# pre-conditions-start

Requires(Acc(list_pred(num)))

Requires(((0) <= (i)) and ((i) <= (j)) and ((j) <= (len(num))))

# pre-conditions-end

# pure-start

if i == j:

return 0

else:

return (1 if IsPrimeHexDigit(num[j - 1]) else 0) + count__prime__hex__digits__rec(i, j - 1, num)

# pre-conditions-start

Requires(Acc(list_pred(s)))

# pre-conditions-end

# post-conditions-start

Ensures(Acc(list_pred(s)))

Ensures((Result()) == (count__prime__hex__digits__rec(0, len(s), s)))

Ensures(((0) <= (Result())) and ((Result()) <= (len(s))))

# post-conditions-end

# impl-start

count : int = 0

d_1_i_ : int = 0

while (d_1_i_) < (len(s)):

# invariants-start

Invariant(Acc(list_pred(s)))

Invariant(((0) <= (d_1_i_)) and ((d_1_i_) <= (len(s))))

Invariant((count) == (count__prime__hex__digits__rec(0, d_1_i_, s)))

Invariant(count >= 0 and count <= d_1_i_)

Invariant(Forall(int, lambda x : (Implies(x >= 0 and x < len(s), (count__prime__hex__digits__rec(0, x + 1, s)) == ((count__prime__hex__digits__rec(0, x, s) + ((1 if IsPrimeHexDigit((s)[x]) else 0))))), [[count__prime__hex__digits__rec(0, x + 1, s)]])))

# invariants-end

# assert-start

Assert((count__prime__hex__digits__rec(0, d_1_i_ + 1, s)) == ((count__prime__hex__digits__rec(0, d_1_i_, s) + ((1 if IsPrimeHexDigit((s)[d_1_i_]) else 0)))))

# assert-end

count = (count) + ((1 if IsPrimeHexDigit((s)[d_1_i_]) else 0))

d_1_i_ = (d_1_i_) + (1)

return count