%   ORIGINAL: h4/path/labels__unfold
% 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/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ 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/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/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/combin/I__o__ID_c1: !f. h4/combin/o f h4/combin/I = f
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/option/OPTION__MAP__EQ__SOME: !y x f. h4/option/OPTION__MAP f x = h4/option/SOME y <=> (?z. x = h4/option/SOME z /\ y = f z)
% Assm: h4/option/OPTION__MAP__EQ__NONE__both__ways_c0: !x f. h4/option/OPTION__MAP f x = h4/option/NONE <=> x = h4/option/NONE
% Assm: h4/option/OPTION__MAP__COMPOSE: !x g f. h4/option/OPTION__MAP f (h4/option/OPTION__MAP g x) = h4/option/OPTION__MAP (h4/combin/o f g) x
% 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/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/llist/LMAP__LUNFOLD: !s g f. h4/llist/LMAP f (h4/llist/LUNFOLD g s) = h4/llist/LUNFOLD (\s0. h4/option/OPTION__MAP (h4/pair/UNCURRY (\x y. h4/pair/_2C x (f y))) (g s0)) s
% Assm: h4/path/path__rep__bijections__thm_c1: !r. h4/path/fromPath (h4/path/toPath r) = r
% Assm: h4/path/labels__LMAP: !p. h4/path/labels p = h4/llist/LMAP h4/pair/FST (h4/pair/SND (h4/path/fromPath p))
% Assm: h4/path/unfold__def: !s proj f. h4/path/unfold proj f s = h4/path/toPath (h4/pair/_2C (proj s) (h4/llist/LUNFOLD (\s0. h4/option/OPTION__MAP (h4/pair/UNCURRY (\next__s lbl. h4/pair/_2C next__s (h4/pair/_2C lbl (proj next__s)))) (f s0)) s))
% Goal: !s proj f. h4/path/labels (h4/path/unfold proj f s) = h4/llist/LUNFOLD f 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_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ 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_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_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_combins_Iu_u_ou_u_IDu_c1]: !f. h4/combin/o f h4/combin/I = f
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_options_OPTIONu_u_MAPu_u_EQu_u_SOME]: !y x f. h4/option/OPTION__MAP f x = h4/option/SOME y <=> (?z. x = h4/option/SOME z /\ y = happ f z)
% Assm [h4s_options_OPTIONu_u_MAPu_u_EQu_u_NONEu_u_bothu_u_waysu_c0]: !x f. h4/option/OPTION__MAP f x = h4/option/NONE <=> x = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_MAPu_u_COMPOSE]: !x g f. h4/option/OPTION__MAP f (h4/option/OPTION__MAP g x) = h4/option/OPTION__MAP (h4/combin/o f g) x
% 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_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_pairs_FST0]: !y x. happ h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_llists_LMAPu_u_LUNFOLD]: !_2. (!x f y. happ (happ (happ _2 x) f) y = h4/pair/_2C x (happ f y)) ==> (!_1. (!f x. happ (happ _1 f) x = happ (happ _2 x) f) ==> (!_0. (!f g s0. happ (happ (happ _0 f) g) s0 = h4/option/OPTION__MAP (h4/pair/UNCURRY (happ _1 f)) (happ g s0)) ==> (!s g f. h4/llist/LMAP f (h4/llist/LUNFOLD g s) = h4/llist/LUNFOLD (happ (happ _0 f) g) s)))
% Assm [h4s_paths_pathu_u_repu_u_bijectionsu_u_thmu_c1]: !r. h4/path/fromPath (h4/path/toPath r) = r
% Assm [h4s_paths_labelsu_u_LMAP]: !p. h4/path/labels p = h4/llist/LMAP h4/pair/FST (h4/pair/SND (h4/path/fromPath p))
% Assm [h4s_paths_unfoldu_u_def]: !_2. (!proj next__s lbl. happ (happ (happ _2 proj) next__s) lbl = h4/pair/_2C next__s (h4/pair/_2C lbl (happ proj next__s))) ==> (!_1. (!proj next__s. happ (happ _1 proj) next__s = happ (happ _2 proj) next__s) ==> (!_0. (!proj f s0. happ (happ (happ _0 proj) f) s0 = h4/option/OPTION__MAP (h4/pair/UNCURRY (happ _1 proj)) (happ f s0)) ==> (!s proj f. h4/path/unfold proj f s = h4/path/toPath (h4/pair/_2C (happ proj s) (h4/llist/LUNFOLD (happ (happ _0 proj) f) s)))))
% Goal: !s proj f. h4/path/labels (h4/path/unfold proj f s) = h4/llist/LUNFOLD f s
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_Q143710,TV_Q143706]: ![V_f, V_g]: (![V_x]: s(TV_Q143706,happ(s(t_fun(TV_Q143710,TV_Q143706),V_f),s(TV_Q143710,V_x))) = s(TV_Q143706,happ(s(t_fun(TV_Q143710,TV_Q143706),V_g),s(TV_Q143710,V_x))) => s(t_fun(TV_Q143710,TV_Q143706),V_f) = s(t_fun(TV_Q143710,TV_Q143706),V_g))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
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_FORALLu_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_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> ~ (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_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,f0))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => p(s(t_bool,f0))))).
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_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_combins_Iu_u_ou_u_IDu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)).
fof(ah4s_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_options_OPTIONu_u_MAPu_u_EQu_u_SOME, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_f]: (s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),V_x))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_y))) <=> ?[V_z]: (s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_z))) & s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_z)))))).
fof(ah4s_options_OPTIONu_u_MAPu_u_EQu_u_NONEu_u_bothu_u_waysu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: (s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_map(s(t_fun(TV_u_27b,TV_u_27a),V_f),s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> s(t_h4s_options_option(TV_u_27b),V_x) = s(t_h4s_options_option(TV_u_27b),h4s_options_none))).
fof(ah4s_options_OPTIONu_u_MAPu_u_COMPOSE, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_g, V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(t_h4s_options_option(TV_u_27a),V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(t_h4s_options_option(TV_u_27a),V_x)))).
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_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),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(TV_u_27a,V_x)).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(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(TV_u_27b,V_y)).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),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(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_llists_LMAPu_u_LUNFOLD, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_uu_2]: (![V_x, V_f, V_y]: s(t_h4s_pairs_prod(TV_u_27c,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),happ(s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),V_uu_2),s(TV_u_27c,V_x))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_y))) = s(t_h4s_pairs_prod(TV_u_27c,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27c,V_x),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))) => ![V_uu_1]: (![V_f, V_x]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),V_uu_1),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27c,V_x))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),happ(s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),V_uu_2),s(TV_u_27c,V_x))),s(t_fun(TV_u_27a,TV_u_27b),V_f))) => ![V_uu_0]: (![V_f, V_g, V_s0]: s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),happ(s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),V_g))),s(TV_u_27c,V_s0))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),h4s_pairs_uncurry(s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),V_uu_1),s(t_fun(TV_u_27a,TV_u_27b),V_f))))),s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a)),happ(s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),V_g),s(TV_u_27c,V_s0))))) => ![V_s, V_g, V_f]: s(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_lunfold(s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),V_g),s(TV_u_27c,V_s))))) = s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lunfold(s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b)))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27a))),V_g))),s(TV_u_27c,V_s))))))).
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_labelsu_u_LMAP, axiom, ![TV_u_27a,TV_u_27b]: ![V_p]: s(t_h4s_llists_llist(TV_u_27b),h4s_paths_labels(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lmap(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),TV_u_27b),h4s_pairs_fst),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_unfoldu_u_def, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_2]: (![V_proj, V_nextu_u_s, V_lbl]: s(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),V_uu_2),s(t_fun(TV_u_27c,TV_u_27a),V_proj))),s(TV_u_27c,V_nextu_u_s))),s(TV_u_27b,V_lbl))) = s(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27c,V_nextu_u_s),s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27b,V_lbl),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_proj),s(TV_u_27c,V_nextu_u_s))))))) => ![V_uu_1]: (![V_proj, V_nextu_u_s]: s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),V_uu_1),s(t_fun(TV_u_27c,TV_u_27a),V_proj))),s(TV_u_27c,V_nextu_u_s))) = s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),V_uu_2),s(t_fun(TV_u_27c,TV_u_27a),V_proj))),s(TV_u_27c,V_nextu_u_s))) => ![V_uu_0]: (![V_proj, V_f, V_s0]: s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),happ(s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))))),V_uu_0),s(t_fun(TV_u_27c,TV_u_27a),V_proj))),s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),V_f))),s(TV_u_27c,V_s0))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_options_optionu_u_map(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27b),t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))),h4s_pairs_uncurry(s(t_fun(TV_u_27c,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),V_uu_1),s(t_fun(TV_u_27c,TV_u_27a),V_proj))))),s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b)),happ(s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),V_f),s(TV_u_27c,V_s0))))) => ![V_s, V_proj, V_f]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_unfold(s(t_fun(TV_u_27c,TV_u_27a),V_proj),s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),V_f),s(TV_u_27c,V_s))) = 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,happ(s(t_fun(TV_u_27c,TV_u_27a),V_proj),s(TV_u_27c,V_s))),s(t_h4s_llists_llist(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_llists_lunfold(s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a))))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,t_h4s_pairs_prod(TV_u_27b,TV_u_27a)))))),V_uu_0),s(t_fun(TV_u_27c,TV_u_27a),V_proj))),s(t_fun(TV_u_27c,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27c,TV_u_27b))),V_f))),s(TV_u_27c,V_s))))))))))).
fof(ch4s_paths_labelsu_u_unfold, conjecture, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_s, V_proj, V_f]: s(t_h4s_llists_llist(TV_u_27c),h4s_paths_labels(s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_unfold(s(t_fun(TV_u_27a,TV_u_27b),V_proj),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f),s(TV_u_27a,V_s))))) = s(t_h4s_llists_llist(TV_u_27c),h4s_llists_lunfold(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,TV_u_27c))),V_f),s(TV_u_27a,V_s)))).
