| Who Cites count pairs? |
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count_pairs | Def count(x < y in L | P(x;y))
Def == sum(if (i< j)![](FONT/and.png) P(L[i];L[j]) 1 else 0 fi | i < ||L||; j < ||L||) |
| | Thm* T:Type, L:T List, P:(T![](FONT/dash.png) T![](FONT/dash.png) ![](FONT/then_med.png) ). count(x < y in L | P(x,y)) ![](FONT/nat.png) |
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select | Def l[i] == hd(nth_tl(i;l)) |
| | Thm* A:Type, l:A List, n: . 0 n ![](FONT/eq.png) n<||l|| ![](FONT/eq.png) l[n] A |
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nth_tl | Def nth_tl(n;as) == if n![](FONT/le.png) 0 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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le_int | Def i![](FONT/le.png) j == ![](FONT/not.png) j< i |
| | Thm* i,j: . (i![](FONT/le.png) j) ![](FONT/bool.png) |
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lt_int | Def i< j == if i<j true ; false fi |
| | Thm* i,j: . (i< j) ![](FONT/bool.png) |
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band | Def p![](FONT/and.png) q == if p q else false fi |
| | Thm* p,q: . (p![](FONT/and.png) q) ![](FONT/bool.png) |
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length | Def ||as|| == Case of as; nil 0 ; a.as' ||as'||+1 (recursive) |
| | Thm* A:Type, l:A List. ||l|| ![](FONT/int.png) |
| | Thm* ||nil|| ![](FONT/int.png) |
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double_sum | Def sum(f(x;y) | x < n; y < m) == sum(sum(f(x;y) | y < m) | x < n) |
| | Thm* n,m: , f:( n![](FONT/dash.png) ![](FONT/then_med.png) m![](FONT/dash.png) ![](FONT/then_med.png) ). sum(f(x,y) | x < n; y < m) ![](FONT/int.png) |
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hd | Def hd(l) == Case of l; nil "?" ; h.t h |
| | Thm* A:Type, l:A List. ||l|| 1 ![](FONT/eq.png) hd(l) A |
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sum | Def sum(f(x) | x < k) == primrec(k;0; x,n. n+f(x)) |
| | Thm* n: , f:( n![](FONT/dash.png) ![](FONT/then_med.png) ). sum(f(x) | x < n) ![](FONT/int.png) |
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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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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:( n![](FONT/dash.png) T![](FONT/dash.png) T). primrec(n;b;c) T |
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bnot | Def ![](FONT/not.png) b == if b false else true fi |
| | Thm* b: . ![](FONT/not.png) b ![](FONT/bool.png) |
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eq_int | Def i= j == if i=j true ; false fi |
| | Thm* i,j: . (i= j) ![](FONT/bool.png) |