(* Title: Set/Set.ML ID: $Id: Set.ML,v 1.9 2005/09/17 15:35:28 wenzelm Exp $ *) val [prem] = goal (the_context ()) "[| P(a) |] ==> a : {x. P(x)}"; by (rtac (mem_Collect_iff RS iffD2) 1); by (rtac prem 1); qed "CollectI"; val prems = goal (the_context ()) "[| a : {x. P(x)} |] ==> P(a)"; by (resolve_tac (prems RL [mem_Collect_iff RS iffD1]) 1); qed "CollectD"; val CollectE = make_elim CollectD; val [prem] = goal (the_context ()) "[| !!x. x:A <-> x:B |] ==> A = B"; by (rtac (set_extension RS iffD2) 1); by (rtac (prem RS allI) 1); qed "set_ext"; (*** Bounded quantifiers ***) val prems = goalw (the_context ()) [Ball_def] "[| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)"; by (REPEAT (ares_tac (prems @ [allI,impI]) 1)); qed "ballI"; val [major,minor] = goalw (the_context ()) [Ball_def] "[| ALL x:A. P(x); x:A |] ==> P(x)"; by (rtac (minor RS (major RS spec RS mp)) 1); qed "bspec"; val major::prems = goalw (the_context ()) [Ball_def] "[| ALL 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 ALL 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 (the_context ()) [Bex_def] "[| P(x); x:A |] ==> EX x:A. P(x)"; by (REPEAT (ares_tac (prems @ [exI,conjI]) 1)); qed "bexI"; qed_goal "bexCI" (the_context ()) "[| EX x:A. ~P(x) ==> P(a); a:A |] ==> EX x:A. P(x)" (fn prems=> [ (rtac classical 1), (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1)) ]); val major::prems = goalw (the_context ()) [Bex_def] "[| EX 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!*) val prems = goal (the_context ()) "(ALL x:A. True) <-> True"; by (REPEAT (ares_tac [TrueI,ballI,iffI] 1)); qed "ball_rew"; (** Congruence rules **) val prems = goal (the_context ()) "[| A=A'; !!x. x:A' ==> P(x) <-> P'(x) |] ==> \ \ (ALL x:A. P(x)) <-> (ALL x:A'. P'(x))"; by (resolve_tac (prems RL [ssubst,iffD2]) 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 (the_context ()) "[| A=A'; !!x. x:A' ==> P(x) <-> P'(x) |] ==> \ \ (EX x:A. P(x)) <-> (EX x:A'. P'(x))"; by (resolve_tac (prems RL [ssubst,iffD2]) 1); by (REPEAT (etac bexE 1 ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1)); qed "bex_cong"; (*** Rules for subsets ***) val prems = goalw (the_context ()) [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 (the_context ()) [subset_def] "[| A <= B; c:A |] ==> c:B"; by (rtac (major RS bspec) 1); by (resolve_tac prems 1); qed "subsetD"; (*Classical elimination rule*) val major::prems = goalw (the_context ()) [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" (the_context ()) "A <= A" (fn _=> [ (REPEAT (ares_tac [subsetI] 1)) ]); Goal "[| A<=B; B<=C |] ==> A<=C"; by (rtac subsetI 1); by (REPEAT (eresolve_tac [asm_rl, subsetD] 1)); qed "subset_trans"; (*** Rules for equality ***) (*Anti-symmetry of the subset relation*) val prems = goal (the_context ()) "[| A <= B; B <= A |] ==> A = B"; by (rtac (iffI RS set_ext) 1); by (REPEAT (ares_tac (prems RL [subsetD]) 1)); qed "subset_antisym"; val equalityI = subset_antisym; (* Equality rules from ZF set theory -- are they appropriate here? *) val prems = goal (the_context ()) "A = B ==> A<=B"; by (resolve_tac (prems RL [subst]) 1); by (rtac subset_refl 1); qed "equalityD1"; val prems = goal (the_context ()) "A = B ==> B<=A"; by (resolve_tac (prems RL [subst]) 1); by (rtac subset_refl 1); qed "equalityD2"; val prems = goal (the_context ()) "[| A = B; [| A<=B; B<=A |] ==> P |] ==> P"; by (resolve_tac prems 1); by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1)); qed "equalityE"; val major::prems = goal (the_context ()) "[| 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 (the_context ()) "[| p:A; !!z. z:A ==> p=z --> R |] ==> R"; by (rtac mp 1); by (REPEAT (resolve_tac (refl::prems) 1)); qed "setup_induction"; Goal "{x. x:A} = A"; by (REPEAT (ares_tac [equalityI,subsetI,CollectI] 1 ORELSE etac CollectD 1)); qed "trivial_set"; (*** Rules for binary union -- Un ***) val prems = goalw (the_context ()) [Un_def] "c:A ==> c : A Un B"; by (REPEAT (resolve_tac (prems @ [CollectI,disjI1]) 1)); qed "UnI1"; val prems = goalw (the_context ()) [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" (the_context ()) "(~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 (the_context ()) [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"; (*** Rules for small intersection -- Int ***) val prems = goalw (the_context ()) [Int_def] "[| c:A; c:B |] ==> c : A Int B"; by (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1)); qed "IntI"; val [major] = goalw (the_context ()) [Int_def] "c : A Int B ==> c:A"; by (rtac (major RS CollectD RS conjunct1) 1); qed "IntD1"; val [major] = goalw (the_context ()) [Int_def] "c : A Int B ==> c:B"; by (rtac (major RS CollectD RS conjunct2) 1); qed "IntD2"; val [major,minor] = goal (the_context ()) "[| 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"; (*** Rules for set complement -- Compl ***) val prems = goalw (the_context ()) [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 (the_context ()) [Compl_def] "[| c : Compl(A) |] ==> ~c:A"; by (rtac (major RS CollectD) 1); qed "ComplD"; val ComplE = make_elim ComplD; (*** Empty sets ***) Goalw [empty_def] "{x. False} = {}"; by (rtac refl 1); qed "empty_eq"; val [prem] = goalw (the_context ()) [empty_def] "a : {} ==> P"; by (rtac (prem RS CollectD RS FalseE) 1); qed "emptyD"; val emptyE = make_elim emptyD; val [prem] = goal (the_context ()) "~ A={} ==> (EX x. x:A)"; by (rtac (prem RS swap) 1); by (rtac equalityI 1); by (ALLGOALS (fast_tac (FOL_cs addSIs [subsetI] addSEs [emptyD]))); qed "not_emptyD"; (*** Singleton sets ***) Goalw [singleton_def] "a : {a}"; by (rtac CollectI 1); by (rtac refl 1); qed "singletonI"; val [major] = goalw (the_context ()) [singleton_def] "b : {a} ==> b=a"; by (rtac (major RS CollectD) 1); qed "singletonD"; val singletonE = make_elim singletonD; (*** Unions of families ***) (*The order of the premises presupposes that A is rigid; b may be flexible*) val prems = goalw (the_context ()) [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 (the_context ()) [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 (the_context ()) "[| 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"; (*** Intersections of families -- INTER x:A. B(x) is Inter(B)``A ) *) val prems = goalw (the_context ()) [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 (the_context ()) [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 rule -- does not require proving X:C *) val major::prems = goalw (the_context ()) [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 (the_context ()) "[| 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"; (*** Rules for Unions ***) (*The order of the premises presupposes that C is rigid; A may be flexible*) val prems = goalw (the_context ()) [Union_def] "[| X:C; A:X |] ==> A : Union(C)"; by (REPEAT (resolve_tac (prems @ [UN_I]) 1)); qed "UnionI"; val major::prems = goalw (the_context ()) [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"; (*** Rules for Inter ***) val prems = goalw (the_context ()) [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 (the_context ()) [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 (the_context ()) [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";