# GENERATED by tools/generators/rw.py from ../../../tools/generators/secp256k1_hand/sel.src; edit the .src
import Base
import ../../../spec/lib/common.bend as C
import ../../../spec/crypto/secp256k1/field.bend as FS
import ../../../spec/crypto/secp256k1/curve.bend as CV
import ../../../src/crypto/secp256k1/limbs.bend as L
import ../../../src/crypto/secp256k1/point.bend as P
import ../../lib/logic.bend as Lg
import ../../lib/lemmas/proofs/nat_algebra.bend as NA
import ./fieldops.bend as FO
import ./bitsv.bend as BV
import ./bitsrel.bend as BR
import ./point.bend as PT

# Selection (b ? p : q, b in {0, 1}) and small facts for the point proofs.
def bit01(n: Nat) -> {Nat.is_lt(C.bit(n), 2n) == True{} : Bool}:
  match n:
    case 0n:
      {==}
    case 1n:
      {==}
    case 2n+m:
      bit01(m)

def sb01(i: Nat, +off: Nat, +k: Nat) -> {BV.all01(BR.sb(i, off, k)) == True{} : Bool}:
  match i:
    case 0n:
      {==}
    case 1n+ +j:
      Lg.and_intro(Nat.is_lt(CV.bit(Nat.add(off, j), k), 2n), BV.all01(BR.sb(j, off, k)), bit01(C.high(Nat.add(off, j), k)), sb01(j, off, k))

def sel0(+one: Nat, n: Nat, xs: List<&2, Nat>, ys: List<&2, Nat>, +hx: {FS.limbs16(one, n, xs) == True{} : Bool}, +hy: {FS.limbs16(one, n, ys) == True{} : Bool}) -> {L.select(0n, xs, ys) == ys : List<&2, Nat>}:
  match n xs:
    case 0n Nil{}:
      match ys:
        case Nil{}:
          {==}
        case y <> t:
          Empty.absurd({L.select(0n, [], y <> t) == y <> t : List<&2, Nat>}, Lg.false_true(hy))
    case 0n x <> t:
      Empty.absurd({L.select(0n, x <> t, ys) == ys : List<&2, Nat>}, Lg.false_true(hx))
    case 1n+k Nil{}:
      Empty.absurd({L.select(0n, [], ys) == ys : List<&2, Nat>}, Lg.false_true(hx))
    case 1n+ +k +x <> +xt:
      match ys:
        case Nil{}:
          Empty.absurd({L.select(0n, x <> xt, []) == [] : List<&2, Nat>}, Lg.false_true(hy))
        case +y <> +yt:
          +hxt = Lg.and_right(Nat.is_lt(x, C.shift(16n, one)), FS.limbs16(one, k, xt), hx)
          +hyt = Lg.and_right(Nat.is_lt(y, C.shift(16n, one)), FS.limbs16(one, k, yt), hy)
          %Equal.sym(List<&2, Nat>, L.select(0n, xt, yt), yt, sel0(one, k, xt, yt, hxt, hyt)) : {Nat.add(Nat.mul(x, 0n), Nat.mul(y, 1n)) <> _ == y <> yt : List<&2, Nat>}
          %Equal.sym(Nat, Nat.mul(x, 0n), 0n, NA.mul_zero(x)) : {Nat.add(_, Nat.mul(y, 1n)) <> yt == y <> yt : List<&2, Nat>}
          %Equal.sym(Nat, Nat.mul(y, 1n), y, NA.mul_one(y)) : {Nat.add(0n, _) <> yt == y <> yt : List<&2, Nat>}
          {==}

def sel1(+one: Nat, n: Nat, xs: List<&2, Nat>, ys: List<&2, Nat>, +hx: {FS.limbs16(one, n, xs) == True{} : Bool}, +hy: {FS.limbs16(one, n, ys) == True{} : Bool}) -> {L.select(1n, xs, ys) == xs : List<&2, Nat>}:
  match n xs:
    case 0n Nil{}:
      match ys:
        case Nil{}:
          {==}
        case y <> t:
          Empty.absurd({L.select(1n, [], y <> t) == [] : List<&2, Nat>}, Lg.false_true(hy))
    case 0n x <> t:
      Empty.absurd({L.select(1n, x <> t, ys) == x <> t : List<&2, Nat>}, Lg.false_true(hx))
    case 1n+k Nil{}:
      Empty.absurd({L.select(1n, [], ys) == [] : List<&2, Nat>}, Lg.false_true(hx))
    case 1n+ +k +x <> +xt:
      match ys:
        case Nil{}:
          Empty.absurd({L.select(1n, x <> xt, []) == x <> xt : List<&2, Nat>}, Lg.false_true(hy))
        case +y <> +yt:
          +hxt = Lg.and_right(Nat.is_lt(x, C.shift(16n, one)), FS.limbs16(one, k, xt), hx)
          +hyt = Lg.and_right(Nat.is_lt(y, C.shift(16n, one)), FS.limbs16(one, k, yt), hy)
          %Equal.sym(List<&2, Nat>, L.select(1n, xt, yt), xt, sel1(one, k, xt, yt, hxt, hyt)) : {Nat.add(Nat.mul(x, 1n), Nat.mul(y, 0n)) <> _ == x <> xt : List<&2, Nat>}
          %Equal.sym(Nat, Nat.mul(y, 0n), 0n, NA.mul_zero(y)) : {Nat.add(Nat.mul(x, 1n), _) <> xt == x <> xt : List<&2, Nat>}
          %Equal.sym(Nat, Nat.mul(x, 1n), x, NA.mul_one(x)) : {Nat.add(_, 0n) <> xt == x <> xt : List<&2, Nat>}
          %Equal.sym(Nat, Nat.add(x, 0n), x, NA.add_zero(x)) : {_ <> xt == x <> xt : List<&2, Nat>}
          {==}

def l16(+one: Nat, +x: List<&2, Nat>, +h: {PT.red(one, x) == True{} : Bool}) -> {FS.limbs16(one, 16n, x) == True{} : Bool}:
  Lg.and_left(FS.limbs16(one, 16n, x), Nat.is_lt(FS.value(one, x), FS.prime(one)), h)

def rx(+one: Nat, +x: List<&2, Nat>, +y: List<&2, Nat>, +z: List<&2, Nat>, +h: {PT.pred(one, P.Point{x, y, z}) == True{} : Bool}) -> {FS.limbs16(one, 16n, x) == True{} : Bool}:
  l16(one, x, Lg.and_left(PT.red(one, x), Bool.and(PT.red(one, y), PT.red(one, z)), h))

def ry(+one: Nat, +x: List<&2, Nat>, +y: List<&2, Nat>, +z: List<&2, Nat>, +h: {PT.pred(one, P.Point{x, y, z}) == True{} : Bool}) -> {FS.limbs16(one, 16n, y) == True{} : Bool}:
  l16(one, y, Lg.and_left(PT.red(one, y), PT.red(one, z), Lg.and_right(PT.red(one, x), Bool.and(PT.red(one, y), PT.red(one, z)), h)))

def rz(+one: Nat, +x: List<&2, Nat>, +y: List<&2, Nat>, +z: List<&2, Nat>, +h: {PT.pred(one, P.Point{x, y, z}) == True{} : Bool}) -> {FS.limbs16(one, 16n, z) == True{} : Bool}:
  l16(one, z, Lg.and_right(PT.red(one, y), PT.red(one, z), Lg.and_right(PT.red(one, x), Bool.and(PT.red(one, y), PT.red(one, z)), h)))

# b = 0 picks the second point, b = 1 the first
def psel0(+one: Nat, a: P.Point, b: P.Point, +ha: {PT.pred(one, a) == True{} : Bool}, +hb: {PT.pred(one, b) == True{} : Bool}) -> {P.select(0n, a, b) == b : P.Point}:
  match a b:
    case P.Point{+x1, +y1, +z1} P.Point{+x2, +y2, +z2}:
      %Equal.sym(List<&2, Nat>, L.select(0n, x1, x2), x2, sel0(one, 16n, x1, x2, rx(one, x1, y1, z1, ha), rx(one, x2, y2, z2, hb))) : {P.Point{_, L.select(0n, y1, y2), L.select(0n, z1, z2)} == P.Point{x2, y2, z2} : P.Point}
      %Equal.sym(List<&2, Nat>, L.select(0n, y1, y2), y2, sel0(one, 16n, y1, y2, ry(one, x1, y1, z1, ha), ry(one, x2, y2, z2, hb))) : {P.Point{x2, _, L.select(0n, z1, z2)} == P.Point{x2, y2, z2} : P.Point}
      %Equal.sym(List<&2, Nat>, L.select(0n, z1, z2), z2, sel0(one, 16n, z1, z2, rz(one, x1, y1, z1, ha), rz(one, x2, y2, z2, hb))) : {P.Point{x2, y2, _} == P.Point{x2, y2, z2} : P.Point}
      {==}

def psel1(+one: Nat, a: P.Point, b: P.Point, +ha: {PT.pred(one, a) == True{} : Bool}, +hb: {PT.pred(one, b) == True{} : Bool}) -> {P.select(1n, a, b) == a : P.Point}:
  match a b:
    case P.Point{+x1, +y1, +z1} P.Point{+x2, +y2, +z2}:
      %Equal.sym(List<&2, Nat>, L.select(1n, x1, x2), x1, sel1(one, 16n, x1, x2, rx(one, x1, y1, z1, ha), rx(one, x2, y2, z2, hb))) : {P.Point{_, L.select(1n, y1, y2), L.select(1n, z1, z2)} == P.Point{x1, y1, z1} : P.Point}
      %Equal.sym(List<&2, Nat>, L.select(1n, y1, y2), y1, sel1(one, 16n, y1, y2, ry(one, x1, y1, z1, ha), ry(one, x2, y2, z2, hb))) : {P.Point{x1, _, L.select(1n, z1, z2)} == P.Point{x1, y1, z1} : P.Point}
      %Equal.sym(List<&2, Nat>, L.select(1n, z1, z2), z1, sel1(one, 16n, z1, z2, rz(one, x1, y1, z1, ha), rz(one, x2, y2, z2, hb))) : {P.Point{x1, y1, _} == P.Point{x1, y1, z1} : P.Point}
      {==}

def one_lt(+one: Nat, +h1: {one == 1n : Nat}) -> {Nat.is_lt(1n, FS.prime(one)) == True{} : Bool}:
  FO.small_lt_p(one, h1, 1n, {==})

def lt2(c: Nat) -> {Nat.is_lt(2n+c, 2n) == False{} : Bool}:
  match c:
    case 0n:
      {==}
    case 1n+d:
      {==}

def not01(+c: Nat, +h: {Nat.is_lt(2n+c, 2n) == True{} : Bool}) -> Empty:
  Lg.false_true(Equal.trans(Bool, False{}, Nat.is_lt(2n+c, 2n), True{}, Equal.sym(Bool, Nat.is_lt(2n+c, 2n), False{}, lt2(c)), h))

def red_of(+one: Nat, +x: List<&2, Nat>, +hl: {FS.limbs16(one, 16n, x) == True{} : Bool}, +hb: {Nat.is_lt(FS.value(one, x), FS.prime(one)) == True{} : Bool}) -> {PT.red(one, x) == True{} : Bool}:
  Lg.and_intro(FS.limbs16(one, 16n, x), Nat.is_lt(FS.value(one, x), FS.prime(one)), hl, hb)

def sl16(+one: Nat, +x: List<&2, Nat>, +h: {FS.reduced(one, FS.order(one), x) == True{} : Bool}) -> {FS.limbs16(one, 16n, x) == True{} : Bool}:
  Lg.and_left(FS.limbs16(one, 16n, x), Nat.is_lt(FS.value(one, x), FS.order(one)), h)