%   ORIGINAL: h4/intto/apintto__thm
% 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> 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/relation/transitive__def: !R. h4/relation/transitive R <=> (!x y z. R x y /\ R y z ==> R x z)
% Assm: h4/relation/irreflexive__def: !R. h4/relation/irreflexive R <=> (!x. ~R x x)
% Assm: h4/relation/trichotomous0: !R. h4/relation/trichotomous R <=> (!a b. R a b \/ R b a \/ a = b)
% Assm: h4/relation/StrongLinearOrder0: !R. h4/relation/StrongLinearOrder R <=> h4/relation/StrongOrder R /\ h4/relation/trichotomous R
% Assm: h4/integer/INT__LT__TOTAL: !y x. x = y \/ h4/integer/int__lt x y \/ h4/integer/int__lt y x
% Assm: h4/integer/INT__LT__REFL: !x. ~h4/integer/int__lt x x
% Assm: h4/integer/INT__LT__TRANS: !z y x. h4/integer/int__lt x y /\ h4/integer/int__lt y z ==> h4/integer/int__lt x z
% Assm: h4/toto/TO__apto__TO__ID: !r. h4/toto/TotOrd r <=> h4/toto/apto (h4/toto/TO r) = r
% Assm: h4/toto/TotOrd__TO__of__Strong: !r. h4/relation/StrongLinearOrder r ==> h4/toto/TotOrd (h4/toto/TO__of__LinearOrder r)
% Assm: h4/toto/StrongOrder__ALT: !Z. h4/relation/StrongOrder Z <=> h4/relation/irreflexive Z /\ h4/relation/transitive Z
% Assm: h4/intto/intOrd0: h4/intto/intOrd = h4/toto/TO__of__LinearOrder h4/integer/int__lt
% Assm: h4/intto/intto0: h4/intto/intto = h4/toto/TO h4/intto/intOrd
% Goal: h4/toto/apto h4/intto/intto = h4/intto/intOrd
%   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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> 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_relations_transitiveu_u_def]: !R. h4/relation/transitive R <=> (!x y z. happ (happ R x) y /\ happ (happ R y) z ==> happ (happ R x) z)
% Assm [h4s_relations_irreflexiveu_u_def]: !R. h4/relation/irreflexive R <=> (!x. ~happ (happ R x) x)
% Assm [h4s_relations_trichotomous0]: !R. h4/relation/trichotomous R <=> (!a b. happ (happ R a) b \/ happ (happ R b) a \/ a = b)
% Assm [h4s_relations_StrongLinearOrder0]: !R. h4/relation/StrongLinearOrder R <=> h4/relation/StrongOrder R /\ h4/relation/trichotomous R
% Assm [h4s_integers_INTu_u_LTu_u_TOTAL]: !y x. x = y \/ happ (happ h4/integer/int__lt x) y \/ happ (happ h4/integer/int__lt y) x
% Assm [h4s_integers_INTu_u_LTu_u_REFL]: !x. ~happ (happ h4/integer/int__lt x) x
% Assm [h4s_integers_INTu_u_LTu_u_TRANS]: !z y x. happ (happ h4/integer/int__lt x) y /\ happ (happ h4/integer/int__lt y) z ==> happ (happ h4/integer/int__lt x) z
% Assm [h4s_totos_TOu_u_aptou_u_TOu_u_ID]: !r. h4/toto/TotOrd r <=> h4/toto/apto (h4/toto/TO r) = r
% Assm [h4s_totos_TotOrdu_u_TOu_u_ofu_u_Strong]: !r. h4/relation/StrongLinearOrder r ==> h4/toto/TotOrd (h4/toto/TO__of__LinearOrder r)
% Assm [h4s_totos_StrongOrderu_u_ALT]: !Z. h4/relation/StrongOrder Z <=> h4/relation/irreflexive Z /\ h4/relation/transitive Z
% Assm [h4s_inttos_intOrd0]: h4/intto/intOrd = h4/toto/TO__of__LinearOrder h4/integer/int__lt
% Assm [h4s_inttos_intto0]: h4/intto/intto = h4/toto/TO h4/intto/intOrd
% Goal: h4/toto/apto h4/intto/intto = h4/intto/intOrd
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_Q115692,TV_Q115688]: ![V_f, V_g]: (![V_x]: s(TV_Q115688,happ(s(t_fun(TV_Q115692,TV_Q115688),V_f),s(TV_Q115692,V_x))) = s(TV_Q115688,happ(s(t_fun(TV_Q115692,TV_Q115688),V_g),s(TV_Q115692,V_x))) => s(t_fun(TV_Q115692,TV_Q115688),V_f) = s(t_fun(TV_Q115692,TV_Q115688),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_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ORu_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_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_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_relations_transitiveu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_x, V_y, V_z]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_z))))))).
fof(ah4s_relations_irreflexiveu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_irreflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_trichotomous0, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_trichotomous(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_a, V_b]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_a))),s(TV_u_27a,V_b)))) | (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_b))),s(TV_u_27a,V_a)))) | s(TV_u_27a,V_a) = s(TV_u_27a,V_b))))).
fof(ah4s_relations_StrongLinearOrder0, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_stronglinearorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_strongorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & p(s(t_bool,h4s_relations_trichotomous(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))).
fof(ah4s_integers_INTu_u_LTu_u_TOTAL, axiom, ![V_y, V_x]: (s(t_h4s_integers_int,V_x) = s(t_h4s_integers_int,V_y) | (p(s(t_bool,happ(s(t_fun(t_h4s_integers_int,t_bool),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_bool)),h4s_integers_intu_u_lt),s(t_h4s_integers_int,V_x))),s(t_h4s_integers_int,V_y)))) | p(s(t_bool,happ(s(t_fun(t_h4s_integers_int,t_bool),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_bool)),h4s_integers_intu_u_lt),s(t_h4s_integers_int,V_y))),s(t_h4s_integers_int,V_x))))))).
fof(ah4s_integers_INTu_u_LTu_u_REFL, axiom, ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(t_h4s_integers_int,t_bool),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_bool)),h4s_integers_intu_u_lt),s(t_h4s_integers_int,V_x))),s(t_h4s_integers_int,V_x)))))).
fof(ah4s_integers_INTu_u_LTu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,happ(s(t_fun(t_h4s_integers_int,t_bool),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_bool)),h4s_integers_intu_u_lt),s(t_h4s_integers_int,V_x))),s(t_h4s_integers_int,V_y)))) & p(s(t_bool,happ(s(t_fun(t_h4s_integers_int,t_bool),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_bool)),h4s_integers_intu_u_lt),s(t_h4s_integers_int,V_y))),s(t_h4s_integers_int,V_z))))) => p(s(t_bool,happ(s(t_fun(t_h4s_integers_int,t_bool),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_bool)),h4s_integers_intu_u_lt),s(t_h4s_integers_int,V_x))),s(t_h4s_integers_int,V_z)))))).
fof(ah4s_totos_TOu_u_aptou_u_TOu_u_ID, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_totos_totord(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_totos_cpn)),V_r)))) <=> s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_totos_cpn)),h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),h4s_totos_to(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_totos_cpn)),V_r))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_totos_cpn)),V_r))).
fof(ah4s_totos_TotOrdu_u_TOu_u_ofu_u_Strong, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_relations_stronglinearorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r)))) => p(s(t_bool,h4s_totos_totord(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_totos_cpn)),h4s_totos_tou_u_ofu_u_linearorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r)))))))).
fof(ah4s_totos_StrongOrderu_u_ALT, axiom, ![TV_u_27a]: ![V_Z]: (p(s(t_bool,h4s_relations_strongorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_Z)))) <=> (p(s(t_bool,h4s_relations_irreflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_Z)))) & p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_Z))))))).
fof(ah4s_inttos_intOrd0, axiom, s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_totos_cpn)),h4s_inttos_intord) = s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_totos_cpn)),h4s_totos_tou_u_ofu_u_linearorder(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_bool)),h4s_integers_intu_u_lt)))).
fof(ah4s_inttos_intto0, axiom, s(t_h4s_totos_toto(t_h4s_integers_int),h4s_inttos_intto) = s(t_h4s_totos_toto(t_h4s_integers_int),h4s_totos_to(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_totos_cpn)),h4s_inttos_intord)))).
fof(ch4s_inttos_apinttou_u_thm, conjecture, s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_totos_cpn)),h4s_totos_apto(s(t_h4s_totos_toto(t_h4s_integers_int),h4s_inttos_intto))) = s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_totos_cpn)),h4s_inttos_intord)).
