Phc.div10_lt: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/phc.bend as Phc
Lines 1271-1279 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/phc.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/phc.bend&start_line=1271&end_line=1279

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/math/natural/arith.bend as NR

# n / 10 < n for n >= 1.
def div10_lt(+n: Nat, +h: {Nat.is_le(1n, n) == True{} : Bool}) -> {Nat.is_lt(Nat.div(n, 10n), n) == True{} : Bool}:
  match n:
    case 0n:
      Empty.absurd({Nat.is_lt(Nat.div(0n, 10n), 0n) == True{} : Bool}, L.false_true(h))
    case 1n+ +m:
      +e = NR.le_div(9n, 1n+m, 1n+m)
      +f = N.lt_not_le(m, Nat.add(9n, Nat.mul(m, 10n)), lt9(m))
      N.not_le_lt(1n+m, Nat.div(1n+m, 10n), Equal.trans(Bool, Nat.is_le(1n+m, Nat.div(1n+m, 10n)), Nat.is_le(Nat.mul(1n+m, 10n), 1n+m), False{}, e, f))

Uses (14):
  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.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.div
  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.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/argon2/phc.bend&name=lt9
  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/nat.bend&name=lt_not_le
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=not_le_lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend&name=le_div