Suppose you are given the following axioms:
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$0 \leq 3$ .
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$7 \leq 9$ .
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${\forall,x;;} ; ; x \leq x$ .
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${\forall,x;;} ; ; x \leq x+0$ .
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${\forall,x;;} ; ; x+0 \leq x$ .
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${\forall,x,y;;} ; ; x+y \leq y+x$ .
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${\forall,w,x,y,z;;} ; ; w \leq y$ $\wedge$ $x \leq z$ ${:;{\Rightarrow}:;}$ $w+x \leq y+z$ .
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${\forall,x,y,z;;} ; ; x \leq y \wedge y \leq z : {:;{\Rightarrow}:;}: x \leq z$
1. Give a backward-chaining proof of the sentence $7 \leq 3+9$. (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.
- Give a forward-chaining proof of the sentence
$7 \leq 3+9$ . Again, show only the steps that lead to success.