author | ballarin |
Thu, 02 Mar 2017 21:16:02 +0100 | |
changeset 65099 | 30d0b2f1df76 |
child 66453 | cc19f7ca2ed6 |
permissions | -rw-r--r-- |
65099
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Knaster-Tarski fixed point theorem and Galois Connections.
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(* Title: HOL/Algebra/Order.thy |
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Knaster-Tarski fixed point theorem and Galois Connections.
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Author: Clemens Ballarin, started 7 November 2003 |
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Knaster-Tarski fixed point theorem and Galois Connections.
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Copyright: Clemens Ballarin |
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Knaster-Tarski fixed point theorem and Galois Connections.
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|
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Knaster-Tarski fixed point theorem and Galois Connections.
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Most congruence rules by Stephan Hohe. |
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With additional contributions from Alasdair Armstrong and Simon Foster. |
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Knaster-Tarski fixed point theorem and Galois Connections.
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*) |
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Knaster-Tarski fixed point theorem and Galois Connections.
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|
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Knaster-Tarski fixed point theorem and Galois Connections.
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theory Order |
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Knaster-Tarski fixed point theorem and Galois Connections.
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imports |
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Knaster-Tarski fixed point theorem and Galois Connections.
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"~~/src/HOL/Library/FuncSet" |
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Congruence |
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Knaster-Tarski fixed point theorem and Galois Connections.
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begin |
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Knaster-Tarski fixed point theorem and Galois Connections.
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|
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Knaster-Tarski fixed point theorem and Galois Connections.
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section \<open>Orders\<close> |
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|
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subsection \<open>Partial Orders\<close> |
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record 'a gorder = "'a eq_object" + |
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le :: "['a, 'a] => bool" (infixl "\<sqsubseteq>\<index>" 50) |
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Knaster-Tarski fixed point theorem and Galois Connections.
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|
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abbreviation inv_gorder :: "_ \<Rightarrow> 'a gorder" where |
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"inv_gorder L \<equiv> |
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\<lparr> carrier = carrier L, |
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eq = op .=\<^bsub>L\<^esub>, |
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le = (\<lambda> x y. y \<sqsubseteq>\<^bsub>L \<^esub>x) \<rparr>" |
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Knaster-Tarski fixed point theorem and Galois Connections.
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lemma inv_gorder_inv: |
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"inv_gorder (inv_gorder L) = L" |
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by simp |
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locale weak_partial_order = equivalence L for L (structure) + |
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assumes le_refl [intro, simp]: |
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"x \<in> carrier L ==> x \<sqsubseteq> x" |
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and weak_le_antisym [intro]: |
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"[| x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier L; y \<in> carrier L |] ==> x .= y" |
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and le_trans [trans]: |
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"[| x \<sqsubseteq> y; y \<sqsubseteq> z; x \<in> carrier L; y \<in> carrier L; z \<in> carrier L |] ==> x \<sqsubseteq> z" |
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and le_cong: |
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"\<lbrakk> x .= y; z .= w; x \<in> carrier L; y \<in> carrier L; z \<in> carrier L; w \<in> carrier L \<rbrakk> \<Longrightarrow> |
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x \<sqsubseteq> z \<longleftrightarrow> y \<sqsubseteq> w" |
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|
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Knaster-Tarski fixed point theorem and Galois Connections.
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definition |
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Knaster-Tarski fixed point theorem and Galois Connections.
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lless :: "[_, 'a, 'a] => bool" (infixl "\<sqsubset>\<index>" 50) |
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where "x \<sqsubset>\<^bsub>L\<^esub> y \<longleftrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> y & x .\<noteq>\<^bsub>L\<^esub> y" |
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Knaster-Tarski fixed point theorem and Galois Connections.
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|
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Knaster-Tarski fixed point theorem and Galois Connections.
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Knaster-Tarski fixed point theorem and Galois Connections.
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subsubsection \<open>The order relation\<close> |
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49 |
|
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context weak_partial_order |
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begin |
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Knaster-Tarski fixed point theorem and Galois Connections.
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|
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Knaster-Tarski fixed point theorem and Galois Connections.
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lemma le_cong_l [intro, trans]: |
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"\<lbrakk> x .= y; y \<sqsubseteq> z; x \<in> carrier L; y \<in> carrier L; z \<in> carrier L \<rbrakk> \<Longrightarrow> x \<sqsubseteq> z" |
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55 |
by (auto intro: le_cong [THEN iffD2]) |
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Knaster-Tarski fixed point theorem and Galois Connections.
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|
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Knaster-Tarski fixed point theorem and Galois Connections.
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lemma le_cong_r [intro, trans]: |
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"\<lbrakk> x \<sqsubseteq> y; y .= z; x \<in> carrier L; y \<in> carrier L; z \<in> carrier L \<rbrakk> \<Longrightarrow> x \<sqsubseteq> z" |
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by (auto intro: le_cong [THEN iffD1]) |
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60 |
|
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Knaster-Tarski fixed point theorem and Galois Connections.
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lemma weak_refl [intro, simp]: "\<lbrakk> x .= y; x \<in> carrier L; y \<in> carrier L \<rbrakk> \<Longrightarrow> x \<sqsubseteq> y" |
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by (simp add: le_cong_l) |
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Knaster-Tarski fixed point theorem and Galois Connections.
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|
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Knaster-Tarski fixed point theorem and Galois Connections.
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end |
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Knaster-Tarski fixed point theorem and Galois Connections.
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65 |
|
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Knaster-Tarski fixed point theorem and Galois Connections.
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lemma weak_llessI: |
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Knaster-Tarski fixed point theorem and Galois Connections.
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fixes R (structure) |
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68 |
assumes "x \<sqsubseteq> y" and "~(x .= y)" |
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shows "x \<sqsubset> y" |
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using assms unfolding lless_def by simp |
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Knaster-Tarski fixed point theorem and Galois Connections.
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71 |
|
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Knaster-Tarski fixed point theorem and Galois Connections.
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lemma lless_imp_le: |
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Knaster-Tarski fixed point theorem and Galois Connections.
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fixes R (structure) |
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Knaster-Tarski fixed point theorem and Galois Connections.
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74 |
assumes "x \<sqsubset> y" |
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Knaster-Tarski fixed point theorem and Galois Connections.
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75 |
shows "x \<sqsubseteq> y" |
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76 |
using assms unfolding lless_def by simp |
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Knaster-Tarski fixed point theorem and Galois Connections.
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77 |
|
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Knaster-Tarski fixed point theorem and Galois Connections.
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78 |
lemma weak_lless_imp_not_eq: |
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Knaster-Tarski fixed point theorem and Galois Connections.
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79 |
fixes R (structure) |
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Knaster-Tarski fixed point theorem and Galois Connections.
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80 |
assumes "x \<sqsubset> y" |
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81 |
shows "\<not> (x .= y)" |
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Knaster-Tarski fixed point theorem and Galois Connections.
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82 |
using assms unfolding lless_def by simp |
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Knaster-Tarski fixed point theorem and Galois Connections.
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83 |
|
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Knaster-Tarski fixed point theorem and Galois Connections.
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84 |
lemma weak_llessE: |
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Knaster-Tarski fixed point theorem and Galois Connections.
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85 |
fixes R (structure) |
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86 |
assumes p: "x \<sqsubset> y" and e: "\<lbrakk>x \<sqsubseteq> y; \<not> (x .= y)\<rbrakk> \<Longrightarrow> P" |
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Knaster-Tarski fixed point theorem and Galois Connections.
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shows "P" |
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88 |
using p by (blast dest: lless_imp_le weak_lless_imp_not_eq e) |
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89 |
|
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90 |
lemma (in weak_partial_order) lless_cong_l [trans]: |
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91 |
assumes xx': "x .= x'" |
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92 |
and xy: "x' \<sqsubset> y" |
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93 |
and carr: "x \<in> carrier L" "x' \<in> carrier L" "y \<in> carrier L" |
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94 |
shows "x \<sqsubset> y" |
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95 |
using assms unfolding lless_def by (auto intro: trans sym) |
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96 |
|
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97 |
lemma (in weak_partial_order) lless_cong_r [trans]: |
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98 |
assumes xy: "x \<sqsubset> y" |
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99 |
and yy': "y .= y'" |
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100 |
and carr: "x \<in> carrier L" "y \<in> carrier L" "y' \<in> carrier L" |
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101 |
shows "x \<sqsubset> y'" |
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102 |
using assms unfolding lless_def by (auto intro: trans sym) (*slow*) |
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Knaster-Tarski fixed point theorem and Galois Connections.
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103 |
|
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Knaster-Tarski fixed point theorem and Galois Connections.
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104 |
|
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105 |
lemma (in weak_partial_order) lless_antisym: |
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106 |
assumes "a \<in> carrier L" "b \<in> carrier L" |
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107 |
and "a \<sqsubset> b" "b \<sqsubset> a" |
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108 |
shows "P" |
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109 |
using assms |
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110 |
by (elim weak_llessE) auto |
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111 |
|
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112 |
lemma (in weak_partial_order) lless_trans [trans]: |
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113 |
assumes "a \<sqsubset> b" "b \<sqsubset> c" |
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114 |
and carr[simp]: "a \<in> carrier L" "b \<in> carrier L" "c \<in> carrier L" |
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115 |
shows "a \<sqsubset> c" |
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116 |
using assms unfolding lless_def by (blast dest: le_trans intro: sym) |
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117 |
|
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118 |
lemma weak_partial_order_subset: |
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119 |
assumes "weak_partial_order L" "A \<subseteq> carrier L" |
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120 |
shows "weak_partial_order (L\<lparr> carrier := A \<rparr>)" |
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121 |
proof - |
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122 |
interpret L: weak_partial_order L |
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123 |
by (simp add: assms) |
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124 |
interpret equivalence "(L\<lparr> carrier := A \<rparr>)" |
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125 |
by (simp add: L.equivalence_axioms assms(2) equivalence_subset) |
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126 |
show ?thesis |
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127 |
apply (unfold_locales, simp_all) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
128 |
using assms(2) apply auto[1] |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
129 |
using assms(2) apply auto[1] |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
130 |
apply (meson L.le_trans assms(2) contra_subsetD) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
131 |
apply (meson L.le_cong assms(2) subsetCE) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
132 |
done |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
133 |
qed |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
134 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
135 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
136 |
subsubsection \<open>Upper and lower bounds of a set\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
137 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
138 |
definition |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
139 |
Upper :: "[_, 'a set] => 'a set" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
140 |
where "Upper L A = {u. (ALL x. x \<in> A \<inter> carrier L --> x \<sqsubseteq>\<^bsub>L\<^esub> u)} \<inter> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
141 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
142 |
definition |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
143 |
Lower :: "[_, 'a set] => 'a set" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
144 |
where "Lower L A = {l. (ALL x. x \<in> A \<inter> carrier L --> l \<sqsubseteq>\<^bsub>L\<^esub> x)} \<inter> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
145 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
146 |
lemma Upper_closed [intro!, simp]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
147 |
"Upper L A \<subseteq> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
148 |
by (unfold Upper_def) clarify |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
149 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
150 |
lemma Upper_memD [dest]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
151 |
fixes L (structure) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
152 |
shows "[| u \<in> Upper L A; x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> u \<and> u \<in> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
153 |
by (unfold Upper_def) blast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
154 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
155 |
lemma (in weak_partial_order) Upper_elemD [dest]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
156 |
"[| u .\<in> Upper L A; u \<in> carrier L; x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> u" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
157 |
unfolding Upper_def elem_def |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
158 |
by (blast dest: sym) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
159 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
160 |
lemma Upper_memI: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
161 |
fixes L (structure) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
162 |
shows "[| !! y. y \<in> A ==> y \<sqsubseteq> x; x \<in> carrier L |] ==> x \<in> Upper L A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
163 |
by (unfold Upper_def) blast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
164 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
165 |
lemma (in weak_partial_order) Upper_elemI: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
166 |
"[| !! y. y \<in> A ==> y \<sqsubseteq> x; x \<in> carrier L |] ==> x .\<in> Upper L A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
167 |
unfolding Upper_def by blast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
168 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
169 |
lemma Upper_antimono: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
170 |
"A \<subseteq> B ==> Upper L B \<subseteq> Upper L A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
171 |
by (unfold Upper_def) blast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
172 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
173 |
lemma (in weak_partial_order) Upper_is_closed [simp]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
174 |
"A \<subseteq> carrier L ==> is_closed (Upper L A)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
175 |
by (rule is_closedI) (blast intro: Upper_memI)+ |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
176 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
177 |
lemma (in weak_partial_order) Upper_mem_cong: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
178 |
assumes a'carr: "a' \<in> carrier L" and Acarr: "A \<subseteq> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
179 |
and aa': "a .= a'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
180 |
and aelem: "a \<in> Upper L A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
181 |
shows "a' \<in> Upper L A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
182 |
proof (rule Upper_memI[OF _ a'carr]) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
183 |
fix y |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
184 |
assume yA: "y \<in> A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
185 |
hence "y \<sqsubseteq> a" by (intro Upper_memD[OF aelem, THEN conjunct1] Acarr) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
186 |
also note aa' |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
187 |
finally |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
188 |
show "y \<sqsubseteq> a'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
189 |
by (simp add: a'carr subsetD[OF Acarr yA] subsetD[OF Upper_closed aelem]) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
190 |
qed |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
191 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
192 |
lemma (in weak_partial_order) Upper_cong: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
193 |
assumes Acarr: "A \<subseteq> carrier L" and A'carr: "A' \<subseteq> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
194 |
and AA': "A {.=} A'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
195 |
shows "Upper L A = Upper L A'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
196 |
unfolding Upper_def |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
197 |
apply rule |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
198 |
apply (rule, clarsimp) defer 1 |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
199 |
apply (rule, clarsimp) defer 1 |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
200 |
proof - |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
201 |
fix x a' |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
202 |
assume carr: "x \<in> carrier L" "a' \<in> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
203 |
and a'A': "a' \<in> A'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
204 |
assume aLxCond[rule_format]: "\<forall>a. a \<in> A \<and> a \<in> carrier L \<longrightarrow> a \<sqsubseteq> x" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
205 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
206 |
from AA' and a'A' have "\<exists>a\<in>A. a' .= a" by (rule set_eqD2) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
207 |
from this obtain a |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
208 |
where aA: "a \<in> A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
209 |
and a'a: "a' .= a" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
210 |
by auto |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
211 |
note [simp] = subsetD[OF Acarr aA] carr |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
212 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
213 |
note a'a |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
214 |
also have "a \<sqsubseteq> x" by (simp add: aLxCond aA) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
215 |
finally show "a' \<sqsubseteq> x" by simp |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
216 |
next |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
217 |
fix x a |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
218 |
assume carr: "x \<in> carrier L" "a \<in> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
219 |
and aA: "a \<in> A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
220 |
assume a'LxCond[rule_format]: "\<forall>a'. a' \<in> A' \<and> a' \<in> carrier L \<longrightarrow> a' \<sqsubseteq> x" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
221 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
222 |
from AA' and aA have "\<exists>a'\<in>A'. a .= a'" by (rule set_eqD1) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
223 |
from this obtain a' |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
224 |
where a'A': "a' \<in> A'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
225 |
and aa': "a .= a'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
226 |
by auto |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
227 |
note [simp] = subsetD[OF A'carr a'A'] carr |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
228 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
229 |
note aa' |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
230 |
also have "a' \<sqsubseteq> x" by (simp add: a'LxCond a'A') |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
231 |
finally show "a \<sqsubseteq> x" by simp |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
232 |
qed |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
233 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
234 |
lemma Lower_closed [intro!, simp]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
235 |
"Lower L A \<subseteq> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
236 |
by (unfold Lower_def) clarify |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
237 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
238 |
lemma Lower_memD [dest]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
239 |
fixes L (structure) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
240 |
shows "[| l \<in> Lower L A; x \<in> A; A \<subseteq> carrier L |] ==> l \<sqsubseteq> x \<and> l \<in> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
241 |
by (unfold Lower_def) blast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
242 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
243 |
lemma Lower_memI: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
244 |
fixes L (structure) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
245 |
shows "[| !! y. y \<in> A ==> x \<sqsubseteq> y; x \<in> carrier L |] ==> x \<in> Lower L A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
246 |
by (unfold Lower_def) blast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
247 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
248 |
lemma Lower_antimono: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
249 |
"A \<subseteq> B ==> Lower L B \<subseteq> Lower L A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
250 |
by (unfold Lower_def) blast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
251 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
252 |
lemma (in weak_partial_order) Lower_is_closed [simp]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
253 |
"A \<subseteq> carrier L \<Longrightarrow> is_closed (Lower L A)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
254 |
by (rule is_closedI) (blast intro: Lower_memI dest: sym)+ |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
255 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
256 |
lemma (in weak_partial_order) Lower_mem_cong: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
257 |
assumes a'carr: "a' \<in> carrier L" and Acarr: "A \<subseteq> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
258 |
and aa': "a .= a'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
259 |
and aelem: "a \<in> Lower L A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
260 |
shows "a' \<in> Lower L A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
261 |
using assms Lower_closed[of L A] |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
262 |
by (intro Lower_memI) (blast intro: le_cong_l[OF aa'[symmetric]]) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
263 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
264 |
lemma (in weak_partial_order) Lower_cong: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
265 |
assumes Acarr: "A \<subseteq> carrier L" and A'carr: "A' \<subseteq> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
266 |
and AA': "A {.=} A'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
267 |
shows "Lower L A = Lower L A'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
268 |
unfolding Lower_def |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
269 |
apply rule |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
270 |
apply clarsimp defer 1 |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
271 |
apply clarsimp defer 1 |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
272 |
proof - |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
273 |
fix x a' |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
274 |
assume carr: "x \<in> carrier L" "a' \<in> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
275 |
and a'A': "a' \<in> A'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
276 |
assume "\<forall>a. a \<in> A \<and> a \<in> carrier L \<longrightarrow> x \<sqsubseteq> a" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
277 |
hence aLxCond: "\<And>a. \<lbrakk>a \<in> A; a \<in> carrier L\<rbrakk> \<Longrightarrow> x \<sqsubseteq> a" by fast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
278 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
279 |
from AA' and a'A' have "\<exists>a\<in>A. a' .= a" by (rule set_eqD2) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
280 |
from this obtain a |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
281 |
where aA: "a \<in> A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
282 |
and a'a: "a' .= a" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
283 |
by auto |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
284 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
285 |
from aA and subsetD[OF Acarr aA] |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
286 |
have "x \<sqsubseteq> a" by (rule aLxCond) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
287 |
also note a'a[symmetric] |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
288 |
finally |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
289 |
show "x \<sqsubseteq> a'" by (simp add: carr subsetD[OF Acarr aA]) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
290 |
next |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
291 |
fix x a |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
292 |
assume carr: "x \<in> carrier L" "a \<in> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
293 |
and aA: "a \<in> A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
294 |
assume "\<forall>a'. a' \<in> A' \<and> a' \<in> carrier L \<longrightarrow> x \<sqsubseteq> a'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
295 |
hence a'LxCond: "\<And>a'. \<lbrakk>a' \<in> A'; a' \<in> carrier L\<rbrakk> \<Longrightarrow> x \<sqsubseteq> a'" by fast+ |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
296 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
297 |
from AA' and aA have "\<exists>a'\<in>A'. a .= a'" by (rule set_eqD1) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
298 |
from this obtain a' |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
299 |
where a'A': "a' \<in> A'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
300 |
and aa': "a .= a'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
301 |
by auto |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
302 |
from a'A' and subsetD[OF A'carr a'A'] |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
303 |
have "x \<sqsubseteq> a'" by (rule a'LxCond) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
304 |
also note aa'[symmetric] |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
305 |
finally show "x \<sqsubseteq> a" by (simp add: carr subsetD[OF A'carr a'A']) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
306 |
qed |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
307 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
308 |
text \<open>Jacobson: Theorem 8.1\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
309 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
310 |
lemma Lower_empty [simp]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
311 |
"Lower L {} = carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
312 |
by (unfold Lower_def) simp |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
313 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
314 |
lemma Upper_empty [simp]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
315 |
"Upper L {} = carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
316 |
by (unfold Upper_def) simp |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
317 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
318 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
319 |
subsubsection \<open>Least and greatest, as predicate\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
320 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
321 |
definition |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
322 |
least :: "[_, 'a, 'a set] => bool" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
323 |
where "least L l A \<longleftrightarrow> A \<subseteq> carrier L & l \<in> A & (ALL x : A. l \<sqsubseteq>\<^bsub>L\<^esub> x)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
324 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
325 |
definition |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
326 |
greatest :: "[_, 'a, 'a set] => bool" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
327 |
where "greatest L g A \<longleftrightarrow> A \<subseteq> carrier L & g \<in> A & (ALL x : A. x \<sqsubseteq>\<^bsub>L\<^esub> g)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
328 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
329 |
text (in weak_partial_order) \<open>Could weaken these to @{term "l \<in> carrier L \<and> l |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
330 |
.\<in> A"} and @{term "g \<in> carrier L \<and> g .\<in> A"}.\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
331 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
332 |
lemma least_closed [intro, simp]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
333 |
"least L l A ==> l \<in> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
334 |
by (unfold least_def) fast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
335 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
336 |
lemma least_mem: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
337 |
"least L l A ==> l \<in> A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
338 |
by (unfold least_def) fast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
339 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
340 |
lemma (in weak_partial_order) weak_least_unique: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
341 |
"[| least L x A; least L y A |] ==> x .= y" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
342 |
by (unfold least_def) blast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
343 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
344 |
lemma least_le: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
345 |
fixes L (structure) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
346 |
shows "[| least L x A; a \<in> A |] ==> x \<sqsubseteq> a" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
347 |
by (unfold least_def) fast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
348 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
349 |
lemma (in weak_partial_order) least_cong: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
350 |
"[| x .= x'; x \<in> carrier L; x' \<in> carrier L; is_closed A |] ==> least L x A = least L x' A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
351 |
by (unfold least_def) (auto dest: sym) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
352 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
353 |
abbreviation is_lub :: "[_, 'a, 'a set] => bool" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
354 |
where "is_lub L x A \<equiv> least L x (Upper L A)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
355 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
356 |
text (in weak_partial_order) \<open>@{const least} is not congruent in the second parameter for |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
357 |
@{term "A {.=} A'"}\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
358 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
359 |
lemma (in weak_partial_order) least_Upper_cong_l: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
360 |
assumes "x .= x'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
361 |
and "x \<in> carrier L" "x' \<in> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
362 |
and "A \<subseteq> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
363 |
shows "least L x (Upper L A) = least L x' (Upper L A)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
364 |
apply (rule least_cong) using assms by auto |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
365 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
366 |
lemma (in weak_partial_order) least_Upper_cong_r: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
367 |
assumes Acarrs: "A \<subseteq> carrier L" "A' \<subseteq> carrier L" (* unneccessary with current Upper? *) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
368 |
and AA': "A {.=} A'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
369 |
shows "least L x (Upper L A) = least L x (Upper L A')" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
370 |
apply (subgoal_tac "Upper L A = Upper L A'", simp) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
371 |
by (rule Upper_cong) fact+ |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
372 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
373 |
lemma least_UpperI: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
374 |
fixes L (structure) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
375 |
assumes above: "!! x. x \<in> A ==> x \<sqsubseteq> s" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
376 |
and below: "!! y. y \<in> Upper L A ==> s \<sqsubseteq> y" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
377 |
and L: "A \<subseteq> carrier L" "s \<in> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
378 |
shows "least L s (Upper L A)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
379 |
proof - |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
380 |
have "Upper L A \<subseteq> carrier L" by simp |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
381 |
moreover from above L have "s \<in> Upper L A" by (simp add: Upper_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
382 |
moreover from below have "ALL x : Upper L A. s \<sqsubseteq> x" by fast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
383 |
ultimately show ?thesis by (simp add: least_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
384 |
qed |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
385 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
386 |
lemma least_Upper_above: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
387 |
fixes L (structure) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
388 |
shows "[| least L s (Upper L A); x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> s" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
389 |
by (unfold least_def) blast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
390 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
391 |
lemma greatest_closed [intro, simp]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
392 |
"greatest L l A ==> l \<in> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
393 |
by (unfold greatest_def) fast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
394 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
395 |
lemma greatest_mem: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
396 |
"greatest L l A ==> l \<in> A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
397 |
by (unfold greatest_def) fast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
398 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
399 |
lemma (in weak_partial_order) weak_greatest_unique: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
400 |
"[| greatest L x A; greatest L y A |] ==> x .= y" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
401 |
by (unfold greatest_def) blast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
402 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
403 |
lemma greatest_le: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
404 |
fixes L (structure) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
405 |
shows "[| greatest L x A; a \<in> A |] ==> a \<sqsubseteq> x" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
406 |
by (unfold greatest_def) fast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
407 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
408 |
lemma (in weak_partial_order) greatest_cong: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
409 |
"[| x .= x'; x \<in> carrier L; x' \<in> carrier L; is_closed A |] ==> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
410 |
greatest L x A = greatest L x' A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
411 |
by (unfold greatest_def) (auto dest: sym) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
412 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
413 |
abbreviation is_glb :: "[_, 'a, 'a set] => bool" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
414 |
where "is_glb L x A \<equiv> greatest L x (Lower L A)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
415 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
416 |
text (in weak_partial_order) \<open>@{const greatest} is not congruent in the second parameter for |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
417 |
@{term "A {.=} A'"} \<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
418 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
419 |
lemma (in weak_partial_order) greatest_Lower_cong_l: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
420 |
assumes "x .= x'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
421 |
and "x \<in> carrier L" "x' \<in> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
422 |
and "A \<subseteq> carrier L" (* unneccessary with current Lower *) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
423 |
shows "greatest L x (Lower L A) = greatest L x' (Lower L A)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
424 |
apply (rule greatest_cong) using assms by auto |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
425 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
426 |
lemma (in weak_partial_order) greatest_Lower_cong_r: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
427 |
assumes Acarrs: "A \<subseteq> carrier L" "A' \<subseteq> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
428 |
and AA': "A {.=} A'" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
429 |
shows "greatest L x (Lower L A) = greatest L x (Lower L A')" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
430 |
apply (subgoal_tac "Lower L A = Lower L A'", simp) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
431 |
by (rule Lower_cong) fact+ |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
432 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
433 |
lemma greatest_LowerI: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
434 |
fixes L (structure) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
435 |
assumes below: "!! x. x \<in> A ==> i \<sqsubseteq> x" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
436 |
and above: "!! y. y \<in> Lower L A ==> y \<sqsubseteq> i" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
437 |
and L: "A \<subseteq> carrier L" "i \<in> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
438 |
shows "greatest L i (Lower L A)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
439 |
proof - |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
440 |
have "Lower L A \<subseteq> carrier L" by simp |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
441 |
moreover from below L have "i \<in> Lower L A" by (simp add: Lower_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
442 |
moreover from above have "ALL x : Lower L A. x \<sqsubseteq> i" by fast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
443 |
ultimately show ?thesis by (simp add: greatest_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
444 |
qed |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
445 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
446 |
lemma greatest_Lower_below: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
447 |
fixes L (structure) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
448 |
shows "[| greatest L i (Lower L A); x \<in> A; A \<subseteq> carrier L |] ==> i \<sqsubseteq> x" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
449 |
by (unfold greatest_def) blast |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
450 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
451 |
lemma Lower_dual [simp]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
452 |
"Lower (inv_gorder L) A = Upper L A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
453 |
by (simp add:Upper_def Lower_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
454 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
455 |
lemma Upper_dual [simp]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
456 |
"Upper (inv_gorder L) A = Lower L A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
457 |
by (simp add:Upper_def Lower_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
458 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
459 |
lemma least_dual [simp]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
460 |
"least (inv_gorder L) x A = greatest L x A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
461 |
by (simp add:least_def greatest_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
462 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
463 |
lemma greatest_dual [simp]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
464 |
"greatest (inv_gorder L) x A = least L x A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
465 |
by (simp add:least_def greatest_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
466 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
467 |
lemma (in weak_partial_order) dual_weak_order: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
468 |
"weak_partial_order (inv_gorder L)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
469 |
apply (unfold_locales) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
470 |
apply (simp_all) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
471 |
apply (metis sym) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
472 |
apply (metis trans) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
473 |
apply (metis weak_le_antisym) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
474 |
apply (metis le_trans) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
475 |
apply (metis le_cong_l le_cong_r sym) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
476 |
done |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
477 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
478 |
lemma dual_weak_order_iff: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
479 |
"weak_partial_order (inv_gorder A) \<longleftrightarrow> weak_partial_order A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
480 |
proof |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
481 |
assume "weak_partial_order (inv_gorder A)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
482 |
then interpret dpo: weak_partial_order "inv_gorder A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
483 |
rewrites "carrier (inv_gorder A) = carrier A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
484 |
and "le (inv_gorder A) = (\<lambda> x y. le A y x)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
485 |
and "eq (inv_gorder A) = eq A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
486 |
by (simp_all) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
487 |
show "weak_partial_order A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
488 |
by (unfold_locales, auto intro: dpo.sym dpo.trans dpo.le_trans) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
489 |
next |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
490 |
assume "weak_partial_order A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
491 |
thus "weak_partial_order (inv_gorder A)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
492 |
by (metis weak_partial_order.dual_weak_order) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
493 |
qed |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
494 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
495 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
496 |
subsubsection \<open>Intervals\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
497 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
498 |
definition |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
499 |
at_least_at_most :: "('a, 'c) gorder_scheme \<Rightarrow> 'a => 'a => 'a set" ("(1\<lbrace>_.._\<rbrace>\<index>)") |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
500 |
where "\<lbrace>l..u\<rbrace>\<^bsub>A\<^esub> = {x \<in> carrier A. l \<sqsubseteq>\<^bsub>A\<^esub> x \<and> x \<sqsubseteq>\<^bsub>A\<^esub> u}" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
501 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
502 |
context weak_partial_order |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
503 |
begin |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
504 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
505 |
lemma at_least_at_most_upper [dest]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
506 |
"x \<in> \<lbrace>a..b\<rbrace> \<Longrightarrow> x \<sqsubseteq> b" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
507 |
by (simp add: at_least_at_most_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
508 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
509 |
lemma at_least_at_most_lower [dest]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
510 |
"x \<in> \<lbrace>a..b\<rbrace> \<Longrightarrow> a \<sqsubseteq> x" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
511 |
by (simp add: at_least_at_most_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
512 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
513 |
lemma at_least_at_most_closed: "\<lbrace>a..b\<rbrace> \<subseteq> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
514 |
by (auto simp add: at_least_at_most_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
515 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
516 |
lemma at_least_at_most_member [intro]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
517 |
"\<lbrakk> x \<in> carrier L; a \<sqsubseteq> x; x \<sqsubseteq> b \<rbrakk> \<Longrightarrow> x \<in> \<lbrace>a..b\<rbrace>" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
518 |
by (simp add: at_least_at_most_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
519 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
520 |
end |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
521 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
522 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
523 |
subsubsection \<open>Isotone functions\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
524 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
525 |
definition isotone :: "('a, 'c) gorder_scheme \<Rightarrow> ('b, 'd) gorder_scheme \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
526 |
where |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
527 |
"isotone A B f \<equiv> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
528 |
weak_partial_order A \<and> weak_partial_order B \<and> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
529 |
(\<forall>x\<in>carrier A. \<forall>y\<in>carrier A. x \<sqsubseteq>\<^bsub>A\<^esub> y \<longrightarrow> f x \<sqsubseteq>\<^bsub>B\<^esub> f y)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
530 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
531 |
lemma isotoneI [intro?]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
532 |
fixes f :: "'a \<Rightarrow> 'b" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
533 |
assumes "weak_partial_order L1" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
534 |
"weak_partial_order L2" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
535 |
"(\<And>x y. \<lbrakk> x \<in> carrier L1; y \<in> carrier L1; x \<sqsubseteq>\<^bsub>L1\<^esub> y \<rbrakk> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
536 |
\<Longrightarrow> f x \<sqsubseteq>\<^bsub>L2\<^esub> f y)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
537 |
shows "isotone L1 L2 f" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
538 |
using assms by (auto simp add:isotone_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
539 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
540 |
abbreviation Monotone :: "('a, 'b) gorder_scheme \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool" ("Mono\<index>") |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
541 |
where "Monotone L f \<equiv> isotone L L f" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
542 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
543 |
lemma use_iso1: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
544 |
"\<lbrakk>isotone A A f; x \<in> carrier A; y \<in> carrier A; x \<sqsubseteq>\<^bsub>A\<^esub> y\<rbrakk> \<Longrightarrow> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
545 |
f x \<sqsubseteq>\<^bsub>A\<^esub> f y" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
546 |
by (simp add: isotone_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
547 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
548 |
lemma use_iso2: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
549 |
"\<lbrakk>isotone A B f; x \<in> carrier A; y \<in> carrier A; x \<sqsubseteq>\<^bsub>A\<^esub> y\<rbrakk> \<Longrightarrow> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
550 |
f x \<sqsubseteq>\<^bsub>B\<^esub> f y" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
551 |
by (simp add: isotone_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
552 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
553 |
lemma iso_compose: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
554 |
"\<lbrakk>f \<in> carrier A \<rightarrow> carrier B; isotone A B f; g \<in> carrier B \<rightarrow> carrier C; isotone B C g\<rbrakk> \<Longrightarrow> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
555 |
isotone A C (g \<circ> f)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
556 |
by (simp add: isotone_def, safe, metis Pi_iff) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
557 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
558 |
lemma (in weak_partial_order) inv_isotone [simp]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
559 |
"isotone (inv_gorder A) (inv_gorder B) f = isotone A B f" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
560 |
by (auto simp add:isotone_def dual_weak_order dual_weak_order_iff) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
561 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
562 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
563 |
subsubsection \<open>Idempotent functions\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
564 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
565 |
definition idempotent :: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
566 |
"('a, 'b) gorder_scheme \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool" ("Idem\<index>") where |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
567 |
"idempotent L f \<equiv> \<forall>x\<in>carrier L. f (f x) .=\<^bsub>L\<^esub> f x" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
568 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
569 |
lemma (in weak_partial_order) idempotent: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
570 |
"\<lbrakk> Idem f; x \<in> carrier L \<rbrakk> \<Longrightarrow> f (f x) .= f x" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
571 |
by (auto simp add: idempotent_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
572 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
573 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
574 |
subsubsection \<open>Order embeddings\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
575 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
576 |
definition order_emb :: "('a, 'c) gorder_scheme \<Rightarrow> ('b, 'd) gorder_scheme \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
577 |
where |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
578 |
"order_emb A B f \<equiv> weak_partial_order A |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
579 |
\<and> weak_partial_order B |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
580 |
\<and> (\<forall>x\<in>carrier A. \<forall>y\<in>carrier A. f x \<sqsubseteq>\<^bsub>B\<^esub> f y \<longleftrightarrow> x \<sqsubseteq>\<^bsub>A\<^esub> y )" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
581 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
582 |
lemma order_emb_isotone: "order_emb A B f \<Longrightarrow> isotone A B f" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
583 |
by (auto simp add: isotone_def order_emb_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
584 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
585 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
586 |
subsubsection \<open>Commuting functions\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
587 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
588 |
definition commuting :: "('a, 'c) gorder_scheme \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool" where |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
589 |
"commuting A f g = (\<forall>x\<in>carrier A. (f \<circ> g) x .=\<^bsub>A\<^esub> (g \<circ> f) x)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
590 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
591 |
subsection \<open>Partial orders where \<open>eq\<close> is the Equality\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
592 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
593 |
locale partial_order = weak_partial_order + |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
594 |
assumes eq_is_equal: "op .= = op =" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
595 |
begin |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
596 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
597 |
declare weak_le_antisym [rule del] |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
598 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
599 |
lemma le_antisym [intro]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
600 |
"[| x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier L; y \<in> carrier L |] ==> x = y" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
601 |
using weak_le_antisym unfolding eq_is_equal . |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
602 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
603 |
lemma lless_eq: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
604 |
"x \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y & x \<noteq> y" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
605 |
unfolding lless_def by (simp add: eq_is_equal) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
606 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
607 |
lemma set_eq_is_eq: "A {.=} B \<longleftrightarrow> A = B" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
608 |
by (auto simp add: set_eq_def elem_def eq_is_equal) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
609 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
610 |
end |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
611 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
612 |
lemma (in partial_order) dual_order: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
613 |
"partial_order (inv_gorder L)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
614 |
proof - |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
615 |
interpret dwo: weak_partial_order "inv_gorder L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
616 |
by (metis dual_weak_order) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
617 |
show ?thesis |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
618 |
by (unfold_locales, simp add:eq_is_equal) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
619 |
qed |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
620 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
621 |
lemma dual_order_iff: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
622 |
"partial_order (inv_gorder A) \<longleftrightarrow> partial_order A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
623 |
proof |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
624 |
assume assm:"partial_order (inv_gorder A)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
625 |
then interpret po: partial_order "inv_gorder A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
626 |
rewrites "carrier (inv_gorder A) = carrier A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
627 |
and "le (inv_gorder A) = (\<lambda> x y. le A y x)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
628 |
and "eq (inv_gorder A) = eq A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
629 |
by (simp_all) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
630 |
show "partial_order A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
631 |
apply (unfold_locales, simp_all) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
632 |
apply (metis po.sym, metis po.trans) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
633 |
apply (metis po.weak_le_antisym, metis po.le_trans) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
634 |
apply (metis (full_types) po.eq_is_equal, metis po.eq_is_equal) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
635 |
done |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
636 |
next |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
637 |
assume "partial_order A" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
638 |
thus "partial_order (inv_gorder A)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
639 |
by (metis partial_order.dual_order) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
640 |
qed |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
641 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
642 |
text \<open>Least and greatest, as predicate\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
643 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
644 |
lemma (in partial_order) least_unique: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
645 |
"[| least L x A; least L y A |] ==> x = y" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
646 |
using weak_least_unique unfolding eq_is_equal . |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
647 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
648 |
lemma (in partial_order) greatest_unique: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
649 |
"[| greatest L x A; greatest L y A |] ==> x = y" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
650 |
using weak_greatest_unique unfolding eq_is_equal . |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
651 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
652 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
653 |
subsection \<open>Bounded Orders\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
654 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
655 |
definition |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
656 |
top :: "_ => 'a" ("\<top>\<index>") where |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
657 |
"\<top>\<^bsub>L\<^esub> = (SOME x. greatest L x (carrier L))" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
658 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
659 |
definition |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
660 |
bottom :: "_ => 'a" ("\<bottom>\<index>") where |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
661 |
"\<bottom>\<^bsub>L\<^esub> = (SOME x. least L x (carrier L))" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
662 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
663 |
locale weak_partial_order_bottom = weak_partial_order L for L (structure) + |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
664 |
assumes bottom_exists: "\<exists> x. least L x (carrier L)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
665 |
begin |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
666 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
667 |
lemma bottom_least: "least L \<bottom> (carrier L)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
668 |
proof - |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
669 |
obtain x where "least L x (carrier L)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
670 |
by (metis bottom_exists) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
671 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
672 |
thus ?thesis |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
673 |
by (auto intro:someI2 simp add: bottom_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
674 |
qed |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
675 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
676 |
lemma bottom_closed [simp, intro]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
677 |
"\<bottom> \<in> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
678 |
by (metis bottom_least least_mem) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
679 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
680 |
lemma bottom_lower [simp, intro]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
681 |
"x \<in> carrier L \<Longrightarrow> \<bottom> \<sqsubseteq> x" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
682 |
by (metis bottom_least least_le) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
683 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
684 |
end |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
685 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
686 |
locale weak_partial_order_top = weak_partial_order L for L (structure) + |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
687 |
assumes top_exists: "\<exists> x. greatest L x (carrier L)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
688 |
begin |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
689 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
690 |
lemma top_greatest: "greatest L \<top> (carrier L)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
691 |
proof - |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
692 |
obtain x where "greatest L x (carrier L)" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
693 |
by (metis top_exists) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
694 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
695 |
thus ?thesis |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
696 |
by (auto intro:someI2 simp add: top_def) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
697 |
qed |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
698 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
699 |
lemma top_closed [simp, intro]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
700 |
"\<top> \<in> carrier L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
701 |
by (metis greatest_mem top_greatest) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
702 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
703 |
lemma top_higher [simp, intro]: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
704 |
"x \<in> carrier L \<Longrightarrow> x \<sqsubseteq> \<top>" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
705 |
by (metis greatest_le top_greatest) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
706 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
707 |
end |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
708 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
709 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
710 |
subsection \<open>Total Orders\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
711 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
712 |
locale weak_total_order = weak_partial_order + |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
713 |
assumes total: "\<lbrakk> x \<in> carrier L; y \<in> carrier L \<rbrakk> \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
714 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
715 |
text \<open>Introduction rule: the usual definition of total order\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
716 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
717 |
lemma (in weak_partial_order) weak_total_orderI: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
718 |
assumes total: "!!x y. \<lbrakk> x \<in> carrier L; y \<in> carrier L \<rbrakk> \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
719 |
shows "weak_total_order L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
720 |
by unfold_locales (rule total) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
721 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
722 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
723 |
subsection \<open>Total orders where \<open>eq\<close> is the Equality\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
724 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
725 |
locale total_order = partial_order + |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
726 |
assumes total_order_total: "\<lbrakk> x \<in> carrier L; y \<in> carrier L \<rbrakk> \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
727 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
728 |
sublocale total_order < weak?: weak_total_order |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
729 |
by unfold_locales (rule total_order_total) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
730 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
731 |
text \<open>Introduction rule: the usual definition of total order\<close> |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
732 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
733 |
lemma (in partial_order) total_orderI: |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
734 |
assumes total: "!!x y. \<lbrakk> x \<in> carrier L; y \<in> carrier L \<rbrakk> \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
735 |
shows "total_order L" |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
736 |
by unfold_locales (rule total) |
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
737 |
|
30d0b2f1df76
Knaster-Tarski fixed point theorem and Galois Connections.
ballarin
parents:
diff
changeset
|
738 |
end |