Difference between revisions of "Inference Rules"

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{{RRRow}}|*||<font size="-2"> CUT </font>|| <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
 
{{RRRow}}|*||<font size="-2"> CUT </font>|| <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
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{{RRRow}}|*||<font size="-2"> EXISTS_INST </font>|| <math>\frac{\textbf{H} \;\;\vdash \;\; {WD}(E) \qquad \textbf{H} , [x \bcmeq E]\textbf{P} \;\;\vdash \;\; \textbf{G}}{\textbf{H}, \exists x \qdot \textbf{P}  \;\;\vdash\;\; \textbf{G}}</math> || <math>x</math> is instantiated with <math>E</math> || M
  
 
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Revision as of 14:26, 3 July 2009

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


* 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


* 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


* 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} \  \  \  \  \ldots \  \  \  \   \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{P},\; \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{Q},\; \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_L \frac{\textbf{H},\; G=E  
,\;\textbf{P}(F)\;\;\vdash\;\;\textbf{Q} \qquad \textbf{H},\; \neg\,(G=E)  
,\;\textbf{P}(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" M


* OV_R \frac{\textbf{H},\; G=E \;\;\vdash\;\;\textbf{Q}(F) 
\qquad \textbf{H},\; \neg\,(G=E)  \;\;\vdash\;\;\textbf{Q}(f(G))}{\textbf{H} 
\;\;\vdash \;\; \textbf{Q}((f\ovl\{E \mapsto F\})(G)) \ \ \ \ \ } the \ovl operator must appear at the "top level" M


* 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}(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" M


* 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}(f(G))}{\textbf{H} \;\;\vdash \;\; \textbf{Q}((f\ovl g)(G)) \ \ \ \ \ } the \ovl operator must appear at the "top level" M


* 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^{-1} must appear at the "top level". Moreover A and B denote some type. Similar left distribution rules exist M


* DIS_SETMINUS_R \frac{\textbf{H} \;\;\vdash\;\; f \in A \pfun B    \qquad\textbf{H} \;\;\vdash\;\;\textbf{Q}(f^{-1}[S] \setminus f^{-1}[T]) }{\textbf{H} \;\;\vdash \;\; \textbf{Q}(f^{-1}[S \setminus T]) \ \ \ \ \ } the occurrence of f^{-1} must appear at the "top level". Moreover A and B denote some type. Similar left distribution rules exist 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". A similar left simplification rule exists. 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". A similar left simplification rule exists. M


* FIN_SUBSETEQ_R \frac{\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 \;\; 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]) \ \ \ \ \ \ \ } the user has to write the set corresponding to S  \pfun T in the editing area of the Proof Control Window M


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


* FIN_REL_DOM_R \frac{\textbf{H} \;\;\vdash \;\; \finite\,(r) }{\textbf{H} \;\;\vdash \;\; \finite\,(\dom(r)) \ \ \ \ \ \ \ } the user has to write the set corresponding to S  \pfun T in the editing area of the Proof Control Window M


* FIN_FUN1_R \frac{\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 \;\; 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 \;\; 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 \;\; 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 \;\; 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}(a-b+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


* CARD_SUBSETEQ \frac{\textbf{H}  \;\;\vdash \;\; \textbf{P}(S \subseteq T)  }{\textbf{H} \;\;\vdash\;\; \textbf{P}(\card\,(S) \leq \card(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


* 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} , [x \bcmeq E]\textbf{P} \;\;\vdash \;\; \textbf{G}}{\textbf{H}, \exists x \qdot \textbf{P}  \;\;\vdash\;\; \textbf{G}} x is instantiated with E M