:: SCMFSA_7 semantic presentation
theorem Th1: :: SCMFSA_7:1
Lemma2:
for b1, b2, b3 being Nat holds
( b1 >= b3 & b2 >= b3 & b1 -' b3 = b2 -' b3 implies b1 = b2 )
by BINARITH:49;
Lemma3:
for b1, b2 being natural number holds
( b1 >= b2 implies b1 -' b2 = b1 - b2 )
by BINARITH:50;
Lemma4:
for b1, b2 being Integer holds
( b1 < b2 implies b1 <= b2 - 1 )
by INT_1:79;
theorem Th2: :: SCMFSA_7:2
canceled;
theorem Th3: :: SCMFSA_7:3
canceled;
theorem Th4: :: SCMFSA_7:4
canceled;
theorem Th5: :: SCMFSA_7:5
canceled;
theorem Th6: :: SCMFSA_7:6
canceled;
theorem Th7: :: SCMFSA_7:7
canceled;
theorem Th8: :: SCMFSA_7:8
canceled;
theorem Th9: :: SCMFSA_7:9
canceled;
theorem Th10: :: SCMFSA_7:10
canceled;
Lemma5:
for b1, b2, b3 being FinSequence holds
( ((len b1) + (len b2)) + (len b3) = len ((b1 ^ b2) ^ b3) & ((len b1) + (len b2)) + (len b3) = len (b1 ^ (b2 ^ b3)) & (len b1) + ((len b2) + (len b3)) = len (b1 ^ (b2 ^ b3)) & (len b1) + ((len b2) + (len b3)) = len ((b1 ^ b2) ^ b3) )
Lemma6:
for b1, b2, b3, b4 being FinSequence holds
( ((b1 ^ b2) ^ b3) ^ b4 = (b1 ^ b2) ^ (b3 ^ b4) & ((b1 ^ b2) ^ b3) ^ b4 = b1 ^ ((b2 ^ b3) ^ b4) & ((b1 ^ b2) ^ b3) ^ b4 = b1 ^ (b2 ^ (b3 ^ b4)) & ((b1 ^ b2) ^ b3) ^ b4 = (b1 ^ (b2 ^ b3)) ^ b4 )
Lemma7:
for b1, b2, b3, b4, b5 being FinSequence holds
( (((b1 ^ b2) ^ b3) ^ b4) ^ b5 = ((b1 ^ b2) ^ b3) ^ (b4 ^ b5) & (((b1 ^ b2) ^ b3) ^ b4) ^ b5 = (b1 ^ b2) ^ ((b3 ^ b4) ^ b5) & (((b1 ^ b2) ^ b3) ^ b4) ^ b5 = (b1 ^ b2) ^ (b3 ^ (b4 ^ b5)) & (((b1 ^ b2) ^ b3) ^ b4) ^ b5 = b1 ^ (((b2 ^ b3) ^ b4) ^ b5) & (((b1 ^ b2) ^ b3) ^ b4) ^ b5 = b1 ^ ((b2 ^ b3) ^ (b4 ^ b5)) & (((b1 ^ b2) ^ b3) ^ b4) ^ b5 = b1 ^ (b2 ^ ((b3 ^ b4) ^ b5)) & (((b1 ^ b2) ^ b3) ^ b4) ^ b5 = b1 ^ (b2 ^ (b3 ^ (b4 ^ b5))) & (((b1 ^ b2) ^ b3) ^ b4) ^ b5 = ((b1 ^ b2) ^ (b3 ^ b4)) ^ b5 & (((b1 ^ b2) ^ b3) ^ b4) ^ b5 = (b1 ^ ((b2 ^ b3) ^ b4)) ^ b5 & (((b1 ^ b2) ^ b3) ^ b4) ^ b5 = (b1 ^ (b2 ^ (b3 ^ b4))) ^ b5 & (((b1 ^ b2) ^ b3) ^ b4) ^ b5 = b1 ^ ((b2 ^ (b3 ^ b4)) ^ b5) )
theorem Th11: :: SCMFSA_7:11
canceled;
theorem Th12: :: SCMFSA_7:12
theorem Th13: :: SCMFSA_7:13
theorem Th14: :: SCMFSA_7:14
theorem Th15: :: SCMFSA_7:15
theorem Th16: :: SCMFSA_7:16
theorem Th17: :: SCMFSA_7:17
theorem Th18: :: SCMFSA_7:18
theorem Th19: :: SCMFSA_7:19
theorem Th20: :: SCMFSA_7:20
theorem Th21: :: SCMFSA_7:21
theorem Th22: :: SCMFSA_7:22
theorem Th23: :: SCMFSA_7:23
deffunc H1( Nat) -> Element of the Instruction-Locations of SCM+FSA = insloc (a1 -' 1);
:: deftheorem Def1 defines Load SCMFSA_7:def 1 :
theorem Th24: :: SCMFSA_7:24
canceled;
theorem Th25: :: SCMFSA_7:25
theorem Th26: :: SCMFSA_7:26
theorem Th27: :: SCMFSA_7:27
canceled;
theorem Th28: :: SCMFSA_7:28
theorem Th29: :: SCMFSA_7:29
theorem Th30: :: SCMFSA_7:30
theorem Th31: :: SCMFSA_7:31
theorem Th32: :: SCMFSA_7:32
:: deftheorem Def2 defines := SCMFSA_7:def 2 :
definition
let c
1 be
Int-Location ;
let c
2 be
Integer;
func aSeq c
1,c
2 -> FinSequence of the
Instructions of
SCM+FSA means :
Def3:
:: SCMFSA_7:def 3
ex b
1 being
Nat st
( b
1 + 1
= a
2 & a
3 = <*(a1 := (intloc 0))*> ^ (b1 |-> (AddTo a1,(intloc 0))) )
if a
2 > 0
otherwise ex b
1 being
Nat st
( b
1 + a
2 = 1 & a
3 = <*(a1 := (intloc 0))*> ^ (b1 |-> (SubFrom a1,(intloc 0))) );
existence
( not ( c2 > 0 & ( for b1 being FinSequence of the Instructions of SCM+FSA
for b2 being Nat holds
not ( b2 + 1 = c2 & b1 = <*(c1 := (intloc 0))*> ^ (b2 |-> (AddTo c1,(intloc 0))) ) ) ) & not ( not c2 > 0 & ( for b1 being FinSequence of the Instructions of SCM+FSA
for b2 being Nat holds
not ( b2 + c2 = 1 & b1 = <*(c1 := (intloc 0))*> ^ (b2 |-> (SubFrom c1,(intloc 0))) ) ) ) )
uniqueness
for b1, b2 being FinSequence of the Instructions of SCM+FSA holds
( ( c2 > 0 & ex b3 being Nat st
( b3 + 1 = c2 & b1 = <*(c1 := (intloc 0))*> ^ (b3 |-> (AddTo c1,(intloc 0))) ) & ex b3 being Nat st
( b3 + 1 = c2 & b2 = <*(c1 := (intloc 0))*> ^ (b3 |-> (AddTo c1,(intloc 0))) ) implies b1 = b2 ) & ( not c2 > 0 & ex b3 being Nat st
( b3 + c2 = 1 & b1 = <*(c1 := (intloc 0))*> ^ (b3 |-> (SubFrom c1,(intloc 0))) ) & ex b3 being Nat st
( b3 + c2 = 1 & b2 = <*(c1 := (intloc 0))*> ^ (b3 |-> (SubFrom c1,(intloc 0))) ) implies b1 = b2 ) )
;
correctness
consistency
for b1 being FinSequence of the Instructions of SCM+FSA holds
verum;
;
end;
:: deftheorem Def3 defines aSeq SCMFSA_7:def 3 :
theorem Th33: :: SCMFSA_7:33
definition
let c
1 be
FinSeq-Location ;
let c
2 be
FinSequence of
INT ;
func aSeq c
1,c
2 -> FinSequence of the
Instructions of
SCM+FSA means :
Def4:
:: SCMFSA_7:def 4
ex b
1 being
FinSequence of the
Instructions of
SCM+FSA * st
(
len b
1 = len a
2 & ( for b
2 being
Nat holds
not ( 1
<= b
2 & b
2 <= len a
2 & ( for b
3 being
Integer holds
not ( b
3 = a
2 . b
2 & b
1 . b
2 = ((aSeq (intloc 1),b2) ^ (aSeq (intloc 2),b3)) ^ <*(a1,(intloc 1) := (intloc 2))*> ) ) ) ) & a
3 = FlattenSeq b
1 );
existence
ex b1 being FinSequence of the Instructions of SCM+FSA ex b2 being FinSequence of the Instructions of SCM+FSA * st
( len b2 = len c2 & ( for b3 being Nat holds
not ( 1 <= b3 & b3 <= len c2 & ( for b4 being Integer holds
not ( b4 = c2 . b3 & b2 . b3 = ((aSeq (intloc 1),b3) ^ (aSeq (intloc 2),b4)) ^ <*(c1,(intloc 1) := (intloc 2))*> ) ) ) ) & b1 = FlattenSeq b2 )
uniqueness
for b1, b2 being FinSequence of the Instructions of SCM+FSA holds
( ex b3 being FinSequence of the Instructions of SCM+FSA * st
( len b3 = len c2 & ( for b4 being Nat holds
not ( 1 <= b4 & b4 <= len c2 & ( for b5 being Integer holds
not ( b5 = c2 . b4 & b3 . b4 = ((aSeq (intloc 1),b4) ^ (aSeq (intloc 2),b5)) ^ <*(c1,(intloc 1) := (intloc 2))*> ) ) ) ) & b1 = FlattenSeq b3 ) & ex b3 being FinSequence of the Instructions of SCM+FSA * st
( len b3 = len c2 & ( for b4 being Nat holds
not ( 1 <= b4 & b4 <= len c2 & ( for b5 being Integer holds
not ( b5 = c2 . b4 & b3 . b4 = ((aSeq (intloc 1),b4) ^ (aSeq (intloc 2),b5)) ^ <*(c1,(intloc 1) := (intloc 2))*> ) ) ) ) & b2 = FlattenSeq b3 ) implies b1 = b2 )
correctness
;
end;
:: deftheorem Def4 defines aSeq SCMFSA_7:def 4 :
:: deftheorem Def5 defines := SCMFSA_7:def 5 :
theorem Th34: :: SCMFSA_7:34
theorem Th35: :: SCMFSA_7:35
theorem Th36: :: SCMFSA_7:36
theorem Th37: :: SCMFSA_7:37
theorem Th38: :: SCMFSA_7:38
theorem Th39: :: SCMFSA_7:39