10751
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(* Title : Zorn.ML
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ID : $Id$
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Author : Jacques D. Fleuriot
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Copyright : 1998 University of Cambridge
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Description : Zorn's Lemma -- adapted proofs from lcp's ZF/Zorn.ML
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*)
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(*---------------------------------------------------------------
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Section 1. Mathematical Preamble
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---------------------------------------------------------------*)
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Goal "(ALL x:C. x<=A | B<=x) ==> Union(C)<=A | B<=Union(C)";
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by (Blast_tac 1);
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qed "Union_lemma0";
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(*-- similar to subset_cs in ZF/subset.thy --*)
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bind_thms ("thissubset_SIs",
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[subset_refl,Union_least, UN_least, Un_least,
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Inter_greatest, Int_greatest,
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Un_upper1, Un_upper2, Int_lower1, Int_lower2]);
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(*A claset for subset reasoning*)
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val thissubset_cs = claset()
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delrules [subsetI, subsetCE]
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addSIs thissubset_SIs
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addIs [Union_upper, Inter_lower];
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(* increasingD2 of ZF/Zorn.ML *)
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Goalw [succ_def] "x <= succ S x";
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by (rtac (split_if RS iffD2) 1);
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by (auto_tac (claset(),simpset() addsimps [super_def,
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maxchain_def,psubset_def]));
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by (rtac swap 1 THEN assume_tac 1);
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by (rtac someI2 1);
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by (ALLGOALS(Blast_tac));
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qed "Abrial_axiom1";
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val [TFin_succI, Pow_TFin_UnionI] = TFin.intrs;
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val TFin_UnionI = PowI RS Pow_TFin_UnionI;
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bind_thm ("TFin_succI", TFin_succI);
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bind_thm ("Pow_TFin_UnionI", Pow_TFin_UnionI);
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bind_thm ("TFin_UnionI", TFin_UnionI);
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val major::prems = Goal
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"[| n : TFin S; \
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\ !!x. [| x: TFin S; P(x) |] ==> P(succ S x); \
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\ !!Y. [| Y <= TFin S; Ball Y P |] ==> P(Union Y) |] \
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\ ==> P(n)";
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by (rtac (major RS TFin.induct) 1);
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by (ALLGOALS (fast_tac (claset() addIs prems)));
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qed "TFin_induct";
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(*Perform induction on n, then prove the major premise using prems. *)
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fun TFin_ind_tac a prems i =
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EVERY [induct_thm_tac TFin_induct a i,
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rename_last_tac a ["1"] (i+1),
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rename_last_tac a ["2"] (i+2),
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ares_tac prems i];
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Goal "x <= y ==> x <= succ S y";
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by (etac (Abrial_axiom1 RSN (2,subset_trans)) 1);
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qed "succ_trans";
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(*Lemma 1 of section 3.1*)
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Goal "[| n: TFin S; m: TFin S; \
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\ ALL x: TFin S. x <= m --> x = m | succ S x <= m \
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\ |] ==> n <= m | succ S m <= n";
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by (etac TFin_induct 1);
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by (etac Union_lemma0 2); (*or just Blast_tac*)
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by (blast_tac (thissubset_cs addIs [succ_trans]) 1);
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qed "TFin_linear_lemma1";
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(* Lemma 2 of section 3.2 *)
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Goal "m: TFin S ==> ALL n: TFin S. n<=m --> n=m | succ S n<=m";
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by (etac TFin_induct 1);
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by (rtac (impI RS ballI) 1);
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(*case split using TFin_linear_lemma1*)
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by (res_inst_tac [("n1","n"), ("m1","x")]
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(TFin_linear_lemma1 RS disjE) 1 THEN REPEAT (assume_tac 1));
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by (dres_inst_tac [("x","n")] bspec 1 THEN assume_tac 1);
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by (blast_tac (thissubset_cs addIs [succ_trans]) 1);
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by (REPEAT (ares_tac [disjI1,equalityI] 1));
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(*second induction step*)
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by (rtac (impI RS ballI) 1);
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by (rtac (Union_lemma0 RS disjE) 1);
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by (rtac disjI2 3);
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by (REPEAT (ares_tac [disjI1,equalityI] 2));
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by (rtac ballI 1);
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by (ball_tac 1);
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by (set_mp_tac 1);
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by (res_inst_tac [("n1","n"), ("m1","x")]
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(TFin_linear_lemma1 RS disjE) 1 THEN REPEAT (assume_tac 1));
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by (blast_tac thissubset_cs 1);
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by (rtac (Abrial_axiom1 RS subset_trans RS disjI1) 1);
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by (assume_tac 1);
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qed "TFin_linear_lemma2";
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(*a more convenient form for Lemma 2*)
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Goal "[| n<=m; m: TFin S; n: TFin S |] ==> n=m | succ S n<=m";
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by (rtac (TFin_linear_lemma2 RS bspec RS mp) 1);
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by (REPEAT (assume_tac 1));
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qed "TFin_subsetD";
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(*Consequences from section 3.3 -- Property 3.2, the ordering is total*)
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Goal "[| m: TFin S; n: TFin S|] ==> n<=m | m<=n";
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by (rtac (TFin_linear_lemma2 RSN (3,TFin_linear_lemma1) RS disjE) 1);
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by (REPEAT (assume_tac 1) THEN etac disjI2 1);
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by (blast_tac (thissubset_cs addIs [Abrial_axiom1 RS subset_trans]) 1);
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qed "TFin_subset_linear";
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(*Lemma 3 of section 3.3*)
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Goal "[| n: TFin S; m: TFin S; m = succ S m |] ==> n<=m";
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by (etac TFin_induct 1);
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by (dtac TFin_subsetD 1);
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by (REPEAT (assume_tac 1));
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by (fast_tac (claset() addEs [ssubst]) 1);
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by (blast_tac (thissubset_cs) 1);
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qed "eq_succ_upper";
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(*Property 3.3 of section 3.3*)
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Goal "m: TFin S ==> (m = succ S m) = (m = Union(TFin S))";
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by (rtac iffI 1);
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by (rtac (Union_upper RS equalityI) 1);
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by (rtac (eq_succ_upper RS Union_least) 2);
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by (REPEAT (assume_tac 1));
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by (etac ssubst 1);
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by (rtac (Abrial_axiom1 RS equalityI) 1);
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by (blast_tac (thissubset_cs addIs [TFin_UnionI, TFin_succI]) 1);
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qed "equal_succ_Union";
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(*-------------------------------------------------------------------------
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Section 4. Hausdorff's Theorem: every set contains a maximal chain
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NB: We assume the partial ordering is <=, the subset relation!
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-------------------------------------------------------------------------*)
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Goalw [chain_def] "({} :: 'a set set) : chain S";
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by (Auto_tac);
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qed "empty_set_mem_chain";
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Goalw [super_def] "super S c <= chain S";
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by (Fast_tac 1);
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qed "super_subset_chain";
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Goalw [maxchain_def] "maxchain S <= chain S";
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by (Fast_tac 1);
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qed "maxchain_subset_chain";
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Goalw [succ_def] "c ~: chain S ==> succ S c = c";
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by (fast_tac (claset() addSIs [if_P]) 1);
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qed "succI1";
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Goalw [succ_def] "c: maxchain S ==> succ S c = c";
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by (fast_tac (claset() addSIs [if_P]) 1);
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qed "succI2";
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Goalw [succ_def] "c: chain S - maxchain S ==> \
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\ succ S c = (@c'. c': super S c)";
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by (fast_tac (claset() addSIs [if_not_P]) 1);
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qed "succI3";
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Goal "c: chain S - maxchain S ==> ? d. d: super S c";
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by (rewrite_goals_tac [super_def,maxchain_def]);
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by (Auto_tac);
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qed "mem_super_Ex";
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Goal "c: chain S - maxchain S ==> \
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\ (@c'. c': super S c): super S c";
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by (etac (mem_super_Ex RS exE) 1);
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by (rtac someI2 1);
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by (Auto_tac);
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qed "select_super";
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Goal "c: chain S - maxchain S ==> \
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\ (@c'. c': super S c) ~= c";
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by (rtac notI 1);
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by (dtac select_super 1);
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by (asm_full_simp_tac (simpset() addsimps [super_def,psubset_def]) 1);
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qed "select_not_equals";
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Goal "c: chain S - maxchain S ==> \
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\ succ S c ~= c";
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by (ftac succI3 1);
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by (Asm_simp_tac 1);
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by (rtac select_not_equals 1);
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by (assume_tac 1);
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qed "succ_not_equals";
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Goal "c: TFin S ==> (c :: 'a set set): chain S";
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by (etac TFin_induct 1);
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by (asm_simp_tac (simpset() addsimps [succ_def,
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select_super RS (super_subset_chain RS subsetD)]
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addsplits [split_if]) 1);
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by (rewtac chain_def);
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by (rtac CollectI 1);
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by Safe_tac;
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by (dtac bspec 1 THEN assume_tac 1);
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by (res_inst_tac [("m1","Xa"), ("n1","X")] (TFin_subset_linear RS disjE) 2);
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by (ALLGOALS(Blast_tac));
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qed "TFin_chain_lemm4";
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Goal "EX c. (c :: 'a set set): maxchain S";
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by (res_inst_tac [("x", "Union(TFin S)")] exI 1);
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by (rtac classical 1);
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by (subgoal_tac "succ S (Union(TFin S)) = Union(TFin S)" 1);
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by (resolve_tac [equal_succ_Union RS iffD2 RS sym] 2);
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by (resolve_tac [subset_refl RS TFin_UnionI] 2);
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by (rtac refl 2);
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by (cut_facts_tac [subset_refl RS TFin_UnionI RS TFin_chain_lemm4] 1);
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by (dtac (DiffI RS succ_not_equals) 1);
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by (ALLGOALS(Blast_tac));
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qed "Hausdorff";
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(*---------------------------------------------------------------
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Section 5. Zorn's Lemma: if all chains have upper bounds
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there is a maximal element
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----------------------------------------------------------------*)
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Goalw [chain_def]
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"[| c: chain S; z: S; \
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\ ALL x:c. x<=(z:: 'a set) |] ==> {z} Un c : chain S";
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by (Blast_tac 1);
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qed "chain_extend";
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Goalw [chain_def] "[| c: chain S; x: c |] ==> x <= Union(c)";
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by (Auto_tac);
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qed "chain_Union_upper";
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Goalw [chain_def] "c: chain S ==> ! x: c. x <= Union(c)";
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by (Auto_tac);
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qed "chain_ball_Union_upper";
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Goal "[| c: maxchain S; u: S; Union(c) <= u |] ==> Union(c) = u";
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by (rtac ccontr 1);
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by (asm_full_simp_tac (simpset() addsimps [maxchain_def]) 1);
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by (etac conjE 1);
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by (subgoal_tac "({u} Un c): super S c" 1);
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by (Asm_full_simp_tac 1);
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by (rewrite_tac [super_def,psubset_def]);
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by (blast_tac (claset() addIs [chain_extend] addDs [chain_Union_upper]) 1);
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qed "maxchain_Zorn";
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Goal "ALL c: chain S. Union(c): S ==> \
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\ EX y: S. ALL z: S. y <= z --> y = z";
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by (cut_facts_tac [Hausdorff,maxchain_subset_chain] 1);
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by (etac exE 1);
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by (dtac subsetD 1 THEN assume_tac 1);
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by (dtac bspec 1 THEN assume_tac 1);
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by (res_inst_tac [("x","Union(c)")] bexI 1);
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by (rtac ballI 1 THEN rtac impI 1);
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by (blast_tac (claset() addSDs [maxchain_Zorn]) 1);
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by (assume_tac 1);
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qed "Zorn_Lemma";
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(*-------------------------------------------------------------
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Alternative version of Zorn's Lemma
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--------------------------------------------------------------*)
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Goal "ALL (c:: 'a set set): chain S. EX y : S. ALL x : c. x <= y ==> \
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\ EX y : S. ALL x : S. (y :: 'a set) <= x --> y = x";
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by (cut_facts_tac [Hausdorff,maxchain_subset_chain] 1);
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by (EVERY1[etac exE, dtac subsetD, assume_tac]);
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by (EVERY1[dtac bspec, assume_tac, etac bexE]);
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by (res_inst_tac [("x","y")] bexI 1);
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by (assume_tac 2);
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by (EVERY1[rtac ballI, rtac impI, rtac ccontr]);
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by (forw_inst_tac [("z","x")] chain_extend 1);
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by (assume_tac 1 THEN Blast_tac 1);
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by (rewrite_tac [maxchain_def,super_def,psubset_def]);
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by (blast_tac (claset() addSEs [equalityCE]) 1);
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qed "Zorn_Lemma2";
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(** misc. lemmas **)
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Goalw [chain_def] "[| c : chain S; x: c; y: c |] ==> x <= y | y <= x";
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by (Blast_tac 1);
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qed "chainD";
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Goalw [chain_def] "!!(c :: 'a set set). c: chain S ==> c <= S";
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by (Blast_tac 1);
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qed "chainD2";
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(* proved elsewhere? *)
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Goal "x : Union(c) ==> EX m:c. x:m";
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by (Blast_tac 1);
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qed "mem_UnionD";
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