Fieldops.subl_v: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/fieldops.bend as Fieldops
Lines 66-75 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/fieldops.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/fieldops.bend&start_line=66&end_line=75

import Base
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/lib/u32.bend as U
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
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/poly.bend as PL

# one limb: (x + k) - y with y <= k, no wrap
def subl_v(+one: Nat, +h1: {one == 1n : Nat}, +x: U32, +y: U32, +k: U32, +bx: Nat, +by: Nat, +bk: Nat, +hx: {Nat.is_le(v(x), bx) == True{} : Bool}, +hy: {Nat.is_le(v(y), by) == True{} : Bool}, +hk: {Nat.is_le(v(k), bk) == True{} : Bool}, +hg: {Nat.is_le(by, v(k)) == True{} : Bool}, +hb: {Nat.is_lt(Nat.add(bx, bk), WD.sc(32n, one)) == True{} : Bool}) -> {Nat.add(v(U32.sub(U32.add(x, k), y)), v(y)) == Nat.add(v(x), v(k)) : Nat}:
  +ea = PL.addl_v(one, h1, x, k, bx, bk, hx, hk, hb)
  +yk = N.le_trans(v(y), by, v(k), hy, hg)
  +yl = N.le_trans(v(y), v(k), Nat.add(v(x), v(k)), yk, PL.le_add_l(v(x), v(k)))
  +yl2 = L.subst(Nat, z => {Nat.is_le(v(y), z) == True{} : Bool}, Nat.add(v(x), v(k)), v(U32.add(x, k)), Equal.sym(Nat, v(U32.add(x, k)), Nat.add(v(x), v(k)), ea), yl)
  +es = U.sub_nat(U32.add(x, k), y, yl2)
  +e1 = M.cong_l(v(U32.sub(U32.add(x, k), y)), Nat.sub(v(U32.add(x, k)), v(y)), v(y), es)
  +e2 = Equal.trans(Nat, Nat.add(Nat.sub(v(U32.add(x, k)), v(y)), v(y)), Nat.add(v(y), Nat.sub(v(U32.add(x, k)), v(y))), v(U32.add(x, k)), NA.add_comm(Nat.sub(v(U32.add(x, k)), v(y)), v(y)), N.sub_add(v(U32.add(x, k)), v(y), yl2))
  Equal.trans(Nat, Nat.add(v(U32.sub(U32.add(x, k), y)), v(y)), Nat.add(Nat.sub(v(U32.add(x, k)), v(y)), v(y)), Nat.add(v(x), v(k)), e1, Equal.trans(Nat, Nat.add(Nat.sub(v(U32.add(x, k)), v(y)), v(y)), v(U32.add(x, k)), Nat.add(v(x), v(k)), e2, ea))

Uses (21):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.trans
  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=Nat.sub
  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=Base/base.bend&name=U32.sub
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/fieldops.bend&name=v
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/num.bend&name=cong_l
  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=le_add_l
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_comm
  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_trans
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=sub_add
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/u32.bend&name=sub_nat
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend&name=sc