<s>
Structural	B-Algorithm
induction	I-Algorithm
is	O
a	O
proof	O
method	O
that	O
is	O
used	O
in	O
mathematical	O
logic	O
(	O
e.g.	O
,	O
in	O
the	O
proof	O
of	O
Łoś	O
 '	O
theorem	O
)	O
,	O
computer	B-General_Concept
science	I-General_Concept
,	O
graph	O
theory	O
,	O
and	O
some	O
other	O
mathematical	O
fields	O
.	O
</s>
<s>
It	O
is	O
a	O
generalization	O
of	O
mathematical	B-Algorithm
induction	I-Algorithm
over	I-Algorithm
natural	I-Algorithm
numbers	I-Algorithm
and	O
can	O
be	O
further	O
generalized	O
to	O
arbitrary	O
Noetherian	B-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
Structural	B-Algorithm
recursion	I-Algorithm
is	O
a	O
recursion	O
method	O
bearing	O
the	O
same	O
relationship	O
to	O
structural	B-Algorithm
induction	I-Algorithm
as	O
ordinary	O
recursion	O
bears	O
to	O
ordinary	O
mathematical	B-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
A	O
well-founded	B-Algorithm
partial	O
order	O
is	O
defined	O
on	O
the	O
structures	O
(	O
"	O
subformula	O
"	O
for	O
formulas	O
,	O
"	O
sublist	O
"	O
for	O
lists	O
,	O
and	O
"	O
subtree	O
"	O
for	O
trees	O
)	O
.	O
</s>
<s>
The	O
structural	B-Algorithm
induction	I-Algorithm
proof	O
is	O
a	O
proof	O
that	O
the	O
proposition	O
holds	O
for	O
all	O
the	O
minimal	O
structures	O
and	O
that	O
if	O
it	O
holds	O
for	O
the	O
immediate	O
substructures	O
of	O
a	O
certain	O
structure	O
,	O
then	O
it	O
must	O
hold	O
for	O
also	O
.	O
</s>
<s>
(	O
Formally	O
speaking	O
,	O
this	O
then	O
satisfies	O
the	O
premises	O
of	O
an	O
axiom	O
of	O
well-founded	B-Algorithm
induction	I-Algorithm
,	O
which	O
asserts	O
that	O
these	O
two	O
conditions	O
are	O
sufficient	O
for	O
the	O
proposition	O
to	O
hold	O
for	O
all	O
.	O
)	O
</s>
<s>
Structural	B-Algorithm
recursion	I-Algorithm
is	O
usually	O
proved	O
correct	O
by	O
structural	B-Algorithm
induction	I-Algorithm
;	O
in	O
particularly	O
easy	O
cases	O
,	O
the	O
inductive	O
step	O
is	O
often	O
left	O
out	O
.	O
</s>
<s>
A	O
structural	B-Algorithm
induction	I-Algorithm
proof	O
of	O
some	O
proposition	O
then	O
consists	O
of	O
two	O
parts	O
:	O
A	O
proof	O
that	O
is	O
true	O
and	O
a	O
proof	O
that	O
if	O
is	O
true	O
for	O
some	O
list	O
,	O
and	O
if	O
is	O
the	O
tail	O
of	O
list	O
,	O
then	O
must	O
also	O
be	O
true	O
.	O
</s>
<s>
In	O
those	O
cases	O
,	O
a	O
structural	B-Algorithm
induction	I-Algorithm
proof	O
of	O
some	O
proposition	O
then	O
consists	O
of	O
:	O
</s>
<s>
As	O
an	O
example	O
,	O
the	O
property	O
"	O
An	O
ancestor	O
tree	O
extending	O
over	O
generations	O
shows	O
at	O
most	O
persons	O
"	O
can	O
be	O
proven	O
by	O
structural	B-Algorithm
induction	I-Algorithm
as	O
follows	O
:	O
</s>
<s>
the	O
induction	B-Algorithm
hypothesis	O
)	O
.	O
</s>
<s>
Hence	O
,	O
by	O
structural	B-Algorithm
induction	I-Algorithm
,	O
each	O
ancestor	O
tree	O
satisfies	O
the	O
property	O
.	O
</s>
<s>
We	O
will	O
prove	O
this	O
by	O
structural	B-Algorithm
induction	I-Algorithm
on	O
lists	O
.	O
</s>
<s>
The	O
induction	B-Algorithm
hypothesis	O
is	O
that	O
is	O
true	O
for	O
all	O
values	O
of	O
when	O
is	O
:	O
</s>
<s>
Thus	O
,	O
from	O
structural	B-Algorithm
induction	I-Algorithm
,	O
we	O
obtain	O
that	O
is	O
true	O
for	O
all	O
lists	O
.	O
</s>
<s>
Just	O
as	O
standard	O
mathematical	B-Algorithm
induction	I-Algorithm
is	O
equivalent	O
to	O
the	O
well-ordering	B-Algorithm
principle	I-Algorithm
,	O
structural	B-Algorithm
induction	I-Algorithm
is	O
also	O
equivalent	O
to	O
a	O
well-ordering	B-Algorithm
principle	I-Algorithm
.	O
</s>
<s>
If	O
the	O
set	O
of	O
all	O
structures	O
of	O
a	O
certain	O
kind	O
admits	O
a	O
well-founded	B-Algorithm
partial	O
order	O
,	O
then	O
every	O
nonempty	O
subset	O
must	O
have	O
a	O
minimal	O
element	O
.	O
</s>
<s>
(	O
This	O
is	O
the	O
definition	O
of	O
"	O
well-founded	B-Algorithm
"	O
.	O
)	O
</s>
