proofs/crypto/secp256k1/point.bend source
proofs/crypto/secp256k1/point.bend on the hub · documented module
# GENERATED by tools/generators/rw.py from ../../../tools/generators/secp256k1_hand/point.src; edit the .srcimport Baseimport ../../../spec/crypto/secp256k1/field.bend as FSimport ../../../spec/crypto/secp256k1/curve.bend as CVimport ../../../src/crypto/secp256k1/field.bend as Fimport ../../../src/crypto/secp256k1/point.bend as Pimport ../../lib/logic.bend as Lgimport ./fieldops.bend as FO# The group operations of src/crypto/secp256k1/point.bend against# spec/crypto/secp256k1/curve.bend: the register programs (RCB 2016# Algorithms 7 and 9) run on field elements compute, register by register,# the values the specification's programs compute on the natural numbers.def pr(+one: Nat) -> Nat: FS.prime(one)def red(+one: Nat, +x: List<&2, Nat>) -> Bool: FS.reduced(one, FS.prime(one), x)def vals(+one: Nat, rs: List<&2, List<&2, Nat>>) -> List<&2, Nat>: match rs: case Nil{}: Nil{} case r <> t: FS.value(one, r) <> vals(one, t)def allred(+one: Nat, rs: List<&2, List<&2, Nat>>) -> Bool: match rs: case Nil{}: True{} case r <> t: Bool.and(red(one, r), allred(one, t))def cop(op: P.Op) -> CV.SOp: match op: case P.OAdd{i, j}: CV.SAdd{i, j} case P.OSub{i, j}: CV.SSub{i, j} case P.OMul{i, j}: CV.SMul{i, j}def cprog(ops: List<&2, P.Op>) -> List<&2, CV.SOp>: match ops: case Nil{}: Nil{} case o <> t: cop(o) <> cprog(t)# ---- registers ----def zero_r(+one: Nat, +h1: {one == 1n : Nat}) -> {red(one, F.zero()) == True{} : Bool}: FO.small_r(one, h1, 0n, FO.small_lt_p(one, h1, 0n, {==}))def zero_v(+one: Nat, +h1: {one == 1n : Nat}) -> {FS.value(one, F.zero()) == 0n : Nat}: FO.small_val(one, h1, 0n, FO.small_lt_p(one, h1, 0n, {==}))def nth_r(+one: Nat, +h1: {one == 1n : Nat}, i: Nat, rs: List<&2, List<&2, Nat>>, +h: {allred(one, rs) == True{} : Bool}) -> {red(one, P.nth(i, rs)) == True{} : Bool}: match i rs: case 0n Nil{}: zero_r(one, h1) case 1n+k Nil{}: zero_r(one, h1) case 0n +r <> +t: Lg.and_left(red(one, r), allred(one, t), h) case 1n+ +k +r <> +t: nth_r(one, h1, k, t, Lg.and_right(red(one, r), allred(one, t), h))def nth_v(+one: Nat, +h1: {one == 1n : Nat}, i: Nat, rs: List<&2, List<&2, Nat>>) -> {FS.value(one, P.nth(i, rs)) == CV.nth(i, vals(one, rs)) : Nat}: match i rs: case 0n Nil{}: zero_v(one, h1) case 1n+k Nil{}: zero_v(one, h1) case 0n r <> t: {==} case 1n+ +k r <> +t: nth_v(one, h1, k, t)def exec_r(+one: Nat, +h1: {one == 1n : Nat}, op: P.Op, +rs: List<&2, List<&2, Nat>>, +h: {allred(one, rs) == True{} : Bool}) -> {red(one, P.exec(op, rs)) == True{} : Bool}: match op: case P.OAdd{+i, +j}: FO.add_r(one, h1, P.nth(i, rs), P.nth(j, rs), nth_r(one, h1, i, rs, h), nth_r(one, h1, j, rs, h)) case P.OSub{+i, +j}: FO.sub_r(one, h1, P.nth(i, rs), P.nth(j, rs), nth_r(one, h1, i, rs, h), nth_r(one, h1, j, rs, h)) case P.OMul{+i, +j}: FO.mul_r(one, h1, P.nth(i, rs), P.nth(j, rs), nth_r(one, h1, i, rs, h), nth_r(one, h1, j, rs, h))def exec_v(+one: Nat, +h1: {one == 1n : Nat}, op: P.Op, +rs: List<&2, List<&2, Nat>>, +h: {allred(one, rs) == True{} : Bool}) -> {FS.value(one, P.exec(op, rs)) == CV.exec(FS.prime(one), cop(op), vals(one, rs)) : Nat}: match op: case P.OAdd{+i, +j}: %Equal.sym(Nat, FS.value(one, F.add(P.nth(i, rs), P.nth(j, rs))), FS.madd(FS.prime(one), FS.value(one, P.nth(i, rs)), FS.value(one, P.nth(j, rs))), FO.add_v(one, h1, P.nth(i, rs), P.nth(j, rs), nth_r(one, h1, i, rs, h), nth_r(one, h1, j, rs, h))) : {_ == FS.madd(FS.prime(one), CV.nth(i, vals(one, rs)), CV.nth(j, vals(one, rs))) : Nat} %Equal.sym(Nat, FS.value(one, P.nth(i, rs)), CV.nth(i, vals(one, rs)), nth_v(one, h1, i, rs)) : {FS.madd(FS.prime(one), _, FS.value(one, P.nth(j, rs))) == FS.madd(FS.prime(one), CV.nth(i, vals(one, rs)), CV.nth(j, vals(one, rs))) : Nat} %Equal.sym(Nat, FS.value(one, P.nth(j, rs)), CV.nth(j, vals(one, rs)), nth_v(one, h1, j, rs)) : {FS.madd(FS.prime(one), CV.nth(i, vals(one, rs)), _) == FS.madd(FS.prime(one), CV.nth(i, vals(one, rs)), CV.nth(j, vals(one, rs))) : Nat} {==} case P.OSub{+i, +j}: %Equal.sym(Nat, FS.value(one, F.sub(P.nth(i, rs), P.nth(j, rs))), FS.msub(FS.prime(one), FS.value(one, P.nth(i, rs)), FS.value(one, P.nth(j, rs))), FO.sub_v(one, h1, P.nth(i, rs), P.nth(j, rs), nth_r(one, h1, i, rs, h), nth_r(one, h1, j, rs, h))) : {_ == FS.msub(FS.prime(one), CV.nth(i, vals(one, rs)), CV.nth(j, vals(one, rs))) : Nat} %Equal.sym(Nat, FS.value(one, P.nth(i, rs)), CV.nth(i, vals(one, rs)), nth_v(one, h1, i, rs)) : {FS.msub(FS.prime(one), _, FS.value(one, P.nth(j, rs))) == FS.msub(FS.prime(one), CV.nth(i, vals(one, rs)), CV.nth(j, vals(one, rs))) : Nat} %Equal.sym(Nat, FS.value(one, P.nth(j, rs)), CV.nth(j, vals(one, rs)), nth_v(one, h1, j, rs)) : {FS.msub(FS.prime(one), CV.nth(i, vals(one, rs)), _) == FS.msub(FS.prime(one), CV.nth(i, vals(one, rs)), CV.nth(j, vals(one, rs))) : Nat} {==} case P.OMul{+i, +j}: %Equal.sym(Nat, FS.value(one, F.mul(P.nth(i, rs), P.nth(j, rs))), FS.mmul(FS.prime(one), FS.value(one, P.nth(i, rs)), FS.value(one, P.nth(j, rs))), FO.mul_v(one, h1, P.nth(i, rs), P.nth(j, rs), nth_r(one, h1, i, rs, h), nth_r(one, h1, j, rs, h))) : {_ == FS.mmul(FS.prime(one), CV.nth(i, vals(one, rs)), CV.nth(j, vals(one, rs))) : Nat} %Equal.sym(Nat, FS.value(one, P.nth(i, rs)), CV.nth(i, vals(one, rs)), nth_v(one, h1, i, rs)) : {FS.mmul(FS.prime(one), _, FS.value(one, P.nth(j, rs))) == FS.mmul(FS.prime(one), CV.nth(i, vals(one, rs)), CV.nth(j, vals(one, rs))) : Nat} %Equal.sym(Nat, FS.value(one, P.nth(j, rs)), CV.nth(j, vals(one, rs)), nth_v(one, h1, j, rs)) : {FS.mmul(FS.prime(one), CV.nth(i, vals(one, rs)), _) == FS.mmul(FS.prime(one), CV.nth(i, vals(one, rs)), CV.nth(j, vals(one, rs))) : Nat} {==}def vals_app(+one: Nat, rs: List<&2, List<&2, Nat>>, +x: List<&2, Nat>) -> {vals(one, List.append(&2, List<&2, Nat>, rs, [x])) == List.append(&2, Nat, vals(one, rs), [FS.value(one, x)]) : List<&2, Nat>}: match rs: case Nil{}: {==} case +r <> +t: Equal.cong(List<&2, Nat>, List<&2, Nat>, l => FS.value(one, r) <> l, vals(one, List.append(&2, List<&2, Nat>, t, [x])), List.append(&2, Nat, vals(one, t), [FS.value(one, x)]), vals_app(one, t, x))def allred_app(+one: Nat, rs: List<&2, List<&2, Nat>>, +x: List<&2, Nat>, +h: {allred(one, rs) == True{} : Bool}, +hx: {red(one, x) == True{} : Bool}) -> {allred(one, List.append(&2, List<&2, Nat>, rs, [x])) == True{} : Bool}: match rs: case Nil{}: Lg.and_intro(red(one, x), True{}, hx, {==}) case +r <> +t: Lg.and_intro(red(one, r), allred(one, List.append(&2, List<&2, Nat>, t, [x])), Lg.and_left(red(one, r), allred(one, t), h), allred_app(one, t, x, Lg.and_right(red(one, r), allred(one, t), h), hx))def run_go_r(+one: Nat, +h1: {one == 1n : Nat}, prog: List<&2, P.Op>, +rs: List<&2, List<&2, Nat>>, +h: {allred(one, rs) == True{} : Bool}) -> {allred(one, P.run_go(prog, rs)) == True{} : Bool}: match prog: case Nil{}: h case +op <> +t: run_go_r(one, h1, t, List.append(&2, List<&2, Nat>, rs, [P.exec(op, rs)]), allred_app(one, rs, P.exec(op, rs), h, exec_r(one, h1, op, rs, h)))def run_go_v(+one: Nat, +h1: {one == 1n : Nat}, prog: List<&2, P.Op>, +rs: List<&2, List<&2, Nat>>, +h: {allred(one, rs) == True{} : Bool}) -> {vals(one, P.run_go(prog, rs)) == CV.run_go(FS.prime(one), cprog(prog), vals(one, rs)) : List<&2, Nat>}: match prog: case Nil{}: {==} case +op <> +t: +rs2 = List.append(&2, List<&2, Nat>, rs, [P.exec(op, rs)]) +h2 = allred_app(one, rs, P.exec(op, rs), h, exec_r(one, h1, op, rs, h)) %Equal.sym(List<&2, Nat>, vals(one, P.run_go(t, List.append(&2, List<&2, Nat>, rs, [P.exec(op, rs)]))), CV.run_go(FS.prime(one), cprog(t), vals(one, List.append(&2, List<&2, Nat>, rs, [P.exec(op, rs)]))), run_go_v(one, h1, t, List.append(&2, List<&2, Nat>, rs, [P.exec(op, rs)]), h2)) : {_ == CV.run_go(FS.prime(one), cprog(t), List.append(&2, Nat, vals(one, rs), [CV.exec(FS.prime(one), cop(op), vals(one, rs))])) : List<&2, Nat>} %Equal.sym(List<&2, Nat>, vals(one, List.append(&2, List<&2, Nat>, rs, [P.exec(op, rs)])), List.append(&2, Nat, vals(one, rs), [FS.value(one, P.exec(op, rs))]), vals_app(one, rs, P.exec(op, rs))) : {CV.run_go(FS.prime(one), cprog(t), _) == CV.run_go(FS.prime(one), cprog(t), List.append(&2, Nat, vals(one, rs), [CV.exec(FS.prime(one), cop(op), vals(one, rs))])) : List<&2, Nat>} %Equal.sym(Nat, FS.value(one, P.exec(op, rs)), CV.exec(FS.prime(one), cop(op), vals(one, rs)), exec_v(one, h1, op, rs, h)) : {CV.run_go(FS.prime(one), cprog(t), List.append(&2, Nat, vals(one, rs), [_])) == CV.run_go(FS.prime(one), cprog(t), List.append(&2, Nat, vals(one, rs), [CV.exec(FS.prime(one), cop(op), vals(one, rs))])) : List<&2, Nat>} {==}# run looks at its first register firstdef run_is(prog: List<&2, P.Op>, rs: List<&2, List<&2, Nat>>) -> {P.run(prog, rs) == P.run_go(prog, rs) : List<&2, List<&2, Nat>>}: match rs: case Nil{}: {==} case Nil{} <> t: {==} case (y <> u) <> t: {==}def srun_is(+p: Nat, prog: List<&2, CV.SOp>, rs: List<&2, Nat>) -> {CV.run(p, prog, rs) == CV.run_go(p, prog, rs) : List<&2, Nat>}: match rs: case Nil{}: {==} case 0n <> t: {==} case (1n+k) <> t: {==}def run_r(+one: Nat, +h1: {one == 1n : Nat}, +prog: List<&2, P.Op>, +rs: List<&2, List<&2, Nat>>, +h: {allred(one, rs) == True{} : Bool}) -> {allred(one, P.run(prog, rs)) == True{} : Bool}: %Equal.sym(List<&2, List<&2, Nat>>, P.run(prog, rs), P.run_go(prog, rs), run_is(prog, rs)) : {allred(one, _) == True{} : Bool} run_go_r(one, h1, prog, rs, h)def run_v(+one: Nat, +h1: {one == 1n : Nat}, +prog: List<&2, P.Op>, +rs: List<&2, List<&2, Nat>>, +h: {allred(one, rs) == True{} : Bool}) -> {vals(one, P.run(prog, rs)) == CV.run(FS.prime(one), cprog(prog), vals(one, rs)) : List<&2, Nat>}: %Equal.sym(List<&2, List<&2, Nat>>, P.run(prog, rs), P.run_go(prog, rs), run_is(prog, rs)) : {vals(one, _) == CV.run(FS.prime(one), cprog(prog), vals(one, rs)) : List<&2, Nat>} %Equal.sym(List<&2, Nat>, CV.run(FS.prime(one), cprog(prog), vals(one, rs)), CV.run_go(FS.prime(one), cprog(prog), vals(one, rs)), srun_is(FS.prime(one), cprog(prog), vals(one, rs))) : {vals(one, P.run_go(prog, rs)) == _ : List<&2, Nat>} run_go_v(one, h1, prog, rs, h)# ---- points ----def pv(+one: Nat, p: P.Point) -> CV.SPoint: match p: case P.Point{x, y, z}: CV.SPoint{FS.value(one, x), FS.value(one, y), FS.value(one, z)}def pred(+one: Nat, p: P.Point) -> Bool: match p: case P.Point{x, y, z}: Bool.and(red(one, x), Bool.and(red(one, y), red(one, z)))def b3_r(+one: Nat, +h1: {one == 1n : Nat}) -> {red(one, P.b3()) == True{} : Bool}: FO.small_r(one, h1, 21n, FO.small_lt_p(one, h1, 21n, {==}))def b3_v(+one: Nat, +h1: {one == 1n : Nat}) -> {FS.value(one, P.b3()) == 21n : Nat}: FO.small_val(one, h1, 21n, FO.small_lt_p(one, h1, 21n, {==}))def add_prog_eq() -> {cprog(P.add_prog()) == CV.add_prog() : List<&2, CV.SOp>}: {==}def dbl_prog_eq() -> {cprog(P.dbl_prog()) == CV.dbl_prog() : List<&2, CV.SOp>}: {==}def regs7(+one: Nat, +h1: {one == 1n : Nat}, +x1: List<&2, Nat>, +y1: List<&2, Nat>, +z1: List<&2, Nat>, +x2: List<&2, Nat>, +y2: List<&2, Nat>, +z2: List<&2, Nat>, +h: {Bool.and(red(one, x1), Bool.and(red(one, y1), red(one, z1))) == True{} : Bool}, +g: {Bool.and(red(one, x2), Bool.and(red(one, y2), red(one, z2))) == True{} : Bool}) -> {allred(one, [x1, y1, z1, x2, y2, z2, P.b3()]) == True{} : Bool}: +hx = Lg.and_left(red(one, x1), Bool.and(red(one, y1), red(one, z1)), h) +hy = Lg.and_left(red(one, y1), red(one, z1), Lg.and_right(red(one, x1), Bool.and(red(one, y1), red(one, z1)), h)) +hz = Lg.and_right(red(one, y1), red(one, z1), Lg.and_right(red(one, x1), Bool.and(red(one, y1), red(one, z1)), h)) +gx = Lg.and_left(red(one, x2), Bool.and(red(one, y2), red(one, z2)), g) +gy = Lg.and_left(red(one, y2), red(one, z2), Lg.and_right(red(one, x2), Bool.and(red(one, y2), red(one, z2)), g)) +gz = Lg.and_right(red(one, y2), red(one, z2), Lg.and_right(red(one, x2), Bool.and(red(one, y2), red(one, z2)), g)) +hb = b3_r(one, h1) Lg.and_intro(red(one, x1), allred(one, [y1, z1, x2, y2, z2, P.b3()]), hx, Lg.and_intro(red(one, y1), allred(one, [z1, x2, y2, z2, P.b3()]), hy, Lg.and_intro(red(one, z1), allred(one, [x2, y2, z2, P.b3()]), hz, Lg.and_intro(red(one, x2), allred(one, [y2, z2, P.b3()]), gx, Lg.and_intro(red(one, y2), allred(one, [z2, P.b3()]), gy, Lg.and_intro(red(one, z2), allred(one, [P.b3()]), gz, Lg.and_intro(red(one, P.b3()), True{}, hb, {==})))))))def vals7(+one: Nat, +a: List<&2, Nat>, +b: List<&2, Nat>, +c: List<&2, Nat>, +d: List<&2, Nat>, +e: List<&2, Nat>, +f: List<&2, Nat>, +g: List<&2, Nat>) -> {vals(one, [a, b, c, d, e, f, g]) == [FS.value(one, a), FS.value(one, b), FS.value(one, c), FS.value(one, d), FS.value(one, e), FS.value(one, f), FS.value(one, g)] : List<&2, Nat>}: {==}def padd_v(+one: Nat, +h1: {one == 1n : Nat}, a: P.Point, b: P.Point, +ha: {pred(one, a) == True{} : Bool}, +hb: {pred(one, b) == True{} : Bool}) -> {pv(one, P.add(a, b)) == CV.padd(FS.prime(one), pv(one, a), pv(one, b)) : CV.SPoint}: match a b: case P.Point{+x1, +y1, +z1} P.Point{+x2, +y2, +z2}: +hr = regs7(one, h1, x1, y1, z1, x2, y2, z2, ha, hb) %Equal.sym(Nat, FS.value(one, P.nth(33n, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()]))), CV.nth(33n, vals(one, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()]))), nth_v(one, h1, 33n, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()]))) : {CV.SPoint{_, FS.value(one, P.nth(36n, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()]))), FS.value(one, P.nth(39n, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()])))} == CV.add_out(CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), CV.b3()])) : CV.SPoint} %Equal.sym(Nat, FS.value(one, P.nth(36n, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()]))), CV.nth(36n, vals(one, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()]))), nth_v(one, h1, 36n, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()]))) : {CV.SPoint{CV.nth(33n, vals(one, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()]))), _, FS.value(one, P.nth(39n, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()])))} == CV.add_out(CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), CV.b3()])) : CV.SPoint} %Equal.sym(Nat, FS.value(one, P.nth(39n, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()]))), CV.nth(39n, vals(one, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()]))), nth_v(one, h1, 39n, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()]))) : {CV.SPoint{CV.nth(33n, vals(one, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()]))), CV.nth(36n, vals(one, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()]))), _} == CV.add_out(CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, Nat>, vals(one, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()])), CV.run(FS.prime(one), cprog(P.add_prog()), vals(one, [x1, y1, z1, x2, y2, z2, P.b3()])), run_v(one, h1, P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()], hr)) : {CV.SPoint{CV.nth(33n, _), CV.nth(36n, vals(one, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()]))), CV.nth(39n, vals(one, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()])))} == CV.add_out(CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, Nat>, vals(one, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()])), CV.run(FS.prime(one), cprog(P.add_prog()), vals(one, [x1, y1, z1, x2, y2, z2, P.b3()])), run_v(one, h1, P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()], hr)) : {CV.SPoint{CV.nth(33n, CV.run(FS.prime(one), cprog(P.add_prog()), vals(one, [x1, y1, z1, x2, y2, z2, P.b3()]))), CV.nth(36n, _), CV.nth(39n, vals(one, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()])))} == CV.add_out(CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, Nat>, vals(one, P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()])), CV.run(FS.prime(one), cprog(P.add_prog()), vals(one, [x1, y1, z1, x2, y2, z2, P.b3()])), run_v(one, h1, P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()], hr)) : {CV.SPoint{CV.nth(33n, CV.run(FS.prime(one), cprog(P.add_prog()), vals(one, [x1, y1, z1, x2, y2, z2, P.b3()]))), CV.nth(36n, CV.run(FS.prime(one), cprog(P.add_prog()), vals(one, [x1, y1, z1, x2, y2, z2, P.b3()]))), CV.nth(39n, _)} == CV.add_out(CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, CV.SOp>, cprog(P.add_prog()), CV.add_prog(), add_prog_eq()) : {CV.SPoint{CV.nth(33n, CV.run(FS.prime(one), _, vals(one, [x1, y1, z1, x2, y2, z2, P.b3()]))), CV.nth(36n, CV.run(FS.prime(one), cprog(P.add_prog()), vals(one, [x1, y1, z1, x2, y2, z2, P.b3()]))), CV.nth(39n, CV.run(FS.prime(one), cprog(P.add_prog()), vals(one, [x1, y1, z1, x2, y2, z2, P.b3()])))} == CV.add_out(CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, CV.SOp>, cprog(P.add_prog()), CV.add_prog(), add_prog_eq()) : {CV.SPoint{CV.nth(33n, CV.run(FS.prime(one), CV.add_prog(), vals(one, [x1, y1, z1, x2, y2, z2, P.b3()]))), CV.nth(36n, CV.run(FS.prime(one), _, vals(one, [x1, y1, z1, x2, y2, z2, P.b3()]))), CV.nth(39n, CV.run(FS.prime(one), cprog(P.add_prog()), vals(one, [x1, y1, z1, x2, y2, z2, P.b3()])))} == CV.add_out(CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, CV.SOp>, cprog(P.add_prog()), CV.add_prog(), add_prog_eq()) : {CV.SPoint{CV.nth(33n, CV.run(FS.prime(one), CV.add_prog(), vals(one, [x1, y1, z1, x2, y2, z2, P.b3()]))), CV.nth(36n, CV.run(FS.prime(one), CV.add_prog(), vals(one, [x1, y1, z1, x2, y2, z2, P.b3()]))), CV.nth(39n, CV.run(FS.prime(one), _, vals(one, [x1, y1, z1, x2, y2, z2, P.b3()])))} == CV.add_out(CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, Nat>, vals(one, [x1, y1, z1, x2, y2, z2, P.b3()]), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), FS.value(one, P.b3())], vals7(one, x1, y1, z1, x2, y2, z2, P.b3())) : {CV.SPoint{CV.nth(33n, CV.run(FS.prime(one), CV.add_prog(), _)), CV.nth(36n, CV.run(FS.prime(one), CV.add_prog(), vals(one, [x1, y1, z1, x2, y2, z2, P.b3()]))), CV.nth(39n, CV.run(FS.prime(one), CV.add_prog(), vals(one, [x1, y1, z1, x2, y2, z2, P.b3()])))} == CV.add_out(CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, Nat>, vals(one, [x1, y1, z1, x2, y2, z2, P.b3()]), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), FS.value(one, P.b3())], vals7(one, x1, y1, z1, x2, y2, z2, P.b3())) : {CV.SPoint{CV.nth(33n, CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), FS.value(one, P.b3())])), CV.nth(36n, CV.run(FS.prime(one), CV.add_prog(), _)), CV.nth(39n, CV.run(FS.prime(one), CV.add_prog(), vals(one, [x1, y1, z1, x2, y2, z2, P.b3()])))} == CV.add_out(CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, Nat>, vals(one, [x1, y1, z1, x2, y2, z2, P.b3()]), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), FS.value(one, P.b3())], vals7(one, x1, y1, z1, x2, y2, z2, P.b3())) : {CV.SPoint{CV.nth(33n, CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), FS.value(one, P.b3())])), CV.nth(36n, CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), FS.value(one, P.b3())])), CV.nth(39n, CV.run(FS.prime(one), CV.add_prog(), _))} == CV.add_out(CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), CV.b3()])) : CV.SPoint} %Equal.sym(Nat, FS.value(one, P.b3()), 21n, b3_v(one, h1)) : {CV.SPoint{CV.nth(33n, CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), _])), CV.nth(36n, CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), FS.value(one, P.b3())])), CV.nth(39n, CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), FS.value(one, P.b3())]))} == CV.add_out(CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), CV.b3()])) : CV.SPoint} %Equal.sym(Nat, FS.value(one, P.b3()), 21n, b3_v(one, h1)) : {CV.SPoint{CV.nth(33n, CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), 21n])), CV.nth(36n, CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), _])), CV.nth(39n, CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), FS.value(one, P.b3())]))} == CV.add_out(CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), CV.b3()])) : CV.SPoint} %Equal.sym(Nat, FS.value(one, P.b3()), 21n, b3_v(one, h1)) : {CV.SPoint{CV.nth(33n, CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), 21n])), CV.nth(36n, CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), 21n])), CV.nth(39n, CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), _]))} == CV.add_out(CV.run(FS.prime(one), CV.add_prog(), [FS.value(one, x1), FS.value(one, y1), FS.value(one, z1), FS.value(one, x2), FS.value(one, y2), FS.value(one, z2), CV.b3()])) : CV.SPoint} {==}def regs4(+one: Nat, +h1: {one == 1n : Nat}, +x: List<&2, Nat>, +y: List<&2, Nat>, +z: List<&2, Nat>, +h: {Bool.and(red(one, x), Bool.and(red(one, y), red(one, z))) == True{} : Bool}) -> {allred(one, [x, y, z, P.b3()]) == True{} : Bool}: +hx = Lg.and_left(red(one, x), Bool.and(red(one, y), red(one, z)), h) +hy = Lg.and_left(red(one, y), red(one, z), Lg.and_right(red(one, x), Bool.and(red(one, y), red(one, z)), h)) +hz = Lg.and_right(red(one, y), red(one, z), Lg.and_right(red(one, x), Bool.and(red(one, y), red(one, z)), h)) +hb = b3_r(one, h1) Lg.and_intro(red(one, x), allred(one, [y, z, P.b3()]), hx, Lg.and_intro(red(one, y), allred(one, [z, P.b3()]), hy, Lg.and_intro(red(one, z), allred(one, [P.b3()]), hz, Lg.and_intro(red(one, P.b3()), True{}, hb, {==}))))def vals4(+one: Nat, +a: List<&2, Nat>, +b: List<&2, Nat>, +c: List<&2, Nat>, +d: List<&2, Nat>) -> {vals(one, [a, b, c, d]) == [FS.value(one, a), FS.value(one, b), FS.value(one, c), FS.value(one, d)] : List<&2, Nat>}: {==}def pdbl_v(+one: Nat, +h1: {one == 1n : Nat}, a: P.Point, +ha: {pred(one, a) == True{} : Bool}) -> {pv(one, P.dbl(a)) == CV.pdbl(FS.prime(one), pv(one, a)) : CV.SPoint}: match a: case P.Point{+x, +y, +z}: +hr = regs4(one, h1, x, y, z, ha) %Equal.sym(Nat, FS.value(one, P.nth(21n, P.run(P.dbl_prog(), [x, y, z, P.b3()]))), CV.nth(21n, vals(one, P.run(P.dbl_prog(), [x, y, z, P.b3()]))), nth_v(one, h1, 21n, P.run(P.dbl_prog(), [x, y, z, P.b3()]))) : {CV.SPoint{_, FS.value(one, P.nth(18n, P.run(P.dbl_prog(), [x, y, z, P.b3()]))), FS.value(one, P.nth(13n, P.run(P.dbl_prog(), [x, y, z, P.b3()])))} == CV.dbl_out(CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), CV.b3()])) : CV.SPoint} %Equal.sym(Nat, FS.value(one, P.nth(18n, P.run(P.dbl_prog(), [x, y, z, P.b3()]))), CV.nth(18n, vals(one, P.run(P.dbl_prog(), [x, y, z, P.b3()]))), nth_v(one, h1, 18n, P.run(P.dbl_prog(), [x, y, z, P.b3()]))) : {CV.SPoint{CV.nth(21n, vals(one, P.run(P.dbl_prog(), [x, y, z, P.b3()]))), _, FS.value(one, P.nth(13n, P.run(P.dbl_prog(), [x, y, z, P.b3()])))} == CV.dbl_out(CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), CV.b3()])) : CV.SPoint} %Equal.sym(Nat, FS.value(one, P.nth(13n, P.run(P.dbl_prog(), [x, y, z, P.b3()]))), CV.nth(13n, vals(one, P.run(P.dbl_prog(), [x, y, z, P.b3()]))), nth_v(one, h1, 13n, P.run(P.dbl_prog(), [x, y, z, P.b3()]))) : {CV.SPoint{CV.nth(21n, vals(one, P.run(P.dbl_prog(), [x, y, z, P.b3()]))), CV.nth(18n, vals(one, P.run(P.dbl_prog(), [x, y, z, P.b3()]))), _} == CV.dbl_out(CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, Nat>, vals(one, P.run(P.dbl_prog(), [x, y, z, P.b3()])), CV.run(FS.prime(one), cprog(P.dbl_prog()), vals(one, [x, y, z, P.b3()])), run_v(one, h1, P.dbl_prog(), [x, y, z, P.b3()], hr)) : {CV.SPoint{CV.nth(21n, _), CV.nth(18n, vals(one, P.run(P.dbl_prog(), [x, y, z, P.b3()]))), CV.nth(13n, vals(one, P.run(P.dbl_prog(), [x, y, z, P.b3()])))} == CV.dbl_out(CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, Nat>, vals(one, P.run(P.dbl_prog(), [x, y, z, P.b3()])), CV.run(FS.prime(one), cprog(P.dbl_prog()), vals(one, [x, y, z, P.b3()])), run_v(one, h1, P.dbl_prog(), [x, y, z, P.b3()], hr)) : {CV.SPoint{CV.nth(21n, CV.run(FS.prime(one), cprog(P.dbl_prog()), vals(one, [x, y, z, P.b3()]))), CV.nth(18n, _), CV.nth(13n, vals(one, P.run(P.dbl_prog(), [x, y, z, P.b3()])))} == CV.dbl_out(CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, Nat>, vals(one, P.run(P.dbl_prog(), [x, y, z, P.b3()])), CV.run(FS.prime(one), cprog(P.dbl_prog()), vals(one, [x, y, z, P.b3()])), run_v(one, h1, P.dbl_prog(), [x, y, z, P.b3()], hr)) : {CV.SPoint{CV.nth(21n, CV.run(FS.prime(one), cprog(P.dbl_prog()), vals(one, [x, y, z, P.b3()]))), CV.nth(18n, CV.run(FS.prime(one), cprog(P.dbl_prog()), vals(one, [x, y, z, P.b3()]))), CV.nth(13n, _)} == CV.dbl_out(CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, CV.SOp>, cprog(P.dbl_prog()), CV.dbl_prog(), dbl_prog_eq()) : {CV.SPoint{CV.nth(21n, CV.run(FS.prime(one), _, vals(one, [x, y, z, P.b3()]))), CV.nth(18n, CV.run(FS.prime(one), cprog(P.dbl_prog()), vals(one, [x, y, z, P.b3()]))), CV.nth(13n, CV.run(FS.prime(one), cprog(P.dbl_prog()), vals(one, [x, y, z, P.b3()])))} == CV.dbl_out(CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, CV.SOp>, cprog(P.dbl_prog()), CV.dbl_prog(), dbl_prog_eq()) : {CV.SPoint{CV.nth(21n, CV.run(FS.prime(one), CV.dbl_prog(), vals(one, [x, y, z, P.b3()]))), CV.nth(18n, CV.run(FS.prime(one), _, vals(one, [x, y, z, P.b3()]))), CV.nth(13n, CV.run(FS.prime(one), cprog(P.dbl_prog()), vals(one, [x, y, z, P.b3()])))} == CV.dbl_out(CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, CV.SOp>, cprog(P.dbl_prog()), CV.dbl_prog(), dbl_prog_eq()) : {CV.SPoint{CV.nth(21n, CV.run(FS.prime(one), CV.dbl_prog(), vals(one, [x, y, z, P.b3()]))), CV.nth(18n, CV.run(FS.prime(one), CV.dbl_prog(), vals(one, [x, y, z, P.b3()]))), CV.nth(13n, CV.run(FS.prime(one), _, vals(one, [x, y, z, P.b3()])))} == CV.dbl_out(CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, Nat>, vals(one, [x, y, z, P.b3()]), [FS.value(one, x), FS.value(one, y), FS.value(one, z), FS.value(one, P.b3())], vals4(one, x, y, z, P.b3())) : {CV.SPoint{CV.nth(21n, CV.run(FS.prime(one), CV.dbl_prog(), _)), CV.nth(18n, CV.run(FS.prime(one), CV.dbl_prog(), vals(one, [x, y, z, P.b3()]))), CV.nth(13n, CV.run(FS.prime(one), CV.dbl_prog(), vals(one, [x, y, z, P.b3()])))} == CV.dbl_out(CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, Nat>, vals(one, [x, y, z, P.b3()]), [FS.value(one, x), FS.value(one, y), FS.value(one, z), FS.value(one, P.b3())], vals4(one, x, y, z, P.b3())) : {CV.SPoint{CV.nth(21n, CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), FS.value(one, P.b3())])), CV.nth(18n, CV.run(FS.prime(one), CV.dbl_prog(), _)), CV.nth(13n, CV.run(FS.prime(one), CV.dbl_prog(), vals(one, [x, y, z, P.b3()])))} == CV.dbl_out(CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), CV.b3()])) : CV.SPoint} %Equal.sym(List<&2, Nat>, vals(one, [x, y, z, P.b3()]), [FS.value(one, x), FS.value(one, y), FS.value(one, z), FS.value(one, P.b3())], vals4(one, x, y, z, P.b3())) : {CV.SPoint{CV.nth(21n, CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), FS.value(one, P.b3())])), CV.nth(18n, CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), FS.value(one, P.b3())])), CV.nth(13n, CV.run(FS.prime(one), CV.dbl_prog(), _))} == CV.dbl_out(CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), CV.b3()])) : CV.SPoint} %Equal.sym(Nat, FS.value(one, P.b3()), 21n, b3_v(one, h1)) : {CV.SPoint{CV.nth(21n, CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), _])), CV.nth(18n, CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), FS.value(one, P.b3())])), CV.nth(13n, CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), FS.value(one, P.b3())]))} == CV.dbl_out(CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), CV.b3()])) : CV.SPoint} %Equal.sym(Nat, FS.value(one, P.b3()), 21n, b3_v(one, h1)) : {CV.SPoint{CV.nth(21n, CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), 21n])), CV.nth(18n, CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), _])), CV.nth(13n, CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), FS.value(one, P.b3())]))} == CV.dbl_out(CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), CV.b3()])) : CV.SPoint} %Equal.sym(Nat, FS.value(one, P.b3()), 21n, b3_v(one, h1)) : {CV.SPoint{CV.nth(21n, CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), 21n])), CV.nth(18n, CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), 21n])), CV.nth(13n, CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), _]))} == CV.dbl_out(CV.run(FS.prime(one), CV.dbl_prog(), [FS.value(one, x), FS.value(one, y), FS.value(one, z), CV.b3()])) : CV.SPoint} {==}# the results are reduceddef pr3(+one: Nat, +a: List<&2, Nat>, +b: List<&2, Nat>, +c: List<&2, Nat>, +ha: {red(one, a) == True{} : Bool}, +hb: {red(one, b) == True{} : Bool}, +hc: {red(one, c) == True{} : Bool}) -> {pred(one, P.Point{a, b, c}) == True{} : Bool}: Lg.and_intro(red(one, a), Bool.and(red(one, b), red(one, c)), ha, Lg.and_intro(red(one, b), red(one, c), hb, hc))def padd_r(+one: Nat, +h1: {one == 1n : Nat}, a: P.Point, b: P.Point, +ha: {pred(one, a) == True{} : Bool}, +hb: {pred(one, b) == True{} : Bool}) -> {pred(one, P.add(a, b)) == True{} : Bool}: match a b: case P.Point{+x1, +y1, +z1} P.Point{+x2, +y2, +z2}: +hr = run_r(one, h1, P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()], regs7(one, h1, x1, y1, z1, x2, y2, z2, ha, hb)) +rs = P.run(P.add_prog(), [x1, y1, z1, x2, y2, z2, P.b3()]) pr3(one, P.nth(33n, rs), P.nth(36n, rs), P.nth(39n, rs), nth_r(one, h1, 33n, rs, hr), nth_r(one, h1, 36n, rs, hr), nth_r(one, h1, 39n, rs, hr))def pdbl_r(+one: Nat, +h1: {one == 1n : Nat}, a: P.Point, +ha: {pred(one, a) == True{} : Bool}) -> {pred(one, P.dbl(a)) == True{} : Bool}: match a: case P.Point{+x, +y, +z}: +hr = run_r(one, h1, P.dbl_prog(), [x, y, z, P.b3()], regs4(one, h1, x, y, z, ha)) +rs = P.run(P.dbl_prog(), [x, y, z, P.b3()]) pr3(one, P.nth(21n, rs), P.nth(18n, rs), P.nth(13n, rs), nth_r(one, h1, 21n, rs, hr), nth_r(one, h1, 18n, rs, hr), nth_r(one, h1, 13n, rs, hr))