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author | ballarin |

Tue, 31 Jul 2007 14:18:24 +0200 | |

changeset 24087 | eb025d149a34 |

parent 24086 | 21900a460ba4 |

child 24088 | c2d8270e53a5 |

Proper interpretation of total orders in lattices.

--- a/src/HOL/Algebra/Lattice.thy Tue Jul 31 13:31:01 2007 +0200 +++ b/src/HOL/Algebra/Lattice.thy Tue Jul 31 14:18:24 2007 +0200 @@ -656,7 +656,7 @@ subsection {* Total Orders *} -locale total_order = lattice + +locale total_order = partial_order + assumes total: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x" text {* Introduction rule: the usual definition of total order *} @@ -664,50 +664,52 @@ lemma (in partial_order) total_orderI: assumes total: "!!x y. [| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x" shows "total_order L" -proof intro_locales - show "lattice_axioms L" - proof (rule lattice_axioms.intro) - fix x y - assume L: "x \<in> carrier L" "y \<in> carrier L" - show "EX s. least L s (Upper L {x, y})" - proof - - note total L - moreover - { - assume "x \<sqsubseteq> y" - with L have "least L y (Upper L {x, y})" - by (rule_tac least_UpperI) auto - } - moreover - { - assume "y \<sqsubseteq> x" - with L have "least L x (Upper L {x, y})" - by (rule_tac least_UpperI) auto - } - ultimately show ?thesis by blast - qed - next - fix x y - assume L: "x \<in> carrier L" "y \<in> carrier L" - show "EX i. greatest L i (Lower L {x, y})" - proof - - note total L - moreover - { - assume "y \<sqsubseteq> x" - with L have "greatest L y (Lower L {x, y})" - by (rule_tac greatest_LowerI) auto - } - moreover - { - assume "x \<sqsubseteq> y" - with L have "greatest L x (Lower L {x, y})" - by (rule_tac greatest_LowerI) auto - } - ultimately show ?thesis by blast - qed + by unfold_locales (rule total) + +text {* Total orders are lattices. *} + +interpretation total_order < lattice +proof unfold_locales + fix x y + assume L: "x \<in> carrier L" "y \<in> carrier L" + show "EX s. least L s (Upper L {x, y})" + proof - + note total L + moreover + { + assume "x \<sqsubseteq> y" + with L have "least L y (Upper L {x, y})" + by (rule_tac least_UpperI) auto + } + moreover + { + assume "y \<sqsubseteq> x" + with L have "least L x (Upper L {x, y})" + by (rule_tac least_UpperI) auto + } + ultimately show ?thesis by blast qed -qed (rule total total_order_axioms.intro)+ +next + fix x y + assume L: "x \<in> carrier L" "y \<in> carrier L" + show "EX i. greatest L i (Lower L {x, y})" + proof - + note total L + moreover + { + assume "y \<sqsubseteq> x" + with L have "greatest L y (Lower L {x, y})" + by (rule_tac greatest_LowerI) auto + } + moreover + { + assume "x \<sqsubseteq> y" + with L have "greatest L x (Lower L {x, y})" + by (rule_tac greatest_LowerI) auto + } + ultimately show ?thesis by blast + qed +qed subsection {* Complete lattices *}