def mul_le

bend-collections-laws-crypto@1.0.0.0 · proofs/lib/arith.bend:72

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
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA

72def mul_le(+a: Nat, +b: Nat, +c: Nat, +h: {Nat.is_le(a, b) == True{} : Bool}) -> {Nat.is_le(Nat.mul(a, c), Nat.mul(b, c)) == True{} : Bool}:
73  +e = Nat.sub(b, a)
74  +hb = N.sub_add(b, a, h)
75  +m = Equal.trans(Nat, Nat.add(Nat.mul(a, c), Nat.mul(e, c)), Nat.mul(Nat.add(a, e), c), Nat.mul(b, c), Equal.sym(Nat, Nat.mul(Nat.add(a, e), c), Nat.add(Nat.mul(a, c), Nat.mul(e, c)), NA.mul_add_right(a, e, c)), Equal.cong(Nat, Nat, z => Nat.mul(z, c), Nat.add(a, e), b, hb))
76  L.subst(Nat, z => {Nat.is_le(Nat.mul(a, c), z) == True{} : Bool}, Nat.add(Nat.mul(a, c), Nat.mul(e, c)), Nat.mul(b, c), m, N.le_add_right(Nat.mul(a, c), Nat.mul(e, c)))