28 lines
No EOL
732 B
Text
28 lines
No EOL
732 B
Text
import macros.
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theorem and_comm (a b : Bool) : (a ∧ b) → (b ∧ a)
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:= λ H_ab, and::intro (and::elimr H_ab) (and::eliml H_ab).
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theorem or_comm (a b : Bool) : (a ∨ b) → (b ∨ a)
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:= λ H_ab,
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or::elim H_ab
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(λ H_a, or::intror b H_a)
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(λ H_b, or::introl H_b a).
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-- (em a) is the excluded middle a ∨ ¬a
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-- (mt H H_na) is Modus Tollens with
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-- H : (a → b) → a)
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-- H_na : ¬a
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-- produces
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-- ¬(a → b)
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--
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-- not::imp::eliml applied to
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-- (MT H H_na) : ¬(a → b)
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-- produces
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-- a
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theorem pierce (a b : Bool) : ((a → b) → a) → a
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:= λ H, or::elim (em a)
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(λ H_a, H_a)
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(λ H_na, not::imp::eliml (mt H H_na)).
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print environment 3. |