Thm* Y( f,x. if x= 0 1 else x f(x-1) fi)     | [sfa_doc_ycomb_factorial_typing] |
Thm* ( f,x. if x= 0 1 else x f(x-1) fi) ( k: . ( k  )  (k+1)  ) | [sfa_doc_factbody_polytyping] |
Thm* ( x.x) = ( x.if x<0 0 ; x fi)     | [sfa_doc_funeq_example1] |
Thm* ( n.n< 0) = ( n.false )     | [sfa_doc_sqtype_ctr_example_part2] |
Thm* x: . y: . y y x & x<(y+1) (y+1) | [sfa_doc_nat_sqrt] |
Thm* A:Type, a:A, e:(A A), i: . i steps of e from a A | [sfa_doc_sequence_rel_wf] |
Thm* ( x.Case of x
Thm* ( x.Canil 0
Thm* ( x.Cau.v, rec:r InjCase(if u<r inl(*) ; inr( .*) fi; a. r; b. u+1))
Thm* =
Thm* ext{sfa_doc_find_small_greater_bound}
Thm* ( k: . x:( k List) {y: (k+1)| y greater-bounds x }) | [sfa_doc_find_small_greater_bound_extr_eq] |
Thm* ( k,x. k)
Thm* =
Thm* ext{sfa_doc_find_greater_bound_stupidly}
Thm* k:  x:( k List) {y: (k+1)| y greater-bounds x } | [sfa_doc_find_greater_bound_stupidly_extr_eq] |
Thm* k:  x:( k List) {y: (k+1)| y greater-bounds x } | [sfa_doc_find_greater_bound_stupidly] |
Thm* k: . x:( k List) {y: (k+1)| y greater-bounds x } | [sfa_doc_find_small_greater_bound] |
Thm* f,g:(   ).
Thm* ( x: . f zero beyond x) & ( x: . g zero beyond x)
Thm* 
Thm* ( x: . ( i.if i even f(i) else g(i) fi) zero beyond x) | [sfa_doc_alternate_zero_beyond] |
Thm* a,b: . max(a;b)  | [sfa_doc_max_int_wf] |
Thm* x: , f:(   ). f zero beyond x Prop | [sfa_doc_props_as_types_example1_wf] |
Thm* P Q  ( x: . (x = 0  P) & (x 0  Q)) | [sfa_doc_or_dec] |
Thm* f:(   ).
Thm* ( n: . f(n) = 0)
Thm* 
Thm* ( n: . if f(n)= 0 False else C:Prop{1}. C fi) Prop{1} | [sfa_doc_predicativity_sample8] |
Thm* A:Type, P:(A Prop), n: , X:(A^n). ( u in X: A^n. P(u)) Prop | [sfa_doc_ntuple_contains_wf] |
Thm* n: . n 2  (A^n) = A (A^(n-1)) | [sfa_doc_ntuple_is_prod] |
Thm* A:Type, n: . (A^n) Type | [sfa_doc_ntuple_wf] |
Thm* x:{x: | (x rem 2) = 0 }. x!  | [sfa_doc_factorial2_wf] |
Thm* x: . x!  | [sfa_doc_factorial_wf] |
Thm* k: . 0<k & (2 k)<1  2<k | [sfa_doc_cand_example] |
Thm* f:{f:(   )| x: . f(x) }, i: . i<mu(f)  f(i) | [kleene_minimize_is_lb] |
Thm* f:{f:(   )| x: . f(x) }. f(mu(f)) | [kleene_minimize_is_fp] |
Thm* mu {f:(   )| x: . f(x) }   | [kleene_minimize_wf] |
Thm* f:{f:(   )| x: . f(x) }. f(0)  ( x.f(1+x)) {f:(   )| x: . f(x) } | [kleene_tail] |
Def f zero beyond x == y: . x<y  f(y) = 0  | [sfa_doc_props_as_types_example1] |
Def sfa_doc_oparm_sample{2:n}(A) == A  | [sfa_doc_oparm_sample2] |
Def sfa_doc_oparm_sample{1:n}(A) ==   A | [sfa_doc_oparm_sample1] |
Def Induction for x: . P(x) == P(0) & ( x: . P(x)  P(x+1))  ( x: . P(x)) | [sfa_doc_indscheme] |
Def u in X: A^n. P(u)
Def == n = 1 & P(X) n 2 & (X/a,rest. P(a) ( u in rest: A^(n-1). P(u)))
Def (recursive) | [sfa_doc_ntuple_contains] |