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e63eb7f | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 | import OAI.Combinatorics.EditApproximation.Geometry.RankedMemoWork
import OAI.Combinatorics.EditApproximation.Geometry.DependencyExpansion
namespace OAI
/-! The dependency tree here is obtained from the requests made by the
finite adaptive program at its actual answers. Local depth/index progress
therefore bounds execution, rather than an independently supplied tree. -/
namespace EditApproximation
open Finset
section Queries
variable {κ : Type*} (depth index : κ → ℕ)
(program : (key : κ) → FiniteQuery
{child : κ // depth child + index child < depth key + index key})
def queryRequestList (key : κ) : List κ :=
((program key).keys (fun child =>
rankedProgramValue (fun x => depth x + index x) program child.val)).map Subtype.val
def querySameRequests (key : κ) : List κ :=
(queryRequestList depth index program key).filter (fun child => !(index child < index key))
def queryEarlierRequests (key : κ) : List κ :=
(queryRequestList depth index program key).filter (fun child => index child < index key)
theorem queryRequestList_cost (key : κ) (weight : κ → ℕ) :
(program key).cost
(fun child => rankedProgramValue (fun x => depth x + index x) program child.val)
(fun child => weight child.val) =
((queryRequestList depth index program key).map weight).sum := by
rw [FiniteQuery.cost_eq_sum]
simp only [queryRequestList, List.map_map, Function.comp_def]
theorem sum_map_partition (predicate : κ → Bool) (weight : κ → ℕ) (keys : List κ) :
((keys.filter (fun key => !predicate key)).map weight).sum +
((keys.filter predicate).map weight).sum = (keys.map weight).sum := by
induction keys with
| nil => rfl
| cons key keys ih =>
cases hp : predicate key <;> simp only [List.filter_cons, hp, Bool.not_false,
Bool.not_true, Bool.false_eq_true, reduceIte, List.map_cons, List.sum_cons]
all_goals omega
variable (progress : ∀ key child, child ∈ queryRequestList depth index program key →
(depth child < depth key ∧ index child ≤ index key) ∨
(depth child ≤ depth key ∧ index child < index key))
include progress
theorem querySameRequests_progress (key : κ)
(i : Fin (querySameRequests depth index program key).length) :
depth ((querySameRequests depth index program key).get i) < depth key ∧
index ((querySameRequests depth index program key).get i) ≤ index key := by
have hm := List.mem_filter.mp (List.get_mem (querySameRequests depth index program key) i)
have hnot : ¬ index ((querySameRequests depth index program key).get i) < index key := by
simpa using hm.2
rcases progress key _ hm.1 with h | h
· exact h
· exact False.elim (hnot h.2)
theorem queryEarlierRequests_progress (key : κ)
(i : Fin (queryEarlierRequests depth index program key).length) :
depth ((queryEarlierRequests depth index program key).get i) ≤ depth key ∧
index ((queryEarlierRequests depth index program key).get i) < index key := by
have hm := List.mem_filter.mp (List.get_mem (queryEarlierRequests depth index program key) i)
have hi : index ((queryEarlierRequests depth index program key).get i) < index key := by
simpa only [decide_eq_true_eq] using hm.2
refine ⟨?_, hi⟩
rcases progress key _ hm.1 with h | h
· exact h.1.le
· exact h.1
def queryDependencyExpansion (root : κ) : DependencyExpansion :=
DependencyExpansion.build depth index
(fun key => (querySameRequests depth index program key).length)
(fun key => (queryEarlierRequests depth index program key).length)
(fun key i => (querySameRequests depth index program key).get i)
(fun key i => (queryEarlierRequests depth index program key).get i)
(querySameRequests_progress depth index program progress)
(queryEarlierRequests_progress depth index program progress) root
theorem queryDependencyExpansion_depth (root : κ) :
(queryDependencyExpansion depth index program progress root).depth = depth root :=
DependencyExpansion.build_depth _ _ _ _ _ _ _ _ root
theorem queryDependencyExpansion_index (root : κ) :
(queryDependencyExpansion depth index program progress root).rank = index root :=
DependencyExpansion.build_rank _ _ _ _ _ _ _ _ root
theorem queryDependencyExpansion_valid (a b : ℕ)
(hsame : ∀ key, (querySameRequests depth index program key).length ≤ a)
(hearlier : ∀ key, (queryEarlierRequests depth index program key).length ≤ b) (root : κ) :
DependencyExpansion.Valid a b (queryDependencyExpansion depth index program progress root) :=
DependencyExpansion.build_valid _ _ _ _ _ _ _ _ a b hsame hearlier root
theorem queryDependencyExpansion_nodes (root : κ) :
(queryDependencyExpansion depth index program progress root).nodeCount =
rankedUncachedCalls (fun key => depth key + index key) program root := by
have hall : ∀ level root, depth root + index root = level →
(queryDependencyExpansion depth index program progress root).nodeCount =
rankedUncachedCalls (fun key => depth key + index key) program root := by
intro level
induction level using Nat.strong_induction_on with
| h level ih =>
intro root hroot
conv_lhs => unfold queryDependencyExpansion
rw [DependencyExpansion.build]
change 1 + (∑ i : Fin (querySameRequests depth index program root).length,
(queryDependencyExpansion depth index program progress
((querySameRequests depth index program root).get i)).nodeCount) +
(∑ i : Fin (queryEarlierRequests depth index program root).length,
(queryDependencyExpansion depth index program progress
((queryEarlierRequests depth index program root).get i)).nodeCount) = _
have hs (i : Fin (querySameRequests depth index program root).length) :=
querySameRequests_progress depth index program progress root i
have he (i : Fin (queryEarlierRequests depth index program root).length) :=
queryEarlierRequests_progress depth index program progress root i
have hsame : (∑ i : Fin (querySameRequests depth index program root).length,
(queryDependencyExpansion depth index program progress
((querySameRequests depth index program root).get i)).nodeCount) =
(((querySameRequests depth index program root).map
(rankedUncachedCalls (fun key => depth key + index key) program)).sum) := by
calc
_ = ∑ i : Fin (querySameRequests depth index program root).length,
rankedUncachedCalls (fun key => depth key + index key) program
((querySameRequests depth index program root).get i) := by
apply sum_congr rfl
intro i hi
apply ih _ _ _ rfl
rw [← hroot]
exact Nat.add_lt_add_of_lt_of_le (hs i).1 (hs i).2
_ = _ := Fin.sum_univ_fun_getElem _ _
have hearlier : (∑ i : Fin (queryEarlierRequests depth index program root).length,
(queryDependencyExpansion depth index program progress
((queryEarlierRequests depth index program root).get i)).nodeCount) =
(((queryEarlierRequests depth index program root).map
(rankedUncachedCalls (fun key => depth key + index key) program)).sum) := by
calc
_ = ∑ i : Fin (queryEarlierRequests depth index program root).length,
rankedUncachedCalls (fun key => depth key + index key) program
((queryEarlierRequests depth index program root).get i) := by
apply sum_congr rfl
intro i hi
apply ih _ _ _ rfl
rw [← hroot]
exact Nat.add_lt_add_of_le_of_lt (he i).1 (he i).2
_ = _ := Fin.sum_univ_fun_getElem _ _
rw [hsame, hearlier, rankedUncachedCalls, queryRequestList_cost]
have hp := sum_map_partition (fun child => decide (index child < index root))
(rankedUncachedCalls (fun key => depth key + index key) program)
(queryRequestList depth index program root)
change 1 + _ + _ = _ + 1
change _ + _ = _ at hp
dsimp only [querySameRequests, queryEarlierRequests]
omega
exact hall (depth root + index root) root rfl
theorem rankedMemoEvaluate_dependency_bound (a b bits : ℕ) (encode : κ → ℕ)
(hinjective : Function.Injective encode) (hbounded : ∀ key, encode key < 2 ^ bits)
(hsame : ∀ key, (querySameRequests depth index program key).length ≤ a)
(hearlier : ∀ key, (queryEarlierRequests depth index program key).length ≤ b) (root : κ) :
(rankedMemoEvaluate (fun key => depth key + index key) program bits encode root .empty).requests ≤
dependencyPathBudget (depth root) (index root) a b := by
calc
_ ≤ rankedUncachedCalls (fun key => depth key + index key) program root :=
rankedMemoEvaluate_requests_le _ _ bits encode hinjective hbounded root .empty
(memoSound_empty bits encode _)
_ = (queryDependencyExpansion depth index program progress root).nodeCount :=
(queryDependencyExpansion_nodes depth index program progress root).symm
_ ≤ dependencyPathBudget (depth root) (index root) a b := by
simpa only [queryDependencyExpansion_depth, queryDependencyExpansion_index] using
DependencyExpansion.nodeCount_le_budget a b _
(queryDependencyExpansion_valid depth index program progress a b hsame hearlier root)
end Queries
end EditApproximation
end OAI
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