--- a/src/HOLCF/Porder.ML Wed Mar 02 12:06:15 2005 +0100
+++ b/src/HOLCF/Porder.ML Wed Mar 02 22:30:00 2005 +0100
@@ -1,182 +1,46 @@
-(* Title: HOLCF/Porder
- ID: $Id$
- Author: Franz Regensburger
-Conservative extension of theory Porder0 by constant definitions
-*)
-
-(* ------------------------------------------------------------------------ *)
-(* lubs are unique *)
-(* ------------------------------------------------------------------------ *)
-
-
-Goalw [is_lub_def, is_ub_def]
- "[| S <<| x ; S <<| y |] ==> x=y";
-by (blast_tac (claset() addIs [antisym_less]) 1);
-qed "unique_lub";
-
-(* ------------------------------------------------------------------------ *)
-(* chains are monotone functions *)
-(* ------------------------------------------------------------------------ *)
-
-Goalw [chain_def] "chain F ==> x<y --> F x<<F y";
-by (induct_tac "y" 1);
-by Auto_tac;
-by (blast_tac (claset() addIs [trans_less]) 2);
-by (blast_tac (claset() addSEs [less_SucE]) 1);
-qed_spec_mp "chain_mono";
-
-Goal "[| chain F; x <= y |] ==> F x << F y";
-by (dtac le_imp_less_or_eq 1);
-by (blast_tac (claset() addIs [chain_mono]) 1);
-qed "chain_mono3";
-
-
-(* ------------------------------------------------------------------------ *)
-(* The range of a chain is a totally ordered << *)
-(* ------------------------------------------------------------------------ *)
-
-Goalw [tord_def] "chain(F) ==> tord(range(F))";
-by Safe_tac;
-by (rtac nat_less_cases 1);
-by (ALLGOALS (fast_tac (claset() addIs [chain_mono])));
-qed "chain_tord";
-
-
-(* ------------------------------------------------------------------------ *)
-(* technical lemmas about lub and is_lub *)
-(* ------------------------------------------------------------------------ *)
-bind_thm("lub",lub_def RS meta_eq_to_obj_eq);
-
-Goal "EX x. M <<| x ==> M <<| lub(M)";
-by (asm_full_simp_tac (simpset() addsimps [lub, some_eq_ex]) 1);
-bind_thm ("lubI", exI RS result());
-
-Goal "M <<| l ==> lub(M) = l";
-by (rtac unique_lub 1);
-by (stac lub 1);
-by (etac someI 1);
-by (atac 1);
-qed "thelubI";
-
-
-Goal "lub{x} = x";
-by (simp_tac (simpset() addsimps [thelubI,is_lub_def,is_ub_def]) 1);
-qed "lub_singleton";
-Addsimps [lub_singleton];
-
-(* ------------------------------------------------------------------------ *)
-(* access to some definition as inference rule *)
-(* ------------------------------------------------------------------------ *)
-
-Goalw [is_lub_def] "S <<| x ==> S <| x";
-by Auto_tac;
-qed "is_lubD1";
-
-Goalw [is_lub_def] "[| S <<| x; S <| u |] ==> x << u";
-by Auto_tac;
-qed "is_lub_lub";
-
-val prems = Goalw [is_lub_def]
- "[| S <| x; !!u. S <| u ==> x << u |] ==> S <<| x";
-by (blast_tac (claset() addIs prems) 1);
-qed "is_lubI";
-
-Goalw [chain_def] "chain F ==> F(i) << F(Suc(i))";
-by Auto_tac;
-qed "chainE";
+(* legacy ML bindings *)
-val prems = Goalw [chain_def] "(!!i. F i << F(Suc i)) ==> chain F";
-by (blast_tac (claset() addIs prems) 1);
-qed "chainI";
-
-Goal "chain Y ==> chain (%i. Y (i + j))";
-by (rtac chainI 1);
-by (Clarsimp_tac 1);
-by (etac chainE 1);
-qed "chain_shift";
-
-(* ------------------------------------------------------------------------ *)
-(* technical lemmas about (least) upper bounds of chains *)
-(* ------------------------------------------------------------------------ *)
-
-Goalw [is_ub_def] "range S <| x ==> S(i) << x";
-by (Blast_tac 1);
-qed "ub_rangeD";
-
-val prems = Goalw [is_ub_def] "(!!i. S i << x) ==> range S <| x";
-by (blast_tac (claset() addIs prems) 1);
-qed "ub_rangeI";
-
-bind_thm ("is_ub_lub", is_lubD1 RS ub_rangeD);
-(* range(?S1) <<| ?x1 ==> ?S1(?x) << ?x1 *)
-
-
-(* ------------------------------------------------------------------------ *)
-(* results about finite chains *)
-(* ------------------------------------------------------------------------ *)
-
-Goalw [max_in_chain_def]
- "[| chain C; max_in_chain i C|] ==> range C <<| C i";
-by (rtac is_lubI 1);
-by (rtac ub_rangeI 1);
-by (res_inst_tac [("m","i")] nat_less_cases 1);
-by (rtac (antisym_less_inverse RS conjunct2) 1);
-by (etac (disjI1 RS less_or_eq_imp_le RS rev_mp) 1);
-by (etac spec 1);
-by (rtac (antisym_less_inverse RS conjunct2) 1);
-by (etac (disjI2 RS less_or_eq_imp_le RS rev_mp) 1);
-by (etac spec 1);
-by (etac chain_mono 1);
-by (atac 1);
-by (etac (ub_rangeD) 1);
-qed "lub_finch1";
+val is_ub_def = thm "is_ub_def";
+val is_lub_def = thm "is_lub_def";
+val tord_def = thm "tord_def";
+val chain_def = thm "chain_def";
+val max_in_chain_def = thm "max_in_chain_def";
+val finite_chain_def = thm "finite_chain_def";
+val lub_def = thm "lub_def";
+val unique_lub = thm "unique_lub";
+val chain_mono = thm "chain_mono";
+val chain_mono3 = thm "chain_mono3";
+val chain_tord = thm "chain_tord";
+val lub = thm "lub";
+val lubI = thm "lubI";
+val thelubI = thm "thelubI";
+val lub_singleton = thm "lub_singleton";
+val is_lubD1 = thm "is_lubD1";
+val is_lub_lub = thm "is_lub_lub";
+val is_lubI = thm "is_lubI";
+val chainE = thm "chainE";
+val chainI = thm "chainI";
+val chain_shift = thm "chain_shift";
+val ub_rangeD = thm "ub_rangeD";
+val ub_rangeI = thm "ub_rangeI";
+val is_ub_lub = thm "is_ub_lub";
+val lub_finch1 = thm "lub_finch1";
+val lub_finch2 = thm "lub_finch2";
+val bin_chain = thm "bin_chain";
+val bin_chainmax = thm "bin_chainmax";
+val lub_bin_chain = thm "lub_bin_chain";
+val lub_chain_maxelem = thm "lub_chain_maxelem";
+val lub_const = thm "lub_const";
-Goalw [finite_chain_def]
- "finite_chain(C) ==> range(C) <<| C(@ i. max_in_chain i C)";
-by (rtac lub_finch1 1);
-by (best_tac (claset() addIs [someI]) 2);
-by (Blast_tac 1);
-qed "lub_finch2";
-
-
-Goal "x<<y ==> chain (%i. if i=0 then x else y)";
-by (rtac chainI 1);
-by (induct_tac "i" 1);
-by Auto_tac;
-qed "bin_chain";
-
-Goalw [max_in_chain_def,le_def]
- "x<<y ==> max_in_chain (Suc 0) (%i. if (i=0) then x else y)";
-by (rtac allI 1);
-by (induct_tac "j" 1);
-by Auto_tac;
-qed "bin_chainmax";
-
-Goal "x << y ==> range(%i::nat. if (i=0) then x else y) <<| y";
-by (res_inst_tac [("s","if (Suc 0) = 0 then x else y")] subst 1
- THEN rtac lub_finch1 2);
-by (etac bin_chain 2);
-by (etac bin_chainmax 2);
-by (Simp_tac 1);
-qed "lub_bin_chain";
-
-(* ------------------------------------------------------------------------ *)
-(* the maximal element in a chain is its lub *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "[| Y i = c; ALL i. Y i<<c |] ==> lub(range Y) = c";
-by (blast_tac (claset() addDs [ub_rangeD]
- addIs [thelubI, is_lubI, ub_rangeI]) 1);
-qed "lub_chain_maxelem";
-
-(* ------------------------------------------------------------------------ *)
-(* the lub of a constant chain is the constant *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "range(%x. c) <<| c";
-by (blast_tac (claset() addDs [ub_rangeD] addIs [is_lubI, ub_rangeI]) 1);
-qed "lub_const";
-
-
-
+structure Porder =
+struct
+ val thy = the_context ();
+ val is_ub_def = is_ub_def;
+ val is_lub_def = is_lub_def;
+ val tord_def = tord_def;
+ val chain_def = chain_def;
+ val max_in_chain_def = max_in_chain_def;
+ val finite_chain_def = finite_chain_def;
+ val lub_def = lub_def;
+end;
--- a/src/HOLCF/Porder.thy Wed Mar 02 12:06:15 2005 +0100
+++ b/src/HOLCF/Porder.thy Wed Mar 02 22:30:00 2005 +0100
@@ -1,11 +1,12 @@
(* Title: HOLCF/porder.thy
ID: $Id$
Author: Franz Regensburger
+ License: GPL (GNU GENERAL PUBLIC LICENSE)
Conservative extension of theory Porder0 by constant definitions
*)
-Porder = Porder0 +
+theory Porder = Porder0:
consts
"<|" :: "['a set,'a::po] => bool" (infixl 55)
@@ -25,25 +26,217 @@
syntax (xsymbols)
- "LUB " :: "[idts, 'a] => 'a" ("(3\\<Squnion>_./ _)"[0,10] 10)
+ "LUB " :: "[idts, 'a] => 'a" ("(3\<Squnion>_./ _)"[0,10] 10)
defs
(* class definitions *)
-is_ub_def "S <| x == ! y. y:S --> y<<x"
-is_lub_def "S <<| x == S <| x & (!u. S <| u --> x << u)"
+is_ub_def: "S <| x == ! y. y:S --> y<<x"
+is_lub_def: "S <<| x == S <| x & (!u. S <| u --> x << u)"
(* Arbitrary chains are total orders *)
-tord_def "tord S == !x y. x:S & y:S --> (x<<y | y<<x)"
+tord_def: "tord S == !x y. x:S & y:S --> (x<<y | y<<x)"
(* Here we use countable chains and I prefer to code them as functions! *)
-chain_def "chain F == !i. F i << F (Suc i)"
+chain_def: "chain F == !i. F i << F (Suc i)"
(* finite chains, needed for monotony of continouous functions *)
-max_in_chain_def "max_in_chain i C == ! j. i <= j --> C(i) = C(j)"
-finite_chain_def "finite_chain C == chain(C) & (? i. max_in_chain i C)"
+max_in_chain_def: "max_in_chain i C == ! j. i <= j --> C(i) = C(j)"
+finite_chain_def: "finite_chain C == chain(C) & (? i. max_in_chain i C)"
+
+lub_def: "lub S == (@x. S <<| x)"
+
+(* Title: HOLCF/Porder
+ ID: $Id$
+ Author: Franz Regensburger
+ License: GPL (GNU GENERAL PUBLIC LICENSE)
+
+Conservative extension of theory Porder0 by constant definitions
+*)
+
+(* ------------------------------------------------------------------------ *)
+(* lubs are unique *)
+(* ------------------------------------------------------------------------ *)
+
+
+lemma unique_lub:
+ "[| S <<| x ; S <<| y |] ==> x=y"
+apply (unfold is_lub_def is_ub_def)
+apply (blast intro: antisym_less)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* chains are monotone functions *)
+(* ------------------------------------------------------------------------ *)
+
+lemma chain_mono [rule_format]: "chain F ==> x<y --> F x<<F y"
+apply (unfold chain_def)
+apply (induct_tac "y")
+apply auto
+prefer 2 apply (blast intro: trans_less)
+apply (blast elim!: less_SucE)
+done
+
+lemma chain_mono3: "[| chain F; x <= y |] ==> F x << F y"
+apply (drule le_imp_less_or_eq)
+apply (blast intro: chain_mono)
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* The range of a chain is a totally ordered << *)
+(* ------------------------------------------------------------------------ *)
+
+lemma chain_tord: "chain(F) ==> tord(range(F))"
+apply (unfold tord_def)
+apply safe
+apply (rule nat_less_cases)
+apply (fast intro: chain_mono)+
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* technical lemmas about lub and is_lub *)
+(* ------------------------------------------------------------------------ *)
+lemmas lub = lub_def [THEN meta_eq_to_obj_eq, standard]
+
+lemma lubI[OF exI]: "EX x. M <<| x ==> M <<| lub(M)"
+apply (simp add: lub some_eq_ex)
+done
+
+lemma thelubI: "M <<| l ==> lub(M) = l"
+apply (rule unique_lub)
+apply (subst lub)
+apply (erule someI)
+apply assumption
+done
+
+
+lemma lub_singleton: "lub{x} = x"
+apply (simp (no_asm) add: thelubI is_lub_def is_ub_def)
+done
+declare lub_singleton [simp]
+
+(* ------------------------------------------------------------------------ *)
+(* access to some definition as inference rule *)
+(* ------------------------------------------------------------------------ *)
+
+lemma is_lubD1: "S <<| x ==> S <| x"
+apply (unfold is_lub_def)
+apply auto
+done
+
+lemma is_lub_lub: "[| S <<| x; S <| u |] ==> x << u"
+apply (unfold is_lub_def)
+apply auto
+done
+
+lemma is_lubI:
+ "[| S <| x; !!u. S <| u ==> x << u |] ==> S <<| x"
+apply (unfold is_lub_def)
+apply blast
+done
-lub_def "lub S == (@x. S <<| x)"
+lemma chainE: "chain F ==> F(i) << F(Suc(i))"
+apply (unfold chain_def)
+apply auto
+done
+
+lemma chainI: "(!!i. F i << F(Suc i)) ==> chain F"
+apply (unfold chain_def)
+apply blast
+done
+
+lemma chain_shift: "chain Y ==> chain (%i. Y (i + j))"
+apply (rule chainI)
+apply clarsimp
+apply (erule chainE)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* technical lemmas about (least) upper bounds of chains *)
+(* ------------------------------------------------------------------------ *)
+
+lemma ub_rangeD: "range S <| x ==> S(i) << x"
+apply (unfold is_ub_def)
+apply blast
+done
+
+lemma ub_rangeI: "(!!i. S i << x) ==> range S <| x"
+apply (unfold is_ub_def)
+apply blast
+done
+
+lemmas is_ub_lub = is_lubD1 [THEN ub_rangeD, standard]
+(* range(?S1) <<| ?x1 ==> ?S1(?x) << ?x1 *)
+
+
+(* ------------------------------------------------------------------------ *)
+(* results about finite chains *)
+(* ------------------------------------------------------------------------ *)
+
+lemma lub_finch1:
+ "[| chain C; max_in_chain i C|] ==> range C <<| C i"
+apply (unfold max_in_chain_def)
+apply (rule is_lubI)
+apply (rule ub_rangeI)
+apply (rule_tac m = "i" in nat_less_cases)
+apply (rule antisym_less_inverse [THEN conjunct2])
+apply (erule disjI1 [THEN less_or_eq_imp_le, THEN rev_mp])
+apply (erule spec)
+apply (rule antisym_less_inverse [THEN conjunct2])
+apply (erule disjI2 [THEN less_or_eq_imp_le, THEN rev_mp])
+apply (erule spec)
+apply (erule chain_mono)
+apply assumption
+apply (erule ub_rangeD)
+done
+
+lemma lub_finch2:
+ "finite_chain(C) ==> range(C) <<| C(@ i. max_in_chain i C)"
+apply (unfold finite_chain_def)
+apply (rule lub_finch1)
+prefer 2 apply (best intro: someI)
+apply blast
+done
+
+
+lemma bin_chain: "x<<y ==> chain (%i. if i=0 then x else y)"
+apply (rule chainI)
+apply (induct_tac "i")
+apply auto
+done
+
+lemma bin_chainmax:
+ "x<<y ==> max_in_chain (Suc 0) (%i. if (i=0) then x else y)"
+apply (unfold max_in_chain_def le_def)
+apply (rule allI)
+apply (induct_tac "j")
+apply auto
+done
+
+lemma lub_bin_chain: "x << y ==> range(%i::nat. if (i=0) then x else y) <<| y"
+apply (rule_tac s = "if (Suc 0) = 0 then x else y" in subst , rule_tac [2] lub_finch1)
+apply (erule_tac [2] bin_chain)
+apply (erule_tac [2] bin_chainmax)
+apply (simp (no_asm))
+done
+
+(* ------------------------------------------------------------------------ *)
+(* the maximal element in a chain is its lub *)
+(* ------------------------------------------------------------------------ *)
+
+lemma lub_chain_maxelem: "[| Y i = c; ALL i. Y i<<c |] ==> lub(range Y) = c"
+apply (blast dest: ub_rangeD intro: thelubI is_lubI ub_rangeI)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* the lub of a constant chain is the constant *)
+(* ------------------------------------------------------------------------ *)
+
+lemma lub_const: "range(%x. c) <<| c"
+apply (blast dest: ub_rangeD intro: is_lubI ub_rangeI)
+done
end
--- a/src/HOLCF/Porder0.ML Wed Mar 02 12:06:15 2005 +0100
+++ b/src/HOLCF/Porder0.ML Wed Mar 02 22:30:00 2005 +0100
@@ -1,34 +1,10 @@
-(* Title: HOLCF/Porder0.ML
- ID: $Id$
- Author: Oscar Slotosch
-derive the characteristic axioms for the characteristic constants
-*)
-
-AddIffs [refl_less];
-
-(* ------------------------------------------------------------------------ *)
-(* minimal fixes least element *)
-(* ------------------------------------------------------------------------ *)
-Goal "!x::'a::po. uu<<x ==> uu=(@u.!y. u<<y)";
-by (blast_tac (claset() addIs [someI2,antisym_less]) 1);
-bind_thm ("minimal2UU", allI RS result());
+(* legacy ML bindings *)
-(* ------------------------------------------------------------------------ *)
-(* the reverse law of anti--symmetrie of << *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "(x::'a::po)=y ==> x << y & y << x";
-by (Blast_tac 1);
-qed "antisym_less_inverse";
-
-
-Goal "[| (a::'a::po) << b; c << a; b << d|] ==> c << d";
-by (etac trans_less 1);
-by (etac trans_less 1);
-by (atac 1);
-qed "box_less";
-
-Goal "((x::'a::po)=y) = (x << y & y << x)";
-by (fast_tac (HOL_cs addSEs [antisym_less_inverse] addSIs [antisym_less]) 1);
-qed "po_eq_conv";
+val refl_less = thm "refl_less";
+val antisym_less = thm "antisym_less";
+val trans_less = thm "trans_less";
+val minimal2UU = thm "minimal2UU";
+val antisym_less_inverse = thm "antisym_less_inverse";
+val box_less = thm "box_less";
+val po_eq_conv = thm "po_eq_conv";
--- a/src/HOLCF/Porder0.thy Wed Mar 02 12:06:15 2005 +0100
+++ b/src/HOLCF/Porder0.thy Wed Mar 02 22:30:00 2005 +0100
@@ -1,11 +1,12 @@
(* Title: HOLCF/Porder0.thy
ID: $Id$
Author: Franz Regensburger
+ License: GPL (GNU GENERAL PUBLIC LICENSE)
Definition of class porder (partial order).
*)
-Porder0 = Main +
+theory Porder0 = Main:
(* introduce a (syntactic) class for the constant << *)
axclass sq_ord < type
@@ -15,14 +16,41 @@
"<<" :: "['a,'a::sq_ord] => bool" (infixl 55)
syntax (xsymbols)
- "op <<" :: "['a,'a::sq_ord] => bool" (infixl "\\<sqsubseteq>" 55)
+ "op <<" :: "['a,'a::sq_ord] => bool" (infixl "\<sqsubseteq>" 55)
axclass po < sq_ord
(* class axioms: *)
-refl_less "x << x"
-antisym_less "[|x << y; y << x |] ==> x = y"
-trans_less "[|x << y; y << z |] ==> x << z"
-
+refl_less: "x << x"
+antisym_less: "[|x << y; y << x |] ==> x = y"
+trans_less: "[|x << y; y << z |] ==> x << z"
+
+declare refl_less [iff]
+
+(* ------------------------------------------------------------------------ *)
+(* minimal fixes least element *)
+(* ------------------------------------------------------------------------ *)
+lemma minimal2UU[OF allI] : "!x::'a::po. uu<<x ==> uu=(@u.!y. u<<y)"
+apply (blast intro: someI2 antisym_less)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* the reverse law of anti--symmetrie of << *)
+(* ------------------------------------------------------------------------ *)
+
+lemma antisym_less_inverse: "(x::'a::po)=y ==> x << y & y << x"
+apply blast
+done
+
+
+lemma box_less: "[| (a::'a::po) << b; c << a; b << d|] ==> c << d"
+apply (erule trans_less)
+apply (erule trans_less)
+apply assumption
+done
+
+lemma po_eq_conv: "((x::'a::po)=y) = (x << y & y << x)"
+apply (fast elim!: antisym_less_inverse intro!: antisym_less)
+done
end