Article SCMFSA10, MML version 4.99.1005
:: SCMFSA10:funcnot 1 => SCMFSA10:func 1
definition
let a1, a2 be Int-Location;
let a3, a4 be integer set;
redefine func (a1,a2)-->(a3,a4) -> finite Element of sproduct the Object-Kind of SCM+FSA;
end;
:: SCMFSA10:th 3
theorem
for b1 being Int-Location holds
not b1 in NAT;
:: SCMFSA10:th 4
theorem
for b1 being FinSeq-Location holds
not b1 in NAT;
:: SCMFSA10:th 5
theorem
SCM+FSA-Data-Loc <> NAT;
:: SCMFSA10:th 6
theorem
SCM+FSA-Data*-Loc <> NAT;
:: SCMFSA10:th 7
theorem
for b1 being Element of the carrier of SCM+FSA
st b1 <> IC SCM+FSA & not b1 in NAT & b1 is not Int-Location
holds b1 is FinSeq-Location;
:: SCMFSA10:condreg 1
registration
cluster -> natural (Element of NAT);
end;
:: SCMFSA10:th 9
theorem
for b1, b2 being Int-Location holds
b1 := b2 = [1,<*b1,b2*>];
:: SCMFSA10:th 10
theorem
for b1, b2 being Int-Location holds
AddTo(b1,b2) = [2,<*b1,b2*>];
:: SCMFSA10:th 11
theorem
for b1, b2 being Int-Location holds
SubFrom(b1,b2) = [3,<*b1,b2*>];
:: SCMFSA10:th 12
theorem
for b1, b2 being Int-Location holds
MultBy(b1,b2) = [4,<*b1,b2*>];
:: SCMFSA10:th 13
theorem
for b1, b2 being Int-Location holds
Divide(b1,b2) = [5,<*b1,b2*>];
:: SCMFSA10:th 14
theorem
for b1 being Instruction-Location of SCM+FSA holds
goto b1 = [6,<*b1*>];
:: SCMFSA10:th 15
theorem
for b1 being Int-Location
for b2 being Instruction-Location of SCM+FSA holds
b1 =0_goto b2 = [7,<*b2,b1*>];
:: SCMFSA10:th 16
theorem
for b1 being Int-Location
for b2 being Instruction-Location of SCM+FSA holds
b1 >0_goto b2 = [8,<*b2,b1*>];
:: SCMFSA10:th 17
theorem
AddressPart halt SCM+FSA = {};
:: SCMFSA10:th 18
theorem
for b1, b2 being Int-Location holds
AddressPart (b1 := b2) = <*b1,b2*>;
:: SCMFSA10:th 19
theorem
for b1, b2 being Int-Location holds
AddressPart AddTo(b1,b2) = <*b1,b2*>;
:: SCMFSA10:th 20
theorem
for b1, b2 being Int-Location holds
AddressPart SubFrom(b1,b2) = <*b1,b2*>;
:: SCMFSA10:th 21
theorem
for b1, b2 being Int-Location holds
AddressPart MultBy(b1,b2) = <*b1,b2*>;
:: SCMFSA10:th 22
theorem
for b1, b2 being Int-Location holds
AddressPart Divide(b1,b2) = <*b1,b2*>;
:: SCMFSA10:th 23
theorem
for b1 being Instruction-Location of SCM+FSA holds
AddressPart goto b1 = <*b1*>;
:: SCMFSA10:th 24
theorem
for b1 being Instruction-Location of SCM+FSA
for b2 being Int-Location holds
AddressPart (b2 =0_goto b1) = <*b1,b2*>;
:: SCMFSA10:th 25
theorem
for b1 being Instruction-Location of SCM+FSA
for b2 being Int-Location holds
AddressPart (b2 >0_goto b1) = <*b1,b2*>;
:: SCMFSA10:th 26
theorem
for b1, b2 being Int-Location
for b3 being FinSeq-Location holds
AddressPart (b1 :=(b3,b2)) = <*b1,b3,b2*>;
:: SCMFSA10:th 27
theorem
for b1, b2 being Int-Location
for b3 being FinSeq-Location holds
AddressPart ((b3,b1):= b2) = <*b2,b3,b1*>;
:: SCMFSA10:th 28
theorem
for b1 being Int-Location
for b2 being FinSeq-Location holds
AddressPart (b1 :=len b2) = <*b1,b2*>;
:: SCMFSA10:th 29
theorem
for b1 being Int-Location
for b2 being FinSeq-Location holds
AddressPart (b2 :=<0,...,0> b1) = <*b1,b2*>;
:: SCMFSA10:th 30
theorem
for b1 being Element of InsCodes SCM+FSA
st b1 = {}
holds AddressParts b1 = {{}};
:: SCMFSA10:funcreg 1
registration
let a1 be Element of InsCodes SCM+FSA;
cluster AddressParts a1 -> non empty;
end;
:: SCMFSA10:th 31
theorem
for b1 being Element of InsCodes SCM+FSA
st b1 = 1
holds proj1 product" AddressParts b1 = {1,2};
:: SCMFSA10:th 32
theorem
for b1 being Element of InsCodes SCM+FSA
st b1 = 2
holds proj1 product" AddressParts b1 = {1,2};
:: SCMFSA10:th 33
theorem
for b1 being Element of InsCodes SCM+FSA
st b1 = 3
holds proj1 product" AddressParts b1 = {1,2};
:: SCMFSA10:th 34
theorem
for b1 being Element of InsCodes SCM+FSA
st b1 = 4
holds proj1 product" AddressParts b1 = {1,2};
:: SCMFSA10:th 35
theorem
for b1 being Element of InsCodes SCM+FSA
st b1 = 5
holds proj1 product" AddressParts b1 = {1,2};
:: SCMFSA10:th 36
theorem
for b1 being Element of InsCodes SCM+FSA
st b1 = 6
holds proj1 product" AddressParts b1 = {1};
:: SCMFSA10:th 37
theorem
for b1 being Element of InsCodes SCM+FSA
st b1 = 7
holds proj1 product" AddressParts b1 = {1,2};
:: SCMFSA10:th 38
theorem
for b1 being Element of InsCodes SCM+FSA
st b1 = 8
holds proj1 product" AddressParts b1 = {1,2};
:: SCMFSA10:th 39
theorem
for b1 being Element of InsCodes SCM+FSA
st b1 = 9
holds proj1 product" AddressParts b1 = {1,2,3};
:: SCMFSA10:th 40
theorem
for b1 being Element of InsCodes SCM+FSA
st b1 = 10
holds proj1 product" AddressParts b1 = {1,2,3};
:: SCMFSA10:th 41
theorem
for b1 being Element of InsCodes SCM+FSA
st b1 = 11
holds proj1 product" AddressParts b1 = {1,2};
:: SCMFSA10:th 42
theorem
for b1 being Element of InsCodes SCM+FSA
st b1 = 12
holds proj1 product" AddressParts b1 = {1,2};
:: SCMFSA10:th 43
theorem
for b1, b2 being Int-Location holds
(product" AddressParts InsCode (b1 := b2)) . 1 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 44
theorem
for b1, b2 being Int-Location holds
(product" AddressParts InsCode (b1 := b2)) . 2 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 45
theorem
for b1, b2 being Int-Location holds
(product" AddressParts InsCode AddTo(b1,b2)) . 1 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 46
theorem
for b1, b2 being Int-Location holds
(product" AddressParts InsCode AddTo(b1,b2)) . 2 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 47
theorem
for b1, b2 being Int-Location holds
(product" AddressParts InsCode SubFrom(b1,b2)) . 1 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 48
theorem
for b1, b2 being Int-Location holds
(product" AddressParts InsCode SubFrom(b1,b2)) . 2 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 49
theorem
for b1, b2 being Int-Location holds
(product" AddressParts InsCode MultBy(b1,b2)) . 1 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 50
theorem
for b1, b2 being Int-Location holds
(product" AddressParts InsCode MultBy(b1,b2)) . 2 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 51
theorem
for b1, b2 being Int-Location holds
(product" AddressParts InsCode Divide(b1,b2)) . 1 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 52
theorem
for b1, b2 being Int-Location holds
(product" AddressParts InsCode Divide(b1,b2)) . 2 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 53
theorem
for b1 being Instruction-Location of SCM+FSA holds
(product" AddressParts InsCode goto b1) . 1 = NAT;
:: SCMFSA10:th 54
theorem
for b1 being Instruction-Location of SCM+FSA
for b2 being Int-Location holds
(product" AddressParts InsCode (b2 =0_goto b1)) . 1 = NAT;
:: SCMFSA10:th 55
theorem
for b1 being Instruction-Location of SCM+FSA
for b2 being Int-Location holds
(product" AddressParts InsCode (b2 =0_goto b1)) . 2 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 56
theorem
for b1 being Instruction-Location of SCM+FSA
for b2 being Int-Location holds
(product" AddressParts InsCode (b2 >0_goto b1)) . 1 = NAT;
:: SCMFSA10:th 57
theorem
for b1 being Instruction-Location of SCM+FSA
for b2 being Int-Location holds
(product" AddressParts InsCode (b2 >0_goto b1)) . 2 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 58
theorem
for b1, b2 being Int-Location
for b3 being FinSeq-Location holds
(product" AddressParts InsCode (b1 :=(b3,b2))) . 1 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 59
theorem
for b1, b2 being Int-Location
for b3 being FinSeq-Location holds
(product" AddressParts InsCode (b1 :=(b3,b2))) . 2 = SCM+FSA-Data*-Loc;
:: SCMFSA10:th 60
theorem
for b1, b2 being Int-Location
for b3 being FinSeq-Location holds
(product" AddressParts InsCode (b1 :=(b3,b2))) . 3 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 61
theorem
for b1, b2 being Int-Location
for b3 being FinSeq-Location holds
(product" AddressParts InsCode ((b3,b1):= b2)) . 1 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 62
theorem
for b1, b2 being Int-Location
for b3 being FinSeq-Location holds
(product" AddressParts InsCode ((b3,b1):= b2)) . 2 = SCM+FSA-Data*-Loc;
:: SCMFSA10:th 63
theorem
for b1, b2 being Int-Location
for b3 being FinSeq-Location holds
(product" AddressParts InsCode ((b3,b1):= b2)) . 3 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 64
theorem
for b1 being Int-Location
for b2 being FinSeq-Location holds
(product" AddressParts InsCode (b1 :=len b2)) . 1 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 65
theorem
for b1 being Int-Location
for b2 being FinSeq-Location holds
(product" AddressParts InsCode (b1 :=len b2)) . 2 = SCM+FSA-Data*-Loc;
:: SCMFSA10:th 66
theorem
for b1 being Int-Location
for b2 being FinSeq-Location holds
(product" AddressParts InsCode (b2 :=<0,...,0> b1)) . 1 = SCM+FSA-Data-Loc;
:: SCMFSA10:th 67
theorem
for b1 being Int-Location
for b2 being FinSeq-Location holds
(product" AddressParts InsCode (b2 :=<0,...,0> b1)) . 2 = SCM+FSA-Data*-Loc;
:: SCMFSA10:th 68
theorem
for b1 being Instruction-Location of SCM+FSA holds
NIC(halt SCM+FSA,b1) = {b1};
:: SCMFSA10:funcreg 2
registration
cluster JUMP halt SCM+FSA -> empty;
end;
:: SCMFSA10:th 69
theorem
for b1 being Instruction-Location of SCM+FSA
for b2, b3 being Int-Location holds
NIC(b2 := b3,b1) = {Next b1};
:: SCMFSA10:funcreg 3
registration
let a1, a2 be Int-Location;
cluster JUMP (a1 := a2) -> empty;
end;
:: SCMFSA10:th 70
theorem
for b1 being Instruction-Location of SCM+FSA
for b2, b3 being Int-Location holds
NIC(AddTo(b2,b3),b1) = {Next b1};
:: SCMFSA10:funcreg 4
registration
let a1, a2 be Int-Location;
cluster JUMP AddTo(a1,a2) -> empty;
end;
:: SCMFSA10:th 71
theorem
for b1 being Instruction-Location of SCM+FSA
for b2, b3 being Int-Location holds
NIC(SubFrom(b2,b3),b1) = {Next b1};
:: SCMFSA10:funcreg 5
registration
let a1, a2 be Int-Location;
cluster JUMP SubFrom(a1,a2) -> empty;
end;
:: SCMFSA10:th 72
theorem
for b1 being Instruction-Location of SCM+FSA
for b2, b3 being Int-Location holds
NIC(MultBy(b2,b3),b1) = {Next b1};
:: SCMFSA10:funcreg 6
registration
let a1, a2 be Int-Location;
cluster JUMP MultBy(a1,a2) -> empty;
end;
:: SCMFSA10:th 73
theorem
for b1 being Instruction-Location of SCM+FSA
for b2, b3 being Int-Location holds
NIC(Divide(b2,b3),b1) = {Next b1};
:: SCMFSA10:funcreg 7
registration
let a1, a2 be Int-Location;
cluster JUMP Divide(a1,a2) -> empty;
end;
:: SCMFSA10:th 74
theorem
for b1, b2 being Instruction-Location of SCM+FSA holds
NIC(goto b1,b2) = {b1};
:: SCMFSA10:th 75
theorem
for b1 being Instruction-Location of SCM+FSA holds
JUMP goto b1 = {b1};
:: SCMFSA10:funcreg 8
registration
let a1 be Instruction-Location of SCM+FSA;
cluster JUMP goto a1 -> non empty trivial;
end;
:: SCMFSA10:th 76
theorem
for b1, b2 being Instruction-Location of SCM+FSA
for b3 being Int-Location holds
NIC(b3 =0_goto b1,b2) = {b1,Next b2};
:: SCMFSA10:th 77
theorem
for b1 being Instruction-Location of SCM+FSA
for b2 being Int-Location holds
JUMP (b2 =0_goto b1) = {b1};
:: SCMFSA10:funcreg 9
registration
let a1 be Int-Location;
let a2 be Instruction-Location of SCM+FSA;
cluster JUMP (a1 =0_goto a2) -> non empty trivial;
end;
:: SCMFSA10:th 78
theorem
for b1, b2 being Instruction-Location of SCM+FSA
for b3 being Int-Location holds
NIC(b3 >0_goto b1,b2) = {b1,Next b2};
:: SCMFSA10:th 79
theorem
for b1 being Instruction-Location of SCM+FSA
for b2 being Int-Location holds
JUMP (b2 >0_goto b1) = {b1};
:: SCMFSA10:funcreg 10
registration
let a1 be Int-Location;
let a2 be Instruction-Location of SCM+FSA;
cluster JUMP (a1 >0_goto a2) -> non empty trivial;
end;
:: SCMFSA10:th 80
theorem
for b1 being Instruction-Location of SCM+FSA
for b2, b3 being Int-Location
for b4 being FinSeq-Location holds
NIC(b2 :=(b4,b3),b1) = {Next b1};
:: SCMFSA10:funcreg 11
registration
let a1, a2 be Int-Location;
let a3 be FinSeq-Location;
cluster JUMP (a1 :=(a3,a2)) -> empty;
end;
:: SCMFSA10:th 81
theorem
for b1 being Instruction-Location of SCM+FSA
for b2, b3 being Int-Location
for b4 being FinSeq-Location holds
NIC((b4,b2):= b3,b1) = {Next b1};
:: SCMFSA10:funcreg 12
registration
let a1, a2 be Int-Location;
let a3 be FinSeq-Location;
cluster JUMP ((a3,a2):= a1) -> empty;
end;
:: SCMFSA10:th 82
theorem
for b1 being Instruction-Location of SCM+FSA
for b2 being Int-Location
for b3 being FinSeq-Location holds
NIC(b2 :=len b3,b1) = {Next b1};
:: SCMFSA10:funcreg 13
registration
let a1 be Int-Location;
let a2 be FinSeq-Location;
cluster JUMP (a1 :=len a2) -> empty;
end;
:: SCMFSA10:th 83
theorem
for b1 being Instruction-Location of SCM+FSA
for b2 being Int-Location
for b3 being FinSeq-Location holds
NIC(b3 :=<0,...,0> b2,b1) = {Next b1};
:: SCMFSA10:funcreg 14
registration
let a1 be Int-Location;
let a2 be FinSeq-Location;
cluster JUMP (a2 :=<0,...,0> a1) -> empty;
end;
:: SCMFSA10:th 84
theorem
for b1 being Instruction-Location of SCM+FSA holds
SUCC b1 = {b1,Next b1};
:: SCMFSA10:th 85
theorem
for b1 being IL-Function of NAT,SCM+FSA
st for b2 being Element of NAT holds
b1 . b2 = insloc b2
holds b1 is bijective(NAT, NAT) &
(for b2 being Element of NAT holds
b1 . (b2 + 1) in SUCC (b1 . b2) &
(for b3 being Element of NAT
st b1 . b3 in SUCC (b1 . b2)
holds b2 <= b3));
:: SCMFSA10:funcreg 15
registration
cluster SCM+FSA -> strict standard;
end;
:: SCMFSA10:th 86
theorem
for b1 being natural set holds
il.(SCM+FSA,b1) = insloc b1;
:: SCMFSA10:th 87
theorem
for b1 being natural set holds
Next il.(SCM+FSA,b1) = il.(SCM+FSA,b1 + 1);
:: SCMFSA10:th 88
theorem
for b1 being Instruction-Location of SCM+FSA holds
Next b1 = NextLoc b1;
:: SCMFSA10:funcreg 16
registration
cluster (halt SCM+FSA) `1 -> jump-only;
end;
:: SCMFSA10:funcreg 17
registration
cluster halt SCM+FSA -> jump-only;
end;
:: SCMFSA10:funcreg 18
registration
let a1 be Instruction-Location of SCM+FSA;
cluster (goto a1) `1 -> jump-only;
end;
:: SCMFSA10:funcreg 19
registration
let a1 be Instruction-Location of SCM+FSA;
cluster goto a1 -> jump-only non sequential non ins-loc-free;
end;
:: SCMFSA10:funcreg 20
registration
let a1 be Int-Location;
let a2 be Instruction-Location of SCM+FSA;
cluster (a1 =0_goto a2) `1 -> jump-only;
end;
:: SCMFSA10:funcreg 21
registration
let a1 be Int-Location;
let a2 be Instruction-Location of SCM+FSA;
cluster (a1 >0_goto a2) `1 -> jump-only;
end;
:: SCMFSA10:funcreg 22
registration
let a1 be Int-Location;
let a2 be Instruction-Location of SCM+FSA;
cluster a1 =0_goto a2 -> jump-only non sequential non ins-loc-free;
end;
:: SCMFSA10:funcreg 23
registration
let a1 be Int-Location;
let a2 be Instruction-Location of SCM+FSA;
cluster a1 >0_goto a2 -> jump-only non sequential non ins-loc-free;
end;
:: SCMFSA10:funcreg 24
registration
let a1, a2 be Int-Location;
cluster (a1 := a2) `1 -> non jump-only;
end;
:: SCMFSA10:funcreg 25
registration
let a1, a2 be Int-Location;
cluster (AddTo(a1,a2)) `1 -> non jump-only;
end;
:: SCMFSA10:funcreg 26
registration
let a1, a2 be Int-Location;
cluster (SubFrom(a1,a2)) `1 -> non jump-only;
end;
:: SCMFSA10:funcreg 27
registration
let a1, a2 be Int-Location;
cluster (MultBy(a1,a2)) `1 -> non jump-only;
end;
:: SCMFSA10:funcreg 28
registration
let a1, a2 be Int-Location;
cluster (Divide(a1,a2)) `1 -> non jump-only;
end;
:: SCMFSA10:funcreg 29
registration
let a1, a2 be Int-Location;
cluster a1 := a2 -> non jump-only sequential;
end;
:: SCMFSA10:funcreg 30
registration
let a1, a2 be Int-Location;
cluster AddTo(a1,a2) -> non jump-only sequential;
end;
:: SCMFSA10:funcreg 31
registration
let a1, a2 be Int-Location;
cluster SubFrom(a1,a2) -> non jump-only sequential;
end;
:: SCMFSA10:funcreg 32
registration
let a1, a2 be Int-Location;
cluster MultBy(a1,a2) -> non jump-only sequential;
end;
:: SCMFSA10:funcreg 33
registration
let a1, a2 be Int-Location;
cluster Divide(a1,a2) -> non jump-only sequential;
end;
:: SCMFSA10:funcreg 34
registration
let a1, a2 be Int-Location;
let a3 be FinSeq-Location;
cluster (a2 :=(a3,a1)) `1 -> non jump-only;
end;
:: SCMFSA10:funcreg 35
registration
let a1, a2 be Int-Location;
let a3 be FinSeq-Location;
cluster ((a3,a1):= a2) `1 -> non jump-only;
end;
:: SCMFSA10:funcreg 36
registration
let a1, a2 be Int-Location;
let a3 be FinSeq-Location;
cluster a2 :=(a3,a1) -> non jump-only sequential;
end;
:: SCMFSA10:funcreg 37
registration
let a1, a2 be Int-Location;
let a3 be FinSeq-Location;
cluster (a3,a1):= a2 -> non jump-only sequential;
end;
:: SCMFSA10:funcreg 38
registration
let a1 be Int-Location;
let a2 be FinSeq-Location;
cluster (a1 :=len a2) `1 -> non jump-only;
end;
:: SCMFSA10:funcreg 39
registration
let a1 be Int-Location;
let a2 be FinSeq-Location;
cluster (a2 :=<0,...,0> a1) `1 -> non jump-only;
end;
:: SCMFSA10:funcreg 40
registration
let a1 be Int-Location;
let a2 be FinSeq-Location;
cluster a1 :=len a2 -> non jump-only sequential;
end;
:: SCMFSA10:funcreg 41
registration
let a1 be Int-Location;
let a2 be FinSeq-Location;
cluster a2 :=<0,...,0> a1 -> non jump-only sequential;
end;
:: SCMFSA10:funcreg 42
registration
cluster SCM+FSA -> strict homogeneous with_explicit_jumps without_implicit_jumps;
end;
:: SCMFSA10:funcreg 43
registration
cluster SCM+FSA -> strict regular;
end;
:: SCMFSA10:th 89
theorem
for b1 being Instruction-Location of SCM+FSA
for b2 being natural set holds
IncAddr(goto b1,b2) = goto il.(SCM+FSA,(locnum b1) + b2);
:: SCMFSA10:th 90
theorem
for b1 being Instruction-Location of SCM+FSA
for b2 being natural set
for b3 being Int-Location holds
IncAddr(b3 =0_goto b1,b2) = b3 =0_goto il.(SCM+FSA,(locnum b1) + b2);
:: SCMFSA10:th 91
theorem
for b1 being Instruction-Location of SCM+FSA
for b2 being natural set
for b3 being Int-Location holds
IncAddr(b3 >0_goto b1,b2) = b3 >0_goto il.(SCM+FSA,(locnum b1) + b2);
:: SCMFSA10:funcreg 44
registration
cluster SCM+FSA -> strict IC-good Exec-preserving;
end;