def d_pos

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/f64sqv.bend:179

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/spec/math/w64.bend as SW
import bend-collections-laws-math@1.0.0.0/src/math/u64.bend as WU
import bend-collections-laws-math@1.0.0.0/src/math/natural.bend as M
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/math/typed/width.bend as WW
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64sqa.bend as AQ

179# 2 s > 0, as nh >= 2^60
180def d_pos(+one: Nat, +h1: {one == 1n : Nat}, +nh: WU.U64, +h62: {C.fits(62n, SW.value(nh)) == True{} : Bool}, +h60: {C.fits(60n, SW.value(nh)) == False{} : Bool}) -> {Nat.is_lt(0n, Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))) == True{} : Bool}:
181  +h1n = N.le_trans(1n, C.shift(60n, one), SW.value(nh), L.subst(Nat, o => {Nat.is_le(o, C.shift(60n, one)) == True{} : Bool}, one, 1n, h1, WW.shift_ge(60n, one)), vn60(one, h1, nh, h62, h60))
182  +hs = AQ.isqrt_ge(SW.value(nh), 1n, h1n)
183  N.lt_le_trans(0n, 1n, Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh))), {==}, N.le_trans(1n, M.isqrt(SW.value(nh)), Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh))), hs, N.le_add_right(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))))