| import OAI.Computability.MatchingCount.RawContraction |
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| namespace OAI |
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| section |
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| namespace BinaryMatching.Raw.Compiler |
| open StackCode StackCode.Poly |
| @[fun_prop] theorem doubleNatSum {α : Type} [Coding α] [Inhabited α] |
| (n : α → ℕ) (f : (α×ℕ)×ℕ → ℕ) (hn : Poly n) (hf : Poly f) : |
| Poly (fun a => (tab (n a) (fun i => (tab (n a) (fun j => f ((a,i),j))).sum)).sum) := by |
| apply Poly.comp List.sum _ Poly.listNatSum |
| apply Poly.rawTab n (fun z : α×ℕ => (tab (n z.1) (fun j => f (z,j))).sum) hn |
| apply Poly.comp List.sum _ Poly.listNatSum |
| exact Poly.rawTab (fun z : α×ℕ => n z.1) f (by fun_prop) hf |
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| @[fun_prop] theorem outside_poly : Poly (fun z : ℕ×Cut => outside z.1 z.2) := by |
| unfold outside |
| exact Poly.findWith (fun z : ℕ×Cut => List.range z.1) |
| (fun z : (ℕ×Cut)×ℕ => !mem z.1.2 z.2) (by fun_prop) (by fun_prop) |
| @[fun_prop] theorem contractMap_poly : Poly (fun z : Cut×(ℕ×ℕ) => contractMap z.1 z.2.1 z.2.2) := by |
| unfold contractMap |
| fun_prop |
| @[fun_prop] theorem contract_poly : Poly (fun z : ℕ×(Matrix ℕ×(Cut×ℕ)) => contract z.1 z.2.1 z.2.2.1 z.2.2.2) := by |
| unfold contract |
| apply Poly.rawMatrix Prod.fst |
| (fun z : ((ℕ×(Matrix ℕ×(Cut×ℕ)))×ℕ)×ℕ => if z.1.2=z.2 then 0 else |
| (tab z.1.1.1 (fun u => (tab z.1.1.1 (fun v => |
| if contractMap z.1.1.2.2.1 z.1.1.2.2.2 u=z.1.2 ∧ contractMap z.1.1.2.2.1 z.1.1.2.2.2 v=z.2 |
| then get z.1.1.2.1 u v else 0)).sum)).sum) |
| · fun_prop |
| · have hs := doubleNatSum |
| (fun z : ((ℕ×(Matrix ℕ×(Cut×ℕ)))×ℕ)×ℕ => z.1.1.1) |
| (fun z : ((((ℕ×(Matrix ℕ×(Cut×ℕ)))×ℕ)×ℕ)×ℕ)×ℕ => |
| if decide (contractMap z.1.1.1.1.2.2.1 z.1.1.1.1.2.2.2 z.1.2=z.1.1.1.2) && |
| decide (contractMap z.1.1.1.1.2.2.1 z.1.1.1.1.2.2.2 z.2=z.1.1.2) |
| then get z.1.1.1.1.2.1 z.1.2 z.2 else 0) |
| (by fun_prop) (by fun_prop) |
| have h : Poly (fun z : ((ℕ×(Matrix ℕ×(Cut×ℕ)))×ℕ)×ℕ => if decide (z.1.2=z.2) then 0 else |
| (tab z.1.1.1 (fun u => (tab z.1.1.1 (fun v => |
| if decide (contractMap z.1.1.2.2.1 z.1.1.2.2.2 u=z.1.2) && decide (contractMap z.1.1.2.2.1 z.1.1.2.2.2 v=z.2) |
| then get z.1.1.2.1 u v else 0)).sum)).sum) := by |
| fun_prop |
| exact h.congr (fun z => by simp only [decide_eq_true_eq,Bool.and_eq_true]) |
| @[fun_prop] theorem lift_poly : Poly (fun z : ℕ×(Cut×(ℕ×Cut)) => lift z.1 z.2.1 z.2.2.1 z.2.2.2) := by |
| unfold lift |
| exact Poly.rawTab Prod.fst |
| (fun z : (ℕ×(Cut×(ℕ×Cut)))×ℕ => mem z.1.2.2.2 (contractMap z.1.2.1 z.1.2.2.1 z.2)) |
| (by fun_prop) (by fun_prop) |
| @[fun_prop] theorem cheaper_poly : Poly (fun z : ℕ×(Matrix ℕ×(Cut×Cut)) => cheaper z.1 z.2.1 z.2.2.1 z.2.2.2) := by |
| unfold cheaper |
| have h : Poly (fun z : ℕ×(Matrix ℕ×(Cut×Cut)) => |
| if decide (cutCapacity z.1 z.2.1 z.2.2.1≤cutCapacity z.1 z.2.1 z.2.2.2) then z.2.2.1 else z.2.2.2) := by fun_prop |
| exact h.congr (fun _ => by simp) |
| end BinaryMatching.Raw.Compiler |
| end |
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| end OAI |
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