Proof of Lemma 4.1 Let \(\pi ^\prime \) be an \(\infty \) -proof of \(\varGamma \Rightarrow \varDelta , p\) and \(\pi ^{\prime \prime }\) be an \(\infty \) -proof of \(p, \varGamma \Rightarrow \varDelta \) .

We define \(\mathcal R_{p}(\pi ^\prime ,\pi ^{\prime \prime })\) by induction on \(|\pi ^\prime |\) .

If \(|\pi ^\prime |=0\) , then \(\varGamma \Rightarrow \varDelta , p\) is an initial sequent. Suppose that \(\varGamma \Rightarrow \varDelta \) is also an initial sequent. Then \(\mathcal R_{p}(\pi ^\prime ,\pi ^{\prime \prime })\) is defined as the \(\infty \) -proof consisting only of this initial sequent. Otherwise, \(\varGamma \) has the form \(p,\varPhi \) , and \(\pi ^{\prime \prime }\) is an \(\infty \) -proof of \(p,p,\varPhi \Rightarrow \varDelta \) . Applying the nonexpansive mapping \( acl _p\) from Lemma 3.6 , we put \(\mathcal R_{p}(\pi ^\prime ,\pi ^{\prime \prime }) := acl _p (\pi ^{\prime \prime })\) .

Now suppose that \(|\pi ^\prime |>0\) . We consider the last application of an inference rule in \(\pi ^\prime \) .

Case 1. The

\(\infty \) -proof

\(\pi ^\prime \) has the form

where

\(A\rightarrow B,\varSigma = \varDelta \) . Notice that

\(|\pi ^\prime _0 |< |\pi ^\prime |\) . In addition,

\(\pi ^{\prime \prime }\) is an

\(\infty \) -proof of

\(p,\varGamma \Rightarrow A\rightarrow B,\varSigma \) . We define

\(\mathcal R_{p}(\pi ^\prime ,\pi ^{\prime \prime }) \) as

where \( i _{A\rightarrow B}\) is a nonexpansive mapping from Lemma 3.5 .

Case 2. The

\(\infty \) -proof

\(\pi ^\prime \) has the form

where

\(\varSigma , A\rightarrow B = \varGamma \) . We see that

\(|\pi ^\prime _0 |< |\pi ^\prime |\) and

\(|\pi ^\prime _1 |< |\pi ^\prime |\) . Also,

\(\pi ^{\prime \prime }\) is an

\(\infty \) -proof of

\(p,\varSigma , A\rightarrow B\Rightarrow \varDelta \) . We define

\(\mathcal R_{p}(\pi ^\prime ,\pi ^{\prime \prime }) \) as

where \( li _{A\rightarrow B}\) and \( ri _{A\rightarrow B}\) are nonexpansive mappings from Lemma 3.5 .

Case 3. The

\(\infty \) -proof

\(\pi ^\prime \) has the form

where

\(\varSigma , \Box \, A = \varGamma \) . We have that

\(|\pi ^\prime |< |\pi |\) . Define

\(\mathcal R_{p}(\pi ^\prime ,\pi ^{\prime \prime }) \) as

where \( wk _{A, \emptyset }\) is the nonexpansive mapping from Lemma 3.4 .

Case 4. Now consider the final case when

\(\pi ^\prime \) has the form

where

\(\varPhi , \Box \varPi = \varGamma \) and

\(\Box \, A, \varSigma =\varDelta \) . Notice that

\(|\pi ^\prime _0 |< |\pi ^\prime |\) . In addition,

\(\pi ^{\prime \prime }\) is an

\(\infty \) -proof of

\(p,\varPhi , \Box \varPi \Rightarrow \Box \, A, \varSigma \) . We define

\(\mathcal R_{p}(\pi ^\prime ,\pi ^{\prime \prime }) \) as

where \( li _{\, \Box \, A}\) is a nonexpansive mapping from Lemma 3.5 .

The mapping

\(\mathcal {R}_p\) is well defined. It remains to check that

\(\mathcal {R}_p\) is nonexpansive, i.e. for any

\(n\in \mathbb {N}\) and any

\(\pi ^\prime \) ,

\(\pi ^{\prime \prime }\) ,

\(\tau ^\prime \) ,

\(\tau ^{\prime \prime }\) from

\( \mathcal P_0\) $$(\pi ^\prime \sim _n \tau ^\prime \wedge \pi ^{\prime \prime } \sim _n \tau ^{\prime \prime }) \Rightarrow \mathcal {R}_p(\pi ^\prime , \pi ^{\prime \prime }) \sim _n \mathcal {R}_p(\tau ^\prime , \tau ^{\prime \prime })\;. $$

This condition is checked by structural induction on the inductively defined relation

\(\pi ^\prime \sim _n \tau ^\prime \) in a straightforward way. So we omit further details.

\(\square \) Proof of Lemma 4.2 Let \(\pi ^\prime \) be an \(\infty \) -proof of \(\varGamma \Rightarrow \varDelta , \Box B\) and \(\pi ^{\prime \prime }\) be an \(\infty \) -proof of \(\Box B, \varGamma \Rightarrow \varDelta \) .

We define \(\mathcal {R}_{\Box B}(\pi ^\prime ,\pi ^{\prime \prime })\) by induction on \(|\pi ^\prime |+ |\pi ^{\prime \prime } |\) .

If \(|\pi ^\prime |=0\) or \(|\pi ^{\prime \prime } |=0\) , then \(\varGamma \Rightarrow \varDelta \) is an initial sequent. Then \(\mathcal R_{\Box B}(\pi ^\prime ,\pi ^{\prime \prime })\) is defined as the \(\infty \) -proof consisting only of this initial sequent.

The only interesting cases are when the formula \(\Box B\) is the principal formula in both \(\pi ^\prime \) and \(\pi ^{\prime \prime }\) .

So the

\(\infty \) -proof

\(\pi ^\prime \) has the form

The cases for the \(\infty \) -proof \(\pi ^{\prime \prime }\) are the following:

Case 1. The

\(\infty \) -proof

\(\pi ^{\prime \prime }\) has the form

Since that

\(|\pi ^{\prime \prime }_0 |< |\pi ^{\prime \prime } |\) , we can define

\(\mathcal R_{\Box B}(\pi ^\prime ,\pi ^{\prime \prime }) \) as

$$\mathcal R_{B}(\pi ^\prime _0,\mathcal R_{\Box B}( wk_{B,\emptyset } (\pi ^\prime ),\pi ^{\prime \prime }_0)).$$

where

\( wk _{-,-}\) is a nonexpansive mapping from Lemma

3.4 .

Case 2. The

\(\infty \) -proof

\(\pi ^{\prime \prime }\) has the form

Since

\(|\pi ^{\prime \prime }_0 |< |\pi ^{\prime \prime } |\) and the sequent

\(\varPhi ^\prime , \Box \varPi ^\prime \Rightarrow \Box C, \varSigma ^\prime \) , the sequent

\(\varPhi , \Box \varPi \Rightarrow \varSigma \) , and the sequent

\(\varGamma \Rightarrow \varDelta \) are equal, we can define

\(\mathcal R_{\Box B}(\pi ^\prime ,\pi ^{\prime \prime }) \) as

where \( wk _{-,-}\) is a nonexpansive mapping from Lemma 3.4 and \( li _{\, \Box \, A}\) is a nonexpansive mapping from Lemma 3.5 . Since the instance of the rule \(\mathsf {cut}\) is not in the main fragment, this proof is in \(\mathcal P_1\) . \(\square \)