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Exercise 9.11

Suppose you are given the following axioms:

  1. $0 \leq 3$.
  2. $7 \leq 9$.
  3. ${\forall\,x\;\;} \; \; x \leq x$.
  4. ${\forall\,x\;\;} \; \; x \leq x+0$.
  5. ${\forall\,x\;\;} \; \; x+0 \leq x$.
  6. ${\forall\,x,y\;\;} \; \; x+y \leq y+x$.
  7. ${\forall\,w,x,y,z\;\;} \; \; w \leq y$ $\wedge$ $x \leq z$ ${:\;{\Rightarrow}:\;}$ $w+x \leq y+z$.
  8. ${\forall\,x,y,z\;\;} \; \; x \leq y \wedge y \leq z : {:\;{\Rightarrow}:\;}: x \leq z$

  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.

  2. Give a forward-chaining proof of the sentence $7 \leq 3+9$. Again, show only the steps that lead to success.

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