Cong.c_add: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/cong.bend as Cong
Lines 29-41 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/cong.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/cong.bend&start_line=29&end_line=41

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend as NR

def c_add(+pp: Nat, +a: Nat, +a2: Nat, +b: Nat, +b2: Nat, +ha: ceq(pp, a, a2), +hb: ceq(pp, b, b2)) -> ceq(pp, Nat.add(a, b), Nat.add(a2, b2)):
  +n = {1n+pp : Nat}
  +e1 = Equal.sym(Nat, Nat.mod(Nat.add(Nat.mod(a, n), b), n), Nat.mod(Nat.add(a, b), n), NR.mod_add_l(pp, a, b))
  +e2 = Equal.cong(Nat, Nat, t => Nat.mod(Nat.add(t, b), n), Nat.mod(a, n), Nat.mod(a2, n), ha)
  +e3 = NR.mod_add_l(pp, a2, b)
  +e4 = Equal.sym(Nat, Nat.mod(Nat.add(a2, Nat.mod(b, n)), n), Nat.mod(Nat.add(a2, b), n), NR.mod_add_r(pp, a2, b))
  +e5 = Equal.cong(Nat, Nat, t => Nat.mod(Nat.add(a2, t), n), Nat.mod(b, n), Nat.mod(b2, n), hb)
  +e6 = NR.mod_add_r(pp, a2, b2)
  Equal.trans(Nat, Nat.mod(Nat.add(a, b), n), Nat.mod(Nat.add(Nat.mod(a, n), b), n), Nat.mod(Nat.add(a2, b2), n), e1,
    Equal.trans(Nat, Nat.mod(Nat.add(Nat.mod(a, n), b), n), Nat.mod(Nat.add(Nat.mod(a2, n), b), n), Nat.mod(Nat.add(a2, b2), n), e2,
      Equal.trans(Nat, Nat.mod(Nat.add(Nat.mod(a2, n), b), n), Nat.mod(Nat.add(a2, b), n), Nat.mod(Nat.add(a2, b2), n), e3,
        Equal.trans(Nat, Nat.mod(Nat.add(a2, b), n), Nat.mod(Nat.add(a2, Nat.mod(b, n)), n), Nat.mod(Nat.add(a2, b2), n), e4,
          Equal.trans(Nat, Nat.mod(Nat.add(a2, Nat.mod(b, n)), n), Nat.mod(Nat.add(a2, Nat.mod(b2, n)), n), Nat.mod(Nat.add(a2, b2), n), e5, e6)))))

Uses (9):
  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.mod
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/cong.bend&name=ceq
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend&name=mod_add_l
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend&name=mod_add_r