## The rightmost bit trick

A smart and efficient code to compute the rightmost bit of a given bit vector. This case study is detailed in Inria research report 8821.

**Authors:** Clément Fumex / Claude Marché

**Topics:** Ghost code / Bitwise operations

**Tools:** Why3

**References:** ProofInUse joint laboratory

see also the index (by topic, by tool, by reference, by year)

module Rmbt use int.Int use ref.Ref use bv.BV64 let ghost rightmost_position_set (a : t) : t requires { a <> zeros } ensures { ult result (64:t) } ensures { eq_sub_bv a zeros zeros result } ensures { nth_bv a result } = let i = ref zeros in while ult !i (64:t) && not (nth_bv a !i) do variant {64 - t'int !i} invariant {eq_sub_bv a zeros zeros !i} i := add !i BV64.one done; !i let rightmost_bit_trick (x: t) (ghost p : ref int) : t requires { x <> zeros } writes { p } ensures { 0 <= !p < 64 } ensures { eq_sub x zeros 0 !p } ensures { nth x !p } ensures { eq_sub result zeros 0 !p } ensures { eq_sub result zeros (!p+1) (63 - !p) } ensures { nth result !p } = let ghost p_bv = rightmost_position_set x in ghost p := t'int p_bv; assert { nth_bv (neg x) p_bv }; bw_and x (neg x) end

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# Why3 Proof Results for Project "rightmostbittrick"

## Theory "rightmostbittrick.Rmbt": fully verified

Obligations | CVC4 1.5 | CVC5 1.0.5 | Z3 4.12.2 | |

VC for rightmost_position_set | --- | --- | --- | |

split_goal_right | ||||

loop invariant init | --- | --- | 0.01 | |

loop variant decrease | --- | --- | 0.03 | |

loop invariant preservation | --- | --- | 0.02 | |

postcondition | --- | --- | 0.01 | |

postcondition | --- | --- | 0.01 | |

postcondition | --- | --- | 0.01 | |

VC for rightmost_bit_trick | --- | --- | --- | |

split_goal_right | ||||

precondition | --- | --- | 0.01 | |

assertion | --- | --- | 0.02 | |

postcondition | --- | --- | 0.02 | |

postcondition | --- | --- | 0.02 | |

postcondition | --- | 0.03 | --- | |

postcondition | --- | 0.79 | --- | |

postcondition | 0.47 | --- | --- | |

postcondition | --- | 0.55 | --- |