PL 2020-2 QE Proof

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original question

Claim :

Proof by induct on :

• :

  the goal become ,In this case by inversion precedence in goal with respect to rule Prefix we can know, can only be 0 and at this time claim holds

• :

  by rewriting the second definition of sort, then the goal become:

  .

  by rule Prefix we can know, can only be and we can always find exists . And union something is always larger than itself so claim holds

• Similar operation to before, the goal become:

  .

  by rule Choice1 and Choice2 we can see, we can always find a to let precedence hold, and can become either or . union of set is monotonic, of course larger. Clearly claim holds.

• The goal become:

  similar inversion, this time we have 3 interesting case

  • by rule Par1 : we can have , which means by 4th rule of sort we can have . By inductive hypothesis we can know if is the derivation of then we will have so clearly final goal is true
  • Par2 and Par3 are similar. the only difference in Par3 is both and cause a shrink the size of set.

• The goal become:

  By rule Restrict we can know can become , according to the last rule of sort we can rewrite in the goal as . And according to inductive hypothesis, we can know . So clearly claim hold.

All inductive cases is proved.

Q.E.D