876aa20ad6
Also prove a theorem similar to Lemma 7.3.1 There are still some sorry's in hit.suspension
47 lines
1.4 KiB
Text
47 lines
1.4 KiB
Text
/-
|
|
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,
|
|
do 3 focus (intro b;cases b;all_goals (exact idp))
|
|
--should information be propagated with all_goals?
|
|
-- all_goals (intro b;cases b),
|
|
-- all_goals (exact idp)
|
|
-- all_goals (focus (intro b;cases b;all_goals (exact idp))),
|
|
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
|