begin
Lm1:
for S being non empty 1-sorted holds dom ((SCM-VAL S) * SCM-OK) = SCM-Memory
definition
canceled;canceled;canceled;canceled;canceled;canceled;canceled;let R be
Ring;
let x be
Element of
SCM-Instr R;
let s be
SCM-State of
R;
func SCM-Exec-Res (
x,
s)
-> SCM-State of
R equals
SCM-Chg (
(SCM-Chg (s,(x address_1),(s . (x address_2)))),
(succ (IC s)))
if ex
mk,
ml being
Element of
SCM-Data-Loc st
x = [1,{},<*mk,ml*>] SCM-Chg (
(SCM-Chg (s,(x address_1),((s . (x address_1)) + (s . (x address_2))))),
(succ (IC s)))
if ex
mk,
ml being
Element of
SCM-Data-Loc st
x = [2,{},<*mk,ml*>] SCM-Chg (
(SCM-Chg (s,(x address_1),((s . (x address_1)) - (s . (x address_2))))),
(succ (IC s)))
if ex
mk,
ml being
Element of
SCM-Data-Loc st
x = [3,{},<*mk,ml*>] SCM-Chg (
(SCM-Chg (s,(x address_1),((s . (x address_1)) * (s . (x address_2))))),
(succ (IC s)))
if ex
mk,
ml being
Element of
SCM-Data-Loc st
x = [4,{},<*mk,ml*>] SCM-Chg (
s,
(x jump_address))
if ex
mk being
Element of
NAT st
x = [6,<*mk*>,{}] SCM-Chg (
s,
(IFEQ ((s . (x cond_address)),(0. R),(x cjump_address),(succ (IC s)))))
if ex
mk being
Element of
NAT ex
ml being
Element of
SCM-Data-Loc st
x = [7,<*mk*>,<*ml*>] SCM-Chg (
(SCM-Chg (s,(x const_address),(x const_value))),
(succ (IC s)))
if ex
mk being
Element of
SCM-Data-Loc ex
r being
Element of
R st
x = [5,{},<*mk,r*>] otherwise s;
coherence
( ( ex mk, ml being Element of SCM-Data-Loc st x = [1,{},<*mk,ml*>] implies SCM-Chg ((SCM-Chg (s,(x address_1),(s . (x address_2)))),(succ (IC s))) is SCM-State of R ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [2,{},<*mk,ml*>] implies SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) + (s . (x address_2))))),(succ (IC s))) is SCM-State of R ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [3,{},<*mk,ml*>] implies SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) - (s . (x address_2))))),(succ (IC s))) is SCM-State of R ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [4,{},<*mk,ml*>] implies SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) * (s . (x address_2))))),(succ (IC s))) is SCM-State of R ) & ( ex mk being Element of NAT st x = [6,<*mk*>,{}] implies SCM-Chg (s,(x jump_address)) is SCM-State of R ) & ( ex mk being Element of NAT ex ml being Element of SCM-Data-Loc st x = [7,<*mk*>,<*ml*>] implies SCM-Chg (s,(IFEQ ((s . (x cond_address)),(0. R),(x cjump_address),(succ (IC s))))) is SCM-State of R ) & ( ex mk being Element of SCM-Data-Loc ex r being Element of R st x = [5,{},<*mk,r*>] implies SCM-Chg ((SCM-Chg (s,(x const_address),(x const_value))),(succ (IC s))) is SCM-State of R ) & ( ( for mk, ml being Element of SCM-Data-Loc holds not x = [1,{},<*mk,ml*>] ) & ( for mk, ml being Element of SCM-Data-Loc holds not x = [2,{},<*mk,ml*>] ) & ( for mk, ml being Element of SCM-Data-Loc holds not x = [3,{},<*mk,ml*>] ) & ( for mk, ml being Element of SCM-Data-Loc holds not x = [4,{},<*mk,ml*>] ) & ( for mk being Element of NAT holds not x = [6,<*mk*>,{}] ) & ( for mk being Element of NAT
for ml being Element of SCM-Data-Loc holds not x = [7,<*mk*>,<*ml*>] ) & ( for mk being Element of SCM-Data-Loc
for r being Element of R holds not x = [5,{},<*mk,r*>] ) implies s is SCM-State of R ) )
;
consistency
for b1 being SCM-State of R holds
( ( ex mk, ml being Element of SCM-Data-Loc st x = [1,{},<*mk,ml*>] & ex mk, ml being Element of SCM-Data-Loc st x = [2,{},<*mk,ml*>] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),(s . (x address_2)))),(succ (IC s))) iff b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) + (s . (x address_2))))),(succ (IC s))) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [1,{},<*mk,ml*>] & ex mk, ml being Element of SCM-Data-Loc st x = [3,{},<*mk,ml*>] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),(s . (x address_2)))),(succ (IC s))) iff b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) - (s . (x address_2))))),(succ (IC s))) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [1,{},<*mk,ml*>] & ex mk, ml being Element of SCM-Data-Loc st x = [4,{},<*mk,ml*>] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),(s . (x address_2)))),(succ (IC s))) iff b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) * (s . (x address_2))))),(succ (IC s))) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [1,{},<*mk,ml*>] & ex mk being Element of NAT st x = [6,<*mk*>,{}] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),(s . (x address_2)))),(succ (IC s))) iff b1 = SCM-Chg (s,(x jump_address)) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [1,{},<*mk,ml*>] & ex mk being Element of NAT ex ml being Element of SCM-Data-Loc st x = [7,<*mk*>,<*ml*>] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),(s . (x address_2)))),(succ (IC s))) iff b1 = SCM-Chg (s,(IFEQ ((s . (x cond_address)),(0. R),(x cjump_address),(succ (IC s))))) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [1,{},<*mk,ml*>] & ex mk being Element of SCM-Data-Loc ex r being Element of R st x = [5,{},<*mk,r*>] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),(s . (x address_2)))),(succ (IC s))) iff b1 = SCM-Chg ((SCM-Chg (s,(x const_address),(x const_value))),(succ (IC s))) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [2,{},<*mk,ml*>] & ex mk, ml being Element of SCM-Data-Loc st x = [3,{},<*mk,ml*>] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) + (s . (x address_2))))),(succ (IC s))) iff b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) - (s . (x address_2))))),(succ (IC s))) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [2,{},<*mk,ml*>] & ex mk, ml being Element of SCM-Data-Loc st x = [4,{},<*mk,ml*>] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) + (s . (x address_2))))),(succ (IC s))) iff b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) * (s . (x address_2))))),(succ (IC s))) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [2,{},<*mk,ml*>] & ex mk being Element of NAT st x = [6,<*mk*>,{}] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) + (s . (x address_2))))),(succ (IC s))) iff b1 = SCM-Chg (s,(x jump_address)) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [2,{},<*mk,ml*>] & ex mk being Element of NAT ex ml being Element of SCM-Data-Loc st x = [7,<*mk*>,<*ml*>] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) + (s . (x address_2))))),(succ (IC s))) iff b1 = SCM-Chg (s,(IFEQ ((s . (x cond_address)),(0. R),(x cjump_address),(succ (IC s))))) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [2,{},<*mk,ml*>] & ex mk being Element of SCM-Data-Loc ex r being Element of R st x = [5,{},<*mk,r*>] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) + (s . (x address_2))))),(succ (IC s))) iff b1 = SCM-Chg ((SCM-Chg (s,(x const_address),(x const_value))),(succ (IC s))) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [3,{},<*mk,ml*>] & ex mk, ml being Element of SCM-Data-Loc st x = [4,{},<*mk,ml*>] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) - (s . (x address_2))))),(succ (IC s))) iff b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) * (s . (x address_2))))),(succ (IC s))) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [3,{},<*mk,ml*>] & ex mk being Element of NAT st x = [6,<*mk*>,{}] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) - (s . (x address_2))))),(succ (IC s))) iff b1 = SCM-Chg (s,(x jump_address)) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [3,{},<*mk,ml*>] & ex mk being Element of NAT ex ml being Element of SCM-Data-Loc st x = [7,<*mk*>,<*ml*>] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) - (s . (x address_2))))),(succ (IC s))) iff b1 = SCM-Chg (s,(IFEQ ((s . (x cond_address)),(0. R),(x cjump_address),(succ (IC s))))) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [3,{},<*mk,ml*>] & ex mk being Element of SCM-Data-Loc ex r being Element of R st x = [5,{},<*mk,r*>] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) - (s . (x address_2))))),(succ (IC s))) iff b1 = SCM-Chg ((SCM-Chg (s,(x const_address),(x const_value))),(succ (IC s))) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [4,{},<*mk,ml*>] & ex mk being Element of NAT st x = [6,<*mk*>,{}] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) * (s . (x address_2))))),(succ (IC s))) iff b1 = SCM-Chg (s,(x jump_address)) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [4,{},<*mk,ml*>] & ex mk being Element of NAT ex ml being Element of SCM-Data-Loc st x = [7,<*mk*>,<*ml*>] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) * (s . (x address_2))))),(succ (IC s))) iff b1 = SCM-Chg (s,(IFEQ ((s . (x cond_address)),(0. R),(x cjump_address),(succ (IC s))))) ) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [4,{},<*mk,ml*>] & ex mk being Element of SCM-Data-Loc ex r being Element of R st x = [5,{},<*mk,r*>] implies ( b1 = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) * (s . (x address_2))))),(succ (IC s))) iff b1 = SCM-Chg ((SCM-Chg (s,(x const_address),(x const_value))),(succ (IC s))) ) ) & ( ex mk being Element of NAT st x = [6,<*mk*>,{}] & ex mk being Element of NAT ex ml being Element of SCM-Data-Loc st x = [7,<*mk*>,<*ml*>] implies ( b1 = SCM-Chg (s,(x jump_address)) iff b1 = SCM-Chg (s,(IFEQ ((s . (x cond_address)),(0. R),(x cjump_address),(succ (IC s))))) ) ) & ( ex mk being Element of NAT st x = [6,<*mk*>,{}] & ex mk being Element of SCM-Data-Loc ex r being Element of R st x = [5,{},<*mk,r*>] implies ( b1 = SCM-Chg (s,(x jump_address)) iff b1 = SCM-Chg ((SCM-Chg (s,(x const_address),(x const_value))),(succ (IC s))) ) ) & ( ex mk being Element of NAT ex ml being Element of SCM-Data-Loc st x = [7,<*mk*>,<*ml*>] & ex mk being Element of SCM-Data-Loc ex r being Element of R st x = [5,{},<*mk,r*>] implies ( b1 = SCM-Chg (s,(IFEQ ((s . (x cond_address)),(0. R),(x cjump_address),(succ (IC s))))) iff b1 = SCM-Chg ((SCM-Chg (s,(x const_address),(x const_value))),(succ (IC s))) ) ) )
by XTUPLE_0:3;
end;
::
deftheorem defines
SCM-Exec-Res SCMRING1:def 14 :
for R being Ring
for x being Element of SCM-Instr R
for s being SCM-State of R holds
( ( ex mk, ml being Element of SCM-Data-Loc st x = [1,{},<*mk,ml*>] implies SCM-Exec-Res (x,s) = SCM-Chg ((SCM-Chg (s,(x address_1),(s . (x address_2)))),(succ (IC s))) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [2,{},<*mk,ml*>] implies SCM-Exec-Res (x,s) = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) + (s . (x address_2))))),(succ (IC s))) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [3,{},<*mk,ml*>] implies SCM-Exec-Res (x,s) = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) - (s . (x address_2))))),(succ (IC s))) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [4,{},<*mk,ml*>] implies SCM-Exec-Res (x,s) = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) * (s . (x address_2))))),(succ (IC s))) ) & ( ex mk being Element of NAT st x = [6,<*mk*>,{}] implies SCM-Exec-Res (x,s) = SCM-Chg (s,(x jump_address)) ) & ( ex mk being Element of NAT ex ml being Element of SCM-Data-Loc st x = [7,<*mk*>,<*ml*>] implies SCM-Exec-Res (x,s) = SCM-Chg (s,(IFEQ ((s . (x cond_address)),(0. R),(x cjump_address),(succ (IC s))))) ) & ( ex mk being Element of SCM-Data-Loc ex r being Element of R st x = [5,{},<*mk,r*>] implies SCM-Exec-Res (x,s) = SCM-Chg ((SCM-Chg (s,(x const_address),(x const_value))),(succ (IC s))) ) & ( ( for mk, ml being Element of SCM-Data-Loc holds not x = [1,{},<*mk,ml*>] ) & ( for mk, ml being Element of SCM-Data-Loc holds not x = [2,{},<*mk,ml*>] ) & ( for mk, ml being Element of SCM-Data-Loc holds not x = [3,{},<*mk,ml*>] ) & ( for mk, ml being Element of SCM-Data-Loc holds not x = [4,{},<*mk,ml*>] ) & ( for mk being Element of NAT holds not x = [6,<*mk*>,{}] ) & ( for mk being Element of NAT
for ml being Element of SCM-Data-Loc holds not x = [7,<*mk*>,<*ml*>] ) & ( for mk being Element of SCM-Data-Loc
for r being Element of R holds not x = [5,{},<*mk,r*>] ) implies SCM-Exec-Res (x,s) = s ) );