%   ORIGINAL: h4/ring/plus__zero__right
% 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/ring/is__ring__def: !r. h4/ring/is__ring r <=> (!n m. h4/ring/ring__RP r n m = h4/ring/ring__RP r m n) /\ (!n m p. h4/ring/ring__RP r n (h4/ring/ring__RP r m p) = h4/ring/ring__RP r (h4/ring/ring__RP r n m) p) /\ (!n m. h4/ring/ring__RM r n m = h4/ring/ring__RM r m n) /\ (!n m p. h4/ring/ring__RM r n (h4/ring/ring__RM r m p) = h4/ring/ring__RM r (h4/ring/ring__RM r n m) p) /\ (!n. h4/ring/ring__RP r (h4/ring/ring__R0 r) n = n) /\ (!n. h4/ring/ring__RM r (h4/ring/ring__R1 r) n = n) /\ (!n. h4/ring/ring__RP r n (h4/ring/ring__RN r n) = h4/ring/ring__R0 r) /\ (!n m p. h4/ring/ring__RM r (h4/ring/ring__RP r n m) p = h4/ring/ring__RP r (h4/ring/ring__RM r n p) (h4/ring/ring__RM r m p))
% Goal: !r. h4/ring/is__ring r ==> (!n. h4/ring/ring__RP r n (h4/ring/ring__R0 r) = 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_rings_isu_u_ringu_u_def]: !r. h4/ring/is__ring r <=> (!n m. h4/ring/ring__RP r n m = h4/ring/ring__RP r m n) /\ (!n m p. h4/ring/ring__RP r n (h4/ring/ring__RP r m p) = h4/ring/ring__RP r (h4/ring/ring__RP r n m) p) /\ (!n m. h4/ring/ring__RM r n m = h4/ring/ring__RM r m n) /\ (!n m p. h4/ring/ring__RM r n (h4/ring/ring__RM r m p) = h4/ring/ring__RM r (h4/ring/ring__RM r n m) p) /\ (!n. h4/ring/ring__RP r (h4/ring/ring__R0 r) n = n) /\ (!n. h4/ring/ring__RM r (h4/ring/ring__R1 r) n = n) /\ (!n. h4/ring/ring__RP r n (h4/ring/ring__RN r n) = h4/ring/ring__R0 r) /\ (!n m p. h4/ring/ring__RM r (h4/ring/ring__RP r n m) p = h4/ring/ring__RP r (h4/ring/ring__RM r n p) (h4/ring/ring__RM r m p))
% Goal: !r. h4/ring/is__ring r ==> (!n. h4/ring/ring__RP r n (h4/ring/ring__R0 r) = n)
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_Q296965,TV_Q296961]: ![V_f, V_g]: (![V_x]: s(TV_Q296961,happ(s(t_fun(TV_Q296965,TV_Q296961),V_f),s(TV_Q296965,V_x))) = s(TV_Q296961,happ(s(t_fun(TV_Q296965,TV_Q296961),V_g),s(TV_Q296965,V_x))) => s(t_fun(TV_Q296965,TV_Q296961),V_f) = s(t_fun(TV_Q296965,TV_Q296961),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_rings_isu_u_ringu_u_def, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) <=> (![V_n, V_m]: s(TV_u_27a,h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_n),s(TV_u_27a,V_m))) = s(TV_u_27a,h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_m),s(TV_u_27a,V_n))) & (![V_n, V_m, V_p]: s(TV_u_27a,h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_n),s(TV_u_27a,h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_m),s(TV_u_27a,V_p))))) = s(TV_u_27a,h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_n),s(TV_u_27a,V_m))),s(TV_u_27a,V_p))) & (![V_n, V_m]: s(TV_u_27a,h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_n),s(TV_u_27a,V_m))) = s(TV_u_27a,h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_m),s(TV_u_27a,V_n))) & (![V_n, V_m, V_p]: s(TV_u_27a,h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_n),s(TV_u_27a,h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_m),s(TV_u_27a,V_p))))) = s(TV_u_27a,h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_n),s(TV_u_27a,V_m))),s(TV_u_27a,V_p))) & (![V_n]: s(TV_u_27a,h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))) = s(TV_u_27a,V_n) & (![V_n]: s(TV_u_27a,h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_n))) = s(TV_u_27a,V_n) & (![V_n]: s(TV_u_27a,h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_n),s(TV_u_27a,h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_n))))) = s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))) & ![V_n, V_m, V_p]: s(TV_u_27a,h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_n),s(TV_u_27a,V_m))),s(TV_u_27a,V_p))) = s(TV_u_27a,h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_n),s(TV_u_27a,V_p))),s(TV_u_27a,h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_m),s(TV_u_27a,V_p)))))))))))))).
fof(ch4s_rings_plusu_u_zerou_u_right, conjecture, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_n]: s(TV_u_27a,h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r),s(TV_u_27a,V_n),s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))))) = s(TV_u_27a,V_n))).
