Sqrtn.succ_mul_succ: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/natural/sqrtn.bend as Sqrtn
Lines 22-25 of bend-collections-laws-math@1.0.0.0/proofs/math/natural/sqrtn.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/sqrtn.bend&start_line=22&end_line=25

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 A

# (1 + u)(1 + v) == 1 + ((u + v) + u v)
def succ_mul_succ(+u: Nat, +v: Nat) -> {Nat.mul(1n+u, 1n+v) == 1n+Nat.add(Nat.add(u, v), Nat.mul(u, v)) : Nat}:
  %Equal.sym(Nat, Nat.mul(u, 1n+v), Nat.add(u, Nat.mul(u, v)), A.mul_succ(u, v)) : {Nat.add(1n+v, _) == 1n+Nat.add(Nat.add(u, v), Nat.mul(u, v)) : Nat}
  N.succ_cong(Nat.add(v, Nat.add(u, Nat.mul(u, v))), Nat.add(Nat.add(u, v), Nat.mul(u, v)), Equal.trans(Nat, Nat.add(v, Nat.add(u, Nat.mul(u, v))), Nat.add(u, Nat.add(v, Nat.mul(u, v))), Nat.add(Nat.add(u, v), Nat.mul(u, v)), A.add_swap(v, u, Nat.mul(u, v)), Equal.sym(Nat, Nat.add(Nat.add(u, v), Nat.mul(u, v)), Nat.add(u, Nat.add(v, Nat.mul(u, v))), A.add_assoc(u, v, Nat.mul(u, v)))))

Uses (9):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.trans
  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.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_assoc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_swap
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_succ
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=succ_cong