%   ORIGINAL: h4/prelim/list__compare__def_c0
% 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/prelim/list__compare__curried__def: !x2 x1 x. h4/prelim/list__compare x x1 x2 = h4/prelim/list__compare__tupled (h4/pair/_2C x (h4/pair/_2C x1 x2))
% Assm: h4/prelim/EQUAL__def: h4/prelim/EQUAL = h4/prelim/num2ordering (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/prelim/list__compare__tupled__primitive__def: h4/prelim/list__compare__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!y x l2 l1 cmp. R (h4/pair/_2C cmp (h4/pair/_2C l1 l2)) (h4/pair/_2C cmp (h4/pair/_2C (h4/list/CONS x l1) (h4/list/CONS y l2)))))) (\list__compare__tupled a. h4/pair/pair__CASE a (\cmp v1. h4/pair/pair__CASE v1 (\v2 v3. h4/list/list__CASE v2 (h4/list/list__CASE v3 (h4/combin/I h4/prelim/EQUAL) (\v10 v11. h4/combin/I h4/prelim/LESS)) (\x l1. h4/list/list__CASE v3 (h4/combin/I h4/prelim/GREATER) (\y l2. h4/combin/I (h4/prelim/compare (cmp x y) h4/prelim/LESS (list__compare__tupled (h4/pair/_2C cmp (h4/pair/_2C l1 l2))) h4/prelim/GREATER))))))
% Assm: h4/prelim/ordering2num__thm_c1: h4/prelim/ordering2num h4/prelim/EQUAL = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/prelim/list__compare__ind: !P. (!cmp. P cmp h4/list/NIL h4/list/NIL) /\ (!cmp v8 v9. P cmp h4/list/NIL (h4/list/CONS v8 v9)) /\ (!cmp v4 v5. P cmp (h4/list/CONS v4 v5) h4/list/NIL) /\ (!cmp x l1 y l2. P cmp l1 l2 ==> P cmp (h4/list/CONS x l1) (h4/list/CONS y l2)) ==> (!v v1 v2. P v v1 v2)
% Assm: h4/prelim/ordering__EQ__ordering: !a_27 a. a = a_27 <=> h4/prelim/ordering2num a = h4/prelim/ordering2num a_27
% Assm: h4/prelim/ordering2num__thm_c0: h4/prelim/ordering2num h4/prelim/LESS = h4/num/0
% Assm: h4/prelim/ordering2num__thm_c2: h4/prelim/ordering2num h4/prelim/GREATER = h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/numeral/numeral__eq_c1: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__distrib_c19: !n m. h4/arithmetic/NUMERAL n = h4/arithmetic/NUMERAL m <=> n = m
% Assm: h4/numeral/numeral__eq_c4: !n m. h4/arithmetic/BIT1 n = h4/arithmetic/BIT2 m <=> F
% Assm: h4/numeral/numeral__distrib_c18: !n. h4/num/0 = h4/arithmetic/NUMERAL n <=> n = h4/arithmetic/ZERO
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/prelim/ordering__case__def_c1: !v2 v1 v0. h4/prelim/ordering__CASE h4/prelim/EQUAL v0 v1 v2 = v1
% Assm: h4/prelim/ordering__eq__dec_c6: h4/prelim/GREATER = h4/prelim/EQUAL <=> F
% Assm: h4/bool/TRUTH: T
% Assm: h4/prelim/ordering__Axiom: !x2 x1 x0. ?f. f h4/prelim/LESS = x0 /\ f h4/prelim/EQUAL = x1 /\ f h4/prelim/GREATER = x2
% Assm: h4/prelim/ordering__distinct_c2: ~(h4/prelim/EQUAL = h4/prelim/GREATER)
% Assm: h4/prelim/compare__def_c1: !lt gt eq. h4/prelim/compare h4/prelim/EQUAL lt eq gt = eq
% Assm: h4/prelim/ordering__distinct_c0: ~(h4/prelim/LESS = h4/prelim/EQUAL)
% Assm: h4/prelim/num2ordering__thm_c1: h4/prelim/num2ordering (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = h4/prelim/EQUAL
% Assm: h4/numeral/numeral__eq_c3: !n. h4/arithmetic/BIT2 n = h4/arithmetic/ZERO <=> F
% Assm: h4/prelim/ordering__eq__dec_c4: h4/prelim/EQUAL = h4/prelim/LESS <=> F
% Assm: h4/prelim/ordering__eq__dec_c3: h4/prelim/EQUAL = h4/prelim/GREATER <=> F
% Assm: h4/prelim/ordering__case__cong: !v2_27 v2 v1_27 v1 v0_27 v0 M_27 M. M = M_27 /\ (M_27 = h4/prelim/LESS ==> v0 = v0_27) /\ (M_27 = h4/prelim/EQUAL ==> v1 = v1_27) /\ (M_27 = h4/prelim/GREATER ==> v2 = v2_27) ==> h4/prelim/ordering__CASE M v0 v1 v2 = h4/prelim/ordering__CASE M_27 v0_27 v1_27 v2_27
% Assm: h4/prelim/ordering__nchotomy: !a. a = h4/prelim/LESS \/ a = h4/prelim/EQUAL \/ a = h4/prelim/GREATER
% Assm: h4/prelim/ordering__eq__dec_c1: h4/prelim/LESS = h4/prelim/EQUAL <=> F
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/numeral/numeral__lt_c3: !n m. h4/prim__rec/_3C (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m) <=> h4/prim__rec/_3C n m
% Assm: h4/numeral/numeral__distrib_c22: !n m. h4/prim__rec/_3C (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) <=> h4/prim__rec/_3C 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/numeral/numeral__lt_c0: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT1 n) <=> T
% 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/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/prelim/ordering2num__num2ordering: !r. h4/prim__rec/_3C r (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) <=> h4/prelim/ordering2num (h4/prelim/num2ordering r) = r
% Assm: h4/numeral/numeral__lt_c2: !n. h4/prim__rec/_3C n h4/arithmetic/ZERO <=> F
% Assm: h4/bool/LET__DEF: h4/bool/LET = (\f x. f x)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/prelim/ordering__CASE0: !x v2 v1 v0. h4/prelim/ordering__CASE x v0 v1 v2 = (\m. h4/bool/COND (h4/prim__rec/_3C m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) v0 (h4/bool/COND (m = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) v1 v2)) (h4/prelim/ordering2num x)
% Assm: h4/numeral/numeral__distrib_c26: !n. h4/arithmetic/_3C_3D h4/num/0 n <=> T
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/numeral/numeral__distrib_c28: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/numeral/numeral__lte_c3: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/numeral/numeral__lte_c5: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 n) (h4/arithmetic/BIT1 m) <=> ~h4/arithmetic/_3C_3D m n
% Assm: h4/numeral/numeral__lte_c0: !n. h4/arithmetic/_3C_3D h4/arithmetic/ZERO n <=> T
% Assm: h4/bool/EQ__EXPAND: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% 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/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/LESS__EQ: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% 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/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% 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/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/list/list__nchotomy: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm: h4/list/list__size__def_c1: !f a1 a0. h4/list/list__size f (h4/list/CONS a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (f a0) (h4/list/list__size f a1))
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/LEFT__OR__EXISTS__THM: !Q P. (?x. P x) \/ Q <=> (?x. P x \/ Q)
% Assm: h4/numeral/numeral__distrib_c21: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/bool/EXISTS__OR__THM: !Q P. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/prim__rec/WF__LESS: h4/relation/WF h4/prim__rec/_3C
% Assm: h4/relation/WF__inv__image: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% 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/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/prelim/GREATER__def: h4/prelim/GREATER = h4/prelim/num2ordering (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))
% Assm: h4/numeral/iDUB__removal_c2: h4/numeral/iDUB h4/arithmetic/ZERO = h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__sub: !n m. h4/arithmetic/NUMERAL (h4/arithmetic/_2D n m) = h4/bool/COND (h4/prim__rec/_3C m n) (h4/arithmetic/NUMERAL (h4/numeral/iSUB T n m)) h4/num/0
% Assm: h4/numeral/iSUB__THM_c8: !n m. h4/numeral/iSUB T (h4/arithmetic/BIT2 n) (h4/arithmetic/BIT1 m) = h4/arithmetic/BIT1 (h4/numeral/iSUB T n m)
% Assm: h4/prelim/num2ordering__ONTO: !a. ?r. a = h4/prelim/num2ordering r /\ h4/prim__rec/_3C r (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm: h4/numeral/iDUB__removal_c0: !n. h4/numeral/iDUB (h4/arithmetic/BIT1 n) = h4/arithmetic/BIT2 (h4/numeral/iDUB n)
% Assm: h4/numeral/iSUB__THM_c3: !n m. h4/numeral/iSUB T (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m) = h4/numeral/iDUB (h4/numeral/iSUB T n m)
% Assm: h4/numeral/iSUB__THM_c1: !n. h4/numeral/iSUB T n h4/arithmetic/ZERO = n
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/numeral/numeral__lte_c2: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 n) h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__lt_c5: !n m. h4/prim__rec/_3C (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT2 m) <=> ~h4/prim__rec/_3C m n
% Assm: h4/prelim/LESS__def: h4/prelim/LESS = h4/prelim/num2ordering h4/num/0
% Assm: h4/numeral/numeral__add_c2: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__add_c0: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm: h4/numeral/numeral__distrib_c8: !n m. h4/arithmetic/_2D (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/arithmetic/_2D n m)
% Assm: h4/numeral/numeral__distrib_c20: !n. h4/prim__rec/_3C n h4/num/0 <=> F
% Assm: h4/numeral/numeral__distrib_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__distrib_c2: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/arithmetic/SUB__LEFT__LESS: !p n m. h4/prim__rec/_3C m (h4/arithmetic/_2D n p) <=> h4/prim__rec/_3C (h4/arithmetic/_2B m p) n
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/arithmetic/SUB__LEFT__EQ: !p n m. m = h4/arithmetic/_2D n p <=> h4/arithmetic/_2B m p = n \/ h4/arithmetic/_3C_3D m h4/num/0 /\ h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/NOT__NUM__EQ: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/arithmetic/NOT__LEQ: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/arithmetic/EQ__LESS__EQ: !n m. m = n <=> h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/prelim/ordering__case__def_c2: !v2 v1 v0. h4/prelim/ordering__CASE h4/prelim/GREATER v0 v1 v2 = v2
% Assm: h4/prelim/ordering__case__def_c0: !v2 v1 v0. h4/prelim/ordering__CASE h4/prelim/LESS v0 v1 v2 = v0
% Goal: !cmp. h4/prelim/list__compare cmp h4/list/NIL h4/list/NIL = h4/prelim/EQUAL
%   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_prelims_listu_u_compareu_u_curriedu_u_def]: !x2 x1 x. h4/prelim/list__compare x x1 x2 = happ h4/prelim/list__compare__tupled (h4/pair/_2C x (h4/pair/_2C x1 x2))
% Assm [h4s_prelims_EQUALu_u_def]: h4/prelim/EQUAL = h4/prelim/num2ordering (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_prelims_listu_u_compareu_u_tupledu_u_primitiveu_u_def]: !_12. (!x y list__compare__tupled cmp l1 l2. happ (happ (happ (happ (happ (happ _12 x) y) list__compare__tupled) cmp) l1) l2 = h4/combin/I (h4/prelim/compare (happ (happ cmp x) y) h4/prelim/LESS (happ list__compare__tupled (h4/pair/_2C cmp (h4/pair/_2C l1 l2))) h4/prelim/GREATER)) ==> (!_11. (!x list__compare__tupled cmp l1 y. happ (happ (happ (happ (happ _11 x) list__compare__tupled) cmp) l1) y = happ (happ (happ (happ (happ _12 x) y) list__compare__tupled) cmp) l1) ==> (!_10. (!v3 x list__compare__tupled cmp l1. happ (happ (happ (happ (happ _10 v3) x) list__compare__tupled) cmp) l1 = h4/list/list__CASE v3 (h4/combin/I h4/prelim/GREATER) (happ (happ (happ (happ _11 x) list__compare__tupled) cmp) l1)) ==> (!_9. (!v3 list__compare__tupled cmp x. happ (happ (happ (happ _9 v3) list__compare__tupled) cmp) x = happ (happ (happ (happ _10 v3) x) list__compare__tupled) cmp) ==> (!_8. (!v11. happ _8 v11 = h4/combin/I h4/prelim/LESS) ==> (!_7. (!v10. happ _7 v10 = _8) ==> (!_6. (!v2 list__compare__tupled cmp v3. happ (happ (happ (happ _6 v2) list__compare__tupled) cmp) v3 = h4/list/list__CASE v2 (h4/list/list__CASE v3 (h4/combin/I h4/prelim/EQUAL) _7) (happ (happ (happ _9 v3) list__compare__tupled) cmp)) ==> (!_5. (!list__compare__tupled cmp v2. happ (happ (happ _5 list__compare__tupled) cmp) v2 = happ (happ (happ _6 v2) list__compare__tupled) cmp) ==> (!_4. (!list__compare__tupled cmp v1. happ (happ (happ _4 list__compare__tupled) cmp) v1 = h4/pair/pair__CASE v1 (happ (happ _5 list__compare__tupled) cmp)) ==> (!_3. (!list__compare__tupled cmp. happ (happ _3 list__compare__tupled) cmp = happ (happ _4 list__compare__tupled) cmp) ==> (!_2. (!list__compare__tupled a. happ (happ _2 list__compare__tupled) a = h4/pair/pair__CASE a (happ _3 list__compare__tupled)) ==> (!_1. (!list__compare__tupled. happ _1 list__compare__tupled = happ _2 list__compare__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!y x l2 l1 cmp. happ (happ R (h4/pair/_2C cmp (h4/pair/_2C l1 l2))) (h4/pair/_2C cmp (h4/pair/_2C (h4/list/CONS x l1) (h4/list/CONS y l2))))) ==> h4/prelim/list__compare__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))))))
% Assm [h4s_prelims_ordering2numu_u_thmu_c1]: h4/prelim/ordering2num h4/prelim/EQUAL = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_prelims_listu_u_compareu_u_ind]: !P. (!cmp. happ (happ (happ P cmp) h4/list/NIL) h4/list/NIL) /\ (!cmp v8 v9. happ (happ (happ P cmp) h4/list/NIL) (h4/list/CONS v8 v9)) /\ (!cmp v4 v5. happ (happ (happ P cmp) (h4/list/CONS v4 v5)) h4/list/NIL) /\ (!cmp x l1 y l2. happ (happ (happ P cmp) l1) l2 ==> happ (happ (happ P cmp) (h4/list/CONS x l1)) (h4/list/CONS y l2)) ==> (!v v1 v2. happ (happ (happ P v) v1) v2)
% Assm [h4s_prelims_orderingu_u_EQu_u_ordering]: !a_27 a. a = a_27 <=> h4/prelim/ordering2num a = h4/prelim/ordering2num a_27
% Assm [h4s_prelims_ordering2numu_u_thmu_c0]: h4/prelim/ordering2num h4/prelim/LESS = h4/num/0
% Assm [h4s_prelims_ordering2numu_u_thmu_c2]: h4/prelim/ordering2num h4/prelim/GREATER = h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_numerals_numeralu_u_equ_c1]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_distribu_c19]: !n m. h4/arithmetic/NUMERAL n = h4/arithmetic/NUMERAL m <=> n = m
% Assm [h4s_numerals_numeralu_u_equ_c4]: !n m. h4/arithmetic/BIT1 n = h4/arithmetic/BIT2 m <=> F
% Assm [h4s_numerals_numeralu_u_distribu_c18]: !n. h4/num/0 = h4/arithmetic/NUMERAL n <=> n = h4/arithmetic/ZERO
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_prelims_orderingu_u_caseu_u_defu_c1]: !v2 v1 v0. h4/prelim/ordering__CASE h4/prelim/EQUAL v0 v1 v2 = v1
% Assm [h4s_prelims_orderingu_u_equ_u_decu_c6]: h4/prelim/GREATER = h4/prelim/EQUAL <=> F
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_prelims_orderingu_u_Axiom]: !x2 x1 x0. ?f. happ f h4/prelim/LESS = x0 /\ happ f h4/prelim/EQUAL = x1 /\ happ f h4/prelim/GREATER = x2
% Assm [h4s_prelims_orderingu_u_distinctu_c2]: ~(h4/prelim/EQUAL = h4/prelim/GREATER)
% Assm [h4s_prelims_compareu_u_defu_c1]: !lt gt eq. h4/prelim/compare h4/prelim/EQUAL lt eq gt = eq
% Assm [h4s_prelims_orderingu_u_distinctu_c0]: ~(h4/prelim/LESS = h4/prelim/EQUAL)
% Assm [h4s_prelims_num2orderingu_u_thmu_c1]: h4/prelim/num2ordering (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = h4/prelim/EQUAL
% Assm [h4s_numerals_numeralu_u_equ_c3]: !n. h4/arithmetic/BIT2 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_prelims_orderingu_u_equ_u_decu_c4]: h4/prelim/EQUAL = h4/prelim/LESS <=> F
% Assm [h4s_prelims_orderingu_u_equ_u_decu_c3]: h4/prelim/EQUAL = h4/prelim/GREATER <=> F
% Assm [h4s_prelims_orderingu_u_caseu_u_cong]: !v2_27 v2 v1_27 v1 v0_27 v0 M_27 M. M = M_27 /\ (M_27 = h4/prelim/LESS ==> v0 = v0_27) /\ (M_27 = h4/prelim/EQUAL ==> v1 = v1_27) /\ (M_27 = h4/prelim/GREATER ==> v2 = v2_27) ==> h4/prelim/ordering__CASE M v0 v1 v2 = h4/prelim/ordering__CASE M_27 v0_27 v1_27 v2_27
% Assm [h4s_prelims_orderingu_u_nchotomy]: !a. a = h4/prelim/LESS \/ a = h4/prelim/EQUAL \/ a = h4/prelim/GREATER
% Assm [h4s_prelims_orderingu_u_equ_u_decu_c1]: h4/prelim/LESS = h4/prelim/EQUAL <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_numerals_numeralu_u_ltu_c3]: !n m. happ (happ h4/prim__rec/_3C (h4/arithmetic/BIT1 n)) (h4/arithmetic/BIT1 m) <=> happ (happ h4/prim__rec/_3C n) m
% Assm [h4s_numerals_numeralu_u_distribu_c22]: !n m. happ (happ h4/prim__rec/_3C (h4/arithmetic/NUMERAL n)) (h4/arithmetic/NUMERAL m) <=> happ (happ h4/prim__rec/_3C 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_numerals_numeralu_u_ltu_c0]: !n. happ (happ h4/prim__rec/_3C h4/arithmetic/ZERO) (h4/arithmetic/BIT1 n) <=> T
% 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_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_prelims_ordering2numu_u_num2ordering]: !r. happ (happ h4/prim__rec/_3C r) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) <=> h4/prelim/ordering2num (h4/prelim/num2ordering r) = r
% Assm [h4s_numerals_numeralu_u_ltu_c2]: !n. happ (happ h4/prim__rec/_3C n) h4/arithmetic/ZERO <=> F
% Assm [h4s_bools_LETu_u_DEF]: !x x. h4/bool/LET x x = happ x x
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_prelims_orderingu_u_CASE0]: !x v2 v1 v0. ?v. (v <=> h4/prelim/ordering2num x = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) /\ h4/prelim/ordering__CASE x v0 v1 v2 = h4/bool/COND (happ (happ h4/prim__rec/_3C (h4/prelim/ordering2num x)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) v0 (h4/bool/COND v v1 v2)
% Assm [h4s_numerals_numeralu_u_distribu_c26]: !n. h4/arithmetic/_3C_3D h4/num/0 n <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_numerals_numeralu_u_distribu_c28]: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_numerals_numeralu_u_lteu_c3]: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_numerals_numeralu_u_lteu_c5]: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 n) (h4/arithmetic/BIT1 m) <=> ~h4/arithmetic/_3C_3D m n
% Assm [h4s_numerals_numeralu_u_lteu_c0]: !n. h4/arithmetic/_3C_3D h4/arithmetic/ZERO n <=> T
% Assm [h4s_bools_EQu_u_EXPAND]: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% 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_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_LESSu_u_EQ]: !n m. happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% 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_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% 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_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_sats_NOTu_u_NOT]: !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_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_lists_listu_u_nchotomy]: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm [h4s_lists_listu_u_sizeu_u_defu_c1]: !f a1 a0. h4/list/list__size f (h4/list/CONS a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (happ f a0) (h4/list/list__size f a1))
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM]: !Q P. (?x. happ P x) \/ Q <=> (?x. happ P x \/ Q)
% Assm [h4s_numerals_numeralu_u_distribu_c21]: !n. happ (happ h4/prim__rec/_3C h4/num/0) (h4/arithmetic/NUMERAL n) <=> happ (happ h4/prim__rec/_3C h4/arithmetic/ZERO) n
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. happ P x /\ Q) <=> (?x. happ P x) /\ Q
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% 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_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_bools_EXISTSu_u_ORu_u_THM]: !Q P. (?x. happ P x \/ happ Q x) <=> (?x. happ P x) \/ (?x. happ Q x)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_primu_u_recs_WFu_u_LESS]: h4/relation/WF h4/prim__rec/_3C
% 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_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_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_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_prelims_GREATERu_u_def]: h4/prelim/GREATER = h4/prelim/num2ordering (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))
% Assm [h4s_numerals_iDUBu_u_removalu_c2]: h4/numeral/iDUB h4/arithmetic/ZERO = h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_sub]: !n m. h4/arithmetic/NUMERAL (h4/arithmetic/_2D n m) = h4/bool/COND (happ (happ h4/prim__rec/_3C m) n) (h4/arithmetic/NUMERAL (h4/numeral/iSUB T n m)) h4/num/0
% Assm [h4s_numerals_iSUBu_u_THMu_c8]: !n m. h4/numeral/iSUB T (h4/arithmetic/BIT2 n) (h4/arithmetic/BIT1 m) = h4/arithmetic/BIT1 (h4/numeral/iSUB T n m)
% Assm [h4s_prelims_num2orderingu_u_ONTO]: !a. ?r. a = h4/prelim/num2ordering r /\ happ (happ h4/prim__rec/_3C r) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm [h4s_numerals_iDUBu_u_removalu_c0]: !n. h4/numeral/iDUB (h4/arithmetic/BIT1 n) = h4/arithmetic/BIT2 (h4/numeral/iDUB n)
% Assm [h4s_numerals_iSUBu_u_THMu_c3]: !n m. h4/numeral/iSUB T (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m) = h4/numeral/iDUB (h4/numeral/iSUB T n m)
% Assm [h4s_numerals_iSUBu_u_THMu_c1]: !n. h4/numeral/iSUB T n h4/arithmetic/ZERO = n
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_numerals_numeralu_u_lteu_c2]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_ltu_c5]: !n m. happ (happ h4/prim__rec/_3C (h4/arithmetic/BIT1 n)) (h4/arithmetic/BIT2 m) <=> ~happ (happ h4/prim__rec/_3C m) n
% Assm [h4s_prelims_LESSu_u_def]: h4/prelim/LESS = h4/prelim/num2ordering h4/num/0
% Assm [h4s_numerals_numeralu_u_addu_c2]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_addu_c0]: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm [h4s_numerals_numeralu_u_distribu_c8]: !n m. h4/arithmetic/_2D (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/arithmetic/_2D n m)
% Assm [h4s_numerals_numeralu_u_distribu_c20]: !n. happ (happ h4/prim__rec/_3C n) h4/num/0 <=> F
% Assm [h4s_numerals_numeralu_u_distribu_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_distribu_c2]: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_LESS]: !p n m. happ (happ h4/prim__rec/_3C m) (h4/arithmetic/_2D n p) <=> happ (happ h4/prim__rec/_3C (h4/arithmetic/_2B m p)) n
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_EQ]: !p n m. m = h4/arithmetic/_2D n p <=> h4/arithmetic/_2B m p = n \/ h4/arithmetic/_3C_3D m h4/num/0 /\ h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_NOTu_u_NUMu_u_EQ]: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_arithmetics_NOTu_u_LEQ]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_arithmetics_EQu_u_LESSu_u_EQ]: !n m. m = n <=> h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm [h4s_prelims_orderingu_u_caseu_u_defu_c2]: !v2 v1 v0. h4/prelim/ordering__CASE h4/prelim/GREATER v0 v1 v2 = v2
% Assm [h4s_prelims_orderingu_u_caseu_u_defu_c0]: !v2 v1 v0. h4/prelim/ordering__CASE h4/prelim/LESS v0 v1 v2 = v0
% Goal: !cmp. h4/prelim/list__compare cmp h4/list/NIL h4/list/NIL = h4/prelim/EQUAL
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_Q1278938,TV_Q1278934]: ![V_f, V_g]: (![V_x]: s(TV_Q1278934,happ(s(t_fun(TV_Q1278938,TV_Q1278934),V_f),s(TV_Q1278938,V_x))) = s(TV_Q1278934,happ(s(t_fun(TV_Q1278938,TV_Q1278934),V_g),s(TV_Q1278938,V_x))) => s(t_fun(TV_Q1278938,TV_Q1278934),V_f) = s(t_fun(TV_Q1278938,TV_Q1278934),V_g))).
fof(ah4s_prelims_listu_u_compareu_u_curriedu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x2, V_x1, V_x]: s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_x),s(t_h4s_lists_list(TV_u_27a),V_x1),s(t_h4s_lists_list(TV_u_27b),V_x2))) = s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),h4s_prelims_listu_u_compareu_u_tupled),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_x),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_x1),s(t_h4s_lists_list(TV_u_27b),V_x2)))))))).
fof(ah4s_prelims_EQUALu_u_def, axiom, s(t_h4s_prelims_ordering,h4s_prelims_equal) = s(t_h4s_prelims_ordering,h4s_prelims_num2ordering(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_prelims_listu_u_compareu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_12]: (![V_x, V_y, V_listu_u_compareu_u_tupled, V_cmp, V_l1, V_l2]: s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)))))),V_uu_12),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27b),V_l2))) = s(t_h4s_prelims_ordering,h4s_combins_i(s(t_h4s_prelims_ordering,h4s_prelims_compare(s(t_h4s_prelims_ordering,happ(s(t_fun(TV_u_27b,t_h4s_prelims_ordering),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))),s(t_h4s_prelims_ordering,h4s_prelims_less),s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2))))))),s(t_h4s_prelims_ordering,h4s_prelims_greater))))) => ![V_uu_11]: (![V_x, V_listu_u_compareu_u_tupled, V_cmp, V_l1, V_y]: s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering))))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)))))),V_uu_11),s(TV_u_27a,V_x))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(TV_u_27b,V_y))) = s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)))))),V_uu_12),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))),s(t_h4s_lists_list(TV_u_27a),V_l1))) => ![V_uu_10]: (![V_v3, V_x, V_listu_u_compareu_u_tupled, V_cmp, V_l1]: s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering)))),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27a,t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering))))),V_uu_10),s(t_h4s_lists_list(TV_u_27b),V_v3))),s(TV_u_27a,V_x))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))),s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(t_h4s_prelims_ordering,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27b),V_v3),s(t_h4s_prelims_ordering,h4s_combins_i(s(t_h4s_prelims_ordering,h4s_prelims_greater))),s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering))))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)))))),V_uu_11),s(TV_u_27a,V_x))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))),s(t_h4s_lists_list(TV_u_27a),V_l1))))) => ![V_uu_9]: (![V_v3, V_listu_u_compareu_u_tupled, V_cmp, V_x]: s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering)))),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering))))),V_uu_9),s(t_h4s_lists_list(TV_u_27b),V_v3))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))),s(TV_u_27a,V_x))) = s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering)))),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27a,t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering))))),V_uu_10),s(t_h4s_lists_list(TV_u_27b),V_v3))),s(TV_u_27a,V_x))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))) => ![V_uu_8]: (![V_v11]: s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering),V_uu_8),s(t_h4s_lists_list(TV_u_27b),V_v11))) = s(t_h4s_prelims_ordering,h4s_combins_i(s(t_h4s_prelims_ordering,h4s_prelims_less))) => ![V_uu_7]: (![V_v10]: s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)),V_uu_7),s(TV_u_27b,V_v10))) = s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering),V_uu_8) => ![V_uu_6]: (![V_v2, V_listu_u_compareu_u_tupled, V_cmp, V_v3]: s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)))),V_uu_6),s(t_h4s_lists_list(TV_u_27a),V_v2))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))),s(t_h4s_lists_list(TV_u_27b),V_v3))) = s(t_h4s_prelims_ordering,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),V_v2),s(t_h4s_prelims_ordering,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27b),V_v3),s(t_h4s_prelims_ordering,h4s_combins_i(s(t_h4s_prelims_ordering,h4s_prelims_equal))),s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)),V_uu_7))),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering)))),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_prelims_ordering))))),V_uu_9),s(t_h4s_lists_list(TV_u_27b),V_v3))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))))) => ![V_uu_5]: (![V_listu_u_compareu_u_tupled, V_cmp, V_v2]: s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))),s(t_h4s_lists_list(TV_u_27a),V_v2))) = s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)))),V_uu_6),s(t_h4s_lists_list(TV_u_27a),V_v2))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))) => ![V_uu_4]: (![V_listu_u_compareu_u_tupled, V_cmp, V_v1]: s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_prelims_ordering),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_prelims_ordering)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_prelims_ordering))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),V_v1))) = s(t_h4s_prelims_ordering,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),V_v1),s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_prelims_ordering)))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))))) => ![V_uu_3]: (![V_listu_u_compareu_u_tupled, V_cmp]: s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_prelims_ordering),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_prelims_ordering)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_prelims_ordering))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))) = s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_prelims_ordering),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_prelims_ordering)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_prelims_ordering))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))) => ![V_uu_2]: (![V_listu_u_compareu_u_tupled, V_a]: s(t_h4s_prelims_ordering,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering)),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),V_a))) = s(t_h4s_prelims_ordering,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),V_a),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_prelims_ordering)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_prelims_ordering))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))))) => ![V_uu_1]: (![V_listu_u_compareu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering)),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering)),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),V_listu_u_compareu_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_bool)),V_R)))) & ![V_y, V_x, V_l2, V_l1, V_cmp]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_bool)),V_R),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2))))))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_y),s(t_h4s_lists_list(TV_u_27b),V_l2)))))))))))) => s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),h4s_prelims_listu_u_compareu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),t_h4s_prelims_ordering)),V_uu_1))))))))))))))))).
fof(ah4s_prelims_ordering2numu_u_thmu_c1, axiom, s(t_h4s_nums_num,h4s_prelims_ordering2num(s(t_h4s_prelims_ordering,h4s_prelims_equal))) = 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_prelims_listu_u_compareu_u_ind, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: ((![V_cmp]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool))),V_P),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)))) & (![V_cmp, V_v8, V_v9]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool))),V_P),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_v8),s(t_h4s_lists_list(TV_u_27b),V_v9)))))) & (![V_cmp, V_v4, V_v5]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool))),V_P),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_v4),s(t_h4s_lists_list(TV_u_27a),V_v5))))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)))) & ![V_cmp, V_x, V_l1, V_y, V_l2]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool))),V_P),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27b),V_l2)))) => p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool))),V_P),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l1))))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_y),s(t_h4s_lists_list(TV_u_27b),V_l2)))))))))) => ![V_v, V_v1, V_v2]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool))),V_P),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_v))),s(t_h4s_lists_list(TV_u_27a),V_v1))),s(t_h4s_lists_list(TV_u_27b),V_v2)))))).
fof(ah4s_prelims_orderingu_u_EQu_u_ordering, axiom, ![V_au_27, V_a]: (s(t_h4s_prelims_ordering,V_a) = s(t_h4s_prelims_ordering,V_au_27) <=> s(t_h4s_nums_num,h4s_prelims_ordering2num(s(t_h4s_prelims_ordering,V_a))) = s(t_h4s_nums_num,h4s_prelims_ordering2num(s(t_h4s_prelims_ordering,V_au_27))))).
fof(ah4s_prelims_ordering2numu_u_thmu_c0, axiom, s(t_h4s_nums_num,h4s_prelims_ordering2num(s(t_h4s_prelims_ordering,h4s_prelims_less))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_prelims_ordering2numu_u_thmu_c2, axiom, s(t_h4s_nums_num,h4s_prelims_ordering2num(s(t_h4s_prelims_ordering,h4s_prelims_greater))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero)))))).
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_numerals_numeralu_u_equ_c1, 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_zero) <=> p(s(t_bool,f)))).
fof(ah4s_numerals_numeralu_u_distribu_c19, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,V_m))).
fof(ah4s_numerals_numeralu_u_equ_c4, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_m))) <=> p(s(t_bool,f)))).
fof(ah4s_numerals_numeralu_u_distribu_c18, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_nums_0) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_zero))).
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_prelims_orderingu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v2, V_v1, V_v0]: s(TV_u_27a,h4s_prelims_orderingu_u_case(s(t_h4s_prelims_ordering,h4s_prelims_equal),s(TV_u_27a,V_v0),s(TV_u_27a,V_v1),s(TV_u_27a,V_v2))) = s(TV_u_27a,V_v1)).
fof(ah4s_prelims_orderingu_u_equ_u_decu_c6, axiom, (s(t_h4s_prelims_ordering,h4s_prelims_greater) = s(t_h4s_prelims_ordering,h4s_prelims_equal) <=> p(s(t_bool,f)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_prelims_orderingu_u_Axiom, axiom, ![TV_u_27a]: ![V_x2, V_x1, V_x0]: ?[V_f]: (s(TV_u_27a,happ(s(t_fun(t_h4s_prelims_ordering,TV_u_27a),V_f),s(t_h4s_prelims_ordering,h4s_prelims_less))) = s(TV_u_27a,V_x0) & (s(TV_u_27a,happ(s(t_fun(t_h4s_prelims_ordering,TV_u_27a),V_f),s(t_h4s_prelims_ordering,h4s_prelims_equal))) = s(TV_u_27a,V_x1) & s(TV_u_27a,happ(s(t_fun(t_h4s_prelims_ordering,TV_u_27a),V_f),s(t_h4s_prelims_ordering,h4s_prelims_greater))) = s(TV_u_27a,V_x2)))).
fof(ah4s_prelims_orderingu_u_distinctu_c2, axiom, ~ (s(t_h4s_prelims_ordering,h4s_prelims_equal) = s(t_h4s_prelims_ordering,h4s_prelims_greater))).
fof(ah4s_prelims_compareu_u_defu_c1, axiom, ![TV_u_27a]: ![V_lt, V_gt, V_eq]: s(TV_u_27a,h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_equal),s(TV_u_27a,V_lt),s(TV_u_27a,V_eq),s(TV_u_27a,V_gt))) = s(TV_u_27a,V_eq)).
fof(ah4s_prelims_orderingu_u_distinctu_c0, axiom, ~ (s(t_h4s_prelims_ordering,h4s_prelims_less) = s(t_h4s_prelims_ordering,h4s_prelims_equal))).
fof(ah4s_prelims_num2orderingu_u_thmu_c1, axiom, s(t_h4s_prelims_ordering,h4s_prelims_num2ordering(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_prelims_ordering,h4s_prelims_equal)).
fof(ah4s_numerals_numeralu_u_equ_c3, 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_zero) <=> p(s(t_bool,f)))).
fof(ah4s_prelims_orderingu_u_equ_u_decu_c4, axiom, (s(t_h4s_prelims_ordering,h4s_prelims_equal) = s(t_h4s_prelims_ordering,h4s_prelims_less) <=> p(s(t_bool,f)))).
fof(ah4s_prelims_orderingu_u_equ_u_decu_c3, axiom, (s(t_h4s_prelims_ordering,h4s_prelims_equal) = s(t_h4s_prelims_ordering,h4s_prelims_greater) <=> p(s(t_bool,f)))).
fof(ah4s_prelims_orderingu_u_caseu_u_cong, axiom, ![TV_u_27a]: ![V_v2u_27, V_v2, V_v1u_27, V_v1, V_v0u_27, V_v0, V_Mu_27, V_M]: ((s(t_h4s_prelims_ordering,V_M) = s(t_h4s_prelims_ordering,V_Mu_27) & ((s(t_h4s_prelims_ordering,V_Mu_27) = s(t_h4s_prelims_ordering,h4s_prelims_less) => s(TV_u_27a,V_v0) = s(TV_u_27a,V_v0u_27)) & ((s(t_h4s_prelims_ordering,V_Mu_27) = s(t_h4s_prelims_ordering,h4s_prelims_equal) => s(TV_u_27a,V_v1) = s(TV_u_27a,V_v1u_27)) & (s(t_h4s_prelims_ordering,V_Mu_27) = s(t_h4s_prelims_ordering,h4s_prelims_greater) => s(TV_u_27a,V_v2) = s(TV_u_27a,V_v2u_27))))) => s(TV_u_27a,h4s_prelims_orderingu_u_case(s(t_h4s_prelims_ordering,V_M),s(TV_u_27a,V_v0),s(TV_u_27a,V_v1),s(TV_u_27a,V_v2))) = s(TV_u_27a,h4s_prelims_orderingu_u_case(s(t_h4s_prelims_ordering,V_Mu_27),s(TV_u_27a,V_v0u_27),s(TV_u_27a,V_v1u_27),s(TV_u_27a,V_v2u_27))))).
fof(ah4s_prelims_orderingu_u_nchotomy, axiom, ![V_a]: (s(t_h4s_prelims_ordering,V_a) = s(t_h4s_prelims_ordering,h4s_prelims_less) | (s(t_h4s_prelims_ordering,V_a) = s(t_h4s_prelims_ordering,h4s_prelims_equal) | s(t_h4s_prelims_ordering,V_a) = s(t_h4s_prelims_ordering,h4s_prelims_greater)))).
fof(ah4s_prelims_orderingu_u_equ_u_decu_c1, axiom, (s(t_h4s_prelims_ordering,h4s_prelims_less) = s(t_h4s_prelims_ordering,h4s_prelims_equal) <=> p(s(t_bool,f)))).
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_numerals_numeralu_u_ltu_c3, axiom, ![V_n, V_m]: 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_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))) = 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_numerals_numeralu_u_distribu_c22, axiom, ![V_n, V_m]: 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_arithmetics_numeral(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = 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_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_numerals_numeralu_u_ltu_c0, axiom, ![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,h4s_arithmetics_zero))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))) = s(t_bool,t)).
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_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_prelims_ordering2numu_u_num2ordering, axiom, ![V_r]: (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_r))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))) <=> s(t_h4s_nums_num,h4s_prelims_ordering2num(s(t_h4s_prelims_ordering,h4s_prelims_num2ordering(s(t_h4s_nums_num,V_r))))) = s(t_h4s_nums_num,V_r))).
fof(ah4s_numerals_numeralu_u_ltu_c2, axiom, ![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,h4s_arithmetics_zero))) = s(t_bool,f)).
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_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_prelims_orderingu_u_CASE0, axiom, ![TV_u_27a]: ![V_x, V_v2, V_v1, V_v0]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,h4s_prelims_ordering2num(s(t_h4s_prelims_ordering,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(TV_u_27a,h4s_prelims_orderingu_u_case(s(t_h4s_prelims_ordering,V_x),s(TV_u_27a,V_v0),s(TV_u_27a,V_v1),s(TV_u_27a,V_v2))) = s(TV_u_27a,h4s_bools_cond(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_prelims_ordering2num(s(t_h4s_prelims_ordering,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(TV_u_27a,V_v0),s(TV_u_27a,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27a,V_v1),s(TV_u_27a,V_v2))))))).
fof(ah4s_numerals_numeralu_u_distribu_c26, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))) = s(t_bool,t)).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_numerals_numeralu_u_distribu_c28, axiom, ![V_n, V_m]: 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_arithmetics_numeral(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_numerals_numeralu_u_lteu_c3, axiom, ![V_n, V_m]: 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_bit1(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_numerals_numeralu_u_lteu_c5, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,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))))))).
fof(ah4s_numerals_numeralu_u_lteu_c0, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))) = s(t_bool,t)).
fof(ah4s_bools_EQu_u_EXPAND, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) | (~ (p(s(t_bool,V_t1))) & ~ (p(s(t_bool,V_t2))))))).
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_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_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_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
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_arithmetics_LESSu_u_EQ, axiom, ![V_n, V_m]: 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_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
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_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_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_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_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_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_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_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_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, 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,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
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_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![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_0))) = s(t_h4s_nums_num,V_n)).
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_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_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_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_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_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_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_lists_listu_u_nchotomy, axiom, ![TV_u_27a]: ![V_l]: (s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) | ?[V_h, V_t]: s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t))))).
fof(ah4s_lists_listu_u_sizeu_u_defu_c1, axiom, ![TV_u_27a]: ![V_f, V_a1, V_a0]: s(t_h4s_nums_num,h4s_lists_listu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))))) = 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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_a0))),s(t_h4s_nums_num,h4s_lists_listu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(TV_u_27a),V_a1)))))))).
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_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> ?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_numerals_numeralu_u_distribu_c21, axiom, ![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,h4s_nums_0))),s(t_h4s_nums_num,h4s_arithmetics_numeral(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,h4s_arithmetics_zero))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_bools_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_arithmetics_ADDu_u_0, 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_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_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_EXISTSu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),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)))) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ?[V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_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_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_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_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_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_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_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_prelims_GREATERu_u_def, axiom, s(t_h4s_prelims_ordering,h4s_prelims_greater) = s(t_h4s_prelims_ordering,h4s_prelims_num2ordering(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
fof(ah4s_numerals_iDUBu_u_removalu_c2, axiom, s(t_h4s_nums_num,h4s_numerals_idub(s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_h4s_nums_num,h4s_arithmetics_zero)).
fof(ah4s_numerals_numeralu_u_sub, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_bools_cond(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,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_numerals_isub(s(t_bool,t),s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))),s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_numerals_iSUBu_u_THMu_c8, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_isub(s(t_bool,t),s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_numerals_isub(s(t_bool,t),s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_prelims_num2orderingu_u_ONTO, axiom, ![V_a]: ?[V_r]: (s(t_h4s_prelims_ordering,V_a) = s(t_h4s_prelims_ordering,h4s_prelims_num2ordering(s(t_h4s_nums_num,V_r))) & 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_r))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))))).
fof(ah4s_numerals_iDUBu_u_removalu_c0, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numerals_idub(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_numerals_idub(s(t_h4s_nums_num,V_n)))))).
fof(ah4s_numerals_iSUBu_u_THMu_c3, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_isub(s(t_bool,t),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_numerals_idub(s(t_h4s_nums_num,h4s_numerals_isub(s(t_bool,t),s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_numerals_iSUBu_u_THMu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numerals_isub(s(t_bool,t),s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_bools_CONJu_u_ASSOC, 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_numerals_numeralu_u_lteu_c2, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_numerals_numeralu_u_ltu_c5, 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,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_arithmetics_bit2(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,V_m))),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_prelims_LESSu_u_def, axiom, s(t_h4s_prelims_ordering,h4s_prelims_less) = s(t_h4s_prelims_ordering,h4s_prelims_num2ordering(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_numerals_numeralu_u_addu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_numerals_iz(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_numerals_numeralu_u_addu_c0, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_distribu_c8, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(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_numerals_numeralu_u_distribu_c20, axiom, ![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,h4s_nums_0))) = s(t_bool,f)).
fof(ah4s_numerals_numeralu_u_distribu_c17, axiom, ![V_n]: (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_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_zero))).
fof(ah4s_numerals_numeralu_u_distribu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_numerals_iz(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_arithmetics_SUBu_u_LEFTu_u_LESS, axiom, ![V_p, V_n, V_m]: 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,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_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_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))),s(t_h4s_nums_num,V_n)))).
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_LEFTu_u_EQ, axiom, ![V_p, V_n, V_m]: (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_p))) <=> (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_h4s_nums_num,V_n) | (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0)))) & 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_NUMu_u_EQ, axiom, ![V_n, V_m]: (~ (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,h4s_nums_suc(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,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))))))).
fof(ah4s_arithmetics_NOTu_u_LEQ, 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,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,V_m)))))).
fof(ah4s_arithmetics_EQu_u_LESSu_u_EQ, axiom, ![V_n, V_m]: (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_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_arithmetics_ADDu_u_ASSOC, axiom, ![V_p, 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_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(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,V_p)))).
fof(ah4s_prelims_orderingu_u_caseu_u_defu_c2, axiom, ![TV_u_27a]: ![V_v2, V_v1, V_v0]: s(TV_u_27a,h4s_prelims_orderingu_u_case(s(t_h4s_prelims_ordering,h4s_prelims_greater),s(TV_u_27a,V_v0),s(TV_u_27a,V_v1),s(TV_u_27a,V_v2))) = s(TV_u_27a,V_v2)).
fof(ah4s_prelims_orderingu_u_caseu_u_defu_c0, axiom, ![TV_u_27a]: ![V_v2, V_v1, V_v0]: s(TV_u_27a,h4s_prelims_orderingu_u_case(s(t_h4s_prelims_ordering,h4s_prelims_less),s(TV_u_27a,V_v0),s(TV_u_27a,V_v1),s(TV_u_27a,V_v2))) = s(TV_u_27a,V_v0)).
fof(ch4s_prelims_listu_u_compareu_u_defu_c0, conjecture, ![TV_u_27a,TV_u_27b]: ![V_cmp]: s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_prelims_ordering)),V_cmp),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))) = s(t_h4s_prelims_ordering,h4s_prelims_equal)).
