let A be free preIfWhileAlgebra; :: thesis: for C, I1, I2 being Element of A
for n being Nat st if-then-else C,I1,I2 in (ElementaryInstructions A) |^ n holds
ex i being Nat st
( n = i + 1 & C in (ElementaryInstructions A) |^ i & I1 in (ElementaryInstructions A) |^ i & I2 in (ElementaryInstructions A) |^ i )
set B = ElementaryInstructions A;
A1:
ElementaryInstructions A is GeneratorSet of A
by Def25;
let C, I1, I2 be Element of A; :: thesis: for n being Nat st if-then-else C,I1,I2 in (ElementaryInstructions A) |^ n holds
ex i being Nat st
( n = i + 1 & C in (ElementaryInstructions A) |^ i & I1 in (ElementaryInstructions A) |^ i & I2 in (ElementaryInstructions A) |^ i )
if-then-else C,I1,I2 nin ElementaryInstructions A
by Th51;
then A2:
if-then-else C,I1,I2 nin (ElementaryInstructions A) |^ 0
by Th18;
let n be Nat; :: thesis: ( if-then-else C,I1,I2 in (ElementaryInstructions A) |^ n implies ex i being Nat st
( n = i + 1 & C in (ElementaryInstructions A) |^ i & I1 in (ElementaryInstructions A) |^ i & I2 in (ElementaryInstructions A) |^ i ) )
assume A3:
if-then-else C,I1,I2 in (ElementaryInstructions A) |^ n
; :: thesis: ex i being Nat st
( n = i + 1 & C in (ElementaryInstructions A) |^ i & I1 in (ElementaryInstructions A) |^ i & I2 in (ElementaryInstructions A) |^ i )
then consider i being Nat such that
A4:
n = i + 1
by A2, NAT_1:6;
take
i
; :: thesis: ( n = i + 1 & C in (ElementaryInstructions A) |^ i & I1 in (ElementaryInstructions A) |^ i & I2 in (ElementaryInstructions A) |^ i )
thus
n = i + 1
by A4; :: thesis: ( C in (ElementaryInstructions A) |^ i & I1 in (ElementaryInstructions A) |^ i & I2 in (ElementaryInstructions A) |^ i )
A5:
dom (Den (In 3,(dom the charact of A)),A) = 3 -tuples_on the carrier of A
by Th47;
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 3,(dom the charact of A)
proof
let o be
OperSymbol of
A;
:: thesis: for p being FinSequence st p in dom (Den o,A) & (Den o,A) . p in ElementaryInstructions A holds
o <> In 3,(dom the charact of A)let p be
FinSequence;
:: thesis: ( p in dom (Den o,A) & (Den o,A) . p in ElementaryInstructions A implies o <> In 3,(dom the charact of A) )
assume A7:
(
p in dom (Den o,A) &
(Den o,A) . p in ElementaryInstructions A &
o = In 3,
(dom the charact of A) )
;
:: thesis: contradiction
then consider a,
b,
c being
set such that A8:
(
a in the
carrier of
A &
b in the
carrier of
A &
c in the
carrier of
A &
p = <*a,b,c*> )
by A5, CATALG_1:12;
reconsider a =
a,
b =
b,
c =
c as
Element of
A by A8;
(Den o,A) . p = if-then-else a,
b,
c
by A7, A8;
hence
contradiction
by A7, Th51;
:: thesis: verum
end;
<*C,I1,I2*> in dom (Den (In 3,(dom the charact of A)),A)
by A5, CATALG_1:12;
then
rng <*C,I1,I2*> c= (ElementaryInstructions A) |^ i
by A1, A3, A4, A6, Th39;
then
( {C,I1,I2} c= (ElementaryInstructions A) |^ i & C in {C,I1,I2} & I1 in {C,I1,I2} & I2 in {C,I1,I2} )
by ENUMSET1:def 1, FINSEQ_2:148;
hence
( C in (ElementaryInstructions A) |^ i & I1 in (ElementaryInstructions A) |^ i & I2 in (ElementaryInstructions A) |^ i )
; :: thesis: verum