Explain the essence and the mode of operation of function defined in the image below

Compare it with function defined below

List and explain all essential similarities and differences between these two.

To test whether a Church numeral is zero, we must find some appropriate pair of arguments that will give us back this information-specifically, we must apply our numeral to a pair of terms zz and ss such that applying ss to zz one or more times yields fls, while not applying it at all yields tru. Clearly, we should take zz to be just tru. For ss, we use a function that throws away its argument and always returns fls: pair co Co pair co Ci opy/1 pairci c2 copy/1 pair cz c3 copy/1 pair ca ca Figure 5-1: The predecessor function's "inner loop" iszro CI iszro (times co c) Surprisingly, subtraction using Church numerals is quite a bit more difficult than addition. It can be done using the following rather tricky "predecessor function," which, given Co as argument, returns co and, given , returns c: zz-pair co Co ss. p. pair (snd p) (plus cl (snd p)); prd Am. fst (m ss zz); This definition works by using m as a function to apply m copies of the func tion ss to the starting value zz. Each copy of ss takes a pair of numerals pair c.cj as its argument and yields pair cj Cj+1 as its result (see Figure 5. 1) So applying ss, m times, to pair co Co yields pair co Co when m 0 and pair Cm-1 Cm when m is positive. In both cases, the predecessor of m is found in the first component. cc-An. As. Az. s (n s z): The term scc is a combinator that takes a Church numeral n and returns another Church numeral that is, it yields a function that takes arguments s and z and applies s repeatedly to z. We get the right number of applications of s to z by first passing s and z as arguments to n, and then explicitly applying s one more time to the result. To test whether a Church numeral is zero, we must find some appropriate pair of arguments that will give us back this information-specifically, we must apply our numeral to a pair of terms zz and ss such that applying ss to zz one or more times yields fls, while not applying it at all yields tru. Clearly, we should take zz to be just tru. For ss, we use a function that throws away its argument and always returns fls: pair co Co pair co Ci opy/1 pairci c2 copy/1 pair cz c3 copy/1 pair ca ca Figure 5-1: The predecessor function's "inner loop" iszro CI iszro (times co c) Surprisingly, subtraction using Church numerals is quite a bit more difficult than addition. It can be done using the following rather tricky "predecessor function," which, given Co as argument, returns co and, given , returns c: zz-pair co Co ss. p. pair (snd p) (plus cl (snd p)); prd Am. fst (m ss zz); This definition works by using m as a function to apply m copies of the func tion ss to the starting value zz. Each copy of ss takes a pair of numerals pair c.cj as its argument and yields pair cj Cj+1 as its result (see Figure 5. 1) So applying ss, m times, to pair co Co yields pair co Co when m 0 and pair Cm-1 Cm when m is positive. In both cases, the predecessor of m is found in the first component. cc-An. As. Az. s (n s z): The term scc is a combinator that takes a Church numeral n and returns another Church numeral that is, it yields a function that takes arguments s and z and applies s repeatedly to z. We get the right number of applications of s to z by first passing s and z as arguments to n, and then explicitly applying s one more time to the result