def dbig

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/natfuel.bend:420

import Base
import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as C
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/spec/math/natural.bend as S
import bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend as FA

420# 1 + k <= j <= n gives 2^k <= n.desc(j)
421def dbig(+k: Nat, +n: Nat, +j: Nat, +hj: {Nat.is_le(1n+k, j) == True{} : Bool}, +hjn: {Nat.is_le(j, n) == True{} : Bool}) -> {Nat.is_le(C.pow2(k), S.desc(n, j)) == True{} : Bool}:
422  +h1 = L.subst(Nat, z => {Nat.is_le(z, S.desc(n, j)) == True{} : Bool}, S.desc(j, j), S.factorial(j), FA.desc_self(j), dmono1(j, n, j, hjn))
423  N.le_trans(C.pow2(k), S.factorial(j), S.desc(n, j), fbig(k, j, hj), h1)