PALPHA

PairRules.PALPHA : term -> term -> thm

Proves equality of paired alpha-equivalent terms.

When applied to a pair of terms t1 and t1' which are alpha-equivalent, ALPHA returns the theorem |- t1 = t1'.

   -------------  PALPHA "t1" "t1'"
    |- t1 = t1'

The difference between PALPHA and ALPHA is that PALPHA is prepared to consider pair structures of different structure to be alpha-equivalent. In its most trivial case this means that PALPHA can consider a variable and a pair to alpha-equivalent.

Failure

Fails unless the terms provided are alpha-equivalent.

Example

- PALPHA (Term `\(x:'a,y:'a). (x,y)`) (Term`\xy:'a#'a. xy`);
> val it = |- (\(x,y). (x,y)) = (\xy. xy) : thm

Comments

Alpha-converting a paired abstraction to a nonpaired abstraction can introduce instances of the terms FST and SND. A paired abstraction and a nonpaired abstraction will be considered equivalent by PALPHA if the nonpaired abstraction contains all those instances of FST and SND present in the paired abstraction, plus the minimum additional instances of FST and SND. For example:

   - PALPHA
      (Term `\(x:'a,y:'b). (f x y (x,y)):'c`)
      (Term `\xy:'a#'b. (f (FST xy) (SND xy) xy):'c`);
   > val it = |- (\(x,y). f x y (x,y)) = (\xy. f (FST xy) (SND xy) xy) : thm

   - PALPHA
      (Term `\(x:'a,y:'b). (f x y (x,y)):'c`)
      (Term `\xy:'a#'b. (f (FST xy) (SND xy) (FST xy, SND xy)):'c`)
     handle e => Raise e;

   Exception raised at ??.failwith:
   PALPHA
   ! Uncaught exception:
   ! HOL_ERR

See also

Thm.ALPHA, Term.aconv, PairRules.PALPHA_CONV, PairRules.GEN_PALPHA_CONV