variable N : Type variable h : N -> N -> N -- Specialize congruence theorem for h-applications theorem congrh {a1 a2 b1 b2 : N} (H1 : a1 = b1) (H2 : a2 = b2) : (h a1 a2) = (h b1 b2) := congr (congr (refl h) H1) H2 -- Declare some variables variable a : N variable b : N variable c : N variable d : N variable e : N -- Add axioms stating facts about these variables axiom H1 : (a = b ∧ b = c) ∨ (d = c ∧ a = d) axiom H2 : b = e -- Proof that (h a b) = (h c e) theorem T1 : (h a b) = (h c e) := or_elim H1 (λ C1, congrh (trans (and_eliml C1) (and_elimr C1)) H2) (λ C2, congrh (trans (and_elimr C2) (and_eliml C2)) H2) -- We can use theorem T1 to prove other theorems theorem T2 : (h a (h a b)) = (h a (h c e)) := congrh (refl a) T1 -- Display the last two objects (i.e., theorems) added to the environment print environment 2 -- print implicit arguments set_option lean::pp::implicit true set_option pp::width 150 print environment 2