%   ORIGINAL: h4/quotient__pair/PAIR__REL__THM
% 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/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/quotient__pair/PAIR__REL: !R2 R1. h4/quotient__pair/_23_23_23 R1 R2 = h4/pair/UNCURRY (\a b. h4/pair/UNCURRY (\c d. R1 a c /\ R2 b d))
% Goal: !d c b a R2 R1. h4/quotient__pair/_23_23_23 R1 R2 (h4/pair/_2C a b) (h4/pair/_2C c d) <=> R1 a c /\ R2 b d
%   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_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_quotientu_u_pairs_PAIRu_u_REL]: !_3. (!R1 a c R2 b d. happ (happ (happ (happ (happ (happ _3 R1) a) c) R2) b) d <=> happ (happ R1 a) c /\ happ (happ R2 b) d) ==> (!_2. (!R1 a R2 b c. happ (happ (happ (happ (happ _2 R1) a) R2) b) c = happ (happ (happ (happ (happ _3 R1) a) c) R2) b) ==> (!_1. (!R1 a R2 b. happ (happ (happ (happ _1 R1) a) R2) b = h4/pair/UNCURRY (happ (happ (happ (happ _2 R1) a) R2) b)) ==> (!_0. (!R1 R2 a. happ (happ (happ _0 R1) R2) a = happ (happ (happ _1 R1) a) R2) ==> (!R2 R1. h4/quotient__pair/_23_23_23 R1 R2 = h4/pair/UNCURRY (happ (happ _0 R1) R2)))))
% Goal: !d c b a R2 R1. happ (happ (h4/quotient__pair/_23_23_23 R1 R2) (h4/pair/_2C a b)) (h4/pair/_2C c d) <=> happ (happ R1 a) c /\ happ (happ R2 b) d
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_Q236970,TV_Q236966]: ![V_f, V_g]: (![V_x]: s(TV_Q236966,happ(s(t_fun(TV_Q236970,TV_Q236966),V_f),s(TV_Q236970,V_x))) = s(TV_Q236966,happ(s(t_fun(TV_Q236970,TV_Q236966),V_g),s(TV_Q236970,V_x))) => s(t_fun(TV_Q236970,TV_Q236966),V_f) = s(t_fun(TV_Q236970,TV_Q236966),V_g))).
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_quotientu_u_pairs_PAIRu_u_REL, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b,TV_u_27d]: ![V_uu_3]: (![V_R1, V_a, V_c, V_R2, V_b, V_d]: (p(s(t_bool,happ(s(t_fun(TV_u_27d,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool))),happ(s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)))))),V_uu_3),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1))),s(TV_u_27a,V_a))),s(TV_u_27c,V_c))),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))),s(TV_u_27b,V_b))),s(TV_u_27d,V_d)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1),s(TV_u_27a,V_a))),s(TV_u_27c,V_c)))) & p(s(t_bool,happ(s(t_fun(TV_u_27d,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2),s(TV_u_27b,V_b))),s(TV_u_27d,V_d)))))) => ![V_uu_2]: (![V_R1, V_a, V_R2, V_b, V_c]: s(t_fun(TV_u_27d,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool)))))),V_uu_2),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1))),s(TV_u_27a,V_a))),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))),s(TV_u_27b,V_b))),s(TV_u_27c,V_c))) = s(t_fun(TV_u_27d,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool))),happ(s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)))))),V_uu_3),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1))),s(TV_u_27a,V_a))),s(TV_u_27c,V_c))),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))),s(TV_u_27b,V_b))) => ![V_uu_1]: (![V_R1, V_a, V_R2, V_b]: s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool))))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1))),s(TV_u_27a,V_a))),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))),s(TV_u_27b,V_b))) = s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool),h4s_pairs_uncurry(s(t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool)))))),V_uu_2),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1))),s(TV_u_27a,V_a))),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))),s(TV_u_27b,V_b))))) => ![V_uu_0]: (![V_R1, V_R2, V_a]: s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool))))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))),s(TV_u_27a,V_a))) = s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool))))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1))),s(TV_u_27a,V_a))),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))) => ![V_R2, V_R1]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)),h4s_quotientu_u_pairs_u_23u_23u_23(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool))))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2)))))))))).
fof(ch4s_quotientu_u_pairs_PAIRu_u_RELu_u_THM, conjecture, ![TV_u_27a,TV_u_27c,TV_u_27b,TV_u_27d]: ![V_d, V_c, V_b, V_a, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)),h4s_quotientu_u_pairs_u_23u_23u_23(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))),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(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),h4s_pairs_u_2c(s(TV_u_27c,V_c),s(TV_u_27d,V_d)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1),s(TV_u_27a,V_a))),s(TV_u_27c,V_c)))) & p(s(t_bool,happ(s(t_fun(TV_u_27d,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2),s(TV_u_27b,V_b))),s(TV_u_27d,V_d))))))).
