Exercise 9.12

Suppose you are given the following axioms:

1. $0 \leq 4$.

1. $5 \leq 9$.

1. ${\forall\,x\;\;} \; \; x \leq x$.

1. ${\forall\,x\;\;} \; \; x \leq x+0$.

1. ${\forall\,x\;\;} \; \; x+0 \leq x$.

1. ${\forall\,x,y\;\;} \; \; x+y \leq y+x$.

1. ${\forall\,w,x,y,z\;\;} \; \; w \leq y$ $\wedge$ $x \leq z {:\;{\Rightarrow}:\;}$ $w+x \leq y+z$.

1. ${\forall\,x,y,z\;\;} \; \; x \leq y \wedge y \leq z : {:\;{\Rightarrow}:\;}: x \leq z$

1. Give a backward-chaining proof of the sentence $5 \leq 4+9$. (Be sure, of course, to use only the axioms given here, not anything else you may know about arithmetic.) Show only the steps that leads to success, not the irrelevant steps.

2. Give a forward-chaining proof of the sentence $5 \leq 4+9$. Again, show only the steps that lead to success.