Fround.rs_c: 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 360-374 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=360&end_line=374

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/src/math/f64.bend as F
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/proofs/math/typed/width.bend as WW
import bend-collections-laws-math@1.0.0.0/src/math/natural.bend as M

def rs_c(+s: Bool, +k: Nat, +m: Nat, +x: Nat, +z: Bool, +hz: {Nat.is_eq(m, 0n) == z : Bool}) -> {SF.round(s, C.shift(k, m), x) == SF.round(s, m, Nat.add(x, k)) : F.F64}:
  match z:
    case True{}:
      +e0 = Equal.trans(Bool, Nat.is_eq(C.shift(k, m), 0n), Nat.is_eq(m, 0n), True{}, WW.shift_eq0(k, m), hz)
      Equal.trans(F.F64, SF.round(s, C.shift(k, m), x), SF.zero(s), SF.round(s, m, Nat.add(x, k)), Equal.cong(Bool, F.F64, t => SF.pick(F.F64, t, SF.zero(s), SF.round_u(s, C.shift(k, m), x, Nat.max(Nat.sub(Nat.add(x, M.bit_length(C.shift(k, m))), 53n), Nat.sub(SF.zb(), 1074n)))), Nat.is_eq(C.shift(k, m), 0n), True{}, e0), Equal.sym(F.F64, SF.round(s, m, Nat.add(x, k)), SF.zero(s), Equal.cong(Bool, F.F64, t => SF.pick(F.F64, t, SF.zero(s), SF.round_u(s, m, Nat.add(x, k), Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n)))), Nat.is_eq(m, 0n), True{}, hz)))
    case False{}:
      +B = M.bit_length(m)
      +e0 = Equal.trans(Bool, Nat.is_eq(C.shift(k, m), 0n), Nat.is_eq(m, 0n), False{}, WW.shift_eq0(k, m), hz)
      +l1 = Equal.cong(Bool, F.F64, t => SF.pick(F.F64, t, SF.zero(s), SF.round_u(s, C.shift(k, m), x, Nat.max(Nat.sub(Nat.add(x, M.bit_length(C.shift(k, m))), 53n), Nat.sub(SF.zb(), 1074n)))), Nat.is_eq(C.shift(k, m), 0n), False{}, e0)
      +ea = Equal.trans(Nat, Nat.add(x, Nat.add(B, k)), Nat.add(x, Nat.add(k, B)), Nat.add(Nat.add(x, k), B), Equal.cong(Nat, Nat, z => Nat.add(x, z), Nat.add(B, k), Nat.add(k, B), NA.add_comm(B, k)), Equal.sym(Nat, Nat.add(Nat.add(x, k), B), Nat.add(x, Nat.add(k, B)), NA.add_assoc(x, k, B)))
      +eu = Equal.trans(Nat, Nat.max(Nat.sub(Nat.add(x, M.bit_length(C.shift(k, m))), 53n), Nat.sub(SF.zb(), 1074n)), Nat.max(Nat.sub(Nat.add(x, Nat.add(B, k)), 53n), Nat.sub(SF.zb(), 1074n)), Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n)), Equal.cong(Nat, Nat, w => Nat.max(Nat.sub(Nat.add(x, w), 53n), Nat.sub(SF.zb(), 1074n)), M.bit_length(C.shift(k, m)), Nat.add(B, k), bl_shift(k, m, hz)), Equal.cong(Nat, Nat, w => Nat.max(Nat.sub(w, 53n), Nat.sub(SF.zb(), 1074n)), Nat.add(x, Nat.add(B, k)), Nat.add(Nat.add(x, k), B), ea))
      +l2 = Equal.cong(Nat, F.F64, w => SF.round_u(s, C.shift(k, m), x, w), Nat.max(Nat.sub(Nat.add(x, M.bit_length(C.shift(k, m))), 53n), Nat.sub(SF.zb(), 1074n)), Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n)), eu)
      +l3 = Equal.cong(Nat, F.F64, w => SF.pack(s, w, Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n))), SF.pick(Nat, Nat.is_le(x, Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n))), SF.rne(C.shift(k, m), Nat.sub(Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n)), x)), C.shift(Nat.sub(x, Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n))), C.shift(k, m))), SF.pick(Nat, Nat.is_le(Nat.add(x, k), Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n))), SF.rne(m, Nat.sub(Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n)), Nat.add(x, k))), C.shift(Nat.sub(Nat.add(x, k), Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n))), m)), qeq_c(k, m, x, Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n)), Nat.is_le(x, Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n))), {==}, Nat.is_le(Nat.add(x, k), Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n))), {==}))
      +r1 = Equal.cong(Bool, F.F64, t => SF.pick(F.F64, t, SF.zero(s), SF.round_u(s, m, Nat.add(x, k), Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n)))), Nat.is_eq(m, 0n), False{}, hz)
      Equal.trans(F.F64, SF.round(s, C.shift(k, m), x), SF.round_u(s, m, Nat.add(x, k), Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n))), SF.round(s, m, Nat.add(x, k)), Equal.trans(F.F64, SF.round(s, C.shift(k, m), x), SF.round_u(s, C.shift(k, m), x, Nat.max(Nat.sub(Nat.add(x, M.bit_length(C.shift(k, m))), 53n), Nat.sub(SF.zb(), 1074n))), SF.round_u(s, m, Nat.add(x, k), Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n))), l1, Equal.trans(F.F64, SF.round_u(s, C.shift(k, m), x, Nat.max(Nat.sub(Nat.add(x, M.bit_length(C.shift(k, m))), 53n), Nat.sub(SF.zb(), 1074n))), SF.round_u(s, C.shift(k, m), x, Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n))), SF.round_u(s, m, Nat.add(x, k), Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n))), l2, l3)), Equal.sym(F.F64, SF.round(s, m, Nat.add(x, k)), SF.round_u(s, m, Nat.add(x, k), Nat.max(Nat.sub(Nat.add(Nat.add(x, k), M.bit_length(m)), 53n), Nat.sub(SF.zb(), 1074n))), r1))

Uses (25):
  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.is_eq
  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.max
  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_assoc
  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/math/random/fround.bend&name=bl_shift
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/fround.bend&name=qeq_c
  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=shift
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/f64.bend&name=pack
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/f64.bend&name=pick
  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=round
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/f64.bend&name=round_u
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/f64.bend&name=zb
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/f64.bend&name=zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/f64.bend&name=F64
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/natural.bend&name=bit_length