Presburger Arithmetic and Its Use in Verification

B.4.
PAGENERATOR.FS (EXCERPT)
B.4 PAGenerator.fs (excerpt)
// Generate PA Fragments for testing Cooper algorithm
module PAGenerator
// Formulate pigeon hole principle in propositional logic (N pigeons and K holes)
let rec tab n f =
match n with
| xwhenx []
| _ −> fn:: tab (n−1) f
// Pigeon i is in hole k
let Xik= A( L ("x(" + string i + "," + string k + ")"), true)
let Yik= A( C(Zero, var ("x" + string i + "_" + string k + ""), LE), true)
let Zik= A( C(Zero, var ("x" + string i + "_" + string k + ""), EQ), true)
let Tik= A( D(2, var ("x" + string i + "_" + string k + "")), true)
// If pigeon i is in hole k so no one else is in hole k
let Fpred(i, k) (N, K) =(pred i k) =>(And (tab N (fun i’ −> if i = i’ then TT else
Not (pred i’ k))))
// Apply F for all i and k
let Fall pred (N, K) =And (tab N (fun i −> And (tab K (fun k −> Fpred(i, k) (N, K)))
))
// Pigeon i has assigned a hole
let Gpredi(N, K) =Or (tab K (fun k −> pred i k))
// All pigeons have been assigned holes
let Gall pred (N, K) =And (tab N (fun i −> Gpredi(N, K)))
// A pigeon is only in one hole
let Hpred(i, k) (N, K) =(pred i k) =>(And (tab K (fun k’ −> if k = k’ then TT else
Not (pred i k’))))
// Every pigeon has exactly one hole
let Hall pred (N, K) =And (tab N (fun i −> And (tab K (fun k −> Hpred(i, k) (N, K)))
))
let pigeon pred (N, K) =And [Fall pred (N, K) ;Gall pred (N, K) ;Hall pred (N, K)]
let pigeonY = pigeon Y
let pigeonZ = pigeon Z
let pigeonT = pigeon T
let generateFormula pigeonPred (N, K, Q) =
let rec takeLast(ls, q) =
if List.length ls