author  haftmann 
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parent 30304  d8e4cd2ac2a1 
child 30531  ab3d61baf66a 
permissions  rwrr 
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(* Title: HOL/Set.thy 
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Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel 
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*) 
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header {* Set theory for higherorder logic *} 
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theory Set 
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imports Lattices 
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begin 
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subsection {* Basic operations *} 
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subsubsection {* Comprehension and membership *} 

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text {* A set in HOL is simply a predicate. *} 
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global 
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types 'a set = "'a => bool" 
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consts 
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Collect :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set" 
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"op :" :: "'a => 'a set => bool" 

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local 
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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}" == "Collect (%x. P)" 

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notation 
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"op :" ("op :") and 
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"op :" ("(_/ : _)" [50, 51] 50) 
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abbreviation 
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"not_mem x A == ~ (x : A)"  "nonmembership" 
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notation 
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not_mem ("op ~:") and 
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not_mem ("(_/ ~: _)" [50, 51] 50) 
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notation (xsymbols) 
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"op :" ("op \<in>") and 
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"op :" ("(_/ \<in> _)" [50, 51] 50) and 
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not_mem ("op \<notin>") and 
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not_mem ("(_/ \<notin> _)" [50, 51] 50) 
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notation (HTML output) 
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"op :" ("op \<in>") and 
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"op :" ("(_/ \<in> _)" [50, 51] 50) and 
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not_mem ("op \<notin>") and 
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not_mem ("(_/ \<notin> _)" [50, 51] 50) 
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defs 
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Collect_def [code]: "Collect P \<equiv> P" 

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mem_def [code]: "x \<in> S \<equiv> S x" 

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text {* Relating predicates and sets *} 

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lemma mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)" 

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by (simp add: Collect_def mem_def) 

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lemma Collect_mem_eq [simp]: "{x. x:A} = A" 

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by (simp add: Collect_def mem_def) 

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lemma CollectI: "P(a) ==> a : {x. P(x)}" 

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by simp 

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lemma CollectD: "a : {x. P(x)} ==> P(a)" 

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by simp 

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lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}" 

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by simp 

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lemmas CollectE = CollectD [elim_format] 

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lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B" 

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apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals]) 

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apply (rule Collect_mem_eq) 

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apply (rule Collect_mem_eq) 

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done 

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(* Due to Brian Huffman *) 

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lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))" 

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by(auto intro:set_ext) 

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lemma equalityCE [elim]: 

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"A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P" 

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by blast 

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lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)" 

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by simp 

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lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)" 

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by simp 

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subsubsection {* Subset relation, empty and universal set *} 

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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 empty :: "'a set" ("{}") where 
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"empty \<equiv> {x. False}" 

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definition UNIV :: "'a set" where 

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"UNIV \<equiv> {x. True}" 

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lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B" 

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by (auto simp add: mem_def intro: predicate1I) 

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text {* 

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\medskip Map the type @{text "'a set => anything"} to just @{typ 

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'a}; for overloading constants whose first argument has type @{typ 

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"'a set"}. 

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*} 

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lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B" 

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 {* Rule in Modus Ponens style. *} 

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by (unfold mem_def) blast 

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declare subsetD [intro?]  FIXME 

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lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B" 

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 {* The same, with reversed premises for use with @{text erule}  

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cf @{text rev_mp}. *} 

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by (rule subsetD) 

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declare rev_subsetD [intro?]  FIXME 

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text {* 

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\medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}. 

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*} 

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ML {* 

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fun impOfSubs th = th RSN (2, @{thm rev_subsetD}) 

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*} 

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lemma subsetCE [elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P" 

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 {* Classical elimination rule. *} 

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by (unfold mem_def) blast 

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text {* 

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\medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and 

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creates the assumption @{prop "c \<in> B"}. 

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*} 

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ML {* 

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fun set_mp_tac i = etac @{thm subsetCE} i THEN mp_tac i 

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*} 

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lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A" 

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by blast 

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lemma subset_refl [simp,atp]: "A \<subseteq> A" 

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by fast 

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lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C" 

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by blast 

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lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B" 

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 {* Antisymmetry of the subset relation. *} 

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by (iprover intro: set_ext subsetD) 

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text {* 

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\medskip Equality rules from ZF set theory  are they appropriate 

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here? 

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*} 

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lemma equalityD1: "A = B ==> A \<subseteq> B" 

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by (simp add: subset_refl) 

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lemma equalityD2: "A = B ==> B \<subseteq> A" 

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by (simp add: subset_refl) 

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text {* 

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\medskip Be careful when adding this to the claset as @{text 

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subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{} 

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\<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}! 

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*} 

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lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P" 

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by (simp add: subset_refl) 

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

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by (simp add: empty_def) 

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

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by simp 

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

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 {* One effect is to delete the ASSUMPTION @{prop "{} \<subseteq> A"} *} 

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by blast 

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lemma bot_set_eq: "bot = {}" 

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by (iprover intro!: subset_antisym empty_subsetI bot_least) 

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

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by blast 

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

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 {* Use for reasoning about disjointness *} 

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by blast 

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lemma UNIV_I [simp]: "x : UNIV" 

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by (simp add: UNIV_def) 

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declare UNIV_I [intro]  {* unsafe makes it less likely to cause problems *} 

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lemma UNIV_witness [intro?]: "EX x. x : UNIV" 

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by simp 

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lemma subset_UNIV [simp]: "A \<subseteq> UNIV" 

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by (rule subsetI) (rule UNIV_I) 

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lemma top_set_eq: "top = UNIV" 

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by (iprover intro!: subset_antisym subset_UNIV top_greatest) 

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lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A" 

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by auto 

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lemma UNIV_not_empty [iff]: "UNIV ~= {}" 

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by blast 

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lemma psubsetI [intro!,noatp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B" 

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by (unfold less_le) blast 

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lemma psubsetE [elim!,noatp]: 

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"[A \<subset> B; [A \<subseteq> B; ~ (B\<subseteq>A)] ==> R] ==> R" 

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by (unfold less_le) blast 

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lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)" 

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by (simp only: less_le) 

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lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B" 

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by (simp add: psubset_eq) 

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lemma psubset_trans: "[ A \<subset> B; B \<subset> C ] ==> A \<subset> C" 

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apply (unfold less_le) 

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apply (auto dest: subset_antisym) 

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done 

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lemma psubsetD: "[ A \<subset> B; c \<in> A ] ==> c \<in> B" 

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apply (unfold less_le) 

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apply (auto dest: subsetD) 

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done 

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lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C" 

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by (auto simp add: psubset_eq) 

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lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C" 

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by (auto simp add: psubset_eq) 

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lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)" 

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by blast 

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lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)" 

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by blast 

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subsubsection {* Intersection and union *} 

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definition Int :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where 

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"A Int B \<equiv> {x. x \<in> A \<and> x \<in> B}" 

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definition Un :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where 

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"A Un B \<equiv> {x. x \<in> A \<or> x \<in> B}" 

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notation (xsymbols) 

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"Int" (infixl "\<inter>" 70) and 

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"Un" (infixl "\<union>" 65) 

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notation (HTML output) 

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"Int" (infixl "\<inter>" 70) and 

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"Un" (infixl "\<union>" 65) 

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

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by (unfold Int_def) blast 

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

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by simp 

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

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by simp 

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

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by simp 

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

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by simp 

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lemma Un_iff [simp]: "(c : A Un B) = (c:A  c:B)" 

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by (unfold Un_def) blast 

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lemma UnI1 [elim?]: "c:A ==> c : A Un B" 

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by simp 

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lemma UnI2 [elim?]: "c:B ==> c : A Un B" 

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by simp 

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text {* 

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\medskip Classical introduction rule: no commitment to @{prop A} vs 

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@{prop B}. 

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*} 

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lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B" 

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by auto 

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lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P" 

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by (unfold Un_def) blast 

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lemma Int_lower1: "A \<inter> B \<subseteq> A" 

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by blast 

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lemma Int_lower2: "A \<inter> B \<subseteq> B" 

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by blast 

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lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B" 

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by blast 

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lemma inf_set_eq: "inf A B = A \<inter> B" 

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apply (rule subset_antisym) 

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apply (rule Int_greatest) 

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apply (rule inf_le1) 

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apply (rule inf_le2) 

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apply (rule inf_greatest) 

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apply (rule Int_lower1) 

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apply (rule Int_lower2) 

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done 

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lemma Un_upper1: "A \<subseteq> A \<union> B" 

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by blast 

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lemma Un_upper2: "B \<subseteq> A \<union> B" 

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by blast 

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lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C" 

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by blast 

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lemma sup_set_eq: "sup A B = A \<union> B" 

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apply (rule subset_antisym) 

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apply (rule sup_least) 

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apply (rule Un_upper1) 

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apply (rule Un_upper2) 

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apply (rule Un_least) 

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apply (rule sup_ge1) 

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apply (rule sup_ge2) 

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done 

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lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}" 

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by blast 

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lemma Collect_disj_eq: "{x. P x  Q x} = {x. P x} \<union> {x. Q x}" 

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by blast 

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lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})" 

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by blast 

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lemma not_psubset_empty [iff]: "\<not> (A \<subset> {})" 

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by (unfold less_le) blast 

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lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})" 

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 {* supersedes @{text "Collect_False_empty"} *} 

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by auto 

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subsubsection {* Complement and set difference *} 

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instantiation bool :: minus 

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begin 

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definition 

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bool_diff_def: "A  B \<longleftrightarrow> A \<and> \<not> B" 

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instance .. 

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end 

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instantiation "fun" :: (type, minus) minus 

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begin 

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definition 

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fun_diff_def: "A  B = (\<lambda>x. A x  B x)" 

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instance .. 

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end 

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instantiation bool :: uminus 

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begin 

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definition 

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bool_Compl_def: " A \<longleftrightarrow> \<not> A" 

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instance .. 

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end 

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instantiation "fun" :: (type, uminus) uminus 

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begin 

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definition 

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fun_Compl_def: " A = (\<lambda>x.  A x)" 

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instance .. 

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end 

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lemma Compl_iff [simp]: "(c \<in> A) = (c \<notin> A)" 

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by (simp add: mem_def fun_Compl_def bool_Compl_def) 

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lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> A" 

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by (unfold mem_def fun_Compl_def bool_Compl_def) blast 

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text {* 

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\medskip This form, with negated conclusion, works well with the 

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Classical prover. Negated assumptions behave like formulae on the 

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right side of the notional turnstile ... *} 

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lemma ComplD [dest!]: "c : A ==> c~:A" 

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by (simp add: mem_def fun_Compl_def bool_Compl_def) 

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lemmas ComplE = ComplD [elim_format] 

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lemma Compl_eq: " A = {x. ~ x : A}" by blast 

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lemma Diff_iff [simp]: "(c : A  B) = (c:A & c~:B)" 

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by (simp add: mem_def fun_diff_def bool_diff_def) 

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lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A  B" 

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by simp 

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lemma DiffD1: "c : A  B ==> c : A" 

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by simp 

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lemma DiffD2: "c : A  B ==> c : B ==> P" 

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by simp 

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lemma DiffE [elim!]: "c : A  B ==> (c:A ==> c~:B ==> P) ==> P" 

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by simp 

479 

480 
lemma set_diff_eq: "A  B = {x. x : A & ~ x : B}" by blast 

481 

482 
lemma Compl_eq_Diff_UNIV: "A = (UNIV  A)" 

483 
by blast 

484 

485 
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B  A)" 

486 
by (unfold less_le) blast 

487 

488 
lemma Diff_subset: "A  B \<subseteq> A" 

489 
by blast 

490 

491 
lemma Diff_subset_conv: "(A  B \<subseteq> C) = (A \<subseteq> B \<union> C)" 

492 
by blast 

493 

494 
lemma Collect_imp_eq: "{x. P x > Q x} = {x. P x} \<union> {x. Q x}" 

495 
by blast 

496 

497 
lemma Collect_neg_eq: "{x. \<not> P x} =  {x. P x}" 

498 
by blast 

499 

500 

501 
subsubsection {* Set enumerations *} 

502 

503 
global 

504 

505 
consts 

506 
insert :: "'a => 'a set => 'a set" 

507 

508 
local 

509 

510 
defs 

511 
insert_def: "insert a B == {x. x=a} Un B" 

512 

513 
syntax 

514 
"@Finset" :: "args => 'a set" ("{(_)}") 

515 

516 
translations 

517 
"{x, xs}" == "insert x {xs}" 

518 
"{x}" == "insert x {}" 

519 

520 
lemma insert_iff [simp]: "(a : insert b A) = (a = b  a:A)" 

521 
by (unfold insert_def) blast 

522 

523 
lemma insertI1: "a : insert a B" 

524 
by simp 

525 

526 
lemma insertI2: "a : B ==> a : insert b B" 

527 
by simp 

528 

529 
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P" 

530 
by (unfold insert_def) blast 

531 

532 
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B" 

533 
 {* Classical introduction rule. *} 

534 
by auto 

535 

536 
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A  {x} \<subseteq> B else A \<subseteq> B)" 

537 
by auto 

538 

539 
lemma set_insert: 

540 
assumes "x \<in> A" 

541 
obtains B where "A = insert x B" and "x \<notin> B" 

542 
proof 

543 
from assms show "A = insert x (A  {x})" by blast 

544 
next 

545 
show "x \<notin> A  {x}" by blast 

546 
qed 

547 

548 
lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)" 

549 
by auto 

550 

551 
lemma insert_is_Un: "insert a A = {a} Un A" 

552 
 {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *} 

553 
by blast 

554 

555 
lemma insert_not_empty [simp]: "insert a A \<noteq> {}" 

556 
by blast 

557 

558 
lemmas empty_not_insert = insert_not_empty [symmetric, standard] 

559 
declare empty_not_insert [simp] 

560 

561 
lemma insert_absorb: "a \<in> A ==> insert a A = A" 

562 
 {* @{text "[simp]"} causes recursive calls when there are nested inserts *} 

563 
 {* with \emph{quadratic} running time *} 

564 
by blast 

565 

566 
lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A" 

567 
by blast 

568 

569 
lemma insert_commute: "insert x (insert y A) = insert y (insert x A)" 

570 
by blast 

571 

572 
lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)" 

573 
by blast 

574 

575 
lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B" 

576 
 {* use new @{text B} rather than @{text "A  {a}"} to avoid infinite unfolding *} 

577 
apply (rule_tac x = "A  {a}" in exI, blast) 

578 
done 

579 

580 
lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a > P u}" 

581 
by auto 

582 

583 
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)" 

584 
by blast 

585 

586 
lemma insert_disjoint [simp,noatp]: 

587 
"(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})" 

588 
"({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)" 

589 
by auto 

590 

591 
lemma disjoint_insert [simp,noatp]: 

592 
"(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})" 

593 
"({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)" 

594 
by auto 

595 

596 
text {* Singletons, using insert *} 

597 

598 
lemma singletonI [intro!,noatp]: "a : {a}" 

599 
 {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *} 

600 
by (rule insertI1) 

601 

602 
lemma singletonD [dest!,noatp]: "b : {a} ==> b = a" 

603 
by blast 

604 

605 
lemmas singletonE = singletonD [elim_format] 

606 

607 
lemma singleton_iff: "(b : {a}) = (b = a)" 

608 
by blast 

609 

610 
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b" 

611 
by blast 

612 

613 
lemma singleton_insert_inj_eq [iff,noatp]: 

614 
"({b} = insert a A) = (a = b & A \<subseteq> {b})" 

615 
by blast 

616 

617 
lemma singleton_insert_inj_eq' [iff,noatp]: 

618 
"(insert a A = {b}) = (a = b & A \<subseteq> {b})" 

619 
by blast 

620 

621 
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {}  A = {x}" 

622 
by fast 

623 

624 
lemma singleton_conv [simp]: "{x. x = a} = {a}" 

625 
by blast 

626 

627 
lemma singleton_conv2 [simp]: "{x. a = x} = {a}" 

628 
by blast 

629 

630 
lemma diff_single_insert: "A  {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B" 

631 
by blast 

632 

633 
lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d  a=d & b=c)" 

634 
by (blast elim: equalityE) 

635 

636 
lemma psubset_insert_iff: 

637 
"(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)" 

638 
by (auto simp add: less_le subset_insert_iff) 

639 

640 
lemma subset_insertI: "B \<subseteq> insert a B" 

641 
by (rule subsetI) (erule insertI2) 

642 

643 
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B" 

644 
by blast 

645 

646 
lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)" 

647 
by blast 

648 

649 

650 
subsubsection {* Bounded quantifiers and operators *} 

651 

652 
global 

653 

654 
consts 

655 
Ball :: "'a set => ('a => bool) => bool"  "bounded universal quantifiers" 

656 
Bex :: "'a set => ('a => bool) => bool"  "bounded existential quantifiers" 

657 
Bex1 :: "'a set => ('a => bool) => bool"  "bounded unique existential quantifiers" 

658 

659 
local 

660 

661 
defs 

662 
Ball_def: "Ball A P == ALL x. x:A > P(x)" 

663 
Bex_def: "Bex A P == EX x. x:A & P(x)" 

664 
Bex1_def: "Bex1 A P == EX! x. x:A & P(x)" 

665 

666 
syntax 

667 
"_Ball" :: "pttrn => 'a set => bool => bool" ("(3ALL _:_./ _)" [0, 0, 10] 10) 

668 
"_Bex" :: "pttrn => 'a set => bool => bool" ("(3EX _:_./ _)" [0, 0, 10] 10) 

669 
"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3EX! _:_./ _)" [0, 0, 10] 10) 

670 
"_Bleast" :: "id => 'a set => bool => 'a" ("(3LEAST _:_./ _)" [0, 0, 10] 10) 

671 

672 
syntax (HOL) 

673 
"_Ball" :: "pttrn => 'a set => bool => bool" ("(3! _:_./ _)" [0, 0, 10] 10) 

674 
"_Bex" :: "pttrn => 'a set => bool => bool" ("(3? _:_./ _)" [0, 0, 10] 10) 

675 
"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3?! _:_./ _)" [0, 0, 10] 10) 

676 

677 
syntax (xsymbols) 

678 
"_Ball" :: "pttrn => 'a set => bool => bool" ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10) 

679 
"_Bex" :: "pttrn => 'a set => bool => bool" ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10) 

680 
"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10) 

681 
"_Bleast" :: "id => 'a set => bool => 'a" ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10) 

682 

683 
syntax (HTML output) 

684 
"_Ball" :: "pttrn => 'a set => bool => bool" ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10) 

685 
"_Bex" :: "pttrn => 'a set => bool => bool" ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10) 

686 
"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10) 

687 

688 
translations 

689 
"ALL x:A. P" == "Ball A (%x. P)" 

690 
"EX x:A. P" == "Bex A (%x. P)" 

691 
"EX! x:A. P" == "Bex1 A (%x. P)" 

692 
"LEAST x:A. P" => "LEAST x. x:A & P" 

14804
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Corrected printer bug for bounded quantifiers Q x<=y. P
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parents:
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changeset

693 

19656
09be06943252
tuned concrete syntax  abbreviation/const_syntax;
wenzelm
parents:
19637
diff
changeset

694 
syntax (output) 
14804
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parents:
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diff
changeset

695 
"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10) 
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
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changeset

696 
"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10) 
8de39d3e8eb6
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changeset

697 
"_setleAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10) 
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
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parents:
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diff
changeset

698 
"_setleEx" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19870
diff
changeset

699 
"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3EX! _<=_./ _)" [0, 0, 10] 10) 
14804
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

700 

8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
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parents:
14752
diff
changeset

701 
syntax (xsymbols) 
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

702 
"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subset>_./ _)" [0, 0, 10] 10) 
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

703 
"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subset>_./ _)" [0, 0, 10] 10) 
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

704 
"_setleAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10) 
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

705 
"_setleEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10) 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19870
diff
changeset

706 
"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10) 
14804
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

707 

19656
09be06943252
tuned concrete syntax  abbreviation/const_syntax;
wenzelm
parents:
19637
diff
changeset

708 
syntax (HOL output) 
14804
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

709 
"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10) 
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

710 
"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10) 
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

711 
"_setleAll" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10) 
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

712 
"_setleEx" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19870
diff
changeset

713 
"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3?! _<=_./ _)" [0, 0, 10] 10) 
14804
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

714 

8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

715 
syntax (HTML output) 
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

716 
"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subset>_./ _)" [0, 0, 10] 10) 
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

717 
"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subset>_./ _)" [0, 0, 10] 10) 
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

718 
"_setleAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10) 
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

719 
"_setleEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10) 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19870
diff
changeset

720 
"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10) 
14804
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

721 

8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

722 
translations 
30352  723 
"\<forall>A\<subset>B. P" => "ALL A. A \<subset> B > P" 
724 
"\<exists>A\<subset>B. P" => "EX A. A \<subset> B & P" 

725 
"\<forall>A\<subseteq>B. P" => "ALL A. A \<subseteq> B > P" 

726 
"\<exists>A\<subseteq>B. P" => "EX A. A \<subseteq> B & P" 

727 
"\<exists>!A\<subseteq>B. P" => "EX! A. A \<subseteq> B & P" 

14804
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

728 

8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

729 
print_translation {* 
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
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diff
changeset

730 
let 
22377  731 
val Type (set_type, _) = @{typ "'a set"}; 
732 
val All_binder = Syntax.binder_name @{const_syntax "All"}; 

733 
val Ex_binder = Syntax.binder_name @{const_syntax "Ex"}; 

734 
val impl = @{const_syntax "op >"}; 

735 
val conj = @{const_syntax "op &"}; 

736 
val sbset = @{const_syntax "subset"}; 

737 
val sbset_eq = @{const_syntax "subset_eq"}; 

21819  738 

739 
val trans = 

740 
[((All_binder, impl, sbset), "_setlessAll"), 

741 
((All_binder, impl, sbset_eq), "_setleAll"), 

742 
((Ex_binder, conj, sbset), "_setlessEx"), 

743 
((Ex_binder, conj, sbset_eq), "_setleEx")]; 

744 

745 
fun mk v v' c n P = 

746 
if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v  _ => false) n) 

747 
then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match; 

748 

749 
fun tr' q = (q, 

750 
fn [Const ("_bound", _) $ Free (v, Type (T, _)), Const (c, _) $ (Const (d, _) $ (Const ("_bound", _) $ Free (v', _)) $ n) $ P] => 

751 
if T = (set_type) then case AList.lookup (op =) trans (q, c, d) 

752 
of NONE => raise Match 

753 
 SOME l => mk v v' l n P 

754 
else raise Match 

755 
 _ => raise Match); 

14804
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

756 
in 
21819  757 
[tr' All_binder, tr' Ex_binder] 
14804
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
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parents:
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diff
changeset

758 
end 
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset

759 
*} 
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
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diff
changeset

760 

11979  761 
text {* 
762 
\medskip Translate between @{text "{e  x1...xn. P}"} and @{text 

763 
"{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is 

764 
only translated if @{text "[0..n] subset bvs(e)"}. 

765 
*} 

766 

30352  767 
syntax 
768 
"@SetCompr" :: "'a => idts => bool => 'a set" ("(1{_ /_./ _})") 

769 
"@Collect" :: "idt => 'a set => bool => 'a set" ("(1{_ :/ _./ _})") 

770 

771 
syntax (xsymbols) 

772 
"@Collect" :: "idt => 'a set => bool => 'a set" ("(1{_ \<in>/ _./ _})") 

773 

774 
translations 

775 
"{x:A. P}" => "{x. x:A & P}" 

776 

11979  777 
parse_translation {* 
778 
let 

779 
val ex_tr = snd (mk_binder_tr ("EX ", "Ex")); 

3947  780 

11979  781 
fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1 
782 
 nvars _ = 1; 

783 

784 
fun setcompr_tr [e, idts, b] = 

785 
let 

786 
val eq = Syntax.const "op =" $ Bound (nvars idts) $ e; 

787 
val P = Syntax.const "op &" $ eq $ b; 

788 
val exP = ex_tr [idts, P]; 

17784  789 
in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end; 
11979  790 

791 
in [("@SetCompr", setcompr_tr)] end; 

792 
*} 

923  793 

13763
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset

794 
print_translation {* 
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset

795 
let 
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset

796 
val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY")); 
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset

797 

f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset

798 
fun setcompr_tr' [Abs (abs as (_, _, P))] = 
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
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diff
changeset

799 
let 
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset

800 
fun check (Const ("Ex", _) $ Abs (_, _, P), n) = check (P, n + 1) 
f94b569cd610
added print translations tha avoid eta contraction for important binders.
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diff
changeset

801 
 check (Const ("op &", _) $ (Const ("op =", _) $ Bound m $ e) $ P, n) = 
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
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diff
changeset

802 
n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso 
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset

803 
((0 upto (n  1)) subset add_loose_bnos (e, 0, [])) 
13764  804 
 check _ = false 
923  805 

11979  806 
fun tr' (_ $ abs) = 
807 
let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs] 

808 
in Syntax.const "@SetCompr" $ e $ idts $ Q end; 

13763
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset

809 
in if check (P, 0) then tr' P 
15535  810 
else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs 
811 
val M = Syntax.const "@Coll" $ x $ t 

812 
in case t of 

813 
Const("op &",_) 

814 
$ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A) 

815 
$ P => 

816 
if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M 

817 
 _ => M 

818 
end 

13763
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset

819 
end; 
11979  820 
in [("Collect", setcompr_tr')] end; 
821 
*} 

822 

823 
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x" 

824 
by (simp add: Ball_def) 

825 

826 
lemmas strip = impI allI ballI 

827 

828 
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x" 

829 
by (simp add: Ball_def) 

830 

831 
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q" 

832 
by (unfold Ball_def) blast 

22139  833 

834 
ML {* bind_thm ("rev_ballE", permute_prems 1 1 @{thm ballE}) *} 

11979  835 

836 
text {* 

837 
\medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and 

838 
@{prop "a:A"}; creates assumption @{prop "P a"}. 

839 
*} 

840 

841 
ML {* 

22139  842 
fun ball_tac i = etac @{thm ballE} i THEN contr_tac (i + 1) 
11979  843 
*} 
844 

845 
text {* 

846 
Gives better instantiation for bound: 

847 
*} 

848 

26339  849 
declaration {* fn _ => 
850 
Classical.map_cs (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1)) 

11979  851 
*} 
852 

853 
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x" 

854 
 {* Normally the best argument order: @{prop "P x"} constrains the 

855 
choice of @{prop "x:A"}. *} 

856 
by (unfold Bex_def) blast 

857 

13113  858 
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x" 
11979  859 
 {* The best argument order when there is only one @{prop "x:A"}. *} 
860 
by (unfold Bex_def) blast 

861 

862 
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x" 

863 
by (unfold Bex_def) blast 

864 

865 
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q" 

866 
by (unfold Bex_def) blast 

867 

868 
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) > P)" 

869 
 {* Trival rewrite rule. *} 

870 
by (simp add: Ball_def) 

871 

872 
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)" 

873 
 {* Dual form for existentials. *} 

874 
by (simp add: Bex_def) 

875 

876 
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)" 

877 
by blast 

878 

879 
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)" 

880 
by blast 

881 

882 
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)" 

883 
by blast 

884 

885 
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)" 

886 
by blast 

887 

888 
lemma ball_one_point1 [simp]: "(ALL x:A. x = a > P x) = (a:A > P a)" 

889 
by blast 

890 

891 
lemma ball_one_point2 [simp]: "(ALL x:A. a = x > P x) = (a:A > P a)" 

892 
by blast 

893 

26480  894 
ML {* 
13462  895 
local 
22139  896 
val unfold_bex_tac = unfold_tac @{thms "Bex_def"}; 
18328  897 
fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac; 
11979  898 
val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac; 
899 

22139  900 
val unfold_ball_tac = unfold_tac @{thms "Ball_def"}; 
18328  901 
fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac; 
11979  902 
val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac; 
903 
in 

18328  904 
val defBEX_regroup = Simplifier.simproc (the_context ()) 
13462  905 
"defined BEX" ["EX x:A. P x & Q x"] rearrange_bex; 
18328  906 
val defBALL_regroup = Simplifier.simproc (the_context ()) 
13462  907 
"defined BALL" ["ALL x:A. P x > Q x"] rearrange_ball; 
11979  908 
end; 
13462  909 

910 
Addsimprocs [defBALL_regroup, defBEX_regroup]; 

11979  911 
*} 
912 

30352  913 
text {* 
914 
\medskip Etacontracting these four rules (to remove @{text P}) 

915 
causes them to be ignored because of their interaction with 

916 
congruence rules. 

917 
*} 

918 

919 
lemma ball_UNIV [simp]: "Ball UNIV P = All P" 

920 
by (simp add: Ball_def) 

921 

922 
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P" 

923 
by (simp add: Bex_def) 

924 

925 
lemma ball_empty [simp]: "Ball {} P = True" 

926 
by (simp add: Ball_def) 

927 

928 
lemma bex_empty [simp]: "Bex {} P = False" 

929 
by (simp add: Bex_def) 

930 

931 
text {* Congruence rules *} 

11979  932 

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

933 
lemma ball_cong: 
11979  934 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
935 
(ALL x:A. P x) = (ALL x:B. Q x)" 

936 
by (simp add: Ball_def) 

937 

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

938 
lemma strong_ball_cong [cong]: 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

939 
"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

940 
(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

941 
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

942 

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

943 
lemma bex_cong: 
11979  944 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
945 
(EX x:A. P x) = (EX x:B. Q x)" 

946 
by (simp add: Bex_def cong: conj_cong) 

1273  947 

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

948 
lemma strong_bex_cong [cong]: 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

949 
"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

950 
(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

951 
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

952 

26800  953 
lemma subset_eq: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast 
11979  954 

30352  955 
lemma atomize_ball: 
956 
"(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)" 

957 
by (simp only: Ball_def atomize_all atomize_imp) 

958 

959 
lemmas [symmetric, rulify] = atomize_ball 

960 
and [symmetric, defn] = atomize_ball 

961 

962 

963 
subsubsection {* Image of a set under a function. *} 

964 

11979  965 
text {* 
30352  966 
Frequently @{term b} does not have the syntactic form of @{term "f x"}. 
11979  967 
*} 
968 

30352  969 
global 
970 

971 
consts 

972 
image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90) 

973 

974 
local 

975 

976 
defs 

977 
image_def [noatp]: "f`A == {y. EX x:A. y = f(x)}" 

978 

979 
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A" 

980 
by (unfold image_def) blast 

981 

982 
lemma imageI: "x : A ==> f x : f ` A" 

983 
by (rule image_eqI) (rule refl) 

984 

985 
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A" 

986 
 {* This version's more effective when we already have the 

987 
required @{term x}. *} 

988 
by (unfold image_def) blast 

989 

990 
lemma imageE [elim!]: 

991 
"b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P" 

992 
 {* The etaexpansion gives variablename preservation. *} 

993 
by (unfold image_def) blast 

994 

995 
lemma image_Un: "f`(A Un B) = f`A Un f`B" 

11979  996 
by blast 
997 

30352  998 
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)" 
999 
by blast 

1000 

1001 
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)" 

1002 
 {* This rewrite rule would confuse users if made default. *} 

11979  1003 
by blast 
923  1004 

30352  1005 
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)" 
1006 
apply safe 

1007 
prefer 2 apply fast 

1008 
apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast) 

13865  1009 
done 
1010 

30352  1011 
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B" 
1012 
 {* Replaces the three steps @{text subsetI}, @{text imageE}, 

1013 
@{text hypsubst}, but breaks too many existing proofs. *} 

1014 
by blast 

1015 

1016 
lemma image_empty [simp]: "f`{} = {}" 

1017 
by blast 

1018 

1019 
lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)" 

1020 
by blast 

1021 

1022 
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}" 

1023 
by auto 

1024 

1025 
lemma image_constant_conv: "(%x. c) ` A = (if A = {} then {} else {c})" 

1026 
by auto 

1027 

1028 
lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A" 

1029 
by blast 

1030 

1031 
lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A" 

1032 
by blast 

1033 

1034 
lemma image_is_empty [iff]: "(f`A = {}) = (A = {})" 

1035 
by blast 

1036 

1037 

1038 
lemma image_Collect [noatp]: "f ` {x. P x} = {f x  x. P x}" 

1039 
 {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS, 

1040 
with its implicit quantifier and conjunction. Also image enjoys better 

1041 
equational properties than does the RHS. *} 

1042 
by blast 

1043 

1044 
lemma if_image_distrib [simp]: 

1045 
"(\<lambda>x. if P x then f x else g x) ` S 

1046 
= (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))" 

1047 
by (auto simp add: image_def) 

1048 

1049 
lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N" 

1050 
by (simp add: image_def) 

1051 

1052 

1053 
subsection {* Set reasoning tools *} 

11979  1054 

1055 
text {* 

30352  1056 
Rewrite rules for boolean casesplitting: faster than @{text 
1057 
"split_if [split]"}. 

11979  1058 
*} 
1059 

30352  1060 
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q > x = b) & (~ Q > y = b))" 
1061 
by (rule split_if) 

1062 

1063 
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q > a = x) & (~ Q > a = y))" 

1064 
by (rule split_if) 

2388  1065 

11979  1066 
text {* 
30352  1067 
Split ifs on either side of the membership relation. Not for @{text 
1068 
"[simp]"}  can cause goals to blow up! 

1069 
*} 

1070 

1071 
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q > x : b) & (~ Q > y : b))" 

1072 
by (rule split_if) 

1073 

1074 
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q > a : x) & (~ Q > a : y))" 

1075 
by (rule split_if [where P="%S. a : S"]) 

1076 

1077 
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2 

1078 

1079 
(*Would like to add these, but the existing code only searches for the 

1080 
outerlevel constant, which in this case is just "op :"; we instead need 

1081 
to use termnets to associate patterns with rules. Also, if a rule fails to 

1082 
apply, then the formula should be kept. 

1083 
[("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]), 

1084 
("Int", [IntD1,IntD2]), 

1085 
("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])] 

1086 
*) 

1087 

1088 
ML {* 

1089 
val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs; 

1090 
*} 

1091 
declaration {* fn _ => 

1092 
Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs)) 

11979  1093 
*} 
1094 

30352  1095 
text {* Transitivity rules for calculational reasoning *} 
1096 

1097 
lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B" 

1098 
by (rule subsetD) 

1099 

1100 
lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B" 

1101 
by (rule subsetD) 

1102 

1103 
lemmas basic_trans_rules [trans] = 

1104 
order_trans_rules set_rev_mp set_mp 

1105 

1106 

1107 
subsection {* Complete lattices *} 

1108 

1109 
notation 

1110 
less_eq (infix "\<sqsubseteq>" 50) and 

1111 
less (infix "\<sqsubset>" 50) and 

1112 
inf (infixl "\<sqinter>" 70) and 

1113 
sup (infixl "\<squnion>" 65) 

1114 

1115 
class complete_lattice = lattice + bot + top + 

1116 
fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900) 

1117 
and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900) 

1118 
assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x" 

1119 
and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A" 

1120 
assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A" 

1121 
and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z" 

1122 
begin 

1123 

1124 
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}" 

1125 
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) 

1126 

1127 
lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}" 

1128 
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) 

1129 

1130 
lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}" 

1131 
unfolding Sup_Inf by auto 

1132 

1133 
lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}" 

1134 
unfolding Inf_Sup by auto 

1135 

1136 
lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A" 

1137 
by (auto intro: antisym Inf_greatest Inf_lower) 

1138 

1139 
lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A" 

1140 
by (auto intro: antisym Sup_least Sup_upper) 

1141 

1142 
lemma Inf_singleton [simp]: 

1143 
"\<Sqinter>{a} = a" 

1144 
by (auto intro: antisym Inf_lower Inf_greatest) 

1145 

1146 
lemma Sup_singleton [simp]: 

1147 
"\<Squnion>{a} = a" 

1148 
by (auto intro: antisym Sup_upper Sup_least) 

1149 

1150 
lemma Inf_insert_simp: 

1151 
"\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)" 

1152 
by (cases "A = {}") (simp_all, simp add: Inf_insert) 

1153 

1154 
lemma Sup_insert_simp: 

1155 
"\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)" 

1156 
by (cases "A = {}") (simp_all, simp add: Sup_insert) 

1157 

1158 
lemma Inf_binary: 

1159 
"\<Sqinter>{a, b} = a \<sqinter> b" 

1160 
by (simp add: Inf_insert_simp) 

1161 

1162 
lemma Sup_binary: 

1163 
"\<Squnion>{a, b} = a \<squnion> b" 

1164 
by (simp add: Sup_insert_simp) 

1165 

1166 
lemma bot_def: 

1167 
"bot = \<Squnion>{}" 

1168 
by (auto intro: antisym Sup_least) 

1169 

1170 
lemma top_def: 

1171 
"top = \<Sqinter>{}" 

1172 
by (auto intro: antisym Inf_greatest) 

1173 

1174 
lemma sup_bot [simp]: 

1175 
"x \<squnion> bot = x" 

1176 
using bot_least [of x] by (simp add: le_iff_sup sup_commute) 

1177 

1178 
lemma inf_top [simp]: 

1179 
"x \<sqinter> top = x" 

1180 
using top_greatest [of x] by (simp add: le_iff_inf inf_commute) 

1181 

1182 
definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where 

1183 
"SUPR A f == \<Squnion> (f ` A)" 

1184 

1185 
definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where 

1186 
"INFI A f == \<Sqinter> (f ` A)" 

1187 

1188 
end 

1189 

1190 
syntax 

1191 
"_SUP1" :: "pttrns => 'b => 'b" ("(3SUP _./ _)" [0, 10] 10) 

1192 
"_SUP" :: "pttrn => 'a set => 'b => 'b" ("(3SUP _:_./ _)" [0, 10] 10) 

1193 
"_INF1" :: "pttrns => 'b => 'b" ("(3INF _./ _)" [0, 10] 10) 

1194 
"_INF" :: "pttrn => 'a set => 'b => 'b" ("(3INF _:_./ _)" [0, 10] 10) 

1195 

1196 
translations 

1197 
"SUP x y. B" == "SUP x. SUP y. B" 

1198 
"SUP x. B" == "CONST SUPR CONST UNIV (%x. B)" 

1199 
"SUP x. B" == "SUP x:CONST UNIV. B" 

1200 
"SUP x:A. B" == "CONST SUPR A (%x. B)" 

1201 
"INF x y. B" == "INF x. INF y. B" 

1202 
"INF x. B" == "CONST INFI CONST UNIV (%x. B)" 

1203 
"INF x. B" == "INF x:CONST UNIV. B" 

1204 
"INF x:A. B" == "CONST INFI A (%x. B)" 

1205 

1206 
(* To avoid etacontraction of body: *) 

1207 
print_translation {* 

1208 
let 

1209 
fun btr' syn (A :: Abs abs :: ts) = 

1210 
let val (x,t) = atomic_abs_tr' abs 

1211 
in list_comb (Syntax.const syn $ x $ A $ t, ts) end 

1212 
val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const 

1213 
in 

1214 
[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")] 

1215 
end 

1216 
*} 

1217 

1218 
context complete_lattice 

1219 
begin 

1220 

1221 
lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)" 

1222 
by (auto simp add: SUPR_def intro: Sup_upper) 

1223 

1224 
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u" 

1225 
by (auto simp add: SUPR_def intro: Sup_least) 

1226 

1227 
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i" 

1228 
by (auto simp add: INFI_def intro: Inf_lower) 

1229 

1230 
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)" 

1231 
by (auto simp add: INFI_def intro: Inf_greatest) 

1232 

1233 
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M" 

1234 
by (auto intro: antisym SUP_leI le_SUPI) 

1235 

1236 
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M" 

1237 
by (auto intro: antisym INF_leI le_INFI) 

1238 

1239 
end 

1240 

1241 
subsubsection {* Bool as complete lattice *} 

1242 

1243 
instantiation bool :: complete_lattice 

1244 
begin 

1245 

1246 
definition 

1247 
Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)" 

1248 

1249 
definition 

1250 
Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)" 

1251 

1252 
instance 

1253 
by intro_classes (auto simp add: Inf_bool_def Sup_bool_def le_bool_def) 

1254 

1255 
end 

1256 

1257 
lemma Inf_empty_bool [simp]: 

1258 
"\<Sqinter>{}" 

1259 
unfolding Inf_bool_def by auto 

1260 

1261 
lemma not_Sup_empty_bool [simp]: 

1262 
"\<not> Sup {}" 

1263 
unfolding Sup_bool_def by auto 

1264 

1265 

1266 
subsubsection {* Fun as complete lattice *} 

1267 

1268 
instantiation "fun" :: (type, complete_lattice) complete_lattice 

1269 
begin 

1270 

1271 
definition 

1272 
Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})" 

1273 

1274 
definition 

1275 
Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})" 

1276 

1277 
instance 

1278 
by intro_classes 

1279 
(auto simp add: Inf_fun_def Sup_fun_def le_fun_def 

1280 
intro: Inf_lower Sup_upper Inf_greatest Sup_least) 

1281 

1282 
end 

1283 

1284 
lemma Inf_empty_fun: 

1285 
"\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})" 

1286 
by rule (auto simp add: Inf_fun_def) 

1287 

1288 
lemma Sup_empty_fun: 

1289 
"\<Squnion>{} = (\<lambda>_. \<Squnion>{})" 

1290 
by rule (auto simp add: Sup_fun_def) 

1291 

1292 
no_notation 

1293 
less_eq (infix "\<sqsubseteq>" 50) and 

1294 
less (infix "\<sqsubset>" 50) and 

1295 
inf (infixl "\<sqinter>" 70) and 

1296 
sup (infixl "\<squnion>" 65) and 

1297 
Inf ("\<Sqinter>_" [900] 900) and 

1298 
Sup ("\<Squnion>_" [900] 900) 

1299 

1300 

1301 
subsection {* Further operations *} 

1302 

1303 
subsubsection {* Big families as specialisation of lattice operations *} 

1304 

1305 
definition INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where 

1306 
"INTER A B \<equiv> {y. \<forall>x\<in>A. y \<in> B x}" 

1307 

1308 
definition UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where 

1309 
"UNION A B \<equiv> {y. \<exists>x\<in>A. y \<in> B x}" 

1310 

1311 
definition Inter :: "'a set set \<Rightarrow> 'a set" where 

1312 
"Inter S \<equiv> INTER S (\<lambda>x. x)" 

1313 

1314 
definition Union :: "'a set set \<Rightarrow> 'a set" where 

1315 
"Union S \<equiv> UNION S (\<lambda>x. x)" 

1316 

1317 
notation (xsymbols) 

1318 
Inter ("\<Inter>_" [90] 90) and 

1319 
Union ("\<Union>_" [90] 90) 

1320 

1321 
syntax 

1322 
"@INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10) 

1323 
"@UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10) 

1324 
"@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 10] 10) 

1325 
"@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [0, 10] 10) 

1326 

1327 
syntax (xsymbols) 

1328 
"@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10) 

1329 
"@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10) 

1330 
"@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 10] 10) 

1331 
"@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 10] 10) 

1332 

1333 
syntax (latex output) 

1334 
"@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) 

1335 
"@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) 

1336 
"@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10) 

1337 
"@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10) 

1338 

1339 
translations 

1340 
"INT x y. B" == "INT x. INT y. B" 

1341 
"INT x. B" == "CONST INTER CONST UNIV (%x. B)" 

1342 
"INT x. B" == "INT x:CONST UNIV. B" 

1343 
"INT x:A. B" == "CONST INTER A (%x. B)" 

1344 
"UN x y. B" == "UN x. UN y. B" 

1345 
"UN x. B" == "CONST UNION CONST UNIV (%x. B)" 

1346 
"UN x. B" == "UN x:CONST UNIV. B" 

1347 
"UN x:A. B" == "CONST UNION A (%x. B)" 

923  1348 

11979  1349 
text {* 
30352  1350 
Note the difference between ordinary xsymbol syntax of indexed 
1351 
unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}) 

1352 
and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The 

1353 
former does not make the index expression a subscript of the 

1354 
union/intersection symbol because this leads to problems with nested 

1355 
subscripts in Proof General. 

11979  1356 
*} 
1357 

30352  1358 
(* To avoid etacontraction of body: *) 
1359 
(*FIXME integrate with / factor out from similar situations*) 

1360 
print_translation {* 

1361 
let 

1362 
fun btr' syn [A, Abs abs] = 

1363 
let val (x, t) = atomic_abs_tr' abs 

1364 
in Syntax.const syn $ x $ A $ t end 

1365 
in 

1366 
[(@{const_syntax Ball}, btr' "_Ball"), (@{const_syntax Bex}, btr' "_Bex"), 

1367 
(@{const_syntax UNION}, btr' "@UNION"),(@{const_syntax INTER}, btr' "@INTER")] 

1368 
end 

1369 
*} 

11979  1370 

1371 
subsubsection {* Unions of families *} 

1372 

1373 
text {* 

1374 
@{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}. 

1375 
*} 

1376 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

1377 
declare UNION_def [noatp] 