author  haftmann 
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parent 33271  7be66dee1a5a 
child 33594  357f74e0090c 
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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header {* Cartesian products *} 
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theory Product_Type 
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imports Inductive 
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uses 
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("Tools/split_rule.ML") 
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("Tools/inductive_set.ML") 
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("Tools/inductive_realizer.ML") 
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("Tools/Datatype/datatype_realizer.ML") 
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begin 
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subsection {* @{typ bool} is a datatype *} 
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rep_datatype True False by (auto intro: bool_induct) 
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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]: "eq_class.eq False P \<longleftrightarrow> \<not> P" 
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and [code]: "eq_class.eq True P \<longleftrightarrow> P" 

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and [code]: "eq_class.eq P False \<longleftrightarrow> \<not> P" 

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and [code]: "eq_class.eq P True \<longleftrightarrow> P" 

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and [code nbe]: "eq_class.eq P P \<longleftrightarrow> True" 
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by (simp_all add: eq) 
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code_const "eq_class.eq \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool" 
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(Haskell infixl 4 "==") 
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code_instance bool :: eq 
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(Haskell ) 
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subsection {* Unit *} 
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typedef unit = "{True}" 

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proof 

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show "True : ?unit" .. 
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qed 
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definition 
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Unity :: unit ("'(')") 
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where 
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"() = Abs_unit True" 
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lemma unit_eq [noatp]: "u = ()" 
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by (induct u) (simp add: unit_def Unity_def) 
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text {* 

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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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*} 

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ML {* 
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val unit_eq_proc = 
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let val unit_meta_eq = mk_meta_eq @{thm unit_eq} in 
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Simplifier.simproc @{theory} "unit_eq" ["x::unit"] 
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(fn _ => fn _ => fn t => if HOLogic.is_unit t then NONE else SOME unit_meta_eq) 
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end; 
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Addsimprocs [unit_eq_proc]; 

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*} 

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rep_datatype "()" by simp 
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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 {* 

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This rewrite counters the effect of @{text unit_eq_proc} 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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*} 

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lemma unit_abs_eta_conv [simp,noatp]: "(%u::unit. f ()) = f" 
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by (rule ext) simp 
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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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text {* code generator setup *} 
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lemma [code]: 
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"eq_class.eq (u\<Colon>unit) v \<longleftrightarrow> True" unfolding eq unit_eq [of u] unit_eq [of v] by rule+ 
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code_type unit 
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(SML "unit") 
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(OCaml "unit") 
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(Haskell "()") 
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code_instance unit :: eq 
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(Haskell ) 
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code_const "eq_class.eq \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool" 
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(Haskell infixl 4 "==") 
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code_const Unity 
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(SML "()") 
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(OCaml "()") 
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(Haskell "()") 
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code_reserved SML 
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unit 
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code_reserved OCaml 
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unit 
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subsection {* Pairs *} 
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subsubsection {* Product type, basic operations and concrete syntax *} 
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definition 
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Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" 
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where 
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"Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)" 
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global 

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typedef (Prod) 

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('a, 'b) "*" (infixr "*" 20) 
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= "{f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}" 
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proof 
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fix a b show "Pair_Rep a b \<in> ?Prod" 
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by rule+ 
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qed 
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syntax (xsymbols) 
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"*" :: "[type, type] => type" ("(_ \<times>/ _)" [21, 20] 20) 
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syntax (HTML output) 
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"*" :: "[type, type] => type" ("(_ \<times>/ _)" [21, 20] 20) 
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consts 

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Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" 
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fst :: "'a \<times> 'b \<Rightarrow> 'a" 
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snd :: "'a \<times> 'b \<Rightarrow> 'b" 
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split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" 
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curry :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" 
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local 
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19535  156 
defs 
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Pair_def: "Pair a b == Abs_Prod (Pair_Rep a b)" 

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fst_def: "fst p == THE a. EX b. p = Pair a b" 

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snd_def: "snd p == THE b. EX a. p = Pair a b" 

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split_def: "split == (%c p. c (fst p) (snd p))" 
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curry_def: "curry == (%c x y. c (Pair x y))" 
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text {* 
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Patterns  extends predefined type @{typ pttrn} used in 

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abstractions. 

166 
*} 

10213  167 

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nonterminals 

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tuple_args patterns 

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syntax 

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

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

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"_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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"_patterns" :: "[pttrn, patterns] => patterns" ("_,/ _") 
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179 
translations 

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"(x, y)" == "Pair x y" 

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"_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))" 

182 
"%(x,y,zs).b" == "split(%x (y,zs).b)" 

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"%(x,y).b" == "split(%x y. b)" 

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"_abs (Pair x y) t" => "%(x,y).t" 

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(* The last rule accommodates tuples in `case C ... (x,y) ... => ...' 

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The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *) 

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(* reconstructs pattern from (nested) splits, avoiding etacontraction of body*) 
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(* works best with enclosing "let", if "let" does not avoid etacontraction *) 

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print_translation {* 

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let fun split_tr' [Abs (x,T,t as (Abs abs))] = 

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

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let val (y,t') = atomic_abs_tr' abs; 

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val (x',t'') = atomic_abs_tr' (x,T,t'); 

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in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end 

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 split_tr' [Abs (x,T,(s as Const ("split",_)$t))] = 

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

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let val (Const ("_abs",_)$(Const ("_pattern",_)$y$z)$t') = split_tr' [t]; 

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val (x',t'') = atomic_abs_tr' (x,T,t'); 

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in Syntax.const "_abs"$ 

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(Syntax.const "_pattern"$x'$(Syntax.const "_patterns"$y$z))$t'' end 

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 split_tr' [Const ("split",_)$t] = 

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

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split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *) 

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 split_tr' [Const ("_abs",_)$x_y$(Abs abs)] = 

207 
(* split (%pttrn z. t) => %(pttrn,z). t *) 

208 
let val (z,t) = atomic_abs_tr' abs; 

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in Syntax.const "_abs" $ (Syntax.const "_pattern" $x_y$z) $ t end 

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 split_tr' _ = raise Match; 

211 
in [("split", split_tr')] 

212 
end 

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*} 

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(* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
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typed_print_translation {* 
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let 
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fun split_guess_names_tr' _ T [Abs (x,_,Abs _)] = raise Match 
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 split_guess_names_tr' _ T [Abs (x,xT,t)] = 
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(case (head_of t) of 
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Const ("split",_) => raise Match 
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 _ => let 
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val (_::yT::_) = binder_types (domain_type T) handle Bind => raise Match; 
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val (y,t') = atomic_abs_tr' ("y",yT,(incr_boundvars 1 t)$Bound 0); 
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val (x',t'') = atomic_abs_tr' (x,xT,t'); 
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in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end) 
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 split_guess_names_tr' _ T [t] = 
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(case (head_of t) of 
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Const ("split",_) => raise Match 
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 _ => let 
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val (xT::yT::_) = binder_types (domain_type T) handle Bind => raise Match; 
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val (y,t') = 
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atomic_abs_tr' ("y",yT,(incr_boundvars 2 t)$Bound 1$Bound 0); 
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val (x',t'') = atomic_abs_tr' ("x",xT,t'); 
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in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end) 
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 split_guess_names_tr' _ _ _ = raise Match; 
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in [("split", split_guess_names_tr')] 
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end 
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*} 
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10213  241 

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text {* Towards a datatype declaration *} 
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lemma surj_pair [simp]: "EX x y. p = (x, y)" 
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apply (unfold Pair_def) 
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apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE]) 
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apply (erule exE, erule exE, rule exI, rule exI) 
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apply (rule Rep_Prod_inverse [symmetric, THEN trans]) 
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apply (erule arg_cong) 
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done 
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lemma PairE [cases type: *]: 
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obtains x y where "p = (x, y)" 
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using surj_pair [of p] by blast 
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lemma ProdI: "Pair_Rep a b \<in> Prod" 
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unfolding Prod_def by rule+ 
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lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' \<Longrightarrow> a = a' \<and> b = b'" 
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unfolding Pair_Rep_def by (drule fun_cong, drule fun_cong) blast 
10213  261 

11838  262 
lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod" 
263 
apply (rule inj_on_inverseI) 

264 
apply (erule Abs_Prod_inverse) 

265 
done 

266 

267 
lemma Pair_inject: 

18372  268 
assumes "(a, b) = (a', b')" 
269 
and "a = a' ==> b = b' ==> R" 

270 
shows R 

271 
apply (insert prems [unfolded Pair_def]) 

272 
apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE]) 

273 
apply (assumption  rule ProdI)+ 

274 
done 

10213  275 

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rep_datatype (prod) Pair 
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proof  
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fix P p 
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assume "\<And>x y. P (x, y)" 
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then show "P p" by (cases p) simp 
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qed (auto elim: Pair_inject) 
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lemmas Pair_eq = prod.inject 
11838  284 

22886  285 
lemma fst_conv [simp, code]: "fst (a, b) = a" 
19535  286 
unfolding fst_def by blast 
11838  287 

22886  288 
lemma snd_conv [simp, code]: "snd (a, b) = b" 
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subsubsection {* Basic rules and proof tools *} 
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11838  294 
lemma fst_eqD: "fst (x, y) = a ==> x = a" 
295 
by simp 

296 

297 
lemma snd_eqD: "snd (x, y) = a ==> y = a" 

298 
by simp 

299 

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lemma pair_collapse [simp]: "(fst p, snd p) = p" 
11838  301 
by (cases p) simp 
302 

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lemmas surjective_pairing = pair_collapse [symmetric] 
11838  304 

305 
lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))" 

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proof 
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fix a b 
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assume "!!x. PROP P x" 
19535  309 
then show "PROP P (a, b)" . 
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next 
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fix x 
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assume "!!a b. PROP P (a, b)" 
19535  313 
from `PROP P (fst x, snd x)` show "PROP P x" by simp 
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qed 
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11838  316 
text {* 
317 
The rule @{thm [source] split_paired_all} does not work with the 

318 
Simplifier because it also affects premises in congrence rules, 

319 
where this can lead to premises of the form @{text "!!a b. ... = 

320 
?P(a, b)"} which cannot be solved by reflexivity. 

321 
*} 

322 

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lemmas split_tupled_all = split_paired_all unit_all_eq2 
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26480  325 
ML {* 
11838  326 
(* replace parameters of product type by individual component parameters *) 
327 
val safe_full_simp_tac = generic_simp_tac true (true, false, false); 

328 
local (* filtering with exists_paired_all is an essential optimization *) 

16121  329 
fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) = 
11838  330 
can HOLogic.dest_prodT T orelse exists_paired_all t 
331 
 exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u 

332 
 exists_paired_all (Abs (_, _, t)) = exists_paired_all t 

333 
 exists_paired_all _ = false; 

334 
val ss = HOL_basic_ss 

26340  335 
addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}] 
11838  336 
addsimprocs [unit_eq_proc]; 
337 
in 

338 
val split_all_tac = SUBGOAL (fn (t, i) => 

339 
if exists_paired_all t then safe_full_simp_tac ss i else no_tac); 

340 
val unsafe_split_all_tac = SUBGOAL (fn (t, i) => 

341 
if exists_paired_all t then full_simp_tac ss i else no_tac); 

342 
fun split_all th = 

26340  343 
if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th; 
11838  344 
end; 
26340  345 
*} 
11838  346 

26340  347 
declaration {* fn _ => 
348 
Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac)) 

16121  349 
*} 
11838  350 

351 
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))" 

352 
 {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *} 

353 
by fast 

354 

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lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))" 
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by fast 
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lemma Pair_fst_snd_eq: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t" 
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by (cases s, cases t) simp 
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lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q" 
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by (simp add: Pair_fst_snd_eq) 
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subsubsection {* @{text split} and @{text curry} *} 
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28562  367 
lemma split_conv [simp, code]: "split f (a, b) = f a b" 
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by (simp add: split_def) 
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28562  370 
lemma curry_conv [simp, code]: "curry f a b = f (a, b)" 
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by (simp add: curry_def) 
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lemmas split = split_conv  {* for backwards compatibility *} 
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lemma splitI: "f a b \<Longrightarrow> split f (a, b)" 
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by (rule split_conv [THEN iffD2]) 
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lemma splitD: "split f (a, b) \<Longrightarrow> f a b" 
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by (rule split_conv [THEN iffD1]) 
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lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b" 
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by (simp add: curry_def) 
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383 

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lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)" 
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by (simp add: curry_def) 
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lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q" 
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by (simp add: curry_def) 
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14189  390 
lemma curry_split [simp]: "curry (split f) = f" 
391 
by (simp add: curry_def split_def) 

392 

393 
lemma split_curry [simp]: "split (curry f) = f" 

394 
by (simp add: curry_def split_def) 

395 

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lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id" 
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by (simp add: split_def id_def) 
11838  398 

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lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f" 
31775  400 
 {* Subsumes the old @{text split_Pair} when @{term f} is the identity Datatype. *} 
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by (rule ext) auto 
11838  402 

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lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)" 
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by (cases x) simp 
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lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p" 
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unfolding split_def .. 
11838  408 

409 
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))" 

410 
 {* Can't be added to simpset: loops! *} 

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by (simp add: split_eta) 
11838  412 

413 
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))" 

414 
by (simp add: split_def) 

415 

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lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q" 
11838  417 
 {* Prevents simplification of @{term c}: much faster *} 
418 
by (erule arg_cong) 

419 

420 
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g" 

421 
by (simp add: split_eta) 

422 

423 
text {* 

424 
Simplification procedure for @{thm [source] cond_split_eta}. Using 

425 
@{thm [source] split_eta} as a rewrite rule is not general enough, 

426 
and using @{thm [source] cond_split_eta} directly would render some 

427 
existing proofs very inefficient; similarly for @{text 

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split_beta}. 
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429 
*} 
11838  430 

26480  431 
ML {* 
11838  432 

433 
local 

18328  434 
val cond_split_eta_ss = HOL_basic_ss addsimps [thm "cond_split_eta"] 
11838  435 
fun Pair_pat k 0 (Bound m) = (m = k) 
436 
 Pair_pat k i (Const ("Pair", _) $ Bound m $ t) = i > 0 andalso 

437 
m = k+i andalso Pair_pat k (i1) t 

438 
 Pair_pat _ _ _ = false; 

439 
fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t 

440 
 no_args k i (t $ u) = no_args k i t andalso no_args k i u 

441 
 no_args k i (Bound m) = m < k orelse m > k+i 

442 
 no_args _ _ _ = true; 

15531  443 
fun split_pat tp i (Abs (_,_,t)) = if tp 0 i t then SOME (i,t) else NONE 
11838  444 
 split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t 
15531  445 
 split_pat tp i _ = NONE; 
20044
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446 
fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] [] 
13480
bb72bd43c6c3
use Tactic.prove instead of prove_goalw_cterm in internal proofs!
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447 
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs))) 
18328  448 
(K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1))); 
11838  449 

450 
fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t 

451 
 beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse 

452 
(beta_term_pat k i t andalso beta_term_pat k i u) 

453 
 beta_term_pat k i t = no_args k i t; 

454 
fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg 

455 
 eta_term_pat _ _ _ = false; 

456 
fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t) 

457 
 subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg 

458 
else (subst arg k i t $ subst arg k i u) 

459 
 subst arg k i t = t; 

20044
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460 
fun beta_proc ss (s as Const ("split", _) $ Abs (_, _, t) $ arg) = 
11838  461 
(case split_pat beta_term_pat 1 t of 
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462 
SOME (i,f) => SOME (metaeq ss s (subst arg 0 i f)) 
15531  463 
 NONE => NONE) 
20044
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464 
 beta_proc _ _ = NONE; 
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465 
fun eta_proc ss (s as Const ("split", _) $ Abs (_, _, t)) = 
11838  466 
(case split_pat eta_term_pat 1 t of 
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467 
SOME (_,ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end)) 
15531  468 
 NONE => NONE) 
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469 
 eta_proc _ _ = NONE; 
11838  470 
in 
32010  471 
val split_beta_proc = Simplifier.simproc @{theory} "split_beta" ["split f z"] (K beta_proc); 
472 
val split_eta_proc = Simplifier.simproc @{theory} "split_eta" ["split f"] (K eta_proc); 

11838  473 
end; 
474 

475 
Addsimprocs [split_beta_proc, split_eta_proc]; 

476 
*} 

477 

26798
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478 
lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)" 
11838  479 
by (subst surjective_pairing, rule split_conv) 
480 

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481 
lemma split_split [noatp]: "R(split c p) = (ALL x y. p = (x, y) > R(c x y))" 
11838  482 
 {* For use with @{text split} and the Simplifier. *} 
15481  483 
by (insert surj_pair [of p], clarify, simp) 
11838  484 

485 
text {* 

486 
@{thm [source] split_split} could be declared as @{text "[split]"} 

487 
done after the Splitter has been speeded up significantly; 

488 
precompute the constants involved and don't do anything unless the 

489 
current goal contains one of those constants. 

490 
*} 

491 

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492 
lemma split_split_asm [noatp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))" 
14208  493 
by (subst split_split, simp) 
11838  494 

495 

496 
text {* 

497 
\medskip @{term split} used as a logical connective or set former. 

498 

499 
\medskip These rules are for use with @{text blast}; could instead 

500 
call @{text simp} using @{thm [source] split} as rewrite. *} 

501 

502 
lemma splitI2: "!!p. [ !!a b. p = (a, b) ==> c a b ] ==> split c p" 

503 
apply (simp only: split_tupled_all) 

504 
apply (simp (no_asm_simp)) 

505 
done 

506 

507 
lemma splitI2': "!!p. [ !!a b. (a, b) = p ==> c a b x ] ==> split c p x" 

508 
apply (simp only: split_tupled_all) 

509 
apply (simp (no_asm_simp)) 

510 
done 

511 

512 
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" 

513 
by (induct p) (auto simp add: split_def) 

514 

515 
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" 

516 
by (induct p) (auto simp add: split_def) 

517 

518 
lemma splitE2: 

519 
"[ Q (split P z); !!x y. [z = (x, y); Q (P x y)] ==> R ] ==> R" 

520 
proof  

521 
assume q: "Q (split P z)" 

522 
assume r: "!!x y. [z = (x, y); Q (P x y)] ==> R" 

523 
show R 

524 
apply (rule r surjective_pairing)+ 

525 
apply (rule split_beta [THEN subst], rule q) 

526 
done 

527 
qed 

528 

529 
lemma splitD': "split R (a,b) c ==> R a b c" 

530 
by simp 

531 

532 
lemma mem_splitI: "z: c a b ==> z: split c (a, b)" 

533 
by simp 

534 

535 
lemma mem_splitI2: "!!p. [ !!a b. p = (a, b) ==> z: c a b ] ==> z: split c p" 

14208  536 
by (simp only: split_tupled_all, simp) 
11838  537 

18372  538 
lemma mem_splitE: 
539 
assumes major: "z: split c p" 

540 
and cases: "!!x y. [ p = (x,y); z: c x y ] ==> Q" 

541 
shows Q 

542 
by (rule major [unfolded split_def] cases surjective_pairing)+ 

11838  543 

544 
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!] 

545 
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!] 

546 

26340  547 
ML {* 
11838  548 
local (* filtering with exists_p_split is an essential optimization *) 
16121  549 
fun exists_p_split (Const ("split",_) $ _ $ (Const ("Pair",_)$_$_)) = true 
11838  550 
 exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u 
551 
 exists_p_split (Abs (_, _, t)) = exists_p_split t 

552 
 exists_p_split _ = false; 

16121  553 
val ss = HOL_basic_ss addsimps [thm "split_conv"]; 
11838  554 
in 
555 
val split_conv_tac = SUBGOAL (fn (t, i) => 

556 
if exists_p_split t then safe_full_simp_tac ss i else no_tac); 

557 
end; 

26340  558 
*} 
559 

11838  560 
(* This prevents applications of splitE for already splitted arguments leading 
561 
to quite timeconsuming computations (in particular for nested tuples) *) 

26340  562 
declaration {* fn _ => 
563 
Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac)) 

16121  564 
*} 
11838  565 

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566 
lemma split_eta_SetCompr [simp,noatp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P" 
18372  567 
by (rule ext) fast 
11838  568 

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569 
lemma split_eta_SetCompr2 [simp,noatp]: "(%u. EX x y. u = (x, y) & P x y) = split P" 
18372  570 
by (rule ext) fast 
11838  571 

572 
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)" 

573 
 {* Allows simplifications of nested splits in case of independent predicates. *} 

18372  574 
by (rule ext) blast 
11838  575 

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

576 
(* Do NOT make this a simp rule as it 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
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diff
changeset

577 
a) only helps in special situations 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
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parents:
14208
diff
changeset

578 
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

579 
*) 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
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diff
changeset

580 
lemma split_comp_eq: 
20415  581 
fixes f :: "'a => 'b => 'c" and g :: "'d => 'a" 
582 
shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))" 

18372  583 
by (rule ext) auto 
14101  584 

26358
d6a508c16908
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haftmann
parents:
26340
diff
changeset

585 
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

586 
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

587 
apply auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

588 
done 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

589 

11838  590 
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)" 
591 
by blast 

592 

593 
(* 

594 
the following would be slightly more general, 

595 
but cannot be used as rewrite rule: 

596 
### Cannot add premise as rewrite rule because it contains (type) unknowns: 

597 
### ?y = .x 

598 
Goal "[ P y; !!x. P x ==> x = y ] ==> (@(x',y). x = x' & P y) = (x,y)" 

14208  599 
by (rtac some_equality 1) 
600 
by ( Simp_tac 1) 

601 
by (split_all_tac 1) 

602 
by (Asm_full_simp_tac 1) 

11838  603 
qed "The_split_eq"; 
604 
*) 

605 

606 
text {* 

607 
Setup of internal @{text split_rule}. 

608 
*} 

609 

25511  610 
definition 
611 
internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" 

612 
where 

11032  613 
"internal_split == split" 
614 

615 
lemma internal_split_conv: "internal_split c (a, b) = c a b" 

616 
by (simp only: internal_split_def split_conv) 

617 

618 
hide const internal_split 

619 

11025
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
10289
diff
changeset

620 
use "Tools/split_rule.ML" 
11032  621 
setup SplitRule.setup 
10213  622 

24699
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diff
changeset

623 
lemmas prod_caseI = prod.cases [THEN iffD2, standard] 
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haftmann
parents:
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changeset

624 

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changeset

625 
lemma prod_caseI2: "!!p. [ !!a b. p = (a, b) ==> c a b ] ==> prod_case c p" 
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parents:
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changeset

626 
by auto 
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haftmann
parents:
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diff
changeset

627 

c6674504103f
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haftmann
parents:
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changeset

628 
lemma prod_caseI2': "!!p. [ !!a b. (a, b) = p ==> c a b x ] ==> prod_case c p x" 
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haftmann
parents:
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diff
changeset

629 
by (auto simp: split_tupled_all) 
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haftmann
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changeset

630 

c6674504103f
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changeset

631 
lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" 
c6674504103f
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haftmann
parents:
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changeset

632 
by (induct p) auto 
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haftmann
parents:
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changeset

633 

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634 
lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" 
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changeset

635 
by (induct p) auto 
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parents:
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diff
changeset

636 

c6674504103f
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637 
lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))" 
c6674504103f
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parents:
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diff
changeset

638 
by (simp add: expand_fun_eq) 
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haftmann
parents:
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changeset

639 

c6674504103f
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640 
declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!] 
c6674504103f
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haftmann
parents:
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changeset

641 
declare prod_caseE' [elim!] prod_caseE [elim!] 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

642 

24844
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

643 
lemma prod_case_split: 
24699
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haftmann
parents:
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diff
changeset

644 
"prod_case = split" 
c6674504103f
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haftmann
parents:
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changeset

645 
by (auto simp add: expand_fun_eq) 
c6674504103f
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haftmann
parents:
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changeset

646 

26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset

647 
lemma prod_case_beta: 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset

648 
"prod_case f p = f (fst p) (snd p)" 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset

649 
unfolding prod_case_split split_beta .. 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset

650 

24699
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651 

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652 
subsection {* Further cases/induct rules for tuples *} 
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parents:
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653 

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654 
lemma prod_cases3 [cases type]: 
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parents:
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655 
obtains (fields) a b c where "y = (a, b, c)" 
c6674504103f
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haftmann
parents:
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diff
changeset

656 
by (cases y, case_tac b) blast 
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parents:
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changeset

657 

c6674504103f
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658 
lemma prod_induct3 [case_names fields, induct type]: 
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parents:
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659 
"(!!a b c. P (a, b, c)) ==> P x" 
c6674504103f
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parents:
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changeset

660 
by (cases x) blast 
c6674504103f
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changeset

661 

c6674504103f
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662 
lemma prod_cases4 [cases type]: 
c6674504103f
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663 
obtains (fields) a b c d where "y = (a, b, c, d)" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
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changeset

664 
by (cases y, case_tac c) blast 
c6674504103f
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haftmann
parents:
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665 

c6674504103f
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666 
lemma prod_induct4 [case_names fields, induct type]: 
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parents:
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667 
"(!!a b c d. P (a, b, c, d)) ==> P x" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
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diff
changeset

668 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
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changeset

669 

c6674504103f
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670 
lemma prod_cases5 [cases type]: 
c6674504103f
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671 
obtains (fields) a b c d e where "y = (a, b, c, d, e)" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
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diff
changeset

672 
by (cases y, case_tac d) blast 
c6674504103f
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haftmann
parents:
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changeset

673 

c6674504103f
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parents:
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674 
lemma prod_induct5 [case_names fields, induct type]: 
c6674504103f
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parents:
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675 
"(!!a b c d e. P (a, b, c, d, e)) ==> P x" 
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datatype interpretators for size and datatype_realizer
haftmann
parents:
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changeset

676 
by (cases x) blast 
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parents:
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changeset

677 

c6674504103f
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678 
lemma prod_cases6 [cases type]: 
c6674504103f
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679 
obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)" 
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24286
diff
changeset

680 
by (cases y, case_tac e) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

681 

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

682 
lemma prod_induct6 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

683 
"(!!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

684 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

685 

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

686 
lemma prod_cases7 [cases type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

687 
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

688 
by (cases y, case_tac f) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

689 

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

690 
lemma prod_induct7 [case_names fields, induct type]: 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

691 
"(!!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

692 
by (cases x) blast 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

693 

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

694 

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

695 
subsubsection {* Derived operations *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

696 

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

697 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

698 
The compositionuncurry combinator. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

699 
*} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

700 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

701 
notation fcomp (infixl "o>" 60) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

702 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

703 
definition 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

704 
scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "o\<rightarrow>" 60) 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

705 
where 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

706 
"f o\<rightarrow> g = (\<lambda>x. split g (f x))" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

707 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

708 
lemma scomp_apply: "(f o\<rightarrow> g) x = split g (f x)" 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

709 
by (simp add: scomp_def) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

710 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

711 
lemma Pair_scomp: "Pair x o\<rightarrow> f = f x" 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

712 
by (simp add: expand_fun_eq scomp_apply) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

713 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

714 
lemma scomp_Pair: "x o\<rightarrow> Pair = x" 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

715 
by (simp add: expand_fun_eq scomp_apply) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

716 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

717 
lemma scomp_scomp: "(f o\<rightarrow> g) o\<rightarrow> h = f o\<rightarrow> (\<lambda>x. g x o\<rightarrow> h)" 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

718 
by (simp add: expand_fun_eq split_twice scomp_def) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

719 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

720 
lemma scomp_fcomp: "(f o\<rightarrow> g) o> h = f o\<rightarrow> (\<lambda>x. g x o> h)" 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

721 
by (simp add: expand_fun_eq scomp_apply fcomp_def split_def) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

722 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

723 
lemma fcomp_scomp: "(f o> g) o\<rightarrow> h = f o> (g o\<rightarrow> h)" 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

724 
by (simp add: expand_fun_eq scomp_apply fcomp_apply) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

725 

31202
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset

726 
code_const scomp 
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset

727 
(Eval infixl 3 "#>") 
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset

728 

26588
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

729 
no_notation fcomp (infixl "o>" 60) 
d83271bfaba5
removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents:
26480
diff
changeset

730 
no_notation scomp (infixl "o\<rightarrow>" 60) 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

731 

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

732 

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

733 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

734 
@{term prod_fun}  action of the product functor upon 
31775  735 
Datatypes. 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

736 
*} 
21195  737 

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

738 
definition prod_fun :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where 
28562  739 
[code del]: "prod_fun 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

740 

28562  741 
lemma prod_fun [simp, code]: "prod_fun f g (a, b) = (f a, g b)" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

742 
by (simp add: prod_fun_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

743 

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

744 
lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

745 
by (rule ext) auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

746 

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

747 
lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

748 
by (rule ext) auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

749 

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

750 
lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

751 
apply (rule image_eqI) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

752 
apply (rule prod_fun [symmetric], assumption) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

753 
done 
21195  754 

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

755 
lemma prod_fun_imageE [elim!]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

756 
assumes major: "c: (prod_fun f g)`r" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

757 
and cases: "!!x y. [ c=(f(x),g(y)); (x,y):r ] ==> P" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

758 
shows P 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

759 
apply (rule major [THEN imageE]) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

760 
apply (rule_tac p = x in PairE) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

761 
apply (rule cases) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

762 
apply (blast intro: prod_fun) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

763 
apply blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

764 
done 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

765 

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

766 
definition 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

767 
apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

768 
where 
28562  769 
[code del]: "apfst f = prod_fun f id" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

770 

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

771 
definition 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

772 
apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

773 
where 
28562  774 
[code del]: "apsnd f = prod_fun id f" 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

775 

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

776 
lemma apfst_conv [simp, code]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

777 
"apfst f (x, y) = (f x, y)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

778 
by (simp add: apfst_def) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

779 

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

780 
lemma upd_snd_conv [simp, code]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

781 
"apsnd f (x, y) = (x, f y)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

782 
by (simp add: apsnd_def) 
21195  783 

784 

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

785 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

786 
Disjoint union of a family of sets  Sigma. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

787 
*} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

788 

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

789 
definition Sigma :: "['a set, 'a => 'b set] => ('a \<times> 'b) set" where 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

790 
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

791 

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

792 
abbreviation 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

793 
Times :: "['a set, 'b set] => ('a * 'b) set" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

794 
(infixr "<*>" 80) where 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

795 
"A <*> B == Sigma A (%_. B)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

796 

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

797 
notation (xsymbols) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

798 
Times (infixr "\<times>" 80) 
15394  799 

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

800 
notation (HTML output) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

801 
Times (infixr "\<times>" 80) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

802 

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

803 
syntax 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

804 
"@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10) 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

805 

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

806 
translations 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

807 
"SIGMA x:A. B" == "Product_Type.Sigma A (%x. B)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

808 

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

809 
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

810 
by (unfold Sigma_def) blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

811 

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

812 
lemma SigmaE [elim!]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

813 
"[ c: Sigma A B; 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

814 
!!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

815 
] ==> P" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

816 
 {* The general elimination rule. *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

817 
by (unfold Sigma_def) blast 
20588  818 

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

819 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

820 
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

821 
eigenvariables. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

822 
*} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

823 

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

824 
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

825 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

826 

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

827 
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

828 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

829 

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

830 
lemma SigmaE2: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

831 
"[ (a, b) : Sigma A B; 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

832 
[ a:A; b:B(a) ] ==> P 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

833 
] ==> P" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

834 
by blast 
20588  835 

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

836 
lemma Sigma_cong: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

837 
"\<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

838 
\<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

839 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

840 

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

841 
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

842 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

843 

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

844 
lemma Sigma_empty1 [simp]: "Sigma {} B = {}" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

845 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

846 

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

847 
lemma Sigma_empty2 [simp]: "A <*> {} = {}" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

848 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

849 

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

850 
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

851 
by auto 
21908  852 

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

853 
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

854 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

855 

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

856 
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

857 
by auto 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

858 

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

859 
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

860 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

861 

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

862 
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

863 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

864 

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

865 
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

866 
by (blast elim: equalityE) 
20588  867 

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

868 
lemma SetCompr_Sigma_eq: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

869 
"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

870 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

871 

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

872 
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

873 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

874 

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

875 
lemma UN_Times_distrib: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

876 
"(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)" 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

877 
 {* Suggested by Pierre Chartier *} 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

878 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

879 

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

880 
lemma split_paired_Ball_Sigma [simp,noatp]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

881 
"(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

882 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

883 

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

884 
lemma split_paired_Bex_Sigma [simp,noatp]: 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

885 
"(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

886 
by blast 
21908  887 

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

888 
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

889 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

890 

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

891 
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

892 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

893 

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

894 
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

895 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

896 

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

897 
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

898 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

899 

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

900 
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

901 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

902 

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

903 
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

904 
by blast 
21908  905 

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

906 
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

907 
by blast 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

908 

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

909 
text {* 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

910 
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

911 
matching, especially when the rules are reoriented. 
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

912 
*} 
21908  913 

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

914 
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)" 
28719  915 
by blast 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

916 

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

917 
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)" 
28719  918 
by blast 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

919 

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

920 
lemma Times_Diff_distrib1: "(A  B) <*> C = (A <*> C)  (B <*> C)" 
28719  921 
by blast 
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset

922 

28719  923 
lemma insert_times_insert[simp]: 
924 
"insert a A \<times> insert b B = 

925 
insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)" 

926 
by blast 

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

927 

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

928 
lemma vimage_Times: "f ` (A \<times> B) = ((fst \<circ> f) ` A) \<inter> ((snd \<circ> f) ` B)" 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33089
diff
changeset

929 
by (auto, rule_tac p = "f x" in PairE, auto) 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33089
diff
changeset

930 

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

931 
subsubsection {* Code generator setup *} 
21908  932 

28562  933 
lemma [code]: 
28346
b8390cd56b8f
discontinued special treatment of op = vs. eq_class.eq
haftmann
parents:
28262
diff
changeset

934 
"eq_class.eq (x1\<Colon>'a\<Colon>eq, y1\<Colon>'b\<Colon>eq) (x2, y2) \<longleftrightarrow> x1 = x2 \<and> y1 = y2" by (simp add: eq) 
20588  935 

24844
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

936 
lemma split_case_cert: 
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

937 
assumes "CASE \<equiv> split f" 
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

938 
shows "CASE (a, b) \<equiv> f a b" 
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

939 
using assms by simp 
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

940 

98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

941 
setup {* 
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

942 
Code.add_case @{thm split_case_cert} 
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

943 
*} 
98c006a30218
certificates for code generator case expressions
haftmann
parents:
24699
diff
changeset

944 

21908  945 
code_type * 
946 
(SML infix 2 "*") 

947 
(OCaml infix 2 "*") 

948 
(Haskell "!((_),/ (_))") 

949 

20588  950 
code_instance * :: eq 
951 
(Haskell ) 

952 

28346
b8390cd56b8f
discontinued special treatment of op = vs. eq_class.eq
haftmann
parents:
28262
diff
changeset

953 
code_const "eq_class.eq \<Colon> 'a\<Colon>eq \<times> 'b\<Colon>eq \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" 
20588  954 
(Haskell infixl 4 "==") 
955 

21908  956 
code_const Pair 
957 
(SML "!((_),/ (_))") 

958 
(OCaml "!((_),/ (_))") 

959 
(Haskell "!((_),/ (_))") 

20588  960 

22389  961 
code_const fst and snd 
962 
(Haskell "fst" and "snd") 

963 

15394  964 
types_code 
965 
"*" ("(_ */ _)") 

16770
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset

966 
attach (term_of) {* 
25885  967 
fun term_of_id_42 aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y; 
16770
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset

968 
*} 
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset

969 
attach (test) {* 
25885  970 
fun gen_id_42 aG aT bG bT i = 
971 
let 

972 
val (x, t) = aG i; 

973 
val (y, u) = bG i 

974 
in ((x, y), fn () => HOLogic.pair_const aT bT $ t () $ u ()) end; 

16770
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset

975 
*} 
15394  976 

18706
1e7562c7afe6
Reinserted consts_code declaration accidentally deleted
berghofe
parents:
18702
diff
changeset

977 
consts_code 
1e7562c7afe6
Reinserted consts_code declaration accidentally deleted
berghofe
parents:
18702
diff
changeset

978 
"Pair" ("(_,/ _)") 
1e7562c7afe6
Reinserted consts_code declaration accidentally deleted
berghofe
parents:
18702
diff
changeset

979 

21908  980 
setup {* 
981 

982 
let 

18013  983 

19039  984 
fun strip_abs_split 0 t = ([], t) 
985 
 strip_abs_split i (Abs (s, T, t)) = 

18013  986 
let 
987 
val s' = Codegen.new_name t s; 

988 
val v = Free (s', T) 

19039  989 
in apfst (cons v) (strip_abs_split (i1) (subst_bound (v, t))) end 
990 
 strip_abs_split i (u as Const ("split", _) $ t) = (case strip_abs_split (i+1) t of 

15394  991 
(v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u) 
992 
 _ => ([], u)) 

30604  993 
 strip_abs_split i t = 
994 
strip_abs_split i (Abs ("x", hd (binder_types (fastype_of t)), t $ Bound 0)); 

18013  995 

28537
1e84256d1a8a
established canonical argument order in SML code generators
haftmann
parents:
28346
diff
changeset

996 
fun let_codegen thy defs dep thyname brack t gr = (case strip_comb t of 
16634  997 
(t1 as Const ("Let", _), t2 :: t3 :: ts) => 
15394  998 
let 
999 
fun dest_let (l as Const ("Let", _) $ t $ u) = 

19039  1000 
(case strip_abs_split 1 u of 
15394  1001 
([p], u') => apfst (cons (p, t)) (dest_let u') 
1002 
 _ => ([], l)) 

1003 
 dest_let t = ([], t); 

28537
1e84256d1a8a
established canonical argument order in SML code generators
haftmann
parents:
28346
diff
changeset

1004 
fun mk_code (l, r) gr = 
15394  1005 
let 
28537
1e84256d1a8a
established canonical argument order in SML code generators
haftmann
parents:
28346
diff
changeset

1006 
val (pl, gr1) = Codegen.invoke_codegen thy defs dep thyname false l gr; 
1e84256d1a8a
established canonical argument order in SML code generators
haftmann
parents:
28346
diff
changeset

1007 
val (pr, gr2) = Codegen.invoke_codegen thy defs dep thyname false r gr1; 
1e84256d1a8a
established canonical argument order in SML code generators
haftmann
parents:
28346
diff
changeset

1008 
in ((pl, pr), gr2) end 
16634  1009 
in case dest_let (t1 $ t2 $ t3) of 
15531  1010 
([], _) => NONE 
15394  1011 
 (ps, u) => 
1012 
let 

28537
1e84256d1a8a
established canonical argument order in SML code generators
haftmann
parents:
28346
diff
changeset

1013 
val (qs, gr1) = fold_map mk_code ps gr; 
1e84256d1a8a
established canonical argument order in SML code generators
haftmann
parents:
28346
diff
changeset

1014 
val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1; 
1e84256d1a8a
established canonical argument order in SML code generators
haftmann
parents:
28346
diff
changeset

1015 
val (pargs, gr3) = fold_map 
1e84256d1a8a
established canonical argument order in SML code generators
haftmann
parents:
28346
diff
changeset

1016 
(Codegen.invoke_codegen thy defs dep thyname true) ts gr2 
15394  1017 
in 
28537
1e84256d1a8a
established canonical argument order in SML code generators
haftmann
parents:
28346
diff
changeset

1018 
SOME (Codegen.mk_app brack 
32952  1019 
(Pretty.blk (0, [Codegen.str "let ", Pretty.blk (0, flat 
26975
103dca19ef2e
Replaced Pretty.str and Pretty.string_of by specific functions (from Codegen) that
berghofe
parents:
26798
diff
changeset

1020 
(separate [Codegen.str ";", Pretty.brk 1] (map (fn (pl, pr) => 
103dca19ef2e
Replaced Pretty.str and Pretty.string_of by specific functions (from Codegen) that
berghofe
parents:
26798
diff
changeset

1021 
[Pretty.block [Codegen.str "val ", pl, Codegen.str " =", 
16634  1022 
Pretty.brk 1, pr]]) qs))), 
26975
103dca19ef2e
Replaced Pretty.str and Pretty.string_of by specific functions (from Codegen) that
berghofe
parents:
26798
diff
changeset

1023 
Pretty.brk 1, Codegen.str "in ", pu, 
28537
1e84256d1a8a
established canonical argument order in SML code generators
haftmann
parents:
28346
diff
changeset

1024 
Pretty.brk 1, Codegen.str "end"])) pargs, gr3) 
15394  1025 
end 
1026 
end 

16634  1027 
 _ => NONE); 
15394  1028 

28537
1e84256d1a8a
established canonical argument order in SML code generators
haftmann
parents:
28346
diff
changeset

1029 
fun split_codegen thy defs dep thyname brack t gr = (case strip_comb t of 
16634  1030 
(t1 as Const ("split", _), t2 :: ts) => 
30604  1031 
let 
1032 
val ([p], u) = strip_abs_split 1 (t1 $ t2); 

1033 
val (q, gr1) = Codegen.invoke_codegen thy defs dep thyname false p gr; 

1034 
val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1; 

1035 
val (pargs, gr3) = fold_map 

1036 
(Codegen.invoke_codegen thy defs dep thyname true) ts gr2 

1037 
in 

1038 
SOME (Codegen.mk_app brack 

1039 
(Pretty.block [Codegen.str "(fn ", q, Codegen.str " =>", 

1040 
Pretty.brk 1, pu, Codegen.str ")"]) pargs, gr2) 

1041 
end 

16634  1042 
 _ => NONE); 
15394  1043 

21908  1044 
in 
1045 

20105  1046 
Codegen.add_codegen "let_codegen" let_codegen 
1047 
#> Codegen.add_codegen "split_codegen" split_codegen 

15394  1048 

21908  1049 
end 
15394  1050 
*} 
1051 

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

1052 

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

1053 
subsection {* Legacy bindings *} 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1054 

21908  1055 
ML {* 
15404  1056 
val Collect_split = thm "Collect_split"; 
1057 
val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1"; 

1058 
val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2"; 

1059 
val PairE = thm "PairE"; 

1060 
val Pair_Rep_inject = thm "Pair_Rep_inject"; 

1061 
val Pair_def = thm "Pair_def"; 

27104
791607529f6d
rep_datatype command now takes list of constructors as input arguments
haftmann
parents:
26975
diff
changeset

1062 
val Pair_eq = @{thm "prod.inject"}; 
15404  1063 
val Pair_fst_snd_eq = thm "Pair_fst_snd_eq"; 
1064 
val ProdI = thm "ProdI"; 

1065 
val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq"; 

1066 
val SigmaD1 = thm "SigmaD1"; 

1067 
val SigmaD2 = thm "SigmaD2"; 

1068 
val SigmaE = thm "SigmaE"; 

1069 
val SigmaE2 = thm "SigmaE2"; 

1070 
val SigmaI = thm "SigmaI"; 

1071 
val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1"; 

1072 
val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2"; 

1073 
val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1"; 

1074 
val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2"; 

1075 
val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1"; 

1076 
val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2"; 

1077 
val Sigma_Union = thm "Sigma_Union"; 

1078 
val Sigma_def = thm "Sigma_def"; 

1079 
val Sigma_empty1 = thm "Sigma_empty1"; 

1080 
val Sigma_empty2 = thm "Sigma_empty2"; 

1081 
val Sigma_mono = thm "Sigma_mono"; 

1082 
val The_split = thm "The_split"; 

1083 
val The_split_eq = thm "The_split_eq"; 

1084 
val The_split_eq = thm "The_split_eq"; 

1085 
val Times_Diff_distrib1 = thm "Times_Diff_distrib1"; 

1086 
val Times_Int_distrib1 = thm "Times_Int_distrib1"; 

1087 
val Times_Un_distrib1 = thm "Times_Un_distrib1"; 

1088 
val Times_eq_cancel2 = thm "Times_eq_cancel2"; 

1089 
val Times_subset_cancel2 = thm "Times_subset_cancel2"; 

1090 
val UNIV_Times_UNIV = thm "UNIV_Times_UNIV"; 

1091 
val UN_Times_distrib = thm "UN_Times_distrib"; 

1092 
val Unity_def = thm "Unity_def"; 

1093 
val cond_split_eta = thm "cond_split_eta"; 

1094 
val fst_conv = thm "fst_conv"; 

1095 
val fst_def = thm "fst_def"; 

1096 
val fst_eqD = thm "fst_eqD"; 

1097 
val inj_on_Abs_Prod = thm "inj_on_Abs_Prod"; 

1098 
val mem_Sigma_iff = thm "mem_Sigma_iff"; 

1099 
val mem_splitE = thm "mem_splitE"; 

1100 
val mem_splitI = thm "mem_splitI"; 

1101 
val mem_splitI2 = thm "mem_splitI2"; 

1102 
val prod_eqI = thm "prod_eqI"; 

1103 
val prod_fun = thm "prod_fun"; 

1104 
val prod_fun_compose = thm "prod_fun_compose"; 

1105 
val prod_fun_def = thm "prod_fun_def"; 

1106 
val prod_fun_ident = thm "prod_fun_ident"; 

1107 
val prod_fun_imageE = thm "prod_fun_imageE"; 

1108 
val prod_fun_imageI = thm "prod_fun_imageI"; 

27104
791607529f6d
rep_datatype command now takes list of constructors as input arguments
haftmann
parents:
26975
diff
changeset

1109 
val prod_induct = thm "prod.induct"; 
15404  1110 
val snd_conv = thm "snd_conv"; 
1111 
val snd_def = thm "snd_def"; 

1112 
val snd_eqD = thm "snd_eqD"; 

1113 
val split = thm "split"; 

1114 
val splitD = thm "splitD"; 

1115 
val splitD' = thm "splitD'"; 

1116 
val splitE = thm "splitE"; 

1117 
val splitE' = thm "splitE'"; 

1118 
val splitE2 = thm "splitE2"; 

1119 
val splitI = thm "splitI"; 

1120 
val splitI2 = thm "splitI2"; 

1121 
val splitI2' = thm "splitI2'"; 

1122 
val split_beta = thm "split_beta"; 

1123 
val split_conv = thm "split_conv"; 

1124 
val split_def = thm "split_def"; 

1125 
val split_eta = thm "split_eta"; 

1126 
val split_eta_SetCompr = thm "split_eta_SetCompr"; 

1127 
val split_eta_SetCompr2 = thm "split_eta_SetCompr2"; 

1128 
val split_paired_All = thm "split_paired_All"; 

1129 
val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma"; 

1130 
val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma"; 

1131 
val split_paired_Ex = thm "split_paired_Ex"; 

1132 
val split_paired_The = thm "split_paired_The"; 

1133 
val split_paired_all = thm "split_paired_all"; 

1134 
val split_part = thm "split_part"; 

1135 
val split_split = thm "split_split"; 

1136 
val split_split_asm = thm "split_split_asm"; 

1137 
val split_tupled_all = thms "split_tupled_all"; 

1138 
val split_weak_cong = thm "split_weak_cong"; 

1139 
val surj_pair = thm "surj_pair"; 

1140 
val surjective_pairing = thm "surjective_pairing"; 

1141 
val unit_abs_eta_conv = thm "unit_abs_eta_conv"; 

1142 
val unit_all_eq1 = thm "unit_all_eq1"; 

1143 
val unit_all_eq2 = thm "unit_all_eq2"; 

1144 
val unit_eq = thm "unit_eq"; 

1145 
*} 

1146 

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

1147 

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

1148 
subsection {* Further inductive packages *} 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1149 

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

1150 
use "Tools/inductive_realizer.ML" 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1151 
setup InductiveRealizer.setup 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1152 

31723
f5cafe803b55
discontinued ancient tradition to suffix certain ML module names with "_package"
haftmann
parents:
31667
diff
changeset

1153 
use "Tools/inductive_set.ML" 
f5cafe803b55
discontinued ancient tradition to suffix certain ML module names with "_package"
haftmann
parents:
31667
diff
changeset

1154 
setup Inductive_Set.setup 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1155 

31775  1156 
use "Tools/Datatype/datatype_realizer.ML" 
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1157 
setup DatatypeRealizer.setup 
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset

1158 

10213  1159 
end 