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2f887db | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 | import Mathlib.Data.Finset.Lattice.Fold
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Prod
import Mathlib.Basic.Real.Basic
import Mathlib.Tactic.NormNum
import OAI.Computability.PerfectCompleteness.Model
import OAI.Computability.PerfectCompleteness.Reduction.Target
namespace OAI
section
namespace PerfectCompleteness
namespace Instance
theorem countSatisfied_le_maxSatisfied {q : Nat} (G : Instance q) (s : Labeling G) :
countSatisfied s.1 s.2 G.edges ≤ G.maxSatisfied := by
exact Finset.le_sup (f := fun s : Labeling G => countSatisfied s.1 s.2 G.edges)
(Finset.mem_univ s)
theorem maxSatisfied_le_length {q : Nat} (G : Instance q) :
G.maxSatisfied ≤ G.edges.length := by
apply Finset.sup_le
intro s _
exact countSatisfied_le_length s.1 s.2 G.edges
theorem maxSatisfied_attained {q : Nat} (G : Instance q) (hq : 0 < q) :
∃ s : Labeling G, countSatisfied s.1 s.2 G.edges = G.maxSatisfied := by
have hne : (Finset.univ : Finset (Labeling G)).Nonempty :=
⟨(fun _ => ⟨0, by omega⟩, fun _ => ⟨0, hq⟩), Finset.mem_univ _⟩
obtain ⟨s, _, h⟩ := Finset.exists_mem_eq_sup Finset.univ hne
(fun s : Labeling G => countSatisfied s.1 s.2 G.edges)
exact ⟨s, h.symm⟩
theorem value_nonneg {q : Nat} (G : Instance q) : 0 ≤ G.value :=
div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
theorem value_le_one {q : Nat} (G : Instance q) : G.value ≤ 1 := by
have hlen : (0 : ℝ) < G.edges.length := by exact_mod_cast G.edgeCount_positive
rw [value, div_le_iff₀ hlen, one_mul]
exact_mod_cast G.maxSatisfied_le_length
theorem rate_le_value {q : Nat} (G : Instance q) (s : Labeling G) :
(countSatisfied s.1 s.2 G.edges : ℝ) / G.edges.length ≤ G.value := by
apply div_le_div_of_nonneg_right
· exact_mod_cast G.countSatisfied_le_maxSatisfied s
· exact Nat.cast_nonneg _
theorem perfectlyComplete_iff_value_eq_one {q : Nat} (G : Instance q) (hq : 0 < q) :
PerfectlyComplete G ↔ G.value = 1 := by
have hlen : (0 : ℝ) < G.edges.length := by exact_mod_cast G.edgeCount_positive
constructor
· rintro ⟨s, hs⟩
have hc := (countSatisfied_eq_length_iff s.1 s.2 G.edges).2 hs
have hm : G.maxSatisfied = G.edges.length :=
le_antisymm G.maxSatisfied_le_length (hc ▸ G.countSatisfied_le_maxSatisfied s)
simp [value, hm, ne_of_gt hlen]
· intro hv
obtain ⟨s, hs⟩ := G.maxSatisfied_attained hq
have hm : G.maxSatisfied = G.edges.length := by
have hreal : (G.maxSatisfied : ℝ) = G.edges.length := by
have := (div_eq_one_iff_eq (ne_of_gt hlen)).1 hv
exact this
exact_mod_cast hreal
exact ⟨s, (countSatisfied_eq_length_iff s.1 s.2 G.edges).1 (hs.trans hm)⟩
theorem soundAt_iff_value_le {q : Nat} (G : Instance q) (hq : 0 < q)
(d : RationalThreshold) :
SoundAt d G ↔ G.value ≤ (d.numerator : ℝ) / d.denominator := by
have hlen : (0 : ℝ) < G.edges.length := by exact_mod_cast G.edgeCount_positive
have hd : (0 : ℝ) < d.denominator := by
exact_mod_cast (lt_trans d.positive d.belowOne)
have hrat : ∀ s : Labeling G,
d.denominator * countSatisfied s.1 s.2 G.edges ≤ d.numerator * G.edges.length ↔
(countSatisfied s.1 s.2 G.edges : ℝ) / G.edges.length ≤
(d.numerator : ℝ) / d.denominator := by
intro s
rw [div_le_div_iff₀ hlen hd]
norm_cast
rw [Nat.mul_comm (countSatisfied _ _ _) d.denominator]
constructor
· intro h
obtain ⟨s, hs⟩ := G.maxSatisfied_attained hq
have := (hrat s).1 (h s)
simpa [value, hs] using this
· intro h s
exact (hrat s).2 ((G.rate_le_value s).trans h)
end Instance
end PerfectCompleteness
end
end OAI
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