src/HOL/WF.ML
changeset 972 e61b058d58d2
parent 950 323f8ca4587a
child 1264 3eb91524b938
     1.1 --- a/src/HOL/WF.ML	Thu Mar 23 15:39:13 1995 +0100
     1.2 +++ b/src/HOL/WF.ML	Fri Mar 24 12:30:35 1995 +0100
     1.3 @@ -14,7 +14,7 @@
     1.4  (*Restriction to domain A.  If r is well-founded over A then wf(r)*)
     1.5  val [prem1,prem2] = goalw WF.thy [wf_def]
     1.6   "[| r <= Sigma A (%u.A);  \
     1.7 -\    !!x P. [| ! x. (! y. <y,x> : r --> P(y)) --> P(x);  x:A |] ==> P(x) |]  \
     1.8 +\    !!x P. [| ! x. (! y. (y,x) : r --> P(y)) --> P(x);  x:A |] ==> P(x) |]  \
     1.9  \ ==>  wf(r)";
    1.10  by (strip_tac 1);
    1.11  by (rtac allE 1);
    1.12 @@ -24,7 +24,7 @@
    1.13  
    1.14  val major::prems = goalw WF.thy [wf_def]
    1.15      "[| wf(r);          \
    1.16 -\       !!x.[| ! y. <y,x>: r --> P(y) |] ==> P(x) \
    1.17 +\       !!x.[| ! y. (y,x): r --> P(y) |] ==> P(x) \
    1.18  \    |]  ==>  P(a)";
    1.19  by (rtac (major RS spec RS mp RS spec) 1);
    1.20  by (fast_tac (HOL_cs addEs prems) 1);
    1.21 @@ -36,14 +36,14 @@
    1.22  	   rename_last_tac a ["1"] (i+1),
    1.23  	   ares_tac prems i];
    1.24  
    1.25 -val prems = goal WF.thy "[| wf(r);  <a,x>:r;  <x,a>:r |] ==> P";
    1.26 -by (subgoal_tac "! x. <a,x>:r --> <x,a>:r --> P" 1);
    1.27 +val prems = goal WF.thy "[| wf(r);  (a,x):r;  (x,a):r |] ==> P";
    1.28 +by (subgoal_tac "! x. (a,x):r --> (x,a):r --> P" 1);
    1.29  by (fast_tac (HOL_cs addIs prems) 1);
    1.30  by (wf_ind_tac "a" prems 1);
    1.31  by (fast_tac set_cs 1);
    1.32  qed "wf_asym";
    1.33  
    1.34 -val prems = goal WF.thy "[| wf(r);  <a,a>: r |] ==> P";
    1.35 +val prems = goal WF.thy "[| wf(r);  (a,a): r |] ==> P";
    1.36  by (rtac wf_asym 1);
    1.37  by (REPEAT (resolve_tac prems 1));
    1.38  qed "wf_anti_refl";
    1.39 @@ -68,12 +68,12 @@
    1.40  (*This rewrite rule works upon formulae; thus it requires explicit use of
    1.41    H_cong to expose the equality*)
    1.42  goalw WF.thy [cut_def]
    1.43 -    "(cut f r x = cut g r x) = (!y. <y,x>:r --> f(y)=g(y))";
    1.44 +    "(cut f r x = cut g r x) = (!y. (y,x):r --> f(y)=g(y))";
    1.45  by(simp_tac (HOL_ss addsimps [expand_fun_eq]
    1.46                      setloop (split_tac [expand_if])) 1);
    1.47  qed "cut_cut_eq";
    1.48  
    1.49 -goalw WF.thy [cut_def] "!!x. <x,a>:r ==> (cut f r a)(x) = f(x)";
    1.50 +goalw WF.thy [cut_def] "!!x. (x,a):r ==> (cut f r a)(x) = f(x)";
    1.51  by(asm_simp_tac HOL_ss 1);
    1.52  qed "cut_apply";
    1.53  
    1.54 @@ -81,12 +81,12 @@
    1.55  (*** is_recfun ***)
    1.56  
    1.57  goalw WF.thy [is_recfun_def,cut_def]
    1.58 -    "!!f. [| is_recfun r a H f;  ~<b,a>:r |] ==> f(b) = (@z.True)";
    1.59 +    "!!f. [| is_recfun r a H f;  ~(b,a):r |] ==> f(b) = (@z.True)";
    1.60  by (etac ssubst 1);
    1.61  by(asm_simp_tac HOL_ss 1);
    1.62  qed "is_recfun_undef";
    1.63  
    1.64 -(*eresolve_tac transD solves <a,b>:r using transitivity AT MOST ONCE
    1.65 +(*eresolve_tac transD solves (a,b):r using transitivity AT MOST ONCE
    1.66    mp amd allE  instantiate induction hypotheses*)
    1.67  fun indhyp_tac hyps =
    1.68      ares_tac (TrueI::hyps) ORELSE' 
    1.69 @@ -104,7 +104,7 @@
    1.70  
    1.71  val prems = goalw WF.thy [is_recfun_def,cut_def]
    1.72      "[| wf(r);  trans(r);  is_recfun r a H f;  is_recfun r b H g |] ==> \
    1.73 -    \ <x,a>:r --> <x,b>:r --> f(x)=g(x)";
    1.74 +    \ (x,a):r --> (x,b):r --> f(x)=g(x)";
    1.75  by (cut_facts_tac prems 1);
    1.76  by (etac wf_induct 1);
    1.77  by (REPEAT (rtac impI 1 ORELSE etac ssubst 1));
    1.78 @@ -115,7 +115,7 @@
    1.79  
    1.80  val prems as [wfr,transr,recfa,recgb,_] = goalw WF.thy [cut_def]
    1.81      "[| wf(r);  trans(r); \
    1.82 -\       is_recfun r a H f;  is_recfun r b H g;  <b,a>:r |] ==> \
    1.83 +\       is_recfun r a H f;  is_recfun r b H g;  (b,a):r |] ==> \
    1.84  \    cut f r b = g";
    1.85  val gundef = recgb RS is_recfun_undef
    1.86  and fisg   = recgb RS (recfa RS (transr RS (wfr RS is_recfun_equal)));
    1.87 @@ -150,13 +150,13 @@
    1.88  
    1.89  (*Beware incompleteness of unification!*)
    1.90  val prems = goal WF.thy
    1.91 -    "[| wf(r);  trans(r);  <c,a>:r;  <c,b>:r |] \
    1.92 +    "[| wf(r);  trans(r);  (c,a):r;  (c,b):r |] \
    1.93  \    ==> the_recfun r a H c = the_recfun r b H c";
    1.94  by (DEPTH_SOLVE (ares_tac (prems@[is_recfun_equal,unfold_the_recfun]) 1));
    1.95  qed "the_recfun_equal";
    1.96  
    1.97  val prems = goal WF.thy
    1.98 -    "[| wf(r); trans(r); <b,a>:r |] \
    1.99 +    "[| wf(r); trans(r); (b,a):r |] \
   1.100  \    ==> cut (the_recfun r a H) r b = the_recfun r b H";
   1.101  by (REPEAT (ares_tac (prems@[is_recfun_cut,unfold_the_recfun]) 1));
   1.102  qed "the_recfun_cut";