module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 518,
"column": 2
} | {
"line": 518,
"column": 41
} | {
"line": 518,
"column": 42
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Inhabited A\nG : Type u_2\ninst✝² : Monoid G\ninst✝¹ : IsLeftCancelMul G\ninst✝ : DiscreteTopology A\np : Pattern A G\ng : G\n⊢ IsOpen[Pi.topologicalSpace] {x | p.mulOccursInAt x g}",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [... | [
"A : Type u_1\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Inhabited A\nG : Type u_2\ninst✝² : Monoid G\ninst✝¹ : IsLeftCancelMul G\ninst✝ : DiscreteTopology A\np : Pattern A G\ng : G\n⊢ IsOpen[Pi.topologicalSpace] (cylinder (Finset.image (fun x ↦ g * x) p.support) (p.mulShift g))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 544,
"column": 2
} | {
"line": 544,
"column": 40
} | {
"line": 544,
"column": 41
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Inhabited A\nG : Type u_2\ninst✝² : Monoid G\ninst✝¹ : IsLeftCancelMul G\ninst✝ : DiscreteTopology A\nF : Set (Pattern A G)\nh_eq : {x | ∀ p ∈ F, ∀ (v : G), ¬p.mulOccursInAt x v} = ⋂ p ∈ F, ⋂ v, {x | ¬p.mulOccursInAt x v}\np : Pattern A G\nhp : p ∈ F\... | [
"A : Type u_1\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Inhabited A\nG : Type u_2\ninst✝² : Monoid G\ninst✝¹ : IsLeftCancelMul G\ninst✝ : DiscreteTopology A\nF : Set (Pattern A G)\nh_eq : {x | ∀ p ∈ F, ∀ (v : G), ¬p.mulOccursInAt x v} = ⋂ p ∈ F, ⋂ v, {x | ¬p.mulOccursInAt x v}\np : Pattern A G\nhp : p ∈ F\nv : G\nthis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 550,
"column": 2
} | {
"line": 550,
"column": 41
} | {
"line": 550,
"column": 42
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Inhabited A\nG : Type u_2\ninst✝² : Monoid G\ninst✝¹ : IsLeftCancelMul G\ninst✝ : T1Space A\np : Pattern A G\ng : G\n⊢ IsClosed[Pi.topologicalSpace] {x | p.mulOccursInAt x g}",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
... | [
"A : Type u_1\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Inhabited A\nG : Type u_2\ninst✝² : Monoid G\ninst✝¹ : IsLeftCancelMul G\ninst✝ : T1Space A\np : Pattern A G\ng : G\n⊢ IsClosed[Pi.topologicalSpace] (cylinder (Finset.image (fun x ↦ g * x) p.support) (p.mulShift g))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Extreme | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 59
} | {
"line": 104,
"column": 60
} | [
{
"pp": "X : Type u_1\nm : MeasurableSpace X\nμ : Measure X\nf : X → X\ninst✝ : IsProbabilityMeasure μ\nhμ : Ergodic f μ\n⊢ μ ∈ extremePoints ℝ≥0∞ {ν | MeasurePreserving f ν ν ∧ IsProbabilityMeasure ν}",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"MeasureTheory.MeasurePreserving",... | [
"X : Type u_1\nm : MeasurableSpace X\nμ : Measure X\nf : X → X\ninst✝ : IsProbabilityMeasure μ\nhμ : Ergodic f μ\n⊢ μ ∈ extremePoints ℝ≥0∞ {ν | MeasurePreserving f ν ν ∧ ν univ = 1}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.TopologicalEntropy.DynamicalEntourage | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 47
} | {
"line": 84,
"column": 2
} | [
{
"pp": "X : Type u_1\nT : X → X\nU : SetRel X X\ninst✝ : UniformSpace X\nh : Continuous[inst✝.toTopologicalSpace, inst✝.toTopologicalSpace] T\nU_uni : U ∈ 𝓤 X\nn : ℕ\nx : X\nk : ℕ\nb✝ : k ∈ Ico 0 n\n⊢ ball x ((map T T)^[↑⟨k, b✝⟩] ⁻¹' U) ∈ 𝓝 x",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants"... | [
"X : Type u_1\nT : X → X\nU : SetRel X X\ninst✝ : UniformSpace X\nh : Continuous[inst✝.toTopologicalSpace, inst✝.toTopologicalSpace] T\nU_uni : U ∈ 𝓤 X\nn : ℕ\nx : X\nk : ℕ\nb✝ : k ∈ Ico 0 n\n⊢ T^[k] ⁻¹' ball (T^[k] x) U ∈ 𝓝 x"
] | simp only [map_iterate, _root_.ball_preimage] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Dynamics.TopologicalEntropy.NetEntropy | {
"line": 137,
"column": 57
} | {
"line": 137,
"column": 68
} | {
"line": 137,
"column": 69
} | [
{
"pp": "X : Type u_1\nT : X → X\nF : Set X\nU : SetRel X X\nn : ℕ\nh✝ : netMaxcard T F U n < ⊤\nk : ℕ\nk_max : ↑k = sSup (WithTop.some '' Finset.card '' {s | IsDynNetIn T F U n ↑s})\nthis : netMaxcard T F U n = sSup (WithTop.some '' Finset.card '' {s | IsDynNetIn T F U n ↑s})\ns : Finset X\nh : IsDynNetIn T F ... | [
"X : Type u_1\nT : X → X\nF : Set X\nU : SetRel X X\nn : ℕ\nh✝ : netMaxcard T F U n < ⊤\nk : ℕ\nk_max : ↑k = sSup (WithTop.some '' Finset.card '' {s | IsDynNetIn T F U n ↑s})\nthis : netMaxcard T F U n = sSup (WithTop.some '' Finset.card '' {s | IsDynNetIn T F U n ↑s})\ns : Finset X\nh : IsDynNetIn T F U n ↑s\na : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.TopologicalEntropy.NetEntropy | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 54
} | {
"line": 173,
"column": 2
} | [
{
"pp": "X : Type u_1\nT : X → X\nF : Set X\nU : SetRel X X\nn : ℕ\n⊢ 1 ≤ netMaxcard T F U n ↔ F.Nonempty",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instCompleteLinearOrderENat",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"ChainCompletePart... | [
"X : Type u_1\nT : X → X\nF : Set X\nU : SetRel X X\nn : ℕ\n⊢ netMaxcard T F U n ≠ 0 ↔ F ≠ ∅"
] | rw [Order.one_le_iff_ne_zero, nonempty_iff_ne_empty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 382,
"column": 53
} | {
"line": 382,
"column": 92
} | {
"line": 383,
"column": 4
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\ns : Set α\nhs : MeasurableSet s\n⊢ (ν.withDensity (μ.rnDeriv ν)).real s = μ.real s",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\ns : Set α\nhs : MeasurableSet s\n⊢ μ.real s = μ.real s"
] | Measure.withDensity_rnDeriv_eq _ _ hμν, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Dynamics.TopologicalEntropy.CoverEntropy | {
"line": 177,
"column": 75
} | {
"line": 177,
"column": 86
} | {
"line": 177,
"column": 87
} | [
{
"pp": "X : Type u_1\nT : X → X\nU : SetRel X X\nF : Set X\nm : ℕ\nF_inv : MapsTo T F F\ninst✝ : U.IsSymm\nn : ℕ\ns : Finset X\nh : IsDynCoverOf T F U m ↑s\nx✝¹ : Nonempty X\ns_nemp : s.Nonempty\nx : X\nx_F : x ∈ F\nm_pos : m > 0\ndyncover : (Fin n → ↥s) → X\nh_dyncover :\n ∀ (t : Fin n → ↥s),\n ⋂ k, T^[m ... | [
"X : Type u_1\nT : X → X\nU : SetRel X X\nF : Set X\nm : ℕ\nF_inv : MapsTo T F F\ninst✝ : U.IsSymm\nn : ℕ\ns : Finset X\nh : IsDynCoverOf T F U m ↑s\nx✝¹ : Nonempty X\ns_nemp : s.Nonempty\nx : X\nx_F : x ∈ F\nm_pos : m > 0\ndyncover : (Fin n → ↥s) → X\nh_dyncover :\n ∀ (t : Fin n → ↥s),\n ⋂ k, T^[m * ↑k] ⁻¹' ba... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.TopologicalEntropy.CoverEntropy | {
"line": 197,
"column": 64
} | {
"line": 197,
"column": 75
} | {
"line": 197,
"column": 76
} | [
{
"pp": "X : Type u_1\nT : X → X\nU : SetRel X X\nF : Set X\ninst✝ : UniformSpace X\nF_comp : IsCompact F\nF_inv : MapsTo T F F\nU_uni : U ∈ 𝓤 X\nn : ℕ\nV : SetRel X X\nV_uni : V ∈ 𝓤 X\nV_symm : V.IsSymm\nV_U : V ○ V ⊆ U\ns : Finset X\nleft✝ : ∀ x ∈ s, x ∈ F\ns_cover : F ⊆ ⋃ x ∈ s, ball x V\n⊢ F ⊆ ⋃ y ∈ ↑s, b... | [
"X : Type u_1\nT : X → X\nU : SetRel X X\nF : Set X\ninst✝ : UniformSpace X\nF_comp : IsCompact F\nF_inv : MapsTo T F F\nU_uni : U ∈ 𝓤 X\nn : ℕ\nV : SetRel X X\nV_uni : V ∈ 𝓤 X\nV_symm : V.IsSymm\nV_U : V ○ V ⊆ U\ns : Finset X\nleft✝ : ∀ x ∈ s, x ∈ F\ns_cover : F ⊆ ⋃ x ∈ s, ball x V\n⊢ F ⊆ ⋃ y ∈ s, ball y V"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 464,
"column": 2
} | {
"line": 465,
"column": 30
} | {
"line": 466,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\n⊢ μ.rnDeriv ν =ᵐ[ν] fun x ↦ μ.rnDeriv (μ + ν) x / ν.rnDeriv (μ + ν) x",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"ENNReal.instAdd",
"Meas... | [
"α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\na : α\nha1 : ν.rnDeriv (μ + ν) a = (μ.rnDeriv ν a + 1)⁻¹\nha2 : μ.rnDeriv (μ + ν) a = μ.rnDeriv ν a / (μ.rnDeriv ν a + 1)\nha_lt_top : μ.rnDeriv ν a < ∞\n⊢ μ.rnDeriv ν a = μ.rnDeriv (μ + ν) a / ν.rnDeriv (μ + ν) a"... | filter_upwards [rnDeriv_add_self ν μ, rnDeriv_self_add μ ν, μ.rnDeriv_lt_top ν]
with a ha1 ha2 ha_lt_top | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 650,
"column": 7
} | {
"line": 650,
"column": 18
} | {
"line": 650,
"column": 19
} | [
{
"pp": "G : Type u_3\ninst✝⁶ : Group G\nmG : MeasurableSpace G\ninst✝⁵ : MeasurableMul₂ G\ninst✝⁴ : MeasurableInv G\nμ : Measure G\ninst✝³ : μ.IsMulLeftInvariant\ninst✝² : SFinite μ\nν₁ ν₂ : Measure G\ninst✝¹ : ν₁.HaveLebesgueDecomposition μ\ninst✝ : ν₂.HaveLebesgueDecomposition μ\nhν₁ : ν₁ ≪ μ\nhν₂ : ν₂ ≪ μ\n... | [
"G : Type u_3\ninst✝⁶ : Group G\nmG : MeasurableSpace G\ninst✝⁵ : MeasurableMul₂ G\ninst✝⁴ : MeasurableInv G\nμ : Measure G\ninst✝³ : μ.IsMulLeftInvariant\ninst✝² : SFinite μ\nν₁ ν₂ : Measure G\ninst✝¹ : ν₁.HaveLebesgueDecomposition μ\ninst✝ : ν₂.HaveLebesgueDecomposition μ\nhν₁ : ν₁ ≪ μ\nhν₂ : ν₂ ≪ μ\n⊢ ν₁ ∗ₘ ν₂ =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.Cardinality | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 28
} | {
"line": 66,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\nL : Type u\ninst✝⁴ : CommRing L\ninst✝³ : IsDomain L\ninst✝² : Algebra R L\ninst✝¹ : IsTorsionFree R L\ninst✝ : Algebra.IsAlgebraic R L\n⊢ #L ≤ #((p : R[X]) × { x // x ∈ p.aroots L })",
"ppTerm": "?m.33",
"assigned": false,
"usedConstant... | [
"R : Type u\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\nL : Type u\ninst✝⁴ : CommRing L\ninst✝³ : IsDomain L\ninst✝² : Algebra R L\ninst✝¹ : IsTorsionFree R L\ninst✝ : Algebra.IsAlgebraic R L\n⊢ #L ≤ #((p : R[X]) × { x // x ∈ p.aroots L })"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.Cardinality | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 28
} | {
"line": 72,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\nL : Type u\ninst✝⁴ : CommRing L\ninst✝³ : IsDomain L\ninst✝² : Algebra R L\ninst✝¹ : IsTorsionFree R L\ninst✝ : Algebra.IsAlgebraic R L\n⊢ #L ≤ max #R ℵ₀",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"R : Type u\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\nL : Type u\ninst✝⁴ : CommRing L\ninst✝³ : IsDomain L\ninst✝² : Algebra R L\ninst✝¹ : IsTorsionFree R L\ninst✝ : Algebra.IsAlgebraic R L\n⊢ #L ≤ max #R ℵ₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Classification | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 17
} | {
"line": 99,
"column": 18
} | [
{
"pp": "R : Type u\nK : Type v\ninst✝³ : CommRing R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsAlgClosed K\nι : Type w\nv : ι → K\nhv : IsTranscendenceBasis R v\nthis : IsAlgClosure (↥(Algebra.adjoin R (Set.range v))) K := isAlgClosure_of_transcendence_basis v hv\n⊢ Cardinal.lift.{max u w, v} #K ≤ Card... | [
"R : Type u\nK : Type v\ninst✝³ : CommRing R\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsAlgClosed K\nι : Type w\nv : ι → K\nhv : IsTranscendenceBasis R v\nthis : IsAlgClosure (↥(Algebra.adjoin R (Set.range v))) K := isAlgClosure_of_transcendence_basis v hv\n⊢ #K ≤ #↥(Algebra.adjoin R (Set.range v)) ∨ #K ≤ ℵ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Classification | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 13
} | {
"line": 115,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nK' : Type u\ninst✝² : Field K'\ninst✝¹ : Algebra R K'\ninst✝ : IsAlgClosed K'\nι' : Type u\nv' : ι' → K'\nhv : IsTranscendenceBasis R v'\n⊢ #K' ≤ max (max #R #ι') ℵ₀",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toS... | [
"R : Type u\ninst✝³ : CommRing R\nK' : Type u\ninst✝² : Field K'\ninst✝¹ : Algebra R K'\ninst✝ : IsAlgClosed K'\nι' : Type u\nv' : ι' → K'\nhv : IsTranscendenceBasis R v'\n⊢ (#K' ≤ #R ∨ #K' ≤ #ι') ∨ #K' ≤ ℵ₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Classification | {
"line": 130,
"column": 45
} | {
"line": 130,
"column": 56
} | {
"line": 130,
"column": 57
} | [
{
"pp": "R : Type u\nK : Type v\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsAlgClosed K\nι : Type w\nv : ι → K\ninst✝ : Nontrivial R\nhv : IsTranscendenceBasis R v\nhR : #R ≤ ℵ₀\nhK : ℵ₀ < #K\nh : Cardinal.lift.{max u v, w} #ι < ℵ₀\n⊢ Cardinal.lift.{max u v, w} #ι ≤ ℵ₀",
"ppTerm... | [
"R : Type u\nK : Type v\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsAlgClosed K\nι : Type w\nv : ι → K\ninst✝ : Nontrivial R\nhv : IsTranscendenceBasis R v\nhR : #R ≤ ℵ₀\nhK : ℵ₀ < #K\nh : Cardinal.lift.{max u v, w} #ι < ℵ₀\n⊢ #ι ≤ ℵ₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Classification | {
"line": 140,
"column": 29
} | {
"line": 140,
"column": 40
} | {
"line": 140,
"column": 41
} | [
{
"pp": "R : Type u\nK : Type v\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsAlgClosed K\nι : Type w\nv : ι → K\ninst✝ : Nontrivial R\nhv : IsTranscendenceBasis R v\nhR : #R ≤ ℵ₀\nhK : ℵ₀ < #K\nthis : ℵ₀ ≤ Cardinal.lift.{max u v, w} #ι\n⊢ Cardinal.lift.{max v w, u} #R ≤ ℵ₀",
"ppT... | [
"R : Type u\nK : Type v\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsAlgClosed K\nι : Type w\nv : ι → K\ninst✝ : Nontrivial R\nhv : IsTranscendenceBasis R v\nhR : #R ≤ ℵ₀\nhK : ℵ₀ < #K\nthis : ℵ₀ ≤ Cardinal.lift.{max u v, w} #ι\n⊢ #R ≤ ℵ₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Classification | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 13
} | {
"line": 151,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\nK' : Type u\ninst✝³ : Field K'\ninst✝² : Algebra R K'\ninst✝¹ : IsAlgClosed K'\nι' : Type u\nv' : ι' → K'\ninst✝ : Nontrivial R\nhv : IsTranscendenceBasis R v'\nhR : #R ≤ ℵ₀\nhK : ℵ₀ < #K'\n⊢ #K' = #ι'",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants":... | [
"R : Type u\ninst✝⁴ : CommRing R\nK' : Type u\ninst✝³ : Field K'\ninst✝² : Algebra R K'\ninst✝¹ : IsAlgClosed K'\nι' : Type u\nv' : ι' → K'\ninst✝ : Nontrivial R\nhv : IsTranscendenceBasis R v'\nhR : #R ≤ ℵ₀\nhK : ℵ₀ < #K'\n⊢ #K' = #ι'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.LanguageMap | {
"line": 423,
"column": 21
} | {
"line": 423,
"column": 39
} | {
"line": 423,
"column": 40
} | [
{
"pp": "L : Language\nL' : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type w'\nie : IsEmpty α\n⊢ ((LHom.id L).sumElim (LHom.ofIsEmpty (constantsOn α) L)).comp (L.lhomWithConstants α) = LHom.id L",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FirstOrder.Langua... | [
"L : Language\nL' : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type w'\nie : IsEmpty α\n⊢ ((LHom.id L).sumElim (LHom.ofIsEmpty (constantsOn α) L)).comp LHom.sumInl = LHom.id L"
] | lhomWithConstants, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Syntax | {
"line": 438,
"column": 4
} | {
"line": 438,
"column": 57
} | {
"line": 439,
"column": 4
} | [
{
"pp": "case rel\nL : Language\nα : Type u'\nk n✝¹ l✝ : ℕ\nR✝ : L.Relations l✝\nts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝¹)\nm✝ n✝ : ℕ\nkm✝ : n✝¹ ≤ m✝\nmn✝ : m✝ ≤ n✝\n⊢ rel R✝ (Term.relabel (Sum.map id (Fin.castLE mn✝)) ∘ Term.relabel (Sum.map id (Fin.castLE km✝)) ∘ ts✝) =\n rel R✝ (Term.relabel (Sum.map id (Fin.ca... | [
"case rel\nL : Language\nα : Type u'\nk n✝¹ l✝ : ℕ\nR✝ : L.Relations l✝\nts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝¹)\nm✝ n✝ : ℕ\nkm✝ : n✝¹ ≤ m✝\nmn✝ : m✝ ≤ n✝\n⊢ rel R✝ (Term.relabel (Sum.map id (Fin.castLE mn✝) ∘ Sum.map id (Fin.castLE km✝)) ∘ ts✝) =\n rel R✝ (Term.relabel (Sum.map id (Fin.castLE ⋯)) ∘ ts✝)"
] | rw [← Function.comp_assoc, Term.relabel_comp_relabel] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.Syntax | {
"line": 907,
"column": 6
} | {
"line": 907,
"column": 34
} | {
"line": 908,
"column": 6
} | [
{
"pp": "case refine_3\nL : Language\nα : Type u'\ns : Set α\ni j : α\n⊢ (∃ i_1, (∃ (x : i ∈ s), ⟨i, ⋯⟩ ∈ i_1) ∧ ∃ (x : j ∈ s), ⟨j, ⋯⟩ ∈ i_1) → i ∈ s ∧ j ∈ s",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Iff.of_eq",
"Finset",
"Membership.mem",
... | [
"case refine_3\nL : Language\nα : Type u'\ns : Set α\ni j : α\nt : Finset ↑s\nis : i ∈ s\nh✝¹ : ⟨i, ⋯⟩ ∈ t\njs : j ∈ s\nh✝ : ⟨j, ⋯⟩ ∈ t\n⊢ i ∈ s ∧ j ∈ s"
] | rintro ⟨t, ⟨is, _⟩, ⟨js, _⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.ModelTheory.Semantics | {
"line": 448,
"column": 39
} | {
"line": 448,
"column": 49
} | {
"line": 448,
"column": 49
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nv : β → M\nv' : α → M\nn✝ : ℕ\nt₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)\nf : ↥(equal t₁✝ t₂✝).freeVarFinset → β\nxs : Fin n✝ → M\nhv' : ∀ (a : ↥(equal t₁✝ t₂✝).freeVarFinset), v (f a) = v' ↑a\n⊢ ∀ (a : ↥t₁✝.v... | [] | simp [hv'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Semantics | {
"line": 448,
"column": 39
} | {
"line": 448,
"column": 49
} | {
"line": 448,
"column": 49
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nv : β → M\nv' : α → M\nn✝ : ℕ\nt₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)\nf : ↥(equal t₁✝ t₂✝).freeVarFinset → β\nxs : Fin n✝ → M\nhv' : ∀ (a : ↥(equal t₁✝ t₂✝).freeVarFinset), v (f a) = v' ↑a\n⊢ ∀ (a : ↥t₁✝.v... | [] | simp [hv'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Semantics | {
"line": 448,
"column": 39
} | {
"line": 448,
"column": 49
} | {
"line": 448,
"column": 49
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nv : β → M\nv' : α → M\nn✝ : ℕ\nt₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)\nf : ↥(equal t₁✝ t₂✝).freeVarFinset → β\nxs : Fin n✝ → M\nhv' : ∀ (a : ↥(equal t₁✝ t₂✝).freeVarFinset), v (f a) = v' ↑a\n⊢ ∀ (a : ↥t₁✝.v... | [] | simp [hv'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Semantics | {
"line": 448,
"column": 83
} | {
"line": 448,
"column": 93
} | {
"line": 448,
"column": 93
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nv : β → M\nv' : α → M\nn✝ : ℕ\nt₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)\nf : ↥(equal t₁✝ t₂✝).freeVarFinset → β\nxs : Fin n✝ → M\nhv' : ∀ (a : ↥(equal t₁✝ t₂✝).freeVarFinset), v (f a) = v' ↑a\n⊢ ∀ (a : ↥t₂✝.v... | [] | simp [hv'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Semantics | {
"line": 448,
"column": 83
} | {
"line": 448,
"column": 93
} | {
"line": 448,
"column": 93
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nv : β → M\nv' : α → M\nn✝ : ℕ\nt₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)\nf : ↥(equal t₁✝ t₂✝).freeVarFinset → β\nxs : Fin n✝ → M\nhv' : ∀ (a : ↥(equal t₁✝ t₂✝).freeVarFinset), v (f a) = v' ↑a\n⊢ ∀ (a : ↥t₂✝.v... | [] | simp [hv'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Semantics | {
"line": 448,
"column": 83
} | {
"line": 448,
"column": 93
} | {
"line": 448,
"column": 93
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nv : β → M\nv' : α → M\nn✝ : ℕ\nt₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)\nf : ↥(equal t₁✝ t₂✝).freeVarFinset → β\nxs : Fin n✝ → M\nhv' : ∀ (a : ↥(equal t₁✝ t₂✝).freeVarFinset), v (f a) = v' ↑a\n⊢ ∀ (a : ↥t₂✝.v... | [] | simp [hv'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Semantics | {
"line": 453,
"column": 39
} | {
"line": 453,
"column": 49
} | {
"line": 453,
"column": 49
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nv : β → M\nv' : α → M\nn✝ l✝ : ℕ\nR✝ : L.Relations l✝\nts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝)\nf : ↥(rel R✝ ts✝).freeVarFinset → β\nxs : Fin n✝ → M\nhv' : ∀ (a : ↥(rel R✝ ts✝).freeVarFinset), v (f a) =... | [] | simp [hv'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Semantics | {
"line": 453,
"column": 39
} | {
"line": 453,
"column": 49
} | {
"line": 453,
"column": 49
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nv : β → M\nv' : α → M\nn✝ l✝ : ℕ\nR✝ : L.Relations l✝\nts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝)\nf : ↥(rel R✝ ts✝).freeVarFinset → β\nxs : Fin n✝ → M\nhv' : ∀ (a : ↥(rel R✝ ts✝).freeVarFinset), v (f a) =... | [] | simp [hv'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Semantics | {
"line": 453,
"column": 39
} | {
"line": 453,
"column": 49
} | {
"line": 453,
"column": 49
} | [
{
"pp": "L : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nv : β → M\nv' : α → M\nn✝ l✝ : ℕ\nR✝ : L.Relations l✝\nts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝)\nf : ↥(rel R✝ ts✝).freeVarFinset → β\nxs : Fin n✝ → M\nhv' : ∀ (a : ↥(rel R✝ ts✝).freeVarFinset), v (f a) =... | [] | simp [hv'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Semantics | {
"line": 457,
"column": 22
} | {
"line": 457,
"column": 32
} | {
"line": 458,
"column": 2
} | [
{
"pp": "case imp\nL : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nv : β → M\nv' : α → M\nn✝ : ℕ\nf₁✝ f₂✝ : L.BoundedFormula α n✝\nih1 :\n ∀ {f : ↥f₁✝.freeVarFinset → β} {xs : Fin n✝ → M},\n (∀ (a : ↥f₁✝.freeVarFinset), v (f a) = v' ↑a) → ((f₁✝.restr... | [] | simp [hv'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Semantics | {
"line": 457,
"column": 22
} | {
"line": 457,
"column": 32
} | {
"line": 458,
"column": 2
} | [
{
"pp": "case imp\nL : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nv : β → M\nv' : α → M\nn✝ : ℕ\nf₁✝ f₂✝ : L.BoundedFormula α n✝\nih1 :\n ∀ {f : ↥f₁✝.freeVarFinset → β} {xs : Fin n✝ → M},\n (∀ (a : ↥f₁✝.freeVarFinset), v (f a) = v' ↑a) → ((f₁✝.restr... | [] | simp [hv'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.ElementaryMaps | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 16
} | {
"line": 200,
"column": 2
} | [
{
"pp": "L : Language\nM : Type u_1\nN : Type u_2\nP : Type u_3\nQ : Type u_4\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\ninst✝¹ : L.Structure P\ninst✝ : L.Structure Q\nf : M ↪ₑ[L] N\nA : Set M\n⊢ M ↪ₑ[L[[↑A]]] f.toEmbedding.withConstants A",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants"... | [
"L : Language\nM : Type u_1\nN : Type u_2\nP : Type u_3\nQ : Type u_4\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\ninst✝¹ : L.Structure P\ninst✝ : L.Structure Q\nf : M ↪ₑ[L] N\nA : Set M\n⊢ ∀ ⦃n : ℕ⦄ (φ : L[[↑A]].Formula (Fin n)) (x : Fin n → M), φ.Realize (⇑f ∘ x) ↔ φ.Realize x"
] | refine ⟨f, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.ModelTheory.Semantics | {
"line": 461,
"column": 14
} | {
"line": 461,
"column": 24
} | {
"line": 463,
"column": 0
} | [
{
"pp": "case all\nL : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nβ : Type v'\ninst✝ : DecidableEq α\nn : ℕ\nv : β → M\nv' : α → M\nn✝ : ℕ\nf✝ : L.BoundedFormula α (n✝ + 1)\nih3 :\n ∀ {f : ↥f✝.freeVarFinset → β} {xs : Fin (n✝ + 1) → M},\n (∀ (a : ↥f✝.freeVarFinset), v (f a) = v' ↑a) → ((f✝.r... | [] | simp [hv'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Substructures | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 18
} | {
"line": 344,
"column": 19
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\np : M → Prop\nx : M\ns : Set M\nhs : (closure L).toFun s = ⊤\nHs : ∀ x ∈ s, p x\nHfun : ∀ {n : ℕ} (f : L.Functions n), ClosedUnder f (setOf p)\nthis : ∀ x ∈ (closure L).toFun s, p x\n⊢ p x",
"ppTerm": "?m.37",
"assigned": false,
"usedConstant... | [
"L : Language\nM : Type w\ninst✝ : L.Structure M\np : M → Prop\nx : M\ns : Set M\nhs : (closure L).toFun s = ⊤\nHs : ∀ x ∈ s, p x\nHfun : ∀ {n : ℕ} (f : L.Functions n), ClosedUnder f (setOf p)\nthis : ∀ x ∈ (closure L).toFun s, p x\n⊢ p x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Substructures | {
"line": 393,
"column": 4
} | {
"line": 393,
"column": 55
} | {
"line": 393,
"column": 56
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nι : Type u_3\nhι : Nonempty ι\nS : ι → L.Substructure M\nhS : Directed (fun x1 x2 ↦ x1 ≤ x2) S\nx : M\nthis : x ∈ (closure L).toFun (⋃ i, ↑(S i)) → ∃ i, x ∈ S i\n⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i",
"ppTerm": "?m.60",
"assigned": false,
"usedConst... | [
"L : Language\nM : Type w\ninst✝ : L.Structure M\nι : Type u_3\nhι : Nonempty ι\nS : ι → L.Substructure M\nhS : Directed (fun x1 x2 ↦ x1 ≤ x2) S\nx : M\nthis : x ∈ (closure L).toFun (⋃ i, ↑(S i)) → ∃ i, x ∈ S i\n⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Encoding | {
"line": 244,
"column": 8
} | {
"line": 244,
"column": 54
} | {
"line": 245,
"column": 4
} | [
{
"pp": "case h\nL : Language\nα : Type u'\nl✝ : List ((n : ℕ) × L.BoundedFormula α n)\nn φ_n φ_l : ℕ\nφ_R : L.Relations φ_l\nts : Fin φ_l → L.Term (α ⊕ Fin φ_n)\nl : List ((k : ℕ) × L.Term (α ⊕ Fin k) ⊕ (n : ℕ) × L.Relations n ⊕ ℕ)\ni : Fin φ_l\n⊢ ↑i < (List.map (Sum.getLeft? ∘ fun i ↦ Sum.inl ⟨φ_n, ts i⟩) (fi... | [] | simp only [length_map, length_finRange, is_lt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Encoding | {
"line": 244,
"column": 8
} | {
"line": 244,
"column": 54
} | {
"line": 245,
"column": 4
} | [
{
"pp": "case h\nL : Language\nα : Type u'\nl✝ : List ((n : ℕ) × L.BoundedFormula α n)\nn φ_n φ_l : ℕ\nφ_R : L.Relations φ_l\nts : Fin φ_l → L.Term (α ⊕ Fin φ_n)\nl : List ((k : ℕ) × L.Term (α ⊕ Fin k) ⊕ (n : ℕ) × L.Relations n ⊕ ℕ)\ni : Fin φ_l\n⊢ ↑i < (List.map (Sum.getLeft? ∘ fun i ↦ Sum.inl ⟨φ_n, ts i⟩) (fi... | [] | simp only [length_map, length_finRange, is_lt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Encoding | {
"line": 244,
"column": 8
} | {
"line": 244,
"column": 54
} | {
"line": 245,
"column": 4
} | [
{
"pp": "case h\nL : Language\nα : Type u'\nl✝ : List ((n : ℕ) × L.BoundedFormula α n)\nn φ_n φ_l : ℕ\nφ_R : L.Relations φ_l\nts : Fin φ_l → L.Term (α ⊕ Fin φ_n)\nl : List ((k : ℕ) × L.Term (α ⊕ Fin k) ⊕ (n : ℕ) × L.Relations n ⊕ ℕ)\ni : Fin φ_l\n⊢ ↑i < (List.map (Sum.getLeft? ∘ fun i ↦ Sum.inl ⟨φ_n, ts i⟩) (fi... | [] | simp only [length_map, length_finRange, is_lt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Semantics | {
"line": 581,
"column": 6
} | {
"line": 581,
"column": 14
} | {
"line": 581,
"column": 15
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nβ : Type v'\nφ : L.Formula α\ng : α → β\nv : β → M\n⊢ (relabel g φ).Realize v ↔ φ.Realize (v ∘ g)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Inhabited.default",
"congrArg",
"Pi.un... | [
"L : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nβ : Type v'\nφ : L.Formula α\ng : α → β\nv : β → M\n⊢ BoundedFormula.Realize (relabel g φ) v default ↔ φ.Realize (v ∘ g)"
] | Realize, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Semantics | {
"line": 581,
"column": 15
} | {
"line": 581,
"column": 23
} | {
"line": 581,
"column": 24
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nβ : Type v'\nφ : L.Formula α\ng : α → β\nv : β → M\n⊢ BoundedFormula.Realize (relabel g φ) v default ↔ φ.Realize (v ∘ g)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Inhabited.default",
"... | [
"L : Language\nM : Type w\ninst✝ : L.Structure M\nα : Type u'\nβ : Type v'\nφ : L.Formula α\ng : α → β\nv : β → M\n⊢ BoundedFormula.Realize (relabel g φ) v default ↔ BoundedFormula.Realize φ (v ∘ g) default"
] | Realize, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Encoding | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 30
} | {
"line": 296,
"column": 2
} | [
{
"pp": "L : Language\nα : Type u'\n⊢ lift.{max (max u u') v, max (max u u') v} #((k : ℕ) × L.Term (α ⊕ Fin k) ⊕ (n : ℕ) × L.Relations n ⊕ ℕ) ≤\n max ℵ₀ (lift.{max (max u u') v, max (max u u') v} (lift.{max u v, u'} #α + lift.{u', max u v} L.card)) ∧\n ℵ₀ ≤ max ℵ₀ (lift.{max (max u u') v, max (max u u')... | [
"L : Language\nα : Type u'\n⊢ lift.{max (max u u') v, max (max u u') v} #((k : ℕ) × L.Term (α ⊕ Fin k) ⊕ (n : ℕ) × L.Relations n ⊕ ℕ) ≤\n max ℵ₀ (lift.{max (max u u') v, max (max u u') v} (lift.{max u v, u'} #α + lift.{u', max u v} L.card))"
] | refine ⟨?_, le_max_left _ _⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.ModelTheory.ElementarySubstructures | {
"line": 190,
"column": 11
} | {
"line": 190,
"column": 38
} | {
"line": 190,
"column": 39
} | [
{
"pp": "case e'_8\nL : Language\nM : Type u_1\ninst✝ : L.Structure M\nA : Set M\nhA : MeetsDefinable A\nn : ℕ\nφ : L.BoundedFormula Empty (n + 1)\nx : Fin n → ↥((closure L).toFun A)\na : M\nhφ : φ.Realize default (Fin.snoc (Subtype.val ∘ x) a)\nD : Set M := {y | φ.Realize default (Fin.snoc (Subtype.val ∘ x) y)... | [
"case e'_8.last\nL : Language\nM : Type u_1\ninst✝ : L.Structure M\nA : Set M\nhA : MeetsDefinable A\nn : ℕ\nφ : L.BoundedFormula Empty (n + 1)\nx : Fin n → ↥((closure L).toFun A)\na : M\nhφ : φ.Realize default (Fin.snoc (Subtype.val ∘ x) a)\nD : Set M := {y | φ.Realize default (Fin.snoc (Subtype.val ∘ x) y)}\nhD_n... | cases i using Fin.lastCases | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.ModelTheory.ElementarySubstructures | {
"line": 215,
"column": 4
} | {
"line": 216,
"column": 58
} | {
"line": 217,
"column": 8
} | [
{
"pp": "L : Language\nM : Type u_1\ninst✝ : L.Structure M\nS : L.ElementarySubstructure M\nD : Set M\nx : M\nhx : x ∈ D\nφ : L[[↑↑S]].Formula (Fin 1)\nhφ : {x | x 0 ∈ D} = setOf φ.Realize\nhφx : φ.Realize ![x]\nψ : L[[↑↑S]].Sentence := iExs (Fin 1) (relabel Sum.inr φ)\n⊢ M ⊨ ψ",
"ppTerm": "?m.53",
"ass... | [
"L : Language\nM : Type u_1\ninst✝ : L.Structure M\nS : L.ElementarySubstructure M\nD : Set M\nx : M\nhx : x ∈ D\nφ : L[[↑↑S]].Formula (Fin 1)\nhφ : {x | x 0 ∈ D} = setOf φ.Realize\nhφx : φ.Realize ![x]\nψ : L[[↑↑S]].Sentence := iExs (Fin 1) (relabel Sum.inr φ)\n⊢ ∃ i, φ.Realize i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Semantics | {
"line": 949,
"column": 2
} | {
"line": 949,
"column": 71
} | {
"line": 949,
"column": 72
} | [
{
"pp": "L : Language\nM : Type w\ninst✝³ : L.Structure M\nα : Type u'\ninst✝² : DecidableEq α\ninst✝¹ : L[[α]].Structure M\ninst✝ : (L.lhomWithConstants α).IsExpansionOn M\nφ : L.Formula α\nh : M ⊨ equivSentence φ\n⊢ Realize (BoundedFormula.restrictFreeVar φ id) fun a ↦ ↑(L.con ↑a)",
"ppTerm": "?m.27",
... | [
"L : Language\nM : Type w\ninst✝³ : L.Structure M\nα : Type u'\ninst✝² : DecidableEq α\ninst✝¹ : L[[α]].Structure M\ninst✝ : (L.lhomWithConstants α).IsExpansionOn M\nφ : L.Formula α\nh : M ⊨ equivSentence φ\n⊢ BoundedFormula.Realize φ (fun a ↦ ↑(L.con a)) default"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Semantics | {
"line": 961,
"column": 12
} | {
"line": 962,
"column": 15
} | {
"line": 962,
"column": 16
} | [
{
"pp": "L : Language\nM : Type w\ninst✝² : L.Structure M\nα : Type u'\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty M\nφ : L.Formula α\nv : α → M\nhv : M ⊨ equivSentence φ\n⊢ BoundedFormula.Realize φ v default",
"ppTerm": "?m.54",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"L : Language\nM : Type w\ninst✝² : L.Structure M\nα : Type u'\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty M\nφ : L.Formula α\nv : α → M\nhv : M ⊨ equivSentence φ\n⊢ BoundedFormula.Realize φ v default"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Semantics | {
"line": 970,
"column": 4
} | {
"line": 970,
"column": 57
} | {
"line": 971,
"column": 6
} | [
{
"pp": "case mpr\nL : Language\nM : Type w\ninst✝² : L.Structure M\nα : Type u'\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty M\nφ : L.Formula α\nh : M ⊨ φ.exClosure\nv : ↥(BoundedFormula.freeVarFinset φ) → M\nhv : Realize (BoundedFormula.restrictFreeVar φ id) v\nv' : α → M := fun a ↦ if hmem : a ∈ BoundedFormula.... | [
"case mpr\nL : Language\nM : Type w\ninst✝² : L.Structure M\nα : Type u'\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty M\nφ : L.Formula α\nh : M ⊨ φ.exClosure\nv : ↥(BoundedFormula.freeVarFinset φ) → M\nhv : Realize (BoundedFormula.restrictFreeVar φ id) v\nv' : α → M := fun a ↦ if hmem : a ∈ BoundedFormula.freeVarFinse... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Semantics | {
"line": 999,
"column": 6
} | {
"line": 999,
"column": 38
} | {
"line": 1000,
"column": 6
} | [
{
"pp": "case all.mpr\nL : Language\nM : Type w\nN : Type u_1\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\nα : Type u'\nn : ℕ\nF : Type u_4\ninst✝¹ : EquivLike F M N\ninst✝ : L.StrongHomClass F M N\ng : F\nv : α → M\nn✝ : ℕ\nf✝ : L.BoundedFormula α (n✝ + 1)\nih3 : ∀ {xs : Fin (n✝ + 1) → M}, f✝.Realize (⇑g ∘... | [
"case all.mpr\nL : Language\nM : Type w\nN : Type u_1\ninst✝³ : L.Structure M\ninst✝² : L.Structure N\nα : Type u'\nn : ℕ\nF : Type u_4\ninst✝¹ : EquivLike F M N\ninst✝ : L.StrongHomClass F M N\ng : F\nv : α → M\nn✝ : ℕ\nf✝ : L.BoundedFormula α (n✝ + 1)\nih3 : ∀ {xs : Fin (n✝ + 1) → M}, f✝.Realize (⇑g ∘ v) (⇑g ∘ xs... | have h' := h (EquivLike.inv g a) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.ModelTheory.ElementarySubstructures | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 18
} | {
"line": 233,
"column": 19
} | [
{
"pp": "L : Language\nM : Type u_1\ninst✝ : L.Structure M\nS : L.ElementarySubstructure M\nD : Set M\nx : M\nhx : x ∈ D\nφ : L[[↑↑S]].Formula (Fin 1)\nhφ : {x | x 0 ∈ D} = setOf φ.Realize\nhφx : φ.Realize ![x]\nψ : L[[↑↑S]].Sentence := iExs (Fin 1) (relabel Sum.inr φ)\nhψM : M ⊨ ψ\nv' : Fin 1 → ↥S\nhv' : φ.Rea... | [
"L : Language\nM : Type u_1\ninst✝ : L.Structure M\nS : L.ElementarySubstructure M\nD : Set M\nx : M\nhx : x ∈ D\nφ : L[[↑↑S]].Formula (Fin 1)\nhφ : {x | x 0 ∈ D} = setOf φ.Realize\nhφx : φ.Realize ![x]\nψ : L[[↑↑S]].Sentence := iExs (Fin 1) (relabel Sum.inr φ)\nhψM : M ⊨ ψ\nv' : Fin 1 → ↥S\nhv' : φ.Realize v'\nhv'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Semantics | {
"line": 1040,
"column": 90
} | {
"line": 1042,
"column": 67
} | {
"line": 1044,
"column": 0
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nr : L.Relations 2\n⊢ M ⊨ r.symmetric ↔ Std.Symm fun x y ↦ RelMap r ![x, y]",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"FirstOrder.Language.BoundedFormula.imp",
"FirstOrder.Language.Sentence.Realize",
"Eq.mpr",... | [] | by
rw [symm_def]
exact forall₂_congr fun _ _ ↦ imp_congr realize_rel₂ realize_rel₂ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Semantics | {
"line": 1076,
"column": 22
} | {
"line": 1076,
"column": 33
} | {
"line": 1076,
"column": 34
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nn : ℕ\nxs : Fin n → M\nh :\n ∀ (φ : L.BoundedFormula Empty n) (x x_1 : Fin n),\n (decide ¬x = x_1) = true → ∼(var (Sum.inr x) =' var (Sum.inr x_1)) = φ → φ.Realize default xs\ni j : Fin n\nij : i ≠ j\n⊢ (decide ¬i = j) = true",
"ppTerm": "?m.70",... | [
"L : Language\nM : Type w\ninst✝ : L.Structure M\nn : ℕ\nxs : Fin n → M\nh :\n ∀ (φ : L.BoundedFormula Empty n) (x x_1 : Fin n),\n (decide ¬x = x_1) = true → ∼(var (Sum.inr x) =' var (Sum.inr x_1)) = φ → φ.Realize default xs\ni j : Fin n\nij : i ≠ j\n⊢ ¬i = j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Semantics | {
"line": 1078,
"column": 4
} | {
"line": 1078,
"column": 15
} | {
"line": 1078,
"column": 16
} | [
{
"pp": "case refine_2\nL : Language\nM : Type w\ninst✝ : L.Structure M\nn : ℕ\nxs : Nonempty (Fin n ↪ M)\ni j : Fin n\nij : (decide ¬i = j) = true\n⊢ (∼(var (Sum.inr i) =' var (Sum.inr j))).Realize default ⇑xs.some",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Function.instEm... | [
"case refine_2\nL : Language\nM : Type w\ninst✝ : L.Structure M\nn : ℕ\nxs : Nonempty (Fin n ↪ M)\ni j : Fin n\nij : (decide ¬i = j) = true\n⊢ ¬i = j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Skolem | {
"line": 153,
"column": 6
} | {
"line": 153,
"column": 47
} | {
"line": 154,
"column": 6
} | [
{
"pp": "case refine_2\nL : Language\nM : Type w\ninst✝ : L.Structure M\ns : Set M\nκ : Cardinal.{w'}\nh3 : lift.{w', max u v} L.card ≤ lift.{max u v, w'} κ\nh4 : lift.{w, w'} κ ≤ lift.{w', w} #M\ns' : Set (ULift.{w', w} M)\nh2 : lift.{w', w} #↑s ≤ #↑s'\nh1 : ℵ₀ ≤ #↑s'\nhs' : #↑s' = lift.{w, w'} κ\nthis : Nonem... | [
"case refine_2\nL : Language\nM : Type w\ninst✝ : L.Structure M\ns : Set M\nκ : Cardinal.{w'}\nh3 : lift.{w', max u v} L.card ≤ lift.{max u v, w'} κ\nh4 : lift.{w, w'} κ ≤ lift.{w', w} #M\ns' : Set (ULift.{w', w} M)\nh2 : lift.{w', w} #↑s ≤ #↑s'\nh1 : ℵ₀ ≤ #↑s'\nhs' : #↑s' = lift.{w, w'} κ\nthis : Nonempty M\nh : l... | refine _root_.trans (lift_le.{w}.2 h3) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.ModelTheory.Definability | {
"line": 519,
"column": 2
} | {
"line": 523,
"column": 74
} | {
"line": 524,
"column": 2
} | [
{
"pp": "M : Type u_1\nL : Language\ninst✝¹ : L.Structure M\nA : Set M\nα : Type u_4\nβ : Type u_5\ninst✝ : Finite β\nF : (α → M) → β → M\nhF : DefinableMap L A F\nS : Set (β → M)\nhS : A.Definable L S\n⊢ A.Definable L (F ⁻¹' S)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"M : Type u_1\nL : Language\ninst✝¹ : L.Structure M\nA : Set M\nα : Type u_4\nβ : Type u_5\ninst✝ : Finite β\nF : (α → M) → β → M\nhF : DefinableMap L A F\nS : Set (β → M)\nhS : A.Definable L S\nh_graph : A.Definable L {w | ∀ (i : β), F (w ∘ Sum.inl) i = w (Sum.inr i)}\n⊢ A.Definable L (F ⁻¹' S)"
] | have h_graph : A.Definable L { w : α ⊕ β → M | ∀ i, F (w ∘ Sum.inl) i = w (Sum.inr i) } := by
rw [setOf_forall]
refine definable_iInter_of_finite fun i => ?_
simpa [tupleGraph] using!
(hF i).preimage_comp (fun | none => Sum.inr i | some j => Sum.inl j) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.MvPolynomial.FreeCommRing | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 64
} | {
"line": 68,
"column": 65
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nR : Type u_3\ninst✝² : DecidableEq κ\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nmonoms : ι → Finset (κ →₀ ℕ)\np : { p // ∀ (i : ι), (p i).support ⊆ monoms i }\ni : ι\nm : κ →₀ ℕ\nhm : m ∉ monoms i\nthis : m ∉ (↑p i).support\n⊢ 0 = (↑p i).coeff m",
"ppTerm": "?m.117",
... | [
"ι : Type u_1\nκ : Type u_2\nR : Type u_3\ninst✝² : DecidableEq κ\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nmonoms : ι → Finset (κ →₀ ℕ)\np : { p // ∀ (i : ι), (p i).support ⊆ monoms i }\ni : ι\nm : κ →₀ ℕ\nhm : m ∉ monoms i\nthis : m ∉ (↑p i).support\n⊢ 0 = (↑p i).coeff m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Satisfiability | {
"line": 159,
"column": 6
} | {
"line": 160,
"column": 57
} | {
"line": 162,
"column": 0
} | [
{
"pp": "L : Language\nα : Type w\nT : L.Theory\ns : Set α\nM : Type w'\ninst✝² : L.Structure M\ninst✝¹ : M ⊨ T\ninst✝ : Infinite M\n⊢ Monotone fun i ↦ (fun a ↦ ↑a) '' ↑i",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice... | [] | exact Monotone.comp (g := Set.image ((↑) : s → α)) (f := ((↑) : Finset s → Set s))
Set.monotone_image fun _ _ => Finset.coe_subset.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.ModelTheory.Satisfiability | {
"line": 379,
"column": 4
} | {
"line": 379,
"column": 15
} | {
"line": 379,
"column": 16
} | [
{
"pp": "case mp\nL : Language\nT : L.Theory\nφ : L.Sentence\nthis : DecidableEq L.Sentence := Classical.decEq L.Sentence\nh : ∀ (T0 : Finset L.Sentence), ↑T0 ⊆ T ∪ {Formula.not φ} → IsSatisfiable ↑T0\nT0 : Finset L.Sentence\nhT0 : ↑T0 ⊆ T\n⊢ (↑T0 ∪ {Formula.not φ}).IsSatisfiable",
"ppTerm": "?mp",
"ass... | [
"case mp\nL : Language\nT : L.Theory\nφ : L.Sentence\nthis : DecidableEq L.Sentence := Classical.decEq L.Sentence\nh : ∀ (T0 : Finset L.Sentence), ↑T0 ⊆ T ∪ {Formula.not φ} → IsSatisfiable ↑T0\nT0 : Finset L.Sentence\nhT0 : ↑T0 ⊆ T\n⊢ IsSatisfiable (insert (Formula.not φ) ↑T0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Satisfiability | {
"line": 381,
"column": 8
} | {
"line": 382,
"column": 60
} | {
"line": 382,
"column": 60
} | [
{
"pp": "L : Language\nT : L.Theory\nφ : L.Sentence\nthis : DecidableEq L.Sentence := Classical.decEq L.Sentence\nh : ∀ (T0 : Finset L.Sentence), ↑T0 ⊆ T ∪ {Formula.not φ} → IsSatisfiable ↑T0\nT0 : Finset L.Sentence\nhT0 : ↑T0 ⊆ T\n⊢ ↑(T0 ∪ {Formula.not φ}) ⊆ T ∪ {Formula.not φ}",
"ppTerm": "?m.41",
"as... | [] | simp only [Finset.coe_union, Finset.coe_singleton]
exact Set.union_subset_union hT0 (Set.Subset.refl _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Satisfiability | {
"line": 381,
"column": 8
} | {
"line": 382,
"column": 60
} | {
"line": 382,
"column": 60
} | [
{
"pp": "L : Language\nT : L.Theory\nφ : L.Sentence\nthis : DecidableEq L.Sentence := Classical.decEq L.Sentence\nh : ∀ (T0 : Finset L.Sentence), ↑T0 ⊆ T ∪ {Formula.not φ} → IsSatisfiable ↑T0\nT0 : Finset L.Sentence\nhT0 : ↑T0 ⊆ T\n⊢ ↑(T0 ∪ {Formula.not φ}) ⊆ T ∪ {Formula.not φ}",
"ppTerm": "?m.41",
"as... | [] | simp only [Finset.coe_union, Finset.coe_singleton]
exact Set.union_subset_union hT0 (Set.Subset.refl _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Satisfiability | {
"line": 385,
"column": 10
} | {
"line": 385,
"column": 21
} | {
"line": 385,
"column": 22
} | [
{
"pp": "L : Language\nT : L.Theory\nφ : L.Sentence\nthis : DecidableEq L.Sentence := Classical.decEq L.Sentence\nh : ∀ (T0 : Finset L.Sentence), ↑T0 ⊆ T → (↑T0 ∪ {Formula.not φ}).IsSatisfiable\nT0 : Finset L.Sentence\nhT0 : ↑T0 ⊆ T ∪ {Formula.not φ}\n⊢ ↑(T0.erase (Formula.not φ)) ⊆ T",
"ppTerm": "?m.72",
... | [
"L : Language\nT : L.Theory\nφ : L.Sentence\nthis : DecidableEq L.Sentence := Classical.decEq L.Sentence\nh : ∀ (T0 : Finset L.Sentence), ↑T0 ⊆ T → (↑T0 ∪ {Formula.not φ}).IsSatisfiable\nT0 : Finset L.Sentence\nhT0 : ↑T0 ⊆ T ∪ {Formula.not φ}\n⊢ ↑T0 ⊆ insert (Formula.not φ) T"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Algebra.Field.IsAlgClosed | {
"line": 197,
"column": 29
} | {
"line": 197,
"column": 67
} | {
"line": 197,
"column": 68
} | [
{
"pp": "φ : Language.ring.Sentence\nT0 : Finset Language.ring.Sentence\nhT0 : ↑T0 ⊆ Theory.ACF 0\nh : ↑T0 ⊨ᵇ φ\np : ℕ\nhp : p ∈ {q | Nat.Prime q}\nq : ℕ\nproperty✝ : Nat.Prime q\nhq : ⟨q, property✝⟩ ∉ {⟨p, hp⟩}\nK : Type\nstruc✝ : Language.ring.Structure K\nis_model✝ : K ⊨ Theory.ACF ↑⟨q, property✝⟩\nnonempty'... | [
"φ : Language.ring.Sentence\nT0 : Finset Language.ring.Sentence\nhT0 : ↑T0 ⊆ Theory.ACF 0\nh : ↑T0 ⊨ᵇ φ\np : ℕ\nhp : p ∈ {q | Nat.Prime q}\nq : ℕ\nproperty✝ : Nat.Prime q\nhq : ⟨q, property✝⟩ ∉ {⟨p, hp⟩}\nK : Type\nstruc✝ : Language.ring.Structure K\nis_model✝ : K ⊨ Theory.ACF ↑⟨q, property✝⟩\nnonempty'✝ : Nonempty... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Algebra.Field.IsAlgClosed | {
"line": 225,
"column": 2
} | {
"line": 226,
"column": 79
} | {
"line": 227,
"column": 4
} | [
{
"pp": "φ : Language.ring.Sentence\n⊢ ¬Theory.ACF 0 ⊨ᵇ φ → ¬{p | Theory.ACF ↑p ⊨ᵇ φ}.Infinite",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FirstOrder.Language.ring",
"Nat.Prime",
"setOf",
"FirstOrder.Language.Theory.ACF",
"FirstOrder.Langu... | [
"φ : Language.ring.Sentence\n⊢ ¬Theory.ACF 0 ⊨ᵇ φ → {p | Theory.ACF ↑p ⊨ᵇ φ}.Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.AxGrothendieck | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 43
} | {
"line": 87,
"column": 44
} | [
{
"pp": "case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nS : Set (ι → R)\nhm : Set.MapsTo (fun v i ↦ (eval v) (ps i)) S S\nhinj : Set.InjOn (fun v i ↦... | [
"case intro\nι : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Finite K\ninst✝² : CommRing R\ninst✝¹ : Finite ι\ninst✝ : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : ι → MvPolynomial ι R\nS : Set (ι → R)\nhm : Set.MapsTo (fun v i ↦ (eval v) (ps i)) S S\nhinj : Set.InjOn (fun v i ↦ (eval v) (p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.AxGrothendieck | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 49
} | {
"line": 166,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝³ : Finite α\nK : Type u_3\ninst✝² : Field K\ninst✝¹ : CompatibleRing K\ninst✝ : Finite ι\nφ : ring.Formula (α ⊕ ι)\nmons : ι → Finset (ι →₀ ℕ)\ninjOnAlt : ∀ {S : Set (ι → K)} (f : (ι → K) → ι → K), Set.InjOn f S ↔ ∀ (x y : ι → K), x ∈ S → y ∈ S → f x = f y → x = y\n⊢ (... | [
"ι : Type u_1\nα : Type u_2\ninst✝³ : Finite α\nK : Type u_3\ninst✝² : Field K\ninst✝¹ : CompatibleRing K\ninst✝ : Finite ι\nφ : ring.Formula (α ⊕ ι)\nmons : ι → Finset (ι →₀ ℕ)\ninjOnAlt : ∀ {S : Set (ι → K)} (f : (ι → K) → ι → K), Set.InjOn f S ↔ ∀ (x y : ι → K), x ∈ S → y ∈ S → f x = f y → x = y\n⊢ (∀ (fa : ⋯) (... | simp +singlePass only [← Sum.elim_comp_inl_inr] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Localization.Cardinality | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 13
} | {
"line": 49,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝³ : CommSemiring R\nL : Type u\ninst✝² : CommSemiring L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\n⊢ #L ≤ #R",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝³ : CommSemiring R\nL : Type u\ninst✝² : CommSemiring L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\n⊢ #L ≤ #R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Cardinality | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 13
} | {
"line": 79,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nL : Type u\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\nhS : S ≤ R⁰\n⊢ #L = #R",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝³ : CommRing R\nL : Type u\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\nhS : S ≤ R⁰\n⊢ #L = #R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Cardinality | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 36
} | {
"line": 81,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nL : Type u\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\nhS : S ≤ R⁰\n⊢ #L = #R",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"IsLocalization.lift_cardinalMk",
"Cardinal",
"congrArg",
... | [] | simpa using lift_cardinalMk L S hS | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Localization.Cardinality | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 36
} | {
"line": 81,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nL : Type u\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\nhS : S ≤ R⁰\n⊢ #L = #R",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"IsLocalization.lift_cardinalMk",
"Cardinal",
"congrArg",
... | [] | simpa using lift_cardinalMk L S hS | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Cardinality | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 36
} | {
"line": 81,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nL : Type u\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\nS : Submonoid R\ninst✝ : IsLocalization S L\nhS : S ≤ R⁰\n⊢ #L = #R",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"IsLocalization.lift_cardinalMk",
"Cardinal",
"congrArg",
... | [] | simpa using lift_cardinalMk L S hS | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.OreLocalization.Cardinality | {
"line": 92,
"column": 2
} | {
"line": 107,
"column": 7
} | {
"line": 109,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type v\ninst✝ : MulAction R X\nhc : ∀ (s s' : ↥S), Commute s s'\n⊢ #(OreLocalization S X) ≤ lift.{u, v} #X",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"OreLocalization.oreDiv_eq_iff",
... | [] | rcases finite_or_infinite X with _ | _
· have := lift_mk_le_lift_mk_of_surjective (oreDiv_one_surjective_of_finite_right S X)
rwa [lift_umax.{v, u}, lift_id'] at this
have key (x : X) (s s' : S) (h : s • x = s' • x) (hc : Commute s s') : x /ₒ s = x /ₒ s' := by
rw [oreDiv_eq_iff]
refine ⟨s, s'.1, h, ?_⟩
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.OreLocalization.Cardinality | {
"line": 92,
"column": 2
} | {
"line": 107,
"column": 7
} | {
"line": 109,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type v\ninst✝ : MulAction R X\nhc : ∀ (s s' : ↥S), Commute s s'\n⊢ #(OreLocalization S X) ≤ lift.{u, v} #X",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"OreLocalization.oreDiv_eq_iff",
... | [] | rcases finite_or_infinite X with _ | _
· have := lift_mk_le_lift_mk_of_surjective (oreDiv_one_surjective_of_finite_right S X)
rwa [lift_umax.{v, u}, lift_id'] at this
have key (x : X) (s s' : S) (h : s • x = s' • x) (hc : Commute s s') : x /ₒ s = x /ₒ s' := by
rw [oreDiv_eq_iff]
refine ⟨s, s'.1, h, ?_⟩
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Divisibility | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 96
} | {
"line": 92,
"column": 4
} | [
{
"pp": "a : Cardinal.{u}\nha : ℵ₀ ≤ a\nh : ∀ ⦃a_1 b : Cardinal.{u}⦄, a = a_1 * b → IsUnit a_1 ∨ IsUnit b\n⊢ False",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : Cardinal.{u}\nha : ℵ₀ ≤ a\nh : ∀ ⦃a_1 b : Cardinal.{u}⦄, a = a_1 * b → IsUnit a_1 ∨ IsUnit b\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.AxGrothendieck | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 13
} | {
"line": 247,
"column": 14
} | [
{
"pp": "K : Type u_1\nι : Type u_2\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : Finite ι\np : ι → MvPolynomial ι K\n⊢ (Function.Injective fun v i ↦ (eval v) (p i)) → Function.Surjective fun v i ↦ (eval v) (p i)",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"K : Type u_1\nι : Type u_2\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : Finite ι\np : ι → MvPolynomial ι K\n⊢ (Function.Injective fun v i ↦ (eval v) (p i)) → Function.Surjective fun v i ↦ (eval v) (p i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Cardinality | {
"line": 68,
"column": 4
} | {
"line": 69,
"column": 56
} | {
"line": 69,
"column": 57
} | [
{
"pp": "case inl\nα : Type u\nh : Fintype α\n⊢ Nonempty (Field α) ↔ ℵ₀ ≤ #α ∨ ∃ n, #α = ↑n ∧ IsPrimePow n",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"eq_false",
"Cardinal",
"congrArg",
... | [
"case inl\nα : Type u\nh : Fintype α\n⊢ Nonempty (Field α) ↔ IsPrimePow ‖α‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Cardinality | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 68
} | {
"line": 70,
"column": 69
} | [
{
"pp": "case inr\nα : Type u\nh : Infinite α\n⊢ Nonempty (Field α) ↔ ℵ₀ ≤ #α ∨ ∃ n, #α = ↑n ∧ IsPrimePow n",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.FieldTheory.Cardinality.0.Field.nonempty_iff._simp_1_3",
"Cardinal",
"congrArg",
... | [
"case inr\nα : Type u\nh : Infinite α\n⊢ Nonempty (Field α)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Differential.Liouville | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 57
} | {
"line": 93,
"column": 4
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝⁸ : Field F\ninst✝⁷ : Field K\ninst✝⁶ : Differential F\ninst✝⁵ : Differential K\ninst✝⁴ : Algebra F K\ninst✝³ : DifferentialAlgebra F K\ninst✝² : CharZero F\nB : IntermediateField F K\ninst✝¹ : FiniteDimensional F ↥B\ninst : IsLiouville F K\na : F\nι : Type\ninst✝ : Fin... | [
"F : Type u_1\nK : Type u_2\ninst✝⁸ : Field F\ninst✝⁷ : Field K\ninst✝⁶ : Differential F\ninst✝⁵ : Differential K\ninst✝⁴ : Algebra F K\ninst✝³ : DifferentialAlgebra F K\ninst✝² : CharZero F\nB : IntermediateField F K\ninst✝¹ : FiniteDimensional F ↥B\ninst : IsLiouville F K\na : F\nι : Type\ninst✝ : Fintype ι\nc : ... | apply inst.isLiouville a ι c hc (B.val ∘ u) (B.val v) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.FieldTheory.Differential.Liouville | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 89
} | {
"line": 117,
"column": 90
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝¹² : Field F\ninst✝¹¹ : Field K\ninst✝¹⁰ : Differential F\ninst✝⁹ : Differential K\ninst✝⁸ : Algebra F K\ninst✝⁷ : DifferentialAlgebra F K\ninst✝⁶ : CharZero F\nK' : Type u_3\ninst✝⁵ : Field K'\ninst✝⁴ : Differential K'\ninst✝³ : Algebra F K'\ninst✝² : DifferentialAlgeb... | [
"F : Type u_1\nK : Type u_2\ninst✝¹² : Field F\ninst✝¹¹ : Field K\ninst✝¹⁰ : Differential F\ninst✝⁹ : Differential K\ninst✝⁸ : Algebra F K\ninst✝⁷ : DifferentialAlgebra F K\ninst✝⁶ : CharZero F\nK' : Type u_3\ninst✝⁵ : Field K'\ninst✝⁴ : Differential K'\ninst✝³ : Algebra F K'\ninst✝² : DifferentialAlgebra F K'\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Extension | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 13
} | {
"line": 150,
"column": 14
} | [
{
"pp": "k : Type u_1\ninst✝⁶ : Field k\ninst✝⁵ : Finite k\np : ℕ\ninst✝⁴ : Fact (Nat.Prime p)\ninst✝³ : CharP k p\nl : Type u_2\ninst✝² : Field l\ninst✝¹ : Algebra k l\ninst✝ : Finite l\ng : Gal(l/k)\nn : ℕ := Module.finrank k l\nthis : NeZero n\ni : ℕ\nleft✝ : i < n\nhi : Extension.frob k p n ^ i = (algEquivE... | [
"k : Type u_1\ninst✝⁶ : Field k\ninst✝⁵ : Finite k\np : ℕ\ninst✝⁴ : Fact (Nat.Prime p)\ninst✝³ : CharP k p\nl : Type u_2\ninst✝² : Field l\ninst✝¹ : Algebra k l\ninst✝ : Finite l\ng : Gal(l/k)\nn : ℕ := Module.finrank k l\nthis : NeZero n\ni : ℕ\nleft✝ : i < n\nhi : Extension.frob k p n ^ i = (algEquivExtension k p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Extension | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 33
} | {
"line": 176,
"column": 2
} | [
{
"pp": "case inr.inl.refine_1\nK : Type u_2\ninst✝ : Field K\nn : ℕ\nf : K[X]\nhi : Irreducible f\nh : f ∣ X ^ Nat.card K ^ n - X\nhn : n ≠ 0\nh✝ : Finite K\np : ℕ\nhp : CharP K p\nthis✝ : Fact (Nat.Prime p)\nthis : NeZero n\n⊢ (Polynomial.map (algebraMap K (Extension K p n)) (X ^ Nat.card K ^ n - X)).Splits",... | [] | apply IsSplittingField.splits | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.FieldTheory.Finite.Extension | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 33
} | {
"line": 176,
"column": 2
} | [
{
"pp": "case inr.inl.refine_1\nK : Type u_2\ninst✝ : Field K\nn : ℕ\nf : K[X]\nhi : Irreducible f\nh : f ∣ X ^ Nat.card K ^ n - X\nhn : n ≠ 0\nh✝ : Finite K\np : ℕ\nhp : CharP K p\nthis✝ : Fact (Nat.Prime p)\nthis : NeZero n\n⊢ (Polynomial.map (algebraMap K (Extension K p n)) (X ^ Nat.card K ^ n - X)).Splits",... | [] | apply IsSplittingField.splits | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Finite.Extension | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 33
} | {
"line": 176,
"column": 2
} | [
{
"pp": "case inr.inl.refine_1\nK : Type u_2\ninst✝ : Field K\nn : ℕ\nf : K[X]\nhi : Irreducible f\nh : f ∣ X ^ Nat.card K ^ n - X\nhn : n ≠ 0\nh✝ : Finite K\np : ℕ\nhp : CharP K p\nthis✝ : Fact (Nat.Prime p)\nthis : NeZero n\n⊢ (Polynomial.map (algebraMap K (Extension K p n)) (X ^ Nat.card K ^ n - X)).Splits",... | [] | apply IsSplittingField.splits | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 138,
"column": 6
} | {
"line": 138,
"column": 91
} | {
"line": 140,
"column": 0
} | [
{
"pp": "case intro.refine_2\nK : Type u_1\nσ : Type u_2\ninst✝² : Field K\ninst✝¹ : Fintype K\ninst✝ : Finite σ\nval✝ : Fintype σ\ne : (σ → K) → K\nx✝ : e ∈ ⊤\nn : σ → K\n⊢ e n * (eval n) (indicator n) = e n",
"ppTerm": "?intro.refine_2",
"assigned": true,
"usedConstants": [
"MvPolynomial.eva... | [] | aesop (add simp [eval_indicator_apply_eq_zero, eval_indicator_apply_eq_one, eq_comm]) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 138,
"column": 6
} | {
"line": 138,
"column": 91
} | {
"line": 140,
"column": 0
} | [
{
"pp": "case intro.refine_2.h₀\nK : Type u_1\nσ : Type u_2\ninst✝² : Field K\ninst✝¹ : Fintype K\ninst✝ : Finite σ\nval✝ : Fintype σ\ne : (σ → K) → K\nx✝ : e ∈ ⊤\nn : σ → K\n⊢ ∀ b ∈ Finset.univ, b ≠ n → e b * (eval n) (indicator b) = 0",
"ppTerm": "?intro.refine_2.h₀",
"assigned": true,
"usedConsta... | [] | aesop (add simp [eval_indicator_apply_eq_zero, eval_indicator_apply_eq_one, eq_comm]) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 138,
"column": 6
} | {
"line": 138,
"column": 91
} | {
"line": 140,
"column": 0
} | [
{
"pp": "case intro.refine_2.h₁\nK : Type u_1\nσ : Type u_2\ninst✝² : Field K\ninst✝¹ : Fintype K\ninst✝ : Finite σ\nval✝ : Fintype σ\ne : (σ → K) → K\nx✝ : e ∈ ⊤\nn : σ → K\n⊢ n ∉ Finset.univ → e n * (eval n) (indicator n) = 0",
"ppTerm": "?intro.refine_2.h₁",
"assigned": true,
"usedConstants": [
... | [] | aesop (add simp [eval_indicator_apply_eq_zero, eval_indicator_apply_eq_one, eq_comm]) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.FieldTheory.Finite.Polynomial | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 27
} | {
"line": 204,
"column": 28
} | [
{
"pp": "case intro\nσ K : Type u\ninst✝² : Fintype K\ninst✝¹ : Field K\ninst✝ : Finite σ\nval✝ : Fintype σ\n⊢ Module.rank K (R σ K) < ℵ₀",
"ppTerm": "?intro",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"congrArg",
"AddCommGroup.toAddCom... | [
"case intro\nσ K : Type u\ninst✝² : Fintype K\ninst✝¹ : Field K\ninst✝ : Finite σ\nval✝ : Fintype σ\n⊢ ↑(Fintype.card (σ → K)) < ℵ₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Connected | {
"line": 27,
"column": 6
} | {
"line": 27,
"column": 72
} | {
"line": 27,
"column": 72
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nh : TotallyDisconnectedSpace X\nx : X\n⊢ IsClosed[inst✝] {x}",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"totallyDisconnectedSpace_iff_connectedComponent_singleton",
"congrArg",
"Set.instSingletonSet",... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nh : TotallyDisconnectedSpace X\nx : X\n⊢ IsClosed[inst✝] (connectedComponent x)"
] | ← totallyDisconnectedSpace_iff_connectedComponent_singleton.mp h x | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Differential.Liouville | {
"line": 187,
"column": 6
} | {
"line": 193,
"column": 30
} | {
"line": 195,
"column": 0
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝⁹ : Field F\ninst✝⁸ : Field K\ninst✝⁷ : Differential F\ninst✝⁶ : Differential K\ninst✝⁵ : Algebra F K\ninst✝⁴ : DifferentialAlgebra F K\ninst✝³ : CharZero F\ninst✝² : FiniteDimensional F K\ninst✝¹ : IsGalois F K\na : F\nι : Type\ninst✝ : Fintype ι\nc : ι → F\nhc : ∀ (x ... | [] | · rcongr e
apply_fun e at h
simp only [AlgEquiv.commutes, map_add, map_sum, map_mul] at h
convert! h using 2
· rcongr x
simp [logDeriv, algEquiv_deriv']
· rw [algEquiv_deriv'] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.Galois.NormalBasis | {
"line": 54,
"column": 35
} | {
"line": 54,
"column": 64
} | {
"line": 54,
"column": 64
} | [
{
"pp": "case e'_4\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Finite L\nthis✝ : Finite K\nthis : Fintype K\nx : L\nhx :\n Ideal.span {X ^ finrank K L - 1} =\n (toSpanSingleton K[X] (AEval' (frobeniusAlgHom K L).toLinearMap)\n ((AEval'.of (frobeniusA... | [
"case e'_4\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Finite L\nthis✝ : Finite K\nthis : Fintype K\nx : L\nhx :\n Ideal.span {X ^ finrank K L - 1} =\n (toSpanSingleton K[X] (AEval' (frobeniusAlgHom K L).toLinearMap)\n ((AEval'.of (frobeniusAlgHom K L).t... | LinearMap.coe_restrictScalars | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.CardinalEmb | {
"line": 137,
"column": 8
} | {
"line": 138,
"column": 83
} | {
"line": 139,
"column": 8
} | [
{
"pp": "case inl\nF : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nrank_inf : Fact (ℵ₀ ≤ Module.rank F E)\ninst✝ : Algebra.IsAlgebraic F E\ni : (Module.rank F E).ord.ToType\nih : (y : (Module.rank F E).ord.ToType) → y < i → (Module.rank F E).ord.ToType\ns : Set E := failed to p... | [
"case inl\nF : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nrank_inf : Fact (ℵ₀ ≤ Module.rank F E)\ninst✝ : Algebra.IsAlgebraic F E\ni : (Module.rank F E).ord.ToType\nih : (y : (Module.rank F E).ord.ToType) → y < i → (Module.rank F E).ord.ToType\ns : Set E := failed to pretty print ... | have : FiniteDimensional F (adjoin F s) :=
finiteDimensional_adjoin fun x _ ↦ (IsAlgebraic.isAlgebraic x).isIntegral | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.CardinalEmb | {
"line": 149,
"column": 70
} | {
"line": 151,
"column": 71
} | {
"line": 153,
"column": 0
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nrank_inf : Fact (ℵ₀ ≤ Module.rank F E)\ninst✝ : Algebra.IsAlgebraic F E\ni : (Module.rank F E).ord.ToType\n⊢ IsLeast {k | b k ∉ adjoin F (⇑b ∘ φ '' Iio i)} (φ i)",
"ppTerm": "?m.50",
"assigned": true,
"usedCon... | [] | by
rw [image_eq_range, leastExt, wellFounded_lt.fix_eq]
exact ⟨wellFounded_lt.min_mem _ _, fun _ ↦ (wellFounded_lt.min_le ·)⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Galois.Profinite | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 23
} | {
"line": 316,
"column": 24
} | [
{
"pp": "case h\nk : Type u_3\nK : Type u_4\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nH : Set Gal(K/k)\nL : FiniteGaloisIntermediateField k K\nle : ↑L.fixingSubgroup ⊆ H\n⊢ (fun a ↦ (mulEquivToLimit k K) a) '' ↑L.fixingSubgroup ⊆ (fun a ↦ (mulEquivToLimit k K).toEquiv a) '... | [
"case h\nk : Type u_3\nK : Type u_4\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nH : Set Gal(K/k)\nL : FiniteGaloisIntermediateField k K\nle : ↑L.fixingSubgroup ⊆ H\n⊢ ↑L.fixingSubgroup ⊆ (fun a ↦ (mulEquivToLimit k K) a) ⁻¹' (fun a ↦ (mulEquivToLimit k K) a) '' H"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PerfectClosure | {
"line": 352,
"column": 17
} | {
"line": 353,
"column": 38
} | {
"line": 353,
"column": 38
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ × K\nn : ℕ\nih : mk K p x ^ n = mk K p (x.1, x.2 ^ n)\n⊢ (⇑(frobenius K p))^[(x.1, x.2 ^ n * x.2).1 + 0]\n ((x.1, x.2 ^ n).1 + x.1,\n (⇑(frobenius K p))^[x.1] (x.1, x.2 ^ n).2 * (⇑(frobenius K p)... | [] | simp_rw [iterate_frobenius, add_zero, mul_pow, ← pow_mul,
← pow_add, mul_assoc, ← pow_add] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.FieldTheory.PerfectClosure | {
"line": 352,
"column": 17
} | {
"line": 353,
"column": 38
} | {
"line": 353,
"column": 38
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ × K\nn : ℕ\nih : mk K p x ^ n = mk K p (x.1, x.2 ^ n)\n⊢ (⇑(frobenius K p))^[(x.1, x.2 ^ n * x.2).1 + 0]\n ((x.1, x.2 ^ n).1 + x.1,\n (⇑(frobenius K p))^[x.1] (x.1, x.2 ^ n).2 * (⇑(frobenius K p)... | [] | simp_rw [iterate_frobenius, add_zero, mul_pow, ← pow_mul,
← pow_add, mul_assoc, ← pow_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.PerfectClosure | {
"line": 352,
"column": 17
} | {
"line": 353,
"column": 38
} | {
"line": 353,
"column": 38
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ × K\nn : ℕ\nih : mk K p x ^ n = mk K p (x.1, x.2 ^ n)\n⊢ (⇑(frobenius K p))^[(x.1, x.2 ^ n * x.2).1 + 0]\n ((x.1, x.2 ^ n).1 + x.1,\n (⇑(frobenius K p))^[x.1] (x.1, x.2 ^ n).2 * (⇑(frobenius K p)... | [] | simp_rw [iterate_frobenius, add_zero, mul_pow, ← pow_mul,
← pow_add, mul_assoc, ← pow_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.PerfectClosure | {
"line": 374,
"column": 4
} | {
"line": 374,
"column": 64
} | {
"line": 374,
"column": 65
} | [
{
"pp": "case mp\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y z : ℕ\nH : (⇑(frobenius K p))^[(0, ↑y).1 + z] (0, ↑x).2 = (⇑(frobenius K p))^[(0, ↑x).1 + z] (0, ↑y).2\n⊢ ↑x = ↑y",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": ... | [
"case mp\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y z : ℕ\nH : (⇑(frobenius K p))^[(0, ↑y).1 + z] (0, ↑x).2 = (⇑(frobenius K p))^[(0, ↑x).1 + z] (0, ↑y).2\n⊢ ↑x = ↑y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PerfectClosure | {
"line": 405,
"column": 21
} | {
"line": 405,
"column": 78
} | {
"line": 405,
"column": 79
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝¹ : PerfectClosure K p\nx : ℕ × K\nx✝ : IsNilpotent (mk K p x)\nn m : ℕ\nh : (iterateFrobenius K p m) (x.2 ^ p ^ n) = 0\n⊢ (iterateFrobenius K p (n + m)) x.2 = 0",
"ppTerm": "?m.96",
"assigned": true,
... | [
"K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝¹ : PerfectClosure K p\nx : ℕ × K\nx✝ : IsNilpotent (mk K p x)\nn m : ℕ\nh : (iterateFrobenius K p m) (x.2 ^ p ^ n) = 0\n⊢ (x.2 ^ p ^ n) ^ p ^ m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsPerfectClosure | {
"line": 165,
"column": 15
} | {
"line": 165,
"column": 51
} | {
"line": 165,
"column": 52
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : CommSemiring K\ninst✝¹ : CommSemiring L\ni : K →+* L\np : ℕ\ninst✝ : IsPRadical i p\nx : K\nn : ℕ\nh : x ^ p ^ n = 0\n⊢ i x ^ p ^ n = 0",
"ppTerm": "?m.141",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\nL : Type u_2\ninst✝² : CommSemiring K\ninst✝¹ : CommSemiring L\ni : K →+* L\np : ℕ\ninst✝ : IsPRadical i p\nx : K\nn : ℕ\nh : x ^ p ^ n = 0\n⊢ i x ^ p ^ n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsPerfectClosure | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 59
} | {
"line": 181,
"column": 60
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring K\ninst✝³ : CommSemiring L\ninst✝² : CommSemiring M\ni : K →+* L\nf : L →+* M\np : ℕ\ninst✝¹ : IsPRadical i p\ninst✝ : IsPRadical f p\nx : K\nh : i x ∈ RingHom.ker f\n⊢ x ∈ pNilradical K p",
"ppTerm": "?m.164",
"assigned": false,
... | [
"K : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring K\ninst✝³ : CommSemiring L\ninst✝² : CommSemiring M\ni : K →+* L\nf : L →+* M\np : ℕ\ninst✝¹ : IsPRadical i p\ninst✝ : IsPRadical f p\nx : K\nh : i x ∈ RingHom.ker f\n⊢ x ∈ pNilradical K p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsRealClosed.Basic | {
"line": 80,
"column": 40
} | {
"line": 80,
"column": 51
} | {
"line": 80,
"column": 52
} | [
{
"pp": "R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nn : ℕ\nhn : Odd n\nr : R\nhr : (X ^ n - C x).IsRoot r\n⊢ r ^ n - x = 0",
"ppTerm": "?m.55",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝¹ : Field R\ninst✝ : IsRealClosed R\nx : R\nn : ℕ\nhn : Odd n\nr : R\nhr : (X ^ n - C x).IsRoot r\n⊢ r ^ n - x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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