simplification

new_system2.tex

@@ -366,13 +366,11 @@ TODO: Algorithmic rules
 \\
   \Infer[Pair]
   {%\forall i\in I.\ \Gamma\bvdash{x_1}{t_i}\{(t_i^j,\Gamma_{1,i}^j)\}_{j\in J_i}\\
-  \Gamma\vdash x_1:t_\circ\\
-  \Gamma\vdash x_2:s_\circ\\
-  \forall i\in I.\ \Gamma_1^i = \Gamma\subst{x_1}{t_\circ\land t_i}\\
+   \forall i\in I.\ \Gamma_1^i = \Gamma\subst{x_1}{t_i}\\
    %\forall i\in I.\ \Gamma\bvdash{x_2}{s_i}\{(s_i^k,\Gamma_{2,i}^k)\}_{k\in K_i}
-   \forall i\in I.\ \Gamma_2^i = \Gamma\subst{x_2}{s_\circ\land s_i}\\
+   \forall i\in I.\ \Gamma_2^i = \Gamma\subst{x_2}{s_i}\\
    }
-  {\Gamma\bvdash {(x_1,x_2)} {\bigcup_{i\in I}(t_i,s_i)} \bigcup_{i\in I}\{(\pair{(t_\circ\land t_i)}{(s_\circ\land s_i)}, \Gamma_1^i\land\Gamma_2^i)\}}
+  {\Gamma\bvdash {(x_1,x_2)} {\bigcup_{i\in I}(t_i,s_i)} \bigcup_{i\in I}\{(\pair{t_i}{s_i}, \Gamma_1^i\land\Gamma_2^i)\}}
   { }
   \\
   \Infer[Proj]