law word_adc_exact_sym

bend-mathlib@0.8.1.0 · u32.bend:1004

import Base
import bend-mathlib@0.8.1.0/nat.bend as MNat

1004# The adder is exact below the overflow: a + b + carry-in < 2^n implies to_nat (adc a b False c) = a + b + carry-in, reversed to rewrite toward the simple side.
1005law word_adc_exact_sym:
1006  for n: Nat
1007  for a: Word(n)
1008  for b: Word(n)
1009  for c: Bool
1010  for h: MNat.lt(Nat.add(Nat.add(Word.to_nat(n, a), Word.to_nat(n, b)), Bool.pick(Nat, c, 1n, 0n)), Nat.pow(2n, n))
1011  {Nat.add(Nat.add(Word.to_nat(n, a), Word.to_nat(n, b)), Bool.pick(Nat, c, 1n, 0n)) == Word.to_nat(n, Word.adc(n, a, b, False{}, c)) : Nat}