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(* Title: HOL/Library/List_lexord.thy
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ID: $Id$
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Author: Norbert Voelker
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*)
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header {* Lexicographic order on lists *}
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theory List_lexord
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imports Main
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begin
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instance list :: (ord) ord ..
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defs (overloaded)
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list_le_def: "(xs::('a::ord) list) \<le> ys \<equiv> (xs < ys \<or> xs = ys)"
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list_less_def: "(xs::('a::ord) list) < ys \<equiv> (xs, ys) \<in> lexord {(u,v). u < v}"
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lemmas list_ord_defs = list_less_def list_le_def
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instance list :: (order) order
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apply (intro_classes, unfold list_ord_defs)
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apply (rule disjI2, safe)
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apply (blast intro: lexord_trans transI order_less_trans)
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apply (rule_tac r1 = "{(a::'a,b). a < b}" in lexord_irreflexive [THEN notE])
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apply simp
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apply (blast intro: lexord_trans transI order_less_trans)
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apply (rule_tac r1 = "{(a::'a,b). a < b}" in lexord_irreflexive [THEN notE])
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apply simp
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apply assumption
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done
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instance list::(linorder)linorder
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apply (intro_classes, unfold list_le_def list_less_def, safe)
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apply (cut_tac x = x and y = y and r = "{(a,b). a < b}" in lexord_linear)
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apply force
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apply simp
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done
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lemma not_less_Nil[simp]: "~(x < [])"
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by (unfold list_less_def) simp
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lemma Nil_less_Cons[simp]: "[] < a # x"
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by (unfold list_less_def) simp
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lemma Cons_less_Cons[simp]: "(a # x < b # y) = (a < b | a = b & x < y)"
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by (unfold list_less_def) simp
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lemma le_Nil[simp]: "(x <= []) = (x = [])"
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by (unfold list_ord_defs, cases x) auto
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lemma Nil_le_Cons [simp]: "([] <= x)"
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by (unfold list_ord_defs, cases x) auto
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lemma Cons_le_Cons[simp]: "(a # x <= b # y) = (a < b | a = b & x <= y)"
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by (unfold list_ord_defs) auto
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end
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