(* 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
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 (\x y. (x, y) \ 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 \ A <*> B;
!!x P. [|\x. (\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) \ r"
obtains "(x, a) \ 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) \ 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 \ R `` A"
shows "A = {}"
proof -
{ fix x
from wf have "x \ A"
proof induct
fix x assume "\y. (y, x) \ R \ y \ A"
then have "x \ R `` A" by blast
with a show "x \ A" by blast
qed
} thus ?thesis by auto
qed
lemma wfI_pf:
assumes a: "\A. A \ R `` A \ A = {}"
shows "wf R"
proof (rule wfUNIVI)
fix P :: "'a \ bool" and x
let ?A = "{x. \ P x}"
assume "\x. (\y. (y, x) \ R \ P y) \ P x"
then have "?A \ 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 \ Q"
obtains z where "z \ Q" "\y. (y, z) \ R \ y \ Q"
using Q wfE_pf[OF wf, of Q] by blast
lemma wfI_min:
assumes a: "\x Q. x \ Q \ \z\Q. \y. (y, z) \ R \ y \ Q"
shows "wf R"
proof (rule wfI_pf)
fix A assume b: "A \ R `` A"
{ fix x assume "x \ A"
from a[OF this] b have "False" by blast
}
thus "A = {}" by blast
qed
lemma wf_eq_minimal: "wf r = (\Q x. x\Q --> (\z\Q. \y. (y,z)\r --> y\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 = `\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 (\x y. False)"
proof -
have "wfP bot" by (fact wf_empty [to_pred bot_empty_eq2])
then show ?thesis by (simp add: bot_fun_def)
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 \ R `` A"
then have "A \ (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_map_pair_image: "[| wf r; inj f |] ==> wf(map_pair 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 \ R"
shows "wf (R \ S)"
proof (rule wfI_min)
fix x :: 'a and Q
let ?Q' = "{x \ Q. \y. (y, x) \ R \ y \ Q}"
assume "x \ Q"
obtain a where "a \ ?Q'"
by (rule wfE_min [OF `wf R` `x \ Q`]) blast
with `wf S`
obtain z where "z \ ?Q'" and zmin: "\y. (y, z) \ S \ y \ ?Q'" by (erule wfE_min)
{
fix y assume "(y, z) \ S"
then have "y \ ?Q'" by (rule zmin)
have "y \ Q"
proof
assume "y \ Q"
with `y \ ?Q'`
obtain w where "(w, y) \ R" and "w \ Q" by auto
from `(w, y) \ R` `(y, z) \ S` have "(w, z) \ R O S" by (rule relcompI)
with `R O S \ R` have "(w, z) \ R" ..
with `z \ ?Q'` have "w \ Q" by blast
with `w \ Q` show False by contradiction
qed
}
with `z \ ?Q'` show "\z\Q. \y. (y, z) \ R \ S \ y \ 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:
"\i. wfP (r i) \ \i j. r i \ r j \ inf (DomainP (r i)) (RangeP (r j)) = bot \ wfP (SUPR UNIV r)"
apply (rule wf_UN[to_pred])
apply simp_all
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)"
using wf_UN[of R "\i. i"] by (simp add: SUP_def)
(*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 \ S) = wf (R O R \ S O R \ S)" (is "wf ?A = wf ?B")
proof
assume "wf ?A"
with wf_trancl have wfT: "wf (?A^+)" .
moreover have "?B \ ?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 \ Q"
with `wf ?B`
obtain z where "z \ Q" and "\y. (y, z) \ ?B \ y \ Q"
by (erule wfE_min)
then have A1: "\y. (y, z) \ R O R \ y \ Q"
and A2: "\y. (y, z) \ S O R \ y \ Q"
and A3: "\y. (y, z) \ S \ y \ Q"
by auto
show "\z\Q. \y. (y, z) \ ?A \ y \ Q"
proof (cases "\y. (y, z) \ R \ y \ Q")
case True
with `z \ Q` A3 show ?thesis by blast
next
case False
then obtain z' where "z'\Q" "(z', z) \ R" by blast
have "\y. (y, z') \ ?A \ y \ Q"
proof (intro allI impI)
fix y assume "(y, z') \ ?A"
then show "y \ Q"
proof
assume "(y, z') \ R"
then have "(y, z) \ R O R" using `(z', z) \ R` ..
with A1 show "y \ Q" .
next
assume "(y, z') \ S"
then have "(y, z) \ S O R" using `(z', z) \ R` ..
with A2 show "y \ Q" .
qed
qed
with `z' \ 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 *}
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]
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 < = (\m n. n = Suc m)^++"
proof (rule ext, rule ext, rule iffI)
fix n m :: nat
assume "m < n"
then show "(\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 "(\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) \ pred_nat^+ \ m < n"
unfolding less_nat_rel pred_nat_def trancl_def by simp
lemma pred_nat_trancl_eq_le:
"(m, n) \ pred_nat^* \ m \ 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 '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\\)"
abbreviation
termi :: "('a * 'a) set => 'a set" where
"termi r \ acc (r\)"
lemmas accpI = accp.accI
text {* Induction rules *}
theorem accp_induct:
assumes major: "accp r a"
assumes hyp: "!!x. accp r x ==> \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: "\ accp R x"
obtains z where "R z x" and "\ accp R z"
proof -
assume a: "\z. \R z x; \ accp R z\ \ thesis"
show thesis
proof (cases "\z. R z x \ accp R z")
case True
hence "\z. R z x \ 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 "\ 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: "\x. accp r x ==> wfP r"
apply (rule wfPUNIVI)
apply (rule_tac P=P in 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 = (\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 \ R2"
shows "accp R2 \ accp R1"
proof (rule predicate1I)
fix x assume "accp R2 x"
then show "accp R1 x"
proof (induct x)
fix x
assume ih: "\y. R2 y x \ 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 \ accp R"
and dcl: "\x z. \D x; R z x\ \ D z"
and "D x"
and istep: "\x. \D x; (\z. R z x \ P z)\ \ 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 "\y. R y x \ D y \ 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]
lemmas acc_subset_induct = accp_subset_induct [to_set]
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, code_unfold]: "((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
lemma wf_if_measure: fixes f :: "'a \ nat"
shows "(!!x. P x \ f(g x) < f x) \ wf {(y,x). P x \ y = g x}"
apply(insert wf_measure[of f])
apply(simp only: measure_def inv_image_def less_than_def less_eq)
apply(erule wf_subset)
apply auto
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 \ (a = a' \ (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 \ nat) \ ('a \ 'a) set \ ('a \ 'a) set" (infixr "<*mlex*>" 80)
where
"f <*mlex*> R = inv_image (less_than <*lex*> R) (%x. (f x, x))"
lemma wf_mlex: "wf R \ wf (f <*mlex*> R)"
unfolding mlex_prod_def
by auto
lemma mlex_less: "f x < f y \ (x, y) \ f <*mlex*> R"
unfolding mlex_prod_def by simp
lemma mlex_leq: "f x \ f y \ (x, y) \ R \ (x, y) \ 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) \ 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 \ 'a) set \ ('a set \ 'a set) set"
for R :: "('a \ 'a) set"
where
max_extI[intro]: "finite X \ finite Y \ Y \ {} \ (\x. x \ X \ \y\Y. (x, y) \ R) \ (X, Y) \ 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 \ acc (max_ext r)" (is "_ \ ?W")
proof cases
assume "finite M"
thus ?thesis
proof (induct M)
show "{} \ ?W"
by (rule accI) (auto elim: max_ext.cases)
next
fix M a assume "M \ ?W" "finite M"
with wf show "insert a M \ ?W"
proof (induct arbitrary: M)
fix M a
assume "M \ ?W" and [intro]: "finite M"
assume hyp: "\b M. (b, a) \ r \ M \