Download Simple Type Theory as a Clausal Theory

Simple Type Theory as a Clausal Theory
Gilles Dowek
École polytechnique and INRIA
LIX, École polytechnique, 91128 Palaiseau Cedex, France.
[email protected], http://www.lix.polytechnique.fr/~dowek
1
Introduction
Deduction modulo is an extension of first-order predicate logic where axioms are
replaced by rewrite rules. For instance, the axiom P ⇔ (Q ⇒ R) is replaced
by the rule P −→ (Q ⇒ R). These rules define an equivalence relation and,
in a proof, a proposition can be replaced by an equivalent one at any time. A
presentation of Simple Type Theory in Deduction modulo, called HOL, has been
given in [3].
Polarized deduction modulo [1] is an extension of Deduction modulo where
possibly different rewrite rules apply to the negative and positive occurrences
of atomic propositions. Like any theory expressed in Deduction modulo, Simple
Type Theory can be expressed in Polarized deduction modulo. Each rule just
needs to be duplicated in a negative and a positive instance.
A rewrite system in Polarized deduction modulo is said to be clausal when
negative rules rewrite atomic propositions to clausal propositions and positive
rules rewrite atomic propositions to the negation of clausal propositions. This
way, clauses rewrite to clauses, which is a useful property in automated theorem
proving [2].
We give in this note a presentation of Simple Type Theory as a clausal rewrite
system in Polarized deduction modulo, called HOL± .
This system is does not have the cut elimination property in general but cut
elimination holds for sequents well-formed in the language of HOL and, for such
sequents, provability in HOL± and in HOL are equivalent.
2
Polarized deduction modulo
Definition 1 (Polarized rewrite system). A polarized rewrite system is a
triple R = hE, R− , R+ i where E is a set of equations between terms, R− and
R+ are sets of rewrite rules whose left hand sides are atomic propositions and
right hand sides are arbitrary propositions. The rules of R− are called negative
and those of R+ are called positive.
Definition 2 (Polarized rewriting). Let R = hE, R− , R+ i be a polarized
rewrite system. We define the equivalence relation =E as the congruence on
terms generated by the equations of E. We then define the one step negative and
positive rewriting relations −→− and −→+ as follows.
– If ti =E t′ then P (t1 , . . . , ti , . . . , tn ) −→− P (t1 , . . . , t′ , . . . , tn )
and P (t1 , . . . , ti , . . . , tn ) −→+ P (t1 , . . . , t′ , . . . , tn ).
– If P −→ A is a rule of R− and σ is a substitution then σP −→− σA.
If P −→ A is a rule of R+ and σ is a substitution then σP −→+ σA.
– If A −→+ A′ then ¬A −→− ¬A′ . If A −→− A′ then ¬A −→+ ¬A′ .
– If (A −→− A′ and B = B ′ ) or (A = A′ and B −→− B ′ ), then
A ∧ B −→− A′ ∧ B ′ and A ∨ B −→− A′ ∨ B ′ .
If (A −→+ A′ and B = B ′ ) or (A = A′ and B −→+ B ′ ), then
A ∧ B −→+ A′ ∧ B ′ and A ∨ B −→+ A′ ∨ B ′ .
– If (A −→+ A′ and B = B ′ ) or (A = A′ and B −→− B ′ ), then
A ⇒ B −→− A′ ⇒ B ′ .
If (A −→− A′ and B = B ′ ) or (A = A′ and B −→+ B ′ ), then
A ⇒ B −→+ A′ ⇒ B ′ .
– If A −→− A′ then ∀x A −→− ∀x A′ and ∃x A −→− ∃x A′ .
If A −→+ A′ then ∀x A −→+ ∀x A′ and ∃x A −→+ ∃x A′ .
We define the sequent one step term rewriting relation −→ as follows.
– If A −→− A′ then (Γ, A ⊢ ∆) −→ (Γ, A′ ⊢ ∆).
– If A −→+ A′ then (Γ ⊢ A, ∆) −→ (Γ ⊢ A′ , ∆).
As usual, if R is any binary relation, we write R∗ for its reflexive-transitive
closure. The rules of Polarized sequent calculus modulo are those of Fig. 1. Proof
checking is decidable when the relations −→∗− and −→∗+ are. The usual, non
polarized, Deduction modulo can be recovered by taking R− = R+ and predicate
logic by taking E = R− = R+ = ∅.
A theory is a pair hR, T i formed with a polarized rewrite system R and a set
of axioms T . We say that the sequent Γ ⊢ ∆ is provable in the theory hR, T i,
or that it is provable in T modulo R, if there exists a finite subset T ′ of T such
that the sequent Γ, T ′ ⊢ ∆ is provable in Polarized sequent calculus modulo R.
When T is empty, we simply say that the sequent Γ ⊢ ∆ is provable modulo R.
When R is empty, we say that the sequent Γ ⊢ ∆ is provable in T in predicate
logic.
As discussed in [1], rewriting, in general, has two properties. First, it is oriented and, for instance, the proposition x ∈ P(y) rewrites to ∀z (z ∈ x ⇒ z ∈ y),
but ∀z (z ∈ x ⇒ z ∈ y) does not rewrite to x ∈ P(y). Then, rewriting preserves
provability. For instance, the proposition x ∈ P(y) rewrites to ∀z (z ∈ x ⇒ z ∈ y)
that is provably equivalent. Thus, we can always transform the proposition
x ∈ P(y) into ∀z (z ∈ x ⇒ z ∈ y) and we never need to backtrack on this
operation. When rewriting is polarized, the first property is kept, but not the
second. For instance, if we have the negative rule P −→ Q, the sequent P ⊢ P
can be proved with the axiom rule, but its normal form Q ⊢ P cannot.
Definition 3 (Literal, Clausal proposition). A proposition is a literal if it
is either atomic of the negation of an atomic proposition. A proposition is clausal
if it is ⊥ or of the form ∀x1 . . . ∀xp (L1 ∨ . . . ∨ Ln ) where L1 , . . . , Ln are literals
and x1 , . . . , xp variables.
A⊢B
axiom if A −→∗− P, B −→∗+ P and P atomic
Γ, B ⊢ ∆ Γ ⊢ C, ∆ cut if A −→∗ B, A −→∗ C
−
+
Γ ⊢∆
Γ, B, C ⊢ ∆ contr-left if A −→∗ B, A −→∗ C
−
−
Γ, A ⊢ ∆
Γ ⊢ B, C, ∆ contr-right if A −→∗ B, A −→∗ C
+
+
Γ ⊢ A, ∆
Γ ⊢∆
Γ, A ⊢ ∆ weak-left
Γ ⊢ ∆ weak-right
Γ ⊢ A, ∆
Γ ⊢ A, ∆
Γ, A ⊢ ∆
⊤-right if A −→∗+ ⊤
⊥-left if A −→∗− ⊥
Γ ⊢ B, ∆ ¬-left if A −→∗ ¬B
−
Γ, A ⊢ ∆
Γ, B ⊢ ∆ ¬-right if A −→∗ ¬B
+
Γ ⊢ A, ∆
Γ, B, C ⊢ ∆ ∧-left if A −→∗ (B ∧ C)
−
Γ, A ⊢ ∆
Γ ⊢ B, ∆ Γ ⊢ C, ∆ ∧-right if A −→∗ (B ∧ C)
+
Γ ⊢ A, ∆
Γ, B ⊢ ∆ Γ, C ⊢ ∆ ∨-left if A −→∗ (B ∨ C)
−
Γ, A ⊢ ∆
Γ ⊢ B, C, ∆ ∨-right if A −→∗ (B ∨ C)
+
Γ ⊢ A, ∆
Γ ⊢ B, ∆ Γ, C ⊢ ∆ ⇒-left if A −→∗ (B ⇒ C)
−
Γ, A ⊢ ∆
Γ, B ⊢ C, ∆ ⇒-right if A −→∗ (B ⇒ C)
+
Γ ⊢ A, ∆
Γ, C ⊢ ∆ hx, B, ti ∀-left if A −→∗ ∀x B, (t/x)B −→∗ C
−
−
Γ, A ⊢ ∆
Γ ⊢ B, ∆ hx, Bi ∀-right if A −→∗ ∀x B, x 6∈ F V (Γ ∆)
+
Γ ⊢ A, ∆
Γ, B ⊢ ∆ hx, Bi ∃-left if A −→∗ ∃x B, x 6∈ F V (Γ ∆)
−
Γ, A ⊢ ∆
Γ ⊢ C, ∆ hx, B, ti ∃-right if A −→∗ ∃x B, (t/x)B −→∗ C
+
+
Γ ⊢ A, ∆
Fig. 1. Polarized sequent calculus modulo
Definition 4 (Clausal rewrite system). A rewrite system is clausal if negative rules rewrite atomic propositions to a clausal propositions and positive rules
atomic propositions to negations of clausal propositions.
3
Equivalence
We want to show that rewrite rules build-in axioms, i.e. that for each rewrite
system R, there is a set of axioms UR such that a sequent is provable modulo R
if and only if it is provable in UR in predicate logic. The set of axioms we wish
to consider contains for each equational axiom t = u of E, the universal closure
of the proposition t = u, for each rule P −→ A of R− , the universal closure of
the proposition P ⇒ A, and for each rule P −→ A of R+ , the universal closure
of the proposition A ⇒ P . A problem is that the language we start with need
not contain an equality predicate. Thus, we must first add such a predicate and
the axioms of equality and prove that this extension is conservative.
Definition 5 (Compatibility). Let R be a polarized rewrite system and T and
U be two sets of axioms. The theory hR, T i is compatible with U if
–
–
–
–
if
if
if
if
A −→∗− B in R, then ⊢ A ⇒ B is provable in U in predicate logic,
A −→∗+ B in R, then ⊢ B ⇒ A is provable in U in predicate logic,
A ∈ T , then ⊢ A is provable in U in predicate logic,
A ∈ U, then ⊢ A is provable in T modulo R.
Proposition 1 (Equivalence). Let R be a polarized rewrite system and T and
U be two sets of axioms such that the theory hR, T i is compatible with U, then a
sequent is provable in T modulo R, if and only if it is provable in U in predicate
logic.
Proof. If the sequent Γ ⊢ ∆ is provable in U in predicate logic, there exists a
finite subset U ′ of U such that Γ, U ′ ⊢ ∆ is provable in predicate logic and hence
modulo R. Each Ui in U ′ is provable in T modulo R, thus, for each Ui , there
exists a finite subset Ti′ of T such that Ti′ ⊢ Ui is provable modulo R. Let T ′
be the union of all the Ti ’s. Using the cut rule, we build a proof of Γ, T ′ ⊢ ∆
modulo R. Thus, the sequent Γ ⊢ ∆ is provable in T modulo R.
The converse is a simple induction over proof structure.
Definition 6. Let R be a polarized rewrite system. Let AR be the set of axioms
containing
– for each pair of propositions A and B such that A −→∗− B, the universal
closure of A ⇒ B,
– for each pair of propositions A and B such that A −→∗+ B, the universal
closure of B ⇒ A.
Proposition 2. Let R be a polarized rewrite system and T be a set of axioms.
Then, the theory hR, T i and the set of axioms AR ∪ T are compatible.
Proof. If A −→∗− B in R, then the universal closure of A ⇒ B is an element
of AR . Thus, the sequent ⊢ A ⇒ B is provable in AR ∪ T in predicate logic. If
A −→∗+ B in R, then the universal closure of B ⇒ A is an element of AR . Thus,
the sequent ⊢ B ⇒ A is provable in AR ∪ T in predicate logic. If A ∈ T , then
A ∈ AR ∪ T and thus the sequent ⊢ A is provable in AR ∪ T in predicate logic.
Conversely, if A ∈ T , then the sequent ⊢ A is provable in T modulo R and
if A ∈ AR , then the sequent ⊢ A is provable in T modulo R.
Definition 7 (Model). Let R be a polarized rewrite system and T be a set of
axioms, a model of the theory hR, T i is a model of the set of axioms AR ∪ T .
Proposition 3 (Soundness and completeness). A sequent Γ ⊢ ∆ is provable in T modulo R if and only if valid in all models of hR, T i.
Proof. By Propositions 1 and 2, the sequent Γ ⊢ ∆ is provable in T modulo
R if and only if it is provable in AR ∪ T . By the soundness and completeness
theorem of predicate logic it is provable in AR ∪ T if and only if it is valid in all
models of AR ∪ T , i.e. in all models of hR, T i.
Definition 8 (Equality model). Let R be a polarized rewrite system and T
be a set of axioms. An equality model of hR, T i is a model where if t =E u then
for all φ, JtKφ = JuKφ .
Proposition 4 (Soundness and completeness for equality models). A
sequent Γ ⊢ ∆ is provable in T modulo R if and only if valid in all equality
models of hR, T i.
Proof. All we need to prove is that for each model M of hR, T i we can build
an equality model of hR, T i. Let M be a model of hR, T i. We write MT for
the domain of M of sort T , fˆ for the interpretation of the function symbol
f and P̂ for the interpretation of the predicate symbol P . For each sort T ,
we define the relation ∼T on the elements of MT , by a ∼T b if and only if
there exists two terms t and u of sort T and a valuation φ such that t =E u,
a = JtKφ and b = JuKφ . This relation is obviously an equivalence relation and it
is compatible with the interpretation of all the function symbols. To prove that
it is compatible with the denotation of the predicate symbols, we remark that if
t =E u then P (t1 , ..., t, ..., tn ) −→− P (t1 , ..., u, ..., tn ) and P (t1 , ..., t, ..., tn ) −→+
P (t1 , ..., u, ..., tn ), thus the proposition P (t1 , ..., t, ..., tn ) ⇔ P (t1 , ..., u, ..., tn ) is
provable modulo R and thus valid in M. We finally define a model M′ by taking
M′T = MT / ≡T and by interpreting the function symbol f by the function fˆ/ ≡
and the predicate symbol f by the function P̂ / ≡. The propositions valid in the
models M and M′ are the same.
Definition 9. Let L be a language containing an equality predicate in each sort.
Let R be a polarized rewrite system in L. Let UR be the set of axioms containing
– the axioms of equality for L,
– for each equational axiom t = u of E, the universal closure of the proposition
t = u,
– for each rule P −→ A of R− , the universal closure of the proposition P ⇒ A,
– for each rule P −→ A of R+ , the universal closure of the proposition A ⇒ P .
Proposition 5. Let L be a language containing an equality predicate in each
sort. Let Eq be the axioms of equality for L. Let R be a polarized rewrite system
in L. Then, the theory hR, Eqi and the set of axioms UR are compatible.
Proof. It is routine to check that if A −→∗− B in R, then the sequent ⊢ A ⇒ B
is provable in UR in predicate logic, and if A −→∗+ B in R, then the sequent
⊢ B ⇒ A is provable in UR in predicate logic. If A is an axiom of Eq, then it is
an axiom of UR , hence the sequent ⊢ A is provable in UR in predicate logic.
Conversely, we check, considering each of the four cases, that if A ∈ UR , then
the sequent ⊢ A is provable in Eq modulo R.
Proposition 6. Let R be a polarized rewrite system in a language L. Let L′
be the language obtained by adding an equality symbol in each sort of L. Let
Eq be the axioms of equality for L′ . Then, the theory hR, Eqi is a conservative
extension of R, i.e. a sequent Γ ⊢ ∆ of L is provable modulo R if and only if it
is provable in Eq modulo R.
Proof. An equality model of R extends to an equality model of hR, Eqi by interpreting equality by equality.
Remark that this proof would not go through if we did not consider equality
models. Indeed if t =E u, then t = t −→− t = u and if t = u were not valid in
the model, it would not be a model of the proposition t = t ⇒ t = u.
Proposition 7. Let L be a language and R be a polarized rewrite system in L.
Let L′ be the language obtained by adding an equality symbol in each sort of L.
Then, a sequent Γ ⊢ ∆ of L is provable modulo R if and only if it is provable in
UR .
Proof. Let Eq be the axioms of equality for L′ . By Proposition 6, the sequent
Γ ⊢ ∆ is provable modulo R if and only if it is provable in Eq modulo R and by
Propositions 1 and 5 it is provable in Eq modulo R if and only if it is provable
in UR .
4
Simple Type Theory as a clausal rewrite system
A presentation of Simple Type Theory in non polarized deduction modulo has
been given in [3]. To define it in polarized deduction modulo we just duplicate
each rule. We also consider an extension of the system presented in [3] with rules
expressing the existence of a non surjective injection Succ of type ι → ι, that
allow to prove the “axiom” of infinity.
Definition 10 (The theory HOL).
The sorts are simple types, inductively defined by
– ι and o are sorts,
– if T and U are sorts then T → U is a sort.
As usual, we write T1 → . . . → Tn → U for T1 → (. . . → (Tn → U ) . . . ). The
language contains
– for each pair of sorts T, U , a constant KT,U of sort T → U → T ,
– for each triple of sorts T, U, V , a constant ST,U,V of sort (T → U → V ) →
(T → U ) → T → V ,
– a constant ∨˙ or sort o → o → o,
– a constant ¬˙ or sort o → o,
– for each sort T , a constant ∀˙ T of sort (T → o) → o,
– a constant 0 of sort ι, two constants Succ and Pred of sort ι → ι, and a
constant Null of sort ι → o,
– for each pair of sorts T, U , a function symbol αT,U of rank hT → U, T, U i,
– a predicate symbol ε of rank hoi.
As usual, we write (t u) for αT,U (t, u) and (t u1 . . . un ) for (. . . (t u1 ) . . . un ).
The rewrite rules are
(KT,U x y) =E x
(ST,U,V x y z) =E (x z (y z))
(Pred (Succ x)) =E x
ε(x ∨˙ y) −→−
ε(¬˙ x) −→−
ε(∀˙ T x) −→−
ε(Null (S x)) −→−
ε(Null 0) −→−
(ε(x) ∨ ε(y))
¬ε(x)
∀y ε(x y)
⊥
⊤
ε(x ∨˙ y) −→+
ε(¬˙ x) −→+
ε(∀˙ T x) −→+
ε(Null (S x)) −→+
ε(Null 0) −→+
(ε(x) ∨ ε(y))
¬ε(x)
∀y ε(x y)
⊥
⊤
The theory HOL is not clausal. We now define a clausal theory HOL± and
prove it is equivalent to HOL.
Definition 11 (The theory HOL± ). The sorts are the same as those of HOL.
The symbols are the same as those of HOL and, for each sort T , a function
symbol HT of sort (T → o) → T . The rewrite rules are
(KT,U x y) =E x
(ST,U,V x y z) =E (x z (y z))
(Pred (Succ x)) =E x
ε(x ∨˙ y) −→− (ε(x) ∨ ε(y))
ε(¬˙ x) −→− ¬ε(x)
ε(∀˙ T x) −→− ∀y ε(x y)
ε(Null (S x)) −→− ⊥
ε(x ∨˙
ε(x ∨˙
ε(¬˙
ε(∀˙ T
y) −→+
y) −→+
x) −→+
x) −→+
¬¬ε(x)
¬¬ε(y)
¬ε(x)
¬¬ε(x HT (x))
ε(Null 0) −→+ ¬⊥
Proposition 8. If a sequent, containing no occurrence of the symbols HT , has
a proof in HOL± , then it has a proof in HOL.
Proof. Using Proposition 7, all we need to prove is that the theory UHOL± is a
conservative extension of UHOL .
The theories UHOL± and UHOL differ on three points. First, the theory
UHOL± contains the axioms ∀x∀y (¬¬ε(x) ⇒ ε(x ∨˙ y)) and ∀x∀y (¬¬ε(y) ⇒
ε(x ∨˙ y)) while the theory UHOL contains the axiom ∀x∀y ((ε(x) ∨ ε(y)) ⇒
ε(x ∨˙ y)). But the conjunction of the two axioms of UHOL± is equivalent to that
of UHOL .
Second, the theory UHOL contains two axioms ε(Null 0) ⇒ ⊤ and ∀x (⊥ ⇒
ε(Null (S x))). But these axioms are trivially provable in predicate logic and
they can be eliminated.
Third, the theory UHOL± contains the axiom ∀x (¬¬ε(x HT (x)) ⇒ ε(∀˙ T x))
and the axioms of equality for the symbols HT and the theory UHOL the axiom
∀x ((∀y ε(x y)) ⇒ ε(∀˙ T x)). But the axiom of UHOL± is equivalent to the
Skolemization of that of UHOL .
Thus, using Skolem theorem for classical logic with equality, we get that
UHOL± is a conservative extension of UHOL .
Trivially, if a sequent Γ ⊢ ∆, containing no occurrence of the symbols HT ,
has a cut free proof in HOL± , it has a proof in HOL± and thus it has a proof
in HOL. Using the cut elimination theorem for HOL, we get that it has a cut
free proof in HOL. We now want to prove the converse, i.e. that if a sequent,
containing no occurrence of the symbols HT , has a cut free proof in HOL, it has
a cut free proof in HOL± .
Proposition 9. If (Γ ⊢ ∆) −→∗ (Γ ′ ⊢ ∆′ ) and Γ ′ ⊢ ∆′ has a cut free proof
modulo R then Γ ⊢ ∆ has a cut free proof modulo R of the same size.
Proof. By induction over proof structure.
Proposition 10. If a sequent containing no occurrence of the symbols HT , has
a cut free proof in HOL, it has a cut free proof in HOL± .
Proof. Let Γ ⊢ ∆ be a sequent that has a cut free proof in HOL. By induction
on the size of this proof, we build a cut free proof of this sequent in HOL± . We
give only two cases.
– If the proof has the form
π
Γ ⊢ B, C, ∆
∨-right
Γ ⊢ A, ∆
with A −→HOL
(B ∨ C), then either A = (B ′ ∨ C ′ ) or A is atomic.
+
C. By Proposition 9,
B, C ′ −→HOL
In the first case we have B ′ −→HOL
+
+
′
′
the sequent Γ ⊢ B , C , ∆ has a cut free proof of the same size, by induction
hypothesis it has a cut free proof in HOL± and we conclude with the ∨-right
rule.
In the second, consider a reduction sequence from A to B ∨ C and in this
reduction sequence, the last atomic proposition A′ and its successor B ′ ∨ C ′ .
We have A −→HOL∗
A′ −→HOL
(B ′ ∨ C ′ ) −→HOL∗
(B ∨ C). As A′ is
+
+
+
′
˙
atomic and A′ −→HOL
(B
∨
C),
we
have
A
=
ε(t
∨
u),
B ′ = ε(t), and
+
HOL∗
C ′ = ε(u). As (ε(t) ∨ ε(u)) −→HOL∗
(B
∨
C),
we
have
ε(t)
−→
B and
+
+
ε(u) −→HOL∗
C.
By
Proposition
9,
the
sequent
Γ
⊢
ε(t),
ε(u),
∆
has
a cut
+
free proof of the same size in HOL and by induction hypothesis, it has a
cut free proof in HOL± . As A and ε(t ∨˙ u) are atomic and A −→HOL∗
+
±
HOL± ∗
ε(t ∨˙ u), we have A −→+
ε(t ∨˙ u). Then, ε(t ∨˙ u) −→HOL
¬¬ε(t)
+
±
±
±
∗
∗
and ε(t ∨˙ u) −→HOL
¬¬ε(u). Thus, A −→HOL
¬¬ε(t) and A −→HOL
+
+
+
¬¬ε(u). We build a cut free proof of Γ ⊢ A, ∆ in HOL± with the rules
contraction-right, ¬-right, and ¬-left and the proof of Γ ⊢ ε(t), ε(u), ∆.
– If the proof has the form
π
Γ ⊢ B, ∆
∀-right
Γ ⊢ A, ∆
with A −→HOL
∀x B, then either A = ∀x B ′ or A is atomic.
+
In the first case we have B ′ −→HOL
B. By Proposition 9, the sequent Γ ⊢
+
B ′ , ∆ has a cut free proof of the same size, by induction hypothesis it has a
cut free proof in HOL± and we conclude with the ∀-right rule.
In the second, consider a reduction sequence from A to ∀x B and in this
reduction sequence, the last atomic proposition A′ and its successor ∀x B ′ .
We have A −→HOL∗
A′ −→HOL
∀x B ′ −→HOL∗
∀x B. As A′ is atomic and
+
+
+
′
HOL
′
′
˙
A −→+
∀x B, we have A = ε(∀T t) and B = ε(t x). As ∀x ε(t x) −→HOL∗
+
∀x B, we have ε(t x) −→HOL∗
B. By Proposition 9, the sequent Γ ⊢ ε(t x), ∆
+
has a cut free proof of the same size in HOL and by induction hypothesis,
it has a cut free proof in HOL± . By substituting the term HT (t) for the
variable x in this proof, we get a proof of the sequent Γ ⊢ ε(t HT (t))
in HOL± . As A and ε(∀˙ T t) are atomic and A −→HOL∗
ε(∀˙ T t), we
+
±
±
∗
have A −→HOL
ε(∀˙ T t). Then, ε(∀˙ T t) −→HOL
¬¬ε(t HT (t)). Thus,
+
+
HOL± ∗
A −→+
¬¬ε(t HT (t)). We build a cut free proof of Γ ⊢ A, ∆ in HOL±
with the rules ¬-right and ¬-left and the proof of Γ ⊢ ε(t HT (t)), ∆.
Proposition 11. For a sequent containing no occurrence of the symbols HT the
following conditions are equivalent
1.
2.
3.
4.
the sequent has a proof in HOL± ,
it has a proof in HOL,
it has a cut free proof in HOL,
it has a cut free proof in HOL± .
Proof. 1. ⇒ 2. is Proposition 8, 2. ⇒ 3. is the cut elimination for HOL [6, 7, 5]
(see also [4]), 3. ⇒ 4. is Proposition 10, 4. ⇒ 1. is trivial.
Notice that HOL± does not have the cut elimination property in general.
For instance, the sequent ε(x HT (x)) ⊢ ∀y ε(x y) has a proof with a cut (on
ε(∀˙ T x)) but no cut free proof. Yet, for sequents well-formed in the language of
HOL (i.e. containing no symbols HT ), the cut elimination property holds and
provability is equivalent to provability in HOL.
References
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Aspects of Computer Science, Lecture Notes in Computer Science 2285, SpringerVerlag, 2002, pp. 50-64.
2. G. Dowek, Polarized Resolution Modulo, manuscript, 2009.
3. G. Dowek, Th. Hardin, and C. Kirchner, HOL-lambda-sigma: an intentional firstorder expression of higher-order logic, Mathematical Structures in Computer Science, 11, 2001, pp. 1-25.
4. G. Dowek and B. Werner, Proof normalization modulo, The Journal of Symbolic
Logic, 68, 4, 2003, pp. 1289-1316.
5. J.-Y. Girard, Une extension de l’interprétation de Gödel à l’analyse et son application à l’élimination des coupures dans l’analyse et la théorie des types, J.E. Fenstad
(Ed.) Second Scandinavian Logic Symposium, North-Holland, 1970.
6. D. Prawitz. Hauptsatz for higher order logic. The Journal of Symbolic Logic,
33:452–457, 1968.
7. M. o. Takahashi. A proof of cut-elimination theorem in simple type theory. Journal
of the Mathematical Society of Japan, 19:399–410, 1967.