# GENERATED by tools/generators/rw.py from ../../../tools/generators/secp256k1_hand/fieldops.src; edit the .src
import Base
import ../../../spec/lib/common.bend as C
import ../../../spec/crypto/secp256k1/field.bend as FS
import ../../../src/crypto/secp256k1/limbs.bend as L
import ../../../src/crypto/secp256k1/field.bend as F
import ../../lib/nat.bend as N
import ../../lib/logic.bend as Lg
import ../../lib/arith.bend as AR
import ../../lib/lemmas/proofs/nat_algebra.bend as NA
import ../../math/natural/arith.bend as NR
import ./limbs.bend as LV
import ./fprot.bend as FR
import ./ineq.bend as I
import ./bounds.bend as B
import ./reducek.bend as RK
import ./consts.bend as K
import ./negate.bend as NG
import ./listops.bend as LO

# GF(p) of src/crypto/secp256k1/field.bend against spec/crypto/secp256k1/field.bend:
# on reduced inputs (16 limbs below 2^16, value below p) every operation
# returns a reduced element whose value is the specified one, exactly.

# ---- sizes ----

def tt_lt(+one: Nat, +h1: {one == 1n : Nat}, +x: List<&2, Nat>, +h: {FS.limbs16(one, 16n, x) == True{} : Bool}) -> {Nat.is_lt(LV.ev(x), C.shift(256n, one)) == True{} : Bool}:
  I.lt_subst_r(LV.ev(x), LV.pos(16n, one), C.shift(256n, one), B.pos16(one), B.limbs_lt(one, h1, 16n, x, h))

def three_le(+one: Nat, +h1: {one == 1n : Nat}) -> {Nat.is_le(3n, C.shift(256n, one)) == True{} : Bool}:
  K.small_le(one, h1, 256n, 3n, {==})

# below 3 2^256 is below 2^512
def lt3(+one: Nat, +h1: {one == 1n : Nat}, +x: Nat, +hx: {Nat.is_lt(x, Nat.add(C.shift(256n, one), Nat.add(C.shift(256n, one), C.shift(256n, one)))) == True{} : Bool}) -> {Nat.is_lt(x, Nat.mul(C.shift(256n, one), C.shift(256n, one))) == True{} : Bool}:
  +t = C.shift(256n, one)
  +e3 = Equal.trans(Nat, Nat.add(t, Nat.add(t, t)), Nat.add(t, Nat.add(t, Nat.add(t, 0n))), Nat.mul(3n, t), Equal.cong(Nat, Nat, z => Nat.add(t, Nat.add(t, z)), t, Nat.add(t, 0n), Equal.sym(Nat, Nat.add(t, 0n), t, NA.add_zero(t))), {==})
  N.lt_le_trans(x, Nat.mul(3n, t), Nat.mul(t, t), I.lt_subst_r(x, Nat.add(t, Nat.add(t, t)), Nat.mul(3n, t), e3, hx), AR.mul_le(3n, t, t, three_le(one, h1)))

def sum_lt(+one: Nat, +h1: {one == 1n : Nat}, +a: Nat, +b: Nat, +ha: {Nat.is_lt(a, C.shift(256n, one)) == True{} : Bool}, +hb: {Nat.is_lt(b, C.shift(256n, one)) == True{} : Bool}) -> {Nat.is_lt(Nat.add(a, b), Nat.mul(C.shift(256n, one), C.shift(256n, one))) == True{} : Bool}:
  +t = C.shift(256n, one)
  lt3(one, h1, Nat.add(a, b), I.lt_le_add(a, b, t, Nat.add(t, t), ha, N.le_trans(b, t, Nat.add(t, t), N.lt_le(b, t, hb), N.le_add_right(t, t))))

def mul_lt(+a: Nat, +b: Nat, +t: Nat, +ha: {Nat.is_lt(a, t) == True{} : Bool}, +hb: {Nat.is_lt(b, t) == True{} : Bool}) -> {Nat.is_lt(Nat.mul(a, b), Nat.mul(t, t)) == True{} : Bool}:
  match t:
    case 0n:
      Empty.absurd({Nat.is_lt(Nat.mul(a, b), Nat.mul(0n, 0n)) == True{} : Bool}, N.lt_zero_absurd(a, ha))
    case 1n+ +tp:
      +s1 = I.le_mul2(a, b, tp, 1n+tp, N.lt_succ_le(a, tp, ha), N.lt_le(b, 1n+tp, hb))
      +mm = Nat.mul(tp, 1n+tp)
      +s2 = I.lt_subst_r(mm, Nat.add(mm, 1n+tp), Nat.add(1n+tp, mm), NA.add_comm(mm, 1n+tp), I.lt_add_pos(mm, 1n+tp, {==}))
      N.le_lt_trans(Nat.mul(a, b), mm, Nat.add(1n+tp, mm), s1, s2)

# ---- the reduction mod p ----

def red_e(+one: Nat, +h1: {one == 1n : Nat}, +xs: List<&2, Nat>, +hx: {Nat.is_lt(LV.ev(xs), Nat.mul(C.shift(256n, one), C.shift(256n, one))) == True{} : Bool}) -> {LV.ev(F.reduce(xs)) == Nat.mod(LV.ev(xs), 1n+K.pp(one)) : Nat}:
  RK.reduce3_vv(one, h1, K.pp(one), F.c(), K.hm_p(one, h1), K.cp_cc(one, h1), K.cp_hb(one, h1), xs, hx)

def red_l(+one: Nat, +h1: {one == 1n : Nat}, +xs: List<&2, Nat>, +hx: {Nat.is_lt(LV.ev(xs), Nat.mul(C.shift(256n, one), C.shift(256n, one))) == True{} : Bool}) -> {FS.limbs16(one, 16n, F.reduce(xs)) == True{} : Bool}:
  RK.reduce3_lv(one, h1, F.c(), K.cp_cc(one, h1), K.cp_hb(one, h1), xs, hx)

# the reduced value below p, in the specification's terms
def red_r(+one: Nat, +h1: {one == 1n : Nat}, +xs: List<&2, Nat>, +hx: {Nat.is_lt(LV.ev(xs), Nat.mul(C.shift(256n, one), C.shift(256n, one))) == True{} : Bool}) -> {FS.reduced(one, FS.prime(one), F.reduce(xs)) == True{} : Bool}:
  +lt = I.lt_subst_l(Nat.mod(LV.ev(xs), 1n+K.pp(one)), LV.ev(F.reduce(xs)), 1n+K.pp(one), Equal.sym(Nat, LV.ev(F.reduce(xs)), Nat.mod(LV.ev(xs), 1n+K.pp(one)), red_e(one, h1, xs, hx)), NR.dm_lt(K.pp(one), LV.ev(xs)))
  +lt2 = I.lt_subst_l(LV.ev(F.reduce(xs)), FS.value(one, F.reduce(xs)), 1n+K.pp(one), Equal.sym(Nat, FS.value(one, F.reduce(xs)), LV.ev(F.reduce(xs)), LV.ev_value(one, h1, F.reduce(xs))), lt)
  +lt3 = I.lt_subst_r(FS.value(one, F.reduce(xs)), 1n+K.pp(one), FS.prime(one), Equal.sym(Nat, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1)), lt2)
  Lg.and_intro(FS.limbs16(one, 16n, F.reduce(xs)), Nat.is_lt(FS.value(one, F.reduce(xs)), FS.prime(one)), red_l(one, h1, xs, hx), lt3)

def red_v(+one: Nat, +h1: {one == 1n : Nat}, +xs: List<&2, Nat>, +hx: {Nat.is_lt(LV.ev(xs), Nat.mul(C.shift(256n, one), C.shift(256n, one))) == True{} : Bool}) -> {FS.value(one, F.reduce(xs)) == Nat.mod(LV.ev(xs), FS.prime(one)) : Nat}:
  %Equal.sym(Nat, FS.value(one, F.reduce(xs)), LV.ev(F.reduce(xs)), LV.ev_value(one, h1, F.reduce(xs))) : {_ == Nat.mod(LV.ev(xs), FS.prime(one)) : Nat}
  %Equal.sym(Nat, LV.ev(F.reduce(xs)), Nat.mod(LV.ev(xs), 1n+K.pp(one)), red_e(one, h1, xs, hx)) : {_ == Nat.mod(LV.ev(xs), FS.prime(one)) : Nat}
  %Equal.sym(Nat, 1n+K.pp(one), FS.prime(one), Equal.sym(Nat, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1))) : {Nat.mod(LV.ev(xs), _) == Nat.mod(LV.ev(xs), FS.prime(one)) : Nat}
  {==}

# ---- reduced inputs ----

def r16(+one: Nat, +x: List<&2, Nat>, +h: {FS.reduced(one, FS.prime(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 rlt(+one: Nat, +h1: {one == 1n : Nat}, +x: List<&2, Nat>, +h: {FS.reduced(one, FS.prime(one), x) == True{} : Bool}) -> {Nat.is_lt(LV.ev(x), 1n+K.pp(one)) == True{} : Bool}:
  +l = Lg.and_right(FS.limbs16(one, 16n, x), Nat.is_lt(FS.value(one, x), FS.prime(one)), h)
  +l2 = I.lt_subst_l(FS.value(one, x), LV.ev(x), FS.prime(one), LV.ev_value(one, h1, x), l)
  I.lt_subst_r(LV.ev(x), FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1), l2)

def rtt(+one: Nat, +h1: {one == 1n : Nat}, +x: List<&2, Nat>, +h: {FS.reduced(one, FS.prime(one), x) == True{} : Bool}) -> {Nat.is_lt(LV.ev(x), C.shift(256n, one)) == True{} : Bool}:
  tt_lt(one, h1, x, r16(one, x, h))

# ---- add, mul ----

def add_hx(+one: Nat, +h1: {one == 1n : Nat}, +a: List<&2, Nat>, +b: List<&2, Nat>, +ha: {FS.reduced(one, FS.prime(one), a) == True{} : Bool}, +hb: {FS.reduced(one, FS.prime(one), b) == True{} : Bool}) -> {Nat.is_lt(LV.ev(L.add(a, b)), Nat.mul(C.shift(256n, one), C.shift(256n, one))) == True{} : Bool}:
  %Equal.sym(Nat, LV.ev(L.add(a, b)), Nat.add(LV.ev(a), LV.ev(b)), LV.add_v(a, b)) : {Nat.is_lt(_, Nat.mul(C.shift(256n, one), C.shift(256n, one))) == True{} : Bool}
  sum_lt(one, h1, LV.ev(a), LV.ev(b), rtt(one, h1, a, ha), rtt(one, h1, b, hb))

def add_v(+one: Nat, +h1: {one == 1n : Nat}, +a: List<&2, Nat>, +b: List<&2, Nat>, +ha: {FS.reduced(one, FS.prime(one), a) == True{} : Bool}, +hb: {FS.reduced(one, FS.prime(one), b) == True{} : Bool}) -> {FS.value(one, F.add(a, b)) == FS.madd(FS.prime(one), FS.value(one, a), FS.value(one, b)) : Nat}:
  %Equal.sym(List<&2, Nat>, F.add(a, b), F.add_u(a, b), FR.f_add(a, b)) : {FS.value(one, _) == FS.madd(FS.prime(one), FS.value(one, a), FS.value(one, b)) : Nat}
  %Equal.sym(Nat, FS.value(one, F.reduce(L.add(a, b))), Nat.mod(LV.ev(L.add(a, b)), FS.prime(one)), red_v(one, h1, L.add(a, b), add_hx(one, h1, a, b, ha, hb))) : {_ == Nat.mod(Nat.add(FS.value(one, a), FS.value(one, b)), FS.prime(one)) : Nat}
  %Equal.sym(Nat, LV.ev(L.add(a, b)), Nat.add(LV.ev(a), LV.ev(b)), LV.add_v(a, b)) : {Nat.mod(_, FS.prime(one)) == Nat.mod(Nat.add(FS.value(one, a), FS.value(one, b)), FS.prime(one)) : Nat}
  %Equal.sym(Nat, LV.ev(a), FS.value(one, a), Equal.sym(Nat, FS.value(one, a), LV.ev(a), LV.ev_value(one, h1, a))) : {Nat.mod(Nat.add(_, LV.ev(b)), FS.prime(one)) == Nat.mod(Nat.add(FS.value(one, a), FS.value(one, b)), FS.prime(one)) : Nat}
  %Equal.sym(Nat, LV.ev(b), FS.value(one, b), Equal.sym(Nat, FS.value(one, b), LV.ev(b), LV.ev_value(one, h1, b))) : {Nat.mod(Nat.add(FS.value(one, a), _), FS.prime(one)) == Nat.mod(Nat.add(FS.value(one, a), FS.value(one, b)), FS.prime(one)) : Nat}
  {==}

def add_r(+one: Nat, +h1: {one == 1n : Nat}, +a: List<&2, Nat>, +b: List<&2, Nat>, +ha: {FS.reduced(one, FS.prime(one), a) == True{} : Bool}, +hb: {FS.reduced(one, FS.prime(one), b) == True{} : Bool}) -> {FS.reduced(one, FS.prime(one), F.add(a, b)) == True{} : Bool}:
  %Equal.sym(List<&2, Nat>, F.add(a, b), F.add_u(a, b), FR.f_add(a, b)) : {FS.reduced(one, FS.prime(one), _) == True{} : Bool}
  red_r(one, h1, L.add(a, b), add_hx(one, h1, a, b, ha, hb))

def mul_hx(+one: Nat, +h1: {one == 1n : Nat}, +a: List<&2, Nat>, +b: List<&2, Nat>, +ha: {FS.reduced(one, FS.prime(one), a) == True{} : Bool}, +hb: {FS.reduced(one, FS.prime(one), b) == True{} : Bool}) -> {Nat.is_lt(LV.ev(L.conv(a, b)), Nat.mul(C.shift(256n, one), C.shift(256n, one))) == True{} : Bool}:
  %Equal.sym(Nat, LV.ev(L.conv(a, b)), Nat.mul(LV.ev(a), LV.ev(b)), LV.conv_v(a, b)) : {Nat.is_lt(_, Nat.mul(C.shift(256n, one), C.shift(256n, one))) == True{} : Bool}
  mul_lt(LV.ev(a), LV.ev(b), C.shift(256n, one), rtt(one, h1, a, ha), rtt(one, h1, b, hb))

def mul_v(+one: Nat, +h1: {one == 1n : Nat}, +a: List<&2, Nat>, +b: List<&2, Nat>, +ha: {FS.reduced(one, FS.prime(one), a) == True{} : Bool}, +hb: {FS.reduced(one, FS.prime(one), b) == True{} : Bool}) -> {FS.value(one, F.mul(a, b)) == FS.mmul(FS.prime(one), FS.value(one, a), FS.value(one, b)) : Nat}:
  %Equal.sym(List<&2, Nat>, F.mul(a, b), F.mul_u(a, b), FR.f_mul(a, b)) : {FS.value(one, _) == FS.mmul(FS.prime(one), FS.value(one, a), FS.value(one, b)) : Nat}
  %Equal.sym(Nat, FS.value(one, F.reduce(L.conv(a, b))), Nat.mod(LV.ev(L.conv(a, b)), FS.prime(one)), red_v(one, h1, L.conv(a, b), mul_hx(one, h1, a, b, ha, hb))) : {_ == Nat.mod(Nat.mul(FS.value(one, a), FS.value(one, b)), FS.prime(one)) : Nat}
  %Equal.sym(Nat, LV.ev(L.conv(a, b)), Nat.mul(LV.ev(a), LV.ev(b)), LV.conv_v(a, b)) : {Nat.mod(_, FS.prime(one)) == Nat.mod(Nat.mul(FS.value(one, a), FS.value(one, b)), FS.prime(one)) : Nat}
  %Equal.sym(Nat, LV.ev(a), FS.value(one, a), Equal.sym(Nat, FS.value(one, a), LV.ev(a), LV.ev_value(one, h1, a))) : {Nat.mod(Nat.mul(_, LV.ev(b)), FS.prime(one)) == Nat.mod(Nat.mul(FS.value(one, a), FS.value(one, b)), FS.prime(one)) : Nat}
  %Equal.sym(Nat, LV.ev(b), FS.value(one, b), Equal.sym(Nat, FS.value(one, b), LV.ev(b), LV.ev_value(one, h1, b))) : {Nat.mod(Nat.mul(FS.value(one, a), _), FS.prime(one)) == Nat.mod(Nat.mul(FS.value(one, a), FS.value(one, b)), FS.prime(one)) : Nat}
  {==}

def mul_r(+one: Nat, +h1: {one == 1n : Nat}, +a: List<&2, Nat>, +b: List<&2, Nat>, +ha: {FS.reduced(one, FS.prime(one), a) == True{} : Bool}, +hb: {FS.reduced(one, FS.prime(one), b) == True{} : Bool}) -> {FS.reduced(one, FS.prime(one), F.mul(a, b)) == True{} : Bool}:
  %Equal.sym(List<&2, Nat>, F.mul(a, b), F.mul_u(a, b), FR.f_mul(a, b)) : {FS.reduced(one, FS.prime(one), _) == True{} : Bool}
  red_r(one, h1, L.conv(a, b), mul_hx(one, h1, a, b, ha, hb))

# ---- sub, neg ----

def negr_e(+one: Nat, +h1: {one == 1n : Nat}, +b: List<&2, Nat>, +hb: {FS.reduced(one, FS.prime(one), b) == True{} : Bool}) -> {Nat.add(LV.ev(L.neg_raw(F.c(), b)), LV.ev(b)) == 1n+K.pp(one) : Nat}:
  NG.neg_vv(one, h1, K.pp(one), F.c(), K.hm_p(one, h1), b, rlt(one, h1, b, hb))

def negr_le(+one: Nat, +h1: {one == 1n : Nat}, +b: List<&2, Nat>, +hb: {FS.reduced(one, FS.prime(one), b) == True{} : Bool}) -> {Nat.is_le(LV.ev(L.neg_raw(F.c(), b)), C.shift(256n, one)) == True{} : Bool}:
  +m = {1n+K.pp(one) : Nat}
  +nb = LV.ev(L.neg_raw(F.c(), b))
  +mt = I.le_subst_r(m, Nat.add(m, FS.cp(one)), C.shift(256n, one), K.hm_p(one, h1), N.le_add_right(m, FS.cp(one)))
  N.le_trans(nb, m, C.shift(256n, one), I.le_subst_r(nb, Nat.add(nb, LV.ev(b)), m, negr_e(one, h1, b, hb), N.le_add_right(nb, LV.ev(b))), mt)

def sub_hx(+one: Nat, +h1: {one == 1n : Nat}, +a: List<&2, Nat>, +b: List<&2, Nat>, +ha: {FS.reduced(one, FS.prime(one), a) == True{} : Bool}, +hb: {FS.reduced(one, FS.prime(one), b) == True{} : Bool}) -> {Nat.is_lt(LV.ev(L.add(a, L.neg_raw(F.c(), b))), Nat.mul(C.shift(256n, one), C.shift(256n, one))) == True{} : Bool}:
  %Equal.sym(Nat, LV.ev(L.add(a, L.neg_raw(F.c(), b))), Nat.add(LV.ev(a), LV.ev(L.neg_raw(F.c(), b))), LV.add_v(a, L.neg_raw(F.c(), b))) : {Nat.is_lt(_, Nat.mul(C.shift(256n, one), C.shift(256n, one))) == True{} : Bool}
  lt3(one, h1, Nat.add(LV.ev(a), LV.ev(L.neg_raw(F.c(), b))), I.lt_le_add(LV.ev(a), LV.ev(L.neg_raw(F.c(), b)), C.shift(256n, one), Nat.add(C.shift(256n, one), C.shift(256n, one)), rtt(one, h1, a, ha), N.le_trans(LV.ev(L.neg_raw(F.c(), b)), C.shift(256n, one), Nat.add(C.shift(256n, one), C.shift(256n, one)), negr_le(one, h1, b, hb), N.le_add_right(C.shift(256n, one), C.shift(256n, one)))))

# F.sub looks at a first; a reduced a has 16 limbs
def sub_go(+one: Nat, +h1: {one == 1n : Nat}, a: List<&2, Nat>, +b: List<&2, Nat>, +ha: {FS.reduced(one, FS.prime(one), a) == True{} : Bool}) -> {F.sub(a, b) == F.reduce(L.add(a, L.neg_raw(F.c(), b))) : List<&2, Nat>}:
  match a:
    case Nil{}:
      Empty.absurd({F.sub([], b) == F.reduce(L.add([], L.neg_raw(F.c(), b))) : List<&2, Nat>}, Lg.false_true(Lg.and_left(FS.limbs16(one, 16n, []), Nat.is_lt(FS.value(one, []), FS.prime(one)), ha)))
    case x <> t:
      {==}

def sub_v(+one: Nat, +h1: {one == 1n : Nat}, +a: List<&2, Nat>, +b: List<&2, Nat>, +ha: {FS.reduced(one, FS.prime(one), a) == True{} : Bool}, +hb: {FS.reduced(one, FS.prime(one), b) == True{} : Bool}) -> {FS.value(one, F.sub(a, b)) == FS.msub(FS.prime(one), FS.value(one, a), FS.value(one, b)) : Nat}:
  %Equal.sym(List<&2, Nat>, F.sub(a, b), F.reduce(L.add(a, L.neg_raw(F.c(), b))), sub_go(one, h1, a, b, ha)) : {FS.value(one, _) == FS.msub(FS.prime(one), FS.value(one, a), FS.value(one, b)) : Nat}
  %Equal.sym(Nat, FS.value(one, F.reduce(L.add(a, L.neg_raw(F.c(), b)))), Nat.mod(LV.ev(L.add(a, L.neg_raw(F.c(), b))), FS.prime(one)), red_v(one, h1, L.add(a, L.neg_raw(F.c(), b)), sub_hx(one, h1, a, b, ha, hb))) : {_ == Nat.mod(Nat.sub(Nat.add(FS.value(one, a), FS.prime(one)), Nat.mod(FS.value(one, b), FS.prime(one))), FS.prime(one)) : Nat}
  %Equal.sym(Nat, LV.ev(L.add(a, L.neg_raw(F.c(), b))), Nat.add(LV.ev(a), LV.ev(L.neg_raw(F.c(), b))), LV.add_v(a, L.neg_raw(F.c(), b))) : {Nat.mod(_, FS.prime(one)) == Nat.mod(Nat.sub(Nat.add(FS.value(one, a), FS.prime(one)), Nat.mod(FS.value(one, b), FS.prime(one))), FS.prime(one)) : Nat}
  %Equal.sym(Nat, FS.value(one, a), LV.ev(a), LV.ev_value(one, h1, a)) : {Nat.mod(Nat.add(LV.ev(a), LV.ev(L.neg_raw(F.c(), b))), FS.prime(one)) == Nat.mod(Nat.sub(Nat.add(_, FS.prime(one)), Nat.mod(FS.value(one, b), FS.prime(one))), FS.prime(one)) : Nat}
  %Equal.sym(Nat, FS.value(one, b), LV.ev(b), LV.ev_value(one, h1, b)) : {Nat.mod(Nat.add(LV.ev(a), LV.ev(L.neg_raw(F.c(), b))), FS.prime(one)) == Nat.mod(Nat.sub(Nat.add(LV.ev(a), FS.prime(one)), Nat.mod(_, FS.prime(one))), FS.prime(one)) : Nat}
  %Equal.sym(Nat, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1)) : {Nat.mod(Nat.add(LV.ev(a), LV.ev(L.neg_raw(F.c(), b))), _) == Nat.mod(Nat.sub(Nat.add(LV.ev(a), FS.prime(one)), Nat.mod(LV.ev(b), FS.prime(one))), FS.prime(one)) : Nat}
  %Equal.sym(Nat, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1)) : {Nat.mod(Nat.add(LV.ev(a), LV.ev(L.neg_raw(F.c(), b))), 1n+K.pp(one)) == Nat.mod(Nat.sub(Nat.add(LV.ev(a), _), Nat.mod(LV.ev(b), FS.prime(one))), FS.prime(one)) : Nat}
  %Equal.sym(Nat, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1)) : {Nat.mod(Nat.add(LV.ev(a), LV.ev(L.neg_raw(F.c(), b))), 1n+K.pp(one)) == Nat.mod(Nat.sub(Nat.add(LV.ev(a), 1n+K.pp(one)), Nat.mod(LV.ev(b), _)), FS.prime(one)) : Nat}
  %Equal.sym(Nat, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1)) : {Nat.mod(Nat.add(LV.ev(a), LV.ev(L.neg_raw(F.c(), b))), 1n+K.pp(one)) == Nat.mod(Nat.sub(Nat.add(LV.ev(a), 1n+K.pp(one)), Nat.mod(LV.ev(b), 1n+K.pp(one))), _) : Nat}
  NG.sub_mod(K.pp(one), LV.ev(a), LV.ev(b), LV.ev(L.neg_raw(F.c(), b)), rlt(one, h1, b, hb), negr_e(one, h1, b, hb))

def sub_r(+one: Nat, +h1: {one == 1n : Nat}, +a: List<&2, Nat>, +b: List<&2, Nat>, +ha: {FS.reduced(one, FS.prime(one), a) == True{} : Bool}, +hb: {FS.reduced(one, FS.prime(one), b) == True{} : Bool}) -> {FS.reduced(one, FS.prime(one), F.sub(a, b)) == True{} : Bool}:
  %Equal.sym(List<&2, Nat>, F.sub(a, b), F.reduce(L.add(a, L.neg_raw(F.c(), b))), sub_go(one, h1, a, b, ha)) : {FS.reduced(one, FS.prime(one), _) == True{} : Bool}
  red_r(one, h1, L.add(a, L.neg_raw(F.c(), b)), sub_hx(one, h1, a, b, ha, hb))

def neg_hx(+one: Nat, +h1: {one == 1n : Nat}, +b: List<&2, Nat>, +hb: {FS.reduced(one, FS.prime(one), b) == True{} : Bool}) -> {Nat.is_lt(LV.ev(L.neg_raw(F.c(), b)), Nat.mul(C.shift(256n, one), C.shift(256n, one))) == True{} : Bool}:
  +t = C.shift(256n, one)
  lt3(one, h1, LV.ev(L.neg_raw(F.c(), b)), N.le_lt_trans(LV.ev(L.neg_raw(F.c(), b)), t, Nat.add(t, Nat.add(t, t)), negr_le(one, h1, b, hb), I.lt_add_pos(t, Nat.add(t, t), N.lt_le_trans(0n, t, Nat.add(t, t), K.shift_pos(one, h1, 256n), N.le_add_right(t, t)))))

def neg_v(+one: Nat, +h1: {one == 1n : Nat}, +b: List<&2, Nat>, +hb: {FS.reduced(one, FS.prime(one), b) == True{} : Bool}) -> {FS.value(one, F.neg(b)) == FS.mneg(FS.prime(one), FS.value(one, b)) : Nat}:
  %Equal.sym(List<&2, Nat>, F.neg(b), F.neg_u(b), FR.f_neg(b)) : {FS.value(one, _) == FS.mneg(FS.prime(one), FS.value(one, b)) : Nat}
  %Equal.sym(Nat, FS.value(one, F.reduce(L.neg_raw(F.c(), b))), Nat.mod(LV.ev(L.neg_raw(F.c(), b)), FS.prime(one)), red_v(one, h1, L.neg_raw(F.c(), b), neg_hx(one, h1, b, hb))) : {_ == Nat.mod(Nat.sub(Nat.add(0n, FS.prime(one)), Nat.mod(FS.value(one, b), FS.prime(one))), FS.prime(one)) : Nat}
  %Equal.sym(Nat, FS.value(one, b), LV.ev(b), LV.ev_value(one, h1, b)) : {Nat.mod(LV.ev(L.neg_raw(F.c(), b)), FS.prime(one)) == Nat.mod(Nat.sub(Nat.add(0n, FS.prime(one)), Nat.mod(_, FS.prime(one))), FS.prime(one)) : Nat}
  %Equal.sym(Nat, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1)) : {Nat.mod(LV.ev(L.neg_raw(F.c(), b)), _) == Nat.mod(Nat.sub(Nat.add(0n, FS.prime(one)), Nat.mod(LV.ev(b), FS.prime(one))), FS.prime(one)) : Nat}
  %Equal.sym(Nat, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1)) : {Nat.mod(LV.ev(L.neg_raw(F.c(), b)), 1n+K.pp(one)) == Nat.mod(Nat.sub(Nat.add(0n, _), Nat.mod(LV.ev(b), FS.prime(one))), FS.prime(one)) : Nat}
  %Equal.sym(Nat, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1)) : {Nat.mod(LV.ev(L.neg_raw(F.c(), b)), 1n+K.pp(one)) == Nat.mod(Nat.sub(Nat.add(0n, 1n+K.pp(one)), Nat.mod(LV.ev(b), _)), FS.prime(one)) : Nat}
  %Equal.sym(Nat, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1)) : {Nat.mod(LV.ev(L.neg_raw(F.c(), b)), 1n+K.pp(one)) == Nat.mod(Nat.sub(Nat.add(0n, 1n+K.pp(one)), Nat.mod(LV.ev(b), 1n+K.pp(one))), _) : Nat}
  NG.sub_mod(K.pp(one), 0n, LV.ev(b), LV.ev(L.neg_raw(F.c(), b)), rlt(one, h1, b, hb), negr_e(one, h1, b, hb))

def neg_r(+one: Nat, +h1: {one == 1n : Nat}, +b: List<&2, Nat>, +hb: {FS.reduced(one, FS.prime(one), b) == True{} : Bool}) -> {FS.reduced(one, FS.prime(one), F.neg(b)) == True{} : Bool}:
  %Equal.sym(List<&2, Nat>, F.neg(b), F.neg_u(b), FR.f_neg(b)) : {FS.reduced(one, FS.prime(one), _) == True{} : Bool}
  red_r(one, h1, L.neg_raw(F.c(), b), neg_hx(one, h1, b, hb))

# ---- in the limb value ev ----

def mul_e(+one: Nat, +h1: {one == 1n : Nat}, +a: List<&2, Nat>, +b: List<&2, Nat>, +ha: {FS.reduced(one, FS.prime(one), a) == True{} : Bool}, +hb: {FS.reduced(one, FS.prime(one), b) == True{} : Bool}) -> {LV.ev(F.mul(a, b)) == Nat.mod(Nat.mul(LV.ev(a), LV.ev(b)), 1n+K.pp(one)) : Nat}:
  %Equal.sym(List<&2, Nat>, F.mul(a, b), F.mul_u(a, b), FR.f_mul(a, b)) : {LV.ev(_) == Nat.mod(Nat.mul(LV.ev(a), LV.ev(b)), 1n+K.pp(one)) : Nat}
  %Equal.sym(Nat, LV.ev(F.reduce(L.conv(a, b))), Nat.mod(LV.ev(L.conv(a, b)), 1n+K.pp(one)), red_e(one, h1, L.conv(a, b), mul_hx(one, h1, a, b, ha, hb))) : {_ == Nat.mod(Nat.mul(LV.ev(a), LV.ev(b)), 1n+K.pp(one)) : Nat}
  %Equal.sym(Nat, LV.ev(L.conv(a, b)), Nat.mul(LV.ev(a), LV.ev(b)), LV.conv_v(a, b)) : {Nat.mod(_, 1n+K.pp(one)) == Nat.mod(Nat.mul(LV.ev(a), LV.ev(b)), 1n+K.pp(one)) : Nat}
  {==}

# ---- small constants ----

def small_r(+one: Nat, +h1: {one == 1n : Nat}, +k: Nat, +hk: {Nat.is_lt(k, FS.prime(one)) == True{} : Bool}) -> {FS.reduced(one, FS.prime(one), F.small(k)) == True{} : Bool}:
  +hk2 = I.lt_subst_r(k, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1), hk)
  +hkt = N.lt_le_trans(k, 1n+K.pp(one), C.shift(256n, one), hk2, I.le_subst_r(1n+K.pp(one), Nat.add(1n+K.pp(one), FS.cp(one)), C.shift(256n, one), K.hm_p(one, h1), N.le_add_right(1n+K.pp(one), FS.cp(one))))
  +ev = LO.small_v(one, h1, k, hkt)
  +lt = I.lt_subst_l(k, FS.value(one, F.small(k)), FS.prime(one), Equal.sym(Nat, FS.value(one, F.small(k)), k, Equal.trans(Nat, FS.value(one, F.small(k)), LV.ev(F.small(k)), k, LV.ev_value(one, h1, F.small(k)), ev)), hk)
  Lg.and_intro(FS.limbs16(one, 16n, F.small(k)), Nat.is_lt(FS.value(one, F.small(k)), FS.prime(one)), LO.small_l(one, h1, k), lt)

def small_val(+one: Nat, +h1: {one == 1n : Nat}, +k: Nat, +hk: {Nat.is_lt(k, FS.prime(one)) == True{} : Bool}) -> {FS.value(one, F.small(k)) == k : Nat}:
  +hk2 = I.lt_subst_r(k, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1), hk)
  +hkt = N.lt_le_trans(k, 1n+K.pp(one), C.shift(256n, one), hk2, I.le_subst_r(1n+K.pp(one), Nat.add(1n+K.pp(one), FS.cp(one)), C.shift(256n, one), K.hm_p(one, h1), N.le_add_right(1n+K.pp(one), FS.cp(one))))
  Equal.trans(Nat, FS.value(one, F.small(k)), LV.ev(F.small(k)), k, LV.ev_value(one, h1, F.small(k)), LO.small_v(one, h1, k, hkt))

# m + c = h + h and c <= h: h <= m
def le_cancel_r(+x: Nat, +y: Nat, +c: Nat) -> {Nat.is_le(Nat.add(x, c), Nat.add(y, c)) == Nat.is_le(x, y) : Bool}:
  %Equal.sym(Nat, Nat.add(x, c), Nat.add(c, x), NA.add_comm(x, c)) : {Nat.is_le(_, Nat.add(y, c)) == Nat.is_le(x, y) : Bool}
  %Equal.sym(Nat, Nat.add(y, c), Nat.add(c, y), NA.add_comm(y, c)) : {Nat.is_le(Nat.add(c, x), _) == Nat.is_le(x, y) : Bool}
  NR.le_add_cancel(c, x, y)

# small k < 2^32 is below p
def lt_p_half(+h: Nat, +m: Nat, +c: Nat, +e: {Nat.add(m, c) == Nat.add(h, h) : Nat}, +hc: {Nat.is_le(c, h) == True{} : Bool}) -> {Nat.is_le(h, m) == True{} : Bool}:
  +s1 = N.le_add_left(c, h, h, hc)
  +s2 = I.le_subst_r(Nat.add(h, c), Nat.add(h, h), Nat.add(m, c), Equal.sym(Nat, Nat.add(m, c), Nat.add(h, h), e), s1)
  +s3 = Equal.trans(Bool, Nat.is_le(h, m), Nat.is_le(Nat.add(h, c), Nat.add(m, c)), True{}, Equal.sym(Bool, Nat.is_le(Nat.add(h, c), Nat.add(m, c)), Nat.is_le(h, m), le_cancel_r(h, m, c)), s2)
  s3

def small_lt_p(+one: Nat, +h1: {one == 1n : Nat}, +k: Nat, +hk: {Nat.is_lt(k, C.shift(10n, 1n)) == True{} : Bool}) -> {Nat.is_lt(k, FS.prime(one)) == True{} : Bool}:
  +hk1 = I.lt_subst_r(k, C.shift(10n, 1n), C.shift(10n, one), Equal.cong(Nat, Nat, z => C.shift(10n, z), 1n, one, Equal.sym(Nat, one, 1n, h1)), hk)
  +s1 = N.lt_le_trans(k, C.shift(10n, one), C.shift(255n, one), hk1, K.shift_le_k(10n, 255n, one, 245n, {==}))
  +pp = K.pp(one)
  # 2^255 <= p: p + c = 2^256 = 2^255 + 2^255 and c <= 2^33 <= 2^255
  +c = FS.cp(one)
  +e1 = Equal.trans(Nat, Nat.add(1n+pp, c), C.shift(256n, one), Nat.add(C.shift(255n, one), C.shift(255n, one)), K.hm_p(one, h1), Equal.sym(Nat, Nat.add(C.shift(255n, one), C.shift(255n, one)), C.shift(256n, one), K.shift_twice(255n, one)))
  +ch = N.le_trans(c, C.shift(33n, one), C.shift(255n, one), K.cp_le(one, h1), K.shift_le_k(33n, 255n, one, 222n, {==}))
  +s2 = lt_p_half(C.shift(255n, one), 1n+pp, c, e1, ch)
  I.lt_subst_r(k, 1n+pp, FS.prime(one), Equal.sym(Nat, FS.prime(one), 1n+pp, K.prime_eq(one, h1)), N.lt_le_trans(k, C.shift(255n, one), 1n+pp, s1, s2))