Arith.mul_le: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/lib/arith.bend as Arith
Lines 72-76 of bend-collections-laws-crypto@1.0.0.0/proofs/lib/arith.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/arith.bend&start_line=72&end_line=76

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

def 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}:
  +e = Nat.sub(b, a)
  +hb = N.sub_add(b, a, h)
  +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))
  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)))

Uses (13):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  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.is_le
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.sub
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_add_right
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend&name=subst
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=le_add_right
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=sub_add