%   ORIGINAL: h4/string/EXPLODE__DEST__STRING
% 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/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f 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/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/string/EXPLODE__def_c0: h4/string/EXPLODE h4/list/NIL = h4/list/NIL
% Assm: h4/string/EXPLODE__def_c1: !s c. h4/string/EXPLODE (h4/list/CONS c s) = h4/list/CONS c (h4/string/EXPLODE s)
% Assm: h4/string/EXPLODE__11: !s2 s1. h4/string/EXPLODE s1 = h4/string/EXPLODE s2 <=> s1 = s2
% Assm: h4/string/DEST__STRING__def_c0: h4/string/DEST__STRING h4/list/NIL = h4/option/NONE
% Assm: h4/string/DEST__STRING__def_c1: !rst c. h4/string/DEST__STRING (h4/list/CONS c rst) = h4/option/SOME (h4/pair/_2C c rst)
% Goal: !s. h4/string/EXPLODE s = h4/option/option__CASE (h4/string/DEST__STRING s) h4/list/NIL (\v. h4/pair/pair__CASE v (\c t. h4/list/CONS c (h4/string/EXPLODE t)))
%   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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_options_optionu_u_caseu_u_defu_c1]: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = happ f 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_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_strings_EXPLODEu_u_defu_c0]: h4/string/EXPLODE h4/list/NIL = h4/list/NIL
% Assm [h4s_strings_EXPLODEu_u_defu_c1]: !s c. h4/string/EXPLODE (h4/list/CONS c s) = h4/list/CONS c (h4/string/EXPLODE s)
% Assm [h4s_strings_EXPLODEu_u_11]: !s2 s1. h4/string/EXPLODE s1 = h4/string/EXPLODE s2 <=> s1 = s2
% Assm [h4s_strings_DESTu_u_STRINGu_u_defu_c0]: h4/string/DEST__STRING h4/list/NIL = h4/option/NONE
% Assm [h4s_strings_DESTu_u_STRINGu_u_defu_c1]: !rst c. h4/string/DEST__STRING (h4/list/CONS c rst) = h4/option/SOME (h4/pair/_2C c rst)
% Goal: !_2. (!c t. happ (happ _2 c) t = h4/list/CONS c (h4/string/EXPLODE t)) ==> (!_1. (!c. happ _1 c = happ _2 c) ==> (!_0. (!v. happ _0 v = h4/pair/pair__CASE v _1) ==> (!s. h4/string/EXPLODE s = h4/option/option__CASE (h4/string/DEST__STRING s) h4/list/NIL _0)))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q212316,TV_Q212312]: ![V_f, V_g]: (![V_x]: s(TV_Q212312,happ(s(t_fun(TV_Q212316,TV_Q212312),V_f),s(TV_Q212316,V_x))) = s(TV_Q212312,happ(s(t_fun(TV_Q212316,TV_Q212312),V_g),s(TV_Q212316,V_x))) => s(t_fun(TV_Q212316,TV_Q212312),V_f) = s(t_fun(TV_Q212316,TV_Q212312),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) & 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,t0)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) <=> p(s(t_bool,V_t)))).
fof(ah4s_options_optionu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_options_optionu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),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_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_strings_EXPLODEu_u_defu_c0, axiom, s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_explode(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_EXPLODEu_u_defu_c1, axiom, ![V_s, V_c]: s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_explode(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),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)))))).
fof(ah4s_strings_EXPLODEu_u_11, axiom, ![V_s2, V_s1]: (s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_explode(s(t_h4s_lists_list(t_h4s_strings_char),V_s1))) = s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_explode(s(t_h4s_lists_list(t_h4s_strings_char),V_s2))) <=> s(t_h4s_lists_list(t_h4s_strings_char),V_s1) = s(t_h4s_lists_list(t_h4s_strings_char),V_s2))).
fof(ah4s_strings_DESTu_u_STRINGu_u_defu_c0, axiom, s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char))),h4s_strings_destu_u_string(s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_nil))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char))),h4s_options_none)).
fof(ah4s_strings_DESTu_u_STRINGu_u_defu_c1, axiom, ![V_rst, V_c]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char))),h4s_strings_destu_u_string(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_rst))))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char))),h4s_options_some(s(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char)),h4s_pairs_u_2c(s(t_h4s_strings_char,V_c),s(t_h4s_lists_list(t_h4s_strings_char),V_rst)))))).
fof(ch4s_strings_EXPLODEu_u_DESTu_u_STRING, conjecture, ![V_uu_2]: (![V_c, V_t]: s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),V_uu_2),s(t_h4s_strings_char,V_c))),s(t_h4s_lists_list(t_h4s_strings_char),V_t))) = 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_t))))) => ![V_uu_1]: (![V_c]: s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),V_uu_1),s(t_h4s_strings_char,V_c))) = s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),V_uu_2),s(t_h4s_strings_char,V_c))) => ![V_uu_0]: (![V_v]: s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_strings_char)),V_uu_0),s(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char)),V_v))) = s(t_h4s_lists_list(t_h4s_strings_char),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char)),V_v),s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),V_uu_1))) => ![V_s]: 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),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char))),h4s_strings_destu_u_string(s(t_h4s_lists_list(t_h4s_strings_char),V_s))),s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_nil),s(t_fun(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_strings_char)),V_uu_0))))))).
