%   ORIGINAL: h4/numRing/num__ring__thms_c48
% 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_c0: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring0 a a0 f f0) = a
% Assm: h4/canonical/interp__cs__def_c0: !vm sr. h4/canonical/interp__cs sr vm h4/canonical/Nil__monom = h4/semi__ring/semi__ring__SR0 sr
% Assm: h4/canonical/interp__cs__def_c1: !vm t sr l. h4/canonical/interp__cs sr vm (h4/canonical/Cons__varlist l t) = h4/canonical/ics__aux sr vm (h4/canonical/interp__vl sr vm l) t
% Assm: h4/canonical/interp__cs__def_c2: !vm t sr l c. h4/canonical/interp__cs sr vm (h4/canonical/Cons__monom c l t) = h4/canonical/ics__aux sr vm (h4/canonical/interp__m sr vm c l) t
% Assm: h4/numRing/num__interp__cs__def: h4/numRing/num__interp__cs = h4/canonical/interp__cs (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__ics__aux__def: h4/numRing/num__ics__aux = h4/canonical/ics__aux (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__interp__m__def: h4/numRing/num__interp__m = h4/canonical/interp__m (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/numRing/num__interp__vl__def: h4/numRing/num__interp__vl = h4/canonical/interp__vl (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Goal: !vm t l. h4/numRing/num__interp__cs vm (h4/canonical/Cons__varlist l t) = h4/numRing/num__ics__aux vm (h4/numRing/num__interp__vl vm l) 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_c0]: !f0 f a0 a. h4/semi__ring/semi__ring__SR0 (h4/semi__ring/semi__ring0 a a0 f f0) = a
% Assm [h4s_canonicals_interpu_u_csu_u_defu_c0]: !vm sr. happ (happ (h4/canonical/interp__cs sr) vm) h4/canonical/Nil__monom = h4/semi__ring/semi__ring__SR0 sr
% Assm [h4s_canonicals_interpu_u_csu_u_defu_c1]: !vm t sr l. happ (happ (h4/canonical/interp__cs sr) vm) (h4/canonical/Cons__varlist l t) = happ (happ (happ (h4/canonical/ics__aux sr) vm) (happ (happ (h4/canonical/interp__vl sr) vm) l)) t
% Assm [h4s_canonicals_interpu_u_csu_u_defu_c2]: !vm t sr l c. happ (happ (h4/canonical/interp__cs sr) vm) (h4/canonical/Cons__monom c l t) = happ (happ (happ (h4/canonical/ics__aux sr) vm) (happ (happ (happ (h4/canonical/interp__m sr) vm) c) l)) t
% Assm [h4s_numRings_numu_u_interpu_u_csu_u_def]: h4/numRing/num__interp__cs = h4/canonical/interp__cs (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_icsu_u_auxu_u_def]: h4/numRing/num__ics__aux = h4/canonical/ics__aux (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_interpu_u_mu_u_def]: h4/numRing/num__interp__m = h4/canonical/interp__m (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_numRings_numu_u_interpu_u_vlu_u_def]: h4/numRing/num__interp__vl = h4/canonical/interp__vl (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Goal: !vm t l. happ (happ h4/numRing/num__interp__cs vm) (h4/canonical/Cons__varlist l t) = happ (happ (happ h4/numRing/num__ics__aux vm) (happ (happ h4/numRing/num__interp__vl vm) l)) 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_Q279623,TV_Q279619]: ![V_f, V_g]: (![V_x]: s(TV_Q279619,happ(s(t_fun(TV_Q279623,TV_Q279619),V_f),s(TV_Q279623,V_x))) = s(TV_Q279619,happ(s(t_fun(TV_Q279623,TV_Q279619),V_g),s(TV_Q279623,V_x))) => s(t_fun(TV_Q279623,TV_Q279619),V_f) = s(t_fun(TV_Q279623,TV_Q279619),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_c0, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(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(TV_u_27a,V_a)).
fof(ah4s_canonicals_interpu_u_csu_u_defu_c0, axiom, ![TV_u_27a]: ![V_vm, V_sr]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom))) = s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr0(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr)))).
fof(ah4s_canonicals_interpu_u_csu_u_defu_c1, axiom, ![TV_u_27a]: ![V_vm, V_t, V_sr, V_l]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),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(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a))),h4s_canonicals_icsu_u_aux(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),h4s_canonicals_interpu_u_vl(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t)))).
fof(ah4s_canonicals_interpu_u_csu_u_defu_c2, axiom, ![TV_u_27a]: ![V_vm, V_t, V_sr, V_l, V_c]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_cs(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),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(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a))),h4s_canonicals_icsu_u_aux(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_canonicals_interpu_u_m(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),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)))).
fof(ah4s_numRings_numu_u_interpu_u_csu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_nums_num)),h4s_numrings_numu_u_interpu_u_cs) = s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_nums_num)),h4s_canonicals_interpu_u_cs(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(ah4s_numRings_numu_u_icsu_u_auxu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_nums_num))),h4s_numrings_numu_u_icsu_u_aux) = s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_nums_num))),h4s_canonicals_icsu_u_aux(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(ah4s_numRings_numu_u_interpu_u_mu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num))),h4s_numrings_numu_u_interpu_u_m) = s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num))),h4s_canonicals_interpu_u_m(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(ah4s_numRings_numu_u_interpu_u_vlu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num)),h4s_numrings_numu_u_interpu_u_vl) = s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num)),h4s_canonicals_interpu_u_vl(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_c48, conjecture, ![V_vm, V_t, V_l]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_nums_num),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_nums_num)),h4s_numrings_numu_u_interpu_u_cs),s(t_h4s_quotes_varmap(t_h4s_nums_num),V_vm))),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_nums_num,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),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_nums_num)),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),t_h4s_nums_num))),h4s_numrings_numu_u_icsu_u_aux),s(t_h4s_quotes_varmap(t_h4s_nums_num),V_vm))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num)),h4s_numrings_numu_u_interpu_u_vl),s(t_h4s_quotes_varmap(t_h4s_nums_num),V_vm))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_nums_num),V_t)))).
