# Inference Rules

From Event-B

Revision as of 08:25, 6 September 2010 by imported>Nicolas (simplified representation of rule DATATYPE_INDUCTION (the meaning remains the same))

CAUTION! Any modification to this page shall be announced on the User mailing list!

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)#Inference_Rules.

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

* | HYP |
A
| ||

* | HYP_OR |
A
| ||

* | CNTR |
A
| ||

* | FALSE_HYP |
A
| ||

* | TRUE_GOAL |
A
| ||

* | FUN_GOAL |
where and denote types and is one of , , , , , , . | A
| |

FUN_IMAGE_GOAL |
where denotes a set of relations (any arrow) and occurs at the top level | M
| ||

FUN_GOAL_REC |
where and denote types, denotes a set of relations (any arrow) and is one of , , , , , , . | A
| ||

* | DBL_HYP |
A
| ||

* | AND_L |
A
| ||

* | AND_R |
A
| ||

IMP_L1 |
A
| |||

* | IMP_R |
A
| ||

* | IMP_AND_L |
A
| ||

* | IMP_OR_L |
A
| ||

* | AUTO_MH |
A
| ||

* | NEG_IN_L |
A
| ||

* | NEG_IN_R |
A
| ||

* | XST_L |
A
| ||

* | ALL_R |
A
| ||

* | EQL_LR |
is a variable which is not free in | A
| |

* | EQL_RL |
is a variable which is not free in | A
| |

SUBSET_INTER |
the operator must appear at the "top level" | A
| ||

IN_INTER |
the operator must appear at the "top level" | A
| ||

NOTIN_INTER |
the operator must appear at the "top level" | A
| ||

* | FIN_L_LOWER_BOUND_L |
The goal is discharged | A
| |

* | FIN_L_LOWER_BOUND_R |
The goal is discharged | A
| |

* | FIN_L_UPPER_BOUND_L |
The goal is discharged | A
| |

* | FIN_L_UPPER_BOUND_R |
The goal is discharged | A
| |

* | CONTRADICT_L |
M
| ||

* | CONTRADICT_R |
M
| ||

* | CASE |
M
| ||

* | MH |
M
| ||

* | HM |
M
| ||

EQV |
M
| |||

* | OV_SETENUM_L |
the operator must appear at the "top level" | A
| |

* | OV_SETENUM_R |
the operator must appear at the "top level" | A
| |

* | OV_L |
the operator must appear at the "top level" | A
| |

* | OV_R |
the operator must appear at the "top level" | A
| |

* | DIS_BINTER_R |
the occurrence of must appear at the "top level". Moreover and denote some type. | M
| |

* | DIS_BINTER_L |
the occurrence of must appear at the "top level". Moreover and denote some type. | M
| |

* | DIS_SETMINUS_R |
the occurrence of must appear at the "top level". Moreover and denote some type. | M
| |

* | DIS_SETMINUS_L |
the occurrence of must appear at the "top level". Moreover and denote some type. | M
| |

* | SIM_REL_IMAGE_R |
the occurrence of must appear at the "top level". | M
| |

* | SIM_REL_IMAGE_L |
the occurrence of must appear at the "top level". | M
| |

* | SIM_FCOMP_R |
the occurrence of must appear at the "top level". | M
| |

* | SIM_FCOMP_L |
the occurrence of must appear at the "top level". | M
| |

* | FIN_SUBSETEQ_R |
the user has to write the set corresponding to in the editing area of the Proof Control Window | M
| |

* | FIN_BINTER_R |
M
| ||

* | FIN_SETMINUS_R |
M
| ||

* | FIN_REL_R |
the user has to write the set corresponding to in the editing area of the Proof Control Window | M
| |

* | FIN_REL_IMG_R |
M
| ||

* | FIN_REL_RAN_R |
M
| ||

* | FIN_REL_DOM_R |
M
| ||

* | FIN_FUN1_R |
the user has to write the set corresponding to in the editing area of the Proof Control Window | M
| |

* | FIN_FUN2_R |
the user has to write the set corresponding to in the editing area of the Proof Control Window | M
| |

* | FIN_FUN_IMG_R |
the user has to write the set corresponding to in the editing area of the Proof Control Window | M
| |

* | FIN_FUN_RAN_R |
the user has to write the set corresponding to in the editing area of the Proof Control Window | M
| |

* | FIN_FUN_DOM_R |
the user has to write the set corresponding to in the editing area of the Proof Control Window | M
| |

* | LOWER_BOUND_L |
must not contain any bound variable | M
| |

* | LOWER_BOUND_R |
must not contain any bound variable | M
| |

* | UPPER_BOUND_L |
must not contain any bound variable | M
| |

* | UPPER_BOUND_R |
must not contain any bound variable | M
| |

* | FIN_LT_0 |
M
| ||

* | FIN_GE_0 |
M
| ||

CARD_INTERV |
must appear at "top-level" | M
| ||

CARD_EMPTY_INTERV |
must appear at "top-level" | M
| ||

* | DERIV_LE_CARD |
and bear the same type | M
| |

* | DERIV_GE_CARD |
and bear the same type | M
| |

* | DERIV_LT_CARD |
and bear the same type | M
| |

* | DERIV_GT_CARD |
and bear the same type | M
| |

* | DERIV_EQUAL_CARD |
and bear the same type | M
| |

SIMP_CARD_SETMINUS_L |
must appear at "top-level" | M | ||

SIMP_CARD_SETMINUS_R |
must appear at "top-level" | M
| ||

SIMP_CARD_CPROD_L |
must appear at "top-level" | M | ||

SIMP_CARD_CPROD_R |
must appear at "top-level" | M
| ||

* | FORALL_INST |
is instantiated with | M
| |

* | FORALL_INST_MP |
is instantiated with and a Modus Ponens is applied | M
| |

* | CUT |
hypothesis is added | M
| |

* | EXISTS_INST |
is instantiated with | M
| |

* | DISTINCT_CASE |
case distinction on predicate | M
| |

ONE_POINT_L |
The rule can be applied with as well as with | A
| ||

ONE_POINT_R |
The rule can be applied with as well as with | A
| ||

DATATYPE_DISTINCT_CASE |
where has a datatype as type and appears free in , has constructors , parameters are introduced as fresh identifiers | M
| ||

DATATYPE_INDUCTION |
where has inductive datatype as type and appears free in ; are the inductive parameters (if any); an antecedent is created for every constructor of ; all parameters are introduced as fresh identifiers | M |