(* Title: ZF/ZF.ML
ID: $Id$
Author: Lawrence C Paulson and Martin D Coen, CU Computer Laboratory
Copyright 1994 University of Cambridge
Basic introduction and elimination rules for Zermelo-Fraenkel Set Theory
*)
open ZF;
(*Useful examples: singletonI RS subst_elem, subst_elem RSN (2,IntI) *)
goal ZF.thy "!!a b A. [| b:A; a=b |] ==> a:A";
by (etac ssubst 1);
by (assume_tac 1);
val subst_elem = result();
(*** Bounded universal quantifier ***)
qed_goalw "ballI" ZF.thy [Ball_def]
"[| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)"
(fn prems=> [ (REPEAT (ares_tac (prems @ [allI,impI]) 1)) ]);
qed_goalw "bspec" 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) ]);
qed_goalw "ballE" 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*)
qed_goal "rev_bspec" 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?*)
qed_goal "rev_ballE" 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)) ]);
AddSIs [ballI];
AddEs [rev_ballE];
(*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!*)
qed_goal "ball_triv" ZF.thy "(ALL x:A. P) <-> ((EX x. x:A) --> P)"
(fn _=> [ simp_tac (simpset() addsimps [Ball_def]) 1 ]);
Addsimps [ball_triv];
(*Congruence rule for rewriting*)
qed_goalw "ball_cong" 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 ***)
qed_goalw "bexI" ZF.thy [Bex_def]
"!!P. [| P(x); x: A |] ==> EX x:A. P(x)"
(fn _=> [ Blast_tac 1 ]);
(*Not of the general form for such rules; ~EX has become ALL~ *)
qed_goal "bexCI" 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)) ]);
qed_goalw "bexE" 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)) ]);
AddIs [bexI];
AddSEs [bexE];
(*We do not even have (EX x:A. True) <-> True unless A is nonempty!!*)
qed_goal "bex_triv" ZF.thy "(EX x:A. P) <-> ((EX x. x:A) & P)"
(fn _=> [ simp_tac (simpset() addsimps [Bex_def]) 1 ]);
Addsimps [bex_triv];
qed_goalw "bex_cong" 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) ]);
Addcongs [ball_cong, bex_cong];
(*** Rules for subsets ***)
qed_goalw "subsetI" 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] *)
qed_goalw "subsetD" 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*)
qed_goalw "subsetCE" 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)) ]);
AddSIs [subsetI];
AddEs [subsetCE, subsetD];
(*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*)
qed_goal "rev_subsetD" ZF.thy "!!A B c. [| c:A; A<=B |] ==> c:B"
(fn _=> [ Blast_tac 1 ]);
qed_goal "contra_subsetD" ZF.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"
(fn _=> [ Blast_tac 1 ]);
qed_goal "rev_contra_subsetD" ZF.thy "!!c. [| c ~: B; A <= B |] ==> c ~: A"
(fn _=> [ Blast_tac 1 ]);
qed_goal "subset_refl" ZF.thy "A <= A"
(fn _=> [ Blast_tac 1 ]);
Addsimps [subset_refl];
qed_goal "subset_trans" ZF.thy "!!A B C. [| A<=B; B<=C |] ==> A<=C"
(fn _=> [ Blast_tac 1 ]);
(*Useful for proving A<=B by rewriting in some cases*)
qed_goalw "subset_iff" ZF.thy [subset_def,Ball_def]
"A<=B <-> (ALL x. x:A --> x:B)"
(fn _=> [ (rtac iff_refl 1) ]);
(*** Rules for equality ***)
(*Anti-symmetry of the subset relation*)
qed_goal "equalityI" ZF.thy "[| A <= B; B <= A |] ==> A = B"
(fn prems=> [ (REPEAT (resolve_tac (prems@[conjI, extension RS iffD2]) 1)) ]);
AddIs [equalityI];
qed_goal "equality_iffI" 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)) ]);
bind_thm ("equalityD1", extension RS iffD1 RS conjunct1);
bind_thm ("equalityD2", extension RS iffD1 RS conjunct2);
qed_goal "equalityE" ZF.thy
"[| A = B; [| A<=B; B<=A |] ==> P |] ==> P"
(fn prems=>
[ (DEPTH_SOLVE (resolve_tac (prems@[equalityD1,equalityD2]) 1)) ]);
qed_goal "equalityCE" 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)" ??*)
qed_goal "setup_induction" 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 ***)
qed_goalw "Replace_iff" 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. *)
qed_goal "ReplaceI" ZF.thy
"[| P(x,b); x: A; !!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. *)
qed_goal "ReplaceE" 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)) ]);
(*As above but without the (generally useless) 3rd assumption*)
qed_goal "ReplaceE2" ZF.thy
"[| b : {y. x:A, P(x,y)}; \
\ !!x. [| x: A; P(x,b) |] ==> R \
\ |] ==> R"
(fn major::prems=>
[ (rtac (major RS ReplaceE) 1),
(REPEAT (ares_tac prems 1)) ]);
AddIs [ReplaceI];
AddSEs [ReplaceE2];
qed_goal "Replace_cong" 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);
Addcongs [Replace_cong];
(*** Rules for RepFun ***)
qed_goalw "RepFunI" 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*)
qed_goal "RepFun_eqI" ZF.thy
"!!b a f. [| b=f(a); a : A |] ==> b : {f(x). x:A}"
(fn _ => [ etac ssubst 1, etac RepFunI 1 ]);
qed_goalw "RepFunE" 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)) ]);
AddIs [RepFun_eqI];
AddSEs [RepFunE];
qed_goalw "RepFun_cong" ZF.thy [RepFun_def]
"[| A=B; !!x. x:B ==> f(x)=g(x) |] ==> RepFun(A,f) = RepFun(B,g)"
(fn prems=> [ (simp_tac (simpset() addsimps prems) 1) ]);
Addcongs [RepFun_cong];
qed_goalw "RepFun_iff" ZF.thy [Bex_def]
"b : {f(x). x:A} <-> (EX x:A. b=f(x))"
(fn _ => [Blast_tac 1]);
goal ZF.thy "{x. x:A} = A";
by (Blast_tac 1);
qed "triv_RepFun";
Addsimps [RepFun_iff, triv_RepFun];
(*** Rules for Collect -- forming a subset by separation ***)
(*Separation is derivable from Replacement*)
qed_goalw "separation" ZF.thy [Collect_def]
"a : {x:A. P(x)} <-> a:A & P(a)"
(fn _=> [Blast_tac 1]);
Addsimps [separation];
qed_goal "CollectI" ZF.thy
"!!P. [| a:A; P(a) |] ==> a : {x:A. P(x)}"
(fn _=> [ Asm_simp_tac 1 ]);
qed_goal "CollectE" 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)) ]);
qed_goal "CollectD1" ZF.thy "!!P. a : {x:A. P(x)} ==> a:A"
(fn _=> [ (etac CollectE 1), (assume_tac 1) ]);
qed_goal "CollectD2" ZF.thy "!!P. a : {x:A. P(x)} ==> P(a)"
(fn _=> [ (etac CollectE 1), (assume_tac 1) ]);
qed_goalw "Collect_cong" ZF.thy [Collect_def]
"[| A=B; !!x. x:B ==> P(x) <-> Q(x) |] ==> Collect(A,P) = Collect(B,Q)"
(fn prems=> [ (simp_tac (simpset() addsimps prems) 1) ]);
AddSIs [CollectI];
AddSEs [CollectE];
Addcongs [Collect_cong];
(*** Rules for Unions ***)
Addsimps [Union_iff];
(*The order of the premises presupposes that C is rigid; A may be flexible*)
qed_goal "UnionI" ZF.thy "!!C. [| B: C; A: B |] ==> A: Union(C)"
(fn _=> [ Simp_tac 1, Blast_tac 1 ]);
qed_goal "UnionE" 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 Unions of families ***)
(* UN x:A. B(x) abbreviates Union({B(x). x:A}) *)
qed_goalw "UN_iff" ZF.thy [Bex_def]
"b : (UN x:A. B(x)) <-> (EX x:A. b : B(x))"
(fn _=> [ Simp_tac 1, Blast_tac 1 ]);
Addsimps [UN_iff];
(*The order of the premises presupposes that A is rigid; b may be flexible*)
qed_goal "UN_I" ZF.thy "!!A B. [| a: A; b: B(a) |] ==> b: (UN x:A. B(x))"
(fn _=> [ Simp_tac 1, Blast_tac 1 ]);
qed_goal "UN_E" 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)) ]);
qed_goal "UN_cong" ZF.thy
"[| A=B; !!x. x:B ==> C(x)=D(x) |] ==> (UN x:A. C(x)) = (UN x:B. D(x))"
(fn prems=> [ (simp_tac (simpset() addsimps prems) 1) ]);
(*No "Addcongs [UN_cong]" because UN is a combination of constants*)
(* UN_E appears before UnionE so that it is tried first, to avoid expensive
calls to hyp_subst_tac. Cannot include UN_I as it is unsafe: would enlarge
the search space.*)
AddIs [UnionI];
AddSEs [UN_E];
AddSEs [UnionE];
(*** Rules for Inter ***)
(*Not obviously useful towards proving InterI, InterD, InterE*)
qed_goalw "Inter_iff" ZF.thy [Inter_def,Ball_def]
"A : Inter(C) <-> (ALL x:C. A: x) & (EX x. x:C)"
(fn _=> [ Simp_tac 1, Blast_tac 1 ]);
(* Intersection is well-behaved only if the family is non-empty! *)
qed_goal "InterI" ZF.thy
"[| !!x. x: C ==> A: x; EX c. c:C |] ==> A : Inter(C)"
(fn prems=> [ (simp_tac (simpset() addsimps [Inter_iff]) 1),
blast_tac (claset() addIs 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". *)
qed_goalw "InterD" ZF.thy [Inter_def]
"!!C. [| A : Inter(C); B : C |] ==> A : B"
(fn _=> [ Blast_tac 1 ]);
(*"Classical" elimination rule -- does not require exhibiting B:C *)
qed_goalw "InterE" ZF.thy [Inter_def]
"[| A : Inter(C); B~:C ==> R; A:B ==> R |] ==> R"
(fn major::prems=>
[ (rtac (major RS CollectD2 RS ballE) 1),
(REPEAT (eresolve_tac prems 1)) ]);
AddSIs [InterI];
AddEs [InterD, InterE];
(*** Rules for Intersections of families ***)
(* INT x:A. B(x) abbreviates Inter({B(x). x:A}) *)
qed_goalw "INT_iff" ZF.thy [Inter_def]
"b : (INT x:A. B(x)) <-> (ALL x:A. b : B(x)) & (EX x. x:A)"
(fn _=> [ Simp_tac 1, Best_tac 1 ]);
qed_goal "INT_I" ZF.thy
"[| !!x. x: A ==> b: B(x); a: A |] ==> b: (INT x:A. B(x))"
(fn prems=> [ blast_tac (claset() addIs prems) 1 ]);
qed_goal "INT_E" 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) ]);
qed_goal "INT_cong" ZF.thy
"[| A=B; !!x. x:B ==> C(x)=D(x) |] ==> (INT x:A. C(x)) = (INT x:B. D(x))"
(fn prems=> [ (simp_tac (simpset() addsimps prems) 1) ]);
(*No "Addcongs [INT_cong]" because INT is a combination of constants*)
(*** Rules for Powersets ***)
qed_goal "PowI" ZF.thy "A <= B ==> A : Pow(B)"
(fn [prem]=> [ (rtac (prem RS (Pow_iff RS iffD2)) 1) ]);
qed_goal "PowD" ZF.thy "A : Pow(B) ==> A<=B"
(fn [major]=> [ (rtac (major RS (Pow_iff RS iffD1)) 1) ]);
AddSIs [PowI];
AddSDs [PowD];
(*** Rules for the empty set ***)
(*The set {x:0.False} is empty; by foundation it equals 0
See Suppes, page 21.*)
qed_goal "not_mem_empty" ZF.thy "a ~: 0"
(fn _=>
[ (cut_facts_tac [foundation] 1),
(best_tac (claset() addDs [equalityD2]) 1) ]);
bind_thm ("emptyE", not_mem_empty RS notE);
Addsimps [not_mem_empty];
AddSEs [emptyE];
qed_goal "empty_subsetI" ZF.thy "0 <= A"
(fn _=> [ Blast_tac 1 ]);
Addsimps [empty_subsetI];
qed_goal "equals0I" ZF.thy "[| !!y. y:A ==> False |] ==> A=0"
(fn prems=> [ blast_tac (claset() addDs prems) 1 ]);
qed_goal "equals0D" ZF.thy "!!P. [| A=0; a:A |] ==> P"
(fn _=> [ Full_simp_tac 1, Blast_tac 1 ]);
qed_goal "not_emptyI" ZF.thy "!!A a. a:A ==> A ~= 0"
(fn _=> [ Blast_tac 1 ]);
qed_goal "not_emptyE" ZF.thy "[| A ~= 0; !!x. x:A ==> R |] ==> R"
(fn [major,minor]=>
[ rtac ([major, equals0I] MRS swap) 1,
swap_res_tac [minor] 1,
assume_tac 1 ]);
(*** Cantor's Theorem: There is no surjection from a set to its powerset. ***)
val cantor_cs = FOL_cs (*precisely the rules needed for the proof*)
addSIs [ballI, CollectI, PowI, subsetI] addIs [bexI]
addSEs [CollectE, equalityCE];
(*The search is undirected; similar proof attempts may fail.
b represents ANY map, such as (lam x:A.b(x)): A->Pow(A). *)
qed_goal "cantor" ZF.thy "EX S: Pow(A). ALL x:A. b(x) ~= S"
(fn _ => [best_tac cantor_cs 1]);
(*Lemma for the inductive definition in Zorn.thy*)
qed_goal "Union_in_Pow" ZF.thy
"!!Y. Y : Pow(Pow(A)) ==> Union(Y) : Pow(A)"
(fn _ => [Blast_tac 1]);