def s_hi

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/f64sqv.bend:95

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/logic.bend as L
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64sqa.bend as AQ

95def s_hi(+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}, +r0w: WU.U64, +hR0: {SW.value(r0w) == C.shift(32n, Nat.add(one, M.isqrt(SW.value(nh)))) : Nat}, +hhi: {U32.is_zero(X.hi(r0w)) == False{} : Bool}) -> {Nat.is_lt(M.isqrt(C.shift(64n, SW.value(nh))), SW.value(r0w)) == True{} : Bool}:
96  +l1 = AQ.lt_sh2(32n, SW.value(nh), Nat.mul(Nat.add(one, M.isqrt(SW.value(nh))), Nat.add(one, M.isqrt(SW.value(nh)))), ilt1(one, h1, nh, h62, h60))
97  +e1 = Equal.trans(Nat, Nat.mul(SW.value(r0w), SW.value(r0w)), Nat.mul(C.shift(32n, Nat.add(one, M.isqrt(SW.value(nh)))), C.shift(32n, Nat.add(one, M.isqrt(SW.value(nh))))), C.shift(32n, C.shift(32n, Nat.mul(Nat.add(one, M.isqrt(SW.value(nh))), Nat.add(one, M.isqrt(SW.value(nh)))))), Equal.trans(Nat, Nat.mul(SW.value(r0w), SW.value(r0w)), Nat.mul(C.shift(32n, Nat.add(one, M.isqrt(SW.value(nh)))), SW.value(r0w)), Nat.mul(C.shift(32n, Nat.add(one, M.isqrt(SW.value(nh)))), C.shift(32n, Nat.add(one, M.isqrt(SW.value(nh))))), Equal.cong(Nat, Nat, z => Nat.mul(z, SW.value(r0w)), SW.value(r0w), C.shift(32n, Nat.add(one, M.isqrt(SW.value(nh)))), hR0), Equal.cong(Nat, Nat, z => Nat.mul(C.shift(32n, Nat.add(one, M.isqrt(SW.value(nh)))), z), SW.value(r0w), C.shift(32n, Nat.add(one, M.isqrt(SW.value(nh)))), hR0)), AQ.sh_sq(32n, Nat.add(one, M.isqrt(SW.value(nh)))))
98  AQ.isqrt_lt(C.shift(64n, SW.value(nh)), SW.value(r0w), L.subst(Nat, z => {Nat.is_lt(z, Nat.mul(SW.value(r0w), SW.value(r0w))) == True{} : Bool}, C.shift(32n, C.shift(32n, SW.value(nh))), C.shift(64n, SW.value(nh)), Equal.sym(Nat, C.shift(64n, SW.value(nh)), C.shift(32n, C.shift(32n, SW.value(nh))), nn_eq(nh)), L.subst(Nat, z => {Nat.is_lt(C.shift(32n, C.shift(32n, SW.value(nh))), z) == True{} : Bool}, C.shift(32n, C.shift(32n, Nat.mul(Nat.add(one, M.isqrt(SW.value(nh))), Nat.add(one, M.isqrt(SW.value(nh)))))), Nat.mul(SW.value(r0w), SW.value(r0w)), Equal.sym(Nat, Nat.mul(SW.value(r0w), SW.value(r0w)), C.shift(32n, C.shift(32n, Nat.mul(Nat.add(one, M.isqrt(SW.value(nh))), Nat.add(one, M.isqrt(SW.value(nh)))))), e1), l1)))