The admissibility of cut (also called cut elimination) is one of the most important metatheorems of an object logic sequent calculus. It is the key theorem for showing the equivalence of natural deduction and the cut-free sequent calculus, which has several corollaries, including the existence of normal forms for every natural deduction term, and the consistency of the logic. This tutorial shows how to prove admissibility of cut for a simple intuitionistic sequent calculus in Twelf. We will make use of lexicographic orderings in the termination argument.

##  Cut

For a sequent calculus Γ C, where Γ holds hypotheses and the proposition C is the conclusion, the statement of cut is as follows: If Γ A and Γ,A C then Γ C.

Often, logicians will include cut in the sequent calculus as a rule:

${ { \Gamma \Rightarrow A \qquad \Gamma, A \Rightarrow C }\over{ \Gamma \Rightarrow C } }{\quad \rm cut }$

They then prove that any proof that uses the cut rule can be transformed into a proof without cut (thus, "cut elimination"). In Twelf, it will be simpler to prove cut as a metatheorem ("admissibility of cut") over a sequent calculus with no cut rule.

##  Sequent calculus

Suppose we have the following sequent calculus:

 ${ { \qquad }\over{ \Gamma, C \Rightarrow C } }{\quad \rm init }$ ${ { \qquad }\over{\Gamma \Rightarrow \top} }{\quad \top\rm R}$ ${ { \Gamma \Rightarrow A \qquad \Gamma \Rightarrow B }\over{ \Gamma \Rightarrow A \land B } }{\quad \land\rm R }$ ${ { \Gamma, A \land B, A, B \Rightarrow C }\over{ \Gamma, A \land B \Rightarrow C } }{\quad \land\rm L }$ ${ { \Gamma, A \Rightarrow B }\over{ \Gamma \Rightarrow A \supset B } }{\quad \supset\rm R }$ ${ { \Gamma, A \supset B \Rightarrow A \qquad \Gamma, A \supset B, B \Rightarrow C}\over{ \Gamma, A \supset B \Rightarrow C } }{\quad \supset\rm L }$

The logic supports only the , ∧, and ⊃ connectives to simplify this tutorial. In LF, we encode the sequent calculus as follows:

prop : type.    %name prop A.
top : prop.
imp : prop -> prop -> prop.
and : prop -> prop -> prop.

hypotheses
hyp : prop -> type.

G |- C
conc : prop -> type.

init : hyp A -> conc A.
topR : conc top.
andL : (hyp A -> hyp B -> conc C) ->
(hyp (and A B) -> conc C).
andR : conc A -> conc B ->
conc (and A B).
impL : conc A -> (hyp B -> conc C) ->
(hyp (imp A B) -> conc C).
impR : (hyp A -> conc B) ->
conc (imp A B).

The sequent is represented as the judgment conc, which is hypothetical in hypotheses hyp. A rule init allows us to use a hypothesis to satisfy a conclusion.

##  Statement and proof of cut

cut : {A: prop}
{D: conc A}
{E: hyp A -> conc C}
{F: conc C}
type.
%mode cut +A +D +E -F.

We will refer to the input derivations as D and E throughout this tutorial and the output as F. Because we will induct on the structure of the type A, we make it an explicit argument.

There are nominally about n² cases to consider in this proof; for each of the n rules that can be at the bottom of D, n rules can be used at the bottom of E. In actuality, two factors reduce the number of cases we'll make: some combinations of rules are incompatible (Twelf's excludes them via unification, so we don't even need to write them down) and some cases can be done schematically by using universal quantification.

###  Initial cuts

If either derivation is an instance of the init rule, we are done:

initD : cut A (init Ha) ([Ha] E Ha) (E Ha).

initE : cut A D ([Ha] init Ha) D.
closed : cut A D ([Ha] E') E'.

If D is the init rule used to conclude conc A from hyp A, then Γ includes hyp A already, so we can satisfy the hypothesis in E by simply using the A that is already around (case initD).

If E is the init rule used to conclude conc C from hyp A (so A = C), then D is a proof of Γ C, so F is just D (case initE).

It's also possible that E is the init rule used to conclude conc C from some other hypothesis C in Γ. In this case, E doesn't use the hypothesis A at all, so E can be strengthened to produce the output derivation F which doesn't depend on A. In fact, we can easily generalize this case to any derivation E that does not use the hypothesis A; this is the closed case. (When we write ([Ha] E'), the variable E'—which is implicitly quantified at the outside of the goal—cannot depend on the lambda-bound variable Ha. This is how we indicate the derivation is closed with respect to its argument.)

###  Principal cuts

The most interesting cases are those where the cut formula A is concluded with a right rule in D, and used with a left rule in E. These are known as principal cuts.

There is no principal cut for because there is no left rule, so the easiest one will be A ∧ B:

andC : cut (and A B) (andR D1 D2) ([Hab : hyp (and A B)] andL ([Ha : hyp A] [Hb : hyp B] E' Hab Ha Hb) Hab) F
<- ({Ha} {Hb} cut (and A B) (andR D1 D2) ([Hab] E' Hab Ha Hb) (F1 Ha Hb))
<- ({Hb} cut A D1 ([Ha] F1 Ha Hb) (F2 Hb))
<- cut B D2 ([Hb] F2 Hb) F.

We identify the case where D is an instance of andR, and E is andL acting on the hypothesis of the cut formula. (The hypothetical derivation E is represented as a LF function taking the hypothesis A ∧ B; we identify the case where this specific hypothesis is used by andL by passing that bound variable (Hab) to the andL constant.) We have a subderivation E' of conc C under hypotheses hyp (and A B), hyp A, and hyp B. We must eliminate each of these to produce F. In the first subgoal we eliminate the and A B hypothesis. D itself is a derivation of conc (and A B) in Γ, so we want to appeal to the IH on D and E'. There are a few things to notice. First, we did not name D (indeed there is no way to do so) because we pattern matched against it as an application of andR; therefore, in the inductive call we have to build up D again as andR D1 D2. Second, E' is not of the correct type for the inductive call, because it is a curried function of three arguments, not one. We therefore must make the subgoal higher order: we hypothesize the existence of Ha and Hb (of type hyp A and hyp B respectively). We then form the term of type hyp (and A B) -> conc C by abstracting the hypothesis we seek to eliminate (Hab) and applying E' to it and the Π-bound Ha and Hb. Because this inductive call is in a context including Ha and Hb, the resulting derivation F1 depends on those two variables as well.

We then want to do the same thing to eliminate the hypotheses for A and B. In the second subgoal, we prove that for all derivations of hyp B (Π-bound variable Hb), we can cut D1 with the result of the previous induction F1 to get F2, which only depends on Hb. Finally, in the third subgoal we eliminate the hyp B hypothesis to produce a derivation of conc C in the ambient Γ with no extra assumptions, which is what we need for F.

We must pay close attention to the induction order. The first inductive call is well-founded because it is on a subderivation E'. The output derivation F1 may be larger, however, and it is used as an input to the second inductive call. Observe, however, that the cut formula A is a subderivation of the original cut formula and A B. Ultimately, our induction order will be lexicographic. Either D or E will get smaller (with the other staying the same size), or they both may get larger but A will get smaller. We will specify this induction measure in the %total declaration when we check the metatheorem below.

impC : cut (imp A B) (impR ([Ha] D' Ha))
([Hab] impL (E1 Hab : conc A) ([Hb] E2 Hab Hb : conc C) Hab)
F
<- cut (imp A B) (impR D') E1 (F1 : conc A)
<- ({Hb:hyp B} cut (imp A B) (impR D') ([Hab] E2 Hab Hb) (F2 Hb : conc C))
<- cut A F1 D' (F3 : conc B)
<- cut B F3 F2 F.

The impC case works in much the same way. We first cut the A ⊃ B hypothesis from both subderivations E1 and E2. We then have F1 : conc A and F2 : hyp B -> conc C. To get F3 : conc B we cut the hyp A from D' : hyp A -> conc B using F1 (note that in this inductive call, derivations from the "D side" and "E side" have switched roles!). Finally, we cut the hyp B from F2 to get the result F.

###  Left-commutative cuts

The remainder of the cases are commutative cases where we search for initial and principal cuts to apply. The first set are the left commutative cases, where the rule used to derive D is a left rule.

andLLC : cut A (andL ([Ha][Hb] D' Ha Hb) Hab) E
(andL ([Ha][Hb] F' Ha Hb) Hab)
<- ({Ha}{Hb} cut A (D' Ha Hb) E (F' Ha Hb)).

impLLC : cut A (impL D1 ([Hb] D2 Hb) Hi) E
(impL D1 F2 Hi)
<- ({Hb} cut A (D2 Hb) E (F2 Hb)).

In the andLLC, D : conc A is an instance of andL. We proceed by hypothesizing the Ha : hyp A1 and Hb : hyp B1 that D' : hyp A1 -> hyp B1 -> conc A depends on, so that we may cut it against E. The resulting derivation F' then also depends on a hyp A1 and hyp B1; we wrap it with an instance of the andL rule. In these commutative cases, we find rules that do not involve the cut formula, work underneath them inductively, and then re-apply the rule to the cut-free proof we obtained. The impLLC case follows the same pattern.

###  Right-commutative cuts

The right commutative cuts are those where the rule at the bottom of E is not a left rule acting on the cut formula. The first two cases are where E is a right rule:

andRRC : cut A D ([Ha] andR (E1 Ha) (E2 Ha)) (andR F1 F2)
<- cut A D E1 F1
<- cut A D E2 F2.

impRRC : cut A D ([Ha] impR ([H1] E1 Ha H1)) (impR ([H1] F1 H1))
<- ({H1} cut A D ([Ha] E1 Ha H1) (F1 H1)).

In the andRRC case, we simply apply straightforward induction. E consists of two subderivations, E1 and E2, each of which may use the cut formula A. We inductively cut out uses of A, and then rebuild the derivation with the impR rule. The impRRC rule is similar.

The final two cases are uses of the left rules, but where they act on a hypothesis different from the cut hypothesis:

andLRC : cut A (D : conc A) ([Ha] andL ([H1] [H2] E' Ha H1 H2) Hp)
(andL ([H1][H2] F' H1 H2) Hp)
<- ({H1}{H2} cut A D ([Ha] E' Ha H1 H2) (F' H1 H2)).

impLRC : cut A (D : conc A) ([Ha] impL (E1 Ha) ([Hb] E2 Ha Hb) Hi)
(impL F1 ([Hb] F2 Hb) Hi)
<- cut A D ([Ha] E1 Ha) F1
<- ({Hb} cut A D ([Ha] E2 Ha Hb) (F2 Hb)).

The andLRC case is similar to the principal cut case andC, but here the andL in E acts on some hypothesis Hp : and A1 B1 instead of the cut hypothesis Ha : hyp A. To proceed, we simply eliminate uses of Ha within E' inductively, and then reapply the andL rule to the resulting derivation. We follow a similar pattern for the impLRC case.

##  The metatheorem

We can now state and check the metatheorem: that for every ground arguments A, D and E to the cut relation, there exists a derivation F that concludes C without using the hypothesis A. Our %worlds declaration for cut is only that hypotheses may appear in the context:

%block hyp : some {A : prop} block {H : hyp A}.
%worlds (hyp) (cut _ _ _ _).

We check the totality of cut using a lexicographic induction order, as discussed above. We write {A [D E]} to mean that either the cut formula A gets smaller, or it stays the same size and one of D or E gets smaller with the other staying the same size.

%total {A [D E]} (cut A D E F).

We proved the admissibility of cut for a small sequent calculus with only a few connectives. Experience shows that this same strategy works for more complex logics. For instance, the case study on Classical S5 includes a cut theorem for a classical modal logic. It also formalizes the translations between natural deduction and the sequent calculus.