%   ORIGINAL: h4/gcd/BINARY__GCD_c1
% 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/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/arithmetic/MULT__COMM: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm: h4/arithmetic/ODD__EVEN: !n. h4/arithmetic/ODD n <=> ~h4/arithmetic/EVEN n
% Assm: h4/arithmetic/EVEN__EXISTS: !n. h4/arithmetic/EVEN n <=> (?m. n = h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) m)
% Assm: h4/arithmetic/MULT__DIV: !q n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/DIV (h4/arithmetic/_2A q n) n = q
% Assm: h4/divides/divides__def: !b a. h4/divides/divides a b <=> (?q. b = h4/arithmetic/_2A q a)
% Assm: h4/divides/PRIME__2: h4/divides/prime (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))
% Assm: h4/divides/PRIME__POS: !p. h4/divides/prime p ==> h4/prim__rec/_3C h4/num/0 p
% Assm: h4/gcd/GCD__SYM: !b a. h4/gcd/gcd a b = h4/gcd/gcd b a
% Assm: h4/gcd/PRIME__GCD: !p b. h4/divides/prime p ==> h4/divides/divides p b \/ h4/gcd/gcd p b = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/gcd/GCD__COMMON__FACTOR: !n m k. h4/gcd/gcd (h4/arithmetic/_2A k m) (h4/arithmetic/_2A k n) = h4/arithmetic/_2A k (h4/gcd/gcd m n)
% Assm: h4/gcd/GCD__CANCEL__MULT: !n m k. h4/gcd/gcd m k = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) ==> h4/gcd/gcd m (h4/arithmetic/_2A k n) = h4/gcd/gcd m n
% Goal: !n m. h4/arithmetic/EVEN m /\ h4/arithmetic/ODD n ==> h4/gcd/gcd m n = h4/gcd/gcd (h4/arithmetic/DIV m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) 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_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_arithmetics_MULTu_u_COMM]: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm [h4s_arithmetics_ODDu_u_EVEN]: !n. h4/arithmetic/ODD n <=> ~h4/arithmetic/EVEN n
% Assm [h4s_arithmetics_EVENu_u_EXISTS]: !n. h4/arithmetic/EVEN n <=> (?m. n = h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) m)
% Assm [h4s_arithmetics_MULTu_u_DIV]: !q n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/DIV (h4/arithmetic/_2A q n) n = q
% Assm [h4s_dividess_dividesu_u_def]: !b a. h4/divides/divides a b <=> (?q. b = h4/arithmetic/_2A q a)
% Assm [h4s_dividess_PRIMEu_u_2]: h4/divides/prime (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))
% Assm [h4s_dividess_PRIMEu_u_POS]: !p. h4/divides/prime p ==> h4/prim__rec/_3C h4/num/0 p
% Assm [h4s_gcds_GCDu_u_SYM]: !b a. h4/gcd/gcd a b = h4/gcd/gcd b a
% Assm [h4s_gcds_PRIMEu_u_GCD]: !p b. h4/divides/prime p ==> h4/divides/divides p b \/ h4/gcd/gcd p b = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_gcds_GCDu_u_COMMONu_u_FACTOR]: !n m k. h4/gcd/gcd (h4/arithmetic/_2A k m) (h4/arithmetic/_2A k n) = h4/arithmetic/_2A k (h4/gcd/gcd m n)
% Assm [h4s_gcds_GCDu_u_CANCELu_u_MULT]: !n m k. h4/gcd/gcd m k = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) ==> h4/gcd/gcd m (h4/arithmetic/_2A k n) = h4/gcd/gcd m n
% Goal: !n m. h4/arithmetic/EVEN m /\ h4/arithmetic/ODD n ==> h4/gcd/gcd m n = h4/gcd/gcd (h4/arithmetic/DIV m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) 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_Q332027,TV_Q332023]: ![V_f, V_g]: (![V_x]: s(TV_Q332023,happ(s(t_fun(TV_Q332027,TV_Q332023),V_f),s(TV_Q332027,V_x))) = s(TV_Q332023,happ(s(t_fun(TV_Q332027,TV_Q332023),V_g),s(TV_Q332027,V_x))) => s(t_fun(TV_Q332027,TV_Q332023),V_f) = s(t_fun(TV_Q332027,TV_Q332023),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> 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_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_arithmetics_MULTu_u_COMM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
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_EXISTS, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n)))) <=> ?[V_m]: s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))))).
fof(ah4s_arithmetics_MULTu_u_DIV, axiom, ![V_q, V_n]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n)))) => s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_q))).
fof(ah4s_dividess_dividesu_u_def, axiom, ![V_b, V_a]: (p(s(t_bool,h4s_dividess_divides(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,V_b)))) <=> ?[V_q]: s(t_h4s_nums_num,V_b) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,V_a))))).
fof(ah4s_dividess_PRIMEu_u_2, axiom, p(s(t_bool,h4s_dividess_prime(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))).
fof(ah4s_dividess_PRIMEu_u_POS, axiom, ![V_p]: (p(s(t_bool,h4s_dividess_prime(s(t_h4s_nums_num,V_p)))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_gcds_GCDu_u_SYM, axiom, ![V_b, V_a]: s(t_h4s_nums_num,h4s_gcds_gcd(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,V_b))) = s(t_h4s_nums_num,h4s_gcds_gcd(s(t_h4s_nums_num,V_b),s(t_h4s_nums_num,V_a)))).
fof(ah4s_gcds_PRIMEu_u_GCD, axiom, ![V_p, V_b]: (p(s(t_bool,h4s_dividess_prime(s(t_h4s_nums_num,V_p)))) => (p(s(t_bool,h4s_dividess_divides(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_b)))) | s(t_h4s_nums_num,h4s_gcds_gcd(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_b))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
fof(ah4s_gcds_GCDu_u_COMMONu_u_FACTOR, axiom, ![V_n, V_m, V_k]: s(t_h4s_nums_num,h4s_gcds_gcd(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,h4s_gcds_gcd(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_gcds_GCDu_u_CANCELu_u_MULT, axiom, ![V_n, V_m, V_k]: (s(t_h4s_nums_num,h4s_gcds_gcd(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_k))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))) => s(t_h4s_nums_num,h4s_gcds_gcd(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_gcds_gcd(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))).
fof(ch4s_gcds_BINARYu_u_GCDu_c1, conjecture, ![V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_m)))) & p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n))))) => s(t_h4s_nums_num,h4s_gcds_gcd(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_gcds_gcd(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_n))))).
