Sqrtn.isqrt_le: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/sqrtn.bend as Sqrtn
Lines 142-146 of bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/sqrtn.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/sqrtn.bend&start_line=142&end_line=146

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/lemmas/proofs/nat_algebra.bend as A
import bend-collections-laws-crypto@1.0.0.0/src/math/natural.bend as M
import bend-collections-laws-crypto@1.0.0.0/src/math/pow2.bend as P2

# isqrt(n)^2 <= n   (Mathlib Nat.sqrt_le')
def isqrt_le(+n: Nat) -> {Nat.is_le(Nat.pow(M.isqrt(n), 2n), n) == True{} : Bool}:
  +g = P2.pow2t(1n+Nat.div(M.bit_length(n), 2n))
  %Equal.sym(Nat, Nat.pow(M.isqrt(n), 2n), Nat.mul(M.isqrt(n), M.isqrt(n)), Equal.cong(Nat, Nat, z => Nat.mul(M.isqrt(n), z), Nat.mul(M.isqrt(n), 1n), M.isqrt(n), A.mul_one(M.isqrt(n)))) : {Nat.is_le(_, n) == True{} : Bool}
  iter_sq(g, n, g, M.sqrt_next(n, g), Nat.is_lt(M.sqrt_next(n, g), g), {==}, {==}, N.le_refl(g))

Uses (16):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.cong
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  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=Base/base.bend&name=Nat.pow
  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/nat.bend&name=le_refl
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/sqrtn.bend&name=iter_sq
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/math/natural.bend&name=bit_length
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/math/natural.bend&name=isqrt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/math/natural.bend&name=sqrt_next
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/math/pow2.bend&name=pow2t