Scalar.addg_val: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar.bend as Scalar
Lines 177-182 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar.bend&start_line=177&end_line=182

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/crypto/curve25519/field.bend as FS
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/crypto/curve25519/num.bend as M
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/limbs.bend as LM
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/consts.bend as K

def addg_val(+one: Nat, +h1: {one == 1n : Nat}, +mm: Nat, +h2: {Nat.is_le(Nat.add(1n+mm, 1n+mm), T(one)) == True{} : Bool}, +cp: List<&2, U32>, +hk: {LM.okb(32n, cp, 255n) == True{} : Bool}, +hv: {Nat.add(K.valo(one, cp), 1n+mm) == T(one) : Nat}, +a: List<&2, U32>, +b: List<&2, U32>, +ha: {LM.okb(32n, a, 255n) == True{} : Bool}, +hb: {LM.okb(32n, b, 255n) == True{} : Bool}, +la: {Nat.is_lt(FS.value(a), 1n+mm) == True{} : Bool}, +lb: {Nat.is_lt(FS.value(b), 1n+mm) == True{} : Bool}) -> {FS.value(addg(a, b, cp)) == Nat.mod(Nat.add(FS.value(a), FS.value(b)), 1n+mm) : Nat}:
  +s = F.cr_limbs(F.carry(F.addl(a, b), 0))
  +es = sum_val(one, h1, mm, h2, a, b, ha, hb, la, lb)
  +h2m = L.subst(Nat, z => {Nat.is_lt(z, Nat.add(1n+mm, 1n+mm)) == True{} : Bool}, Nat.add(FS.value(a), FS.value(b)), FS.value(s), Equal.sym(Nat, FS.value(s), Nat.add(FS.value(a), FS.value(b)), es), N.lt_le_trans(Nat.add(FS.value(a), FS.value(b)), Nat.add(1n+mm, FS.value(b)), Nat.add(1n+mm, 1n+mm), N.lt_add_r2(FS.value(a), 1n+mm, FS.value(b), la), N.lt_le(Nat.add(1n+mm, FS.value(b)), Nat.add(1n+mm, 1n+mm), N.lt_add_left(FS.value(b), 1n+mm, 1n+mm, lb))))
  +e1 = cs_val_by(one, h1, mm, s, cp, sum_okb(one, h1, a, b, ha, hb), hk, hv, h2m, Nat.is_lt(v(LM.cc(F.carry(F.addl(s, cp), 0))), one), {==})
  Equal.trans(Nat, FS.value(addg(a, b, cp)), Nat.mod(FS.value(s), 1n+mm), Nat.mod(Nat.add(FS.value(a), FS.value(b)), 1n+mm), e1, M.cong_mod(1n+mm, FS.value(s), Nat.add(FS.value(a), FS.value(b)), es))

Uses (29):
  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=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=Nat.mod
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/consts.bend&name=valo
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/limbs.bend&name=cc
  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/num.bend&name=cong_mod
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar.bend&name=T
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar.bend&name=addg
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar.bend&name=cs_val_by
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar.bend&name=sum_okb
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar.bend&name=sum_val
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar.bend&name=v
  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=lt_add_left
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=lt_add_r2
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=lt_le
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=lt_le_trans
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/crypto/curve25519/field.bend&name=value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/field.bend&name=addl
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/field.bend&name=carry
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/field.bend&name=cr_limbs