%   ORIGINAL: h4/defCNF/DEF__SNOC
% 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/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/ADD__CLAUSES_c2: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/ADD__CLAUSES_c3: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/list/LENGTH0_c0: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm: h4/list/LENGTH0_c1: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH 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/list/SNOC0_c0: !x. h4/list/SNOC x h4/list/NIL = h4/list/CONS x h4/list/NIL
% Assm: h4/list/SNOC0_c1: !x_27 x l. h4/list/SNOC x (h4/list/CONS x_27 l) = h4/list/CONS x_27 (h4/list/SNOC x l)
% Assm: h4/defCNF/DEF__def_c0: !v n. h4/defCNF/DEF v n h4/list/NIL <=> T
% Assm: h4/defCNF/DEF__def_c1: !xs x v n. h4/defCNF/DEF v n (h4/list/CONS x xs) <=> h4/defCNF/UNIQUE v n x /\ h4/defCNF/DEF v (h4/num/SUC n) xs
% Goal: !x v n l. h4/defCNF/DEF v n (h4/list/SNOC x l) <=> h4/defCNF/DEF v n l /\ h4/defCNF/UNIQUE v (h4/arithmetic/_2B n (h4/list/LENGTH l)) 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_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c2]: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c3]: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_lists_LENGTH0u_c0]: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm [h4s_lists_LENGTH0u_c1]: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH 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_lists_SNOC0u_c0]: !x. h4/list/SNOC x h4/list/NIL = h4/list/CONS x h4/list/NIL
% Assm [h4s_lists_SNOC0u_c1]: !x_27 x l. h4/list/SNOC x (h4/list/CONS x_27 l) = h4/list/CONS x_27 (h4/list/SNOC x l)
% Assm [h4s_defCNFs_DEFu_u_defu_c0]: !v n. h4/defCNF/DEF v n h4/list/NIL <=> T
% Assm [h4s_defCNFs_DEFu_u_defu_c1]: !xs x v n. h4/defCNF/DEF v n (h4/list/CONS x xs) <=> h4/defCNF/UNIQUE v n x /\ h4/defCNF/DEF v (h4/num/SUC n) xs
% Goal: !x v n l. h4/defCNF/DEF v n (h4/list/SNOC x l) <=> h4/defCNF/DEF v n l /\ h4/defCNF/UNIQUE v (h4/arithmetic/_2B n (h4/list/LENGTH l)) 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_Q172233,TV_Q172229]: ![V_f, V_g]: (![V_x]: s(TV_Q172229,happ(s(t_fun(TV_Q172233,TV_Q172229),V_f),s(TV_Q172233,V_x))) = s(TV_Q172229,happ(s(t_fun(TV_Q172233,TV_Q172229),V_g),s(TV_Q172233,V_x))) => s(t_fun(TV_Q172233,TV_Q172229),V_f) = s(t_fun(TV_Q172233,TV_Q172229),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_CONJu_u_ASSOC, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) & (p(s(t_bool,V_t2)) & p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) & p(s(t_bool,V_t3))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_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_arithmetics_ADDu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c3, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_lists_LENGTH0u_c0, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_lists_LENGTH0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_nums_num,h4s_lists_length(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))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_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_lists_SNOC0u_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))).
fof(ah4s_lists_SNOC0u_c1, axiom, ![TV_u_27a]: ![V_xu_27, V_x, V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_xu_27),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_xu_27),s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_defCNFs_DEFu_u_defu_c0, axiom, ![V_v, V_n]: s(t_bool,h4s_defcnfs_def(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),h4s_lists_nil))) = s(t_bool,t)).
fof(ah4s_defCNFs_DEFu_u_defu_c1, axiom, ![V_xs, V_x, V_v, V_n]: (p(s(t_bool,h4s_defcnfs_def(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),h4s_lists_cons(s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),V_x),s(t_h4s_lists_list(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),V_xs)))))) <=> (p(s(t_bool,h4s_defcnfs_unique(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),V_x)))) & p(s(t_bool,h4s_defcnfs_def(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),V_xs))))))).
fof(ch4s_defCNFs_DEFu_u_SNOC, conjecture, ![V_x, V_v, V_n, V_l]: (p(s(t_bool,h4s_defcnfs_def(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),h4s_lists_snoc(s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),V_x),s(t_h4s_lists_list(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),V_l)))))) <=> (p(s(t_bool,h4s_defcnfs_def(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),V_l)))) & p(s(t_bool,h4s_defcnfs_unique(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),V_l))))),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),V_x))))))).
