%   ORIGINAL: h4/finite__map/fmap__rel__FEMPTY
% 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !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/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/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/finite__map/FDOM__FEMPTY: h4/finite__map/FDOM h4/finite__map/FEMPTY = h4/pred__set/EMPTY
% Assm: h4/finite__map/fmap__rel__def: !f2 f1 R. h4/finite__map/fmap__rel R f1 f2 <=> h4/finite__map/FDOM f2 = h4/finite__map/FDOM f1 /\ (!x. h4/bool/IN x (h4/finite__map/FDOM f1) ==> R (h4/finite__map/FAPPLY f1 x) (h4/finite__map/FAPPLY f2 x))
% Goal: !R. h4/finite__map/fmap__rel R h4/finite__map/FEMPTY h4/finite__map/FEMPTY
%   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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !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_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_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_finiteu_u_maps_FDOMu_u_FEMPTY]: h4/finite__map/FDOM h4/finite__map/FEMPTY = h4/pred__set/EMPTY
% Assm [h4s_finiteu_u_maps_fmapu_u_relu_u_def]: !f2 f1 R. h4/finite__map/fmap__rel R f1 f2 <=> h4/finite__map/FDOM f2 = h4/finite__map/FDOM f1 /\ (!x. h4/bool/IN x (h4/finite__map/FDOM f1) ==> happ (happ R (h4/finite__map/FAPPLY f1 x)) (h4/finite__map/FAPPLY f2 x))
% Goal: !R. h4/finite__map/fmap__rel R h4/finite__map/FEMPTY h4/finite__map/FEMPTY
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_Q76942,TV_Q76938]: ![V_f, V_g]: (![V_x]: s(TV_Q76938,happ(s(t_fun(TV_Q76942,TV_Q76938),V_f),s(TV_Q76942,V_x))) = s(TV_Q76938,happ(s(t_fun(TV_Q76942,TV_Q76938),V_g),s(TV_Q76942,V_x))) => s(t_fun(TV_Q76942,TV_Q76938),V_f) = s(t_fun(TV_Q76942,TV_Q76938),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_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_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_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_predu_u_sets_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_finiteu_u_maps_FDOMu_u_FEMPTY, axiom, ![TV_u_27b,TV_u_27a]: s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fempty))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_finiteu_u_maps_fmapu_u_relu_u_def, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_f2, V_f1, V_R]: (p(s(t_bool,h4s_finiteu_u_maps_fmapu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),V_f1),s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),V_f2)))) <=> (s(t_fun(TV_u_27c,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),V_f2))) = s(t_fun(TV_u_27c,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),V_f1))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27c,V_x),s(t_fun(TV_u_27c,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),V_f1)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),V_f1),s(TV_u_27c,V_x))))),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),V_f2),s(TV_u_27c,V_x)))))))))).
fof(ch4s_finiteu_u_maps_fmapu_u_relu_u_FEMPTY, conjecture, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_R]: p(s(t_bool,h4s_finiteu_u_maps_fmapu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27a),h4s_finiteu_u_maps_fempty),s(t_h4s_finiteu_u_maps_fmap(TV_u_27c,TV_u_27b),h4s_finiteu_u_maps_fempty))))).
