author | berghofe |
Fri, 13 Oct 2006 18:10:16 +0200 | |
changeset 21017 | 5693e4471c2b |
parent 17589 | 58eeffd73be1 |
child 21312 | 1d39091a3208 |
permissions | -rw-r--r-- |
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(* Title: HOL/FixedPoint.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Author: Stefan Berghofer, TU Muenchen |
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Copyright 1992 University of Cambridge |
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*) |
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header{* Fixed Points and the Knaster-Tarski Theorem*} |
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theory FixedPoint |
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imports Product_Type |
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begin |
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subsection {* Complete lattices *} |
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consts |
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Meet :: "'a::order set \<Rightarrow> 'a" |
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Join :: "'a::order set \<Rightarrow> 'a" |
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defs Join_def: "Join A == Meet {b. \<forall>a \<in> A. a <= b}" |
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axclass comp_lat < order |
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Meet_lower: "x \<in> A \<Longrightarrow> Meet A <= x" |
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Meet_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z <= x) \<Longrightarrow> z <= Meet A" |
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theorem Join_upper: "(x::'a::comp_lat) \<in> A \<Longrightarrow> x <= Join A" |
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by (auto simp: Join_def intro: Meet_greatest) |
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theorem Join_least: "(\<And>x::'a::comp_lat. x \<in> A \<Longrightarrow> x <= z) \<Longrightarrow> Join A <= z" |
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by (auto simp: Join_def intro: Meet_lower) |
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text {* A complete lattice is a lattice *} |
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lemma is_meet_Meet: "is_meet (\<lambda>(x::'a::comp_lat) y. Meet {x, y})" |
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by (auto simp: is_meet_def intro: Meet_lower Meet_greatest) |
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lemma is_join_Join: "is_join (\<lambda>(x::'a::comp_lat) y. Join {x, y})" |
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by (auto simp: is_join_def intro: Join_upper Join_least) |
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instance comp_lat < lorder |
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proof |
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from is_meet_Meet show "\<exists>m::'a\<Rightarrow>'a\<Rightarrow>'a. is_meet m" by iprover |
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from is_join_Join show "\<exists>j::'a\<Rightarrow>'a\<Rightarrow>'a. is_join j" by iprover |
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qed |
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lemma mono_join: "mono f \<Longrightarrow> join (f A) (f B) <= f (join A B)" |
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by (auto simp add: mono_def intro: join_imp_le join_left_le join_right_le) |
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lemma mono_meet: "mono f \<Longrightarrow> f (meet A B) <= meet (f A) (f B)" |
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by (auto simp add: mono_def intro: meet_imp_le meet_left_le meet_right_le) |
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subsection {* Some instances of the type class of complete lattices *} |
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subsubsection {* Booleans *} |
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instance bool :: ord .. |
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defs |
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le_bool_def: "P <= Q == P \<longrightarrow> Q" |
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less_bool_def: "P < Q == (P::bool) <= Q \<and> P \<noteq> Q" |
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theorem le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P <= Q" |
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by (simp add: le_bool_def) |
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theorem le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P <= Q" |
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by (simp add: le_bool_def) |
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theorem le_boolE: "P <= Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R" |
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by (simp add: le_bool_def) |
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theorem le_boolD: "P <= Q \<Longrightarrow> P \<longrightarrow> Q" |
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by (simp add: le_bool_def) |
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instance bool :: order |
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apply intro_classes |
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apply (unfold le_bool_def less_bool_def) |
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apply iprover+ |
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done |
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defs Meet_bool_def: "Meet A == ALL x:A. x" |
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instance bool :: comp_lat |
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apply intro_classes |
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apply (unfold Meet_bool_def) |
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apply (iprover intro!: le_boolI elim: ballE) |
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apply (iprover intro!: ballI le_boolI elim: ballE le_boolE) |
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done |
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theorem meet_bool_eq: "meet P Q = (P \<and> Q)" |
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apply (rule order_antisym) |
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apply (rule le_boolI) |
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apply (rule conjI) |
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apply (rule le_boolE) |
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apply (rule meet_left_le) |
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apply assumption+ |
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apply (rule le_boolE) |
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apply (rule meet_right_le) |
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apply assumption+ |
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apply (rule meet_imp_le) |
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apply (rule le_boolI) |
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apply (erule conjunct1) |
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apply (rule le_boolI) |
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apply (erule conjunct2) |
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done |
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theorem join_bool_eq: "join P Q = (P \<or> Q)" |
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apply (rule order_antisym) |
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apply (rule join_imp_le) |
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apply (rule le_boolI) |
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apply (erule disjI1) |
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apply (rule le_boolI) |
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apply (erule disjI2) |
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apply (rule le_boolI) |
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apply (erule disjE) |
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apply (rule le_boolE) |
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apply (rule join_left_le) |
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apply assumption+ |
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apply (rule le_boolE) |
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apply (rule join_right_le) |
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apply assumption+ |
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done |
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theorem Join_bool_eq: "Join A = (EX x:A. x)" |
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apply (rule order_antisym) |
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apply (rule Join_least) |
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apply (rule le_boolI) |
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apply (erule bexI, assumption) |
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apply (rule le_boolI) |
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apply (erule bexE) |
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apply (rule le_boolE) |
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apply (rule Join_upper) |
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apply assumption+ |
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done |
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subsubsection {* Functions *} |
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137 |
|
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138 |
instance "fun" :: (type, ord) ord .. |
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139 |
|
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140 |
defs |
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le_fun_def: "f <= g == \<forall>x. f x <= g x" |
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142 |
less_fun_def: "f < g == (f::'a\<Rightarrow>'b::ord) <= g \<and> f \<noteq> g" |
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143 |
|
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144 |
theorem le_funI: "(\<And>x. f x <= g x) \<Longrightarrow> f <= g" |
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145 |
by (simp add: le_fun_def) |
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146 |
|
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147 |
theorem le_funE: "f <= g \<Longrightarrow> (f x <= g x \<Longrightarrow> P) \<Longrightarrow> P" |
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148 |
by (simp add: le_fun_def) |
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149 |
|
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|
150 |
theorem le_funD: "f <= g \<Longrightarrow> f x <= g x" |
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151 |
by (simp add: le_fun_def) |
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152 |
|
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|
153 |
text {* |
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|
154 |
Handy introduction and elimination rules for @{text "\<le>"} |
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155 |
on unary and binary predicates |
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156 |
*} |
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|
157 |
|
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|
158 |
lemma predicate1I [intro]: |
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|
159 |
assumes PQ: "\<And>x. P x \<Longrightarrow> Q x" |
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|
160 |
shows "P \<le> Q" |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
161 |
apply (rule le_funI) |
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|
162 |
apply (rule le_boolI) |
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|
163 |
apply (rule PQ) |
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|
164 |
apply assumption |
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|
165 |
done |
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|
166 |
|
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|
167 |
lemma predicate1D [elim]: "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x" |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
168 |
apply (erule le_funE) |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
169 |
apply (erule le_boolE) |
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|
170 |
apply assumption+ |
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|
171 |
done |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
172 |
|
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Generalized gfp and lfp to arbitrary complete lattices.
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|
173 |
lemma predicate2I [intro]: |
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|
174 |
assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y" |
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changeset
|
175 |
shows "P \<le> Q" |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
176 |
apply (rule le_funI)+ |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
177 |
apply (rule le_boolI) |
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|
178 |
apply (rule PQ) |
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|
179 |
apply assumption |
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|
180 |
done |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
181 |
|
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Generalized gfp and lfp to arbitrary complete lattices.
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|
182 |
lemma predicate2D [elim]: "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y" |
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|
183 |
apply (erule le_funE)+ |
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|
184 |
apply (erule le_boolE) |
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|
185 |
apply assumption+ |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
186 |
done |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
187 |
|
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Generalized gfp and lfp to arbitrary complete lattices.
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|
188 |
instance "fun" :: (type, order) order |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
189 |
apply intro_classes |
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changeset
|
190 |
apply (rule le_funI) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
191 |
apply (rule order_refl) |
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parents:
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|
192 |
apply (rule le_funI) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
193 |
apply (erule le_funE)+ |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
194 |
apply (erule order_trans) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
195 |
apply assumption |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
196 |
apply (rule ext) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
197 |
apply (erule le_funE)+ |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
198 |
apply (erule order_antisym) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
199 |
apply assumption |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
200 |
apply (simp add: less_fun_def) |
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Generalized gfp and lfp to arbitrary complete lattices.
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changeset
|
201 |
done |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
202 |
|
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Generalized gfp and lfp to arbitrary complete lattices.
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|
203 |
defs Meet_fun_def: "Meet A == (\<lambda>x. Meet {y. EX f:A. y = f x})" |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
204 |
|
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
205 |
instance "fun" :: (type, comp_lat) comp_lat |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
206 |
apply intro_classes |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
207 |
apply (unfold Meet_fun_def) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
208 |
apply (rule le_funI) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
209 |
apply (rule Meet_lower) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
210 |
apply (rule CollectI) |
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
211 |
apply (rule bexI) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
212 |
apply (rule refl) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
213 |
apply assumption |
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
214 |
apply (rule le_funI) |
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
215 |
apply (rule Meet_greatest) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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diff
changeset
|
216 |
apply (erule CollectE) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
217 |
apply (erule bexE) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
218 |
apply (iprover elim: le_funE) |
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Generalized gfp and lfp to arbitrary complete lattices.
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changeset
|
219 |
done |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
220 |
|
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
221 |
theorem meet_fun_eq: "meet f g = (\<lambda>x. meet (f x) (g x))" |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
222 |
apply (rule order_antisym) |
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
223 |
apply (rule le_funI) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
224 |
apply (rule meet_imp_le) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
225 |
apply (rule le_funD [OF meet_left_le]) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
226 |
apply (rule le_funD [OF meet_right_le]) |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
227 |
apply (rule meet_imp_le) |
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
228 |
apply (rule le_funI) |
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
229 |
apply (rule meet_left_le) |
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
230 |
apply (rule le_funI) |
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
231 |
apply (rule meet_right_le) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
232 |
done |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
233 |
|
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
234 |
theorem join_fun_eq: "join f g = (\<lambda>x. join (f x) (g x))" |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
235 |
apply (rule order_antisym) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
236 |
apply (rule join_imp_le) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
237 |
apply (rule le_funI) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
238 |
apply (rule join_left_le) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
239 |
apply (rule le_funI) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
240 |
apply (rule join_right_le) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
241 |
apply (rule le_funI) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
242 |
apply (rule join_imp_le) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
243 |
apply (rule le_funD [OF join_left_le]) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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|
244 |
apply (rule le_funD [OF join_right_le]) |
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
245 |
done |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
246 |
|
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
247 |
theorem Join_fun_eq: "Join A = (\<lambda>x. Join {y::'a::comp_lat. EX f:A. y = f x})" |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
248 |
apply (rule order_antisym) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
249 |
apply (rule Join_least) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
250 |
apply (rule le_funI) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
251 |
apply (rule Join_upper) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
252 |
apply fast |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
253 |
apply (rule le_funI) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
254 |
apply (rule Join_least) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
255 |
apply (erule CollectE) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
256 |
apply (erule bexE) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
257 |
apply (drule le_funD [OF Join_upper]) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
258 |
apply simp |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
259 |
done |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
260 |
|
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
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|
261 |
subsubsection {* Sets *} |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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diff
changeset
|
262 |
|
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
263 |
defs Meet_set_def: "Meet S == \<Inter>S" |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
264 |
|
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
265 |
instance set :: (type) comp_lat |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
266 |
by intro_classes (auto simp add: Meet_set_def) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
267 |
|
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theorem meet_set_eq: "meet A B = A \<inter> B" |
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apply (rule subset_antisym) |
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apply (rule Int_greatest) |
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271 |
apply (rule meet_left_le) |
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apply (rule meet_right_le) |
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apply (rule meet_imp_le) |
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274 |
apply (rule Int_lower1) |
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275 |
apply (rule Int_lower2) |
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276 |
done |
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277 |
|
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theorem join_set_eq: "join A B = A \<union> B" |
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279 |
apply (rule subset_antisym) |
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280 |
apply (rule join_imp_le) |
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281 |
apply (rule Un_upper1) |
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282 |
apply (rule Un_upper2) |
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Generalized gfp and lfp to arbitrary complete lattices.
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283 |
apply (rule Un_least) |
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284 |
apply (rule join_left_le) |
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285 |
apply (rule join_right_le) |
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|
286 |
done |
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Generalized gfp and lfp to arbitrary complete lattices.
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287 |
|
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288 |
theorem Join_set_eq: "Join S = \<Union>S" |
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289 |
apply (rule subset_antisym) |
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290 |
apply (rule Join_least) |
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Generalized gfp and lfp to arbitrary complete lattices.
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291 |
apply (erule Union_upper) |
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292 |
apply (rule Union_least) |
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293 |
apply (erule Join_upper) |
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|
294 |
done |
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Generalized gfp and lfp to arbitrary complete lattices.
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295 |
|
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296 |
|
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subsection {* Least and greatest fixed points *} |
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298 |
|
17006 | 299 |
constdefs |
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lfp :: "(('a::comp_lat) => 'a) => 'a" |
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"lfp f == Meet {u. f u <= u}" --{*least fixed point*} |
17006 | 302 |
|
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gfp :: "(('a::comp_lat) => 'a) => 'a" |
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304 |
"gfp f == Join {u. u <= f u}" --{*greatest fixed point*} |
17006 | 305 |
|
306 |
||
307 |
subsection{*Proof of Knaster-Tarski Theorem using @{term lfp}*} |
|
308 |
||
309 |
||
310 |
text{*@{term "lfp f"} is the least upper bound of |
|
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311 |
the set @{term "{u. f(u) \<le> u}"} *} |
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312 |
|
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Generalized gfp and lfp to arbitrary complete lattices.
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313 |
lemma lfp_lowerbound: "f A \<le> A ==> lfp f \<le> A" |
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Generalized gfp and lfp to arbitrary complete lattices.
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314 |
by (auto simp add: lfp_def intro: Meet_lower) |
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Generalized gfp and lfp to arbitrary complete lattices.
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315 |
|
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Generalized gfp and lfp to arbitrary complete lattices.
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316 |
lemma lfp_greatest: "(!!u. f u \<le> u ==> A \<le> u) ==> A \<le> lfp f" |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
317 |
by (auto simp add: lfp_def intro: Meet_greatest) |
17006 | 318 |
|
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Generalized gfp and lfp to arbitrary complete lattices.
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319 |
lemma lfp_lemma2: "mono f ==> f (lfp f) \<le> lfp f" |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
320 |
by (iprover intro: lfp_greatest order_trans monoD lfp_lowerbound) |
17006 | 321 |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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|
322 |
lemma lfp_lemma3: "mono f ==> lfp f \<le> f (lfp f)" |
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Generalized gfp and lfp to arbitrary complete lattices.
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323 |
by (iprover intro: lfp_lemma2 monoD lfp_lowerbound) |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
324 |
|
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Generalized gfp and lfp to arbitrary complete lattices.
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|
325 |
lemma lfp_unfold: "mono f ==> lfp f = f (lfp f)" |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
326 |
by (iprover intro: order_antisym lfp_lemma2 lfp_lemma3) |
17006 | 327 |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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|
328 |
subsection{*General induction rules for least fixed points*} |
17006 | 329 |
|
21017
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|
330 |
theorem lfp_induct: |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
331 |
assumes mono: "mono f" and ind: "f (meet (lfp f) P) <= P" |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
332 |
shows "lfp f <= P" |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
333 |
proof - |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
334 |
have "meet (lfp f) P <= lfp f" by (rule meet_left_le) |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
335 |
with mono have "f (meet (lfp f) P) <= f (lfp f)" .. |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
336 |
also from mono have "f (lfp f) = lfp f" by (rule lfp_unfold [symmetric]) |
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Generalized gfp and lfp to arbitrary complete lattices.
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337 |
finally have "f (meet (lfp f) P) <= lfp f" . |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
338 |
from this and ind have "f (meet (lfp f) P) <= meet (lfp f) P" by (rule meet_imp_le) |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
339 |
hence "lfp f <= meet (lfp f) P" by (rule lfp_lowerbound) |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
340 |
also have "meet (lfp f) P <= P" by (rule meet_right_le) |
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Generalized gfp and lfp to arbitrary complete lattices.
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341 |
finally show ?thesis . |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
342 |
qed |
17006 | 343 |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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344 |
lemma lfp_induct_set: |
17006 | 345 |
assumes lfp: "a: lfp(f)" |
346 |
and mono: "mono(f)" |
|
347 |
and indhyp: "!!x. [| x: f(lfp(f) Int {x. P(x)}) |] ==> P(x)" |
|
348 |
shows "P(a)" |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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|
349 |
by (rule lfp_induct [THEN subsetD, THEN CollectD, OF mono _ lfp]) |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
350 |
(auto simp: meet_set_eq intro: indhyp) |
17006 | 351 |
|
352 |
||
353 |
text{*Version of induction for binary relations*} |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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|
354 |
lemmas lfp_induct2 = lfp_induct_set [of "(a,b)", split_format (complete)] |
17006 | 355 |
|
356 |
||
357 |
lemma lfp_ordinal_induct: |
|
358 |
assumes mono: "mono f" |
|
359 |
shows "[| !!S. P S ==> P(f S); !!M. !S:M. P S ==> P(Union M) |] |
|
360 |
==> P(lfp f)" |
|
361 |
apply(subgoal_tac "lfp f = Union{S. S \<subseteq> lfp f & P S}") |
|
362 |
apply (erule ssubst, simp) |
|
363 |
apply(subgoal_tac "Union{S. S \<subseteq> lfp f & P S} \<subseteq> lfp f") |
|
364 |
prefer 2 apply blast |
|
365 |
apply(rule equalityI) |
|
366 |
prefer 2 apply assumption |
|
367 |
apply(drule mono [THEN monoD]) |
|
368 |
apply (cut_tac mono [THEN lfp_unfold], simp) |
|
369 |
apply (rule lfp_lowerbound, auto) |
|
370 |
done |
|
371 |
||
372 |
||
373 |
text{*Definition forms of @{text lfp_unfold} and @{text lfp_induct}, |
|
374 |
to control unfolding*} |
|
375 |
||
376 |
lemma def_lfp_unfold: "[| h==lfp(f); mono(f) |] ==> h = f(h)" |
|
377 |
by (auto intro!: lfp_unfold) |
|
378 |
||
379 |
lemma def_lfp_induct: |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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|
380 |
"[| A == lfp(f); mono(f); |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
381 |
f (meet A P) \<le> P |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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diff
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|
382 |
|] ==> A \<le> P" |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
383 |
by (blast intro: lfp_induct) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
384 |
|
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
385 |
lemma def_lfp_induct_set: |
17006 | 386 |
"[| A == lfp(f); mono(f); a:A; |
387 |
!!x. [| x: f(A Int {x. P(x)}) |] ==> P(x) |
|
388 |
|] ==> P(a)" |
|
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
389 |
by (blast intro: lfp_induct_set) |
17006 | 390 |
|
391 |
(*Monotonicity of lfp!*) |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
392 |
lemma lfp_mono: "(!!Z. f Z \<le> g Z) ==> lfp f \<le> lfp g" |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
393 |
by (rule lfp_lowerbound [THEN lfp_greatest], blast intro: order_trans) |
17006 | 394 |
|
395 |
||
396 |
subsection{*Proof of Knaster-Tarski Theorem using @{term gfp}*} |
|
397 |
||
398 |
||
399 |
text{*@{term "gfp f"} is the greatest lower bound of |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
400 |
the set @{term "{u. u \<le> f(u)}"} *} |
17006 | 401 |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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diff
changeset
|
402 |
lemma gfp_upperbound: "X \<le> f X ==> X \<le> gfp f" |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
403 |
by (auto simp add: gfp_def intro: Join_upper) |
17006 | 404 |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
405 |
lemma gfp_least: "(!!u. u \<le> f u ==> u \<le> X) ==> gfp f \<le> X" |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
406 |
by (auto simp add: gfp_def intro: Join_least) |
17006 | 407 |
|
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
408 |
lemma gfp_lemma2: "mono f ==> gfp f \<le> f (gfp f)" |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
409 |
by (iprover intro: gfp_least order_trans monoD gfp_upperbound) |
17006 | 410 |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
411 |
lemma gfp_lemma3: "mono f ==> f (gfp f) \<le> gfp f" |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
412 |
by (iprover intro: gfp_lemma2 monoD gfp_upperbound) |
17006 | 413 |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
414 |
lemma gfp_unfold: "mono f ==> gfp f = f (gfp f)" |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
415 |
by (iprover intro: order_antisym gfp_lemma2 gfp_lemma3) |
17006 | 416 |
|
417 |
subsection{*Coinduction rules for greatest fixed points*} |
|
418 |
||
419 |
text{*weak version*} |
|
420 |
lemma weak_coinduct: "[| a: X; X \<subseteq> f(X) |] ==> a : gfp(f)" |
|
421 |
by (rule gfp_upperbound [THEN subsetD], auto) |
|
422 |
||
423 |
lemma weak_coinduct_image: "!!X. [| a : X; g`X \<subseteq> f (g`X) |] ==> g a : gfp f" |
|
424 |
apply (erule gfp_upperbound [THEN subsetD]) |
|
425 |
apply (erule imageI) |
|
426 |
done |
|
427 |
||
428 |
lemma coinduct_lemma: |
|
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
429 |
"[| X \<le> f (join X (gfp f)); mono f |] ==> join X (gfp f) \<le> f (join X (gfp f))" |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
430 |
apply (frule gfp_lemma2) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
431 |
apply (drule mono_join) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
432 |
apply (rule join_imp_le) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
433 |
apply assumption |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
434 |
apply (rule order_trans) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
435 |
apply (rule order_trans) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
436 |
apply assumption |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
437 |
apply (rule join_right_le) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
438 |
apply assumption |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
439 |
done |
17006 | 440 |
|
441 |
text{*strong version, thanks to Coen and Frost*} |
|
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
442 |
lemma coinduct_set: "[| mono(f); a: X; X \<subseteq> f(X Un gfp(f)) |] ==> a : gfp(f)" |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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443 |
by (blast intro: weak_coinduct [OF _ coinduct_lemma, simplified join_set_eq]) |
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444 |
|
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lemma coinduct: "[| mono(f); X \<le> f (join X (gfp f)) |] ==> X \<le> gfp(f)" |
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446 |
apply (rule order_trans) |
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447 |
apply (rule join_left_le) |
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448 |
apply (erule gfp_upperbound [OF coinduct_lemma]) |
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apply assumption |
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done |
17006 | 451 |
|
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lemma gfp_fun_UnI2: "[| mono(f); a: gfp(f) |] ==> a: f(X Un gfp(f))" |
|
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by (blast dest: gfp_lemma2 mono_Un) |
|
454 |
||
455 |
subsection{*Even Stronger Coinduction Rule, by Martin Coen*} |
|
456 |
||
457 |
text{* Weakens the condition @{term "X \<subseteq> f(X)"} to one expressed using both |
|
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@{term lfp} and @{term gfp}*} |
|
459 |
||
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lemma coinduct3_mono_lemma: "mono(f) ==> mono(%x. f(x) Un X Un B)" |
|
17589 | 461 |
by (iprover intro: subset_refl monoI Un_mono monoD) |
17006 | 462 |
|
463 |
lemma coinduct3_lemma: |
|
464 |
"[| X \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f))); mono(f) |] |
|
465 |
==> lfp(%x. f(x) Un X Un gfp(f)) \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f)))" |
|
466 |
apply (rule subset_trans) |
|
467 |
apply (erule coinduct3_mono_lemma [THEN lfp_lemma3]) |
|
468 |
apply (rule Un_least [THEN Un_least]) |
|
469 |
apply (rule subset_refl, assumption) |
|
470 |
apply (rule gfp_unfold [THEN equalityD1, THEN subset_trans], assumption) |
|
471 |
apply (rule monoD, assumption) |
|
472 |
apply (subst coinduct3_mono_lemma [THEN lfp_unfold], auto) |
|
473 |
done |
|
474 |
||
475 |
lemma coinduct3: |
|
476 |
"[| mono(f); a:X; X \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f))) |] ==> a : gfp(f)" |
|
477 |
apply (rule coinduct3_lemma [THEN [2] weak_coinduct]) |
|
478 |
apply (rule coinduct3_mono_lemma [THEN lfp_unfold, THEN ssubst], auto) |
|
479 |
done |
|
480 |
||
481 |
||
482 |
text{*Definition forms of @{text gfp_unfold} and @{text coinduct}, |
|
483 |
to control unfolding*} |
|
484 |
||
485 |
lemma def_gfp_unfold: "[| A==gfp(f); mono(f) |] ==> A = f(A)" |
|
486 |
by (auto intro!: gfp_unfold) |
|
487 |
||
488 |
lemma def_coinduct: |
|
21017
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489 |
"[| A==gfp(f); mono(f); X \<le> f(join X A) |] ==> X \<le> A" |
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|
490 |
by (iprover intro!: coinduct) |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
491 |
|
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492 |
lemma def_coinduct_set: |
17006 | 493 |
"[| A==gfp(f); mono(f); a:X; X \<subseteq> f(X Un A) |] ==> a: A" |
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|
494 |
by (auto intro!: coinduct_set) |
17006 | 495 |
|
496 |
(*The version used in the induction/coinduction package*) |
|
497 |
lemma def_Collect_coinduct: |
|
498 |
"[| A == gfp(%w. Collect(P(w))); mono(%w. Collect(P(w))); |
|
499 |
a: X; !!z. z: X ==> P (X Un A) z |] ==> |
|
500 |
a : A" |
|
21017
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501 |
apply (erule def_coinduct_set, auto) |
17006 | 502 |
done |
503 |
||
504 |
lemma def_coinduct3: |
|
505 |
"[| A==gfp(f); mono(f); a:X; X \<subseteq> f(lfp(%x. f(x) Un X Un A)) |] ==> a: A" |
|
506 |
by (auto intro!: coinduct3) |
|
507 |
||
508 |
text{*Monotonicity of @{term gfp}!*} |
|
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509 |
lemma gfp_mono: "(!!Z. f Z \<le> g Z) ==> gfp f \<le> gfp g" |
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|
510 |
by (rule gfp_upperbound [THEN gfp_least], blast intro: order_trans) |
17006 | 511 |
|
512 |
||
513 |
ML |
|
514 |
{* |
|
515 |
val lfp_def = thm "lfp_def"; |
|
516 |
val lfp_lowerbound = thm "lfp_lowerbound"; |
|
517 |
val lfp_greatest = thm "lfp_greatest"; |
|
518 |
val lfp_unfold = thm "lfp_unfold"; |
|
519 |
val lfp_induct = thm "lfp_induct"; |
|
520 |
val lfp_induct2 = thm "lfp_induct2"; |
|
521 |
val lfp_ordinal_induct = thm "lfp_ordinal_induct"; |
|
522 |
val def_lfp_unfold = thm "def_lfp_unfold"; |
|
523 |
val def_lfp_induct = thm "def_lfp_induct"; |
|
21017
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|
524 |
val def_lfp_induct_set = thm "def_lfp_induct_set"; |
17006 | 525 |
val lfp_mono = thm "lfp_mono"; |
526 |
val gfp_def = thm "gfp_def"; |
|
527 |
val gfp_upperbound = thm "gfp_upperbound"; |
|
528 |
val gfp_least = thm "gfp_least"; |
|
529 |
val gfp_unfold = thm "gfp_unfold"; |
|
530 |
val weak_coinduct = thm "weak_coinduct"; |
|
531 |
val weak_coinduct_image = thm "weak_coinduct_image"; |
|
532 |
val coinduct = thm "coinduct"; |
|
533 |
val gfp_fun_UnI2 = thm "gfp_fun_UnI2"; |
|
534 |
val coinduct3 = thm "coinduct3"; |
|
535 |
val def_gfp_unfold = thm "def_gfp_unfold"; |
|
536 |
val def_coinduct = thm "def_coinduct"; |
|
537 |
val def_Collect_coinduct = thm "def_Collect_coinduct"; |
|
538 |
val def_coinduct3 = thm "def_coinduct3"; |
|
539 |
val gfp_mono = thm "gfp_mono"; |
|
21017
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540 |
val le_funI = thm "le_funI"; |
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541 |
val le_boolI = thm "le_boolI"; |
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Generalized gfp and lfp to arbitrary complete lattices.
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542 |
val le_boolI' = thm "le_boolI'"; |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
543 |
val meet_fun_eq = thm "meet_fun_eq"; |
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|
544 |
val meet_bool_eq = thm "meet_bool_eq"; |
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|
545 |
val le_funE = thm "le_funE"; |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
546 |
val le_boolE = thm "le_boolE"; |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
547 |
val le_boolD = thm "le_boolD"; |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
548 |
val le_bool_def = thm "le_bool_def"; |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
549 |
val le_fun_def = thm "le_fun_def"; |
17006 | 550 |
*} |
551 |
||
552 |
end |