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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text {* 
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Simproc for pulling @{text "x=t"} in @{text "{x. \<dots> & x=t & \<dots>}"} 
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to the front (and similarly for @{text "t=x"}): 
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*} 
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setup {* 
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let 
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val Coll_perm_tac = rtac @{thm Collect_cong} 1 THEN rtac @{thm iffI} 1 THEN 
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ALLGOALS(EVERY'[REPEAT_DETERM o (etac @{thm conjE}), 
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DEPTH_SOLVE_1 o (ares_tac [@{thm conjI}])]) 
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val defColl_regroup = Simplifier.simproc @{theory} 
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"defined Collect" ["{x. P x & Q x}"] 
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(Quantifier1.rearrange_Coll Coll_perm_tac) 
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in 
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Simplifier.map_simpset (fn ss => ss addsimprocs [defColl_regroup]) 
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end 
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*} 
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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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bot_set_eq [symmetric]: "{} = bot" 
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definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where 

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

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

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translations 

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

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"{x}" == "CONST insert x {}" 

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subsection {* Subsets and bounded quantifiers *} 

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abbreviation 

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

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"subset \<equiv> less" 

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abbreviation 

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

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"subset_eq \<equiv> less_eq" 

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notation (output) 

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subset ("op <") and 

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

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subset_eq ("op <=") and 

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

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notation (xsymbols) 

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subset ("op \<subset>") and 

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

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subset_eq ("op \<subseteq>") and 

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

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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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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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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  240 
val Type (set_type, _) = @{typ "'a set"}; 
241 
val All_binder = Syntax.binder_name @{const_syntax "All"}; 

242 
val Ex_binder = Syntax.binder_name @{const_syntax "Ex"}; 

243 
val impl = @{const_syntax "op >"}; 

244 
val conj = @{const_syntax "op &"}; 

245 
val sbset = @{const_syntax "subset"}; 

246 
val sbset_eq = @{const_syntax "subset_eq"}; 

21819  247 

248 
val trans = 

249 
[((All_binder, impl, sbset), "_setlessAll"), 

250 
((All_binder, impl, sbset_eq), "_setleAll"), 

251 
((Ex_binder, conj, sbset), "_setlessEx"), 

252 
((Ex_binder, conj, sbset_eq), "_setleEx")]; 

253 

254 
fun mk v v' c n P = 

255 
if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v  _ => false) n) 

256 
then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match; 

257 

258 
fun tr' q = (q, 

259 
fn [Const ("_bound", _) $ Free (v, Type (T, _)), Const (c, _) $ (Const (d, _) $ (Const ("_bound", _) $ Free (v', _)) $ n) $ P] => 

260 
if T = (set_type) then case AList.lookup (op =) trans (q, c, d) 

261 
of NONE => raise Match 

262 
 SOME l => mk v v' l n P 

263 
else raise Match 

264 
 _ => raise Match); 

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in 
21819  266 
[tr' All_binder, tr' Ex_binder] 
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end 
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*} 
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11979  271 
text {* 
272 
\medskip Translate between @{text "{e  x1...xn. P}"} and @{text 

273 
"{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is 

274 
only translated if @{text "[0..n] subset bvs(e)"}. 

275 
*} 

276 

277 
parse_translation {* 

278 
let 

279 
val ex_tr = snd (mk_binder_tr ("EX ", "Ex")); 

3947  280 

11979  281 
fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1 
282 
 nvars _ = 1; 

283 

284 
fun setcompr_tr [e, idts, b] = 

285 
let 

286 
val eq = Syntax.const "op =" $ Bound (nvars idts) $ e; 

287 
val P = Syntax.const "op &" $ eq $ b; 

288 
val exP = ex_tr [idts, P]; 

17784  289 
in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end; 
11979  290 

291 
in [("@SetCompr", setcompr_tr)] end; 

292 
*} 

923  293 

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print_translation {* [ 
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Syntax.preserve_binder_abs2_tr' @{const_syntax Ball} "_Ball", 
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Syntax.preserve_binder_abs2_tr' @{const_syntax Bex} "_Bex" 
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] *}  {* to avoid etacontraction of body *} 
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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  309 
 check _ = false 
923  310 

11979  311 
fun tr' (_ $ abs) = 
312 
let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs] 

313 
in Syntax.const "@SetCompr" $ e $ idts $ Q end; 

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in if check (P, 0) then tr' P 
15535  315 
else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs 
316 
val M = Syntax.const "@Coll" $ x $ t 

317 
in case t of 

318 
Const("op &",_) 

319 
$ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A) 

320 
$ P => 

321 
if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M 

322 
 _ => M 

323 
end 

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end; 
11979  325 
in [("Collect", setcompr_tr')] end; 
326 
*} 

327 

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setup {* 
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let 
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val unfold_bex_tac = unfold_tac @{thms "Bex_def"}; 
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fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac; 
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val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac; 
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val unfold_ball_tac = unfold_tac @{thms "Ball_def"}; 
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fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac; 
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val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac; 
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val defBEX_regroup = Simplifier.simproc @{theory} 
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"defined BEX" ["EX x:A. P x & Q x"] rearrange_bex; 
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val defBALL_regroup = Simplifier.simproc @{theory} 
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"defined BALL" ["ALL x:A. P x > Q x"] rearrange_ball; 
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in 
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Simplifier.map_simpset (fn ss => ss addsimprocs [defBALL_regroup, defBEX_regroup]) 
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end 
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*} 
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344 

11979  345 
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x" 
346 
by (simp add: Ball_def) 

347 

348 
lemmas strip = impI allI ballI 

349 

350 
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x" 

351 
by (simp add: Ball_def) 

352 

353 
text {* 

354 
Gives better instantiation for bound: 

355 
*} 

356 

26339  357 
declaration {* fn _ => 
358 
Classical.map_cs (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1)) 

11979  359 
*} 
360 

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ML {* 
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structure Simpdata = 
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struct 
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364 

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365 
open Simpdata; 
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val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs; 
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end; 
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open Simpdata; 
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*} 
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373 

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declaration {* fn _ => 
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Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs)) 
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*} 
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377 

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lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q" 
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by (unfold Ball_def) blast 
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380 

11979  381 
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x" 
382 
 {* Normally the best argument order: @{prop "P x"} constrains the 

383 
choice of @{prop "x:A"}. *} 

384 
by (unfold Bex_def) blast 

385 

13113  386 
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x" 
11979  387 
 {* The best argument order when there is only one @{prop "x:A"}. *} 
388 
by (unfold Bex_def) blast 

389 

390 
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x" 

391 
by (unfold Bex_def) blast 

392 

393 
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q" 

394 
by (unfold Bex_def) blast 

395 

396 
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) > P)" 

397 
 {* Trival rewrite rule. *} 

398 
by (simp add: Ball_def) 

399 

400 
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)" 

401 
 {* Dual form for existentials. *} 

402 
by (simp add: Bex_def) 

403 

404 
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)" 

405 
by blast 

406 

407 
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)" 

408 
by blast 

409 

410 
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)" 

411 
by blast 

412 

413 
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)" 

414 
by blast 

415 

416 
lemma ball_one_point1 [simp]: "(ALL x:A. x = a > P x) = (a:A > P a)" 

417 
by blast 

418 

419 
lemma ball_one_point2 [simp]: "(ALL x:A. a = x > P x) = (a:A > P a)" 

420 
by blast 

421 

422 

32081  423 
text {* Congruence rules *} 
11979  424 

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

425 
lemma ball_cong: 
11979  426 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
427 
(ALL x:A. P x) = (ALL x:B. Q x)" 

428 
by (simp add: Ball_def) 

429 

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

430 
lemma strong_ball_cong [cong]: 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

431 
"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

432 
(ALL x:A. P x) = (ALL x:B. Q x)" 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

433 
by (simp add: simp_implies_def Ball_def) 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

434 

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

435 
lemma bex_cong: 
11979  436 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
437 
(EX x:A. P x) = (EX x:B. Q x)" 

438 
by (simp add: Bex_def cong: conj_cong) 

1273  439 

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

440 
lemma strong_bex_cong [cong]: 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

441 
"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

442 
(EX x:A. P x) = (EX x:B. Q x)" 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

443 
by (simp add: simp_implies_def Bex_def cong: conj_cong) 
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset

444 

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

445 

32081  446 
subsection {* Basic operations *} 
447 

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

448 
subsubsection {* Subsets *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

449 

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

450 
lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

451 
by (auto simp add: mem_def intro: predicate1I) 
30352  452 

11979  453 
text {* 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

454 
\medskip Map the type @{text "'a set => anything"} to just @{typ 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

455 
'a}; for overloading constants whose first argument has type @{typ 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

456 
"'a set"}. 
11979  457 
*} 
458 

30596  459 
lemma subsetD [elim, intro?]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B" 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

460 
 {* Rule in Modus Ponens style. *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

461 
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

462 

30596  463 
lemma rev_subsetD [intro?]: "c \<in> A ==> A \<subseteq> B ==> c \<in> B" 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

464 
 {* The same, with reversed premises for use with @{text erule}  
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

465 
cf @{text rev_mp}. *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

466 
by (rule subsetD) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

467 

11979  468 
text {* 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

469 
\medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}. 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

470 
*} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

471 

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

472 
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

473 
 {* Classical elimination rule. *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

474 
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

475 

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

476 
lemma subset_eq: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast 
2388  477 

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

478 
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

479 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

480 

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

481 
lemma subset_refl [simp,atp]: "A \<subseteq> A" 
32081  482 
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

483 

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

484 
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C" 
32081  485 
by (fact order_trans) 
486 

487 
lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B" 

488 
by (rule subsetD) 

489 

490 
lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B" 

491 
by (rule subsetD) 

492 

493 
lemmas basic_trans_rules [trans] = 

494 
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

495 

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

496 

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

497 
subsubsection {* Equality *} 
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 
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

500 
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

501 
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

502 
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

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

504 

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

505 
(* Due to Brian Huffman *) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

506 
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

507 
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

508 

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

509 
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

510 
 {* 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

511 
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

512 

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

513 
text {* 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

514 
\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

515 
here? 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

516 
*} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

517 

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

518 
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

519 
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

520 

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

521 
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

522 
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

523 

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

524 
text {* 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

525 
\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

526 
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

527 
\<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}! 
30352  528 
*} 
529 

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

530 
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

531 
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

532 

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

533 
lemma equalityCE [elim]: 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

534 
"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

535 
by blast 
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 
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

538 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

539 

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

540 
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

541 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

542 

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

543 

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

544 
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

545 

32081  546 
definition UNIV :: "'a set" where 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

547 
top_set_eq [symmetric]: "UNIV = top" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

548 

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

549 
lemma UNIV_def: 
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset

550 
"UNIV = {x. True}" 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

551 
by (simp add: top_set_eq [symmetric] top_fun_eq top_bool_eq Collect_def) 
32081  552 

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

553 
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

554 
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

555 

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

556 
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

557 

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

558 
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

559 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

560 

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

561 
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

562 
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

563 

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

564 
text {* 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

565 
\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

566 
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

567 
congruence rules. 
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 

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

570 
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

571 
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

572 

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

573 
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

574 
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

575 

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

576 
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

577 
by auto 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

578 

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

579 

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

580 
subsubsection {* The empty set *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

581 

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

582 
lemma empty_def: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

583 
"{} = {x. False}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

584 
by (simp add: bot_set_eq [symmetric] bot_fun_eq bot_bool_eq Collect_def) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

585 

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

586 
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

587 
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

588 

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

589 
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

590 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

591 

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

592 
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

593 
 {* 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

594 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

595 

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

596 
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

597 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

598 

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

599 
lemma equals0D: "A = {} ==> a \<notin> A" 
32082  600 
 {* Use for reasoning about disjointness: @{text "A Int B = {}"} *} 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

601 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

602 

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

603 
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

604 
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

605 

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

606 
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

607 
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

608 

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

609 
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

610 
by (blast elim: equalityE) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

611 

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 
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

614 

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

615 
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

616 
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

617 

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

618 
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

619 
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

620 

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

621 
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

622 
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

623 

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

624 
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

625 
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

626 

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

627 
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

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

629 

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

630 
lemma 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

631 
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

632 

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

633 

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

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

635 

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

636 
lemma 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

637 
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

638 

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

639 
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

640 
by (unfold mem_def fun_Compl_def bool_Compl_def) blast 
923  641 

11979  642 
text {* 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

643 
\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

644 
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

645 
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

646 

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

647 
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

648 
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

649 

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

650 
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

651 

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

652 
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

653 

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

654 

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

655 
subsubsection {* Binary union  Un *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

656 

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

657 
definition union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

658 
sup_set_eq [symmetric]: "A Un B = sup A B" 
32081  659 

660 
notation (xsymbols) 

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

661 
union (infixl "\<union>" 65) 
32081  662 

663 
notation (HTML output) 

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

664 
union (infixl "\<union>" 65) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

665 

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

666 
lemma Un_def: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

667 
"A \<union> B = {x. x \<in> A \<or> x \<in> B}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

668 
by (simp add: sup_fun_eq sup_bool_eq sup_set_eq [symmetric] Collect_def mem_def) 
32081  669 

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

670 
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

671 
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

672 

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

673 
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

674 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

675 

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

676 
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

677 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

678 

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

679 
text {* 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

680 
\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

681 
@{prop B}. 
11979  682 
*} 
683 

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

684 
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

685 
by auto 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

686 

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

687 
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

688 
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

689 

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

690 
lemma insert_def: "insert a B = {x. x = a} \<union> B" 
32081  691 
by (simp add: Collect_def mem_def insert_compr Un_def) 
692 

693 
lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)" 

694 
apply (fold sup_set_eq) 

695 
apply (erule mono_sup) 

696 
done 

697 

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

698 

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

699 
subsubsection {* Binary intersection  Int *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

700 

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

701 
definition inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

702 
inf_set_eq [symmetric]: "A Int B = inf A B" 
32081  703 

704 
notation (xsymbols) 

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

705 
inter (infixl "\<inter>" 70) 
32081  706 

707 
notation (HTML output) 

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

708 
inter (infixl "\<inter>" 70) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

709 

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

710 
lemma Int_def: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

711 
"A \<inter> B = {x. x \<in> A \<and> x \<in> B}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

712 
by (simp add: inf_fun_eq inf_bool_eq inf_set_eq [symmetric] Collect_def mem_def) 
32081  713 

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

714 
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

715 
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

716 

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

717 
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

718 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

719 

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

720 
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

721 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

722 

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

723 
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

724 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

725 

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

726 
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

727 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

728 

32081  729 
lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B" 
730 
apply (fold inf_set_eq) 

731 
apply (erule mono_inf) 

732 
done 

733 

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

734 

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

735 
subsubsection {* Set difference *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

736 

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

737 
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

738 
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

739 

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

740 
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

741 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

742 

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

743 
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

744 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

745 

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

746 
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

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

748 

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

749 
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

750 
by simp 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

751 

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

752 
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

753 

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

754 
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

755 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

756 

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

757 

31456  758 
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

759 

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

760 
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

761 
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

762 

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

763 
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

764 
by simp 
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 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

767 
by simp 
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 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

770 
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

771 

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

772 
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

773 
 {* Classical introduction rule. *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

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

775 

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

776 
lemma 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

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

778 

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

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

780 
assumes "x \<in> A" 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

781 
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

782 
proof 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

783 
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

784 
next 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

785 
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

786 
qed 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

787 

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

788 
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

789 
by auto 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

790 

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

791 
subsubsection {* Singletons, using insert *} 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

792 

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

793 
lemma 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

794 
 {* 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

795 
by (rule insertI1) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

796 

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

797 
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

798 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

799 

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

800 
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

801 

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

802 
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

803 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

804 

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

805 
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

806 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

807 

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

808 
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

809 
"({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

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_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

813 
"(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

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

815 

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

816 
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

817 
by fast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

818 

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

819 
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

820 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

821 

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

822 
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

823 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

824 

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

825 
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

826 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

827 

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

828 
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

829 
by (blast elim: equalityE) 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

830 

11979  831 

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

832 
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

833 

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 
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

835 
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

836 
*} 
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 
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

839 
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

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 
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

842 
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

843 
"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

844 

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 
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

846 
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

847 

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 
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

849 
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

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 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

852 
 {* 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

853 
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

854 
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

855 

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 
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

857 
"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

858 
 {* 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

859 
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

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 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

862 
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

863 

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 
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

865 
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

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_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

868 
 {* 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

869 
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

870 

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

871 
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

872 
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

873 
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

874 
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

875 
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

876 

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 
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

878 
 {* 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

879 
@{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

880 
by blast 
11979  881 

882 
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

883 
\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

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 

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

886 
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

887 
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

888 

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

889 
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

890 
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

891 

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

892 
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

893 
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

894 

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

895 

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

896 
subsubsection {* Some rules with @{text "if"} *} 
32081  897 

898 
text{* Elimination of @{text"{x. \<dots> & x=t & \<dots>}"}. *} 

899 

900 
lemma Collect_conv_if: "{x. x=a & P x} = (if P a then {a} else {})" 

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

901 
by auto 
32081  902 

903 
lemma Collect_conv_if2: "{x. a=x & P x} = (if P a then {a} else {})" 

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

904 
by auto 
32081  905 

906 
text {* 

907 
Rewrite rules for boolean casesplitting: faster than @{text 

908 
"split_if [split]"}. 

909 
*} 

910 

911 
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q > x = b) & (~ Q > y = b))" 

912 
by (rule split_if) 

913 

914 
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q > a = x) & (~ Q > a = y))" 

915 
by (rule split_if) 

916 

917 
text {* 

918 
Split ifs on either side of the membership relation. Not for @{text 

919 
"[simp]"}  can cause goals to blow up! 

920 
*} 

921 

922 
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q > x : b) & (~ Q > y : b))" 

923 
by (rule split_if) 

924 

925 
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q > a : x) & (~ Q > a : y))" 

926 
by (rule split_if [where P="%S. a : S"]) 

927 

928 
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2 

929 

930 
(*Would like to add these, but the existing code only searches for the 

931 
outerlevel constant, which in this case is just "op :"; we instead need 

932 
to use termnets to associate patterns with rules. Also, if a rule fails to 

933 
apply, then the formula should be kept. 

934 
[("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]), 

935 
("Int", [IntD1,IntD2]), 

936 
("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])] 

937 
*) 

938 

939 

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

940 
subsection {* Further operations and lemmas *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

941 

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

942 
subsubsection {* The ``proper subset'' relation *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

943 

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

944 
lemma psubsetI [intro!,noatp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

945 
by (unfold less_le) blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

946 

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

947 
lemma psubsetE [elim!,noatp]: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

948 
"[A \<subset> B; [A \<subseteq> B; ~ (B\<subseteq>A)] ==> R] ==> R" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

949 
by (unfold less_le) blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

950 

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

951 
lemma psubset_insert_iff: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

952 
"(A \<subset> insert x B) = (if x \<in> B then A \<subset> B else if x \<in> A then A  {x} \<subset> B else A \<subseteq> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

953 
by (auto simp add: less_le subset_insert_iff) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

954 

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

955 
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

956 
by (simp only: less_le) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

957 

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

958 
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

959 
by (simp add: psubset_eq) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

960 

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

961 
lemma psubset_trans: "[ A \<subset> B; B \<subset> C ] ==> A \<subset> C" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

962 
apply (unfold less_le) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

963 
apply (auto dest: subset_antisym) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

964 
done 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

965 

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

966 
lemma psubsetD: "[ A \<subset> B; c \<in> A ] ==> c \<in> B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

967 
apply (unfold less_le) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

968 
apply (auto dest: subsetD) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

969 
done 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

970 

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

971 
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

972 
by (auto simp add: psubset_eq) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

973 

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

974 
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

975 
by (auto simp add: psubset_eq) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

976 

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

977 
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B  A)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

978 
by (unfold less_le) blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

979 

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

980 
lemma atomize_ball: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

981 
"(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

982 
by (simp only: Ball_def atomize_all atomize_imp) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

983 

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

984 
lemmas [symmetric, rulify] = atomize_ball 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

985 
and [symmetric, defn] = atomize_ball 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

986 

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

987 
subsubsection {* Derived rules involving subsets. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

988 

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

989 
text {* @{text insert}. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

990 

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

991 
lemma subset_insertI: "B \<subseteq> insert a B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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diff
changeset

992 
by (rule subsetI) (erule insertI2) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

993 

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

994 
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

995 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

996 

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

997 
lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

998 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

999 

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

1000 

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

1001 
text {* \medskip Finite Union  the least upper bound of two sets. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1002 

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

1003 
lemma Un_upper1: "A \<subseteq> A \<union> B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1004 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1005 

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

1006 
lemma Un_upper2: "B \<subseteq> A \<union> B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1007 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1008 

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

1009 
lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1010 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1011 

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

1012 

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

1013 
text {* \medskip Finite Intersection  the greatest lower bound of two sets. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1014 

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

1015 
lemma Int_lower1: "A \<inter> B \<subseteq> A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

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

1017 

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

1018 
lemma Int_lower2: "A \<inter> B \<subseteq> B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1019 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1020 

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

1021 
lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1022 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1023 

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

1024 

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

1025 
text {* \medskip Set difference. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1026 

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

1027 
lemma Diff_subset: "A  B \<subseteq> A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1028 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1029 

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

1030 
lemma Diff_subset_conv: "(A  B \<subseteq> C) = (A \<subseteq> B \<union> C)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1031 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1032 

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

1033 

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

1034 
subsubsection {* Equalities involving union, intersection, inclusion, etc. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1035 

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

1036 
text {* @{text "{}"}. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1037 

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

1038 
lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1039 
 {* supersedes @{text "Collect_False_empty"} *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1040 
by auto 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1041 

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

1042 
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1043 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1044 

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

1045 
lemma not_psubset_empty [iff]: "\<not> (A < {})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1046 
by (unfold less_le) blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1047 

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

1048 
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1049 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1050 

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

1051 
lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1052 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1053 

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

1054 
lemma Collect_neg_eq: "{x. \<not> P x} =  {x. P x}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1055 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1056 

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

1057 
lemma Collect_disj_eq: "{x. P x  Q x} = {x. P x} \<union> {x. Q x}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1058 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1059 

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

1060 
lemma Collect_imp_eq: "{x. P x > Q x} = {x. P x} \<union> {x. Q x}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1061 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1062 

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

1063 
lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1064 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1065 

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

1066 

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

1067 
text {* \medskip @{text insert}. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1068 

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

1069 
lemma insert_is_Un: "insert a A = {a} Un A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1070 
 {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1071 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1072 

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

1073 
lemma insert_not_empty [simp]: "insert a A \<noteq> {}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1074 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1075 

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

1076 
lemmas empty_not_insert = insert_not_empty [symmetric, standard] 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1077 
declare empty_not_insert [simp] 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1078 

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

1079 
lemma insert_absorb: "a \<in> A ==> insert a A = A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1080 
 {* @{text "[simp]"} causes recursive calls when there are nested inserts *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1081 
 {* with \emph{quadratic} running time *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1082 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1083 

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

1084 
lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1085 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1086 

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

1087 
lemma insert_commute: "insert x (insert y A) = insert y (insert x A)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1088 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1089 

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

1090 
lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1091 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1092 

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

1093 
lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1094 
 {* use new @{text B} rather than @{text "A  {a}"} to avoid infinite unfolding *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1095 
apply (rule_tac x = "A  {a}" in exI, blast) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1096 
done 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1097 

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

1098 
lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a > P u}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1099 
by auto 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1100 

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

1101 
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1102 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1103 

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

1104 
lemma insert_disjoint [simp,noatp]: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1105 
"(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1106 
"({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

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

1108 

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

1109 
lemma disjoint_insert [simp,noatp]: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1110 
"(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1111 
"({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1112 
by auto 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1113 

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

1114 
text {* \medskip @{text image}. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1115 

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

1116 
lemma image_empty [simp]: "f`{} = {}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1117 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1118 

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

1119 
lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1120 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1121 

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

1122 
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1123 
by auto 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1124 

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

1125 
lemma image_constant_conv: "(%x. c) ` A = (if A = {} then {} else {c})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1126 
by auto 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1127 

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

1128 
lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1129 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1130 

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

1131 
lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1132 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1133 

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

1134 
lemma image_is_empty [iff]: "(f`A = {}) = (A = {})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1135 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1136 

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

1137 
lemma empty_is_image[iff]: "({} = f ` A) = (A = {})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

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

1139 

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

1140 

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

1141 
lemma image_Collect [noatp]: "f ` {x. P x} = {f x  x. P x}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1142 
 {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS, 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1143 
with its implicit quantifier and conjunction. Also image enjoys better 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1144 
equational properties than does the RHS. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1145 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1146 

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

1147 
lemma if_image_distrib [simp]: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1148 
"(\<lambda>x. if P x then f x else g x) ` S 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1149 
= (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1150 
by (auto simp add: image_def) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1151 

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

1152 
lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1153 
by (simp add: image_def) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1154 

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

1155 

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

1156 
text {* \medskip @{text range}. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1157 

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

1158 
lemma full_SetCompr_eq [noatp]: "{u. \<exists>x. u = f x} = range f" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1159 
by auto 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1160 

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

1161 
lemma range_composition: "range (\<lambda>x. f (g x)) = f`range g" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1162 
by (subst image_image, simp) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1163 

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

1164 

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

1165 
text {* \medskip @{text Int} *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1166 

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

1167 
lemma Int_absorb [simp]: "A \<inter> A = A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1168 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1169 

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

1170 
lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1171 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1172 

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

1173 
lemma Int_commute: "A \<inter> B = B \<inter> A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

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

1175 

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

1176 
lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1177 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1178 

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

1179 
lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1180 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1181 

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

1182 
lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1183 
 {* Intersection is an ACoperator *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1184 

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

1185 
lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1186 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1187 

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

1188 
lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1189 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1190 

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

1191 
lemma Int_empty_left [simp]: "{} \<inter> B = {}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1192 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1193 

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

1194 
lemma Int_empty_right [simp]: "A \<inter> {} = {}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1195 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1196 

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

1197 
lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1198 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1199 

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

1200 
lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1201 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1202 

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

1203 
lemma Int_UNIV_left [simp]: "UNIV \<inter> B = B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1204 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1205 

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

1206 
lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1207 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1208 

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

1209 
lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1210 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1211 

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

1212 
lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1213 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1214 

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

1215 
lemma Int_UNIV [simp,noatp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1216 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1217 

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

1218 
lemma Int_subset_iff [simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1219 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1220 

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

1221 
lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1222 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1223 

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

1224 

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

1225 
text {* \medskip @{text Un}. *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1226 

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

1227 
lemma Un_absorb [simp]: "A \<union> A = A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1228 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1229 

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

1230 
lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1231 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1232 

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

1233 
lemma Un_commute: "A \<union> B = B \<union> A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1234 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1235 

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

1236 
lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1237 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1238 

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

1239 
lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1240 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1241 

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

1242 
lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1243 
 {* Union is an ACoperator *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1244 

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

1245 
lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1246 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1247 

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

1248 
lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1249 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1250 

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

1251 
lemma Un_empty_left [simp]: "{} \<union> B = B" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1252 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1253 

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

1254 
lemma Un_empty_right [simp]: "A \<union> {} = A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1255 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1256 

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

1257 
lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1258 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1259 

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

1260 
lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1261 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1262 

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

1263 
lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1264 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1265 

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

1266 
lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1267 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1268 

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

1269 
lemma Int_insert_left: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1270 
"(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

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

1272 

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

1273 
lemma Int_insert_right: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1274 
"A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1275 
by auto 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1276 

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

1277 
lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1278 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1279 

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

1280 
lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1281 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1282 

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

1283 
lemma Un_Int_crazy: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1284 
"(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1285 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1286 

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

1287 
lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1288 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1289 

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

1290 
lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1291 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
