src/CCL/Fix.ML

 open Fix;
 
 (*** Fixed Point Induction ***)
 
 val [base,step,incl] = goalw Fix.thy [INCL_def]
-    "[| P(bot);  !!x.P(x) ==> P(f(x));  INCL(P) |] ==> P(fix(f))";
+    "[| P(bot);  !!x. P(x) ==> P(f(x));  INCL(P) |] ==> P(fix(f))";
 by (rtac (incl RS spec RS mp) 1);
 by (rtac (Nat_ind RS ballI) 1 THEN atac 1);
 by (ALLGOALS (simp_tac term_ss));
 by (REPEAT (ares_tac [base,step] 1));
 qed "fix_ind";

      "INCL(P) <-> (ALL f. (ALL n:Nat. P(f ^ n ` bot)) --> P(fix(f)))";
 by (rtac iff_refl 1);
 qed "inclXH";
 
 val prems = goal Fix.thy
-     "[| !!f.ALL n:Nat.P(f^n`bot) ==> P(fix(f)) |] ==> INCL(%x.P(x))";
+     "[| !!f. ALL n:Nat. P(f^n`bot) ==> P(fix(f)) |] ==> INCL(%x. P(x))";
 by (fast_tac (term_cs addIs (prems @ [XH_to_I inclXH])) 1);
 qed "inclI";
 
 val incl::prems = goal Fix.thy
-     "[| INCL(P);  !!n.n:Nat ==> P(f^n`bot) |] ==> P(fix(f))";
+     "[| INCL(P);  !!n. n:Nat ==> P(f^n`bot) |] ==> P(fix(f))";
 by (fast_tac (term_cs addIs ([ballI RS (incl RS (XH_to_D inclXH) RS spec RS mp)] 
                        @ prems)) 1);
 qed "inclD";
 
 val incl::prems = goal Fix.thy
-     "[| INCL(P);  (ALL n:Nat.P(f^n`bot))-->P(fix(f)) ==> R |] ==> R";
+     "[| INCL(P);  (ALL n:Nat. P(f^n`bot))-->P(fix(f)) ==> R |] ==> R";
 by (fast_tac (term_cs addIs ([incl RS inclD] @ prems)) 1);
 qed "inclE";
 
 
 (*** Lemmas for Inclusive Predicates ***)

 by (assume_tac 2);
 by (stac napplyBzero 1);
 by (rtac po_cong 1 THEN rtac po_bot 1);
 qed "npo_INCL";
 
-val prems = goal Fix.thy "[| INCL(P);  INCL(Q) |] ==> INCL(%x.P(x) & Q(x))";
+val prems = goal Fix.thy "[| INCL(P);  INCL(Q) |] ==> INCL(%x. P(x) & Q(x))";
 by (fast_tac (set_cs addSIs ([inclI] @ (prems RL [inclD]))) 1);;
 qed "conj_INCL";
 
-val prems = goal Fix.thy "[| !!a.INCL(P(a)) |] ==> INCL(%x.ALL a.P(a,x))";
+val prems = goal Fix.thy "[| !!a. INCL(P(a)) |] ==> INCL(%x. ALL a. P(a,x))";
 by (fast_tac (set_cs addSIs ([inclI] @ (prems RL [inclD]))) 1);;
 qed "all_INCL";
 
-val prems = goal Fix.thy "[| !!a.a:A ==> INCL(P(a)) |] ==> INCL(%x.ALL a:A.P(a,x))";
+val prems = goal Fix.thy "[| !!a. a:A ==> INCL(P(a)) |] ==> INCL(%x. ALL a:A. P(a,x))";
 by (fast_tac (set_cs addSIs ([inclI] @ (prems RL [inclD]))) 1);;
 qed "ball_INCL";
 
-goal Fix.thy "INCL(%x.a(x) = (b(x)::'a::prog))";
+goal Fix.thy "INCL(%x. a(x) = (b(x)::'a::prog))";
 by (simp_tac (term_ss addsimps [eq_iff]) 1);
 by (REPEAT (resolve_tac [conj_INCL,po_INCL] 1));
 qed "eq_INCL";
 
 (*** Derivation of Reachability Condition ***)

 
 goal Fix.thy "idgen(fix(idgen)) = fix(idgen)";
 by (rtac (fixB RS sym) 1);
 qed "fix_idgenfp";
 
-goalw Fix.thy [idgen_def] "idgen(lam x.x) = lam x.x";
+goalw Fix.thy [idgen_def] "idgen(lam x. x) = lam x. x";
 by (simp_tac term_ss 1);
 by (rtac (term_case RS allI) 1);
 by (ALLGOALS (simp_tac term_ss));
 qed "id_idgenfp";
 

     in map mk_thm
           [    "idgen(d) = d ==> d ` bot = bot",
                "idgen(d) = d ==> d ` true = true",
                "idgen(d) = d ==> d ` false = false",
                "idgen(d) = d ==> d ` <a,b> = <d ` a,d ` b>",
-               "idgen(d) = d ==> d ` (lam x.f(x)) = lam x.d ` f(x)"]
+               "idgen(d) = d ==> d ` (lam x. f(x)) = lam x. d ` f(x)"]
     end;
 
 (* Proof of Reachability law - show that fix and lam x.x both give LEAST fixed points 
                                of idgen and hence are they same *)
 
 val [p1,p2,p3] = goal CCL.thy
-    "[| ALL x.t ` x [= u ` x;  EX f.t=lam x.f(x);  EX f.u=lam x.f(x) |] ==> t [= u";
+    "[| ALL x. t ` x [= u ` x;  EX f. t=lam x. f(x);  EX f. u=lam x. f(x) |] ==> t [= u";
 by (stac (p2 RS cond_eta) 1);
 by (stac (p3 RS cond_eta) 1);
 by (rtac (p1 RS (po_lam RS iffD2)) 1);
 qed "po_eta";
 

 by (rtac refl 1);
 qed "po_eta_lemma";
 
 val [prem] = goal Fix.thy
     "idgen(d) = d ==> \
-\      {p.EX a b.p=<a,b> & (EX t.a=fix(idgen) ` t & b = d ` t)} <=   \
-\      POgen({p.EX a b.p=<a,b> & (EX t.a=fix(idgen) ` t  & b = d ` t)})";
+\      {p. EX a b. p=<a,b> & (EX t. a=fix(idgen) ` t & b = d ` t)} <=   \
+\      POgen({p. EX a b. p=<a,b> & (EX t. a=fix(idgen) ` t  & b = d ` t)})";
 by (REPEAT (step_tac term_cs 1));
 by (term_case_tac "t" 1);
 by (ALLGOALS (simp_tac (term_ss addsimps (POgenXH::([prem,fix_idgenfp] RL idgen_lemmas)))));
 by (ALLGOALS (fast_tac set_cs));
 qed "lemma1";

 by (ALLGOALS (fast_tac (term_cs addIs [prem,po_eta_lemma,fix_idgenfp])));
 qed "fix_least_idgen";
 
 val [prem] = goal Fix.thy
     "idgen(d) = d ==> \
-\      {p.EX a b.p=<a,b> & b = d ` a} <= POgen({p.EX a b.p=<a,b> & b = d ` a})";
+\      {p. EX a b. p=<a,b> & b = d ` a} <= POgen({p. EX a b. p=<a,b> & b = d ` a})";
 by (REPEAT (step_tac term_cs 1));
 by (term_case_tac "a" 1);
 by (ALLGOALS (simp_tac (term_ss addsimps (POgenXH::([prem] RL idgen_lemmas)))));
 by (ALLGOALS (fast_tac set_cs));
 qed "lemma2";
 
 val [prem] = goal Fix.thy
-    "idgen(d) = d ==> lam x.x [= d";
+    "idgen(d) = d ==> lam x. x [= d";
 by (rtac (allI RS po_eta) 1);
 by (rtac (lemma2 RSN(2,po_coinduct)) 1);
 by (simp_tac term_ss 1);
 by (ALLGOALS (fast_tac (term_cs addIs [prem,po_eta_lemma,fix_idgenfp])));
 qed "id_least_idgen";
 
-goal Fix.thy  "fix(idgen) = lam x.x";
+goal Fix.thy  "fix(idgen) = lam x. x";
 by (fast_tac (term_cs addIs [eq_iff RS iffD2,
                              id_idgenfp RS fix_least_idgen,
                              fix_idgenfp RS id_least_idgen]) 1);
 qed "reachability";
 
 (********)
 
-val [prem] = goal Fix.thy "f = lam x.x ==> f`t = t";
+val [prem] = goal Fix.thy "f = lam x. x ==> f`t = t";
 by (rtac (prem RS sym RS subst) 1);
 by (rtac applyB 1);
 qed "id_apply";
 
 val prems = goal Fix.thy
      "[| P(bot);  P(true);  P(false);  \
 \        !!x y.[| P(x);  P(y) |] ==> P(<x,y>);  \
-\        !!u.(!!x.P(u(x))) ==> P(lam x.u(x));  INCL(P) |] ==> \
+\        !!u.(!!x. P(u(x))) ==> P(lam x. u(x));  INCL(P) |] ==> \
 \     P(t)";
 by (rtac (reachability RS id_apply RS subst) 1);
 by (res_inst_tac [("x","t")] spec 1);
 by (rtac fix_ind 1);
 by (rewtac idgen_def);