fix

new_system3.tex

@@ -141,7 +141,7 @@ TODO: Theorems and proof: no need for rule Inter, etc.
 \\
 \Infer[Proj*]
 {\Gamma(x)\equiv\textstyle\bigvee_{i\in I}\pair{t_i}{s_i}\\
-\forall i\in I.\ \Gamma\subst{x}{\pair{t_i}{s_i}}\vdash_\ct \ct[\pi_i x]: \{(u_j,\Gamma_j)\}_{j\in J_i}
+\forall i\in I.\ \Gamma\subst{x}{\pair{t_i}{s_i}}\vdash_{[]} \ct[\pi_i x]: \{(u_j,\Gamma_j)\}_{j\in J_i}
 }
 {\Gamma \vdash_\ct \pi_i x: \{(u_j,\Gamma_j)\,\alt\,i\in I,\ j\in J_i\}}
 { }
@@ -150,8 +150,8 @@ TODO: Theorems and proof: no need for rule Inter, etc.
 {\Gamma(x) = t\\
 t\land(\pair{\Any}{\Any})\neq\Empty\\
 t\land\neg(\pair{\Any}{\Any})\neq\Empty\\
-\Gamma\subst{x}{t\land(\pair{\Any}{\Any})}\vdash_\ct \ct[\pi_i x]: \{(u_i,\Gamma_i)\}_{i\in I}\\
-\Gamma\subst{x}{t\land\neg(\pair{\Any}{\Any})}\vdash_\ct \ct[\pi_i x]: \{(u_j,\Gamma_j)\}_{j\in J}
+\Gamma\subst{x}{t\land(\pair{\Any}{\Any})}\vdash_{[]} \ct[\pi_i x]: \{(u_i,\Gamma_i)\}_{i\in I}\\
+\Gamma\subst{x}{t\land\neg(\pair{\Any}{\Any})}\vdash_{[]} \ct[\pi_i x]: \{(u_j,\Gamma_j)\}_{j\in J}
 }
 {\Gamma \vdash_\ct \pi_i x: \{(u_i,\Gamma_i)\}_{i\in I}\cup\{(u_j,\Gamma_j)\}_{j\in J}}
 { }
@@ -175,7 +175,7 @@ t\land\neg(\pair{\Any}{\Any})\neq\Empty\\
   \Gamma(x_1)\equiv \textstyle\bigwedge_{i\in I}\arrow {s_i}{t_i}\\
   \Gamma(x_2)=s\\
   s\leq \textstyle\bigvee_{i\in I}s_i\\
-  \forall i\in I.\ \Gamma\subst{x_2}{s\land s_i}\vdash_\ct \ct[{x_1}{x_2}]: \{(u_j,\Gamma_j)\}_{j\in J_i}
+  \forall i\in I.\ \Gamma\subst{x_2}{s\land s_i}\vdash_{[]} \ct[{x_1}{x_2}]: \{(u_j,\Gamma_j)\}_{j\in J_i}
 }
 { \Gamma \vdash_\ct {x_1}{x_2}: \{(u_j,\Gamma_j)\,\alt\,i\in I,\ j\in J_i\} }
 { }
@@ -185,23 +185,23 @@ t\land\neg(\pair{\Any}{\Any})\neq\Empty\\
   \Gamma(x_1)\equiv \textstyle\bigwedge_{i\in I}\arrow {s_i}{t_i}\\
   s_\circ=\textstyle\bigvee_{i\in I}s_i\\
   \Gamma(x_2)=s\\s\land s_\circ\neq\Empty\\s\land \neg s_\circ\neq\Empty\\
-  \Gamma\subst{x_2}{s\land s_\circ}\vdash_\ct \ct[{x_1}{x_2}]: \{(u_i,\Gamma_i)\}_{i\in I}\\
-  \Gamma\subst{x_2}{s\land \neg s_\circ}\vdash_\ct \ct[{x_1}{x_2}]: \{(u_j,\Gamma_j)\}_{j\in J}
+  \Gamma\subst{x_2}{s\land s_\circ}\vdash_{[]} \ct[{x_1}{x_2}]: \{(u_i,\Gamma_i)\}_{i\in I}\\
+  \Gamma\subst{x_2}{s\land \neg s_\circ}\vdash_{[]} \ct[{x_1}{x_2}]: \{(u_j,\Gamma_j)\}_{j\in J}
 }
 { \Gamma \vdash_\ct {x_1}{x_2}: \{(u_i,\Gamma_i)\}_{i\in I}\cup\{(u_j,\Gamma_j)\}_{j\in J} }
 { }
 \\
 \Infer[AppL*]
 {\Gamma(x_1)\equiv\textstyle\bigvee_{i\in I}t_i\leq\arrow{\Empty}{\Any}\\
-\forall i\in I.\ \Gamma\subst{x_1}{t_i}\vdash_\ct \ct[{x_1}{x_2}]: \{(u_j,\Gamma_j)\}_{j\in J_i}
+\forall i\in I.\ \Gamma\subst{x_1}{t_i}\vdash_{[]} \ct[{x_1}{x_2}]: \{(u_j,\Gamma_j)\}_{j\in J_i}
 }
 {\Gamma \vdash_\ct {x_1}{x_2}: \{(u_j,\Gamma_j)\,\alt\,i\in I,\ j\in J_i\}}
 { }
 \\
 \Infer[AppLDom]
 {\Gamma(x_1) = t\\t\land (\arrow{\Empty}{\Any})\neq\Empty\\t\land \neg (\arrow{\Empty}{\Any})\neq\Empty\\
-\Gamma\subst{x_1}{t\land(\arrow{\Empty}{\Any})}\vdash_\ct \ct[{x_1}{x_2}]: \{(u_i,\Gamma_i)\}_{i\in I}\\
-\Gamma\subst{x_1}{t\land\neg(\arrow{\Empty}{\Any})}\vdash_\ct \ct[{x_1}{x_2}]: \{(u_j,\Gamma_j)\}_{j\in J}
+\Gamma\subst{x_1}{t\land(\arrow{\Empty}{\Any})}\vdash_{[]} \ct[{x_1}{x_2}]: \{(u_i,\Gamma_i)\}_{i\in I}\\
+\Gamma\subst{x_1}{t\land\neg(\arrow{\Empty}{\Any})}\vdash_{[]} \ct[{x_1}{x_2}]: \{(u_j,\Gamma_j)\}_{j\in J}
 }
 {\Gamma \vdash_\ct {x_1}{x_2}: \{(u_i,\Gamma_i)\}_{i\in I}\cup\{(u_j,\Gamma_j)\}_{j\in J}}
 { }
@@ -224,8 +224,8 @@ t\land\neg(\pair{\Any}{\Any})\neq\Empty\\
 \\
 \Infer[Case*]
 {\Gamma(x) = t_\circ\\
-\Gamma\subst{x}{t_\circ\land t}\vdash_\ct \ct[\tcase {x} t {e_1}{e_2}]: \{(u_i,\Gamma_i)\}_{i\in I}\\
-\Gamma\subst{x}{t_\circ\land\neg t}\vdash_\ct \ct[\tcase {x} t {e_1}{e_2}]: \{(u_j,\Gamma_j)\}_{j\in J}
+\Gamma\subst{x}{t_\circ\land t}\vdash_{[]} \ct[\tcase {x} t {e_1}{e_2}]: \{(u_i,\Gamma_i)\}_{i\in I}\\
+\Gamma\subst{x}{t_\circ\land\neg t}\vdash_{[]} \ct[\tcase {x} t {e_1}{e_2}]: \{(u_j,\Gamma_j)\}_{j\in J}
 }
 {\Gamma\vdash_\ct \tcase {x} t {e_1}{e_2}: \{(u_i,\Gamma_i)\}_{i\in I}\cup\{(u_j,\Gamma_j)\}_{j\in J}}
 { }