def sub_half

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/combnat.bend:41

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA

41# 1 + 2 i == i + (1 + i), so (1 + 2 i) - i == 1 + i
42def sub_half(+i: Nat) -> {Nat.sub(1n+Nat.double(i), i) == 1n+i : Nat}:
43  +e = Equal.trans(Nat, 1n+Nat.double(i), 1n+Nat.add(i, i), Nat.add(i, 1n+i), Equal.cong(Nat, Nat, t => 1n+t, Nat.double(i), Nat.add(i, i), NA.double_self(i)), Equal.sym(Nat, Nat.add(i, 1n+i), 1n+Nat.add(i, i), NA.add_succ(i, i)))
44  Equal.trans(Nat, Nat.sub(1n+Nat.double(i), i), Nat.sub(Nat.add(i, 1n+i), i), 1n+i, Equal.cong(Nat, Nat, t => Nat.sub(t, i), 1n+Nat.double(i), Nat.add(i, 1n+i), e), N.add_sub_cancel(i, 1n+i))