Natfuel.dstep: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/natfuel.bend as Natfuel
Lines 401-403 of bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/natfuel.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/natfuel.bend&start_line=401&end_line=403

import Base
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
import bend-collections-laws-crypto@1.0.0.0/spec/math/natural.bend as S
import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/fact.bend as FA
import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/roots.bend as RT

def dstep(+n: Nat, +s: Nat, +h: {Nat.is_lt(s, n) == True{} : Bool}) -> {Nat.is_le(S.desc(n, s), S.desc(n, 1n+s)) == True{} : Bool}:
  +pos = L.subst(Nat, z => {Nat.is_lt(0n, z) == True{} : Bool}, 1n+Nat.sub(n, 1n+s), Nat.sub(n, s), Equal.sym(Nat, Nat.sub(n, s), 1n+Nat.sub(n, 1n+s), FA.sub_lt_succ(n, s, h)), {==})
  L.subst(Nat, z => {Nat.is_le(S.desc(n, s), z) == True{} : Bool}, Nat.mul(S.desc(n, s), Nat.sub(n, s)), Nat.mul(Nat.sub(n, s), S.desc(n, s)), NA.mul_comm(S.desc(n, s), Nat.sub(n, s)), RT.le_mul_pos(S.desc(n, s), Nat.sub(n, s), pos))

Uses (12):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  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.is_lt
  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_comm
  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/math/natural/fact.bend&name=sub_lt_succ
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/roots.bend&name=le_mul_pos
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/math/natural.bend&name=desc