module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Dynamics.Ergodic.Ergodic | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 18
} | {
"line": 76,
"column": 19
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns : Set α\nf : α → α\nμ : Measure α\ninst✝ : IsProbabilityMeasure μ\nhf : PreErgodic f μ\nhs : MeasurableSet s\nhs' : f ⁻¹' s = s\n⊢ μ s = 0 ∨ μ s = 1",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"α : Type u_1\nm : MeasurableSpace α\ns : Set α\nf : α → α\nμ : Measure α\ninst✝ : IsProbabilityMeasure μ\nhf : PreErgodic f μ\nhs : MeasurableSet s\nhs' : f ⁻¹' s = s\n⊢ μ s = 0 ∨ μ s = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Action.Regular | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 19
} | {
"line": 38,
"column": 20
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : MeasurableMul₂ G\ninst✝² : MeasurableInv G\nμ : Measure G\ninst✝¹ : SFinite μ\ninst✝ : μ.IsMulLeftInvariant\ns : Set G\nhsm : MeasurableSet s\nhs : ∀ (g : G), (fun x ↦ g • x) ⁻¹' s =ᵐ[μ] s\nhμs : ∃ᵐ (x : G) ∂μ, x ∈ s\na : G\nhas : a ∈... | [
"G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : MeasurableMul₂ G\ninst✝² : MeasurableInv G\nμ : Measure G\ninst✝¹ : SFinite μ\ninst✝ : μ.IsMulLeftInvariant\ns : Set G\nhsm : MeasurableSet s\nhs : ∀ (g : G), (fun x ↦ g • x) ⁻¹' s =ᵐ[μ] s\nhμs : ∃ᵐ (x : G) ∂μ, x ∈ s\na : G\nhas : a ∈ s\nha : ∀ᵐ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Action.Regular | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 19
} | {
"line": 53,
"column": 20
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : MeasurableMul₂ G\ninst✝² : MeasurableInv G\nμ : Measure G\ninst✝¹ : SFinite μ\ninst✝ : μ.IsMulRightInvariant\ns : Set G\nhsm : MeasurableSet s\nhs : ∀ (g : Gᵐᵒᵖ), (fun x ↦ g • x) ⁻¹' s =ᵐ[μ] s\nhμs : ∃ᵐ (x : G) ∂μ, x ∈ s\na : G\nhas :... | [
"G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : MeasurableMul₂ G\ninst✝² : MeasurableInv G\nμ : Measure G\ninst✝¹ : SFinite μ\ninst✝ : μ.IsMulRightInvariant\ns : Set G\nhsm : MeasurableSet s\nhs : ∀ (g : Gᵐᵒᵖ), (fun x ↦ g • x) ⁻¹' s =ᵐ[μ] s\nhμs : ∃ᵐ (x : G) ∂μ, x ∈ s\na : G\nhas : a ∈ s\nha :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Action.Basic | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 36
} | {
"line": 72,
"column": 37
} | [
{
"pp": "G : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : ErgodicSMul G α μ\ns : Set α\nhm : NullMeasurableSet s μ\nh : ∀ (g : G), g • s =ᵐ[μ] s\ng : G\n⊢ (fun x ↦ g • x) ⁻¹' s =ᵐ[μ] s",
"ppTerm": "?m.30",
"assigned": true,
"usedCons... | [
"G : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : ErgodicSMul G α μ\ns : Set α\nhm : NullMeasurableSet s μ\nh : ∀ (g : G), g • s =ᵐ[μ] s\ng : G\n⊢ g⁻¹ • s =ᵐ[μ] s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 204,
"column": 35
} | {
"line": 204,
"column": 46
} | {
"line": 204,
"column": 47
} | [
{
"pp": "f✝ g : CircleDeg1Lift\nf : CircleDeg1Liftˣ\na✝ b✝ : ℝ\nh :\n { toFun := ⇑↑f, invFun := ⇑↑f⁻¹, left_inv := ⋯, right_inv := ⋯ } a✝ ≤\n { toFun := ⇑↑f, invFun := ⇑↑f⁻¹, left_inv := ⋯, right_inv := ⋯ } b✝\n⊢ a✝ ≤ b✝",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars... | [
"f✝ g : CircleDeg1Lift\nf : CircleDeg1Liftˣ\na✝ b✝ : ℝ\nh :\n { toFun := ⇑↑f, invFun := ⇑↑f⁻¹, left_inv := ⋯, right_inv := ⋯ } a✝ ≤\n { toFun := ⇑↑f, invFun := ⇑↑f⁻¹, left_inv := ⋯, right_inv := ⋯ } b✝\n⊢ a✝ ≤ b✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 293,
"column": 2
} | {
"line": 293,
"column": 48
} | {
"line": 293,
"column": 49
} | [
{
"pp": "f : CircleDeg1Lift\nn : ℕ\n⊢ Function.Commute ⇑f fun x ↦ ↑n + x",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : CircleDeg1Lift\nn : ℕ\n⊢ Function.Commute ⇑f fun x ↦ ↑n + x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 304,
"column": 17
} | {
"line": 304,
"column": 45
} | {
"line": 304,
"column": 46
} | [
{
"pp": "f : CircleDeg1Lift\nn : ℕ\n⊢ Function.Commute ⇑f fun x ↦ x + ↑-[n+1]",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"AddMonoid.toAddSemigro... | [
"f : CircleDeg1Lift\nn : ℕ\n⊢ Function.Commute ⇑f fun x ↦ x + (-1 + -↑n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 307,
"column": 2
} | {
"line": 307,
"column": 39
} | {
"line": 307,
"column": 40
} | [
{
"pp": "f : CircleDeg1Lift\nn : ℤ\n⊢ Function.Commute ⇑f fun x ↦ ↑n + x",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : CircleDeg1Lift\nn : ℤ\n⊢ Function.Commute ⇑f fun x ↦ ↑n + x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 477,
"column": 6
} | {
"line": 477,
"column": 34
} | {
"line": 477,
"column": 35
} | [
{
"pp": "f : CircleDeg1Lift\n⊢ Tendsto (fun x ↦ x - 1) atTop atTop",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder",
"AddGroupWithOne.t... | [
"f : CircleDeg1Lift\n⊢ Tendsto (fun x ↦ x + -1) atTop atTop"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 495,
"column": 2
} | {
"line": 495,
"column": 52
} | {
"line": 496,
"column": 4
} | [
{
"pp": "f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nh : f x ≤ x + ↑m\nn : ℕ\n⊢ (⇑f)^[n] x ≤ x + ↑n * ↑m",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nh : f x ≤ x + ↑m\nn : ℕ\n⊢ (⇑f)^[n] x ≤ x + ↑n * ↑m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 500,
"column": 2
} | {
"line": 500,
"column": 52
} | {
"line": 501,
"column": 4
} | [
{
"pp": "f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nh : x + ↑m ≤ f x\nn : ℕ\n⊢ x + ↑n * ↑m ≤ (⇑f)^[n] x",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nh : x + ↑m ≤ f x\nn : ℕ\n⊢ x + ↑n * ↑m ≤ (⇑f)^[n] x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 505,
"column": 2
} | {
"line": 505,
"column": 52
} | {
"line": 505,
"column": 53
} | [
{
"pp": "f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nh : f x = x + ↑m\nn : ℕ\n⊢ (⇑f)^[n] x = x + ↑n * ↑m",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nh : f x = x + ↑m\nn : ℕ\n⊢ (⇑f)^[n] x = x + ↑n * ↑m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 509,
"column": 2
} | {
"line": 509,
"column": 52
} | {
"line": 510,
"column": 4
} | [
{
"pp": "f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nn : ℕ\nhn : 0 < n\n⊢ (⇑f)^[n] x ≤ x + ↑n * ↑m ↔ f x ≤ x + ↑m",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nn : ℕ\nhn : 0 < n\n⊢ (⇑f)^[n] x ≤ x + ↑n * ↑m ↔ f x ≤ x + ↑m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 514,
"column": 2
} | {
"line": 514,
"column": 52
} | {
"line": 515,
"column": 4
} | [
{
"pp": "f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nn : ℕ\nhn : 0 < n\n⊢ (⇑f)^[n] x < x + ↑n * ↑m ↔ f x < x + ↑m",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nn : ℕ\nhn : 0 < n\n⊢ (⇑f)^[n] x < x + ↑n * ↑m ↔ f x < x + ↑m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 519,
"column": 2
} | {
"line": 519,
"column": 52
} | {
"line": 520,
"column": 4
} | [
{
"pp": "f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nn : ℕ\nhn : 0 < n\n⊢ (⇑f)^[n] x = x + ↑n * ↑m ↔ f x = x + ↑m",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nn : ℕ\nhn : 0 < n\n⊢ (⇑f)^[n] x = x + ↑n * ↑m ↔ f x = x + ↑m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 524,
"column": 2
} | {
"line": 524,
"column": 27
} | {
"line": 524,
"column": 28
} | [
{
"pp": "f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nn : ℕ\nhn : 0 < n\n⊢ x + ↑n * ↑m ≤ (⇑f)^[n] x ↔ x + ↑m ≤ f x",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nn : ℕ\nhn : 0 < n\n⊢ x + ↑n * ↑m ≤ (⇑f)^[n] x ↔ x + ↑m ≤ f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 528,
"column": 2
} | {
"line": 528,
"column": 27
} | {
"line": 528,
"column": 28
} | [
{
"pp": "f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nn : ℕ\nhn : 0 < n\n⊢ x + ↑n * ↑m < (⇑f)^[n] x ↔ x + ↑m < f x",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nn : ℕ\nhn : 0 < n\n⊢ x + ↑n * ↑m < (⇑f)^[n] x ↔ x + ↑m < f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 567,
"column": 4
} | {
"line": 567,
"column": 64
} | {
"line": 568,
"column": 6
} | [
{
"pp": "f : CircleDeg1Lift\nτ' : ℝ\nh : Tendsto (fun n ↦ (⇑f)^[n] 0 / ↑n) atTop (𝓝 τ')\n⊢ Tendsto f.transnumAuxSeq atTop (𝓝 τ')",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Real.instZero",
"congrArg",
"Nat.instMonoid"... | [
"f : CircleDeg1Lift\nτ' : ℝ\nh : Tendsto (fun n ↦ (⇑f)^[n] 0 / ↑n) atTop (𝓝 τ')\n⊢ Tendsto (fun n ↦ (⇑f)^[2 ^ n] 0 / 2 ^ n) atTop (𝓝 τ')"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Action.OfMinimal | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 36
} | {
"line": 117,
"column": 37
} | [
{
"pp": "M : Type u_1\nX : Type u_2\ninst✝¹⁵ : Monoid M\ninst✝¹⁴ : SMul M X\ninst✝¹³ : TopologicalSpace X\ninst✝¹² : R1Space X\ninst✝¹¹ : MeasurableSpace X\ninst✝¹⁰ : BorelSpace X\nμ : Measure X\ninst✝⁹ : IsFiniteMeasure μ\ninst✝⁸ : μ.InnerRegular\nN : Type u_3\ninst✝⁷ : MulAction M N\ninst✝⁶ : Monoid N\ninst✝⁵... | [
"M : Type u_1\nX : Type u_2\ninst✝¹⁵ : Monoid M\ninst✝¹⁴ : SMul M X\ninst✝¹³ : TopologicalSpace X\ninst✝¹² : R1Space X\ninst✝¹¹ : MeasurableSpace X\ninst✝¹⁰ : BorelSpace X\nμ : Measure X\ninst✝⁹ : IsFiniteMeasure μ\ninst✝⁸ : μ.InnerRegular\nN : Type u_3\ninst✝⁷ : MulAction M N\ninst✝⁶ : Monoid N\ninst✝⁵ : Topologic... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Action.OfMinimal | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 15
} | {
"line": 122,
"column": 16
} | [
{
"pp": "M : Type u_1\nX : Type u_2\ninst✝¹⁵ : Monoid M\ninst✝¹⁴ : SMul M X\ninst✝¹³ : TopologicalSpace X\ninst✝¹² : R1Space X\ninst✝¹¹ : MeasurableSpace X\ninst✝¹⁰ : BorelSpace X\nμ : Measure X\ninst✝⁹ : IsFiniteMeasure μ\ninst✝⁸ : μ.InnerRegular\nN : Type u_3\ninst✝⁷ : MulAction M N\ninst✝⁶ : Monoid N\ninst✝⁵... | [
"M : Type u_1\nX : Type u_2\ninst✝¹⁵ : Monoid M\ninst✝¹⁴ : SMul M X\ninst✝¹³ : TopologicalSpace X\ninst✝¹² : R1Space X\ninst✝¹¹ : MeasurableSpace X\ninst✝¹⁰ : BorelSpace X\nμ : Measure X\ninst✝⁹ : IsFiniteMeasure μ\ninst✝⁸ : μ.InnerRegular\nN : Type u_3\ninst✝⁷ : MulAction M N\ninst✝⁶ : Monoid N\ninst✝⁵ : Topologic... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Action.OfMinimal | {
"line": 119,
"column": 4
} | {
"line": 122,
"column": 20
} | {
"line": 124,
"column": 0
} | [
{
"pp": "M : Type u_1\nX : Type u_2\ninst✝¹⁵ : Monoid M\ninst✝¹⁴ : SMul M X\ninst✝¹³ : TopologicalSpace X\ninst✝¹² : R1Space X\ninst✝¹¹ : MeasurableSpace X\ninst✝¹⁰ : BorelSpace X\nμ : Measure X\ninst✝⁹ : IsFiniteMeasure μ\ninst✝⁸ : μ.InnerRegular\nN : Type u_3\ninst✝⁷ : MulAction M N\ninst✝⁶ : Monoid N\ninst✝⁵... | [] | refine aeconst_of_dense_setOfPred_preimage_smul_ae (M := N) hsm.nullMeasurableSet ?_
refine (MulAction.dense_orbit M 1).mono ?_
rintro _ ⟨g, rfl⟩
simpa using hs g | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Dynamics.Ergodic.Action.OfMinimal | {
"line": 119,
"column": 4
} | {
"line": 122,
"column": 20
} | {
"line": 124,
"column": 0
} | [
{
"pp": "M : Type u_1\nX : Type u_2\ninst✝¹⁵ : Monoid M\ninst✝¹⁴ : SMul M X\ninst✝¹³ : TopologicalSpace X\ninst✝¹² : R1Space X\ninst✝¹¹ : MeasurableSpace X\ninst✝¹⁰ : BorelSpace X\nμ : Measure X\ninst✝⁹ : IsFiniteMeasure μ\ninst✝⁸ : μ.InnerRegular\nN : Type u_3\ninst✝⁷ : MulAction M N\ninst✝⁶ : Monoid N\ninst✝⁵... | [] | refine aeconst_of_dense_setOfPred_preimage_smul_ae (M := N) hsm.nullMeasurableSet ?_
refine (MulAction.dense_orbit M 1).mono ?_
rintro _ ⟨g, rfl⟩
simpa using hs g | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Dynamics.Ergodic.Action.OfMinimal | {
"line": 152,
"column": 2
} | {
"line": 156,
"column": 52
} | {
"line": 158,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹¹ : Group G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : ContinuousInv G\nX : Type u_2\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : R1Space X\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : BorelSpace X\ninst✝⁴ : MulAction G X\ninst✝³ : ContinuousSMul G X\ng : G\nhg : DenseRange fun x ↦ g ^ x\nμ : Measu... | [] | borelize G
refine ⟨measurePreserving_smul _ _, ⟨fun s hsm hs ↦ ?_⟩⟩
refine aeconst_of_dense_aestabilizer_smul hsm.nullMeasurableSet (hg.mono ?_)
rw [← Subgroup.coe_zpowers, SetLike.coe_subset_coe, ← Subgroup.zpowers_inv, Subgroup.zpowers_le,
MulAction.mem_aestabilizer, ← preimage_smul, hs] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Dynamics.Ergodic.Action.OfMinimal | {
"line": 152,
"column": 2
} | {
"line": 156,
"column": 52
} | {
"line": 158,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹¹ : Group G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : ContinuousInv G\nX : Type u_2\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : R1Space X\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : BorelSpace X\ninst✝⁴ : MulAction G X\ninst✝³ : ContinuousSMul G X\ng : G\nhg : DenseRange fun x ↦ g ^ x\nμ : Measu... | [] | borelize G
refine ⟨measurePreserving_smul _ _, ⟨fun s hsm hs ↦ ?_⟩⟩
refine aeconst_of_dense_aestabilizer_smul hsm.nullMeasurableSet (hg.mono ?_)
rw [← Subgroup.coe_zpowers, SetLike.coe_subset_coe, ← Subgroup.zpowers_inv, Subgroup.zpowers_le,
MulAction.mem_aestabilizer, ← preimage_smul, hs] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 872,
"column": 4
} | {
"line": 872,
"column": 15
} | {
"line": 872,
"column": 16
} | [
{
"pp": "f₁ f₂ : CircleDeg1Liftˣ\nh : τ ↑f₁ = τ ↑f₂\nthis :\n ∀ (n : Multiplicative ℤ),\n τ (((Units.coeHom CircleDeg1Lift).comp ((zpowersHom CircleDeg1Liftˣ) f₁)) n) =\n τ (((Units.coeHom CircleDeg1Lift).comp ((zpowersHom CircleDeg1Liftˣ) f₂)) n)\nF : CircleDeg1Lift\nhF :\n ∀ (g : Multiplicative ℤ),\... | [
"f₁ f₂ : CircleDeg1Liftˣ\nh : τ ↑f₁ = τ ↑f₂\nthis :\n ∀ (n : Multiplicative ℤ),\n τ (((Units.coeHom CircleDeg1Lift).comp ((zpowersHom CircleDeg1Liftˣ) f₁)) n) =\n τ (((Units.coeHom CircleDeg1Lift).comp ((zpowersHom CircleDeg1Liftˣ) f₂)) n)\nF : CircleDeg1Lift\nhF :\n ∀ (g : Multiplicative ℤ),\n Semicon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Action.OfMinimal | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 90
} | {
"line": 181,
"column": 91
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : IsTopologicalGroup G\ninst✝² : MeasurableSpace G\ninst✝¹ : OpensMeasurableSpace G\nμ : Measure G\ninst✝ : μ.IsOpenPosMeasure\ng : G\nhg : Ergodic (fun x ↦ g * x) μ\na : G\nh : (range fun x ↦ g ^ x)ᶜ ∈ 𝓝 a\nthis :\n Tendsto\n (fu... | [
"G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : IsTopologicalGroup G\ninst✝² : MeasurableSpace G\ninst✝¹ : OpensMeasurableSpace G\nμ : Measure G\ninst✝ : μ.IsOpenPosMeasure\ng : G\nhg : Ergodic (fun x ↦ g * x) μ\na : G\nh : (range fun x ↦ g ^ x)ᶜ ∈ 𝓝 a\nthis :\n Tendsto\n (fun x ↦\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.AddCircle | {
"line": 66,
"column": 30
} | {
"line": 66,
"column": 75
} | {
"line": 66,
"column": 76
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : ℕ := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * ↑n))\nhI : I =ᵐ[volume] B\nhn : 1 ≤ ↑n\ng : AddCircle T\nhg : g ∈ G\nhg' : ⟨g, hg⟩ ≠ 0\... | [
"T : ℝ\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : ℕ := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * ↑n))\nhI : I =ᵐ[volume] B\nhn : 1 ≤ ↑n\ng : AddCircle T\nhg : g ∈ G\nhg' : ⟨g, hg⟩ ≠ 0\n⊢ ¬g = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.CountableSeparating | {
"line": 42,
"column": 6
} | {
"line": 42,
"column": 33
} | {
"line": 42,
"column": 34
} | [
{
"pp": "case refine_3.inr\nX : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : LinearOrder X\ninst✝¹ : OrderTopology X\ninst✝ : SecondCountableTopology X\ns✝ s : Set X\nhsc : s.Countable\nhsd : Dense s\nt : Set X := s ∪ {x | ∃ y, y ⋖ x}\nx y : X\nh : ∀ s ∈ Iio '' t, x ∈ s ↔ y ∈ s\nhne : x ≠ y\nthis : ∀ (x y : ... | [
"case refine_3.inr\nX : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : LinearOrder X\ninst✝¹ : OrderTopology X\ninst✝ : SecondCountableTopology X\ns✝ s : Set X\nhsc : s.Countable\nhsd : Dense s\nt : Set X := s ∪ {x | ∃ y, y ⋖ x}\nx y : X\nh : ∀ s ∈ Iio '' t, x ∈ s ↔ y ∈ s\nhne : x ≠ y\nthis : ∀ (x y : X), (∀ s ∈ I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.CountableSeparating | {
"line": 45,
"column": 65
} | {
"line": 45,
"column": 76
} | {
"line": 45,
"column": 77
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : LinearOrder X\ninst✝¹ : OrderTopology X\ninst✝ : SecondCountableTopology X\ns✝ s : Set X\nhsc : s.Countable\nhsd : Dense s\nt : Set X := s ∪ {x | ∃ y, y ⋖ x}\nx y : X\nh : ∀ s ∈ Iio '' t, x ∈ s ↔ y ∈ s\nhne : x ≠ y\nhlt : x < y\nhe : Ioo x y = ∅\n⊢ ∀ ... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : LinearOrder X\ninst✝¹ : OrderTopology X\ninst✝ : SecondCountableTopology X\ns✝ s : Set X\nhsc : s.Countable\nhsd : Dense s\nt : Set X := s ∪ {x | ∃ y, y ⋖ x}\nx y : X\nh : ∀ s ∈ Iio '' t, x ∈ s ↔ y ∈ s\nhne : x ≠ y\nhlt : x < y\nhe : Ioo x y = ∅\n⊢ ∀ ⦃c : X⦄, x <... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.CountableSeparating | {
"line": 49,
"column": 6
} | {
"line": 49,
"column": 35
} | {
"line": 49,
"column": 36
} | [
{
"pp": "case inr\nX : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : LinearOrder X\ninst✝¹ : OrderTopology X\ninst✝ : SecondCountableTopology X\ns✝ s : Set X\nhsc : s.Countable\nhsd : Dense s\nt : Set X := s ∪ {x | ∃ y, y ⋖ x}\nx y : X\nh : ∀ s ∈ Iio '' t, x ∈ s ↔ y ∈ s\nhne✝ : x ≠ y\nhlt : x < y\nhne : (Ioo ... | [
"case inr\nX : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : LinearOrder X\ninst✝¹ : OrderTopology X\ninst✝ : SecondCountableTopology X\ns✝ s : Set X\nhsc : s.Countable\nhsd : Dense s\nt : Set X := s ∪ {x | ∃ y, y ⋖ x}\nx y : X\nh : ∀ s ∈ Iio '' t, x ∈ s ↔ y ∈ s\nhne✝ : x ≠ y\nhlt : x < y\nhne : (Ioo x y).Nonempt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.CountableSeparating | {
"line": 56,
"column": 31
} | {
"line": 56,
"column": 73
} | {
"line": 56,
"column": 74
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : LinearOrder X\ninst✝¹ : OrderTopology X\ninst✝ : SecondCountableTopology X\ns : Set X\nt : Set (Set X)\nhtc : t.Countable\nht_sub : ∀ s ∈ t, s ∈ range Ioi\nht : ∀ x ∈ s, ∀ y ∈ s, (∀ s ∈ t, x ∈ s ↔ y ∈ s) → x = y\n⊢ ∀ s ∈ compl '' t, s ∈ range Iic",
... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : LinearOrder X\ninst✝¹ : OrderTopology X\ninst✝ : SecondCountableTopology X\ns : Set X\nt : Set (Set X)\nhtc : t.Countable\nht_sub : ∀ s ∈ t, s ∈ range Ioi\nht : ∀ x ∈ s, ∀ y ∈ s, (∀ s ∈ t, x ∈ s ↔ y ∈ s) → x = y\n⊢ ∀ a ∈ t, ∃ y, Iic y = aᶜ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Extreme | {
"line": 51,
"column": 4
} | {
"line": 52,
"column": 60
} | {
"line": 53,
"column": 4
} | [
{
"pp": "case inr\nX : Type u_1\nm : MeasurableSpace X\nμ : Measure X\nf : X → X\nc : ℝ≥0∞\nhc : c ≠ ∞\nhf : MeasurePreserving f μ μ\nhc₀ : c ≠ 0\nthis✝ : IsFiniteMeasure μ\nS : Set (Measure X) := {ν | MeasurePreserving f ν ν ∧ ν univ = c}\nh : μ ∈ extremePoints ℝ≥0∞ S\nthis : ∀ {s : Set X}, MeasurableSet s → f... | [
"case inr\nX : Type u_1\nm : MeasurableSpace X\nμ : Measure X\nf : X → X\nc : ℝ≥0∞\nhc : c ≠ ∞\nhf : MeasurePreserving f μ μ\nhc₀ : c ≠ 0\nthis✝ : IsFiniteMeasure μ\nS : Set (Measure X) := {ν | MeasurePreserving f ν ν ∧ ν univ = c}\nh : μ ∈ extremePoints ℝ≥0∞ S\nthis : ∀ {s : Set X}, MeasurableSet s → f ⁻¹' s = s →... | obtain ⟨hs, hs'⟩ : μ s ≠ 0 ∧ μ sᶜ ≠ 0 := by
simpa [eventuallyConst_set, ae_iff, and_comm] using! H | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Dynamics.FixedPoints.Support | {
"line": 40,
"column": 2
} | {
"line": 40,
"column": 50
} | {
"line": 41,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nf : X → X\n⊢ HasCompactFixedSupport f ↔ ∃ K, IsClosed K ∧ IsCompact K ∧ (fixedPoints f)ᶜ ⊆ K",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Compl.compl",
"Function.fixedPoints",
"Exists",
"Function.HasCompactFixed... | [
"case refine_1\nX : Type u_1\ninst✝ : TopologicalSpace X\nf : X → X\nhf : HasCompactFixedSupport f\n⊢ ∃ K, IsClosed K ∧ IsCompact K ∧ (fixedPoints f)ᶜ ⊆ K",
"case refine_2\nX : Type u_1\ninst✝ : TopologicalSpace X\nf : X → X\nx✝ : ∃ K, IsClosed K ∧ IsCompact K ∧ (fixedPoints f)ᶜ ⊆ K\nK : Set X\nhK₁ : IsClosed K\n... | refine ⟨fun hf ↦ ?_, fun ⟨K, hK₁, hK₂, hf⟩ ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Dynamics.Ergodic.Extreme | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 47
} | {
"line": 67,
"column": 48
} | [
{
"pp": "X : Type u_1\nm : MeasurableSpace X\nμ : Measure X\nf : X → X\nh : μ ∈ extremePoints ℝ≥0∞ {ν | MeasurePreserving f ν ν ∧ IsProbabilityMeasure ν}\n⊢ μ ∈ extremePoints ℝ≥0∞ {ν | MeasurePreserving f ν ν ∧ ν univ = 1}",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"X : Type u_1\nm : MeasurableSpace X\nμ : Measure X\nf : X → X\nh : μ ∈ extremePoints ℝ≥0∞ {ν | MeasurePreserving f ν ν ∧ IsProbabilityMeasure ν}\n⊢ μ ∈ extremePoints ℝ≥0∞ {ν | MeasurePreserving f ν ν ∧ ν univ = 1}"
] | 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.SymbolicDynamics.Basic | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 38
} | {
"line": 216,
"column": 39
} | [
{
"pp": "A : Type u_1\nG : Type u_2\ninst✝¹ : TopologicalSpace A\ninst✝ : DiscreteTopology A\nU : Finset G\nx : G → A\n⊢ IsOpen[Pi.topologicalSpace] (cylinder U x)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.topologicalSpace",
"congrArg",
"Finset",
... | [
"A : Type u_1\nG : Type u_2\ninst✝¹ : TopologicalSpace A\ninst✝ : DiscreteTopology A\nU : Finset G\nx : G → A\n⊢ IsOpen[Pi.topologicalSpace] ((↑U).pi fun i ↦ {x i})"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 38
} | {
"line": 221,
"column": 39
} | [
{
"pp": "A : Type u_1\nG : Type u_2\ninst✝¹ : TopologicalSpace A\ninst✝ : T1Space A\nU : Finset G\nx : G → A\n⊢ IsClosed[Pi.topologicalSpace] (cylinder U x)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.topologicalSpace",
"congrArg",
"Finset",
"... | [
"A : Type u_1\nG : Type u_2\ninst✝¹ : TopologicalSpace A\ninst✝ : T1Space A\nU : Finset G\nx : G → A\n⊢ IsClosed[Pi.topologicalSpace] ((↑U).pi fun i ↦ {x i})"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 381,
"column": 6
} | {
"line": 381,
"column": 36
} | {
"line": 381,
"column": 37
} | [
{
"pp": "A : Type u_1\ninst✝² : Inhabited A\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : IsLeftCancelMul G\np : Pattern A G\nv h : G\nhmem : h ∈ Finset.image (fun x ↦ v * x) p.support\n⊢ ∃ w ∈ p.support, v * w = h",
"ppTerm": "?m.45",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"us... | [
"A : Type u_1\ninst✝² : Inhabited A\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : IsLeftCancelMul G\np : Pattern A G\nv h : G\nhmem : h ∈ Finset.image (fun x ↦ v * x) p.support\n⊢ ∃ w ∈ p.support, v * w = h"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 377,
"column": 2
} | {
"line": 384,
"column": 17
} | {
"line": 386,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝² : Inhabited A\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : IsLeftCancelMul G\np : Pattern A G\nv : G\n⊢ G → A",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Inhabited.default",
"HMul.hMul",
"Monoid.toMulOneClass",
"Finset",
"Classic... | [] | intro h
if hmem : h ∈ p.support.image (v * ·) then
-- package existence of a preimage under (v * ·)
let ex : ∃ w, w ∈ p.support ∧ v * w = h := by
simpa [Finset.mem_image] using hmem
exact p.config (Classical.choose ex)
else
exact default | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 377,
"column": 2
} | {
"line": 384,
"column": 17
} | {
"line": 386,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝² : Inhabited A\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : IsLeftCancelMul G\np : Pattern A G\nv : G\n⊢ G → A",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Inhabited.default",
"HMul.hMul",
"Monoid.toMulOneClass",
"Finset",
"Classic... | [] | intro h
if hmem : h ∈ p.support.image (v * ·) then
-- package existence of a preimage under (v * ·)
let ex : ∃ w, w ∈ p.support ∧ v * w = h := by
simpa [Finset.mem_image] using hmem
exact p.config (Classical.choose ex)
else
exact default | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 423,
"column": 4
} | {
"line": 423,
"column": 34
} | {
"line": 423,
"column": 35
} | [
{
"pp": "A : Type u_1\ninst✝² : Inhabited A\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : IsLeftCancelMul G\np : Pattern A G\nv w : G\nhw : w ∈ p.support\nhmem : v * w ∈ Finset.image (fun x ↦ v * x) p.support\n⊢ ∃ w' ∈ p.support, v * w' = v * w",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
... | [
"A : Type u_1\ninst✝² : Inhabited A\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : IsLeftCancelMul G\np : Pattern A G\nv w : G\nhw : w ∈ p.support\nhmem : v * w ∈ Finset.image (fun x ↦ v * x) p.support\n⊢ w ∈ p.support"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 464,
"column": 2
} | {
"line": 464,
"column": 38
} | {
"line": 464,
"column": 39
} | [
{
"pp": "A : Type u_3\nG : Type u_4\ninst✝¹ : Inhabited A\ninst✝ : Monoid G\nF : Set (Pattern A G)\nh : G\nx : G → A\np : Pattern A G\nhp : p ∈ F\ng : G\nhx : p.mulOccursInAt (mulShift h x) g\n⊢ p.mulOccursInAt x (h * g)",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"A : Type u_3\nG : Type u_4\ninst✝¹ : Inhabited A\ninst✝ : Monoid G\nF : Set (Pattern A G)\nh : G\nx : G → A\np : Pattern A G\nhp : p ∈ F\ng : G\nhx : p.mulOccursInAt (mulShift h x) g\n⊢ p.mulOccursInAt x (h * g)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 500,
"column": 4
} | {
"line": 500,
"column": 86
} | {
"line": 500,
"column": 87
} | [
{
"pp": "case mp\nA : Type u_1\ninst✝² : Inhabited A\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : IsLeftCancelMul G\np : Pattern A G\ng : G\nx : G → A\nH : x ∈ {x | p.mulOccursInAt x g}\nw : G\nhw : w ∈ p.support\nhu : g * w ∈ Finset.image (fun x ↦ g * x) p.support\nhx : x (g * w) = p.config w\n⊢ x (g * w) = p.mul... | [
"case mp\nA : Type u_1\ninst✝² : Inhabited A\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : IsLeftCancelMul G\np : Pattern A G\ng : G\nx : G → A\nH : x ∈ {x | p.mulOccursInAt x g}\nw : G\nhw : w ∈ p.support\nhu : g * w ∈ Finset.image (fun x ↦ g * x) p.support\nhx : x (g * w) = p.config w\n⊢ x (g * w) = p.config w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 507,
"column": 4
} | {
"line": 507,
"column": 86
} | {
"line": 507,
"column": 87
} | [
{
"pp": "case mpr\nA : Type u_1\ninst✝² : Inhabited A\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : IsLeftCancelMul G\np : Pattern A G\ng : G\nx : G → A\nH : x ∈ cylinder (Finset.image (fun x ↦ g * x) p.support) (p.mulShift g)\nu : G\nhu : u ∈ p.support\nhx : x (g * u) = p.mulShift g (g * u)\n⊢ x (g * u) = p.config... | [
"case mpr\nA : Type u_1\ninst✝² : Inhabited A\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : IsLeftCancelMul G\np : Pattern A G\ng : G\nx : G → A\nH : x ∈ cylinder (Finset.image (fun x ↦ g * x) p.support) (p.mulShift g)\nu : G\nhu : u ∈ p.support\nhx : x (g * u) = p.mulShift g (g * u)\n⊢ x (g * u) = p.config u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 521,
"column": 2
} | {
"line": 521,
"column": 41
} | {
"line": 521,
"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.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 23
} | {
"line": 168,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nν : Measure α\ninst✝¹ : μ.HaveLebesgueDecomposition ν\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nhf : AEMeasurable f ν\nhf_ne_zero : ∀ᵐ (x : α) ∂ν, f x ≠ 0\nhf_ne_top : ∀ᵐ (x : α) ∂ν, f x ≠ ∞\nthis : SigmaFinite (ν.withDensity f)\ns : Set α\nh... | [] | exact hf.restrict | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 23
} | {
"line": 168,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nν : Measure α\ninst✝¹ : μ.HaveLebesgueDecomposition ν\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nhf : AEMeasurable f ν\nhf_ne_zero : ∀ᵐ (x : α) ∂ν, f x ≠ 0\nhf_ne_top : ∀ᵐ (x : α) ∂ν, f x ≠ ∞\nthis : SigmaFinite (ν.withDensity f)\ns : Set α\nh... | [] | exact hf.restrict | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 23
} | {
"line": 168,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nν : Measure α\ninst✝¹ : μ.HaveLebesgueDecomposition ν\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nhf : AEMeasurable f ν\nhf_ne_zero : ∀ᵐ (x : α) ∂ν, f x ≠ 0\nhf_ne_top : ∀ᵐ (x : α) ∂ν, f x ≠ ∞\nthis : SigmaFinite (ν.withDensity f)\ns : Set α\nh... | [] | exact hf.restrict | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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.SymbolicDynamics.Basic | {
"line": 547,
"column": 2
} | {
"line": 547,
"column": 40
} | {
"line": 547,
"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": 553,
"column": 2
} | {
"line": 553,
"column": 41
} | {
"line": 553,
"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.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.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.Subset | {
"line": 154,
"column": 2
} | {
"line": 163,
"column": 29
} | {
"line": 165,
"column": 0
} | [
{
"pp": "X : Type u_1\nT : X → X\nF G : Set X\nU : SetRel X X\nn : ℕ\n⊢ coverMincard T (F ∪ G) U n ≤ coverMincard T F U n + coverMincard T G U n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Dynamics.IsDynCoverOf.union",
"Eq.mpr",
"WithTop.coe_mono",
"Preorder.to... | [] | rcases eq_top_or_lt_top (coverMincard T F U n) with hF | hF
· rw [hF, top_add]; exact le_top
rcases eq_top_or_lt_top (coverMincard T G U n) with hG | hG
· rw [hG, add_top]; exact le_top
obtain ⟨s, s_cover, s_coverMincard⟩ := (coverMincard_finite_iff T F U n).1 hF
obtain ⟨t, t_cover, t_coverMincard⟩ := (coverM... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Dynamics.TopologicalEntropy.Subset | {
"line": 154,
"column": 2
} | {
"line": 163,
"column": 29
} | {
"line": 165,
"column": 0
} | [
{
"pp": "X : Type u_1\nT : X → X\nF G : Set X\nU : SetRel X X\nn : ℕ\n⊢ coverMincard T (F ∪ G) U n ≤ coverMincard T F U n + coverMincard T G U n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Dynamics.IsDynCoverOf.union",
"Eq.mpr",
"WithTop.coe_mono",
"Preorder.to... | [] | rcases eq_top_or_lt_top (coverMincard T F U n) with hF | hF
· rw [hF, top_add]; exact le_top
rcases eq_top_or_lt_top (coverMincard T G U n) with hG | hG
· rw [hG, add_top]; exact le_top
obtain ⟨s, s_cover, s_coverMincard⟩ := (coverMincard_finite_iff T F U n).1 hF
obtain ⟨t, t_cover, t_coverMincard⟩ := (coverM... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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": 41,
"column": 15
} | {
"line": 45,
"column": 38
} | {
"line": 46,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\nL : Type v\ninst✝⁴ : CommRing L\ninst✝³ : IsDomain L\ninst✝² : Algebra R L\ninst✝¹ : IsTorsionFree R L\ninst✝ : Algebra.IsAlgebraic R L\nx y : L\n⊢ (fun x ↦\n let p := Classical.indefiniteDescription (fun x_1 ↦ x_1 ≠ 0 ∧ (Polynomial.aeval x... | [] | by
intro h
simp only [Set.coe_ofPred, ne_eq, Set.mem_ofPred_eq, Sigma.mk.inj_iff] at h
refine (Subtype.heq_iff_coe_eq ?_).1 h.2
simp only [h.1, forall_true_iff] | [anonymous] | Lean.Parser.Term.byTactic |
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": 439,
"column": 4
} | {
"line": 439,
"column": 57
} | {
"line": 440,
"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": 908,
"column": 6
} | {
"line": 908,
"column": 34
} | {
"line": 909,
"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": 451,
"column": 39
} | {
"line": 451,
"column": 49
} | {
"line": 451,
"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": 451,
"column": 39
} | {
"line": 451,
"column": 49
} | {
"line": 451,
"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": 451,
"column": 39
} | {
"line": 451,
"column": 49
} | {
"line": 451,
"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": 451,
"column": 83
} | {
"line": 451,
"column": 93
} | {
"line": 451,
"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": 451,
"column": 83
} | {
"line": 451,
"column": 93
} | {
"line": 451,
"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": 451,
"column": 83
} | {
"line": 451,
"column": 93
} | {
"line": 451,
"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": 456,
"column": 39
} | {
"line": 456,
"column": 49
} | {
"line": 456,
"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": 456,
"column": 39
} | {
"line": 456,
"column": 49
} | {
"line": 456,
"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": 456,
"column": 39
} | {
"line": 456,
"column": 49
} | {
"line": 456,
"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.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": 460,
"column": 22
} | {
"line": 460,
"column": 32
} | {
"line": 461,
"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": 460,
"column": 22
} | {
"line": 460,
"column": 32
} | {
"line": 461,
"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": 464,
"column": 14
} | {
"line": 464,
"column": 24
} | {
"line": 466,
"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": 345,
"column": 2
} | {
"line": 345,
"column": 18
} | {
"line": 345,
"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 (ofPred p)\nthis : ∀ x ∈ (closure L).toFun s, p x\n⊢ p x",
"ppTerm": "?m.37",
"assigned": false,
"usedConstan... | [
"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 (ofPred p)\nthis : ∀ x ∈ (closure L).toFun s, p x\n⊢ p x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Substructures | {
"line": 394,
"column": 4
} | {
"line": 394,
"column": 55
} | {
"line": 394,
"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.Semantics | {
"line": 584,
"column": 6
} | {
"line": 584,
"column": 14
} | {
"line": 584,
"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": 584,
"column": 15
} | {
"line": 584,
"column": 23
} | {
"line": 584,
"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.Substructures | {
"line": 403,
"column": 35
} | {
"line": 405,
"column": 94
} | {
"line": 407,
"column": 0
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nS : Set (L.Substructure M)\nSne : S.Nonempty\nhS : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) S\nx : M\n⊢ x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"FirstOrder.Language.Substructure.mem_iSup_of_directed... | [] | by
have : Nonempty S := Sne.to_subtype
simp only [sSup_eq_iSup', mem_iSup_of_directed hS.directed_val, Subtype.exists, exists_prop] | [anonymous] | Lean.Parser.Term.byTactic |
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.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": 214,
"column": 4
} | {
"line": 215,
"column": 58
} | {
"line": 216,
"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} = ofPred φ.Realize\nhφx : φ.Realize ![x]\nψ : L[[↑↑S]].Sentence := iExs (Fin 1) (relabel Sum.inr φ)\n⊢ M ⊨ ψ",
"ppTerm": "?m.52",
"as... | [
"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} = ofPred 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.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": 232,
"column": 2
} | {
"line": 232,
"column": 18
} | {
"line": 232,
"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} = ofPred 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.Re... | [
"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} = ofPred 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": 956,
"column": 2
} | {
"line": 956,
"column": 71
} | {
"line": 956,
"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": 968,
"column": 12
} | {
"line": 969,
"column": 15
} | {
"line": 969,
"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": 976,
"column": 4
} | {
"line": 976,
"column": 57
} | {
"line": 977,
"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": 1005,
"column": 6
} | {
"line": 1005,
"column": 38
} | {
"line": 1006,
"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__ |
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