%   ORIGINAL: h4/path/okpath__thm_c0
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EXISTS__REFL: !a. ?x. x = a
% Assm: h4/path/stopped__at__11: !y x. h4/path/stopped__at x = h4/path/stopped__at y <=> x = y
% Assm: h4/path/stopped__at__not__pcons_c0: !y x r p. ~(h4/path/stopped__at x = h4/path/pcons y r p)
% Assm: h4/path/okpath__cases: !x R. h4/path/okpath R x <=> (?x_27. x = h4/path/stopped__at x_27) \/ (?x_27 r p. x = h4/path/pcons x_27 r p /\ R x_27 r (h4/path/first p) /\ h4/path/okpath R p)
% Goal: !x R. h4/path/okpath R (h4/path/stopped__at 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_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EXISTSu_u_REFL]: !a. ?x. x = a
% Assm [h4s_paths_stoppedu_u_atu_u_11]: !y x. h4/path/stopped__at x = h4/path/stopped__at y <=> x = y
% Assm [h4s_paths_stoppedu_u_atu_u_notu_u_pconsu_c0]: !y x r p. ~(h4/path/stopped__at x = h4/path/pcons y r p)
% Assm [h4s_paths_okpathu_u_cases]: !x R. h4/path/okpath R x <=> (?x_27. x = h4/path/stopped__at x_27) \/ (?x_27 r p. x = h4/path/pcons x_27 r p /\ happ (happ (happ R x_27) r) (h4/path/first p) /\ h4/path/okpath R p)
% Goal: !x R. h4/path/okpath R (h4/path/stopped__at 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_Q143210,TV_Q143206]: ![V_f, V_g]: (![V_x]: s(TV_Q143206,happ(s(t_fun(TV_Q143210,TV_Q143206),V_f),s(TV_Q143210,V_x))) = s(TV_Q143206,happ(s(t_fun(TV_Q143210,TV_Q143206),V_g),s(TV_Q143210,V_x))) => s(t_fun(TV_Q143210,TV_Q143206),V_f) = s(t_fun(TV_Q143210,TV_Q143206),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_EXISTSu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (?[V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
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_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EXISTSu_u_REFL, axiom, ![TV_u_27a]: ![V_a]: ?[V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_a)).
fof(ah4s_paths_stoppedu_u_atu_u_11, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_paths_stoppedu_u_atu_u_notu_u_pconsu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_r, V_p]: ~ (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_y),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))).
fof(ah4s_paths_okpathu_u_cases, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_R]: (p(s(t_bool,h4s_paths_okpath(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_x)))) <=> (?[V_xu_27]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_xu_27))) | ?[V_xu_27, V_r, V_p]: (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_xu_27),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R),s(TV_u_27a,V_xu_27))),s(TV_u_27b,V_r))),s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))) & p(s(t_bool,h4s_paths_okpath(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))))))).
fof(ch4s_paths_okpathu_u_thmu_c0, conjecture, ![TV_u_27b,TV_u_27a]: ![V_x, V_R]: p(s(t_bool,h4s_paths_okpath(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_R),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))))).
