%   ORIGINAL: h4/path/first__plink
% 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__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% 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/path/FORALL__path: !P. (!p. P p) <=> (!x. P (h4/path/stopped__at x)) /\ (!x r p. P (h4/path/pcons x r p))
% Assm: h4/path/first__thm_c0: !x. h4/path/first (h4/path/stopped__at x) = x
% Assm: h4/path/first__thm_c1: !x r p. h4/path/first (h4/path/pcons x r p) = x
% 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/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: !p2 p1. h4/path/last p1 = h4/path/first p2 ==> h4/path/first (h4/path/plink p1 p2) = h4/path/first p1
%   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_CLAUSESu_c1]: !t. t ==> T <=> T
% 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_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_paths_FORALLu_u_path]: !P. (!p. happ P p) <=> (!x. happ P (h4/path/stopped__at x)) /\ (!x r p. happ P (h4/path/pcons x r p))
% Assm [h4s_paths_firstu_u_thmu_c0]: !x. h4/path/first (h4/path/stopped__at x) = x
% Assm [h4s_paths_firstu_u_thmu_c1]: !x r p. h4/path/first (h4/path/pcons x r p) = x
% 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_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: !p2 p1. h4/path/last p1 = h4/path/first p2 ==> h4/path/first (h4/path/plink p1 p2) = h4/path/first p1
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_Q143335,TV_Q143331]: ![V_f, V_g]: (![V_x]: s(TV_Q143331,happ(s(t_fun(TV_Q143335,TV_Q143331),V_f),s(TV_Q143335,V_x))) = s(TV_Q143331,happ(s(t_fun(TV_Q143335,TV_Q143331),V_g),s(TV_Q143335,V_x))) => s(t_fun(TV_Q143335,TV_Q143331),V_f) = s(t_fun(TV_Q143335,TV_Q143331),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> 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_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_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_paths_FORALLu_u_path, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (![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)))) <=> (![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,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))))))))).
fof(ah4s_paths_firstu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_paths_first(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_firstu_u_thmu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_r, V_p]: s(TV_u_27a,h4s_paths_first(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,V_x)).
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_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_firstu_u_plink, conjecture, ![TV_u_27a,TV_u_27b]: ![V_p2, V_p1]: (s(TV_u_27a,h4s_paths_last(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1))) = s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p2))) => s(TV_u_27a,h4s_paths_first(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))))) = s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p1))))).
