Inference Rules

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Revision as of 16:56, 25 August 2010 by imported>Laurent (Changed REC_FUN_GOAL to FUN_GOAL_REC and fixed typing.)
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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} \;\;\vdash \;\; \textbf{P}} A


*
HYP_OR
\frac{}{\textbf{H},\textbf{Q} \;\;\vdash \;\; \textbf{P} \lor \ldots \lor  \textbf{Q} \lor \ldots \lor \textbf{R}} A


*
CNTR
\frac{}{\textbf{H},\;\textbf{P},\;\neg\,\textbf{P} \;\;\vdash \;\; \textbf{Q}} 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_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\}  \; \; \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\}  \; \; \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})} the \binter operator must appear at the "top level" 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})} the \binter operator must appear at the "top level" 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})} the \binter operator must appear at the "top level" 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


*
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}} the \ovl operator must appear at the "top level" 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))} the \ovl operator must appear at the "top level" 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}} the \ovl operator must appear at the "top level" 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))} the \ovl operator must appear at the "top level" 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])} the occurrence of f must appear at the "top level". Moreover A and B denote some type. 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}} the occurrence of f must appear at the "top level". Moreover A and B denote some type. 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])} the occurrence of f must appear at the "top level". Moreover A and B denote some type. 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}} the occurrence of f must appear at the "top level". Moreover A and B denote some type. 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\} ])} the occurrence of f must appear at the "top level". 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} } the occurrence of f must appear at the "top level". 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))} the occurrence of f \fcomp g must appear at the "top level". 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}} the occurrence of f \fcomp g must appear at the "top level". 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_SETMINUS_R
\frac{\textbf{H} \;\;\vdash 
\;\;\finite\,(S)}{\textbf{H} \;\;\vdash \;\; \finite\,(S \;\setminus\; T)} M


*
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_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))} \card (a \upto b) must appear at "top-level" 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}} \card (a \upto b) must appear at "top-level" 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}} \card (S \setminus  T) must appear at "top-level" 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))} \card (S \setminus  T) must appear at "top-level" 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}} \card (S \cprod  T) must appear at "top-level" 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))} \card (S \cprod  T) must appear at "top-level" 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


*
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


DATATYPE_DISTINCT_CASE
\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


DATATYPE_INDUCTION
\frac{\textbf{H}, x=c_N(\ldots) \;\;\vdash \;\; \textbf{P}(c_N(\ldots)) \qquad  \textbf{H}, x=c_I(p_N, p_I),\textbf{P}(p_I)  \;\;\vdash \;\; \textbf{P}(c_I(p_N, p_I)) }{ \textbf{H}  \;\;\vdash\;\;  \textbf{P}(x) } (N = Non inductive; I = Inductive) where x has inductive datatype DT as type and appears free in \textbf{P}; c are constructors of DT; an antecedent is created for each c_N and each c_I, an hypothesis \textbf{P}(p_I) is added for each p_I; all parameters are introduced as fresh identifiers M