Definitions bundle OAIQ3Ff001 of the split of OpenAI, Invariant-projection counterexamples for every irrational rotation

import Mathlib

section
-- module Solutions.OAIQ3_FF.Compat
namespace OAI

/-! Compatibility with the newer Lean core OpenAI's code was written against: lemmas it uses by
names this environment (Lean 4.33.1, Mathlib 0df444a) does not have yet. Each is the
corresponding `if_pos` / `if_neg` / `dif_pos` / `dif_neg`. The `dite` forms are stated as
`dite p a b`, not with the `if hp : p then …` notation: its eta-expanded lambdas form is a higher-order pattern
that `rw` cannot find (DepthThree, PeriodicTiling 2026-10-25). -/
section OAICompat434
/-- Mathlib's newer `TensorProduct.inductionOn` (no `zero` case: it is the `tmul` case at
`0 ⊗ₜ 0`); ours has only `TensorProduct.induction_on`, which asks for one. Not named
`TensorProduct.inductionOn`: inside `namespace OAI` that would make `open TensorProduct` open
`OAI.TensorProduct` instead of Mathlib's. -/
@[elab_as_elim]
theorem tensorInductionOn {R M N : Type*} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N]
    [Module R M] [Module R N] {motive : TensorProduct R M N → Prop} (z : TensorProduct R M N)
    (tmul : ∀ x y, motive (TensorProduct.tmul R x y))
    (add : ∀ x y, motive x → motive y → motive (x + y)) : motive z :=
  TensorProduct.induction_on z (by simpa using tmul 0 0) tmul add
/-- Mathlib's newer case names for `FreeGroup.induction_on` (`one`, where ours says `C1`). -/
@[elab_as_elim]
theorem freeGroupInductionOn {α : Type*} {C : FreeGroup α → Prop} (z : FreeGroup α) (one : C 1)
    (of : ∀ x, C (FreeGroup.of x)) (inv_of : ∀ x, C (FreeGroup.of x) → C (FreeGroup.of x)⁻¹)
    (mul : ∀ x y, C x → C y → C (x * y)) : C z :=
  FreeGroup.induction_on z one of inv_of mul
/-- The first projection of a probability measure on a product: found by instance search in d13f23b
(total `Measure.map`), here a measurable map. -/
instance isProbabilityMeasure_map_fst {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
    {μ : MeasureTheory.Measure (α × β)} [MeasureTheory.IsProbabilityMeasure μ] :
    MeasureTheory.IsProbabilityMeasure (μ.map Prod.fst) :=
  MeasureTheory.Measure.isProbabilityMeasure_map measurable_fst.aemeasurable
instance isProbabilityMeasure_map_snd {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
    {μ : MeasureTheory.Measure (α × β)} [MeasureTheory.IsProbabilityMeasure μ] :
    MeasureTheory.IsProbabilityMeasure (μ.map Prod.snd) :=
  MeasureTheory.Measure.isProbabilityMeasure_map measurable_snd.aemeasurable
instance isProbabilityMeasure_map_subtype_val {α : Type*} [MeasurableSpace α] {p : α → Prop}
    {μ : MeasureTheory.Measure {a // p a}} [MeasureTheory.IsProbabilityMeasure μ] :
    MeasureTheory.IsProbabilityMeasure (μ.map Subtype.val) :=
  MeasureTheory.Measure.isProbabilityMeasure_map measurable_subtype_coe.aemeasurable
/-- Mathlib d13f23b's `YoungDiagram.card_transpose`. -/
@[simp]
theorem _root_.YoungDiagram.card_transpose (μ : YoungDiagram) : μ.transpose.card = μ.card := by
  unfold YoungDiagram.card
  apply Finset.card_bij (fun c _ => c.swap)
  · intro c hc
    have : c ∈ μ.transpose := hc
    rw [YoungDiagram.mem_transpose] at this
    exact this
  · intro a _ b _ h
    simpa using h
  · intro b hb
    refine ⟨b.swap, ?_, by simp⟩
    show b.swap ∈ μ.transpose
    rw [YoungDiagram.mem_transpose]
    simpa using hb
/-- For a measurable equivalence `e`, `IsProbabilityMeasure (μ.map e)` is found by instance search
(Mathlib d13f23b finds it for every map `f`: there `Measure.map` of a non-measurable map is a Dirac
mass, here it is `0`, so only the measurable case can be shimmed). -/
instance isProbabilityMeasure_map_measurableEquiv {α β : Type*} [MeasurableSpace α]
    [MeasurableSpace β] {μ : MeasureTheory.Measure α} [MeasureTheory.IsProbabilityMeasure μ]
    (e : α ≃ᵐ β) : MeasureTheory.IsProbabilityMeasure (μ.map e) :=
  MeasureTheory.Measure.isProbabilityMeasure_map e.measurable.aemeasurable
end OAICompat434

end OAI
end

section
-- module Solutions.OAIQ3_FF.FiniteFactor_CrossedProduct_Powers
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

namespace FiniteFactor
open MeasureTheory Filter
open scoped ENNReal Topology
noncomputable section

def sourceShiftNIndex {ι : Type*} (n : ℕ) (z : (ι → ℝ) × ℤ) : (ι → ℝ) × ℤ :=
  (z.1, z.2-(n : ℤ))

end
end FiniteFactor

end OAI
end

section
-- module Solutions.OAIQ3_FF.FiniteFactor_Diffuse_OperatorCesaro
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section
open Filter
open scoped Topology BigOperators
namespace Nonhyperlinear.Diffuseness
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

/-- Cesàro means of the powers of a bounded operator. -/
def cesaro (S : H →L[ℂ] H) (n : ℕ) : H →L[ℂ] H :=
  ((n+1 : ℕ) : ℂ)⁻¹ • ∑ k ∈ Finset.range (n+1), S^k

end Nonhyperlinear.Diffuseness

end

end OAI
end

section
-- module Solutions.OAIQ3_FF.FiniteFactor_Diffuse_OperatorSupport
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section
open Filter
open scoped Topology
namespace Nonhyperlinear.Diffuseness
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

/-- The actual orthogonal projection onto the closed range of an operator. -/
def rangeSupport (T : H →L[ℂ] H) : H →L[ℂ] H :=
  T.range.topologicalClosure.starProjection

end Nonhyperlinear.Diffuseness

end

end OAI
end

section
-- module Solutions.OAIQ3_FF.FiniteFactor_Diffuse_SupportTrace
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section
open Filter
open scoped Topology BigOperators
namespace Nonhyperlinear.Diffuseness
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

def vectorFunctional (v:H) : (H→L[ℂ]H)→ₗ[ℂ]ℂ where
  toFun T := inner ℂ v (T v)
  map_add' T S := inner_add_right v (T v) (S v)
  map_smul' c T := inner_smul_right v (T v) c

end Nonhyperlinear.Diffuseness

end

end OAI
end