%   ORIGINAL: h4/arithmetic/ODD__ADD
% 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
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/arithmetic/ODD__EVEN: !n. h4/arithmetic/ODD n <=> ~h4/arithmetic/EVEN n
% Assm: h4/arithmetic/EVEN__ADD: !n m. h4/arithmetic/EVEN (h4/arithmetic/_2B m n) <=> h4/arithmetic/EVEN m <=> h4/arithmetic/EVEN n
% Goal: !n m. h4/arithmetic/ODD (h4/arithmetic/_2B m n) <=> ~(h4/arithmetic/ODD m <=> h4/arithmetic/ODD n)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_arithmetics_ODDu_u_EVEN]: !n. h4/arithmetic/ODD n <=> ~h4/arithmetic/EVEN n
% Assm [h4s_arithmetics_EVENu_u_ADD]: !n m. h4/arithmetic/EVEN (h4/arithmetic/_2B m n) <=> h4/arithmetic/EVEN m <=> h4/arithmetic/EVEN n
% Goal: !n m. h4/arithmetic/ODD (h4/arithmetic/_2B m n) <=> ~(h4/arithmetic/ODD m <=> h4/arithmetic/ODD n)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q308473,TV_Q308469]: ![V_f, V_g]: (![V_x]: s(TV_Q308469,happ(s(t_fun(TV_Q308473,TV_Q308469),V_f),s(TV_Q308473,V_x))) = s(TV_Q308469,happ(s(t_fun(TV_Q308473,TV_Q308469),V_g),s(TV_Q308473,V_x))) => s(t_fun(TV_Q308473,TV_Q308469),V_f) = s(t_fun(TV_Q308473,TV_Q308469),V_g))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_arithmetics_ODDu_u_EVEN, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n)))) <=> ~ (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_EVENu_u_ADD, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))) <=> s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_m))) = s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n))))).
fof(ch4s_arithmetics_ODDu_u_ADD, conjecture, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))) <=> ~ (s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_m))) = s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n)))))).
