%   ORIGINAL: h4/sptree/mk__BN__def_c4
% 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/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/sptree/spt__case__def_c0: !v f2 f1 f. h4/sptree/spt__CASE h4/sptree/LN v f f1 f2 = v
% Assm: h4/sptree/spt__case__def_c1: !v f2 f1 f a. h4/sptree/spt__CASE (h4/sptree/LS a) v f f1 f2 = f a
% Assm: h4/sptree/spt__case__def_c2: !v f2 f1 f a1 a0. h4/sptree/spt__CASE (h4/sptree/BN a0 a1) v f f1 f2 = f1 a0 a1
% Assm: h4/sptree/spt__case__def_c3: !v f2 f1 f a2 a1 a0. h4/sptree/spt__CASE (h4/sptree/BS a0 a1 a2) v f f1 f2 = f2 a0 a1 a2
% Assm: h4/sptree/mk__BN__tupled__primitive__def: h4/sptree/mk__BN__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R)) (\mk__BN__tupled a. h4/pair/pair__CASE a (\v v1. h4/sptree/spt__CASE v (h4/sptree/spt__CASE v1 (h4/combin/I h4/sptree/LN) (\v20. h4/combin/I (h4/sptree/BN h4/sptree/LN (h4/sptree/LS v20))) (\v21 v22. h4/combin/I (h4/sptree/BN h4/sptree/LN (h4/sptree/BN v21 v22))) (\v23 v24 v25. h4/combin/I (h4/sptree/BN h4/sptree/LN (h4/sptree/BS v23 v24 v25)))) (\v8. h4/combin/I (h4/sptree/BN (h4/sptree/LS v8) v1)) (\v9 v10. h4/combin/I (h4/sptree/BN (h4/sptree/BN v9 v10) v1)) (\v11 v12 v13. h4/combin/I (h4/sptree/BN (h4/sptree/BS v11 v12 v13) v1))))
% Assm: h4/sptree/mk__BN__curried__def: !x1 x. h4/sptree/mk__BN x x1 = h4/sptree/mk__BN__tupled (h4/pair/_2C x x1)
% Goal: !v2 t2. h4/sptree/mk__BN (h4/sptree/LS v2) t2 = h4/sptree/BN (h4/sptree/LS v2) t2
%   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_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_sptrees_sptu_u_caseu_u_defu_c0]: !v f2 f1 f. h4/sptree/spt__CASE h4/sptree/LN v f f1 f2 = v
% Assm [h4s_sptrees_sptu_u_caseu_u_defu_c1]: !v f2 f1 f a. h4/sptree/spt__CASE (h4/sptree/LS a) v f f1 f2 = happ f a
% Assm [h4s_sptrees_sptu_u_caseu_u_defu_c2]: !v f2 f1 f a1 a0. h4/sptree/spt__CASE (h4/sptree/BN a0 a1) v f f1 f2 = happ (happ f1 a0) a1
% Assm [h4s_sptrees_sptu_u_caseu_u_defu_c3]: !v f2 f1 f a2 a1 a0. h4/sptree/spt__CASE (h4/sptree/BS a0 a1 a2) v f f1 f2 = happ (happ (happ f2 a0) a1) a2
% Assm [h4s_sptrees_mku_u_BNu_u_tupledu_u_primitiveu_u_def]: !_16. (!v11 v12 v1 v13. happ (happ (happ (happ _16 v11) v12) v1) v13 = h4/combin/I (h4/sptree/BN (h4/sptree/BS v11 v12 v13) v1)) ==> (!_15. (!v11 v1 v12. happ (happ (happ _15 v11) v1) v12 = happ (happ (happ _16 v11) v12) v1) ==> (!_14. (!v1 v11. happ (happ _14 v1) v11 = happ (happ _15 v11) v1) ==> (!_13. (!v9 v1 v10. happ (happ (happ _13 v9) v1) v10 = h4/combin/I (h4/sptree/BN (h4/sptree/BN v9 v10) v1)) ==> (!_12. (!v1 v9. happ (happ _12 v1) v9 = happ (happ _13 v9) v1) ==> (!_11. (!v1 v8. happ (happ _11 v1) v8 = h4/combin/I (h4/sptree/BN (h4/sptree/LS v8) v1)) ==> (!_10. (!v23 v24 v25. happ (happ (happ _10 v23) v24) v25 = h4/combin/I (h4/sptree/BN h4/sptree/LN (h4/sptree/BS v23 v24 v25))) ==> (!_9. (!v23 v24. happ (happ _9 v23) v24 = happ (happ _10 v23) v24) ==> (!_8. (!v23. happ _8 v23 = happ _9 v23) ==> (!_7. (!v21 v22. happ (happ _7 v21) v22 = h4/combin/I (h4/sptree/BN h4/sptree/LN (h4/sptree/BN v21 v22))) ==> (!_6. (!v21. happ _6 v21 = happ _7 v21) ==> (!_5. (!v20. happ _5 v20 = h4/combin/I (h4/sptree/BN h4/sptree/LN (h4/sptree/LS v20))) ==> (!_4. (!v v1. happ (happ _4 v) v1 = h4/sptree/spt__CASE v (h4/sptree/spt__CASE v1 (h4/combin/I h4/sptree/LN) _5 _6 _8) (happ _11 v1) (happ _12 v1) (happ _14 v1)) ==> (!_3. (!v. happ _3 v = happ _4 v) ==> (!_2. (!a. happ _2 a = h4/pair/pair__CASE a _3) ==> (!_1. (!mk__BN__tupled. happ _1 mk__BN__tupled = _2) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R) ==> h4/sptree/mk__BN__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))))))))))
% Assm [h4s_sptrees_mku_u_BNu_u_curriedu_u_def]: !x1 x. h4/sptree/mk__BN x x1 = happ h4/sptree/mk__BN__tupled (h4/pair/_2C x x1)
% Goal: !v2 t2. h4/sptree/mk__BN (h4/sptree/LS v2) t2 = h4/sptree/BN (h4/sptree/LS v2) t2
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_Q81200,TV_Q81196]: ![V_f, V_g]: (![V_x]: s(TV_Q81196,happ(s(t_fun(TV_Q81200,TV_Q81196),V_f),s(TV_Q81200,V_x))) = s(TV_Q81196,happ(s(t_fun(TV_Q81200,TV_Q81196),V_g),s(TV_Q81200,V_x))) => s(t_fun(TV_Q81200,TV_Q81196),V_f) = s(t_fun(TV_Q81200,TV_Q81196),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_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_sptrees_sptu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f2, V_f1, V_f]: s(TV_u_27b,h4s_sptrees_sptu_u_case(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),V_f1),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b))),V_f2))) = s(TV_u_27b,V_v)).
fof(ah4s_sptrees_sptu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f2, V_f1, V_f, V_a]: s(TV_u_27b,h4s_sptrees_sptu_u_case(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ls(s(TV_u_27a,V_a))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),V_f1),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b))),V_f2))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_a)))).
fof(ah4s_sptrees_sptu_u_caseu_u_defu_c2, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f2, V_f1, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_sptrees_sptu_u_case(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bn(s(t_h4s_sptrees_spt(TV_u_27a),V_a0),s(t_h4s_sptrees_spt(TV_u_27a),V_a1))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),V_f1),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b))),V_f2))) = s(TV_u_27b,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),V_f1),s(t_h4s_sptrees_spt(TV_u_27a),V_a0))),s(t_h4s_sptrees_spt(TV_u_27a),V_a1)))).
fof(ah4s_sptrees_sptu_u_caseu_u_defu_c3, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f2, V_f1, V_f, V_a2, V_a1, V_a0]: s(TV_u_27b,h4s_sptrees_sptu_u_case(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bs(s(t_h4s_sptrees_spt(TV_u_27a),V_a0),s(TV_u_27a,V_a1),s(t_h4s_sptrees_spt(TV_u_27a),V_a2))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),V_f1),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b))),V_f2))) = s(TV_u_27b,happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),TV_u_27b))),V_f2),s(t_h4s_sptrees_spt(TV_u_27a),V_a0))),s(TV_u_27a,V_a1))),s(t_h4s_sptrees_spt(TV_u_27a),V_a2)))).
fof(ah4s_sptrees_mku_u_BNu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a]: ![V_uu_16]: (![V_v11, V_v12, V_v1, V_v13]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),V_uu_16),s(t_h4s_sptrees_spt(TV_u_27a),V_v11))),s(TV_u_27a,V_v12))),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))),s(t_h4s_sptrees_spt(TV_u_27a),V_v13))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_combins_i(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bn(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bs(s(t_h4s_sptrees_spt(TV_u_27a),V_v11),s(TV_u_27a,V_v12),s(t_h4s_sptrees_spt(TV_u_27a),V_v13))),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))))) => ![V_uu_15]: (![V_v11, V_v1, V_v12]: s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),V_uu_15),s(t_h4s_sptrees_spt(TV_u_27a),V_v11))),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))),s(TV_u_27a,V_v12))) = s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),V_uu_16),s(t_h4s_sptrees_spt(TV_u_27a),V_v11))),s(TV_u_27a,V_v12))),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))) => ![V_uu_14]: (![V_v1, V_v11]: s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),V_uu_14),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))),s(t_h4s_sptrees_spt(TV_u_27a),V_v11))) = s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),V_uu_15),s(t_h4s_sptrees_spt(TV_u_27a),V_v11))),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))) => ![V_uu_13]: (![V_v9, V_v1, V_v10]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_13),s(t_h4s_sptrees_spt(TV_u_27a),V_v9))),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))),s(t_h4s_sptrees_spt(TV_u_27a),V_v10))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_combins_i(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bn(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bn(s(t_h4s_sptrees_spt(TV_u_27a),V_v9),s(t_h4s_sptrees_spt(TV_u_27a),V_v10))),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))))) => ![V_uu_12]: (![V_v1, V_v9]: s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_12),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))),s(t_h4s_sptrees_spt(TV_u_27a),V_v9))) = s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_13),s(t_h4s_sptrees_spt(TV_u_27a),V_v9))),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))) => ![V_uu_11]: (![V_v1, V_v8]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),V_uu_11),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))),s(TV_u_27a,V_v8))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_combins_i(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bn(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ls(s(TV_u_27a,V_v8))),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))))) => ![V_uu_10]: (![V_v23, V_v24, V_v25]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_10),s(t_h4s_sptrees_spt(TV_u_27a),V_v23))),s(TV_u_27a,V_v24))),s(t_h4s_sptrees_spt(TV_u_27a),V_v25))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_combins_i(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bn(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bs(s(t_h4s_sptrees_spt(TV_u_27a),V_v23),s(TV_u_27a,V_v24),s(t_h4s_sptrees_spt(TV_u_27a),V_v25))))))) => ![V_uu_9]: (![V_v23, V_v24]: s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_9),s(t_h4s_sptrees_spt(TV_u_27a),V_v23))),s(TV_u_27a,V_v24))) = s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_10),s(t_h4s_sptrees_spt(TV_u_27a),V_v23))),s(TV_u_27a,V_v24))) => ![V_uu_8]: (![V_v23]: s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_8),s(t_h4s_sptrees_spt(TV_u_27a),V_v23))) = s(t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_9),s(t_h4s_sptrees_spt(TV_u_27a),V_v23))) => ![V_uu_7]: (![V_v21, V_v22]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),V_uu_7),s(t_h4s_sptrees_spt(TV_u_27a),V_v21))),s(t_h4s_sptrees_spt(TV_u_27a),V_v22))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_combins_i(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bn(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bn(s(t_h4s_sptrees_spt(TV_u_27a),V_v21),s(t_h4s_sptrees_spt(TV_u_27a),V_v22))))))) => ![V_uu_6]: (![V_v21]: s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),V_uu_6),s(t_h4s_sptrees_spt(TV_u_27a),V_v21))) = s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),V_uu_7),s(t_h4s_sptrees_spt(TV_u_27a),V_v21))) => ![V_uu_5]: (![V_v20]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),V_uu_5),s(TV_u_27a,V_v20))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_combins_i(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bn(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ls(s(TV_u_27a,V_v20))))))) => ![V_uu_4]: (![V_v, V_v1]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),V_uu_4),s(t_h4s_sptrees_spt(TV_u_27a),V_v))),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_sptu_u_case(s(t_h4s_sptrees_spt(TV_u_27a),V_v),s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_sptu_u_case(s(t_h4s_sptrees_spt(TV_u_27a),V_v1),s(t_h4s_sptrees_spt(TV_u_27a),h4s_combins_i(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln))),s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),V_uu_5),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),V_uu_6),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_8))),s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),V_uu_11),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_12),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))))),V_uu_14),s(t_h4s_sptrees_spt(TV_u_27a),V_v1))))) => ![V_uu_3]: (![V_v]: s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),V_uu_3),s(t_h4s_sptrees_spt(TV_u_27a),V_v))) = s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),V_uu_4),s(t_h4s_sptrees_spt(TV_u_27a),V_v))) => ![V_uu_2]: (![V_a]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),V_uu_2),s(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),V_a))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),V_a),s(t_fun(t_h4s_sptrees_spt(TV_u_27a),t_fun(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a))),V_uu_3))) => ![V_uu_1]: (![V_mku_u_BNu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),V_mku_u_BNu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),V_uu_2) => ![V_uu_0]: (![V_R]: s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_bool)),V_R))) = s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_bool)),V_R))) => s(t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_mku_u_bnu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a))),V_uu_1))))))))))))))))))))).
fof(ah4s_sptrees_mku_u_BNu_u_curriedu_u_def, axiom, ![TV_u_27a]: ![V_x1, V_x]: s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_mku_u_bn(s(t_h4s_sptrees_spt(TV_u_27a),V_x),s(t_h4s_sptrees_spt(TV_u_27a),V_x1))) = s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_mku_u_bnu_u_tupled),s(t_h4s_pairs_prod(t_h4s_sptrees_spt(TV_u_27a),t_h4s_sptrees_spt(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_sptrees_spt(TV_u_27a),V_x),s(t_h4s_sptrees_spt(TV_u_27a),V_x1)))))).
fof(ch4s_sptrees_mku_u_BNu_u_defu_c4, conjecture, ![TV_u_27a]: ![V_v2, V_t2]: s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_mku_u_bn(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ls(s(TV_u_27a,V_v2))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_bn(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ls(s(TV_u_27a,V_v2))),s(t_h4s_sptrees_spt(TV_u_27a),V_t2)))).
