import Base
import ../../../spec/crypto/aes/aes.bend as S
import ../../lib/logic.bend as L

# Generated by tools/generators/aes/sbox.py. FIPS 197 section 4.4: the
# multiplicative inverse b of a nonzero byte a satisfies a * b = {01}; the
# specification computes it as a^254 (spec/crypto/aes/aes.bend). Proved here
# for every U32 (both functions read only its low 8 bits): one closed
# evaluation of 13 field products per byte value.

law inverse_unit_bits:
  for +b0: Bool
  for +b1: Bool
  for +b2: Bool
  for +b3: Bool
  for +b4: Bool
  for +b5: Bool
  for +b6: Bool
  for +b7: Bool
  for +b8: Bool
  for +b9: Bool
  for +b10: Bool
  for +b11: Bool
  for +b12: Bool
  for +b13: Bool
  for +b14: Bool
  for +b15: Bool
  for +b16: Bool
  for +b17: Bool
  for +b18: Bool
  for +b19: Bool
  for +b20: Bool
  for +b21: Bool
  for +b22: Bool
  for +b23: Bool
  for +b24: Bool
  for +b25: Bool
  for +b26: Bool
  for +b27: Bool
  for +b28: Bool
  for +b29: Bool
  for +b30: Bool
  for +b31: Bool
  for +h: {U32.is_eq(U32.and(255, U32{WCon{b0, WCon{b1, WCon{b2, WCon{b3, WCon{b4, WCon{b5, WCon{b6, WCon{b7, WCon{b8, WCon{b9, WCon{b10, WCon{b11, WCon{b12, WCon{b13, WCon{b14, WCon{b15, WCon{b16, WCon{b17, WCon{b18, WCon{b19, WCon{b20, WCon{b21, WCon{b22, WCon{b23, WCon{b24, WCon{b25, WCon{b26, WCon{b27, WCon{b28, WCon{b29, WCon{b30, WCon{b31, WNil{}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}), 0) == False{} : Bool}
  {S.mul(U32{WCon{b0, WCon{b1, WCon{b2, WCon{b3, WCon{b4, WCon{b5, WCon{b6, WCon{b7, WCon{b8, WCon{b9, WCon{b10, WCon{b11, WCon{b12, WCon{b13, WCon{b14, WCon{b15, WCon{b16, WCon{b17, WCon{b18, WCon{b19, WCon{b20, WCon{b21, WCon{b22, WCon{b23, WCon{b24, WCon{b25, WCon{b26, WCon{b27, WCon{b28, WCon{b29, WCon{b30, WCon{b31, WNil{}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}, S.inverse(U32{WCon{b0, WCon{b1, WCon{b2, WCon{b3, WCon{b4, WCon{b5, WCon{b6, WCon{b7, WCon{b8, WCon{b9, WCon{b10, WCon{b11, WCon{b12, WCon{b13, WCon{b14, WCon{b15, WCon{b16, WCon{b17, WCon{b18, WCon{b19, WCon{b20, WCon{b21, WCon{b22, WCon{b23, WCon{b24, WCon{b25, WCon{b26, WCon{b27, WCon{b28, WCon{b29, WCon{b30, WCon{b31, WNil{}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}})) == 1 : U32}

def inverse_unit_bits(b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, b30, b31, h):
  match b0 b1 b2 b3 b4 b5 b6 b7:
    case False{} False{} False{} False{} False{} False{} False{} False{}:
      Empty.absurd({S.mul(U32{WCon{False{}, WCon{False{}, WCon{False{}, WCon{False{}, WCon{False{}, WCon{False{}, WCon{False{}, WCon{False{}, WCon{b8, WCon{b9, WCon{b10, WCon{b11, WCon{b12, WCon{b13, WCon{b14, WCon{b15, WCon{b16, WCon{b17, WCon{b18, WCon{b19, WCon{b20, WCon{b21, WCon{b22, WCon{b23, WCon{b24, WCon{b25, WCon{b26, WCon{b27, WCon{b28, WCon{b29, WCon{b30, WCon{b31, WNil{}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}, S.inverse(U32{WCon{False{}, WCon{False{}, WCon{False{}, WCon{False{}, WCon{False{}, WCon{False{}, WCon{False{}, WCon{False{}, WCon{b8, WCon{b9, WCon{b10, WCon{b11, WCon{b12, WCon{b13, WCon{b14, WCon{b15, WCon{b16, WCon{b17, WCon{b18, WCon{b19, WCon{b20, WCon{b21, WCon{b22, WCon{b23, WCon{b24, WCon{b25, WCon{b26, WCon{b27, WCon{b28, WCon{b29, WCon{b30, WCon{b31, WNil{}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}})) == 1 : U32}, L.true_false(h))
    case True{} False{} False{} False{} False{} False{} False{} False{}:
      {==}
    case False{} True{} False{} False{} False{} False{} False{} False{}:
      {==}
    case True{} True{} False{} False{} False{} False{} False{} False{}:
      {==}
    case False{} False{} True{} False{} False{} False{} False{} False{}:
      {==}
    case True{} False{} True{} False{} False{} False{} False{} False{}:
      {==}
    case False{} True{} True{} False{} False{} False{} False{} False{}:
      {==}
    case True{} True{} True{} False{} False{} False{} False{} False{}:
      {==}
    case False{} False{} False{} True{} False{} False{} False{} False{}:
      {==}
    case True{} False{} False{} True{} False{} False{} False{} False{}:
      {==}
    case False{} True{} False{} True{} False{} False{} False{} False{}:
      {==}
    case True{} True{} False{} True{} False{} False{} False{} False{}:
      {==}
    case False{} False{} True{} True{} False{} False{} False{} False{}:
      {==}
    case True{} False{} True{} True{} False{} False{} False{} False{}:
      {==}
    case False{} True{} True{} True{} False{} False{} False{} False{}:
      {==}
    case True{} True{} True{} True{} False{} False{} False{} False{}:
      {==}
    case False{} False{} False{} False{} True{} False{} False{} False{}:
      {==}
    case True{} False{} False{} False{} True{} False{} False{} False{}:
      {==}
    case False{} True{} False{} False{} True{} False{} False{} False{}:
      {==}
    case True{} True{} False{} False{} True{} False{} False{} False{}:
      {==}
    case False{} False{} True{} False{} True{} False{} False{} False{}:
      {==}
    case True{} False{} True{} False{} True{} False{} False{} False{}:
      {==}
    case False{} True{} True{} False{} True{} False{} False{} False{}:
      {==}
    case True{} True{} True{} False{} True{} False{} False{} False{}:
      {==}
    case False{} False{} False{} True{} True{} False{} False{} False{}:
      {==}
    case True{} False{} False{} True{} True{} False{} False{} False{}:
      {==}
    case False{} True{} False{} True{} True{} False{} False{} False{}:
      {==}
    case True{} True{} False{} True{} True{} False{} False{} False{}:
      {==}
    case False{} False{} True{} True{} True{} False{} False{} False{}:
      {==}
    case True{} False{} True{} True{} True{} False{} False{} False{}:
      {==}
    case False{} True{} True{} True{} True{} False{} False{} False{}:
      {==}
    case True{} True{} True{} True{} True{} False{} False{} False{}:
      {==}
    case False{} False{} False{} False{} False{} True{} False{} False{}:
      {==}
    case True{} False{} False{} False{} False{} True{} False{} False{}:
      {==}
    case False{} True{} False{} False{} False{} True{} False{} False{}:
      {==}
    case True{} True{} False{} False{} False{} True{} False{} False{}:
      {==}
    case False{} False{} True{} False{} False{} True{} False{} False{}:
      {==}
    case True{} False{} True{} False{} False{} True{} False{} False{}:
      {==}
    case False{} True{} True{} False{} False{} True{} False{} False{}:
      {==}
    case True{} True{} True{} False{} False{} True{} False{} False{}:
      {==}
    case False{} False{} False{} True{} False{} True{} False{} False{}:
      {==}
    case True{} False{} False{} True{} False{} True{} False{} False{}:
      {==}
    case False{} True{} False{} True{} False{} True{} False{} False{}:
      {==}
    case True{} True{} False{} True{} False{} True{} False{} False{}:
      {==}
    case False{} False{} True{} True{} False{} True{} False{} False{}:
      {==}
    case True{} False{} True{} True{} False{} True{} False{} False{}:
      {==}
    case False{} True{} True{} True{} False{} True{} False{} False{}:
      {==}
    case True{} True{} True{} True{} False{} True{} False{} False{}:
      {==}
    case False{} False{} False{} False{} True{} True{} False{} False{}:
      {==}
    case True{} False{} False{} False{} True{} True{} False{} False{}:
      {==}
    case False{} True{} False{} False{} True{} True{} False{} False{}:
      {==}
    case True{} True{} False{} False{} True{} True{} False{} False{}:
      {==}
    case False{} False{} True{} False{} True{} True{} False{} False{}:
      {==}
    case True{} False{} True{} False{} True{} True{} False{} False{}:
      {==}
    case False{} True{} True{} False{} True{} True{} False{} False{}:
      {==}
    case True{} True{} True{} False{} True{} True{} False{} False{}:
      {==}
    case False{} False{} False{} True{} True{} True{} False{} False{}:
      {==}
    case True{} False{} False{} True{} True{} True{} False{} False{}:
      {==}
    case False{} True{} False{} True{} True{} True{} False{} False{}:
      {==}
    case True{} True{} False{} True{} True{} True{} False{} False{}:
      {==}
    case False{} False{} True{} True{} True{} True{} False{} False{}:
      {==}
    case True{} False{} True{} True{} True{} True{} False{} False{}:
      {==}
    case False{} True{} True{} True{} True{} True{} False{} False{}:
      {==}
    case True{} True{} True{} True{} True{} True{} False{} False{}:
      {==}
    case False{} False{} False{} False{} False{} False{} True{} False{}:
      {==}
    case True{} False{} False{} False{} False{} False{} True{} False{}:
      {==}
    case False{} True{} False{} False{} False{} False{} True{} False{}:
      {==}
    case True{} True{} False{} False{} False{} False{} True{} False{}:
      {==}
    case False{} False{} True{} False{} False{} False{} True{} False{}:
      {==}
    case True{} False{} True{} False{} False{} False{} True{} False{}:
      {==}
    case False{} True{} True{} False{} False{} False{} True{} False{}:
      {==}
    case True{} True{} True{} False{} False{} False{} True{} False{}:
      {==}
    case False{} False{} False{} True{} False{} False{} True{} False{}:
      {==}
    case True{} False{} False{} True{} False{} False{} True{} False{}:
      {==}
    case False{} True{} False{} True{} False{} False{} True{} False{}:
      {==}
    case True{} True{} False{} True{} False{} False{} True{} False{}:
      {==}
    case False{} False{} True{} True{} False{} False{} True{} False{}:
      {==}
    case True{} False{} True{} True{} False{} False{} True{} False{}:
      {==}
    case False{} True{} True{} True{} False{} False{} True{} False{}:
      {==}
    case True{} True{} True{} True{} False{} False{} True{} False{}:
      {==}
    case False{} False{} False{} False{} True{} False{} True{} False{}:
      {==}
    case True{} False{} False{} False{} True{} False{} True{} False{}:
      {==}
    case False{} True{} False{} False{} True{} False{} True{} False{}:
      {==}
    case True{} True{} False{} False{} True{} False{} True{} False{}:
      {==}
    case False{} False{} True{} False{} True{} False{} True{} False{}:
      {==}
    case True{} False{} True{} False{} True{} False{} True{} False{}:
      {==}
    case False{} True{} True{} False{} True{} False{} True{} False{}:
      {==}
    case True{} True{} True{} False{} True{} False{} True{} False{}:
      {==}
    case False{} False{} False{} True{} True{} False{} True{} False{}:
      {==}
    case True{} False{} False{} True{} True{} False{} True{} False{}:
      {==}
    case False{} True{} False{} True{} True{} False{} True{} False{}:
      {==}
    case True{} True{} False{} True{} True{} False{} True{} False{}:
      {==}
    case False{} False{} True{} True{} True{} False{} True{} False{}:
      {==}
    case True{} False{} True{} True{} True{} False{} True{} False{}:
      {==}
    case False{} True{} True{} True{} True{} False{} True{} False{}:
      {==}
    case True{} True{} True{} True{} True{} False{} True{} False{}:
      {==}
    case False{} False{} False{} False{} False{} True{} True{} False{}:
      {==}
    case True{} False{} False{} False{} False{} True{} True{} False{}:
      {==}
    case False{} True{} False{} False{} False{} True{} True{} False{}:
      {==}
    case True{} True{} False{} False{} False{} True{} True{} False{}:
      {==}
    case False{} False{} True{} False{} False{} True{} True{} False{}:
      {==}
    case True{} False{} True{} False{} False{} True{} True{} False{}:
      {==}
    case False{} True{} True{} False{} False{} True{} True{} False{}:
      {==}
    case True{} True{} True{} False{} False{} True{} True{} False{}:
      {==}
    case False{} False{} False{} True{} False{} True{} True{} False{}:
      {==}
    case True{} False{} False{} True{} False{} True{} True{} False{}:
      {==}
    case False{} True{} False{} True{} False{} True{} True{} False{}:
      {==}
    case True{} True{} False{} True{} False{} True{} True{} False{}:
      {==}
    case False{} False{} True{} True{} False{} True{} True{} False{}:
      {==}
    case True{} False{} True{} True{} False{} True{} True{} False{}:
      {==}
    case False{} True{} True{} True{} False{} True{} True{} False{}:
      {==}
    case True{} True{} True{} True{} False{} True{} True{} False{}:
      {==}
    case False{} False{} False{} False{} True{} True{} True{} False{}:
      {==}
    case True{} False{} False{} False{} True{} True{} True{} False{}:
      {==}
    case False{} True{} False{} False{} True{} True{} True{} False{}:
      {==}
    case True{} True{} False{} False{} True{} True{} True{} False{}:
      {==}
    case False{} False{} True{} False{} True{} True{} True{} False{}:
      {==}
    case True{} False{} True{} False{} True{} True{} True{} False{}:
      {==}
    case False{} True{} True{} False{} True{} True{} True{} False{}:
      {==}
    case True{} True{} True{} False{} True{} True{} True{} False{}:
      {==}
    case False{} False{} False{} True{} True{} True{} True{} False{}:
      {==}
    case True{} False{} False{} True{} True{} True{} True{} False{}:
      {==}
    case False{} True{} False{} True{} True{} True{} True{} False{}:
      {==}
    case True{} True{} False{} True{} True{} True{} True{} False{}:
      {==}
    case False{} False{} True{} True{} True{} True{} True{} False{}:
      {==}
    case True{} False{} True{} True{} True{} True{} True{} False{}:
      {==}
    case False{} True{} True{} True{} True{} True{} True{} False{}:
      {==}
    case True{} True{} True{} True{} True{} True{} True{} False{}:
      {==}
    case False{} False{} False{} False{} False{} False{} False{} True{}:
      {==}
    case True{} False{} False{} False{} False{} False{} False{} True{}:
      {==}
    case False{} True{} False{} False{} False{} False{} False{} True{}:
      {==}
    case True{} True{} False{} False{} False{} False{} False{} True{}:
      {==}
    case False{} False{} True{} False{} False{} False{} False{} True{}:
      {==}
    case True{} False{} True{} False{} False{} False{} False{} True{}:
      {==}
    case False{} True{} True{} False{} False{} False{} False{} True{}:
      {==}
    case True{} True{} True{} False{} False{} False{} False{} True{}:
      {==}
    case False{} False{} False{} True{} False{} False{} False{} True{}:
      {==}
    case True{} False{} False{} True{} False{} False{} False{} True{}:
      {==}
    case False{} True{} False{} True{} False{} False{} False{} True{}:
      {==}
    case True{} True{} False{} True{} False{} False{} False{} True{}:
      {==}
    case False{} False{} True{} True{} False{} False{} False{} True{}:
      {==}
    case True{} False{} True{} True{} False{} False{} False{} True{}:
      {==}
    case False{} True{} True{} True{} False{} False{} False{} True{}:
      {==}
    case True{} True{} True{} True{} False{} False{} False{} True{}:
      {==}
    case False{} False{} False{} False{} True{} False{} False{} True{}:
      {==}
    case True{} False{} False{} False{} True{} False{} False{} True{}:
      {==}
    case False{} True{} False{} False{} True{} False{} False{} True{}:
      {==}
    case True{} True{} False{} False{} True{} False{} False{} True{}:
      {==}
    case False{} False{} True{} False{} True{} False{} False{} True{}:
      {==}
    case True{} False{} True{} False{} True{} False{} False{} True{}:
      {==}
    case False{} True{} True{} False{} True{} False{} False{} True{}:
      {==}
    case True{} True{} True{} False{} True{} False{} False{} True{}:
      {==}
    case False{} False{} False{} True{} True{} False{} False{} True{}:
      {==}
    case True{} False{} False{} True{} True{} False{} False{} True{}:
      {==}
    case False{} True{} False{} True{} True{} False{} False{} True{}:
      {==}
    case True{} True{} False{} True{} True{} False{} False{} True{}:
      {==}
    case False{} False{} True{} True{} True{} False{} False{} True{}:
      {==}
    case True{} False{} True{} True{} True{} False{} False{} True{}:
      {==}
    case False{} True{} True{} True{} True{} False{} False{} True{}:
      {==}
    case True{} True{} True{} True{} True{} False{} False{} True{}:
      {==}
    case False{} False{} False{} False{} False{} True{} False{} True{}:
      {==}
    case True{} False{} False{} False{} False{} True{} False{} True{}:
      {==}
    case False{} True{} False{} False{} False{} True{} False{} True{}:
      {==}
    case True{} True{} False{} False{} False{} True{} False{} True{}:
      {==}
    case False{} False{} True{} False{} False{} True{} False{} True{}:
      {==}
    case True{} False{} True{} False{} False{} True{} False{} True{}:
      {==}
    case False{} True{} True{} False{} False{} True{} False{} True{}:
      {==}
    case True{} True{} True{} False{} False{} True{} False{} True{}:
      {==}
    case False{} False{} False{} True{} False{} True{} False{} True{}:
      {==}
    case True{} False{} False{} True{} False{} True{} False{} True{}:
      {==}
    case False{} True{} False{} True{} False{} True{} False{} True{}:
      {==}
    case True{} True{} False{} True{} False{} True{} False{} True{}:
      {==}
    case False{} False{} True{} True{} False{} True{} False{} True{}:
      {==}
    case True{} False{} True{} True{} False{} True{} False{} True{}:
      {==}
    case False{} True{} True{} True{} False{} True{} False{} True{}:
      {==}
    case True{} True{} True{} True{} False{} True{} False{} True{}:
      {==}
    case False{} False{} False{} False{} True{} True{} False{} True{}:
      {==}
    case True{} False{} False{} False{} True{} True{} False{} True{}:
      {==}
    case False{} True{} False{} False{} True{} True{} False{} True{}:
      {==}
    case True{} True{} False{} False{} True{} True{} False{} True{}:
      {==}
    case False{} False{} True{} False{} True{} True{} False{} True{}:
      {==}
    case True{} False{} True{} False{} True{} True{} False{} True{}:
      {==}
    case False{} True{} True{} False{} True{} True{} False{} True{}:
      {==}
    case True{} True{} True{} False{} True{} True{} False{} True{}:
      {==}
    case False{} False{} False{} True{} True{} True{} False{} True{}:
      {==}
    case True{} False{} False{} True{} True{} True{} False{} True{}:
      {==}
    case False{} True{} False{} True{} True{} True{} False{} True{}:
      {==}
    case True{} True{} False{} True{} True{} True{} False{} True{}:
      {==}
    case False{} False{} True{} True{} True{} True{} False{} True{}:
      {==}
    case True{} False{} True{} True{} True{} True{} False{} True{}:
      {==}
    case False{} True{} True{} True{} True{} True{} False{} True{}:
      {==}
    case True{} True{} True{} True{} True{} True{} False{} True{}:
      {==}
    case False{} False{} False{} False{} False{} False{} True{} True{}:
      {==}
    case True{} False{} False{} False{} False{} False{} True{} True{}:
      {==}
    case False{} True{} False{} False{} False{} False{} True{} True{}:
      {==}
    case True{} True{} False{} False{} False{} False{} True{} True{}:
      {==}
    case False{} False{} True{} False{} False{} False{} True{} True{}:
      {==}
    case True{} False{} True{} False{} False{} False{} True{} True{}:
      {==}
    case False{} True{} True{} False{} False{} False{} True{} True{}:
      {==}
    case True{} True{} True{} False{} False{} False{} True{} True{}:
      {==}
    case False{} False{} False{} True{} False{} False{} True{} True{}:
      {==}
    case True{} False{} False{} True{} False{} False{} True{} True{}:
      {==}
    case False{} True{} False{} True{} False{} False{} True{} True{}:
      {==}
    case True{} True{} False{} True{} False{} False{} True{} True{}:
      {==}
    case False{} False{} True{} True{} False{} False{} True{} True{}:
      {==}
    case True{} False{} True{} True{} False{} False{} True{} True{}:
      {==}
    case False{} True{} True{} True{} False{} False{} True{} True{}:
      {==}
    case True{} True{} True{} True{} False{} False{} True{} True{}:
      {==}
    case False{} False{} False{} False{} True{} False{} True{} True{}:
      {==}
    case True{} False{} False{} False{} True{} False{} True{} True{}:
      {==}
    case False{} True{} False{} False{} True{} False{} True{} True{}:
      {==}
    case True{} True{} False{} False{} True{} False{} True{} True{}:
      {==}
    case False{} False{} True{} False{} True{} False{} True{} True{}:
      {==}
    case True{} False{} True{} False{} True{} False{} True{} True{}:
      {==}
    case False{} True{} True{} False{} True{} False{} True{} True{}:
      {==}
    case True{} True{} True{} False{} True{} False{} True{} True{}:
      {==}
    case False{} False{} False{} True{} True{} False{} True{} True{}:
      {==}
    case True{} False{} False{} True{} True{} False{} True{} True{}:
      {==}
    case False{} True{} False{} True{} True{} False{} True{} True{}:
      {==}
    case True{} True{} False{} True{} True{} False{} True{} True{}:
      {==}
    case False{} False{} True{} True{} True{} False{} True{} True{}:
      {==}
    case True{} False{} True{} True{} True{} False{} True{} True{}:
      {==}
    case False{} True{} True{} True{} True{} False{} True{} True{}:
      {==}
    case True{} True{} True{} True{} True{} False{} True{} True{}:
      {==}
    case False{} False{} False{} False{} False{} True{} True{} True{}:
      {==}
    case True{} False{} False{} False{} False{} True{} True{} True{}:
      {==}
    case False{} True{} False{} False{} False{} True{} True{} True{}:
      {==}
    case True{} True{} False{} False{} False{} True{} True{} True{}:
      {==}
    case False{} False{} True{} False{} False{} True{} True{} True{}:
      {==}
    case True{} False{} True{} False{} False{} True{} True{} True{}:
      {==}
    case False{} True{} True{} False{} False{} True{} True{} True{}:
      {==}
    case True{} True{} True{} False{} False{} True{} True{} True{}:
      {==}
    case False{} False{} False{} True{} False{} True{} True{} True{}:
      {==}
    case True{} False{} False{} True{} False{} True{} True{} True{}:
      {==}
    case False{} True{} False{} True{} False{} True{} True{} True{}:
      {==}
    case True{} True{} False{} True{} False{} True{} True{} True{}:
      {==}
    case False{} False{} True{} True{} False{} True{} True{} True{}:
      {==}
    case True{} False{} True{} True{} False{} True{} True{} True{}:
      {==}
    case False{} True{} True{} True{} False{} True{} True{} True{}:
      {==}
    case True{} True{} True{} True{} False{} True{} True{} True{}:
      {==}
    case False{} False{} False{} False{} True{} True{} True{} True{}:
      {==}
    case True{} False{} False{} False{} True{} True{} True{} True{}:
      {==}
    case False{} True{} False{} False{} True{} True{} True{} True{}:
      {==}
    case True{} True{} False{} False{} True{} True{} True{} True{}:
      {==}
    case False{} False{} True{} False{} True{} True{} True{} True{}:
      {==}
    case True{} False{} True{} False{} True{} True{} True{} True{}:
      {==}
    case False{} True{} True{} False{} True{} True{} True{} True{}:
      {==}
    case True{} True{} True{} False{} True{} True{} True{} True{}:
      {==}
    case False{} False{} False{} True{} True{} True{} True{} True{}:
      {==}
    case True{} False{} False{} True{} True{} True{} True{} True{}:
      {==}
    case False{} True{} False{} True{} True{} True{} True{} True{}:
      {==}
    case True{} True{} False{} True{} True{} True{} True{} True{}:
      {==}
    case False{} False{} True{} True{} True{} True{} True{} True{}:
      {==}
    case True{} False{} True{} True{} True{} True{} True{} True{}:
      {==}
    case False{} True{} True{} True{} True{} True{} True{} True{}:
      {==}
    case True{} True{} True{} True{} True{} True{} True{} True{}:
      {==}

law inverse_unit:
  for +x: U32
  for +h: {U32.is_eq(U32.and(255, x), 0) == False{} : Bool}
  {S.mul(x, S.inverse(x)) == 1 : U32}

def inverse_unit(x, h):
  match x:
    case U32{WCon{b0, WCon{b1, WCon{b2, WCon{b3, WCon{b4, WCon{b5, WCon{b6, WCon{b7, WCon{b8, WCon{b9, WCon{b10, WCon{b11, WCon{b12, WCon{b13, WCon{b14, WCon{b15, WCon{b16, WCon{b17, WCon{b18, WCon{b19, WCon{b20, WCon{b21, WCon{b22, WCon{b23, WCon{b24, WCon{b25, WCon{b26, WCon{b27, WCon{b28, WCon{b29, WCon{b30, WCon{b31, WNil{}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}:
      inverse_unit_bits(b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, b30, b31, h)

law inverse_zero:
  {S.inverse(0) == 0 : U32}

def inverse_zero():
  {==}