25 lines
826 B
Text
25 lines
826 B
Text
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import tactic
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using Nat
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-- Manual proof using symmetry, transitivity, distributivity and 1*x=x.
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theorem T1 (a b c : Nat)
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(H1 : a = b + c) -- First hypothesis
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(H2 : b = c) -- Second hypothesis
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: a = 2*c -- Conclusion
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:= calc a = b + c : H1
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... = c + c : { H2 }
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... = 1*c + 1*c : { symm (mul_onel c) }
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... = (1 + 1)*c : symm (distributel 1 1 c)
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... = 2*c : refl (2*c)
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add_rewrite mul_onel
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-- The simplifier can already compress the proof above.
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theorem T2 (a b c : Nat)
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(H1 : a = b + c) -- first hypothesis
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(H2 : b = c) -- second hypothesis
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: a = 2*c
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:= calc a = 1*c + 1*c : by simp
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... = (1 + 1)*c : symm (distributel 1 1 c)
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... = 2*c : refl (2*c)
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