Empty Set Rewrite Rules: Difference between revisions
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imported>Josselin Update introduction |
imported>Josselin Added rules SIMP_BINTER_SING_EQUAL_EMPTY, SIMP_BINTER_SETMINUS_EQUAL_EMPTY |
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{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_EQUAL_COMPSET}}||<math> \{ x \qdot P(x) \mid E \} = \emptyset \;\;\defi\;\; \forall x\qdot \lnot\, P(x) </math>|| || A | {{RRRow}}|*||{{Rulename|SIMP_SPECIAL_EQUAL_COMPSET}}||<math> \{ x \qdot P(x) \mid E \} = \emptyset \;\;\defi\;\; \forall x\qdot \lnot\, P(x) </math>|| || A | ||
{{RRRow}}|||{{Rulename|SIMP_BINTER_EQUAL_TYPE}}||<math> A \binter \ldots \binter B = \mathit{Ty} \;\;\defi\;\; A = \mathit{Ty} \land \ldots \land B = \mathit{Ty} </math>|| where <math>\mathit{Ty}</math> is a type expression || A | {{RRRow}}|||{{Rulename|SIMP_BINTER_EQUAL_TYPE}}||<math> A \binter \ldots \binter B = \mathit{Ty} \;\;\defi\;\; A = \mathit{Ty} \land \ldots \land B = \mathit{Ty} </math>|| where <math>\mathit{Ty}</math> is a type expression || A | ||
{{RRRow}}|||{{Rulename|SIMP_BINTER_SING_EQUAL_EMPTY}}||<math> A \binter \{ a \} = \emptyset \;\;\defi\;\; \lnot\, a \in A </math>|| || A | |||
{{RRRow}}|||{{Rulename|SIMP_BINTER_SETMINUS_EQUAL_EMPTY}}||<math> (A \setminus B) \binter C = \emptyset \;\;\defi\;\; (A \binter C) \setminus B = \emptyset</math>|| || A | |||
{{RRRow}}|||{{Rulename|SIMP_BUNION_EQUAL_EMPTY}}||<math> A \bunion \ldots \bunion B = \emptyset \;\;\defi\;\; A = \emptyset \land \ldots \land B = \emptyset </math>|| || A | {{RRRow}}|||{{Rulename|SIMP_BUNION_EQUAL_EMPTY}}||<math> A \bunion \ldots \bunion B = \emptyset \;\;\defi\;\; A = \emptyset \land \ldots \land B = \emptyset </math>|| || A | ||
{{RRRow}}|||{{Rulename|SIMP_SETMINUS_EQUAL_EMPTY}}||<math> A \setminus B = \emptyset \;\;\defi\;\; A \subseteq B </math>|| || A | {{RRRow}}|||{{Rulename|SIMP_SETMINUS_EQUAL_EMPTY}}||<math> A \setminus B = \emptyset \;\;\defi\;\; A \subseteq B </math>|| || A |
Revision as of 13:27, 22 May 2013
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 are also applicable to predicates of the form and , as these predicates are equivalent. All rewrite rules that match the pattern are also applicable to predicates of the form and , as these predicates are equivalent.
Name | Rule | Side Condition | A/M | |
---|---|---|---|---|
* | DEF_SPECIAL_NOT_EQUAL |
where is not free in | M | |
SIMP_SETENUM_EQUAL_EMPTY |
A | |||
* | SIMP_SPECIAL_EQUAL_COMPSET |
A | ||
SIMP_BINTER_EQUAL_TYPE |
where is a type expression | A | ||
SIMP_BINTER_SING_EQUAL_EMPTY |
A | |||
SIMP_BINTER_SETMINUS_EQUAL_EMPTY |
A | |||
SIMP_BUNION_EQUAL_EMPTY |
A | |||
SIMP_SETMINUS_EQUAL_EMPTY |
A | |||
SIMP_SETMINUS_EQUAL_TYPE |
where is a type expression | A | ||
SIMP_POW_EQUAL_EMPTY |
A | |||
SIMP_POW1_EQUAL_EMPTY |
A | |||
SIMP_KINTER_EQUAL_TYPE |
where is a type expression | A | ||
SIMP_KUNION_EQUAL_EMPTY |
A | |||
SIMP_QINTER_EQUAL_TYPE |
where is a type expression | A | ||
SIMP_QUNION_EQUAL_EMPTY |
A | |||
SIMP_NATURAL_EQUAL_EMPTY |
A | |||
SIMP_NATURAL1_EQUAL_EMPTY |
A | |||
* | SIMP_TYPE_EQUAL_EMPTY |
where is a type expression | A | |
SIMP_CPROD_EQUAL_EMPTY |
A | |||
SIMP_CPROD_EQUAL_TYPE |
where is a type expression equal to | A | ||
SIMP_UPTO_EQUAL_EMPTY |
A | |||
SIMP_UPTO_EQUAL_INTEGER |
A | |||
SIMP_UPTO_EQUAL_NATURAL |
A | |||
SIMP_UPTO_EQUAL_NATURAL1 |
A | |||
* | SIMP_SPECIAL_EQUAL_REL |
idem for operators | A | |
SIMP_TYPE_EQUAL_REL |
where is a type expression equal to | A | ||
* | SIMP_SPECIAL_EQUAL_RELDOM |
idem for operator | A | |
SIMP_TYPE_EQUAL_RELDOMRAN |
where is a type expression, idem for operator | A | ||
SIMP_SREL_EQUAL_EMPTY |
A | |||
SIMP_STREL_EQUAL_EMPTY |
A | |||
SIMP_DOM_EQUAL_EMPTY |
A | |||
SIMP_RAN_EQUAL_EMPTY |
A | |||
SIMP_FCOMP_EQUAL_EMPTY |
A | |||
SIMP_BCOMP_EQUAL_EMPTY |
A | |||
SIMP_DOMRES_EQUAL_EMPTY |
A | |||
SIMP_DOMRES_EQUAL_TYPE |
where is a type expression equal to | A | ||
SIMP_DOMSUB_EQUAL_EMPTY |
A | |||
SIMP_DOMSUB_EQUAL_TYPE |
where is a type expression | A | ||
SIMP_RANRES_EQUAL_EMPTY |
A | |||
SIMP_RANRES_EQUAL_TYPE |
where is a type expression equal to | A | ||
SIMP_RANSUB_EQUAL_EMPTY |
A | |||
SIMP_RANSUB_EQUAL_TYPE |
where is a type expression | A | ||
SIMP_CONVERSE_EQUAL_EMPTY |
A | |||
SIMP_CONVERSE_EQUAL_TYPE |
where is a type expression | A | ||
SIMP_RELIMAGE_EQUAL_EMPTY |
A | |||
SIMP_OVERL_EQUAL_EMPTY |
A | |||
SIMP_DPROD_EQUAL_EMPTY |
A | |||
SIMP_DPROD_EQUAL_TYPE |
where is a type expression equal to | A | ||
SIMP_PPROD_EQUAL_EMPTY |
A | |||
SIMP_PPROD_EQUAL_TYPE |
where is a type expression equal to | A | ||
SIMP_ID_EQUAL_EMPTY |
A | |||
SIMP_PRJ1_EQUAL_EMPTY |
A | |||
SIMP_PRJ2_EQUAL_EMPTY |
A |