module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 379,
"column": 6
} | {
"line": 386,
"column": 10
} | [
{
"pp": "case refine_1\nd x y z w : ℕ\n⊢ SqLe x 1 y d → SqLe z 1 w d → SqLe (x * w + y * z) d (x * z + d * y * w) 1",
"usedConstants": [
"Mathlib.Tactic.Ring.mul_pp_pf_overlap",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.N... | intro xy zw
have :=
Int.mul_nonneg (sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le xy))
(sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le zw))
refine Int.le_of_ofNat_le_ofNat (le_of_sub_nonneg ?_)
convert this using 1
simp only [one_mul, Int.natCast_add, Int.natCast_mul]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 379,
"column": 6
} | {
"line": 386,
"column": 10
} | [
{
"pp": "case refine_2\nd x y z w : ℕ\n⊢ SqLe x 1 y d → SqLe w d z 1 → SqLe (x * z + d * y * w) 1 (x * w + y * z) d",
"usedConstants": [
"Mathlib.Tactic.Ring.mul_pp_pf_overlap",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.N... | intro xy zw
have :=
Int.mul_nonneg (sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le xy))
(sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le zw))
refine Int.le_of_ofNat_le_ofNat (le_of_sub_nonneg ?_)
convert this using 1
simp only [one_mul, Int.natCast_add, Int.natCast_mul]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 379,
"column": 6
} | {
"line": 386,
"column": 10
} | [
{
"pp": "case refine_2\nd x y z w : ℕ\n⊢ SqLe x 1 y d → SqLe w d z 1 → SqLe (x * z + d * y * w) 1 (x * w + y * z) d",
"usedConstants": [
"Mathlib.Tactic.Ring.mul_pp_pf_overlap",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.N... | intro xy zw
have :=
Int.mul_nonneg (sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le xy))
(sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le zw))
refine Int.le_of_ofNat_le_ofNat (le_of_sub_nonneg ?_)
convert this using 1
simp only [one_mul, Int.natCast_add, Int.natCast_mul]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 379,
"column": 6
} | {
"line": 386,
"column": 10
} | [
{
"pp": "case refine_3\nd x y z w : ℕ\n⊢ SqLe y d x 1 → SqLe z 1 w d → SqLe (x * z + d * y * w) 1 (x * w + y * z) d",
"usedConstants": [
"Mathlib.Tactic.Ring.mul_pp_pf_overlap",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.N... | intro xy zw
have :=
Int.mul_nonneg (sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le xy))
(sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le zw))
refine Int.le_of_ofNat_le_ofNat (le_of_sub_nonneg ?_)
convert this using 1
simp only [one_mul, Int.natCast_add, Int.natCast_mul]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 379,
"column": 6
} | {
"line": 386,
"column": 10
} | [
{
"pp": "case refine_3\nd x y z w : ℕ\n⊢ SqLe y d x 1 → SqLe z 1 w d → SqLe (x * z + d * y * w) 1 (x * w + y * z) d",
"usedConstants": [
"Mathlib.Tactic.Ring.mul_pp_pf_overlap",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.N... | intro xy zw
have :=
Int.mul_nonneg (sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le xy))
(sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le zw))
refine Int.le_of_ofNat_le_ofNat (le_of_sub_nonneg ?_)
convert this using 1
simp only [one_mul, Int.natCast_add, Int.natCast_mul]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 379,
"column": 6
} | {
"line": 386,
"column": 10
} | [
{
"pp": "case refine_4\nd x y z w : ℕ\n⊢ SqLe y d x 1 → SqLe w d z 1 → SqLe (x * w + y * z) d (x * z + d * y * w) 1",
"usedConstants": [
"Mathlib.Tactic.Ring.mul_pp_pf_overlap",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.N... | intro xy zw
have :=
Int.mul_nonneg (sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le xy))
(sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le zw))
refine Int.le_of_ofNat_le_ofNat (le_of_sub_nonneg ?_)
convert this using 1
simp only [one_mul, Int.natCast_add, Int.natCast_mul]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 379,
"column": 6
} | {
"line": 386,
"column": 10
} | [
{
"pp": "case refine_4\nd x y z w : ℕ\n⊢ SqLe y d x 1 → SqLe w d z 1 → SqLe (x * w + y * z) d (x * z + d * y * w) 1",
"usedConstants": [
"Mathlib.Tactic.Ring.mul_pp_pf_overlap",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.N... | intro xy zw
have :=
Int.mul_nonneg (sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le xy))
(sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le zw))
refine Int.le_of_ofNat_le_ofNat (le_of_sub_nonneg ?_)
convert this using 1
simp only [one_mul, Int.natCast_add, Int.natCast_mul]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 828,
"column": 69
} | {
"line": 828,
"column": 72
} | [
{
"pp": "a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\nrem : b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 ... | i0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 517,
"column": 10
} | {
"line": 517,
"column": 74
} | [
{
"pp": "ξ : ℝ\nu : ℤ\nih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = ξ.convergent n\nleft✝ : IsCoprime u ↑1\nh₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹\nht : ξ < ↑u\nh₁ : ↑u + -(1 / 2) < ξ\nhξ₁ : ⌊ξ⌋ = u - 1\nHξ : ξ ≠ ↑⌊ξ⌋\n⊢ 2⁻¹ < fract ξ",
"usedConstants": [
"sub_a... | rw [fract, hξ₁, cast_sub, cast_one, lt_sub_iff_add_lt', sub_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.MulChar.Lemmas | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Finite Rˣ\nχ : MulChar R ℂ\na : R\n⊢ star (χ a) = χ⁻¹ a",
"usedConstants": [
"Complex.commRing",
"congrArg",
"CommSemiring.toSemiring",
"RingHom",
"CommSemiring.toCommMonoidWithZero",
"NormedField.toField",
"MulCha... | simp only [RCLike.star_def, ← star_eq_inv, star_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.MulChar.Lemmas | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Finite Rˣ\nχ : MulChar R ℂ\na : R\n⊢ star (χ a) = χ⁻¹ a",
"usedConstants": [
"Complex.commRing",
"congrArg",
"CommSemiring.toSemiring",
"RingHom",
"CommSemiring.toCommMonoidWithZero",
"NormedField.toField",
"MulCha... | simp only [RCLike.star_def, ← star_eq_inv, star_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.MulChar.Lemmas | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Finite Rˣ\nχ : MulChar R ℂ\na : R\n⊢ star (χ a) = χ⁻¹ a",
"usedConstants": [
"Complex.commRing",
"congrArg",
"CommSemiring.toSemiring",
"RingHom",
"CommSemiring.toCommMonoidWithZero",
"NormedField.toField",
"MulCha... | simp only [RCLike.star_def, ← star_eq_inv, star_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 101,
"column": 2
} | {
"line": 102,
"column": 73
} | [
{
"pp": "k : ℕ\na t : ℝ\nht : 0 < t\nthis : Summable fun n ↦ ↑n ^ k * rexp (-π * (↑n + a) ^ 2 * t)\n⊢ ∀ᶠ (i : ℕ) in atTop, ‖f_nat k a t i‖ ≤ 2 ^ k * (↑i ^ k * rexp (-π * (↑i + a) ^ 2 * t))",
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminor... | simp_rw [← mul_assoc, f_nat, norm_mul, norm_eq_abs, abs_exp,
mul_le_mul_iff_of_pos_right (exp_pos _), ← mul_pow, abs_pow, two_mul] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 128,
"column": 24
} | {
"line": 128,
"column": 33
} | [
{
"pp": "case h.e'_3.h.e'_3\nt : ℝ\nht : 0 < t\n⊢ rexp (-π * (0 * 0) * t) + ∑' (b : ℕ), f_nat 0 0 t (b + 1) - 1 = F_nat 0 1 t",
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"MulZeroClass.toMul",
"Real.... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 128,
"column": 34
} | {
"line": 128,
"column": 43
} | [
{
"pp": "case h.e'_3.h.e'_3\nt : ℝ\nht : 0 < t\n⊢ rexp (-π * 0 * t) + ∑' (b : ℕ), f_nat 0 0 t (b + 1) - 1 = F_nat 0 1 t",
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"MulZeroClass.toMul",
"Real.instZe... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 74,
"column": 42
} | {
"line": 74,
"column": 64
} | [
{
"pp": "ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nq : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhq : ∀ (i : ι), a i = 0 ∨ 0 < q i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-π * q i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / q i ^ s.re\ni : ι\n⊢ a i = 0 ∨ 0 < π * q i",
"usedConstants": []
}
] | rcases hq i with h | h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 28
} | [
{
"pp": "ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nq : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhq : ∀ (i : ι), a i = 0 ∨ 0 < q i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-π * q i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / q i ^ s.re\nhp : ∀ (i : ι), a i = 0 ∨ 0 < π * q i\ni : ι\n⊢ a i / ↑(π * q ... | rcases hq i with h | h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 173,
"column": 10
} | {
"line": 173,
"column": 18
} | [
{
"pp": "case h.e'_6\na : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\nh0' : ‖rexp (-π * t)‖ < 1\n⊢ rexp (-π * (a ^ 2 + 1) * t) / (1 - rexp (-π * t)) ^ 2 =\n rexp (-π * a ^ 2 * t) * (rexp (-π * t) / (1 - rexp (-π * t)) ^ 2)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Real",
"... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 27
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\n⊢ (fun x ↦ (P.ε * ↑(x ^ (-P.k))) • P.g₀) =O[𝓝[>] 0] fun x ↦ x ^ (-P.k)",
"usedConstants": [
"Asymptotics.isBigO_norm_norm",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSemino... | rw [← isBigO_norm_norm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : StrongFEPair E\ns : ℂ\n⊢ P.Λ (↑P.k - s) = P.ε • P.symm.Λ s",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"DistribMulAction.toDistribSMul",
"WeakFEPair.f",
"NormedSpace.toModule",
... | P.Λ_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 28
} | [
{
"pp": "ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nq : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhq : ∀ (i : ι), a i = 0 ∨ 0 < q i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-π * q i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / q i ^ s.re\nhp : ∀ (i : ι), a i = 0 ∨ 0 < π * q i\ni : ι\n⊢ ‖a i‖ / (π * q... | rcases hq i with h | h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 29
} | [
{
"pp": "case h.e'_2.h.e'_5.h.negSucc\nk : ℕ\na : ℝ\nha : a ∈ Icc 0 1\nt : ℝ\nht : 0 < t\na✝ : ℕ\n⊢ f_int k a t (Int.negSucc a✝) = Int.rec (f_nat k a t) (f_nat k (1 - a) t) (Int.negSucc a✝)",
"usedConstants": [
"HurwitzKernelBounds.f_int",
"Eq.mpr",
"Real",
"HurwitzKernelBounds.f_int... | · rw [f_int_negSucc _ ha.2] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 16
} | [
{
"pp": "S T : ℝ\nhT : 0 < T\nz τ : ℂ\nhz : |z.im| ≤ S\nhτ : T ≤ τ.im\nn : ℤ\n⊢ -(2 * S * ↑|n|) ≤ 2 * ↑n * z.im",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Int.cast",
"Eq.mpr",
"Semigroup.toMul",
"Real",
"NonUnitalCommRing.toNonUnita... | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 30
} | [
{
"pp": "a : ℝ\nha : a ∈ Ico 0 1\np : ℝ\nhp : 0 < p\nhp' : (fun t ↦ F_nat 0 a t - if a = 0 then 1 else 0) =O[atTop] fun t ↦ rexp (-p * t)\nq : ℝ\nhq : 0 < q\nhq' : (fun t ↦ F_nat 0 (1 - a) t) =O[atTop] fun t ↦ rexp (-q * t)\n⊢ ∃ p, 0 < p ∧ (fun t ↦ F_int 0 (↑a) t - if a = 0 then 1 else 0) =O[atTop] fun t ↦ rexp... | refine ⟨_, lt_min hp hq, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 30
} | [
{
"pp": "a : ℝ\nha : a ∈ Ico 0 1\np : ℝ\nhp : 0 < p\nhp' : F_nat 1 a =O[atTop] fun t ↦ rexp (-p * t)\nq : ℝ\nhq : 0 < q\nhq' : F_nat 1 (1 - a) =O[atTop] fun t ↦ rexp (-q * t)\n⊢ ∃ p, 0 < p ∧ F_int 1 ↑a =O[atTop] fun t ↦ rexp (-p * t)",
"usedConstants": [
"Real",
"HMul.hMul",
"Real.instZero... | refine ⟨_, lt_min hp hq, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 144,
"column": 73
} | {
"line": 144,
"column": 82
} | [
{
"pp": "case refine_2.inr\nz τ : ℂ\nhτ✝ : τ.im ≤ 0\nhτ : τ.im = 0\nh : Summable fun x ↦ rexp (-π * ↑x ^ 2 * 0 - 2 * π * ↑x * z.im)\n⊢ False",
"usedConstants": [
"Int.cast",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"Real.instZero",
... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 354,
"column": 6
} | {
"line": 354,
"column": 18
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\neval_fst_CLM : (ℂ × ℂ →L[ℂ] ℂ) →L[ℂ] ℂ := { toFun := fun f ↦ f (1, 0), map_add' := ⋯, map_smul' := ⋯, cont := ⋯ }\nstep1 : HasSum (fun n ↦ (jacobiTheta₂_term_fderiv n z τ) (1, 0)) ((jacobiTheta₂_fderiv z τ) (1, 0))\nstep2 : ∀ (n : ℤ), (jacobiTheta₂_term_fderiv n z τ) (1, 0) = ja... | funext step2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 341,
"column": 2
} | {
"line": 357,
"column": 32
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\n⊢ HasDerivAt (fun x ↦ jacobiTheta₂ x τ) (jacobiTheta₂' z τ) z",
"usedConstants": [
"ContinuousLinearMap.comp",
"IsModuleTopology.toContinuousSMul",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Int.cast",
"InnerProductSpace.toNor... | let eval_fst_CLM : (ℂ × ℂ →L[ℂ] ℂ) →L[ℂ] ℂ :=
{ toFun := fun f ↦ f (1, 0)
cont := continuous_id'.clm_apply continuous_const
map_add' := by simp only [ContinuousLinearMap.add_apply, forall_const]
map_smul' := by simp }
have step1 : HasSum (fun n ↦ (jacobiTheta₂_term_fderiv n z τ) (1, 0))
((jacobiTh... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 341,
"column": 2
} | {
"line": 357,
"column": 32
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\n⊢ HasDerivAt (fun x ↦ jacobiTheta₂ x τ) (jacobiTheta₂' z τ) z",
"usedConstants": [
"ContinuousLinearMap.comp",
"IsModuleTopology.toContinuousSMul",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Int.cast",
"InnerProductSpace.toNor... | let eval_fst_CLM : (ℂ × ℂ →L[ℂ] ℂ) →L[ℂ] ℂ :=
{ toFun := fun f ↦ f (1, 0)
cont := continuous_id'.clm_apply continuous_const
map_add' := by simp only [ContinuousLinearMap.add_apply, forall_const]
map_smul' := by simp }
have step1 : HasSum (fun n ↦ (jacobiTheta₂_term_fderiv n z τ) (1, 0))
((jacobiTh... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 384,
"column": 51
} | {
"line": 384,
"column": 59
} | [
{
"pp": "z τ : ℂ\nn : ℤ\nthis : cexp (↑π * I * ↑n ^ 2 * 2) = 1\n⊢ cexp (2 * ↑π * I * ↑n * z) * cexp (↑π * I * ↑n ^ 2 * (τ + 2)) =\n cexp (2 * ↑π * I * ↑n * z) * cexp (↑π * I * ↑n ^ 2 * τ)",
"usedConstants": [
"Distrib.leftDistribClass",
"Int.cast",
"Eq.mpr",
"NonUnitalCommRing.toN... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 391,
"column": 30
} | {
"line": 391,
"column": 38
} | [
{
"pp": "z τ : ℂ\nn : ℤ\n⊢ cexp (2 * ↑π * I * ↑n * (z + 1) + ↑π * I * ↑n ^ 2 * τ) = cexp (2 * ↑π * I * ↑n * z + ↑π * I * ↑n ^ 2 * τ)",
"usedConstants": [
"Distrib.leftDistribClass",
"Int.cast",
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hM... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 419,
"column": 51
} | {
"line": 419,
"column": 59
} | [
{
"pp": "z τ : ℂ\nn : ℤ\nthis : cexp (↑π * I * ↑n ^ 2 * 2) = 1\n⊢ 2 * ↑π * I * ↑n * (cexp (2 * ↑π * I * ↑n * z) * cexp (↑π * I * ↑n ^ 2 * (τ + 2))) =\n 2 * ↑π * I * ↑n * (cexp (2 * ↑π * I * ↑n * z) * cexp (↑π * I * ↑n ^ 2 * τ))",
"usedConstants": [
"Distrib.leftDistribClass",
"Int.cast",
... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 433,
"column": 64
} | {
"line": 433,
"column": 73
} | [
{
"pp": "case inl\nz τ : ℂ\nhτ : τ.im ≤ 0\n⊢ 0 = cexp (-↑π * I * (τ + 2 * z)) * (0 - 2 * ↑π * I * 0)",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.instNorme... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 439,
"column": 6
} | {
"line": 439,
"column": 14
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\nn : ℤ\n| (2 * ↑π * I * (↑n + 1) - 2 * ↑π * I) *\n cexp (2 * ↑π * I * (↑n + 1) * z + ↑π * I * (↑n + 1) ^ 2 * τ + -↑π * I * (τ + 2 * z))",
"usedConstants": [
"Distrib.leftDistribClass",
"Int.cast",
"Semigroup.toMul",
"NonUnitalCommRing.toNonUnitalNon... | mul_add, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 209,
"column": 4
} | {
"line": 209,
"column": 13
} | [
{
"pp": "a t : ℝ\nht : 0 < t\nthis :\n HasSum\n (fun n ↦\n cexp (2 * ↑π * I * ↑a * ↑↑(n + 1)) * ↑(rexp (-π * ↑↑(n + 1) ^ 2 * t)) +\n cexp (2 * ↑π * I * ↑a * ↑(-↑(n + 1))) * ↑(rexp (-π * ↑(-↑(n + 1)) ^ 2 * t)))\n (↑(cosKernel (↑a) t) + cexp (2 * ↑π * I * ↑a * 0) * ↑(rexp (-π * 0 * t)) -\n ... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 88,
"column": 41
} | {
"line": 88,
"column": 50
} | [
{
"pp": "case inl\nz : ℂ\n⊢ 0 = -2 * ↑π / (-I * 0) ^ (3 / 2) * jacobiTheta₂'' z (-1 / 0)",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"Nat.instAtLeastTwoHA... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 101,
"column": 60
} | {
"line": 101,
"column": 70
} | [
{
"pp": "case inr.e_a\nz τ : ℂ\nhτ : τ ≠ 0\naux1 : -2 * ↑π / (2 * ↑π * I) = I\n⊢ -(I *\n (cexp (-↑π * I * z ^ 2 / τ) *\n (jacobiTheta₂' (z / τ) (-1 / τ) - 2 * ↑π * I * z * jacobiTheta₂ (z / τ) (-1 / τ)))) =\n I *\n (cexp (↑π * I * z ^ 2 * (-1 / τ)) *\n (-jacobiTheta₂' (z / τ) (-1 ... | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 101,
"column": 71
} | {
"line": 101,
"column": 81
} | [
{
"pp": "case inr.e_a\nz τ : ℂ\nhτ : τ ≠ 0\naux1 : -2 * ↑π / (2 * ↑π * I) = I\n⊢ I *\n -(cexp (-↑π * I * z ^ 2 / τ) *\n (jacobiTheta₂' (z / τ) (-1 / τ) - 2 * ↑π * I * z * jacobiTheta₂ (z / τ) (-1 / τ))) =\n I *\n (cexp (↑π * I * z ^ 2 * (-1 / τ)) *\n (-jacobiTheta₂' (z / τ) (-1 / τ)... | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 13
} | [
{
"pp": "case H\nx : ℝ\nhx : x ≤ 0\na' : ℝ\n⊢ cexp (-↑π * ↑a' ^ 2 * ↑x) * (↑a' * 0) = 0",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"Complex.instNormedField",
"Comple... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 163,
"column": 11
} | {
"line": 163,
"column": 27
} | [
{
"pp": "case H\na : ℝ\n⊢ ContinuousOn (fun x ↦ ↑(oddKernel (↑a) x)) (Ioi 0)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Set.Ioi",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.instZero",
"HurwitzZeta.oddKernel_def'",
... | oddKernel_def' a | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 504,
"column": 75
} | {
"line": 504,
"column": 84
} | [
{
"pp": "case inl\nz τ : ℂ\nhτ : τ.im ≤ 0\n⊢ 0 = 1 / (-I * τ) ^ (1 / 2) * cexp (-↑π * I * z ^ 2 / τ) / τ * (0 - 2 * ↑π * I * z * 0)",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"MulZeroClass.toMul",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 519,
"column": 4
} | {
"line": 519,
"column": 54
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\nhτ' : 0 < (-1 / τ).im\nhj : HasDerivAt (fun w ↦ jacobiTheta₂ (w / τ) (-1 / τ)) (1 / τ * jacobiTheta₂' (z / τ) (-1 / τ)) z\n⊢ deriv (fun x ↦ jacobiTheta₂ x τ) z =\n deriv (fun z ↦ 1 / (-I * τ) ^ (1 / 2) * cexp (-↑π * I * z ^ 2 / τ) * jacobiTheta₂ (z / τ) (-1 / τ)) z",
"use... | rw [funext (jacobiTheta₂_functional_equation · τ)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 519,
"column": 4
} | {
"line": 519,
"column": 54
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\nhτ' : 0 < (-1 / τ).im\nhj : HasDerivAt (fun w ↦ jacobiTheta₂ (w / τ) (-1 / τ)) (1 / τ * jacobiTheta₂' (z / τ) (-1 / τ)) z\n⊢ deriv (fun x ↦ jacobiTheta₂ x τ) z =\n deriv (fun z ↦ 1 / (-I * τ) ^ (1 / 2) * cexp (-↑π * I * z ^ 2 / τ) * jacobiTheta₂ (z / τ) (-1 / τ)) z",
"use... | rw [funext (jacobiTheta₂_functional_equation · τ)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 519,
"column": 4
} | {
"line": 519,
"column": 54
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\nhτ' : 0 < (-1 / τ).im\nhj : HasDerivAt (fun w ↦ jacobiTheta₂ (w / τ) (-1 / τ)) (1 / τ * jacobiTheta₂' (z / τ) (-1 / τ)) z\n⊢ deriv (fun x ↦ jacobiTheta₂ x τ) z =\n deriv (fun z ↦ 1 / (-I * τ) ^ (1 / 2) * cexp (-↑π * I * z ^ 2 / τ) * jacobiTheta₂ (z / τ) (-1 / τ)) z",
"use... | rw [funext (jacobiTheta₂_functional_equation · τ)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 417,
"column": 41
} | {
"line": 417,
"column": 50
} | [
{
"pp": "a : ℝ\ns : ℂ\nhs : 1 < s.re\nthis :\n HasSum\n (fun n ↦\n (s + 1).Gammaℝ * -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑↑n) / ↑|↑n| ^ s / 2 +\n (s + 1).Gammaℝ * -I * ↑(-↑n).sign * cexp (2 * ↑π * I * ↑a * ↑(-↑n)) / ↑|(-↑n)| ^ s / 2)\n (completedSinZeta (↑a) s + (s + 1).Gammaℝ * -I * 0 *... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 511,
"column": 27
} | {
"line": 511,
"column": 85
} | [
{
"pp": "case inr\na : ℝ\nha : a ∈ Icc 0 1\ns : ℂ\nhs : 1 < s.re\nn : ℕ\nb : ℝ\nhb✝ : 0 ≤ b\nhb : 0 = ↑n + b\n⊢ ↑(SignType.sign 0) / 0 ^ s = 1 / 0 ^ s",
"usedConstants": [
"Iff.mpr",
"SignType.cast",
"Eq.mpr",
"False",
"Real",
"Preorder.toLT",
"instHDiv",
"Rea... | zero_cpow ((not_lt.mpr zero_le_one) ∘ (zero_re ▸ · ▸ hs)), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 534,
"column": 4
} | {
"line": 534,
"column": 20
} | [
{
"pp": "case inl\na : ℝ\ns : ℂ\nhs : 1 < s.re\nthis :\n HasSum\n (fun n ↦\n -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑n) / ↑n ^ s / 2 +\n -(-I * ↑(↑n).sign) * cexp (-(2 * ↑π * I * ↑a * ↑n)) / ↑n ^ s / 2)\n (sinZeta (↑a) s)\nn : ℕ\nh : n ≠ 0\n⊢ (cexp (-(2 * ↑π * ↑a * ↑n * I)) * I - cexp (2 *... | congr 5 <;> ring | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 555,
"column": 95
} | {
"line": 558,
"column": 47
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\n⊢ hurwitzZetaOdd a (1 - s) = 2 * (2 * ↑π) ^ (-s) * Complex.Gamma s * Complex.sin (↑π * s / 2) * sinZeta a s",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"NonAssocSemiring.toAddCommMon... | by
rw [← Gammaℂ, hurwitzZetaOdd, (by ring : 1 - s + 1 = 2 - s), div_eq_mul_inv,
inv_Gammaℝ_two_sub hs, completedHurwitzZetaOdd_one_sub, sinZeta, ← div_eq_mul_inv,
← mul_div_assoc, ← mul_div_assoc, mul_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 409,
"column": 2
} | {
"line": 409,
"column": 35
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\n⊢ L ((fun n ↦ χ ↑n) * fun n ↦ ↑(Λ n)) s = -deriv (L fun n ↦ χ ↑n) s / L (fun n ↦ χ ↑n) s",
"usedConstants": [
"instOfNatNat",
"Nat",
"eq_or_ne",
"OfNat.ofNat"
]
}
] | rcases eq_or_ne N 0 with rfl | hN | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.RingTheory.Radical.Basic | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 36
} | [
{
"pp": "case inr.inl\nM : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na : M\nha : a ≠ 0\nhc : IsRelPrime a 0\n⊢ primeFactors (a * 0) = (primeFactors a).disjUnion (primeFactors 0) ⋯",
"usedConstants": [
"congrArg",
"IsUnit",
... | rw [isRelPrime_zero_right] at hc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Radical.Basic | {
"line": 275,
"column": 63
} | {
"line": 278,
"column": 43
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na b : M\n⊢ radical a ∣ radical b ↔ primeFactors a ⊆ primeFactors b",
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",
"Multiset.toFinset",
"Eq.mpr",
... | by
classical
rw [radical_dvd_radical_iff_normalizedFactors_subset_normalizedFactors, primeFactors,
primeFactors, Multiset.toFinset_subset] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Radical.Basic | {
"line": 399,
"column": 2
} | {
"line": 399,
"column": 46
} | [
{
"pp": "E : Type u_1\ninst✝² : EuclideanDomain E\ninst✝¹ : NormalizationMonoid E\ninst✝ : UniqueFactorizationMonoid E\na b : E\nh : IsCoprime (radical a * EuclideanDomain.divRadical a) (radical b * EuclideanDomain.divRadical b)\n⊢ IsCoprime (EuclideanDomain.divRadical a) (EuclideanDomain.divRadical b)",
"u... | exact h.of_mul_left_right.of_mul_right_right | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 54
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : DecidableEq α\ns : Finset α\ni : α → ℕ\np : α\nhps : p ∉ s\nis_prime : ∀ q ∈ insert p s, Prime q\nis_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'\nhp : Prime p\nd : α\nhdprod : d ∣ ∏ p' ∈ s, p' ... | obtain ⟨q, q_mem, rfl⟩ := Multiset.mem_map.mp q_mem' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.FLT.Basic | {
"line": 223,
"column": 2
} | {
"line": 235,
"column": 44
} | [
{
"pp": "n : ℕ\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : IsDomain R\ninst✝¹ : DecidableEq R\ninst✝ : NormalizedGCDMonoid R\nhn : ∀ (a b c : R), a ≠ 0 → b ≠ 0 → c ≠ 0 → {a, b, c}.gcd id = 1 → a ^ n + b ^ n ≠ c ^ n\n⊢ FermatLastTheoremWith R n",
"usedConstants": [
"Distrib.leftDistribClass",
... | intro a b c ha hb hc habc
let s : Finset R := {a, b, c}; let d := s.gcd id
obtain ⟨A, hA⟩ : d ∣ a := gcd_dvd (by simp [s])
obtain ⟨B, hB⟩ : d ∣ b := gcd_dvd (by simp [s])
obtain ⟨C, hC⟩ : d ∣ c := gcd_dvd (by simp [s])
simp only [hA, hB, hC, mul_ne_zero_iff, mul_pow] at ha hb hc habc
rw [← mul_add, mul_righ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.FLT.Basic | {
"line": 223,
"column": 2
} | {
"line": 235,
"column": 44
} | [
{
"pp": "n : ℕ\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : IsDomain R\ninst✝¹ : DecidableEq R\ninst✝ : NormalizedGCDMonoid R\nhn : ∀ (a b c : R), a ≠ 0 → b ≠ 0 → c ≠ 0 → {a, b, c}.gcd id = 1 → a ^ n + b ^ n ≠ c ^ n\n⊢ FermatLastTheoremWith R n",
"usedConstants": [
"Distrib.leftDistribClass",
... | intro a b c ha hb hc habc
let s : Finset R := {a, b, c}; let d := s.gcd id
obtain ⟨A, hA⟩ : d ∣ a := gcd_dvd (by simp [s])
obtain ⟨B, hB⟩ : d ∣ b := gcd_dvd (by simp [s])
obtain ⟨C, hC⟩ : d ∣ c := gcd_dvd (by simp [s])
simp only [hA, hB, hC, mul_ne_zero_iff, mul_pow] at ha hb hc habc
rw [← mul_add, mul_righ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 128,
"column": 2
} | {
"line": 129,
"column": 18
} | [
{
"pp": "case inr.inl\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhx : x % 2 = 1\nhy : y % 2 = 0\n⊢ x % 2 = 0 ∧ y % 2 = 1 ∨ x % 2 = 1 ∧ y % 2 = 0",
"usedConstants": [
"instHMod",
"Int",
"HMod.hMod",
"And",
"instOfNat",
"And.intro",
"Int.instMod",
... | · right
exact ⟨hx, hy⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative | {
"line": 125,
"column": 13
} | {
"line": 125,
"column": 22
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_3\ninst✝ : CommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, IsRel... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FLT.MasonStothers | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 41
} | [
{
"pp": "k : Type u_1\ninst✝¹ : Field k\ninst✝ : DecidableEq k\na b c : k[X]\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\nhab : IsCoprime a b\nhsum : a + b + c = 0\nw : k[X] := a.wronskian b\nwab : w = a.wronskian b\nhbc : IsCoprime b c\nhsum' : -(b + c) = a\n⊢ IsCoprime c a",
"usedConstants": [
"AddGroup.toS... | rw [← hsum', IsCoprime.neg_right_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 92
} | [
{
"pp": "m n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (m ^ 2 + n ^ 2) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ m ^ 2 + n ^ 2\nh2m : ↑p ∣ 2 * m ^ 2\nh2n : ↑p ∣ 2 * n ^ 2\n⊢ False",
"usedConstants": [
"Dvd.dvd",
"instOfNatNat",
... | have hmc : p = 2 ∨ p ∣ Int.natAbs m := prime_two_or_dvd_of_dvd_two_mul_pow_self_two hp h2m | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 330,
"column": 4
} | {
"line": 330,
"column": 41
} | [
{
"pp": "case mp.h\nS : Type u_1\ninst✝⁴ : CommRing S\ninst✝³ : Nontrivial S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\nI : Ideal S\nhI : absNorm I = 0\nx : S\nhx : x ∈ I\n⊢ span {x} ≤ I",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
... | rwa [Ideal.span_singleton_le_iff_mem] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 354,
"column": 6
} | {
"line": 360,
"column": 19
} | [
{
"pp": "case inr.refine_1\nS : Type u_1\ninst✝⁵ : CommRing S\ninst✝⁴ : Nontrivial S\ninst✝³ : IsDedekindDomain S\ninst✝² : Free ℤ S\ninst✝¹ : Module.Finite ℤ S\ninst✝ : CharZero S\nn : ℕ\nhn : n > 0\nf : Ideal S → Ideal (S ⧸ span {↑n}) := fun I ↦ map (Quotient.mk (span {↑n})) I\n⊢ (f '' {I | absNorm I = n}).Fi... | suffices Finite (S ⧸ @Ideal.span S _ {↑n}) by
let g := ((↑) : Ideal (S ⧸ @Ideal.span S _ {↑n}) → Set (S ⧸ @Ideal.span S _ {↑n}))
refine Set.Finite.of_finite_image (f := g) ?_ SetLike.coe_injective.injOn
exact Set.Finite.subset Set.finite_univ (Set.subset_univ _)
rw [← absNorm_ne_zero_iff, ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 354,
"column": 6
} | {
"line": 360,
"column": 19
} | [
{
"pp": "case inr.refine_1\nS : Type u_1\ninst✝⁵ : CommRing S\ninst✝⁴ : Nontrivial S\ninst✝³ : IsDedekindDomain S\ninst✝² : Free ℤ S\ninst✝¹ : Module.Finite ℤ S\ninst✝ : CharZero S\nn : ℕ\nhn : n > 0\nf : Ideal S → Ideal (S ⧸ span {↑n}) := fun I ↦ map (Quotient.mk (span {↑n})) I\n⊢ (f '' {I | absNorm I = n}).Fi... | suffices Finite (S ⧸ @Ideal.span S _ {↑n}) by
let g := ((↑) : Ideal (S ⧸ @Ideal.span S _ {↑n}) → Set (S ⧸ @Ideal.span S _ {↑n}))
refine Set.Finite.of_finite_image (f := g) ?_ SetLike.coe_injective.injOn
exact Set.Finite.subset Set.finite_univ (Set.subset_univ _)
rw [← absNorm_ne_zero_iff, ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 590,
"column": 12
} | {
"line": 590,
"column": 17
} | [
{
"pp": "case h.inl.inr\nx y z : ℤ\nh : PythagoreanTriple x y z\nh_coprime : x.gcd y = 1\nh_parity : x % 2 = 1\nm n : ℤ\nht3 : m.gcd n = 1\nht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0\nhm : 0 ≤ m\nh_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n\nh_neg : z = -(m ^ 2 + n ^ 2)\n⊢ 0 < z → False",
"usedConstan... | h_neg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 615,
"column": 12
} | {
"line": 615,
"column": 17
} | [
{
"pp": "case h.inl.right.inr\nx y z : ℤ\nh : PythagoreanTriple x y z\nh_coprime : x.gcd y = 1\nh_parity : x % 2 = 1\nm n : ℤ\nht3 : m.gcd n = 1\nht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0\nhm : m < 0\nh_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n\nh_neg : z = -(m ^ 2 + n ^ 2)\n⊢ 0 < z → False",
"usedC... | h_neg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 266,
"column": 8
} | {
"line": 266,
"column": 14
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nhn : 3 ≤ n\nchn : ↑n ≠ 0\na b c a' b' : k[X]\nd : k[X] := gcd a b\neq_a : a = a' * d\neq_b : b = b' * d\nhd : d ≠ 0\nc' : k[X]\nheq : a' ^ n + b' ^ n = c' ^ n\nhc : d ≠ 0 ∧ c' ≠ 0\neq_c : c = c' * d\nca' : k\nha : d ≠ 0 ∧ ca' ≠ 0\nha' : C ca' = a'\ncb' : k\nhb : d ... | ← hc', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 206,
"column": 2
} | {
"line": 208,
"column": 67
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nφ : K →+* ℂ\nhφ : ComplexEmbedding.IsReal (mk φ).embedding\n⊢ ComplexEmbedding.IsReal φ",
"usedConstants": [
"NumberField.ComplexEmbedding.conjugate",
"NumberField.ComplexEmbedding.IsReal",
"congrArg",
"NumberField.InfinitePlace.mk_embedding",
... | cases mk_eq_iff.mp (mk_embedding (mk φ)) with
| inl h => rwa [h] at hφ
| inr h => rwa [← ComplexEmbedding.isReal_conjugate_iff, h] at hφ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 317,
"column": 2
} | {
"line": 318,
"column": 46
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nα : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : NumberField K\nf : InfinitePlace K → α\n⊢ ∏ w, f w = (∏ w, f ↑w) * ∏ w, f ↑w",
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"Equiv.instEquivLike",
"HMul.hMul",
"Finset.univ",
"Number... | rw [← Equiv.prod_comp (Equiv.subtypeEquivRight (fun _ ↦ not_isReal_iff_isComplex))]
simp [Fintype.prod_subtype_mul_prod_subtype] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 317,
"column": 2
} | {
"line": 318,
"column": 46
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nα : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : NumberField K\nf : InfinitePlace K → α\n⊢ ∏ w, f w = (∏ w, f ↑w) * ∏ w, f ↑w",
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"Equiv.instEquivLike",
"HMul.hMul",
"Finset.univ",
"Number... | rw [← Equiv.prod_comp (Equiv.subtypeEquivRight (fun _ ↦ not_isReal_iff_isComplex))]
simp [Fintype.prod_subtype_mul_prod_subtype] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 484,
"column": 55
} | {
"line": 484,
"column": 64
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh : finrank ℚ K = 1\nthis : nrRealPlaces K + 2 * 0 = 1\n⊢ nrRealPlaces K = 1",
"usedConstants": [
"Nat.instMulZeroClass",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"Eq.mp",
"instOfNatNat",
"NumberF... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 203,
"column": 2
} | {
"line": 204,
"column": 49
} | [
{
"pp": "case h.refine_1\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nζ : (𝓞 K)ˣ\nh : ζ ^ torsionOrder K = 1\n⊢ ζ ∈ CommGroup.torsion (𝓞 K)ˣ",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"NumberField.Units.torsionOrder_pos",
"NumberField.instCommRingRingOfIntegers",
... | · rw [CommGroup.mem_torsion, isOfFinOrder_iff_pow_eq_one]
exact ⟨torsionOrder K, torsionOrder_pos K, h⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex | {
"line": 97,
"column": 74
} | {
"line": 97,
"column": 83
} | [
{
"pp": "K : Type u_2\ninst✝¹ : Field K\ninst✝ : NumberField K\nh : IsTotallyReal K\n⊢ nrRealPlaces K + 2 * 0 = nrRealPlaces K",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"id",
"instMulNat",
"instOfNatN... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 481,
"column": 13
} | {
"line": 481,
"column": 19
} | [
{
"pp": "case e_s\nk : Type u_1\ninst✝⁵ : Field k\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra k K\ninst✝² : NumberField K\ninst✝¹ : NumberField k\ninst✝ : IsGalois k K\nw : InfinitePlace K\nhw : IsUnramifiedIn K ((fun x ↦ x.comap (algebraMap k K)) w)\n⊢ {a ∈ {w | IsUnramified k w} | a.comap (algebraMap k ... | ext w' | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 504,
"column": 13
} | {
"line": 504,
"column": 19
} | [
{
"pp": "case e_s\nk : Type u_1\ninst✝⁵ : Field k\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra k K\ninst✝² : NumberField K\ninst✝¹ : NumberField k\ninst✝ : IsGalois k K\nw : InfinitePlace K\nhw : ¬IsUnramifiedIn K ((fun x ↦ x.comap (algebraMap k K)) w)\n⊢ {a ∉ {w | IsUnramified k w} | a.comap (algebraMap k... | ext w' | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 80,
"column": 2
} | {
"line": 88,
"column": 30
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\n⊢ p.ramificationIdx P = n",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Semiring.toModule",
"IsScal... | let Q : ℕ → Prop := fun m => ∀ k : ℕ, map f p ≤ P ^ k → k ≤ m
have : Q n := by
intro k hk
refine le_of_not_gt fun hnk => ?_
exact hgt (hk.trans (Ideal.pow_le_pow_right hnk))
rw [ramificationIdx_eq_find ⟨n, this⟩]
refine le_antisymm (Nat.find_min' _ this) (le_of_not_gt fun h : Nat.find _ < n => ?_)
o... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 80,
"column": 2
} | {
"line": 88,
"column": 30
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\n⊢ p.ramificationIdx P = n",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Semiring.toModule",
"IsScal... | let Q : ℕ → Prop := fun m => ∀ k : ℕ, map f p ≤ P ^ k → k ≤ m
have : Q n := by
intro k hk
refine le_of_not_gt fun hnk => ?_
exact hgt (hk.trans (Ideal.pow_le_pow_right hnk))
rw [ramificationIdx_eq_find ⟨n, this⟩]
refine le_antisymm (Nat.find_min' _ this) (le_of_not_gt fun h : Nat.find _ < n => ?_)
o... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 95,
"column": 4
} | {
"line": 97,
"column": 55
} | [
{
"pp": "case succ\nR : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nn : ℕ\nhgt : ¬map f p ≤ P ^ (n + 1)\n⊢ p.ramificationIdx P ≤ n",
"usedConstants": [
"Preorder.toLT",
"Semiring.toModule",
"IsScalarTower.right",
"CommS... | have : ∀ k, map f p ≤ P ^ k → k ≤ n := by
refine fun k hk => le_of_not_gt fun hnk => ?_
exact hgt (hk.trans (Ideal.pow_le_pow_right hnk)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 285,
"column": 71
} | {
"line": 285,
"column": 80
} | [
{
"pp": "case neg\nR : Type u\ninst✝⁶ : CommRing R\nS : Type v\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsDedekindDomain S\ninst✝² : IsDedekindDomain R\ninst✝¹ : FaithfulSMul R S\nv : Ideal R\nw : Ideal S\np : Ideal R\nhv : Irreducible v\nhp : Prime p\nhw : Irreducible w\nhw_bot : w ≠ ⊥\ninst✝ : w.L... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 304,
"column": 86
} | {
"line": 304,
"column": 94
} | [
{
"pp": "case h₃\nR : Type u\ninst✝⁶ : CommRing R\nS : Type v\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsDedekindDomain S\ninst✝² : IsDedekindDomain R\ninst✝¹ : FaithfulSMul R S\nv : Ideal R\nw : Ideal S\nhv : Irreducible v\nhw : Irreducible w\nhw_bot : w ≠ ⊥\ninst✝ : w.LiesOver v\nI p : Ideal R\nhI... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 608,
"column": 4
} | {
"line": 609,
"column": 22
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nB : ℝ\nh : minkowskiBound K I ≤ volume (convexBodySum K B)\nhB : 0 ≤ B\nh1 : 0 < (↑(finrank ℚ K))⁻¹\nh2 : 0 ≤ B / ↑(finrank ℚ K)\nh_fund :\n IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalI... | simp_rw [← prod_eq_abs_norm, rpow_natCast]
exact le_of_eq rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 608,
"column": 4
} | {
"line": 609,
"column": 22
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nB : ℝ\nh : minkowskiBound K I ≤ volume (convexBodySum K B)\nhB : 0 ≤ B\nh1 : 0 < (↑(finrank ℚ K))⁻¹\nh2 : 0 ≤ B / ↑(finrank ℚ K)\nh_fund :\n IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalI... | simp_rw [← prod_eq_abs_norm, rpow_natCast]
exact le_of_eq rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.RamificationInertia.Galois | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 30
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\nG : Type u_3\ninst✝⁸ : Group G\ninst✝⁷ : Finite G\ninst✝⁶ : MulSemiringAction G B\ninst✝⁵ : IsGaloisGroup G A B\np : Ideal A\ninst✝⁴ : p.IsMaximal\ninst✝³ : IsDomain A\ninst✝² : IsTorsionFree A B\ninst✝¹ : Mod... | exact inertiaDeg_ne_zero _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.RamificationInertia.Galois | {
"line": 352,
"column": 2
} | {
"line": 353,
"column": 60
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nG : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Group G\ninst✝⁹ : MulSemiringAction G S\ninst✝⁸ : IsGaloisGroup G R S\ninst✝⁷ : Finite G\ninst✝⁶ : IsDedekindDomain R\ninst✝⁵ : IsDedekindDomain S\ninst✝⁴ : Module.Finite R S\ninst✝³ :... | rw [card_stabilizer_eq_card_inertia_mul_finrank p P, card_inertia_eq_ramificationIdxIn p hp,
inertiaDegIn_eq_inertiaDeg p P G, inertiaDeg_algebraMap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.ClassNumber | {
"line": 86,
"column": 14
} | {
"line": 86,
"column": 23
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nJ : ↥(Ideal (𝓞 K))⁰\nhJ : ClassGroup.mk0 J = C⁻¹\na : 𝓞 K\nha : a ∈ ↑↑J\nh_nm :\n ↑|(Algebra.norm ℚ) ((Algebra.linearMap (𝓞 K) K) a)| ≤\n ↑(FractionalIdeal.absNorm ↑((FractionalIdeal.mk0 K) J)) * (4 / π) ^ nrComplexPla... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 8
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nh_zero : (commMap K) x = 0\nh_mem : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))\n⊢ x = 0",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Pi.addCommMonoid",
"Complex.instNormedAddCom... | ext1 φ | Lean.Elab.Tactic.Ext._aux_Init_Ext___macroRules_Lean_Elab_Tactic_Ext_tacticExt1____1 | Lean.Elab.Tactic.Ext.tacticExt1___ |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 540,
"column": 56
} | {
"line": 540,
"column": 91
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nφ : K →+* ℂ\nhφ : ComplexEmbedding.IsReal φ\n⊢ (fun c ↦\n Sum.casesOn (motive := fun t ↦ c = t → K →+* ℂ) c (fun w h ↦ (↑w).embedding)\n (fun wj h ↦\n Prod.casesOn wj fun w j ↦ if j = 0 then (↑w).embedding else ComplexEmb... | simp [embedding_mk_eq_of_isReal hφ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 540,
"column": 56
} | {
"line": 540,
"column": 91
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nφ : K →+* ℂ\nhφ : ComplexEmbedding.IsReal φ\n⊢ (fun c ↦\n Sum.casesOn (motive := fun t ↦ c = t → K →+* ℂ) c (fun w h ↦ (↑w).embedding)\n (fun wj h ↦\n Prod.casesOn wj fun w j ↦ if j = 0 then (↑w).embedding else ComplexEmb... | simp [embedding_mk_eq_of_isReal hφ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 540,
"column": 56
} | {
"line": 540,
"column": 91
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nφ : K →+* ℂ\nhφ : ComplexEmbedding.IsReal φ\n⊢ (fun c ↦\n Sum.casesOn (motive := fun t ↦ c = t → K →+* ℂ) c (fun w h ↦ (↑w).embedding)\n (fun wj h ↦\n Prod.casesOn wj fun w j ↦ if j = 0 then (↑w).embedding else ComplexEmb... | simp [embedding_mk_eq_of_isReal hφ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 583,
"column": 10
} | {
"line": 583,
"column": 21
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ ∏ _k, 2⁻¹ * I = (2⁻¹ * I) ^ Fintype.card { w // w.IsComplex }",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Finset.univ",
"Complex.commRing",
"NumberField.InfinitePlace.NumberField.InfinitePlace.fintype",
"c... | prod_const, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 367,
"column": 4
} | {
"line": 367,
"column": 85
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI : FractionalIdeal R⁰ K\nhI : I ≠ 0\na : R\nJ : Ideal R\na₁ : R := choose ⋯\nJ₁ : Ideal R := choose ⋯\nh_aJ : ↑(J₁ * Ideal.span {a}... | rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 379,
"column": 4
} | {
"line": 379,
"column": 85
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R⁰ K\nhI : I ≠ 0\nhI' : I' ≠ 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a ≠ 0\nh... | rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 383,
"column": 4
} | {
"line": 383,
"column": 85
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R⁰ K\nhI : I ≠ 0\nhI' : I' ≠ 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a ≠ 0\nh... | rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Int | {
"line": 78,
"column": 4
} | {
"line": 84,
"column": 32
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Ring R\nI : Ideal R\nh : absNorm (under ℤ I) = 0\n⊢ absNorm (under ℤ I) = sInf {d | 0 < d ∧ ↑d ∈ I}",
"usedConstants": [
"Int.instAddCommGroup",
"Int.cast",
"Eq.mpr",
"Nat.instMulZeroOneClass",
"Int.cast_natCast",
"False",
"D... | have : {d : ℕ | 0 < d ∧ ↑d ∈ I} = ∅ := by
refine Set.eq_empty_of_forall_notMem ?_
intro x ⟨hx₁, hx₂⟩
rw [← cast_natCast, cast_mem_ideal_iff, h, natCast_dvd_natCast, Nat.zero_dvd] at hx₂
rw [Nat.pos_iff_ne_zero] at hx₁
exact hx₁ hx₂
rw [h, this, Nat.sInf_empty] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Int | {
"line": 78,
"column": 4
} | {
"line": 84,
"column": 32
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Ring R\nI : Ideal R\nh : absNorm (under ℤ I) = 0\n⊢ absNorm (under ℤ I) = sInf {d | 0 < d ∧ ↑d ∈ I}",
"usedConstants": [
"Int.instAddCommGroup",
"Int.cast",
"Eq.mpr",
"Nat.instMulZeroOneClass",
"Int.cast_natCast",
"False",
"D... | have : {d : ℕ | 0 < d ∧ ↑d ∈ I} = ∅ := by
refine Set.eq_empty_of_forall_notMem ?_
intro x ⟨hx₁, hx₂⟩
rw [← cast_natCast, cast_mem_ideal_iff, h, natCast_dvd_natCast, Nat.zero_dvd] at hx₂
rw [Nat.pos_iff_ne_zero] at hx₁
exact hx₁ hx₂
rw [h, this, Nat.sInf_empty] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Int | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 17
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Ring R\nI : Ideal R\nh : ¬absNorm (under ℤ I) = 0\nh₁ : absNorm (under ℤ I) ∈ {d | 0 < d ∧ ↑d ∈ I}\n⊢ absNorm (under ℤ I) ≤ sInf {d | 0 < d ∧ ↑d ∈ I}",
"usedConstants": [
"Int.instAddCommGroup",
"Nat.instMulZeroOneClass",
"Semiring.toModule",
... | by_contra! h₀ | Mathlib.Tactic.ByContra._aux_Mathlib_Tactic_ByContra___macroRules_Mathlib_Tactic_ByContra_byContra!_1 | Mathlib.Tactic.ByContra.byContra! |
Mathlib.RingTheory.Ideal.Int | {
"line": 100,
"column": 47
} | {
"line": 100,
"column": 62
} | [
{
"pp": "S : Type u_2\ninst✝² : CommRing S\ninst✝¹ : IsDedekindDomain S\ninst✝ : Module.Free ℤ S\nI : Ideal S\nh✝ : Finite (S ⧸ I)\nthis : Fintype (S ⧸ I)\nd : ℕ\nh : ∀ (x : S ⧸ I), (Ideal.Quotient.mk I) ↑d * x = 0\n⊢ (Ideal.Quotient.mk I) ↑d = 0",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.... | simpa using h 1 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Ideal.Int | {
"line": 100,
"column": 47
} | {
"line": 100,
"column": 62
} | [
{
"pp": "S : Type u_2\ninst✝² : CommRing S\ninst✝¹ : IsDedekindDomain S\ninst✝ : Module.Free ℤ S\nI : Ideal S\nh✝ : Finite (S ⧸ I)\nthis : Fintype (S ⧸ I)\nd : ℕ\nh : ∀ (x : S ⧸ I), (Ideal.Quotient.mk I) ↑d * x = 0\n⊢ (Ideal.Quotient.mk I) ↑d = 0",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.... | simpa using h 1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Int | {
"line": 100,
"column": 47
} | {
"line": 100,
"column": 62
} | [
{
"pp": "S : Type u_2\ninst✝² : CommRing S\ninst✝¹ : IsDedekindDomain S\ninst✝ : Module.Free ℤ S\nI : Ideal S\nh✝ : Finite (S ⧸ I)\nthis : Fintype (S ⧸ I)\nd : ℕ\nh : ∀ (x : S ⧸ I), (Ideal.Quotient.mk I) ↑d * x = 0\n⊢ (Ideal.Quotient.mk I) ↑d = 0",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.... | simpa using h 1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 228,
"column": 69
} | {
"line": 228,
"column": 89
} | [
{
"pp": "R : Type u_1\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : IsDomain R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : IsIntegrallyClosed R\ninst✝¹³ : IsIntegrallyClosed S\ninst✝¹² : Algebra R S\ninst✝¹¹ : Module.Finite R S\ninst✝¹⁰ : IsTorsionFree R S\nT : Type u_4\ninst✝⁹ : CommRing T\ninst✝⁸ ... | rw [intNorm_intNorm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 228,
"column": 69
} | {
"line": 228,
"column": 89
} | [
{
"pp": "R : Type u_1\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : IsDomain R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : IsIntegrallyClosed R\ninst✝¹³ : IsIntegrallyClosed S\ninst✝¹² : Algebra R S\ninst✝¹¹ : Module.Finite R S\ninst✝¹⁰ : IsTorsionFree R S\nT : Type u_4\ninst✝⁹ : CommRing T\ninst✝⁸ ... | rw [intNorm_intNorm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 228,
"column": 69
} | {
"line": 228,
"column": 89
} | [
{
"pp": "R : Type u_1\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : IsDomain R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : IsIntegrallyClosed R\ninst✝¹³ : IsIntegrallyClosed S\ninst✝¹² : Algebra R S\ninst✝¹¹ : Module.Finite R S\ninst✝¹⁰ : IsTorsionFree R S\nT : Type u_4\ninst✝⁹ : CommRing T\ninst✝⁸ ... | rw [intNorm_intNorm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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