%   ORIGINAL: h4/path/pmap__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/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/pair/CLOSED__PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/pair/PAIR__MAP__THM: !y x g f. h4/pair/_23_23 f g (h4/pair/_2C x y) = h4/pair/_2C (f x) (g y)
% Assm: h4/llist/LMAP0_c0: !f. h4/llist/LMAP f h4/llist/LNIL = h4/llist/LNIL
% Assm: h4/path/path__rep__bijections__thm_c1: !r. h4/path/fromPath (h4/path/toPath r) = r
% Assm: h4/path/toPath__11: !r_27 r. h4/path/toPath r = h4/path/toPath r_27 <=> r = r_27
% Assm: h4/path/stopped__at__def: !x. h4/path/stopped__at x = h4/path/toPath (h4/pair/_2C x h4/llist/LNIL)
% Assm: h4/path/pmap__def: !p g f. h4/path/pmap f g p = h4/path/toPath (h4/pair/_23_23 f (h4/llist/LMAP (h4/pair/_23_23 g f)) (h4/path/fromPath p))
% Goal: !x g f. h4/path/pmap f g (h4/path/stopped__at x) = h4/path/stopped__at (f 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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_pairs_CLOSEDu_u_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_pairs_PAIRu_u_MAPu_u_THM]: !y x g f. happ (h4/pair/_23_23 f g) (h4/pair/_2C x y) = h4/pair/_2C (happ f x) (happ g y)
% Assm [h4s_llists_LMAP0u_c0]: !f. happ (h4/llist/LMAP f) h4/llist/LNIL = h4/llist/LNIL
% Assm [h4s_paths_pathu_u_repu_u_bijectionsu_u_thmu_c1]: !r. h4/path/fromPath (h4/path/toPath r) = r
% Assm [h4s_paths_toPathu_u_11]: !r_27 r. h4/path/toPath r = h4/path/toPath r_27 <=> r = r_27
% Assm [h4s_paths_stoppedu_u_atu_u_def]: !x. h4/path/stopped__at x = h4/path/toPath (h4/pair/_2C x h4/llist/LNIL)
% Assm [h4s_paths_pmapu_u_def]: !p g f. h4/path/pmap f g p = h4/path/toPath (happ (h4/pair/_23_23 f (h4/llist/LMAP (h4/pair/_23_23 g f))) (h4/path/fromPath p))
% Goal: !x g f. h4/path/pmap f g (h4/path/stopped__at x) = h4/path/stopped__at (happ f x)
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_Q140851,TV_Q140847]: ![V_f, V_g]: (![V_x]: s(TV_Q140847,happ(s(t_fun(TV_Q140851,TV_Q140847),V_f),s(TV_Q140851,V_x))) = s(TV_Q140847,happ(s(t_fun(TV_Q140851,TV_Q140847),V_g),s(TV_Q140851,V_x))) => s(t_fun(TV_Q140851,TV_Q140847),V_f) = s(t_fun(TV_Q140851,TV_Q140847),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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_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_pairs_CLOSEDu_u_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_pairs_PAIRu_u_MAPu_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27d,TV_u_27c]: ![V_y, V_x, V_g, V_f]: s(t_h4s_pairs_prod(TV_u_27b,TV_u_27d),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_h4s_pairs_prod(TV_u_27b,TV_u_27d)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27d),V_g))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27c,V_y))))) = s(t_h4s_pairs_prod(TV_u_27b,TV_u_27d),h4s_pairs_u_2c(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_g),s(TV_u_27c,V_y)))))).
fof(ah4s_llists_LMAP0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_llists_llist(TV_u_27b),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_h4s_llists_llist(TV_u_27b)),h4s_llists_lmap(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lnil)).
fof(ah4s_paths_pathu_u_repu_u_bijectionsu_u_thmu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_r]: s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_paths_frompath(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_topath(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_r))))) = s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_r)).
fof(ah4s_paths_toPathu_u_11, axiom, ![TV_u_27b,TV_u_27a]: ![V_ru_27, V_r]: (s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_topath(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_r))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_topath(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_ru_27))) <=> s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_r) = s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),V_ru_27))).
fof(ah4s_paths_stoppedu_u_atu_u_def, axiom, ![TV_u_27b,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_x))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_topath(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_llists_lnil)))))).
fof(ah4s_paths_pmapu_u_def, axiom, ![TV_u_27d,TV_u_27c,TV_u_27a,TV_u_27b]: ![V_p, V_g, V_f]: s(t_h4s_paths_path(TV_u_27c,TV_u_27d),h4s_paths_pmap(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(t_h4s_paths_path(TV_u_27c,TV_u_27d),h4s_paths_topath(s(t_h4s_pairs_prod(TV_u_27c,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27d,TV_u_27c))),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),t_h4s_pairs_prod(TV_u_27c,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27d,TV_u_27c)))),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27d,TV_u_27c))),h4s_llists_lmap(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_h4s_pairs_prod(TV_u_27d,TV_u_27c)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_fun(TV_u_27a,TV_u_27c),V_f))))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_paths_frompath(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))))).
fof(ch4s_paths_pmapu_u_thmu_c0, conjecture, ![TV_u_27d,TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_pmap(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27d,TV_u_27c),V_g),s(t_h4s_paths_path(TV_u_27a,TV_u_27d),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_stoppedu_u_at(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))).
