def cong_add_r

bend-collections-laws-crypto@1.0.0.0 · proofs/math/natural/inverse.bend:31

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

31def cong_add_r(+mp: Nat, +x: Nat, +y: Nat, +y2: Nat, +e: {Nat.mod(y, 1n+mp) == Nat.mod(y2, 1n+mp) : Nat}) -> {Nat.mod(Nat.add(x, y), 1n+mp) == Nat.mod(Nat.add(x, y2), 1n+mp) : Nat}:
32  Equal.trans(Nat, Nat.mod(Nat.add(x, y), 1n+mp), Nat.mod(Nat.add(x, Nat.mod(y, 1n+mp)), 1n+mp), Nat.mod(Nat.add(x, y2), 1n+mp), Equal.sym(Nat, Nat.mod(Nat.add(x, Nat.mod(y, 1n+mp)), 1n+mp), Nat.mod(Nat.add(x, y), 1n+mp), R.mod_add_r(mp, x, y)),
33    Equal.trans(Nat, Nat.mod(Nat.add(x, Nat.mod(y, 1n+mp)), 1n+mp), Nat.mod(Nat.add(x, Nat.mod(y2, 1n+mp)), 1n+mp), Nat.mod(Nat.add(x, y2), 1n+mp), Equal.cong(Nat, Nat, z => Nat.mod(Nat.add(x, z), 1n+mp), Nat.mod(y, 1n+mp), Nat.mod(y2, 1n+mp), e), R.mod_add_r(mp, x, y2)))