Egcd.alg_neg: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/number/egcd.bend as Egcd
Lines 55-62 of bend-collections-laws-math@1.0.0.0/proofs/math/number/egcd.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/number/egcd.bend&start_line=55&end_line=62

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

# child r y == g + b x  gives  (q b + r) y == g + b (x + q y)
def alg_neg(+g: Nat, +x: Nat, +y: Nat, +q: Nat, +b: Nat, +r: Nat, +ih: {Nat.mul(r, y) == Nat.add(g, Nat.mul(b, x)) : Nat}) -> {Nat.mul(Nat.add(Nat.mul(q, b), r), y) == Nat.add(g, Nat.mul(b, Nat.add(x, Nat.mul(q, y)))) : Nat}:
  %Equal.sym(Nat, Nat.mul(Nat.add(Nat.mul(q, b), r), y), Nat.add(Nat.mul(b, Nat.mul(q, y)), Nat.mul(r, y)), spread(q, b, r, y)) : {_ == Nat.add(g, Nat.mul(b, Nat.add(x, Nat.mul(q, y)))) : Nat}
  %Equal.sym(Nat, Nat.mul(r, y), Nat.add(g, Nat.mul(b, x)), ih) : {Nat.add(Nat.mul(b, Nat.mul(q, y)), _) == Nat.add(g, Nat.mul(b, Nat.add(x, Nat.mul(q, y)))) : Nat}
  %Equal.sym(Nat, Nat.mul(b, Nat.add(x, Nat.mul(q, y))), Nat.add(Nat.mul(b, x), Nat.mul(b, Nat.mul(q, y))), A.mul_add_left(b, x, Nat.mul(q, y))) : {Nat.add(Nat.mul(b, Nat.mul(q, y)), Nat.add(g, Nat.mul(b, x))) == Nat.add(g, _) : Nat}
  %Equal.sym(Nat, Nat.add(Nat.mul(b, Nat.mul(q, y)), Nat.add(g, Nat.mul(b, x))), Nat.add(g, Nat.add(Nat.mul(b, Nat.mul(q, y)), Nat.mul(b, x))), A.add_swap(Nat.mul(b, Nat.mul(q, y)), g, Nat.mul(b, x))) : {_ == Nat.add(g, Nat.add(Nat.mul(b, x), Nat.mul(b, Nat.mul(q, y)))) : Nat}
  %Equal.sym(Nat, Nat.add(Nat.mul(b, Nat.mul(q, y)), Nat.mul(b, x)), Nat.add(Nat.mul(b, x), Nat.mul(b, Nat.mul(q, y))), A.add_comm(Nat.mul(b, Nat.mul(q, y)), Nat.mul(b, x))) : {Nat.add(g, _) == Nat.add(g, Nat.add(Nat.mul(b, x), Nat.mul(b, Nat.mul(q, y)))) : Nat}
  {==}

Uses (8):
  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.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_comm
  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_add_left
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/number/egcd.bend&name=spread