(* ID: $Id$
Author: Tobias Nipkow
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Author: Konrad Slind, Alexander Krauss
Copyright 1992-2008 University of Cambridge and TU Muenchen
*)
header {*Well-founded Recursion*}
theory Wellfounded
imports Finite_Set Nat
uses ("Tools/function_package/size.ML")
begin
subsection {* Basic Definitions *}
inductive
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 {(x, y). r x y}"
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 {(x, y). r x y}"
class wellorder = linorder +
assumes wf: "wf {(x, y). x < y}"
lemma wfP_wf_eq [pred_set_conv]: "wfP (\<lambda>x y. (x, y) \<in> r) = wf 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)"
unfolding wf_def by 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"
unfolding wf_def by blast
lemma wf_induct:
"[| wf(r);
!!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)
|] ==> P(a)"
unfolding wf_def by 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, induct set: wfP]
lemma wf_not_sym: "wf r ==> (a, x) : r ==> (x, a) ~: r"
by (induct a arbitrary: x set: wf) 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]
subsection {* Basic Results *}
text{*transitive closure of a well-founded relation is well-founded! *}
lemma wf_trancl:
assumes "wf r"
shows "wf (r^+)"
proof -
{
fix P and x
assume induct_step: "!!x. (!!y. (y, x) : r^+ ==> P y) ==> P x"
have "P x"
proof (rule induct_step)
fix y assume "(y, x) : r^+"
with `wf r` show "P y"
proof (induct x arbitrary: y)
case (less x)
note hyp = `\<And>x' y'. (x', x) : r ==> (y', x') : r^+ ==> P y'`
from `(y, x) : r^+` show "P y"
proof cases
case base
show "P y"
proof (rule induct_step)
fix y' assume "(y', y) : r^+"
with `(y, x) : r` show "P y'" by (rule hyp [of y y'])
qed
next
case step
then obtain x' where "(x', x) : r" and "(y, x') : r^+" by simp
then show "P y" by (rule hyp [of x' y])
qed
qed
qed
} then show ?thesis unfolding wf_def by blast
qed
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
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"
then show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q"
unfolding wf_def
by (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])
then have "\<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
then show "P x" by simp
qed
qed
lemma wfE_min:
assumes "wf R" "x \<in> Q"
obtains z where "z \<in> Q" "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
using assms 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 bot_empty_eq2, simplified bot_fun_eq bot_bool_eq]
lemma wf_Int1: "wf r ==> wf (r Int r')"
apply (erule wf_subset)
apply (rule Int_lower1)
done
lemma wf_Int2: "wf r ==> wf (r' Int r)"
apply (erule wf_subset)
apply (rule Int_lower2)
done
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
subsection {* Well-Foundedness Results for Unions *}
lemma wf_union_compatible:
assumes "wf R" "wf S"
assumes "S O R \<subseteq> R"
shows "wf (R \<union> S)"
proof (rule wfI_min)
fix x :: 'a and Q
let ?Q' = "{x \<in> Q. \<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> Q}"
assume "x \<in> Q"
obtain a where "a \<in> ?Q'"
by (rule wfE_min [OF `wf R` `x \<in> Q`]) blast
with `wf S`
obtain z where "z \<in> ?Q'" and zmin: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> ?Q'" by (erule wfE_min)
{
fix y assume "(y, z) \<in> S"
then have "y \<notin> ?Q'" by (rule zmin)
have "y \<notin> Q"
proof
assume "y \<in> Q"
with `y \<notin> ?Q'`
obtain w where "(w, y) \<in> R" and "w \<in> Q" by auto
from `(w, y) \<in> R` `(y, z) \<in> S` have "(w, z) \<in> S O R" by (rule rel_compI)
with `S O R \<subseteq> R` have "(w, z) \<in> R" ..
with `z \<in> ?Q'` have "w \<notin> Q" by blast
with `w \<in> Q` show False by contradiction
qed
}
with `z \<in> ?Q'` show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<union> S \<longrightarrow> y \<notin> Q" by blast
qed
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. {(x, y). r i x y}",
to_pred SUP_UN_eq2 bot_empty_eq pred_equals_eq, 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)"
using wf_union_compatible[of s r]
by (auto simp: Un_ac)
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)
then have 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"
then show "y \<notin> Q"
proof
assume "(y, z') \<in> R"
then have "(y, z) \<in> R O R" using `(z', z) \<in> R` ..
with A1 show "y \<notin> Q" .
next
assume "(y, z') \<in> S"
then have "(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 (rule 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
text{* Wellfoundedness of finite acyclic relations*}
lemma finite_acyclic_wf [rule_format]: "finite r ==> acyclic r --> wf r"
apply (erule finite_induct, blast)
apply (simp (no_asm_simp) only: split_tupled_all)
apply simp
done
lemma finite_acyclic_wf_converse: "[|finite r; acyclic r|] ==> wf (r^-1)"
apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
apply (erule acyclic_converse [THEN iffD2])
done
lemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r"
by (blast intro: finite_acyclic_wf wf_acyclic)
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
subsection {* Code generator setup *}
consts_code
"wfrec" ("\<module>wfrec?")
attach {*
fun wfrec f x = f (wfrec f) x;
*}
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
subsection {* @{typ nat} is well-founded *}
lemma less_nat_rel: "op < = (\<lambda>m n. n = Suc m)^++"
proof (rule ext, rule ext, rule iffI)
fix n m :: nat
assume "m < n"
then show "(\<lambda>m n. n = Suc m)^++ m n"
proof (induct n)
case 0 then show ?case by auto
next
case (Suc n) then show ?case
by (auto simp add: less_Suc_eq_le le_less intro: tranclp.trancl_into_trancl)
qed
next
fix n m :: nat
assume "(\<lambda>m n. n = Suc m)^++ m n"
then show "m < n"
by (induct n)
(simp_all add: less_Suc_eq_le reflexive le_less)
qed
definition
pred_nat :: "(nat * nat) set" where
"pred_nat = {(m, n). n = Suc m}"
definition
less_than :: "(nat * nat) set" where
"less_than = pred_nat^+"
lemma less_eq: "(m, n) \<in> pred_nat^+ \<longleftrightarrow> m < n"
unfolding less_nat_rel pred_nat_def trancl_def by simp
lemma pred_nat_trancl_eq_le:
"(m, n) \<in> pred_nat^* \<longleftrightarrow> m \<le> n"
unfolding less_eq rtrancl_eq_or_trancl by auto
lemma wf_pred_nat: "wf pred_nat"
apply (unfold wf_def pred_nat_def, clarify)
apply (induct_tac x, blast+)
done
lemma wf_less_than [iff]: "wf less_than"
by (simp add: less_than_def wf_pred_nat [THEN wf_trancl])
lemma trans_less_than [iff]: "trans less_than"
by (simp add: less_than_def trans_trancl)
lemma less_than_iff [iff]: "((x,y): less_than) = (x<y)"
by (simp add: less_than_def less_eq)
lemma wf_less: "wf {(x, y::nat). x < y}"
using wf_less_than by (simp add: less_than_def less_eq [symmetric])
text {* Type @{typ nat} is a wellfounded order *}
instance nat :: wellorder
by intro_classes
(assumption |
rule le_refl le_trans le_anti_sym nat_less_le nat_le_linear wf_less)+
text {* @{text LEAST} theorems for type @{typ nat}*}
lemma Least_Suc:
"[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
apply (case_tac "n", auto)
apply (frule LeastI)
apply (drule_tac P = "%x. P (Suc x) " in LeastI)
apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
apply (erule_tac [2] Least_le)
apply (case_tac "LEAST x. P x", auto)
apply (drule_tac P = "%x. P (Suc x) " in Least_le)
apply (blast intro: order_antisym)
done
lemma Least_Suc2:
"[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
apply (erule (1) Least_Suc [THEN ssubst])
apply simp
done
lemma ex_least_nat_le: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not>P i) & P(k)"
apply (cases n)
apply blast
apply (rule_tac x="LEAST k. P(k)" in exI)
apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
done
lemma ex_least_nat_less: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not>P i) & P(k+1)"
apply (cases n)
apply blast
apply (frule (1) ex_least_nat_le)
apply (erule exE)
apply (case_tac k)
apply simp
apply (rename_tac k1)
apply (rule_tac x=k1 in exI)
apply fastsimp
done
subsection {* Accessible Part *}
text {*
Inductive definition of the accessible part @{term "acc r"} of a
relation; see also \cite{paulin-tlca}.
*}
inductive_set
acc :: "('a * 'a) set => 'a set"
for r :: "('a * 'a) set"
where
accI: "(!!y. (y, x) : r ==> y : acc r) ==> x : acc r"
abbreviation
termip :: "('a => 'a => bool) => 'a => bool" where
"termip r == accp (r\<inverse>\<inverse>)"
abbreviation
termi :: "('a * 'a) set => 'a set" where
"termi r == acc (r\<inverse>)"
lemmas accpI = accp.accI
text {* Induction rules *}
theorem accp_induct:
assumes major: "accp r a"
assumes hyp: "!!x. accp r x ==> \<forall>y. r y x --> P y ==> P x"
shows "P a"
apply (rule major [THEN accp.induct])
apply (rule hyp)
apply (rule accp.accI)
apply fast
apply fast
done
theorems accp_induct_rule = accp_induct [rule_format, induct set: accp]
theorem accp_downward: "accp r b ==> r a b ==> accp r a"
apply (erule accp.cases)
apply fast
done
lemma not_accp_down:
assumes na: "\<not> accp R x"
obtains z where "R z x" and "\<not> accp R z"
proof -
assume a: "\<And>z. \<lbrakk>R z x; \<not> accp R z\<rbrakk> \<Longrightarrow> thesis"
show thesis
proof (cases "\<forall>z. R z x \<longrightarrow> accp R z")
case True
hence "\<And>z. R z x \<Longrightarrow> accp R z" by auto
hence "accp R x"
by (rule accp.accI)
with na show thesis ..
next
case False then obtain z where "R z x" and "\<not> accp R z"
by auto
with a show thesis .
qed
qed
lemma accp_downwards_aux: "r\<^sup>*\<^sup>* b a ==> accp r a --> accp r b"
apply (erule rtranclp_induct)
apply blast
apply (blast dest: accp_downward)
done
theorem accp_downwards: "accp r a ==> r\<^sup>*\<^sup>* b a ==> accp r b"
apply (blast dest: accp_downwards_aux)
done
theorem accp_wfPI: "\<forall>x. accp r x ==> wfP r"
apply (rule wfPUNIVI)
apply (induct_tac P x rule: accp_induct)
apply blast
apply blast
done
theorem accp_wfPD: "wfP r ==> accp r x"
apply (erule wfP_induct_rule)
apply (rule accp.accI)
apply blast
done
theorem wfP_accp_iff: "wfP r = (\<forall>x. accp r x)"
apply (blast intro: accp_wfPI dest: accp_wfPD)
done
text {* Smaller relations have bigger accessible parts: *}
lemma accp_subset:
assumes sub: "R1 \<le> R2"
shows "accp R2 \<le> accp R1"
proof (rule predicate1I)
fix x assume "accp R2 x"
then show "accp R1 x"
proof (induct x)
fix x
assume ih: "\<And>y. R2 y x \<Longrightarrow> accp R1 y"
with sub show "accp R1 x"
by (blast intro: accp.accI)
qed
qed
text {* This is a generalized induction theorem that works on
subsets of the accessible part. *}
lemma accp_subset_induct:
assumes subset: "D \<le> accp R"
and dcl: "\<And>x z. \<lbrakk>D x; R z x\<rbrakk> \<Longrightarrow> D z"
and "D x"
and istep: "\<And>x. \<lbrakk>D x; (\<And>z. R z x \<Longrightarrow> P z)\<rbrakk> \<Longrightarrow> P x"
shows "P x"
proof -
from subset and `D x`
have "accp R x" ..
then show "P x" using `D x`
proof (induct x)
fix x
assume "D x"
and "\<And>y. R y x \<Longrightarrow> D y \<Longrightarrow> P y"
with dcl and istep show "P x" by blast
qed
qed
text {* Set versions of the above theorems *}
lemmas acc_induct = accp_induct [to_set]
lemmas acc_induct_rule = acc_induct [rule_format, induct set: acc]
lemmas acc_downward = accp_downward [to_set]
lemmas not_acc_down = not_accp_down [to_set]
lemmas acc_downwards_aux = accp_downwards_aux [to_set]
lemmas acc_downwards = accp_downwards [to_set]
lemmas acc_wfI = accp_wfPI [to_set]
lemmas acc_wfD = accp_wfPD [to_set]
lemmas wf_acc_iff = wfP_accp_iff [to_set]
lemmas acc_subset = accp_subset [to_set pred_subset_eq]
lemmas acc_subset_induct = accp_subset_induct [to_set pred_subset_eq]
subsection {* Tools for building wellfounded relations *}
text {* Inverse Image *}
lemma wf_inv_image [simp,intro!]: "wf(r) ==> wf(inv_image r (f::'a=>'b))"
apply (simp (no_asm_use) add: inv_image_def wf_eq_minimal)
apply clarify
apply (subgoal_tac "EX (w::'b) . w : {w. EX (x::'a) . x: Q & (f x = w) }")
prefer 2 apply (blast del: allE)
apply (erule allE)
apply (erule (1) notE impE)
apply blast
done
lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)"
by (auto simp:inv_image_def)
text {* Measure functions into @{typ nat} *}
definition measure :: "('a => nat) => ('a * 'a)set"
where "measure == inv_image less_than"
lemma in_measure[simp]: "((x,y) : measure f) = (f x < f y)"
by (simp add:measure_def)
lemma wf_measure [iff]: "wf (measure f)"
apply (unfold measure_def)
apply (rule wf_less_than [THEN wf_inv_image])
done
text{* Lexicographic combinations *}
definition
lex_prod :: "[('a*'a)set, ('b*'b)set] => (('a*'b)*('a*'b))set"
(infixr "<*lex*>" 80)
where
"ra <*lex*> rb == {((a,b),(a',b')). (a,a') : ra | a=a' & (b,b') : rb}"
lemma wf_lex_prod [intro!]: "[| wf(ra); wf(rb) |] ==> wf(ra <*lex*> rb)"
apply (unfold wf_def lex_prod_def)
apply (rule allI, rule impI)
apply (simp (no_asm_use) only: split_paired_All)
apply (drule spec, erule mp)
apply (rule allI, rule impI)
apply (drule spec, erule mp, blast)
done
lemma in_lex_prod[simp]:
"(((a,b),(a',b')): r <*lex*> s) = ((a,a'): r \<or> (a = a' \<and> (b, b') : s))"
by (auto simp:lex_prod_def)
text{* @{term "op <*lex*>"} preserves transitivity *}
lemma trans_lex_prod [intro!]:
"[| trans R1; trans R2 |] ==> trans (R1 <*lex*> R2)"
by (unfold trans_def lex_prod_def, blast)
text {* lexicographic combinations with measure functions *}
definition
mlex_prod :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" (infixr "<*mlex*>" 80)
where
"f <*mlex*> R = inv_image (less_than <*lex*> R) (%x. (f x, x))"
lemma wf_mlex: "wf R \<Longrightarrow> wf (f <*mlex*> R)"
unfolding mlex_prod_def
by auto
lemma mlex_less: "f x < f y \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
unfolding mlex_prod_def by simp
lemma mlex_leq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
unfolding mlex_prod_def by auto
text {* proper subset relation on finite sets *}
definition finite_psubset :: "('a set * 'a set) set"
where "finite_psubset == {(A,B). A < B & finite B}"
lemma wf_finite_psubset: "wf(finite_psubset)"
apply (unfold finite_psubset_def)
apply (rule wf_measure [THEN wf_subset])
apply (simp add: measure_def inv_image_def less_than_def less_eq)
apply (fast elim!: psubset_card_mono)
done
lemma trans_finite_psubset: "trans finite_psubset"
by (simp add: finite_psubset_def less_le trans_def, blast)
text {*Wellfoundedness of @{text same_fst}*}
definition
same_fst :: "('a => bool) => ('a => ('b * 'b)set) => (('a*'b)*('a*'b))set"
where
"same_fst P R == {((x',y'),(x,y)) . x'=x & P x & (y',y) : R x}"
--{*For @{text rec_def} declarations where the first n parameters
stay unchanged in the recursive call.
See @{text "Library/While_Combinator.thy"} for an application.*}
lemma same_fstI [intro!]:
"[| P x; (y',y) : R x |] ==> ((x,y'),(x,y)) : same_fst P R"
by (simp add: same_fst_def)
lemma wf_same_fst:
assumes prem: "(!!x. P x ==> wf(R x))"
shows "wf(same_fst P R)"
apply (simp cong del: imp_cong add: wf_def same_fst_def)
apply (intro strip)
apply (rename_tac a b)
apply (case_tac "wf (R a)")
apply (erule_tac a = b in wf_induct, blast)
apply (blast intro: prem)
done
subsection{*Weakly decreasing sequences (w.r.t. some well-founded order)
stabilize.*}
text{*This material does not appear to be used any longer.*}
lemma lemma1: "[| ALL i. (f (Suc i), f i) : r^* |] ==> (f (i+k), f i) : r^*"
apply (induct_tac "k", simp_all)
apply (blast intro: rtrancl_trans)
done
lemma lemma2: "[| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]
==> ALL m. f m = x --> (EX i. ALL k. f (m+i+k) = f (m+i))"
apply (erule wf_induct, clarify)
apply (case_tac "EX j. (f (m+j), f m) : r^+")
apply clarify
apply (subgoal_tac "EX i. ALL k. f ((m+j) +i+k) = f ( (m+j) +i) ")
apply clarify
apply (rule_tac x = "j+i" in exI)
apply (simp add: add_ac, blast)
apply (rule_tac x = 0 in exI, clarsimp)
apply (drule_tac i = m and k = k in lemma1)
apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
done
lemma wf_weak_decr_stable: "[| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]
==> EX i. ALL k. f (i+k) = f i"
apply (drule_tac x = 0 in lemma2 [THEN spec], auto)
done
(* special case of the theorem above: <= *)
lemma weak_decr_stable:
"ALL i. f (Suc i) <= ((f i)::nat) ==> EX i. ALL k. f (i+k) = f i"
apply (rule_tac r = pred_nat in wf_weak_decr_stable)
apply (simp add: pred_nat_trancl_eq_le)
apply (intro wf_trancl wf_pred_nat)
done
subsection {* size of a datatype value *}
use "Tools/function_package/size.ML"
setup Size.setup
lemma nat_size [simp, code func]: "size (n\<Colon>nat) = n"
by (induct n) simp_all
end