Proof theory

103 Pages·1994·0.534 MB·English

by  Schwichtenberg H.

Proof Theory
Helmut Schwichtenberg
Mathematisches Institut der Universit¨at Mu¨nchen
Sommersemester 1994
Preface
These notes represent the content of a lecture course on proof theory given during Som-
mersemester 1994 at the Mathematisches Institut der Universit¨at Mu¨nchen. They are still
rather sketchy and will have to undergo many more revisions.
For their help in preparing these notes I would like to thank Ulrich Berger, Michael
Bopp, Felix Joachimski, Ralph Matthes, Karl–Heinz Niggl, Jaco van de Pol and Robert
St¨ark.
Mu¨nchen, July 1994
Helmut Schwichtenberg
ii
Contents
Part I: Preliminaries
1 Typed languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Natural deduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Hilbert style systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 The strong existential quantifier . . . . . . . . . . . . . . . . . . . . . . . 38
6 Realizing terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Part II: Computational content of proofs
8 A–translation with program extraction and the direct method . . . . . . . . . . 54
9 The root example; refinements . . . . . . . . . . . . . . . . . . . . . . . 63
Part III: Classifying arithmetical proofs
10 Ordinals below epsilon zero . . . . . . . . . . . . . . . . . . . . . . . . 73
11 Provability of initial cases of transfinite induction . . . . . . . . . . . . . . 76
12 Normalization for arithmetic with the omega rule . . . . . . . . . . . . . . 80
13 Unprovable initial cases of transfinite induction . . . . . . . . . . . . . . . 83
14 Normalization for arithmetic is impossible . . . . . . . . . . . . . . . . . . 86
Appendix
15 Permutative conversions . . . . . . . . . . . . . . . . . . . . . . . . . . 87
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
iii
Part I: Preliminaries
1 Typed languages
Let us first fix our language . Let G be a set of ground types (e.g. nat and boole). Types
L
(also called object types or simple types) are formed from G by the operations ρ σ and
→
ρ σ. The level of a type ρ is defined by
×
lev(ι) := 0 for any ground type ι,
lev(ρ σ) := max(lev(ρ)+1,lev(σ)),
→
lev(ρ σ) := max(lev(ρ),lev(σ)).
×
For any type ρ let a countable infinite set of variables of type ρ be given. We denote
variables of type ρ by xρ,yρ,.... We also assume that a set of constants denoted cρ
C
is given, each of an arbitrary type ρ. Furthermore, we assume that a set of function
F
symbols denoted f is given, each of a “functionality” (ρ ,...,ρ ) σ. They are intended
1 m
→
to denote external functionals, which need not be represented by objects in the model.
Note that it is a consequence of this that we can not freely abstract variables in a term, for
otherwise we could form λxρf(x) for an arbitrary function symbol f, and hence f would
be an object of our model.
We define inductively terms tρ of type ρ, the set FV(tρ) of variables free in tρ and the
set nonabs(tρ) of non–abstractable variables in FV(tρ).
xρ and cρ are terms of type ρ, FV(xρ) = xρ , FV(cρ) = , nonabs(xρ) = ,
• { } ∅ ∅
nonabs(cρ) = .
∅
Iftρ1,...,tρn aretermsandf afunctionsymboloffunctionality(ρ ,...,ρ ) σ,
• 1 n ∈ F 1 m →
then f(t ,...,t ) is a term of type σ. FV(f(t ,...,t )) = FV(t ) FV(t ),
1 n 1 n 1 n
∪ ··· ∪
nonabs(f(t ,...,t )) = FV(t ) FV(t )
1 n 1 n
∪···∪
If t is a term of type σ and xρ nonabs(t), then λxρt is a term of type ρ σ.
• 6∈ →
FV(λxρt) = FV(t) xρ , nonabs(λxρt) = nonabs(t).
\{ }
If t is a term of type ρ σ and s is a term of type ρ, then ts is a term of type σ.
• →
FV(ts) = FV(t) FV(s), nonabs(ts) = nonabs(t) nonabs(s).
∪ ∪
If t is a term of type ρ for i 0,1 , then t ,t is a term of type ρ ρ .
i i 0 1 0 1
• ∈ { } h i ×
FV( t ,t ) = FV(t ) FV(t ), nonabs( t ,t ) = nonabs(t ) nonabs(t ).
0 1 0 1 0 1 0 1
h i ∪ h i ∪
If t is a term of type ρ ρ and i 0,1 , then π (t) is a term of type ρ . FV(π (t)) =
0 1 i i i
• × ∈ { }
FV(t), nonabs(π (t)) = nonabs(t).
i
We now fix our notion of a formula. Let an additional ground type be given, to be viewed
◦
as the type of propositions. is not to be used in object types ρ,σ, . We assume that
◦ ···
2 1 Typed languages
a set of relation symbols R of functionalities (ρ ,...,ρ ) is given. 0–ary relation
1 n
P → ◦
symbols are called propositional symbols. Formulas are defined inductively by
If tρ1,...,tρn are terms and R a relation symbol of functionality (ρ ,...,ρ ) ,
• 1 n ∈ P 1 n → ◦
then R(t ,...,t ) is a formula.
1 n
(to be read “falsity”) is a formula.
• ⊥
If A and B are formulas, then A B is a formula.
• →
If A and B are formulas, then A B is a formula.
• ∧
If A is a formula and xρ is a variable, then xρA is a formula.
• ∀
R(t ,...,t ) and are called atomic formulas or atoms. Note that is not to a propo-
1 n
⊥ ⊥
sitional symbol.
We use (also with indices)
r,s,t for terms,
x,y,z for variables,
a,b,c for constants,
P,Q,R for relation symbols,
f,g,h for function symbols,
A,B,C for formulas.
Negation, disjunction and the existential quantifier are defined by
A := A ,
¬ → ⊥
A B := A B ,
∨ ¬ ∧¬ → ⊥
xA := x A.
∃ ¬∀ ¬
The set FV(A) of variables free in an –formula A is defined as usual:
L
FV(R(t ,...,t )) := vars(t ) ... vars(t ),
1 n 1 n
∪ ∪
FV( ) := .
⊥ ∅
FV(A B) := FV(A) FV(B).
→ ∪
FV(A B) := FV(A) FV(B).
∧ ∪
FV( xA) := FV(A) x .
∀ \{ }
A term t and a formula A are called closed, if FV(t) = or FV(A) = , respectively. We
∅ ∅
write t[x ,...,x ] or A[x ,...,x ] to indicate that all variables free in t or A are in the
1 n 1 n
list x ,...,x .
1 n
1 Typed languages 3
For simplicity we identify terms and formulas which differ only by the names of bound
variables. This makes it possible to define substitution r[tρ/xρ] and A[tρ/xρ] of a term tρ
for a variable xρ in a particularly simple fashion.
For a term r, variable xρ and a term tρ we define the result of substituting tρ for xρ
in r inductively on r:
t, falls x = y;
y[t/x] :=
y, sonst.
(cid:26)
c[t/x] := c.
f(t ,...,t )[t/x] := f(t [t/x],...,t [t/x]),
1 n 1 n
(λyρr)[t/x] := λyρr[t/x], where y = x and y FV(t),
6 6∈
(rs)[t/x] := r[t/x]s[t/x],
t ,t [t/x] := t [t/x],t [t/x] ,
0 1 0 1
h i h i
π (r)[t/x] := π (r[t/x]),
i i
and siminarlyfor a formula A, variablexρ and a term tρ we define the result of substituting
tρ for xρ in A inductively on A:
R(t ,...,t )[t/x] := R(t [t/x],...,t [t/x]),
1 n 1 n
[t/x] := ,
⊥ ⊥
(A B)[t/x] := A[t/x] B[t/x].
→ →
(A B)[t/x] := A[t/x] B[t/x].
∧ ∧
( yA)[t/x] := yA[t/x], where y = x and y FV(t).
∀ ∀ 6 6∈
We now want to formulate a general notion of a structure or model for our typed
languages, called environment model in [3].
= ((D ) ,I ,I ) is called an –structure, if D is a nonempty set for any type ρ,
ρ ρ 0 1 ρ
M L
I is a mapping assigning to any constant symbol cρ an object I (c) D and to any
0 0 ρ
∈ C ∈
function symbol f of functionality (ρ ,...,ρ ) σ a function
1 n
∈ F →
I (f):D D D ,
0 ρ1 ×···× ρn → σ
and I is a mapping assigning to any relation symbol R of functionality (ρ ,...,ρ )
1 1 n
∈ P →
a relation
◦
I (R) D D .
1 ⊆ ρ1 ×···× ρn
In case n = 0 we have I (R) true,false . An –environment model is given by an
1
∈ { } L
–structure = ((D ) ,I ,I ) and in addition
ρ ρ 0 1
L M
(i) bijections Φ :D [D D ] between D and some set [D D ] of
ρ,σ ρ→σ ρ σ ρ→σ ρ σ
→ → →
functions from D to D ,
ρ σ
(ii) bijections Ψ :D D D , and
ρ,σ ρ×σ ρ σ
→ ×
4 1 Typed languages
(iii) a mapping assigning to any term t and any environment U (i.e. mapping from
the set of variables into D such that for each xρ we have U(xρ) D ) an element
ρ ρ ∈ ρ
[[tρ]] D such that the following holds.
U ρ
∈ S
[[xρ]] = U(xρ),
U
[[cρ]] = I (cρ),
U 0
[[f(t ,...,t )]] = I (f)([[t ]] ,...,[[t ]] ),
1 n U 0 1 U n U
[[λxρt]] = Φ−1(f), where f(a) = [[t]] for all a D ,
U ρ,σ U[a/xρ] ∈ ρ
[[ts]] = (Φ ([[t]] ))([[s]] ),
U ρ,σ U U
[[ t ,t ]] = Ψ−1 ([[[t ]] ,[[t ]] ]),
h 0 1i U ρ,σ 0 U 1 U
[[π (t)]] = (Ψ ([[t]] )) for i 0,1 ,
i U ρ,σ U i
∈ { }
Note that for a given –structure = ((D ) ,I ,I ) and bijections Φ and Ψ it is
ρ ρ 0 1 ρ,σ ρ,σ
L M
in general not possible to define [[t]] by the requirements above. The reason is that the
U
function f:D D defined by f(a) = [[t]] may not be in the range [D D ]
ρ σ U[a/xρ] ρ σ
→ →
of Φ , and hence we cannot define [[λxρt]] by Φ−1(f). — A trivial way out of this
ρ,σ U ρ,σ
difficulty is of course to let [D D ] be the set of all functions from D to D . Hence
ρ σ ρ σ
→
we obtain the following trivial
Example of an environment model. Let D for any ground type ι be an arbitrary
ι
nonempty set. Choose D to be the set of all functions from D to D , and D =
ρ→σ ρ σ ρ×σ
D D . Ifwe then takeΦ and Ψ to beidentities, we candefine [[t]] by theequations
ρ σ ρ,σ ρ,σ U
×
above.
There are more interesting examples of environment models, most notably the model
of the partial continuous function due to Scott [27] and Ersov [9]; see also [24] for an
exposition.
We will usually be somewhat sloppy with our notation and leave out Φ and Ψ. So we
write
[[t]] [[s]] instead of (Φ ([[t]] ))([[s]] ),
U U ρ,σ U U
([[t]] ) instead of (Ψ ([[t]] )) ,
U i ρ,σ U i
[[[t ]] ,[[t ]] ] instead of Ψ−1 ([[[t ]] ,[[t ]] ]).
0 U 1 U ρ,σ 0 U 1 U
We often write fM for the interpretation I (f) of a function symbol and RM for the
0
interpretation I (R) of a relation symbol.
1
For any environment model , environment U and formula A we define a model
M
relation = A[U] by induction on A.
M |
= (R(t ,...,t ))[U] [[t ]] ,...,[[t ]] RM.
1 n 1 U n U
M | ⇐⇒ ∈
= [U]
M 6| ⊥
= (A B)[U] if = A[U], then = B[U].
M | → ⇐⇒ M | M |
= (A B)[U] = A[U] and = B[U].
M | ∧ ⇐⇒ M | M |
= ( xA)[U] for all a we have = A[U[a/x]].
M | ∀ ⇐⇒ ∈ |M| M |
1 Typed languages 5
Coincidence Lemma. (i) If U(x) = U′(x) for all x FV(t), then [[t]]U = [[t]]U′.
∈
(ii) If U(x) = U′(x) for all x FV(A), then = A[U] = A[U′].
∈ M | ⇐⇒ M |
Proof by induction on t and A. (cid:3)
Substitution Lemma. (i) [[r[t/x]]]U = [[r]]U′ with U′ = U[[[t]]U/x].
(ii) = (A[t/x])[U] = A[U′] with U′ = U[[[t]] /x].
U
M | ⇐⇒ M |
Proof by induction on r and A, using the Coincidence Lemma. (cid:3)
2 Natural deduction
As our deductive formalism we use the system of natural deduction introduced by Gerhard
Gentzen in [11]. It consists of the following introduction and elimination rules for ,
→ ∧
and .
∀
For any –formula A let countably many assumption variables of type A be given.
L
We use uA, vA, wA to denote assumption variables of type A.
Later we will define substitution d[t/x] of object terms t for object variables x in
derivation terms d. We want to avoid that different assumption variables are identified by
R(t) R(x)
a substitution of object terms, as e.g. in u ,u [t/x]. Therefore we assume that for
h 0 0 i
any two assumption variables uA, uB in a derivation d in case A = B we also have i = j
i j 6 6
(cf. [33], p. 30).
The notions of a derivation term dA in minimal logic and its set FA(dA) of free
assumption variables are defined inductively by
(A) uA is a derivation term with FA(uA) = uA .
{ }
( +) If dB is a derivation term, then
→
(λuAdB)A→B
is a derivation term with FA(λuAdB) = FA(dB) uA .
\{ }
( −) If dA→B and eA are derivation terms, then
→
(dA→BeA)B
is a derivation term with FA(dA→BeA) = FA(dA→B) FA(eA).
∪
( +) If dB and eB are derivation terms, then
∧
dA,eB A∧B
h i
is a derivation term with FA( dA,eB A∧B) := FA(dA) FA(eB).
h i ∪
( −) If dA0∧A1 is a derivation term, then
∧
π (dA0∧A1)Ai
i
is a derivation term with FA(π (dA0∧A1)Ai) := FA(dA0∧A1) (for i 0,1 ).
i
∈ { }
( +) If dA is a derivation term and xρ FV(B) uB FA(dA) , then
∀ 6∈ { | ∈ }
S(λxρdA)∀xρA
2 Natural deduction 7
is a derivation term with FA(λxρdA) = FA(dA).
( −) If d∀xρA is a derivation term and tρ is a term, then
∀
(d∀xρAtρ)A[t/xρ]
is a derivation term with FA(d∀xρAtρ) = FA(d∀xρA).
It is sometimes useful to display derivation terms in the following graphical fashion.
( +)
→
u:A
|
B
+ u
A B →
→
( −)
→
| |
A B A
→ −
B →
( +)
∧
| |
A A
0 1 +
A A ∧
0 1
∧
( −)
∧i
|
A A
0 ∧ 1 −
A ∧i
i
( +)
∀
|
A
+x
xA∀
∀
provided x does not occur free in any open assumption of the given derivation of A.
( −)
∀
|
xA t
∀ −
A[t/x] ∀