File size: 3,851 Bytes
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 | import OAI.Combinatorics.EditApproximation.Geometry.ActionLower
import OAI.Combinatorics.EditApproximation.Geometry.BalancedIntervals
import OAI.Combinatorics.EditApproximation.Geometry.StateGeometry
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
/-!
# Connection values at labeled physical children
Child table values retain their child labels even when two source
substrings coincide. The edit and shift lower bounds are proved for
the actual finite sum of these labeled values.
-/
namespace EditApproximation
open Finset EditDistortion
noncomputable def endpointActionValue {n M : ℕ} (q : TargetInterval n)
(states : Fin M → TargetInterval n) (value : Fin M → TargetInterval n → ℝ) : ℝ :=
endpointActionConnection q states + ∑ i, value i (states i)
theorem actual_endpoint_action_lower {α : Type*} (source target : List α)
{M : ℕ} (hM : 0 < M) (parent : TargetInterval source.length)
(q : TargetInterval target.length) (states : Fin M → TargetInterval target.length)
(a : ℝ) (ha : 1 ≤ a) (value : Fin M → TargetInterval target.length → ℝ)
(hcost : ∀ i, (stateEditCost
(substring source (sourceChild parent i).lo (sourceChild parent i).hi) target (states i) : ℝ) / a ≤
value i (states i))
(hgap : ∀ i, |stateLengthGap
(substring source (sourceChild parent i).lo (sourceChild parent i).hi) target (states i)| ≤
value i (states i)) :
max ((stateEditCost (substring source parent.lo parent.hi) target q : ℝ) / a)
|stateLengthGap (substring source parent.lo parent.hi) target q| ≤
endpointActionValue q states value := by
let piece := fun i : Fin M =>
substring source (sourceChild parent i).lo (sourceChild parent i).hi
let children := List.ofFn fun i : Fin M => (piece i, states i)
have hsource : (List.ofFn piece).flatten = substring source parent.lo parent.hi :=
sourceChild_partition source hM parent
have htarget : (children.map Prod.snd) = List.ofFn states := by
simp only [children, List.map_ofFn, Function.comp_def]
have hchildSource : (children.map Prod.fst).flatten = substring source parent.lo parent.hi := by
simpa only [children, List.map_ofFn, Function.comp_def] using hsource
have hconnection := child_connection_bound target children q
rw [hchildSource, htarget] at hconnection
simp only [children, List.map_ofFn, List.sum_ofFn, Function.comp_def] at hconnection
have hconnectionR : (stateEditCost (substring source parent.lo parent.hi) target q : ℝ) ≤
(∑ i, (stateEditCost (piece i) target (states i) : ℝ)) + endpointActionConnection q states := by
exact_mod_cast hconnection
have hshift := child_shift_connection_bound target children q
rw [hchildSource, htarget] at hshift
simp only [children, List.map_ofFn, List.sum_ofFn, Function.comp_def] at hshift
have hgapSum : (∑ i, |signedLengthGap (piece i)
(substring target (states i).lo (states i).hi)|) ≤ ∑ i, value i (states i) := by
apply sum_le_sum
intro i hi
rw [← stateLengthGap_actual]
exact hgap i
have hcostSum : (∑ i, (stateEditCost (piece i) target (states i) : ℝ)) / a ≤
∑ i, value i (states i) := by
rw [sum_div]
exact sum_le_sum (fun i _ => hcost i)
have hconn0 : (0 : ℝ) ≤ endpointActionConnection q states := Nat.cast_nonneg _
have ha0 : 0 < a := by linarith
have hconnDiv : (endpointActionConnection q states : ℝ) / a ≤ endpointActionConnection q states := by
apply (div_le_iff₀ ha0).2
nlinarith
apply max_le
· have hdivide := div_le_div_of_nonneg_right hconnectionR ha0.le
rw [add_div] at hdivide
unfold endpointActionValue
linarith
· rw [stateLengthGap_actual]
unfold endpointActionValue endpointActionConnection
linarith
end EditApproximation
end OAI
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