import Mathlib
import OAI.Geometry.CAT0Fillings.Model
import OAI.Geometry.CAT0Fillings.Currents.MassBounds

namespace OAI

section

section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal

namespace CAT0Fillings

attribute [local instance] Classical.propDecidable

universe u

end CAT0Fillings

open Filter Set
open scoped Topology NNReal
open Set Filter MeasureTheory TopologicalSpace
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric
open scoped Topology Pointwise NNReal
open Set MeasureTheory
open scoped RealInnerProductSpace
open Matrix
open scoped RealInnerProductSpace MatrixOrder

namespace CAT0Fillings
open Set MeasureTheory Filter
open scoped ENNReal NNReal Topology

section SumControls
variable {X : Type*} [MetricSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X]
  {k : ℕ} {T : Functional X k} {S : ℕ → Functional X k}
  {μ : ℕ → Measure X} [IsFiniteMeasure (Measure.sum μ)]

omit [CompactSpace X] in
lemma controls_sum_general (hμ : ∀ i, Controls (S i) (μ i))
    (hT : ∀ b π, T b π = ∑' i, S i b π) : Controls T (Measure.sum μ) := by
  intro b π hb hπ
  have hbi : Integrable (fun x => |b x|) (Measure.sum μ) :=
    (hb.toBoundedContinuous.integrable (Measure.sum μ)).abs
  have hs : Summable (fun i => ∫ x, |b x| ∂μ i) := by
    simpa only [Real.norm_eq_abs,abs_abs] using hbi.summable_integral
  have hsa : Summable (fun i => S i b π) :=
    hs.of_norm_bounded (fun i => by simpa only [Real.norm_eq_abs] using hμ i b π hb hπ)
  rw [hT b π,integral_sum_measure hbi]
  calc
    |∑' i, S i b π| ≤ ∑' i, |S i b π| := by
      simpa only [Real.norm_eq_abs] using
        (norm_tsum_le_tsum_norm (f := fun i => S i b π) hsa.norm)
    _ ≤ ∑' i, ∫ x, |b x| ∂μ i := hsa.abs.tsum_le_tsum (fun i => hμ i b π hb hπ) hs

end SumControls

end CAT0Fillings

namespace CAT0Fillings
attribute [local instance] Classical.propDecidable

attribute [local instance] Classical.propDecidable

attribute [local instance] Classical.propDecidable

attribute [local instance] Classical.propDecidable

attribute [local instance] Classical.propDecidable

attribute [local instance] Classical.propDecidable


attribute [local instance] Classical.propDecidable
open Set Metric
open scoped NNReal

end CAT0Fillings

namespace CAT0Fillings
open Set MeasureTheory Filter
open scoped Topology NNReal ENNReal

universe u
namespace CurrentOperations
attribute [local instance] Classical.propDecidable

variable {X : Type u} {Y : Type*} [MetricSpace X] [MetricSpace Y]
  [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y]

end CurrentOperations
end CAT0Fillings
end
end

end OAI