Thm* L:T List, P:(T T  ).
Thm* ( x,y:T. P(x,y)  P(y,x))
Thm* 
Thm* ( L':T List.
Thm* ((L guarded_permutation(T; L,i. P(L[i],L[(i+1)])) L')
Thm* (& ( i: (||L'||-1). P(L'[i],L'[(i+1)]))) | [partial_sort_exists_2] |
Thm* P:(L:(T List)  (||L||-1)  ), m:((T List)  ).
Thm* ( L:T List, i: (||L||-1).
Thm* (P(L,i)  P(swap(L;i;i+1),i) & m(swap(L;i;i+1))<m(L))
Thm* 
Thm* ( L:T List.
Thm* ( L':T List.
Thm* ((L guarded_permutation(T; L,i. P(L,i)) L') & ( i: (||L'||-1). P(L',i))) | [partial_sort_exists] |
Thm* P:(T  ), T':Type, f:({x:T| P(x) } T'), L:T List, x:T'.
Thm* (x mapfilter(f;P;L))  ( y:T. (y L) & P(y) & x = f(y)) | [member_map_filter] |
Thm* T:Type, P:(T  ), T':Type, f:({x:T| P(x) } T'), L:T List.
Thm* mapfilter(f;P;L) T' List | [mapfilter_wf] |
Thm* l:T List, P:(T  ), x,y:T.
Thm* x before y filter(P;l)  P(x) & P(y) & x before y l | [l_before_filter] |
Thm* L1,L2:T List, P:(T  ). L2 filter(P;L1)  L2 L1 & ( x L2.P(x)) | [sublist_filter] |
Thm* L:T List, P:(T  ). ( x L.P(x))  reduce( x,y. P(x) y;true ;L) | [l_all_reduce] |
Thm* P:(T  ), L:T List. ( x L. P(x))  (filter(P;L) ~ nil) | [filter_is_nil] |
Thm* P:(T  ), L:T List. ( x L.P(x))  filter(P;L) = L | [filter_trivial2] |
Thm* P:(T  ), L:T List. ( x L.P(x))  (filter(P;L) ~ L) | [filter_trivial] |