%   ORIGINAL: h4/poset/gfp__unique
% 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/IMP__CLAUSES_c1: !t. t ==> T <=> 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/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/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/poset/poset__def: !s r. h4/poset/poset (h4/pair/_2C s r) <=> (?x. s x) /\ (!x. s x ==> r x x) /\ (!x y. s x /\ s y /\ r x y /\ r y x ==> x = y) /\ (!x y z. s x /\ s y /\ s z /\ r x y /\ r y z ==> r x z)
% Assm: h4/poset/gfp__def: !x s r f. h4/poset/gfp (h4/pair/_2C s r) f x <=> s x /\ f x = x /\ (!y. s y /\ r y (f y) ==> r y x)
% Goal: !x_27 x p f. h4/poset/poset p /\ h4/poset/gfp p f x /\ h4/poset/gfp p f x_27 ==> x = x_27
%   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_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> 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_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_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_posets_posetu_u_def]: !s r. h4/poset/poset (h4/pair/_2C s r) <=> (?x. happ s x) /\ (!x. happ s x ==> happ (happ r x) x) /\ (!x y. happ s x /\ happ s y /\ happ (happ r x) y /\ happ (happ r y) x ==> x = y) /\ (!x y z. happ s x /\ happ s y /\ happ s z /\ happ (happ r x) y /\ happ (happ r y) z ==> happ (happ r x) z)
% Assm [h4s_posets_gfpu_u_def]: !x s r f. h4/poset/gfp (h4/pair/_2C s r) f x <=> happ s x /\ happ f x = x /\ (!y. happ s y /\ happ (happ r y) (happ f y) ==> happ (happ r y) x)
% Goal: !x_27 x p f. h4/poset/poset p /\ h4/poset/gfp p f x /\ h4/poset/gfp p f x_27 ==> x = x_27
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_Q95863,TV_Q95859]: ![V_f, V_g]: (![V_x]: s(TV_Q95859,happ(s(t_fun(TV_Q95863,TV_Q95859),V_f),s(TV_Q95863,V_x))) = s(TV_Q95859,happ(s(t_fun(TV_Q95863,TV_Q95859),V_g),s(TV_Q95863,V_x))) => s(t_fun(TV_Q95863,TV_Q95859),V_f) = s(t_fun(TV_Q95863,TV_Q95859),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_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_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_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_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_posets_posetu_u_def, axiom, ![TV_u_27a]: ![V_s, V_r]: (p(s(t_bool,h4s_posets_poset(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r)))))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x)))) & (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r),s(TV_u_27a,V_x))),s(TV_u_27a,V_x))))) & (![V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_y)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r),s(TV_u_27a,V_y))),s(TV_u_27a,V_x))))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)) & ![V_x, V_y, V_z]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_y)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_z)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))))))))).
fof(ah4s_posets_gfpu_u_def, axiom, ![TV_u_27a]: ![V_x, V_s, V_r, V_f]: (p(s(t_bool,h4s_posets_gfp(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r))),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x)))) & (s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x) & ![V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r),s(TV_u_27a,V_y))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_y))))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r),s(TV_u_27a,V_y))),s(TV_u_27a,V_x))))))))).
fof(ch4s_posets_gfpu_u_unique, conjecture, ![TV_u_27a]: ![V_xu_27, V_x, V_p, V_f]: ((p(s(t_bool,h4s_posets_poset(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),V_p)))) & (p(s(t_bool,h4s_posets_gfp(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),V_p),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x)))) & p(s(t_bool,h4s_posets_gfp(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),V_p),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_xu_27)))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27))).
