%   ORIGINAL: h4/fixedPoint/fnsum__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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/pred__set/UNION__ASSOC: !u t s. h4/pred__set/UNION s (h4/pred__set/UNION t u) = h4/pred__set/UNION (h4/pred__set/UNION s t) u
% Assm: h4/fixedPoint/fnsum__def: !f2 f1 X. h4/fixedPoint/fnsum f1 f2 X = h4/pred__set/UNION (f1 X) (f2 X)
% Goal: !h g f. h4/fixedPoint/fnsum f (h4/fixedPoint/fnsum g h) = h4/fixedPoint/fnsum (h4/fixedPoint/fnsum f g) h
%   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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_predu_u_sets_UNIONu_u_ASSOC]: !u t s. h4/pred__set/UNION s (h4/pred__set/UNION t u) = h4/pred__set/UNION (h4/pred__set/UNION s t) u
% Assm [h4s_fixedPoints_fnsumu_u_def]: !f2 f1 X. happ (h4/fixedPoint/fnsum f1 f2) X = h4/pred__set/UNION (happ f1 X) (happ f2 X)
% Goal: !h g f. h4/fixedPoint/fnsum f (h4/fixedPoint/fnsum g h) = h4/fixedPoint/fnsum (h4/fixedPoint/fnsum f g) h
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q351298,TV_Q351294]: ![V_f, V_g]: (![V_x]: s(TV_Q351294,happ(s(t_fun(TV_Q351298,TV_Q351294),V_f),s(TV_Q351298,V_x))) = s(TV_Q351294,happ(s(t_fun(TV_Q351298,TV_Q351294),V_g),s(TV_Q351298,V_x))) => s(t_fun(TV_Q351298,TV_Q351294),V_f) = s(t_fun(TV_Q351298,TV_Q351294),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_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_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_predu_u_sets_UNIONu_u_ASSOC, axiom, ![TV_u_27a]: ![V_u, V_t, V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_u))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))),s(t_fun(TV_u_27a,t_bool),V_u)))).
fof(ah4s_fixedPoints_fnsumu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_f2, V_f1, V_X]: s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),h4s_fixedpoints_fnsum(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),V_f1),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),V_f2))),s(TV_u_27b,V_X))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),V_f1),s(TV_u_27b,V_X))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),V_f2),s(TV_u_27b,V_X)))))).
fof(ch4s_fixedPoints_fnsumu_u_ASSOC, conjecture, ![TV_u_27a,TV_u_27b]: ![V_h, V_g, V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_fixedpoints_fnsum(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_fixedpoints_fnsum(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_g),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_h))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_fixedpoints_fnsum(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_fixedpoints_fnsum(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_g))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_h)))).
