%   ORIGINAL: h4/quotient__option/OPTION__REL__EQ
% 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/ETA__AX: !t. (\x. t x) = t
% 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/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/option/option__CLAUSES_c0: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/option/option__CLAUSES_c2: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/option/option__CLAUSES_c3: !x. ~(h4/option/SOME x = h4/option/NONE)
% Assm: h4/quotient__option/OPTION__REL__def_c0: !R. h4/option/OPTREL R h4/option/NONE h4/option/NONE <=> T
% Assm: h4/quotient__option/OPTION__REL__def_c1: !x R. h4/option/OPTREL R (h4/option/SOME x) h4/option/NONE <=> F
% Assm: h4/quotient__option/OPTION__REL__def_c2: !y R. h4/option/OPTREL R h4/option/NONE (h4/option/SOME y) <=> F
% Assm: h4/quotient__option/OPTION__REL__def_c3: !y x R. h4/option/OPTREL R (h4/option/SOME x) (h4/option/SOME y) <=> R x y
% Goal: h4/option/OPTREL $equals = $equals
%   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_ETAu_u_AX]: !t x. happ t x = happ t x
% 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_NOTu_u_CLAUSESu_c2]: ~F <=> 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_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_options_optionu_u_CLAUSESu_c0]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_options_optionu_u_CLAUSESu_c2]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_options_optionu_u_CLAUSESu_c3]: !x. ~(h4/option/SOME x = h4/option/NONE)
% Assm [h4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c0]: !R. happ (happ (h4/option/OPTREL R) h4/option/NONE) h4/option/NONE <=> T
% Assm [h4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c1]: !x R. happ (happ (h4/option/OPTREL R) (h4/option/SOME x)) h4/option/NONE <=> F
% Assm [h4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c2]: !y R. happ (happ (h4/option/OPTREL R) h4/option/NONE) (h4/option/SOME y) <=> F
% Assm [h4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c3]: !y x R. happ (happ (h4/option/OPTREL R) (h4/option/SOME x)) (h4/option/SOME y) <=> happ (happ R x) y
% Goal: h4/option/OPTREL $equals = $equals
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_Q205297,TV_Q205293]: ![V_f, V_g]: (![V_x]: s(TV_Q205293,happ(s(t_fun(TV_Q205297,TV_Q205293),V_f),s(TV_Q205297,V_x))) = s(TV_Q205293,happ(s(t_fun(TV_Q205297,TV_Q205293),V_g),s(TV_Q205297,V_x))) => s(t_fun(TV_Q205297,TV_Q205293),V_f) = s(t_fun(TV_Q205297,TV_Q205293),V_g))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> 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_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_options_optionu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_options_optionu_u_CLAUSESu_c2, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_options_optionu_u_CLAUSESu_c3, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))).
fof(ah4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c0, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_h4s_options_option(TV_u_27a),t_bool)),h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_bool,t)).
fof(ah4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c1, axiom, ![TV_u_27a]: ![V_x, V_R]: s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_h4s_options_option(TV_u_27a),t_bool)),h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_bool,f)).
fof(ah4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c2, axiom, ![TV_u_27a]: ![V_y, V_R]: s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_h4s_options_option(TV_u_27a),t_bool)),h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))))) = s(t_bool,f)).
fof(ah4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c3, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_h4s_options_option(TV_u_27a),t_bool)),h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))).
fof(ch4s_quotientu_u_options_OPTIONu_u_RELu_u_EQ, conjecture, ![TV_u_27a]: s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_h4s_options_option(TV_u_27a),t_bool)),h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals))) = s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_h4s_options_option(TV_u_27a),t_bool)),d_equals)).
