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## Sorted Lists

Some theorems about sorted lists, based on the above implementation of permutations.

``` ** Module lib_list_sorted Imports lib_list_PermII
* lengthisahomomorphismcons = ... :
{A|Type}{l|list A}{n|nat}(Eq (suc n) (length l))->
Ex ([a:A]Eq l (cons a (tail l)))
Rlist = ... : {A|Type}(Rel A A)->A->Pred (list A)
Sorted = ... : {A|Type}(Rel A A)->Pred (list A)
SortSpec = ... : {A|Type}(Rel A A)->Rel (list A) (list A)
* sortedlistinduction = ... :
{A|Type}{R:Rel A A}{B|Type}{phi:Rel (list A) B}{n|B}
{c|A->(list A)->B->B}(phi (nil A) n)->
({a|A}{l|list A}{b|B}(Sorted R (cons a l))->(phi l b)->
phi (cons a l) (c a l b))->{l:list A}(Sorted R l)->
phi l (list_rec n ([a:A][l'12:list A][b:B]c a l'12 b) l)
* nilSorted = ... : {A|Type}{R:Rel A A}Sorted R (nil A)
* nilRlist = ... : {A|Type}{R:Rel A A}{c|A}Rlist R c (nil A)
* heredSortedlemma = ... :
{A|Type}{R:Rel A A}{b|A}{n|list A}(Sorted R (cons b n))->Sorted R n
* SortedImpliesRlist = ... :
{A|Type}{R:Rel A A}{b|A}{n|list A}(Sorted R (cons b n))->Rlist R b n
* heredRlistlemma = ... :
{A|Type}{R:Rel A A}{b,c|A}{n|list A}(Rlist R c (cons b n))->
Rlist R c n
* heredRlist1 = ... :
{A|Type}{R:Rel A A}{c|A}{m,n|list A}(Rlist R c (append m n))->
Rlist R c n
* heredRlist2 = ... :
{A|Type}{R:Rel A A}{c|A}{m,n|list A}(Rlist R c (append m n))->
Rlist R c m
* appclRlist = ... :
{A|Type}{R:Rel A A}{c|A}{m,n|list A}(Rlist R c m)->(Rlist R c n)->
Rlist R c (append m n)
* heredSorted = ... :
{A|Type}{R:Rel A A}{m,n|list A}(Sorted R (append m n))->
and (Sorted R m) (Sorted R n)
* RlistIsMonotone = ... :
{A|Type}{R:Rel A A}(trans R)->{b,c|A}{m|list A}(R c b)->
(Rlist R b m)->Rlist R c m
* PermPreservesRlist = ... :
{A|Type}{R:Rel A A}{c|A}{m,n|list A}(Perm m n)->(Rlist R c m)->
Rlist R c n
{A|Type}{R:Rel A A}(refl R)->{b,c|A}{m,n|list A}
(Sorted R (cons b m))->(Sorted R (cons c n))->
(Perm (cons b m) (cons c n))->R b c