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
>>
>>
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