| | Who Cites count pairs? |
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| count_pairs | Def count(x < y in L | P(x;y)) == sum(if (i <  j) P(L[i];L[j]) 1 else 0 fi | i < ||L||; j < ||L||) |
| | | Thm*  T:Type, L:T List, P:(T  T ). count(x < y in L | P(x,y))   |
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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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| 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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| 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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| band | Def pq == if p q else false fi |
| | | Thm* p,q:. (pq) |
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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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| double_sum | Def sum(f(x;y) | x < n; y < m) == sum(sum(f(x;y) | y < m) | x < n) |
| | | Thm* n,m:, f:(nm). sum(f(x,y) | x < n; y < m) |
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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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| sum | Def sum(f(x) | x < k) == primrec(k;0; x,n. n+f(x)) |
| | | Thm* n:, f:(n). sum(f(x) | x < n) |
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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:(nTT). primrec(n;b;c) T |
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| bnot | Def b == if b false else true fi |
| | | Thm* b:. b |
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| eq_int | Def i=j == if i=j true ; false fi |
| | | Thm* i,j:. (i=j) |
