def fits_add1

bend-collections-laws-math@1.0.0.0 · proofs/math/random/fround.bend:57

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/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend as WW

57def fits_add1(+k: Nat, +a: Nat, +b: Nat, +ha: {C.fits(k, a) == True{} : Bool}, +hb: {C.fits(k, b) == True{} : Bool}) -> {C.fits(1n+k, Nat.add(a, b)) == True{} : Bool}:
58  +P = C.pow2(k)
59  +h1 = N.lt_add_r2(a, P, b, WW.lt_of_fits(k, a, ha))
60  +h2 = N.lt_add_left(b, P, P, WW.lt_of_fits(k, b, hb))
61  +h3 = N.lt_trans(Nat.add(a, b), Nat.add(P, b), Nat.add(P, P), h1, h2)
62  +h4 = L.subst(Nat, z => {Nat.is_lt(Nat.add(a, b), z) == True{} : Bool}, Nat.add(P, P), Nat.double(P), Equal.sym(Nat, Nat.double(P), Nat.add(P, P), NA.double_self(P)), h3)
63  WW.fits_of_lt(1n+k, Nat.add(a, b), h4)