Automated Formal Static Analysis and Retrieval of Source Code - JKU

2.4. IMPLEMENTATION AND EXAMPLES 25
The simplified list of verification conditions is:
T rue, T rue,
a ≠ 0 =⇒ ( x 12 = (− b a ) ⇒ (a ∗ x 12 + b = 0) ) ∧
(a ≠ 0 =⇒
(
(a ∗ x12 + b = 0) ⇒ x 12 = (− b a )) )
2.4.3 Solving Second Degree Equations
1. Program[”SecondDegreeSolver-calling-FirstDegreeSolver”,SDS[↓ a,↓ b,↓ c],
2. Module[{delta ,sqrtDelta,x1,x2,sol},
3. If[a==0,
4. sol := FDS[b,c]; Return[sol],
4. delta :=(b*b)-(4*a*c);
5. If[delta =0,
6. x1 := -(b/(2*a));x2:=x1,
7. If[delta > 0,
8. sqrtDelta:=Sqrt[delta]; x1 := (-b+sqrtDelta)/(2*a); x2 := (-b-sqrtDelta)/(2*a),
9. sqrtDelta := Sqrt[-delta ]; x1 := (-b-(i*sqrtDelta))/(2*a); x2:= (-b+(i*sqrtDelta))/(2*a)]];
10. sol:={x1, x2};Return[sol]]],
11. Pre ← T rue,
12. Post ← ∀((x ∈ y) ⇐⇒ (a ∗ x 2 + b ∗ x + c = 0))
x
We display the verification conditions after simplification.
T rue,
(
(x ∗ 1 = F DS[b, c]) ⇒ (b ∗ x ∗ 1 + c = 0) ∧ )
(b ∗ x ∗ 1 + c = 0) ⇒ (x ∗ 1 = F DS[b, c]) =⇒
(
(x 4 = F DS[b, c]) ⇒ (b ∗ x 4 + c = 0) ∧ )
(b ∗ x 4 + c = 0) ⇒ (x 4 = F DS[b, c]) ,
T rue, T rue, T rue, T rue, T rue, T rue, T rue,
(
a ≠ 0 ∧ (b ∗ b − 4 ∗ a ∗ c = 0) =⇒ (a ∗ x 2 5 + b ∗ x 5 + c = 0) ⇒ (x 5 = − b
2 ∗ a )∨
((a ∗ x 2 5 + b ∗ x 5 + c = 0) ⇒ (x 5 = −
b
2 ∗ a )) ∧ (((x 5 = − b
2 ∗ a )) ⇒ (a ∗ x2 5 + b ∗ x 5 + c = 0))∨
((x 5 = (−
b
)
2 ∗ a )) ⇒ (a ∗ x2 5 + b ∗ x 5 + c = 0)) ,
a ≠ 0 ∧ b ∗ b − 4 ∗ a ∗ c > 0 ∧ (−b ∗ b) + 4 ∗ a ∗ c > 0 =⇒ b ∗ b − 4 ∗ a ∗ c ≥ 0,
T rue, T rue, T rue, T rue, T rue, T rue, T rue, T rue, T rue,
a ≠ 0 ∧ b ∗ b − 4 ∗ a ∗ c > 0 ∧ (−b ∗ b) + 4 ∗ a ∗ c > 0 =⇒
(x 6 = (−b) + (√ b ∗ b − 4 ∗ a ∗ c)
2 ∗ a
(
(((a ∗ x 2 6 + b ∗ x 6 + c = 0) ⇒
)) ∨ ((a ∗ x 2 6 + b ∗ x 6 + c = 0) ⇒ (x 6 = (−b) − (√ b ∗ b − 4 ∗ a ∗ c)
))) ∧
2 ∗ a
(((x 6 = (−b) + (√ b ∗ b − 4 ∗ a ∗ c)
) ⇒ (a ∗ x 2 6 + b ∗ x 6 + c = 0))∨
2 ∗ a
)
((x 6 = (−b) − (√ b ∗ b − 4 ∗ a ∗ c)
) ⇒ (a ∗ x 2 6 + b ∗ x 6 + c = 0)))
2 ∗ a
,