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|
// Datatypes
datatype Bool { False; True; }
datatype Nat { Zero; Suc(Nat); }
datatype List { Nil; Cons(Nat, List); }
datatype Pair { Pair(Nat, Nat); }
datatype PList { PNil; PCons(Pair, PList); }
datatype Tree { Leaf; Node(Tree, Nat, Tree); }
// Boolean functions
function not(b: Bool): Bool
{
match b
case False => #Bool.True
case True => #Bool.False
}
function and(a: Bool, b: Bool): Bool
{
if a == #Bool.True && b == #Bool.True then #Bool.True else #Bool.False
}
// Natural number functions
function add(x: Nat, y: Nat): Nat
{
match x
case Zero => y
case Suc(w) => #Nat.Suc(add(w, y))
}
function minus(x: Nat, y: Nat): Nat
{
match x
case Zero => #Nat.Zero
case Suc(a) => match y
case Zero => x
case Suc(b) => minus(a, b)
}
function eq(x: Nat, y: Nat): Bool
{
match x
case Zero => (match y
case Zero => #Bool.True
case Suc(b) => #Bool.False)
case Suc(a) => (match y
case Zero => #Bool.False
case Suc(b) => eq(a, b))
}
function leq(x: Nat, y: Nat): Bool
{
match x
case Zero => #Bool.True
case Suc(a) => match y
case Zero => #Bool.False
case Suc(b) => leq(a, b)
}
function less(x: Nat, y: Nat): Bool
{
match y
case Zero => #Bool.False
case Suc(b) => match x
case Zero => #Bool.True
case Suc(a) => less(a, b)
}
function min(x: Nat, y: Nat): Nat
{
match x
case Zero => #Nat.Zero
case Suc(a) => match y
case Zero => #Nat.Zero
case Suc(b) => #Nat.Suc(min(a, b))
}
function max(x: Nat, y: Nat): Nat
{
match x
case Zero => y
case Suc(a) => match y
case Zero => x
case Suc(b) => #Nat.Suc(max(a, b))
}
// List functions
function concat(xs: List, ys: List): List
{
match xs
case Nil => ys
case Cons(x,tail) => #List.Cons(x, concat(tail, ys))
}
function mem(x: Nat, xs: List): Bool
{
match xs
case Nil => #Bool.False
case Cons(y, ys) => if x == y then #Bool.True else mem(x, ys)
}
function delete(n: Nat, xs: List): List
{
match xs
case Nil => #List.Nil
case Cons(y, ys) =>
if y == n then delete(n, ys) else #List.Cons(y, delete(n, ys))
}
function drop(n: Nat, xs: List): List
{
match n
case Zero => xs
case Suc(m) => match xs
case Nil => #List.Nil
case Cons(x, tail) => drop(m, tail)
}
function take(n: Nat, xs: List): List
{
match n
case Zero => #List.Nil
case Suc(m) => match xs
case Nil => #List.Nil
case Cons(x, tail) => #List.Cons(x, take(m, tail))
}
function len(xs: List): Nat
{
match xs
case Nil => #Nat.Zero
case Cons(y, ys) => #Nat.Suc(len(ys))
}
function count(x: Nat, xs: List): Nat
{
match xs
case Nil => #Nat.Zero
case Cons(y, ys) =>
if x == y then #Nat.Suc(count(x, ys)) else count(x, ys)
}
function last(xs: List): Nat
{
match xs
case Nil => #Nat.Zero
case Cons(y, ys) => match ys
case Nil => y
case Cons(z, zs) => last(ys)
}
function mapF(xs: List): List
{
match xs
case Nil => #List.Nil
case Cons(y, ys) => #List.Cons(HardcodedUninterpretedFunction(y), mapF(ys))
}
function HardcodedUninterpretedFunction(n: Nat): Nat;
function takeWhileAlways(hardcodedResultOfP: Bool, xs: List): List
{
match xs
case Nil => #List.Nil
case Cons(y, ys) =>
if whilePredicate(hardcodedResultOfP, y) == #Bool.True
then #List.Cons(y, takeWhileAlways(hardcodedResultOfP, ys))
else #List.Nil
}
function whilePredicate(result: Bool, arg: Nat): Bool { result }
function dropWhileAlways(hardcodedResultOfP: Bool, xs: List): List
{
match xs
case Nil => #List.Nil
case Cons(y, ys) =>
if whilePredicate(hardcodedResultOfP, y) == #Bool.True
then dropWhileAlways(hardcodedResultOfP, ys)
else #List.Cons(y, ys)
}
function filterP(xs: List): List
{
match xs
case Nil => #List.Nil
case Cons(y, ys) =>
if HardcodedUninterpretedPredicate(y) == #Bool.True
then #List.Cons(y, filterP(ys))
else filterP(ys)
}
function HardcodedUninterpretedPredicate(n: Nat): Bool;
function insort(n: Nat, xs: List): List
{
match xs
case Nil => #List.Cons(n, #List.Nil)
case Cons(y, ys) =>
if leq(n, y) == #Bool.True
then #List.Cons(n, #List.Cons(y, ys))
else #List.Cons(y, ins(n, ys))
}
function ins(n: Nat, xs: List): List
{
match xs
case Nil => #List.Cons(n, #List.Nil)
case Cons(y, ys) =>
if less(n, y) == #Bool.True
then #List.Cons(n, #List.Cons(y, ys))
else #List.Cons(y, ins(n, ys))
}
function ins1(n: Nat, xs: List): List
{
match xs
case Nil => #List.Cons(n, #List.Nil)
case Cons(y, ys) =>
if n == y
then #List.Cons(y, ys)
else #List.Cons(y, ins1(n, ys))
}
function sort(xs: List): List
{
match xs
case Nil => #List.Nil
case Cons(y, ys) => insort(y, sort(ys))
}
// Pair list functions
function zip(a: List, b: List): PList
{
match a
case Nil => #PList.PNil
case Cons(x, xs) => match b
case Nil => #PList.PNil
case Cons(y, ys) => #PList.PCons(#Pair.Pair(x, y), zip(xs, ys))
}
function zipConcat(x: Nat, xs: List, more: List): PList
{
match more
case Nil => #PList.PNil
case Cons(y, ys) => #PList.PCons(#Pair.Pair(x, y), zip(xs, ys))
}
// Binary tree functions
function height(t: Tree): Nat
{
match t
case Leaf => #Nat.Zero
case Node(l, x, r) => #Nat.Suc(max(height(l), height(r)))
}
function mirror(t: Tree): Tree
{
match t
case Leaf => #Tree.Leaf
case Node(l, x, r) => #Tree.Node(mirror(r), x, mirror(l))
}
// The theorems to be proved
ghost method P1()
ensures (forall n, xs :: concat(take(n, xs), drop(n, xs)) == xs);
{
}
ghost method P2()
ensures (forall n, xs, ys :: add(count(n, xs), count(n, ys)) == count(n, (concat(xs, ys))));
{
}
ghost method P3()
ensures (forall n, xs, ys :: leq(count(n, xs), count(n, concat(xs, ys))) == #Bool.True);
{
}
ghost method P4()
ensures (forall n, xs :: add(#Nat.Suc(#Nat.Zero), count(n, xs)) == count(n, #List.Cons(n, xs)));
{
}
ghost method P5()
ensures (forall n, xs, x ::
add(#Nat.Suc(#Nat.Zero), count(n, xs)) == count(n, #List.Cons(x, xs))
==> n == x);
{
}
ghost method P6()
ensures (forall m, n :: minus(n, add(n, m)) == #Nat.Zero);
{
}
ghost method P7()
ensures (forall m, n :: minus(add(n, m), n) == m);
{
}
ghost method P8()
ensures (forall k, m, n :: minus(add(k, m), add(k, n)) == minus(m, n));
{
}
ghost method P9()
ensures (forall i, j, k :: minus(minus(i, j), k) == minus(i, add(j, k)));
{
}
ghost method P10()
ensures (forall m :: minus(m, m) == #Nat.Zero);
{
}
ghost method P11()
ensures (forall xs :: drop(#Nat.Zero, xs) == xs);
{
}
ghost method P12()
ensures (forall n, xs :: drop(n, mapF(xs)) == mapF(drop(n, xs)));
{
}
ghost method P13()
ensures (forall n, x, xs :: drop(#Nat.Suc(n), #List.Cons(x, xs)) == drop(n, xs));
{
}
ghost method P14()
ensures (forall xs, ys :: filterP(concat(xs, ys)) == concat(filterP(xs), filterP(ys)));
{
}
ghost method P15()
ensures (forall x, xs :: len(ins(x, xs)) == #Nat.Suc(len(xs)));
{
}
ghost method P16()
ensures (forall x, xs :: xs == #List.Nil ==> last(#List.Cons(x, xs)) == x);
{
}
ghost method P17()
ensures (forall n :: leq(n, #Nat.Zero) == #Bool.True <==> n == #Nat.Zero);
{
}
ghost method P18()
ensures (forall i, m :: less(i, #Nat.Suc(add(i, m))) == #Bool.True);
{
}
ghost method P19()
ensures (forall n, xs :: len(drop(n, xs)) == minus(len(xs), n));
{
}
ghost method P20()
ensures (forall xs :: len(sort(xs)) == len(xs));
{
assume false; // Dafny is not able to verify it automatically
}
ghost method P21()
ensures (forall n, m :: leq(n, add(n, m)) == #Bool.True);
{
}
ghost method P22()
ensures (forall a, b, c :: max(max(a, b), c) == max(a, max(b, c)));
{
}
ghost method P23()
ensures (forall a, b :: max(a, b) == max(b, a));
{
}
ghost method P24()
ensures (forall a, b :: max(a, b) == a <==> leq(b, a) == #Bool.True);
{
}
ghost method P25()
ensures (forall a, b :: max(a, b) == b <==> leq(a, b) == #Bool.True);
{
}
ghost method P26()
ensures (forall x, xs, ys :: mem(x, xs) == #Bool.True ==> mem(x, concat(xs, ys)) == #Bool.True);
{
}
ghost method P27()
ensures (forall x, xs, ys :: mem(x, ys) == #Bool.True ==> mem(x, concat(xs, ys)) == #Bool.True);
{
}
ghost method P28()
ensures (forall x, xs :: mem(x, concat(xs, #List.Cons(x, #List.Nil))) == #Bool.True);
{
}
ghost method P29()
ensures (forall x, xs :: mem(x, ins1(x, xs)) == #Bool.True);
{
}
ghost method P30()
ensures (forall x, xs :: mem(x, ins(x, xs)) == #Bool.True);
{
}
ghost method P31()
ensures (forall a, b, c :: min(min(a, b), c) == min(a, min(b, c)));
{
}
ghost method P32()
ensures (forall a, b :: min(a, b) == min(b, a));
{
}
ghost method P33()
ensures (forall a, b :: min(a, b) == a <==> leq(a, b) == #Bool.True);
{
}
ghost method P34()
ensures (forall a, b :: min(a, b) == b <==> leq(b, a) == #Bool.True);
{
}
ghost method P35()
ensures (forall xs :: dropWhileAlways(#Bool.False, xs) == xs);
{
}
ghost method P36()
ensures (forall xs :: takeWhileAlways(#Bool.True, xs) == xs);
{
}
ghost method P37()
ensures (forall x, xs :: not(mem(x, delete(x, xs))) == #Bool.True);
{
}
ghost method P38()
ensures (forall n, xs :: count(n, concat(xs, #List.Cons(n, #List.Nil))) == #Nat.Suc(count(n, xs)));
{
}
ghost method P39()
ensures (forall n, x, xs ::
add(count(n, #List.Cons(x, #List.Nil)), count(n, xs)) == count(n, #List.Cons(x, xs)));
{
}
ghost method P40()
ensures (forall xs :: take(#Nat.Zero, xs) == #List.Nil);
{
}
ghost method P41()
ensures (forall n, xs :: take(n, mapF(xs)) == mapF(take(n, xs)));
{
}
ghost method P42()
ensures (forall n, x, xs :: take(#Nat.Suc(n), #List.Cons(x, xs)) == #List.Cons(x, take(n, xs)));
{
}
ghost method P43(p: Bool)
// this is an approximation of the actual problem 43
ensures (forall xs :: concat(takeWhileAlways(p, xs), dropWhileAlways(p, xs)) == xs);
{
}
ghost method P44()
ensures (forall x, xs, ys :: zip(#List.Cons(x, xs), ys) == zipConcat(x, xs, ys));
{
}
ghost method P45()
ensures (forall x, xs, y, ys ::
zip(#List.Cons(x, xs), #List.Cons(y, ys)) ==
#PList.PCons(#Pair.Pair(x, y), zip(xs, ys)));
{
}
ghost method P46()
ensures (forall ys :: zip(#List.Nil, ys) == #PList.PNil);
{
}
ghost method P47()
ensures (forall a :: height(mirror(a)) == height(a));
{
assume false; // Dafny is not able to verify it automatically
}
// ...
ghost method P54()
ensures (forall m, n :: minus(add(m, n), n) == m);
{
assume false; // Dafny is not able to verify it automatically
}
ghost method P65()
ensures (forall i, m :: less(i, #Nat.Suc(add(m, i))) == #Bool.True);
{
assume false; // Dafny is not able to verify it automatically
}
ghost method P67()
ensures (forall m, n :: leq(n, add(m, n)) == #Bool.True);
{
assume false; // Dafny is not able to verify it automatically
}
|