%   ORIGINAL: h4/Coder/wf__coder0
% 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> 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/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/Coder/wf__coder__def: !p e d. h4/Coder/wf__coder (h4/pair/_2C p (h4/pair/_2C e d)) <=> h4/Encode/wf__pred p /\ h4/Encode/wf__encoder p e /\ d = h4/Decode/enc2dec p e
% Assm: h4/Coder/domain__def: !p e d. h4/Coder/domain (h4/pair/_2C p (h4/pair/_2C e d)) = p
% Assm: h4/Coder/encoder__def: !p e d. h4/Coder/encoder (h4/pair/_2C p (h4/pair/_2C e d)) = e
% Assm: h4/Coder/decoder__def: !p e d. h4/Coder/decoder (h4/pair/_2C p (h4/pair/_2C e d)) = h4/Coder/decode p d
% Assm: h4/Coder/decode__encode: !x p e. h4/Encode/wf__encoder p e /\ p x ==> h4/Coder/decode p (h4/Decode/enc2dec p e) (e x) = x
% Goal: !c. h4/Coder/wf__coder c ==> (!x. h4/Coder/domain c x ==> h4/Coder/decoder c (h4/Coder/encoder c x) = x)
%   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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> 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_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_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_Coders_wfu_u_coderu_u_def]: !p e d. h4/Coder/wf__coder (h4/pair/_2C p (h4/pair/_2C e d)) <=> h4/Encode/wf__pred p /\ h4/Encode/wf__encoder p e /\ d = h4/Decode/enc2dec p e
% Assm [h4s_Coders_domainu_u_def]: !p e d. h4/Coder/domain (h4/pair/_2C p (h4/pair/_2C e d)) = p
% Assm [h4s_Coders_encoderu_u_def]: !p e d. h4/Coder/encoder (h4/pair/_2C p (h4/pair/_2C e d)) = e
% Assm [h4s_Coders_decoderu_u_def]: !p e d. h4/Coder/decoder (h4/pair/_2C p (h4/pair/_2C e d)) = h4/Coder/decode p d
% Assm [h4s_Coders_decodeu_u_encode]: !x p e. h4/Encode/wf__encoder p e /\ happ p x ==> happ (h4/Coder/decode p (h4/Decode/enc2dec p e)) (happ e x) = x
% Goal: !c. h4/Coder/wf__coder c ==> (!x. happ (h4/Coder/domain c) x ==> happ (h4/Coder/decoder c) (happ (h4/Coder/encoder c) x) = x)
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_Q17326,TV_Q17322]: ![V_f, V_g]: (![V_x]: s(TV_Q17322,happ(s(t_fun(TV_Q17326,TV_Q17322),V_f),s(TV_Q17326,V_x))) = s(TV_Q17322,happ(s(t_fun(TV_Q17326,TV_Q17322),V_g),s(TV_Q17326,V_x))) => s(t_fun(TV_Q17326,TV_Q17322),V_f) = s(t_fun(TV_Q17326,TV_Q17322),V_g))).
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_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_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_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_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_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_Coders_wfu_u_coderu_u_def, axiom, ![TV_u_27a]: ![V_p, V_e, V_d]: (p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d)))))))) <=> (p(s(t_bool,h4s_encodes_wfu_u_pred(s(t_fun(TV_u_27a,t_bool),V_p)))) & (p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)))) & s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e))))))).
fof(ah4s_Coders_domainu_u_def, axiom, ![TV_u_27a]: ![V_p, V_e, V_d]: s(t_fun(TV_u_27a,t_bool),h4s_coders_domain(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d))))))) = s(t_fun(TV_u_27a,t_bool),V_p)).
fof(ah4s_Coders_encoderu_u_def, axiom, ![TV_u_27a]: ![V_p, V_e, V_d]: s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),h4s_coders_encoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d))))))) = s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)).
fof(ah4s_Coders_decoderu_u_def, axiom, ![TV_u_27a]: ![V_p, V_e, V_d]: s(t_fun(t_h4s_lists_list(t_bool),TV_u_27a),h4s_coders_decoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d))))))) = s(t_fun(t_h4s_lists_list(t_bool),TV_u_27a),h4s_coders_decode(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d)))).
fof(ah4s_Coders_decodeu_u_encode, axiom, ![TV_u_27a]: ![V_x, V_p, V_e]: ((p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_x))))) => s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_bool),TV_u_27a),h4s_coders_decode(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e))))),s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x))).
fof(ch4s_Coders_wfu_u_coder0, conjecture, ![TV_u_27a]: ![V_c]: (p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c)))) => ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),h4s_coders_domain(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))),s(TV_u_27a,V_x)))) => s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_bool),TV_u_27a),h4s_coders_decoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))),s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),h4s_coders_encoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)))).
