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