# Arithmetic Rewrite Rules

From Event-B

Revision as of 15:36, 17 March 2011 by imported>Laurent (Added rule SIMP_CARD_SETMINUS suggested by Alexei.)

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.

Name | Rule | Side Condition | A/M | |
---|---|---|---|---|

* | SIMP_SPECIAL_MOD_0 |
A | ||

* | SIMP_SPECIAL_MOD_1 |
A | ||

* | SIMP_MIN_SING |
where is a single expression | A | |

* | SIMP_MAX_SING |
where is a single expression | A | |

* | SIMP_MIN_NATURAL |
A | ||

* | SIMP_MIN_NATURAL1 |
A | ||

* | SIMP_MIN_BUNION_SING |
A | ||

* | SIMP_MAX_BUNION_SING |
A | ||

* | SIMP_MIN_UPTO |
A | ||

* | SIMP_MAX_UPTO |
A | ||

* | SIMP_LIT_MIN |
where and are literals and | A | |

* | SIMP_LIT_MAX |
where and are literals and | A | |

* | SIMP_SPECIAL_CARD |
A | ||

* | SIMP_CARD_SING |
where is a single expression | A | |

* | SIMP_SPECIAL_EQUAL_CARD |
A | ||

* | SIMP_CARD_POW |
A | ||

* | SIMP_CARD_BUNION |
A | ||

SIMP_CARD_SETMINUS |
with hypotheses and either or | A | ||

* | SIMP_CARD_CONVERSE |
A | ||

* | SIMP_CARD_ID |
where has type | A | |

* | SIMP_CARD_ID_DOMRES |
A | ||

* | SIMP_CARD_PRJ1 |
where has type | A | |

* | SIMP_CARD_PRJ2 |
where has type | A | |

* | SIMP_CARD_PRJ1_DOMRES |
A | ||

* | SIMP_CARD_PRJ2_DOMRES |
A | ||

* | SIMP_CARD_LAMBDA |
where is a maplet combination of bound identifiers and expressions that are not bound by the comprehension set (i.e., is syntactically injective) and all identifiers bound by the comprehension set that occur in also occur in | A | |

* | SIMP_LIT_CARD_UPTO |
where and are literals and | A | |

SIMP_TYPE_CARD |
where is a carrier set containing elements | A | ||

* | SIMP_LIT_GE_CARD_1 |
A | ||

* | SIMP_LIT_LE_CARD_1 |
A | ||

* | SIMP_LIT_LE_CARD_0 |
A | ||

* | SIMP_LIT_GE_CARD_0 |
A | ||

* | SIMP_LIT_GT_CARD_0 |
A | ||

* | SIMP_LIT_LT_CARD_0 |
A | ||

* | SIMP_LIT_EQUAL_CARD_1 |
A | ||

* | SIMP_CARD_NATURAL |
A | ||

* | SIMP_CARD_NATURAL1 |
A | ||

* | SIMP_LIT_IN_NATURAL |
where is a non-negative literal | A | |

* | SIMP_SPECIAL_IN_NATURAL1 |
A | ||

* | SIMP_LIT_IN_NATURAL1 |
where is a positive literal | A | |

* | SIMP_LIT_UPTO |
where and are literals and | A | |

* | SIMP_LIT_IN_MINUS_NATURAL |
where is a positive literal | A | |

* | SIMP_LIT_IN_MINUS_NATURAL1 |
where is a non-negative literal | A | |

* | DEF_IN_NATURAL |
M | ||

* | DEF_IN_NATURAL1 |
M | ||

* | SIMP_LIT_EQUAL_KBOOL_TRUE |
A | ||

* | SIMP_LIT_EQUAL_KBOOL_FALSE |
A | ||

DEF_EQUAL_MIN |
where non free in | M | ||

DEF_EQUAL_MAX |
where non free in | M | ||

* | SIMP_SPECIAL_PLUS |
A | ||

* | SIMP_SPECIAL_MINUS_R |
A | ||

* | SIMP_SPECIAL_MINUS_L |
A | ||

* | SIMP_MINUS_MINUS |
A | ||

* | SIMP_MINUS_UNMINUS |
where is a unary minus expression or a negative literal | M | |

* | SIMP_MULTI_MINUS |
A | ||

* | SIMP_MULTI_MINUS_PLUS_L |
M | ||

* | SIMP_MULTI_MINUS_PLUS_R |
M | ||

* | SIMP_MULTI_MINUS_PLUS_PLUS |
M | ||

* | SIMP_MULTI_PLUS_MINUS |
M | ||

* | SIMP_MULTI_ARITHREL_PLUS_PLUS |
where the root relation ( here) is one of | M | |

* | SIMP_MULTI_ARITHREL_PLUS_R |
where the root relation ( here) is one of | M | |

* | SIMP_MULTI_ARITHREL_PLUS_L |
where the root relation ( here) is one of | M | |

* | SIMP_MULTI_ARITHREL_MINUS_MINUS_R |
where the root relation ( here) is one of | M | |

* | SIMP_MULTI_ARITHREL_MINUS_MINUS_L |
where the root relation ( here) is one of | M | |

* | SIMP_SPECIAL_PROD_0 |
A | ||

* | SIMP_SPECIAL_PROD_1 |
A | ||

* | SIMP_SPECIAL_PROD_MINUS_EVEN |
if an even number of | A | |

* | SIMP_SPECIAL_PROD_MINUS_ODD |
if an odd number of | A | |

* | SIMP_LIT_MINUS |
where is a literal | A | |

* | SIMP_LIT_EQUAL |
where and are literals | A | |

* | SIMP_LIT_LE |
where and are literals | A | |

* | SIMP_LIT_LT |
where and are literals | A | |

* | SIMP_LIT_GE |
where and are literals | A | |

* | SIMP_LIT_GT |
where and are literals | A | |

* | SIMP_DIV_MINUS |
A | ||

* | SIMP_SPECIAL_DIV_1 |
A | ||

* | SIMP_SPECIAL_DIV_0 |
A | ||

* | SIMP_SPECIAL_EXPN_1_R |
A | ||

* | SIMP_SPECIAL_EXPN_1_L |
A | ||

* | SIMP_SPECIAL_EXPN_0 |
A | ||

* | SIMP_MULTI_LE |
A | ||

* | SIMP_MULTI_LT |
A | ||

* | SIMP_MULTI_GE |
A | ||

* | SIMP_MULTI_GT |
A | ||

* | SIMP_MULTI_DIV |
A | ||

* | SIMP_MULTI_DIV_PROD |
A | ||

* | SIMP_MULTI_MOD |
A | ||

DISTRI_PROD_PLUS |
M | |||

DISTRI_PROD_MINUS |
M | |||

DERIV_NOT_EQUAL |
and must be of Integer type | M |