%   ORIGINAL: h4/sptree/fromAList__def_c0
% 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/TRUTH: T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/relation/WF__inv__image: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% 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/prim__rec/WF__LESS: h4/relation/WF h4/prim__rec/_3C
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__COMM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/list/list__case__def_c0: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm: h4/list/list__case__def_c1: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = f a0 a1
% Assm: h4/list/list__size__def_c1: !f a1 a0. h4/list/list__size f (h4/list/CONS a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (f a0) (h4/list/list__size f a1))
% Assm: h4/sptree/fromAList__primitive__def: h4/sptree/fromAList = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!y x xs. R xs (h4/list/CONS (h4/pair/_2C x y) xs)))) (\fromAList a. h4/list/list__CASE a (h4/combin/I h4/sptree/LN) (\v xs. h4/pair/pair__CASE v (\x y. h4/combin/I (h4/sptree/insert x y (fromAList xs)))))
% Goal: h4/sptree/fromAList h4/list/NIL = h4/sptree/LN
%   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_TRUTH]: T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (h4/relation/inv__image R f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% Assm [h4s_relations_WFu_u_invu_u_image]: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm [h4s_relations_RESTRICTu_u_LEMMA]: !z y f R. happ (happ R y) z ==> happ (h4/relation/RESTRICT f R z) y = happ f y
% 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_primu_u_recs_WFu_u_LESS]: h4/relation/WF h4/prim__rec/_3C
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% 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_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_COMM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_lists_listu_u_caseu_u_defu_c0]: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm [h4s_lists_listu_u_caseu_u_defu_c1]: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = happ (happ f a0) a1
% Assm [h4s_lists_listu_u_sizeu_u_defu_c1]: !f a1 a0. h4/list/list__size f (h4/list/CONS a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (happ f a0) (h4/list/list__size f a1))
% Assm [h4s_sptrees_fromAListu_u_primitiveu_u_def]: !_6. (!x fromAList xs y. happ (happ (happ (happ _6 x) fromAList) xs) y = h4/combin/I (h4/sptree/insert x y (happ fromAList xs))) ==> (!_5. (!fromAList xs x. happ (happ (happ _5 fromAList) xs) x = happ (happ (happ _6 x) fromAList) xs) ==> (!_4. (!v fromAList xs. happ (happ (happ _4 v) fromAList) xs = h4/pair/pair__CASE v (happ (happ _5 fromAList) xs)) ==> (!_3. (!fromAList v. happ (happ _3 fromAList) v = happ (happ _4 v) fromAList) ==> (!_2. (!fromAList a. happ (happ _2 fromAList) a = h4/list/list__CASE a (h4/combin/I h4/sptree/LN) (happ _3 fromAList)) ==> (!_1. (!fromAList. happ _1 fromAList = happ _2 fromAList) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!y x xs. happ (happ R xs) (h4/list/CONS (h4/pair/_2C x y) xs))) ==> h4/sptree/fromAList = h4/relation/WFREC (h4/min/_40 _0) _1))))))
% Goal: happ h4/sptree/fromAList h4/list/NIL = h4/sptree/LN
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_Q84346,TV_Q84342]: ![V_f, V_g]: (![V_x]: s(TV_Q84342,happ(s(t_fun(TV_Q84346,TV_Q84342),V_f),s(TV_Q84346,V_x))) = s(TV_Q84342,happ(s(t_fun(TV_Q84346,TV_Q84342),V_g),s(TV_Q84346,V_x))) => s(t_fun(TV_Q84346,TV_Q84342),V_f) = s(t_fun(TV_Q84346,TV_Q84342),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_TRUTH, axiom, p(s(t_bool,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_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
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_invu_u_imageu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_x, V_xi_]: 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)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = 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_R),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_f),s(TV_u_27a,V_xi_)))))).
fof(ah4s_relations_WFu_u_invu_u_image, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R)))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f)))))))).
fof(ah4s_relations_RESTRICTu_u_LEMMA, axiom, ![TV_u_27b,TV_u_27a]: ![V_z, V_y, V_f, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))) => s(TV_u_27b,happ(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_z))),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))).
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_primu_u_recs_WFu_u_LESS, axiom, p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c))))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
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_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_COMM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(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)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_lists_listu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_lists_listu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1)))).
fof(ah4s_lists_listu_u_sizeu_u_defu_c1, axiom, ![TV_u_27a]: ![V_f, V_a1, V_a0]: s(t_h4s_nums_num,h4s_lists_listu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_a0))),s(t_h4s_nums_num,h4s_lists_listu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(TV_u_27a),V_a1)))))))).
fof(ah4s_sptrees_fromAListu_u_primitiveu_u_def, axiom, ![TV_u_27a]: ![V_uu_6]: (![V_x, V_fromAList, V_xs, V_y]: 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_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))))),V_uu_6),s(t_h4s_nums_num,V_x))),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),V_fromAList))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),V_xs))),s(TV_u_27a,V_y))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_combins_i(s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_insert(s(t_h4s_nums_num,V_x),s(TV_u_27a,V_y),s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),V_fromAList),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),V_xs))))))) => ![V_uu_5]: (![V_fromAList, V_xs, V_x]: s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))))),V_uu_5),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),V_fromAList))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),V_xs))),s(t_h4s_nums_num,V_x))) = s(t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))))),V_uu_6),s(t_h4s_nums_num,V_x))),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),V_fromAList))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),V_xs))) => ![V_uu_4]: (![V_v, V_fromAList, V_xs]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_4),s(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),V_v))),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),V_fromAList))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),V_xs))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),V_v),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_sptrees_spt(TV_u_27a))))),V_uu_5),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),V_fromAList))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),V_xs))))) => ![V_uu_3]: (![V_fromAList, V_v]: s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),V_fromAList))),s(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),V_v))) = s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_4),s(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),V_v))),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),V_fromAList))) => ![V_uu_2]: (![V_fromAList, V_a]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),V_fromAList))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),V_a))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_lists_listu_u_case(s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),V_a),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(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),V_fromAList))))) => ![V_uu_1]: (![V_fromAList]: s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a))),V_uu_1),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),V_fromAList))) = s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),V_fromAList))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_bool)),V_R)))) & ![V_y, V_x, V_xs]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_bool),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_bool)),V_R),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),V_xs))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),h4s_lists_cons(s(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_nums_num,V_x),s(TV_u_27a,V_y))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),V_xs)))))))) => s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_fromalist) = s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),h4s_relations_wfrec(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a))),V_uu_1))))))))))).
fof(ch4s_sptrees_fromAListu_u_defu_c0, conjecture, ![TV_u_27a]: s(t_h4s_sptrees_spt(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),t_h4s_sptrees_spt(TV_u_27a)),h4s_sptrees_fromalist),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),h4s_lists_nil))) = s(t_h4s_sptrees_spt(TV_u_27a),h4s_sptrees_ln)).
