author | wenzelm |
Thu, 30 May 2013 20:38:50 +0200 | |
changeset 52252 | 81fcc11d8c65 |
parent 51489 | f738e6dbd844 |
child 52729 | 412c9e0381a1 |
permissions | -rw-r--r-- |
50634 | 1 |
(* Author: Alessandro Coglio *) |
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theory Finite_Lattice |
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51115
7dbd6832a689
consolidation of library theories on product orders
haftmann
parents:
50634
diff
changeset
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imports Product_Order |
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begin |
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text {* A non-empty finite lattice is a complete lattice. |
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Since types are never empty in Isabelle/HOL, |
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a type of classes @{class finite} and @{class lattice} |
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should also have class @{class complete_lattice}. |
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A type class is defined |
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that extends classes @{class finite} and @{class lattice} |
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with the operators @{const bot}, @{const top}, @{const Inf}, and @{const Sup}, |
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along with assumptions that define these operators |
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in terms of the ones of classes @{class finite} and @{class lattice}. |
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The resulting class is a subclass of @{class complete_lattice}. |
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Classes @{class bot} and @{class top} already include assumptions that suffice |
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to define the operators @{const bot} and @{const top} (as proved below), |
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and so no explicit assumptions on these two operators are needed |
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in the following type class.% |
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\footnote{The Isabelle/HOL library does not provide |
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syntactic classes for the operators @{const bot} and @{const top}.} *} |
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class finite_lattice_complete = finite + lattice + bot + top + Inf + Sup + |
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assumes Inf_def: "Inf A = Finite_Set.fold inf top A" |
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assumes Sup_def: "Sup A = Finite_Set.fold sup bot A" |
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-- "No explicit assumptions on @{const bot} or @{const top}." |
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instance finite_lattice_complete \<subseteq> bounded_lattice .. |
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-- "This subclass relation eases the proof of the next two lemmas." |
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lemma finite_lattice_complete_bot_def: |
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"(bot::'a::finite_lattice_complete) = \<Sqinter>\<^bsub>fin\<^esub>UNIV" |
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by (metis finite_UNIV sup_Inf_absorb sup_bot_left iso_tuple_UNIV_I) |
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-- "Derived definition of @{const bot}." |
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lemma finite_lattice_complete_top_def: |
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"(top::'a::finite_lattice_complete) = \<Squnion>\<^bsub>fin\<^esub>UNIV" |
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by (metis finite_UNIV inf_Sup_absorb inf_top_left iso_tuple_UNIV_I) |
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-- "Derived definition of @{const top}." |
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lemma finite_lattice_complete_Inf_empty: |
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"Inf {} = (top :: 'a::finite_lattice_complete)" |
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by (simp add: Inf_def) |
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lemma finite_lattice_complete_Sup_empty: |
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"Sup {} = (bot :: 'a::finite_lattice_complete)" |
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by (simp add: Sup_def) |
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lemma finite_lattice_complete_Inf_insert: |
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fixes A :: "'a::finite_lattice_complete set" |
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shows "Inf (insert x A) = inf x (Inf A)" |
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proof - |
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interpret comp_fun_idem "inf :: 'a \<Rightarrow> _" by (fact comp_fun_idem_inf) |
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show ?thesis by (simp add: Inf_def) |
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qed |
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lemma finite_lattice_complete_Sup_insert: |
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fixes A :: "'a::finite_lattice_complete set" |
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shows "Sup (insert x A) = sup x (Sup A)" |
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proof - |
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interpret comp_fun_idem "sup :: 'a \<Rightarrow> _" by (fact comp_fun_idem_sup) |
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show ?thesis by (simp add: Sup_def) |
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qed |
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text {* The definitional assumptions |
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on the operators @{const Inf} and @{const Sup} |
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of class @{class finite_lattice_complete} |
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ensure that they yield infimum and supremum, |
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as required for a complete lattice. *} |
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lemma finite_lattice_complete_Inf_lower: |
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"(x::'a::finite_lattice_complete) \<in> A \<Longrightarrow> Inf A \<le> x" |
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using finite [of A] by (induct A) (auto simp add: finite_lattice_complete_Inf_insert intro: le_infI2) |
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lemma finite_lattice_complete_Inf_greatest: |
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"\<forall>x::'a::finite_lattice_complete \<in> A. z \<le> x \<Longrightarrow> z \<le> Inf A" |
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using finite [of A] by (induct A) (auto simp add: finite_lattice_complete_Inf_empty finite_lattice_complete_Inf_insert) |
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lemma finite_lattice_complete_Sup_upper: |
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"(x::'a::finite_lattice_complete) \<in> A \<Longrightarrow> Sup A \<ge> x" |
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using finite [of A] by (induct A) (auto simp add: finite_lattice_complete_Sup_insert intro: le_supI2) |
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lemma finite_lattice_complete_Sup_least: |
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"\<forall>x::'a::finite_lattice_complete \<in> A. z \<ge> x \<Longrightarrow> z \<ge> Sup A" |
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using finite [of A] by (induct A) (auto simp add: finite_lattice_complete_Sup_empty finite_lattice_complete_Sup_insert) |
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instance finite_lattice_complete \<subseteq> complete_lattice |
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proof |
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qed (auto simp: |
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finite_lattice_complete_Inf_lower |
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finite_lattice_complete_Inf_greatest |
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finite_lattice_complete_Sup_upper |
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finite_lattice_complete_Sup_least) |
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text {* The product of two finite lattices is already a finite lattice. *} |
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lemma finite_Inf_prod: |
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"Inf(A::('a::finite_lattice_complete \<times> 'b::finite_lattice_complete) set) = |
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Finite_Set.fold inf top A" |
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by (metis Inf_fold_inf finite_code) |
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lemma finite_Sup_prod: |
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"Sup (A::('a::finite_lattice_complete \<times> 'b::finite_lattice_complete) set) = |
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Finite_Set.fold sup bot A" |
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by (metis Sup_fold_sup finite_code) |
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instance prod :: |
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(finite_lattice_complete, finite_lattice_complete) finite_lattice_complete |
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proof qed (auto simp: finite_Inf_prod finite_Sup_prod) |
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text {* Functions with a finite domain and with a finite lattice as codomain |
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already form a finite lattice. *} |
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lemma finite_Inf_fun: |
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"Inf (A::('a::finite \<Rightarrow> 'b::finite_lattice_complete) set) = |
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Finite_Set.fold inf top A" |
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by (metis Inf_fold_inf finite_code) |
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lemma finite_Sup_fun: |
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"Sup (A::('a::finite \<Rightarrow> 'b::finite_lattice_complete) set) = |
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Finite_Set.fold sup bot A" |
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by (metis Sup_fold_sup finite_code) |
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instance "fun" :: (finite, finite_lattice_complete) finite_lattice_complete |
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proof qed (auto simp: finite_Inf_fun finite_Sup_fun) |
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subsection {* Finite Distributive Lattices *} |
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text {* A finite distributive lattice is a complete lattice |
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whose @{const inf} and @{const sup} operators |
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distribute over @{const Sup} and @{const Inf}. *} |
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class finite_distrib_lattice_complete = |
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distrib_lattice + finite_lattice_complete |
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lemma finite_distrib_lattice_complete_sup_Inf: |
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"sup (x::'a::finite_distrib_lattice_complete) (Inf A) = (INF y:A. sup x y)" |
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apply (rule finite_induct) |
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apply (metis finite_code) |
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apply (metis INF_empty Inf_empty sup_top_right) |
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apply (metis INF_insert Inf_insert sup_inf_distrib1) |
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done |
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lemma finite_distrib_lattice_complete_inf_Sup: |
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"inf (x::'a::finite_distrib_lattice_complete) (Sup A) = (SUP y:A. inf x y)" |
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apply (rule finite_induct) |
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apply (metis finite_code) |
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apply (metis SUP_empty Sup_empty inf_bot_right) |
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apply (metis SUP_insert Sup_insert inf_sup_distrib1) |
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done |
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instance finite_distrib_lattice_complete \<subseteq> complete_distrib_lattice |
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proof |
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qed (auto simp: |
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finite_distrib_lattice_complete_sup_Inf |
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finite_distrib_lattice_complete_inf_Sup) |
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text {* The product of two finite distributive lattices |
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is already a finite distributive lattice. *} |
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instance prod :: |
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(finite_distrib_lattice_complete, finite_distrib_lattice_complete) |
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finite_distrib_lattice_complete |
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.. |
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text {* Functions with a finite domain |
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and with a finite distributive lattice as codomain |
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already form a finite distributive lattice. *} |
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instance "fun" :: |
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(finite, finite_distrib_lattice_complete) finite_distrib_lattice_complete |
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.. |
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subsection {* Linear Orders *} |
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text {* A linear order is a distributive lattice. |
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Since in Isabelle/HOL |
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a subclass must have all the parameters of its superclasses, |
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class @{class linorder} cannot be a subclass of @{class distrib_lattice}. |
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So class @{class linorder} is extended with |
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the operators @{const inf} and @{const sup}, |
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along with assumptions that define these operators |
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in terms of the ones of class @{class linorder}. |
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The resulting class is a subclass of @{class distrib_lattice}. *} |
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class linorder_lattice = linorder + inf + sup + |
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assumes inf_def: "inf x y = (if x \<le> y then x else y)" |
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assumes sup_def: "sup x y = (if x \<ge> y then x else y)" |
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text {* The definitional assumptions |
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on the operators @{const inf} and @{const sup} |
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of class @{class linorder_lattice} |
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ensure that they yield infimum and supremum, |
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and that they distribute over each other, |
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as required for a distributive lattice. *} |
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lemma linorder_lattice_inf_le1: "inf (x::'a::linorder_lattice) y \<le> x" |
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unfolding inf_def by (metis (full_types) linorder_linear) |
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lemma linorder_lattice_inf_le2: "inf (x::'a::linorder_lattice) y \<le> y" |
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unfolding inf_def by (metis (full_types) linorder_linear) |
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lemma linorder_lattice_inf_greatest: |
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"(x::'a::linorder_lattice) \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> inf y z" |
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unfolding inf_def by (metis (full_types)) |
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lemma linorder_lattice_sup_ge1: "sup (x::'a::linorder_lattice) y \<ge> x" |
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unfolding sup_def by (metis (full_types) linorder_linear) |
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lemma linorder_lattice_sup_ge2: "sup (x::'a::linorder_lattice) y \<ge> y" |
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unfolding sup_def by (metis (full_types) linorder_linear) |
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lemma linorder_lattice_sup_least: |
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"(x::'a::linorder_lattice) \<ge> y \<Longrightarrow> x \<ge> z \<Longrightarrow> x \<ge> sup y z" |
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by (auto simp: sup_def) |
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lemma linorder_lattice_sup_inf_distrib1: |
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"sup (x::'a::linorder_lattice) (inf y z) = inf (sup x y) (sup x z)" |
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by (auto simp: inf_def sup_def) |
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instance linorder_lattice \<subseteq> distrib_lattice |
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proof |
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qed (auto simp: |
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linorder_lattice_inf_le1 |
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linorder_lattice_inf_le2 |
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linorder_lattice_inf_greatest |
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linorder_lattice_sup_ge1 |
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linorder_lattice_sup_ge2 |
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linorder_lattice_sup_least |
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linorder_lattice_sup_inf_distrib1) |
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subsection {* Finite Linear Orders *} |
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text {* A (non-empty) finite linear order is a complete linear order. *} |
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class finite_linorder_complete = linorder_lattice + finite_lattice_complete |
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instance finite_linorder_complete \<subseteq> complete_linorder .. |
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text {* A (non-empty) finite linear order is a complete lattice |
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whose @{const inf} and @{const sup} operators |
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distribute over @{const Sup} and @{const Inf}. *} |
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instance finite_linorder_complete \<subseteq> finite_distrib_lattice_complete .. |
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end |