def dm_hx

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/w64dm.bend:197

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/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/w64mul.bend as W64M
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64add.bend as WA
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/u32laws.bend as LW

197# a < 2^32 b for b >= 2^32
198def dm_hx(+one: Nat, +h1: {one == 1n : Nat}, +al: U32, +ah: U32, +bl: U32, +bh: U32, +hz: {U32.is_zero(bh) == False{} : Bool}) -> {Nat.is_lt(X96(WU.U64{al, ah}, 0), Nat.mul(C.shift(32n, one), SW.value(WU.U64{bl, bh}))) == True{} : Bool}:
199  +PP = C.shift(32n, one)
200  +vbv = SW.value(WU.U64{bl, bh})
201  +hbh = L.subst(Nat, z => {Nat.is_le(z, v(bh)) == True{} : Bool}, 1n, one, Equal.sym(Nat, one, 1n, h1), WA.pos_ne(v(bh), Equal.trans(Bool, Nat.is_eq(v(bh), 0n), U32.is_zero(bh), False{}, Equal.sym(Bool, U32.is_zero(bh), Nat.is_eq(v(bh), 0n), LW.zero_nat(bh)), hz)))
202  +hPb = N.le_trans(PP, C.shift(32n, v(bh)), vbv, WW.shift_mono(32n, one, v(bh), hbh), L.subst(Nat, z => {Nat.is_le(C.shift(32n, v(bh)), z) == True{} : Bool}, Nat.add(C.shift(32n, v(bh)), v(bl)), vbv, N.add_comm(C.shift(32n, v(bh)), v(bl)), N.le_add_right(C.shift(32n, v(bh)), v(bl))))
203  +eS = Equal.trans(Nat, C.shift(64n, one), C.shift(32n, PP), Nat.mul(PP, PP), WW.shift_comp(32n, 32n, one), WW.shift_mul_one(32n, one, h1, PP))
204  +ha = L.subst(Nat, z => {Nat.is_lt(SW.value(WU.U64{al, ah}), z) == True{} : Bool}, C.shift(64n, one), Nat.mul(PP, PP), eS, WW.lt_one(64n, one, h1, SW.value(WU.U64{al, ah}), fit64(WU.U64{al, ah})))
205  +hX0 = N.lt_le_trans(SW.value(WU.U64{al, ah}), Nat.mul(PP, PP), Nat.mul(PP, vbv), ha, W64M.le_mul_r(PP, PP, vbv, hPb))
206  L.subst(Nat, z => {Nat.is_lt(z, Nat.mul(PP, vbv)) == True{} : Bool}, SW.value(WU.U64{al, ah}), X96(WU.U64{al, ah}, 0), Equal.sym(Nat, X96(WU.U64{al, ah}, 0), SW.value(WU.U64{al, ah}), x96_0(WU.U64{al, ah})), hX0)