%   ORIGINAL: h4/state__transformer/JOIN__UNIT
% 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/K__DEF: h4/combin/K = (\x y. x)
% Assm: h4/combin/S__DEF: h4/combin/S = (\f g x. f x (g x))
% Assm: h4/combin/I__DEF: h4/combin/I = h4/combin/S h4/combin/K h4/combin/K
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/state__transformer/JOIN__DEF: !z. h4/state__transformer/JOIN z = h4/state__transformer/BIND z h4/combin/I
% Assm: h4/state__transformer/BIND__LEFT__UNIT: !x k. h4/state__transformer/BIND (h4/state__transformer/UNIT x) k = k x
% Goal: h4/combin/o h4/state__transformer/JOIN h4/state__transformer/UNIT = h4/combin/I
%   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_Ku_u_DEF]: !x x. happ (happ h4/combin/K x) x = x
% Assm [h4s_combins_Su_u_DEF]: !x x x. happ (h4/combin/S x x) x = happ (happ x x) (happ x x)
% Assm [h4s_combins_Iu_u_DEF]: h4/combin/I = h4/combin/S h4/combin/K h4/combin/K
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_stateu_u_transformers_JOINu_u_DEF]: !z. happ h4/state__transformer/JOIN z = h4/state__transformer/BIND z h4/combin/I
% Assm [h4s_stateu_u_transformers_BINDu_u_LEFTu_u_UNIT]: !x k. h4/state__transformer/BIND (happ h4/state__transformer/UNIT x) k = happ k x
% Goal: h4/combin/o h4/state__transformer/JOIN h4/state__transformer/UNIT = h4/combin/I
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_Q351726,TV_Q351722]: ![V_f, V_g]: (![V_x]: s(TV_Q351722,happ(s(t_fun(TV_Q351726,TV_Q351722),V_f),s(TV_Q351726,V_x))) = s(TV_Q351722,happ(s(t_fun(TV_Q351726,TV_Q351722),V_g),s(TV_Q351726,V_x))) => s(t_fun(TV_Q351726,TV_Q351722),V_f) = s(t_fun(TV_Q351726,TV_Q351722),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_Ku_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),h4s_combins_k),s(TV_u_27a,V_x))),s(TV_u_27b,V_x0))) = s(TV_u_27a,V_x)).
fof(ah4s_combins_Su_u_DEF, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_x0, V_x1]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_s(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(t_fun(TV_u_27a,TV_u_27b),V_x0))),s(TV_u_27a,V_x1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27a,V_x1))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x0),s(TV_u_27a,V_x1)))))).
fof(ah4s_combins_Iu_u_DEF, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i) = s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_s(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27a),TV_u_27a)),h4s_combins_k),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_combins_k)))).
fof(ah4s_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_stateu_u_transformers_JOINu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_z]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_stateu_u_transformers_join),s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),V_z))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),V_z),s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_combins_i)))).
fof(ah4s_stateu_u_transformers_BINDu_u_LEFTu_u_UNIT, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_k]: s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_stateu_u_transformers_unit),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),V_k))) = s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),V_k),s(TV_u_27a,V_x)))).
fof(ch4s_stateu_u_transformers_JOINu_u_UNIT, conjecture, ![TV_u_27b,TV_u_27a]: s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_stateu_u_transformers_join),s(t_fun(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_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27a))),h4s_stateu_u_transformers_unit))) = s(t_fun(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_combins_i)).
