oopart of a total order

Johan G. F. Belinfante2009 May 25

In[1]:= SetDirectory@"l:"D;

Lemma. An element v of fix[oopart[to[x]]] is minimal. That is, there is no element u less than v.

In[5]:= HMap@implies@#, equal@u, vDD &, SubstTest@member, pair@u, vD,composite@id@yD, tD, y fix@oopart@tDDDD . t to@xDL MapNotNot

Out[5]= or@equal@u, vD, not@member@v, fix@oopart@to@xDDDDD,not@member@pair@u, vD, to@xDDDD True

In[6]:= H% . 8u u_, v v_, x x_ not@member@pair@v, uD, to@xDDD,p6 not@member@pair@u, vD, cartsq@fix@to@xDDDDD fix@to@xDD

In[13]:= Map@not, SubstTest@and, implies@p1, or@p3, p4DD,implies@and@p1, p4D, p2D, implies@and@p1, p3D, p2D, not@implies@p1, p2DD,8p1 not@empty@oopart@to@xDDDD, p2 member@to@xD, range@SINGLETONDD,p3 empty@to@xDD, p4 > equal@fix@oopart@to@xDDD, fix@to@xDDD subclass@to@xD, IdD, p3 > not@equal@0, oopart@to@xDDDD

In[22]:= H% . x x_L . Equal SetDelayed

Combining the theorem of the preceding section with this converse yields a logical equivalence that can be made into arewrite rule.

Theorem.

In[23]:= equiv@empty@oopart@to@xDDD, not@member@to@xD, range@SINGLETONDDDD

Out[23]= True

In[24]:= equal@0, oopart@to@x_DDD := not@member@to@xD, range@SINGLETONDDD

replacement for the removed rule

The removed rewrite rule can now be replaced by a more direct rule.

In[25]:= SubstTest@empty, fix@oopart@po@tDDD, t to@xDD Reverse

Out[25]= equal@0, fix@oopart@to@xDDDD not@member@to@xD, range@SINGLETONDDD

In[26]:= equal@0, fix@oopart@to@x_DDDD := not@member@to@xD, range@SINGLETONDDD

oo-to-0.nb 4