def mle2

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/natlight.bend:26

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend as L
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/natfuel.bend as NF

26def mle2(+a: Nat, +b: Nat, +c: Nat, +d: Nat, +hab: {Nat.is_le(a, b) == True{} : Bool}, +hcd: {Nat.is_le(c, d) == True{} : Bool}) -> {Nat.is_le(Nat.mul(a, c), Nat.mul(b, d)) == True{} : Bool}:
27  +l1 = NF.mul_le_r(a, c, d, hcd)
28  +l2 = L.subst(Nat, z => {Nat.is_le(z, Nat.mul(d, b)) == True{} : Bool}, Nat.mul(d, a), Nat.mul(a, d), NA.mul_comm(d, a), NF.mul_le_r(d, a, b, hab))
29  N.le_trans(Nat.mul(a, c), Nat.mul(a, d), Nat.mul(b, d), l1, L.subst(Nat, z => {Nat.is_le(Nat.mul(a, d), z) == True{} : Bool}, Nat.mul(d, b), Nat.mul(b, d), NA.mul_comm(d, b), l2))