Fround.bl_shift: 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 315-325 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=315&end_line=325

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/proofs/lib/nat.bend as N
import bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend as L
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-math@1.0.0.0/src/math/natural.bend as M
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64bl.bend as FO

def bl_shift(+k: Nat, +m: Nat, +hz: {Nat.is_eq(m, 0n) == False{} : Bool}) -> {M.bit_length(C.shift(k, m)) == Nat.add(M.bit_length(m), k) : Nat}:
  +B = M.bit_length(m)
  +j = Nat.sub(B, 1n)
  +ej = N.sub_add(B, 1n, FO.bl_pos(m, hz, B, {==}))
  +nf = Equal.trans(Bool, C.fits(Nat.add(k, j), C.shift(k, m)), C.fits(j, m), False{}, fits_sh(k, j, m), bl_nfit(m, hz))
  +f0 = L.subst(Nat, z => {C.fits(z, m) == True{} : Bool}, B, 1n+j, Equal.sym(Nat, 1n+j, B, ej), bl_fit(m))
  +f1 = Equal.trans(Bool, C.fits(Nat.add(k, 1n+j), C.shift(k, m)), C.fits(1n+j, m), True{}, fits_sh(k, 1n+j, m), f0)
  +f2 = L.subst(Nat, z => {C.fits(z, C.shift(k, m)) == True{} : Bool}, Nat.add(k, 1n+j), 1n+Nat.add(k, j), N.add_succ(k, j), f1)
  +eb = bl_c(Nat.add(k, j), C.shift(k, m), nf, f2, Nat.cmp(M.bit_length(C.shift(k, m)), 1n+Nat.add(k, j)), {==})
  +er = Equal.trans(Nat, Nat.add(B, k), Nat.add(1n+j, k), 1n+Nat.add(k, j), Equal.cong(Nat, Nat, z => Nat.add(z, k), B, 1n+j, Equal.sym(Nat, 1n+j, B, ej)), Equal.cong(Nat, Nat, z => 1n+z, Nat.add(j, k), Nat.add(k, j), NA.add_comm(j, k)))
  Equal.trans(Nat, M.bit_length(C.shift(k, m)), 1n+Nat.add(k, j), Nat.add(B, k), eb, Equal.sym(Nat, Nat.add(B, k), 1n+Nat.add(k, j), er))

Uses (21):
  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=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.cmp
  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.sub
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_comm
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend&name=subst
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=add_succ
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=sub_add
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/fround.bend&name=bl_c
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/fround.bend&name=bl_fit
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/fround.bend&name=bl_nfit
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/fround.bend&name=fits_sh
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64bl.bend&name=bl_pos
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/lib/common.bend&name=fits
  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/src/math/natural.bend&name=bit_length