def one_lt_sc

bend-collections-laws-crypto@1.0.0.0 · proofs/math/typed/w64mul.bend:232

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/word.bend as WD

232def one_lt_sc(+k: Nat, +one: Nat, +h1: {one == 1n : Nat}) -> {Nat.is_lt(1n, WD.sc(1n+k, one)) == True{} : Bool}:
233  +p = L.subst(Nat, o => {Nat.is_lt(0n, WD.sc(k, o)) == True{} : Bool}, 1n, one, Equal.sym(Nat, one, 1n, h1), WD.sc_pos(k))
234  N.lt_le_trans(1n, 1n+WD.sc(k, one), Nat.double(WD.sc(k, one)), N.lt_add_left(0n, WD.sc(k, one), 1n, p), N.double_succ_le(WD.sc(k, one), N.lt_succ_le_succ(0n, WD.sc(k, one), p)))