def s0_63

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/f64sqr.bend:212

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/f64sqv.bend as QV

212# S0 < 2^63: q < 2^32 and s + 1 <= 2^31
213def s0_63(+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, +hQ: {Nat.is_lt(v(q), C.shift(32n, one)) == True{} : Bool}) -> {Nat.is_lt(Nat.add(v(q), C.shift(32n, M.isqrt(SW.value(nh)))), C.shift(63n, one)) == True{} : Bool}:
214  +l1 = N.lt_le_trans(Nat.add(v(q), C.shift(32n, M.isqrt(SW.value(nh)))), Nat.add(C.shift(32n, one), C.shift(32n, M.isqrt(SW.value(nh)))), Nat.add(C.shift(32n, one), C.shift(32n, M.isqrt(SW.value(nh)))), N.lt_add_r2(v(q), C.shift(32n, one), C.shift(32n, M.isqrt(SW.value(nh))), hQ), N.le_refl(Nat.add(C.shift(32n, one), C.shift(32n, M.isqrt(SW.value(nh))))))
215  +e1 = Equal.sym(Nat, C.shift(32n, Nat.add(one, M.isqrt(SW.value(nh)))), Nat.add(C.shift(32n, one), C.shift(32n, M.isqrt(SW.value(nh)))), WW.shift_add(32n, one, M.isqrt(SW.value(nh))))
216  +l0 = L.subst(Nat, o => {Nat.is_le(Nat.add(o, M.isqrt(SW.value(nh))), C.shift(31n, one)) == True{} : Bool}, 1n, one, Equal.sym(Nat, one, 1n, h1), N.lt_succ_le_succ(M.isqrt(SW.value(nh)), C.shift(31n, one), QV.iv31(one, h1, nh, h62, h60)))
217  +l2 = WW.shift_mono(32n, Nat.add(one, M.isqrt(SW.value(nh))), C.shift(31n, one), l0)
218  +l3 = L.subst(Nat, z => {Nat.is_le(C.shift(32n, Nat.add(one, M.isqrt(SW.value(nh)))), z) == True{} : Bool}, C.shift(32n, C.shift(31n, one)), C.shift(63n, one), Equal.sym(Nat, C.shift(63n, one), C.shift(32n, C.shift(31n, one)), WW.shift_comp(32n, 31n, one)), l2)
219  N.lt_le_trans(Nat.add(v(q), C.shift(32n, M.isqrt(SW.value(nh)))), Nat.add(C.shift(32n, one), C.shift(32n, M.isqrt(SW.value(nh)))), C.shift(63n, one), l1, L.subst(Nat, z => {Nat.is_le(z, C.shift(63n, one)) == True{} : Bool}, C.shift(32n, Nat.add(one, M.isqrt(SW.value(nh)))), Nat.add(C.shift(32n, one), C.shift(32n, M.isqrt(SW.value(nh)))), Equal.sym(Nat, Nat.add(C.shift(32n, one), C.shift(32n, M.isqrt(SW.value(nh)))), C.shift(32n, Nat.add(one, M.isqrt(SW.value(nh)))), e1), l3))