%   ORIGINAL: h4/quotient__sum/SUM__REL__def_c3
% 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/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/sum/sum__case__def_c0: !x f1 f. h4/sum/sum__CASE (h4/sum/INL x) f f1 = f x
% Assm: h4/sum/sum__case__def_c1: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = f1 y
% Assm: h4/relation/WF__EMPTY__REL: h4/relation/WF h4/relation/EMPTY__REL
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/quotient__sum/SUM__REL__tupled__primitive__def: h4/quotient__sum/SUM__REL__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R)) (\SUM__REL__tupled a. h4/pair/pair__CASE a (\R1 v1. h4/pair/pair__CASE v1 (\R2 v3. h4/pair/pair__CASE v3 (\v4 v5. h4/sum/sum__CASE v4 (\a1. h4/sum/sum__CASE v5 (\a2. h4/combin/I (R1 a1 a2)) (\b2_27. h4/combin/I F)) (\b1. h4/sum/sum__CASE v5 (\a2_27. h4/combin/I F) (\b2. h4/combin/I (R2 b1 b2)))))))
% Assm: h4/quotient__sum/SUM__REL__curried__def: !x3 x2 x1 x. h4/quotient__sum/_2B_2B_2B x x1 x2 x3 <=> h4/quotient__sum/SUM__REL__tupled (h4/pair/_2C x (h4/pair/_2C x1 (h4/pair/_2C x2 x3)))
% Goal: !b1 a2 R2 R1. h4/quotient__sum/_2B_2B_2B R1 R2 (h4/sum/INR b1) (h4/sum/INL a2) <=> F
%   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_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_sums_sumu_u_caseu_u_defu_c0]: !x f1 f. h4/sum/sum__CASE (h4/sum/INL x) f f1 = happ f x
% Assm [h4s_sums_sumu_u_caseu_u_defu_c1]: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = happ f1 y
% Assm [h4s_relations_WFu_u_EMPTYu_u_REL]: h4/relation/WF h4/relation/EMPTY__REL
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_quotientu_u_sums_SUMu_u_RELu_u_tupledu_u_primitiveu_u_def]: !_14. (!R2 b1 b2. happ (happ (happ _14 R2) b1) b2 <=> h4/combin/I (happ (happ R2 b1) b2)) ==> (!_13. (!a2_27. happ _13 a2_27 <=> h4/combin/I F) ==> (!_12. (!v5 R2 b1. happ (happ (happ _12 v5) R2) b1 <=> h4/sum/sum__CASE v5 _13 (happ (happ _14 R2) b1)) ==> (!_11. (!b2_27. happ _11 b2_27 <=> h4/combin/I F) ==> (!_10. (!R1 a1 a2. happ (happ (happ _10 R1) a1) a2 <=> h4/combin/I (happ (happ R1 a1) a2)) ==> (!_9. (!v5 R1 a1. happ (happ (happ _9 v5) R1) a1 <=> h4/sum/sum__CASE v5 (happ (happ _10 R1) a1) _11) ==> (!_8. (!v4 R1 R2 v5. happ (happ (happ (happ _8 v4) R1) R2) v5 <=> h4/sum/sum__CASE v4 (happ (happ _9 v5) R1) (happ (happ _12 v5) R2)) ==> (!_7. (!R1 R2 v4. happ (happ (happ _7 R1) R2) v4 = happ (happ (happ _8 v4) R1) R2) ==> (!_6. (!R1 R2 v3. happ (happ (happ _6 R1) R2) v3 <=> h4/pair/pair__CASE v3 (happ (happ _7 R1) R2)) ==> (!_5. (!R1 R2. happ (happ _5 R1) R2 = happ (happ _6 R1) R2) ==> (!_4. (!R1 v1. happ (happ _4 R1) v1 <=> h4/pair/pair__CASE v1 (happ _5 R1)) ==> (!_3. (!R1. happ _3 R1 = happ _4 R1) ==> (!_2. (!a. happ _2 a <=> h4/pair/pair__CASE a _3) ==> (!_1. (!SUM__REL__tupled. happ _1 SUM__REL__tupled = _2) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R) ==> h4/quotient__sum/SUM__REL__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))))))))
% Assm [h4s_quotientu_u_sums_SUMu_u_RELu_u_curriedu_u_def]: !x3 x2 x1 x. h4/quotient__sum/_2B_2B_2B x x1 x2 x3 <=> happ h4/quotient__sum/SUM__REL__tupled (h4/pair/_2C x (h4/pair/_2C x1 (h4/pair/_2C x2 x3)))
% Goal: !b1 a2 R2 R1. h4/quotient__sum/_2B_2B_2B R1 R2 (h4/sum/INR b1) (h4/sum/INL a2) <=> F
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_Q344056,TV_Q344052]: ![V_f, V_g]: (![V_x]: s(TV_Q344052,happ(s(t_fun(TV_Q344056,TV_Q344052),V_f),s(TV_Q344056,V_x))) = s(TV_Q344052,happ(s(t_fun(TV_Q344056,TV_Q344052),V_g),s(TV_Q344056,V_x))) => s(t_fun(TV_Q344056,TV_Q344052),V_f) = s(t_fun(TV_Q344056,TV_Q344052),V_g))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
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_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_sums_sumu_u_caseu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_sums_sumu_u_caseu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_y, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f1),s(TV_u_27b,V_y)))).
fof(ah4s_relations_WFu_u_EMPTYu_u_REL, axiom, ![TV_u_27a]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_emptyu_u_rel))))).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: 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),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),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_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_quotientu_u_sums_SUMu_u_RELu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_14]: (![V_R2, V_b1, V_b2]: 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)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))),V_uu_14),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_b1))),s(TV_u_27b,V_b2))) = s(t_bool,h4s_combins_i(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,V_b1))),s(TV_u_27b,V_b2))))) => ![V_uu_13]: (![V_a2u_27]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_uu_13),s(TV_u_27a,V_a2u_27))) = s(t_bool,h4s_combins_i(s(t_bool,f))) => ![V_uu_12]: (![V_v5, V_R2, V_b1]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_bool))),V_uu_12),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v5))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_b1))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v5),s(t_fun(TV_u_27a,t_bool),V_uu_13),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))),V_uu_14),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_b1))))) => ![V_uu_11]: (![V_b2u_27]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_uu_11),s(TV_u_27b,V_b2u_27))) = s(t_bool,h4s_combins_i(s(t_bool,f))) => ![V_uu_10]: (![V_R1, V_a1, V_a2]: 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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),V_uu_10),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_a1))),s(TV_u_27a,V_a2))) = s(t_bool,h4s_combins_i(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_a1))),s(TV_u_27a,V_a2))))) => ![V_uu_9]: (![V_v5, V_R1, V_a1]: 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)),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool))),V_uu_9),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v5))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_a1))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v5),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,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_fun(TV_u_27a,t_bool))),V_uu_10),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_a1))),s(t_fun(TV_u_27b,t_bool),V_uu_11))) => ![V_uu_8]: (![V_v4, V_R1, V_R2, V_v5]: s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool)))),V_uu_8),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v4))),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_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v5))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v4),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)),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool))),V_uu_9),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v5))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_bool))),V_uu_12),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v5))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))) => ![V_uu_7]: (![V_R1, V_R2, V_v4]: s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool)))),V_uu_7),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_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v4))) = s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool)))),V_uu_8),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v4))),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))) => ![V_uu_6]: (![V_R1, V_R2, V_v3]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool))),V_uu_6),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_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_v3))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_v3),s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool)))),V_uu_7),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))))) => ![V_uu_5]: (![V_R1, V_R2]: s(t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool))),V_uu_5),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(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool))),V_uu_6),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))) => ![V_uu_4]: (![V_R1, V_v1]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),t_bool)),V_uu_4),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),V_v1))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),V_v1),s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool))),V_uu_5),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))))) => ![V_uu_3]: (![V_R1]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),t_bool)),V_uu_3),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),t_bool)),V_uu_4),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))) => ![V_uu_2]: (![V_a]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),V_uu_2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),V_a))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),V_a),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),t_bool)),V_uu_3))) => ![V_uu_1]: (![V_SUMu_u_RELu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool)),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),V_SUMu_u_RELu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),V_uu_2) => ![V_uu_0]: (![V_R]: s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool)),V_R))) = s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool)),V_R))) => s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),h4s_quotientu_u_sums_sumu_u_relu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool)),V_uu_1))))))))))))))))))).
fof(ah4s_quotientu_u_sums_SUMu_u_RELu_u_curriedu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x3, V_x2, V_x1, V_x]: s(t_bool,h4s_quotientu_u_sums_u_2bu_2bu_2b(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_x),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_x1),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_x2),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_x3))) = s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),h4s_quotientu_u_sums_sumu_u_relu_u_tupled),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_x),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_x1),s(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_x2),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_x3)))))))))).
fof(ch4s_quotientu_u_sums_SUMu_u_RELu_u_defu_c3, conjecture, ![TV_u_27b,TV_u_27a]: ![V_b1, V_a2, V_R2, V_R1]: s(t_bool,h4s_quotientu_u_sums_u_2bu_2bu_2b(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_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_b1))),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_a2))))) = s(t_bool,f)).
