Theorem and_comm (a b : Bool) : (a ∧ b) ⇒ (b ∧ a) :=
    Discharge (λ H_ab, Conj (Conjunct2 H_ab) (Conjunct1 H_ab))

Theorem or_comm (a b : Bool) : (a ∨ b) ⇒ (b ∨ a) :=
    Discharge (λ H_ab, DisjCases H_ab
                                 (λ H_a, Disj2 b H_a)
                                 (λ H_b, Disj1 H_b a))

(* ---------------------------------
(EM a) is the excluded middle  a ∨ ¬a
(MT H H_na) is Modus Tollens with
       H    : (a ⇒ b) ⇒ a)
       H_na : ¬a
produces
       ¬(a ⇒ b)

NotImp1 applied to
        (MT H H_na) : ¬(a ⇒ b)
produces
        a
----------------------------------- *)
Theorem pierce (a b : Bool) : ((a ⇒ b) ⇒ a) ⇒ a :=
    Discharge (λ H, DisjCases (EM a)
                              (λ H_a, H_a)
                              (λ H_na, NotImp1 (MT H H_na)))

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