%   ORIGINAL: h4/set__relation/reln__to__rel__inv
% 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/set__relation/in__rel__to__reln: !xy R. h4/bool/IN xy (h4/set__relation/rel__to__reln R) <=> R (h4/pair/FST xy) (h4/pair/SND xy)
% Assm: h4/set__relation/reln__to__rel__app: !y x r. h4/set__relation/reln__to__rel r x y <=> h4/bool/IN (h4/pair/_2C x y) r
% Goal: !r. h4/set__relation/rel__to__reln (h4/set__relation/reln__to__rel r) = r
%   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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_setu_u_relations_inu_u_relu_u_tou_u_reln]: !xy R. h4/bool/IN xy (h4/set__relation/rel__to__reln R) <=> happ (happ R (h4/pair/FST xy)) (h4/pair/SND xy)
% Assm [h4s_setu_u_relations_relnu_u_tou_u_relu_u_app]: !y x r. happ (happ (h4/set__relation/reln__to__rel r) x) y <=> h4/bool/IN (h4/pair/_2C x y) r
% Goal: !r. h4/set__relation/rel__to__reln (h4/set__relation/reln__to__rel r) = r
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_Q139218,TV_Q139214]: ![V_f, V_g]: (![V_x]: s(TV_Q139214,happ(s(t_fun(TV_Q139218,TV_Q139214),V_f),s(TV_Q139218,V_x))) = s(TV_Q139214,happ(s(t_fun(TV_Q139218,TV_Q139214),V_g),s(TV_Q139218,V_x))) => s(t_fun(TV_Q139218,TV_Q139214),V_f) = s(t_fun(TV_Q139218,TV_Q139214),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_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_PAIR, axiom, ![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,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_setu_u_relations_inu_u_relu_u_tou_u_reln, axiom, ![TV_u_27a,TV_u_27b]: ![V_xy, V_R]: s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_xy),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_setu_u_relations_relu_u_tou_u_reln(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R))))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_xy))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_xy)))))).
fof(ah4s_setu_u_relations_relnu_u_tou_u_relu_u_app, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_r]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_setu_u_relations_relnu_u_tou_u_rel(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(t_bool,h4s_bools_in(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_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r)))).
fof(ch4s_setu_u_relations_relnu_u_tou_u_relu_u_inv, conjecture, ![TV_u_27a,TV_u_27b]: ![V_r]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_setu_u_relations_relu_u_tou_u_reln(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_setu_u_relations_relnu_u_tou_u_rel(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r)).
