# Theory Recursion_Thms

```section‹Some enhanced theorems on recursion›

theory Recursion_Thms
imports "Eclose_Absolute"

begin

hide_const (open) Order.pred

― ‹Removing arities from inherited simpset›
declare arity_And [simp del] arity_Or[simp del] arity_Implies[simp del]
arity_Exists[simp del] arity_Iff[simp del]
arity_subset_fm [simp del] arity_ordinal_fm[simp del] arity_transset_fm[simp del]

text‹We prove results concerning definitions by well-founded
recursion on some relation \<^term>‹R› and its transitive closure
\<^term>‹R^*››

lemma fld_restrict_eq : "a ∈ A ⟹ (r ∩ A×A)-``{a} = (r-``{a} ∩ A)"
by(force)

lemma fld_restrict_mono : "relation(r) ⟹ A ⊆ B ⟹ r ∩ A×A ⊆ r ∩ B×B"
by(auto)

lemma fld_restrict_dom :
assumes "relation(r)" "domain(r) ⊆ A" "range(r)⊆ A"
shows "r∩ A×A = r"
proof (rule equalityI,blast,rule subsetI)
{ fix x
assume xr: "x ∈ r"
from xr assms have "∃ a b . x = ⟨a,b⟩" by (simp add: relation_def)
then obtain a b where "⟨a,b⟩ ∈ r" "⟨a,b⟩ ∈ r∩A×A" "x ∈ r∩A×A"
using assms xr
by force
then have "x∈ r ∩ A×A" by simp
}
then show "x ∈ r ⟹ x∈ r∩A×A" for x .
qed

definition tr_down :: "[i,i] ⇒ i"
where "tr_down(r,a) = (r^+)-``{a}"

lemma tr_downD : "x ∈ tr_down(r,a) ⟹ ⟨x,a⟩ ∈ r^+"

lemma pred_down : "relation(r) ⟹ r-``{a} ⊆ tr_down(r,a)"

lemma tr_down_mono : "relation(r) ⟹ x ∈ r-``{a} ⟹ tr_down(r,x) ⊆ tr_down(r,a)"

lemma rest_eq :
assumes "relation(r)" and "r-``{a} ⊆ B" and "a ∈ B"
shows "r-``{a} = (r∩B×B)-``{a}"
proof (intro equalityI subsetI)
fix x
assume "x ∈ r-``{a}"
then
have "x ∈ B" using assms by (simp add: subsetD)
from ‹x∈ r-``{a}›
have "⟨x,a⟩ ∈ r" using underD by simp
then
show "x ∈ (r∩B×B)-``{a}" using ‹x∈B› ‹a∈B› underI by simp
next
from assms
show "x ∈ r -`` {a}" if  "x ∈ (r ∩ B×B) -`` {a}" for x
using vimage_mono that by auto
qed

lemma wfrec_restr_eq : "r' = r ∩ A×A ⟹ wfrec[A](r,a,H) = wfrec(r',a,H)"

lemma wfrec_restr :
assumes rr: "relation(r)" and wfr:"wf(r)"
shows  "a ∈ A ⟹ tr_down(r,a) ⊆ A ⟹ wfrec(r,a,H) = wfrec[A](r,a,H)"
proof (induct a arbitrary:A rule:wf_induct_raw[OF wfr] )
case (1 a)
have wfRa : "wf[A](r)"
using wf_subset wfr wf_on_def Int_lower1 by simp
from pred_down rr
have "r -`` {a} ⊆ tr_down(r, a)" .
with 1
have "r-``{a} ⊆ A" by (force simp add: subset_trans)
{
fix x
assume x_a : "x ∈ r-``{a}"
with ‹r-``{a} ⊆ A›
have "x ∈ A" ..
from pred_down rr
have b : "r -``{x} ⊆ tr_down(r,x)" .
then
have "tr_down(r,x) ⊆ tr_down(r,a)"
using tr_down_mono x_a rr by simp
with 1
have "tr_down(r,x) ⊆ A" using subset_trans by force
have "⟨x,a⟩ ∈ r" using x_a  underD by simp
with 1 ‹tr_down(r,x) ⊆ A› ‹x ∈ A›
have "wfrec(r,x,H) = wfrec[A](r,x,H)" by simp
}
then
have "x∈ r-``{a} ⟹ wfrec(r,x,H) =  wfrec[A](r,x,H)" for x  .
then
have Eq1 :"(λ x ∈ r-``{a} . wfrec(r,x,H)) = (λ x ∈ r-``{a} . wfrec[A](r,x,H))"
using lam_cong by simp

from assms
have "wfrec(r,a,H) = H(a,λ x ∈ r-``{a} . wfrec(r,x,H))" by (simp add:wfrec)
also
have "... = H(a,λ x ∈ r-``{a} . wfrec[A](r,x,H))"
using assms Eq1 by simp
also from 1 ‹r-``{a} ⊆ A›
have "... = H(a,λ x ∈ (r∩A×A)-``{a} . wfrec[A](r,x,H))"
using assms rest_eq  by simp
also from ‹a∈A›
have "... = H(a,λ x ∈ (r-``{a})∩A . wfrec[A](r,x,H))"
using fld_restrict_eq by simp
also from ‹a∈A› ‹wf[A](r)›
have "... = wfrec[A](r,a,H)" using wfrec_on by simp
finally show ?case .
qed

lemmas wfrec_tr_down = wfrec_restr[OF _ _ _ subset_refl]

lemma wfrec_trans_restr : "relation(r) ⟹ wf(r) ⟹ trans(r) ⟹ r-``{a}⊆A ⟹ a ∈ A ⟹
wfrec(r, a, H) = wfrec[A](r, a, H)"
by(subgoal_tac "tr_down(r,a) ⊆ A",auto simp add : wfrec_restr tr_down_def trancl_eq_r)

lemma field_trancl : "field(r^+) = field(r)"
by (blast intro: r_into_trancl dest!: trancl_type [THEN subsetD])

definition
Rrel :: "[i⇒i⇒o,i] ⇒ i" where
"Rrel(R,A) ≡ {z∈A×A. ∃x y. z = ⟨x, y⟩ ∧ R(x,y)}"

lemma RrelI : "x ∈ A ⟹ y ∈ A ⟹ R(x,y) ⟹ ⟨x,y⟩ ∈ Rrel(R,A)"
unfolding Rrel_def by simp

lemma Rrel_mem: "Rrel(mem,x) = Memrel(x)"
unfolding Rrel_def Memrel_def ..

lemma relation_Rrel: "relation(Rrel(R,d))"
unfolding Rrel_def relation_def by simp

lemma field_Rrel: "field(Rrel(R,d)) ⊆  d"
unfolding Rrel_def by auto

lemma Rrel_mono : "A ⊆ B ⟹ Rrel(R,A) ⊆ Rrel(R,B)"
unfolding Rrel_def by blast

lemma Rrel_restr_eq : "Rrel(R,A) ∩ B×B = Rrel(R,A∩B)"
unfolding Rrel_def by blast

― ‹We obtain this lemmas as a consequence of the previous one;
alternatively it can be obtained using @{thm [source] Ordinal.Memrel_type}›
lemma field_Memrel : "field(Memrel(A)) ⊆ A"
using Rrel_mem field_Rrel by blast

lemma restrict_trancl_Rrel:
assumes "R(w,y)"
shows "restrict(f,Rrel(R,d)-``{y})`w
= restrict(f,(Rrel(R,d)^+)-``{y})`w"
proof (cases "y∈d")
let ?r="Rrel(R,d)" and ?s="(Rrel(R,d))^+"
case True
show ?thesis
proof (cases "w∈d")
case True
with ‹y∈d› assms
have "⟨w,y⟩∈?r"
unfolding Rrel_def by blast
then
have "⟨w,y⟩∈?s"
using r_subset_trancl[of ?r] relation_Rrel[of R d] by blast
with ‹⟨w,y⟩∈?r›
have "w∈?r-``{y}" "w∈?s-``{y}"
using vimage_singleton_iff by simp_all
then
show ?thesis by simp
next
case False
then
have "w∉domain(restrict(f,?r-``{y}))"
using subsetD[OF field_Rrel[of R d]] by auto
moreover from ‹w∉d›
have "w∉domain(restrict(f,?s-``{y}))"
using subsetD[OF field_Rrel[of R d], of w] field_trancl[of ?r]
fieldI1[of w y ?s] by auto
ultimately
have "restrict(f,?r-``{y})`w = 0" "restrict(f,?s-``{y})`w = 0"
unfolding apply_def by auto
then show ?thesis by simp
qed
next
let ?r="Rrel(R,d)"
let ?s="?r^+"
case False
then
have "?r-``{y}=0"
unfolding Rrel_def by blast
then
have "w∉?r-``{y}" by simp
with ‹y∉d› assms
have "y∉field(?s)"
using field_trancl subsetD[OF field_Rrel[of R d]] by force
then
have "w∉?s-``{y}"
using vimage_singleton_iff by blast
with ‹w∉?r-``{y}›
show ?thesis by simp
qed

lemma restrict_trans_eq:
assumes "w ∈ y"
shows "restrict(f,Memrel(eclose({x}))-``{y})`w
= restrict(f,(Memrel(eclose({x}))^+)-``{y})`w"
using assms restrict_trancl_Rrel[of mem ] Rrel_mem by (simp)

lemma wf_eq_trancl:
assumes "⋀ f y . H(y,restrict(f,R-``{y})) = H(y,restrict(f,R^+-``{y}))"
shows  "wfrec(R, x, H) = wfrec(R^+, x, H)" (is "wfrec(?r,_,_) = wfrec(?r',_,_)")
proof -
have "wfrec(R, x, H) = wftrec(?r^+, x, λy f. H(y, restrict(f,?r-``{y})))"
unfolding wfrec_def ..
also
have " ... = wftrec(?r^+, x, λy f. H(y, restrict(f,(?r^+)-``{y})))"
using assms by simp
also
have " ... =  wfrec(?r^+, x, H)"
unfolding wfrec_def using trancl_eq_r[OF relation_trancl trans_trancl] by simp
finally
show ?thesis .
qed

lemma transrec_equal_on_Ord:
assumes
"⋀x f . Ord(x) ⟹ foo(x,f) = bar(x,f)"
"Ord(α)"
shows
"transrec(α, foo) = transrec(α, bar)"
proof -
have "transrec(β,foo) = transrec(β,bar)" if "Ord(β)" for β
using that
proof (induct rule:trans_induct)
case (step β)
have "transrec(β, foo) = foo(β, λx∈β. transrec(x, foo))"
using def_transrec[of "λx. transrec(x, foo)" foo] by blast
also from assms and step
have " … = bar(β, λx∈β. transrec(x, foo))"
by simp
also from step
have " … = bar(β, λx∈β. transrec(x, bar))"
by (auto)
also
have " … = transrec(β, bar)"
using def_transrec[of "λx. transrec(x, bar)" bar, symmetric]
by blast
finally
show "transrec(β, foo) = transrec(β, bar)" .
qed
with assms
show ?thesis by simp
qed

― ‹Next theorem is very similar to @{thm [source] transrec_equal_on_Ord}›
lemma (in M_eclose) transrec_equal_on_M:
assumes
"⋀x f . M(x) ⟹ M(f) ⟹ foo(x,f) = bar(x,f)"
"⋀β. M(β) ⟹ transrec_replacement(M,is_foo,β)" "relation2(M,is_foo,foo)"
"strong_replacement(M, λx y. y = ⟨x, transrec(x, foo)⟩)"
"∀x[M]. ∀g[M]. function(g) ⟶ M(foo(x,g))"
"M(α)" "Ord(α)"
shows
"transrec(α, foo) = transrec(α, bar)"
proof -
have "M(transrec(x, foo))" if "Ord(x)" and "M(x)" for x
using that assms transrec_closed[of is_foo]
by simp
have "transrec(β,foo) = transrec(β,bar)" "M(transrec(β,foo))" if "Ord(β)" "M(β)" for β
using that
proof (induct rule:trans_induct)
case (step β)
moreover
assume "M(β)"
moreover
note ‹Ord(β)⟹ M(β) ⟹ M(transrec(β, foo))›
ultimately
show "M(transrec(β, foo))" by blast
with step ‹M(β)› ‹⋀x. Ord(x)⟹ M(x) ⟹ M(transrec(x, foo))›
‹strong_replacement(M, λx y. y = ⟨x, transrec(x, foo)⟩)›
have "M(λx∈β. transrec(x, foo))"
using Ord_in_Ord transM[of _ β]
by (rule_tac lam_closed) auto
have "transrec(β, foo) = foo(β, λx∈β. transrec(x, foo))"
using def_transrec[of "λx. transrec(x, foo)" foo] by blast
also from assms and ‹M(λx∈β. transrec(x, foo))› ‹M(β)›
have " … = bar(β, λx∈β. transrec(x, foo))"
by simp
also from step and ‹M(β)›
have " … = bar(β, λx∈β. transrec(x, bar))"
using transM[of _ β] by (auto)
also
have " … = transrec(β, bar)"
using def_transrec[of "λx. transrec(x, bar)" bar, symmetric]
by blast
finally
show "transrec(β, foo) = transrec(β, bar)" .
qed
with assms
show ?thesis by simp
qed

lemma ordermap_restr_eq:
assumes "well_ord(X,r)"
shows "ordermap(X, r) = ordermap(X, r ∩ X×X)"
proof -
let ?A="λx . Order.pred(X, x, r)"
let ?B="λx . Order.pred(X, x, r ∩ X × X)"
let ?F="λx f. f `` ?A(x)"
let ?G="λx f. f `` ?B(x)"
let ?P="λ z. z∈X ⟶ wfrec(r ∩ X × X,z,λx f. f `` ?A(x)) = wfrec(r ∩ X × X,z,λx f. f `` ?B(x))"
have pred_eq:
"Order.pred(X, x, r ∩ X × X) = Order.pred(X, x, r)" if "x∈X" for x
unfolding Order.pred_def using that by auto
from assms
have wf_onX:"wf(r ∩ X × X)" unfolding well_ord_def wf_on_def by simp
{
have "?P(z)" for z
proof(induct rule:wf_induct[where P="?P",OF wf_onX])
case (1 x)
{
assume "x∈X"
from 1
have lam_eq:
"(λw∈(r ∩ X × X) -`` {x}. wfrec(r ∩ X × X, w, ?F)) =
(λw∈(r ∩ X × X) -`` {x}. wfrec(r ∩ X × X, w, ?G))" (is "?L=?R")
proof -
have "wfrec(r ∩ X × X, w, ?F) = wfrec(r ∩ X × X, w, ?G)" if "w∈(r∩X×X)-``{x}" for w
using 1 that by auto
then show ?thesis using lam_cong[OF refl] by simp
qed
then
have "wfrec(r ∩ X × X, x, ?F) = ?L `` ?A(x)"
using wfrec[OF wf_onX,of x ?F] by simp
also have "... =  ?R `` ?B(x)"
using lam_eq pred_eq[OF ‹x∈_›] by simp
also
have "... = wfrec(r ∩ X × X, x, ?G)"
using wfrec[OF wf_onX,of x ?G] by simp
finally
have "wfrec(r ∩ X × X, x, ?F) = wfrec(r ∩ X × X, x, ?G)" by simp
}
then
show ?case by simp
qed
}
then
show ?thesis
unfolding ordermap_def wfrec_on_def using Int_ac by simp
qed

end
```