:: Two Programs for {\bf SCM}. Part II - Proofs
:: by Grzegorz Bancerek and Piotr Rudnicki
::
:: Copyright (c) 1993-2021 Association of Mizar Users

Lm1: 0 = [\(log (2,1))/]
proof end;

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 )

proof end;

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

proof end;

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 )

proof end;

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

proof end;

Lm6: for N being Nat st N <> 0 holds
(6 * N) - 4 > 0

proof end;

Lm7:
by AMI_3:10;

Lm8:
by AMI_3:10;

Lm9:
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:
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 the InstructionsF of SCM equals :: FIB_FUSC:def 1
(((((((<%((dl. 1) >0_goto 2)%> ^ <%()%>) ^ <%((dl. 3) := ())%>) ^ <%(SubFrom ((dl. 1),()))%>) ^ <%((dl. 1) =0_goto 1)%>) ^ <%((dl. 4) := (dl. 2))%>) ^ <%((dl. 2) := (dl. 3))%>) ^ <%(AddTo ((dl. 3),(dl. 4)))%>) ^ <%()%>;
coherence
(((((((<%((dl. 1) >0_goto 2)%> ^ <%()%>) ^ <%((dl. 3) := ())%>) ^ <%(SubFrom ((dl. 1),()))%>) ^ <%((dl. 1) =0_goto 1)%>) ^ <%((dl. 4) := (dl. 2))%>) ^ <%((dl. 2) := (dl. 3))%>) ^ <%(AddTo ((dl. 3),(dl. 4)))%>) ^ <%()%> is XFinSequence of the InstructionsF of SCM
;
end;

:: deftheorem defines Fib_Program FIB_FUSC:def 1 :
Fib_Program = (((((((<%((dl. 1) >0_goto 2)%> ^ <%()%>) ^ <%((dl. 3) := ())%>) ^ <%(SubFrom ((dl. 1),()))%>) ^ <%((dl. 1) =0_goto 1)%>) ^ <%((dl. 4) := (dl. 2))%>) ^ <%((dl. 2) := (dl. 3))%>) ^ <%(AddTo ((dl. 3),(dl. 4)))%>) ^ <%()%>;

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 )

proof end;

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 )
proof end;

:: 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 ) )
proof end;
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 the InstructionsF of SCM equals :: FIB_FUSC:def 3
(((((((<%((dl. 1) =0_goto 8)%> ^ <%((dl. 4) := ())%>) ^ <%(Divide ((dl. 1),(dl. 4)))%>) ^ <%((dl. 4) =0_goto 6)%>) ^ <%(AddTo ((dl. 3),(dl. 2)))%>) ^ ) ^ <%(AddTo ((dl. 2),(dl. 3)))%>) ^ ) ^ <%()%>;
coherence
(((((((<%((dl. 1) =0_goto 8)%> ^ <%((dl. 4) := ())%>) ^ <%(Divide ((dl. 1),(dl. 4)))%>) ^ <%((dl. 4) =0_goto 6)%>) ^ <%(AddTo ((dl. 3),(dl. 2)))%>) ^ ) ^ <%(AddTo ((dl. 2),(dl. 3)))%>) ^ ) ^ <%()%> is XFinSequence of the InstructionsF of SCM
;
end;

:: deftheorem defines Fusc_Program FIB_FUSC:def 3 :
Fusc_Program = (((((((<%((dl. 1) =0_goto 8)%> ^ <%((dl. 4) := ())%>) ^ <%(Divide ((dl. 1),(dl. 4)))%>) ^ <%((dl. 4) =0_goto 6)%>) ^ <%(AddTo ((dl. 3),(dl. 2)))%>) ^ ) ^ <%(AddTo ((dl. 2),(dl. 3)))%>) ^ ) ^ <%()%>;

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 )
proof end;

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 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) ) )
proof end;

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 ) )
proof end;

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 ) )
proof end;