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d4317d7 | 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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 | import OAI.MeasureTheory.Falconer.Estimates.ActualDirectBuffer
import OAI.MeasureTheory.Falconer.Recursion.ActualPolicyPayments
import OAI.MeasureTheory.Falconer.Recursion.InheritedPolicyState
namespace OAI
namespace PlanarFalconer
theorem remaining_descriptor_companion (A : ℕ → ℝ) {N q a₀ t₀ a : ℕ}
(hq : 0 < q) {d : PolicyDescriptor}
(hd : d ∈ remainingPolicyDescriptors A N q a₀ t₀ a) :
PolicyDescriptor.parallel (max a (descriptorAnchor d-descriptorLength d))
(descriptorAnchor d) ∈ remainingPolicyDescriptors A N q a₀ t₀ a := by
classical
obtain ⟨hd,hap⟩ := Finset.mem_filter.mp hd
obtain ⟨z,hz,hd⟩ := Finset.mem_biUnion.mp hd
have hz' := Finset.mem_filter.mp hz
have huN : z.2 ≤ N := Nat.le_of_lt_succ (Finset.mem_range.mp (Finset.mem_product.mp hz'.1).2)
have hp := originDescriptor_anchor A hd
have hl := originDescriptor_length A hq huN hz'.2.2.2 hd
have hLp := originDescriptor_length_le_anchor A hd
apply Finset.mem_filter.mpr
refine ⟨Finset.mem_biUnion.mpr ⟨z,hz,?_⟩,hap⟩
apply Finset.mem_union_left
apply Finset.mem_image.mpr
refine ⟨max a (descriptorAnchor d-descriptorLength d),Finset.mem_Ico.mpr ⟨?_,?_⟩,?_⟩
· rw [← hp]
omega
· omega
· rw [hp]
structure DirectPolicyWitness (A : ℕ → ℝ) (N q a₀ t₀ a v : ℕ) (sig : ℝ)
(d : PolicyDescriptor) : Prop where
mem : d ∈ remainingPolicyDescriptors A N q a₀ t₀ a
maximal : descriptorLength d = (remainingPolicyDescriptors A N q a₀ t₀ a).sup descriptorLength
buffer : sourceFarBuffer N (descriptorLength d) sig = descriptorLength d
direct_depth : v ≤ a+2*descriptorLength d
anchor_gt : a < descriptorAnchor d
anchor_lt : descriptorAnchor d < v
width : descriptorAnchor d+10*q ≤ v
companion : PolicyDescriptor.parallel (max a (descriptorAnchor d-descriptorLength d))
(descriptorAnchor d) ∈ remainingPolicyDescriptors A N q a₀ t₀ a
g_lt : max a (descriptorAnchor d-descriptorLength d) < descriptorAnchor d
g_slack : ((max a (descriptorAnchor d-descriptorLength d):ℕ):ℝ)-a ≤ sig*N
frequency_slack : 10*sig*N ≤ (v:ℝ)-descriptorAnchor d
depth_scale : ((descriptorAnchor d-a:ℕ):ℝ) ≤ (v:ℝ)-descriptorAnchor d+2*sig*N
length_scale : (descriptorLength d:ℝ) ≤ (v:ℝ)-descriptorAnchor d+2*sig*N
midpoint : 2*descriptorAnchor d ≤ a+v+2*q
shrinks : (remainingPolicyDescriptors A N q a₀ t₀ (descriptorAnchor d)).card <
(remainingPolicyDescriptors A N q a₀ t₀ a).card
payment : potential A a v (10*q)+height A a (descriptorAnchor d) ≤
potential A (descriptorAnchor d) v (10*q)
theorem actual_direct_policy_witness (A : ℕ → ℝ) {N q a₀ t₀ a v : ℕ} {sig : ℝ}
(hsig : 0 ≤ sig) (hq : 0 < q) (hceil : ⌈10*sig*(N:ℝ)⌉₊ < q)
(hslack : sig*(N:ℝ) ≤ (q:ℝ)) (hslack10 : 10*sig*(N:ℝ) ≤ (q:ℝ))
(ht : (t₀:ℝ) ≤ (v:ℝ)+sig*N) (hw : a+100*q < v)
(hdirect : v-2*sourceFarBuffer N
((remainingPolicyDescriptors A N q a₀ t₀ a).sup descriptorLength) sig ≤ a) :
∃ d : PolicyDescriptor, DirectPolicyWitness A N q a₀ t₀ a v sig d := by
classical
let S := remainingPolicyDescriptors A N q a₀ t₀ a
have hb := direct_buffer_eq_length hceil hw hdirect
have hS : S.Nonempty := by
by_contra h
have he := Finset.not_nonempty_iff_eq_empty.mp h
have hz : S.sup descriptorLength = 0 := by rw [he]; rfl
exact (Nat.ne_of_gt hb.2.2) hz
obtain ⟨d,hd,hmax⟩ := Finset.exists_mem_eq_sup S hS descriptorLength
dsimp only [S] at hmax
have hmax' := hmax.symm
have hdir : v ≤ a+2*descriptorLength d := by simpa only [hmax] using hb.2.1
have hmem := Finset.mem_filter.mp hd
have hap : a < descriptorAnchor d := hmem.2
obtain ⟨z,hz,hzd⟩ := Finset.mem_biUnion.mp hmem.1
have hz' := Finset.mem_filter.mp hz
have huN : z.2 ≤ N := Nat.le_of_lt_succ (Finset.mem_range.mp (Finset.mem_product.mp hz'.1).2)
have hl := originDescriptor_length A hq huN hz'.2.2.2 hzd
have hLp := originDescriptor_length_le_anchor A hzd
have hpt : descriptorAnchor d+descriptorLength d ≤ t₀ := hl.2.2.trans hz'.2.2.1
have hptR : (descriptorAnchor d:ℝ)+(descriptorLength d:ℝ) ≤ (t₀:ℝ) := by exact_mod_cast hpt
have hsigN : 0 ≤ sig*(N:ℝ) := mul_nonneg hsig (Nat.cast_nonneg _)
have hdirR : (v:ℝ) ≤ (a:ℝ)+2*(descriptorLength d:ℝ) := by exact_mod_cast hdir
have hmidR : 2*(descriptorAnchor d:ℝ) ≤ (a:ℝ)+v+2*q := by linarith
have hmid : 2*descriptorAnchor d ≤ a+v+2*q := by exact_mod_cast hmidR
have hwidth : descriptorAnchor d+10*q ≤ v := by omega
have hwidthR : (descriptorAnchor d:ℝ)+10*(q:ℝ) ≤ (v:ℝ) := by exact_mod_cast hwidth
have hn : ((descriptorAnchor d-a:ℕ):ℝ) = (descriptorAnchor d:ℝ)-a := Nat.cast_sub hap.le
have hgs : ((descriptorAnchor d-descriptorLength d:ℕ):ℝ) =
(descriptorAnchor d:ℝ)-descriptorLength d := Nat.cast_sub hLp
refine ⟨d, {
mem := hd
maximal := hmax'
buffer := ?_
direct_depth := hdir
anchor_gt := hap
anchor_lt := by omega
width := hwidth
companion := remaining_descriptor_companion A hq hd
g_lt := by omega
g_slack := ?_
frequency_slack := by linarith
depth_scale := ?_
length_scale := ?_
midpoint := hmid
shrinks := reached_anchor_strictly_reduces_list A N q a₀ t₀ hap.le hd le_rfl
payment := (actual_direct_policy_payment A hq hw hmem.1 hap hdir hmid).2 }⟩
· simpa only [← hmax] using hb.1
· rw [Nat.cast_max,hgs,sub_le_iff_le_add]
apply max_le <;> linarith [show 0 ≤ sig*(N:ℝ) by positivity]
· rw [hn]
linarith
· have haR : (a:ℝ) < (descriptorAnchor d:ℝ) := Nat.cast_lt.mpr hap
linarith
end PlanarFalconer
end OAI
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