PL 2015 QE Proof

PL QE2015

Claim

For all expression , state and , and integer .

If then there is a state , .

Proof

First if we take a close look of the rule here, this semantic is a undeterministic semantic, which means each original step may cause 2 different state, to be more clear in claim actually means a set of state because in rule Choice ' s side condition we can clear see 2 different state generated at same time. In following proof we will assume this. If you think in claim just means a particular state then this claim not hold when in rule Choice contain environment change expression.

Induction on derivation of original semantic ():

• Num , Var are trivial base case, are not changed which means there is always (satisfy ANum and AVar )make claim holds

• Inc is also a trivial base case. we can always find make claim holds.

• Add is an inductive case, in this case by this rule we can know , we can know in the (similar for )created by the step result of , there must exist some , which can step to. so all precedence in rule aAdd is true, and we can apply this step rule to find some

• Choice is interesting case.

  by inductive hypothesis we can know that if then we can have , meanwhile by the rule of this semantic it is not hard to see, this semantic has progress properties, which means express will not stuck at somewhere so we can always evaluate have . Then we can apply rule aChoice and if we let in the inference of aChoice , and let in Choice rule we can found, there is always a , 2 different semantic agree on, and at this time we can find a . Claim hold.

so because of above all cases we can see claim holds.