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


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