# Re: [ProofPower] ProofPower and Discrete Math

```I have made some more progress in applying ProofPower to forward proofs in
discrete math. e.g. I can prove DeMorgan's theorem successful
(propositional calculus).  I would like to use the newly proved theorem in
other proofs now.  What is the best strategy to do that?  Is it to create
an SML function that takes an argument of type TERM and returns a THM?
Could you give an example of a proved THM intended to be used as a new rule
of inference?```
```
I also discovered the  not_or_thm in the zed theory, which is DeMorgan's
Law, but that does not seem to be a function, so I am not sure how to apply
it in a forward proof.

I appreciate any pointers in the right direction (or reference to which
manual might enlighten me). Thanks, Dave

On Wed, Aug 17, 2016 at 11:06 AM, David Topham <dtop...@ohlone.edu> wrote:

> Thank you both (Rob and Roger) for your "above and beyond the call of
> duty" help in getting me through my attempts to approach using ProofPower
> to teach Discrete Math. I understand your comments that backward proof is
> more economical than forward proof. Yet, I don't think the students will
> learn how to formulate proof thinking skills if ProofPower does it for
> them! True, that learning how to use the mechanized assistance is
> practical, but we (students) need to understand what constitutes valid
> reasoning so have to see the tedious steps one-at-a-time.  (e.g. I tried "a
> (prove_tac[]);" and it works perfectly, but does not show me all the steps
> that were used (is there a way for it to do that?))
>
> I believe the majority of courses teaching DM don't even try to go beyond
> the paper and pencil proofs--maybe that is good enough, but I can't help
> but think that exposure to professional tools (i.e. ProofPower) that force
> us to think carefully about what we are doing and to have the computer not
> allow us to proceed on shaky grounds is beneficial.
>
> Even though ProofPower is designed for math pros and is difficult to
> approach as beginning math novices, it is very well designed in its
> interface and ease of use. I am drawn to it by integration of "literate"
> approaches to documentation, ability to enter math notation, support for
> SML, and Z too (which is my next goal once basic Prop/Pred Calc is under
> our belts). Overall we should be able to derive correct programs and/or
> prove algorithms.
>
> My idea is that application of proofs to software design is the main goal
> of a Discrete Math course (recently changed to Discrete Structures by the
> powers that be).  I do recognize the value of your time, and will resist
> asking each time I get stuck unless I am unable to continue.
> -Dave
>
>
> On Wed, Aug 17, 2016 at 3:02 AM, Rob Arthan <r...@lemma-one.com> wrote:
>
>> David,
>>
>> I endorse what Roger said about forward v. backwards proof.
>> There is definitely a tension between teaching proof theory
>> and teaching practical mechanised proof.
>>
>> For the record, the problem was that you had the two theorem
>> arguments of ∃_elim the wrong way round and you seemed
>> to have misunderstood the role of the term argument: it is
>> the variable that is free in the assumption of the second theorem
>> that is going to be discharged by the first theorem (the one
>> with the existential conclusion). Here is the complete proof
>> with the output you should see in the comments.
>>
>> val L1 = asm_rule ⌜p(x,y):BOOL⌝;
>> (* val L1 = p (x, y) ⊢ p (x, y): THM *)
>> val L2 = ⌜∃x:'a⦁p(x,y)⌝;
>> (* val L2 = ⌜∃ x⦁ p (x, y)⌝: TERM *)
>> val L3 = ∃_intro L2 L1;
>> (* val L3 = p (x, y) ⊢ ∃ x⦁ p (x, y): THM *)
>> val L4 = ⌜∃y:'b⦁∃x:'a⦁p(x,y)⌝;
>> (* val L4 = ⌜∃ y x⦁ p (x, y)⌝: TERM *)
>> val L5 = ∃_intro L4 L3;
>> (* val L5 = p (x, y) ⊢ ∃ y x⦁ p (x, y): THM *)
>> val L6 = asm_rule ⌜∃y:'b⦁p(x,y)⌝;
>> (* val L6 = ∃ y⦁ p (x, y) ⊢ ∃ y⦁ p (x, y): THM *)
>> val L7 = ∃_elim ⌜y:'b⌝ L6 L5;
>> (* val L7 = ∃ y⦁ p (x, y) ⊢ ∃ y x⦁ p (x, y): THM *)
>> val L8 = asm_rule ⌜∃x:'a⦁ ∃y:'b⦁p(x,y)⌝;
>> (* val L8 = ∃ x y⦁ p (x, y) ⊢ ∃ x y⦁ p (x, y): THM *)
>> val L9 = ∃_elim ⌜x:'a⌝ L8 L7;
>> (* val L9 = ∃ x y⦁ p (x, y) ⊢ ∃ y x⦁ p (x, y): THM *)
>> val L10 = ⇒_intro ⌜∃x:'a⦁ ∃y:'b⦁p(x,y)⌝ L9;
>> (* val L10 = ⊢ (∃ x y⦁ p (x, y)) ⇒ (∃ y x⦁ p (x, y)): THM *)
>>
>> The documentation and the error messages talk about varstructs
>> meaning either variables or  things formed from variables using pairing
>> (like ∃_elim will work on theorems with conclusions like ∃(x, y)⦁ p (x,
>> y)).
>> You can just read “variable” for “varstruct” in simple examples.
>>
>> Regards,
>>
>> Rob.
>>
>> On 16 Aug 2016, at 21:24, David Topham <dtop...@ohlone.edu> wrote:
>>
>> Since the slides for this book use slightly different notation, I am back
>> to trying to implement the proofs in the main book:  UsingZ from
>> www.usingz.com  (in text link, it is zedbook)
>>
>> On page 42, the proof is using nested existentials, and I am trying
>> to get past my lack of understanding in applying E-elim
>> (Roger already helped me with E-intro)
>>
>> Here are two of my attempts (using ASCII since I can't attach pdf here)
>> val L1 = asm_rule %<%p(x,y):BOOL%>%;
>> val L2 = %<%%exists%x:'a%spot%p(x,y)%>%;
>> val L3 = %exists%_intro L2 L1;
>> val L4 = %<%%exists%y:'b%spot%%exists%x:'a%spot%p(x,y)%>%;
>> val L5 = %exists%_intro L4 L3;
>> val L6 = asm_rule %<%%exists%y:'b%spot%p(x,y)%>%;
>> val L7 = %exists%_elim L4 L5 L6;
>>
>> val L1 = asm_rule %<%p:BOOL%>%;
>> val L2 = %<%%exists%x:'a%spot%p%>%;
>> val L3 = %exists%_intro L2 L1;
>> val L4 = %<%%exists%y:'b%spot%%exists%x:'a%spot%p%>%;
>> val L5 = %exists%_intro L4 L3;
>> val L6 = asm_rule %<%%exists%y:'b%spot%p%>%;
>> val L7 = %exists%_elim L4 L5 L6;
>>
>> The error I get is "does not match the bound varstruct"
>>
>> Does anyone have a suggestion to get me past this roadblock?
>>
>> -Dave
>>
>> On Sun, Aug 14, 2016 at 2:21 AM, Roger Bishop Jones <r...@rbjones.com>
>> wrote:
>>
>>>
>>> On 14/08/2016 08:44, David Topham wrote:
>>>
>>>> Thanks Roger, I am using slides he distributes.  He  has false
>>>> introduction rules starting on page 24 (attached).
>>>> Sorry about my poor example, please ignore that since is a confused use
>>>> of this technique anyway!  -Dave
>>>>
>>>> Looks like he changed the name.
>>>
>>> In fact the original name (the one he uses in the book) is good in
>>> ProofPower.
>>> ¬_elim is available in ProofPower and does what you want (though it is
>>> sligftly more general, it proves anything from a contradiction so you have
>>> to tell it what result you are after).
>>> Details in reference manual.
>>>
>>> Roger
>>>
>>>
>>> This message did not originate from Ohlone College  and must be viewed
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>>> information could be intercepted, corrupted, lost, destroyed, arrive late
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>>>
>>> Desk at itserviced...@ohlone.edu  or (510) 659-7333.
>>>
>>>
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>>
>>
>
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