def s0_2

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/f64sqr.bend:221

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/math/typed/f64sqa.bend as AQ
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64sqv.bend as QV

221# 2 <= S0 (S0 >= isqrt(N) >= 2^62)
222def s0_2(+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}, +q: U32, +u: U32, +hc: {Nat.add(Nat.mul(v(q), Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))), v(u)) == C.shift(32n, Nat.sub(SW.value(nh), Nat.mul(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh))))) : Nat}, +hU: {Nat.is_le(v(u), Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))) == True{} : Bool}, +hQ: {Nat.is_lt(v(q), C.shift(32n, one)) == True{} : Bool}) -> {Nat.is_le(2n, Nat.add(v(q), C.shift(32n, M.isqrt(SW.value(nh))))) == True{} : Bool}:
223  +hT = N.lt_succ_le(M.isqrt(C.shift(64n, SW.value(nh))), Nat.add(v(q), C.shift(32n, M.isqrt(SW.value(nh)))), AQ.isqrt_lt(C.shift(64n, SW.value(nh)), 1n+Nat.add(v(q), C.shift(32n, M.isqrt(SW.value(nh)))), lo_bound(M.isqrt(SW.value(nh)), Nat.sub(SW.value(nh), Nat.mul(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))), SW.value(nh), v(q), v(u), Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh))), QV.hn_v(one, h1, nh, h62, h60), {==}, hc, hU)))
224  N.le_trans(2n, C.shift(62n, one), Nat.add(v(q), C.shift(32n, M.isqrt(SW.value(nh)))), two62(one, h1), N.le_trans(C.shift(62n, one), M.isqrt(C.shift(64n, SW.value(nh))), Nat.add(v(q), C.shift(32n, M.isqrt(SW.value(nh)))), QV.s62(one, h1, nh, h62, h60), hT))