let A be free preIfWhileAlgebra; for C, I being Element of A
for n being Nat st while C,I in (ElementaryInstructions A) |^ n holds
ex i being Nat st
( n = i + 1 & C in (ElementaryInstructions A) |^ i & I in (ElementaryInstructions A) |^ i )
set B = ElementaryInstructions A;
A1:
ElementaryInstructions A is GeneratorSet of A
by Def25;
let C, I be Element of A; for n being Nat st while C,I in (ElementaryInstructions A) |^ n holds
ex i being Nat st
( n = i + 1 & C in (ElementaryInstructions A) |^ i & I in (ElementaryInstructions A) |^ i )
while C,I nin ElementaryInstructions A
by Th52;
then A2:
while C,I nin (ElementaryInstructions A) |^ 0
by Th18;
let n be Nat; ( while C,I in (ElementaryInstructions A) |^ n implies ex i being Nat st
( n = i + 1 & C in (ElementaryInstructions A) |^ i & I in (ElementaryInstructions A) |^ i ) )
assume A3:
while C,I in (ElementaryInstructions A) |^ n
; ex i being Nat st
( n = i + 1 & C in (ElementaryInstructions A) |^ i & I in (ElementaryInstructions A) |^ i )
then consider i being Nat such that
A4:
n = i + 1
by A2, NAT_1:6;
take
i
; ( n = i + 1 & C in (ElementaryInstructions A) |^ i & I in (ElementaryInstructions A) |^ i )
thus
n = i + 1
by A4; ( C in (ElementaryInstructions A) |^ i & I in (ElementaryInstructions A) |^ i )
A5:
dom (Den (In 4,(dom the charact of A)),A) = 2 -tuples_on the carrier of A
by Th48;
A6:
for o being OperSymbol of A
for p being FinSequence st p in dom (Den o,A) & (Den o,A) . p in ElementaryInstructions A holds
o <> In 4,(dom the charact of A)
proof
let o be
OperSymbol of
A;
for p being FinSequence st p in dom (Den o,A) & (Den o,A) . p in ElementaryInstructions A holds
o <> In 4,(dom the charact of A)let p be
FinSequence;
( p in dom (Den o,A) & (Den o,A) . p in ElementaryInstructions A implies o <> In 4,(dom the charact of A) )
assume that A7:
p in dom (Den o,A)
and A8:
(Den o,A) . p in ElementaryInstructions A
and A9:
o = In 4,
(dom the charact of A)
;
contradiction
consider a,
b being
set such that A10:
a in the
carrier of
A
and A11:
b in the
carrier of
A
and A12:
p = <*a,b*>
by A5, A7, A9, FINSEQ_2:157;
reconsider a =
a,
b =
b as
Element of
A by A10, A11;
(Den o,A) . p = while a,
b
by A9, A12;
hence
contradiction
by A8, Th52;
verum
end;
<*C,I*> in dom (Den (In 4,(dom the charact of A)),A)
by A5, FINSEQ_2:157;
then
rng <*C,I*> c= (ElementaryInstructions A) |^ i
by A1, A3, A4, A6, Th39;
then
{C,I} c= (ElementaryInstructions A) |^ i
by FINSEQ_2:147;
hence
( C in (ElementaryInstructions A) |^ i & I in (ElementaryInstructions A) |^ i )
by ZFMISC_1:38; verum