%   ORIGINAL: h4/path/finite__path__end__cases
% 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_c0: !t. T /\ t <=> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> 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/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/EXISTS__REFL: !a. ?x. x = a
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/path/stopped__at__11: !y x. h4/path/stopped__at x = h4/path/stopped__at y <=> x = y
% Assm: h4/path/pcons__11: !y x s r q p. h4/path/pcons x r p = h4/path/pcons y s q <=> x = y /\ r = s /\ p = q
% 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/last__thm_c0: !x. h4/path/last (h4/path/stopped__at x) = x
% Assm: h4/path/last__thm_c1: !x r p. h4/path/last (h4/path/pcons x r p) = h4/path/last p
% Assm: h4/path/finite__path__ind: !P. (!x. P (h4/path/stopped__at x)) /\ (!x r p. h4/path/finite p /\ P p ==> P (h4/path/pcons x r p)) ==> (!q. h4/path/finite q ==> P q)
% Assm: h4/path/plink__def_c0: !x p. h4/path/plink (h4/path/stopped__at x) p = p
% Assm: h4/path/plink__def_c1: !x r p2 p1. h4/path/plink (h4/path/pcons x r p1) p2 = h4/path/pcons x r (h4/path/plink p1 p2)
% Goal: !p. h4/path/finite p ==> (?x. p = h4/path/stopped__at x) \/ (?p_27 l s. p = h4/path/plink p_27 (h4/path/pcons (h4/path/last p_27) l (h4/path/stopped__at s)))
%   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_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> 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_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_EXISTSu_u_REFL]: !a. ?x. x = a
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% 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_pconsu_u_11]: !y x s r q p. h4/path/pcons x r p = h4/path/pcons y s q <=> x = y /\ r = s /\ p = q
% 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_lastu_u_thmu_c0]: !x. h4/path/last (h4/path/stopped__at x) = x
% Assm [h4s_paths_lastu_u_thmu_c1]: !x r p. h4/path/last (h4/path/pcons x r p) = h4/path/last p
% Assm [h4s_paths_finiteu_u_pathu_u_ind]: !P. (!x. happ P (h4/path/stopped__at x)) /\ (!x r p. h4/path/finite p /\ happ P p ==> happ P (h4/path/pcons x r p)) ==> (!q. h4/path/finite q ==> happ P q)
% Assm [h4s_paths_plinku_u_defu_c0]: !x p. h4/path/plink (h4/path/stopped__at x) p = p
% Assm [h4s_paths_plinku_u_defu_c1]: !x r p2 p1. h4/path/plink (h4/path/pcons x r p1) p2 = h4/path/pcons x r (h4/path/plink p1 p2)
% Goal: !p. h4/path/finite p ==> (?x. p = h4/path/stopped__at x) \/ (?p_27 l s. p = h4/path/plink p_27 (h4/path/pcons (h4/path/last p_27) l (h4/path/stopped__at s)))
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_Q143960,TV_Q143956]: ![V_f, V_g]: (![V_x]: s(TV_Q143956,happ(s(t_fun(TV_Q143960,TV_Q143956),V_f),s(TV_Q143960,V_x))) = s(TV_Q143956,happ(s(t_fun(TV_Q143960,TV_Q143956),V_g),s(TV_Q143960,V_x))) => s(t_fun(TV_Q143960,TV_Q143956),V_f) = s(t_fun(TV_Q143960,TV_Q143956),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_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
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_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
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_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_ANDu_u_IMPu_u_INTRO, 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_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
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_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
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_pconsu_u_11, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_s, V_r, V_q, V_p]: (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = 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_s),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) & (s(TV_u_27b,V_r) = s(TV_u_27b,V_s) & s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q))))).
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_lastu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_paths_last(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_paths_lastu_u_thmu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p]: s(TV_u_27a,h4s_paths_last(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27a,h4s_paths_last(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))).
fof(ah4s_paths_finiteu_u_pathu_u_ind, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: ((![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x)))))) & ![V_x, V_r, V_p]: ((p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))) & p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) => p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))))) => ![V_q]: (p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q)))) => p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q))))))).
fof(ah4s_paths_plinku_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_p]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_plink(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),V_p))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)).
fof(ah4s_paths_plinku_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p2, V_p1]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_plink(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_plink(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2)))))).
fof(ch4s_paths_finiteu_u_pathu_u_endu_u_cases, conjecture, ![TV_u_27b,TV_u_27a]: ![V_p]: (p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))) => (?[V_x]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))) | ?[V_pu_27, V_l, V_s]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_plink(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_pu_27),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,h4s_paths_last(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_pu_27))),s(TV_u_27b,V_l),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_s)))))))))).
