(* ID: $Id$
Author: Tobias Nipkow
Copyright 1992 University of Cambridge
*)
header {*Well-founded Recursion*}
theory Wellfounded_Recursion
imports Transitive_Closure
begin
inductive2
wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b => bool"
for R :: "('a * 'a) set"
and F :: "('a => 'b) => 'a => 'b"
where
wfrecI: "ALL z. (z, x) : R --> wfrec_rel R F z (g z) ==>
wfrec_rel R F x (F g x)"
constdefs
wf :: "('a * 'a)set => bool"
"wf(r) == (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"
wfP :: "('a => 'a => bool) => bool"
"wfP r == wf (Collect2 r)"
acyclic :: "('a*'a)set => bool"
"acyclic r == !x. (x,x) ~: r^+"
cut :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b"
"cut f r x == (%y. if (y,x):r then f y else arbitrary)"
adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool"
"adm_wf R F == ALL f g x.
(ALL z. (z, x) : R --> f z = g z) --> F f x = F g x"
wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b"
[code func del]: "wfrec R F == %x. THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"
abbreviation acyclicP :: "('a => 'a => bool) => bool" where
"acyclicP r == acyclic (Collect2 r)"
class wellorder = linorder +
assumes wf: "wf {(x, y). x \<sqsubset> y}"
lemma wfP_wf_eq [pred_set_conv]: "wfP (member2 r) = wf r"
by (simp add: wfP_def)
lemma wf_implies_wfP: "wf r \<Longrightarrow> wfP (member2 r)"
by (simp add: wfP_def)
lemma wfP_implies_wf: "wfP r \<Longrightarrow> wf (Collect2 r)"
by (simp add: wfP_def)
lemma wfUNIVI:
"(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"
by (unfold wf_def, blast)
lemmas wfPUNIVI = wfUNIVI [to_pred]
text{*Restriction to domain @{term A} and range @{term B}. If @{term r} is
well-founded over their intersection, then @{term "wf r"}*}
lemma wfI:
"[| r \<subseteq> A <*> B;
!!x P. [|\<forall>x. (\<forall>y. (y,x) : r --> P y) --> P x; x : A; x : B |] ==> 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)
lemmas wfP_induct = wf_induct [to_pred]
lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]
lemmas wfP_induct_rule =
wf_induct_rule [to_pred, consumes 1, case_names less, induct set: wfP]
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
lemmas wfP_trancl = wf_trancl [to_pred]
lemma wf_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"
apply (subst trancl_converse [symmetric])
apply (erule wf_trancl)
done
subsubsection{*Other simple well-foundedness results*}
text{*Minimal-element characterization of well-foundedness*}
lemma wf_eq_minimal: "wf r = (\<forall>Q x. x\<in>Q --> (\<exists>z\<in>Q. \<forall>y. (y,z)\<in>r --> y\<notin>Q))"
proof (intro iffI strip)
fix Q::"'a set" and x
assume "wf r" and "x \<in> Q"
thus "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q"
by (unfold wf_def,
blast dest: spec [of _ "%x. x\<in>Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y,z) \<in> r \<longrightarrow> y\<notin>Q)"])
next
assume 1: "\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q)"
show "wf r"
proof (rule wfUNIVI)
fix P :: "'a \<Rightarrow> bool" and x
assume 2: "\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x"
let ?Q = "{x. \<not> P x}"
have "x \<in> ?Q \<longrightarrow> (\<exists>z\<in>?Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> ?Q)"
by (rule 1 [THEN spec, THEN spec])
hence "\<not> P x \<longrightarrow> (\<exists>z. \<not> P z \<and> (\<forall>y. (y, z) \<in> r \<longrightarrow> P y))" by simp
with 2 have "\<not> P x \<longrightarrow> (\<exists>z. \<not> P z \<and> P z)" by fast
thus "P x" by simp
qed
qed
lemma wfE_min:
assumes p:"wf R" "x \<in> Q"
obtains z where "z \<in> Q" "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
using p
unfolding wf_eq_minimal
by blast
lemma wfI_min:
"(\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q)
\<Longrightarrow> wf R"
unfolding wf_eq_minimal
by blast
lemmas wfP_eq_minimal = wf_eq_minimal [to_pred]
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
lemmas wfP_subset = wf_subset [to_pred]
text{*Well-foundedness of the empty relation*}
lemma wf_empty [iff]: "wf({})"
by (simp add: wf_def)
lemmas wfP_empty [iff] = wf_empty [to_pred]
lemma wf_Int1: "wf r ==> wf (r Int r')"
by (erule wf_subset, rule Int_lower1)
lemma wf_Int2: "wf r ==> wf (r' Int r)"
by (erule wf_subset, rule Int_lower2)
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 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
lemmas wfP_SUP = wf_UN [where I=UNIV and r="\<lambda>i. Collect2 (r i)",
to_pred member2_SUP, simplified, standard]
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
lemma wf_union_merge:
"wf (R \<union> S) = wf (R O R \<union> R O S \<union> S)" (is "wf ?A = wf ?B")
proof
assume "wf ?A"
with wf_trancl have wfT: "wf (?A^+)" .
moreover have "?B \<subseteq> ?A^+"
by (subst trancl_unfold, subst trancl_unfold) blast
ultimately show "wf ?B" by (rule wf_subset)
next
assume "wf ?B"
show "wf ?A"
proof (rule wfI_min)
fix Q :: "'a set" and x
assume "x \<in> Q"
with `wf ?B`
obtain z where "z \<in> Q" and "\<And>y. (y, z) \<in> ?B \<Longrightarrow> y \<notin> Q"
by (erule wfE_min)
hence A1: "\<And>y. (y, z) \<in> R O R \<Longrightarrow> y \<notin> Q"
and A2: "\<And>y. (y, z) \<in> R O S \<Longrightarrow> y \<notin> Q"
and A3: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> Q"
by auto
show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> ?A \<longrightarrow> y \<notin> Q"
proof (cases "\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q")
case True
with `z \<in> Q` A3 show ?thesis by blast
next
case False
then obtain z' where "z'\<in>Q" "(z', z) \<in> R" by blast
have "\<forall>y. (y, z') \<in> ?A \<longrightarrow> y \<notin> Q"
proof (intro allI impI)
fix y assume "(y, z') \<in> ?A"
thus "y \<notin> Q"
proof
assume "(y, z') \<in> R"
hence "(y, z) \<in> R O R" using `(z', z) \<in> R` ..
with A1 show "y \<notin> Q" .
next
assume "(y, z') \<in> S"
hence "(y, z) \<in> R O S" using `(z', z) \<in> R` ..
with A2 show "y \<notin> Q" .
qed
qed
with `z' \<in> Q` show ?thesis ..
qed
qed
qed
lemma wf_comp_self: "wf R = wf (R O R)" (* special case *)
by (fact wf_union_merge[where S = "{}", simplified])
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
lemmas wfP_acyclicP = wf_acyclic [to_pred]
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)
lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
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. wfrec_rel R F x y"
apply (simp add: adm_wf_def)
apply (erule_tac a=x in wf_induct)
apply (rule ex1I)
apply (rule_tac g = "%x. THE y. wfrec_rel R F x y" 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 {* Code generator setup *}
consts_code
"wfrec" ("\<module>wfrec?")
attach {*
fun wfrec f x = f (wfrec f) x;
*}
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]
-- "The following 3 lemmas are due to Brian Huffman"
lemma LeastI_ex: "EX x::'a::wellorder. P x ==> P (Least P)"
apply (erule exE)
apply (erule LeastI)
done
lemma LeastI2:
"[| P (a::'a::wellorder); !!x. P x ==> Q x |] ==> Q (Least P)"
by (blast intro: LeastI)
lemma LeastI2_ex:
"[| EX a::'a::wellorder. P a; !!x. P x ==> Q x |] ==> Q (Least P)"
by (blast intro: LeastI_ex)
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