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header {* Order on product types *}
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theory Precedence_ord
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imports Main
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begin
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datatype precedence = Prc nat nat
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instantiation precedence :: order
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begin
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definition
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precedence_le_def: "x \<le> y \<longleftrightarrow> (case (x, y) of
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(Prc fx sx, Prc fy sy) \<Rightarrow>
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fx < fy \<or> (fx \<le> fy \<and> sy \<le> sx))"
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definition
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precedence_less_def: "x < y \<longleftrightarrow> (case (x, y) of
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(Prc fx sx, Prc fy sy) \<Rightarrow>
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fx < fy \<or> (fx \<le> fy \<and> sy < sx))"
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instance
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proof
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qed (auto simp: precedence_le_def precedence_less_def
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intro: order_trans split:precedence.splits)
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end
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instance precedence :: preorder ..
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instance precedence :: linorder proof
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qed (auto simp: precedence_le_def precedence_less_def
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intro: order_trans split:precedence.splits)
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end
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