%   ORIGINAL: h4/integer/TINT__EQ__EQUIV
% 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/IMP__CLAUSES_c3: !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/integer/TINT__EQ__REFL: !x. h4/integer/tint__eq x x
% Assm: h4/integer/TINT__EQ__SYM: !y x. h4/integer/tint__eq x y <=> h4/integer/tint__eq y x
% Assm: h4/integer/TINT__EQ__TRANS: !z y x. h4/integer/tint__eq x y /\ h4/integer/tint__eq y z ==> h4/integer/tint__eq x z
% Goal: !q p. h4/integer/tint__eq p q <=> h4/integer/tint__eq p = h4/integer/tint__eq q
%   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_IMPu_u_CLAUSESu_c3]: !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_integers_TINTu_u_EQu_u_REFL]: !x. happ (h4/integer/tint__eq x) x
% Assm [h4s_integers_TINTu_u_EQu_u_SYM]: !y x. happ (h4/integer/tint__eq x) y <=> happ (h4/integer/tint__eq y) x
% Assm [h4s_integers_TINTu_u_EQu_u_TRANS]: !z y x. happ (h4/integer/tint__eq x) y /\ happ (h4/integer/tint__eq y) z ==> happ (h4/integer/tint__eq x) z
% Goal: !q p. happ (h4/integer/tint__eq p) q <=> h4/integer/tint__eq p = h4/integer/tint__eq q
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_Q700,TV_Q696]: ![V_f, V_g]: (![V_x]: s(TV_Q696,happ(s(t_fun(TV_Q700,TV_Q696),V_f),s(TV_Q700,V_x))) = s(TV_Q696,happ(s(t_fun(TV_Q700,TV_Q696),V_g),s(TV_Q700,V_x))) => s(t_fun(TV_Q700,TV_Q696),V_f) = s(t_fun(TV_Q700,TV_Q696),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_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_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_integers_TINTu_u_EQu_u_REFL, axiom, ![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_integers_tintu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x))))).
fof(ah4s_integers_TINTu_u_EQu_u_SYM, axiom, ![V_y, V_x]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_integers_tintu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y))) = s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_integers_tintu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x)))).
fof(ah4s_integers_TINTu_u_EQu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_integers_tintu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_integers_tintu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_z))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_integers_tintu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_z)))))).
fof(ch4s_integers_TINTu_u_EQu_u_EQUIV, conjecture, ![V_q, V_p]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_integers_tintu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q)))) <=> s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_integers_tintu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_integers_tintu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q))))).
