is mentioned by

Thm* f:(ABA). (a:A, b1,b2:B. f(a,b1) = f(a,b2) b1 = b2) (a,a':A. Dec(b:B. a' = f(a,b))) Graph(a:A -- > f(a,b) | b:B) Graph(a,a':A. b:B. a' = f(a,b))	[fun-graph-rel-graph]
Thm* R1,R2:(TTProp). (x,y:T. R1(x,y) R2(x,y)) Graph(x,y:T. R1(x,y)) Graph(x,y:T. R2(x,y))	[rel-graph_functionality]
Thm* G,H:Graph. G H H G	[graph-isomorphic_inversion]
Thm* G,H,K:Graph. G H H K G K	[graph-isomorphic_transitivity]
Thm* G,H:Graph. G = H G H	[graph-isomorphic_weakening]
Thm* For any graph L1,L2:V List, s:Traversal. L1-- > *L2 dfsl-traversal(the_graph;L1;s) dfsl-traversal(the_graph;L1 @ L2;s)	[dfsl-traversal-append]
Thm* For any graph L:V List, i:V, s:Traversal. (inr(i) s) dfl-traversal(the_graph;L;s) dfl-traversal(the_graph;L;[inl(i) / s])	[dfl-traversal-consl]
Thm* For any graph L:V List, i:V, s:Traversal. (j:V. (inr(j) s) (inl(j) s) j-the_graph- > *i) L-- > *i dfl-traversal(the_graph;L;s) dfl-traversal(the_graph;L;[inr(i) / s])	[dfl-traversal-consr]
Thm* For any graph L1,L2:V List, s:Traversal. L1-- > *L2 dfl-traversal(the_graph;L2;s) dfl-traversal(the_graph;L1;s)	[dfl-traversal-connect]
Thm* G:Graph. (x,y:Vertices(G). Dec(x = y)) (s:traversal(G). paren(Vertices(G);s) no_repeats(Vertices(G)+Vertices(G);s) df-traversal(G;s) depthfirst-traversal(G;s))	[paren-df-traversal]
Thm* For any graph i:V, s:Traversal. (j:V. (inr(j) s) (inl(j) s) j-the_graph- > *i) df-traversal(the_graph;s) df-traversal(the_graph;[inr(i) / s])	[df-traversal-cons2]
Thm* For any graph A,B,C:V List. A-- > *C B-- > *C A @ B-- > *C	[list-list-connect-append2]
Thm* For any graph A,B:V List. A = B A-- > *B	[list-list-connect_weakening]
Thm* For any graph A,B:V List. B A A-- > *B	[list-list-connect-iseg]
Thm* For any graph A,B,C:V List. A-- > *B B-- > *C A-- > *C	[list-list-connect_transitivity]
Thm* For any graph (x,y:V. Dec(x = y)) (x,y:V. x-the_graph- > *y x = y (z:V. z = x & x-the_graph- > z & z-the_graph- > *y))	[connect-iff]
Thm* For any graph p:V List. 1 < ||p|| path(the_graph;p) path(the_graph;tl(p))	[path-tl]
Thm* For any graph x,y:V. x-the_graph- > y x-the_graph- > *y	[edge-connect]
Thm* For any graph x,y:V. x = y x-the_graph- > *y	[connect_weakening]
Thm* For any graph x,y,z:V. x-the_graph- > *y y-the_graph- > *z x-the_graph- > *z	[connect_transitivity]
Thm* (x,y:T. Dec(x = y)) (s:(T+T) List, i:T. Dec((inl(i) s)))	[decidable__l_member_paren]
Thm* s:(T+T) List. paren(T;s) no_repeats(T+T;s) (s1,s2,s3:(T+T) List, x:T. s = (s1 @ [inl(x)] @ s2 @ [inr(x)] @ s3) paren(T;s2))	[paren_interval]
Thm* s:(T+T) List. paren(T;s) (i:T, s1,s2:(T+T) List. s = (s1 @ [inl(i)] @ s2) (inr(i) s2))	[paren_order1]
Thm* s',s3:(T+T) List. paren(T;s') paren(T;s3) paren(T;s3 @ s')	[append_paren]
Thm* s':(T+T) List. paren(T;s') (i:T. (inr(i) s') (inl(i) s'))	[paren_balance2]
Thm* E:(TT). (x,y:T. E(x,y) x = y) (i:T, s:(T+T) List. member-left-paren(x,y.E(x,y);i;s) (inl(i) s))	[assert-member-left-paren]
Thm* E:(TT). (x,y:T. E(x,y) x = y) (i:T, s:(T+T) List. member-right-paren(x,y.E(x,y);i;s) (inr(i) s))	[assert-member-right-paren]
Thm* E:(TT). (x,y:T. E(x,y) x = y) (i:T, s:(T+T) List. member-paren(x,y.E(x,y);i;s) (inl(i) s) (inr(i) s))	[assert-member-paren]
Thm* s':(T+T) List. paren(T;s') (i:T. (inl(i) s') (inr(i) s'))	[paren_balance1]
Thm* P:(((T+T) List)Prop). P(nil) (s1,s2:(T+T) List. P(s1) P(s2) paren(T;s1) paren(T;s2) P(s1 @ s2)) (s:(T+T) List, t:T. P(s) paren(T;s) P([inl(t)] @ s @ [inr(t)])) (s:(T+T) List. paren(T;s) P(s))	[paren_induction]
Thm* k:, L: List. 0 < ||L|| (r:. r- > L^k)	[Ramsey]
Thm* r,k:, L,R: List. 2k ||R|| = ||L|| (i:||L||. 0 < L[i] R[i]- > L[i--]^k) r- > R^k-1 r+1- > L^k	[Ramsey-recursion]
Thm* L: List, i,j:||L||. 0 < L[i] L[i--][j] = if j=i L[j]-1 else L[j] fi	[list-dec-select]
Thm* L: List, i:||L||. 0 < L[i] ||L[i--]|| = ||L||	[list-dec-length]
Thm* L: List, i:||L||. 0 < L[i] L[i--] List	[list-dec_wf]
Thm* k:, L: List, r1,r2:. r1r2 r1- > L^k r2- > L^k	[arrows-monotone1]
Thm* n,m:, f:(nm). Inj(n; m; f) nm	[pigeon-hole]
Thm* n,k:, c:(nk). p:(k( List)). sum(||p(j)|| | j < k) = n & (j:k, x,y:||p(j)||. x < y (p(j))[x] > (p(j))[y]) & (j:k, x:||p(j)||. (p(j))[x] < n & c((p(j))[x]) = j)	[finite-partition]
Thm* n,m:, f:((n-1)(m-1)). Inj((n-1); (m-1); f) Inj(n; m; f[n-1:=m-1])	[fappend-inject]
Thm* n,m:, f:((n-1)(m-1)). increasing(f;n-1) increasing(f[n-1:=m-1];n)	[fappend-increasing]
Thm* p:. prime(p) (m,n:. mm = pnn n = 0)	[sqrt_prime_irrational]
Def topsortedl(the_graph;L;s) == (i,j:Vertices(the_graph). j = i i-the_graph- > *j i before j s) & (i,j,k:Vertices(the_graph). k-the_graph- > *j k-the_graph- > *i (k':Vertices(the_graph). k' before k L k'-the_graph- > *i) i before j s)	[topsortedl]
Def dfsl-traversal(the_graph;L;s) == df-traversal(the_graph;s) & (i:Vertices(the_graph). (inl(i) s) L-the_graph- > *i) & ((i:Vertices(the_graph). L-the_graph- > *i non-trivial-loop(the_graph;i)) (L1,L2:Vertices(the_graph) List. L = (L1 @ L2) (s1,s2:traversal(the_graph). s = (s2 @ s1) traversal(the_graph) & paren(Vertices(the_graph);s1) & paren(Vertices(the_graph);s2) & (j:Vertices(the_graph). ((inl(j) s1) L1-the_graph- > *j) & ((inl(j) s2) L2-the_graph- > *j & L1-the_graph- > *j)))))	[dfsl-traversal]
Def dfl-traversal(the_graph;L;s) == (i:Vertices(the_graph), s1,s2:traversal(the_graph). s = (s1 @ [inr(i)] @ s2) traversal(the_graph) (j:Vertices(the_graph). (inr(j) s2) (inl(j) s2) j-the_graph- > *i)) & (j:Vertices(the_graph). (inr(j) s) L-the_graph- > *j) & (i:Vertices(the_graph), s1,s2:traversal(the_graph). (j:Vertices(the_graph). i-the_graph- > *j non-trivial-loop(the_graph;j)) s = (s1 @ [inl(i)] @ s2) traversal(the_graph) L-the_graph- > *i)	[dfl-traversal]
Def topsorted(the_graph;s) == i,j:Vertices(the_graph). i-the_graph- > *j i = j i before j s	[topsorted]
Def depthfirst-traversal(the_graph;s) == i:Vertices(the_graph), s1,s2:traversal(the_graph). (j:Vertices(the_graph). i-the_graph- > *j non-trivial-loop(the_graph;j)) s = (s1 @ [inl(i)] @ s2) traversal(the_graph) (j:Vertices(the_graph). j = i i-the_graph- > *j (inl(j) s2))	[depthfirst-traversal]
Def df-traversal(G;s) == (i:Vertices(G), s1,s2:traversal(G). s = (s1 @ [inr(i)] @ s2) traversal(G) (j:Vertices(G). (inr(j) s2) (inl(j) s2) j-G- > *i)) & (i:Vertices(G), s1,s2:traversal(G). (j:Vertices(G). i-G- > *j non-trivial-loop(G;j)) s = (s1 @ [inl(i)] @ s2) traversal(G) (j:Vertices(G). i-G- > *j (inr(j) s2)))	[df-traversal]
Def r- > L^k == n:. rn (G:({s:(n List)| ||s|| = k & (x,y:||s||. x < y s[x] < s[y]) }||L||). c:||L||, f:(L[c]n). increasing(f;L[c]) & (s:L[c] List. ||s|| = k (x,y:||s||. x < y s[x] < s[y]) G(map(f;s)) = c))	[arrows]

In prior sections: core well fnd int 1 bool 1 int 2 union rel 1 num thy 1 fun 1 list 1 sqequal 1 prog 1 mb basic mb nat mb list 1 mb list 2 graph 1 1

Try larger context: Graphs