By John Harrison PhD, MA (auth.)

This publication discusses using the genuine numbers in theorem proving. Typ ically, theorem provers in basic terms aid a number of 'discrete' datatypes resembling the ordinary numbers. but the availability of the genuine numbers opens up many fascinating and critical program components, reminiscent of the verification of waft ing aspect and hybrid platforms. It additionally permits the formalization of many extra branches of classical arithmetic, that's relatively correct for makes an attempt to inject extra rigour into computing device algebra platforms. Our paintings is performed in a model of the HOL theorem prover. We de scribe the rigorous definitional development of the true numbers, utilizing a brand new model of Cantor's technique, and the formalization of a good portion of genuine research. We additionally describe a sophisticated derived selection method for the 'Tarski subset' of genuine algebra in addition to a few extra modest yet virtually invaluable instruments for automating particular calculations and regimen linear mathematics reasoning. ultimately, we think of in additional element attention-grabbing program components. We talk about the desirability of mixing the rigour of theorem provers with the ability and comfort of laptop algebra platforms, and clarify a mode we've utilized in perform to accomplish this. We then movement directly to the verification of floating element undefined. After a cautious dialogue of attainable correctness standards, we document on case experiences, one related to a transcendental function.

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**Additional resources for Theorem Proving with the Real Numbers**

**Example text**

A::; b vb::; a. Proof. lmb n - nbml ::; B(m + n) Now suppose it is not the case that a ::; b Vb::; a. In that case, there are m and n, which we may assume without loss of generality to be nonzero, with: bm > am + (A + B) so man + nbm > mbn + nam + (A + B)(m + n) However this contradicts the fact that: so the theorem is true. 2 Injecting the naturals The natural injection ~ from N can be defined as follows; evidently it yields a nearly-additive function. ~(n)i = ni For brevity, we will denote the injection of n by n* rather than ~(n).

Constructing the Real Numbers for a generic pair of type bijections then instantiated with each call (similar remarks apply to other general theorems that we use). VP. (Vx : p. P(mk(R x))) ¢:> (Va: a. P(a)) The proof is rather straightforward, since precisely everything in a is an isomorphic image of an R-equivalence class. We also prove the same thing for the existential quantifier. Now, simply higher order rewriting with the derived theorems from the function-lifting stage together with these quantifier theorems gives the required result.

Conway (1976) emphasizes the difficulty of constructing IR from Q by Dedekind cuts: Nobody can seriously pretend that he has ever discussed even eight cases in such a theorem - yet I have seen a presentation in which one theorem actually had 64 cases ... Of course an elegant treatment will manage to discuss several cases at once, but one has to work very hard to find such a treatment. He advocates instead following the path on the lattice diagram through Q+ and IR+, at least if Dedekind's method is to be used.