(* Title: HOL/Product_Type.thy Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1992 University of Cambridge *) header {* Cartesian products *} theory Product_Type imports Typedef Inductive Fun keywords "inductive_set" "coinductive_set" :: thy_decl begin subsection {* @{typ bool} is a datatype *} free_constructors case_bool for True | False by auto text {* Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}. *} setup {* Sign.mandatory_path "old" *} old_rep_datatype True False by (auto intro: bool_induct) setup {* Sign.parent_path *} text {* But erase the prefix for properties that are not generated by @{text free_constructors}. *} setup {* Sign.mandatory_path "bool" *} lemmas induct = old.bool.induct lemmas inducts = old.bool.inducts lemmas rec = old.bool.rec lemmas simps = bool.distinct bool.case bool.rec setup {* Sign.parent_path *} declare case_split [cases type: bool] -- "prefer plain propositional version" lemma shows [code]: "HOL.equal False P \ \ P" and [code]: "HOL.equal True P \ P" and [code]: "HOL.equal P False \ \ P" and [code]: "HOL.equal P True \ P" and [code nbe]: "HOL.equal P P \ True" by (simp_all add: equal) lemma If_case_cert: assumes "CASE \ (\b. If b f g)" shows "(CASE True \ f) &&& (CASE False \ g)" using assms by simp_all setup {* Code.add_case @{thm If_case_cert} *} code_printing constant "HOL.equal :: bool \ bool \ bool" \ (Haskell) infix 4 "==" | class_instance "bool" :: "equal" \ (Haskell) - subsection {* The @{text unit} type *} typedef unit = "{True}" by auto definition Unity :: unit ("'(')") where "() = Abs_unit True" lemma unit_eq [no_atp]: "u = ()" by (induct u) (simp add: Unity_def) text {* Simplification procedure for @{thm [source] unit_eq}. Cannot use this rule directly --- it loops! *} simproc_setup unit_eq ("x::unit") = {* fn _ => fn _ => fn ct => if HOLogic.is_unit (term_of ct) then NONE else SOME (mk_meta_eq @{thm unit_eq}) *} free_constructors case_unit for "()" by auto text {* Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}. *} setup {* Sign.mandatory_path "old" *} old_rep_datatype "()" by simp setup {* Sign.parent_path *} text {* But erase the prefix for properties that are not generated by @{text free_constructors}. *} setup {* Sign.mandatory_path "unit" *} lemmas induct = old.unit.induct lemmas inducts = old.unit.inducts lemmas rec = old.unit.rec lemmas simps = unit.case unit.rec setup {* Sign.parent_path *} 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 simproc @{text unit_eq} 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]: "(%u::unit. f ()) = f" by (rule ext) simp lemma UNIV_unit: "UNIV = {()}" by auto instantiation unit :: default begin definition "default = ()" instance .. end instantiation unit :: "{complete_boolean_algebra, complete_linorder, wellorder}" begin definition less_eq_unit :: "unit \ unit \ bool" where "(_::unit) \ _ \ True" lemma less_eq_unit [iff]: "(u::unit) \ v" by (simp add: less_eq_unit_def) definition less_unit :: "unit \ unit \ bool" where "(_::unit) < _ \ False" lemma less_unit [iff]: "\ (u::unit) < v" by (simp_all add: less_eq_unit_def less_unit_def) definition bot_unit :: unit where [code_unfold]: "\ = ()" definition top_unit :: unit where [code_unfold]: "\ = ()" definition inf_unit :: "unit \ unit \ unit" where [simp]: "_ \ _ = ()" definition sup_unit :: "unit \ unit \ unit" where [simp]: "_ \ _ = ()" definition Inf_unit :: "unit set \ unit" where [simp]: "\_ = ()" definition Sup_unit :: "unit set \ unit" where [simp]: "\_ = ()" definition uminus_unit :: "unit \ unit" where [simp]: "- _ = ()" declare less_eq_unit_def [abs_def, code_unfold] less_unit_def [abs_def, code_unfold] inf_unit_def [abs_def, code_unfold] sup_unit_def [abs_def, code_unfold] Inf_unit_def [abs_def, code_unfold] Sup_unit_def [abs_def, code_unfold] uminus_unit_def [abs_def, code_unfold] instance by intro_classes auto end lemma [code]: "HOL.equal (u\unit) v \ True" unfolding equal unit_eq [of u] unit_eq [of v] by rule+ code_printing type_constructor unit \ (SML) "unit" and (OCaml) "unit" and (Haskell) "()" and (Scala) "Unit" | constant Unity \ (SML) "()" and (OCaml) "()" and (Haskell) "()" and (Scala) "()" | class_instance unit :: equal \ (Haskell) - | constant "HOL.equal :: unit \ unit \ bool" \ (Haskell) infix 4 "==" code_reserved SML unit code_reserved OCaml unit code_reserved Scala Unit subsection {* The product type *} subsubsection {* Type definition *} definition Pair_Rep :: "'a \ 'b \ 'a \ 'b \ bool" where "Pair_Rep a b = (\x y. x = a \ y = b)" definition "prod = {f. \a b. f = Pair_Rep (a\'a) (b\'b)}" typedef ('a, 'b) prod (infixr "*" 20) = "prod :: ('a \ 'b \ bool) set" unfolding prod_def by auto type_notation (xsymbols) "prod" ("(_ \/ _)" [21, 20] 20) type_notation (HTML output) "prod" ("(_ \/ _)" [21, 20] 20) definition Pair :: "'a \ 'b \ 'a \ 'b" where "Pair a b = Abs_prod (Pair_Rep a b)" lemma prod_cases: "(\a b. P (Pair a b)) \ P p" by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def) free_constructors case_prod for Pair fst snd proof - fix P :: bool and p :: "'a \ 'b" show "(\x1 x2. p = Pair x1 x2 \ P) \ P" by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def) next fix a c :: 'a and b d :: 'b have "Pair_Rep a b = Pair_Rep c d \ a = c \ b = d" by (auto simp add: Pair_Rep_def fun_eq_iff) moreover have "Pair_Rep a b \ prod" and "Pair_Rep c d \ prod" by (auto simp add: prod_def) ultimately show "Pair a b = Pair c d \ a = c \ b = d" by (simp add: Pair_def Abs_prod_inject) qed text {* Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}. *} setup {* Sign.mandatory_path "old" *} old_rep_datatype Pair by (erule prod_cases) (rule prod.inject) setup {* Sign.parent_path *} text {* But erase the prefix for properties that are not generated by @{text free_constructors}. *} setup {* Sign.mandatory_path "prod" *} declare old.prod.inject[iff del] lemmas induct = old.prod.induct lemmas inducts = old.prod.inducts lemmas rec = old.prod.rec lemmas simps = prod.inject prod.case prod.rec setup {* Sign.parent_path *} declare prod.case [nitpick_simp del] declare prod.case_cong_weak [cong del] subsubsection {* Tuple syntax *} abbreviation (input) split :: "('a \ 'b \ 'c) \ 'a \ 'b \ 'c" where "split \ case_prod" text {* Patterns -- extends pre-defined type @{typ pttrn} used in abstractions. *} nonterminal tuple_args and 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)" == "CONST Pair x y" "_pattern x y" => "CONST Pair x y" "_patterns x y" => "CONST Pair x y" "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))" "%(x, y, zs). b" == "CONST case_prod (%x (y, zs). b)" "%(x, y). b" == "CONST case_prod (%x y. b)" "_abs (CONST 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 *} (*reconstruct 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') = Syntax_Trans.atomic_abs_tr' abs; val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t'); in Syntax.const @{syntax_const "_abs"} $ (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' end | split_tr' [Abs (x, T, (s as Const (@{const_syntax case_prod}, _) $ t))] = (* split (%x. (split (%y z. t))) => %(x,y,z). t *) let val Const (@{syntax_const "_abs"}, _) $ (Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' = split_tr' [t]; val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t'); in Syntax.const @{syntax_const "_abs"} $ (Syntax.const @{syntax_const "_pattern"} $ x' $ (Syntax.const @{syntax_const "_patterns"} $ y $ z)) $ t'' end | split_tr' [Const (@{const_syntax case_prod}, _) $ 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 (@{syntax_const "_abs"}, _) $ x_y $ Abs abs] = (* split (%pttrn z. t) => %(pttrn,z). t *) let val (z, t) = Syntax_Trans.atomic_abs_tr' abs in Syntax.const @{syntax_const "_abs"} $ (Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t end | split_tr' _ = raise Match; in [(@{const_syntax case_prod}, K 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 (@{const_syntax case_prod}, _) => raise Match | _ => let val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; val (y, t') = Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0); val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, xT, t'); in Syntax.const @{syntax_const "_abs"} $ (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' end) | split_guess_names_tr' T [t] = (case head_of t of Const (@{const_syntax case_prod}, _) => raise Match | _ => let val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; val (y, t') = Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0); val (x', t'') = Syntax_Trans.atomic_abs_tr' ("x", xT, t'); in Syntax.const @{syntax_const "_abs"} $ (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' end) | split_guess_names_tr' _ _ = raise Match; in [(@{const_syntax case_prod}, K split_guess_names_tr')] end *} subsubsection {* Code generator setup *} code_printing type_constructor prod \ (SML) infix 2 "*" and (OCaml) infix 2 "*" and (Haskell) "!((_),/ (_))" and (Scala) "((_),/ (_))" | constant Pair \ (SML) "!((_),/ (_))" and (OCaml) "!((_),/ (_))" and (Haskell) "!((_),/ (_))" and (Scala) "!((_),/ (_))" | class_instance prod :: equal \ (Haskell) - | constant "HOL.equal :: 'a \ 'b \ 'a \ 'b \ bool" \ (Haskell) infix 4 "==" subsubsection {* Fundamental operations and properties *} lemma Pair_inject: assumes "(a, b) = (a', b')" and "a = a' ==> b = b' ==> R" shows R using assms by simp lemma surj_pair [simp]: "EX x y. p = (x, y)" by (cases p) simp code_printing constant fst \ (Haskell) "fst" | constant snd \ (Haskell) "snd" lemma case_prod_unfold [nitpick_unfold]: "case_prod = (%c p. c (fst p) (snd p))" by (simp add: fun_eq_iff split: prod.split) lemma fst_eqD: "fst (x, y) = a ==> x = a" by simp lemma snd_eqD: "snd (x, y) = a ==> y = a" by simp lemmas surjective_pairing = prod.collapse [symmetric] lemma prod_eq_iff: "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: prod_eq_iff) lemma split_conv [simp, code]: "split f (a, b) = f a b" by (fact prod.case) 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 split_Pair [simp]: "(\(x, y). (x, y)) = id" by (simp add: fun_eq_iff split: prod.split) lemma split_eta: "(\(x, y). f (x, y)) = f" -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *} by (simp add: fun_eq_iff split: prod.split) 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" by (cases p) simp lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))" by (simp add: case_prod_unfold) lemma split_weak_cong: "p = q \ split c p = split c q" -- {* Prevents simplification of @{term c}: much faster *} by (fact prod.case_cong_weak) lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g" by (simp add: split_eta) lemma split_paired_all [no_atp]: "(!!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 lemma case_prod_distrib: "f (case x of (x, y) \ g x y) = (case x of (x, y) \ f (g x y))" by (cases x) simp 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 *) local (* filtering with exists_paired_all is an essential optimization *) fun exists_paired_all (Const (@{const_name Pure.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 = simpset_of (put_simpset HOL_basic_ss @{context} addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}] addsimprocs [@{simproc unit_eq}]); in fun split_all_tac ctxt = SUBGOAL (fn (t, i) => if exists_paired_all t then safe_full_simp_tac (put_simpset ss ctxt) i else no_tac); fun unsafe_split_all_tac ctxt = SUBGOAL (fn (t, i) => if exists_paired_all t then full_simp_tac (put_simpset ss ctxt) i else no_tac); fun split_all ctxt th = if exists_paired_all (Thm.prop_of th) then full_simplify (put_simpset ss ctxt) th else th; end; *} setup {* map_theory_claset (fn ctxt => ctxt addSbefore ("split_all_tac", split_all_tac)) *} lemma split_paired_All [simp, no_atp]: "(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, no_atp]: "(EX x. P x) = (EX a b. P (a, b))" by fast lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))" -- {* Can't be added to simpset: loops! *} 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 = simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms cond_split_eta}); fun Pair_pat k 0 (Bound m) = (m = k) | Pair_pat k i (Const (@{const_name 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 (@{const_name case_prod}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t | split_pat tp i _ = NONE; fun metaeq ctxt lhs rhs = mk_meta_eq (Goal.prove ctxt [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))) (K (simp_tac (put_simpset cond_split_eta_ss ctxt) 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; in fun beta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t) $ arg) = (case split_pat beta_term_pat 1 t of SOME (i, f) => SOME (metaeq ctxt s (subst arg 0 i f)) | NONE => NONE) | beta_proc _ _ = NONE; fun eta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t)) = (case split_pat eta_term_pat 1 t of SOME (_, ft) => SOME (metaeq ctxt s (let val (f $ arg) = ft in f end)) | NONE => NONE) | eta_proc _ _ = NONE; end; *} simproc_setup split_beta ("split f z") = {* fn _ => fn ctxt => fn ct => beta_proc ctxt (term_of ct) *} simproc_setup split_eta ("split f") = {* fn _ => fn ctxt => fn ct => eta_proc ctxt (term_of ct) *} lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)" by (subst surjective_pairing, rule split_conv) lemma split_beta': "(\(x,y). f x y) = (\x. f (fst x) (snd x))" by (auto simp: fun_eq_iff) lemma split_split [no_atp]: "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 [no_atp]: "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] prod.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 lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" by (induct p) auto 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 case_prod_unfold] 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 (@{const_name case_prod},_) $ _ $ (Const (@{const_name 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; in fun split_conv_tac ctxt = SUBGOAL (fn (t, i) => if exists_p_split t then safe_full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms split_conv}) 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) *) setup {* map_theory_claset (fn ctxt => ctxt addSbefore ("split_conv_tac", split_conv_tac)) *} lemma split_eta_SetCompr [simp, no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P" by (rule ext) fast lemma split_eta_SetCompr2 [simp, no_atp]: "(%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}. *} lemmas case_prodI = prod.case [THEN iffD2] lemma case_prodI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> case_prod c p" by (fact splitI2) lemma case_prodI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> case_prod c p x" by (fact splitI2') lemma case_prodE: "case_prod c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" by (fact splitE) lemma case_prodE': "case_prod c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" by (fact splitE') declare case_prodI [intro!] lemma case_prod_beta: "case_prod f p = f (fst p) (snd p)" by (fact split_beta) 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 lemma split_def: "split = (\c p. c (fst p) (snd p))" by (fact case_prod_unfold) 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) ML_file "Tools/split_rule.ML" setup Split_Rule.setup hide_const internal_split subsubsection {* Derived operations *} definition curry :: "('a \ 'b \ 'c) \ 'a \ 'b \ 'c" where "curry = (\c x y. c (x, y))" lemma curry_conv [simp, code]: "curry f a b = f (a, b)" by (simp add: curry_def) 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 curry_K: "curry (\x. c) = (\x y. c)" by(simp add: fun_eq_iff) text {* The composition-uncurry combinator. *} notation fcomp (infixl "\>" 60) definition scomp :: "('a \ 'b \ 'c) \ ('b \ 'c \ 'd) \ 'a \ 'd" (infixl "\\" 60) where "f \\ g = (\x. case_prod g (f x))" lemma scomp_unfold: "scomp = (\f g x. g (fst (f x)) (snd (f x)))" by (simp add: fun_eq_iff scomp_def case_prod_unfold) lemma scomp_apply [simp]: "(f \\ g) x = case_prod g (f x)" by (simp add: scomp_unfold case_prod_unfold) lemma Pair_scomp: "Pair x \\ f = f x" by (simp add: fun_eq_iff) lemma scomp_Pair: "x \\ Pair = x" by (simp add: fun_eq_iff) lemma scomp_scomp: "(f \\ g) \\ h = f \\ (\x. g x \\ h)" by (simp add: fun_eq_iff scomp_unfold) lemma scomp_fcomp: "(f \\ g) \> h = f \\ (\x. g x \> h)" by (simp add: fun_eq_iff scomp_unfold fcomp_def) lemma fcomp_scomp: "(f \> g) \\ h = f \> (g \\ h)" by (simp add: fun_eq_iff scomp_unfold) code_printing constant scomp \ (Eval) infixl 3 "#->" no_notation fcomp (infixl "\>" 60) no_notation scomp (infixl "\\" 60) text {* @{term map_prod} --- action of the product functor upon functions. *} definition map_prod :: "('a \ 'c) \ ('b \ 'd) \ 'a \ 'b \ 'c \ 'd" where "map_prod f g = (\(x, y). (f x, g y))" lemma map_prod_simp [simp, code]: "map_prod f g (a, b) = (f a, g b)" by (simp add: map_prod_def) functor map_prod: map_prod by (auto simp add: split_paired_all) lemma fst_map_prod [simp]: "fst (map_prod f g x) = f (fst x)" by (cases x) simp_all lemma snd_prod_fun [simp]: "snd (map_prod f g x) = g (snd x)" by (cases x) simp_all lemma fst_comp_map_prod [simp]: "fst \ map_prod f g = f \ fst" by (rule ext) simp_all lemma snd_comp_map_prod [simp]: "snd \ map_prod f g = g \ snd" by (rule ext) simp_all lemma map_prod_compose: "map_prod (f1 o f2) (g1 o g2) = (map_prod f1 g1 o map_prod f2 g2)" by (rule ext) (simp add: map_prod.compositionality comp_def) lemma map_prod_ident [simp]: "map_prod (%x. x) (%y. y) = (%z. z)" by (rule ext) (simp add: map_prod.identity) lemma map_prod_imageI [intro]: "(a, b) \ R \ (f a, g b) \ map_prod f g ` R" by (rule image_eqI) simp_all lemma prod_fun_imageE [elim!]: assumes major: "c \ map_prod f g ` R" and cases: "\x y. c = (f x, g y) \ (x, y) \ R \ P" shows P apply (rule major [THEN imageE]) apply (case_tac x) apply (rule cases) apply simp_all done definition apfst :: "('a \ 'c) \ 'a \ 'b \ 'c \ 'b" where "apfst f = map_prod f id" definition apsnd :: "('b \ 'c) \ 'a \ 'b \ 'a \ 'c" where "apsnd f = map_prod id f" lemma apfst_conv [simp, code]: "apfst f (x, y) = (f x, y)" by (simp add: apfst_def) lemma apsnd_conv [simp, code]: "apsnd f (x, y) = (x, f y)" by (simp add: apsnd_def) lemma fst_apfst [simp]: "fst (apfst f x) = f (fst x)" by (cases x) simp lemma fst_comp_apfst [simp]: "fst \ apfst f = f \ fst" by (simp add: fun_eq_iff) lemma fst_apsnd [simp]: "fst (apsnd f x) = fst x" by (cases x) simp lemma fst_comp_apsnd [simp]: "fst \ apsnd f = fst" by (simp add: fun_eq_iff) lemma snd_apfst [simp]: "snd (apfst f x) = snd x" by (cases x) simp lemma snd_comp_apfst [simp]: "snd \ apfst f = snd" by (simp add: fun_eq_iff) lemma snd_apsnd [simp]: "snd (apsnd f x) = f (snd x)" by (cases x) simp lemma snd_comp_apsnd [simp]: "snd \ apsnd f = f \ snd" by (simp add: fun_eq_iff) lemma apfst_compose: "apfst f (apfst g x) = apfst (f \ g) x" by (cases x) simp lemma apsnd_compose: "apsnd f (apsnd g x) = apsnd (f \ g) x" by (cases x) simp lemma apfst_apsnd [simp]: "apfst f (apsnd g x) = (f (fst x), g (snd x))" by (cases x) simp lemma apsnd_apfst [simp]: "apsnd f (apfst g x) = (g (fst x), f (snd x))" by (cases x) simp lemma apfst_id [simp] : "apfst id = id" by (simp add: fun_eq_iff) lemma apsnd_id [simp] : "apsnd id = id" by (simp add: fun_eq_iff) lemma apfst_eq_conv [simp]: "apfst f x = apfst g x \ f (fst x) = g (fst x)" by (cases x) simp lemma apsnd_eq_conv [simp]: "apsnd f x = apsnd g x \ f (snd x) = g (snd x)" by (cases x) simp lemma apsnd_apfst_commute: "apsnd f (apfst g p) = apfst g (apsnd f p)" by simp context begin local_setup {* Local_Theory.map_naming (Name_Space.mandatory_path "prod") *} definition swap :: "'a \ 'b \ 'b \ 'a" where "swap p = (snd p, fst p)" end lemma swap_simp [simp]: "prod.swap (x, y) = (y, x)" by (simp add: prod.swap_def) lemma swap_swap [simp]: "prod.swap (prod.swap p) = p" by (cases p) simp lemma swap_comp_swap [simp]: "prod.swap \ prod.swap = id" by (simp add: fun_eq_iff) lemma pair_in_swap_image [simp]: "(y, x) \ prod.swap ` A \ (x, y) \ A" by (auto intro!: image_eqI) lemma inj_swap [simp]: "inj_on prod.swap A" by (rule inj_onI) auto lemma swap_inj_on: "inj_on (\(i, j). (j, i)) A" by (rule inj_onI) auto lemma surj_swap [simp]: "surj prod.swap" by (rule surjI [of _ prod.swap]) simp lemma bij_swap [simp]: "bij prod.swap" by (simp add: bij_def) lemma case_swap [simp]: "(case prod.swap p of (y, x) \ f x y) = (case p of (x, y) \ f x y)" by (cases p) simp 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) hide_const (open) Times syntax "_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10) translations "SIGMA x:A. B" == "CONST 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, no_atp]: "(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, no_atp]: "(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 (fact Sigma_Un_distrib1) lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)" by (fact Sigma_Int_distrib1) lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)" by (fact Sigma_Diff_distrib1) lemma Times_empty[simp]: "A \ B = {} \ A = {} \ B = {}" by auto lemma times_eq_iff: "A \ B = C \ D \ A = C \ B = D \ ((A = {} \ B = {}) \ (C = {} \ D = {}))" by auto lemma fst_image_times[simp]: "fst ` (A \ B) = (if B = {} then {} else A)" by force lemma snd_image_times[simp]: "snd ` (A \ B) = (if A = {} then {} else B)" by force lemma vimage_fst: "fst -` A = A \ UNIV" by auto lemma vimage_snd: "snd -` A = UNIV \ A" by auto lemma insert_times_insert[simp]: "insert a A \ insert b B = insert (a,b) (A \ insert b B \ insert a A \ B)" by blast lemma vimage_Times: "f -` (A \ B) = ((fst \ f) -` A) \ ((snd \ f) -` B)" apply auto apply (case_tac "f x") apply auto done lemma times_Int_times: "A \ B \ C \ D = (A \ C) \ (B \ D)" by auto lemma product_swap: "prod.swap ` (A \ B) = B \ A" by (auto simp add: set_eq_iff) lemma swap_product: "(\(i, j). (j, i)) ` (A \ B) = B \ A" by (auto simp add: set_eq_iff) lemma image_split_eq_Sigma: "(\x. (f x, g x)) ` A = Sigma (f ` A) (\x. g ` (f -` {x} \ A))" proof (safe intro!: imageI) fix a b assume *: "a \ A" "b \ A" and eq: "f a = f b" show "(f b, g a) \ (\x. (f x, g x)) ` A" using * eq[symmetric] by auto qed simp_all definition product :: "'a set \ 'b set \ ('a \ 'b) set" where [code_abbrev]: "product A B = A \ B" hide_const (open) product lemma member_product: "x \ Product_Type.product A B \ x \ A \ B" by (simp add: product_def) text {* The following @{const map_prod} lemmas are due to Joachim Breitner: *} lemma map_prod_inj_on: assumes "inj_on f A" and "inj_on g B" shows "inj_on (map_prod f g) (A \ B)" proof (rule inj_onI) fix x :: "'a \ 'c" and y :: "'a \ 'c" assume "x \ A \ B" hence "fst x \ A" and "snd x \ B" by auto assume "y \ A \ B" hence "fst y \ A" and "snd y \ B" by auto assume "map_prod f g x = map_prod f g y" hence "fst (map_prod f g x) = fst (map_prod f g y)" by (auto) hence "f (fst x) = f (fst y)" by (cases x,cases y,auto) with `inj_on f A` and `fst x \ A` and `fst y \ A` have "fst x = fst y" by (auto dest:dest:inj_onD) moreover from `map_prod f g x = map_prod f g y` have "snd (map_prod f g x) = snd (map_prod f g y)" by (auto) hence "g (snd x) = g (snd y)" by (cases x,cases y,auto) with `inj_on g B` and `snd x \ B` and `snd y \ B` have "snd x = snd y" by (auto dest:dest:inj_onD) ultimately show "x = y" by(rule prod_eqI) qed lemma map_prod_surj: fixes f :: "'a \ 'b" and g :: "'c \ 'd" assumes "surj f" and "surj g" shows "surj (map_prod f g)" unfolding surj_def proof fix y :: "'b \ 'd" from `surj f` obtain a where "fst y = f a" by (auto elim:surjE) moreover from `surj g` obtain b where "snd y = g b" by (auto elim:surjE) ultimately have "(fst y, snd y) = map_prod f g (a,b)" by auto thus "\x. y = map_prod f g x" by auto qed lemma map_prod_surj_on: assumes "f ` A = A'" and "g ` B = B'" shows "map_prod f g ` (A \ B) = A' \ B'" unfolding image_def proof(rule set_eqI,rule iffI) fix x :: "'a \ 'c" assume "x \ {y\'a \ 'c. \x\'b \ 'd\A \ B. y = map_prod f g x}" then obtain y where "y \ A \ B" and "x = map_prod f g y" by blast from `image f A = A'` and `y \ A \ B` have "f (fst y) \ A'" by auto moreover from `image g B = B'` and `y \ A \ B` have "g (snd y) \ B'" by auto ultimately have "(f (fst y), g (snd y)) \ (A' \ B')" by auto with `x = map_prod f g y` show "x \ A' \ B'" by (cases y, auto) next fix x :: "'a \ 'c" assume "x \ A' \ B'" hence "fst x \ A'" and "snd x \ B'" by auto from `image f A = A'` and `fst x \ A'` have "fst x \ image f A" by auto then obtain a where "a \ A" and "fst x = f a" by (rule imageE) moreover from `image g B = B'` and `snd x \ B'` obtain b where "b \ B" and "snd x = g b" by auto ultimately have "(fst x, snd x) = map_prod f g (a,b)" by auto moreover from `a \ A` and `b \ B` have "(a , b) \ A \ B" by auto ultimately have "\y \ A \ B. x = map_prod f g y" by auto thus "x \ {x. \y \ A \ B. x = map_prod f g y}" by auto qed subsection {* Simproc for rewriting a set comprehension into a pointfree expression *} ML_file "Tools/set_comprehension_pointfree.ML" setup {* Code_Preproc.map_pre (fn ctxt => ctxt addsimprocs [Raw_Simplifier.make_simproc {name = "set comprehension", lhss = [@{cpat "Collect ?P"}], proc = K Set_Comprehension_Pointfree.code_simproc, identifier = []}]) *} subsection {* Inductively defined sets *} (* simplify {(x1, ..., xn). (x1, ..., xn) : S} to S *) simproc_setup Collect_mem ("Collect t") = {* fn _ => fn ctxt => fn ct => (case term_of ct of S as Const (@{const_name Collect}, Type (@{type_name fun}, [_, T])) $ t => let val (u, _, ps) = HOLogic.strip_psplits t in (case u of (c as Const (@{const_name Set.member}, _)) $ q $ S' => (case try (HOLogic.strip_ptuple ps) q of NONE => NONE | SOME ts => if not (Term.is_open S') andalso ts = map Bound (length ps downto 0) then let val simp = full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm split_paired_all}, @{thm split_conv}]) 1 in SOME (Goal.prove ctxt [] [] (Const (@{const_name Pure.eq}, T --> T --> propT) $ S $ S') (K (EVERY [rtac eq_reflection 1, rtac @{thm subset_antisym} 1, rtac subsetI 1, dtac CollectD 1, simp, rtac subsetI 1, rtac CollectI 1, simp]))) end else NONE) | _ => NONE) end | _ => NONE) *} ML_file "Tools/inductive_set.ML" subsection {* Legacy theorem bindings and duplicates *} lemma PairE: obtains x y where "p = (x, y)" by (fact prod.exhaust) lemmas Pair_eq = prod.inject lemmas fst_conv = prod.sel(1) lemmas snd_conv = prod.sel(2) lemmas pair_collapse = prod.collapse lemmas split = split_conv lemmas Pair_fst_snd_eq = prod_eq_iff hide_const (open) prod end