module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.MonoidAlgebra.Grading | {
"line": 175,
"column": 6
} | {
"line": 177,
"column": 70
} | [
{
"pp": "M : Type u_1\nι : Type u_2\nR : Type u_3\ninst✝³ : AddMonoid M\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : CommSemiring R\nf : M →+ ι\n⊢ (coeAlgHom (gradeBy R ⇑f)).comp (decomposeAux f) = AlgHom.id R R[M]",
"usedConstants": [
"Subtype.coe_mk",
"Eq.mpr",
"NonAssocSemirin... | ext : 4
dsimp
rw [decomposeAux_single, DirectSum.coeAlgHom_of, Subtype.coe_mk] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 60
} | [
{
"pp": "case neg.hbc\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nh✝ : Nontrivial ↥L\nI : Type u_1 := Free.ChooseBasisIndex ℤ ↥L\nthis : Fintype I\nb : Basis I ℤ ↥L := Free.chooseBasis ℤ ↥L\nd : ℕ := Fint... | refine Real.self_le_rpow_of_one_le (not_le.mp hA').le ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.Nat.Factorial.DoubleFactorial | {
"line": 79,
"column": 4
} | {
"line": 81,
"column": 7
} | [
{
"pp": "n : ℕ\n⊢ (2 * (n + 1) + 1)‼ = ∏ i ∈ Finset.range (n + 1), (2 * (i + 1) + 1)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormNum.isNat_add",
"Mathlib.Tactic.Ring.Common.mul_congr",
"HMul.hMul",
"Mathlib.Tactic.Ring.Commo... | rw [Finset.prod_range_succ, ← doubleFactorial_eq_prod_odd _, mul_comm (2 * n + 1)‼,
(by ring : 2 * (n + 1) + 1 = 2 * n + 1 + 2)]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Factorial.DoubleFactorial | {
"line": 79,
"column": 4
} | {
"line": 81,
"column": 7
} | [
{
"pp": "n : ℕ\n⊢ (2 * (n + 1) + 1)‼ = ∏ i ∈ Finset.range (n + 1), (2 * (i + 1) + 1)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormNum.isNat_add",
"Mathlib.Tactic.Ring.Common.mul_congr",
"HMul.hMul",
"Mathlib.Tactic.Ring.Commo... | rw [Finset.prod_range_succ, ← doubleFactorial_eq_prod_odd _, mul_comm (2 * n + 1)‼,
(by ring : 2 * (n + 1) + 1 = 2 * n + 1 + 2)]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Comap | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 30
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nυ : Type u_3\nR : Type u_4\ninst✝ : CommSemiring R\nf : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R\nx : τ → R\n⊢ comap ((↑f).comp ↑f.symm) x = x",
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"AlgEquiv.toAlgHom",
"AlgEquiv.... | apply comap_eq_id_of_eq_id | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.MvPolynomial.Comap | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 30
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nυ : Type u_3\nR : Type u_4\ninst✝ : CommSemiring R\nf : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R\nx : σ → R\n⊢ comap ((↑f.symm).comp ↑f) x = x",
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"AlgEquiv.toAlgHom",
"AlgEquiv.... | apply comap_eq_id_of_eq_id | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.MvPolynomial.Division | {
"line": 203,
"column": 18
} | {
"line": 203,
"column": 32
} | [
{
"pp": "case mpr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr : R\ni j : σ →₀ ℕ\nd : R\nh : r * d = 0\n⊢ (monomial i) r ∣ (monomial j) 0",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Dvd.dvd",
"Semiring.toModule",
"Add... | monomial_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.MvPolynomial.Division | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 42
} | [
{
"pp": "case mpr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr : R\ni j : σ →₀ ℕ\nd : R\nh : r * d = 0\n⊢ (monomial i) r ∣ (monomial j) (r * d)",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Semigroup.toMul",
"Dvd.dvd",
... | · simp_rw [h, monomial_zero, dvd_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 445,
"column": 8
} | {
"line": 445,
"column": 32
} | [
{
"pp": "x✝ : ℕ\nl : List ℕ\nsuccEmb : ℕ ↪ ℕ := addRightEmbedding 1\nthis :\n (Finsupp.single 0 x✝ + Finsupp.embDomain succEmb l.toFinsupp).update 0 0 =\n (Finsupp.embDomain succEmb l.toFinsupp).update 0 0\nx : ℕ\n⊢ (Finsupp.embDomain succEmb l.toFinsupp) (x + 1) = l[x]?.getD 0",
"usedConstants": [
... | Finsupp.embDomain_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.GameAdd | {
"line": 84,
"column": 20
} | {
"line": 84,
"column": 22
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nrα : α → α → Prop\nrβ : β → β → Prop\nx✝¹ x✝ : α × β\nh : RProd rα rβ x✝¹ x✝\na₁✝ : α\nb₁✝ : β\na₂✝ : α\nb₂✝ : β\n⊢ rα a₁✝ a₂✝ → rβ b₁✝ b₂✝ → Relation.TransGen (GameAdd rα rβ) (a₁✝, b₁✝) (a₂✝, b₂✝)",
"usedConstants": []
}
] | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Data.DFinsupp.WellFounded | {
"line": 124,
"column": 10
} | {
"line": 124,
"column": 31
} | [
{
"pp": "case empty\nι : Type u_1\nα : ι → Type u_2\ninst✝² : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\nhbot : ∀ ⦃i : ι⦄ ⦃a : α i⦄, ¬s i a 0\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → (x : α i) → Decidable (x ≠ 0)\nx : Π₀ (i : ι), α i\nht : x.support = ∅\n⊢ (∀ i ∈ ∅, Acc (DFinsupp.... | support_eq_empty.1 ht | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finsupp.MonomialOrder.DegLex | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 63
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\nx : DegLex (α →₀ ℕ)\n⊢ toDegLex 0 ≤ x",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidHom.instAddMonoidHomClass",
"Equiv.instEquivLike",
"congrArg",
"Lex",
"AddMonoid.to... | simp only [le_iff, ofDegLex_toDegLex, toLex_zero, map_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 159,
"column": 4
} | {
"line": 159,
"column": 9
} | [
{
"pp": "case inl\nR : Type u_1\nσ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\np : MvPolynomial σ R\nn : σ →₀ ℕ\nhR : Subsingleton R\n⊢ ∃ m, m ≤ n",
"usedConstants": [
"Finsupp.instLE",
"le_refl",
"Nat.instMulZeroClass",
"LE.le",
"instLENat",
"Nat.instPreor... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Data.Nat.Factorization.PrimePow | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 22
} | [
{
"pp": "case mp\np k : ℕ\nhp : Nat.Prime p\nhk : 0 < k\n⊢ ∀ (y : ℕ), (fun p_1 ↦ Nat.Prime p_1 ∧ p_1 ∣ p ^ k) y → y = p",
"usedConstants": [
"Nat"
]
}
] | rintro q ⟨hq, hq'⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 765,
"column": 2
} | {
"line": 765,
"column": 34
} | [
{
"pp": "case inr\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np q : MvPolynomial σ R\nthis :\n ∀ {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [inst : CommSemiring R] [NoZeroDivisors R]\n (p q : MvPolynomial σ R), p ≠ 0 ∧ q ≠ 0 → m.degree (m.lea... | · obtain rfl | rfl := h <;> simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Nat.Factorization.PrimePow | {
"line": 156,
"column": 27
} | {
"line": 159,
"column": 83
} | [
{
"pp": "x✝ : Primes × ℕ\np : Primes\nk : ℕ\n⊢ (fun n ↦ (⟨(↑n).minFac, ⋯⟩, (↑n).factorization (↑n).minFac - 1)) ((fun pk ↦ ⟨↑pk.1 ^ (pk.2 + 1), ⋯⟩) (p, k)) = (p, k)",
"usedConstants": [
"Finsupp.instFunLike",
"Subtype.mk.congr_simp",
"Nat.instMulZeroClass",
"Nat.instOrderedSub",
... | by
simp only [p.prop.pow_minFac k.add_one_ne_zero, Subtype.coe_eta, factorization_pow, p.prop,
Prime.factorization, Finsupp.smul_single, smul_eq_mul, mul_one, Finsupp.single_add,
Finsupp.coe_add, Pi.add_apply, Finsupp.single_eq_same, add_tsub_cancel_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nf : ℕ → R\nx y : ℕ\nhx : ¬x = 0\nhy : ¬y = 0\nhxy : x.Coprime y\nhxy₀ : x * y ≠ 0\n⊢ ∏ p ∈ (x * y).primeFactors, f p = (∏ p ∈ x.primeFactors, f p) * ∏ p ∈ y.primeFactors, f p",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
... | primeFactors_mul hx hy, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 127,
"column": 40
} | {
"line": 129,
"column": 6
} | [
{
"pp": "⊢ pow 0 = ζ",
"usedConstants": [
"MulOne.toOne",
"Nat.instMulZeroClass",
"ArithmeticFunction.ext",
"ArithmeticFunction.instFunLikeNat",
"Monoid.toMulOneClass",
"congrArg",
"ArithmeticFunction.zeta",
"Nat.instMonoid",
"instOfNatNat",
"pow_z... | by
ext n
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 131,
"column": 39
} | {
"line": 133,
"column": 6
} | [
{
"pp": "⊢ pow 1 = ArithmeticFunction.id",
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"ArithmeticFunction.ext",
"ArithmeticFunction.instFunLikeNat",
"Nat.instOne",
"congrArg",
"Nat.instMonoid",
"false_and",
"one_ne_zero._simp_1",
"instOfNa... | by
ext n
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 242,
"column": 49
} | {
"line": 244,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf g : ArithmeticFunction ℕ\n⊢ ↑(f * g) = ↑f * ↑g",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroClass",
"ArithmeticFunction.ext",
"HMul.hMul",
"Nat.divisorsAntidiagonal",
"ArithmeticFunction.instF... | by
ext n
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 248,
"column": 50
} | {
"line": 250,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nf g : ArithmeticFunction ℤ\n⊢ ↑(f * g) = ↑f * ↑g",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Int.instAddCommMonoid",
"ArithmeticFunction.ext",
"HMul.hMul",
"Nat.divisorsAntidiagonal",
"ArithmeticFunction.i... | by
ext n
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 378,
"column": 29
} | {
"line": 378,
"column": 53
} | [
{
"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 (f a * ∏ x ∈ s, f x) = ω (f a) + ∑ x ∈ s, ω (f x)",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZe... | cardDistinctFactors_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Antidiag.Nat | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 26
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\nb : ℕ × ℕ\nhb : b ∈ {x ∈ n.divisors ×ˢ n.divisors | x.1.lcm x.2 = n}\n⊢ ∃ a, ∃ (ha : a ∈ finMulAntidiag 3 n), f a ha = b",
"usedConstants": [
"Nat.gcd",
"Prod.fst",
"Nat",
"Prod.snd"
]
}
] | let g := b.fst.gcd b.snd | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Order.Archimedean.IndicatorCard | {
"line": 75,
"column": 12
} | {
"line": 75,
"column": 36
} | [
{
"pp": "α : Type u_1\nR : Type u_2\ninst✝⁴ : AddCommMonoid R\ninst✝³ : PartialOrder R\ninst✝² : IsOrderedAddMonoid R\ninst✝¹ : AddLeftStrictMono R\ninst✝ : Archimedean R\nr : R\nh : 0 < r\ns : ℕ → Set α\n⊢ limsup s atTop = {ω | Tendsto (fun n ↦ ∑ k ∈ Finset.range n, (s k).indicator (fun x ↦ r) ω) atTop atTop}"... | ← Nat.cofinite_eq_atTop, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Rearrangement | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 34
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁸ : Semiring α\ninst✝⁷ : LinearOrder α\ninst✝⁶ : IsStrictOrderedRing α\ninst✝⁵ : ExistsAddOfLE α\ninst✝⁴ : AddCommMonoid β\ninst✝³ : LinearOrder β\ninst✝² : IsOrderedCancelAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulStrictMono α β\ns : Finset ι\nσ : P... | obtain rfl | hxy := eq_or_ne x y | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 133,
"column": 20
} | {
"line": 137,
"column": 42
} | [
{
"pp": "M✝ : Type u_1\ninst✝⁴ : Group M✝\ninst✝³ : Lattice M✝\nM : Type u_2\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na✝ b✝ : M\na b c : MulArchimedeanOrder M\n⊢ a ≤ b → b ≤ c → a ≤ c",
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"HMul.hMul",
... | by
intro ⟨m, hm⟩ ⟨n, hn⟩
use m * n
rw [pow_mul]
exact hn.trans (pow_le_pow_left' hm n) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 671,
"column": 2
} | {
"line": 671,
"column": 11
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na : M\nc : MulArchimedeanClass M\nhA : c ≠ ⊤\n⊢ a ∈ c.ballSubgroup ↔ c < mk a",
"usedConstants": [
"False",
"Preorder.toLT",
"UpperSet",
"eq_false",
"congrArg",
"UpperSet.instS... | simp [hA] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 671,
"column": 2
} | {
"line": 671,
"column": 11
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na : M\nc : MulArchimedeanClass M\nhA : c ≠ ⊤\n⊢ a ∈ c.ballSubgroup ↔ c < mk a",
"usedConstants": [
"False",
"Preorder.toLT",
"UpperSet",
"eq_false",
"congrArg",
"UpperSet.instS... | simp [hA] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 671,
"column": 2
} | {
"line": 671,
"column": 11
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na : M\nc : MulArchimedeanClass M\nhA : c ≠ ⊤\n⊢ a ∈ c.ballSubgroup ↔ c < mk a",
"usedConstants": [
"False",
"Preorder.toLT",
"UpperSet",
"eq_false",
"congrArg",
"UpperSet.instS... | simp [hA] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Floor.Div | {
"line": 227,
"column": 30
} | {
"line": 227,
"column": 65
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : (i : ι) → AddCommMonoid (π i)\ninst✝² : (i : ι) → PartialOrder (π i)\ninst✝¹ : (i : ι) → SMulZeroClass α (π i)\ninst✝ : (i : ι) → CeilDiv α (π i)\na : α\nha : a ≤ 0\nf : (i : ι) → π i... | ext i; exact ceilDiv_of_nonpos ha _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Floor.Div | {
"line": 227,
"column": 30
} | {
"line": 227,
"column": 65
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : (i : ι) → AddCommMonoid (π i)\ninst✝² : (i : ι) → PartialOrder (π i)\ninst✝¹ : (i : ι) → SMulZeroClass α (π i)\ninst✝ : (i : ι) → CeilDiv α (π i)\na : α\nha : a ≤ 0\nf : (i : ι) → π i... | ext i; exact ceilDiv_of_nonpos ha _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Floor.Div | {
"line": 266,
"column": 30
} | {
"line": 266,
"column": 65
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMulZeroClass α β\ninst✝ : CeilDiv α β\nf✝ : ι →₀ β\na✝ a : α\nha : a ≤ 0\nf : ι →₀ β\n⊢ mapRange (fun x ↦ x ⌈/⌉ a) ⋯ f = 0",
"usedConstants": [
... | ext i; exact ceilDiv_of_nonpos ha _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Floor.Div | {
"line": 266,
"column": 30
} | {
"line": 266,
"column": 65
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMulZeroClass α β\ninst✝ : CeilDiv α β\nf✝ : ι →₀ β\na✝ a : α\nha : a ≤ 0\nf : ι →₀ β\n⊢ mapRange (fun x ↦ x ⌈/⌉ a) ⋯ f = 0",
"usedConstants": [
... | ext i; exact ceilDiv_of_nonpos ha _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Floor.Semifield | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 24
} | [
{
"pp": "case inr\nK : Type u_2\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsOrderedRing K\ninst✝ : FloorSemiring K\na b : K\nhb : 1 < b\nhba✝ : ↑⌈(b - 1)⁻¹⌉₊ / b < a\nhba : (b - 1)⁻¹ ≤ a\n⊢ ↑⌈a⌉₊ < b * a",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"sub_p... | rw [← sub_pos] at hb | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Group.Cone | {
"line": 57,
"column": 29
} | {
"line": 57,
"column": 74
} | [
{
"pp": "G : Type u_1\ninst✝ : CommGroup G\np q : GroupCone G\nh : (fun C ↦ C.carrier) p = (fun C ↦ C.carrier) q\n⊢ p = q",
"usedConstants": [
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"Submonoid.toSubse... | cases p; cases q; congr; exact SetLike.ext' h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Cone | {
"line": 57,
"column": 29
} | {
"line": 57,
"column": 74
} | [
{
"pp": "G : Type u_1\ninst✝ : CommGroup G\np q : GroupCone G\nh : (fun C ↦ C.carrier) p = (fun C ↦ C.carrier) q\n⊢ p = q",
"usedConstants": [
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"Submonoid.toSubse... | cases p; cases q; congr; exact SetLike.ext' h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Interval.Set.SuccPred | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 77
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : One α\ninst✝¹ : Sub α\ninst✝ : PredSubOrder α\na b : α\nh : a ≤ b\nha : ¬IsMin a\n⊢ insert a (Ioc a b) = Ioc (a - 1) b",
"usedConstants": [
"Set.Ioc",
"PredSubOrder.toPredOrder",
"congrArg",
"PartialOrder.toPreorder",
"HSu... | simpa [pred_eq_sub_one] using insert_Ioc_left_eq_Ioc_pred_of_not_isMin h ha | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Interval.Set.SuccPred | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 77
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : One α\ninst✝¹ : Sub α\ninst✝ : PredSubOrder α\na b : α\nh : a ≤ b\nha : ¬IsMin a\n⊢ insert a (Ioc a b) = Ioc (a - 1) b",
"usedConstants": [
"Set.Ioc",
"PredSubOrder.toPredOrder",
"congrArg",
"PartialOrder.toPreorder",
"HSu... | simpa [pred_eq_sub_one] using insert_Ioc_left_eq_Ioc_pred_of_not_isMin h ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Interval.Set.SuccPred | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 77
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : One α\ninst✝¹ : Sub α\ninst✝ : PredSubOrder α\na b : α\nh : a ≤ b\nha : ¬IsMin a\n⊢ insert a (Ioc a b) = Ioc (a - 1) b",
"usedConstants": [
"Set.Ioc",
"PredSubOrder.toPredOrder",
"congrArg",
"PartialOrder.toPreorder",
"HSu... | simpa [pred_eq_sub_one] using insert_Ioc_left_eq_Ioc_pred_of_not_isMin h ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Interval.Basic | {
"line": 495,
"column": 4
} | {
"line": 495,
"column": 25
} | [
{
"pp": "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).... | exact neg_add_rev _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Interval.Basic | {
"line": 495,
"column": 4
} | {
"line": 495,
"column": 25
} | [
{
"pp": "case refine_2\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.2 = (-t + -s).... | exact neg_add_rev _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 376,
"column": 2
} | {
"line": 376,
"column": 21
} | [
{
"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⊢ ¬x.coeff x.order = 0",
"usedConstants": [
"HahnSeries.support",
"Iff.mpr",
"Eq.mpr",
"HahnSeries.order",
"congrArg",
"PartialOrder.toPreorder",
"i... | rw [order_of_ne hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 399,
"column": 2
} | {
"line": 399,
"column": 21
} | [
{
"pp": "case inr\nΓ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : Zero Γ\nx : R⟦Γ⟧\ni : Γ\nhx : x ≠ 0\nhi : i ∈ x.support\n⊢ ¬i < x.order",
"usedConstants": [
"HahnSeries.support",
"Iff.mpr",
"Eq.mpr",
"HahnSeries.order",
"Preorder.toLT",
"c... | rw [order_of_ne hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 395,
"column": 2
} | {
"line": 400,
"column": 34
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : Zero Γ\nx : R⟦Γ⟧\ni : Γ\nhi : i < x.order\n⊢ x.coeff i = 0",
"usedConstants": [
"HahnSeries.support",
"Iff.mpr",
"Eq.mpr",
"HahnSeries.order",
"Preorder.toLT",
"congrArg",
"Partia... | rcases eq_or_ne x 0 with (rfl | hx)
· simp
contrapose! hi
rw [← mem_support] at hi
rw [order_of_ne hx]
exact Set.IsWF.not_lt_min _ _ hi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 395,
"column": 2
} | {
"line": 400,
"column": 34
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : Zero Γ\nx : R⟦Γ⟧\ni : Γ\nhi : i < x.order\n⊢ x.coeff i = 0",
"usedConstants": [
"HahnSeries.support",
"Iff.mpr",
"Eq.mpr",
"HahnSeries.order",
"Preorder.toLT",
"congrArg",
"Partia... | rcases eq_or_ne x 0 with (rfl | hx)
· simp
contrapose! hi
rw [← mem_support] at hi
rw [order_of_ne hx]
exact Set.IsWF.not_lt_min _ _ hi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Addition | {
"line": 273,
"column": 4
} | {
"line": 273,
"column": 85
} | [
{
"pp": "case inr\nR : Type u_8\nΓ : Type u_9\ninst✝² : LinearOrder Γ\ninst✝¹ : Zero Γ\ninst✝ : AddCancelCommMonoid R\nx y : R⟦Γ⟧\nhxy : x = y + (single x.order) x.leadingCoeff\nhy : y ≠ 0\nthis : x.order ≠ y.order\ng : Γ\nhg : g ∈ y.support\nhgx : g ≠ x.order\n⊢ g ∈ x.support",
"usedConstants": [
"Eq... | have : x.coeff g = (y + (single x.order) x.leadingCoeff).coeff g := by rw [← hxy] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.HahnSeries.Addition | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 18
} | [
{
"pp": "case coeff.h\nΓ : 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✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\nx y : R⟦Γ⟧\nx✝ : Γ\n⊢ (x + y).coeff x✝ = (y + x).coeff x✝",
"usedConstants": [
"AddMonoid.toAddZeroClass",
"AddZeroClass.toA... | apply add_comm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 214,
"column": 8
} | {
"line": 214,
"column": 28
} | [
{
"pp": "case hi\nΓ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : LinearOrder R\ninst✝¹ : AddCommGroup R\ninst✝ : IsOrderedAddMonoid R\nx y : Lex R⟦Γ⟧\nh : (ofLex x).orderTop = (ofLex y).orderTop\nhy : y ≠ 0\nhx : x ≠ 0\nh' : (ofLex |x|).orderTop = (ofLex |y|).orderTop\nn : ℕ\nhn : |(ofLex y).leadi... | simpa [← h] using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 217,
"column": 8
} | {
"line": 217,
"column": 22
} | [
{
"pp": "case hi\nΓ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : LinearOrder R\ninst✝¹ : AddCommGroup R\ninst✝ : IsOrderedAddMonoid R\nx y : Lex R⟦Γ⟧\nh : (ofLex x).orderTop = (ofLex y).orderTop\nhy : y ≠ 0\nhx : x ≠ 0\nh' : (ofLex |x|).orderTop = (ofLex |y|).orderTop\nn : ℕ\nhn : |(ofLex y).leadi... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Ring.Cone | {
"line": 39,
"column": 29
} | {
"line": 39,
"column": 74
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np q : RingCone R\nh : (fun C ↦ C.carrier) p = (fun C ↦ C.carrier) q\n⊢ p = q",
"usedConstants": [
"NegZeroClass.toNeg",
"Subsemiring.instSetLike",
"Submonoid.toSubsemigroup",
"Membership.mem",
"Eq.rec",
"MulOne.toMul",
"Subtrac... | cases p; cases q; congr; exact SetLike.ext' h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Cone | {
"line": 39,
"column": 29
} | {
"line": 39,
"column": 74
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np q : RingCone R\nh : (fun C ↦ C.carrier) p = (fun C ↦ C.carrier) q\n⊢ p = q",
"usedConstants": [
"NegZeroClass.toNeg",
"Subsemiring.instSetLike",
"Submonoid.toSubsemigroup",
"Membership.mem",
"Eq.rec",
"MulOne.toMul",
"Subtrac... | cases p; cases q; congr; exact SetLike.ext' h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.IsNonarchimedean | {
"line": 68,
"column": 2
} | {
"line": 69,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\nF : Type u_2\nα : Type u_3\ninst✝³ : AddGroupWithOne α\ninst✝² : FunLike F α R\ninst✝¹ : AddGroupSeminormClass F α R\ninst✝ : OneHomClass F α R\nf : F\nhna : IsNonarchimedean ⇑f\nn : ℤ\n⊢ f ↑n ≤ 1",
"usedCons... | obtain ⟨a, rfl | rfl⟩ := Int.eq_nat_or_neg n <;>
simp [apply_natCast_le_one hna] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Order.Ring.IsNonarchimedean | {
"line": 68,
"column": 2
} | {
"line": 69,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\nF : Type u_2\nα : Type u_3\ninst✝³ : AddGroupWithOne α\ninst✝² : FunLike F α R\ninst✝¹ : AddGroupSeminormClass F α R\ninst✝ : OneHomClass F α R\nf : F\nhna : IsNonarchimedean ⇑f\nn : ℤ\n⊢ f ↑n ≤ 1",
"usedCons... | obtain ⟨a, rfl | rfl⟩ := Int.eq_nat_or_neg n <;>
simp [apply_natCast_le_one hna] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.IsNonarchimedean | {
"line": 68,
"column": 2
} | {
"line": 69,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\nF : Type u_2\nα : Type u_3\ninst✝³ : AddGroupWithOne α\ninst✝² : FunLike F α R\ninst✝¹ : AddGroupSeminormClass F α R\ninst✝ : OneHomClass F α R\nf : F\nhna : IsNonarchimedean ⇑f\nn : ℤ\n⊢ f ↑n ≤ 1",
"usedCons... | obtain ⟨a, rfl | rfl⟩ := Int.eq_nat_or_neg n <;>
simp [apply_natCast_le_one hna] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 54
} | [
{
"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... | simpa [baseEmbedding] using seed.hahnCoeff_apply h c | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 54
} | [
{
"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... | simpa [baseEmbedding] using seed.hahnCoeff_apply h c | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 54
} | [
{
"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... | simpa [baseEmbedding] using seed.hahnCoeff_apply h c | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 57,
"column": 29
} | {
"line": 57,
"column": 74
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\np q : RingPreordering R\nh : (fun P ↦ (↑P).carrier) p = (fun P ↦ (↑P).carrier) q\n⊢ p = q",
"usedConstants": [
"NegZeroClass.toNeg",
"CommRing",
"Subsemiring.instSetLike",
"RingPreordering.toSubsemiring",
"CommSemiring.toSemiring",
... | cases p; cases q; congr; exact SetLike.ext' h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 57,
"column": 29
} | {
"line": 57,
"column": 74
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\np q : RingPreordering R\nh : (fun P ↦ (↑P).carrier) p = (fun P ↦ (↑P).carrier) q\n⊢ p = q",
"usedConstants": [
"NegZeroClass.toNeg",
"CommRing",
"Subsemiring.instSetLike",
"RingPreordering.toSubsemiring",
"CommSemiring.toSemiring",
... | cases p; cases q; congr; exact SetLike.ext' h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Archimedean | {
"line": 120,
"column": 19
} | {
"line": 120,
"column": 28
} | [
{
"pp": "case mk.mk\nR : Type u_1\ninst✝² : LinearOrder R\ninst✝¹ : CommRing R\ninst✝ : IsStrictOrderedRing R\nx : R\nhx : x ≠ 0\ny z : R\nhyz : (fun x_1 ↦ mk x + x_1) (mk y) = (fun x_1 ↦ mk x + x_1) (mk z)\n⊢ mk y = mk z",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Eq.... | | mk z => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 465,
"column": 6
} | {
"line": 466,
"column": 38
} | [
{
"pp": "case h.mp\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nr : R\nx : R⟦Γ⟧\na b : Γ\nhr : ¬r = 0\nhx : ¬x.coeff a = 0\na1 a2 : Γ\n⊢ a1 ∈ x.support ∧ a2 = b ∧ a1 + a2 = a + b → a1 = a ∧ a2 = b",
... | rintro ⟨_, rfl, h1⟩
exact ⟨add_right_cancel h1, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 465,
"column": 6
} | {
"line": 466,
"column": 38
} | [
{
"pp": "case h.mp\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nr : R\nx : R⟦Γ⟧\na b : Γ\nhr : ¬r = 0\nhx : ¬x.coeff a = 0\na1 a2 : Γ\n⊢ a1 ∈ x.support ∧ a2 = b ∧ a1 + a2 = a + b → a1 = a ∧ a2 = b",
... | rintro ⟨_, rfl, h1⟩
exact ⟨add_right_cancel h1, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 64
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nh : ∀ (a b : R), -(a * b) ∈ P → a ∈ P ∨ b ∈ P\nthis✝⁴ : HasMemOrNegMem P\nx y : R\nx✝ : x * y ∈ P.support\nthis✝³ : ¬(x ∈ P.support ∨ y ∈ P.support)\nthis✝² : -(-x * y) ∈ P → -x ∈ P ∨ y ∈ P\nthis✝¹ : -(-x * -y) ∈ P → -x ∈ P ∨ -y ∈ ... | cases (by aesop : x ∈ P ∨ -x ∈ P) <;> simp_all [mem_support] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 351,
"column": 6
} | {
"line": 351,
"column": 44
} | [
{
"pp": "case neg.refine_1\nK : Type u\ninst✝ : Field K\nR S : ValuationSubring K\nh : R ≤ S\nx : K\nhx : x ∈ S\nthis : x ≠ 0\nhr : x⁻¹ ∈ R\n⊢ IsUnit ⟨x⁻¹, ⋯⟩",
"usedConstants": [
"SubmonoidClass.instIsDedekindFiniteMonoidSubtypeMem",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid... | refine .of_mul_eq_one (⟨x, hx⟩ : S) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 519,
"column": 4
} | {
"line": 519,
"column": 58
} | [
{
"pp": "case mp\nK : Type u\ninst✝ : Field K\nA B : ValuationSubring K\nh : A.unitGroup ≤ B.unitGroup\nx : K\nhx : x ∈ A\n⊢ x ∈ B",
"usedConstants": [
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"Div... | rw [← A.valuation_le_one_iff x, le_iff_lt_or_eq] at hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 666,
"column": 4
} | {
"line": 667,
"column": 50
} | [
{
"pp": "case pos\nK : Type u\ninst✝ : Field K\nA B : ValuationSubring K\nh : B.principalUnitGroup ≤ A.principalUnitGroup\nx : K\nhx : x ∈ A\nh_1 : ¬x = 0\nh_2 : x⁻¹ + 1 = 0\n⊢ x ∈ B",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonUnitalCommRi... | · rw [add_eq_zero_iff_eq_neg, inv_eq_iff_eq_inv, inv_neg, inv_one] at h_2
simpa only [h_2] using B.neg_mem _ B.one_mem | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Pointwise.Stabilizer | {
"line": 53,
"column": 54
} | {
"line": 57,
"column": 31
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nf : G →* H\ns : Set G\n⊢ Subgroup.map f (stabilizer G s) ≤ stabilizer H (⇑f '' s)",
"usedConstants": [
"Eq.mpr",
"MonoidHom.instMonoidHomClass",
"instHSMul",
"MonoidHom.instFunLike",
"instSMulOfMul",
... | by
rintro a
simp only [Subgroup.mem_map, mem_stabilizer_iff, forall_exists_index, and_imp]
rintro a ha rfl
rw [← image_smul_distrib, ha] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Pointwise.Stabilizer | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 76
} | [
{
"pp": "case h.H\nG : Type u_1\ninst✝ : CommGroup G\ns : Set G\na : G\n⊢ ↑a ∈ stabilizer (G ⧸ stabilizer G s) (QuotientGroup.mk '' s) ↔ ↑a ∈ ⊥",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"InvOneClass.toOne",
"CommMonoid.toCommSemigroup",
"DivInvOneMonoid.toInvOneClass",
... | simp only [mem_stabilizer_iff, Subgroup.mem_bot, QuotientGroup.eq_one_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.CoeffList | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 20
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : DecidableEq R\nP : R[X]\n⊢ P.coeffList.length = if P = 0 then 0 else P.natDegree + 1",
"usedConstants": [
"Polynomial.instDecidableEq",
"instOfNatNat",
"Polynomial",
"dite",
"instHAdd",
"HAdd.hAdd",
"Polynomial.coe... | by_cases h : P = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 400,
"column": 2
} | {
"line": 400,
"column": 54
} | [
{
"pp": "K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx : K\nh : 0 ≤ x\n⊢ 0 ≤ stdPart x",
"usedConstants": [
"IsDomain.to_noZeroDivisors",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"Field.toDivisionRing",
... | obtain hx | hx := eq_or_ne (ArchimedeanClass.mk x) 0 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Polynomial.Degree.CardPowDegree | {
"line": 59,
"column": 26
} | {
"line": 66,
"column": 92
} | [
{
"pp": "Fq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n\nthis : DecidableEq Fq := Classical.decEq Fq\np q : Fq[X]\n⊢ (if p + q = 0 then 0 else ↑(Fintype.card Fq) ^ (p + q).natDegree) ≤\n (if p = 0 then 0 else ↑(Fintype.car... | by
by_cases hp : p = 0; · simp [hp]
by_cases hq : q = 0; · simp [hq]
by_cases hpq : p + q = 0
· simp only [hpq, hp, hq, if_true, if_false]
exact add_nonneg (pow_pos _).le (pow_pos _).le
simp only [hpq, hp, hq, if_false]
exact le_trans (pow_right_mono₀ (by lia) (Polynomial.nat... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 112,
"column": 16
} | {
"line": 112,
"column": 53
} | [
{
"pp": "case succ.refine_1.last\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⊢ (Fin.snoc r (q %ₘ g) (Fin.last n)).degree < g.degree",
"usedConstants... | simpa using degree_modByMonic_lt q hg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 112,
"column": 16
} | {
"line": 112,
"column": 53
} | [
{
"pp": "case succ.refine_1.last\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⊢ (Fin.snoc r (q %ₘ g) (Fin.last n)).degree < g.degree",
"usedConstants... | simpa using degree_modByMonic_lt q hg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 112,
"column": 16
} | {
"line": 112,
"column": 53
} | [
{
"pp": "case succ.refine_1.last\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⊢ (Fin.snoc r (q %ₘ g) (Fin.last n)).degree < g.degree",
"usedConstants... | simpa using degree_modByMonic_lt q hg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.List.Destutter | {
"line": 284,
"column": 4
} | {
"line": 284,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\nr : α → α → Prop\ninst✝ : Std.Antisymm r\nx y : α\nxs : List α\nh : (∀ (a' : α), a' ∈ y :: xs → r x a') ∧ Pairwise r (y :: xs)\n⊢ (if x ≠ y then x :: (y :: xs).dedup else destutter (fun x1 x2 ↦ x1 ≠ x2) (x :: xs)) = (x :: y :: xs).dedup",
"usedConstants": [
... | obtain rfl | hxy := eq_or_ne x y | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Polynomial.Homogenize | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nn : ℕ\nq : MvPolynomial (Fin 2) R\nhq : q.IsHomogeneous n\n⊢ ∑ x ∈ q.support, MvPolynomial.C (MvPolynomial.coeff x q) * (x.prod fun i k ↦ ![X, 1] i ^ k).homogenize n =\n ∑ v ∈ q.support, (MvPolynomial.monomial v) (MvPolynomial.coeff v q)",
"usedConstants": [... | refine Finset.sum_congr rfl fun m hm ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Polynomial.SumIteratedDerivative | {
"line": 232,
"column": 31
} | {
"line": 232,
"column": 45
} | [
{
"pp": "R : 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 : ∀... | natDegree_pow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.SumIteratedDerivative | {
"line": 243,
"column": 2
} | {
"line": 249,
"column": 36
} | [
{
"pp": "case inr.refine_2\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 ⋯.choo... | · congr 2
· apply sum_eq_zero
exact fun x hx => aeval_iterate_derivative_of_lt p _ r hp (mem_range.mp hx)
· rw [← aeval_iterate_derivative_self _ _ _ hp]
· rw [smul_sum, sum_congr rfl]
intro k hk
exact hc k (mem_Ico.mp hk).1 r | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.QuadraticAlgebra.Defs | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 13
} | [
{
"pp": "R : Type u_1\na b : R\ninst✝¹ : AddCommMonoidWithOne R\nn : ℕ\ninst✝ : n.AtLeastTwo\n⊢ QuadraticAlgebra.C (OfNat.ofNat n) = OfNat.ofNat n",
"usedConstants": [
"QuadraticAlgebra.re",
"QuadraticAlgebra.ext",
"QuadraticAlgebra",
"QuadraticAlgebra.instAddCommMonoidWithOne",
... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.QuadraticAlgebra.Defs | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 13
} | [
{
"pp": "R : Type u_1\na b : R\ninst✝¹ : AddCommMonoidWithOne R\nn : ℕ\ninst✝ : n.AtLeastTwo\n⊢ QuadraticAlgebra.C (OfNat.ofNat n) = OfNat.ofNat n",
"usedConstants": [
"QuadraticAlgebra.re",
"QuadraticAlgebra.ext",
"QuadraticAlgebra",
"QuadraticAlgebra.instAddCommMonoidWithOne",
... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.QuadraticAlgebra.Defs | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 13
} | [
{
"pp": "R : Type u_1\na b : R\ninst✝¹ : AddCommMonoidWithOne R\nn : ℕ\ninst✝ : n.AtLeastTwo\n⊢ QuadraticAlgebra.C (OfNat.ofNat n) = OfNat.ofNat n",
"usedConstants": [
"QuadraticAlgebra.re",
"QuadraticAlgebra.ext",
"QuadraticAlgebra",
"QuadraticAlgebra.instAddCommMonoidWithOne",
... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 99,
"column": 4
} | {
"line": 101,
"column": 9
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : LinearOrder R\nP : R[X]\nh : P ≠ 0\nhpz : ¬P = 0\nhsl : SignType.sign P.leadingCoeff ≠ 0\nc : R\ncs : List R\nh_eL : P.eraseLead.coeffList = c :: cs\nh₁ : SignType.sign c ≠ 0\n⊢ c = P.eraseLead.leadingCoeff",
"usedConstants": [
"Polynomial.coeffList_... | have h_eL : eraseLead P ≠ 0 := by simp [← coeffList_eq_nil, h_eL]
obtain ⟨ls, hls⟩ := coeffList_eq_cons_leadingCoeff h_eL
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 99,
"column": 4
} | {
"line": 101,
"column": 9
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : LinearOrder R\nP : R[X]\nh : P ≠ 0\nhpz : ¬P = 0\nhsl : SignType.sign P.leadingCoeff ≠ 0\nc : R\ncs : List R\nh_eL : P.eraseLead.coeffList = c :: cs\nh₁ : SignType.sign c ≠ 0\n⊢ c = P.eraseLead.leadingCoeff",
"usedConstants": [
"Polynomial.coeffList_... | have h_eL : eraseLead P ≠ 0 := by simp [← coeffList_eq_nil, h_eL]
obtain ⟨ls, hls⟩ := coeffList_eq_cons_leadingCoeff h_eL
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.QuadraticAlgebra.NormDeterminant | {
"line": 30,
"column": 2
} | {
"line": 32,
"column": 8
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\na b : R\nz : QuadraticAlgebra R a b\n⊢ ((LinearMap.toMatrix (basis a b) (basis a b)) (DistribSMul.toLinearMap R (QuadraticAlgebra R a b) z)).det = norm z",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"QuadraticAlgebra.re",
"Eq.mpr... | have : !![z.re, a * z.im; z.im, z.re + b * z.im].det = z.norm := by
simp [norm]
ring | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 153,
"column": 51
} | {
"line": 160,
"column": 22
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nη : R\nP : R[X]\nhx : η ≠ 0\n⊢ (C η * P).signVariations = P.signVariations",
"usedConstants": [
"Left.neg_pos_iff._simp_1",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Polynomial.C",
"... | by
wlog! hx2 : 0 < η
· simpa [lt_of_le_of_ne hx2, hx] using this (η := -η) (P := -P)
rw [signVariations, signVariations]
rw [coeffList_C_mul _ (lt_or_lt_iff_ne.mp (.inr hx2)), ← List.comp_map]
congr 5
funext
simp [hx2, sign_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 123,
"column": 18
} | {
"line": 123,
"column": 28
} | [
{
"pp": "case e_a\nK : Type u_1\nR : Type u_2\na b : R\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nu : { u // u * u = a • 1 + b • u }\nz w : QuadraticAlgebra R a b\n⊢ (z.re * w.re) • 1 + ((z.re * w.im) • ↑u + (z.im * w.re) • ↑u) = (z.re * w.re) • 1 + (z.re * w.im + z.im * w.re)... | ← add_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Quandle | {
"line": 170,
"column": 53
} | {
"line": 171,
"column": 67
} | [
{
"pp": "S : Type u_1\ninst✝ : UnitalShelf S\nx y z : S\n⊢ (x ◃ y) ◃ z = x ◃ y ◃ z",
"usedConstants": [
"Eq.mpr",
"Shelf.self_distrib",
"congrArg",
"UnitalShelf.toShelf",
"id",
"UnitalShelf.act_self_act_eq",
"Shelf.act",
"Eq.refl",
"Eq",
"UnitalShe... | by
rw [self_distrib, self_distrib, act_act_self_eq, act_self_act_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 350,
"column": 24
} | {
"line": 350,
"column": 53
} | [
{
"pp": "K : Type u_1\nR : Type u_2\na✝ b✝ : R\ninst✝ : Field K\na b : K\nHab : Fact (∀ (r : K), r ^ 2 ≠ a + b * r)\nq : ℚ≥0\nx : QuadraticAlgebra K a b\n⊢ q • x = ↑q * x",
"usedConstants": [
"QuadraticAlgebra.re",
"QuadraticAlgebra.instSMul",
"Semigroup.toMul",
"QuadraticAlgebra.ext... | ext <;> simp [NNRat.smul_def] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 350,
"column": 24
} | {
"line": 350,
"column": 53
} | [
{
"pp": "K : Type u_1\nR : Type u_2\na✝ b✝ : R\ninst✝ : Field K\na b : K\nHab : Fact (∀ (r : K), r ^ 2 ≠ a + b * r)\nq : ℚ≥0\nx : QuadraticAlgebra K a b\n⊢ q • x = ↑q * x",
"usedConstants": [
"QuadraticAlgebra.re",
"QuadraticAlgebra.instSMul",
"Semigroup.toMul",
"QuadraticAlgebra.ext... | ext <;> simp [NNRat.smul_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 350,
"column": 24
} | {
"line": 350,
"column": 53
} | [
{
"pp": "K : Type u_1\nR : Type u_2\na✝ b✝ : R\ninst✝ : Field K\na b : K\nHab : Fact (∀ (r : K), r ^ 2 ≠ a + b * r)\nq : ℚ≥0\nx : QuadraticAlgebra K a b\n⊢ q • x = ↑q * x",
"usedConstants": [
"QuadraticAlgebra.re",
"QuadraticAlgebra.instSMul",
"Semigroup.toMul",
"QuadraticAlgebra.ext... | ext <;> simp [NNRat.smul_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.QuaternionBasis | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 16
} | [
{
"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\n⊢ q.i * q.k = c₁ • q.j + c₂ • q.k",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"HMul.hMul",
"congrArg",
"QuaternionAlgebra.Basis.i_mul_j",
"A... | ← i_mul_j, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.QuaternionBasis | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 16
} | [
{
"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\n⊢ q.k * q.i = -c₁ • q.j",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"instHSMul",
"HMul.hMul",
"congrArg",
"QuaternionAlgebra.Basis.... | ← i_mul_j, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 367,
"column": 6
} | {
"line": 369,
"column": 72
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nη : R\nhη : 0 < η\nP : R[X]\nhP : P ≠ 0\nh_lC : 0 < P.leadingCoeff\nh_mul : (X - C η) * P ≠ 0\nh_deg_mul : ((X - C η) * P).natDegree = P.natDegree + 1\nd : ℕ\nhd : P.natDegree = d + 1\ns_nC_mul : SignType\nhs_nC_mul :... | suffices SignType.sign ((X - C η) * P).nextCoeff = -1 by
simp +decide [signVariations_eq_eraseLead_add_ite h_mul, h_lC,
leadingCoeff_eraseLead_eq_nextCoeff, ← sign_eq_zero_iff, this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Algebra.QuaternionBasis | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 16
} | [
{
"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\n⊢ q.k * q.j = c₃ • q.i",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"HMul.hMul",
"congrArg",
"QuaternionAlgebra.Basis.i_mul_j",
"Algebra.toSM... | ← i_mul_j, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.QuaternionBasis | {
"line": 101,
"column": 6
} | {
"line": 101,
"column": 16
} | [
{
"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\n⊢ q.j * q.k = (c₂ * c₃) • 1 - c₃ • q.i",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"congrArg",
"C... | ← i_mul_j, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.QuaternionBasis | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 16
} | [
{
"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\n⊢ q.k * q.k = -((c₁ * c₃) • 1)",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"instHSMul",
"HMul.hMul",
"congrArg",
"CommSemiring.toSe... | ← i_mul_j, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.QuaternionBasis | {
"line": 106,
"column": 58
} | {
"line": 106,
"column": 68
} | [
{
"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\n⊢ q.i * ((c₂ • q.j - q.k) * q.j) = -((c₁ * c₃) • 1)",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
"instHSMul",
"HMul.hMu... | ← i_mul_j, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.QuaternionBasis | {
"line": 161,
"column": 33
} | {
"line": 161,
"column": 53
} | [
{
"pp": "case a.inr.inl\nR : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nc₁ c₂ c₃ : R\nB : Basis A c₁ c₂ c₃\n⊢ B.j ∈ ↑B.liftHom.range",
"usedConstants": [
"QuaternionAlgebra.Basis.self",
"RingHom",
"QuaternionAlgebra.instAlgebra",
"QuaternionAlg... | use (Basis.self R).j | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Star.CHSH | {
"line": 148,
"column": 2
} | {
"line": 149,
"column": 6
} | [
{
"pp": "⊢ (√2)⁻¹ * (√2)⁻¹ = 2⁻¹",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"Monoid... | rw [← mul_inv]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Star.CHSH | {
"line": 148,
"column": 2
} | {
"line": 149,
"column": 6
} | [
{
"pp": "⊢ (√2)⁻¹ * (√2)⁻¹ = 2⁻¹",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"Monoid... | rw [← mul_inv]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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