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
*
SIMP_SPECIAL_COND_BTRUE
<math> \operatorname{COND}(\btrue, E_1, E_2) \;\;\defi\;\; E_1 </math> A
*
SIMP_SPECIAL_COND_BFALSE
<math> \operatorname{COND}(\bfalse, E_1, E_2) \;\;\defi\;\; E_2 </math> A
*
SIMP_MULTI_COND
<math> \operatorname{COND}(C, E, E) \;\;\defi\;\; E </math> A


Inference Rules

  Name Rule Side Condition A/M
*
DATATYPE_DISTINCT_CASE
<math>\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} }</math> where <math>x</math> has a datatype <math>DT</math> as type and appears free in <math>\textbf{G}</math>, <math>DT</math> has constructors <math>c_1, \ldots, c_n</math>, parameters <math>p_{ij}</math> are introduced as fresh identifiers M


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