File size: 3,365 Bytes
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 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | import Mathlib
import OAI.Combinatorics.MatchingCount.Arithmetic.TreeNumberOps
import OAI.Combinatorics.MatchingCount.Arithmetic.SubtractProc
namespace OAI
namespace MatchingFPRAS.TreeArithmetic
open RegisterProc TreeData TreeMachine Turing
variable {R : Type} [DecidableEq R]
def sub (i j : R) (hij : i≠j) : Proc (K R) :=
seq (decode i) (seq (decode j)
(seq (reindex BinaryArithmeticProc.sub (pairEmbedding i j hij))
(seq (normalize i) (seq (encode i) (atom (.push (.inl j) (fun _=>false) .halt))))))
theorem sub_bits_exec (E : Env R) (i j : R) (hij : i≠j) (A B : List Bool)
(hm : E i=bits A) (hn : E j=bits B) (hle : StackAddition.value B≤StackAddition.value A) :
Exec (sub i j hij) (encodeEnv E) (encodeEnv (putPair E i j (nat (StackAddition.value A-StackAddition.value B)) .nil))
(9*(A.length+B.length)+20) := by
let S:=(StackSubtract.subDigits false A B).2
have h₁:=on_left TreeBits.decodeForward E i j hij (TreeData.encode (bits A)) (TreeData.encode (bits B)) A _
(TreeBits.decodeForward_exec A [])
have h₂:=on_right TreeBits.decodeForward E i j hij A (TreeData.encode (bits B)) B _
(TreeBits.decodeForward_exec B [])
have h₃:=on_pair BinaryArithmeticProc.sub E i j hij A B S [] _ (BinaryArithmeticProc.sub_exec A B)
have h₄:=on_left BinaryNormalize.program E i j hij S [] (BinaryNormalize.normalize S) _
(BinaryNormalize.exec S [])
have h₅:=on_left TreeBits.encodeForward E i j hij (BinaryNormalize.normalize S) []
(TreeData.encode (bits (BinaryNormalize.normalize S))) _ (TreeBits.encodeForward_exec _ [])
have h₆ : Exec (atom (.push (.inl j) (fun _=>false) .halt))
(rawPair E i j (TreeData.encode (bits (BinaryNormalize.normalize S))) [])
(rawPair E i j (TreeData.encode (bits (BinaryNormalize.normalize S))) [false]) 1 := by
apply exec_atom
simp only [TM2.stepAux,rawPair_right]
congr 1
simp [rawPair]
have hh:=h₁.seq (h₂.seq (h₃.seq (h₄.seq (h₅.seq h₆))))
have hS : BinaryNormalize.normalize S=(StackAddition.value A-StackAddition.value B).bits := by
rw [BinaryNormalize.normalize_bits]
congr 1
simpa only [Bool.toNat_false,Nat.add_zero,Nat.sub_zero] using
(StackSubtract.subDigits_correct A B false (by simpa using hle)).2
rw [hS] at hh
have hstart : rawPair E i j (TreeData.encode (bits A)) (TreeData.encode (bits B))=encodeEnv E := by
rw [←hm,←hn]
exact rawPair_same E i j
have hend : rawPair E i j (TreeData.encode (bits (StackAddition.value A-StackAddition.value B).bits)) [false]=
encodeEnv (putPair E i j (nat (StackAddition.value A-StackAddition.value B)) .nil) := rawPair_encode E i j (nat (StackAddition.value A-StackAddition.value B)) .nil
rw [hstart,hend] at hh
apply hh.mono
have hl:=StackSubtract.subDigits_length A B false
have hl' := BinaryNormalize.normalize_length S
rw [hS] at hl'
dsimp only [S] at hl hl' ⊢
omega
lemma sub_exec (E : Env R) (i j : R) (hij : i≠j) (m n : ℕ)
(hi : E i=nat m) (hj : E j=nat n) (hle : n ≤ m) :
Exec (sub i j hij) (encodeEnv E) (encodeEnv (putPair E i j (nat (m-n)) .nil))
(9*(m.bits.length+n.bits.length)+20) := by
simpa only [StackAddition.value_bits] using sub_bits_exec E i j hij m.bits n.bits hi hj
(by simpa only [StackAddition.value_bits] using hle)
end MatchingFPRAS.TreeArithmetic
end OAI
|