Thm* InvFuns(A;A;f;g) (i:. InvFuns(A;A;f{i};g{i})) | [compose_iter_inverses] |
Thm* Bij(A; A; f) (i:. Bij(A; A; f{i})) | [compose_iter_bijection] |
Thm* Surj(A; A; f) (i:. Surj(A; A; f{i})) | [compose_iter_surjection] |
Thm* f:(A inj A), i:. f{i} A inj A | [compose_iter_injection2] |
Thm* k:, f:(k inj k). i:. f{i} = Id kk | [iter_perm_cycles_uniform2] |
Thm* k:, f:(k inj k). i:. u:k. f{i}(u) = u | [iter_perm_cycles_uniform] |
Thm* k:, f:(k inj k), u:k. i:. f{i}(u) = u | [compose_iter_inj_cycles] |
Thm* Inj(A; A; f) (i:. Inj(A; A; f{i})) | [compose_iter_injection] |
Thm* f:(AA), i:. f{i} = f{i-1} o f | [compose_iter_pos_comp2] |
Thm* f:(AA), i:, x:A. f{i}(x) = f{i-1}(f(x)) | [compose_iter_pos2] |
Thm* x:A, f:(AA), i,j:. f{i}{j}(x) = f{ij}(x) | [compose_iter_prod2] |
Thm* x:A, f:(AA), i,j,k:. k = ij f{i}{j}(x) = f{k}(x) A | [compose_iter_prod] |
Thm* f:(AA), k:, i:{0...k}. f{k} = f{i} o f{k-i} | [compose_iter_sum_comp_rw] |
Thm* x:A, f:(AA), k:, i:{0...k}. f{k}(x) = f{i}(f{k-i}(x)) | [compose_iter_sum_rw] |
Thm* f:(AA), i,j,k:. k = i+j f{i} o f{j} = f{k} | [compose_iter_sum_comp] |
Thm* x:A, f:(AA), i,j,k:. k = i+j f{i}(f{j}(x)) = f{k}(x) | [compose_iter_sum] |
Thm* i:. Id{i} = Id AA | [compose_iter_id] |
Thm* f:(AA), i:. f{i} = f o f{i-1} | [compose_iter_pos_comp] |
Thm* f:(AA), u:A. f(u) = u (i:. f{i}(u) = u) | [compose_iter_point_id] |
Thm* f:(AA), i:, x:A. f{i}(x) = f(f{i-1}(x)) | [compose_iter_pos] |
Thm* f:(AA). f{1} = f | [compose_iter_once] |
Thm* f:(AA). f{0} = Id | [compose_iter_zero_id] |
Thm* f:(AA), x:A. f{0}(x) = x | [compose_iter_zero] |
Def nat_to_nat_pair(i) == next_nat_pair{i}(<0,0>) | [nat_to_nat_pair] |