(* Title: HOL/Product_Type.thy ID: $Id$ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1992 University of Cambridge *) header {* Cartesian products *} theory Product_Type imports Inductive uses ("Tools/split_rule.ML") ("Tools/inductive_set_package.ML") ("Tools/inductive_realizer.ML") ("Tools/datatype_realizer.ML") begin subsection {* @{typ bool} is a datatype *} rep_datatype bool distinct True_not_False False_not_True induction bool_induct declare case_split [cases type: bool] -- "prefer plain propositional version" lemma [code func]: shows "False = P \ \ P" and "True = P \ P" and "P = False \ \ P" and "P = True \ P" by simp_all code_const "op = \ bool \ bool \ bool" (Haskell infixl 4 "==") code_instance bool :: eq (Haskell -) subsection {* Unit *} typedef unit = "{True}" proof show "True : ?unit" .. qed definition Unity :: unit ("'(')") where "() = Abs_unit True" lemma unit_eq [noatp]: "u = ()" by (induct u) (simp add: unit_def Unity_def) text {* Simplification procedure for @{thm [source] unit_eq}. Cannot use this rule directly --- it loops! *} ML {* val unit_eq_proc = let val unit_meta_eq = mk_meta_eq @{thm unit_eq} in Simplifier.simproc @{theory} "unit_eq" ["x::unit"] (fn _ => fn _ => fn t => if HOLogic.is_unit t then NONE else SOME unit_meta_eq) end; Addsimprocs [unit_eq_proc]; *} lemma unit_induct [noatp,induct type: unit]: "P () ==> P x" by simp rep_datatype unit induction unit_induct lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()" by simp lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P" by (rule triv_forall_equality) text {* This rewrite counters the effect of @{text unit_eq_proc} on @{term [source] "%u::unit. f u"}, replacing it by @{term [source] f} rather than by @{term [source] "%u. f ()"}. *} lemma unit_abs_eta_conv [simp,noatp]: "(%u::unit. f ()) = f" by (rule ext) simp text {* code generator setup *} instance unit :: eq .. lemma [code func]: "(u\unit) = v \ True" unfolding unit_eq [of u] unit_eq [of v] by rule+ code_type unit (SML "unit") (OCaml "unit") (Haskell "()") code_instance unit :: eq (Haskell -) code_const "op = \ unit \ unit \ bool" (Haskell infixl 4 "==") code_const Unity (SML "()") (OCaml "()") (Haskell "()") code_reserved SML unit code_reserved OCaml unit subsection {* Pairs *} subsubsection {* Product type, basic operations and concrete syntax *} definition Pair_Rep :: "'a \ 'b \ 'a \ 'b \ bool" where "Pair_Rep a b = (\x y. x = a \ y = b)" global typedef (Prod) ('a, 'b) "*" (infixr "*" 20) = "{f. \a b. f = Pair_Rep (a\'a) (b\'b)}" proof fix a b show "Pair_Rep a b \ ?Prod" by rule+ qed syntax (xsymbols) "*" :: "[type, type] => type" ("(_ \/ _)" [21, 20] 20) syntax (HTML output) "*" :: "[type, type] => type" ("(_ \/ _)" [21, 20] 20) consts Pair :: "'a \ 'b \ 'a \ 'b" fst :: "'a \ 'b \ 'a" snd :: "'a \ 'b \ 'b" split :: "('a \ 'b \ 'c) \ 'a \ 'b \ 'c" curry :: "('a \ 'b \ 'c) \ 'a \ 'b \ 'c" local defs Pair_def: "Pair a b == Abs_Prod (Pair_Rep a b)" fst_def: "fst p == THE a. EX b. p = Pair a b" snd_def: "snd p == THE b. EX a. p = Pair a b" split_def: "split == (%c p. c (fst p) (snd p))" curry_def: "curry == (%c x y. c (Pair x y))" text {* Patterns -- extends pre-defined type @{typ pttrn} used in abstractions. *} nonterminals tuple_args patterns syntax "_tuple" :: "'a => tuple_args => 'a * 'b" ("(1'(_,/ _'))") "_tuple_arg" :: "'a => tuple_args" ("_") "_tuple_args" :: "'a => tuple_args => tuple_args" ("_,/ _") "_pattern" :: "[pttrn, patterns] => pttrn" ("'(_,/ _')") "" :: "pttrn => patterns" ("_") "_patterns" :: "[pttrn, patterns] => patterns" ("_,/ _") translations "(x, y)" == "Pair x y" "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))" "%(x,y,zs).b" == "split(%x (y,zs).b)" "%(x,y).b" == "split(%x y. b)" "_abs (Pair x y) t" => "%(x,y).t" (* The last rule accommodates tuples in `case C ... (x,y) ... => ...' The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *) (* reconstructs pattern from (nested) splits, avoiding eta-contraction of body*) (* works best with enclosing "let", if "let" does not avoid eta-contraction *) print_translation {* let fun split_tr' [Abs (x,T,t as (Abs abs))] = (* split (%x y. t) => %(x,y) t *) let val (y,t') = atomic_abs_tr' abs; val (x',t'') = atomic_abs_tr' (x,T,t'); in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end | split_tr' [Abs (x,T,(s as Const ("split",_)$t))] = (* split (%x. (split (%y z. t))) => %(x,y,z). t *) let val (Const ("_abs",_)$(Const ("_pattern",_)$y$z)$t') = split_tr' [t]; val (x',t'') = atomic_abs_tr' (x,T,t'); in Syntax.const "_abs"$ (Syntax.const "_pattern"$x'$(Syntax.const "_patterns"$y$z))$t'' end | split_tr' [Const ("split",_)$t] = (* split (split (%x y z. t)) => %((x,y),z). t *) split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *) | split_tr' [Const ("_abs",_)$x_y$(Abs abs)] = (* split (%pttrn z. t) => %(pttrn,z). t *) let val (z,t) = atomic_abs_tr' abs; in Syntax.const "_abs" $ (Syntax.const "_pattern" $x_y$z) $ t end | split_tr' _ = raise Match; in [("split", split_tr')] end *} (* print "split f" as "\(x,y). f x y" and "split (\x. f x)" as "\(x,y). f x y" *) typed_print_translation {* let fun split_guess_names_tr' _ T [Abs (x,_,Abs _)] = raise Match | split_guess_names_tr' _ T [Abs (x,xT,t)] = (case (head_of t) of Const ("split",_) => raise Match | _ => let val (_::yT::_) = binder_types (domain_type T) handle Bind => raise Match; val (y,t') = atomic_abs_tr' ("y",yT,(incr_boundvars 1 t)$Bound 0); val (x',t'') = atomic_abs_tr' (x,xT,t'); in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end) | split_guess_names_tr' _ T [t] = (case (head_of t) of Const ("split",_) => raise Match | _ => let val (xT::yT::_) = binder_types (domain_type T) handle Bind => raise Match; val (y,t') = atomic_abs_tr' ("y",yT,(incr_boundvars 2 t)$Bound 1$Bound 0); val (x',t'') = atomic_abs_tr' ("x",xT,t'); in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end) | split_guess_names_tr' _ _ _ = raise Match; in [("split", split_guess_names_tr')] end *} text {* Towards a datatype declaration *} lemma surj_pair [simp]: "EX x y. p = (x, y)" apply (unfold Pair_def) apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE]) apply (erule exE, erule exE, rule exI, rule exI) apply (rule Rep_Prod_inverse [symmetric, THEN trans]) apply (erule arg_cong) done lemma PairE [cases type: *]: obtains x y where "p = (x, y)" using surj_pair [of p] by blast lemma prod_induct [induct type: *]: "(\a b. P (a, b)) \ P x" by (cases x) simp lemma ProdI: "Pair_Rep a b \ Prod" unfolding Prod_def by rule+ lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' \ a = a' \ b = b'" unfolding Pair_Rep_def by (drule fun_cong, drule fun_cong) blast lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod" apply (rule inj_on_inverseI) apply (erule Abs_Prod_inverse) done lemma Pair_inject: assumes "(a, b) = (a', b')" and "a = a' ==> b = b' ==> R" shows R apply (insert prems [unfolded Pair_def]) apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE]) apply (assumption | rule ProdI)+ done lemma Pair_eq [iff]: "((a, b) = (a', b')) = (a = a' & b = b')" by (blast elim!: Pair_inject) lemma fst_conv [simp, code]: "fst (a, b) = a" unfolding fst_def by blast lemma snd_conv [simp, code]: "snd (a, b) = b" unfolding snd_def by blast rep_datatype prod inject Pair_eq induction prod_induct subsubsection {* Basic rules and proof tools *} lemma fst_eqD: "fst (x, y) = a ==> x = a" by simp lemma snd_eqD: "snd (x, y) = a ==> y = a" by simp lemma pair_collapse [simp]: "(fst p, snd p) = p" by (cases p) simp lemmas surjective_pairing = pair_collapse [symmetric] lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))" proof fix a b assume "!!x. PROP P x" then show "PROP P (a, b)" . next fix x assume "!!a b. PROP P (a, b)" from `PROP P (fst x, snd x)` show "PROP P x" by simp qed text {* The rule @{thm [source] split_paired_all} does not work with the Simplifier because it also affects premises in congrence rules, where this can lead to premises of the form @{text "!!a b. ... = ?P(a, b)"} which cannot be solved by reflexivity. *} lemmas split_tupled_all = split_paired_all unit_all_eq2 ML {* (* replace parameters of product type by individual component parameters *) val safe_full_simp_tac = generic_simp_tac true (true, false, false); local (* filtering with exists_paired_all is an essential optimization *) fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) = can HOLogic.dest_prodT T orelse exists_paired_all t | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u | exists_paired_all (Abs (_, _, t)) = exists_paired_all t | exists_paired_all _ = false; val ss = HOL_basic_ss addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}] addsimprocs [unit_eq_proc]; in val split_all_tac = SUBGOAL (fn (t, i) => if exists_paired_all t then safe_full_simp_tac ss i else no_tac); val unsafe_split_all_tac = SUBGOAL (fn (t, i) => if exists_paired_all t then full_simp_tac ss i else no_tac); fun split_all th = if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th; end; *} declaration {* fn _ => Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac)) *} lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))" -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *} by fast lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))" by fast lemma Pair_fst_snd_eq: "s = t \ fst s = fst t \ snd s = snd t" by (cases s, cases t) simp lemma prod_eqI [intro?]: "fst p = fst q \ snd p = snd q \ p = q" by (simp add: Pair_fst_snd_eq) subsubsection {* @{text split} and @{text curry} *} lemma split_conv [simp, code func]: "split f (a, b) = f a b" by (simp add: split_def) lemma curry_conv [simp, code func]: "curry f a b = f (a, b)" by (simp add: curry_def) lemmas split = split_conv -- {* for backwards compatibility *} lemma splitI: "f a b \ split f (a, b)" by (rule split_conv [THEN iffD2]) lemma splitD: "split f (a, b) \ f a b" by (rule split_conv [THEN iffD1]) lemma curryI [intro!]: "f (a, b) \ curry f a b" by (simp add: curry_def) lemma curryD [dest!]: "curry f a b \ f (a, b)" by (simp add: curry_def) lemma curryE: "curry f a b \ (f (a, b) \ Q) \ Q" by (simp add: curry_def) lemma curry_split [simp]: "curry (split f) = f" by (simp add: curry_def split_def) lemma split_curry [simp]: "split (curry f) = f" by (simp add: curry_def split_def) lemma split_Pair [simp]: "(\(x, y). (x, y)) = id" by (simp add: split_def id_def) lemma split_eta: "(\(x, y). f (x, y)) = f" -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *} by (rule ext) auto lemma split_comp: "split (f \ g) x = f (g (fst x)) (snd x)" by (cases x) simp lemma split_twice: "split f (split g p) = split (\x y. split f (g x y)) p" unfolding split_def .. lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))" -- {* Can't be added to simpset: loops! *} by (simp add: split_eta) lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))" by (simp add: split_def) lemma split_weak_cong: "p = q \ split c p = split c q" -- {* Prevents simplification of @{term c}: much faster *} by (erule arg_cong) lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g" by (simp add: split_eta) text {* Simplification procedure for @{thm [source] cond_split_eta}. Using @{thm [source] split_eta} as a rewrite rule is not general enough, and using @{thm [source] cond_split_eta} directly would render some existing proofs very inefficient; similarly for @{text split_beta}. *} ML {* local val cond_split_eta_ss = HOL_basic_ss addsimps [thm "cond_split_eta"] fun Pair_pat k 0 (Bound m) = (m = k) | Pair_pat k i (Const ("Pair", _) $ Bound m $ t) = i > 0 andalso m = k+i andalso Pair_pat k (i-1) t | Pair_pat _ _ _ = false; fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t | no_args k i (t $ u) = no_args k i t andalso no_args k i u | no_args k i (Bound m) = m < k orelse m > k+i | no_args _ _ _ = true; fun split_pat tp i (Abs (_,_,t)) = if tp 0 i t then SOME (i,t) else NONE | split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t | split_pat tp i _ = NONE; fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs))) (K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1))); fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t | beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u) | beta_term_pat k i t = no_args k i t; fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg | eta_term_pat _ _ _ = false; fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t) | subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg else (subst arg k i t $ subst arg k i u) | subst arg k i t = t; fun beta_proc ss (s as Const ("split", _) $ Abs (_, _, t) $ arg) = (case split_pat beta_term_pat 1 t of SOME (i,f) => SOME (metaeq ss s (subst arg 0 i f)) | NONE => NONE) | beta_proc _ _ = NONE; fun eta_proc ss (s as Const ("split", _) $ Abs (_, _, t)) = (case split_pat eta_term_pat 1 t of SOME (_,ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end)) | NONE => NONE) | eta_proc _ _ = NONE; in val split_beta_proc = Simplifier.simproc @{theory} "split_beta" ["split f z"] (K beta_proc); val split_eta_proc = Simplifier.simproc @{theory} "split_eta" ["split f"] (K eta_proc); end; Addsimprocs [split_beta_proc, split_eta_proc]; *} lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)" by (subst surjective_pairing, rule split_conv) lemma split_split [noatp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))" -- {* For use with @{text split} and the Simplifier. *} by (insert surj_pair [of p], clarify, simp) text {* @{thm [source] split_split} could be declared as @{text "[split]"} done after the Splitter has been speeded up significantly; precompute the constants involved and don't do anything unless the current goal contains one of those constants. *} lemma split_split_asm [noatp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))" by (subst split_split, simp) text {* \medskip @{term split} used as a logical connective or set former. \medskip These rules are for use with @{text blast}; could instead call @{text simp} using @{thm [source] split} as rewrite. *} lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p" apply (simp only: split_tupled_all) apply (simp (no_asm_simp)) done lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x" apply (simp only: split_tupled_all) apply (simp (no_asm_simp)) done lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" by (induct p) (auto simp add: split_def) lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" by (induct p) (auto simp add: split_def) lemma splitE2: "[| Q (split P z); !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R" proof - assume q: "Q (split P z)" assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R" show R apply (rule r surjective_pairing)+ apply (rule split_beta [THEN subst], rule q) done qed lemma splitD': "split R (a,b) c ==> R a b c" by simp lemma mem_splitI: "z: c a b ==> z: split c (a, b)" by simp lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p" by (simp only: split_tupled_all, simp) lemma mem_splitE: assumes major: "z: split c p" and cases: "!!x y. [| p = (x,y); z: c x y |] ==> Q" shows Q by (rule major [unfolded split_def] cases surjective_pairing)+ declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!] declare mem_splitE [elim!] splitE' [elim!] splitE [elim!] ML {* local (* filtering with exists_p_split is an essential optimization *) fun exists_p_split (Const ("split",_) $ _ $ (Const ("Pair",_)$_$_)) = true | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u | exists_p_split (Abs (_, _, t)) = exists_p_split t | exists_p_split _ = false; val ss = HOL_basic_ss addsimps [thm "split_conv"]; in val split_conv_tac = SUBGOAL (fn (t, i) => if exists_p_split t then safe_full_simp_tac ss i else no_tac); end; *} (* This prevents applications of splitE for already splitted arguments leading to quite time-consuming computations (in particular for nested tuples) *) declaration {* fn _ => Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac)) *} lemma split_eta_SetCompr [simp,noatp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P" by (rule ext) fast lemma split_eta_SetCompr2 [simp,noatp]: "(%u. EX x y. u = (x, y) & P x y) = split P" by (rule ext) fast lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)" -- {* Allows simplifications of nested splits in case of independent predicates. *} by (rule ext) blast (* Do NOT make this a simp rule as it a) only helps in special situations b) can lead to nontermination in the presence of split_def *) lemma split_comp_eq: fixes f :: "'a => 'b => 'c" and g :: "'d => 'a" shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))" by (rule ext) auto lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A" apply (rule_tac x = "(a, b)" in image_eqI) apply auto done lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)" by blast (* the following would be slightly more general, but cannot be used as rewrite rule: ### Cannot add premise as rewrite rule because it contains (type) unknowns: ### ?y = .x Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)" by (rtac some_equality 1) by ( Simp_tac 1) by (split_all_tac 1) by (Asm_full_simp_tac 1) qed "The_split_eq"; *) text {* Setup of internal @{text split_rule}. *} definition internal_split :: "('a \ 'b \ 'c) \ 'a \ 'b \ 'c" where "internal_split == split" lemma internal_split_conv: "internal_split c (a, b) = c a b" by (simp only: internal_split_def split_conv) hide const internal_split use "Tools/split_rule.ML" setup SplitRule.setup lemmas prod_caseI = prod.cases [THEN iffD2, standard] lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p" by auto lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x" by (auto simp: split_tupled_all) lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" by (induct p) auto lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" by (induct p) auto lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))" by (simp add: expand_fun_eq) declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!] declare prod_caseE' [elim!] prod_caseE [elim!] lemma prod_case_split: "prod_case = split" by (auto simp add: expand_fun_eq) lemma prod_case_beta: "prod_case f p = f (fst p) (snd p)" unfolding prod_case_split split_beta .. subsection {* Further cases/induct rules for tuples *} lemma prod_cases3 [cases type]: obtains (fields) a b c where "y = (a, b, c)" by (cases y, case_tac b) blast lemma prod_induct3 [case_names fields, induct type]: "(!!a b c. P (a, b, c)) ==> P x" by (cases x) blast lemma prod_cases4 [cases type]: obtains (fields) a b c d where "y = (a, b, c, d)" by (cases y, case_tac c) blast lemma prod_induct4 [case_names fields, induct type]: "(!!a b c d. P (a, b, c, d)) ==> P x" by (cases x) blast lemma prod_cases5 [cases type]: obtains (fields) a b c d e where "y = (a, b, c, d, e)" by (cases y, case_tac d) blast lemma prod_induct5 [case_names fields, induct type]: "(!!a b c d e. P (a, b, c, d, e)) ==> P x" by (cases x) blast lemma prod_cases6 [cases type]: obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)" by (cases y, case_tac e) blast lemma prod_induct6 [case_names fields, induct type]: "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x" by (cases x) blast lemma prod_cases7 [cases type]: obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)" by (cases y, case_tac f) blast lemma prod_induct7 [case_names fields, induct type]: "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x" by (cases x) blast subsubsection {* Derived operations *} text {* The composition-uncurry combinator. *} notation fcomp (infixl "o>" 60) definition scomp :: "('a \ 'b \ 'c) \ ('b \ 'c \ 'd) \ 'a \ 'd" (infixl "o\" 60) where "f o\ g = (\x. split g (f x))" lemma scomp_apply: "(f o\ g) x = split g (f x)" by (simp add: scomp_def) lemma Pair_scomp: "Pair x o\ f = f x" by (simp add: expand_fun_eq scomp_apply) lemma scomp_Pair: "x o\ Pair = x" by (simp add: expand_fun_eq scomp_apply) lemma scomp_scomp: "(f o\ g) o\ h = f o\ (\x. g x o\ h)" by (simp add: expand_fun_eq split_twice scomp_def) lemma scomp_fcomp: "(f o\ g) o> h = f o\ (\x. g x o> h)" by (simp add: expand_fun_eq scomp_apply fcomp_def split_def) lemma fcomp_scomp: "(f o> g) o\ h = f o> (g o\ h)" by (simp add: expand_fun_eq scomp_apply fcomp_apply) no_notation fcomp (infixl "o>" 60) no_notation scomp (infixl "o\" 60) text {* @{term prod_fun} --- action of the product functor upon functions. *} definition prod_fun :: "('a \ 'c) \ ('b \ 'd) \ 'a \ 'b \ 'c \ 'd" where [code func del]: "prod_fun f g = (\(x, y). (f x, g y))" lemma prod_fun [simp, code func]: "prod_fun f g (a, b) = (f a, g b)" by (simp add: prod_fun_def) lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)" by (rule ext) auto lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)" by (rule ext) auto lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r" apply (rule image_eqI) apply (rule prod_fun [symmetric], assumption) done lemma prod_fun_imageE [elim!]: assumes major: "c: (prod_fun f g)`r" and cases: "!!x y. [| c=(f(x),g(y)); (x,y):r |] ==> P" shows P apply (rule major [THEN imageE]) apply (rule_tac p = x in PairE) apply (rule cases) apply (blast intro: prod_fun) apply blast done definition apfst :: "('a \ 'c) \ 'a \ 'b \ 'c \ 'b" where [code func del]: "apfst f = prod_fun f id" definition apsnd :: "('b \ 'c) \ 'a \ 'b \ 'a \ 'c" where [code func del]: "apsnd f = prod_fun id f" lemma apfst_conv [simp, code]: "apfst f (x, y) = (f x, y)" by (simp add: apfst_def) lemma upd_snd_conv [simp, code]: "apsnd f (x, y) = (x, f y)" by (simp add: apsnd_def) text {* Disjoint union of a family of sets -- Sigma. *} definition Sigma :: "['a set, 'a => 'b set] => ('a \ 'b) set" where Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}" abbreviation Times :: "['a set, 'b set] => ('a * 'b) set" (infixr "<*>" 80) where "A <*> B == Sigma A (%_. B)" notation (xsymbols) Times (infixr "\" 80) notation (HTML output) Times (infixr "\" 80) syntax "@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10) translations "SIGMA x:A. B" == "Product_Type.Sigma A (%x. B)" lemma SigmaI [intro!]: "[| a:A; b:B(a) |] ==> (a,b) : Sigma A B" by (unfold Sigma_def) blast lemma SigmaE [elim!]: "[| c: Sigma A B; !!x y.[| x:A; y:B(x); c=(x,y) |] ==> P |] ==> P" -- {* The general elimination rule. *} by (unfold Sigma_def) blast text {* Elimination of @{term "(a, b) : A \ B"} -- introduces no eigenvariables. *} lemma SigmaD1: "(a, b) : Sigma A B ==> a : A" by blast lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a" by blast lemma SigmaE2: "[| (a, b) : Sigma A B; [| a:A; b:B(a) |] ==> P |] ==> P" by blast lemma Sigma_cong: "\A = B; !!x. x \ B \ C x = D x\ \ (SIGMA x: A. C x) = (SIGMA x: B. D x)" by auto lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D" by blast lemma Sigma_empty1 [simp]: "Sigma {} B = {}" by blast lemma Sigma_empty2 [simp]: "A <*> {} = {}" by blast lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV" by auto lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)" by auto lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV" by auto lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))" by blast lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)" by blast lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)" by (blast elim: equalityE) lemma SetCompr_Sigma_eq: "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))" by blast lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q" by blast lemma UN_Times_distrib: "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)" -- {* Suggested by Pierre Chartier *} by blast lemma split_paired_Ball_Sigma [simp,noatp]: "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))" by blast lemma split_paired_Bex_Sigma [simp,noatp]: "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))" by blast lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))" by blast lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))" by blast lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))" by blast lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))" by blast lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))" by blast lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))" by blast lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)" by blast text {* Non-dependent versions are needed to avoid the need for higher-order matching, especially when the rules are re-oriented. *} lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)" by blast lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)" by blast lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)" by blast subsubsection {* Code generator setup *} instance * :: (eq, eq) eq .. lemma [code func]: "(x1\'a\eq, y1\'b\eq) = (x2, y2) \ x1 = x2 \ y1 = y2" by auto lemma split_case_cert: assumes "CASE \ split f" shows "CASE (a, b) \ f a b" using assms by simp setup {* Code.add_case @{thm split_case_cert} *} code_type * (SML infix 2 "*") (OCaml infix 2 "*") (Haskell "!((_),/ (_))") code_instance * :: eq (Haskell -) code_const "op = \ 'a\eq \ 'b\eq \ 'a \ 'b \ bool" (Haskell infixl 4 "==") code_const Pair (SML "!((_),/ (_))") (OCaml "!((_),/ (_))") (Haskell "!((_),/ (_))") code_const fst and snd (Haskell "fst" and "snd") types_code "*" ("(_ */ _)") attach (term_of) {* fun term_of_id_42 aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y; *} attach (test) {* fun gen_id_42 aG aT bG bT i = let val (x, t) = aG i; val (y, u) = bG i in ((x, y), fn () => HOLogic.pair_const aT bT $ t () $ u ()) end; *} consts_code "Pair" ("(_,/ _)") setup {* let fun strip_abs_split 0 t = ([], t) | strip_abs_split i (Abs (s, T, t)) = let val s' = Codegen.new_name t s; val v = Free (s', T) in apfst (cons v) (strip_abs_split (i-1) (subst_bound (v, t))) end | strip_abs_split i (u as Const ("split", _) $ t) = (case strip_abs_split (i+1) t of (v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u) | _ => ([], u)) | strip_abs_split i t = ([], t); fun let_codegen thy defs gr dep thyname brack t = (case strip_comb t of (t1 as Const ("Let", _), t2 :: t3 :: ts) => let fun dest_let (l as Const ("Let", _) $ t $ u) = (case strip_abs_split 1 u of ([p], u') => apfst (cons (p, t)) (dest_let u') | _ => ([], l)) | dest_let t = ([], t); fun mk_code (gr, (l, r)) = let val (gr1, pl) = Codegen.invoke_codegen thy defs dep thyname false (gr, l); val (gr2, pr) = Codegen.invoke_codegen thy defs dep thyname false (gr1, r); in (gr2, (pl, pr)) end in case dest_let (t1 $ t2 $ t3) of ([], _) => NONE | (ps, u) => let val (gr1, qs) = foldl_map mk_code (gr, ps); val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u); val (gr3, pargs) = foldl_map (Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts) in SOME (gr3, Codegen.mk_app brack (Pretty.blk (0, [Codegen.str "let ", Pretty.blk (0, List.concat (separate [Codegen.str ";", Pretty.brk 1] (map (fn (pl, pr) => [Pretty.block [Codegen.str "val ", pl, Codegen.str " =", Pretty.brk 1, pr]]) qs))), Pretty.brk 1, Codegen.str "in ", pu, Pretty.brk 1, Codegen.str "end"])) pargs) end end | _ => NONE); fun split_codegen thy defs gr dep thyname brack t = (case strip_comb t of (t1 as Const ("split", _), t2 :: ts) => (case strip_abs_split 1 (t1 $ t2) of ([p], u) => let val (gr1, q) = Codegen.invoke_codegen thy defs dep thyname false (gr, p); val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u); val (gr3, pargs) = foldl_map (Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts) in SOME (gr2, Codegen.mk_app brack (Pretty.block [Codegen.str "(fn ", q, Codegen.str " =>", Pretty.brk 1, pu, Codegen.str ")"]) pargs) end | _ => NONE) | _ => NONE); in Codegen.add_codegen "let_codegen" let_codegen #> Codegen.add_codegen "split_codegen" split_codegen end *} subsection {* Legacy bindings *} ML {* val Collect_split = thm "Collect_split"; val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1"; val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2"; val PairE = thm "PairE"; val Pair_Rep_inject = thm "Pair_Rep_inject"; val Pair_def = thm "Pair_def"; val Pair_eq = thm "Pair_eq"; val Pair_fst_snd_eq = thm "Pair_fst_snd_eq"; val ProdI = thm "ProdI"; val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq"; val SigmaD1 = thm "SigmaD1"; val SigmaD2 = thm "SigmaD2"; val SigmaE = thm "SigmaE"; val SigmaE2 = thm "SigmaE2"; val SigmaI = thm "SigmaI"; val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1"; val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2"; val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1"; val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2"; val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1"; val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2"; val Sigma_Union = thm "Sigma_Union"; val Sigma_def = thm "Sigma_def"; val Sigma_empty1 = thm "Sigma_empty1"; val Sigma_empty2 = thm "Sigma_empty2"; val Sigma_mono = thm "Sigma_mono"; val The_split = thm "The_split"; val The_split_eq = thm "The_split_eq"; val The_split_eq = thm "The_split_eq"; val Times_Diff_distrib1 = thm "Times_Diff_distrib1"; val Times_Int_distrib1 = thm "Times_Int_distrib1"; val Times_Un_distrib1 = thm "Times_Un_distrib1"; val Times_eq_cancel2 = thm "Times_eq_cancel2"; val Times_subset_cancel2 = thm "Times_subset_cancel2"; val UNIV_Times_UNIV = thm "UNIV_Times_UNIV"; val UN_Times_distrib = thm "UN_Times_distrib"; val Unity_def = thm "Unity_def"; val cond_split_eta = thm "cond_split_eta"; val fst_conv = thm "fst_conv"; val fst_def = thm "fst_def"; val fst_eqD = thm "fst_eqD"; val inj_on_Abs_Prod = thm "inj_on_Abs_Prod"; val mem_Sigma_iff = thm "mem_Sigma_iff"; val mem_splitE = thm "mem_splitE"; val mem_splitI = thm "mem_splitI"; val mem_splitI2 = thm "mem_splitI2"; val prod_eqI = thm "prod_eqI"; val prod_fun = thm "prod_fun"; val prod_fun_compose = thm "prod_fun_compose"; val prod_fun_def = thm "prod_fun_def"; val prod_fun_ident = thm "prod_fun_ident"; val prod_fun_imageE = thm "prod_fun_imageE"; val prod_fun_imageI = thm "prod_fun_imageI"; val prod_induct = thm "prod_induct"; val snd_conv = thm "snd_conv"; val snd_def = thm "snd_def"; val snd_eqD = thm "snd_eqD"; val split = thm "split"; val splitD = thm "splitD"; val splitD' = thm "splitD'"; val splitE = thm "splitE"; val splitE' = thm "splitE'"; val splitE2 = thm "splitE2"; val splitI = thm "splitI"; val splitI2 = thm "splitI2"; val splitI2' = thm "splitI2'"; val split_beta = thm "split_beta"; val split_conv = thm "split_conv"; val split_def = thm "split_def"; val split_eta = thm "split_eta"; val split_eta_SetCompr = thm "split_eta_SetCompr"; val split_eta_SetCompr2 = thm "split_eta_SetCompr2"; val split_paired_All = thm "split_paired_All"; val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma"; val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma"; val split_paired_Ex = thm "split_paired_Ex"; val split_paired_The = thm "split_paired_The"; val split_paired_all = thm "split_paired_all"; val split_part = thm "split_part"; val split_split = thm "split_split"; val split_split_asm = thm "split_split_asm"; val split_tupled_all = thms "split_tupled_all"; val split_weak_cong = thm "split_weak_cong"; val surj_pair = thm "surj_pair"; val surjective_pairing = thm "surjective_pairing"; val unit_abs_eta_conv = thm "unit_abs_eta_conv"; val unit_all_eq1 = thm "unit_all_eq1"; val unit_all_eq2 = thm "unit_all_eq2"; val unit_eq = thm "unit_eq"; *} subsection {* Further inductive packages *} use "Tools/inductive_realizer.ML" setup InductiveRealizer.setup use "Tools/inductive_set_package.ML" setup InductiveSetPackage.setup use "Tools/datatype_realizer.ML" setup DatatypeRealizer.setup end