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\"Nil\" as \"[]\"\ncon \"Cons\" infix rightassoc 5 as \":\"\nvar \"ThPermTrans\" hide 3\nvar \"ThPermCons\" hide 3\nvar \"ThPermInsert\" hide 2\nvar \"ThInsertInSorted\" hide 2\nvar \"Permutation\" infix 4 as \"~\" with symbolfont\nvar \"List\" mixfix 0 as \"[_]\"\nvar \"IsLeFirst\" hide 3\nvar \"singleton\" hide 1 mixfix 0 as \"[_]\"\nvar \"listInd\" hide 2\nvar \"SortLemma\" hide 2\n #-}"),PT (Pn 24013 486 387) (TS ";" 8),PT (Pn 24015 501 1) (T_Comment "{-# GF Eng insert x xs = mkPN ([\"the list with\"]++x.s!pnv++[\"inserted into\"]++xs.s!pnv) #-}"),PT (Pn 24106 501 93) (TS ";" 8),PT (Pn 24108 502 1) (T_Comment "{-# GF Eng IsLeAll x xs = mkSent (x.s!pnv++[\"is less than or equal to all the elements of\"]++xs.s!pnv) #-}"),PT (Pn 24214 502 108) (TS ";" 8),PT (Pn 24216 503 1) (T_Comment "{-# GF Eng IsLeFirst a x xs ax p =\n  mkText (ax.s!text++\".\"++p.s!text++\n          [\". This means that \"]++\n          a.s!pnv++[\"is less than or equal to all the elements of the list with\"]++\n          x.s!pnv++[\"prepended to\"]++xs.s!pnv) #-}"),PT (Pn 24457 503 243) (TS ";" 8),PT (Pn 24459 508 1) (T_Comment "{-# GF Eng SortLemma x xs' h =\n  mkText (h.s!text++[\". Use SortLemma\"]) #-}"),PT (Pn 24534 508 77) (TS ";" 8),PT (Pn 24536 510 1) (T_Comment "{-# GF Eng ThInsertInSorted x xs p =\n  mkText (p.s!text++[\". Use the correctness of insertion\"]) #-}"),PT (Pn 24636 510 102) (TS ";" 8),PT (Pn 24638 512 1) (T_Comment "{-# GF Eng ThPermNil  = mkThm [\"the permutation of empty list theorem\"] #-}"),PT (Pn 24713 512 77) (TS ";" 8),PT (Pn 24715 513 1) (T_Comment "{-# GF Eng ThPermCons x ys zs p =\n  mkText (p.s!text++\n          [\". We can now use the theorem about prepending an element to permutations\"]) #-}"),PT (Pn 24861 513 148) (TS ";" 8),PT (Pn 24863 516 1) (T_Comment "{-# GF Eng ThPermTrans xs ys zs xy yz = \n  mkText (indent(\"\183\"++xy.s!text++\".\"++newParagraph++\"\183\"++yz.s!text)++\n          [\". By the transitivity of the permutation relation,\"]++\n          zs.s!pnv++\"is a permutation of\"++xs.s!pnv) #-}"),PT (Pn 25097 516 236) (TS ";" 8),PT (Pn 25099 520 1) (T_Comment "{-# GF Eng ThPermInsert x xs = mkThm [\"the theorem that inserting yields a permutation of prepending\"] #-}"),PT (Pn 25205 520 108) (TS ";" 8),PT (Pn 25207 521 1) (T_Comment "{-# GF Sve IsLeAll x xs = mkSent (x.s!pn++[\"\228r mindre \228n alla element i\"]++xs.s!pn) #-}"),PT (Pn 25294 521 89) (TS ";" 8),PT (Pn 25296 522 1) (T_Comment "{-# GF Sve CCCons x xs = mkPN ([\"listan med\"]++x.s!pn++[\"insatt f\246rst i\"]++xs.s!pn) en #-}"),PT (Pn 25386 522 92) (TS ";" 8),PT (Pn 25388 523 1) (T_Comment "{-# GF Sve singleton A x = mkPN ([\"enelementslistan med\"]++x.s!pn) en #-}"),PT (Pn 25461 523 75) (TS ";" 8),PT (Pn 25463 524 1) (T_Comment "{-# GF Sve CCNil  = mkPN [\"den tomma listan\"] en #-}"),PT (Pn 25515 524 54) (TS ";" 8),PT (Pn 25517 525 1) (T_Comment "{-# GF Sve List A = {s = tbl {n => (\"list\"+nomReg1!n)++\"av\" ++ A.s ! (cn pl)} ; form = CN en} #-}"),PT (Pn 25614 525 99) (TS ";" 8),PT (Pn 25616 526 1) (T_Comment "{-# GF Eng ThSortIsCorrect xs = mkThm([\"the correctness of insertion sort applied to\"]++xs.s!pnv) #-}"),PT (Pn 25717 526 103) (TS ";" 8),PT (Pn 25719 527 1) (T_Comment "{-# GF Eng ThSorting  = mkThm [\"a sorting theorem\"] #-}"),PT (Pn 25774 527 57) (TS ";" 8),PT (Pn 25776 528 1) (T_Comment "{-# GF Eng SortSpec xs ys = mkSent(ys.s!pnv++[\"is a sorted version of\"]++xs.s!pnv) #-}"),PT (Pn 25862 528 88) (TS ";" 8),PT (Pn 25864 529 1) (T_Comment "{-# GF Eng sort xs = mkPN([\"insertion sort applied to\"]++xs.s!pnv) #-}"),PT (Pn 25934 529 72) (TS ";" 8),PT (Pn 25936 530 1) (T_Comment "{-# GF Eng sortfun xs = mkPN([\"insertion sort\"]) #-}"),PT (Pn 25988 530 54) (TS ";" 8),PT (Pn 25990 531 1) (T_Comment "{-# GF Eng List A = mkCN(tbl {n=>(\"list\"+nomReg!n)++\"of\"++A.s!(cn pl)}) #-}"),PT (Pn 26065 531 77) (TS ";" 8),PT (Pn 26067 532 1) (T_Comment "{-# GF Eng Permutation xs ys = mkSent(ys.s!pnv++[\"is a permutation of\"]++xs.s!pnv) #-}"),PT (Pn 26153 532 88) (TS ";" 8),PT (Pn 26155 533 1) (T_Comment "{-# GF Eng count x xs = mkPN([\"the number of occurences of\"]++x.s!pnv++\"in\"++xs.s!pnv) #-}"),PT (Pn 26245 533 92) (TS ";" 8),PT (Pn 26247 534 1) (T_Comment "{-# GF Eng singleton A x = mkPN([\"the singleton list containing\"]++x.s!pnv) #-}"),PT (Pn 26326 534 81) (TS ";" 8),PT (Pn 26328 535 1) (T_Comment "{-# GF Eng IsSorted xs = mkSent (xs.s!pnv ++ [\"is sorted\"]) #-}"),PT (Pn 26391 535 65) (TS ";" 8),PT (Pn 26393 536 1) (T_Comment "{-# GF Eng CCCons x xs = mkPN ([\"the list with\"]++x.s!pnv++[\"prepended to\"]++xs.s!pnv) #-}"),PT (Pn 26483 536 92) (TS ";" 8),PT (Pn 26485 537 1) (T_Comment "{-# GF Eng CCNil  = mkPN [\"the empty list\"] #-}"),PT (Pn 26532 537 49) (TS ";" 8)]
syntax error at line 1 before {- \section{Introduction}

In this document, we prove the correctness of insertion sort.

\subsection{Preliminaries}

In the sections below, we rely on various definitions stated in the
following separate documents:
-} ; {- \section{Miscellaneuos definitions and lemmas}

Here we define what a list is, what it means for a list to be sorted (ordered)
and how to insert a new element at the right position in a sorted list.

-} ;