def sc_sub

bend-collections-laws-math@1.0.0.0 · proofs/lib/arith.bend:151

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/u32alg.bend as A

151# 2^k (a - b) == 2^k a - 2^k b
152def sc_sub(+k: Nat, +a: Nat, +b: Nat) -> {sc(k, Nat.sub(a, b)) == Nat.sub(sc(k, a), sc(k, b)) : Nat}:
153  match a b:
154    case 0n _:
155      %Equal.sym(Nat, Nat.sub(0n, b), 0n, zsub(b)) : {sc(k, _) == Nat.sub(sc(k, 0n), sc(k, b)) : Nat}
156      %Equal.sym(Nat, sc(k, 0n), 0n, A.sc_zero(k)) : {_ == Nat.sub(sc(k, 0n), sc(k, b)) : Nat}
157      %Equal.sym(Nat, sc(k, 0n), 0n, A.sc_zero(k)) : {0n == Nat.sub(_, sc(k, b)) : Nat}
158      Equal.sym(Nat, Nat.sub(0n, sc(k, b)), 0n, zsub(sc(k, b)))
159    case 1n+p 0n:
160      %Equal.sym(Nat, sc(k, 0n), 0n, A.sc_zero(k)) : {sc(k, 1n+p) == Nat.sub(sc(k, 1n+p), _) : Nat}
161      Equal.sym(Nat, Nat.sub(sc(k, 1n+p), 0n), sc(k, 1n+p), N.sub_zero(sc(k, 1n+p)))
162    case 1n+p 1n+q:
163      %Equal.sym(Nat, sc(k, 1n+p), Nat.add(sc(k, 1n), sc(k, p)), A.sc_succ(k, p)) : {sc(k, Nat.sub(p, q)) == Nat.sub(_, sc(k, 1n+q)) : Nat}
164      %Equal.sym(Nat, sc(k, 1n+q), Nat.add(sc(k, 1n), sc(k, q)), A.sc_succ(k, q)) : {sc(k, Nat.sub(p, q)) == Nat.sub(Nat.add(sc(k, 1n), sc(k, p)), _) : Nat}
165      %Equal.sym(Nat, Nat.sub(Nat.add(sc(k, 1n), sc(k, p)), Nat.add(sc(k, 1n), sc(k, q))), Nat.sub(sc(k, p), sc(k, q)), sub_cancel_l(sc(k, 1n), sc(k, p), sc(k, q))) : {sc(k, Nat.sub(p, q)) == _ : Nat}
166      sc_sub(k, p, q)