module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.AlgebraicIndependent.Transcendental | {
"line": 243,
"column": 6
} | {
"line": 243,
"column": 36
} | {
"line": 243,
"column": 36
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nA : Type v\nx : ι → A\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nhx : AlgebraicIndependent R x\ninst✝ : Nontrivial A\ns : Set ι\ni : ι\n⊢ Transcendental (↥(adjoin R (x '' s))) (x i) ↔ i ∉ s",
"ppTerm": "?m.28",
"assigned": true,
"usedConstan... | [
"ι : Type u_1\nR : Type u_3\nA : Type v\nx : ι → A\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nhx : AlgebraicIndependent R x\ninst✝ : Nontrivial A\ns : Set ι\ni : ι\n⊢ Transcendental (↥(adjoin R (x '' s))) (x i) ↔ Disjoint s {i}"
] | ← Set.disjoint_singleton_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 91
} | {
"line": 151,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\ns : Set α\ng : G\n⊢ s ∈ fixedBy (Set α) g ↔ ∀ (x : α), g • x ∈ s ↔ x ∈ s",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHSMul",
"congrArg",
"Mul... | [] | simp_rw [mem_fixedBy, ← eq_inv_smul_iff, Set.ext_iff, Set.mem_inv_smul_set_iff, Iff.comm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 91
} | {
"line": 151,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\ns : Set α\ng : G\n⊢ s ∈ fixedBy (Set α) g ↔ ∀ (x : α), g • x ∈ s ↔ x ∈ s",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHSMul",
"congrArg",
"Mul... | [] | simp_rw [mem_fixedBy, ← eq_inv_smul_iff, Set.ext_iff, Set.mem_inv_smul_set_iff, Iff.comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 91
} | {
"line": 151,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\ns : Set α\ng : G\n⊢ s ∈ fixedBy (Set α) g ↔ ∀ (x : α), g • x ∈ s ↔ x ∈ s",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHSMul",
"congrArg",
"Mul... | [] | simp_rw [mem_fixedBy, ← eq_inv_smul_iff, Set.ext_iff, Set.mem_inv_smul_set_iff, Iff.comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 220,
"column": 2
} | {
"line": 222,
"column": 38
} | {
"line": 224,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\ng h : G\ncomm : Commute g h\n⊢ fixedBy α g ∈ fixedBy (Set α) h",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Semigroup.toMul",
"instHSMul",
"HMul.hMul",
... | [] | ext x
rw [Set.mem_smul_set_iff_inv_smul_mem, mem_fixedBy, ← mul_smul, comm.inv_right, mul_smul,
smul_left_cancel_iff, mem_fixedBy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 220,
"column": 2
} | {
"line": 222,
"column": 38
} | {
"line": 224,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\ng h : G\ncomm : Commute g h\n⊢ fixedBy α g ∈ fixedBy (Set α) h",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Semigroup.toMul",
"instHSMul",
"HMul.hMul",
... | [] | ext x
rw [Set.mem_smul_set_iff_inv_smul_mem, mem_fixedBy, ← mul_smul, comm.inv_right, mul_smul,
smul_left_cancel_iff, mem_fixedBy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 60
} | {
"line": 126,
"column": 60
} | [
{
"pp": "ι : Type u\nR : Type u_1\nA : Type w\nx : ι → A\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nind : AlgebraicIndependent R x\nalg : Algebra.IsAlgebraic (↥(adjoin R (range x))) A\ns : Set A\nind_s : AlgebraicIndepOn R _root_.id s\nhxs : range x ⊆ s\nhxs' : ¬range... | [] | rw [← range_comp, val_comp_inclusion, Subtype.range_val] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 60
} | {
"line": 126,
"column": 60
} | [
{
"pp": "ι : Type u\nR : Type u_1\nA : Type w\nx : ι → A\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nind : AlgebraicIndependent R x\nalg : Algebra.IsAlgebraic (↥(adjoin R (range x))) A\ns : Set A\nind_s : AlgebraicIndepOn R _root_.id s\nhxs : range x ⊆ s\nhxs' : ¬range... | [] | rw [← range_comp, val_comp_inclusion, Subtype.range_val] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 60
} | {
"line": 126,
"column": 60
} | [
{
"pp": "ι : Type u\nR : Type u_1\nA : Type w\nx : ι → A\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nind : AlgebraicIndependent R x\nalg : Algebra.IsAlgebraic (↥(adjoin R (range x))) A\ns : Set A\nind_s : AlgebraicIndepOn R _root_.id s\nhxs : range x ⊆ s\nhxs' : ¬range... | [] | rw [← range_comp, val_comp_inclusion, Subtype.range_val] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 407,
"column": 2
} | {
"line": 408,
"column": 65
} | {
"line": 410,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : IsDomain A\ninst✝ : FaithfulSMul R A\ns : Set A\n⊢ Algebra.IsAlgebraic (↥(adjoin R s)) A ↔ ∃ t ⊆ s, IsTranscendenceBasis R Subtype.val",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [... | [] | simp_rw [← matroid_spanning_iff, ← matroid_isBase_iff, and_comm (a := _ ⊆ s)]
exact Matroid.spanning_iff_exists_isBase_subset (subset_univ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 407,
"column": 2
} | {
"line": 408,
"column": 65
} | {
"line": 410,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : IsDomain A\ninst✝ : FaithfulSMul R A\ns : Set A\n⊢ Algebra.IsAlgebraic (↥(adjoin R s)) A ↔ ∃ t ⊆ s, IsTranscendenceBasis R Subtype.val",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [... | [] | simp_rw [← matroid_spanning_iff, ← matroid_isBase_iff, and_comm (a := _ ⊆ s)]
exact Matroid.spanning_iff_exists_isBase_subset (subset_univ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.FilterBasis | {
"line": 225,
"column": 45
} | {
"line": 225,
"column": 65
} | {
"line": 225,
"column": 65
} | [
{
"pp": "G : Type u\ninst✝ : Group G\nt : TopologicalSpace G\nF : GroupFilterBasis G\nhG : F.topology = t\n⊢ ⋂₀ F.sets ⊆ {1} ∧ {1} ⊆ ⋂₀ F.sets ↔ ⋂₀ F.sets ⊆ {1}",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"ChainCompletePartialOrder.instO... | [
"G : Type u\ninst✝ : Group G\nt : TopologicalSpace G\nF : GroupFilterBasis G\nhG : F.topology = t\n⊢ ⋂₀ F.sets ⊆ {1} → {1} ⊆ ⋂₀ F.sets"
] | and_iff_left_iff_imp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Galois.Basic | {
"line": 479,
"column": 2
} | {
"line": 479,
"column": 61
} | {
"line": 480,
"column": 2
} | [
{
"pp": "case adjoin_rootSet'\nF : 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\nh1 : F⟮α⟯ = ⊤\n⊢ Algebra.adjoin F ((minpoly F α).rootSet E) = ⊤",
"ppTerm": "?adjoin_rootSet'",
"assigned": true,
"usedCon... | [
"case adjoin_rootSet'\nF : 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\nh1 : F⟮α⟯ = ⊤\n⊢ F⟮α⟯.toSubalgebra ≤ Algebra.adjoin F ((minpoly F α).rootSet E)"
] | rw [eq_top_iff, ← IntermediateField.top_toSubalgebra, ← h1] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.SeparableDegree | {
"line": 441,
"column": 4
} | {
"line": 453,
"column": 16
} | {
"line": 454,
"column": 2
} | [
{
"pp": "case pos\nF : Type u\ninst✝ : Field F\nf g : F[X]\nh : f * g = 0\n⊢ (f * g).natSepDegree = f.natSepDegree + g.natSepDegree ↔ f = 0 ∧ g = 0 ∨ IsCoprime f g",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.m... | [] | rw [mul_eq_zero] at h
wlog hf : f = 0 generalizing f g
· simpa only [mul_comm, add_comm, and_comm,
isCoprime_comm] using this g f h.symm (h.resolve_left hf)
rw [hf, zero_mul, natSepDegree_zero, zero_add, isCoprime_zero_left, isUnit_iff, eq_comm,
natSepDegree_eq_zero_iff, natDegree_eq_zero]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.SeparableDegree | {
"line": 441,
"column": 4
} | {
"line": 453,
"column": 16
} | {
"line": 454,
"column": 2
} | [
{
"pp": "case pos\nF : Type u\ninst✝ : Field F\nf g : F[X]\nh : f * g = 0\n⊢ (f * g).natSepDegree = f.natSepDegree + g.natSepDegree ↔ f = 0 ∧ g = 0 ∨ IsCoprime f g",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.m... | [] | rw [mul_eq_zero] at h
wlog hf : f = 0 generalizing f g
· simpa only [mul_comm, add_comm, and_comm,
isCoprime_comm] using this g f h.symm (h.resolve_left hf)
rw [hf, zero_mul, natSepDegree_zero, zero_add, isCoprime_zero_left, isUnit_iff, eq_comm,
natSepDegree_eq_zero_iff, natDegree_eq_zero]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Galois.Basic | {
"line": 518,
"column": 30
} | {
"line": 518,
"column": 51
} | {
"line": 518,
"column": 52
} | [
{
"pp": "F : Type u_1\ninst✝⁶ : Field F\nE : Type u_2\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : p.Separable\nK : Type u_3\ninst✝³ : Field K\ninst✝² : Algebra F K\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nx : E\nhx : x ∈... | [
"F : Type u_1\ninst✝⁶ : Field F\nE : Type u_2\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : p.Separable\nK : Type u_3\ninst✝³ : Field K\ninst✝² : Algebra F K\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nx : E\nhx : x ∈ p.aroots E\... | Polynomial.eval₂_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.FilterBasis | {
"line": 391,
"column": 21
} | {
"line": 391,
"column": 40
} | {
"line": 391,
"column": 40
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : TopologicalSpace R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nB : ModuleFilterBasis R M\ninst✝ : IsTopologicalRing R\nB' : AddGroupFilterBasis M := B.toAddGroupFilterBasis\nx✝² : TopologicalSpace M := B'.topology\nx✝¹ : IsTopologicalAddGroup... | [] | simpa using! B.smul | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Topology.Algebra.FilterBasis | {
"line": 391,
"column": 21
} | {
"line": 391,
"column": 40
} | {
"line": 391,
"column": 40
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : TopologicalSpace R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nB : ModuleFilterBasis R M\ninst✝ : IsTopologicalRing R\nB' : AddGroupFilterBasis M := B.toAddGroupFilterBasis\nx✝² : TopologicalSpace M := B'.topology\nx✝¹ : IsTopologicalAddGroup... | [] | simpa using! B.smul | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.FilterBasis | {
"line": 391,
"column": 21
} | {
"line": 391,
"column": 40
} | {
"line": 391,
"column": 40
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : TopologicalSpace R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nB : ModuleFilterBasis R M\ninst✝ : IsTopologicalRing R\nB' : AddGroupFilterBasis M := B.toAddGroupFilterBasis\nx✝² : TopologicalSpace M := B'.topology\nx✝¹ : IsTopologicalAddGroup... | [] | simpa using! B.smul | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.SeparableDegree | {
"line": 667,
"column": 2
} | {
"line": 667,
"column": 71
} | {
"line": 669,
"column": 0
} | [
{
"pp": "case convert_2.refine_2\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\nx✝ : ∃ n y, (Polynomial.aeval x) (X ^ q ^ n - C y) = 0\nn : ℕ\ny : F\nh : (Polynomial.aeval x) (X ^ q ^ n - C y) = 0\nhnezero : X ^ q ^ n - C y ≠... | [] | exact minpoly.natDegree_pos <| IsAlgebraic.isIntegral ⟨_, hnezero, h⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.SeparableClosure | {
"line": 246,
"column": 2
} | {
"line": 248,
"column": 24
} | {
"line": 249,
"column": 2
} | [
{
"pp": "case refine_1\nF : 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✝\nK L : IntermediateField F E\nle : K ≤ L\nx : E\nhx : x ∈ (fun K ↦ IntermediateField.restrictScalars F (separableClosure (↥K) E)) K\n⊢ x ∈ (fun K ↦ Inter... | [
"case refine_2\nF : 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✝\nK : IntermediateField F E\nx : E\nhx :\n x ∈\n IntermediateField.restrictScalars F\n (separableClosure (↥(IntermediateField.restrictScalars F (separableClo... | · let _ := (inclusion le).toAlgebra
have : IsScalarTower K L E := .of_algebraMap_eq' rfl
exact hx.tower_top _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.SeparableDegree | {
"line": 902,
"column": 27
} | {
"line": 902,
"column": 32
} | {
"line": 903,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\nx✝ : PerfectField F\nf : F[X]\nhf : f.natSepDegree = 1\nh : ∀ {f : F[X]}, Irreducible f → f.Separable\n⊢ f ≠ 0",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg",
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 82,
"column": 6
} | {
"line": 82,
"column": 83
} | {
"line": 84,
"column": 0
} | [
{
"pp": "case h\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nσ : Gal(K/k)\nh : σ ∈ (↑L.fixingSubgroup)ᶜ\ny : K\nyL : y ∈ ↑L\nne : y ∉ fun a ↦ σ • a = a\n⊢ 1 ∈ ↑(adjoin k {y}).fixingSubgroup ∧ (fun x ↦ σ • x) 1 = σ",
"... | [] | simp only [SetLike.mem_coe, smul_eq_mul, mul_one, and_true, Subgroup.one_mem] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.IsSepClosed | {
"line": 183,
"column": 65
} | {
"line": 183,
"column": 70
} | {
"line": 183,
"column": 70
} | [
{
"pp": "k : Type u\ninst✝ : Field k\nH : ∀ (p : k[X]), p.Monic → Irreducible p → p.Separable → ∃ x, eval x p = 0\np : k[X]\nhp : Irreducible p\nhs : p.Separable\n⊢ IsUnit (C p.leadingCoeff⁻¹)",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"inv_eq_zero._simp_1",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.FieldTheory.IsSepClosed | {
"line": 183,
"column": 65
} | {
"line": 183,
"column": 70
} | {
"line": 183,
"column": 70
} | [
{
"pp": "k : Type u\ninst✝ : Field k\nH : ∀ (p : k[X]), p.Monic → Irreducible p → p.Separable → ∃ x, eval x p = 0\np : k[X]\nhp : Irreducible p\nhs : p.Separable\n⊢ IsUnit (C p.leadingCoeff⁻¹)",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"inv_eq_zero._simp_1",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.IsSepClosed | {
"line": 183,
"column": 65
} | {
"line": 183,
"column": 70
} | {
"line": 183,
"column": 70
} | [
{
"pp": "k : Type u\ninst✝ : Field k\nH : ∀ (p : k[X]), p.Monic → Irreducible p → p.Separable → ∃ x, eval x p = 0\np : k[X]\nhp : Irreducible p\nhs : p.Separable\n⊢ IsUnit (C p.leadingCoeff⁻¹)",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"inv_eq_zero._simp_1",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 210,
"column": 15
} | {
"line": 210,
"column": 54
} | {
"line": 210,
"column": 54
} | [
{
"pp": "case heval.h\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\nthis : F... | [
"case heval.h\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\nthis : Function.Inje... | map_eq_zero_iff σ σ.toRingHom.injective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 219,
"column": 51
} | {
"line": 220,
"column": 74
} | {
"line": 221,
"column": 2
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Ring E\ninst✝² : IsDomain E\ninst✝¹ : Algebra F E\nq : ℕ\ninst✝ : ExpChar F q\nh : ∀ (x : E), ∃ n, x ^ q ^ n ∈ (algebraMap F E).range\nx : E\nhdeg : (minpoly F x).natSepDegree = 1\nh' : ¬IsIntegral F x\n⊢ False",
"ppTerm": "?m.140",
"assigned":... | [] | by
simp only [minpoly.eq_zero h', natSepDegree_zero, zero_ne_one] at hdeg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 449,
"column": 4
} | {
"line": 449,
"column": 24
} | {
"line": 449,
"column": 24
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\n⊢ Subsingleton (Emb F E) ∧ Nonempty (Emb F E) ↔ Subsingleton (Emb F E)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"And",
"Iff",
"pr... | [
"F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\n⊢ Subsingleton (Emb F E) → Nonempty (Emb F E)"
] | and_iff_left_iff_imp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 23
} | {
"line": 106,
"column": 2
} | [
{
"pp": "case h\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nhx : ∃ s, Nonempty (Basis (↥s) K ↥K⟮x⟯)\n⊢ IsIntegral K x",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"instSMulOfMul",
"CommSemiring.toSemiring",
"IntermediateFie... | [
"case h\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\ns : Finset ↥K⟮x⟯\nb : Basis (↥s) K ↥K⟮x⟯\n⊢ IsIntegral K x"
] | obtain ⟨s, ⟨b⟩⟩ := hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Invariant.Basic | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 53
} | {
"line": 61,
"column": 53
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\nG : Type u_3\ninst✝³ : Group G\ninst✝² : MulSemiringAction G B\ninst✝¹ : SMulCommClass G A B\nH : Subgroup G\ninst✝ : H.Normal\na b : G\nhb : { unop' := b } ∈ H.op\nc : ↥(FixedPoints.subring B ↥H)\n⊢ ↑⟨(fun m ↦ ... | [] | simpa [mul_smul] using! congr(a • $(c.2 ⟨b, hb⟩)) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.RingTheory.Invariant.Basic | {
"line": 111,
"column": 2
} | {
"line": 113,
"column": 51
} | {
"line": 115,
"column": 0
} | [
{
"pp": "B : Type u_2\nG : Type u_3\ninst✝³ : CommRing B\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G B\ninst✝ : Fintype G\nb : B\n⊢ eval b (charpoly G b) = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Finset.mem_univ",
"Eq.mpr",
"Polynomial.C",
"Polynomial... | [] | rw [charpoly_eq, eval_prod]
apply Finset.prod_eq_zero (Finset.mem_univ (1 : G))
rw [one_smul, eval_sub, eval_C, eval_X, sub_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Invariant.Basic | {
"line": 111,
"column": 2
} | {
"line": 113,
"column": 51
} | {
"line": 115,
"column": 0
} | [
{
"pp": "B : Type u_2\nG : Type u_3\ninst✝³ : CommRing B\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G B\ninst✝ : Fintype G\nb : B\n⊢ eval b (charpoly G b) = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Finset.mem_univ",
"Eq.mpr",
"Polynomial.C",
"Polynomial... | [] | rw [charpoly_eq, eval_prod]
apply Finset.prod_eq_zero (Finset.mem_univ (1 : G))
rw [one_smul, eval_sub, eval_C, eval_X, sub_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ZMod.ValMinAbs | {
"line": 172,
"column": 2
} | {
"line": 175,
"column": 9
} | {
"line": 177,
"column": 0
} | [
{
"pp": "n a : ℕ\nha : a ≤ n / 2\n⊢ (↑a).valMinAbs = ↑a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"instHDiv",
"ZMod.commRing",
"ZMod.valMinAbs_def_pos",
"congrArg",
"ZMod.valMinAbs",
"AddGroupWithOne.toAddMonoidWithOne"... | [] | cases n
· simp
· simp [valMinAbs_def_pos, val_natCast, Nat.mod_eq_of_lt (ha.trans_lt <| Nat.div_lt_self' _ 0),
ha] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ZMod.ValMinAbs | {
"line": 172,
"column": 2
} | {
"line": 175,
"column": 9
} | {
"line": 177,
"column": 0
} | [
{
"pp": "n a : ℕ\nha : a ≤ n / 2\n⊢ (↑a).valMinAbs = ↑a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"instHDiv",
"ZMod.commRing",
"ZMod.valMinAbs_def_pos",
"congrArg",
"ZMod.valMinAbs",
"AddGroupWithOne.toAddMonoidWithOne"... | [] | cases n
· simp
· simp [valMinAbs_def_pos, val_natCast, Nat.mod_eq_of_lt (ha.trans_lt <| Nat.div_lt_self' _ 0),
ha] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Trace.Basic | {
"line": 445,
"column": 58
} | {
"line": 446,
"column": 79
} | {
"line": 448,
"column": 0
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Algebra K L\nκ : Type w\nE : Type z\ninst✝⁴ : Field E\ninst✝³ : Algebra K E\ninst✝² : Module.Finite K L\ninst✝¹ : Algebra.IsSeparable K L\ninst✝ : IsAlgClosed E\nb : κ → L\n⊢ (traceMatrix K b).map ⇑(algebraMap K E) = embeddingsMat... | [] | by
ext (i j); simp [trace_eq_sum_embeddings, embeddingsMatrix, Matrix.mul_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Trace.Basic | {
"line": 513,
"column": 2
} | {
"line": 513,
"column": 63
} | {
"line": 514,
"column": 2
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : FiniteDimensional K L\ntfae_1_to_3 : Algebra.IsSeparable K L → (traceForm K L).Nondegenerate\n⊢ [Algebra.IsSeparable K L, Algebra.trace K L ≠ 0, (traceForm K L).Nondegenerate].TFAE",
"ppTerm": "?m.51",
... | [
"K : Type u_4\nL : Type u_5\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : FiniteDimensional K L\ntfae_1_to_3 : Algebra.IsSeparable K L → (traceForm K L).Nondegenerate\ntfae_3_to_2 : (traceForm K L).Nondegenerate → Algebra.trace K L ≠ 0\n⊢ [Algebra.IsSeparable K L, Algebra.trace K L ≠ 0, (traceF... | tfae_have 3 → 2 := fun H₁ H₂ ↦ H₁.ne_zero (by ext; simp [H₂]) | Mathlib.Tactic.TFAE._aux_Mathlib_Tactic_TFAE___macroRules_Mathlib_Tactic_TFAE_tfaeHave_1 | Mathlib.Tactic.TFAE.tfaeHave |
Mathlib.RingTheory.Int.Basic | {
"line": 47,
"column": 2
} | {
"line": 49,
"column": 20
} | {
"line": 50,
"column": 2
} | [
{
"pp": "a b c : ℤ\nh : a.gcd b = 1\nheq : a * b = c ^ 2\n⊢ ∃ a0, a = a0 ^ 2 ∨ a = -a0 ^ 2",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Int.gcd",
"Eq.mpr",
"congrArg",
"IsUnit",
"CommSemiring.toCommMonoidWithZero",
"id",
"instOfNatNat",
"... | [
"a b c : ℤ\nh : a.gcd b = 1\nheq : a * b = c ^ 2\nh' : IsUnit (GCDMonoid.gcd a b)\n⊢ ∃ a0, a = a0 ^ 2 ∨ a = -a0 ^ 2"
] | have h' : IsUnit (GCDMonoid.gcd a b) := by
rw [← coe_gcd, h, Int.ofNat_one]
exact isUnit_one | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Quotient.Pi | {
"line": 55,
"column": 8
} | {
"line": 55,
"column": 25
} | {
"line": 55,
"column": 25
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝⁶ : CommRing R\nMs : ι → Type u_3\ninst✝⁵ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁴ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\nq : Submodule R N\nf :... | [
"ι : Type u_1\nR : Type u_2\ninst✝⁶ : CommRing R\nMs : ι → Type u_3\ninst✝⁵ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁴ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\nq : Submodule R N\nf : (i : ι) → M... | Pi.single_eq_same | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Quotient.Pi | {
"line": 115,
"column": 27
} | {
"line": 115,
"column": 44
} | {
"line": 115,
"column": 44
} | [
{
"pp": "case pos\nι : Type u_1\nR : Type u_2\ninst✝⁴ : CommRing R\nMs : ι → Type u_3\ninst✝³ : (i : ι) → AddCommGroup (Ms i)\ninst✝² : (i : ι) → Module R (Ms i)\np : (i : ι) → Submodule R (Ms i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ni : ι\nx' : Ms i\n⊢ Quotient.mk x' = Pi.single i (Quotient.mk x') i",
... | [
"case pos\nι : Type u_1\nR : Type u_2\ninst✝⁴ : CommRing R\nMs : ι → Type u_3\ninst✝³ : (i : ι) → AddCommGroup (Ms i)\ninst✝² : (i : ι) → Module R (Ms i)\np : (i : ι) → Submodule R (Ms i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ni : ι\nx' : Ms i\n⊢ Quotient.mk x' = Quotient.mk x'"
] | Pi.single_eq_same | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 387,
"column": 6
} | {
"line": 391,
"column": 82
} | {
"line": 392,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : Fintype K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Finite L\nthis : Fintype L\nm : ℕ\nlt : m < Module.finrank K L\npos : 0 < m\neq : frobeniusAlgHom K L ^ m = 1\nx : L\nx✝ : x ∈ univ.val\n⊢ x ∈ (X ^ q ^ m - X).roots",
"ppTerm": "?m.82",... | [] | simp_rw [mem_roots', IsRoot, eval_sub, eval_pow, eval_X]
have := DFunLike.congr_fun eq x
rw [AlgHom.coe_pow, coe_frobeniusAlgHom, pow_iterate, AlgHom.one_apply, ← sub_eq_zero] at this
refine ⟨fun h ↦ ?_, this⟩
simpa [Fintype.one_lt_card.ne, pos.ne, eqComm] using congr_arg (coeff · 1) h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Finite.Basic | {
"line": 387,
"column": 6
} | {
"line": 391,
"column": 82
} | {
"line": 392,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : Fintype K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Finite L\nthis : Fintype L\nm : ℕ\nlt : m < Module.finrank K L\npos : 0 < m\neq : frobeniusAlgHom K L ^ m = 1\nx : L\nx✝ : x ∈ univ.val\n⊢ x ∈ (X ^ q ^ m - X).roots",
"ppTerm": "?m.82",... | [] | simp_rw [mem_roots', IsRoot, eval_sub, eval_pow, eval_X]
have := DFunLike.congr_fun eq x
rw [AlgHom.coe_pow, coe_frobeniusAlgHom, pow_iterate, AlgHom.one_apply, ← sub_eq_zero] at this
refine ⟨fun h ↦ ?_, this⟩
simpa [Fintype.one_lt_card.ne, pos.ne, eqComm] using congr_arg (coeff · 1) h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Finite.Basic | {
"line": 492,
"column": 2
} | {
"line": 493,
"column": 46
} | {
"line": 495,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\nf : K[X]\np : ℕ\nhp✝ : CharP K p\nn : ℕ\nnpos : 0 < n\nhp : Nat.Prime p\nhn : q = p ^ n\nthis : Fact (Nat.Prime p)\n⊢ (expand K q) f = f ^ q",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.t... | [] | rw [hn, ← map_iterateFrobenius_expand, iterateFrobenius_eq_pow,
frobenius_pow hn, RingHom.one_def, map_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Finite.Basic | {
"line": 504,
"column": 2
} | {
"line": 507,
"column": 20
} | {
"line": 508,
"column": 2
} | [
{
"pp": "case inl\nhp : Fact (Nat.Prime 2)\nx : ZMod 2\n⊢ ∃ a b, a ^ 2 + b ^ 2 = x",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"one_pow",
"instNeZeroNatHAdd_1",
"MulOne.toOne",
"False",
"IsDomain.to_noZeroDivisors",
"Fintype.elems",
"Nat.le_refl... | [
"case inr\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ZMod p\nhp_odd : p % 2 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = x"
] | · change Fin 2 at x
fin_cases x
· use 0; simp
· use 0, 1; simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Norm.Basic | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 23
} | {
"line": 155,
"column": 2
} | [
{
"pp": "case h\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nhx : ∃ s, Nonempty (Basis (↥s) K ↥K⟮x⟯)\n⊢ IsIntegral K x",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"instSMulOfMul",
"CommSemiring.toSemiring",
"IntermediateFie... | [
"case h\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\ns : Finset ↥K⟮x⟯\nb : Basis (↥s) K ↥K⟮x⟯\n⊢ IsIntegral K x"
] | obtain ⟨s, ⟨b⟩⟩ := hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Norm.Basic | {
"line": 196,
"column": 56
} | {
"line": 209,
"column": 34
} | {
"line": 211,
"column": 0
} | [
{
"pp": "K : Type u_4\nL : Type u_5\nF : Type u_6\ninst✝¹¹ : Field K\ninst✝¹⁰ : Field L\ninst✝⁹ : Field F\ninst✝⁸ : Algebra K L\ninst✝⁷ : Algebra K F\nE : Type u_7\ninst✝⁶ : Field E\ninst✝⁵ : Algebra K E\ninst✝⁴ : Algebra L F\ninst✝³ : IsScalarTower K L F\ninst✝² : IsAlgClosed E\ninst✝¹ : Algebra.IsSeparable K ... | [] | by
haveI : FiniteDimensional L F := FiniteDimensional.right K L F
haveI : Algebra.IsSeparable L F := Algebra.isSeparable_tower_top_of_isSeparable K L F
letI : Fintype (L →ₐ[K] E) := PowerBasis.AlgHom.fintype pb
rw [Fintype.prod_equiv algHomEquivSigma (fun σ : F →ₐ[K] E => _) fun σ => σ.1 pb.gen,
← Finset.un... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Invariant.Basic | {
"line": 455,
"column": 2
} | {
"line": 473,
"column": 22
} | {
"line": 475,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nk : Type u_3\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B\ninst✝¹¹ : Algebra A B\nG : Type u_4\ninst✝¹⁰ : Finite G\ninst✝⁹ : Group G\ninst✝⁸ : MulSemiringAction G B\ninst✝⁷ : Algebra.IsInvariant A B G\nP : Ideal A\nQ : Ideal B\ninst✝⁶ : Q.LiesOver P\ninst✝⁵ : CommRing k\ninst... | [] | let f : (B ⧸ Q) →ₐ[A ⧸ P] k := IsScalarTower.toAlgHom _ _ _
have hf : Function.Injective f := FaithfulSMul.algebraMap_injective _ _
obtain ⟨τ₁, h₁⟩ := Ideal.Quotient.exists_algHom_fixedPoint_quotient_under G P Q σ.toAlgHom
obtain ⟨τ₂, h₂⟩ := Ideal.Quotient.exists_algHom_fixedPoint_quotient_under G P Q σ.symm.toAl... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Invariant.Basic | {
"line": 455,
"column": 2
} | {
"line": 473,
"column": 22
} | {
"line": 475,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nk : Type u_3\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B\ninst✝¹¹ : Algebra A B\nG : Type u_4\ninst✝¹⁰ : Finite G\ninst✝⁹ : Group G\ninst✝⁸ : MulSemiringAction G B\ninst✝⁷ : Algebra.IsInvariant A B G\nP : Ideal A\nQ : Ideal B\ninst✝⁶ : Q.LiesOver P\ninst✝⁵ : CommRing k\ninst... | [] | let f : (B ⧸ Q) →ₐ[A ⧸ P] k := IsScalarTower.toAlgHom _ _ _
have hf : Function.Injective f := FaithfulSMul.algebraMap_injective _ _
obtain ⟨τ₁, h₁⟩ := Ideal.Quotient.exists_algHom_fixedPoint_quotient_under G P Q σ.toAlgHom
obtain ⟨τ₂, h₂⟩ := Ideal.Quotient.exists_algHom_fixedPoint_quotient_under G P Q σ.symm.toAl... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.RamificationInertia.Inertia | {
"line": 117,
"column": 21
} | {
"line": 117,
"column": 60
} | {
"line": 117,
"column": 60
} | [
{
"pp": "case neg\nR : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal R\nS₁ : Type u_1\ninst✝¹ : CommRing S₁\ninst✝ : Algebra R S₁\ne : S ≃ₐ[R] S₁\nP : Ideal S₁\nhe : comap f (comap e P) = p ↔ comap (algebraMap R S₁) P = p\nh : ¬P.LiesOver p\n⊢ (if hPp : comap f (c... | [
"case neg\nR : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal R\nS₁ : Type u_1\ninst✝¹ : CommRing S₁\ninst✝ : Algebra R S₁\ne : S ≃ₐ[R] S₁\nP : Ideal S₁\nhe : comap f (comap e P) = p ↔ comap (algebraMap R S₁) P = p\nh : ¬P.LiesOver p\n⊢ 0 = p.inertiaDeg' P"
] | dif_neg (fun eq => h ⟨(he.mp eq).symm⟩) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 337,
"column": 2
} | {
"line": 338,
"column": 21
} | {
"line": 340,
"column": 0
} | [
{
"pp": "case mpr\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\nI : Ideal S\n⊢ I = ⊥ → absNorm I = 0",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroOneClass",
"Semiring.toModule",
"Ideal.absN... | [] | · rintro rfl
exact absNorm_bot | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 458,
"column": 2
} | {
"line": 459,
"column": 29
} | {
"line": 460,
"column": 2
} | [
{
"pp": "S : Type u_1\ninst✝⁴ : CommRing S\ninst✝³ : IsDedekindDomain S\ninst✝² : Free ℤ S\ninst✝¹ : Module.Finite ℤ S\nn : ℕ\ninst✝ : CharZero S\nthis✝ : Finite { I // I ∈ (Ideal S)⁰ ∧ absNorm I ≤ n }\nthis : Finite { I // I ∉ (Ideal S)⁰ ∧ absNorm I ≤ n }\n⊢ Nat.card { I // absNorm I ≤ n } = Nat.card { I // ab... | [
"S : Type u_1\ninst✝⁴ : CommRing S\ninst✝³ : IsDedekindDomain S\ninst✝² : Free ℤ S\ninst✝¹ : Module.Finite ℤ S\nn : ℕ\ninst✝ : CharZero S\nthis✝ : Finite { I // I ∈ (Ideal S)⁰ ∧ absNorm I ≤ n }\nthis : Finite { I // I ∉ (Ideal S)⁰ ∧ absNorm I ≤ n }\n⊢ Nat.card { I // absNorm I ≤ n } = Nat.card { x // x ∈ (Ideal S)⁰... | rw [Nat.card_congr (Equiv.subtypeSubtypeEquivSubtypeInter (fun I ↦ I ∈ (Ideal S)⁰)
(fun I ↦ absNorm I ≤ n))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.JacobsonSpace | {
"line": 63,
"column": 86
} | {
"line": 63,
"column": 91
} | {
"line": 63,
"column": 91
} | [
{
"pp": "case e'_3\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nhs : s.Finite\nhs' : s ⊆ closedPoints X\nx : ↑s\ne_1✝ : ↑s = { x // x ∈ s }\n⊢ {x} = Subtype.val ⁻¹' {↑x}",
"ppTerm": "?e'_3",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"congrArg",
"Subty... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.JacobsonSpace | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 61
} | {
"line": 181,
"column": 2
} | [
{
"pp": "X : Type u_2\ninst✝ : TopologicalSpace X\nι : Type u_1\nU : ι → Opens X\nhU : IsOpenCover U\nH : ∀ (i : ι), JacobsonSpace ↥(U i)\nZ : Set X\nhZ' : IsLocallyClosed Z\ni : ι\nx : X\nhx : x ∈ Z\nhx' : x ∈ ↑(U i)\ny : ↥(U i)\nhy : y ∈ Subtype.val ⁻¹' Z\nhy' : y ∈ closedPoints ↥(U i)\n⊢ (Z ∩ closedPoints X)... | [
"X : Type u_2\ninst✝ : TopologicalSpace X\nι : Type u_1\nU : ι → Opens X\nhU : IsOpenCover U\nH : ∀ (i : ι), JacobsonSpace ↥(U i)\nZ : Set X\nhZ' : IsLocallyClosed Z\ni : ι\nx : X\nhx : x ∈ Z\nhx' : x ∈ ↑(U i)\ny : ↥(U i)\nhy : y ∈ Subtype.val ⁻¹' Z\nhy' : y ∈ closedPoints ↥(U i)\nj : ι\n⊢ IsClosed[instTopologicalS... | refine ⟨y, hy, hU.isClosed_iff_coe_preimage.mpr fun j ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.LocalRing.Length | {
"line": 110,
"column": 4
} | {
"line": 112,
"column": 96
} | {
"line": 113,
"column": 4
} | [
{
"pp": "case pos\nA : Type u_1\nB : Type u_2\nM : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsLocalRing A\ninst✝⁵ : IsLocalRing B\ninst✝⁴ : Algebra A B\ninst✝³ : IsLocalHom (algebraMap A B)\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : Flat A B\nh : IsFiniteLength A M\ns : Compositi... | [
"case pos\nA : Type u_1\nB : Type u_2\nM : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsLocalRing A\ninst✝⁵ : IsLocalRing B\ninst✝⁴ : Algebra A B\ninst✝³ : IsLocalHom (algebraMap A B)\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : Flat A B\nh : IsFiniteLength A M\ns : CompositionSeries (Su... | suffices ∀ k, length B ((s k).baseChange B) =
k * length B (B ⧸ (maximalIdeal A).map (algebraMap A B)) by
rw [← Fin.val_last s.length, ← this, ← RelSeries.last, hs_top, baseChange_top, length_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 41
} | {
"line": 93,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m ↦ ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\nthis : Q n\n⊢ p.ramificationIdx' P = n",
"ppTerm": "?m.70",
"a... | [
"R : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nn : ℕ\nhle : map f p ≤ P ^ n\nhgt : ¬map f p ≤ P ^ (n + 1)\nQ : ℕ → Prop := fun m ↦ ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m\nthis : Q n\n⊢ Nat.find ⋯ = n"
] | rw [ramificationIdx'_eq_find ⟨n, this⟩] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 43
} | {
"line": 108,
"column": 4
} | [
{
"pp": "case succ\nR : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nn : ℕ\nhgt : ¬map f p ≤ P ^ (n + 1)\nthis : ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ n\n⊢ p.ramificationIdx' P ≤ n",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
... | [
"case succ\nR : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nn : ℕ\nhgt : ¬map f p ≤ P ^ (n + 1)\nthis : ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ n\n⊢ Nat.find ⋯ ≤ n"
] | rw [ramificationIdx'_eq_find ⟨n, this⟩] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 370,
"column": 10
} | {
"line": 370,
"column": 15
} | {
"line": 371,
"column": 2
} | [
{
"pp": "case h₁\nR : Type u\ninst✝⁶ : CommRing R\nS : Type v\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsDedekindDomain S\ninst✝² : IsDedekindDomain R\ninst✝¹ : FaithfulSMul R S\nv : Ideal R\nw : Ideal S\nhv : Irreducible v\nhw : Irreducible w\nhw_bot : w ≠ ⊥\ninst✝ : w.LiesOver v\nh : 0 ≠ ⊥\n⊢ emul... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 370,
"column": 10
} | {
"line": 370,
"column": 15
} | {
"line": 371,
"column": 2
} | [
{
"pp": "case h₁\nR : Type u\ninst✝⁶ : CommRing R\nS : Type v\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsDedekindDomain S\ninst✝² : IsDedekindDomain R\ninst✝¹ : FaithfulSMul R S\nv : Ideal R\nw : Ideal S\nhv : Irreducible v\nhw : Irreducible w\nhw_bot : w ≠ ⊥\ninst✝ : w.LiesOver v\nh : 0 ≠ ⊥\n⊢ emul... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 370,
"column": 10
} | {
"line": 370,
"column": 15
} | {
"line": 371,
"column": 2
} | [
{
"pp": "case h₁\nR : Type u\ninst✝⁶ : CommRing R\nS : Type v\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsDedekindDomain S\ninst✝² : IsDedekindDomain R\ninst✝¹ : FaithfulSMul R S\nv : Ideal R\nw : Ideal S\nhv : Irreducible v\nhw : Irreducible w\nhw_bot : w ≠ ⊥\ninst✝ : w.LiesOver v\nh : 0 ≠ ⊥\n⊢ emul... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Unramified.Basic | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 73
} | {
"line": 226,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹⁰ : CommRing R\nA : Type u_2\ninst✝⁹ : CommRing A\ninst✝⁸ : Algebra R A\nB : Type u_3\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallyUnramified R A\ninst✝² : FormallyUnramified A B\nC : Type u_3\ninst✝¹ : CommRing C\n... | [
"R : Type u_1\ninst✝¹⁰ : CommRing R\nA : Type u_2\ninst✝⁹ : CommRing A\ninst✝⁸ : Algebra R A\nB : Type u_3\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallyUnramified R A\ninst✝² : FormallyUnramified A B\nC : Type u_3\ninst✝¹ : CommRing C\ninst✝ : Alge... | let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Unramified.Finite | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 34
} | {
"line": 112,
"column": 4
} | [
{
"pp": "case h_zero\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nI : Type u_4\ninst✝ : DecidableEq I\nb : Basis I R S\nf : I →₀ S\nx : S\na : I → I →₀ R := fun i ↦ b.repr (b i * x)\ni : I\n⊢ ∀ (a : I), 0 • b a ⊗ₜ[R] b i = 0",
"ppTerm": "?h_zero",
"assigne... | [
"case h_add\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nI : Type u_4\ninst✝ : DecidableEq I\nb : Basis I R S\nf : I →₀ S\nx : S\na : I → I →₀ R := fun i ↦ b.repr (b i * x)\ni : I\n⊢ ∀ (a : I) (b₁ b₂ : R), (b₁ + b₂) • b a ⊗ₜ[R] b i = b₁ • b a ⊗ₜ[R] b i + b₂ • b a ⊗ₜ[R... | · intro; simp only [zero_smul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Smooth.Basic | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 12
} | {
"line": 101,
"column": 2
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : FormallySmooth R A\nI : Ideal B\nhI : I ^ 2 = ⊥\nf : A →ₐ[R] B ⧸ I\nP : Generators R A A := Generators.self R A\nhP : Function.Injective ⇑P.toExtensio... | [] | simp [P] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.Basic | {
"line": 117,
"column": 2
} | {
"line": 118,
"column": 69
} | {
"line": 119,
"column": 2
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\nB : Type u_1\nP✝ : Type u_2\nC : Type u_3\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : CommRing C\ninst✝² : Algebra R C\ninst✝¹ : CommRing P✝\ninst✝ : Algebra R P✝\nσ : Type u_4\nP : Generators R (MvPolynomi... | [
"R : Type u\nA : Type v\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\nB : Type u_1\nP✝ : Type u_2\nC : Type u_3\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : CommRing C\ninst✝² : Algebra R C\ninst✝¹ : CommRing P✝\ninst✝ : Algebra R P✝\nσ : Type u_4\nP : Generators R (MvPolynomial σ R) σ :=... | have : Subsingleton ↥P.toExtension.ker :=
Submodule.subsingleton_iff_eq_bot.mpr Generators.ker_mvPolynomial | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Unramified.Finite | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 18
} | {
"line": 233,
"column": 0
} | [
{
"pp": "case h\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FormallyUnramified R S\ninst✝¹ : EssFiniteType R S\ninst✝ : Free R S\nI : Type u_2 := ⋯\nb : Basis I R S := ⋯\nf : I →₀ S\nhf : elem R S = f.sum fun i x ↦ x ⊗ₜ[R] b i\nx : S\na : I → I →₀ R := ⋯... | [] | use j, hj, i, hi | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Smooth.Basic | {
"line": 466,
"column": 4
} | {
"line": 466,
"column": 95
} | {
"line": 467,
"column": 2
} | [
{
"pp": "case refine_1\nR✝ : Type u\nA✝ : Type v\ninst✝¹⁴ : CommRing R✝\ninst✝¹³ : CommRing A✝\ninst✝¹² : Algebra R✝ A✝\nB✝ : Type u_1\nP : Type u_2\nC✝ : Type u_3\ninst✝¹¹ : CommRing B✝\ninst✝¹⁰ : Algebra R✝ B✝\ninst✝⁹ : CommRing C✝\ninst✝⁸ : Algebra R✝ C✝\ninst✝⁷ : CommRing P\ninst✝⁶ : Algebra R✝ P\nR : Type ... | [] | exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Smooth.Basic | {
"line": 466,
"column": 4
} | {
"line": 466,
"column": 95
} | {
"line": 467,
"column": 2
} | [
{
"pp": "case refine_1\nR✝ : Type u\nA✝ : Type v\ninst✝¹⁴ : CommRing R✝\ninst✝¹³ : CommRing A✝\ninst✝¹² : Algebra R✝ A✝\nB✝ : Type u_1\nP : Type u_2\nC✝ : Type u_3\ninst✝¹¹ : CommRing B✝\ninst✝¹⁰ : Algebra R✝ B✝\ninst✝⁹ : CommRing C✝\ninst✝⁸ : Algebra R✝ C✝\ninst✝⁷ : CommRing P\ninst✝⁶ : Algebra R✝ P\nR : Type ... | [] | exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Smooth.Basic | {
"line": 466,
"column": 4
} | {
"line": 466,
"column": 95
} | {
"line": 467,
"column": 2
} | [
{
"pp": "case refine_1\nR✝ : Type u\nA✝ : Type v\ninst✝¹⁴ : CommRing R✝\ninst✝¹³ : CommRing A✝\ninst✝¹² : Algebra R✝ A✝\nB✝ : Type u_1\nP : Type u_2\nC✝ : Type u_3\ninst✝¹¹ : CommRing B✝\ninst✝¹⁰ : Algebra R✝ B✝\ninst✝⁹ : CommRing C✝\ninst✝⁸ : Algebra R✝ C✝\ninst✝⁷ : CommRing P\ninst✝⁶ : Algebra R✝ P\nR : Type ... | [] | exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Unramified.Field | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 83
} | {
"line": 76,
"column": 2
} | [
{
"pp": "K : Type u_1\nA : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra K A\ninst✝³ : FormallyUnramified K A\ninst✝² : EssFiniteType K A\ninst✝¹ : IsAlgClosed K\ninst✝ : IsLocalRing A\nthis✝ : Module.Finite K A\nthis : IsArtinianRing A\nhA : IsNilpotent (IsLocalRing.maximalIdeal A)\ne : K ≃... | [
"K : Type u_1\nA : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra K A\ninst✝³ : FormallyUnramified K A\ninst✝² : EssFiniteType K A\ninst✝¹ : IsAlgClosed K\ninst✝ : IsLocalRing A\nthis✝ : Module.Finite K A\nthis : IsArtinianRing A\nhA : IsNilpotent (IsLocalRing.maximalIdeal A)\ne : K ≃ₐ[K] A ⧸ IsL... | let f : A ⧸ IsLocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap.comp (sec K A _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 263,
"column": 6
} | {
"line": 263,
"column": 23
} | {
"line": 263,
"column": 24
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nQ : Generators S T ι\nr : S\n⊢ (δAux R Q) (C r) = 1 ⊗ₜ[S] (D R S) r",
"ppTerm": "?m.56",
... | [
"R : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nQ : Generators S T ι\nr : S\n⊢ (δAux R Q) ((monomial 0) r) = 1 ⊗ₜ[S] (D R S) r"
] | ← monomial_zero', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RamificationInertia.Ramification | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 81
} | {
"line": 262,
"column": 2
} | [
{
"pp": "case neg\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nq : Ideal S\nG : Type u_4\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G S\ninst✝ : SMulCommClass G R S\ng : G\nhq : ¬q.IsPrime\n⊢ (g • q).ramificationIdx R = q.ramificationIdx R",
"ppTerm": "?neg... | [
"case pos\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nq : Ideal S\nG : Type u_4\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G S\ninst✝ : SMulCommClass G R S\ng : G\nhq : q.IsPrime\n⊢ (g • q).ramificationIdx R = q.ramificationIdx R"
] | · rw [ramificationIdx_of_not_isPrime, ramificationIdx_of_not_isPrime] <;> simpa | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Enumerative.InclusionExclusion | {
"line": 61,
"column": 28
} | {
"line": 61,
"column": 33
} | {
"line": 62,
"column": 4
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ns : Finset ι\nS : ι → Set α\na : α\nha : a ∉ ⋃ i ∈ s, S i\ni : ι\nhi : i ∈ s\n⊢ a ∉ S i",
"ppTerm": "?m.106",
"assigned": true,
"usedConstants": [
"not_exists._simp_1",
"False",
"eq_false",
"congrArg",
"Finset",
"Set.mem_iUnion... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Combinatorics.Enumerative.InclusionExclusion | {
"line": 139,
"column": 8
} | {
"line": 139,
"column": 13
} | {
"line": 140,
"column": 6
} | [
{
"pp": "case pos.e_s\nι : Type u_1\nα : Type u_2\nG : Type u_3\ninst✝¹ : AddCommGroup G\ninst✝ : DecidableEq α\ns : Finset ι\nS : ι → Finset α\nf : α → G\nt : Finset ι\na✝ : t ∈ s.powerset\ni : ι\nhi : i ∈ t\n⊢ t.inf' ⋯ S = {x ∈ s.biUnion S | ∀ i ∈ t, x ∈ S i}",
"ppTerm": "?pos.e_s✝",
"assigned": true,... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 38
} | {
"line": 133,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I ≠ 0\n⊢ HasFiniteMulSupport fun v ↦ ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk I).factors)",
"ppTerm": "?m.40",
... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I ≠ 0\n⊢ {x | ↑x.asIdeal ^ ↑((Associates.mk x.asIdeal).count (Associates.mk I).factors) ≠ 1}.Finite"
] | rw [HasFiniteMulSupport, mulSupport] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 38
} | {
"line": 145,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I ≠ 0\n⊢ HasFiniteMulSupport fun v ↦ ↑v.asIdeal ^ (-↑((Associates.mk v.asIdeal).count (Associates.mk I).factors))",
"ppTerm": "?m.42",
... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I ≠ 0\n⊢ {x | ↑x.asIdeal ^ (-↑((Associates.mk x.asIdeal).count (Associates.mk I).factors)) ≠ 1}.Finite"
] | rw [HasFiniteMulSupport, mulSupport] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 145,
"column": 11
} | {
"line": 145,
"column": 20
} | {
"line": 145,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I ≠ 0\n⊢ {x | ↑x.asIdeal ^ (-↑((Associates.mk x.asIdeal).count (Associates.mk I).factors)) ≠ 1}.Finite",
"ppTerm": "?m.52",
"assign... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I ≠ 0\n⊢ {x | (↑x.asIdeal ^ ↑((Associates.mk x.asIdeal).count (Associates.mk I).factors))⁻¹ ≠ 1}.Finite"
] | zpow_neg, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.Archimedean.Real.Basic | {
"line": 45,
"column": 67
} | {
"line": 45,
"column": 89
} | {
"line": 47,
"column": 0
} | [
{
"pp": "f : ℕ → ℚ\nH : IsCauSeq abs fun i ↦ ↑(f i)\nε : ℚ\nε0 : ε > 0\ni : ℕ\nhi : ∀ j ≥ i, |↑(f j) - ↑(f i)| < ↑ε\nj : ℕ\nij : j ≥ i\n⊢ |f j - f i| < ε",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"Rat.instSub",
"Real.partialOrder",
"Real",
"Preorder.toLT",
... | [] | exact mod_cast hi _ ij | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 131,
"column": 6
} | {
"line": 131,
"column": 11
} | {
"line": 132,
"column": 4
} | [
{
"pp": "case h\nA : Type u_1\nM : Type u_2\ninst✝² : CommRing A\nI : Ideal A\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nN₁ N₂ : Submodule A M\nhD : Disjoint N₁ N₂\ny : M\nhymem : y ∈ N₁\nn₁ : ℕ\nz : M\nhzmem : z ∈ N₂\nn₂ : ℕ\na : A\nhz : a • z = 0\nhy : a • y = 0\nha : a ∈ I ^ max n₁ n₂\n⊢ a • (y + z) = 0",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 156,
"column": 6
} | {
"line": 156,
"column": 11
} | {
"line": 157,
"column": 4
} | [
{
"pp": "case mp\nA : Type u_1\nM : Type u_2\ninst✝³ : CommRing A\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : IsDedekindDomain A\nh : IsTorsion A M\nx✝ : M\na : A\nha : a ∈ A⁰\nhmem : x✝ ∈ torsionBySet A M ↑(⨅ i, i.maxPowDividing (span {a}))\nha0 : span {a} ≠ ⊥\nthis : Fintype ↑(mulSupport fun v ↦ v.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 156,
"column": 6
} | {
"line": 156,
"column": 11
} | {
"line": 157,
"column": 4
} | [
{
"pp": "case mp\nA : Type u_1\nM : Type u_2\ninst✝³ : CommRing A\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : IsDedekindDomain A\nh : IsTorsion A M\nx✝ : M\na : A\nha : a ∈ A⁰\nhmem : x✝ ∈ torsionBySet A M ↑(⨅ i, i.maxPowDividing (span {a}))\nha0 : span {a} ≠ ⊥\nthis : Fintype ↑(mulSupport fun v ↦ v.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 156,
"column": 6
} | {
"line": 156,
"column": 11
} | {
"line": 157,
"column": 4
} | [
{
"pp": "case mp\nA : Type u_1\nM : Type u_2\ninst✝³ : CommRing A\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : IsDedekindDomain A\nh : IsTorsion A M\nx✝ : M\na : A\nha : a ∈ A⁰\nhmem : x✝ ∈ torsionBySet A M ↑(⨅ i, i.maxPowDividing (span {a}))\nha0 : span {a} ≠ ⊥\nthis : Fintype ↑(mulSupport fun v ↦ v.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 11
} | {
"line": 220,
"column": 4
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI : Ideal R\nh0 : I ≠ 0\nx : R\n⊢ x ∈ ⨅ i, i.maxPowDividing I → x ∈ ⨅ i ∈ Finite.toFinset ⋯, i.maxPowDividing I",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Submodule",
"MulOne.toOne",
"Fun... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 11
} | {
"line": 220,
"column": 4
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI : Ideal R\nh0 : I ≠ 0\nx : R\n⊢ x ∈ ⨅ i, i.maxPowDividing I → x ∈ ⨅ i ∈ Finite.toFinset ⋯, i.maxPowDividing I",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Submodule",
"MulOne.toOne",
"Fun... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 11
} | {
"line": 220,
"column": 4
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI : Ideal R\nh0 : I ≠ 0\nx : R\n⊢ x ∈ ⨅ i, i.maxPowDividing I → x ∈ ⨅ i ∈ Finite.toFinset ⋯, i.maxPowDividing I",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Submodule",
"MulOne.toOne",
"Fun... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.NNReal.Defs | {
"line": 713,
"column": 27
} | {
"line": 713,
"column": 40
} | {
"line": 713,
"column": 40
} | [
{
"pp": "a b c : ℝ≥0\nh : a ≠ 0\n⊢ b = c ∨ a = 0 ↔ b = c",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"NNReal",
"NNReal.instZero",
"Iff",
"propext",
"NNReal.instSemiring",
"Zero.toOfNat0",
"Or",
... | [
"a b c : ℝ≥0\nh : a ≠ 0\n⊢ b = c ↔ b = c"
] | or_iff_left h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.NNReal.Defs | {
"line": 939,
"column": 4
} | {
"line": 939,
"column": 9
} | {
"line": 941,
"column": 0
} | [
{
"pp": "case inl\nr✝ : ℝ\nr : ℝ≥0\n⊢ ∃ x, - -↑r = ↑x ∨ - -↑r = -↑x",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"congrArg",
"Exists",
"neg_neg",
"NonUnitalN... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.NNReal.Defs | {
"line": 939,
"column": 4
} | {
"line": 939,
"column": 9
} | {
"line": 941,
"column": 0
} | [
{
"pp": "case inr\nr✝ : ℝ\nr : ℝ≥0\n⊢ ∃ x, -↑r = ↑x ∨ -↑r = -↑x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"congrArg",
"neg_inj._simp_1",
"Exists",
"exists... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 38
} | {
"line": 291,
"column": 2
} | [
{
"pp": "E : Type u_3\ninst✝ : Group E\ns : Set (GroupSeminorm E)\nhs : BddAbove s\nx : E\n⊢ (sSup s) x = sSup ((fun x_1 ↦ x_1 x) '' s)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"GroupSeminorm.instSupSet",
"congrArg",
"iSup",
"Mem... | [
"E : Type u_3\ninst✝ : Group E\ns : Set (GroupSeminorm E)\nhs : BddAbove s\nx : E\n⊢ sSup (range fun p ↦ ↑p x) = sSup ((fun x_1 ↦ x_1 x) '' s)"
] | rw [coe_sSup_apply hs, ← sSup_range] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 298,
"column": 20
} | {
"line": 298,
"column": 36
} | {
"line": 298,
"column": 36
} | [
{
"pp": "E : Type u_3\ninst✝ : Group E\nι : Type u_6\nf : ι → GroupSeminorm E\nh : BddAbove (range f)\nx : E\n⊢ (sSup (range fun i ↦ f i)) x = ⨆ i, (f i) x",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"GroupSeminorm.instSupSet",
"congrArg",
... | [
"E : Type u_3\ninst✝ : Group E\nι : Type u_6\nf : ι → GroupSeminorm E\nh : BddAbove (range f)\nx : E\n⊢ ⨆ p, ↑p x = ⨆ i, (f i) x"
] | coe_sSup_apply h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 562,
"column": 2
} | {
"line": 562,
"column": 38
} | {
"line": 563,
"column": 2
} | [
{
"pp": "E : Type u_3\ninst✝ : AddGroup E\ns : Set (NonarchAddGroupSeminorm E)\nhs : BddAbove s\nx : E\n⊢ (sSup s) x = sSup ((fun x_1 ↦ x_1 x) '' s)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"iSup",
"NonarchAddGroupSeminorm.... | [
"E : Type u_3\ninst✝ : AddGroup E\ns : Set (NonarchAddGroupSeminorm E)\nhs : BddAbove s\nx : E\n⊢ sSup (range fun p ↦ ↑p x) = sSup ((fun x_1 ↦ x_1 x) '' s)"
] | rw [coe_sSup_apply hs, ← sSup_range] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 569,
"column": 20
} | {
"line": 569,
"column": 36
} | {
"line": 569,
"column": 36
} | [
{
"pp": "E : Type u_3\ninst✝ : AddGroup E\nι : Type u_6\nf : ι → NonarchAddGroupSeminorm E\nh : BddAbove (range f)\nx : E\n⊢ (sSup (range fun i ↦ f i)) x = ⨆ i, (f i) x",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"iSup",
"Nona... | [
"E : Type u_3\ninst✝ : AddGroup E\nι : Type u_6\nf : ι → NonarchAddGroupSeminorm E\nh : BddAbove (range f)\nx : E\n⊢ ⨆ p, ↑p x = ⨆ i, (f i) x"
] | coe_sSup_apply h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 208,
"column": 72
} | {
"line": 208,
"column": 77
} | {
"line": 208,
"column": 77
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDedekindDomain A\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝³ : AddCommGroup M₁\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module A M₁\ninst✝ : Module A M₂\nhM₁ : IsTorsion A M₁\nP : HeightOneSpectrum A\nφ : M₁ →ₗ[A] M₂\nhf✝ : Surjective ⇑φ\nb : M₁\nhy : φ b ∈ primaryCom... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 208,
"column": 72
} | {
"line": 208,
"column": 77
} | {
"line": 208,
"column": 77
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDedekindDomain A\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝³ : AddCommGroup M₁\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module A M₁\ninst✝ : Module A M₂\nhM₁ : IsTorsion A M₁\nP : HeightOneSpectrum A\nφ : M₁ →ₗ[A] M₂\nhf✝ : Surjective ⇑φ\nb : M₁\nhy : φ b ∈ primaryCom... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 208,
"column": 72
} | {
"line": 208,
"column": 77
} | {
"line": 208,
"column": 77
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDedekindDomain A\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝³ : AddCommGroup M₁\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module A M₁\ninst✝ : Module A M₂\nhM₁ : IsTorsion A M₁\nP : HeightOneSpectrum A\nφ : M₁ →ₗ[A] M₂\nhf✝ : Surjective ⇑φ\nb : M₁\nhy : φ b ∈ primaryCom... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENNReal.Operations | {
"line": 444,
"column": 2
} | {
"line": 444,
"column": 45
} | {
"line": 445,
"column": 2
} | [
{
"pp": "a b c : ℝ≥0∞\nha : a ≠ ∞\nh : b ≤ a\n⊢ a - c - (a - b) = b - c",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Ne",
"ne_top_of_le_ne_top",
"ENNReal",
"ENNReal.instPartialOrder",
"ENNReal.instTop",
"Top.top",
"ENNReal.instOrderTop"
],
... | [
"a b c : ℝ≥0∞\nha : a ≠ ∞\nh : b ≤ a\nhb : b ≠ ∞\n⊢ a - c - (a - b) = b - c"
] | have hb : b ≠ ∞ := ne_top_of_le_ne_top ha h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.ENNReal.Operations | {
"line": 451,
"column": 4
} | {
"line": 451,
"column": 20
} | {
"line": 453,
"column": 0
} | [
{
"pp": "case coe\na b : ℝ≥0\nh : ↑b ≤ ↑a\nx✝ : ℝ≥0\n⊢ b ≤ a",
"ppTerm": "?coe✝",
"assigned": true,
"usedConstants": [
"ENNReal.ofNNReal",
"PartialOrder.toPreorder",
"Preorder.toLE",
"cast",
"NNReal",
"LE.le",
"NNReal.instPartialOrder",
"ENNReal.coe_le... | [] | exact mod_cast h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.ENNReal.Operations | {
"line": 549,
"column": 59
} | {
"line": 550,
"column": 42
} | {
"line": 552,
"column": 0
} | [
{
"pp": "ι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\n⊢ a + iInf f = ⨅ b, a + f b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"iInf",
"ENNReal.instAddCommMonoid",
"congrArg",
"id",
"ENNReal.iInf_add",
"ConditionallyC... | [] | by
rw [add_comm, iInf_add]; simp [add_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.EReal.Basic | {
"line": 128,
"column": 39
} | {
"line": 128,
"column": 44
} | {
"line": 129,
"column": 2
} | [
{
"pp": "case bot\np : EReal → Prop\nhr : p ⊥\n⊢ p ⊥ ∨ p ⊤ ∨ ∃ r, p ↑r",
"ppTerm": "?bot",
"assigned": true,
"usedConstants": [
"Real",
"congrArg",
"true_or",
"EReal",
"Exists",
"instTopEReal",
"Bot.bot",
"True",
"eq_true",
"of_eq_true",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.EReal.Basic | {
"line": 128,
"column": 39
} | {
"line": 128,
"column": 44
} | {
"line": 129,
"column": 2
} | [
{
"pp": "case coe\np : EReal → Prop\na✝ : ℝ\nhr : p ↑a✝\n⊢ p ⊥ ∨ p ⊤ ∨ ∃ r, p ↑r",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"Real",
"EReal",
"Exists",
"instTopEReal",
"Bot.bot",
"Exists.intro",
"Or",
"Top.top",
"instBotEReal",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.EReal.Basic | {
"line": 128,
"column": 39
} | {
"line": 128,
"column": 44
} | {
"line": 129,
"column": 2
} | [
{
"pp": "case top\np : EReal → Prop\nhr : p ⊤\n⊢ p ⊥ ∨ p ⊤ ∨ ∃ r, p ↑r",
"ppTerm": "?top",
"assigned": true,
"usedConstants": [
"Real",
"congrArg",
"true_or",
"EReal",
"Exists",
"instTopEReal",
"Bot.bot",
"True",
"eq_true",
"of_eq_true",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 488,
"column": 21
} | {
"line": 488,
"column": 57
} | {
"line": 488,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv w : HeightOneSpectrum R\nhw : w ≠ v\nhw_fact : ↑w.asIdeal = spanSingleton R⁰ ((algebraMap R K) 1)⁻¹ * ↑w.asIdeal\nhw_ne_zero : ↑w.asIdeal ≠ 0\nhv : Irreduc... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv w : HeightOneSpectrum R\nhw : w ≠ v\nhw_fact : ↑w.asIdeal = spanSingleton R⁰ ((algebraMap R K) 1)⁻¹ * ↑w.asIdeal\nhw_ne_zero : ↑w.asIdeal ≠ 0\nhv : Irreducible (Associ... | ← pow_one (Associates.mk w.asIdeal), | Lean.Elab.Tactic.evalRewriteSeq | null |
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