John, thanks for your responses!  It's a great pleasure to be logged
in at the same time as you (because my wife is on vacation).  I think
I can largely solve my locale/axiom problems by using HOL, which I
just started learning about, in the sets section of your tutorial.  

Thanks for correcting me, that was really dumb of me.  This works fine:

parse_as_prefix "elt_inv";;
let left_inv = new_axiom
  `!x . (left_inv x) prod x = unit`;;

Or even this, although I don't understand the priority biz:

let left_inv = new_axiom
  `!x . left_inv x prod x = unit`;;

I should have guessed the lambda_calculus connection maybe, and did I
tell you I wrote a paper on  the lambda_calculus, 
http://www.math.northwestern.edu/~richter/lambda-calc.pdf
giving a simpler proof than Barendregt's of the Standard Reduction
Theorem in the Lambda Calculus, which explains in what sense the
Lambda Calculus is a compiler.  

-- 
Best,
Bill 

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