:: SCM_1 semantic presentation
theorem Th1: :: SCM_1:1
theorem Th2: :: SCM_1:2
theorem Th3: :: SCM_1:3
theorem Th4: :: SCM_1:4
theorem Th5: :: SCM_1:5
canceled;
theorem Th6: :: SCM_1:6
canceled;
theorem Th7: :: SCM_1:7
:: deftheorem Def1 defines State-consisting SCM_1:def 1 :
theorem Th8: :: SCM_1:8
theorem Th9: :: SCM_1:9
theorem Th10: :: SCM_1:10
theorem Th11: :: SCM_1:11
theorem Th12: :: SCM_1:12
theorem Th13: :: SCM_1:13
theorem Th14: :: SCM_1:14
for
I1,
I2,
I3,
I4,
I5,
I6,
I7,
I8,
I9 being
Instruction of
SCM for
i1,
i2,
i3,
i4 being
Integer for
il being
Element of
NAT for
s being
State-consisting of
il,0,0,
(((((((<*I1*> ^ <*I2*>) ^ <*I3*>) ^ <*I4*>) ^ <*I5*>) ^ <*I6*>) ^ <*I7*>) ^ <*I8*>) ^ <*I9*>,
((<*i1*> ^ <*i2*>) ^ <*i3*>) ^ <*i4*> holds
(
IC s = il. il &
s . (il. 0) = I1 &
s . (il. 1) = I2 &
s . (il. 2) = I3 &
s . (il. 3) = I4 &
s . (il. 4) = I5 &
s . (il. 5) = I6 &
s . (il. 6) = I7 &
s . (il. 7) = I8 &
s . (il. 8) = I9 &
s . (dl. 0) = i1 &
s . (dl. 1) = i2 &
s . (dl. 2) = i3 &
s . (dl. 3) = i4 )
theorem Th15: :: SCM_1:15
:: deftheorem Def2 defines Complexity SCM_1:def 2 :
theorem Th16: :: SCM_1:16
theorem Th17: :: SCM_1:17
Lemma94:
for n being Element of NAT holds Next (il. n) = il. (n + 1)
Lemma96:
for k being Element of NAT
for s being State of SCM holds (Computation s) . (k + 1) = Exec (CurInstr ((Computation s) . k)),((Computation s) . k)
E97:
now
let k be
Element of
NAT ,
n be
Element of
NAT ;
let s be
State of
SCM ;
let a be
Data-Location ,
b be
Data-Location ;
assume E23:
IC ((Computation s) . k) = il. n
;
E25:
((Computation s) . k) . (il. n) = s . (il. n)
by AMI_1:54;
set csk =
(Computation s) . k;
set csk1 =
(Computation s) . (k + 1);
E41:
((Computation s) . k) . 0
= il. n
by , ;
assume E43:
(
s . (il. n) = a := b or
s . (il. n) = AddTo a,
b or
s . (il. n) = SubFrom a,
b or
s . (il. n) = MultBy a,
b or (
a <> b &
s . (il. n) = Divide a,
b ) )
;
thus E44:
(Computation s) . (k + 1) =
Exec (CurInstr ((Computation s) . k)),
((Computation s) . k)
by
.=
Exec (s . (il. n)),
((Computation s) . k)
by , Th1,
;
thus IC ((Computation s) . (k + 1)) =
Next (IC ((Computation s) . k))
by , Th2, AMI_3:8, AMI_3:9, AMI_3:10, AMI_3:11, AMI_3:12
.=
il. (n + 1)
by ,
;
end;
theorem Th18: :: SCM_1:18
theorem Th19: :: SCM_1:19
theorem Th20: :: SCM_1:20
theorem Th21: :: SCM_1:21
theorem Th22: :: SCM_1:22
theorem Th23: :: SCM_1:23
theorem Th24: :: SCM_1:24
theorem Th25: :: SCM_1:25
theorem Th26: :: SCM_1:26
theorem Th27: :: SCM_1:27
theorem Th28: :: SCM_1:28
theorem Th29: :: SCM_1:29
theorem Th30: :: SCM_1:30
for
I1,
I2,
I3,
I4,
I5,
I6,
I7,
I8,
I9 being
Instruction of
SCM for
i1,
i2,
i3,
i4 being
Integer for
il being
Element of
NAT for
s being
State of
SCM st
IC s = il. il &
s . (il. 0) = I1 &
s . (il. 1) = I2 &
s . (il. 2) = I3 &
s . (il. 3) = I4 &
s . (il. 4) = I5 &
s . (il. 5) = I6 &
s . (il. 6) = I7 &
s . (il. 7) = I8 &
s . (il. 8) = I9 &
s . (dl. 0) = i1 &
s . (dl. 1) = i2 &
s . (dl. 2) = i3 &
s . (dl. 3) = i4 holds
s is
State-consisting of
il,0,0,
(((((((<*I1*> ^ <*I2*>) ^ <*I3*>) ^ <*I4*>) ^ <*I5*>) ^ <*I6*>) ^ <*I7*>) ^ <*I8*>) ^ <*I9*>,
((<*i1*> ^ <*i2*>) ^ <*i3*>) ^ <*i4*>
theorem Th31: :: SCM_1:31
theorem Th32: :: SCM_1:32
theorem Th33: :: SCM_1:33
theorem Th34: :: SCM_1:34
theorem Th35: :: SCM_1:35
theorem Th36: :: SCM_1:36
theorem Th37: :: SCM_1:37
theorem Th38: :: SCM_1:38
theorem Th39: :: SCM_1:39