Scalar3.reduce_m: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar3.bend as Scalar3
Lines 94-101 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar3.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar3.bend&start_line=94&end_line=101

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend as C
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/src/crypto/curve25519/x25519.bend as X
import bend-collections-laws-crypto@1.0.0.0/src/crypto/ed25519/scalar.bend as ESC
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/u32alg.bend as A
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/limbs.bend as LM
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar2.bend as S2

def reduce_m(+one: Nat, +h1: {one == 1n : Nat}, +bs: List<&2, U32>, +hb: {LM.lea(bs, 255n) == True{} : Bool}) -> {FS.value(ESC.reduce(bs)) == Nat.mod(FS.value(bs), 1n+mmL(one)) : Nat}:
  +mm = mmL(one)
  +n = X.bitlen(bs)
  +V = FS.value(bs)
  %Equal.sym(List<&2, U32>, ESC.mac(n, bs, F.one(), F.zero()), S2.macg(n, bs, F.one(), F.zero(), ESC.comp_l()), S2.mac_eq(n, bs, F.one(), F.zero())) : {FS.value(_) == Nat.mod(V, 1n+mm) : Nat}
  +e = S2.mac_val(one, h1, mm, h2L(one, h1), ESC.comp_l(), cp_okb(), hvL(one, h1), n, bs, F.one(), F.zero(), hb, {==}, one_lt(one, h1), {==}, {==})
  +e2 = Equal.trans(Nat, Nat.add(WD.sc(n, 0n), Nat.mul(C.low(n, V), 1n)), Nat.add(0n, Nat.mul(C.low(n, V), 1n)), V, M.cong_l(WD.sc(n, 0n), 0n, Nat.mul(C.low(n, V), 1n), A.sc_zero(n)), Equal.trans(Nat, Nat.mul(C.low(n, V), 1n), C.low(n, V), V, NA.mul_one(C.low(n, V)), S2.low_all(bs, hb)))
  Equal.trans(Nat, FS.value(S2.macg(n, bs, F.one(), F.zero(), ESC.comp_l())), Nat.mod(Nat.add(WD.sc(n, 0n), Nat.mul(C.low(n, V), 1n)), 1n+mm), Nat.mod(V, 1n+mm), e, M.cong_mod(1n+mm, Nat.add(WD.sc(n, 0n), Nat.mul(C.low(n, V), 1n)), V, e2))

Uses (32):
  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.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/limbs.bend&name=lea
  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/ed25519/scalar2.bend&name=low_all
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar2.bend&name=mac_eq
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar2.bend&name=mac_val
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar2.bend&name=macg
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar3.bend&name=cp_okb
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar3.bend&name=h2L
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar3.bend&name=hvL
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar3.bend&name=mmL
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/ed25519/scalar3.bend&name=one_lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_one
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/u32alg.bend&name=sc_zero
  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/spec/crypto/curve25519/field.bend&name=value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=low
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/field.bend&name=one
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/field.bend&name=zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/x25519.bend&name=bitlen
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/ed25519/scalar.bend&name=comp_l
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/ed25519/scalar.bend&name=mac
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/ed25519/scalar.bend&name=reduce