Prove two algorithms are identical - algorithm

Question

I am supposed to show that two algorithms execute identical statements in identical order. One is a tail recursive version of the other. They are written in Eiffel.
tail_rec(x:instance_type):result_type is
local
y:instance_type;
do
if b(x) then
Result :=c(x)
else
y:=d(x);
Result:=tail_rec(y)
end
end
Then the non-tail recursive version.
non_rec(x:instance_type):result_type is
do
from until b(x) loop
x:= d(x)
end;
Result:= c(x)
end
where b(x), c(x), and d(x) are any functions of type BOOLEAN, result_type, and instance_type respectively.
How are these two algorithms similar and how do they execute identical statements in identical order?

Answer 1

By substituting all flow-of-control constructs with gotos, (essentially turning the code from Eiffel into pseudocode,) and allowing if statements to execute only gotos, it can be shown that both functions end up consisting of the exact same set of instructions.
Let me begin by copying the original tail_rec here for convenience:
tail_rec(x:instance_type):result_type is
local
y:instance_type;
do
if b(x) then
Result := c(x)
else
y := d(x);
Result := tail_rec(y)
end
end
First, get rid of Eiffel's silly Result := construct and replace it with return for convenience. (Otherwise, we are going to have to add more gotos, and frankly, the fewer the better.)
tail_rec(x:instance_type):result_type is
local
y:instance_type;
do
if b(x) then
return c(x)
end
y := d(x);
return tail_rec(y)
end
Substitute the if-then-end with if-then goto:
tail_rec(x:instance_type):result_type is
local
y:instance_type;
do
if not b(x) then goto label1
return c(x)
label1:
y := d(x);
return tail_rec(y)
end
Replace tail recursion with another goto:
tail_rec(x:instance_type):result_type is
do
label0:
if not b(x) then goto label1
return c(x)
label1:
x := d(x);
goto label0
end
Replace if not b(x) with if b(x):
tail_rec(x:instance_type):result_type is
do
label0:
if b(x) then goto label1
x := d(x);
goto label0
label1:
return c(x)
end
Similar transformations to tail_rec should turn it into the exact same thing.