translk.tex

\allowdisplaybreaks

\chapter{Translations of classical logic}\label{translations-of-classical-logic}

\section{\texorpdfstring{T-translation \(A\imp B \mapsto \oc{\wn{A}}\limp\wn{B}\)}{T-translation A\textbackslash{}imp B \textbackslash{}mapsto \textbackslash{}oc\{\textbackslash{}wn\{A\}\}\textbackslash{}limp\textbackslash{}wn\{B\}}}\label{t-translation-aimp-b-mapsto-ocwnalimpwnb}

Formulas are translated as:
\begin{align*}
X^T & = X \\
(A\imp B)^T & = \oc{\wn{A^T}}\limp\wn{B^T} \\
(A\wedge B)^T & = \wn{A^T} \with \wn{B^T} \\
T^T & = \top \\
(A\vee B)^T & = \wn{A^T}\parr\wn{B^T} \\
F^T & = \bot \\
(\neg A)^T & = \wn{\oc{(A^T)\orth}} \\
(\forall\xi A)^T & = \forall\xi \wn{A^T} \\
(\exists\xi A)^T & = \exists\xi \oc{\wn{A^T}}
\end{align*}

This is extended to sequents by
\((\Gamma\vdash\Delta)^T = \oc{\wn{\Gamma^T}}\vdash\wn{\Delta^T}\).

This allows one to translate the rules of classical logic into linear logic:
\begin{gather*}
\LabelRule{\rulename{ax}}
\NulRule{A\vdash A}
\DisplayProof
\qquad\mapsto\qquad
\LabelRule{\rulename{ax}}
\NulRule{\wn{A^T}\vdash\wn{A^T}}
\LabelRule{\oc L}
\UnaRule{\oc{\wn{A^T}}\vdash\wn{A^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash A,\Delta}
\AxRule{\Gamma',A\vdash\Delta'}
\LabelRule{\rulename{cut}}
\BinRule{\Gamma,\Gamma'\vdash\Delta,\Delta'}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\oc{\wn{\Gamma^T}}\vdash\wn{A^T},\wn{\Delta^T}}
\LabelRule{\oc R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash\oc{\wn{A^T}},\wn{\Delta^T}}
\AxRule{\oc{\wn{\Gamma'^T}},\oc{\wn{A^T}}\vdash\wn{\Delta'^T}}
\LabelRule{\rulename{cut}}
\BinRule{\oc{\wn{\Gamma^T}},\oc{\wn{\Gamma'^T}}\vdash\wn{\Delta^T},\wn{\Delta'^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma,A,A\vdash\Delta}
\LabelRule{c L}
\UnaRule{\Gamma,A\vdash\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}},\oc{\wn{A^T}}\vdash\wn{\Delta^T}}
\LabelRule{\oc c L}
\UnaRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}}\vdash\wn{\Delta^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash A,A,\Delta}
\LabelRule{c R}
\UnaRule{\Gamma\vdash A,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\oc{\wn{\Gamma^T}}\vdash\wn{A^T},\wn{A^T},\wn{\Delta^T}}
\LabelRule{\wn c R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash\wn{A^T},\wn{\Delta^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash\Delta}
\LabelRule{w L}
\UnaRule{\Gamma,A\vdash\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\oc{\wn{\Gamma^T}}\vdash\wn{\Delta^T}}
\LabelRule{\oc w L}
\UnaRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}}\vdash\wn{\Delta^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash\Delta}
\LabelRule{w R}
\UnaRule{\Gamma\vdash A,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\oc{\wn{\Gamma^T}}\vdash\wn{\Delta^T}}
\LabelRule{\wn w R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash\wn{A^T},\wn{\Delta^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma,A\vdash B,\Delta}
\LabelRule{\imp R}
\UnaRule{\Gamma\vdash A\imp B,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}}\vdash\wn{B^T},\wn{\Delta^T}}
\LabelRule{\limp R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash \oc{\wn{A^T}}\limp\wn{B^T},\wn{\Delta^T}}
\LabelRule{\wn d R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{(\oc{\wn{A^T}}\limp\wn{B^T})},\wn{\Delta^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash A,\Delta}
\AxRule{\Gamma',B\vdash\Delta'}
\LabelRule{\imp L}
\BinRule{\Gamma,\Gamma',A\imp B\vdash\Delta,\Delta'}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\oc{\wn{\Gamma^T}}\vdash\wn{A^T},\wn{\Delta^T}}
\LabelRule{\oc R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash\oc{\wn{A^T}},\wn{\Delta^T}}
\LabelRule{\rulename{ax}}
\NulRule{\wn{B^T}\vdash\wn{B^T}}
\LabelRule{\limp L}
\BinRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}}\limp\wn{B^T}\vdash\wn{B^T},\wn{\Delta^T}}
\LabelRule{\wn L}
\UnaRule{\oc{\wn{\Gamma^T}},\wn{(\oc{\wn{A^T}}\limp\wn{B^T})}\vdash\wn{B^T},\wn{\Delta^T}}
\LabelRule{\oc d L}
\UnaRule{\oc{\wn{\Gamma^T}},\oc{\wn{(\oc{\wn{A^T}}\limp\wn{B^T})}}\vdash\wn{B^T},\wn{\Delta^T}}
\LabelRule{\oc R}
\UnaRule{\oc{\wn{\Gamma^T}},\oc{\wn{(\oc{\wn{A^T}}\limp\wn{B^T})}}\vdash\oc{\wn{B^T}},\wn{\Delta^T}}
\AxRule{\oc{\wn{\Gamma'^T}},\oc{\wn{B^T}}\vdash\wn{\Delta'^T}}
\LabelRule{\rulename{cut}}
\BinRule{\oc{\wn{\Gamma^T}},\oc{\wn{\Gamma'^T}},\oc{\wn{(\oc{\wn{A^T}}\limp\wn{B^T})}}\vdash\wn{\Delta^T},\wn{\Delta'^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash A,\Delta}
\AxRule{\Gamma\vdash B,\Delta}
\LabelRule{\wedge R}
\BinRule{\Gamma\vdash A\wedge B,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\oc{\wn{\Gamma^T}}\vdash \wn{A^T},\wn{\Delta^T}}
\AxRule{\oc{\wn{\Gamma^T}}\vdash \wn{B^T},\wn{\Delta^T}}
\LabelRule{\with R}
\BinRule{\oc{\wn{\Gamma^T}}\vdash \wn{A^T}\with \wn{B^T},\wn{\Delta^T}}
\LabelRule{\wn d R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{(\wn{A^T}\with \wn{B^T})},\wn{\Delta^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma,A\vdash \Delta}
\LabelRule{\wedge_1 L}
\UnaRule{\Gamma,A\wedge B\vdash \Delta}
\DisplayProof
\qquad\mapsto\qquad
\LabelRule{\rulename{ax}}
\NulRule{\wn{A^T}\vdash \wn{A^T}}
\LabelRule{\with_1 L}
\UnaRule{\wn{A^T}\with \wn{B^T}\vdash \wn{A^T}}
\LabelRule{\wn L}
\UnaRule{\wn{(\wn{A^T}\with \wn{B^T})}\vdash \wn{A^T}}
\LabelRule{\oc d L}
\UnaRule{\oc{\wn{(\wn{A^T}\with \wn{B^T})}}\vdash \wn{A^T}}
\LabelRule{\oc R}
\UnaRule{\oc{\wn{(\wn{A^T}\with \wn{B^T})}}\vdash \oc{\wn{A^T}}}
\AxRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}}\vdash \wn{\Delta^T}}
\LabelRule{\rulename{cut}}
\BinRule{\oc{\wn{\Gamma^T}},\oc{\wn{(\wn{A^T}\with \wn{B^T})}}\vdash \wn{\Delta^T}}
\DisplayProof
\\[2ex]
\LabelRule{T R}
\NulRule{\Gamma\vdash T,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\LabelRule{\top R}
\NulRule{\oc{\wn{\Gamma^T}}\vdash \top,\wn{\Delta^T}}
\LabelRule{\wn d R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{\top},\wn{\Delta^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash A,B,\Delta}
\LabelRule{\vee R}
\UnaRule{\Gamma\vdash A\vee B,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\oc{\wn{\Gamma^T}}\vdash \wn{A^T},\wn{B^T},\wn{\Delta^T}}
\LabelRule{\parr R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{A^T}\parr\wn{B^T},\wn{\Delta^T}}
\LabelRule{\wn d R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{(\wn{A^T}\parr\wn{B^T})},\wn{\Delta^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma,A\vdash \Delta}
\AxRule{\Gamma',B\vdash \Delta'}
\LabelRule{\vee L}
\BinRule{\Gamma,\Gamma',A\vee B\vdash \Delta,\Delta'}
\DisplayProof
\qquad\mapsto\qquad
\LabelRule{\rulename{ax}}
\NulRule{\wn{A^T}\vdash \wn{A^T}}
\LabelRule{\rulename{ax}}
\NulRule{\wn{B^T}\vdash \wn{B^T}}
\LabelRule{\parr L}
\BinRule{\wn{A^T}\parr \wn{B^T}\vdash \wn{A^T},\wn{B^T}}
\LabelRule{\wn L}
\UnaRule{\wn{(\wn{A^T}\parr \wn{B^T})}\vdash \wn{A^T},\wn{B^T}}
\LabelRule{\oc d L}
\UnaRule{\oc{\wn{(\wn{A^T}\parr \wn{B^T})}}\vdash \wn{A^T},\wn{B^T}}
\LabelRule{\oc R}
\UnaRule{\oc{\wn{(\wn{A^T}\parr \wn{B^T})}}\vdash \wn{A^T},\oc{\wn{B^T}}}
\AxRule{\oc{\wn{\Gamma'^T}},\oc{\wn{B^T}}\vdash \wn{\Delta'^T}}
\LabelRule{\rulename{cut}}
\BinRule{\oc{\wn{\Gamma'^T}},\oc{\wn{(\wn{A^T}\parr \wn{B^T})}}\vdash \wn{A^T},\wn{\Delta'^T}}
\LabelRule{\oc R}
\UnaRule{\oc{\wn{\Gamma'^T}},\oc{\wn{(\wn{A^T}\parr \wn{B^T})}}\vdash \oc{\wn{A^T}},\wn{\Delta'^T}}
\AxRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}}\vdash \wn{\Delta^T}}
\LabelRule{\rulename{cut}}
\BinRule{\oc{\wn{\Gamma^T}},\oc{\wn{\Gamma'^T}},\oc{\wn{(\wn{A^T}\parr \wn{B^T})}}\vdash \wn{\Delta^T},\wn{\Delta'^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash \Delta}
\LabelRule{F R}
\UnaRule{\Gamma\vdash F,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\oc{\wn{\Gamma^T}}\vdash \wn{\Delta^T}}
\LabelRule{\bot R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash \bot,\wn{\Delta^T}}
\LabelRule{\wn R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{\bot},\wn{\Delta^T}}
\DisplayProof
\\[2ex]
\LabelRule{F L}
\NulRule{F\vdash {}}
\DisplayProof
\qquad\mapsto\qquad
\LabelRule{\bot L}
\NulRule{\bot\vdash {}}
\LabelRule{\wn L}
\UnaRule{\wn{\bot}\vdash {}}
\LabelRule{\oc d L}
\UnaRule{\oc{\wn{\bot}}\vdash {}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma,A\vdash \Delta}
\LabelRule{\neg R}
\UnaRule{\Gamma\vdash \neg A,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}}\vdash \wn{\Delta^T}}
\LabelRule{(.)\orth R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{\oc{(A^T)\orth}},\wn{\Delta^T}}
\LabelRule{\wn d R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{\wn{\oc{(A^T)\orth}}},\wn{\Delta^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash A,\Delta}
\LabelRule{\neg L}
\UnaRule{\Gamma,\neg A\vdash \Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\oc{\wn{\Gamma^T}}\vdash \wn{A^T},\wn{\Delta^T}}
\LabelRule{(.)\orth L}
\UnaRule{\oc{\wn{\Gamma^T}},\oc{(A^T)\orth}\vdash \wn{\Delta^T}}
\LabelRule{\wn L}
\UnaRule{\oc{\wn{\Gamma^T}},\wn{\oc{(A^T)\orth}}\vdash \wn{\Delta^T}}
\LabelRule{\wn L}
\UnaRule{\oc{\wn{\Gamma^T}},\wn{\wn{\oc{(A^T)\orth}}}\vdash \wn{\Delta^T}}
\LabelRule{\oc d L}
\UnaRule{\oc{\wn{\Gamma^T}},\oc{\wn{\wn{\oc{(A^T)\orth}}}}\vdash \wn{\Delta^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash A,\Delta}
\LabelRule{\forall R}
\UnaRule{\Gamma\vdash \forall\xi A,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\oc{\wn{\Gamma^T}}\vdash \wn{A^T},\wn{\Delta^T}}
\LabelRule{\forall R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash \forall\xi \wn{A^T},\wn{\Delta^T}}
\LabelRule{\wn d R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{\forall\xi \wn{A^T}},\wn{\Delta^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma,A[\tau/\xi]\vdash \Delta}
\LabelRule{\forall L}
\UnaRule{\Gamma,\forall\xi A\vdash \Delta}
\DisplayProof
\qquad\mapsto\qquad
\LabelRule{\rulename{ax}}
\NulRule{\wn{A^T}[\tau^T/\xi]\vdash \wn{A^T}[\tau^T/\xi]}
\LabelRule{\forall L}
\UnaRule{\forall\xi \wn{A^T}\vdash \wn{A^T}[\tau^T/\xi]}
\LabelRule{\wn L}
\UnaRule{\wn{\forall\xi \wn{A^T}}\vdash \wn{A^T}[\tau^T/\xi]}
\LabelRule{\oc d L}
\UnaRule{\oc{\wn{\forall\xi \wn{A^T}}}\vdash \wn{A^T}[\tau^T/\xi]}
\LabelRule{\oc R}
\UnaRule{\oc{\wn{\forall\xi \wn{A^T}}}\vdash \oc{\wn{A^T}}[\tau^T/\xi]}
\AxRule{\oc{\wn{\Gamma^T}},\oc{\wn{(A^T[\tau^T/\xi])}}\vdash \wn{\Delta^T}}
\LabelRule{\rulename{cut}}
\BinRule{\oc{\wn{\Gamma^T}},\oc{\wn{\forall\xi \wn{A^T}}}\vdash \wn{\Delta^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash A[\tau/\xi],\Delta}
\LabelRule{\exists R}
\UnaRule{\Gamma\vdash \exists\xi A,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\oc{\wn{\Gamma^T}}\vdash \wn{A^T[\tau^T/\xi]},\wn{\Delta^T}}
\LabelRule{\oc R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash \oc{\wn{A^T[\tau^T/\xi]}},\wn{\Delta^T}}
\LabelRule{\exists R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash \exists\xi \oc{\wn{A^T}},\wn{\Delta^T}}
\LabelRule{\wn d R}
\UnaRule{\oc{\wn{\Gamma^T}}\vdash \wn{\exists\xi \oc{\wn{A^T}}},\wn{\Delta^T}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma,A\vdash \Delta}
\LabelRule{\exists L}
\UnaRule{\Gamma,\exists\xi A\vdash \Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\oc{\wn{\Gamma^T}},\oc{\wn{A^T}}\vdash \wn{\Delta^T}}
\LabelRule{\exists L}
\UnaRule{\oc{\wn{\Gamma^T}},\exists\xi\oc{\wn{A^T}}\vdash \wn{\Delta^T}}
\LabelRule{\wn L}
\UnaRule{\oc{\wn{\Gamma^T}},\wn{\exists\xi\oc{\wn{A^T}}}\vdash \wn{\Delta^T}}
\LabelRule{\oc d L}
\UnaRule{\oc{\wn{\Gamma^T}},\oc{\wn{\exists\xi\oc{\wn{A^T}}}}\vdash \wn{\Delta^T}}
\DisplayProof
\end{gather*}

\subsection{Alternative presentation}\label{alternative-presentation}

It is also possible to define \(A^{\overline{T}}\) by:
\begin{align*}
X^{\overline{T}} & = \wn{X} \\
(A\imp B)^{\overline{T}} & = \wn{(\oc{A^{\overline{T}}}\limp B^{\overline{T}})} \\
(A\wedge B)^{\overline{T}} & = \wn{(A^{\overline{T}} \with B^{\overline{T}})} \\
T^{\overline{T}} & = \wn{\top} \\
(A\vee B)^{\overline{T}} & = \wn{(A^{\overline{T}}\parr B^{\overline{T}})} \\
F^{\overline{T}} & = \wn{\bot} \\
(\neg A)^{\overline{T}} & = \wn{\wn{(A^{\overline{T}})\orth}} \\
(\forall\xi A)^{\overline{T}} & = \wn{\forall\xi A^{\overline{T}}} \\
(\exists\xi A)^{\overline{T}} & = \wn{\exists\xi \oc{A^{\overline{T}}}}
\end{align*}

If we define
\((\Gamma\vdash\Delta)^{\overline{T}} = \oc{\Gamma^{\overline{T}}}\vdash \Delta^{\overline{T}}\),
we have \((\Gamma\vdash\Delta)^{\overline{T}} = (\Gamma\vdash\Delta)^T\)
and thus we obtain the same translation of proofs.

\section{\texorpdfstring{Q-translation \(A\imp B \mapsto \oc{(A\limp\wn{B})}\)}{Q-translation A\textbackslash{}imp B \textbackslash{}mapsto \textbackslash{}oc\{(A\textbackslash{}limp\textbackslash{}wn\{B\})\}}}\label{q-translation-aimp-b-mapsto-ocalimpwnb}

Formulas are translated as:
\begin{align*}
X^Q & = \oc{X} \\
(A\imp B)^Q & = \oc{(A^Q\limp\wn{B^Q})} \\
(A\wedge B)^Q & = \oc{(A^Q \tens B^Q)} \\
T^Q & = \oc{\one} \\
(A\vee B)^Q & = \oc{(A^Q\plus B^Q)} \\
F^Q & = \oc{\zero} \\
(\neg A)^Q & = \oc{(A^Q)\orth} \\
(\forall\xi A)^Q & = \oc{\forall\xi \wn{A^Q}} \\
(\exists\xi A)^Q & = \oc{\exists\xi A^Q}
\end{align*}

The translation of any formula starts with \(\oc\), we define
\(A^{\underline{Q}}\) such that \(A^Q=\oc{A^{\underline{Q}}}\).

The translation of sequents is
\((\Gamma\vdash\Delta)^Q = \Gamma^Q\vdash\wn{\Delta^Q}\).

This allows one to translate the rules of classical logic into linear logic:
\begin{gather*}
\LabelRule{\rulename{ax}}
\NulRule{A\vdash A}
\DisplayProof
\qquad\mapsto\qquad
\LabelRule{\rulename{ax}}
\NulRule{A^Q\vdash A^Q}
\LabelRule{\wn d R}
\UnaRule{A^Q\vdash \wn{A^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash A,\Delta}
\AxRule{\Gamma',A\vdash \Delta'}
\LabelRule{\rulename{cut}}
\BinRule{\Gamma,\Gamma'\vdash \Delta,\Delta'}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q\vdash \wn{A^Q},\wn{\Delta^Q}}
\AxRule{\Gamma'^Q,A^Q\vdash \wn{\Delta'^Q}}
\LabelRule{\wn L}
\UnaRule{\Gamma'^Q,\wn{A^Q}\vdash \wn{\Delta'^Q}}
\LabelRule{\rulename{cut}}
\BinRule{\Gamma^Q,\Gamma'^Q\vdash \wn{\Delta^Q},\wn{\Delta'^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma,A,A\vdash \Delta}
\LabelRule{c L}
\UnaRule{\Gamma,A\vdash \Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q,A^Q,A^Q\vdash \wn{\Delta^Q}}
\LabelRule{\oc c L}
\UnaRule{\Gamma^Q,A^Q\vdash \wn{\Delta^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash A,A,\Delta}
\LabelRule{c R}
\UnaRule{\Gamma\vdash A,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q\vdash \wn{A^Q},\wn{A^Q},\wn{\Delta^Q}}
\LabelRule{\wn c R}
\UnaRule{\Gamma^Q\vdash \wn{A^Q},\wn{\Delta^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash \Delta}
\LabelRule{w L}
\UnaRule{\Gamma,A\vdash \Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q\vdash \wn{\Delta^Q}}
\LabelRule{\oc w L}
\UnaRule{\Gamma^Q,A^Q\vdash \wn{\Delta^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash \Delta}
\LabelRule{w R}
\UnaRule{\Gamma\vdash A,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q\vdash \wn{\Delta^Q}}
\LabelRule{\wn w R}
\UnaRule{\Gamma^Q\vdash \wn{A^Q},\wn{\Delta^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma,A\vdash B,\Delta}
\LabelRule{\imp R}
\UnaRule{\Gamma\vdash A\imp B,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q,A^Q\vdash \wn{B^Q},\wn{\Delta^Q}}
\LabelRule{\limp R}
\UnaRule{\Gamma^Q\vdash A^Q\limp \wn{B^Q},\wn{\Delta^Q}}
\LabelRule{\oc R}
\UnaRule{\Gamma^Q\vdash \oc{(A^Q\limp \wn{B^Q})},\wn{\Delta^Q}}
\LabelRule{\wn d R}
\UnaRule{\Gamma^Q\vdash \wn{\oc{(A^Q\limp \wn{B^Q})}},\wn{\Delta^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash A,\Delta}
\AxRule{\Gamma',B\vdash \Delta'}
\LabelRule{\imp L}
\BinRule{\Gamma,\Gamma',A\imp B\vdash \Delta,\Delta'}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q\vdash \wn{A^Q},\wn{\Delta^Q}}
\LabelRule{\rulename{ax}}
\NulRule{A^Q\vdash A^Q}
\AxRule{\Gamma'^Q,B^Q\vdash \wn{\Delta'^Q}}
\LabelRule{\wn L}
\UnaRule{\Gamma'^Q,\wn{B^Q}\vdash \wn{\Delta'^Q}}
\LabelRule{\limp L}
\BinRule{\Gamma'^Q,A^Q\limp \wn{B^Q},A^Q\vdash \wn{\Delta'^Q}}
\LabelRule{\oc d L}
\UnaRule{\Gamma'^Q,\oc{(A^Q\limp \wn{B^Q})},A^Q\vdash \wn{\Delta'^Q}}
\LabelRule{\wn L}
\UnaRule{\Gamma'^Q,\oc{(A^Q\limp \wn{B^Q})},\wn{A^Q}\vdash \wn{\Delta'^Q}}
\LabelRule{\rulename{cut}}
\BinRule{\Gamma^Q,\Gamma'^Q,\oc{(A^Q\limp \wn{B^Q})}\vdash \wn{\Delta^Q},\wn{\Delta'^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash A,\Delta}
\AxRule{\Gamma'\vdash B,\Delta'}
\LabelRule{\wedge R}
\BinRule{\Gamma,\Gamma'\vdash A\wedge B,\Delta,\Delta'}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q\vdash \wn{A^Q},\wn{\Delta^Q}}
\AxRule{\Gamma'^Q\vdash \wn{B^Q},\wn{\Delta'^Q}}
\LabelRule{\rulename{ax}}
\NulRule{A^Q\vdash A^Q}
\LabelRule{\rulename{ax}}
\NulRule{B^Q\vdash B^Q}
\LabelRule{\tens R}
\BinRule{A^Q,B^Q\vdash A^Q\tens B^Q}
\LabelRule{\oc R}
\UnaRule{A^Q,B^Q\vdash \oc{(A^Q\tens B^Q)}}
\LabelRule{\wn d R}
\UnaRule{A^Q,B^Q\vdash \wn{\oc{(A^Q\tens B^Q)}}}
\LabelRule{\wn L}
\UnaRule{A^Q,\wn{B^Q}\vdash \wn{\oc{(A^Q\tens B^Q)}}}
\LabelRule{\rulename{cut}}
\BinRule{\Gamma'^Q,A^Q\vdash \wn{\oc{(A^Q\tens B^Q)}},\wn{\Delta'^Q}}
\LabelRule{\wn L}
\UnaRule{\Gamma'^Q,\wn{A^Q}\vdash \wn{\oc{(A^Q\tens B^Q)}},\wn{\Delta'^Q}}
\LabelRule{\rulename{cut}}
\BinRule{\Gamma^Q,\Gamma'^Q\vdash \wn{\oc{(A^Q\tens B^Q)}},\wn{\Delta^Q},\wn{\Delta'^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma,A,B\vdash \Delta}
\LabelRule{\wedge L}
\UnaRule{\Gamma,A\wedge B\vdash \Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q,A^Q,B^Q\vdash \wn{\Delta^Q}}
\LabelRule{\tens L}
\UnaRule{\Gamma^Q,A^Q\tens B^Q\vdash \wn{\Delta^Q}}
\LabelRule{\oc d L}
\UnaRule{\Gamma^Q,\oc{(A^Q\tens B^Q)}\vdash \wn{\Delta^Q}}
\DisplayProof
\\[2ex]
\LabelRule{T R}
\NulRule{{}\vdash T}
\DisplayProof
\qquad\mapsto\qquad
\LabelRule{\one R}
\NulRule{{}\vdash \one}
\LabelRule{\oc R}
\UnaRule{{}\vdash \oc{\one}}
\LabelRule{\wn d R}
\UnaRule{{}\vdash \wn{\oc{\one}}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash \Delta}
\LabelRule{T L}
\UnaRule{\Gamma,T\vdash \Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q\vdash \wn{\Delta^Q}}
\LabelRule{\one L}
\UnaRule{\Gamma^Q,\one\vdash \wn{\Delta^Q}}
\LabelRule{\oc d L}
\UnaRule{\Gamma^Q,\oc{\one}\vdash \wn{\Delta^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash A,\Delta}
\LabelRule{\vee_1 R}
\UnaRule{\Gamma\vdash A\vee B,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q\vdash \wn{A^Q},\wn{\Delta^Q}}
\LabelRule{\rulename{ax}}
\NulRule{A^Q\vdash A^Q}
\LabelRule{\plus_1 R}
\UnaRule{A^Q\vdash A^Q\plus B^Q}
\LabelRule{\oc R}
\UnaRule{A^Q\vdash \oc{(A^Q\plus B^Q)}}
\LabelRule{\wn d R}
\UnaRule{A^Q\vdash \wn{\oc{(A^Q\plus B^Q)}}}
\LabelRule{\wn L}
\UnaRule{\wn{A^Q}\vdash \wn{\oc{(A^Q\plus B^Q)}}}
\LabelRule{\rulename{cut}}
\BinRule{\Gamma^Q\vdash \wn{\oc{(A^Q\plus B^Q)}},\wn{\Delta^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma,A\vdash \Delta}
\AxRule{\Gamma,B\vdash \Delta}
\LabelRule{\vee L}
\BinRule{\Gamma,A\vee B\vdash \Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q,A^Q\vdash \wn{\Delta^Q}}
\AxRule{\Gamma^Q,B^Q\vdash \wn{\Delta^Q}}
\LabelRule{\plus L}
\BinRule{\Gamma^Q,A^Q\plus B^Q\vdash \wn{\Delta^Q}}
\LabelRule{\oc d L}
\UnaRule{\Gamma^Q,\oc{(A^Q\plus B^Q)}\vdash \wn{\Delta^Q}}
\DisplayProof
\\[2ex]
\LabelRule{F L}
\NulRule{\Gamma,F\vdash \Delta}
\DisplayProof
\qquad\mapsto\qquad
\LabelRule{\zero L}
\NulRule{\Gamma^Q,\zero\vdash \wn{\Delta^Q}}
\LabelRule{\oc d L}
\UnaRule{\Gamma^Q,\oc{\zero}\vdash \wn{\Delta^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma,A\vdash \Delta}
\LabelRule{\neg R}
\UnaRule{\Gamma\vdash \neg A,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q,A^Q\vdash \wn{\Delta^Q}}
\LabelRule{(.)\orth R}
\UnaRule{\Gamma^Q\vdash (A^Q)\orth,\wn{\Delta^Q}}
\LabelRule{\oc R}
\UnaRule{\Gamma^Q\vdash \oc{(A^Q)\orth},\wn{\Delta^Q}}
\LabelRule{\wn d R}
\UnaRule{\Gamma^Q\vdash \wn{\oc{(A^Q)\orth}},\wn{\Delta^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash A,\Delta}
\LabelRule{\neg L}
\UnaRule{\Gamma,\neg A\vdash \Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q\vdash \wn{A^Q},\wn{\Delta^Q}}
\LabelRule{(.)\orth L}
\UnaRule{\Gamma^Q,\oc{(A^Q)\orth}\vdash \wn{\Delta^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash A,\Delta}
\LabelRule{\forall R}
\UnaRule{\Gamma\vdash \forall\xi A,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q\vdash \wn{A^Q},\wn{\Delta^Q}}
\LabelRule{\forall R}
\UnaRule{\Gamma^Q\vdash \forall\xi \wn{A^Q},\wn{\Delta^Q}}
\LabelRule{\oc R}
\UnaRule{\Gamma^Q\vdash \oc{\forall\xi \wn{A^Q}},\wn{\Delta^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma,A[\tau/\xi]\vdash \Delta}
\LabelRule{\forall L}
\UnaRule{\Gamma,\forall\xi A\vdash \Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q,A^Q[\tau^{\underline{Q}}/\xi]\vdash \wn{\Delta^Q}}
\LabelRule{\wn L}
\UnaRule{\Gamma^Q,\wn{A^Q[\tau^{\underline{Q}}/\xi]}\vdash \wn{\Delta^Q}}
\LabelRule{\forall L}
\UnaRule{\Gamma^Q,\forall\xi \wn{A^Q}\vdash \wn{\Delta^Q}}
\LabelRule{\oc d L}
\UnaRule{\Gamma^Q,\oc{\forall\xi \wn{A^Q}}\vdash \wn{\Delta^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma\vdash A[\tau/\xi],\Delta}
\LabelRule{\exists R}
\UnaRule{\Gamma\vdash \exists\xi A,\Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q\vdash \wn{A^Q[\tau^{\underline{Q}}/\xi]},\wn{\Delta^Q}}
\LabelRule{\rulename{ax}}
\NulRule{A^Q[\tau^{\underline{Q}}/\xi]\vdash A^Q[\tau^{\underline{Q}}/\xi]}
\LabelRule{\exists R}
\UnaRule{A^Q[\tau^{\underline{Q}}/\xi]\vdash \exists\xi A^Q}
\LabelRule{\oc R}
\UnaRule{A^Q[\tau^{\underline{Q}}/\xi]\vdash \oc{\exists\xi A^Q}}
\LabelRule{\wn d R}
\UnaRule{A^Q[\tau^{\underline{Q}}/\xi]\vdash \wn{\oc{\exists\xi A^Q}}}
\LabelRule{\wn L}
\UnaRule{\wn{A^Q[\tau^{\underline{Q}}/\xi]}\vdash \wn{\oc{\exists\xi A^Q}}}
\LabelRule{\rulename{cut}}
\BinRule{\Gamma^Q\vdash \wn{\oc{\exists\xi A^Q}},\wn{\Delta^Q}}
\DisplayProof
\\[2ex]
\AxRule{\Gamma,A\vdash \Delta}
\LabelRule{\exists L}
\UnaRule{\Gamma,\exists\xi A\vdash \Delta}
\DisplayProof
\qquad\mapsto\qquad
\AxRule{\Gamma^Q,A^Q\vdash \wn{\Delta^Q}}
\LabelRule{\exists L}
\UnaRule{\Gamma^Q,\exists\xi A^Q\vdash \wn{\Delta^Q}}
\LabelRule{\oc d L}
\UnaRule{\Gamma^Q,\oc{\exists\xi A^Q}\vdash \wn{\Delta^Q}}
\DisplayProof
\end{gather*}
We use \((A[\tau/\xi])^Q=A^Q[\tau^{\underline{Q}}/\xi]\).

\subsection{Alternative presentation}\label{alternative-presentation-1}

It is also possible to define \(A^{\underline{Q}}\) as the primitive construction.
\begin{align*}
X^{\underline{Q}} & = X \\
(A\imp B)^{\underline{Q}} & = \oc{A^{\underline{Q}}}\limp\wn{\oc{B^{\underline{Q}}}} \\
(A\wedge B)^{\underline{Q}} & = \oc{A^{\underline{Q}}}\tens\oc{B^{\underline{Q}}} \\
T^{\underline{Q}} & = \one \\
(A\vee B)^{\underline{Q}} & = \oc{A^{\underline{Q}}}\plus\oc{B^{\underline{Q}}} \\
F^{\underline{Q}} & = \zero \\
(\neg A)^{\underline{Q}} & = \wn{(A^{\underline{Q}})\orth} \\
(\forall\xi A)^{\underline{Q}} & = \forall\xi \wn{\oc{A^{\underline{Q}}}} \\
(\exists\xi A)^{\underline{Q}} & = \exists\xi \oc{A^{\underline{Q}}}
\end{align*}

If we define
\((\Gamma\vdash\Delta)^{\underline{Q}} = \oc{\Gamma^{\underline{Q}}}\vdash\wn{\oc{\Delta^{\underline{Q}}}}\),
we have
\((\Gamma\vdash\Delta)^{\underline{Q}} = (\Gamma\vdash\Delta)^Q\) and
thus we obtain the same translation of proofs.


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