.rayleighQuotient x|\nnonneg : 0 ≤ M\nx : E\nhM : |T.rayleighQuotient x| ≤ M\nhx : ¬0 < ‖x‖ ^ 2\n⊢ |re ⟪T x, x⟫_𝕜| ≤ M * ‖x‖ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 128,
"column": 6
} | {
"line": 128,
"column": 14
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →L[𝕜] E\nhT : (↑T).IsSymmetric\nM : ℝ := ⨆ x, |T.rayleighQuotient x|\nnonneg : 0 ≤ M\nx : E\nhM : |T.rayleighQuotient x| ≤ M\nhx : ¬0 < ‖x‖ ^ 2\n⊢ |re ⟪T x, x⟫_𝕜| ≤ M * ‖x‖ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Fin.Tuple.Sort | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 42
} | [
{
"pp": "n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nx✝ : ↥(graph f)\nx : α\ni : Fin n\nh : (x, i) ∈ graph f\n⊢ (fun i ↦ ⟨(f i, i), ⋯⟩) ((fun p ↦ (↑p).2) ⟨(x, i), h⟩) = ⟨(x, i), h⟩",
"usedConstants": [
"Eq.mpr",
"Tuple.graphEquiv₁._proof_4",
"Finset.univ",
"LinearOrder... | simpa [graph, eq_comm, eqComm] using h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.Fin.Tuple.Sort | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 42
} | [
{
"pp": "n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nx✝ : ↥(graph f)\nx : α\ni : Fin n\nh : (x, i) ∈ graph f\n⊢ (fun i ↦ ⟨(f i, i), ⋯⟩) ((fun p ↦ (↑p).2) ⟨(x, i), h⟩) = ⟨(x, i), h⟩",
"usedConstants": [
"Eq.mpr",
"Tuple.graphEquiv₁._proof_4",
"Finset.univ",
"LinearOrder... | simpa [graph, eq_comm, eqComm] using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fin.Tuple.Sort | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 42
} | [
{
"pp": "n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nx✝ : ↥(graph f)\nx : α\ni : Fin n\nh : (x, i) ∈ graph f\n⊢ (fun i ↦ ⟨(f i, i), ⋯⟩) ((fun p ↦ (↑p).2) ⟨(x, i), h⟩) = ⟨(x, i), h⟩",
"usedConstants": [
"Eq.mpr",
"Tuple.graphEquiv₁._proof_4",
"Finset.univ",
"LinearOrder... | simpa [graph, eq_comm, eqComm] using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 628,
"column": 6
} | {
"line": 628,
"column": 55
} | [
{
"pp": "F : Type u_3\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\nμ : Measure E\ninst✝¹ : μ.IsAddHaarMeasure\ninst✝ : FiniteDimensional ℝ F\nu :... | apply eLpNorm_le_nnreal_smul_eLpNorm_of_ae_le_mul | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 59
} | [
{
"pp": "m : Type um\nn : Type un\nR : Type uR\ninst✝² : Fintype n\ninst✝¹ : CommRing R\ninst✝ : DecidableEq n\nA : Matrix n n R\nB : Matrix m n R\nhA : IsUnit A.det\nv : n → R\n⊢ ∃ a, A.mulVecLin a = v",
"usedConstants": [
"Pi.Function.module",
"Semiring.toModule",
"Pi.addCommMonoid",
... | exact ⟨(A⁻¹).mulVecLin v, by simp [mul_nonsing_inv _ hA]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Matrix.HermitianFunctionalCalculus | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 72
} | [
{
"pp": "case inr\nn : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.IsHermitian\ninst✝ : Nontrivial (Matrix n n 𝕜)\na : Matrix n n 𝕜\nha : IsSelfAdjoint a\nh : Nonempty n\n⊢ (spectrum ℝ a).Nonempty",
"usedConstants": [
"NormedComm... | · exact spectrum_real_eq_range_eigenvalues ha ▸ Set.range_nonempty _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Module.LinearPMap | {
"line": 82,
"column": 8
} | {
"line": 82,
"column": 12
} | [
{
"pp": "R : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : Module R E\ninst✝⁷ : Module R F\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : ContinuousAdd E\ninst✝³ : ContinuousAdd F\ninst✝² : TopologicalSpace R\ninst✝¹ ... | ← hf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.SingularValues | {
"line": 173,
"column": 2
} | {
"line": 174,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : FiniteDimensional 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] F\n⊢ T.singularValues.support.card... | have hS : ∀ m ∈ T.singularValues.support, m < finrank 𝕜 E := by
grind [singularValues_of_finrank_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 197,
"column": 4
} | {
"line": 197,
"column": 43
} | [
{
"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\n⊢ T.IsSymmetric →\n ((∀ (x : F), 0 ≤ re ⟪(↑f.toLinearEquiv ∘ₗ T ∘ₗ ↑f.symm.to... | LinearIsometryEquiv.toLinearEquiv_symm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 383,
"column": 2
} | {
"line": 384,
"column": 90
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] E\nhT : T.IsPositive\nS : E →ₗ[𝕜]... | simpa [← isPositive_toContinuousLinearMap_iff] using
((T.isPositive_toContinuousLinearMap_iff.mpr hT).conj_adjoint S.toContinuousLinearMap) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 409,
"column": 41
} | {
"line": 409,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nU : Submodule 𝕜 E\nT : E →L[𝕜] E\nhT : T.IsPositive\ninst✝ : U.HasOrthogonalProjection\n⊢ True ∧ True",
"usedConstants": [
"and_self",
"And",
"True",
"of_eq_tru... | and_self | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 417,
"column": 41
} | {
"line": 417,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →L[𝕜] E\nhT : T.IsPositive\nU : Submodule 𝕜 E\ninst✝ : U.HasOrthogonalProjection\n⊢ True ∧ True",
"usedConstants": [
"and_self",
"And",
"True",
"of_eq_tru... | and_self | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 479,
"column": 2
} | {
"line": 479,
"column": 95
} | [
{
"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\np : E →L[𝕜] E\nhp : IsIdempotentElem p\n⊢ p.IsPositive ↔ IsSelfAdjoint p",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Is... | rw [← isPositive_toLinearMap_iff, IsIdempotentElem.isPositive_iff_isSymmetric hp.toLinearMap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.LocallyConvex.AbsConvexOpen | {
"line": 99,
"column": 58
} | {
"line": 103,
"column": 88
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ (gaugeSeminormFamily 𝕜 E s).ball 0 1 = ↑s",
"usedConstants": [
... | by
dsimp only [gaugeSeminormFamily]
rw [Seminorm.ball_zero_eq]
simp_rw [gaugeSeminorm_toFun]
exact gauge_lt_one_eq_self_of_isOpen (s.coe_convex.lift ℝ) s.coe_zero_mem s.coe_isOpen | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Reproducing | {
"line": 270,
"column": 2
} | {
"line": 271,
"column": 66
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace 𝕜 V\nH : Type u_4\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : RKHS 𝕜 H X V\ninst✝² : CompleteSpace H\ninst✝¹ : CompleteSpace V\nK : Matrix X X (V →L[�... | simp only [inner_H₀_def, RCLike.star_def, mul_zero, zero_mul,
Finsupp.sum_single_index, mul_one, map_zero, map_one, one_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 12
} | [
{
"pp": "case «0».«0»\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nh : ⟪x, y⟫ = 0\nhij : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩\n⊢ ⟪![x, y] ((fun i ↦ i) ⟨0, ⋯⟩), ![x, y] ((fun i ↦ i) ⟨0, ⋯⟩)⟫ = 0",
"usedC... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 12
} | [
{
"pp": "case «0».«0»\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nh : ⟪x, y⟫ = 0\nhij : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩\n⊢ ⟪![x, y] ((fun i ↦ i) ⟨0, ⋯⟩), ![x, y] ((fun i ↦ i) ⟨0, ⋯⟩)⟫ = 0",
"usedC... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 12
} | [
{
"pp": "case «0».«0»\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nh : ⟪x, y⟫ = 0\nhij : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩\n⊢ ⟪![x, y] ((fun i ↦ i) ⟨0, ⋯⟩), ![x, y] ((fun i ↦ i) ⟨0, ⋯⟩)⟫ = 0",
"usedC... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 145,
"column": 4
} | {
"line": 145,
"column": 12
} | [
{
"pp": "case «1».«1»\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nh : ⟪x, y⟫ = 0\nhij : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩\n⊢ ⟪![x, y] ((fun i ↦ i) ⟨1, ⋯⟩), ![x, y] ((fun i ↦ i) ⟨1, ⋯⟩)⟫ = 0",
"usedC... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 145,
"column": 4
} | {
"line": 145,
"column": 12
} | [
{
"pp": "case «1».«1»\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nh : ⟪x, y⟫ = 0\nhij : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩\n⊢ ⟪![x, y] ((fun i ↦ i) ⟨1, ⋯⟩), ![x, y] ((fun i ↦ i) ⟨1, ⋯⟩)⟫ = 0",
"usedC... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 145,
"column": 4
} | {
"line": 145,
"column": 12
} | [
{
"pp": "case «1».«1»\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nh : ⟪x, y⟫ = 0\nhij : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩\n⊢ ⟪![x, y] ((fun i ↦ i) ⟨1, ⋯⟩), ![x, y] ((fun i ↦ i) ⟨1, ⋯⟩)⟫ = 0",
"usedC... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 336,
"column": 40
} | {
"line": 336,
"column": 48
} | [
{
"pp": "case «0».«0»\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx : E\nhx : x ≠ 0\nhij : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩\n⊢ ⟪![x, o.rightAngleRotation x] ((fun i ↦ i) ⟨0, ⋯⟩), ![x, o.rightAngleRotation x] ((... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 336,
"column": 40
} | {
"line": 336,
"column": 48
} | [
{
"pp": "case «0».«1»\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx : E\nhx : x ≠ 0\nhij : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩\n⊢ ⟪![x, o.rightAngleRotation x] ((fun i ↦ i) ⟨0, ⋯⟩), ![x, o.rightAngleRotation x] ((... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 336,
"column": 40
} | {
"line": 336,
"column": 48
} | [
{
"pp": "case «1».«0»\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx : E\nhx : x ≠ 0\nhij : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩\n⊢ ⟪![x, o.rightAngleRotation x] ((fun i ↦ i) ⟨1, ⋯⟩), ![x, o.rightAngleRotation x] ((... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 336,
"column": 40
} | {
"line": 336,
"column": 48
} | [
{
"pp": "case «1».«1»\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx : E\nhx : x ≠ 0\nhij : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩\n⊢ ⟪![x, o.rightAngleRotation x] ((fun i ↦ i) ⟨1, ⋯⟩), ![x, o.rightAngleRotation x] ((... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 395,
"column": 4
} | {
"line": 395,
"column": 12
} | [
{
"pp": "case neg.mp\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nhx : ¬x = 0\na : ℝ := ((o.basisRightAngleRotation x hx).repr y) 0\nb : ℝ := ((o.basisRightAngleRotation x hx).repr y) 1\n⊢ 0 ≤ a * ‖x‖ ^ 2 ∧ b ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 525,
"column": 2
} | {
"line": 525,
"column": 48
} | [
{
"pp": "z w : ℂ\n⊢ ⟪Complex.orientation.rightAngleRotation z, w⟫ = ⟪I * z, w⟫",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"LinearIsometryEquiv.instEquivLike",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"Algebra.to_smulCommClass",
"Semiring.to... | rw [Orientation.inner_rightAngleRotation_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.MellinTransform | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 90
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nhab : b < a\nhf : f =O[atTop] fun x ↦ x ^ (-a)\nt : ℝ\nht : 0 < t\n⊢ t ^ (a - b) • t ^ (-a) = t ^ (-b)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
... | rw [smul_eq_mul, ← rpow_add ht, ← sub_eq_add_neg, sub_eq_add_neg a, add_sub_cancel_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.MellinTransform | {
"line": 380,
"column": 8
} | {
"line": 380,
"column": 26
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b : ℝ\nf : ℝ → E\ns : ℂ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x ↦ x ^ (-a)\nhs_top : s.re < a\nhf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)\nhs_bot : b < s.re\nF : ℂ → ℝ → E := fun z t ↦ ↑t... | simp_rw [mul_comm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
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