module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.ZMod.Basic | {
"line": 939,
"column": 63
} | {
"line": 939,
"column": 69
} | {
"line": 939,
"column": 69
} | [
{
"pp": "m n : ℕ\n⊢ ∀ (a b : (ZMod 2)ˣ), a = b",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"ZMod.commRing",
"CommSemiring.toSemiring",
"ZMod.fintype",
"ZMod.decidableEq",
"Units",
"instFintypeUnitsOfDecidableEq",
"id",... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ZMod.Basic | {
"line": 939,
"column": 63
} | {
"line": 939,
"column": 69
} | {
"line": 939,
"column": 69
} | [
{
"pp": "m n : ℕ\n⊢ ∀ (a b : (ZMod 2)ˣ), a = b",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"ZMod.commRing",
"CommSemiring.toSemiring",
"ZMod.fintype",
"ZMod.decidableEq",
"Units",
"instFintypeUnitsOfDecidableEq",
"id",... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ZMod.Basic | {
"line": 987,
"column": 4
} | {
"line": 989,
"column": 39
} | {
"line": 990,
"column": 4
} | [
{
"pp": "case succ.mp.zero\nn✝ : ℕ\na : ZMod (n✝ + 1)\nhe : 2 * a.val = (n✝ + 1) * 0\n⊢ a = 0 ∨ 2 * a.val = n✝ + 1",
"ppTerm": "?succ.mp.zero",
"assigned": true,
"usedConstants": [
"Nat.instCanonicallyOrderedAdd",
"Nat.instMulZeroClass",
"HMul.hMul",
"ZMod.commRing",
"M... | [
"case succ.mp.succ\nn✝ : ℕ\na : ZMod (n✝ + 1)\nm : ℕ\nhe : 2 * a.val = (n✝ + 1) * (m + 1)\n⊢ a = 0 ∨ 2 * a.val = n✝ + 1"
] | · rw [mul_zero, mul_eq_zero] at he
rcases he with (⟨⟨⟩⟩ | he)
exact Or.inl (a.val_eq_zero.1 he) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Nat.Digits.Defs | {
"line": 117,
"column": 25
} | {
"line": 117,
"column": 31
} | {
"line": 117,
"column": 31
} | [
{
"pp": "n✝ : ℕ\nh : 1 < 0\n⊢ ¬1 < 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Digits.Defs | {
"line": 117,
"column": 25
} | {
"line": 117,
"column": 31
} | {
"line": 117,
"column": 31
} | [
{
"pp": "n✝ : ℕ\nh : 1 < 0\n⊢ ¬1 < 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Digits.Defs | {
"line": 117,
"column": 25
} | {
"line": 117,
"column": 31
} | {
"line": 117,
"column": 31
} | [
{
"pp": "n✝ : ℕ\nh : 1 < 0\n⊢ ¬1 < 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Digits.Defs | {
"line": 118,
"column": 25
} | {
"line": 118,
"column": 31
} | {
"line": 118,
"column": 31
} | [
{
"pp": "n✝ : ℕ\nh : 1 < 1\n⊢ ¬1 < 1",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Digits.Defs | {
"line": 118,
"column": 25
} | {
"line": 118,
"column": 31
} | {
"line": 118,
"column": 31
} | [
{
"pp": "n✝ : ℕ\nh : 1 < 1\n⊢ ¬1 < 1",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Digits.Defs | {
"line": 118,
"column": 25
} | {
"line": 118,
"column": 31
} | {
"line": 118,
"column": 31
} | [
{
"pp": "n✝ : ℕ\nh : 1 < 1\n⊢ ¬1 < 1",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ZMod.Basic | {
"line": 1087,
"column": 2
} | {
"line": 1087,
"column": 63
} | {
"line": 1088,
"column": 2
} | [
{
"pp": "case inr\nn : ℕ\nx : ℤ\nhl : x.natAbs ≤ n / 2\nm : ℤ\nhm : m ≠ 0\n⊢ n / 2 + n / 2 ≤ (↑n * m + x - x).natAbs",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"Int.natAbs_mul",
"congrArg",
"AddMonoid.toAddZeroClass... | [
"case inr\nn : ℕ\nx : ℤ\nhl : x.natAbs ≤ n / 2\nm : ℤ\nhm : m ≠ 0\n⊢ n / 2 + n / 2 ≤ n * m.natAbs"
] | rw [add_sub_cancel_right, Int.natAbs_mul, Int.natAbs_natCast] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Nat.Digits.Lemmas | {
"line": 31,
"column": 2
} | {
"line": 34,
"column": 18
} | {
"line": 35,
"column": 2
} | [
{
"pp": "b : ℕ\nl : List ℕ\n⊢ (List.zipWith (fun a i ↦ a * b ^ (i + 1)) l (List.range l.length)).sum =\n b * (List.zipWith (fun a i ↦ a * b ^ i) l (List.range l.length)).sum",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"List.zipWith",
"Nat.instMulZeroClass",
"HMul.h... | [
"b : ℕ\nl : List ℕ\n⊢ List.zipWith (fun a i ↦ a * b ^ (i + 1)) l (List.range l.length) =\n List.zipWith (fun a i ↦ b * (a * b ^ i)) l (List.range l.length)"
] | suffices
l.zipWith (fun a i : ℕ => a * b ^ (i + 1)) (List.range l.length) =
l.zipWith (fun a i => b * (a * b ^ i)) (List.range l.length)
by simp [this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Data.Nat.Factorization.Defs | {
"line": 133,
"column": 29
} | {
"line": 133,
"column": 68
} | {
"line": 135,
"column": 0
} | [
{
"pp": "n p : ℕ\nhp : ¬Prime p\n⊢ n.factorization p = 0",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"_private.Mathlib.Data.Nat.Factorization.Defs.0.Nat.factorization_eq_zero_of_not_prime._simp_1_1",
"False",
"Nat.instMulZeroClass",
"... | [] | by simp [factorization_eq_zero_iff, hp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Factorization.Defs | {
"line": 149,
"column": 7
} | {
"line": 149,
"column": 48
} | {
"line": 149,
"column": 48
} | [
{
"pp": "p r i : ℕ\nhr : ¬p ∣ r\n⊢ ¬p ∣ p * i + r",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"Dvd.intro",
"id",
"instMulNat",
"Nat.instSemigroup",
"instHAdd",
... | [
"p r i : ℕ\nhr : ¬p ∣ r\n⊢ ¬p ∣ r"
] | ← Nat.dvd_add_iff_right (Dvd.intro i rfl) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Factorization.Defs | {
"line": 164,
"column": 45
} | {
"line": 164,
"column": 80
} | {
"line": 164,
"column": 80
} | [
{
"pp": "case refine_1\nd n : ℕ\nhd : d ≠ 0\nhn : n ≠ 0\nhdn : d.factorization ≤ n.factorization\n⊢ d ∣ n.factorization.prod fun x1 x2 ↦ x1 ^ x2",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Dvd.dvd",
"congrArg",
"Nat.i... | [
"case refine_1\nd n : ℕ\nhd : d ≠ 0\nhn : n ≠ 0\nhdn : d.factorization ≤ n.factorization\n⊢ (d.factorization.prod fun x1 x2 ↦ x1 ^ x2) ∣ n.factorization.prod fun x1 x2 ↦ x1 ^ x2"
] | ← prod_factorization_pow_eq_self hd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Multiplicity | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 8
} | {
"line": 109,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Monoid α\na b : α\nh : ¬FiniteMultiplicity a b\n⊢ WithTop.untopD 1 ⊤ = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"instTopENat",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"ENat",
"Boo... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Factorization.Basic | {
"line": 250,
"column": 36
} | {
"line": 250,
"column": 77
} | {
"line": 250,
"column": 77
} | [
{
"pp": "k m p : ℕ\nhp : Prime p\n⊢ m / p ^ m.factorization p = m ↔ m = 0 ∨ ¬p ∣ m",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",
"Dvd.dvd",
"instHDiv",
"congrArg",
"Nat.instMonoid",
"... | [
"k m p : ℕ\nhp : Prime p\n⊢ m = 0 ∨ ¬p ∣ m ↔ m = 0 ∨ ¬p ∣ m"
] | ordCompl_eq_self_iff_zero_or_not_dvd m hp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Multiplicity | {
"line": 509,
"column": 2
} | {
"line": 511,
"column": 20
} | {
"line": 513,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Semiring α\np a b : α\nhf : FiniteMultiplicity p (a + b)\n⊢ FiniteMultiplicity p a ∨ FiniteMultiplicity p b",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Distrib.leftDistribClass",
"False",
... | [] | by_contra! nh
obtain ⟨c, hc⟩ := hf
simp_all [dvd_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Multiplicity | {
"line": 509,
"column": 2
} | {
"line": 511,
"column": 20
} | {
"line": 513,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Semiring α\np a b : α\nhf : FiniteMultiplicity p (a + b)\n⊢ FiniteMultiplicity p a ∨ FiniteMultiplicity p b",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Distrib.leftDistribClass",
"False",
... | [] | by_contra! nh
obtain ⟨c, hc⟩ := hf
simp_all [dvd_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Factorization.Basic | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 45
} | {
"line": 346,
"column": 2
} | [
{
"pp": "case inr.inr\nn d : ℕ\nhn : n ≠ 0\nhd : d ≠ 0\n⊢ (∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ n) → d ∣ n",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",
"Nat.Prime",
"Dvd.dvd",
"congrArg... | [
"case inr.inr\nn d : ℕ\nhn : n ≠ 0\nhd : d ≠ 0\n⊢ (∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ n) → ∀ (p : ℕ), Prime p → d.factorization p ≤ n.factorization p"
] | rw [← factorization_prime_le_iff_dvd hd hn] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Nat.Digits.Lemmas | {
"line": 439,
"column": 37
} | {
"line": 442,
"column": 7
} | {
"line": 444,
"column": 0
} | [
{
"pp": "b : ℕ\nhb : 1 < b\n⊢ fixedLengthDigits hb 0 = {[]}",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Nat.instMonoid",
"Nat.ofDigits",
"setOf",
"Set.fintypeLTNat",
"Finset.ext",
"Membership.mem",
... | [] | by
ext
simp [fixedLengthDigits]
grind | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Factorization.Basic | {
"line": 470,
"column": 2
} | {
"line": 470,
"column": 39
} | {
"line": 471,
"column": 2
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\nm : ℕ\npr : n < m\n⊢ (n.factorization.prod fun x1 x2 ↦ x1 ^ x2) = ∏ p ∈ Finset.range m with Prime p, p ^ padicValNat p n",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZeroClass",
"Nat.Prime",
"Nat.inst... | [
"case h\nn : ℕ\nhn : n ≠ 0\nm : ℕ\npr : n < m\n⊢ n.factorization.support ⊆ {p ∈ Finset.range m | Prime p}",
"case hg\nn : ℕ\nhn : n ≠ 0\nm : ℕ\npr : n < m\n⊢ ∀ x ∈ {p ∈ Finset.range m | Prime p} \\ n.factorization.support, x ^ padicValNat x n = 1",
"case hfg\nn : ℕ\nhn : n ≠ 0\nm : ℕ\npr : n < m\n⊢ ∀ x ∈ n.fact... | apply Finset.prod_subset_one_on_sdiff | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Multiplicity | {
"line": 752,
"column": 30
} | {
"line": 752,
"column": 41
} | {
"line": 752,
"column": 41
} | [
{
"pp": "p a b : ℕ\nhp : p ≠ 1\nhle : multiplicity p a ≤ multiplicity p b\nhab : a.Coprime b\nnh : 0 < multiplicity p a\nda : p ∣ a\ndb : p ∣ b\nthis : p ∣ 1\n⊢ False",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"congrArg",
"Nat.dvd_one",
"Eq.mp",
... | [
"p a b : ℕ\nhp : p ≠ 1\nhle : multiplicity p a ≤ multiplicity p b\nhab : a.Coprime b\nnh : 0 < multiplicity p a\nda : p ∣ a\ndb : p ∣ b\nthis : p = 1\n⊢ False"
] | Nat.dvd_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Multiplicity | {
"line": 747,
"column": 2
} | {
"line": 753,
"column": 15
} | {
"line": 755,
"column": 0
} | [
{
"pp": "p a b : ℕ\nhp : p ≠ 1\nhle : multiplicity p a ≤ multiplicity p b\nhab : a.Coprime b\n⊢ multiplicity p a = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Dvd.dvd",
"_private.Mathlib.RingTheory.Multiplicity.0.multiplicity_eq_zero_of_coprime._simp_1_1... | [] | apply Nat.eq_zero_of_not_pos
intro nh
have da : p ∣ a := by simpa [multiplicity_eq_zero] using nh.ne.symm
have db : p ∣ b := by simpa [multiplicity_eq_zero] using (nh.trans_le hle).ne.symm
have := Nat.dvd_gcd da db
rw [Coprime.gcd_eq_one hab, Nat.dvd_one] at this
exact hp this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Multiplicity | {
"line": 747,
"column": 2
} | {
"line": 753,
"column": 15
} | {
"line": 755,
"column": 0
} | [
{
"pp": "p a b : ℕ\nhp : p ≠ 1\nhle : multiplicity p a ≤ multiplicity p b\nhab : a.Coprime b\n⊢ multiplicity p a = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Dvd.dvd",
"_private.Mathlib.RingTheory.Multiplicity.0.multiplicity_eq_zero_of_coprime._simp_1_1... | [] | apply Nat.eq_zero_of_not_pos
intro nh
have da : p ∣ a := by simpa [multiplicity_eq_zero] using nh.ne.symm
have db : p ∣ b := by simpa [multiplicity_eq_zero] using (nh.trans_le hle).ne.symm
have := Nat.dvd_gcd da db
rw [Coprime.gcd_eq_one hab, Nat.dvd_one] at this
exact hp this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Multiplicity | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 14
} | {
"line": 144,
"column": 4
} | [
{
"pp": "n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\n⊢ ∀ m ∈ Ico (p * n + 1) (p * (n + 1)), emultiplicity p m = 0",
"ppTerm": "?m.198",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Fins... | [
"n p : ℕ\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 ≤ p\nh1 : 1 ≤ p * n + 1\nh2 : p * n + 1 ≤ p * (n + 1)\nh3 : p * n + 1 ≤ p * (n + 1) + 1\nm : ℕ\nhm : m ∈ Ico (p * n + 1) (p * (n + 1))\n⊢ emultiplicity p m = 0"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.GroupTheory.OrderOfElement | {
"line": 648,
"column": 13
} | {
"line": 648,
"column": 37
} | {
"line": 648,
"column": 37
} | [
{
"pp": "G : Type u_1\ninst✝ : LeftCancelMonoid G\na : G\nh : (↑(powers a)).Finite\nn : ℕ\n⊢ a ^ n ∈ ↑(powers a)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"Monoid.toMulOneClass",
"_private.Mathlib.GroupTheory.OrderOfElement.0.finite_powers... | [] | by simp [mem_powers_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.OrderOfElement | {
"line": 730,
"column": 13
} | {
"line": 730,
"column": 37
} | {
"line": 730,
"column": 37
} | [
{
"pp": "G : Type u_1\ninst✝ : RightCancelMonoid G\na : G\nh : (↑(powers a)).Finite\nn : ℕ\n⊢ a ^ n ∈ ↑(powers a)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"Monoid.toMulOneClass",
"_private.Mathlib.GroupTheory.OrderOfElement.0.RightCancelM... | [] | by simp [mem_powers_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.OrderOfElement | {
"line": 769,
"column": 2
} | {
"line": 769,
"column": 48
} | {
"line": 770,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nx : G\ni : ℤ\n⊢ ↑(orderOf x) ∣ i ↔ x ^ i = 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"DivInvMonoid.toZPow",
"Group.toDivisionMonoid",
"Divisi... | [
"case inl\nG : Type u_1\ninst✝ : Group G\nx : G\ni : ℕ\n⊢ ↑(orderOf x) ∣ ↑i ↔ x ^ ↑i = 1",
"case inr\nG : Type u_1\ninst✝ : Group G\nx : G\ni : ℕ\n⊢ ↑(orderOf x) ∣ -↑i ↔ x ^ (-↑i) = 1"
] | rcases Int.eq_nat_or_neg i with ⟨i, rfl | rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Data.Nat.Multiplicity | {
"line": 283,
"column": 8
} | {
"line": 283,
"column": 14
} | {
"line": 284,
"column": 4
} | [
{
"pp": "case pos\nh2 : _root_.Prime 2\nb : Bool\nih : 0 ≠ 0 → emultiplicity 2 0! < ↑0\nh : b = true\n⊢ Prime 2",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Multiplicity | {
"line": 283,
"column": 8
} | {
"line": 283,
"column": 14
} | {
"line": 284,
"column": 4
} | [
{
"pp": "case pos\nh2 : _root_.Prime 2\nb : Bool\nih : 0 ≠ 0 → emultiplicity 2 0! < ↑0\nh : b = true\n⊢ Prime 2",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Multiplicity | {
"line": 283,
"column": 8
} | {
"line": 283,
"column": 14
} | {
"line": 284,
"column": 4
} | [
{
"pp": "case pos\nh2 : _root_.Prime 2\nb : Bool\nih : 0 ≠ 0 → emultiplicity 2 0! < ↑0\nh : b = true\n⊢ Prime 2",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Inductions | {
"line": 114,
"column": 8
} | {
"line": 114,
"column": 78
} | {
"line": 115,
"column": 8
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nhp0 : p ≠ 0\nthis : Nontrivial R\nh : p.degree ≤ 0\nh' : C (p.coeff 0) ≠ 0\n⊢ p.divX.degree < (p.divX * X + C (p.coeff 0)).degree",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Polynomi... | [
"R : Type u\ninst✝ : Semiring R\np : R[X]\nhp0 : p ≠ 0\nthis : Nontrivial R\nh : p.degree ≤ 0\nh' : C (p.coeff 0) ≠ 0\n⊢ ⊥ < (C ((C (p.coeff 0)).coeff 0)).degree"
] | rw [eq_C_of_degree_le_zero h, divX_C, degree_zero, zero_mul, zero_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Inductions | {
"line": 123,
"column": 24
} | {
"line": 123,
"column": 30
} | {
"line": 124,
"column": 12
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nhp0 : p ≠ 0\nthis✝ : Nontrivial R\nh : ¬p.degree ≤ 0\nhXp0 : p.divX ≠ 0\nthis : p.divX.leadingCoeff * X.leadingCoeff ≠ 0\n⊢ 0 < 1",
"ppTerm": "?m.230",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Nat.instMulZeroClass",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Algebra.Polynomial.Inductions | {
"line": 123,
"column": 24
} | {
"line": 123,
"column": 30
} | {
"line": 124,
"column": 12
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nhp0 : p ≠ 0\nthis✝ : Nontrivial R\nh : ¬p.degree ≤ 0\nhXp0 : p.divX ≠ 0\nthis : p.divX.leadingCoeff * X.leadingCoeff ≠ 0\n⊢ 0 < 1",
"ppTerm": "?m.230",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Nat.instMulZeroClass",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Inductions | {
"line": 123,
"column": 24
} | {
"line": 123,
"column": 30
} | {
"line": 124,
"column": 12
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nhp0 : p ≠ 0\nthis✝ : Nontrivial R\nh : ¬p.degree ≤ 0\nhXp0 : p.divX ≠ 0\nthis : p.divX.leadingCoeff * X.leadingCoeff ≠ 0\n⊢ 0 < 1",
"ppTerm": "?m.230",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Nat.instMulZeroClass",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Inductions | {
"line": 169,
"column": 71
} | {
"line": 169,
"column": 77
} | {
"line": 169,
"column": 77
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nP : R[X] → Prop\np : R[X]\nh0 : 0 < p.degree\nhC : ∀ {a : R}, a ≠ 0 → P (C a * X)\nhX : ∀ {p : R[X]}, 0 < p.degree → P p → P (p * X)\nhadd : ∀ {p : R[X]} {a : R}, 0 < p.degree → P p → P (p + C a)\nh : 0 < ⊥\n⊢ ¬0 < ⊥",
"ppTerm": "?m.145",
"assigned": true,
"u... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Algebra.Polynomial.Inductions | {
"line": 169,
"column": 71
} | {
"line": 169,
"column": 77
} | {
"line": 169,
"column": 77
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nP : R[X] → Prop\np : R[X]\nh0 : 0 < p.degree\nhC : ∀ {a : R}, a ≠ 0 → P (C a * X)\nhX : ∀ {p : R[X]}, 0 < p.degree → P p → P (p * X)\nhadd : ∀ {p : R[X]} {a : R}, 0 < p.degree → P p → P (p + C a)\nh : 0 < ⊥\n⊢ ¬0 < ⊥",
"ppTerm": "?m.145",
"assigned": true,
"u... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Inductions | {
"line": 169,
"column": 71
} | {
"line": 169,
"column": 77
} | {
"line": 169,
"column": 77
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nP : R[X] → Prop\np : R[X]\nh0 : 0 < p.degree\nhC : ∀ {a : R}, a ≠ 0 → P (C a * X)\nhX : ∀ {p : R[X]}, 0 < p.degree → P p → P (p * X)\nhadd : ∀ {p : R[X]} {a : R}, 0 < p.degree → P p → P (p + C a)\nh : 0 < ⊥\n⊢ ¬0 < ⊥",
"ppTerm": "?m.145",
"assigned": true,
"u... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.RingDivision | {
"line": 200,
"column": 11
} | {
"line": 200,
"column": 48
} | {
"line": 200,
"column": 48
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np q : R[X]\nx : R\nhpq : eval x (p /ₘ (X - C x) ^ rootMultiplicity x p) * eval x (q /ₘ (X - C x) ^ rootMultiplicity x q) ≠ 0\n⊢ rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
... | [
"R : Type u\ninst✝ : CommRing R\np q : R[X]\nx : R\nhpq : (p.comp (X + C x)).trailingCoeff * (q.comp (X + C x)).trailingCoeff ≠ 0\n⊢ rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q"
] | eval_divByMonic_eq_trailingCoeff_comp | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.RingDivision | {
"line": 312,
"column": 66
} | {
"line": 312,
"column": 72
} | {
"line": 312,
"column": 72
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\np : R[X]\nhp : p ≠ 0\nthis : Decidable (∃ x, p.IsRoot x)\nh : ∃ x, p.IsRoot x\nx : R\nhx : p.IsRoot x\nhpd : 0 < p.degree\nhd0 : p /ₘ (X - C x) ≠ 0\n⊢ 0 < 1",
"ppTerm": "?m.99",
"assigned": true,
"usedConstants": [... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Algebra.Polynomial.RingDivision | {
"line": 312,
"column": 66
} | {
"line": 312,
"column": 72
} | {
"line": 312,
"column": 72
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\np : R[X]\nhp : p ≠ 0\nthis : Decidable (∃ x, p.IsRoot x)\nh : ∃ x, p.IsRoot x\nx : R\nhx : p.IsRoot x\nhpd : 0 < p.degree\nhd0 : p /ₘ (X - C x) ≠ 0\n⊢ 0 < 1",
"ppTerm": "?m.99",
"assigned": true,
"usedConstants": [... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.RingDivision | {
"line": 312,
"column": 66
} | {
"line": 312,
"column": 72
} | {
"line": 312,
"column": 72
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\np : R[X]\nhp : p ≠ 0\nthis : Decidable (∃ x, p.IsRoot x)\nh : ∃ x, p.IsRoot x\nx : R\nhx : p.IsRoot x\nhpd : 0 < p.degree\nhd0 : p /ₘ (X - C x) ≠ 0\n⊢ 0 < 1",
"ppTerm": "?m.99",
"assigned": true,
"usedConstants": [... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Expand | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 17
} | {
"line": 88,
"column": 18
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\np : ℕ\nf : R[X]\n⊢ derivative ((expand R p) f) = (expand R p) (derivative f) * (↑p * X ^ (p - 1))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Polynomial.C",
"Semiring.toModule",
... | [
"R : Type u\ninst✝ : CommSemiring R\np : ℕ\nf : R[X]\n⊢ derivative (eval₂ C (X ^ p) f) = eval₂ C (X ^ p) (derivative f) * (↑p * X ^ (p - 1))"
] | coe_expand, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.RingDivision | {
"line": 330,
"column": 12
} | {
"line": 330,
"column": 37
} | {
"line": 330,
"column": 38
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\np : R[X]\nhp : p ≠ 0\nthis : Decidable (∃ x, p.IsRoot x)\nh : ∃ x, p.IsRoot x\nx : R\nhx : p.IsRoot x\nhpd : 0 < p.degree\nhd0 : p /ₘ (X - C x) ≠ 0\nwf : (p /ₘ (X - C x)).degree < p.degree\nt : Multiset R\nhtd : ↑t.card ≤ (p /... | [
"R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\np : R[X]\nhp : p ≠ 0\nthis : Decidable (∃ x, p.IsRoot x)\nh : ∃ x, p.IsRoot x\nx : R\nhx : p.IsRoot x\nhpd : 0 < p.degree\nhd0 : p /ₘ (X - C x) ≠ 0\nwf : (p /ₘ (X - C x)).degree < p.degree\nt : Multiset R\nhtd : ↑t.card ≤ (p /ₘ (X - C x))... | rootMultiplicity_X_sub_C, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Expand | {
"line": 136,
"column": 12
} | {
"line": 136,
"column": 23
} | {
"line": 136,
"column": 24
} | [
{
"pp": "case inl\nR : Type u\ninst✝ : CommSemiring R\np : ℕ\nf : R[X]\nhp : p = 0\n⊢ ((expand R 0) f).natDegree = f.natDegree * 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
... | [
"case inl\nR : Type u\ninst✝ : CommSemiring R\np : ℕ\nf : R[X]\nhp : p = 0\n⊢ (eval₂ C (X ^ 0) f).natDegree = f.natDegree * 0"
] | coe_expand, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 112,
"column": 41
} | {
"line": 112,
"column": 66
} | {
"line": 112,
"column": 66
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np q : R[X]\nh : q.degree ≤ p.degree ∧ p ≠ 0\nhq : q.Monic\nhp : p.leadingCoeff ≠ 0\nhq0 : q ≠ 0\n⊢ ↑q.natDegree ≤ p.degree",
"ppTerm": "?m.114",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"Eq.mpr",
"WithBot",
"Polynom... | [
"R : Type u\ninst✝ : Ring R\np q : R[X]\nh : q.degree ≤ p.degree ∧ p ≠ 0\nhq : q.Monic\nhp : p.leadingCoeff ≠ 0\nhq0 : q ≠ 0\n⊢ q.degree ≤ p.degree"
] | ← degree_eq_natDegree hq0 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 117,
"column": 67
} | {
"line": 117,
"column": 81
} | {
"line": 117,
"column": 81
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np q : R[X]\nh : q.degree ≤ p.degree ∧ p ≠ 0\nhq : q.Monic\nhp : p.leadingCoeff ≠ 0\nhq0 : q ≠ 0\nhlt : q.natDegree ≤ p.natDegree\n⊢ p.leadingCoeff = (C p.leadingCoeff).leadingCoeff",
"ppTerm": "?m.170",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"R : Type u\ninst✝ : Ring R\np q : R[X]\nh : q.degree ≤ p.degree ∧ p ≠ 0\nhq : q.Monic\nhp : p.leadingCoeff ≠ 0\nhq0 : q ≠ 0\nhlt : q.natDegree ≤ p.natDegree\n⊢ p.leadingCoeff = p.leadingCoeff"
] | leadingCoeff_C | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 159,
"column": 10
} | {
"line": 159,
"column": 44
} | {
"line": 160,
"column": 10
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : ¬(q.degree ≤ p.degree ∧ p ≠ 0)\n⊢ ¬q.degree ≤ p.degree → (p %ₘ q).degree < q.degree",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"WithBot.instPreord... | [
"R : Type u\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : ¬(q.degree ≤ p.degree ∧ p ≠ 0)\n⊢ ¬q.degree ≤ p.degree →\n (if h : q.Monic then\n (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then\n have z := C p.leadingCoeff * X ^ (p.nat... | unfold modByMonic divModByMonicAux | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.Algebra.Polynomial.Div | {
"line": 165,
"column": 10
} | {
"line": 165,
"column": 44
} | {
"line": 166,
"column": 10
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : ¬(q.degree ≤ p.degree ∧ p ≠ 0)\nhp : ¬p ≠ 0\n⊢ (p %ₘ q).degree < q.degree",
"ppTerm": "?m.129",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
... | [
"R : Type u\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : ¬(q.degree ≤ p.degree ∧ p ≠ 0)\nhp : ¬p ≠ 0\n⊢ (if h : q.Monic then\n (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then\n have z := C p.leadingCoeff * X ^ (p.natDegree - q.natDegre... | unfold modByMonic divModByMonicAux | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.RingTheory.Algebraic.Defs | {
"line": 58,
"column": 6
} | {
"line": 58,
"column": 21
} | {
"line": 58,
"column": 22
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nx : A\n⊢ Transcendental R x ↔ ∀ (p : R[X]), (aeval x) p = 0 → p = 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsAlgebraic",
"congrArg",
"CommSemiring.toSemir... | [
"R : Type u\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nx : A\n⊢ ¬IsAlgebraic R x ↔ ∀ (p : R[X]), (aeval x) p = 0 → p = 0"
] | Transcendental, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 194,
"column": 10
} | {
"line": 194,
"column": 44
} | {
"line": 194,
"column": 44
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np : R[X]\nthis : DecidableEq R := Classical.decEq R\nh : ¬Monic 0\n⊢ p %ₘ 0 = p",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
"instDecidableNot",
"WithBot",
"dite_co... | [
"R : Type u\ninst✝ : Ring R\np : R[X]\nthis : DecidableEq R := Classical.decEq R\nh : ¬Monic 0\n⊢ (if h : Monic 0 then\n (if h_1 : degree 0 ≤ p.degree ∧ p ≠ 0 then\n have z := C p.leadingCoeff * X ^ (p.natDegree - natDegree 0);\n have _wf := ⋯;\n have dm := (p - 0 * z).divModByMonicA... | unfold modByMonic divModByMonicAux | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.Algebra.Polynomial.Div | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 38
} | {
"line": 214,
"column": 38
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\np q : R[X]\ninst✝ : Nontrivial R\nhq : q.Monic\nh : p.degree < q.degree\nthis : ¬q.degree ≤ p.degree\n⊢ p %ₘ q = p",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
"instDecidableN... | [
"R : Type u\ninst✝¹ : Ring R\np q : R[X]\ninst✝ : Nontrivial R\nhq : q.Monic\nh : p.degree < q.degree\nthis : ¬q.degree ≤ p.degree\n⊢ (if h : q.Monic then\n (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then\n have z := C p.leadingCoeff * X ^ (p.natDegree - q.natDegree);\n have _wf := ⋯;\n h... | unfold modByMonic divModByMonicAux | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.RingTheory.Polynomial.Tower | {
"line": 92,
"column": 32
} | {
"line": 92,
"column": 48
} | {
"line": 92,
"column": 49
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : Multiset A\nx : A\nhx : x ∈ s\np : R[X]\nhp : (mapAlg R A) p = (Multiset.map (fun x ↦ X - C x) s).prod\n⊢ (aeval x) (map (algebraMap R A) p) = 0",
"ppTerm": "?m.55",
"assigned": true,
"usedCon... | [
"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : Multiset A\nx : A\nhx : x ∈ s\np : R[X]\nhp : (mapAlg R A) p = (Multiset.map (fun x ↦ X - C x) s).prod\n⊢ (aeval x) ((mapAlg R A) p) = 0"
] | ← mapAlg_eq_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Expand | {
"line": 277,
"column": 4
} | {
"line": 277,
"column": 8
} | {
"line": 278,
"column": 4
} | [
{
"pp": "case succ\nR : Type u\ninst✝¹ : CommSemiring R\np : ℕ\ninst✝ : ExpChar R p\nf : R[X]\nk : ℕ\nn_ih : map (iterateFrobenius R p k) ((expand R (p ^ k)) f) = f ^ p ^ k\n⊢ map (iterateFrobenius R p (k + 1)) ((expand R (p ^ (k + 1))) f) = f ^ p ^ (k + 1)",
"ppTerm": "?succ",
"assigned": true,
"us... | [
"case succ\nR : Type u\ninst✝¹ : CommSemiring R\np : ℕ\ninst✝ : ExpChar R p\nf : R[X]\nk : ℕ\nn_ih : map (iterateFrobenius R p k) ((expand R (p ^ k)) f) = f ^ p ^ k\n⊢ f ^ p ^ (k + 1) = map (iterateFrobenius R p (k + 1)) ((expand R (p ^ (k + 1))) f)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Polynomial.Derivative | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 64
} | {
"line": 295,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nk : ℕ\n⊢ (⇑derivative)^[k] p = ∑ x ∈ ((⇑derivative)^[k] p).support, C ((x + k).descFactorial k • p.coeff (x + k)) * X ^ x",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Polynomial.C",
... | [
"R : Type u\ninst✝ : Semiring R\np : R[X]\nk : ℕ\n⊢ ∑ i ∈ ((⇑derivative)^[k] p).support, C (((⇑derivative)^[k] p).coeff i) * X ^ i =\n ∑ x ∈ ((⇑derivative)^[k] p).support, C ((x + k).descFactorial k • p.coeff (x + k)) * X ^ x"
] | conv_lhs => rw [(derivative^[k] p).as_sum_support_C_mul_X_pow] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.Order.Filter.Finite | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 33
} | {
"line": 102,
"column": 4
} | [
{
"pp": "case mp\nα : Type u\nι : Type u_2\ns : ι → Filter α\nU : Set α\nI : Set ι\nIfin : I.Finite\nσ : ↑{i | i ∈ I} → Set (Set α)\nσfin : ∀ (i : ↑{i | i ∈ I}), (σ i).Finite\nσsub : ∀ (i : ↑{i | i ∈ I}), σ i ⊆ (s ↑i).sets\ntsub : ⋃ i, σ i ⊆ ⋃ i, (s i).sets\ntfin : (⋃ i, σ i).Finite\ntinter : ⋂ i, ⋂₀ σ i ⊆ U\nV... | [
"case mp\nα : Type u\nι : Type u_2\ns : ι → Filter α\nU : Set α\nI : Set ι\nIfin : I.Finite\nσ : ↑{i | i ∈ I} → Set (Set α)\nσfin : ∀ (i : ↑{i | i ∈ I}), (σ i).Finite\nσsub : ∀ (i : ↑{i | i ∈ I}), σ i ⊆ (s ↑i).sets\ntsub : ⋃ i, σ i ⊆ ⋃ i, (s i).sets\ntfin : (⋃ i, σ i).Finite\ntinter : ⋂ i, ⋂₀ σ i ⊆ U\nV : ↑{i | i ∈... | refine ⟨I, Ifin, V, V_in, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.Filter.Finite | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 28
} | {
"line": 114,
"column": 2
} | [
{
"pp": "α : Type u\nι : Type u_2\ns : ι → Filter α\nU : Set α\n⊢ (∃ I, I.Finite ∧ ∃ V, (∀ (i : ↑I), V i ∈ s ↑i) ∧ U = ⋂ i, V i) →\n ∃ I, I.Finite ∧ ∃ V, (∀ (i : ι), V i ∈ s i) ∧ (∀ i ∉ I, V i = univ) ∧ U = ⋂ x, V ↑x ∧ U = ⋂ i, V i",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"... | [
"α : Type u\nι : Type u_2\ns : ι → Filter α\nI : Set ι\nIf : I.Finite\nV : ↑I → Set α\nhV : ∀ (i : ↑I), V i ∈ s ↑i\n⊢ ∃ I_1,\n I_1.Finite ∧\n ∃ V_1, (∀ (i : ι), V_1 i ∈ s i) ∧ (∀ i ∉ I_1, V_1 i = univ) ∧ ⋂ i, V i = ⋂ x, V_1 ↑x ∧ ⋂ i, V i = ⋂ i, V_1 i"
] | rintro ⟨I, If, V, hV, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Polynomial.Div | {
"line": 477,
"column": 6
} | {
"line": 477,
"column": 50
} | {
"line": 477,
"column": 51
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np : R[X]\na : R\nn : ℕ\nh : p.natDegree ≤ n\n⊢ (p /ₘ (X - C a)).coeff n = ∑ i ∈ Icc (n + 1) p.natDegree, a ^ (i - (n + 1)) * p.coeff i",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Polynomial.C",
"HMul.hMu... | [
"R : Type u\ninst✝ : Ring R\np : R[X]\na : R\nn : ℕ\nh : p.natDegree ≤ n\n⊢ (p /ₘ (X - C a)).coeff n = ∑ i ∈ ∅, a ^ (i - (n + 1)) * p.coeff i"
] | Icc_eq_empty (Nat.lt_succ_iff.mpr h).not_ge, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Derivative | {
"line": 437,
"column": 4
} | {
"line": 437,
"column": 67
} | {
"line": 438,
"column": 4
} | [
{
"pp": "case inr\nR : Type u\ninst✝¹ : Semiring R\np : R[X]\ninst✝ : IsAddTorsionFree R\nhp : derivative p = 0\nhp' : p ≠ 0\nf_nat_degree_pos : 0 < p.natDegree\nm : ℕ := p.natDegree - 1\n⊢ False",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Nat.instCanonicallyOrderedAdd",
"N... | [
"case inr\nR : Type u\ninst✝¹ : Semiring R\np : R[X]\ninst✝ : IsAddTorsionFree R\nhp : derivative p = 0\nhp' : p ≠ 0\nf_nat_degree_pos : 0 < p.natDegree\nm : ℕ := p.natDegree - 1\nhm : m + 1 = p.natDegree\n⊢ False"
] | have hm : m + 1 = p.natDegree := tsub_add_cancel_of_le (by lia) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Polynomial.Div | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 8
} | {
"line": 510,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np : R[X]\na : R\nh0 : p ≠ 0\nthis : Nontrivial R\n⊢ 0 < 1",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Nat.instMulZeroClass",
"WithBot",
"Preorder.toLT",
"of_decide_eq_true",
"Nat.instOne",... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Order.Filter.Pi | {
"line": 61,
"column": 2
} | {
"line": 63,
"column": 36
} | {
"line": 65,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\nf : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\nI : Set ι\nhI : I.Finite\nh : ∀ i ∈ I, s i ∈ f i\n⊢ I.pi s ∈ pi f",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"Set.iInter... | [] | rw [pi_def, biInter_eq_iInter]
refine mem_iInf_of_iInter hI (fun i => ?_) Subset.rfl
exact preimage_mem_comap (h i i.2) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.Pi | {
"line": 61,
"column": 2
} | {
"line": 63,
"column": 36
} | {
"line": 65,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\nf : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\nI : Set ι\nhI : I.Finite\nh : ∀ i ∈ I, s i ∈ f i\n⊢ I.pi s ∈ pi f",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"Set.iInter... | [] | rw [pi_def, biInter_eq_iInter]
refine mem_iInf_of_iInter hI (fun i => ?_) Subset.rfl
exact preimage_mem_comap (h i i.2) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Div | {
"line": 661,
"column": 71
} | {
"line": 661,
"column": 77
} | {
"line": 661,
"column": 77
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np : R[X]\na : R\nhp : p ≠ 0\nthis✝ : Nontrivial R\nq : R[X]\nhq : p /ₘ (X - C a) ^ rootMultiplicity a p = (X - C a) * q\nthis : (X - C a) ^ (multiplicity (X - C a) p + 1) * q = p\n⊢ 0 < 1",
"ppTerm": "?m.149",
"assigned": true,
"usedConstants": [
"WithB... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Algebra.Polynomial.Div | {
"line": 790,
"column": 41
} | {
"line": 790,
"column": 47
} | {
"line": 790,
"column": 47
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\np : R[X]\ninst✝ : IsDomain R\nhi : Irreducible p\nx : R\nhx : p.IsRoot x\ng : R[X]\nhg : p = (X - C x) * g\nthis : IsUnit (X - C x) ∨ IsUnit g\nh : IsUnit (X - C x)\nh₁ : (X - C x).degree = 1\nh₂ : 1 = 0\n⊢ ¬1 = 0",
"ppTerm": "?m.118",
"assigned": true,
"use... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Algebra.Polynomial.Div | {
"line": 790,
"column": 41
} | {
"line": 790,
"column": 47
} | {
"line": 790,
"column": 47
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\np : R[X]\ninst✝ : IsDomain R\nhi : Irreducible p\nx : R\nhx : p.IsRoot x\ng : R[X]\nhg : p = (X - C x) * g\nthis : IsUnit (X - C x) ∨ IsUnit g\nh : IsUnit (X - C x)\nh₁ : (X - C x).degree = 1\nh₂ : 1 = 0\n⊢ ¬1 = 0",
"ppTerm": "?m.118",
"assigned": true,
"use... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Div | {
"line": 790,
"column": 41
} | {
"line": 790,
"column": 47
} | {
"line": 790,
"column": 47
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\np : R[X]\ninst✝ : IsDomain R\nhi : Irreducible p\nx : R\nhx : p.IsRoot x\ng : R[X]\nhg : p = (X - C x) * g\nthis : IsUnit (X - C x) ∨ IsUnit g\nh : IsUnit (X - C x)\nh₁ : (X - C x).degree = 1\nh₂ : 1 = 0\n⊢ ¬1 = 0",
"ppTerm": "?m.118",
"assigned": true,
"use... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Heyting.Boundary | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 27
} | {
"line": 109,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ ∂ b ≤ ∂ (a ⊔ b) ⊔ ∂ (a ⊓ b)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPart... | [
"α : Type u_1\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ ∂ b ≤ ∂ (b ⊔ a) ⊔ ∂ (b ⊓ a)"
] | rw [sup_comm a, inf_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Derivative | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 14
} | {
"line": 620,
"column": 0
} | [
{
"pp": "case neg\nR : Type u\nι : Type y\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq ι\ns✝ : Multiset ι\nf : ι → R[X]\ni : ι\ns : Multiset ι\nh :\n derivative (Multiset.map f s).prod =\n (Multiset.map (fun i ↦ (Multiset.map f (s.erase i)).prod * derivative (f i)) s).sum\nj : ι\nhj : j ∈ s\nhij : ¬i = j\n... | [] | · simp [hij] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Filter.Prod | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 67
} | {
"line": 316,
"column": 67
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nf : Filter α\ng : Filter β\nh : Filter γ\nk : Filter δ\n⊢ map (fun p ↦ ((p.1.1, p.2.1), p.1.2, p.2.2)) ((f ×ˢ g) ×ˢ h ×ˢ k) = (f ×ˢ h) ×ˢ g ×ˢ k",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SProd.spr... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nf : Filter α\ng : Filter β\nh : Filter γ\nk : Filter δ\n⊢ comap ((Prod.fst ∘ Prod.fst) ∘ fun p ↦ ((p.1.1, p.2.1), p.1.2, p.2.2)) f ⊓\n comap ((Prod.snd ∘ Prod.fst) ∘ fun p ↦ ((p.1.1, p.2.1), p.1.2, p.2.2)) g ⊓\n (comap ((Prod.fst ∘ Prod.snd) ∘ fun... | simp_rw [map_swap4_eq_comap, prod_eq_inf, comap_inf, comap_comap] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.Polynomial.Derivative | {
"line": 703,
"column": 28
} | {
"line": 718,
"column": 86
} | {
"line": 720,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nk : ℕ\nP : R[X]\nh : P.degree < ↑k\n⊢ (⇑derivative)^[k] P = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Iff.mpr",
"Polynomial.derivative",
"WithBot.instPreorder",
"Mathlib.Tactic.Ring.Comm... | [] | by
induction k generalizing P
case zero => exact degree_eq_bot.mp <| WithBot.lt_coe_bot.mp h
case succ k ind =>
by_cases P = 0
case pos hP => simp [hP]
case neg hP =>
rw [Function.iterate_add_apply, Function.iterate_one]
by_cases derivative P = 0
case pos hP' => simp [hP']
case... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.BigOperators.Associated | {
"line": 77,
"column": 8
} | {
"line": 77,
"column": 26
} | {
"line": 77,
"column": 26
} | [
{
"pp": "M₀ : Type u_3\ninst✝¹ : CommMonoidWithZero M₀\ninst✝ : IsCancelMulZero M₀\np : M₀\nhp : Prime p\ns✝ : Multiset M₀\na : M₀\ns : Multiset M₀\nih : (∀ r ∈ s, Prime r) → p ∣ s.prod → ∃ q ∈ s, p ~ᵤ q\nhs : ∀ r ∈ a ::ₘ s, Prime r\nhps : p ∣ (a ::ₘ s).prod\n⊢ ∃ q ∈ a ::ₘ s, p ~ᵤ q",
"ppTerm": "?m.32",
... | [
"M₀ : Type u_3\ninst✝¹ : CommMonoidWithZero M₀\ninst✝ : IsCancelMulZero M₀\np : M₀\nhp : Prime p\ns✝ : Multiset M₀\na : M₀\ns : Multiset M₀\nih : (∀ r ∈ s, Prime r) → p ∣ s.prod → ∃ q ∈ s, p ~ᵤ q\nhs : ∀ r ∈ a ::ₘ s, Prime r\nhps : p ∣ a * s.prod\n⊢ ∃ q ∈ a ::ₘ s, p ~ᵤ q"
] | Multiset.prod_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Derivative | {
"line": 727,
"column": 4
} | {
"line": 752,
"column": 23
} | {
"line": 754,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nS : Finset R\nk : ℕ\nind : k ≤ #S → (⇑derivative)^[k] (∏ a ∈ S, (X - C a)) = ↑k ! * ∑ T ∈ powersetCard (#S - k) S, ∏ a ∈ T, (X - C a)\nhk : k + 1 ≤ #S\n⊢ (⇑derivative)^[k + 1] (∏ a ∈ S, (X - C a)) = ↑(k + 1)! * ∑ T ∈ powersetCard (#S - (k + 1)) S, ∏ a ∈ T, (X - C a)",
... | [] | specialize ind (Nat.le_of_succ_le hk)
nth_rewrite 1 [add_comm]
rw [Function.iterate_add_apply, Function.iterate_one, ind, ← nsmul_eq_mul, derivative_smul,
nsmul_eq_mul, derivative_sum, Nat.factorial_succ, mul_comm (k + 1), Nat.cast_mul, mul_assoc]
congr 1
calc
∑ T ∈ S.powersetCard (#S - k), ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Derivative | {
"line": 727,
"column": 4
} | {
"line": 752,
"column": 23
} | {
"line": 754,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nS : Finset R\nk : ℕ\nind : k ≤ #S → (⇑derivative)^[k] (∏ a ∈ S, (X - C a)) = ↑k ! * ∑ T ∈ powersetCard (#S - k) S, ∏ a ∈ T, (X - C a)\nhk : k + 1 ≤ #S\n⊢ (⇑derivative)^[k + 1] (∏ a ∈ S, (X - C a)) = ↑(k + 1)! * ∑ T ∈ powersetCard (#S - (k + 1)) S, ∏ a ∈ T, (X - C a)",
... | [] | specialize ind (Nat.le_of_succ_le hk)
nth_rewrite 1 [add_comm]
rw [Function.iterate_add_apply, Function.iterate_one, ind, ← nsmul_eq_mul, derivative_smul,
nsmul_eq_mul, derivative_sum, Nat.factorial_succ, mul_comm (k + 1), Nat.cast_mul, mul_assoc]
congr 1
calc
∑ T ∈ S.powersetCard (#S - k), ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Derivative | {
"line": 765,
"column": 4
} | {
"line": 765,
"column": 65
} | {
"line": 766,
"column": 2
} | [
{
"pp": "case neg\nR : Type u\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nP : R[X]\nh : P ∣ derivative P\nhP : ¬P = 0\n⊢ derivative P = 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Semiring.toModule",
"LinearMap.instFunLike",
"Poly... | [] | exact eq_zero_of_dvd_of_degree_lt h (degree_derivative_lt hP) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.BigOperators.Associated | {
"line": 129,
"column": 8
} | {
"line": 129,
"column": 26
} | {
"line": 129,
"column": 26
} | [
{
"pp": "case cons\nM₀ : Type u_3\ninst✝² : CommMonoidWithZero M₀\ninst✝¹ : IsCancelMulZero M₀\ninst✝ : (a : M₀) → DecidablePred (Associated a)\na : M₀\ns : Multiset M₀\ninduct : ∀ (n : M₀), (∀ a ∈ s, Prime a) → (∀ a ∈ s, a ∣ n) → (∀ (a : M₀), countP (Associated a) s ≤ 1) → s.prod ∣ n\nn : M₀\nh : ∀ a_1 ∈ a ::ₘ... | [
"case cons\nM₀ : Type u_3\ninst✝² : CommMonoidWithZero M₀\ninst✝¹ : IsCancelMulZero M₀\ninst✝ : (a : M₀) → DecidablePred (Associated a)\na : M₀\ns : Multiset M₀\ninduct : ∀ (n : M₀), (∀ a ∈ s, Prime a) → (∀ a ∈ s, a ∣ n) → (∀ (a : M₀), countP (Associated a) s ≤ 1) → s.prod ∣ n\nn : M₀\nh : ∀ a_1 ∈ a ::ₘ s, Prime a_... | Multiset.prod_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors | {
"line": 116,
"column": 16
} | {
"line": 116,
"column": 34
} | {
"line": 116,
"column": 34
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\na p : α\nha0 : a ≠ 0\nhp : Irreducible p\nx✝ : p ∣ a\nb : α\nhb : a = p * b\nhb0 : b ≠ 0\n⊢ p * b ~ᵤ (p ::ₘ normalizedFactors b).prod",
"ppTerm": "?m.112",
"assigned": true,
"us... | [
"α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\na p : α\nha0 : a ≠ 0\nhp : Irreducible p\nx✝ : p ∣ a\nb : α\nhb : a = p * b\nhb0 : b ≠ 0\n⊢ p * b ~ᵤ p * (normalizedFactors b).prod"
] | Multiset.prod_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 171,
"column": 22
} | {
"line": 171,
"column": 40
} | {
"line": 171,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\neif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a\nuif :\n ∀ (f g : Multiset α),\n (∀ x ∈ f, Irreducible x) → (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g\np : α\nthis : Decidable... | [
"α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\neif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a\nuif :\n ∀ (f g : Multiset α),\n (∀ x ∈ f, Irreducible x) → (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g\np : α\nthis : DecidableEq α := Clas... | Multiset.prod_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Roots | {
"line": 190,
"column": 19
} | {
"line": 190,
"column": 44
} | {
"line": 190,
"column": 45
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nr s : R\n⊢ rootMultiplicity s (X - C r) = count s {r}",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"CommSemiring.toSemiring",
"HSub.hSub",
"Classical.p... | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nr s : R\n⊢ (if s = r then 1 else 0) = count s {r}"
] | rootMultiplicity_X_sub_C, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 20
} | {
"line": 290,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\np q : Multiset (Associates α)\n⊢ ∀ (s s_1 : Multiset α),\n (∀ a ∈ map (Quot.mk ⇑(Associated.setoid α)) s_1, Irreducible a) →\n (∀ a ∈ map (Quot.mk ⇑(Associated.setoid α)) s, Irreducible a) →\n (map (Quot.mk... | [
"α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\np q : Multiset (Associates α)\ns t : Multiset α\nhs : ∀ a ∈ map (Quot.mk ⇑(Associated.setoid α)) t, Irreducible a\nht : ∀ a ∈ map (Quot.mk ⇑(Associated.setoid α)) s, Irreducible a\neq : (map (Quot.mk ⇑(Associated.setoid α)) t).prod =... | intro s t hs ht eq | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Polynomial.Roots | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 57
} | {
"line": 365,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nr : R\n⊢ nthRoots 0 r = 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Polynomial.roots",
"AddGroupWithOne.toAddGroup",
"congrArg",
... | [] | simp only [pow_zero, nthRoots, ← C_1, ← C_sub, roots_C] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Roots | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 57
} | {
"line": 365,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nr : R\n⊢ nthRoots 0 r = 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Polynomial.roots",
"AddGroupWithOne.toAddGroup",
"congrArg",
... | [] | simp only [pow_zero, nthRoots, ← C_1, ← C_sub, roots_C] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Roots | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 57
} | {
"line": 365,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nr : R\n⊢ nthRoots 0 r = 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Polynomial.roots",
"AddGroupWithOne.toAddGroup",
"congrArg",
... | [] | simp only [pow_zero, nthRoots, ← C_1, ← C_sub, roots_C] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Content | {
"line": 128,
"column": 6
} | {
"line": 128,
"column": 14
} | {
"line": 129,
"column": 2
} | [
{
"pp": "case succ.mpr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\na b : ℕ\nh1 : ¬p.coeff b = 0\nh2 : b.succ = a + 1\n⊢ ¬p.coeff b = 0",
"ppTerm": "?succ.mpr",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h1"
],
"usedGoals": []
}
] | [] | apply h1 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 352,
"column": 41
} | {
"line": 352,
"column": 81
} | {
"line": 352,
"column": 82
} | [
{
"pp": "case h₁\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\npf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a\na b : α\nane0 : a ≠ 0\nc : α\nhc : ¬IsUnit c\nb_eq : b = a * c\nh : ¬b = 0\ncne0 : c ≠ 0\n⊢ (Classical.choose ⋯).prod ~ᵤ a",
"ppTerm": "?h₁",
"assigned... | [
"α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\npf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a\na b : α\nane0 : a ≠ 0\nc : α\nhc : ¬IsUnit c\nb_eq : b = a * c\nh : ¬b = 0\ncne0 : c ≠ 0\n⊢ a ≠ 0"
] | apply (Classical.choose_spec (pf _ _)).2 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 352,
"column": 41
} | {
"line": 352,
"column": 81
} | {
"line": 352,
"column": 82
} | [
{
"pp": "case h₂\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\npf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a\na b : α\nane0 : a ≠ 0\nc : α\nhc : ¬IsUnit c\nb_eq : b = a * c\nh : ¬b = 0\ncne0 : c ≠ 0\n⊢ (Classical.choose ⋯).prod ~ᵤ c",
"ppTerm": "?h₂",
"assigned... | [
"α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\npf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a\na b : α\nane0 : a ≠ 0\nc : α\nhc : ¬IsUnit c\nb_eq : b = a * c\nh : ¬b = 0\ncne0 : c ≠ 0\n⊢ c ≠ 0"
] | apply (Classical.choose_spec (pf _ _)).2 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Polynomial.Roots | {
"line": 451,
"column": 48
} | {
"line": 451,
"column": 85
} | {
"line": 452,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nh₁ : X ^ 2 - C 1 = (X + C 1) * (X - C 1)\nh : X ^ 2 - C 1 = 0\n⊢ False",
"ppTerm": "?m.115",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
... | [] | by simpa using congrArg (coeff · 0) h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 131,
"column": 8
} | {
"line": 131,
"column": 71
} | {
"line": 131,
"column": 71
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np : R[X]\nt : R\nh : p ≠ 0\n⊢ ↑1! ∈ nonZeroDivisors R",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
"Membership.mem",
"nonZe... | [] | rw [Nat.factorial_one, Nat.cast_one]; exact Submonoid.one_mem _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 131,
"column": 8
} | {
"line": 131,
"column": 71
} | {
"line": 131,
"column": 71
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np : R[X]\nt : R\nh : p ≠ 0\n⊢ ↑1! ∈ nonZeroDivisors R",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
"Membership.mem",
"nonZe... | [] | rw [Nat.factorial_one, Nat.cast_one]; exact Submonoid.one_mem _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 464,
"column": 6
} | {
"line": 464,
"column": 32
} | {
"line": 465,
"column": 6
} | [
{
"pp": "case neg\nR : Type u_2\ninst✝¹ : CommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\na p : R\np_prime : Prime p\na' b' c' : R\ncoprime : IsRelPrime a' b'\na_ne_zero : c' * a' ≠ 0\nih_a : c' * a' ≠ 0 → ∀ (b : R), ∃ a'_1 b' c'_1, IsRelPrime a'_1 b' ∧ c'_1 * a'_1 = c' * a' ∧ c'_1 * b' = b\npa_ne_ze... | [
"case neg\nR : Type u_2\ninst✝¹ : CommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\na p : R\np_prime : Prime p\na' b' c' : R\ncoprime : IsRelPrime a' b'\na_ne_zero : c' * a' ≠ 0\nih_a : c' * a' ≠ 0 → ∀ (b : R), ∃ a'_1 b' c'_1, IsRelPrime a'_1 b' ∧ c'_1 * a'_1 = c' * a' ∧ c'_1 * b' = b\npa_ne_zero : p * (c'... | intro q q_dvd_pa' q_dvd_b' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Polynomial.Roots | {
"line": 639,
"column": 7
} | {
"line": 639,
"column": 65
} | {
"line": 641,
"column": 0
} | [
{
"pp": "R : Type u\na : R\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nhp : p ≠ C a\nx✝ : R\n⊢ x✝ ∈ (fun x ↦ eval x p) ⁻¹' {a} ↔ x✝ ∈ (p - C a).rootSet R",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Polynomial.C",
... | [] | simp [mem_rootSet_of_ne (sub_ne_zero.mpr hp), sub_eq_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 413,
"column": 2
} | {
"line": 416,
"column": 60
} | {
"line": 418,
"column": 0
} | [
{
"pp": "R : Type u\nk : Type y\ninst✝¹ : Field R\np q : R[X]\ninst✝ : Field k\nf : R →+* k\n⊢ map f (p / q) = map f p / map f q",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"RingHom.instRingHomClass",
"Polynomial.div_def",
"Polyno... | [] | if hq0 : q = 0 then simp [hq0]
else
rw [div_def, div_def, Polynomial.map_mul, map_divByMonic f (monic_mul_leadingCoeff_inv hq0),
Polynomial.map_mul, map_C, leadingCoeff_map, map_inv₀] | Lean.Parser.Tactic._aux_Init_TacticsExtra___macroRules_Lean_Parser_Tactic_tacDepIfThenElse_1 | Lean.Parser.Tactic.tacDepIfThenElse |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 413,
"column": 2
} | {
"line": 416,
"column": 60
} | {
"line": 418,
"column": 0
} | [
{
"pp": "R : Type u\nk : Type y\ninst✝¹ : Field R\np q : R[X]\ninst✝ : Field k\nf : R →+* k\n⊢ map f (p / q) = map f p / map f q",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"RingHom.instRingHomClass",
"Polynomial.div_def",
"Polyno... | [] | if hq0 : q = 0 then simp [hq0]
else
rw [div_def, div_def, Polynomial.map_mul, map_divByMonic f (monic_mul_leadingCoeff_inv hq0),
Polynomial.map_mul, map_C, leadingCoeff_map, map_inv₀] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 413,
"column": 2
} | {
"line": 416,
"column": 60
} | {
"line": 418,
"column": 0
} | [
{
"pp": "R : Type u\nk : Type y\ninst✝¹ : Field R\np q : R[X]\ninst✝ : Field k\nf : R →+* k\n⊢ map f (p / q) = map f p / map f q",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"RingHom.instRingHomClass",
"Polynomial.div_def",
"Polyno... | [] | if hq0 : q = 0 then simp [hq0]
else
rw [div_def, div_def, Polynomial.map_mul, map_divByMonic f (monic_mul_leadingCoeff_inv hq0),
Polynomial.map_mul, map_C, leadingCoeff_map, map_inv₀] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Roots | {
"line": 902,
"column": 2
} | {
"line": 904,
"column": 53
} | {
"line": 906,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommRing A\ninst✝ : CommRing B\np : A[X]\nf : A →+* B\nhmap : map f p ≠ 0\na : A\n⊢ rootMultiplicity a p ≤ rootMultiplicity (f a) (map f p)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.... | [] | rw [le_rootMultiplicity_iff hmap]
refine _root_.trans ?_ (_root_.map_dvd (mapRingHom f) (pow_rootMultiplicity_dvd p a))
rw [map_pow, map_sub, coe_mapRingHom, map_X, map_C] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Roots | {
"line": 902,
"column": 2
} | {
"line": 904,
"column": 53
} | {
"line": 906,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommRing A\ninst✝ : CommRing B\np : A[X]\nf : A →+* B\nhmap : map f p ≠ 0\na : A\n⊢ rootMultiplicity a p ≤ rootMultiplicity (f a) (map f p)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.... | [] | rw [le_rootMultiplicity_iff hmap]
refine _root_.trans ?_ (_root_.map_dvd (mapRingHom f) (pow_rootMultiplicity_dvd p a))
rw [map_pow, map_sub, coe_mapRingHom, map_X, map_C] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Roots | {
"line": 944,
"column": 2
} | {
"line": 944,
"column": 79
} | {
"line": 945,
"column": 2
} | [
{
"pp": "case pos\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : IsDomain A\ninst✝ : IsDomain B\np : A[X]\nf : A →+* B\nhf : Function.Injective ⇑f\nhp0 : p = 0\n⊢ Multiset.map (⇑f) p.roots ≤ (map f p).roots",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
... | [
"case neg\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : IsDomain A\ninst✝ : IsDomain B\np : A[X]\nf : A →+* B\nhf : Function.Injective ⇑f\nhp0 : ¬p = 0\n⊢ Multiset.map (⇑f) p.roots ≤ (map f p).roots"
] | · simp only [hp0, roots_zero, Multiset.map_zero, Polynomial.map_zero, le_rfl] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 574,
"column": 37
} | {
"line": 575,
"column": 43
} | {
"line": 577,
"column": 0
} | [
{
"pp": "R : Type u\na : R\ninst✝ : Field R\np : R[X]\n⊢ p / C a = p * C a⁻¹",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Polynomial.C",
"Polynomial.instOne",
"instHDiv",
"NonUnitalCommRing.toNon... | [] | by
simpa [mul_comm] using div_C_mul (q := 1) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid | {
"line": 41,
"column": 64
} | {
"line": 41,
"column": 75
} | {
"line": 42,
"column": 6
} | [
{
"pp": "α✝ : Type u_1\nα : Type u_2\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na b c : α\nhac : a ∣ c\nhab : a ∣ b\n⊢ Associates.mk a ≤ Associates.mk c ⊓ Associates.mk b",
"ppTerm": "?m.119",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
... | [
"α✝ : Type u_1\nα : Type u_2\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na b c : α\nhac : a ∣ c\nhab : a ∣ b\n⊢ Associates.mk a ≤ Associates.mk c ∧ Associates.mk a ≤ Associates.mk b"
] | le_inf_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid | {
"line": 67,
"column": 25
} | {
"line": 67,
"column": 36
} | {
"line": 67,
"column": 37
} | [
{
"pp": "α✝ : Type u_1\nα : Type u_2\ninst✝² : CommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : NormalizationMonoid α\na b c : α\nhac : a ∣ c\nhab : a ∣ b\n⊢ Associates.mk a ≤ Associates.mk c ⊓ Associates.mk b",
"ppTerm": "?m.154",
"assigned": true,
"usedConstants": [
"CommM... | [
"α✝ : Type u_1\nα : Type u_2\ninst✝² : CommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : NormalizationMonoid α\na b c : α\nhac : a ∣ c\nhab : a ∣ b\n⊢ Associates.mk a ≤ Associates.mk c ∧ Associates.mk a ≤ Associates.mk b"
] | le_inf_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 729,
"column": 2
} | {
"line": 729,
"column": 80
} | {
"line": 730,
"column": 2
} | [
{
"pp": "case inr\nR : Type u\ninst✝ : Field R\np₁ p₂ q : R[X]\nh : q ∣ p₁ - p₂\nhq : q ≠ 0\n⊢ p₁ %ₘ (q * C q.leadingCoeff⁻¹) = p₂ %ₘ (q * C q.leadingCoeff⁻¹)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemirin... | [
"case inr\nR : Type u\ninst✝ : Field R\np₁ p₂ q : R[X]\nh : q ∣ p₁ - p₂\nhq : q ≠ 0\n⊢ q * C q.leadingCoeff⁻¹ ∣ p₁ - p₂"
] | apply Polynomial.modByMonic_eq_of_dvd_sub (by simp [Polynomial.Monic.def, hq]) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
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