| import OAI.Computability.SpaceSimulation.Epoch.Nodes |
|
|
| namespace OAI |
|
|
| namespace Fifths.Tour |
| variable (D : Tour) |
|
|
| attribute [local irreducible] Fifths.Circuit.price |
| theorem coins_array {e i} (l : e<D.E) (h : i < D.N) : |
| D.coins (D.mkV e i) = e*D.per := by |
| simp [coins, D.ae l h,D.ac l h,h] |
|
|
| theorem coins_history {e i} (l : e<D.E) (h : i ≤ D.k) : |
| D.coins (D.hV e i) = e*D.per + D.coinsHard + (i+1)*D.coinsEasy := by |
| simp [coins,D.he l h,D.hc l h,D.hr l h] |
|
|
| -- Both kinds at once. Charge a retained Y only on a call into an array |
| -- column of this epoch. |
| theorem supplied (v : D.Vertex) (p : D.baseOf v = false) : |
| D.coins (D.header v) + D.coinsEasy ≤ D.coins v ∧ |
| (∀ i < D.N, D.coins (D.childrenOf v i) + D.circuit.price (D.guesses v) ≤ D.coins v) := by |
| rcases D.either v with ⟨e,l,j,b,rfl⟩|⟨e,l,j,b,rfl⟩ |
| · cases e with |
| | zero => simp [baseOf,D.ae l b,D.ac l b,b.le] at p |
| | succ e => |
| have h : e<D.E := by omega |
| simp only [coins_array D l b,guesses,header,childrenOf,D.ac l b,D.ae l b,b,b.le, |
| ↓reduceIte,Nat.add_sub_cancel,coins_array D h b,coins_history D h le_rfl,show D.circuit.price 0 = D.coinsEasy from rfl] |
| constructor |
| · unfold per |
| nlinarith |
| · intro i _ |
| unfold per |
| nlinarith |
| · cases j with |
| | zero => |
| cases e with |
| | zero => simp [baseOf,D.he l b,D.hc l b] at p |
| | succ e => |
| have f : D.col (D.hV (e+1) 0) = D.N := D.hc l b |
| have h : e<D.E := by omega |
| simp only [coins_history D l b,guesses,header,childrenOf,f,D.he l b,le_refl, |
| lt_self_iff_false,↓reduceIte,Nat.add_sub_cancel,coins_history D h le_rfl,show D.circuit.price 0 = D.coinsEasy from rfl] |
| constructor |
| · unfold per; nlinarith |
| · intro i _ |
| unfold per; nlinarith |
| | succ j => |
| have f : D.col (D.hV e (j+1)) > D.N := by rw [D.hc l b]; omega |
| have q : j ≤ D.k := by omega |
| simp only [header,guesses,childrenOf,D.he l b,D.hr l b,not_le.mpr f,Nat.not_lt.mpr f.le, |
| Ne.symm f.ne,↓reduceIte,Nat.add_sub_cancel,coins_history D l b,coins_history D l q, |
| show D.circuit.price D.guessLength = D.coinsHard from rfl] |
| constructor |
| · nlinarith |
| · intro i u; rw [D.coins_array l u]; omega |
|
|
| theorem covers : D.circuit.Meter D.coins where |
| cheap := fun v h => (D.supplied v h).1 |
| expensive := fun v h => (D.supplied v h).2 |
|
|
| theorem coins_total (v : D.Vertex) : D.coins v ≤ D.E * D.per := by |
| obtain (⟨e,l,j,b,rfl⟩|⟨e,l,j,b,rfl⟩) := D.either v |
| · rw [D.coins_array l b] |
| exact Nat.mul_le_mul_right _ (Nat.le_of_lt l) |
| · rw [D.coins_history l b] |
| calc |
| _ ≤ (e+1)*D.per := by unfold per; nlinarith |
| _ ≤ _ := Nat.mul_le_mul_right _ l |
| end Fifths.Tour |
|
|
| end OAI |
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|
