21 lines
567 B
Text
21 lines
567 B
Text
import data.nat
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open nat
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namespace foo
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definition lt.trans {a b c : nat} (H₁ : a < b) (H₂ : b < c) : a < c :=
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have aux : a < b → a < c, from
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le.rec_on H₂
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(λ h₁, lt.step h₁)
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(λ b₁ bb₁ ih h₁, by constructor; exact ih h₁),
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aux H₁
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definition succ_lt_succ {a b : nat} (H : a < b) : succ a < succ b :=
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le.rec_on H
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(by constructor)
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(λ b hlt ih, lt.trans ih (by constructor))
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definition lt_of_succ_lt {a b : nat} (H : succ a < b) : a < b :=
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le.rec_on H
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(by constructor; constructor)
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(λ b h ih, by constructor; exact ih)
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end foo
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