author  haftmann 
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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 {* Sets as predicates *} 
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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 => bool) => 'a set"  "comprehension" 
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"op :" :: "'a => 'a set => bool"  "membership" 
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local 
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notation 
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"op :" ("op :") and 
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"op :" ("(_/ : _)" [50, 51] 50) 
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defs 
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mem_def [code]: "x : S == S x" 
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Collect_def [code]: "Collect P == P" 
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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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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}" == "Collect (%x. P)" 
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syntax 
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"@SetCompr" :: "'a => idts => bool => 'a set" ("(1{_ /_./ _})") 

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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 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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text {* Set enumerations *} 

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

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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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global 
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consts 
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Ball :: "'a set => ('a => bool) => bool"  "bounded universal quantifiers" 
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Bex :: "'a set => ('a => bool) => bool"  "bounded existential quantifiers" 
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Bex1 :: "'a set => ('a => bool) => bool"  "bounded unique existential quantifiers" 
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local 
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defs 
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Ball_def: "Ball A P == ALL x. x:A > P(x)" 
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Bex_def: "Bex A P == EX x. x:A & P(x)" 
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Bex1_def: "Bex1 A P == EX! x. x:A & P(x)" 
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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" == "Ball A (%x. P)" 
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"EX x:A. P" == "Bex A (%x. P)" 
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"EX! x:A. P" == "Bex1 A (%x. 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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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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205 

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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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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 
22377  222 
val Type (set_type, _) = @{typ "'a set"}; 
223 
val All_binder = Syntax.binder_name @{const_syntax "All"}; 

224 
val Ex_binder = Syntax.binder_name @{const_syntax "Ex"}; 

225 
val impl = @{const_syntax "op >"}; 

226 
val conj = @{const_syntax "op &"}; 

227 
val sbset = @{const_syntax "subset"}; 

228 
val sbset_eq = @{const_syntax "subset_eq"}; 

21819  229 

230 
val trans = 

231 
[((All_binder, impl, sbset), "_setlessAll"), 

232 
((All_binder, impl, sbset_eq), "_setleAll"), 

233 
((Ex_binder, conj, sbset), "_setlessEx"), 

234 
((Ex_binder, conj, sbset_eq), "_setleEx")]; 

235 

236 
fun mk v v' c n P = 

237 
if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v  _ => false) n) 

238 
then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match; 

239 

240 
fun tr' q = (q, 

241 
fn [Const ("_bound", _) $ Free (v, Type (T, _)), Const (c, _) $ (Const (d, _) $ (Const ("_bound", _) $ Free (v', _)) $ n) $ P] => 

242 
if T = (set_type) then case AList.lookup (op =) trans (q, c, d) 

243 
of NONE => raise Match 

244 
 SOME l => mk v v' l n P 

245 
else raise Match 

246 
 _ => raise Match); 

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in 
21819  248 
[tr' All_binder, tr' Ex_binder] 
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end 
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*} 
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251 

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11979  253 
text {* 
254 
\medskip Translate between @{text "{e  x1...xn. P}"} and @{text 

255 
"{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is 

256 
only translated if @{text "[0..n] subset bvs(e)"}. 

257 
*} 

258 

259 
parse_translation {* 

260 
let 

261 
val ex_tr = snd (mk_binder_tr ("EX ", "Ex")); 

3947  262 

11979  263 
fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1 
264 
 nvars _ = 1; 

265 

266 
fun setcompr_tr [e, idts, b] = 

267 
let 

268 
val eq = Syntax.const "op =" $ Bound (nvars idts) $ e; 

269 
val P = Syntax.const "op &" $ eq $ b; 

270 
val exP = ex_tr [idts, P]; 

17784  271 
in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end; 
11979  272 

273 
in [("@SetCompr", setcompr_tr)] end; 

274 
*} 

923  275 

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(* To avoid etacontraction of body: *) 
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print_translation {* 
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let 
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fun btr' syn [A, Abs abs] = 
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let val (x, t) = atomic_abs_tr' abs 
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in Syntax.const syn $ x $ A $ t end 
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in [(@{const_syntax Ball}, btr' "_Ball"), (@{const_syntax Bex}, btr' "_Bex")] end 
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283 
*} 
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284 

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285 
print_translation {* 
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let 
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val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY")); 
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fun setcompr_tr' [Abs (abs as (_, _, P))] = 
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let 
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fun check (Const ("Ex", _) $ Abs (_, _, P), n) = check (P, n + 1) 
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 check (Const ("op &", _) $ (Const ("op =", _) $ Bound m $ e) $ P, n) = 
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n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso 
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((0 upto (n  1)) subset add_loose_bnos (e, 0, [])) 
13764  295 
 check _ = false 
923  296 

11979  297 
fun tr' (_ $ abs) = 
298 
let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs] 

299 
in Syntax.const "@SetCompr" $ e $ idts $ Q end; 

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in if check (P, 0) then tr' P 
15535  301 
else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs 
302 
val M = Syntax.const "@Coll" $ x $ t 

303 
in case t of 

304 
Const("op &",_) 

305 
$ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A) 

306 
$ P => 

307 
if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M 

308 
 _ => M 

309 
end 

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310 
end; 
11979  311 
in [("Collect", setcompr_tr')] end; 
312 
*} 

313 

314 
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x" 

315 
by (simp add: Ball_def) 

316 

317 
lemmas strip = impI allI ballI 

318 

319 
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x" 

320 
by (simp add: Ball_def) 

321 

322 
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q" 

323 
by (unfold Ball_def) blast 

22139  324 

31945  325 
ML {* bind_thm ("rev_ballE", Thm.permute_prems 1 1 @{thm ballE}) *} 
11979  326 

327 
text {* 

328 
\medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and 

329 
@{prop "a:A"}; creates assumption @{prop "P a"}. 

330 
*} 

331 

332 
ML {* 

22139  333 
fun ball_tac i = etac @{thm ballE} i THEN contr_tac (i + 1) 
11979  334 
*} 
335 

336 
text {* 

337 
Gives better instantiation for bound: 

338 
*} 

339 

26339  340 
declaration {* fn _ => 
341 
Classical.map_cs (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1)) 

11979  342 
*} 
343 

344 
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x" 

345 
 {* Normally the best argument order: @{prop "P x"} constrains the 

346 
choice of @{prop "x:A"}. *} 

347 
by (unfold Bex_def) blast 

348 

13113  349 
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x" 
11979  350 
 {* The best argument order when there is only one @{prop "x:A"}. *} 
351 
by (unfold Bex_def) blast 

352 

353 
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x" 

354 
by (unfold Bex_def) blast 

355 

356 
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q" 

357 
by (unfold Bex_def) blast 

358 

359 
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) > P)" 

360 
 {* Trival rewrite rule. *} 

361 
by (simp add: Ball_def) 

362 

363 
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)" 

364 
 {* Dual form for existentials. *} 

365 
by (simp add: Bex_def) 

366 

367 
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)" 

368 
by blast 

369 

370 
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)" 

371 
by blast 

372 

373 
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)" 

374 
by blast 

375 

376 
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)" 

377 
by blast 

378 

379 
lemma ball_one_point1 [simp]: "(ALL x:A. x = a > P x) = (a:A > P a)" 

380 
by blast 

381 

382 
lemma ball_one_point2 [simp]: "(ALL x:A. a = x > P x) = (a:A > P a)" 

383 
by blast 

384 

26480  385 
ML {* 
13462  386 
local 
22139  387 
val unfold_bex_tac = unfold_tac @{thms "Bex_def"}; 
18328  388 
fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac; 
11979  389 
val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac; 
390 

22139  391 
val unfold_ball_tac = unfold_tac @{thms "Ball_def"}; 
18328  392 
fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac; 
11979  393 
val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac; 
394 
in 

32010  395 
val defBEX_regroup = Simplifier.simproc @{theory} 
13462  396 
"defined BEX" ["EX x:A. P x & Q x"] rearrange_bex; 
32010  397 
val defBALL_regroup = Simplifier.simproc @{theory} 
13462  398 
"defined BALL" ["ALL x:A. P x > Q x"] rearrange_ball; 
11979  399 
end; 
13462  400 

401 
Addsimprocs [defBALL_regroup, defBEX_regroup]; 

11979  402 
*} 
403 

32081  404 
text {* Congruence rules *} 
11979  405 

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406 
lemma ball_cong: 
11979  407 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
408 
(ALL x:A. P x) = (ALL x:B. Q x)" 

409 
by (simp add: Ball_def) 

410 

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411 
lemma strong_ball_cong [cong]: 
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412 
"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> 
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413 
(ALL x:A. P x) = (ALL x:B. Q x)" 
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414 
by (simp add: simp_implies_def Ball_def) 
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415 

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416 
lemma bex_cong: 
11979  417 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
418 
(EX x:A. P x) = (EX x:B. Q x)" 

419 
by (simp add: Bex_def cong: conj_cong) 

1273  420 

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421 
lemma strong_bex_cong [cong]: 
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422 
"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> 
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423 
(EX x:A. P x) = (EX x:B. Q x)" 
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424 
by (simp add: simp_implies_def Bex_def cong: conj_cong) 
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425 

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426 

32081  427 
subsection {* Basic operations *} 
428 

30531
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429 
subsubsection {* Subsets *} 
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430 

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431 
lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B" 
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432 
by (auto simp add: mem_def intro: predicate1I) 
30352  433 

11979  434 
text {* 
30531
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435 
\medskip Map the type @{text "'a set => anything"} to just @{typ 
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436 
'a}; for overloading constants whose first argument has type @{typ 
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437 
"'a set"}. 
11979  438 
*} 
439 

30596  440 
lemma subsetD [elim, intro?]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B" 
30531
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441 
 {* Rule in Modus Ponens style. *} 
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442 
by (unfold mem_def) blast 
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443 

30596  444 
lemma rev_subsetD [intro?]: "c \<in> A ==> A \<subseteq> B ==> c \<in> B" 
30531
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445 
 {* The same, with reversed premises for use with @{text erule}  
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446 
cf @{text rev_mp}. *} 
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447 
by (rule subsetD) 
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448 

11979  449 
text {* 
30531
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450 
\medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}. 
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451 
*} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

452 

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

453 
ML {* 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

454 
fun impOfSubs th = th RSN (2, @{thm rev_subsetD}) 
11979  455 
*} 
456 

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

457 
lemma subsetCE [elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

458 
 {* Classical elimination rule. *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

459 
by (unfold mem_def) blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

460 

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

461 
lemma subset_eq: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast 
2388  462 

11979  463 
text {* 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

464 
\medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

465 
creates the assumption @{prop "c \<in> B"}. 
30352  466 
*} 
467 

468 
ML {* 

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

469 
fun set_mp_tac i = etac @{thm subsetCE} i THEN mp_tac i 
11979  470 
*} 
471 

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

472 
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

473 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

474 

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

475 
lemma subset_refl [simp,atp]: "A \<subseteq> A" 
32081  476 
by (fact order_refl) 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

477 

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

478 
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C" 
32081  479 
by (fact order_trans) 
480 

481 
lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B" 

482 
by (rule subsetD) 

483 

484 
lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B" 

485 
by (rule subsetD) 

486 

487 
lemmas basic_trans_rules [trans] = 

488 
order_trans_rules set_rev_mp set_mp 

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

489 

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

490 

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

491 
subsubsection {* Equality *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

492 

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

493 
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

494 
apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals]) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

495 
apply (rule Collect_mem_eq) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

496 
apply (rule Collect_mem_eq) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

497 
done 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

498 

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

499 
(* Due to Brian Huffman *) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

500 
lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

501 
by(auto intro:set_ext) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

502 

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

503 
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

504 
 {* Antisymmetry of the subset relation. *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

505 
by (iprover intro: set_ext subsetD) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

506 

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

507 
text {* 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

508 
\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

509 
here? 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

510 
*} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

511 

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

512 
lemma equalityD1: "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

513 
by (simp add: subset_refl) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

514 

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

515 
lemma equalityD2: "A = B ==> B \<subseteq> A" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

516 
by (simp add: subset_refl) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

517 

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

518 
text {* 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

519 
\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

520 
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

521 
\<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}! 
30352  522 
*} 
523 

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

524 
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

525 
by (simp add: subset_refl) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

526 

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

527 
lemma equalityCE [elim]: 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

528 
"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

529 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

530 

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

531 
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

532 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

533 

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

534 
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

535 
by simp 
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 

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

538 
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

539 

32081  540 
definition UNIV :: "'a set" where 
541 
"UNIV \<equiv> {x. True}" 

542 

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

543 
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

544 
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

545 

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

546 
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

547 

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

548 
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

549 
by simp 
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 subset_UNIV [simp]: "A \<subseteq> UNIV" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

552 
by (rule subsetI) (rule UNIV_I) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

553 

32081  554 
lemma top_set_eq: "top = UNIV" 
555 
by (iprover intro!: subset_antisym subset_UNIV top_greatest) 

556 

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

557 
text {* 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

558 
\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

559 
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

560 
congruence rules. 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

561 
*} 
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 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

564 
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

565 

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

566 
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

567 
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

568 

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

569 
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

570 
by auto 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

571 

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

574 

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

575 
lemma empty_iff [simp]: "(c : {}) = False" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

576 
by (simp add: empty_def) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

577 

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

578 
lemma emptyE [elim!]: "a : {} ==> P" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

579 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

580 

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

581 
lemma empty_subsetI [iff]: "{} \<subseteq> A" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

582 
 {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

583 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

584 

32081  585 
lemma bot_set_eq: "bot = {}" 
586 
by (iprover intro!: subset_antisym empty_subsetI bot_least) 

587 

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

588 
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

589 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

590 

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

591 
lemma equals0D: "A = {} ==> a \<notin> A" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

592 
 {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

593 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

594 

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

595 
lemma ball_empty [simp]: "Ball {} P = True" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

596 
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

597 

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

598 
lemma bex_empty [simp]: "Bex {} P = False" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

599 
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

600 

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

601 
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

602 
by (blast elim: equalityE) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

603 

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

604 

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

605 
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

606 

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

607 
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

608 
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

609 

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

610 
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

611 
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

612 

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

613 
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

614 
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

615 

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

616 
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

617 
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

618 

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

619 
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

620 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

621 

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

622 
lemma Pow_top: "A \<in> Pow A" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

623 
by (simp add: subset_refl) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

624 

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

625 

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

626 
subsubsection {* Set complement *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

627 

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

628 
lemma Compl_iff [simp]: "(c \<in> A) = (c \<notin> A)" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

629 
by (simp add: mem_def fun_Compl_def bool_Compl_def) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

630 

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

631 
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> A" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

632 
by (unfold mem_def fun_Compl_def bool_Compl_def) blast 
923  633 

11979  634 
text {* 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

635 
\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

636 
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

637 
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

638 

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

639 
lemma ComplD [dest!]: "c : A ==> c~:A" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

640 
by (simp add: mem_def fun_Compl_def bool_Compl_def) 
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 
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

643 

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

644 
lemma Compl_eq: " A = {x. ~ x : A}" by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

645 

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

646 

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

647 
subsubsection {* Binary union  Un *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

648 

32081  649 
definition Un :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where 
650 
"A Un B \<equiv> {x. x \<in> A \<or> x \<in> B}" 

651 

652 
notation (xsymbols) 

653 
"Un" (infixl "\<union>" 65) 

654 

655 
notation (HTML output) 

656 
"Un" (infixl "\<union>" 65) 

657 

658 
lemma sup_set_eq: "sup A B = A \<union> B" 

659 
by (simp add: sup_fun_eq sup_bool_eq Un_def Collect_def mem_def) 

660 

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

661 
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

662 
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

663 

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

664 
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

665 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

666 

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

667 
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

668 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

669 

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

670 
text {* 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

671 
\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

672 
@{prop B}. 
11979  673 
*} 
674 

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

675 
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

676 
by auto 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

677 

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

678 
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

679 
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

680 

32081  681 
lemma insert_def: "insert a B \<equiv> {x. x = a} \<union> B" 
682 
by (simp add: Collect_def mem_def insert_compr Un_def) 

683 

684 
lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)" 

685 
apply (fold sup_set_eq) 

686 
apply (erule mono_sup) 

687 
done 

688 

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

689 

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

690 
subsubsection {* Binary intersection  Int *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

691 

32081  692 
definition Int :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where 
693 
"A Int B \<equiv> {x. x \<in> A \<and> x \<in> B}" 

694 

695 
notation (xsymbols) 

696 
"Int" (infixl "\<inter>" 70) 

697 

698 
notation (HTML output) 

699 
"Int" (infixl "\<inter>" 70) 

700 

701 
lemma inf_set_eq: "inf A B = A \<inter> B" 

702 
by (simp add: inf_fun_eq inf_bool_eq Int_def Collect_def mem_def) 

703 

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

704 
lemma Int_iff [simp]: "(c : A Int 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

705 
by (unfold Int_def) blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

706 

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

707 
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

708 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

709 

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

710 
lemma IntD1: "c : A Int B ==> c:A" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

711 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

712 

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

713 
lemma IntD2: "c : A Int B ==> c:B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

714 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

715 

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

716 
lemma IntE [elim!]: "c : A Int 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

717 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

718 

32081  719 
lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B" 
720 
apply (fold inf_set_eq) 

721 
apply (erule mono_inf) 

722 
done 

723 

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

724 

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

725 
subsubsection {* Set difference *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

726 

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

727 
lemma Diff_iff [simp]: "(c : A  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

728 
by (simp add: mem_def fun_diff_def bool_diff_def) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

729 

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

730 
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

731 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

732 

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

733 
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

734 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

735 

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

736 
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

737 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

738 

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

739 
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

740 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

741 

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

742 
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

743 

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

744 
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

745 
by blast 
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 

31456  748 
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

749 

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

750 
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

751 
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

752 

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

753 
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

754 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

755 

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

756 
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

757 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

758 

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

759 
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

760 
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

761 

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

762 
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

763 
 {* Classical introduction rule. *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

764 
by auto 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

765 

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

766 
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

767 
by auto 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

768 

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

769 
lemma set_insert: 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

770 
assumes "x \<in> A" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

771 
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

772 
proof 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

773 
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

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

775 
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

776 
qed 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

777 

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

778 
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

779 
by auto 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

780 

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

781 
subsubsection {* Singletons, using insert *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

782 

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

783 
lemma singletonI [intro!,noatp]: "a : {a}" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

784 
 {* 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

785 
by (rule insertI1) 
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 
lemma singletonD [dest!,noatp]: "b : {a} ==> b = a" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

788 
by blast 
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 
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

791 

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

792 
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

793 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

794 

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

795 
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

796 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

797 

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

798 
lemma singleton_insert_inj_eq [iff,noatp]: 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

799 
"({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

800 
by 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 singleton_insert_inj_eq' [iff,noatp]: 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

803 
"(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

804 
by blast 
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_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

807 
by fast 
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 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

810 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

811 

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

812 
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

813 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

814 

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

815 
lemma diff_single_insert: "A  {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

816 
by blast 
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 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

819 
by (blast elim: equalityE) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

820 

11979  821 

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

822 
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

823 

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

824 
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

825 
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

826 
*} 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

827 

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

828 
definition image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90) 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

829 
image_def [noatp]: "f ` A = {y. EX x:A. y = 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

830 

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

831 
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

832 
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

833 
"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

834 

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

835 
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

836 
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

837 

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

838 
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

839 
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

840 

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

841 
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

842 
 {* 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

843 
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

844 
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

845 

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

846 
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

847 
"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

848 
 {* 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

849 
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

850 

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

851 
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

852 
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

853 

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

854 
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

855 
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

856 

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

857 
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> 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

858 
 {* 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

859 
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

860 

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

861 
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

862 
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

863 
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

864 
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

865 
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

866 

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

867 
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

868 
 {* 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

869 
@{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

870 
by blast 
11979  871 

872 
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

873 
\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

874 
*} 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

875 

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

876 
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

877 
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

878 

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

879 
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

880 
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

881 

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 
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

883 
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

884 

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 

32081  886 
subsubsection {* Some proof tools *} 
887 

888 
text{* Elimination of @{text"{x. \<dots> & x=t & \<dots>}"}. *} 

889 

890 
lemma Collect_conv_if: "{x. x=a & P x} = (if P a then {a} else {})" 

891 
by auto 

892 

893 
lemma Collect_conv_if2: "{x. a=x & P x} = (if P a then {a} else {})" 

894 
by auto 

895 

896 
text {* 

897 
Simproc for pulling @{text "x=t"} in @{text "{x. \<dots> & x=t & \<dots>}"} 

898 
to the front (and similarly for @{text "t=x"}): 

899 
*} 

900 

901 
ML{* 

902 
local 

903 
val Coll_perm_tac = rtac @{thm Collect_cong} 1 THEN rtac @{thm iffI} 1 THEN 

904 
ALLGOALS(EVERY'[REPEAT_DETERM o (etac @{thm conjE}), 

905 
DEPTH_SOLVE_1 o (ares_tac [@{thm conjI}])]) 

906 
in 

907 
val defColl_regroup = Simplifier.simproc @{theory} 

908 
"defined Collect" ["{x. P x & Q x}"] 

909 
(Quantifier1.rearrange_Coll Coll_perm_tac) 

910 
end; 

911 

912 
Addsimprocs [defColl_regroup]; 

913 
*} 

914 

915 
text {* 

916 
Rewrite rules for boolean casesplitting: faster than @{text 

917 
"split_if [split]"}. 

918 
*} 

919 

920 
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q > x = b) & (~ Q > y = b))" 

921 
by (rule split_if) 

922 

923 
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q > a = x) & (~ Q > a = y))" 

924 
by (rule split_if) 

925 

926 
text {* 

927 
Split ifs on either side of the membership relation. Not for @{text 

928 
"[simp]"}  can cause goals to blow up! 

929 
*} 

930 

931 
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q > x : b) & (~ Q > y : b))" 

932 
by (rule split_if) 

933 

934 
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q > a : x) & (~ Q > a : y))" 

935 
by (rule split_if [where P="%S. a : S"]) 

936 

937 
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2 

938 

939 
(*Would like to add these, but the existing code only searches for the 

940 
outerlevel constant, which in this case is just "op :"; we instead need 

941 
to use termnets to associate patterns with rules. Also, if a rule fails to 

942 
apply, then the formula should be kept. 

943 
[("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]), 

944 
("Int", [IntD1,IntD2]), 

945 
("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])] 

946 
*) 

947 

948 
ML {* 

949 
val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs; 

950 
*} 

951 
declaration {* fn _ => 

952 
Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs)) 

953 
*} 

954 

955 

32077
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956 
subsection {* Complete lattices *} 
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957 

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958 
notation 
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959 
less_eq (infix "\<sqsubseteq>" 50) and 
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960 
less (infix "\<sqsubset>" 50) and 
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961 
inf (infixl "\<sqinter>" 70) and 
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962 
sup (infixl "\<squnion>" 65) 
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963 

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964 
class complete_lattice = lattice + bot + top + 
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965 
fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900) 
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966 
and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900) 
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967 
assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x" 
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968 
and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A" 
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969 
assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A" 
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970 
and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z" 
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971 
begin 
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972 

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973 
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}" 
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974 
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) 
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975 

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976 
lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}" 
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977 
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) 
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haftmann
parents:
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diff
changeset

978 

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979 
lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}" 
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980 
unfolding Sup_Inf by (auto simp add: UNIV_def) 
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parents:
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981 

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982 
lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}" 
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983 
unfolding Inf_Sup by (auto simp add: UNIV_def) 
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984 

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985 
lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A" 
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parents:
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986 
by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower) 
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haftmann
parents:
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diff
changeset

987 

3698947146b2
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parents:
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988 
lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A" 
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989 
by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper) 
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haftmann
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990 

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991 
lemma Inf_singleton [simp]: 
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parents:
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992 
"\<Sqinter>{a} = a" 
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haftmann
parents:
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diff
changeset

993 
by (auto intro: antisym Inf_lower Inf_greatest) 
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haftmann
parents:
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diff
changeset

994 

3698947146b2
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parents:
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995 
lemma Sup_singleton [simp]: 
3698947146b2
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haftmann
parents:
32064
diff
changeset

996 
"\<Squnion>{a} = a" 
3698947146b2
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haftmann
parents:
32064
diff
changeset

997 
by (auto intro: antisym Sup_upper Sup_least) 
3698947146b2
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haftmann
parents:
32064
diff
changeset

998 

3698947146b2
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haftmann
parents:
32064
diff
changeset

999 
lemma Inf_insert_simp: 
3698947146b2
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haftmann
parents:
32064
diff
changeset

1000 
"\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)" 
3698947146b2
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haftmann
parents:
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diff
changeset

1001 
by (cases "A = {}") (simp_all, simp add: Inf_insert) 
3698947146b2
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haftmann
parents:
32064
diff
changeset

1002 

3698947146b2
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haftmann
parents:
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diff
changeset

1003 
lemma Sup_insert_simp: 
3698947146b2
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haftmann
parents:
32064
diff
changeset

1004 
"\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)" 
3698947146b2
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haftmann
parents:
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diff
changeset

1005 
by (cases "A = {}") (simp_all, simp add: Sup_insert) 
3698947146b2
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haftmann
parents:
32064
diff
changeset

1006 

3698947146b2
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haftmann
parents:
32064
diff
changeset

1007 
lemma Inf_binary: 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
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1008 
"\<Sqinter>{a, b} = a \<sqinter> b" 
3698947146b2
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haftmann
parents:
32064
diff
changeset

1009 
by (auto simp add: Inf_insert_simp) 
3698947146b2
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haftmann
parents:
32064
diff
changeset

1010 

3698947146b2
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haftmann
parents:
32064
diff
changeset

1011 
lemma Sup_binary: 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
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1012 
"\<Squnion>{a, b} = a \<squnion> b" 
3698947146b2
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haftmann
parents:
32064
diff
changeset

1013 
by (auto simp add: Sup_insert_simp) 
3698947146b2
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haftmann
parents:
32064
diff
changeset

1014 

3698947146b2
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haftmann
parents:
32064
diff
changeset

1015 
lemma bot_def: 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
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parents:
32064
diff
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1016 
"bot = \<Squnion>{}" 
3698947146b2
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haftmann
parents:
32064
diff
changeset

1017 
by (auto intro: antisym Sup_least) 
3698947146b2
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haftmann
parents:
32064
diff
changeset

1018 

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

1019 
lemma top_def: 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
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1020 
"top = \<Sqinter>{}" 
3698947146b2
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haftmann
parents:
32064
diff
changeset

1021 
by (auto intro: antisym Inf_greatest) 
3698947146b2
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haftmann
parents:
32064
diff
changeset

1022 

3698947146b2
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haftmann
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32064
diff
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1023 
lemma sup_bot [simp]: 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
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1024 
"x \<squnion> bot = x" 
3698947146b2
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haftmann
parents:
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diff
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1025 
using bot_least [of x] by (simp add: le_iff_sup sup_commute) 
3698947146b2
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haftmann
parents:
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diff
changeset

1026 

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

1027 
lemma inf_top [simp]: 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
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diff
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1028 
"x \<sqinter> top = x" 
3698947146b2
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haftmann
parents:
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diff
changeset

1029 
using top_greatest [of x] by (simp add: le_iff_inf inf_commute) 
3698947146b2
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haftmann
parents:
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diff
changeset

1030 

3698947146b2
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diff
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1031 
definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where 
3698947146b2
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1032 
"SUPR A f == \<Squnion> (f ` A)" 
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haftmann
parents:
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diff
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1033 

3698947146b2
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1034 
definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where 
3698947146b2
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1035 
"INFI A f == \<Sqinter> (f ` A)" 
3698947146b2
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haftmann
parents:
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diff
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1036 

3698947146b2
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diff
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1037 
end 
3698947146b2
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haftmann
parents:
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diff
changeset

1038 

3698947146b2
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haftmann
parents:
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diff
changeset

1039 
syntax 
3698947146b2
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1040 
"_SUP1" :: "pttrns => 'b => 'b" ("(3SUP _./ _)" [0, 10] 10) 
3698947146b2
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1041 
"_SUP" :: "pttrn => 'a set => 'b => 'b" ("(3SUP _:_./ _)" [0, 10] 10) 
3698947146b2
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1042 
"_INF1" :: "pttrns => 'b => 'b" ("(3INF _./ _)" [0, 10] 10) 
3698947146b2
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1043 
"_INF" :: "pttrn => 'a set => 'b => 'b" ("(3INF _:_./ _)" [0, 10] 10) 
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haftmann
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1044 

3698947146b2
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haftmann
parents:
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1045 
translations 
3698947146b2
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haftmann
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1046 
"SUP x y. B" == "SUP x. SUP y. B" 
3698947146b2
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haftmann
parents:
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1047 
"SUP x. B" == "CONST SUPR CONST UNIV (%x. B)" 
3698947146b2
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haftmann
parents:
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1048 
"SUP x. B" == "SUP x:CONST UNIV. B" 
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parents:
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1049 
"SUP x:A. B" == "CONST SUPR A (%x. B)" 
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parents:
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1050 
"INF x y. B" == "INF x. INF y. B" 
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parents:
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1051 
"INF x. B" == "CONST INFI CONST UNIV (%x. B)" 
3698947146b2
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haftmann
parents:
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1052 
"INF x. B" == "INF x:CONST UNIV. B" 
3698947146b2
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haftmann
parents:
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1053 
"INF x:A. B" == "CONST INFI A (%x. B)" 
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haftmann
parents:
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diff
changeset

1054 

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1055 
(* To avoid etacontraction of body: *) 
3698947146b2
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haftmann
parents:
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diff
changeset

1056 
print_translation {* 
3698947146b2
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haftmann
parents:
32064
diff
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1057 
let 
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1058 
fun btr' syn (A :: Abs abs :: ts) = 
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1059 
let val (x,t) = atomic_abs_tr' abs 
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1060 
in list_comb (Syntax.const syn $ x $ A $ t, ts) end 
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1061 
val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const 
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1062 
in 
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1063 
[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")] 
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1064 
end 
11979  1065 
*} 
1066 

32077
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1067 
context complete_lattice 
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1068 
begin 
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1069 

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1070 
lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)" 
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1071 
by (auto simp add: SUPR_def intro: Sup_upper) 
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1072 

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1073 
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u" 
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1074 
by (auto simp add: SUPR_def intro: Sup_least) 
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1075 

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1076 
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i" 
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1077 
by (auto simp add: INFI_def intro: Inf_lower) 
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1078 

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1079 
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)" 
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1080 
by (auto simp add: INFI_def intro: Inf_greatest) 
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1081 

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1082 
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M" 
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1083 
by (auto intro: antisym SUP_leI le_SUPI) 
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1084 

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1085 
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M" 
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1086 
by (auto intro: antisym INF_leI le_INFI) 
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1087 

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1088 
end 
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1089 

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1090 

32081  1091 
subsubsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *} 
32077
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1092 

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1093 
instantiation bool :: complete_lattice 
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1094 
begin 
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1095 

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1096 
definition 
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1097 
Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)" 
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1098 

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1099 
definition 
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1100 
Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)" 
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1101 

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1102 
instance proof 
3698947146b2
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1103 
qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def) 
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1104 

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1105 
end 
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1106 

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1107 
lemma Inf_empty_bool [simp]: 
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1108 
"\<Sqinter>{}" 
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1109 
unfolding Inf_bool_def by auto 
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1110 

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1111 
lemma not_Sup_empty_bool [simp]: 
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1112 
"\<not> \<Squnion>{}" 
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1113 
unfolding Sup_bool_def by auto 
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1114 

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1115 
instantiation "fun" :: (type, complete_lattice) complete_lattice 
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1116 
begin 
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1117 

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1118 
definition 
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1119 
Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})" 
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1120 

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1121 
definition 
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1122 
Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})" 
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1123 

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1124 
instance proof 
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1125 
qed (auto simp add: Inf_fun_def Sup_fun_def le_fun_def 
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1126 
intro: Inf_lower Sup_upper Inf_greatest Sup_least) 
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1127 

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1128 
end 
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1129 

3698947146b2
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1130 
lemma Inf_empty_fun: 
3698947146b2
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1131 
"\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})" 
3698947146b2
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haftmann
parents:
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1132 
by rule (simp add: Inf_fun_def, simp add: empty_def) 
3698947146b2
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haftmann
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1133 

3698947146b2
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haftmann
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1134 
lemma Sup_empty_fun: 
3698947146b2
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1135 
"\<Squnion>{} = (\<lambda>_. \<Squnion>{})" 
3698947146b2
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haftmann
parents:
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1136 
by rule (simp add: Sup_fun_def, simp add: empty_def) 
3698947146b2
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haftmann
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1137 

3698947146b2
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haftmann
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1138 

32081  1139 
subsubsection {* Unions of families *} 
32077
3698947146b2
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1140 

3698947146b2
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1141 
definition UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where 
3698947146b2
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1142 
"UNION A B \<equiv> {y. \<exists>x\<in>A. y \<in> B x}" 
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1143 

3698947146b2
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1144 
syntax 
3698947146b2
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1145 
"@UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10) 
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1146 
"@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [0, 10] 10) 
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1147 

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1148 
syntax (xsymbols) 
3698947146b2
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diff
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1149 
"@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10) 
3698947146b2
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1150 
"@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 10] 10) 
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1151 

3698947146b2
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1152 
syntax (latex output) 
3698947146b2
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haftmann
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1153 
"@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) 
3698947146b2
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1154 
"@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10) 
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1155 

3698947146b2
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1156 
translations 
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1157 
"UN x y. B" == "UN x. UN y. B" 
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1158 
"UN x. B" == "CONST UNION CONST UNIV (%x. B)" 
3698947146b2
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haftmann
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1159 
"UN x. B" == "UN x:CONST UNIV. B" 
3698947146b2
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1160 
"UN x:A. B" == "CONST UNION A (%x. B)" 
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haftmann
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1161 

3698947146b2
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haftmann
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1162 
text {* 
3698947146b2
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haftmann
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1163 
Note the difference between ordinary xsymbol syntax of indexed 
3698947146b2
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haftmann
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1164 
unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}) 
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1165 
and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The 
3698947146b2
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haftmann
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1166 
former does not make the index expression a subscript of the 
3698947146b2
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haftmann
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1167 
union/intersection symbol because this leads to problems with nested 
3698947146b2
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haftmann
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1168 
subscripts in Proof General. 
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haftmann
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1169 
*} 
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haftmann
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1170 

3698947146b2
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1171 
(* To avoid etacontraction of body: *) 
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haftmann
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32064
diff
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1172 
print_translation {* 
3698947146b2
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haftmann
parents:
32064
diff
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1173 
let 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
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changeset

1174 
fun btr' syn [A, Abs abs] = 
3698947146b2
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haftmann
parents:
32064
diff
changeset

1175 
let val (x, t) = atomic_abs_tr' abs 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

1176 
in Syntax.const syn $ x $ A $ t end 
32081  1177 
in [(@{const_syntax UNION}, btr' "@UNION")] end 
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

1178 
*} 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

1179 

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

1180 
declare UNION_def [noatp] 
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

1181 

11979  1182 
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)" 
1183 
by (unfold UNION_def) blast 

1184 

1185 
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)" 

1186 
 {* The order of the premises presupposes that @{term A} is rigid; 

1187 
@{term b} may be flexible. *} 

1188 
by auto 

1189 

1190 
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R" 

1191 
by (unfold UNION_def) blast 

923  1192 

11979  1193 
lemma UN_cong [cong]: 
1194 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" 

1195 
by (simp add: UNION_def) 

1196 

29691  1197 
lemma strong_UN_cong: 
1198 
"A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" 

1199 
by (simp add: UNION_def simp_implies_def) 

1200 

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

1201 
lemma image_eq_UN: "f`A = (UN x:A. {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

1202 
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

1203 

11979  1204 

1205 
subsubsection {* Intersections of families *} 

1206 

32081  1207 
definition INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where 
1208 
"INTER A B \<equiv> {y. \<forall>x\<in>A. y \<in> B x}" 

1209 

1210 
syntax 

1211 
"@INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10) 

1212 
"@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 10] 10) 

1213 

1214 
syntax (xsymbols) 

1215 
"@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10) 

1216 
"@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 10] 10) 

1217 

1218 
syntax (latex output) 

1219 
"@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) 

1220 
"@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10) 

1221 

1222 
translations 

1223 
"INT x y. B" == "INT x. INT y. B" 

1224 
"INT x. B" == "CONST INTER CONST UNIV (%x. B)" 

1225 
"INT x. B" == "INT x:CONST UNIV. B" 

1226 
"INT x:A. B" == "CONST INTER A (%x. B)" 

1227 

1228 
(* To avoid etacontraction of body: *) 

1229 
print_translation {* 

1230 
let 

1231 
fun btr' syn [A, Abs abs] = 

1232 
let val (x, t) = atomic_abs_tr' abs 

1233 
in Syntax.const syn $ x $ A $ t end 

1234 
in [(@{const_syntax INTER}, btr' "@INTER")] end 

1235 
*} 

1236 

11979  1237 
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)" 
1238 
by (unfold INTER_def) blast 

923  1239 

11979  1240 
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)" 
1241 
by (unfold INTER_def) blast 

1242 

1243 
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a" 

1244 
by auto 

1245 

1246 
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R" 

1247 
 {* "Classical" elimination  by the Excluded Middle on @{prop "a:A"}. *} 

1248 
by (unfold INTER_def) blast 

1249 

1250 
lemma INT_cong [cong]: 

1251 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)" 

1252 
by (simp add: INTER_def) 

7238
36e58620ffc8
replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents:
5931
diff
changeset

1253 

923  1254 

11979  1255 
subsubsection {* Union *} 
1256 

32081  1257 
definition Union :: "'a set set \<Rightarrow> 'a set" where 
1258 
"Union S \<equiv> UNION S (\<lambda>x. x)" 

1259 

1260 
notation (xsymbols) 

1261 
Union ("\<Union>_" [90] 90) 

1262 

1263 
lemma Union_image_eq [simp]: 

1264 
"\<Union>(B`A) = (\<Union>x\<in>A. B x)" 

1265 
by (auto simp add: Union_def UNION_def image_def) 

1266 

1267 
lemma Union_eq: 

1268 
"\<Union>A = {x. \<exists>B \<in> A. x \<in> B}" 

1269 
by (simp add: Union_def UNION_def) 

1270 

1271 
lemma Sup_set_eq: 

1272 
"\<Squnion>S = \<Union>S" 

1273 
proof (rule set_ext) 

1274 
fix x 

1275 
have "(\<exists>Q\<in>{P. \<exists>A\<in>S. P \<longleftrightarrow> x \<in> A}. Q) \<longleftrightarrow> (\<exists>A\<in>S. x \<in> A)" 

1276 
by auto 

1277 
then show "x \<in> \<Squnion>S \<longleftrightarrow> x \<in> \<Union>S" 

1278 
by (simp add: Union_eq Sup_fun_def Sup_bool_def) (simp add: mem_def) 

1279 
qed 

1280 

1281 
lemma SUPR_set_eq: 

1282 
"(SUP x:S. f x) = (\<Union>x\<in>S. f x)" 

1283 
by (simp add: SUPR_def Sup_set_eq) 

1284 

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

1285 
lemma Union_iff [simp,noatp]: "(A : Union C) = (EX X:C. A:X)" 
11979  1286 
by (unfold Union_def) blast 
1287 

1288 
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C" 

1289 
 {* The order of the premises presupposes that @{term C} is rigid; 

1290 
@{term A} may be flexible. *} 

1291 
by auto 

1292 

1293 
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R" 

1294 
by (unfold Union_def) blast 

1295 

1296 

1297 
subsubsection {* Inter *} 

1298 

32081  1299 
definition Inter :: "'a set set \<Rightarrow> 'a set" where 
1300 
"Inter S \<equiv> INTER S (\<lambda>x. x)" 

1301 

1302 
notation (xsymbols) 

1303 
Inter ("\<Inter>_" [90] 90) 

1304 

1305 
lemma Inter_image_eq [simp]: 

1306 
"\<Inter>(B`A) = (\<Inter>x\<in>A. B x)" 

1307 
by (auto simp add: Inter_def INTER_def image_def) 

1308 

1309 
lemma Inter_eq: 

1310 
"\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}" 

1311 
by (simp add: Inter_def INTER_def) 

1312 

1313 
lemma Inf_set_eq: 

1314 
"\<Sqinter>S = \<Inter>S" 

1315 
proof (rule set_ext) 

1316 
fix x 

1317 
have "(\<forall>Q\<in>{P. \<exists>A\<in>S. P \<longleftrightarrow> x \<in> A}. Q) \<longleftrightarrow> (\<forall>A\<in>S. x \<in> A)" 

1318 
by auto 

1319 
then show "x \<in> \<Sqinter>S \<longleftrightarrow> x \<in> \<Inter>S" 

1320 
by (simp add: Inter_ 