def lo_w

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/f64sqv.bend:190

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/w64.bend as X
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

190def lo_w(+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}) -> {v(X.lo(X.sub(nh, X.mul32(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh)))))) == Nat.sub(SW.value(nh), Nat.mul(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))) : Nat}:
191  +hr = N.le_lt_trans(Nat.sub(SW.value(nh), Nat.mul(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))), Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh))), C.shift(32n, one), r_le(one, h1, nh, h62, h60), d_lt(one, h1, nh, h62, h60))
192  +f = WW.fits_one(32n, one, h1, SW.value(X.sub(nh, X.mul32(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh))))), L.subst(Nat, z => {Nat.is_lt(z, C.shift(32n, one)) == True{} : Bool}, Nat.sub(SW.value(nh), Nat.mul(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))), SW.value(X.sub(nh, X.mul32(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh))))), Equal.sym(Nat, SW.value(X.sub(nh, X.mul32(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh))))), Nat.sub(SW.value(nh), Nat.mul(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))), w_v(one, h1, nh, h62, h60)), hr))
193  Equal.trans(Nat, v(X.lo(X.sub(nh, X.mul32(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh)))))), SW.value(X.sub(nh, X.mul32(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh))))), Nat.sub(SW.value(nh), Nat.mul(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))), lo_val(X.sub(nh, X.mul32(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh)))), f), w_v(one, h1, nh, h62, h60))