:: Two Programs for {\bf SCM}. Part II - Proofs :: by Grzegorz Bancerek and Piotr Rudnicki :: :: Received October 8, 1993 :: Copyright (c) 1993-2012 Association of Mizar Users begin Lm1: 0 = [\(log (2,1))/] proofend; Lm2: for nn9 being Element of NAT st nn9 > 0 holds ( [\(log (2,nn9))/] is Element of NAT & (6 * ([\(log (2,nn9))/] + 1)) + 1 > 0 ) proofend; Lm3: for nn, nn9 being Element of NAT st nn = (2 * nn9) + 1 & nn9 > 0 holds 6 + ((6 * ([\(log (2,nn9))/] + 1)) + 1) = (6 * ([\(log (2,nn))/] + 1)) + 1 proofend; Lm4: for n being Element of NAT st n > 0 holds ( log (2,(2 * n)) = 1 + (log (2,n)) & log (2,(2 * n)) = (log (2,n)) + 1 ) proofend; Lm5: for nn, nn9 being Element of NAT st nn = 2 * nn9 & nn9 > 0 holds 6 + ((6 * ([\(log (2,nn9))/] + 1)) + 1) = (6 * ([\(log (2,nn))/] + 1)) + 1 proofend; Lm6: for N being Element of NAT st N <> 0 holds (6 * N) - 4 > 0 proofend; Lm7: dl. 0 <> dl. 1 by AMI_3:10; Lm8: dl. 0 <> dl. 2 by AMI_3:10; Lm9: dl. 0 <> dl. 3 by AMI_3:10; Lm10: dl. 1 <> dl. 2 by AMI_3:10; Lm11: dl. 1 <> dl. 3 by AMI_3:10; Lm12: dl. 2 <> dl. 3 by AMI_3:10; Lm13: dl. 0 <> dl. 4 by AMI_3:10; Lm14: dl. 1 <> dl. 4 by AMI_3:10; Lm15: dl. 2 <> dl. 4 by AMI_3:10; Lm16: dl. 3 <> dl. 4 by AMI_3:10; definition func Fib_Program -> XFinSequence of equals :: FIB_FUSC:def 1 (((((((<%((dl. 1) >0_goto 2)%> ^ <%(halt SCM)%>) ^ <%((dl. 3) := (dl. 0))%>) ^ <%(SubFrom ((dl. 1),(dl. 0)))%>) ^ <%((dl. 1) =0_goto 1)%>) ^ <%((dl. 4) := (dl. 2))%>) ^ <%((dl. 2) := (dl. 3))%>) ^ <%(AddTo ((dl. 3),(dl. 4)))%>) ^ <%(SCM-goto 3)%>; coherence (((((((<%((dl. 1) >0_goto 2)%> ^ <%(halt SCM)%>) ^ <%((dl. 3) := (dl. 0))%>) ^ <%(SubFrom ((dl. 1),(dl. 0)))%>) ^ <%((dl. 1) =0_goto 1)%>) ^ <%((dl. 4) := (dl. 2))%>) ^ <%((dl. 2) := (dl. 3))%>) ^ <%(AddTo ((dl. 3),(dl. 4)))%>) ^ <%(SCM-goto 3)%> is XFinSequence of ; end; :: deftheorem defines Fib_Program FIB_FUSC:def_1_:_ Fib_Program = (((((((<%((dl. 1) >0_goto 2)%> ^ <%(halt SCM)%>) ^ <%((dl. 3) := (dl. 0))%>) ^ <%(SubFrom ((dl. 1),(dl. 0)))%>) ^ <%((dl. 1) =0_goto 1)%>) ^ <%((dl. 4) := (dl. 2))%>) ^ <%((dl. 2) := (dl. 3))%>) ^ <%(AddTo ((dl. 3),(dl. 4)))%>) ^ <%(SCM-goto 3)%>; Lm17: for F being NAT -defined the InstructionsF of SCM -valued total Function for I1, I2, I3, I4, I5, I6, I7, I8, I9 being Instruction of SCM st (((((((<%I1%> ^ <%I2%>) ^ <%I3%>) ^ <%I4%>) ^ <%I5%>) ^ <%I6%>) ^ <%I7%>) ^ <%I8%>) ^ <%I9%> c= F holds ( F . 0 = I1 & F . 1 = I2 & F . 2 = I3 & F . 3 = I4 & F . 4 = I5 & F . 5 = I6 & F . 6 = I7 & F . 7 = I8 & F . 8 = I9 ) proofend; theorem :: FIB_FUSC:1 for F being NAT -defined the InstructionsF of SCM -valued total Function st Fib_Program c= F holds for N being Element of NAT for s being 0 -started State-consisting of <%1,N,0,0%> holds ( F halts_on s & ( N = 0 implies LifeSpan (F,s) = 1 ) & ( N > 0 implies LifeSpan (F,s) = (6 * N) - 2 ) & (Result (F,s)) . (dl. 3) = Fib N ) proofend; :: Fusc definition let i be Integer; func Fusc' i -> Element of NAT means :Def2: :: FIB_FUSC:def 2 ( ex n being Element of NAT st ( i = n & it = Fusc n ) or ( i is not Element of NAT & it = 0 ) ); existence ex b1 being Element of NAT st ( ex n being Element of NAT st ( i = n & b1 = Fusc n ) or ( i is not Element of NAT & b1 = 0 ) ) proofend; uniqueness for b1, b2 being Element of NAT st ( ex n being Element of NAT st ( i = n & b1 = Fusc n ) or ( i is not Element of NAT & b1 = 0 ) ) & ( ex n being Element of NAT st ( i = n & b2 = Fusc n ) or ( i is not Element of NAT & b2 = 0 ) ) holds b1 = b2 ; end; :: deftheorem Def2 defines Fusc' FIB_FUSC:def_2_:_ for i being Integer for b2 being Element of NAT holds ( b2 = Fusc' i iff ( ex n being Element of NAT st ( i = n & b2 = Fusc n ) or ( i is not Element of NAT & b2 = 0 ) ) ); definition func Fusc_Program -> XFinSequence of equals :: FIB_FUSC:def 3 (((((((<%((dl. 1) =0_goto 8)%> ^ <%((dl. 4) := (dl. 0))%>) ^ <%(Divide ((dl. 1),(dl. 4)))%>) ^ <%((dl. 4) =0_goto 6)%>) ^ <%(AddTo ((dl. 3),(dl. 2)))%>) ^ <%(SCM-goto 0)%>) ^ <%(AddTo ((dl. 2),(dl. 3)))%>) ^ <%(SCM-goto 0)%>) ^ <%(halt SCM)%>; coherence (((((((<%((dl. 1) =0_goto 8)%> ^ <%((dl. 4) := (dl. 0))%>) ^ <%(Divide ((dl. 1),(dl. 4)))%>) ^ <%((dl. 4) =0_goto 6)%>) ^ <%(AddTo ((dl. 3),(dl. 2)))%>) ^ <%(SCM-goto 0)%>) ^ <%(AddTo ((dl. 2),(dl. 3)))%>) ^ <%(SCM-goto 0)%>) ^ <%(halt SCM)%> is XFinSequence of ; end; :: deftheorem defines Fusc_Program FIB_FUSC:def_3_:_ Fusc_Program = (((((((<%((dl. 1) =0_goto 8)%> ^ <%((dl. 4) := (dl. 0))%>) ^ <%(Divide ((dl. 1),(dl. 4)))%>) ^ <%((dl. 4) =0_goto 6)%>) ^ <%(AddTo ((dl. 3),(dl. 2)))%>) ^ <%(SCM-goto 0)%>) ^ <%(AddTo ((dl. 2),(dl. 3)))%>) ^ <%(SCM-goto 0)%>) ^ <%(halt SCM)%>; theorem :: FIB_FUSC:2 for F being NAT -defined the InstructionsF of SCM -valued total Function st Fusc_Program c= F holds for N being Element of NAT st N > 0 holds for s being 0 -started State-consisting of <%2,N,1,0%> holds ( F halts_on s & (Result (F,s)) . (dl. 3) = Fusc N & LifeSpan (F,s) = (6 * ([\(log (2,N))/] + 1)) + 1 ) proofend; theorem :: FIB_FUSC:3 for F being NAT -defined the InstructionsF of SCM -valued total Function st Fib_Program c= F holds for N, k, Fk, Fk1 being Element of NAT for s being 3 -started State-consisting of <%1,N,Fk,Fk1%> st N > 0 & Fk = Fib k & Fk1 = Fib (k + 1) holds ( F halts_on s & LifeSpan (F,s) = (6 * N) - 4 & ex m being Element of NAT st ( m = (k + N) - 1 & (Result (F,s)) . (dl. 2) = Fib m & (Result (F,s)) . (dl. 3) = Fib (m + 1) ) ) proofend; theorem Th4: :: FIB_FUSC:4 for F being NAT -defined the InstructionsF of SCM -valued total Function st Fusc_Program c= F holds for n, N, A, B being Element of NAT for s being 0 -started State-consisting of <%2,n,A,B%> st N > 0 & Fusc N = (A * (Fusc n)) + (B * (Fusc (n + 1))) holds ( F halts_on s & (Result (F,s)) . (dl. 3) = Fusc N & ( n = 0 implies LifeSpan (F,s) = 1 ) & ( n > 0 implies LifeSpan (F,s) = (6 * ([\(log (2,n))/] + 1)) + 1 ) ) proofend; theorem :: FIB_FUSC:5 for F being NAT -defined the InstructionsF of SCM -valued total Function st Fusc_Program c= F holds for N being Element of NAT st N > 0 holds for s being 0 -started State-consisting of <%2,N,1,0%> holds ( F halts_on s & (Result (F,s)) . (dl. 3) = Fusc N & ( N = 0 implies LifeSpan (F,s) = 1 ) & ( N > 0 implies LifeSpan (F,s) = (6 * ([\(log (2,N))/] + 1)) + 1 ) ) proofend;