let N be non empty with_zero set ; :: thesis: for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N

for p being NAT -defined the InstructionsF of b_{1} -valued non halt-free Function

for d being FinPartState of S holds

( d is Autonomy of p iff p,d computes {} .--> {} )

let S be non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N; :: thesis: for p being NAT -defined the InstructionsF of S -valued non halt-free Function

for d being FinPartState of S holds

( d is Autonomy of p iff p,d computes {} .--> {} )

let p be NAT -defined the InstructionsF of S -valued non halt-free Function; :: thesis: for d being FinPartState of S holds

( d is Autonomy of p iff p,d computes {} .--> {} )

let d be FinPartState of S; :: thesis: ( d is Autonomy of p iff p,d computes {} .--> {} )

thus ( d is Autonomy of p implies p,d computes {} .--> {} ) :: thesis: ( p,d computes {} .--> {} implies d is Autonomy of p )

assume p,d computes {} .--> {} ; :: thesis: d is Autonomy of p

then ex s being FinPartState of S st

( {} = s & d +* s is Autonomy of p & ({} .--> {}) . s c= Result (p,(d +* s)) ) by A6;

hence d is Autonomy of p ; :: thesis: verum

for p being NAT -defined the InstructionsF of b

for d being FinPartState of S holds

( d is Autonomy of p iff p,d computes {} .--> {} )

let S be non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N; :: thesis: for p being NAT -defined the InstructionsF of S -valued non halt-free Function

for d being FinPartState of S holds

( d is Autonomy of p iff p,d computes {} .--> {} )

let p be NAT -defined the InstructionsF of S -valued non halt-free Function; :: thesis: for d being FinPartState of S holds

( d is Autonomy of p iff p,d computes {} .--> {} )

let d be FinPartState of S; :: thesis: ( d is Autonomy of p iff p,d computes {} .--> {} )

thus ( d is Autonomy of p implies p,d computes {} .--> {} ) :: thesis: ( p,d computes {} .--> {} implies d is Autonomy of p )

proof

A6:
{} in dom ({} .--> {})
by TARSKI:def 1;
assume A2:
d is Autonomy of p
; :: thesis: p,d computes {} .--> {}

let x be set ; :: according to EXTPRO_1:def 14 :: thesis: ( x in dom ({} .--> {}) implies ex s being FinPartState of S st

( x = s & d +* s is Autonomy of p & ({} .--> {}) . s c= Result (p,(d +* s)) ) )

assume A3: x in dom ({} .--> {}) ; :: thesis: ex s being FinPartState of S st

( x = s & d +* s is Autonomy of p & ({} .--> {}) . s c= Result (p,(d +* s)) )

then x = {} by TARSKI:def 1;

then reconsider s = x as FinPartState of S by FUNCT_1:104, RELAT_1:171;

take s ; :: thesis: ( x = s & d +* s is Autonomy of p & ({} .--> {}) . s c= Result (p,(d +* s)) )

A5: d +* {} = d ;

thus x = s ; :: thesis: ( d +* s is Autonomy of p & ({} .--> {}) . s c= Result (p,(d +* s)) )

thus d +* s is Autonomy of p by A2, A3, A5, TARSKI:def 1; :: thesis: ({} .--> {}) . s c= Result (p,(d +* s))

({} .--> {}) . s = {} ;

hence ({} .--> {}) . s c= Result (p,(d +* s)) by XBOOLE_1:2; :: thesis: verum

end;let x be set ; :: according to EXTPRO_1:def 14 :: thesis: ( x in dom ({} .--> {}) implies ex s being FinPartState of S st

( x = s & d +* s is Autonomy of p & ({} .--> {}) . s c= Result (p,(d +* s)) ) )

assume A3: x in dom ({} .--> {}) ; :: thesis: ex s being FinPartState of S st

( x = s & d +* s is Autonomy of p & ({} .--> {}) . s c= Result (p,(d +* s)) )

then x = {} by TARSKI:def 1;

then reconsider s = x as FinPartState of S by FUNCT_1:104, RELAT_1:171;

take s ; :: thesis: ( x = s & d +* s is Autonomy of p & ({} .--> {}) . s c= Result (p,(d +* s)) )

A5: d +* {} = d ;

thus x = s ; :: thesis: ( d +* s is Autonomy of p & ({} .--> {}) . s c= Result (p,(d +* s)) )

thus d +* s is Autonomy of p by A2, A3, A5, TARSKI:def 1; :: thesis: ({} .--> {}) . s c= Result (p,(d +* s))

({} .--> {}) . s = {} ;

hence ({} .--> {}) . s c= Result (p,(d +* s)) by XBOOLE_1:2; :: thesis: verum

assume p,d computes {} .--> {} ; :: thesis: d is Autonomy of p

then ex s being FinPartState of S st

( {} = s & d +* s is Autonomy of p & ({} .--> {}) . s c= Result (p,(d +* s)) ) by A6;

hence d is Autonomy of p ; :: thesis: verum