F64rtools.a1: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64rtools.bend as F64rtools
Lines 91-94 of bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64rtools.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64rtools.bend&start_line=91&end_line=94

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

def a1(+x: Nat, +k: Nat, +u: Nat, +h: {Nat.is_le(Nat.add(x, k), u) == True{} : Bool}) -> {Nat.sub(u, x) == Nat.add(Nat.sub(u, Nat.add(x, k)), k) : Nat}:
  +r = Nat.sub(u, Nat.add(x, k))
  +eu = N.sub_add(u, Nat.add(x, k), h)
  Equal.trans(Nat, Nat.sub(u, x), Nat.sub(Nat.add(Nat.add(x, k), r), x), Nat.add(r, k), Equal.cong(Nat, Nat, z => Nat.sub(z, x), u, Nat.add(Nat.add(x, k), r), Equal.sym(Nat, Nat.add(Nat.add(x, k), r), u, eu)), Equal.trans(Nat, Nat.sub(Nat.add(Nat.add(x, k), r), x), Nat.sub(Nat.add(x, Nat.add(k, r)), x), Nat.add(r, k), Equal.cong(Nat, Nat, z => Nat.sub(z, x), Nat.add(Nat.add(x, k), r), Nat.add(x, Nat.add(k, r)), NA.add_assoc(x, k, r)), Equal.trans(Nat, Nat.sub(Nat.add(x, Nat.add(k, r)), x), Nat.add(k, r), Nat.add(r, k), N.add_sub_cancel(x, Nat.add(k, r)), NA.add_comm(k, r))))

Uses (12):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.cong
  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.is_le
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.sub
  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_comm
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=add_sub_cancel
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=sub_add