%   ORIGINAL: 'h4/thm/numeral/bit_induction_'
% 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: !P. P 'h4/const/arithmetic/ZERO' /\ (!n. P n ==> P ('h4/const/arithmetic/BIT1' n)) /\ (!n. P n ==> P ('h4/const/arithmetic/BIT2' n)) ==> (!n. P 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: !P. happ P 'h4/const/arithmetic/ZERO' /\ (!n. happ P n ==> happ P ('h4/const/arithmetic/BIT1' n)) /\ (!n. happ P n ==> happ P ('h4/const/arithmetic/BIT2' n)) ==> (!n. happ P 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_3f44209,V_3f44205]: ![F, G]: (![X]: s(V_3f44205,happ(s(fun(V_3f44209,V_3f44205),F),s(V_3f44209,X))) = s(V_3f44205,happ(s(fun(V_3f44209,V_3f44205),G),s(V_3f44209,X))) => s(fun(V_3f44209,V_3f44205),F) = s(fun(V_3f44209,V_3f44205),G))).
fof('h4/thm/numeral/bit_induction_', conjecture, ![P]: ((p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num','h4/const/arithmetic/ZERO')))) & (![N]: (p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num',N)))) => p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num',N))))))) & ![N]: (p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num',N)))) => p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num',N))))))))) => ![N]: p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num',N)))))).
