module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 467,
"column": 2
} | {
"line": 467,
"column": 25
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.φ 3 = C X * C W.Ψ₃ ^ 2 - C W.preΨ₄ * W.ψ₂ ^ 2",
"usedConstants": [
"Polynomial.C",
"Semigroup.toMul",
"WeierstrassCurve.ψ",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Monoid.toMulOneClass",
"con... | simp [φ, mul_assoc, sq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 560,
"column": 2
} | {
"line": 560,
"column": 31
} | [
{
"pp": "R : Type r\nS : Type s\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\nW : WeierstrassCurve R\ninst✝⁸ : Algebra R S\nA : Type u\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra S A\ninst✝⁴ : IsScalarTower R S A\nB : Type v\ninst✝³ : CommRing B\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ ... | rw [← map_Ψ₃, map_baseChange] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 560,
"column": 2
} | {
"line": 560,
"column": 31
} | [
{
"pp": "R : Type r\nS : Type s\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\nW : WeierstrassCurve R\ninst✝⁸ : Algebra R S\nA : Type u\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra S A\ninst✝⁴ : IsScalarTower R S A\nB : Type v\ninst✝³ : CommRing B\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ ... | rw [← map_Ψ₃, map_baseChange] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 560,
"column": 2
} | {
"line": 560,
"column": 31
} | [
{
"pp": "R : Type r\nS : Type s\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\nW : WeierstrassCurve R\ninst✝⁸ : Algebra R S\nA : Type u\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra S A\ninst✝⁴ : IsScalarTower R S A\nB : Type v\ninst✝³ : CommRing B\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ ... | rw [← map_Ψ₃, map_baseChange] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 218,
"column": 51
} | {
"line": 218,
"column": 56
} | [
{
"pp": "case even.right\nR : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\ndm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=\n fun {m n} {p q} ↦ natDegree_mul_le_of_le\ndp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p}... | h₁.2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 575,
"column": 55
} | {
"line": 577,
"column": 7
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : W.IsCharTwoJEqZeroNF\n⊢ W.b₈ = -W.a₄ ^ 2",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.single_pow",
"NonUnitalCommRing.toNonUnitalN... | by
rw [b₈, a₁_of_isCharTwoJEqZeroNF, a₂_of_isCharTwoJEqZeroNF]
ring1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 228,
"column": 74
} | {
"line": 228,
"column": 79
} | [
{
"pp": "case odd.right\nR : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\ndm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=\n fun {m n} {p q} ↦ natDegree_mul_le_of_le\ndp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ... | h₁.2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 366,
"column": 65
} | {
"line": 366,
"column": 74
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx₁ y₁ : F\nh₁ : W.Equation x₁ y₁\nsup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c\nh₂ : W.Equation x₁ y₁\nhy : y₁ ≠ W.negY x₁ y₁\ny : F := (y₁ - W.negY x₁ y₁) ^ 2\nhxy : (y₁ - W.negY x₁ y₁) ^ 2 ≠ 0\n⊢ 1 ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 340,
"column": 4
} | {
"line": 341,
"column": 55
} | [
{
"pp": "case succ.right\nR : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\ndp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := ⋯\nh : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n) := ⋯\nn : ℕ\nhd : (n + 1) ^ 2 - 1 = 2 * expDegree ... | rw [coeff_mul_add_eq_of_natDegree_le (dp h.1), coeff_pow_of_natDegree_le h.1, h.2,
apply_ite₂ coeff, coeff_Ψ₂Sq, coeff_one_zero, hc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 91
} | [
{
"pp": "case succ.succ.left\nR : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\ndm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=\n fun {m n} {p q} ↦ natDegree_mul_le_of_le\ndp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n}... | refine natDegree_sub_le_of_le (dm (dm natDegree_X_le (dp h.1)) ?_) (dm (dm h.1 h.1) ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 22
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Affine R\np q : R[X]\n⊢ ((CoordinateRing.basis W').repr (p • 1) + (CoordinateRing.basis W').repr (q • (mk W') Y)) 0 *\n ((CoordinateRing.basis W').repr ((q * (X ^ 3 + C W'.a₂ * X ^ 2 + C W'.a₄ * X + C W'.a₆)) • 1) +\n (CoordinateRing.basis W').repr... | Finsupp.add_apply, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.Normal.Closure | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 37
} | [
{
"pp": "case inl\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra F K\ninst✝ : Algebra F L\nh : Normal F L\nh✝ : IsEmpty (K →ₐ[F] L)\n⊢ Normal F ↥(normalClosure F K L)",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
... | rw [normalClosure, iSup_of_empty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 96,
"column": 14
} | {
"line": 96,
"column": 29
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\ng : G\na : α\nh : ∀ (j : ℤ), a ∈ fixedBy α (g ^ j)\n⊢ a ∈ fixedBy α g",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"DivInvMonoid.toZPow",
"MulAction.fixedBy",
"Membership.mem",
... | simpa using h 1 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 96,
"column": 14
} | {
"line": 96,
"column": 29
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\ng : G\na : α\nh : ∀ (j : ℤ), a ∈ fixedBy α (g ^ j)\n⊢ a ∈ fixedBy α g",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"DivInvMonoid.toZPow",
"MulAction.fixedBy",
"Membership.mem",
... | simpa using h 1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 96,
"column": 14
} | {
"line": 96,
"column": 29
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\ng : G\na : α\nh : ∀ (j : ℤ), a ∈ fixedBy α (g ^ j)\n⊢ a ∈ fixedBy α g",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"DivInvMonoid.toZPow",
"MulAction.fixedBy",
"Membership.mem",
... | simpa using h 1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 141,
"column": 2
} | {
"line": 145,
"column": 45
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsSeparable F E\nhα : IsIntegral F α\nhβ : IsIntegral F β\nf : F[X] := minpoly F α\ng : F[X] := minpoly F β\nιFE : F →+* E := algebraMap F E\nιEE' : E →+* (Polynomial.map ... | have h_root : h.eval β = 0 := by
apply eval_gcd_eq_zero
· rw [eval_comp, eval_sub, eval_mul, eval_C, eval_C, eval_X, eval_map_algebraMap, ←
Algebra.smul_def, add_sub_cancel_right, minpoly.aeval]
· rw [eval_map_algebraMap, minpoly.aeval] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 33
} | [
{
"pp": "case h.a\nF : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Finite (IntermediateField F E)\nf : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯\nx y : F\nhneq : ¬x = y\nheq : F⟮α + x • β⟯ = F⟮α + y • β⟯\nαxβ_in_K : α + x • β ∈ ... | simp only [← heq] at αyβ_in_K | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 735,
"column": 6
} | {
"line": 735,
"column": 71
} | [
{
"pp": "case neg\nR : Type r\nS : Type s\nA F : Type u\nB K : Type v\nL : Type w\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Field F\ninst✝⁴ : Field K\ninst✝³ : Field L\nW' : Affine R\nW : Affine F\ninst✝² : DecidableEq F\ninst✝¹ : DecidableEq K\ninst✝ : Decida... | exact (CoordinateRing.mk_XYIdeal'_mul_mk_XYIdeal' h₁ h₂ hxy).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 758,
"column": 4
} | {
"line": 758,
"column": 36
} | [
{
"pp": "case mp\nF : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nP : W.Point\nhP : toClass P = 0\n⊢ P = 0",
"usedConstants": []
}
] | rcases P with (_ | ⟨_, _, h, _⟩) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 825,
"column": 2
} | {
"line": 825,
"column": 97
} | [
{
"pp": "case some.some\nR : Type r\nS : Type s\nF : Type u\nK : Type v\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Field F\ninst✝⁹ : Field K\nW' : Affine R\ninst✝⁸ : DecidableEq F\ninst✝⁷ : DecidableEq K\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R F\ninst✝⁴ : Algebra S F\ninst✝³ : IsScalarTower R S F\... | · simpa only [some.injEq] using ⟨f.injective (some.inj h).left, f.injective (some.inj h).right⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.Galois.Basic | {
"line": 164,
"column": 2
} | {
"line": 167,
"column": 27
} | [
{
"pp": "F : Type u_1\nE : Type u_3\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\n⊢ IsGalois (↥⊥) E ↔ IsGalois F E",
"usedConstants": [
"IsGalois.tower_top_intermediateField",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"IntermediateField",
"OrderBot.to... | constructor
· intro h
exact IsGalois.tower_top_of_isGalois (⊥ : IntermediateField F E) F E
· intro h; infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Galois.Basic | {
"line": 164,
"column": 2
} | {
"line": 167,
"column": 27
} | [
{
"pp": "F : Type u_1\nE : Type u_3\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\n⊢ IsGalois (↥⊥) E ↔ IsGalois F E",
"usedConstants": [
"IsGalois.tower_top_intermediateField",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"IntermediateField",
"OrderBot.to... | constructor
· intro h
exact IsGalois.tower_top_of_isGalois (⊥ : IntermediateField F E) F E
· intro h; infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 248,
"column": 61
} | {
"line": 248,
"column": 70
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 255,
"column": 62
} | {
"line": 255,
"column": 71
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 273,
"column": 59
} | {
"line": 273,
"column": 68
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 277,
"column": 37
} | {
"line": 277,
"column": 46
} | [
{
"pp": "F : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero hchar2\nt... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 280,
"column": 60
} | {
"line": 280,
"column": 69
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 436,
"column": 64
} | {
"line": 436,
"column": 73
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Jacobian R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhPz : P z = 0\nhPx : P x = 0\nhP : 2 * 0 ≠ 0\n⊢ False",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"MulZeroClass.t... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 301,
"column": 33
} | {
"line": 301,
"column": 42
} | [
{
"pp": "F : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero hchar2\nt... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 100,
"column": 17
} | {
"line": 100,
"column": 26
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nhPz : P z = 0\n⊢ -P y - W'.a₁ * P x * 0 - W'.a₃ * 0 ^ 3 = -P y",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRi... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 100,
"column": 61
} | {
"line": 100,
"column": 70
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nhPz : P z = 0\n⊢ -P y - W'.a₃ * 0 = -P y",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"MulZeroCla... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 411,
"column": 17
} | {
"line": 411,
"column": 26
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nP Q : Fin 3 → F\nhQz : Q z = 0\n⊢ -((P y * 0 ^ 3 - Q y * P z ^ 3) / (P z * 0)) = 0",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCom... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 151,
"column": 59
} | {
"line": 151,
"column": 68
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP : Fin 3 → R\nhP : W'.Equation P\n⊢ ![0, -W'.dblZ P, 0] = ![-W'.dblZ P * ![0, 1, 0] x, -W'.dblZ P * ![0, 1, 0] y, -W'.dblZ P * ![0, 1, 0] z]",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"NonUnitalCommRing.toNonUnitalNonAs... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.OpenSubgroup | {
"line": 545,
"column": 2
} | {
"line": 549,
"column": 16
} | [
{
"pp": "case h\nG : Type u_2\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nW : Set G\nWClopen : IsClopen W\neinW : 1 ∈ W\nV : Set G\nhV : mulInvClosureNhd V W\nS : Subgroup G := { carrier := ⋃ n, V ^ (n + 1), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }... | have (n : ℕ) : V ^ (n + 1) ⊆ W * V ^ (n + 1) := by
intro x xin
rw [Set.mem_mul]
use 1, einW, x, xin
rw [one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Algebra.GroupCompletion | {
"line": 124,
"column": 10
} | {
"line": 124,
"column": 56
} | [
{
"pp": "M : Type u_1\nR : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝² : UniformSpace α\ninst✝¹ : AddGroup α\ninst✝ : IsUniformAddGroup α\nn : ℕ\na✝ : Completion α\na : α\n⊢ (n + 1) • ↑a = n • ↑a + ↑a",
"usedConstants": [
"Eq.mpr",
"UniformSpace.Completion.coe'",
"instHSMul",
"AddMo... | rw [← coe_smul, succ_nsmul, coe_add, coe_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.GroupCompletion | {
"line": 124,
"column": 10
} | {
"line": 124,
"column": 56
} | [
{
"pp": "M : Type u_1\nR : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝² : UniformSpace α\ninst✝¹ : AddGroup α\ninst✝ : IsUniformAddGroup α\nn : ℕ\na✝ : Completion α\na : α\n⊢ (n + 1) • ↑a = n • ↑a + ↑a",
"usedConstants": [
"Eq.mpr",
"UniformSpace.Completion.coe'",
"instHSMul",
"AddMo... | rw [← coe_smul, succ_nsmul, coe_add, coe_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.GroupCompletion | {
"line": 124,
"column": 10
} | {
"line": 124,
"column": 56
} | [
{
"pp": "M : Type u_1\nR : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝² : UniformSpace α\ninst✝¹ : AddGroup α\ninst✝ : IsUniformAddGroup α\nn : ℕ\na✝ : Completion α\na : α\n⊢ (n + 1) • ↑a = n • ↑a + ↑a",
"usedConstants": [
"Eq.mpr",
"UniformSpace.Completion.coe'",
"instHSMul",
"AddMo... | rw [← coe_smul, succ_nsmul, coe_add, coe_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.UniformRing | {
"line": 284,
"column": 14
} | {
"line": 286,
"column": 43
} | [
{
"pp": "α : Type u_1\ninst✝⁹ : UniformSpace α\ninst✝⁸ : Semiring α\nβ : Type u_2\ninst✝⁷ : UniformSpace β\ninst✝⁶ : Semiring β\ninst✝⁵ : IsTopologicalSemiring β\nγ : Type u_3\ninst✝⁴ : UniformSpace γ\ninst✝³ : Semiring γ\ninst✝² : IsTopologicalSemiring γ\ninst✝¹ : T2Space γ\ninst✝ : CompleteSpace γ\ni : α →+* ... | by
convert IsDenseInducing.extend_eq (ue.isDenseInducing dr) hf.continuous 1
exacts [i.map_one.symm, f.map_one.symm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 12
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nthis : LinearOrderedCommGroupWithZero (ValueGroup₀ v) := instLinearOrderedCommGroupWithZero\nγ γ₀ : (ValueGroup₀ v)ˣ\nh : γ₀ * γ₀ ≤ γ\n⊢ ∃ j,\n ↑(v.ltAddSubgroup ((Units.map ↑embedding) j)) * ↑(v... | use γ₀ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 303,
"column": 93
} | {
"line": 306,
"column": 40
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nr : ValueGroup₀ v\n⊢ IsClosed {x | v.restrict x = r}",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWit... | by
rcases eq_or_ne r 0 with rfl | hr
· simpa using isClosed_closedBall R 0
exact isClopen_sphere _ hr |>.isClosed | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.BilinearForm.DualLattice | {
"line": 114,
"column": 39
} | {
"line": 114,
"column": 48
} | [
{
"pp": "case intro.a.h\nR : Type u_4\nS : Type u_2\nM : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Field S\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : IsScalarTower R S M\nB : BilinForm S M\nι : Type u_1\ninst✝¹ : Finite ι\ninst✝ : DecidableEq ι\nhB : B.Nonde... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.GaussLemma | {
"line": 222,
"column": 2
} | {
"line": 234,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsFractionRing R K\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : K[X]\nh0 : p ≠ 0\nh : IsUnit (integerNormalization R⁰ p).primPart\n⊢ IsUnit p",
"usedConstants": [
"Iff.mpr",
"Ad... | rcases isUnit_iff.1 h with ⟨_, ⟨u, rfl⟩, hu⟩
obtain ⟨c, c0, hc⟩ := integerNormalization_spec R⁰ p
rw [Algebra.smul_def, algebraMap_apply] at hc
apply isUnit_of_mul_isUnit_right
rw [← hc, (integerNormalization R⁰ p).eq_C_content_mul_primPart, ← hu, ← map_mul, isUnit_iff]
refine
⟨algebraMap R K ((integerNor... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.GaussLemma | {
"line": 222,
"column": 2
} | {
"line": 234,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsFractionRing R K\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : K[X]\nh0 : p ≠ 0\nh : IsUnit (integerNormalization R⁰ p).primPart\n⊢ IsUnit p",
"usedConstants": [
"Iff.mpr",
"Ad... | rcases isUnit_iff.1 h with ⟨_, ⟨u, rfl⟩, hu⟩
obtain ⟨c, c0, hc⟩ := integerNormalization_spec R⁰ p
rw [Algebra.smul_def, algebraMap_apply] at hc
apply isUnit_of_mul_isUnit_right
rw [← hc, (integerNormalization R⁰ p).eq_C_content_mul_primPart, ← hu, ← map_mul, isUnit_iff]
refine
⟨algebraMap R K ((integerNor... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 547,
"column": 4
} | {
"line": 547,
"column": 60
} | [
{
"pp": "case mk\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx y z : R\nb c : ValueGroupWithZero R\nhbc : b ≤ c\na₁ : R\na₂ : ↥(posSubmonoid R)\nhab : ValueGroupWithZero.mk a₁ a₂ ≤ b\n⊢ ValueGroupWithZero.mk a₁ a₂ ≤ c",
"usedConstants": [
"CommSemiring.toSemiring",
"Membership.me... | induction b using ValueGroupWithZero.ind with | mk b₁ b₂
=> _ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 60,
"column": 88
} | {
"line": 62,
"column": 99
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : IsDomain R\ninst✝⁷ : Algebra R S\nK : Type u_3\ninst✝⁶ : Field K\ninst✝⁵ : Algebra R K\ninst✝⁴ : IsFractionRing R K\ninst✝³ : IsIntegrallyClosed R\ninst✝² : IsDomain S\ninst✝¹ : Algebra K S\ninst✝ : IsScalarTower R K S\ns :... | by
let L := FractionRing S
rw [← isIntegrallyClosed_eq_field_fractions K L hs, algebraMap_eq (IsFractionRing.injective S L)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 166,
"column": 36
} | {
"line": 166,
"column": 48
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\ns : S\np : R[X]\nhp : (Polynomial.aeval s) p = 0\nh₀ : p ≠ 0\npmin : ∀ (q : R[X]), q.Monic → (Polynomial.aeval ... | by simp [h₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1121,
"column": 14
} | {
"line": 1121,
"column": 45
} | [
{
"pp": "R✝ : Type u_1\ninst✝⁵ : CommRing R✝\ninst✝⁴ : ValuativeRel R✝\nR : Type u_2\nΓ : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\n⊢ ValueGroupWithZero.lift (fun r s ↦ (restrict₀ v) r / (restrict₀ v) ↑s) ⋯ 1 = 1",... | by simp [ValueGroup₀.restrict₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 102,
"column": 48
} | {
"line": 102,
"column": 57
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhP : W'.Equation P\nhPz : P z = 0\n⊢ -P y - W'.a₁ * 0 - W'.a₃ * 0 = -P y",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.h... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 102,
"column": 68
} | {
"line": 102,
"column": 77
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhP : W'.Equation P\nhPz : P z = 0\n⊢ -P y - W'.a₃ * 0 = -P y",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AlgebraicIndependent.Adjoin | {
"line": 47,
"column": 20
} | {
"line": 47,
"column": 28
} | [
{
"pp": "case hC.a\nι : Type u_1\nF : Type u_2\nE : Type u_3\nx : ι → E\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nhx : AlgebraicIndependent F x\ni : FractionRing (MvPolynomial ι F) →ₐ[F] E := IsFractionRing.liftAlgHom ⋯\nx✝ : F\n⊢ (((↑i).comp (algebraMap (MvPolynomial ι F) (FractionRing (MvPolyn... | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.AlgebraicIndependent.Adjoin | {
"line": 47,
"column": 20
} | {
"line": 47,
"column": 28
} | [
{
"pp": "case hX\nι : Type u_1\nF : Type u_2\nE : Type u_3\nx : ι → E\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nhx : AlgebraicIndependent F x\ni : FractionRing (MvPolynomial ι F) →ₐ[F] E := IsFractionRing.liftAlgHom ⋯\ni✝ : ι\n⊢ ((↑i).comp (algebraMap (MvPolynomial ι F) (FractionRing (MvPolynomi... | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 849,
"column": 84
} | {
"line": 850,
"column": 36
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhX : X ⊆ M.E\n⊢ M.IsBasis I X ↔ M.Indep I ∧ I ⊆ X ∧ ∀ (J : Set α), M.Indep J → I ⊆ J → J ⊆ X → I = J",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Iff.rfl",
"and_iff_left",
"Matroid.Indep",
"id",
"M... | by
rw [isBasis_iff', and_iff_left hX] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 232,
"column": 8
} | {
"line": 232,
"column": 59
} | [
{
"pp": "α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug :\n ∀ ⦃I J : Set α⦄, Indep I → I.Finite → Indep J → J.Finite → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nindep_compact : ∀ (I : Set α), (∀ J ⊆ I, J.... | refine hIe <| indep_compact _ fun J hJss hJfin ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 234,
"column": 19
} | {
"line": 234,
"column": 24
} | [
{
"pp": "α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug :\n ∀ ⦃I J : Set α⦄, Indep I → I.Finite → Indep J → J.Finite → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nindep_compact : ∀ (I : Set α), (∀ J ⊆ I, J.... | hImax | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 28
} | [
{
"pp": "α : Type u_1\n⊢ (emptyOn α)✶ = emptyOn α",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Matroid.dual",
"id",
"Matroid.emptyOn",
"propext",
"Set.instEmptyCollection",
"EmptyCollection.emptyCollection",
"Eq.symm",
"Eq",
... | rw [← ground_eq_empty_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 28
} | [
{
"pp": "α : Type u_1\nM : Matroid α\n⊢ M = emptyOn α ∨ M.Nonempty",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"id",
"Matroid.emptyOn",
"propext",
"Set.instEmptyCollection",
"Or",
"EmptyCollection.emptyCollection",
"Eq.symm",
"Eq"... | rw [← ground_eq_empty_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 28
} | [
{
"pp": "α : Type u_1\ninst✝ : IsEmpty α\nM : Matroid α\n⊢ M = emptyOn α",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"id",
"Matroid.emptyOn",
"propext",
"Set.instEmptyCollection",
"EmptyCollection.emptyCollection",
"Eq.symm",
"Eq",
... | rw [← ground_eq_empty_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Map | {
"line": 251,
"column": 15
} | {
"line": 251,
"column": 35
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nN : Matroid β\nf : α → β\n⊢ N.comap f ↾ f ⁻¹' N.E = N.comap f",
"usedConstants": [
"Eq.mpr",
"Matroid.restrict_eq_self_iff",
"congrArg",
"Matroid.E",
"id",
"Set.preimage",
"propext",
"Matroid.comap",
"Matroid.restrict... | restrict_eq_self_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Map | {
"line": 403,
"column": 2
} | {
"line": 403,
"column": 53
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nM : Matroid α\nhf : InjOn f M.E\nB : Set α\nhB : B ⊆ M.E\n⊢ (∃ B₀, M.IsBase B₀ ∧ f '' B = f '' B₀) ↔ M.IsBase B",
"usedConstants": [
"Exists",
"Matroid.IsBase",
"And",
"And.intro",
"Iff.intro",
"Exists.intro",
"Set.ima... | refine ⟨fun ⟨J, hJ, hIJ⟩ ↦ ?_, fun h ↦ ⟨B, h, rfl⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Matroid.Map | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 74
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nI : Set α\nM : Matroid α\nX : Set α\nhIX : M.IsBasis I X\nf : α → β\nhf : InjOn f M.E\ne : α\nhe : e ∈ X\nhe' : f e ∉ f '' I\nhss : insert e I ⊆ M.E\n⊢ (M.map f hf).Dep (insert (f e) (f '' I))",
"usedConstants": [
"Eq.mpr",
"Matroid.Dep",
"_private.Math... | rw [← not_indep_iff (by simpa [← image_insert_eq] using image_mono hss)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Map | {
"line": 536,
"column": 42
} | {
"line": 538,
"column": 20
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Matroid α\nB : Set α\nhB : M.IsBase B\nf : α ↪ β\n⊢ (M.mapEmbedding f).IsBase (⇑f '' B)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.mapEmbedding",
"Exists",
"Matroid.IsBase",
"id",
"Matroid.mapEmbedding._proof_1",
... | by
rw [Matroid.mapEmbedding, map_isBase_iff]
exact ⟨B, hB, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Map | {
"line": 676,
"column": 52
} | {
"line": 676,
"column": 87
} | [
{
"pp": "α : Type u_1\nM N : Matroid α\nhN : N.E = M.E\nh : M.restrictSubtype M.E = N.restrictSubtype M.E\nI : Set α\nhI : I ⊆ M.E\n⊢ (N.restrictSubtype M.E).Indep (M.E ↓∩ I) ↔ N.Indep I",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Membership.mem",
"Matroid.Indep",
... | restrictSubtype_indep_iff_of_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 450,
"column": 4
} | {
"line": 450,
"column": 13
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhP : W'.Equation P\nhPz : P z = 0\n⊢ ![0, P y ^ 4, 0] = ![P y ^ 4 * ![0, 1, 0] x, P y ^ 4 * ![0, 1, 0] y, P y ^ 4 * ![0, 1, 0] z]",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNon... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 456,
"column": 43
} | {
"line": 456,
"column": 52
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Projective F\nP Q : Fin 3 → F\nhP : W.Equation P\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z = Q y * P z\nhy' : P y * Q z = W.negY Q * P z\n⊢ ![0, W.dblU P, 0] = ![W.dblU P * ![0, 1, 0] x, W.dblU P * ![0, 1, 0] y, W.dblU P * ![0, 1, 0] z]",... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 132,
"column": 2
} | {
"line": 134,
"column": 46
} | [
{
"pp": "α✝ : Type u_1\nM✝ : Matroid α✝\nC✝ : Set α✝\nα : Type u_1\nM : Matroid α\nC : Set α\nhCE : C ⊆ M.E\n⊢ M.IsCircuit C ↔ M.Dep C ∧ ∀ e ∈ C, M.Indep (C \\ {e})",
"usedConstants": [
"Matroid.Dep",
"Classical.not_not._simp_1",
"congrArg",
"Set.minimal_iff_forall_diff_singleton",
... | simp [isCircuit_iff_minimal_not_indep hCE, ← not_indep_iff hCE,
minimal_iff_forall_diff_singleton (P := (¬ M.Indep ·))
(fun _ _ hY hYX hX ↦ hY <| hX.subset hYX)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 487,
"column": 4
} | {
"line": 487,
"column": 31
} | [
{
"pp": "case inl\nα : Type u_2\nM : Matroid α\nJs : Set (Set α)\nhne : Js.Nonempty\nhiI : M.Indep (⋂₀ Js)\ne : α\nhe : ∀ i ∈ Js, e ∈ M.closure i\nhe' : M.Indep (insert e (⋂₀ Js))\nJ : Set α\nheJ : insert e (⋂₀ Js) ⊆ J\nf : α\nhI : M.Indep (insert f (J \\ {e}))\nhiX : ⋂₀ Js ⊆ insert f (J \\ {e})\nheEI : e ∈ M.E... | exact hIs.trans diff_subset | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 502,
"column": 2
} | {
"line": 502,
"column": 42
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nC : Set α\ninst✝ : M.Finitary\nhC : M.IsCircuit C\nJ : Set α\nhJC : J ⊆ C\nhJfin : J.Finite\nhJ : ¬M.Indep J\n⊢ C.Finite",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.Finite",
"id",
"Matroid.IsCircuit.eq_of_not_indep_subset",
"Eq.sy... | rwa [← hC.eq_of_not_indep_subset hJ hJC] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 873,
"column": 2
} | {
"line": 873,
"column": 95
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nS B : Set α\nhS : M.Spanning S\n⊢ M.IsBasis B S ↔ M.IsBase B ∧ B ⊆ S",
"usedConstants": [
"Matroid.IsBase",
"HasSubset.Subset",
"Matroid.IsBasis.subset",
"And",
"And.right",
"And.left",
"And.intro",
"Iff.intro",
"Mat... | refine ⟨fun h ↦ ⟨?_, h.subset⟩, fun h ↦ h.1.indep.isBasis_of_subset_of_subset_closure h.2 ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 907,
"column": 19
} | {
"line": 907,
"column": 73
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nX B : Set α\nhB : M.IsBase B\nhX : M.Spanning X\nhXB : X ⊆ B\nB' : Set α\nhB' : M.IsBase B'\nhB'X : B' ⊆ X\n⊢ B ⊆ X",
"usedConstants": [
"Eq.mpr",
"congrArg",
"HasSubset.Subset.trans",
"id",
"Set.instIsTransSubset",
"HasSubset.Subset"... | by rwa [← hB'.eq_of_subset_isBase hB (hB'X.trans hXB)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 612,
"column": 45
} | {
"line": 612,
"column": 77
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nK X : Set α\nhX : ¬M.Spanning (M.E \\ X)\n⊢ ¬M.Spanning (M.E \\ (X ∩ M.E))",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"id",
"Set.instInter",
"Inter.inter",
"SDiff.sdiff",
"Set.diff_inter_self_eq_diff",
"... | by rwa [diff_inter_self_eq_diff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 931,
"column": 19
} | {
"line": 931,
"column": 46
} | [
{
"pp": "case h\nα : Type u_2\nM : Matroid α\nX R I : Set α\nhI : (M ↾ R).IsBasis' I X\nhI' : M.IsBasis' I (X ∩ R)\nhIR : I ⊆ R\ne : α\n⊢ e ∈ (M ↾ R).E ∧ ((M ↾ R).Indep (insert e I) → e ∈ I) ↔ e ∈ M.closure I ∧ e ∈ R ∨ e ∈ R \\ M.E",
"usedConstants": [
"Eq.mpr",
"Matroid.Indep.mem_closure_iff'",... | hI'.indep.mem_closure_iff', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 970,
"column": 62
} | {
"line": 970,
"column": 89
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nM : Matroid β\nf : α → β\nX I : Set α\nhI : (M.comap f).IsBasis' I X\nhI' : M.IsBasis' (f '' I) (f '' X)\nhIinj : InjOn f I\n⊢ ∀ (x : α), f x ∈ M.E ∧ ((M.comap f).Indep (insert x I) → x ∈ I) ↔ f x ∈ M.closure (f '' I)",
"usedConstants": [
"Eq.mpr",
"Matroid.I... | hI'.indep.mem_closure_iff', | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 583,
"column": 51
} | {
"line": 583,
"column": 96
} | [
{
"pp": "α : Type u_1\nX : Set α\nβ : Type u_2\nf : α → β\nM : Matroid α\nhf : InjOn f M.E\nhX : X ⊆ M.E\nI : Set α\nhI : M.IsBasis I X\n⊢ (f '' I).encard = I.encard",
"usedConstants": [
"Eq.mpr",
"Set.encard",
"congrArg",
"Matroid.E",
"id",
"Matroid.Indep.subset_ground",... | (hf.mono hI.indep.subset_ground).encard_image | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 639,
"column": 26
} | {
"line": 639,
"column": 35
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ Disjoint (B ∩ X) (M.E \\ B)",
"usedConstants": [
"Eq.mpr",
"CompleteLattice.instOmegaCompletePartial... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 639,
"column": 26
} | {
"line": 639,
"column": 35
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ Disjoint (B ∩ X) (M.E \\ B)",
"usedConstants": [
"Eq.mpr",
"CompleteLattice.instOmegaCompletePartial... | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 639,
"column": 26
} | {
"line": 639,
"column": 35
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ Disjoint (B ∩ X) (M.E \\ B)",
"usedConstants": [
"Eq.mpr",
"CompleteLattice.instOmegaCompletePartial... | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 639,
"column": 60
} | {
"line": 639,
"column": 69
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ Disjoint (M.E \\ B ∩ (M.E \\ X)) X",
"usedConstants": [
"Eq.mpr",
"CompleteLattice.instOmegaComplete... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 639,
"column": 60
} | {
"line": 639,
"column": 69
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ Disjoint (M.E \\ B ∩ (M.E \\ X)) X",
"usedConstants": [
"Eq.mpr",
"CompleteLattice.instOmegaComplete... | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 639,
"column": 60
} | {
"line": 639,
"column": 69
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ Disjoint (M.E \\ B ∩ (M.E \\ X)) X",
"usedConstants": [
"Eq.mpr",
"CompleteLattice.instOmegaComplete... | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 640,
"column": 24
} | {
"line": 640,
"column": 33
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ B ∩ X ∪ M.E \\ B = M.E \\ B ∩ (M.E \\ X) ∪ X",
"usedConstants": [
"Iff.mpr",
"_private.Mathlib.Combi... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 640,
"column": 24
} | {
"line": 640,
"column": 33
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ B ∩ X ∪ M.E \\ B = M.E \\ B ∩ (M.E \\ X) ∪ X",
"usedConstants": [
"Iff.mpr",
"_private.Mathlib.Combi... | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 640,
"column": 24
} | {
"line": 640,
"column": 33
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ B ∩ X ∪ M.E \\ B = M.E \\ B ∩ (M.E \\ X) ∪ X",
"usedConstants": [
"Iff.mpr",
"_private.Mathlib.Combi... | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 353,
"column": 51
} | {
"line": 353,
"column": 74
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nx : K\n⊢ embedding (v.restrict x) = v x",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
... | Valuation.restrict_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Algebraic.MvPolynomial | {
"line": 70,
"column": 23
} | {
"line": 71,
"column": 41
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\ni : σ\nf : R[X]\nhf : Transcendental R f\nthis : Transcendental (↥(supported R ∅)) ((Polynomial.aeval (X i)) f)\ng : R ≃ₐ[R] ↥(supported R ∅) := (Algebra.botEquivOfInjective ⋯).symm.trans ((supported R ∅).equivOfEq ⊥ ⋯).symm\n⊢ ¬IsAlgebraic R ((Polynomial... | ← isAlgebraic_ringHom_iff_of_comp_eq g (RingHom.id (MvPolynomial σ R))
Function.injective_id (by ext1; rfl), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 425,
"column": 49
} | {
"line": 425,
"column": 72
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\n⊢ embedding (extensionValuation.restrict ↑⋯.choose) = embedding 1",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinearOr... | Valuation.restrict_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 454,
"column": 10
} | {
"line": 457,
"column": 46
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\na b : ValueGroup₀ v\nx : K := ⋯.choose\nhx_def : x = ⋯.choose\nhx : (restrict₀ v) x = a\ny : K := ⋯.choose\nhy_def : y = ⋯.choose\nhy : (restrict₀ v) y = b\nxy : K := ⋯.choose\nhxy_def ... | · ext
simp only [Units.val_mk0]
rw [Units.val_mk0, ← map_mul, ← v.restrict_inj, map_mul]
simp [v.restrict_def, hx, hy, hxy] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.AlgebraicIndependent.Basic | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 92
} | [
{
"pp": "R : Type u_2\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nA' : Type v\ninst✝¹ : CommRing A'\ninst✝ : Algebra R A'\nf : A →ₐ[R] A'\nhf : Surjective ⇑f\n⊢ trdeg R A' ≤ trdeg R A",
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Cardinal.li... | rw [← (trdeg R A).lift_id, ← (trdeg R A').lift_id]; exact lift_trdeg_le_of_surjective f hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AlgebraicIndependent.Basic | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 92
} | [
{
"pp": "R : Type u_2\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nA' : Type v\ninst✝¹ : CommRing A'\ninst✝ : Algebra R A'\nf : A →ₐ[R] A'\nhf : Surjective ⇑f\n⊢ trdeg R A' ≤ trdeg R A",
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Cardinal.li... | rw [← (trdeg R A).lift_id, ← (trdeg R A').lift_id]; exact lift_trdeg_le_of_surjective f hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 787,
"column": 46
} | {
"line": 787,
"column": 55
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Projective F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\n⊢ ![0, addU P Q, 0] = ![addU P Q * ![0, 1, 0] x, addU P Q * ![0, 1, 0] y, addU P Q * ![0, 1, 0] z]",
"usedConstants": [
"Eq.mpr",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 534,
"column": 44
} | {
"line": 534,
"column": 67
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\ns : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nthis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.map ↑embeddi... | Valuation.restrict_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 540,
"column": 16
} | {
"line": 540,
"column": 39
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\ns : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nthis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.ma... | Valuation.restrict_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 551,
"column": 16
} | {
"line": 551,
"column": 39
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\ns : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nthis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.ma... | Valuation.restrict_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 331,
"column": 27
} | {
"line": 331,
"column": 66
} | [
{
"pp": "case inl.ha\nR : Type u_1\nA : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : FaithfulSMul R A\ninst✝ : NoZeroDivisors A\ns t : Set A\na : A\nh : (matroid R A).IsBasis s t\nh✝ : Subsingleton A\n⊢ ¬IsAlgebraic (↥(adjoin R s)) a",
"usedConstants": [
"Subalgebra... | apply is_transcendental_of_subsingleton | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 331,
"column": 27
} | {
"line": 331,
"column": 66
} | [
{
"pp": "case inl.hb\nR : Type u_1\nA : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : FaithfulSMul R A\ninst✝ : NoZeroDivisors A\ns t : Set A\na : A\nh : (matroid R A).IsBasis s t\nh✝ : Subsingleton A\n⊢ ¬IsAlgebraic (↥(adjoin R t)) a",
"usedConstants": [
"Subalgebra... | apply is_transcendental_of_subsingleton | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Trace.Defs | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 95
} | [
{
"pp": "case pos\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : StrongRankCondition R\ninst✝ : Free R S\nx : R\nH : ∃ s, Nonempty (Basis (↥s) R S)\n⊢ (trace R S) ((algebraMap R S) x) = finrank R S • x",
"usedConstants": [
"Eq.mpr",
"Exists.... | rw [trace_algebraMap_of_basis H.choose_spec.some, finrank_eq_card_basis H.choose_spec.some] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Trace.Defs | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 95
} | [
{
"pp": "case pos\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : StrongRankCondition R\ninst✝ : Free R S\nx : R\nH : ∃ s, Nonempty (Basis (↥s) R S)\n⊢ (trace R S) ((algebraMap R S) x) = finrank R S • x",
"usedConstants": [
"Eq.mpr",
"Exists.... | rw [trace_algebraMap_of_basis H.choose_spec.some, finrank_eq_card_basis H.choose_spec.some] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Trace.Defs | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 95
} | [
{
"pp": "case pos\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : StrongRankCondition R\ninst✝ : Free R S\nx : R\nH : ∃ s, Nonempty (Basis (↥s) R S)\n⊢ (trace R S) ((algebraMap R S) x) = finrank R S • x",
"usedConstants": [
"Eq.mpr",
"Exists.... | rw [trace_algebraMap_of_basis H.choose_spec.some, finrank_eq_card_basis H.choose_spec.some] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Trace.Defs | {
"line": 162,
"column": 28
} | {
"line": 162,
"column": 63
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Free R S\ninst✝² : Free R T\ninst✝¹ : Module.Finite R S\ninst✝ : Module.Finite R T\np : S × T\n⊢ (trace R (S × T)) p = ((trace R S).coprod (trace... | rw [coprod_apply, trace_prod_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Trace.Defs | {
"line": 162,
"column": 28
} | {
"line": 162,
"column": 63
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Free R S\ninst✝² : Free R T\ninst✝¹ : Module.Finite R S\ninst✝ : Module.Finite R T\np : S × T\n⊢ (trace R (S × T)) p = ((trace R S).coprod (trace... | rw [coprod_apply, trace_prod_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Trace.Defs | {
"line": 162,
"column": 28
} | {
"line": 162,
"column": 63
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Free R S\ninst✝² : Free R T\ninst✝¹ : Module.Finite R S\ninst✝ : Module.Finite R T\np : S × T\n⊢ (trace R (S × T)) p = ((trace R S).coprod (trace... | rw [coprod_apply, trace_prod_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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