%   ORIGINAL: h4/string/IMPLODE__EQ__THM_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/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> 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/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/list/list__nchotomy: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm: h4/list/CONS__11: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm: h4/list/NOT__NIL__CONS: !a1 a0. ~(h4/list/NIL = h4/list/CONS a0 a1)
% Assm: h4/string/IMPLODE__def_c0: h4/string/IMPLODE h4/list/NIL = h4/list/NIL
% Assm: h4/string/IMPLODE__def_c1: !cs c. h4/string/IMPLODE (h4/list/CONS c cs) = h4/list/CONS c (h4/string/IMPLODE cs)
% Assm: h4/string/IMPLODE__EXPLODE: !s. h4/string/IMPLODE (h4/string/EXPLODE s) = s
% Assm: h4/string/EXPLODE__IMPLODE: !cs. h4/string/EXPLODE (h4/string/IMPLODE cs) = cs
% Goal: !s l c. h4/string/IMPLODE l = h4/list/CONS c s <=> l = h4/list/CONS c (h4/string/EXPLODE s)
%   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_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> 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_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_lists_listu_u_nchotomy]: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm [h4s_lists_CONSu_u_11]: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm [h4s_lists_NOTu_u_NILu_u_CONS]: !a1 a0. ~(h4/list/NIL = h4/list/CONS a0 a1)
% Assm [h4s_strings_IMPLODEu_u_defu_c0]: h4/string/IMPLODE h4/list/NIL = h4/list/NIL
% Assm [h4s_strings_IMPLODEu_u_defu_c1]: !cs c. h4/string/IMPLODE (h4/list/CONS c cs) = h4/list/CONS c (h4/string/IMPLODE cs)
% Assm [h4s_strings_IMPLODEu_u_EXPLODE]: !s. h4/string/IMPLODE (h4/string/EXPLODE s) = s
% Assm [h4s_strings_EXPLODEu_u_IMPLODE]: !cs. h4/string/EXPLODE (h4/string/IMPLODE cs) = cs
% Goal: !s l c. h4/string/IMPLODE l = h4/list/CONS c s <=> l = h4/list/CONS c (h4/string/EXPLODE s)
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_Q212291,TV_Q212287]: ![V_f, V_g]: (![V_x]: s(TV_Q212287,happ(s(t_fun(TV_Q212291,TV_Q212287),V_f),s(TV_Q212291,V_x))) = s(TV_Q212287,happ(s(t_fun(TV_Q212291,TV_Q212287),V_g),s(TV_Q212291,V_x))) => s(t_fun(TV_Q212291,TV_Q212287),V_f) = s(t_fun(TV_Q212291,TV_Q212287),V_g))).
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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,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_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_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_lists_listu_u_nchotomy, axiom, ![TV_u_27a]: ![V_l]: (s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) | ?[V_h, V_t]: s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t))))).
fof(ah4s_lists_CONSu_u_11, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (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_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0u_27),s(t_h4s_lists_list(TV_u_27a),V_a1u_27))) <=> (s(TV_u_27a,V_a0) = s(TV_u_27a,V_a0u_27) & s(t_h4s_lists_list(TV_u_27a),V_a1) = s(t_h4s_lists_list(TV_u_27a),V_a1u_27)))).
fof(ah4s_lists_NOTu_u_NILu_u_CONS, axiom, ![TV_u_27a]: ![V_a1, V_a0]: ~ (s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = 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))))).
fof(ah4s_strings_IMPLODEu_u_defu_c0, axiom, s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_implode(s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_nil))) = s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_nil)).
fof(ah4s_strings_IMPLODEu_u_defu_c1, axiom, ![V_cs, V_c]: s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_implode(s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_cons(s(t_h4s_strings_char,V_c),s(t_h4s_lists_list(t_h4s_strings_char),V_cs))))) = s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_cons(s(t_h4s_strings_char,V_c),s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_implode(s(t_h4s_lists_list(t_h4s_strings_char),V_cs)))))).
fof(ah4s_strings_IMPLODEu_u_EXPLODE, axiom, ![V_s]: s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_implode(s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_explode(s(t_h4s_lists_list(t_h4s_strings_char),V_s))))) = s(t_h4s_lists_list(t_h4s_strings_char),V_s)).
fof(ah4s_strings_EXPLODEu_u_IMPLODE, axiom, ![V_cs]: s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_explode(s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_implode(s(t_h4s_lists_list(t_h4s_strings_char),V_cs))))) = s(t_h4s_lists_list(t_h4s_strings_char),V_cs)).
fof(ch4s_strings_IMPLODEu_u_EQu_u_THMu_c1, conjecture, ![V_s, V_l, V_c]: (s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_implode(s(t_h4s_lists_list(t_h4s_strings_char),V_l))) = s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_cons(s(t_h4s_strings_char,V_c),s(t_h4s_lists_list(t_h4s_strings_char),V_s))) <=> s(t_h4s_lists_list(t_h4s_strings_char),V_l) = s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_cons(s(t_h4s_strings_char,V_c),s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_explode(s(t_h4s_lists_list(t_h4s_strings_char),V_s))))))).
