%   ORIGINAL: h4/state__transformer/mapM__cons
% 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/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/list/MAP0_c1: !t h f. h4/list/MAP f (h4/list/CONS h t) = h4/list/CONS (f h) (h4/list/MAP f t)
% Assm: h4/list/FOLDR0_c1: !x l f e. h4/list/FOLDR f e (h4/list/CONS x l) = f x (h4/list/FOLDR f e l)
% Assm: h4/state__transformer/sequence__def: h4/state__transformer/sequence = h4/list/FOLDR (\m ms. h4/state__transformer/BIND m (\x. h4/state__transformer/BIND ms (\xs. h4/state__transformer/UNIT (h4/list/CONS x xs)))) (h4/state__transformer/UNIT h4/list/NIL)
% Assm: h4/state__transformer/mapM__def: !f. h4/state__transformer/mapM f = h4/combin/o h4/state__transformer/sequence (h4/list/MAP f)
% Goal: !xs x f. h4/state__transformer/mapM f (h4/list/CONS x xs) = h4/state__transformer/BIND (f x) (\y. h4/state__transformer/BIND (h4/state__transformer/mapM f xs) (\ys. h4/state__transformer/UNIT (h4/list/CONS y ys)))
%   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_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_lists_MAP0u_c1]: !t h f. happ (h4/list/MAP f) (h4/list/CONS h t) = h4/list/CONS (happ f h) (happ (h4/list/MAP f) t)
% Assm [h4s_lists_FOLDR0u_c1]: !x l f e. happ (h4/list/FOLDR f e) (h4/list/CONS x l) = happ (happ f x) (happ (h4/list/FOLDR f e) l)
% Assm [h4s_stateu_u_transformers_sequenceu_u_def]: !_3. (!x xs. happ (happ _3 x) xs = h4/state__transformer/UNIT (h4/list/CONS x xs)) ==> (!_2. (!ms x. happ (happ _2 ms) x = h4/state__transformer/BIND ms (happ _3 x)) ==> (!_1. (!m ms. happ (happ _1 m) ms = h4/state__transformer/BIND m (happ _2 ms)) ==> (!_0. (!m. happ _0 m = happ _1 m) ==> h4/state__transformer/sequence = h4/list/FOLDR _0 (h4/state__transformer/UNIT h4/list/NIL))))
% Assm [h4s_stateu_u_transformers_mapMu_u_def]: !f. h4/state__transformer/mapM f = h4/combin/o h4/state__transformer/sequence (h4/list/MAP f)
% Goal: !_1. (!y ys. happ (happ _1 y) ys = h4/state__transformer/UNIT (h4/list/CONS y ys)) ==> (!_0. (!f xs y. happ (happ (happ _0 f) xs) y = h4/state__transformer/BIND (happ (h4/state__transformer/mapM f) xs) (happ _1 y)) ==> (!xs x f. happ (h4/state__transformer/mapM f) (h4/list/CONS x xs) = h4/state__transformer/BIND (happ f x) (happ (happ _0 f) xs)))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q351951,TV_Q351947]: ![V_f, V_g]: (![V_x]: s(TV_Q351947,happ(s(t_fun(TV_Q351951,TV_Q351947),V_f),s(TV_Q351951,V_x))) = s(TV_Q351947,happ(s(t_fun(TV_Q351951,TV_Q351947),V_g),s(TV_Q351951,V_x))) => s(t_fun(TV_Q351951,TV_Q351947),V_f) = s(t_fun(TV_Q351951,TV_Q351947),V_g))).
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_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_lists_MAP0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_h, V_f]: s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),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_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_FOLDR0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f, V_e]: s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e))),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),V_l))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e))),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_stateu_u_transformers_sequenceu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_3]: (![V_x, V_xs]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_3),s(TV_u_27b,V_x))),s(t_h4s_lists_list(TV_u_27b),V_xs))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),h4s_stateu_u_transformers_unit(s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_x),s(t_h4s_lists_list(TV_u_27b),V_xs))))) => ![V_uu_2]: (![V_ms, V_x]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_2),s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_ms))),s(TV_u_27b,V_x))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_ms),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_3),s(TV_u_27b,V_x))))) => ![V_uu_1]: (![V_m, V_ms]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_1),s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_m))),s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_ms))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_m),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_2),s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_ms))))) => ![V_uu_0]: (![V_m]: s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_m))) = s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_1),s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_m))) => s(t_fun(t_h4s_lists_list(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))),h4s_stateu_u_transformers_sequence) = s(t_fun(t_h4s_lists_list(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))),h4s_lists_foldr(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),h4s_stateu_u_transformers_unit(s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)))))))))).
fof(ah4s_stateu_u_transformers_mapMu_u_def, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_f]: s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27c),TV_u_27b))),h4s_stateu_u_transformers_mapm(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),V_f))) = s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27c),TV_u_27b))),h4s_combins_o(s(t_fun(t_h4s_lists_list(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27c),TV_u_27b))),h4s_stateu_u_transformers_sequence),s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),h4s_lists_map(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),V_f)))))).
fof(ch4s_stateu_u_transformers_mapMu_u_cons, conjecture, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_uu_1]: (![V_y, V_ys]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_1),s(TV_u_27b,V_y))),s(t_h4s_lists_list(TV_u_27b),V_ys))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),h4s_stateu_u_transformers_unit(s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_y),s(t_h4s_lists_list(TV_u_27b),V_ys))))) => ![V_uu_0]: (![V_f, V_xs, V_y]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(TV_u_27c),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),t_fun(t_h4s_lists_list(TV_u_27c),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_f))),s(t_h4s_lists_list(TV_u_27c),V_xs))),s(TV_u_27b,V_y))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))),h4s_stateu_u_transformers_mapm(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_f))),s(t_h4s_lists_list(TV_u_27c),V_xs))),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_1),s(TV_u_27b,V_y))))) => ![V_xs, V_x, V_f]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27c),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))),h4s_stateu_u_transformers_mapm(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_f))),s(t_h4s_lists_list(TV_u_27c),h4s_lists_cons(s(TV_u_27c,V_x),s(t_h4s_lists_list(TV_u_27c),V_xs))))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_f),s(TV_u_27c,V_x))),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(TV_u_27c),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),t_fun(t_h4s_lists_list(TV_u_27c),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),TV_u_27a))))),V_uu_0),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_f))),s(t_h4s_lists_list(TV_u_27c),V_xs)))))))).
