Proof:   et   o  be a set of subgoals,    be a node in the divisibilit   tree of   o, C(   ) be a strongl  valid set of orderings of ...

If there is no min consistent ordering of   , then C(  ) is valid, b  the definition of a valid set ( ection . ).

therwise, there e ists at least one minimal ordering of  0, binder consistent with ver ordering in  ( (  ))  C(  ) is either    contradicting or blockwise e uivalent to some member of C ( ). To prove that C ( ) is valid, we must show that it contains an ordering N, which is binder consistent with some minimal ordering of o.

et be a minimal ordering of  0, binder consistent with  .  et   N be the pro ection of    on   . If    N    C(   ), we are done (  N
C( ). x cannot be contradicting (it is min consistent to ), therefore it must be blockwise e uivalent to some N C( ).   locks of  N are not violated in   , since mmmmmer of the 0" is well defined.     is minimal, since is minimal and N and N are blockwise e uivalent. is binder consistent with N since was binder consistent with N . Thereupon and N satisf the re uirements of validit  (  N
□
- N !

## T Heore M  3

Let 0 be a set of subgoals. For each node - of the divisibility tree of 0, Algorithm 6 creates a strongly valid candidate set of orderings. Proof: induction on the height of s subtree. Inductive base:    is a leaf node, which means that   (  ) is independent under   (  ).

The candidate set of contains one element, whose subgoals are sorted between 11 orderings that belong to  ( (  )) 
hence are contradicting ( emma  ), or are sorted b    , and hence are blockwise e uivalent to the candidate ( orollar   ).   onse uently valid.

Inductive hypothesis: or all children of , lgorithm produces strongl valid candi date sets.

Inductive step: n internal node in a divisibilit tree is either an N node or an node.

1. I is an N node, etc. 1. 2. I h be the children of . irst we show that
( ) is strongl  valid.

e ( ). Or all learn to be the protection of et N on i. The set of protections and 2 minutes can belong to one of the three following t  pes, with  regard to   N 
(a) The sets of the first t pe contain at least one contradicting pro ection. In such a case N is contradicting too. ssume the contrar there e ists a minimal ordering of o, binder consistent with N et ni be an contradicting pro ection. ince is consistent with N , it is also consistent