| import OAI.Combinatorics.EditApproximation.Binary.BinaryGuardSpace |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryBandReads |
| import OAI.Combinatorics.EditApproximation.Definitions |
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| namespace OAI |
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| namespace EditApproximation.BinaryFraction |
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| theorem anchorGuardAllocation_le_work {nx ny M : ℕ} (parent : TargetInterval nx) (r : TargetInterval ny) |
| (b : ℕ) (states : Fin M → TargetInterval ny) (i : Fin M) : |
| anchorGuardAllocation parent r b states i ≤ |
| (translatedEndpointWithWork r.lo parent.lo (sourceChild parent i).lo).2 + |
| (translatedEndpointWithWork r.lo parent.lo (sourceChild parent i).hi).2 + |
| (binaryNaturalMulWithWork 6 b).2 + |
| (naturalDistanceLeWithWork (translatedEndpointWithWork r.lo parent.lo (sourceChild parent i).lo).1 |
| (states i).lo (bitWordValue (binaryNaturalMulWithWork 6 b).1)).2 + |
| (naturalDistanceLeWithWork (translatedEndpointWithWork r.lo parent.lo (sourceChild parent i).hi).1 |
| (states i).hi (bitWordValue (binaryNaturalMulWithWork 6 b).1)).2 + 3 := by |
| have hl := translatedEndpointAllocation_le_work r.lo parent.lo (sourceChild parent i).lo |
| have hr := translatedEndpointAllocation_le_work r.lo parent.lo (sourceChild parent i).hi |
| have hm := binaryNaturalMulAllocation_le_work 6 b |
| have hld := naturalDistanceLeAllocation_le_work |
| (translatedEndpointWithWork r.lo parent.lo (sourceChild parent i).lo).1 |
| (states i).lo (bitWordValue (binaryNaturalMulWithWork 6 b).1) |
| have hrd := naturalDistanceLeAllocation_le_work |
| (translatedEndpointWithWork r.lo parent.lo (sourceChild parent i).hi).1 |
| (states i).hi (bitWordValue (binaryNaturalMulWithWork 6 b).1) |
| unfold anchorGuardAllocation |
| dsimp only |
| omega |
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| theorem bandPredicateReadAllocation_le_work {nx ny M : ℕ} (parent : TargetInterval nx) (r : TargetInterval ny) |
| (b : ℕ) (tau theta velocity : BinaryFraction) |
| (actions : List (Fin M → TargetInterval ny)) (hne : actions ≠ []) |
| (read : Fin M → TargetInterval ny → BinaryFraction × ℕ) |
| (allocation : Fin M → TargetInterval ny → ℕ) (states : Fin M → TargetInterval ny) |
| (h : ∀ action ∈ actions, ∀ i, allocation i (action i) ≤ (read i (action i)).2) : |
| bandPredicateReadAllocation parent r b tau theta velocity actions hne read allocation states ≤ |
| (bandPredicateReadWithWork parent r b tau theta velocity actions hne read states).2 := by |
| have hz := zeroConnectionAllocation_le_work r.lo r.hi (List.ofFn states) |
| have ha := allAllocation_with_spine (fun i : Fin M => |
| let leftAnchor := translatedEndpointWithWork r.lo parent.lo (sourceChild parent i).lo |
| let rightAnchor := translatedEndpointWithWork r.lo parent.lo (sourceChild parent i).hi |
| let radius := binaryNaturalMulWithWork 6 b |
| let left := naturalDistanceLeWithWork leftAnchor.1 (states i).lo (bitWordValue radius.1) |
| let right := naturalDistanceLeWithWork rightAnchor.1 (states i).hi (bitWordValue radius.1) |
| (left.1 && right.1, leftAnchor.2 + rightAnchor.2 + radius.2 + left.2 + right.2 + 3)) |
| (anchorGuardAllocation parent r b states) (List.finRange M) |
| (fun i _ => anchorGuardAllocation_le_work parent r b states i) |
| have ht := bandToleranceAllocation_le_work tau theta b M |
| have hc := allAllocation_le_work |
| (cutGuardReadWithWork parent r actions hne read states velocity (bandToleranceWithWork tau theta b M).1) |
| (cutGuardReadAllocation parent r actions hne read allocation states velocity (bandToleranceWithWork tau theta b M).1) |
| (List.finRange M) (fun i _ => cutGuardReadAllocation_le_work parent r actions hne read allocation states velocity _ i h) |
| simp only [List.length_finRange] at ha |
| unfold bandPredicateReadAllocation bandPredicateReadWithWork |
| dsimp only at ha hc ⊢ |
| omega |
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| end EditApproximation.BinaryFraction |
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| end OAI |
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