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.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.ΨSq 2 = W.Ψ₂Sq",
"usedConstants": [
"one_pow",
"MulOne.toOne",
"Polynomial.instOne",
"HMul.hMul",
"even_two._simp_1",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"Add... | simp [ΨSq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.ΨSq 2 = W.Ψ₂Sq",
"usedConstants": [
"one_pow",
"MulOne.toOne",
"Polynomial.instOne",
"HMul.hMul",
"even_two._simp_1",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"Add... | simp [ΨSq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\nn : ℤ\n⊢ W.ΨSq (-n) = W.ΨSq n",
"usedConstants": [
"Int.instAddCommGroup",
"Polynomial.instOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Polynomial.instNeg",
"HMul.hMul",
"even_two._simp_1",
... | simp [ΨSq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\nn : ℤ\n⊢ W.ΨSq (-n) = W.ΨSq n",
"usedConstants": [
"Int.instAddCommGroup",
"Polynomial.instOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Polynomial.instNeg",
"HMul.hMul",
"even_two._simp_1",
... | simp [ΨSq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\nn : ℤ\n⊢ W.ΨSq (-n) = W.ΨSq n",
"usedConstants": [
"Int.instAddCommGroup",
"Polynomial.instOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Polynomial.instNeg",
"HMul.hMul",
"even_two._simp_1",
... | simp [ΨSq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 227,
"column": 2
} | {
"line": 240,
"column": 9
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nb c d : R\nm : ℤ\n⊢ preNormEDS b c d (2 * m + 1) =\n (preNormEDS b c d (m + 2) * preNormEDS b c d m ^ 3 * if Even m then b else 1) -\n preNormEDS b c d (m - 1) * preNormEDS b c d (m + 1) ^ 3 * if Even m then 1 else b",
"usedConstants": [
"Int.instAddCom... | induction m using Int.negInduction with
| nat m =>
rcases m with _ | _ | _
iterate 2 simp
simp_rw [Nat.cast_succ, Int.add_sub_cancel, Int.even_add_one, not_not, Int.even_coe_nat]
norm_cast
simpa only [preNormEDS_ofNat] using preNormEDS'_odd ..
| neg ih m =>
rcases m with _ | m
· simp
... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 227,
"column": 2
} | {
"line": 240,
"column": 9
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nb c d : R\nm : ℤ\n⊢ preNormEDS b c d (2 * m + 1) =\n (preNormEDS b c d (m + 2) * preNormEDS b c d m ^ 3 * if Even m then b else 1) -\n preNormEDS b c d (m - 1) * preNormEDS b c d (m + 1) ^ 3 * if Even m then 1 else b",
"usedConstants": [
"Int.instAddCom... | induction m using Int.negInduction with
| nat m =>
rcases m with _ | _ | _
iterate 2 simp
simp_rw [Nat.cast_succ, Int.add_sub_cancel, Int.even_add_one, not_not, Int.even_coe_nat]
norm_cast
simpa only [preNormEDS_ofNat] using preNormEDS'_odd ..
| neg ih m =>
rcases m with _ | m
· simp
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 227,
"column": 2
} | {
"line": 240,
"column": 9
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nb c d : R\nm : ℤ\n⊢ preNormEDS b c d (2 * m + 1) =\n (preNormEDS b c d (m + 2) * preNormEDS b c d m ^ 3 * if Even m then b else 1) -\n preNormEDS b c d (m - 1) * preNormEDS b c d (m + 1) ^ 3 * if Even m then 1 else b",
"usedConstants": [
"Int.instAddCom... | induction m using Int.negInduction with
| nat m =>
rcases m with _ | _ | _
iterate 2 simp
simp_rw [Nat.cast_succ, Int.add_sub_cancel, Int.even_add_one, not_not, Int.even_coe_nat]
norm_cast
simpa only [preNormEDS_ofNat] using preNormEDS'_odd ..
| neg ih m =>
rcases m with _ | m
· simp
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 337,
"column": 6
} | {
"line": 337,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nW : WeierstrassCurve R\ninst✝¹ : W.IsCharThreeJNeZeroNF\ninst✝ : CharP R 3\n⊢ W.c₄ = W.a₂ ^ 2",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toA... | c₄_of_isCharThreeJNeZeroNF | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Solvable | {
"line": 172,
"column": 2
} | {
"line": 174,
"column": 41
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : IsSolvable G\nH : Subgroup G\nhH : H ≠ ⊥\n⊢ ⁅H, H⁆ < H",
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"MonoidHom.range",
"Preorder.toLT",
"subgroup_solvable_of_solvable",
"Subgroup.map",
"congrArg",
"Subgroup.subt... | rw [← nontrivial_iff_ne_bot] at hH
rw [← H.range_subtype, MonoidHom.range_eq_map, ← map_commutator, map_subtype_lt_map_subtype]
exact commutator_lt_top_of_nontrivial H | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Solvable | {
"line": 172,
"column": 2
} | {
"line": 174,
"column": 41
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : IsSolvable G\nH : Subgroup G\nhH : H ≠ ⊥\n⊢ ⁅H, H⁆ < H",
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"MonoidHom.range",
"Preorder.toLT",
"subgroup_solvable_of_solvable",
"Subgroup.map",
"congrArg",
"Subgroup.subt... | rw [← nontrivial_iff_ne_bot] at hH
rw [← H.range_subtype, MonoidHom.range_eq_map, ← map_commutator, map_subtype_lt_map_subtype]
exact commutator_lt_top_of_nontrivial H | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1053,
"column": 19
} | {
"line": 1053,
"column": 55
} | [
{
"pp": "I : Type u\ninst✝³ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝² : IsCofiltered I\ninst✝¹ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)\ni : I\nU : (D.obj i).Opens\nhU' : IsCompact ↑U\nhU : IsAffineOpen (c.π.app i ⁻¹ᵁ U)\nt... | by simp [← Scheme.Hom.comp_preimage] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Normal.Basic | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 34
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝² : Field F\ninst✝¹ : Field K\ninst✝ : Algebra F K\nι : Type u_3\nt : ι → IntermediateField F K\nh : ∀ (i : ι), Normal F ↥(t i)\nx : ↥(⨆ i, t i)\ns : Finset ((i : ι) × ↥(t i))\nhx : ↑x ∈ ⨆ i ∈ s, adjoin F ((minpoly F i.snd).rootSet K)\nE : IntermediateField F K := ⨆ i ∈... | rwa [Polynomial.map_map] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.FieldTheory.Normal.Closure | {
"line": 169,
"column": 69
} | {
"line": 172,
"column": 54
} | [
{
"pp": "F : 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\n⊢ Normal F ↥(normalClosure F K L)",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"NonUnitalCommRing.toNonUnitalNon... | by
obtain _ | φ := isEmpty_or_nonempty (K →ₐ[F] L)
· rw [normalClosure, iSup_of_empty]; exact Normal.of_algEquiv (botEquiv F L).symm
· exact (isNormalClosure_normalClosure F K L).normal | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Normal.Closure | {
"line": 249,
"column": 4
} | {
"line": 249,
"column": 32
} | [
{
"pp": "case refine_2\nF : Type u_1\nL : Type u_3\ninst✝³ : Field F\ninst✝² : Field L\ninst✝¹ : Algebra F L\nK : IntermediateField F L\ninst✝ : Normal F L\nf : Gal(L/F)\n⊢ map (↑f) K ≤ ⨆ f, map f K",
"usedConstants": [
"IntermediateField.instPartialOrder",
"le_rfl",
"AlgEquiv.toAlgHom",
... | exact le_iSup_of_le f le_rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.Normal.Closure | {
"line": 249,
"column": 4
} | {
"line": 249,
"column": 32
} | [
{
"pp": "case refine_2\nF : Type u_1\nL : Type u_3\ninst✝³ : Field F\ninst✝² : Field L\ninst✝¹ : Algebra F L\nK : IntermediateField F L\ninst✝ : Normal F L\nf : Gal(L/F)\n⊢ map (↑f) K ≤ ⨆ f, map f K",
"usedConstants": [
"IntermediateField.instPartialOrder",
"le_rfl",
"AlgEquiv.toAlgHom",
... | exact le_iSup_of_le f le_rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Normal.Closure | {
"line": 249,
"column": 4
} | {
"line": 249,
"column": 32
} | [
{
"pp": "case refine_2\nF : Type u_1\nL : Type u_3\ninst✝³ : Field F\ninst✝² : Field L\ninst✝¹ : Algebra F L\nK : IntermediateField F L\ninst✝ : Normal F L\nf : Gal(L/F)\n⊢ map (↑f) K ≤ ⨆ f, map f K",
"usedConstants": [
"IntermediateField.instPartialOrder",
"le_rfl",
"AlgEquiv.toAlgHom",
... | exact le_iSup_of_le f le_rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1199,
"column": 4
} | {
"line": 1204,
"column": 33
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : LocallyOfFinitePresentation f\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.ob... | choose k ak hk hk' using fun j ↦ exists_π_app_comp_eq_of_locallyOfFinitePresentation_of_isAffine
_ (t𝒱 j) (f.resLE _ _ j.1.1.2.prop.2) _ (isLimitOpensCone D c hc i' (𝒱' j))
(a.resLE _ _ ((h𝒱'𝒰 _).trans_le j.1.prop.2)) (by
ext k
simp [t𝒱, Hom.resLE_comp_resLE, show c.π.app k.left ≫ t.app k.l... | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.FieldTheory.Galois.Basic | {
"line": 114,
"column": 36
} | {
"line": 114,
"column": 38
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nhα : F⟮α⟯ = ⊤\ne : E\n⊢ e ∈ F⟮α⟯",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrA... | hα | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Int.WithZero | {
"line": 54,
"column": 6
} | {
"line": 54,
"column": 46
} | [
{
"pp": "case pos.inr\ne : ℝ≥0\nhe : e ≠ 0\nx y : WithZero (Multiplicative ℤ)\nhxy : x * y = 0\nhy : y = 0\n⊢ (if hx : x * y = 0 then 0 else e ^ toAdd (unzero hx)) =\n (if hx : x = 0 then 0 else e ^ toAdd (unzero hx)) * if hx : y = 0 then 0 else e ^ toAdd (unzero hx)",
"usedConstants": [
"Eq.mpr",
... | · rw [dif_pos hxy, dif_pos hy, mul_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.OpenSubgroup | {
"line": 530,
"column": 4
} | {
"line": 530,
"column": 37
} | [
{
"pp": "G : 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' := ⋯ }\n⊢ IsOp... | refine isOpen_iUnion (fun n ↦ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 165,
"column": 6
} | {
"line": 172,
"column": 24
} | [
{
"pp": "case mpr\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\na : A\ns : Set A\n⊢ (∃ i, {b | b - a ∈ B i} ⊆ s) → ∃ i ∈ hB.toRingFilterBasis.toAddGroupFilterBasis, (fun y ↦ a + y) '' i ⊆ s",
"usedConstants": [
"Eq.mpr",
"SetL... | rintro ⟨i, hi⟩
use B i
constructor
· use i
· rw [image_subset_iff]
rintro b b_in
apply hi
simpa using b_in | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 165,
"column": 6
} | {
"line": 172,
"column": 24
} | [
{
"pp": "case mpr\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\na : A\ns : Set A\n⊢ (∃ i, {b | b - a ∈ B i} ⊆ s) → ∃ i ∈ hB.toRingFilterBasis.toAddGroupFilterBasis, (fun y ↦ a + y) '' i ⊆ s",
"usedConstants": [
"Eq.mpr",
"SetL... | rintro ⟨i, hi⟩
use B i
constructor
· use i
· rw [image_subset_iff]
rintro b b_in
apply hi
simpa using b_in | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx y : R\nhx : 0 <ᵥ x\nhy : x * y ≤ᵥ 0\n⊢ y ≤ᵥ 0",
"usedConstants": [
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp",
"ValuativeRel.vle",
"MulZeroClass.mul_zero",
"instDistribOfSemiring"... | rw [show (0 : R) = x * 0 by simp, mul_comm x y, mul_comm x 0] at hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 518,
"column": 6
} | {
"line": 518,
"column": 50
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx✝ y✝ z✝ x y z w : R\nt s u v : ↥(posSubmonoid R)\nh₁ : x * ↑t ≤ᵥ y * ↑s\nh₂ : y * ↑s ≤ᵥ x * ↑t\nh₃ : z * ↑u ≤ᵥ w * ↑v\nh₄ : w * ↑v ≤ᵥ z * ↑u\nhw : ¬w ≤ᵥ 0\nhz : z ≤ᵥ 0\n⊢ w * ↑v ≤ᵥ 0 * ↑v",
"usedConstants": [
"Eq.mpr",
... | simpa using h₄.trans (mul_vle_mul_left hz u) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 548,
"column": 4
} | {
"line": 548,
"column": 32
} | [
{
"pp": "case mk.mk.mk\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx y z a₁ : R\na₂ : ↥(posSubmonoid R)\nb₁ : R\nb₂ : ↥(posSubmonoid R)\nhab : a₁ * ↑b₂ ≤ᵥ b₁ * ↑a₂\nc₁ : R\nc₂ : ↥(posSubmonoid R)\nhbc : b₁ * ↑c₂ ≤ᵥ c₁ * ↑b₂\n⊢ a₁ * ↑c₂ ≤ᵥ c₁ * ↑a₂",
"usedConstants": [
"HMul.hMul",
... | apply vle_mul_cancel b₂.prop | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1087,
"column": 2
} | {
"line": 1087,
"column": 48
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : IsDiscrete R\ninst✝ : IsNontrivial R\na : ValueGroupWithZero R\nha : ¬a = 0\n⊢ (uniformizer R)⁻¹ ≤ a ↔ 1 < a",
"usedConstants": [
"Iff.mpr",
"ValuativeRel.instLinearOrderValueGroupWithZero",
"Preorder.t... | replace ha : 0 < a := bot_lt_iff_ne_bot.mpr ha | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 110,
"column": 8
} | {
"line": 111,
"column": 34
} | [
{
"pp": "case h\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx : R\nγ : (ValueGroup₀ v)ˣ\nHx : v x = 0\ny : R\na✝ : y ∈ ↑(v.ltAddSubgroup ((Units.map ↑embedding) 1))\n⊢ y ∈ (fun x_1 ↦ x_1 * x) ⁻¹' ↑(v.ltAddSubgroup ((Units.map ↑embedding) γ))",
"u... | simp only [coe_ltAddSubgroup, preimage_setOf_eq, mem_setOf_eq, Valuation.map_mul, Hx,
mul_zero, Units.zero_lt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.UniformField | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 85
} | [
{
"pp": "K✝ : Type u_1\ninst✝⁴ : Field K✝\ninst✝³ : UniformSpace K✝\nL : Type u_2\ninst✝² : Field L\ninst✝¹ : UniformSpace L\ninst✝ : CompletableTopField L\nK : Subfield L\nF : Filter ↥K\nF_cau : Cauchy F\ninf_F : 𝓝 0 ⊓ F = ⊥\ni : ↥K →+* L := K.subtype\n⊢ Cauchy (map (fun x ↦ x⁻¹) F)",
"usedConstants": [
... | have hi : IsUniformInducing i := isUniformEmbedding_subtype_val.isUniformInducing | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 584,
"column": 6
} | {
"line": 584,
"column": 18
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhQ : W'.Equation Q\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\n⊢ W'.negAddY P Q * (P z * Q z) ^ 3 = (P y * Q z ^ 3 - Q y * P z ^ 3) ^ 3",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"AddGroupWithOne.toAdd... | negAddY_eq', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.BilinearForm.DualLattice | {
"line": 76,
"column": 9
} | {
"line": 76,
"column": 37
} | [
{
"pp": "R : Type ?u.28442\nS : Type ?u.28445\nM : Type ?u.28448\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\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R S\nN : Submodule R M\nx ... | by simp [← Algebra.smul_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.BilinearForm.DualLattice | {
"line": 80,
"column": 7
} | {
"line": 80,
"column": 35
} | [
{
"pp": "R : Type ?u.28442\nS : Type ?u.28445\nM : Type ?u.28448\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\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R S\nN : Submodule R M\nr ... | by simp [← Algebra.smul_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 31
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\n⊢ IsClosed[_i.toTopologicalSpace] {x | v.restrict x ≤ 1}",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
... | exact isClosed_closedBall _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.BilinearForm.DualLattice | {
"line": 122,
"column": 2
} | {
"line": 124,
"column": 32
} | [
{
"pp": "R : Type u_4\nS : Type u_2\nM : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : Field S\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Algebra R S\ninst✝³ : Module R M\ninst✝² : Module S M\ninst✝¹ : IsScalarTower R S M\nB : BilinForm S M\nι : Type u_1\ninst✝ : Finite ι\nhB : B.Nondegenerate\nb : Basis ι S M\n⊢ B.dualSubmo... | letI := b.finiteDimensional_of_finite
rw [dualSubmodule_span_of_basis _ hB.flip, dualSubmodule_span_of_basis B hB,
dualBasis_dualBasis_flip hB] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.BilinearForm.DualLattice | {
"line": 122,
"column": 2
} | {
"line": 124,
"column": 32
} | [
{
"pp": "R : Type u_4\nS : Type u_2\nM : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : Field S\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Algebra R S\ninst✝³ : Module R M\ninst✝² : Module S M\ninst✝¹ : IsScalarTower R S M\nB : BilinForm S M\nι : Type u_1\ninst✝ : Finite ι\nhB : B.Nondegenerate\nb : Basis ι S M\n⊢ B.dualSubmo... | letI := b.finiteDimensional_of_finite
rw [dualSubmodule_span_of_basis _ hB.flip, dualSubmodule_span_of_basis B hB,
dualBasis_dualBasis_flip hB] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 693,
"column": 2
} | {
"line": 694,
"column": 14
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Jacobian 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 ^ 2 = Q x * P z ^ 2\n⊢ W.addXYZ P Q = addU P Q • ![1, 1, 0]",
"usedConstants": [
"instHSMul",
"WeierstrassCurve.Jacobian.addY",
"HMul.hM... | simp [addXYZ, addX_of_X_eq hP hQ hPz hQz hx, addY_of_X_eq hP hQ hPz hQz hx, addZ_of_X_eq hx,
smul_fin3] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 693,
"column": 2
} | {
"line": 694,
"column": 14
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Jacobian 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 ^ 2 = Q x * P z ^ 2\n⊢ W.addXYZ P Q = addU P Q • ![1, 1, 0]",
"usedConstants": [
"instHSMul",
"WeierstrassCurve.Jacobian.addY",
"HMul.hM... | simp [addXYZ, addX_of_X_eq hP hQ hPz hQz hx, addY_of_X_eq hP hQ hPz hQz hx, addZ_of_X_eq hx,
smul_fin3] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 693,
"column": 2
} | {
"line": 694,
"column": 14
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Jacobian 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 ^ 2 = Q x * P z ^ 2\n⊢ W.addXYZ P Q = addU P Q • ![1, 1, 0]",
"usedConstants": [
"instHSMul",
"WeierstrassCurve.Jacobian.addY",
"HMul.hM... | simp [addXYZ, addX_of_X_eq hP hQ hPz hQz hx, addY_of_X_eq hP hQ hPz hQz hx, addZ_of_X_eq hx,
smul_fin3] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 222,
"column": 11
} | {
"line": 222,
"column": 25
} | [
{
"pp": "case hcard.refine_2\nK : Type u_2\nL : Type u_3\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Algebra K L\nE : Type u_1\ninst✝⁴ : Field E\ninst✝³ : Algebra K E\ninst✝² : IsAlgClosed E\ninst✝¹ : FiniteDimensional K L\ninst✝ : Algebra.IsSeparable K L\nx : L\nhxL : K[x] = ⊤\nr : ℕ\nhr : r < finrank K L\nt... | natDegree_pow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 419,
"column": 20
} | {
"line": 419,
"column": 55
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nB₁ B₂ : Set α\nhB₁ : M.IsBase B₁\nhB₂ : M.IsBase B₂\n⊢ B₁.encard.toNat = B₂.ncard",
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Matroid.IsBase.encard_eq_encard_of_isBase",
"congrArg",
"id",
"Nat",
"ENat",
"ENat.toNat",
... | hB₁.encard_eq_encard_of_isBase hB₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 107,
"column": 2
} | {
"line": 111,
"column": 73
} | [
{
"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]\nhp0 : p ≠ 0\nhp : (Polynomial.aeval s) p = 0\n⊢ (minpoly R s).degree ≤ p.degree",
"usedCon... | by_cases! hs : ¬IsIntegral R s
· simp [minpoly, hs]
rw [degree_eq_natDegree (minpoly.ne_zero hs), degree_eq_natDegree hp0]
norm_cast
exact natDegree_le_of_dvd ((isIntegrallyClosed_dvd_iff hs _).mp hp) hp0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 107,
"column": 2
} | {
"line": 111,
"column": 73
} | [
{
"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]\nhp0 : p ≠ 0\nhp : (Polynomial.aeval s) p = 0\n⊢ (minpoly R s).degree ≤ p.degree",
"usedCon... | by_cases! hs : ¬IsIntegral R s
· simp [minpoly, hs]
rw [degree_eq_natDegree (minpoly.ne_zero hs), degree_eq_natDegree hp0]
norm_cast
exact natDegree_le_of_dvd ((isIntegrallyClosed_dvd_iff hs _).mp hp) hp0 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 767,
"column": 21
} | {
"line": 769,
"column": 27
} | [
{
"pp": "α : Type u_1\nM✝ M : Matroid α\nE : Set α\nIsBase Indep : Set α → Prop\nhE : E = M.E\nhB : ∀ (B : Set α), IsBase B ↔ M.IsBase B\nhI : ∀ (I : Set α), Indep I ↔ M.Indep I\n⊢ ExchangeProperty IsBase",
"usedConstants": [
"Eq.mpr",
"Matroid.ExchangeProperty",
"congrArg",
"Matroid... | by
simp_rw [show IsBase = M.IsBase from funext (by simp [hB])]
exact M.isBase_exchange | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Dual | {
"line": 207,
"column": 78
} | {
"line": 208,
"column": 53
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nh : M✶.RankPos\n⊢ ¬M.IsBase M.E",
"usedConstants": [
"Matroid.rankPos_iff",
"congrArg",
"Matroid.E",
"Matroid.dual",
"Matroid.IsBase",
"Eq.mp",
"Matroid.dual_isBase_iff",
"HasSubset.Subset",
"Set.diff_empty",
"... | by
rwa [rankPos_iff, dual_isBase_iff, diff_empty] at h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Minor.Restrict | {
"line": 361,
"column": 97
} | {
"line": 362,
"column": 39
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\nN : Matroid α\nhI : M.Indep I\nhNM : N ≤r M\nhIN : I ⊆ N.E\n⊢ N.Indep I",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Matroid.Indep",
"id",
"HasSubset.Subset",
"Matroid.IsRestriction",
"And.casesOn",
... | by
obtain ⟨R, -, rfl⟩ := hNM; simpa [hI] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Minor.Restrict | {
"line": 441,
"column": 19
} | {
"line": 441,
"column": 49
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X J : Set α\ne : α\nhIX : M.IsBasis I X\nhJX : M.IsBasis J X\nhe : e ∈ I \\ J\ny : α\nhy : y ∈ J \\ I\nh : (M ↾ X).IsBase (insert y (I \\ {e}))\n⊢ M.IsBasis (insert y (I \\ {e})) X",
"usedConstants": [
"Matroid.IsBasis.subset_ground",
"congrArg",
"Ma... | rwa [isBase_restrict_iff] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Combinatorics.Matroid.Minor.Restrict | {
"line": 441,
"column": 19
} | {
"line": 441,
"column": 49
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X J : Set α\ne : α\nhIX : M.IsBasis I X\nhJX : M.IsBasis J X\nhe : e ∈ I \\ J\ny : α\nhy : y ∈ J \\ I\nh : (M ↾ X).IsBase (insert y (I \\ {e}))\n⊢ M.IsBasis (insert y (I \\ {e})) X",
"usedConstants": [
"Matroid.IsBasis.subset_ground",
"congrArg",
"Ma... | rwa [isBase_restrict_iff] at h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Restrict | {
"line": 441,
"column": 19
} | {
"line": 441,
"column": 49
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X J : Set α\ne : α\nhIX : M.IsBasis I X\nhJX : M.IsBasis J X\nhe : e ∈ I \\ J\ny : α\nhy : y ∈ J \\ I\nh : (M ↾ X).IsBase (insert y (I \\ {e}))\n⊢ M.IsBasis (insert y (I \\ {e})) X",
"usedConstants": [
"Matroid.IsBasis.subset_ground",
"congrArg",
"Ma... | rwa [isBase_restrict_iff] at h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 302,
"column": 4
} | {
"line": 302,
"column": 9
} | [
{
"pp": "α : Type u_1\nP : Set α → Prop\nX : Set α\nn : ℕ\nI : Set α\nhI : P I\nhIX : I ⊆ X\nhP : ∀ (Y : Set α), P Y → ∃ n₀, Y.encard = ↑n₀ ∧ n₀ ≤ n\n⊢ ∃ x, ∀ y ∈ ncard '' {Y | P Y ∧ I ⊆ Y ∧ Y ⊆ X}, y ≤ x",
"usedConstants": [
"setOf",
"Preorder.toLE",
"Membership.mem",
"HasSubset.Sub... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Combinatorics.Matroid.Map | {
"line": 349,
"column": 35
} | {
"line": 349,
"column": 99
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I✝ : Set α\nM✝ : Matroid α\nN : Matroid β\nM : Matroid α\nf : α → β\nhf : InjOn f M.E\nI : Set β\n⊢ (∃ I₀, M.Indep (Subtype.val '' I₀) ∧ I = f '' Subtype.val '' I₀) ↔ ∃ I₀, M.Indep I₀ ∧ I = f '' I₀",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | Subtype.exists_set_subtype (p := fun J ↦ M.Indep J ∧ I = f '' J) | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Matroid.Map | {
"line": 423,
"column": 2
} | {
"line": 423,
"column": 20
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Matroid α\nI X : Set α\nf : α → β\nhf : InjOn f M.E\nhI : I ⊆ M.E\nhX : X ⊆ M.E\nh : (M.map f hf).IsBasis (f '' I) (f '' X)\nI' : Set α\nhI' : M.Indep I'\nhII' : I = I'\n⊢ M.IsBasis I X",
"usedConstants": []
}
] | obtain rfl := hII' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 249,
"column": 13
} | {
"line": 249,
"column": 26
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nE : Set α\nh : M = loopyOn E\n⊢ M.loops = E ∧ M.E = E",
"usedConstants": [
"Eq.mpr",
"Matroid.loopyOn",
"congrArg",
"Matroid.E",
"id",
"And",
"Set.instEmptyCollection",
"Matroid.closure",
"Matroid.loops.eq_1",
... | rw [h, loops] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 338,
"column": 31
} | {
"line": 340,
"column": 43
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nC : Set α\nhC : M.IsCircuit C\ne : α\n⊢ C ⊆ M.closure (C \\ {e})",
"usedConstants": [
"Eq.mpr",
"Matroid.IsCircuit.closure_diff_singleton_eq",
"congrArg",
"Set.instSingletonSet",
"id",
"Matroid.subset_closure",
"HasSubset.Subset... | by
rw [hC.closure_diff_singleton_eq]
exact M.subset_closure _ hC.subset_ground | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 398,
"column": 11
} | {
"line": 398,
"column": 57
} | [
{
"pp": "case inr\nα : Type u_1\nM : Matroid α\ne f : α\nhe : M.IsNonloop e\nhf : M.IsNonloop f\nhne : e ≠ f\n⊢ M.closure {e} = M.closure {f} ↔ e = f ∨ ¬M.Indep {e, f}",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.Indep",
"Set.instSingletonSet",
"id",
"Insert.insert... | he.closure_eq_closure_iff_isCircuit_of_ne hne, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 368,
"column": 4
} | {
"line": 368,
"column": 18
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nI : Set α\nx : α\nhI : M.Indep I\n⊢ ¬M.Indep (insert x I) ∧ x ∈ M.E ∨ x ∈ I ↔ x ∈ M.E ∧ (M.Indep (insert x I) → x ∈ I)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Membership.mem",
"Matroid.Indep",
"id",
"Insert.inse... | imp_iff_not_or | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 504,
"column": 2
} | {
"line": 504,
"column": 21
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nι : Type u_4\nA : Set ι\nhA : A.Nonempty\nI : ι → Set α\nh : M.Indep (⋃ i ∈ A, I i)\n⊢ M.closure (⋂ i ∈ A, I i) = ⋂ i ∈ A, M.closure (I i)",
"usedConstants": [
"Set.Elem",
"Set.Nonempty.coe_sort",
"Nonempty"
]
}
] | have := hA.coe_sort | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 320,
"column": 41
} | {
"line": 321,
"column": 66
} | [
{
"pp": "α : Type u_1\nX Y : Set α\nM : Matroid α\nh : Y ⊆ X\n⊢ M.eRk X ≤ M.eRk Y + M.eRk (X \\ Y)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion",
"instAddENat",
"id",
"LE.le",
"Matroid.eRk_union_le_eRk_add_eRk",
"instLEENat",
"Set.union_diff... | by
nth_rw 1 [← union_diff_cancel h]; apply eRk_union_le_eRk_add_eRk | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Rank.Cardinal | {
"line": 296,
"column": 4
} | {
"line": 296,
"column": 28
} | [
{
"pp": "α : Type u\nM : Matroid α\ninst✝ : M.InvariantCardinalRank\nX Y Ii : Set α\nhIi : M.IsBasis' Ii (X ∩ Y)\nIX : Set α\nhIX : M.IsBasis' IX X\nhIX' : Ii ⊆ IX\nIY : Set α\nhIY : M.IsBasis' IY Y\nhIY' : Ii ⊆ IY\n⊢ #↑Ii + M.cRk (IX ∪ IY) ≤ #↑IX + M.cRk Y",
"usedConstants": [
"Eq.mpr",
"Cardin... | ← hIY.cardinalMk_eq_cRk, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Algebraic.MvPolynomial | {
"line": 89,
"column": 4
} | {
"line": 92,
"column": 25
} | [
{
"pp": "case refine_2\nσ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\ni : σ\ns : Set σ\nf : R[X]\nh : Transcendental (↥(supported R s)) ((Polynomial.aeval (X i)) f)\n⊢ Transcendental R f",
"usedConstants": [
"Subalgebra.instSetLike",
"Finsupp.instAddZeroClass",
"Eq... | rw [← transcendental_polynomial_aeval_X_iff R i]
refine h.restrictScalars fun _ _ heq ↦ MvPolynomial.C_injective σ R ?_
simp_rw [← MvPolynomial.algebraMap_eq]
exact congr($(heq).1) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Algebraic.MvPolynomial | {
"line": 89,
"column": 4
} | {
"line": 92,
"column": 25
} | [
{
"pp": "case refine_2\nσ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\ni : σ\ns : Set σ\nf : R[X]\nh : Transcendental (↥(supported R s)) ((Polynomial.aeval (X i)) f)\n⊢ Transcendental R f",
"usedConstants": [
"Subalgebra.instSetLike",
"Finsupp.instAddZeroClass",
"Eq... | rw [← transcendental_polynomial_aeval_X_iff R i]
refine h.restrictScalars fun _ _ heq ↦ MvPolynomial.C_injective σ R ?_
simp_rw [← MvPolynomial.algebraMap_eq]
exact congr($(heq).1) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 900,
"column": 45
} | {
"line": 900,
"column": 59
} | [
{
"pp": "α : Type u_2\nM M' : Matroid α\nh : M.E = M'.E\nhsp : ∀ S ⊆ M.E, M.Spanning S ↔ M'.Spanning S\nS : Set α\nhSE : S ⊆ M.E\n⊢ M.Spanning S ↔ M'.Spanning S",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"id",
"Iff",
"Matroid.Spanning",
"propext",
"... | rw [hsp _ hSE] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 900,
"column": 45
} | {
"line": 900,
"column": 59
} | [
{
"pp": "α : Type u_2\nM M' : Matroid α\nh : M.E = M'.E\nhsp : ∀ S ⊆ M.E, M.Spanning S ↔ M'.Spanning S\nS : Set α\nhSE : S ⊆ M.E\n⊢ M.Spanning S ↔ M'.Spanning S",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"id",
"Iff",
"Matroid.Spanning",
"propext",
"... | rw [hsp _ hSE] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 900,
"column": 45
} | {
"line": 900,
"column": 59
} | [
{
"pp": "α : Type u_2\nM M' : Matroid α\nh : M.E = M'.E\nhsp : ∀ S ⊆ M.E, M.Spanning S ↔ M'.Spanning S\nS : Set α\nhSE : S ⊆ M.E\n⊢ M.Spanning S ↔ M'.Spanning S",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"id",
"Iff",
"Matroid.Spanning",
"propext",
"... | rw [hsp _ hSE] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AlgebraicIndependent.Transcendental | {
"line": 97,
"column": 54
} | {
"line": 97,
"column": 71
} | [
{
"pp": "R : Type u_3\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ ¬trdeg R A = 0 ↔ ¬Algebra.IsAlgebraic R A",
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"id",
"Algebra.IsAlgebraic",
"Algebra.trdeg",
"Iff",
"propext",... | trdeg_eq_zero_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.SeparableDegree | {
"line": 115,
"column": 2
} | {
"line": 122,
"column": 44
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nq : ℕ\nhq : NeZero q\ninst✝ : CharP F q\ng g' : F[X]\nm m' : ℕ\nh_expand : (expand F (q ^ m)) g = (expand F (q ^ m')) g'\nhg : g.Separable\nhg' : g'.Separable\n⊢ g.natDegree = g'.natDegree",
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
... | wlog hm : m ≤ m'
· exact (this q g' g m' m h_expand.symm hg' hg (le_of_not_ge hm)).symm
obtain ⟨s, rfl⟩ := exists_add_of_le hm
rw [pow_add, expand_mul, expand_inj (pow_pos (NeZero.pos q) m)] at h_expand
subst h_expand
rcases isUnit_or_eq_zero_of_separable_expand q s (NeZero.pos q) hg with (h | rfl)
· rw [na... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.SeparableDegree | {
"line": 115,
"column": 2
} | {
"line": 122,
"column": 44
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nq : ℕ\nhq : NeZero q\ninst✝ : CharP F q\ng g' : F[X]\nm m' : ℕ\nh_expand : (expand F (q ^ m)) g = (expand F (q ^ m')) g'\nhg : g.Separable\nhg' : g'.Separable\n⊢ g.natDegree = g'.natDegree",
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
... | wlog hm : m ≤ m'
· exact (this q g' g m' m h_expand.symm hg' hg (le_of_not_ge hm)).symm
obtain ⟨s, rfl⟩ := exists_add_of_le hm
rw [pow_add, expand_mul, expand_inj (pow_pos (NeZero.pos q) m)] at h_expand
subst h_expand
rcases isUnit_or_eq_zero_of_separable_expand q s (NeZero.pos q) hg with (h | rfl)
· rw [na... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 353,
"column": 32
} | {
"line": 353,
"column": 50
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nx : K\n⊢ embedding (extension ↑x) = v x",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
... | extension_extends, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 503,
"column": 12
} | {
"line": 503,
"column": 30
} | [
{
"pp": "K : 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) ⋯.choose = a := Exists.choose_spec (restrict₀_surjective v a)\ny : K := ⋯.choose\nhab : extension ↑x = extension ... | extension_extends, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 503,
"column": 31
} | {
"line": 503,
"column": 49
} | [
{
"pp": "K : 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) ⋯.choose = a := Exists.choose_spec (restrict₀_surjective v a)\ny : K := ⋯.choose\nhab : v.restrict x = extension ... | extension_extends, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 510,
"column": 50
} | {
"line": 510,
"column": 68
} | [
{
"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\nhab : extensionValuation.restrict ↑⋯.choose = extensionValuation.restrict ↑⋯.choose\nx : K := ⋯.choose\nhx_def : x = ⋯.choose\nhx : (restrict₀ v) ⋯.choose = a := Ex... | extension_extends, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 421,
"column": 6
} | {
"line": 421,
"column": 25
} | [
{
"pp": "ι : Type u\nR : Type u_1\nA : Type w\nx : ι → A\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Nontrivial R\ninst✝ : NoZeroDivisors A\nhx : IsTranscendenceBasis R x\nthis : FaithfulSMul R A\n⊢ lift.{w, u} #ι = lift.{u, w} (trdeg R A)",
"usedConstants": [
"Eq.mpr",
... | ← matroid_cRank_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 516,
"column": 6
} | {
"line": 516,
"column": 25
} | [
{
"pp": "ι : Type u\nR : Type u_1\nA : Type w\nx : ι → A\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Nontrivial R\ninst✝ : NoZeroDivisors A\nhx : AlgebraicIndependent R x\nfin : trdeg R A < ℵ₀\nle : lift.{u, w} (trdeg R A) ≤ lift.{w, u} #ι\nthis : FaithfulSMul R A\n⊢ IsTranscendenc... | ← matroid_cRank_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 549,
"column": 41
} | {
"line": 551,
"column": 31
} | [
{
"pp": "R : Type u_1\nS : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Nontrivial R\nA : Type v\ninst✝⁶ : CommRing A\ninst✝⁵ : NoZeroDivisors A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S A\ninst✝² : FaithfulSMul R S\ninst✝¹ : FaithfulSMul S A\ninst✝ : IsScalarTower R S A\n⊢ ... | by
rw [← (trdeg R S).lift_id, ← (trdeg S A).lift_id, ← (trdeg R A).lift_id]
exact lift_trdeg_add_eq R S A | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Trace.Basic | {
"line": 190,
"column": 6
} | {
"line": 190,
"column": 91
} | [
{
"pp": "case pos\nA : Type u_7\nB : Type u_8\nC : Type u_9\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing C\ninst✝¹ : Algebra A C\ninst✝ : Algebra B C\ne : A ≃+* B\nhe : (algebraMap B C).comp ↑e = algebraMap A C\nx : C\ns : Finset C\nb : Basis (↥s) B C\nthis✝ : Algebra A B := (↑e).toAlgebra\nthis... | Algebra.trace_eq_matrix_trace (b.mapCoeffs e.symm (by simp [Algebra.smul_def, ← he])) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 674,
"column": 4
} | {
"line": 674,
"column": 9
} | [
{
"pp": "case inl.tmul\nk : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁶ : Field k\ninst✝⁵ : Field K\ninst✝⁴ : Algebra k K\ninst✝³ : CommRing R\ninst✝² : Algebra k R\ninst✝¹ : IsPurelyInseparable k K\nq : ℕ\ninst✝ : ExpChar k q\na✝ : Nontrivial (R ⊗[k] K)\nhq : Nat.Prime q\nx : R\ny : K\nn : ℕ\na : k\nha : (alg... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Trace.Basic | {
"line": 494,
"column": 15
} | {
"line": 496,
"column": 25
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\nι : Type w\ninst✝² : Algebra.IsSeparable K L\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι K L\nthis : FiniteDimensional K L\npb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L\n⊢ (pb.basis.toMat... | by
simp only [Basis.toMatrix_mul_toMatrix_flip, Matrix.transpose_one, Matrix.mul_one,
Matrix.det_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.SeparableClosure | {
"line": 105,
"column": 61
} | {
"line": 107,
"column": 38
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK : Type w\ninst✝¹ : Field K\ninst✝ : Algebra F K\ni : E →ₐ[F] K\n⊢ comap i (separableClosure F K) = separableClosure F E",
"usedConstants": [
"map_mem_separableClosure_iff",
"IntermediateField.ext",
... | by
ext x
exact map_mem_separableClosure_iff i | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.SeparableClosure | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 48
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.EssFiniteType F E\nd : Finset E → ℕ := fun s ↦ finInsepDegree (↥(adjoin F ↑s)) E\n⊢ {s | IsTranscendenceBasis F Subtype.val}.Nonempty",
"usedConstants": [
"Lattice.toSemilatticeSup",
"Inter... | have ⟨s, hs⟩ := IntermediateField.fg_top F E | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.SeparableDegree | {
"line": 577,
"column": 4
} | {
"line": 577,
"column": 38
} | [
{
"pp": "case zero\nF : Type u\ninst✝² : Field F\nf : F[X]\nhm : f.Monic\nhi : Irreducible f\nh : f.natSepDegree = 1\nn : ℕ\ny : F\ninst✝¹ : CharZero F\ninst✝ : ExpChar F 1\nhf : f = X ^ 1 ^ n - C y\n⊢ ∃ n y, (n = 0 ∨ y ∉ (frobenius F 1).range) ∧ f = X ^ 1 ^ n - C y",
"usedConstants": [
"one_pow",
... | simp_rw [one_pow, pow_one] at hf ⊢ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.FieldTheory.SeparableDegree | {
"line": 673,
"column": 4
} | {
"line": 673,
"column": 9
} | [
{
"pp": "case refine_1\nF : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Ring E\ninst✝¹ : IsDomain E\ninst✝ : Algebra F E\nq : ℕ\nhF : ExpChar F q\nx : E\nthis✝ : ExpChar E q\nthis : ExpChar E[X] q\nh✝ : (minpoly F x).natSepDegree = 1\nn : ℕ\ny : F\nh : minpoly F x = X ^ q ^ n - C y\nhx : (algebraMap F E) y =... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.FieldTheory.SeparableDegree | {
"line": 670,
"column": 2
} | {
"line": 675,
"column": 57
} | [
{
"pp": "case refine_1\nF : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Ring E\ninst✝¹ : IsDomain E\ninst✝ : Algebra F E\nq : ℕ\nhF : ExpChar F q\nx : E\nthis✝ : ExpChar E q\nthis : ExpChar E[X] q\nh : (minpoly F x).natSepDegree = 1\n⊢ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n",
... | · obtain ⟨n, y, h⟩ := (natSepDegree_eq_one_iff_eq_X_pow_sub_C q).1 h
have hx := congr_arg (Polynomial.aeval x) h.symm
rw [minpoly.aeval, map_sub, map_pow, aeval_X, aeval_C, sub_eq_zero, eq_comm] at hx
use n
rw [h, Polynomial.map_sub, Polynomial.map_pow, map_X, map_C, hx, map_pow,
← sub_pow_expChar... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 7
} | [
{
"pp": "Γ : Type u_1\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝ : Field K\nv : Valuation K Γ\nhv : v.IsRankOneDiscrete\nr : ↥v.valuationSubring\nhr : r ≠ 0\nπ : v.Uniformizer\nhr₀ : v ↑r ≠ 0\nvr : Γˣ := Units.mk0 (v ↑r) hr₀\nhvr_def : vr = Units.mk0 (v ↑r) hr₀\nm : ℤ\nhm : Units.mk0 (v ↑π.... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.Algebra.Valued.WithVal | {
"line": 440,
"column": 4
} | {
"line": 444,
"column": 28
} | [
{
"pp": "R✝ : Type u_1\nΓ₀ : Type u_2\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\nR : Type u_3\ninst✝ : Ring R\nv : Valuation R Γ₀\na b : ValueGroup₀ Valued.v\n⊢ (WithZero.map' ↑(valueGroupEquiv v)) a ≤ (WithZero.map' ↑(valueGroupEquiv v)) b ↔ a ≤ b",
"usedConstants": [
"GroupWithZero.toMonoidWithZer... | match a, b with
| 0, 0 => simp
| 0, .coe _ => simp
| .coe _, 0 => simp
| .coe a, .coe b => simp | Lean.Elab.Tactic.evalMatch | Lean.Parser.Tactic.match |
Mathlib.Topology.Algebra.Valued.WithVal | {
"line": 440,
"column": 4
} | {
"line": 444,
"column": 28
} | [
{
"pp": "R✝ : Type u_1\nΓ₀ : Type u_2\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\nR : Type u_3\ninst✝ : Ring R\nv : Valuation R Γ₀\na b : ValueGroup₀ Valued.v\n⊢ (WithZero.map' ↑(valueGroupEquiv v)) a ≤ (WithZero.map' ↑(valueGroupEquiv v)) b ↔ a ≤ b",
"usedConstants": [
"GroupWithZero.toMonoidWithZer... | match a, b with
| 0, 0 => simp
| 0, .coe _ => simp
| .coe _, 0 => simp
| .coe a, .coe b => simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.WithVal | {
"line": 440,
"column": 4
} | {
"line": 444,
"column": 28
} | [
{
"pp": "R✝ : Type u_1\nΓ₀ : Type u_2\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\nR : Type u_3\ninst✝ : Ring R\nv : Valuation R Γ₀\na b : ValueGroup₀ Valued.v\n⊢ (WithZero.map' ↑(valueGroupEquiv v)) a ≤ (WithZero.map' ↑(valueGroupEquiv v)) b ↔ a ≤ b",
"usedConstants": [
"GroupWithZero.toMonoidWithZer... | match a, b with
| 0, 0 => simp
| 0, .coe _ => simp
| .coe _, 0 => simp
| .coe a, .coe b => simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 305,
"column": 32
} | {
"line": 305,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nr : R\nhr : r ≠ 0\nhv : v.asIdeal = Ideal.span {r}\n⊢ exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r})).factors)) = exp (-1)",
"usedConstants": [
"Eq.mpr",
"Associates.mk",
... | ← hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 237,
"column": 2
} | {
"line": 238,
"column": 25
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nm n : σ →₀ ℕ\nφ : MvPowerSeries σ R\na : R\nthis : ∀ p ∈ antidiagonal m, (coeff p.1) ((monomial n) a) * (coeff p.2) φ ≠ 0 → p.1 = n\n⊢ (coeff m) ((monomial n) a * φ) = if n ≤ m then a * (coeff (m - n)) φ else 0",
"usedConstants": [
"Finsupp.inst... | rw [coeff_mul, ← Finset.sum_filter_of_ne this, Finset.filter_fst_eq_antidiagonal _ n,
Finset.sum_ite_index] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 44
} | [
{
"pp": "G : Type u_2\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : Nontrivial G\ng : G\n⊢ ({x | g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) ↔ Nonempty (G ≃*o Multiplicative ℤ)",
"usedConstants": [
"Additive",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | let e : G ≃o Additive G := OrderIso.refl G | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 544,
"column": 2
} | {
"line": 544,
"column": 36
} | [
{
"pp": "case h\nσ : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\nn : σ →₀ ℕ\na : R\nm : σ →₀ ℕ\n⊢ (coeff m) ((map f) ((monomial n) a)) = (coeff m) ((monomial n) (f a))",
"usedConstants": [
"RingHom.instRingHomClass",
"Nat.instMulZeroClass",
"... | simp [coeff_monomial, apply_ite f] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 556,
"column": 42
} | {
"line": 558,
"column": 6
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝² : Semiring R\nS₁ : Type u_5\nS₂ : Type u_6\ninst✝¹ : CommSemiring S₁\ninst✝ : CommSemiring S₂\nf : R →+* S₁\ng : S₁ →+* S₂\np : MvPowerSeries σ R\n⊢ (map g) ((map f) p) = (map (g.comp f)) p",
"usedConstants": [
"Nat.instMulZeroClass",
"Semiring.toModul... | by
ext n
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 529,
"column": 6
} | {
"line": 529,
"column": 41
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx : K\nhv : (valuation K v) x ≤ 1\nn : R\nd : ↥v.asIdeal.primeCompl\nhnd : x * (algebraMap R K) n = (algebraMap R K) ↑d\n⊢... | ← valuation_of_algebraMap (K := K), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 529,
"column": 42
} | {
"line": 529,
"column": 77
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx : K\nhv : (valuation K v) x ≤ 1\nn : R\nd : ↥v.asIdeal.primeCompl\nhnd : x * (algebraMap R K) n = (algebraMap R K) ↑d\n⊢... | ← valuation_of_algebraMap (K := K), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 552,
"column": 4
} | {
"line": 553,
"column": 86
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\na b : R\nhv : v.intValuation b ≤ v.intValuation a\nγ : Multiplicative ℤ\nha : ¬a = 0\nhvaz : v.intValuation a ≠ 0\nhγz : ↑γ ≠ 0\nn : ℕ\nhna : exp (-↑n) < v.intValuation a\nhnγ : exp (-↑n) < ↑γ\nhvn : emult... | have hb : b ∈ v.asIdeal ^ multiplicity v.asIdeal (Ideal.span {a}) := by
rwa [← intValuation_le_pow_iff_mem, ← v.intValuation_eq_exp_neg_multiplicity ha] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 571,
"column": 6
} | {
"line": 571,
"column": 41
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx : K\nhv : (valuation K v) x ≤ 1\nγ : (WithZero (Multiplicative ℤ))ˣ\nn d : R\nhd✝ : d ∈ v.asIdeal.primeCompl\nhd : v.intVa... | ← valuation_of_algebraMap (K := K), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 421,
"column": 2
} | {
"line": 424,
"column": 25
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nφ : R⟦X⟧\n⊢ φ = (mk fun p ↦ (coeff (p + 1)) φ) * X + C (constantCoeff φ)",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"False",
"PowerSeries.coeff_succ_mul_X",
"RingHomClass.toAddMonoidHomC... | ext (_ | n)
· simp
· simp only [coeff_succ_mul_X, coeff_mk, map_add, coeff_C, n.succ_ne_zero,
if_false, add_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 421,
"column": 2
} | {
"line": 424,
"column": 25
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nφ : R⟦X⟧\n⊢ φ = (mk fun p ↦ (coeff (p + 1)) φ) * X + C (constantCoeff φ)",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"False",
"PowerSeries.coeff_succ_mul_X",
"RingHomClass.toAddMonoidHomC... | ext (_ | n)
· simp
· simp only [coeff_succ_mul_X, coeff_mk, map_add, coeff_C, n.succ_ne_zero,
if_false, add_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 703,
"column": 19
} | {
"line": 704,
"column": 17
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDedekindDomain R\nK : Type u_2\nS : Type u_3\ninst✝³ : Field K\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nr : R\nx : ↥(adicCompletionIntegers K v)\n⊢ { toFun := fun r ↦ ⟨↑((WithVal.equiv (valuation K... | by
rw [mul_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 582,
"column": 2
} | {
"line": 584,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf : R⟦X⟧\na b : R\n⊢ (rescale b) ((rescale a) f) = (rescale (a * b)) f",
"usedConstants": [
"Eq.mpr",
"PowerSeries.coeff_rescale",
"Semigroup.toMul",
"Semiring.toModule",
"HMul.hMul",
"Monoid.toMulOneClass",
"CommSemiri... | ext n
simp_rw [coeff_rescale]
rw [mul_pow, mul_comm _ (b ^ n), mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 582,
"column": 2
} | {
"line": 584,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf : R⟦X⟧\na b : R\n⊢ (rescale b) ((rescale a) f) = (rescale (a * b)) f",
"usedConstants": [
"Eq.mpr",
"PowerSeries.coeff_rescale",
"Semigroup.toMul",
"Semiring.toModule",
"HMul.hMul",
"Monoid.toMulOneClass",
"CommSemiri... | ext n
simp_rw [coeff_rescale]
rw [mul_pow, mul_comm _ (b ^ n), mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 763,
"column": 4
} | {
"line": 764,
"column": 13
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\na : adicCompletion K v\nha : a ∈ adicCompletionIntegers K v\n⊢ ∃ b ∈ R⁰, a * ↑b ∈ adicCompletionIntegers K v",
"usedCo... | use 1
simp [ha] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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