| | Who Cites swap? |
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| swap | Def swap(L;i;j) == (L o (i, j)) |
| | | Thm*  T:Type, L:T List, i,j: ||L||. swap(L;i;j)  T List |
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| flip | Def (i, j)(x) == if x= i j ;x=ji else x fi |
| | | Thm* k:, i,j:k. (i, j) k  k |
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| permute_list | Def (L o f) == mklist(||L||; i.L[(f(i))]) |
| | | Thm* T:Type, L:T List, f:(||L||||L||). (L o f) T List |
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| mklist | Def mklist(n;f) == primrec(n;nil;i,l. l @ [(f(i))]) |
| | | Thm* T:Type, n:, f:(nT). mklist(n;f) T List |
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| primrec | Def primrec(n;b;c) == if n=0 b else c(n-1,primrec(n-1;b;c)) fi (recursive) |
| | | Thm* T:Type, n:, b:T, c:(nTT). primrec(n;b;c) T |
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| eq_int | Def i=j == if i=j true ; false fi |
| | | Thm* i,j: . (i=j)  |
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| select | Def l[i] == hd(nth_tl(i;l)) |
| | | Thm* A:Type, l:A List, n:. 0 n   n < ||l|| l[n] A |
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| length | Def ||as|| == Case of as; nil 0 ; a.as' ||as'||+1 (recursive) |
| | | Thm* A:Type, l:A List. ||l|| |
| | | Thm* ||nil|| |
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| nth_tl | Def nth_tl(n;as) == if n0 as else nth_tl(n-1;tl(as)) fi (recursive) |
| | | Thm* A:Type, as:A List, i:. nth_tl(i;as) A List |
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| hd | Def hd(l) == Case of l; nil "?" ; h.t h |
| | | Thm* A:Type, l:A List. ||l|| 1 hd(l) A |
| | | Thm* A:Type, l:A List . hd(l) A |
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| append | Def as @ bs == Case of as; nil bs ; a.as' [a / (as' @ bs)] (recursive) |
| | | Thm* T:Type, as,bs:T List. (as @ bs) T List |
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| tl | Def tl(l) == Case of l; nil nil ; h.t t |
| | | Thm* A:Type, l:A List. tl(l) A List |
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| le_int | Def ij ==  j < i |
| | | Thm* i,j:. (ij) |
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| lt_int | Def i < j == if i < j true ; false fi |
| | | Thm* i,j:. (i < j) |
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| bnot | Def b == if b false else true fi |
| | | Thm* b:. b |
