Fround.rne_exact: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/random/fround.bend as Fround
Lines 270-285 of bend-collections-laws-math@1.0.0.0/proofs/math/random/fround.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/fround.bend&start_line=270&end_line=285

import Base
import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-math@1.0.0.0/spec/math/f64.bend as SF
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend as WW

def rne_exact(+k: Nat, +m: Nat) -> {SF.rne(C.shift(k, m), k) == m : Nat}:
  match k:
    case 0n:
      {==}
    case 1n+ +i:
      +S = C.shift(1n+i, m)
      +h = C.shift(i, 1n)
      +eq = WW.high_u(1n+i, 0n, m, fz(1n+i))
      +el = WW.low_u(1n+i, 0n, m, fz(1n+i))
      +e1 = Equal.cong(Nat, Nat, z => SF.rne_up(z, C.low(1n+i, S), h), C.high(1n+i, S), m, eq)
      +e2 = Equal.cong(Nat, Nat, z => SF.rne_up(m, z, h), C.low(1n+i, S), 0n, el)
      +hz = N.is_eq_sym_false(h, 0n, Equal.trans(Bool, Nat.is_eq(h, 0n), Nat.is_eq(1n, 0n), False{}, WW.shift_eq0(i, 1n), {==}))
      +e3 = Equal.cong(Bool, Nat, t => Nat.add(m, SF.b2n(Bool.or(t, Bool.and(Nat.is_eq(0n, h), Nat.is_eq(Nat.mod(m, 2n), 1n))))), Nat.is_lt(h, 0n), False{}, lt0f(h))
      +e4 = Equal.cong(Bool, Nat, t => Nat.add(m, SF.b2n(Bool.or(False{}, Bool.and(t, Nat.is_eq(Nat.mod(m, 2n), 1n))))), Nat.is_eq(0n, h), False{}, hz)
      +e5 = N.add_zero(m)
      Equal.trans(Nat, SF.rne_up(C.high(1n+i, S), C.low(1n+i, S), h), SF.rne_up(m, C.low(1n+i, S), h), m, e1, Equal.trans(Nat, SF.rne_up(m, C.low(1n+i, S), h), SF.rne_up(m, 0n, h), m, e2, Equal.trans(Nat, SF.rne_up(m, 0n, h), Nat.add(m, SF.b2n(Bool.or(False{}, Bool.and(Nat.is_eq(0n, h), Nat.is_eq(Nat.mod(m, 2n), 1n))))), m, e3, Equal.trans(Nat, Nat.add(m, SF.b2n(Bool.or(False{}, Bool.and(Nat.is_eq(0n, h), Nat.is_eq(Nat.mod(m, 2n), 1n))))), Nat.add(m, 0n), m, e4, e5))))

Uses (23):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool.and
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool.or
  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.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.is_eq
  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=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=add_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=is_eq_sym_false
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/fround.bend&name=fz
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/fround.bend&name=lt0f
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend&name=high_u
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend&name=low_u
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend&name=shift_eq0
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/lib/common.bend&name=high
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/lib/common.bend&name=low
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/lib/common.bend&name=shift
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/f64.bend&name=b2n
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/f64.bend&name=rne
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/f64.bend&name=rne_up