%   ORIGINAL: 'h4/thm/numeral/bit_initiality_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !zf b1f b2f. ?f. f 'h4/const/arithmetic/ZERO' = zf /\ (!n. f ('h4/const/arithmetic/BIT1' n) = b1f n (f n)) /\ (!n. f ('h4/const/arithmetic/BIT2' n) = b2f n (f n))
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !zf b1f b2f. ?f. happ f 'h4/const/arithmetic/ZERO' = zf /\ (!n. happ f ('h4/const/arithmetic/BIT1' n) = happ (happ b1f n) (happ f n)) /\ (!n. happ f ('h4/const/arithmetic/BIT2' n) = happ (happ b2f n) (happ f n))
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f44184,V_3f44180]: ![F, G]: (![X]: s(V_3f44180,happ(s(fun(V_3f44184,V_3f44180),F),s(V_3f44184,X))) = s(V_3f44180,happ(s(fun(V_3f44184,V_3f44180),G),s(V_3f44184,X))) => s(fun(V_3f44184,V_3f44180),F) = s(fun(V_3f44184,V_3f44180),G))).
fof('h4/thm/numeral/bit_initiality_', conjecture, ![A]: ![Zf, B1f, B2f]: ?[F]: (s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num','h4/const/arithmetic/ZERO'))) = s(A,Zf) & (![N]: s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num',N))))) = s(A,happ(s(fun(A,A),happ(s(fun('h4/type/num/num',fun(A,A)),B1f),s('h4/type/num/num',N))),s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num',N))))) & ![N]: s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num',N))))) = s(A,happ(s(fun(A,A),happ(s(fun('h4/type/num/num',fun(A,A)),B2f),s('h4/type/num/num',N))),s(A,happ(s(fun('h4/type/num/num',A),F),s('h4/type/num/num',N)))))))).
