%   ORIGINAL: h4/canonical/ivl__aux__ok
% 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/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/list/list__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/semi__ring/mult__one__right: !r. h4/semi__ring/is__semi__ring r ==> (!n. h4/semi__ring/semi__ring__SRM r n (h4/semi__ring/semi__ring__SR1 r) = n)
% Assm: h4/canonical/ivl__aux__def_c0: !x vm sr. h4/canonical/ivl__aux sr vm x h4/list/NIL = h4/quote/varmap__find x vm
% Assm: h4/canonical/ivl__aux__def_c1: !x_27 x vm t_27 sr. h4/canonical/ivl__aux sr vm x (h4/list/CONS x_27 t_27) = h4/semi__ring/semi__ring__SRM sr (h4/quote/varmap__find x vm) (h4/canonical/ivl__aux sr vm x_27 t_27)
% Assm: h4/canonical/interp__vl__def_c0: !vm sr. h4/canonical/interp__vl sr vm h4/list/NIL = h4/semi__ring/semi__ring__SR1 sr
% Assm: h4/canonical/interp__vl__def_c1: !x vm t sr. h4/canonical/interp__vl sr vm (h4/list/CONS x t) = h4/canonical/ivl__aux sr vm x t
% Goal: !sr. h4/semi__ring/is__semi__ring sr ==> (!vm v i. h4/canonical/ivl__aux sr vm i v = h4/semi__ring/semi__ring__SRM sr (h4/quote/varmap__find i vm) (h4/canonical/interp__vl sr vm v))
%   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_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_lists_listu_u_induction]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_semiu_u_rings_multu_u_oneu_u_right]: !r. h4/semi__ring/is__semi__ring r ==> (!n. h4/semi__ring/semi__ring__SRM r n (h4/semi__ring/semi__ring__SR1 r) = n)
% Assm [h4s_canonicals_ivlu_u_auxu_u_defu_c0]: !x vm sr. h4/canonical/ivl__aux sr vm x h4/list/NIL = h4/quote/varmap__find x vm
% Assm [h4s_canonicals_ivlu_u_auxu_u_defu_c1]: !x_27 x vm t_27 sr. h4/canonical/ivl__aux sr vm x (h4/list/CONS x_27 t_27) = h4/semi__ring/semi__ring__SRM sr (h4/quote/varmap__find x vm) (h4/canonical/ivl__aux sr vm x_27 t_27)
% Assm [h4s_canonicals_interpu_u_vlu_u_defu_c0]: !vm sr. h4/canonical/interp__vl sr vm h4/list/NIL = h4/semi__ring/semi__ring__SR1 sr
% Assm [h4s_canonicals_interpu_u_vlu_u_defu_c1]: !x vm t sr. h4/canonical/interp__vl sr vm (h4/list/CONS x t) = h4/canonical/ivl__aux sr vm x t
% Goal: !sr. h4/semi__ring/is__semi__ring sr ==> (!vm v i. h4/canonical/ivl__aux sr vm i v = h4/semi__ring/semi__ring__SRM sr (h4/quote/varmap__find i vm) (h4/canonical/interp__vl sr vm v))
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_Q235634,TV_Q235630]: ![V_f, V_g]: (![V_x]: s(TV_Q235630,happ(s(t_fun(TV_Q235634,TV_Q235630),V_f),s(TV_Q235634,V_x))) = s(TV_Q235630,happ(s(t_fun(TV_Q235634,TV_Q235630),V_g),s(TV_Q235634,V_x))) => s(t_fun(TV_Q235634,TV_Q235630),V_f) = s(t_fun(TV_Q235634,TV_Q235630),V_g))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
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_lists_listu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_semiu_u_rings_multu_u_oneu_u_right, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r)))) => ![V_n]: s(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_r),s(TV_u_27a,V_n),s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_r))))) = s(TV_u_27a,V_n))).
fof(ah4s_canonicals_ivlu_u_auxu_u_defu_c0, axiom, ![TV_u_27a]: ![V_x, V_vm, V_sr]: s(TV_u_27a,h4s_canonicals_ivlu_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(t_h4s_quotes_index,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))) = s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_vm)))).
fof(ah4s_canonicals_ivlu_u_auxu_u_defu_c1, axiom, ![TV_u_27a]: ![V_xu_27, V_x, V_vm, V_tu_27, V_sr]: s(TV_u_27a,h4s_canonicals_ivlu_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(t_h4s_quotes_index,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_xu_27),s(t_h4s_lists_list(t_h4s_quotes_index),V_tu_27))))) = s(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,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(TV_u_27a,h4s_canonicals_ivlu_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(t_h4s_quotes_index,V_xu_27),s(t_h4s_lists_list(t_h4s_quotes_index),V_tu_27)))))).
fof(ah4s_canonicals_interpu_u_vlu_u_defu_c0, axiom, ![TV_u_27a]: ![V_vm, V_sr]: s(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),h4s_lists_nil))) = s(TV_u_27a,h4s_semiu_u_rings_semiu_u_ringu_u_sr1(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr)))).
fof(ah4s_canonicals_interpu_u_vlu_u_defu_c1, axiom, ![TV_u_27a]: ![V_x, V_vm, V_t, V_sr]: s(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),h4s_lists_cons(s(t_h4s_quotes_index,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),V_t))))) = s(TV_u_27a,h4s_canonicals_ivlu_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(t_h4s_quotes_index,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),V_t)))).
fof(ch4s_canonicals_ivlu_u_auxu_u_ok, conjecture, ![TV_u_27a]: ![V_sr]: (p(s(t_bool,h4s_semiu_u_rings_isu_u_semiu_u_ring(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr)))) => ![V_vm, V_v, V_i]: s(TV_u_27a,h4s_canonicals_ivlu_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(t_h4s_quotes_index,V_i),s(t_h4s_lists_list(t_h4s_quotes_index),V_v))) = s(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,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(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_v))))))).
