def add_le

bend-collections-laws-crypto@1.0.0.0 · proofs/math/hash/hash.bend:25

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/logic.bend as L

25def add_le(+a: Nat, +b: Nat, +c: Nat, +d: Nat, +h1: {Nat.is_le(a, b) == True{} : Bool}, +h2: {Nat.is_le(c, d) == True{} : Bool}) -> {Nat.is_le(Nat.add(a, c), Nat.add(b, d)) == True{} : Bool}:
26  +l1 = L.subst(Nat, z => {Nat.is_le(z, Nat.add(c, b)) == True{} : Bool}, Nat.add(c, a), Nat.add(a, c), N.add_comm(c, a), N.le_add_left(a, b, c, h1))
27  +l2 = L.subst(Nat, z => {Nat.is_le(Nat.add(a, c), z) == True{} : Bool}, Nat.add(c, b), Nat.add(b, c), N.add_comm(c, b), l1)
28  N.le_trans(Nat.add(a, c), Nat.add(b, c), Nat.add(b, d), l2, N.le_add_left(c, d, b, h2))