token translations: context dependent, result Pretty.T;
added Markup.fixed (analogous to Markup.const);
tuned;
(* ID: $Id$
Author: Tobias Nipkow
Copyright 1992 University of Cambridge
*)
header {*Well-founded Recursion*}
theory Wellfounded_Recursion
imports Transitive_Closure Nat
uses ("Tools/function_package/size.ML")
begin
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]
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
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"
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
subsubsection {* 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, 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
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
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 {* Complete induction, aka course-of-values induction *}
lemma nat_less_induct:
assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
apply (induct n rule: wf_induct [OF wf_pred_nat [THEN wf_trancl]])
apply (rule assms)
apply (unfold less_eq [symmetric], assumption)
done
lemmas less_induct = nat_less_induct [rule_format, case_names less]
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)+
lemma nat_induct2: "[|P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))|] ==> P n"
apply (rule nat_less_induct)
apply (case_tac n)
apply (case_tac [2] nat)
apply (blast intro: less_trans)+
done
text {* The method of infinite descent, frequently used in number theory.
Provided by Roelof Oosterhuis.
$P(n)$ is true for all $n\in\mathbb{N}$ if
\begin{itemize}
\item case ``0'': given $n=0$ prove $P(n)$,
\item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
a smaller integer $m$ such that $\neg P(m)$.
\end{itemize} *}
lemma infinite_descent0[case_names 0 smaller]:
"\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n"
by (induct n rule: less_induct, case_tac "n>0", auto)
text{* A compact version without explicit base case: *}
lemma infinite_descent:
"\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow> \<exists>m<n. \<not> P m \<rbrakk> \<Longrightarrow> P n"
by (induct n rule: less_induct, auto)
text {*
Infinite descent using a mapping to $\mathbb{N}$:
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
\begin{itemize}
\item case ``0'': given $V(x)=0$ prove $P(x)$,
\item case ``smaller'': given $V(x)>0$ and $\neg P(x)$ prove there exists a $y \in D$ such that $V(y)<V(x)$ and $~\neg P(y)$.
\end{itemize}
NB: the proof also shows how to use the previous lemma. *}
corollary infinite_descent0_measure [case_names 0 smaller]:
assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
and A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
shows "P x"
proof -
obtain n where "n = V x" by auto
moreover have "\<And>x. V x = n \<Longrightarrow> P x"
proof (induct n rule: infinite_descent0)
case 0 -- "i.e. $V(x) = 0$"
with A0 show "P x" by auto
next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
case (smaller n)
then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
then show ?case by auto
qed
ultimately show "P x" by auto
qed
text{* Again, without explicit base case: *}
lemma infinite_descent_measure:
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
proof -
from assms obtain n where "n = V x" by auto
moreover have "!!x. V x = n \<Longrightarrow> P x"
proof (induct n rule: infinite_descent, auto)
fix x assume "\<not> P x"
with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
qed
ultimately show "P x" by auto
qed
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 {* 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