(* Title: ZF/zf.ML
ID: $Id$
Author: Lawrence C Paulson and Martin D Coen, CU Computer Laboratory
Copyright 1992 University of Cambridge
Basic introduction and elimination rules for Zermelo-Fraenkel Set Theory
*)
open ZF;
signature ZF_LEMMAS =
sig
val ballE : thm
val ballI : thm
val ball_cong : thm
val ball_simp : thm
val ball_tac : int -> tactic
val bexCI : thm
val bexE : thm
val bexI : thm
val bex_cong : thm
val bspec : thm
val CollectD1 : thm
val CollectD2 : thm
val CollectE : thm
val CollectI : thm
val Collect_cong : thm
val emptyE : thm
val empty_subsetI : thm
val equalityCE : thm
val equalityD1 : thm
val equalityD2 : thm
val equalityE : thm
val equalityI : thm
val equality_iffI : thm
val equals0D : thm
val equals0I : thm
val ex1_functional : thm
val InterD : thm
val InterE : thm
val InterI : thm
val INT_E : thm
val INT_I : thm
val lemmas_cs : claset
val PowD : thm
val PowI : thm
val RepFunE : thm
val RepFunI : thm
val RepFun_eqI : thm
val RepFun_cong : thm
val ReplaceE : thm
val ReplaceI : thm
val Replace_iff : thm
val Replace_cong : thm
val rev_ballE : thm
val rev_bspec : thm
val rev_subsetD : thm
val separation : thm
val setup_induction : thm
val set_mp_tac : int -> tactic
val subsetCE : thm
val subsetD : thm
val subsetI : thm
val subset_refl : thm
val subset_trans : thm
val UnionE : thm
val UnionI : thm
val UN_E : thm
val UN_I : thm
end;
structure ZF_Lemmas : ZF_LEMMAS =
struct
(*** Bounded universal quantifier ***)
val ballI = prove_goalw ZF.thy [Ball_def]
"[| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)"
(fn prems=> [ (REPEAT (ares_tac (prems @ [allI,impI]) 1)) ]);
val bspec = prove_goalw ZF.thy [Ball_def]
"[| ALL x:A. P(x); x: A |] ==> P(x)"
(fn major::prems=>
[ (rtac (major RS spec RS mp) 1),
(resolve_tac prems 1) ]);
val ballE = prove_goalw ZF.thy [Ball_def]
"[| ALL x:A. P(x); P(x) ==> Q; x~:A ==> Q |] ==> Q"
(fn major::prems=>
[ (rtac (major RS allE) 1),
(REPEAT (eresolve_tac (prems@[asm_rl,impCE]) 1)) ]);
(*Used in the datatype package*)
val rev_bspec = prove_goal ZF.thy
"!!x A P. [| x: A; ALL x:A. P(x) |] ==> P(x)"
(fn _ =>
[ REPEAT (ares_tac [bspec] 1) ]);
(*Instantiates x first: better for automatic theorem proving?*)
val rev_ballE = prove_goal ZF.thy
"[| ALL x:A. P(x); x~:A ==> Q; P(x) ==> Q |] ==> Q"
(fn major::prems=>
[ (rtac (major RS ballE) 1),
(REPEAT (eresolve_tac prems 1)) ]);
(*Takes assumptions ALL x:A.P(x) and a:A; creates assumption P(a)*)
val ball_tac = dtac bspec THEN' assume_tac;
(*Trival rewrite rule; (ALL x:A.P)<->P holds only if A is nonempty!*)
val ball_simp = prove_goal ZF.thy "(ALL x:A. True) <-> True"
(fn _=> [ (REPEAT (ares_tac [TrueI,ballI,iffI] 1)) ]);
(*Congruence rule for rewriting*)
val ball_cong = prove_goalw ZF.thy [Ball_def]
"[| A=A'; !!x. x:A' ==> P(x) <-> P'(x) |] ==> Ball(A,P) <-> Ball(A',P')"
(fn prems=> [ (simp_tac (FOL_ss addsimps prems) 1) ]);
(*** Bounded existential quantifier ***)
val bexI = prove_goalw ZF.thy [Bex_def]
"[| P(x); x: A |] ==> EX x:A. P(x)"
(fn prems=> [ (REPEAT (ares_tac (prems @ [exI,conjI]) 1)) ]);
(*Not of the general form for such rules; ~EX has become ALL~ *)
val bexCI = prove_goal ZF.thy
"[| ALL 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 bexE = prove_goalw ZF.thy [Bex_def]
"[| EX x:A. P(x); !!x. [| x:A; P(x) |] ==> Q \
\ |] ==> Q"
(fn major::prems=>
[ (rtac (major RS exE) 1),
(REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1)) ]);
(*We do not even have (EX x:A. True) <-> True unless A is nonempty!!*)
val bex_cong = prove_goalw ZF.thy [Bex_def]
"[| A=A'; !!x. x:A' ==> P(x) <-> P'(x) \
\ |] ==> Bex(A,P) <-> Bex(A',P')"
(fn prems=> [ (simp_tac (FOL_ss addsimps prems addcongs [conj_cong]) 1) ]);
(*** Rules for subsets ***)
val subsetI = prove_goalw ZF.thy [subset_def]
"(!!x.x:A ==> x:B) ==> A <= B"
(fn prems=> [ (REPEAT (ares_tac (prems @ [ballI]) 1)) ]);
(*Rule in Modus Ponens style [was called subsetE] *)
val subsetD = prove_goalw ZF.thy [subset_def] "[| A <= B; c:A |] ==> c:B"
(fn major::prems=>
[ (rtac (major RS bspec) 1),
(resolve_tac prems 1) ]);
(*Classical elimination rule*)
val subsetCE = prove_goalw ZF.thy [subset_def]
"[| A <= B; c~:A ==> P; c:B ==> P |] ==> P"
(fn major::prems=>
[ (rtac (major RS ballE) 1),
(REPEAT (eresolve_tac prems 1)) ]);
(*Takes assumptions A<=B; c:A and creates the assumption c:B *)
val set_mp_tac = dtac subsetD THEN' assume_tac;
(*Sometimes useful with premises in this order*)
val rev_subsetD = prove_goal ZF.thy "!!A B c. [| c:A; A<=B |] ==> c:B"
(fn _=> [REPEAT (ares_tac [subsetD] 1)]);
val subset_refl = prove_goal ZF.thy "A <= A"
(fn _=> [ (rtac subsetI 1), atac 1 ]);
val subset_trans = prove_goal ZF.thy "[| A<=B; B<=C |] ==> A<=C"
(fn prems=> [ (REPEAT (ares_tac ([subsetI]@(prems RL [subsetD])) 1)) ]);
(*** Rules for equality ***)
(*Anti-symmetry of the subset relation*)
val equalityI = prove_goal ZF.thy "[| A <= B; B <= A |] ==> A = B"
(fn prems=> [ (REPEAT (resolve_tac (prems@[conjI, extension RS iffD2]) 1)) ]);
val equality_iffI = prove_goal ZF.thy "(!!x. x:A <-> x:B) ==> A = B"
(fn [prem] =>
[ (rtac equalityI 1),
(REPEAT (ares_tac [subsetI, prem RS iffD1, prem RS iffD2] 1)) ]);
val equalityD1 = prove_goal ZF.thy "A = B ==> A<=B"
(fn prems=>
[ (rtac (extension RS iffD1 RS conjunct1) 1),
(resolve_tac prems 1) ]);
val equalityD2 = prove_goal ZF.thy "A = B ==> B<=A"
(fn prems=>
[ (rtac (extension RS iffD1 RS conjunct2) 1),
(resolve_tac prems 1) ]);
val equalityE = prove_goal ZF.thy
"[| A = B; [| A<=B; B<=A |] ==> P |] ==> P"
(fn prems=>
[ (DEPTH_SOLVE (resolve_tac (prems@[equalityD1,equalityD2]) 1)) ]);
val equalityCE = prove_goal ZF.thy
"[| A = B; [| c:A; c:B |] ==> P; [| c~:A; c~:B |] ==> P |] ==> P"
(fn major::prems=>
[ (rtac (major RS equalityE) 1),
(REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1)) ]);
(*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.
Would it be better to do subgoal_tac "ALL z. p = f(z) --> R(z)" ??*)
val setup_induction = prove_goal ZF.thy
"[| p: A; !!z. z: A ==> p=z --> R |] ==> R"
(fn prems=>
[ (rtac mp 1),
(REPEAT (resolve_tac (refl::prems) 1)) ]);
(*** Rules for Replace -- the derived form of replacement ***)
val ex1_functional = prove_goal ZF.thy
"[| EX! z. P(a,z); P(a,b); P(a,c) |] ==> b = c"
(fn prems=>
[ (cut_facts_tac prems 1),
(best_tac FOL_dup_cs 1) ]);
val Replace_iff = prove_goalw ZF.thy [Replace_def]
"b : {y. x:A, P(x,y)} <-> (EX x:A. P(x,b) & (ALL y. P(x,y) --> y=b))"
(fn _=>
[ (rtac (replacement RS iff_trans) 1),
(REPEAT (ares_tac [refl,bex_cong,iffI,ballI,allI,conjI,impI,ex1I] 1
ORELSE eresolve_tac [conjE, spec RS mp, ex1_functional] 1)) ]);
(*Introduction; there must be a unique y such that P(x,y), namely y=b. *)
val ReplaceI = prove_goal ZF.thy
"[| x: A; P(x,b); !!y. P(x,y) ==> y=b |] ==> \
\ b : {y. x:A, P(x,y)}"
(fn prems=>
[ (rtac (Replace_iff RS iffD2) 1),
(REPEAT (ares_tac (prems@[bexI,conjI,allI,impI]) 1)) ]);
(*Elimination; may asssume there is a unique y such that P(x,y), namely y=b. *)
val ReplaceE = prove_goal ZF.thy
"[| b : {y. x:A, P(x,y)}; \
\ !!x. [| x: A; P(x,b); ALL y. P(x,y)-->y=b |] ==> R \
\ |] ==> R"
(fn prems=>
[ (rtac (Replace_iff RS iffD1 RS bexE) 1),
(etac conjE 2),
(REPEAT (ares_tac prems 1)) ]);
val Replace_cong = prove_goal ZF.thy
"[| A=B; !!x y. x:B ==> P(x,y) <-> Q(x,y) |] ==> \
\ Replace(A,P) = Replace(B,Q)"
(fn prems=>
let val substprems = prems RL [subst, ssubst]
and iffprems = prems RL [iffD1,iffD2]
in [ (rtac equalityI 1),
(REPEAT (eresolve_tac (substprems@[asm_rl, ReplaceE, spec RS mp]) 1
ORELSE resolve_tac [subsetI, ReplaceI] 1
ORELSE (resolve_tac iffprems 1 THEN assume_tac 2))) ]
end);
(*** Rules for RepFun ***)
val RepFunI = prove_goalw ZF.thy [RepFun_def]
"!!a A. a : A ==> f(a) : {f(x). x:A}"
(fn _ => [ (REPEAT (ares_tac [ReplaceI,refl] 1)) ]);
(*Useful for coinduction proofs*)
val RepFun_eqI = prove_goal ZF.thy
"!!b a f. [| b=f(a); a : A |] ==> b : {f(x). x:A}"
(fn _ => [ etac ssubst 1, etac RepFunI 1 ]);
val RepFunE = prove_goalw ZF.thy [RepFun_def]
"[| b : {f(x). x:A}; \
\ !!x.[| x:A; b=f(x) |] ==> P |] ==> \
\ P"
(fn major::prems=>
[ (rtac (major RS ReplaceE) 1),
(REPEAT (ares_tac prems 1)) ]);
val RepFun_cong = prove_goalw ZF.thy [RepFun_def]
"[| A=B; !!x. x:B ==> f(x)=g(x) |] ==> RepFun(A,f) = RepFun(B,g)"
(fn prems=> [ (simp_tac (FOL_ss addcongs [Replace_cong] addsimps prems) 1) ]);
(*** Rules for Collect -- forming a subset by separation ***)
(*Separation is derivable from Replacement*)
val separation = prove_goalw ZF.thy [Collect_def]
"a : {x:A. P(x)} <-> a:A & P(a)"
(fn _=> [ (fast_tac (FOL_cs addIs [bexI,ReplaceI]
addSEs [bexE,ReplaceE]) 1) ]);
val CollectI = prove_goal ZF.thy
"[| a:A; P(a) |] ==> a : {x:A. P(x)}"
(fn prems=>
[ (rtac (separation RS iffD2) 1),
(REPEAT (resolve_tac (prems@[conjI]) 1)) ]);
val CollectE = prove_goal ZF.thy
"[| a : {x:A. P(x)}; [| a:A; P(a) |] ==> R |] ==> R"
(fn prems=>
[ (rtac (separation RS iffD1 RS conjE) 1),
(REPEAT (ares_tac prems 1)) ]);
val CollectD1 = prove_goal ZF.thy "a : {x:A. P(x)} ==> a:A"
(fn [major]=>
[ (rtac (major RS CollectE) 1),
(assume_tac 1) ]);
val CollectD2 = prove_goal ZF.thy "a : {x:A. P(x)} ==> P(a)"
(fn [major]=>
[ (rtac (major RS CollectE) 1),
(assume_tac 1) ]);
val Collect_cong = prove_goalw ZF.thy [Collect_def]
"[| A=B; !!x. x:B ==> P(x) <-> Q(x) |] ==> Collect(A,P) = Collect(B,Q)"
(fn prems=> [ (simp_tac (FOL_ss addcongs [Replace_cong] addsimps prems) 1) ]);
(*** Rules for Unions ***)
(*The order of the premises presupposes that C is rigid; A may be flexible*)
val UnionI = prove_goal ZF.thy "[| B: C; A: B |] ==> A: Union(C)"
(fn prems=>
[ (resolve_tac [union_iff RS iffD2] 1),
(REPEAT (resolve_tac (prems @ [bexI]) 1)) ]);
val UnionE = prove_goal ZF.thy
"[| A : Union(C); !!B.[| A: B; B: C |] ==> R |] ==> R"
(fn prems=>
[ (resolve_tac [union_iff RS iffD1 RS bexE] 1),
(REPEAT (ares_tac prems 1)) ]);
(*** Rules for Inter ***)
(*Not obviously useful towards proving InterI, InterD, InterE*)
val Inter_iff = prove_goalw ZF.thy [Inter_def,Ball_def]
"A : Inter(C) <-> (ALL x:C. A: x) & (EX x. x:C)"
(fn _=> [ (rtac (separation RS iff_trans) 1),
(fast_tac (FOL_cs addIs [UnionI] addSEs [UnionE]) 1) ]);
(* Intersection is well-behaved only if the family is non-empty! *)
val InterI = prove_goalw ZF.thy [Inter_def]
"[| !!x. x: C ==> A: x; c:C |] ==> A : Inter(C)"
(fn prems=>
[ (DEPTH_SOLVE (ares_tac ([CollectI,UnionI,ballI] @ prems) 1)) ]);
(*A "destruct" rule -- every B in C contains A as an element, but
A:B can hold when B:C does not! This rule is analogous to "spec". *)
val InterD = prove_goalw ZF.thy [Inter_def]
"[| A : Inter(C); B : C |] ==> A : B"
(fn [major,minor]=>
[ (rtac (major RS CollectD2 RS bspec) 1),
(rtac minor 1) ]);
(*"Classical" elimination rule -- does not require exhibiting B:C *)
val InterE = prove_goalw ZF.thy [Inter_def]
"[| A : Inter(C); A:B ==> R; B~:C ==> R |] ==> R"
(fn major::prems=>
[ (rtac (major RS CollectD2 RS ballE) 1),
(REPEAT (eresolve_tac prems 1)) ]);
(*** Rules for Unions of families ***)
(* UN x:A. B(x) abbreviates Union({B(x). x:A}) *)
(*The order of the premises presupposes that A is rigid; b may be flexible*)
val UN_I = prove_goal ZF.thy "[| a: A; b: B(a) |] ==> b: (UN x:A. B(x))"
(fn prems=>
[ (REPEAT (resolve_tac (prems@[UnionI,RepFunI]) 1)) ]);
val UN_E = prove_goal ZF.thy
"[| b : (UN x:A. B(x)); !!x.[| x: A; b: B(x) |] ==> R |] ==> R"
(fn major::prems=>
[ (rtac (major RS UnionE) 1),
(REPEAT (eresolve_tac (prems@[asm_rl, RepFunE, subst]) 1)) ]);
(*** Rules for Intersections of families ***)
(* INT x:A. B(x) abbreviates Inter({B(x). x:A}) *)
val INT_I = prove_goal ZF.thy
"[| !!x. x: A ==> b: B(x); a: A |] ==> b: (INT x:A. B(x))"
(fn prems=>
[ (REPEAT (ares_tac (prems@[InterI,RepFunI]) 1
ORELSE eresolve_tac [RepFunE,ssubst] 1)) ]);
val INT_E = prove_goal ZF.thy
"[| b : (INT x:A. B(x)); a: A |] ==> b : B(a)"
(fn [major,minor]=>
[ (rtac (major RS InterD) 1),
(rtac (minor RS RepFunI) 1) ]);
(*** Rules for Powersets ***)
val PowI = prove_goal ZF.thy "A <= B ==> A : Pow(B)"
(fn [prem]=> [ (rtac (prem RS (power_set RS iffD2)) 1) ]);
val PowD = prove_goal ZF.thy "A : Pow(B) ==> A<=B"
(fn [major]=> [ (rtac (major RS (power_set RS iffD1)) 1) ]);
(*** Rules for the empty set ***)
(*The set {x:0.False} is empty; by foundation it equals 0
See Suppes, page 21.*)
val emptyE = prove_goal ZF.thy "a:0 ==> P"
(fn [major]=>
[ (rtac (foundation RS disjE) 1),
(etac (equalityD2 RS subsetD RS CollectD2 RS FalseE) 1),
(rtac major 1),
(etac bexE 1),
(etac (CollectD2 RS FalseE) 1) ]);
val empty_subsetI = prove_goal ZF.thy "0 <= A"
(fn _ => [ (REPEAT (ares_tac [equalityI,subsetI,emptyE] 1)) ]);
val equals0I = prove_goal ZF.thy "[| !!y. y:A ==> False |] ==> A=0"
(fn prems=>
[ (REPEAT (ares_tac (prems@[empty_subsetI,subsetI,equalityI]) 1
ORELSE eresolve_tac (prems RL [FalseE]) 1)) ]);
val equals0D = prove_goal ZF.thy "[| A=0; a:A |] ==> P"
(fn [major,minor]=>
[ (rtac (minor RS (major RS equalityD1 RS subsetD RS emptyE)) 1) ]);
val lemmas_cs = FOL_cs
addSIs [ballI, InterI, CollectI, PowI, subsetI]
addIs [bexI, UnionI, ReplaceI, RepFunI]
addSEs [bexE, make_elim PowD, UnionE, ReplaceE, RepFunE,
CollectE, emptyE]
addEs [rev_ballE, InterD, make_elim InterD, subsetD, subsetCE];
end;
open ZF_Lemmas;