Exercise 5: A simple FOL, Sequent Based, prover

Overview

Play with a backwards prover implementing FOL in LK. This prover uses explicit terms for all on the left and exists on the right.

Learning Objectives

• Learn about FOL

Getting Started

Code is here.

Alternate resource: Check out Larry Paulson's FOL theorem prover linked from this page: http://www.cl.cam.ac.uk/teaching/1011/LogicProof/

A short demo:

Fire up ghci; load FOL/LK.hs

We prove example formula (forall x. P(x)) --> (forall y. P(F(y))) as follows:

m1 = (replayM s1 [impliesR 1, allR 1, allL (toTerm "F(?b)") 1, axiom 1 1])

As in the other backwards provers, the list is a list of rules. We begin with implies on the right (impliesR 1): This turns the sequent with only an implication

|- ((ALL x. P(x)) --> (ALL y. P(F(y))))

(ALL x. P(x)) |- (ALL y. P(F(y)))

We continue with all on the right (allR 1), which introduces a new variable ?b, that is fresh:

(ALL x. P(x)) |- P(F(?b)), (ALL y. P(F(y)))

Next we apply exists on the right, providing a term as required by the rule:

P(F(?b)) |- P(F(?b)), (ALL y. P(F(y)))

This is followed with all on the left (allL (toTerm "F(?b)") 1), which also requires a term. We choose "F(?b)" .

We complete the proof with the axiom rule (axiom 1 1), which leaves us with:

QED

What to do

1. Work some examples (at least 3). You can find some in Smullyan or in Coble, or make up your own.
2. Study the rules al,ar,ol,ar,il,ir, etc. (lines 198-225) that add some search.
3. convince your self that they will work
4. then redo at least 2 of your examples.

What to turn in.

• Create a file (that imports LK.hs) with your examples, and submit it to D2l
• Be prepared to talk about your solution in class.