(* Title: HOL/set ID: $Id$ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1991 University of Cambridge For set.thy. Set theory for higher-order logic. A set is simply a predicate. *) open Set; section "Relating predicates and sets"; val [prem] = goal Set.thy "P(a) ==> a : {x.P(x)}"; by (rtac (mem_Collect_eq RS ssubst) 1); by (rtac prem 1); qed "CollectI"; val prems = goal Set.thy "[| a : {x.P(x)} |] ==> P(a)"; by (resolve_tac (prems RL [mem_Collect_eq RS subst]) 1); qed "CollectD"; val [prem] = goal Set.thy "[| !!x. (x:A) = (x:B) |] ==> A = B"; by (rtac (prem RS ext RS arg_cong RS box_equals) 1); by (rtac Collect_mem_eq 1); by (rtac Collect_mem_eq 1); qed "set_ext"; val [prem] = goal Set.thy "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}"; by (rtac (prem RS ext RS arg_cong) 1); qed "Collect_cong"; val CollectE = make_elim CollectD; section "Bounded quantifiers"; val prems = goalw Set.thy [Ball_def] "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)"; by (REPEAT (ares_tac (prems @ [allI,impI]) 1)); qed "ballI"; val [major,minor] = goalw Set.thy [Ball_def] "[| ! x:A. P(x); x:A |] ==> P(x)"; by (rtac (minor RS (major RS spec RS mp)) 1); qed "bspec"; val major::prems = goalw Set.thy [Ball_def] "[| ! x:A. P(x); P(x) ==> Q; x~:A ==> Q |] ==> Q"; by (rtac (major RS spec RS impCE) 1); by (REPEAT (eresolve_tac prems 1)); qed "ballE"; (*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*) fun ball_tac i = etac ballE i THEN contr_tac (i+1); val prems = goalw Set.thy [Bex_def] "[| P(x); x:A |] ==> ? x:A. P(x)"; by (REPEAT (ares_tac (prems @ [exI,conjI]) 1)); qed "bexI"; qed_goal "bexCI" Set.thy "[| ! x:A. ~P(x) ==> P(a); a:A |] ==> ? x:A.P(x)" (fn prems=> [ (rtac classical 1), (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1)) ]); val major::prems = goalw Set.thy [Bex_def] "[| ? x:A. P(x); !!x. [| x:A; P(x) |] ==> Q |] ==> Q"; by (rtac (major RS exE) 1); by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1)); qed "bexE"; (*Trival rewrite rule; (! x:A.P)=P holds only if A is nonempty!*) goalw Set.thy [Ball_def] "(! x:A. True) = True"; by (Simp_tac 1); qed "ball_True"; (*Dual form for existentials*) goalw Set.thy [Bex_def] "(? x:A. False) = False"; by (Simp_tac 1); qed "bex_False"; Addsimps [ball_True, bex_False]; (** Congruence rules **) val prems = goal Set.thy "[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ \ (! x:A. P(x)) = (! x:B. Q(x))"; by (resolve_tac (prems RL [ssubst]) 1); by (REPEAT (ares_tac [ballI,iffI] 1 ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1)); qed "ball_cong"; val prems = goal Set.thy "[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ \ (? x:A. P(x)) = (? x:B. Q(x))"; by (resolve_tac (prems RL [ssubst]) 1); by (REPEAT (etac bexE 1 ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1)); qed "bex_cong"; section "Subsets"; val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B"; by (REPEAT (ares_tac (prems @ [ballI]) 1)); qed "subsetI"; (*Rule in Modus Ponens style*) val major::prems = goalw Set.thy [subset_def] "[| A <= B; c:A |] ==> c:B"; by (rtac (major RS bspec) 1); by (resolve_tac prems 1); qed "subsetD"; (*The same, with reversed premises for use with etac -- cf rev_mp*) qed_goal "rev_subsetD" Set.thy "[| c:A; A <= B |] ==> c:B" (fn prems=> [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]); (*Converts A<=B to x:A ==> x:B*) fun impOfSubs th = th RSN (2, rev_subsetD); qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A" (fn prems=> [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]); qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B; A <= B |] ==> c ~: A" (fn prems=> [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]); (*Classical elimination rule*) val major::prems = goalw Set.thy [subset_def] "[| A <= B; c~:A ==> P; c:B ==> P |] ==> P"; by (rtac (major RS ballE) 1); by (REPEAT (eresolve_tac prems 1)); qed "subsetCE"; (*Takes assumptions A<=B; c:A and creates the assumption c:B *) fun set_mp_tac i = etac subsetCE i THEN mp_tac i; qed_goal "subset_refl" Set.thy "A <= (A::'a set)" (fn _=> [ (REPEAT (ares_tac [subsetI] 1)) ]); val prems = goal Set.thy "[| A<=B; B<=C |] ==> A<=(C::'a set)"; by (cut_facts_tac prems 1); by (REPEAT (ares_tac [subsetI] 1 ORELSE set_mp_tac 1)); qed "subset_trans"; section "Equality"; (*Anti-symmetry of the subset relation*) val prems = goal Set.thy "[| A <= B; B <= A |] ==> A = (B::'a set)"; by (rtac (iffI RS set_ext) 1); by (REPEAT (ares_tac (prems RL [subsetD]) 1)); qed "subset_antisym"; val equalityI = subset_antisym; AddSIs [equalityI]; (* Equality rules from ZF set theory -- are they appropriate here? *) val prems = goal Set.thy "A = B ==> A<=(B::'a set)"; by (resolve_tac (prems RL [subst]) 1); by (rtac subset_refl 1); qed "equalityD1"; val prems = goal Set.thy "A = B ==> B<=(A::'a set)"; by (resolve_tac (prems RL [subst]) 1); by (rtac subset_refl 1); qed "equalityD2"; val prems = goal Set.thy "[| A = B; [| A<=B; B<=(A::'a set) |] ==> P |] ==> P"; by (resolve_tac prems 1); by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1)); qed "equalityE"; val major::prems = goal Set.thy "[| A = B; [| c:A; c:B |] ==> P; [| c~:A; c~:B |] ==> P |] ==> P"; by (rtac (major RS equalityE) 1); by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1)); qed "equalityCE"; (*Lemma for creating induction formulae -- for "pattern matching" on p To make the induction hypotheses usable, apply "spec" or "bspec" to put universal quantifiers over the free variables in p. *) val prems = goal Set.thy "[| p:A; !!z. z:A ==> p=z --> R |] ==> R"; by (rtac mp 1); by (REPEAT (resolve_tac (refl::prems) 1)); qed "setup_induction"; section "Set complement -- Compl"; val prems = goalw Set.thy [Compl_def] "[| c:A ==> False |] ==> c : Compl(A)"; by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1)); qed "ComplI"; (*This form, with negated conclusion, works well with the Classical prover. Negated assumptions behave like formulae on the right side of the notional turnstile...*) val major::prems = goalw Set.thy [Compl_def] "[| c : Compl(A) |] ==> c~:A"; by (rtac (major RS CollectD) 1); qed "ComplD"; val ComplE = make_elim ComplD; qed_goal "Compl_iff" Set.thy "(c : Compl(A)) = (c~:A)" (fn _ => [ (fast_tac (!claset addSIs [ComplI] addSEs [ComplE]) 1) ]); section "Binary union -- Un"; val prems = goalw Set.thy [Un_def] "c:A ==> c : A Un B"; by (REPEAT (resolve_tac (prems @ [CollectI,disjI1]) 1)); qed "UnI1"; val prems = goalw Set.thy [Un_def] "c:B ==> c : A Un B"; by (REPEAT (resolve_tac (prems @ [CollectI,disjI2]) 1)); qed "UnI2"; (*Classical introduction rule: no commitment to A vs B*) qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B" (fn prems=> [ (rtac classical 1), (REPEAT (ares_tac (prems@[UnI1,notI]) 1)), (REPEAT (ares_tac (prems@[UnI2,notE]) 1)) ]); val major::prems = goalw Set.thy [Un_def] "[| c : A Un B; c:A ==> P; c:B ==> P |] ==> P"; by (rtac (major RS CollectD RS disjE) 1); by (REPEAT (eresolve_tac prems 1)); qed "UnE"; qed_goal "Un_iff" Set.thy "(c : A Un B) = (c:A | c:B)" (fn _ => [ (fast_tac (!claset addSIs [UnCI] addSEs [UnE]) 1) ]); section "Binary intersection -- Int"; val prems = goalw Set.thy [Int_def] "[| c:A; c:B |] ==> c : A Int B"; by (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1)); qed "IntI"; val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:A"; by (rtac (major RS CollectD RS conjunct1) 1); qed "IntD1"; val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:B"; by (rtac (major RS CollectD RS conjunct2) 1); qed "IntD2"; val [major,minor] = goal Set.thy "[| c : A Int B; [| c:A; c:B |] ==> P |] ==> P"; by (rtac minor 1); by (rtac (major RS IntD1) 1); by (rtac (major RS IntD2) 1); qed "IntE"; qed_goal "Int_iff" Set.thy "(c : A Int B) = (c:A & c:B)" (fn _ => [ (fast_tac (!claset addSIs [IntI] addSEs [IntE]) 1) ]); section "Set difference"; qed_goalw "DiffI" Set.thy [set_diff_def] "[| c : A; c ~: B |] ==> c : A - B" (fn prems=> [ (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1)) ]); qed_goalw "DiffD1" Set.thy [set_diff_def] "c : A - B ==> c : A" (fn [major]=> [ (rtac (major RS CollectD RS conjunct1) 1) ]); qed_goalw "DiffD2" Set.thy [set_diff_def] "[| c : A - B; c : B |] ==> P" (fn [major,minor]=> [rtac (minor RS (major RS CollectD RS conjunct2 RS notE)) 1]); qed_goal "DiffE" Set.thy "[| c : A - B; [| c:A; c~:B |] ==> P |] ==> P" (fn prems=> [ (resolve_tac prems 1), (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]); qed_goal "Diff_iff" Set.thy "(c : A-B) = (c:A & c~:B)" (fn _ => [ (fast_tac (!claset addSIs [DiffI] addSEs [DiffE]) 1) ]); section "The empty set -- {}"; qed_goalw "emptyE" Set.thy [empty_def] "a:{} ==> P" (fn [prem] => [rtac (prem RS CollectD RS FalseE) 1]); qed_goal "empty_subsetI" Set.thy "{} <= A" (fn _ => [ (REPEAT (ares_tac [equalityI,subsetI,emptyE] 1)) ]); qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}" (fn prems=> [ (REPEAT (ares_tac (prems@[empty_subsetI,subsetI,equalityI]) 1 ORELSE eresolve_tac (prems RL [FalseE]) 1)) ]); qed_goal "equals0D" Set.thy "[| A={}; a:A |] ==> P" (fn [major,minor]=> [ (rtac (minor RS (major RS equalityD1 RS subsetD RS emptyE)) 1) ]); qed_goal "empty_iff" Set.thy "(c : {}) = False" (fn _ => [ (fast_tac (!claset addSEs [emptyE]) 1) ]); goal Set.thy "Ball {} P = True"; by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Ball_def, empty_def]) 1); qed "ball_empty"; goal Set.thy "Bex {} P = False"; by (simp_tac (HOL_ss addsimps [mem_Collect_eq, Bex_def, empty_def]) 1); qed "bex_empty"; Addsimps [ball_empty, bex_empty]; section "Augmenting a set -- insert"; qed_goalw "insertI1" Set.thy [insert_def] "a : insert a B" (fn _ => [rtac (CollectI RS UnI1) 1, rtac refl 1]); qed_goalw "insertI2" Set.thy [insert_def] "a : B ==> a : insert b B" (fn [prem]=> [ (rtac (prem RS UnI2) 1) ]); qed_goalw "insertE" Set.thy [insert_def] "[| a : insert b A; a=b ==> P; a:A ==> P |] ==> P" (fn major::prems=> [ (rtac (major RS UnE) 1), (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]); qed_goal "insert_iff" Set.thy "a : insert b A = (a=b | a:A)" (fn _ => [fast_tac (!claset addIs [insertI1,insertI2] addSEs [insertE]) 1]); (*Classical introduction rule*) qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B" (fn [prem]=> [ (rtac (disjCI RS (insert_iff RS iffD2)) 1), (etac prem 1) ]); section "Singletons, using insert"; qed_goal "singletonI" Set.thy "a : {a}" (fn _=> [ (rtac insertI1 1) ]); goalw Set.thy [insert_def] "!!a. b : {a} ==> b=a"; by (fast_tac (!claset addSEs [emptyE,CollectE,UnE]) 1); qed "singletonD"; bind_thm ("singletonE", make_elim singletonD); qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" (fn _ => [ rtac iffI 1, etac singletonD 1, hyp_subst_tac 1, rtac singletonI 1]); val [major] = goal Set.thy "{a}={b} ==> a=b"; by (rtac (major RS equalityD1 RS subsetD RS singletonD) 1); by (rtac singletonI 1); qed "singleton_inject"; section "The universal set -- UNIV"; qed_goal "UNIV_I" Set.thy "x : UNIV" (fn _ => [rtac ComplI 1, etac emptyE 1]); qed_goal "subset_UNIV" Set.thy "A <= UNIV" (fn _ => [rtac subsetI 1, rtac UNIV_I 1]); section "Unions of families -- UNION x:A. B(x) is Union(B``A)"; (*The order of the premises presupposes that A is rigid; b may be flexible*) val prems = goalw Set.thy [UNION_def] "[| a:A; b: B(a) |] ==> b: (UN x:A. B(x))"; by (REPEAT (resolve_tac (prems @ [bexI,CollectI]) 1)); qed "UN_I"; val major::prems = goalw Set.thy [UNION_def] "[| b : (UN x:A. B(x)); !!x.[| x:A; b: B(x) |] ==> R |] ==> R"; by (rtac (major RS CollectD RS bexE) 1); by (REPEAT (ares_tac prems 1)); qed "UN_E"; val prems = goal Set.thy "[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ \ (UN x:A. C(x)) = (UN x:B. D(x))"; by (REPEAT (etac UN_E 1 ORELSE ares_tac ([UN_I,equalityI,subsetI] @ (prems RL [equalityD1,equalityD2] RL [subsetD])) 1)); qed "UN_cong"; section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)"; val prems = goalw Set.thy [INTER_def] "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))"; by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1)); qed "INT_I"; val major::prems = goalw Set.thy [INTER_def] "[| b : (INT x:A. B(x)); a:A |] ==> b: B(a)"; by (rtac (major RS CollectD RS bspec) 1); by (resolve_tac prems 1); qed "INT_D"; (*"Classical" elimination -- by the Excluded Middle on a:A *) val major::prems = goalw Set.thy [INTER_def] "[| b : (INT x:A. B(x)); b: B(a) ==> R; a~:A ==> R |] ==> R"; by (rtac (major RS CollectD RS ballE) 1); by (REPEAT (eresolve_tac prems 1)); qed "INT_E"; val prems = goal Set.thy "[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ \ (INT x:A. C(x)) = (INT x:B. D(x))"; by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI])); by (REPEAT (dtac INT_D 1 ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1)); qed "INT_cong"; section "Unions over a type; UNION1(B) = Union(range(B))"; (*The order of the premises presupposes that A is rigid; b may be flexible*) val prems = goalw Set.thy [UNION1_def] "b: B(x) ==> b: (UN x. B(x))"; by (REPEAT (resolve_tac (prems @ [TrueI, CollectI RS UN_I]) 1)); qed "UN1_I"; val major::prems = goalw Set.thy [UNION1_def] "[| b : (UN x. B(x)); !!x. b: B(x) ==> R |] ==> R"; by (rtac (major RS UN_E) 1); by (REPEAT (ares_tac prems 1)); qed "UN1_E"; section "Intersections over a type; INTER1(B) = Inter(range(B))"; val prems = goalw Set.thy [INTER1_def] "(!!x. b: B(x)) ==> b : (INT x. B(x))"; by (REPEAT (ares_tac (INT_I::prems) 1)); qed "INT1_I"; val [major] = goalw Set.thy [INTER1_def] "b : (INT x. B(x)) ==> b: B(a)"; by (rtac (TrueI RS (CollectI RS (major RS INT_D))) 1); qed "INT1_D"; section "Union"; (*The order of the premises presupposes that C is rigid; A may be flexible*) val prems = goalw Set.thy [Union_def] "[| X:C; A:X |] ==> A : Union(C)"; by (REPEAT (resolve_tac (prems @ [UN_I]) 1)); qed "UnionI"; val major::prems = goalw Set.thy [Union_def] "[| A : Union(C); !!X.[| A:X; X:C |] ==> R |] ==> R"; by (rtac (major RS UN_E) 1); by (REPEAT (ares_tac prems 1)); qed "UnionE"; section "Inter"; val prems = goalw Set.thy [Inter_def] "[| !!X. X:C ==> A:X |] ==> A : Inter(C)"; by (REPEAT (ares_tac ([INT_I] @ prems) 1)); qed "InterI"; (*A "destruct" rule -- every X in C contains A as an element, but A:X can hold when X:C does not! This rule is analogous to "spec". *) val major::prems = goalw Set.thy [Inter_def] "[| A : Inter(C); X:C |] ==> A:X"; by (rtac (major RS INT_D) 1); by (resolve_tac prems 1); qed "InterD"; (*"Classical" elimination rule -- does not require proving X:C *) val major::prems = goalw Set.thy [Inter_def] "[| A : Inter(C); A:X ==> R; X~:C ==> R |] ==> R"; by (rtac (major RS INT_E) 1); by (REPEAT (eresolve_tac prems 1)); qed "InterE"; section "The Powerset operator -- Pow"; qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)" (fn _ => [ (etac CollectI 1) ]); qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B) ==> A<=B" (fn _=> [ (etac CollectD 1) ]); val Pow_bottom = empty_subsetI RS PowI; (* {}: Pow(B) *) val Pow_top = subset_refl RS PowI; (* A : Pow(A) *) (*** Set reasoning tools ***) val mem_simps = [ Un_iff, Int_iff, Compl_iff, Diff_iff, singleton_iff, mem_Collect_eq]; (*Not for Addsimps -- it can cause goals to blow up!*) goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))"; by (simp_tac (!simpset setloop split_tac [expand_if]) 1); qed "mem_if"; val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs; simpset := !simpset addsimps mem_simps addcongs [ball_cong,bex_cong] setmksimps (mksimps mksimps_pairs);