1.1 --- a/src/LCF/LCF.thy	Mon Mar 19 21:25:15 2012 +0100
     1.2 +++ b/src/LCF/LCF.thy	Mon Mar 19 21:49:52 2012 +0100
     1.3 @@ -44,77 +44,83 @@
     1.4   COND   :: "[tr,'a,'a] => 'a"   ("(_ =>/ (_ |/ _))" [60,60,60] 60)
     1.5   less   :: "['a,'a] => o"       (infixl "<<" 50)
     1.6  
     1.7 -axioms
     1.8 +axiomatization where
     1.9    (** DOMAIN THEORY **)
    1.10  
    1.11 -  eq_def:        "x=y == x << y & y << x"
    1.12 +  eq_def:        "x=y == x << y & y << x" and
    1.13  
    1.14 -  less_trans:    "[| x << y; y << z |] ==> x << z"
    1.15 +  less_trans:    "[| x << y; y << z |] ==> x << z" and
    1.16  
    1.17 -  less_ext:      "(ALL x. f(x) << g(x)) ==> f << g"
    1.18 +  less_ext:      "(ALL x. f(x) << g(x)) ==> f << g" and
    1.19  
    1.20 -  mono:          "[| f << g; x << y |] ==> f(x) << g(y)"
    1.21 +  mono:          "[| f << g; x << y |] ==> f(x) << g(y)" and
    1.22  
    1.23 -  minimal:       "UU << x"
    1.24 +  minimal:       "UU << x" and
    1.25  
    1.26 -  FIX_eq:        "f(FIX(f)) = FIX(f)"
    1.27 +  FIX_eq:        "\<And>f. f(FIX(f)) = FIX(f)"
    1.28  
    1.29 +axiomatization where
    1.30    (** TR **)
    1.31  
    1.32 -  tr_cases:      "p=UU | p=TT | p=FF"
    1.33 +  tr_cases:      "p=UU | p=TT | p=FF" and
    1.34  
    1.35 -  not_TT_less_FF: "~ TT << FF"
    1.36 -  not_FF_less_TT: "~ FF << TT"
    1.37 -  not_TT_less_UU: "~ TT << UU"
    1.38 -  not_FF_less_UU: "~ FF << UU"
    1.39 +  not_TT_less_FF: "~ TT << FF" and
    1.40 +  not_FF_less_TT: "~ FF << TT" and
    1.41 +  not_TT_less_UU: "~ TT << UU" and
    1.42 +  not_FF_less_UU: "~ FF << UU" and
    1.43  
    1.44 -  COND_UU:       "UU => x | y  =  UU"
    1.45 -  COND_TT:       "TT => x | y  =  x"
    1.46 +  COND_UU:       "UU => x | y  =  UU" and
    1.47 +  COND_TT:       "TT => x | y  =  x" and
    1.48    COND_FF:       "FF => x | y  =  y"
    1.49  
    1.50 +axiomatization where
    1.51    (** PAIRS **)
    1.52  
    1.53 -  surj_pairing:  "<FST(z),SND(z)> = z"
    1.54 +  surj_pairing:  "<FST(z),SND(z)> = z" and
    1.55  
    1.56 -  FST:   "FST(<x,y>) = x"
    1.57 +  FST:   "FST(<x,y>) = x" and
    1.58    SND:   "SND(<x,y>) = y"
    1.59  
    1.60 +axiomatization where
    1.61    (*** STRICT SUM ***)
    1.62  
    1.63 -  INL_DEF: "~x=UU ==> ~INL(x)=UU"
    1.64 -  INR_DEF: "~x=UU ==> ~INR(x)=UU"
    1.65 +  INL_DEF: "~x=UU ==> ~INL(x)=UU" and
    1.66 +  INR_DEF: "~x=UU ==> ~INR(x)=UU" and
    1.67  
    1.68 -  INL_STRICT: "INL(UU) = UU"
    1.69 -  INR_STRICT: "INR(UU) = UU"
    1.70 +  INL_STRICT: "INL(UU) = UU" and
    1.71 +  INR_STRICT: "INR(UU) = UU" and
    1.72  
    1.73 -  WHEN_UU:  "WHEN(f,g,UU) = UU"
    1.74 -  WHEN_INL: "~x=UU ==> WHEN(f,g,INL(x)) = f(x)"
    1.75 -  WHEN_INR: "~x=UU ==> WHEN(f,g,INR(x)) = g(x)"
    1.76 +  WHEN_UU:  "WHEN(f,g,UU) = UU" and
    1.77 +  WHEN_INL: "~x=UU ==> WHEN(f,g,INL(x)) = f(x)" and
    1.78 +  WHEN_INR: "~x=UU ==> WHEN(f,g,INR(x)) = g(x)" and
    1.79  
    1.80    SUM_EXHAUSTION:
    1.81      "z = UU | (EX x. ~x=UU & z = INL(x)) | (EX y. ~y=UU & z = INR(y))"
    1.82  
    1.83 +axiomatization where
    1.84    (** VOID **)
    1.85  
    1.86    void_cases:    "(x::void) = UU"
    1.87  
    1.88    (** INDUCTION **)
    1.89  
    1.90 +axiomatization where
    1.91    induct:        "[| adm(P); P(UU); ALL x. P(x) --> P(f(x)) |] ==> P(FIX(f))"
    1.92  
    1.93 +axiomatization where
    1.94    (** Admissibility / Chain Completeness **)
    1.95    (* All rules can be found on pages 199--200 of Larry's LCF book.
    1.96       Note that "easiness" of types is not taken into account
    1.97       because it cannot be expressed schematically; flatness could be. *)
    1.98  
    1.99 -  adm_less:      "adm(%x. t(x) << u(x))"
   1.100 -  adm_not_less:  "adm(%x.~ t(x) << u)"
   1.101 -  adm_not_free:  "adm(%x. A)"
   1.102 -  adm_subst:     "adm(P) ==> adm(%x. P(t(x)))"
   1.103 -  adm_conj:      "[| adm(P); adm(Q) |] ==> adm(%x. P(x)&Q(x))"
   1.104 -  adm_disj:      "[| adm(P); adm(Q) |] ==> adm(%x. P(x)|Q(x))"
   1.105 -  adm_imp:       "[| adm(%x.~P(x)); adm(Q) |] ==> adm(%x. P(x)-->Q(x))"
   1.106 -  adm_all:       "(!!y. adm(P(y))) ==> adm(%x. ALL y. P(y,x))"
   1.107 +  adm_less:      "\<And>t u. adm(%x. t(x) << u(x))" and
   1.108 +  adm_not_less:  "\<And>t u. adm(%x.~ t(x) << u)" and
   1.109 +  adm_not_free:  "\<And>A. adm(%x. A)" and
   1.110 +  adm_subst:     "\<And>P t. adm(P) ==> adm(%x. P(t(x)))" and
   1.111 +  adm_conj:      "\<And>P Q. [| adm(P); adm(Q) |] ==> adm(%x. P(x)&Q(x))" and
   1.112 +  adm_disj:      "\<And>P Q. [| adm(P); adm(Q) |] ==> adm(%x. P(x)|Q(x))" and
   1.113 +  adm_imp:       "\<And>P Q. [| adm(%x.~P(x)); adm(Q) |] ==> adm(%x. P(x)-->Q(x))" and
   1.114 +  adm_all:       "\<And>P. (!!y. adm(P(y))) ==> adm(%x. ALL y. P(y,x))"
   1.115  
   1.116  
   1.117  lemma eq_imp_less1: "x = y ==> x << y"

     2.1 --- a/src/LCF/ex/Ex1.thy	Mon Mar 19 21:25:15 2012 +0100
     2.2 +++ b/src/LCF/ex/Ex1.thy	Mon Mar 19 21:49:52 2012 +0100
     2.3 @@ -4,15 +4,14 @@
     2.4  imports LCF
     2.5  begin
     2.6  
     2.7 -consts
     2.8 -  P     :: "'a => tr"
     2.9 -  G     :: "'a => 'a"
    2.10 -  H     :: "'a => 'a"
    2.11 +axiomatization
    2.12 +  P     :: "'a => tr" and
    2.13 +  G     :: "'a => 'a" and
    2.14 +  H     :: "'a => 'a" and
    2.15    K     :: "('a => 'a) => ('a => 'a)"
    2.16 -
    2.17 -axioms
    2.18 -  P_strict:     "P(UU) = UU"
    2.19 -  K:            "K = (%h x. P(x) => x | h(h(G(x))))"
    2.20 +where
    2.21 +  P_strict:     "P(UU) = UU" and
    2.22 +  K:            "K = (%h x. P(x) => x | h(h(G(x))))" and
    2.23    H:            "H = FIX(K)"
    2.24  
    2.25  
    2.26 @@ -30,11 +29,11 @@
    2.27  
    2.28  lemma H_idemp_lemma: "ALL x. H(FIX(K,x)) = FIX(K,x)"
    2.29    apply (tactic {* induct_tac @{context} "K" 1 *})
    2.30 -  apply (simp (no_asm))
    2.31 -  apply (simp (no_asm) split: COND_cases_iff)
    2.32 +  apply simp
    2.33 +  apply (simp split: COND_cases_iff)
    2.34    apply (intro strip)
    2.35    apply (subst H_unfold)
    2.36 -  apply (simp (no_asm_simp))
    2.37 +  apply simp
    2.38    done
    2.39  
    2.40  lemma H_idemp: "ALL x. H(H(x)) = H(x)"

     3.1 --- a/src/LCF/ex/Ex2.thy	Mon Mar 19 21:25:15 2012 +0100
     3.2 +++ b/src/LCF/ex/Ex2.thy	Mon Mar 19 21:49:52 2012 +0100
     3.3 @@ -4,16 +4,15 @@
     3.4  imports LCF
     3.5  begin
     3.6  
     3.7 -consts
     3.8 -  P     :: "'a => tr"
     3.9 -  F     :: "'a => 'a"
    3.10 -  G     :: "'a => 'a"
    3.11 -  H     :: "'a => 'b => 'b"
    3.12 +axiomatization
    3.13 +  P     :: "'a => tr" and
    3.14 +  F     :: "'b => 'b" and
    3.15 +  G     :: "'a => 'a" and
    3.16 +  H     :: "'a => 'b => 'b" and
    3.17    K     :: "('a => 'b => 'b) => ('a => 'b => 'b)"
    3.18 -
    3.19 -axioms
    3.20 -  F_strict:     "F(UU) = UU"
    3.21 -  K:            "K = (%h x y. P(x) => y | F(h(G(x),y)))"
    3.22 +where
    3.23 +  F_strict:     "F(UU) = UU" and
    3.24 +  K:            "K = (%h x y. P(x) => y | F(h(G(x),y)))" and
    3.25    H:            "H = FIX(K)"
    3.26  
    3.27  declare F_strict [simp] K [simp]
    3.28 @@ -21,8 +20,8 @@
    3.29  lemma example: "ALL x. F(H(x::'a,y::'b)) = H(x,F(y))"
    3.30    apply (simplesubst H)
    3.31    apply (tactic {* induct_tac @{context} "K:: ('a=>'b=>'b) => ('a=>'b=>'b)" 1 *})
    3.32 -  apply (simp (no_asm))
    3.33 -  apply (simp (no_asm_simp) split: COND_cases_iff)
    3.34 +  apply simp
    3.35 +  apply (simp split: COND_cases_iff)
    3.36    done
    3.37  
    3.38  end

     4.1 --- a/src/LCF/ex/Ex3.thy	Mon Mar 19 21:25:15 2012 +0100
     4.2 +++ b/src/LCF/ex/Ex3.thy	Mon Mar 19 21:49:52 2012 +0100
     4.3 @@ -4,20 +4,19 @@
     4.4  imports LCF
     4.5  begin
     4.6  
     4.7 -consts
     4.8 -  s     :: "'a => 'a"
     4.9 +axiomatization
    4.10 +  s     :: "'a => 'a" and
    4.11    p     :: "'a => 'a => 'a"
    4.12 -
    4.13 -axioms
    4.14 -  p_strict:     "p(UU) = UU"
    4.15 +where
    4.16 +  p_strict:     "p(UU) = UU" and
    4.17    p_s:          "p(s(x),y) = s(p(x,y))"
    4.18  
    4.19  declare p_strict [simp] p_s [simp]
    4.20  
    4.21  lemma example: "p(FIX(s),y) = FIX(s)"
    4.22    apply (tactic {* induct_tac @{context} "s" 1 *})
    4.23 -  apply (simp (no_asm))
    4.24 -  apply (simp (no_asm))
    4.25 +  apply simp
    4.26 +  apply simp
    4.27    done
    4.28  
    4.29  end