import OAI.Combinatorics.ExpectationThreshold.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic

namespace OAI

/-!
# Elementary bounds for the actual expectation thresholds

When the empty set is absent from the family, zero is feasible for both
thresholds. Their feasible sets are bounded by one, and indicator covers
give the integral-to-fractional comparison.
-/

namespace TalagrandThreshold

noncomputable section

variable {α : Type*} [Fintype α] [DecidableEq α]

omit [Fintype α] [DecidableEq α] in
theorem integralFeasible_subset_Icc (F : Family α) :
    integralFeasible F ⊆ Set.Icc (0 : ℝ) 1 :=
  fun _ hp => ⟨hp.1, hp.2.1⟩

omit [DecidableEq α] in
theorem fractionalFeasible_subset_Icc (F : Family α) :
    fractionalFeasible F ⊆ Set.Icc (0 : ℝ) 1 :=
  fun _ hp => ⟨hp.1, hp.2.1⟩

omit [Fintype α] [DecidableEq α] in
theorem integralFeasible_bddAbove (F : Family α) : BddAbove (integralFeasible F) :=
  ⟨1, fun _ hp => hp.2.1⟩

omit [DecidableEq α] in
theorem fractionalFeasible_bddAbove (F : Family α) : BddAbove (fractionalFeasible F) :=
  ⟨1, fun _ hp => hp.2.1⟩

omit [Fintype α] [DecidableEq α] in
theorem zero_mem_integralFeasible (F : Family α) (hF : (∅ : Finset α) ∉ F) :
    (0 : ℝ) ∈ integralFeasible F :=
  ⟨le_rfl, zero_le_one, small_zero F hF⟩

theorem zero_mem_fractionalFeasible (F : Family α) (hF : (∅ : Finset α) ∉ F) :
    (0 : ℝ) ∈ fractionalFeasible F :=
  ⟨le_rfl, zero_le_one, fractionalSmall_zero F hF⟩

omit [Fintype α] [DecidableEq α] in
theorem integralFeasible_nonempty (F : Family α) (hF : (∅ : Finset α) ∉ F) :
    (integralFeasible F).Nonempty :=
  ⟨0, zero_mem_integralFeasible F hF⟩

theorem fractionalFeasible_nonempty (F : Family α) (hF : (∅ : Finset α) ∉ F) :
    (fractionalFeasible F).Nonempty :=
  ⟨0, zero_mem_fractionalFeasible F hF⟩

omit [Fintype α] [DecidableEq α] in
/-- Integral feasibility is preserved by lowering the nonnegative density. -/
theorem integralFeasible_mono {F : Family α} {p r : ℝ}
    (hr : r ∈ integralFeasible F) (hp : 0 ≤ p) (hpr : p ≤ r) :
    p ∈ integralFeasible F :=
  ⟨hp, hpr.trans hr.2.1, hr.2.2.mono hp hpr⟩

omit [DecidableEq α] in
/-- Fractional feasibility is preserved by lowering the nonnegative density. -/
theorem fractionalFeasible_mono {F : Family α} {p r : ℝ}
    (hr : r ∈ fractionalFeasible F) (hp : 0 ≤ p) (hpr : p ≤ r) :
    p ∈ fractionalFeasible F :=
  ⟨hp, hpr.trans hr.2.1, hr.2.2.mono hp hpr⟩

theorem integralFeasible_subset_fractionalFeasible (F : Family α) :
    integralFeasible F ⊆ fractionalFeasible F :=
  fun _ hp => ⟨hp.1, hp.2.1, hp.2.2.fractionalSmall⟩

omit [Fintype α] [DecidableEq α] in
theorem le_q_of_mem {F : Family α} {p : ℝ} (hp : p ∈ integralFeasible F) :
    p ≤ q F :=
  le_csSup (integralFeasible_bddAbove F) hp

omit [DecidableEq α] in
theorem le_qf_of_mem {F : Family α} {p : ℝ} (hp : p ∈ fractionalFeasible F) :
    p ≤ qf F :=
  le_csSup (fractionalFeasible_bddAbove F) hp

omit [Fintype α] [DecidableEq α] in
theorem le_q_of_small {F : Family α} {p : ℝ}
    (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (hp : Small p F) : p ≤ q F :=
  le_q_of_mem ⟨hp0, hp1, hp⟩

omit [DecidableEq α] in
theorem le_qf_of_fractionalSmall {F : Family α} {p : ℝ}
    (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (hp : FractionalSmall p F) : p ≤ qf F :=
  le_qf_of_mem ⟨hp0, hp1, hp⟩

omit [Fintype α] [DecidableEq α] in
theorem q_nonneg (F : Family α) (hF : (∅ : Finset α) ∉ F) : 0 ≤ q F :=
  le_q_of_mem (zero_mem_integralFeasible F hF)

theorem qf_nonneg (F : Family α) (hF : (∅ : Finset α) ∉ F) : 0 ≤ qf F :=
  le_qf_of_mem (zero_mem_fractionalFeasible F hF)

omit [Fintype α] [DecidableEq α] in
theorem q_le_one (F : Family α) (hF : (∅ : Finset α) ∉ F) : q F ≤ 1 :=
  csSup_le (integralFeasible_nonempty F hF) (fun _ hp => hp.2.1)

theorem qf_le_one (F : Family α) (hF : (∅ : Finset α) ∉ F) : qf F ≤ 1 :=
  csSup_le (fractionalFeasible_nonempty F hF) (fun _ hp => hp.2.1)

theorem q_le_qf (F : Family α) (hF : (∅ : Finset α) ∉ F) : q F ≤ qf F :=
  csSup_le_csSup (fractionalFeasible_bddAbove F) (integralFeasible_nonempty F hF)
    (integralFeasible_subset_fractionalFeasible F)

end
end TalagrandThreshold

end OAI