%   ORIGINAL: h4/Encode/wf__pred__bnum0
% 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__CLAUSES_c1: !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/LESS__MOD: !n k. h4/prim__rec/_3C k n ==> h4/arithmetic/MOD k n = k
% Assm: h4/Encode/collision__free__def: !p m. h4/Encode/collision__free m p <=> (!x y. p x /\ p y /\ h4/arithmetic/MOD x (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) m) = h4/arithmetic/MOD y (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) m) ==> x = y)
% Assm: h4/Encode/wf__pred__bnum__def: !p m. h4/Encode/wf__pred__bnum m p <=> h4/Encode/wf__pred p /\ (!x. p x ==> h4/prim__rec/_3C x (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) m))
% Goal: !p m. h4/Encode/wf__pred__bnum m p ==> h4/Encode/collision__free m p
%   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_CLAUSESu_c1]: !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_LESSu_u_MOD]: !n k. h4/prim__rec/_3C k n ==> h4/arithmetic/MOD k n = k
% Assm [h4s_Encodes_collisionu_u_freeu_u_def]: !p m. h4/Encode/collision__free m p <=> (!x y. happ p x /\ happ p y /\ h4/arithmetic/MOD x (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) m) = h4/arithmetic/MOD y (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) m) ==> x = y)
% Assm [h4s_Encodes_wfu_u_predu_u_bnumu_u_def]: !p m. h4/Encode/wf__pred__bnum m p <=> h4/Encode/wf__pred p /\ (!x. happ p x ==> h4/prim__rec/_3C x (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) m))
% Goal: !p m. h4/Encode/wf__pred__bnum m p ==> h4/Encode/collision__free m p
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_Q204310,TV_Q204306]: ![V_f, V_g]: (![V_x]: s(TV_Q204306,happ(s(t_fun(TV_Q204310,TV_Q204306),V_f),s(TV_Q204310,V_x))) = s(TV_Q204306,happ(s(t_fun(TV_Q204310,TV_Q204306),V_g),s(TV_Q204310,V_x))) => s(t_fun(TV_Q204310,TV_Q204306),V_f) = s(t_fun(TV_Q204310,TV_Q204306),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,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_LESSu_u_MOD, axiom, ![V_n, V_k]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n)))) => s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_k))).
fof(ah4s_Encodes_collisionu_u_freeu_u_def, axiom, ![V_p, V_m]: (p(s(t_bool,h4s_encodes_collisionu_u_free(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_bool),V_p)))) <=> ![V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_h4s_nums_num,V_x)))) & (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_h4s_nums_num,V_y)))) & s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,h4s_arithmetics_exp(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))))) = s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_y),s(t_h4s_nums_num,h4s_arithmetics_exp(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))))))) => s(t_h4s_nums_num,V_x) = s(t_h4s_nums_num,V_y)))).
fof(ah4s_Encodes_wfu_u_predu_u_bnumu_u_def, axiom, ![V_p, V_m]: (p(s(t_bool,h4s_encodes_wfu_u_predu_u_bnum(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_bool),V_p)))) <=> (p(s(t_bool,h4s_encodes_wfu_u_pred(s(t_fun(t_h4s_nums_num,t_bool),V_p)))) & ![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_h4s_nums_num,V_x)))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,h4s_arithmetics_exp(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(ch4s_Encodes_wfu_u_predu_u_bnum0, conjecture, ![V_p, V_m]: (p(s(t_bool,h4s_encodes_wfu_u_predu_u_bnum(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_bool),V_p)))) => p(s(t_bool,h4s_encodes_collisionu_u_free(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_bool),V_p)))))).
