%   ORIGINAL: h4/set__relation/nth__min__def__compute_c1
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/set__relation/nth__min__def_c1: !s r_27 r n. h4/set__relation/nth__min r_27 (h4/pair/_2C s r) (h4/num/SUC n) = h4/bool/LET (\min. h4/bool/COND (min = h4/option/NONE) h4/option/NONE (h4/set__relation/nth__min r_27 (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) n)) (h4/set__relation/get__min r_27 (h4/pair/_2C s r))
% Assm: h4/set__relation/nth__min__tupled__primitive__def: h4/set__relation/nth__min__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!n r s r_27 min. min = h4/set__relation/get__min r_27 (h4/pair/_2C s r) /\ ~(min = h4/option/NONE) ==> R (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) n)) (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C s r) (h4/num/SUC n)))))) (\nth__min__tupled a. h4/pair/pair__CASE a (\r_27 v1. h4/pair/pair__CASE v1 (\v2 v3. h4/pair/pair__CASE v2 (\s r. h4/arithmetic/num__CASE v3 (h4/combin/I (h4/set__relation/get__min r_27 (h4/pair/_2C s r))) (\n. h4/combin/I (h4/bool/LET (\min. h4/bool/COND (min = h4/option/NONE) h4/option/NONE (nth__min__tupled (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) n)))) (h4/set__relation/get__min r_27 (h4/pair/_2C s r))))))))
% Assm: h4/set__relation/nth__min__ind: !P. (!r_27 s r. P r_27 (h4/pair/_2C s r) h4/num/0) /\ (!r_27 s r n. (!min. min = h4/set__relation/get__min r_27 (h4/pair/_2C s r) /\ ~(min = h4/option/NONE) ==> P r_27 (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) n) ==> P r_27 (h4/pair/_2C s r) (h4/num/SUC n)) ==> (!v v1 v2 v3. P v (h4/pair/_2C v1 v2) v3)
% Assm: h4/arithmetic/SUC__ELIM__NUMERALS: !g f. (!n. g (h4/num/SUC n) = f n (h4/num/SUC n)) <=> (!n. g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = f (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) /\ (!n. g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)))
% Assm: h4/set__relation/nth__min__def_c0: !s r_27 r. h4/set__relation/nth__min r_27 (h4/pair/_2C s r) h4/num/0 = h4/set__relation/get__min r_27 (h4/pair/_2C s r)
% Assm: h4/set__relation/nth__min__def__compute_c0: !s r_27 r. h4/set__relation/nth__min r_27 (h4/pair/_2C s r) h4/num/0 = h4/set__relation/get__min r_27 (h4/pair/_2C s r)
% Assm: h4/bool/TRUTH: T
% Assm: h4/option/THE__DEF: !x. h4/option/THE (h4/option/SOME x) = x
% Assm: h4/set__relation/get__min__def: !s r_27 r. h4/set__relation/get__min r_27 (h4/pair/_2C s r) = h4/bool/LET (\mins. h4/bool/COND (h4/pred__set/SING mins) (h4/option/SOME (h4/pred__set/CHOICE mins)) h4/option/NONE) (h4/set__relation/minimal__elements (h4/set__relation/minimal__elements s r) r_27)
% Assm: h4/set__relation/nth__min__curried__def: !x2 x1 x. h4/set__relation/nth__min x x1 x2 = h4/set__relation/nth__min__tupled (h4/pair/_2C x (h4/pair/_2C x1 x2))
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/relation/WF__inv__image: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/prim__rec/WF__LESS: h4/relation/WF h4/prim__rec/_3C
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/pred__set/chooser__def__compute_c1: !s n. h4/pred__set/chooser s (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = h4/pred__set/chooser (h4/pred__set/REST s) (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm: h4/numpair/tri__def__compute_c1: !n. h4/numpair/tri (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/numpair/tri (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm: h4/arithmetic/EXP__SUB__NUMERAL: !x n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/DIV (h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x))) n = h4/arithmetic/EXP n (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ h4/arithmetic/DIV (h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 x))) n = h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x))
% Assm: h4/option/OPTION__MAP2__DEF: !y x f. h4/option/OPTION__MAP2 f x y = h4/bool/COND (h4/option/IS__SOME x /\ h4/option/IS__SOME y) (h4/option/SOME (f (h4/option/THE x) (h4/option/THE y))) h4/option/NONE
% Assm: h4/option/option__case__compute: !x f e. h4/option/option__CASE x e f = h4/bool/COND (h4/option/IS__SOME x) (f (h4/option/THE x)) e
% Assm: h4/option/option__Axiom: !f e. ?fn. fn h4/option/NONE = e /\ (!x. fn (h4/option/SOME x) = f x)
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/option/option__CLAUSES_c8: !x. h4/option/IS__SOME x ==> h4/option/SOME (h4/option/THE x) = x
% Assm: h4/numpair/tri__def_c0: h4/numpair/tri h4/num/0 = h4/num/0
% Assm: h4/numpair/tri__def_c1: !n. h4/numpair/tri (h4/num/SUC n) = h4/arithmetic/_2B (h4/num/SUC n) (h4/numpair/tri n)
% Assm: h4/pred__set/chooser__def_c0: !s. h4/pred__set/chooser s h4/num/0 = h4/pred__set/CHOICE s
% Assm: h4/pred__set/chooser__def_c1: !s n. h4/pred__set/chooser s (h4/num/SUC n) = h4/pred__set/chooser (h4/pred__set/REST s) n
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% 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/arithmetic/num__case__def_c0: !v f. h4/arithmetic/num__CASE h4/num/0 v f = v
% Assm: h4/arithmetic/num__case__def_c1: !v n f. h4/arithmetic/num__CASE (h4/num/SUC n) v f = f n
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/LET__CONG: !g f N M. M = N /\ (!x. x = N ==> f x = g x) ==> h4/bool/LET f M = h4/bool/LET g N
% Assm: h4/option/IS__SOME__DEF_c1: h4/option/IS__SOME h4/option/NONE <=> F
% Assm: h4/option/IS__SOME__DEF_c0: !x. h4/option/IS__SOME (h4/option/SOME x) <=> T
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/arithmetic/ADD__CLAUSES_c1: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/ADD__CLAUSES_c3: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/BIT10: !n. h4/arithmetic/BIT1 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC h4/num/0))
% Assm: h4/arithmetic/NUMERAL__DEF: !x. h4/arithmetic/NUMERAL x = x
% Assm: h4/arithmetic/num__CASES: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Assm: h4/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/option/option__CLAUSES_c1: !x. h4/option/THE (h4/option/SOME x) = x
% Assm: h4/option/option__CLAUSES_c12: !x f e. h4/option/IS__SOME x ==> h4/option/option__CASE x e f = f (h4/option/THE x)
% Assm: h4/arithmetic/BIT20: !n. h4/arithmetic/BIT2 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC (h4/num/SUC h4/num/0)))
% Assm: h4/arithmetic/ALT__ZERO: h4/arithmetic/ZERO = h4/num/0
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/relation/WF__INDUCTION__THM: !R. h4/relation/WF R ==> (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P x))
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/arithmetic/SUB__0_c1: !m. h4/arithmetic/_2D m h4/num/0 = m
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/arithmetic/SUB__MONO__EQ: !n m. h4/arithmetic/_2D (h4/num/SUC n) (h4/num/SUC m) = h4/arithmetic/_2D n m
% 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/option/NOT__SOME__NONE: !x. ~(h4/option/SOME x = h4/option/NONE)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/bool/LET__DEF: h4/bool/LET = (\f x. f x)
% Assm: h4/numeral/TWO__EXP__THM_c1: !n. h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = h4/arithmetic/NUMERAL (h4/numeral/texp__help (h4/prim__rec/PRE (h4/arithmetic/BIT1 n)) h4/arithmetic/ZERO)
% Assm: h4/pred__set/chooser__def__compute_c2: !s n. h4/pred__set/chooser s (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = h4/pred__set/chooser (h4/pred__set/REST s) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))
% Assm: h4/arithmetic/NUMERAL__MULT__EQ__DIV_c0: !z y x. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x)) y = h4/arithmetic/NUMERAL z <=> y = h4/arithmetic/DIV (h4/arithmetic/NUMERAL z) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x)) /\ h4/arithmetic/MOD (h4/arithmetic/NUMERAL z) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x)) = h4/num/0
% Assm: h4/numeral/numeral__funpow_c1: !x n f. h4/arithmetic/FUNPOW f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) x = h4/arithmetic/FUNPOW f (h4/prim__rec/PRE (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) (f x)
% Assm: h4/numeral/numeral__distrib_c9: !n. h4/arithmetic/EXP h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = h4/num/0
% Assm: h4/arithmetic/LESS__EQ__MONO: !n m. h4/arithmetic/_3C_3D (h4/num/SUC n) (h4/num/SUC m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/EXP__1_c1: !n. h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = n
% Assm: h4/arithmetic/EXP__SUB: !q p n. h4/prim__rec/_3C h4/num/0 n /\ h4/arithmetic/_3C_3D q p ==> h4/arithmetic/EXP n (h4/arithmetic/_2D p q) = h4/arithmetic/DIV (h4/arithmetic/EXP n p) (h4/arithmetic/EXP n q)
% Assm: h4/arithmetic/ODD__EXISTS: !n. h4/arithmetic/ODD n <=> (?m. n = h4/num/SUC (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) m))
% Assm: h4/arithmetic/EVEN__EXISTS: !n. h4/arithmetic/EVEN n <=> (?m. n = h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) m)
% Assm: h4/arithmetic/EVEN__OR__ODD: !n. h4/arithmetic/EVEN n \/ h4/arithmetic/ODD n
% Assm: h4/arithmetic/TIMES2: !n. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n = h4/arithmetic/_2B n n
% Assm: h4/prim__rec/PRE0_c1: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm: h4/arithmetic/MULT__EQ__DIV: !z y x. h4/prim__rec/_3C h4/num/0 x ==> (h4/arithmetic/_2A x y = z <=> y = h4/arithmetic/DIV z x /\ h4/arithmetic/MOD z x = h4/num/0)
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/numeral/texp__help0: !n. h4/numeral/texp__help n h4/num/0 = h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) (h4/arithmetic/_2B n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm: h4/arithmetic/ADD__CLAUSES_c2: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/FUNPOW0_c1: !x n f. h4/arithmetic/FUNPOW f (h4/num/SUC n) x = h4/arithmetic/FUNPOW f n (f x)
% Assm: h4/arithmetic/FUNPOW0_c0: !x f. h4/arithmetic/FUNPOW f h4/num/0 x = x
% Assm: h4/arithmetic/MULT__CLAUSES_c1: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm: h4/arithmetic/SUB__0_c0: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% Assm: h4/arithmetic/MULT__CLAUSES_c0: !m. h4/arithmetic/_2A h4/num/0 m = h4/num/0
% Assm: h4/arithmetic/LESS__0__CASES: !m. h4/num/0 = m \/ h4/prim__rec/_3C h4/num/0 m
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/arithmetic/ODD0_c0: h4/arithmetic/ODD h4/num/0 <=> F
% Assm: h4/arithmetic/GREATER__DEF: !n m. h4/arithmetic/_3E m n <=> h4/prim__rec/_3C n m
% Assm: h4/arithmetic/LESS__OR__EQ: !n m. h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C m n \/ m = n
% Assm: h4/arithmetic/EXP0_c1: !n m. h4/arithmetic/EXP m (h4/num/SUC n) = h4/arithmetic/_2A m (h4/arithmetic/EXP m n)
% Assm: h4/arithmetic/EVEN0_c0: h4/arithmetic/EVEN h4/num/0 <=> T
% Assm: h4/arithmetic/GREATER__OR__EQ: !n m. h4/arithmetic/_3E_3D m n <=> h4/arithmetic/_3E m n \/ m = n
% Assm: h4/arithmetic/EXP0_c0: !m. h4/arithmetic/EXP m h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Goal: !s r_27 r n. h4/set__relation/nth__min r_27 (h4/pair/_2C s r) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = h4/bool/LET (\min. h4/bool/COND (min = h4/option/NONE) h4/option/NONE (h4/set__relation/nth__min r_27 (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))) (h4/set__relation/get__min r_27 (h4/pair/_2C s r))
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_setu_u_relations_nthu_u_minu_u_defu_c1]: !_0. (!r_27 s r n min. ?v. (v <=> min = h4/option/NONE) /\ happ (happ (happ (happ (happ _0 r_27) s) r) n) min = h4/bool/COND v h4/option/NONE (h4/set__relation/nth__min r_27 (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) n)) ==> (!s r_27 r n. h4/set__relation/nth__min r_27 (h4/pair/_2C s r) (h4/num/SUC n) = h4/bool/LET (happ (happ (happ (happ _0 r_27) s) r) n) (h4/set__relation/get__min r_27 (h4/pair/_2C s r)))
% Assm [h4s_setu_u_relations_nthu_u_minu_u_tupledu_u_primitiveu_u_def]: !_10. (!nth__min__tupled r_27 s r n min. ?v. (v <=> min = h4/option/NONE) /\ happ (happ (happ (happ (happ (happ _10 nth__min__tupled) r_27) s) r) n) min = h4/bool/COND v h4/option/NONE (happ nth__min__tupled (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) n)))) ==> (!_9. (!nth__min__tupled r_27 s r n. happ (happ (happ (happ (happ _9 nth__min__tupled) r_27) s) r) n = h4/combin/I (h4/bool/LET (happ (happ (happ (happ (happ _10 nth__min__tupled) r_27) s) r) n) (h4/set__relation/get__min r_27 (h4/pair/_2C s r)))) ==> (!_8. (!v3 nth__min__tupled r_27 s r. happ (happ (happ (happ (happ _8 v3) nth__min__tupled) r_27) s) r = h4/arithmetic/num__CASE v3 (h4/combin/I (h4/set__relation/get__min r_27 (h4/pair/_2C s r))) (happ (happ (happ (happ _9 nth__min__tupled) r_27) s) r)) ==> (!_7. (!v3 nth__min__tupled r_27 s. happ (happ (happ (happ _7 v3) nth__min__tupled) r_27) s = happ (happ (happ (happ _8 v3) nth__min__tupled) r_27) s) ==> (!_6. (!v2 nth__min__tupled r_27 v3. happ (happ (happ (happ _6 v2) nth__min__tupled) r_27) v3 = h4/pair/pair__CASE v2 (happ (happ (happ _7 v3) nth__min__tupled) r_27)) ==> (!_5. (!nth__min__tupled r_27 v2. happ (happ (happ _5 nth__min__tupled) r_27) v2 = happ (happ (happ _6 v2) nth__min__tupled) r_27) ==> (!_4. (!nth__min__tupled r_27 v1. happ (happ (happ _4 nth__min__tupled) r_27) v1 = h4/pair/pair__CASE v1 (happ (happ _5 nth__min__tupled) r_27)) ==> (!_3. (!nth__min__tupled r_27. happ (happ _3 nth__min__tupled) r_27 = happ (happ _4 nth__min__tupled) r_27) ==> (!_2. (!nth__min__tupled a. happ (happ _2 nth__min__tupled) a = h4/pair/pair__CASE a (happ _3 nth__min__tupled)) ==> (!_1. (!nth__min__tupled. happ _1 nth__min__tupled = happ _2 nth__min__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!n r s r_27 min. min = h4/set__relation/get__min r_27 (h4/pair/_2C s r) /\ ~(min = h4/option/NONE) ==> happ (happ R (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) n))) (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C s r) (h4/num/SUC n))))) ==> h4/set__relation/nth__min__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))))
% Assm [h4s_setu_u_relations_nthu_u_minu_u_ind]: !P. (!r_27 s r. happ (happ (happ P r_27) (h4/pair/_2C s r)) h4/num/0) /\ (!r_27 s r n. (!min. min = h4/set__relation/get__min r_27 (h4/pair/_2C s r) /\ ~(min = h4/option/NONE) ==> happ (happ (happ P r_27) (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r)) n) ==> happ (happ (happ P r_27) (h4/pair/_2C s r)) (h4/num/SUC n)) ==> (!v v1 v2 v3. happ (happ (happ P v) (h4/pair/_2C v1 v2)) v3)
% Assm [h4s_arithmetics_SUCu_u_ELIMu_u_NUMERALS]: !g f. (!n. happ g (h4/num/SUC n) = happ (happ f n) (h4/num/SUC n)) <=> (!n. happ g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = happ (happ f (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) /\ (!n. happ g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = happ (happ f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)))
% Assm [h4s_setu_u_relations_nthu_u_minu_u_defu_c0]: !s r_27 r. h4/set__relation/nth__min r_27 (h4/pair/_2C s r) h4/num/0 = h4/set__relation/get__min r_27 (h4/pair/_2C s r)
% Assm [h4s_setu_u_relations_nthu_u_minu_u_defu_u_computeu_c0]: !s r_27 r. h4/set__relation/nth__min r_27 (h4/pair/_2C s r) h4/num/0 = h4/set__relation/get__min r_27 (h4/pair/_2C s r)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_options_THEu_u_DEF]: !x. h4/option/THE (h4/option/SOME x) = x
% Assm [h4s_setu_u_relations_getu_u_minu_u_def]: !_0. (!mins. happ _0 mins = h4/bool/COND (h4/pred__set/SING mins) (h4/option/SOME (h4/pred__set/CHOICE mins)) h4/option/NONE) ==> (!s r_27 r. h4/set__relation/get__min r_27 (h4/pair/_2C s r) = h4/bool/LET _0 (h4/set__relation/minimal__elements (h4/set__relation/minimal__elements s r) r_27))
% Assm [h4s_setu_u_relations_nthu_u_minu_u_curriedu_u_def]: !x2 x1 x. h4/set__relation/nth__min x x1 x2 = happ h4/set__relation/nth__min__tupled (h4/pair/_2C x (h4/pair/_2C x1 x2))
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_relations_WFu_u_invu_u_image]: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_primu_u_recs_WFu_u_LESS]: h4/relation/WF h4/prim__rec/_3C
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (h4/relation/inv__image R f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% Assm [h4s_predu_u_sets_chooseru_u_defu_u_computeu_c1]: !s n. h4/pred__set/chooser s (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = h4/pred__set/chooser (h4/pred__set/REST s) (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm [h4s_numpairs_triu_u_defu_u_computeu_c1]: !n. h4/numpair/tri (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/numpair/tri (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm [h4s_arithmetics_EXPu_u_SUBu_u_NUMERAL]: !x n. happ (happ h4/prim__rec/_3C h4/num/0) n ==> h4/arithmetic/DIV (h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x))) n = h4/arithmetic/EXP n (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ h4/arithmetic/DIV (h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 x))) n = h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x))
% Assm [h4s_options_OPTIONu_u_MAP2u_u_DEF]: !y x f. ?v. (v <=> h4/option/IS__SOME x /\ h4/option/IS__SOME y) /\ h4/option/OPTION__MAP2 f x y = h4/bool/COND v (h4/option/SOME (happ (happ f (h4/option/THE x)) (h4/option/THE y))) h4/option/NONE
% Assm [h4s_options_optionu_u_caseu_u_compute]: !x f e. h4/option/option__CASE x e f = h4/bool/COND (h4/option/IS__SOME x) (happ f (h4/option/THE x)) e
% Assm [h4s_options_optionu_u_Axiom]: !f e. ?fn. happ fn h4/option/NONE = e /\ (!x. happ fn (h4/option/SOME x) = happ f x)
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_options_optionu_u_CLAUSESu_c8]: !x. h4/option/IS__SOME x ==> h4/option/SOME (h4/option/THE x) = x
% Assm [h4s_numpairs_triu_u_defu_c0]: h4/numpair/tri h4/num/0 = h4/num/0
% Assm [h4s_numpairs_triu_u_defu_c1]: !n. h4/numpair/tri (h4/num/SUC n) = h4/arithmetic/_2B (h4/num/SUC n) (h4/numpair/tri n)
% Assm [h4s_predu_u_sets_chooseru_u_defu_c0]: !s. h4/pred__set/chooser s h4/num/0 = h4/pred__set/CHOICE s
% Assm [h4s_predu_u_sets_chooseru_u_defu_c1]: !s n. h4/pred__set/chooser s (h4/num/SUC n) = h4/pred__set/chooser (h4/pred__set/REST s) n
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% 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_arithmetics_numu_u_caseu_u_defu_c0]: !v f. h4/arithmetic/num__CASE h4/num/0 v f = v
% Assm [h4s_arithmetics_numu_u_caseu_u_defu_c1]: !v n f. h4/arithmetic/num__CASE (h4/num/SUC n) v f = happ f n
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_relations_RESTRICTu_u_LEMMA]: !z y f R. happ (happ R y) z ==> happ (h4/relation/RESTRICT f R z) y = happ f y
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_bools_LETu_u_CONG]: !g f N M. M = N /\ (!x. x = N ==> happ f x = happ g x) ==> h4/bool/LET f M = h4/bool/LET g N
% Assm [h4s_options_ISu_u_SOMEu_u_DEFu_c1]: h4/option/IS__SOME h4/option/NONE <=> F
% Assm [h4s_options_ISu_u_SOMEu_u_DEFu_c0]: !x. h4/option/IS__SOME (h4/option/SOME x) <=> T
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c3]: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_arithmetics_BIT10]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC h4/num/0))
% Assm [h4s_arithmetics_NUMERALu_u_DEF]: !x. h4/arithmetic/NUMERAL x = x
% Assm [h4s_arithmetics_numu_u_CASES]: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm [h4s_options_optionu_u_caseu_u_defu_c1]: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = happ f x
% Assm [h4s_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_options_optionu_u_CLAUSESu_c1]: !x. h4/option/THE (h4/option/SOME x) = x
% Assm [h4s_options_optionu_u_CLAUSESu_c12]: !x f e. h4/option/IS__SOME x ==> h4/option/option__CASE x e f = happ f (h4/option/THE x)
% Assm [h4s_arithmetics_BIT20]: !n. h4/arithmetic/BIT2 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC (h4/num/SUC h4/num/0)))
% Assm [h4s_arithmetics_ALTu_u_ZERO]: h4/arithmetic/ZERO = h4/num/0
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_relations_WFu_u_INDUCTIONu_u_THM]: !R. h4/relation/WF R ==> (!P. (!x. (!y. happ (happ R y) x ==> happ P y) ==> happ P x) ==> (!x. happ P x))
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_arithmetics_SUBu_u_0u_c1]: !m. h4/arithmetic/_2D m h4/num/0 = m
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_arithmetics_SUBu_u_MONOu_u_EQ]: !n m. h4/arithmetic/_2D (h4/num/SUC n) (h4/num/SUC m) = h4/arithmetic/_2D n m
% 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_options_NOTu_u_SOMEu_u_NONE]: !x. ~(h4/option/SOME x = h4/option/NONE)
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_bools_LETu_u_DEF]: !x x. h4/bool/LET x x = happ x x
% Assm [h4s_numerals_TWOu_u_EXPu_u_THMu_c1]: !n. h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = h4/arithmetic/NUMERAL (h4/numeral/texp__help (h4/prim__rec/PRE (h4/arithmetic/BIT1 n)) h4/arithmetic/ZERO)
% Assm [h4s_predu_u_sets_chooseru_u_defu_u_computeu_c2]: !s n. h4/pred__set/chooser s (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = h4/pred__set/chooser (h4/pred__set/REST s) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))
% Assm [h4s_arithmetics_NUMERALu_u_MULTu_u_EQu_u_DIVu_c0]: !z y x. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x)) y = h4/arithmetic/NUMERAL z <=> y = h4/arithmetic/DIV (h4/arithmetic/NUMERAL z) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x)) /\ h4/arithmetic/MOD (h4/arithmetic/NUMERAL z) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x)) = h4/num/0
% Assm [h4s_numerals_numeralu_u_funpowu_c1]: !x n f. h4/arithmetic/FUNPOW f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) x = h4/arithmetic/FUNPOW f (h4/prim__rec/PRE (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) (happ f x)
% Assm [h4s_numerals_numeralu_u_distribu_c9]: !n. h4/arithmetic/EXP h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = h4/num/0
% Assm [h4s_arithmetics_LESSu_u_EQu_u_MONO]: !n m. h4/arithmetic/_3C_3D (h4/num/SUC n) (h4/num/SUC m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_EXPu_u_1u_c1]: !n. h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = n
% Assm [h4s_arithmetics_EXPu_u_SUB]: !q p n. happ (happ h4/prim__rec/_3C h4/num/0) n /\ h4/arithmetic/_3C_3D q p ==> h4/arithmetic/EXP n (h4/arithmetic/_2D p q) = h4/arithmetic/DIV (h4/arithmetic/EXP n p) (h4/arithmetic/EXP n q)
% Assm [h4s_arithmetics_ODDu_u_EXISTS]: !n. h4/arithmetic/ODD n <=> (?m. n = h4/num/SUC (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) m))
% Assm [h4s_arithmetics_EVENu_u_EXISTS]: !n. h4/arithmetic/EVEN n <=> (?m. n = h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) m)
% Assm [h4s_arithmetics_EVENu_u_ORu_u_ODD]: !n. h4/arithmetic/EVEN n \/ h4/arithmetic/ODD n
% Assm [h4s_arithmetics_TIMES2]: !n. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n = h4/arithmetic/_2B n n
% Assm [h4s_primu_u_recs_PRE0u_c1]: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm [h4s_arithmetics_MULTu_u_EQu_u_DIV]: !z y x. happ (happ h4/prim__rec/_3C h4/num/0) x ==> (h4/arithmetic/_2A x y = z <=> y = h4/arithmetic/DIV z x /\ h4/arithmetic/MOD z x = h4/num/0)
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. happ (happ h4/prim__rec/_3C h4/num/0) (h4/num/SUC n)
% Assm [h4s_numerals_texpu_u_help0]: !n. h4/numeral/texp__help n h4/num/0 = h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) (h4/arithmetic/_2B n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c2]: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_arithmetics_FUNPOW0u_c1]: !x n f. h4/arithmetic/FUNPOW f (h4/num/SUC n) x = h4/arithmetic/FUNPOW f n (happ f x)
% Assm [h4s_arithmetics_FUNPOW0u_c0]: !x f. h4/arithmetic/FUNPOW f h4/num/0 x = x
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm [h4s_arithmetics_SUBu_u_0u_c0]: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2A h4/num/0 m = h4/num/0
% Assm [h4s_arithmetics_LESSu_u_0u_u_CASES]: !m. h4/num/0 = m \/ happ (happ h4/prim__rec/_3C h4/num/0) m
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_arithmetics_ODD0u_c0]: h4/arithmetic/ODD h4/num/0 <=> F
% Assm [h4s_arithmetics_GREATERu_u_DEF]: !n m. h4/arithmetic/_3E m n <=> happ (happ h4/prim__rec/_3C n) m
% Assm [h4s_arithmetics_LESSu_u_ORu_u_EQ]: !n m. h4/arithmetic/_3C_3D m n <=> happ (happ h4/prim__rec/_3C m) n \/ m = n
% Assm [h4s_arithmetics_EXP0u_c1]: !n m. h4/arithmetic/EXP m (h4/num/SUC n) = h4/arithmetic/_2A m (h4/arithmetic/EXP m n)
% Assm [h4s_arithmetics_EVEN0u_c0]: h4/arithmetic/EVEN h4/num/0 <=> T
% Assm [h4s_arithmetics_GREATERu_u_ORu_u_EQ]: !n m. h4/arithmetic/_3E_3D m n <=> h4/arithmetic/_3E m n \/ m = n
% Assm [h4s_arithmetics_EXP0u_c0]: !m. h4/arithmetic/EXP m h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Goal: !_0. (!r_27 s r n min. ?v. (v <=> min = h4/option/NONE) /\ happ (happ (happ (happ (happ _0 r_27) s) r) n) min = h4/bool/COND v h4/option/NONE (h4/set__relation/nth__min r_27 (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))) ==> (!s r_27 r n. h4/set__relation/nth__min r_27 (h4/pair/_2C s r) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = h4/bool/LET (happ (happ (happ (happ _0 r_27) s) r) n) (h4/set__relation/get__min r_27 (h4/pair/_2C s r)))
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_Q1225542,TV_Q1225538]: ![V_f, V_g]: (![V_x]: s(TV_Q1225538,happ(s(t_fun(TV_Q1225542,TV_Q1225538),V_f),s(TV_Q1225542,V_x))) = s(TV_Q1225538,happ(s(t_fun(TV_Q1225542,TV_Q1225538),V_g),s(TV_Q1225542,V_x))) => s(t_fun(TV_Q1225542,TV_Q1225538),V_f) = s(t_fun(TV_Q1225542,TV_Q1225538),V_g))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_setu_u_relations_nthu_u_minu_u_defu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_ru_27, V_s, V_r, V_n, V_min]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_options_option(TV_u_27a),V_min) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)) & s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))),s(t_h4s_options_option(TV_u_27a),V_min))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_nthu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_min))))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n)))))) => ![V_s, V_ru_27, V_r, V_n]: s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_nthu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_let(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))),s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))))).
fof(ah4s_setu_u_relations_nthu_u_minu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a]: ![V_uu_10]: (![V_nthu_u_minu_u_tupled, V_ru_27, V_s, V_r, V_n, V_min]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_options_option(TV_u_27a),V_min) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)) & s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))))),V_uu_10),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))),s(t_h4s_options_option(TV_u_27a),V_min))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled),s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),h4s_pairs_u_2c(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_min))))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n)))))))))) => ![V_uu_9]: (![V_nthu_u_minu_u_tupled, V_ru_27, V_s, V_r, V_n]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))))),V_uu_9),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))) = s(t_h4s_options_option(TV_u_27a),h4s_combins_i(s(t_h4s_options_option(TV_u_27a),h4s_bools_let(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))))),V_uu_10),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))),s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))))) => ![V_uu_8]: (![V_v3, V_nthu_u_minu_u_tupled, V_ru_27, V_s, V_r]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))))),V_uu_8),s(t_h4s_nums_num,V_v3))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))) = s(t_h4s_options_option(TV_u_27a),h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,V_v3),s(t_h4s_options_option(TV_u_27a),h4s_combins_i(s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))),s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))))),V_uu_9),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))) => ![V_uu_7]: (![V_v3, V_nthu_u_minu_u_tupled, V_ru_27, V_s]: s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))))),V_uu_7),s(t_h4s_nums_num,V_v3))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))))),V_uu_8),s(t_h4s_nums_num,V_v3))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))) => ![V_uu_6]: (![V_v2, V_nthu_u_minu_u_tupled, V_ru_27, V_v3]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),V_uu_6),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),V_v2))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_h4s_nums_num,V_v3))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),V_v2),s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))))),V_uu_7),s(t_h4s_nums_num,V_v3))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))))) => ![V_uu_5]: (![V_nthu_u_minu_u_tupled, V_ru_27, V_v2]: s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),V_v2))) = s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),V_uu_6),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),V_v2))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))) => ![V_uu_4]: (![V_nthu_u_minu_u_tupled, V_ru_27, V_v1]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),V_v1))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),V_v1),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))))) => ![V_uu_3]: (![V_nthu_u_minu_u_tupled, V_ru_27]: s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))) = s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))) => ![V_uu_2]: (![V_nthu_u_minu_u_tupled, V_a]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),V_a))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),V_a),s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))))) => ![V_uu_1]: (![V_nthu_u_minu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),V_R)))) & ![V_n, V_r, V_s, V_ru_27, V_min]: ((s(t_h4s_options_option(TV_u_27a),V_min) = s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))) & ~ (s(t_h4s_options_option(TV_u_27a),V_min) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),V_R),s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),h4s_pairs_u_2c(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_min))))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))))))),s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),h4s_pairs_u_2c(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))))))))) => s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),h4s_setu_u_relations_nthu_u_minu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a))),V_uu_1))))))))))))))).
fof(ah4s_setu_u_relations_nthu_u_minu_u_ind, axiom, ![TV_u_27a]: ![V_P]: ((![V_ru_27, V_s, V_r]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_bool))),V_P),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_ru_27, V_s, V_r, V_n]: (![V_min]: ((s(t_h4s_options_option(TV_u_27a),V_min) = s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))) & ~ (s(t_h4s_options_option(TV_u_27a),V_min) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_bool))),V_P),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_min))))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))),s(t_h4s_nums_num,V_n))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_bool))),V_P),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) => ![V_v, V_v1, V_v2, V_v3]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_bool))),V_P),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_v))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_v1),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_v2))))),s(t_h4s_nums_num,V_v3)))))).
fof(ah4s_arithmetics_SUCu_u_ELIMu_u_NUMERALS, axiom, ![TV_u_27a]: ![V_g, V_f]: (![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) <=> (![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n)))))))))).
fof(ah4s_setu_u_relations_nthu_u_minu_u_defu_c0, axiom, ![TV_u_27a]: ![V_s, V_ru_27, V_r]: s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_nthu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))).
fof(ah4s_setu_u_relations_nthu_u_minu_u_defu_u_computeu_c0, axiom, ![TV_u_27a]: ![V_s, V_ru_27, V_r]: s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_nthu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_options_THEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_setu_u_relations_getu_u_minu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_mins]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_options_option(TV_u_27a)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_mins))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,h4s_predu_u_sets_sing(s(t_fun(TV_u_27a,t_bool),V_mins))),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_mins))))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) => ![V_s, V_ru_27, V_r]: s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_let(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_options_option(TV_u_27a)),V_uu_0),s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_minimalu_u_elements(s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_minimalu_u_elements(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))))))).
fof(ah4s_setu_u_relations_nthu_u_minu_u_curriedu_u_def, axiom, ![TV_u_27a]: ![V_x2, V_x1, V_x]: s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_nthu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_x),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),V_x1),s(t_h4s_nums_num,V_x2))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),h4s_setu_u_relations_nthu_u_minu_u_tupled),s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),h4s_pairs_u_2c(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_x),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),V_x1),s(t_h4s_nums_num,V_x2)))))))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_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_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_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_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_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_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_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,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,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_bools_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_relations_WFu_u_invu_u_image, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R)))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f)))))))).
fof(ah4s_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, 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_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_primu_u_recs_WFu_u_LESS, axiom, p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c))))).
fof(ah4s_bools_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_relations_invu_u_imageu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_x, V_xi_]: 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)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),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_xi_)))))).
fof(ah4s_predu_u_sets_chooseru_u_defu_u_computeu_c1, axiom, ![TV_u_27a]: ![V_s, V_n]: s(TV_u_27a,h4s_predu_u_sets_chooser(s(t_fun(TV_u_27a,t_bool),V_s),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) = s(TV_u_27a,h4s_predu_u_sets_chooser(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))).
fof(ah4s_numpairs_triu_u_defu_u_computeu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numpairs_tri(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_numpairs_tri(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))))).
fof(ah4s_arithmetics_EXPu_u_SUBu_u_NUMERAL, axiom, ![V_x, V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,V_n)))) => (s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_x))))))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_x))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))) & s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_x))))))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_x)))))))))).
fof(ah4s_options_OPTIONu_u_MAP2u_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_y, V_x, V_f]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27b),V_x)))) & p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27c),V_y)))))) & s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_map2(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(t_h4s_options_option(TV_u_27b),V_x),s(t_h4s_options_option(TV_u_27c),V_y))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,h4s_options_the(s(t_h4s_options_option(TV_u_27b),V_x))))),s(TV_u_27c,h4s_options_the(s(t_h4s_options_option(TV_u_27c),V_y))))))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))))).
fof(ah4s_options_optionu_u_caseu_u_compute, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f, V_e]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(TV_u_27b,V_e),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(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,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_x))))),s(TV_u_27b,V_e)))).
fof(ah4s_options_optionu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27b,happ(s(t_fun(t_h4s_options_option(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(TV_u_27b,V_e) & ![V_x]: s(TV_u_27b,happ(s(t_fun(t_h4s_options_option(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_options_option(TV_u_27a),h4s_options_some(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_x))))).
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_options_optionu_u_CLAUSESu_c8, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),V_x)))) => s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_x))))) = s(t_h4s_options_option(TV_u_27a),V_x))).
fof(ah4s_numpairs_triu_u_defu_c0, axiom, s(t_h4s_nums_num,h4s_numpairs_tri(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_numpairs_triu_u_defu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numpairs_tri(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_numpairs_tri(s(t_h4s_nums_num,V_n)))))).
fof(ah4s_predu_u_sets_chooseru_u_defu_c0, axiom, ![TV_u_27a]: ![V_s]: s(TV_u_27a,h4s_predu_u_sets_chooser(s(t_fun(TV_u_27a,t_bool),V_s),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s)))).
fof(ah4s_predu_u_sets_chooseru_u_defu_c1, axiom, ![TV_u_27a]: ![V_s, V_n]: s(TV_u_27a,h4s_predu_u_sets_chooser(s(t_fun(TV_u_27a,t_bool),V_s),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,h4s_predu_u_sets_chooser(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_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_arithmetics_numu_u_caseu_u_defu_c0, axiom, ![TV_u_27a]: ![V_v, V_f]: s(TV_u_27a,h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_v),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))) = s(TV_u_27a,V_v)).
fof(ah4s_arithmetics_numu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v, V_n, V_f]: s(TV_u_27a,h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_v),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_relations_RESTRICTu_u_LEMMA, axiom, ![TV_u_27b,TV_u_27a]: ![V_z, V_y, V_f, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_z))),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_LETu_u_CONG, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f, V_N, V_M]: ((s(TV_u_27a,V_M) = s(TV_u_27a,V_N) & ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_N) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))) => s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_M))) = s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_N))))).
fof(ah4s_options_ISu_u_SOMEu_u_DEFu_c1, axiom, ![TV_u_27a]: s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_bool,f)).
fof(ah4s_options_ISu_u_SOMEu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_bool,t)).
fof(ah4s_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c3, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_BIT10, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))))))).
fof(ah4s_arithmetics_NUMERALu_u_DEF, axiom, ![V_x]: s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,V_x)).
fof(ah4s_arithmetics_numu_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | ?[V_n]: s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))).
fof(ah4s_options_optionu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_options_optionu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_options_optionu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_options_optionu_u_CLAUSESu_c12, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f, V_e]: (p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),V_x)))) => s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(TV_u_27b,V_e),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_x))))))).
fof(ah4s_arithmetics_BIT20, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))))))))).
fof(ah4s_arithmetics_ALTu_u_ZERO, axiom, s(t_h4s_nums_num,h4s_arithmetics_zero) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_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_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_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_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_relations_WFu_u_INDUCTIONu_u_THM, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_P]: (![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_R),s(TV_u_27a,V_y))),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))))) => 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_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_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_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_arithmetics_SUBu_u_0u_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
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_arithmetics_SUBu_u_MONOu_u_EQ, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
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_options_NOTu_u_SOMEu_u_NONE, 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_none))).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, 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_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_bools_LETu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0)))).
fof(ah4s_numerals_TWOu_u_EXPu_u_THMu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_numerals_texpu_u_help(s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_arithmetics_zero)))))).
fof(ah4s_predu_u_sets_chooseru_u_defu_u_computeu_c2, axiom, ![TV_u_27a]: ![V_s, V_n]: s(TV_u_27a,h4s_predu_u_sets_chooser(s(t_fun(TV_u_27a,t_bool),V_s),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))))) = s(TV_u_27a,h4s_predu_u_sets_chooser(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_arithmetics_NUMERALu_u_MULTu_u_EQu_u_DIVu_c0, axiom, ![V_z, V_y, V_x]: (s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_x))))),s(t_h4s_nums_num,V_y))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_z))) <=> (s(t_h4s_nums_num,V_y) = s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_z))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_x))))))) & s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_z))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_x))))))) = s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_numerals_numeralu_u_funpowu_c1, axiom, ![TV_u_27a]: ![V_x, V_n, V_f]: s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(TV_u_27a,V_x))) = s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x)))))).
fof(ah4s_numerals_numeralu_u_distribu_c9, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_LESSu_u_EQu_u_MONO, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_EXPu_u_1u_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_arithmetics_EXPu_u_SUB, axiom, ![V_q, V_p, V_n]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,V_p))))) => s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_q))))) = s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_q))))))).
fof(ah4s_arithmetics_ODDu_u_EXISTS, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n)))) <=> ?[V_m]: s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))))))).
fof(ah4s_arithmetics_EVENu_u_EXISTS, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n)))) <=> ?[V_m]: s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))))).
fof(ah4s_arithmetics_EVENu_u_ORu_u_ODD, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n)))) | p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_TIMES2, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_n)))).
fof(ah4s_primu_u_recs_PRE0u_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_MULTu_u_EQu_u_DIV, axiom, ![V_z, V_y, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,V_x)))) => (s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_y))) = s(t_h4s_nums_num,V_z) <=> (s(t_h4s_nums_num,V_y) = s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_z),s(t_h4s_nums_num,V_x))) & s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_z),s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,h4s_nums_0))))).
fof(ah4s_primu_u_recs_LESSu_u_0, axiom, ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_numerals_texpu_u_help0, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numerals_texpu_u_help(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_FUNPOW0u_c1, axiom, ![TV_u_27a]: ![V_x, V_n, V_f]: s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x))) = s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_n),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x)))))).
fof(ah4s_arithmetics_FUNPOW0u_c0, axiom, ![TV_u_27a]: ![V_x, V_f]: s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_SUBu_u_0u_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_LESSu_u_0u_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,h4s_nums_0) = s(t_h4s_nums_num,V_m) | p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,V_m)))))).
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_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_arithmetics_ODD0u_c0, axiom, s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,f)).
fof(ah4s_arithmetics_GREATERu_u_DEF, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3e(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_LESSu_u_ORu_u_EQ, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(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),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))) | s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_EXP0u_c1, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_EVEN0u_c0, axiom, s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,t)).
fof(ah4s_arithmetics_GREATERu_u_ORu_u_EQ, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3eu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) <=> (p(s(t_bool,h4s_arithmetics_u_3e(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) | s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_EXP0u_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))).
fof(ch4s_setu_u_relations_nthu_u_minu_u_defu_u_computeu_c1, conjecture, ![TV_u_27a]: ![V_uu_0]: (![V_ru_27, V_s, V_r, V_n, V_min]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_options_option(TV_u_27a),V_min) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)) & s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))),s(t_h4s_options_option(TV_u_27a),V_min))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_nthu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_min))))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))))) => ![V_s, V_ru_27, V_r, V_n]: s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_nthu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_let(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))),s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))))).
