%   ORIGINAL: h4/quotient__pair/COMMA__PRS
% 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/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/quotient/QUOTIENT__ABS__REP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. abs (rep a) = a)
% Assm: h4/pair/PAIR__MAP__THM: !y x g f. h4/pair/_23_23 f g (h4/pair/_2C x y) = h4/pair/_2C (f x) (g y)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/quotient/FUN__MAP: !g f. h4/quotient/_2D_2D_3E f g = (\h x. g (h (f x)))
% Assm: h4/quotient/FUN__REL: !g f R2 R1. h4/quotient/_3D_3D_3D_3E R1 R2 f g <=> (!x y. R1 x y ==> R2 (f x) (g y))
% Assm: h4/quotient/FUN__MAP__THM: !x h g f. h4/quotient/_2D_2D_3E f g h x = g (h (f x))
% Assm: h4/pair/PAIR__MAP: !p g f. h4/pair/_23_23 f g p = h4/pair/_2C (f (h4/pair/FST p)) (g (h4/pair/SND p))
% Assm: h4/quotient/FUN__REL__MP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. h4/quotient/_3D_3D_3D_3E R1 R2 f g /\ R1 x y ==> R2 (f x) (g y)))
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% 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))
% Assm: h4/sum/sum__Axiom: !g f. ?h. (!x. h (h4/sum/INL x) = f x) /\ (!y. h (h4/sum/INR y) = g y)
% Assm: h4/quotient/FUN__REL__EQUALS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g. h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2) f /\ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2) g ==> (h4/quotient/_2D_2D_3E rep1 abs2 f = h4/quotient/_2D_2D_3E rep1 abs2 g <=> (!x y. R1 x y ==> R2 (f x) (g y)))))
% Assm: h4/quotient__pair/PAIR__REL__THM: !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
% Assm: h4/combin/W__THM: !x f. h4/combin/W f x = f x x
% Assm: h4/quotient/K__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!x y. h4/combin/K x y = abs1 (h4/combin/K (rep1 x) (rep2 y))))
% Assm: h4/quotient/W__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. h4/combin/W f x = abs2 (h4/combin/W (h4/quotient/_2D_2D_3E abs1 (h4/quotient/_2D_2D_3E abs1 rep2) f) (rep1 x))))
% Assm: h4/quotient/literal__case__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. h4/bool/literal__case f x = abs2 (h4/bool/literal__case (h4/quotient/_2D_2D_3E abs1 rep2 f) (rep1 x))))
% Assm: h4/quotient/LET__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. h4/bool/LET f x = abs2 (h4/bool/LET (h4/quotient/_2D_2D_3E abs1 rep2 f) (rep1 x))))
% Assm: h4/quotient/C__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!R3 abs3 rep3. h4/quotient/QUOTIENT R3 abs3 rep3 ==> (!f x y. h4/combin/C f x y = abs3 (h4/combin/C (h4/quotient/_2D_2D_3E abs1 (h4/quotient/_2D_2D_3E abs2 rep3) f) (rep2 x) (rep1 y)))))
% Assm: h4/quotient/APPLY__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. f x = abs2 (h4/quotient/_2D_2D_3E abs1 rep2 f (rep1 x))))
% Assm: h4/quotient/QUOTIENT__def: !rep abs R. h4/quotient/QUOTIENT R abs rep <=> (!a. abs (rep a) = a) /\ (!a. R (rep a) (rep a)) /\ (!r s. R r s <=> R r r /\ R s s /\ abs r = abs s)
% Assm: h4/bool/literal__case__DEF: h4/bool/literal__case = (\f x. f x)
% Assm: h4/bool/LET__DEF: h4/bool/LET = (\f x. f x)
% Assm: h4/combin/C__THM: !y x f. h4/combin/C f x y = f y x
% Assm: h4/combin/K__THM: !y x. h4/combin/K x y = x
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/quotient/FUN__QUOTIENT: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient/_3D_3D_3D_3E R1 R2) (h4/quotient/_2D_2D_3E rep1 abs2) (h4/quotient/_2D_2D_3E abs1 rep2))
% Assm: h4/quotient__pair/PAIR__QUOTIENT: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient__pair/_23_23_23 R1 R2) (h4/pair/_23_23 abs1 abs2) (h4/pair/_23_23 rep1 rep2))
% Assm: h4/quotient/LAMBDA__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f. (\x. f x) = h4/quotient/_2D_2D_3E rep1 abs2 (\x. rep2 (f (abs1 x)))))
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/ind__type/ISO__FUN: !g_27 g f_27 f. h4/ind__type/ISO f f_27 /\ h4/ind__type/ISO g g_27 ==> h4/ind__type/ISO (\h a_27. g (h (f_27 a_27))) (\h a. g_27 (h (f a)))
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/quotient/respects__def: h4/quotient/respects = h4/combin/W
% Assm: h4/quotient/FUN__REL__IMP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g. h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2) f /\ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2) g /\ h4/quotient/_2D_2D_3E rep1 abs2 f = h4/quotient/_2D_2D_3E rep1 abs2 g ==> (!x y. R1 x y ==> R2 (f x) (g y))))
% Assm: h4/quotient/APPLY__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. h4/quotient/_3D_3D_3D_3E R1 R2 f g /\ R1 x y ==> R2 (f x) (g y)))
% Assm: h4/quotient/LAMBDA__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f1 f2. h4/quotient/_3D_3D_3D_3E R1 R2 f1 f2 ==> h4/quotient/_3D_3D_3D_3E R1 R2 (\x. f1 x) (\y. f2 y)))
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/quotient/QUOTIENT__REP__REFL: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. R (rep a) (rep a))
% Assm: h4/quotient__pair/SND__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!p. h4/pair/SND p = abs2 (h4/pair/SND (h4/pair/_23_23 rep1 rep2 p))))
% Assm: h4/quotient__pair/FST__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!p. h4/pair/FST p = abs1 (h4/pair/FST (h4/pair/_23_23 rep1 rep2 p))))
% Assm: h4/sum/SUM__MAP__def_c1: !g f b. h4/sum/_2B_2B f g (h4/sum/INR b) = h4/sum/INR (g b)
% Assm: h4/sum/SUM__MAP__def_c0: !g f a. h4/sum/_2B_2B f g (h4/sum/INL a) = h4/sum/INL (f a)
% Assm: h4/pair/FST__PAIR__MAP: !p g f. h4/pair/FST (h4/pair/_23_23 f g p) = f (h4/pair/FST p)
% Assm: h4/quotient__pair/PAIR__MAP__I: h4/pair/_23_23 h4/combin/I h4/combin/I = h4/combin/I
% Assm: h4/pair/SND__PAIR__MAP: !p g f. h4/pair/SND (h4/pair/_23_23 f g p) = g (h4/pair/SND p)
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/quotient/QUOTIENT__REL__REP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a b. R (rep a) (rep b) <=> a = b)
% Assm: h4/quotient/LAMBDA__REP__ABS__RSP: !rep2 rep1 f2 f1 abs2 abs1 REL2 REL1. ((!r r_27. REL1 r r_27 ==> REL1 r (rep1 (abs1 r_27))) /\ (!r r_27. REL2 r r_27 ==> REL2 r (rep2 (abs2 r_27)))) /\ h4/quotient/_3D_3D_3D_3E REL1 REL2 f1 f2 ==> h4/quotient/_3D_3D_3D_3E REL1 REL2 f1 (h4/quotient/_2D_2D_3E abs1 rep2 (h4/quotient/_2D_2D_3E rep1 abs2 f2))
% Assm: h4/quotient/LET__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. h4/quotient/_3D_3D_3D_3E R1 R2 f g /\ R1 x y ==> R2 (h4/bool/LET f x) (h4/bool/LET g y)))
% Assm: h4/quotient/literal__case__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. h4/quotient/_3D_3D_3D_3E R1 R2 f g /\ R1 x y ==> R2 (h4/bool/literal__case f x) (h4/bool/literal__case g y)))
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/quotient/QUOTIENT__REL: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. R r s <=> R r r /\ R s s /\ abs r = abs s)
% Assm: h4/quotient/FUN__REL__EQ__REL: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g. h4/quotient/_3D_3D_3D_3E R1 R2 f g <=> h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2) f /\ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2) g /\ h4/quotient/_2D_2D_3E rep1 abs2 f = h4/quotient/_2D_2D_3E rep1 abs2 g))
% Assm: h4/quotient/RES__ABSTRACT__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f1 f2. h4/quotient/_3D_3D_3D_3E R1 R2 f1 f2 ==> h4/quotient/_3D_3D_3D_3E R1 R2 (h4/bool/RES__ABSTRACT (h4/quotient/respects R1) f1) (h4/bool/RES__ABSTRACT (h4/quotient/respects R1) f2)))
% Assm: h4/quotient/C__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!R3 abs3 rep3. h4/quotient/QUOTIENT R3 abs3 rep3 ==> (!f1 f2 x1 x2 y1 y2. h4/quotient/_3D_3D_3D_3E R1 (h4/quotient/_3D_3D_3D_3E R2 R3) f1 f2 /\ R2 x1 x2 /\ R1 y1 y2 ==> R3 (h4/combin/C f1 x1 y1) (h4/combin/C f2 x2 y2))))
% Assm: h4/quotient/o__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!R3 abs3 rep3. h4/quotient/QUOTIENT R3 abs3 rep3 ==> (!f1 f2 g1 g2. h4/quotient/_3D_3D_3D_3E R2 R3 f1 f2 /\ h4/quotient/_3D_3D_3D_3E R1 R2 g1 g2 ==> h4/quotient/_3D_3D_3D_3E R1 R3 (h4/combin/o f1 g1) (h4/combin/o f2 g2))))
% Assm: h4/quotient/o__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!R3 abs3 rep3. h4/quotient/QUOTIENT R3 abs3 rep3 ==> (!f g. h4/combin/o f g = h4/quotient/_2D_2D_3E rep1 abs3 (h4/combin/o (h4/quotient/_2D_2D_3E abs2 rep3 f) (h4/quotient/_2D_2D_3E abs1 rep2 g)))))
% Assm: h4/quotient/QUOTIENT__REL__ABS__EQ: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. R r r ==> R s s ==> (R r s <=> abs r = abs s))
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/quotient/K__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!x1 x2 y1 y2. R1 x1 x2 /\ R2 y1 y2 ==> R1 (h4/combin/K x1 y1) (h4/combin/K x2 y2)))
% Assm: h4/quotient/W__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f1 f2 x1 x2. h4/quotient/_3D_3D_3D_3E R1 (h4/quotient/_3D_3D_3D_3E R1 R2) f1 f2 /\ R1 x1 x2 ==> R2 (h4/combin/W f1 x1) (h4/combin/W f2 x2)))
% Assm: h4/quotient/EXISTS__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $exists f <=> h4/bool/RES__EXISTS (h4/quotient/respects R) (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/quotient/COND__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a b c. h4/bool/COND a b c = abs (h4/bool/COND a (rep b) (rep c)))
% Assm: h4/quotient/EQUALS__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x y. x = y <=> R (rep x) (rep y))
% Assm: h4/quotient/I__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!e. h4/combin/I e = abs (h4/combin/I (rep e)))
% Assm: h4/quotient/ABSTRACT__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f. f = h4/quotient/_2D_2D_3E rep1 abs2 (h4/bool/RES__ABSTRACT (h4/quotient/respects R1) (h4/quotient/_2D_2D_3E abs1 rep2 f))))
% Assm: h4/quotient/RES__FORALL__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!P f. h4/bool/RES__FORALL P f <=> h4/bool/RES__FORALL (h4/quotient/_2D_2D_3E abs h4/combin/I P) (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/quotient/FORALL__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $forall f <=> h4/bool/RES__FORALL (h4/quotient/respects R) (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/quotient/RES__EXISTS__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!P f. h4/bool/RES__EXISTS P f <=> h4/bool/RES__EXISTS (h4/quotient/_2D_2D_3E abs h4/combin/I P) (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/quotient/EXISTS__UNIQUE__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. h4/bool/_3F_21 f <=> h4/quotient/RES__EXISTS__EQUIV R (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/ind__type/ISO0: !g f. h4/ind__type/ISO f g <=> (!x. f (g x) = x) /\ (!y. g (f y) = y)
% 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/sum/SUM__MAP__CASE: !z g f. h4/sum/_2B_2B f g z = h4/sum/sum__CASE z (h4/combin/o h4/sum/INL f) (h4/combin/o h4/sum/INR g)
% Assm: h4/res__quan/RES__ABSTRACT: !x p m. h4/bool/IN x p ==> h4/bool/RES__ABSTRACT p m x = m x
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/bool/COND__RAND: !y x f b. f (h4/bool/COND b x y) = h4/bool/COND b (f x) (f y)
% Assm: h4/res__quan/RES__FORALL: !f P. h4/bool/RES__FORALL P f <=> (!x. h4/bool/IN x P ==> f x)
% Assm: h4/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/res__quan/RES__EXISTS: !f P. h4/bool/RES__EXISTS P f <=> (?x. h4/bool/IN x P /\ f x)
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/quotient/QUOTIENT__REL__ABS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. R r s ==> abs r = abs s)
% Assm: h4/quotient/QUOTIENT__REP__ABS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r. R r r ==> R (rep (abs r)) r)
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/sum/sum__case__def_c1: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = f1 y
% Assm: h4/sum/sum__case__def_c0: !x f1 f. h4/sum/sum__CASE (h4/sum/INL x) f f1 = f x
% Assm: h4/sum/FORALL__SUM: !P. (!s. P s) <=> (!x. P (h4/sum/INL x)) /\ (!y. P (h4/sum/INR y))
% Assm: h4/sum/INR__INL__11_c1: !y x. h4/sum/INR x = h4/sum/INR y <=> x = y
% Assm: h4/sum/INR__INL__11_c0: !y x. h4/sum/INL x = h4/sum/INL y <=> x = y
% Assm: h4/bool/FORALL__DEF: $forall = (\P. P = (\x. T))
% Assm: h4/bool/EXISTS__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> x = y))
% Assm: h4/quotient/RES__EXISTS__EQUIV0: !m R. h4/quotient/RES__EXISTS__EQUIV R m <=> h4/bool/RES__EXISTS (h4/quotient/respects R) (\x. m x) /\ h4/bool/RES__FORALL (h4/quotient/respects R) (\x. h4/bool/RES__FORALL (h4/quotient/respects R) (\y. m x /\ m y ==> R x y))
% Goal: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!a b. h4/pair/_2C a b = h4/pair/_23_23 abs1 abs2 (h4/pair/_2C (rep1 a) (rep2 b))))
%   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_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_quotients_QUOTIENTu_u_ABSu_u_REP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. happ abs (happ rep a) = a)
% Assm [h4s_pairs_PAIRu_u_MAPu_u_THM]: !y x g f. happ (h4/pair/_23_23 f g) (h4/pair/_2C x y) = h4/pair/_2C (happ f x) (happ g y)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_quotients_FUNu_u_MAP]: !g f x x. happ (happ (h4/quotient/_2D_2D_3E f g) x) x = happ g (happ x (happ f x))
% Assm [h4s_quotients_FUNu_u_REL]: !g f R2 R1. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g <=> (!x y. happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y))
% Assm [h4s_quotients_FUNu_u_MAPu_u_THM]: !x h g f. happ (happ (h4/quotient/_2D_2D_3E f g) h) x = happ g (happ h (happ f x))
% Assm [h4s_pairs_PAIRu_u_MAP]: !p g f. happ (h4/pair/_23_23 f g) p = h4/pair/_2C (happ f (h4/pair/FST p)) (happ g (h4/pair/SND p))
% Assm [h4s_quotients_FUNu_u_RELu_u_MP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g /\ happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y)))
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% 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)))))
% Assm [h4s_sums_sumu_u_Axiom]: !g f. ?h. (!x. happ h (happ h4/sum/INL x) = happ f x) /\ (!y. happ h (happ h4/sum/INR y) = happ g y)
% Assm [h4s_quotients_FUNu_u_RELu_u_EQUALS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g. happ (happ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2)) f /\ happ (happ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2)) g ==> (happ (h4/quotient/_2D_2D_3E rep1 abs2) f = happ (h4/quotient/_2D_2D_3E rep1 abs2) g <=> (!x y. happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y)))))
% Assm [h4s_quotientu_u_pairs_PAIRu_u_RELu_u_THM]: !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
% Assm [h4s_combins_Wu_u_THM]: !x f. happ (happ h4/combin/W f) x = happ (happ f x) x
% Assm [h4s_quotients_Ku_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!x y. h4/combin/K x y = happ abs1 (h4/combin/K (happ rep1 x) (happ rep2 y))))
% Assm [h4s_quotients_Wu_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. happ (happ h4/combin/W f) x = happ abs2 (happ (happ h4/combin/W (happ (h4/quotient/_2D_2D_3E abs1 (h4/quotient/_2D_2D_3E abs1 rep2)) f)) (happ rep1 x))))
% Assm [h4s_quotients_literalu_u_caseu_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. h4/bool/literal__case f x = happ abs2 (h4/bool/literal__case (happ (h4/quotient/_2D_2D_3E abs1 rep2) f) (happ rep1 x))))
% Assm [h4s_quotients_LETu_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. h4/bool/LET f x = happ abs2 (h4/bool/LET (happ (h4/quotient/_2D_2D_3E abs1 rep2) f) (happ rep1 x))))
% Assm [h4s_quotients_Cu_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!R3 abs3 rep3. h4/quotient/QUOTIENT R3 abs3 rep3 ==> (!f x y. h4/combin/C f x y = happ abs3 (h4/combin/C (happ (h4/quotient/_2D_2D_3E abs1 (h4/quotient/_2D_2D_3E abs2 rep3)) f) (happ rep2 x) (happ rep1 y)))))
% Assm [h4s_quotients_APPLYu_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. happ f x = happ abs2 (happ (happ (h4/quotient/_2D_2D_3E abs1 rep2) f) (happ rep1 x))))
% Assm [h4s_quotients_QUOTIENTu_u_def]: !rep abs R. h4/quotient/QUOTIENT R abs rep <=> (!a. happ abs (happ rep a) = a) /\ (!a. happ (happ R (happ rep a)) (happ rep a)) /\ (!r s. happ (happ R r) s <=> happ (happ R r) r /\ happ (happ R s) s /\ happ abs r = happ abs s)
% Assm [h4s_bools_literalu_u_caseu_u_DEF]: !x x. h4/bool/literal__case x x = happ x x
% Assm [h4s_bools_LETu_u_DEF]: !x x. h4/bool/LET x x = happ x x
% Assm [h4s_combins_Cu_u_THM]: !y x f. h4/combin/C f x y = happ (happ f y) x
% Assm [h4s_combins_Ku_u_THM]: !y x. h4/combin/K x y = x
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_quotients_FUNu_u_QUOTIENT]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient/_3D_3D_3D_3E R1 R2) (h4/quotient/_2D_2D_3E rep1 abs2) (h4/quotient/_2D_2D_3E abs1 rep2))
% Assm [h4s_quotientu_u_pairs_PAIRu_u_QUOTIENT]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient__pair/_23_23_23 R1 R2) (h4/pair/_23_23 abs1 abs2) (h4/pair/_23_23 rep1 rep2))
% Assm [h4s_quotients_LAMBDAu_u_PRS]: !_0. (!rep2 f abs1 x. happ (happ (happ (happ _0 rep2) f) abs1) x = happ rep2 (happ f (happ abs1 x))) ==> (!rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. happ f x = happ (happ (h4/quotient/_2D_2D_3E rep1 abs2) (happ (happ (happ _0 rep2) f) abs1)) x)))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_indu_u_types_ISOu_u_FUN]: !_3. (!g_27 h f a. happ (happ (happ (happ _3 g_27) h) f) a = happ g_27 (happ h (happ f a))) ==> (!_2. (!g_27 f h. happ (happ (happ _2 g_27) f) h = happ (happ (happ _3 g_27) h) f) ==> (!_1. (!g h f_27 a_27. happ (happ (happ (happ _1 g) h) f_27) a_27 = happ g (happ h (happ f_27 a_27))) ==> (!_0. (!g f_27 h. happ (happ (happ _0 g) f_27) h = happ (happ (happ _1 g) h) f_27) ==> (!g_27 g f_27 f. h4/ind__type/ISO f f_27 /\ h4/ind__type/ISO g g_27 ==> h4/ind__type/ISO (happ (happ _0 g) f_27) (happ (happ _2 g_27) f)))))
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_quotients_respectsu_u_def]: h4/quotient/respects = h4/combin/W
% Assm [h4s_quotients_FUNu_u_RELu_u_IMP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g. happ (happ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2)) f /\ happ (happ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2)) g /\ happ (h4/quotient/_2D_2D_3E rep1 abs2) f = happ (h4/quotient/_2D_2D_3E rep1 abs2) g ==> (!x y. happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y))))
% Assm [h4s_quotients_APPLYu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g /\ happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y)))
% Assm [h4s_quotients_LAMBDAu_u_RSP]: !_0. (!f1 x. happ (happ _0 f1) x = happ f1 x) ==> (!rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f1 f2. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f1) f2 ==> happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) (happ _0 f1)) (happ _0 f2))))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_quotients_QUOTIENTu_u_REPu_u_REFL]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. happ (happ R (happ rep a)) (happ rep a))
% Assm [h4s_quotientu_u_pairs_SNDu_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!p. h4/pair/SND p = happ abs2 (h4/pair/SND (happ (h4/pair/_23_23 rep1 rep2) p))))
% Assm [h4s_quotientu_u_pairs_FSTu_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!p. h4/pair/FST p = happ abs1 (h4/pair/FST (happ (h4/pair/_23_23 rep1 rep2) p))))
% Assm [h4s_sums_SUMu_u_MAPu_u_defu_c1]: !g f b. h4/sum/_2B_2B f g (happ h4/sum/INR b) = happ h4/sum/INR (happ g b)
% Assm [h4s_sums_SUMu_u_MAPu_u_defu_c0]: !g f a. h4/sum/_2B_2B f g (happ h4/sum/INL a) = happ h4/sum/INL (happ f a)
% Assm [h4s_pairs_FSTu_u_PAIRu_u_MAP]: !p g f. h4/pair/FST (happ (h4/pair/_23_23 f g) p) = happ f (h4/pair/FST p)
% Assm [h4s_quotientu_u_pairs_PAIRu_u_MAPu_u_I]: h4/pair/_23_23 h4/combin/I h4/combin/I = h4/combin/I
% Assm [h4s_pairs_SNDu_u_PAIRu_u_MAP]: !p g f. h4/pair/SND (happ (h4/pair/_23_23 f g) p) = happ g (h4/pair/SND p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_quotients_QUOTIENTu_u_RELu_u_REP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a b. happ (happ R (happ rep a)) (happ rep b) <=> a = b)
% Assm [h4s_quotients_LAMBDAu_u_REPu_u_ABSu_u_RSP]: !rep2 rep1 f2 f1 abs2 abs1 REL2 REL1. ((!r r_27. happ (happ REL1 r) r_27 ==> happ (happ REL1 r) (happ rep1 (happ abs1 r_27))) /\ (!r r_27. happ (happ REL2 r) r_27 ==> happ (happ REL2 r) (happ rep2 (happ abs2 r_27)))) /\ happ (happ (h4/quotient/_3D_3D_3D_3E REL1 REL2) f1) f2 ==> happ (happ (h4/quotient/_3D_3D_3D_3E REL1 REL2) f1) (happ (h4/quotient/_2D_2D_3E abs1 rep2) (happ (h4/quotient/_2D_2D_3E rep1 abs2) f2))
% Assm [h4s_quotients_LETu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g /\ happ (happ R1 x) y ==> happ (happ R2 (h4/bool/LET f x)) (h4/bool/LET g y)))
% Assm [h4s_quotients_literalu_u_caseu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g /\ happ (happ R1 x) y ==> happ (happ R2 (h4/bool/literal__case f x)) (h4/bool/literal__case g y)))
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_quotients_QUOTIENTu_u_REL]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. happ (happ R r) s <=> happ (happ R r) r /\ happ (happ R s) s /\ happ abs r = happ abs s)
% Assm [h4s_quotients_FUNu_u_RELu_u_EQu_u_REL]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g <=> happ (happ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2)) f /\ happ (happ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2)) g /\ happ (h4/quotient/_2D_2D_3E rep1 abs2) f = happ (h4/quotient/_2D_2D_3E rep1 abs2) g))
% Assm [h4s_quotients_RESu_u_ABSTRACTu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f1 f2. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f1) f2 ==> happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) (h4/bool/RES__ABSTRACT (happ h4/quotient/respects R1) f1)) (h4/bool/RES__ABSTRACT (happ h4/quotient/respects R1) f2)))
% Assm [h4s_quotients_Cu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!R3 abs3 rep3. h4/quotient/QUOTIENT R3 abs3 rep3 ==> (!f1 f2 x1 x2 y1 y2. happ (happ (h4/quotient/_3D_3D_3D_3E R1 (h4/quotient/_3D_3D_3D_3E R2 R3)) f1) f2 /\ happ (happ R2 x1) x2 /\ happ (happ R1 y1) y2 ==> happ (happ R3 (h4/combin/C f1 x1 y1)) (h4/combin/C f2 x2 y2))))
% Assm [h4s_quotients_ou_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!R3 abs3 rep3. h4/quotient/QUOTIENT R3 abs3 rep3 ==> (!f1 f2 g1 g2. happ (happ (h4/quotient/_3D_3D_3D_3E R2 R3) f1) f2 /\ happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) g1) g2 ==> happ (happ (h4/quotient/_3D_3D_3D_3E R1 R3) (h4/combin/o f1 g1)) (h4/combin/o f2 g2))))
% Assm [h4s_quotients_ou_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!R3 abs3 rep3. h4/quotient/QUOTIENT R3 abs3 rep3 ==> (!f g. h4/combin/o f g = happ (h4/quotient/_2D_2D_3E rep1 abs3) (h4/combin/o (happ (h4/quotient/_2D_2D_3E abs2 rep3) f) (happ (h4/quotient/_2D_2D_3E abs1 rep2) g)))))
% Assm [h4s_quotients_QUOTIENTu_u_RELu_u_ABSu_u_EQ]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. happ (happ R r) r ==> happ (happ R s) s ==> (happ (happ R r) s <=> happ abs r = happ abs s))
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_quotients_Ku_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!x1 x2 y1 y2. happ (happ R1 x1) x2 /\ happ (happ R2 y1) y2 ==> happ (happ R1 (h4/combin/K x1 y1)) (h4/combin/K x2 y2)))
% Assm [h4s_quotients_Wu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f1 f2 x1 x2. happ (happ (h4/quotient/_3D_3D_3D_3E R1 (h4/quotient/_3D_3D_3D_3E R1 R2)) f1) f2 /\ happ (happ R1 x1) x2 ==> happ (happ R2 (happ (happ h4/combin/W f1) x1)) (happ (happ h4/combin/W f2) x2)))
% Assm [h4s_quotients_EXISTSu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $exists f <=> h4/bool/RES__EXISTS (happ h4/quotient/respects R) (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_quotients_CONDu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a b c. h4/bool/COND a b c = happ abs (h4/bool/COND a (happ rep b) (happ rep c)))
% Assm [h4s_quotients_EQUALSu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x y. x = y <=> happ (happ R (happ rep x)) (happ rep y))
% Assm [h4s_quotients_Iu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!e. happ h4/combin/I e = happ abs (happ h4/combin/I (happ rep e)))
% Assm [h4s_quotients_ABSTRACTu_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f. f = happ (h4/quotient/_2D_2D_3E rep1 abs2) (h4/bool/RES__ABSTRACT (happ h4/quotient/respects R1) (happ (h4/quotient/_2D_2D_3E abs1 rep2) f))))
% Assm [h4s_quotients_RESu_u_FORALLu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!P f. h4/bool/RES__FORALL P f <=> h4/bool/RES__FORALL (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) P) (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_quotients_FORALLu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $forall f <=> h4/bool/RES__FORALL (happ h4/quotient/respects R) (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_quotients_RESu_u_EXISTSu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!P f. h4/bool/RES__EXISTS P f <=> h4/bool/RES__EXISTS (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) P) (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_quotients_EXISTSu_u_UNIQUEu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. h4/bool/_3F_21 f <=> h4/quotient/RES__EXISTS__EQUIV R (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_indu_u_types_ISO0]: !g f. h4/ind__type/ISO f g <=> (!x. happ f (happ g x) = x) /\ (!y. happ g (happ f y) = y)
% 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_sums_SUMu_u_MAPu_u_CASE]: !z g f. h4/sum/_2B_2B f g z = h4/sum/sum__CASE z (h4/combin/o h4/sum/INL f) (h4/combin/o h4/sum/INR g)
% Assm [h4s_resu_u_quans_RESu_u_ABSTRACT]: !x p m. h4/bool/IN x p ==> happ (h4/bool/RES__ABSTRACT p m) x = happ m x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_bools_CONDu_u_RAND]: !y x f b. happ f (h4/bool/COND b x y) = h4/bool/COND b (happ f x) (happ f y)
% Assm [h4s_resu_u_quans_RESu_u_FORALL]: !f P. h4/bool/RES__FORALL P f <=> (!x. h4/bool/IN x P ==> happ f x)
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_resu_u_quans_RESu_u_EXISTS]: !f P. h4/bool/RES__EXISTS P f <=> (?x. h4/bool/IN x P /\ happ f x)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_quotients_QUOTIENTu_u_RELu_u_ABS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. happ (happ R r) s ==> happ abs r = happ abs s)
% Assm [h4s_quotients_QUOTIENTu_u_REPu_u_ABS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r. happ (happ R r) r ==> happ (happ R (happ rep (happ abs r))) r)
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_sums_sumu_u_caseu_u_defu_c1]: !y f1 f. h4/sum/sum__CASE (happ h4/sum/INR y) f f1 = happ f1 y
% Assm [h4s_sums_sumu_u_caseu_u_defu_c0]: !x f1 f. h4/sum/sum__CASE (happ h4/sum/INL x) f f1 = happ f x
% Assm [h4s_sums_FORALLu_u_SUM]: !P. (!s. happ P s) <=> (!x. happ P (happ h4/sum/INL x)) /\ (!y. happ P (happ h4/sum/INR y))
% Assm [h4s_sums_INRu_u_INLu_u_11u_c1]: !y x. happ h4/sum/INR x = happ h4/sum/INR y <=> x = y
% Assm [h4s_sums_INRu_u_INLu_u_11u_c0]: !y x. happ h4/sum/INL x = happ h4/sum/INL y <=> x = y
% Assm [h4s_bools_FORALLu_u_DEF]: !x. $forall x <=> (!x. happ x x <=> T)
% Assm [h4s_bools_EXISTSu_u_UNIQUEu_u_DEF]: !x. h4/bool/_3F_21 x <=> $exists x /\ (!x y. happ x x /\ happ x y ==> x = y)
% Assm [h4s_quotients_RESu_u_EXISTSu_u_EQUIV0]: !_2. (!m R x y. happ (happ (happ (happ _2 m) R) x) y <=> happ m x /\ happ m y ==> happ (happ R x) y) ==> (!_1. (!m R x. happ (happ (happ _1 m) R) x <=> h4/bool/RES__FORALL (happ h4/quotient/respects R) (happ (happ (happ _2 m) R) x)) ==> (!_0. (!m x. happ (happ _0 m) x <=> happ m x) ==> (!m R. h4/quotient/RES__EXISTS__EQUIV R m <=> h4/bool/RES__EXISTS (happ h4/quotient/respects R) (happ _0 m) /\ h4/bool/RES__FORALL (happ h4/quotient/respects R) (happ _0 (happ (happ _1 m) R)))))
% Goal: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!a b. h4/pair/_2C a b = happ (h4/pair/_23_23 abs1 abs2) (h4/pair/_2C (happ rep1 a) (happ rep2 b))))
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_Q1329771,TV_Q1329767]: ![V_f, V_g]: (![V_x]: s(TV_Q1329767,happ(s(t_fun(TV_Q1329771,TV_Q1329767),V_f),s(TV_Q1329771,V_x))) = s(TV_Q1329767,happ(s(t_fun(TV_Q1329771,TV_Q1329767),V_g),s(TV_Q1329771,V_x))) => s(t_fun(TV_Q1329771,TV_Q1329767),V_f) = s(t_fun(TV_Q1329771,TV_Q1329767),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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_quotients_QUOTIENTu_u_ABSu_u_REP, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a))).
fof(ah4s_pairs_PAIRu_u_MAPu_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27d,TV_u_27c]: ![V_y, V_x, V_g, V_f]: s(t_h4s_pairs_prod(TV_u_27b,TV_u_27d),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_h4s_pairs_prod(TV_u_27b,TV_u_27d)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27d),V_g))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27c,V_y))))) = s(t_h4s_pairs_prod(TV_u_27b,TV_u_27d),h4s_pairs_u_2c(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_g),s(TV_u_27c,V_y)))))).
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_quotients_FUNu_u_MAP, axiom, ![TV_u_27d,TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x, V_x0]: s(TV_u_27d,happ(s(t_fun(TV_u_27a,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27a,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27c,TV_u_27b),V_x))),s(TV_u_27a,V_x0))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_x),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x0)))))))).
fof(ah4s_quotients_FUNu_u_REL, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) <=> ![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_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y))))))))).
fof(ah4s_quotients_FUNu_u_MAPu_u_THM, axiom, ![TV_u_27d,TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_h, V_g, V_f]: s(TV_u_27d,happ(s(t_fun(TV_u_27a,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27a,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27c,TV_u_27b),V_h))),s(TV_u_27a,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_h),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x)))))))).
fof(ah4s_pairs_PAIRu_u_MAP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_p, V_g, V_f]: s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(TV_u_27c,TV_u_27d)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))) = s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),h4s_pairs_u_2c(s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))))),s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))))))).
fof(ah4s_quotients_FUNu_u_RELu_u_MP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g, V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) & 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_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y)))))))))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(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_27b,V_y)).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(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_27a,V_x)).
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(ah4s_sums_sumu_u_Axiom, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_g, V_f]: ?[V_h]: (![V_x]: s(TV_u_27c,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_x))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x))) & ![V_y]: s(TV_u_27c,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(TV_u_27b,V_y))))).
fof(ah4s_quotients_FUNu_u_RELu_u_EQUALS, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_respects),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,TV_u_27b),V_f)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_respects),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,TV_u_27b),V_g))))) => (s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),V_g))) <=> ![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_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y)))))))))))).
fof(ah4s_quotientu_u_pairs_PAIRu_u_RELu_u_THM, axiom, ![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))))))).
fof(ah4s_combins_Wu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_w),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_x)))).
fof(ah4s_quotients_Ku_u_PRS, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b,TV_u_27d]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_x, V_y]: s(TV_u_27c,h4s_combins_k(s(TV_u_27c,V_x),s(TV_u_27d,V_y))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(TV_u_27a,h4s_combins_k(s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(TV_u_27c,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_rep2),s(TV_u_27d,V_y)))))))))).
fof(ah4s_quotients_Wu_u_PRS, axiom, ![TV_u_27b,TV_u_27d,TV_u_27a,TV_u_27c]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_x]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27c,TV_u_27d)),t_fun(TV_u_27c,TV_u_27d)),h4s_combins_w),s(t_fun(TV_u_27c,t_fun(TV_u_27c,TV_u_27d)),V_f))),s(TV_u_27c,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_w),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27c,TV_u_27d)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b))),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))))),s(t_fun(TV_u_27c,t_fun(TV_u_27c,TV_u_27d)),V_f))))),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(TV_u_27c,V_x)))))))))).
fof(ah4s_quotients_literalu_u_caseu_u_PRS, axiom, ![TV_u_27b,TV_u_27d,TV_u_27a,TV_u_27c]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_x]: s(TV_u_27d,h4s_bools_literalu_u_case(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(TV_u_27c,V_x)))))))))).
fof(ah4s_quotients_LETu_u_PRS, axiom, ![TV_u_27b,TV_u_27d,TV_u_27a,TV_u_27c]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_x]: s(TV_u_27d,h4s_bools_let(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(TV_u_27c,V_x)))))))))).
fof(ah4s_quotients_Cu_u_PRS, axiom, ![TV_u_27c,TV_u_27f,TV_u_27b,TV_u_27e,TV_u_27a,TV_u_27d]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27d),V_abs1),s(t_fun(TV_u_27d,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27e),V_abs2),s(t_fun(TV_u_27e,TV_u_27b),V_rep2)))) => ![V_R3, V_abs3, V_rep3]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27c,t_fun(TV_u_27c,t_bool)),V_R3),s(t_fun(TV_u_27c,TV_u_27f),V_abs3),s(t_fun(TV_u_27f,TV_u_27c),V_rep3)))) => ![V_f, V_x, V_y]: s(TV_u_27f,h4s_combins_c(s(t_fun(TV_u_27d,t_fun(TV_u_27e,TV_u_27f)),V_f),s(TV_u_27e,V_x),s(TV_u_27d,V_y))) = s(TV_u_27f,happ(s(t_fun(TV_u_27c,TV_u_27f),V_abs3),s(TV_u_27c,h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27d,t_fun(TV_u_27e,TV_u_27f)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27d),V_abs1),s(t_fun(t_fun(TV_u_27e,TV_u_27f),t_fun(TV_u_27b,TV_u_27c)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27b,TV_u_27e),V_abs2),s(t_fun(TV_u_27f,TV_u_27c),V_rep3))))),s(t_fun(TV_u_27d,t_fun(TV_u_27e,TV_u_27f)),V_f))),s(TV_u_27b,happ(s(t_fun(TV_u_27e,TV_u_27b),V_rep2),s(TV_u_27e,V_x))),s(TV_u_27a,happ(s(t_fun(TV_u_27d,TV_u_27a),V_rep1),s(TV_u_27d,V_y))))))))))).
fof(ah4s_quotients_APPLYu_u_PRS, axiom, ![TV_u_27b,TV_u_27d,TV_u_27a,TV_u_27c]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_x]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(TV_u_27c,V_x)))))))))).
fof(ah4s_quotients_QUOTIENTu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) <=> (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & (![V_a]: 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,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a)))))) & ![V_r, V_s]: (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_r))),s(TV_u_27a,V_s)))) <=> (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_r))),s(TV_u_27a,V_r)))) & (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_s))),s(TV_u_27a,V_s)))) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_s)))))))))).
fof(ah4s_bools_literalu_u_caseu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0)))).
fof(ah4s_bools_LETu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0)))).
fof(ah4s_combins_Cu_u_THM, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27b,V_x),s(TV_u_27a,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_y))),s(TV_u_27b,V_x)))).
fof(ah4s_combins_Ku_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_combins_k(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(TV_u_27a,V_x)).
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_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_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_quotients_FUNu_u_QUOTIENT, axiom, ![TV_u_27a,TV_u_27c,TV_u_27d,TV_u_27b]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))))))))).
fof(ah4s_quotientu_u_pairs_PAIRu_u_QUOTIENT, axiom, ![TV_u_27c,TV_u_27a,TV_u_27d,TV_u_27b]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),h4s_quotientu_u_pairs_u_23u_23u_23(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(TV_u_27c,TV_u_27d)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))))))))).
fof(ah4s_quotients_LAMBDAu_u_PRS, axiom, ![TV_u_27b,TV_u_27d,TV_u_27a,TV_u_27c]: ![V_uu_0]: (![V_rep2, V_f, V_abs1, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_abs1))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_rep2),s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(TV_u_27a,V_x))))))) => ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_x]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_abs1))))),s(TV_u_27c,V_x))))))).
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_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_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_indu_u_types_ISOu_u_FUN, axiom, ![TV_u_27d,TV_u_27b,TV_u_27a,TV_u_27c]: ![V_uu_3]: (![V_gu_27, V_h, V_f, V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_3),s(t_fun(TV_u_27d,TV_u_27b),V_gu_27))),s(t_fun(TV_u_27c,TV_u_27d),V_h))),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(TV_u_27a,V_a))) = s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_gu_27),s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_h),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_a))))))) => ![V_uu_2]: (![V_gu_27, V_f, V_h]: s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)))),V_uu_2),s(t_fun(TV_u_27d,TV_u_27b),V_gu_27))),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(t_fun(TV_u_27c,TV_u_27d),V_h))) = s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_3),s(t_fun(TV_u_27d,TV_u_27b),V_gu_27))),s(t_fun(TV_u_27c,TV_u_27d),V_h))),s(t_fun(TV_u_27a,TV_u_27c),V_f))) => ![V_uu_1]: (![V_g, V_h, V_fu_27, V_au_27]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d)))),V_uu_1),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27a,TV_u_27b),V_h))),s(t_fun(TV_u_27c,TV_u_27a),V_fu_27))),s(TV_u_27c,V_au_27))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_h),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_fu_27),s(TV_u_27c,V_au_27))))))) => ![V_uu_0]: (![V_g, V_fu_27, V_h]: s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)))),V_uu_0),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27c,TV_u_27a),V_fu_27))),s(t_fun(TV_u_27a,TV_u_27b),V_h))) = s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d)))),V_uu_1),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27a,TV_u_27b),V_h))),s(t_fun(TV_u_27c,TV_u_27a),V_fu_27))) => ![V_gu_27, V_g, V_fu_27, V_f]: ((p(s(t_bool,h4s_indu_u_types_iso(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_fu_27)))) & p(s(t_bool,h4s_indu_u_types_iso(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_fun(TV_u_27d,TV_u_27b),V_gu_27))))) => p(s(t_bool,h4s_indu_u_types_iso(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)))),V_uu_0),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27c,TV_u_27a),V_fu_27))),s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)))),V_uu_2),s(t_fun(TV_u_27d,TV_u_27b),V_gu_27))),s(t_fun(TV_u_27a,TV_u_27c),V_f)))))))))))).
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_quotients_respectsu_u_def, axiom, ![TV_u_27a,TV_u_27b]: s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_respects) = s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_w)).
fof(ah4s_quotients_FUNu_u_RELu_u_IMP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_respects),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,TV_u_27b),V_f)))) & (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_respects),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) & s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),V_g))))) => ![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_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y))))))))))).
fof(ah4s_quotients_APPLYu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g, V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) & 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_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y)))))))))).
fof(ah4s_quotients_LAMBDAu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_f1, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(TV_u_27a,V_x))) => ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f1, V_f2]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(t_fun(TV_u_27a,TV_u_27b),V_f2)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f1))))),s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f2))))))))))).
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_quotients_QUOTIENTu_u_REPu_u_REFL, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_a]: 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,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a)))))))).
fof(ah4s_quotientu_u_pairs_SNDu_u_PRS, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c,TV_u_27d]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_p]: s(TV_u_27d,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),V_p))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),V_p)))))))))).
fof(ah4s_quotientu_u_pairs_FSTu_u_PRS, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c,TV_u_27d]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_p]: s(TV_u_27c,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),V_p))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),V_p)))))))))).
fof(ah4s_sums_SUMu_u_MAPu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27c,TV_u_27d,TV_u_27b]: ![V_g, V_f, V_b]: s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),h4s_sums_u_2bu_2b(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_b))))) = s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),happ(s(t_fun(TV_u_27d,t_h4s_sums_sum(TV_u_27c,TV_u_27d)),h4s_sums_inr),s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,V_b)))))).
fof(ah4s_sums_SUMu_u_MAPu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27d,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_a]: s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),h4s_sums_u_2bu_2b(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_a))))) = s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),happ(s(t_fun(TV_u_27c,t_h4s_sums_sum(TV_u_27c,TV_u_27d)),h4s_sums_inl),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_a)))))).
fof(ah4s_pairs_FSTu_u_PAIRu_u_MAP, axiom, ![TV_u_27d,TV_u_27c,TV_u_27a,TV_u_27b]: ![V_p, V_g, V_f]: s(TV_u_27c,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(TV_u_27c,TV_u_27d)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_quotientu_u_pairs_PAIRu_u_MAPu_u_I, axiom, ![TV_u_27a,TV_u_27b]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(t_fun(TV_u_27b,TV_u_27b),h4s_combins_i))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_combins_i)).
fof(ah4s_pairs_SNDu_u_PAIRu_u_MAP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_p, V_g, V_f]: s(TV_u_27d,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(TV_u_27c,TV_u_27d)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))))).
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_quotients_QUOTIENTu_u_RELu_u_REP, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_a, V_b]: (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,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_b)))))) <=> s(TV_u_27b,V_a) = s(TV_u_27b,V_b)))).
fof(ah4s_quotients_LAMBDAu_u_REPu_u_ABSu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_rep2, V_rep1, V_f2, V_f1, V_abs2, V_abs1, V_REL2, V_REL1]: (((![V_r, V_ru_27]: (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_REL1),s(TV_u_27a,V_r))),s(TV_u_27a,V_ru_27)))) => 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_REL1),s(TV_u_27a,V_r))),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(TV_u_27a,V_ru_27))))))))) & ![V_r, V_ru_27]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_REL2),s(TV_u_27b,V_r))),s(TV_u_27b,V_ru_27)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_REL2),s(TV_u_27b,V_r))),s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_rep2),s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(TV_u_27b,V_ru_27)))))))))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_REL1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_REL2))),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(t_fun(TV_u_27a,TV_u_27b),V_f2))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_REL1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_REL2))),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),V_f2)))))))))).
fof(ah4s_quotients_LETu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g, V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) & 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_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y)))))))))).
fof(ah4s_quotients_literalu_u_caseu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g, V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) & 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_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y)))))))))).
fof(ah4s_predu_u_sets_SPECIFICATION, axiom, ![TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
fof(ah4s_quotients_QUOTIENTu_u_REL, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_r, V_s]: (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_r))),s(TV_u_27a,V_s)))) <=> (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_r))),s(TV_u_27a,V_r)))) & (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_s))),s(TV_u_27a,V_s)))) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_s)))))))).
fof(ah4s_quotients_FUNu_u_RELu_u_EQu_u_REL, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) <=> (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_respects),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,TV_u_27b),V_f)))) & (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_respects),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) & s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),V_g))))))))).
fof(ah4s_quotients_RESu_u_ABSTRACTu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f1, V_f2]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(t_fun(TV_u_27a,TV_u_27b),V_f2)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),h4s_bools_resu_u_abstract(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27a,TV_u_27b),V_f1))))),s(t_fun(TV_u_27a,TV_u_27b),h4s_bools_resu_u_abstract(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27a,TV_u_27b),V_f2)))))))))).
fof(ah4s_quotients_Cu_u_RSP, axiom, ![TV_u_27d,TV_u_27e,TV_u_27f,TV_u_27c,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27d),V_abs1),s(t_fun(TV_u_27d,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27e),V_abs2),s(t_fun(TV_u_27e,TV_u_27b),V_rep2)))) => ![V_R3, V_abs3, V_rep3]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27c,t_fun(TV_u_27c,t_bool)),V_R3),s(t_fun(TV_u_27c,TV_u_27f),V_abs3),s(t_fun(TV_u_27f,TV_u_27c),V_rep3)))) => ![V_f1, V_f2, V_x1, V_x2, V_y1, V_y2]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(TV_u_27b,TV_u_27c),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27c,t_fun(TV_u_27c,t_bool)),V_R3))))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f1))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f2)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,V_x1))),s(TV_u_27b,V_x2)))) & 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_R1),s(TV_u_27a,V_y1))),s(TV_u_27a,V_y2)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27c,t_bool)),V_R3),s(TV_u_27c,h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f1),s(TV_u_27b,V_x1),s(TV_u_27a,V_y1))))),s(TV_u_27c,h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f2),s(TV_u_27b,V_x2),s(TV_u_27a,V_y2))))))))))).
fof(ah4s_quotients_ou_u_RSP, axiom, ![TV_u_27d,TV_u_27e,TV_u_27f,TV_u_27c,TV_u_27a,TV_u_27b]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27d),V_abs1),s(t_fun(TV_u_27d,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27e),V_abs2),s(t_fun(TV_u_27e,TV_u_27b),V_rep2)))) => ![V_R3, V_abs3, V_rep3]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27c,t_fun(TV_u_27c,t_bool)),V_R3),s(t_fun(TV_u_27c,TV_u_27f),V_abs3),s(t_fun(TV_u_27f,TV_u_27c),V_rep3)))) => ![V_f1, V_f2, V_g1, V_g2]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_bool),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(TV_u_27b,TV_u_27c),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27c,t_fun(TV_u_27c,t_bool)),V_R3))),s(t_fun(TV_u_27b,TV_u_27c),V_f1))),s(t_fun(TV_u_27b,TV_u_27c),V_f2)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_g1))),s(t_fun(TV_u_27a,TV_u_27b),V_g2))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27a,TV_u_27c),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27c,t_fun(TV_u_27c,t_bool)),V_R3))),s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27c),V_f1),s(t_fun(TV_u_27a,TV_u_27b),V_g1))))),s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27c),V_f2),s(t_fun(TV_u_27a,TV_u_27b),V_g2))))))))))).
fof(ah4s_quotients_ou_u_PRS, axiom, ![TV_u_27c,TV_u_27f,TV_u_27a,TV_u_27b,TV_u_27d,TV_u_27e]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27d),V_abs1),s(t_fun(TV_u_27d,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27e),V_abs2),s(t_fun(TV_u_27e,TV_u_27b),V_rep2)))) => ![V_R3, V_abs3, V_rep3]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27c,t_fun(TV_u_27c,t_bool)),V_R3),s(t_fun(TV_u_27c,TV_u_27f),V_abs3),s(t_fun(TV_u_27f,TV_u_27c),V_rep3)))) => ![V_f, V_g]: s(t_fun(TV_u_27d,TV_u_27f),h4s_combins_o(s(t_fun(TV_u_27e,TV_u_27f),V_f),s(t_fun(TV_u_27d,TV_u_27e),V_g))) = s(t_fun(TV_u_27d,TV_u_27f),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27d,TV_u_27f)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27d,TV_u_27a),V_rep1),s(t_fun(TV_u_27c,TV_u_27f),V_abs3))),s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27e,TV_u_27f),t_fun(TV_u_27b,TV_u_27c)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27b,TV_u_27e),V_abs2),s(t_fun(TV_u_27f,TV_u_27c),V_rep3))),s(t_fun(TV_u_27e,TV_u_27f),V_f))),s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27e),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27d),V_abs1),s(t_fun(TV_u_27e,TV_u_27b),V_rep2))),s(t_fun(TV_u_27d,TV_u_27e),V_g))))))))))).
fof(ah4s_quotients_QUOTIENTu_u_RELu_u_ABSu_u_EQ, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_r, V_s]: (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_r))),s(TV_u_27a,V_r)))) => (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_s))),s(TV_u_27a,V_s)))) => (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_r))),s(TV_u_27a,V_s)))) <=> s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_s)))))))).
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_quotients_Ku_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_x1, V_x2, V_y1, V_y2]: ((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_R1),s(TV_u_27a,V_x1))),s(TV_u_27a,V_x2)))) & p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,V_y1))),s(TV_u_27b,V_y2))))) => 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_R1),s(TV_u_27a,h4s_combins_k(s(TV_u_27a,V_x1),s(TV_u_27b,V_y1))))),s(TV_u_27a,h4s_combins_k(s(TV_u_27a,V_x2),s(TV_u_27b,V_y2)))))))))).
fof(ah4s_quotients_Wu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f1, V_f2, V_x1, V_x2]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f2)))) & 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_R1),s(TV_u_27a,V_x1))),s(TV_u_27a,V_x2))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_w),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f1))),s(TV_u_27a,V_x1))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_w),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f2))),s(TV_u_27a,V_x2)))))))))).
fof(ah4s_quotients_EXISTSu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f]: s(t_bool,d_exists(s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_quotients_CONDu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_a, V_b, V_c]: s(TV_u_27b,h4s_bools_cond(s(t_bool,V_a),s(TV_u_27b,V_b),s(TV_u_27b,V_c))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,h4s_bools_cond(s(t_bool,V_a),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_b))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_c))))))))).
fof(ah4s_quotients_EQUALSu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x, V_y]: (s(TV_u_27b,V_x) = s(TV_u_27b,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,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_x))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_y))))))))).
fof(ah4s_quotients_Iu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_e]: s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),h4s_combins_i),s(TV_u_27b,V_e))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_e))))))))).
fof(ah4s_quotients_ABSTRACTu_u_PRS, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c,TV_u_27d]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f]: s(t_fun(TV_u_27c,TV_u_27d),V_f) = s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),h4s_bools_resu_u_abstract(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f)))))))))).
fof(ah4s_quotients_RESu_u_FORALLu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_P, V_f]: s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27b,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_quotients_FORALLu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f]: s(t_bool,d_forall(s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_quotients_RESu_u_EXISTSu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_P, V_f]: s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27b,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_quotients_EXISTSu_u_UNIQUEu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f]: s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_quotients_resu_u_existsu_u_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
fof(ah4s_indu_u_types_ISO0, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (p(s(t_bool,h4s_indu_u_types_iso(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27b,TV_u_27a),V_g)))) <=> (![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_g),s(TV_u_27b,V_x))))) = s(TV_u_27b,V_x) & ![V_y]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))) = s(TV_u_27a,V_y)))).
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_sums_SUMu_u_MAPu_u_CASE, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b,TV_u_27d]: ![V_z, V_g, V_f]: s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),h4s_sums_u_2bu_2b(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_z))) = s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_z),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27c,TV_u_27d)),h4s_combins_o(s(t_fun(TV_u_27c,t_h4s_sums_sum(TV_u_27c,TV_u_27d)),h4s_sums_inl),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27c,TV_u_27d)),h4s_combins_o(s(t_fun(TV_u_27d,t_h4s_sums_sum(TV_u_27c,TV_u_27d)),h4s_sums_inr),s(t_fun(TV_u_27b,TV_u_27d),V_g)))))).
fof(ah4s_resu_u_quans_RESu_u_ABSTRACT, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_p, V_m]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_p)))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_bools_resu_u_abstract(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_m))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_m),s(TV_u_27a,V_x))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_bools_CONDu_u_RAND, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_f, V_b]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27a,V_x),s(TV_u_27a,V_y))))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))))).
fof(ah4s_resu_u_quans_RESu_u_FORALL, axiom, ![TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_f)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_pairs_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (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_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(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_resu_u_quans_RESu_u_EXISTS, axiom, ![TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_f)))) <=> ?[V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_quotients_QUOTIENTu_u_RELu_u_ABS, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_r, V_s]: (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_r))),s(TV_u_27a,V_s)))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_s)))))).
fof(ah4s_quotients_QUOTIENTu_u_REPu_u_ABS, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_r]: (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_r))),s(TV_u_27a,V_r)))) => 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,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))))),s(TV_u_27a,V_r))))))).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) | ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_IMPu_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_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (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,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_sums_sumu_u_caseu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_y, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f1),s(TV_u_27b,V_y)))).
fof(ah4s_sums_sumu_u_caseu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_sums_FORALLu_u_SUM, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (![V_s]: p(s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_s)))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_x)))))) & ![V_y]: p(s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))))))))).
fof(ah4s_sums_INRu_u_INLu_u_11u_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_x))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))) <=> s(TV_u_27b,V_x) = s(TV_u_27b,V_y))).
fof(ah4s_sums_INRu_u_INLu_u_11u_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_x))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_bools_FORALLu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,d_forall(s(t_fun(TV_u_27a,t_bool),V_x)))) <=> ![V_x0]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_x0))) = s(t_bool,t))).
fof(ah4s_bools_EXISTSu_u_UNIQUEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),V_x)))) <=> (p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x)))) & ![V_x0, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_x0)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_y))))) => s(TV_u_27a,V_x0) = s(TV_u_27a,V_y))))).
fof(ah4s_quotients_RESu_u_EXISTSu_u_EQUIV0, axiom, ![TV_u_27a]: ![V_uu_2]: (![V_m, V_R, 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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)))),V_uu_2),s(t_fun(TV_u_27a,t_bool),V_m))),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),V_m),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_m),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)))))) => ![V_uu_1]: (![V_m, V_R, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_m))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)))),V_uu_2),s(t_fun(TV_u_27a,t_bool),V_m))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))))) => ![V_uu_0]: (![V_m, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_m))),s(TV_u_27a,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_m),s(TV_u_27a,V_x))) => ![V_m, V_R]: (p(s(t_bool,h4s_quotients_resu_u_existsu_u_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_m)))) <=> (p(s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_m)))))) & p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_m))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))))))))))).
fof(ch4s_quotientu_u_pairs_COMMAu_u_PRS, conjecture, ![TV_u_27a,TV_u_27c,TV_u_27b,TV_u_27d]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_a, V_b]: s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),h4s_pairs_u_2c(s(TV_u_27c,V_a),s(TV_u_27d,V_b))) = s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(TV_u_27c,TV_u_27d)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(TV_u_27c,V_a))),s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_rep2),s(TV_u_27d,V_b)))))))))).
