Proof tree example

See proof-tree for an introduction to proof trees, and for a list of related topics. Here we present a detailed example followed by a shorter example that illustrates proof by induction.

Consider the guard proof for the definition of a function

( DEFUN CANCEL_EQUAL_PLUS ...) 18 Goal preprocess | <18 subgoals>

This is to be read as follows.

At this stage of the proof we have encountered the top-level goal, named "Goal", which generated 18 subgoals using the ``preprocess'' process. We have not yet begun to work on those subgoals.

The corresponding message from the ordinary prover output is:

By case analysis we reduce the conjecture to the following 18 conjectures.

Note that the field just before the name of the goal (

The next proof tree display is:

( DEFUN CANCEL_EQUAL_PLUS ...) 18 Goal preprocess 1 | Subgoal 18 simp | | <1 subgoal> | <17 more subgoals>

which indicates that at this point, the prover has used the simplification (``simp'') process on Subgoal 18 to create one subgoal (``<1 subgoal>''). The vertical bar (``|'') below ``Subgoal 18'', accompanied by the line below it, signifies that there are 17 siblings of Subgoal 18 that remain to be processed.

The next proof tree displayed is:

( DEFUN CANCEL_EQUAL_PLUS ...) 18 Goal preprocess 1 | Subgoal 18 simp c 2 | | Subgoal 18' ELIM | | | <2 subgoals> | <17 more subgoals>

Let us focus on the fourth line of this display:

c 2 | | Subgoal 18' ELIM

The ``c'' field marks this goal as a ``checkpoint'', i.e., a goal worthy of careful scrutiny. In fact, any goal that creates children by a process other than ``preprocess'' or ``simp'' is marked as a checkpoint. In this case, the destructor-elimination (``elim'') process has been used to create subgoals of this goal. The indentation shows that this goal, Subgoal 18', is a child of Subgoal 18. The number ``2'' indicates that 2 subgoals have been created (by elim). Note that this information is consistent with the line just below it, which says ``<2 subgoals>''.

Finally, the last line of this proof tree display,

| <17 more subgoals>

is connected by vertical bars (``|'') up to the string

The next proof tree display differs from the previous one only in that now, Subgoal 18' has only one more subgoal to be processed.

( DEFUN CANCEL_EQUAL_PLUS ...) 18 Goal preprocess 1 | Subgoal 18 simp c 2 | | Subgoal 18' ELIM | | | <1 more subgoal> | <17 more subgoals>

Note that the word ``more'' in ``<1 more subgoal>'' tells us that there was originally more than one subgoal of Subgoal 18. In fact that information already follows from the line above, which (as previously explained) says that Subgoal 18' originally created 2 subgoals.

The next proof tree display occurs when the prover completes the proof of that ``1 more subgoal'' referred to above.

( DEFUN CANCEL_EQUAL_PLUS ...) 18 Goal preprocess | <17 more subgoals>

Then, Subgoal 17 is processed and creates one subgoal, by simplification:

( DEFUN CANCEL_EQUAL_PLUS ...) 18 Goal preprocess 1 | Subgoal 17 simp | | <1 subgoal> | <16 more subgoals>

... and so on.

Later in the proof one might find the following successive proof tree displays.

( DEFUN CANCEL_EQUAL_PLUS ...) 18 Goal preprocess | <9 more subgoals> ( DEFUN CANCEL_EQUAL_PLUS ...) 18 Goal preprocess 0 | Subgoal 9 simp (FORCED) | <8 more subgoals>

These displays tell us that Subgoal 9 simplified to

In fact, the proof tree displayed at the end of the ``main proof''(the 0-th forcing round) is as follows.

( DEFUN CANCEL_EQUAL_PLUS ...) 18 Goal preprocess 0 | Subgoal 9 simp (FORCED) 0 | Subgoal 8 simp (FORCED) 0 | Subgoal 7 simp (FORCED) 0 | Subgoal 6 simp (FORCED) 0 | Subgoal 4 simp (FORCED) 0 | Subgoal 3 simp (FORCED)

This is followed by the following proof tree display at the start of the forcing round.

18 Goal preprocess 0 | Subgoal 9 simp (FORCED [1]Subgoal 4) 0 | Subgoal 8 simp (FORCED [1]Subgoal 6) 0 | Subgoal 7 simp (FORCED [1]Subgoal 1) 0 | Subgoal 6 simp (FORCED [1]Subgoal 3) 0 | Subgoal 4 simp (FORCED [1]Subgoal 5) 0 | Subgoal 3 simp (FORCED [1]Subgoal 2) ++++++++++++++++++++++++++++++ 6 [1]Goal FORCING-ROUND 2 | [1]Subgoal 6 preprocess | | <2 subgoals> | <5 more subgoals>

This display shows which goals to ``blame'' for the existence of each goal in the forcing round. For example, Subgoal 9 is to blame for the creation of [1]Subgoal 4.

Actually, there is no real goal named

6 [1]Goal FORCING-ROUND

appears in the proof tree display to suggest a ``parent'' of the six top-level goals in that forcing round. As usual, the numeric field before the goal name contains the original number of children of that (virtual, in this case) goal — in this case, 6.

In our example proof, Subgoal 6 eventually gets proved, without doing any further forcing. At that point, the proof tree display looks as follows.

( DEFUN CANCEL_EQUAL_PLUS ...) 18 Goal preprocess 0 | Subgoal 9 simp (FORCED [1]Subgoal 4) 0 | Subgoal 7 simp (FORCED [1]Subgoal 1) 0 | Subgoal 6 simp (FORCED [1]Subgoal 3) 0 | Subgoal 4 simp (FORCED [1]Subgoal 5) 0 | Subgoal 3 simp (FORCED [1]Subgoal 2) ++++++++++++++++++++++++++++++ 6 [1]Goal FORCING-ROUND | <5 more subgoals>

Notice that the line for Subgoal 8,

0 | Subgoal 8 simp (FORCED [1]Subgoal 6)

no longer appears. That is because the goal [1]Subgoal 6 has been proved, along with all its children; and hence, the proof of Subgoal 8 no longer depends on any further reasoning.

The final two proof tree displays in our example are as follows.

( DEFUN CANCEL_EQUAL_PLUS ...) 18 Goal preprocess 0 | Subgoal 7 simp (FORCED [1]Subgoal 1) ++++++++++++++++++++++++++++++ 6 [1]Goal FORCING-ROUND 2 | [1]Subgoal 1 preprocess 1 | | [1]Subgoal 1.1 preprocess 1 | | | [1]Subgoal 1.1' simp c 3 | | | | [1]Subgoal 1.1'' ELIM | | | | | <1 more subgoal> ( DEFUN CANCEL_EQUAL_PLUS ...) <<PROOF TREE IS EMPTY>>

The explanation for the empty proof tree is simple: once [1]Subgoal 1.1.1 was proved, nothing further remained to be proved. In fact, the much sought-after ``Q.E.D.'' appeared shortly after the final proof tree was displayed.

Let us conclude with a final, brief example that illustrates proof by induction. Partway through the proof one might come across the following proof tree display.

( DEFTHM PLUS-TREE-DEL ...) 1 Goal preprocess 2 | Goal' simp c 0 | | Subgoal 2 PUSH *1 | | <1 more subgoal>

This display says that in the attempt to prove a theorem called

( DEFTHM PLUS-TREE-DEL ...) 1 Goal preprocess 2 | Goal' simp c 0 | | Subgoal 2 PUSH *1 ( DEFTHM PLUS-TREE-DEL ...) 1 Goal preprocess 2 | Goal' simp c 0 | | Subgoal 2 PUSH *1 ++++++++++++++++++++++++++++++ c 6 *1 INDUCT | <5 more subgoals>

The separator ``+++++...'' says that we are beginning another trip through the waterfall. In fact this trip is for a proof by induction (as opposed to a forcing round), as indicated by the word ``INDUCT''. Apparently *1.6 was proved immediately, because it was not even displayed; a goal is only displayed when there is some work left to do either on it or on some goal that it brought (perhaps indirectly) into existence.

Once a proof by induction is completed, the ``PUSH'' line that refers to that proof is eliminated (``pruned''). So for example, when the present proof by induction is completed, the line

c 0 | | Subgoal 2 PUSH *1

is eliminated, which in fact causes the lines above it to be eliminated (since they no longer refer to unproved children). Hence, at that point one might expect to see:

( DEFTHM PLUS-TREE-DEL ...) <<PROOF TREE IS EMPTY>>

However, if the proof by induction of *1 necessitates further proofs by induction or a forcing round, then this ``pruning'' will not yet be done.