Require Import UniMath.Foundations.Init.
Require Import UniMath.Foundations.Sets.

Definition andb : bool -> bool -> bool.
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Definition orb : bool -> bool -> bool.
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Definition implb : bool -> bool -> bool.
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Lemma andb_is_associative :
  ∏ b1 b2 b3 : bool, andb (andb b1 b2) b3 = andb b1 (andb b2 b3).
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Lemma orb_is_associative :
  ∏ b1 b2 b3 : bool, orb (orb b1 b2) b3 = orb b1 (orb b2 b3).
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Lemma andb_is_commutative :
  ∏ b1 b2 : bool, andb b1 b2 = andb b2 b1.
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Lemma orb_is_commutative :
  ∏ b1 b2 : bool, orb b1 b2 = orb b2 b1.
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