%   ORIGINAL: h4/defCNF/OK__def_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/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/defCNF/OK__tupled__primitive__def: h4/defCNF/OK__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R)) (\OK__tupled a_27. h4/pair/pair__CASE a_27 (\n v1. h4/pair/pair__CASE v1 (\conn v3. h4/pair/pair__CASE v3 (\v4 v5. h4/sum/sum__CASE v4 (\i. h4/sum/sum__CASE v5 (\j. h4/combin/I (h4/prim__rec/_3C i n /\ h4/prim__rec/_3C j n)) (\b. h4/combin/I (h4/prim__rec/_3C i n))) (\a. h4/sum/sum__CASE v5 (\j_27. h4/combin/I (h4/prim__rec/_3C j_27 n)) (\b_27. h4/combin/I T))))))
% Assm: h4/defCNF/OK__curried__def: !x1 x. h4/defCNF/OK x x1 <=> h4/defCNF/OK__tupled (h4/pair/_2C x x1)
% Goal: !n i conn b. h4/defCNF/OK n (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INL i) (h4/sum/INR b))) <=> h4/prim__rec/_3C i 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_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_defCNFs_OKu_u_tupledu_u_primitiveu_u_def]: !_14. (!b_27. happ _14 b_27 <=> h4/combin/I T) ==> (!_13. (!n j_27. happ (happ _13 n) j_27 <=> h4/combin/I (h4/prim__rec/_3C j_27 n)) ==> (!_12. (!v5 n a. happ (happ (happ _12 v5) n) a <=> h4/sum/sum__CASE v5 (happ _13 n) _14) ==> (!_11. (!i n b. happ (happ (happ _11 i) n) b <=> h4/combin/I (h4/prim__rec/_3C i n)) ==> (!_10. (!i n j. ?v. (v <=> h4/prim__rec/_3C i n /\ h4/prim__rec/_3C j n) /\ (happ (happ (happ _10 i) n) j <=> h4/combin/I v)) ==> (!_9. (!v5 n i. happ (happ (happ _9 v5) n) i <=> h4/sum/sum__CASE v5 (happ (happ _10 i) n) (happ (happ _11 i) n)) ==> (!_8. (!v4 n v5. happ (happ (happ _8 v4) n) v5 <=> h4/sum/sum__CASE v4 (happ (happ _9 v5) n) (happ (happ _12 v5) n)) ==> (!_7. (!n v4. happ (happ _7 n) v4 = happ (happ _8 v4) n) ==> (!_6. (!n v3. happ (happ _6 n) v3 <=> h4/pair/pair__CASE v3 (happ _7 n)) ==> (!_5. (!n conn. happ (happ _5 n) conn = happ _6 n) ==> (!_4. (!n v1. happ (happ _4 n) v1 <=> h4/pair/pair__CASE v1 (happ _5 n)) ==> (!_3. (!n. happ _3 n = happ _4 n) ==> (!_2. (!a_27. happ _2 a_27 <=> h4/pair/pair__CASE a_27 _3) ==> (!_1. (!OK__tupled. happ _1 OK__tupled = _2) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R) ==> h4/defCNF/OK__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))))))))
% Assm [h4s_defCNFs_OKu_u_curriedu_u_def]: !x1 x. h4/defCNF/OK x x1 <=> happ h4/defCNF/OK__tupled (h4/pair/_2C x x1)
% Goal: !n i conn b. h4/defCNF/OK n (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INL i) (h4/sum/INR b))) <=> h4/prim__rec/_3C i 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_Q172155,TV_Q172151]: ![V_f, V_g]: (![V_x]: s(TV_Q172151,happ(s(t_fun(TV_Q172155,TV_Q172151),V_f),s(TV_Q172155,V_x))) = s(TV_Q172151,happ(s(t_fun(TV_Q172155,TV_Q172151),V_g),s(TV_Q172155,V_x))) => s(t_fun(TV_Q172155,TV_Q172151),V_f) = s(t_fun(TV_Q172155,TV_Q172151),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_defCNFs_OKu_u_tupledu_u_primitiveu_u_def, axiom, ![V_uu_14]: (![V_bu_27]: s(t_bool,happ(s(t_fun(t_bool,t_bool),V_uu_14),s(t_bool,V_bu_27))) = s(t_bool,h4s_combins_i(s(t_bool,t))) => ![V_uu_13]: (![V_n, V_ju_27]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_13),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_ju_27))) = s(t_bool,h4s_combins_i(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_ju_27),s(t_h4s_nums_num,V_n))))) => ![V_uu_12]: (![V_v5, V_n, V_a]: s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool)),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool))),V_uu_12),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v5))),s(t_h4s_nums_num,V_n))),s(t_bool,V_a))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v5),s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_13),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_bool),V_uu_14))) => ![V_uu_11]: (![V_i, V_n, V_b]: s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool))),V_uu_11),s(t_h4s_nums_num,V_i))),s(t_h4s_nums_num,V_n))),s(t_bool,V_b))) = s(t_bool,h4s_combins_i(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,V_n))))) => ![V_uu_10]: (![V_i, V_n, V_j]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_j),s(t_h4s_nums_num,V_n)))))) & s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool))),V_uu_10),s(t_h4s_nums_num,V_i))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_j))) = s(t_bool,h4s_combins_i(s(t_bool,V_v)))) => ![V_uu_9]: (![V_v5, V_n, V_i]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool))),V_uu_9),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v5))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_i))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v5),s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool))),V_uu_10),s(t_h4s_nums_num,V_i))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool))),V_uu_11),s(t_h4s_nums_num,V_i))),s(t_h4s_nums_num,V_n))))) => ![V_uu_8]: (![V_v4, V_n, V_v5]: s(t_bool,happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))),V_uu_8),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v4))),s(t_h4s_nums_num,V_n))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v5))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v4),s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool))),V_uu_9),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v5))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool)),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool))),V_uu_12),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v5))),s(t_h4s_nums_num,V_n))))) => ![V_uu_7]: (![V_n, V_v4]: s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))),V_uu_7),s(t_h4s_nums_num,V_n))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v4))) = s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))),V_uu_8),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v4))),s(t_h4s_nums_num,V_n))) => ![V_uu_6]: (![V_n, V_v3]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)),V_uu_6),s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),V_v3))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),V_v3),s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))),V_uu_7),s(t_h4s_nums_num,V_n))))) => ![V_uu_5]: (![V_n, V_conn]: s(t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool))),V_uu_5),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))) = s(t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)),V_uu_6),s(t_h4s_nums_num,V_n))) => ![V_uu_4]: (![V_n, V_v1]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),V_uu_4),s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),V_v1))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),V_v1),s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool))),V_uu_5),s(t_h4s_nums_num,V_n))))) => ![V_uu_3]: (![V_n]: s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),V_uu_3),s(t_h4s_nums_num,V_n))) = s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),V_uu_4),s(t_h4s_nums_num,V_n))) => ![V_uu_2]: (![V_au_27]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool),V_uu_2),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),V_au_27))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),V_au_27),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),V_uu_3))) => ![V_uu_1]: (![V_OKu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool)),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool),V_OKu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool),V_uu_2) => ![V_uu_0]: (![V_R]: s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool)),V_R))) = s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool)),V_R))) => s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool),h4s_defcnfs_oku_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool)),V_uu_1))))))))))))))))))).
fof(ah4s_defCNFs_OKu_u_curriedu_u_def, axiom, ![V_x1, V_x]: s(t_bool,h4s_defcnfs_ok(s(t_h4s_nums_num,V_x),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),V_x1))) = s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool),h4s_defcnfs_oku_u_tupled),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_x),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),V_x1)))))).
fof(ch4s_defCNFs_OKu_u_defu_c1, conjecture, ![V_n, V_i, V_conn, V_b]: s(t_bool,h4s_defcnfs_ok(s(t_h4s_nums_num,V_n),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),h4s_pairs_u_2c(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_pairs_u_2c(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),h4s_sums_inl(s(t_h4s_nums_num,V_i))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),h4s_sums_inr(s(t_bool,V_b))))))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,V_n)))).
