41 lines
1.1 KiB
Text
41 lines
1.1 KiB
Text
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/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: types.bool
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Authors: Floris van Doorn
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Theorems about the booleans
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-/
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open is_equiv eq equiv function is_trunc
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namespace bool
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definition ff_ne_tt : ff = tt → empty
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| [none]
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definition is_equiv_bnot [instance] [priority 500] : is_equiv bnot :=
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begin
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fapply is_equiv.mk,
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exact bnot,
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do 3 focus (intro b;cases b;all_goals (exact idp))
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--should information be propagated with all_goals?
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-- all_goals (intro b;cases b),
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-- all_goals (exact idp)
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-- all_goals (focus (intro b;cases b;all_goals (exact idp))),
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end
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definition equiv_bnot : bool ≃ bool := equiv.mk bnot _
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definition eq_bnot : bool = bool := ua equiv_bnot
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definition eq_bnot_ne_idp : eq_bnot ≠ idp :=
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assume H : eq_bnot = idp,
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assert H2 : bnot = id, from !cast_ua_fn⁻¹ ⬝ ap cast H,
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absurd (ap10 H2 tt) ff_ne_tt
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definition not_is_hset_type : ¬is_hset Type₀ :=
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assume H : is_hset Type₀,
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absurd !is_hset.elim eq_bnot_ne_idp
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end bool
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