Poly.addl_okb: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/poly.bend as Poly
Lines 137-155 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/poly.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/poly.bend&start_line=137&end_line=155

import Base
import bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/field.bend as F
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend as L
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend as WD
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/num.bend as M
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/limbs.bend as LM

# n limbs and at most n limbs give n limbs
def addl_okb(+one: Nat, +h1: {one == 1n : Nat}, +n: Nat, +xs: List<&2, U32>, +ys: List<&2, U32>, +a: Nat, +b: Nat, +hx: {LM.okb(n, xs, a) == True{} : Bool}, +hy: {LM.lea(ys, b) == True{} : Bool}, +hl: {len_le(n, ys) == True{} : Bool}, +hab: {Nat.is_lt(Nat.add(a, b), WD.sc(32n, one)) == True{} : Bool}) -> {LM.okb(n, F.addl(xs, ys), Nat.add(a, b)) == True{} : Bool}:
  match n xs ys:
    case 0n Nil{} Nil{}:
      {==}
    case 0n Nil{} Con{y, yt}:
      Empty.absurd({LM.okb(0n, Con{y, yt}, Nat.add(a, b)) == True{} : Bool}, L.false_true(hl))
    case 0n Con{x, xt} _:
      Empty.absurd({LM.okb(0n, F.addl(Con{x, xt}, ys), Nat.add(a, b)) == True{} : Bool}, L.false_true(hx))
    case 1n+m Nil{} _:
      Empty.absurd({LM.okb(1n+m, ys, Nat.add(a, b)) == True{} : Bool}, L.false_true(hx))
    case 1n+ +m Con{+x, +xt} Nil{}:
      okb_mono(1n+m, Con{x, xt}, a, Nat.add(a, b), N.le_add_right(a, b), hx)
    case 1n+ +m Con{+x, +xt} Con{+y, +yt}:
      +hxh = LM.okb_head(m, x, xt, a, hx)
      +hyh = LM.lea_head(y, yt, b, hy)
      +ev = addl_v(one, h1, x, y, a, b, hxh, hyh, hab)
      +hd = L.subst(Nat, z => {Nat.is_le(z, Nat.add(a, b)) == True{} : Bool}, Nat.add(v(x), v(y)), v(U32.add(x, y)), Equal.sym(Nat, v(U32.add(x, y)), Nat.add(v(x), v(y)), ev), M.le_add2(v(x), v(y), a, b, hxh, hyh))
      L.and_intro(Nat.is_le(v(U32.add(x, y)), Nat.add(a, b)), LM.okb(m, F.addl(xt, yt), Nat.add(a, b)), hd, addl_okb(one, h1, m, xt, yt, a, b, LM.okb_tail(m, x, xt, a, hx), LM.lea_tail(y, yt, b, hy), hl, hab))

Uses (27):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Empty.absurd
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=List
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.add
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_le
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.add
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/limbs.bend&name=lea
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/limbs.bend&name=lea_head
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/limbs.bend&name=lea_tail
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/limbs.bend&name=okb
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/limbs.bend&name=okb_head
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/limbs.bend&name=okb_tail
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/num.bend&name=le_add2
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/poly.bend&name=addl_v
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/poly.bend&name=len_le
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/poly.bend&name=okb_mono
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/poly.bend&name=v
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend&name=and_intro
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend&name=false_true
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend&name=subst
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=le_add_right
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend&name=sc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/field.bend&name=addl