%   ORIGINAL: h4/Decode/dec2enc__decode__bool
% 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/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/Encode/wf__encode__bool: !p. h4/Encode/wf__encoder p h4/Encode/encode__bool
% Assm: h4/Decode/dec2enc__enc2dec: !x p e. h4/Encode/wf__encoder p e /\ p x ==> h4/Decode/dec2enc (h4/Decode/enc2dec p e) x = e x
% Assm: h4/Decode/decode__bool__def: !p. h4/Decode/decode__bool p = h4/Decode/enc2dec p h4/Encode/encode__bool
% Goal: !x p. p x ==> h4/Decode/dec2enc (h4/Decode/decode__bool p) x = h4/Encode/encode__bool 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_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_Encodes_wfu_u_encodeu_u_bool]: !p. h4/Encode/wf__encoder p h4/Encode/encode__bool
% Assm [h4s_Decodes_dec2encu_u_enc2dec]: !x p e. h4/Encode/wf__encoder p e /\ happ p x ==> h4/Decode/dec2enc (h4/Decode/enc2dec p e) x = happ e x
% Assm [h4s_Decodes_decodeu_u_boolu_u_def]: !p. h4/Decode/decode__bool p = h4/Decode/enc2dec p h4/Encode/encode__bool
% Goal: !x p. happ p x ==> h4/Decode/dec2enc (h4/Decode/decode__bool p) x = happ h4/Encode/encode__bool 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_Q135047,TV_Q135043]: ![V_f, V_g]: (![V_x]: s(TV_Q135043,happ(s(t_fun(TV_Q135047,TV_Q135043),V_f),s(TV_Q135047,V_x))) = s(TV_Q135043,happ(s(t_fun(TV_Q135047,TV_Q135043),V_g),s(TV_Q135047,V_x))) => s(t_fun(TV_Q135047,TV_Q135043),V_f) = s(t_fun(TV_Q135047,TV_Q135043),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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_Encodes_wfu_u_encodeu_u_bool, axiom, ![V_p]: p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(t_bool,t_bool),V_p),s(t_fun(t_bool,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_bool))))).
fof(ah4s_Decodes_dec2encu_u_enc2dec, 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(t_h4s_lists_list(t_bool),h4s_decodes_dec2enc(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(TV_u_27a,V_x))) = 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))))).
fof(ah4s_Decodes_decodeu_u_boolu_u_def, axiom, ![V_p]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_bool,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_bool(s(t_fun(t_bool,t_bool),V_p))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_bool,t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(t_bool,t_bool),V_p),s(t_fun(t_bool,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_bool)))).
fof(ch4s_Decodes_dec2encu_u_decodeu_u_bool, conjecture, ![V_x, V_p]: (p(s(t_bool,happ(s(t_fun(t_bool,t_bool),V_p),s(t_bool,V_x)))) => s(t_h4s_lists_list(t_bool),h4s_decodes_dec2enc(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_bool,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_bool(s(t_fun(t_bool,t_bool),V_p))),s(t_bool,V_x))) = s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_bool,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_bool),s(t_bool,V_x))))).
