def dig_q

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/f64divd.bend:26

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/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/width.bend as WW
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/u32laws.bend as LW
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/w64mm.bend as MM

26# the quotient digit
27def dig_q(+one: Nat, +h1: {one == 1n : Nat}, +rl: U32, +rh: U32, +ml: U32, +mh: U32, +e: U32, +hx: {Nat.is_lt(SW.value(WU.U64{rl, rh}), SW.value(WU.U64{ml, mh})) == True{} : Bool}, +hz: {U32.is_zero(mh) == False{} : Bool}, +he: {Nat.is_lt(DM.X96(WU.U64{0, rl}, rh), Nat.mul(1n+v(e), SW.value(WU.U64{ml, mh}))) == True{} : Bool}) -> {v(X.q_start(WU.U64{0, rl}, rh, WU.U64{ml, mh}, e)) == Nat.div(C.shift(32n, SW.value(WU.U64{rl, rh})), SW.value(WU.U64{ml, mh})) : Nat}:
28  +PP = C.shift(32n, one)
29  +ex = MM.x96_eq(0, rl, rh)
30  +h0 = N.lt_add_r2(v(0), PP, C.shift(32n, SW.value(WU.U64{rl, rh})), WW.lt_one(32n, one, h1, v(0), LW.vb(0)))
31  +e1 = Equal.sym(Nat, C.shift(32n, Nat.add(one, SW.value(WU.U64{rl, rh}))), Nat.add(PP, C.shift(32n, SW.value(WU.U64{rl, rh}))), WW.shift_add(32n, one, SW.value(WU.U64{rl, rh})))
32  +h2 = WW.shift_mono(32n, Nat.add(one, SW.value(WU.U64{rl, rh})), SW.value(WU.U64{ml, mh}), L.subst(Nat, z => {Nat.is_le(Nat.add(z, SW.value(WU.U64{rl, rh})), 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{rl, rh}), SW.value(WU.U64{ml, mh}), hx)))
33  +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))
34  +hY = N.lt_le_trans(Nat.add(v(0), C.shift(32n, SW.value(WU.U64{rl, rh}))), C.shift(32n, Nat.add(one, SW.value(WU.U64{rl, rh}))), Nat.mul(PP, SW.value(WU.U64{ml, mh})), L.subst(Nat, z => {Nat.is_lt(Nat.add(v(0), C.shift(32n, SW.value(WU.U64{rl, rh}))), z) == True{} : Bool}, Nat.add(PP, C.shift(32n, SW.value(WU.U64{rl, rh}))), C.shift(32n, Nat.add(one, SW.value(WU.U64{rl, rh}))), e1, h0), L.subst(Nat, z => {Nat.is_le(C.shift(32n, Nat.add(one, SW.value(WU.U64{rl, rh}))), z) == True{} : Bool}, C.shift(32n, SW.value(WU.U64{ml, mh})), Nat.mul(PP, SW.value(WU.U64{ml, mh})), e2, h2))
35  +hX = L.subst(Nat, z => {Nat.is_lt(z, Nat.mul(PP, SW.value(WU.U64{ml, mh}))) == True{} : Bool}, Nat.add(v(0), C.shift(32n, SW.value(WU.U64{rl, rh}))), DM.X96(WU.U64{0, rl}, rh), Equal.sym(Nat, DM.X96(WU.U64{0, rl}, rh), Nat.add(v(0), C.shift(32n, SW.value(WU.U64{rl, rh}))), ex), hY)
36  +bp = Nat.sub(SW.value(WU.U64{ml, mh}), 1n)
37  +hB = Equal.sym(Nat, 1n+bp, SW.value(WU.U64{ml, mh}), N.sub_add(SW.value(WU.U64{ml, mh}), 1n, N.le_trans(1n, 1n+SW.value(WU.U64{rl, rh}), SW.value(WU.U64{ml, mh}), N.zero_le(SW.value(WU.U64{rl, rh})), N.lt_succ_le_succ(SW.value(WU.U64{rl, rh}), SW.value(WU.U64{ml, mh}), hx))))
38  pr = DM.qs_pair(one, h1, WU.U64{0, rl}, rh, WU.U64{ml, mh}, e, 4294967295, {==}, hX, he)
39  pq = DM.qs_fin2(DM.X96(WU.U64{0, rl}, rh), v(DM.QS(WU.U64{0, rl}, rh, WU.U64{ml, mh}, e, 4294967295)), SW.value(WU.U64{ml, mh}), bp, hB, pr)
40  +dq = DM.pr1({Nat.div(DM.X96(WU.U64{0, rl}, rh), 1n+bp) == v(DM.QS(WU.U64{0, rl}, rh, WU.U64{ml, mh}, e, 4294967295)) : Nat}, {Nat.mod(DM.X96(WU.U64{0, rl}, rh), 1n+bp) == Nat.sub(DM.X96(WU.U64{0, rl}, rh), Nat.mul(v(DM.QS(WU.U64{0, rl}, rh, WU.U64{ml, mh}, e, 4294967295)), 1n+bp)) : Nat}, pq)
41  +eq = Equal.cong(U32, Nat, w => v(w), X.q_start(WU.U64{0, rl}, rh, WU.U64{ml, mh}, e), DM.QS(WU.U64{0, rl}, rh, WU.U64{ml, mh}, e, 4294967295), DM.impl_qs(WU.U64{0, rl}, rh, WU.U64{ml, mh}, e))
42  +ed = Equal.trans(Nat, Nat.div(DM.X96(WU.U64{0, rl}, rh), 1n+bp), Nat.div(C.shift(32n, SW.value(WU.U64{rl, rh})), 1n+bp), Nat.div(C.shift(32n, SW.value(WU.U64{rl, rh})), SW.value(WU.U64{ml, mh})), Equal.cong(Nat, Nat, w => Nat.div(w, 1n+bp), DM.X96(WU.U64{0, rl}, rh), C.shift(32n, SW.value(WU.U64{rl, rh})), ex), Equal.cong(Nat, Nat, w => Nat.div(C.shift(32n, SW.value(WU.U64{rl, rh})), w), 1n+bp, SW.value(WU.U64{ml, mh}), Equal.sym(Nat, SW.value(WU.U64{ml, mh}), 1n+bp, hB)))
43  Equal.trans(Nat, v(X.q_start(WU.U64{0, rl}, rh, WU.U64{ml, mh}, e)), v(DM.QS(WU.U64{0, rl}, rh, WU.U64{ml, mh}, e, 4294967295)), Nat.div(C.shift(32n, SW.value(WU.U64{rl, rh})), SW.value(WU.U64{ml, mh})), eq, Equal.trans(Nat, v(DM.QS(WU.U64{0, rl}, rh, WU.U64{ml, mh}, e, 4294967295)), Nat.div(DM.X96(WU.U64{0, rl}, rh), 1n+bp), Nat.div(C.shift(32n, SW.value(WU.U64{rl, rh})), SW.value(WU.U64{ml, mh})), Equal.sym(Nat, Nat.div(DM.X96(WU.U64{0, rl}, rh), 1n+bp), v(DM.QS(WU.U64{0, rl}, rh, WU.U64{ml, mh}, e, 4294967295)), dq), ed))