module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.MvPolynomial.Expand | {
"line": 96,
"column": 4
} | {
"line": 97,
"column": 37
} | {
"line": 99,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\nn : ℕ\nhn : 0 < n\ng g' : MvPolynomial σ R\nH : (expand n) g = (expand n) g'\nd : σ →₀ ℕ\n⊢ coeff d g = coeff d g'",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Nat.instMulZeroClass",
... | [] | rw [← coeff_expand_smul _ (n.ne_zero_iff_zero_lt.mpr hn), H, coeff_expand_smul _
(n.ne_zero_iff_zero_lt.mpr hn)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MvPolynomial.Division | {
"line": 340,
"column": 2
} | {
"line": 379,
"column": 36
} | {
"line": 381,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np q : MvPolynomial σ R\ninst✝ : IsCancelMulZero R\nn : σ →₀ ℕ\n⊢ p ∣ (monomial n) 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = (monomial m) 1 * r",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Finsupp.instCanonicallyOrderedAddOfAddLef... | [] | rcases subsingleton_or_nontrivial R with hR | hR
· simp only [Subsingleton.elim _ p, dvd_refl, and_self, and_true, exists_const, true_iff]
refine ⟨n, le_refl n⟩
suffices ∀ (d) (n : σ →₀ ℕ) (hd : n.degree = d) (p q : MvPolynomial σ R),
p ∣ monomial n 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = monomial m 1 * r from t... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Division | {
"line": 340,
"column": 2
} | {
"line": 379,
"column": 36
} | {
"line": 381,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np q : MvPolynomial σ R\ninst✝ : IsCancelMulZero R\nn : σ →₀ ℕ\n⊢ p ∣ (monomial n) 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = (monomial m) 1 * r",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Finsupp.instCanonicallyOrderedAddOfAddLef... | [] | rcases subsingleton_or_nontrivial R with hR | hR
· simp only [Subsingleton.elim _ p, dvd_refl, and_self, and_true, exists_const, true_iff]
refine ⟨n, le_refl n⟩
suffices ∀ (d) (n : σ →₀ ℕ) (hd : n.degree = d) (p q : MvPolynomial σ R),
p ∣ monomial n 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = monomial m 1 * r from t... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.GameAdd | {
"line": 151,
"column": 55
} | {
"line": 151,
"column": 96
} | {
"line": 151,
"column": 96
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nrα✝ : α → α → Prop\nrβ : β → β → Prop\na : α\nb : β\nrα : α → α → Prop\na₁ b₁ a₂ b₂ : α\n⊢ (Prod.GameAdd rα rα (a₁, b₁) (a₂, b₂) ∨ Prod.GameAdd rα rα (b₁, a₁) (a₂, b₂)) =\n (Prod.GameAdd rα rα (b₁, a₁) (a₂, b₂) ∨ Prod.GameAdd rα rα (a₁, b₁) (a₂, b₂)) ∧\n (Prod.GameAd... | [
"α : Type u_1\nβ : Type u_2\nrα✝ : α → α → Prop\nrβ : β → β → Prop\na : α\nb : β\nrα : α → α → Prop\na₁ b₁ a₂ b₂ : α\n⊢ (Prod.GameAdd rα rα (a₁, b₁) (a₂, b₂) ∨ Prod.GameAdd rα rα (b₁, a₁) (a₂, b₂)) =\n (Prod.GameAdd rα rα (b₁, a₁) (a₂, b₂) ∨ Prod.GameAdd rα rα (a₁, b₁) (a₂, b₂)) ∧\n (Prod.GameAdd rα rα (a₁,... | Prod.gameAdd_swap_swap_mk _ _ a₁ b₂ b₁ a₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.GameAdd | {
"line": 155,
"column": 97
} | {
"line": 157,
"column": 5
} | {
"line": 159,
"column": 0
} | [
{
"pp": "α : Type u_1\nrα : α → α → Prop\n⊢ ∀ {x y : α × α}, GameAdd rα s(x.1, x.2) s(y.1, y.2) ↔ Prod.GameAdd rα rα x y ∨ Prod.GameAdd rα rα x.swap y",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Sym2.mk",
"Prod.GameAdd",
"Iff.rfl",
"Prod.mk",
"Prod.fst",
... | [] | by
rintro ⟨_, _⟩ ⟨_, _⟩
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 497,
"column": 4
} | {
"line": 497,
"column": 59
} | {
"line": 498,
"column": 4
} | [
{
"pp": "case succ\nm : Multiset ℕ\nk : ℕ\nhk : (k • m).multinomial = (Nat.multinomial (Finset.range k) fun x ↦ m.sum) * m.multinomial ^ k\n⊢ (m.sum + k • m.sum).choose m.sum * m.multinomial *\n ((Nat.multinomial (Finset.range k) fun x ↦ m.sum) * m.multinomial ^ k) =\n ((m.sum + ∑ i ∈ Finset.range k, m.... | [
"case succ\nm : Multiset ℕ\nk : ℕ\nhk : (k • m).multinomial = (Nat.multinomial (Finset.range k) fun x ↦ m.sum) * m.multinomial ^ k\n⊢ (m.sum + k * m.sum).choose m.sum * m.multinomial *\n ((Nat.multinomial (Finset.range k) fun x ↦ m.sum) * m.multinomial ^ k) =\n ((m.sum + k * m.sum).choose m.sum * Nat.multin... | simp [smul_eq_mul, Finset.sum_const, Finset.card_range] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 61,
"column": 9
} | {
"line": 61,
"column": 43
} | {
"line": 61,
"column": 43
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\nn : σ\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nι : Type u_3\nf : ι → MvPolynomial σ R\nnontrivial : Nontrivial (MvPolynomial σ R)\na : ι\ns : Finset ι\na_not_mem : a ∉ s\nih : (∀ i ∈ s, f i ≠ 0) → degreeOf n (∏ i ∈ s, f i) = ∑ i ∈ s, degreeOf n (f i)\nha : ¬f a = ... | [] | by rw [prod_ne_zero_iff]; exact hs | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 157,
"column": 2
} | {
"line": 163,
"column": 30
} | {
"line": 165,
"column": 0
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\np : MvPolynomial σ R\nn : σ →₀ ℕ\n⊢ p ∣ (monomial n) 1 ↔ ∃ m u, m ≤ n ∧ IsUnit u ∧ p = (monomial m) u",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"Finsupp.instL... | [] | rcases subsingleton_or_nontrivial R with hR | hR
· suffices ∃ m, m ≤ n by simpa [Subsingleton.elim _ p]
use n
rw [dvd_monomial_iff_exists (one_ne_zero' R)]
apply exists_congr
intro m
simp_rw [isUnit_iff_dvd_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 157,
"column": 2
} | {
"line": 163,
"column": 30
} | {
"line": 165,
"column": 0
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\np : MvPolynomial σ R\nn : σ →₀ ℕ\n⊢ p ∣ (monomial n) 1 ↔ ∃ m u, m ≤ n ∧ IsUnit u ∧ p = (monomial m) u",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"Finsupp.instL... | [] | rcases subsingleton_or_nontrivial R with hR | hR
· suffices ∃ m, m ≤ n by simpa [Subsingleton.elim _ p]
use n
rw [dvd_monomial_iff_exists (one_ne_zero' R)]
apply exists_congr
intro m
simp_rw [isUnit_iff_dvd_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 159,
"column": 30
} | {
"line": 159,
"column": 41
} | {
"line": 159,
"column": 41
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Subsingleton R\nf : MvPolynomial σ R\n⊢ m.degree 0 = 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"congrArg... | [
"σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Subsingleton R\nf : MvPolynomial σ R\n⊢ 0 = 0"
] | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 582,
"column": 23
} | {
"line": 582,
"column": 34
} | {
"line": 582,
"column": 34
} | [
{
"pp": "case neg\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : IsReduced R\nf : MvPolynomial σ R\nn : ℕ\nhf : f = 0\nhn : ¬n = 0\n⊢ m.degree 0 = 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddM... | [
"case neg\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : IsReduced R\nf : MvPolynomial σ R\nn : ℕ\nhf : f = 0\nhn : ¬n = 0\n⊢ 0 = 0"
] | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 893,
"column": 34
} | {
"line": 895,
"column": 18
} | {
"line": 897,
"column": 0
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nf : MvPolynomial σ R\n⊢ m.degree (-f) = m.degree f",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Nat.instMulZeroClass",
"Lattice.toSemilatticeSup",
... | [] | by
unfold degree
rw [support_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 975,
"column": 17
} | {
"line": 975,
"column": 54
} | {
"line": 975,
"column": 55
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nf g : MvPolynomial σ R\n⊢ m.leadingCoeff g * coeff (m.degree f) f -\n coeff (m.degree f ⊔ m.degree g) ((monomial (m.degree f ⊔ m.degree g - m.degree g)) (m.leadingCoeff f) * g) =\n 0",
"ppTerm": "?m.56",
"assigned": tr... | [
"σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nf g : MvPolynomial σ R\n⊢ m.leadingCoeff g * coeff (m.degree f) f -\n coeff (m.degree f ⊔ m.degree g - m.degree g + m.degree g)\n ((monomial (m.degree f ⊔ m.degree g - m.degree g)) (m.leadingCoeff f) * g) =\n 0"
] | ← tsub_add_cancel_of_le le_sup_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Zeta | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 16
} | {
"line": 62,
"column": 16
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : MulAction R M\nf : ArithmeticFunction M\nx : ℕ\n⊢ (↑ζ • f) x = ∑ i ∈ x.divisors, f i",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [
"R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : MulAction R M\nf : ArithmeticFunction M\nx : ℕ\n⊢ ∑ x ∈ x.divisorsAntidiagonal, ↑ζ x.1 • f x.2 = ∑ i ∈ x.divisors, f i"
] | smul_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Zeta | {
"line": 197,
"column": 3
} | {
"line": 197,
"column": 19
} | {
"line": 197,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\n⊢ (f.pmul g) 1 = 1",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"ArithmeticFunction.pmul",
"MulOne.toOne",
"HMul.hMul",
"ArithmeticFunctio... | [] | by simp [hf, hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.Zeta | {
"line": 204,
"column": 3
} | {
"line": 204,
"column": 19
} | {
"line": 204,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\n⊢ (f.pdiv g) 1 = 1",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"False",
"instHDiv"... | [] | by simp [hf, hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Factorization.PrimePow | {
"line": 62,
"column": 47
} | {
"line": 62,
"column": 82
} | {
"line": 62,
"column": 82
} | [
{
"pp": "n : ℕ\nhn : 2 ≤ n\n⊢ ¬n.primeFactors.card = 1 ↔ n.primeFactors.Nontrivial",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.one_lt_card_iff_nontrivial",
"congrArg",
"id",
"instOfNatNat",
"Iff",
"Nat",
"LT.lt",
"... | [
"n : ℕ\nhn : 2 ≤ n\n⊢ ¬n.primeFactors.card = 1 ↔ 1 < n.primeFactors.card"
] | ← Finset.one_lt_card_iff_nontrivial | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 175,
"column": 82
} | {
"line": 176,
"column": 38
} | {
"line": 178,
"column": 0
} | [
{
"pp": "k n : ℕ\n⊢ (σ k) n = ∑ d ∈ n.divisors, (n / d) ^ k",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"instHDiv",
"ArithmeticFunction.instFunLikeNat",
"ArithmeticFunction.sigma_apply",
"congrArg",
"Nat.instMo... | [] | by
rw [sigma_apply, ← sum_div_divisors] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 76
} | {
"line": 323,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ Ω n = n.factorization.sum fun x k ↦ k",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLikeNat",
"congrArg",
"Finset",
"_private.Mathlib.NumberTheory.ArithmeticFun... | [] | simp [cardFactors_apply, ← List.sum_toFinset_count_eq_length, Finsupp.sum] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 76
} | {
"line": 323,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ Ω n = n.factorization.sum fun x k ↦ k",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLikeNat",
"congrArg",
"Finset",
"_private.Mathlib.NumberTheory.ArithmeticFun... | [] | simp [cardFactors_apply, ← List.sum_toFinset_count_eq_length, Finsupp.sum] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 76
} | {
"line": 323,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ Ω n = n.factorization.sum fun x k ↦ k",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLikeNat",
"congrArg",
"Finset",
"_private.Mathlib.NumberTheory.ArithmeticFun... | [] | simp [cardFactors_apply, ← List.sum_toFinset_count_eq_length, Finsupp.sum] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 378,
"column": 4
} | {
"line": 378,
"column": 57
} | {
"line": 379,
"column": 4
} | [
{
"pp": "case cons\nι : Type u_2\nf : ι → ℕ\na : ι\ns : Finset ι\nha : a ∉ s\nih : (↑s).Pairwise (Coprime on f) → ω (∏ i ∈ s, f i) = ∑ i ∈ s, ω (f i)\nh : (↑(cons a s ha)).Pairwise (Coprime on f)\n⊢ ω (∏ i ∈ cons a s ha, f i) = ∑ i ∈ cons a s ha, ω (f i)",
"ppTerm": "?cons",
"assigned": true,
"usedC... | [
"case cons\nι : Type u_2\nf : ι → ℕ\na : ι\ns : Finset ι\nha : a ∉ s\nih : (↑s).Pairwise (Coprime on f) → ω (∏ i ∈ s, f i) = ∑ i ∈ s, ω (f i)\nh : (↑(cons a s ha)).Pairwise (Coprime on f)\n⊢ (↑s).Pairwise (Coprime on f)",
"case cons\nι : Type u_2\nf : ι → ℕ\na : ι\ns : Finset ι\nha : a ∉ s\nih : (↑s).Pairwise (Co... | rw [prod_cons, sum_cons, cardDistinctFactors_mul, ih] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.CompleteField | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 21
} | {
"line": 67,
"column": 4
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : Field α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : IsStrictOrderedRing α\n⊢ ∀ (x : α), ∃ n, x < ↑n",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Mathlib.... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : Field α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : IsStrictOrderedRing α\nx : α\nh : ∀ (n : ℕ), ↑n ≤ x\n⊢ False"
] | by_contra! ⟨x, h⟩ | Mathlib.Tactic.ByContra._aux_Mathlib_Tactic_ByContra___macroRules_Mathlib_Tactic_ByContra_byContra!_1 | Mathlib.Tactic.ByContra.byContra! |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 522,
"column": 27
} | {
"line": 522,
"column": 88
} | {
"line": 522,
"column": 89
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : ℕ\nha : ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 ≠ 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2... | [
"R : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : ℕ\nha : ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 ≠ 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : ((a1... | cop.coprime_mul_left.coprime_mul_right_right.symm.gcd_eq_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 525,
"column": 10
} | {
"line": 525,
"column": 71
} | {
"line": 525,
"column": 72
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : ℕ\nha : ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 ≠ 0\nhb : ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 ≠ 0\nc1 c2 d1 d2 : ℕ\ncop : (((c1, c2), d1, d2).1.1 * ((c1,... | [
"R : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : ℕ\nha : ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 ≠ 0\nhb : ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 ≠ 0\nc1 c2 d1 d2 : ℕ\ncop : (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2... | cop.coprime_mul_left.coprime_mul_right_right.symm.gcd_eq_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.CompleteField | {
"line": 248,
"column": 6
} | {
"line": 248,
"column": 16
} | {
"line": 248,
"column": 16
} | [
{
"pp": "case inr\nα : Type u_2\nβ : Type u_3\ninst✝⁶ : Field α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsStrictOrderedRing α\ninst✝³ : Field β\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : Archimedean α\na : α\nha : 0 < a\nb : β\nhba : b < inducedMap α β a * inducedMap α β a\n... | [
"case inr\nα : Type u_2\nβ : Type u_3\ninst✝⁶ : Field α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsStrictOrderedRing α\ninst✝³ : Field β\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : Archimedean α\na : α\nha : 0 < a\nb : β\nhba : b < inducedMap α β a * inducedMap α β a\nhb : 0 ≤ b\n... | ← cast_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 628,
"column": 2
} | {
"line": 632,
"column": 42
} | {
"line": 634,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nι : Type u_2\nf : ι → ArithmeticFunction R\ns : Finset ι\nhf : ∀ i ∈ s, (f i).IsMultiplicative\n⊢ (∏ i ∈ s, f i).IsMultiplicative",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"F... | [] | induction s using Finset.cons_induction
case empty => simp
case cons a s ha ih =>
rw [Finset.prod_cons]
exact (hf a (by grind)).mul (by grind) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 628,
"column": 2
} | {
"line": 632,
"column": 42
} | {
"line": 634,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nι : Type u_2\nf : ι → ArithmeticFunction R\ns : Finset ι\nhf : ∀ i ∈ s, (f i).IsMultiplicative\n⊢ (∏ i ∈ s, f i).IsMultiplicative",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"F... | [] | induction s using Finset.cons_induction
case empty => simp
case cons a s ha ih =>
rw [Finset.prod_cons]
exact (hf a (by grind)).mul (by grind) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 532,
"column": 2
} | {
"line": 532,
"column": 38
} | {
"line": 534,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝⁵ : CommGroup M\ninst✝⁴ : LinearOrder M\ninst✝³ : IsOrderedMonoid M\na b : M\nN : Type u_2\ninst✝² : CommGroup N\ninst✝¹ : LinearOrder N\ninst✝ : IsOrderedMonoid N\nf : M →*o N\nh : mk a = mk b\n⊢ mk (f a) = mk (f b)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants":... | [] | rw [← orderHom_mk, ← orderHom_mk, h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 532,
"column": 2
} | {
"line": 532,
"column": 38
} | {
"line": 534,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝⁵ : CommGroup M\ninst✝⁴ : LinearOrder M\ninst✝³ : IsOrderedMonoid M\na b : M\nN : Type u_2\ninst✝² : CommGroup N\ninst✝¹ : LinearOrder N\ninst✝ : IsOrderedMonoid N\nf : M →*o N\nh : mk a = mk b\n⊢ mk (f a) = mk (f b)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants":... | [] | rw [← orderHom_mk, ← orderHom_mk, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 532,
"column": 2
} | {
"line": 532,
"column": 38
} | {
"line": 534,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝⁵ : CommGroup M\ninst✝⁴ : LinearOrder M\ninst✝³ : IsOrderedMonoid M\na b : M\nN : Type u_2\ninst✝² : CommGroup N\ninst✝¹ : LinearOrder N\ninst✝ : IsOrderedMonoid N\nf : M →*o N\nh : mk a = mk b\n⊢ mk (f a) = mk (f b)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants":... | [] | rw [← orderHom_mk, ← orderHom_mk, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Group.Ideal | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 27
} | {
"line": 45,
"column": 4
} | [
{
"pp": "case mp\nM : Type u_1\ninst✝³ : CommMonoid M\ninst✝² : PartialOrder M\ninst✝¹ : WellQuasiOrderedLE M\ninst✝ : CanonicallyOrderedMul M\nI : SemigroupIdeal M\nhpwo : {x | x ∈ I}.IsPWO\nx : M\nhx : x ∈ I\nz : M\nhz : ∃ c, x = c * z\nhz' : Minimal (fun x ↦ x ∈ {x | x ∈ I}) z\n⊢ ∃ y z, Minimal (fun x ↦ x ∈ ... | [
"case mp\nM : Type u_1\ninst✝³ : CommMonoid M\ninst✝² : PartialOrder M\ninst✝¹ : WellQuasiOrderedLE M\ninst✝ : CanonicallyOrderedMul M\nI : SemigroupIdeal M\nhpwo : {x | x ∈ I}.IsPWO\nz : M\nhz' : Minimal (fun x ↦ x ∈ {x | x ∈ I}) z\ny : M\nhx : y * z ∈ I\n⊢ ∃ y_1 z_1, Minimal (fun x ↦ x ∈ I) z_1 ∧ y_1 * z_1 = y * ... | rcases hz with ⟨y, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.Order.Group.Int.Sum | {
"line": 50,
"column": 12
} | {
"line": 50,
"column": 14
} | {
"line": 50,
"column": 14
} | [
{
"pp": "case refine_1\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nx : ℕ\n⊢ x ∈ range #s → c - ↑x ∈ Ioc (c - ↑(#s)) c",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"Int",
"Finset.range",
"Finset.instSetLike",
"Nat",
... | [
"case refine_1\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nx : ℕ\nmx : x ∈ range #s\n⊢ c - ↑x ∈ Ioc (c - ↑(#s)) c"
] | mx | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Order.Group.Int.Sum | {
"line": 51,
"column": 12
} | {
"line": 51,
"column": 14
} | {
"line": 51,
"column": 15
} | [
{
"pp": "case refine_2\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nx : ℕ\n⊢ x ∈ ↑(range #s) → ∀ ⦃x₂ : ℕ⦄, x₂ ∈ ↑(range #s) → (fun x ↦ c - ↑x) x = (fun x ↦ c - ↑x) x₂ → x = x₂",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"Int",
"Fins... | [
"case refine_2\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nx : ℕ\nmx : x ∈ ↑(range #s)\n⊢ ∀ ⦃x₂ : ℕ⦄, x₂ ∈ ↑(range #s) → (fun x ↦ c - ↑x) x = (fun x ↦ c - ↑x) x₂ → x = x₂"
] | mx | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Order.Group.Int.Sum | {
"line": 52,
"column": 12
} | {
"line": 52,
"column": 14
} | {
"line": 52,
"column": 14
} | [
{
"pp": "case refine_3\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nx : ℤ\n⊢ x ∈ ↑(Ioc (c - ↑(#s)) c) → x ∈ (fun x ↦ c - ↑x) '' ↑(range #s)",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"Finset",
"PartialOrder.toPreorder",
"HSub.hSub",
"Membership.mem",
"S... | [
"case refine_3\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nx : ℤ\nmx : x ∈ ↑(Ioc (c - ↑(#s)) c)\n⊢ x ∈ (fun x ↦ c - ↑x) '' ↑(range #s)"
] | mx | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Order.GroupWithZero.Bounds | {
"line": 27,
"column": 2
} | {
"line": 27,
"column": 59
} | {
"line": 28,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : Nonempty α\ninst✝² : Preorder β\ninst✝¹ : Zero β\ninst✝ : Preorder γ\nf : α → β\ng : β → γ\nhf : BddAbove (range f)\nhf0 : 0 ≤ f\nhg : MonotoneOn g {x | 0 ≤ x}\n⊢ BddAbove (g '' range f)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : Nonempty α\ninst✝² : Preorder β\ninst✝¹ : Zero β\ninst✝ : Preorder γ\nf : α → β\ng : β → γ\nhf : BddAbove (range f)\nhf0 : 0 ≤ f\nhg : MonotoneOn g {x | 0 ≤ x}\n⊢ (upperBounds (range f) ∩ {x | 0 ≤ x}).Nonempty"
] | apply hg.map_bddAbove (by rintro x ⟨a, rfl⟩; exact hf0 a) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Order.Group.Int.Sum | {
"line": 76,
"column": 12
} | {
"line": 76,
"column": 14
} | {
"line": 76,
"column": 14
} | [
{
"pp": "case refine_1\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, c ≤ x\nx : ℕ\n⊢ x ∈ range #s → c + ↑x ∈ Ico c (c + ↑(#s))",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"Int",
"Finset.range",
"Finset.instSetLike",
"Nat",
... | [
"case refine_1\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, c ≤ x\nx : ℕ\nmx : x ∈ range #s\n⊢ c + ↑x ∈ Ico c (c + ↑(#s))"
] | mx | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Order.Group.Int.Sum | {
"line": 77,
"column": 12
} | {
"line": 77,
"column": 14
} | {
"line": 77,
"column": 15
} | [
{
"pp": "case refine_2\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, c ≤ x\nx : ℕ\n⊢ x ∈ ↑(range #s) → ∀ ⦃x₂ : ℕ⦄, x₂ ∈ ↑(range #s) → (fun x ↦ c + ↑x) x = (fun x ↦ c + ↑x) x₂ → x = x₂",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"Int",
"Fins... | [
"case refine_2\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, c ≤ x\nx : ℕ\nmx : x ∈ ↑(range #s)\n⊢ ∀ ⦃x₂ : ℕ⦄, x₂ ∈ ↑(range #s) → (fun x ↦ c + ↑x) x = (fun x ↦ c + ↑x) x₂ → x = x₂"
] | mx | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Order.Group.Int.Sum | {
"line": 78,
"column": 12
} | {
"line": 78,
"column": 14
} | {
"line": 78,
"column": 14
} | [
{
"pp": "case refine_3\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, c ≤ x\nx : ℤ\n⊢ x ∈ ↑(Ico c (c + ↑(#s))) → x ∈ (fun x ↦ c + ↑x) '' ↑(range #s)",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"Finset",
"PartialOrder.toPreorder",
"Membership.mem",
"SemilatticeInf.toPar... | [
"case refine_3\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, c ≤ x\nx : ℤ\nmx : x ∈ ↑(Ico c (c + ↑(#s)))\n⊢ x ∈ (fun x ↦ c + ↑x) '' ↑(range #s)"
] | mx | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Order.Interval.Set.SuccPred | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 70
} | {
"line": 55,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : One α\ninst✝¹ : Add α\ninst✝ : SuccAddOrder α\na : α\nha : ¬IsMax a\nb : α\n⊢ Icc (a + 1) b = Ioc a b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"Order.succ",
"Order.succ_eq_add_one",
"congrArg"... | [] | simpa [succ_eq_add_one] using Icc_succ_left_eq_Ioc_of_not_isMax ha b | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Interval.Set.SuccPred | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 70
} | {
"line": 55,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : One α\ninst✝¹ : Add α\ninst✝ : SuccAddOrder α\na : α\nha : ¬IsMax a\nb : α\n⊢ Icc (a + 1) b = Ioc a b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"Order.succ",
"Order.succ_eq_add_one",
"congrArg"... | [] | simpa [succ_eq_add_one] using Icc_succ_left_eq_Ioc_of_not_isMax ha b | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Interval.Set.SuccPred | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 70
} | {
"line": 55,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : One α\ninst✝¹ : Add α\ninst✝ : SuccAddOrder α\na : α\nha : ¬IsMax a\nb : α\n⊢ Icc (a + 1) b = Ioc a b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"Order.succ",
"Order.succ_eq_add_one",
"congrArg"... | [] | simpa [succ_eq_add_one] using Icc_succ_left_eq_Ioc_of_not_isMax ha b | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Module.Archimedean | {
"line": 38,
"column": 51
} | {
"line": 38,
"column": 62
} | {
"line": 38,
"column": 63
} | [
{
"pp": "M : Type u_1\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LinearOrder M\ninst✝⁶ : IsOrderedAddMonoid M\nK : Type u_2\ninst✝⁵ : Ring K\ninst✝⁴ : LinearOrder K\ninst✝³ : IsOrderedRing K\ninst✝² : Archimedean K\ninst✝¹ : Module K M\ninst✝ : PosSMulMono K M\na : M\nk : K\nh : k ≠ 0\nthis : Nontrivial K\nm : ℕ\nhm : ... | [
"M : Type u_1\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LinearOrder M\ninst✝⁶ : IsOrderedAddMonoid M\nK : Type u_2\ninst✝⁵ : Ring K\ninst✝⁴ : LinearOrder K\ninst✝³ : IsOrderedRing K\ninst✝² : Archimedean K\ninst✝¹ : Module K M\ninst✝ : PosSMulMono K M\na : M\nk : K\nh : k ≠ 0\nthis : Nontrivial K\nm : ℕ\nhm : 1 ≤ m • |k|\... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Interval.Basic | {
"line": 490,
"column": 4
} | {
"line": 490,
"column": 48
} | {
"line": 490,
"column": 49
} | [
{
"pp": "ι : Type u_1\nα✝ : Type u_2\ninst✝⁵ : CommGroup α✝\ninst✝⁴ : PartialOrder α✝\ninst✝³ : IsOrderedMonoid α✝\ns✝ t✝ : NonemptyInterval α✝\nα : Type u\ninst✝² : AddCommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedAddMonoid α\ns t : NonemptyInterval α\n⊢ s - t = s + -t",
"ppTerm": "?m.32",
"as... | [
"case refine_1\nι : Type u_1\nα✝ : Type u_2\ninst✝⁵ : CommGroup α✝\ninst✝⁴ : PartialOrder α✝\ninst✝³ : IsOrderedMonoid α✝\ns✝ t✝ : NonemptyInterval α✝\nα : Type u\ninst✝² : AddCommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedAddMonoid α\ns t : NonemptyInterval α\n⊢ (s - t).toProd.1 = (s + -t).toProd.1",
"ca... | refine NonemptyInterval.ext (Prod.ext ?_ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Order.Interval.Basic | {
"line": 494,
"column": 4
} | {
"line": 494,
"column": 48
} | {
"line": 494,
"column": 49
} | [
{
"pp": "ι : Type u_1\nα✝ : Type u_2\ninst✝⁵ : CommGroup α✝\ninst✝⁴ : PartialOrder α✝\ninst✝³ : IsOrderedMonoid α✝\ns✝ t✝ : NonemptyInterval α✝\nα : Type u\ninst✝² : AddCommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedAddMonoid α\ns t : NonemptyInterval α\n⊢ -(s + t) = -t + -s",
"ppTerm": "?m.69",
... | [
"case refine_1\nι : Type u_1\nα✝ : Type u_2\ninst✝⁵ : CommGroup α✝\ninst✝⁴ : PartialOrder α✝\ninst✝³ : IsOrderedMonoid α✝\ns✝ t✝ : NonemptyInterval α✝\nα : Type u\ninst✝² : AddCommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedAddMonoid α\ns t : NonemptyInterval α\n⊢ (-(s + t)).toProd.1 = (-t + -s).toProd.1",
... | refine NonemptyInterval.ext (Prod.ext ?_ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Order.Interval.Basic | {
"line": 497,
"column": 4
} | {
"line": 497,
"column": 72
} | {
"line": 498,
"column": 4
} | [
{
"pp": "ι : Type u_1\nα✝ : Type u_2\ninst✝⁵ : CommGroup α✝\ninst✝⁴ : PartialOrder α✝\ninst✝³ : IsOrderedMonoid α✝\ns✝ t✝ : NonemptyInterval α✝\nα : Type u\ninst✝² : AddCommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedAddMonoid α\ns t : NonemptyInterval α\nh : s + t = 0\n⊢ -s = t",
"ppTerm": "?m.88",
... | [
"ι : Type u_1\nα✝ : Type u_2\ninst✝⁵ : CommGroup α✝\ninst✝⁴ : PartialOrder α✝\ninst✝³ : IsOrderedMonoid α✝\ns t : NonemptyInterval α✝\nα : Type u\ninst✝² : AddCommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedAddMonoid α\na b : α\nhab : a + b = 0\nh : pure a + pure b = 0\n⊢ -pure a = pure b"
] | obtain ⟨a, b, rfl, rfl, hab⟩ := NonemptyInterval.add_eq_zero_iff.1 h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Order.Interval.Basic | {
"line": 505,
"column": 4
} | {
"line": 505,
"column": 48
} | {
"line": 505,
"column": 49
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\ns✝ t✝ s t : NonemptyInterval α\n⊢ s / t = s * t⁻¹",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"instDivNonemptyInterval",
"instHDiv",
... | [
"case refine_1\nι : Type u_1\nα : Type u_2\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\ns✝ t✝ s t : NonemptyInterval α\n⊢ (s / t).toProd.1 = (s * t⁻¹).toProd.1",
"case refine_2\nι : Type u_1\nα : Type u_2\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\ns✝ t... | refine NonemptyInterval.ext (Prod.ext ?_ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Order.Interval.Basic | {
"line": 509,
"column": 4
} | {
"line": 509,
"column": 48
} | {
"line": 509,
"column": 49
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\ns✝ t✝ s t : NonemptyInterval α\n⊢ (s * t)⁻¹ = t⁻¹ * s⁻¹",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"HMul.hMul",
"PartialOrder.toPreorder"... | [
"case refine_1\nι : Type u_1\nα : Type u_2\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\ns✝ t✝ s t : NonemptyInterval α\n⊢ (s * t)⁻¹.toProd.1 = (t⁻¹ * s⁻¹).toProd.1",
"case refine_2\nι : Type u_1\nα : Type u_2\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\n... | refine NonemptyInterval.ext (Prod.ext ?_ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 403,
"column": 36
} | {
"line": 404,
"column": 51
} | {
"line": 406,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : Zero Γ\nx : R⟦Γ⟧\nhx : x ≠ 0\n⊢ 0 < x.orderTop ↔ 0 < x.order",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"Iff.mpr",
"HahnSeries.order",
"Preorder.toLT",
... | [] | by
simp_all [orderTop_of_ne_zero hx, order_of_ne hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.HahnSeries.Addition | {
"line": 464,
"column": 4
} | {
"line": 464,
"column": 19
} | {
"line": 466,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nS : Type u_4\nU : Type u_5\nV✝ : Type u_6\nα : Type u_7\ninst✝³ : PartialOrder Γ\nV : Type u_8\ninst✝² : Monoid R\ninst✝¹ : AddMonoid V\ninst✝ : DistribMulAction R V\nx✝³ x✝² : R\nx✝¹ : V⟦Γ⟧\nx✝ : Γ\n⊢ ((x✝³ * x✝²) • x✝¹).coeff x✝ = (x✝³ • x✝² • x✝¹).coeff x✝"... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 394,
"column": 4
} | {
"line": 394,
"column": 87
} | {
"line": 395,
"column": 4
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : PartialOrder R\nΓ' : Type u_3\ninst✝¹ : LinearOrder Γ'\nf : Γ ↪o Γ'\ninst✝ : Zero R\na b : Lex R⟦Γ⟧\n⊢ (∃ i,\n (∀ j < i,\n (ofLex ({ toFun := fun a ↦ toLex (embDomain f (ofLex a)), inj' := ⋯ } a)).coeff j =\n (o... | [
"Γ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : PartialOrder R\nΓ' : Type u_3\ninst✝¹ : LinearOrder Γ'\nf : Γ ↪o Γ'\ninst✝ : Zero R\na b : Lex R⟦Γ⟧\n⊢ (∃ i,\n (∀ j < i, (embDomain f (ofLex a)).coeff j = (embDomain f (ofLex b)).coeff j) ∧\n (embDomain f (ofLex a)).coeff i < (embDomain f... | simp only [Function.Embedding.coeFn_mk, ofLex_toLex, EmbeddingLike.apply_eq_iff_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 294,
"column": 49
} | {
"line": 296,
"column": 21
} | {
"line": 299,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹¹ : DivisionRing K\ninst✝¹⁰ : LinearOrder K\ninst✝⁹ : IsOrderedRing K\ninst✝⁸ : Archimedean K\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : LinearOrder M\ninst✝⁵ : IsOrderedAddMonoid M\ninst✝⁴ : Module K M\ninst✝³ : IsOrderedModule K M\nR : Type u_3\ninst✝² : AddCommGroup R\ninst... | [] | by
apply (seed.strictMono_coeff (f.support.min' hsupport))
simpa using! this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 529,
"column": 6
} | {
"line": 529,
"column": 63
} | {
"line": 530,
"column": 2
} | [
{
"pp": "case neg.inr\nK : Type u\ninst✝ : Field K\nA B : ValuationSubring K\nh : A.unitGroup ≤ B.unitGroup\nx : K\nh_1 : ¬x = 0\nh_2 : ¬1 + x = 0\nhx : A.valuation x = 1\nthis : Units.mk0 x h_1 ∈ B.unitGroup\n⊢ x ∈ B",
"ppTerm": "?neg.inr✝",
"assigned": true,
"usedConstants": [
"Units.val",
... | [] | exact SetLike.coe_mem (B.unitGroupMulEquiv ⟨_, this⟩ : B) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 531,
"column": 4
} | {
"line": 531,
"column": 33
} | {
"line": 532,
"column": 4
} | [
{
"pp": "case mpr\nK : Type u\ninst✝ : Field K\nA B : ValuationSubring K\nh : A ≤ B\nx : Kˣ\nhx : A.valuation ↑x = 1\n⊢ x ∈ B.unitGroup",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Units.val",
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinearOr... | [
"case mpr\nK : Type u\ninst✝ : Field K\nA B : ValuationSubring K\nh : A ≤ B\nx : Kˣ\nhx : (A.mapOfLE B h) (A.valuation ↑x) = (A.mapOfLE B h) 1\n⊢ x ∈ B.unitGroup"
] | apply_fun A.mapOfLE B h at hx | Mathlib.Tactic._aux_Mathlib_Tactic_ApplyFun___elabRules_Mathlib_Tactic_applyFun_1 | Mathlib.Tactic.applyFun |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 349,
"column": 54
} | {
"line": 349,
"column": 65
} | {
"line": 349,
"column": 65
} | [
{
"pp": "K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx y : K\nhx : 0 ≤ mk x\nhy : 0 ≤ mk y\n⊢ stdPart x + stdPart (-y) = stdPart x + -stdPart y",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
... | [
"K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx y : K\nhx : 0 ≤ mk x\nhy : 0 ≤ mk y\n⊢ stdPart x + -stdPart y = stdPart x + -stdPart y",
"K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx y : K\nhx : 0 ≤ mk x\nhy : 0 ≤ mk y\n⊢ 0 ≤ mk (-y)"
] | stdPart_neg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 353,
"column": 47
} | {
"line": 353,
"column": 58
} | {
"line": 353,
"column": 58
} | [
{
"pp": "K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx y : K\nhx : 0 < mk x\n⊢ stdPart (-y) = -stdPart y",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"AddGroupWithOne.toAddGroup",
... | [
"K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx y : K\nhx : 0 < mk x\n⊢ -stdPart y = -stdPart y"
] | stdPart_neg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 427,
"column": 97
} | {
"line": 429,
"column": 34
} | {
"line": 431,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx : K\nf : ℝ →+*o K\nr : ℝ\nhx : 0 ≤ mk x\nh : stdPart x < r\n⊢ x < f r",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"neg_lt_neg_i... | [] | by
rw [← neg_lt_neg_iff, ← map_neg]
apply lt_of_lt_stdPart <;> simpa | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.CoeffList | {
"line": 109,
"column": 4
} | {
"line": 110,
"column": 54
} | {
"line": 112,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : Semiring R\nx : R\nhx : x ≠ 0\nn : ℕ\nh : ¬(monomial n) x = 0\nk : ℕ\nh₁✝¹ : k + 1 < ((monomial n) x).coeffList.length\nh₁✝ : k + 1 < (x :: List.replicate n 0).length\nh₁ : k + 1 < n + 1\nthis : ((monomial n) x).natDegree.succ = n + 1\n⊢ ((monomial n) x).coeffList.get ⟨... | [] | simpa [coeffList, withBotSucc_degree_eq_natDegree_add_one h]
using Polynomial.coeff_monomial_of_ne _ (by lia) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Derivation.MapCoeffs | {
"line": 103,
"column": 2
} | {
"line": 108,
"column": 7
} | {
"line": 110,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Algebra R A\nB : Type u_4\nM' : Type u_5\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra R B\ninst✝⁶ : Algebra A B\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module B M'\ninst✝³ : Module R M'\ninst✝² : Module A M'\ninst✝¹ : IsScalarTower ... | [] | convert! apply_aeval_eq' (d.compAlgebraMap A) d LinearMap.id _ x p
· apply Finsupp.ext
intro x
rfl
· intro a
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Derivation.MapCoeffs | {
"line": 103,
"column": 2
} | {
"line": 108,
"column": 7
} | {
"line": 110,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Algebra R A\nB : Type u_4\nM' : Type u_5\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra R B\ninst✝⁶ : Algebra A B\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module B M'\ninst✝³ : Module R M'\ninst✝² : Module A M'\ninst✝¹ : IsScalarTower ... | [] | convert! apply_aeval_eq' (d.compAlgebraMap A) d LinearMap.id _ x p
· apply Finsupp.ext
intro x
rfl
· intro a
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | {
"line": 89,
"column": 18
} | {
"line": 89,
"column": 26
} | {
"line": 89,
"column": 27
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : Semiring R\np : R[X]\nm : ℕ\nhp : p.IsMonicOfDegree m\nn : ℕ\nih : (p ^ n).IsMonicOfDegree (m * n)\n⊢ (p ^ n * p).IsMonicOfDegree (m * (n + 1))",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
... | [
"case succ\nR : Type u_1\ninst✝ : Semiring R\np : R[X]\nm : ℕ\nhp : p.IsMonicOfDegree m\nn : ℕ\nih : (p ^ n).IsMonicOfDegree (m * n)\n⊢ (p ^ n * p).IsMonicOfDegree (m * n + m * 1)"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | {
"line": 241,
"column": 4
} | {
"line": 243,
"column": 81
} | {
"line": 245,
"column": 0
} | [] | [] | _ ≤ max (C a * X).natDegree (C b).natDegree := natDegree_sub_le ..
_ = (C a * X).natDegree := by simp
_ < 2 := natDegree_C_mul_le .. |>.trans natDegree_X_le |>.trans_lt one_lt_two | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Algebra.Polynomial.DenomsClearable | {
"line": 61,
"column": 47
} | {
"line": 61,
"column": 55
} | {
"line": 61,
"column": 56
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝¹ : Semiring R\ninst✝ : CommSemiring K\ni : R →+* K\na b : R\nN : ℕ\nf g : R[X]\nx✝¹ : DenomsClearable a b N f i\nx✝ : DenomsClearable a b N g i\nDf : R\nbf : K\nbfu : bf * i b = 1\nHf : i Df = i b ^ N * eval (i a * bf) (Polynomial.map i f)\nDg : R\nbg : K\nbgu : bg * i... | [
"R : Type u_1\nK : Type u_2\ninst✝¹ : Semiring R\ninst✝ : CommSemiring K\ni : R →+* K\na b : R\nN : ℕ\nf g : R[X]\nx✝¹ : DenomsClearable a b N f i\nx✝ : DenomsClearable a b N g i\nDf : R\nbf : K\nbfu : bf * i b = 1\nHf : i Df = i b ^ N * eval (i a * bf) (Polynomial.map i f)\nDg : R\nbg : K\nbgu : bg * i b = 1\nHg :... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Mirror | {
"line": 60,
"column": 2
} | {
"line": 65,
"column": 84
} | {
"line": 67,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]\n⊢ p.mirror.natDegree = p.natDegree",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"N... | [] | by_cases hp : p = 0
· rw [hp, mirror_zero]
nontriviality R
rw [mirror, natDegree_mul', reverse_natDegree, natDegree_X_pow,
tsub_add_cancel_of_le p.natTrailingDegree_le_natDegree]
rwa [leadingCoeff_X_pow, mul_one, reverse_leadingCoeff, Ne, trailingCoeff_eq_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Mirror | {
"line": 60,
"column": 2
} | {
"line": 65,
"column": 84
} | {
"line": 67,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]\n⊢ p.mirror.natDegree = p.natDegree",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"N... | [] | by_cases hp : p = 0
· rw [hp, mirror_zero]
nontriviality R
rw [mirror, natDegree_mul', reverse_natDegree, natDegree_X_pow,
tsub_add_cancel_of_le p.natTrailingDegree_le_natDegree]
rwa [leadingCoeff_X_pow, mul_one, reverse_leadingCoeff, Ne, trailingCoeff_eq_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Mirror | {
"line": 170,
"column": 12
} | {
"line": 170,
"column": 21
} | {
"line": 170,
"column": 22
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : Semiring R\np q : R[X]\ninst✝ : NoZeroDivisors R\nhp : ¬p = 0\nhq : q = 0\n⊢ (p * 0).mirror = p.mirror * mirror 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
... | [
"case pos\nR : Type u_1\ninst✝¹ : Semiring R\np q : R[X]\ninst✝ : NoZeroDivisors R\nhp : ¬p = 0\nhq : q = 0\n⊢ mirror 0 = p.mirror * mirror 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 113,
"column": 63
} | {
"line": 113,
"column": 76
} | {
"line": 114,
"column": 8
} | [
{
"pp": "case succ.refine_2\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nf g : R[X]\nhg : g.Monic\nn : ℕ\nq : R[X]\nr : Fin n → R[X]\nhr : ∀ (i : Fin n), (r i).degree < g.degree\nhf : f = q * g ^ n + ∑ i, r i * g ^ ↑i\n⊢ q * g ^ n + ∑ i, r i * g ^ ↑i =\n q %ₘ g * g ^ ↑(Fin.last n) + (q /ₘ g * g ... | [
"case succ.refine_2\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nf g : R[X]\nhg : g.Monic\nn : ℕ\nq : R[X]\nr : Fin n → R[X]\nhr : ∀ (i : Fin n), (r i).degree < g.degree\nhf : f = q * g ^ n + ∑ i, r i * g ^ ↑i\n⊢ q * g ^ n + ∑ i, r i * g ^ ↑i =\n q %ₘ g * g ^ n + (q /ₘ g * g ^ (n + 1) + ∑ i, Fin.sno... | Fin.val_last, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 128,
"column": 22
} | {
"line": 128,
"column": 36
} | {
"line": 128,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ng : R[X]\nhg : g.Monic\nq₁ q₂ : R[X]\nr₁ r₂ : Fin 0 → R[X]\nhr₁ : ∀ (i : Fin 0), (r₁ i).degree < g.degree\nhr₂ : ∀ (i : Fin 0), (r₂ i).degree < g.degree\nhf : q₁ * g ^ 0 + ∑ i, r₁ i * g ^ ↑i = q₂ * g ^ 0 + ∑ i, r₂ i * g ^ ↑i\n⊢ q₁ = q₂",
"ppTerm": "?m.92",
"ass... | [] | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 128,
"column": 22
} | {
"line": 128,
"column": 36
} | {
"line": 128,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ng : R[X]\nhg : g.Monic\nq₁ q₂ : R[X]\nr₁ r₂ : Fin 0 → R[X]\nhr₁ : ∀ (i : Fin 0), (r₁ i).degree < g.degree\nhr₂ : ∀ (i : Fin 0), (r₂ i).degree < g.degree\nhf : q₁ * g ^ 0 + ∑ i, r₁ i * g ^ ↑i = q₂ * g ^ 0 + ∑ i, r₂ i * g ^ ↑i\n⊢ q₁ = q₂",
"ppTerm": "?m.92",
"ass... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 128,
"column": 22
} | {
"line": 128,
"column": 36
} | {
"line": 128,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ng : R[X]\nhg : g.Monic\nq₁ q₂ : R[X]\nr₁ r₂ : Fin 0 → R[X]\nhr₁ : ∀ (i : Fin 0), (r₁ i).degree < g.degree\nhr₂ : ∀ (i : Fin 0), (r₂ i).degree < g.degree\nhf : q₁ * g ^ 0 + ∑ i, r₁ i * g ^ ↑i = q₂ * g ^ 0 + ∑ i, r₂ i * g ^ ↑i\n⊢ q₁ = q₂",
"ppTerm": "?m.92",
"ass... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 204,
"column": 78
} | {
"line": 204,
"column": 87
} | {
"line": 204,
"column": 88
} | [
{
"pp": "case monomial.succ\nR : Type u_1\ninst✝⁴ : Semiring R\np : R[X]\nS : Type u_2\ninst✝³ : NonAssocSemiring S\ninst✝² : Module R S\ninst✝¹ : Pow S ℕ\ninst✝ : NatPowAssoc S\na : R\nn : ℕ\n⊢ a • (0 ^ n * 0) = 0 • 1",
"ppTerm": "?monomial.succ",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"case monomial.succ\nR : Type u_1\ninst✝⁴ : Semiring R\np : R[X]\nS : Type u_2\ninst✝³ : NonAssocSemiring S\ninst✝² : Module R S\ninst✝¹ : Pow S ℕ\ninst✝ : NatPowAssoc S\na : R\nn : ℕ\n⊢ a • 0 = 0 • 1"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 207,
"column": 13
} | {
"line": 207,
"column": 27
} | {
"line": 208,
"column": 2
} | [
{
"pp": "case empty\nR : Type u_1\ninst✝¹ : CommRing R\nι : Type u_2\ninst✝ : DecidableEq ι\ng : ι → R[X]\nq₁ q₂ : R[X]\nr₁ r₂ : ι → R[X]\nhg : ∀ i ∈ ∅, (g i).Monic\nhgg : (↑∅).Pairwise fun i j ↦ IsCoprime (g i) (g j)\nhr₁ : ∀ i ∈ ∅, (r₁ i).degree < (g i).degree\nhr₂ : ∀ i ∈ ∅, (r₂ i).degree < (g i).degree\nhf ... | [] | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 207,
"column": 13
} | {
"line": 207,
"column": 27
} | {
"line": 208,
"column": 2
} | [
{
"pp": "case empty\nR : Type u_1\ninst✝¹ : CommRing R\nι : Type u_2\ninst✝ : DecidableEq ι\ng : ι → R[X]\nq₁ q₂ : R[X]\nr₁ r₂ : ι → R[X]\nhg : ∀ i ∈ ∅, (g i).Monic\nhgg : (↑∅).Pairwise fun i j ↦ IsCoprime (g i) (g j)\nhr₁ : ∀ i ∈ ∅, (r₁ i).degree < (g i).degree\nhr₂ : ∀ i ∈ ∅, (r₂ i).degree < (g i).degree\nhf ... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 207,
"column": 13
} | {
"line": 207,
"column": 27
} | {
"line": 208,
"column": 2
} | [
{
"pp": "case empty\nR : Type u_1\ninst✝¹ : CommRing R\nι : Type u_2\ninst✝ : DecidableEq ι\ng : ι → R[X]\nq₁ q₂ : R[X]\nr₁ r₂ : ι → R[X]\nhg : ∀ i ∈ ∅, (g i).Monic\nhgg : (↑∅).Pairwise fun i j ↦ IsCoprime (g i) (g j)\nhr₁ : ∀ i ∈ ∅, (r₁ i).degree < (g i).degree\nhr₂ : ∀ i ∈ ∅, (r₂ i).degree < (g i).degree\nhf ... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 175,
"column": 11
} | {
"line": 175,
"column": 20
} | {
"line": 175,
"column": 21
} | [
{
"pp": "k m n : ℕ\nhkm : k < m\nhmn : m < n\nx y z : ℤ\nhp : #{k, m, n} = 3 ∧ ∀ k_1 ∈ {k, m, n}, IsUnit ((C x * X ^ k + C y * X ^ m + C z * X ^ n).coeff k_1)\nhx : IsUnit (x + y * 0 + z * 0)\nhy : IsUnit (x * 0 + y + z * 0)\nhz : IsUnit (x * 0 + y * 0 + z)\n⊢ (C x * X ^ k + C y * X ^ m + C z * X ^ n).IsUnitTri... | [
"k m n : ℕ\nhkm : k < m\nhmn : m < n\nx y z : ℤ\nhp : #{k, m, n} = 3 ∧ ∀ k_1 ∈ {k, m, n}, IsUnit ((C x * X ^ k + C y * X ^ m + C z * X ^ n).coeff k_1)\nhx : IsUnit (x + 0 + 0)\nhy : IsUnit (0 + y + 0)\nhz : IsUnit (0 + 0 + z)\n⊢ (C x * X ^ k + C y * X ^ m + C z * X ^ n).IsUnitTrinomial"
] | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 188,
"column": 62
} | {
"line": 188,
"column": 95
} | {
"line": 189,
"column": 6
} | [
{
"pp": "case refine_1\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\n⊢ (trinomial k m n ↑u ↑v ↑w).coeff k ^ 2 +\n ((trinomial k m n ↑u ↑v ↑w).coeff m ^ 2 + (trinomial k m n ↑u ↑v ↑w).coeff n ^ 2) =\n 3",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Units.val",
... | [
"case refine_1\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\n⊢ (trinomial k m n ↑u ↑v ↑w).coeff k ^ 2 + ((trinomial k m n ↑u ↑v ↑w).coeff m ^ 2 + ↑w ^ 2) = 3"
] | trinomial_leading_coeff' hkm hmn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 212,
"column": 60
} | {
"line": 212,
"column": 68
} | {
"line": 212,
"column": 69
} | [
{
"pp": "p : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ C ↑v * ((monomial (m + n)) ↑u + (monomial (n - m + k + n)) ↑w) =\n {\n toFinsupp :=\n Finsupp.filter (fun x ↦ x ∈ Set.Ioo (k + n) (n + n))\n ((monomial k) ↑u * ((... | [
"p : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ C ↑v * (monomial (m + n)) ↑u + C ↑v * (monomial (n - m + k + n)) ↑w =\n {\n toFinsupp :=\n Finsupp.filter (fun x ↦ x ∈ Set.Ioo (k + n) (n + n))\n ((monomial k) ↑u * (monomial... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 227,
"column": 37
} | {
"line": 227,
"column": 45
} | {
"line": 227,
"column": 46
} | [
{
"pp": "case cons\nR : Type u_1\ninst✝¹ : CommRing R\nι : Type u_2\ninst✝ : DecidableEq ι\ng : ι → R[X]\nq₁ q₂ : R[X]\nr₁ r₂ : ι → R[X]\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n (∀ i ∈ s, (g i).Monic) →\n ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →\n (∀ i ∈ s, (r₁ i).degree < (g i).degree) →\n ... | [
"case cons\nR : Type u_1\ninst✝¹ : CommRing R\nι : Type u_2\ninst✝ : DecidableEq ι\ng : ι → R[X]\nq₁ q₂ : R[X]\nr₁ r₂ : ι → R[X]\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n (∀ i ∈ s, (g i).Monic) →\n ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →\n (∀ i ∈ s, (r₁ i).degree < (g i).degree) →\n (∀ i ∈ ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.SumIteratedDerivative | {
"line": 236,
"column": 14
} | {
"line": 236,
"column": 23
} | {
"line": 236,
"column": 24
} | [
{
"pp": "case hq\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_3\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Nontrivial A\ninst✝ : NoZeroDivisors A\np : R[X]\nq : ℕ\nhq : 0 < q\ninj_amap : Function.Injective ⇑(algebraMap R A)\np0 : p ≠ 0\nc : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choose else 0\... | [
"case hq\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_3\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Nontrivial A\ninst✝ : NoZeroDivisors A\np : R[X]\nq : ℕ\nhq : 0 < q\ninj_amap : Function.Injective ⇑(algebraMap R A)\np0 : p ≠ 0\nc : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choose else 0\nc_le : ∀ (k... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.QuadraticAlgebra.Defs | {
"line": 261,
"column": 31
} | {
"line": 261,
"column": 46
} | {
"line": 263,
"column": 0
} | [
{
"pp": "case re\nR : Type u_1\nS : Type u_2\nT : Type u_3\na b r : R\nx y : QuadraticAlgebra R a b\ninst✝¹ : Monoid S\ninst✝ : MulAction S R\nx✝² x✝¹ : S\nx✝ : QuadraticAlgebra R a b\n⊢ ((x✝² * x✝¹) • x✝).re = (x✝² • x✝¹ • x✝).re",
"ppTerm": "?re",
"assigned": true,
"usedConstants": [
"Quadra... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.QuadraticAlgebra.Defs | {
"line": 261,
"column": 31
} | {
"line": 261,
"column": 46
} | {
"line": 263,
"column": 0
} | [
{
"pp": "case im\nR : Type u_1\nS : Type u_2\nT : Type u_3\na b r : R\nx y : QuadraticAlgebra R a b\ninst✝¹ : Monoid S\ninst✝ : MulAction S R\nx✝² x✝¹ : S\nx✝ : QuadraticAlgebra R a b\n⊢ ((x✝² * x✝¹) • x✝).im = (x✝² • x✝¹ • x✝).im",
"ppTerm": "?im",
"assigned": true,
"usedConstants": [
"Quadra... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 91,
"column": 31
} | {
"line": 93,
"column": 32
} | {
"line": 95,
"column": 0
} | [
{
"pp": "R : Type u_2\na b : R\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf g : QuadraticAlgebra R a b →ₐ[R] A\nh : f ω = g ω\n⊢ f = g",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"QuadraticAlgebra... | [] | by
ext ⟨x, y⟩
simp [mk_eq_add_smul_omega, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 251,
"column": 18
} | {
"line": 251,
"column": 27
} | {
"line": 251,
"column": 28
} | [
{
"pp": "case cons\nR : Type u_1\ninst✝¹ : CommRing R\nι : Type u_2\ninst✝ : DecidableEq ι\ng : ι → R[X]\nq₁ q₂ : R[X]\nr₁ r₂ : ι → R[X]\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n (∀ i ∈ s, (g i).Monic) →\n ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →\n (∀ i ∈ s, (r₁ i).degree < (g i).degree) →\n ... | [
"case cons\nR : Type u_1\ninst✝¹ : CommRing R\nι : Type u_2\ninst✝ : DecidableEq ι\ng : ι → R[X]\nq₁ q₂ : R[X]\nr₁ r₂ : ι → R[X]\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n (∀ i ∈ s, (g i).Monic) →\n ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →\n (∀ i ∈ s, (r₁ i).degree < (g i).degree) →\n (∀ i ∈ ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 61
} | {
"line": 248,
"column": 4
} | [
{
"pp": "case h.left\nR : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nP : R[X]\nη : R\nh₁ : 0 < P.leadingCoeff\nh₃ : P ≠ 0\nh₄ : P.eraseLead.natDegree + 1 = P.natDegree\nh₅ : X - C η ≠ 0\nh₆ : P.eraseLead ≠ 0\nd : ℕ\nhd : P.natDegree = 0 + d + 1\nh₂ : P.eraseLead.leadingCoe... | [
"case h.left\nR : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nP : R[X]\nη : R\nh₁ : 0 < P.leadingCoeff\nh₃ : P ≠ 0\nh₄ : P.eraseLead.natDegree + 1 = P.natDegree\nh₅ : X - C η ≠ 0\nh₆ : P.eraseLead ≠ 0\nd : ℕ\nhd : P.natDegree = 0 + d + 1\nh₂ : P.eraseLead.leadingCoeff = P.nextC... | have := leadingCoeff_monic_mul (q := P) (monic_X_sub_C η) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.QuaternionBasis | {
"line": 127,
"column": 20
} | {
"line": 127,
"column": 28
} | {
"line": 127,
"column": 29
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nc₁ c₂ c₃ : R\nq : Basis A c₁ c₂ c₃\nx y : ℍ[R,c₁,c₂,c₃]\n⊢ (x * y).re • 1 + (x * y).imI • q.i + (x * y).imJ • q.j + (x * y).imK • q.k =\n x.re • 1 * (y.re • 1 + y.imI • q.i + y.imJ • q.j + y.imK • q.k) +\n ... | [
"R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nc₁ c₂ c₃ : R\nq : Basis A c₁ c₂ c₃\nx y : ℍ[R,c₁,c₂,c₃]\n⊢ (x * y).re • 1 + (x * y).imI • q.i + (x * y).imJ • q.j + (x * y).imK • q.k =\n x.re • 1 * y.re • 1 + x.re • 1 * y.imI • q.i + x.re • 1 * y.imJ • q.j + x.re • 1 * y.im... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 388,
"column": 2
} | {
"line": 388,
"column": 68
} | {
"line": 390,
"column": 2
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nn✝ : ℕ\nih : ∀ (P : R[X]), Multiset.countP (fun x ↦ 0 < x) P.roots = n✝ → n✝ ≤ P.signVariations\nP : R[X]\nh : Multiset.countP (fun x ↦ 0 < x) P.roots = n✝ + 1\nhp : P ≠ 0\nη : R\nη_root : η ∈ P.roots\n... | [
"case succ\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nn✝ : ℕ\nih : ∀ (P : R[X]), Multiset.countP (fun x ↦ 0 < x) P.roots = n✝ → n✝ ≤ P.signVariations\nη : R\nη_pos : 0 < η\nQ : R[X]\nh : Multiset.countP (fun x ↦ 0 < x) ((X - C η) * Q).roots = n✝ + 1\nhp : (X - C η) * ... | obtain ⟨Q, rfl⟩ := dvd_iff_isRoot.mpr (isRoot_of_mem_roots η_root) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Ring.WithZero | {
"line": 25,
"column": 4
} | {
"line": 25,
"column": 45
} | {
"line": 27,
"column": 0
} | [
{
"pp": "case coe.coe.coe\nα : Type u_1\ninst✝² : Mul α\ninst✝¹ : Add α\ninst✝ : LeftDistribClass α\na✝² a✝¹ a✝ : α\n⊢ ↑a✝² * (↑a✝¹ + ↑a✝) = ↑a✝² * ↑a✝¹ + ↑a✝² * ↑a✝",
"ppTerm": "?coe.coe.coe",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Option.some",
"congr_arg",
"ins... | [] | exact congr_arg some (left_distrib _ _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.SkewMonoidAlgebra.Single | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 27
} | {
"line": 61,
"column": 2
} | [
{
"pp": "M : Type u_4\nα : Type u_5\ninst✝ : AddCommMonoid M\na : α\nf : SkewMonoidAlgebra M α\n⊢ single a (f.coeff a) + (erase a) f = f",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddZeroClass",
"SkewMonoidAlgebra.instAddMonoid",
"AddZeroClass.toAddZero"... | [
"M : Type u_4\nα : Type u_5\ninst✝ : AddCommMonoid M\na : α\nf : SkewMonoidAlgebra M α\n⊢ (single a (f.coeff a) + (erase a) f).toFinsupp = f.toFinsupp"
] | apply toFinsupp_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.SkewMonoidAlgebra.Lift | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 27
} | {
"line": 125,
"column": 2
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝ : AddCommMonoid k\nf : G ≃ H\nl : SkewMonoidAlgebra k G\n⊢ equivMapDomain f l = (mapDomain ⇑f) l",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"SkewMonoidAlgebra.mapDomain",
"SkewMonoidAlgebra.equivMapDomain",
"Eq... | [
"k : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝ : AddCommMonoid k\nf : G ≃ H\nl : SkewMonoidAlgebra k G\n⊢ (equivMapDomain f l).toFinsupp = ((mapDomain ⇑f) l).toFinsupp"
] | apply toFinsupp_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.SkewMonoidAlgebra.Single | {
"line": 71,
"column": 4
} | {
"line": 78,
"column": 47
} | {
"line": 80,
"column": 0
} | [
{
"pp": "M : Type u_4\nα : Type u_5\ninst✝ : AddCommMonoid M\np : SkewMonoidAlgebra M α → Prop\nf✝ : SkewMonoidAlgebra M α\nh0 : p 0\nha : ∀ (a : α) (b : M) (f : SkewMonoidAlgebra M α), a ∉ f.support → b ≠ 0 → p f → p (single a b + f)\ns✝ : Finset α\na : α\ns : Finset α\nhas : a ∉ s\nih : ∀ (f : SkewMonoidAlgeb... | [] | suffices p (single a (f.coeff a) + f.erase a) by rwa [single_add_erase] at this
classical
apply ha
· rw [support_erase, Finset.mem_erase]
exact fun H ↦ H.1 rfl
· simp only [← mem_support_iff, hf, Finset.mem_cons_self]
· apply ih
rw [support_erase, hf, Finset.erase_cons] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.SkewMonoidAlgebra.Single | {
"line": 71,
"column": 4
} | {
"line": 78,
"column": 47
} | {
"line": 80,
"column": 0
} | [
{
"pp": "M : Type u_4\nα : Type u_5\ninst✝ : AddCommMonoid M\np : SkewMonoidAlgebra M α → Prop\nf✝ : SkewMonoidAlgebra M α\nh0 : p 0\nha : ∀ (a : α) (b : M) (f : SkewMonoidAlgebra M α), a ∉ f.support → b ≠ 0 → p f → p (single a b + f)\ns✝ : Finset α\na : α\ns : Finset α\nhas : a ∉ s\nih : ∀ (f : SkewMonoidAlgeb... | [] | suffices p (single a (f.coeff a) + f.erase a) by rwa [single_add_erase] at this
classical
apply ha
· rw [support_erase, Finset.mem_erase]
exact fun H ↦ H.1 rfl
· simp only [← mem_support_iff, hf, Finset.mem_cons_self]
· apply ih
rw [support_erase, hf, Finset.erase_cons] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.SkewMonoidAlgebra.Lift | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 27
} | {
"line": 143,
"column": 2
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝ : AddCommMonoid k\nf : G ≃ H\na : G\nb : k\n⊢ equivMapDomain f (single a b) = single (f a) b",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"SkewMonoidAlgebra.equivMapDomain",
"Equiv.instEquivLike",
"Equiv",
... | [
"k : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝ : AddCommMonoid k\nf : G ≃ H\na : G\nb : k\n⊢ (equivMapDomain f (single a b)).toFinsupp = (single (f a) b).toFinsupp"
] | apply toFinsupp_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.SkewMonoidAlgebra.Single | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 27
} | {
"line": 97,
"column": 2
} | [
{
"pp": "M : Type u_4\nα : Type u_5\ninst✝ : AddCommMonoid M\na : α\nf : α →₀ M\n⊢ { toFinsupp := f }.update a ({ toFinsupp := f }.coeff a) = { toFinsupp := f }",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAddZero",
"SkewMono... | [
"M : Type u_4\nα : Type u_5\ninst✝ : AddCommMonoid M\na : α\nf : α →₀ M\n⊢ ({ toFinsupp := f }.update a ({ toFinsupp := f }.coeff a)).toFinsupp = { toFinsupp := f }.toFinsupp"
] | apply toFinsupp_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 419,
"column": 56
} | {
"line": 419,
"column": 65
} | {
"line": 419,
"column": 66
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\np q : SkewPolynomial R\ninst✝ : MulSemiringAction (Multiplicative ℕ) R\nh : p ≠ q\nh01 : 0 = 1\n⊢ p * 0 = q * 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Multiplicative... | [
"R : Type u_1\ninst✝¹ : Semiring R\np q : SkewPolynomial R\ninst✝ : MulSemiringAction (Multiplicative ℕ) R\nh : p ≠ q\nh01 : 0 = 1\n⊢ 0 = q * 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 612,
"column": 2
} | {
"line": 612,
"column": 64
} | {
"line": 614,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np : SkewPolynomial R\nn : ℕ\n⊢ (-p).coeff n = -p.coeff n",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"AddGroupWithOne.toAddGro... | [] | simp [← add_eq_zero_iff_eq_neg, ← coeff_add, neg_add_cancel p] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 612,
"column": 2
} | {
"line": 612,
"column": 64
} | {
"line": 614,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np : SkewPolynomial R\nn : ℕ\n⊢ (-p).coeff n = -p.coeff n",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"AddGroupWithOne.toAddGro... | [] | simp [← add_eq_zero_iff_eq_neg, ← coeff_add, neg_add_cancel p] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 612,
"column": 2
} | {
"line": 612,
"column": 64
} | {
"line": 614,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np : SkewPolynomial R\nn : ℕ\n⊢ (-p).coeff n = -p.coeff n",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"AddGroupWithOne.toAddGro... | [] | simp [← add_eq_zero_iff_eq_neg, ← coeff_add, neg_add_cancel p] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 718,
"column": 2
} | {
"line": 719,
"column": 38
} | {
"line": 721,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : SkewPolynomial R\nn : ℕ\n⊢ p.update n 0 = erase n p",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"SkewPolynomial.erase",
"id",
"SkewPolynomial.coeff_update_apply",
"SkewPolynomial.ext... | [] | ext
rw [coeff_update_apply, coeff_erase] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 718,
"column": 2
} | {
"line": 719,
"column": 38
} | {
"line": 721,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : SkewPolynomial R\nn : ℕ\n⊢ p.update n 0 = erase n p",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"SkewPolynomial.erase",
"id",
"SkewPolynomial.coeff_update_apply",
"SkewPolynomial.ext... | [] | ext
rw [coeff_update_apply, coeff_erase] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 27
} | {
"line": 387,
"column": 27
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝¹ : AddCommMonoid k\nk' : Type u_3\nG' : Type u_4\ninst✝ : AddCommMonoid k'\nf : G →₀ k\ng : G → k → G' →₀ k'\n⊢ { toFinsupp := f.sum g } = { toFinsupp := f }.sum fun x1 x2 ↦ { toFinsupp := g x1 x2 }",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
... | [
"k : Type u_1\nG : Type u_2\ninst✝¹ : AddCommMonoid k\nk' : Type u_3\nG' : Type u_4\ninst✝ : AddCommMonoid k'\nf : G →₀ k\ng : G → k → G' →₀ k'\n⊢ { toFinsupp := f.sum g }.toFinsupp = ({ toFinsupp := f }.sum fun x1 x2 ↦ { toFinsupp := g x1 x2 }).toFinsupp"
] | apply toFinsupp_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
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