%   ORIGINAL: h4/path/pmap__thm_c1
% 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/pair/FST__PAIR__MAP: !p g f. h4/pair/FST (h4/pair/_23_23 f g p) = f (h4/pair/FST p)
% Assm: h4/pair/SND__PAIR__MAP: !p g f. h4/pair/SND (h4/pair/_23_23 f g p) = g (h4/pair/SND p)
% Assm: h4/llist/LCONS__11: !t2 t1 h2 h1. h4/llist/LCONS h1 t1 = h4/llist/LCONS h2 t2 <=> h1 = h2 /\ t1 = t2
% Assm: h4/llist/LMAP0_c1: !t h f. h4/llist/LMAP f (h4/llist/LCONS h t) = h4/llist/LCONS (f h) (h4/llist/LMAP f t)
% 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/first__def: !p. h4/path/first p = h4/pair/FST (h4/path/fromPath p)
% Assm: h4/path/pcons__def: !x r p. h4/path/pcons x r p = h4/path/toPath (h4/pair/_2C x (h4/llist/LCONS (h4/pair/_2C r (h4/path/first p)) (h4/pair/SND (h4/path/fromPath p))))
% 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 r p g f. h4/path/pmap f g (h4/path/pcons x r p) = h4/path/pcons (f x) (g r) (h4/path/pmap f g p)
%   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_pairs_FSTu_u_PAIRu_u_MAP]: !p g f. h4/pair/FST (happ (h4/pair/_23_23 f g) p) = happ f (h4/pair/FST p)
% Assm [h4s_pairs_SNDu_u_PAIRu_u_MAP]: !p g f. h4/pair/SND (happ (h4/pair/_23_23 f g) p) = happ g (h4/pair/SND p)
% Assm [h4s_llists_LCONSu_u_11]: !t2 t1 h2 h1. h4/llist/LCONS h1 t1 = h4/llist/LCONS h2 t2 <=> h1 = h2 /\ t1 = t2
% Assm [h4s_llists_LMAP0u_c1]: !t h f. happ (h4/llist/LMAP f) (h4/llist/LCONS h t) = h4/llist/LCONS (happ f h) (happ (h4/llist/LMAP f) t)
% 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_firstu_u_def]: !p. h4/path/first p = h4/pair/FST (h4/path/fromPath p)
% Assm [h4s_paths_pconsu_u_def]: !x r p. h4/path/pcons x r p = h4/path/toPath (h4/pair/_2C x (h4/llist/LCONS (h4/pair/_2C r (h4/path/first p)) (h4/pair/SND (h4/path/fromPath p))))
% 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 r p g f. h4/path/pmap f g (h4/path/pcons x r p) = h4/path/pcons (happ f x) (happ g r) (h4/path/pmap f g p)
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_Q140876,TV_Q140872]: ![V_f, V_g]: (![V_x]: s(TV_Q140872,happ(s(t_fun(TV_Q140876,TV_Q140872),V_f),s(TV_Q140876,V_x))) = s(TV_Q140872,happ(s(t_fun(TV_Q140876,TV_Q140872),V_g),s(TV_Q140876,V_x))) => s(t_fun(TV_Q140876,TV_Q140872),V_f) = s(t_fun(TV_Q140876,TV_Q140872),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_pairs_FSTu_u_PAIRu_u_MAP, axiom, ![TV_u_27d,TV_u_27c,TV_u_27a,TV_u_27b]: ![V_p, V_g, V_f]: s(TV_u_27c,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(TV_u_27c,TV_u_27d)),h4s_pairs_u_23u_23(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_pairs_prod(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_pairs_SNDu_u_PAIRu_u_MAP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_p, V_g, V_f]: s(TV_u_27d,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(TV_u_27c,TV_u_27d)),h4s_pairs_u_23u_23(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_pairs_prod(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_llists_LCONSu_u_11, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_h2, V_h1]: (s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h1),s(t_h4s_llists_llist(TV_u_27a),V_t1))) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h2),s(t_h4s_llists_llist(TV_u_27a),V_t2))) <=> (s(TV_u_27a,V_h1) = s(TV_u_27a,V_h2) & s(t_h4s_llists_llist(TV_u_27a),V_t1) = s(t_h4s_llists_llist(TV_u_27a),V_t2)))).
fof(ah4s_llists_LMAP0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_h, 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_lcons(s(TV_u_27a,V_h),s(t_h4s_llists_llist(TV_u_27a),V_t))))) = s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lcons(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_h))),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),V_t)))))).
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_firstu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_p]: s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(TV_u_27a,h4s_pairs_fst(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(ah4s_paths_pconsu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, 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_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_lcons(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),h4s_pairs_u_2c(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))))),s(t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_pairs_snd(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(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_c1, conjecture, ![TV_u_27b,TV_u_27c,TV_u_27a,TV_u_27d]: ![V_x, V_r, V_p, 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_pcons(s(TV_u_27a,V_x),s(TV_u_27d,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27d),V_p))))) = s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_pcons(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_27c,happ(s(t_fun(TV_u_27d,TV_u_27c),V_g),s(TV_u_27d,V_r))),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),V_p)))))).
