*** empty log message ***
authornipkow
Tue Jan 09 15:32:27 2001 +0100 (2001-01-09)
changeset 10834a7897aebbffc
parent 10833 c0844a30ea4e
child 10835 f4745d77e620
*** empty log message ***
src/HOL/Hyperreal/HRealAbs.ML
src/HOL/Hyperreal/HSeries.ML
src/HOL/Hyperreal/HSeries.thy
src/HOL/Hyperreal/HyperDef.ML
src/HOL/Hyperreal/HyperDef.thy
src/HOL/Hyperreal/HyperNat.ML
src/HOL/Hyperreal/HyperNat.thy
src/HOL/Hyperreal/HyperPow.ML
src/HOL/Hyperreal/HyperPow.thy
src/HOL/Hyperreal/Lim.ML
src/HOL/Hyperreal/NSA.ML
src/HOL/Hyperreal/NatStar.ML
src/HOL/Hyperreal/NatStar.thy
src/HOL/Hyperreal/SEQ.ML
src/HOL/Hyperreal/Star.ML
src/HOL/Hyperreal/Star.thy
src/HOL/IMPP/Hoare.ML
src/HOL/IMPP/Hoare.thy
src/HOL/Induct/LList.ML
src/HOL/Induct/LList.thy
src/HOL/Integ/Equiv.ML
src/HOL/Integ/Equiv.thy
src/HOL/Integ/IntDef.ML
src/HOL/Integ/IntDef.thy
src/HOL/Integ/int_arith1.ML
src/HOL/Lattice/Bounds.thy
src/HOL/Lattice/CompleteLattice.thy
src/HOL/Lattice/Orders.thy
src/HOL/Lex/Automata.ML
src/HOL/Lex/Automata.thy
src/HOL/Lex/NA.thy
src/HOL/Lex/NAe.ML
src/HOL/Lex/NAe.thy
src/HOL/Lex/RegExp2NA.thy
src/HOL/Lex/RegExp2NAe.thy
src/HOL/MicroJava/BV/JVM.thy
src/HOL/MicroJava/J/JBasis.ML
src/HOL/NumberTheory/BijectionRel.ML
src/HOL/NumberTheory/EulerFermat.ML
src/HOL/NumberTheory/EulerFermat.thy
src/HOL/NumberTheory/WilsonBij.ML
src/HOL/Real/PNat.ML
src/HOL/Real/PNat.thy
src/HOL/Real/PRat.ML
src/HOL/Real/PRat.thy
src/HOL/Real/PReal.ML
src/HOL/Real/RealDef.ML
src/HOL/Real/RealDef.thy
src/HOL/Real/RealInt.ML
src/HOL/Real/RealInt.thy
src/HOL/UNITY/Channel.ML
src/HOL/UNITY/Channel.thy
src/HOL/UNITY/ELT.ML
src/HOL/UNITY/ELT.thy
src/HOL/UNITY/Extend.ML
src/HOL/UNITY/Extend.thy
src/HOL/UNITY/FP.ML
src/HOL/UNITY/Lift_prog.ML
src/HOL/UNITY/Lift_prog.thy
src/HOL/UNITY/PPROD.ML
src/HOL/UNITY/PriorityAux.ML
src/HOL/UNITY/PriorityAux.thy
src/HOL/UNITY/Project.ML
src/HOL/UNITY/Reach.ML
src/HOL/UNITY/Rename.ML
src/HOL/UNITY/SubstAx.ML
src/HOL/UNITY/Token.ML
src/HOL/UNITY/UNITY.ML
src/HOL/UNITY/UNITY.thy
src/HOL/UNITY/WFair.ML
src/HOL/UNITY/WFair.thy
src/HOL/ex/Multiquote.thy
src/HOL/ex/PiSets.ML
src/HOL/ex/Tarski.ML
src/HOL/ex/Tarski.thy
src/HOL/ex/set.ML
src/HOLCF/Cfun1.ML
src/HOLCF/Cfun1.thy
src/HOLCF/Cfun2.ML
src/HOLCF/Cfun3.ML
src/HOLCF/Cfun3.thy
src/HOLCF/Cprod3.ML
src/HOLCF/Cprod3.thy
src/HOLCF/Fix.ML
src/HOLCF/Fix.thy
src/HOLCF/Lift.ML
src/HOLCF/Lift3.ML
src/HOLCF/Lift3.thy
src/HOLCF/Sprod3.ML
src/HOLCF/Sprod3.thy
src/HOLCF/Ssum0.ML
src/HOLCF/Ssum0.thy
src/HOLCF/Ssum3.ML
src/HOLCF/Ssum3.thy
src/HOLCF/Tr.ML
src/HOLCF/Tr.thy
src/HOLCF/Up1.ML
src/HOLCF/Up1.thy
src/HOLCF/Up3.ML
src/HOLCF/Up3.thy
     1.1 --- a/src/HOL/Hyperreal/HRealAbs.ML	Tue Jan 09 15:29:17 2001 +0100
     1.2 +++ b/src/HOL/Hyperreal/HRealAbs.ML	Tue Jan 09 15:32:27 2001 +0100
     1.3 @@ -19,8 +19,8 @@
     1.4  Addsimps [hrabs_number_of];
     1.5  
     1.6  Goalw [hrabs_def]
     1.7 -     "abs (Abs_hypreal (hyprel ``` {X})) = \
     1.8 -\     Abs_hypreal(hyprel ``` {%n. abs (X n)})";
     1.9 +     "abs (Abs_hypreal (hyprel `` {X})) = \
    1.10 +\     Abs_hypreal(hyprel `` {%n. abs (X n)})";
    1.11  by (auto_tac (claset(),
    1.12                simpset_of HyperDef.thy 
    1.13                    addsimps [hypreal_zero_def, hypreal_le,hypreal_minus]));
    1.14 @@ -232,7 +232,7 @@
    1.15  (*------------------------------------------------------------*)
    1.16  
    1.17  Goalw [hypreal_of_nat_def, hypreal_of_real_def, real_of_nat_def] 
    1.18 -     "hypreal_of_nat  m = Abs_hypreal(hyprel```{%n. real_of_nat m})";
    1.19 +     "hypreal_of_nat  m = Abs_hypreal(hyprel``{%n. real_of_nat m})";
    1.20  by Auto_tac;
    1.21  qed "hypreal_of_nat_iff";
    1.22  
     2.1 --- a/src/HOL/Hyperreal/HSeries.ML	Tue Jan 09 15:29:17 2001 +0100
     2.2 +++ b/src/HOL/Hyperreal/HSeries.ML	Tue Jan 09 15:32:27 2001 +0100
     2.3 @@ -8,14 +8,14 @@
     2.4  Goalw [sumhr_def]
     2.5       "sumhr(M,N,f) =  \
     2.6  \       Abs_hypreal(UN X:Rep_hypnat(M). UN Y: Rep_hypnat(N). \
     2.7 -\         hyprel ```{%n::nat. sumr (X n) (Y n) f})";
     2.8 +\         hyprel ``{%n::nat. sumr (X n) (Y n) f})";
     2.9  by (Auto_tac);
    2.10  qed "sumhr_iff";
    2.11  
    2.12  Goalw [sumhr_def]
    2.13 -     "sumhr(Abs_hypnat(hypnatrel```{%n. M n}), \
    2.14 -\           Abs_hypnat(hypnatrel```{%n. N n}), f) = \
    2.15 -\     Abs_hypreal(hyprel ``` {%n. sumr (M n) (N n) f})";
    2.16 +     "sumhr(Abs_hypnat(hypnatrel``{%n. M n}), \
    2.17 +\           Abs_hypnat(hypnatrel``{%n. N n}), f) = \
    2.18 +\     Abs_hypreal(hyprel `` {%n. sumr (M n) (N n) f})";
    2.19  by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);
    2.20  by (Auto_tac THEN Ultra_tac 1);
    2.21  qed "sumhr";
    2.22 @@ -27,7 +27,7 @@
    2.23  Goalw [sumhr_def]
    2.24        "sumhr p = (%(M,N,f). Abs_hypreal(UN X:Rep_hypnat(M). \
    2.25  \                     UN Y: Rep_hypnat(N). \
    2.26 -\           hyprel ```{%n::nat. sumr (X n) (Y n) f})) p";
    2.27 +\           hyprel ``{%n::nat. sumr (X n) (Y n) f})) p";
    2.28  by (res_inst_tac [("p","p")] PairE 1);
    2.29  by (res_inst_tac [("p","y")] PairE 1);
    2.30  by (Auto_tac);
     3.1 --- a/src/HOL/Hyperreal/HSeries.thy	Tue Jan 09 15:29:17 2001 +0100
     3.2 +++ b/src/HOL/Hyperreal/HSeries.thy	Tue Jan 09 15:32:27 2001 +0100
     3.3 @@ -15,7 +15,7 @@
     3.4     "sumhr p
     3.5         == Abs_hypreal(UN X:Rep_hypnat(fst p). 
     3.6                UN Y: Rep_hypnat(fst(snd p)).
     3.7 -              hyprel```{%n::nat. sumr (X n) (Y n) (snd(snd p))})"
     3.8 +              hyprel``{%n::nat. sumr (X n) (Y n) (snd(snd p))})"
     3.9  
    3.10  constdefs
    3.11     NSsums  :: [nat=>real,real] => bool     (infixr 80)
     4.1 --- a/src/HOL/Hyperreal/HyperDef.ML	Tue Jan 09 15:29:17 2001 +0100
     4.2 +++ b/src/HOL/Hyperreal/HyperDef.ML	Tue Jan 09 15:32:27 2001 +0100
     4.3 @@ -206,11 +206,11 @@
     4.4  	      simpset() addsimps [FreeUltrafilterNat_Nat_set]));
     4.5  qed "equiv_hyprel";
     4.6  
     4.7 -(* (hyprel ``` {x} = hyprel ``` {y}) = ((x,y) : hyprel) *)
     4.8 +(* (hyprel `` {x} = hyprel `` {y}) = ((x,y) : hyprel) *)
     4.9  bind_thm ("equiv_hyprel_iff",
    4.10      	  [equiv_hyprel, UNIV_I, UNIV_I] MRS eq_equiv_class_iff);
    4.11  
    4.12 -Goalw  [hypreal_def,hyprel_def,quotient_def] "hyprel```{x}:hypreal";
    4.13 +Goalw  [hypreal_def,hyprel_def,quotient_def] "hyprel``{x}:hypreal";
    4.14  by (Blast_tac 1);
    4.15  qed "hyprel_in_hypreal";
    4.16  
    4.17 @@ -230,7 +230,7 @@
    4.18  by (rtac Rep_hypreal_inverse 1);
    4.19  qed "inj_Rep_hypreal";
    4.20  
    4.21 -Goalw [hyprel_def] "x: hyprel ``` {x}";
    4.22 +Goalw [hyprel_def] "x: hyprel `` {x}";
    4.23  by (Step_tac 1);
    4.24  by (auto_tac (claset() addSIs [FreeUltrafilterNat_Nat_set], simpset()));
    4.25  qed "lemma_hyprel_refl";
    4.26 @@ -267,7 +267,7 @@
    4.27  qed "inj_hypreal_of_real";
    4.28  
    4.29  val [prem] = goal (the_context ())
    4.30 -    "(!!x y. z = Abs_hypreal(hyprel```{x}) ==> P) ==> P";
    4.31 +    "(!!x y. z = Abs_hypreal(hyprel``{x}) ==> P) ==> P";
    4.32  by (res_inst_tac [("x1","z")] 
    4.33      (rewrite_rule [hypreal_def] Rep_hypreal RS quotientE) 1);
    4.34  by (dres_inst_tac [("f","Abs_hypreal")] arg_cong 1);
    4.35 @@ -278,13 +278,13 @@
    4.36  (**** hypreal_minus: additive inverse on hypreal ****)
    4.37  
    4.38  Goalw [congruent_def]
    4.39 -  "congruent hyprel (%X. hyprel```{%n. - (X n)})";
    4.40 +  "congruent hyprel (%X. hyprel``{%n. - (X n)})";
    4.41  by Safe_tac;
    4.42  by (ALLGOALS Ultra_tac);
    4.43  qed "hypreal_minus_congruent";
    4.44  
    4.45  Goalw [hypreal_minus_def]
    4.46 -   "- (Abs_hypreal(hyprel```{%n. X n})) = Abs_hypreal(hyprel ``` {%n. -(X n)})";
    4.47 +   "- (Abs_hypreal(hyprel``{%n. X n})) = Abs_hypreal(hyprel `` {%n. -(X n)})";
    4.48  by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);
    4.49  by (simp_tac (simpset() addsimps 
    4.50        [hyprel_in_hypreal RS Abs_hypreal_inverse,
    4.51 @@ -322,20 +322,20 @@
    4.52  (**** hyperreal addition: hypreal_add  ****)
    4.53  
    4.54  Goalw [congruent2_def]
    4.55 -    "congruent2 hyprel (%X Y. hyprel```{%n. X n + Y n})";
    4.56 +    "congruent2 hyprel (%X Y. hyprel``{%n. X n + Y n})";
    4.57  by Safe_tac;
    4.58  by (ALLGOALS(Ultra_tac));
    4.59  qed "hypreal_add_congruent2";
    4.60  
    4.61  Goalw [hypreal_add_def]
    4.62 -  "Abs_hypreal(hyprel```{%n. X n}) + Abs_hypreal(hyprel```{%n. Y n}) = \
    4.63 -\  Abs_hypreal(hyprel```{%n. X n + Y n})";
    4.64 +  "Abs_hypreal(hyprel``{%n. X n}) + Abs_hypreal(hyprel``{%n. Y n}) = \
    4.65 +\  Abs_hypreal(hyprel``{%n. X n + Y n})";
    4.66  by (simp_tac (simpset() addsimps 
    4.67           [[equiv_hyprel, hypreal_add_congruent2] MRS UN_equiv_class2]) 1);
    4.68  qed "hypreal_add";
    4.69  
    4.70 -Goal "Abs_hypreal(hyprel```{%n. X n}) - Abs_hypreal(hyprel```{%n. Y n}) = \
    4.71 -\     Abs_hypreal(hyprel```{%n. X n - Y n})";
    4.72 +Goal "Abs_hypreal(hyprel``{%n. X n}) - Abs_hypreal(hyprel``{%n. Y n}) = \
    4.73 +\     Abs_hypreal(hyprel``{%n. X n - Y n})";
    4.74  by (simp_tac (simpset() addsimps 
    4.75           [hypreal_diff_def, hypreal_minus,hypreal_add]) 1);
    4.76  qed "hypreal_diff";
    4.77 @@ -496,14 +496,14 @@
    4.78  (**** hyperreal multiplication: hypreal_mult  ****)
    4.79  
    4.80  Goalw [congruent2_def]
    4.81 -    "congruent2 hyprel (%X Y. hyprel```{%n. X n * Y n})";
    4.82 +    "congruent2 hyprel (%X Y. hyprel``{%n. X n * Y n})";
    4.83  by Safe_tac;
    4.84  by (ALLGOALS(Ultra_tac));
    4.85  qed "hypreal_mult_congruent2";
    4.86  
    4.87  Goalw [hypreal_mult_def]
    4.88 -  "Abs_hypreal(hyprel```{%n. X n}) * Abs_hypreal(hyprel```{%n. Y n}) = \
    4.89 -\  Abs_hypreal(hyprel```{%n. X n * Y n})";
    4.90 +  "Abs_hypreal(hyprel``{%n. X n}) * Abs_hypreal(hyprel``{%n. Y n}) = \
    4.91 +\  Abs_hypreal(hyprel``{%n. X n * Y n})";
    4.92  by (simp_tac (simpset() addsimps 
    4.93        [[equiv_hyprel, hypreal_mult_congruent2] MRS UN_equiv_class2]) 1);
    4.94  qed "hypreal_mult";
    4.95 @@ -622,13 +622,13 @@
    4.96  (**** multiplicative inverse on hypreal ****)
    4.97  
    4.98  Goalw [congruent_def]
    4.99 -  "congruent hyprel (%X. hyprel```{%n. if X n = #0 then #0 else inverse(X n)})";
   4.100 +  "congruent hyprel (%X. hyprel``{%n. if X n = #0 then #0 else inverse(X n)})";
   4.101  by (Auto_tac THEN Ultra_tac 1);
   4.102  qed "hypreal_inverse_congruent";
   4.103  
   4.104  Goalw [hypreal_inverse_def]
   4.105 -      "inverse (Abs_hypreal(hyprel```{%n. X n})) = \
   4.106 -\      Abs_hypreal(hyprel ``` {%n. if X n = #0 then #0 else inverse(X n)})";
   4.107 +      "inverse (Abs_hypreal(hyprel``{%n. X n})) = \
   4.108 +\      Abs_hypreal(hyprel `` {%n. if X n = #0 then #0 else inverse(X n)})";
   4.109  by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);
   4.110  by (simp_tac (simpset() addsimps 
   4.111     [hyprel_in_hypreal RS Abs_hypreal_inverse,
   4.112 @@ -800,8 +800,8 @@
   4.113    makes many of them more straightforward. 
   4.114   -------------------------------------------------------*)
   4.115  Goalw [hypreal_less_def]
   4.116 -      "(Abs_hypreal(hyprel```{%n. X n}) < \
   4.117 -\           Abs_hypreal(hyprel```{%n. Y n})) = \
   4.118 +      "(Abs_hypreal(hyprel``{%n. X n}) < \
   4.119 +\           Abs_hypreal(hyprel``{%n. Y n})) = \
   4.120  \      ({n. X n < Y n} : FreeUltrafilterNat)";
   4.121  by (auto_tac (claset() addSIs [lemma_hyprel_refl], simpset()));
   4.122  by (Ultra_tac 1);
   4.123 @@ -840,7 +840,7 @@
   4.124                           Trichotomy of the hyperreals
   4.125    --------------------------------------------------------------------------------*)
   4.126  
   4.127 -Goalw [hyprel_def] "EX x. x: hyprel ``` {%n. #0}";
   4.128 +Goalw [hyprel_def] "EX x. x: hyprel `` {%n. #0}";
   4.129  by (res_inst_tac [("x","%n. #0")] exI 1);
   4.130  by (Step_tac 1);
   4.131  by (auto_tac (claset() addSIs [FreeUltrafilterNat_Nat_set], simpset()));
   4.132 @@ -958,8 +958,8 @@
   4.133  (*------ hypreal le iff reals le a.e ------*)
   4.134  
   4.135  Goalw [hypreal_le_def,real_le_def]
   4.136 -      "(Abs_hypreal(hyprel```{%n. X n}) <= \
   4.137 -\           Abs_hypreal(hyprel```{%n. Y n})) = \
   4.138 +      "(Abs_hypreal(hyprel``{%n. X n}) <= \
   4.139 +\           Abs_hypreal(hyprel``{%n. Y n})) = \
   4.140  \      ({n. X n <= Y n} : FreeUltrafilterNat)";
   4.141  by (auto_tac (claset(),simpset() addsimps [hypreal_less]));
   4.142  by (ALLGOALS(Ultra_tac));
     5.1 --- a/src/HOL/Hyperreal/HyperDef.thy	Tue Jan 09 15:29:17 2001 +0100
     5.2 +++ b/src/HOL/Hyperreal/HyperDef.thy	Tue Jan 09 15:32:27 2001 +0100
     5.3 @@ -37,28 +37,28 @@
     5.4  defs
     5.5  
     5.6    hypreal_zero_def
     5.7 -  "0 == Abs_hypreal(hyprel```{%n::nat. (#0::real)})"
     5.8 +  "0 == Abs_hypreal(hyprel``{%n::nat. (#0::real)})"
     5.9  
    5.10    hypreal_one_def
    5.11 -  "1hr == Abs_hypreal(hyprel```{%n::nat. (#1::real)})"
    5.12 +  "1hr == Abs_hypreal(hyprel``{%n::nat. (#1::real)})"
    5.13  
    5.14    (* an infinite number = [<1,2,3,...>] *)
    5.15    omega_def
    5.16 -  "whr == Abs_hypreal(hyprel```{%n::nat. real_of_nat (Suc n)})"
    5.17 +  "whr == Abs_hypreal(hyprel``{%n::nat. real_of_nat (Suc n)})"
    5.18      
    5.19    (* an infinitesimal number = [<1,1/2,1/3,...>] *)
    5.20    epsilon_def
    5.21 -  "ehr == Abs_hypreal(hyprel```{%n::nat. inverse (real_of_nat (Suc n))})"
    5.22 +  "ehr == Abs_hypreal(hyprel``{%n::nat. inverse (real_of_nat (Suc n))})"
    5.23  
    5.24    hypreal_minus_def
    5.25 -  "- P == Abs_hypreal(UN X: Rep_hypreal(P). hyprel```{%n::nat. - (X n)})"
    5.26 +  "- P == Abs_hypreal(UN X: Rep_hypreal(P). hyprel``{%n::nat. - (X n)})"
    5.27  
    5.28    hypreal_diff_def 
    5.29    "x - y == x + -(y::hypreal)"
    5.30  
    5.31    hypreal_inverse_def
    5.32    "inverse P == Abs_hypreal(UN X: Rep_hypreal(P). 
    5.33 -                    hyprel```{%n. if X n = #0 then #0 else inverse (X n)})"
    5.34 +                    hyprel``{%n. if X n = #0 then #0 else inverse (X n)})"
    5.35  
    5.36    hypreal_divide_def
    5.37    "P / Q::hypreal == P * inverse Q"
    5.38 @@ -66,17 +66,17 @@
    5.39  constdefs
    5.40  
    5.41    hypreal_of_real  :: real => hypreal                 
    5.42 -  "hypreal_of_real r         == Abs_hypreal(hyprel```{%n::nat. r})"
    5.43 +  "hypreal_of_real r         == Abs_hypreal(hyprel``{%n::nat. r})"
    5.44  
    5.45  defs 
    5.46  
    5.47    hypreal_add_def  
    5.48    "P + Q == Abs_hypreal(UN X:Rep_hypreal(P). UN Y:Rep_hypreal(Q).
    5.49 -                hyprel```{%n::nat. X n + Y n})"
    5.50 +                hyprel``{%n::nat. X n + Y n})"
    5.51  
    5.52    hypreal_mult_def  
    5.53    "P * Q == Abs_hypreal(UN X:Rep_hypreal(P). UN Y:Rep_hypreal(Q).
    5.54 -                hyprel```{%n::nat. X n * Y n})"
    5.55 +                hyprel``{%n::nat. X n * Y n})"
    5.56  
    5.57    hypreal_less_def
    5.58    "P < (Q::hypreal) == EX X Y. X: Rep_hypreal(P) & 
     6.1 --- a/src/HOL/Hyperreal/HyperNat.ML	Tue Jan 09 15:29:17 2001 +0100
     6.2 +++ b/src/HOL/Hyperreal/HyperNat.ML	Tue Jan 09 15:32:27 2001 +0100
     6.3 @@ -65,7 +65,7 @@
     6.4  val equiv_hypnatrel_iff =
     6.5      [UNIV_I, UNIV_I] MRS (equiv_hypnatrel RS eq_equiv_class_iff);
     6.6  
     6.7 -Goalw  [hypnat_def,hypnatrel_def,quotient_def] "hypnatrel```{x}:hypnat";
     6.8 +Goalw  [hypnat_def,hypnatrel_def,quotient_def] "hypnatrel``{x}:hypnat";
     6.9  by (Blast_tac 1);
    6.10  qed "hypnatrel_in_hypnat";
    6.11  
    6.12 @@ -85,7 +85,7 @@
    6.13  by (rtac Rep_hypnat_inverse 1);
    6.14  qed "inj_Rep_hypnat";
    6.15  
    6.16 -Goalw [hypnatrel_def] "x: hypnatrel ``` {x}";
    6.17 +Goalw [hypnatrel_def] "x: hypnatrel `` {x}";
    6.18  by (Step_tac 1);
    6.19  by Auto_tac;
    6.20  qed "lemma_hypnatrel_refl";
    6.21 @@ -121,7 +121,7 @@
    6.22  qed "inj_hypnat_of_nat";
    6.23  
    6.24  val [prem] = Goal
    6.25 -    "(!!x. z = Abs_hypnat(hypnatrel```{x}) ==> P) ==> P";
    6.26 +    "(!!x. z = Abs_hypnat(hypnatrel``{x}) ==> P) ==> P";
    6.27  by (res_inst_tac [("x1","z")] 
    6.28      (rewrite_rule [hypnat_def] Rep_hypnat RS quotientE) 1);
    6.29  by (dres_inst_tac [("f","Abs_hypnat")] arg_cong 1);
    6.30 @@ -133,14 +133,14 @@
    6.31     Addition for hyper naturals: hypnat_add 
    6.32   -----------------------------------------------------------*)
    6.33  Goalw [congruent2_def]
    6.34 -     "congruent2 hypnatrel (%X Y. hypnatrel```{%n. X n + Y n})";
    6.35 +     "congruent2 hypnatrel (%X Y. hypnatrel``{%n. X n + Y n})";
    6.36  by Safe_tac;
    6.37  by (ALLGOALS(Fuf_tac));
    6.38  qed "hypnat_add_congruent2";
    6.39  
    6.40  Goalw [hypnat_add_def]
    6.41 -  "Abs_hypnat(hypnatrel```{%n. X n}) + Abs_hypnat(hypnatrel```{%n. Y n}) = \
    6.42 -\  Abs_hypnat(hypnatrel```{%n. X n + Y n})";
    6.43 +  "Abs_hypnat(hypnatrel``{%n. X n}) + Abs_hypnat(hypnatrel``{%n. Y n}) = \
    6.44 +\  Abs_hypnat(hypnatrel``{%n. X n + Y n})";
    6.45  by (asm_simp_tac
    6.46      (simpset() addsimps [[equiv_hypnatrel, hypnat_add_congruent2] 
    6.47       MRS UN_equiv_class2]) 1);
    6.48 @@ -186,14 +186,14 @@
    6.49     Subtraction for hyper naturals: hypnat_minus
    6.50   -----------------------------------------------------------*)
    6.51  Goalw [congruent2_def]
    6.52 -    "congruent2 hypnatrel (%X Y. hypnatrel```{%n. X n - Y n})";
    6.53 +    "congruent2 hypnatrel (%X Y. hypnatrel``{%n. X n - Y n})";
    6.54  by Safe_tac;
    6.55  by (ALLGOALS(Fuf_tac));
    6.56  qed "hypnat_minus_congruent2";
    6.57   
    6.58  Goalw [hypnat_minus_def]
    6.59 -  "Abs_hypnat(hypnatrel```{%n. X n}) - Abs_hypnat(hypnatrel```{%n. Y n}) = \
    6.60 -\  Abs_hypnat(hypnatrel```{%n. X n - Y n})";
    6.61 +  "Abs_hypnat(hypnatrel``{%n. X n}) - Abs_hypnat(hypnatrel``{%n. Y n}) = \
    6.62 +\  Abs_hypnat(hypnatrel``{%n. X n - Y n})";
    6.63  by (asm_simp_tac
    6.64      (simpset() addsimps [[equiv_hypnatrel, hypnat_minus_congruent2] 
    6.65       MRS UN_equiv_class2]) 1);
    6.66 @@ -273,14 +273,14 @@
    6.67     Multiplication for hyper naturals: hypnat_mult
    6.68   -----------------------------------------------------------*)
    6.69  Goalw [congruent2_def]
    6.70 -    "congruent2 hypnatrel (%X Y. hypnatrel```{%n. X n * Y n})";
    6.71 +    "congruent2 hypnatrel (%X Y. hypnatrel``{%n. X n * Y n})";
    6.72  by Safe_tac;
    6.73  by (ALLGOALS(Fuf_tac));
    6.74  qed "hypnat_mult_congruent2";
    6.75  
    6.76  Goalw [hypnat_mult_def]
    6.77 -  "Abs_hypnat(hypnatrel```{%n. X n}) * Abs_hypnat(hypnatrel```{%n. Y n}) = \
    6.78 -\  Abs_hypnat(hypnatrel```{%n. X n * Y n})";
    6.79 +  "Abs_hypnat(hypnatrel``{%n. X n}) * Abs_hypnat(hypnatrel``{%n. Y n}) = \
    6.80 +\  Abs_hypnat(hypnatrel``{%n. X n * Y n})";
    6.81  by (asm_simp_tac
    6.82      (simpset() addsimps [[equiv_hypnatrel,hypnat_mult_congruent2] MRS
    6.83       UN_equiv_class2]) 1);
    6.84 @@ -475,8 +475,8 @@
    6.85  (* See comments in HYPER for corresponding thm *)
    6.86  
    6.87  Goalw [hypnat_less_def]
    6.88 -      "(Abs_hypnat(hypnatrel```{%n. X n}) < \
    6.89 -\           Abs_hypnat(hypnatrel```{%n. Y n})) = \
    6.90 +      "(Abs_hypnat(hypnatrel``{%n. X n}) < \
    6.91 +\           Abs_hypnat(hypnatrel``{%n. Y n})) = \
    6.92  \      ({n. X n < Y n} : FreeUltrafilterNat)";
    6.93  by (auto_tac (claset() addSIs [lemma_hypnatrel_refl],simpset()));
    6.94  by (Fuf_tac 1);
    6.95 @@ -527,7 +527,7 @@
    6.96  (*---------------------------------------------------------------------------------
    6.97                     Trichotomy of the hyper naturals
    6.98    --------------------------------------------------------------------------------*)
    6.99 -Goalw [hypnatrel_def] "EX x. x: hypnatrel ``` {%n. 0}";
   6.100 +Goalw [hypnatrel_def] "EX x. x: hypnatrel `` {%n. 0}";
   6.101  by (res_inst_tac [("x","%n. 0")] exI 1);
   6.102  by (Step_tac 1);
   6.103  by Auto_tac;
   6.104 @@ -620,8 +620,8 @@
   6.105  
   6.106  (*------ hypnat le iff nat le a.e ------*)
   6.107  Goalw [hypnat_le_def,le_def]
   6.108 -      "(Abs_hypnat(hypnatrel```{%n. X n}) <= \
   6.109 -\           Abs_hypnat(hypnatrel```{%n. Y n})) = \
   6.110 +      "(Abs_hypnat(hypnatrel``{%n. X n}) <= \
   6.111 +\           Abs_hypnat(hypnatrel``{%n. Y n})) = \
   6.112  \      ({n. X n <= Y n} : FreeUltrafilterNat)";
   6.113  by (auto_tac (claset() addSDs [FreeUltrafilterNat_Compl_mem],
   6.114      simpset() addsimps [hypnat_less]));
   6.115 @@ -833,7 +833,7 @@
   6.116                Existence of infinite hypernatural number
   6.117   ---------------------------------------------------------------------------------*)
   6.118  
   6.119 -Goal "hypnatrel```{%n::nat. n} : hypnat";
   6.120 +Goal "hypnatrel``{%n::nat. n} : hypnat";
   6.121  by Auto_tac;
   6.122  qed "hypnat_omega";
   6.123  
   6.124 @@ -936,23 +936,23 @@
   6.125  by Auto_tac;
   6.126  qed "SHNat_iff";
   6.127  
   6.128 -Goalw [SHNat_def] "hypnat_of_nat ``(UNIV::nat set) = SNat";
   6.129 +Goalw [SHNat_def] "hypnat_of_nat `(UNIV::nat set) = SNat";
   6.130  by Auto_tac;
   6.131  qed "hypnat_of_nat_image";
   6.132  
   6.133 -Goalw [SHNat_def] "inv hypnat_of_nat ``SNat = (UNIV::nat set)";
   6.134 +Goalw [SHNat_def] "inv hypnat_of_nat `SNat = (UNIV::nat set)";
   6.135  by Auto_tac;
   6.136  by (rtac (inj_hypnat_of_nat RS inv_f_f RS subst) 1);
   6.137  by (Blast_tac 1);
   6.138  qed "inv_hypnat_of_nat_image";
   6.139  
   6.140  Goalw [SHNat_def] 
   6.141 -     "[| EX x. x: P; P <= SNat |] ==> EX Q. P = hypnat_of_nat `` Q";
   6.142 +     "[| EX x. x: P; P <= SNat |] ==> EX Q. P = hypnat_of_nat ` Q";
   6.143  by (Best_tac 1); 
   6.144  qed "SHNat_hypnat_of_nat_image";
   6.145  
   6.146  Goalw [SHNat_def] 
   6.147 -      "SNat = hypnat_of_nat `` (UNIV::nat set)";
   6.148 +      "SNat = hypnat_of_nat ` (UNIV::nat set)";
   6.149  by Auto_tac;
   6.150  qed "SHNat_hypnat_of_nat_iff";
   6.151  
   6.152 @@ -1066,7 +1066,7 @@
   6.153       "HNatInfinite = {N. ALL n:SNat. n < N}";
   6.154  by (Step_tac 1);
   6.155  by (dres_inst_tac [("x","Abs_hypnat \
   6.156 -\        (hypnatrel ``` {%n. N})")] bspec 2);
   6.157 +\        (hypnatrel `` {%n. N})")] bspec 2);
   6.158  by (res_inst_tac [("z","x")] eq_Abs_hypnat 1);
   6.159  by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
   6.160  by (auto_tac (claset(),simpset() addsimps [hypnat_less_iff]));
   6.161 @@ -1215,14 +1215,14 @@
   6.162      Embedding of the hypernaturals into the hyperreal
   6.163   --------------------------------------------------------------*)
   6.164  
   6.165 -Goal "(Ya : hyprel ```{%n. f(n)}) = \
   6.166 +Goal "(Ya : hyprel ``{%n. f(n)}) = \
   6.167  \     ({n. f n = Ya n} : FreeUltrafilterNat)";
   6.168  by Auto_tac;
   6.169  qed "lemma_hyprel_FUFN";
   6.170  
   6.171  Goalw [hypreal_of_hypnat_def]
   6.172 -      "hypreal_of_hypnat (Abs_hypnat(hypnatrel```{%n. X n})) = \
   6.173 -\         Abs_hypreal(hyprel ``` {%n. real_of_nat (X n)})";
   6.174 +      "hypreal_of_hypnat (Abs_hypnat(hypnatrel``{%n. X n})) = \
   6.175 +\         Abs_hypreal(hyprel `` {%n. real_of_nat (X n)})";
   6.176  by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);
   6.177  by (auto_tac (claset()
   6.178                    addEs [FreeUltrafilterNat_Int RS FreeUltrafilterNat_subset],
     7.1 --- a/src/HOL/Hyperreal/HyperNat.thy	Tue Jan 09 15:29:17 2001 +0100
     7.2 +++ b/src/HOL/Hyperreal/HyperNat.thy	Tue Jan 09 15:32:27 2001 +0100
     7.3 @@ -25,7 +25,7 @@
     7.4  
     7.5    (* embedding the naturals in the hypernaturals *)
     7.6    hypnat_of_nat   :: nat => hypnat
     7.7 -  "hypnat_of_nat m  == Abs_hypnat(hypnatrel```{%n::nat. m})"
     7.8 +  "hypnat_of_nat m  == Abs_hypnat(hypnatrel``{%n::nat. m})"
     7.9  
    7.10    (* hypernaturals as members of the hyperreals; the set is defined as  *)
    7.11    (* the nonstandard extension of set of naturals embedded in the reals *)
    7.12 @@ -39,7 +39,7 @@
    7.13    (* explicit embedding of the hypernaturals in the hyperreals *)    
    7.14    hypreal_of_hypnat :: hypnat => hypreal
    7.15    "hypreal_of_hypnat N  == Abs_hypreal(UN X: Rep_hypnat(N). 
    7.16 -                            hyprel```{%n::nat. real_of_nat (X n)})"
    7.17 +                            hyprel``{%n::nat. real_of_nat (X n)})"
    7.18    
    7.19  defs
    7.20  
    7.21 @@ -53,23 +53,23 @@
    7.22  
    7.23    (** hypernatural arithmetic **)
    7.24    
    7.25 -  hypnat_zero_def      "0 == Abs_hypnat(hypnatrel```{%n::nat. 0})"
    7.26 -  hypnat_one_def       "1hn == Abs_hypnat(hypnatrel```{%n::nat. 1})"
    7.27 +  hypnat_zero_def      "0 == Abs_hypnat(hypnatrel``{%n::nat. 0})"
    7.28 +  hypnat_one_def       "1hn == Abs_hypnat(hypnatrel``{%n::nat. 1})"
    7.29  
    7.30    (* omega is in fact an infinite hypernatural number = [<1,2,3,...>] *)
    7.31 -  hypnat_omega_def     "whn == Abs_hypnat(hypnatrel```{%n::nat. n})"
    7.32 +  hypnat_omega_def     "whn == Abs_hypnat(hypnatrel``{%n::nat. n})"
    7.33   
    7.34    hypnat_add_def  
    7.35    "P + Q == Abs_hypnat(UN X:Rep_hypnat(P). UN Y:Rep_hypnat(Q).
    7.36 -                hypnatrel```{%n::nat. X n + Y n})"
    7.37 +                hypnatrel``{%n::nat. X n + Y n})"
    7.38  
    7.39    hypnat_mult_def  
    7.40    "P * Q == Abs_hypnat(UN X:Rep_hypnat(P). UN Y:Rep_hypnat(Q).
    7.41 -                hypnatrel```{%n::nat. X n * Y n})"
    7.42 +                hypnatrel``{%n::nat. X n * Y n})"
    7.43  
    7.44    hypnat_minus_def  
    7.45    "P - Q == Abs_hypnat(UN X:Rep_hypnat(P). UN Y:Rep_hypnat(Q).
    7.46 -                hypnatrel```{%n::nat. X n - Y n})"
    7.47 +                hypnatrel``{%n::nat. X n - Y n})"
    7.48  
    7.49    hypnat_less_def
    7.50    "P < (Q::hypnat) == EX X Y. X: Rep_hypnat(P) & 
     8.1 --- a/src/HOL/Hyperreal/HyperPow.ML	Tue Jan 09 15:29:17 2001 +0100
     8.2 +++ b/src/HOL/Hyperreal/HyperPow.ML	Tue Jan 09 15:32:27 2001 +0100
     8.3 @@ -178,7 +178,7 @@
     8.4                simpset() addsimps [hypreal_mult_less_mono2]));
     8.5  qed_spec_mp "hrealpow_Suc_le";
     8.6  
     8.7 -Goal "Abs_hypreal(hyprel```{%n. X n}) ^ m = Abs_hypreal(hyprel```{%n. (X n) ^ m})";
     8.8 +Goal "Abs_hypreal(hyprel``{%n. X n}) ^ m = Abs_hypreal(hyprel``{%n. (X n) ^ m})";
     8.9  by (induct_tac "m" 1);
    8.10  by (auto_tac (claset(),
    8.11                simpset() delsimps [one_eq_numeral_1]
    8.12 @@ -221,14 +221,14 @@
    8.13   --------------------------------------------------------------*)
    8.14  Goalw [congruent_def]
    8.15       "congruent hyprel \
    8.16 -\    (%X Y. hyprel```{%n. ((X::nat=>real) n ^ (Y::nat=>nat) n)})";
    8.17 +\    (%X Y. hyprel``{%n. ((X::nat=>real) n ^ (Y::nat=>nat) n)})";
    8.18  by (safe_tac (claset() addSIs [ext]));
    8.19  by (ALLGOALS(Fuf_tac));
    8.20  qed "hyperpow_congruent";
    8.21  
    8.22  Goalw [hyperpow_def]
    8.23 -  "Abs_hypreal(hyprel```{%n. X n}) pow Abs_hypnat(hypnatrel```{%n. Y n}) = \
    8.24 -\  Abs_hypreal(hyprel```{%n. X n ^ Y n})";
    8.25 +  "Abs_hypreal(hyprel``{%n. X n}) pow Abs_hypnat(hypnatrel``{%n. Y n}) = \
    8.26 +\  Abs_hypreal(hyprel``{%n. X n ^ Y n})";
    8.27  by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);
    8.28  by (auto_tac (claset() addSIs [lemma_hyprel_refl,bexI],
    8.29      simpset() addsimps [hyprel_in_hypreal RS 
     9.1 --- a/src/HOL/Hyperreal/HyperPow.thy	Tue Jan 09 15:29:17 2001 +0100
     9.2 +++ b/src/HOL/Hyperreal/HyperPow.thy	Tue Jan 09 15:32:27 2001 +0100
     9.3 @@ -32,5 +32,5 @@
     9.4    hyperpow_def
     9.5    "(R::hypreal) pow (N::hypnat) 
     9.6        == Abs_hypreal(UN X:Rep_hypreal(R). UN Y: Rep_hypnat(N).
     9.7 -             hyprel```{%n::nat. (X n) ^ (Y n)})"
     9.8 +             hyprel``{%n::nat. (X n) ^ (Y n)})"
     9.9  end
    10.1 --- a/src/HOL/Hyperreal/Lim.ML	Tue Jan 09 15:29:17 2001 +0100
    10.2 +++ b/src/HOL/Hyperreal/Lim.ML	Tue Jan 09 15:32:27 2001 +0100
    10.3 @@ -220,7 +220,7 @@
    10.4  by (fold_tac [real_le_def]);
    10.5  by (dtac lemma_skolemize_LIM2 1);
    10.6  by (Step_tac 1);
    10.7 -by (dres_inst_tac [("x","Abs_hypreal(hyprel```{X})")] spec 1);
    10.8 +by (dres_inst_tac [("x","Abs_hypreal(hyprel``{X})")] spec 1);
    10.9  by (asm_full_simp_tac
   10.10      (simpset() addsimps [starfun, hypreal_minus, 
   10.11                           hypreal_of_real_def,hypreal_add]) 1);
   10.12 @@ -726,8 +726,8 @@
   10.13  by (fold_tac [real_le_def]);
   10.14  by (dtac lemma_skolemize_LIM2u 1);
   10.15  by (Step_tac 1);
   10.16 -by (dres_inst_tac [("x","Abs_hypreal(hyprel```{X})")] spec 1);
   10.17 -by (dres_inst_tac [("x","Abs_hypreal(hyprel```{Y})")] spec 1);
   10.18 +by (dres_inst_tac [("x","Abs_hypreal(hyprel``{X})")] spec 1);
   10.19 +by (dres_inst_tac [("x","Abs_hypreal(hyprel``{Y})")] spec 1);
   10.20  by (asm_full_simp_tac
   10.21      (simpset() addsimps [starfun, hypreal_minus,hypreal_add]) 1);
   10.22  by Auto_tac;
    11.1 --- a/src/HOL/Hyperreal/NSA.ML	Tue Jan 09 15:29:17 2001 +0100
    11.2 +++ b/src/HOL/Hyperreal/NSA.ML	Tue Jan 09 15:32:27 2001 +0100
    11.3 @@ -94,18 +94,18 @@
    11.4  by Auto_tac;
    11.5  qed "SReal_iff";
    11.6  
    11.7 -Goalw [SReal_def] "hypreal_of_real ``(UNIV::real set) = SReal";
    11.8 +Goalw [SReal_def] "hypreal_of_real `(UNIV::real set) = SReal";
    11.9  by Auto_tac;
   11.10  qed "hypreal_of_real_image";
   11.11  
   11.12 -Goalw [SReal_def] "inv hypreal_of_real ``SReal = (UNIV::real set)";
   11.13 +Goalw [SReal_def] "inv hypreal_of_real `SReal = (UNIV::real set)";
   11.14  by Auto_tac;
   11.15  by (rtac (inj_hypreal_of_real RS inv_f_f RS subst) 1);
   11.16  by (Blast_tac 1);
   11.17  qed "inv_hypreal_of_real_image";
   11.18  
   11.19  Goalw [SReal_def] 
   11.20 -      "[| EX x. x: P; P <= SReal |] ==> EX Q. P = hypreal_of_real `` Q";
   11.21 +      "[| EX x. x: P; P <= SReal |] ==> EX Q. P = hypreal_of_real ` Q";
   11.22  by (Best_tac 1); 
   11.23  qed "SReal_hypreal_of_real_image";
   11.24  
   11.25 @@ -140,13 +140,13 @@
   11.26      lifting of ub and property of lub
   11.27   -------------------------------------------------------*)
   11.28  Goalw [isUb_def,setle_def] 
   11.29 -      "(isUb (SReal) (hypreal_of_real `` Q) (hypreal_of_real Y)) = \
   11.30 +      "(isUb (SReal) (hypreal_of_real ` Q) (hypreal_of_real Y)) = \
   11.31  \      (isUb (UNIV :: real set) Q Y)";
   11.32  by Auto_tac;
   11.33  qed "hypreal_of_real_isUb_iff";
   11.34  
   11.35  Goalw [isLub_def,leastP_def] 
   11.36 -     "isLub SReal (hypreal_of_real `` Q) (hypreal_of_real Y) \
   11.37 +     "isLub SReal (hypreal_of_real ` Q) (hypreal_of_real Y) \
   11.38  \     ==> isLub (UNIV :: real set) Q Y";
   11.39  by (auto_tac (claset() addIs [hypreal_of_real_isUb_iff RS iffD2],
   11.40                simpset() addsimps [hypreal_of_real_isUb_iff, setge_def]));
   11.41 @@ -154,7 +154,7 @@
   11.42  
   11.43  Goalw [isLub_def,leastP_def] 
   11.44        "isLub (UNIV :: real set) Q Y \
   11.45 -\      ==> isLub SReal (hypreal_of_real `` Q) (hypreal_of_real Y)";
   11.46 +\      ==> isLub SReal (hypreal_of_real ` Q) (hypreal_of_real Y)";
   11.47  by (auto_tac (claset(),
   11.48                simpset() addsimps [hypreal_of_real_isUb_iff, setge_def]));
   11.49  by (forw_inst_tac [("x2","x")] (isUbD2a RS (SReal_iff RS iffD1) RS exE) 1);
   11.50 @@ -163,7 +163,7 @@
   11.51  by (auto_tac (claset(), simpset() addsimps [hypreal_of_real_isUb_iff]));
   11.52  qed "hypreal_of_real_isLub2";
   11.53  
   11.54 -Goal "(isLub SReal (hypreal_of_real `` Q) (hypreal_of_real Y)) = \
   11.55 +Goal "(isLub SReal (hypreal_of_real ` Q) (hypreal_of_real Y)) = \
   11.56  \     (isLub (UNIV :: real set) Q Y)";
   11.57  by (blast_tac (claset() addIs [hypreal_of_real_isLub1,
   11.58                                 hypreal_of_real_isLub2]) 1);
   11.59 @@ -2059,7 +2059,7 @@
   11.60         Omega is a member of HInfinite
   11.61   -----------------------------------------------*)
   11.62  
   11.63 -Goal "hyprel```{%n::nat. real_of_nat (Suc n)} : hypreal";
   11.64 +Goal "hyprel``{%n::nat. real_of_nat (Suc n)} : hypreal";
   11.65  by Auto_tac;
   11.66  qed "hypreal_omega";
   11.67  
   11.68 @@ -2192,7 +2192,7 @@
   11.69      |X(n) - x| < 1/n ==> [<X n>] - hypreal_of_real x|: Infinitesimal 
   11.70   -----------------------------------------------------*)
   11.71  Goal "ALL n. abs(X n + -x) < inverse(real_of_nat(Suc n)) \ 
   11.72 -\    ==> Abs_hypreal(hyprel```{X}) + -hypreal_of_real x : Infinitesimal";
   11.73 +\    ==> Abs_hypreal(hyprel``{X}) + -hypreal_of_real x : Infinitesimal";
   11.74  by (auto_tac (claset() addSIs [bexI] 
   11.75             addDs [rename_numerals FreeUltrafilterNat_inverse_real_of_posnat,
   11.76                    FreeUltrafilterNat_all,FreeUltrafilterNat_Int] 
   11.77 @@ -2203,21 +2203,21 @@
   11.78  qed "real_seq_to_hypreal_Infinitesimal";
   11.79  
   11.80  Goal "ALL n. abs(X n + -x) < inverse(real_of_nat(Suc n)) \ 
   11.81 -\     ==> Abs_hypreal(hyprel```{X}) @= hypreal_of_real x";
   11.82 +\     ==> Abs_hypreal(hyprel``{X}) @= hypreal_of_real x";
   11.83  by (rtac (inf_close_minus_iff RS ssubst) 1);
   11.84  by (rtac (mem_infmal_iff RS subst) 1);
   11.85  by (etac real_seq_to_hypreal_Infinitesimal 1);
   11.86  qed "real_seq_to_hypreal_inf_close";
   11.87  
   11.88  Goal "ALL n. abs(x + -X n) < inverse(real_of_nat(Suc n)) \ 
   11.89 -\              ==> Abs_hypreal(hyprel```{X}) @= hypreal_of_real x";
   11.90 +\              ==> Abs_hypreal(hyprel``{X}) @= hypreal_of_real x";
   11.91  by (asm_full_simp_tac (simpset() addsimps [abs_minus_add_cancel,
   11.92          real_seq_to_hypreal_inf_close]) 1);
   11.93  qed "real_seq_to_hypreal_inf_close2";
   11.94  
   11.95  Goal "ALL n. abs(X n + -Y n) < inverse(real_of_nat(Suc n)) \ 
   11.96 -\     ==> Abs_hypreal(hyprel```{X}) + \
   11.97 -\         -Abs_hypreal(hyprel```{Y}) : Infinitesimal";
   11.98 +\     ==> Abs_hypreal(hyprel``{X}) + \
   11.99 +\         -Abs_hypreal(hyprel``{Y}) : Infinitesimal";
  11.100  by (auto_tac (claset() addSIs [bexI] 
  11.101                    addDs [rename_numerals 
  11.102                           FreeUltrafilterNat_inverse_real_of_posnat,
    12.1 --- a/src/HOL/Hyperreal/NatStar.ML	Tue Jan 09 15:29:17 2001 +0100
    12.2 +++ b/src/HOL/Hyperreal/NatStar.ML	Tue Jan 09 15:32:27 2001 +0100
    12.3 @@ -117,7 +117,7 @@
    12.4           simpset()));
    12.5  qed "NatStar_mem";
    12.6  
    12.7 -Goalw [starsetNat_def] "hypnat_of_nat `` A <= *sNat* A";
    12.8 +Goalw [starsetNat_def] "hypnat_of_nat ` A <= *sNat* A";
    12.9  by (auto_tac (claset(), simpset() addsimps [hypnat_of_nat_def]));
   12.10  by (blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 1);
   12.11  qed "NatStar_hypreal_of_real_image_subset";
   12.12 @@ -128,7 +128,7 @@
   12.13  qed "NatStar_SHNat_subset";
   12.14  
   12.15  Goalw [starsetNat_def] 
   12.16 -     "*sNat* X Int SNat = hypnat_of_nat `` X";
   12.17 +     "*sNat* X Int SNat = hypnat_of_nat ` X";
   12.18  by (auto_tac (claset(),
   12.19           simpset() addsimps 
   12.20             [hypnat_of_nat_def,SHNat_def]));
   12.21 @@ -140,7 +140,7 @@
   12.22  by (Auto_tac);
   12.23  qed "NatStar_hypreal_of_real_Int";
   12.24  
   12.25 -Goal "x ~: hypnat_of_nat `` A ==> ALL y: A. x ~= hypnat_of_nat y";
   12.26 +Goal "x ~: hypnat_of_nat ` A ==> ALL y: A. x ~= hypnat_of_nat y";
   12.27  by (Auto_tac);
   12.28  qed "lemma_not_hypnatA";
   12.29  
   12.30 @@ -188,15 +188,15 @@
   12.31  qed "starfunNat2_n_starfunNat2";
   12.32  
   12.33  Goalw [congruent_def] 
   12.34 -      "congruent hypnatrel (%X. hypnatrel```{%n. f (X n)})";
   12.35 +      "congruent hypnatrel (%X. hypnatrel``{%n. f (X n)})";
   12.36  by (safe_tac (claset()));
   12.37  by (ALLGOALS(Fuf_tac));
   12.38  qed "starfunNat_congruent";
   12.39  
   12.40  (* f::nat=>real *)
   12.41  Goalw [starfunNat_def]
   12.42 -      "(*fNat* f) (Abs_hypnat(hypnatrel```{%n. X n})) = \
   12.43 -\      Abs_hypreal(hyprel ``` {%n. f (X n)})";
   12.44 +      "(*fNat* f) (Abs_hypnat(hypnatrel``{%n. X n})) = \
   12.45 +\      Abs_hypreal(hyprel `` {%n. f (X n)})";
   12.46  by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);
   12.47  by (simp_tac (simpset() addsimps 
   12.48     [hyprel_in_hypreal RS Abs_hypreal_inverse]) 1);
   12.49 @@ -205,8 +205,8 @@
   12.50  
   12.51  (* f::nat=>nat *)
   12.52  Goalw [starfunNat2_def]
   12.53 -      "(*fNat2* f) (Abs_hypnat(hypnatrel```{%n. X n})) = \
   12.54 -\      Abs_hypnat(hypnatrel ``` {%n. f (X n)})";
   12.55 +      "(*fNat2* f) (Abs_hypnat(hypnatrel``{%n. X n})) = \
   12.56 +\      Abs_hypnat(hypnatrel `` {%n. f (X n)})";
   12.57  by (res_inst_tac [("f","Abs_hypnat")] arg_cong 1);
   12.58  by (simp_tac (simpset() addsimps 
   12.59     [hypnatrel_in_hypnat RS Abs_hypnat_inverse,
   12.60 @@ -413,14 +413,14 @@
   12.61       Internal functions - some redundancy with *fNat* now
   12.62   ---------------------------------------------------------*)
   12.63  Goalw [congruent_def] 
   12.64 -      "congruent hypnatrel (%X. hypnatrel```{%n. f n (X n)})";
   12.65 +      "congruent hypnatrel (%X. hypnatrel``{%n. f n (X n)})";
   12.66  by (safe_tac (claset()));
   12.67  by (ALLGOALS(Fuf_tac));
   12.68  qed "starfunNat_n_congruent";
   12.69  
   12.70  Goalw [starfunNat_n_def]
   12.71 -     "(*fNatn* f) (Abs_hypnat(hypnatrel```{%n. X n})) = \
   12.72 -\     Abs_hypreal(hyprel ``` {%n. f n (X n)})";
   12.73 +     "(*fNatn* f) (Abs_hypnat(hypnatrel``{%n. X n})) = \
   12.74 +\     Abs_hypreal(hyprel `` {%n. f n (X n)})";
   12.75  by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);
   12.76  by Auto_tac;
   12.77  by (Ultra_tac 1);
   12.78 @@ -468,7 +468,7 @@
   12.79  by (auto_tac (claset(), simpset() addsimps [starfunNat_n, hypreal_minus]));
   12.80  qed "starfunNat_n_minus";
   12.81  
   12.82 -Goal "(*fNatn* f) (hypnat_of_nat n) = Abs_hypreal(hyprel ``` {%i. f i n})";
   12.83 +Goal "(*fNatn* f) (hypnat_of_nat n) = Abs_hypreal(hyprel `` {%i. f i n})";
   12.84  by (auto_tac (claset(), simpset() addsimps [starfunNat_n,hypnat_of_nat_def]));
   12.85  qed "starfunNat_n_eq";
   12.86  Addsimps [starfunNat_n_eq];
    13.1 --- a/src/HOL/Hyperreal/NatStar.thy	Tue Jan 09 15:29:17 2001 +0100
    13.2 +++ b/src/HOL/Hyperreal/NatStar.thy	Tue Jan 09 15:32:27 2001 +0100
    13.3 @@ -23,10 +23,10 @@
    13.4      (* star transform of functions f:Nat --> Real *)
    13.5  
    13.6      starfunNat :: (nat => real) => hypnat => hypreal        ("*fNat* _" [80] 80)
    13.7 -    "*fNat* f  == (%x. Abs_hypreal(UN X: Rep_hypnat(x). hyprel```{%n. f (X n)}))" 
    13.8 +    "*fNat* f  == (%x. Abs_hypreal(UN X: Rep_hypnat(x). hyprel``{%n. f (X n)}))" 
    13.9  
   13.10      starfunNat_n :: (nat => (nat => real)) => hypnat => hypreal        ("*fNatn* _" [80] 80)
   13.11 -    "*fNatn* F  == (%x. Abs_hypreal(UN X: Rep_hypnat(x). hyprel```{%n. (F n)(X n)}))" 
   13.12 +    "*fNatn* F  == (%x. Abs_hypreal(UN X: Rep_hypnat(x). hyprel``{%n. (F n)(X n)}))" 
   13.13  
   13.14      InternalNatFuns :: (hypnat => hypreal) set
   13.15      "InternalNatFuns == {X. EX F. X = *fNatn* F}"
   13.16 @@ -34,10 +34,10 @@
   13.17      (* star transform of functions f:Nat --> Nat *)
   13.18  
   13.19      starfunNat2 :: (nat => nat) => hypnat => hypnat        ("*fNat2* _" [80] 80)
   13.20 -    "*fNat2* f  == (%x. Abs_hypnat(UN X: Rep_hypnat(x). hypnatrel```{%n. f (X n)}))" 
   13.21 +    "*fNat2* f  == (%x. Abs_hypnat(UN X: Rep_hypnat(x). hypnatrel``{%n. f (X n)}))" 
   13.22  
   13.23      starfunNat2_n :: (nat => (nat => nat)) => hypnat => hypnat        ("*fNat2n* _" [80] 80)
   13.24 -    "*fNat2n* F  == (%x. Abs_hypnat(UN X: Rep_hypnat(x). hypnatrel```{%n. (F n)(X n)}))" 
   13.25 +    "*fNat2n* F  == (%x. Abs_hypnat(UN X: Rep_hypnat(x). hypnatrel``{%n. (F n)(X n)}))" 
   13.26  
   13.27      InternalNatFuns2 :: (hypnat => hypnat) set
   13.28      "InternalNatFuns2 == {X. EX F. X = *fNat2n* F}"
    14.1 --- a/src/HOL/Hyperreal/SEQ.ML	Tue Jan 09 15:29:17 2001 +0100
    14.2 +++ b/src/HOL/Hyperreal/SEQ.ML	Tue Jan 09 15:32:27 2001 +0100
    14.3 @@ -141,7 +141,7 @@
    14.4  
    14.5  (* thus, the sequence defines an infinite hypernatural! *)
    14.6  Goal "ALL n. n <= f n \
    14.7 -\         ==> Abs_hypnat (hypnatrel ``` {f}) : HNatInfinite";
    14.8 +\         ==> Abs_hypnat (hypnatrel `` {f}) : HNatInfinite";
    14.9  by (auto_tac (claset(),simpset() addsimps [HNatInfinite_FreeUltrafilterNat_iff]));
   14.10  by (EVERY[rtac bexI 1, rtac lemma_hypnatrel_refl 2, Step_tac 1]);
   14.11  by (etac FreeUltrafilterNat_NSLIMSEQ 1);
   14.12 @@ -156,7 +156,7 @@
   14.13  val lemmaLIM2 = result();
   14.14  
   14.15  Goal "[| #0 < r; ALL n. r <= abs (X (f n) + - L); \
   14.16 -\          (*fNat* X) (Abs_hypnat (hypnatrel ``` {f})) + \
   14.17 +\          (*fNat* X) (Abs_hypnat (hypnatrel `` {f})) + \
   14.18  \          - hypreal_of_real  L @= 0 |] ==> False";
   14.19  by (auto_tac (claset(),simpset() addsimps [starfunNat,
   14.20      mem_infmal_iff RS sym,hypreal_of_real_def,
   14.21 @@ -178,7 +178,7 @@
   14.22  by (Step_tac 1);
   14.23  (* skolemization step *)
   14.24  by (dtac choice 1 THEN Step_tac 1);
   14.25 -by (dres_inst_tac [("x","Abs_hypnat(hypnatrel```{f})")] bspec 1);
   14.26 +by (dres_inst_tac [("x","Abs_hypnat(hypnatrel``{f})")] bspec 1);
   14.27  by (dtac (inf_close_minus_iff RS iffD1) 2);
   14.28  by (fold_tac [real_le_def]);
   14.29  by (blast_tac (claset() addIs [HNatInfinite_NSLIMSEQ]) 1);
   14.30 @@ -511,7 +511,7 @@
   14.31  val lemmaNSBseq2 = result();
   14.32  
   14.33  Goal "ALL N. real_of_nat(Suc N) < abs (X (f N)) \
   14.34 -\         ==>  Abs_hypreal(hyprel```{X o f}) : HInfinite";
   14.35 +\         ==>  Abs_hypreal(hyprel``{X o f}) : HInfinite";
   14.36  by (auto_tac (claset(),
   14.37                simpset() addsimps [HInfinite_FreeUltrafilterNat_iff,o_def]));
   14.38  by (EVERY[rtac bexI 1, rtac lemma_hyprel_refl 2, Step_tac 1]);
   14.39 @@ -542,7 +542,7 @@
   14.40  val lemma_finite_NSBseq2 = result();
   14.41  
   14.42  Goal "ALL N. real_of_nat(Suc N) < abs (X (f N)) \
   14.43 -\     ==> Abs_hypnat(hypnatrel```{f}) : HNatInfinite";
   14.44 +\     ==> Abs_hypnat(hypnatrel``{f}) : HNatInfinite";
   14.45  by (auto_tac (claset(),
   14.46                simpset() addsimps [HNatInfinite_FreeUltrafilterNat_iff]));
   14.47  by (EVERY[rtac bexI 1, rtac lemma_hypnatrel_refl 2, Step_tac 1]);
   14.48 @@ -795,7 +795,7 @@
   14.49  (*-------------------------------
   14.50        Standard def => NS def
   14.51   -------------------------------*)
   14.52 -Goal "Abs_hypnat (hypnatrel ``` {x}) : HNatInfinite \
   14.53 +Goal "Abs_hypnat (hypnatrel `` {x}) : HNatInfinite \
   14.54  \         ==> {n. M <= x n} : FreeUltrafilterNat";
   14.55  by (auto_tac (claset(),
   14.56                simpset() addsimps [HNatInfinite_FreeUltrafilterNat_iff]));
   14.57 @@ -843,7 +843,7 @@
   14.58      step_tac (claset() addSDs [all_conj_distrib RS iffD1]) 1);
   14.59  by (REPEAT(dtac HNatInfinite_NSLIMSEQ 1));
   14.60  by (dtac bspec 1 THEN assume_tac 1);
   14.61 -by (dres_inst_tac [("x","Abs_hypnat (hypnatrel ``` {fa})")] bspec 1 
   14.62 +by (dres_inst_tac [("x","Abs_hypnat (hypnatrel `` {fa})")] bspec 1 
   14.63      THEN auto_tac (claset(), simpset() addsimps [starfunNat]));
   14.64  by (dtac (inf_close_minus_iff RS iffD1) 1);
   14.65  by (dtac (mem_infmal_iff RS iffD2) 1);
   14.66 @@ -1334,7 +1334,7 @@
   14.67                   Hyperreals and Sequences
   14.68   ---------------------------------------------------------------***)
   14.69  (*** A bounded sequence is a finite hyperreal ***)
   14.70 -Goal "NSBseq X ==> Abs_hypreal(hyprel```{X}) : HFinite";
   14.71 +Goal "NSBseq X ==> Abs_hypreal(hyprel``{X}) : HFinite";
   14.72  by (auto_tac (claset() addSIs [bexI,lemma_hyprel_refl] addIs 
   14.73         [FreeUltrafilterNat_all RS FreeUltrafilterNat_subset],
   14.74         simpset() addsimps [HFinite_FreeUltrafilterNat_iff,
   14.75 @@ -1343,7 +1343,7 @@
   14.76  
   14.77  (*** A sequence converging to zero defines an infinitesimal ***)
   14.78  Goalw [NSLIMSEQ_def] 
   14.79 -      "X ----NS> #0 ==> Abs_hypreal(hyprel```{X}) : Infinitesimal";
   14.80 +      "X ----NS> #0 ==> Abs_hypreal(hyprel``{X}) : Infinitesimal";
   14.81  by (dres_inst_tac [("x","whn")] bspec 1);
   14.82  by (simp_tac (simpset() addsimps [HNatInfinite_whn]) 1);
   14.83  by (auto_tac (claset(),
    15.1 --- a/src/HOL/Hyperreal/Star.ML	Tue Jan 09 15:29:17 2001 +0100
    15.2 +++ b/src/HOL/Hyperreal/Star.ML	Tue Jan 09 15:32:27 2001 +0100
    15.3 @@ -83,12 +83,12 @@
    15.4  by (auto_tac (claset() addIs [FreeUltrafilterNat_subset],simpset()));
    15.5  qed "STAR_mem";
    15.6  
    15.7 -Goalw [starset_def] "hypreal_of_real `` A <= *s* A";
    15.8 +Goalw [starset_def] "hypreal_of_real ` A <= *s* A";
    15.9  by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_def]));
   15.10  by (blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 1);
   15.11  qed "STAR_hypreal_of_real_image_subset";
   15.12  
   15.13 -Goalw [starset_def] "*s* X Int SReal = hypreal_of_real `` X";
   15.14 +Goalw [starset_def] "*s* X Int SReal = hypreal_of_real ` X";
   15.15  by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_def,SReal_def]));
   15.16  by (fold_tac [hypreal_of_real_def]);
   15.17  by (rtac imageI 1 THEN rtac ccontr 1);
   15.18 @@ -98,7 +98,7 @@
   15.19  by (Auto_tac);
   15.20  qed "STAR_hypreal_of_real_Int";
   15.21  
   15.22 -Goal "x ~: hypreal_of_real `` A ==> ALL y: A. x ~= hypreal_of_real y";
   15.23 +Goal "x ~: hypreal_of_real ` A ==> ALL y: A. x ~= hypreal_of_real y";
   15.24  by (Auto_tac);
   15.25  qed "lemma_not_hyprealA";
   15.26  
   15.27 @@ -108,7 +108,7 @@
   15.28  
   15.29  Goalw [starset_def]
   15.30      "ALL n. (X n) ~: M \
   15.31 -\         ==> Abs_hypreal(hyprel```{X}) ~: *s* M";
   15.32 +\         ==> Abs_hypreal(hyprel``{X}) ~: *s* M";
   15.33  by (Auto_tac THEN rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
   15.34  by (Auto_tac);
   15.35  qed "STAR_real_seq_to_hypreal";
   15.36 @@ -193,14 +193,14 @@
   15.37      Nonstandard extension of functions- congruence 
   15.38   -----------------------------------------------------------------------*) 
   15.39  
   15.40 -Goalw [congruent_def] "congruent hyprel (%X. hyprel```{%n. f (X n)})";
   15.41 +Goalw [congruent_def] "congruent hyprel (%X. hyprel``{%n. f (X n)})";
   15.42  by (safe_tac (claset()));
   15.43  by (ALLGOALS(Fuf_tac));
   15.44  qed "starfun_congruent";
   15.45  
   15.46  Goalw [starfun_def]
   15.47 -      "(*f* f) (Abs_hypreal(hyprel```{%n. X n})) = \
   15.48 -\      Abs_hypreal(hyprel ``` {%n. f (X n)})";
   15.49 +      "(*f* f) (Abs_hypreal(hyprel``{%n. X n})) = \
   15.50 +\      Abs_hypreal(hyprel `` {%n. f (X n)})";
   15.51  by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);
   15.52  by (simp_tac (simpset() addsimps 
   15.53     [hyprel_in_hypreal RS Abs_hypreal_inverse,[equiv_hyprel,
   15.54 @@ -417,8 +417,8 @@
   15.55     applied entrywise to equivalence class representative.
   15.56     This is easily proved using starfun and ns extension thm
   15.57   ------------------------------------------------------------*)
   15.58 -Goal "abs (Abs_hypreal (hyprel ``` {X})) = \
   15.59 -\                 Abs_hypreal(hyprel ``` {%n. abs (X n)})";
   15.60 +Goal "abs (Abs_hypreal (hyprel `` {X})) = \
   15.61 +\                 Abs_hypreal(hyprel `` {%n. abs (X n)})";
   15.62  by (simp_tac (simpset() addsimps [starfun_rabs_hrabs RS sym,starfun]) 1);
   15.63  qed "hypreal_hrabs";
   15.64  
   15.65 @@ -470,7 +470,7 @@
   15.66  by (Fuf_tac 1);
   15.67  qed  "Infinitesimal_FreeUltrafilterNat_iff2";
   15.68  
   15.69 -Goal "(Abs_hypreal(hyprel```{X}) @= Abs_hypreal(hyprel```{Y})) = \
   15.70 +Goal "(Abs_hypreal(hyprel``{X}) @= Abs_hypreal(hyprel``{Y})) = \
   15.71  \     (ALL m. {n. abs (X n + - Y n) < \
   15.72  \                 inverse(real_of_nat(Suc m))} : FreeUltrafilterNat)";
   15.73  by (rtac (inf_close_minus_iff RS ssubst) 1);
   15.74 @@ -485,6 +485,6 @@
   15.75  Goal "inj starfun";
   15.76  by (rtac injI 1);
   15.77  by (rtac ext 1 THEN rtac ccontr 1);
   15.78 -by (dres_inst_tac [("x","Abs_hypreal(hyprel ```{%n. xa})")] fun_cong 1);
   15.79 +by (dres_inst_tac [("x","Abs_hypreal(hyprel ``{%n. xa})")] fun_cong 1);
   15.80  by (auto_tac (claset(),simpset() addsimps [starfun]));
   15.81  qed "inj_starfun";
    16.1 --- a/src/HOL/Hyperreal/Star.thy	Tue Jan 09 15:29:17 2001 +0100
    16.2 +++ b/src/HOL/Hyperreal/Star.thy	Tue Jan 09 15:32:27 2001 +0100
    16.3 @@ -25,11 +25,11 @@
    16.4                          ((y = (F x)) = ({n. Y n = f(X n)} : FreeUltrafilterNat)))"
    16.5      
    16.6      starfun :: (real => real) => hypreal => hypreal        ("*f* _" [80] 80)
    16.7 -    "*f* f  == (%x. Abs_hypreal(UN X: Rep_hypreal(x). hyprel```{%n. f(X n)}))" 
    16.8 +    "*f* f  == (%x. Abs_hypreal(UN X: Rep_hypreal(x). hyprel``{%n. f(X n)}))" 
    16.9  
   16.10      (* internal functions *)
   16.11      starfun_n :: (nat => (real => real)) => hypreal => hypreal        ("*fn* _" [80] 80)
   16.12 -    "*fn* F  == (%x. Abs_hypreal(UN X: Rep_hypreal(x). hyprel```{%n. (F n)(X n)}))" 
   16.13 +    "*fn* F  == (%x. Abs_hypreal(UN X: Rep_hypreal(x). hyprel``{%n. (F n)(X n)}))" 
   16.14  
   16.15      InternalFuns :: (hypreal => hypreal) set
   16.16      "InternalFuns == {X. EX F. X = *fn* F}"
    17.1 --- a/src/HOL/IMPP/Hoare.ML	Tue Jan 09 15:29:17 2001 +0100
    17.2 +++ b/src/HOL/IMPP/Hoare.ML	Tue Jan 09 15:32:27 2001 +0100
    17.3 @@ -64,8 +64,8 @@
    17.4  by (Fast_tac 1);
    17.5  qed "conseq2";
    17.6  
    17.7 -Goal "[| G Un (%p. {P p}.      BODY p  .{Q p})``Procs  \
    17.8 -\         ||- (%p. {P p}. the (body p) .{Q p})``Procs; \
    17.9 +Goal "[| G Un (%p. {P p}.      BODY p  .{Q p})`Procs  \
   17.10 +\         ||- (%p. {P p}. the (body p) .{Q p})`Procs; \
   17.11  \   pn:Procs |] ==> G|-{P pn}. BODY pn .{Q pn}";
   17.12  bd hoare_derivs.Body 1;
   17.13  be hoare_derivs.weaken 1;
   17.14 @@ -129,7 +129,7 @@
   17.15  
   17.16  Goal "[| finite U; \
   17.17  \ !p. G |-     {P' p}.c0 p.{Q' p}       --> G |-     {P p}.c0 p.{Q p} |] ==> \
   17.18 -\     G||-(%p. {P' p}.c0 p.{Q' p}) `` U --> G||-(%p. {P p}.c0 p.{Q p}) `` U";
   17.19 +\     G||-(%p. {P' p}.c0 p.{Q' p}) ` U --> G||-(%p. {P p}.c0 p.{Q p}) ` U";
   17.20  be finite_induct 1;
   17.21  by (ALLGOALS Clarsimp_tac);
   17.22  bd derivs_insertD 1;
   17.23 @@ -154,9 +154,9 @@
   17.24  qed "Loop_sound_lemma";
   17.25  
   17.26  Goalw [hoare_valids_def]
   17.27 -   "[| G Un (%pn. {P pn}.      BODY pn  .{Q pn})``Procs \
   17.28 -\        ||=(%pn. {P pn}. the (body pn) .{Q pn})``Procs |] ==> \
   17.29 -\       G||=(%pn. {P pn}.      BODY pn  .{Q pn})``Procs";
   17.30 +   "[| G Un (%pn. {P pn}.      BODY pn  .{Q pn})`Procs \
   17.31 +\        ||=(%pn. {P pn}. the (body pn) .{Q pn})`Procs |] ==> \
   17.32 +\       G||=(%pn. {P pn}.      BODY pn  .{Q pn})`Procs";
   17.33  br allI 1;
   17.34  by (induct_tac "n" 1);
   17.35  by  (fast_tac (claset() addIs [Body_triple_valid_0]) 1);
   17.36 @@ -255,12 +255,12 @@
   17.37  \ !!G pn. P (insert (mgt_call pn) G) {mgt(the(body pn))} ==> P G {mgt_call pn};\
   17.38  \         !!G c. [| wt c; !pn:U. P G {mgt_call pn} |] ==> P G {mgt c}; \
   17.39  \         !!pn. pn : U ==> wt (the (body pn)); \
   17.40 -\         finite U; uG = mgt_call``U |] ==> \
   17.41 +\         finite U; uG = mgt_call`U |] ==> \
   17.42  \ !G. G <= uG --> n <= card uG --> card G = card uG - n --> (!c. wt c --> P G {mgt c})";
   17.43  by (cut_facts_tac (premises()) 1);
   17.44  by (induct_tac "n" 1);
   17.45  by  (ALLGOALS Clarsimp_tac);
   17.46 -by  (subgoal_tac "G = mgt_call `` U" 1);
   17.47 +by  (subgoal_tac "G = mgt_call ` U" 1);
   17.48  by   (asm_simp_tac (simpset() addsimps [card_seteq, finite_imageI]) 2);
   17.49  by  (Asm_full_simp_tac 1);
   17.50  by  (eresolve_tac (tl(tl(premises()))(*MGF_lemma1*)) 1);
   17.51 @@ -308,9 +308,9 @@
   17.52  (* Version: simultaneous recursion in call rule *)
   17.53  
   17.54  (* finiteness not really necessary here *)
   17.55 -Goalw [MGT_def]     "[| G Un (%pn. {=}.      BODY pn  .{->})``Procs \
   17.56 -\                         ||-(%pn. {=}. the (body pn) .{->})``Procs; \
   17.57 -\ finite Procs |] ==>   G ||-(%pn. {=}.      BODY pn  .{->})``Procs";
   17.58 +Goalw [MGT_def]     "[| G Un (%pn. {=}.      BODY pn  .{->})`Procs \
   17.59 +\                         ||-(%pn. {=}. the (body pn) .{->})`Procs; \
   17.60 +\ finite Procs |] ==>   G ||-(%pn. {=}.      BODY pn  .{->})`Procs";
   17.61  br hoare_derivs.Body 1;
   17.62  be finite_pointwise 1;
   17.63  ba  2;
   17.64 @@ -321,8 +321,8 @@
   17.65  
   17.66  (* requires empty, insert, com_det *)
   17.67  Goal "[| state_not_singleton; WT_bodies; \
   17.68 -\ F<=(%pn. {=}.the (body pn).{->})``dom body |] ==> \
   17.69 -\    (%pn. {=}.     BODY pn .{->})``dom body||-F";
   17.70 +\ F<=(%pn. {=}.the (body pn).{->})`dom body |] ==> \
   17.71 +\    (%pn. {=}.     BODY pn .{->})`dom body||-F";
   17.72  by (ftac finite_subset 1);
   17.73  br  (finite_dom_body RS finite_imageI) 1;
   17.74  by (rotate_tac 2 1);
   17.75 @@ -344,7 +344,7 @@
   17.76  ba   1;
   17.77  ba  2;
   17.78  by (Clarsimp_tac 1);
   17.79 -by (subgoal_tac "{}||-(%pn. {=}. BODY pn .{->})``dom body" 1);
   17.80 +by (subgoal_tac "{}||-(%pn. {=}. BODY pn .{->})`dom body" 1);
   17.81  be  hoare_derivs.weaken 1;
   17.82  by  (fast_tac (claset() addIs [domI]) 1);
   17.83  br (finite_dom_body RSN (2,MGT_Body)) 1;
    18.1 --- a/src/HOL/IMPP/Hoare.thy	Tue Jan 09 15:29:17 2001 +0100
    18.2 +++ b/src/HOL/IMPP/Hoare.thy	Tue Jan 09 15:32:27 2001 +0100
    18.3 @@ -88,9 +88,9 @@
    18.4  	  |-{P}.  the (body pn) .{Q} ==>
    18.5  	 G|-{P}.       BODY pn  .{Q}"
    18.6  *)
    18.7 -  Body	"[| G Un (%p. {P p}.      BODY p  .{Q p})``Procs
    18.8 -	      ||-(%p. {P p}. the (body p) .{Q p})``Procs |]
    18.9 -	==>  G||-(%p. {P p}.      BODY p  .{Q p})``Procs"
   18.10 +  Body	"[| G Un (%p. {P p}.      BODY p  .{Q p})`Procs
   18.11 +	      ||-(%p. {P p}. the (body p) .{Q p})`Procs |]
   18.12 +	==>  G||-(%p. {P p}.      BODY p  .{Q p})`Procs"
   18.13  
   18.14    Call	   "G|-{P}. BODY pn .{%Z s. Q Z (setlocs s (getlocs s')[X::=s<Res>])}
   18.15  	==> G|-{%Z s. s'=s & P Z (setlocs s newlocs[Loc Arg::=a s])}.
    19.1 --- a/src/HOL/Induct/LList.ML	Tue Jan 09 15:29:17 2001 +0100
    19.2 +++ b/src/HOL/Induct/LList.ML	Tue Jan 09 15:32:27 2001 +0100
    19.3 @@ -420,7 +420,7 @@
    19.4  \    f(NIL)=g(NIL);                                             \
    19.5  \    !!x l. [| x:A;  l: llist(A) |] ==>                         \
    19.6  \           (f(CONS x l),g(CONS x l)) :                         \
    19.7 -\               LListD_Fun (diag A) ((%u.(f(u),g(u)))``llist(A) Un  \
    19.8 +\               LListD_Fun (diag A) ((%u.(f(u),g(u)))`llist(A) Un  \
    19.9  \                                   diag(llist(A)))             \
   19.10  \ |] ==> f(M) = g(M)";
   19.11  by (rtac LList_equalityI 1);
   19.12 @@ -461,7 +461,7 @@
   19.13  qed "Lmap_type";
   19.14  
   19.15  (*This type checking rule synthesises a sufficiently large set for f*)
   19.16 -Goal "M: llist(A) ==> Lmap f M: llist(f``A)";
   19.17 +Goal "M: llist(A) ==> Lmap f M: llist(f`A)";
   19.18  by (etac Lmap_type 1);
   19.19  by (etac imageI 1);
   19.20  qed "Lmap_type2";
   19.21 @@ -541,7 +541,7 @@
   19.22  
   19.23  (*strong co-induction: bisimulation and case analysis on one variable*)
   19.24  Goal "[| M: llist(A); N: llist(A) |] ==> Lappend M N: llist(A)";
   19.25 -by (res_inst_tac [("X", "(%u. Lappend u N)``llist(A)")] llist_coinduct 1);
   19.26 +by (res_inst_tac [("X", "(%u. Lappend u N)`llist(A)")] llist_coinduct 1);
   19.27  by (etac imageI 1);
   19.28  by (rtac image_subsetI 1);
   19.29  by (eres_inst_tac [("aa", "x")] llist.elim 1);
   19.30 @@ -623,21 +623,21 @@
   19.31  by (Fast_tac 1);
   19.32  qed "LListD_Fun_subset_Times_llist";
   19.33  
   19.34 -Goal "prod_fun Rep_LList Rep_LList `` r <= \
   19.35 +Goal "prod_fun Rep_LList Rep_LList ` r <= \
   19.36  \    (llist(range Leaf)) <*> (llist(range Leaf))";
   19.37  by (fast_tac (claset() delrules [image_subsetI]
   19.38  		       addIs [Rep_LList RS LListD]) 1);
   19.39  qed "subset_Times_llist";
   19.40  
   19.41  Goal "r <= (llist(range Leaf)) <*> (llist(range Leaf)) ==> \
   19.42 -\    prod_fun (Rep_LList o Abs_LList) (Rep_LList o Abs_LList) `` r <= r";
   19.43 +\    prod_fun (Rep_LList o Abs_LList) (Rep_LList o Abs_LList) ` r <= r";
   19.44  by Safe_tac;
   19.45  by (etac (subsetD RS SigmaE2) 1);
   19.46  by (assume_tac 1);
   19.47  by (asm_simp_tac (simpset() addsimps [LListI RS Abs_LList_inverse]) 1);
   19.48  qed "prod_fun_lemma";
   19.49  
   19.50 -Goal "prod_fun Rep_LList  Rep_LList `` range(%x. (x, x)) = \
   19.51 +Goal "prod_fun Rep_LList  Rep_LList ` range(%x. (x, x)) = \
   19.52  \    diag(llist(range Leaf))";
   19.53  by (rtac equalityI 1);
   19.54  by (Blast_tac 1);
   19.55 @@ -659,7 +659,7 @@
   19.56  Goalw [llistD_Fun_def]
   19.57      "[| (l1,l2) : r;  r <= llistD_Fun(r Un range(%x.(x,x))) |] ==> l1=l2";
   19.58  by (rtac (inj_Rep_LList RS injD) 1);
   19.59 -by (res_inst_tac [("r", "prod_fun Rep_LList Rep_LList ``r"),
   19.60 +by (res_inst_tac [("r", "prod_fun Rep_LList Rep_LList `r"),
   19.61                    ("A", "range(Leaf)")] 
   19.62          LList_equalityI 1);
   19.63  by (etac prod_fun_imageI 1);
    20.1 --- a/src/HOL/Induct/LList.thy	Tue Jan 09 15:29:17 2001 +0100
    20.2 +++ b/src/HOL/Induct/LList.thy	Tue Jan 09 15:32:27 2001 +0100
    20.3 @@ -20,7 +20,7 @@
    20.4    llistD_Fun_def
    20.5     "llistD_Fun(r) ==    
    20.6         {(LNil,LNil)}  Un        
    20.7 -       (UN x. (split(%l1 l2.(LCons(x,l1),LCons(x,l2))))``r)"
    20.8 +       (UN x. (split(%l1 l2.(LCons(x,l1),LCons(x,l2))))`r)"
    20.9  *)
   20.10  
   20.11  LList = Main + SList +
   20.12 @@ -87,9 +87,9 @@
   20.13  
   20.14    llistD_Fun :: "('a llist * 'a llist)set => ('a llist * 'a llist)set"
   20.15      "llistD_Fun(r) ==    
   20.16 -        prod_fun Abs_LList Abs_LList ``         
   20.17 +        prod_fun Abs_LList Abs_LList `         
   20.18                  LListD_Fun (diag(range Leaf))   
   20.19 -                            (prod_fun Rep_LList Rep_LList `` r)"
   20.20 +                            (prod_fun Rep_LList Rep_LList ` r)"
   20.21  
   20.22  
   20.23  
    21.1 --- a/src/HOL/Integ/Equiv.ML	Tue Jan 09 15:29:17 2001 +0100
    21.2 +++ b/src/HOL/Integ/Equiv.ML	Tue Jan 09 15:32:27 2001 +0100
    21.3 @@ -37,33 +37,33 @@
    21.4  
    21.5  (*Lemma for the next result*)
    21.6  Goalw [equiv_def,trans_def,sym_def]
    21.7 -     "[| equiv A r;  (a,b): r |] ==> r```{a} <= r```{b}";
    21.8 +     "[| equiv A r;  (a,b): r |] ==> r``{a} <= r``{b}";
    21.9  by (Blast_tac 1);
   21.10  qed "equiv_class_subset";
   21.11  
   21.12 -Goal "[| equiv A r;  (a,b): r |] ==> r```{a} = r```{b}";
   21.13 +Goal "[| equiv A r;  (a,b): r |] ==> r``{a} = r``{b}";
   21.14  by (REPEAT (ares_tac [equalityI, equiv_class_subset] 1));
   21.15  by (rewrite_goals_tac [equiv_def,sym_def]);
   21.16  by (Blast_tac 1);
   21.17  qed "equiv_class_eq";
   21.18  
   21.19 -Goalw [equiv_def,refl_def] "[| equiv A r;  a: A |] ==> a: r```{a}";
   21.20 +Goalw [equiv_def,refl_def] "[| equiv A r;  a: A |] ==> a: r``{a}";
   21.21  by (Blast_tac 1);
   21.22  qed "equiv_class_self";
   21.23  
   21.24  (*Lemma for the next result*)
   21.25  Goalw [equiv_def,refl_def]
   21.26 -    "[| equiv A r;  r```{b} <= r```{a};  b: A |] ==> (a,b): r";
   21.27 +    "[| equiv A r;  r``{b} <= r``{a};  b: A |] ==> (a,b): r";
   21.28  by (Blast_tac 1);
   21.29  qed "subset_equiv_class";
   21.30  
   21.31 -Goal "[| r```{a} = r```{b};  equiv A r;  b: A |] ==> (a,b): r";
   21.32 +Goal "[| r``{a} = r``{b};  equiv A r;  b: A |] ==> (a,b): r";
   21.33  by (REPEAT (ares_tac [equalityD2, subset_equiv_class] 1));
   21.34  qed "eq_equiv_class";
   21.35  
   21.36 -(*thus r```{a} = r```{b} as well*)
   21.37 +(*thus r``{a} = r``{b} as well*)
   21.38  Goalw [equiv_def,trans_def,sym_def]
   21.39 -     "[| equiv A r;  x: (r```{a} Int r```{b}) |] ==> (a,b): r";
   21.40 +     "[| equiv A r;  x: (r``{a} Int r``{b}) |] ==> (a,b): r";
   21.41  by (Blast_tac 1);
   21.42  qed "equiv_class_nondisjoint";
   21.43  
   21.44 @@ -71,12 +71,12 @@
   21.45  by (Blast_tac 1);
   21.46  qed "equiv_type";
   21.47  
   21.48 -Goal "equiv A r ==> ((x,y): r) = (r```{x} = r```{y} & x:A & y:A)";
   21.49 +Goal "equiv A r ==> ((x,y): r) = (r``{x} = r``{y} & x:A & y:A)";
   21.50  by (blast_tac (claset() addSIs [equiv_class_eq]
   21.51  	                addDs [eq_equiv_class, equiv_type]) 1);
   21.52  qed "equiv_class_eq_iff";
   21.53  
   21.54 -Goal "[| equiv A r;  x: A;  y: A |] ==> (r```{x} = r```{y}) = ((x,y): r)";
   21.55 +Goal "[| equiv A r;  x: A;  y: A |] ==> (r``{x} = r``{y}) = ((x,y): r)";
   21.56  by (blast_tac (claset() addSIs [equiv_class_eq]
   21.57  	                addDs [eq_equiv_class, equiv_type]) 1);
   21.58  qed "eq_equiv_class_iff";
   21.59 @@ -85,12 +85,12 @@
   21.60  
   21.61  (** Introduction/elimination rules -- needed? **)
   21.62  
   21.63 -Goalw [quotient_def] "x:A ==> r```{x}: A//r";
   21.64 +Goalw [quotient_def] "x:A ==> r``{x}: A//r";
   21.65  by (Blast_tac 1);
   21.66  qed "quotientI";
   21.67  
   21.68  val [major,minor] = Goalw [quotient_def]
   21.69 -    "[| X:(A//r);  !!x. [| X = r```{x};  x:A |] ==> P |]  \
   21.70 +    "[| X:(A//r);  !!x. [| X = r``{x};  x:A |] ==> P |]  \
   21.71  \    ==> P";
   21.72  by (resolve_tac [major RS UN_E] 1);
   21.73  by (rtac minor 1);
   21.74 @@ -127,13 +127,13 @@
   21.75  
   21.76  (*Conversion rule*)
   21.77  Goal "[| equiv A r;  congruent r b;  a: A |] \
   21.78 -\     ==> (UN x:r```{a}. b(x)) = b(a)";
   21.79 +\     ==> (UN x:r``{a}. b(x)) = b(a)";
   21.80  by (rtac (equiv_class_self RS UN_constant_eq) 1 THEN REPEAT (assume_tac 1));
   21.81  by (rewrite_goals_tac [equiv_def,congruent_def,sym_def]);
   21.82  by (blast_tac (claset() delrules [equalityI]) 1);
   21.83  qed "UN_equiv_class";
   21.84  
   21.85 -(*type checking of  UN x:r``{a}. b(x) *)
   21.86 +(*type checking of  UN x:r`{a}. b(x) *)
   21.87  val prems = Goalw [quotient_def]
   21.88      "[| equiv A r;  congruent r b;  X: A//r;     \
   21.89  \       !!x.  x : A ==> b(x) : B |]             \
   21.90 @@ -171,7 +171,7 @@
   21.91  
   21.92  Goalw [congruent_def]
   21.93      "[| equiv A r;  congruent2 r b;  a: A |] ==> \
   21.94 -\    congruent r (%x1. UN x2:r```{a}. b x1 x2)";
   21.95 +\    congruent r (%x1. UN x2:r``{a}. b x1 x2)";
   21.96  by (Clarify_tac 1);
   21.97  by (rtac (equiv_type RS subsetD RS SigmaE2) 1 THEN REPEAT (assume_tac 1));
   21.98  by (asm_simp_tac (simpset() addsimps [UN_equiv_class,
   21.99 @@ -181,7 +181,7 @@
  21.100  qed "congruent2_implies_congruent_UN";
  21.101  
  21.102  Goal "[| equiv A r;  congruent2 r b;  a1: A;  a2: A |]  \
  21.103 -\    ==> (UN x1:r```{a1}. UN x2:r```{a2}. b x1 x2) = b a1 a2";
  21.104 +\    ==> (UN x1:r``{a1}. UN x2:r``{a2}. b x1 x2) = b a1 a2";
  21.105  by (asm_simp_tac (simpset() addsimps [UN_equiv_class,
  21.106                                       congruent2_implies_congruent,
  21.107                                       congruent2_implies_congruent_UN]) 1);
    22.1 --- a/src/HOL/Integ/Equiv.thy	Tue Jan 09 15:29:17 2001 +0100
    22.2 +++ b/src/HOL/Integ/Equiv.thy	Tue Jan 09 15:32:27 2001 +0100
    22.3 @@ -12,7 +12,7 @@
    22.4      "equiv A r == refl A r & sym(r) & trans(r)"
    22.5  
    22.6    quotient :: "['a set, ('a*'a) set] => 'a set set"  (infixl "'/'/" 90) 
    22.7 -    "A//r == UN x:A. {r```{x}}"      (*set of equiv classes*)
    22.8 +    "A//r == UN x:A. {r``{x}}"      (*set of equiv classes*)
    22.9  
   22.10    congruent  :: "[('a*'a) set, 'a=>'b] => bool"
   22.11      "congruent r b  == ALL y z. (y,z):r --> b(y)=b(z)"
    23.1 --- a/src/HOL/Integ/IntDef.ML	Tue Jan 09 15:29:17 2001 +0100
    23.2 +++ b/src/HOL/Integ/IntDef.ML	Tue Jan 09 15:32:27 2001 +0100
    23.3 @@ -23,7 +23,7 @@
    23.4  	  [equiv_intrel, UNIV_I, UNIV_I] MRS eq_equiv_class_iff);
    23.5  
    23.6  Goalw [Integ_def,intrel_def,quotient_def]
    23.7 -     "intrel```{(x,y)}:Integ";
    23.8 +     "intrel``{(x,y)}:Integ";
    23.9  by (Fast_tac 1);
   23.10  qed "intrel_in_integ";
   23.11  
   23.12 @@ -58,18 +58,18 @@
   23.13  (**** zminus: unary negation on Integ ****)
   23.14  
   23.15  Goalw [congruent_def, intrel_def]
   23.16 -     "congruent intrel (%(x,y). intrel```{(y,x)})";
   23.17 +     "congruent intrel (%(x,y). intrel``{(y,x)})";
   23.18  by (auto_tac (claset(), simpset() addsimps add_ac));
   23.19  qed "zminus_congruent";
   23.20  
   23.21  Goalw [zminus_def]
   23.22 -      "- Abs_Integ(intrel```{(x,y)}) = Abs_Integ(intrel ``` {(y,x)})";
   23.23 +      "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})";
   23.24  by (simp_tac (simpset() addsimps 
   23.25  	      [equiv_intrel RS UN_equiv_class, zminus_congruent]) 1);
   23.26  qed "zminus";
   23.27  
   23.28  (*Every integer can be written in the form Abs_Integ(...) *)
   23.29 -val [prem] = Goal "(!!x y. z = Abs_Integ(intrel```{(x,y)}) ==> P) ==> P";
   23.30 +val [prem] = Goal "(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P";
   23.31  by (res_inst_tac [("x1","z")] 
   23.32      (rewrite_rule [Integ_def] Rep_Integ RS quotientE) 1);
   23.33  by (dres_inst_tac [("f","Abs_Integ")] arg_cong 1);
   23.34 @@ -114,8 +114,8 @@
   23.35  (**** zadd: addition on Integ ****)
   23.36  
   23.37  Goalw [zadd_def]
   23.38 -  "Abs_Integ(intrel```{(x1,y1)}) + Abs_Integ(intrel```{(x2,y2)}) = \
   23.39 -\  Abs_Integ(intrel```{(x1+x2, y1+y2)})";
   23.40 +  "Abs_Integ(intrel``{(x1,y1)}) + Abs_Integ(intrel``{(x2,y2)}) = \
   23.41 +\  Abs_Integ(intrel``{(x1+x2, y1+y2)})";
   23.42  by (asm_simp_tac (simpset() addsimps [UN_UN_split_split_eq]) 1);
   23.43  by (stac (equiv_intrel RS UN_equiv_class2) 1);
   23.44  by (auto_tac (claset(), simpset() addsimps [congruent2_def]));
   23.45 @@ -232,7 +232,7 @@
   23.46  (*Congruence property for multiplication*)
   23.47  Goal "congruent2 intrel \
   23.48  \       (%p1 p2. (%(x1,y1). (%(x2,y2).   \
   23.49 -\                   intrel```{(x1*x2 + y1*y2, x1*y2 + y1*x2)}) p2) p1)";
   23.50 +\                   intrel``{(x1*x2 + y1*y2, x1*y2 + y1*x2)}) p2) p1)";
   23.51  by (rtac (equiv_intrel RS congruent2_commuteI) 1);
   23.52  by (pair_tac "w" 2);
   23.53  by (ALLGOALS Clarify_tac);
   23.54 @@ -249,8 +249,8 @@
   23.55  qed "zmult_congruent2";
   23.56  
   23.57  Goalw [zmult_def]
   23.58 -   "Abs_Integ((intrel```{(x1,y1)})) * Abs_Integ((intrel```{(x2,y2)})) =   \
   23.59 -\   Abs_Integ(intrel ``` {(x1*x2 + y1*y2, x1*y2 + y1*x2)})";
   23.60 +   "Abs_Integ((intrel``{(x1,y1)})) * Abs_Integ((intrel``{(x2,y2)})) =   \
   23.61 +\   Abs_Integ(intrel `` {(x1*x2 + y1*y2, x1*y2 + y1*x2)})";
   23.62  by (asm_simp_tac
   23.63      (simpset() addsimps [UN_UN_split_split_eq, zmult_congruent2,
   23.64  			 equiv_intrel RS UN_equiv_class2]) 1);
    24.1 --- a/src/HOL/Integ/IntDef.thy	Tue Jan 09 15:29:17 2001 +0100
    24.2 +++ b/src/HOL/Integ/IntDef.thy	Tue Jan 09 15:32:27 2001 +0100
    24.3 @@ -19,12 +19,12 @@
    24.4  
    24.5  defs
    24.6    zminus_def
    24.7 -    "- Z == Abs_Integ(UN (x,y):Rep_Integ(Z). intrel```{(y,x)})"
    24.8 +    "- Z == Abs_Integ(UN (x,y):Rep_Integ(Z). intrel``{(y,x)})"
    24.9  
   24.10  constdefs
   24.11  
   24.12    int :: nat => int
   24.13 -  "int m == Abs_Integ(intrel ``` {(m,0)})"
   24.14 +  "int m == Abs_Integ(intrel `` {(m,0)})"
   24.15  
   24.16    neg   :: int => bool
   24.17    "neg(Z) == EX x y. x<y & (x,y::nat):Rep_Integ(Z)"
   24.18 @@ -40,7 +40,7 @@
   24.19    zadd_def
   24.20     "z + w == 
   24.21         Abs_Integ(UN (x1,y1):Rep_Integ(z). UN (x2,y2):Rep_Integ(w).   
   24.22 -		 intrel```{(x1+x2, y1+y2)})"
   24.23 +		 intrel``{(x1+x2, y1+y2)})"
   24.24  
   24.25    zdiff_def "z - (w::int) == z + (-w)"
   24.26  
   24.27 @@ -51,6 +51,6 @@
   24.28    zmult_def
   24.29     "z * w == 
   24.30         Abs_Integ(UN (x1,y1):Rep_Integ(z). UN (x2,y2):Rep_Integ(w).   
   24.31 -		 intrel```{(x1*x2 + y1*y2, x1*y2 + y1*x2)})"
   24.32 +		 intrel``{(x1*x2 + y1*y2, x1*y2 + y1*x2)})"
   24.33  
   24.34  end
    25.1 --- a/src/HOL/Integ/int_arith1.ML	Tue Jan 09 15:29:17 2001 +0100
    25.2 +++ b/src/HOL/Integ/int_arith1.ML	Tue Jan 09 15:32:27 2001 +0100
    25.3 @@ -436,7 +436,7 @@
    25.4      simpset = simpset addsimps add_rules
    25.5                        addsimprocs simprocs
    25.6                        addcongs [if_weak_cong]}),
    25.7 -  arith_inj_const ("IntDef.int", HOLogic.natT --> Type("IntDef.int",[])),
    25.8 +  arith_inj_const ("IntDef.int", HOLogic.natT --> HOLogic.intT),
    25.9    arith_discrete ("IntDef.int", true)];
   25.10  
   25.11  end;
    26.1 --- a/src/HOL/Lattice/Bounds.thy	Tue Jan 09 15:29:17 2001 +0100
    26.2 +++ b/src/HOL/Lattice/Bounds.thy	Tue Jan 09 15:32:27 2001 +0100
    26.3 @@ -100,11 +100,11 @@
    26.4    by (simp add: is_inf_def is_sup_def dual_all [symmetric] dual_leq)
    26.5  
    26.6  theorem dual_Inf [iff?]:
    26.7 -    "is_Inf (dual `` A) (dual sup) = is_Sup A sup"
    26.8 +    "is_Inf (dual ` A) (dual sup) = is_Sup A sup"
    26.9    by (simp add: is_Inf_def is_Sup_def dual_all [symmetric] dual_leq)
   26.10  
   26.11  theorem dual_Sup [iff?]:
   26.12 -    "is_Sup (dual `` A) (dual inf) = is_Inf A inf"
   26.13 +    "is_Sup (dual ` A) (dual inf) = is_Inf A inf"
   26.14    by (simp add: is_Inf_def is_Sup_def dual_all [symmetric] dual_leq)
   26.15  
   26.16  
   26.17 @@ -169,8 +169,8 @@
   26.18    assume sup: "is_Sup A sup" and sup': "is_Sup A sup'"
   26.19    have "dual sup = dual sup'"
   26.20    proof (rule is_Inf_uniq)
   26.21 -    from sup show "is_Inf (dual `` A) (dual sup)" ..
   26.22 -    from sup' show "is_Inf (dual `` A) (dual sup')" ..
   26.23 +    from sup show "is_Inf (dual ` A) (dual sup)" ..
   26.24 +    from sup' show "is_Inf (dual ` A) (dual sup')" ..
   26.25    qed
   26.26    thus "sup = sup'" ..
   26.27  qed
   26.28 @@ -268,9 +268,9 @@
   26.29  
   26.30  theorem is_Sup_binary: "is_Sup {x, y} sup = is_sup x y sup"
   26.31  proof -
   26.32 -  have "is_Sup {x, y} sup = is_Inf (dual `` {x, y}) (dual sup)"
   26.33 +  have "is_Sup {x, y} sup = is_Inf (dual ` {x, y}) (dual sup)"
   26.34      by (simp only: dual_Inf)
   26.35 -  also have "dual `` {x, y} = {dual x, dual y}"
   26.36 +  also have "dual ` {x, y} = {dual x, dual y}"
   26.37      by simp
   26.38    also have "is_Inf \<dots> (dual sup) = is_inf (dual x) (dual y) (dual sup)"
   26.39      by (rule is_Inf_binary)
   26.40 @@ -312,12 +312,12 @@
   26.41  theorem Sup_Inf: "is_Sup {b. \<forall>a \<in> A. b \<sqsubseteq> a} inf \<Longrightarrow> is_Inf A inf"
   26.42  proof -
   26.43    assume "is_Sup {b. \<forall>a \<in> A. b \<sqsubseteq> a} inf"
   26.44 -  hence "is_Inf (dual `` {b. \<forall>a \<in> A. dual a \<sqsubseteq> dual b}) (dual inf)"
   26.45 +  hence "is_Inf (dual ` {b. \<forall>a \<in> A. dual a \<sqsubseteq> dual b}) (dual inf)"
   26.46      by (simp only: dual_Inf dual_leq)
   26.47 -  also have "dual `` {b. \<forall>a \<in> A. dual a \<sqsubseteq> dual b} = {b'. \<forall>a' \<in> dual `` A. a' \<sqsubseteq> b'}"
   26.48 +  also have "dual ` {b. \<forall>a \<in> A. dual a \<sqsubseteq> dual b} = {b'. \<forall>a' \<in> dual ` A. a' \<sqsubseteq> b'}"
   26.49      by (auto iff: dual_ball dual_Collect)  (* FIXME !? *)
   26.50    finally have "is_Inf \<dots> (dual inf)" .
   26.51 -  hence "is_Sup (dual `` A) (dual inf)"
   26.52 +  hence "is_Sup (dual ` A) (dual inf)"
   26.53      by (rule Inf_Sup)
   26.54    thus ?thesis ..
   26.55  qed
    27.1 --- a/src/HOL/Lattice/CompleteLattice.thy	Tue Jan 09 15:29:17 2001 +0100
    27.2 +++ b/src/HOL/Lattice/CompleteLattice.thy	Tue Jan 09 15:32:27 2001 +0100
    27.3 @@ -201,8 +201,8 @@
    27.4    fix A' :: "'a::complete_lattice dual set"
    27.5    show "\<exists>inf'. is_Inf A' inf'"
    27.6    proof -
    27.7 -    have "\<exists>sup. is_Sup (undual `` A') sup" by (rule ex_Sup)
    27.8 -    hence "\<exists>sup. is_Inf (dual `` undual `` A') (dual sup)" by (simp only: dual_Inf)
    27.9 +    have "\<exists>sup. is_Sup (undual ` A') sup" by (rule ex_Sup)
   27.10 +    hence "\<exists>sup. is_Inf (dual ` undual ` A') (dual sup)" by (simp only: dual_Inf)
   27.11      thus ?thesis by (simp add: dual_ex [symmetric] image_compose [symmetric])
   27.12    qed
   27.13  qed
   27.14 @@ -212,17 +212,17 @@
   27.15    other.
   27.16  *}
   27.17  
   27.18 -theorem dual_Meet [intro?]: "dual (\<Sqinter>A) = \<Squnion>(dual `` A)"
   27.19 +theorem dual_Meet [intro?]: "dual (\<Sqinter>A) = \<Squnion>(dual ` A)"
   27.20  proof -
   27.21 -  from is_Inf_Meet have "is_Sup (dual `` A) (dual (\<Sqinter>A))" ..
   27.22 -  hence "\<Squnion>(dual `` A) = dual (\<Sqinter>A)" ..
   27.23 +  from is_Inf_Meet have "is_Sup (dual ` A) (dual (\<Sqinter>A))" ..
   27.24 +  hence "\<Squnion>(dual ` A) = dual (\<Sqinter>A)" ..
   27.25    thus ?thesis ..
   27.26  qed
   27.27  
   27.28 -theorem dual_Join [intro?]: "dual (\<Squnion>A) = \<Sqinter>(dual `` A)"
   27.29 +theorem dual_Join [intro?]: "dual (\<Squnion>A) = \<Sqinter>(dual ` A)"
   27.30  proof -
   27.31 -  from is_Sup_Join have "is_Inf (dual `` A) (dual (\<Squnion>A))" ..
   27.32 -  hence "\<Sqinter>(dual `` A) = dual (\<Squnion>A)" ..
   27.33 +  from is_Sup_Join have "is_Inf (dual ` A) (dual (\<Squnion>A))" ..
   27.34 +  hence "\<Sqinter>(dual ` A) = dual (\<Squnion>A)" ..
   27.35    thus ?thesis ..
   27.36  qed
   27.37  
   27.38 @@ -306,8 +306,8 @@
   27.39  theorem Join_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B"
   27.40  proof -
   27.41    assume "A \<subseteq> B"
   27.42 -  hence "dual `` A \<subseteq> dual `` B" by blast
   27.43 -  hence "\<Sqinter>(dual `` B) \<sqsubseteq> \<Sqinter>(dual `` A)" by (rule Meet_subset_antimono)
   27.44 +  hence "dual ` A \<subseteq> dual ` B" by blast
   27.45 +  hence "\<Sqinter>(dual ` B) \<sqsubseteq> \<Sqinter>(dual ` A)" by (rule Meet_subset_antimono)
   27.46    hence "dual (\<Squnion>B) \<sqsubseteq> dual (\<Squnion>A)" by (simp only: dual_Join)
   27.47    thus ?thesis by (simp only: dual_leq)
   27.48  qed
   27.49 @@ -352,9 +352,9 @@
   27.50  
   27.51  theorem Join_Un: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
   27.52  proof -
   27.53 -  have "dual (\<Squnion>(A \<union> B)) = \<Sqinter>(dual `` A \<union> dual `` B)"
   27.54 +  have "dual (\<Squnion>(A \<union> B)) = \<Sqinter>(dual ` A \<union> dual ` B)"
   27.55      by (simp only: dual_Join image_Un)
   27.56 -  also have "\<dots> = \<Sqinter>(dual `` A) \<sqinter> \<Sqinter>(dual `` B)"
   27.57 +  also have "\<dots> = \<Sqinter>(dual ` A) \<sqinter> \<Sqinter>(dual ` B)"
   27.58      by (rule Meet_Un)
   27.59    also have "\<dots> = dual (\<Squnion>A \<squnion> \<Squnion>B)"
   27.60      by (simp only: dual_join dual_Join)
    28.1 --- a/src/HOL/Lattice/Orders.thy	Tue Jan 09 15:29:17 2001 +0100
    28.2 +++ b/src/HOL/Lattice/Orders.thy	Tue Jan 09 15:32:27 2001 +0100
    28.3 @@ -91,15 +91,15 @@
    28.4  lemma dual_equality [iff?]: "(dual x = dual y) = (x = y)"
    28.5    by simp
    28.6  
    28.7 -lemma dual_ball [iff?]: "(\<forall>x \<in> A. P (dual x)) = (\<forall>x' \<in> dual `` A. P x')"
    28.8 +lemma dual_ball [iff?]: "(\<forall>x \<in> A. P (dual x)) = (\<forall>x' \<in> dual ` A. P x')"
    28.9  proof
   28.10    assume a: "\<forall>x \<in> A. P (dual x)"
   28.11 -  show "\<forall>x' \<in> dual `` A. P x'"
   28.12 +  show "\<forall>x' \<in> dual ` A. P x'"
   28.13    proof
   28.14 -    fix x' assume x': "x' \<in> dual `` A"
   28.15 +    fix x' assume x': "x' \<in> dual ` A"
   28.16      have "undual x' \<in> A"
   28.17      proof -
   28.18 -      from x' have "undual x' \<in> undual `` dual `` A" by simp
   28.19 +      from x' have "undual x' \<in> undual ` dual ` A" by simp
   28.20        thus "undual x' \<in> A" by (simp add: image_compose [symmetric])
   28.21      qed
   28.22      with a have "P (dual (undual x'))" ..
   28.23 @@ -107,16 +107,16 @@
   28.24      finally show "P x'" .
   28.25    qed
   28.26  next
   28.27 -  assume a: "\<forall>x' \<in> dual `` A. P x'"
   28.28 +  assume a: "\<forall>x' \<in> dual ` A. P x'"
   28.29    show "\<forall>x \<in> A. P (dual x)"
   28.30    proof
   28.31      fix x assume "x \<in> A"
   28.32 -    hence "dual x \<in> dual `` A" by simp
   28.33 +    hence "dual x \<in> dual ` A" by simp
   28.34      with a show "P (dual x)" ..
   28.35    qed
   28.36  qed
   28.37  
   28.38 -lemma range_dual [simp]: "dual `` UNIV = UNIV"
   28.39 +lemma range_dual [simp]: "dual ` UNIV = UNIV"
   28.40  proof (rule surj_range)
   28.41    have "\<And>x'. dual (undual x') = x'" by simp
   28.42    thus "surj dual" by (rule surjI)
   28.43 @@ -124,7 +124,7 @@
   28.44  
   28.45  lemma dual_all [iff?]: "(\<forall>x. P (dual x)) = (\<forall>x'. P x')"
   28.46  proof -
   28.47 -  have "(\<forall>x \<in> UNIV. P (dual x)) = (\<forall>x' \<in> dual `` UNIV. P x')"
   28.48 +  have "(\<forall>x \<in> UNIV. P (dual x)) = (\<forall>x' \<in> dual ` UNIV. P x')"
   28.49      by (rule dual_ball)
   28.50    thus ?thesis by simp
   28.51  qed
    29.1 --- a/src/HOL/Lex/Automata.ML	Tue Jan 09 15:29:17 2001 +0100
    29.2 +++ b/src/HOL/Lex/Automata.ML	Tue Jan 09 15:32:27 2001 +0100
    29.3 @@ -7,7 +7,7 @@
    29.4  (*** Equivalence of NA and DA ***)
    29.5  
    29.6  Goalw [na2da_def]
    29.7 - "!Q. DA.delta (na2da A) w Q = Union(NA.delta A w `` Q)";
    29.8 + "!Q. DA.delta (na2da A) w Q = Union(NA.delta A w ` Q)";
    29.9  by (induct_tac "w" 1);
   29.10   by Auto_tac;
   29.11  qed_spec_mp "DA_delta_is_lift_NA_delta";
   29.12 @@ -21,7 +21,7 @@
   29.13  (*** Direct equivalence of NAe and DA ***)
   29.14  
   29.15  Goalw [nae2da_def]
   29.16 - "!Q. (eps A)^* ``` (DA.delta (nae2da A) w Q) = steps A w ``` Q";
   29.17 + "!Q. (eps A)^* `` (DA.delta (nae2da A) w Q) = steps A w `` Q";
   29.18  by (induct_tac "w" 1);
   29.19   by (Simp_tac 1);
   29.20  by (asm_full_simp_tac (simpset() addsimps [step_def]) 1);
   29.21 @@ -29,7 +29,7 @@
   29.22  qed_spec_mp "espclosure_DA_delta_is_steps";
   29.23  
   29.24  Goalw [nae2da_def]
   29.25 - "fin (nae2da A) Q = (? q : (eps A)^* ``` Q. fin A q)";
   29.26 + "fin (nae2da A) Q = (? q : (eps A)^* `` Q. fin A q)";
   29.27  by (Simp_tac 1);
   29.28  val lemma = result();
   29.29  
    30.1 --- a/src/HOL/Lex/Automata.thy	Tue Jan 09 15:29:17 2001 +0100
    30.2 +++ b/src/HOL/Lex/Automata.thy	Tue Jan 09 15:32:27 2001 +0100
    30.3 @@ -10,11 +10,11 @@
    30.4  
    30.5  constdefs
    30.6   na2da :: ('a,'s)na => ('a,'s set)da
    30.7 -"na2da A == ({start A}, %a Q. Union(next A a `` Q), %Q. ? q:Q. fin A q)"
    30.8 +"na2da A == ({start A}, %a Q. Union(next A a ` Q), %Q. ? q:Q. fin A q)"
    30.9  
   30.10   nae2da :: ('a,'s)nae => ('a,'s set)da
   30.11  "nae2da A == ({start A},
   30.12 -              %a Q. Union(next A (Some a) `` ((eps A)^* ``` Q)),
   30.13 -              %Q. ? p: (eps A)^* ``` Q. fin A p)"
   30.14 +              %a Q. Union(next A (Some a) ` ((eps A)^* `` Q)),
   30.15 +              %Q. ? p: (eps A)^* `` Q. fin A p)"
   30.16  
   30.17  end
    31.1 --- a/src/HOL/Lex/NA.thy	Tue Jan 09 15:29:17 2001 +0100
    31.2 +++ b/src/HOL/Lex/NA.thy	Tue Jan 09 15:32:27 2001 +0100
    31.3 @@ -13,7 +13,7 @@
    31.4  consts delta :: "('a,'s)na => 'a list => 's => 's set"
    31.5  primrec
    31.6  "delta A []    p = {p}"
    31.7 -"delta A (a#w) p = Union(delta A w `` next A a p)"
    31.8 +"delta A (a#w) p = Union(delta A w ` next A a p)"
    31.9  
   31.10  constdefs
   31.11   accepts ::   ('a,'s)na => 'a list => bool
    32.1 --- a/src/HOL/Lex/NAe.ML	Tue Jan 09 15:29:17 2001 +0100
    32.2 +++ b/src/HOL/Lex/NAe.ML	Tue Jan 09 15:32:27 2001 +0100
    32.3 @@ -39,7 +39,7 @@
    32.4  AddIffs [in_steps_append];
    32.5  
    32.6  (* Equivalence of steps and delta
    32.7 -(* Use "(? x : f `` A. P x) = (? a:A. P(f x))" ?? *)
    32.8 +(* Use "(? x : f ` A. P x) = (? a:A. P(f x))" ?? *)
    32.9  Goal "!p. (p,q) : steps A w = (q : delta A w p)";
   32.10  by (induct_tac "w" 1);
   32.11   by (Simp_tac 1);
    33.1 --- a/src/HOL/Lex/NAe.thy	Tue Jan 09 15:29:17 2001 +0100
    33.2 +++ b/src/HOL/Lex/NAe.thy	Tue Jan 09 15:32:27 2001 +0100
    33.3 @@ -25,7 +25,7 @@
    33.4  (* not really used:
    33.5  consts delta :: "('a,'s)nae => 'a list => 's => 's set"
    33.6  primrec
    33.7 -"delta A [] s = (eps A)^* ``` {s}"
    33.8 -"delta A (a#w) s = lift(delta A w) (lift(next A (Some a)) ((eps A)^* ``` {s}))"
    33.9 +"delta A [] s = (eps A)^* `` {s}"
   33.10 +"delta A (a#w) s = lift(delta A w) (lift(next A (Some a)) ((eps A)^* `` {s}))"
   33.11  *)
   33.12  end
    34.1 --- a/src/HOL/Lex/RegExp2NA.thy	Tue Jan 09 15:29:17 2001 +0100
    34.2 +++ b/src/HOL/Lex/RegExp2NA.thy	Tue Jan 09 15:32:27 2001 +0100
    34.3 @@ -12,7 +12,7 @@
    34.4  types 'a bitsNA = ('a,bool list)na
    34.5  
    34.6  syntax "##" :: 'a => 'a list set => 'a list set (infixr 65)
    34.7 -translations "x ## S" == "Cons x `` S"
    34.8 +translations "x ## S" == "Cons x ` S"
    34.9  
   34.10  constdefs
   34.11   atom  :: 'a => 'a bitsNA
    35.1 --- a/src/HOL/Lex/RegExp2NAe.thy	Tue Jan 09 15:29:17 2001 +0100
    35.2 +++ b/src/HOL/Lex/RegExp2NAe.thy	Tue Jan 09 15:32:27 2001 +0100
    35.3 @@ -12,7 +12,7 @@
    35.4  types 'a bitsNAe = ('a,bool list)nae
    35.5  
    35.6  syntax "##" :: 'a => 'a list set => 'a list set (infixr 65)
    35.7 -translations "x ## S" == "Cons x `` S"
    35.8 +translations "x ## S" == "Cons x ` S"
    35.9  
   35.10  constdefs
   35.11   atom  :: 'a => 'a bitsNAe
    36.1 --- a/src/HOL/MicroJava/BV/JVM.thy	Tue Jan 09 15:29:17 2001 +0100
    36.2 +++ b/src/HOL/MicroJava/BV/JVM.thy	Tue Jan 09 15:32:27 2001 +0100
    36.3 @@ -187,7 +187,7 @@
    36.4    have "set pTs \<subseteq> types G"
    36.5      by auto
    36.6  
    36.7 -  hence "OK `` set pTs \<subseteq> err (types G)"
    36.8 +  hence "OK ` set pTs \<subseteq> err (types G)"
    36.9      by auto
   36.10  
   36.11    with instrs maxr isclass 
   36.12 @@ -329,7 +329,7 @@
   36.13    have "set pTs \<subseteq> types G"
   36.14      by auto
   36.15  
   36.16 -  hence "OK `` set pTs \<subseteq> err (types G)"
   36.17 +  hence "OK ` set pTs \<subseteq> err (types G)"
   36.18      by auto
   36.19  
   36.20    with instrs isclass 
    37.1 --- a/src/HOL/MicroJava/J/JBasis.ML	Tue Jan 09 15:29:17 2001 +0100
    37.2 +++ b/src/HOL/MicroJava/J/JBasis.ML	Tue Jan 09 15:32:27 2001 +0100
    37.3 @@ -25,8 +25,8 @@
    37.4  
    37.5  section "unique";
    37.6  
    37.7 -Goal "(x, y) : set l --> x : fst `` set l";
    37.8 -by (induct_tac "l" 1);
    37.9 +Goal "(x, y) : set xys --> x : fst ` set xys";
   37.10 +by (induct_tac "xys" 1);
   37.11  by  Auto_tac;
   37.12  qed_spec_mp "fst_in_set_lemma";
   37.13  
    38.1 --- a/src/HOL/NumberTheory/BijectionRel.ML	Tue Jan 09 15:29:17 2001 +0100
    38.2 +++ b/src/HOL/NumberTheory/BijectionRel.ML	Tue Jan 09 15:32:27 2001 +0100
    38.3 @@ -33,12 +33,12 @@
    38.4  val lemma_induct = result();
    38.5  
    38.6  Goalw [inj_on_def] 
    38.7 -      "[| A <= B; a ~: A ; a : B; inj_on f B |] ==> (f a) ~: f``A";
    38.8 +      "[| A <= B; a ~: A ; a : B; inj_on f B |] ==> (f a) ~: f`A";
    38.9  by Auto_tac;
   38.10  val lemma = result();
   38.11  
   38.12  Goal "[| ALL a. a:A --> P a (f a); inj_on f A; finite A; F <= A |] \
   38.13 -\    ==> (F,f``F) : bijR P";
   38.14 +\    ==> (F,f`F) : bijR P";
   38.15  by (res_inst_tac [("F","F"),("A","A")] lemma_induct 1);
   38.16  by (rtac finite_subset 1);
   38.17  by Auto_tac;
   38.18 @@ -48,7 +48,7 @@
   38.19  val lemma = result();
   38.20  
   38.21  Goal "[| ALL a. a:A --> P a (f a); inj_on f A; finite A |] \
   38.22 -\    ==> (A,f``A) : bijR P";
   38.23 +\    ==> (A,f`A) : bijR P";
   38.24  by (rtac lemma 1);
   38.25  by Auto_tac;
   38.26  qed "inj_func_bijR";
    39.1 --- a/src/HOL/NumberTheory/EulerFermat.ML	Tue Jan 09 15:29:17 2001 +0100
    39.2 +++ b/src/HOL/NumberTheory/EulerFermat.ML	Tue Jan 09 15:32:27 2001 +0100
    39.3 @@ -121,7 +121,7 @@
    39.4  by Auto_tac;
    39.5  qed_spec_mp "RRset_gcd";
    39.6  
    39.7 -Goal "[| A : RsetR m;  #0<m; zgcd(x, m) = #1 |] ==> (%a. a*x)``A : RsetR m";
    39.8 +Goal "[| A : RsetR m;  #0<m; zgcd(x, m) = #1 |] ==> (%a. a*x)`A : RsetR m";
    39.9  by (etac RsetR.induct 1);
   39.10  by (ALLGOALS Simp_tac);
   39.11  by (rtac RsetR.insert 1);
   39.12 @@ -196,7 +196,7 @@
   39.13  by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zcong_sym])));
   39.14  qed "RRset2norRR_inj";
   39.15  
   39.16 -Goal "[| #1<m; is_RRset A m |] ==> (RRset2norRR A m)``A = (norRRset m)";
   39.17 +Goal "[| #1<m; is_RRset A m |] ==> (RRset2norRR A m)`A = (norRRset m)";
   39.18  by (rtac card_seteq 1);
   39.19  by (stac card_image 3);
   39.20  by (rtac RRset2norRR_inj 4);
   39.21 @@ -207,11 +207,11 @@
   39.22  by (auto_tac (claset(),simpset() addsimps [RsetR_fin,Bnor_fin]));
   39.23  qed "RRset2norRR_eq_norR";
   39.24  
   39.25 -Goalw [inj_on_def] "[| a ~: A ; inj f |] ==> (f a) ~: f``A";
   39.26 +Goalw [inj_on_def] "[| a ~: A ; inj f |] ==> (f a) ~: f`A";
   39.27  by Auto_tac;
   39.28  val lemma = result();
   39.29  
   39.30 -Goal "x~=#0 ==> a<m --> setprod ((%a. a*x) `` BnorRset(a,m)) = \
   39.31 +Goal "x~=#0 ==> a<m --> setprod ((%a. a*x) ` BnorRset(a,m)) = \
   39.32  \     setprod (BnorRset(a,m)) * x^card(BnorRset(a,m))";
   39.33  by (induct_thm_tac BnorRset_induct "a m" 1);
   39.34  by (stac BnorRset_eq 2);
    40.1 --- a/src/HOL/NumberTheory/EulerFermat.thy	Tue Jan 09 15:29:17 2001 +0100
    40.2 +++ b/src/HOL/NumberTheory/EulerFermat.thy	Tue Jan 09 15:32:27 2001 +0100
    40.3 @@ -30,7 +30,7 @@
    40.4  defs
    40.5    norRRset_def "norRRset m   == BnorRset (m-#1,m)"
    40.6  
    40.7 -  noXRRset_def "noXRRset m x == (%a. a*x)``(norRRset m)"
    40.8 +  noXRRset_def "noXRRset m x == (%a. a*x)`(norRRset m)"
    40.9  
   40.10    phi_def      "phi m == card (norRRset m)"
   40.11  
    41.1 --- a/src/HOL/NumberTheory/WilsonBij.ML	Tue Jan 09 15:29:17 2001 +0100
    41.2 +++ b/src/HOL/NumberTheory/WilsonBij.ML	Tue Jan 09 15:32:27 2001 +0100
    41.3 @@ -127,7 +127,7 @@
    41.4  by (auto_tac (claset() addIs [d22set_g_1,d22set_le,l1,l2,l3,l4],simpset()));
    41.5  qed "inv_inj";
    41.6  
    41.7 -Goal "p : zprime ==> (inv p)``(d22set (p-#2)) = (d22set (p-#2))";
    41.8 +Goal "p : zprime ==> (inv p)`(d22set (p-#2)) = (d22set (p-#2))";
    41.9  by (rtac endo_inj_surj 1);
   41.10  by (rtac d22set_fin 1);
   41.11  by (etac inv_inj 2);
   41.12 @@ -141,7 +141,7 @@
   41.13  
   41.14  Goalw [reciR_def] "p:zprime \
   41.15  \    ==> (d22set(p-#2),d22set(p-#2)) : (bijR (reciR p))";
   41.16 -by (res_inst_tac [("s","(d22set(p-#2),(inv p)``(d22set(p-#2)))")] subst 1);
   41.17 +by (res_inst_tac [("s","(d22set(p-#2),(inv p)`(d22set(p-#2)))")] subst 1);
   41.18  by (asm_simp_tac (simpset() addsimps [inv_d22set_d22set]) 1);
   41.19  by (rtac inj_func_bijR 1);
   41.20  by (rtac d22set_fin 3);
    42.1 --- a/src/HOL/Real/PNat.ML	Tue Jan 09 15:29:17 2001 +0100
    42.2 +++ b/src/HOL/Real/PNat.ML	Tue Jan 09 15:32:27 2001 +0100
    42.3 @@ -6,7 +6,7 @@
    42.4  The positive naturals -- proofs mainly as in theory Nat.
    42.5  *)
    42.6  
    42.7 -Goal "mono(%X. {1} Un (Suc``X))";
    42.8 +Goal "mono(%X. {1} Un Suc`X)";
    42.9  by (REPEAT (ares_tac [monoI, subset_refl, image_mono, Un_mono] 1));
   42.10  qed "pnat_fun_mono";
   42.11  
    43.1 --- a/src/HOL/Real/PNat.thy	Tue Jan 09 15:29:17 2001 +0100
    43.2 +++ b/src/HOL/Real/PNat.thy	Tue Jan 09 15:32:27 2001 +0100
    43.3 @@ -9,7 +9,7 @@
    43.4  PNat = Main +
    43.5  
    43.6  typedef
    43.7 -  pnat = "lfp(%X. {1} Un (Suc``X))"   (lfp_def)
    43.8 +  pnat = "lfp(%X. {1} Un Suc`X)"   (lfp_def)
    43.9  
   43.10  instance
   43.11     pnat :: {ord, plus, times}
    44.1 --- a/src/HOL/Real/PRat.ML	Tue Jan 09 15:29:17 2001 +0100
    44.2 +++ b/src/HOL/Real/PRat.ML	Tue Jan 09 15:32:27 2001 +0100
    44.3 @@ -61,7 +61,7 @@
    44.4  
    44.5  bind_thm ("equiv_ratrel_iff", [equiv_ratrel, UNIV_I, UNIV_I] MRS eq_equiv_class_iff);
    44.6  
    44.7 -Goalw  [prat_def,ratrel_def,quotient_def] "ratrel```{(x,y)}:prat";
    44.8 +Goalw  [prat_def,ratrel_def,quotient_def] "ratrel``{(x,y)}:prat";
    44.9  by (Blast_tac 1);
   44.10  qed "ratrel_in_prat";
   44.11  
   44.12 @@ -95,7 +95,7 @@
   44.13  qed "inj_prat_of_pnat";
   44.14  
   44.15  val [prem] = Goal
   44.16 -    "(!!x y. z = Abs_prat(ratrel```{(x,y)}) ==> P) ==> P";
   44.17 +    "(!!x y. z = Abs_prat(ratrel``{(x,y)}) ==> P) ==> P";
   44.18  by (res_inst_tac [("x1","z")] 
   44.19      (rewrite_rule [prat_def] Rep_prat RS quotientE) 1);
   44.20  by (dres_inst_tac [("f","Abs_prat")] arg_cong 1);
   44.21 @@ -106,12 +106,12 @@
   44.22  
   44.23  (**** qinv: inverse on prat ****)
   44.24  
   44.25 -Goalw [congruent_def] "congruent ratrel (%(x,y). ratrel```{(y,x)})";
   44.26 +Goalw [congruent_def] "congruent ratrel (%(x,y). ratrel``{(y,x)})";
   44.27  by (auto_tac (claset(), simpset() addsimps [pnat_mult_commute]));  
   44.28  qed "qinv_congruent";
   44.29  
   44.30  Goalw [qinv_def]
   44.31 -      "qinv (Abs_prat(ratrel```{(x,y)})) = Abs_prat(ratrel ``` {(y,x)})";
   44.32 +      "qinv (Abs_prat(ratrel``{(x,y)})) = Abs_prat(ratrel `` {(y,x)})";
   44.33  by (simp_tac (simpset() addsimps 
   44.34  	      [equiv_ratrel RS UN_equiv_class, qinv_congruent]) 1);
   44.35  qed "qinv";
   44.36 @@ -145,7 +145,7 @@
   44.37  qed "prat_add_congruent2_lemma";
   44.38  
   44.39  Goal "congruent2 ratrel (%p1 p2.                  \
   44.40 -\        (%(x1,y1). (%(x2,y2). ratrel```{(x1*y2 + x2*y1, y1*y2)}) p2) p1)";
   44.41 +\        (%(x1,y1). (%(x2,y2). ratrel``{(x1*y2 + x2*y1, y1*y2)}) p2) p1)";
   44.42  by (rtac (equiv_ratrel RS congruent2_commuteI) 1);
   44.43  by (auto_tac (claset() delrules [equalityI],
   44.44                simpset() addsimps [prat_add_congruent2_lemma]));
   44.45 @@ -153,8 +153,8 @@
   44.46  qed "prat_add_congruent2";
   44.47  
   44.48  Goalw [prat_add_def]
   44.49 -   "Abs_prat((ratrel```{(x1,y1)})) + Abs_prat((ratrel```{(x2,y2)})) =   \
   44.50 -\   Abs_prat(ratrel ``` {(x1*y2 + x2*y1, y1*y2)})";
   44.51 +   "Abs_prat((ratrel``{(x1,y1)})) + Abs_prat((ratrel``{(x2,y2)})) =   \
   44.52 +\   Abs_prat(ratrel `` {(x1*y2 + x2*y1, y1*y2)})";
   44.53  by (simp_tac (simpset() addsimps [UN_UN_split_split_eq, prat_add_congruent2, 
   44.54  				  equiv_ratrel RS UN_equiv_class2]) 1);
   44.55  qed "prat_add";
   44.56 @@ -189,7 +189,7 @@
   44.57  
   44.58  Goalw [congruent2_def]
   44.59      "congruent2 ratrel (%p1 p2.                  \
   44.60 -\         (%(x1,y1). (%(x2,y2). ratrel```{(x1*x2, y1*y2)}) p2) p1)";
   44.61 +\         (%(x1,y1). (%(x2,y2). ratrel``{(x1*x2, y1*y2)}) p2) p1)";
   44.62  (*Proof via congruent2_commuteI seems longer*)
   44.63  by (Clarify_tac 1);
   44.64  by (asm_simp_tac (simpset() addsimps [pnat_mult_assoc]) 1);
   44.65 @@ -200,8 +200,8 @@
   44.66  qed "pnat_mult_congruent2";
   44.67  
   44.68  Goalw [prat_mult_def]
   44.69 -  "Abs_prat(ratrel```{(x1,y1)}) * Abs_prat(ratrel```{(x2,y2)}) = \
   44.70 -\  Abs_prat(ratrel```{(x1*x2, y1*y2)})";
   44.71 +  "Abs_prat(ratrel``{(x1,y1)}) * Abs_prat(ratrel``{(x2,y2)}) = \
   44.72 +\  Abs_prat(ratrel``{(x1*x2, y1*y2)})";
   44.73  by (asm_simp_tac 
   44.74      (simpset() addsimps [UN_UN_split_split_eq, pnat_mult_congruent2,
   44.75  			 equiv_ratrel RS UN_equiv_class2]) 1);
   44.76 @@ -389,7 +389,7 @@
   44.77  Goal "!(q::prat). EX x. x + x = q";
   44.78  by (rtac allI 1);
   44.79  by (res_inst_tac [("z","q")] eq_Abs_prat 1);
   44.80 -by (res_inst_tac [("x","Abs_prat (ratrel ``` {(x, y+y)})")] exI 1);
   44.81 +by (res_inst_tac [("x","Abs_prat (ratrel `` {(x, y+y)})")] exI 1);
   44.82  by (auto_tac (claset(),
   44.83  	      simpset() addsimps 
   44.84                [prat_add,pnat_mult_assoc RS sym,pnat_add_mult_distrib,
   44.85 @@ -398,7 +398,7 @@
   44.86  
   44.87  Goal "EX (x::prat). x + x = q";
   44.88  by (res_inst_tac [("z","q")] eq_Abs_prat 1);
   44.89 -by (res_inst_tac [("x","Abs_prat (ratrel ``` {(x, y+y)})")] exI 1);
   44.90 +by (res_inst_tac [("x","Abs_prat (ratrel `` {(x, y+y)})")] exI 1);
   44.91  by (auto_tac (claset(),simpset() addsimps 
   44.92                [prat_add,pnat_mult_assoc RS sym,pnat_add_mult_distrib,
   44.93                 pnat_add_mult_distrib2]));
   44.94 @@ -454,7 +454,7 @@
   44.95  
   44.96  (* lemma for proving $< is linear *)
   44.97  Goalw [prat_def,prat_less_def] 
   44.98 -      "ratrel ``` {(x, y * ya)} : {p::(pnat*pnat).True}//ratrel";
   44.99 +      "ratrel `` {(x, y * ya)} : {p::(pnat*pnat).True}//ratrel";
  44.100  by (asm_full_simp_tac (simpset() addsimps [ratrel_def,quotient_def]) 1);
  44.101  by (Blast_tac 1);
  44.102  qed "lemma_prat_less_linear";
  44.103 @@ -470,15 +470,15 @@
  44.104  by (cut_inst_tac  [("z1.0","x*ya"), ("z2.0","xa*y")] pnat_linear_Ex_eq 1);
  44.105  by (EVERY1[etac disjE,etac exE]);
  44.106  by (eres_inst_tac 
  44.107 -    [("x","Abs_prat(ratrel```{(xb,ya*y)})")] allE 1);
  44.108 +    [("x","Abs_prat(ratrel``{(xb,ya*y)})")] allE 1);
  44.109  by (asm_full_simp_tac 
  44.110      (simpset() addsimps [prat_add, pnat_mult_assoc 
  44.111       RS sym,pnat_add_mult_distrib RS sym]) 1);
  44.112  by (EVERY1[asm_full_simp_tac (simpset() addsimps pnat_mult_ac),
  44.113      etac disjE, assume_tac, etac exE]);
  44.114 -by (thin_tac "!T. Abs_prat (ratrel ``` {(x, y)}) + T ~= \
  44.115 -\     Abs_prat (ratrel ``` {(xa, ya)})" 1);
  44.116 -by (eres_inst_tac [("x","Abs_prat(ratrel```{(xb,y*ya)})")] allE 1);
  44.117 +by (thin_tac "!T. Abs_prat (ratrel `` {(x, y)}) + T ~= \
  44.118 +\     Abs_prat (ratrel `` {(xa, ya)})" 1);
  44.119 +by (eres_inst_tac [("x","Abs_prat(ratrel``{(xb,y*ya)})")] allE 1);
  44.120  by (asm_full_simp_tac (simpset() addsimps [prat_add,
  44.121        pnat_mult_assoc RS sym,pnat_add_mult_distrib RS sym]) 1);
  44.122  by (asm_full_simp_tac (simpset() addsimps pnat_mult_ac) 1);
  44.123 @@ -696,12 +696,12 @@
  44.124  
  44.125  (*** lemmas required for lemma_gleason9_34 in PReal : w*y > y/z ***)
  44.126  Goalw [prat_of_pnat_def] 
  44.127 -      "Abs_prat(ratrel```{(x,y)}) = (prat_of_pnat x)*qinv(prat_of_pnat y)";
  44.128 +      "Abs_prat(ratrel``{(x,y)}) = (prat_of_pnat x)*qinv(prat_of_pnat y)";
  44.129  by (auto_tac (claset(),simpset() addsimps [prat_mult,qinv,pnat_mult_1_left,
  44.130      pnat_mult_1]));
  44.131  qed "Abs_prat_mult_qinv";
  44.132  
  44.133 -Goal "Abs_prat(ratrel```{(x,y)}) <= Abs_prat(ratrel```{(x,Abs_pnat 1)})";
  44.134 +Goal "Abs_prat(ratrel``{(x,y)}) <= Abs_prat(ratrel``{(x,Abs_pnat 1)})";
  44.135  by (simp_tac (simpset() addsimps [Abs_prat_mult_qinv]) 1);
  44.136  by (rtac prat_mult_left_le2_mono1 1);
  44.137  by (rtac qinv_prat_le 1);
  44.138 @@ -713,7 +713,7 @@
  44.139      pSuc_is_plus_one,pnat_one_def,prat_of_pnat_add]));
  44.140  qed "lemma_Abs_prat_le1";
  44.141  
  44.142 -Goal "Abs_prat(ratrel```{(x,Abs_pnat 1)}) <= Abs_prat(ratrel```{(x*y,Abs_pnat 1)})";
  44.143 +Goal "Abs_prat(ratrel``{(x,Abs_pnat 1)}) <= Abs_prat(ratrel``{(x*y,Abs_pnat 1)})";
  44.144  by (simp_tac (simpset() addsimps [Abs_prat_mult_qinv]) 1);
  44.145  by (rtac prat_mult_le2_mono1 1);
  44.146  by (pnat_ind_tac "y" 1);
  44.147 @@ -726,19 +726,19 @@
  44.148  			prat_of_pnat_add,prat_of_pnat_mult]));
  44.149  qed "lemma_Abs_prat_le2";
  44.150  
  44.151 -Goal "Abs_prat(ratrel```{(x,z)}) <= Abs_prat(ratrel```{(x*y,Abs_pnat 1)})";
  44.152 +Goal "Abs_prat(ratrel``{(x,z)}) <= Abs_prat(ratrel``{(x*y,Abs_pnat 1)})";
  44.153  by (fast_tac (claset() addIs [prat_le_trans,
  44.154  			      lemma_Abs_prat_le1,lemma_Abs_prat_le2]) 1);
  44.155  qed "lemma_Abs_prat_le3";
  44.156  
  44.157 -Goal "Abs_prat(ratrel```{(x*y,Abs_pnat 1)}) * Abs_prat(ratrel```{(w,x)}) = \
  44.158 -\         Abs_prat(ratrel```{(w*y,Abs_pnat 1)})";
  44.159 +Goal "Abs_prat(ratrel``{(x*y,Abs_pnat 1)}) * Abs_prat(ratrel``{(w,x)}) = \
  44.160 +\         Abs_prat(ratrel``{(w*y,Abs_pnat 1)})";
  44.161  by (full_simp_tac (simpset() addsimps [prat_mult,
  44.162      pnat_mult_1,pnat_mult_1_left] @ pnat_mult_ac) 1);
  44.163  qed "pre_lemma_gleason9_34";
  44.164  
  44.165 -Goal "Abs_prat(ratrel```{(y*x,Abs_pnat 1*y)}) = \
  44.166 -\         Abs_prat(ratrel```{(x,Abs_pnat 1)})";
  44.167 +Goal "Abs_prat(ratrel``{(y*x,Abs_pnat 1*y)}) = \
  44.168 +\         Abs_prat(ratrel``{(x,Abs_pnat 1)})";
  44.169  by (auto_tac (claset(),
  44.170  	      simpset() addsimps [pnat_mult_1,pnat_mult_1_left] @ pnat_mult_ac));
  44.171  qed "pre_lemma_gleason9_34b";
    45.1 --- a/src/HOL/Real/PRat.thy	Tue Jan 09 15:29:17 2001 +0100
    45.2 +++ b/src/HOL/Real/PRat.thy	Tue Jan 09 15:32:27 2001 +0100
    45.3 @@ -20,20 +20,20 @@
    45.4  constdefs
    45.5  
    45.6    prat_of_pnat :: pnat => prat           
    45.7 -  "prat_of_pnat m == Abs_prat(ratrel```{(m,Abs_pnat 1)})"
    45.8 +  "prat_of_pnat m == Abs_prat(ratrel``{(m,Abs_pnat 1)})"
    45.9  
   45.10    qinv      :: prat => prat
   45.11 -  "qinv(Q)  == Abs_prat(UN (x,y):Rep_prat(Q). ratrel```{(y,x)})" 
   45.12 +  "qinv(Q)  == Abs_prat(UN (x,y):Rep_prat(Q). ratrel``{(y,x)})" 
   45.13  
   45.14  defs
   45.15  
   45.16    prat_add_def  
   45.17    "P + Q == Abs_prat(UN (x1,y1):Rep_prat(P). UN (x2,y2):Rep_prat(Q).
   45.18 -		     ratrel```{(x1*y2 + x2*y1, y1*y2)})"
   45.19 +		     ratrel``{(x1*y2 + x2*y1, y1*y2)})"
   45.20  
   45.21    prat_mult_def  
   45.22    "P * Q == Abs_prat(UN (x1,y1):Rep_prat(P). UN (x2,y2):Rep_prat(Q).
   45.23 -		     ratrel```{(x1*x2, y1*y2)})"
   45.24 +		     ratrel``{(x1*x2, y1*y2)})"
   45.25   
   45.26    (*** Gleason p. 119 ***)
   45.27    prat_less_def
    46.1 --- a/src/HOL/Real/PReal.ML	Tue Jan 09 15:29:17 2001 +0100
    46.2 +++ b/src/HOL/Real/PReal.ML	Tue Jan 09 15:32:27 2001 +0100
    46.3 @@ -582,10 +582,10 @@
    46.4     prat_of_pnat_add,prat_add_assoc RS sym]));
    46.5  qed "lemma1_gleason9_34";
    46.6  
    46.7 -Goal "Abs_prat (ratrel ``` {(y, z)}) < xb + \
    46.8 -\         Abs_prat (ratrel ``` {(x*y, Abs_pnat 1)})*Abs_prat (ratrel ``` {(w, x)})";
    46.9 -by (res_inst_tac [("j","Abs_prat (ratrel ``` {(x * y, Abs_pnat 1)}) *\
   46.10 -\                   Abs_prat (ratrel ``` {(w, x)})")] prat_le_less_trans 1);
   46.11 +Goal "Abs_prat (ratrel `` {(y, z)}) < xb + \
   46.12 +\         Abs_prat (ratrel `` {(x*y, Abs_pnat 1)})*Abs_prat (ratrel `` {(w, x)})";
   46.13 +by (res_inst_tac [("j","Abs_prat (ratrel `` {(x * y, Abs_pnat 1)}) *\
   46.14 +\                   Abs_prat (ratrel `` {(w, x)})")] prat_le_less_trans 1);
   46.15  by (rtac prat_self_less_add_right 2);
   46.16  by (auto_tac (claset() addIs [lemma_Abs_prat_le3],
   46.17      simpset() addsimps [prat_mult,pre_lemma_gleason9_34b,pnat_mult_assoc]));
    47.1 --- a/src/HOL/Real/RealDef.ML	Tue Jan 09 15:29:17 2001 +0100
    47.2 +++ b/src/HOL/Real/RealDef.ML	Tue Jan 09 15:32:27 2001 +0100
    47.3 @@ -57,11 +57,11 @@
    47.4                         addSEs [sym, preal_trans_lemma]) 1);
    47.5  qed "equiv_realrel";
    47.6  
    47.7 -(* (realrel ``` {x} = realrel ``` {y}) = ((x,y) : realrel) *)
    47.8 +(* (realrel `` {x} = realrel `` {y}) = ((x,y) : realrel) *)
    47.9  bind_thm ("equiv_realrel_iff",
   47.10      	  [equiv_realrel, UNIV_I, UNIV_I] MRS eq_equiv_class_iff);
   47.11  
   47.12 -Goalw  [real_def,realrel_def,quotient_def] "realrel```{(x,y)}:real";
   47.13 +Goalw  [real_def,realrel_def,quotient_def] "realrel``{(x,y)}:real";
   47.14  by (Blast_tac 1);
   47.15  qed "realrel_in_real";
   47.16  
   47.17 @@ -95,7 +95,7 @@
   47.18  qed "inj_real_of_preal";
   47.19  
   47.20  val [prem] = Goal
   47.21 -    "(!!x y. z = Abs_real(realrel```{(x,y)}) ==> P) ==> P";
   47.22 +    "(!!x y. z = Abs_real(realrel``{(x,y)}) ==> P) ==> P";
   47.23  by (res_inst_tac [("x1","z")] 
   47.24      (rewrite_rule [real_def] Rep_real RS quotientE) 1);
   47.25  by (dres_inst_tac [("f","Abs_real")] arg_cong 1);
   47.26 @@ -107,13 +107,13 @@
   47.27  (**** real_minus: additive inverse on real ****)
   47.28  
   47.29  Goalw [congruent_def]
   47.30 -  "congruent realrel (%p. (%(x,y). realrel```{(y,x)}) p)";
   47.31 +  "congruent realrel (%p. (%(x,y). realrel``{(y,x)}) p)";
   47.32  by (Clarify_tac 1); 
   47.33  by (asm_full_simp_tac (simpset() addsimps [preal_add_commute]) 1);
   47.34  qed "real_minus_congruent";
   47.35  
   47.36  Goalw [real_minus_def]
   47.37 -      "- (Abs_real(realrel```{(x,y)})) = Abs_real(realrel ``` {(y,x)})";
   47.38 +      "- (Abs_real(realrel``{(x,y)})) = Abs_real(realrel `` {(y,x)})";
   47.39  by (res_inst_tac [("f","Abs_real")] arg_cong 1);
   47.40  by (simp_tac (simpset() addsimps 
   47.41     [realrel_in_real RS Abs_real_inverse,
   47.42 @@ -150,7 +150,7 @@
   47.43  (*** Congruence property for addition ***)
   47.44  Goalw [congruent2_def]
   47.45      "congruent2 realrel (%p1 p2.                  \
   47.46 -\         (%(x1,y1). (%(x2,y2). realrel```{(x1+x2, y1+y2)}) p2) p1)";
   47.47 +\         (%(x1,y1). (%(x2,y2). realrel``{(x1+x2, y1+y2)}) p2) p1)";
   47.48  by (Clarify_tac 1); 
   47.49  by (asm_simp_tac (simpset() addsimps [preal_add_assoc]) 1);
   47.50  by (res_inst_tac [("z1.1","x1a")] (preal_add_left_commute RS ssubst) 1);
   47.51 @@ -159,8 +159,8 @@
   47.52  qed "real_add_congruent2";
   47.53  
   47.54  Goalw [real_add_def]
   47.55 -  "Abs_real(realrel```{(x1,y1)}) + Abs_real(realrel```{(x2,y2)}) = \
   47.56 -\  Abs_real(realrel```{(x1+x2, y1+y2)})";
   47.57 +  "Abs_real(realrel``{(x1,y1)}) + Abs_real(realrel``{(x2,y2)}) = \
   47.58 +\  Abs_real(realrel``{(x1+x2, y1+y2)})";
   47.59  by (simp_tac (simpset() addsimps 
   47.60                [[equiv_realrel, real_add_congruent2] MRS UN_equiv_class2]) 1);
   47.61  qed "real_add";
   47.62 @@ -301,7 +301,7 @@
   47.63  
   47.64  Goal 
   47.65      "congruent2 realrel (%p1 p2.                  \
   47.66 -\         (%(x1,y1). (%(x2,y2). realrel```{(x1*x2 + y1*y2, x1*y2+x2*y1)}) p2) p1)";
   47.67 +\         (%(x1,y1). (%(x2,y2). realrel``{(x1*x2 + y1*y2, x1*y2+x2*y1)}) p2) p1)";
   47.68  by (rtac (equiv_realrel RS congruent2_commuteI) 1);
   47.69  by (Clarify_tac 1); 
   47.70  by (rewtac split_def);
   47.71 @@ -310,8 +310,8 @@
   47.72  qed "real_mult_congruent2";
   47.73  
   47.74  Goalw [real_mult_def]
   47.75 -   "Abs_real((realrel```{(x1,y1)})) * Abs_real((realrel```{(x2,y2)})) =   \
   47.76 -\   Abs_real(realrel ``` {(x1*x2+y1*y2,x1*y2+x2*y1)})";
   47.77 +   "Abs_real((realrel``{(x1,y1)})) * Abs_real((realrel``{(x2,y2)})) =   \
   47.78 +\   Abs_real(realrel `` {(x1*x2+y1*y2,x1*y2+x2*y1)})";
   47.79  by (simp_tac (simpset() addsimps
   47.80                 [[equiv_realrel, real_mult_congruent2] MRS UN_equiv_class2]) 1);
   47.81  qed "real_mult";
   47.82 @@ -451,7 +451,7 @@
   47.83  
   47.84  (*** existence of inverse ***)
   47.85  (** lemma -- alternative definition of 0 **)
   47.86 -Goalw [real_zero_def] "0 = Abs_real (realrel ``` {(x, x)})";
   47.87 +Goalw [real_zero_def] "0 = Abs_real (realrel `` {(x, x)})";
   47.88  by (auto_tac (claset(),simpset() addsimps [preal_add_commute]));
   47.89  qed "real_zero_iff";
   47.90  
   47.91 @@ -461,10 +461,10 @@
   47.92  by (cut_inst_tac [("r1.0","xa"),("r2.0","y")] preal_linear 1);
   47.93  by (auto_tac (claset() addSDs [preal_less_add_left_Ex],
   47.94             simpset() addsimps [real_zero_iff RS sym]));
   47.95 -by (res_inst_tac [("x","Abs_real (realrel ``` \
   47.96 +by (res_inst_tac [("x","Abs_real (realrel `` \
   47.97  \   {(preal_of_prat(prat_of_pnat 1p),pinv(D)+\
   47.98  \    preal_of_prat(prat_of_pnat 1p))})")] exI 1);
   47.99 -by (res_inst_tac [("x","Abs_real (realrel ``` \
  47.100 +by (res_inst_tac [("x","Abs_real (realrel `` \
  47.101  \   {(pinv(D)+preal_of_prat(prat_of_pnat 1p),\
  47.102  \    preal_of_prat(prat_of_pnat 1p))})")] exI 2);
  47.103  by (auto_tac (claset(),
  47.104 @@ -716,13 +716,13 @@
  47.105  
  47.106  Goalw [real_of_preal_def]
  47.107        "!!(x::preal). y < x ==> \
  47.108 -\      EX m. Abs_real (realrel ``` {(x,y)}) = real_of_preal m";
  47.109 +\      EX m. Abs_real (realrel `` {(x,y)}) = real_of_preal m";
  47.110  by (auto_tac (claset() addSDs [preal_less_add_left_Ex],
  47.111      simpset() addsimps preal_add_ac));
  47.112  qed "real_of_preal_ExI";
  47.113  
  47.114  Goalw [real_of_preal_def]
  47.115 -      "!!(x::preal). EX m. Abs_real (realrel ``` {(x,y)}) = \
  47.116 +      "!!(x::preal). EX m. Abs_real (realrel `` {(x,y)}) = \
  47.117  \                    real_of_preal m ==> y < x";
  47.118  by (auto_tac (claset(),
  47.119  	      simpset() addsimps 
  47.120 @@ -731,7 +731,7 @@
  47.121      [preal_add_assoc RS sym,preal_self_less_add_left]) 1);
  47.122  qed "real_of_preal_ExD";
  47.123  
  47.124 -Goal "(EX m. Abs_real (realrel ``` {(x,y)}) = real_of_preal m) = (y < x)";
  47.125 +Goal "(EX m. Abs_real (realrel `` {(x,y)}) = real_of_preal m) = (y < x)";
  47.126  by (blast_tac (claset() addSIs [real_of_preal_ExI,real_of_preal_ExD]) 1);
  47.127  qed "real_of_preal_iff";
  47.128  
    48.1 --- a/src/HOL/Real/RealDef.thy	Tue Jan 09 15:29:17 2001 +0100
    48.2 +++ b/src/HOL/Real/RealDef.thy	Tue Jan 09 15:32:27 2001 +0100
    48.3 @@ -31,14 +31,14 @@
    48.4  defs
    48.5  
    48.6    real_zero_def  
    48.7 -  "0 == Abs_real(realrel```{(preal_of_prat(prat_of_pnat 1p),
    48.8 +  "0 == Abs_real(realrel``{(preal_of_prat(prat_of_pnat 1p),
    48.9                                  preal_of_prat(prat_of_pnat 1p))})"
   48.10    real_one_def   
   48.11 -  "1r == Abs_real(realrel```{(preal_of_prat(prat_of_pnat 1p) + 
   48.12 +  "1r == Abs_real(realrel``{(preal_of_prat(prat_of_pnat 1p) + 
   48.13              preal_of_prat(prat_of_pnat 1p),preal_of_prat(prat_of_pnat 1p))})"
   48.14  
   48.15    real_minus_def
   48.16 -  "- R ==  Abs_real(UN (x,y):Rep_real(R). realrel```{(y,x)})"
   48.17 +  "- R ==  Abs_real(UN (x,y):Rep_real(R). realrel``{(y,x)})"
   48.18  
   48.19    real_diff_def
   48.20    "R - (S::real) == R + - S"
   48.21 @@ -53,7 +53,7 @@
   48.22  
   48.23    real_of_preal :: preal => real            
   48.24    "real_of_preal m     ==
   48.25 -           Abs_real(realrel```{(m+preal_of_prat(prat_of_pnat 1p),
   48.26 +           Abs_real(realrel``{(m+preal_of_prat(prat_of_pnat 1p),
   48.27                                 preal_of_prat(prat_of_pnat 1p))})"
   48.28  
   48.29    real_of_posnat :: nat => real             
   48.30 @@ -66,11 +66,11 @@
   48.31  
   48.32    real_add_def  
   48.33    "P+Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
   48.34 -                   (%(x1,y1). (%(x2,y2). realrel```{(x1+x2, y1+y2)}) p2) p1)"
   48.35 +                   (%(x1,y1). (%(x2,y2). realrel``{(x1+x2, y1+y2)}) p2) p1)"
   48.36    
   48.37    real_mult_def  
   48.38    "P*Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
   48.39 -                   (%(x1,y1). (%(x2,y2). realrel```{(x1*x2+y1*y2,x1*y2+x2*y1)})
   48.40 +                   (%(x1,y1). (%(x2,y2). realrel``{(x1*x2+y1*y2,x1*y2+x2*y1)})
   48.41  		   p2) p1)"
   48.42  
   48.43    real_less_def
    49.1 --- a/src/HOL/Real/RealInt.ML	Tue Jan 09 15:29:17 2001 +0100
    49.2 +++ b/src/HOL/Real/RealInt.ML	Tue Jan 09 15:32:27 2001 +0100
    49.3 @@ -7,7 +7,7 @@
    49.4  
    49.5  
    49.6  Goalw [congruent_def]
    49.7 -  "congruent intrel (%p. (%(i,j). realrel ``` \
    49.8 +  "congruent intrel (%p. (%(i,j). realrel `` \
    49.9  \  {(preal_of_prat (prat_of_pnat (pnat_of_nat i)), \
   49.10  \    preal_of_prat (prat_of_pnat (pnat_of_nat j)))}) p)";
   49.11  by (auto_tac (claset(),
   49.12 @@ -16,8 +16,8 @@
   49.13  qed "real_of_int_congruent";
   49.14  
   49.15  Goalw [real_of_int_def]
   49.16 -   "real_of_int (Abs_Integ (intrel ``` {(i, j)})) = \
   49.17 -\     Abs_real(realrel ``` \
   49.18 +   "real_of_int (Abs_Integ (intrel `` {(i, j)})) = \
   49.19 +\     Abs_real(realrel `` \
   49.20  \       {(preal_of_prat (prat_of_pnat (pnat_of_nat i)), \
   49.21  \         preal_of_prat (prat_of_pnat (pnat_of_nat j)))})";
   49.22  by (res_inst_tac [("f","Abs_real")] arg_cong 1);
    50.1 --- a/src/HOL/Real/RealInt.thy	Tue Jan 09 15:29:17 2001 +0100
    50.2 +++ b/src/HOL/Real/RealInt.thy	Tue Jan 09 15:32:27 2001 +0100
    50.3 @@ -9,7 +9,7 @@
    50.4  
    50.5  constdefs 
    50.6     real_of_int :: int => real
    50.7 -   "real_of_int z == Abs_real(UN (i,j): Rep_Integ z. realrel ```
    50.8 +   "real_of_int z == Abs_real(UN (i,j): Rep_Integ z. realrel ``
    50.9                       {(preal_of_prat(prat_of_pnat(pnat_of_nat i)),
   50.10                         preal_of_prat(prat_of_pnat(pnat_of_nat j)))})"
   50.11  
    51.1 --- a/src/HOL/UNITY/Channel.ML	Tue Jan 09 15:29:17 2001 +0100
    51.2 +++ b/src/HOL/UNITY/Channel.ML	Tue Jan 09 15:32:27 2001 +0100
    51.3 @@ -23,7 +23,7 @@
    51.4  by Auto_tac;
    51.5  qed_spec_mp "minSet_nonempty";
    51.6  
    51.7 -Goal "F : (minSet -`` {Some x}) leadsTo (minSet -`` (Some``greaterThan x))";
    51.8 +Goal "F : (minSet -` {Some x}) leadsTo (minSet -` (Some`greaterThan x))";
    51.9  by (rtac leadsTo_weaken 1);
   51.10  by (res_inst_tac [("x1","x")] ([UC2, UC1] MRS psp) 1);
   51.11  by Safe_tac;
   51.12 @@ -32,7 +32,7 @@
   51.13  qed "minSet_greaterThan";
   51.14  
   51.15  (*The induction*)
   51.16 -Goal "F : (UNIV-{{}}) leadsTo (minSet -`` (Some``atLeast y))";
   51.17 +Goal "F : (UNIV-{{}}) leadsTo (minSet -` (Some`atLeast y))";
   51.18  by (rtac leadsTo_weaken_R 1);
   51.19  by (res_inst_tac  [("l", "y"), ("f", "the o minSet"), ("B", "{}")]
   51.20       greaterThan_bounded_induct 1);
    52.1 --- a/src/HOL/UNITY/Channel.thy	Tue Jan 09 15:29:17 2001 +0100
    52.2 +++ b/src/HOL/UNITY/Channel.thy	Tue Jan 09 15:32:27 2001 +0100
    52.3 @@ -21,10 +21,10 @@
    52.4  
    52.5  rules
    52.6  
    52.7 -  UC1  "F : (minSet -`` {Some x}) co (minSet -`` (Some``atLeast x))"
    52.8 +  UC1  "F : (minSet -` {Some x}) co (minSet -` (Some`atLeast x))"
    52.9  
   52.10    (*  UC1  "F : {s. minSet s = x} co {s. x <= minSet s}"  *)
   52.11  
   52.12 -  UC2  "F : (minSet -`` {Some x}) leadsTo {s. x ~: s}"
   52.13 +  UC2  "F : (minSet -` {Some x}) leadsTo {s. x ~: s}"
   52.14  
   52.15  end
    53.1 --- a/src/HOL/UNITY/ELT.ML	Tue Jan 09 15:29:17 2001 +0100
    53.2 +++ b/src/HOL/UNITY/ELT.ML	Tue Jan 09 15:32:27 2001 +0100
    53.3 @@ -15,7 +15,7 @@
    53.4  
    53.5  Goalw [givenBy_def] "(givenBy v) = {A. ALL x:A. ALL y. v x = v y --> y: A}";
    53.6  by Safe_tac;
    53.7 -by (res_inst_tac [("x", "v `` ?u")] image_eqI 2);
    53.8 +by (res_inst_tac [("x", "v ` ?u")] image_eqI 2);
    53.9  by Auto_tac;
   53.10  qed "givenBy_eq_all";
   53.11  
   53.12 @@ -307,7 +307,7 @@
   53.13  (*??IS THIS NEEDED?? or is it just an example of what's provable??*)
   53.14  Goal "[| F: (A leadsTo[givenBy v] B);  G : preserves v;  \
   53.15  \        F Join G : stable C |] \
   53.16 -\     ==> F Join G : ((C Int A) leadsTo[(%D. C Int D) `` givenBy v] B)";
   53.17 +\     ==> F Join G : ((C Int A) leadsTo[(%D. C Int D) ` givenBy v] B)";
   53.18  by (etac leadsETo_induct 1);
   53.19  by (stac Int_Union 3);
   53.20  by (blast_tac (claset() addIs [leadsETo_UN]) 3);
   53.21 @@ -368,7 +368,7 @@
   53.22  
   53.23  Goalw [LeadsETo_def]
   53.24       "A LeadsTo[CC] B = \
   53.25 -\       {F. F : (reachable F Int A) leadsTo[(%C. reachable F Int C) `` CC] \
   53.26 +\       {F. F : (reachable F Int A) leadsTo[(%C. reachable F Int C) ` CC] \
   53.27  \       (reachable F Int B)}";
   53.28  by (blast_tac (claset() addDs [e_psp_stable2] addIs [leadsETo_weaken]) 1);
   53.29  qed "LeadsETo_eq_leadsETo";
   53.30 @@ -467,7 +467,7 @@
   53.31  
   53.32  (*givenBy laws that need to be in the locale*)
   53.33  
   53.34 -Goal "givenBy (v o f) = extend_set h `` (givenBy v)";
   53.35 +Goal "givenBy (v o f) = extend_set h ` (givenBy v)";
   53.36  by (simp_tac (simpset() addsimps [givenBy_eq_Collect]) 1);
   53.37  by (Deepen_tac 0 1);
   53.38  qed "givenBy_o_eq_extend_set";
   53.39 @@ -483,7 +483,7 @@
   53.40  qed "extend_set_givenBy_I";
   53.41  
   53.42  Goal "F : A leadsTo[CC] B \
   53.43 -\     ==> extend h F : (extend_set h A) leadsTo[extend_set h `` CC] \
   53.44 +\     ==> extend h F : (extend_set h A) leadsTo[extend_set h ` CC] \
   53.45  \                      (extend_set h B)";
   53.46  by (etac leadsETo_induct 1);
   53.47  by (asm_simp_tac (simpset() addsimps [leadsETo_UN, extend_set_Union]) 3);
   53.48 @@ -531,11 +531,11 @@
   53.49  qed "preserves_o_project_transient_empty";
   53.50  
   53.51  Goal "[| extend h F Join G : stable C;  \
   53.52 -\        F Join project h C G : (project_set h C Int A) leadsTo[(%D. project_set h C Int D)``givenBy v] B;  \
   53.53 +\        F Join project h C G : (project_set h C Int A) leadsTo[(%D. project_set h C Int D)`givenBy v] B;  \
   53.54  \        G : preserves (v o f) |] \
   53.55  \     ==> extend h F Join G : \
   53.56  \           (C Int extend_set h (project_set h C Int A)) \
   53.57 -\           leadsTo[(%D. C Int extend_set h D)``givenBy v]  (extend_set h B)";
   53.58 +\           leadsTo[(%D. C Int extend_set h D)`givenBy v]  (extend_set h B)";
   53.59  by (etac leadsETo_induct 1);
   53.60  by (asm_simp_tac (simpset() delsimps UN_simps
   53.61  		  addsimps [Int_UN_distrib, leadsETo_UN, extend_set_Union]) 3);
   53.62 @@ -560,10 +560,10 @@
   53.63  Goal "[| extend h F Join G : stable C;  \
   53.64  \        F Join project h C G : \
   53.65  \            (project_set h C Int A) \
   53.66 -\            leadsTo[(%D. project_set h C Int D)``givenBy v] B;  \
   53.67 +\            leadsTo[(%D. project_set h C Int D)`givenBy v] B;  \
   53.68  \        G : preserves (v o f) |] \
   53.69  \     ==> extend h F Join G : (C Int extend_set h A) \
   53.70 -\           leadsTo[(%D. C Int extend_set h D)``givenBy v] (extend_set h B)";
   53.71 +\           leadsTo[(%D. C Int extend_set h D)`givenBy v] (extend_set h B)";
   53.72  by (rtac (lemma RS leadsETo_weaken) 1);
   53.73  by (auto_tac (claset(), 
   53.74  	      simpset() addsimps [split_extended_all]));
   53.75 @@ -630,7 +630,7 @@
   53.76  
   53.77  Goal "[| extend h F Join G : stable C;  \
   53.78  \        extend h F Join G : \
   53.79 -\          (C Int A) leadsTo[(%D. C Int D)``givenBy f]  B |]  \
   53.80 +\          (C Int A) leadsTo[(%D. C Int D)`givenBy f]  B |]  \
   53.81  \ ==> F Join project h C G  \
   53.82  \   : (project_set h C Int project_set h (C Int A)) leadsTo (project_set h B)";
   53.83  by (etac leadsETo_induct 1);
    54.1 --- a/src/HOL/UNITY/ELT.thy	Tue Jan 09 15:29:17 2001 +0100
    54.2 +++ b/src/HOL/UNITY/ELT.thy	Tue Jan 09 15:32:27 2001 +0100
    54.3 @@ -44,7 +44,7 @@
    54.4    
    54.5    (*the set of all sets determined by f alone*)
    54.6    givenBy :: "['a => 'b] => 'a set set"
    54.7 -    "givenBy f == range (%B. f-`` B)"
    54.8 +    "givenBy f == range (%B. f-` B)"
    54.9  
   54.10    (*visible version of the LEADS-TO relation*)
   54.11    leadsETo :: "['a set, 'a set set, 'a set] => 'a program set"
   54.12 @@ -54,6 +54,6 @@
   54.13    LeadsETo :: "['a set, 'a set set, 'a set] => 'a program set"
   54.14                                          ("(3_/ LeadsTo[_]/ _)" [80,0,80] 80)
   54.15      "LeadsETo A CC B ==
   54.16 -      {F. F : (reachable F Int A) leadsTo[(%C. reachable F Int C) `` CC] B}"
   54.17 +      {F. F : (reachable F Int A) leadsTo[(%C. reachable F Int C) ` CC] B}"
   54.18  
   54.19  end
    55.1 --- a/src/HOL/UNITY/Extend.ML	Tue Jan 09 15:29:17 2001 +0100
    55.2 +++ b/src/HOL/UNITY/Extend.ML	Tue Jan 09 15:32:27 2001 +0100
    55.3 @@ -50,7 +50,7 @@
    55.4  
    55.5  Goalw [Restrict_def, image_def]
    55.6       "[| s : RR;  Restrict A r = Restrict A s |] \
    55.7 -\     ==> Restrict A r : Restrict A `` RR";
    55.8 +\     ==> Restrict A r : Restrict A ` RR";
    55.9  by Auto_tac;
   55.10  qed "Restrict_imageI";
   55.11  
   55.12 @@ -58,14 +58,14 @@
   55.13  by (Blast_tac 1);
   55.14  qed "Domain_Restrict";
   55.15  
   55.16 -Goal "(Restrict A r) ``` B = r ``` (A Int B)";
   55.17 +Goal "(Restrict A r) `` B = r `` (A Int B)";
   55.18  by (Blast_tac 1);
   55.19  qed "Image_Restrict";
   55.20  
   55.21  Addsimps [Domain_Restrict, Image_Restrict];
   55.22  
   55.23  
   55.24 -Goal "f Id = Id ==> insert Id (f``Acts F) = f `` Acts F";
   55.25 +Goal "f Id = Id ==> insert Id (f`Acts F) = f ` Acts F";
   55.26  by (blast_tac (claset() addIs [sym RS image_eqI]) 1);
   55.27  qed "insert_Id_image_Acts";
   55.28  
   55.29 @@ -211,7 +211,7 @@
   55.30  (*** project_set: basic properties ***)
   55.31  
   55.32  (*project_set is simply image!*)
   55.33 -Goal "project_set h C = f `` C";
   55.34 +Goal "project_set h C = f ` C";
   55.35  by (auto_tac (claset() addIs [f_h_eq RS sym], 
   55.36  	      simpset() addsimps [split_extended_all]));
   55.37  qed "project_set_eq";
   55.38 @@ -308,7 +308,7 @@
   55.39  qed "inj_extend_act";
   55.40  
   55.41  Goalw [extend_set_def, extend_act_def]
   55.42 -     "extend_act h act ``` (extend_set h A) = extend_set h (act ``` A)";
   55.43 +     "extend_act h act `` (extend_set h A) = extend_set h (act `` A)";
   55.44  by (Force_tac 1);
   55.45  qed "extend_act_Image";
   55.46  Addsimps [extend_act_Image];
   55.47 @@ -363,17 +363,17 @@
   55.48  qed "Init_project";
   55.49  Addsimps [Init_project];
   55.50  
   55.51 -Goal "Acts (extend h F) = (extend_act h `` Acts F)";
   55.52 +Goal "Acts (extend h F) = (extend_act h ` Acts F)";
   55.53  by (simp_tac (simpset() addsimps [extend_def, insert_Id_image_Acts]) 1);
   55.54  qed "Acts_extend";
   55.55  Addsimps [Acts_extend];
   55.56  
   55.57 -Goal "AllowedActs (extend h F) = project_act h -`` AllowedActs F";
   55.58 +Goal "AllowedActs (extend h F) = project_act h -` AllowedActs F";
   55.59  by (simp_tac (simpset() addsimps [extend_def, insert_absorb]) 1);
   55.60  qed "AllowedActs_extend";
   55.61  Addsimps [AllowedActs_extend];
   55.62  
   55.63 -Goal "Acts(project h C F) = insert Id (project_act h `` Restrict C `` Acts F)";
   55.64 +Goal "Acts(project h C F) = insert Id (project_act h ` Restrict C ` Acts F)";
   55.65  by (auto_tac (claset(), 
   55.66  	      simpset() addsimps [project_def, image_iff]));
   55.67  qed "Acts_project";
   55.68 @@ -381,7 +381,7 @@
   55.69  
   55.70  Goal "AllowedActs(project h C F) = \
   55.71  \       {act. Restrict (project_set h C) act \
   55.72 -\              : project_act h `` Restrict C `` AllowedActs F}";
   55.73 +\              : project_act h ` Restrict C ` AllowedActs F}";
   55.74  by (simp_tac (simpset() addsimps [project_def, image_iff]) 1);
   55.75  by (stac insert_absorb 1);
   55.76  by (auto_tac (claset() addSIs [inst "x" "Id" bexI], 
   55.77 @@ -389,7 +389,7 @@
   55.78  qed "AllowedActs_project";
   55.79  Addsimps [AllowedActs_project];
   55.80  
   55.81 -Goal "Allowed (extend h F) = project h UNIV -`` Allowed F";
   55.82 +Goal "Allowed (extend h F) = project h UNIV -` Allowed F";
   55.83  by (simp_tac (simpset() addsimps [AllowedActs_extend, Allowed_def]) 1);
   55.84  by (Blast_tac 1); 
   55.85  qed "Allowed_extend";
   55.86 @@ -422,8 +422,8 @@
   55.87  qed "project_act_Restrict_Id_eq";
   55.88  
   55.89  Goal "project h C (extend h F) = \
   55.90 -\     mk_program (Init F, Restrict (project_set h C) `` Acts F, \
   55.91 -\                 {act. Restrict (project_set h C) act : project_act h `` Restrict C `` (project_act h -`` AllowedActs F)})";
   55.92 +\     mk_program (Init F, Restrict (project_set h C) ` Acts F, \
   55.93 +\                 {act. Restrict (project_set h C) act : project_act h ` Restrict C ` (project_act h -` AllowedActs F)})";
   55.94  by (rtac program_equalityI 1);
   55.95  by (asm_simp_tac (simpset() addsimps [image_eq_UN]) 2);
   55.96  by (Simp_tac 1);
   55.97 @@ -761,14 +761,14 @@
   55.98  qed "OK_extend_iff";
   55.99  
  55.100  Goal "F : X guarantees Y ==> \
  55.101 -\     extend h F : (extend h `` X) guarantees (extend h `` Y)";
  55.102 +\     extend h F : (extend h ` X) guarantees (extend h ` Y)";
  55.103  by (rtac guaranteesI 1);
  55.104  by (Clarify_tac 1);
  55.105  by (blast_tac (claset() addDs [ok_extend_imp_ok_project, 
  55.106                                 extend_Join_eq_extend_D, guaranteesD]) 1);
  55.107  qed "guarantees_imp_extend_guarantees";
  55.108  
  55.109 -Goal "extend h F : (extend h `` X) guarantees (extend h `` Y) \
  55.110 +Goal "extend h F : (extend h ` X) guarantees (extend h ` Y) \
  55.111  \     ==> F : X guarantees Y";
  55.112  by (auto_tac (claset(), simpset() addsimps [guar_def]));
  55.113  by (dres_inst_tac [("x", "extend h G")] spec 1);
  55.114 @@ -778,7 +778,7 @@
  55.115                           inj_extend RS inj_image_mem_iff]) 1);
  55.116  qed "extend_guarantees_imp_guarantees";
  55.117  
  55.118 -Goal "(extend h F : (extend h `` X) guarantees (extend h `` Y)) = \
  55.119 +Goal "(extend h F : (extend h ` X) guarantees (extend h ` Y)) = \
  55.120  \    (F : X guarantees Y)";
  55.121  by (blast_tac (claset() addIs [guarantees_imp_extend_guarantees,
  55.122  			       extend_guarantees_imp_guarantees]) 1);
    56.1 --- a/src/HOL/UNITY/Extend.thy	Tue Jan 09 15:29:17 2001 +0100
    56.2 +++ b/src/HOL/UNITY/Extend.thy	Tue Jan 09 15:32:27 2001 +0100
    56.3 @@ -21,7 +21,7 @@
    56.4       (*Using the locale constant "f", this is  f (h (x,y))) = x*)
    56.5    
    56.6    extend_set :: "['a*'b => 'c, 'a set] => 'c set"
    56.7 -    "extend_set h A == h `` (A <*> UNIV)"
    56.8 +    "extend_set h A == h ` (A <*> UNIV)"
    56.9  
   56.10    project_set :: "['a*'b => 'c, 'c set] => 'a set"
   56.11      "project_set h C == {x. EX y. h(x,y) : C}"
   56.12 @@ -34,16 +34,16 @@
   56.13  
   56.14    extend :: "['a*'b => 'c, 'a program] => 'c program"
   56.15      "extend h F == mk_program (extend_set h (Init F),
   56.16 -			       extend_act h `` Acts F,
   56.17 -			       project_act h -`` AllowedActs F)"
   56.18 +			       extend_act h ` Acts F,
   56.19 +			       project_act h -` AllowedActs F)"
   56.20  
   56.21    (*Argument C allows weak safety laws to be projected*)
   56.22    project :: "['a*'b => 'c, 'c set, 'c program] => 'a program"
   56.23      "project h C F ==
   56.24         mk_program (project_set h (Init F),
   56.25 -		   project_act h `` Restrict C `` Acts F,
   56.26 +		   project_act h ` Restrict C ` Acts F,
   56.27  		   {act. Restrict (project_set h C) act :
   56.28 -		         project_act h `` Restrict C `` AllowedActs F})"
   56.29 +		         project_act h ` Restrict C ` AllowedActs F})"
   56.30  
   56.31  locale Extend =
   56.32    fixes 
    57.1 --- a/src/HOL/UNITY/FP.ML	Tue Jan 09 15:29:17 2001 +0100
    57.2 +++ b/src/HOL/UNITY/FP.ML	Tue Jan 09 15:32:27 2001 +0100
    57.3 @@ -49,11 +49,11 @@
    57.4  qed "FP_weakest";
    57.5  
    57.6  Goalw [FP_def, stable_def, constrains_def]
    57.7 -    "-(FP F) = (UN act: Acts F. -{s. act```{s} <= {s}})";
    57.8 +    "-(FP F) = (UN act: Acts F. -{s. act``{s} <= {s}})";
    57.9  by (Blast_tac 1);
   57.10  qed "Compl_FP";
   57.11  
   57.12 -Goal "A - (FP F) = (UN act: Acts F. A - {s. act```{s} <= {s}})";
   57.13 +Goal "A - (FP F) = (UN act: Acts F. A - {s. act``{s} <= {s}})";
   57.14  by (simp_tac (simpset() addsimps [Diff_eq, Compl_FP]) 1);
   57.15  qed "Diff_FP";
   57.16  
    58.1 --- a/src/HOL/UNITY/Lift_prog.ML	Tue Jan 09 15:29:17 2001 +0100
    58.2 +++ b/src/HOL/UNITY/Lift_prog.ML	Tue Jan 09 15:32:27 2001 +0100
    58.3 @@ -214,14 +214,14 @@
    58.4  (** guarantees **)
    58.5  
    58.6  Goalw [lift_def]
    58.7 -     "(lift i F : (lift i `` X) guarantees (lift i `` Y)) = \
    58.8 +     "(lift i F : (lift i ` X) guarantees (lift i ` Y)) = \
    58.9  \     (F : X guarantees Y)";
   58.10  by (stac (bij_lift_map RS rename_rename_guarantees_eq RS sym) 1);
   58.11  by (asm_simp_tac (simpset() addsimps [o_def]) 1);
   58.12  qed "lift_lift_guarantees_eq";
   58.13  
   58.14  Goal "(lift i F : X guarantees Y) = \
   58.15 -\     (F : (rename (drop_map i) `` X) guarantees (rename (drop_map i) `` Y))";
   58.16 +\     (F : (rename (drop_map i) ` X) guarantees (rename (drop_map i) ` Y))";
   58.17  by (asm_simp_tac 
   58.18      (simpset() addsimps [bij_lift_map RS rename_guarantees_eq_rename_inv,
   58.19  			 lift_def]) 1);
   58.20 @@ -255,11 +255,11 @@
   58.21  
   58.22  (*A set of the form (A <*> UNIV) ignores the second (dummy) state component*)
   58.23  
   58.24 -Goal "(f o fst) -`` A = (f-``A) <*> UNIV";
   58.25 +Goal "(f o fst) -` A = (f-`A) <*> UNIV";
   58.26  by Auto_tac;
   58.27  qed "vimage_o_fst_eq";
   58.28  
   58.29 -Goal "(sub i -``A) <*> UNIV = lift_set i (A <*> UNIV)";
   58.30 +Goal "(sub i -`A) <*> UNIV = lift_set i (A <*> UNIV)";
   58.31  by Auto_tac;
   58.32  qed "vimage_sub_eq_lift_set";
   58.33  
   58.34 @@ -356,7 +356,7 @@
   58.35  qed "lift_transient_eq_disj";
   58.36  
   58.37  (*USELESS??*)
   58.38 -Goal "lift_map i `` (A <*> UNIV) = \
   58.39 +Goal "lift_map i ` (A <*> UNIV) = \
   58.40  \     (UN s:A. UN f. {insert_map i s f}) <*> UNIV";
   58.41  by (auto_tac (claset() addSIs [bexI, image_eqI],
   58.42                simpset() addsimps [lift_map_def]));
   58.43 @@ -475,12 +475,12 @@
   58.44                           UNION_OK_lift_I]) 1); 
   58.45  qed "OK_lift_I";
   58.46  
   58.47 -Goal "Allowed (lift i F) = lift i `` (Allowed F)";
   58.48 +Goal "Allowed (lift i F) = lift i ` (Allowed F)";
   58.49  by (simp_tac (simpset() addsimps [lift_def, Allowed_rename]) 1); 
   58.50  qed "Allowed_lift"; 
   58.51  Addsimps [Allowed_lift];
   58.52  
   58.53 -Goal "lift i `` preserves v = preserves (v o drop_map i)";
   58.54 +Goal "lift i ` preserves v = preserves (v o drop_map i)";
   58.55  by (simp_tac (simpset() addsimps [rename_image_preserves, lift_def, 
   58.56                                    inv_lift_map_eq]) 1); 
   58.57  qed "lift_image_preserves";
    59.1 --- a/src/HOL/UNITY/Lift_prog.thy	Tue Jan 09 15:29:17 2001 +0100
    59.2 +++ b/src/HOL/UNITY/Lift_prog.thy	Tue Jan 09 15:32:27 2001 +0100
    59.3 @@ -25,7 +25,7 @@
    59.4      "drop_map i == %(g, uu). (g i, (delete_map i g, uu))"
    59.5  
    59.6    lift_set :: "[nat, ('b * ((nat=>'b) * 'c)) set] => ((nat=>'b) * 'c) set"
    59.7 -    "lift_set i A == lift_map i `` A"
    59.8 +    "lift_set i A == lift_map i ` A"
    59.9  
   59.10    lift :: "[nat, ('b * ((nat=>'b) * 'c)) program] => ((nat=>'b) * 'c) program"
   59.11      "lift i == rename (lift_map i)"
    60.1 --- a/src/HOL/UNITY/PPROD.ML	Tue Jan 09 15:29:17 2001 +0100
    60.2 +++ b/src/HOL/UNITY/PPROD.ML	Tue Jan 09 15:32:27 2001 +0100
    60.3 @@ -127,7 +127,7 @@
    60.4  by (asm_simp_tac (simpset() addsimps [guarantees_JN_I]) 1);
    60.5  qed "guarantees_PLam_I";
    60.6  
    60.7 -Goal "Allowed (PLam I F) = (INT i:I. lift i `` Allowed(F i))";
    60.8 +Goal "Allowed (PLam I F) = (INT i:I. lift i ` Allowed(F i))";
    60.9  by (simp_tac (simpset() addsimps [PLam_def]) 1); 
   60.10  qed "Allowed_PLam";
   60.11  Addsimps [Allowed_PLam];
    61.1 --- a/src/HOL/UNITY/PriorityAux.ML	Tue Jan 09 15:29:17 2001 +0100
    61.2 +++ b/src/HOL/UNITY/PriorityAux.ML	Tue Jan 09 15:32:27 2001 +0100
    61.3 @@ -20,14 +20,14 @@
    61.4  (* The equalities (above i r = {}) = (A i r = {}) 
    61.5     and (reach i r = {}) = (R i r) rely on the following theorem  *)
    61.6  
    61.7 -Goal "((r^+)```{i} = {}) = (r```{i} = {})";
    61.8 +Goal "((r^+)``{i} = {}) = (r``{i} = {})";
    61.9  by Auto_tac;
   61.10  by (etac trancl_induct 1);
   61.11  by Auto_tac;
   61.12  qed "image0_trancl_iff_image0_r";
   61.13  
   61.14  (* Another form usefull in some situation *)
   61.15 -Goal "(r```{i}={}) = (ALL x. ((i,x):r^+) = False)";
   61.16 +Goal "(r``{i}={}) = (ALL x. ((i,x):r^+) = False)";
   61.17  by Auto_tac;
   61.18  by (dtac (image0_trancl_iff_image0_r RS ssubst) 1);
   61.19  by Auto_tac;
   61.20 @@ -76,7 +76,7 @@
   61.21  
   61.22  (* Lemma 2 *)
   61.23  Goal 
   61.24 -"(z, i):r^+ ==> (ALL y. (y, z):r --> (y, i)~:r^+) = ((r^-1)```{z}={})";
   61.25 +"(z, i):r^+ ==> (ALL y. (y, z):r --> (y, i)~:r^+) = ((r^-1)``{z}={})";
   61.26  by Auto_tac;
   61.27  by (forw_inst_tac [("r", "r")] trancl_into_trancl2 1);
   61.28  by Auto_tac;
   61.29 @@ -86,7 +86,7 @@
   61.30   "acyclic r ==> A i r~={}-->(EX j:above i r. A j r = {})";
   61.31  by (full_simp_tac (simpset() 
   61.32              addsimps [acyclic_eq_wf, wf_eq_minimal]) 1);
   61.33 -by (dres_inst_tac [("x", "((r^-1)^+)```{i}")] spec 1);
   61.34 +by (dres_inst_tac [("x", "((r^-1)^+)``{i}")] spec 1);
   61.35  by Auto_tac;
   61.36  by (rotate_tac ~1 1);
   61.37  by (asm_full_simp_tac (simpset() 
    62.1 --- a/src/HOL/UNITY/PriorityAux.thy	Tue Jan 09 15:29:17 2001 +0100
    62.2 +++ b/src/HOL/UNITY/PriorityAux.thy	Tue Jan 09 15:32:27 2001 +0100
    62.3 @@ -18,20 +18,20 @@
    62.4  
    62.5    (* Neighbors of a vertex i *)
    62.6    neighbors :: "[vertex, (vertex*vertex)set]=>vertex set"
    62.7 - "neighbors i r == ((r Un r^-1)```{i}) - {i}"
    62.8 + "neighbors i r == ((r Un r^-1)``{i}) - {i}"
    62.9  
   62.10    R :: "[vertex, (vertex*vertex)set]=>vertex set"
   62.11 -  "R i r == r```{i}"
   62.12 +  "R i r == r``{i}"
   62.13  
   62.14    A :: "[vertex, (vertex*vertex)set]=>vertex set"
   62.15 -  "A i r == (r^-1)```{i}"
   62.16 +  "A i r == (r^-1)``{i}"
   62.17  
   62.18    (* reachable and above vertices: the original notation was R* and A* *)  
   62.19    reach :: "[vertex, (vertex*vertex)set]=> vertex set"
   62.20 -  "reach i r == (r^+)```{i}"
   62.21 +  "reach i r == (r^+)``{i}"
   62.22  
   62.23    above :: "[vertex, (vertex*vertex)set]=> vertex set"
   62.24 -  "above i r == ((r^-1)^+)```{i}"  
   62.25 +  "above i r == ((r^-1)^+)``{i}"  
   62.26  
   62.27    reverse :: "[vertex, (vertex*vertex) set]=>(vertex*vertex)set"
   62.28    "reverse i r == (r - {(x,y). x=i | y=i} Int r) Un ({(x,y). x=i|y=i} Int r)^-1"
    63.1 --- a/src/HOL/UNITY/Project.ML	Tue Jan 09 15:29:17 2001 +0100
    63.2 +++ b/src/HOL/UNITY/Project.ML	Tue Jan 09 15:32:27 2001 +0100
    63.3 @@ -384,7 +384,7 @@
    63.4       "[| G : transient (C Int extend_set h A);  G : stable C |]  \
    63.5  \     ==> project h C G : transient (project_set h C Int A)";
    63.6  by (auto_tac (claset(), simpset() addsimps [Domain_project_act]));
    63.7 -by (subgoal_tac "act ``` (C Int extend_set h A) <= - extend_set h A" 1);
    63.8 +by (subgoal_tac "act `` (C Int extend_set h A) <= - extend_set h A" 1);
    63.9  by (asm_full_simp_tac 
   63.10      (simpset() addsimps [stable_def, constrains_def]) 2);
   63.11  by (Blast_tac 2);
   63.12 @@ -502,8 +502,8 @@
   63.13  
   63.14  
   63.15  Goalw [project_set_def, extend_set_def, project_act_def]
   63.16 -     "act ``` (C Int extend_set h A) <= B \
   63.17 -\     ==> project_act h (Restrict C act) ``` (project_set h C Int A) \
   63.18 +     "act `` (C Int extend_set h A) <= B \
   63.19 +\     ==> project_act h (Restrict C act) `` (project_set h C Int A) \
   63.20  \         <= project_set h B";
   63.21  by (Blast_tac 1);
   63.22  qed "act_subset_imp_project_act_subset";
   63.23 @@ -512,9 +512,9 @@
   63.24    property upwards.  The hard part would be to 
   63.25    show that G's action has a big enough domain.*)
   63.26  Goal "[| act: Acts G;       \
   63.27 -\        (project_act h (Restrict C act))``` \
   63.28 +\        (project_act h (Restrict C act))`` \
   63.29  \             (project_set h C Int A - B) <= -(project_set h C Int A - B) |] \
   63.30 -\     ==> act```(C Int extend_set h A - extend_set h B) \
   63.31 +\     ==> act``(C Int extend_set h A - extend_set h B) \
   63.32  \           <= -(C Int extend_set h A - extend_set h B)"; 
   63.33  by (auto_tac (claset(), 
   63.34       simpset() addsimps [project_set_def, extend_set_def, project_act_def]));  
   63.35 @@ -535,8 +535,8 @@
   63.36  				  extend_set_Diff_distrib RS sym]));
   63.37  by (dtac act_subset_imp_project_act_subset 1);
   63.38  by (subgoal_tac
   63.39 -    "project_act h (Restrict C act) ``` (project_set h C Int (A - B)) = {}" 1);
   63.40 -by (REPEAT (thin_tac "?r```?A <= ?B" 1));
   63.41 +    "project_act h (Restrict C act) `` (project_set h C Int (A - B)) = {}" 1);
   63.42 +by (REPEAT (thin_tac "?r``?A <= ?B" 1));
   63.43  by (rewrite_goals_tac [project_set_def, extend_set_def, project_act_def]);
   63.44  by (Blast_tac 2);
   63.45  by (rtac ccontr 1);
    64.1 --- a/src/HOL/UNITY/Reach.ML	Tue Jan 09 15:29:17 2001 +0100
    64.2 +++ b/src/HOL/UNITY/Reach.ML	Tue Jan 09 15:32:27 2001 +0100
    64.3 @@ -108,7 +108,7 @@
    64.4  	      simpset() addsimps [fun_upd_idem]));
    64.5  qed "metric_le";
    64.6  
    64.7 -Goal "Rprg : ((metric-``{m}) - fixedpoint) LeadsTo (metric-``(lessThan m))";
    64.8 +Goal "Rprg : ((metric-`{m}) - fixedpoint) LeadsTo (metric-`(lessThan m))";
    64.9  by (simp_tac (simpset() addsimps [Diff_fixedpoint]) 1);
   64.10  by (rtac LeadsTo_UN 1);
   64.11  by Auto_tac;
   64.12 @@ -120,7 +120,7 @@
   64.13  	      simpset()));
   64.14  qed "LeadsTo_Diff_fixedpoint";
   64.15  
   64.16 -Goal "Rprg : (metric-``{m}) LeadsTo (metric-``(lessThan m) Un fixedpoint)";
   64.17 +Goal "Rprg : (metric-`{m}) LeadsTo (metric-`(lessThan m) Un fixedpoint)";
   64.18  by (rtac ([LeadsTo_Diff_fixedpoint RS LeadsTo_weaken_R,
   64.19  	   subset_imp_LeadsTo] MRS LeadsTo_Diff) 1);
   64.20  by Auto_tac;
    65.1 --- a/src/HOL/UNITY/Rename.ML	Tue Jan 09 15:29:17 2001 +0100
    65.2 +++ b/src/HOL/UNITY/Rename.ML	Tue Jan 09 15:32:27 2001 +0100
    65.3 @@ -26,18 +26,18 @@
    65.4  by (etac surj_f_inv_f 1);
    65.5  qed "fst_o_inv_eq_inv";
    65.6  
    65.7 -Goal "bij h ==> z : h``A = (inv h z : A)";
    65.8 +Goal "bij h ==> z : h`A = (inv h z : A)";
    65.9  by (auto_tac (claset() addSIs [image_eqI],
   65.10  	      simpset() addsimps [bij_is_inj, bij_is_surj RS surj_f_inv_f]));
   65.11  qed "mem_rename_set_iff";
   65.12  
   65.13 -Goal "extend_set (%(x,u). h x) A = h``A";
   65.14 +Goal "extend_set (%(x,u). h x) A = h`A";
   65.15  by (auto_tac (claset() addSIs [image_eqI],
   65.16  	      simpset() addsimps [extend_set_def]));
   65.17  qed "extend_set_eq_image";
   65.18  Addsimps [extend_set_eq_image];
   65.19  
   65.20 -Goalw [rename_def] "Init (rename h F) = h``(Init F)";
   65.21 +Goalw [rename_def] "Init (rename h F) = h`(Init F)";
   65.22  by (Simp_tac 1);
   65.23  qed "Init_rename";
   65.24  
   65.25 @@ -145,7 +145,7 @@
   65.26  by (asm_simp_tac (simpset() addsimps [export extend_inverse]) 1); 
   65.27  qed "inv_project_eq";
   65.28  
   65.29 -Goal "bij h ==> Allowed (rename h F) = rename h `` Allowed F";
   65.30 +Goal "bij h ==> Allowed (rename h F) = rename h ` Allowed F";
   65.31  by (asm_simp_tac (simpset() addsimps [rename_def, export Allowed_extend]) 1);
   65.32  by (stac bij_vimage_eq_inv_image 1); 
   65.33  by (rtac bij_project 1); 
   65.34 @@ -209,17 +209,17 @@
   65.35  (*** Strong Safety: co, stable ***)
   65.36  
   65.37  Goalw [rename_def]
   65.38 -     "bij h ==> (rename h F : (h``A) co (h``B)) = (F : A co B)";
   65.39 +     "bij h ==> (rename h F : (h`A) co (h`B)) = (F : A co B)";
   65.40  by (REPEAT (stac (extend_set_eq_image RS sym) 1));
   65.41  by (etac (good_map_bij RS export extend_constrains) 1);
   65.42  qed "rename_constrains";
   65.43  
   65.44  Goalw [stable_def]
   65.45 -     "bij h ==> (rename h F : stable (h``A)) = (F : stable A)";
   65.46 +     "bij h ==> (rename h F : stable (h`A)) = (F : stable A)";
   65.47  by (asm_simp_tac (simpset() addsimps [rename_constrains]) 1);
   65.48  qed "rename_stable";
   65.49  
   65.50 -Goal "bij h ==> (rename h F : invariant (h``A)) = (F : invariant A)";
   65.51 +Goal "bij h ==> (rename h F : invariant (h`A)) = (F : invariant A)";
   65.52  by (asm_simp_tac (simpset() addsimps [invariant_def, rename_stable,
   65.53  				      bij_is_inj RS inj_image_subset_iff]) 1);
   65.54  qed "rename_invariant";
   65.55 @@ -234,22 +234,22 @@
   65.56  (*** Weak Safety: Co, Stable ***)
   65.57  
   65.58  Goalw [rename_def]
   65.59 -     "bij h ==> reachable (rename h F) = h `` (reachable F)";
   65.60 +     "bij h ==> reachable (rename h F) = h ` (reachable F)";
   65.61  by (asm_simp_tac (simpset() addsimps [export reachable_extend_eq]) 1);
   65.62  qed "reachable_rename_eq";
   65.63  
   65.64 -Goal "bij h ==> (rename h F : (h``A) Co (h``B)) = (F : A Co B)";
   65.65 +Goal "bij h ==> (rename h F : (h`A) Co (h`B)) = (F : A Co B)";
   65.66  by (asm_simp_tac
   65.67      (simpset() addsimps [Constrains_def, reachable_rename_eq, 
   65.68  			 rename_constrains, bij_is_inj, image_Int RS sym]) 1);
   65.69  qed "rename_Constrains";
   65.70  
   65.71  Goalw [Stable_def]
   65.72 -     "bij h ==> (rename h F : Stable (h``A)) = (F : Stable A)";
   65.73 +     "bij h ==> (rename h F : Stable (h`A)) = (F : Stable A)";
   65.74  by (asm_simp_tac (simpset() addsimps [rename_Constrains]) 1);
   65.75  qed "rename_Stable";
   65.76  
   65.77 -Goal "bij h ==> (rename h F : Always (h``A)) = (F : Always A)";
   65.78 +Goal "bij h ==> (rename h F : Always (h`A)) = (F : Always A)";
   65.79  by (asm_simp_tac (simpset() addsimps [Always_def, rename_Stable,
   65.80  				      bij_is_inj RS inj_image_subset_iff]) 1);
   65.81  qed "rename_Always";
   65.82 @@ -264,32 +264,32 @@
   65.83  (*** Progress: transient, ensures ***)
   65.84  
   65.85  Goalw [rename_def]
   65.86 -     "bij h ==> (rename h F : transient (h``A)) = (F : transient A)";
   65.87 +     "bij h ==> (rename h F : transient (h`A)) = (F : transient A)";
   65.88  by (stac (extend_set_eq_image RS sym) 1);
   65.89  by (etac (good_map_bij RS export extend_transient) 1);
   65.90  qed "rename_transient";
   65.91  
   65.92  Goalw [rename_def]
   65.93 -     "bij h ==> (rename h F : (h``A) ensures (h``B)) = (F : A ensures B)";
   65.94 +     "bij h ==> (rename h F : (h`A) ensures (h`B)) = (F : A ensures B)";
   65.95  by (REPEAT (stac (extend_set_eq_image RS sym) 1));
   65.96  by (etac (good_map_bij RS export extend_ensures) 1);
   65.97  qed "rename_ensures";
   65.98  
   65.99  Goalw [rename_def]
  65.100 -     "bij h ==> (rename h F : (h``A) leadsTo (h``B)) = (F : A leadsTo B)";
  65.101 +     "bij h ==> (rename h F : (h`A) leadsTo (h`B)) = (F : A leadsTo B)";
  65.102  by (REPEAT (stac (extend_set_eq_image RS sym) 1));
  65.103  by (etac (good_map_bij RS export extend_leadsTo) 1);
  65.104  qed "rename_leadsTo";
  65.105  
  65.106  Goalw [rename_def]
  65.107 -     "bij h ==> (rename h F : (h``A) LeadsTo (h``B)) = (F : A LeadsTo B)";
  65.108 +     "bij h ==> (rename h F : (h`A) LeadsTo (h`B)) = (F : A LeadsTo B)";
  65.109  by (REPEAT (stac (extend_set_eq_image RS sym) 1));
  65.110  by (etac (good_map_bij RS export extend_LeadsTo) 1);
  65.111  qed "rename_LeadsTo";
  65.112  
  65.113  Goalw [rename_def]
  65.114 -     "bij h ==> (rename h F : (rename h `` X) guarantees \
  65.115 -\                             (rename h `` Y)) = \
  65.116 +     "bij h ==> (rename h F : (rename h ` X) guarantees \
  65.117 +\                             (rename h ` Y)) = \
  65.118  \               (F : X guarantees Y)";
  65.119  by (stac (good_map_bij RS export extend_guarantees_eq RS sym) 1);
  65.120  by (assume_tac 1);
  65.121 @@ -297,8 +297,8 @@
  65.122  qed "rename_rename_guarantees_eq";
  65.123  
  65.124  Goal "bij h ==> (rename h F : X guarantees Y) = \
  65.125 -\               (F : (rename (inv h) `` X) guarantees \
  65.126 -\                    (rename (inv h) `` Y))";
  65.127 +\               (F : (rename (inv h) ` X) guarantees \
  65.128 +\                    (rename (inv h) ` Y))";
  65.129  by (stac (rename_rename_guarantees_eq RS sym) 1);
  65.130  by (assume_tac 1);
  65.131  by (asm_simp_tac
  65.132 @@ -336,47 +336,47 @@
  65.133  	 (auto_tac (claset() addSIs [surj_rename RS surj_f_inv_f RS sym],
  65.134  	      simpset() addsimps ths))];
  65.135  
  65.136 -Goal "bij h ==> rename h `` (A co B) = (h `` A) co (h``B)";
  65.137 +Goal "bij h ==> rename h ` (A co B) = (h ` A) co (h`B)";
  65.138  by (rename_image_tac [rename_constrains]);
  65.139  qed "rename_image_constrains";
  65.140  
  65.141 -Goal "bij h ==> rename h `` stable A = stable (h `` A)";
  65.142 +Goal "bij h ==> rename h ` stable A = stable (h ` A)";
  65.143  by (rename_image_tac [rename_stable]);
  65.144  qed "rename_image_stable";
  65.145  
  65.146 -Goal "bij h ==> rename h `` increasing func = increasing (func o inv h)";
  65.147 +Goal "bij h ==> rename h ` increasing func = increasing (func o inv h)";
  65.148  by (rename_image_tac [rename_increasing, o_def, bij_is_inj]);
  65.149  qed "rename_image_increasing";
  65.150  
  65.151 -Goal "bij h ==> rename h `` invariant A = invariant (h `` A)";
  65.152 +Goal "bij h ==> rename h ` invariant A = invariant (h ` A)";
  65.153  by (rename_image_tac [rename_invariant]);
  65.154  qed "rename_image_invariant";
  65.155  
  65.156 -Goal "bij h ==> rename h `` (A Co B) = (h `` A) Co (h``B)";
  65.157 +Goal "bij h ==> rename h ` (A Co B) = (h ` A) Co (h`B)";
  65.158  by (rename_image_tac [rename_Constrains]);
  65.159  qed "rename_image_Constrains";
  65.160  
  65.161 -Goal "bij h ==> rename h `` preserves v = preserves (v o inv h)";
  65.162 +Goal "bij h ==> rename h ` preserves v = preserves (v o inv h)";
  65.163  by (asm_simp_tac (simpset() addsimps [o_def, rename_image_stable,
  65.164                      preserves_def, bij_image_INT, bij_image_Collect_eq]) 1); 
  65.165  qed "rename_image_preserves";
  65.166  
  65.167 -Goal "bij h ==> rename h `` Stable A = Stable (h `` A)";
  65.168 +Goal "bij h ==> rename h ` Stable A = Stable (h ` A)";
  65.169  by (rename_image_tac [rename_Stable]);
  65.170  qed "rename_image_Stable";
  65.171  
  65.172 -Goal "bij h ==> rename h `` Increasing func = Increasing (func o inv h)";
  65.173 +Goal "bij h ==> rename h ` Increasing func = Increasing (func o inv h)";
  65.174  by (rename_image_tac [rename_Increasing, o_def, bij_is_inj]);
  65.175  qed "rename_image_Increasing";
  65.176  
  65.177 -Goal "bij h ==> rename h `` Always A = Always (h `` A)";
  65.178 +Goal "bij h ==> rename h ` Always A = Always (h ` A)";
  65.179  by (rename_image_tac [rename_Always]);
  65.180  qed "rename_image_Always";
  65.181  
  65.182 -Goal "bij h ==> rename h `` (A leadsTo B) = (h `` A) leadsTo (h``B)";
  65.183 +Goal "bij h ==> rename h ` (A leadsTo B) = (h ` A) leadsTo (h`B)";
  65.184  by (rename_image_tac [rename_leadsTo]);
  65.185  qed "rename_image_leadsTo";
  65.186  
  65.187 -Goal "bij h ==> rename h `` (A LeadsTo B) = (h `` A) LeadsTo (h``B)";
  65.188 +Goal "bij h ==> rename h ` (A LeadsTo B) = (h ` A) LeadsTo (h`B)";
  65.189  by (rename_image_tac [rename_LeadsTo]);
  65.190  qed "rename_image_LeadsTo";
    66.1 --- a/src/HOL/UNITY/SubstAx.ML	Tue Jan 09 15:29:17 2001 +0100
    66.2 +++ b/src/HOL/UNITY/SubstAx.ML	Tue Jan 09 15:32:27 2001 +0100
    66.3 @@ -313,8 +313,8 @@
    66.4  
    66.5  (** Meta or object quantifier ????? **)
    66.6  Goal "[| wf r;     \
    66.7 -\        ALL m. F : (A Int f-``{m}) LeadsTo                     \
    66.8 -\                           ((A Int f-``(r^-1 ``` {m})) Un B) |] \
    66.9 +\        ALL m. F : (A Int f-`{m}) LeadsTo                     \
   66.10 +\                           ((A Int f-`(r^-1 `` {m})) Un B) |] \
   66.11  \     ==> F : A LeadsTo B";
   66.12  by (full_simp_tac (simpset() addsimps [LeadsTo_eq_leadsTo]) 1);
   66.13  by (etac leadsTo_wf_induct 1);
   66.14 @@ -323,9 +323,9 @@
   66.15  
   66.16  
   66.17  Goal "[| wf r;     \
   66.18 -\        ALL m:I. F : (A Int f-``{m}) LeadsTo                   \
   66.19 -\                             ((A Int f-``(r^-1 ``` {m})) Un B) |] \
   66.20 -\     ==> F : A LeadsTo ((A - (f-``I)) Un B)";
   66.21 +\        ALL m:I. F : (A Int f-`{m}) LeadsTo                   \
   66.22 +\                             ((A Int f-`(r^-1 `` {m})) Un B) |] \
   66.23 +\     ==> F : A LeadsTo ((A - (f-`I)) Un B)";
   66.24  by (etac LeadsTo_wf_induct 1);
   66.25  by Safe_tac;
   66.26  by (case_tac "m:I" 1);
   66.27 @@ -335,7 +335,7 @@
   66.28  
   66.29  
   66.30  val prems = 
   66.31 -Goal "(!!m::nat. F : (A Int f-``{m}) LeadsTo ((A Int f-``(lessThan m)) Un B)) \
   66.32 +Goal "(!!m::nat. F : (A Int f-`{m}) LeadsTo ((A Int f-`(lessThan m)) Un B)) \
   66.33  \     ==> F : A LeadsTo B";
   66.34  by (rtac (wf_less_than RS LeadsTo_wf_induct) 1);
   66.35  by (auto_tac (claset() addIs prems, simpset()));
   66.36 @@ -353,17 +353,17 @@
   66.37  by (auto_tac (claset(), simpset() addsimps [nat_eq_iff, nat_less_iff]));
   66.38  qed "integ_0_le_induct";
   66.39  
   66.40 -Goal "!!l::nat. [| ALL m:(greaterThan l). F : (A Int f-``{m}) LeadsTo   \
   66.41 -\                                        ((A Int f-``(lessThan m)) Un B) |] \
   66.42 -\           ==> F : A LeadsTo ((A Int (f-``(atMost l))) Un B)";
   66.43 +Goal "!!l::nat. [| ALL m:(greaterThan l). F : (A Int f-`{m}) LeadsTo   \
   66.44 +\                                        ((A Int f-`(lessThan m)) Un B) |] \
   66.45 +\           ==> F : A LeadsTo ((A Int (f-`(atMost l))) Un B)";
   66.46  by (simp_tac (HOL_ss addsimps [Diff_eq RS sym, vimage_Compl, Compl_greaterThan RS sym]) 1);
   66.47  by (rtac (wf_less_than RS Bounded_induct) 1);
   66.48  by (Asm_simp_tac 1);
   66.49  qed "LessThan_bounded_induct";
   66.50  
   66.51 -Goal "!!l::nat. [| ALL m:(lessThan l). F : (A Int f-``{m}) LeadsTo   \
   66.52 -\                              ((A Int f-``(greaterThan m)) Un B) |] \
   66.53 -\     ==> F : A LeadsTo ((A Int (f-``(atLeast l))) Un B)";
   66.54 +Goal "!!l::nat. [| ALL m:(lessThan l). F : (A Int f-`{m}) LeadsTo   \
   66.55 +\                              ((A Int f-`(greaterThan m)) Un B) |] \
   66.56 +\     ==> F : A LeadsTo ((A Int (f-`(atLeast l))) Un B)";
   66.57  by (res_inst_tac [("f","f"),("f1", "%k. l - k")]
   66.58      (wf_less_than RS wf_inv_image RS LeadsTo_wf_induct) 1);
   66.59  by (simp_tac (simpset() addsimps [inv_image_def, Image_singleton]) 1);
    67.1 --- a/src/HOL/UNITY/Token.ML	Tue Jan 09 15:29:17 2001 +0100
    67.2 +++ b/src/HOL/UNITY/Token.ML	Tue Jan 09 15:32:27 2001 +0100
    67.3 @@ -74,8 +74,7 @@
    67.4  by (auto_tac (claset(), simpset() addsimps [HasTok_def, nodeOrder_eq]));
    67.5  qed "TR7_aux";
    67.6  
    67.7 -Goal "({s. token s < N} Int token -`` {m}) = \
    67.8 -\     (if m<N then token -`` {m} else {})";
    67.9 +Goal "({s. token s < N} Int token -` {m}) = (if m<N then token -` {m} else {})";
   67.10  by Auto_tac;
   67.11  val token_lemma = result();
   67.12  
    68.1 --- a/src/HOL/UNITY/UNITY.ML	Tue Jan 09 15:29:17 2001 +0100
    68.2 +++ b/src/HOL/UNITY/UNITY.ML	Tue Jan 09 15:32:27 2001 +0100
    68.3 @@ -383,17 +383,17 @@
    68.4  qed "Un_Diff_Diff";
    68.5  Addsimps [Un_Diff_Diff];
    68.6  
    68.7 -Goal "Union(B) Int A = Union((%C. C Int A)``B)";
    68.8 +Goal "Union(B) Int A = Union((%C. C Int A)`B)";
    68.9  by (Blast_tac 1);
   68.10  qed "Int_Union_Union";
   68.11  
   68.12  (** Needed for WF reasoning in WFair.ML **)
   68.13  
   68.14 -Goal "less_than ``` {k} = greaterThan k";
   68.15 +Goal "less_than `` {k} = greaterThan k";
   68.16  by (Blast_tac 1);
   68.17  qed "Image_less_than";
   68.18  
   68.19 -Goal "less_than^-1 ``` {k} = lessThan k";
   68.20 +Goal "less_than^-1 `` {k} = lessThan k";
   68.21  by (Blast_tac 1);
   68.22  qed "Image_inverse_less_than";
   68.23  
    69.1 --- a/src/HOL/UNITY/UNITY.thy	Tue Jan 09 15:29:17 2001 +0100
    69.2 +++ b/src/HOL/UNITY/UNITY.thy	Tue Jan 09 15:32:27 2001 +0100
    69.3 @@ -51,7 +51,7 @@
    69.4  
    69.5  
    69.6  defs
    69.7 -  constrains_def "A co B == {F. ALL act: Acts F. act```A <= B}"
    69.8 +  constrains_def "A co B == {F. ALL act: Acts F. act``A <= B}"
    69.9  
   69.10    unless_def     "A unless B == (A-B) co (A Un B)"
   69.11  
    70.1 --- a/src/HOL/UNITY/WFair.ML	Tue Jan 09 15:29:17 2001 +0100
    70.2 +++ b/src/HOL/UNITY/WFair.ML	Tue Jan 09 15:32:27 2001 +0100
    70.3 @@ -27,14 +27,14 @@
    70.4  qed "transient_strengthen";
    70.5  
    70.6  Goalw [transient_def]
    70.7 -    "[| act: Acts F;  A <= Domain act;  act```A <= -A |] ==> F : transient A";
    70.8 +    "[| act: Acts F;  A <= Domain act;  act``A <= -A |] ==> F : transient A";
    70.9  by (Blast_tac 1);
   70.10  qed "transientI";
   70.11  
   70.12  val major::prems = 
   70.13  Goalw [transient_def]
   70.14      "[| F : transient A;  \
   70.15 -\       !!act. [| act: Acts F;  A <= Domain act;  act```A <= -A |] ==> P |] \
   70.16 +\       !!act. [| act: Acts F;  A <= Domain act;  act``A <= -A |] ==> P |] \
   70.17  \    ==> P";
   70.18  by (rtac (major RS CollectD RS bexE) 1);
   70.19  by (blast_tac (claset() addIs prems) 1);
   70.20 @@ -361,11 +361,11 @@
   70.21  (** The most general rule: r is any wf relation; f is any variant function **)
   70.22  
   70.23  Goal "[| wf r;     \
   70.24 -\        ALL m. F : (A Int f-``{m}) leadsTo                     \
   70.25 -\                   ((A Int f-``(r^-1 ``` {m})) Un B) |] \
   70.26 -\     ==> F : (A Int f-``{m}) leadsTo B";
   70.27 +\        ALL m. F : (A Int f-`{m}) leadsTo                     \
   70.28 +\                   ((A Int f-`(r^-1 `` {m})) Un B) |] \
   70.29 +\     ==> F : (A Int f-`{m}) leadsTo B";
   70.30  by (eres_inst_tac [("a","m")] wf_induct 1);
   70.31 -by (subgoal_tac "F : (A Int (f -`` (r^-1 ``` {x}))) leadsTo B" 1);
   70.32 +by (subgoal_tac "F : (A Int (f -` (r^-1 `` {x}))) leadsTo B" 1);
   70.33  by (stac vimage_eq_UN 2);
   70.34  by (asm_simp_tac (HOL_ss addsimps (UN_simps RL [sym])) 2);
   70.35  by (blast_tac (claset() addIs [leadsTo_UN]) 2);
   70.36 @@ -375,8 +375,8 @@
   70.37  
   70.38  (** Meta or object quantifier ? **)
   70.39  Goal "[| wf r;     \
   70.40 -\        ALL m. F : (A Int f-``{m}) leadsTo                     \
   70.41 -\                   ((A Int f-``(r^-1 ``` {m})) Un B) |] \
   70.42 +\        ALL m. F : (A Int f-`{m}) leadsTo                     \
   70.43 +\                   ((A Int f-`(r^-1 `` {m})) Un B) |] \
   70.44  \     ==> F : A leadsTo B";
   70.45  by (res_inst_tac [("t", "A")] subst 1);
   70.46  by (rtac leadsTo_UN 2);
   70.47 @@ -387,9 +387,9 @@
   70.48  
   70.49  
   70.50  Goal "[| wf r;     \
   70.51 -\        ALL m:I. F : (A Int f-``{m}) leadsTo                   \
   70.52 -\                     ((A Int f-``(r^-1 ``` {m})) Un B) |] \
   70.53 -\     ==> F : A leadsTo ((A - (f-``I)) Un B)";
   70.54 +\        ALL m:I. F : (A Int f-`{m}) leadsTo                   \
   70.55 +\                     ((A Int f-`(r^-1 `` {m})) Un B) |] \
   70.56 +\     ==> F : A leadsTo ((A - (f-`I)) Un B)";
   70.57  by (etac leadsTo_wf_induct 1);
   70.58  by Safe_tac;
   70.59  by (case_tac "m:I" 1);
   70.60 @@ -398,9 +398,9 @@
   70.61  qed "bounded_induct";
   70.62  
   70.63  
   70.64 -(*Alternative proof is via the lemma F : (A Int f-``(lessThan m)) leadsTo B*)
   70.65 +(*Alternative proof is via the lemma F : (A Int f-`(lessThan m)) leadsTo B*)
   70.66  val prems = 
   70.67 -Goal "[| !!m::nat. F : (A Int f-``{m}) leadsTo ((A Int f-``{..m(}) Un B) |] \
   70.68 +Goal "[| !!m::nat. F : (A Int f-`{m}) leadsTo ((A Int f-`{..m(}) Un B) |] \
   70.69  \     ==> F : A leadsTo B";
   70.70  by (rtac (wf_less_than RS leadsTo_wf_induct) 1);
   70.71  by (Asm_simp_tac 1);
   70.72 @@ -408,8 +408,8 @@
   70.73  qed "lessThan_induct";
   70.74  
   70.75  Goal "!!l::nat. [| ALL m:(greaterThan l).    \
   70.76 -\           F : (A Int f-``{m}) leadsTo ((A Int f-``(lessThan m)) Un B) |] \
   70.77 -\     ==> F : A leadsTo ((A Int (f-``(atMost l))) Un B)";
   70.78 +\           F : (A Int f-`{m}) leadsTo ((A Int f-`(lessThan m)) Un B) |] \
   70.79 +\     ==> F : A leadsTo ((A Int (f-`(atMost l))) Un B)";
   70.80  by (simp_tac (HOL_ss addsimps [Diff_eq RS sym, vimage_Compl, 
   70.81  			       Compl_greaterThan RS sym]) 1);
   70.82  by (rtac (wf_less_than RS bounded_induct) 1);
   70.83 @@ -417,8 +417,8 @@
   70.84  qed "lessThan_bounded_induct";
   70.85  
   70.86  Goal "!!l::nat. [| ALL m:(lessThan l).    \
   70.87 -\           F : (A Int f-``{m}) leadsTo ((A Int f-``(greaterThan m)) Un B) |] \
   70.88 -\     ==> F : A leadsTo ((A Int (f-``(atLeast l))) Un B)";
   70.89 +\           F : (A Int f-`{m}) leadsTo ((A Int f-`(greaterThan m)) Un B) |] \
   70.90 +\     ==> F : A leadsTo ((A Int (f-`(atLeast l))) Un B)";
   70.91  by (res_inst_tac [("f","f"),("f1", "%k. l - k")]
   70.92      (wf_less_than RS wf_inv_image RS leadsTo_wf_induct) 1);
   70.93  by (simp_tac (simpset() addsimps [inv_image_def, Image_singleton]) 1);
    71.1 --- a/src/HOL/UNITY/WFair.thy	Tue Jan 09 15:29:17 2001 +0100
    71.2 +++ b/src/HOL/UNITY/WFair.thy	Tue Jan 09 15:32:27 2001 +0100
    71.3 @@ -15,7 +15,7 @@
    71.4    (*This definition specifies weak fairness.  The rest of the theory
    71.5      is generic to all forms of fairness.*)
    71.6    transient :: "'a set => 'a program set"
    71.7 -    "transient A == {F. EX act: Acts F. A <= Domain act & act```A <= -A}"
    71.8 +    "transient A == {F. EX act: Acts F. A <= Domain act & act``A <= -A}"
    71.9  
   71.10    ensures :: "['a set, 'a set] => 'a program set"       (infixl 60)
   71.11      "A ensures B == (A-B co A Un B) Int transient (A-B)"
    72.1 --- a/src/HOL/ex/Multiquote.thy	Tue Jan 09 15:29:17 2001 +0100
    72.2 +++ b/src/HOL/ex/Multiquote.thy	Tue Jan 09 15:32:27 2001 +0100
    72.3 @@ -13,7 +13,7 @@
    72.4  
    72.5  syntax
    72.6    "_quote" :: "'b => ('a => 'b)"	     (".'(_')." [0] 1000)
    72.7 -  "_antiquote" :: "('a => 'b) => 'b"         ("`_" [1000] 1000)
    72.8 +  "_antiquote" :: "('a => 'b) => 'b"         ("_" [1000] 1000)
    72.9  
   72.10  parse_translation {*
   72.11    let
   72.12 @@ -35,14 +35,14 @@
   72.13  
   72.14  text {* basic examples *}
   72.15  term ".(a + b + c)."
   72.16 -term ".(a + b + c + `x + `y + 1)."
   72.17 -term ".(`(f w) + `x)."
   72.18 -term ".(f `x `y z)."
   72.19 +term ".(a + b + c + x + y + 1)."
   72.20 +term ".((f w) + x)."
   72.21 +term ".(f x y z)."
   72.22  
   72.23  text {* advanced examples *}
   72.24 -term ".(.(` `x + `y).)."
   72.25 -term ".(.(` `x + `y). o `f)."
   72.26 -term ".(`(f o `g))."
   72.27 -term ".(.( ` `(f o `g) ).)."
   72.28 +term ".(.( x + y).)."
   72.29 +term ".(.( x + y). o f)."
   72.30 +term ".((f o g))."
   72.31 +term ".(.(  (f o g) ).)."
   72.32  
   72.33  end
    73.1 --- a/src/HOL/ex/PiSets.ML	Tue Jan 09 15:29:17 2001 +0100
    73.2 +++ b/src/HOL/ex/PiSets.ML	Tue Jan 09 15:32:27 2001 +0100
    73.3 @@ -38,7 +38,7 @@
    73.4  
    73.5  
    73.6  
    73.7 -Goal "PiBij A B `` (Pi A B) = Graph A B";
    73.8 +Goal "PiBij A B ` (Pi A B) = Graph A B";
    73.9  by (rtac equalityI 1);
   73.10  by (force_tac (claset(), simpset() addsimps [image_def,PiBij_in_Graph]) 1);
   73.11  by (rtac subsetI 1);
    74.1 --- a/src/HOL/ex/Tarski.ML	Tue Jan 09 15:29:17 2001 +0100
    74.2 +++ b/src/HOL/ex/Tarski.ML	Tue Jan 09 15:32:27 2001 +0100
    74.3 @@ -400,7 +400,7 @@
    74.4  by (simp_tac (simpset() addsimps PO_simp) 1);
    74.5  qed "CLF_E2";
    74.6  
    74.7 -Goal "f : CLF ``` {cl} ==> f : CLF ``` {dual cl}";
    74.8 +Goal "f : CLF `` {cl} ==> f : CLF `` {dual cl}";
    74.9  by (afs [CLF_def, CL_dualCL, monotone_dual] 1); 
   74.10  by (afs [dualA_iff] 1);
   74.11  qed "CLF_dual";
    75.1 --- a/src/HOL/ex/Tarski.thy	Tue Jan 09 15:29:17 2001 +0100
    75.2 +++ b/src/HOL/ex/Tarski.thy	Tue Jan 09 15:32:27 2001 +0100
    75.3 @@ -94,7 +94,7 @@
    75.4    "@SL"  :: "['a set, 'a potype] => bool" ("_ <<= _" [51,50]50)
    75.5  
    75.6  translations
    75.7 -  "S <<= cl" == "S : sublattice ``` {cl}"
    75.8 +  "S <<= cl" == "S : sublattice `` {cl}"
    75.9  
   75.10  constdefs
   75.11    dual :: "'a potype => 'a potype"
   75.12 @@ -121,7 +121,7 @@
   75.13    f :: "'a => 'a"
   75.14    P :: "'a set"
   75.15  assumes 
   75.16 -  f_cl "f : CLF```{cl}"
   75.17 +  f_cl "f : CLF``{cl}"
   75.18  defines
   75.19    P_def "P == fix f A"
   75.20  
    76.1 --- a/src/HOL/ex/set.ML	Tue Jan 09 15:29:17 2001 +0100
    76.2 +++ b/src/HOL/ex/set.ML	Tue Jan 09 15:32:27 2001 +0100
    76.3 @@ -25,12 +25,12 @@
    76.4  (** Examples for the Blast_tac paper **)
    76.5  
    76.6  (*Union-image, called Un_Union_image on equalities.ML*)
    76.7 -Goal "(UN x:C. f(x) Un g(x)) = Union(f``C)  Un  Union(g``C)";
    76.8 +Goal "(UN x:C. f(x) Un g(x)) = Union(f`C)  Un  Union(g`C)";
    76.9  by (Blast_tac 1);
   76.10  qed "";
   76.11  
   76.12  (*Inter-image, called Int_Inter_image on equalities.ML*)
   76.13 -Goal "(INT x:C. f(x) Int g(x)) = Inter(f``C) Int Inter(g``C)";
   76.14 +Goal "(INT x:C. f(x) Int g(x)) = Inter(f`C) Int Inter(g`C)";
   76.15  by (Blast_tac 1);
   76.16  qed "";
   76.17  
   76.18 @@ -83,24 +83,24 @@
   76.19  
   76.20  (*** The Schroeder-Berstein Theorem ***)
   76.21  
   76.22 -Goal "[| -(f``X) = g``(-X);  f(a)=g(b);  a:X |] ==> b:X";
   76.23 +Goal "[| -(f`X) = g`(-X);  f(a)=g(b);  a:X |] ==> b:X";
   76.24  by (Blast_tac 1);
   76.25  qed "disj_lemma";
   76.26  
   76.27 -Goal "-(f``X) = g``(-X) ==> surj(%z. if z:X then f(z) else g(z))";
   76.28 +Goal "-(f`X) = g`(-X) ==> surj(%z. if z:X then f(z) else g(z))";
   76.29  by (asm_simp_tac (simpset() addsimps [surj_def]) 1);
   76.30  by (Blast_tac 1);
   76.31  qed "surj_if_then_else";
   76.32  
   76.33  Goalw [inj_on_def]
   76.34 -     "[| inj_on f X;  inj_on g (-X);  -(f``X) = g``(-X); \
   76.35 +     "[| inj_on f X;  inj_on g (-X);  -(f`X) = g`(-X); \
   76.36  \        h = (%z. if z:X then f(z) else g(z)) |]       \
   76.37  \     ==> inj(h) & surj(h)";
   76.38  by (asm_simp_tac (simpset() addsimps [surj_if_then_else]) 1);
   76.39  by (blast_tac (claset() addDs [disj_lemma, sym]) 1);
   76.40  qed "bij_if_then_else";
   76.41  
   76.42 -Goal "EX X. X = - (g``(- (f``X)))";
   76.43 +Goal "EX X. X = - (g`(- (f`X)))";
   76.44  by (rtac exI 1);
   76.45  by (rtac lfp_unfold 1);
   76.46  by (REPEAT (ares_tac [monoI, image_mono, Compl_anti_mono] 1));
    77.1 --- a/src/HOLCF/Cfun1.ML	Tue Jan 09 15:29:17 2001 +0100
    77.2 +++ b/src/HOLCF/Cfun1.ML	Tue Jan 09 15:32:27 2001 +0100
    77.3 @@ -50,15 +50,15 @@
    77.4  (* lemmas about application of continuous functions                         *)
    77.5  (* ------------------------------------------------------------------------ *)
    77.6  
    77.7 -Goal "[| f=g; x=y |] ==> f`x = g`y";
    77.8 +Goal "[| f=g; x=y |] ==> f$x = g$y";
    77.9  by (Asm_simp_tac 1);
   77.10  qed "cfun_cong";
   77.11  
   77.12 -Goal "f=g ==> f`x = g`x";
   77.13 +Goal "f=g ==> f$x = g$x";
   77.14  by (Asm_simp_tac 1);
   77.15  qed "cfun_fun_cong";
   77.16  
   77.17 -Goal "x=y ==> f`x = f`y";
   77.18 +Goal "x=y ==> f$x = f$y";
   77.19  by (Asm_simp_tac 1);
   77.20  qed "cfun_arg_cong";
   77.21  
   77.22 @@ -77,7 +77,7 @@
   77.23  (* simplification of application                                            *)
   77.24  (* ------------------------------------------------------------------------ *)
   77.25  
   77.26 -Goal "cont f ==> (Abs_CFun f)`x = f x";
   77.27 +Goal "cont f ==> (Abs_CFun f)$x = f x";
   77.28  by (etac (Abs_Cfun_inverse2 RS fun_cong) 1);
   77.29  qed "Cfunapp2";
   77.30  
   77.31 @@ -85,7 +85,7 @@
   77.32  (* beta - equality for continuous functions                                 *)
   77.33  (* ------------------------------------------------------------------------ *)
   77.34  
   77.35 -Goal "cont(c1) ==> (LAM x .c1 x)`u = c1 u";
   77.36 +Goal "cont(c1) ==> (LAM x .c1 x)$u = c1 u";
   77.37  by (rtac Cfunapp2 1);
   77.38  by (atac 1);
   77.39  qed "beta_cfun";
    78.1 --- a/src/HOLCF/Cfun1.thy	Tue Jan 09 15:29:17 2001 +0100
    78.2 +++ b/src/HOLCF/Cfun1.thy	Tue Jan 09 15:32:27 2001 +0100
    78.3 @@ -17,7 +17,7 @@
    78.4  instance "->"  :: (cpo,cpo)sq_ord
    78.5  
    78.6  syntax
    78.7 -	Rep_CFun  :: "('a -> 'b)=>('a => 'b)" ("_`_" [999,1000] 999)
    78.8 +	Rep_CFun  :: "('a -> 'b)=>('a => 'b)" ("_$_" [999,1000] 999)
    78.9                                                  (* application      *)
   78.10          Abs_CFun  :: "('a => 'b)=>('a -> 'b)"     (binder "LAM " 10)
   78.11                                                  (* abstraction      *)
    79.1 --- a/src/HOLCF/Cfun2.ML	Tue Jan 09 15:29:17 2001 +0100
    79.2 +++ b/src/HOLCF/Cfun2.ML	Tue Jan 09 15:32:27 2001 +0100
    79.3 @@ -63,10 +63,10 @@
    79.4  (* ------------------------------------------------------------------------ *)
    79.5  
    79.6  bind_thm ("cont_cfun_arg", (cont_Rep_CFun2 RS contE RS spec RS mp));
    79.7 -(* chain(?x1) ==> range (%i. ?fo3`(?x1 i)) <<| ?fo3`(lub (range ?x1))    *)
    79.8 +(* chain(?x1) ==> range (%i. ?fo3$(?x1 i)) <<| ?fo3$(lub (range ?x1))    *)
    79.9   
   79.10  bind_thm ("contlub_cfun_arg", (contlub_Rep_CFun2 RS contlubE RS spec RS mp));
   79.11 -(* chain(?x1) ==> ?fo4`(lub (range ?x1)) = lub (range (%i. ?fo4`(?x1 i))) *)
   79.12 +(* chain(?x1) ==> ?fo4$(lub (range ?x1)) = lub (range (%i. ?fo4$(?x1 i))) *)
   79.13  
   79.14  
   79.15  (* ------------------------------------------------------------------------ *)
   79.16 @@ -83,7 +83,7 @@
   79.17  (* monotonicity of application Rep_CFun in mixfix syntax [_]_                   *)
   79.18  (* ------------------------------------------------------------------------ *)
   79.19  
   79.20 -Goal  "f1 << f2 ==> f1`x << f2`x";
   79.21 +Goal  "f1 << f2 ==> f1$x << f2$x";
   79.22  by (res_inst_tac [("x","x")] spec 1);
   79.23  by (rtac (less_fun RS subst) 1);
   79.24  by (etac (monofun_Rep_CFun1 RS monofunE RS spec RS spec RS mp) 1);
   79.25 @@ -91,20 +91,20 @@
   79.26  
   79.27  
   79.28  bind_thm ("monofun_cfun_arg", monofun_Rep_CFun2 RS monofunE RS spec RS spec RS mp);
   79.29 -(* ?x2 << ?x1 ==> ?fo5`?x2 << ?fo5`?x1                                      *)
   79.30 +(* ?x2 << ?x1 ==> ?fo5$?x2 << ?fo5$?x1                                      *)
   79.31  
   79.32  (* ------------------------------------------------------------------------ *)
   79.33  (* monotonicity of Rep_CFun in both arguments in mixfix syntax [_]_             *)
   79.34  (* ------------------------------------------------------------------------ *)
   79.35  
   79.36 -Goal "[|f1<<f2;x1<<x2|] ==> f1`x1 << f2`x2";
   79.37 +Goal "[|f1<<f2;x1<<x2|] ==> f1$x1 << f2$x2";
   79.38  by (rtac trans_less 1);
   79.39  by (etac monofun_cfun_arg 1);
   79.40  by (etac monofun_cfun_fun 1);
   79.41  qed "monofun_cfun";
   79.42  
   79.43  
   79.44 -Goal "f`x = UU ==> f`UU = UU";
   79.45 +Goal "f$x = UU ==> f$UU = UU";
   79.46  by (rtac (eq_UU_iff RS iffD2) 1);
   79.47  by (etac subst 1);
   79.48  by (rtac (minimal RS monofun_cfun_arg) 1);
   79.49 @@ -116,13 +116,13 @@
   79.50  (* use MF2 lemmas from Cont.ML                                              *)
   79.51  (* ------------------------------------------------------------------------ *)
   79.52  
   79.53 -Goal "chain(Y) ==> chain(%i. f`(Y i))";
   79.54 +Goal "chain(Y) ==> chain(%i. f$(Y i))";
   79.55  by (etac (monofun_Rep_CFun2 RS ch2ch_MF2R) 1);
   79.56  qed "ch2ch_Rep_CFunR";
   79.57  
   79.58  
   79.59  bind_thm ("ch2ch_Rep_CFunL", monofun_Rep_CFun1 RS ch2ch_MF2L);
   79.60 -(* chain(?F) ==> chain (%i. ?F i`?x)                                  *)
   79.61 +(* chain(?F) ==> chain (%i. ?F i$?x)                                  *)
   79.62  
   79.63  
   79.64  (* ------------------------------------------------------------------------ *)
   79.65 @@ -130,7 +130,7 @@
   79.66  (* use MF2 lemmas from Cont.ML                                              *)
   79.67  (* ------------------------------------------------------------------------ *)
   79.68  
   79.69 -Goal "chain(F) ==> monofun(% x. lub(range(% j.(F j)`x)))";
   79.70 +Goal "chain(F) ==> monofun(% x. lub(range(% j.(F j)$x)))";
   79.71  by (rtac lub_MF2_mono 1);
   79.72  by (rtac monofun_Rep_CFun1 1);
   79.73  by (rtac (monofun_Rep_CFun2 RS allI) 1);
   79.74 @@ -143,8 +143,8 @@
   79.75  (* ------------------------------------------------------------------------ *)
   79.76  
   79.77  Goal "[| chain(F); chain(Y) |] ==>\
   79.78 -\               lub(range(%j. lub(range(%i. F(j)`(Y i))))) =\
   79.79 -\               lub(range(%i. lub(range(%j. F(j)`(Y i)))))";
   79.80 +\               lub(range(%j. lub(range(%i. F(j)$(Y i))))) =\
   79.81 +\               lub(range(%i. lub(range(%j. F(j)$(Y i)))))";
   79.82  by (rtac ex_lubMF2 1);
   79.83  by (rtac monofun_Rep_CFun1 1);
   79.84  by (rtac (monofun_Rep_CFun2 RS allI) 1);
   79.85 @@ -156,7 +156,7 @@
   79.86  (* the lub of a chain of cont. functions is continuous                      *)
   79.87  (* ------------------------------------------------------------------------ *)
   79.88  
   79.89 -Goal "chain(F) ==> cont(% x. lub(range(% j. F(j)`x)))";
   79.90 +Goal "chain(F) ==> cont(% x. lub(range(% j. F(j)$x)))";
   79.91  by (rtac monocontlub2cont 1);
   79.92  by (etac lub_cfun_mono 1);
   79.93  by (rtac contlubI 1);
   79.94 @@ -171,7 +171,7 @@
   79.95  (* type 'a -> 'b is chain complete                                          *)
   79.96  (* ------------------------------------------------------------------------ *)
   79.97  
   79.98 -Goal "chain(CCF) ==> range(CCF) <<| (LAM x. lub(range(% i. CCF(i)`x)))";
   79.99 +Goal "chain(CCF) ==> range(CCF) <<| (LAM x. lub(range(% i. CCF(i)$x)))";
  79.100  by (rtac is_lubI 1);
  79.101  by (rtac ub_rangeI 1);
  79.102  by (stac less_cfun 1);
  79.103 @@ -189,7 +189,7 @@
  79.104  
  79.105  bind_thm ("thelub_cfun", lub_cfun RS thelubI);
  79.106  (* 
  79.107 -chain(?CCF1) ==>  lub (range ?CCF1) = (LAM x. lub (range (%i. ?CCF1 i`x)))
  79.108 +chain(?CCF1) ==>  lub (range ?CCF1) = (LAM x. lub (range (%i. ?CCF1 i$x)))
  79.109  *)
  79.110  
  79.111  Goal "chain(CCF::nat=>('a->'b)) ==> ? x. range(CCF) <<| x";
  79.112 @@ -202,7 +202,7 @@
  79.113  (* Extensionality in 'a -> 'b                                               *)
  79.114  (* ------------------------------------------------------------------------ *)
  79.115  
  79.116 -val prems = Goal "(!!x. f`x = g`x) ==> f = g";
  79.117 +val prems = Goal "(!!x. f$x = g$x) ==> f = g";
  79.118  by (res_inst_tac [("t","f")] (Rep_Cfun_inverse RS subst) 1);
  79.119  by (res_inst_tac [("t","g")] (Rep_Cfun_inverse RS subst) 1);
  79.120  by (res_inst_tac [("f","Abs_CFun")] arg_cong 1);
  79.121 @@ -227,7 +227,7 @@
  79.122  (* Extenionality wrt. << in 'a -> 'b                                        *)
  79.123  (* ------------------------------------------------------------------------ *)
  79.124  
  79.125 -val prems = Goal "(!!x. f`x << g`x) ==> f << g";
  79.126 +val prems = Goal "(!!x. f$x << g$x) ==> f << g";
  79.127  by (res_inst_tac [("t","f")] (Rep_Cfun_inverse RS subst) 1);
  79.128  by (res_inst_tac [("t","g")] (Rep_Cfun_inverse RS subst) 1);
  79.129  by (rtac semi_monofun_Abs_CFun 1);
    80.1 --- a/src/HOLCF/Cfun3.ML	Tue Jan 09 15:29:17 2001 +0100
    80.2 +++ b/src/HOLCF/Cfun3.ML	Tue Jan 09 15:32:27 2001 +0100
    80.3 @@ -47,7 +47,7 @@
    80.4  
    80.5  Goal 
    80.6  "chain(FY) ==>\
    80.7 -\ lub(range FY)`x = lub(range (%i. FY(i)`x))";
    80.8 +\ lub(range FY)$x = lub(range (%i. FY(i)$x))";
    80.9  by (rtac trans 1);
   80.10  by (etac (contlub_Rep_CFun1 RS contlubE RS spec RS mp RS fun_cong) 1);
   80.11  by (stac thelub_fun 1);
   80.12 @@ -58,7 +58,7 @@
   80.13  
   80.14  Goal 
   80.15  "chain(FY) ==>\
   80.16 -\ range(%i. FY(i)`x) <<| lub(range FY)`x";
   80.17 +\ range(%i. FY(i)$x) <<| lub(range FY)$x";
   80.18  by (rtac thelubE 1);
   80.19  by (etac ch2ch_Rep_CFunL 1);
   80.20  by (etac (contlub_cfun_fun RS sym) 1);
   80.21 @@ -71,7 +71,7 @@
   80.22  
   80.23  Goal 
   80.24  "[|chain(FY);chain(TY)|] ==>\
   80.25 -\ (lub(range FY))`(lub(range TY)) = lub(range(%i. FY(i)`(TY i)))";
   80.26 +\ (lub(range FY))$(lub(range TY)) = lub(range(%i. FY(i)$(TY i)))";
   80.27  by (rtac contlub_CF2 1);
   80.28  by (rtac cont_Rep_CFun1 1);
   80.29  by (rtac allI 1);
   80.30 @@ -82,7 +82,7 @@
   80.31  
   80.32  Goal 
   80.33  "[|chain(FY);chain(TY)|] ==>\
   80.34 -\ range(%i.(FY i)`(TY i)) <<| (lub (range FY))`(lub(range TY))";
   80.35 +\ range(%i.(FY i)$(TY i)) <<| (lub (range FY))$(lub(range TY))";
   80.36  by (rtac thelubE 1);
   80.37  by (rtac (monofun_Rep_CFun1 RS ch2ch_MF2LR) 1);
   80.38  by (rtac allI 1);
   80.39 @@ -98,7 +98,7 @@
   80.40  (* cont2cont lemma for Rep_CFun                                               *)
   80.41  (* ------------------------------------------------------------------------ *)
   80.42  
   80.43 -Goal "[|cont(%x. ft x);cont(%x. tt x)|] ==> cont(%x. (ft x)`(tt x))";
   80.44 +Goal "[|cont(%x. ft x);cont(%x. tt x)|] ==> cont(%x. (ft x)$(tt x))";
   80.45  by (best_tac (claset() addIs [cont2cont_app2, cont_const, cont_Rep_CFun1,
   80.46  	                      cont_Rep_CFun2]) 1);
   80.47  qed "cont2cont_Rep_CFun";
   80.48 @@ -162,7 +162,7 @@
   80.49  (* function application _[_]  is strict in its first arguments              *)
   80.50  (* ------------------------------------------------------------------------ *)
   80.51  
   80.52 -Goal "(UU::'a::cpo->'b)`x = (UU::'b)";
   80.53 +Goal "(UU::'a::cpo->'b)$x = (UU::'b)";
   80.54  by (stac inst_cfun_pcpo 1);
   80.55  by (stac beta_cfun 1);
   80.56  by (Simp_tac 1);
   80.57 @@ -180,7 +180,7 @@
   80.58  qed "Istrictify1";
   80.59  
   80.60  Goalw [Istrictify_def]
   80.61 -        "~x=UU ==> Istrictify(f)(x)=f`x";
   80.62 +        "~x=UU ==> Istrictify(f)(x)=f$x";
   80.63  by (Asm_simp_tac 1);
   80.64  qed "Istrictify2";
   80.65  
   80.66 @@ -251,7 +251,7 @@
   80.67  by (asm_simp_tac (simpset() addsimps [Istrictify1, chain_UU_I_inverse, chain_UU_I, Istrictify1]) 1);
   80.68  by (stac Istrictify2 1);
   80.69  by (atac 1);
   80.70 -by (res_inst_tac [("s","lub(range(%i. f`(Y i)))")] trans 1);
   80.71 +by (res_inst_tac [("s","lub(range(%i. f$(Y i)))")] trans 1);
   80.72  by (rtac contlub_cfun_arg 1);
   80.73  by (atac 1);
   80.74  by (rtac lub_equal2 1);
   80.75 @@ -271,7 +271,7 @@
   80.76          (monofun_Istrictify2 RS monocontlub2cont)); 
   80.77  
   80.78  
   80.79 -Goalw [strictify_def] "strictify`f`UU=UU";
   80.80 +Goalw [strictify_def] "strictify$f$UU=UU";
   80.81  by (stac beta_cfun 1);
   80.82  by (simp_tac (simpset() addsimps [cont_Istrictify2,cont_Istrictify1, cont2cont_CF1L]) 1);
   80.83  by (stac beta_cfun 1);
   80.84 @@ -279,7 +279,7 @@
   80.85  by (rtac Istrictify1 1);
   80.86  qed "strictify1";
   80.87  
   80.88 -Goalw [strictify_def] "~x=UU ==> strictify`f`x=f`x";
   80.89 +Goalw [strictify_def] "~x=UU ==> strictify$f$x=f$x";
   80.90  by (stac beta_cfun 1);
   80.91  by (simp_tac (simpset() addsimps [cont_Istrictify2,cont_Istrictify1, cont2cont_CF1L]) 1);
   80.92  by (stac beta_cfun 1);
   80.93 @@ -307,7 +307,7 @@
   80.94  (* ------------------------------------------------------------------------ *)
   80.95  
   80.96  Goal "chain (Y::nat => 'a::cpo->'b::chfin) \
   80.97 -\     ==> !s. ? n. lub(range(Y))`s = Y n`s";
   80.98 +\     ==> !s. ? n. lub(range(Y))$s = Y n$s";
   80.99  by (rtac allI 1);
  80.100  by (stac contlub_cfun_fun 1);
  80.101  by (atac 1);
  80.102 @@ -320,21 +320,21 @@
  80.103  (* ------------------------------------------------------------------------ *)
  80.104  
  80.105  Goal  
  80.106 -"!!f g.[|!y. f`(g`y)=(y::'b) ; !x. g`(f`x)=(x::'a) |] \
  80.107 -\ ==> f`UU=UU & g`UU=UU";
  80.108 +"!!f g.[|!y. f$(g$y)=(y::'b) ; !x. g$(f$x)=(x::'a) |] \
  80.109 +\ ==> f$UU=UU & g$UU=UU";
  80.110  by (rtac conjI 1);
  80.111  by (rtac UU_I 1);
  80.112 -by (res_inst_tac [("s","f`(g`(UU::'b))"),("t","UU::'b")] subst 1);
  80.113 +by (res_inst_tac [("s","f$(g$(UU::'b))"),("t","UU::'b")] subst 1);
  80.114  by (etac spec 1);
  80.115  by (rtac (minimal RS monofun_cfun_arg) 1);
  80.116  by (rtac UU_I 1);
  80.117 -by (res_inst_tac [("s","g`(f`(UU::'a))"),("t","UU::'a")] subst 1);
  80.118 +by (res_inst_tac [("s","g$(f$(UU::'a))"),("t","UU::'a")] subst 1);
  80.119  by (etac spec 1);
  80.120  by (rtac (minimal RS monofun_cfun_arg) 1);
  80.121  qed "iso_strict";
  80.122  
  80.123  
  80.124 -Goal "[|!x. rep`(ab`x)=x;!y. ab`(rep`y)=y; z~=UU|] ==> rep`z ~= UU";
  80.125 +Goal "[|!x. rep$(ab$x)=x;!y. ab$(rep$y)=y; z~=UU|] ==> rep$z ~= UU";
  80.126  by (etac contrapos_nn 1);
  80.127  by (dres_inst_tac [("f","ab")] cfun_arg_cong 1);
  80.128  by (etac box_equals 1);
  80.129 @@ -343,7 +343,7 @@
  80.130  by (atac 1);
  80.131  qed "isorep_defined";
  80.132  
  80.133 -Goal "[|!x. rep`(ab`x) = x;!y. ab`(rep`y)=y ; z~=UU|] ==> ab`z ~= UU";
  80.134 +Goal "[|!x. rep$(ab$x) = x;!y. ab$(rep$y)=y ; z~=UU|] ==> ab$z ~= UU";
  80.135  by (etac contrapos_nn 1);
  80.136  by (dres_inst_tac [("f","rep")] cfun_arg_cong 1);
  80.137  by (etac box_equals 1);
  80.138 @@ -357,19 +357,19 @@
  80.139  (* ------------------------------------------------------------------------ *)
  80.140  
  80.141  Goal "!!f g.[|! Y::nat=>'a. chain Y --> (? n. max_in_chain n Y); \
  80.142 -\ !y. f`(g`y)=(y::'b) ; !x. g`(f`x)=(x::'a::chfin) |] \
  80.143 +\ !y. f$(g$y)=(y::'b) ; !x. g$(f$x)=(x::'a::chfin) |] \
  80.144  \ ==> ! Y::nat=>'b. chain Y --> (? n. max_in_chain n Y)";
  80.145  by (rewtac max_in_chain_def);
  80.146  by (strip_tac 1);
  80.147  by (rtac exE 1);
  80.148 -by (res_inst_tac [("P","chain(%i. g`(Y i))")] mp 1);
  80.149 +by (res_inst_tac [("P","chain(%i. g$(Y i))")] mp 1);
  80.150  by (etac spec 1);
  80.151  by (etac ch2ch_Rep_CFunR 1);
  80.152  by (rtac exI 1);
  80.153  by (strip_tac 1);
  80.154 -by (res_inst_tac [("s","f`(g`(Y x))"),("t","Y(x)")] subst 1);
  80.155 +by (res_inst_tac [("s","f$(g$(Y x))"),("t","Y(x)")] subst 1);
  80.156  by (etac spec 1);
  80.157 -by (res_inst_tac [("s","f`(g`(Y j))"),("t","Y(j)")] subst 1);
  80.158 +by (res_inst_tac [("s","f$(g$(Y j))"),("t","Y(j)")] subst 1);
  80.159  by (etac spec 1);
  80.160  by (rtac cfun_arg_cong 1);
  80.161  by (rtac mp 1);
  80.162 @@ -379,25 +379,25 @@
  80.163  
  80.164  
  80.165  Goal "!!f g.[|!x y::'a. x<<y --> x=UU | x=y; \
  80.166 -\ !y. f`(g`y)=(y::'b); !x. g`(f`x)=(x::'a)|] ==> !x y::'b. x<<y --> x=UU | x=y";
  80.167 +\ !y. f$(g$y)=(y::'b); !x. g$(f$x)=(x::'a)|] ==> !x y::'b. x<<y --> x=UU | x=y";
  80.168  by (strip_tac 1);
  80.169  by (rtac disjE 1);
  80.170 -by (res_inst_tac [("P","g`x<<g`y")] mp 1);
  80.171 +by (res_inst_tac [("P","g$x<<g$y")] mp 1);
  80.172  by (etac monofun_cfun_arg 2);
  80.173  by (dtac spec 1);
  80.174  by (etac spec 1);
  80.175  by (rtac disjI1 1);
  80.176  by (rtac trans 1);
  80.177 -by (res_inst_tac [("s","f`(g`x)"),("t","x")] subst 1);
  80.178 +by (res_inst_tac [("s","f$(g$x)"),("t","x")] subst 1);
  80.179  by (etac spec 1);
  80.180  by (etac cfun_arg_cong 1);
  80.181  by (rtac (iso_strict RS conjunct1) 1);
  80.182  by (atac 1);
  80.183  by (atac 1);
  80.184  by (rtac disjI2 1);
  80.185 -by (res_inst_tac [("s","f`(g`x)"),("t","x")] subst 1);
  80.186 +by (res_inst_tac [("s","f$(g$x)"),("t","x")] subst 1);
  80.187  by (etac spec 1);
  80.188 -by (res_inst_tac [("s","f`(g`y)"),("t","y")] subst 1);
  80.189 +by (res_inst_tac [("s","f$(g$y)"),("t","y")] subst 1);
  80.190  by (etac spec 1);
  80.191  by (etac cfun_arg_cong 1);
  80.192  qed "flat2flat";
  80.193 @@ -406,19 +406,19 @@
  80.194  (* a result about functions with flat codomain                               *)
  80.195  (* ------------------------------------------------------------------------- *)
  80.196  
  80.197 -Goal "f`(x::'a)=(c::'b::flat) ==> f`(UU::'a)=(UU::'b) | (!z. f`(z::'a)=c)";
  80.198 -by (case_tac "f`(x::'a)=(UU::'b)" 1);
  80.199 +Goal "f$(x::'a)=(c::'b::flat) ==> f$(UU::'a)=(UU::'b) | (!z. f$(z::'a)=c)";
  80.200 +by (case_tac "f$(x::'a)=(UU::'b)" 1);
  80.201  by (rtac disjI1 1);
  80.202  by (rtac UU_I 1);
  80.203 -by (res_inst_tac [("s","f`(x)"),("t","UU::'b")] subst 1);
  80.204 +by (res_inst_tac [("s","f$(x)"),("t","UU::'b")] subst 1);
  80.205  by (atac 1);
  80.206  by (rtac (minimal RS monofun_cfun_arg) 1);
  80.207 -by (case_tac "f`(UU::'a)=(UU::'b)" 1);
  80.208 +by (case_tac "f$(UU::'a)=(UU::'b)" 1);
  80.209  by (etac disjI1 1);
  80.210  by (rtac disjI2 1);
  80.211  by (rtac allI 1);
  80.212  by (hyp_subst_tac 1);
  80.213 -by (res_inst_tac [("a","f`(UU::'a)")] (refl RS box_equals) 1);
  80.214 +by (res_inst_tac [("a","f$(UU::'a)")] (refl RS box_equals) 1);
  80.215  by (res_inst_tac [("fo5","f")] ((minimal RS monofun_cfun_arg) RS (ax_flat RS spec RS spec RS mp) RS disjE) 1);
  80.216  by (contr_tac 1);
  80.217  by (atac 1);
  80.218 @@ -433,13 +433,13 @@
  80.219  (* ------------------------------------------------------------------------ *)
  80.220  
  80.221  
  80.222 -Goalw [ID_def] "ID`x=x";
  80.223 +Goalw [ID_def] "ID$x=x";
  80.224  by (stac beta_cfun 1);
  80.225  by (rtac cont_id 1);
  80.226  by (rtac refl 1);
  80.227  qed "ID1";
  80.228  
  80.229 -Goalw [oo_def] "(f oo g)=(LAM x. f`(g`x))";
  80.230 +Goalw [oo_def] "(f oo g)=(LAM x. f$(g$x))";
  80.231  by (stac beta_cfun 1);
  80.232  by (Simp_tac 1);
  80.233  by (stac beta_cfun 1);
  80.234 @@ -447,7 +447,7 @@
  80.235  by (rtac refl 1);
  80.236  qed "cfcomp1";
  80.237  
  80.238 -Goal  "(f oo g)`x=f`(g`x)";
  80.239 +Goal  "(f oo g)$x=f$(g$x)";
  80.240  by (stac cfcomp1 1);
  80.241  by (stac beta_cfun 1);
  80.242  by (Simp_tac 1);
  80.243 @@ -481,7 +481,7 @@
  80.244  
  80.245  Goal "f oo (g oo h) = (f oo g) oo h";
  80.246  by (rtac ext_cfun 1);
  80.247 -by (res_inst_tac [("s","f`(g`(h`x))")] trans  1);
  80.248 +by (res_inst_tac [("s","f$(g$(h$x))")] trans  1);
  80.249  by (stac cfcomp2 1);
  80.250  by (stac cfcomp2 1);
  80.251  by (rtac refl 1);
    81.1 --- a/src/HOLCF/Cfun3.thy	Tue Jan 09 15:29:17 2001 +0100
    81.2 +++ b/src/HOLCF/Cfun3.thy	Tue Jan 09 15:32:27 2001 +0100
    81.3 @@ -19,7 +19,7 @@
    81.4          strictify    :: "('a->'b)->'a->'b"
    81.5  defs
    81.6  
    81.7 -Istrictify_def  "Istrictify f x == if x=UU then UU else f`x"    
    81.8 +Istrictify_def  "Istrictify f x == if x=UU then UU else f$x"    
    81.9  strictify_def   "strictify == (LAM f x. Istrictify f x)"
   81.10  
   81.11  consts
   81.12 @@ -28,11 +28,11 @@
   81.13  
   81.14  syntax  "@oo"   :: "('b->'c)=>('a->'b)=>'a->'c" ("_ oo _" [101,100] 100)
   81.15       
   81.16 -translations    "f1 oo f2" == "cfcomp`f1`f2"
   81.17 +translations    "f1 oo f2" == "cfcomp$f1$f2"
   81.18  
   81.19  defs
   81.20  
   81.21    ID_def        "ID ==(LAM x. x)"
   81.22 -  oo_def        "cfcomp == (LAM f g x. f`(g`x))" 
   81.23 +  oo_def        "cfcomp == (LAM f g x. f$(g$x))" 
   81.24  
   81.25  end
    82.1 --- a/src/HOLCF/Cprod3.ML	Tue Jan 09 15:29:17 2001 +0100
    82.2 +++ b/src/HOLCF/Cprod3.ML	Tue Jan 09 15:32:27 2001 +0100
    82.3 @@ -125,7 +125,7 @@
    82.4  (* ------------------------------------------------------------------------ *)
    82.5  
    82.6  Goalw [cpair_def]
    82.7 -        "(LAM x y.(x,y))`a`b = (a,b)";
    82.8 +        "(LAM x y.(x,y))$a$b = (a,b)";
    82.9  by (stac beta_cfun 1);
   82.10  by (simp_tac (simpset() addsimps [cont_pair1,cont_pair2,cont2cont_CF1L]) 1);
   82.11  by (stac beta_cfun 1);
   82.12 @@ -170,7 +170,7 @@
   82.13  qed "cprodE";
   82.14  
   82.15  Goalw [cfst_def,cpair_def] 
   82.16 -        "cfst`<x,y>=x";
   82.17 +        "cfst$<x,y>=x";
   82.18  by (stac beta_cfun_cprod 1);
   82.19  by (stac beta_cfun 1);
   82.20  by (rtac cont_fst 1);
   82.21 @@ -178,22 +178,22 @@
   82.22  qed "cfst2";
   82.23  
   82.24  Goalw [csnd_def,cpair_def] 
   82.25 -        "csnd`<x,y>=y";
   82.26 +        "csnd$<x,y>=y";
   82.27  by (stac beta_cfun_cprod 1);
   82.28  by (stac beta_cfun 1);
   82.29  by (rtac cont_snd 1);
   82.30  by (Simp_tac  1);
   82.31  qed "csnd2";
   82.32  
   82.33 -Goal "cfst`UU = UU";
   82.34 +Goal "cfst$UU = UU";
   82.35  by (simp_tac (HOL_ss addsimps [inst_cprod_pcpo2,cfst2]) 1);
   82.36  qed "cfst_strict";
   82.37  
   82.38 -Goal "csnd`UU = UU";
   82.39 +Goal "csnd$UU = UU";
   82.40  by (simp_tac (HOL_ss addsimps [inst_cprod_pcpo2,csnd2]) 1);
   82.41  qed "csnd_strict";
   82.42  
   82.43 -Goalw [cfst_def,csnd_def,cpair_def] "<cfst`p , csnd`p> = p";
   82.44 +Goalw [cfst_def,csnd_def,cpair_def] "<cfst$p , csnd$p> = p";
   82.45  by (stac beta_cfun_cprod 1);
   82.46  by (stac beta_cfun 1);
   82.47  by (rtac cont_snd 1);
   82.48 @@ -212,7 +212,7 @@
   82.49  
   82.50  Goalw [cfst_def,csnd_def,cpair_def]
   82.51  "[|chain(S)|] ==> range(S) <<| \
   82.52 -\ <(lub(range(%i. cfst`(S i)))) , lub(range(%i. csnd`(S i)))>";
   82.53 +\ <(lub(range(%i. cfst$(S i)))) , lub(range(%i. csnd$(S i)))>";
   82.54  by (stac beta_cfun_cprod 1);
   82.55  by (stac (beta_cfun RS ext) 1);
   82.56  by (rtac cont_snd 1);
   82.57 @@ -226,17 +226,17 @@
   82.58  (*
   82.59  chain ?S1 ==>
   82.60   lub (range ?S1) =
   82.61 - <lub (range (%i. cfst`(?S1 i))), lub (range (%i. csnd`(?S1 i)))>" 
   82.62 + <lub (range (%i. cfst$(?S1 i))), lub (range (%i. csnd$(?S1 i)))>" 
   82.63  *)
   82.64  Goalw [csplit_def]
   82.65 -        "csplit`f`<x,y> = f`x`y";
   82.66 +        "csplit$f$<x,y> = f$x$y";
   82.67  by (stac beta_cfun 1);
   82.68  by (Simp_tac 1);
   82.69  by (simp_tac (simpset() addsimps [cfst2,csnd2]) 1);
   82.70  qed "csplit2";
   82.71  
   82.72  Goalw [csplit_def]
   82.73 -  "csplit`cpair`z=z";
   82.74 +  "csplit$cpair$z=z";
   82.75  by (stac beta_cfun 1);
   82.76  by (Simp_tac 1);
   82.77  by (simp_tac (simpset() addsimps [surjective_pairing_Cprod2]) 1);
    83.1 --- a/src/HOLCF/Cprod3.thy	Tue Jan 09 15:29:17 2001 +0100
    83.2 +++ b/src/HOLCF/Cprod3.thy	Tue Jan 09 15:32:27 2001 +0100
    83.3 @@ -22,13 +22,13 @@
    83.4  
    83.5  translations
    83.6          "<x, y, z>"   == "<x, <y, z>>"
    83.7 -        "<x, y>"      == "cpair`x`y"
    83.8 +        "<x, y>"      == "cpair$x$y"
    83.9  
   83.10  defs
   83.11  cpair_def       "cpair  == (LAM x y.(x,y))"
   83.12  cfst_def        "cfst   == (LAM p. fst(p))"
   83.13  csnd_def        "csnd   == (LAM p. snd(p))"      
   83.14 -csplit_def      "csplit == (LAM f p. f`(cfst`p)`(csnd`p))"
   83.15 +csplit_def      "csplit == (LAM f p. f$(cfst$p)$(csnd$p))"
   83.16  
   83.17  
   83.18  
   83.19 @@ -43,7 +43,7 @@
   83.20  
   83.21  constdefs
   83.22    CLet           :: "'a -> ('a -> 'b) -> 'b"
   83.23 -  "CLet == LAM s f. f`s"
   83.24 +  "CLet == LAM s f. f$s"
   83.25  
   83.26  
   83.27  (* syntax for Let *)
   83.28 @@ -59,7 +59,7 @@
   83.29  
   83.30  translations
   83.31    "_CLet (_Cbinds b bs) e"  == "_CLet b (_CLet bs e)"
   83.32 -  "Let x = a in e"          == "CLet`a`(LAM x. e)"
   83.33 +  "Let x = a in e"          == "CLet$a$(LAM x. e)"
   83.34  
   83.35  
   83.36  (* syntax for LAM <x,y,z>.e *)
   83.37 @@ -68,9 +68,9 @@
   83.38    "_LAM"    :: "[patterns, 'a => 'b] => ('a -> 'b)"  ("(3LAM <_>./ _)" [0, 10] 10)
   83.39  
   83.40  translations
   83.41 -  "LAM <x,y,zs>.b"        == "csplit`(LAM x. LAM <y,zs>.b)"
   83.42 -  "LAM <x,y>. LAM zs. b"  <= "csplit`(LAM x y zs. b)"
   83.43 -  "LAM <x,y>.b"           == "csplit`(LAM x y. b)"
   83.44 +  "LAM <x,y,zs>.b"        == "csplit$(LAM x. LAM <y,zs>.b)"
   83.45 +  "LAM <x,y>. LAM zs. b"  <= "csplit$(LAM x y zs. b)"
   83.46 +  "LAM <x,y>.b"           == "csplit$(LAM x y. b)"
   83.47  
   83.48  syntax (symbols)
   83.49    "_LAM"    :: "[patterns, 'a => 'b] => ('a -> 'b)"  ("(3\\<Lambda>()<_>./ _)" [0, 10] 10)
    84.1 --- a/src/HOLCF/Fix.ML	Tue Jan 09 15:29:17 2001 +0100
    84.2 +++ b/src/HOLCF/Fix.ML	Tue Jan 09 15:32:27 2001 +0100
    84.3 @@ -10,7 +10,7 @@
    84.4  (* derive inductive properties of iterate from primitive recursion          *)
    84.5  (* ------------------------------------------------------------------------ *)
    84.6  
    84.7 -Goal "iterate (Suc n) F x = iterate n F (F`x)";
    84.8 +Goal "iterate (Suc n) F x = iterate n F (F$x)";
    84.9  by (induct_tac "n" 1);
   84.10  by Auto_tac;  
   84.11  qed "iterate_Suc2";
   84.12 @@ -20,7 +20,7 @@
   84.13  (* This property is essential since monotonicity of iterate makes no sense  *)
   84.14  (* ------------------------------------------------------------------------ *)
   84.15  
   84.16 -Goalw [chain]  "x << F`x ==> chain (%i. iterate i F x)";
   84.17 +Goalw [chain]  "x << F$x ==> chain (%i. iterate i F x)";
   84.18  by (strip_tac 1);
   84.19  by (induct_tac "i" 1);
   84.20  by Auto_tac;  
   84.21 @@ -40,7 +40,7 @@
   84.22  (* ------------------------------------------------------------------------ *)
   84.23  
   84.24  
   84.25 -Goalw [Ifix_def] "Ifix F =F`(Ifix F)";
   84.26 +Goalw [Ifix_def] "Ifix F =F$(Ifix F)";
   84.27  by (stac contlub_cfun_arg 1);
   84.28  by (rtac chain_iterate 1);
   84.29  by (rtac antisym_less 1);
   84.30 @@ -61,7 +61,7 @@
   84.31  qed "Ifix_eq";
   84.32  
   84.33  
   84.34 -Goalw [Ifix_def] "F`x=x ==> Ifix(F) << x";
   84.35 +Goalw [Ifix_def] "F$x=x ==> Ifix(F) << x";
   84.36  by (rtac is_lub_thelub 1);
   84.37  by (rtac chain_iterate 1);
   84.38  by (rtac ub_rangeI 1);
   84.39 @@ -250,19 +250,19 @@
   84.40  (* propagate properties of Ifix to its continuous counterpart               *)
   84.41  (* ------------------------------------------------------------------------ *)
   84.42  
   84.43 -Goalw [fix_def] "fix`F = F`(fix`F)";
   84.44 +Goalw [fix_def] "fix$F = F$(fix$F)";
   84.45  by (asm_simp_tac (simpset() addsimps [cont_Ifix]) 1);
   84.46  by (rtac Ifix_eq 1);
   84.47  qed "fix_eq";
   84.48  
   84.49 -Goalw [fix_def] "F`x = x ==> fix`F << x";
   84.50 +Goalw [fix_def] "F$x = x ==> fix$F << x";
   84.51  by (asm_simp_tac (simpset() addsimps [cont_Ifix]) 1);
   84.52  by (etac Ifix_least 1);
   84.53  qed "fix_least";
   84.54  
   84.55  
   84.56  Goal
   84.57 -"[| F`x = x; !z. F`z = z --> x << z |] ==> x = fix`F";
   84.58 +"[| F$x = x; !z. F$z = z --> x << z |] ==> x = fix$F";
   84.59  by (rtac antisym_less 1);
   84.60  by (etac allE 1);
   84.61  by (etac mp 1);
   84.62 @@ -271,22 +271,22 @@
   84.63  qed "fix_eqI";
   84.64  
   84.65  
   84.66 -Goal "f == fix`F ==> f = F`f";
   84.67 +Goal "f == fix$F ==> f = F$f";
   84.68  by (asm_simp_tac (simpset() addsimps [fix_eq RS sym]) 1);
   84.69  qed "fix_eq2";
   84.70  
   84.71 -Goal "f == fix`F ==> f`x = F`f`x";
   84.72 +Goal "f == fix$F ==> f$x = F$f$x";
   84.73  by (etac (fix_eq2 RS cfun_fun_cong) 1);
   84.74  qed "fix_eq3";
   84.75  
   84.76  fun fix_tac3 thm i  = ((rtac trans i) THEN (rtac (thm RS fix_eq3) i)); 
   84.77  
   84.78 -Goal "f = fix`F ==> f = F`f";
   84.79 +Goal "f = fix$F ==> f = F$f";
   84.80  by (hyp_subst_tac 1);
   84.81  by (rtac fix_eq 1);
   84.82  qed "fix_eq4";
   84.83  
   84.84 -Goal "f = fix`F ==> f`x = F`f`x";
   84.85 +Goal "f = fix$F ==> f$x = F$f$x";
   84.86  by (rtac trans 1);
   84.87  by (etac (fix_eq4 RS cfun_fun_cong) 1);
   84.88  by (rtac refl 1);
   84.89 @@ -294,7 +294,7 @@
   84.90  
   84.91  fun fix_tac5 thm i  = ((rtac trans i) THEN (rtac (thm RS fix_eq5) i)); 
   84.92  
   84.93 -(* proves the unfolding theorem for function equations f = fix`... *)
   84.94 +(* proves the unfolding theorem for function equations f = fix$... *)
   84.95  fun fix_prover thy fixeq s = prove_goal thy s (fn prems => [
   84.96          (rtac trans 1),
   84.97          (rtac (fixeq RS fix_eq4) 1),
   84.98 @@ -303,7 +303,7 @@
   84.99          (Simp_tac 1)
  84.100          ]);
  84.101  
  84.102 -(* proves the unfolding theorem for function definitions f == fix`... *)
  84.103 +(* proves the unfolding theorem for function definitions f == fix$... *)
  84.104  fun fix_prover2 thy fixdef s = prove_goal thy s (fn prems => [
  84.105          (rtac trans 1),
  84.106          (rtac (fix_eq2) 1),
  84.107 @@ -335,7 +335,7 @@
  84.108  (* direct connection between fix and iteration without Ifix                 *)
  84.109  (* ------------------------------------------------------------------------ *)
  84.110  
  84.111 -Goalw [fix_def] "fix`F = lub(range(%i. iterate i F UU))";
  84.112 +Goalw [fix_def] "fix$F = lub(range(%i. iterate i F UU))";
  84.113  by (fold_goals_tac [Ifix_def]);
  84.114  by (asm_simp_tac (simpset() addsimps [cont_Ifix]) 1);
  84.115  qed "fix_def2";
  84.116 @@ -379,7 +379,7 @@
  84.117  (* ------------------------------------------------------------------------ *)
  84.118  
  84.119  val major::prems = Goal
  84.120 -     "[| adm(P); P(UU); !!x. P(x) ==> P(F`x)|] ==> P(fix`F)";
  84.121 +     "[| adm(P); P(UU); !!x. P(x) ==> P(F$x)|] ==> P(fix$F)";
  84.122  by (stac fix_def2 1);
  84.123  by (rtac (major RS admD) 1);
  84.124  by (rtac chain_iterate 1);
  84.125 @@ -389,8 +389,8 @@
  84.126  by (asm_simp_tac (simpset() addsimps (iterate_Suc::prems)) 1);
  84.127  qed "fix_ind";
  84.128  
  84.129 -val prems = Goal "[| f == fix`F; adm(P); \
  84.130 -\       P(UU); !!x. P(x) ==> P(F`x)|] ==> P f";
  84.131 +val prems = Goal "[| f == fix$F; adm(P); \
  84.132 +\       P(UU); !!x. P(x) ==> P(F$x)|] ==> P f";
  84.133  by (cut_facts_tac prems 1);
  84.134  by (asm_simp_tac HOL_ss 1);
  84.135  by (etac fix_ind 1);
  84.136 @@ -402,7 +402,7 @@
  84.137  (* computational induction for weak admissible formulae                     *)
  84.138  (* ------------------------------------------------------------------------ *)
  84.139  
  84.140 -Goal "[| admw(P); !n. P(iterate n F UU)|] ==> P(fix`F)";
  84.141 +Goal "[| admw(P); !n. P(iterate n F UU)|] ==> P(fix$F)";
  84.142  by (stac fix_def2 1);
  84.143  by (rtac (admw_def2 RS iffD1 RS spec RS mp) 1);
  84.144  by (atac 1);
  84.145 @@ -410,7 +410,7 @@
  84.146  by (etac spec 1);
  84.147  qed "wfix_ind";
  84.148  
  84.149 -Goal "[| f == fix`F; admw(P); \
  84.150 +Goal "[| f == fix$F; admw(P); \
  84.151  \       !n. P(iterate n F UU) |] ==> P f";
  84.152  by (asm_simp_tac HOL_ss 1);
  84.153  by (etac wfix_ind 1);
  84.154 @@ -440,7 +440,7 @@
  84.155  (* some lemmata for functions with flat/chfin domain/range types	    *)
  84.156  (* ------------------------------------------------------------------------ *)
  84.157  
  84.158 -val _ = goalw thy [adm_def] "adm (%(u::'a::cpo->'b::chfin). P(u`s))";
  84.159 +val _ = goalw thy [adm_def] "adm (%(u::'a::cpo->'b::chfin). P(u$s))";
  84.160  by (strip_tac 1);
  84.161  by (dtac chfin_Rep_CFunR 1);
  84.162  by (eres_inst_tac [("x","s")] allE 1);
    85.1 --- a/src/HOLCF/Fix.thy	Tue Jan 09 15:29:17 2001 +0100
    85.2 +++ b/src/HOLCF/Fix.thy	Tue Jan 09 15:32:27 2001 +0100
    85.3 @@ -20,7 +20,7 @@
    85.4  
    85.5  primrec
    85.6    iterate_0   "iterate 0 F x = x"
    85.7 -  iterate_Suc "iterate (Suc n) F x  = F`(iterate n F x)"
    85.8 +  iterate_Suc "iterate (Suc n) F x  = F$(iterate n F x)"
    85.9  
   85.10  defs
   85.11  
    86.1 --- a/src/HOLCF/Lift.ML	Tue Jan 09 15:29:17 2001 +0100
    86.2 +++ b/src/HOLCF/Lift.ML	Tue Jan 09 15:32:27 2001 +0100
    86.3 @@ -81,25 +81,25 @@
    86.4  (* ---------------------------------------------------------- *)
    86.5  
    86.6  
    86.7 -Goal "flift1 f`(Def x) = (f x)";
    86.8 +Goal "flift1 f$(Def x) = (f x)";
    86.9  by (simp_tac (simpset() addsimps [flift1_def]) 1);
   86.10  qed"flift1_Def";
   86.11  
   86.12 -Goal "flift2 f`(Def x) = Def (f x)";
   86.13 +Goal "flift2 f$(Def x) = Def (f x)";
   86.14  by (simp_tac (simpset() addsimps [flift2_def]) 1);
   86.15  qed"flift2_Def";
   86.16  
   86.17 -Goal "flift1 f`UU = UU";
   86.18 +Goal "flift1 f$UU = UU";
   86.19  by (simp_tac (simpset() addsimps [flift1_def]) 1);
   86.20  qed"flift1_UU";
   86.21  
   86.22 -Goal "flift2 f`UU = UU";
   86.23 +Goal "flift2 f$UU = UU";
   86.24  by (simp_tac (simpset() addsimps [flift2_def]) 1);
   86.25  qed"flift2_UU";
   86.26  
   86.27  Addsimps [flift1_Def,flift2_Def,flift1_UU,flift2_UU];
   86.28  
   86.29 -Goal "x~=UU ==> (flift2 f)`x~=UU";
   86.30 +Goal "x~=UU ==> (flift2 f)$x~=UU";
   86.31  by (def_tac 1);
   86.32  qed"flift2_nUU";
   86.33  
    87.1 --- a/src/HOLCF/Lift3.ML	Tue Jan 09 15:29:17 2001 +0100
    87.2 +++ b/src/HOLCF/Lift3.ML	Tue Jan 09 15:32:27 2001 +0100
    87.3 @@ -134,13 +134,13 @@
    87.4  
    87.5  (* Two specific lemmas for the combination of LCF and HOL terms *)
    87.6  
    87.7 -Goal "[|cont g; cont f|] ==> cont(%x. ((f x)`(g x)) s)";
    87.8 +Goal "[|cont g; cont f|] ==> cont(%x. ((f x)$(g x)) s)";
    87.9  by (rtac cont2cont_CF1L 1);
   87.10  by (REPEAT (resolve_tac cont_lemmas1 1));
   87.11  by Auto_tac;
   87.12  qed"cont_Rep_CFun_app";
   87.13  
   87.14 -Goal "[|cont g; cont f|] ==> cont(%x. ((f x)`(g x)) s t)";
   87.15 +Goal "[|cont g; cont f|] ==> cont(%x. ((f x)$(g x)) s t)";
   87.16  by (rtac cont2cont_CF1L 1);
   87.17  by (etac cont_Rep_CFun_app 1);
   87.18  by (assume_tac 1);
    88.1 --- a/src/HOLCF/Lift3.thy	Tue Jan 09 15:29:17 2001 +0100
    88.2 +++ b/src/HOLCF/Lift3.thy	Tue Jan 09 15:32:27 2001 +0100
    88.3 @@ -29,9 +29,9 @@
    88.4                     Undef => UU
    88.5                   | Def y => Def (f y)))"
    88.6   liftpair_def
    88.7 -  "liftpair x  == (case (cfst`x) of 
    88.8 +  "liftpair x  == (case (cfst$x) of 
    88.9                    Undef  => UU
   88.10 -                | Def x1 => (case (csnd`x) of 
   88.11 +                | Def x1 => (case (csnd$x) of 
   88.12                                 Undef => UU
   88.13                               | Def x2 => Def (x1,x2)))"
   88.14  
    89.1 --- a/src/HOLCF/Sprod3.ML	Tue Jan 09 15:29:17 2001 +0100
    89.2 +++ b/src/HOLCF/Sprod3.ML	Tue Jan 09 15:32:27 2001 +0100
    89.3 @@ -240,7 +240,7 @@
    89.4  (* ------------------------------------------------------------------------ *)
    89.5  
    89.6  Goalw [spair_def]
    89.7 -        "(LAM x y. Ispair x y)`a`b = Ispair a b";
    89.8 +        "(LAM x y. Ispair x y)$a$b = Ispair a b";
    89.9  by (stac beta_cfun 1);
   89.10  by (simp_tac (simpset() addsimps [cont_Ispair2, cont_Ispair1, cont2cont_CF1L]) 1);
   89.11  by (stac beta_cfun 1);
   89.12 @@ -343,7 +343,7 @@
   89.13  
   89.14  
   89.15  Goalw [sfst_def] 
   89.16 -        "p=UU==>sfst`p=UU";
   89.17 +        "p=UU==>sfst$p=UU";
   89.18  by (stac beta_cfun 1);
   89.19  by (rtac cont_Isfst 1);
   89.20  by (rtac strict_Isfst 1);
   89.21 @@ -352,7 +352,7 @@
   89.22  qed "strict_sfst";
   89.23  
   89.24  Goalw [sfst_def,spair_def] 
   89.25 -        "sfst`(:UU,y:) = UU";
   89.26 +        "sfst$(:UU,y:) = UU";
   89.27  by (stac beta_cfun_sprod 1);
   89.28  by (stac beta_cfun 1);
   89.29  by (rtac cont_Isfst 1);
   89.30 @@ -360,7 +360,7 @@
   89.31  qed "strict_sfst1";
   89.32   
   89.33  Goalw [sfst_def,spair_def] 
   89.34 -        "sfst`(:x,UU:) = UU";
   89.35 +        "sfst$(:x,UU:) = UU";
   89.36  by (stac beta_cfun_sprod 1);
   89.37  by (stac beta_cfun 1);
   89.38  by (rtac cont_Isfst 1);
   89.39 @@ -368,7 +368,7 @@
   89.40  qed "strict_sfst2";
   89.41  
   89.42  Goalw [ssnd_def] 
   89.43 -        "p=UU==>ssnd`p=UU";
   89.44 +        "p=UU==>ssnd$p=UU";
   89.45  by (stac beta_cfun 1);
   89.46  by (rtac cont_Issnd 1);
   89.47  by (rtac strict_Issnd 1);
   89.48 @@ -377,7 +377,7 @@
   89.49  qed "strict_ssnd";
   89.50  
   89.51  Goalw [ssnd_def,spair_def] 
   89.52 -        "ssnd`(:UU,y:) = UU";
   89.53 +        "ssnd$(:UU,y:) = UU";
   89.54  by (stac beta_cfun_sprod 1);
   89.55  by (stac beta_cfun 1);
   89.56  by (rtac cont_Issnd 1);
   89.57 @@ -385,7 +385,7 @@
   89.58  qed "strict_ssnd1";
   89.59  
   89.60  Goalw [ssnd_def,spair_def] 
   89.61 -        "ssnd`(:x,UU:) = UU";
   89.62 +        "ssnd$(:x,UU:) = UU";
   89.63  by (stac beta_cfun_sprod 1);
   89.64  by (stac beta_cfun 1);
   89.65  by (rtac cont_Issnd 1);
   89.66 @@ -393,7 +393,7 @@
   89.67  qed "strict_ssnd2";
   89.68  
   89.69  Goalw [sfst_def,spair_def] 
   89.70 -        "y~=UU ==>sfst`(:x,y:)=x";
   89.71 +        "y~=UU ==>sfst$(:x,y:)=x";
   89.72  by (stac beta_cfun_sprod 1);
   89.73  by (stac beta_cfun 1);
   89.74  by (rtac cont_Isfst 1);
   89.75 @@ -401,7 +401,7 @@
   89.76  qed "sfst2";
   89.77  
   89.78  Goalw [ssnd_def,spair_def] 
   89.79 -        "x~=UU ==>ssnd`(:x,y:)=y";
   89.80 +        "x~=UU ==>ssnd$(:x,y:)=y";
   89.81  by (stac beta_cfun_sprod 1);
   89.82  by (stac beta_cfun 1);
   89.83  by (rtac cont_Issnd 1);
   89.84 @@ -410,7 +410,7 @@
   89.85  
   89.86  
   89.87  Goalw [sfst_def,ssnd_def,spair_def]
   89.88 -        "p~=UU ==> sfst`p ~=UU & ssnd`p ~=UU";
   89.89 +        "p~=UU ==> sfst$p ~=UU & ssnd$p ~=UU";
   89.90  by (stac beta_cfun 1);
   89.91  by (rtac cont_Issnd 1);
   89.92  by (stac beta_cfun 1);
   89.93 @@ -420,7 +420,7 @@
   89.94  by (atac 1);
   89.95  qed "defined_sfstssnd";
   89.96   
   89.97 -Goalw [sfst_def,ssnd_def,spair_def] "(:sfst`p , ssnd`p:) = p";
   89.98 +Goalw [sfst_def,ssnd_def,spair_def] "(:sfst$p , ssnd$p:) = p";
   89.99  by (stac beta_cfun_sprod 1);
  89.100  by (stac beta_cfun 1);
  89.101  by (rtac cont_Issnd 1);
  89.102 @@ -431,7 +431,7 @@
  89.103  
  89.104  Goalw [sfst_def,ssnd_def,spair_def]
  89.105  "chain(S) ==> range(S) <<| \
  89.106 -\              (: lub(range(%i. sfst`(S i))), lub(range(%i. ssnd`(S i))) :)";
  89.107 +\              (: lub(range(%i. sfst$(S i))), lub(range(%i. ssnd$(S i))) :)";
  89.108  by (stac beta_cfun_sprod 1);
  89.109  by (stac (beta_cfun RS ext) 1);
  89.110  by (rtac cont_Issnd 1);
  89.111 @@ -445,11 +445,11 @@
  89.112  (*
  89.113   "chain ?S1 ==>
  89.114   lub (range ?S1) =
  89.115 - (:lub (range (%i. sfst`(?S1 i))), lub (range (%i. ssnd`(?S1 i))):)" : thm
  89.116 + (:lub (range (%i. sfst$(?S1 i))), lub (range (%i. ssnd$(?S1 i))):)" : thm
  89.117  *)
  89.118  
  89.119  Goalw [ssplit_def]
  89.120 -        "ssplit`f`UU=UU";
  89.121 +        "ssplit$f$UU=UU";
  89.122  by (stac beta_cfun 1);
  89.123  by (Simp_tac 1);
  89.124  by (stac strictify1 1);
  89.125 @@ -457,7 +457,7 @@
  89.126  qed "ssplit1";
  89.127  
  89.128  Goalw [ssplit_def]
  89.129 -        "[|x~=UU;y~=UU|] ==> ssplit`f`(:x,y:)= f`x`y";
  89.130 +        "[|x~=UU;y~=UU|] ==> ssplit$f$(:x,y:)= f$x$y";
  89.131  by (stac beta_cfun 1);
  89.132  by (Simp_tac 1);
  89.133  by (stac strictify2 1);
  89.134 @@ -475,7 +475,7 @@
  89.135  
  89.136  
  89.137  Goalw [ssplit_def]
  89.138 -  "ssplit`spair`z=z";
  89.139 +  "ssplit$spair$z=z";
  89.140  by (stac beta_cfun 1);
  89.141  by (Simp_tac 1);
  89.142  by (case_tac "z=UU" 1);
    90.1 --- a/src/HOLCF/Sprod3.thy	Tue Jan 09 15:29:17 2001 +0100
    90.2 +++ b/src/HOLCF/Sprod3.thy	Tue Jan 09 15:32:27 2001 +0100
    90.3 @@ -21,12 +21,12 @@
    90.4  
    90.5  translations
    90.6          "(:x, y, z:)"   == "(:x, (:y, z:):)"
    90.7 -        "(:x, y:)"      == "spair`x`y"
    90.8 +        "(:x, y:)"      == "spair$x$y"
    90.9  
   90.10  defs
   90.11  spair_def       "spair  == (LAM x y. Ispair x y)"
   90.12  sfst_def        "sfst   == (LAM p. Isfst p)"
   90.13  ssnd_def        "ssnd   == (LAM p. Issnd p)"     
   90.14 -ssplit_def      "ssplit == (LAM f. strictify`(LAM p. f`(sfst`p)`(ssnd`p)))"
   90.15 +ssplit_def      "ssplit == (LAM f. strictify$(LAM p. f$(sfst$p)$(ssnd$p)))"
   90.16  
   90.17  end
    91.1 --- a/src/HOLCF/Ssum0.ML	Tue Jan 09 15:29:17 2001 +0100
    91.2 +++ b/src/HOLCF/Ssum0.ML	Tue Jan 09 15:32:27 2001 +0100
    91.3 @@ -237,7 +237,7 @@
    91.4  
    91.5  
    91.6  Goalw [Iwhen_def]
    91.7 -        "x~=UU ==> Iwhen f g (Isinl x) = f`x";
    91.8 +        "x~=UU ==> Iwhen f g (Isinl x) = f$x";
    91.9  by (rtac some_equality 1);
   91.10  by (fast_tac HOL_cs  2);
   91.11  by (rtac conjI 1);
   91.12 @@ -260,7 +260,7 @@
   91.13  qed "Iwhen2";
   91.14  
   91.15  Goalw [Iwhen_def]
   91.16 -        "y~=UU ==> Iwhen f g (Isinr y) = g`y";
   91.17 +        "y~=UU ==> Iwhen f g (Isinr y) = g$y";
   91.18  by (rtac some_equality 1);
   91.19  by (fast_tac HOL_cs  2);
   91.20  by (rtac conjI 1);
    92.1 --- a/src/HOLCF/Ssum0.thy	Tue Jan 09 15:29:17 2001 +0100
    92.2 +++ b/src/HOLCF/Ssum0.thy	Tue Jan 09 15:32:27 2001 +0100
    92.3 @@ -31,7 +31,7 @@
    92.4  
    92.5    Iwhen_def     "Iwhen(f)(g)(s) == @z.
    92.6                                      (s=Isinl(UU) --> z=UU)
    92.7 -                        &(!a. a~=UU & s=Isinl(a) --> z=f`a)  
    92.8 -                        &(!b. b~=UU & s=Isinr(b) --> z=g`b)"  
    92.9 +                        &(!a. a~=UU & s=Isinl(a) --> z=f$a)  
   92.10 +                        &(!b. b~=UU & s=Isinr(b) --> z=g$b)"  
   92.11  
   92.12  end
    93.1 --- a/src/HOLCF/Ssum3.ML	Tue Jan 09 15:29:17 2001 +0100
    93.2 +++ b/src/HOLCF/Ssum3.ML	Tue Jan 09 15:32:27 2001 +0100
    93.3 @@ -319,43 +319,43 @@
    93.4  (* continuous versions of lemmas for 'a ++ 'b                               *)
    93.5  (* ------------------------------------------------------------------------ *)
    93.6  
    93.7 -Goalw [sinl_def] "sinl`UU =UU";
    93.8 +Goalw [sinl_def] "sinl$UU =UU";
    93.9  by (simp_tac (Ssum0_ss addsimps [cont_Isinl]) 1);
   93.10  by (rtac (inst_ssum_pcpo RS sym) 1);
   93.11  qed "strict_sinl";
   93.12  Addsimps [strict_sinl];
   93.13  
   93.14 -Goalw [sinr_def] "sinr`UU=UU";
   93.15 +Goalw [sinr_def] "sinr$UU=UU";
   93.16  by (simp_tac (Ssum0_ss addsimps [cont_Isinr]) 1);
   93.17  by (rtac (inst_ssum_pcpo RS sym) 1);
   93.18  qed "strict_sinr";
   93.19  Addsimps [strict_sinr];
   93.20  
   93.21  Goalw [sinl_def,sinr_def] 
   93.22 -        "sinl`a=sinr`b ==> a=UU & b=UU";
   93.23 +        "sinl$a=sinr$b ==> a=UU & b=UU";
   93.24  by (auto_tac (claset() addSDs [noteq_IsinlIsinr], simpset()));
   93.25  qed "noteq_sinlsinr";
   93.26  
   93.27  Goalw [sinl_def,sinr_def] 
   93.28 -        "sinl`a1=sinl`a2==> a1=a2";
   93.29 +        "sinl$a1=sinl$a2==> a1=a2";
   93.30  by Auto_tac;
   93.31  qed "inject_sinl";
   93.32  
   93.33  Goalw [sinl_def,sinr_def] 
   93.34 -        "sinr`a1=sinr`a2==> a1=a2";
   93.35 +        "sinr$a1=sinr$a2==> a1=a2";
   93.36  by Auto_tac;
   93.37  qed "inject_sinr";
   93.38  
   93.39  AddSDs [inject_sinl, inject_sinr];
   93.40  
   93.41 -Goal "x~=UU ==> sinl`x ~= UU";
   93.42 +Goal "x~=UU ==> sinl$x ~= UU";
   93.43  by (etac contrapos_nn 1);
   93.44  by (rtac inject_sinl 1);
   93.45  by Auto_tac;
   93.46  qed "defined_sinl";
   93.47  Addsimps [defined_sinl];
   93.48  
   93.49 -Goal "x~=UU ==> sinr`x ~= UU";
   93.50 +Goal "x~=UU ==> sinr$x ~= UU";
   93.51  by (etac contrapos_nn 1);
   93.52  by (rtac inject_sinr 1);
   93.53  by Auto_tac;
   93.54 @@ -363,7 +363,7 @@
   93.55  Addsimps [defined_sinr];
   93.56  
   93.57  Goalw [sinl_def,sinr_def] 
   93.58 -        "z=UU | (? a. z=sinl`a & a~=UU) | (? b. z=sinr`b & b~=UU)";
   93.59 +        "z=UU | (? a. z=sinl$a & a~=UU) | (? b. z=sinr$b & b~=UU)";
   93.60  by (asm_simp_tac (Ssum0_ss addsimps [cont_Isinr,cont_Isinl]) 1);
   93.61  by (stac inst_ssum_pcpo 1);
   93.62  by (rtac Exh_Ssum 1);
   93.63 @@ -372,8 +372,8 @@
   93.64  
   93.65  val [major,prem2,prem3] = Goalw [sinl_def,sinr_def] 
   93.66          "[|p=UU ==> Q ;\
   93.67 -\       !!x.[|p=sinl`x; x~=UU |] ==> Q;\
   93.68 -\       !!y.[|p=sinr`y; y~=UU |] ==> Q|] ==> Q";
   93.69 +\       !!x.[|p=sinl$x; x~=UU |] ==> Q;\
   93.70 +\       !!y.[|p=sinr$y; y~=UU |] ==> Q|] ==> Q";
   93.71  by (rtac (major RS IssumE) 1);
   93.72  by (stac inst_ssum_pcpo 1);
   93.73  by (atac 1);
   93.74 @@ -387,8 +387,8 @@
   93.75  
   93.76  
   93.77  val [preml,premr] = Goalw [sinl_def,sinr_def] 
   93.78 -      "[|!!x.[|p=sinl`x|] ==> Q;\
   93.79 -\        !!y.[|p=sinr`y|] ==> Q|] ==> Q";
   93.80 +      "[|!!x.[|p=sinl$x|] ==> Q;\
   93.81 +\        !!y.[|p=sinr$y|] ==> Q|] ==> Q";
   93.82  by (rtac IssumE2 1);
   93.83  by (rtac preml 1);
   93.84  by (rtac premr 2);
   93.85 @@ -399,7 +399,7 @@
   93.86                  cont_Iwhen3,cont2cont_CF1L]) 1)); 
   93.87  
   93.88  Goalw [sscase_def,sinl_def,sinr_def] 
   93.89 -        "sscase`f`g`UU = UU";
   93.90 +        "sscase$f$g$UU = UU";
   93.91  by (stac inst_ssum_pcpo 1);
   93.92  by (stac beta_cfun 1);
   93.93  by tac;
   93.94 @@ -416,7 +416,7 @@
   93.95                  cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1));
   93.96  
   93.97  Goalw [sscase_def,sinl_def,sinr_def] 
   93.98 -        "x~=UU==> sscase`f`g`(sinl`x) = f`x";
   93.99 +        "x~=UU==> sscase$f$g$(sinl$x) = f$x";
  93.100  by (stac beta_cfun 1);
  93.101  by tac;
  93.102  by (stac beta_cfun 1);
  93.103 @@ -430,7 +430,7 @@
  93.104  Addsimps [sscase2];
  93.105  
  93.106  Goalw [sscase_def,sinl_def,sinr_def] 
  93.107 -        "x~=UU==> sscase`f`g`(sinr`x) = g`x";
  93.108 +        "x~=UU==> sscase$f$g$(sinr$x) = g$x";
  93.109  by (stac beta_cfun 1);
  93.110  by tac;
  93.111  by (stac beta_cfun 1);
  93.112 @@ -445,7 +445,7 @@
  93.113  
  93.114  
  93.115  Goalw [sinl_def,sinr_def] 
  93.116 -        "(sinl`x << sinl`y) = (x << y)";
  93.117 +        "(sinl$x << sinl$y) = (x << y)";
  93.118  by (stac beta_cfun 1);
  93.119  by tac;
  93.120  by (stac beta_cfun 1);
  93.121 @@ -454,7 +454,7 @@
  93.122  qed "less_ssum4a";
  93.123  
  93.124  Goalw [sinl_def,sinr_def] 
  93.125 -        "(sinr`x << sinr`y) = (x << y)";
  93.126 +        "(sinr$x << sinr$y) = (x << y)";
  93.127  by (stac beta_cfun 1);
  93.128  by tac;
  93.129  by (stac beta_cfun 1);
  93.130 @@ -463,7 +463,7 @@
  93.131  qed "less_ssum4b";
  93.132  
  93.133  Goalw [sinl_def,sinr_def] 
  93.134 -        "(sinl`x << sinr`y) = (x = UU)";
  93.135 +        "(sinl$x << sinr$y) = (x = UU)";
  93.136  by (stac beta_cfun 1);
  93.137  by tac;
  93.138  by (stac beta_cfun 1);
  93.139 @@ -472,7 +472,7 @@
  93.140  qed "less_ssum4c";
  93.141  
  93.142  Goalw [sinl_def,sinr_def] 
  93.143 -        "(sinr`x << sinl`y) = (x = UU)";
  93.144 +        "(sinr$x << sinl$y) = (x = UU)";
  93.145  by (stac beta_cfun 1);
  93.146  by tac;
  93.147  by (stac beta_cfun 1);
  93.148 @@ -481,15 +481,15 @@
  93.149  qed "less_ssum4d";
  93.150  
  93.151  Goalw [sinl_def,sinr_def] 
  93.152 -        "chain(Y) ==> (!i.? x.(Y i)=sinl`x)|(!i.? y.(Y i)=sinr`y)";
  93.153 +        "chain(Y) ==> (!i.? x.(Y i)=sinl$x)|(!i.? y.(Y i)=sinr$y)";
  93.154  by (asm_simp_tac (Ssum0_ss addsimps [cont_Isinr,cont_Isinl]) 1);
  93.155  by (etac ssum_lemma4 1);
  93.156  qed "ssum_chainE";
  93.157  
  93.158  
  93.159  Goalw [sinl_def,sinr_def,sscase_def] 
  93.160 -"[| chain(Y); !i.? x. Y(i) = sinl`x |] ==>\ 
  93.161 -\   lub(range(Y)) = sinl`(lub(range(%i. sscase`(LAM x. x)`(LAM y. UU)`(Y i))))";
  93.162 +"[| chain(Y); !i.? x. Y(i) = sinl$x |] ==>\ 
  93.163 +\   lub(range(Y)) = sinl$(lub(range(%i. sscase$(LAM x. x)$(LAM y. UU)$(Y i))))";
  93.164  by (stac beta_cfun 1);
  93.165  by tac;
  93.166  by (stac beta_cfun 1);
  93.167 @@ -510,8 +510,8 @@
  93.168  qed "thelub_ssum2a";
  93.169  
  93.170  Goalw [sinl_def,sinr_def,sscase_def] 
  93.171 -"[| chain(Y); !i.? x. Y(i) = sinr`x |] ==>\ 
  93.172 -\   lub(range(Y)) = sinr`(lub(range(%i. sscase`(LAM y. UU)`(LAM x. x)`(Y i))))";
  93.173 +"[| chain(Y); !i.? x. Y(i) = sinr$x |] ==>\ 
  93.174 +\   lub(range(Y)) = sinr$(lub(range(%i. sscase$(LAM y. UU)$(LAM x. x)$(Y i))))";
  93.175  by (stac beta_cfun 1);
  93.176  by tac;
  93.177  by (stac beta_cfun 1);
  93.178 @@ -532,22 +532,22 @@
  93.179  qed "thelub_ssum2b";
  93.180  
  93.181  Goalw [sinl_def,sinr_def] 
  93.182 -        "[| chain(Y); lub(range(Y)) = sinl`x|] ==> !i.? x. Y(i)=sinl`x";
  93.183 +        "[| chain(Y); lub(range(Y)) = sinl$x|] ==> !i.? x. Y(i)=sinl$x";
  93.184  by (asm_simp_tac (Ssum0_ss addsimps [cont_Isinr,cont_Isinl,cont_Iwhen1,cont_Iwhen2, cont_Iwhen3]) 1);
  93.185  by (etac ssum_lemma9 1);
  93.186  by (asm_simp_tac (Ssum0_ss addsimps [cont_Isinr,cont_Isinl,cont_Iwhen1,cont_Iwhen2, cont_Iwhen3]) 1);
  93.187  qed "thelub_ssum2a_rev";
  93.188  
  93.189  Goalw [sinl_def,sinr_def] 
  93.190 -     "[| chain(Y); lub(range(Y)) = sinr`x|] ==> !i.? x. Y(i)=sinr`x";
  93.191 +     "[| chain(Y); lub(range(Y)) = sinr$x|] ==> !i.? x. Y(i)=sinr$x";
  93.192  by (asm_simp_tac (Ssum0_ss addsimps [cont_Isinr,cont_Isinl,cont_Iwhen1,cont_Iwhen2, cont_Iwhen3]) 1);
  93.193  by (etac ssum_lemma10 1);
  93.194  by (asm_simp_tac (Ssum0_ss addsimps [cont_Isinr,cont_Isinl,cont_Iwhen1,cont_Iwhen2, cont_Iwhen3]) 1);
  93.195  qed "thelub_ssum2b_rev";
  93.196  
  93.197  Goal "chain(Y) ==>\ 
  93.198 -\   lub(range(Y)) = sinl`(lub(range(%i. sscase`(LAM x. x)`(LAM y. UU)`(Y i))))\
  93.199 -\ | lub(range(Y)) = sinr`(lub(range(%i. sscase`(LAM y. UU)`(LAM x. x)`(Y i))))";
  93.200 +\   lub(range(Y)) = sinl$(lub(range(%i. sscase$(LAM x. x)$(LAM y. UU)$(Y i))))\
  93.201 +\ | lub(range(Y)) = sinr$(lub(range(%i. sscase$(LAM y. UU)$(LAM x. x)$(Y i))))";
  93.202  by (rtac (ssum_chainE RS disjE) 1);
  93.203  by (atac 1);
  93.204  by (rtac disjI1 1);
  93.205 @@ -558,7 +558,7 @@
  93.206  by (atac 1);
  93.207  qed "thelub_ssum3";
  93.208  
  93.209 -Goal "sscase`sinl`sinr`z=z";
  93.210 +Goal "sscase$sinl$sinr$z=z";
  93.211  by (res_inst_tac [("p","z")] ssumE 1);
  93.212  by Auto_tac;
  93.213  qed "sscase4";
    94.1 --- a/src/HOLCF/Ssum3.thy	Tue Jan 09 15:29:17 2001 +0100
    94.2 +++ b/src/HOLCF/Ssum3.thy	Tue Jan 09 15:32:27 2001 +0100
    94.3 @@ -22,6 +22,6 @@
    94.4  sscase_def      "sscase   == (LAM f g s. Iwhen(f)(g)(s))"
    94.5  
    94.6  translations
    94.7 -"case s of sinl`x => t1 | sinr`y => t2" == "sscase`(LAM x. t1)`(LAM y. t2)`s"
    94.8 +"case s of sinl$x => t1 | sinr$y => t2" == "sscase$(LAM x. t1)$(LAM y. t2)$s"
    94.9  
   94.10  end
    95.1 --- a/src/HOLCF/Tr.ML	Tue Jan 09 15:29:17 2001 +0100
    95.2 +++ b/src/HOLCF/Tr.ML	Tue Jan 09 15:32:27 2001 +0100
    95.3 @@ -72,9 +72,9 @@
    95.4  			"(y orelse y) = y"]);
    95.5  
    95.6  bind_thms ("neg_thms", map prover [
    95.7 -                        "neg`TT = FF",
    95.8 -                        "neg`FF = TT",
    95.9 -                        "neg`UU = UU"
   95.10 +                        "neg$TT = FF",
   95.11 +                        "neg$FF = TT",
   95.12 +                        "neg$UU = UU"
   95.13                          ]);
   95.14  
   95.15  bind_thms ("ifte_thms", map prover [
    96.1 --- a/src/HOLCF/Tr.thy	Tue Jan 09 15:29:17 2001 +0100
    96.2 +++ b/src/HOLCF/Tr.thy	Tue Jan 09 15:32:27 2001 +0100
    96.3 @@ -27,14 +27,14 @@
    96.4          "@orelse"       :: "tr => tr => tr" ("_ orelse _"  [31,30] 30)
    96.5   
    96.6  translations 
    96.7 -	     "x andalso y" == "trand`x`y"
    96.8 -             "x orelse y"  == "tror`x`y"
    96.9 -             "If b then e1 else e2 fi" == "Icifte`b`e1`e2"
   96.10 +	     "x andalso y" == "trand$x$y"
   96.11 +             "x orelse y"  == "tror$x$y"
   96.12 +             "If b then e1 else e2 fi" == "Icifte$b$e1$e2"
   96.13  defs
   96.14    TT_def      "TT==Def True"
   96.15    FF_def      "FF==Def False"
   96.16    neg_def     "neg == flift2 Not"
   96.17 -  ifte_def    "Icifte == (LAM b t e. flift1(%b. if b then t else e)`b)"
   96.18 +  ifte_def    "Icifte == (LAM b t e. flift1(%b. if b then t else e)$b)"
   96.19    andalso_def "trand == (LAM x y. If x then y else FF fi)"
   96.20    orelse_def  "tror == (LAM x y. If x then TT else y fi)"
   96.21    If2_def     "If2 Q x y == If Q then x else y fi"
    97.1 --- a/src/HOLCF/Up1.ML	Tue Jan 09 15:29:17 2001 +0100
    97.2 +++ b/src/HOLCF/Up1.ML	Tue Jan 09 15:32:27 2001 +0100
    97.3 @@ -64,7 +64,7 @@
    97.4  qed "Ifup1";
    97.5  
    97.6  Goalw [Ifup_def,Iup_def]
    97.7 -        "Ifup(f)(Iup(x))=f`x";
    97.8 +        "Ifup(f)(Iup(x))=f$x";
    97.9  by (stac Abs_Up_inverse2 1);
   97.10  by (stac sum_case_Inr 1);
   97.11  by (rtac refl 1);
    98.1 --- a/src/HOLCF/Up1.thy	Tue Jan 09 15:29:17 2001 +0100
    98.2 +++ b/src/HOLCF/Up1.thy	Tue Jan 09 15:32:27 2001 +0100
    98.3 @@ -22,7 +22,7 @@
    98.4  
    98.5  defs
    98.6    Iup_def     "Iup x == Abs_Up(Inr(x))"
    98.7 -  Ifup_def    "Ifup(f)(x)== case Rep_Up(x) of Inl(y) => UU | Inr(z) => f`z"
    98.8 +  Ifup_def    "Ifup(f)(x)== case Rep_Up(x) of Inl(y) => UU | Inr(z) => f$z"
    98.9    less_up_def "(op <<) == (%x1 x2. case Rep_Up(x1) of                 
   98.10                 Inl(y1) => True          
   98.11               | Inr(y2) => (case Rep_Up(x2) of Inl(z1) => False       
    99.1 --- a/src/HOLCF/Up3.ML	Tue Jan 09 15:29:17 2001 +0100
    99.2 +++ b/src/HOLCF/Up3.ML	Tue Jan 09 15:32:27 2001 +0100
    99.3 @@ -133,23 +133,23 @@
    99.4  (* continuous versions of lemmas for ('a)u                                  *)
    99.5  (* ------------------------------------------------------------------------ *)
    99.6  
    99.7 -Goalw [up_def] "z = UU | (EX x. z = up`x)";
    99.8 +Goalw [up_def] "z = UU | (EX x. z = up$x)";
    99.9  by (simp_tac (Up0_ss addsimps [cont_Iup]) 1);
   99.10  by (stac inst_up_pcpo 1);
   99.11  by (rtac Exh_Up 1);
   99.12  qed "Exh_Up1";
   99.13  
   99.14 -Goalw [up_def] "up`x=up`y ==> x=y";
   99.15 +Goalw [up_def] "up$x=up$y ==> x=y";
   99.16  by (rtac inject_Iup 1);
   99.17  by Auto_tac;
   99.18  qed "inject_up";
   99.19  
   99.20 -Goalw [up_def] " up`x ~= UU";
   99.21 +Goalw [up_def] " up$x ~= UU";
   99.22  by Auto_tac;
   99.23  qed "defined_up";
   99.24  
   99.25  val prems = Goalw [up_def] 
   99.26 -        "[| p=UU ==> Q; !!x. p=up`x==>Q|] ==>Q";
   99.27 +        "[| p=UU ==> Q; !!x. p=up$x==>Q|] ==>Q";
   99.28  by (rtac upE 1);
   99.29  by (resolve_tac prems 1);
   99.30  by (etac (inst_up_pcpo RS ssubst) 1);
   99.31 @@ -160,7 +160,7 @@
   99.32  val tac = (simp_tac (simpset() addsimps [cont_Iup,cont_Ifup1,
   99.33                  cont_Ifup2,cont2cont_CF1L]) 1);
   99.34  
   99.35 -Goalw [up_def,fup_def] "fup`f`UU=UU";
   99.36 +Goalw [up_def,fup_def] "fup$f$UU=UU";
   99.37  by (stac inst_up_pcpo 1);
   99.38  by (stac beta_cfun 1);
   99.39  by tac;
   99.40 @@ -169,7 +169,7 @@
   99.41  by (simp_tac (Up0_ss addsimps [cont_Iup,cont_Ifup1,cont_Ifup2]) 1);
   99.42  qed "fup1";
   99.43  
   99.44 -Goalw [up_def,fup_def] "fup`f`(up`x)=f`x";
   99.45 +Goalw [up_def,fup_def] "fup$f$(up$x)=f$x";
   99.46  by (stac beta_cfun 1);
   99.47  by (rtac cont_Iup 1);
   99.48  by (stac beta_cfun 1);
   99.49 @@ -179,20 +179,20 @@
   99.50  by (simp_tac (Up0_ss addsimps [cont_Iup,cont_Ifup1,cont_Ifup2]) 1);
   99.51  qed "fup2";
   99.52  
   99.53 -Goalw [up_def,fup_def] "~ up`x << UU";
   99.54 +Goalw [up_def,fup_def] "~ up$x << UU";
   99.55  by (simp_tac (Up0_ss addsimps [cont_Iup]) 1);
   99.56  by (rtac less_up3b 1);
   99.57  qed "less_up4b";
   99.58  
   99.59  Goalw [up_def,fup_def]
   99.60 -         "(up`x << up`y) = (x<<y)";
   99.61 +         "(up$x << up$y) = (x<<y)";
   99.62  by (simp_tac (Up0_ss addsimps [cont_Iup]) 1);
   99.63  by (rtac less_up2c 1);
   99.64  qed "less_up4c";
   99.65  
   99.66  Goalw [up_def,fup_def] 
   99.67 -"[| chain(Y); EX i x. Y(i) = up`x |] ==>\
   99.68 -\      lub(range(Y)) = up`(lub(range(%i. fup`(LAM x. x)`(Y i))))";
   99.69 +"[| chain(Y); EX i x. Y(i) = up$x |] ==>\
   99.70 +\      lub(range(Y)) = up$(lub(range(%i. fup$(LAM x. x)$(Y i))))";
   99.71  by (stac beta_cfun 1);
   99.72  by tac;
   99.73  by (stac beta_cfun 1);
   99.74 @@ -213,7 +213,7 @@
   99.75  
   99.76  
   99.77  Goalw [up_def,fup_def] 
   99.78 -"[| chain(Y); ! i x. Y(i) ~= up`x |] ==> lub(range(Y)) = UU";
   99.79 +"[| chain(Y); ! i x. Y(i) ~= up$x |] ==> lub(range(Y)) = UU";
   99.80  by (stac inst_up_pcpo 1);
   99.81  by (rtac thelub_up1b 1);
   99.82  by (atac 1);
   99.83 @@ -224,7 +224,7 @@
   99.84  qed "thelub_up2b";
   99.85  
   99.86  
   99.87 -Goal "(EX x. z = up`x) = (z~=UU)";
   99.88 +Goal "(EX x. z = up$x) = (z~=UU)";
   99.89  by (rtac iffI 1);
   99.90  by (etac exE 1);
   99.91  by (hyp_subst_tac 1);
   99.92 @@ -236,7 +236,7 @@
   99.93  qed "up_lemma2";
   99.94  
   99.95  
   99.96 -Goal "[| chain(Y); lub(range(Y)) = up`x |] ==> EX i x. Y(i) = up`x";
   99.97 +Goal "[| chain(Y); lub(range(Y)) = up$x |] ==> EX i x. Y(i) = up$x";
   99.98  by (rtac exE 1);
   99.99  by (rtac chain_UU_I_inverse2 1);
  99.100  by (rtac (up_lemma2 RS iffD1) 1);
  99.101 @@ -246,14 +246,14 @@
  99.102  by (atac 1);
  99.103  qed "thelub_up2a_rev";
  99.104  
  99.105 -Goal "[| chain(Y); lub(range(Y)) = UU |] ==> ! i x.  Y(i) ~= up`x";
  99.106 +Goal "[| chain(Y); lub(range(Y)) = UU |] ==> ! i x.  Y(i) ~= up$x";
  99.107  by (blast_tac (claset() addSDs [chain_UU_I RS spec, 
  99.108                                  exI RS (up_lemma2 RS iffD1)]) 1);
  99.109  qed "thelub_up2b_rev";
  99.110  
  99.111  
  99.112  Goal "chain(Y) ==> lub(range(Y)) = UU | \
  99.113 -\                  lub(range(Y)) = up`(lub(range(%i. fup`(LAM x. x)`(Y i))))";
  99.114 +\                  lub(range(Y)) = up$(lub(range(%i. fup$(LAM x. x)$(Y i))))";
  99.115  by (rtac disjE 1);
  99.116  by (rtac disjI1 2);
  99.117  by (rtac thelub_up2b 2);
  99.118 @@ -266,7 +266,7 @@
  99.119  by (fast_tac HOL_cs 1);
  99.120  qed "thelub_up3";
  99.121  
  99.122 -Goal "fup`up`x=x";
  99.123 +Goal "fup$up$x=x";
  99.124  by (res_inst_tac [("p","x")] upE1 1);
  99.125  by (asm_simp_tac ((simpset_of Cfun3.thy) addsimps [fup1,fup2]) 1);
  99.126  by (asm_simp_tac ((simpset_of Cfun3.thy) addsimps [fup1,fup2]) 1);
   100.1 --- a/src/HOLCF/Up3.thy	Tue Jan 09 15:29:17 2001 +0100
   100.2 +++ b/src/HOLCF/Up3.thy	Tue Jan 09 15:32:27 2001 +0100
   100.3 @@ -19,7 +19,7 @@
   100.4         "fup == (LAM f p. Ifup(f)(p))"
   100.5  
   100.6  translations
   100.7 -"case l of up`x => t1" == "fup`(LAM x. t1)`l"
   100.8 +"case l of up$x => t1" == "fup$(LAM x. t1)$l"
   100.9  
  100.10  end
  100.11