--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Real/Hyperreal/Zorn.ML Fri Nov 27 11:24:27 1998 +0100
@@ -0,0 +1,293 @@
+(* Title : Zorn.ML
+ Author : Jacques D. Fleuriot
+ Copyright : 1998 University of Cambridge
+ Description : Zorn's Lemma -- adapted proofs from lcp's ZF/Zorn.ML
+*)
+
+open Zorn;
+
+(*---------------------------------------------------------------
+ Section 1. Mathematical Preamble
+ ---------------------------------------------------------------*)
+
+Goal "(ALL x:C. x<=A | B<=x) ==> Union(C)<=A | B<=Union(C)";
+by (Blast_tac 1);
+qed "Union_lemma0";
+
+(*-- similar to subset_cs in ZF/subset.thy --*)
+val thissubset_SIs =
+ [subset_refl,Union_least, UN_least, Un_least,
+ Inter_greatest, Int_greatest,
+ Un_upper1, Un_upper2, Int_lower1, Int_lower2];
+
+
+(*A claset for subset reasoning*)
+val thissubset_cs = claset()
+ delrules [subsetI, subsetCE]
+ addSIs thissubset_SIs
+ addIs [Union_upper, Inter_lower];
+
+(* increasingD2 of ZF/Zorn.ML *)
+Goalw [succ_def] "x <= succ S x";
+by (rtac (expand_if RS iffD2) 1);
+by (auto_tac (claset(),simpset() addsimps [super_def,
+ maxchain_def,psubset_def]));
+by (rtac swap 1 THEN assume_tac 1);
+by (rtac selectI2 1);
+by (ALLGOALS(Blast_tac));
+qed "Abrial_axiom1";
+
+val [TFin_succI, Pow_TFin_UnionI] = TFin.intrs;
+val TFin_UnionI = PowI RS Pow_TFin_UnionI;
+
+val major::prems = Goal
+ "[| n : TFin S; \
+\ !!x. [| x: TFin S; P(x) |] ==> P(succ S x); \
+\ !!Y. [| Y <= TFin S; Ball Y P |] ==> P(Union Y) |] \
+\ ==> P(n)";
+by (rtac (major RS TFin.induct) 1);
+by (ALLGOALS (fast_tac (claset() addIs prems)));
+qed "TFin_induct";
+
+(*Perform induction on n, then prove the major premise using prems. *)
+fun TFin_ind_tac a prems i =
+ EVERY [res_inst_tac [("n",a)] TFin_induct i,
+ rename_last_tac a ["1"] (i+1),
+ rename_last_tac a ["2"] (i+2),
+ ares_tac prems i];
+
+Goal "x <= y ==> x <= succ S y";
+by (etac (Abrial_axiom1 RSN (2,subset_trans)) 1);
+qed "succ_trans";
+
+(*Lemma 1 of section 3.1*)
+Goal "[| n: TFin S; m: TFin S; \
+\ ALL x: TFin S. x <= m --> x = m | succ S x <= m \
+\ |] ==> n <= m | succ S m <= n";
+by (etac TFin_induct 1);
+by (etac Union_lemma0 2); (*or just Blast_tac*)
+by (blast_tac (thissubset_cs addIs [succ_trans]) 1);
+qed "TFin_linear_lemma1";
+
+(* Lemma 2 of section 3.2 *)
+Goal "m: TFin S ==> ALL n: TFin S. n<=m --> n=m | succ S n<=m";
+by (etac TFin_induct 1);
+by (rtac (impI RS ballI) 1);
+(*case split using TFin_linear_lemma1*)
+by (res_inst_tac [("n1","n"), ("m1","x")]
+ (TFin_linear_lemma1 RS disjE) 1 THEN REPEAT (assume_tac 1));
+by (dres_inst_tac [("x","n")] bspec 1 THEN assume_tac 1);
+by (blast_tac (thissubset_cs addIs [succ_trans]) 1);
+by (REPEAT (ares_tac [disjI1,equalityI] 1));
+(*second induction step*)
+by (rtac (impI RS ballI) 1);
+by (rtac (Union_lemma0 RS disjE) 1);
+by (rtac disjI2 3);
+by (REPEAT (ares_tac [disjI1,equalityI] 2));
+by (rtac ballI 1);
+by (ball_tac 1);
+by (set_mp_tac 1);
+by (res_inst_tac [("n1","n"), ("m1","x")]
+ (TFin_linear_lemma1 RS disjE) 1 THEN REPEAT (assume_tac 1));
+by (blast_tac thissubset_cs 1);
+by (rtac (Abrial_axiom1 RS subset_trans RS disjI1) 1);
+by (assume_tac 1);
+qed "TFin_linear_lemma2";
+
+(*a more convenient form for Lemma 2*)
+Goal "[| n<=m; m: TFin S; n: TFin S |] ==> n=m | succ S n<=m";
+by (rtac (TFin_linear_lemma2 RS bspec RS mp) 1);
+by (REPEAT (assume_tac 1));
+qed "TFin_subsetD";
+
+(*Consequences from section 3.3 -- Property 3.2, the ordering is total*)
+Goal "[| m: TFin S; n: TFin S|] ==> n<=m | m<=n";
+by (rtac (TFin_linear_lemma2 RSN (3,TFin_linear_lemma1) RS disjE) 1);
+by (REPEAT (assume_tac 1) THEN etac disjI2 1);
+by (blast_tac (thissubset_cs addIs [Abrial_axiom1 RS subset_trans]) 1);
+qed "TFin_subset_linear";
+
+(*Lemma 3 of section 3.3*)
+Goal "[| n: TFin S; m: TFin S; m = succ S m |] ==> n<=m";
+by (etac TFin_induct 1);
+by (dtac TFin_subsetD 1);
+by (REPEAT (assume_tac 1));
+by (fast_tac (claset() addEs [ssubst]) 1);
+by (blast_tac (thissubset_cs) 1);
+qed "eq_succ_upper";
+
+(*Property 3.3 of section 3.3*)
+Goal "m: TFin S ==> (m = succ S m) = (m = Union(TFin S))";
+by (rtac iffI 1);
+by (rtac (Union_upper RS equalityI) 1);
+by (rtac (eq_succ_upper RS Union_least) 2);
+by (REPEAT (assume_tac 1));
+by (etac ssubst 1);
+by (rtac (Abrial_axiom1 RS equalityI) 1);
+by (blast_tac (thissubset_cs addIs [TFin_UnionI, TFin_succI]) 1);
+qed "equal_succ_Union";
+
+(*-------------------------------------------------------------------------
+ Section 4. Hausdorff's Theorem: every set contains a maximal chain
+ NB: We assume the partial ordering is <=, the subset relation!
+ -------------------------------------------------------------------------*)
+
+Goalw [chain_def] "({} :: 'a set set) : chain S";
+by (Auto_tac);
+qed "empty_set_mem_chain";
+
+Goalw [super_def] "super S c <= chain S";
+by (Fast_tac 1);
+qed "super_subset_chain";
+
+Goalw [maxchain_def] "maxchain S <= chain S";
+by (Fast_tac 1);
+qed "maxchain_subset_chain";
+
+Goalw [succ_def] "c ~: chain S ==> succ S c = c";
+by (fast_tac (claset() addSIs [if_P]) 1);
+qed "succI1";
+
+Goalw [succ_def] "c: maxchain S ==> succ S c = c";
+by (fast_tac (claset() addSIs [if_P]) 1);
+qed "succI2";
+
+Goalw [succ_def] "c: chain S - maxchain S ==> \
+\ succ S c = (@c'. c': super S c)";
+by (fast_tac (claset() addSIs [if_not_P]) 1);
+qed "succI3";
+
+Goal "c: chain S - maxchain S ==> ? d. d: super S c";
+by (rewrite_goals_tac [super_def,maxchain_def]);
+by (Auto_tac);
+qed "mem_super_Ex";
+
+Goal "c: chain S - maxchain S ==> \
+\ (@c'. c': super S c): super S c";
+by (etac (mem_super_Ex RS exE) 1);
+by (rtac selectI2 1);
+by (Auto_tac);
+qed "select_super";
+
+Goal "c: chain S - maxchain S ==> \
+\ (@c'. c': super S c) ~= c";
+by (rtac notI 1);
+by (dtac select_super 1);
+by (asm_full_simp_tac (simpset() addsimps [super_def,psubset_def]) 1);
+qed "select_not_equals";
+
+Goal "c: chain S - maxchain S ==> \
+\ succ S c ~= c";
+by (forward_tac [succI3] 1);
+by (Asm_simp_tac 1);
+by (rtac select_not_equals 1);
+by (assume_tac 1);
+qed "succ_not_equals";
+
+Goal "c: TFin S ==> (c :: 'a set set): chain S";
+by (etac TFin_induct 1);
+by (asm_simp_tac (simpset() addsimps [succ_def,
+ select_super RS (super_subset_chain RS subsetD)]
+ setloop split_tac [expand_if]) 1);
+by (rewtac chain_def);
+by (rtac CollectI 1);
+by (safe_tac(claset()));
+by (dtac bspec 1 THEN assume_tac 1);
+by (res_inst_tac [("m1","Xa"), ("n1","X")] (TFin_subset_linear RS disjE) 2);
+by (ALLGOALS(Blast_tac));
+qed "TFin_chain_lemm4";
+
+Goal "EX c. (c :: 'a set set): maxchain S";
+by (res_inst_tac [("x", "Union(TFin S)")] exI 1);
+by (rtac classical 1);
+by (subgoal_tac "succ S (Union(TFin S)) = Union(TFin S)" 1);
+by (resolve_tac [equal_succ_Union RS iffD2 RS sym] 2);
+by (resolve_tac [subset_refl RS TFin_UnionI] 2);
+by (rtac refl 2);
+by (cut_facts_tac [subset_refl RS TFin_UnionI RS TFin_chain_lemm4] 1);
+by (dtac (DiffI RS succ_not_equals) 1);
+by (ALLGOALS(Blast_tac));
+qed "Hausdorff";
+
+
+(*---------------------------------------------------------------
+ Section 5. Zorn's Lemma: if all chains have upper bounds
+ there is a maximal element
+ ----------------------------------------------------------------*)
+Goalw [chain_def]
+ "[| c: chain S; z: S; \
+\ ALL x:c. x<=(z:: 'a set) |] ==> {z} Un c : chain S";
+by (Blast_tac 1);
+qed "chain_extend";
+
+Goalw [chain_def] "[| c: chain S; x: c |] ==> x <= Union(c)";
+by (Auto_tac);
+qed "chain_Union_upper";
+
+Goalw [chain_def] "c: chain S ==> ! x: c. x <= Union(c)";
+by (Auto_tac);
+qed "chain_ball_Union_upper";
+
+Goal "[| c: maxchain S; u: S; Union(c) <= u |] ==> Union(c) = u";
+by (rtac ccontr 1);
+by (asm_full_simp_tac (simpset() addsimps [maxchain_def]) 1);
+by (etac conjE 1);
+by (subgoal_tac "({u} Un c): super S c" 1);
+by (Asm_full_simp_tac 1);
+by (rewrite_tac [super_def,psubset_def]);
+by (safe_tac (claset()));
+by (fast_tac (claset() addEs [chain_extend]) 1);
+by (subgoal_tac "u ~: c" 1);
+by (blast_tac (claset() addEs [equalityE]) 1);
+by (blast_tac (claset() addDs [chain_Union_upper]) 1);
+qed "maxchain_Zorn";
+
+Goal "ALL c: chain S. Union(c): S ==> \
+\ EX y: S. ALL z: S. y <= z --> y = z";
+by (cut_facts_tac [Hausdorff,maxchain_subset_chain] 1);
+by (etac exE 1);
+by (dtac subsetD 1 THEN assume_tac 1);
+by (dtac bspec 1 THEN assume_tac 1);
+by (res_inst_tac [("x","Union(c)")] bexI 1);
+by (rtac ballI 1 THEN rtac impI 1);
+by (blast_tac (claset() addSDs [maxchain_Zorn]) 1);
+by (assume_tac 1);
+qed "Zorn_Lemma";
+
+(*-------------------------------------------------------------
+ Alternative version of Zorn's Lemma
+ --------------------------------------------------------------*)
+Goal "ALL (c:: 'a set set): chain S. EX y : S. ALL x : c. x <= y ==> \
+\ EX y : S. ALL x : S. (y :: 'a set) <= x --> y = x";
+by (cut_facts_tac [Hausdorff,maxchain_subset_chain] 1);
+by (EVERY1[etac exE, dtac subsetD, assume_tac]);
+by (EVERY1[dtac bspec, assume_tac, etac bexE]);
+by (res_inst_tac [("x","y")] bexI 1);
+by (assume_tac 2);
+by (EVERY1[rtac ballI, rtac impI, rtac ccontr]);
+by (forw_inst_tac [("z","x")] chain_extend 1);
+by (assume_tac 1 THEN Blast_tac 1);
+by (rewrite_tac [maxchain_def,super_def,psubset_def]);
+by (Step_tac 1);
+by (eres_inst_tac [("c","{x} Un c")] equalityCE 1);
+by (Step_tac 1);
+by (subgoal_tac "x ~: c" 1);
+by (blast_tac (claset() addEs [equalityE]) 1);
+by (Blast_tac 1);
+qed "Zorn_Lemma2";
+
+(** misc. lemmas **)
+
+Goalw [chain_def] "[| c : chain S; x: c; y: c |] ==> x <= y | y <= x";
+by (Blast_tac 1);
+qed "chainD";
+
+Goalw [chain_def] "!!(c :: 'a set set). c: chain S ==> c <= S";
+by (Blast_tac 1);
+qed "chainD2";
+
+(* proved elsewhere? *)
+Goal "x : Union(c) ==> EX m:c. x:m";
+by (Blast_tac 1);
+qed "mem_UnionD";
+