%   ORIGINAL: h4/numRing/num__ring__thms_c24
% 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/semi__ring/semi__ring__accessors_c3: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring0 a a0 f f0) = f0
% Assm: h4/canonical/canonical__sum__scalar__def_c0: !t sr l c0 c. h4/canonical/canonical__sum__scalar sr c0 (h4/canonical/Cons__monom c l t) = h4/canonical/Cons__monom (h4/semi__ring/semi__ring__SRM sr c0 c) l (h4/canonical/canonical__sum__scalar sr c0 t)
% Assm: h4/canonical/canonical__sum__scalar__def_c1: !t sr l c0. h4/canonical/canonical__sum__scalar sr c0 (h4/canonical/Cons__varlist l t) = h4/canonical/Cons__monom c0 l (h4/canonical/canonical__sum__scalar sr c0 t)
% Assm: h4/canonical/canonical__sum__scalar__def_c2: !sr c0. h4/canonical/canonical__sum__scalar sr c0 h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm: h4/numRing/num__canonical__sum__scalar__def: h4/numRing/num__canonical__sum__scalar = h4/canonical/canonical__sum__scalar (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Goal: !t l c0. h4/numRing/num__canonical__sum__scalar c0 (h4/canonical/Cons__varlist l t) = h4/canonical/Cons__monom c0 l (h4/numRing/num__canonical__sum__scalar c0 t)
%   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_semiu_u_rings_semiu_u_ringu_u_accessorsu_c3]: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring0 a a0 f f0) = f0
% Assm [h4s_canonicals_canonicalu_u_sumu_u_scalaru_u_defu_c0]: !t sr l c0 c. happ (happ (h4/canonical/canonical__sum__scalar sr) c0) (h4/canonical/Cons__monom c l t) = h4/canonical/Cons__monom (happ (happ (h4/semi__ring/semi__ring__SRM sr) c0) c) l (happ (happ (h4/canonical/canonical__sum__scalar sr) c0) t)
% Assm [h4s_canonicals_canonicalu_u_sumu_u_scalaru_u_defu_c1]: !t sr l c0. happ (happ (h4/canonical/canonical__sum__scalar sr) c0) (h4/canonical/Cons__varlist l t) = h4/canonical/Cons__monom c0 l (happ (happ (h4/canonical/canonical__sum__scalar sr) c0) t)
% Assm [h4s_canonicals_canonicalu_u_sumu_u_scalaru_u_defu_c2]: !sr c0. happ (happ (h4/canonical/canonical__sum__scalar sr) c0) h4/canonical/Nil__monom = h4/canonical/Nil__monom
% Assm [h4s_numRings_numu_u_canonicalu_u_sumu_u_scalaru_u_def]: h4/numRing/num__canonical__sum__scalar = h4/canonical/canonical__sum__scalar (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Goal: !t l c0. happ (happ h4/numRing/num__canonical__sum__scalar c0) (h4/canonical/Cons__varlist l t) = h4/canonical/Cons__monom c0 l (happ (happ h4/numRing/num__canonical__sum__scalar c0) t)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q279015,TV_Q279011]: ![V_f, V_g]: (![V_x]: s(TV_Q279011,happ(s(t_fun(TV_Q279015,TV_Q279011),V_f),s(TV_Q279015,V_x))) = s(TV_Q279011,happ(s(t_fun(TV_Q279015,TV_Q279011),V_g),s(TV_Q279015,V_x))) => s(t_fun(TV_Q279015,TV_Q279011),V_f) = s(t_fun(TV_Q279015,TV_Q279011),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_semiu_u_rings_semiu_u_ringu_u_accessorsu_c3, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_semiu_u_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)).
fof(ah4s_canonicals_canonicalu_u_sumu_u_scalaru_u_defu_c0, axiom, ![TV_u_27a]: ![V_t, V_sr, V_l, V_c0, V_c]: 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(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_scalar(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,V_c0))),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),V_l),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,V_c0))),s(TV_u_27a,V_c))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l),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(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_scalar(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,V_c0))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))))).
fof(ah4s_canonicals_canonicalu_u_sumu_u_scalaru_u_defu_c1, axiom, ![TV_u_27a]: ![V_t, V_sr, V_l, V_c0]: 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(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_scalar(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,V_c0))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l),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(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_scalar(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,V_c0))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))))).
fof(ah4s_canonicals_canonicalu_u_sumu_u_scalaru_u_defu_c2, axiom, ![TV_u_27a]: ![V_sr, V_c0]: 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(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_scalar(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,V_c0))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom)).
fof(ah4s_numRings_numu_u_canonicalu_u_sumu_u_scalaru_u_def, axiom, s(t_fun(t_h4s_nums_num,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_numrings_numu_u_canonicalu_u_sumu_u_scalar) = s(t_fun(t_h4s_nums_num,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_canonicals_canonicalu_u_sumu_u_scalar(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ch4s_numRings_numu_u_ringu_u_thmsu_c24, conjecture, ![V_t, V_l, V_c0]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_numrings_numu_u_canonicalu_u_sumu_u_scalar),s(t_h4s_nums_num,V_c0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),h4s_canonicals_consu_u_monom(s(t_h4s_nums_num,V_c0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num))),h4s_numrings_numu_u_canonicalu_u_sumu_u_scalar),s(t_h4s_nums_num,V_c0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),V_t)))))).
