--- a/src/HOL/IsaMakefile Mon Nov 29 11:25:32 2004 +0100
+++ b/src/HOL/IsaMakefile Mon Nov 29 14:02:55 2004 +0100
@@ -112,7 +112,7 @@
Tools/specification_package.ML \
Tools/split_rule.ML Tools/typedef_package.ML \
Transitive_Closure.thy Transitive_Closure.ML Typedef.thy \
- Wellfounded_Recursion.ML Wellfounded_Recursion.thy Wellfounded_Relations.ML \
+ Wellfounded_Recursion.thy Wellfounded_Relations.ML \
Wellfounded_Relations.thy arith_data.ML antisym_setup.ML \
blastdata.ML cladata.ML \
document/root.tex hologic.ML simpdata.ML thy_syntax.ML
--- a/src/HOL/MicroJava/BV/Listn.thy Mon Nov 29 11:25:32 2004 +0100
+++ b/src/HOL/MicroJava/BV/Listn.thy Mon Nov 29 14:02:55 2004 +0100
@@ -185,19 +185,19 @@
done
lemma in_list_Suc_iff:
- "(xs : list (Suc n) A) = (? y:A. ? ys:list n A. xs = y#ys)"
+ "(xs : list (Suc n) A) = (\<exists>y\<in> A. \<exists>ys\<in> list n A. xs = y#ys)"
apply (unfold list_def)
apply (case_tac "xs")
apply auto
done
lemma Cons_in_list_Suc [iff]:
- "(x#xs : list (Suc n) A) = (x:A & xs : list n A)";
+ "(x#xs : list (Suc n) A) = (x\<in> A & xs : list n A)";
apply (simp add: in_list_Suc_iff)
done
lemma list_not_empty:
- "? a. a:A \<Longrightarrow> ? xs. xs : list n A";
+ "\<exists>a. a\<in> A \<Longrightarrow> \<exists>xs. xs : list n A";
apply (induct "n")
apply simp
apply (simp add: in_list_Suc_iff)
@@ -248,7 +248,7 @@
lemma listt_update_in_list [simp, intro!]:
- "\<lbrakk> xs : list n A; x:A \<rbrakk> \<Longrightarrow> xs[i := x] : list n A"
+ "\<lbrakk> xs : list n A; x\<in> A \<rbrakk> \<Longrightarrow> xs[i := x] : list n A"
apply (unfold list_def)
apply simp
done
@@ -306,7 +306,7 @@
done
lemma (in semilat) list_update_incr [rule_format]:
- "x:A \<Longrightarrow> set xs <= A \<longrightarrow>
+ "x\<in> A \<Longrightarrow> set xs <= A \<longrightarrow>
(!i. i<size xs \<longrightarrow> xs <=[r] xs[i := x +_f xs!i])"
apply (unfold unfold_lesub_list)
apply (simp add: Listn.le_def list_all2_conv_all_nth)
@@ -330,8 +330,6 @@
apply (rename_tac m n)
apply (case_tac "m=n")
apply simp
- apply (rule conjI)
- apply (fast intro!: equals0I dest: not_sym)
apply (fast intro!: equals0I dest: not_sym)
apply clarify
apply (rename_tac n)
@@ -342,15 +340,15 @@
apply (simp (no_asm) add: length_Suc_conv cong: conj_cong)
apply clarify
apply (rename_tac M m)
-apply (case_tac "? x xs. size xs = k & x#xs : M")
+apply (case_tac "\<exists>x xs. size xs = k & x#xs : M")
prefer 2
apply (erule thin_rl)
apply (erule thin_rl)
apply blast
-apply (erule_tac x = "{a. ? xs. size xs = k & a#xs:M}" in allE)
+apply (erule_tac x = "{a. \<exists>xs. size xs = k & a#xs:M}" in allE)
apply (erule impE)
apply blast
-apply (thin_tac "? x xs. ?P x xs")
+apply (thin_tac "\<exists>x xs. ?P x xs")
apply clarify
apply (rename_tac maxA xs)
apply (erule_tac x = "{ys. size ys = size xs & maxA#ys : M}" in allE)
@@ -435,8 +433,8 @@
done
lemma lift2_le_ub:
- "\<lbrakk> semilat(err A, Err.le r, lift2 f); x:A; y:A; x +_f y = OK z;
- u:A; x <=_r u; y <=_r u \<rbrakk> \<Longrightarrow> z <=_r u"
+ "\<lbrakk> semilat(err A, Err.le r, lift2 f); x\<in> A; y\<in> A; x +_f y = OK z;
+ u\<in> A; x <=_r u; y <=_r u \<rbrakk> \<Longrightarrow> z <=_r u"
apply (unfold semilat_Def plussub_def err_def)
apply (simp add: lift2_def)
apply clarify
@@ -464,16 +462,16 @@
done
lemma lift2_eq_ErrD:
- "\<lbrakk> x +_f y = Err; semilat(err A, Err.le r, lift2 f); x:A; y:A \<rbrakk>
- \<Longrightarrow> ~(? u:A. x <=_r u & y <=_r u)"
+ "\<lbrakk> x +_f y = Err; semilat(err A, Err.le r, lift2 f); x\<in> A; y\<in> A \<rbrakk>
+ \<Longrightarrow> ~(\<exists>u\<in> A. x <=_r u & y <=_r u)"
by (simp add: OK_plus_OK_eq_Err_conv [THEN iffD1])
lemma coalesce_eq_Err_D [rule_format]:
"\<lbrakk> semilat(err A, Err.le r, lift2 f) \<rbrakk>
- \<Longrightarrow> !xs. xs:list n A \<longrightarrow> (!ys. ys:list n A \<longrightarrow>
+ \<Longrightarrow> !xs. xs\<in> list n A \<longrightarrow> (!ys. ys\<in> list n A \<longrightarrow>
coalesce (xs +[f] ys) = Err \<longrightarrow>
- ~(? zs:list n A. xs <=[r] zs & ys <=[r] zs))"
+ ~(\<exists>zs\<in> list n A. xs <=[r] zs & ys <=[r] zs))"
apply (induct n)
apply simp
apply clarify
@@ -484,14 +482,14 @@
done
lemma closed_err_lift2_conv:
- "closed (err A) (lift2 f) = (!x:A. !y:A. x +_f y : err A)"
+ "closed (err A) (lift2 f) = (\<forall>x\<in> A. \<forall>y\<in> A. x +_f y : err A)"
apply (unfold closed_def)
apply (simp add: err_def)
done
lemma closed_map2_list [rule_format]:
"closed (err A) (lift2 f) \<Longrightarrow>
- !xs. xs : list n A \<longrightarrow> (!ys. ys : list n A \<longrightarrow>
+ \<forall>xs. xs : list n A \<longrightarrow> (\<forall>ys. ys : list n A \<longrightarrow>
map2 f xs ys : list n (err A))"
apply (unfold map2_def)
apply (induct n)
--- a/src/HOL/Nat.ML Mon Nov 29 11:25:32 2004 +0100
+++ b/src/HOL/Nat.ML Mon Nov 29 14:02:55 2004 +0100
@@ -25,10 +25,8 @@
bind_thm ("nat_case_0", nat_case_0);
bind_thm ("nat_case_Suc", nat_case_Suc);
-val LeastI = thm "LeastI";
val Least_Suc = thm "Least_Suc";
val Least_Suc2 = thm "Least_Suc2";
-val Least_le = thm "Least_le";
val One_nat_def = thm "One_nat_def";
val Suc_Suc_eq = thm "Suc_Suc_eq";
val Suc_def = thm "Suc_def";
@@ -218,7 +216,6 @@
val not_leE = thm "not_leE";
val not_le_iff_less = thm "not_le_iff_less";
val not_less0 = thm "not_less0";
-val not_less_Least = thm "not_less_Least";
val not_less_eq = thm "not_less_eq";
val not_less_iff_le = thm "not_less_iff_le";
val not_less_less_Suc_eq = thm "not_less_less_Suc_eq";
--- a/src/HOL/Nat.thy Mon Nov 29 11:25:32 2004 +0100
+++ b/src/HOL/Nat.thy Mon Nov 29 14:02:55 2004 +0100
@@ -564,11 +564,7 @@
apply (blast intro: less_trans)+
done
-subsection {* @{text LEAST} theorems for type @{typ nat} by specialization *}
-
-lemmas LeastI = wellorder_LeastI
-lemmas Least_le = wellorder_Least_le
-lemmas not_less_Least = wellorder_not_less_Least
+subsection {* @{text LEAST} theorems for type @{typ nat}*}
lemma Least_Suc:
"[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
--- a/src/HOL/Wellfounded_Recursion.ML Mon Nov 29 11:25:32 2004 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,333 +0,0 @@
-(* Title: HOL/Wellfounded_Recursion.ML
- ID: $Id$
- Author: Tobias Nipkow, with minor changes by Konrad Slind
- Copyright 1992 University of Cambridge/1995 TU Munich
-
-Wellfoundedness, induction, and recursion
-*)
-
-Goal "x = y ==> H x z = H y z";
-by (Asm_simp_tac 1);
-val H_cong2 = (*freeze H!*)
- read_instantiate [("H","H")] (result());
-
-val [prem] = Goalw [wf_def]
- "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)";
-by (Clarify_tac 1);
-by (rtac prem 1);
-by (assume_tac 1);
-qed "wfUNIVI";
-
-(*Restriction to domain A. If r is well-founded over A then wf(r)*)
-val [prem1,prem2] = Goalw [wf_def]
- "[| r <= A <*> A; \
-\ !!x P. [| ALL x. (ALL y. (y,x) : r --> P y) --> P x; x:A |] ==> P x |] \
-\ ==> wf r";
-by (cut_facts_tac [prem1] 1);
-by (blast_tac (claset() addIs [prem2]) 1);
-qed "wfI";
-
-val major::prems = Goalw [wf_def]
- "[| wf(r); \
-\ !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x) \
-\ |] ==> P(a)";
-by (rtac (major RS spec RS mp RS spec) 1);
-by (blast_tac (claset() addIs prems) 1);
-qed "wf_induct";
-
-(*Perform induction on i, then prove the wf(r) subgoal using prems. *)
-fun wf_ind_tac a prems i =
- EVERY [res_inst_tac [("a",a)] wf_induct i,
- rename_last_tac a ["1"] (i+1),
- ares_tac prems i];
-
-Goal "wf(r) ==> ALL x. (a,x):r --> (x,a)~:r";
-by (wf_ind_tac "a" [] 1);
-by (Blast_tac 1);
-qed_spec_mp "wf_not_sym";
-
-(* [| wf r; ~Z ==> (a,x) : r; (x,a) ~: r ==> Z |] ==> Z *)
-bind_thm ("wf_asym", cla_make_elim wf_not_sym);
-
-Goal "wf(r) ==> (a,a) ~: r";
-by (blast_tac (claset() addEs [wf_asym]) 1);
-qed "wf_not_refl";
-Addsimps [wf_not_refl];
-
-(* [| wf r; (a,a) ~: r ==> PROP W |] ==> PROP W *)
-bind_thm ("wf_irrefl", make_elim wf_not_refl);
-
-(*transitive closure of a wf relation is wf! *)
-Goal "wf(r) ==> wf(r^+)";
-by (stac wf_def 1);
-by (Clarify_tac 1);
-(*must retain the universal formula for later use!*)
-by (rtac allE 1 THEN assume_tac 1);
-by (etac mp 1);
-by (eres_inst_tac [("a","x")] wf_induct 1);
-by (blast_tac (claset() addEs [tranclE]) 1);
-qed "wf_trancl";
-
-Goal "wf (r^-1) ==> wf ((r^+)^-1)";
-by (stac (trancl_converse RS sym) 1);
-by (etac wf_trancl 1);
-qed "wf_converse_trancl";
-
-
-(*----------------------------------------------------------------------------
- * Minimal-element characterization of well-foundedness
- *---------------------------------------------------------------------------*)
-
-Goalw [wf_def] "wf r ==> x:Q --> (EX z:Q. ALL y. (y,z):r --> y~:Q)";
-by (dtac spec 1);
-by (etac (mp RS spec) 1);
-by (Blast_tac 1);
-val lemma1 = result();
-
-Goalw [wf_def] "(ALL Q x. x:Q --> (EX z:Q. ALL y. (y,z):r --> y~:Q)) ==> wf r";
-by (Clarify_tac 1);
-by (dres_inst_tac [("x", "{x. ~ P x}")] spec 1);
-by (Blast_tac 1);
-val lemma2 = result();
-
-Goal "wf r = (ALL Q x. x:Q --> (EX z:Q. ALL y. (y,z):r --> y~:Q))";
-by (blast_tac (claset() addSIs [lemma1, lemma2]) 1);
-qed "wf_eq_minimal";
-
-(*---------------------------------------------------------------------------
- * Wellfoundedness of subsets
- *---------------------------------------------------------------------------*)
-
-Goal "[| wf(r); p<=r |] ==> wf(p)";
-by (full_simp_tac (simpset() addsimps [wf_eq_minimal]) 1);
-by (Fast_tac 1);
-qed "wf_subset";
-
-(*---------------------------------------------------------------------------
- * Wellfoundedness of the empty relation.
- *---------------------------------------------------------------------------*)
-
-Goal "wf({})";
-by (simp_tac (simpset() addsimps [wf_def]) 1);
-qed "wf_empty";
-AddIffs [wf_empty];
-
-(*---------------------------------------------------------------------------
- * Wellfoundedness of `insert'
- *---------------------------------------------------------------------------*)
-
-Goal "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)";
-by (rtac iffI 1);
- by (blast_tac (claset() addEs [wf_trancl RS wf_irrefl]
- addIs [rtrancl_into_trancl1,wf_subset,impOfSubs rtrancl_mono]) 1);
-by (asm_full_simp_tac (simpset() addsimps [wf_eq_minimal]) 1);
-by Safe_tac;
-by (EVERY1[rtac allE, assume_tac, etac impE, Blast_tac]);
-by (etac bexE 1);
-by (rename_tac "a" 1 THEN case_tac "a = x" 1);
- by (res_inst_tac [("x","a")]bexI 2);
- by (assume_tac 3);
- by (Blast_tac 2);
-by (case_tac "y:Q" 1);
- by (Blast_tac 2);
-by (res_inst_tac [("x","{z. z:Q & (z,y) : r^*}")] allE 1);
- by (assume_tac 1);
-by (thin_tac "ALL Q. (EX x. x : Q) --> ?P Q" 1); (*essential for speed*)
-(*Blast_tac with new substOccur fails*)
-by (best_tac (claset() addIs [converse_rtrancl_into_rtrancl]) 1);
-qed "wf_insert";
-AddIffs [wf_insert];
-
-(*---------------------------------------------------------------------------
- * Wellfoundedness of `disjoint union'
- *---------------------------------------------------------------------------*)
-
-(*Intuition behind this proof for the case of binary union:
-
- Goal: find an (R u S)-min element of a nonempty subset A.
- by case distinction:
- 1. There is a step a -R-> b with a,b : A.
- Pick an R-min element z of the (nonempty) set {a:A | EX b:A. a -R-> b}.
- By definition, there is z':A s.t. z -R-> z'. Because z is R-min in the
- subset, z' must be R-min in A. Because z' has an R-predecessor, it cannot
- have an S-successor and is thus S-min in A as well.
- 2. There is no such step.
- Pick an S-min element of A. In this case it must be an R-min
- element of A as well.
-
-*)
-
-Goal "[| ALL i:I. wf(r i); \
-\ ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {} & \
-\ Domain(r j) Int Range(r i) = {} \
-\ |] ==> wf(UN i:I. r i)";
-by (asm_full_simp_tac (HOL_basic_ss addsimps [wf_eq_minimal]) 1);
-by (Clarify_tac 1);
-by (rename_tac "A a" 1 THEN case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i" 1);
- by (Asm_full_simp_tac 2);
- by (Best_tac 2); (*much faster than Blast_tac*)
-by (Clarify_tac 1);
-by (EVERY1[dtac bspec, assume_tac,
- eres_inst_tac [("x","{a. a:A & (EX b:A. (b,a) : r i)}")] allE]);
-by (EVERY1[etac allE, etac impE]);
- by (ALLGOALS Blast_tac);
-qed "wf_UN";
-
-Goalw [Union_def]
- "[| ALL r:R. wf r; \
-\ ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {} & \
-\ Domain s Int Range r = {} \
-\ |] ==> wf(Union R)";
-by (blast_tac (claset() addIs [wf_UN]) 1);
-qed "wf_Union";
-
-Goal "[| wf r; wf s; Domain r Int Range s = {}; Domain s Int Range r = {} \
-\ |] ==> wf(r Un s)";
-by (rtac (simplify (simpset()) (read_instantiate[("R","{r,s}")]wf_Union)) 1);
-by (Blast_tac 1);
-by (Blast_tac 1);
-qed "wf_Un";
-
-(*---------------------------------------------------------------------------
- * Wellfoundedness of `image'
- *---------------------------------------------------------------------------*)
-
-Goal "[| wf r; inj f |] ==> wf(prod_fun f f ` r)";
-by (asm_full_simp_tac (HOL_basic_ss addsimps [wf_eq_minimal]) 1);
-by (Clarify_tac 1);
-by (case_tac "EX p. f p : Q" 1);
-by (eres_inst_tac [("x","{p. f p : Q}")]allE 1);
-by (fast_tac (claset() addDs [injD]) 1);
-by (Blast_tac 1);
-qed "wf_prod_fun_image";
-
-(*** acyclic ***)
-
-Goalw [acyclic_def] "ALL x. (x, x) ~: r^+ ==> acyclic r";
-by (assume_tac 1);
-qed "acyclicI";
-
-Goalw [acyclic_def] "wf r ==> acyclic r";
-by (blast_tac (claset() addEs [wf_trancl RS wf_irrefl]) 1);
-qed "wf_acyclic";
-
-Goalw [acyclic_def] "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)";
-by (simp_tac (simpset() addsimps [trancl_insert]) 1);
-by (blast_tac (claset() addIs [rtrancl_trans]) 1);
-qed "acyclic_insert";
-AddIffs [acyclic_insert];
-
-Goalw [acyclic_def] "acyclic(r^-1) = acyclic r";
-by (simp_tac (simpset() addsimps [trancl_converse]) 1);
-qed "acyclic_converse";
-AddIffs [acyclic_converse];
-
-Goalw [acyclic_def,antisym_def] "acyclic r ==> antisym(r^*)";
-by (blast_tac (claset() addEs [rtranclE]
- addIs [rtrancl_into_trancl1,rtrancl_trancl_trancl]) 1);
-qed "acyclic_impl_antisym_rtrancl";
-
-(* Other direction:
-acyclic = no loops
-antisym = only self loops
-Goalw [acyclic_def,antisym_def] "antisym(r^* ) ==> acyclic(r - Id)";
-==> "antisym(r^* ) = acyclic(r - Id)";
-*)
-
-Goalw [acyclic_def] "[| acyclic s; r <= s |] ==> acyclic r";
-by (blast_tac (claset() addIs [trancl_mono]) 1);
-qed "acyclic_subset";
-
-(** cut **)
-
-(*This rewrite rule works upon formulae; thus it requires explicit use of
- H_cong to expose the equality*)
-Goalw [cut_def] "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))";
-by (simp_tac (HOL_ss addsimps [expand_fun_eq]) 1);
-qed "cuts_eq";
-
-Goalw [cut_def] "(x,a):r ==> (cut f r a)(x) = f(x)";
-by (asm_simp_tac HOL_ss 1);
-qed "cut_apply";
-
-(*** Inductive characterization of wfrec combinator; for details see: ***)
-(*** John Harrison, "Inductive definitions: automation and application" ***)
-
-Goalw [adm_wf_def]
- "[| adm_wf R F; wf R |] ==> EX! y. (x, y) : wfrec_rel R F";
-by (wf_ind_tac "x" [] 1);
-by (rtac ex1I 1);
-by (res_inst_tac [("g","%x. THE y. (x, y) : wfrec_rel R F")] wfrec_rel.wfrecI 1);
-by (fast_tac (claset() addSDs [theI']) 1);
-by (etac wfrec_rel.elim 1);
-by (Asm_full_simp_tac 1);
-byev [etac allE 1, etac allE 1, etac allE 1, etac mp 1];
-by (fast_tac (claset() addIs [the_equality RS sym]) 1);
-qed "wfrec_unique";
-
-Goalw [adm_wf_def] "adm_wf R (%f x. F (cut f R x) x)";
-by (strip_tac 1);
-by (rtac (cuts_eq RS iffD2 RS subst) 1);
-by (atac 1);
-by (rtac refl 1);
-qed "adm_lemma";
-
-Goalw [wfrec_def]
- "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a";
-by (rtac (adm_lemma RS wfrec_unique RS the1_equality) 1);
-by (atac 1);
-by (rtac wfrec_rel.wfrecI 1);
-by (strip_tac 1);
-by (etac (adm_lemma RS wfrec_unique RS theI') 1);
-qed "wfrec";
-
-
-(*---------------------------------------------------------------------------
- * This form avoids giant explosions in proofs. NOTE USE OF ==
- *---------------------------------------------------------------------------*)
-Goal "[| f==wfrec r H; wf(r) |] ==> f(a) = H (cut f r a) a";
-by Auto_tac;
-by (blast_tac (claset() addIs [wfrec]) 1);
-qed "def_wfrec";
-
-
-(**** TFL variants ****)
-
-Goal "ALL R. wf R --> \
-\ (ALL P. (ALL x. (ALL y. (y,x):R --> P y) --> P x) --> (ALL x. P x))";
-by (Clarify_tac 1);
-by (res_inst_tac [("r","R"),("P","P"), ("a","x")] wf_induct 1);
-by (assume_tac 1);
-by (Blast_tac 1);
-qed"tfl_wf_induct";
-
-Goal "ALL f R. (x,a):R --> (cut f R a)(x) = f(x)";
-by (Clarify_tac 1);
-by (rtac cut_apply 1);
-by (assume_tac 1);
-qed"tfl_cut_apply";
-
-Goal "ALL M R f. (f=wfrec R M) --> wf R --> (ALL x. f x = M (cut f R x) x)";
-by (Clarify_tac 1);
-by (etac wfrec 1);
-qed "tfl_wfrec";
-
-(*LEAST and wellorderings*)
-(* ### see also wf_linord_ex_has_least and its consequences in Wellfounded_Relations.ML *)
-
-Goal "P (k::'a::wellorder) --> P (LEAST x. P(x)) & (LEAST x. P(x)) <= k";
-by (res_inst_tac [("a","k")] (wf RS wf_induct) 1);
-by (rtac impI 1);
-by (rtac classical 1);
-by (res_inst_tac [("s","x")] (Least_equality RS ssubst) 1);
-by Auto_tac;
-by (auto_tac (claset(), simpset() addsimps [linorder_not_less RS sym]));
-bind_thm("wellorder_LeastI", result() RS mp RS conjunct1);
-bind_thm("wellorder_Least_le", result() RS mp RS conjunct2);
-
-Goal "[| k < (LEAST x. P x) |] ==> ~P (k::'a::wellorder)";
-by (full_simp_tac (simpset() addsimps [linorder_not_le RS sym]) 1);
-by (etac contrapos_nn 1);
-by (etac wellorder_Least_le 1);
-qed "wellorder_not_less_Least";
-
--- a/src/HOL/Wellfounded_Recursion.thy Mon Nov 29 11:25:32 2004 +0100
+++ b/src/HOL/Wellfounded_Recursion.thy Mon Nov 29 14:02:55 2004 +0100
@@ -1,19 +1,20 @@
-(* Title: HOL/Wellfounded_Recursion.thy
- ID: $Id$
+(* ID: $Id$
Author: Tobias Nipkow
Copyright 1992 University of Cambridge
-
-Well-founded Recursion
*)
-Wellfounded_Recursion = Transitive_Closure +
+header {*Well-founded Recursion*}
+
+theory Wellfounded_Recursion
+imports Transitive_Closure
+begin
consts
wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => ('a * 'b) set"
inductive "wfrec_rel R F"
-intrs
- wfrecI "ALL z. (z, x) : R --> (z, g z) : wfrec_rel R F ==>
+intros
+ wfrecI: "ALL z. (z, x) : R --> (z, g z) : wfrec_rel R F ==>
(x, F g x) : wfrec_rel R F"
constdefs
@@ -33,8 +34,329 @@
wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b"
"wfrec R F == %x. THE y. (x, y) : wfrec_rel R (%f x. F (cut f R x) x)"
-axclass
- wellorder < linorder
- wf "wf {(x,y::'a::ord). x<y}"
+axclass wellorder \<subseteq> linorder
+ wf: "wf {(x,y::'a::ord). x<y}"
+
+
+lemma wfUNIVI:
+ "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"
+by (unfold wf_def, blast)
+
+text{*Restriction to domain @{term A}.
+ If @{term r} is well-founded over @{term A} then @{term "wf r"}*}
+lemma wfI:
+ "[| r <= A <*> A;
+ !!x P. [| ALL x. (ALL y. (y,x) : r --> P y) --> P x; x:A |] ==> P x |]
+ ==> wf r"
+by (unfold wf_def, blast)
+
+lemma wf_induct:
+ "[| wf(r);
+ !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)
+ |] ==> P(a)"
+by (unfold wf_def, blast)
+
+lemma wf_not_sym [rule_format]: "wf(r) ==> ALL x. (a,x):r --> (x,a)~:r"
+by (erule_tac a=a in wf_induct, blast)
+
+(* [| wf r; ~Z ==> (a,x) : r; (x,a) ~: r ==> Z |] ==> Z *)
+lemmas wf_asym = wf_not_sym [elim_format]
+
+lemma wf_not_refl [simp]: "wf(r) ==> (a,a) ~: r"
+by (blast elim: wf_asym)
+
+(* [| wf r; (a,a) ~: r ==> PROP W |] ==> PROP W *)
+lemmas wf_irrefl = wf_not_refl [elim_format]
+
+text{*transitive closure of a well-founded relation is well-founded! *}
+lemma wf_trancl: "wf(r) ==> wf(r^+)"
+apply (subst wf_def, clarify)
+apply (rule allE, assumption)
+ --{*Retains the universal formula for later use!*}
+apply (erule mp)
+apply (erule_tac a = x in wf_induct)
+apply (blast elim: tranclE)
+done
+
+lemma wf_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"
+apply (subst trancl_converse [symmetric])
+apply (erule wf_trancl)
+done
+
+
+subsubsection{*Minimal-element characterization of well-foundedness*}
+
+lemma lemma1: "wf r ==> x:Q --> (EX z:Q. ALL y. (y,z):r --> y~:Q)"
+apply (unfold wf_def)
+apply (drule spec)
+apply (erule mp [THEN spec], blast)
+done
+
+lemma lemma2: "(ALL Q x. x:Q --> (EX z:Q. ALL y. (y,z):r --> y~:Q)) ==> wf r"
+apply (unfold wf_def, clarify)
+apply (drule_tac x = "{x. ~ P x}" in spec, blast)
+done
+
+lemma wf_eq_minimal: "wf r = (ALL Q x. x:Q --> (EX z:Q. ALL y. (y,z):r --> y~:Q))"
+by (blast intro!: lemma1 lemma2)
+
+subsubsection{*Other simple well-foundedness results*}
+
+
+text{*Well-foundedness of subsets*}
+lemma wf_subset: "[| wf(r); p<=r |] ==> wf(p)"
+apply (simp (no_asm_use) add: wf_eq_minimal)
+apply fast
+done
+
+text{*Well-foundedness of the empty relation*}
+lemma wf_empty [iff]: "wf({})"
+by (simp add: wf_def)
+
+text{*Well-foundedness of insert*}
+lemma wf_insert [iff]: "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)"
+apply (rule iffI)
+ apply (blast elim: wf_trancl [THEN wf_irrefl]
+ intro: rtrancl_into_trancl1 wf_subset
+ rtrancl_mono [THEN [2] rev_subsetD])
+apply (simp add: wf_eq_minimal, safe)
+apply (rule allE, assumption, erule impE, blast)
+apply (erule bexE)
+apply (rename_tac "a", case_tac "a = x")
+ prefer 2
+apply blast
+apply (case_tac "y:Q")
+ prefer 2 apply blast
+apply (rule_tac x = "{z. z:Q & (z,y) : r^*}" in allE)
+ apply assumption
+apply (erule_tac V = "ALL Q. (EX x. x : Q) --> ?P Q" in thin_rl)
+ --{*essential for speed*}
+txt{*Blast_tac with new substOccur fails*}
+apply (fast intro: converse_rtrancl_into_rtrancl)
+done
+
+text{*Well-foundedness of image*}
+lemma wf_prod_fun_image: "[| wf r; inj f |] ==> wf(prod_fun f f ` r)"
+apply (simp only: wf_eq_minimal, clarify)
+apply (case_tac "EX p. f p : Q")
+apply (erule_tac x = "{p. f p : Q}" in allE)
+apply (fast dest: inj_onD, blast)
+done
+
+
+subsubsection{*Well-Foundedness Results for Unions*}
+
+text{*Well-foundedness of indexed union with disjoint domains and ranges*}
+
+lemma wf_UN: "[| ALL i:I. wf(r i);
+ ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {}
+ |] ==> wf(UN i:I. r i)"
+apply (simp only: wf_eq_minimal, clarify)
+apply (rename_tac A a, case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i")
+ prefer 2
+ apply force
+apply clarify
+apply (drule bspec, assumption)
+apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r i) }" in allE)
+apply (blast elim!: allE)
+done
+
+lemma wf_Union:
+ "[| ALL r:R. wf r;
+ ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {}
+ |] ==> wf(Union R)"
+apply (simp add: Union_def)
+apply (blast intro: wf_UN)
+done
+
+(*Intuition: we find an (R u S)-min element of a nonempty subset A
+ by case distinction.
+ 1. There is a step a -R-> b with a,b : A.
+ Pick an R-min element z of the (nonempty) set {a:A | EX b:A. a -R-> b}.
+ By definition, there is z':A s.t. z -R-> z'. Because z is R-min in the
+ subset, z' must be R-min in A. Because z' has an R-predecessor, it cannot
+ have an S-successor and is thus S-min in A as well.
+ 2. There is no such step.
+ Pick an S-min element of A. In this case it must be an R-min
+ element of A as well.
+
+*)
+lemma wf_Un:
+ "[| wf r; wf s; Domain r Int Range s = {} |] ==> wf(r Un s)"
+apply (simp only: wf_eq_minimal, clarify)
+apply (rename_tac A a)
+apply (case_tac "EX a:A. EX b:A. (b,a) : r")
+ prefer 2
+ apply simp
+ apply (drule_tac x=A in spec)+
+ apply blast
+apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r) }" in allE)+
+apply (blast elim!: allE)
+done
+
+subsubsection {*acyclic*}
+
+lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
+by (simp add: acyclic_def)
+
+lemma wf_acyclic: "wf r ==> acyclic r"
+apply (simp add: acyclic_def)
+apply (blast elim: wf_trancl [THEN wf_irrefl])
+done
+
+lemma acyclic_insert [iff]:
+ "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
+apply (simp add: acyclic_def trancl_insert)
+apply (blast intro: rtrancl_trans)
+done
+
+lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
+by (simp add: acyclic_def trancl_converse)
+
+lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
+apply (simp add: acyclic_def antisym_def)
+apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
+done
+
+(* Other direction:
+acyclic = no loops
+antisym = only self loops
+Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
+==> antisym( r^* ) = acyclic(r - Id)";
+*)
+
+lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
+apply (simp add: acyclic_def)
+apply (blast intro: trancl_mono)
+done
+
+
+subsection{*Well-Founded Recursion*}
+
+text{*cut*}
+
+lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))"
+by (simp add: expand_fun_eq cut_def)
+
+lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)"
+by (simp add: cut_def)
+
+text{*Inductive characterization of wfrec combinator; for details see:
+John Harrison, "Inductive definitions: automation and application"*}
+
+lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. (x, y) : wfrec_rel R F"
+apply (simp add: adm_wf_def)
+apply (erule_tac a=x in wf_induct)
+apply (rule ex1I)
+apply (rule_tac g = "%x. THE y. (x, y) : wfrec_rel R F" in wfrec_rel.wfrecI)
+apply (fast dest!: theI')
+apply (erule wfrec_rel.cases, simp)
+apply (erule allE, erule allE, erule allE, erule mp)
+apply (fast intro: the_equality [symmetric])
+done
+
+lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"
+apply (simp add: adm_wf_def)
+apply (intro strip)
+apply (rule cuts_eq [THEN iffD2, THEN subst], assumption)
+apply (rule refl)
+done
+
+lemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"
+apply (simp add: wfrec_def)
+apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption)
+apply (rule wfrec_rel.wfrecI)
+apply (intro strip)
+apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
+done
+
+
+text{** This form avoids giant explosions in proofs. NOTE USE OF ==*}
+lemma def_wfrec: "[| f==wfrec r H; wf(r) |] ==> f(a) = H (cut f r a) a"
+apply auto
+apply (blast intro: wfrec)
+done
+
+
+subsection{*Variants for TFL: the Recdef Package*}
+
+lemma tfl_wf_induct: "ALL R. wf R -->
+ (ALL P. (ALL x. (ALL y. (y,x):R --> P y) --> P x) --> (ALL x. P x))"
+apply clarify
+apply (rule_tac r = R and P = P and a = x in wf_induct, assumption, blast)
+done
+
+lemma tfl_cut_apply: "ALL f R. (x,a):R --> (cut f R a)(x) = f(x)"
+apply clarify
+apply (rule cut_apply, assumption)
+done
+
+lemma tfl_wfrec:
+ "ALL M R f. (f=wfrec R M) --> wf R --> (ALL x. f x = M (cut f R x) x)"
+apply clarify
+apply (erule wfrec)
+done
+
+subsection {*LEAST and wellorderings*}
+
+text{* See also @{text wf_linord_ex_has_least} and its consequences in
+ @{text Wellfounded_Relations.ML}*}
+
+lemma wellorder_Least_lemma [rule_format]:
+ "P (k::'a::wellorder) --> P (LEAST x. P(x)) & (LEAST x. P(x)) <= k"
+apply (rule_tac a = k in wf [THEN wf_induct])
+apply (rule impI)
+apply (rule classical)
+apply (rule_tac s = x in Least_equality [THEN ssubst], auto)
+apply (auto simp add: linorder_not_less [symmetric])
+done
+
+lemmas LeastI = wellorder_Least_lemma [THEN conjunct1, standard]
+lemmas Least_le = wellorder_Least_lemma [THEN conjunct2, standard]
+
+lemma not_less_Least: "[| k < (LEAST x. P x) |] ==> ~P (k::'a::wellorder)"
+apply (simp (no_asm_use) add: linorder_not_le [symmetric])
+apply (erule contrapos_nn)
+apply (erule Least_le)
+done
+
+ML
+{*
+val wf_def = thm "wf_def";
+val wfUNIVI = thm "wfUNIVI";
+val wfI = thm "wfI";
+val wf_induct = thm "wf_induct";
+val wf_not_sym = thm "wf_not_sym";
+val wf_asym = thm "wf_asym";
+val wf_not_refl = thm "wf_not_refl";
+val wf_irrefl = thm "wf_irrefl";
+val wf_trancl = thm "wf_trancl";
+val wf_converse_trancl = thm "wf_converse_trancl";
+val wf_eq_minimal = thm "wf_eq_minimal";
+val wf_subset = thm "wf_subset";
+val wf_empty = thm "wf_empty";
+val wf_insert = thm "wf_insert";
+val wf_UN = thm "wf_UN";
+val wf_Union = thm "wf_Union";
+val wf_Un = thm "wf_Un";
+val wf_prod_fun_image = thm "wf_prod_fun_image";
+val acyclicI = thm "acyclicI";
+val wf_acyclic = thm "wf_acyclic";
+val acyclic_insert = thm "acyclic_insert";
+val acyclic_converse = thm "acyclic_converse";
+val acyclic_impl_antisym_rtrancl = thm "acyclic_impl_antisym_rtrancl";
+val acyclic_subset = thm "acyclic_subset";
+val cuts_eq = thm "cuts_eq";
+val cut_apply = thm "cut_apply";
+val wfrec_unique = thm "wfrec_unique";
+val wfrec = thm "wfrec";
+val def_wfrec = thm "def_wfrec";
+val tfl_wf_induct = thm "tfl_wf_induct";
+val tfl_cut_apply = thm "tfl_cut_apply";
+val tfl_wfrec = thm "tfl_wfrec";
+val LeastI = thm "LeastI";
+val Least_le = thm "Least_le";
+val not_less_Least = thm "not_less_Least";
+*}
end