%   ORIGINAL: h4/defCNF/UNIQUE__ind
% 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/defCNF/UNIQUE__tupled__primitive__def: h4/defCNF/UNIQUE__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R)) (\UNIQUE__tupled a_27. h4/pair/pair__CASE a_27 (\v v2. h4/pair/pair__CASE v2 (\n v4. h4/pair/pair__CASE v4 (\conn v6. h4/pair/pair__CASE v6 (\v7 v8. h4/sum/sum__CASE v7 (\i. h4/sum/sum__CASE v8 (\j. h4/combin/I (v n <=> conn (v i) (v j))) (\b. h4/combin/I (v n <=> conn (v i) b))) (\a. h4/sum/sum__CASE v8 (\j_27. h4/combin/I (v n <=> conn a (v j_27))) (\b_27. h4/combin/I (v n <=> conn a b_27))))))))
% 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/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/sum/sum__axiom: !g f. h4/bool/_3F_21 (\h. h4/combin/o h h4/sum/INL = f /\ h4/combin/o h h4/sum/INR = g)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/sum/sum__distinct: !y x. ~(h4/sum/INL x = h4/sum/INR y)
% Assm: h4/sum/sum__INDUCT: !P. (!x. P (h4/sum/INL x)) /\ (!y. P (h4/sum/INR y)) ==> (!s. P s)
% Assm: h4/bool/DATATYPE__TAG__THM: !x. h4/bool/DATATYPE x <=> T
% Assm: h4/sum/INR__11: !y x. h4/sum/INR x = h4/sum/INR y <=> x = y
% Assm: h4/sum/INL__11: !y x. h4/sum/INL x = h4/sum/INL y <=> x = y
% Assm: h4/sum/sum__CASES: !ss. (?x. ss = h4/sum/INL x) \/ (?y. ss = h4/sum/INR y)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/sum/FORALL__SUM: !P. (!s. P s) <=> (!x. P (h4/sum/INL x)) /\ (!y. P (h4/sum/INR y))
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% 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/sum__case__def_c1: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = f1 y
% Assm: h4/sum/INR__INL__11_c0: !y x. h4/sum/INL x = h4/sum/INL y <=> x = y
% Assm: h4/sum/INR__INL__11_c1: !y x. h4/sum/INR x = h4/sum/INR y <=> x = y
% Assm: h4/sum/datatype__sum: !sum. h4/bool/DATATYPE (sum h4/sum/INL h4/sum/INR)
% Assm: h4/pred__set/SUM__UNIV: h4/pred__set/UNIV = h4/pred__set/UNION (h4/pred__set/IMAGE h4/sum/INL h4/pred__set/UNIV) (h4/pred__set/IMAGE h4/sum/INR h4/pred__set/UNIV)
% 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/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EXISTS__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> x = y))
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% 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/INR__neq__INL: !v2 v1. ~(h4/sum/INR v2 = h4/sum/INL v1)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% 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/sum/INR__DEF: !e. h4/sum/INR e = h4/sum/ABS__sum (\b x y. y = e /\ ~b)
% Assm: h4/sum/INL__DEF: !e. h4/sum/INL e = h4/sum/ABS__sum (\b x y. x = e /\ b)
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/sum/cond__sum__expand_c3: !z y x P. h4/bool/COND P (h4/sum/INL x) (h4/sum/INR y) = h4/sum/INR z <=> ~P /\ z = y
% Assm: h4/sum/cond__sum__expand_c1: !z y x P. h4/bool/COND P (h4/sum/INR x) (h4/sum/INL y) = h4/sum/INL z <=> ~P /\ z = y
% Assm: h4/sum/cond__sum__expand_c0: !z y x P. h4/bool/COND P (h4/sum/INR x) (h4/sum/INL y) = h4/sum/INR z <=> P /\ z = x
% Assm: h4/sum/cond__sum__expand_c2: !z y x P. h4/bool/COND P (h4/sum/INL x) (h4/sum/INR y) = h4/sum/INL z <=> P /\ z = x
% Assm: h4/sum/sum__distinct1: !y x. ~(h4/sum/INR y = h4/sum/INL x)
% Assm: h4/pred__set/INJ__INR: !t s. (!x. h4/bool/IN x s ==> h4/bool/IN (h4/sum/INR x) t) ==> h4/pred__set/INJ h4/sum/INR s t
% Assm: h4/pred__set/INJ__INL: !t s. (!x. h4/bool/IN x s ==> h4/bool/IN (h4/sum/INL x) t) ==> h4/pred__set/INJ h4/sum/INL s t
% Assm: h4/sum/EXISTS__SUM: !P. (?s. P s) <=> (?x. P (h4/sum/INL x)) \/ (?y. P (h4/sum/INR y))
% Assm: h4/sum/sum__case__cong: !f1_27 f1 f_27 f M_27 M. M = M_27 /\ (!x. M_27 = h4/sum/INL x ==> f x = f_27 x) /\ (!y. M_27 = h4/sum/INR y ==> f1 y = f1_27 y) ==> h4/sum/sum__CASE M f f1 = h4/sum/sum__CASE M_27 f_27 f1_27
% Assm: h4/sum/SUM__MAP: !z g f. h4/sum/_2B_2B f g z = h4/bool/COND (h4/sum/ISL z) (h4/sum/INL (f (h4/sum/OUTL z))) (h4/sum/INR (g (h4/sum/OUTR z)))
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sum/ISL0_c0: !x. h4/sum/ISL (h4/sum/INL x)
% Assm: h4/sum/ISL0_c1: !y. ~h4/sum/ISL (h4/sum/INR y)
% Assm: h4/bool/ABS__SIMP: !t2 t1. (\x. t1) t2 = t1
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/defCNF/UNIQUE__curried__def: !x2 x1 x. h4/defCNF/UNIQUE x x1 x2 <=> h4/defCNF/UNIQUE__tupled (h4/pair/_2C x (h4/pair/_2C x1 x2))
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/sum/IS__SUM__REP0: !f. h4/sum/IS__SUM__REP f <=> (?v1 v2. f = (\b x y. x = v1 /\ b) \/ f = (\b x y. y = v2 /\ ~b))
% Assm: h4/sum/sum__ISO__DEF_c0: !a. h4/sum/ABS__sum (h4/sum/REP__sum a) = a
% Assm: h4/sum/sum__ISO__DEF_c1: !r. h4/sum/IS__SUM__REP r <=> h4/sum/REP__sum (h4/sum/ABS__sum r) = r
% Assm: h4/sum/OUTR0: !x. h4/sum/OUTR (h4/sum/INR x) = x
% Assm: h4/sum/ISR0_c0: !x. h4/sum/ISR (h4/sum/INR x)
% Assm: h4/sum/ISR0_c1: !y. ~h4/sum/ISR (h4/sum/INL y)
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/sum/OUTL0: !x. h4/sum/OUTL (h4/sum/INL x) = x
% Assm: h4/basicSize/sum__size__def_c1: !y g f. h4/basicSize/sum__size f g (h4/sum/INR y) = g y
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/basicSize/sum__size__def_c0: !x g f. h4/basicSize/sum__size f g (h4/sum/INL x) = f x
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/sum/INR0: !x. h4/sum/ISR x ==> h4/sum/INR (h4/sum/OUTR x) = x
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/pred__set/INJ__DEF: !t s f. h4/pred__set/INJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (f x) t) /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s ==> f x = f y ==> x = y)
% Assm: h4/sum/INL0: !x. h4/sum/ISL x ==> h4/sum/INL (h4/sum/OUTL x) = x
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/option/NONE__DEF: h4/option/NONE = h4/option/option__ABS (h4/sum/INR h4/one/one0)
% Assm: h4/option/SOME__DEF: !x. h4/option/SOME x = h4/option/option__ABS (h4/sum/INL x)
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/while/OLEAST__INTRO: !Q P. ((!n. ~P n) ==> Q h4/option/NONE) /\ (!n. P n /\ (!m. h4/prim__rec/_3C m n ==> ~P m) ==> Q (h4/option/SOME n)) ==> Q (h4/while/OLEAST P)
% Assm: h4/pred__set/in__max__set: !s. h4/pred__set/FINITE s ==> (!x. h4/bool/IN x s ==> h4/arithmetic/_3C_3D x (h4/pred__set/MAX__SET s))
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/EXISTS__UNIQUE__THM: !P. h4/bool/_3F_21 (\x. P x) <=> (?x. P x) /\ (!x y. P x /\ P y ==> x = y)
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/SELECT__REFL__2: !x. h4/min/_40 (\y. x = y) = x
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/bool/EXISTS__REFL: !a. ?x. x = a
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% 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/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/pred__set/IN__IMAGE: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = f x /\ h4/bool/IN x s)
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/pred__set/IN__UNION: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm: h4/bool/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/pred__set/FINITE__INDUCT: !P. P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ P s ==> (!e. ~h4/bool/IN e s ==> P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> P s)
% Assm: h4/pred__set/IN__INSERT: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/pred__set/MAX__SET__THM_c1: !s e. h4/pred__set/FINITE s ==> h4/pred__set/MAX__SET (h4/pred__set/INSERT e s) = h4/arithmetic/MAX e (h4/pred__set/MAX__SET s)
% Assm: h4/arithmetic/MAX__LE_c0: !p n m. h4/arithmetic/_3C_3D p (h4/arithmetic/MAX m n) <=> h4/arithmetic/_3C_3D p m \/ h4/arithmetic/_3C_3D p n
% Assm: h4/arithmetic/LESS__EQ__REFL: !m. h4/arithmetic/_3C_3D m m
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/while/OLEAST__def: !P. h4/while/OLEAST P = h4/bool/COND (?n. P n) (h4/option/SOME (h4/while/LEAST (\n. P n))) h4/option/NONE
% Assm: h4/while/LEAST__ELIM: !Q P. (?n. P n) /\ (!n. (!m. h4/prim__rec/_3C m n ==> ~P m) /\ P n ==> Q n) ==> Q (h4/while/LEAST P)
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Goal: !P. (!v n conn i j. P v n (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INL i) (h4/sum/INL j)))) /\ (!v n conn i b. P v n (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INL i) (h4/sum/INR b)))) /\ (!v n conn a j. P v n (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INR a) (h4/sum/INL j)))) /\ (!v n conn a b. P v n (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INR a) (h4/sum/INR b)))) ==> (!v v1 v2 v3 v4. P v v1 (h4/pair/_2C v2 (h4/pair/_2C v3 v4)))
%   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_defCNFs_UNIQUEu_u_tupledu_u_primitiveu_u_def]: !_16. (!v n conn a b_27. ?v. (v <=> happ v n <=> happ (happ conn a) b_27) /\ (happ (happ (happ (happ (happ _16 v) n) conn) a) b_27 <=> h4/combin/I v)) ==> (!_15. (!n conn a v j_27. ?v. (v <=> happ v n <=> happ (happ conn a) (happ v j_27)) /\ (happ (happ (happ (happ (happ _15 n) conn) a) v) j_27 <=> h4/combin/I v)) ==> (!_14. (!v8 v n conn a. happ (happ (happ (happ (happ _14 v8) v) n) conn) a <=> h4/sum/sum__CASE v8 (happ (happ (happ (happ _15 n) conn) a) v) (happ (happ (happ (happ _16 v) n) conn) a)) ==> (!_13. (!n conn v i b. ?v. (v <=> happ v n <=> happ (happ conn (happ v i)) b) /\ (happ (happ (happ (happ (happ _13 n) conn) v) i) b <=> h4/combin/I v)) ==> (!_12. (!n conn i v j. ?v. (v <=> happ v n <=> happ (happ conn (happ v i)) (happ v j)) /\ (happ (happ (happ (happ (happ _12 n) conn) i) v) j <=> h4/combin/I v)) ==> (!_11. (!v8 n conn v i. happ (happ (happ (happ (happ _11 v8) n) conn) v) i <=> h4/sum/sum__CASE v8 (happ (happ (happ (happ _12 n) conn) i) v) (happ (happ (happ (happ _13 n) conn) v) i)) ==> (!_10. (!v7 v n conn v8. happ (happ (happ (happ (happ _10 v7) v) n) conn) v8 <=> h4/sum/sum__CASE v7 (happ (happ (happ (happ _11 v8) n) conn) v) (happ (happ (happ (happ _14 v8) v) n) conn)) ==> (!_9. (!v n conn v7. happ (happ (happ (happ _9 v) n) conn) v7 = happ (happ (happ (happ _10 v7) v) n) conn) ==> (!_8. (!v n conn v6. happ (happ (happ (happ _8 v) n) conn) v6 <=> h4/pair/pair__CASE v6 (happ (happ (happ _9 v) n) conn)) ==> (!_7. (!v n conn. happ (happ (happ _7 v) n) conn = happ (happ (happ _8 v) n) conn) ==> (!_6. (!v n v4. happ (happ (happ _6 v) n) v4 <=> h4/pair/pair__CASE v4 (happ (happ _7 v) n)) ==> (!_5. (!v n. happ (happ _5 v) n = happ (happ _6 v) n) ==> (!_4. (!v v2. happ (happ _4 v) v2 <=> h4/pair/pair__CASE v2 (happ _5 v)) ==> (!_3. (!v. happ _3 v = happ _4 v) ==> (!_2. (!a_27. happ _2 a_27 <=> h4/pair/pair__CASE a_27 _3) ==> (!_1. (!UNIQUE__tupled. happ _1 UNIQUE__tupled = _2) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R) ==> h4/defCNF/UNIQUE__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))))))))))
% 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_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_sums_sumu_u_axiom]: !_0. (!f g h. happ (happ (happ _0 f) g) h <=> h4/combin/o h h4/sum/INL = f /\ h4/combin/o h h4/sum/INR = g) ==> (!g f. h4/bool/_3F_21 (happ (happ _0 f) g))
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_sums_sumu_u_distinct]: !y x. ~(happ h4/sum/INL x = happ h4/sum/INR y)
% Assm [h4s_sums_sumu_u_INDUCT]: !P. (!x. happ P (happ h4/sum/INL x)) /\ (!y. happ P (happ h4/sum/INR y)) ==> (!s. happ P s)
% Assm [h4s_bools_DATATYPEu_u_TAGu_u_THM]: !x. h4/bool/DATATYPE x <=> T
% Assm [h4s_sums_INRu_u_11]: !y x. happ h4/sum/INR x = happ h4/sum/INR y <=> x = y
% Assm [h4s_sums_INLu_u_11]: !y x. happ h4/sum/INL x = happ h4/sum/INL y <=> x = y
% Assm [h4s_sums_sumu_u_CASES]: !ss. (?x. ss = happ h4/sum/INL x) \/ (?y. ss = happ h4/sum/INR y)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t 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_bools_FALSITY]: !t. F ==> t
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% 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_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_INRu_u_INLu_u_11u_c0]: !y x. happ h4/sum/INL x = happ h4/sum/INL y <=> x = 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_datatypeu_u_sum]: !sum. h4/bool/DATATYPE (happ (happ sum h4/sum/INL) h4/sum/INR)
% Assm [h4s_predu_u_sets_SUMu_u_UNIV]: h4/pred__set/UNIV = h4/pred__set/UNION (h4/pred__set/IMAGE h4/sum/INL h4/pred__set/UNIV) (h4/pred__set/IMAGE h4/sum/INR h4/pred__set/UNIV)
% 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_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% 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_INRu_u_nequ_u_INL]: !v2 v1. ~(happ h4/sum/INR v2 = happ h4/sum/INL v1)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% 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_sums_INRu_u_DEF]: !_2. (!e b y. happ (happ (happ _2 e) b) y <=> y = e /\ ~b) ==> (!_1. (!e b x. happ (happ (happ _1 e) b) x = happ (happ _2 e) b) ==> (!_0. (!e b. happ (happ _0 e) b = happ (happ _1 e) b) ==> (!e. happ h4/sum/INR e = h4/sum/ABS__sum (happ _0 e))))
% Assm [h4s_sums_INLu_u_DEF]: !_2. (!x e b y. happ (happ (happ (happ _2 x) e) b) y <=> x = e /\ b) ==> (!_1. (!e b x. happ (happ (happ _1 e) b) x = happ (happ (happ _2 x) e) b) ==> (!_0. (!e b. happ (happ _0 e) b = happ (happ _1 e) b) ==> (!e. happ h4/sum/INL e = h4/sum/ABS__sum (happ _0 e))))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_sums_condu_u_sumu_u_expandu_c3]: !z y x P. h4/bool/COND P (happ h4/sum/INL x) (happ h4/sum/INR y) = happ h4/sum/INR z <=> ~P /\ z = y
% Assm [h4s_sums_condu_u_sumu_u_expandu_c1]: !z y x P. h4/bool/COND P (happ h4/sum/INR x) (happ h4/sum/INL y) = happ h4/sum/INL z <=> ~P /\ z = y
% Assm [h4s_sums_condu_u_sumu_u_expandu_c0]: !z y x P. h4/bool/COND P (happ h4/sum/INR x) (happ h4/sum/INL y) = happ h4/sum/INR z <=> P /\ z = x
% Assm [h4s_sums_condu_u_sumu_u_expandu_c2]: !z y x P. h4/bool/COND P (happ h4/sum/INL x) (happ h4/sum/INR y) = happ h4/sum/INL z <=> P /\ z = x
% Assm [h4s_sums_sumu_u_distinct1]: !y x. ~(happ h4/sum/INR y = happ h4/sum/INL x)
% Assm [h4s_predu_u_sets_INJu_u_INR]: !t s. (!x. h4/bool/IN x s ==> h4/bool/IN (happ h4/sum/INR x) t) ==> h4/pred__set/INJ h4/sum/INR s t
% Assm [h4s_predu_u_sets_INJu_u_INL]: !t s. (!x. h4/bool/IN x s ==> h4/bool/IN (happ h4/sum/INL x) t) ==> h4/pred__set/INJ h4/sum/INL s t
% Assm [h4s_sums_EXISTSu_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_sumu_u_caseu_u_cong]: !f1_27 f1 f_27 f M_27 M. M = M_27 /\ (!x. M_27 = happ h4/sum/INL x ==> happ f x = happ f_27 x) /\ (!y. M_27 = happ h4/sum/INR y ==> happ f1 y = happ f1_27 y) ==> h4/sum/sum__CASE M f f1 = h4/sum/sum__CASE M_27 f_27 f1_27
% Assm [h4s_sums_SUMu_u_MAP]: !z g f. h4/sum/_2B_2B f g z = h4/bool/COND (h4/sum/ISL z) (happ h4/sum/INL (happ f (h4/sum/OUTL z))) (happ h4/sum/INR (happ g (h4/sum/OUTR z)))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sums_ISL0u_c0]: !x. h4/sum/ISL (happ h4/sum/INL x)
% Assm [h4s_sums_ISL0u_c1]: !y. ~h4/sum/ISL (happ h4/sum/INR y)
% Assm [h4s_bools_ABSu_u_SIMP]: !t2 t1. t1 = t1
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_defCNFs_UNIQUEu_u_curriedu_u_def]: !x2 x1 x. h4/defCNF/UNIQUE x x1 x2 <=> happ h4/defCNF/UNIQUE__tupled (h4/pair/_2C x (h4/pair/_2C x1 x2))
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_sums_ISu_u_SUMu_u_REP0]: !f. h4/sum/IS__SUM__REP f <=> (?v1 v2. (!x x x. happ (happ (happ f x) x) x <=> x = v1 /\ x) \/ (!x x x. happ (happ (happ f x) x) x <=> x = v2 /\ ~x))
% Assm [h4s_sums_sumu_u_ISOu_u_DEFu_c0]: !a. h4/sum/ABS__sum (h4/sum/REP__sum a) = a
% Assm [h4s_sums_sumu_u_ISOu_u_DEFu_c1]: !r. h4/sum/IS__SUM__REP r <=> h4/sum/REP__sum (h4/sum/ABS__sum r) = r
% Assm [h4s_sums_OUTR0]: !x. h4/sum/OUTR (happ h4/sum/INR x) = x
% Assm [h4s_sums_ISR0u_c0]: !x. h4/sum/ISR (happ h4/sum/INR x)
% Assm [h4s_sums_ISR0u_c1]: !y. ~h4/sum/ISR (happ h4/sum/INL y)
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_sums_OUTL0]: !x. h4/sum/OUTL (happ h4/sum/INL x) = x
% Assm [h4s_basicSizes_sumu_u_sizeu_u_defu_c1]: !y g f. h4/basicSize/sum__size f g (happ h4/sum/INR y) = happ g y
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_basicSizes_sumu_u_sizeu_u_defu_c0]: !x g f. h4/basicSize/sum__size f g (happ h4/sum/INL x) = happ f x
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_sums_INR0]: !x. h4/sum/ISR x ==> happ h4/sum/INR (h4/sum/OUTR x) = x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_predu_u_sets_INJu_u_DEF]: !t s f. h4/pred__set/INJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (happ f x) t) /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s ==> happ f x = happ f y ==> x = y)
% Assm [h4s_sums_INL0]: !x. h4/sum/ISL x ==> happ h4/sum/INL (h4/sum/OUTL x) = x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_options_NONEu_u_DEF]: h4/option/NONE = h4/option/option__ABS (happ h4/sum/INR h4/one/one0)
% Assm [h4s_options_SOMEu_u_DEF]: !x. h4/option/SOME x = h4/option/option__ABS (happ h4/sum/INL x)
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_whiles_OLEASTu_u_INTRO]: !Q P. ((!n. ~happ P n) ==> happ Q h4/option/NONE) /\ (!n. happ P n /\ (!m. h4/prim__rec/_3C m n ==> ~happ P m) ==> happ Q (h4/option/SOME n)) ==> happ Q (h4/while/OLEAST P)
% Assm [h4s_predu_u_sets_inu_u_maxu_u_set]: !s. h4/pred__set/FINITE s ==> (!x. h4/bool/IN x s ==> h4/arithmetic/_3C_3D x (h4/pred__set/MAX__SET s))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_EXISTSu_u_UNIQUEu_u_THM]: !_0. (!P x. happ (happ _0 P) x <=> happ P x) ==> (!P. h4/bool/_3F_21 (happ _0 P) <=> (?x. happ P x) /\ (!x y. happ P x /\ happ P y ==> x = y))
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_SELECTu_u_REFLu_u_2]: !_0. (!x y. happ (happ _0 x) y <=> x = y) ==> (!x. h4/min/_40 (happ _0 x) = x)
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_bools_EXISTSu_u_REFL]: !a. ?x. x = a
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% 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_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_predu_u_sets_INu_u_IMAGE]: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = happ f x /\ h4/bool/IN x s)
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_predu_u_sets_INu_u_UNION]: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm [h4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM]: !Q P. (!x. happ P x) /\ Q <=> (!x. happ P x /\ Q)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_predu_u_sets_FINITEu_u_INDUCT]: !P. happ P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ happ P s ==> (!e. ~h4/bool/IN e s ==> happ P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> happ P s)
% Assm [h4s_predu_u_sets_INu_u_INSERT]: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_MAXu_u_SETu_u_THMu_c1]: !s e. h4/pred__set/FINITE s ==> h4/pred__set/MAX__SET (h4/pred__set/INSERT e s) = h4/arithmetic/MAX e (h4/pred__set/MAX__SET s)
% Assm [h4s_arithmetics_MAXu_u_LEu_c0]: !p n m. h4/arithmetic/_3C_3D p (h4/arithmetic/MAX m n) <=> h4/arithmetic/_3C_3D p m \/ h4/arithmetic/_3C_3D p n
% Assm [h4s_arithmetics_LESSu_u_EQu_u_REFL]: !m. h4/arithmetic/_3C_3D m m
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_whiles_OLEASTu_u_def]: !_0. (!P n. happ (happ _0 P) n <=> happ P n) ==> (!P. ?v. (v <=> (?n. happ P n)) /\ h4/while/OLEAST P = h4/bool/COND v (h4/option/SOME (h4/while/LEAST (happ _0 P))) h4/option/NONE)
% Assm [h4s_whiles_LEASTu_u_ELIM]: !Q P. (?n. happ P n) /\ (!n. (!m. h4/prim__rec/_3C m n ==> ~happ P m) /\ happ P n ==> happ Q n) ==> happ Q (h4/while/LEAST P)
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Goal: !P. (!v n conn i j. happ (happ (happ P v) n) (h4/pair/_2C conn (h4/pair/_2C (happ h4/sum/INL i) (happ h4/sum/INL j)))) /\ (!v n conn i b. happ (happ (happ P v) n) (h4/pair/_2C conn (h4/pair/_2C (happ h4/sum/INL i) (happ h4/sum/INR b)))) /\ (!v n conn a j. happ (happ (happ P v) n) (h4/pair/_2C conn (h4/pair/_2C (happ h4/sum/INR a) (happ h4/sum/INL j)))) /\ (!v n conn a b. happ (happ (happ P v) n) (h4/pair/_2C conn (h4/pair/_2C (happ h4/sum/INR a) (happ h4/sum/INR b)))) ==> (!v v1 v2 v3 v4. happ (happ (happ P v) v1) (h4/pair/_2C v2 (h4/pair/_2C v3 v4)))
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_Q1260480,TV_Q1260476]: ![V_f, V_g]: (![V_x]: s(TV_Q1260476,happ(s(t_fun(TV_Q1260480,TV_Q1260476),V_f),s(TV_Q1260480,V_x))) = s(TV_Q1260476,happ(s(t_fun(TV_Q1260480,TV_Q1260476),V_g),s(TV_Q1260480,V_x))) => s(t_fun(TV_Q1260480,TV_Q1260476),V_f) = s(t_fun(TV_Q1260480,TV_Q1260476),V_g))).
fof(ah4s_defCNFs_UNIQUEu_u_tupledu_u_primitiveu_u_def, axiom, ![V_uu_16]: (![V_v, V_n, V_conn, V_a, V_bu_27]: ?[V_v0]: ((p(s(t_bool,V_v0)) <=> s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n))) = s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_bool,V_a))),s(t_bool,V_bu_27)))) & s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_bool,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_bool,t_bool)))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_bool,t_bool))))),V_uu_16),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_bool,V_a))),s(t_bool,V_bu_27))) = s(t_bool,h4s_combins_i(s(t_bool,V_v0)))) => ![V_uu_15]: (![V_n, V_conn, V_a, V_v, V_ju_27]: ?[V_v0]: ((p(s(t_bool,V_v0)) <=> s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n))) = s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_bool,V_a))),s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_ju_27)))))) & s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_bool,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))))),V_uu_15),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_bool,V_a))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_ju_27))) = s(t_bool,h4s_combins_i(s(t_bool,V_v0)))) => ![V_uu_14]: (![V_v8, V_v, V_n, V_conn, V_a]: s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_bool)))),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_bool))))),V_uu_14),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v8))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_bool,V_a))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v8),s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_bool,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))))),V_uu_15),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_bool,V_a))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_bool,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_bool,t_bool)))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_bool,t_bool))))),V_uu_16),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_bool,V_a))))) => ![V_uu_13]: (![V_n, V_conn, V_v, V_i, V_b]: ?[V_v0]: ((p(s(t_bool,V_v0)) <=> s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n))) = s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_i))))),s(t_bool,V_b)))) & s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool))),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool))))),V_uu_13),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_i))),s(t_bool,V_b))) = s(t_bool,h4s_combins_i(s(t_bool,V_v0)))) => ![V_uu_12]: (![V_n, V_conn, V_i, V_v, V_j]: ?[V_v0]: ((p(s(t_bool,V_v0)) <=> s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n))) = s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_i))))),s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_j)))))) & s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))))),V_uu_12),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_h4s_nums_num,V_i))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_j))) = s(t_bool,h4s_combins_i(s(t_bool,V_v0)))) => ![V_uu_11]: (![V_v8, V_n, V_conn, V_v, V_i]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))))),V_uu_11),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v8))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_i))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v8),s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))))),V_uu_12),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_h4s_nums_num,V_i))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_fun(t_bool,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool))),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool))))),V_uu_13),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_i))))) => ![V_uu_10]: (![V_v7, V_v, V_n, V_conn, V_v8]: s(t_bool,happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)))),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))))),V_uu_10),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v7))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v8))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v7),s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))))),V_uu_11),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v8))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_fun(t_bool,t_bool),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_bool)))),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_bool))))),V_uu_14),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v8))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))))) => ![V_uu_9]: (![V_v, V_n, V_conn, V_v7]: s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))))),V_uu_9),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v7))) = s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)))),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))))),V_uu_10),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v7))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))) => ![V_uu_8]: (![V_v, V_n, V_conn, V_v6]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)))),V_uu_8),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),V_v6))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),V_v6),s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))))),V_uu_9),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))))) => ![V_uu_7]: (![V_v, V_n, V_conn]: s(t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)))),V_uu_7),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))) = s(t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)))),V_uu_8),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))) => ![V_uu_6]: (![V_v, V_n, V_v4]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_uu_6),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),V_v4))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),V_v4),s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)))),V_uu_7),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))))) => ![V_uu_5]: (![V_v, V_n]: s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_uu_5),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))) = s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_uu_6),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))) => ![V_uu_4]: (![V_v, V_v2]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool)),V_uu_4),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),V_v2))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),V_v2),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_uu_5),s(t_fun(t_h4s_nums_num,t_bool),V_v))))) => ![V_uu_3]: (![V_v]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool)),V_uu_3),s(t_fun(t_h4s_nums_num,t_bool),V_v))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool)),V_uu_4),s(t_fun(t_h4s_nums_num,t_bool),V_v))) => ![V_uu_2]: (![V_au_27]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),V_uu_2),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),V_au_27))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),V_au_27),s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool)),V_uu_3))) => ![V_uu_1]: (![V_UNIQUEu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool)),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),V_UNIQUEu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),V_uu_2) => ![V_uu_0]: (![V_R]: s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool)),V_R))) = s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool)),V_R))) => s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),h4s_defcnfs_uniqueu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool)),V_uu_1))))))))))))))))))))).
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_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sums_sumu_u_axiom, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_0]: (![V_f, V_g, V_h]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(t_fun(TV_u_27b,TV_u_27c),V_g))),s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h)))) <=> (s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_o(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl))) = s(t_fun(TV_u_27a,TV_u_27c),V_f) & s(t_fun(TV_u_27b,TV_u_27c),h4s_combins_o(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr))) = s(t_fun(TV_u_27b,TV_u_27c),V_g))) => ![V_g, V_f]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(t_fun(TV_u_27b,TV_u_27c),V_g)))))))).
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_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_sums_sumu_u_distinct, 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_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_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))))).
fof(ah4s_sums_sumu_u_INDUCT, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: ((![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))))))) => ![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)))))).
fof(ah4s_bools_DATATYPEu_u_TAGu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_bools_datatype(s(TV_u_27a,V_x))) = s(t_bool,t)).
fof(ah4s_sums_INRu_u_11, 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_INLu_u_11, 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_sums_sumu_u_CASES, axiom, ![TV_u_27a,TV_u_27b]: ![V_ss]: (?[V_x]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_ss) = 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]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_ss) = 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_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_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_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_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_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_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_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_datatypeu_u_sum, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_sum]: p(s(t_bool,h4s_bools_datatype(s(TV_u_27c,happ(s(t_fun(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_fun(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),TV_u_27c)),V_sum),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl))),s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr))))))).
fof(ah4s_predu_u_sets_SUMu_u_UNIV, axiom, ![TV_u_27a,TV_u_27b]: s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_univ) = s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))),s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_univ)))))).
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_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_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_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => 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,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
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_INRu_u_nequ_u_INL, axiom, ![TV_u_27b,TV_u_27a]: ![V_v2, V_v1]: ~ (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_v2))) = 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_v1))))).
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_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_sums_INRu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_2]: (![V_e, V_b, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27b,t_bool))),V_uu_2),s(TV_u_27b,V_e))),s(t_bool,V_b))),s(TV_u_27b,V_y)))) <=> (s(TV_u_27b,V_y) = s(TV_u_27b,V_e) & ~ (p(s(t_bool,V_b))))) => ![V_uu_1]: (![V_e, V_b, V_x]: s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_1),s(TV_u_27b,V_e))),s(t_bool,V_b))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27b,t_bool))),V_uu_2),s(TV_u_27b,V_e))),s(t_bool,V_b))) => ![V_uu_0]: (![V_e, V_b]: s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(TV_u_27b,V_e))),s(t_bool,V_b))) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_1),s(TV_u_27b,V_e))),s(t_bool,V_b))) => ![V_e]: 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_e))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_absu_u_sum(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(TV_u_27b,V_e))))))))).
fof(ah4s_sums_INLu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_2]: (![V_x, V_e, V_b, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27b,t_bool)))),V_uu_2),s(TV_u_27a,V_x))),s(TV_u_27a,V_e))),s(t_bool,V_b))),s(TV_u_27b,V_y)))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_e) & p(s(t_bool,V_b)))) => ![V_uu_1]: (![V_e, V_b, V_x]: s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_1),s(TV_u_27a,V_e))),s(t_bool,V_b))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27b,t_bool)))),V_uu_2),s(TV_u_27a,V_x))),s(TV_u_27a,V_e))),s(t_bool,V_b))) => ![V_uu_0]: (![V_e, V_b]: s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(TV_u_27a,V_e))),s(t_bool,V_b))) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_1),s(TV_u_27a,V_e))),s(t_bool,V_b))) => ![V_e]: 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_e))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_absu_u_sum(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(TV_u_27a,V_e))))))))).
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_sums_condu_u_sumu_u_expandu_c3, axiom, ![TV_u_27g,TV_u_27h]: ![V_z, V_y, V_x, V_P]: (s(t_h4s_sums_sum(TV_u_27g,TV_u_27h),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_sums_sum(TV_u_27g,TV_u_27h),happ(s(t_fun(TV_u_27g,t_h4s_sums_sum(TV_u_27g,TV_u_27h)),h4s_sums_inl),s(TV_u_27g,V_x))),s(t_h4s_sums_sum(TV_u_27g,TV_u_27h),happ(s(t_fun(TV_u_27h,t_h4s_sums_sum(TV_u_27g,TV_u_27h)),h4s_sums_inr),s(TV_u_27h,V_y))))) = s(t_h4s_sums_sum(TV_u_27g,TV_u_27h),happ(s(t_fun(TV_u_27h,t_h4s_sums_sum(TV_u_27g,TV_u_27h)),h4s_sums_inr),s(TV_u_27h,V_z))) <=> (~ (p(s(t_bool,V_P))) & s(TV_u_27h,V_z) = s(TV_u_27h,V_y)))).
fof(ah4s_sums_condu_u_sumu_u_expandu_c1, axiom, ![TV_u_27c,TV_u_27d]: ![V_z, V_y, V_x, V_P]: (s(t_h4s_sums_sum(TV_u_27d,TV_u_27c),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_sums_sum(TV_u_27d,TV_u_27c),happ(s(t_fun(TV_u_27c,t_h4s_sums_sum(TV_u_27d,TV_u_27c)),h4s_sums_inr),s(TV_u_27c,V_x))),s(t_h4s_sums_sum(TV_u_27d,TV_u_27c),happ(s(t_fun(TV_u_27d,t_h4s_sums_sum(TV_u_27d,TV_u_27c)),h4s_sums_inl),s(TV_u_27d,V_y))))) = s(t_h4s_sums_sum(TV_u_27d,TV_u_27c),happ(s(t_fun(TV_u_27d,t_h4s_sums_sum(TV_u_27d,TV_u_27c)),h4s_sums_inl),s(TV_u_27d,V_z))) <=> (~ (p(s(t_bool,V_P))) & s(TV_u_27d,V_z) = s(TV_u_27d,V_y)))).
fof(ah4s_sums_condu_u_sumu_u_expandu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_z, V_y, V_x, V_P]: (s(t_h4s_sums_sum(TV_u_27b,TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_sums_sum(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,TV_u_27a)),h4s_sums_inr),s(TV_u_27a,V_x))),s(t_h4s_sums_sum(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27b,TV_u_27a)),h4s_sums_inl),s(TV_u_27b,V_y))))) = s(t_h4s_sums_sum(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,TV_u_27a)),h4s_sums_inr),s(TV_u_27a,V_z))) <=> (p(s(t_bool,V_P)) & s(TV_u_27a,V_z) = s(TV_u_27a,V_x)))).
fof(ah4s_sums_condu_u_sumu_u_expandu_c2, axiom, ![TV_u_27f,TV_u_27e]: ![V_z, V_y, V_x, V_P]: (s(t_h4s_sums_sum(TV_u_27e,TV_u_27f),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_sums_sum(TV_u_27e,TV_u_27f),happ(s(t_fun(TV_u_27e,t_h4s_sums_sum(TV_u_27e,TV_u_27f)),h4s_sums_inl),s(TV_u_27e,V_x))),s(t_h4s_sums_sum(TV_u_27e,TV_u_27f),happ(s(t_fun(TV_u_27f,t_h4s_sums_sum(TV_u_27e,TV_u_27f)),h4s_sums_inr),s(TV_u_27f,V_y))))) = s(t_h4s_sums_sum(TV_u_27e,TV_u_27f),happ(s(t_fun(TV_u_27e,t_h4s_sums_sum(TV_u_27e,TV_u_27f)),h4s_sums_inl),s(TV_u_27e,V_z))) <=> (p(s(t_bool,V_P)) & s(TV_u_27e,V_z) = s(TV_u_27e,V_x)))).
fof(ah4s_sums_sumu_u_distinct1, 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_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))) = 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))))).
fof(ah4s_predu_u_sets_INJu_u_INR, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_s]: (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(t_h4s_sums_sum(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,TV_u_27a)),h4s_sums_inr),s(TV_u_27a,V_x))),s(t_fun(t_h4s_sums_sum(TV_u_27b,TV_u_27a),t_bool),V_t))))) => p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,TV_u_27a)),h4s_sums_inr),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_sums_sum(TV_u_27b,TV_u_27a),t_bool),V_t)))))).
fof(ah4s_predu_u_sets_INJu_u_INL, axiom, ![TV_u_27a,TV_u_27b]: ![V_t, V_s]: (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(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(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_t))))) => p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_t)))))).
fof(ah4s_sums_EXISTSu_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_sumu_u_caseu_u_cong, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_f1u_27, V_f1, V_fu_27, V_f, V_Mu_27, V_M]: ((s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_M) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_Mu_27) & (![V_x]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_Mu_27) = 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))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_fu_27),s(TV_u_27a,V_x)))) & ![V_y]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_Mu_27) = 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_f1),s(TV_u_27b,V_y))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f1u_27),s(TV_u_27b,V_y)))))) => s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_M),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,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_Mu_27),s(t_fun(TV_u_27a,TV_u_27c),V_fu_27),s(t_fun(TV_u_27b,TV_u_27c),V_f1u_27))))).
fof(ah4s_sums_SUMu_u_MAP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![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_bools_cond(s(t_bool,h4s_sums_isl(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_z))),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,h4s_sums_outl(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_z))))))),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,h4s_sums_outr(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_z)))))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sums_ISL0u_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: p(s(t_bool,h4s_sums_isl(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))))))).
fof(ah4s_sums_ISL0u_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_y]: ~ (p(s(t_bool,h4s_sums_isl(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_bools_ABSu_u_SIMP, axiom, ![TV_u_27b,TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,V_t1) = s(TV_u_27a,V_t1)).
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_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_defCNFs_UNIQUEu_u_curriedu_u_def, axiom, ![V_x2, V_x1, V_x]: s(t_bool,h4s_defcnfs_unique(s(t_fun(t_h4s_nums_num,t_bool),V_x),s(t_h4s_nums_num,V_x1),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),V_x2))) = s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),h4s_defcnfs_uniqueu_u_tupled),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_bool),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_x1),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),V_x2)))))))).
fof(ah4s_bools_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_sums_ISu_u_SUMu_u_REP0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: (p(s(t_bool,h4s_sums_isu_u_sumu_u_rep(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_f)))) <=> ?[V_v1, V_v2]: (![V_x, V_x0, V_x1]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_f),s(t_bool,V_x))),s(TV_u_27a,V_x0))),s(TV_u_27b,V_x1)))) <=> (s(TV_u_27a,V_x0) = s(TV_u_27a,V_v1) & p(s(t_bool,V_x)))) | ![V_x, V_x0, V_x1]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_f),s(t_bool,V_x))),s(TV_u_27a,V_x0))),s(TV_u_27b,V_x1)))) <=> (s(TV_u_27b,V_x1) = s(TV_u_27b,V_v2) & ~ (p(s(t_bool,V_x)))))))).
fof(ah4s_sums_sumu_u_ISOu_u_DEFu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_a]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_absu_u_sum(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),h4s_sums_repu_u_sum(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_a))))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_a)).
fof(ah4s_sums_sumu_u_ISOu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_r]: (p(s(t_bool,h4s_sums_isu_u_sumu_u_rep(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_r)))) <=> s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),h4s_sums_repu_u_sum(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_absu_u_sum(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_r))))) = s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_r))).
fof(ah4s_sums_OUTR0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(TV_u_27b,h4s_sums_outr(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(TV_u_27b,V_x)).
fof(ah4s_sums_ISR0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: p(s(t_bool,h4s_sums_isr(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))))))).
fof(ah4s_sums_ISR0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_y]: ~ (p(s(t_bool,h4s_sums_isr(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)))))))).
fof(ah4s_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, 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_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_sums_OUTL0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_sums_outl(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_27a,V_x)).
fof(ah4s_basicSizes_sumu_u_sizeu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_g, V_f]: s(t_h4s_nums_num,h4s_basicsizes_sumu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_fun(TV_u_27b,t_h4s_nums_num),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_y))))) = s(t_h4s_nums_num,happ(s(t_fun(TV_u_27b,t_h4s_nums_num),V_g),s(TV_u_27b,V_y)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
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_basicSizes_sumu_u_sizeu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: s(t_h4s_nums_num,h4s_basicsizes_sumu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_fun(TV_u_27b,t_h4s_nums_num),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_x))))) = s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_bools_EXISTSu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (?[V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_sums_INR0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: (p(s(t_bool,h4s_sums_isr(s(t_h4s_sums_sum(TV_u_27a,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,h4s_sums_outr(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_x))).
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_predu_u_sets_INJu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) <=> (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),V_t))))) & ![V_x, V_y]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s))))) => (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))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))))).
fof(ah4s_sums_INL0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: (p(s(t_bool,h4s_sums_isl(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),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,h4s_sums_outl(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),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_options_NONEu_u_DEF, axiom, ![TV_u_27a]: s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),happ(s(t_fun(t_h4s_ones_one,t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one)),h4s_sums_inr),s(t_h4s_ones_one,h4s_ones_one0)))))).
fof(ah4s_options_SOMEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one)),h4s_sums_inl),s(TV_u_27a,V_x)))))).
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_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_whiles_OLEASTu_u_INTRO, axiom, ![V_Q, V_P]: (((![V_n]: ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n))))) => p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),V_Q),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))))) & ![V_n]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) & ![V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),V_Q),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,V_n)))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),V_Q),s(t_h4s_options_option(t_h4s_nums_num),h4s_whiles_oleast(s(t_fun(t_h4s_nums_num,t_bool),V_P)))))))).
fof(ah4s_predu_u_sets_inu_u_maxu_u_set, axiom, ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => ![V_x]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_x),s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,h4s_predu_u_sets_maxu_u_set(s(t_fun(t_h4s_nums_num,t_bool),V_s))))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EXISTSu_u_UNIQUEu_u_THM, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_P, 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_P))),s(TV_u_27a,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))) => ![V_P]: (p(s(t_bool,h4s_bools_u_3fu_21(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_P)))))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_SELECTu_u_REFLu_u_2, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y)) => ![V_x]: s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x))).
fof(ah4s_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_bools_EXISTSu_u_REFL, axiom, ![TV_u_27a]: ![V_a]: ?[V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_a)).
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_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_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_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_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
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_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_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_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_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_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_predu_u_sets_INu_u_IMAGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_s, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> ?[V_x]: (s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_predu_u_sets_INu_u_UNIV, axiom, ![TV_u_27a]: ![V_x]: p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))))).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_predu_u_sets_INu_u_UNION, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((![V_x]: 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,V_Q))) <=> ![V_x]: (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,V_Q))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((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))))) <=> ![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))))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_predu_u_sets_FINITEu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) & ![V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))) => ![V_e]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))))))) => ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_INu_u_INSERT, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_predu_u_sets_MAXu_u_SETu_u_THMu_c1, axiom, ![V_s, V_e]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => s(t_h4s_nums_num,h4s_predu_u_sets_maxu_u_set(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_insert(s(t_h4s_nums_num,V_e),s(t_fun(t_h4s_nums_num,t_bool),V_s))))) = s(t_h4s_nums_num,h4s_arithmetics_max(s(t_h4s_nums_num,V_e),s(t_h4s_nums_num,h4s_predu_u_sets_maxu_u_set(s(t_fun(t_h4s_nums_num,t_bool),V_s))))))).
fof(ah4s_arithmetics_MAXu_u_LEu_c0, axiom, ![V_p, V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,h4s_arithmetics_max(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))) <=> (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_m)))) | p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_REFL, axiom, ![V_m]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_m))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_whiles_OLEASTu_u_def, axiom, ![V_uu_0]: (![V_P, V_n]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_fun(t_h4s_nums_num,t_bool),V_P))),s(t_h4s_nums_num,V_n))) = s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n))) => ![V_P]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n))))) & s(t_h4s_options_option(t_h4s_nums_num),h4s_whiles_oleast(s(t_fun(t_h4s_nums_num,t_bool),V_P))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_fun(t_h4s_nums_num,t_bool),V_P))))))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none)))))).
fof(ah4s_whiles_LEASTu_u_ELIM, axiom, ![V_Q, V_P]: ((?[V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) & ![V_n]: ((![V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_Q),s(t_h4s_nums_num,V_n)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_Q),s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),V_P)))))))).
fof(ah4s_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_LEFTu_u_ORu_u_OVERu_u_AND, 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_A)) | p(s(t_bool,V_C)))))).
fof(ch4s_defCNFs_UNIQUEu_u_ind, conjecture, ![V_P]: ((![V_v, V_n, V_conn, V_i, V_j]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_P),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),h4s_pairs_u_2c(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_pairs_u_2c(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_sums_inl),s(t_h4s_nums_num,V_i))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_sums_inl),s(t_h4s_nums_num,V_j)))))))))) & (![V_v, V_n, V_conn, V_i, V_b]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_P),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),h4s_pairs_u_2c(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_pairs_u_2c(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_sums_inl),s(t_h4s_nums_num,V_i))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),happ(s(t_fun(t_bool,t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_sums_inr),s(t_bool,V_b)))))))))) & (![V_v, V_n, V_conn, V_a, V_j]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_P),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),h4s_pairs_u_2c(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_pairs_u_2c(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),happ(s(t_fun(t_bool,t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_sums_inr),s(t_bool,V_a))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_sums_inl),s(t_h4s_nums_num,V_j)))))))))) & ![V_v, V_n, V_conn, V_a, V_b]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_P),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),h4s_pairs_u_2c(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_pairs_u_2c(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),happ(s(t_fun(t_bool,t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_sums_inr),s(t_bool,V_a))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),happ(s(t_fun(t_bool,t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_sums_inr),s(t_bool,V_b))))))))))))) => ![V_v, V_v1, V_v2, V_v3, V_v4]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_P),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_v1))),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),h4s_pairs_u_2c(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_v2),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_pairs_u_2c(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v3),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v4)))))))))).
