/- Copyright (c) 2015 Floris van Doorn. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Floris van Doorn Theorems about the booleans -/ open is_equiv eq equiv function is_trunc option unit namespace bool definition ff_ne_tt : ff = tt → empty | [none] definition is_equiv_bnot [instance] [priority 500] : is_equiv bnot := begin fapply is_equiv.mk, exact bnot, all_goals (intro b;cases b), do 6 reflexivity -- all_goals (focus (intro b;cases b;all_goals reflexivity)), end definition equiv_bnot : bool ≃ bool := equiv.mk bnot _ definition eq_bnot : bool = bool := ua equiv_bnot definition eq_bnot_ne_idp : eq_bnot ≠ idp := assume H : eq_bnot = idp, assert H2 : bnot = id, from !cast_ua_fn⁻¹ ⬝ ap cast H, absurd (ap10 H2 tt) ff_ne_tt definition bool_equiv_option_unit : bool ≃ option unit := begin fapply equiv.MK, { intro b, cases b, exact none, exact some star}, { intro u, cases u, exact ff, exact tt}, { intro u, cases u with u, reflexivity, cases u, reflexivity}, { intro b, cases b, reflexivity, reflexivity}, end end bool