Pow.sub_add_r: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/pow.bend as Pow
Lines 37-42 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/pow.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/pow.bend&start_line=37&end_line=42

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/lemmas/proofs/nat_algebra.bend as NA

# (a + c) - b == a + (c - b) for b <= c
def sub_add_r(+a: Nat, +c: Nat, +b: Nat, +h: {Nat.is_le(b, c) == True{} : Bool}) -> {Nat.sub(Nat.add(a, c), b) == Nat.add(a, Nat.sub(c, b)) : Nat}:
  +d = Nat.sub(c, b)
  +e1 = Equal.cong(Nat, Nat, t => Nat.sub(Nat.add(a, t), b), c, Nat.add(b, d), Equal.sym(Nat, Nat.add(b, d), c, N.sub_add(c, b, h)))
  +e2 = Equal.cong(Nat, Nat, t => Nat.sub(t, b), Nat.add(a, Nat.add(b, d)), Nat.add(b, Nat.add(a, d)), NA.add_swap(a, b, d))
  Equal.trans(Nat, Nat.sub(Nat.add(a, c), b), Nat.sub(Nat.add(a, Nat.add(b, d)), b), Nat.add(a, d), e1, Equal.trans(Nat, Nat.sub(Nat.add(a, Nat.add(b, d)), b), Nat.sub(Nat.add(b, Nat.add(a, d)), b), Nat.add(a, d), e2, N.add_sub_cancel(b, Nat.add(a, d))))

Uses (11):
  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-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_swap
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=add_sub_cancel
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=sub_add