module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 535,
"column": 18
} | {
"line": 535,
"column": 76
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nA B : ValuationSubring K\nh : A.unitGroup = B.unitGroup\n⊢ A = B",
"usedConstants": [
"Eq.mpr",
"ValuationSubring.instPartialOrder",
"_private.Mathlib.RingTheory.Valuation.ValuationSubring.0.ValuationSubring.unitGroup_injective._simp_1_1",
"Parti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 585,
"column": 18
} | {
"line": 585,
"column": 74
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nA B : ValuationSubring K\nh : A.nonunits = B.nonunits\n⊢ A = B",
"usedConstants": [
"Eq.mpr",
"ValuationSubring.instPartialOrder",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"LE.le",
"_private.Mathlib.RingTheory.Valuation... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 1021,
"column": 4
} | {
"line": 1021,
"column": 38
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝¹³ : DivisionRing K\ninst✝¹² : LinearOrder K\ninst✝¹¹ : IsOrderedRing K\ninst✝¹⁰ : Archimedean K\nM : Type u_2\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : LinearOrder M\ninst✝⁷ : IsOrderedAddMonoid M\ninst✝⁶ : Module K M\ninst✝⁵ : IsOrderedModule K M\nR : Type u_3\ninst✝⁴ : Add... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 654,
"column": 2
} | {
"line": 654,
"column": 35
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : Kˣ\nh : a ∈ A.principalUnitGroup\n⊢ a ∈ A.unitGroup",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 667,
"column": 6
} | {
"line": 667,
"column": 28
} | [
{
"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\n⊢ x ∈ B",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"InvOneClass.toOne",
"Comm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 670,
"column": 6
} | {
"line": 670,
"column": 27
} | [
{
"pp": "case neg\nK : Type u\ninst✝ : Field K\nA B : ValuationSubring K\nh : B.principalUnitGroup ≤ A.principalUnitGroup\nx : K\nhx✝¹ : ¬A.valuation x⁻¹ < 1\nhx✝ : ¬A.valuation (x⁻¹ + 1 - 1) < 1\nh_1 : ¬x = 0\nh_2 : ¬x⁻¹ + 1 = 0\nhx : Units.mk0 (x⁻¹ + 1) h_2 ∉ A.principalUnitGroup\n⊢ Units.mk0 (x⁻¹ + 1) h_2 ∉ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 676,
"column": 2
} | {
"line": 676,
"column": 73
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nA B : ValuationSubring K\nh : A.principalUnitGroup = B.principalUnitGroup\n⊢ A = B",
"usedConstants": [
"Eq.mpr",
"ValuationSubring.instPartialOrder",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"LE.le",
"And",
"Eq",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Pointwise.Stabilizer | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 13
} | [
{
"pp": "G : Type u_1\ninst✝ : CommGroup G\ns : Set G\na : G\nha : a ∈ s\n⊢ a • ↑(stabilizer G s) ⊆ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Pointwise.Stabilizer | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 58
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\ns : Subgroup G\na : G\nh : ∀ (b : G), a • b ∈ ↑s ↔ b ∈ ↑s\n⊢ a ∈ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Pointwise.Stabilizer | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 61
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\ns : Subgroup G\na : G\nh : ∀ (b : G), b * a ∈ s ↔ b ∈ s\n⊢ a ∈ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Pointwise.Stabilizer | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 61
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\ns : Subgroup Gᵐᵒᵖ\na : G\nh : ∀ (b : Gᵐᵒᵖ), a • b ∈ ↑s ↔ b ∈ ↑s\nthis : 1 * op a ∈ s\n⊢ a ∈ s.unop",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 15
} | [
{
"pp": "K : Type u_1\ninst✝⁶ : LinearOrder K\ninst✝⁵ : Field K\ninst✝⁴ : IsOrderedRing K\nx✝ y : K\nR : Type u_2\ninst✝³ : LinearOrder R\ninst✝² : CommRing R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : Archimedean R\nx : FiniteElement K\nn : ℕ\nhn : |ArchimedeanOrder.val (ArchimedeanOrder.of ↑x)| ≤ n • |Archimede... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.CoeffList | {
"line": 95,
"column": 2
} | {
"line": 96,
"column": 70
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nP : R[X]\nh : P.nextCoeff ≠ 0\nh₂ : P ≠ 0\nh₃ : ¬P.eraseLead = 0\n⊢ P.leadingCoeff :: P.eraseLead.coeffList = P.coeffList",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Polynomial.natDegree_eraseLead_add_one",
"List.map",
"WithBot.succ",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.CoeffList | {
"line": 109,
"column": 4
} | {
"line": 110,
"column": 11
} | [
{
"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 using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.CoeffList | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nP : R[X]\nh : P ≠ 0\nhdp : ¬P.natDegree = 0\nhep : P.eraseLead = 0\n⊢ (monomial P.natDegree) P.leadingCoeff = P",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 22
} | [
{
"pp": "K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx : FiniteElement K\n⊢ mk x = 0 ↔ 0 < ArchimedeanClass.mk ↑x",
"usedConstants": [
"AddValuation.toValuation",
"IsDomain.to_noZeroDivisors",
"Preorder.toLT",
"NonUnitalCommRing.toNonUnitalNonAsso... | apply mk_eq_mk.trans | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Polynomial.CoeffList | {
"line": 143,
"column": 32
} | {
"line": 143,
"column": 87
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nP : R[X]\nh : P ≠ 0\nhdp : ¬P.natDegree = 0\nhep : ¬P.eraseLead = 0\nh₁ : P.degree.succ = P.natDegree + 1\nh₂ : P.eraseLead.degree.succ = P.eraseLead.natDegree + 1\nn : ℕ\nhn : P.natDegree = P.eraseLead.natDegree + 1 + n\nk : ℕ\nhkd : k + 1 < P.natDegree + 1\ndk : ℕ\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.CoeffList | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 43
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\nP : R[X]\nh : P ≠ 0\nhdp : ¬P.natDegree = 0\nhep : ¬P.eraseLead = 0\nh₁ : P.degree.succ = P.natDegree + 1\nh₂ : P.eraseLead.degree.succ = P.eraseLead.natDegree + 1\nn : ℕ\nhn : P.natDegree = P.eraseLead.natDegree + 1 + n\nk : ℕ\nhkd : k + 1 < P.natDegree + 1\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.CoeffMem | {
"line": 44,
"column": 6
} | {
"line": 44,
"column": 17
} | [
{
"pp": "case neg\nR : Type u_2\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\np q : S[X]\nhq : q.Monic\ni : ℕ\nH₀ : ∀ (i : ℕ), p.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p)\nhpq : ¬(q.degree ≤ p.degree ∧ p ≠ 0)\n⊢ (0, p).1.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p) ∧ (0, p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.CoeffMem | {
"line": 47,
"column": 4
} | {
"line": 47,
"column": 24
} | [
{
"pp": "case pos\nR : Type u_2\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\np q : S[X]\nhq : q.Monic\ni : ℕ\nH₀ : ∀ (i : ℕ), p.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p)\nhpq : q.degree ≤ p.degree ∧ p ≠ 0\nr : S[X]\nhr : p - q * (C p.leadingCoeff * X ^ (deg(p) - deg(q))) = r\... | by_cases hr' : r = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 13
} | [
{
"pp": "K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx : K\nh : x ≤ 0\n⊢ stdPart x ≤ 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 425,
"column": 4
} | {
"line": 425,
"column": 15
} | [
{
"pp": "case hneg\nK : 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 : r < stdPart x\n⊢ f (r - stdPart x) ≤ 0",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"IsDomain.to_noZeroDivisors",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 452,
"column": 47
} | {
"line": 452,
"column": 58
} | [
{
"pp": "K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nf : ℝ →+*o K\nx : K\nhx : 0 ≤ mk x\na : ℤ\nha : ↑a < x\nb : ℤ\nhb : x < ↑b\n⊢ ↑b ∈ {r | x < f r}",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real",
"Preorder.toLT",
"congrArg",
"Part... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 456,
"column": 44
} | {
"line": 456,
"column": 55
} | [
{
"pp": "K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nf : ℝ →+*o K\nx : K\nhx : 0 ≤ mk x\na : ℤ\nha : ↑a < x\nb : ℤ\nhb : x < ↑b\nhn : {r | x < f r}.Nonempty\nr : ℝ\nhr : r ∈ {r | x < f r}\nhra : r < ↑a\n⊢ (↑↑f.toRingHom).toFun ↑a < (↑↑f.toRingHom).toFun r",
"usedConstant... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 458,
"column": 6
} | {
"line": 458,
"column": 17
} | [
{
"pp": "case inl.hl\nK : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nf : ℝ →+*o K\nx : K\nhx : 0 ≤ mk x\na : ℤ\nha : ↑a < x\nb : ℤ\nhb✝ : x < ↑b\nhn : {r | x < f r}.Nonempty\nhb : BddBelow {r | x < f r}\nr : ℝ\nhr : r < sInf {r | x < f r}\n⊢ f r ≤ x",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 15
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nhp : p.IsMonicOfDegree n\nhq : q.natDegree < n\nH : Subsingleton R\n⊢ (p + q).IsMonicOfDegree n",
"usedConstants": [
"Eq.mpr",
"id",
"instOfNatNat",
"Polynomial.instAdd",
"Polynomial",
"instHAdd",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nn : ℕ\n⊢ ((monomial n) 1).IsMonicOfDegree n",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"congrArg",
"LinearMap.instFunLike",
"Polynomial.monomial",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Bivariate | {
"line": 266,
"column": 2
} | {
"line": 266,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\n⊢ swap (C f) = map C f",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.eval",
"Algebra.algebraMap",
"congrArg",
"CommSemiring.toSemiring",
"AlgHom",
"AlgHom.funLike",
"Polynomial.algebra... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | {
"line": 277,
"column": 4
} | {
"line": 277,
"column": 15
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : CommRing R\np : R[X]\nn : ℕ\nhp : p.IsMonicOfDegree n\nr : R\nH : Subsingleton R\n⊢ ((aeval (X + C r)) p).IsMonicOfDegree n",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"CommSemiring.toSemiring",
"AlgHom",
"AlgHom.funLike",
"Poly... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Mirror | {
"line": 151,
"column": 66
} | {
"line": 151,
"column": 78
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nhn : n ∈ Finset.range (p.natDegree + p.natTrailingDegree).succ\n⊢ p.coeff (n, (revAt (p.natDegree + p.natTrailingDegree)) n).1 *\n p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) (n, (revAt (p.natDegree + p.natTrailingDegree)) n).2) =\n p.... | revAt_invol, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Mirror | {
"line": 158,
"column": 6
} | {
"line": 158,
"column": 49
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : Semiring R\np : R[X]\ninst✝ : NoZeroDivisors R\nhp : ¬p = 0\n⊢ (p * p.mirror).natDegree = 2 * p.natDegree",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Polynomial.natDegree_mul",
"congrArg",
"Polynomial.mirror",
"mt",
"id",
... | natDegree_mul hp (mt mirror_eq_zero.mp hp), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Destutter | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 36
} | [
{
"pp": "α : Type u_1\nR : α → α → Prop\ninst✝ : DecidableRel R\nl : List α\na b : α\nhab : R a b\n⊢ IsChain R (a :: destutter' R b l)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 111,
"column": 18
} | {
"line": 111,
"column": 29
} | [
{
"pp": "case succ.refine_1.cast\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\ni : Fin n\n⊢ (Fin.snoc r (q %ₘ g) i.castSucc).degree < g.degree",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 112,
"column": 16
} | {
"line": 112,
"column": 27
} | [
{
"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 | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Destutter | {
"line": 257,
"column": 6
} | {
"line": 257,
"column": 44
} | [
{
"pp": "case pos\nα : Type u_1\nR : α → α → Prop\ninst✝¹ : DecidableRel R\ninst✝ : IsEquiv α Rᶜ\nl₁ : List α\nb : α\nl₂ : List α\na : α\nhl : l₁ <+ l₂\nhl₁ : IsChain R (a :: l₁)\nhab : R a b\n⊢ (a :: l₁).length ≤ (destutter R (a :: b :: l₂)).length",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Destutter | {
"line": 258,
"column": 6
} | {
"line": 258,
"column": 44
} | [
{
"pp": "case neg\nα : Type u_1\nR : α → α → Prop\ninst✝¹ : DecidableRel R\ninst✝ : IsEquiv α Rᶜ\nl₁ : List α\nb : α\nl₂ : List α\na : α\nhl : l₁ <+ l₂\nhl₁ : IsChain R (a :: l₁)\nhab : ¬R a b\n⊢ (a :: l₁).length ≤ (destutter R (a :: b :: l₂)).length",
"usedConstants": [
"Eq.mpr",
"eq_false",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Destutter | {
"line": 261,
"column": 4
} | {
"line": 262,
"column": 11
} | [
{
"pp": "α : Type u_1\nR : α → α → Prop\ninst✝¹ : DecidableRel R\ninst✝ : IsEquiv α Rᶜ\nb : α\nl₁ l₂ : List α\na : α\nhl : l₁ <+ l₂\nhl₁ : IsChain R (a :: b :: l₁)\n⊢ (a :: b :: l₁).length ≤ (destutter R (a :: b :: l₂)).length",
"usedConstants": [
"Eq.mpr",
"congrArg",
"List.destutter",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Destutter | {
"line": 285,
"column": 6
} | {
"line": 285,
"column": 17
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝¹ : DecidableEq α\nr : α → α → Prop\ninst✝ : Std.Antisymm r\nx : α\nxs : List α\nh : (∀ (a' : α), a' ∈ x :: xs → r x a') ∧ Pairwise r (x :: xs)\n⊢ (if x ≠ x then x :: (x :: xs).dedup else destutter (fun x1 x2 ↦ x1 ≠ x2) (x :: xs)) = (x :: x :: xs).dedup",
"usedConstants... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 128,
"column": 22
} | {
"line": 128,
"column": 33
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 137,
"column": 19
} | {
"line": 137,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ng : R[X]\nhg : g.Monic\nn : ℕ\nih :\n ∀ {q₁ q₂ : R[X]} {r₁ r₂ : Fin n → R[X]},\n (∀ (i : Fin n), (r₁ i).degree < g.degree) →\n (∀ (i : Fin n), (r₂ i).degree < g.degree) →\n q₁ * g ^ n + ∑ i, r₁ i * g ^ ↑i = q₂ * g ^ n + ∑ i, r₂ i * g ^ ↑i → q₁ = q₂ ∧ r₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 138,
"column": 19
} | {
"line": 138,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ng : R[X]\nhg : g.Monic\nn : ℕ\nih :\n ∀ {q₁ q₂ : R[X]} {r₁ r₂ : Fin n → R[X]},\n (∀ (i : Fin n), (r₁ i).degree < g.degree) →\n (∀ (i : Fin n), (r₂ i).degree < g.degree) →\n q₁ * g ^ n + ∑ i, r₁ i * g ^ ↑i = q₂ * g ^ n + ∑ i, r₂ i * g ^ ↑i → q₁ = q₂ ∧ r₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 139,
"column": 66
} | {
"line": 139,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ng : R[X]\nhg : g.Monic\nn : ℕ\nih :\n ∀ {q₁ q₂ : R[X]} {r₁ r₂ : Fin n → R[X]},\n (∀ (i : Fin n), (r₁ i).degree < g.degree) →\n (∀ (i : Fin n), (r₂ i).degree < g.degree) →\n q₁ * g ^ n + ∑ i, r₁ i * g ^ ↑i = q₂ * g ^ n + ∑ i, r₂ i * g ^ ↑i → q₁ = q₂ ∧ r₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 140,
"column": 66
} | {
"line": 140,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ng : R[X]\nhg : g.Monic\nn : ℕ\nih :\n ∀ {q₁ q₂ : R[X]} {r₁ r₂ : Fin n → R[X]},\n (∀ (i : Fin n), (r₁ i).degree < g.degree) →\n (∀ (i : Fin n), (r₂ i).degree < g.degree) →\n q₁ * g ^ n + ∑ i, r₁ i * g ^ ↑i = q₂ * g ^ n + ∑ i, r₂ i * g ^ ↑i → q₁ = q₂ ∧ r₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Homogenize | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nc : R\nn : ℕ\n⊢ (C c).homogenize n = MvPolynomial.C c * MvPolynomial.X 1 ^ n",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Polynomial.C",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Semiring.toModule",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Homogenize | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nn : ℕ\n⊢ homogenize 1 n = MvPolynomial.X 1 ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Homogenize | {
"line": 177,
"column": 42
} | {
"line": 177,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nn : ℕ\nq : MvPolynomial (Fin 2) R\nhq : q.IsHomogeneous n\nm : Fin 2 →₀ ℕ\nhm : m ∈ q.support\n⊢ MvPolynomial.coeff m q ≠ 0",
"usedConstants": [
"CommSemiring.toSemiring",
"id",
"Ne",
"instOfNatNat",
"MvPolynomial.coeff",
"Na... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Homogenize | {
"line": 261,
"column": 4
} | {
"line": 261,
"column": 37
} | [
{
"pp": "case «1»\nR : Type u_1\ninst✝ : CommSemiring R\np : R[X]\n⊢ (p.toTupleMvPolynomial ((fun i ↦ i) ⟨1, ⋯⟩)).IsHomogeneous p.natDegree",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Nat.le_refl",
"congrArg",
"CommSemiring.toSemir... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Homogenize | {
"line": 292,
"column": 4
} | {
"line": 292,
"column": 83
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝¹ : CommSemiring R\nM : Type u_2\ninst✝ : AddCommMonoid M\np : R[X]\nf : R → M\ns : Fin 2 →₀ ℕ\nhs : s ∈ (p.homogenize p.natDegree).support\n⊢ (fun s ↦ s 0) s ∈ p.support",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Homogenize | {
"line": 293,
"column": 4
} | {
"line": 294,
"column": 11
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝¹ : CommSemiring R\nM : Type u_2\ninst✝ : AddCommMonoid M\np : R[X]\nf : R → M\nn : ℕ\nhn : n ∈ p.support\n⊢ (fun n ↦ fun₀ | 0 => n | 1 => p.natDegree - n) n ∈ (p.homogenize p.natDegree).support",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 207,
"column": 13
} | {
"line": 207,
"column": 24
} | [
{
"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 | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 13
} | [
{
"pp": "case h\nR : Type u_3\ninst✝ : Semiring R\nr : R\nx✝ : R[X]\n⊢ (leval r) x✝ = x✝.smeval r",
"usedConstants": [
"MonoidWithZero.toMulActionWithZero",
"Semiring.toModule",
"LinearMap.instFunLike",
"id",
"Polynomial.leval",
"LinearMap",
"Polynomial",
"Mon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 293,
"column": 20
} | {
"line": 293,
"column": 49
} | [
{
"pp": "case monomial\nR : Type u_1\ninst✝⁴ : Semiring R\np : R[X]\nS : Type u_2\ninst✝³ : Semiring S\ninst✝² : Module R S\ninst✝¹ : IsScalarTower R S S\ninst✝ : SMulCommClass R S S\nx y : S\nhc : Commute x y\nn : ℕ\na : R\n⊢ Commute (((monomial n) a).smeval x) y",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 13
} | [
{
"pp": "hp : IsUnitTrinomial 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 182,
"column": 2
} | {
"line": 197,
"column": 31
} | [
{
"pp": "p : ℤ[X]\n⊢ p.IsUnitTrinomial ↔ (p * p.mirror).coeff (((p * p.mirror).natDegree + (p * p.mirror).natTrailingDegree) / 2) = 3",
"usedConstants": [
"Iff.mpr",
"Int.sq_eq_one_of_sq_le_three",
"Distrib.leftDistribClass",
"Units.val",
"Eq.mpr",
"Int.instAddCommMonoid"... | rw [natDegree_mul_mirror, natTrailingDegree_mul_mirror, ← mul_add,
Nat.mul_div_right _ zero_lt_two, coeff_mul_mirror]
refine ⟨?_, fun hp => ?_⟩
· rintro ⟨k, m, n, hkm, hmn, u, v, w, rfl⟩
rw [sum_def, trinomial_support hkm hmn u.ne_zero v.ne_zero w.ne_zero,
sum_insert (mt mem_insert.mp (not_or_intro hk... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 182,
"column": 2
} | {
"line": 197,
"column": 31
} | [
{
"pp": "p : ℤ[X]\n⊢ p.IsUnitTrinomial ↔ (p * p.mirror).coeff (((p * p.mirror).natDegree + (p * p.mirror).natTrailingDegree) / 2) = 3",
"usedConstants": [
"Iff.mpr",
"Int.sq_eq_one_of_sq_le_three",
"Distrib.leftDistribClass",
"Units.val",
"Eq.mpr",
"Int.instAddCommMonoid"... | rw [natDegree_mul_mirror, natTrailingDegree_mul_mirror, ← mul_add,
Nat.mul_div_right _ zero_lt_two, coeff_mul_mirror]
refine ⟨?_, fun hp => ?_⟩
· rintro ⟨k, m, n, hkm, hmn, u, v, w, rfl⟩
rw [sum_def, trinomial_support hkm hmn u.ne_zero v.ne_zero w.ne_zero,
sum_insert (mt mem_insert.mp (not_or_intro hk... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 396,
"column": 4
} | {
"line": 397,
"column": 70
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : Algebra R[X] K\ninst✝ : FaithfulSMul R[X] K\nι : Type u_3\ns : Finset ι\ng : ι → R[X]\nhg : ∀ i ∈ s, (g i).Monic\nhgg : (↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)\nn : ι → ℕ\ngi : ι → K\nhgi : ∀ i ∈ s, gi i * (algebraMap ... | obtain ⟨hq, hr⟩ := quo_mul_prod_pow_add_sum_rem_mul_prod_pow_unique hg hgg
(fun i hi j => hr₁ i hi j.rev) (fun i hi j => hr₂ i hi j.rev) hf | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Polynomial.SumIteratedDerivative | {
"line": 252,
"column": 8
} | {
"line": 252,
"column": 18
} | [
{
"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... | mem_inter, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.SumIteratedDerivative | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : Nontrivial R\ninst✝ : NoZeroDivisors R\np : R[X]\nq : ℕ\nhq : 0 < q\n⊢ ∃ gp,\n gp.natDegree ≤ p.natDegree - q ∧\n ∀ (r : R) {p' : R[X]},\n p = (X - C r) ^ (q - 1) * p' → eval r (sumIDeriv p) = (q - 1)! • eval r p' + q ! • eval r gp",
"usedC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 434,
"column": 24
} | {
"line": 434,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R[X] K\ninst✝ : FaithfulSMul R[X] K\nf : R[X]\nι : Type u_3\ng : ι → R[X]\ns : Finset ι\nhg : ∀ i ∈ s, (g i).Monic\nhcop : (↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)\nthis : Nontrivial R\ni : ι\nhi : i ∈ s\n⊢ (algebr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 461,
"column": 24
} | {
"line": 461,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R[X] K\ninst✝ : FaithfulSMul R[X] K\nι : Type u_3\ng : ι → R[X]\ns : Finset ι\nhg : ∀ i ∈ s, (g i).Monic\nhcop : (↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)\nq₁ q₂ : R[X]\nr₁ r₂ : ι → R[X]\nhr₁ : ∀ i ∈ s, (r₁ i).degre... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 484,
"column": 2
} | {
"line": 484,
"column": 78
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R[X] K\ninst✝ : FaithfulSMul R[X] K\nf g₁ g₂ : R[X]\nhg₁ : g₁.Monic\nhg₂ : g₂.Monic\nhcoprime : IsCoprime g₁ g₂\ng : Bool → R[X] := fun t ↦ Bool.rec g₂ g₁ t\n⊢ ∃ q r₁ r₂, r₁.degree < g₁.degree ∧ r₂.degree < g₂.degree ∧ ... | have hg (i : Bool) (_ : i ∈ Finset.univ) : (g i).Monic := Bool.rec hg₂ hg₁ i | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 489,
"column": 2
} | {
"line": 489,
"column": 28
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R[X] K\ninst✝ : FaithfulSMul R[X] K\nf g₁ g₂ : R[X]\nhg₁ : g₁.Monic\nhg₂ : g₂.Monic\nhcoprime✝ : IsCoprime g₁ g₂\ng : Bool → R[X] := fun t ↦ Bool.rec g₂ g₁ t\nhg : ∀ i ∈ Finset.univ, (g i).Monic\nhcoprime : (↑Finset.uni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 78
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R[X] K\ninst✝ : FaithfulSMul R[X] K\ng₁ g₂ : R[X]\nhg₁ : g₁.Monic\nhg₂ : g₂.Monic\nhcoprime : IsCoprime g₁ g₂\nq₁ q₂ r₁₁ r₁₂ r₂₁ r₂₂ : R[X]\nhr₁₁ : r₁₁.degree < g₁.degree\nhr₁₂ : r₁₂.degree < g₁.degree\nhr₂₁ : r₂₁.degre... | have hg (i : Bool) (_ : i ∈ Finset.univ) : (g i).Monic := Bool.rec hg₂ hg₁ i | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 515,
"column": 2
} | {
"line": 515,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R[X] K\ninst✝ : FaithfulSMul R[X] K\ng₁ g₂ : R[X]\nhg₁ : g₁.Monic\nhg₂ : g₂.Monic\nhcoprime✝ : IsCoprime g₁ g₂\nq₁ q₂ r₁₁ r₁₂ r₂₁ r₂₂ : R[X]\nhr₁₁ : r₁₁.degree < g₁.degree\nhr₁₂ : r₁₂.degree < g₁.degree\nhr₂₁ : r₂₁.degr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.QuadraticAlgebra.Defs | {
"line": 242,
"column": 32
} | {
"line": 242,
"column": 53
} | [
{
"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✝² : SMul S R\ninst✝¹ : SMul T R\ns✝ : S\ninst✝ : SMulCommClass S T R\ns : S\nt : T\nz : QuadraticAlgebra R a b\n⊢ (s • t • z).re = (t • s • z).re",
"usedConstants": [
"QuadraticAlgebra.re",
... | exact smul_comm _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.QuadraticAlgebra.Defs | {
"line": 242,
"column": 32
} | {
"line": 242,
"column": 53
} | [
{
"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✝² : SMul S R\ninst✝¹ : SMul T R\ns✝ : S\ninst✝ : SMulCommClass S T R\ns : S\nt : T\nz : QuadraticAlgebra R a b\n⊢ (s • t • z).im = (t • s • z).im",
"usedConstants": [
"QuadraticAlgebra.im",
... | exact smul_comm _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 40
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nη : R\nP : R[X]\nhx : η ≠ 0\nthis :\n ∀ {R : Type u_1} [inst : Ring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R] {η : R} (P : R[X]),\n η ≠ 0 → 0 < η → (C η * P).signVariations = P.signVariations\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 126,
"column": 14
} | {
"line": 126,
"column": 27
} | [
{
"pp": "K : 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 + z.im * w.re) • ↑u + (z.im * w.im) • (↑u * ↑u) =\n (z.re * w.re) • 1 + (z.re * w.im ... | simp [u.prop] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 126,
"column": 14
} | {
"line": 126,
"column": 27
} | [
{
"pp": "K : 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 + z.im * w.re) • ↑u + (z.im * w.im) • (↑u * ↑u) =\n (z.re * w.re) • 1 + (z.re * w.im ... | simp [u.prop] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 126,
"column": 14
} | {
"line": 126,
"column": 27
} | [
{
"pp": "K : 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 + z.im * w.re) • ↑u + (z.im * w.im) • (↑u * ↑u) =\n (z.re * w.re) • 1 + (z.re * w.im ... | simp [u.prop] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Quandle | {
"line": 264,
"column": 2
} | {
"line": 264,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝ : Rack R\nx y : R\nh : (op x ◃ op x) ◃ op y = op x ◃ op y\n⊢ (x ◃⁻¹ x) ◃⁻¹ y = x ◃⁻¹ y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Quandle | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝ : Rack R\nx y : R\nh : (op x ◃ op x) ◃⁻¹ op y = op x ◃⁻¹ op y\n⊢ (x ◃⁻¹ x) ◃ y = x ◃ y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Quandle | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝ : Rack R\nx y : R\nh : op x ◃ op x = op y ◃ op y ↔ op x = op y\n⊢ x ◃⁻¹ x = y ◃⁻¹ y ↔ x = y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 220,
"column": 2
} | {
"line": 222,
"column": 7
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nη : R\nd : ℕ\nc : R\nhc : c ≠ 0\nhη : 0 < η\nh₁ : ((X - C η) * (monomial d) c).nextCoeff = -(η * c)\nh₂ : ((X - C η) * (monomial d) c).eraseLead ≠ 0\nh₃ : SignType.sign c ≠ SignType.sign (-(η * c))\n⊢ ((monomial d) c)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 305,
"column": 4
} | {
"line": 305,
"column": 15
} | [
{
"pp": "case h.inr\nR : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nη : R\nhη : 0 < η\nd : ℕ\nih : ∀ m < d, ∀ {P : R[X]}, P ≠ 0 → P.natDegree = m → P.signVariations + 1 ≤ ((X - C η) * P).signVariations\nP : R[X]\nhP : P ≠ 0\nhd : P.natDegree = d\nH : ∀ {P : R[X]}, P ≠ 0 → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 328,
"column": 33
} | {
"line": 328,
"column": 60
} | [
{
"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\nih : ∀ m < d + 1, ∀ {P : R[X]}, P ≠ 0 → P.natDegree = m → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Action.Pointwise.Finset | {
"line": 39,
"column": 87
} | {
"line": 40,
"column": 54
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝⁵ : Semiring R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommMonoid M\ninst✝² : DecidableEq M\ninst✝¹ : Module R M\ninst✝ : IsTorsionFree R M\ns : Finset R\nt : Finset M\n⊢ 0 ∈ s • t ↔ 0 ∈ s ∧ t.Nonempty ∨ 0 ∈ t ∧ s.Nonempty",
"usedConstants": [
"Eq.mpr",
"instH... | by
rw [← mem_coe, coe_smul, Set.zero_mem_smul_iff]; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.Ext | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 26
} | [
{
"pp": "R : Type u\ninst₁ inst₂ : NonAssocSemiring R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_mul : HMul.hMul = HMul.hMul\n⊢ inst₁.toNonUnitalNonAssocSemiring = inst₂.toNonUnitalNonAssocSemiring",
"usedConstants": [
"NonAssocSemiring.toNonUnitalNonAssocSemiring",
"NonUnitalNonAssocSemiring.ext"
]
... | ext : 1 <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Ring.Ext | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 26
} | [
{
"pp": "R : Type u\ninst₁ inst₂ : NonAssocSemiring R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_mul : HMul.hMul = HMul.hMul\nh : inst₁.toNonUnitalNonAssocSemiring = inst₂.toNonUnitalNonAssocSemiring\nh_zero : Zero.zero = Zero.zero\nh_one' : inst₁.toMulZeroOneClass.toOne = inst₂.toMulZeroOneClass.toOne\nh_one : One.one ... | ext : 1 <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Ring.Ext | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 26
} | [
{
"pp": "R : Type u\ninst₁ inst₂ : NonUnitalRing R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_mul : HMul.hMul = HMul.hMul\n⊢ inst₁.toNonUnitalNonAssocRing = inst₂.toNonUnitalNonAssocRing",
"usedConstants": [
"NonUnitalRing.toNonUnitalNonAssocRing",
"NonUnitalNonAssocRing.ext"
]
}
] | ext : 1 <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Ring.Ext | {
"line": 253,
"column": 4
} | {
"line": 253,
"column": 26
} | [
{
"pp": "R : Type u\ninst₁ inst₂ : NonAssocRing R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_mul : HMul.hMul = HMul.hMul\n⊢ inst₁.toNonUnitalNonAssocRing = inst₂.toNonUnitalNonAssocRing",
"usedConstants": [
"NonAssocRing.toNonUnitalNonAssocRing",
"NonUnitalNonAssocRing.ext"
]
}
] | ext : 1 <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Ring.Ext | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 26
} | [
{
"pp": "R : Type u\ninst₁ inst₂ : NonAssocRing R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_mul : HMul.hMul = HMul.hMul\nh₁ : inst₁.toNonUnitalNonAssocRing = inst₂.toNonUnitalNonAssocRing\n⊢ inst₁.toNonAssocSemiring = inst₂.toNonAssocSemiring",
"usedConstants": [
"NonAssocSemiring.ext",
"NonAssocRing.to... | ext : 1 <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Ring.Ext | {
"line": 285,
"column": 4
} | {
"line": 285,
"column": 26
} | [
{
"pp": "R : Type u\ninst₁ inst₂ : Semiring R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_mul : HMul.hMul = HMul.hMul\n⊢ inst₁.toAddCommMonoid = inst₂.toAddCommMonoid",
"usedConstants": [
"Semiring.toAddCommMonoid",
"AddCommMonoid.ext"
]
}
] | ext : 1 <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Ring.Ext | {
"line": 287,
"column": 4
} | {
"line": 287,
"column": 26
} | [
{
"pp": "R : Type u\ninst₁ inst₂ : Semiring R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_mul : HMul.hMul = HMul.hMul\nh₀ : inst₁.toAddCommMonoid = inst₂.toAddCommMonoid\n⊢ inst₁.toNonUnitalSemiring = inst₂.toNonUnitalSemiring",
"usedConstants": [
"NonUnitalSemiring.ext",
"Semiring.toNonUnitalSemiring"
... | ext : 1 <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Ring.Ext | {
"line": 289,
"column": 4
} | {
"line": 289,
"column": 26
} | [
{
"pp": "R : Type u\ninst₁ inst₂ : Semiring R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_mul : HMul.hMul = HMul.hMul\nh₀ : inst₁.toAddCommMonoid = inst₂.toAddCommMonoid\nh₁ : inst₁.toNonUnitalSemiring = inst₂.toNonUnitalSemiring\n⊢ inst₁.toNonAssocSemiring = inst₂.toNonAssocSemiring",
"usedConstants": [
"NonAs... | ext : 1 <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Ring.Ext | {
"line": 321,
"column": 4
} | {
"line": 321,
"column": 26
} | [
{
"pp": "R : Type u\ninst₁ inst₂ : Ring R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_mul : HMul.hMul = HMul.hMul\n⊢ inst₁.toSemiring = inst₂.toSemiring",
"usedConstants": [
"Semiring.ext",
"Ring.toSemiring"
]
}
] | ext : 1 <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Ring.Ext | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 26
} | [
{
"pp": "R : Type u\ninst₁ inst₂ : Ring R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_mul : HMul.hMul = HMul.hMul\nh₁ : inst₁.toSemiring = inst₂.toSemiring\n⊢ toNonAssocRing = toNonAssocRing",
"usedConstants": [
"NonAssocRing.ext",
"Ring.toNonAssocRing"
]
}
] | ext : 1 <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Ring.Semireal.Defs | {
"line": 43,
"column": 30
} | {
"line": 43,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝³ : AddGroup R\ninst✝² : One R\ninst✝¹ : Mul R\ninst✝ : IsSemireal R\nx✝ : IsSumSq (-1)\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Semireal.Defs | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 17
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : IsSemireal R\nn : ℕ\nhn : 1 + ↑n = 0\n⊢ n + 1 = 0",
"usedConstants": [
"Eq.mpr",
"False",
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg",
"Nat.add_eq_zero_iff._simp_1",
"id",
"one_ne_zero._s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 63,
"column": 8
} | {
"line": 63,
"column": 19
} | [
{
"pp": "case sq_add.inr.inl\nR : Type u_1\ninst✝ : NonUnitalNonAssocSemiring R\ns a : R\nne_a : a ≠ 0\nhs✝ : IsSumSq 0\nih : 0 ≠ 0 → IsSumNonzeroSq 0\nhs : a * a + 0 ≠ 0\n⊢ IsSumNonzeroSq (a * a + 0)",
"usedConstants": [
"Eq.mpr",
"IsSumNonzeroSq",
"HMul.hMul",
"congrArg",
"Ad... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 125,
"column": 53
} | {
"line": 125,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocSemiring R\nh : ∀ {s a : R}, IsSumSq s → a * a + s = 0 → a = 0\nx a✝ : R\nha : a✝ ≠ 0\nhc : a✝ * a✝ = 0\n⊢ a✝ * a✝ + 0 = 0",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"AddMonoid.toAddZeroClass",
"NonUnitalNonAssocSemir... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.IsFormallyReal | {
"line": 136,
"column": 38
} | {
"line": 136,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsFormallyReal R\nx : R\nhx : x ^ 2 = 0\nhc : x ≠ 0\n⊢ IsSumNonzeroSq 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Support | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 52
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝³ : Monoid G\ninst✝² : Semiring k\ninst✝¹ : MulSemiringAction G k\nf : SkewMonoidAlgebra k G\ninst✝ : DecidableEq G\nr : k\nx : G\nlx : IsLeftRegular x\nhrx : ∀ (y : k), r * x • y = 0 ↔ y = 0\ny : G\nhy : y ∈ image (fun x_1 ↦ x * x_1) f.support\n⊢ ∃ a ∈ f.support, x * a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Support | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 52
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝³ : Monoid G\ninst✝² : Semiring k\ninst✝¹ : MulSemiringAction G k\nf : SkewMonoidAlgebra k G\ninst✝ : DecidableEq G\nr : k\nx : G\nrx : IsRightRegular x\nhrx : ∀ (g : G) (y : k), y * g • r = 0 ↔ y = 0\ny : G\nhy : y ∈ image (fun x_1 ↦ x_1 * x) f.support\n⊢ ∃ a ∈ f.suppo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 66
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np q : SkewPolynomial R\n⊢ (p + q).support ⊆ p.support ∪ q.support",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Equiv.instEquivLike",
"Finset.instUnion",
"congrArg",
"Finset",
"Finset.map_subset... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.CentroidHom | {
"line": 496,
"column": 23
} | {
"line": 496,
"column": 39
} | [
{
"pp": "α : Type u_5\ninst✝ : NonUnitalNonAssocSemiring α\na : α\nhc : R a = L a\nT : CentroidHom α\nhT : (toEndRingHom α) T = L a\ne1 : ∀ (d : α), T d = a * d\ne2 : ∀ (d : α), T d = d * a\n⊢ ∀ (b c : α), a * (b * c) = a * b * c",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.CentroidHom | {
"line": 497,
"column": 24
} | {
"line": 497,
"column": 40
} | [
{
"pp": "α : Type u_5\ninst✝ : NonUnitalNonAssocSemiring α\na : α\nhc : R a = L a\nT : CentroidHom α\nhT : (toEndRingHom α) T = L a\ne1 : ∀ (d : α), T d = a * d\ne2 : ∀ (d : α), T d = d * a\n⊢ ∀ (a_1 b : α), a_1 * b * a = a_1 * (b * a)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.CHSH | {
"line": 129,
"column": 4
} | {
"line": 130,
"column": 11
} | [
{
"pp": "R : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : Algebra ℝ R\ninst✝ : IsOrderedModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nP : R := 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁\nidem : P * P = 4 * P\nidem' : P = (1 / 4) • (P ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.CHSH | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 50
} | [
{
"pp": "case a\nR : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : Algebra ℝ R\ninst✝ : IsOrderedModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nP : R := ⋯\ni₁ : 0 ≤ P\n⊢ 0 ≤ 2 - (A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁)",
"usedConsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.CHSH | {
"line": 195,
"column": 37
} | {
"line": 195,
"column": 64
} | [
{
"pp": "R : Type u\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : Algebra ℝ R\ninst✝¹ : IsOrderedModule ℝ R\ninst✝ : StarModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nM : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = (↑m * a) • x\nP : R := (√2)⁻¹ • (A₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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