%   ORIGINAL: h4/patricia/ptree__case__cong
% 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/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/patricia/ptree__repfns_c0: !a. h4/patricia/_20_40ind__typepatricia3 (h4/patricia/_20_40ind__typepatricia4 a) = a
% Assm: h4/patricia/ptree__repfns_c1: !r. (\a0_27. !_27ptree_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (\n. h4/ind__type/BOTTOM) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C a00 (h4/pair/_2C a10 h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a0 a1) \/ (?a0 a1 a2 a3. a0_270 = (\a00 a10 a20 a30. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C a00 (h4/pair/_2C h4/bool/ARB a10)) (h4/ind__type/FCONS a20 (h4/ind__type/FCONS a30 (\n. h4/ind__type/BOTTOM)))) a0 a1 a2 a3 /\ _27ptree_27 a2 /\ _27ptree_27 a3) ==> _27ptree_27 a0_270) ==> _27ptree_27 a0_27) r <=> h4/patricia/_20_40ind__typepatricia4 (h4/patricia/_20_40ind__typepatricia3 r) = r
% Assm: h4/patricia/hidden____20__40ind____typepatricia0____def: h4/patricia/_20_40ind__typepatricia0 = h4/patricia/_20_40ind__typepatricia3 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (\n. h4/ind__type/BOTTOM))
% Assm: h4/patricia/hidden____20__40ind____typepatricia1____def: h4/patricia/_20_40ind__typepatricia1 = (\a0 a1. h4/patricia/_20_40ind__typepatricia3 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C a00 (h4/pair/_2C a10 h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a0 a1))
% Assm: h4/patricia/hidden____20__40ind____typepatricia2____def: h4/patricia/_20_40ind__typepatricia2 = (\a0 a1 a2 a3. h4/patricia/_20_40ind__typepatricia3 ((\a00 a10 a20 a30. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C a00 (h4/pair/_2C h4/bool/ARB a10)) (h4/ind__type/FCONS a20 (h4/ind__type/FCONS a30 (\n. h4/ind__type/BOTTOM)))) a0 a1 (h4/patricia/_20_40ind__typepatricia4 a2) (h4/patricia/_20_40ind__typepatricia4 a3)))
% Assm: h4/patricia/Empty0: h4/patricia/Empty = h4/patricia/_20_40ind__typepatricia0
% Assm: h4/patricia/Leaf0: h4/patricia/Leaf = h4/patricia/_20_40ind__typepatricia1
% Assm: h4/patricia/Branch0: h4/patricia/Branch = h4/patricia/_20_40ind__typepatricia2
% Assm: h4/patricia/ptree__case__def_c0: !v f1 f. h4/patricia/ptree__CASE h4/patricia/Empty v f f1 = v
% Assm: h4/patricia/ptree__case__def_c1: !v f1 f a1 a0. h4/patricia/ptree__CASE (h4/patricia/Leaf a0 a1) v f f1 = f a0 a1
% Assm: h4/patricia/ptree__case__def_c2: !v f1 f a3 a2 a1 a0. h4/patricia/ptree__CASE (h4/patricia/Branch a0 a1 a2 a3) v f f1 = f1 a0 a1 a2 a3
% Goal: !v_27 v f1_27 f1 f_27 f M_27 M. M = M_27 /\ (M_27 = h4/patricia/Empty ==> v = v_27) /\ (!a0 a1. M_27 = h4/patricia/Leaf a0 a1 ==> f a0 a1 = f_27 a0 a1) /\ (!a0 a1 a2 a3. M_27 = h4/patricia/Branch a0 a1 a2 a3 ==> f1 a0 a1 a2 a3 = f1_27 a0 a1 a2 a3) ==> h4/patricia/ptree__CASE M v f f1 = h4/patricia/ptree__CASE M_27 v_27 f_27 f1_27
%   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_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_patricias_ptreeu_u_repfnsu_c0]: !a. h4/patricia/_20_40ind__typepatricia3 (h4/patricia/_20_40ind__typepatricia4 a) = a
% Assm [h4s_patricias_ptreeu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27ptree_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) _0 \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C a0 (h4/pair/_2C a1 h4/bool/ARB)) _0) \/ (?a0 a1 a2 a3. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C a0 (h4/pair/_2C h4/bool/ARB a1)) (h4/ind__type/FCONS a2 (h4/ind__type/FCONS a3 _0)) /\ happ _27ptree_27 a2 /\ happ _27ptree_27 a3) ==> happ _27ptree_27 a0_270) ==> happ _27ptree_27 r) <=> h4/patricia/_20_40ind__typepatricia4 (h4/patricia/_20_40ind__typepatricia3 r) = r)
% Assm [h4s_patricias_hiddenu_u_u_u_20u_u_40indu_u_u_u_typepatricia0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/patricia/_20_40ind__typepatricia0 = h4/patricia/_20_40ind__typepatricia3 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) _0)
% Assm [h4s_patricias_hiddenu_u_u_u_20u_u_40indu_u_u_u_typepatricia1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/patricia/_20_40ind__typepatricia1 x) x = h4/patricia/_20_40ind__typepatricia3 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C x (h4/pair/_2C x h4/bool/ARB)) _0))
% Assm [h4s_patricias_hiddenu_u_u_u_20u_u_40indu_u_u_u_typepatricia2u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x' x x'. happ (happ (happ (happ h4/patricia/_20_40ind__typepatricia2 x) x') x) x' = h4/patricia/_20_40ind__typepatricia3 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C x (h4/pair/_2C h4/bool/ARB x')) (h4/ind__type/FCONS (h4/patricia/_20_40ind__typepatricia4 x) (h4/ind__type/FCONS (h4/patricia/_20_40ind__typepatricia4 x') _0))))
% Assm [h4s_patricias_Empty0]: h4/patricia/Empty = h4/patricia/_20_40ind__typepatricia0
% Assm [h4s_patricias_Leaf0]: h4/patricia/Leaf = h4/patricia/_20_40ind__typepatricia1
% Assm [h4s_patricias_Branch0]: h4/patricia/Branch = h4/patricia/_20_40ind__typepatricia2
% Assm [h4s_patricias_ptreeu_u_caseu_u_defu_c0]: !v f1 f. h4/patricia/ptree__CASE h4/patricia/Empty v f f1 = v
% Assm [h4s_patricias_ptreeu_u_caseu_u_defu_c1]: !v f1 f a1 a0. h4/patricia/ptree__CASE (happ (happ h4/patricia/Leaf a0) a1) v f f1 = happ (happ f a0) a1
% Assm [h4s_patricias_ptreeu_u_caseu_u_defu_c2]: !v f1 f a3 a2 a1 a0. h4/patricia/ptree__CASE (happ (happ (happ (happ h4/patricia/Branch a0) a1) a2) a3) v f f1 = happ (happ (happ (happ f1 a0) a1) a2) a3
% Goal: !v_27 v f1_27 f1 f_27 f M_27 M. M = M_27 /\ (M_27 = h4/patricia/Empty ==> v = v_27) /\ (!a0 a1. M_27 = happ (happ h4/patricia/Leaf a0) a1 ==> happ (happ f a0) a1 = happ (happ f_27 a0) a1) /\ (!a0 a1 a2 a3. M_27 = happ (happ (happ (happ h4/patricia/Branch a0) a1) a2) a3 ==> happ (happ (happ (happ f1 a0) a1) a2) a3 = happ (happ (happ (happ f1_27 a0) a1) a2) a3) ==> h4/patricia/ptree__CASE M v f f1 = h4/patricia/ptree__CASE M_27 v_27 f_27 f1_27
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q285148,TV_Q285144]: ![V_f, V_g]: (![V_x]: s(TV_Q285144,happ(s(t_fun(TV_Q285148,TV_Q285144),V_f),s(TV_Q285148,V_x))) = s(TV_Q285144,happ(s(t_fun(TV_Q285148,TV_Q285144),V_g),s(TV_Q285148,V_x))) => s(t_fun(TV_Q285148,TV_Q285144),V_f) = s(t_fun(TV_Q285148,TV_Q285144),V_g))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (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_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => 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,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_patricias_ptreeu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_u_20u_40indu_u_typepatricia3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_patricias_u_20u_40indu_u_typepatricia4(s(t_h4s_patricias_ptree(TV_u_27a),V_a))))) = s(t_h4s_patricias_ptree(TV_u_27a),V_a)).
fof(ah4s_patricias_ptreeu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27ptreeu_27]: (![V_a0u_270]: ((s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_bools_arb),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_nums_num,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0))) | (?[V_a0, V_a1]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),h4s_pairs_u_2c(s(TV_u_27a,V_a1),s(t_h4s_nums_num,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0))) | ?[V_a0, V_a1, V_a2, V_a3]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_nums_num,V_a1))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a2),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a3),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),t_bool),V_uu_27ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),t_bool),V_uu_27ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a3)))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),t_bool),V_uu_27ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),t_bool),V_uu_27ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_patricias_u_20u_40indu_u_typepatricia4(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_u_20u_40indu_u_typepatricia3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_r)))).
fof(ah4s_patricias_hiddenu_u_u_u_20u_u_40indu_u_u_u_typepatricia0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_bottom) => s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_u_20u_40indu_u_typepatricia0) = s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_u_20u_40indu_u_typepatricia3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_bools_arb),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_nums_num,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0))))))).
fof(ah4s_patricias_hiddenu_u_u_u_20u_u_40indu_u_u_u_typepatricia1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_bottom) => ![V_x, V_x0]: s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a))),h4s_patricias_u_20u_40indu_u_typepatricia1),s(t_h4s_nums_num,V_x))),s(TV_u_27a,V_x0))) = s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_u_20u_40indu_u_typepatricia3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_x),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),h4s_pairs_u_2c(s(TV_u_27a,V_x0),s(t_h4s_nums_num,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0))))))).
fof(ah4s_patricias_hiddenu_u_u_u_20u_u_40indu_u_u_u_typepatricia2u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_bottom) => ![V_x, V_xi_, V_x0, V_xi_0]: s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(TV_u_27a)),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(TV_u_27a))))),h4s_patricias_u_20u_40indu_u_typepatricia2),s(t_h4s_nums_num,V_x))),s(t_h4s_nums_num,V_xi_))),s(t_h4s_patricias_ptree(TV_u_27a),V_x0))),s(t_h4s_patricias_ptree(TV_u_27a),V_xi_0))) = s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_u_20u_40indu_u_typepatricia3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_x),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_nums_num,V_xi_))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_patricias_u_20u_40indu_u_typepatricia4(s(t_h4s_patricias_ptree(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_patricias_u_20u_40indu_u_typepatricia4(s(t_h4s_patricias_ptree(TV_u_27a),V_xi_0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0))))))))))).
fof(ah4s_patricias_Empty0, axiom, ![TV_u_27a]: s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty) = s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_u_20u_40indu_u_typepatricia0)).
fof(ah4s_patricias_Leaf0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a))),h4s_patricias_leaf) = s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a))),h4s_patricias_u_20u_40indu_u_typepatricia1)).
fof(ah4s_patricias_Branch0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(TV_u_27a))))),h4s_patricias_branch) = s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(TV_u_27a))))),h4s_patricias_u_20u_40indu_u_typepatricia2)).
fof(ah4s_patricias_ptreeu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f1, V_f]: s(TV_u_27b,h4s_patricias_ptreeu_u_case(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty),s(TV_u_27b,V_v),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_f),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)))),V_f1))) = s(TV_u_27b,V_v)).
fof(ah4s_patricias_ptreeu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f1, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_patricias_ptreeu_u_case(s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a))),h4s_patricias_leaf),s(t_h4s_nums_num,V_a0))),s(TV_u_27a,V_a1))),s(TV_u_27b,V_v),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_f),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)))),V_f1))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_f),s(t_h4s_nums_num,V_a0))),s(TV_u_27a,V_a1)))).
fof(ah4s_patricias_ptreeu_u_caseu_u_defu_c2, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f1, V_f, V_a3, V_a2, V_a1, V_a0]: s(TV_u_27b,h4s_patricias_ptreeu_u_case(s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(TV_u_27a)),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(TV_u_27a))))),h4s_patricias_branch),s(t_h4s_nums_num,V_a0))),s(t_h4s_nums_num,V_a1))),s(t_h4s_patricias_ptree(TV_u_27a),V_a2))),s(t_h4s_patricias_ptree(TV_u_27a),V_a3))),s(TV_u_27b,V_v),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_f),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)))),V_f1))) = s(TV_u_27b,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)))),V_f1),s(t_h4s_nums_num,V_a0))),s(t_h4s_nums_num,V_a1))),s(t_h4s_patricias_ptree(TV_u_27a),V_a2))),s(t_h4s_patricias_ptree(TV_u_27a),V_a3)))).
fof(ch4s_patricias_ptreeu_u_caseu_u_cong, conjecture, ![TV_u_27a,TV_u_27b]: ![V_vu_27, V_v, V_f1u_27, V_f1, V_fu_27, V_f, V_Mu_27, V_M]: ((s(t_h4s_patricias_ptree(TV_u_27a),V_M) = s(t_h4s_patricias_ptree(TV_u_27a),V_Mu_27) & ((s(t_h4s_patricias_ptree(TV_u_27a),V_Mu_27) = s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty) => s(TV_u_27b,V_v) = s(TV_u_27b,V_vu_27)) & (![V_a0, V_a1]: (s(t_h4s_patricias_ptree(TV_u_27a),V_Mu_27) = s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a))),h4s_patricias_leaf),s(t_h4s_nums_num,V_a0))),s(TV_u_27a,V_a1))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_f),s(t_h4s_nums_num,V_a0))),s(TV_u_27a,V_a1))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_fu_27),s(t_h4s_nums_num,V_a0))),s(TV_u_27a,V_a1)))) & ![V_a0, V_a1, V_a2, V_a3]: (s(t_h4s_patricias_ptree(TV_u_27a),V_Mu_27) = s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(TV_u_27a)),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(TV_u_27a))))),h4s_patricias_branch),s(t_h4s_nums_num,V_a0))),s(t_h4s_nums_num,V_a1))),s(t_h4s_patricias_ptree(TV_u_27a),V_a2))),s(t_h4s_patricias_ptree(TV_u_27a),V_a3))) => s(TV_u_27b,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)))),V_f1),s(t_h4s_nums_num,V_a0))),s(t_h4s_nums_num,V_a1))),s(t_h4s_patricias_ptree(TV_u_27a),V_a2))),s(t_h4s_patricias_ptree(TV_u_27a),V_a3))) = s(TV_u_27b,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)))),V_f1u_27),s(t_h4s_nums_num,V_a0))),s(t_h4s_nums_num,V_a1))),s(t_h4s_patricias_ptree(TV_u_27a),V_a2))),s(t_h4s_patricias_ptree(TV_u_27a),V_a3))))))) => s(TV_u_27b,h4s_patricias_ptreeu_u_case(s(t_h4s_patricias_ptree(TV_u_27a),V_M),s(TV_u_27b,V_v),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_f),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)))),V_f1))) = s(TV_u_27b,h4s_patricias_ptreeu_u_case(s(t_h4s_patricias_ptree(TV_u_27a),V_Mu_27),s(TV_u_27b,V_vu_27),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_fu_27),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)))),V_f1u_27))))).
