Arith.sc_sub: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/lib/arith.bend as Arith
Lines 151-166 of bend-collections-laws-crypto@1.0.0.0/proofs/lib/arith.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/arith.bend&start_line=151&end_line=166

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/u32alg.bend as A

# 2^k (a - b) == 2^k a - 2^k b
def sc_sub(+k: Nat, +a: Nat, +b: Nat) -> {sc(k, Nat.sub(a, b)) == Nat.sub(sc(k, a), sc(k, b)) : Nat}:
  match a b:
    case 0n _:
      %Equal.sym(Nat, Nat.sub(0n, b), 0n, zsub(b)) : {sc(k, _) == Nat.sub(sc(k, 0n), sc(k, b)) : Nat}
      %Equal.sym(Nat, sc(k, 0n), 0n, A.sc_zero(k)) : {_ == Nat.sub(sc(k, 0n), sc(k, b)) : Nat}
      %Equal.sym(Nat, sc(k, 0n), 0n, A.sc_zero(k)) : {0n == Nat.sub(_, sc(k, b)) : Nat}
      Equal.sym(Nat, Nat.sub(0n, sc(k, b)), 0n, zsub(sc(k, b)))
    case 1n+p 0n:
      %Equal.sym(Nat, sc(k, 0n), 0n, A.sc_zero(k)) : {sc(k, 1n+p) == Nat.sub(sc(k, 1n+p), _) : Nat}
      Equal.sym(Nat, Nat.sub(sc(k, 1n+p), 0n), sc(k, 1n+p), N.sub_zero(sc(k, 1n+p)))
    case 1n+p 1n+q:
      %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}
      %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}
      %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}
      sc_sub(k, p, q)

Uses (10):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.add
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.sub
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/arith.bend&name=sc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/arith.bend&name=sub_cancel_l
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/arith.bend&name=zsub
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=sub_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/u32alg.bend&name=sc_succ
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/u32alg.bend&name=sc_zero