Scalar.cs_val_by: 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 121-133 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=121&end_line=133

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/logic.bend as L
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/arith.bend as AR
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/math/natural/arith.bend as NR
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/reduce.bend as RD
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/consts.bend as K
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/freeze.bend as FZ

def cs_val_by(+one: Nat, +h1: {one == 1n : Nat}, +mm: Nat, +x: List<&2, U32>, +cp: List<&2, U32>, +hx: {LM.okb(32n, x, 255n) == True{} : Bool}, +hk: {LM.okb(32n, cp, 255n) == True{} : Bool}, +hv: {Nat.add(K.valo(one, cp), 1n+mm) == T(one) : Nat}, +h2m: {Nat.is_lt(FS.value(x), Nat.add(1n+mm, 1n+mm)) == True{} : Bool}, +b: Bool, +hb: {Nat.is_lt(v(LM.cc(F.carry(F.addl(x, cp), 0))), one) == b : Bool}) -> {FS.value(FZ.csg(x, cp)) == Nat.mod(FS.value(x), 1n+mm) : Nat}:
  match b:
    case True{}:
      %Equal.sym(List<&2, U32>, FZ.csg(x, cp), x, cs_mz(one, h1, x, cp, hx, hk, hb)) : {FS.value(_) == Nat.mod(FS.value(x), 1n+mm) : Nat}
      Equal.sym(Nat, Nat.mod(FS.value(x), 1n+mm), FS.value(x), NR.mod_of(0n, mm, FS.value(x), mz(one, h1, mm, x, cp, hx, hk, hv, hb)))
    case False{}:
      +s = LM.cl(F.carry(F.addl(x, cp), 0))
      %Equal.sym(List<&2, U32>, FZ.csg(x, cp), s, cs_mo(one, h1, mm, x, cp, hx, hk, hv, hb)) : {FS.value(_) == Nat.mod(FS.value(x), 1n+mm) : Nat}
      +e1 = mo(one, h1, mm, x, cp, hx, hk, hv, hb)
      +l1 = L.subst(Nat, z => {Nat.is_lt(z, Nat.add(1n+mm, 1n+mm)) == True{} : Bool}, FS.value(x), Nat.add(FS.value(s), 1n+mm), Equal.sym(Nat, Nat.add(FS.value(s), 1n+mm), FS.value(x), e1), h2m)
      +sl = RD.lt_cancel_r(FS.value(s), 1n+mm, 1n+mm, l1)
      +e2 = Equal.trans(Nat, FS.value(x), Nat.add(FS.value(s), 1n+mm), Nat.add(Nat.mul(1n, 1n+mm), FS.value(s)), Equal.sym(Nat, Nat.add(FS.value(s), 1n+mm), FS.value(x), e1), Equal.trans(Nat, Nat.add(FS.value(s), 1n+mm), Nat.add(1n+mm, FS.value(s)), Nat.add(Nat.mul(1n, 1n+mm), FS.value(s)), NA.add_comm(FS.value(s), 1n+mm), M.cong_l(1n+mm, Nat.mul(1n, 1n+mm), FS.value(s), Equal.sym(Nat, Nat.mul(1n, 1n+mm), 1n+mm, AR.one_mul(1n+mm)))))
      Equal.sym(Nat, Nat.mod(FS.value(x), 1n+mm), FS.value(s), Equal.trans(Nat, Nat.mod(FS.value(x), 1n+mm), Nat.mod(Nat.add(Nat.mul(1n, 1n+mm), FS.value(s)), 1n+mm), FS.value(s), M.cong_mod(1n+mm, FS.value(x), Nat.add(Nat.mul(1n, 1n+mm), FS.value(s)), e2), NR.mod_of(1n, mm, FS.value(s), sl)))

Uses (31):
  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_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=Nat.mul
  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/freeze.bend&name=csg
  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=cl
  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_l
  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/curve25519/reduce.bend&name=lt_cancel_r
  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=cs_mo
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar.bend&name=cs_mz
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar.bend&name=mo
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar.bend&name=mz
  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/arith.bend&name=one_mul
  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/math/natural/arith.bend&name=mod_of
  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