Poly.sc_mul_l: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/poly.bend as Poly
Lines 183-187 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/poly.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/poly.bend&start_line=183&end_line=187

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend as WD
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/arith.bend as AR
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/num.bend as M

def sc_mul_l(+k: Nat, +a: Nat, +x: Nat) -> {WD.sc(k, Nat.mul(a, x)) == Nat.mul(a, WD.sc(k, x)) : Nat}:
  +e1 = M.cong_sc(k, Nat.mul(a, x), Nat.mul(x, a), NA.mul_comm(a, x))
  +e2 = AR.sc_mul(k, x, a)
  +e3 = NA.mul_comm(WD.sc(k, x), a)
  Equal.trans(Nat, WD.sc(k, Nat.mul(a, x)), WD.sc(k, Nat.mul(x, a)), Nat.mul(a, WD.sc(k, x)), e1, Equal.trans(Nat, WD.sc(k, Nat.mul(x, a)), Nat.mul(WD.sc(k, x), a), Nat.mul(a, WD.sc(k, x)), e2, e3))

Uses (7):
  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.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/num.bend&name=cong_sc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/arith.bend&name=sc_mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_comm
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend&name=sc