author | paulson |
Mon, 09 Nov 2009 15:50:15 +0000 | |
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parent 33045 | 2b3694001c48 |
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permissions | -rw-r--r-- |
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(* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel *) |
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header {* Set theory for higher-order 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)" -- "non-membership" |
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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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abbreviation empty :: "'a set" ("{}") where |
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"{} \<equiv> bot" |
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definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where |
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insert_compr: "insert a B = {x. x = a \<or> x \<in> B}" |
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syntax |
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"@Finset" :: "args => 'a set" ("{(_)}") |
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translations |
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"{x, xs}" == "CONST insert x {xs}" |
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"{x}" == "CONST insert x {}" |
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subsection {* Subsets and bounded quantifiers *} |
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abbreviation |
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subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where |
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"subset \<equiv> less" |
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abbreviation |
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subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where |
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"subset_eq \<equiv> less_eq" |
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notation (output) |
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subset ("op <") and |
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subset ("(_/ < _)" [50, 51] 50) and |
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subset_eq ("op <=") and |
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subset_eq ("(_/ <= _)" [50, 51] 50) |
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notation (xsymbols) |
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subset ("op \<subset>") and |
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subset ("(_/ \<subset> _)" [50, 51] 50) and |
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subset_eq ("op \<subseteq>") and |
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subset_eq ("(_/ \<subseteq> _)" [50, 51] 50) |
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notation (HTML output) |
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subset ("op \<subset>") and |
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subset ("(_/ \<subset> _)" [50, 51] 50) and |
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subset_eq ("op \<subseteq>") and |
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subset_eq ("(_/ \<subseteq> _)" [50, 51] 50) |
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abbreviation (input) |
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supset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where |
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"supset \<equiv> greater" |
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abbreviation (input) |
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supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where |
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"supset_eq \<equiv> greater_eq" |
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notation (xsymbols) |
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supset ("op \<supset>") and |
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supset ("(_/ \<supset> _)" [50, 51] 50) and |
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supset_eq ("op \<supseteq>") and |
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supset_eq ("(_/ \<supseteq> _)" [50, 51] 50) |
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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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188 |
"_Bleast" :: "id => 'a set => bool => 'a" ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10) |
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189 |
|
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|
190 |
syntax (HTML output) |
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|
191 |
"_Ball" :: "pttrn => 'a set => bool => bool" ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10) |
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|
192 |
"_Bex" :: "pttrn => 'a set => bool => bool" ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10) |
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193 |
"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10) |
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|
194 |
|
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195 |
translations |
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196 |
"ALL x:A. P" == "Ball A (%x. P)" |
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197 |
"EX x:A. P" == "Bex A (%x. P)" |
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198 |
"EX! x:A. P" == "Bex1 A (%x. P)" |
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199 |
"LEAST x:A. P" => "LEAST x. x:A & P" |
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200 |
|
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|
201 |
syntax (output) |
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|
202 |
"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10) |
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|
203 |
"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10) |
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|
204 |
"_setleAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10) |
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|
205 |
"_setleEx" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) |
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|
206 |
"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3EX! _<=_./ _)" [0, 0, 10] 10) |
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|
207 |
|
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|
208 |
syntax (xsymbols) |
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|
209 |
"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subset>_./ _)" [0, 0, 10] 10) |
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|
210 |
"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subset>_./ _)" [0, 0, 10] 10) |
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|
211 |
"_setleAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10) |
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|
212 |
"_setleEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10) |
20217
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|
213 |
"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10) |
14804
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|
214 |
|
19656
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wenzelm
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changeset
|
215 |
syntax (HOL output) |
14804
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changeset
|
216 |
"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10) |
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changeset
|
217 |
"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10) |
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|
218 |
"_setleAll" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10) |
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|
219 |
"_setleEx" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) |
20217
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|
220 |
"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3?! _<=_./ _)" [0, 0, 10] 10) |
14804
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|
221 |
|
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|
222 |
syntax (HTML output) |
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|
223 |
"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subset>_./ _)" [0, 0, 10] 10) |
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changeset
|
224 |
"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subset>_./ _)" [0, 0, 10] 10) |
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changeset
|
225 |
"_setleAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10) |
8de39d3e8eb6
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changeset
|
226 |
"_setleEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10) |
20217
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linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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changeset
|
227 |
"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10) |
14804
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|
228 |
|
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|
229 |
translations |
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
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parents:
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changeset
|
230 |
"\<forall>A\<subset>B. P" => "ALL A. A \<subset> B --> P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
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parents:
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diff
changeset
|
231 |
"\<exists>A\<subset>B. P" => "EX A. A \<subset> B & P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
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diff
changeset
|
232 |
"\<forall>A\<subseteq>B. P" => "ALL A. A \<subseteq> B --> P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
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changeset
|
233 |
"\<exists>A\<subseteq>B. P" => "EX A. A \<subseteq> B & P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
234 |
"\<exists>!A\<subseteq>B. P" => "EX! A. A \<subseteq> B & P" |
14804
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Corrected printer bug for bounded quantifiers Q x<=y. P
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|
235 |
|
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
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|
236 |
print_translation {* |
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|
237 |
let |
22377 | 238 |
val Type (set_type, _) = @{typ "'a set"}; |
239 |
val All_binder = Syntax.binder_name @{const_syntax "All"}; |
|
240 |
val Ex_binder = Syntax.binder_name @{const_syntax "Ex"}; |
|
241 |
val impl = @{const_syntax "op -->"}; |
|
242 |
val conj = @{const_syntax "op &"}; |
|
243 |
val sbset = @{const_syntax "subset"}; |
|
244 |
val sbset_eq = @{const_syntax "subset_eq"}; |
|
21819 | 245 |
|
246 |
val trans = |
|
247 |
[((All_binder, impl, sbset), "_setlessAll"), |
|
248 |
((All_binder, impl, sbset_eq), "_setleAll"), |
|
249 |
((Ex_binder, conj, sbset), "_setlessEx"), |
|
250 |
((Ex_binder, conj, sbset_eq), "_setleEx")]; |
|
251 |
||
252 |
fun mk v v' c n P = |
|
253 |
if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n) |
|
254 |
then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match; |
|
255 |
||
256 |
fun tr' q = (q, |
|
257 |
fn [Const ("_bound", _) $ Free (v, Type (T, _)), Const (c, _) $ (Const (d, _) $ (Const ("_bound", _) $ Free (v', _)) $ n) $ P] => |
|
258 |
if T = (set_type) then case AList.lookup (op =) trans (q, c, d) |
|
259 |
of NONE => raise Match |
|
260 |
| SOME l => mk v v' l n P |
|
261 |
else raise Match |
|
262 |
| _ => raise Match); |
|
14804
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changeset
|
263 |
in |
21819 | 264 |
[tr' All_binder, tr' Ex_binder] |
14804
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Corrected printer bug for bounded quantifiers Q x<=y. P
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|
265 |
end |
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
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|
266 |
*} |
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
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changeset
|
267 |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
268 |
|
11979 | 269 |
text {* |
270 |
\medskip Translate between @{text "{e | x1...xn. P}"} and @{text |
|
271 |
"{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is |
|
272 |
only translated if @{text "[0..n] subset bvs(e)"}. |
|
273 |
*} |
|
274 |
||
275 |
parse_translation {* |
|
276 |
let |
|
277 |
val ex_tr = snd (mk_binder_tr ("EX ", "Ex")); |
|
3947 | 278 |
|
11979 | 279 |
fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1 |
280 |
| nvars _ = 1; |
|
281 |
||
282 |
fun setcompr_tr [e, idts, b] = |
|
283 |
let |
|
284 |
val eq = Syntax.const "op =" $ Bound (nvars idts) $ e; |
|
285 |
val P = Syntax.const "op &" $ eq $ b; |
|
286 |
val exP = ex_tr [idts, P]; |
|
17784 | 287 |
in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end; |
11979 | 288 |
|
289 |
in [("@SetCompr", setcompr_tr)] end; |
|
290 |
*} |
|
923 | 291 |
|
32120
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
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diff
changeset
|
292 |
print_translation {* [ |
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
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diff
changeset
|
293 |
Syntax.preserve_binder_abs2_tr' @{const_syntax Ball} "_Ball", |
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
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diff
changeset
|
294 |
Syntax.preserve_binder_abs2_tr' @{const_syntax Bex} "_Bex" |
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
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diff
changeset
|
295 |
] *} -- {* to avoid eta-contraction of body *} |
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
296 |
|
13763
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
297 |
print_translation {* |
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
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diff
changeset
|
298 |
let |
f94b569cd610
added print translations tha avoid eta contraction for important binders.
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diff
changeset
|
299 |
val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY")); |
f94b569cd610
added print translations tha avoid eta contraction for important binders.
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diff
changeset
|
300 |
|
f94b569cd610
added print translations tha avoid eta contraction for important binders.
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diff
changeset
|
301 |
fun setcompr_tr' [Abs (abs as (_, _, P))] = |
f94b569cd610
added print translations tha avoid eta contraction for important binders.
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parents:
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diff
changeset
|
302 |
let |
f94b569cd610
added print translations tha avoid eta contraction for important binders.
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parents:
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diff
changeset
|
303 |
fun check (Const ("Ex", _) $ Abs (_, _, P), n) = check (P, n + 1) |
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
304 |
| check (Const ("op &", _) $ (Const ("op =", _) $ Bound m $ e) $ P, n) = |
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
305 |
n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso |
33038 | 306 |
subset (op =) (0 upto (n - 1), add_loose_bnos (e, 0, [])) |
13764 | 307 |
| check _ = false |
923 | 308 |
|
11979 | 309 |
fun tr' (_ $ abs) = |
310 |
let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs] |
|
311 |
in Syntax.const "@SetCompr" $ e $ idts $ Q end; |
|
13763
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
312 |
in if check (P, 0) then tr' P |
15535 | 313 |
else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs |
314 |
val M = Syntax.const "@Coll" $ x $ t |
|
315 |
in case t of |
|
316 |
Const("op &",_) |
|
317 |
$ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A) |
|
318 |
$ P => |
|
319 |
if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M |
|
320 |
| _ => M |
|
321 |
end |
|
13763
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
322 |
end; |
11979 | 323 |
in [("Collect", setcompr_tr')] end; |
324 |
*} |
|
325 |
||
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
326 |
setup {* |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
327 |
let |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
328 |
val unfold_bex_tac = unfold_tac @{thms "Bex_def"}; |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
329 |
fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac; |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
330 |
val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac; |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
331 |
val unfold_ball_tac = unfold_tac @{thms "Ball_def"}; |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
332 |
fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac; |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
333 |
val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac; |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
334 |
val defBEX_regroup = Simplifier.simproc @{theory} |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
335 |
"defined BEX" ["EX x:A. P x & Q x"] rearrange_bex; |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
336 |
val defBALL_regroup = Simplifier.simproc @{theory} |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
337 |
"defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball; |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
338 |
in |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
339 |
Simplifier.map_simpset (fn ss => ss addsimprocs [defBALL_regroup, defBEX_regroup]) |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
340 |
end |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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changeset
|
341 |
*} |
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Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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parents:
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diff
changeset
|
342 |
|
11979 | 343 |
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x" |
344 |
by (simp add: Ball_def) |
|
345 |
||
346 |
lemmas strip = impI allI ballI |
|
347 |
||
348 |
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x" |
|
349 |
by (simp add: Ball_def) |
|
350 |
||
351 |
text {* |
|
352 |
Gives better instantiation for bound: |
|
353 |
*} |
|
354 |
||
26339 | 355 |
declaration {* fn _ => |
356 |
Classical.map_cs (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1)) |
|
11979 | 357 |
*} |
358 |
||
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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parents:
32115
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changeset
|
359 |
ML {* |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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parents:
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changeset
|
360 |
structure Simpdata = |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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parents:
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diff
changeset
|
361 |
struct |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
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diff
changeset
|
362 |
|
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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parents:
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diff
changeset
|
363 |
open Simpdata; |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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parents:
32115
diff
changeset
|
364 |
|
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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parents:
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|
365 |
val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs; |
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parents:
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changeset
|
366 |
|
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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parents:
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changeset
|
367 |
end; |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
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changeset
|
368 |
|
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
369 |
open Simpdata; |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
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diff
changeset
|
370 |
*} |
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Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
371 |
|
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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parents:
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changeset
|
372 |
declaration {* fn _ => |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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parents:
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changeset
|
373 |
Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs)) |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
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diff
changeset
|
374 |
*} |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
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diff
changeset
|
375 |
|
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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parents:
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changeset
|
376 |
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q" |
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
377 |
by (unfold Ball_def) blast |
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Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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parents:
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diff
changeset
|
378 |
|
11979 | 379 |
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x" |
380 |
-- {* Normally the best argument order: @{prop "P x"} constrains the |
|
381 |
choice of @{prop "x:A"}. *} |
|
382 |
by (unfold Bex_def) blast |
|
383 |
||
13113 | 384 |
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x" |
11979 | 385 |
-- {* The best argument order when there is only one @{prop "x:A"}. *} |
386 |
by (unfold Bex_def) blast |
|
387 |
||
388 |
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x" |
|
389 |
by (unfold Bex_def) blast |
|
390 |
||
391 |
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q" |
|
392 |
by (unfold Bex_def) blast |
|
393 |
||
394 |
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)" |
|
395 |
-- {* Trival rewrite rule. *} |
|
396 |
by (simp add: Ball_def) |
|
397 |
||
398 |
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)" |
|
399 |
-- {* Dual form for existentials. *} |
|
400 |
by (simp add: Bex_def) |
|
401 |
||
402 |
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)" |
|
403 |
by blast |
|
404 |
||
405 |
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)" |
|
406 |
by blast |
|
407 |
||
408 |
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)" |
|
409 |
by blast |
|
410 |
||
411 |
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)" |
|
412 |
by blast |
|
413 |
||
414 |
lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)" |
|
415 |
by blast |
|
416 |
||
417 |
lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)" |
|
418 |
by blast |
|
419 |
||
420 |
||
32081 | 421 |
text {* Congruence rules *} |
11979 | 422 |
|
16636
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
423 |
lemma ball_cong: |
11979 | 424 |
"A = B ==> (!!x. x:B ==> P x = Q x) ==> |
425 |
(ALL x:A. P x) = (ALL x:B. Q x)" |
|
426 |
by (simp add: Ball_def) |
|
427 |
||
16636
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
428 |
lemma strong_ball_cong [cong]: |
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
429 |
"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
|
430 |
(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
|
431 |
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
|
432 |
|
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
433 |
lemma bex_cong: |
11979 | 434 |
"A = B ==> (!!x. x:B ==> P x = Q x) ==> |
435 |
(EX x:A. P x) = (EX x:B. Q x)" |
|
436 |
by (simp add: Bex_def cong: conj_cong) |
|
1273 | 437 |
|
16636
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
438 |
lemma strong_bex_cong [cong]: |
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
439 |
"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
|
440 |
(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
|
441 |
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
|
442 |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
443 |
|
32081 | 444 |
subsection {* Basic operations *} |
445 |
||
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
446 |
subsubsection {* Subsets *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
447 |
|
33022
c95102496490
Removal of the unused atpset concept, the atp attribute and some related code.
paulson
parents:
32888
diff
changeset
|
448 |
lemma subsetI [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B) \<Longrightarrow> A \<subseteq> B" |
32888 | 449 |
unfolding mem_def by (rule le_funI, rule le_boolI) |
30352 | 450 |
|
11979 | 451 |
text {* |
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
452 |
\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
|
453 |
'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
|
454 |
"'a set"}. |
11979 | 455 |
*} |
456 |
||
30596 | 457 |
lemma subsetD [elim, intro?]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B" |
32888 | 458 |
unfolding mem_def by (erule le_funE, erule le_boolE) |
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
459 |
-- {* 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
|
460 |
|
33044 | 461 |
lemma rev_subsetD [noatp,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
|
462 |
-- {* 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
|
463 |
cf @{text rev_mp}. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
464 |
by (rule subsetD) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
465 |
|
11979 | 466 |
text {* |
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
467 |
\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
|
468 |
*} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
469 |
|
33044 | 470 |
lemma subsetCE [noatp,elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P" |
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
471 |
-- {* Classical elimination rule. *} |
32888 | 472 |
unfolding mem_def by (blast dest: le_funE le_boolE) |
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
473 |
|
33044 | 474 |
lemma subset_eq [noatp]: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast |
2388 | 475 |
|
33044 | 476 |
lemma contra_subsetD [noatp]: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A" |
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
477 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
478 |
|
33022
c95102496490
Removal of the unused atpset concept, the atp attribute and some related code.
paulson
parents:
32888
diff
changeset
|
479 |
lemma subset_refl [simp]: "A \<subseteq> A" |
32081 | 480 |
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
|
481 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
482 |
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C" |
32081 | 483 |
by (fact order_trans) |
484 |
||
485 |
lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B" |
|
486 |
by (rule subsetD) |
|
487 |
||
488 |
lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B" |
|
489 |
by (rule subsetD) |
|
490 |
||
33044 | 491 |
lemma eq_mem_trans: "a=b ==> b \<in> A ==> a \<in> A" |
492 |
by simp |
|
493 |
||
32081 | 494 |
lemmas basic_trans_rules [trans] = |
33044 | 495 |
order_trans_rules set_rev_mp set_mp eq_mem_trans |
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
496 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
497 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
498 |
subsubsection {* Equality *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
499 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
500 |
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
|
501 |
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
|
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 |
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
|
504 |
done |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
505 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
506 |
(* Due to Brian Huffman *) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
507 |
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
|
508 |
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
|
509 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
510 |
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
|
511 |
-- {* Anti-symmetry of the subset relation. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
512 |
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
|
513 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
514 |
text {* |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
515 |
\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
|
516 |
here? |
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 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
519 |
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
|
520 |
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
|
521 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
522 |
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
|
523 |
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
|
524 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
525 |
text {* |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
526 |
\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
|
527 |
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
|
528 |
\<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}! |
30352 | 529 |
*} |
530 |
||
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
531 |
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
|
532 |
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
|
533 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
534 |
lemma equalityCE [elim]: |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
535 |
"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
|
536 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
537 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
538 |
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
|
539 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
540 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
541 |
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
|
542 |
by simp |
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 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
545 |
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
|
546 |
|
32264
0be31453f698
Set.UNIV and Set.empty are mere abbreviations for top and bot
haftmann
parents:
32139
diff
changeset
|
547 |
abbreviation UNIV :: "'a set" where |
0be31453f698
Set.UNIV and Set.empty are mere abbreviations for top and bot
haftmann
parents:
32139
diff
changeset
|
548 |
"UNIV \<equiv> top" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
549 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
550 |
lemma UNIV_def: |
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
551 |
"UNIV = {x. True}" |
32264
0be31453f698
Set.UNIV and Set.empty are mere abbreviations for top and bot
haftmann
parents:
32139
diff
changeset
|
552 |
by (simp add: top_fun_eq top_bool_eq Collect_def) |
32081 | 553 |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
554 |
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
|
555 |
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
|
556 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
557 |
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
|
558 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
559 |
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
|
560 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
561 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
562 |
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
|
563 |
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
|
564 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
565 |
text {* |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
566 |
\medskip Eta-contracting 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
|
567 |
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
|
568 |
congruence rules. |
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 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
571 |
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
|
572 |
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
|
573 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
574 |
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
|
575 |
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
|
576 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
577 |
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
|
578 |
by auto |
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 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
581 |
subsubsection {* The empty set *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
582 |
|
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
|
583 |
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
|
584 |
"{} = {x. False}" |
32264
0be31453f698
Set.UNIV and Set.empty are mere abbreviations for top and bot
haftmann
parents:
32139
diff
changeset
|
585 |
by (simp add: bot_fun_eq bot_bool_eq Collect_def) |
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
|
586 |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
587 |
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
|
588 |
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
|
589 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
590 |
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
|
591 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
592 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
593 |
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
|
594 |
-- {* 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
|
595 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
596 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
597 |
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
|
598 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
599 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
600 |
lemma equals0D: "A = {} ==> a \<notin> A" |
32082 | 601 |
-- {* 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
|
602 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
603 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
604 |
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
|
605 |
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
|
606 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
607 |
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
|
608 |
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
|
609 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
610 |
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
|
611 |
by (blast elim: equalityE) |
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 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
614 |
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
|
615 |
|
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
|
616 |
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
|
617 |
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
|
618 |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
619 |
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
|
620 |
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
|
621 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
622 |
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
|
623 |
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
|
624 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
625 |
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
|
626 |
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
|
627 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
628 |
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
|
629 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
630 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
631 |
lemma 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
|
632 |
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
|
633 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
634 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
635 |
subsubsection {* Set complement *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
636 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
637 |
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
|
638 |
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
|
639 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
640 |
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
|
641 |
by (unfold mem_def fun_Compl_def bool_Compl_def) blast |
923 | 642 |
|
11979 | 643 |
text {* |
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
644 |
\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
|
645 |
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
|
646 |
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
|
647 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
648 |
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
|
649 |
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
|
650 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
651 |
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
|
652 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
653 |
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
|
654 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
655 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
656 |
subsubsection {* Binary union -- Un *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
657 |
|
32683
7c1fe854ca6a
inter and union are mere abbreviations for inf and sup
haftmann
parents:
32456
diff
changeset
|
658 |
abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where |
7c1fe854ca6a
inter and union are mere abbreviations for inf and sup
haftmann
parents:
32456
diff
changeset
|
659 |
"op Un \<equiv> sup" |
32081 | 660 |
|
661 |
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
|
662 |
union (infixl "\<union>" 65) |
32081 | 663 |
|
664 |
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
|
665 |
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
|
666 |
|
f645b51e8e54
set 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 |
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
|
668 |
"A \<union> B = {x. x \<in> A \<or> x \<in> B}" |
32683
7c1fe854ca6a
inter and union are mere abbreviations for inf and sup
haftmann
parents:
32456
diff
changeset
|
669 |
by (simp add: sup_fun_eq sup_bool_eq Collect_def mem_def) |
32081 | 670 |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
671 |
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
|
672 |
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
|
673 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
674 |
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
|
675 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
676 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
677 |
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
|
678 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
679 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
680 |
text {* |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
681 |
\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
|
682 |
@{prop B}. |
11979 | 683 |
*} |
684 |
||
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
685 |
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
|
686 |
by auto |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
687 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
688 |
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
|
689 |
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
|
690 |
|
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
691 |
lemma insert_def: "insert a B = {x. x = a} \<union> B" |
32081 | 692 |
by (simp add: Collect_def mem_def insert_compr Un_def) |
693 |
||
694 |
lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)" |
|
32683
7c1fe854ca6a
inter and union are mere abbreviations for inf and sup
haftmann
parents:
32456
diff
changeset
|
695 |
by (fact mono_sup) |
32081 | 696 |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
697 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
698 |
subsubsection {* Binary intersection -- Int *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
699 |
|
32683
7c1fe854ca6a
inter and union are mere abbreviations for inf and sup
haftmann
parents:
32456
diff
changeset
|
700 |
abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where |
7c1fe854ca6a
inter and union are mere abbreviations for inf and sup
haftmann
parents:
32456
diff
changeset
|
701 |
"op Int \<equiv> inf" |
32081 | 702 |
|
703 |
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
|
704 |
inter (infixl "\<inter>" 70) |
32081 | 705 |
|
706 |
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
|
707 |
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
|
708 |
|
f645b51e8e54
set 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 |
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
|
710 |
"A \<inter> B = {x. x \<in> A \<and> x \<in> B}" |
32683
7c1fe854ca6a
inter and union are mere abbreviations for inf and sup
haftmann
parents:
32456
diff
changeset
|
711 |
by (simp add: inf_fun_eq inf_bool_eq Collect_def mem_def) |
32081 | 712 |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
713 |
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
|
714 |
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
|
715 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
716 |
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
|
717 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
718 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
719 |
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
|
720 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
721 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
722 |
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
|
723 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
724 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
725 |
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
|
726 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
727 |
|
32081 | 728 |
lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B" |
32683
7c1fe854ca6a
inter and union are mere abbreviations for inf and sup
haftmann
parents:
32456
diff
changeset
|
729 |
by (fact mono_inf) |
32081 | 730 |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
731 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
732 |
subsubsection {* Set difference *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
733 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
734 |
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
|
735 |
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
|
736 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
737 |
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
|
738 |
by simp |
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 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
|
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 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
|
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 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
|
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 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
|
750 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
751 |
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
|
752 |
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 |
|
31456 | 755 |
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
|
756 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
757 |
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
|
758 |
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
|
759 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
760 |
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
|
761 |
by simp |
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 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
|
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 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
|
767 |
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
|
768 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
769 |
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
|
770 |
-- {* Classical introduction rule. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
771 |
by auto |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
772 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
773 |
lemma 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
|
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 set_insert: |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
777 |
assumes "x \<in> A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
778 |
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
|
779 |
proof |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
780 |
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
|
781 |
next |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
782 |
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
|
783 |
qed |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
784 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
785 |
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
|
786 |
by auto |
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 |
subsubsection {* Singletons, using insert *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
789 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
790 |
lemma 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
|
791 |
-- {* 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
|
792 |
by (rule insertI1) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
793 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
794 |
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
|
795 |
by blast |
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 |
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
|
798 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
799 |
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
|
800 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
801 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
802 |
lemma singleton_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
|
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_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
|
806 |
"({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
|
807 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
808 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
809 |
lemma singleton_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
|
810 |
"(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
|
811 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
812 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
813 |
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
|
814 |
by fast |
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 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
|
817 |
by blast |
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_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
|
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 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
|
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 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
|
826 |
by (blast elim: equalityE) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
827 |
|
11979 | 828 |
|
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
|
829 |
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
|
830 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
831 |
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
|
832 |
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
|
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 |
|
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 |
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
|
836 |
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
|
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 |
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
|
839 |
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
|
840 |
"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
|
841 |
|
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 |
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
|
843 |
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
|
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 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
|
846 |
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
|
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 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
|
849 |
-- {* 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
|
850 |
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
|
851 |
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
|
852 |
|
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 |
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
|
854 |
"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
|
855 |
-- {* The eta-expansion gives variable-name 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
|
856 |
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
|
857 |
|
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 |
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
|
859 |
by blast |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
860 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
861 |
lemma 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
|
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_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
|
865 |
-- {* 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
|
866 |
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
|
867 |
|
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 |
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
|
869 |
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
|
870 |
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
|
871 |
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
|
872 |
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
|
873 |
|
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 |
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
|
875 |
-- {* 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
|
876 |
@{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
|
877 |
by blast |
11979 | 878 |
|
879 |
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
|
880 |
\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
|
881 |
*} |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
882 |
|
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 |
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
|
884 |
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
|
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 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
|
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 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
|
890 |
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
|
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 |
|
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
893 |
subsubsection {* Some rules with @{text "if"} *} |
32081 | 894 |
|
895 |
text{* Elimination of @{text"{x. \<dots> & x=t & \<dots>}"}. *} |
|
896 |
||
897 |
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
|
898 |
by auto |
32081 | 899 |
|
900 |
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
|
901 |
by auto |
32081 | 902 |
|
903 |
text {* |
|
904 |
Rewrite rules for boolean case-splitting: faster than @{text |
|
905 |
"split_if [split]"}. |
|
906 |
*} |
|
907 |
||
908 |
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))" |
|
909 |
by (rule split_if) |
|
910 |
||
911 |
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))" |
|
912 |
by (rule split_if) |
|
913 |
||
914 |
text {* |
|
915 |
Split ifs on either side of the membership relation. Not for @{text |
|
916 |
"[simp]"} -- can cause goals to blow up! |
|
917 |
*} |
|
918 |
||
919 |
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))" |
|
920 |
by (rule split_if) |
|
921 |
||
922 |
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))" |
|
923 |
by (rule split_if [where P="%S. a : S"]) |
|
924 |
||
925 |
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2 |
|
926 |
||
927 |
(*Would like to add these, but the existing code only searches for the |
|
928 |
outer-level constant, which in this case is just "op :"; we instead need |
|
929 |
to use term-nets to associate patterns with rules. Also, if a rule fails to |
|
930 |
apply, then the formula should be kept. |
|
931 |
[("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]), |
|
932 |
("Int", [IntD1,IntD2]), |
|
933 |
("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])] |
|
934 |
*) |
|
935 |
||
936 |
||
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
|
937 |
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
|
938 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
939 |
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
|
940 |
|
f645b51e8e54
set 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 |
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
|
942 |
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
|
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 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
|
945 |
"[|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
|
946 |
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
|
947 |
|
f645b51e8e54
set 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 |
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
|
949 |
"(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
|
950 |
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
|
951 |
|
f645b51e8e54
set 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 |
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
|
953 |
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
|
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_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
|
956 |
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
|
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_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
|
959 |
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
|
960 |
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
|
961 |
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
|
962 |
|
f645b51e8e54
set 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 |
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
|
964 |
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
|
965 |
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
|
966 |
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
|
967 |
|
f645b51e8e54
set 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 |
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
|
969 |
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
|
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 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
|
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 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
|
975 |
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
|
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 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
|
978 |
"(!!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
|
979 |
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
|
980 |
|
f645b51e8e54
set 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 |
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
|
982 |
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
|
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 |
subsubsection {* Derived rules involving subsets. *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
985 |
|
f645b51e8e54
set 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 |
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
|
987 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
988 |
lemma subset_insertI: "B \<subseteq> insert a B" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
989 |
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
|
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_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
|
992 |
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
|
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_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
|
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 |
|
f645b51e8e54
set 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 |
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
|
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 |
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
|
1001 |
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
|
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_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
|
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_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
|
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 |
|
f645b51e8e54
set 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 |
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
|
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 |
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
|
1013 |
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
|
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_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
|
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_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
|
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 |
|
f645b51e8e54
set 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 |
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
|
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 |
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
|
1025 |
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
|
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_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
|
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 |
|
f645b51e8e54
set 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 |
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
|
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 |
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
|
1034 |
|
f645b51e8e54
set 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 |
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
|
1036 |
-- {* 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
|
1037 |
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
|
1038 |
|
f645b51e8e54
set 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 |
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
|
1040 |
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
|
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 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
|
1043 |
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
|
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 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
|
1046 |
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
|
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 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
|
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 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
|
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_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
|
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_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
|
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_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
|
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 |
|
f645b51e8e54
set 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 |
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
|
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 |
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
|
1067 |
-- {* 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
|
1068 |
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
|
1069 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1070 |
lemma 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
|
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 |
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
|
1074 |
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
|
1075 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1076 |
lemma 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
|
1077 |
-- {* @{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
|
1078 |
-- {* 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
|
1079 |
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
|
1080 |
|
f645b51e8e54
set 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 |
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
|
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_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
|
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_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
|
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 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
|
1091 |
-- {* 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
|
1092 |
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
|
1093 |
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
|
1094 |
|
f645b51e8e54
set 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 |
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
|
1096 |
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
|
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_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
|
1099 |
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
|
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_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
|
1102 |
"(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
|
1103 |
"({} = 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
|
1104 |
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
|
1105 |
|
f645b51e8e54
set 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 |
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
|
1107 |
"(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
|
1108 |
"({} = 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
|
1109 |
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
|
1110 |
|
f645b51e8e54
set 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 |
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
|
1112 |
|
f645b51e8e54
set 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 |
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
|
1114 |
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
|
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_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
|
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_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
|
1120 |
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
|
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_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
|
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_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
|
1126 |
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
|
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 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
|
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 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
|
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 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
|
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 |
|
f645b51e8e54
set 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 |
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
|
1139 |
-- {* 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
|
1140 |
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
|
1141 |
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
|
1142 |
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
|
1143 |
|
f645b51e8e54
set 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 |
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
|
1145 |
"(\<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
|
1146 |
= (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
|
1147 |
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
|
1148 |
|
f645b51e8e54
set 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 |
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
|
1150 |
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
|
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 |
|
f645b51e8e54
set 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 |
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
|
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 |
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
|
1156 |
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
|
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 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
|
1159 |
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
|
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 |
|
f645b51e8e54
set 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 |
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
|
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 |
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
|
1165 |
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
|
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_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
|
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_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
|
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_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
|
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_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
|
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 |
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
|
1180 |
-- {* Intersection is an AC-operator *} |
f645b51e8e54
set 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 |
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
|
1183 |
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
|
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_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
|
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_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
|
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_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
|
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 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
|
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_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
|
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 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
|
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_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
|
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_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
|
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_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
|
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_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
|
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_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
|
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_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
|
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 |
|
f645b51e8e54
set 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 |
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
|
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 |
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
|
1225 |
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
|
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_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
|
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_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
|
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_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
|
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_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
|
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 |
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
|
1240 |
-- {* Union is an AC-operator *} |
f645b51e8e54
set 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 |
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
|
1243 |
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
|
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_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
|
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_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
|
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_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
|
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_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
|
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_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
|
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_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
|
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_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
|
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 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
|
1267 |
"(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
|
1268 |
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
|
1269 |
|
32456
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
32264
diff
changeset
|
1270 |
lemma Int_insert_left_if0[simp]: |
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
32264
diff
changeset
|
1271 |
"a \<notin> C \<Longrightarrow> (insert a B) Int C = B \<inter> C" |
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
32264
diff
changeset
|
1272 |
by auto |
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
32264
diff
changeset
|
1273 |
|
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
32264
diff
changeset
|
1274 |
lemma Int_insert_left_if1[simp]: |
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
32264
diff
changeset
|
1275 |
"a \<in> C \<Longrightarrow> (insert a B) Int C = insert a (B Int C)" |
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
32264
diff
changeset
|
1276 |
by auto |
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
32264
diff
changeset
|
1277 |
|
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
|
1278 |
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
|
1279 |
"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
|
1280 |
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
|
1281 |
|
32456
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
32264
diff
changeset
|
1282 |
lemma Int_insert_right_if0[simp]: |
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
32264
diff
changeset
|
1283 |
"a \<notin> A \<Longrightarrow> A Int (insert a B) = A Int B" |
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
32264
diff
changeset
|
1284 |
by auto |
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
32264
diff
changeset
|
1285 |
|
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
32264
diff
changeset
|
1286 |
lemma Int_insert_right_if1[simp]: |
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
32264
diff
changeset
|
1287 |
"a \<in> A \<Longrightarrow> A Int (insert a B) = insert a (A Int B)" |
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
32264
diff
changeset
|
1288 |
by auto |
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
32264
diff
changeset
|
1289 |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1290 |
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
|
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
|
1292 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1293 |
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
|
1294 |
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
|
1295 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1296 |
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
|
1297 |
"(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
|
1298 |
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
|
1299 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1300 |
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
|
1301 |
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
|
1302 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1303 |
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
|
1304 |
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
|
1305 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1306 |
lemma Un_subset_iff [simp]: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & 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
|
1307 |
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
|
1308 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1309 |
lemma Un_Diff_Int: "(A - B) \<union> (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
|
1310 |
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
|
1311 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1312 |
lemma Diff_Int2: "A \<inter> C - B \<inter> C = A \<inter> C - 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
|
1313 |
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
|
1314 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1315 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1316 |
text {* \medskip Set complement *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1317 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1318 |
lemma Compl_disjoint [simp]: "A \<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
|
1319 |
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
|
1320 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1321 |
lemma Compl_disjoint2 [simp]: "-A \<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
|
1322 |
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
|
1323 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1324 |
lemma Compl_partition: "A \<union> -A = 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
|
1325 |
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
|
1326 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1327 |
lemma Compl_partition2: "-A \<union> A = 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
|
1328 |
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
|
1329 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1330 |
lemma double_complement [simp]: "- (-A) = (A::'a set)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1331 |
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
|
1332 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1333 |
lemma Compl_Un [simp]: "-(A \<union> 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
|
1334 |
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
|
1335 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1336 |
lemma Compl_Int [simp]: "-(A \<inter> 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
|
1337 |
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
|
1338 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1339 |
lemma subset_Compl_self_eq: "(A \<subseteq> -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
|
1340 |
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
|
1341 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1342 |
lemma Un_Int_assoc_eq: "((A \<inter> B) \<union> C = A \<inter> (B \<union> C)) = (C \<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
|
1343 |
-- {* Halmos, Naive Set Theory, page 16. *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1344 |
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
|
1345 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1346 |
lemma Compl_UNIV_eq [simp]: "-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
|
1347 |
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
|
1348 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1349 |
lemma Compl_empty_eq [simp]: "-{} = 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
|
1350 |
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
|
1351 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1352 |
lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (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
|
1353 |
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
|
1354 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1355 |
lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1356 |
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
|
1357 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1358 |
text {* \medskip Bounded quantifiers. |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1359 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1360 |
The following are not added to the default simpset because |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1361 |
(a) they duplicate the body and (b) there are no similar rules for @{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
|
1362 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1363 |
lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) = ((\<forall>x\<in>A. P x) & (\<forall>x\<in>B. 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
|
1364 |
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
|
1365 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1366 |
lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) = ((\<exists>x\<in>A. P x) | (\<exists>x\<in>B. 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
|
1367 |
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
|
1368 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1369 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1370 |
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
|
1371 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1372 |
lemma Diff_eq: "A - 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
|
1373 |
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
|
1374 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1375 |
lemma Diff_eq_empty_iff [simp,noatp]: "(A - 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
|
1376 |
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
|
1377 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1378 |
lemma Diff_cancel [simp]: "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
|
1379 |
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
|
1380 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1381 |
lemma Diff_idemp [simp]: "(A - B) - B = A - (B::'a set)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1382 |
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
|
1383 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1384 |
lemma Diff_triv: "A \<inter> B = {} ==> A - 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
|
1385 |
by (blast elim: equalityE) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1386 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1387 |
lemma empty_Diff [simp]: "{} - 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
|
1388 |
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
|
1389 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1390 |
lemma Diff_empty [simp]: "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
|
1391 |
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
|
1392 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1393 |
lemma Diff_UNIV [simp]: "A - 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
|
1394 |
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
|
1395 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1396 |
lemma Diff_insert0 [simp,noatp]: "x \<notin> A ==> A - insert x 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
|
1397 |
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
|
1398 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1399 |
lemma Diff_insert: "A - insert a B = A - 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
|
1400 |
-- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1401 |
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
|
1402 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1403 |
lemma Diff_insert2: "A - insert a B = A - {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
|
1404 |
-- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1405 |
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
|
1406 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1407 |
lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (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
|
1408 |
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
|
1409 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1410 |
lemma insert_Diff1 [simp]: "x \<in> B ==> insert x A - 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
|
1411 |
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
|
1412 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1413 |
lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert 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
|
1414 |
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
|
1415 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1416 |
lemma insert_Diff: "a \<in> A ==> insert a (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
|
1417 |
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
|
1418 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1419 |
lemma Diff_insert_absorb: "x \<notin> A ==> (insert x A) - {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
|
1420 |
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
|
1421 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1422 |
lemma Diff_disjoint [simp]: "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
|
1423 |
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
|
1424 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1425 |
lemma Diff_partition: "A \<subseteq> B ==> 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
|
1426 |
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
|
1427 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1428 |
lemma double_diff: "A \<subseteq> B ==> B \<subseteq> C ==> B - (C - 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
|
1429 |
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
|
1430 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1431 |
lemma Un_Diff_cancel [simp]: "A \<union> (B - 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
|
1432 |
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
|
1433 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1434 |
lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = 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
|
1435 |
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
|
1436 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1437 |
lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (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
|
1438 |
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
|
1439 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1440 |
lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (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
|
1441 |
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
|
1442 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1443 |
lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - 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
|
1444 |
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
|
1445 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1446 |
lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - 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
|
1447 |
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
|
1448 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1449 |
lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<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
|
1450 |
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
|
1451 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1452 |
lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (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
|
1453 |
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
|
1454 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1455 |
lemma Diff_Compl [simp]: "A - (- 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
|
1456 |
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
|
1457 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1458 |
lemma Compl_Diff_eq [simp]: "- (A - 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
|
1459 |
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
|
1460 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1461 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1462 |
text {* \medskip Quantification over type @{typ bool}. *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1463 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1464 |
lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> 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
|
1465 |
by (cases x) 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
|
1466 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1467 |
lemma all_bool_eq: "(\<forall>b. P b) \<longleftrightarrow> P True \<and> P 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
|
1468 |
by (auto intro: bool_induct) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1469 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1470 |
lemma bool_contrapos: "P x \<Longrightarrow> \<not> P False \<Longrightarrow> P True" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1471 |
by (cases x) 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
|
1472 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1473 |
lemma ex_bool_eq: "(\<exists>b. P b) \<longleftrightarrow> P True \<or> P 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
|
1474 |
by (auto intro: bool_contrapos) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1475 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1476 |
text {* \medskip @{text Pow} *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1477 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1478 |
lemma Pow_empty [simp]: "Pow {} = {{}}" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1479 |
by (auto simp add: Pow_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
|
1480 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1481 |
lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a ` Pow 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
|
1482 |
by (blast intro: image_eqI [where ?x = "u - {a}", 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
|
1483 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1484 |
lemma Pow_Compl: "Pow (- A) = {-B | B. A \<in> Pow 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
|
1485 |
by (blast intro: exI [where ?x = "- u", 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
|
1486 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1487 |
lemma Pow_UNIV [simp]: "Pow 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
|
1488 |
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
|
1489 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1490 |
lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (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
|
1491 |
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
|
1492 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1493 |
lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow 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
|
1494 |
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
|
1495 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1496 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1497 |
text {* \medskip Miscellany. *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1498 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1499 |
lemma set_eq_subset: "(A = B) = (A \<subseteq> B & 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
|
1500 |
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
|
1501 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1502 |
lemma subset_iff: "(A \<subseteq> B) = (\<forall>t. t \<in> A --> t \<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
|
1503 |
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
|
1504 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1505 |
lemma subset_iff_psubset_eq: "(A \<subseteq> B) = ((A \<subset> 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
|
1506 |
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
|
1507 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1508 |
lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> 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
|
1509 |
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
|
1510 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1511 |
lemma ex_in_conv: "(\<exists>x. x \<in> 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
|
1512 |
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
|
1513 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1514 |
lemma distinct_lemma: "f x \<noteq> f y ==> 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
|
1515 |
by iprover |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1516 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1517 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1518 |
subsubsection {* Monotonicity of various operations *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1519 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1520 |
lemma image_mono: "A \<subseteq> B ==> f`A \<subseteq> 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
|
1521 |
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
|
1522 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1523 |
lemma Pow_mono: "A \<subseteq> B ==> Pow A \<subseteq> Pow 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
|
1524 |
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
|
1525 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1526 |
lemma insert_mono: "C \<subseteq> D ==> insert a C \<subseteq> insert a D" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1527 |
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
|
1528 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1529 |
lemma Un_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<union> B \<subseteq> C \<union> D" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1530 |
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
|
1531 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1532 |
lemma Int_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1533 |
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
|
1534 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1535 |
lemma Diff_mono: "A \<subseteq> C ==> D \<subseteq> B ==> A - B \<subseteq> C - D" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1536 |
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
|
1537 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1538 |
lemma Compl_anti_mono: "A \<subseteq> B ==> -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
|
1539 |
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
|
1540 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1541 |
text {* \medskip Monotonicity of implications. *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1542 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1543 |
lemma in_mono: "A \<subseteq> B ==> x \<in> A --> 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
|
1544 |
apply (rule impI) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1545 |
apply (erule subsetD, assumption) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1546 |
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
|
1547 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1548 |
lemma conj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 & P2) --> (Q1 & Q2)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1549 |
by iprover |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1550 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1551 |
lemma disj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 | P2) --> (Q1 | Q2)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1552 |
by iprover |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1553 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1554 |
lemma imp_mono: "Q1 --> P1 ==> P2 --> Q2 ==> (P1 --> P2) --> (Q1 --> Q2)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1555 |
by iprover |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1556 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1557 |
lemma imp_refl: "P --> P" .. |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1558 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1559 |
lemma ex_mono: "(!!x. P x --> Q x) ==> (EX x. P x) --> (EX 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
|
1560 |
by iprover |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1561 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1562 |
lemma all_mono: "(!!x. P x --> Q x) ==> (ALL x. P x) --> (ALL 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
|
1563 |
by iprover |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1564 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1565 |
lemma Collect_mono: "(!!x. P x --> Q x) ==> Collect P \<subseteq> Collect Q" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1566 |
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
|
1567 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1568 |
lemma Int_Collect_mono: |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1569 |
"A \<subseteq> B ==> (!!x. x \<in> A ==> P x --> Q x) ==> A \<inter> Collect P \<subseteq> B \<inter> Collect Q" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1570 |
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
|
1571 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1572 |
lemmas basic_monos = |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1573 |
subset_refl imp_refl disj_mono conj_mono |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1574 |
ex_mono Collect_mono in_mono |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1575 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1576 |
lemma eq_to_mono: "a = b ==> c = d ==> b --> d ==> 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
|
1577 |
by iprover |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1578 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1579 |
lemma eq_to_mono2: "a = b ==> c = d ==> ~ b --> ~ d ==> ~ 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
|
1580 |
by iprover |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1581 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1582 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1583 |
subsubsection {* Inverse image of a function *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1584 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1585 |
constdefs |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1586 |
vimage :: "('a => 'b) => 'b set => 'a set" (infixr "-`" 90) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1587 |
[code del]: "f -` B == {x. f x : 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
|
1588 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1589 |
lemma vimage_eq [simp]: "(a : f -` B) = (f 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
|
1590 |
by (unfold vimage_def) 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
|
1591 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1592 |
lemma vimage_singleton_eq: "(a : f -` {b}) = (f 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
|
1593 |
by 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
|
1594 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1595 |
lemma vimageI [intro]: "f a = b ==> b:B ==> 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
|
1596 |
by (unfold vimage_def) 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
|
1597 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1598 |
lemma vimageI2: "f a : A ==> 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
|
1599 |
by (unfold vimage_def) fast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1600 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1601 |
lemma vimageE [elim!]: "a: f -` B ==> (!!x. f a = x ==> x:B ==> P) ==> P" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1602 |
by (unfold vimage_def) 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
|
1603 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1604 |
lemma vimageD: "a : f -` A ==> 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
|
1605 |
by (unfold vimage_def) fast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1606 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1607 |
lemma vimage_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
|
1608 |
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
|
1609 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1610 |
lemma vimage_Compl: "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
|
1611 |
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
|
1612 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1613 |
lemma vimage_Un [simp]: "f -` (A Un B) = (f -` A) Un (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
|
1614 |
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
|
1615 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1616 |
lemma vimage_Int [simp]: "f -` (A Int B) = (f -` A) Int (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
|
1617 |
by fast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1618 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1619 |
lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f 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
|
1620 |
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
|
1621 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1622 |
lemma vimage_Collect: "(!!x. P (f x) = Q x) ==> f -` (Collect P) = Collect Q" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1623 |
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
|
1624 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1625 |
lemma vimage_insert: "f-`(insert a B) = (f-`{a}) Un (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
|
1626 |
-- {* NOT suitable for rewriting because of the recurrence of @{term "{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
|
1627 |
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
|
1628 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1629 |
lemma vimage_Diff: "f -` (A - B) = (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
|
1630 |
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
|
1631 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1632 |
lemma vimage_UNIV [simp]: "f -` 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
|
1633 |
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
|
1634 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1635 |
lemma vimage_mono: "A \<subseteq> B ==> f -` A \<subseteq> 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
|
1636 |
-- {* monotonicity *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1637 |
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
|
1638 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1639 |
lemma vimage_image_eq [noatp]: "f -` (f ` A) = {y. EX x:A. f x = f 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
|
1640 |
by (blast intro: sym) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1641 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1642 |
lemma image_vimage_subset: "f ` (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
|
1643 |
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
|
1644 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1645 |
lemma image_vimage_eq [simp]: "f ` (f -` A) = A Int 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
|
1646 |
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
|
1647 |
|
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
33045
diff
changeset
|
1648 |
lemma vimage_const [simp]: "((\<lambda>x. c) -` A) = (if c \<in> A then UNIV else {})" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
33045
diff
changeset
|
1649 |
by auto |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
33045
diff
changeset
|
1650 |
|
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
33045
diff
changeset
|
1651 |
lemma vimage_if [simp]: "((\<lambda>x. if x \<in> B then c else d) -` A) = |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
33045
diff
changeset
|
1652 |
(if c \<in> A then (if d \<in> A then UNIV else B) |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
33045
diff
changeset
|
1653 |
else if d \<in> A then -B else {})" |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
33045
diff
changeset
|
1654 |
by (auto simp add: vimage_def) |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
33045
diff
changeset
|
1655 |
|
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
|
1656 |
lemma image_Int_subset: "f`(A Int B) <= f`A Int 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
|
1657 |
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
|
1658 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1659 |
lemma image_diff_subset: "f`A - f`B <= f`(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
|
1660 |
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
|
1661 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1662 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1663 |
subsubsection {* Getting the Contents of a Singleton Set *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1664 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1665 |
definition contents :: "'a set \<Rightarrow> 'a" 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
|
1666 |
[code del]: "contents X = (THE x. X = {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
|
1667 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1668 |
lemma contents_eq [simp]: "contents {x} = 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
|
1669 |
by (simp add: contents_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
|
1670 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1671 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1672 |
subsubsection {* Least value operator *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1673 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1674 |
lemma Least_mono: |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1675 |
"mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= 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
|
1676 |
==> (LEAST y. y : f ` S) = f (LEAST x. 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
|
1677 |
-- {* Courtesy of Stephan Merz *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1678 |
apply clarify |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1679 |
apply (erule_tac P = "%x. x : S" in LeastI2_order, fast) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1680 |
apply (rule LeastI2_order) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1681 |
apply (auto elim: monoD intro!: order_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
|
1682 |
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
|
1683 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
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|
1684 |
subsection {* Misc *} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1685 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1686 |
text {* Rudimentary code generation *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1687 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1688 |
lemma insert_code [code]: "insert y A x \<longleftrightarrow> y = x \<or> A x" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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changeset
|
1689 |
by (auto simp add: insert_compr Collect_def mem_def) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1690 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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|
1691 |
lemma vimage_code [code]: "(f -` A) x = A (f x)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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changeset
|
1692 |
by (simp add: vimage_def Collect_def mem_def) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1693 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1694 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1695 |
text {* Misc theorem and ML bindings *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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changeset
|
1696 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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|
1697 |
lemmas equalityI = subset_antisym |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1698 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1699 |
ML {* |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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|
1700 |
val Ball_def = @{thm Ball_def} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1701 |
val Bex_def = @{thm Bex_def} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1702 |
val CollectD = @{thm CollectD} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1703 |
val CollectE = @{thm CollectE} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1704 |
val CollectI = @{thm CollectI} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1705 |
val Collect_conj_eq = @{thm Collect_conj_eq} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1706 |
val Collect_mem_eq = @{thm Collect_mem_eq} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1707 |
val IntD1 = @{thm IntD1} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1708 |
val IntD2 = @{thm IntD2} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1709 |
val IntE = @{thm IntE} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1710 |
val IntI = @{thm IntI} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1711 |
val Int_Collect = @{thm Int_Collect} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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|
1712 |
val UNIV_I = @{thm UNIV_I} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1713 |
val UNIV_witness = @{thm UNIV_witness} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1714 |
val UnE = @{thm UnE} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1715 |
val UnI1 = @{thm UnI1} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1716 |
val UnI2 = @{thm UnI2} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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|
1717 |
val ballE = @{thm ballE} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1718 |
val ballI = @{thm ballI} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1719 |
val bexCI = @{thm bexCI} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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changeset
|
1720 |
val bexE = @{thm bexE} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1721 |
val bexI = @{thm bexI} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1722 |
val bex_triv = @{thm bex_triv} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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|
1723 |
val bspec = @{thm bspec} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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|
1724 |
val contra_subsetD = @{thm contra_subsetD} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1725 |
val distinct_lemma = @{thm distinct_lemma} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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|
1726 |
val eq_to_mono = @{thm eq_to_mono} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1727 |
val eq_to_mono2 = @{thm eq_to_mono2} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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|
1728 |
val equalityCE = @{thm equalityCE} |
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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|
1729 |
val equalityD1 = @{thm equalityD1} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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|
1730 |
val equalityD2 = @{thm equalityD2} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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|
1731 |
val equalityE = @{thm equalityE} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1732 |
val equalityI = @{thm equalityI} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1733 |
val imageE = @{thm imageE} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1734 |
val imageI = @{thm imageI} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1735 |
val image_Un = @{thm image_Un} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1736 |
val image_insert = @{thm image_insert} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1737 |
val insert_commute = @{thm insert_commute} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1738 |
val insert_iff = @{thm insert_iff} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1739 |
val mem_Collect_eq = @{thm mem_Collect_eq} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
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parents:
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changeset
|
1740 |
val rangeE = @{thm rangeE} |
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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changeset
|
1741 |
val rangeI = @{thm rangeI} |
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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changeset
|
1742 |
val range_eqI = @{thm range_eqI} |
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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changeset
|
1743 |
val subsetCE = @{thm subsetCE} |
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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changeset
|
1744 |
val subsetD = @{thm subsetD} |
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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changeset
|
1745 |
val subsetI = @{thm subsetI} |
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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changeset
|
1746 |
val subset_refl = @{thm subset_refl} |
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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changeset
|
1747 |
val subset_trans = @{thm subset_trans} |
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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changeset
|
1748 |
val vimageD = @{thm vimageD} |
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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changeset
|
1749 |
val vimageE = @{thm vimageE} |
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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changeset
|
1750 |
val vimageI = @{thm vimageI} |
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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changeset
|
1751 |
val vimageI2 = @{thm vimageI2} |
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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changeset
|
1752 |
val vimage_Collect = @{thm vimage_Collect} |
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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changeset
|
1753 |
val vimage_Int = @{thm vimage_Int} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
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changeset
|
1754 |
val vimage_Un = @{thm vimage_Un} |
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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changeset
|
1755 |
*} |
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
|
1756 |
|
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
|
1757 |
end |