# Difference between revisions of "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
$\frac{}{\textbf{H},\textbf{P}^{\dagger} \;\;\vdash \;\; \textbf{P}}$ see below for $\textbf{P}^{\dagger}$ A

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

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

*
FALSE_HYP
$\frac{}{\textbf{H},\bfalse \;\;\vdash \;\; \textbf{P}}$ A

*
TRUE_GOAL
$\frac{}{\textbf{H} \;\;\vdash \;\; \btrue}$ A

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

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

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

*
DBL_HYP
$\frac{\textbf{H},\;\textbf{P} \;\;\vdash \;\; \textbf{Q}}{\textbf{H},\;\textbf{P},\;\textbf{P} \;\;\vdash \;\; \textbf{Q}}$ A

*
AND_L
$\frac{\textbf{H},\textbf{P},\textbf{Q} \; \; \vdash \; \; \textbf{R}}{\textbf{H},\; \textbf{P} \land \textbf{Q} \; \; \vdash \; \; \textbf{R}}$ A

*
AND_R
$\frac{\textbf{H} \; \; \vdash \; \; \textbf{P} \qquad \textbf{H} \; \; \vdash \; \; \textbf{Q}}{\textbf{H} \; \; \vdash \; \; \textbf{P} \; \land \; \textbf{Q}}$ A

IMP_L1
$\frac{\textbf{H},\; \textbf{Q},\; \textbf{P} \land \ldots \land \textbf{R} \limp \textbf{S} \;\;\vdash \;\; \textbf{T}}{\textbf{H},\; \textbf{Q},\; \textbf{P} \land \ldots \land \textbf{Q} \land \ldots \land \textbf{R} \limp \textbf{S} \;\;\vdash \;\; \textbf{T} }$ A

*
IMP_R
$\frac{\textbf{H}, \textbf{P} \;\;\vdash \;\; \textbf{Q}}{\textbf{H} \;\;\vdash \;\; \textbf{P} \limp \textbf{Q}}$ A

*
IMP_AND_L
$\frac{\textbf{H},\textbf{P} \limp \textbf{Q}, \textbf{P} \limp \textbf{R}\;\;\vdash \;\; \textbf{S}}{\textbf{H},\;\textbf{P} \limp \textbf{Q} \land \textbf{R} \;\;\vdash \;\; \textbf{S}}$ A

*
IMP_OR_L
$\frac{ \textbf{H},\textbf{P} \limp \textbf{R}, \textbf{Q} \limp \textbf{R}\;\;\vdash \;\; \textbf{S} }{\textbf{H},\;\textbf{P} \lor \textbf{Q} \limp \textbf{R} \;\;\vdash \;\; \textbf{S}}$ A

*
AUTO_MH
$\frac{ \textbf{H},\textbf{P},\;\textbf{Q}\limp \textbf{R}\;\;\vdash \;\; \textbf{S} }{\textbf{H},\;\textbf{P},\; \textbf{P} \land \textbf{Q} \limp \textbf{R} \;\;\vdash \;\; \textbf{S}}$ A

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

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

*
XST_L
$\frac{\textbf{H},\; \textbf{P(x)} \; \; \vdash \; \; \textbf{Q} }{ \textbf{H},\; \exists \, \textbf{x}\, \qdot\, \textbf{P(x)} \; \; \vdash \; \; \textbf{Q} }$ A

*
ALL_R
$\frac{\textbf{H}\; \; \vdash \; \; \textbf{P(x)} }{ \textbf{H} \; \; \vdash \; \; \forall \textbf{x}\, \qdot\, \textbf{P(x)} }$ A

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

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

SUBSET_INTER
$\frac{\textbf{H},\;\textbf{T} \subseteq \textbf{U} \;\;\vdash \;\; \textbf{G}(\textbf{S} \binter \dots \binter \textbf{T} \binter \dots \binter \textbf{V})} {\textbf{H},\;\textbf{T} \subseteq \textbf{U} \;\;\vdash \;\; \textbf{G}(\textbf{S} \binter \dots \binter \textbf{T} \binter \dots \binter \textbf{U} \binter \dots \binter \textbf{V})}$ where $\mathbf{T}$ and $\mathbf{U}$ are not bound by $\mathbf{G}$ A

IN_INTER
$\frac{\textbf{H},\;\textbf{E} \in \textbf{T} \;\;\vdash \;\; \textbf{G}(\textbf{S} \binter \dots \binter \{\textbf{E}\} \binter \dots \binter \textbf{U})} {\textbf{H},\;\textbf{E} \in \textbf{T} \;\;\vdash \;\; \textbf{G}(\textbf{S} \binter \dots \binter \{\textbf{E}\} \binter \dots \binter \textbf{T} \binter \dots \binter \textbf{U})}$ where $\mathbf{E}$ and $\mathbf{T}$ are not bound by $\mathbf{G}$ A

NOTIN_INTER
$\frac{\textbf{H},\;\lnot\;\textbf{E} \in \textbf{T} \;\;\vdash \;\; \textbf{G}(\emptyset)} {\textbf{H},\;\lnot\;\textbf{E} \in \textbf{T} \;\;\vdash \;\; \textbf{G}(\textbf{S} \binter \dots \binter \{\textbf{E}\} \binter \dots \binter \textbf{T} \binter \dots \binter \textbf{U})}$ where $\mathbf{E}$ and $\mathbf{T}$ are not bound by $\mathbf{G}$ A

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

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

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

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

*
CONTRADICT_L
$\frac{\textbf{H},\;\neg\,\textbf{Q} \;\;\vdash \;\; \neg\,\textbf{P}}{\textbf{H},\;\textbf{P} \;\;\vdash \;\; \textbf{Q}}$ M

*
CONTRADICT_R
$\frac{\textbf{H},\;\neg\,\textbf{Q} \;\;\vdash \;\; \bfalse}{\textbf{H} \;\;\vdash \;\; \textbf{Q}}$ M

*
CASE
$\frac{\textbf{H}, \; \textbf{P} \; \; \vdash \; \; \textbf{R} \qquad\ldots\qquad \textbf{H}, \; \textbf{Q} \; \; \vdash \; \; \textbf{R} }{\textbf{H},\; \textbf{P} \lor \ldots \lor \textbf{Q} \; \; \vdash \; \; \textbf{R} }$ M

*
IMP_CASE
$\frac{\textbf{H}, \; \lnot\textbf{P} \; \; \vdash \; \; \textbf{R} \qquad \textbf{H}, \; \textbf{Q} \; \; \vdash \; \; \textbf{R} }{\textbf{H},\; \textbf{P} \limp\textbf{Q} \; \; \vdash \; \; \textbf{R} }$ M

*
MH
$\frac{\textbf{H} \;\;\vdash\;\;\textbf{P} \qquad \textbf{H},\; \textbf{Q} \;\;\vdash \;\; \textbf{R} }{\textbf{H},\;\textbf{P} \limp \textbf{Q} \;\;\vdash \;\; \textbf{R}}$ M

*
HM
$\frac{\textbf{H} \;\;\vdash\;\;\neg\,\textbf{Q} \qquad \textbf{H},\; \neg\,\textbf{P} \;\;\vdash \;\; \textbf{R} }{\textbf{H},\;\textbf{P} \limp \textbf{Q} \;\;\vdash \;\; \textbf{R}}$ M

EQV
$\frac{\textbf{H(Q)},\; \textbf{P} \leqv \textbf{Q} \;\;\vdash\;\; \textbf{G(Q)}}{\textbf{H(P)},\;\textbf{P} \leqv \textbf{Q} \;\;\vdash \;\; \textbf{G(P)}}$ M

*
OV_SETENUM_L
$\frac{\textbf{H},\; G=E ,\;\textbf{P}(F)\;\;\vdash\;\;\textbf{Q} \qquad \textbf{H},\; \neg\,(G=E) ,\;\textbf{P}((\{E\}) \domsub f)(G))\;\;\vdash\;\;\textbf{Q}}{\textbf{H},\;\textbf{P}((f\ovl\{E \mapsto F\})(G)) \;\;\vdash \;\; \textbf{Q}}$ where $\mathbf{P}$ is WD strict A

*
OV_SETENUM_R
$\frac{\textbf{H},\; G=E \;\;\vdash\;\;\textbf{Q}(F) \qquad \textbf{H},\; \neg\,(G=E) \;\;\vdash\;\;\textbf{Q}((\{E\}) \domsub f)(G))}{\textbf{H} \;\;\vdash \;\; \textbf{Q}((f\ovl\{E \mapsto F\})(G))}$ where $\mathbf{Q}$ is WD strict A

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

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

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

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

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

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

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

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

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

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

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

*
FIN_BINTER_R
$\frac{\textbf{H} \;\;\vdash \;\;\finite\,(S) \;\lor\;\ldots \;\lor\; \finite\,(T)}{\textbf{H} \;\;\vdash \;\; \finite\,(S \;\binter\;\ldots \;\binter\; T)}$ M

FIN_KINTER_R
$\frac{\textbf{H} \;\;\vdash \;\;\exists s\, \qdot\, s \in S \land \finite\,(s)}{\textbf{H} \;\;\vdash \;\; \finite\,(\inter(S))}$ where $s$ is fresh M

FIN_QINTER_R
$\frac{\textbf{H} \;\;\vdash \;\;\exists s\, \qdot\, P \land \finite\,(E)}{\textbf{H} \;\;\vdash \;\; \finite\,(\Inter s\,\qdot\,P\,\mid\,E)}$ M

*
FIN_SETMINUS_R
$\frac{\textbf{H} \;\;\vdash \;\;\finite\,(S)}{\textbf{H} \;\;\vdash \;\; \finite\,(S \;\setminus\; T)}$ M

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

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

*
FIN_REL_IMG_R
$\frac{\textbf{H} \;\;\vdash \;\; \finite\,(r) }{\textbf{H} \;\;\vdash \;\; \finite\,(r[s])}$ M

*
FIN_REL_RAN_R
$\frac{\textbf{H} \;\;\vdash \;\; \finite\,(r) }{\textbf{H} \;\;\vdash \;\; \finite\,(\ran(r))}$ M

*
FIN_REL_DOM_R
$\frac{\textbf{H} \;\;\vdash \;\; \finite\,(r) }{\textbf{H} \;\;\vdash \;\; \finite\,(\dom(r))}$ M

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Those following rules have been implemented in the reasoner GeneralizedModusPonens.

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

Thos following rules have been implemented in the MembershipGoal reasoner.

Name Rule Side Condition A/M

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

Those corresponds to variants of $\textbf{P}$ and $\lnot \textbf{P}$. We consider $\textbf{P} \limp \textbf{P}^{\dagger}$ and $\lnot \textbf{P} \limp \textbf{nP}^{\dagger}$. Moreover, the default value of $\textbf{P}^{\dagger}$ is $\textbf{P}$ and this of $\textbf{nP}^{\dagger}$ is $\lnot \textbf{P}$ .

$\textbf{P}$ $\textbf{P}^{\dagger}$ $\textbf{nP}^{\dagger}$
$a = b$ $b = a, a \le b, b \ge a , a \ge b, b \le a$ $\lnot b = a$
$a < b$ $b > a, a \le b, b \ge a, \lnot a = b, \lnot b = a$ $a > b, a \ge b, b \le a, a = b, b = a$
$a > b$ $b < a, a \ge b, b \le a, \lnot a = b, \lnot b = a$ $a < b, a \le b, b \ge a, a = b, b = a$
$a \le b$ $b \ge a$ $a > b, b < a$
$a \ge b$ $b \le a$ $a < b, b > a$
$a \subseteq b$ $\lnot b \subset a$ $b \subset a$
$a \subset b$ $a \subseteq b, \lnot b \subset a, \lnot b \subseteq a$
$\lnot a = b$ $b = a$