Extension Proof Rules

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Rules that are marked with a * in the first column are implemented in the latest version of Rodin. Rules without a * are planned to be implemented in future versions. Other conventions used in these tables are described in The_Proving_Perspective_(Rodin_User_Manual)#Rewrite_Rules.

Rewrite Rules

  Name Rule Side Condition A/M
  \operatorname{COND}(\btrue, E_1, E_2)  \;\;\defi\;\;  E_1 A
  \operatorname{COND}(\bfalse, E_1, E_2)  \;\;\defi\;\;  E_2 A
  \operatorname{COND}(C, E, E)  \;\;\defi\;\;  E A

Inference Rules

  Name Rule Side Condition A/M
\frac{\textbf{H}, x=c_1(p_{11}, \ldots, p_{1k}) \;\;\vdash \;\; \textbf{G} \qquad \ldots \qquad \textbf{H}, x=c_n(p_{n1}, \ldots, p_{nl}) \;\;\vdash \;\; \textbf{G} }{ \textbf{H}  \;\;\vdash\;\;  \textbf{G} } where x has a datatype DT as type and appears free in \textbf{G}, DT has constructors c_1, \ldots, c_n, parameters p_{ij} are introduced as fresh identifiers M

\frac{ \textbf{H}, x=c_1(p_1, \ldots, p_k),\textbf{P}(p_{I_1}), \ldots, \textbf{P}(p_{I_l})  \;\;\vdash \;\; \textbf{P}(x)  \qquad \ldots \qquad}{ \textbf{H}  \;\;\vdash\;\;  \textbf{P}(x) } where x has inductive datatype DT as type and appears free in \textbf{P}; \{p_{I_1}, \ldots, p_{I_l}\} \subseteq \{p_1, \ldots, p_k\} are the inductive parameters (if any); an antecedent is created for every constructor c_i of DT; all parameters are introduced as fresh identifiers; examples here M