(* Title: HOL/Set.thy ID: $Id: Set.thy,v 1.101 2005/09/29 10:43:40 paulson Exp $ Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel *) header {* Set theory for higher-order logic *} theory Set imports LOrder begin text {* A set in HOL is simply a predicate. *} subsection {* Basic syntax *} global typedecl 'a set arities set :: (type) type consts "{}" :: "'a set" ("{}") UNIV :: "'a set" insert :: "'a => 'a set => 'a set" Collect :: "('a => bool) => 'a set" -- "comprehension" Int :: "'a set => 'a set => 'a set" (infixl 70) Un :: "'a set => 'a set => 'a set" (infixl 65) UNION :: "'a set => ('a => 'b set) => 'b set" -- "general union" INTER :: "'a set => ('a => 'b set) => 'b set" -- "general intersection" Union :: "'a set set => 'a set" -- "union of a set" Inter :: "'a set set => 'a set" -- "intersection of a set" Pow :: "'a set => 'a set set" -- "powerset" Ball :: "'a set => ('a => bool) => bool" -- "bounded universal quantifiers" Bex :: "'a set => ('a => bool) => bool" -- "bounded existential quantifiers" image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90) syntax "op :" :: "'a => 'a set => bool" ("op :") consts "op :" :: "'a => 'a set => bool" ("(_/ : _)" [50, 51] 50) -- "membership" local instance set :: (type) "{ord, minus}" .. subsection {* Additional concrete syntax *} syntax range :: "('a => 'b) => 'b set" -- "of function" "op ~:" :: "'a => 'a set => bool" ("op ~:") -- "non-membership" "op ~:" :: "'a => 'a set => bool" ("(_/ ~: _)" [50, 51] 50) "@Finset" :: "args => 'a set" ("{(_)}") "@Coll" :: "pttrn => bool => 'a set" ("(1{_./ _})") "@SetCompr" :: "'a => idts => bool => 'a set" ("(1{_ |/_./ _})") "@Collect" :: "idt => 'a set => bool => 'a set" ("(1{_ :/ _./ _})") "@INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" 10) "@UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" 10) "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" 10) "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" 10) "_Ball" :: "pttrn => 'a set => bool => bool" ("(3ALL _:_./ _)" [0, 0, 10] 10) "_Bex" :: "pttrn => 'a set => bool => bool" ("(3EX _:_./ _)" [0, 0, 10] 10) syntax (HOL) "_Ball" :: "pttrn => 'a set => bool => bool" ("(3! _:_./ _)" [0, 0, 10] 10) "_Bex" :: "pttrn => 'a set => bool => bool" ("(3? _:_./ _)" [0, 0, 10] 10) translations "range f" == "f`UNIV" "x ~: y" == "~ (x : y)" "{x, xs}" == "insert x {xs}" "{x}" == "insert x {}" "{x. P}" == "Collect (%x. P)" "{x:A. P}" => "{x. x:A & P}" "UN x y. B" == "UN x. UN y. B" "UN x. B" == "UNION UNIV (%x. B)" "UN x. B" == "UN x:UNIV. B" "INT x y. B" == "INT x. INT y. B" "INT x. B" == "INTER UNIV (%x. B)" "INT x. B" == "INT x:UNIV. B" "UN x:A. B" == "UNION A (%x. B)" "INT x:A. B" == "INTER A (%x. B)" "ALL x:A. P" == "Ball A (%x. P)" "EX x:A. P" == "Bex A (%x. P)" syntax (output) "_setle" :: "'a set => 'a set => bool" ("op <=") "_setle" :: "'a set => 'a set => bool" ("(_/ <= _)" [50, 51] 50) "_setless" :: "'a set => 'a set => bool" ("op <") "_setless" :: "'a set => 'a set => bool" ("(_/ < _)" [50, 51] 50) syntax (xsymbols) "_setle" :: "'a set => 'a set => bool" ("op \") "_setle" :: "'a set => 'a set => bool" ("(_/ \ _)" [50, 51] 50) "_setless" :: "'a set => 'a set => bool" ("op \") "_setless" :: "'a set => 'a set => bool" ("(_/ \ _)" [50, 51] 50) "op Int" :: "'a set => 'a set => 'a set" (infixl "\" 70) "op Un" :: "'a set => 'a set => 'a set" (infixl "\" 65) "op :" :: "'a => 'a set => bool" ("op \") "op :" :: "'a => 'a set => bool" ("(_/ \ _)" [50, 51] 50) "op ~:" :: "'a => 'a set => bool" ("op \") "op ~:" :: "'a => 'a set => bool" ("(_/ \ _)" [50, 51] 50) Union :: "'a set set => 'a set" ("\_" [90] 90) Inter :: "'a set set => 'a set" ("\_" [90] 90) "_Ball" :: "pttrn => 'a set => bool => bool" ("(3\_\_./ _)" [0, 0, 10] 10) "_Bex" :: "pttrn => 'a set => bool => bool" ("(3\_\_./ _)" [0, 0, 10] 10) syntax (HTML output) "_setle" :: "'a set => 'a set => bool" ("op \") "_setle" :: "'a set => 'a set => bool" ("(_/ \ _)" [50, 51] 50) "_setless" :: "'a set => 'a set => bool" ("op \") "_setless" :: "'a set => 'a set => bool" ("(_/ \ _)" [50, 51] 50) "op Int" :: "'a set => 'a set => 'a set" (infixl "\" 70) "op Un" :: "'a set => 'a set => 'a set" (infixl "\" 65) "op :" :: "'a => 'a set => bool" ("op \") "op :" :: "'a => 'a set => bool" ("(_/ \ _)" [50, 51] 50) "op ~:" :: "'a => 'a set => bool" ("op \") "op ~:" :: "'a => 'a set => bool" ("(_/ \ _)" [50, 51] 50) "_Ball" :: "pttrn => 'a set => bool => bool" ("(3\_\_./ _)" [0, 0, 10] 10) "_Bex" :: "pttrn => 'a set => bool => bool" ("(3\_\_./ _)" [0, 0, 10] 10) syntax (xsymbols) "@Collect" :: "idt => 'a set => bool => 'a set" ("(1{_ \/ _./ _})") "@UNION1" :: "pttrns => 'b set => 'b set" ("(3\_./ _)" 10) "@INTER1" :: "pttrns => 'b set => 'b set" ("(3\_./ _)" 10) "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\_\_./ _)" 10) "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\_\_./ _)" 10) (* syntax (xsymbols) "@UNION1" :: "pttrns => 'b set => 'b set" ("(3\(00\<^bsub>_\<^esub>)/ _)" 10) "@INTER1" :: "pttrns => 'b set => 'b set" ("(3\(00\<^bsub>_\<^esub>)/ _)" 10) "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\(00\<^bsub>_\_\<^esub>)/ _)" 10) "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\(00\<^bsub>_\_\<^esub>)/ _)" 10) *) syntax (latex output) "@UNION1" :: "pttrns => 'b set => 'b set" ("(3\(00\<^bsub>_\<^esub>)/ _)" 10) "@INTER1" :: "pttrns => 'b set => 'b set" ("(3\(00\<^bsub>_\<^esub>)/ _)" 10) "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\(00\<^bsub>_\_\<^esub>)/ _)" 10) "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\(00\<^bsub>_\_\<^esub>)/ _)" 10) text{* Note the difference between ordinary xsymbol syntax of indexed unions and intersections (e.g.\ @{text"\a\<^isub>1\A\<^isub>1. B"}) and their \LaTeX\ rendition: @{term"\a\<^isub>1\A\<^isub>1. B"}. The former does not make the index expression a subscript of the union/intersection symbol because this leads to problems with nested subscripts in Proof General. *} translations "op \" => "op <= :: _ set => _ set => bool" "op \" => "op < :: _ set => _ set => bool" typed_print_translation {* let fun le_tr' _ (Type ("fun", (Type ("set", _) :: _))) ts = list_comb (Syntax.const "_setle", ts) | le_tr' _ _ _ = raise Match; fun less_tr' _ (Type ("fun", (Type ("set", _) :: _))) ts = list_comb (Syntax.const "_setless", ts) | less_tr' _ _ _ = raise Match; in [("op <=", le_tr'), ("op <", less_tr')] end *} subsubsection "Bounded quantifiers" syntax "_setlessAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10) "_setlessEx" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10) "_setleAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10) "_setleEx" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) syntax (xsymbols) "_setlessAll" :: "[idt, 'a, bool] => bool" ("(3\_\_./ _)" [0, 0, 10] 10) "_setlessEx" :: "[idt, 'a, bool] => bool" ("(3\_\_./ _)" [0, 0, 10] 10) "_setleAll" :: "[idt, 'a, bool] => bool" ("(3\_\_./ _)" [0, 0, 10] 10) "_setleEx" :: "[idt, 'a, bool] => bool" ("(3\_\_./ _)" [0, 0, 10] 10) syntax (HOL) "_setlessAll" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10) "_setlessEx" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10) "_setleAll" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10) "_setleEx" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) syntax (HTML output) "_setlessAll" :: "[idt, 'a, bool] => bool" ("(3\_\_./ _)" [0, 0, 10] 10) "_setlessEx" :: "[idt, 'a, bool] => bool" ("(3\_\_./ _)" [0, 0, 10] 10) "_setleAll" :: "[idt, 'a, bool] => bool" ("(3\_\_./ _)" [0, 0, 10] 10) "_setleEx" :: "[idt, 'a, bool] => bool" ("(3\_\_./ _)" [0, 0, 10] 10) translations "\A\B. P" => "ALL A. A \ B --> P" "\A\B. P" => "EX A. A \ B & P" "\A\B. P" => "ALL A. A \ B --> P" "\A\B. P" => "EX A. A \ B & P" print_translation {* let fun all_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), Const("op -->",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = (if v=v' andalso T="set" then Syntax.const "_setlessAll" $ Syntax.mark_bound v' $ n $ P else raise Match) | all_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), Const("op -->",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = (if v=v' andalso T="set" then Syntax.const "_setleAll" $ Syntax.mark_bound v' $ n $ P else raise Match); fun ex_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), Const("op &",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = (if v=v' andalso T="set" then Syntax.const "_setlessEx" $ Syntax.mark_bound v' $ n $ P else raise Match) | ex_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), Const("op &",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = (if v=v' andalso T="set" then Syntax.const "_setleEx" $ Syntax.mark_bound v' $ n $ P else raise Match) in [("ALL ", all_tr'), ("EX ", ex_tr')] end *} text {* \medskip Translate between @{text "{e | x1...xn. P}"} and @{text "{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is only translated if @{text "[0..n] subset bvs(e)"}. *} parse_translation {* let val ex_tr = snd (mk_binder_tr ("EX ", "Ex")); fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1 | nvars _ = 1; fun setcompr_tr [e, idts, b] = let val eq = Syntax.const "op =" $ Bound (nvars idts) $ e; val P = Syntax.const "op &" $ eq $ b; val exP = ex_tr [idts, P]; in Syntax.const "Collect" $ Abs ("", dummyT, exP) end; in [("@SetCompr", setcompr_tr)] end; *} (* To avoid eta-contraction of body: *) print_translation {* let fun btr' syn [A,Abs abs] = let val (x,t) = atomic_abs_tr' abs in Syntax.const syn $ x $ A $ t end in [("Ball", btr' "_Ball"),("Bex", btr' "_Bex"), ("UNION", btr' "@UNION"),("INTER", btr' "@INTER")] end *} print_translation {* let val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY")); fun setcompr_tr' [Abs (abs as (_, _, P))] = let fun check (Const ("Ex", _) $ Abs (_, _, P), n) = check (P, n + 1) | check (Const ("op &", _) $ (Const ("op =", _) $ Bound m $ e) $ P, n) = n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso ((0 upto (n - 1)) subset add_loose_bnos (e, 0, [])) | check _ = false fun tr' (_ $ abs) = let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs] in Syntax.const "@SetCompr" $ e $ idts $ Q end; in if check (P, 0) then tr' P else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs val M = Syntax.const "@Coll" $ x $ t in case t of Const("op &",_) $ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A) $ P => if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M | _ => M end end; in [("Collect", setcompr_tr')] end; *} subsection {* Rules and definitions *} text {* Isomorphisms between predicates and sets. *} axioms mem_Collect_eq: "(a : {x. P(x)}) = P(a)" Collect_mem_eq: "{x. x:A} = A" finalconsts Collect "op :" defs Ball_def: "Ball A P == ALL x. x:A --> P(x)" Bex_def: "Bex A P == EX x. x:A & P(x)" defs (overloaded) subset_def: "A <= B == ALL x:A. x:B" psubset_def: "A < B == (A::'a set) <= B & ~ A=B" Compl_def: "- A == {x. ~x:A}" set_diff_def: "A - B == {x. x:A & ~x:B}" defs Un_def: "A Un B == {x. x:A | x:B}" Int_def: "A Int B == {x. x:A & x:B}" INTER_def: "INTER A B == {y. ALL x:A. y: B(x)}" UNION_def: "UNION A B == {y. EX x:A. y: B(x)}" Inter_def: "Inter S == (INT x:S. x)" Union_def: "Union S == (UN x:S. x)" Pow_def: "Pow A == {B. B <= A}" empty_def: "{} == {x. False}" UNIV_def: "UNIV == {x. True}" insert_def: "insert a B == {x. x=a} Un B" image_def: "f`A == {y. EX x:A. y = f(x)}" subsection {* Lemmas and proof tool setup *} subsubsection {* Relating predicates and sets *} declare mem_Collect_eq [iff] Collect_mem_eq [simp] lemma CollectI: "P(a) ==> a : {x. P(x)}" by simp lemma CollectD: "a : {x. P(x)} ==> P(a)" by simp lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}" by simp lemmas CollectE = CollectD [elim_format] subsubsection {* Bounded quantifiers *} lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x" by (simp add: Ball_def) lemmas strip = impI allI ballI lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x" by (simp add: Ball_def) lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q" by (unfold Ball_def) blast ML {* bind_thm("rev_ballE",permute_prems 1 1 (thm "ballE")) *} text {* \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and @{prop "a:A"}; creates assumption @{prop "P a"}. *} ML {* local val ballE = thm "ballE" in fun ball_tac i = etac ballE i THEN contr_tac (i + 1) end; *} text {* Gives better instantiation for bound: *} ML_setup {* claset_ref() := claset() addbefore ("bspec", datac (thm "bspec") 1); *} lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x" -- {* Normally the best argument order: @{prop "P x"} constrains the choice of @{prop "x:A"}. *} by (unfold Bex_def) blast lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x" -- {* The best argument order when there is only one @{prop "x:A"}. *} by (unfold Bex_def) blast lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x" by (unfold Bex_def) blast lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q" by (unfold Bex_def) blast lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)" -- {* Trival rewrite rule. *} by (simp add: Ball_def) lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)" -- {* Dual form for existentials. *} by (simp add: Bex_def) lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)" by blast lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)" by blast lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)" by blast lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)" by blast lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)" by blast lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)" by blast ML_setup {* local val Ball_def = thm "Ball_def"; val Bex_def = thm "Bex_def"; val simpset = Simplifier.clear_ss HOL_basic_ss; fun unfold_tac ss th = ALLGOALS (full_simp_tac (Simplifier.inherit_bounds ss simpset addsimps [th])); fun prove_bex_tac ss = unfold_tac ss Bex_def THEN Quantifier1.prove_one_point_ex_tac; val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac; fun prove_ball_tac ss = unfold_tac ss Ball_def THEN Quantifier1.prove_one_point_all_tac; val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac; in val defBEX_regroup = Simplifier.simproc (Theory.sign_of (the_context ())) "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex; val defBALL_regroup = Simplifier.simproc (Theory.sign_of (the_context ())) "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball; end; Addsimprocs [defBALL_regroup, defBEX_regroup]; *} subsubsection {* Congruence rules *} lemma ball_cong: "A = B ==> (!!x. x:B ==> P x = Q x) ==> (ALL x:A. P x) = (ALL x:B. Q x)" by (simp add: Ball_def) lemma strong_ball_cong [cong]: "A = B ==> (!!x. x:B =simp=> P x = Q x) ==> (ALL x:A. P x) = (ALL x:B. Q x)" by (simp add: simp_implies_def Ball_def) lemma bex_cong: "A = B ==> (!!x. x:B ==> P x = Q x) ==> (EX x:A. P x) = (EX x:B. Q x)" by (simp add: Bex_def cong: conj_cong) lemma strong_bex_cong [cong]: "A = B ==> (!!x. x:B =simp=> P x = Q x) ==> (EX x:A. P x) = (EX x:B. Q x)" by (simp add: simp_implies_def Bex_def cong: conj_cong) subsubsection {* Subsets *} lemma subsetI [intro!]: "(!!x. x:A ==> x:B) ==> A \ B" by (simp add: subset_def) text {* \medskip Map the type @{text "'a set => anything"} to just @{typ 'a}; for overloading constants whose first argument has type @{typ "'a set"}. *} lemma subsetD [elim]: "A \ B ==> c \ A ==> c \ B" -- {* Rule in Modus Ponens style. *} by (unfold subset_def) blast declare subsetD [intro?] -- FIXME lemma rev_subsetD: "c \ A ==> A \ B ==> c \ B" -- {* The same, with reversed premises for use with @{text erule} -- cf @{text rev_mp}. *} by (rule subsetD) declare rev_subsetD [intro?] -- FIXME text {* \medskip Converts @{prop "A \ B"} to @{prop "x \ A ==> x \ B"}. *} ML {* local val rev_subsetD = thm "rev_subsetD" in fun impOfSubs th = th RSN (2, rev_subsetD) end; *} lemma subsetCE [elim]: "A \ B ==> (c \ A ==> P) ==> (c \ B ==> P) ==> P" -- {* Classical elimination rule. *} by (unfold subset_def) blast text {* \medskip Takes assumptions @{prop "A \ B"}; @{prop "c \ A"} and creates the assumption @{prop "c \ B"}. *} ML {* local val subsetCE = thm "subsetCE" in fun set_mp_tac i = etac subsetCE i THEN mp_tac i end; *} lemma contra_subsetD: "A \ B ==> c \ B ==> c \ A" by blast lemma subset_refl: "A \ A" by fast lemma subset_trans: "A \ B ==> B \ C ==> A \ C" by blast subsubsection {* Equality *} lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B" apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals]) apply (rule Collect_mem_eq) apply (rule Collect_mem_eq) done (* Due to Brian Huffman *) lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))" by(auto intro:set_ext) lemma subset_antisym [intro!]: "A \ B ==> B \ A ==> A = B" -- {* Anti-symmetry of the subset relation. *} by (iprover intro: set_ext subsetD) lemmas equalityI [intro!] = subset_antisym text {* \medskip Equality rules from ZF set theory -- are they appropriate here? *} lemma equalityD1: "A = B ==> A \ B" by (simp add: subset_refl) lemma equalityD2: "A = B ==> B \ A" by (simp add: subset_refl) text {* \medskip Be careful when adding this to the claset as @{text subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{} \ A"} and @{prop "A \ {}"} and then back to @{prop "A = {}"}! *} lemma equalityE: "A = B ==> (A \ B ==> B \ A ==> P) ==> P" by (simp add: subset_refl) lemma equalityCE [elim]: "A = B ==> (c \ A ==> c \ B ==> P) ==> (c \ A ==> c \ B ==> P) ==> P" by blast text {* \medskip Lemma for creating induction formulae -- for "pattern matching" on @{text p}. To make the induction hypotheses usable, apply @{text spec} or @{text bspec} to put universal quantifiers over the free variables in @{text p}. *} lemma setup_induction: "p:A ==> (!!z. z:A ==> p = z --> R) ==> R" by simp lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)" by simp lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)" by simp subsubsection {* The universal set -- UNIV *} lemma UNIV_I [simp]: "x : UNIV" by (simp add: UNIV_def) declare UNIV_I [intro] -- {* unsafe makes it less likely to cause problems *} lemma UNIV_witness [intro?]: "EX x. x : UNIV" by simp lemma subset_UNIV: "A \ UNIV" by (rule subsetI) (rule UNIV_I) text {* \medskip Eta-contracting these two rules (to remove @{text P}) causes them to be ignored because of their interaction with congruence rules. *} lemma ball_UNIV [simp]: "Ball UNIV P = All P" by (simp add: Ball_def) lemma bex_UNIV [simp]: "Bex UNIV P = Ex P" by (simp add: Bex_def) subsubsection {* The empty set *} lemma empty_iff [simp]: "(c : {}) = False" by (simp add: empty_def) lemma emptyE [elim!]: "a : {} ==> P" by simp lemma empty_subsetI [iff]: "{} \ A" -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *} by blast lemma equals0I: "(!!y. y \ A ==> False) ==> A = {}" by blast lemma equals0D: "A = {} ==> a \ A" -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *} by blast lemma ball_empty [simp]: "Ball {} P = True" by (simp add: Ball_def) lemma bex_empty [simp]: "Bex {} P = False" by (simp add: Bex_def) lemma UNIV_not_empty [iff]: "UNIV ~= {}" by (blast elim: equalityE) subsubsection {* The Powerset operator -- Pow *} lemma Pow_iff [iff]: "(A \ Pow B) = (A \ B)" by (simp add: Pow_def) lemma PowI: "A \ B ==> A \ Pow B" by (simp add: Pow_def) lemma PowD: "A \ Pow B ==> A \ B" by (simp add: Pow_def) lemma Pow_bottom: "{} \ Pow B" by simp lemma Pow_top: "A \ Pow A" by (simp add: subset_refl) subsubsection {* Set complement *} lemma Compl_iff [simp]: "(c \ -A) = (c \ A)" by (unfold Compl_def) blast lemma ComplI [intro!]: "(c \ A ==> False) ==> c \ -A" by (unfold Compl_def) blast text {* \medskip This form, with negated conclusion, works well with the Classical prover. Negated assumptions behave like formulae on the right side of the notional turnstile ... *} lemma ComplD [dest!]: "c : -A ==> c~:A" by (unfold Compl_def) blast lemmas ComplE = ComplD [elim_format] subsubsection {* Binary union -- Un *} lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)" by (unfold Un_def) blast lemma UnI1 [elim?]: "c:A ==> c : A Un B" by simp lemma UnI2 [elim?]: "c:B ==> c : A Un B" by simp text {* \medskip Classical introduction rule: no commitment to @{prop A} vs @{prop B}. *} lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B" by auto lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P" by (unfold Un_def) blast subsubsection {* Binary intersection -- Int *} lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)" by (unfold Int_def) blast lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B" by simp lemma IntD1: "c : A Int B ==> c:A" by simp lemma IntD2: "c : A Int B ==> c:B" by simp lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P" by simp subsubsection {* Set difference *} lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)" by (unfold set_diff_def) blast lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B" by simp lemma DiffD1: "c : A - B ==> c : A" by simp lemma DiffD2: "c : A - B ==> c : B ==> P" by simp lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P" by simp subsubsection {* Augmenting a set -- insert *} lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)" by (unfold insert_def) blast lemma insertI1: "a : insert a B" by simp lemma insertI2: "a : B ==> a : insert b B" by simp lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P" by (unfold insert_def) blast lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B" -- {* Classical introduction rule. *} by auto lemma subset_insert_iff: "(A \ insert x B) = (if x:A then A - {x} \ B else A \ B)" by auto subsubsection {* Singletons, using insert *} lemma singletonI [intro!]: "a : {a}" -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *} by (rule insertI1) lemma singletonD [dest!]: "b : {a} ==> b = a" by blast lemmas singletonE = singletonD [elim_format] lemma singleton_iff: "(b : {a}) = (b = a)" by blast lemma singleton_inject [dest!]: "{a} = {b} ==> a = b" by blast lemma singleton_insert_inj_eq [iff]: "({b} = insert a A) = (a = b & A \ {b})" by blast lemma singleton_insert_inj_eq' [iff]: "(insert a A = {b}) = (a = b & A \ {b})" by blast lemma subset_singletonD: "A \ {x} ==> A = {} | A = {x}" by fast lemma singleton_conv [simp]: "{x. x = a} = {a}" by blast lemma singleton_conv2 [simp]: "{x. a = x} = {a}" by blast lemma diff_single_insert: "A - {x} \ B ==> x \ A ==> A \ insert x B" by blast subsubsection {* Unions of families *} text {* @{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}. *} lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)" by (unfold UNION_def) blast lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)" -- {* The order of the premises presupposes that @{term A} is rigid; @{term b} may be flexible. *} by auto lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R" by (unfold UNION_def) blast lemma UN_cong [cong]: "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" by (simp add: UNION_def) subsubsection {* Intersections of families *} text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *} lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)" by (unfold INTER_def) blast lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)" by (unfold INTER_def) blast lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a" by auto lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R" -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *} by (unfold INTER_def) blast lemma INT_cong [cong]: "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)" by (simp add: INTER_def) subsubsection {* Union *} lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)" by (unfold Union_def) blast lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C" -- {* The order of the premises presupposes that @{term C} is rigid; @{term A} may be flexible. *} by auto lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R" by (unfold Union_def) blast subsubsection {* Inter *} lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)" by (unfold Inter_def) blast lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C" by (simp add: Inter_def) text {* \medskip A ``destruct'' rule -- every @{term X} in @{term C} contains @{term A} as an element, but @{prop "A:X"} can hold when @{prop "X:C"} does not! This rule is analogous to @{text spec}. *} lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X" by auto lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R" -- {* ``Classical'' elimination rule -- does not require proving @{prop "X:C"}. *} by (unfold Inter_def) blast text {* \medskip Image of a set under a function. Frequently @{term b} does not have the syntactic form of @{term "f x"}. *} lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A" by (unfold image_def) blast lemma imageI: "x : A ==> f x : f ` A" by (rule image_eqI) (rule refl) lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A" -- {* This version's more effective when we already have the required @{term x}. *} by (unfold image_def) blast lemma imageE [elim!]: "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P" -- {* The eta-expansion gives variable-name preservation. *} by (unfold image_def) blast lemma image_Un: "f`(A Un B) = f`A Un f`B" by blast lemma image_iff: "(z : f`A) = (EX x:A. z = f x)" by blast lemma image_subset_iff: "(f`A \ B) = (\x\A. f x \ B)" -- {* This rewrite rule would confuse users if made default. *} by blast lemma subset_image_iff: "(B \ f`A) = (EX AA. AA \ A & B = f`AA)" apply safe prefer 2 apply fast apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast) done lemma image_subsetI: "(!!x. x \ A ==> f x \ B) ==> f`A \ B" -- {* Replaces the three steps @{text subsetI}, @{text imageE}, @{text hypsubst}, but breaks too many existing proofs. *} by blast text {* \medskip Range of a function -- just a translation for image! *} lemma range_eqI: "b = f x ==> b \ range f" by simp lemma rangeI: "f x \ range f" by simp lemma rangeE [elim?]: "b \ range (\x. f x) ==> (!!x. b = f x ==> P) ==> P" by blast subsubsection {* Set reasoning tools *} text {* Rewrite rules for boolean case-splitting: faster than @{text "split_if [split]"}. *} lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))" by (rule split_if) lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))" by (rule split_if) text {* Split ifs on either side of the membership relation. Not for @{text "[simp]"} -- can cause goals to blow up! *} lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))" by (rule split_if) lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))" by (rule split_if) lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2 lemmas mem_simps = insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff -- {* Each of these has ALREADY been added @{text "[simp]"} above. *} (*Would like to add these, but the existing code only searches for the outer-level constant, which in this case is just "op :"; we instead need to use term-nets to associate patterns with rules. Also, if a rule fails to apply, then the formula should be kept. [("uminus", Compl_iff RS iffD1), ("op -", [Diff_iff RS iffD1]), ("op Int", [IntD1,IntD2]), ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])] *) ML_setup {* val mksimps_pairs = [("Ball", [thm "bspec"])] @ mksimps_pairs; simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs); *} declare subset_UNIV [simp] subset_refl [simp] subsubsection {* The ``proper subset'' relation *} lemma psubsetI [intro!]: "A \ B ==> A \ B ==> A \ B" by (unfold psubset_def) blast lemma psubsetE [elim!]: "[|A \ B; [|A \ B; ~ (B\A)|] ==> R|] ==> R" by (unfold psubset_def) blast lemma psubset_insert_iff: "(A \ insert x B) = (if x \ B then A \ B else if x \ A then A - {x} \ B else A \ B)" by (auto simp add: psubset_def subset_insert_iff) lemma psubset_eq: "(A \ B) = (A \ B & A \ B)" by (simp only: psubset_def) lemma psubset_imp_subset: "A \ B ==> A \ B" by (simp add: psubset_eq) lemma psubset_trans: "[| A \ B; B \ C |] ==> A \ C" apply (unfold psubset_def) apply (auto dest: subset_antisym) done lemma psubsetD: "[| A \ B; c \ A |] ==> c \ B" apply (unfold psubset_def) apply (auto dest: subsetD) done lemma psubset_subset_trans: "A \ B ==> B \ C ==> A \ C" by (auto simp add: psubset_eq) lemma subset_psubset_trans: "A \ B ==> B \ C ==> A \ C" by (auto simp add: psubset_eq) lemma psubset_imp_ex_mem: "A \ B ==> \b. b \ (B - A)" by (unfold psubset_def) blast lemma atomize_ball: "(!!x. x \ A ==> P x) == Trueprop (\x\A. P x)" by (simp only: Ball_def atomize_all atomize_imp) declare atomize_ball [symmetric, rulify] subsection {* Further set-theory lemmas *} subsubsection {* Derived rules involving subsets. *} text {* @{text insert}. *} lemma subset_insertI: "B \ insert a B" apply (rule subsetI) apply (erule insertI2) done lemma subset_insertI2: "A \ B \ A \ insert b B" by blast lemma subset_insert: "x \ A ==> (A \ insert x B) = (A \ B)" by blast text {* \medskip Big Union -- least upper bound of a set. *} lemma Union_upper: "B \ A ==> B \ Union A" by (iprover intro: subsetI UnionI) lemma Union_least: "(!!X. X \ A ==> X \ C) ==> Union A \ C" by (iprover intro: subsetI elim: UnionE dest: subsetD) text {* \medskip General union. *} lemma UN_upper: "a \ A ==> B a \ (\x\A. B x)" by blast lemma UN_least: "(!!x. x \ A ==> B x \ C) ==> (\x\A. B x) \ C" by (iprover intro: subsetI elim: UN_E dest: subsetD) text {* \medskip Big Intersection -- greatest lower bound of a set. *} lemma Inter_lower: "B \ A ==> Inter A \ B" by blast lemma Inter_subset: "[| !!X. X \ A ==> X \ B; A ~= {} |] ==> \A \ B" by blast lemma Inter_greatest: "(!!X. X \ A ==> C \ X) ==> C \ Inter A" by (iprover intro: InterI subsetI dest: subsetD) lemma INT_lower: "a \ A ==> (\x\A. B x) \ B a" by blast lemma INT_greatest: "(!!x. x \ A ==> C \ B x) ==> C \ (\x\A. B x)" by (iprover intro: INT_I subsetI dest: subsetD) text {* \medskip Finite Union -- the least upper bound of two sets. *} lemma Un_upper1: "A \ A \ B" by blast lemma Un_upper2: "B \ A \ B" by blast lemma Un_least: "A \ C ==> B \ C ==> A \ B \ C" by blast text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *} lemma Int_lower1: "A \ B \ A" by blast lemma Int_lower2: "A \ B \ B" by blast lemma Int_greatest: "C \ A ==> C \ B ==> C \ A \ B" by blast text {* \medskip Set difference. *} lemma Diff_subset: "A - B \ A" by blast lemma Diff_subset_conv: "(A - B \ C) = (A \ B \ C)" by blast text {* \medskip Monotonicity. *} lemma mono_Un: "mono f ==> f A \ f B \ f (A \ B)" by (auto simp add: mono_def) lemma mono_Int: "mono f ==> f (A \ B) \ f A \ f B" by (auto simp add: mono_def) subsubsection {* Equalities involving union, intersection, inclusion, etc. *} text {* @{text "{}"}. *} lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})" -- {* supersedes @{text "Collect_False_empty"} *} by auto lemma subset_empty [simp]: "(A \ {}) = (A = {})" by blast lemma not_psubset_empty [iff]: "\ (A < {})" by (unfold psubset_def) blast lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\x. \ P x)" by auto lemma Collect_neg_eq: "{x. \ P x} = - {x. P x}" by blast lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \ {x. Q x}" by blast lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \ {x. Q x}" by blast lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \ {x. Q x}" by blast lemma Collect_all_eq: "{x. \y. P x y} = (\y. {x. P x y})" by blast lemma Collect_ball_eq: "{x. \y\A. P x y} = (\y\A. {x. P x y})" by blast lemma Collect_ex_eq: "{x. \y. P x y} = (\y. {x. P x y})" by blast lemma Collect_bex_eq: "{x. \y\A. P x y} = (\y\A. {x. P x y})" by blast text {* \medskip @{text insert}. *} lemma insert_is_Un: "insert a A = {a} Un A" -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *} by blast lemma insert_not_empty [simp]: "insert a A \ {}" by blast lemmas empty_not_insert = insert_not_empty [symmetric, standard] declare empty_not_insert [simp] lemma insert_absorb: "a \ A ==> insert a A = A" -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *} -- {* with \emph{quadratic} running time *} by blast lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A" by blast lemma insert_commute: "insert x (insert y A) = insert y (insert x A)" by blast lemma insert_subset [simp]: "(insert x A \ B) = (x \ B & A \ B)" by blast lemma mk_disjoint_insert: "a \ A ==> \B. A = insert a B & a \ B" -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *} apply (rule_tac x = "A - {a}" in exI, blast) done lemma insert_Collect: "insert a (Collect P) = {u. u \ a --> P u}" by auto lemma UN_insert_distrib: "u \ A ==> (\x\A. insert a (B x)) = insert a (\x\A. B x)" by blast lemma insert_inter_insert[simp]: "insert a A \ insert a B = insert a (A \ B)" by blast lemma insert_disjoint[simp]: "(insert a A \ B = {}) = (a \ B \ A \ B = {})" "({} = insert a A \ B) = (a \ B \ {} = A \ B)" by auto lemma disjoint_insert[simp]: "(B \ insert a A = {}) = (a \ B \ B \ A = {})" "({} = A \ insert b B) = (b \ A \ {} = A \ B)" by auto text {* \medskip @{text image}. *} lemma image_empty [simp]: "f`{} = {}" by blast lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)" by blast lemma image_constant: "x \ A ==> (\x. c) ` A = {c}" by auto lemma image_image: "f ` (g ` A) = (\x. f (g x)) ` A" by blast lemma insert_image [simp]: "x \ A ==> insert (f x) (f`A) = f`A" by blast lemma image_is_empty [iff]: "(f`A = {}) = (A = {})" by blast lemma image_Collect: "f ` {x. P x} = {f x | x. P x}" -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS, with its implicit quantifier and conjunction. Also image enjoys better equational properties than does the RHS. *} by blast lemma if_image_distrib [simp]: "(\x. if P x then f x else g x) ` S = (f ` (S \ {x. P x})) \ (g ` (S \ {x. \ P x}))" by (auto simp add: image_def) lemma image_cong: "M = N ==> (!!x. x \ N ==> f x = g x) ==> f`M = g`N" by (simp add: image_def) text {* \medskip @{text range}. *} lemma full_SetCompr_eq: "{u. \x. u = f x} = range f" by auto lemma range_composition [simp]: "range (\x. f (g x)) = f`range g" by (subst image_image, simp) text {* \medskip @{text Int} *} lemma Int_absorb [simp]: "A \ A = A" by blast lemma Int_left_absorb: "A \ (A \ B) = A \ B" by blast lemma Int_commute: "A \ B = B \ A" by blast lemma Int_left_commute: "A \ (B \ C) = B \ (A \ C)" by blast lemma Int_assoc: "(A \ B) \ C = A \ (B \ C)" by blast lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute -- {* Intersection is an AC-operator *} lemma Int_absorb1: "B \ A ==> A \ B = B" by blast lemma Int_absorb2: "A \ B ==> A \ B = A" by blast lemma Int_empty_left [simp]: "{} \ B = {}" by blast lemma Int_empty_right [simp]: "A \ {} = {}" by blast lemma disjoint_eq_subset_Compl: "(A \ B = {}) = (A \ -B)" by blast lemma disjoint_iff_not_equal: "(A \ B = {}) = (\x\A. \y\B. x \ y)" by blast lemma Int_UNIV_left [simp]: "UNIV \ B = B" by blast lemma Int_UNIV_right [simp]: "A \ UNIV = A" by blast lemma Int_eq_Inter: "A \ B = \{A, B}" by blast lemma Int_Un_distrib: "A \ (B \ C) = (A \ B) \ (A \ C)" by blast lemma Int_Un_distrib2: "(B \ C) \ A = (B \ A) \ (C \ A)" by blast lemma Int_UNIV [simp]: "(A \ B = UNIV) = (A = UNIV & B = UNIV)" by blast lemma Int_subset_iff [simp]: "(C \ A \ B) = (C \ A & C \ B)" by blast lemma Int_Collect: "(x \ A \ {x. P x}) = (x \ A & P x)" by blast text {* \medskip @{text Un}. *} lemma Un_absorb [simp]: "A \ A = A" by blast lemma Un_left_absorb: "A \ (A \ B) = A \ B" by blast lemma Un_commute: "A \ B = B \ A" by blast lemma Un_left_commute: "A \ (B \ C) = B \ (A \ C)" by blast lemma Un_assoc: "(A \ B) \ C = A \ (B \ C)" by blast lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute -- {* Union is an AC-operator *} lemma Un_absorb1: "A \ B ==> A \ B = B" by blast lemma Un_absorb2: "B \ A ==> A \ B = A" by blast lemma Un_empty_left [simp]: "{} \ B = B" by blast lemma Un_empty_right [simp]: "A \ {} = A" by blast lemma Un_UNIV_left [simp]: "UNIV \ B = UNIV" by blast lemma Un_UNIV_right [simp]: "A \ UNIV = UNIV" by blast lemma Un_eq_Union: "A \ B = \{A, B}" by blast lemma Un_insert_left [simp]: "(insert a B) \ C = insert a (B \ C)" by blast lemma Un_insert_right [simp]: "A \ (insert a B) = insert a (A \ B)" by blast lemma Int_insert_left: "(insert a B) Int C = (if a \ C then insert a (B \ C) else B \ C)" by auto lemma Int_insert_right: "A \ (insert a B) = (if a \ A then insert a (A \ B) else A \ B)" by auto lemma Un_Int_distrib: "A \ (B \ C) = (A \ B) \ (A \ C)" by blast lemma Un_Int_distrib2: "(B \ C) \ A = (B \ A) \ (C \ A)" by blast lemma Un_Int_crazy: "(A \ B) \ (B \ C) \ (C \ A) = (A \ B) \ (B \ C) \ (C \ A)" by blast lemma subset_Un_eq: "(A \ B) = (A \ B = B)" by blast lemma Un_empty [iff]: "(A \ B = {}) = (A = {} & B = {})" by blast lemma Un_subset_iff [simp]: "(A \ B \ C) = (A \ C & B \ C)" by blast lemma Un_Diff_Int: "(A - B) \ (A \ B) = A" by blast text {* \medskip Set complement *} lemma Compl_disjoint [simp]: "A \ -A = {}" by blast lemma Compl_disjoint2 [simp]: "-A \ A = {}" by blast lemma Compl_partition: "A \ -A = UNIV" by blast lemma Compl_partition2: "-A \ A = UNIV" by blast lemma double_complement [simp]: "- (-A) = (A::'a set)" by blast lemma Compl_Un [simp]: "-(A \ B) = (-A) \ (-B)" by blast lemma Compl_Int [simp]: "-(A \ B) = (-A) \ (-B)" by blast lemma Compl_UN [simp]: "-(\x\A. B x) = (\x\A. -B x)" by blast lemma Compl_INT [simp]: "-(\x\A. B x) = (\x\A. -B x)" by blast lemma subset_Compl_self_eq: "(A \ -A) = (A = {})" by blast lemma Un_Int_assoc_eq: "((A \ B) \ C = A \ (B \ C)) = (C \ A)" -- {* Halmos, Naive Set Theory, page 16. *} by blast lemma Compl_UNIV_eq [simp]: "-UNIV = {}" by blast lemma Compl_empty_eq [simp]: "-{} = UNIV" by blast lemma Compl_subset_Compl_iff [iff]: "(-A \ -B) = (B \ A)" by blast lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))" by blast text {* \medskip @{text Union}. *} lemma Union_empty [simp]: "Union({}) = {}" by blast lemma Union_UNIV [simp]: "Union UNIV = UNIV" by blast lemma Union_insert [simp]: "Union (insert a B) = a \ \B" by blast lemma Union_Un_distrib [simp]: "\(A Un B) = \A \ \B" by blast lemma Union_Int_subset: "\(A \ B) \ \A \ \B" by blast lemma Union_empty_conv [iff]: "(\A = {}) = (\x\A. x = {})" by blast lemma empty_Union_conv [iff]: "({} = \A) = (\x\A. x = {})" by blast lemma Union_disjoint: "(\C \ A = {}) = (\B\C. B \ A = {})" by blast text {* \medskip @{text Inter}. *} lemma Inter_empty [simp]: "\{} = UNIV" by blast lemma Inter_UNIV [simp]: "\UNIV = {}" by blast lemma Inter_insert [simp]: "\(insert a B) = a \ \B" by blast lemma Inter_Un_subset: "\A \ \B \ \(A \ B)" by blast lemma Inter_Un_distrib: "\(A \ B) = \A \ \B" by blast lemma Inter_UNIV_conv [iff]: "(\A = UNIV) = (\x\A. x = UNIV)" "(UNIV = \A) = (\x\A. x = UNIV)" by blast+ text {* \medskip @{text UN} and @{text INT}. Basic identities: *} lemma UN_empty [simp]: "(\x\{}. B x) = {}" by blast lemma UN_empty2 [simp]: "(\x\A. {}) = {}" by blast lemma UN_singleton [simp]: "(\x\A. {x}) = A" by blast lemma UN_absorb: "k \ I ==> A k \ (\i\I. A i) = (\i\I. A i)" by auto lemma INT_empty [simp]: "(\x\{}. B x) = UNIV" by blast lemma INT_absorb: "k \ I ==> A k \ (\i\I. A i) = (\i\I. A i)" by blast lemma UN_insert [simp]: "(\x\insert a A. B x) = B a \ UNION A B" by blast lemma UN_Un: "(\i \ A \ B. M i) = (\i\A. M i) \ (\i\B. M i)" by blast lemma UN_UN_flatten: "(\x \ (\y\A. B y). C x) = (\y\A. \x\B y. C x)" by blast lemma UN_subset_iff: "((\i\I. A i) \ B) = (\i\I. A i \ B)" by blast lemma INT_subset_iff: "(B \ (\i\I. A i)) = (\i\I. B \ A i)" by blast lemma INT_insert [simp]: "(\x \ insert a A. B x) = B a \ INTER A B" by blast lemma INT_Un: "(\i \ A \ B. M i) = (\i \ A. M i) \ (\i\B. M i)" by blast lemma INT_insert_distrib: "u \ A ==> (\x\A. insert a (B x)) = insert a (\x\A. B x)" by blast lemma Union_image_eq [simp]: "\(B`A) = (\x\A. B x)" by blast lemma image_Union: "f ` \S = (\x\S. f ` x)" by blast lemma Inter_image_eq [simp]: "\(B`A) = (\x\A. B x)" by blast lemma UN_constant [simp]: "(\y\A. c) = (if A = {} then {} else c)" by auto lemma INT_constant [simp]: "(\y\A. c) = (if A = {} then UNIV else c)" by auto lemma UN_eq: "(\x\A. B x) = \({Y. \x\A. Y = B x})" by blast lemma INT_eq: "(\x\A. B x) = \({Y. \x\A. Y = B x})" -- {* Look: it has an \emph{existential} quantifier *} by blast lemma UNION_empty_conv[iff]: "({} = (UN x:A. B x)) = (\x\A. B x = {})" "((UN x:A. B x) = {}) = (\x\A. B x = {})" by blast+ lemma INTER_UNIV_conv[iff]: "(UNIV = (INT x:A. B x)) = (\x\A. B x = UNIV)" "((INT x:A. B x) = UNIV) = (\x\A. B x = UNIV)" by blast+ text {* \medskip Distributive laws: *} lemma Int_Union: "A \ \B = (\C\B. A \ C)" by blast lemma Int_Union2: "\B \ A = (\C\B. C \ A)" by blast lemma Un_Union_image: "(\x\C. A x \ B x) = \(A`C) \ \(B`C)" -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *} -- {* Union of a family of unions *} by blast lemma UN_Un_distrib: "(\i\I. A i \ B i) = (\i\I. A i) \ (\i\I. B i)" -- {* Equivalent version *} by blast lemma Un_Inter: "A \ \B = (\C\B. A \ C)" by blast lemma Int_Inter_image: "(\x\C. A x \ B x) = \(A`C) \ \(B`C)" by blast lemma INT_Int_distrib: "(\i\I. A i \ B i) = (\i\I. A i) \ (\i\I. B i)" -- {* Equivalent version *} by blast lemma Int_UN_distrib: "B \ (\i\I. A i) = (\i\I. B \ A i)" -- {* Halmos, Naive Set Theory, page 35. *} by blast lemma Un_INT_distrib: "B \ (\i\I. A i) = (\i\I. B \ A i)" by blast lemma Int_UN_distrib2: "(\i\I. A i) \ (\j\J. B j) = (\i\I. \j\J. A i \ B j)" by blast lemma Un_INT_distrib2: "(\i\I. A i) \ (\j\J. B j) = (\i\I. \j\J. A i \ B j)" by blast text {* \medskip Bounded quantifiers. The following are not added to the default simpset because (a) they duplicate the body and (b) there are no similar rules for @{text Int}. *} lemma ball_Un: "(\x \ A \ B. P x) = ((\x\A. P x) & (\x\B. P x))" by blast lemma bex_Un: "(\x \ A \ B. P x) = ((\x\A. P x) | (\x\B. P x))" by blast lemma ball_UN: "(\z \ UNION A B. P z) = (\x\A. \z \ B x. P z)" by blast lemma bex_UN: "(\z \ UNION A B. P z) = (\x\A. \z\B x. P z)" by blast text {* \medskip Set difference. *} lemma Diff_eq: "A - B = A \ (-B)" by blast lemma Diff_eq_empty_iff [simp]: "(A - B = {}) = (A \ B)" by blast lemma Diff_cancel [simp]: "A - A = {}" by blast lemma Diff_idemp [simp]: "(A - B) - B = A - (B::'a set)" by blast lemma Diff_triv: "A \ B = {} ==> A - B = A" by (blast elim: equalityE) lemma empty_Diff [simp]: "{} - A = {}" by blast lemma Diff_empty [simp]: "A - {} = A" by blast lemma Diff_UNIV [simp]: "A - UNIV = {}" by blast lemma Diff_insert0 [simp]: "x \ A ==> A - insert x B = A - B" by blast lemma Diff_insert: "A - insert a B = A - B - {a}" -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *} by blast lemma Diff_insert2: "A - insert a B = A - {a} - B" -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *} by blast lemma insert_Diff_if: "insert x A - B = (if x \ B then A - B else insert x (A - B))" by auto lemma insert_Diff1 [simp]: "x \ B ==> insert x A - B = A - B" by blast lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A" by blast lemma insert_Diff: "a \ A ==> insert a (A - {a}) = A" by blast lemma Diff_insert_absorb: "x \ A ==> (insert x A) - {x} = A" by auto lemma Diff_disjoint [simp]: "A \ (B - A) = {}" by blast lemma Diff_partition: "A \ B ==> A \ (B - A) = B" by blast lemma double_diff: "A \ B ==> B \ C ==> B - (C - A) = A" by blast lemma Un_Diff_cancel [simp]: "A \ (B - A) = A \ B" by blast lemma Un_Diff_cancel2 [simp]: "(B - A) \ A = B \ A" by blast lemma Diff_Un: "A - (B \ C) = (A - B) \ (A - C)" by blast lemma Diff_Int: "A - (B \ C) = (A - B) \ (A - C)" by blast lemma Un_Diff: "(A \ B) - C = (A - C) \ (B - C)" by blast lemma Int_Diff: "(A \ B) - C = A \ (B - C)" by blast lemma Diff_Int_distrib: "C \ (A - B) = (C \ A) - (C \ B)" by blast lemma Diff_Int_distrib2: "(A - B) \ C = (A \ C) - (B \ C)" by blast lemma Diff_Compl [simp]: "A - (- B) = A \ B" by auto lemma Compl_Diff_eq [simp]: "- (A - B) = -A \ B" by blast text {* \medskip Quantification over type @{typ bool}. *} lemma all_bool_eq: "(\b::bool. P b) = (P True & P False)" apply auto apply (tactic {* case_tac "b" 1 *}, auto) done lemma bool_induct: "P True \ P False \ P x" by (rule conjI [THEN all_bool_eq [THEN iffD2], THEN spec]) lemma ex_bool_eq: "(\b::bool. P b) = (P True | P False)" apply auto apply (tactic {* case_tac "b" 1 *}, auto) done lemma Un_eq_UN: "A \ B = (\b. if b then A else B)" by (auto simp add: split_if_mem2) lemma UN_bool_eq: "(\b::bool. A b) = (A True \ A False)" apply auto apply (tactic {* case_tac "b" 1 *}, auto) done lemma INT_bool_eq: "(\b::bool. A b) = (A True \ A False)" apply auto apply (tactic {* case_tac "b" 1 *}, auto) done text {* \medskip @{text Pow} *} lemma Pow_empty [simp]: "Pow {} = {{}}" by (auto simp add: Pow_def) lemma Pow_insert: "Pow (insert a A) = Pow A \ (insert a ` Pow A)" by (blast intro: image_eqI [where ?x = "u - {a}", standard]) lemma Pow_Compl: "Pow (- A) = {-B | B. A \ Pow B}" by (blast intro: exI [where ?x = "- u", standard]) lemma Pow_UNIV [simp]: "Pow UNIV = UNIV" by blast lemma Un_Pow_subset: "Pow A \ Pow B \ Pow (A \ B)" by blast lemma UN_Pow_subset: "(\x\A. Pow (B x)) \ Pow (\x\A. B x)" by blast lemma subset_Pow_Union: "A \ Pow (\A)" by blast lemma Union_Pow_eq [simp]: "\(Pow A) = A" by blast lemma Pow_Int_eq [simp]: "Pow (A \ B) = Pow A \ Pow B" by blast lemma Pow_INT_eq: "Pow (\x\A. B x) = (\x\A. Pow (B x))" by blast text {* \medskip Miscellany. *} lemma set_eq_subset: "(A = B) = (A \ B & B \ A)" by blast lemma subset_iff: "(A \ B) = (\t. t \ A --> t \ B)" by blast lemma subset_iff_psubset_eq: "(A \ B) = ((A \ B) | (A = B))" by (unfold psubset_def) blast lemma all_not_in_conv [iff]: "(\x. x \ A) = (A = {})" by blast lemma ex_in_conv: "(\x. x \ A) = (A \ {})" by blast lemma distinct_lemma: "f x \ f y ==> x \ y" by iprover text {* \medskip Miniscoping: pushing in quantifiers and big Unions and Intersections. *} lemma UN_simps [simp]: "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))" "!!A B C. (UN x:C. A x Un B) = ((if C={} then {} else (UN x:C. A x) Un B))" "!!A B C. (UN x:C. A Un B x) = ((if C={} then {} else A Un (UN x:C. B x)))" "!!A B C. (UN x:C. A x Int B) = ((UN x:C. A x) Int B)" "!!A B C. (UN x:C. A Int B x) = (A Int (UN x:C. B x))" "!!A B C. (UN x:C. A x - B) = ((UN x:C. A x) - B)" "!!A B C. (UN x:C. A - B x) = (A - (INT x:C. B x))" "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)" "!!A B C. (UN z: UNION A B. C z) = (UN x:A. UN z: B(x). C z)" "!!A B f. (UN x:f`A. B x) = (UN a:A. B (f a))" by auto lemma INT_simps [simp]: "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)" "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))" "!!A B C. (INT x:C. A x - B) = (if C={} then UNIV else (INT x:C. A x) - B)" "!!A B C. (INT x:C. A - B x) = (if C={} then UNIV else A - (UN x:C. B x))" "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)" "!!A B C. (INT x:C. A x Un B) = ((INT x:C. A x) Un B)" "!!A B C. (INT x:C. A Un B x) = (A Un (INT x:C. B x))" "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)" "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)" "!!A B f. (INT x:f`A. B x) = (INT a:A. B (f a))" by auto lemma ball_simps [simp]: "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)" "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))" "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))" "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)" "!!P. (ALL x:{}. P x) = True" "!!P. (ALL x:UNIV. P x) = (ALL x. P x)" "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))" "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)" "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)" "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)" "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))" "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)" by auto lemma bex_simps [simp]: "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)" "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))" "!!P. (EX x:{}. P x) = False" "!!P. (EX x:UNIV. P x) = (EX x. P x)" "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))" "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)" "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)" "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)" "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))" "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)" by auto lemma ball_conj_distrib: "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))" by blast lemma bex_disj_distrib: "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))" by blast text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *} lemma UN_extend_simps: "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))" "!!A B C. (UN x:C. A x) Un B = (if C={} then B else (UN x:C. A x Un B))" "!!A B C. A Un (UN x:C. B x) = (if C={} then A else (UN x:C. A Un B x))" "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)" "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)" "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)" "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)" "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)" "!!A B C. (UN x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)" "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)" by auto lemma INT_extend_simps: "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))" "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))" "!!A B C. (INT x:C. A x) - B = (if C={} then UNIV-B else (INT x:C. A x - B))" "!!A B C. A - (UN x:C. B x) = (if C={} then A else (INT x:C. A - B x))" "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))" "!!A B C. ((INT x:C. A x) Un B) = (INT x:C. A x Un B)" "!!A B C. A Un (INT x:C. B x) = (INT x:C. A Un B x)" "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)" "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)" "!!A B f. (INT a:A. B (f a)) = (INT x:f`A. B x)" by auto subsubsection {* Monotonicity of various operations *} lemma image_mono: "A \ B ==> f`A \ f`B" by blast lemma Pow_mono: "A \ B ==> Pow A \ Pow B" by blast lemma Union_mono: "A \ B ==> \A \ \B" by blast lemma Inter_anti_mono: "B \ A ==> \A \ \B" by blast lemma UN_mono: "A \ B ==> (!!x. x \ A ==> f x \ g x) ==> (\x\A. f x) \ (\x\B. g x)" by (blast dest: subsetD) lemma INT_anti_mono: "B \ A ==> (!!x. x \ A ==> f x \ g x) ==> (\x\A. f x) \ (\x\A. g x)" -- {* The last inclusion is POSITIVE! *} by (blast dest: subsetD) lemma insert_mono: "C \ D ==> insert a C \ insert a D" by blast lemma Un_mono: "A \ C ==> B \ D ==> A \ B \ C \ D" by blast lemma Int_mono: "A \ C ==> B \ D ==> A \ B \ C \ D" by blast lemma Diff_mono: "A \ C ==> D \ B ==> A - B \ C - D" by blast lemma Compl_anti_mono: "A \ B ==> -B \ -A" by blast text {* \medskip Monotonicity of implications. *} lemma in_mono: "A \ B ==> x \ A --> x \ B" apply (rule impI) apply (erule subsetD, assumption) done lemma conj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 & P2) --> (Q1 & Q2)" by iprover lemma disj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 | P2) --> (Q1 | Q2)" by iprover lemma imp_mono: "Q1 --> P1 ==> P2 --> Q2 ==> (P1 --> P2) --> (Q1 --> Q2)" by iprover lemma imp_refl: "P --> P" .. lemma ex_mono: "(!!x. P x --> Q x) ==> (EX x. P x) --> (EX x. Q x)" by iprover lemma all_mono: "(!!x. P x --> Q x) ==> (ALL x. P x) --> (ALL x. Q x)" by iprover lemma Collect_mono: "(!!x. P x --> Q x) ==> Collect P \ Collect Q" by blast lemma Int_Collect_mono: "A \ B ==> (!!x. x \ A ==> P x --> Q x) ==> A \ Collect P \ B \ Collect Q" by blast lemmas basic_monos = subset_refl imp_refl disj_mono conj_mono ex_mono Collect_mono in_mono lemma eq_to_mono: "a = b ==> c = d ==> b --> d ==> a --> c" by iprover lemma eq_to_mono2: "a = b ==> c = d ==> ~ b --> ~ d ==> ~ a --> ~ c" by iprover lemma Least_mono: "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)" -- {* Courtesy of Stephan Merz *} apply clarify apply (erule_tac P = "%x. x : S" in LeastI2_order, fast) apply (rule LeastI2_order) apply (auto elim: monoD intro!: order_antisym) done subsection {* Inverse image of a function *} constdefs vimage :: "('a => 'b) => 'b set => 'a set" (infixr "-`" 90) "f -` B == {x. f x : B}" subsubsection {* Basic rules *} lemma vimage_eq [simp]: "(a : f -` B) = (f a : B)" by (unfold vimage_def) blast lemma vimage_singleton_eq: "(a : f -` {b}) = (f a = b)" by simp lemma vimageI [intro]: "f a = b ==> b:B ==> a : f -` B" by (unfold vimage_def) blast lemma vimageI2: "f a : A ==> a : f -` A" by (unfold vimage_def) fast lemma vimageE [elim!]: "a: f -` B ==> (!!x. f a = x ==> x:B ==> P) ==> P" by (unfold vimage_def) blast lemma vimageD: "a : f -` A ==> f a : A" by (unfold vimage_def) fast subsubsection {* Equations *} lemma vimage_empty [simp]: "f -` {} = {}" by blast lemma vimage_Compl: "f -` (-A) = -(f -` A)" by blast lemma vimage_Un [simp]: "f -` (A Un B) = (f -` A) Un (f -` B)" by blast lemma vimage_Int [simp]: "f -` (A Int B) = (f -` A) Int (f -` B)" by fast lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)" by blast lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)" by blast lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)" by blast lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f y)}" by blast lemma vimage_Collect: "(!!x. P (f x) = Q x) ==> f -` (Collect P) = Collect Q" by blast lemma vimage_insert: "f-`(insert a B) = (f-`{a}) Un (f-`B)" -- {* NOT suitable for rewriting because of the recurrence of @{term "{a}"}. *} by blast lemma vimage_Diff: "f -` (A - B) = (f -` A) - (f -` B)" by blast lemma vimage_UNIV [simp]: "f -` UNIV = UNIV" by blast lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})" -- {* NOT suitable for rewriting *} by blast lemma vimage_mono: "A \ B ==> f -` A \ f -` B" -- {* monotonicity *} by blast subsection {* Getting the Contents of a Singleton Set *} constdefs contents :: "'a set => 'a" "contents X == THE x. X = {x}" lemma contents_eq [simp]: "contents {x} = x" by (simp add: contents_def) subsection {* Transitivity rules for calculational reasoning *} lemma set_rev_mp: "x:A ==> A \ B ==> x:B" by (rule subsetD) lemma set_mp: "A \ B ==> x:A ==> x:B" by (rule subsetD) lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c" by (rule subst) lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c" by (rule ssubst) lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c" by (rule subst) lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c" by (rule ssubst) lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==> (!!x y. x < y ==> f x < f y) ==> f a < c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a < b" hence "f a < f b" by (rule r) also assume "f b < c" finally (order_less_trans) show ?thesis . qed lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==> (!!x y. x < y ==> f x < f y) ==> a < f c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a < f b" also assume "b < c" hence "f b < f c" by (rule r) finally (order_less_trans) show ?thesis . qed lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==> (!!x y. x <= y ==> f x <= f y) ==> f a < c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a <= b" hence "f a <= f b" by (rule r) also assume "f b < c" finally (order_le_less_trans) show ?thesis . qed lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==> (!!x y. x < y ==> f x < f y) ==> a < f c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a <= f b" also assume "b < c" hence "f b < f c" by (rule r) finally (order_le_less_trans) show ?thesis . qed lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==> (!!x y. x < y ==> f x < f y) ==> f a < c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a < b" hence "f a < f b" by (rule r) also assume "f b <= c" finally (order_less_le_trans) show ?thesis . qed lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==> (!!x y. x <= y ==> f x <= f y) ==> a < f c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a < f b" also assume "b <= c" hence "f b <= f c" by (rule r) finally (order_less_le_trans) show ?thesis . qed lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==> (!!x y. x <= y ==> f x <= f y) ==> a <= f c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a <= f b" also assume "b <= c" hence "f b <= f c" by (rule r) finally (order_trans) show ?thesis . qed lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==> (!!x y. x <= y ==> f x <= f y) ==> f a <= c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a <= b" hence "f a <= f b" by (rule r) also assume "f b <= c" finally (order_trans) show ?thesis . qed lemma ord_le_eq_subst: "a <= b ==> f b = c ==> (!!x y. x <= y ==> f x <= f y) ==> f a <= c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a <= b" hence "f a <= f b" by (rule r) also assume "f b = c" finally (ord_le_eq_trans) show ?thesis . qed lemma ord_eq_le_subst: "a = f b ==> b <= c ==> (!!x y. x <= y ==> f x <= f y) ==> a <= f c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a = f b" also assume "b <= c" hence "f b <= f c" by (rule r) finally (ord_eq_le_trans) show ?thesis . qed lemma ord_less_eq_subst: "a < b ==> f b = c ==> (!!x y. x < y ==> f x < f y) ==> f a < c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a < b" hence "f a < f b" by (rule r) also assume "f b = c" finally (ord_less_eq_trans) show ?thesis . qed lemma ord_eq_less_subst: "a = f b ==> b < c ==> (!!x y. x < y ==> f x < f y) ==> a < f c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a = f b" also assume "b < c" hence "f b < f c" by (rule r) finally (ord_eq_less_trans) show ?thesis . qed text {* Note that this list of rules is in reverse order of priorities. *} lemmas basic_trans_rules [trans] = order_less_subst2 order_less_subst1 order_le_less_subst2 order_le_less_subst1 order_less_le_subst2 order_less_le_subst1 order_subst2 order_subst1 ord_le_eq_subst ord_eq_le_subst ord_less_eq_subst ord_eq_less_subst forw_subst back_subst rev_mp mp set_rev_mp set_mp order_neq_le_trans order_le_neq_trans order_less_trans order_less_asym' order_le_less_trans order_less_le_trans order_trans order_antisym ord_le_eq_trans ord_eq_le_trans ord_less_eq_trans ord_eq_less_trans trans end