%   ORIGINAL: h4/prim__rec/PRIM__REC__EQN_c1
% 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/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/prim__rec/SIMP__REC__THM_c1: !x m f. h4/prim__rec/SIMP__REC x f (h4/num/SUC m) = f (h4/prim__rec/SIMP__REC x f m)
% Assm: h4/prim__rec/PRIM__REC__FUN0: !x f. h4/prim__rec/PRIM__REC__FUN x f = h4/prim__rec/SIMP__REC (\n. x) (\fun n. f (fun (h4/prim__rec/PRE n)) n)
% Goal: !x n m f. h4/prim__rec/PRIM__REC__FUN x f (h4/num/SUC m) n = f (h4/prim__rec/PRIM__REC__FUN x f m (h4/prim__rec/PRE n)) n
%   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_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_primu_u_recs_SIMPu_u_RECu_u_THMu_c1]: !x m f. happ (h4/prim__rec/SIMP__REC x f) (h4/num/SUC m) = happ f (happ (h4/prim__rec/SIMP__REC x f) m)
% Assm [h4s_primu_u_recs_PRIMu_u_RECu_u_FUN0]: !_2. (!f fun n. happ (happ (happ _2 f) fun) n = happ (happ f (happ fun (h4/prim__rec/PRE n))) n) ==> (!_1. (!f fun. happ (happ _1 f) fun = happ (happ _2 f) fun) ==> (!_0. (!x n. happ (happ _0 x) n = x) ==> (!x f. h4/prim__rec/PRIM__REC__FUN x f = h4/prim__rec/SIMP__REC (happ _0 x) (happ _1 f))))
% Goal: !x n m f. happ (happ (h4/prim__rec/PRIM__REC__FUN x f) (h4/num/SUC m)) n = happ (happ f (happ (happ (h4/prim__rec/PRIM__REC__FUN x f) m) (h4/prim__rec/PRE n))) n
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_Q187995,TV_Q187991]: ![V_f, V_g]: (![V_x]: s(TV_Q187991,happ(s(t_fun(TV_Q187995,TV_Q187991),V_f),s(TV_Q187995,V_x))) = s(TV_Q187991,happ(s(t_fun(TV_Q187995,TV_Q187991),V_g),s(TV_Q187995,V_x))) => s(t_fun(TV_Q187995,TV_Q187991),V_f) = s(t_fun(TV_Q187995,TV_Q187991),V_g))).
fof(ah4s_bools_TRUTH, axiom, 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_primu_u_recs_SIMPu_u_RECu_u_THMu_c1, axiom, ![TV_u_27a]: ![V_x, V_m, V_f]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_primu_u_recs_simpu_u_rec(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_primu_u_recs_simpu_u_rec(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_primu_u_recs_PRIMu_u_RECu_u_FUN0, axiom, ![TV_u_27a]: ![V_uu_2]: (![V_f, V_fun, V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a))),V_uu_2),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun))),s(t_h4s_nums_num,V_n))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun),s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,V_n))))))),s(t_h4s_nums_num,V_n))) => ![V_uu_1]: (![V_f, V_fun]: s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a))),V_uu_1),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun))) = s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a))),V_uu_2),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun))) => ![V_uu_0]: (![V_x, V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_uu_0),s(TV_u_27a,V_x))),s(t_h4s_nums_num,V_n))) = s(TV_u_27a,V_x) => ![V_x, V_f]: s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),h4s_primu_u_recs_primu_u_recu_u_fun(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))) = s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),h4s_primu_u_recs_simpu_u_rec(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_uu_0),s(TV_u_27a,V_x))),s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a))),V_uu_1),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))))))))).
fof(ch4s_primu_u_recs_PRIMu_u_RECu_u_EQNu_c1, conjecture, ![TV_u_27a]: ![V_x, V_n, V_m, V_f]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),h4s_primu_u_recs_primu_u_recu_u_fun(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))),s(t_h4s_nums_num,V_n))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),h4s_primu_u_recs_primu_u_recu_u_fun(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,V_n))))))),s(t_h4s_nums_num,V_n)))).
