Proof Given a sequence-like derivation $\langle \gamma _i:i<t\rangle $ of $\varphi $ from $\Gamma $ , put $V=[t]$ . Let I be the set of $i<t$ such that $\gamma _i$ is not an axiom (from $\Gamma $ , or an instance of (A1) or (A2)); for each $i\in I$ , fix $i_0,i_1<i$ such that $\gamma _i$ is derived from $\gamma _{i_0}$ and $\gamma _{i_1}$ by (MP), and let $E=\{\langle i_j,i\rangle :i\in I,j\in \{0,1\}\}$ . Observe that $\langle V,E\rangle $ is acyclic as $E\subseteq {<}\mathbin \restriction [t]$ . Then $\langle V,E,\langle \gamma _i:i<t\rangle \rangle $ is a dag-like $\mathrm {F}_\to $ -derivation of $\varphi $ from $\Gamma $ , possibly after eliminating nodes from which the root $t-1$ is not reachable.

Conversely, let $\langle V,E,\gamma \rangle $ be a dag-like $\mathrm {F}_\to $ -derivation of $\varphi $ from $\Gamma $ , and $t=|V|$ . Since $\langle V,E\rangle $ is acyclic, we can find an enumeration $V=\{v_i:{i<t}\}$ such that $E\subseteq \{\langle v_i,v_j\rangle :i<j\}$ (a “topological ordering” of $\langle V,E\rangle $ ). The root $\varrho \in V$ is the only node without a successor, hence we must have $\varrho =v_{t-1}$ . Then $\langle \gamma _{v_i}:i<t\rangle $ is a sequence-like $\mathrm {F}_\to $ -derivation of $\varphi $ from $\Gamma $ .

Observation A.3. An $\mathrm {NM}_\to $ - or (dag-like) $\mathrm {F}_\to $ -derivation with formula size r and t lines has size at most $rt$ and at least $\max \{r,t\}$ . A derivation with height h has less than $2^{h+1}$ lines.

Proof The first part is obvious. In a dag with in-degree $2$ and root $\varrho $ , there are at most $2^l$ paths of length l ending in $\varrho $ . Thus, if $\varrho $ is reachable from any node in at most h steps, there are at most $\sum _{l\le h}2^l=2^{h+1}-1$ nodes.

Proof Let $\Pi =\langle \gamma _i:i<t\rangle $ be a fixed $\mathrm {F}_\to ^*$ -derivation of $\varphi $ from $\Gamma $ such that all variables occurring in $\Pi $ are among $\{p_i:i<k\}$ . Then for any substitution $\sigma $ , $\langle \sigma (\gamma _i):i<t\rangle $ is an $\mathrm {F}_\to $ -derivation of $\sigma (\varphi )$ from $\sigma (\Gamma )$ with t lines and size at most $\left \lVert \Pi \right \rVert s$ . The argument for $\mathrm {NM}_\to ^*$ is completely analogous.

Instead of applying the argument directly to $\Gamma \vdash \varphi $ , we may apply it to the $\mathbf {IPC}_\to $ tautology $\vdash \Gamma \to \varphi $ . This yields tree-like proofs of $\sigma (\Gamma \to \varphi )$ with $O(1)$ lines and size $O(s)$ , which we can turn into derivations of $\sigma (\varphi )$ from $\sigma (\Gamma )$ by $|\Gamma |$ applications of (MP)/ $\mathrm {({\to }E)}$ ; this ensures that each axiom from $\sigma (\Gamma )$ is used only once.

Proof Let $\Pi =\langle V,E,\gamma \rangle $ be a dag-like $\mathrm {F}_\to $ -derivation of $\varphi $ from $\Gamma $ . Reinterpreting the (MP)-nodes as $\mathrm {({\to }E)}$ -nodes, $\Pi $ becomes an $\mathrm {NM}_\to $ -derivation from $\Gamma $ plus the instances of (A1) and (A2) that appear in $\Pi $ . By Lemma A.4, each of the latter can be replaced by a tree-like $\mathrm {NM}_\to $ -subproof with $O(1)$ lines (thus height $O(1)$ ) and size linear in the size of the axiom instance, yielding an $\mathrm {NM}_\to $ -derivation of $\varphi $ from $\Gamma $ with the stated parameters.

Proof We arrange the implications in a balanced binary tree with n leaves. Formally, we construct for eachFootnote ⁴ $k\le \lceil \log n\rceil $ and $i<n$ such that $2^k\mid i$ a derivation $\Pi ^k_i$ of $\varphi _i\to \varphi _{\min \{i+2^k,n\}}$ by induction on k as follows: $\Pi ^0_i$ is the trivial derivation of $\varphi _i\to \varphi _{i+1}$ from itself. Let $k<\lceil \log n\rceil $ and $i<n$ be such that $2^{k+1}\mid i$ . If $i+2^k\ge n$ , we put $\Pi ^{k+1}_i=\Pi ^k_i$ ; otherwise, we combine $\Pi ^k_i$ and $\Pi ^k_{i+2^k}$ to $\Pi ^{k+1}_i$ using an instance of the schematic rule

$$\begin{align*}\alpha\to\beta,\beta\to\gamma\vdash\alpha\to\beta,\end{align*}$$

i.e., an $\mathrm {F}_\to ^*$ -derivation of $\varphi _i\to \varphi _{\min \{i+2^{k+1},n\}}$ from $\varphi _i\to \varphi _{i+2^k}$ and $\varphi _{i+2^k}\to \varphi _{\min \{i+2^{k+1},n\}}$ with $O(1)$ lines and size $O(r)$ that uses each assumption only once, which exists by Lemma A.4.

Then $\Pi ^{\lceil \log n\rceil }_0$ is the desired derivation of $\varphi _0\to \varphi _n$ .

Proof We may assume $\Gamma =\langle \alpha _i\rangle _{i<n}$ and $\Delta =\langle \alpha _i\rangle _{i\in I}$ , $I\subseteq [n]$ . For each $i\le n$ , let $\varphi _i$ denote the formula $(\Delta \mathbin \restriction [i]\to \varphi )\to (\Gamma \mathbin \restriction [i]\to \varphi )$ . Then $\varphi _i\to \varphi _{i+1}$ is an instance of one of the schemata

$$ \begin{align*} (\delta\to\gamma)&\to(\delta\to\alpha\to\gamma),\\ (\delta\to\gamma)&\to((\alpha\to\delta)\to\alpha\to\gamma) \end{align*} $$

with $\delta =(\Delta \mathbin \restriction [i]\to \varphi )$ , $\gamma =(\Gamma \mathbin \restriction [i]\to \varphi )$ , and $\alpha =\alpha _i$ , depending on whether $i\in I$ . Thus, it has an $\mathrm {F}_\to ^*$ -proof with $O(1)$ lines and size $O(s)$ by Lemma A.4. Using Lemma A.7, we can combine these proofs to a proof of $\varphi _0\to \varphi _n$ with $O(n)$ lines, height $O(\log n)$ , formula size $O(s)$ , and size $O(sn)$ . Since $\varphi _n$ is (A.1), it remains to detach the $\mathbf {IPC}_\to $ tautology $\varphi _0=(\varphi \to \varphi )$ , which has a proof with $O(1)$ lines and size $O(|\varphi |)$ .

Proof We prove (A.2) using the same strategy as in Lemma A.8: putting

$$\begin{align*}\varphi_i=(\Gamma\mathbin\restriction[i]\to\varphi\to\psi)\to(\Gamma\mathbin\restriction[i]\to\varphi)\to(\Gamma\mathbin\restriction[i]\to\psi)\end{align*}$$

for each $i\le |\Gamma |$ , $\varphi _i\to \varphi _{i+1}$ has a proof with $O(1)$ lines and size $O(s)$ as it is an instance of the schema

$$\begin{align*}(\beta\to\gamma\to\delta)\to((\alpha\to\beta)\to(\alpha\to\gamma)\to(\alpha\to\delta)).\end{align*}$$

These proofs combine to a proof of $\varphi _0\to \varphi _{|\Gamma |}$ with the stated parameters using Lemma A.7. Then $\varphi _{|\Gamma |}$ is (A.2), and $\varphi _0=(\varphi \to \psi )\to (\varphi \to \psi )$ has a short proof.

For (A.3), we put $\varphi _i=((\Gamma \to \varphi )\to \Gamma \mathbin \restriction [i]\to \varphi )\to \Gamma \mathbin \restriction [i]\to (\Gamma \to \varphi )\to \varphi $ . Then $\varphi _0$ is an instance of $\alpha \to \alpha $ , and $\varphi _i\to \varphi _{i+1}$ is an instance of

$$\begin{align*}((\alpha\to\beta)\to\delta)\to(\alpha\to\gamma\to\beta)\to\gamma\to\delta\end{align*}$$

(with $\alpha =(\Gamma \to \varphi )$ , $\beta =(\Gamma \mathbin \restriction [i]\to \varphi )$ , $\gamma =\gamma _i$ , and

$$\begin{align*}\delta=(\Gamma\mathbin\restriction[i]\to(\Gamma\to\varphi)\to\varphi),\end{align*}$$

where $\Gamma =\langle \gamma _i\rangle _{i<|\Gamma |}$ ). Thus, using Lemmas A.4 and A.7, we obtain an $\mathrm {F}_\to ^*$ -proof of $\varphi _{|\Gamma |}$ with $O(n)$ lines, height $O(\log n)$ , formula size $O(s)$ , and size $O(sn)$ . Detaching the premise $(\Gamma \to \varphi )\to \Gamma \to \varphi $ of $\varphi _{|\Gamma |}$ yields (A.3).

(A.4) follows by (MP) from (A.3) and (A.2).

(A.5): We have $(\Gamma \to \Delta \to \varphi )\to (\Delta \to \Gamma \to \Delta \to \Gamma \to \varphi )$ from (A.1), and $(\Delta \to \Gamma \to \Delta \to \Gamma \to \varphi )\to (\Delta \to \Gamma \to \varphi )$ from (A.4), thus we obtain (A.5) when $\Theta =\varnothing $ . The general case follows by applying (A.2).

Proof Let $\Pi =\langle V,E,\gamma \rangle $ be an $\mathrm {NM}_\to $ -derivation of $\varphi $ from $\Gamma $ . Let $\langle \gamma ^{\prime }_i\rangle _{i<t'}$ , $t'\le t$ , be an injective enumeration of the set $\{\gamma _v:v\in V\}$ , and for each $v\in V$ , let $A^{\prime }_v$ denote the sequence $\langle \gamma ^{\prime }_i:i<t',\gamma ^{\prime }_i\in A_v\smallsetminus \Gamma \rangle $ ; notice that $\lVert A^{\prime }_v\rVert \le s$ . We consider the collection of $\mathbf {IPC}_\to $ tautologies $\langle A^{\prime }_v\to \gamma _v:v\in V\rangle $ , and complete it to a valid $\mathrm {F}_\to $ -derivation from $\Gamma $ using Lemmas A.8 and A.9.

In more detail, for every $v\in V$ , we construct an $\mathrm {F}_\to ^*$ -derivation $\Pi _v$ of $A^{\prime }_v\to \gamma _v$ from $\{A^{\prime }_u\to \gamma _u:\langle u,v\rangle \in E\}\cup \Gamma $ with $O(t)$ lines, height $O(\log t)$ , and formula size $O(s)$ . Moreover, each assumption $A^{\prime }_u\to \gamma _u$ is used only once, and the path from it to the conclusion has length $O(1)$ ; both of these properties are obtained by constructing a derivation of $\langle A^{\prime }_u\to \gamma _u\rangle _{\langle u,v\rangle \in E}\to A^{\prime }_v\to \gamma _v$ from $\Gamma $ and applying (MP):

• • If v is a leaf, then either $\gamma _v\in \Gamma $ and $A^{\prime }_v=\varnothing $ , in which case we take the trivial derivation of $\gamma _v$ from itself, or $A^{\prime }_v=\langle \gamma _v\rangle $ , in which case we find an $\mathrm {F}_\to ^*$ -proof of $\gamma _v\to \gamma _v$ with $O(1)$ lines and size $O(s)$ by Lemma A.4.

• • If v is an $\mathrm {({\to }I)}$ -node with premise u, we have $\gamma _v=(\alpha \to \beta )$ and $\gamma _u=\beta $ for some $\alpha $ and $\beta $ , and $A^{\prime }_v=A^{\prime }_u\smallsetminus \{\alpha \}$ as a set. If $\alpha \in A^{\prime }_u$ , then $(A^{\prime }_u\to \beta )\to (A^{\prime }_v\to \alpha \to \beta )$ is an instance of (A.5), otherwise it is an instance of (A.1).

• • If v is an $\mathrm {({\to }E)}$ -node with premises $u_0$ and $u_1$ , then $\gamma _{u_0}=\alpha $ , $\gamma _{u_1}=(\alpha \to \beta )$ , and $\gamma _v=\beta $ for some $\alpha $ and $\beta $ . We have $A^{\prime }_{u_i}\subseteq A^{\prime }_v$ , hence (A.1) gives $\mathrm {F}_\to ^*$ -proofs of $(A^{\prime }_{u_0}\to \alpha )\to A^{\prime }_v\to \alpha $ and $(A^{\prime }_{u_1}\to \alpha \to \beta )\to A^{\prime }_v\to \alpha \to \beta $ . We infer $(A^{\prime }_{u_1}\to \alpha \to \beta )\to (A^{\prime }_{u_0}\to \alpha )\to A^{\prime }_v\to \beta $ using the instance $(A^{\prime }_v\to \alpha \to \beta )\to (A^{\prime }_v\to \alpha )\to A^{\prime }_v\to \beta $ of (A.2) and $O(1)$ additional proof lines by Lemma A.4.

Combining these derivations $\Pi _v$ along the shape of the original derivation $\Pi $ yields an $\mathrm {F}_\to $ -derivation (tree-like if $\Pi $ is tree-like) of $\varphi $ from $\Gamma $ with $O(t^2)$ lines, height $O(h+\log t)=O(h)$ (cf. Observation A.3), formula size $O(s)$ , and size $O(st^2)$ as promised.

Proof Observe that the inductive definition introduces $\land ^{\varphi }_{\vphantom l}$ only when the sequences on both sides are nonempty. Thus, $\mathop {\bigwedge \nolimits ^{\varphi }_{}\Gamma }$ is a binary tree of $\land ^{\varphi }_{\vphantom l}$ with n leaves where every inner node splits, thus there are $n-1$ inner nodes. Since $\alpha $ and $\beta $ occur only once in $\alpha \land ^{\varphi }_{\vphantom l}\beta $ , each node of the tree gives rise to only one subformula of $\mathop {\bigwedge \nolimits ^{\varphi }_{}\Gamma }$ ; thus, $\mathop {\bigwedge \nolimits ^{\varphi }_{}\Gamma }$ consists of one occurrence of each $\alpha _i$ of total size $\lVert \Gamma \rVert $ , and $O(1)$ occurrences of $\varphi $ and $\to $ per each node of the tree of total size $O\big (\left |\varphi \right | n\big )$ .

Proof We construct the proofs by induction on $\lceil \log m\rceil $ . If $m=2$ or $n=1$ , then (A.6) has a proof with $O(1)$ lines and size $O(s+|\varphi |)$ by Lemma A.4. If $I\subseteq [2^k,2^{k+1})$ for some k, we can just apply the induction hypothesis (without changing the proof) to $\Gamma '=\langle \alpha _{2^k+i}:i\in I-2^k\rangle $ and $\Gamma ^{\prime }_u=\Gamma '\mathbin \restriction (I_u-2^k)$ , as $\mathop {\bigwedge \nolimits ^{\varphi }_{}\Gamma _u}=\mathop {\bigwedge \nolimits ^{\varphi }_{}\Gamma ^{\prime }_u}$ .

Assume that $I\subseteq [2^{k+1}]$ and $I^0,I^1\ne \varnothing $ , where $I^0=I\cap [2^k]$ and $I^1=I-2^k$ . For each $u<3$ , put $I^0_u=I_u\cap [2^k]$ and $I^1_u=I_u-2^k$ . Let $\Gamma ^0=\Gamma \mathbin \restriction I^0$ , $\Gamma ^1=\langle \alpha _{2^k+i}:i\in I^1\rangle $ , and $\Gamma ^v_u=\Gamma ^v\mathbin \restriction I^v_u$ for each $v<2$ , $u<3$ . There are proofs of

(A.7) $$ \begin{align} \mathop{\bigwedge\nolimits^{\varphi}\Gamma_u}\to\mathop{\bigwedge\nolimits^{\varphi}\Gamma^v_u},\qquad\mathop{\bigwedge\nolimits^{\varphi}\Gamma^0_u}\to\mathop{\bigwedge\nolimits^{\varphi}\Gamma^1_u}\to\mathop{\bigwedge\nolimits^{\varphi}\Gamma_u} \end{align} $$

with $O(1)$ lines and size $O(s+\left |\varphi \right | n)$ using Lemma A.4: if $I^v_u=\varnothing $ for some $v<2$ , then $\mathop {\bigwedge \nolimits ^{\varphi }_{}\Gamma _u}=\mathop {\bigwedge \nolimits ^{\varphi }_{}\Gamma ^{1-v}_u}$ and $\mathop {\bigwedge \nolimits ^{\varphi }_{}\Gamma ^v_u}=\top $ ; otherwise, $\mathop {\bigwedge \nolimits ^{\varphi }_{}\Gamma _u}$ is $\big (\mathop {\bigwedge \nolimits ^{\varphi }_{}\Gamma ^0_u}\big )\land ^{\varphi }_{\vphantom l}\big (\mathop {\bigwedge \nolimits ^{\varphi }_{}\Gamma ^1_u}\big )$ , thus (A.7) are instances of the valid schemata $(\alpha ^0\to \varphi )\land ^{\varphi }_{\vphantom l}(\alpha ^1\to \varphi )\to (\alpha ^v\to \varphi )$ and $\alpha \to \beta \to \alpha \land ^{\varphi }_{\vphantom l}\beta $ (observe that each $\mathop {\bigwedge \nolimits ^{\varphi }_{}\Gamma ^v_u}$ is of the form $\alpha \to \varphi $ for some formula $\alpha $ ).

Using (A.7), we can construct proofs of

(A.8) $$ \begin{align} \Big(\mathop{\bigwedge\nolimits^{\varphi}\Gamma^0_0}\to\mathop{\bigwedge\nolimits^{\varphi}\Gamma^0_1}\to\mathop{\bigwedge\nolimits^{\varphi}\Gamma^0_2}\Big)&\to \Big(\mathop{\bigwedge\nolimits^{\varphi}\Gamma^1_0}\to\mathop{\bigwedge\nolimits^{\varphi}\Gamma^1_1}\to\mathop{\bigwedge\nolimits^{\varphi}\Gamma^1_2}\Big)\nonumber\\&\to \Big(\mathop{\bigwedge\nolimits^{\varphi}\Gamma_0}\to\mathop{\bigwedge\nolimits^{\varphi}\Gamma_1}\to\mathop{\bigwedge\nolimits^{\varphi}\Gamma_2}\Big) \end{align} $$

with $O(1)$ lines and size $O(s+\left |\varphi \right | n)$ . The induction hypothesis for $\Gamma ^0$ and $\Gamma ^1$ gives us proofs of $\mathop {\bigwedge \nolimits ^{\varphi }_{}\Gamma ^v_0}\to \mathop {\bigwedge \nolimits ^{\varphi }_{}\Gamma ^v_1}\to \mathop {\bigwedge \nolimits ^{\varphi }_{}\Gamma ^v_2}$ for $v<2$ , and these together yield (A.6).

We can imagine the resulting proof as a binary tree of (A.8) inferences. Since each application of (A.8) corresponds to splitting I to two nonempty disjoint subsets, each inner node has two children, and the tree has at most n leaves. Thus, the proof has $O(n)$ lines. Each application of (A.8) also strictly decreases $\lceil \log m\rceil $ , hence the height of the proof is $O(\log m)$ . The formula size is $O(s+\left |\varphi \right | n)$ using Lemma A.12.

As for the size of the proof, the root of the tree contributes $O(s+\left |\varphi \right | n)$ . Its two children contribute $O(s_0+\left |\varphi \right | n_0)$ and $O(s_1+\left |\varphi \right | n_1)$ , where $s_0+s_1=s$ and $n_0+n_1=n$ , thus $O(s+\left |\varphi \right | n)$ together. Continuing the same way, each level of the tree consists of inferences of size $O(s+\left |\varphi \right | n)$ , and there are at most $O(\log m)$ levels, hence the total size is $O\big ((s+\left |\varphi \right | n)\log m\big )$ . (More formally, we can prove such a bound by induction on $\lceil \log m\rceil $ .)

Proof We construct the proofs by induction on $\lceil \log m\rceil $ , similarly to Lemma A.13. If $m=2$ or $n=1$ , then (A.9) has a proof with $O(1)$ lines and size $O(s+|\varphi |+|\psi |)$ by Lemma A.4. If $I\subseteq [2^k,2^{k+1})$ for some k, we can apply the induction hypothesis to $\Gamma '=\langle \alpha _{2^k+i}:i\in I-2^k\rangle $ without changing the proof. If $I\subseteq [2^{k+1}]$ and $I^0,I^1\ne \varnothing $ , where $I^0=I\cap [2^k]$ and $I^1=I-2^k$ , the induction hypothesis applied to $\Gamma ^0=\Gamma \mathbin \restriction I^0$ and $\Gamma ^1=\langle \alpha _{2^k+i}:i\in I^1\rangle $ gives proofs of

$$\begin{align*}\mathop{\bigwedge\nolimits^{\varphi}\Gamma^0}\to\Big(\mathop{\bigwedge\nolimits^{{\psi}}\Gamma^0}\Big)^\varphi,\qquad \mathop{\bigwedge\nolimits^{\varphi}\Gamma^1}\to\Big(\mathop{\bigwedge\nolimits^{{\psi}}\Gamma^1}\Big)^\varphi.\end{align*}$$

These yield (A.9) using an instance of the schema

$$\begin{align*}(\alpha\to\beta^\varphi)\to(\gamma\to\delta^\varphi)\to\alpha\land^{\varphi}_{\vphantom l}\gamma\to(\beta\land^{{\psi}}_{\vphantom l}\delta)^\varphi\end{align*}$$

(we invite the reader to check this is indeed a valid schema).

The resulting proof has the stated size parameters by the same argument as in Lemma A.13.

Proof We use the same notation and argument structure as in the proof of Theorem A.10, but we work with the formulas $\delta _v=\mathop {\bigwedge \nolimits ^{{\gamma _v}}_{}A^{\prime }_v}\to \gamma _v$ in place of $A^{\prime }_v\to \gamma _v$ . Observe $|\delta _v|=O\big (s+\left |{\gamma _v}\right |t\big )=O(rt)$ , where $s=\lVert \Pi \rVert $ .

For each $v\in V$ , we construct an $\mathrm {F}_\to ^*$ -derivation $\Pi _v$ of $\langle \delta _u\rangle _{\langle u,v\rangle \in E}\to \delta _v$ from $\Gamma $ with $O(t)$ lines, height $O(\log t)$ , formula size $O(s+s_vt)=O(rt)$ , and size $O\big ((s+s_vt)\log t\big )$ , where $s_v=|\gamma _v|+\sum _{\langle u,v\rangle \in E}|\gamma _u|$ :

• • The case of v being a leaf is straightforward.

• • If v is an $\mathrm {({\to }I)}$ -node with premise u, we have $\gamma _v=(\alpha \to \beta )$ and $\gamma _u=\beta $ for some $\alpha $ and $\beta $ , and $A^{\prime }_u\subseteq A^{\prime }_v\cup \{\alpha \}$ as a set. Lemma A.13 gives a proof of $\mathop {\bigwedge \nolimits ^{{\beta }}_{}A^{\prime }_v}\to \alpha ^\beta \to \mathop {\bigwedge \nolimits ^{{\beta }}_{}A^{\prime }_u}$ , which (using $\alpha \to \alpha ^\beta $ ) yields $\delta _u\to \mathop {\bigwedge \nolimits ^{{\beta }}_{}A^{\prime }_v}\to \gamma _v$ . Combining this with $\mathop {\bigwedge \nolimits ^{{\gamma _v}}_{}A^{\prime }_v}\to \big (\mathop {\bigwedge \nolimits ^{{\beta }}_{}A^{\prime }_v}\big )^{\gamma _v}$ from Lemma A.14 gives $\delta _u\to \mathop {\bigwedge \nolimits ^{{\gamma _v}}_{}A^{\prime }_v}\to \gamma _v$ , i.e., $\delta _u\to \delta _v$ .

• • If v is an $\mathrm {({\to }E)}$ -node with premises $u_0$ and $u_1$ , then $\gamma _{u_0}=\alpha $ , $\gamma _{u_1}=(\alpha \to \beta )$ , and $\gamma _v=\beta $ for some $\alpha $ and $\beta $ , and $A^{\prime }_{u_i}\subseteq A^{\prime }_v$ . Using Lemmas A.13 and A.14, we obtain proofs of $\delta _{u_0}\to \mathop {\bigwedge \nolimits ^{{\beta }}_{}A^{\prime }_v}\to \alpha ^\beta $ and $\delta _{u_1}\to \mathop {\bigwedge \nolimits ^{{\beta }}_{}A^{\prime }_v}\to (\alpha \to \beta )^\beta $ , which yield $\delta _{u_0}\to \delta _{u_1}\to \mathop {\bigwedge \nolimits ^{{\beta }}_{}A^{\prime }_v}\to \beta $ (i.e., $\delta _{u_0}\to \delta _{u_1}\to \delta _v$ ) using the schema $\alpha ^\beta \to (\alpha \to \beta )^\beta \to \beta $ .

Combining the $\Pi _v$ derivations yields an $\mathrm {F}_\to $ -derivation (tree-like if $\Pi $ is tree-like) of $\varphi $ from $\Gamma $ with $O(t^2)$ lines, height $O(h)$ , formula size $O(rt)$ , and size $O\big (st\log t+\sum _vs_vt\log t)=O(\tilde st\log t)$ .