def bl_shift

bend-collections-laws-math@1.0.0.0 · proofs/math/random/fround.bend:315

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

315def 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}:
316  +B = M.bit_length(m)
317  +j = Nat.sub(B, 1n)
318  +ej = N.sub_add(B, 1n, FO.bl_pos(m, hz, B, {==}))
319  +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))
320  +f0 = L.subst(Nat, z => {C.fits(z, m) == True{} : Bool}, B, 1n+j, Equal.sym(Nat, 1n+j, B, ej), bl_fit(m))
321  +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)
322  +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)
323  +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)), {==})
324  +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)))
325  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))