%   ORIGINAL: 'h4/thm/arithmetic/SUC_ELIM_NUMERALS_'
% 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: !f g. (!n. g ('h4/const/num/SUC' n) = f n ('h4/const/num/SUC' n)) <=> (!n. g ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' n)) = f ('h4/const/arithmetic/-' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' n)) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO'))) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' n))) /\ (!n. g ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' n)) = f ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' n)) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' 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: !f g. (!n. happ g ('h4/const/num/SUC' n) = happ (happ f n) ('h4/const/num/SUC' n)) <=> (!n. happ g ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' n)) = happ (happ f ('h4/const/arithmetic/-' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' n)) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO')))) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' n))) /\ (!n. happ g ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' n)) = happ (happ f ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' n))) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' 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_3f83216,V_3f83212]: ![F, G]: (![X]: s(V_3f83212,happ(s(fun(V_3f83216,V_3f83212),F),s(V_3f83216,X))) = s(V_3f83212,happ(s(fun(V_3f83216,V_3f83212),G),s(V_3f83216,X))) => s(fun(V_3f83216,V_3f83212),F) = s(fun(V_3f83216,V_3f83212),G))).
fof('h4/thm/arithmetic/SUC_ELIM_NUMERALS_', conjecture, ![A]: ![F, G]: (![N]: s(A,happ(s(fun('h4/type/num/num',A),G),s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',N))))) = s(A,happ(s(fun('h4/type/num/num',A),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',A)),F),s('h4/type/num/num',N))),s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',N))))) <=> (![N]: s(A,happ(s(fun('h4/type/num/num',A),G),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num',N))))))) = s(A,happ(s(fun('h4/type/num/num',A),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',A)),F),s('h4/type/num/num','h4/const/arithmetic/-'(s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num',N))))),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num','h4/const/arithmetic/ZERO'))))))))),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num',N))))))) & ![N]: s(A,happ(s(fun('h4/type/num/num',A),G),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num',N))))))) = s(A,happ(s(fun('h4/type/num/num',A),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',A)),F),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num',N))))))),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num',N)))))))))).
