module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.TensorPower.Basic | {
"line": 148,
"column": 2
} | {
"line": 157,
"column": 57
} | {
"line": 159,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\na : ⨂[R]^n M\n⊢ (cast R M ⋯) (GradedMonoid.GMul.mul GradedMonoid.GOne.one a) = a",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"PiTensorProduct.instModule",
"Eq.... | [] | rw [gMul_def, gOne_def]
induction a using PiTensorProduct.induction_on with
| smul_tprod r a =>
rw [TensorProduct.tmul_smul, map_smul, map_smul, ← gMul_def, tprod_mul_tprod, cast_tprod]
congr 2 with i
rw [Fin.elim0_append]
refine congr_arg a (Fin.ext ?_)
simp
| add x y hx hy =>
rw [TensorP... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorPower.Basic | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 20
} | {
"line": 205,
"column": 20
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\n⊢ algebraMap₀ r = r • GradedMonoid.GOne.one",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"PiTensorProduct.instModule",
"Eq.mpr",
"Nat.instMulZeroClass",
... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\n⊢ r • (tprod R) isEmptyElim = r • GradedMonoid.GOne.one"
] | simp [algebraMap₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.TensorPower.Basic | {
"line": 249,
"column": 4
} | {
"line": 249,
"column": 47
} | {
"line": 250,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\nx : GradedMonoid fun i ↦ ⨂[R]^i M\nthis :\n (cast R M ⋯) (GradedMonoid.GMul.mul (algebraMap₀ r) x.snd) =\n (cast R M ⋯) (GradedMonoid.GMul.mul x.snd (algebraMap₀ r))\n⊢ (cast R M ⋯) (GradedMono... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\nx : GradedMonoid fun i ↦ ⨂[R]^i M\nthis :\n (cast R M ⋯) (GradedMonoid.GMul.mul (algebraMap₀ r) x.snd) =\n (cast R M ⋯) (GradedMonoid.GMul.mul x.snd (algebraMap₀ r))\n⊢ (GradedMonoid.mk 0 ((↑algebraMap₀).t... | rw [← LinearEquiv.eq_symm_apply, cast_symm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.TensorPower.Basic | {
"line": 253,
"column": 4
} | {
"line": 253,
"column": 47
} | {
"line": 254,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\nx : GradedMonoid fun i ↦ ⨂[R]^i M\n⊢ (cast R M ⋯) (r • x).snd = (GradedMonoid.mk 0 ((↑algebraMap₀).toAddMonoidHom r) * x).snd",
"ppTerm": "?m.250",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\nx : GradedMonoid fun i ↦ ⨂[R]^i M\n⊢ (r • x).snd = (cast R M ⋯) (GradedMonoid.mk 0 ((↑algebraMap₀).toAddMonoidHom r) * x).snd"
] | rw [← LinearEquiv.eq_symm_apply, cast_symm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.LinearDisjoint | {
"line": 385,
"column": 2
} | {
"line": 385,
"column": 17
} | {
"line": 387,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : Ring S\ninst✝¹ : Algebra R S\nM N : Submodule R S\nH : M.LinearDisjoint N\ninst✝ : Flat R ↥M\nκ : Type u_1\nι : Type u_2\nm : κ → ↥M\nn : ι → ↥N\nhm : Function.Injective ⇑(Finsupp.linearCombination R m)\nhn : Function.Injective ⇑(Finsupp.linearCombi... | [] | rwa [this] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.LinearDisjoint | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 17
} | {
"line": 408,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : Ring S\ninst✝¹ : Algebra R S\nM N : Submodule R S\nH : M.LinearDisjoint N\ninst✝ : Flat R ↥N\nκ : Type u_1\nι : Type u_2\nm : κ → ↥M\nn : ι → ↥N\nhm : Function.Injective ⇑(Finsupp.linearCombination R m)\nhn : Function.Injective ⇑(Finsupp.linearCombi... | [] | rwa [this] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.PiTensorProduct.Basic | {
"line": 680,
"column": 78
} | {
"line": 680,
"column": 83
} | {
"line": 680,
"column": 83
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.PiTensorProduct.Basic | {
"line": 680,
"column": 78
} | {
"line": 680,
"column": 83
} | {
"line": 680,
"column": 83
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.PiTensorProduct.Basic | {
"line": 680,
"column": 78
} | {
"line": 680,
"column": 83
} | {
"line": 680,
"column": 83
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.PiTensorProduct.Basic | {
"line": 680,
"column": 89
} | {
"line": 680,
"column": 94
} | {
"line": 680,
"column": 94
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.PiTensorProduct.Basic | {
"line": 680,
"column": 89
} | {
"line": 680,
"column": 94
} | {
"line": 680,
"column": 94
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.PiTensorProduct.Basic | {
"line": 680,
"column": 89
} | {
"line": 680,
"column": 94
} | {
"line": 680,
"column": 94
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ClassGroup.Basic | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 54
} | {
"line": 98,
"column": 54
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDomain R\n⊢ Subgroup.map (↑unitsMulEquivSubmodule) (toPrincipalIdeal R (FractionRing R)).range =\n (Units.map ↑(Submodule.spanSingleton R)).range",
"ppTerm": "?m.50",
... | [
"R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDomain R\n⊢ Subgroup.map ((↑unitsMulEquivSubmodule).comp (toPrincipalIdeal R (FractionRing R))) ⊤ =\n Subgroup.map (Units.map ↑(Submodule.spanSingleton R)) ⊤"
] | simp_rw [MonoidHom.range_eq_map, Subgroup.map_map] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.PiTensorProduct.Basic | {
"line": 784,
"column": 6
} | {
"line": 784,
"column": 11
} | {
"line": 785,
"column": 4
} | [
{
"pp": "case refine_1.e_a.e_6\nι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝¹⁰ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁹ : (i : ι) → AddCommMonoid (s i)\ninst✝⁸ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nE : Type u_9... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.PiTensorProduct.Basic | {
"line": 857,
"column": 27
} | {
"line": 857,
"column": 32
} | {
"line": 857,
"column": 32
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝¹¹ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝¹⁰ : (i : ι) → AddCommMonoid (s i)\ninst✝⁹ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\nE : Type u_9\ninst✝⁶ : AddCommMono... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.PiTensorProduct.Basic | {
"line": 857,
"column": 27
} | {
"line": 857,
"column": 32
} | {
"line": 857,
"column": 32
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝¹¹ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝¹⁰ : (i : ι) → AddCommMonoid (s i)\ninst✝⁹ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\nE : Type u_9\ninst✝⁶ : AddCommMono... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.PiTensorProduct.Basic | {
"line": 857,
"column": 27
} | {
"line": 857,
"column": 32
} | {
"line": 857,
"column": 32
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝¹¹ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝¹⁰ : (i : ι) → AddCommMonoid (s i)\ninst✝⁹ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\nE : Type u_9\ninst✝⁶ : AddCommMono... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.LocallyConstant.Basic | {
"line": 216,
"column": 22
} | {
"line": 216,
"column": 51
} | {
"line": 218,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\nα : Type u_4\ninst✝ : TopologicalSpace X\n⊢ Function.Injective LocallyConstant.toFun",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"LocallyConstant.mk",
"IsLocallyConstant",
"Eq.rec",
"LocallyConstant",
"Topo... | [] | rintro ⟨_, _⟩ ⟨_, _⟩ _; congr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.LocallyConstant.Basic | {
"line": 216,
"column": 22
} | {
"line": 216,
"column": 51
} | {
"line": 218,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\nα : Type u_4\ninst✝ : TopologicalSpace X\n⊢ Function.Injective LocallyConstant.toFun",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"LocallyConstant.mk",
"IsLocallyConstant",
"Eq.rec",
"LocallyConstant",
"Topo... | [] | rintro ⟨_, _⟩ ⟨_, _⟩ _; congr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.TensorProduct.IsBaseChangePi | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 74
} | {
"line": 57,
"column": 2
} | [
{
"pp": "case intro\nR : Type u_1\nS : Type u_2\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Algebra R S\nι : Type u_3\ninst✝⁶ : Finite ι\nM : ι → Type u_4\nM' : ι → Type u_5\ninst✝⁵ : (i : ι) → AddCommMonoid (M i)\ninst✝⁴ : (i : ι) → AddCommMonoid (M' i)\ninst✝³ : (i : ι) → Module R (M i)\ninst✝... | [
"case intro\nR : Type u_1\nS : Type u_2\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Algebra R S\nι : Type u_3\ninst✝⁶ : Finite ι\nM : ι → Type u_4\nM' : ι → Type u_5\ninst✝⁵ : (i : ι) → AddCommMonoid (M i)\ninst✝⁴ : (i : ι) → AddCommMonoid (M' i)\ninst✝³ : (i : ι) → Module R (M i)\ninst✝² : (i : ι) ... | apply of_equiv <| piRight R S _ M ≪≫ₗ .piCongrRight fun i ↦ (hf i).equiv | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 380,
"column": 76
} | {
"line": 385,
"column": 79
} | {
"line": 387,
"column": 0
} | [
{
"pp": "R : Type uR\nM : Type uM\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Flat R M\ninst✝ : Module.Finite R M\np : PrimeSpectrum R\n⊢ rankAtStalk M p = finrank p.asIdeal.ResidueField (p.asIdeal.Fiber M)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
... | [] | by
let k := p.asIdeal.ResidueField
let e : k ⊗[Localization.AtPrime p.asIdeal] (Localization.AtPrime p.asIdeal ⊗[R] M) ≃ₗ[k]
k ⊗[R] M :=
AlgebraTensorModule.cancelBaseChange _ _ _ _ _
rw [← e.finrank_eq, finrank_baseChange, rankAtStalk_eq_finrank_tensorProduct] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Normal.Closure | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 26
} | {
"line": 104,
"column": 26
} | [
{
"pp": "F : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝⁵ : Field F\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra F K\ninst✝¹ : Algebra F L\ninst✝ : Algebra.IsAlgebraic F K\nsplits : ∀ (x : K), (Polynomial.map (algebraMap F L) (minpoly F x)).Splits\n⊢ IsNormalClosure F K ↥(normalClosure F K L)",
"pp... | [
"F : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝⁵ : Field F\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra F K\ninst✝¹ : Algebra F L\ninst✝ : Algebra.IsAlgebraic F K\nsplits : ∀ (x : K), (Polynomial.map (algebraMap F L) (minpoly F x)).Splits\n⊢ (∀ (x : K), (Polynomial.map (algebraMap F ↥(normalClosure F K L)... | rw [isNormalClosure_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Normal.Closure | {
"line": 151,
"column": 2
} | {
"line": 155,
"column": 16
} | {
"line": 157,
"column": 0
} | [
{
"pp": "F : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra F K\ninst✝ : Algebra F L\nne : Nonempty (K →ₐ[F] L)\nh : Normal F L\n⊢ normalClosure F K L = ⨆ x, adjoin F ((minpoly F x).rootSet L)",
"ppTerm": "?m.38",
"assigned": true,
"usedC... | [] | have ⟨φ⟩ := ne
refine h.toIsAlgebraic.of_injective φ φ.injective
|>.normalClosure_eq_iSup_adjoin_of_splits fun x ↦ ?_
rw [← minpoly.algHom_eq _ φ.injective]
apply h.splits | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Normal.Closure | {
"line": 151,
"column": 2
} | {
"line": 155,
"column": 16
} | {
"line": 157,
"column": 0
} | [
{
"pp": "F : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra F K\ninst✝ : Algebra F L\nne : Nonempty (K →ₐ[F] L)\nh : Normal F L\n⊢ normalClosure F K L = ⨆ x, adjoin F ((minpoly F x).rootSet L)",
"ppTerm": "?m.38",
"assigned": true,
"usedC... | [] | have ⟨φ⟩ := ne
refine h.toIsAlgebraic.of_injective φ φ.injective
|>.normalClosure_eq_iSup_adjoin_of_splits fun x ↦ ?_
rw [← minpoly.algHom_eq _ φ.injective]
apply h.splits | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PicardGroup | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 98
} | {
"line": 120,
"column": 4
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\nP : Type u_2\nQ : Type u_3\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : AddCommMonoid Q\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\ne : M ⊗[R] N ≃ₗ[R] R\nx✝ ... | [
"R : Type u\nM : Type v\nN : Type u_1\nP : Type u_2\nQ : Type u_3\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : AddCommMonoid Q\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\ne : M ⊗[R] N ≃ₗ[R] R\nx✝ : P →ₗ[R] Q\... | simp_rw [rTensorInv, LinearEquiv.coe_trans, LinearMap.comp_apply, LinearEquiv.coe_toLinearMap] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 65,
"column": 2
} | {
"line": 66,
"column": 27
} | {
"line": 67,
"column": 2
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝³ : Field F\nE : Type u_2\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Finite E\nα : Eˣ\nhα : ∀ (x : Eˣ), x ∈ Subgroup.zpowers α\nx : E\nhx : x = 0\n⊢ x ∈ F⟮↑α⟯",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
... | [
"case neg\nF : Type u_1\ninst✝³ : Field F\nE : Type u_2\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Finite E\nα : Eˣ\nhα : ∀ (x : Eˣ), x ∈ Subgroup.zpowers α\nx : E\nhx : ¬x = 0\n⊢ x ∈ F⟮↑α⟯"
] | · rw [hx]
exact F⟮α.val⟯.zero_mem | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.AlgebraicIndependent.Defs | {
"line": 165,
"column": 58
} | {
"line": 165,
"column": 63
} | {
"line": 165,
"column": 63
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_3\nK : Type u_4\nA : Type u_5\nA' : Type u_6\nx : ι → A\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : CommRing A'\ninst✝¹ : Algebra R A\ninst✝ : Algebra R A'\nhx✝ : AlgebraicIndependent R x\ns : Set ι\nhx : AlgebraicIndepOn R x s\n⊢ (range fun i ↦ x ↑i) = x... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.AlgebraicIndependent.Defs | {
"line": 165,
"column": 58
} | {
"line": 165,
"column": 63
} | {
"line": 165,
"column": 63
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_3\nK : Type u_4\nA : Type u_5\nA' : Type u_6\nx : ι → A\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : CommRing A'\ninst✝¹ : Algebra R A\ninst✝ : Algebra R A'\nhx✝ : AlgebraicIndependent R x\ns : Set ι\nhx : AlgebraicIndepOn R x s\n⊢ (range fun i ↦ x ↑i) = x... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AlgebraicIndependent.Defs | {
"line": 165,
"column": 58
} | {
"line": 165,
"column": 63
} | {
"line": 165,
"column": 63
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_3\nK : Type u_4\nA : Type u_5\nA' : Type u_6\nx : ι → A\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : CommRing A'\ninst✝¹ : Algebra R A\ninst✝ : Algebra R A'\nhx✝ : AlgebraicIndependent R x\ns : Set ι\nhx : AlgebraicIndepOn R x s\n⊢ (range fun i ↦ x ↑i) = x... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AlgebraicIndependent.Defs | {
"line": 165,
"column": 58
} | {
"line": 165,
"column": 63
} | {
"line": 165,
"column": 63
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_3\nK : Type u_4\nA : Type u_5\nA' : Type u_6\nx : ι → A\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : CommRing A'\ninst✝¹ : Algebra R A\ninst✝ : Algebra R A'\nhx✝ : AlgebraicIndependent R x\ns : Set ι\nhx : AlgebraicIndepOn R x s\n⊢ (range fun i ↦ x ↑i) = x... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.AlgebraicIndependent.Defs | {
"line": 165,
"column": 58
} | {
"line": 165,
"column": 63
} | {
"line": 165,
"column": 63
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_3\nK : Type u_4\nA : Type u_5\nA' : Type u_6\nx : ι → A\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : CommRing A'\ninst✝¹ : Algebra R A\ninst✝ : Algebra R A'\nhx✝ : AlgebraicIndependent R x\ns : Set ι\nhx : AlgebraicIndepOn R x s\n⊢ (range fun i ↦ x ↑i) = x... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AlgebraicIndependent.Defs | {
"line": 165,
"column": 58
} | {
"line": 165,
"column": 63
} | {
"line": 165,
"column": 63
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_3\nK : Type u_4\nA : Type u_5\nA' : Type u_6\nx : ι → A\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : CommRing A'\ninst✝¹ : Algebra R A\ninst✝ : Algebra R A'\nhx✝ : AlgebraicIndependent R x\ns : Set ι\nhx : AlgebraicIndepOn R x s\n⊢ (range fun i ↦ x ↑i) = x... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 295,
"column": 71
} | {
"line": 295,
"column": 85
} | {
"line": 296,
"column": 4
} | [
{
"pp": "case inr.inr\nα : Type u_1\nIsBase : Set α → Prop\nB₁ B₂ : Set α\nexch : ExchangeProperty IsBase\nhB₁ : IsBase B₁\nhB₂ : IsBase B₂\ne : α\nhe : e ∈ B₂ \\ B₁\nhcard : ((B₂ \\ B₁) \\ {e}).encard < (B₂ \\ B₁).encard\nf : α\nhf : f ∈ B₁ \\ B₂\nhB' : IsBase (insert f (B₂ \\ {e}))\nthis : (insert f (B₂ \\ {e... | [
"case inr.inr\nα : Type u_1\nIsBase : Set α → Prop\nB₁ B₂ : Set α\nexch : ExchangeProperty IsBase\nhB₁ : IsBase B₁\nhB₂ : IsBase B₂\ne : α\nhe : e ∈ B₂ \\ B₁\nhcard : ((B₂ \\ B₁) \\ {e}).encard < (B₂ \\ B₁).encard\nf : α\nhf : f ∈ B₁ \\ B₂\nhB' : IsBase (insert f (B₂ \\ {e}))\nthis : (insert f (B₂ \\ {e}) \\ B₁).en... | ← sdiff_sdiff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PicardGroup | {
"line": 558,
"column": 4
} | {
"line": 558,
"column": 35
} | {
"line": 560,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\nP : Type u_2\nQ : Type u_3\nA✝ : Type u_4\ninst✝¹³ : CommSemiring R\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : AddCommMonoid P\ninst✝⁹ : AddCommMonoid Q\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : Module R Q\ninst✝... | [] | simp_rw [mk_tensor, mk_eq_self] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 556,
"column": 11
} | {
"line": 556,
"column": 26
} | {
"line": 556,
"column": 26
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\n⊢ M.Indep X ∨ ¬M.Indep X ∧ X ⊆ M.E",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"and_iff_left",
"Matroid.Indep",
"id",
"LE.le",
"And",
... | [
"α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\n⊢ M.Indep X ∨ ¬M.Indep X"
] | and_iff_left hX | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 575,
"column": 11
} | {
"line": 575,
"column": 26
} | {
"line": 575,
"column": 26
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\n⊢ ¬M.Indep X ↔ ¬M.Indep X ∧ X ⊆ M.E",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"and_iff_left",
"Matroid.Indep",
"id",
"LE.le",
"And",
... | [
"α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\n⊢ ¬M.Indep X ↔ ¬M.Indep X"
] | and_iff_left hX | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LocalRing.ResidueField.Fiber | {
"line": 217,
"column": 75
} | {
"line": 234,
"column": 39
} | {
"line": 236,
"column": 0
} | [
{
"pp": "R✝ : Type u_1\nS✝ : Type u_2\ninst✝⁶ : CommRing R✝\ninst✝⁵ : CommRing S✝\ninst✝⁴ : Algebra R✝ S✝\np✝ : Ideal R✝\ninst✝³ : p✝.IsPrime\nR : Type u_3\nS : Type u_4\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : PrimeSpectrum R\n⊢ ↑(comap (algebraMap R S) ⁻¹' {p}) ≃ₜ PrimeSpectrum (p.a... | [] | by
letI H : Topology.IsEmbedding (preimageOrderIsoFiber R S p).symm := by
refine (Topology.IsEmbedding.of_comp_iff .subtypeVal).mp ?_
have := PrimeSpectrum.isEmbedding_tensorProductTo_of_surjectiveOnStalks _ S _
(Ideal.surjectiveOnStalks_residueField p.asIdeal)
exact ((Homeomorph.prodUnique _ _).isE... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 169,
"column": 2
} | {
"line": 171,
"column": 9
} | {
"line": 177,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsSeparable F E\nhα : IsIntegral F α\nhβ : IsIntegral F β\nf : F[X] := ⋯\ng : F[X] := ⋯\nιFE : F →+* E := ⋯\nιEE' : E →+* (Polynomial.map ιFE g).SplittingField := ⋯\nc : F... | [] | · simp only [map_comp, Polynomial.map_map, ← IsScalarTower.algebraMap_eq, Polynomial.map_sub,
map_C, AdjoinSimple.algebraMap_gen, Polynomial.map_mul, map_X]
congr | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 85
} | {
"line": 192,
"column": 4
} | [
{
"pp": "case h.a\nF : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Finite (IntermediateField F E)\nf : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯\nx y : F\nhneq : ¬x = y\nheq : F⟮α + x • β⟯ = F⟮α + y • β⟯\nαxβ_in_K : α + x • β ∈ ... | [
"case h.a\nF : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Finite (IntermediateField F E)\nf : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯\nx y : F\nhneq : ¬x = y\nheq : F⟮α + x • β⟯ = F⟮α + y • β⟯\nαxβ_in_K : α + x • β ∈ F⟮α + x • β⟯... | rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at β_in_K | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 795,
"column": 54
} | {
"line": 795,
"column": 63
} | {
"line": 795,
"column": 63
} | [
{
"pp": "α : Type u_1\nM✝ M : Matroid α\nE : Set α\nIsBase Indep : Set α → Prop\nhE : E = M.E\nhB : ∀ (B : Set α), IsBase B ↔ M.IsBase B\nhI : ∀ (I : Set α), Indep I ↔ M.Indep I\n⊢ ∀ (x : Set α), Indep x = M.Indep x",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"congrArg",
"M... | [] | simp [hI] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 795,
"column": 54
} | {
"line": 795,
"column": 63
} | {
"line": 795,
"column": 63
} | [
{
"pp": "α : Type u_1\nM✝ M : Matroid α\nE : Set α\nIsBase Indep : Set α → Prop\nhE : E = M.E\nhB : ∀ (B : Set α), IsBase B ↔ M.IsBase B\nhI : ∀ (I : Set α), Indep I ↔ M.Indep I\n⊢ ∀ (x : Set α), Indep x = M.Indep x",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"congrArg",
"M... | [] | simp [hI] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 795,
"column": 54
} | {
"line": 795,
"column": 63
} | {
"line": 795,
"column": 63
} | [
{
"pp": "α : Type u_1\nM✝ M : Matroid α\nE : Set α\nIsBase Indep : Set α → Prop\nhE : E = M.E\nhB : ∀ (B : Set α), IsBase B ↔ M.IsBase B\nhI : ∀ (I : Set α), Indep I ↔ M.Indep I\n⊢ ∀ (x : Set α), Indep x = M.Indep x",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"congrArg",
"M... | [] | simp [hI] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 294,
"column": 4
} | {
"line": 295,
"column": 82
} | {
"line": 296,
"column": 2
} | [
{
"pp": "case inl\nF : Type u_1\nE : Type u_2\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Finite (IntermediateField F E)\nK : IntermediateField F E\nthis : FiniteDimensional F E\nh✝ : Finite F\n⊢ ∃ α, F⟮α⟯ = K",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Int... | [] | obtain ⟨α, h⟩ := exists_primitive_element_of_finite_bot F K
exact ⟨α, by simpa only [lift_adjoin_simple, lift_top] using congr_arg lift h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 294,
"column": 4
} | {
"line": 295,
"column": 82
} | {
"line": 296,
"column": 2
} | [
{
"pp": "case inl\nF : Type u_1\nE : Type u_2\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Finite (IntermediateField F E)\nK : IntermediateField F E\nthis : FiniteDimensional F E\nh✝ : Finite F\n⊢ ∃ α, F⟮α⟯ = K",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Int... | [] | obtain ⟨α, h⟩ := exists_primitive_element_of_finite_bot F K
exact ⟨α, by simpa only [lift_adjoin_simple, lift_top] using congr_arg lift h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 874,
"column": 20
} | {
"line": 874,
"column": 35
} | {
"line": 874,
"column": 35
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhX : X ⊆ M.E\n⊢ (M.Indep I ∧ I ⊆ X ∧ ∀ ⦃J : Set α⦄, M.Indep J → I ⊆ J → J ⊆ X → I = J) ∧ X ⊆ M.E ↔\n M.Indep I ∧ I ⊆ X ∧ ∀ (J : Set α), M.Indep J → I ⊆ J → J ⊆ X → I = J",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\nM : Matroid α\nI X : Set α\nhX : X ⊆ M.E\n⊢ (M.Indep I ∧ I ⊆ X ∧ ∀ ⦃J : Set α⦄, M.Indep J → I ⊆ J → J ⊆ X → I = J) ↔\n M.Indep I ∧ I ⊆ X ∧ ∀ (J : Set α), M.Indep J → I ⊆ J → J ⊆ X → I = J"
] | and_iff_left hX | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 900,
"column": 15
} | {
"line": 900,
"column": 30
} | {
"line": 900,
"column": 30
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhX : X ⊆ M.E\n⊢ Maximal (fun A ↦ M.Indep A ∧ A ⊆ X) I ∧ X ⊆ M.E ↔ Maximal (fun I ↦ M.Indep I ∧ I ⊆ X) I",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"and_iff_left",
"Matro... | [
"α : Type u_1\nM : Matroid α\nI X : Set α\nhX : X ⊆ M.E\n⊢ Maximal (fun A ↦ M.Indep A ∧ A ⊆ X) I ↔ Maximal (fun I ↦ M.Indep I ∧ I ⊆ X) I"
] | and_iff_left hX | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 914,
"column": 57
} | {
"line": 914,
"column": 77
} | {
"line": 914,
"column": 77
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\n⊢ M.Indep I ∧ (∀ ⦃J : Set α⦄, M.Indep J → I ⊆ J → J ⊆ I → I = J) ∧ I ⊆ M.E ↔ M.Indep I",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Matroid.Indep",
"id",
"LE.le",
... | [
"α : Type u_1\nM : Matroid α\nI : Set α\n⊢ M.Indep I → (∀ ⦃J : Set α⦄, M.Indep J → I ⊆ J → J ⊆ I → I = J) ∧ I ⊆ M.E"
] | and_iff_left_iff_imp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 329,
"column": 4
} | {
"line": 329,
"column": 83
} | {
"line": 331,
"column": 2
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsAlgebraic F E\nthis : FiniteDimensional F E\nα : E\nhprim : F⟮α⟯ = ⊤\nf : F[X] := minpoly F α\nG : Type (max 0 u_2) := { g // g.Monic ∧ g ∣ Polynomial.map (algebraMap F E) f }\nhfin : Finite G\ng : I... | [] | simpa only [adjoin_minpoly_coeff_of_exists_primitive_element F hprim] using heq | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.Matroid.Minor.Restrict | {
"line": 194,
"column": 40
} | {
"line": 194,
"column": 60
} | {
"line": 194,
"column": 60
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nR I X : Set α\nhXR : X ⊆ R\n⊢ M.IsBasis' I X ∧ I ⊆ R ↔ M.IsBasis' I X",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.IsBasis'",
"id",
"LE.le",
"And",
"Iff",
"Set.instLE",
... | [
"α : Type u_1\nM : Matroid α\nR I X : Set α\nhXR : X ⊆ R\n⊢ M.IsBasis' I X → I ⊆ R"
] | and_iff_left_iff_imp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 128,
"column": 40
} | {
"line": 128,
"column": 49
} | {
"line": 128,
"column": 49
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nh : ∀ ⦃y : Set α⦄, M.Indep y → y = ∅\nI : Set α\nx✝ : I ⊆ M.E\nhI : I = ∅\n⊢ M.Indep I",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"congrArg",
"Matroid.Indep",
"True",
"Set.instEmptyCollection",
"of_eq_true",
"Em... | [] | simp [hI] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 128,
"column": 40
} | {
"line": 128,
"column": 49
} | {
"line": 128,
"column": 49
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nh : ∀ ⦃y : Set α⦄, M.Indep y → y = ∅\nI : Set α\nx✝ : I ⊆ M.E\nhI : I = ∅\n⊢ M.Indep I",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"congrArg",
"Matroid.Indep",
"True",
"Set.instEmptyCollection",
"of_eq_true",
"Em... | [] | simp [hI] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 128,
"column": 40
} | {
"line": 128,
"column": 49
} | {
"line": 128,
"column": 49
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nh : ∀ ⦃y : Set α⦄, M.Indep y → y = ∅\nI : Set α\nx✝ : I ⊆ M.E\nhI : I = ∅\n⊢ M.Indep I",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"congrArg",
"Matroid.Indep",
"True",
"Set.instEmptyCollection",
"of_eq_true",
"Em... | [] | simp [hI] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 221,
"column": 58
} | {
"line": 221,
"column": 78
} | {
"line": 221,
"column": 78
} | [
{
"pp": "α : Type u_1\nE I J : Set α\nhIE : I ⊆ E\n⊢ J ⊆ I ∧ J ⊆ E ↔ J ⊆ I",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"LE.le",
"And",
"Iff",
"Set.instLE",
"propext",
"Eq",
"and_iff_left_iff_imp",
... | [
"α : Type u_1\nE I J : Set α\nhIE : I ⊆ E\n⊢ J ⊆ I → J ⊆ E"
] | and_iff_left_iff_imp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Map | {
"line": 180,
"column": 23
} | {
"line": 180,
"column": 43
} | {
"line": 180,
"column": 43
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nI : Set α\nN : Matroid β\nhf : InjOn f (f ⁻¹' N.E)\n⊢ N.Indep (f '' I) ∧ InjOn f I ↔ N.Indep (f '' I)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.Indep",
"id",
"And",
"Iff... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nI : Set α\nN : Matroid β\nhf : InjOn f (f ⁻¹' N.E)\n⊢ N.Indep (f '' I) → InjOn f I"
] | and_iff_left_iff_imp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 279,
"column": 78
} | {
"line": 279,
"column": 83
} | {
"line": 279,
"column": 83
} | [
{
"pp": "α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug :\n ∀ ⦃I J : Set α⦄, Indep I → I.Finite → Indep J → J.Finite → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nindep_compact : ∀ (I : Set α), (∀ J ⊆ I, J.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Combinatorics.Matroid.Map | {
"line": 188,
"column": 23
} | {
"line": 188,
"column": 28
} | {
"line": 190,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nE : Set β\n⊢ ((loopyOn E).comap f).E = f ⁻¹' E ∧ ∀ X ⊆ ((loopyOn E).comap f).E, ((loopyOn E).comap f).Indep X → X = ∅",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Matroid.loopyOn",
"congrArg",
"and_self",
"Matroid.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 279,
"column": 78
} | {
"line": 279,
"column": 83
} | {
"line": 279,
"column": 83
} | [
{
"pp": "α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug :\n ∀ ⦃I J : Set α⦄, Indep I → I.Finite → Indep J → J.Finite → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nindep_compact : ∀ (I : Set α), (∀ J ⊆ I, J.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 279,
"column": 78
} | {
"line": 279,
"column": 83
} | {
"line": 279,
"column": 83
} | [
{
"pp": "α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug :\n ∀ ⦃I J : Set α⦄, Indep I → I.Finite → Indep J → J.Finite → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nindep_compact : ∀ (I : Set α), (∀ J ⊆ I, J.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Dual | {
"line": 239,
"column": 27
} | {
"line": 239,
"column": 42
} | {
"line": 239,
"column": 42
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\n⊢ (∃ B, M.IsBase B ∧ B ⊆ M.E \\ X) ∧ X ⊆ M.E ↔ ∃ B, M.IsBase B ∧ B ⊆ M.E \\ X",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"and_iff_left",
"Exists",
"Mat... | [
"α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\n⊢ (∃ B, M.IsBase B ∧ B ⊆ M.E \\ X) ↔ ∃ B, M.IsBase B ∧ B ⊆ M.E \\ X"
] | and_iff_left hX | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 359,
"column": 8
} | {
"line": 360,
"column": 33
} | {
"line": 361,
"column": 6
} | [
{
"pp": "α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug : ∀ ⦃I J : Set α⦄, Indep I → Indep J → I.encard < J.encard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nindep_bdd : ∃ n, ∀ (I : Set α), Indep I → I.encard ≤ ↑n\nsubset_gr... | [] | obtain ⟨e, heB, heI, hi⟩ := indep_aug hI hBmax.prop hcard
exact ⟨e, ⟨heB, heI⟩, hi⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 359,
"column": 8
} | {
"line": 360,
"column": 33
} | {
"line": 361,
"column": 6
} | [
{
"pp": "α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug : ∀ ⦃I J : Set α⦄, Indep I → Indep J → I.encard < J.encard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nindep_bdd : ∃ n, ∀ (I : Set α), Indep I → I.encard ≤ ↑n\nsubset_gr... | [] | obtain ⟨e, heB, heI, hi⟩ := indep_aug hI hBmax.prop hcard
exact ⟨e, ⟨heB, heI⟩, hi⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Map | {
"line": 310,
"column": 39
} | {
"line": 310,
"column": 59
} | {
"line": 310,
"column": 59
} | [
{
"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\nI : Set ↑M.E\n⊢ M.Indep ↑I ∧ InjOn (fun x ↦ ↑(invFunOn (⇑f) univ x)) (⇑f '' I) ↔ M.Indep (Subtype.val '' I)",
"ppTer... | [
"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\nI : Set ↑M.E\n⊢ M.Indep ↑I → InjOn (fun x ↦ ↑(invFunOn (⇑f) univ x)) (⇑f '' I)"
] | and_iff_left_iff_imp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Map | {
"line": 446,
"column": 70
} | {
"line": 446,
"column": 90
} | {
"line": 446,
"column": 90
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nM : Matroid α\nhf : InjOn f M.E\nB : Set α\nhB : B ⊆ M.E\n⊢ (M.IsBase (M.E \\ B) ∧ ∃ u ⊆ M.E, f '' u = f '' B) ↔ M.IsBase (M.E \\ B)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.E",
"Exists",
"Mat... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nM : Matroid α\nhf : InjOn f M.E\nB : Set α\nhB : B ⊆ M.E\n⊢ M.IsBase (M.E \\ B) → ∃ u ⊆ M.E, f '' u = f '' B"
] | and_iff_left_iff_imp | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 147,
"column": 2
} | {
"line": 152,
"column": 47
} | {
"line": 154,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\ne : α\nhI : M.Indep I\nheI : e ∉ I\nhe : e ∈ M.closure I\nh : ∀ f ∈ I, e ∉ M.closure (I \\ {f})\n⊢ M.IsCircuit (insert e I)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.Dep",
"False",
"eq_false",... | [] | rw [isCircuit_iff_dep_forall_sdiff_singleton_indep, hI.insert_dep_iff, and_iff_right ⟨he, heI⟩]
rintro f (rfl | hfI)
· simpa [heI]
rw [← insert_sdiff_singleton_comm (by rintro rfl; contradiction),
(hI.sdiff _).insert_indep_iff_of_notMem (by simp [heI])]
exact ⟨mem_ground_of_mem_closure he, h f hfI⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 147,
"column": 2
} | {
"line": 152,
"column": 47
} | {
"line": 154,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\ne : α\nhI : M.Indep I\nheI : e ∉ I\nhe : e ∈ M.closure I\nh : ∀ f ∈ I, e ∉ M.closure (I \\ {f})\n⊢ M.IsCircuit (insert e I)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.Dep",
"False",
"eq_false",... | [] | rw [isCircuit_iff_dep_forall_sdiff_singleton_indep, hI.insert_dep_iff, and_iff_right ⟨he, heI⟩]
rintro f (rfl | hfI)
· simpa [heI]
rw [← insert_sdiff_singleton_comm (by rintro rfl; contradiction),
(hI.sdiff _).insert_indep_iff_of_notMem (by simp [heI])]
exact ⟨mem_ground_of_mem_closure he, h f hfI⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 95
} | {
"line": 217,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ne : α\nR : Set α\nhR : R ⊆ M.E\n⊢ (M ↾ R).IsLoop e ↔ e ∈ (M ↾ R).E ∧ M.IsLoop e",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.restrict_isLoop_iff._simp_1",
"congrArg",
"Matroid.E",
"Membership.mem",
... | [
"α : Type u_1\nM : Matroid α\ne : α\nR : Set α\nhR : R ⊆ M.E\n⊢ e ∈ R → e ∉ M.E → M.IsLoop e"
] | simp only [restrict_isLoop_iff, restrict_ground_eq, and_congr_right_iff, or_iff_left_iff_imp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 338,
"column": 2
} | {
"line": 338,
"column": 7
} | {
"line": 340,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\n⊢ ¬((∃ C ⊆ I, M.IsCircuit C) ∧ I ⊆ M.E) ∧ I ⊆ M.E ↔ (∀ C ⊆ I, ¬M.IsCircuit C) ∧ I ⊆ M.E",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"Matroid.E",
"Exists",
"not_true_eq_false",
"LE.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 403,
"column": 4
} | {
"line": 403,
"column": 24
} | {
"line": 403,
"column": 24
} | [
{
"pp": "case inr\nα : Type u_1\nM : Matroid α\ne f : α\nhe : M.IsNonloop e\nhf : M.IsNonloop f\nhne : e ≠ f\n⊢ (¬M.Indep {e, f} ∧ ∀ e_1 ∈ {e, f}, M.Indep ({e, f} \\ {e_1})) ↔ ¬M.Indep {e, f}",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"Matr... | [
"case inr\nα : Type u_1\nM : Matroid α\ne f : α\nhe : M.IsNonloop e\nhf : M.IsNonloop f\nhne : e ≠ f\n⊢ ¬M.Indep {e, f} → ∀ e_1 ∈ {e, f}, M.Indep ({e, f} \\ {e_1})"
] | and_iff_left_iff_imp | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 374,
"column": 44
} | {
"line": 374,
"column": 65
} | {
"line": 374,
"column": 65
} | [
{
"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",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Matroid.IsBasis'.closure_eq_closure",
"Eq.mpr",
"congrArg",
"Membership.mem"... | [
"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 X"
] | hI.closure_eq_closure | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 9
} | {
"line": 125,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nM✝ : Matroid α\nF X Y : Set α\ne f : α\nM : Matroid α\ns : Set ↑(Iic M.E)\nhs : ∀ a ∈ s, M.IsFlat ↑a\nhne : s.Nonempty\nx✝¹ : Nonempty ↑s\nx✝ : α\n⊢ x✝ ∈ ↑(sInf s) ↔ x✝ ∈ ⋂ i, ↑↑i",
"ppTerm": "?m.116",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 142,
"column": 6
} | {
"line": 142,
"column": 14
} | {
"line": 142,
"column": 15
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\n⊢ M.closure X = ⋂₀ {F | M.IsFlat F ∧ X ⊆ F}",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"setOf",
"Matroid.closure.eq_1",
"id",
"LE.le",
"Set... | [
"α : Type u_2\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\n⊢ ⋂₀ {F | M.IsFlat F ∧ X ∩ M.E ⊆ F} = ⋂₀ {F | M.IsFlat F ∧ X ⊆ F}"
] | closure, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Map | {
"line": 666,
"column": 56
} | {
"line": 666,
"column": 74
} | {
"line": 666,
"column": 74
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nB : Set ↑M.E\n⊢ M.IsBasis (Subtype.val '' B) M.E ↔ M.IsBase (Subtype.val '' B)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Matroid.isBasis_ground_iff",
"Membership.mem",
"Matro... | [
"α : Type u_1\nM : Matroid α\nB : Set ↑M.E\n⊢ M.IsBase (Subtype.val '' B) ↔ M.IsBase (Subtype.val '' B)"
] | isBasis_ground_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 14
} | {
"line": 185,
"column": 15
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nX : Set α\n⊢ M.IsFlat (M.closure X)",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"setOf",
"Matroid.closure.eq_1",
"id",
"LE.le",
"Set.instInter",
"Inter.inter",
... | [
"α : Type u_2\nM : Matroid α\nX : Set α\n⊢ M.IsFlat (⋂₀ {F | M.IsFlat F ∧ X ∩ M.E ⊆ F})"
] | closure, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 233,
"column": 2
} | {
"line": 234,
"column": 37
} | {
"line": 236,
"column": 0
} | [
{
"pp": "α : Type u_2\nX : Set α\ne : α\nM : Matroid α\nheX : e ∈ X\nh : e ∈ M.E\n⊢ e ∈ M.closure X",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.inter_subset_right._simp_1",
"congrArg",
"Matroid.E",
"Membership.mem",
"id",
"LE.le"... | [] | rw [← closure_inter_ground]
exact M.mem_closure_of_mem ⟨heX, h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 233,
"column": 2
} | {
"line": 234,
"column": 37
} | {
"line": 236,
"column": 0
} | [
{
"pp": "α : Type u_2\nX : Set α\ne : α\nM : Matroid α\nheX : e ∈ X\nh : e ∈ M.E\n⊢ e ∈ M.closure X",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.inter_subset_right._simp_1",
"congrArg",
"Matroid.E",
"Membership.mem",
"id",
"LE.le"... | [] | rw [← closure_inter_ground]
exact M.mem_closure_of_mem ⟨heX, h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 51
} | {
"line": 111,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhIX : M.IsBasis' I X\n⊢ M.eRk I = M.eRk X",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Matroid.Indep.eRk_eq_encard",
"congrArg",
"Matroid.IsBasis'.encard_eq_eRk",
"id",
"... | [] | rw [← hIX.encard_eq_eRk, hIX.indep.eRk_eq_encard] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 51
} | {
"line": 111,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhIX : M.IsBasis' I X\n⊢ M.eRk I = M.eRk X",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Matroid.Indep.eRk_eq_encard",
"congrArg",
"Matroid.IsBasis'.encard_eq_eRk",
"id",
"... | [] | rw [← hIX.encard_eq_eRk, hIX.indep.eRk_eq_encard] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 51
} | {
"line": 111,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhIX : M.IsBasis' I X\n⊢ M.eRk I = M.eRk X",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Matroid.Indep.eRk_eq_encard",
"congrArg",
"Matroid.IsBasis'.encard_eq_eRk",
"id",
"... | [] | rw [← hIX.encard_eq_eRk, hIX.indep.eRk_eq_encard] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 51
} | {
"line": 114,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhIX : M.IsBasis I X\n⊢ M.eRk I = M.eRk X",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Matroid.Indep.eRk_eq_encard",
"congrArg",
"Matroid.IsBasis.encard_eq_eRk",
"id",
"EN... | [] | rw [← hIX.encard_eq_eRk, hIX.indep.eRk_eq_encard] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 51
} | {
"line": 114,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhIX : M.IsBasis I X\n⊢ M.eRk I = M.eRk X",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Matroid.Indep.eRk_eq_encard",
"congrArg",
"Matroid.IsBasis.encard_eq_eRk",
"id",
"EN... | [] | rw [← hIX.encard_eq_eRk, hIX.indep.eRk_eq_encard] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 51
} | {
"line": 114,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhIX : M.IsBasis I X\n⊢ M.eRk I = M.eRk X",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Matroid.Indep.eRk_eq_encard",
"congrArg",
"Matroid.IsBasis.encard_eq_eRk",
"id",
"EN... | [] | rw [← hIX.encard_eq_eRk, hIX.indep.eRk_eq_encard] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 198,
"column": 6
} | {
"line": 198,
"column": 34
} | {
"line": 198,
"column": 35
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhIX : M.IsBasis' I X\nY : Set α\n⊢ M.eRk (I ∪ Y) = M.eRk (X ∪ Y)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.eRk_union_closure_left_eq",
"Set.instUnion",
"id",
"ENat",
... | [
"α : Type u_1\nM : Matroid α\nI X : Set α\nhIX : M.IsBasis' I X\nY : Set α\n⊢ M.eRk (M.closure I ∪ Y) = M.eRk (X ∪ Y)"
] | ← eRk_union_closure_left_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 373,
"column": 2
} | {
"line": 373,
"column": 7
} | {
"line": 375,
"column": 0
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nI : Set α\nx : α\nhI : M.Indep I\nthis : I ⊆ M.E\n⊢ ¬M.Indep (insert x I) ∧ x ∈ M.E ∨ x ∈ I ↔ x ∈ M.E ∧ (¬M.Indep (insert x I) ∨ x ∈ I)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"eq_false",
"and_true",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 20
} | {
"line": 292,
"column": 0
} | [
{
"pp": "α : Type u_1\ne : α\nM : Matroid α\nX Y : Set α\n⊢ M.eRk (insert e X ∩ insert e Y) + M.eRk (insert e X ∪ insert e Y) ≤ M.eRk (insert e X) + M.eRk (insert e Y)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Insert.insert",
"Set.instInsert",
"Matroid.eRk_submod",... | [] | apply M.eRk_submod | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 20
} | {
"line": 298,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX Y : Set α\n⊢ M.eRk (M.E \\ X ∩ (M.E \\ Y)) + M.eRk (M.E \\ X ∪ M.E \\ Y) ≤ M.eRk (M.E \\ X) + M.eRk (M.E \\ Y)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Matroid.E",
"SDiff.sdiff",
"Matroid.eRk_submod",
"Set.instSDiff",
... | [] | apply M.eRk_submod | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 403,
"column": 39
} | {
"line": 403,
"column": 52
} | {
"line": 403,
"column": 52
} | [
{
"pp": "case inr\nα : Type u_2\nM : Matroid α\ne : α\nI : Set α\nhI : M.Indep I\nh : e ∉ I\n⊢ e ∈ M.E \\ M.closure I ↔ e ∈ M.E \\ M.closure I ∨ e ∈ I",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Membership.mem",
"id",
... | [
"case inr\nα : Type u_2\nM : Matroid α\ne : α\nI : Set α\nhI : M.Indep I\nh : e ∉ I\n⊢ e ∈ M.E \\ M.closure I ↔ e ∈ M.E \\ M.closure I"
] | or_iff_left h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 746,
"column": 2
} | {
"line": 746,
"column": 7
} | {
"line": 748,
"column": 0
} | [
{
"pp": "α✝ : Type u_1\nM✝ : Matroid α✝\ne✝ : α✝\nα : Type u_1\nM : Matroid α\ne : α\nR : Set α\nhRE : R ⊆ M.E\nheR : e ∈ R\n⊢ (∀ ⦃C : Set α⦄, M.IsCircuit C ∧ C ⊆ R → e ∉ C) ↔ (¬∃ C ⊆ insert e R, M.IsCircuit C ∧ e ∈ C) ∧ e ∈ R",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 554,
"column": 76
} | {
"line": 554,
"column": 97
} | {
"line": 554,
"column": 97
} | [
{
"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",
"ppTerm": "?m.89",
"assigned": true,
"usedConstants": [
"Matroid.IsBa... | [
"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 T"
] | hI.closure_eq_closure | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 854,
"column": 51
} | {
"line": 854,
"column": 71
} | {
"line": 854,
"column": 71
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\n⊢ M.Indep I ∧ I ⊆ {e | M.IsNonloop e} ↔ M.Indep I",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"setOf",
"Matroid.Indep",
"id",
"Matroid.IsNonloop",
"LE.le",
"And",
... | [
"α : Type u_1\nM : Matroid α\nI : Set α\n⊢ M.Indep I → I ⊆ {e | M.IsNonloop e}"
] | and_iff_left_iff_imp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 575,
"column": 28
} | {
"line": 575,
"column": 43
} | {
"line": 575,
"column": 43
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\ninst✝ : M.RankFinite\nhX : X ⊆ M.E\n⊢ M.eRank ≤ M.eRk X ∧ X ⊆ M.E ↔ M.eRank ≤ M.eRk X",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"and_iff_left",
"id",
"LE.le",
... | [
"α : Type u_1\nM : Matroid α\nX : Set α\ninst✝ : M.RankFinite\nhX : X ⊆ M.E\n⊢ M.eRank ≤ M.eRk X ↔ M.eRank ≤ M.eRk X"
] | and_iff_left hX | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 677,
"column": 86
} | {
"line": 677,
"column": 91
} | {
"line": 677,
"column": 91
} | [
{
"pp": "α : Type u_2\nM : Matroid α\ne f : α\nB : Set α\nhB : M.IsBasis B M.E\nhe : e ∈ M.closure (insert f B \\ {e} ∩ M.E)\nhf : f ∉ M.E\nheB : e ∈ B\na✝ : α\n⊢ a✝ ∈ insert f B \\ {e} ∩ M.E → a✝ ∈ B \\ {e}",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 690,
"column": 50
} | {
"line": 690,
"column": 71
} | {
"line": 690,
"column": 72
} | [
{
"pp": "case refine_2\nα : Type u_2\nM : Matroid α\nI : Set α\nh : ∀ ⦃e : α⦄, e ∈ I → M.closure (I \\ {e}) ≠ M.closure I\ne : α\nheI : e ∈ I\nhin : e ∈ M.closure (I \\ {e})\n⊢ M.closure (I \\ {e}) = M.closure (insert e (M.closure (I \\ {e})))",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstant... | [
"case refine_2\nα : Type u_2\nM : Matroid α\nI : Set α\nh : ∀ ⦃e : α⦄, e ∈ I → M.closure (I \\ {e}) ≠ M.closure I\ne : α\nheI : e ∈ I\nhin : e ∈ M.closure (I \\ {e})\n⊢ M.closure (I \\ {e}) = M.closure (M.closure (I \\ {e}))"
] | insert_eq_of_mem hin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 880,
"column": 20
} | {
"line": 880,
"column": 38
} | {
"line": 880,
"column": 38
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nS : Set α\nh : M.Spanning S\nB : Set α\nhB : M.IsBasis B S\nhB' : M.IsBasis B M.E\n⊢ ∃ B, M.IsBase B ∧ B ⊆ S",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"congrArg",
"Matroid.E",
"Matroid.isBasis_ground_iff",
"Matroid.IsBase"... | [
"α : Type u_2\nM : Matroid α\nS : Set α\nh : M.Spanning S\nB : Set α\nhB : M.IsBasis B S\nhB' : M.IsBase B\n⊢ ∃ B, M.IsBase B ∧ B ⊆ S"
] | isBasis_ground_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 768,
"column": 38
} | {
"line": 768,
"column": 52
} | {
"line": 768,
"column": 53
} | [
{
"pp": "case inr.inr.inr\nα : Type u_1\nM : Matroid α\nB : Set α\nhB : M.IsBase B\ne f : α\nhe : M✶.Indep (insert f (M✶.E \\ B) \\ {e})\nhne : e ≠ f\nhB'✝ : M✶.IsBase (M✶.E \\ B)\nhfE : f ∈ M.E\nhfB : f ∈ B\nheE : e ∈ M.E\nheB : e ∉ B\nhB' : M.IsBase (M.E \\ (insert f (M✶.E \\ B) \\ {e}))\n⊢ insert e B ⊆ inser... | [
"case inr.inr.inr\nα : Type u_1\nM : Matroid α\nB : Set α\nhB : M.IsBase B\ne f : α\nhe : M✶.Indep (insert f (M✶.E \\ B) \\ {e})\nhne : e ≠ f\nhB'✝ : M✶.IsBase (M✶.E \\ B)\nhfE : f ∈ M.E\nhfB : f ∈ B\nheE : e ∈ M.E\nheB : e ∉ B\nhB' : M.IsBase (M.E \\ (insert f (M✶.E \\ B) \\ {e}))\n⊢ insert e B ⊆ insert e (insert ... | ← sdiff_sdiff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 896,
"column": 34
} | {
"line": 896,
"column": 60
} | {
"line": 896,
"column": 60
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nS : Set α\nhS : S ⊆ M.E\n⊢ M.Spanning S ↔ M.Spanning (M.E \\ (M.E \\ S))",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"id",
"SDiff.sdiff",
"Set.sdiff_sdiff_cancel_left",
"I... | [
"α : Type u_2\nM : Matroid α\nS : Set α\nhS : S ⊆ M.E\n⊢ M.Spanning S ↔ M.Spanning S"
] | sdiff_sdiff_cancel_left hS | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 981,
"column": 2
} | {
"line": 981,
"column": 36
} | {
"line": 982,
"column": 2
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nX R I : Set α\nhI : (M ↾ R).IsBasis' I X\nhI' : M.IsBasis' I (X ∩ R)\nhIR : I ⊆ R\ne : α\n⊢ e ∈ R ∧ (M.Indep (insert e I) ∧ insert e I ⊆ R → e ∈ I) ↔\n (e ∈ M.E ∧ (M.Indep (insert e I) → e ∈ I)) ∧ e ∈ R ∨ e ∈ R ∧ e ∉ M.E",
"ppTerm": "?m.101",
"assigned": true,
... | [
"case pos\nα : Type u_2\nM : Matroid α\nX R I : Set α\nhI : (M ↾ R).IsBasis' I X\nhI' : M.IsBasis' I (X ∩ R)\nhIR : I ⊆ R\ne : α\nhe : M.Indep (insert e I)\n⊢ e ∈ R ∧ (M.Indep (insert e I) ∧ insert e I ⊆ R → e ∈ I) ↔\n (e ∈ M.E ∧ (M.Indep (insert e I) → e ∈ I)) ∧ e ∈ R ∨ e ∈ R ∧ e ∉ M.E",
"case neg\nα : Type u... | by_cases he : M.Indep (insert e I) | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RingTheory.AlgebraicIndependent.Basic | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 31
} | {
"line": 184,
"column": 2
} | [
{
"pp": "ι : Type u\nR : Type u_2\nA : Type v\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nK : Type u_3\ninst✝³ : CommRing K\ninst✝² : Algebra R K\ninst✝¹ : Algebra K A\ninst✝ : IsScalarTower R K A\nhinj : Injective ⇑(algebraMap R K)\nai : AlgebraicIndependent K x\nthis : (aeval x... | [
"ι : Type u\nR : Type u_2\nA : Type v\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nK : Type u_3\ninst✝³ : CommRing K\ninst✝² : Algebra R K\ninst✝¹ : Algebra K A\ninst✝ : IsScalarTower R K A\nhinj : Injective ⇑(algebraMap R K)\nai : AlgebraicIndependent K x\nthis : (aeval x).comp (MvPo... | rw [← this, RingHom.coe_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 1054,
"column": 2
} | {
"line": 1054,
"column": 100
} | {
"line": 1056,
"column": 0
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nR S : Set α\nhSR : S ⊆ R\nhR : R ⊆ M.E\n⊢ (M ↾ R).Spanning S ↔ R ⊆ M.closure S",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Iff.rfl",
"and_iff_left",
"Matroid.restrict_closure_e... | [] | rw [spanning_iff, restrict_ground_eq, and_iff_left hSR, restrict_closure_eq _ hSR, inter_eq_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 1054,
"column": 2
} | {
"line": 1054,
"column": 100
} | {
"line": 1056,
"column": 0
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nR S : Set α\nhSR : S ⊆ R\nhR : R ⊆ M.E\n⊢ (M ↾ R).Spanning S ↔ R ⊆ M.closure S",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Iff.rfl",
"and_iff_left",
"Matroid.restrict_closure_e... | [] | rw [spanning_iff, restrict_ground_eq, and_iff_left hSR, restrict_closure_eq _ hSR, inter_eq_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 1054,
"column": 2
} | {
"line": 1054,
"column": 100
} | {
"line": 1056,
"column": 0
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nR S : Set α\nhSR : S ⊆ R\nhR : R ⊆ M.E\n⊢ (M ↾ R).Spanning S ↔ R ⊆ M.closure S",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Iff.rfl",
"and_iff_left",
"Matroid.restrict_closure_e... | [] | rw [spanning_iff, restrict_ground_eq, and_iff_left hSR, restrict_closure_eq _ hSR, inter_eq_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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