%   ORIGINAL: h4/hreal/HREAL__ADD__ASSOC
% 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/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/hrat/HRAT__ADD__ASSOC: !j i h. h4/hrat/hrat__add h (h4/hrat/hrat__add i j) = h4/hrat/hrat__add (h4/hrat/hrat__add h i) j
% Assm: h4/hreal/hreal__tybij_c1: !r. h4/hreal/isacut r <=> h4/hreal/cut (h4/hreal/hreal0 r) = r
% Assm: h4/hreal/hreal__add0: !Y X. h4/hreal/hreal__add X Y = h4/hreal/hreal0 (\w. ?x y. w = h4/hrat/hrat__add x y /\ h4/hreal/cut X x /\ h4/hreal/cut Y y)
% Assm: h4/hreal/HREAL__ADD__ISACUT: !Y X. h4/hreal/isacut (\w. ?x y. w = h4/hrat/hrat__add x y /\ h4/hreal/cut X x /\ h4/hreal/cut Y y)
% Goal: !Z Y X. h4/hreal/hreal__add X (h4/hreal/hreal__add Y Z) = h4/hreal/hreal__add (h4/hreal/hreal__add X Y) Z
%   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_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_hrats_HRATu_u_ADDu_u_ASSOC]: !j i h. h4/hrat/hrat__add h (h4/hrat/hrat__add i j) = h4/hrat/hrat__add (h4/hrat/hrat__add h i) j
% Assm [h4s_hreals_hrealu_u_tybiju_c1]: !r. h4/hreal/isacut r <=> h4/hreal/cut (h4/hreal/hreal0 r) = r
% Assm [h4s_hreals_hrealu_u_add0]: !_0. (!X Y w. happ (happ (happ _0 X) Y) w <=> (?x y. w = h4/hrat/hrat__add x y /\ happ (h4/hreal/cut X) x /\ happ (h4/hreal/cut Y) y)) ==> (!Y X. h4/hreal/hreal__add X Y = h4/hreal/hreal0 (happ (happ _0 X) Y))
% Assm [h4s_hreals_HREALu_u_ADDu_u_ISACUT]: !_0. (!X Y w. happ (happ (happ _0 X) Y) w <=> (?x y. w = h4/hrat/hrat__add x y /\ happ (h4/hreal/cut X) x /\ happ (h4/hreal/cut Y) y)) ==> (!Y X. h4/hreal/isacut (happ (happ _0 X) Y))
% Goal: !Z Y X. h4/hreal/hreal__add X (h4/hreal/hreal__add Y Z) = h4/hreal/hreal__add (h4/hreal/hreal__add X Y) Z
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_Q254756,TV_Q254752]: ![V_f, V_g]: (![V_x]: s(TV_Q254752,happ(s(t_fun(TV_Q254756,TV_Q254752),V_f),s(TV_Q254756,V_x))) = s(TV_Q254752,happ(s(t_fun(TV_Q254756,TV_Q254752),V_g),s(TV_Q254756,V_x))) => s(t_fun(TV_Q254756,TV_Q254752),V_f) = s(t_fun(TV_Q254756,TV_Q254752),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_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_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_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_hrats_HRATu_u_ADDu_u_ASSOC, axiom, ![V_j, V_i, V_h]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_i),s(t_h4s_hrats_hrat,V_j))))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,V_i))),s(t_h4s_hrats_hrat,V_j)))).
fof(ah4s_hreals_hrealu_u_tybiju_c1, axiom, ![V_r]: (p(s(t_bool,h4s_hreals_isacut(s(t_fun(t_h4s_hrats_hrat,t_bool),V_r)))) <=> s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),V_r))))) = s(t_fun(t_h4s_hrats_hrat,t_bool),V_r))).
fof(ah4s_hreals_hrealu_u_add0, axiom, ![V_uu_0]: (![V_X, V_Y, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_x, V_y]: (s(t_h4s_hrats_hrat,V_w) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))) & (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_y))))))) => ![V_Y, V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y))) = s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))))))).
fof(ah4s_hreals_HREALu_u_ADDu_u_ISACUT, axiom, ![V_uu_0]: (![V_X, V_Y, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_x, V_y]: (s(t_h4s_hrats_hrat,V_w) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))) & (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_y))))))) => ![V_Y, V_X]: p(s(t_bool,h4s_hreals_isacut(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y)))))))).
fof(ch4s_hreals_HREALu_u_ADDu_u_ASSOC, conjecture, ![V_Z, V_Y, V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,V_Z))))) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hreals_hreal,V_Z)))).
