def red_hx

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/w64mm.bend:52

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/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64dm.bend as DM
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

52# x0 + 2^32 xh < 2^32 m
53def red_hx(+one: Nat, +h1: {one == 1n : Nat}, +x0: U32, +y0: U32, +y1: U32, +ml: U32, +mh: U32, +hx: {Nat.is_lt(SW.value(WU.U64{y0, y1}), SW.value(WU.U64{ml, mh})) == True{} : Bool}) -> {Nat.is_lt(DM.X96(WU.U64{x0, y0}, y1), Nat.mul(C.shift(32n, one), SW.value(WU.U64{ml, mh}))) == True{} : Bool}:
54  +PP = C.shift(32n, one)
55  +ex = x96_eq(x0, y0, y1)
56  +h0 = N.lt_add_r2(v(x0), PP, C.shift(32n, SW.value(WU.U64{y0, y1})), WW.lt_one(32n, one, h1, v(x0), LW.vb(x0)))
57  +e1 = Equal.sym(Nat, C.shift(32n, Nat.add(one, SW.value(WU.U64{y0, y1}))), Nat.add(PP, C.shift(32n, SW.value(WU.U64{y0, y1}))), WW.shift_add(32n, one, SW.value(WU.U64{y0, y1})))
58  +h2 = WW.shift_mono(32n, Nat.add(one, SW.value(WU.U64{y0, y1})), SW.value(WU.U64{ml, mh}), L.subst(Nat, z => {Nat.is_le(Nat.add(z, SW.value(WU.U64{y0, y1})), SW.value(WU.U64{ml, mh})) == True{} : Bool}, 1n, one, Equal.sym(Nat, one, 1n, h1), N.lt_succ_le_succ(SW.value(WU.U64{y0, y1}), SW.value(WU.U64{ml, mh}), hx)))
59  +e2 = Equal.trans(Nat, C.shift(32n, SW.value(WU.U64{ml, mh})), Nat.mul(SW.value(WU.U64{ml, mh}), PP), Nat.mul(PP, SW.value(WU.U64{ml, mh})), WW.shift_mul_one(32n, one, h1, SW.value(WU.U64{ml, mh})), NA.mul_comm(SW.value(WU.U64{ml, mh}), PP))
60  +hY = N.lt_le_trans(Nat.add(v(x0), C.shift(32n, SW.value(WU.U64{y0, y1}))), C.shift(32n, Nat.add(one, SW.value(WU.U64{y0, y1}))), Nat.mul(PP, SW.value(WU.U64{ml, mh})), L.subst(Nat, z => {Nat.is_lt(Nat.add(v(x0), C.shift(32n, SW.value(WU.U64{y0, y1}))), z) == True{} : Bool}, Nat.add(PP, C.shift(32n, SW.value(WU.U64{y0, y1}))), C.shift(32n, Nat.add(one, SW.value(WU.U64{y0, y1}))), e1, h0), L.subst(Nat, z => {Nat.is_le(C.shift(32n, Nat.add(one, SW.value(WU.U64{y0, y1}))), z) == True{} : Bool}, C.shift(32n, SW.value(WU.U64{ml, mh})), Nat.mul(PP, SW.value(WU.U64{ml, mh})), e2, h2))
61  L.subst(Nat, z => {Nat.is_lt(z, Nat.mul(PP, SW.value(WU.U64{ml, mh}))) == True{} : Bool}, Nat.add(v(x0), C.shift(32n, SW.value(WU.U64{y0, y1}))), DM.X96(WU.U64{x0, y0}, y1), Equal.sym(Nat, DM.X96(WU.U64{x0, y0}, y1), Nat.add(v(x0), C.shift(32n, SW.value(WU.U64{y0, y1}))), ex), hY)