%   ORIGINAL: h4/ringNorm/spolynom__normalize__def_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/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/canonical/spolynom__normalize__def_c0: !sr i. h4/canonical/spolynom__normalize sr (h4/canonical/SPvar i) = h4/canonical/Cons__varlist (h4/list/CONS i h4/list/NIL) h4/canonical/Nil__monom
% Assm: h4/canonical/spolynom__normalize__def_c1: !sr c. h4/canonical/spolynom__normalize sr (h4/canonical/SPconst c) = h4/canonical/Cons__monom c h4/list/NIL h4/canonical/Nil__monom
% Assm: h4/canonical/spolynom__normalize__def_c2: !sr r l. h4/canonical/spolynom__normalize sr (h4/canonical/SPplus l r) = h4/canonical/canonical__sum__merge sr (h4/canonical/spolynom__normalize sr l) (h4/canonical/spolynom__normalize sr r)
% Assm: h4/canonical/spolynom__normalize__def_c3: !sr r l. h4/canonical/spolynom__normalize sr (h4/canonical/SPmult l r) = h4/canonical/canonical__sum__prod sr (h4/canonical/spolynom__normalize sr l) (h4/canonical/spolynom__normalize sr 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__prod__def: !r. h4/ringNorm/r__canonical__sum__prod r = h4/canonical/canonical__sum__prod (h4/ring/semi__ring__of r)
% Assm: h4/ringNorm/r__canonical__sum__merge__def: !r. h4/ringNorm/r__canonical__sum__merge r = h4/canonical/canonical__sum__merge (h4/ring/semi__ring__of r)
% Goal: !r c. h4/ringNorm/r__spolynom__normalize r (h4/canonical/SPconst c) = h4/canonical/Cons__monom c h4/list/NIL h4/canonical/Nil__monom
%   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_normalizeu_u_defu_c0]: !sr i. happ (h4/canonical/spolynom__normalize sr) (h4/canonical/SPvar i) = h4/canonical/Cons__varlist (h4/list/CONS i h4/list/NIL) h4/canonical/Nil__monom
% Assm [h4s_canonicals_spolynomu_u_normalizeu_u_defu_c1]: !sr c. happ (h4/canonical/spolynom__normalize sr) (h4/canonical/SPconst c) = h4/canonical/Cons__monom c h4/list/NIL h4/canonical/Nil__monom
% Assm [h4s_canonicals_spolynomu_u_normalizeu_u_defu_c2]: !sr r l. happ (h4/canonical/spolynom__normalize sr) (h4/canonical/SPplus l r) = happ (happ (h4/canonical/canonical__sum__merge sr) (happ (h4/canonical/spolynom__normalize sr) l)) (happ (h4/canonical/spolynom__normalize sr) r)
% Assm [h4s_canonicals_spolynomu_u_normalizeu_u_defu_c3]: !sr r l. happ (h4/canonical/spolynom__normalize sr) (h4/canonical/SPmult l r) = happ (happ (h4/canonical/canonical__sum__prod sr) (happ (h4/canonical/spolynom__normalize sr) l)) (happ (h4/canonical/spolynom__normalize sr) 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_produ_u_def]: !r. h4/ringNorm/r__canonical__sum__prod r = h4/canonical/canonical__sum__prod (h4/ring/semi__ring__of r)
% Assm [h4s_ringNorms_ru_u_canonicalu_u_sumu_u_mergeu_u_def]: !r. h4/ringNorm/r__canonical__sum__merge r = h4/canonical/canonical__sum__merge (h4/ring/semi__ring__of r)
% Goal: !r c. happ (h4/ringNorm/r__spolynom__normalize r) (h4/canonical/SPconst c) = h4/canonical/Cons__monom c h4/list/NIL h4/canonical/Nil__monom
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_Q289339,TV_Q289335]: ![V_f, V_g]: (![V_x]: s(TV_Q289335,happ(s(t_fun(TV_Q289339,TV_Q289335),V_f),s(TV_Q289339,V_x))) = s(TV_Q289335,happ(s(t_fun(TV_Q289339,TV_Q289335),V_g),s(TV_Q289339,V_x))) => s(t_fun(TV_Q289339,TV_Q289335),V_f) = s(t_fun(TV_Q289339,TV_Q289335),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_normalizeu_u_defu_c0, axiom, ![TV_u_27a]: ![V_sr, V_i]: 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),h4s_canonicals_spvar(s(t_h4s_quotes_index,V_i))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_i),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom)))).
fof(ah4s_canonicals_spolynomu_u_normalizeu_u_defu_c1, axiom, ![TV_u_27a]: ![V_sr, V_c]: 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),h4s_canonicals_spconst(s(TV_u_27a,V_c))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom)))).
fof(ah4s_canonicals_spolynomu_u_normalizeu_u_defu_c2, axiom, ![TV_u_27a]: ![V_sr, V_r, V_l]: 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),h4s_canonicals_spplus(s(t_h4s_canonicals_spolynom(TV_u_27a),V_l),s(t_h4s_canonicals_spolynom(TV_u_27a),V_r))))) = 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)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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_merge(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_l))))),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_r)))))).
fof(ah4s_canonicals_spolynomu_u_normalizeu_u_defu_c3, axiom, ![TV_u_27a]: ![V_sr, V_r, V_l]: 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),h4s_canonicals_spmult(s(t_h4s_canonicals_spolynom(TV_u_27a),V_l),s(t_h4s_canonicals_spolynom(TV_u_27a),V_r))))) = 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)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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_prod(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_l))))),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_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_produ_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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_prod(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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_prod(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_mergeu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),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_merge(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_normalizeu_u_defu_c1, conjecture, ![TV_u_27a]: ![V_r, V_c]: 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),h4s_canonicals_spconst(s(TV_u_27a,V_c))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom)))).
