def d_lt

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/f64sqv.bend:167

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/lib/lemmas/proofs/nat_algebra.bend as NA

167# 2 s < 2^32
168def d_lt(+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(Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh))), C.shift(32n, one)) == True{} : Bool}:
169  +h = iv31(one, h1, nh, h62, h60)
170  +S31 = C.shift(31n, one)
171  +l2 = N.lt_le_trans(Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh))), Nat.add(S31, M.isqrt(SW.value(nh))), Nat.add(S31, S31), N.lt_add_r2(M.isqrt(SW.value(nh)), S31, M.isqrt(SW.value(nh)), h), N.le_add_left(M.isqrt(SW.value(nh)), S31, S31, N.lt_le(M.isqrt(SW.value(nh)), S31, h)))
172  L.subst(Nat, z => {Nat.is_lt(Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh))), z) == True{} : Bool}, Nat.add(C.shift(31n, one), C.shift(31n, one)), C.shift(32n, one), Equal.sym(Nat, C.shift(32n, one), Nat.add(C.shift(31n, one), C.shift(31n, one)), NA.double_self(C.shift(31n, one))), l2)