| Who Cites swap adjacent? |
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swap_adjacent | Def swap adjacent[P(x;y)](L1,L2)
== i:(||L1||-1). P(L1[i];L1[(i+1)]) & L2 = swap(L1;i;i+1) A List |
| | Thm* A:Type, P:(AAProp). swap adjacent[P(x,y)] (A List)(A List)Prop |
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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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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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select |
Def l[i] == hd(nth_tl(i;l)) |
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Thm* A:Type, l:A List, n:. 0n n < ||l|| l[n] A |
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length |
Def ||as|| == Case of as; nil 0 ; a.as' ||as'||+1 (recursive) |
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Thm* A:Type, l:A List. ||l|| |
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Thm* ||nil|| |
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int_seg |
Def {i..j} == {k:| i k < j } |
| | Thm* m,n:. {m..n} Type |
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flip |
Def (i, j)(x) == if x=ij ;x=ji else x fi |
| | Thm* k:, i,j:k. (i, j) kk |
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nth_tl |
Def nth_tl(n;as) == if n0 as else nth_tl(n-1;tl(as)) fi (recursive) |
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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 |
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Thm* A:Type, l:A List. ||l||1 hd(l) A |
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Thm* A:Type, l:A List. hd(l) A |
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lelt |
Def i j < k == ij & j < k |
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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) |
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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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tl |
Def tl(l) == Case of l; nil nil ; h.t t |
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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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le |
Def AB == B < A |
| | Thm* i,j:. (ij) Prop |
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append |
Def as @ bs == Case of as; nil bs ; a.as' [a / (as' @ bs)] (recursive) |
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Thm* T:Type, as,bs:T List. (as @ bs) T List |
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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 |
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not |
Def A == A False |
| | Thm* A:Prop. (A) Prop |