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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 | import OAI.Combinatorics.EditApproximation.Geometry.GlobalQueryCost
namespace OAI
namespace EditApproximation.FiniteQuery
theorem cost_mono_weights {ι : Type*} (oracle : ι → ℚ) (program : FiniteQuery ι)
(left right : ι → ℕ) (hle : ∀ key, left key ≤ right key) :
program.cost oracle left ≤ program.cost oracle right := by
induction program with
| done result => exact le_rfl
| read key next ih => exact Nat.add_le_add (hle key) (ih (oracle key))
theorem cost_zero {ι : Type*} (oracle : ι → ℚ) (program : FiniteQuery ι) :
program.cost oracle (fun _ => 0) = 0 := by
induction program with
| done result => rfl
| read key next ih => simpa only [cost, Nat.zero_add] using ih (oracle key)
theorem cost_eq_zero_of_keys {ι : Type*} (oracle : ι → ℚ) (program : FiniteQuery ι)
(weight : ι → ℕ) (hzero : ∀ key ∈ program.keys oracle, weight key = 0) :
program.cost oracle weight = 0 :=
(cost_congr_weights oracle program weight (fun _ => 0) hzero).trans (cost_zero oracle program)
end EditApproximation.FiniteQuery
end OAI
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