Prove:

        (d)    ,

with no assumptions.

We prove (d) by the deduction rule.

We assume

        (1)   

and show

        (2)    .

Formula (1) is transformed into:

        (3)    .

We prove (2) by case distinction using (3).

Case  (4) :

Formula (4) is simplified to:

        (6)    .

By (6) we can take appropriate values such that:

        (7)    .

Because (7) matches a part of (2) , we can proceed in several ways.

Alternative proof 1: proved

For proving (2) it is sufficient to prove:

        (8)    .

Formula (8) is true because it is subsumed by (7).

Alternative proof 2: pending

Pending proof of (2).

Case  (5) :

We prove (2) by contradiction.

We assume

        (9)    ,

and show

Formula (9) is simplified to:

        (10)    .

Formula (10) is simplified to:

        (11)    .

Formula (11) is simplified to:

        (12)    .

The conjunction (12) splits into (13) , (14).

Formula (a contradiction) is proved because (14) and (5) are contradictory.

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Converted by Mathematica      February 11, 2000