Verified

System Programming

Foe each of the following sequent, give a formal proof if entailment holds.

-\quad I \operatorname{ExP}(x)-A x ! P(X)

\text { - } \quad \operatorname{Axf}(\mathrm{P}) \rightarrow \mathrm{Q}(\mathrm{x}), \operatorname{ExP}(\mathrm{x}) \vdash \operatorname{Ex}(\mathrm{x})

\operatorname{AxP}(X) \vdash \operatorname{ExP}(x)

\text { - } \quad A x(P(x) \rightarrow Q(x)), A x P(x) \vdash A x(x)

\text { - IAxP(x) } \vdash \operatorname{Ex}[P(x)

Verified

System Programming

Are the following formulas valid, satisfiable, or unsatisfiable?

-\quad P(a) \rightarrow \operatorname{ExP}(x)

-\quad \operatorname{ExP}(x) \rightarrow P(a)

\text { - } \operatorname{ExP}(x) V \operatorname{ExIP}(x) \rightarrow A x P(x)

Verified

System Programming

3. Write behavioral code for up/down 2-bit binary counter (same as q2). Use state machine. Show the state machine diagram and implement code that reflects the state machine. Following is a template that you can use-

Verified

System Programming

2. Write behavioral code for a 2-bit up/down counter. The counter should have synchronous reset. (State Machine is not needed)

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