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cpdt
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ca54c34f
Commit
ca54c34f
authored
Sep 08, 2008
by
Adam Chlipala
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InductiveTypes.v
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src/InductiveTypes.v
View file @
ca54c34f
...
@@ -121,6 +121,11 @@ Definition not (b : bool) : bool :=
...
@@ -121,6 +121,11 @@ Definition not (b : bool) : bool :=
|
false
=>
true
|
false
=>
true
end
.
end
.
(
**
An
alternative
definition
desugars
to
the
above
:
*
)
Definition
not
'
(
b
:
bool
)
:
bool
:=
if
b
then
false
else
true
.
(
**
We
might
want
to
prove
that
[
not
]
is
its
own
inverse
operation
.
*
)
(
**
We
might
want
to
prove
that
[
not
]
is
its
own
inverse
operation
.
*
)
Theorem
not_inverse
:
forall
b
:
bool
,
not
(
not
b
)
=
b
.
Theorem
not_inverse
:
forall
b
:
bool
,
not
(
not
b
)
=
b
.
...
@@ -150,3 +155,19 @@ The first subgoal follows by Coq's rules of computation, so we can dispatch it e
...
@@ -150,3 +155,19 @@ The first subgoal follows by Coq's rules of computation, so we can dispatch it e
Restart
.
Restart
.
destruct
b
;
reflexivity
.
destruct
b
;
reflexivity
.
Qed
.
Qed
.
(
**
Another
theorem
about
booleans
illustrates
another
useful
tactic
.
*
)
Theorem
not_ineq
:
forall
b
:
bool
,
not
b
<>
b
.
destruct
b
;
discriminate
.
Qed
.
(
**
[
discriminate
]
is
used
to
prove
that
two
values
of
an
inductive
type
are
not
equal
,
whenever
the
values
are
formed
with
different
constructors
.
In
this
case
,
the
different
constructors
are
[
true
]
and
[
false
]
.
At
this
point
,
it
is
probably
not
hard
to
guess
what
the
underlying
induction
principle
for
[
bool
]
is
.
*
)
Check
bool_ind
.
(
**
[[
bool_ind
:
forall
P
:
bool
->
Prop
,
P
true
->
P
false
->
forall
b
:
bool
,
P
b
]]
*
)
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