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 k  k | [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:(A A), i: . f{ i} = f{ i-1} o f | [compose_iter_pos_comp2] |
Thm* f:(A A), i: , x:A. f{ i}(x) = f{ i-1}(f(x)) | [compose_iter_pos2] |
Thm* x:A, f:(A A), i,j: . f{ i}{ j}(x) = f{ i j}(x) | [compose_iter_prod2] |
Thm* x:A, f:(A A), i,j,k: . k = i j  f{ i}{ j}(x) = f{ k}(x) A | [compose_iter_prod] |
Thm* f:(A A), k: , i:{0...k}. f{ k} = f{ i} o f{ k-i} | [compose_iter_sum_comp_rw] |
Thm* x:A, f:(A A), k: , i:{0...k}. f{ k}(x) = f{ i}(f{ k-i}(x)) | [compose_iter_sum_rw] |
Thm* f:(A A), i,j,k: . k = i+j  f{ i} o f{ j} = f{ k} | [compose_iter_sum_comp] |
Thm* x:A, f:(A A), i,j,k: . k = i+j  f{ i}(f{ j}(x)) = f{ k}(x) | [compose_iter_sum] |
Thm* i: . Id{ i} = Id A A | [compose_iter_id] |
Thm* f:(A A), i: . f{ i} = f o f{ i-1} | [compose_iter_pos_comp] |
Thm* f:(A A), u:A. f(u) = u  ( i: . f{ i}(u) = u) | [compose_iter_point_id] |
Thm* f:(A A), i: , x:A. f{ i}(x) = f(f{ i-1}(x)) | [compose_iter_pos] |
Thm* f:(A A). f{ 1} = f | [compose_iter_once] |
Thm* f:(A A). f{ 0} = Id | [compose_iter_zero_id] |
Thm* f:(A A), 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] |