(* Title: HOL/Wellfounded.thy
Author: Tobias Nipkow
Author: Lawrence C Paulson
Author: Konrad Slind
Author: Alexander Krauss
*)
header {*Well-founded Recursion*}
theory Wellfounded
imports Transitive_Closure
uses ("Tools/Function/size.ML")
begin
subsection {* Basic Definitions *}
definition wf :: "('a * 'a) set => bool" where
"wf(r) == (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"
definition wfP :: "('a => 'a => bool) => bool" where
"wfP r == wf {(x, y). r 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
lemma wf_asym:
assumes "wf r" "(a, x) \<in> r"
obtains "(x, a) \<notin> r"
by (drule wf_not_sym[OF assms])
lemma wf_not_refl [simp]: "wf r ==> (a, a) ~: r"
by (blast elim: wf_asym)
lemma wf_irrefl: assumes "wf r" obtains "(a, a) \<notin> r"
by (drule wf_not_refl[OF assms])
lemma wf_wellorderI:
assumes wf: "wf {(x::'a::ord, y). x < y}"
assumes lin: "OFCLASS('a::ord, linorder_class)"
shows "OFCLASS('a::ord, wellorder_class)"
using lin by (rule wellorder_class.intro)
(blast intro: class.wellorder_axioms.intro wf_induct_rule [OF wf])
lemma (in wellorder) wf:
"wf {(x, y). x < y}"
unfolding wf_def by (blast intro: less_induct)
subsection {* Basic Results *}
text {* Point-free characterization of well-foundedness *}
lemma wfE_pf:
assumes wf: "wf R"
assumes a: "A \<subseteq> R `` A"
shows "A = {}"
proof -
{ fix x
from wf have "x \<notin> A"
proof induct
fix x assume "\<And>y. (y, x) \<in> R \<Longrightarrow> y \<notin> A"
then have "x \<notin> R `` A" by blast
with a show "x \<notin> A" by blast
qed
} thus ?thesis by auto
qed
lemma wfI_pf:
assumes a: "\<And>A. A \<subseteq> R `` A \<Longrightarrow> A = {}"
shows "wf R"
proof (rule wfUNIVI)
fix P :: "'a \<Rightarrow> bool" and x
let ?A = "{x. \<not> P x}"
assume "\<forall>x. (\<forall>y. (y, x) \<in> R \<longrightarrow> P y) \<longrightarrow> P x"
then have "?A \<subseteq> R `` ?A" by blast
with a show "P x" by blast
qed
text{*Minimal-element characterization of well-foundedness*}
lemma wfE_min:
assumes wf: "wf R" and Q: "x \<in> Q"
obtains z where "z \<in> Q" "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
using Q wfE_pf[OF wf, of Q] by blast
lemma wfI_min:
assumes a: "\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q"
shows "wf R"
proof (rule wfI_pf)
fix A assume b: "A \<subseteq> R `` A"
{ fix x assume "x \<in> A"
from a[OF this] b have "False" by blast
}
thus "A = {}" by blast
qed
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))"
apply auto
apply (erule wfE_min, assumption, blast)
apply (rule wfI_min, auto)
done
lemmas wfP_eq_minimal = wf_eq_minimal [to_pred]
text{* Well-foundedness of transitive closure *}
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 {* 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)
lemma wfP_empty [iff]:
"wfP (\<lambda>x y. False)"
proof -
have "wfP bot" by (fact wf_empty [to_pred bot_empty_eq2])
then show ?thesis by (simp add: bot_fun_eq bot_bool_eq)
qed
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 {* Exponentiation *}
lemma wf_exp:
assumes "wf (R ^^ n)"
shows "wf R"
proof (rule wfI_pf)
fix A assume "A \<subseteq> R `` A"
then have "A \<subseteq> (R ^^ n) `` A" by (induct n) force+
with `wf (R ^^ n)`
show "A = {}" by (rule wfE_pf)
qed
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 "R O S \<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> R O S" by (rule rel_compI)
with `R O S \<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
lemma wfP_SUP:
"\<forall>i. wfP (r i) \<Longrightarrow> \<forall>i j. r i \<noteq> r j \<longrightarrow> inf (DomainP (r i)) (RangeP (r j)) = bot \<Longrightarrow> wfP (SUPR UNIV r)"
by (rule wf_UN [where I=UNIV and r="\<lambda>i. {(x, y). r i x y}", to_pred SUP_UN_eq2])
(simp_all add: Collect_def)
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> S O R \<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> S O R \<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> S O R" 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])
subsection {* Acyclic relations *}
definition acyclic :: "('a * 'a) set => bool" where
"acyclic r == !x. (x,x) ~: r^+"
abbreviation acyclicP :: "('a => 'a => bool) => bool" where
"acyclicP r == acyclic {(x, y). r x y}"
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 {* @{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)
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])
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
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 \<times>'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> (('a \<times> 'b) \<times> ('a \<times> 'b)) set" (infixr "<*lex*>" 80) where
"ra <*lex*> rb = {((a, b), (a', b')). (a, a') \<in> ra \<or> a = a' \<and> (b, b') \<in> 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[simp]: "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)
lemma in_finite_psubset[simp]: "(A, B) \<in> finite_psubset = (A < B & finite B)"
unfolding finite_psubset_def by auto
text {* max- and min-extension of order to finite sets *}
inductive_set max_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set"
for R :: "('a \<times> 'a) set"
where
max_extI[intro]: "finite X \<Longrightarrow> finite Y \<Longrightarrow> Y \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>Y. (x, y) \<in> R) \<Longrightarrow> (X, Y) \<in> max_ext R"
lemma max_ext_wf:
assumes wf: "wf r"
shows "wf (max_ext r)"
proof (rule acc_wfI, intro allI)
fix M show "M \<in> acc (max_ext r)" (is "_ \<in> ?W")
proof cases
assume "finite M"
thus ?thesis
proof (induct M)
show "{} \<in> ?W"
by (rule accI) (auto elim: max_ext.cases)
next
fix M a assume "M \<in> ?W" "finite M"
with wf show "insert a M \<in> ?W"
proof (induct arbitrary: M)
fix M a
assume "M \<in> ?W" and [intro]: "finite M"
assume hyp: "\<And>b M. (b, a) \<in> r \<Longrightarrow> M \<in> ?W \<Longrightarrow> finite M \<Longrightarrow> insert b M \<in> ?W"
{
fix N M :: "'a set"
assume "finite N" "finite M"
then
have "\<lbrakk>M \<in> ?W ; (\<And>y. y \<in> N \<Longrightarrow> (y, a) \<in> r)\<rbrakk> \<Longrightarrow> N \<union> M \<in> ?W"
by (induct N arbitrary: M) (auto simp: hyp)
}
note add_less = this
show "insert a M \<in> ?W"
proof (rule accI)
fix N assume Nless: "(N, insert a M) \<in> max_ext r"
hence asm1: "\<And>x. x \<in> N \<Longrightarrow> (x, a) \<in> r \<or> (\<exists>y \<in> M. (x, y) \<in> r)"
by (auto elim!: max_ext.cases)
let ?N1 = "{ n \<in> N. (n, a) \<in> r }"
let ?N2 = "{ n \<in> N. (n, a) \<notin> r }"
have N: "?N1 \<union> ?N2 = N" by (rule set_ext) auto
from Nless have "finite N" by (auto elim: max_ext.cases)
then have finites: "finite ?N1" "finite ?N2" by auto
have "?N2 \<in> ?W"
proof cases
assume [simp]: "M = {}"
have Mw: "{} \<in> ?W" by (rule accI) (auto elim: max_ext.cases)
from asm1 have "?N2 = {}" by auto
with Mw show "?N2 \<in> ?W" by (simp only:)
next
assume "M \<noteq> {}"
have N2: "(?N2, M) \<in> max_ext r"
by (rule max_extI[OF _ _ `M \<noteq> {}`]) (insert asm1, auto intro: finites)
with `M \<in> ?W` show "?N2 \<in> ?W" by (rule acc_downward)
qed
with finites have "?N1 \<union> ?N2 \<in> ?W"
by (rule add_less) simp
then show "N \<in> ?W" by (simp only: N)
qed
qed
qed
next
assume [simp]: "\<not> finite M"
show ?thesis
by (rule accI) (auto elim: max_ext.cases)
qed
qed
lemma max_ext_additive:
"(A, B) \<in> max_ext R \<Longrightarrow> (C, D) \<in> max_ext R \<Longrightarrow>
(A \<union> C, B \<union> D) \<in> max_ext R"
by (force elim!: max_ext.cases)
definition min_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set" where
"min_ext r = {(X, Y) | X Y. X \<noteq> {} \<and> (\<forall>y \<in> Y. (\<exists>x \<in> X. (x, y) \<in> r))}"
lemma min_ext_wf:
assumes "wf r"
shows "wf (min_ext r)"
proof (rule wfI_min)
fix Q :: "'a set set"
fix x
assume nonempty: "x \<in> Q"
show "\<exists>m \<in> Q. (\<forall> n. (n, m) \<in> min_ext r \<longrightarrow> n \<notin> Q)"
proof cases
assume "Q = {{}}" thus ?thesis by (simp add: min_ext_def)
next
assume "Q \<noteq> {{}}"
with nonempty
obtain e x where "x \<in> Q" "e \<in> x" by force
then have eU: "e \<in> \<Union>Q" by auto
with `wf r`
obtain z where z: "z \<in> \<Union>Q" "\<And>y. (y, z) \<in> r \<Longrightarrow> y \<notin> \<Union>Q"
by (erule wfE_min)
from z obtain m where "m \<in> Q" "z \<in> m" by auto
from `m \<in> Q`
show ?thesis
proof (rule, intro bexI allI impI)
fix n
assume smaller: "(n, m) \<in> min_ext r"
with `z \<in> m` obtain y where y: "y \<in> n" "(y, z) \<in> r" by (auto simp: min_ext_def)
then show "n \<notin> Q" using z(2) by auto
qed
qed
qed
subsection{*Weakly decreasing sequences (w.r.t. some well-founded order)
stabilize.*}
text{*This material does not appear to be used any longer.*}
lemma sequence_trans: "[| ALL i. (f (Suc i), f i) : r^* |] ==> (f (i+k), f i) : r^*"
by (induct k) (auto intro: rtrancl_trans)
lemma wf_weak_decr_stable:
assumes as: "ALL i. (f (Suc i), f i) : r^*" "wf (r^+)"
shows "EX i. ALL k. f (i+k) = f i"
proof -
have lem: "!!x. [| 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 sequence_trans)
apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
done
from lem[OF as, THEN spec, of 0, simplified]
show ?thesis by auto
qed
(* 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/size.ML"
setup Size.setup
lemma size_bool [code]:
"size (b\<Colon>bool) = 0" by (cases b) auto
lemma nat_size [simp, code]: "size (n\<Colon>nat) = n"
by (induct n) simp_all
declare "prod.size" [no_atp]
lemma [code]:
"size (P :: 'a Predicate.pred) = 0" by (cases P) simp
lemma [code]:
"pred_size f P = 0" by (cases P) simp
end