%   ORIGINAL: h4/ringNorm/spolynom__simplify__def
% 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/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/canonical/spolynom__simplify__def: !x sr. h4/canonical/spolynom__simplify sr x = h4/canonical/canonical__sum__simplify sr (h4/canonical/spolynom__normalize sr x)
% Assm: h4/ringNorm/r__spolynom__simplify__def: !r. h4/ringNorm/r__spolynom__simplify r = h4/canonical/spolynom__simplify (h4/ring/semi__ring__of r)
% Assm: h4/ringNorm/r__spolynom__normalize__def: !r. h4/ringNorm/r__spolynom__normalize r = h4/canonical/spolynom__normalize (h4/ring/semi__ring__of r)
% Assm: h4/ringNorm/r__canonical__sum__simplify__def: !r. h4/ringNorm/r__canonical__sum__simplify r = h4/canonical/canonical__sum__simplify (h4/ring/semi__ring__of r)
% Goal: !x r. h4/ringNorm/r__spolynom__simplify r x = h4/ringNorm/r__canonical__sum__simplify r (h4/ringNorm/r__spolynom__normalize r x)
%   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_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_canonicals_spolynomu_u_simplifyu_u_def]: !x sr. happ (h4/canonical/spolynom__simplify sr) x = happ (h4/canonical/canonical__sum__simplify sr) (happ (h4/canonical/spolynom__normalize sr) x)
% Assm [h4s_ringNorms_ru_u_spolynomu_u_simplifyu_u_def]: !r. h4/ringNorm/r__spolynom__simplify r = h4/canonical/spolynom__simplify (h4/ring/semi__ring__of r)
% Assm [h4s_ringNorms_ru_u_spolynomu_u_normalizeu_u_def]: !r. h4/ringNorm/r__spolynom__normalize r = h4/canonical/spolynom__normalize (h4/ring/semi__ring__of r)
% Assm [h4s_ringNorms_ru_u_canonicalu_u_sumu_u_simplifyu_u_def]: !r. h4/ringNorm/r__canonical__sum__simplify r = h4/canonical/canonical__sum__simplify (h4/ring/semi__ring__of r)
% Goal: !x r. happ (h4/ringNorm/r__spolynom__simplify r) x = happ (h4/ringNorm/r__canonical__sum__simplify r) (happ (h4/ringNorm/r__spolynom__normalize r) x)
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_Q289414,TV_Q289410]: ![V_f, V_g]: (![V_x]: s(TV_Q289410,happ(s(t_fun(TV_Q289414,TV_Q289410),V_f),s(TV_Q289414,V_x))) = s(TV_Q289410,happ(s(t_fun(TV_Q289414,TV_Q289410),V_g),s(TV_Q289414,V_x))) => s(t_fun(TV_Q289414,TV_Q289410),V_f) = s(t_fun(TV_Q289414,TV_Q289410),V_g))).
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_canonicals_spolynomu_u_simplifyu_u_def, axiom, ![TV_u_27a]: ![V_x, V_sr]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_canonicals_spolynomu_u_simplify(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_x))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_canonicals_canonicalu_u_sumu_u_simplify(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_canonicals_spolynomu_u_normalize(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_x)))))).
fof(ah4s_ringNorms_ru_u_spolynomu_u_simplifyu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_ru_u_spolynomu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_canonicals_spolynomu_u_simplify(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_rings_semiu_u_ringu_u_of(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_ringNorms_ru_u_spolynomu_u_normalizeu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_ru_u_spolynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_canonicals_spolynomu_u_normalize(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_rings_semiu_u_ringu_u_of(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_ringNorms_ru_u_canonicalu_u_sumu_u_simplifyu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_canonicals_canonicalu_u_sumu_u_simplify(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_rings_semiu_u_ringu_u_of(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ch4s_ringNorms_spolynomu_u_simplifyu_u_def, conjecture, ![TV_u_27a]: ![V_x, V_r]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_ru_u_spolynomu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_x))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_ru_u_spolynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_x)))))).
