Primitive Inductive Theorems Bridge
Implicit Induction Methods and Inductive Theorems
in HigherOrder Rewriting
KUSAKARI Keiichirou,SAKAI Masahiko,SAKABE Toshiki
Graduate School of Information Science,Nagoya University.
fkusakari,sakai,sakabeg@is.nagoyau.ac.jp
Abstract.Automated reasoning of inductive theorems is considered important in program
veri¯cation.To verify inductive theorems automatically,several implicit induction methods
like the inductionless induction and the rewriting induction methods have been proposed.In
studying inductive theorems on higherorder rewritings,we found that the class of the theorems
shown by known implicit induction methods does not coincide with that of inductive theorems,
and the gap between themis a barrier in developing mechanized methods for disproving inductive
theorems.This paper ¯lls this gap by introducing the notion of primitive inductive theorems,
and clarifying the relation between inductive theorems and primitive inductive theorems.Based
on this relation,we achieve mechanized methods for proving and disproving inductive theorems.
Keyword.Algebraic Speci¯cation,HigherOrder Rewriting,SimplyTyped Term Rewriting
System,Primitive Inductive Theorem,Inductive Theorem,Implicit Induction Method.
1 Introduction
Term rewriting systems (TRSs) provide an opera
tional model of functional programming languages.
TRSs also give a theoretical foundation of alge
braic speci¯cation languages [8,9].In algebraic
speci¯cation,many interesting properties of pro
grams can be formally dealt with as inductive the
orems,characterized by the initial algebra seman
tics [10,12,13,14,24].The concept of inductive
theorems is extremely important in practical appli
cations.In fact,most data structures used in func
tional programming are inductive structures such as
list and tree structures.As a result,most proper
ties that a program must guarantee are formalized
as inductive theorems.
In order to verify programs,automated reason
ing for proving and disproving inductive theorems
is very important.Hence many inductive rea
soning methods in TRSs,called implicit induc
tion methods (the inductionless induction and the
rewriting induction methods),have been studied
[6,7,11,16,21,22,23].
Higherorder functions,which can treat functions
as values,provide a facility of highlevel abstraction
and more expressive power.Unfortunately,TRSs
cannot express higherorder functions directly.For
this reason,the ¯rstauthor proposed simplytyped
term rewriting systems (STRSs) [17].In STRSs,
the Mapfunction,which is one of the most typical
higherorder function,is represented as follows:
½
Map(f;Nil)!Nil
Map(f;x::xs)!f(x)::Map(f;xs)
In this paper,we study inductive theorems on
STRSs for automated reasoning and these charac
terization.We found that the class of the theo
rems shown by known implicit induction methods
does not coincide with that of inductive theorems,
and the gap between them is a barrier in devel
oping mechanized methods for disproving inductive
theorems.We ¯ll this gap by introducing the no
tion of primitive inductive theorems,and clarifying
the relation between inductive theorems and primi
tive inductive theorems.Based on this relation,we
achieve mechanized methods for proving and dis
proving inductive theorems.
The main contributions of the paper are:
(1) We give a notion of primitive inductive the
orem (De¯nition 4.1),and characterize induc
tive theorems by primitive ones (Theorem4.6).
(2) We show that the existing implicit induc
tion methods (the inductionless induction [23]
and the rewriting induction methods [16]) can
be naturally extended to STRSs for prov
ing/disproving primitive inductive theorems
(Theorem 5.2,5.7 and 6.2).
(3) We show that the implicit induction meth
ods are also applicable for proving inductive
1
theorems (Theorem 5.2 and 5.7),because ev
ery primitive inductive theorem is an inductive
theorem (Theorem 4.2).
(4) For disproving inductive theorems,implicit
induction methods do not work well because
some inductive theorems are not a primitive
inductive theorem.To overcome the di±culty,
we present a su±cient condition for inductive
theorems to coincide with primitive inductive
theorems (Theorem 4.17).Under this su±
cient condition,the implicit induction method
is applicable for disproving inductive theorems
(Theorem 6.3).
(5) To give a justi¯cation of our de¯nition of induc
tive theorems,we design a higherorder equa
tional logic,and show that the notion of in
ductive theorems is characterized by the initial
extensional semantics (Theorem 7.10).
The remainder of this paper is organized as fol
lows.The next section gives the preliminaries
needed later on.In Section 3,we give the de¯nition
of inductive theorems.In Section 4,we give the
notion of primitive inductive theorems,and char
acterize inductive theorems by primitive inductive
theorems.We also present a su±cient condition for
inductive theorems to coincide with primitive in
ductive theorems.In Section 5,we study automated
reasoning for proving inductive theorems.In Sec
tion 6,we also study automated reasoning for dis
proving inductive theorems.In Section 7,we study
higherorder equational logic,and show that the no
tion of inductive theorems is characterized by the
initial extensional semantics.
2 Preliminaries
We assume that the reader is familiar with notions
of term rewriting systems [4].
2.1 Abstract Reduction System
An abstract reduction system(ARS) is a pair hA;!i
where A is a set and!is a binary relation on A.
The transitivere°exive closure of a binary relation
!is denoted by
¤
!,the transitive closure is de
noted by
+
!,and the transitivere°exivesymmetric
closure is denoted by
¤
$.
Let R = hA;!i be an ARS.An element a 2 A
is said to be a normal form if there exists no b
such that a!b.We denote the set of all normal
forms by NF(R).An ARS R is said to be weakly
normalizing,denoted by WN(R),if 8a 2 A:9b 2
NF(R):a
¤
!b;to be strongly normalizing (terminat
ing),denoted by SN(R),if there exists no in¯nite
sequence a
0
!a
1
!¢ ¢ ¢;to be con°uent,denoted
by CR(R),if a
1
¤
Ãa
¤
!a
2
implies 9b:a
1
¤
!b
¤
Ãa
2
for
all a;a
1
;a
2
2 A.For ARSs R
1
and R
2
,R
2
retro
gresses to R
1
,denoted by RET(R
2
;R
1
),if a
1
!
2
a
2
implies a
1
!
1
b
1
!
¤
2
b
2
Ã
¤
2
a
2
for some b
1
and b
2
.
Note that SN(R) implies WN(R),and CR(R) im
plies that the normal form of an element is unique
if it exists.
2.2 Untyped Term Rewriting Sys
tem
Untyped term rewriting systems (UTRSs) intro
duced in [17]
1
,which can express higherorder func
tions directly,are term rewriting systems without
aritytyped constraints.In this subsection,we in
troduce some notions of UTRSs needed later on.
Let § be a signature,that is a ¯nite set of
function symbols,which are denoted by F;G;:::.
Let V be an enumerable set of variables with
§\V =;.Variables are denoted by x;y;z;f;:::.
Atom is a function or variable symbol denoted by
a;a
0
;:::.The set T(§;V) of (untyped) terms con
structed from § and V is the smallest set such that
a(t
1
;:::;t
n
) 2 T(§;V) whenever a 2 § [ V and
t
1
;:::;t
n
2 T(§;V).If n = 0,we write a instead of
a().The identity of terms is denoted by ´.We of
ten write s(t
1
;:::;t
m
) for a(s
1
;:::;s
n
;t
1
;:::;t
m
),
where s ´ a(s
1
;:::;s
n
).We de¯ne the root symbol
of a term a(t
1
;:::;t
n
) by root(a(t
1
;:::;t
n
)) = a.
Var(t) is the set of variables in t.A term is said to
be closed if no variable occurs in the term.The set
of closed terms is denoted by T(§).The size jtj of t
is the number of function symbols and variables in
t.
A substitution µ is a mapping from variables to
terms.We may write µ as fx
1
:= µ(x
1
);:::;x
n
:=
µ(x
n
)g if fx
1
;:::;x
n
g = fx 2 V j µ(x) 6´
xg.Each substitution µ is naturally extended
over terms,denoted by
^
µ,as
^
µ(F(t
1
;:::;t
n
)) =
F(
^
µ(t
1
);:::;
^
µ(t
n
)) if F 2 §;
^
µ(z(t
1
;:::;t
n
)) =
a
0
(t
0
1
;:::;t
0
m
;
^
µ(t
1
);:::;
^
µ(t
n
)) if z 2 V with µ(z) =
a
0
(t
0
1
;:::;t
0
m
).For simplicity,we identify µ and
^
µ.
We write tµ instead of µ(t).A context is a term
which has a special symbol ¤,called a hole,at a
leaf position.We can also inductively de¯ne context
as follows:¤ is a context;a(:::;t
i¡1
;C
i
[ ];t
i+1
;:::)
is a context if a is an atom,C
i
[ ] is a context and
t
j
2 T(§;V) for any j(6= i).A su±x context is
a term which has the symbol ¤ at the root posi
tion.We can also inductively de¯ne su±x context
as follows:¤ is a su±x context;(S
0
[ ])(u) is a su±x
context if S
0
[ ] is a su±x context and u 2 T(§;V).
For a context C[ ] (a su±x context S[ ]),C[t] (S[t])
1
In [17],UTRSs were called term rewriting systems with
highorder variables (TRSHVs).Since there exists no
\higherorder variable"in untyped systems,we use UTRS
in the paper.
2
denotes the result of placing t in the hole of C[ ]
(S[ ]).A term t
0
is said to be a subterm of a term
t if there exists a context C[ ] such that t ´ C[t
0
].
We denote by Sub(t) all subterms of t,and de¯ne
s B
sub
t by t 2 Sub(s) and s 6´ t.
Example 2.1 For the term t ´ f(A;x) and the
substitution µ = ff:= G(0);x:= Bg,we have tµ ´
G(0;A;B).We can see that F(0;¤) and F(¤;xs)
are contexts,¤(0) and ¤(0;Nil) are su±x contexts,
and ¤ is both a context and a su±x context.We
can also see that C[t] ´ a(t;t
0
) for C[ ] ´ a(¤;t
0
),
and S[a(t)] ´ a(t;t
0
) for S[ ] ´ ¤(t
0
).
An equivalence relation'on terms is called a
congruence relation if it is closed under contexts and
su±x contexts,that is,s't )C[s]'C[t] ^S[s]'
S[t] for any context C[ ] and su±x context S[ ].
A rewrite rule is a pair (l;r) of terms such that
root(l) =2 V and Var(l) ¶ Var(r).We write
l!r for (l;r).An untyped term rewriting sys
tem (UTRS) is a set of rewrite rules.The reduction
relation!
R
of R is de¯ned by s!
R
t i® s ´ C[S[lµ]]
and t ´ C[S[rµ]] for some l!r 2 R,C[ ],S[ ]
and µ.We often omit the subscript R whenever no
confusion arises.
Example 2.2 The Mapfunction is represented by
the following UTRS R:
½
Map(f;Nil)!Nil
Map(f;x::xs)!f(x)::Map(f;xs)
Note that we use the standard representation for list
structures by symbols::and Nil.We often write
lists in in¯x form.For example,::(x;::(y;Nil)) is
written as x::y::Nil.Then we have the following
reduction relation sequence.
Map(F;F(0)::0::Nil)
!
R
F(F(0))::Map(F;0::Nil)
!
R
F(F(0))::F(0)::Map(F;Nil)
!
R
F(F(0))::F(0)::Nil
2.3 SimplyTyped Term Rewriting
System
A simplytyped version of a UTRS is called a
simplytyped term rewriting system (STRS) [17].
A set of basic types (sorts) is denoted by B.The
set T of simple types is generated from B by the
constructor!,that is,T::= B j (T!T ).A type
attachment ¿ is a function from §[V to T.A term
a(t
1
;:::;t
n
) has type ¯ if ¿(a) = (®
1
!(¢ ¢ ¢!
(®
n
!¯) ¢ ¢ ¢ )) and each t
i
has type ®
i
.A term
t 2 T(§;V) is said to be simplytyped if it has a
simpletype.We denote the set of all simplytyped
terms by T
¿
(§;V).We de¯ne the order ord(®) of
a type ® as follows:ord(®) = 1 if ® 2 B,and
ord(®) = max(1 + ®
1
;®
2
) if ® = ®
1
!®
2
.A
variable is said to be a higherorder variable if it
has a simple type ® with ord(®) > 1.We use V
h
to
stand for the set of higherorder variables.A term
t is said to be ground
2
if t is closed and of basic
types.We denote all closed simplytyped terms by
T
¿
(§),all closed terms having a simple type ® by
T
®
(§),all basictyped terms by T
B
(§;V),and all
ground terms by T
B
(§).We also denote the set of
all basictyped subterms of t by Sub
B
(t)
To keep the type consistency,we assume that
¿(x) = ¿(µ(x)) for all x 2 V and substitutions µ.
We also prepare the hole ¤
®
with a simple type ®,
and for each context C[ ] (su±x context S[ ]) with
a hole ¤
®
we assume that ¿(t) = ® whenever we
denote C[t] (S[t]).
A simplytyped rewrite rule is a rewrite rule l!r
such that l and r have the same type.A simply
typed term rewriting system (STRS) is a set of
simplytyped rewrite rules.
Example 2.3 The UTRS representation of the
Mapfunction in Example 2.2 is regarded as an
STRS with the type attachment ¿(Nil) = L,¿(::) =
N!L!L and ¿(Map) = (N!N)!L!L.
For an STRS R,GNF(R),GWN(R),GSN(R)
and GCR(R) are de¯ned as NF(R
0
),WN(R
0
),
SN(R
0
) and CR(R
0
) in ARS R
0
= hT
B
(§);!
R
i,re
spectively.For STRSs R
1
and R
2
,GRET(R
2
;R
1
)
is de¯ned as that of ARSs hT
B
(§);!
R
1
i and
hT
B
(§);!
R
2
i.
2.4 FirstOrder TermRewriting Sys
tem and Equational Logic
The ¯rstorder term rewriting system (TRS) is the
usual term rewriting system.In this subsection,we
introduce some notions needed later on.All results
stated in this subsection can be found in [4].
We suppose that each symbol F 2 § is associated
with a natural number n,denoted by ar(F) = n.
We also suppose that ar(x) = 0 for any variable
x.A term a(t
1
;:::;t
n
) 2 T(§;V) is said to be a
¯rstorder term if ar(a) = n and each t
i
is also
¯rstorder term.We denote the set of ¯rstorder
terms by T
ar
(§;V),and the set of closed ¯rstorder
terms by T
ar
(§).In the ¯rst order setting,closeness
coincides with groundness.Since ¤is the only su±x
2
It is a natural extension of groundness of ¯rstorder
framework,because all ¯rstorder closed terms are basic.It
is also useful for our purpose,because our inductive rea
soning methods on STRSs are based on properties of closed
basictyped terms (cf.Theorem 5.2,5.7 and 6.3).
3
context,an equivalence relation on ¯rstorder terms
is a congruence relation whenever it is closed under
contexts.
A ¯rstorder equation is a pair (s;t) of ¯rstorder
terms,written as s = t.We de¯ne V ar(s = t)
by V ar(s) [ Var(t).Let E be a set of ¯rstorder
equations.We denote by
¤
$
E
the congruence relation
generated from E.A ¯rstorder equation s = t is
said to be a theorem in E,denoted by E`s = t,if
it is deducible by inference rules in Fig.1 except for
(Functionality)rule.A ¯rstorder equation s = t
is said to be an inductive theorem in E,denoted
by E`
ind
s = t,if it is deducible by inference
rules in Fig.1.Note that (Functionality)rule is a
kind of metarule,and proof trees may be in¯nitely
branching.
(Assumption)
s = t
if s = t 2 E
(Re°exivity)
t = t
(Symmetry)
t = s
s = t
(Transitivity)
s = u u = t
s = t
(Substitutivity)
s = t
sµ = tµ
(Monotonicity)
s
1
= t
1
¢ ¢ ¢ s
n
= t
n
F(s
1
;:::;s
n
) = F(t
1
;:::;t
n
)
if ar(F) = n
(Functionality)
8u 2 T
ar
(§):sfz:= ug = tfz:= ug
s = t
Figure 1:Inference rules for the ¯rstorder equa
tional logic
A §algebra A is a pair hA;§
A
i,where A is
a carrier,§
A
is a mapping which maps each
F 2 § of arity n to a nary function F
A
on
A.The term algebra T
ar
(§;V) is the §algebra
hT
ar
(§;V);§
T
ar
(§;V)
i,where each F
T
ar
(§;V)
is de
¯ned as F
T
ar
(§;V)
(t
1
;:::;t
n
)
def
== F(t
1
;:::;t
n
).
In the case of V =;,the term algebra is said to
be the ground term algebra,denoted by T
ar
(§).
Let hA;§
A
i be a §algebra.An equivalence re
lation » on A is said to be a congruence rela
tion if it is monotonic,that is,
V
n
i=1
a
i
» a
0
i
)
F
A
(a
1
;:::;a
n
) » F
A
(a
0
1
;:::;a
0
n
) for all F 2 § with
ar(F) = n.Suppose that hA;§
A
i is a §algebra and
» is a congruence relation on A.We de¯ne [a]
»
=
fa
0
2 A j a
0
» ag and A=»= f[a]
»
j a 2 Ag.We
also de¯ne the quotient algebra A=»= hA=»;§
A=»
i
of A modulo »,where each F
A=»
is de¯ned as
F
A=»
([a
1
]
»
;:::;[a
n
]
»
) = [F
A
(a
1
;:::;a
n
)]
»
.It is
known that the quotient algebra A=»is wellde¯ned
as a §algebra.Clearly,
¤
$
E
is a congruence relation
on T
ar
(§;V),and thus T
ar
(§;V)=
¤
$
E
is a §algebra,
called a quotient term algebra.T
ar
(§)=
¤
$
E
is also a
§algebra,called a quotient ground term algebra.
Let A = hA;§
A
i be a §algebra.An assignment
into A is a mapping ¾ from V to A.The interpreta
tion of a ¯rstorder term t by an assignment ¾ into
A,denoted by [[t]]
¾
,is de¯ned as [[x]]
¾
= ¾(x) for
any x 2 V;[[F(t
1
;:::;t
n
)]]
¾
= F
A
([[t
1
]]
¾
;:::;[[t
n
]]
¾
)
for any F 2 §.
Let s = t be a ¯rstorder equation,A a §algebra
and ¾ an assignment into A.We denote A;¾ j=
s = t if [[s]]
¾
= [[t]]
¾
.We also denote A j= s = t if
[[s]]
¾
= [[t]]
¾
for any assignment ¾ into A.
Let A be a §algebra and E be a set of ¯rstorder
equations.We say that A is a model of E,denoted
by A j= E,if A j= s = t for all s = t 2 E.We say
that s = t is a semantic consequence of E,denoted
by E j= s = t,if A j= s = t for any model A of E.
We denote by Alg
§
(E) the class of all models of E.
Proposition 2.4 ([4]) Let E be a set of ¯rstorder
equations and s = t be a ¯rstorder equation.Then
the following properties are equivalent.
(1) E j= s = t
(2) s
¤
$
E
t
(3) E`s = t
Let A = hA;§
A
i and B = hB;§
B
i be §algebras.
A mapping Á from A to B is said to be a homomor
phism if Á(F
A
(a
1
;:::;a
n
)) = F
B
(Á(a
1
);:::;Á(a
n
))
for any F 2 § and a
i
2 A.We denote by
Hom(A;B) the class of all homomorphisms from
A to B.Let K be a class of §algebra.An algebra
I 2 K is said to be an initial algebra in K if for any
A 2 K there exists a unique homomorphism from I
to A,that is,jHom(I;A)j = 1.
4
Proposition 2.5 ([4]) Let E be a set of ¯rstorder
equations and s = t be a ¯rstorder equation.The
quotient ground termalgebra T
ar
(§)=
¤
$
E
is an initial
algebra in the class Alg
§
(E).Moreover the follow
ing properties are equivalent.
(1) T
ar
(§)=
¤
$
E
j= s = t
(2) sµ
g
¤
$
E
tµ
g
for all ground substitution µ
g
(3) E`
ind
s = t
We often use the property (2) to de¯ne inductive
theorems.
3 Inductive Theorems
Basic ingredients of the theory of algebraic spec
i¯cation are equational logic and its semantics is
based on §algebra [10,12,13,14,24].Algebraic
speci¯cations for higherorder languages were stud
ied in [19,20].In this section,based on Meinke's
formulation [19],we give a syntax of a higherorder
equational logic on STRSs,and de¯ne inductive
theorems in STRSs.
De¯nition 3.1 A simplytyped equation,written
by s = t,is a pair of simplytyped terms with
the same types (¿(s) = ¿(t)).We also denote
V ar(s) [Var(t) by V ar(s = t).
Note that STRSs are often regarded as simply
typed equation sets.
Next we de¯ne theorems and inductive theorems
in the higherorder equational logic by using infer
ence rules displayed in Fig.2.
De¯nition 3.2 Let E be a set of simplytyped
equations.A simplytyped equation s = t is said
to be a theorem in E,denoted by E`s = t,if
it is deducible by the inference rules in Fig.2 ex
cept for (Functionality) and (Extensionality)rules.
A simplytyped equation s = t is said to be an in
ductive theorem in E,denoted by E`
ind
s = t,if it
is deducible by the inference rules in Fig.2.
The di®erences of inference rules between ¯rstorder
and higherorder settings are as follows:(Mono
tonicity) and (Functionality)rules are modi¯ed,
and (Extensionality)rule is added.
Example 3.3 Consider the following STRS Rwith
¿(Rev) = ¿(Frev) = L!L,¿(Ap) = ¿(F) = L!
L!L,¿(::) = N!L!L and ¿(Nil) = L (this
system can also be considered as TRS):
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
Ap(Nil;xs)!xs
Ap(x::xs;ys)!x::Ap(xs;ys)
Rev(Nil)!Nil
Rev(x::xs)!Ap(Rev(xs);x::Nil)
F(Nil;xs)!xs
F(x::xs;ys)!F(xs;x::ys)
Frev(xs)!F(xs;Nil)
Note that the transformation from Rev to Frev is
a typical example of program optimization.
Both Rev and Frev in R represent the same list
reverse function,and we have
R`
ind
Rev(xs) = Frev(xs);
R`
ind
Rev = Frev:
However the notion of theorems do not equate Rev
and Frev,that is,
R 0 Rev(xs) = Frev(xs);
R 0 Rev = Frev:
(Assumption)
s = t
if s = t 2 E
(Re°exivity)
t = t
(Symmetry)
t = s
s = t
(Transitivity)
s = u u = t
s = t
(Substitutivity)
s = t
sµ = tµ
(Monotonicity)
s
0
= t
0
¢ ¢ ¢ s
n
= t
n
s
0
(s
1
;:::;s
n
) = t
0
(t
1
;:::;t
n
)
if s
0
(s
1
;:::;s
n
) 2 T
¿
(§)
(Functionality)
8u 2 T
¿(z)
(§):sfz:= ug = tfz:= ug
s = t
(Extensionality)
s(z) = t(z)
s = t
if z =2 Var(s = t)
Figure 2:Inference rules for the higherorder equa
tional logic
5
4 Primitive Inductive Theorems
In this section,we give the notion of primitive in
ductive theorems,and study the relation between
primitive inductive theorems and inductive theo
rems.
4.1 Characterizing Inductive The
orems by Primitive Inductive
Theorems
The notion of primitive inductive theorems is a nat
ural extension of the property (2) in Proposition
2.5.
De¯nition 4.1 Let R be a set of simplytyped
equations.A simplytyped equation s = t is said to
be a primitive inductive theorem in R,denoted by
R`
pind
s = t,if S
g
[sµ
c
]
¤
$
R
S
g
[tµ
c
] for all closed sub
stitution µ
c
(i.e.8x 2 Var(s = t):µ
c
(x) 2 T
¿
(§))
and ground su±x context S
g
[ ] 2 T
B
(§[ f¤g).We
also denote R`
pind
E if R`
pind
s = t for all
s = t 2 E.
If an equation s = t has a basic type then we
have R`
pind
s = t () sµ
c
¤
$
R
tµ
c
for all closed
substitution µ
c
,since ¤ is the only su±x context
having a basictyped hole.
Theorem 4.2 Let Rbe a set of simplytyped equa
tions and s = t a simplytyped equation.Then
R`
pind
s = t )R`
ind
s = t
Proof.Suppose that S
g
[sµ
c
]
¤
$
R
S
g
[tµ
c
] for all
closed substitution µ
c
and ground su±x context
S
g
[ ].Let X = fu = v j R`
ind
u = vg.From
the de¯nition of!
R
,we have!
R
µ X by rules
(Assumption),(Substitutivity) and (Monotonicity).
Thus it follows that
¤
$
R
µ X from rules (Re
°exivity),(Symmetry) and (Transitivity).Hence
S
g
[sµ
c
] = S
g
[tµ
c
] 2 X.Let S
g
[ ] ´ ¤(u
1
;:::;u
m
)
and S[ ] ´ ¤(z
1
;:::;z
m
) where z
1
;:::;z
m
are fresh
variables.We de¯ne µ
0
c
by µ
0
c
(z
i
) = u
i
(i = 1;:::;n)
and µ
0
c
(z) = µ
c
(z) for the other variables.Then
S
g
[sµ
c
] ´ S[s]µ
0
c
and S
g
[tµ
c
] ´ S[t]µ
0
c
.From
(Functionality)rule,we have S[s] = S[t] 2 X.
From (Extensionality)rule,we have s = t 2 X.
¤
Since the notion of primitive inductive theorems
is a natural extension of the property (2) in Propo
sition 2.5,one might think that the simplytyped
version of Proposition 2.5 would be obtained.How
ever this is not true;the inverse of Theorem4.2 does
not hold.
Example 4.3 Consider again the STRS R shown
in Example 3.3.Suppose that Apply 2 § with
¿(Apply) = (L!L)!L!L.Then we have
R`
ind
Apply(Rev;xs) = Apply(Frev;xs);
R 0
pind
Apply(Rev;xs) = Apply(Frev;xs):
From this example,some readers might guess
that inductive theorems coincide with the mono
tonic closure of primitive inductive theorems.How
ever,this is also not true.
Example 4.4 We consider the following STRS R:
8
<
:
I(x)!x
I
0
(x)!x
Apply(I;x)!I(x)
where ¿(I) = ¿(I
0
) = N!N and ¿(Apply) =
(N!N)!N!N.Suppose that X is
the monotonic closure of primitive inductive the
orems in R,that is,X is the smallest set such
that R`
pind
s = t implies s = t 2 X,and
s
i
= t
i
2 X (i = 0;:::;n) implies s
0
(s
1
;:::;s
n
) =
t
0
(t
1
;:::;t
n
).Then Apply(I
0
;x) = Apply(I;x) 2
X follows from R`
pind
I
0
= I,and Apply(I;x) =
x 2 X holds.However X is not transitive because of
Apply(I
0
;x) = x =2 X.Since R`
ind
Apply(I
0
;x) =
x,the monotonic closure X of primitive inductive
theorems cannot characterize inductive theorems.
De¯nition 4.5 Let R be a set of simplytyped
equations and s = t a simplytyped equation.We
de¯ne R`
1
pind
s = t by R`
pind
s = t;R`
n+1
pind
s = t
by R
0
`
pind
s = t where R
0
= fu = v j R`
n
pind
u =
vg.
Theorem 4.6 Let Rbe a set of simplytyped equa
tions and s = t a simplytyped equation.Then
R`
ind
s = t i® R`
n
pind
s = t for some n.
Proof.We prove that R`
n
pind
s = t implies R`
ind
s = t by induction on n.The case n = 1 is Theorem
4.2.Suppose that n > 1.Let R
0
= fu = v j R`
n¡1
pind
u = vg.Then R
0
`
ind
s = t follows from Theorem
4.2.From the induction hypothesis,R`
ind
R
0
.
Thus we have R`
ind
s = t.
We prove that R`
ind
s = t implies R`
n
pind
s = t
for some n by induction on the depth of the proof
tree of R`
ind
s = t.Suppose that s = t is de
duced by an inference rule from a subproof P
0
,and
E (possibly empty) is the consequence of P
0
.In
the case that the inference rule is either (Assump
tion) or (Re°exivity),R`
pind
s = t.In other cases,
R`
n
pind
E for some n follows from the induction
hypothesis,hence R`
n+1
pind
s = t.¤
6
4.2 Su±cient Condition
We are now going to explore a su±cient condition
for the inverse of Theorem4.2.This condition plays
an important roll for disproving inductive theorems
(see Section 6).
Before we present the condition,we prepare sev
eral notions and lemmas.In the following,we
assume § is partitioned into D and C,that is,
§ = D [ C and D\C =;.Elements of D are
called de¯ned symbols,and those of C are called
constructors.
De¯nition 4.7 A term t 2 T(§) is said to be a
value if root(t
0
) 2 C for any t
0
2 Sub
B
(t).We denote
the set of all values by Val (C;D),and denote the set
of all basictyped values by Val
B
(C;D).An STRS
R is said to be quasireducible,denoted by QR(R),
if any basictyped termF(t
1
;:::;t
n
) is not a normal
form whenever t
1
;:::;t
n
2 Val (C;D) and F 2 D.
De¯nition 4.8 A term t 2 T(C;V
h
) is said to
be a pseudovalue if any variable occurrence is at
a leaf position.We denote all pseudovalues by
PVal (C;V
h
).An STRS R is said to be strongly
quasireducible,denoted by SQR(R),if any basic
typed term F(t
1
;:::;t
n
) is reducible whenever
t
1
;:::;t
n
2 PVal (C;V
h
) and F 2 D.
Example 4.9 Let C = f0;Pg,D = fAddg and
f;g 2 V such that ¿(0) = N,¿(P) = (N!N)!
N!N,¿(Add) = ¿(g) = N!N!N and
¿(f) = N!N.Then 0,Add and P(Add(0);0) are
values;0,g and P(f;0) are pseudovalues;Add(0;0)
is neither a value nor a pseudovalue.
Lemma 4.10 If SQR(R) then GNF(R) µ
Val
B
(C;D).
Proof.From the de¯nitions of GNF and Val
B
,we
have GNF(R) µ T
B
(§) and Val
B
(R) µ T
B
(§).
Hence it su±ces to show t =2 Val
B
(C;D) ) t =2
GNF(R) for all t 2 T
B
(§).
Suppose that t 2 T
B
(§) and t =2 Val
B
(C;D).
Let u be a minimal size term in Sub
B
(t) such that
root(u) = F 2 D.Let u
1
;:::;u
n
be terms such
that u ´ C[u
1
;:::;u
n
],each root(u
i
) is a de¯ned
symbol and C[ ] is a constructor context except
for the root position.By the minimality of u,
each u
i
is not of basic types.Hence there exist
v
1
;:::;v
m
2 PVal (C;V
h
) such that F(v
1
;:::;v
m
) ´
C[z
1
;:::;z
n
] where z
j
are distinct fresh variables.
From SQR(R),C[z
1
;:::;z
n
] is reducible.Hence
C[u
1
;:::;u
n
] is reducible,which implies the re
ducibility of t.Thus t =2 GNF(R).¤
De¯nition 4.11 An STRS R is said to be su±
cient complete,denoted by SC(R),if 8t 2 T
B
(§):
9v 2 Val
B
(C;D):t
¤
!
R
v.
Lemma 4.12 If GWN(R) and SQR(R) then
SC(R) holds.
Proof.Let t 2 T
B
(§).By GWN(R),t
¤
!
R
u
for some u 2 GNF(R).By Lemma 4.10,u 2
Val
B
(C;D).¤
De¯nition 4.13 If ord(¿(c)) · 2 for any c 2 C
then we say that the set C of constructors has a
¯rstorder structure.
Note that for any simplytyped term c(t
1
;:::;t
n
)
with c 2 C,each t
i
is of basic types whenever C has
a ¯rstorder structure.
Lemma 4.14 If C has a ¯rstorder structure then
Val
B
(C;D) = T
B
(C).
Proof.Val
B
(C;D) ¶ T
B
(C) is trivial.Assume that
Val
B
(C;D)nT
B
(C) 6=;.Let t be a minimal size term
in Val
B
(C;D) n T
B
(C),and t ´ a(t
1
;:::;t
n
).We
have a 2 C because of t 2 Val
B
(C;D).Since C has
a ¯rstorder structure,each t
i
is of basic types,and
hence 8i:t
i
2 Val
B
(C;D).From the minimality of
t,each t
i
is in T
B
(C).Hence a(t
1
;:::;t
n
) 2 T
B
(C).
It is a contradiction.¤
Lemma 4.15 Let R be an STRS.If the following
conditions hold:
(a) GSN(R),
(b) GCR(R),
(c) SQR(R),
(d) C has a ¯rstorder structure,and
(e) l =2 T
¿
(C;V) for any l!r 2 R,
then R`
pind
tµ
c
= tµ
0
c
for any t 2 T
B
(§;V) and
any closed substitutions µ
c
and µ
0
c
such that R`
pind
zµ
c
= zµ
0
c
for all z 2 Var(t).
Proof.We prove the claim by induction on tµ
c
with respect to (!
R
[B
sub
)
+
.Note that the well
foundedness on T
B
(§) of (!
R
[B
sub
)
+
is guaranteed
by the condition (a).Let t ´ a(t
1
;:::;t
n
).There
are three cases.
² Suppose that a 2 C.Since each t
i
is of basic
types from the condition (d),we have the induc
tion hypothesis R`
pind
t
i
µ
c
= t
i
µ
0
c
for each i.
Hence t
i
µ
c
¤
$
R
t
i
µ
0
c
,which implies tµ
c
¤
$
R
tµ
0
c
.Thus
R`
pind
tµ
c
= tµ
0
c
.
7
² Suppose that a 2 D.Consider t
i
of basic types.
From the induction hypothesis,R`
pind
t
i
µ
c
=
t
i
µ
0
c
,that is t
i
µ
c
¤
$
R
t
i
µ
0
c
.Since we have SC(R) from
conditions (a) and (c),and Lemma 4.12,there ex
ist ground constructor terms u
i
and v
i
such that
t
i
µ
c
¤
!
R
u
i
and t
i
µ
0
c
¤
!
R
v
i
by the condition (d) and
Lemma 4.14.Here,we have u
i
;v
i
2 GNF(R)
from the condition (e).Hence u
i
´ v
i
2 T
¿
(C)
follows from the condition (b).
For each t
i
of higherorder types,let u
i
´ v
i
´ f
i
where f
i
is a fresh variable.Let ¾
0
= ff
i
:= t
i
j t
i
is of higherorder types g.
Now we have
tµ
c
´ a(t
1
µ
c
;:::;t
n
µ
c
)
¤
!
R
a(u
1
¾
0
µ
c
;:::;u
n
¾
0
µ
c
)
´ a(u
1
;:::;u
n
)¾
0
µ
c
:
By the condition (c),there exist l!r 2 R and ¾
such that l¾ ´ a(u
1
;:::;u
n
).Hence,we have
a(u
1
;:::;u
n
)¾
0
µ
c
´ l¾¾
0
µ
c
!
R
r¾¾
0
µ
c
:
Similarly,we have tµ
0
c
+
!
R
r¾¾
0
µ
0
c
.From the induc
tion hypothesis,we have R`
pind
r¾¾
0
µ
c
= r¾¾
0
µ
0
c
,
which implies R`
pind
tµ
c
= tµ
0
c
.
² Suppose that a 2 V.Let t
0
´ (aµ
c
)(t
1
;:::;t
n
).
We consider the equation t
0
µ
c
= t
0
µ
0
c
.Since
tµ
c
´ t
0
µ
c
and root(t
0
) 2 C [D,R`
pind
t
0
µ
c
= t
0
µ
0
c
follows from previous cases.Hence tµ
c
´ t
0
µ
c
¤
$
R
t
0
µ
0
c
´ (aµ
c
)(t
1
µ
0
c
;:::;t
n
µ
0
c
).Since R`
pind
aµ
c
= aµ
0
c
,by taking a ground su±x context
¤(t
1
µ
0
c
;:::;t
n
µ
0
c
),we have (aµ
c
)(t
1
µ
0
c
;:::;t
n
µ
0
c
)
¤
$
R
(aµ
0
c
)(t
1
µ
0
c
;:::;t
n
µ
0
c
) ´ tµ
0
c
.Hence R`
pind
tµ
c
= tµ
0
c
.¤
Lemma 4.16 Let R be an STRS.Suppose that
conditions (a){(e) in Lemma 4.15 hold.If
R`
pind
s
i
= t
i
(i = 0;1;:::;n) then R`
pind
s
0
(s
1
;:::;s
n
) = t
0
(t
1
;:::;t
n
).
Proof.We show that
S
g
[s
0
(s
1
;:::;s
n
)µ
c
]
¤
$
R
S
g
[t
0
(t
1
;:::;t
n
)µ
c
]
for any closed substitution µ
c
and ground su±x con
text S
g
[ ].Let s
0
i
´ s
i
µ
c
,t
0
i
´ t
i
µ
c
(i = 0;1;:::;n)
and S
g
[ ] ´ ¤(s
0
n+1
;:::;s
0
m
) ´ ¤(t
0
n+1
;:::;t
0
m
).
From the assumption,R`
pind
s
0
i
= t
0
i
for i =
0;1;:::;m.Let µ and µ
0
be substitutions such that
µ(z
i
) = s
0
i
and µ
0
(z
i
) = t
0
i
for each i.Clearly,
µ and µ
0
are closed and for all i = 0;:::;n,
R`
pind
z
i
µ = z
i
µ
0
.From Lemma 4.15,R`
pind
z
0
(z
1
;:::;z
m
)µ = z
0
(z
1
;:::;z
m
)µ
0
,that is,R`
pind
S
g
[s
0
(s
1
;:::;s
n
)µ
c
] = S
g
[t
0
(t
1
;:::;t
n
)µ
c
].Hence
S
g
[s
0
(s
1
;:::;s
n
)µ
c
]
¤
$
R
S
g
[t
0
(t
1
;:::;t
n
)µ
c
].¤
Theorem 4.17 Let R be an STRS.Suppose that
conditions (a){(e) in Lemma 4.15 hold.Then
R`
pind
s = t () R`
ind
s = t
Proof.The ())part is Theorem 4.2.Suppose
R`
ind
s = t.We prove R`
pind
s = t by induction
on the depth of the proof tree of R`
ind
s = t.In the
case that s = t is deduced by (Monotonicity)rule,
R`
pind
s = t follows from the induction hypothesis
and Lemma 4.16.Other cases are routine.¤
This theorem plays an important role in the im
plicit induction methods for disproving inductive
theorems (Theorem 6.3).
5 Proving Inductive Theorems
Implicit induction methods are intended to prove
inductive theorems.In the ¯rstorder setting,two
kind of implicit induction methods are known:in
ductionless induction and rewriting induction [6,7,
11,16,21,22,23].In this section,we formulate
some methods to prove primitive inductive theo
rems using the results in [16,23].These methods
are also successfully applied to prove inductive the
orems using Theorem 4.2.
5.1 Inductionless Induction
In this subsection,we state how to apply the induc
tionless induction method in [23] to STRSs.
Proposition 5.1 ([23]) Let R
1
= hA;!
1
i and
R
2
= hA;!
2
i be ARSs.Suppose that all of the
following conditions hold:
(i)!
1
µ
¤
$
2
(ii) WN(R
1
)
(iii) CR(R
2
)
(iv) NF(R
1
) µ NF(R
2
)
Then we have
¤
$
1
=
¤
$
2
.
Based on this proposition,we show an abstract
theorem for proving inductive theorems.
Theorem 5.2 Let R and R
0
be STRSs,and E be
a set of equations.Suppose that all of the following
conditions hold:
(i)
¤
$
R[E
µ
¤
$
R
0
in T
B
(§)
(ii) GWN(R)
(iii) GCR(R
0
)
8
(iv) GNF(R) µ GNF(R
0
)
Then R`
pind
E and hence R`
ind
E.
Proof.From the condition (i),we have!
R
µ
¤
$
R
0
in T
B
(§).By applying Proposition 5.1 with T
B
(§)
as A,we have
¤
$
R
=
¤
$
R
0
in T
B
(§).Hence
¤
$
E
µ
¤
$
R
in
T
B
(§),which implies R`
pind
E.We have R`
ind
E
by Theorem 4.2.¤
We prepare a lemma for checking GNF(R) µ
GNF(R
0
).
Lemma 5.3 Let R and R
0
be STRSs.Suppose
that for any l!r 2 R
0
there exists l
0
2 Sub
B
(l)
such that root(l
0
) 2 D.Then Val
B
(C;D) µ
GNF(R
0
).Moreover,if SQR(R) additionally holds
then GNF(R) µ GNF(R
0
) holds.
Proof.Assume that t 2 Val
B
(C;D) and t =2
GNF(R
0
).Then t ´ C[S[lµ]] for some
C[ ];S[ ];µ;l!r 2 R.By assumption,there ex
ists l
0
2 Sub
B
(l) such that root(l
0
) 2 D.Then
l
0
µ 2 Sub
B
(t) and root(l
0
µ) 2 D.This contradicts
t =2 Val
B
(C;D).
Suppose that SQR(R) additionally holds.From
Lemma 4.10,GNF(R) µ Val
B
(C;D).Hence
GNF(R) µ GNF(R
0
).¤
Example 5.4 We consider the following STRS R:
8
>
>
<
>
>
:
Map(f;Nil)!Nil
Map(f;x::xs)!f(x)::Map(f;xs)
Ap(Nil;xs)!xs
Ap(x::xs;ys)!x::Ap(xs;ys)
Suppose that C = f0;S;Nil;::g,D = fAp;Mapg
and ¿(0) = N,¿(S) = N!N,¿(Nil) = L,¿(::) =
N!L!L,¿(Ap) = L!L!L and ¿(Map) =
(N!N)!L!L.Based on Theorem 5.2,we
prove that the following equation is an inductive
theorem in R:
Map(f;Ap(xs;ys)) = Ap(Map(f;xs);Map(f;ys))
Let R
0
be the union of R and the above equa
tion.We can prove SN(R) by the recursive path
order in [18],and CR(R
0
) by the critical pair cri
terion.Hence GWN(R) and GCR(R
0
) hold.Since
SQR(R) hold,and for any l!r 2 R
0
there ex
ists l
0
2 Sub
B
(l) such that root(l
0
) 2 D,the in
clusion GNF(R) µ GNF(R
0
) follows from Lemma
5.3.Therefore conditions (i){(iv) of Theorem 5.2
hold and thus the equation above is an inductive
theorem in R.
Example 5.5 We consider the following STRS R:
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
I(x)!x
^(T;y)!y
^(F;y)!F
Ands(Nil)!T
Ands(T::xs)!Ands(xs)
Ands(F::xs)!F
8(p;Nil)!T
8(p;x::xs)!^(p(x);8(p;xs))
Map(f;Nil)!Nil
Map(f;x::xs)!f(x)::Map(f;xs)
Suppose that ¿(T) = ¿(F) = B,¿(Nil) = L,¿(::
) = B!L!L,¿(I) = B!B,¿(^) = B!B!
B,¿(Ands) = L!B,¿(8) = (B!B)!L!B,
and ¿(Map) = (B!B)!L!L.In the way
similar to Example 5.4,we can prove:
R`
ind
8(I;xs) = Ands(Map(I;xs))
5.2 Rewriting Induction
In this subsection,we apply the rewriting induction
method,proposed in [22] and formalized in [16],to
STRSs.
Proposition 5.6 ([16]) Let R
1
= hA;!
1
i and
R
2
= hA;!
2
i be ARSs.Suppose that all of the
following conditions hold:
(i)!
1
µ
+
!
2
(ii) SN(R
2
)
(iii) RET(R
2
;R
1
)
Then we have
¤
$
1
=
¤
$
2
.
Theorem 5.7 Let R and R
0
be STRSs,and E be
a set of equations.Suppose that all of the following
conditions hold:
(i)!
R
µ
+
!
R
0
and
¤
$
E
µ
¤
$
R
0
hold in T
B
(§)
(ii) GSN(R
0
)
(iii) GRET(R
0
;R)
Then R`
pind
E and R`
ind
E.
Proof.By applying Proposition 5.6 with T
B
(§) as
A,we have
¤
$
R
=
¤
$
R
0
in T
B
(§).Hence
¤
$
E
µ
¤
$
R
in
T
B
(§),which implies R`
pind
E.Hence we have
R`
ind
E by Theorem 4.2.¤
By using the rewriting induction,we can also
prove inductive theorems in Example 5.4 and 5.5.
9
6 Disproving Inductive Theorems
For program veri¯cation,not only proving induc
tive theorems but also disproving ones is important.
In this section,we present an automated reasoning
method for disproving inductive theorems,based on
the methods proposed in [11,21] and formalized in
[16].
Proposition 6.1 ([16]) Let R
1
= hA;!
1
i and
R
2
= hA;!
2
i be ARSs.Suppose that all of the
following conditions hold:
(i)!
1
µ
+
!
2
(ii) SN(R
2
)
(iii) CR(R
1
)
(iv) NF(R
1
) * NF(R
2
)
Then we have
¤
$
1
6=
¤
$
2
.
Based on this abstract result,we design implicit
induction methods for disproving primitive induc
tive theorems.
Theorem 6.2 Let R and R
0
be STRSs,and E be
a set of equations.Suppose that all of the following
conditions hold:
(i)!
R
µ
+
!
R
0
and
¤
$
R[E
=
¤
$
R
0
hold in T
B
(§)
(ii) GSN(R
0
)
(iii) GCR(R)
(iv) GNF(R) * GNF(R
0
)
Then we have R 0
pind
E.
Proof.From Proposition 6.1,we have
¤
$
R
6=
¤
$
R
0
in
T
B
(§).Assume that R`
pind
E.Then it is easily
seen that
¤
$
E
µ
¤
$
R
in T
B
(§).Hence
¤
$
R
=
¤
$
R[E
=
¤
$
R
0
in T
B
(§).It is a contradiction.¤
We have already presented a su±cient condition
of primitive inductive theorems to be inductive the
orems (Theorem 4.17).It means that R 0
pind
E
guarantees R 0
ind
E provided that R satis¯es con
ditions (a){(e) of Lemma 4.15.From this fact,we
can use the implicit induction method to disprove
inductive theorems.
Theorem 6.3 Let R and R
0
be STRSs,and E be
a set of equations.Suppose that in addition to the
properties (i),(ii),(iii) and (iv) in Theorem 6.2,all
of the following properties hold:
(v) SQR(R),
(vi) C has a ¯rstorder structure,and
(vii) l =2 T
¿
(C;V) for any l!r 2 R.
Then we have R 0
ind
E.
Proof.From Theorems 6.2 and 4.17.Note that
GSN(R
0
) implies GSN(R).¤
Example 6.4 Let R be the following STRS:
½
Add(x;0)!0
Add(x;S(y))!S(Add(x;y))
where D = fAddg and C = f0;Sg.We prove
R 0
ind
Add(0) = S
based on Theorem 6.3.Let E = fAdd(0) = Sg and
R
0
= R[fAdd(0;z)!S(z);S(0)!0g.Since 8t 2
T
B
(§):Add(0;t) $
E
S(t) and S(0) $
E
Add(0;0)!
R
0,
we have
¤
$
R[E
¶
¤
$
R
0
in T
B
(§).From the construc
tion of R
0
,we have!
R
µ
+
!
R
0
and
¤
$
R[E
µ
¤
$
R
0
hold
in T
B
(§).Hence the condition (i) holds.The
conditions (v),(vi) and (vii) can be easily shown.
We can prove SN(R
0
) by the recursive path order
in [18],and CR(R) by the critical pair criterion.
Hence GSN(R
0
) and GCR(R) hold.The condi
tion (iv) holds because S(0) 2 GNF(R) and S(0) =2
GNF(R
0
).Therefore we have R 0
ind
Add(0) = S.
Example 6.5 Let R be the following STRS:
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
I(x)!x
^(T;y)!y
^(F;y)!F
Ands(Nil)!T
Ands(T::xs)!Ands(xs)
Ands(F::xs)!F
8(p;Nil)!F
8(p;x::xs)!^(p(x);8(p;xs))
Map(f;Nil)!Nil
Map(f;x::xs)!f(x)::Map(f;xs)
This example is slightly di®erent fromExample 5.5,
that is,8(p;Nil)!F
,which is problematic.Here,
we prove
R 0
ind
8(I;xs) = Ands(Map(I;xs))
based on Theorem 6.3.Let E = f8(I;xs) =
Ands(Map(I;xs))g and R
0
= R [ f8(I;xs)!
Ands(Map(I;xs));F!Tg.Since F Ã
R
8(I;Nil)
$
E
Ands(Map(I;Nil))!
R
Ands(Nil)!
R
T,we have
¤
$
R[E
=
¤
$
R
0
in T
B
(§).Hence the condition (i) holds.
The conditions (v),(vi) and (vii) trivially hold.
10
We can prove SN(R
0
) by the recursive path or
der in [18],and CR(R) by the critical pair crite
rion.Hence GSN(R
0
) and GCR(R) hold.The con
dition (iv) holds because F 2 GNF(R) and F =2
GNF(R
0
).Therefore we have R 0
ind
8(I;xs) =
Ands(Map(I;xs)).
7 Characterizing Inductive Theo
rems
Algebraic speci¯cation is based on equational logic
whose semantics is given on §algebra [10,12,13,
14,24].Higherorder theories have also been con
sidered in [19,20].
In Section 3,we gave a syntax of a higherorder
equational logic,and the de¯nition of inductive
theorems based on Meinke's formulation [19].To
justify our de¯nition,we should give a semantic
counter parts and show that our inductive theo
rems can be characterized by the initial extensional
semantics,because our framework (STRSs) is dif
ferent from Meinke's one.
7.1 Syntax
From this subsection to subsection 7.3,we study
higherorder equational logic on untyped systems
(UTRSs).For any set E of untyped equations and
an untyped equation s = t,we de¯ne E`s = t
and E`
ind
s = t as similar to ones in simplytyped
systems except for type constraints.
Here we prepare for a lemma needed later on.
Lemma 7.1 Let E be a set of equations.Then
E`s = t () s
¤
$
E
t.
Proof.()) By induction on the depth of proof
trees.(() It su±ces to show that E`s = t when
ever s!
E
t.Let s ´ C[S[e
1
µ]] and t ´ C[S[e
2
µ]].
The equation e
1
= e
2
is deducible by (Assumption)
rule.e
1
µ = e
2
µ is deduced by (Substitutivity)rule.
S[e
1
µ] = S[e
2
µ] is deduced by (Monotonicity)rule.
Finally,we obtain E`C[S[e
1
µ]] = C[S[e
2
µ]] by
(Monotonicity)rule.¤
7.2 §
@
algebra
We present the semantic counter parts by curried
terms,which is one of the most standard way for
handling higherorder function in ¯rstorder sys
tems.For example,the Mapfunction is repre
sented in the ¯rstorder term rewriting as follows:
8
<
:
@(@(Map;f);Nil)!Nil
@(@(Map;f);@(@(Cons;x);xs))
!@(@(Cons;@(f;x));@(@(Map;f);xs))
De¯nition 7.2 We de¯ne §
@
= §[ f@g,and de
note by T
@
(§;V) the set of ¯rstorder terms gener
ated by §
@
and V under ar(@) = 2 and ar(a) = 0
for any symbol a (6= @).We also denote the term
algebra T
ar
(§
@
;V) by T
@
(§;V).The ground term
algebra T
@
(§;;) is denoted by T
@
(§).
De¯nition 7.3 For any untyped termt 2 T(§;V),
we inductively de¯ne ¯rstorder term t
@
2
T
@
(§;V) as follows:
² a
@
= a for any a 2 §[ V
² a(t
1
;:::;t
n
)
@
= @(a(t
1
;:::;t
n¡1
)
@
;t
@
n
) if
n ¸ 1
We naturally extend the notion over substitutions
as µ
@
(x) = (µ(x))
@
,and over sets of pairs (like
equations or rules) as E
@
= f(s
@
;t
@
) j (s;t) 2 Eg.
We notice that T
@
(§;V) = ft
@
j t 2 T(§;V)g,
and any context on T
@
(§;V) has a form C
@
[S
@
[ ]]
for some context C[ ] and su±x context S[ ] on
T(§;V).
Lemma 7.4 C[S[tµ]]
@
´ C
@
[S
@
[t
@
µ
@
]]
Proof.Firstly we prove (tµ)
@
´ t
@
µ
@
by induc
tion on jtj.Let t ´ a(t
1
;:::;t
n
).The case
n = 0 is trivial.Suppose that n > 0 and
u ´ a(t
1
;:::;t
n¡1
).Then (tµ)
@
´ (u(t
n
)µ)
@
´
((uµ)(t
n
µ))
@
´ @((uµ)
@
;(t
n
µ)
@
) ´ @(u
@
µ
@
;t
@
n
µ
@
)
´ @(u
@
;t
@
n
)µ
@
´ t
@
µ
@
.
We can prove S[tµ]
@
´ S
@
[(tµ)
@
] by induction on
S[ ],and C[S[tµ]]
@
´ C
@
[S[tµ]
@
] by induction on
C[ ].Hence we obtain C[S[tµ]]
@
´ C
@
[S
@
[t
@
µ
@
]].
¤
Lemma 7.5 Let E be a set of equations.Then
s!
E
t () s
@
!
E
@
t
@
.
Proof.()) The claim follows from Lemma 7.4.
(() Let s
@
!
E
@
t
@
.Then s
@
´ C
@
[S
@
[e
@
1
µ
@
]] and
t
@
´ C
@
[S
@
[e
@
2
µ
@
]] for some C
@
[ ],S
@
[ ],µ
@
and
e
@
1
= e
@
2
2 E
@
,because any context on T
@
(§;V)
has a form C
@
[S
@
[ ]].From Lemma 7.4,s ´
C[S[e
1
µ]] and t ´ C[S[e
2
µ]].Hence s!
E
t.¤
De¯nition 7.6 Let s = t be an equation and E
be a set of equations.For any §
@
algebra A,we
denote A j=
@
s = t instead of A j= s
@
= t
@
.We
also denote E j=
@
s = t if A j=
@
s = t for any
model A of E
@
.
Thanks to Lemma 7.1,Lemma 7.5 and Proposi
tion 2.4,we obtain the following theorem:
Theorem 7.7 Let E be a set of equations and s =
t be an equation.Then E`s = t () E j=
@
s =
t.
11
7.3 Initial Extensional Model
In ¯rstorder algebraic speci¯cation,inductive the
orems are characterized by the initial algebra se
mantics,that is,characterized in T
ar
(§)=
¤
$
E
,which
is an initial algebra in Alg
§
(E).
The initial algebra semantics cannot character
ize inductive theorems in higherorder settings,be
cause the extensionality is built in the syntax.In
this subsection,we show that there exists an ini
tial extensional model,which characterizes induc
tive theorems.For any t
@
2 T
@
(§),we denote its
interpretation by [[t
@
]],because [[t
@
]]
¾
is independent
of assignment ¾.
De¯nition 7.8 A §
@
algebra A = hA;§
A
i is said
to be extensional if 8t
@
2 T
@
(§):@
A
(a;[[t
@
]]) =
@
A
(a
0
;[[t
@
]]) )a = a
0
for all a;a
0
2 A.Let s = t be
an equation and E be a set of equations.We denote
by Alg
´
§
@
(E
@
) the class of all extensional model of
E
@
.We denote E j=
´
@
s = t if A j=
@
s = t for all
A 2 Alg
´
§
@
(E
@
).
Theorem 7.9 We de¯ne s
@
'
E
t
@
as E`
ind
s =
t.The quotient ground term algebra T
@
(§)='
E
is
an initial algebra in Alg
´
§
@
(E).
Proof.It follows easily from the de¯nition of in
ductive theorems that'
E
is a congruence re
lation.Hence the quotient algebra T
@
(§)='
E
is wellde¯ned.Let A 2 Alg
´
§
@
(E
@
).Since
T
@
(§) = T
ar
(§
@
),T
@
(§)=
¤
$
E
@
is an initial algebra in
Alg
§
@(E
@
) by Proposition 2.5.Since Alg
´
§
@
(E
@
) µ
Alg
§
@(E
@
),we have the unique homomorphism Ã
from T
@
(§)=
¤
$
E
@
to A.
From Lemmata 7.1 and 7.5,E`s = t ()
s
@
¤
$
E
@
t
@
.Since E`s = t implies E`
ind
s = t,
¤
$
E
@
µ'
E
holds.Hence there exists a projection p 2
Hom(T
@
(§)=
¤
$
E
@
;T
@
(§)='
E
),which maps [t]
¤
$
E
@
to
[t]
'
E
.
Here we show that jHom(T
@
(§)='
E
;A)j · 1.
Assume that Ã
1
;Ã
2
2 Hom(T
@
(§)='
E
;A) such
that Ã
1
6= Ã
2
.Since p is surjective,Ã
1
¢ p 6= Ã
2
¢ p.
Since Ã
1
¢ p;Ã
2
¢ p 2 Hom(T
@
(§)=
¤
$
E
@
;A),Ã
1
¢ p =
Ã = Ã
2
¢ p.It is a contradiction.
It remains to show that there exists a homo
morphism Ã
0
2 Hom(T
@
(§)='
E
;A).For this,it
su±ces to show s
@
'
E
t
@
implies Ã([s
@
µ
@
g
]) =
Ã([t
@
µ
@
g
]) for any ground substitution µ
@
g
,because
it guarantees the existence Ã
0
such that Ã
0
¢ p = Ã.
The proof proceeds by induction on the depth of
proof trees for E`
ind
s = t.We prove only cases
of (Functionality) and (Extensionality).
Suppose that s = t is deduced by (Functionality)
rule.Fromthe induction hypothesis,8u
@
2 T
@
(§):
Ã([s
@
fz:= u
@
g¾
@
g
]) = Ã([t
@
fz:= u
@
g¾
g
]
@
) for
all ground substitution ¾
@
g
.Hence Ã([s
@
µ
@
g
]) =
Ã([t
@
µ
@
g
]) for all ground substitution µ
@
g
.
Suppose that s = t is deduced by
(Extensionality)rule.From the induc
tion hypothesis,9z =2 Var(s = t).
Ã([@(s
@
;z)µ
@
g
]) = Ã([@(t
@
;z)µ
@
g
]) for all ground
substitution µ
@
g
.Since Ã is a homomorphism,
@
A
(Ã([s
@
µ
@
g
]);Ã([zµ
@
g
])) = Ã([@(s
@
µ
@
g
;zµ
@
g
)])
= Ã([@(t
@
µ
@
g
;zµ
@
g
)]) = @
A
(Ã([t
@
µ
@
g
]);Ã([zµ
@
g
]))
for all ground substitution µ
@
g
.Since A is an
extensional model,Ã([s
@
µ
@
g
]) = Ã([t
@
µ
@
g
]) for any
ground substitution µ
@
g
.¤
p
A
T
@
(§)='
E
T
@
(§)=$
¤
E
@
Ã
0
Ã
Figure 3:proof of Theorem 7.9
7.4 SimplyTyped Systems
In algebraic speci¯cation,type information is very
important and useful.In this last subsection,we
discuss how to incorporate our previous results
based on untyped systems into simplytyped sys
tems.Since our higherorder equational logic is de
signed independently of type structure,it is easy
to combine with type systems.To be precise,we
need the following simplytyped constraint for a
§
@
algebra A:
² there exists a type attachment ¿
A
on A such
that
{ ¿
A
(F
A
) = ¿(F) for any F 2 §,and
{ ¿
A
(@
A
(a
1
;a
2
)) = ¯ if ¿
A
(a
1
) = ®!¯
and ¿
A
(a
1
) = ®.
We also restrict ¾ to run over all the assignments
with
² ¿
A
(¾(x)) = ¿(x) for any x 2 V
when de¯ning j=
@
for simply typed case.
Under the simplytyped constraint and the re
striction for assignments,any properties in this sec
tion still holds on simplytyped systems.Hence in
ductive theorems with simplytyped systems can be
characterized by the initial extensional semantics,
and the simplytyped version of Proposition 2.5 is
obtained.
12
Theorem 7.10 Let E be a set of simplytyped
equations and s = t be a simplytyped equation.
We de¯ne s
@
'
E
t
@
as E`
ind
s = t.Then the
quotient ground term algebra T
@
¿
(§)='
E
is an ini
tial algebra in the class of all extensional model of
E.Moreover the following properties are equiva
lent.
(1) T
@
¿
(§)='
E
j=
@
s = t
(2) E`
n
pind
s = t for some n
(3) E`
ind
s = t
This theorem shows that inductive theorems can
be characterized by the initial extensional seman
tics.
8 Concluding Remarks
Under simplytyped systems,higherorder theories
have also been studied in [19,20].Our syntactical
de¯nition is based on one in [19],and §
@
algebra
is based on one in [20].However,for initial algebra
semantics,minimality (w.r.t.subalgebra relation)
is required in [19],and reachability is required in
[20].On the other hand,our algebra does not re
quire any such restrictions.It is a future subject
to study why such a di®erence comes out.More
over,since our algebra is designed independently of
type structure,it is so easy to combine with arbi
trary type systems.It is important to incorporate
not only simplytyped systems but also complicated
ones like polymorphictyped systems.
We would feel that our proving and disproving
methods for primitive inductive theorems (Theorem
5.2,5.7 and 6.3) were natural extensions of results
in [23,16].Since these approach can apply only
for the class of primitive inductive theorems,it is a
critical problem for disproving inductive theorems.
To overcome the di±culty,we presented a su±cient
condition which guarantees that inductive theorems
and primitive ones coincide (Theorem 4.17).The
su±cient condition is indispensable for disproving
inductive theorems (Theorem 6.3).
A higherorder KnuthBendix procedure and its
application to inductionless induction were imple
mented by the ¯rst author Kusakari,as a post
doctorate subtheme at the Japan Advanced Insti
tute of Science and Technology (JAIST) in 1999.
Example 5.4,which presents an application to in
ductionless induction (Theorem 5.2),was presented
at that time.Results similar to these with respect
to inductionless induction were also presented in
2003 by Aoto,Yamada and Toyama [1].Unfor
tunately,all of the above results confused the dif
ference between inductive theorems and primitive
inductive theorems,and erroneously used the term
\inductive theorems".In this paper,we make a
clear distinction between the concepts of inductive
theorems and primitive inductive ones.At the re
quest of Toyama,Kusakari presented a lecture on
the results in this paper to Aoto and Yamada in Au
gust 2003.In the following year,Aoto,Yamada and
Toyama presented results on the automated prov
ing of inductive theorems [2] using a formalization
di®erent from our results.
Acknowledgments
We would like to thank TOYAMA Yoshihito for
fruitful discussion,and the anonymous referees for
their useful and detailed comments.
This work is partly supported by MEXT.KAK
ENHI#15500007 and#16300005,by Arti¯cial In
telligence Research Promotion Foundation,and by
the 21st Century COE Program (Intelligent Media
Integration for Social Infrastructure),Nagoya Uni
versity.
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14
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