Automated Formal Static Analysis and Retrieval of Source Code - JKU

2.3. THE SIMPLIFICATION OF THE VERIFICATION CONDITIONS 19
The truth constants appear, usually, when safety verification conditions are generated in order
to insure the input condition of the basic functions. A list with the possible occurrences of the truth
constants, either on the left-hand-side, either on the right-hand-side of the logical connectives, is
applied recursively to the initial set of verification conditions, until the formulae do not change
anymore.
Example 2.1.
...
(*truth constants elimination on the rhs*)
•[lhs ⇒ True]:→ •[True],
•[lhs ⇒ False]:→ •[Not[lhs]],
•[lhs ⇔ True]:→ •[lhs],
•[lhs ⇔ False]:→ •[Not[lhs]]
...
In the next step we normalize the formulae by transforming the equalities and inequalities such
that they have 0 on the right-hand-side and ≥, >, =, ≠ occur only. Afterwards, the inequalities
and equalities from the assumptions or from the goal are then rearranged such that the order is:
≠, =, ≥ and >.
Some simplifications involving the relationships between ≠, =, ≥, > are done in the next
steps:
Example 2.2.
•[And[pre ,g ≠ 0,body ,g =0,post ]]:→ •[False],
•[And[pre ,g ≠ 0,body ,g ≥ 0,post ]]:→ •[And[pre,g>0,body,post]],
•[And[pre ,g ≠ 0,body ,g >0,post ]]:→ • [And[pre,g>0,body,post]],
...
For the inference rules:
Quantifiers elimination
From the assumptions
From the goal
Φ,φ x←a ⊢Ψ
Φ,∃φ⊢Ψ (∃ ⊢) Φ⊢Ψ,ψ a is new x←t
Φ⊢Ψ,∃ψ
x x
Φ,φ x←t ⊢Ψ
Φ,∀φ⊢Ψ (∀ ⊢) Φ⊢Ψ,ψ t has to be found x←a
Φ⊢Ψ,∀ψ
x x
we introduce new skolem constants and meta-variables, namely:
(⊢ ∃) t has to be found
(⊢ ∀) a is new
• skolem constants for universally quantified formulae in the goal and existentially quantified
formulae in the assumptions;
• meta-variables for universally quantified formulae in the assumptions and existentially quantified
formulae in the goal.