More Mathematical Reasoning (Conditional Statements) Murali Sitaraman Clemson University

Is the code correct for the given spec? Operation Do_Nothing (updates I: Integer); ensures I = #I; Code: If (I < Max_Int()) then Increment(I); Decrement(I); end;

These specs are the same… Operation Do_Nothing (updates I: Integer); ensures I = #I; Operation Do_Nothing (restores I: Integer);

Establish the goals in state-oriented terms using a table Cond Assume Confirm If (I < Max_Int()) 1 Increment(I); 2 Decrement(I) 3 end; 4 I4 = I0

Establish the conditions Cond Assume Confirm If (I < Max_Int()) 1 I0 < max_int Increment(I); 2 I0 < max_int Decrement(I) 3 I0 < max_int end; 4 I4 = I0

Establish sub-goals for different conditions Cond Assume Confirm If (I < Max_Int) 1 I0 < max_int Increment(I); 2 I0 < max_int Decrement(I) 3 I0 < max_int end; 4.1 not (I0 < max_int) I4 = I0 I4 = I0 4.2 I0 < max_int I4 = I3 I4 = I0

Fill in other assumptions and obligations as before… Cond Assume Confirm If (I < Max_Int) 1 I0 < max_int Increment(I); 2 I0 < max_int Decrement(I) 3 I0 < max_int end; 4.1 not (I0 < max_int) I4 = I0 I4 = I0 4.2 I0 < max_int I4 = I3 I4 = I0

Prove the subgoal(s) 4.1 Case: not (I0 < max_int) Prove I4 = I0 True from the assumption 4.2 Case: (I0 < max_int) Prove: I3 = I0 (assumption in state 4) Prove: (I2 - 1) = I0 (assumption in st 3) …

Prove remaining assertions to be confirmed For the condition (I0 < max_int), additional proofs are needed. These proofs of assertions to be confirmed in states 1 and 2 left as exercises

More Mathematical Reasoning Create this example using the web interface, generate VCs, and prove them For recursive implementations Recursive calls are handled just as any other call Need to show termination using a programmer-supplied decreasing “metric” For iterative implementations, invariants and decreasing metrics are used