File size: 3,392 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 68 69 70 71 72 73 | import Mathlib
import OAI.Combinatorics.MatchingCount.Arithmetic.TreeNumberOps
namespace OAI
namespace MatchingFPRAS.BinaryArithmeticProc
open RegisterProc TreeData Turing
def pickLt : Proc (Fin 3) := atom
(.pop 2 (fun _ x=>x) (.pop 2 (fun x _=>x)
(.branch (fun x=>x.getD false)
(.push 0 (fun _=>false) (.push 0 (fun _=>false) (.push 0 (fun _=>true)
(.load (fun _=>none) .halt))))
(.push 0 (fun _=>false) (.load (fun _=>none) .halt)))))
lemma pickLt_exec (c : Ordering) :
Exec pickLt ![[],[],StackCompare.flags c] ![encode (bit (decide (c=.lt))),[],[]] 1 := by
apply exec_atom
cases c <;> simp [TM2.stepAux,StackCompare.flags,encode,bit]
all_goals funext k; fin_cases k <;> simp
def less : Proc (Fin 3) := seq compare pickLt
lemma less_exec (A B : List Bool) :
Exec less ![A,B,[]] ![encode (bit (decide (StackAddition.value A<StackAddition.value B))),[],[]]
(A.length+B.length+2) := by
have hh:=(compare_exec A B).seq (pickLt_exec (Ord.compare (StackAddition.value A) (StackAddition.value B)))
simp only [compare_lt_iff_lt] at hh
exact hh.mono (by omega)
end MatchingFPRAS.BinaryArithmeticProc
namespace MatchingFPRAS.TreeArithmetic
open RegisterProc TreeData TreeMachine Turing
variable {R : Type} [DecidableEq R]
def less (i j : R) (hij : i≠j) : Proc (K R) :=
seq (decode i) (seq (decode j) (seq (reindex BinaryArithmeticProc.less (pairEmbedding i j hij))
(atom (.push (.inl j) (fun _=>false) .halt))))
theorem less_bits_exec (E : Env R) (i j : R) (hij : i≠j) (A B : List Bool)
(hi : E i=bits A) (hj : E j=bits B) :
Exec (less i j hij) (encodeEnv E)
(encodeEnv (putPair E i j (bit (decide (StackAddition.value A<StackAddition.value B))) .nil))
(3*(A.length+B.length)+7) := by
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.less E i j hij A B _ [] _ (BinaryArithmeticProc.less_exec A B)
have h₄ : Exec (atom (.push (.inl j) (fun _=>false) .halt))
(rawPair E i j (TreeData.encode (bit (decide (StackAddition.value A<StackAddition.value B)))) [])
(rawPair E i j (TreeData.encode (bit (decide (StackAddition.value A<StackAddition.value B)))) [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))
have hs : rawPair E i j (TreeData.encode (bits A)) (TreeData.encode (bits B))=encodeEnv E := by
rw [←hi,←hj]; exact rawPair_same E i j
have he : rawPair E i j (TreeData.encode (bit (decide (StackAddition.value A<StackAddition.value B)))) [false]=
encodeEnv (putPair E i j (bit (decide (StackAddition.value A<StackAddition.value B))) .nil) :=
rawPair_encode E i j _ .nil
rw [hs,he] at hh
exact hh.mono (by omega)
lemma less_exec (E : Env R) (i j : R) (hij : i≠j) (m n : ℕ)
(hi : E i=nat m) (hj : E j=nat n) :
Exec (less i j hij) (encodeEnv E) (encodeEnv (putPair E i j (bit (decide (m<n))) .nil))
(3*(m.bits.length+n.bits.length)+7) := by
simpa only [StackAddition.value_bits] using less_bits_exec E i j hij m.bits n.bits hi hj
end MatchingFPRAS.TreeArithmetic
end OAI
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