author  wenzelm 
Sat, 07 Apr 2012 16:41:59 +0200  
changeset 47389  e8552cba702d 
parent 46882  6242b4bc05bc 
child 47398  07bcf80391d0 
permissions  rwrr 
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(* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel *) 
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header {* Set theory for higherorder logic *} 
4 

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theory Set 
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imports Lattices 
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begin 
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subsection {* Sets as predicates *} 
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typedecl 'a set 
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axiomatization Collect :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set"  "comprehension" 
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and member :: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  "membership" 
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where 
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mem_Collect_eq [iff, code_unfold]: "member a (Collect P) = P a" 
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and Collect_mem_eq [simp]: "Collect (\<lambda>x. member x A) = A" 
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notation 
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member ("op :") and 
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member ("(_/ : _)" [50, 51] 50) 

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abbreviation not_member where 
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"not_member x A \<equiv> ~ (x : A)"  "nonmembership" 

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notation 
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not_member ("op ~:") and 
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not_member ("(_/ ~: _)" [50, 51] 50) 

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notation (xsymbols) 
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member ("op \<in>") and 
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member ("(_/ \<in> _)" [50, 51] 50) and 

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not_member ("op \<notin>") and 

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not_member ("(_/ \<notin> _)" [50, 51] 50) 

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notation (HTML output) 
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member ("op \<in>") and 
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member ("(_/ \<in> _)" [50, 51] 50) and 

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not_member ("op \<notin>") and 

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not_member ("(_/ \<notin> _)" [50, 51] 50) 

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text {* Set comprehensions *} 
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syntax 
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"_Coll" :: "pttrn => bool => 'a set" ("(1{_./ _})") 
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translations 
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"{x. P}" == "CONST Collect (%x. P)" 
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syntax 
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"_Collect" :: "idt => 'a set => bool => 'a set" ("(1{_ :/ _./ _})") 
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syntax (xsymbols) 
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"_Collect" :: "idt => 'a set => bool => 'a set" ("(1{_ \<in>/ _./ _})") 
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translations 
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"{x:A. P}" => "{x. x:A & P}" 
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lemma CollectI: "P a \<Longrightarrow> a \<in> {x. P x}" 
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by simp 
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lemma CollectD: "a \<in> {x. P x} \<Longrightarrow> P a" 
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by simp 
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lemma Collect_cong: "(\<And>x. P x = Q x) ==> {x. P x} = {x. Q x}" 
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by simp 
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text {* 
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Simproc for pulling @{text "x=t"} in @{text "{x. \<dots> & x=t & \<dots>}"} 
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to the front (and similarly for @{text "t=x"}): 
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*} 
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simproc_setup defined_Collect ("{x. P x & Q x}") = {* 
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fn _ => 
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Quantifier1.rearrange_Collect 

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(rtac @{thm Collect_cong} 1 THEN 

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rtac @{thm iffI} 1 THEN 
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ALLGOALS 
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(EVERY' [REPEAT_DETERM o etac @{thm conjE}, DEPTH_SOLVE_1 o ares_tac @{thms conjI}])) 

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*} 
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lemmas CollectE = CollectD [elim_format] 
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lemma set_eqI: 
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assumes "\<And>x. x \<in> A \<longleftrightarrow> x \<in> B" 

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shows "A = B" 

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proof  

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from assms have "{x. x \<in> A} = {x. x \<in> B}" by simp 

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then show ?thesis by simp 

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qed 

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lemma set_eq_iff [no_atp]: 

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"A = B \<longleftrightarrow> (\<forall>x. x \<in> A \<longleftrightarrow> x \<in> B)" 

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by (auto intro:set_eqI) 

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text {* Lifting of predicate class instances *} 
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instantiation set :: (type) boolean_algebra 
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begin 
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definition less_eq_set where 
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"A \<le> B \<longleftrightarrow> (\<lambda>x. member x A) \<le> (\<lambda>x. member x B)" 
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definition less_set where 
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"A < B \<longleftrightarrow> (\<lambda>x. member x A) < (\<lambda>x. member x B)" 
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definition inf_set where 
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"A \<sqinter> B = Collect ((\<lambda>x. member x A) \<sqinter> (\<lambda>x. member x B))" 
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definition sup_set where 
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"A \<squnion> B = Collect ((\<lambda>x. member x A) \<squnion> (\<lambda>x. member x B))" 
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definition bot_set where 
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"\<bottom> = Collect \<bottom>" 
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definition top_set where 
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"\<top> = Collect \<top>" 
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definition uminus_set where 
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" A = Collect ( (\<lambda>x. member x A))" 
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definition minus_set where 
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"A  B = Collect ((\<lambda>x. member x A)  (\<lambda>x. member x B))" 
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instance proof 
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qed (simp_all add: less_eq_set_def less_set_def inf_set_def sup_set_def 
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bot_set_def top_set_def uminus_set_def minus_set_def 
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less_le_not_le inf_compl_bot sup_compl_top sup_inf_distrib1 diff_eq 
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set_eqI fun_eq_iff 
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del: inf_apply sup_apply bot_apply top_apply minus_apply uminus_apply) 

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end 
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text {* Set enumerations *} 
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abbreviation empty :: "'a set" ("{}") where 
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"{} \<equiv> bot" 
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definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where 

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insert_compr: "insert a B = {x. x = a \<or> x \<in> B}" 
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syntax 

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"_Finset" :: "args => 'a set" ("{(_)}") 
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translations 
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"{x, xs}" == "CONST insert x {xs}" 
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"{x}" == "CONST insert x {}" 

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subsection {* Subsets and bounded quantifiers *} 

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abbreviation 

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subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where 

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"subset \<equiv> less" 

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abbreviation 

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subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where 

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"subset_eq \<equiv> less_eq" 

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notation (output) 

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subset ("op <") and 

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subset ("(_/ < _)" [50, 51] 50) and 

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subset_eq ("op <=") and 

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subset_eq ("(_/ <= _)" [50, 51] 50) 

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notation (xsymbols) 

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subset ("op \<subset>") and 

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subset ("(_/ \<subset> _)" [50, 51] 50) and 

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subset_eq ("op \<subseteq>") and 

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subset_eq ("(_/ \<subseteq> _)" [50, 51] 50) 

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notation (HTML output) 

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subset ("op \<subset>") and 

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subset ("(_/ \<subset> _)" [50, 51] 50) and 

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subset_eq ("op \<subseteq>") and 

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subset_eq ("(_/ \<subseteq> _)" [50, 51] 50) 

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abbreviation (input) 

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supset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where 

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"supset \<equiv> greater" 

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abbreviation (input) 

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supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where 

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"supset_eq \<equiv> greater_eq" 

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notation (xsymbols) 

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supset ("op \<supset>") and 

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supset ("(_/ \<supset> _)" [50, 51] 50) and 

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supset_eq ("op \<supseteq>") and 

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supset_eq ("(_/ \<supseteq> _)" [50, 51] 50) 

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definition Ball :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where 
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"Ball A P \<longleftrightarrow> (\<forall>x. x \<in> A \<longrightarrow> P x)"  "bounded universal quantifiers" 
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definition Bex :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where 
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"Bex A P \<longleftrightarrow> (\<exists>x. x \<in> A \<and> P x)"  "bounded existential quantifiers" 
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syntax 
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"_Ball" :: "pttrn => 'a set => bool => bool" ("(3ALL _:_./ _)" [0, 0, 10] 10) 
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"_Bex" :: "pttrn => 'a set => bool => bool" ("(3EX _:_./ _)" [0, 0, 10] 10) 
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"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3EX! _:_./ _)" [0, 0, 10] 10) 
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"_Bleast" :: "id => 'a set => bool => 'a" ("(3LEAST _:_./ _)" [0, 0, 10] 10) 
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syntax (HOL) 
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"_Ball" :: "pttrn => 'a set => bool => bool" ("(3! _:_./ _)" [0, 0, 10] 10) 
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"_Bex" :: "pttrn => 'a set => bool => bool" ("(3? _:_./ _)" [0, 0, 10] 10) 
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"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3?! _:_./ _)" [0, 0, 10] 10) 
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syntax (xsymbols) 
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"_Ball" :: "pttrn => 'a set => bool => bool" ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10) 
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"_Bex" :: "pttrn => 'a set => bool => bool" ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10) 
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"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10) 
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"_Bleast" :: "id => 'a set => bool => 'a" ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10) 
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syntax (HTML output) 
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"_Ball" :: "pttrn => 'a set => bool => bool" ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10) 
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"_Bex" :: "pttrn => 'a set => bool => bool" ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10) 
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"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10) 
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translations 
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"ALL x:A. P" == "CONST Ball A (%x. P)" 
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"EX x:A. P" == "CONST Bex A (%x. P)" 

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"EX! x:A. P" => "EX! x. x:A & P" 

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"LEAST x:A. P" => "LEAST x. x:A & P" 
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syntax (output) 
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"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10) 
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"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10) 
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"_setleAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10) 
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"_setleEx" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) 
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"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3EX! _<=_./ _)" [0, 0, 10] 10) 
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syntax (xsymbols) 
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"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subset>_./ _)" [0, 0, 10] 10) 
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"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subset>_./ _)" [0, 0, 10] 10) 
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"_setleAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10) 
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"_setleEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10) 
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"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10) 
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236 

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syntax (HOL output) 
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"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10) 
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"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10) 
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"_setleAll" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10) 
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"_setleEx" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) 
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"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3?! _<=_./ _)" [0, 0, 10] 10) 
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243 

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syntax (HTML output) 
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"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subset>_./ _)" [0, 0, 10] 10) 
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"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subset>_./ _)" [0, 0, 10] 10) 
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"_setleAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10) 
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"_setleEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10) 
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"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10) 
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250 

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251 
translations 
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"\<forall>A\<subset>B. P" => "ALL A. A \<subset> B > P" 
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"\<exists>A\<subset>B. P" => "EX A. A \<subset> B & P" 
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"\<forall>A\<subseteq>B. P" => "ALL A. A \<subseteq> B > P" 
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"\<exists>A\<subseteq>B. P" => "EX A. A \<subseteq> B & P" 
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"\<exists>!A\<subseteq>B. P" => "EX! A. A \<subseteq> B & P" 
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print_translation {* 
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let 
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val All_binder = Mixfix.binder_name @{const_syntax All}; 
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val Ex_binder = Mixfix.binder_name @{const_syntax Ex}; 
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val impl = @{const_syntax HOL.implies}; 
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val conj = @{const_syntax HOL.conj}; 
35115  264 
val sbset = @{const_syntax subset}; 
265 
val sbset_eq = @{const_syntax subset_eq}; 

21819  266 

267 
val trans = 

35115  268 
[((All_binder, impl, sbset), @{syntax_const "_setlessAll"}), 
269 
((All_binder, impl, sbset_eq), @{syntax_const "_setleAll"}), 

270 
((Ex_binder, conj, sbset), @{syntax_const "_setlessEx"}), 

271 
((Ex_binder, conj, sbset_eq), @{syntax_const "_setleEx"})]; 

21819  272 

273 
fun mk v v' c n P = 

274 
if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v  _ => false) n) 

42284  275 
then Syntax.const c $ Syntax_Trans.mark_bound v' $ n $ P else raise Match; 
21819  276 

277 
fun tr' q = (q, 

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fn [Const (@{syntax_const "_bound"}, _) $ Free (v, Type (@{type_name set}, _)), 
35115  279 
Const (c, _) $ 
280 
(Const (d, _) $ (Const (@{syntax_const "_bound"}, _) $ Free (v', _)) $ n) $ P] => 

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(case AList.lookup (op =) trans (q, c, d) of 
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NONE => raise Match 
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 SOME l => mk v v' l n P) 
35115  284 
 _ => raise Match); 
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in 
21819  286 
[tr' All_binder, tr' Ex_binder] 
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end 
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*} 
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289 

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11979  291 
text {* 
292 
\medskip Translate between @{text "{e  x1...xn. P}"} and @{text 

293 
"{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is 

294 
only translated if @{text "[0..n] subset bvs(e)"}. 

295 
*} 

296 

35115  297 
syntax 
298 
"_Setcompr" :: "'a => idts => bool => 'a set" ("(1{_ /_./ _})") 

299 

11979  300 
parse_translation {* 
301 
let 

42284  302 
val ex_tr = snd (Syntax_Trans.mk_binder_tr ("EX ", @{const_syntax Ex})); 
3947  303 

35115  304 
fun nvars (Const (@{syntax_const "_idts"}, _) $ _ $ idts) = nvars idts + 1 
11979  305 
 nvars _ = 1; 
306 

307 
fun setcompr_tr [e, idts, b] = 

308 
let 

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val eq = Syntax.const @{const_syntax HOL.eq} $ Bound (nvars idts) $ e; 
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val P = Syntax.const @{const_syntax HOL.conj} $ eq $ b; 
11979  311 
val exP = ex_tr [idts, P]; 
44241  312 
in Syntax.const @{const_syntax Collect} $ absdummy dummyT exP end; 
11979  313 

35115  314 
in [(@{syntax_const "_Setcompr"}, setcompr_tr)] end; 
11979  315 
*} 
923  316 

35115  317 
print_translation {* 
42284  318 
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Ball} @{syntax_const "_Ball"}, 
319 
Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Bex} @{syntax_const "_Bex"}] 

35115  320 
*}  {* to avoid etacontraction of body *} 
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print_translation {* 
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let 
42284  324 
val ex_tr' = snd (Syntax_Trans.mk_binder_tr' (@{const_syntax Ex}, "DUMMY")); 
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fun setcompr_tr' [Abs (abs as (_, _, P))] = 
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let 
35115  328 
fun check (Const (@{const_syntax Ex}, _) $ Abs (_, _, P), n) = check (P, n + 1) 
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 check (Const (@{const_syntax HOL.conj}, _) $ 
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(Const (@{const_syntax HOL.eq}, _) $ Bound m $ e) $ P, n) = 
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n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso 
33038  332 
subset (op =) (0 upto (n  1), add_loose_bnos (e, 0, [])) 
35115  333 
 check _ = false; 
923  334 

11979  335 
fun tr' (_ $ abs) = 
336 
let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs] 

35115  337 
in Syntax.const @{syntax_const "_Setcompr"} $ e $ idts $ Q end; 
338 
in 

339 
if check (P, 0) then tr' P 

340 
else 

341 
let 

42284  342 
val (x as _ $ Free(xN, _), t) = Syntax_Trans.atomic_abs_tr' abs; 
35115  343 
val M = Syntax.const @{syntax_const "_Coll"} $ x $ t; 
344 
in 

345 
case t of 

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Const (@{const_syntax HOL.conj}, _) $ 
37677  347 
(Const (@{const_syntax Set.member}, _) $ 
35115  348 
(Const (@{syntax_const "_bound"}, _) $ Free (yN, _)) $ A) $ P => 
349 
if xN = yN then Syntax.const @{syntax_const "_Collect"} $ x $ A $ P else M 

350 
 _ => M 

351 
end 

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end; 
35115  353 
in [(@{const_syntax Collect}, setcompr_tr')] end; 
11979  354 
*} 
355 

42455  356 
simproc_setup defined_Bex ("EX x:A. P x & Q x") = {* 
357 
let 

358 
val unfold_bex_tac = unfold_tac @{thms Bex_def}; 

359 
fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac; 

42459  360 
in fn _ => fn ss => Quantifier1.rearrange_bex (prove_bex_tac ss) ss end 
42455  361 
*} 
362 

363 
simproc_setup defined_All ("ALL x:A. P x > Q x") = {* 

364 
let 

365 
val unfold_ball_tac = unfold_tac @{thms Ball_def}; 

366 
fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac; 

42459  367 
in fn _ => fn ss => Quantifier1.rearrange_ball (prove_ball_tac ss) ss end 
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368 
*} 
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369 

11979  370 
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x" 
371 
by (simp add: Ball_def) 

372 

373 
lemmas strip = impI allI ballI 

374 

375 
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x" 

376 
by (simp add: Ball_def) 

377 

378 
text {* 

379 
Gives better instantiation for bound: 

380 
*} 

381 

26339  382 
declaration {* fn _ => 
46459  383 
Classical.map_cs (fn cs => cs addbefore ("bspec", dtac @{thm bspec} THEN' assume_tac)) 
11979  384 
*} 
385 

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ML {* 
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387 
structure Simpdata = 
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388 
struct 
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389 

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390 
open Simpdata; 
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391 

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392 
val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs; 
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393 

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394 
end; 
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395 

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396 
open Simpdata; 
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397 
*} 
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398 

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399 
declaration {* fn _ => 
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Simplifier.map_ss (Simplifier.set_mksimps (mksimps mksimps_pairs)) 
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401 
*} 
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402 

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403 
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q" 
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404 
by (unfold Ball_def) blast 
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405 

11979  406 
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x" 
407 
 {* Normally the best argument order: @{prop "P x"} constrains the 

408 
choice of @{prop "x:A"}. *} 

409 
by (unfold Bex_def) blast 

410 

13113  411 
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x" 
11979  412 
 {* The best argument order when there is only one @{prop "x:A"}. *} 
413 
by (unfold Bex_def) blast 

414 

415 
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x" 

416 
by (unfold Bex_def) blast 

417 

418 
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q" 

419 
by (unfold Bex_def) blast 

420 

421 
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) > P)" 

422 
 {* Trival rewrite rule. *} 

423 
by (simp add: Ball_def) 

424 

425 
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)" 

426 
 {* Dual form for existentials. *} 

427 
by (simp add: Bex_def) 

428 

429 
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)" 

430 
by blast 

431 

432 
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)" 

433 
by blast 

434 

435 
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)" 

436 
by blast 

437 

438 
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)" 

439 
by blast 

440 

441 
lemma ball_one_point1 [simp]: "(ALL x:A. x = a > P x) = (a:A > P a)" 

442 
by blast 

443 

444 
lemma ball_one_point2 [simp]: "(ALL x:A. a = x > P x) = (a:A > P a)" 

445 
by blast 

446 

43818  447 
lemma ball_conj_distrib: 
448 
"(\<forall>x\<in>A. P x \<and> Q x) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>A. Q x))" 

449 
by blast 

450 

451 
lemma bex_disj_distrib: 

452 
"(\<exists>x\<in>A. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>A. Q x))" 

453 
by blast 

454 

11979  455 

32081  456 
text {* Congruence rules *} 
11979  457 

16636
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

458 
lemma ball_cong: 
11979  459 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
460 
(ALL x:A. P x) = (ALL x:B. Q x)" 

461 
by (simp add: Ball_def) 

462 

16636
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

463 
lemma strong_ball_cong [cong]: 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

464 
"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

465 
(ALL x:A. P x) = (ALL x:B. Q x)" 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

466 
by (simp add: simp_implies_def Ball_def) 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

467 

1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

468 
lemma bex_cong: 
11979  469 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
470 
(EX x:A. P x) = (EX x:B. Q x)" 

471 
by (simp add: Bex_def cong: conj_cong) 

1273  472 

16636
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

473 
lemma strong_bex_cong [cong]: 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

474 
"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

475 
(EX x:A. P x) = (EX x:B. Q x)" 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

476 
by (simp add: simp_implies_def Bex_def cong: conj_cong) 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

477 

30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

478 

32081  479 
subsection {* Basic operations *} 
480 

30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

481 
subsubsection {* Subsets *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

482 

33022
c95102496490
Removal of the unused atpset concept, the atp attribute and some related code.
paulson
parents:
32888
diff
changeset

483 
lemma subsetI [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B) \<Longrightarrow> A \<subseteq> B" 
45959
184d36538e51
`set` is now a proper type constructor; added operation for set monad
haftmann
parents:
45909
diff
changeset

484 
by (simp add: less_eq_set_def le_fun_def) 
30352  485 

11979  486 
text {* 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

487 
\medskip Map the type @{text "'a set => anything"} to just @{typ 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

488 
'a}; for overloading constants whose first argument has type @{typ 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

489 
"'a set"}. 
11979  490 
*} 
491 

30596  492 
lemma subsetD [elim, intro?]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B" 
45959
184d36538e51
`set` is now a proper type constructor; added operation for set monad
haftmann
parents:
45909
diff
changeset

493 
by (simp add: less_eq_set_def le_fun_def) 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

494 
 {* Rule in Modus Ponens style. *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

495 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35576
diff
changeset

496 
lemma rev_subsetD [no_atp,intro?]: "c \<in> A ==> A \<subseteq> B ==> c \<in> B" 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

497 
 {* The same, with reversed premises for use with @{text erule}  
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

498 
cf @{text rev_mp}. *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

499 
by (rule subsetD) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

500 

11979  501 
text {* 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

502 
\medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}. 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

503 
*} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

504 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35576
diff
changeset

505 
lemma subsetCE [no_atp,elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P" 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

506 
 {* Classical elimination rule. *} 
45959
184d36538e51
`set` is now a proper type constructor; added operation for set monad
haftmann
parents:
45909
diff
changeset

507 
by (auto simp add: less_eq_set_def le_fun_def) 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

508 

46127  509 
lemma subset_eq [code, no_atp]: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast 
2388  510 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35576
diff
changeset

511 
lemma contra_subsetD [no_atp]: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A" 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

512 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

513 

45121  514 
lemma subset_refl: "A \<subseteq> A" 
515 
by (fact order_refl) (* already [iff] *) 

30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

516 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

517 
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C" 
32081  518 
by (fact order_trans) 
519 

520 
lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B" 

521 
by (rule subsetD) 

522 

523 
lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B" 

524 
by (rule subsetD) 

525 

46146
6baea4fca6bd
incorporated various theorems from theory More_Set into corpus
haftmann
parents:
46137
diff
changeset

526 
lemma subset_not_subset_eq [code]: 
6baea4fca6bd
incorporated various theorems from theory More_Set into corpus
haftmann
parents:
46137
diff
changeset

527 
"A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> \<not> B \<subseteq> A" 
6baea4fca6bd
incorporated various theorems from theory More_Set into corpus
haftmann
parents:
46137
diff
changeset

528 
by (fact less_le_not_le) 
6baea4fca6bd
incorporated various theorems from theory More_Set into corpus
haftmann
parents:
46137
diff
changeset

529 

33044  530 
lemma eq_mem_trans: "a=b ==> b \<in> A ==> a \<in> A" 
531 
by simp 

532 

32081  533 
lemmas basic_trans_rules [trans] = 
33044  534 
order_trans_rules set_rev_mp set_mp eq_mem_trans 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

535 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

536 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

537 
subsubsection {* Equality *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

538 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

539 
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

540 
 {* Antisymmetry of the subset relation. *} 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39213
diff
changeset

541 
by (iprover intro: set_eqI subsetD) 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

542 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

543 
text {* 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

544 
\medskip Equality rules from ZF set theory  are they appropriate 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

545 
here? 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

546 
*} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

547 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

548 
lemma equalityD1: "A = B ==> A \<subseteq> B" 
34209  549 
by simp 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

550 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

551 
lemma equalityD2: "A = B ==> B \<subseteq> A" 
34209  552 
by simp 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

553 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

554 
text {* 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

555 
\medskip Be careful when adding this to the claset as @{text 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

556 
subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

557 
\<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}! 
30352  558 
*} 
559 

30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

560 
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P" 
34209  561 
by simp 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

562 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

563 
lemma equalityCE [elim]: 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

564 
"A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

565 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

566 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

567 
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

568 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

569 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

570 
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

571 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

572 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

573 

41082  574 
subsubsection {* The empty set *} 
575 

576 
lemma empty_def: 

577 
"{} = {x. False}" 

45959
184d36538e51
`set` is now a proper type constructor; added operation for set monad
haftmann
parents:
45909
diff
changeset

578 
by (simp add: bot_set_def bot_fun_def) 
41082  579 

580 
lemma empty_iff [simp]: "(c : {}) = False" 

581 
by (simp add: empty_def) 

582 

583 
lemma emptyE [elim!]: "a : {} ==> P" 

584 
by simp 

585 

586 
lemma empty_subsetI [iff]: "{} \<subseteq> A" 

587 
 {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *} 

588 
by blast 

589 

590 
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}" 

591 
by blast 

592 

593 
lemma equals0D: "A = {} ==> a \<notin> A" 

594 
 {* Use for reasoning about disjointness: @{text "A Int B = {}"} *} 

595 
by blast 

596 

597 
lemma ball_empty [simp]: "Ball {} P = True" 

598 
by (simp add: Ball_def) 

599 

600 
lemma bex_empty [simp]: "Bex {} P = False" 

601 
by (simp add: Bex_def) 

602 

603 

30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

604 
subsubsection {* The universal set  UNIV *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

605 

32264
0be31453f698
Set.UNIV and Set.empty are mere abbreviations for top and bot
haftmann
parents:
32139
diff
changeset

606 
abbreviation UNIV :: "'a set" where 
0be31453f698
Set.UNIV and Set.empty are mere abbreviations for top and bot
haftmann
parents:
32139
diff
changeset

607 
"UNIV \<equiv> top" 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

608 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

609 
lemma UNIV_def: 
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset

610 
"UNIV = {x. True}" 
45959
184d36538e51
`set` is now a proper type constructor; added operation for set monad
haftmann
parents:
45909
diff
changeset

611 
by (simp add: top_set_def top_fun_def) 
32081  612 

30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

613 
lemma UNIV_I [simp]: "x : UNIV" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

614 
by (simp add: UNIV_def) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

615 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

616 
declare UNIV_I [intro]  {* unsafe makes it less likely to cause problems *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

617 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

618 
lemma UNIV_witness [intro?]: "EX x. x : UNIV" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

619 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

620 

45121  621 
lemma subset_UNIV: "A \<subseteq> UNIV" 
622 
by (fact top_greatest) (* already simp *) 

30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

623 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

624 
text {* 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

625 
\medskip Etacontracting these two rules (to remove @{text P}) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

626 
causes them to be ignored because of their interaction with 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

627 
congruence rules. 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

628 
*} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

629 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

630 
lemma ball_UNIV [simp]: "Ball UNIV P = All P" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

631 
by (simp add: Ball_def) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

632 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

633 
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

634 
by (simp add: Bex_def) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

635 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

636 
lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

637 
by auto 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

638 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

639 
lemma UNIV_not_empty [iff]: "UNIV ~= {}" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

640 
by (blast elim: equalityE) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

641 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

642 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

643 
subsubsection {* The Powerset operator  Pow *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

644 

32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

645 
definition Pow :: "'a set => 'a set set" where 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

646 
Pow_def: "Pow A = {B. B \<le> A}" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

647 

30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

648 
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

649 
by (simp add: Pow_def) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

650 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

651 
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

652 
by (simp add: Pow_def) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

653 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

654 
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

655 
by (simp add: Pow_def) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

656 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

657 
lemma Pow_bottom: "{} \<in> Pow B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

658 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

659 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

660 
lemma Pow_top: "A \<in> Pow A" 
34209  661 
by simp 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

662 

40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

663 
lemma Pow_not_empty: "Pow A \<noteq> {}" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

664 
using Pow_top by blast 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

665 

41076
a7fba340058c
primitive definitions of bot/top/inf/sup for bool and fun are named with canonical suffix `_def` rather than `_eq`;
haftmann
parents:
40872
diff
changeset

666 

30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

667 
subsubsection {* Set complement *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

668 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

669 
lemma Compl_iff [simp]: "(c \<in> A) = (c \<notin> A)" 
45959
184d36538e51
`set` is now a proper type constructor; added operation for set monad
haftmann
parents:
45909
diff
changeset

670 
by (simp add: fun_Compl_def uminus_set_def) 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

671 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

672 
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> A" 
45959
184d36538e51
`set` is now a proper type constructor; added operation for set monad
haftmann
parents:
45909
diff
changeset

673 
by (simp add: fun_Compl_def uminus_set_def) blast 
923  674 

11979  675 
text {* 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

676 
\medskip This form, with negated conclusion, works well with the 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

677 
Classical prover. Negated assumptions behave like formulae on the 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

678 
right side of the notional turnstile ... *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

679 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

680 
lemma ComplD [dest!]: "c : A ==> c~:A" 
45959
184d36538e51
`set` is now a proper type constructor; added operation for set monad
haftmann
parents:
45909
diff
changeset

681 
by simp 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

682 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

683 
lemmas ComplE = ComplD [elim_format] 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

684 

45959
184d36538e51
`set` is now a proper type constructor; added operation for set monad
haftmann
parents:
45909
diff
changeset

685 
lemma Compl_eq: " A = {x. ~ x : A}" 
184d36538e51
`set` is now a proper type constructor; added operation for set monad
haftmann
parents:
45909
diff
changeset

686 
by blast 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

687 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

688 

41082  689 
subsubsection {* Binary intersection *} 
690 

691 
abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where 

692 
"op Int \<equiv> inf" 

693 

694 
notation (xsymbols) 

695 
inter (infixl "\<inter>" 70) 

696 

697 
notation (HTML output) 

698 
inter (infixl "\<inter>" 70) 

699 

700 
lemma Int_def: 

701 
"A \<inter> B = {x. x \<in> A \<and> x \<in> B}" 

45959
184d36538e51
`set` is now a proper type constructor; added operation for set monad
haftmann
parents:
45909
diff
changeset

702 
by (simp add: inf_set_def inf_fun_def) 
41082  703 

704 
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)" 

705 
by (unfold Int_def) blast 

706 

707 
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B" 

708 
by simp 

709 

710 
lemma IntD1: "c : A Int B ==> c:A" 

711 
by simp 

712 

713 
lemma IntD2: "c : A Int B ==> c:B" 

714 
by simp 

715 

716 
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P" 

717 
by simp 

718 

719 
lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B" 

720 
by (fact mono_inf) 

721 

722 

723 
subsubsection {* Binary union *} 

30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

724 

32683
7c1fe854ca6a
inter and union are mere abbreviations for inf and sup
haftmann
parents:
32456
diff
changeset

725 
abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where 
41076
a7fba340058c
primitive definitions of bot/top/inf/sup for bool and fun are named with canonical suffix `_def` rather than `_eq`;
haftmann
parents:
40872
diff
changeset

726 
"union \<equiv> sup" 
32081  727 

728 
notation (xsymbols) 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

729 
union (infixl "\<union>" 65) 
32081  730 

731 
notation (HTML output) 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

732 
union (infixl "\<union>" 65) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

733 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

734 
lemma Un_def: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

735 
"A \<union> B = {x. x \<in> A \<or> x \<in> B}" 
45959
184d36538e51
`set` is now a proper type constructor; added operation for set monad
haftmann
parents:
45909
diff
changeset

736 
by (simp add: sup_set_def sup_fun_def) 
32081  737 

30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

738 
lemma Un_iff [simp]: "(c : A Un B) = (c:A  c:B)" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

739 
by (unfold Un_def) blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

740 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

741 
lemma UnI1 [elim?]: "c:A ==> c : A Un B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

742 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

743 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

744 
lemma UnI2 [elim?]: "c:B ==> c : A Un B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

745 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

746 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

747 
text {* 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

748 
\medskip Classical introduction rule: no commitment to @{prop A} vs 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

749 
@{prop B}. 
11979  750 
*} 
751 

30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

752 
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

753 
by auto 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

754 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

755 
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

756 
by (unfold Un_def) blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

757 

32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset

758 
lemma insert_def: "insert a B = {x. x = a} \<union> B" 
45959
184d36538e51
`set` is now a proper type constructor; added operation for set monad
haftmann
parents:
45909
diff
changeset

759 
by (simp add: insert_compr Un_def) 
32081  760 

761 
lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)" 

32683
7c1fe854ca6a
inter and union are mere abbreviations for inf and sup
haftmann
parents:
32456
diff
changeset

762 
by (fact mono_sup) 
32081  763 

30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

764 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

765 
subsubsection {* Set difference *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

766 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

767 
lemma Diff_iff [simp]: "(c : A  B) = (c:A & c~:B)" 
45959
184d36538e51
`set` is now a proper type constructor; added operation for set monad
haftmann
parents:
45909
diff
changeset

768 
by (simp add: minus_set_def fun_diff_def) 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

769 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

770 
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A  B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

771 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

772 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

773 
lemma DiffD1: "c : A  B ==> c : A" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

774 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

775 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

776 
lemma DiffD2: "c : A  B ==> c : B ==> P" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

777 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

778 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

779 
lemma DiffE [elim!]: "c : A  B ==> (c:A ==> c~:B ==> P) ==> P" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

780 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

781 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

782 
lemma set_diff_eq: "A  B = {x. x : A & ~ x : B}" by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

783 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

784 
lemma Compl_eq_Diff_UNIV: "A = (UNIV  A)" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

785 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

786 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

787 

31456  788 
subsubsection {* Augmenting a set  @{const insert} *} 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

789 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

790 
lemma insert_iff [simp]: "(a : insert b A) = (a = b  a:A)" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

791 
by (unfold insert_def) blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

792 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

793 
lemma insertI1: "a : insert a B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

794 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

795 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

796 
lemma insertI2: "a : B ==> a : insert b B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

797 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

798 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

799 
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

800 
by (unfold insert_def) blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

801 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

802 
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

803 
 {* Classical introduction rule. *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

804 
by auto 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

805 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

806 
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A  {x} \<subseteq> B else A \<subseteq> B)" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

807 
by auto 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

808 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

809 
lemma set_insert: 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

810 
assumes "x \<in> A" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

811 
obtains B where "A = insert x B" and "x \<notin> B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

812 
proof 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

813 
from assms show "A = insert x (A  {x})" by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

814 
next 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

815 
show "x \<notin> A  {x}" by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

816 
qed 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

817 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

818 
lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

819 
by auto 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

820 

44744  821 
lemma insert_eq_iff: assumes "a \<notin> A" "b \<notin> B" 
822 
shows "insert a A = insert b B \<longleftrightarrow> 

823 
(if a=b then A=B else \<exists>C. A = insert b C \<and> b \<notin> C \<and> B = insert a C \<and> a \<notin> C)" 

824 
(is "?L \<longleftrightarrow> ?R") 

825 
proof 

826 
assume ?L 

827 
show ?R 

828 
proof cases 

829 
assume "a=b" with assms `?L` show ?R by (simp add: insert_ident) 

830 
next 

831 
assume "a\<noteq>b" 

832 
let ?C = "A  {b}" 

833 
have "A = insert b ?C \<and> b \<notin> ?C \<and> B = insert a ?C \<and> a \<notin> ?C" 

834 
using assms `?L` `a\<noteq>b` by auto 

835 
thus ?R using `a\<noteq>b` by auto 

836 
qed 

837 
next 

46128
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46127
diff
changeset

838 
assume ?R thus ?L by (auto split: if_splits) 
44744  839 
qed 
840 

30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

841 
subsubsection {* Singletons, using insert *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

842 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35576
diff
changeset

843 
lemma singletonI [intro!,no_atp]: "a : {a}" 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

844 
 {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

845 
by (rule insertI1) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

846 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35576
diff
changeset

847 
lemma singletonD [dest!,no_atp]: "b : {a} ==> b = a" 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

848 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

849 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

850 
lemmas singletonE = singletonD [elim_format] 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

851 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

852 
lemma singleton_iff: "(b : {a}) = (b = a)" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

853 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

854 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

855 
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

856 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

857 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35576
diff
changeset

858 
lemma singleton_insert_inj_eq [iff,no_atp]: 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

859 
"({b} = insert a A) = (a = b & A \<subseteq> {b})" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

860 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

861 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35576
diff
changeset

862 
lemma singleton_insert_inj_eq' [iff,no_atp]: 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

863 
"(insert a A = {b}) = (a = b & A \<subseteq> {b})" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

864 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

865 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

866 
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {}  A = {x}" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

867 
by fast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

868 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

869 
lemma singleton_conv [simp]: "{x. x = a} = {a}" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

870 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

871 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

872 
lemma singleton_conv2 [simp]: "{x. a = x} = {a}" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

873 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

874 

46504
cd4832aa2229
removing unnecessary premise from diff_single_insert
bulwahn
parents:
46459
diff
changeset

875 
lemma diff_single_insert: "A  {x} \<subseteq> B ==> A \<subseteq> insert x B" 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

876 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

877 

ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

878 
lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d  a=d & b=c)" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

879 
by (blast elim: equalityE) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

880 

11979  881 

32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

882 
subsubsection {* Image of a set under a function *} 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

883 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

884 
text {* 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

885 
Frequently @{term b} does not have the syntactic form of @{term "f x"}. 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

886 
*} 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

887 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

888 
definition image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90) where 
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35576
diff
changeset

889 
image_def [no_atp]: "f ` A = {y. EX x:A. y = f(x)}" 
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

890 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

891 
abbreviation 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

892 
range :: "('a => 'b) => 'b set" where  "of function" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

893 
"range f == f ` UNIV" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

894 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

895 
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

896 
by (unfold image_def) blast 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

897 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

898 
lemma imageI: "x : A ==> f x : f ` A" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

899 
by (rule image_eqI) (rule refl) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

900 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

901 
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

902 
 {* This version's more effective when we already have the 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

903 
required @{term x}. *} 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

904 
by (unfold image_def) blast 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

905 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

906 
lemma imageE [elim!]: 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

907 
"b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

908 
 {* The etaexpansion gives variablename preservation. *} 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

909 
by (unfold image_def) blast 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

910 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

911 
lemma image_Un: "f`(A Un B) = f`A Un f`B" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

912 
by blast 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

913 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

914 
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

915 
by blast 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

916 

38648
52ea97d95e4b
"no_atp" a few facts that often lead to unsound proofs
blanchet
parents:
37767
diff
changeset

917 
lemma image_subset_iff [no_atp]: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)" 
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

918 
 {* This rewrite rule would confuse users if made default. *} 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

919 
by blast 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

920 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

921 
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

922 
apply safe 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

923 
prefer 2 apply fast 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

924 
apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

925 
done 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

926 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

927 
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

928 
 {* Replaces the three steps @{text subsetI}, @{text imageE}, 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

929 
@{text hypsubst}, but breaks too many existing proofs. *} 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

930 
by blast 
11979  931 

932 
text {* 

32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

933 
\medskip Range of a function  just a translation for image! 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

934 
*} 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

935 

43898  936 
lemma image_ident [simp]: "(%x. x) ` Y = Y" 
937 
by blast 

938 

32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

939 
lemma range_eqI: "b = f x ==> b \<in> range f" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

940 
by simp 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

941 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

942 
lemma rangeI: "f x \<in> range f" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

943 
by simp 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

944 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

945 
lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

946 
by blast 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

947 

32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset

948 
subsubsection {* Some rules with @{text "if"} *} 
32081  949 

950 
text{* Elimination of @{text"{x. \<dots> & x=t & \<dots>}"}. *} 

951 

952 
lemma Collect_conv_if: "{x. x=a & P x} = (if P a then {a} else {})" 

32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset

953 
by auto 
32081  954 

955 
lemma Collect_conv_if2: "{x. a=x & P x} = (if P a then {a} else {})" 

32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset

956 
by auto 
32081  957 

958 
text {* 

959 
Rewrite rules for boolean casesplitting: faster than @{text 

960 
"split_if [split]"}. 

961 
*} 

962 

963 
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q > x = b) & (~ Q > y = b))" 

964 
by (rule split_if) 

965 

966 
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q > a = x) & (~ Q > a = y))" 

967 
by (rule split_if) 

968 

969 
text {* 

970 
Split ifs on either side of the membership relation. Not for @{text 

971 
"[simp]"}  can cause goals to blow up! 

972 
*} 

973 

974 
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q > x : b) & (~ Q > y : b))" 

975 
by (rule split_if) 

976 

977 
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q > a : x) & (~ Q > a : y))" 

978 
by (rule split_if [where P="%S. a : S"]) 

979 

980 
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2 

981 

982 
(*Would like to add these, but the existing code only searches for the 

37677  983 
outerlevel constant, which in this case is just Set.member; we instead need 
32081  984 
to use termnets to associate patterns with rules. Also, if a rule fails to 
985 
apply, then the formula should be kept. 

34974
18b41bba42b5
new theory Algebras.thy for generic algebraic structures
haftmann
parents:
34209
diff
changeset

986 
[("uminus", Compl_iff RS iffD1), ("minus", [Diff_iff RS iffD1]), 
32081  987 
("Int", [IntD1,IntD2]), 
988 
("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])] 

989 
*) 

990 

991 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

992 
subsection {* Further operations and lemmas *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

993 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

994 
subsubsection {* The ``proper subset'' relation *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

995 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35576
diff
changeset

996 
lemma psubsetI [intro!,no_atp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B" 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

997 
by (unfold less_le) blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

998 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35576
diff
changeset

999 
lemma psubsetE [elim!,no_atp]: 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1000 
"[A \<subset> B; [A \<subseteq> B; ~ (B\<subseteq>A)] ==> R] ==> R" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1001 
by (unfold less_le) blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1002 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1003 
lemma psubset_insert_iff: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1004 
"(A \<subset> insert x B) = (if x \<in> B then A \<subset> B else if x \<in> A then A  {x} \<subset> B else A \<subseteq> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1005 
by (auto simp add: less_le subset_insert_iff) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1006 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1007 
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1008 
by (simp only: less_le) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1009 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1010 
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1011 
by (simp add: psubset_eq) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1012 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1013 
lemma psubset_trans: "[ A \<subset> B; B \<subset> C ] ==> A \<subset> C" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1014 
apply (unfold less_le) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1015 
apply (auto dest: subset_antisym) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1016 
done 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1017 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1018 
lemma psubsetD: "[ A \<subset> B; c \<in> A ] ==> c \<in> B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1019 
apply (unfold less_le) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1020 
apply (auto dest: subsetD) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1021 
done 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1022 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1023 
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1024 
by (auto simp add: psubset_eq) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1025 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1026 
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1027 
by (auto simp add: psubset_eq) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1028 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1029 
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B  A)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1030 
by (unfold less_le) blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1031 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1032 
lemma atomize_ball: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1033 
"(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1034 
by (simp only: Ball_def atomize_all atomize_imp) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1035 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1036 
lemmas [symmetric, rulify] = atomize_ball 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1037 
and [symmetric, defn] = atomize_ball 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1038 

40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

1039 
lemma image_Pow_mono: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

1040 
assumes "f ` A \<le> B" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

1041 
shows "(image f) ` (Pow A) \<le> Pow B" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

1042 
using assms by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

1043 

d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

1044 
lemma image_Pow_surj: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

1045 
assumes "f ` A = B" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

1046 
shows "(image f) ` (Pow A) = Pow B" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

1047 
using assms unfolding Pow_def proof(auto) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

1048 
fix Y assume *: "Y \<le> f ` A" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

1049 
obtain X where X_def: "X = {x \<in> A. f x \<in> Y}" by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

1050 
have "f ` X = Y \<and> X \<le> A" unfolding X_def using * by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

1051 
thus "Y \<in> (image f) ` {X. X \<le> A}" by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

1052 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
39910
diff
changeset

1053 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1054 
subsubsection {* Derived rules involving subsets. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1055 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1056 
text {* @{text insert}. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1057 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1058 
lemma subset_insertI: "B \<subseteq> insert a B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1059 
by (rule subsetI) (erule insertI2) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1060 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1061 
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1062 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1063 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1064 
lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1065 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1066 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1067 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1068 
text {* \medskip Finite Union  the least upper bound of two sets. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1069 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1070 
lemma Un_upper1: "A \<subseteq> A \<union> B" 
36009  1071 
by (fact sup_ge1) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1072 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1073 
lemma Un_upper2: "B \<subseteq> A \<union> B" 
36009  1074 
by (fact sup_ge2) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1075 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1076 
lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C" 
36009  1077 
by (fact sup_least) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1078 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1079 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1080 
text {* \medskip Finite Intersection  the greatest lower bound of two sets. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1081 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1082 
lemma Int_lower1: "A \<inter> B \<subseteq> A" 
36009  1083 
by (fact inf_le1) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1084 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1085 
lemma Int_lower2: "A \<inter> B \<subseteq> B" 
36009  1086 
by (fact inf_le2) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1087 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1088 
lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B" 
36009  1089 
by (fact inf_greatest) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1090 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1091 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1092 
text {* \medskip Set difference. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1093 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1094 
lemma Diff_subset: "A  B \<subseteq> A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1095 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1096 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1097 
lemma Diff_subset_conv: "(A  B \<subseteq> C) = (A \<subseteq> B \<union> C)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1098 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1099 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1100 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1101 
subsubsection {* Equalities involving union, intersection, inclusion, etc. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1102 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1103 
text {* @{text "{}"}. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1104 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1105 
lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1106 
 {* supersedes @{text "Collect_False_empty"} *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1107 
by auto 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1108 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1109 
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})" 
45121  1110 
by (fact bot_unique) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1111 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1112 
lemma not_psubset_empty [iff]: "\<not> (A < {})" 
45121  1113 
by (fact not_less_bot) (* FIXME: already simp *) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1114 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1115 
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1116 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1117 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1118 
lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1119 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1120 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1121 
lemma Collect_neg_eq: "{x. \<not> P x} =  {x. P x}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1122 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1123 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1124 
lemma Collect_disj_eq: "{x. P x  Q x} = {x. P x} \<union> {x. Q x}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1125 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1126 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1127 
lemma Collect_imp_eq: "{x. P x > Q x} = {x. P x} \<union> {x. Q x}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1128 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1129 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1130 
lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1131 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1132 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1133 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1134 
text {* \medskip @{text insert}. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1135 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1136 
lemma insert_is_Un: "insert a A = {a} Un A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1137 
 {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1138 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1139 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1140 
lemma insert_not_empty [simp]: "insert a A \<noteq> {}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1141 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1142 

45607  1143 
lemmas empty_not_insert = insert_not_empty [symmetric] 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1144 
declare empty_not_insert [simp] 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1145 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1146 
lemma insert_absorb: "a \<in> A ==> insert a A = A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1147 
 {* @{text "[simp]"} causes recursive calls when there are nested inserts *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1148 
 {* with \emph{quadratic} running time *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1149 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1150 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1151 
lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1152 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1153 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1154 
lemma insert_commute: "insert x (insert y A) = insert y (insert x A)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1155 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1156 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1157 
lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1158 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1159 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1160 
lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1161 
 {* use new @{text B} rather than @{text "A  {a}"} to avoid infinite unfolding *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1162 
apply (rule_tac x = "A  {a}" in exI, blast) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1163 
done 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1164 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1165 
lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a > P u}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1166 
by auto 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1167 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1168 
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1169 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1170 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35576
diff
changeset

1171 
lemma insert_disjoint [simp,no_atp]: 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1172 
"(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1173 
"({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1174 
by auto 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1175 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35576
diff
changeset

1176 
lemma disjoint_insert [simp,no_atp]: 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1177 
"(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1178 
"({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1179 
by auto 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1180 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1181 
text {* \medskip @{text image}. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1182 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1183 
lemma image_empty [simp]: "f`{} = {}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1184 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1185 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1186 
lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1187 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1188 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1189 
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1190 
by auto 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1191 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1192 
lemma image_constant_conv: "(%x. c) ` A = (if A = {} then {} else {c})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1193 
by auto 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1194 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1195 
lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1196 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1197 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1198 
lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1199 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1200 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1201 
lemma image_is_empty [iff]: "(f`A = {}) = (A = {})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1202 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1203 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1204 
lemma empty_is_image[iff]: "({} = f ` A) = (A = {})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1205 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1206 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1207 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35576
diff
changeset

1208 
lemma image_Collect [no_atp]: "f ` {x. P x} = {f x  x. P x}" 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1209 
 {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS, 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1210 
with its implicit quantifier and conjunction. Also image enjoys better 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1211 
equational properties than does the RHS. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1212 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1213 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1214 
lemma if_image_distrib [simp]: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1215 
"(\<lambda>x. if P x then f x else g x) ` S 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1216 
= (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1217 
by (auto simp add: image_def) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1218 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1219 
lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1220 
by (simp add: image_def) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1221 

43898  1222 
lemma image_Int_subset: "f`(A Int B) <= f`A Int f`B" 
1223 
by blast 

1224 

1225 
lemma image_diff_subset: "f`A  f`B <= f`(A  B)" 

1226 
by blast 

1227 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1228 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1229 
text {* \medskip @{text range}. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1230 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35576
diff
changeset

1231 
lemma full_SetCompr_eq [no_atp]: "{u. \<exists>x. u = f x} = range f" 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1232 
by auto 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1233 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1234 
lemma range_composition: "range (\<lambda>x. f (g x)) = f`range g" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1235 
by (subst image_image, simp) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1236 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1237 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1238 
text {* \medskip @{text Int} *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1239 

45121  1240 
lemma Int_absorb: "A \<inter> A = A" 
1241 
by (fact inf_idem) (* already simp *) 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1242 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1243 
lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B" 
36009  1244 
by (fact inf_left_idem) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1245 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1246 
lemma Int_commute: "A \<inter> B = B \<inter> A" 
36009  1247 
by (fact inf_commute) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1248 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1249 
lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)" 
36009  1250 
by (fact inf_left_commute) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1251 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1252 
lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)" 
36009  1253 
by (fact inf_assoc) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1254 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1255 
lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1256 
 {* Intersection is an ACoperator *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1257 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1258 
lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B" 
36009  1259 
by (fact inf_absorb2) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1260 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1261 
lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A" 
36009  1262 
by (fact inf_absorb1) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1263 

45121  1264 
lemma Int_empty_left: "{} \<inter> B = {}" 
1265 
by (fact inf_bot_left) (* already simp *) 

32135
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1266 

45121  1267 
lemma Int_empty_right: "A \<inter> {} = {}" 
1268 
by (fact inf_bot_right) (* already simp *) 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1269 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1270 
lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1271 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1272 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1273 
lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1274 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1275 

45121  1276 
lemma Int_UNIV_left: "UNIV \<inter> B = B" 
1277 
by (fact inf_top_left) (* already simp *) 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1278 

45121  1279 
lemma Int_UNIV_right: "A \<inter> UNIV = A" 
1280 
by (fact inf_top_right) (* already simp *) 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1281 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1282 
lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)" 
36009  1283 
by (fact inf_sup_distrib1) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1284 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

1285 
lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)" 
36009  1286 
by (fact inf_sup_distrib2) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1287 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35576
diff
changeset

1288 
lemma Int_UNIV [simp,no_atp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)" 
45121  1289 
by (fact inf_eq_top_iff) (* already simp *) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1290 

38648
52ea97d95e4b
"no_atp" a few facts that often lead to unsound proofs
blanchet
parents:
37767
diff
changeset

1291 
lemma Int_subset_iff [no_atp, simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)" 
36009 