src/ZF/Fixedpt.ML

     "[| h(D)<=D;  \
 \       !!W X. [| W<=D;  X<=D;  W<=X |] ==> h(W) <= h(X)  \
 \    |] ==> bnd_mono(D,h)";  
 by (REPEAT (ares_tac (prems@[conjI,allI,impI]) 1
      ORELSE etac subset_trans 1));
-val bnd_monoI = result();
+qed "bnd_monoI";
 
 val [major] = goalw Fixedpt.thy [bnd_mono_def] "bnd_mono(D,h) ==> h(D) <= D";
 by (rtac (major RS conjunct1) 1);
-val bnd_monoD1 = result();
+qed "bnd_monoD1";
 
 val major::prems = goalw Fixedpt.thy [bnd_mono_def]
     "[| bnd_mono(D,h);  W<=X;  X<=D |] ==> h(W) <= h(X)";
 by (rtac (major RS conjunct2 RS spec RS spec RS mp RS mp) 1);
 by (REPEAT (resolve_tac prems 1));
-val bnd_monoD2 = result();
+qed "bnd_monoD2";
 
 val [major,minor] = goal Fixedpt.thy
     "[| bnd_mono(D,h);  X<=D |] ==> h(X) <= D";
 by (rtac (major RS bnd_monoD2 RS subset_trans) 1);
 by (rtac (major RS bnd_monoD1) 3);
 by (rtac minor 1);
 by (rtac subset_refl 1);
-val bnd_mono_subset = result();
+qed "bnd_mono_subset";
 
 goal Fixedpt.thy "!!A B. [| bnd_mono(D,h);  A <= D;  B <= D |] ==> \
 \                         h(A) Un h(B) <= h(A Un B)";
 by (REPEAT (ares_tac [Un_upper1, Un_upper2, Un_least] 1
      ORELSE etac bnd_monoD2 1));
-val bnd_mono_Un = result();
+qed "bnd_mono_Un";
 
 (*Useful??*)
 goal Fixedpt.thy "!!A B. [| bnd_mono(D,h);  A <= D;  B <= D |] ==> \
 \                        h(A Int B) <= h(A) Int h(B)";
 by (REPEAT (ares_tac [Int_lower1, Int_lower2, Int_greatest] 1
      ORELSE etac bnd_monoD2 1));
-val bnd_mono_Int = result();
+qed "bnd_mono_Int";
 
 (**** Proof of Knaster-Tarski Theorem for the lfp ****)
 
 (*lfp is contained in each pre-fixedpoint*)
 goalw Fixedpt.thy [lfp_def]
     "!!A. [| h(A) <= A;  A<=D |] ==> lfp(D,h) <= A";
 by (fast_tac ZF_cs 1);
 (*or  by (rtac (PowI RS CollectI RS Inter_lower) 1); *)
-val lfp_lowerbound = result();
+qed "lfp_lowerbound";
 
 (*Unfolding the defn of Inter dispenses with the premise bnd_mono(D,h)!*)
 goalw Fixedpt.thy [lfp_def,Inter_def] "lfp(D,h) <= D";
 by (fast_tac ZF_cs 1);
-val lfp_subset = result();
+qed "lfp_subset";
 
 (*Used in datatype package*)
 val [rew] = goal Fixedpt.thy "A==lfp(D,h) ==> A <= D";
 by (rewtac rew);
 by (rtac lfp_subset 1);
-val def_lfp_subset = result();
+qed "def_lfp_subset";
 
 val prems = goalw Fixedpt.thy [lfp_def]
     "[| h(D) <= D;  !!X. [| h(X) <= X;  X<=D |] ==> A<=X |] ==> \
 \    A <= lfp(D,h)";
 by (rtac (Pow_top RS CollectI RS Inter_greatest) 1);
 by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [CollectE,PowD] 1));
-val lfp_greatest = result();
+qed "lfp_greatest";
 
 val hmono::prems = goal Fixedpt.thy
     "[| bnd_mono(D,h);  h(A)<=A;  A<=D |] ==> h(lfp(D,h)) <= A";
 by (rtac (hmono RS bnd_monoD2 RS subset_trans) 1);
 by (rtac lfp_lowerbound 1);
 by (REPEAT (resolve_tac prems 1));
-val lfp_lemma1 = result();
+qed "lfp_lemma1";
 
 val [hmono] = goal Fixedpt.thy
     "bnd_mono(D,h) ==> h(lfp(D,h)) <= lfp(D,h)";
 by (rtac (bnd_monoD1 RS lfp_greatest) 1);
 by (rtac lfp_lemma1 2);
 by (REPEAT (ares_tac [hmono] 1));
-val lfp_lemma2 = result();
+qed "lfp_lemma2";
 
 val [hmono] = goal Fixedpt.thy
     "bnd_mono(D,h) ==> lfp(D,h) <= h(lfp(D,h))";
 by (rtac lfp_lowerbound 1);
 by (rtac (hmono RS bnd_monoD2) 1);
 by (rtac (hmono RS lfp_lemma2) 1);
 by (rtac (hmono RS bnd_mono_subset) 2);
 by (REPEAT (rtac lfp_subset 1));
-val lfp_lemma3 = result();
+qed "lfp_lemma3";
 
 val prems = goal Fixedpt.thy
     "bnd_mono(D,h) ==> lfp(D,h) = h(lfp(D,h))";
 by (REPEAT (resolve_tac (prems@[equalityI,lfp_lemma2,lfp_lemma3]) 1));
-val lfp_Tarski = result();
+qed "lfp_Tarski";
 
 (*Definition form, to control unfolding*)
 val [rew,mono] = goal Fixedpt.thy
     "[| A==lfp(D,h);  bnd_mono(D,h) |] ==> A = h(A)";
 by (rewtac rew);
 by (rtac (mono RS lfp_Tarski) 1);
-val def_lfp_Tarski = result();
+qed "def_lfp_Tarski";
 
 (*** General induction rule for least fixedpoints ***)
 
 val [hmono,indstep] = goal Fixedpt.thy
     "[| bnd_mono(D,h);  !!x. x : h(Collect(lfp(D,h),P)) ==> P(x) \

 by (rtac CollectI 1);
 by (etac indstep 2);
 by (rtac (hmono RS lfp_lemma2 RS subsetD) 1);
 by (rtac (hmono RS bnd_monoD2 RS subsetD) 1);
 by (REPEAT (ares_tac [Collect_subset, lfp_subset] 1));
-val Collect_is_pre_fixedpt = result();
+qed "Collect_is_pre_fixedpt";
 
 (*This rule yields an induction hypothesis in which the components of a
   data structure may be assumed to be elements of lfp(D,h)*)
 val prems = goal Fixedpt.thy
     "[| bnd_mono(D,h);  a : lfp(D,h);   		\
 \       !!x. x : h(Collect(lfp(D,h),P)) ==> P(x) 	\
 \    |] ==> P(a)";
 by (rtac (Collect_is_pre_fixedpt RS lfp_lowerbound RS subsetD RS CollectD2) 1);
 by (rtac (lfp_subset RS (Collect_subset RS subset_trans)) 3);
 by (REPEAT (ares_tac prems 1));
-val induct = result();
+qed "induct";
 
 (*Definition form, to control unfolding*)
 val rew::prems = goal Fixedpt.thy
     "[| A == lfp(D,h);  bnd_mono(D,h);  a:A;   \
 \       !!x. x : h(Collect(A,P)) ==> P(x) \
 \    |] ==> P(a)";
 by (rtac induct 1);
 by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems) 1));
-val def_induct = result();
+qed "def_induct";
 
 (*This version is useful when "A" is not a subset of D;
   second premise could simply be h(D Int A) <= D or !!X. X<=D ==> h(X)<=D *)
 val [hsub,hmono] = goal Fixedpt.thy
     "[| h(D Int A) <= A;  bnd_mono(D,h) |] ==> lfp(D,h) <= A";
 by (rtac (lfp_lowerbound RS subset_trans) 1);
 by (rtac (hmono RS bnd_mono_subset RS Int_greatest) 1);
 by (REPEAT (resolve_tac [hsub,Int_lower1,Int_lower2] 1));
-val lfp_Int_lowerbound = result();
+qed "lfp_Int_lowerbound";
 
 (*Monotonicity of lfp, where h precedes i under a domain-like partial order
   monotonicity of h is not strictly necessary; h must be bounded by D*)
 val [hmono,imono,subhi] = goal Fixedpt.thy
     "[| bnd_mono(D,h);  bnd_mono(E,i); 		\

 by (rtac imono 1);
 by (rtac (hmono RSN (2, lfp_Int_lowerbound)) 1);
 by (rtac (Int_lower1 RS subhi RS subset_trans) 1);
 by (rtac (imono RS bnd_monoD2 RS subset_trans) 1);
 by (REPEAT (ares_tac [Int_lower2] 1));
-val lfp_mono = result();
+qed "lfp_mono";
 
 (*This (unused) version illustrates that monotonicity is not really needed,
   but both lfp's must be over the SAME set D;  Inter is anti-monotonic!*)
 val [isubD,subhi] = goal Fixedpt.thy
     "[| i(D) <= D;  !!X. X<=D ==> h(X) <= i(X)  |] ==> lfp(D,h) <= lfp(D,i)";
 by (rtac lfp_greatest 1);
 by (rtac isubD 1);
 by (rtac lfp_lowerbound 1);
 by (etac (subhi RS subset_trans) 1);
 by (REPEAT (assume_tac 1));
-val lfp_mono2 = result();
+qed "lfp_mono2";
 
 
 (**** Proof of Knaster-Tarski Theorem for the gfp ****)
 
 (*gfp contains each post-fixedpoint that is contained in D*)
 val prems = goalw Fixedpt.thy [gfp_def]
     "[| A <= h(A);  A<=D |] ==> A <= gfp(D,h)";
 by (rtac (PowI RS CollectI RS Union_upper) 1);
 by (REPEAT (resolve_tac prems 1));
-val gfp_upperbound = result();
+qed "gfp_upperbound";
 
 goalw Fixedpt.thy [gfp_def] "gfp(D,h) <= D";
 by (fast_tac ZF_cs 1);
-val gfp_subset = result();
+qed "gfp_subset";
 
 (*Used in datatype package*)
 val [rew] = goal Fixedpt.thy "A==gfp(D,h) ==> A <= D";
 by (rewtac rew);
 by (rtac gfp_subset 1);
-val def_gfp_subset = result();
+qed "def_gfp_subset";
 
 val hmono::prems = goalw Fixedpt.thy [gfp_def]
     "[| bnd_mono(D,h);  !!X. [| X <= h(X);  X<=D |] ==> X<=A |] ==> \
 \    gfp(D,h) <= A";
 by (fast_tac (subset_cs addIs ((hmono RS bnd_monoD1)::prems)) 1);
-val gfp_least = result();
+qed "gfp_least";
 
 val hmono::prems = goal Fixedpt.thy
     "[| bnd_mono(D,h);  A<=h(A);  A<=D |] ==> A <= h(gfp(D,h))";
 by (rtac (hmono RS bnd_monoD2 RSN (2,subset_trans)) 1);
 by (rtac gfp_subset 3);
 by (rtac gfp_upperbound 2);
 by (REPEAT (resolve_tac prems 1));
-val gfp_lemma1 = result();
+qed "gfp_lemma1";
 
 val [hmono] = goal Fixedpt.thy
     "bnd_mono(D,h) ==> gfp(D,h) <= h(gfp(D,h))";
 by (rtac gfp_least 1);
 by (rtac gfp_lemma1 2);
 by (REPEAT (ares_tac [hmono] 1));
-val gfp_lemma2 = result();
+qed "gfp_lemma2";
 
 val [hmono] = goal Fixedpt.thy
     "bnd_mono(D,h) ==> h(gfp(D,h)) <= gfp(D,h)";
 by (rtac gfp_upperbound 1);
 by (rtac (hmono RS bnd_monoD2) 1);
 by (rtac (hmono RS gfp_lemma2) 1);
 by (REPEAT (rtac ([hmono, gfp_subset] MRS bnd_mono_subset) 1));
-val gfp_lemma3 = result();
+qed "gfp_lemma3";
 
 val prems = goal Fixedpt.thy
     "bnd_mono(D,h) ==> gfp(D,h) = h(gfp(D,h))";
 by (REPEAT (resolve_tac (prems@[equalityI,gfp_lemma2,gfp_lemma3]) 1));
-val gfp_Tarski = result();
+qed "gfp_Tarski";
 
 (*Definition form, to control unfolding*)
 val [rew,mono] = goal Fixedpt.thy
     "[| A==gfp(D,h);  bnd_mono(D,h) |] ==> A = h(A)";
 by (rewtac rew);
 by (rtac (mono RS gfp_Tarski) 1);
-val def_gfp_Tarski = result();
+qed "def_gfp_Tarski";
 
 
 (*** Coinduction rules for greatest fixed points ***)
 
 (*weak version*)
 goal Fixedpt.thy "!!X h. [| a: X;  X <= h(X);  X <= D |] ==> a : gfp(D,h)";
 by (REPEAT (ares_tac [gfp_upperbound RS subsetD] 1));
-val weak_coinduct = result();
+qed "weak_coinduct";
 
 val [subs_h,subs_D,mono] = goal Fixedpt.thy
     "[| X <= h(X Un gfp(D,h));  X <= D;  bnd_mono(D,h) |] ==>  \
 \    X Un gfp(D,h) <= h(X Un gfp(D,h))";
 by (rtac (subs_h RS Un_least) 1);
 by (rtac (mono RS gfp_lemma2 RS subset_trans) 1);
 by (rtac (Un_upper2 RS subset_trans) 1);
 by (rtac ([mono, subs_D, gfp_subset] MRS bnd_mono_Un) 1);
-val coinduct_lemma = result();
+qed "coinduct_lemma";
 
 (*strong version*)
 goal Fixedpt.thy
     "!!X D. [| bnd_mono(D,h);  a: X;  X <= h(X Un gfp(D,h));  X <= D |] ==> \
 \           a : gfp(D,h)";
 by (rtac weak_coinduct 1);
 by (etac coinduct_lemma 2);
 by (REPEAT (ares_tac [gfp_subset, UnI1, Un_least] 1));
-val coinduct = result();
+qed "coinduct";
 
 (*Definition form, to control unfolding*)
 val rew::prems = goal Fixedpt.thy
     "[| A == gfp(D,h);  bnd_mono(D,h);  a: X;  X <= h(X Un A);  X <= D |] ==> \
 \    a : A";
 by (rewtac rew);
 by (rtac coinduct 1);
 by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems) 1));
-val def_coinduct = result();
+qed "def_coinduct";
 
 (*Lemma used immediately below!*)
 val [subsA,XimpP] = goal ZF.thy
     "[| X <= A;  !!z. z:X ==> P(z) |] ==> X <= Collect(A,P)";
 by (rtac (subsA RS subsetD RS CollectI RS subsetI) 1);
 by (assume_tac 1);
 by (etac XimpP 1);
-val subset_Collect = result();
+qed "subset_Collect";
 
 (*The version used in the induction/coinduction package*)
 val prems = goal Fixedpt.thy
     "[| A == gfp(D, %w. Collect(D,P(w)));  bnd_mono(D, %w. Collect(D,P(w)));  \
 \       a: X;  X <= D;  !!z. z: X ==> P(X Un A, z) |] ==> \
 \    a : A";
 by (rtac def_coinduct 1);
 by (REPEAT (ares_tac (subset_Collect::prems) 1));
-val def_Collect_coinduct = result();
+qed "def_Collect_coinduct";
 
 (*Monotonicity of gfp!*)
 val [hmono,subde,subhi] = goal Fixedpt.thy
     "[| bnd_mono(D,h);  D <= E; 		\
 \       !!X. X<=D ==> h(X) <= i(X)  |] ==> gfp(D,h) <= gfp(E,i)";
 by (rtac gfp_upperbound 1);
 by (rtac (hmono RS gfp_lemma2 RS subset_trans) 1);
 by (rtac (gfp_subset RS subhi) 1);
 by (rtac ([gfp_subset, subde] MRS subset_trans) 1);
-val gfp_mono = result();
-
+qed "gfp_mono";
+