# Theory WF

```(*  Title:      ZF/WF.thy
Author:     Tobias Nipkow and Lawrence C Paulson
Copyright   1994  University of Cambridge

Derived first for transitive relations, and finally for arbitrary WF relations
via wf_trancl and trans_trancl.

It is difficult to derive this general case directly, using r^+ instead of
r.  In is_recfun, the two occurrences of the relation must have the same
form.  Inserting r^+ in the_recfun or wftrec yields a recursion rule with
r^+ -`` {a} instead of r-``{a}.  This recursion rule is stronger in
principle, but harder to use, especially to prove wfrec_eclose_eq in
epsilon.ML.  Expanding out the definition of wftrec in wfrec would yield
a mess.
*)

section‹Well-Founded Recursion›

theory WF imports Trancl begin

definition
wf           :: "i⇒o"  where
(*r is a well-founded relation*)
"wf(r) ≡ ∀Z. Z=0 | (∃x∈Z. ∀y. ⟨y,x⟩:r ⟶ ¬ y ∈ Z)"

definition
wf_on        :: "[i,i]⇒o"                      (‹wf[_]'(_')›)  where
(*r is well-founded on A*)
"wf_on(A,r) ≡ wf(r ∩ A*A)"

definition
is_recfun    :: "[i, i, [i,i]⇒i, i] ⇒o"  where
"is_recfun(r,a,H,f) ≡ (f = (λx∈r-``{a}. H(x, restrict(f, r-``{x}))))"

definition
the_recfun   :: "[i, i, [i,i]⇒i] ⇒i"  where
"the_recfun(r,a,H) ≡ (THE f. is_recfun(r,a,H,f))"

definition
wftrec :: "[i, i, [i,i]⇒i] ⇒i"  where
"wftrec(r,a,H) ≡ H(a, the_recfun(r,a,H))"

definition
wfrec :: "[i, i, [i,i]⇒i] ⇒i"  where
(*public version.  Does not require r to be transitive*)
"wfrec(r,a,H) ≡ wftrec(r^+, a, λx f. H(x, restrict(f,r-``{x})))"

definition
wfrec_on     :: "[i, i, i, [i,i]⇒i] ⇒i"       (‹wfrec[_]'(_,_,_')›)  where
"wfrec[A](r,a,H) ≡ wfrec(r ∩ A*A, a, H)"

subsection‹Well-Founded Relations›

subsubsection‹Equivalences between \<^term>‹wf› and \<^term>‹wf_on››

lemma wf_imp_wf_on: "wf(r) ⟹ wf[A](r)"
by (unfold wf_def wf_on_def, force)

lemma wf_on_imp_wf: "⟦wf[A](r); r ⊆ A*A⟧ ⟹ wf(r)"
by (simp add: wf_on_def subset_Int_iff)

lemma wf_on_field_imp_wf: "wf[field(r)](r) ⟹ wf(r)"
by (unfold wf_def wf_on_def, fast)

lemma wf_iff_wf_on_field: "wf(r) ⟷ wf[field(r)](r)"
by (blast intro: wf_imp_wf_on wf_on_field_imp_wf)

lemma wf_on_subset_A: "⟦wf[A](r);  B<=A⟧ ⟹ wf[B](r)"
by (unfold wf_on_def wf_def, fast)

lemma wf_on_subset_r: "⟦wf[A](r); s<=r⟧ ⟹ wf[A](s)"
by (unfold wf_on_def wf_def, fast)

lemma wf_subset: "⟦wf(s); r<=s⟧ ⟹ wf(r)"
by (simp add: wf_def, fast)

subsubsection‹Introduction Rules for \<^term>‹wf_on››

text‹If every non-empty subset of \<^term>‹A› has an \<^term>‹r›-minimal element
then we have \<^term>‹wf[A](r)›.›
lemma wf_onI:
assumes prem: "⋀Z u. ⟦Z<=A;  u ∈ Z;  ∀x∈Z. ∃y∈Z. ⟨y,x⟩:r⟧ ⟹ False"
shows         "wf[A](r)"
unfolding wf_on_def wf_def
apply (rule equals0I [THEN disjCI, THEN allI])
apply (rule_tac Z = Z in prem, blast+)
done

text‹If \<^term>‹r› allows well-founded induction over \<^term>‹A›
then we have \<^term>‹wf[A](r)›.   Premise is equivalent to
\<^prop>‹⋀B. ∀x∈A. (∀y. ⟨y,x⟩: r ⟶ y ∈ B) ⟶ x ∈ B ⟹ A<=B››
lemma wf_onI2:
assumes prem: "⋀y B. ⟦∀x∈A. (∀y∈A. ⟨y,x⟩:r ⟶ y ∈ B) ⟶ x ∈ B;   y ∈ A⟧
⟹ y ∈ B"
shows         "wf[A](r)"
apply (rule wf_onI)
apply (rule_tac c=u in prem [THEN DiffE])
prefer 3 apply blast
apply fast+
done

subsubsection‹Well-founded Induction›

text‹Consider the least \<^term>‹z› in \<^term>‹domain(r)› such that
\<^term>‹P(z)› does not hold...›
lemma wf_induct_raw:
"⟦wf(r);
⋀x.⟦∀y. ⟨y,x⟩: r ⟶ P(y)⟧ ⟹ P(x)⟧
⟹ P(a)"
unfolding wf_def
apply (erule_tac x = "{z ∈ domain(r). ¬ P(z)}" in allE)
apply blast
done

lemmas wf_induct = wf_induct_raw [rule_format, consumes 1, case_names step, induct set: wf]

text‹The form of this rule is designed to match ‹wfI››
lemma wf_induct2:
"⟦wf(r);  a ∈ A;  field(r)<=A;
⋀x.⟦x ∈ A;  ∀y. ⟨y,x⟩: r ⟶ P(y)⟧ ⟹ P(x)⟧
⟹  P(a)"
apply (erule_tac P="a ∈ A" in rev_mp)
apply (erule_tac a=a in wf_induct, blast)
done

lemma field_Int_square: "field(r ∩ A*A) ⊆ A"
by blast

lemma wf_on_induct_raw [consumes 2, induct set: wf_on]:
"⟦wf[A](r);  a ∈ A;
⋀x.⟦x ∈ A;  ∀y∈A. ⟨y,x⟩: r ⟶ P(y)⟧ ⟹ P(x)
⟧  ⟹  P(a)"
unfolding wf_on_def
apply (erule wf_induct2, assumption)
apply (rule field_Int_square, blast)
done

lemma wf_on_induct [consumes 2, case_names step, induct set: wf_on]:
"wf[A](r) ⟹ a ∈ A ⟹ (⋀x. x ∈ A ⟹ (⋀y. y ∈ A ⟹ ⟨y, x⟩ ∈ r ⟹ P(y)) ⟹ P(x)) ⟹ P(a)"
using wf_on_induct_raw [of A r a P] by simp

text‹If \<^term>‹r› allows well-founded induction
then we have \<^term>‹wf(r)›.›
lemma wfI:
"⟦field(r)<=A;
⋀y B. ⟦∀x∈A. (∀y∈A. ⟨y,x⟩:r ⟶ y ∈ B) ⟶ x ∈ B;  y ∈ A⟧
⟹ y ∈ B⟧
⟹  wf(r)"
apply (rule wf_on_subset_A [THEN wf_on_field_imp_wf])
apply (rule wf_onI2)
prefer 2 apply blast
apply blast
done

subsection‹Basic Properties of Well-Founded Relations›

lemma wf_not_refl: "wf(r) ⟹ ⟨a,a⟩ ∉ r"
by (erule_tac a=a in wf_induct, blast)

lemma wf_not_sym [rule_format]: "wf(r) ⟹ ∀x. ⟨a,x⟩:r ⟶ ⟨x,a⟩ ∉ r"
by (erule_tac a=a in wf_induct, blast)

(* @{term"⟦wf(r);  ⟨a,x⟩ ∈ r;  ¬P ⟹ ⟨x,a⟩ ∈ r⟧ ⟹ P"} *)
lemmas wf_asym = wf_not_sym [THEN swap]

lemma wf_on_not_refl: "⟦wf[A](r); a ∈ A⟧ ⟹ ⟨a,a⟩ ∉ r"
by (erule_tac a=a in wf_on_induct, assumption, blast)

lemma wf_on_not_sym:
"⟦wf[A](r);  a ∈ A⟧ ⟹ (⋀b. b∈A ⟹ ⟨a,b⟩:r ⟹ ⟨b,a⟩∉r)"
apply (atomize (full), intro impI)
apply (erule_tac a=a in wf_on_induct, assumption, blast)
done

lemma wf_on_asym:
"⟦wf[A](r);  ¬Z ⟹ ⟨a,b⟩ ∈ r;
⟨b,a⟩ ∉ r ⟹ Z; ¬Z ⟹ a ∈ A; ¬Z ⟹ b ∈ A⟧ ⟹ Z"
by (blast dest: wf_on_not_sym)

(*Needed to prove well_ordI.  Could also reason that wf[A](r) means
wf(r ∩ A*A);  thus wf( (r ∩ A*A)^+ ) and use wf_not_refl *)
lemma wf_on_chain3:
"⟦wf[A](r); ⟨a,b⟩:r; ⟨b,c⟩:r; ⟨c,a⟩:r; a ∈ A; b ∈ A; c ∈ A⟧ ⟹ P"
apply (subgoal_tac "∀y∈A. ∀z∈A. ⟨a,y⟩:r ⟶ ⟨y,z⟩:r ⟶ ⟨z,a⟩:r ⟶ P",
blast)
apply (erule_tac a=a in wf_on_induct, assumption, blast)
done

text‹transitive closure of a WF relation is WF provided
\<^term>‹A› is downward closed›
lemma wf_on_trancl:
"⟦wf[A](r);  r-``A ⊆ A⟧ ⟹ wf[A](r^+)"
apply (rule wf_onI2)
apply (frule bspec [THEN mp], assumption+)
apply (erule_tac a = y in wf_on_induct, assumption)
apply (blast elim: tranclE, blast)
done

lemma wf_trancl: "wf(r) ⟹ wf(r^+)"
apply (simp add: wf_iff_wf_on_field)
apply (rule wf_on_subset_A)
apply (erule wf_on_trancl)
apply blast
apply (rule trancl_type [THEN field_rel_subset])
done

text‹\<^term>‹r-``{a}› is the set of everything under \<^term>‹a› in \<^term>‹r››

lemmas underI = vimage_singleton_iff [THEN iffD2]
lemmas underD = vimage_singleton_iff [THEN iffD1]

subsection‹The Predicate \<^term>‹is_recfun››

lemma is_recfun_type: "is_recfun(r,a,H,f) ⟹ f ∈ r-``{a} -> range(f)"
unfolding is_recfun_def
apply (erule ssubst)
apply (rule lamI [THEN rangeI, THEN lam_type], assumption)
done

lemmas is_recfun_imp_function = is_recfun_type [THEN fun_is_function]

lemma apply_recfun:
"⟦is_recfun(r,a,H,f); ⟨x,a⟩:r⟧ ⟹ f`x = H(x, restrict(f,r-``{x}))"
unfolding is_recfun_def
txt‹replace f only on the left-hand side›
apply (erule_tac P = "λx. t(x) = u" for t u in ssubst)
apply (simp add: underI)
done

lemma is_recfun_equal [rule_format]:
"⟦wf(r);  trans(r);  is_recfun(r,a,H,f);  is_recfun(r,b,H,g)⟧
⟹ ⟨x,a⟩:r ⟶ ⟨x,b⟩:r ⟶ f`x=g`x"
apply (frule_tac f = f in is_recfun_type)
apply (frule_tac f = g in is_recfun_type)
apply (simp add: is_recfun_def)
apply (erule_tac a=x in wf_induct)
apply (intro impI)
apply (elim ssubst)
apply (simp (no_asm_simp) add: vimage_singleton_iff restrict_def)
apply (rule_tac t = "λz. H (x, z)" for x in subst_context)
apply (subgoal_tac "∀y∈r-``{x}. ∀z. ⟨y,z⟩:f ⟷ ⟨y,z⟩:g")
apply (blast dest: transD)
apply (simp add: apply_iff)
apply (blast dest: transD intro: sym)
done

lemma is_recfun_cut:
"⟦wf(r);  trans(r);
is_recfun(r,a,H,f);  is_recfun(r,b,H,g);  ⟨b,a⟩:r⟧
⟹ restrict(f, r-``{b}) = g"
apply (frule_tac f = f in is_recfun_type)
apply (rule fun_extension)
apply (blast dest: transD intro: restrict_type2)
apply (erule is_recfun_type, simp)
apply (blast dest: transD intro: is_recfun_equal)
done

subsection‹Recursion: Main Existence Lemma›

lemma is_recfun_functional:
"⟦wf(r); trans(r); is_recfun(r,a,H,f); is_recfun(r,a,H,g)⟧  ⟹  f=g"
by (blast intro: fun_extension is_recfun_type is_recfun_equal)

lemma the_recfun_eq:
"⟦is_recfun(r,a,H,f);  wf(r);  trans(r)⟧ ⟹ the_recfun(r,a,H) = f"
unfolding the_recfun_def
apply (blast intro: is_recfun_functional)
done

(*If some f satisfies is_recfun(r,a,H,-) then so does the_recfun(r,a,H) *)
lemma is_the_recfun:
"⟦is_recfun(r,a,H,f);  wf(r);  trans(r)⟧
⟹ is_recfun(r, a, H, the_recfun(r,a,H))"
by (simp add: the_recfun_eq)

lemma unfold_the_recfun:
"⟦wf(r);  trans(r)⟧ ⟹ is_recfun(r, a, H, the_recfun(r,a,H))"
apply (rule_tac a=a in wf_induct, assumption)
apply (rename_tac a1)
apply (rule_tac f = "λy∈r-``{a1}. wftrec (r,y,H)" in is_the_recfun)
apply typecheck
unfolding is_recfun_def wftrec_def
― ‹Applying the substitution: must keep the quantified assumption!›
apply (rule lam_cong [OF refl])
apply (drule underD)
apply (fold is_recfun_def)
apply (rule_tac t = "λz. H(x, z)" for x in subst_context)
apply (rule fun_extension)
apply (blast intro: is_recfun_type)
apply (rule lam_type [THEN restrict_type2])
apply blast
apply (blast dest: transD)
apply atomize
apply (frule spec [THEN mp], assumption)
apply (subgoal_tac "⟨xa,a1⟩ ∈ r")
apply (drule_tac x1 = xa in spec [THEN mp], assumption)
apply (simp add: vimage_singleton_iff
apply_recfun is_recfun_cut)
apply (blast dest: transD)
done

subsection‹Unfolding \<^term>‹wftrec(r,a,H)››

lemma the_recfun_cut:
"⟦wf(r);  trans(r);  ⟨b,a⟩:r⟧
⟹ restrict(the_recfun(r,a,H), r-``{b}) = the_recfun(r,b,H)"
by (blast intro: is_recfun_cut unfold_the_recfun)

(*NOT SUITABLE FOR REWRITING: it is recursive!*)
lemma wftrec:
"⟦wf(r);  trans(r)⟧ ⟹
wftrec(r,a,H) = H(a, λx∈r-``{a}. wftrec(r,x,H))"
unfolding wftrec_def
apply (subst unfold_the_recfun [unfolded is_recfun_def])
apply (simp_all add: vimage_singleton_iff [THEN iff_sym] the_recfun_cut)
done

subsubsection‹Removal of the Premise \<^term>‹trans(r)››

(*NOT SUITABLE FOR REWRITING: it is recursive!*)
lemma wfrec:
"wf(r) ⟹ wfrec(r,a,H) = H(a, λx∈r-``{a}. wfrec(r,x,H))"
unfolding wfrec_def
apply (erule wf_trancl [THEN wftrec, THEN ssubst])
apply (rule trans_trancl)
apply (rule vimage_pair_mono [THEN restrict_lam_eq, THEN subst_context])
apply (erule r_into_trancl)
apply (rule subset_refl)
done

(*This form avoids giant explosions in proofs.  NOTE USE OF ≡ *)
lemma def_wfrec:
"⟦⋀x. h(x)≡wfrec(r,x,H);  wf(r)⟧ ⟹
h(a) = H(a, λx∈r-``{a}. h(x))"
apply simp
apply (elim wfrec)
done

lemma wfrec_type:
"⟦wf(r);  a ∈ A;  field(r)<=A;
⋀x u. ⟦x ∈ A;  u ∈ Pi(r-``{x}, B)⟧ ⟹ H(x,u) ∈ B(x)
⟧ ⟹ wfrec(r,a,H) ∈ B(a)"
apply (rule_tac a = a in wf_induct2, assumption+)
apply (subst wfrec, assumption)
apply (simp add: lam_type underD)
done

lemma wfrec_on:
"⟦wf[A](r);  a ∈ A⟧ ⟹
wfrec[A](r,a,H) = H(a, λx∈(r-``{a}) ∩ A. wfrec[A](r,x,H))"
unfolding wf_on_def wfrec_on_def
apply (erule wfrec [THEN trans])
apply (simp add: vimage_Int_square)
done

text‹Minimal-element characterization of well-foundedness›
lemma wf_eq_minimal: "wf(r) ⟷ (∀Q x. x ∈ Q ⟶ (∃z∈Q. ∀y. ⟨y,z⟩:r ⟶ y∉Q))"
unfolding wf_def by blast

end
```