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.Combinatorics.Matroid.Map | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 98
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I : Set α\nM✝ : Matroid α\nN : Matroid β\nM : Matroid α\nf : ↑M.E ↪ β\ne : α\nhe : e ∈ M.E\nx✝¹ : Nonempty ↑M.E\nx✝ : Nonempty α\n⊢ ∀ (I : Set β),\n (M.comapOn (range ⇑f) fun x ↦ ↑(invFunOn (⇑f) univ x)).Indep I ↔\n (fun I ↦ M.Indep (Subtype.v... | simp_rw [comapOn_indep_iff, ← and_assoc, and_congr_left_iff, subset_range_iff_exists_image_eq] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.Matroid.Rank.Finite | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 39
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : M.IsRkFinite X\ne : α\n⊢ M.IsRkFinite (X ∪ {e})",
"usedConstants": [
"Set.instSingletonSet",
"Matroid.IsRkFinite.union",
"Matroid.isRkFinite_singleton",
"Singleton.singleton",
"Set"
]
}
] | exact hX.union M.isRkFinite_singleton | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Map | {
"line": 458,
"column": 54
} | {
"line": 459,
"column": 22
} | [
{
"pp": "α : Type u_1\nM : Matroid α\n⊢ M.map id ⋯ = M",
"usedConstants": [
"Set.injOn_id",
"congrArg",
"and_self",
"Matroid.E",
"Set.image_id'",
"Matroid.map_indep_iff._simp_1",
"_private.Mathlib.Combinatorics.Matroid.Map.0.Matroid.map_id._simp_1_1",
"Members... | by
simp [ext_iff_indep] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 350,
"column": 36
} | {
"line": 350,
"column": 66
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\ne : α\nhe : e ∈ M.closure X\nheX : e ∉ X\nI : Set α\nhI : M.IsBasis' I X\n⊢ e ∈ M.closure I",
"usedConstants": [
"Matroid.IsBasis'.closure_eq_closure",
"Eq.mpr",
"congrArg",
"Membership.mem",
"id",
"Matroid.closure",
... | by rwa [hI.closure_eq_closure] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 530,
"column": 68
} | {
"line": 530,
"column": 98
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX Y : Set α\ninst✝ : M.Finitary\nhX : X.Finite\nhXY : X ⊆ M.closure Y\nT : Set α\nhT : T ⊆ Y\nhTfin : T.Finite\nhXT : X ⊆ M.closure T\nI : Set α\nhI : M.IsBasis' I T\n⊢ X ⊆ M.closure I",
"usedConstants": [
"Matroid.IsBasis'.closure_eq_closure",
"Eq.mpr",
... | by rwa [hI.closure_eq_closure] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 641,
"column": 11
} | {
"line": 641,
"column": 23
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\n⊢ M.closure X ∩ M.coloops = X ∩ M.coloops",
"usedConstants": [
"Eq.mpr",
"Matroid.coloops",
"Membership.mem",
"id",
"_private.Mathlib.Combinatorics.Matroid.Loop.0.Matroid.closure_inter_coloops_eq._simp_1_1",
"Set.instInter"... | Set.ext_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 738,
"column": 2
} | {
"line": 748,
"column": 52
} | [
{
"pp": "α : Type u_1\nM₁ M₂ : Matroid α\nhE : M₁.E = M₂.E\nhl : M₁.loops = M₂.loops\nhc : M₁.coloops = M₂.coloops\nh : ∀ I ⊆ M₁.E, Disjoint I (M₁.loops ∪ M₁.coloops) → (M₁.Indep I ↔ M₂.Indep I)\n⊢ M₁ = M₂",
"usedConstants": [
"Eq.mpr",
"Set.diff_subset",
"Set.disjoint_union_right",
... | refine ext_indep hE fun I hI ↦ ?_
rw [← diff_coloops_indep_iff, ← @diff_coloops_indep_iff _ M₂, ← hc]
obtain hdj | hndj := em (Disjoint I (M₁.loops))
· rw [h _ (diff_subset.trans hI)]
rw [disjoint_union_right]
exact ⟨disjoint_of_subset_left diff_subset hdj, disjoint_sdiff_left⟩
obtain ⟨e, heI, hel : M₁.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 738,
"column": 2
} | {
"line": 748,
"column": 52
} | [
{
"pp": "α : Type u_1\nM₁ M₂ : Matroid α\nhE : M₁.E = M₂.E\nhl : M₁.loops = M₂.loops\nhc : M₁.coloops = M₂.coloops\nh : ∀ I ⊆ M₁.E, Disjoint I (M₁.loops ∪ M₁.coloops) → (M₁.Indep I ↔ M₂.Indep I)\n⊢ M₁ = M₂",
"usedConstants": [
"Eq.mpr",
"Set.diff_subset",
"Set.disjoint_union_right",
... | refine ext_indep hE fun I hI ↦ ?_
rw [← diff_coloops_indep_iff, ← @diff_coloops_indep_iff _ M₂, ← hc]
obtain hdj | hndj := em (Disjoint I (M₁.loops))
· rw [h _ (diff_subset.trans hI)]
rw [disjoint_union_right]
exact ⟨disjoint_of_subset_left diff_subset hdj, disjoint_sdiff_left⟩
obtain ⟨e, heI, hel : M₁.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 853,
"column": 66
} | {
"line": 853,
"column": 74
} | [
{
"pp": "case h.a\nα : Type u_1\nM : Matroid α\ne : α\n⊢ M.IsNonloop e ∧ M.IsNonloop e ↔ M.IsNonloop e",
"usedConstants": [
"Eq.mpr",
"congrArg",
"and_self",
"id",
"Matroid.IsNonloop",
"And",
"Iff",
"Eq"
]
}
] | and_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Rank.Cardinal | {
"line": 140,
"column": 11
} | {
"line": 140,
"column": 27
} | [
{
"pp": "α : Type u\nX Y : Set α\nM : Matroid α\nhYX : Y ⊆ X\naux : ∀ ⦃I : Set α⦄, M.IsBasis' I Y ↔ (M ↾ X).IsBasis' I Y\n⊢ (M ↾ X).cRk Y = M.cRk Y",
"usedConstants": [
"Eq.mpr",
"Cardinal",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"LE.le",
"And",
"Ca... | le_antisymm_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Matroid.Rank.Cardinal | {
"line": 152,
"column": 24
} | {
"line": 152,
"column": 40
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\nM : Matroid α\nhf : InjOn f M.E\nX : Set α\nhX : X ⊆ M.E\n⊢ lift.{u, v} (⨆ B, #↑↑B) = lift.{v, u} (M.cRk X)",
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"iSup",
"PartialOrder.toPreorder",
"Cardinal.lift",
"Preord... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Rank.Cardinal | {
"line": 175,
"column": 24
} | {
"line": 175,
"column": 40
} | [
{
"pp": "α : Type u\nβ : Type v\nM : Matroid β\nf : α → β\nX : Set α\n⊢ lift.{v, u} (⨆ B, #↑↑B) = lift.{u, v} (M.cRk (f '' X))",
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"iSup",
"PartialOrder.toPreorder",
"Cardinal.lift",
"Preorder.toLE",
"Cardinal... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 432,
"column": 25
} | {
"line": 432,
"column": 40
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\nhI : I.Finite\nh : M.Indep I\n⊢ M.eRk I = I.encard",
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Matroid.Indep.eRk_eq_encard",
"congrArg",
"id",
"ENat",
"Matroid.eRk",
"Eq"
]
}
] | h.eRk_eq_encard | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 621,
"column": 4
} | {
"line": 621,
"column": 56
} | [
{
"pp": "case a.hXY.inl\nα : Type u_2\nM : Matroid α\ne : α\nI : Set α\nhI : M.Indep I\na : α\nright✝ : ¬a = e\nhf : a ∈ M.closure I\nhe : e ∈ M.closure (insert a I \\ {e})\n⊢ a ∈ M.closure I",
"usedConstants": []
},
{
"pp": "case a.hXY.inr\nα : Type u_2\nM : Matroid α\ne f : α\nI : Set α\nhI : M.In... | exacts [hf, M.subset_closure _ hI.subset_ground haI] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 618,
"column": 2
} | {
"line": 624,
"column": 76
} | [
{
"pp": "α : Type u_2\nM : Matroid α\ne f : α\nI : Set α\nhI : M.Indep I\nhf : f ∈ M.closure I\nhe : e ∈ M.closure (insert f I \\ {e})\n⊢ M.closure (insert f I \\ {e}) = M.closure I",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.Matroid.Closure.0.Matroid.Indep.closure_insert_diff... | apply subset_antisymm <;> apply closure_subset_closure_of_subset_closure
· simp only [subset_def, mem_diff, mem_insert_iff, mem_singleton_iff]
rintro a (rfl | haI)
exacts [hf, M.subset_closure _ hI.subset_ground haI]
· intro a haI
obtain rfl | ne := eq_or_ne a e
exacts [he, M.mem_closure_of_mem' ⟨.i... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 618,
"column": 2
} | {
"line": 624,
"column": 76
} | [
{
"pp": "α : Type u_2\nM : Matroid α\ne f : α\nI : Set α\nhI : M.Indep I\nhf : f ∈ M.closure I\nhe : e ∈ M.closure (insert f I \\ {e})\n⊢ M.closure (insert f I \\ {e}) = M.closure I",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.Matroid.Closure.0.Matroid.Indep.closure_insert_diff... | apply subset_antisymm <;> apply closure_subset_closure_of_subset_closure
· simp only [subset_def, mem_diff, mem_insert_iff, mem_singleton_iff]
rintro a (rfl | haI)
exacts [hf, M.subset_closure _ hI.subset_ground haI]
· intro a haI
obtain rfl | ne := eq_or_ne a e
exacts [he, M.mem_closure_of_mem' ⟨.i... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 634,
"column": 81
} | {
"line": 634,
"column": 89
} | [
{
"pp": "α : Type u_2\nM : Matroid α\ne f : α\nI : Set α\nhI : M.Indep I\nhfI : f ∈ M.closure I\nhe : e ∈ M.closure (insert f I \\ {e})\nheI : e ∈ I\nh : f ∈ M.closure (I \\ {e})\na✝ : α\n⊢ a✝ ∈ insert f I \\ {e} → a✝ ∈ insert f (I \\ {e})",
"usedConstants": [
"False",
"eq_false",
"and_tru... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 647,
"column": 2
} | {
"line": 647,
"column": 35
} | [
{
"pp": "α : Type u_2\nM : Matroid α\ne f : α\nB : Set α\nhB : M.IsBase B\nhe : e ∈ M.closure (insert f B \\ {e})\nheB : e ∈ insert f B\n⊢ M.IsBase (insert f B \\ {e})",
"usedConstants": [
"congrArg",
"Matroid.E",
"Matroid.isBasis_ground_iff",
"Matroid.IsBase",
"Eq.mp",
"... | rw [← isBasis_ground_iff] at hB ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Rank.Cardinal | {
"line": 371,
"column": 11
} | {
"line": 371,
"column": 28
} | [
{
"pp": "case refine_2\nα : Type u\nM : Matroid α\nh : M.cRank < ℵ₀\nB : Set α\nhB : M.IsBase B\n⊢ ∃ B, M.IsBase B ∧ B.Finite",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finite",
"Set.Finite",
"Exists",
"Matroid.IsBase",
"Set.Elem",
"id",
"_private.Mathl... | ← finite_coe_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.AlgebraicIndependent.Basic | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 20
} | [
{
"pp": "σ : Type u_3\nR : Type u_4\ninst✝ : CommRing R\n⊢ Injective ⇑(AlgHom.id R (MvPolynomial σ R))",
"usedConstants": [
"Function.injective_id",
"CommRing.toCommSemiring",
"MvPolynomial"
]
}
] | exact injective_id | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 972,
"column": 62
} | {
"line": 972,
"column": 74
} | [
{
"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⊢ (M.comap f).closure I = f ⁻¹' M.closure (f '' I)",
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"id",
"Iff",
"Se... | Set.ext_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 1014,
"column": 6
} | {
"line": 1014,
"column": 38
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nR S : Set α\nhSR : S ⊆ R\n⊢ R ∩ M.E ⊆ M.closure (S ∩ R) ↔ R ∩ M.E ⊆ M.closure S",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"id",
"HasSubset.Subset",
"Set.instInter",
"Inter.inter",
"Iff",
"Matroid.closur... | inter_eq_self_of_subset_left hSR | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AlgebraicIndependent.Basic | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 61
} | [
{
"pp": "ι : Type u\nR : Type u_2\nA : Type v\nx : ι → A\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nhx : AlgebraicIndependent R x\n⊢ lift.{v, u} #ι ≤ lift.{u, v} (trdeg R A)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Cardinal",
"congrArg",
... | rw [lift_mk_eq'.mpr ⟨.ofInjective _ hx.injective⟩, lift_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.AlgebraicIndependent.Transcendental | {
"line": 186,
"column": 2
} | {
"line": 189,
"column": 21
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nA : Type v\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : Set ι\n⊢ AlgebraicIndependent R x ↔ (AlgebraicIndependent R fun i ↦ x ↑i) ∧ AlgebraicIndepOn (↥(adjoin R (x '' s))) x sᶜ",
"usedConstants": [
"Subalgebra.instSetLike",
"... | rw [show x '' s = range fun i : s ↦ x i by ext; simp]
convert! ← sumElim_iff
classical apply algebraicIndependent_equiv' ((Equiv.sumComm ..).trans (Equiv.Set.sumCompl ..))
ext (_ | _) <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AlgebraicIndependent.Transcendental | {
"line": 186,
"column": 2
} | {
"line": 189,
"column": 21
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nA : Type v\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : Set ι\n⊢ AlgebraicIndependent R x ↔ (AlgebraicIndependent R fun i ↦ x ↑i) ∧ AlgebraicIndepOn (↥(adjoin R (x '' s))) x sᶜ",
"usedConstants": [
"Subalgebra.instSetLike",
"... | rw [show x '' s = range fun i : s ↦ x i by ext; simp]
convert! ← sumElim_iff
classical apply algebraicIndependent_equiv' ((Equiv.sumComm ..).trans (Equiv.Set.sumCompl ..))
ext (_ | _) <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.SeparableDegree | {
"line": 106,
"column": 35
} | {
"line": 106,
"column": 63
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nf : F[X]\nirred : Irreducible f\ninst✝ : CharZero F\n⊢ (expand F (1 ^ 0)) f = f",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"AlgHom",
"Nat.instMonoid",
"... | by rw [pow_zero, expand_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 486,
"column": 6
} | {
"line": 494,
"column": 39
} | [
{
"pp": "case pos\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... | have h0 : extension ((algebraMap K (hat K)) x) = 0 := by
simp only [extension_eq_zero_iff, map_eq_zero]
rw [← hx_def, ha0] at hx
simpa using hx
simp only [h0, ↓reduceDIte, extension_eq_zero_iff, map_eq_zero, embedding_apply,
left_eq_dite_iff, WithZero.zero_ne_coe, imp_false, not_no... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 486,
"column": 6
} | {
"line": 494,
"column": 39
} | [
{
"pp": "case pos\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... | have h0 : extension ((algebraMap K (hat K)) x) = 0 := by
simp only [extension_eq_zero_iff, map_eq_zero]
rw [← hx_def, ha0] at hx
simpa using hx
simp only [h0, ↓reduceDIte, extension_eq_zero_iff, map_eq_zero, embedding_apply,
left_eq_dite_iff, WithZero.zero_ne_coe, imp_false, not_no... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 337,
"column": 78
} | {
"line": 337,
"column": 90
} | [
{
"pp": "R : 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✝ : IsDomain A\ns B : Set A\nhB : (matroid R A).IsBasis B s\n⊢ {x | (matroid R A).IsBasis B (insert x B)} = ↑(algebraicClosure (↥(adjoin R s)) A)",
"usedConstants": [
"Suba... | Set.ext_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 434,
"column": 4
} | {
"line": 434,
"column": 23
} | [
{
"pp": "ι : Type u\nι' : Type u'\nR : Type u_1\nA : Type w\nx : ι → A\ny : 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\nhy : IsTranscendenceBasis R y\n⊢ lift.{u', max w u} (lift.{u, w} (trdeg R A)) = lift... | ← lift_lift.{w, u}, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 593,
"column": 26
} | {
"line": 593,
"column": 42
} | [
{
"pp": "case inr.inr\nι : Type u\nR : Type u_1\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FaithfulSMul R S\ninst✝ : NoZeroDivisors S\ns : Set ι\ni j : ι\nv : ι → S\nhj✝ : j ∈ insert i s\nH₁ : IsTranscendenceBasis R fun x ↦ v ↑x\nthis✝³ : Nontrivial ↥(adjoin R (v '' (i... | ← image_eq_range | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 118,
"column": 8
} | {
"line": 118,
"column": 51
} | [
{
"pp": "case pos\nK : Type u_4\nL : Type u_5\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nF : Type u_6\ninst✝¹ : Field F\ninst✝ : Algebra K F\nx : L\nhf : (Polynomial.map (algebraMap K F) (minpoly K x)).Splits\ninjKxL : Function.Injective ⇑(algebraMap (↥K⟮x⟯) L)\nhx : IsIntegral K x\n⊢ ((minpoly ... | simp only [AdjoinSimple.algebraMap_gen _ _] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Trace.Basic | {
"line": 118,
"column": 8
} | {
"line": 118,
"column": 51
} | [
{
"pp": "case pos\nK : Type u_4\nL : Type u_5\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nF : Type u_6\ninst✝¹ : Field F\ninst✝ : Algebra K F\nx : L\nhf : (Polynomial.map (algebraMap K F) (minpoly K x)).Splits\ninjKxL : Function.Injective ⇑(algebraMap (↥K⟮x⟯) L)\nhx : IsIntegral K x\n⊢ ((minpoly ... | simp only [AdjoinSimple.algebraMap_gen _ _] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Trace.Basic | {
"line": 118,
"column": 8
} | {
"line": 118,
"column": 51
} | [
{
"pp": "case pos\nK : Type u_4\nL : Type u_5\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nF : Type u_6\ninst✝¹ : Field F\ninst✝ : Algebra K F\nx : L\nhf : (Polynomial.map (algebraMap K F) (minpoly K x)).Splits\ninjKxL : Function.Injective ⇑(algebraMap (↥K⟮x⟯) L)\nhx : IsIntegral K x\n⊢ ((minpoly ... | simp only [AdjoinSimple.algebraMap_gen _ _] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Trace.Basic | {
"line": 118,
"column": 8
} | {
"line": 118,
"column": 51
} | [
{
"pp": "case pos\nK : Type u_4\nL : Type u_5\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nF : Type u_6\ninst✝¹ : Field F\ninst✝ : Algebra K F\nx : L\nhf : (Polynomial.map (algebraMap K F) (minpoly K x)).Splits\ninjKxL : Function.Injective ⇑(algebraMap (↥K⟮x⟯) L)\nhx : IsIntegral K x\n⊢ (Polynomia... | simp only [AdjoinSimple.algebraMap_gen _ _] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Trace.Basic | {
"line": 118,
"column": 8
} | {
"line": 118,
"column": 51
} | [
{
"pp": "case pos\nK : Type u_4\nL : Type u_5\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nF : Type u_6\ninst✝¹ : Field F\ninst✝ : Algebra K F\nx : L\nhf : (Polynomial.map (algebraMap K F) (minpoly K x)).Splits\ninjKxL : Function.Injective ⇑(algebraMap (↥K⟮x⟯) L)\nhx : IsIntegral K x\n⊢ (Polynomia... | simp only [AdjoinSimple.algebraMap_gen _ _] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Trace.Basic | {
"line": 118,
"column": 8
} | {
"line": 118,
"column": 51
} | [
{
"pp": "case pos\nK : Type u_4\nL : Type u_5\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nF : Type u_6\ninst✝¹ : Field F\ninst✝ : Algebra K F\nx : L\nhf : (Polynomial.map (algebraMap K F) (minpoly K x)).Splits\ninjKxL : Function.Injective ⇑(algebraMap (↥K⟮x⟯) L)\nhx : IsIntegral K x\n⊢ (Polynomia... | simp only [AdjoinSimple.algebraMap_gen _ _] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 684,
"column": 4
} | {
"line": 685,
"column": 70
} | [
{
"pp": "case inr\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\nx : R ⊗[k] K\na✝ : Nontrivial (R ⊗[k] K)\ninst✝ : ExpChar k 1\nthis : CharZero k\n⊢ ∃ n, x ^ 1 ^ n ∈ (algebraMap R (... | exact ⟨0, (Algebra.TensorProduct.includeLeft_surjective R _ <|
IsPurelyInseparable.surjective_algebraMap_of_isSeparable k K) _⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.SeparableDegree | {
"line": 457,
"column": 2
} | {
"line": 457,
"column": 26
} | [
{
"pp": "case neg.refine_2\nF : Type u\ninst✝ : Field F\nf g : F[X]\nh : ¬f = 0 ∧ ¬g = 0\n⊢ f = 0 ∧ g = 0 ∨ IsCoprime f g →\n ∀ (a : AlgebraicClosure F), f ≠ 0 ∧ (aeval a) f = 0 → ∀ (b : AlgebraicClosure F), g ≠ 0 ∧ (aeval b) g = 0 → a ≠ b",
"usedConstants": [
"Field.toSemifield",
"Polynomial... | rintro (⟨rfl, rfl⟩ | hc) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.FieldTheory.SeparableDegree | {
"line": 614,
"column": 2
} | {
"line": 614,
"column": 48
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nf : F[X]\nq : ℕ\ninst✝ : ExpChar F q\nhm : f.Monic\n⊢ f.natSepDegree = 1 ↔ ∃ m n y, m ≠ 0 ∧ f = (X ^ q ^ n - C y) ^ m",
"usedConstants": [
"Polynomial.C",
"Nat.instMonoid",
"HSub.hSub",
"RingHom",
"Exists",
"Field.toDivisionRing",
... | refine ⟨fun h ↦ ?_, fun ⟨m, n, y, hm, h⟩ ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.PrincipalIdealDomainOfPrime | {
"line": 30,
"column": 4
} | {
"line": 44,
"column": 88
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\na : R\n⊢ Submodule.IsPrincipal (I ⊔ span {a}) → Submodule.IsPrincipal (Submodule.colon I ↑(span {a})) → Submodule.IsPrincipal I",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Submodule",
"Ideal.mem_span_singleton'",
"Ideal... | intro ⟨x, hx⟩ ⟨y, hy⟩
refine ⟨x * y, le_antisymm ?_ ?_⟩ <;> rw [submodule_span_eq] at *
· intro i hi
have hisup : i ∈ I ⊔ span {a} := mem_sup_left hi
have hasup : a ∈ I ⊔ span {a} := mem_sup_right (mem_span_singleton_self a)
rw [hx, mem_span_singleton'] at hisup hasup
obtain ⟨u, rfl⟩ := ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PrincipalIdealDomainOfPrime | {
"line": 30,
"column": 4
} | {
"line": 44,
"column": 88
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\na : R\n⊢ Submodule.IsPrincipal (I ⊔ span {a}) → Submodule.IsPrincipal (Submodule.colon I ↑(span {a})) → Submodule.IsPrincipal I",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Submodule",
"Ideal.mem_span_singleton'",
"Ideal... | intro ⟨x, hx⟩ ⟨y, hy⟩
refine ⟨x * y, le_antisymm ?_ ?_⟩ <;> rw [submodule_span_eq] at *
· intro i hi
have hisup : i ∈ I ⊔ span {a} := mem_sup_left hi
have hasup : a ∈ I ⊔ span {a} := mem_sup_right (mem_span_singleton_self a)
rw [hx, mem_span_singleton'] at hisup hasup
obtain ⟨u, rfl⟩ := ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 515,
"column": 4
} | {
"line": 515,
"column": 12
} | [
{
"pp": "case pos\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : IsDiscreteValuationRing A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx y : K\nh : (valuation K (maximalIdeal A)) y = (valuation K (maximalIdeal A)) x\nhx : x = 0\n⊢ ∃ u, u • x = y",
"u... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 517,
"column": 4
} | {
"line": 517,
"column": 12
} | [
{
"pp": "case pos\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : IsDiscreteValuationRing A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx y : K\nh : (valuation K (maximalIdeal A)) x = (valuation K (maximalIdeal A)) y\nhx : ¬x = 0\nhy : y = 0\n⊢ ∃ u, u • x... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 517,
"column": 4
} | {
"line": 517,
"column": 12
} | [
{
"pp": "case pos\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : IsDiscreteValuationRing A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx y : K\nh : (valuation K (maximalIdeal A)) x = (valuation K (maximalIdeal A)) y\nhx : ¬x = 0\nhy : y = 0\n⊢ ∃ u, u • x... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 517,
"column": 4
} | {
"line": 517,
"column": 12
} | [
{
"pp": "case pos\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : IsDiscreteValuationRing A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx y : K\nh : (valuation K (maximalIdeal A)) x = (valuation K (maximalIdeal A)) y\nhx : ¬x = 0\nhy : y = 0\n⊢ ∃ u, u • x... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 518,
"column": 73
} | {
"line": 518,
"column": 81
} | [
{
"pp": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : IsDiscreteValuationRing A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx y : K\nh : (valuation K (maximalIdeal A)) x = (valuation K (maximalIdeal A)) y\nhx : ¬x = 0\nhy : ¬y = 0\n⊢ y / x ∈ MonoidHom.m... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 112,
"column": 77
} | {
"line": 123,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nx y : R\n⊢ v.intValuationDef (x * y) = v.intValuationDef x * v.intValuationDef y",
"usedConstants": [
"Int.instAddCommGroup",
"WithZero.exp_add",
"CommMonoidWithZero.toCommMonoid",
"Iff.m... | by
classical
simp only [intValuationDef]
by_cases hx : x = 0
· rw [hx, zero_mul, if_pos rfl, zero_mul]
· by_cases hy : y = 0
· rw [hy, mul_zero, if_pos rfl, mul_zero]
· rw [if_neg hx, if_neg hy, if_neg (mul_ne_zero hx hy), ← exp_add,
← Ideal.span_singleton_mul_span_singleton, ← Associates.mk_m... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 130,
"column": 75
} | {
"line": 130,
"column": 83
} | [
{
"pp": "case inl\nG : Type u_1\nG' : Type u_2\ninst✝⁵ : CommGroup G\ninst✝⁴ : LinearOrder G\ninst✝³ : IsOrderedMonoid G\ninst✝² : CommGroup G'\ninst✝¹ : LinearOrder G'\ninst✝ : IsOrderedMonoid G'\nx : G\ny : G'\nhxy : x = 1 ↔ y = 1\nhx : ¬x = 1\nx' : G := max x x⁻¹\nhx' : x' = max x x⁻¹\nxpos : 1 < x'\ny' : G'... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 130,
"column": 75
} | {
"line": 130,
"column": 83
} | [
{
"pp": "case inr\nG : Type u_1\nG' : Type u_2\ninst✝⁵ : CommGroup G\ninst✝⁴ : LinearOrder G\ninst✝³ : IsOrderedMonoid G\ninst✝² : CommGroup G'\ninst✝¹ : LinearOrder G'\ninst✝ : IsOrderedMonoid G'\nx : G\ny : G'\nhxy : x = 1 ↔ y = 1\nhx : ¬x = 1\nx' : G := max x x⁻¹\nhx' : x' = max x x⁻¹\nxpos : 1 < x'\ny' : G'... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 131,
"column": 75
} | {
"line": 131,
"column": 83
} | [
{
"pp": "case inl\nG : Type u_1\nG' : Type u_2\ninst✝⁵ : CommGroup G\ninst✝⁴ : LinearOrder G\ninst✝³ : IsOrderedMonoid G\ninst✝² : CommGroup G'\ninst✝¹ : LinearOrder G'\ninst✝ : IsOrderedMonoid G'\nx : G\ny : G'\nhxy : x = 1 ↔ y = 1\nhx : ¬x = 1\nx' : G := max x x⁻¹\nhx' : x' = max x x⁻¹\nxpos : 1 < x'\ny' : G'... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 131,
"column": 75
} | {
"line": 131,
"column": 83
} | [
{
"pp": "case inr\nG : Type u_1\nG' : Type u_2\ninst✝⁵ : CommGroup G\ninst✝⁴ : LinearOrder G\ninst✝³ : IsOrderedMonoid G\ninst✝² : CommGroup G'\ninst✝¹ : LinearOrder G'\ninst✝ : IsOrderedMonoid G'\nx : G\ny : G'\nhxy : x = 1 ↔ y = 1\nhx : ¬x = 1\nx' : G := max x x⁻¹\nhx' : x' = max x x⁻¹\nxpos : 1 < x'\ny' : G'... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 198,
"column": 18
} | {
"line": 198,
"column": 74
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nr : R\nhr : r ≠ 0\nhsr : Ideal.span {r} ≠ 0\nhfm : FiniteMultiplicity v.asIdeal (Ideal.span {r})\n⊢ ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r})).factors) =\n ↑(Multiset.count v.asIdeal (Uniq... | Ideal.count_associates_factors_eq hsr v.isPrime v.ne_bot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 30
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nx : R\nhx : x ≠ 0\n⊢ 0 ≤ ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors)",
"usedConstants": [
"Associates.mk",
"CommSemiring.toSemiring",
"Associates.count... | exact Int.natCast_nonneg _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 374,
"column": 33
} | {
"line": 374,
"column": 41
} | [
{
"pp": "R : 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 b : R\nhb : b ≠ 0\nh : b ∈ v.asIdeal → a ∉ v.asIdeal\nhv : b ∈ v.asIdeal\nx✝ : a = 0\n⊢ False",
"usedConstants": [
"Fals... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 374,
"column": 33
} | {
"line": 374,
"column": 41
} | [
{
"pp": "R : 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 b : R\nhb : b ≠ 0\nh : b ∈ v.asIdeal → a ∉ v.asIdeal\nhv : b ∈ v.asIdeal\nx✝ : a = 0\n⊢ False",
"usedConstants": [
"Fals... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 374,
"column": 33
} | {
"line": 374,
"column": 41
} | [
{
"pp": "R : 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 b : R\nhb : b ≠ 0\nh : b ∈ v.asIdeal → a ∉ v.asIdeal\nhv : b ∈ v.asIdeal\nx✝ : a = 0\n⊢ False",
"usedConstants": [
"Fals... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 267,
"column": 4
} | {
"line": 268,
"column": 74
} | [
{
"pp": "case refine_1\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\na : R\nn : σ →₀ ℕ\nh : ∀ (n_1 : σ →₀ ℕ), (coeff n_1) (φ * (monomial n) a) = (coeff n_1) ((monomial n) a * φ)\nm : σ →₀ ℕ\n⊢ Commute ((coeff m) φ) a",
"usedConstants": [
"Nat.instMulZeroClass",
"Semirin... | have := h (m + n)
rwa [coeff_add_mul_monomial, add_comm, coeff_add_monomial_mul] at this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 267,
"column": 4
} | {
"line": 268,
"column": 74
} | [
{
"pp": "case refine_1\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\na : R\nn : σ →₀ ℕ\nh : ∀ (n_1 : σ →₀ ℕ), (coeff n_1) (φ * (monomial n) a) = (coeff n_1) ((monomial n) a * φ)\nm : σ →₀ ℕ\n⊢ Commute ((coeff m) φ) a",
"usedConstants": [
"Nat.instMulZeroClass",
"Semirin... | have := h (m + n)
rwa [coeff_add_mul_monomial, add_comm, coeff_add_monomial_mul] at this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 412,
"column": 41
} | {
"line": 412,
"column": 86
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nφ : MvPowerSeries σ R\na : R\n⊢ (coeff n) (φ * C a) = (coeff n) φ * a",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Nat.instMulZeroClass",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"AddMonoid.toA... | by simpa using coeff_add_mul_monomial n 0 φ a | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 513,
"column": 2
} | {
"line": 516,
"column": 14
} | [
{
"pp": "R : 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\n⊢ ∃ n d, x * (algebraMap R K) ↑d = (algebraMap R K) n ∨ x * (algebraMap R K) n = (algebraMap R K) ↑d",
"usedConstants": [... | obtain (⟨r, hr⟩ | ⟨r, hr⟩) :=
ValuationRing.isInteger_or_isInteger (valuationSubringAtPrime K v) x
<;> obtain ⟨⟨n, d⟩, hnd⟩ := IsLocalization.surj v.asIdeal.primeCompl r
<;> use n, d | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 275,
"column": 83
} | {
"line": 278,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nn : ℕ\nφ : R⟦X⟧\n⊢ (coeff (n + 1)) (X * φ) = (coeff n) φ",
"usedConstants": [
"Eq.mpr",
"Unit.unit",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Nat.instMulZeroClass",
"Semiring.toModule",
"HMul.hMul",
... | by
simp only [coeff, Finsupp.single_add, add_comm n 1]
convert! φ.coeff_add_monomial_mul (single () 1) (single () n) _
rw [one_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 402,
"column": 61
} | {
"line": 410,
"column": 67
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R⟦X⟧\nn d : ℕ\n⊢ (coeff d) (X ^ n * p) = if n ≤ d then (coeff (d - n)) p else 0",
"usedConstants": [
"not_le",
"Eq.mpr",
"le_of_add_le_right",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.i... | by
split_ifs with h
· rw [← tsub_add_cancel_of_le h, coeff_X_pow_mul]
simp
· refine (coeff_mul _ _ _).trans (Finset.sum_eq_zero fun x hx => ?_)
rw [coeff_X_pow, if_neg, zero_mul]
have := mem_antidiagonal.mp hx
rw [add_comm] at this
exact ((le_of_add_le_right this.le).trans_lt <| not_le.mp h).n... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 598,
"column": 2
} | {
"line": 599,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nS : Type u_3\ninst✝³ : CommSemiring A\ninst✝² : Algebra A R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra A S\nφ : R →ₐ[A] S\na : A\nf : R⟦X⟧\n⊢ (rescale ((algebraMap A S) a)) ((map ↑φ) f) = (map ↑φ) ((rescale ((algebraMap A R) a)) f)",
"usedConstant... | convert! rescale_map (φ : R →+* S) _ _
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 598,
"column": 2
} | {
"line": 599,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nS : Type u_3\ninst✝³ : CommSemiring A\ninst✝² : Algebra A R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra A S\nφ : R →ₐ[A] S\na : A\nf : R⟦X⟧\n⊢ (rescale ((algebraMap A S) a)) ((map ↑φ) f) = (map ↑φ) ((rescale ((algebraMap A R) a)) f)",
"usedConstant... | convert! rescale_map (φ : R →+* S) _ _
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 762,
"column": 2
} | {
"line": 762,
"column": 46
} | [
{
"pp": "R : 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\n⊢ ∃ b ∈ R⁰, a * ↑b ∈ adicCompletionIntegers K v",
"usedConstants": [
"Int.instAddCommGroup",
... | by_cases ha : a ∈ v.adicCompletionIntegers K | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 652,
"column": 25
} | {
"line": 652,
"column": 28
} | [
{
"pp": "case inr.succ\nR : Type u_2\ninst✝ : CommSemiring R\nφ : R⟦X⟧\nn' : ℕ\nih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)\nhn : n' + 1 > 0\nh₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ\nh₂ : antidiagonal 1 = {(0, 1), (1, 0)}\n⊢ ∑ p ∈ antidiagonal 1, (coeff p.1) (φ ^ n') * (coe... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 731,
"column": 2
} | {
"line": 731,
"column": 44
} | [
{
"pp": "case h\nσ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\nf : MvPowerSeries σ R\nm : ℕ\nhf : constantCoeff f ^ m = 0\nd : σ →₀ ℕ\nn : ℕ\nhn : m + degree d ≤ n\nk : ℕ →₀ σ →₀ ℕ\nhk : (range n).sum ⇑k = d ∧ k.support ⊆ range n\ns : Finset ℕ := {i ∈ range n | k i = 0}\nhs_def : s = {i ∈ range n | k i = ... | have hs : s ⊆ range n := filter_subset _ _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 754,
"column": 4
} | {
"line": 754,
"column": 18
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\nf : MvPowerSeries σ R\nm : ℕ\nhf : constantCoeff f ^ m = 0\nd : σ →₀ ℕ\nn : ℕ\nhn : m + degree d ≤ n\nk : ℕ →₀ σ →₀ ℕ\nhk : (range n).sum ⇑k = d ∧ k.support ⊆ range n\ns : Finset ℕ := {i ∈ range n | k i = 0}\nhs_def : s = {i ∈ range n | k i = 0}\nhs :... | intro i hi hi' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.MvPowerSeries.Trunc | {
"line": 251,
"column": 62
} | {
"line": 252,
"column": 28
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : DecidableEq σ\ninst✝ : CommSemiring R\nn : σ →₀ ℕ\na : R\np : MvPowerSeries σ R\n⊢ (trunc' R n) (C a * p) = MvPolynomial.C a * (trunc' R n) p",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Nat.instCanonicallyOrderedAdd",
"Finsupp.instLE",
... | by
ext m; simp [coeff_trunc'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPowerSeries.PiTopology | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 27
} | [
{
"pp": "case hc\nσ : Type u_1\nR : Type u_2\ninst✝³ : TopologicalSpace R\ninst✝² : DecidableEq σ\ninst✝¹ : CommSemiring R\ninst✝ : Nonempty σ\nf : MvPowerSeries σ R\nd : σ →₀ ℕ\ns : σ\nh✝ : True\nn : σ →₀ ℕ\nhn : n ≥ d + Finsupp.single s 1\n⊢ d < n",
"usedConstants": [
"Nat.instMulZeroClass",
"... | apply lt_of_lt_of_le _ hn | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.PowerSeries.Trunc | {
"line": 76,
"column": 43
} | {
"line": 76,
"column": 51
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\nn : ℕ\na : R\nm : ℕ\nH : ¬m < n + 1\nh✝ : m = 0\n⊢ 0 = a",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"False",
"Preorder.toLT",
"Nat.instIsOrderedAddMonoid",
"LinearOrderedCommMonoidWithZero.toI... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.PowerSeries.Trunc | {
"line": 76,
"column": 43
} | {
"line": 76,
"column": 51
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\nn : ℕ\na : R\nm : ℕ\nH : ¬m < n + 1\nh✝ : m = 0\n⊢ 0 = a",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"False",
"Preorder.toLT",
"Nat.instIsOrderedAddMonoid",
"LinearOrderedCommMonoidWithZero.toI... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Trunc | {
"line": 76,
"column": 43
} | {
"line": 76,
"column": 51
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\nn : ℕ\na : R\nm : ℕ\nH : ¬m < n + 1\nh✝ : m = 0\n⊢ 0 = a",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"False",
"Preorder.toLT",
"Nat.instIsOrderedAddMonoid",
"LinearOrderedCommMonoidWithZero.toI... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.PiTopology | {
"line": 186,
"column": 2
} | {
"line": 191,
"column": 28
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : TopologicalSpace R\ninst✝ : Semiring R\n⊢ Continuous[inst✝¹, instTopologicalSpace R] ⇑C",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Nat.instMulZeroClass",
"Continuous",
"Semiring.toModule",
"MvPowerSeries.WithPiTopology.instTop... | simp only [continuous_iff_continuousAt]
refine fun r ↦ (tendsto_iff_coeff_tendsto _ _ _).mpr fun d ↦ ?_
simp only [coeff_C]
split_ifs
· exact tendsto_id
· exact tendsto_const_nhds | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.PiTopology | {
"line": 186,
"column": 2
} | {
"line": 191,
"column": 28
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : TopologicalSpace R\ninst✝ : Semiring R\n⊢ Continuous[inst✝¹, instTopologicalSpace R] ⇑C",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Nat.instMulZeroClass",
"Continuous",
"Semiring.toModule",
"MvPowerSeries.WithPiTopology.instTop... | simp only [continuous_iff_continuousAt]
refine fun r ↦ (tendsto_iff_coeff_tendsto _ _ _).mpr fun d ↦ ?_
simp only [coeff_C]
split_ifs
· exact tendsto_id
· exact tendsto_const_nhds | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Trunc | {
"line": 108,
"column": 8
} | {
"line": 108,
"column": 11
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\nn d : ℕ\nh₁ : d < n + 2\nh₂ : d = 1\n⊢ 1 = Polynomial.X.coeff d",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"id",
"instOfNatNat",
"AddCommMonoidWithOne.toAddMonoidWithOne",
... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.LinearTopology | {
"line": 77,
"column": 55
} | {
"line": 78,
"column": 14
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Ring R\nf : MvPowerSeries σ R\nJd : TwoSidedIdeal R × (σ →₀ ℕ)\n⊢ f ∈ basis σ R Jd ↔ ∀ e ≤ Jd.2, (coeff e) f ∈ Jd.1",
"usedConstants": [
"Finsupp.instLE",
"Nat.instMulZeroClass",
"Semiring.toModule",
"Ring.toNonAssocRing",
"congrArg"... | by
simp [basis] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 328,
"column": 41
} | {
"line": 328,
"column": 61
} | [
{
"pp": "case h\nσ : Type u_1\nR : Type u_3\ninst✝² : CommRing R\nS : Type u_5\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nf : MvPowerSeries σ R\ne : σ →₀ ℕ\n⊢ (algebraMap R S) ((coeff e) f) = (coeff e) f • (coeff e) ((monomial e) 1)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoi... | coeff_monomial_same, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 547,
"column": 84
} | {
"line": 560,
"column": 17
} | [
{
"pp": "σ : Type u_1\nR : Type u_9\ninst✝ : CommSemiring R\n⊢ rescale 0 = C.comp constantCoeff",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",
"MvPowerSeries.rescale",
"Semiring.toModule",
"HMul.hMul",
"outParam",
"MulZeroClass.t... | by
classical
ext x n
simp only [rescale, Pi.zero_apply, RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk,
RingHom.coe_comp, Function.comp_apply, coeff_C]
split_ifs with h
· simp [h, coeff_apply, ← @coeff_zero_eq_constantCoeff_apply, coeff_apply]
· simp only [coeff_apply]
convert! zero_mul _
simp ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 72
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nq : ℕ\nhq : 1 < q\nf : PowerSeries R\nk : ℕ\n⊢ ((ofPowerSeries q) f) (q ^ k) = (PowerSeries.coeff k) f",
"usedConstants": [
"instPowNat",
"Eq.mpr",
"Semiring.toModule",
"ArithmeticFunction.instSemiring",
"ArithmeticFunction.instFun... | rw [ofPowerSeries_apply hq, (Nat.pow_right_injective hq).extend_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 72
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nq : ℕ\nhq : 1 < q\nf : PowerSeries R\nk : ℕ\n⊢ ((ofPowerSeries q) f) (q ^ k) = (PowerSeries.coeff k) f",
"usedConstants": [
"instPowNat",
"Eq.mpr",
"Semiring.toModule",
"ArithmeticFunction.instSemiring",
"ArithmeticFunction.instFun... | rw [ofPowerSeries_apply hq, (Nat.pow_right_injective hq).extend_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 72
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nq : ℕ\nhq : 1 < q\nf : PowerSeries R\nk : ℕ\n⊢ ((ofPowerSeries q) f) (q ^ k) = (PowerSeries.coeff k) f",
"usedConstants": [
"instPowNat",
"Eq.mpr",
"Semiring.toModule",
"ArithmeticFunction.instSemiring",
"ArithmeticFunction.instFun... | rw [ofPowerSeries_apply hq, (Nat.pow_right_injective hq).extend_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 146,
"column": 45
} | {
"line": 149,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nq : ℕ\nf : PowerSeries R\n⊢ ((ofPowerSeries q) f) 1 = PowerSeries.constantCoeff f",
"usedConstants": [
"ArithmeticFunction.ofPowerSeries._proof_1",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Nat.instMulZe... | by
by_cases hq : 1 < q
· rw [← pow_zero q, ofPowerSeries_apply_pow hq, PowerSeries.coeff_zero_eq_constantCoeff]
· simp [ofPowerSeries, dif_neg hq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 171,
"column": 77
} | {
"line": 171,
"column": 90
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : CommRing R\nq k : ℕ\nhk : k ≠ 0\nf : PowerSeries R\nhq : 1 < q\ni : ℕ\nhn : ¬k ∣ i\n⊢ 0 (q ^ i) = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"Nat.instMonoid",
"Pi.zero_apply",
"id",
"Pi.instZero",... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 198,
"column": 8
} | {
"line": 198,
"column": 33
} | [
{
"pp": "case pos.succ.succ\nR : Type u_1\ninst✝ : CommRing R\nf : PowerSeries R\nhf : PowerSeries.constantCoeff f = 1\np k : ℕ\nhp : Nat.Prime p\nhk : 0 < k\nn✝¹ n✝ : ℕ\nhmn : (p ^ (n✝¹ + 1)).Coprime (p ^ (n✝ + 1))\n⊢ ((ofPowerSeries p) (PowerSeries.subst (PowerSeries.X ^ k) f)) (p ^ (n✝¹ + 1) * p ^ (n✝ + 1)) ... | · simp [hp.ne_one] at hmn | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 27
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nthis : UniformSpace R := ⋯\n⊢ IsComplete (Set.range DFunLike.coe)",
"usedConstants": [
"Pi.uniformSpace",
"UniformSpace",
"DiscreteUniformity.instCompleteSpace",
"ArithmeticFunction.instFunLikeNat",
"CommSemiring.toSe... | apply IsClosed.isComplete | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 314,
"column": 2
} | {
"line": 324,
"column": 86
} | [
{
"pp": "case pos\nι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nf : ι → ArithmeticFunction R\nhf : ∀ (i : ι), (f i).IsMultiplicative\nhf' : Multipliable f\n⊢ (eulerProduct f).IsMultiplicative",
"usedConstants": [
"And.imp",
"Nat.Coprime",
"MulOne.toOne",
"Filter.EventuallyEq.t... | · have h (s : Finset ι) : (∏ b ∈ s, f b).IsMultiplicative :=
isMultiplicative_finsetProd f s fun i a ↦ hf i
have key := tendsto_iff.mp hf'.hasProd
refine (forall_and.mp h).imp (fun h ↦ ?_) fun h m n hmn ↦ ?_
· specialize key 1
simp_rw [h] at key
rwa [eventually_const, eq_comm] at key
·... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 342,
"column": 33
} | {
"line": 342,
"column": 41
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nq : ι → ℕ\nhq : Northcott q\nf : ι → PowerSeries R\nhf : ∀ (i : ι), PowerSeries.constantCoeff (f i) = 1\nn✝ : ℕ\ni : ι\nn : ℕ\nhi : n + 1 + 1 + 1 ≤ q i\nhqi : 1 < q i\nk : ℕ\nhk : q i ^ k = n + 1 + 1\nh : k = 0\n⊢ False",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 342,
"column": 33
} | {
"line": 342,
"column": 41
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nq : ι → ℕ\nhq : Northcott q\nf : ι → PowerSeries R\nhf : ∀ (i : ι), PowerSeries.constantCoeff (f i) = 1\nn✝ : ℕ\ni : ι\nn : ℕ\nhi : n + 1 + 1 + 1 ≤ q i\nhqi : 1 < q i\nk : ℕ\nhk : q i ^ k = n + 1 + 1\nh : k = 0\n⊢ False",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 342,
"column": 33
} | {
"line": 342,
"column": 41
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nq : ι → ℕ\nhq : Northcott q\nf : ι → PowerSeries R\nhf : ∀ (i : ι), PowerSeries.constantCoeff (f i) = 1\nn✝ : ℕ\ni : ι\nn : ℕ\nhi : n + 1 + 1 + 1 ≤ q i\nhqi : 1 < q i\nk : ℕ\nhk : q i ^ k = n + 1 + 1\nh : k = 0\n⊢ False",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.EGauge | {
"line": 210,
"column": 30
} | {
"line": 210,
"column": 52
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝² : NormedDivisionRing 𝕜\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nc : 𝕜\ns : Set E\nh : c = 0 → s.Nonempty\nx : E\nhc : c ≠ 0\n⊢ ‖c‖ₑ⁻¹ * egauge 𝕜 s (c • x) ≤ egauge 𝕜 s x",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"GroupWithZ... | ← enorm_inv (by simpa) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.EGauge | {
"line": 262,
"column": 4
} | {
"line": 279,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (E i)\ninst✝ : (i : ι) → Module 𝕜 (E i)\nI : Set ι\nhI : I.Finite\nU : (i : ι) → Set (E i)\nhU : ∀ i ∈ I, Balanced 𝕜 (U i)\nx : (i : ι) → E i\nhI₀ : I = univ ∨ (∃ i ∈ I, x i ≠ 0) ∨ (𝓝[≠] 0)... | have hr₀ : 0 < r := hr.bot_lt
rcases I.eq_empty_or_nonempty with rfl | hIne
· obtain hι | hbot : IsEmpty ι ∨ (𝓝[≠] (0 : 𝕜)).NeBot := by simpa [@eq_comm _ ∅] using hI₀
· use 0
simp [@eq_comm _ ∅, hι, hr₀]
· rcases exists_enorm_lt 𝕜 hr₀.ne' with ⟨c₀, hc₀, hc₀r⟩
exact ⟨c₀, .inl hc₀, ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.EGauge | {
"line": 262,
"column": 4
} | {
"line": 279,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (E i)\ninst✝ : (i : ι) → Module 𝕜 (E i)\nI : Set ι\nhI : I.Finite\nU : (i : ι) → Set (E i)\nhU : ∀ i ∈ I, Balanced 𝕜 (U i)\nx : (i : ι) → E i\nhI₀ : I = univ ∨ (∃ i ∈ I, x i ≠ 0) ∨ (𝓝[≠] 0)... | have hr₀ : 0 < r := hr.bot_lt
rcases I.eq_empty_or_nonempty with rfl | hIne
· obtain hι | hbot : IsEmpty ι ∨ (𝓝[≠] (0 : 𝕜)).NeBot := by simpa [@eq_comm _ ∅] using hI₀
· use 0
simp [@eq_comm _ ∅, hι, hr₀]
· rcases exists_enorm_lt 𝕜 hr₀.ne' with ⟨c₀, hc₀, hc₀r⟩
exact ⟨c₀, .inl hc₀, ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.TangentCone.Defs | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 70
} | [
{
"pp": "R : Type u\nE : Type v\ninst✝² : AddCommGroup E\ninst✝¹ : SMul R E\ninst✝ : TopologicalSpace E\ns : Set E\nx y : E\nα : Type u_1\nl : Filter α\nc : α → R\nd : α → E\nhd₀ : Tendsto d l (𝓝 0)\nhds✝ : ∃ᶠ (n : α) in l, x + d n ∈ s\nhds : (l ⊓ 𝓟 {x_1 | x + d x_1 ∈ s}).NeBot\nhcd : Tendsto (fun n ↦ c n • d... | exact ClusterPt.mono (hcd.mono_left inf_le_left).mapClusterPt this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.TangentCone.Basic | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 12
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : DivisionSemiring 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace 𝕜\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousSMul 𝕜 E\ns : Set E\nx y : E\nα : Type u_3\nl : Filter α\ninst✝ : l.NeBot\nc : α → 𝕜\nhc₀ : Tendsto c ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.PowerSeries.Substitution | {
"line": 477,
"column": 16
} | {
"line": 477,
"column": 27
} | [
{
"pp": "case e_a.succ.succ.h₀.zero\nR : Type u_2\ninst✝¹ : CommRing R\nP : R⟦X⟧\nhP : constantCoeff P = 0\ninst✝ : Invertible ((coeff 1) P)\nn : ℕ\nB : R⟦X⟧\nhB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B\nhB' : constantCoeff B = 0\nk : R\nhk : ⅟((coeff 1) P) * (coeff (n + 1 + 1)) (subst B P) = k\ni : ℕ\nhj : 0 ∈... | simp at hj' | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.Substitution | {
"line": 477,
"column": 16
} | {
"line": 477,
"column": 27
} | [
{
"pp": "case e_a.succ.succ.h₀.zero\nR : Type u_2\ninst✝¹ : CommRing R\nP : R⟦X⟧\nhP : constantCoeff P = 0\ninst✝ : Invertible ((coeff 1) P)\nn : ℕ\nB : R⟦X⟧\nhB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B\nhB' : constantCoeff B = 0\nk : R\nhk : ⅟((coeff 1) P) * (coeff (n + 1 + 1)) (subst B P) = k\ni : ℕ\nhj : 0 ∈... | simp at hj' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Substitution | {
"line": 477,
"column": 16
} | {
"line": 477,
"column": 27
} | [
{
"pp": "case e_a.succ.succ.h₀.zero\nR : Type u_2\ninst✝¹ : CommRing R\nP : R⟦X⟧\nhP : constantCoeff P = 0\ninst✝ : Invertible ((coeff 1) P)\nn : ℕ\nB : R⟦X⟧\nhB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B\nhB' : constantCoeff B = 0\nk : R\nhk : ⅟((coeff 1) P) * (coeff (n + 1 + 1)) (subst B P) = k\ni : ℕ\nhj : 0 ∈... | simp at hj' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Const | {
"line": 192,
"column": 32
} | {
"line": 192,
"column": 45
} | [
{
"pp": "case h.h\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace E\nF : Type u_3\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 F\ninst✝ : TopologicalSpace F\ns : Set E\nc : F\nx✝¹ x✝ : E\n⊢ 0 x✝ = (0 x✝¹) x✝",
"usedC... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
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