%   ORIGINAL: h4/patricia__casts/word__ptree__11
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/patricia__casts/word__ptree__repfns_c1: !r. (\a0_27. !_27word__ptree_27. (!a0_270. (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 a10) (\n. h4/ind__type/BOTTOM)) a0 a1) ==> _27word__ptree_27 a0_270) ==> _27word__ptree_27 a0_27) r <=> h4/patricia__casts/_20_40ind__typepatricia__casts2 (h4/patricia__casts/_20_40ind__typepatricia__casts1 r) = r
% Assm: h4/patricia__casts/hidden____20__40ind____typepatricia____casts0____def: h4/patricia__casts/_20_40ind__typepatricia__casts0 = (\a0 a1. h4/patricia__casts/_20_40ind__typepatricia__casts1 ((\a00 a10. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 a10) (\n. h4/ind__type/BOTTOM)) a0 a1))
% Assm: h4/patricia__casts/Word__ptree0: h4/patricia__casts/Word__ptree = h4/patricia__casts/_20_40ind__typepatricia__casts0
% Goal: !a1_27 a1 a0_27 a0. h4/patricia__casts/Word__ptree a0 a1 = h4/patricia__casts/Word__ptree a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm [h4s_patriciau_u_castss_wordu_u_ptreeu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27word__ptree_27. (!a0_270. (?a0 a1. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 a1) _0) ==> happ _27word__ptree_27 a0_270) ==> happ _27word__ptree_27 r) <=> h4/patricia__casts/_20_40ind__typepatricia__casts2 (h4/patricia__casts/_20_40ind__typepatricia__casts1 r) = r)
% Assm [h4s_patriciau_u_castss_hiddenu_u_u_u_20u_u_40indu_u_u_u_typepatriciau_u_u_u_casts0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/patricia__casts/_20_40ind__typepatricia__casts0 x) x = h4/patricia__casts/_20_40ind__typepatricia__casts1 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C x x) _0))
% Assm [h4s_patriciau_u_castss_Wordu_u_ptree0]: h4/patricia__casts/Word__ptree = h4/patricia__casts/_20_40ind__typepatricia__casts0
% Goal: !a1_27 a1 a0_27 a0. happ (happ h4/patricia__casts/Word__ptree a0) a1 = happ (happ h4/patricia__casts/Word__ptree a0_27) a1_27 <=> a0 = a0_27 /\ a1 = a1_27
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_Q204714,TV_Q204710]: ![V_f, V_g]: (![V_x]: s(TV_Q204710,happ(s(t_fun(TV_Q204714,TV_Q204710),V_f),s(TV_Q204714,V_x))) = s(TV_Q204710,happ(s(t_fun(TV_Q204714,TV_Q204710),V_g),s(TV_Q204714,V_x))) => s(t_fun(TV_Q204714,TV_Q204710),V_f) = s(t_fun(TV_Q204714,TV_Q204710),V_g))).
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_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
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_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(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_27a,V_x)).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(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_27b,V_y)).
fof(ah4s_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_patriciau_u_castss_wordu_u_ptreeu_u_repfnsu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27wordu_u_ptreeu_27]: (![V_a0u_270]: (?[V_a0, V_a1]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0),s(t_h4s_patricias_ptree(TV_u_27b),V_a1))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b)))),V_uu_0))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),t_bool),V_uu_27wordu_u_ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),t_bool),V_uu_27wordu_u_ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),h4s_patriciau_u_castss_u_20u_40indu_u_typepatriciau_u_casts2(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),h4s_patriciau_u_castss_u_20u_40indu_u_typepatriciau_u_casts1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),V_r)))).
fof(ah4s_patriciau_u_castss_hiddenu_u_u_u_20u_u_40indu_u_u_u_typepatriciau_u_u_u_casts0u_u_u_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),h4s_indu_u_types_bottom) => ![V_x, V_x0]: s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_u_20u_40indu_u_typepatriciau_u_casts0),s(t_fun(TV_u_27a,t_h4s_ones_one),V_x))),s(t_h4s_patricias_ptree(TV_u_27b),V_x0))) = s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),h4s_patriciau_u_castss_u_20u_40indu_u_typepatriciau_u_casts1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_ones_one),V_x),s(t_h4s_patricias_ptree(TV_u_27b),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b)))),V_uu_0))))))).
fof(ah4s_patriciau_u_castss_Wordu_u_ptree0, axiom, ![TV_u_27a,TV_u_27b]: s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_wordu_u_ptree) = s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_u_20u_40indu_u_typepatriciau_u_casts0)).
fof(ch4s_patriciau_u_castss_wordu_u_ptreeu_u_11, conjecture, ![TV_u_27a,TV_u_27b]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_wordu_u_ptree),s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0))),s(t_h4s_patricias_ptree(TV_u_27b),V_a1))) = s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_wordu_u_ptree),s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0u_27))),s(t_h4s_patricias_ptree(TV_u_27b),V_a1u_27))) <=> (s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0) = s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0u_27) & s(t_h4s_patricias_ptree(TV_u_27b),V_a1) = s(t_h4s_patricias_ptree(TV_u_27b),V_a1u_27)))).
