Equivalences of programs, extra axioms, (derived) context viz. theorems
Where we last left.
- We had a more powerful technique. Program /equivalences/ rather than just boolean expressions.
Booleans, we had counterexamples. So far our claims that we’ve seen also give us some counterexamples.
| Just Booleans | Expressions | |
| Counterexamples | Generated | Can test if supplied |
| Power | Lesser | Greater |
| Decidable? | Yes | No |
| Automated Proofs | Trivially yes | Can be verified |
With greater power comes fewer freebies.
Machine checkable.
- Which is great!
- Important to have a standardized format.
Proof Setup
- Starting w/a given conjecture
- Check the contracts
- Contract completion (if necessary)
- Exportation (if necessary)
- Write out the /context/
- Write out the /derived context/
These last two are new for us today!
Context
- The initial setup that we have. Presumptions.
Contexts vs. Theorem
How do we think about a context vs. a theorem?
We can instantiate a theorem, since it holds generally. The context is fixed in any particular situation.
(implies (endp x) (endp y))
We had better not confuse (endp x) with something that holds for all values of x!
Derived Context.
These are the obvious things that follow from the context + axioms/theorems
Common patterns:
- (endp x), (tlp x): (== x nil)
- (tlp x), (consp x): (tlp (rest x))
- φ1 ∧ … ∧ φn -> ψ: Derive φ1,…,φn and use MP to derive ψ
HW: Examples of such conjectures
The pen-and-paper prover isn’t fully battle-ready.
Upgrades coming, plus more documentation