Turing Completeness

Both language $1$ and $2($shown below$)$ are Turing complete in the sense that they can be used to describe all the

computations that a Turning machine can perform.

Language $1$

boolean expressions

integer variables

integer arithmetic

$($including $++$ operators like those found in $C++)$

sequential execution of statements

if$-$then statements

assignment statements

while$-$loops

Language $2$

boolean expressions

integer values

integer arithmetic

function definition

function call

conditional$-$expressions

recursion

Problem $1(20$ points$)$

Prove that a language is also Turing complete if it $($just$)$ contains the following constructs:

Note that "proof by specific example" is not a valid technique here. $($Hint: Language equivalence$).$ Note that the

"break" command is not part of any of these languages.