%   ORIGINAL: h4/ind__type/INJ__INVERSE2
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/SELECT__REFL: !x. h4/min/_40 (\y. y = x) = x
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Goal: !P. (!x1 y1 x2 y2. P x1 y1 = P x2 y2 <=> x1 = x2 /\ y1 = y2) ==> (?X Y. !x y. X (P x y) = x /\ Y (P x y) = y)
%   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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_SELECTu_u_REFL]: !_0. (!x y. happ (happ _0 x) y <=> y = x) ==> (!x. h4/min/_40 (happ _0 x) = x)
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Goal: !P. (!x1 y1 x2 y2. happ (happ P x1) y1 = happ (happ P x2) y2 <=> x1 = x2 /\ y1 = y2) ==> (?X Y. !x y. happ X (happ (happ P x) y) = x /\ happ Y (happ (happ P x) y) = y)
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_Q253005,TV_Q253001]: ![V_f, V_g]: (![V_x]: s(TV_Q253001,happ(s(t_fun(TV_Q253005,TV_Q253001),V_f),s(TV_Q253005,V_x))) = s(TV_Q253001,happ(s(t_fun(TV_Q253005,TV_Q253001),V_g),s(TV_Q253005,V_x))) => s(t_fun(TV_Q253005,TV_Q253001),V_f) = s(t_fun(TV_Q253005,TV_Q253001),V_g))).
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_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_bools_SELECTu_u_REFL, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, 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_uu_0),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x)) => ![V_x]: s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x))).
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(ch4s_indu_u_types_INJu_u_INVERSE2, conjecture, ![TV_u_27C,TV_u_27A,TV_u_27B]: ![V_P]: (![V_x1, V_y1, V_x2, V_y2]: (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_P),s(TV_u_27A,V_x1))),s(TV_u_27B,V_y1))) = 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_P),s(TV_u_27A,V_x2))),s(TV_u_27B,V_y2))) <=> (s(TV_u_27A,V_x1) = s(TV_u_27A,V_x2) & s(TV_u_27B,V_y1) = s(TV_u_27B,V_y2))) => ?[V_X, V_Y]: ![V_x, V_y]: (s(TV_u_27A,happ(s(t_fun(TV_u_27C,TV_u_27A),V_X),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_P),s(TV_u_27A,V_x))),s(TV_u_27B,V_y))))) = s(TV_u_27A,V_x) & s(TV_u_27B,happ(s(t_fun(TV_u_27C,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_P),s(TV_u_27A,V_x))),s(TV_u_27B,V_y))))) = s(TV_u_27B,V_y)))).
