author  wenzelm 
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permissions  rwrr 
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(* Title: HOL/Product_Type.thy 
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 
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Copyright 1992 University of Cambridge 

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*) 
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section \<open>Cartesian products\<close> 
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theory Product_Type 
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imports Typedef Inductive Fun 
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keywords "inductive_set" "coinductive_set" :: thy_decl 
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begin 
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subsection \<open>@{typ bool} is a datatype\<close> 
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free_constructors case_bool for True  False 
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by auto 
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text \<open>Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}.\<close> 
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setup \<open>Sign.mandatory_path "old"\<close> 
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old_rep_datatype True False by (auto intro: bool_induct) 
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setup \<open>Sign.parent_path\<close> 
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text \<open>But erase the prefix for properties that are not generated by @{text free_constructors}.\<close> 
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setup \<open>Sign.mandatory_path "bool"\<close> 
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lemmas induct = old.bool.induct 
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lemmas inducts = old.bool.inducts 
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lemmas rec = old.bool.rec 
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lemmas simps = bool.distinct bool.case bool.rec 
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setup \<open>Sign.parent_path\<close> 
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declare case_split [cases type: bool] 
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 "prefer plain propositional version" 
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lemma 
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shows [code]: "HOL.equal False P \<longleftrightarrow> \<not> P" 
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and [code]: "HOL.equal True P \<longleftrightarrow> P" 
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and [code]: "HOL.equal P False \<longleftrightarrow> \<not> P" 
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and [code]: "HOL.equal P True \<longleftrightarrow> P" 
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and [code nbe]: "HOL.equal P P \<longleftrightarrow> True" 
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by (simp_all add: equal) 
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lemma If_case_cert: 
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assumes "CASE \<equiv> (\<lambda>b. If b f g)" 
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shows "(CASE True \<equiv> f) &&& (CASE False \<equiv> g)" 
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using assms by simp_all 
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setup \<open>Code.add_case @{thm If_case_cert}\<close> 
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code_printing 
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constant "HOL.equal :: bool \<Rightarrow> bool \<Rightarrow> bool" \<rightharpoonup> (Haskell) infix 4 "==" 
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 class_instance "bool" :: "equal" \<rightharpoonup> (Haskell)  
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Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
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subsection \<open>The @{text unit} type\<close> 
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typedef unit = "{True}" 
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by auto 
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definition Unity :: unit ("'(')") 
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where "() = Abs_unit True" 
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lemma unit_eq [no_atp]: "u = ()" 
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by (induct u) (simp add: Unity_def) 
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text \<open> 
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Simplification procedure for @{thm [source] unit_eq}. Cannot use 
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this rule directly  it loops! 

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\<close> 
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simproc_setup unit_eq ("x::unit") = \<open> 
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fn _ => fn _ => fn ct => 
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if HOLogic.is_unit (Thm.term_of ct) then NONE 
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else SOME (mk_meta_eq @{thm unit_eq}) 
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\<close> 
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free_constructors case_unit for "()" 
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by auto 
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text \<open>Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}.\<close> 
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setup \<open>Sign.mandatory_path "old"\<close> 
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old_rep_datatype "()" by simp 
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setup \<open>Sign.parent_path\<close> 
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text \<open>But erase the prefix for properties that are not generated by @{text free_constructors}.\<close> 
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setup \<open>Sign.mandatory_path "unit"\<close> 
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lemmas induct = old.unit.induct 
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lemmas inducts = old.unit.inducts 
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lemmas rec = old.unit.rec 
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lemmas simps = unit.case unit.rec 
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setup \<open>Sign.parent_path\<close> 
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lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()" 
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by simp 

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lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P" 

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by (rule triv_forall_equality) 

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text \<open> 
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This rewrite counters the effect of simproc @{text unit_eq} on @{term 
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[source] "%u::unit. f u"}, replacing it by @{term [source] 
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f} rather than by @{term [source] "%u. f ()"}. 

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\<close> 
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lemma unit_abs_eta_conv [simp]: "(%u::unit. f ()) = f" 
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by (rule ext) simp 
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lemma UNIV_unit: 
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"UNIV = {()}" by auto 
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instantiation unit :: default 
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begin 

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definition "default = ()" 

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instance .. 

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end 

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instantiation unit :: "{complete_boolean_algebra, complete_linorder, wellorder}" 
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begin 
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definition less_eq_unit :: "unit \<Rightarrow> unit \<Rightarrow> bool" 
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where 
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"(_::unit) \<le> _ \<longleftrightarrow> True" 
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lemma less_eq_unit [iff]: 
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"(u::unit) \<le> v" 
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by (simp add: less_eq_unit_def) 
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definition less_unit :: "unit \<Rightarrow> unit \<Rightarrow> bool" 
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where 
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"(_::unit) < _ \<longleftrightarrow> False" 
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lemma less_unit [iff]: 
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"\<not> (u::unit) < v" 
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by (simp_all add: less_eq_unit_def less_unit_def) 
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definition bot_unit :: unit 
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where 
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[code_unfold]: "\<bottom> = ()" 
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definition top_unit :: unit 
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where 
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[code_unfold]: "\<top> = ()" 
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definition inf_unit :: "unit \<Rightarrow> unit \<Rightarrow> unit" 
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where 
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[simp]: "_ \<sqinter> _ = ()" 
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definition sup_unit :: "unit \<Rightarrow> unit \<Rightarrow> unit" 
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where 
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[simp]: "_ \<squnion> _ = ()" 
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definition Inf_unit :: "unit set \<Rightarrow> unit" 
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where 
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[simp]: "\<Sqinter>_ = ()" 
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definition Sup_unit :: "unit set \<Rightarrow> unit" 
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where 
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[simp]: "\<Squnion>_ = ()" 
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definition uminus_unit :: "unit \<Rightarrow> unit" 
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where 
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[simp]: " _ = ()" 
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declare less_eq_unit_def [abs_def, code_unfold] 
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less_unit_def [abs_def, code_unfold] 
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inf_unit_def [abs_def, code_unfold] 
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sup_unit_def [abs_def, code_unfold] 
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Inf_unit_def [abs_def, code_unfold] 
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Sup_unit_def [abs_def, code_unfold] 
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uminus_unit_def [abs_def, code_unfold] 
57016  185 

186 
instance 

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187 
by intro_classes auto 
57016  188 

189 
end 

190 

28562  191 
lemma [code]: 
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"HOL.equal (u\<Colon>unit) v \<longleftrightarrow> True" unfolding equal unit_eq [of u] unit_eq [of v] by rule+ 
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193 

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194 
code_printing 
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195 
type_constructor unit \<rightharpoonup> 
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(SML) "unit" 
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and (OCaml) "unit" 
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and (Haskell) "()" 
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and (Scala) "Unit" 
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 constant Unity \<rightharpoonup> 
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(SML) "()" 
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and (OCaml) "()" 
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and (Haskell) "()" 
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and (Scala) "()" 
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 class_instance unit :: equal \<rightharpoonup> 
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(Haskell)  
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 constant "HOL.equal :: unit \<Rightarrow> unit \<Rightarrow> bool" \<rightharpoonup> 
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(Haskell) infix 4 "==" 
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209 

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code_reserved SML 
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211 
unit 
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212 

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213 
code_reserved OCaml 
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unit 
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215 

34886  216 
code_reserved Scala 
217 
Unit 

218 

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219 

60758  220 
subsection \<open>The product type\<close> 
10213  221 

60758  222 
subsubsection \<open>Type definition\<close> 
37166  223 

224 
definition Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" where 

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"Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)" 
10213  226 

45696  227 
definition "prod = {f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}" 
228 

49834  229 
typedef ('a, 'b) prod (infixr "*" 20) = "prod :: ('a \<Rightarrow> 'b \<Rightarrow> bool) set" 
45696  230 
unfolding prod_def by auto 
10213  231 

35427  232 
type_notation (xsymbols) 
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"prod" ("(_ \<times>/ _)" [21, 20] 20) 
35427  234 
type_notation (HTML output) 
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235 
"prod" ("(_ \<times>/ _)" [21, 20] 20) 
10213  236 

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definition Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" where 
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238 
"Pair a b = Abs_prod (Pair_Rep a b)" 
37166  239 

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240 
lemma prod_cases: "(\<And>a b. P (Pair a b)) \<Longrightarrow> P p" 
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241 
by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def) 
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242 

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243 
free_constructors case_prod for Pair fst snd 
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244 
proof  
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245 
fix P :: bool and p :: "'a \<times> 'b" 
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246 
show "(\<And>x1 x2. p = Pair x1 x2 \<Longrightarrow> P) \<Longrightarrow> P" 
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247 
by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def) 
37166  248 
next 
249 
fix a c :: 'a and b d :: 'b 

250 
have "Pair_Rep a b = Pair_Rep c d \<longleftrightarrow> a = c \<and> b = d" 

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251 
by (auto simp add: Pair_Rep_def fun_eq_iff) 
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252 
moreover have "Pair_Rep a b \<in> prod" and "Pair_Rep c d \<in> prod" 
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253 
by (auto simp add: prod_def) 
37166  254 
ultimately show "Pair a b = Pair c d \<longleftrightarrow> a = c \<and> b = d" 
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255 
by (simp add: Pair_def Abs_prod_inject) 
37166  256 
qed 
257 

60758  258 
text \<open>Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}.\<close> 
55442  259 

60758  260 
setup \<open>Sign.mandatory_path "old"\<close> 
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261 

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262 
old_rep_datatype Pair 
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263 
by (erule prod_cases) (rule prod.inject) 
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264 

60758  265 
setup \<open>Sign.parent_path\<close> 
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266 

60758  267 
text \<open>But erase the prefix for properties that are not generated by @{text free_constructors}.\<close> 
55442  268 

60758  269 
setup \<open>Sign.mandatory_path "prod"\<close> 
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270 

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271 
declare 
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272 
old.prod.inject[iff del] 
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273 

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274 
lemmas induct = old.prod.induct 
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lemmas inducts = old.prod.inducts 
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lemmas rec = old.prod.rec 
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277 
lemmas simps = prod.inject prod.case prod.rec 
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278 

60758  279 
setup \<open>Sign.parent_path\<close> 
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280 

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281 
declare prod.case [nitpick_simp del] 
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282 
declare prod.case_cong_weak [cong del] 
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283 

37166  284 

60758  285 
subsubsection \<open>Tuple syntax\<close> 
37166  286 

37591  287 
abbreviation (input) split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where 
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"split \<equiv> case_prod" 
19535  289 

60758  290 
text \<open> 
11777  291 
Patterns  extends predefined type @{typ pttrn} used in 
292 
abstractions. 

60758  293 
\<close> 
10213  294 

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295 
nonterminal tuple_args and patterns 
10213  296 

297 
syntax 

298 
"_tuple" :: "'a => tuple_args => 'a * 'b" ("(1'(_,/ _'))") 

299 
"_tuple_arg" :: "'a => tuple_args" ("_") 

300 
"_tuple_args" :: "'a => tuple_args => tuple_args" ("_,/ _") 

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"_pattern" :: "[pttrn, patterns] => pttrn" ("'(_,/ _')") 
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"" :: "pttrn => patterns" ("_") 
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303 
"_patterns" :: "[pttrn, patterns] => patterns" ("_,/ _") 
10213  304 

305 
translations 

35115  306 
"(x, y)" == "CONST Pair x y" 
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307 
"_pattern x y" => "CONST Pair x y" 
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308 
"_patterns x y" => "CONST Pair x y" 
10213  309 
"_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))" 
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"%(x, y, zs). b" == "CONST case_prod (%x (y, zs). b)" 
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311 
"%(x, y). b" == "CONST case_prod (%x y. b)" 
35115  312 
"_abs (CONST Pair x y) t" => "%(x, y). t" 
60758  313 
 \<open>The last rule accommodates tuples in `case C ... (x,y) ... => ...' 
314 
The (x,y) is parsed as `Pair x y' because it is logic, not pttrn\<close> 

10213  315 

35115  316 
(*reconstruct pattern from (nested) splits, avoiding etacontraction of body; 
317 
works best with enclosing "let", if "let" does not avoid etacontraction*) 

60758  318 
print_translation \<open> 
52143  319 
let 
320 
fun split_tr' [Abs (x, T, t as (Abs abs))] = 

321 
(* split (%x y. t) => %(x,y) t *) 

322 
let 

323 
val (y, t') = Syntax_Trans.atomic_abs_tr' abs; 

324 
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t'); 

325 
in 

326 
Syntax.const @{syntax_const "_abs"} $ 

327 
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' 

328 
end 

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329 
 split_tr' [Abs (x, T, (s as Const (@{const_syntax case_prod}, _) $ t))] = 
52143  330 
(* split (%x. (split (%y z. t))) => %(x,y,z). t *) 
331 
let 

332 
val Const (@{syntax_const "_abs"}, _) $ 

333 
(Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' = split_tr' [t]; 

334 
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t'); 

335 
in 

336 
Syntax.const @{syntax_const "_abs"} $ 

337 
(Syntax.const @{syntax_const "_pattern"} $ x' $ 

338 
(Syntax.const @{syntax_const "_patterns"} $ y $ z)) $ t'' 

339 
end 

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340 
 split_tr' [Const (@{const_syntax case_prod}, _) $ t] = 
52143  341 
(* split (split (%x y z. t)) => %((x, y), z). t *) 
342 
split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *) 

343 
 split_tr' [Const (@{syntax_const "_abs"}, _) $ x_y $ Abs abs] = 

344 
(* split (%pttrn z. t) => %(pttrn,z). t *) 

345 
let val (z, t) = Syntax_Trans.atomic_abs_tr' abs in 

346 
Syntax.const @{syntax_const "_abs"} $ 

347 
(Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t 

348 
end 

349 
 split_tr' _ = raise Match; 

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350 
in [(@{const_syntax case_prod}, K split_tr')] end 
60758  351 
\<close> 
14359  352 

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353 
(* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
60758  354 
typed_print_translation \<open> 
52143  355 
let 
356 
fun split_guess_names_tr' T [Abs (x, _, Abs _)] = raise Match 

357 
 split_guess_names_tr' T [Abs (x, xT, t)] = 

358 
(case (head_of t) of 

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359 
Const (@{const_syntax case_prod}, _) => raise Match 
52143  360 
 _ => 
361 
let 

362 
val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; 

363 
val (y, t') = Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0); 

364 
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, xT, t'); 

365 
in 

366 
Syntax.const @{syntax_const "_abs"} $ 

367 
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' 

368 
end) 

369 
 split_guess_names_tr' T [t] = 

370 
(case head_of t of 

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371 
Const (@{const_syntax case_prod}, _) => raise Match 
52143  372 
 _ => 
373 
let 

374 
val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; 

375 
val (y, t') = 

376 
Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0); 

377 
val (x', t'') = Syntax_Trans.atomic_abs_tr' ("x", xT, t'); 

378 
in 

379 
Syntax.const @{syntax_const "_abs"} $ 

380 
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' 

381 
end) 

382 
 split_guess_names_tr' _ _ = raise Match; 

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383 
in [(@{const_syntax case_prod}, K split_guess_names_tr')] end 
60758  384 
\<close> 
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385 

10213  386 

60758  387 
subsubsection \<open>Code generator setup\<close> 
37166  388 

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389 
code_printing 
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type_constructor prod \<rightharpoonup> 
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(SML) infix 2 "*" 
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392 
and (OCaml) infix 2 "*" 
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393 
and (Haskell) "!((_),/ (_))" 
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changeset

394 
and (Scala) "((_),/ (_))" 
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395 
 constant Pair \<rightharpoonup> 
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396 
(SML) "!((_),/ (_))" 
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397 
and (OCaml) "!((_),/ (_))" 
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diff
changeset

398 
and (Haskell) "!((_),/ (_))" 
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parents:
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399 
and (Scala) "!((_),/ (_))" 
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400 
 class_instance prod :: equal \<rightharpoonup> 
6646bb548c6b
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diff
changeset

401 
(Haskell)  
6646bb548c6b
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diff
changeset

402 
 constant "HOL.equal :: 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" \<rightharpoonup> 
6646bb548c6b
migration from code_(consttypeclassinstance) to code_printing and from code_module to code_identifier
haftmann
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52143
diff
changeset

403 
(Haskell) infix 4 "==" 
37166  404 

405 

60758  406 
subsubsection \<open>Fundamental operations and properties\<close> 
11838  407 

49897
cc69be3c8f87
moving Pair_inject from legacy and duplicate section to general section, as Pair_inject was considered a duplicate in e8400e31528a by mistake (cf. communication on dev mailing list)
bulwahn
parents:
49834
diff
changeset

408 
lemma Pair_inject: 
cc69be3c8f87
moving Pair_inject from legacy and duplicate section to general section, as Pair_inject was considered a duplicate in e8400e31528a by mistake (cf. communication on dev mailing list)
bulwahn
parents:
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diff
changeset

409 
assumes "(a, b) = (a', b')" 
cc69be3c8f87
moving Pair_inject from legacy and duplicate section to general section, as Pair_inject was considered a duplicate in e8400e31528a by mistake (cf. communication on dev mailing list)
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changeset

410 
and "a = a' ==> b = b' ==> R" 
cc69be3c8f87
moving Pair_inject from legacy and duplicate section to general section, as Pair_inject was considered a duplicate in e8400e31528a by mistake (cf. communication on dev mailing list)
bulwahn
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diff
changeset

411 
shows R 
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bulwahn
parents:
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diff
changeset

412 
using assms by simp 
cc69be3c8f87
moving Pair_inject from legacy and duplicate section to general section, as Pair_inject was considered a duplicate in e8400e31528a by mistake (cf. communication on dev mailing list)
bulwahn
parents:
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diff
changeset

413 

26358
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haftmann
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changeset

414 
lemma surj_pair [simp]: "EX x y. p = (x, y)" 
37166  415 
by (cases p) simp 
10213  416 

52435
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417 
code_printing 
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418 
constant fst \<rightharpoonup> (Haskell) "fst" 
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419 
 constant snd \<rightharpoonup> (Haskell) "snd" 
26358
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Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
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diff
changeset

420 

55414
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renamed '{prod,sum,bool,unit}_case' to 'case_...'
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421 
lemma case_prod_unfold [nitpick_unfold]: "case_prod = (%c p. c (fst p) (snd p))" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
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parents:
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diff
changeset

422 
by (simp add: fun_eq_iff split: prod.split) 
26358
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423 

11838  424 
lemma fst_eqD: "fst (x, y) = a ==> x = a" 
425 
by simp 

426 

427 
lemma snd_eqD: "snd (x, y) = a ==> y = a" 

428 
by simp 

429 

55393
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se 'wrap_free_constructors' to register 'sum' , 'prod', 'unit', 'bool' with their discriminators/selectors
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54630
diff
changeset

430 
lemmas surjective_pairing = prod.collapse [symmetric] 
11838  431 

44066
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432 
lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t" 
37166  433 
by (cases s, cases t) simp 
434 

435 
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q" 

44066
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huffman
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changeset

436 
by (simp add: prod_eq_iff) 
37166  437 

438 
lemma split_conv [simp, code]: "split f (a, b) = f a b" 

55642
63beb38e9258
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blanchet
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55469
diff
changeset

439 
by (fact prod.case) 
37166  440 

441 
lemma splitI: "f a b \<Longrightarrow> split f (a, b)" 

442 
by (rule split_conv [THEN iffD2]) 

443 

444 
lemma splitD: "split f (a, b) \<Longrightarrow> f a b" 

445 
by (rule split_conv [THEN iffD1]) 

446 

447 
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id" 

39302
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diff
changeset

448 
by (simp add: fun_eq_iff split: prod.split) 
37166  449 

450 
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f" 

60758  451 
 \<open>Subsumes the old @{text split_Pair} when @{term f} is the identity function.\<close> 
39302
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changeset

452 
by (simp add: fun_eq_iff split: prod.split) 
37166  453 

454 
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)" 

455 
by (cases x) simp 

456 

457 
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p" 

458 
by (cases p) simp 

459 

460 
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))" 

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diff
changeset

461 
by (simp add: case_prod_unfold) 
37166  462 

58468  463 
lemmas split_weak_cong = prod.case_cong_weak 
60758  464 
 \<open>Prevents simplification of @{term c}: much faster\<close> 
37166  465 

466 
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g" 

467 
by (simp add: split_eta) 

468 

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diff
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469 
lemma split_paired_all [no_atp]: "(!!x. PROP P x) == (!!a b. PROP P (a, b))" 
11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
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diff
changeset

470 
proof 
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proper proof of split_paired_all (presently unused);
wenzelm
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diff
changeset

471 
fix a b 
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
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diff
changeset

472 
assume "!!x. PROP P x" 
19535  473 
then show "PROP P (a, b)" . 
11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
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diff
changeset

474 
next 
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
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diff
changeset

475 
fix x 
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
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diff
changeset

476 
assume "!!a b. PROP P (a, b)" 
60758  477 
from \<open>PROP P (fst x, snd x)\<close> show "PROP P x" by simp 
11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
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diff
changeset

478 
qed 
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proper proof of split_paired_all (presently unused);
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479 

50104  480 
lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))" 
481 
by (cases x) simp 

482 

60758  483 
text \<open> 
11838  484 
The rule @{thm [source] split_paired_all} does not work with the 
485 
Simplifier because it also affects premises in congrence rules, 

486 
where this can lead to premises of the form @{text "!!a b. ... = 

487 
?P(a, b)"} which cannot be solved by reflexivity. 

60758  488 
\<close> 
11838  489 

26358
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Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
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changeset

490 
lemmas split_tupled_all = split_paired_all unit_all_eq2 
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Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
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changeset

491 

60758  492 
ML \<open> 
11838  493 
(* replace parameters of product type by individual component parameters *) 
494 
local (* filtering with exists_paired_all is an essential optimization *) 

56245  495 
fun exists_paired_all (Const (@{const_name Pure.all}, _) $ Abs (_, T, t)) = 
11838  496 
can HOLogic.dest_prodT T orelse exists_paired_all t 
497 
 exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u 

498 
 exists_paired_all (Abs (_, _, t)) = exists_paired_all t 

499 
 exists_paired_all _ = false; 

51717
9e7d1c139569
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diff
changeset

500 
val ss = 
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changeset

501 
simpset_of 
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changeset

502 
(put_simpset HOL_basic_ss @{context} 
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diff
changeset

503 
addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}] 
9e7d1c139569
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diff
changeset

504 
addsimprocs [@{simproc unit_eq}]); 
11838  505 
in 
51717
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diff
changeset

506 
fun split_all_tac ctxt = SUBGOAL (fn (t, i) => 
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diff
changeset

507 
if exists_paired_all t then safe_full_simp_tac (put_simpset ss ctxt) i else no_tac); 
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
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changeset

508 

9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
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changeset

509 
fun unsafe_split_all_tac ctxt = SUBGOAL (fn (t, i) => 
9e7d1c139569
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changeset

510 
if exists_paired_all t then full_simp_tac (put_simpset ss ctxt) i else no_tac); 
9e7d1c139569
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parents:
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changeset

511 

9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
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changeset

512 
fun split_all ctxt th = 
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changeset

513 
if exists_paired_all (Thm.prop_of th) 
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changeset

514 
then full_simplify (put_simpset ss ctxt) th else th; 
11838  515 
end; 
60758  516 
\<close> 
11838  517 

60758  518 
setup \<open>map_theory_claset (fn ctxt => ctxt addSbefore ("split_all_tac", split_all_tac))\<close> 
11838  519 

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diff
changeset

520 
lemma split_paired_All [simp, no_atp]: "(ALL x. P x) = (ALL a b. P (a, b))" 
60758  521 
 \<open>@{text "[iff]"} is not a good idea because it makes @{text blast} loop\<close> 
11838  522 
by fast 
523 

47740
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46950
diff
changeset

524 
lemma split_paired_Ex [simp, no_atp]: "(EX x. P x) = (EX a b. P (a, b))" 
26358
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Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

525 
by fast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
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26340
diff
changeset

526 

47740
a8989fe9a3a5
added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
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46950
diff
changeset

527 
lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))" 
60758  528 
 \<open>Can't be added to simpset: loops!\<close> 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

529 
by (simp add: split_eta) 
11838  530 

60758  531 
text \<open> 
11838  532 
Simplification procedure for @{thm [source] cond_split_eta}. Using 
533 
@{thm [source] split_eta} as a rewrite rule is not general enough, 

534 
and using @{thm [source] cond_split_eta} directly would render some 

535 
existing proofs very inefficient; similarly for @{text 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

536 
split_beta}. 
60758  537 
\<close> 
11838  538 

60758  539 
ML \<open> 
11838  540 
local 
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset

541 
val cond_split_eta_ss = 
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset

542 
simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms cond_split_eta}); 
35364  543 
fun Pair_pat k 0 (Bound m) = (m = k) 
544 
 Pair_pat k i (Const (@{const_name Pair}, _) $ Bound m $ t) = 

545 
i > 0 andalso m = k + i andalso Pair_pat k (i  1) t 

546 
 Pair_pat _ _ _ = false; 

547 
fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t 

548 
 no_args k i (t $ u) = no_args k i t andalso no_args k i u 

549 
 no_args k i (Bound m) = m < k orelse m > k + i 

550 
 no_args _ _ _ = true; 

551 
fun split_pat tp i (Abs (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE 

55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset

552 
 split_pat tp i (Const (@{const_name case_prod}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t 
35364  553 
 split_pat tp i _ = NONE; 
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset

554 
fun metaeq ctxt lhs rhs = mk_meta_eq (Goal.prove ctxt [] [] 
35364  555 
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))) 
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset

556 
(K (simp_tac (put_simpset cond_split_eta_ss ctxt) 1))); 
11838  557 

35364  558 
fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t 
559 
 beta_term_pat k i (t $ u) = 

560 
Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u) 

561 
 beta_term_pat k i t = no_args k i t; 

562 
fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg 

563 
 eta_term_pat _ _ _ = false; 

11838  564 
fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t) 
35364  565 
 subst arg k i (t $ u) = 
566 
if Pair_pat k i (t $ u) then incr_boundvars k arg 

567 
else (subst arg k i t $ subst arg k i u) 

568 
 subst arg k i t = t; 

43595  569 
in 
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset

570 
fun beta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t) $ arg) = 
11838  571 
(case split_pat beta_term_pat 1 t of 
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset

572 
SOME (i, f) => SOME (metaeq ctxt s (subst arg 0 i f)) 
15531  573 
 NONE => NONE) 
35364  574 
 beta_proc _ _ = NONE; 
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset

575 
fun eta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t)) = 
11838  576 
(case split_pat eta_term_pat 1 t of 
58468  577 
SOME (_, ft) => SOME (metaeq ctxt s (let val f $ _ = ft in f end)) 
15531  578 
 NONE => NONE) 
35364  579 
 eta_proc _ _ = NONE; 
11838  580 
end; 
60758  581 
\<close> 
59582  582 
simproc_setup split_beta ("split f z") = 
60758  583 
\<open>fn _ => fn ctxt => fn ct => beta_proc ctxt (Thm.term_of ct)\<close> 
59582  584 
simproc_setup split_eta ("split f") = 
60758  585 
\<open>fn _ => fn ctxt => fn ct => eta_proc ctxt (Thm.term_of ct)\<close> 
11838  586 

58468  587 
lemmas split_beta [mono] = prod.case_eq_if 
11838  588 

50104  589 
lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))" 
590 
by (auto simp: fun_eq_iff) 

591 

58468  592 
lemmas split_split [no_atp] = prod.split 
60758  593 
 \<open>For use with @{text split} and the Simplifier.\<close> 
11838  594 

60758  595 
text \<open> 
11838  596 
@{thm [source] split_split} could be declared as @{text "[split]"} 
597 
done after the Splitter has been speeded up significantly; 

598 
precompute the constants involved and don't do anything unless the 

599 
current goal contains one of those constants. 

60758  600 
\<close> 
11838  601 

58468  602 
lemmas split_split_asm [no_atp] = prod.split_asm 
11838  603 

60758  604 
text \<open> 
11838  605 
\medskip @{term split} used as a logical connective or set former. 
606 

607 
\medskip These rules are for use with @{text blast}; could instead 

60758  608 
call @{text simp} using @{thm [source] prod.split} as rewrite.\<close> 
11838  609 

610 
lemma splitI2: "!!p. [ !!a b. p = (a, b) ==> c a b ] ==> split c p" 

611 
apply (simp only: split_tupled_all) 

612 
apply (simp (no_asm_simp)) 

613 
done 

614 

615 
lemma splitI2': "!!p. [ !!a b. (a, b) = p ==> c a b x ] ==> split c p x" 

616 
apply (simp only: split_tupled_all) 

617 
apply (simp (no_asm_simp)) 

618 
done 

619 

620 
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" 

37591  621 
by (induct p) auto 
11838  622 

623 
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" 

37591  624 
by (induct p) auto 
11838  625 

626 
lemma splitE2: 

627 
"[ Q (split P z); !!x y. [z = (x, y); Q (P x y)] ==> R ] ==> R" 

628 
proof  

629 
assume q: "Q (split P z)" 

630 
assume r: "!!x y. [z = (x, y); Q (P x y)] ==> R" 

631 
show R 

632 
apply (rule r surjective_pairing)+ 

633 
apply (rule split_beta [THEN subst], rule q) 

634 
done 

635 
qed 

636 

637 
lemma splitD': "split R (a,b) c ==> R a b c" 

638 
by simp 

639 

640 
lemma mem_splitI: "z: c a b ==> z: split c (a, b)" 

641 
by simp 

642 

643 
lemma mem_splitI2: "!!p. [ !!a b. p = (a, b) ==> z: c a b ] ==> z: split c p" 

14208  644 
by (simp only: split_tupled_all, simp) 
11838  645 

18372  646 
lemma mem_splitE: 
58468  647 
assumes "z \<in> split c p" 
648 
obtains x y where "p = (x, y)" and "z \<in> c x y" 

649 
using assms by (rule splitE2) 

11838  650 

651 
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!] 

652 
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!] 

653 

60758  654 
ML \<open> 
11838  655 
local (* filtering with exists_p_split is an essential optimization *) 
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset

656 
fun exists_p_split (Const (@{const_name case_prod},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true 
11838  657 
 exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u 
658 
 exists_p_split (Abs (_, _, t)) = exists_p_split t 

659 
 exists_p_split _ = false; 

660 
in 

51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset

661 
fun split_conv_tac ctxt = SUBGOAL (fn (t, i) => 
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset

662 
if exists_p_split t 
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset

663 
then safe_full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms split_conv}) i 
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset

664 
else no_tac); 
11838  665 
end; 
60758  666 
\<close> 
26340  667 

11838  668 
(* This prevents applications of splitE for already splitted arguments leading 
669 
to quite timeconsuming computations (in particular for nested tuples) *) 

60758  670 
setup \<open>map_theory_claset (fn ctxt => ctxt addSbefore ("split_conv_tac", split_conv_tac))\<close> 
11838  671 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset

672 
lemma split_eta_SetCompr [simp, no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P" 
18372  673 
by (rule ext) fast 
11838  674 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset

675 
lemma split_eta_SetCompr2 [simp, no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P" 
18372  676 
by (rule ext) fast 
11838  677 

678 
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)" 

60758  679 
 \<open>Allows simplifications of nested splits in case of independent predicates.\<close> 
18372  680 
by (rule ext) blast 
11838  681 

14337
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

682 
(* Do NOT make this a simp rule as it 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

683 
a) only helps in special situations 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

684 
b) can lead to nontermination in the presence of split_def 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

685 
*) 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

686 
lemma split_comp_eq: 
20415  687 
fixes f :: "'a => 'b => 'c" and g :: "'d => 'a" 
688 
shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))" 

18372  689 
by (rule ext) auto 
14101  690 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

691 
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

692 
apply (rule_tac x = "(a, b)" in image_eqI) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

693 
apply auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

694 
done 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

695 

11838  696 
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)" 
697 
by blast 

698 

699 
(* 

700 
the following would be slightly more general, 

701 
but cannot be used as rewrite rule: 

702 
### Cannot add premise as rewrite rule because it contains (type) unknowns: 

703 
### ?y = .x 

704 
Goal "[ P y; !!x. P x ==> x = y ] ==> (@(x',y). x = x' & P y) = (x,y)" 

14208  705 
by (rtac some_equality 1) 
706 
by ( Simp_tac 1) 

707 
by (split_all_tac 1) 

708 
by (Asm_full_simp_tac 1) 

11838  709 
qed "The_split_eq"; 
710 
*) 

711 

60758  712 
text \<open> 
11838  713 
Setup of internal @{text split_rule}. 
60758  714 
\<close> 
11838  715 

55642
63beb38e9258
adapted to renaming of datatype 'cases' and 'recs' to 'case' and 'rec'
blanchet
parents:
55469
diff
changeset

716 
lemmas case_prodI = prod.case [THEN iffD2] 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

717 

55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset

718 
lemma case_prodI2: "!!p. [ !!a b. p = (a, b) ==> c a b ] ==> case_prod c p" 
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset

719 
by (fact splitI2) 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

720 

55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset

721 
lemma case_prodI2': "!!p. [ !!a b. (a, b) = p ==> c a b x ] ==> case_prod c p x" 
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset

722 
by (fact splitI2') 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

723 

55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset

724 
lemma case_prodE: "case_prod c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" 
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset

725 
by (fact splitE) 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

726 

55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset

727 
lemma case_prodE': "case_prod c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" 
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset

728 
by (fact splitE') 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

729 

55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset

730 
declare case_prodI [intro!] 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

731 

55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset

732 
lemma case_prod_beta: 
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset

733 
"case_prod f p = f (fst p) (snd p)" 
37591  734 
by (fact split_beta) 
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset

735 

55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
55414
diff
changeset

736 
lemma prod_cases3 [cases type]: 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

737 
obtains (fields) a b c where "y = (a, b, c)" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

738 
by (cases y, case_tac b) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

739 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

740 
lemma prod_induct3 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

741 
"(!!a b c. P (a, b, c)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

742 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

743 

55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
55414
diff
changeset

744 
lemma prod_cases4 [cases type]: 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

745 
obtains (fields) a b c d where "y = (a, b, c, d)" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

746 
by (cases y, case_tac c) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

747 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

748 
lemma prod_induct4 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

749 
"(!!a b c d. P (a, b, c, d)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

750 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

751 

55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
55414
diff
changeset

752 
lemma prod_cases5 [cases type]: 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

753 
obtains (fields) a b c d e where "y = (a, b, c, d, e)" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

754 
by (cases y, case_tac d) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

755 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

756 
lemma prod_induct5 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

757 
"(!!a b c d e. P (a, b, c, d, e)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

758 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

759 

55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
55414
diff
changeset

760 
lemma prod_cases6 [cases type]: 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

761 
obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

762 
by (cases y, case_tac e) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

763 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

764 
lemma prod_induct6 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

765 
"(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

766 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

767 

55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
55414
diff
changeset

768 
lemma prod_cases7 [cases type]: 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

769 
obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

770 
by (cases y, case_tac f) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

771 

c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

772 
lemma prod_induct7 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

773 
"(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

774 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

775 

37166  776 
lemma split_def: 
777 
"split = (\<lambda>c p. c (fst p) (snd p))" 

55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset

778 
by (fact case_prod_unfold) 
37166  779 

780 
definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where 

781 
"internal_split == split" 

782 

783 
lemma internal_split_conv: "internal_split c (a, b) = c a b" 

784 
by (simp only: internal_split_def split_conv) 

785 

48891  786 
ML_file "Tools/split_rule.ML" 
37166  787 

788 
hide_const internal_split 

789 

24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

790 

60758  791 
subsubsection \<open>Derived operations\<close> 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

792 

37387
3581483cca6c
qualified types "+" and nat; qualified constants Ball, Bex, Suc, curry; modernized some specifications
haftmann
parents:
37278
diff
changeset

793 
definition curry :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where 
3581483cca6c
qualified types "+" and nat; qualified constants Ball, Bex, Suc, curry; modernized some specifications
haftmann
parents:
37278
diff
changeset

794 
"curry = (\<lambda>c x y. c (x, y))" 
37166  795 

796 
lemma curry_conv [simp, code]: "curry f a b = f (a, b)" 

797 
by (simp add: curry_def) 

798 

799 
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b" 

800 
by (simp add: curry_def) 

801 

802 
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)" 

803 
by (simp add: curry_def) 

804 

805 
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q" 

806 
by (simp add: curry_def) 

807 

808 
lemma curry_split [simp]: "curry (split f) = f" 

809 
by (simp add: curry_def split_def) 

810 

811 
lemma split_curry [simp]: "split (curry f) = f" 

812 
by (simp add: curry_def split_def) 

813 

54630
9061af4d5ebc
restrict admissibility to nonempty chains to allow more syntaxdirected proof rules
Andreas Lochbihler
parents:
54147
diff
changeset

814 
lemma curry_K: "curry (\<lambda>x. c) = (\<lambda>x y. c)" 
9061af4d5ebc
restrict admissibility to nonempty chains to allow more syntaxdirected proof rules
Andreas Lochbihler
parents:
54147
diff
changeset

815 
by(simp add: fun_eq_iff) 
9061af4d5ebc
restrict admissibility to nonempty chains to allow more syntaxdirected proof rules
Andreas Lochbihler
parents:
54147
diff
changeset

816 

60758  817 
text \<open> 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

818 
The compositionuncurry combinator. 
60758  819 
\<close> 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

820 

37751  821 
notation fcomp (infixl "\<circ>>" 60) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

822 

37751  823 
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where 
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset

824 
"f \<circ>\<rightarrow> g = (\<lambda>x. case_prod g (f x))" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

825 

37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset

826 
lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))" 
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset

827 
by (simp add: fun_eq_iff scomp_def case_prod_unfold) 
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset

828 

55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset

829 
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = case_prod g (f x)" 
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset

830 
by (simp add: scomp_unfold case_prod_unfold) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

831 

37751  832 
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x" 
44921  833 
by (simp add: fun_eq_iff) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

834 

37751  835 
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x" 
44921  836 
by (simp add: fun_eq_iff) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

837 

37751  838 
lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39272
diff
changeset

839 
by (simp add: fun_eq_iff scomp_unfold) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

840 

37751  841 
lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39272
diff
changeset

842 
by (simp add: fun_eq_iff scomp_unfold fcomp_def) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

843 

37751  844 
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)" 
44921  845 
by (simp add: fun_eq_iff scomp_unfold) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

846 

52435
6646bb548c6b
migration from code_(consttypeclassinstance) to code_printing and from code_module to code_identifier
haftmann
parents:
52143
diff
changeset

847 
code_printing 
6646bb548c6b
migration from code_(consttypeclassinstance) to code_printing and from code_module to code_identifier
haftmann
parents:
52143
diff
changeset

848 
constant scomp \<rightharpoonup> (Eval) infixl 3 "#>" 
31202
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset

849 

37751  850 
no_notation fcomp (infixl "\<circ>>" 60) 
851 
no_notation scomp (infixl "\<circ>\<rightarrow>" 60) 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

852 

60758  853 
text \<open> 
55932  854 
@{term map_prod}  action of the product functor upon 
36664
6302f9ad7047
repaired comments where SOMEthing went utterly wrong (cf. 2b04504fcb69)
krauss
parents:
36622
diff
changeset

855 
functions. 
60758  856 
\<close> 
21195  857 

55932  858 
definition map_prod :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where 
859 
"map_prod f g = (\<lambda>(x, y). (f x, g y))" 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

860 

55932  861 
lemma map_prod_simp [simp, code]: 
862 
"map_prod f g (a, b) = (f a, g b)" 

863 
by (simp add: map_prod_def) 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

864 

55932  865 
functor map_prod: map_prod 
44921  866 
by (auto simp add: split_paired_all) 
37278  867 

55932  868 
lemma fst_map_prod [simp]: 
869 
"fst (map_prod f g x) = f (fst x)" 

40607  870 
by (cases x) simp_all 
37278  871 

58916  872 
lemma snd_map_prod [simp]: 
55932  873 
"snd (map_prod f g x) = g (snd x)" 
40607  874 
by (cases x) simp_all 
37278  875 

55932  876 
lemma fst_comp_map_prod [simp]: 
877 
"fst \<circ> map_prod f g = f \<circ> fst" 

40607  878 
by (rule ext) simp_all 
37278  879 

55932  880 
lemma snd_comp_map_prod [simp]: 
881 
"snd \<circ> map_prod f g = g \<circ> snd" 

40607  882 
by (rule ext) simp_all 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

883 

55932  884 
lemma map_prod_compose: 
885 
"map_prod (f1 o f2) (g1 o g2) = (map_prod f1 g1 o map_prod f2 g2)" 

886 
by (rule ext) (simp add: map_prod.compositionality comp_def) 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

887 

55932  888 
lemma map_prod_ident [simp]: 
889 
"map_prod (%x. x) (%y. y) = (%z. z)" 

890 
by (rule ext) (simp add: map_prod.identity) 

40607  891 

55932  892 
lemma map_prod_imageI [intro]: 
893 
"(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_prod f g ` R" 

40607  894 
by (rule image_eqI) simp_all 
21195  895 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

896 
lemma prod_fun_imageE [elim!]: 
55932  897 
assumes major: "c \<in> map_prod f g ` R" 
40607  898 
and cases: "\<And>x y. c = (f x, g y) \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> P" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

899 
shows P 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

900 
apply (rule major [THEN imageE]) 
37166  901 
apply (case_tac x) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

902 
apply (rule cases) 
40607  903 
apply simp_all 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

904 
done 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

905 

37166  906 
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where 
55932  907 
"apfst f = map_prod f id" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

908 

37166  909 
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where 
55932  910 
"apsnd f = map_prod id f" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

911 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

912 
lemma apfst_conv [simp, code]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

913 
"apfst f (x, y) = (f x, y)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

914 
by (simp add: apfst_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

915 

33638
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset

916 
lemma apsnd_conv [simp, code]: 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

917 
"apsnd f (x, y) = (x, f y)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

918 
by (simp add: apsnd_def) 
21195  919 

33594  920 
lemma fst_apfst [simp]: 
921 
"fst (apfst f x) = f (fst x)" 

922 
by (cases x) simp 

923 

51173  924 
lemma fst_comp_apfst [simp]: 
925 
"fst \<circ> apfst f = f \<circ> fst" 

926 
by (simp add: fun_eq_iff) 

927 

33594  928 
lemma fst_apsnd [simp]: 
929 
"fst (apsnd f x) = fst x" 

930 
by (cases x) simp 

931 

51173  932 
lemma fst_comp_apsnd [simp]: 
933 
"fst \<circ> apsnd f = fst" 

934 
by (simp add: fun_eq_iff) 

935 

33594  936 
lemma snd_apfst [simp]: 
937 
"snd (apfst f x) = snd x" 

938 
by (cases x) simp 

939 

51173  940 
lemma snd_comp_apfst [simp]: 
941 
"snd \<circ> apfst f = snd" 

942 
by (simp add: fun_eq_iff) 

943 

33594  944 
lemma snd_apsnd [simp]: 
945 
"snd (apsnd f x) = f (snd x)" 

946 
by (cases x) simp 

947 

51173  948 
lemma snd_comp_apsnd [simp]: 
949 
"snd \<circ> apsnd f = f \<circ> snd" 

950 
by (simp add: fun_eq_iff) 

951 

33594  952 
lemma apfst_compose: 
953 
"apfst f (apfst g x) = apfst (f \<circ> g) x" 

954 
by (cases x) simp 

955 

956 
lemma apsnd_compose: 

957 
"apsnd f (apsnd g x) = apsnd (f \<circ> g) x" 

958 
by (cases x) simp 

959 

960 
lemma apfst_apsnd [simp]: 

961 
"apfst f (apsnd g x) = (f (fst x), g (snd x))" 

962 
by (cases x) simp 

963 

964 
lemma apsnd_apfst [simp]: 

965 
"apsnd f (apfst g x) = (g (fst x), f (snd x))" 

966 
by (cases x) simp 

967 

968 
lemma apfst_id [simp] : 

969 
"apfst id = id" 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39272
diff
changeset

970 
by (simp add: fun_eq_iff) 
33594  971 

972 
lemma apsnd_id [simp] : 

973 
"apsnd id = id" 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39272
diff
changeset

974 
by (simp add: fun_eq_iff) 
33594  975 

976 
lemma apfst_eq_conv [simp]: 

977 
"apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)" 

978 
by (cases x) simp 

979 

980 
lemma apsnd_eq_conv [simp]: 

981 
"apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)" 

982 
by (cases x) simp 

983 

33638
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset

984 
lemma apsnd_apfst_commute: 
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset

985 
"apsnd f (apfst g p) = apfst g (apsnd f p)" 
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset

986 
by simp 
21195  987 

56626  988 
context 
989 
begin 

990 

60758  991 
local_setup \<open>Local_Theory.map_background_naming (Name_Space.mandatory_path "prod")\<close> 
56626  992 

56545  993 
definition swap :: "'a \<times> 'b \<Rightarrow> 'b \<times> 'a" 
994 
where 

995 
"swap p = (snd p, fst p)" 

996 

56626  997 
end 
998 

56545  999 
lemma swap_simp [simp]: 
56626  1000 
"prod.swap (x, y) = (y, x)" 
1001 
by (simp add: prod.swap_def) 

56545  1002 

58195  1003 
lemma swap_swap [simp]: 
1004 
"prod.swap (prod.swap p) = p" 

1005 
by (cases p) simp 

1006 

1007 
lemma swap_comp_swap [simp]: 

1008 
"prod.swap \<circ> prod.swap = id" 

1009 
by (simp add: fun_eq_iff) 

1010 

56545  1011 
lemma pair_in_swap_image [simp]: 
56626  1012 
"(y, x) \<in> prod.swap ` A \<longleftrightarrow> (x, y) \<in> A" 
56545  1013 
by (auto intro!: image_eqI) 
1014 

1015 
lemma inj_swap [simp]: 

56626  1016 
"inj_on prod.swap A" 
1017 
by (rule inj_onI) auto 

1018 

1019 
lemma swap_inj_on: 

1020 
"inj_on (\<lambda>(i, j). (j, i)) A" 

1021 
by (rule inj_onI) auto 

56545  1022 

58195  1023 
lemma surj_swap [simp]: 
1024 
"surj prod.swap" 

1025 
by (rule surjI [of _ prod.swap]) simp 

1026 

1027 
lemma bij_swap [simp]: 

1028 
"bij prod.swap" 

1029 
by (simp add: bij_def) 

1030 

56545  1031 
lemma case_swap [simp]: 
56626  1032 
"(case prod.swap p of (y, x) \<Rightarrow> f x y) = (case p of (x, y) \<Rightarrow> f x y)" 
56545  1033 
by (cases p) simp 
1034 

60758  1035 
text \<open> 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1036 
Disjoint union of a family of sets  Sigma. 
60758  1037 
\<close> 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1038 

45986
c9e50153e5ae
moved various set operations to theory Set (resp. Product_Type)
haftmann
parents:
45696
diff
changeset

1039 
definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" where 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1040 
Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1041 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1042 
abbreviation 
45986
c9e50153e5ae
moved various set operations to theory Set (resp. Product_Type)
haftmann
parents:
45696
diff
changeset

1043 
Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1044 
(infixr "<*>" 80) where 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1045 
"A <*> B == Sigma A (%_. B)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1046 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1047 
notation (xsymbols) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1048 
Times (infixr "\<times>" 80) 
15394  1049 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1050 
notation (HTML output) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1051 
Times (infixr "\<times>" 80) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1052 

45662
4f7c05990420
Hide Product_Type.Times  too precious an identifier
nipkow
parents:
45607
diff
changeset

1053 
hide_const (open) Times 
4f7c05990420
Hide Product_Type.Times  too precious an identifier
nipkow
parents:
45607
diff
changeset

1054 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1055 
syntax 
35115  1056 
"_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1057 
translations 
35115  1058 
"SIGMA x:A. B" == "CONST Sigma A (%x. B)" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1059 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1060 
lemma SigmaI [intro!]: "[ a:A; b:B(a) ] ==> (a,b) : Sigma A B" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1061 
by (unfold Sigma_def) blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1062 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1063 
lemma SigmaE [elim!]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1064 
"[ c: Sigma A B; 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1065 
!!x y.[ x:A; y:B(x); c=(x,y) ] ==> P 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1066 
] ==> P" 
60758  1067 
 \<open>The general elimination rule.\<close> 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1068 
by (unfold Sigma_def) blast 
20588  1069 

60758  1070 
text \<open> 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1071 
Elimination of @{term "(a, b) : A \<times> B"}  introduces no 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1072 
eigenvariables. 
60758  1073 
\<close> 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1074 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1075 
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1076 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1077 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1078 
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1079 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1080 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1081 
lemma SigmaE2: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1082 
"[ (a, b) : Sigma A B; 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1083 
[ a:A; b:B(a) ] ==> P 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1084 
] ==> P" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1085 
by blast 
20588  1086 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1087 
lemma Sigma_cong: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1088 
"\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk> 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1089 
\<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1090 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1091 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1092 
lemma Sigma_mono: "[ A <= C; !!x. x:A ==> B x <= D x ] ==> Sigma A B <= Sigma C D" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1093 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1094 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1095 
lemma Sigma_empty1 [simp]: "Sigma {} B = {}" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1096 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1097 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1098 
lemma Sigma_empty2 [simp]: "A <*> {} = {}" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1099 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1100 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1101 
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1102 
by auto 
21908  1103 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1104 
lemma Compl_Times_UNIV1 [simp]: " (UNIV <*> A) = UNIV <*> (A)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1105 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1106 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1107 
lemma Compl_Times_UNIV2 [simp]: " (A <*> UNIV) = (A) <*> UNIV" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1108 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1109 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1110 
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1111 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1112 

59000  1113 
lemma Sigma_empty_iff: "(SIGMA i:I. X i) = {} \<longleftrightarrow> (\<forall>i\<in>I. X i = {})" 
1114 
by auto 

1115 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1116 
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1117 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1118 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1119 
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1120 
by (blast elim: equalityE) 
20588  1121 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1122 
lemma SetCompr_Sigma_eq: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1123 
"Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1124 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1125 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1126 
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1127 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1128 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1129 
lemma UN_Times_distrib: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1130 
"(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)" 
60758  1131 
 \<open>Suggested by Pierre Chartier\<close> 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1132 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1133 

47740
a8989fe9a3a5
added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents:
46950
diff
changeset

1134 
lemma split_paired_Ball_Sigma [simp, no_atp]: 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1135 
"(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1136 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1137 

47740
a8989fe9a3a5
added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents:
46950
diff
changeset

1138 
lemma split_paired_Bex_Sigma [simp, no_atp]: 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1139 
"(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1140 
by blast 
21908  1141 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1142 
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1143 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1144 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1145 
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1146 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1147 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1148 
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1149 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1150 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1151 
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1152 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1153 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1154 
lemma Sigma_Diff_distrib1: "(SIGMA i:I  J. C(i)) = (SIGMA i:I. C(i))  (SIGMA j:J. C(j))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1155 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1156 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1157 
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i)  B(i)) = (SIGMA i:I. A(i))  (SIGMA i:I. B(i))" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1158 
by blast 
21908  1159 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1160 
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1161 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1162 

60758  1163 
text \<open> 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1164 
Nondependent versions are needed to avoid the need for higherorder 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1165 
matching, especially when the rules are reoriented. 
60758  1166 
\<close> 
21908  1167 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1168 
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)" 
56545  1169 
by (fact Sigma_Un_distrib1) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1170 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1171 
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)" 
56545  1172 
by (fact Sigma_Int_distrib1) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1173 

d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1174 
lemma Times_Diff_distrib1: "(A  B) <*> C = (A <*> C)  (B <*> C)" 
56545  1175 
by (fact Sigma_Diff_distrib1) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1176 

36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset

1177 
lemma Times_empty[simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}" 
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset

1178 
by auto 
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset

1179 

50104  1180 
lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> ((A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))" 
1181 
by auto 

1182 

36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset

1183 
lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)" 
44921  1184 
by force 
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset

1185 

e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset

1186 
lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)" 
44921  1187 
by force 
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset

1188 

56545  1189 
lemma vimage_fst: 
1190 
"fst ` A = A \<times> UNIV" 

1191 
by auto 

1192 

1193 
lemma vimage_snd: 

1194 
"snd ` A = UNIV \<times> A" 

1195 
by auto 

1196 

28719  1197 
lemma insert_times_insert[simp]: 
1198 
"insert a A \<times> insert b B = 

1199 
insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)" 

1200 
by blast 

26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

1201 

33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33089
diff
changeset

1202 
lemma vimage_Times: "f ` (A \<times> B) = ((fst \<circ> f) ` A) \<inter> ((snd \<circ> f) ` B)" 
47988  1203 
apply auto 
1204 
apply (case_tac "f x") 

1205 
apply auto 

1206 
done 

33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33089
diff
changeset

1207 

50104  1208 
lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)" 
1209 
by auto 

1210 

56626  1211 
lemma product_swap: 
1212 
"prod.swap ` (A \<times> B) = B \<times> A" 

1213 
by (auto simp add: set_eq_iff) 

35822  1214 

1215 
lemma swap_product: 

56626  1216 
"(\<lambda>(i, j). (j, i)) ` (A \<times> B) = B \<times> A" 
1217 
by (auto simp add: set_eq_iff) 

35822  1218 

36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset

1219 
lemma image_split_eq_Sigma: 
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset

1220 
"(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f ` {x} \<inter> A))" 
46128
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset

1221 
proof (safe intro!: imageI) 
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset

1222 
fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b" 
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset

1223 
show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A" 
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset

1224 
using * eq[symmetric] by auto 
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset

1225 
qed simp_all 
35822  1226 

60057  1227 
lemma inj_on_apfst [simp]: "inj_on (apfst f) (A \<times> UNIV) \<longleftrightarrow> inj_on f A" 
1228 
by(auto simp add: inj_on_def) 

1229 

1230 
lemma inj_apfst [simp]: "inj (apfst f) \<longleftrightarrow> inj f" 

1231 
using inj_on_apfst[of f UNIV] by simp 

1232 

1233 
lemma inj_on_apsnd [simp]: "inj_on (apsnd f) (UNIV \<times> A) \<longleftrightarrow> inj_on f A" 

1234 
by(auto simp add: inj_on_def) 

1235 

1236 
lemma inj_apsnd [simp]: "inj (apsnd f) \<longleftrightarrow> inj f" 

1237 
using inj_on_apsnd[of f UNIV] by simp 

1238 

46128
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset

1239 
definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where 
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset

1240 
[code_abbrev]: "product A B = A \<times> B" 
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset

1241 

53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset

1242 
hide_const (open) product 
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1243 

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1244 
lemma member_product: 
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1245 
"x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B" 
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1246 
by (simp add: product_def) 
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1247 

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text \<open>The following @{const map_prod} lemmas are due to Joachim Breitner:\<close> 
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lemma map_prod_inj_on: 
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assumes "inj_on f A" and "inj_on g B" 
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shows "inj_on (map_prod f g) (A \<times> B)" 
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proof (rule inj_onI) 
1254 
fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c" 

1255 
assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto 
