author | wenzelm |
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permissions | -rw-r--r-- |
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(* Title: HOL/Library/Finite_Lattice.thy |
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Author: Alessandro Coglio |
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*) |
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theory Finite_Lattice |
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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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class finite_lattice_complete = finite + lattice + bot + top + Inf + Sup + |
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assumes bot_def: "bot = Inf_fin UNIV" |
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assumes top_def: "top = Sup_fin UNIV" |
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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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text {* The definitional assumptions |
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on the operators @{const bot} and @{const top} |
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of class @{class finite_lattice_complete} |
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ensure that they yield bottom and top. *} |
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lemma finite_lattice_complete_bot_least: "(bot::'a::finite_lattice_complete) \<le> x" |
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by (auto simp: bot_def intro: Inf_fin.coboundedI) |
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instance finite_lattice_complete \<subseteq> order_bot |
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by default (auto simp: finite_lattice_complete_bot_least) |
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lemma finite_lattice_complete_top_greatest: "(top::'a::finite_lattice_complete) \<ge> x" |
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by (auto simp: top_def Sup_fin.coboundedI) |
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instance finite_lattice_complete \<subseteq> order_top |
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by default (auto simp: finite_lattice_complete_top_greatest) |
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instance finite_lattice_complete \<subseteq> bounded_lattice .. |
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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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lemma finite_lattice_complete_Inf_empty: "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: "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> _" |
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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> _" |
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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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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] |
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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] |
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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] |
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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] |
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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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finite_lattice_complete_Inf_empty |
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finite_lattice_complete_Sup_empty) |
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text {* The product of two finite lattices is already a finite lattice. *} |
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lemma finite_bot_prod: |
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"(bot :: ('a::finite_lattice_complete \<times> 'b::finite_lattice_complete)) = |
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Inf_fin UNIV" |
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by (metis Inf_fin.coboundedI UNIV_I bot.extremum_uniqueI finite_UNIV) |
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lemma finite_top_prod: |
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"(top :: ('a::finite_lattice_complete \<times> 'b::finite_lattice_complete)) = |
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Sup_fin UNIV" |
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by (metis Sup_fin.coboundedI UNIV_I top.extremum_uniqueI finite_UNIV) |
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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 :: (finite_lattice_complete, finite_lattice_complete) finite_lattice_complete |
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by default (auto simp: finite_bot_prod finite_top_prod 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_bot_fun: "(bot :: ('a::finite \<Rightarrow> 'b::finite_lattice_complete)) = Inf_fin UNIV" |
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by (metis Inf_UNIV Inf_fin_Inf empty_not_UNIV finite_code) |
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lemma finite_top_fun: "(top :: ('a::finite \<Rightarrow> 'b::finite_lattice_complete)) = Sup_fin UNIV" |
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by (metis Sup_UNIV Sup_fin_Sup empty_not_UNIV finite_code) |
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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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by default (auto simp: finite_bot_fun finite_top_fun 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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using finite |
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by (induct A rule: finite_induct) (simp_all add: sup_inf_distrib1) |
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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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A type class is defined |
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that extends class @{class linorder} |
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with 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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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 |
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