fun Even(n:UInt) {
  some m:UInt. n = 2 * m
}

union Int {
  pos(UInt)
  negsuc(UInt)
}

fun Odd(n:UInt) {
  some m:UInt. n = 1 + 2 * m
}

union UInt

fun abs(x:Int)

abs_neg: (all n:Int. abs(- n) = abs(n))

add_commute_uint_int: (all x:UInt, y:Int. x + y = y + x)

fun div2(b:UInt)

fun divides(a:UInt, b:UInt) {
  some k:UInt. a * k = b
}

fun fromNat(Nat) -> UInt

fromNat_add: (all x:Nat, y:Nat. fromNat(x + y) = fromNat(x) + fromNat(y))

fromNat_expt: (all x:Nat, y:Nat. fromNat(x ^ y) = fromNat(x) ^ fromNat(y))

fromNat_injective: (all x:Nat, y:Nat. (if fromNat(x) = fromNat(y) then x = y))

fromNat_mult: (all x:Nat, y:Nat. fromNat(x * y) = fromNat(x) * fromNat(y))

int_add_assoc: (all x:Int, y:Int, z:Int. (x + y) + z = x + (y + z))

int_add_commute: (all x:Int, y:Int. x + y = y + x)

int_add_inverse: (all x:Int. x + - x = +0)

auto int_add_zero

int_add_zero: (all n:Int. n + +0 = n)

int_sub_cancel: (all x:Int. x - x = +0)

auto int_zero_add

int_zero_add: (all n:Int. +0 + n = n)

less_equal_fromNat: (all x:Nat, y:Nat. (if x ≤ y then fromNat(x) ≤ fromNat(y)))

less_equal_toNat: (all x:UInt, y:UInt. (if toNat(x) ≤ toNat(y) then x ≤ y))

less_fromNat: (all x:Nat, y:Nat. (if x < y then fromNat(x) < fromNat(y)))

less_pow_log: (all n:UInt. (if 0 < n then n < 2 ^ (1 + log(n))))

less_toNat: (all x:UInt, y:UInt. (if toNat(x) < toNat(y) then x < y))

auto lit_add_fromNat

lit_add_fromNat: (all x:Nat, y:Nat. fromNat(lit(x)) + fromNat(lit(y)) = fromNat(lit(x) + lit(y)))

auto lit_expt_fromNat

lit_expt_fromNat: (all x:Nat, y:Nat. fromNat(lit(x)) ^ fromNat(lit(y)) = fromNat(lit(x) ^ lit(y)))

auto lit_monus_fromNat

lit_monus_fromNat: (all x:Nat, y:Nat. fromNat(lit(x)) ∸ fromNat(lit(y)) = fromNat(lit(x) ∸ lit(y)))

auto lit_mult_fromNat

lit_mult_fromNat: (all x:Nat, y:Nat. fromNat(lit(x)) * fromNat(lit(y)) = fromNat(lit(x) * lit(y)))

auto lit_pow_mul_r

lit_pow_mul_r: (all m:Nat, n:Nat, o:UInt. fromNat(lit(m)) ^ (fromNat(lit(n)) * o) = (fromNat(lit(m)) ^ fromNat(lit(n))) ^ o)

fun log(b:UInt) {
  cnt_dubs(pred(b))
}

log_pow: (all n:UInt. log(2 ^ n) = n)

fun max(x:UInt, y:UInt) {
  if x < y then
    y
  else
    x
}

fun min(x:UInt, y:UInt) {
  if x < y then
    x
  else
    y
}

neg_distr_add: (all x:Int, y:Int. - (x + y) = - x + - y)

neg_involutive: (all n:Int. - (- n) = n)

neg_zero: - +0 = +0

fun operator %(n:UInt, m:UInt) {
  n ∸ (n / m) * m
}

fun operator *(UInt,UInt) -> UInt

fun operator *(x:Int, y:Int)

fun operator *(n:UInt, m:Int)

fun operator *(n:Int, m:UInt)

fun operator +(UInt,UInt) -> UInt

fun operator +(x:Int, m:Int)

fun operator +(n:UInt, m:Int)

fun operator +(n:Int, m:UInt)

fun operator -(x:Int)

fun operator -(n:UInt)

fun operator -(n:Int, m:Int) {
  n + - m
}

fun operator -(n:UInt, m:Int)

fun operator -(n:Int, m:UInt)

fun operator -(x:UInt, y:UInt)

recfun operator /(n:UInt, m:UInt) -> UInt
measure	n  {
  if n < m then
    0
  else
    if m = 0 then
      0
    else
      1 + (n ∸ m) / m
}

fun operator /(n:Int, m:Int)

fun operator /(n:Int, m:UInt)

fun operator <(UInt,UInt) -> bool

fun operator >(x:UInt, y:UInt) {
  y < x
}

fun operator ^(a:UInt, b:UInt)

fun operator ∸(UInt,UInt) -> UInt

fun operator ≤(x:UInt, y:UInt) {
  (x < y or x = y)
}

fun operator ≤(a:Int, y:Int)

fun operator ≥(x:UInt, y:UInt) {
  y ≤ x
}

fun sqr(a:UInt) {
  a * a
}

suc_uint_monuso: (all x:UInt, y:UInt. (1 + x) ⊝ (1 + y) = x ⊝ y)

fun toNat(UInt) -> Nat

toNat_add: (all x:UInt, y:UInt. toNat(x + y) = toNat(x) + toNat(y))

toNat_expt: (all p:UInt, n:UInt. toNat(n ^ p) = toNat(n) ^ toNat(p))

toNat_less: (all x:UInt, y:UInt. (if x < y then toNat(x) < toNat(y)))

toNat_less_equal: (all x:UInt, y:UInt. (if x ≤ y then toNat(x) ≤ toNat(y)))

toNat_monus: (all x:UInt, y:UInt. toNat(x ∸ y) = toNat(x) ∸ toNat(y))

toNat_mult: (all x:UInt, y:UInt. toNat(x * y) = toNat(x) * toNat(y))

uint_Even_not_Odd: (all n:UInt. (Even(n) ⇔ not Odd(n)))

uint_Even_or_Odd: (all n:UInt. (Even(n) or Odd(n)))

uint_add_assoc: (all x:UInt, y:UInt, z:UInt. (x + y) + z = x + (y + z))

uint_add_both_monus: (all z:UInt, y:UInt, x:UInt. (z + y) ∸ (z + x) = y ∸ x)

uint_add_both_sides_of_equal: (all x:UInt, y:UInt, z:UInt. ((x + y = x + z) ⇔ (y = z)))

auto uint_add_both_sides_of_le_equal

uint_add_both_sides_of_le_equal: (all x:UInt. (all y:UInt, z:UInt. (x + y ≤ x + z) = (y ≤ z)))

uint_add_both_sides_of_less: (all x:UInt, y:UInt, z:UInt. ((x + y < x + z) ⇔ (y < z)))

uint_add_both_sides_of_less_equal: (all x:UInt. (all y:UInt, z:UInt. ((x + y ≤ x + z) ⇔ (y ≤ z))))

auto uint_add_bzero

uint_add_commute: (all x:UInt, y:UInt. x + y = y + x)

uint_add_div_one: (all n:UInt, m:UInt. (if 0 < m then (n + m) / m = 1 + n / m))

uint_add_mono_less: (all a:UInt, b:UInt, c:UInt, d:UInt. (if (a < c and b < d) then a + b < c + d))

uint_add_mono_less_equal: (all a:UInt, b:UInt, c:UInt, d:UInt. (if (a ≤ c and b ≤ d) then a + b ≤ c + d))

auto uint_add_monus_identity

uint_add_monus_identity: (all m:UInt, n:UInt. (m + n) ∸ m = n)

uint_add_one_le_double: (all x:UInt. (if 0 < x then 1 + x ≤ x + x))

uint_add_to_bzero: (all n:UInt, m:UInt. (if n + m = bzero then (n = bzero and m = bzero)))

uint_add_to_zero: (all n:UInt, m:UInt. (if n + m = 0 then (n = 0 and m = 0)))

auto uint_add_zero

uint_add_zero: (all x:UInt. x + 0 = x)

auto uint_bzero_add

uint_bzero_le: (all x:UInt. bzero ≤ x)

uint_bzero_less_one_add: (all n:UInt. bzero < 1 + n)

uint_bzero_monus: (all x:UInt. bzero ∸ x = bzero)

uint_dichotomy: (all x:UInt, y:UInt. (x ≤ y or y < x))

uint_dist_mult_add: (all a:UInt, x:UInt, y:UInt. a * (x + y) = a * x + a * y)

uint_dist_mult_add_right: (all x:UInt, y:UInt, a:UInt. (x + y) * a = x * a + y * a)

uint_div_cancel: (all y:UInt. (if 0 < y then y / y = 1))

uint_div_mod: (all n:UInt, m:UInt. (if 0 < m then (n / m) * m + n % m = n))

uint_div_one: (all n:UInt. n / 1 = n)

uint_divides_mod: (all d:UInt, m:UInt, n:UInt. (if (divides(d, n) and divides(d, m % n) and 0 < n) then divides(d, m)))

auto uint_equal

uint_equal: (all x:Nat, y:Nat. (fromNat(lit(x)) = fromNat(lit(y))) = (x = y))

uint_equal_implies_less_equal: (all x:UInt, y:UInt. (if x = y then x ≤ y))

uint_even_one_odd: (all n:UInt. (if Even(1 + n) then Odd(n)))

uint_expt_log_less_equal: (all n:UInt. (if 0 < n then 2 ^ log(n) ≤ n))

uint_expt_one: (all n:UInt. n ^ 1 = n)

auto uint_expt_suc

uint_expt_suc: (all n:UInt, m:Nat. n ^ fromNat(lit(suc(m))) = n * n ^ fromNat(lit(m)))

uint_expt_two: (all n:UInt. n ^ 2 = n * n)

auto uint_expt_zero

uint_expt_zero: (all n:UInt. n ^ 0 = 1)

uint_fromNat_toNat: (all b:UInt. fromNat(toNat(b)) = b)

uint_induction: (all P:(fn UInt -> bool). (if (P(0) and (all m:UInt. (if P(m) then P(1 + m)))) then (all n:UInt. P(n))))

uint_k_induction: (all P:(fn UInt -> bool), k:UInt. (if (P(k) and (all m:UInt. (if P(m) then P(1 + m)))) then (all n:UInt. (if k ≤ n then P(n)))))

uint_less_add_one_implies_less_equal: (all x:UInt, y:UInt. (if x < 1 + y then x ≤ y))

uint_less_add_pos: (all x:UInt, y:UInt. (if bzero < y then x < x + y))

uint_less_equal_add: (all x:UInt, y:UInt. x ≤ x + y)

uint_less_equal_add_left: (all x:UInt, y:UInt. y ≤ x + y)

uint_less_equal_antisymmetric: (all x:UInt, y:UInt. (if (x ≤ y and y ≤ x) then x = y))

uint_less_equal_refl: (all n:UInt. n ≤ n)

auto uint_less_equal_refl_true

uint_less_equal_refl_true: (all n:UInt. (n ≤ n) = true)

uint_less_equal_trans: (all x:UInt, y:UInt, z:UInt. (if (x ≤ y and y ≤ z) then x ≤ z))

uint_less_equal_zero: (all x:UInt. (if x ≤ 0 then x = 0))

uint_less_implies_less_equal: (all x:UInt, y:UInt. (if x < y then x ≤ y))

uint_less_implies_not_greater: (all x:UInt, y:UInt. (if x < y then not (y < x)))

uint_less_irreflexive: (all x:UInt. not (x < x))

uint_less_is_less_equal: (all x:UInt, y:UInt. (x < y) = (1 + x ≤ y))

uint_less_plus1: (all n:UInt. n < 1 + n)

auto uint_less_plus1_true

uint_less_plus1_true: (all n:UInt. (n < 1 + n) = true)

auto uint_less_refl_false

uint_less_refl_false: (all x:UInt. (x < x) = false)

uint_less_trans: (all x:UInt, y:UInt, z:UInt. (if (x < y and y < z) then x < z))

auto uint_lit_div

uint_lit_div: (all x:Nat, y:Nat. fromNat(lit(x)) / fromNat(lit(y)) = fromNat(lit(x) / lit(y)))

auto uint_lit_less

uint_lit_less: (all x:Nat, y:Nat. (fromNat(lit(x)) < fromNat(lit(y))) = (lit(x) < lit(y)))

auto uint_lit_less_equal

uint_lit_less_equal: (all x:Nat, y:Nat. (fromNat(lit(x)) ≤ fromNat(lit(y))) = (lit(x) ≤ lit(y)))

uint_log_add_le_log_mult: (all m:UInt, n:UInt. log(m) + log(n) ≤ log(m * n))

uint_log_greater_one: (all n:UInt. (if 2 ≤ n then 1 ≤ log(n)))

uint_log_mono: (all x:UInt, y:UInt. (if x ≤ y then log(x) ≤ log(y)))

uint_log_mult_le_log_add: (all m:UInt, n:UInt. log(m * n) ≤ (1 + log(m)) + log(n))

uint_log_pos: (all n:UInt. (if 1 < n then 0 < log(n)))

uint_logn_le_n: (all n:UInt. log(n) ≤ n)

uint_max_assoc: (all x:UInt, y:UInt, z:UInt. max(max(x, y), z) = max(x, max(y, z)))

uint_max_equal_greater_left: (all x:UInt, y:UInt. (if y ≤ x then max(x, y) = x))

uint_max_equal_greater_right: (all x:UInt, y:UInt. (if x ≤ y then max(x, y) = y))

uint_max_less_equal: (all x:UInt, y:UInt, z:UInt. (if (x ≤ z and y ≤ z) then max(x, y) ≤ z))

uint_max_symmetric: (all x:UInt, y:UInt. max(x, y) = max(y, x))

uint_max_zero: (all x:UInt. max(x, 0) = x)

uint_mod_less_divisor: (all n:UInt, m:UInt. (if 0 < m then n % m < m))

uint_mod_one: (all n:UInt. n % 1 = 0)

uint_mod_self_zero: (all y:UInt. y % y = 0)

uint_monus_add_assoc: (all n:UInt, l:UInt, m:UInt. (if m ≤ n then l + (n ∸ m) = (l + n) ∸ m))

uint_monus_add_identity: (all n:UInt. (all m:UInt. (if m ≤ n then m + (n ∸ m) = n)))

uint_monus_bzero: (all n:UInt. n ∸ bzero = n)

auto uint_monus_cancel

uint_monus_cancel: (all n:UInt. n ∸ n = 0)

uint_monus_cancel_bzero: (all n:UInt. n ∸ n = bzero)

uint_monus_monus_eq_monus_add: (all x:UInt, y:UInt, z:UInt. (x ∸ y) ∸ z = x ∸ (y + z))

uint_monus_order: (all x:UInt, y:UInt, z:UInt. (x ∸ y) ∸ z = (x ∸ z) ∸ y)

auto uint_monus_zero

uint_monus_zero: (all n:UInt. n ∸ 0 = n)

uint_mult_assoc: (all m:UInt, n:UInt, o:UInt. (m * n) * o = m * (n * o))

uint_mult_commute: (all m:UInt, n:UInt. m * n = n * m)

uint_mult_div_inverse: (all n:UInt, m:UInt. (if 0 < m then (n * m) / m = n))

uint_mult_mono_le: (all n:UInt, x:UInt, y:UInt. (if x ≤ y then n * x ≤ n * y))

uint_mult_mono_le2: (all n:UInt, x:UInt, m:UInt, y:UInt. (if (n ≤ m and x ≤ y) then n * x ≤ m * y))

auto uint_mult_one

uint_mult_one: (all n:UInt. n * 1 = n)

uint_mult_to_zero: (all n:UInt, m:UInt. (if n * m = 0 then (n = 0 or m = 0)))

auto uint_mult_zero

uint_mult_zero: (all n:UInt. n * 0 = 0)

uint_not_less_equal_iff_greater: (all x:UInt, y:UInt. (not (x ≤ y) ⇔ (y < x)))

uint_not_less_implies_less_equal: (all x:UInt, y:UInt. (if not (x < y) then y ≤ x))

uint_not_less_zero: (all x:UInt. not (x < 0))

uint_not_one_add_le_zero: (all n:UInt. not (1 + n ≤ 0))

uint_not_one_add_zero: (all n:UInt. not (1 + n = 0))

uint_not_zero_pos: (all n:UInt. (if not (n = 0) then 0 < n))

uint_odd_one_even: (all n:UInt. (if Odd(1 + n) then Even(n)))

auto uint_one_add_zero_false

uint_one_add_zero_false: (all n:UInt. (1 + n = 0) = false)

uint_one_expt: (all n:UInt. 1 ^ n = 1)

auto uint_one_mult

uint_one_mult: (all n:UInt. 1 * n = n)

uint_pos_implies_one_le: (all n:UInt. (if 0 < n then 1 ≤ n))

uint_pos_mult_both_sides_of_less: (all n:UInt, x:UInt, y:UInt. (if (0 < n and x < y) then n * x < n * y))

uint_pos_not_zero: (all n:UInt. (if 0 < n then not (n = 0)))

uint_positive_add_one: (all n:UInt. (if 0 < n then some n':UInt. n = 1 + n'))

uint_pow_add_r: (all m:UInt, n:UInt, o:UInt. m ^ (n + o) = m ^ n * m ^ o)

uint_pow_mul_l: (all m:UInt, n:UInt, o:UInt. (m * n) ^ o = m ^ o * n ^ o)

uint_pow_mul_r: (all m:UInt, n:UInt, o:UInt. (m ^ n) ^ o = m ^ (n * o))

uint_pred_monus: (all n:UInt. pred(n) = n ∸ 1)

uint_toNat_fromNat: (all x:Nat. toNat(fromNat(x)) = x)

uint_toNat_injective: (all x:UInt, y:UInt. (if toNat(x) = toNat(y) then x = y))

uint_trichotomy: (all x:UInt, y:UInt. (x < y or x = y or y < x))

uint_two_mult: (all n:UInt. 2 * n = n + n)

auto uint_zero_add

uint_zero_add: (all x:UInt. 0 + x = x)

uint_zero_div: (all x:UInt. (if 0 < x then 0 / x = 0))

uint_zero_le: (all x:UInt. 0 ≤ x)

uint_zero_le_zero: (all x:UInt. (if x ≤ 0 then x = 0))

auto uint_zero_less_equal_true

uint_zero_less_equal_true: (all x:UInt. (0 ≤ x) = true)

uint_zero_less_one_add: (all n:UInt. 0 < 1 + n)

auto uint_zero_less_one_add_true

uint_zero_less_one_add_true: (all n:UInt. (0 < 1 + n) = true)

uint_zero_max: (all x:UInt. max(0, x) = x)

uint_zero_mod: (all x:UInt. 0 % x = 0)

auto uint_zero_mult

uint_zero_mult: (all n:UInt. 0 * n = 0)

uint_zero_or_add_one: (all x:UInt. (x = 0 or some x':UInt. x = 1 + x'))

uint_zero_or_positive: (all x:UInt. (x = 0 or 0 < x))