:: 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 b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Instruction of
SCM for b
10, b
11, b
12, b
13 being
Integerfor b
14 being
Natfor b
15 being
State-consisting of b
14,0,0,
(((((((<*b1*> ^ <*b2*>) ^ <*b3*>) ^ <*b4*>) ^ <*b5*>) ^ <*b6*>) ^ <*b7*>) ^ <*b8*>) ^ <*b9*>,
((<*b10*> ^ <*b11*>) ^ <*b12*>) ^ <*b13*> holds
(
IC b
15 = il. b
14 & b
15 . (il. 0) = b
1 & b
15 . (il. 1) = b
2 & b
15 . (il. 2) = b
3 & b
15 . (il. 3) = b
4 & b
15 . (il. 4) = b
5 & b
15 . (il. 5) = b
6 & b
15 . (il. 6) = b
7 & b
15 . (il. 7) = b
8 & b
15 . (il. 8) = b
9 & b
15 . (dl. 0) = b
10 & b
15 . (dl. 1) = b
11 & b
15 . (dl. 2) = b
12 & b
15 . (dl. 3) = b
13 )
theorem Th15: :: SCM_1:15
:: deftheorem Def2 defines Complexity SCM_1:def 2 :
theorem Th16: :: SCM_1:16
theorem Th17: :: SCM_1:17
Lemma13:
for b1 being Nat holds Next (il. b1) = il. (b1 + 1)
Lemma14:
for b1 being Nat
for b2 being State of SCM holds (Computation b2) . (b1 + 1) = Exec (CurInstr ((Computation b2) . b1)),((Computation b2) . b1)
E15:
now
let c
1, c
2 be
Nat;
let c
3 be
State of
SCM ;
let c
4, c
5 be
Data-Location ;
assume E16:
IC ((Computation c3) . c1) = il. c
2
;
E17:
((Computation c3) . c1) . (il. c2) = c
3 . (il. c2)
by AMI_1:54;
set c
6 =
(Computation c3) . c
1;
set c
7 =
(Computation c3) . (c1 + 1);
E18:
((Computation c3) . c1) . 0
= il. c
2
by E16, Th1;
assume E19:
not ( not c
3 . (il. c2) = c
4 := c
5 & not c
3 . (il. c2) = AddTo c
4,c
5 & not c
3 . (il. c2) = SubFrom c
4,c
5 & not c
3 . (il. c2) = MultBy c
4,c
5 & not ( c
4 <> c
5 & c
3 . (il. c2) = Divide c
4,c
5 ) )
;
thus E20:
(Computation c3) . (c1 + 1) =
Exec (CurInstr ((Computation c3) . c1)),
((Computation c3) . c1)
by Lemma14
.=
Exec (c3 . (il. c2)),
((Computation c3) . c1)
by E17, E18, Th1
;
thus IC ((Computation c3) . (c1 + 1)) =
((Computation c3) . (c1 + 1)) . 0
by Th1
.=
Next (IC ((Computation c3) . c1))
by E19, E20, AMI_3:4, AMI_3:8, AMI_3:9, AMI_3:10, AMI_3:11, AMI_3:12
.=
il. (c2 + 1)
by E16, Lemma13
;
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 b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Instruction of
SCM for b
10, b
11, b
12, b
13 being
Integerfor b
14 being
Natfor b
15 being
State of
SCM holds
(
IC b
15 = il. b
14 & b
15 . (il. 0) = b
1 & b
15 . (il. 1) = b
2 & b
15 . (il. 2) = b
3 & b
15 . (il. 3) = b
4 & b
15 . (il. 4) = b
5 & b
15 . (il. 5) = b
6 & b
15 . (il. 6) = b
7 & b
15 . (il. 7) = b
8 & b
15 . (il. 8) = b
9 & b
15 . (dl. 0) = b
10 & b
15 . (dl. 1) = b
11 & b
15 . (dl. 2) = b
12 & b
15 . (dl. 3) = b
13 implies b
15 is
State-consisting of b
14,0,0,
(((((((<*b1*> ^ <*b2*>) ^ <*b3*>) ^ <*b4*>) ^ <*b5*>) ^ <*b6*>) ^ <*b7*>) ^ <*b8*>) ^ <*b9*>,
((<*b10*> ^ <*b11*>) ^ <*b12*>) ^ <*b13*> )
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