Empty Set Rewrite 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.

All rewrite rules that match the pattern \textbf{P}=\emptyset are also applicable to predicates of the form \textbf{P}\subseteq\emptyset and \emptyset=\textbf{P}, as these predicates are equivalent. All rewrite rules that match the pattern \textbf{P}=\mathit{Ty} are also applicable to predicates of the form \mathit{Ty}\subseteq\textbf{P} and \mathit{Ty}=\textbf{P}, as these predicates are equivalent.


  Name Rule Side Condition A/M
*
DEF_SPECIAL_NOT_EQUAL
  \lnot\, S = \emptyset  \;\;\defi\;\;  \exists x \qdot  x \in  S where x is not free in S M
*
SIMP_SETENUM_EQUAL_EMPTY
  \{ A, \ldots , B\}  = \emptyset \;\;\defi\;\;  \bfalse A
*
SIMP_SPECIAL_EQUAL_COMPSET
  \{  x \qdot  P(x) \mid  E \}  = \emptyset  \;\;\defi\;\;  \forall x\qdot  \lnot\, P(x) A
*
SIMP_BINTER_EQUAL_TYPE
  A \binter \cdots \binter B = \mathit{Ty} \;\;\defi\;\;  A = \mathit{Ty} \land \cdots \land B  = \mathit{Ty} where \mathit{Ty} is a type expression A
*
SIMP_BINTER_SING_EQUAL_EMPTY
A\binter\cdots\binter\{a\}\binter\cdots\binter B  = \emptyset  \;\;\defi\;\;  \lnot\, a \in A\binter\cdots\binter B A
*
SIMP_BINTER_SETMINUS_EQUAL_EMPTY
A\binter\cdots\binter(B\setminus C)\binter\cdots\binter D  = \emptyset  \;\;\defi\;\;  (A\binter\cdots\binter B\binter\cdots\binter D) \setminus C = \emptyset A
*
SIMP_BUNION_EQUAL_EMPTY
  A \bunion \cdots \bunion B = \emptyset \;\;\defi\;\;  A = \emptyset \land \cdots \land B  = \emptyset A
*
SIMP_SETMINUS_EQUAL_EMPTY
  A \setminus  B = \emptyset \;\;\defi\;\;  A \subseteq  B  A
*
SIMP_SETMINUS_EQUAL_TYPE
  A \setminus  B = \mathit{Ty} \;\;\defi\;\;  A = \mathit{Ty} \land B = \emptyset where \mathit{Ty} is a type expression A
*
SIMP_POW_EQUAL_EMPTY
  \pow (S) = \emptyset \;\;\defi\;\;  \bfalse  A
*
SIMP_POW1_EQUAL_EMPTY
  \pown (S) = \emptyset \;\;\defi\;\;  S = \emptyset  A
*
SIMP_KINTER_EQUAL_TYPE
   \inter (S) = \mathit{Ty} \;\;\defi\;\;  S = \{ \mathit{Ty} \}  where \mathit{Ty} is a type expression A
*
SIMP_KUNION_EQUAL_EMPTY
   \union (S) = \emptyset \;\;\defi\;\;  S \subseteq \{ \emptyset \}  A
*
SIMP_QINTER_EQUAL_TYPE
  (\Inter  x\qdot P(x)  \mid  E(x)) = \mathit{Ty} \;\;\defi\;\;  \forall x\qdot  P(x) \limp E(x) = \mathit{Ty} where \mathit{Ty} is a type expression A
*
SIMP_QUNION_EQUAL_EMPTY
  (\Union  x\qdot P(x)  \mid  E(x)) = \emptyset \;\;\defi\;\;  \forall x\qdot  P(x) \limp E(x) = \emptyset A
*
SIMP_NATURAL_EQUAL_EMPTY
  \nat = \emptyset \;\;\defi\;\;  \bfalse A
*
SIMP_NATURAL1_EQUAL_EMPTY
  \natn = \emptyset \;\;\defi\;\;  \bfalse A
*
SIMP_TYPE_EQUAL_EMPTY
 \mathit{Ty} = \emptyset  \;\;\defi\;\;  \bfalse where \mathit{Ty} is a type expression A
*
SIMP_CPROD_EQUAL_EMPTY
  S \cprod T = \emptyset \;\;\defi\;\; S = \emptyset \lor T = \emptyset A
*
SIMP_CPROD_EQUAL_TYPE
  S \cprod T = \mathit{Ty} \;\;\defi\;\; S = \mathit{Ta} \land T = \mathit{Tb} where \mathit{Ty} is a type expression equal to \mathit{Ta} \cprod \mathit{Tb} A
*
SIMP_UPTO_EQUAL_EMPTY
  i \upto j = \emptyset \;\;\defi\;\; i > j A
*
SIMP_UPTO_EQUAL_INTEGER
  i \upto j = \intg \;\;\defi\;\; \bfalse A
*
SIMP_UPTO_EQUAL_NATURAL
  i \upto j = \nat \;\;\defi\;\; \bfalse A
*
SIMP_UPTO_EQUAL_NATURAL1
  i \upto j = \natn \;\;\defi\;\; \bfalse A
*
SIMP_SPECIAL_EQUAL_REL
  A \rel  B = \emptyset  \;\;\defi\;\;  \bfalse idem for operators \pfun  \pinj A
SIMP_TYPE_EQUAL_REL
  A \rel  B = \mathit{Ty}  \;\;\defi\;\;  A = \mathit{Ta} \land B = \mathit{Tb} where \mathit{Ty} is a type expression equal to \mathit{Ta} \cprod \mathit{Tb} A
*
SIMP_SPECIAL_EQUAL_RELDOM
  A \trel  B = \emptyset  \;\;\defi\;\;  \lnot\, A = \emptyset  \land  B = \emptyset idem for operator \tfun A
SIMP_TYPE_EQUAL_RELDOMRAN
  A \trel  B = \mathit{Ty}  \;\;\defi\;\;  \bfalse where \mathit{Ty} is a type expression, idem for operator \srel, \strel, \tfun, \tinj, \psur, \tsur, \tbij A
*
SIMP_SREL_EQUAL_EMPTY
  A \srel B = \emptyset \;\;\defi\;\; A = \emptyset \land  \lnot\,B = \emptyset A
*
SIMP_STREL_EQUAL_EMPTY
  A \strel B = \emptyset \;\;\defi\;\; (A = \emptyset \leqv  \lnot\,B = \emptyset) A
*
SIMP_DOM_EQUAL_EMPTY
  \dom (r) = \emptyset \;\;\defi\;\; r = \emptyset  A
*
SIMP_RAN_EQUAL_EMPTY
  \ran (r) = \emptyset \;\;\defi\;\; r = \emptyset  A
*
SIMP_FCOMP_EQUAL_EMPTY
 p \fcomp q = \emptyset \;\;\defi\;\; \ran (p) \binter \dom (q) = \emptyset  A
*
SIMP_BCOMP_EQUAL_EMPTY
 p \bcomp q = \emptyset \;\;\defi\;\; \ran (q) \binter \dom (p) = \emptyset  A
*
SIMP_DOMRES_EQUAL_EMPTY
 S \domres r = \emptyset \;\;\defi\;\; \dom (r) \binter S = \emptyset  A
*
SIMP_DOMRES_EQUAL_TYPE
 S \domres r = \mathit{Ty} \;\;\defi\;\; S = \mathit{Ta} \land r = \mathit{Ty}  where \mathit{Ty} is a type expression equal to \mathit{Ta} \cprod \mathit{Tb} A
*
SIMP_DOMSUB_EQUAL_EMPTY
 S \domsub r = \emptyset \;\;\defi\;\; \dom (r) \subseteq S A
*
SIMP_DOMSUB_EQUAL_TYPE
 S \domsub r = \mathit{Ty} \;\;\defi\;\; S = \emptyset \land r = \mathit{Ty} where \mathit{Ty} is a type expression A
*
SIMP_RANRES_EQUAL_EMPTY
 r \ranres S = \emptyset \;\;\defi\;\; \ran (r) \binter S = \emptyset A
*
SIMP_RANRES_EQUAL_TYPE
 r \ranres S = \mathit{Ty} \;\;\defi\;\; S = \mathit{Tb} \land r = \mathit{Ty} where \mathit{Ty} is a type expression equal to \mathit{Ta} \cprod \mathit{Tb} A
*
SIMP_RANSUB_EQUAL_EMPTY
 r \ransub S = \emptyset \;\;\defi\;\; \ran (r) \subseteq S A
*
SIMP_RANSUB_EQUAL_TYPE
 r \ransub S = \mathit{Ty} \;\;\defi\;\; S = \emptyset \land r = \mathit{Ty} where \mathit{Ty} is a type expression A
*
SIMP_CONVERSE_EQUAL_EMPTY
 r^{-1} = \emptyset \;\;\defi\;\; r = \emptyset A
*
SIMP_CONVERSE_EQUAL_TYPE
 r^{-1} = \mathit{Ty} \;\;\defi\;\; r = \mathit{Ty}^{-1} where \mathit{Ty} is a type expression A
*
SIMP_RELIMAGE_EQUAL_EMPTY
 r[S] = \emptyset \;\;\defi\;\; S \domres r = \emptyset A
*
SIMP_OVERL_EQUAL_EMPTY
  r \ovl \cdots \ovl s = \emptyset \;\;\defi\;\; r = \emptyset \land \cdots \land s =  \emptyset A
*
SIMP_DPROD_EQUAL_EMPTY
  p \dprod q = \emptyset \;\;\defi\;\; \dom (p) \binter \dom (q) = \emptyset A
*
SIMP_DPROD_EQUAL_TYPE
  p \dprod q = \mathit{Ty} \;\;\defi\;\; p = \mathit{Ta} \cprod \mathit{Tb} \land q = \mathit{Ta} \cprod \mathit{Tc} where \mathit{Ty} is a type expression equal to \mathit{Ta} \cprod (\mathit{Tb} \cprod \mathit{Tc}) A
*
SIMP_PPROD_EQUAL_EMPTY
  p \pprod q = \emptyset \;\;\defi\;\; p = \emptyset \lor q = \emptyset A
*
SIMP_PPROD_EQUAL_TYPE
  p \pprod q = \mathit{Ty} \;\;\defi\;\;  p = \mathit{Ta} \cprod \mathit{Tc} \land q = \mathit{Tb} \cprod \mathit{Td} where \mathit{Ty} is a type expression equal to (\mathit{Ta} \cprod \mathit{Tb}) \cprod (\mathit{Tc} \cprod \mathit{Td}) A
*
SIMP_ID_EQUAL_EMPTY
  \id = \emptyset \;\;\defi\;\; \bfalse A
*
SIMP_PRJ1_EQUAL_EMPTY
  \prjone = \emptyset \;\;\defi\;\; \bfalse A
*
SIMP_PRJ2_EQUAL_EMPTY
  \prjtwo = \emptyset \;\;\defi\;\; \bfalse A