%   ORIGINAL: h4/quotient__pred__set/FINITER__INDUCT
% 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/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> 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_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/res__quan/RES__FORALL: !f P. h4/bool/RES__FORALL P f <=> (!x. h4/bool/IN x P ==> f x)
% Assm: h4/quotient/FUN__REL: !g f R2 R1. h4/quotient/_3D_3D_3D_3E R1 R2 f g <=> (!x y. R1 x y ==> R2 (f x) (g y))
% Assm: h4/quotient/IN__RESPECTS: !x R. h4/bool/IN x (h4/quotient/respects R) <=> R x x
% Assm: h4/quotient__pred__set/FINITER__def: !s R. h4/quotient__pred__set/FINITER R s <=> h4/bool/RES__FORALL (h4/quotient/respects (h4/quotient/_3D_3D_3D_3E (h4/quotient/_3D_3D_3D_3E R $equals) $equals)) (\P. P h4/pred__set/EMPTY /\ h4/bool/RES__FORALL (h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R $equals)) (\s0. P s0 ==> h4/bool/RES__FORALL (h4/quotient/respects R) (\e. P (h4/quotient__pred__set/INSERTR R e s0))) ==> P s)
% Assm: h4/quotient__pred__set/FINITER__EQ: !s2 s1 R. h4/quotient/_3D_3D_3D_3E R $equals s1 s2 ==> (h4/quotient__pred__set/FINITER R s1 <=> h4/quotient__pred__set/FINITER R s2)
% Assm: h4/quotient__pred__set/FINITER__EMPTY: !R. h4/quotient__pred__set/FINITER R h4/pred__set/EMPTY
% Assm: h4/quotient__pred__set/FINITER__INSERTR: !R. h4/bool/RES__FORALL (h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R $equals)) (\s. h4/quotient__pred__set/FINITER R s ==> h4/bool/RES__FORALL (h4/quotient/respects R) (\x. h4/quotient__pred__set/FINITER R (h4/quotient__pred__set/INSERTR R x s)))
% Assm: h4/quotient__pred__set/ABSORPTIONR: !R. h4/bool/RES__FORALL (h4/quotient/respects R) (\x. h4/bool/RES__FORALL (h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R $equals)) (\s. h4/bool/IN x s <=> h4/quotient/_3D_3D_3D_3E R $equals (h4/quotient__pred__set/INSERTR R x s) s))
% Goal: !R. h4/bool/RES__FORALL (h4/quotient/respects (h4/quotient/_3D_3D_3D_3E (h4/quotient/_3D_3D_3D_3E R $equals) $equals)) (\P. P h4/pred__set/EMPTY /\ h4/bool/RES__FORALL (h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R $equals)) (\s. h4/quotient__pred__set/FINITER R s /\ P s ==> h4/bool/RES__FORALL (h4/quotient/respects R) (\e. ~h4/bool/IN e s ==> P (h4/quotient__pred__set/INSERTR R e s))) ==> h4/bool/RES__FORALL (h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R $equals)) (\s. h4/quotient__pred__set/FINITER R s ==> P s))
%   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_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~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_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> 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_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_resu_u_quans_RESu_u_FORALL]: !f P. h4/bool/RES__FORALL P f <=> (!x. h4/bool/IN x P ==> happ f x)
% Assm [h4s_quotients_FUNu_u_REL]: !g f R2 R1. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g <=> (!x y. happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y))
% Assm [h4s_quotients_INu_u_RESPECTS]: !x R. h4/bool/IN x (h4/quotient/respects R) <=> happ (happ R x) x
% Assm [h4s_quotientu_u_predu_u_sets_FINITERu_u_def]: !_2. (!P R s0 e. happ (happ (happ (happ _2 P) R) s0) e <=> happ P (h4/quotient__pred__set/INSERTR R e s0)) ==> (!_1. (!P R s0. happ (happ (happ _1 P) R) s0 <=> happ P s0 ==> h4/bool/RES__FORALL (h4/quotient/respects R) (happ (happ (happ _2 P) R) s0)) ==> (!_0. (!R s P. happ (happ (happ _0 R) s) P <=> happ P h4/pred__set/EMPTY /\ h4/bool/RES__FORALL (h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R $equals)) (happ (happ _1 P) R) ==> happ P s) ==> (!s R. h4/quotient__pred__set/FINITER R s <=> h4/bool/RES__FORALL (h4/quotient/respects (h4/quotient/_3D_3D_3D_3E (h4/quotient/_3D_3D_3D_3E R $equals) $equals)) (happ (happ _0 R) s))))
% Assm [h4s_quotientu_u_predu_u_sets_FINITERu_u_EQ]: !s2 s1 R. happ (happ (h4/quotient/_3D_3D_3D_3E R $equals) s1) s2 ==> (h4/quotient__pred__set/FINITER R s1 <=> h4/quotient__pred__set/FINITER R s2)
% Assm [h4s_quotientu_u_predu_u_sets_FINITERu_u_EMPTY]: !R. h4/quotient__pred__set/FINITER R h4/pred__set/EMPTY
% Assm [h4s_quotientu_u_predu_u_sets_FINITERu_u_INSERTR]: !_1. (!R s x. happ (happ (happ _1 R) s) x <=> h4/quotient__pred__set/FINITER R (h4/quotient__pred__set/INSERTR R x s)) ==> (!_0. (!R s. happ (happ _0 R) s <=> h4/quotient__pred__set/FINITER R s ==> h4/bool/RES__FORALL (h4/quotient/respects R) (happ (happ _1 R) s)) ==> (!R. h4/bool/RES__FORALL (h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R $equals)) (happ _0 R)))
% Assm [h4s_quotientu_u_predu_u_sets_ABSORPTIONR]: !_1. (!R x s. happ (happ (happ _1 R) x) s <=> h4/bool/IN x s <=> happ (happ (h4/quotient/_3D_3D_3D_3E R $equals) (h4/quotient__pred__set/INSERTR R x s)) s) ==> (!_0. (!R x. happ (happ _0 R) x <=> h4/bool/RES__FORALL (h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R $equals)) (happ (happ _1 R) x)) ==> (!R. h4/bool/RES__FORALL (h4/quotient/respects R) (happ _0 R)))
% Goal: !_3. (!R P s. happ (happ (happ _3 R) P) s <=> h4/quotient__pred__set/FINITER R s ==> happ P s) ==> (!_2. (!P R s e. happ (happ (happ (happ _2 P) R) s) e <=> ~h4/bool/IN e s ==> happ P (h4/quotient__pred__set/INSERTR R e s)) ==> (!_1. (!P R s. happ (happ (happ _1 P) R) s <=> h4/quotient__pred__set/FINITER R s /\ happ P s ==> h4/bool/RES__FORALL (h4/quotient/respects R) (happ (happ (happ _2 P) R) s)) ==> (!_0. (!R P. happ (happ _0 R) P <=> happ P h4/pred__set/EMPTY /\ h4/bool/RES__FORALL (h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R $equals)) (happ (happ _1 P) R) ==> h4/bool/RES__FORALL (h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R $equals)) (happ (happ _3 R) P)) ==> (!R. h4/bool/RES__FORALL (h4/quotient/respects (h4/quotient/_3D_3D_3D_3E (h4/quotient/_3D_3D_3D_3E R $equals) $equals)) (happ _0 R)))))
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_Q86428,TV_Q86424]: ![V_f, V_g]: (![V_x]: s(TV_Q86424,happ(s(t_fun(TV_Q86428,TV_Q86424),V_f),s(TV_Q86428,V_x))) = s(TV_Q86424,happ(s(t_fun(TV_Q86428,TV_Q86424),V_g),s(TV_Q86428,V_x))) => s(t_fun(TV_Q86428,TV_Q86424),V_f) = s(t_fun(TV_Q86428,TV_Q86424),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_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (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_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_IMPu_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_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => 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_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_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_resu_u_quans_RESu_u_FORALL, axiom, ![TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_f)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_quotients_FUNu_u_REL, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) <=> ![V_x, 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_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y))))))))).
fof(ah4s_quotients_INu_u_RESPECTS, axiom, ![TV_u_27a]: ![V_x, V_R]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = 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_quotientu_u_predu_u_sets_FINITERu_u_def, axiom, ![TV_u_27a]: ![V_uu_2]: (![V_P, V_R, V_s0, V_e]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)))),V_uu_2),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_s0))),s(TV_u_27a,V_e))) = s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_quotientu_u_predu_u_sets_insertr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s0))))) => ![V_uu_1]: (![V_P, V_R, V_s0]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_bool))),V_uu_1),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_s0)))) <=> (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s0)))) => p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)))),V_uu_2),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_s0)))))))) => ![V_uu_0]: (![V_R, V_s, V_P]: (p(s(t_bool,happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))) <=> ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) & p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_quotients_respects(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals))))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_bool))),V_uu_1),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s)))))) => ![V_s, V_R]: s(t_bool,h4s_quotientu_u_predu_u_sets_finiter(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool),h4s_quotients_respects(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals))))),s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_s))))))))).
fof(ah4s_quotientu_u_predu_u_sets_FINITERu_u_EQ, axiom, ![TV_u_27a]: ![V_s2, V_s1, V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals))),s(t_fun(TV_u_27a,t_bool),V_s1))),s(t_fun(TV_u_27a,t_bool),V_s2)))) => s(t_bool,h4s_quotientu_u_predu_u_sets_finiter(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_s1))) = s(t_bool,h4s_quotientu_u_predu_u_sets_finiter(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_s2))))).
fof(ah4s_quotientu_u_predu_u_sets_FINITERu_u_EMPTY, axiom, ![TV_u_27a]: ![V_R]: p(s(t_bool,h4s_quotientu_u_predu_u_sets_finiter(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))))).
fof(ah4s_quotientu_u_predu_u_sets_FINITERu_u_INSERTR, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_R, V_s, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27a,V_x))) = s(t_bool,h4s_quotientu_u_predu_u_sets_finiter(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),h4s_quotientu_u_predu_u_sets_insertr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) => ![V_uu_0]: (![V_R, V_s]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_s)))) <=> (p(s(t_bool,h4s_quotientu_u_predu_u_sets_finiter(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_s)))))))) => ![V_R]: p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_quotients_respects(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals))))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))))).
fof(ah4s_quotientu_u_predu_u_sets_ABSORPTIONR, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_R, V_x, V_s]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_bool),t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,t_bool),V_s)))) <=> s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals))),s(t_fun(TV_u_27a,t_bool),h4s_quotientu_u_predu_u_sets_insertr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))),s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_uu_0]: (![V_R, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_quotients_respects(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals))))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_bool),t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))))) => ![V_R]: p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))))).
fof(ch4s_quotientu_u_predu_u_sets_FINITERu_u_INDUCT, conjecture, ![TV_u_27a]: ![V_uu_3]: (![V_R, V_P, V_s]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool))),V_uu_3),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),V_s)))) <=> (p(s(t_bool,h4s_quotientu_u_predu_u_sets_finiter(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s)))))) => ![V_uu_2]: (![V_P, V_R, V_s, V_e]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)))),V_uu_2),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27a,V_e)))) <=> (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_quotientu_u_predu_u_sets_insertr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s)))))))) => ![V_uu_1]: (![V_P, V_R, V_s]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_bool))),V_uu_1),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_s)))) <=> ((p(s(t_bool,h4s_quotientu_u_predu_u_sets_finiter(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))) => p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)))),V_uu_2),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_s)))))))) => ![V_uu_0]: (![V_R, V_P]: (p(s(t_bool,happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))) <=> ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) & p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_quotients_respects(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals))))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_bool))),V_uu_1),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))) => p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_quotients_respects(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals))))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool))),V_uu_3),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))))))) => ![V_R]: p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool),h4s_quotients_respects(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals))))),s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))))))).
