lean2/hott/types/bool.hlean

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/-
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 not_is_hset_type : ¬is_hset Type₀ :=
assume H : is_hset Type₀,
absurd !is_hset.elim eq_bnot_ne_idp
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