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e5a97c6 | 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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.FiniteStore
import OAI.Combinatorics.MatchingCount.Arithmetic.BinarySubtract
namespace OAI
namespace MatchingFPRAS.ForwardSubtract
open Turing MachineCost StackAddition StackSubtract
local instance initialControl : Inhabited (Bool⊕Unit) := ⟨.inl false⟩
abbrev Γ := fun _ : Fin 3 => Bool
def program : Bool⊕Unit → TM2.Stmt Γ (Bool⊕Unit) Store :=
StackComposition.sequence StackSubtract.program ForwardAddition.reverseProgram
lemma run_sub (A B : List Bool) :
∃ k≤2*(A.length+B.length)+2,
MachineCost.run (TM2.step program) k ⟨some default,default,stackData A B []⟩=
⟨none,default,stackData ((subDigits false A B).2) [] []⟩ := by
obtain ⟨n,hn,he⟩:=run_subtraction A B [] false default
have he' : MachineCost.run (TM2.step StackSubtract.program) n
(StackSubtract.cfg (some false) A B [] default)=StackSubtract.cfg none [] [] ((subDigits false A B).2).reverse default := by
simpa only [List.append_nil] using he
have hr := ForwardAddition.run_reverse [] [] ((subDigits false A B).2).reverse default
have hr' : MachineCost.run (TM2.step ForwardAddition.reverseProgram) (((subDigits false A B).2).length+1)
⟨some default,default,stackData [] [] ((subDigits false A B).2).reverse⟩=
⟨none,default,stackData ((subDigits false A B).2) [] []⟩ := by
simpa only [ForwardAddition.rcfg,List.length_reverse,List.reverse_reverse,List.append_nil] using hr
obtain ⟨k,hk,hkrun⟩:=StackComposition.sequence_run StackSubtract.program ForwardAddition.reverseProgram
(StackSubtract.cfg (some false) A B [] default) n (((subDigits false A B).2).length+1)
(default : Store) (default : Store) (stackData [] [] ((subDigits false A B).2).reverse)
(stackData ((subDigits false A B).2) [] []) he' hr'
refine ⟨k,?_,?_⟩
· have hl:=subDigits_length A B false
omega
· exact hkrun
end MatchingFPRAS.ForwardSubtract
namespace MatchingFPRAS.BinaryArithmeticProc
open RegisterProc Turing
local instance subEntry : Inhabited (Bool⊕Unit) := ⟨.inl false⟩
def sub : Proc (Fin 3) := FiniteStore.procedure ForwardSubtract.program (default : StackAddition.Store)
lemma sub_exec (A B : List Bool) :
Exec sub ![A,B,[]] ![(StackSubtract.subDigits false A B).2,[],[]] (2*(A.length+B.length)+2) := by
obtain ⟨n,hn,he⟩:=ForwardSubtract.run_sub A B
exact (FiniteStore.exec ForwardSubtract.program default default _ _ n he).mono hn
end MatchingFPRAS.BinaryArithmeticProc
end OAI
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