Inference 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)#Inference_Rules.


  Name Rule Side Condition A/M


*
HYP
<math>\frac{}{\textbf{H},\textbf{P} \;\;\vdash \;\; \textbf{P}^{\dagger}} </math> see below for <math>\textbf{P}^{\dagger}</math> A


*
HYP_OR
<math>\frac{}{\textbf{H},\textbf{Q} \;\;\vdash \;\; \textbf{P} \lor \ldots \lor \textbf{Q}^{\dagger} \lor \ldots \lor \textbf{R}}</math> see below for <math>\textbf{Q}^{\dagger}</math> A


*
CNTR
<math>\frac{}{\textbf{H},\;\textbf{P},\;\textbf{nP}^{\dagger} \;\;\vdash \;\; \textbf{Q}}</math> see below for <math>\textbf{nP}^{\dagger}</math> A


*
FALSE_HYP
<math>\frac{}{\textbf{H},\bfalse \;\;\vdash \;\; \textbf{P}}</math> A


*
TRUE_GOAL
<math>\frac{}{\textbf{H} \;\;\vdash \;\; \btrue}</math> A


*
FUN_GOAL
<math>\frac{}{\textbf{H},\; f\in E\;\mathit{op}\;F \;\;\vdash\;\; f\in T_1\pfun T_2}</math> where <math>T_1</math> and <math>T_2</math> denote types and <math>\mathit{op}</math> is one of <math>\pfun</math>, <math>\tfun</math>, <math>\pinj</math>, <math>\tinj</math>, <math>\psur</math>, <math>\tsur</math>, <math>\tbij</math>. A


*
FUN_IMAGE_GOAL
<math>\frac{\textbf{H},\; f\in S_1\;\mathit{op}\;S_2,\; f(E)\in S_2\;\;\vdash\;\; \mathbf{P}(f(E))}{\textbf{H},\; f\in S_1\;\mathit{op}\;S_2\;\;\vdash\;\; \mathbf{P}(f(E))}</math> where <math>\mathit{op}</math> denotes a set of relations (any arrow) and <math>\mathbf{P}</math> is WD strict M


FUN_GOAL_REC
<math>\frac{}{\textbf{H},\; f\in S_1\;\mathit{op_1}\;(S_2\;\mathit{op_2}\;(\ldots(S_n\;\mathit{op_n}(U\;\mathit{opf}\;V\;))\ldots)) \;\vdash\;\; f(E_1)(E_2)...(E_n)\in T_1\pfun T_2}</math> where <math>T_1</math> and <math>T_2</math> denote types, <math>\mathit{op}</math> denotes a set of relations (any arrow) and <math>\mathit{opf}</math> is one of <math>\pfun</math>, <math>\tfun</math>, <math>\pinj</math>, <math>\tinj</math>, <math>\psur</math>, <math>\tsur</math>, <math>\tbij</math>. A


*
DBL_HYP
<math>\frac{\textbf{H},\;\textbf{P} \;\;\vdash \;\; \textbf{Q}}{\textbf{H},\;\textbf{P},\;\textbf{P} \;\;\vdash \;\; \textbf{Q}}</math> A


*
AND_L
<math>\frac{\textbf{H},\textbf{P},\textbf{Q} \; \; \vdash \; \; \textbf{R}}{\textbf{H},\; \textbf{P} \land \textbf{Q} \; \; \vdash \; \;

\textbf{R}}</math> || || A


*
AND_R
<math>\frac{\textbf{H} \; \; \vdash \; \; \textbf{P} \qquad \textbf{H} \; \; \vdash \; \; \textbf{Q}}{\textbf{H} \; \; \vdash \; \; \textbf{P} \; \land \; \textbf{Q}}</math> A


IMP_L1
<math>\frac{\textbf{H},\; \textbf{Q},\; \textbf{P} \land \ldots \land \textbf{R} \limp \textbf{S} \;\;\vdash \;\; \textbf{T}}{\textbf{H},\; \textbf{Q},\; \textbf{P} \land \ldots \land \textbf{Q} \land \ldots \land \textbf{R} \limp \textbf{S} \;\;\vdash \;\; \textbf{T} }</math> A


*
IMP_R
<math>\frac{\textbf{H}, \textbf{P} \;\;\vdash \;\; \textbf{Q}}{\textbf{H} \;\;\vdash \;\; \textbf{P} \limp \textbf{Q}}</math> A


*
IMP_AND_L
<math>\frac{\textbf{H},\textbf{P} \limp \textbf{Q}, \textbf{P} \limp \textbf{R}\;\;\vdash \;\; \textbf{S}}{\textbf{H},\;\textbf{P} \limp \textbf{Q} \land \textbf{R} \;\;\vdash \;\; \textbf{S}}</math> A


*
IMP_OR_L
<math>\frac{

\textbf{H},\textbf{P} \limp \textbf{R}, \textbf{Q} \limp \textbf{R}\;\;\vdash \;\; \textbf{S} }{\textbf{H},\;\textbf{P} \lor \textbf{Q} \limp \textbf{R} \;\;\vdash \;\; \textbf{S}}</math> || || A


*
AUTO_MH
<math>\frac{

\textbf{H},\textbf{P},\;\textbf{Q}\limp \textbf{R}\;\;\vdash \;\; \textbf{S} }{\textbf{H},\;\textbf{P},\; \textbf{P} \land \textbf{Q} \limp \textbf{R} \;\;\vdash \;\; \textbf{S}}</math> || || A


*
NEG_IN_L
<math>\frac{\textbf{H},\; E \in \{ a,\ldots , c\},\; \neg\, (E=b) \; \; \vdash \; \; \textbf{P} }{\textbf{H},\; E \in \{ a,\ldots , b, \ldots , c\} , \neg \, (E=b) \; \; \vdash \; \; \textbf{P} }</math> A


*
NEG_IN_R
<math>\frac{\textbf{H},\; E \in \{ a,\ldots , c\},\; \neg\, (b=E) \; \; \vdash \; \; \textbf{P} }{\textbf{H},\; E \in \{ a,\ldots , b, \ldots , c\} , \neg \, (b=E) \; \; \vdash \; \; \textbf{P} }</math> A


*
XST_L
<math>\frac{\textbf{H},\; \textbf{P(x)} \; \; \vdash \; \; \textbf{Q}

}{ \textbf{H},\; \exists \, \textbf{x}\, \qdot\, \textbf{P(x)} \; \; \vdash \; \; \textbf{Q} }</math> || || A


*
ALL_R
<math>\frac{\textbf{H}\; \; \vdash \; \; \textbf{P(x)} }{ \textbf{H} \; \; \vdash \; \; \forall \textbf{x}\, \qdot\, \textbf{P(x)} }</math> A


*
EQL_LR
<math>\frac{\textbf{H(E)} \; \; \vdash \; \; \textbf{P(E)} }{\textbf{H(x)},\; x=E \; \; \vdash \; \; \textbf{P(x)} }</math> <math>x</math> is a variable which is not free in <math>E</math> A


*
EQL_RL
<math>\frac{\textbf{H(E)} \; \; \vdash \; \; \textbf{P(E)} }{\textbf{H(x)},\; E=x \; \; \vdash \; \; \textbf{P(x)} }</math> <math>x</math> is a variable which is not free in <math>E</math> A


SUBSET_INTER
<math>\frac{\textbf{H},\;\textbf{T} \subseteq \textbf{U} \;\;\vdash \;\;

\textbf{G}(\textbf{S} \binter \dots \binter \textbf{T} \binter \dots \binter \textbf{V})} {\textbf{H},\;\textbf{T} \subseteq \textbf{U} \;\;\vdash \;\; \textbf{G}(\textbf{S} \binter \dots \binter \textbf{T} \binter \dots \binter \textbf{U} \binter \dots \binter \textbf{V})}</math> || where <math>\mathbf{T}</math> and <math>\mathbf{U}</math> are not bound by <math>\mathbf{G}</math> || A


IN_INTER
<math>\frac{\textbf{H},\;\textbf{E} \in \textbf{T} \;\;\vdash \;\;

\textbf{G}(\textbf{S} \binter \dots \binter \{\textbf{E}\} \binter \dots \binter \textbf{U})} {\textbf{H},\;\textbf{E} \in \textbf{T} \;\;\vdash \;\; \textbf{G}(\textbf{S} \binter \dots \binter \{\textbf{E}\} \binter \dots \binter \textbf{T} \binter \dots \binter \textbf{U})}</math> || where <math>\mathbf{E}</math> and <math>\mathbf{T}</math> are not bound by <math>\mathbf{G}</math> || A


NOTIN_INTER
<math>\frac{\textbf{H},\;\lnot\;\textbf{E} \in \textbf{T} \;\;\vdash \;\;

\textbf{G}(\emptyset)} {\textbf{H},\;\lnot\;\textbf{E} \in \textbf{T} \;\;\vdash \;\; \textbf{G}(\textbf{S} \binter \dots \binter \{\textbf{E}\} \binter \dots \binter \textbf{T} \binter \dots \binter \textbf{U})}</math> || where <math>\mathbf{E}</math> and <math>\mathbf{T}</math> are not bound by <math>\mathbf{G}</math> || A


*
FIN_L_LOWER_BOUND_L
<math>\frac{}{\textbf{H},\;\finite(S) \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; n \leq x)}</math> The goal is discharged A


*
FIN_L_LOWER_BOUND_R
<math>\frac{}{\textbf{H},\;\finite(S) \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; x \geq n)}</math> The goal is discharged A


*
FIN_L_UPPER_BOUND_L
<math>\frac{}{\textbf{H},\;\finite(S) \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; n \geq x)}</math> The goal is discharged A


*
FIN_L_UPPER_BOUND_R
<math>\frac{}{\textbf{H},\;\finite(S) \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; x \leq n)}</math> The goal is discharged A


*
CONTRADICT_L
<math>\frac{\textbf{H},\;\neg\,\textbf{Q} \;\;\vdash \;\; \neg\,\textbf{P}}{\textbf{H},\;\textbf{P} \;\;\vdash \;\; \textbf{Q}}</math> M


*
CONTRADICT_R
<math>\frac{\textbf{H},\;\neg\,\textbf{Q} \;\;\vdash \;\; \bfalse}{\textbf{H} \;\;\vdash \;\; \textbf{Q}}</math> M


*
CASE
<math>\frac{\textbf{H}, \; \textbf{P} \; \; \vdash \; \; \textbf{R} \qquad\ldots\qquad \textbf{H}, \; \textbf{Q} \; \; \vdash \; \; \textbf{R} }{\textbf{H},\; \textbf{P} \lor \ldots \lor \textbf{Q} \; \; \vdash \; \; \textbf{R} }</math> M


*
IMP_CASE
<math>\frac{\textbf{H}, \; \lnot\textbf{P} \; \; \vdash \; \; \textbf{R} \qquad \textbf{H}, \; \textbf{Q} \; \; \vdash \; \; \textbf{R} }{\textbf{H},\; \textbf{P} \limp\textbf{Q} \; \; \vdash \; \; \textbf{R} }</math> M


*
MH
<math>\frac{\textbf{H} \;\;\vdash\;\;\textbf{P} \qquad \textbf{H},\; \textbf{Q} \;\;\vdash \;\; \textbf{R} }{\textbf{H},\;\textbf{P} \limp \textbf{Q} \;\;\vdash \;\; \textbf{R}}</math> M


*
HM
<math>\frac{\textbf{H} \;\;\vdash\;\;\neg\,\textbf{Q} \qquad \textbf{H},\; \neg\,\textbf{P} \;\;\vdash \;\; \textbf{R} }{\textbf{H},\;\textbf{P} \limp \textbf{Q} \;\;\vdash \;\; \textbf{R}}</math> M


EQV_LR
<math>\frac{\textbf{H(Q)},\; \textbf{P} \leqv \textbf{Q}

\;\;\vdash\;\; \textbf{G(Q)}}{\textbf{H(P)},\;\textbf{P} \leqv \textbf{Q} \;\;\vdash \;\; \textbf{G(P)}}</math> || || M


EQV_RL
<math>\frac{\textbf{H(P)},\; \textbf{P} \leqv \textbf{Q}

\;\;\vdash\;\; \textbf{G(P)}}{\textbf{H(Q)},\;\textbf{P} \leqv \textbf{Q} \;\;\vdash \;\; \textbf{G(Q)}}</math> || || M


*
OV_SETENUM_L
<math>\frac{\textbf{H},\; G=E

,\;\textbf{P}(F)\;\;\vdash\;\;\textbf{Q} \qquad \textbf{H},\; \neg\,(G=E) ,\;\textbf{P}((\{E\}) \domsub f)(G))\;\;\vdash\;\;\textbf{Q}}{\textbf{H},\;\textbf{P}((f\ovl\{E \mapsto F\})(G)) \;\;\vdash \;\; \textbf{Q}}</math> || where <math>\mathbf{P}</math> is WD strict || A


*
OV_SETENUM_R
<math>\frac{\textbf{H},\; G=E \;\;\vdash\;\;\textbf{Q}(F)

\qquad \textbf{H},\; \neg\,(G=E) \;\;\vdash\;\;\textbf{Q}((\{E\}) \domsub f)(G))}{\textbf{H} \;\;\vdash \;\; \textbf{Q}((f\ovl\{E \mapsto F\})(G))}</math> || where <math>\mathbf{Q}</math> is WD strict || A


*
OV_L
<math>\frac{\textbf{H},\; G \in \dom(g) ,\;\textbf{P}(g(G))\;\;\vdash\;\;\textbf{Q} \qquad \textbf{H},\; \neg\,G \in \dom(g) ,\;\textbf{P}((\dom(g) \domsub f)(G))\;\;\vdash\;\;\textbf{Q}}{\textbf{H},\;\textbf{P}((f\ovl g)(G)) \;\;\vdash \;\; \textbf{Q}}</math> where <math>\mathbf{P}</math> is WD strict A


*
OV_R
<math>\frac{\textbf{H},\; G \in \dom(g) \;\;\vdash\;\;\textbf{Q}(g(G)) \qquad \textbf{H},\; \neg\, G \in \dom(g) \;\;\vdash\;\;\textbf{Q}((\dom(g) \domsub f)(G))}{\textbf{H} \;\;\vdash \;\; \textbf{Q}((f\ovl g)(G))}</math> where <math>\mathbf{Q}</math> is WD strict A


*
DIS_BINTER_R
<math>\frac{\textbf{H} \;\;\vdash\;\; f^{-1} \in A \pfun B \qquad\textbf{H} \;\;\vdash\;\;\textbf{Q}(f[S] \binter f[T]) }{\textbf{H} \;\;\vdash \;\; \textbf{Q}(f[S \binter T])}</math> where <math>A</math> and <math>B</math> denote types. M


*
DIS_BINTER_L
<math>\frac{\textbf{H} \;\;\vdash\;\; f^{-1} \in A \pfun B \qquad\textbf{H},\;\textbf{Q}(f[S] \binter f[T]) \;\;\vdash\;\;\textbf{G}}{\textbf{H},\; \textbf{Q}(f[S \binter T]) \;\;\vdash \;\; \textbf{G}}</math> where <math>A</math> and <math>B</math> denote types. M


*
DIS_SETMINUS_R
<math>\frac{\textbf{H} \;\;\vdash\;\; f^{-1} \in A \pfun B \qquad\textbf{H} \;\;\vdash\;\;\textbf{Q}(f[S] \setminus f[T]) }{\textbf{H} \;\;\vdash \;\; \textbf{Q}(f[S \setminus T])}</math> where <math>A</math> and <math>B</math> denote types. M


*
DIS_SETMINUS_L
<math>\frac{\textbf{H} \;\;\vdash\;\; f^{-1} \in A \pfun B \qquad\textbf{H},\;\textbf{Q}(f[S] \setminus f[T]) \;\;\vdash\;\; \textbf{G}}{\textbf{H},\; \textbf{Q}(f[S \setminus T]) \;\;\vdash \;\; \textbf{G}}</math> where <math>A</math> and <math>B</math> denote types. M


*
SIM_REL_IMAGE_R
<math>\frac{\textbf{H} \; \; \vdash \; \; {WD}(\textbf{Q}(\{ f(E)\} )) \qquad\textbf{H} \; \; \vdash \; \; \textbf{Q}(\{ f(E)\} ) }{\textbf{H} \; \; \vdash \; \; \textbf{Q}(f[\{ E\} ])} </math> M


*
SIM_REL_IMAGE_L
<math>\frac{\textbf{H} \; \; \vdash \; \; {WD}(\textbf{Q}(\{ f(E)\} )) \qquad\textbf{H},\; \textbf{Q}(\{ f(E)\}) \;\;\vdash\;\; \textbf{G}}{\textbf{H},\; \textbf{Q}(f[\{ E\} ]) \;\;\vdash\;\; \textbf{G} } </math> M


*
SIM_FCOMP_R
<math>\frac{\textbf{H} \;\;\vdash\;\;{WD}(\textbf{Q}(g(f(x)))) \qquad\textbf{H} \;\;\vdash\;\;\textbf{Q}(g(f(x))) }{\textbf{H} \;\;\vdash \;\; \textbf{Q}((f \fcomp g)(x))}</math> M


*
SIM_FCOMP_L
<math>\frac{\textbf{H} \;\;\vdash\;\;{WD}(\textbf{Q}(g(f(x)))) \qquad\textbf{H},\; \textbf{Q}(g(f(x))) \;\;\vdash\;\; \textbf{G}}{\textbf{H},\; \textbf{Q}((f \fcomp g)(x)) \;\;\vdash \;\; \textbf{G}}</math> M


*
FIN_SUBSETEQ_R
<math>\frac{\textbf{H} \;\;\vdash\;\;{WD}(T) \qquad\textbf{H} \;\;\vdash \;\; S \subseteq T \qquad \textbf{H} \;\;\vdash \;\; \finite\,(T)}{\textbf{H} \;\;\vdash \;\; \finite\,(S)}</math> the user has to write the set corresponding to <math>T</math> in the editing area of the Proof Control Window M


*
FIN_BINTER_R
<math>\frac{\textbf{H} \;\;\vdash

\;\;\finite\,(S) \;\lor\;\ldots \;\lor\; \finite\,(T)}{\textbf{H} \;\;\vdash \;\; \finite\,(S \;\binter\;\ldots \;\binter\; T)}</math> || || M


FIN_KINTER_R
<math>\frac{\textbf{H} \;\;\vdash

\;\;\exists s\, \qdot\, s \in S \land \finite\,(s)}{\textbf{H} \;\;\vdash \;\; \finite\,(\inter(S))}</math> || where <math>s</math> is fresh || M


FIN_QINTER_R
<math>\frac{\textbf{H} \;\;\vdash

\;\;\exists s\, \qdot\, P \land \finite\,(E)}{\textbf{H} \;\;\vdash \;\; \finite\,(\Inter s\,\qdot\,P\,\mid\,E)}</math> || || M


*
FIN_SETMINUS_R
<math>\frac{\textbf{H} \;\;\vdash

\;\;\finite\,(S)}{\textbf{H} \;\;\vdash \;\; \finite\,(S \;\setminus\; T)}</math> || || M


FIN_REL
<math>\frac{}{\textbf{H},\; r\in S\;\mathit{op}\;T,\; \finite\,(S),\; \finite\,(T) \;\;\vdash \;\; \finite\,(r)}</math> where <math>\mathit{op}</math> denotes a set of relations (any arrow) A


*
FIN_REL_R
<math>\frac{\textbf{H} \;\;\vdash\;\;{WD}(S\rel T) \qquad\textbf{H} \;\;\vdash \;\; r \;\in\; S \rel T \qquad \textbf{H} \;\;\vdash \;\; \finite\,(S) \qquad \textbf{H} \;\;\vdash \;\; \finite\,(T)}{\textbf{H} \;\;\vdash \;\; \finite\,(r)}</math> the user has to write the set corresponding to <math>S \rel T</math> in the editing area of the Proof Control Window M


*
FIN_REL_IMG_R
<math>\frac{\textbf{H} \;\;\vdash \;\; \finite\,(r) }{\textbf{H} \;\;\vdash \;\; \finite\,(r[s])}</math> M


*
FIN_REL_RAN_R
<math>\frac{\textbf{H} \;\;\vdash \;\; \finite\,(r) }{\textbf{H} \;\;\vdash \;\; \finite\,(\ran(r))}</math> M


*
FIN_REL_DOM_R
<math>\frac{\textbf{H} \;\;\vdash \;\; \finite\,(r) }{\textbf{H} \;\;\vdash \;\; \finite\,(\dom(r))}</math> M


FIN_FUN_DOM
<math>\frac{}{\textbf{H},\; f\in S\;\mathit{op}\;T,\; \finite\,(S) \;\;\vdash \;\; \finite\,(f)}</math> where <math>\mathit{op}</math> is one of <math>\pfun</math>, <math>\tfun</math>, <math>\pinj</math>, <math>\tinj</math>, <math>\psur</math>, <math>\tsur</math>, <math>\tbij</math> A


FIN_FUN_RAN
<math>\frac{}{\textbf{H},\; f\in S\;\mathit{op}\;T,\; \finite\,(T) \;\;\vdash \;\; \finite\,(f)}</math> where <math>\mathit{op}</math> is one of <math>\pinj</math>, <math>\tinj</math>, <math>\tbij</math> A


*
FIN_FUN1_R
<math>\frac{\textbf{H} \;\;\vdash\;\;{WD}(S\pfun T) \qquad\textbf{H} \;\;\vdash \;\; f \;\in\; S \pfun T \qquad \textbf{H} \;\;\vdash \;\; \finite\,(S) }{\textbf{H} \;\;\vdash \;\; \finite\,(f)}</math> the user has to write the set corresponding to <math>S \pfun T</math> in the editing area of the Proof Control Window M


*
FIN_FUN2_R
<math>\frac{\textbf{H} \;\;\vdash\;\;{WD}(S\pfun T) \qquad\textbf{H} \;\;\vdash \;\; f^{-1} \;\in\; S \pfun T \qquad \textbf{H} \;\;\vdash \;\; \finite\,(S) }{\textbf{H} \;\;\vdash \;\; \finite\,(f)}</math> the user has to write the set corresponding to <math>S \pfun T</math> in the editing area of the Proof Control Window M


*
FIN_FUN_IMG_R
<math>\frac{\textbf{H} \;\;\vdash\;\;{WD}(S\pfun T) \qquad\textbf{H} \;\;\vdash \;\; f \;\in\; S \pfun T \qquad \textbf{H} \;\;\vdash \;\; \finite\,(s) }{\textbf{H} \;\;\vdash \;\; \finite\,(f[s])}</math> the user has to write the set corresponding to <math>S \pfun T</math> in the editing area of the Proof Control Window M


*
FIN_FUN_RAN_R
<math>\frac{\textbf{H} \;\;\vdash\;\;{WD}(S\pfun T) \qquad\textbf{H} \;\;\vdash \;\; f \;\in\; S \pfun T \qquad \textbf{H} \;\;\vdash \;\; \finite\,(S) }{\textbf{H} \;\;\vdash \;\; \finite\,(\ran(f))}</math> the user has to write the set corresponding to <math>S \pfun T</math> in the editing area of the Proof Control Window M


*
FIN_FUN_DOM_R
<math>\frac{\textbf{H} \;\;\vdash\;\;{WD}(S\pfun T) \qquad\textbf{H} \;\;\vdash \;\; f^{-1} \;\in\; S \pfun T \qquad \textbf{H} \;\;\vdash \;\; \finite\,(S) }{\textbf{H} \;\;\vdash \;\; \finite\,(\dom(f))}</math> the user has to write the set corresponding to <math>S \pfun T</math> in the editing area of the Proof Control Window M


*
LOWER_BOUND_L
<math>\frac{\textbf{H} \;\;\vdash \;\; \finite(S) }{\textbf{H} \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; n \leq x)}</math> <math>S</math> must not contain any bound variable M


*
LOWER_BOUND_R
<math>\frac{\textbf{H} \;\;\vdash \;\; \finite(S) }{\textbf{H} \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; x \geq n)}</math> <math>S</math> must not contain any bound variable M


*
UPPER_BOUND_L
<math>\frac{\textbf{H} \;\;\vdash \;\; \finite(S) }{\textbf{H} \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; n \geq x)}</math> <math>S</math> must not contain any bound variable M


*
UPPER_BOUND_R
<math>\frac{\textbf{H} \;\;\vdash \;\; \finite(S) }{\textbf{H} \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; x \leq n)}</math> <math>S</math> must not contain any bound variable M


*
FIN_LT_0
<math>\frac{\textbf{H} \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; n \leq x) \qquad \textbf{H} \;\;\vdash \;\; S \subseteq \intg \setminus \natn }{\textbf{H} \;\;\vdash \;\; \finite(S)}</math> M


*
FIN_GE_0
<math>\frac{\textbf{H} \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; x \leq n) \qquad \textbf{H} \;\;\vdash \;\; S \subseteq \nat }{\textbf{H} \;\;\vdash \;\; \finite(S)}</math> M


CARD_INTERV
<math>\frac{\textbf{H},\, a \leq b \;\;\vdash \;\; \textbf{Q}(b-a+1) \qquad \textbf{H},\, b < a \;\;\vdash \;\; \textbf{Q}(0) }{\textbf{H} \;\;\vdash\;\; \textbf{Q}(\card\,(a\upto b))}</math> where <math>\mathbf{Q}</math> is WD strict M


CARD_EMPTY_INTERV
<math>\frac{\textbf{H},\, a \leq b,\,\textbf{P}(b-a+1) \;\;\vdash \;\; \textbf{Q} \qquad \textbf{H},\, b < a ,\, \textbf{P}(0)\;\;\vdash \;\; \textbf{Q} }{\textbf{H},\,\textbf{P}(\card\,(a\upto b)) \;\;\vdash\;\; \textbf{Q}}</math> where <math>\mathbf{P}</math> is WD strict M


*
DERIV_LE_CARD
<math>\frac{\textbf{H} \;\;\vdash\;\; S \subseteq T}{\textbf{H} \;\;\vdash\;\; \card(S) \leq \card(T)}</math> <math>S</math> and <math>T</math> bear the same type M


*
DERIV_GE_CARD
<math>\frac{\textbf{H} \;\;\vdash\;\; T \subseteq S}{\textbf{H} \;\;\vdash\;\; \card(S) \geq \card(T)}</math> <math>S</math> and <math>T</math> bear the same type M


*
DERIV_LT_CARD
<math>\frac{\textbf{H} \;\;\vdash\;\; S \subset T}{\textbf{H} \;\;\vdash\;\; \card(S) < \card(T)}</math> <math>S</math> and <math>T</math> bear the same type M


*
DERIV_GT_CARD
<math>\frac{\textbf{H} \;\;\vdash\;\; T \subset S}{\textbf{H} \;\;\vdash\;\; \card(S) > \card(T)}</math> <math>S</math> and <math>T</math> bear the same type M


*
DERIV_EQUAL_CARD
<math>\frac{\textbf{H} \;\;\vdash\;\; S = T}{\textbf{H} \;\;\vdash\;\; \card(S) = \card(T)}</math> <math>S</math> and <math>T</math> bear the same type M


SIMP_CARD_SETMINUS_L
<math>\frac{\textbf{H},\, \textbf{P}(\card (S \setminus T)) \;\;\vdash\;\; \finite(S) \qquad \textbf{H},\, \textbf{P}(\card(S) - \card(S\binter T)) \;\;\vdash\;\; \textbf{G}}{\textbf{H},\, \textbf{P}(\card (S \setminus T)) \;\;\vdash\;\; \textbf{G}} </math> M
SIMP_CARD_SETMINUS_R
<math>\frac{\textbf{H} \;\;\vdash\;\; \finite(S) \qquad \textbf{H} \;\;\vdash\;\; \textbf{P}(\card(S) - \card(S\binter T))}{\textbf{H} \;\;\vdash\;\; \textbf{P}(\card (S \setminus T))} </math> M


SIMP_CARD_CPROD_L
<math>\frac{\textbf{H},\, \textbf{P}(\card (S \cprod T)) \;\;\vdash\;\; \finite(S) \qquad \textbf{H},\, \textbf{P}(\card (S \cprod T)) \;\;\vdash\;\; \finite(T) \qquad \textbf{H},\, \textbf{P}(\card(S) * \card(T)) \;\;\vdash\;\; \textbf{G}}{\textbf{H},\, \textbf{P}(\card (S \cprod T)) \;\;\vdash\;\; \textbf{G}} </math> M
SIMP_CARD_CPROD_R
<math>\frac{\textbf{H} \;\;\vdash\;\; \finite(S) \qquad \textbf{H} \;\;\vdash\;\; \finite(T) \qquad \textbf{H} \;\;\vdash\;\; \textbf{P}(\card(S) * \card(T))}{\textbf{H} \;\;\vdash\;\; \textbf{P}(\card (S \cprod T))} </math> M


*
FORALL_INST
<math>\frac{\textbf{H} \;\;\vdash \;\; {WD}(E) \qquad \textbf{H} , [x \bcmeq E]\textbf{P} \;\;\vdash \;\; \textbf{G}}{\textbf{H}, \forall x \qdot \textbf{P} \;\;\vdash\;\; \textbf{G}}</math> <math>x</math> is instantiated with <math>E</math> M


*
FORALL_INST_MP
<math>\frac{\textbf{H} \;\;\vdash \;\; {WD}(E) \qquad \textbf{H}, {WD}(E) \;\;\vdash \;\; [x \bcmeq E]\textbf{P} \qquad \textbf{H}, {WD}(E), [x \bcmeq E]\textbf{Q} \;\;\vdash \;\; \textbf{G}}{\textbf{H}, \forall x \qdot \textbf{P} \limp \textbf{Q} \;\;\vdash\;\; \textbf{G}}</math> <math>x</math> is instantiated with <math>E</math> and a Modus Ponens is applied M


*
FORALL_INST_MT
<math>\frac{\textbf{H} \;\;\vdash \;\; {WD}(E) \qquad \textbf{H}, {WD}(E) \;\;\vdash \;\; [x \bcmeq E]\lnot\textbf{Q} \qquad \textbf{H}, {WD}(E), [x \bcmeq E]\lnot\textbf{P} \;\;\vdash \;\; \textbf{G}}{\textbf{H}, \forall x \qdot \textbf{P} \limp \textbf{Q} \;\;\vdash\;\; \textbf{G}}</math> <math>x</math> is instantiated with <math>E</math> and a Modus Tollens is applied M


*
CUT
<math>\frac{\textbf{H} \;\;\vdash \;\; {WD}(\textbf{P}) \qquad \textbf{H}, {WD}(\textbf{P}) \;\;\vdash \;\; \textbf{\textbf{P}} \qquad \textbf{H}, {WD}(\textbf{P}), \textbf{P} \;\;\vdash \;\; \textbf{G}}{\textbf{H} \;\;\vdash\;\; \textbf{G}}</math> hypothesis <math>\textbf{P}</math> is added M


*
EXISTS_INST
<math>\frac{\textbf{H} \;\;\vdash \;\; {WD}(E) \qquad \textbf{H} \;\;\vdash \;\; \textbf{P}(E)}{\textbf{H} \;\;\vdash\;\; \exists x \qdot \textbf{P}(x)}</math> <math>x</math> is instantiated with <math>E</math> M


*
DISTINCT_CASE
<math>\frac{\textbf{H} \;\;\vdash \;\; {WD}(\textbf{P}) \qquad \textbf{H}, {WD}(\textbf{P}), \textbf{P} \;\;\vdash \;\; \textbf{\textbf{G}} \qquad \textbf{H}, {WD}(\textbf{P}), \lnot \textbf{P} \;\;\vdash \;\; \textbf{G}}{\textbf{H} \;\;\vdash\;\; \textbf{G}}</math> case distinction on predicate <math>\textbf{P}</math> M


*
ONE_POINT_L
<math>\frac{\textbf{H} \;\;\vdash \;\; {WD}(E) \qquad \textbf{H}, \forall x, \ldots, \ldots,z \qdot [y \bcmeq E]\textbf{P} \land \ldots \land \ldots \land [y \bcmeq E]\textbf{Q} \limp [y \bcmeq E]\textbf{R} \;\;\vdash \;\; \textbf{G}}{ \textbf{H}, \forall x, \ldots, y, \ldots, z \qdot \textbf{P} \land \ldots \land y = E \land \ldots \land \textbf{Q} \limp \textbf{R} \;\;\vdash\;\; \textbf{G}}</math> The rule can be applied with <math>\forall</math> as well as with <math>\exists</math> A


*
ONE_POINT_R
<math>\frac{\textbf{H} \;\;\vdash \;\; {WD}(E) \qquad \textbf{H} \;\;\vdash \;\; \forall x, \ldots, \ldots,z \qdot [y \bcmeq E]\textbf{P} \land \ldots \land \ldots \land [y \bcmeq E]\textbf{Q} \limp [y \bcmeq E]\textbf{R} }{ \textbf{H} \;\;\vdash\;\; \forall x, \ldots, y, \ldots, z \qdot \textbf{P} \land \ldots \land y = E \land \ldots \land \textbf{Q} \limp \textbf{R} }</math> The rule can be applied with <math>\forall</math> as well as with <math>\exists</math> A


*
SIM_OV_REL
<math> \frac{\textbf{H}\vdash x\in A\qquad \textbf{H}\vdash y\in B}{\textbf{H}, f\in A\;op\; B\vdash f\ovl\left\{x\mapsto y\right\}\in A\rel B} </math> where <math>\mathit{op}</math> is one of <math>\rel</math>, <math>\trel</math>, <math>\srel</math>, <math>\strel</math>, <math>\pfun</math>, <math>\tfun</math>, <math>\pinj</math>, <math>\tinj</math>, <math>\psur</math>, <math>\tsur</math>, <math>\tbij</math> A


*
SIM_OV_TREL
<math> \frac{\textbf{H}\vdash x\in A\qquad \textbf{H}\vdash y\in B}{\textbf{H}, f\in A\;op\; B\vdash f\ovl\left\{x\mapsto y\right\}\in A\trel B} </math> where <math>\mathit{op}</math> is one of <math>\trel</math>, <math>\strel</math>, <math>\tfun</math>,<math>\tinj</math>, <math>\tsur</math>, <math>\tbij</math> A


*
SIM_OV_PFUN
<math> \frac{\textbf{H}\vdash x\in A\qquad \textbf{H}\vdash y\in B}{\textbf{H}, f\in A\;op\; B\vdash f\ovl\left\{x\mapsto y\right\}\in A\pfun B} </math> where <math>\mathit{op}</math> is one of <math>\pfun</math>, <math>\tfun</math>, <math>\pinj</math>, <math>\tinj</math>, <math>\psur</math>, <math>\tsur</math>, <math>\tbij</math> A


*
SIM_OV_TFUN
<math> \frac{\textbf{H}\vdash x\in A\qquad \textbf{H}\vdash y\in B}{\textbf{H}, f\in A\;op\; B\vdash f\ovl\left\{x\mapsto y\right\}\in A\tfun B} </math> where <math>\mathit{op}</math> is one of <math>\tfun</math>, <math>\tinj</math>, <math>\tsur</math>, <math>\tbij</math> A


INDUC_NAT
<math>\frac{\textbf{H} \;\;\vdash \;\; x\in\nat \qquad \textbf{H}, x=0 \;\;\vdash \;\; \textbf{P}(x) \qquad \textbf{H}, n\in\nat, \textbf{P}(n) \;\;\vdash \;\; \textbf{P}(n+1)}{\textbf{H} \;\;\vdash\;\; \textbf{P}(x)}</math> <math>x</math> of type <math>\intg</math> appears free in <math>\textbf{P}</math>; <math>n</math> is introduced as a fresh identifier M


INDUC_NAT_COMPL
<math>\frac{\textbf{H} \;\;\vdash \;\; x\in\nat \qquad \textbf{H} \;\;\vdash \;\; \textbf{P}(0) \qquad \textbf{H}, n\in\nat, \forall k\qdot 0\leq k\land k < n \limp \textbf{P}(k) \;\;\vdash \;\; \textbf{P}(n)}{\textbf{H} \;\;\vdash\;\; \textbf{P}(x)}</math> <math>x</math> of type <math>\intg</math> appears free in <math>\textbf{P}</math>; <math>n</math> is introduced as a fresh identifier M



Those following rules have been implemented in the reasoner GeneralizedModusPonens.

  Name Rule Side Condition A/M
*
GENMP_HYP_HYP
<math> \frac{P,\varphi(\btrue) \vdash G}{P,\varphi(P^{\dagger}) \vdash G} </math> see below for <math> P^{\dagger} </math> A
*
GENMP_NOT_HYP_HYP
<math> \frac{nP^{\dagger},\varphi(\bfalse) \vdash G}{nP^{\dagger},\varphi(P) \vdash G} </math> see below for <math> P^{\dagger} </math> A
*
GENMP_HYP_GOAL
<math> \frac{P \vdash \varphi(\btrue)}{P \vdash \varphi(P^{\dagger})} </math> see below for <math> P^{\dagger} </math> A
*
GENMP_NOT_HYP_GOAL
<math> \frac{nP^{\dagger} \vdash \varphi(\bfalse)}{nP^{\dagger} \vdash \varphi(P)} </math> see below for <math> P^{\dagger} </math> A
*
GENMP_GOAL_HYP
<math> \frac{H,\varphi(\bfalse)\vdash \lnot nG^{\dagger}}{H,\varphi(G)\vdash \lnot nG^{\dagger}} </math> see below for <math> nG^{\dagger} </math> A
*
GENMP_NOT_GOAL_HYP
<math> \frac{H,\varphi(\btrue)\vdash \lnot G}{H,\varphi(G^{\dagger})\vdash \lnot G} </math> see below for <math> G^{\dagger} </math> A
*
GENMP_OR_GOAL_HYP
<math> \frac{H,\varphi(\bfalse)\vdash G_1\lor\cdots\lor \lnot nG_i^{\dagger}\lor\cdots\lor G_n}{H,\varphi(G_i)\vdash G_1\lor\cdots\lor \lnot nG_i^{\dagger}\lor\cdots\lor G_n} </math> see below for <math> nG_i^{\dagger} </math> A
*
GENMP_OR_NOT_GOAL_HYP
<math> \frac{H,\varphi(\btrue)\vdash G_1\lor\cdots\lor\ \lnot G_i\lor\cdots\lor G_n}{H,\varphi(G_i^{\dagger})\vdash G_1\lor\cdots\lor\ \lnot G_i\lor\cdots\lor G_n} </math> see below for <math> G_i^{\dagger} </math> A


Thos following rules have been implemented in the MembershipGoal reasoner.

  Name Rule Side Condition A/M


*
SUBSET_SUBSETEQ
<math> A\subset B\vdash A\subseteq B </math> A
*
DOM_SUBSET
<math> A\subseteq B\vdash \dom(A)\subseteq\dom(B)</math> A
*
RAN_SUBSET
<math> A\subseteq B\vdash \ran(A)\subseteq\ran(B)</math> A
*
EQUAL_SUBSETEQ_LR
<math> A=B\vdash A\subseteq B</math> A
*
EQUAL_SUBSETEQ_RL
<math> A=B\vdash B\subseteq A</math> A
*
IN_DOM_CPROD
<math> x\in\dom(A\cprod B)\vdash x\in A</math> A
*
IN_RAN_CPROD
<math> y\in\ran(A\cprod B)\vdash y\in B</math> A
*
IN_DOM_REL
<math> x\mapsto y\in f\vdash x\in\dom(f)</math> A
*
IN_RAN_REL
<math> x\mapsto y\in f\vdash y\in\ran(f)</math> A
*
SETENUM_SUBSET
<math> \left\{a,\cdots,x,\cdots, z\right\}\subseteq A\vdash x\in A</math> A
*
OVR_RIGHT_SUBSET
<math> f\ovl\cdots\ovl g\ovl\cdots\ovl h\subseteq A\vdash g\ovl\cdots\ovl h\subseteq A</math> A
*
RELSET_SUBSET_CPROD
<math> f\in A\;op\;B\vdash f\subseteq A\cprod B</math> where <math>\mathit{op}</math> is one of <math>\rel</math>, <math>\trel</math>, <math>\srel</math>, <math>\strel</math>, <math>\pfun</math>, <math>\tfun</math>, <math>\pinj</math>, <math>\tinj</math>, <math>\psur</math>, <math>\tsur</math>, <math>\tbij</math> A
*
DERIV_IN_SUBSET
<math> x\in A,\;\; A\subseteq B\vdash x\in B </math> A


The conventions used in this table are described in Variations in HYP, CNTR and GenMP.

<math>\textbf{P}</math> <math>\textbf{P}^{\dagger}</math> <math>\textbf{nP}^{\dagger}</math> Side Condition
<math> a = b </math> <math> a = b, \ \ b = a </math>
<math> a \le b , \ \ b \ge a </math>
<math> a \ge b , \ \ b \le a </math>
<math> \lnot a = b, \ \ \lnot b = a </math>
<math> a > b, \ \ b < a </math>
<math> a < b, \ \ b > a </math>
where a and b are integers
<math> a < b </math> <math> a < b, \ \ b > a </math>
<math> a \le b, \ \ b \ge a </math>
<math> \lnot a = b, \ \ \lnot b = a </math>
<math> a \ge b, \ \ b \le a </math>
<math> a > b, \ \ b < a </math>
<math> a = b, \ \ b = a </math>
<math> a > b </math> <math> a > b, \ \ b < a </math>
<math> a \ge b, \ \ b \le a </math>
<math> \lnot a = b, \ \ \lnot b = a </math>
<math> a \le b, \ \ b \ge a </math>
<math> a < b, \ \ b > a </math>
<math> a = b, \ \ b = a </math>
<math> a \le b </math> <math> a \le b, \ \ b \ge a </math> <math> a > b, \ \ b < a </math>
<math> a \ge b </math> <math> a \ge b, \ \ b \le a </math> <math> a < b, \ \ b > a </math>
<math> \lnot a = b </math> <math> \lnot a = b, \ \ \lnot b = a </math> <math> a = b, \ \ b = a </math>
<math> A = B </math> <math> A = B, \ \ B = A </math>
<math> A \subseteq B, \ \ B \subseteq A </math>
<math> \lnot A \subset B, \ \ \lnot B \subset A </math>
<math> \lnot A = B, \ \ \lnot B = A </math>
<math> \lnot A \subseteq B, \ \ \lnot B \subseteq A </math>
<math> \ \ A \subset B, \ \ B \subset A </math>
where A and B are sets
<math> A \subseteq B </math> <math> A \subseteq B, \lnot B \subset A </math> <math> \lnot A \subseteq B, B \subset A </math>
<math> A \subset B </math> <math> A \subset B, \ \ A \subseteq B </math>
<math> \lnot B \subset A, \ \ \lnot B \subseteq A </math>
<math> \lnot A = B, \ \ \lnot B = A</math>
<math> \lnot A \subset B, \ \ \lnot A \subseteq B </math>
<math> B \subset A, \ \ B \subseteq A </math>
<math> A = B, \ \ B = A </math>
<math> \lnot A = B </math> <math> \lnot A = B, \ \ \lnot B = A </math> <math> A = B, \ \ B = A </math>
<math> \lnot A \subseteq B </math> <math> \lnot A \subseteq B, \ \ \lnot A \subset B </math>
<math> \lnot A = B, \ \ \lnot B = A </math>
<math> A \subseteq B, \ \ A \subset B </math>
<math> A = B, \ \ B = A </math>
<math> \lnot A \subset B </math> <math> \lnot A \subset B </math> <math> A \subset B </math>
<math> e = f </math> <math> e = f, \ \ f = e </math> <math> \lnot e = f, \ \ \lnot f = e </math> where e and f are scalars
<math> \lnot e = f </math> <math> \lnot e = f, \ \ \lnot f = e </math> <math> e = f, \ \ f = e </math>
<math> \textbf{P} </math> <math> \textbf{P} </math> <math> \lnot \textbf{P} </math>
<math> \lnot \textbf{P} </math> <math> \textbf{P} </math>


See also Extension Proof Rules#Inference Rules.