%   ORIGINAL: h4/lbtree/optmin__def_c1
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/lbtree/optmin__curried__def: !x1 x. h4/lbtree/optmin x x1 = h4/lbtree/optmin__tupled (h4/pair/_2C x x1)
% Assm: h4/lbtree/optmin__def_c0: h4/lbtree/optmin h4/option/NONE h4/option/NONE = h4/option/NONE
% Assm: h4/lbtree/optmin__tupled__primitive__def: h4/lbtree/optmin__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R)) (\optmin__tupled a. h4/pair/pair__CASE a (\v v1. h4/option/option__CASE v (h4/option/option__CASE v1 (h4/combin/I h4/option/NONE) (\y. h4/combin/I (h4/option/SOME y))) (\x. h4/option/option__CASE v1 (h4/combin/I (h4/option/SOME x)) (\y_27. h4/combin/I (h4/option/SOME (h4/arithmetic/MIN x y_27))))))
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/while/OLEAST__INTRO: !Q P. ((!n. ~P n) ==> Q h4/option/NONE) /\ (!n. P n /\ (!m. h4/prim__rec/_3C m n ==> ~P m) ==> Q (h4/option/SOME n)) ==> Q (h4/while/OLEAST P)
% Assm: h4/lbtree/optmin__ind: !P. P h4/option/NONE h4/option/NONE /\ (!x. P (h4/option/SOME x) h4/option/NONE) /\ (!y. P h4/option/NONE (h4/option/SOME y)) /\ (!x y. P (h4/option/SOME x) (h4/option/SOME y)) ==> (!v v1. P v v1)
% Assm: h4/while/OLEAST__EQNS_c0: !x. h4/while/OLEAST (\n. n = x) = h4/option/SOME x
% Assm: h4/llist/LFINITE__HAS__LENGTH: !ll. h4/llist/LFINITE ll ==> (?n. h4/llist/LLENGTH ll = h4/option/SOME n)
% Assm: h4/while/OLEAST__EQNS_c1: !x. h4/while/OLEAST (\n. x = n) = h4/option/SOME x
% Assm: h4/bool/TRUTH: T
% Assm: h4/relation/WF__EMPTY__REL: h4/relation/WF h4/relation/EMPTY__REL
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/bool/NOT__DEF: $not = (\t. t ==> F)
% Assm: h4/bool/DISJ__COMM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/llist/LLENGTH0: !ll. h4/llist/LLENGTH ll = h4/bool/COND (h4/llist/LFINITE ll) (h4/option/SOME (h4/min/_40 (\n. h4/llist/llength__rel ll n))) h4/option/NONE
% Assm: h4/bool/EXISTS__REFL: !a. ?x. x = a
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/option/option__case__ID: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/while/OLEAST__def: !P. h4/while/OLEAST P = h4/bool/COND (?n. P n) (h4/option/SOME (h4/while/LEAST (\n. P n))) h4/option/NONE
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% 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/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/EXISTS__OR__THM: !Q P. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% 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/while/LEAST__ELIM: !Q P. (?n. P n) /\ (!n. (!m. h4/prim__rec/_3C m n ==> ~P m) /\ P n ==> Q n) ==> Q (h4/while/LEAST P)
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/DATATYPE__TAG__THM: !x. h4/bool/DATATYPE x <=> T
% Assm: h4/option/option__CLAUSES_c9: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm: h4/option/datatype__option: !option. h4/bool/DATATYPE (option h4/option/NONE h4/option/SOME)
% Assm: h4/set__relation/nat__order__iso__thm: !s f. (!n m. f m = f n /\ ~(f m = h4/option/NONE) ==> m = n) /\ (!x. h4/bool/IN x s ==> (?m. f m = h4/option/SOME x)) /\ (!m x. f m = h4/option/SOME x ==> h4/bool/IN x s) ==> h4/set__relation/linear__order (h4/pred__set/GSPEC (h4/pair/UNCURRY (\x y. h4/pair/_2C (h4/pair/_2C x y) (?m n. h4/arithmetic/_3C_3D m n /\ f m = h4/option/SOME x /\ f n = h4/option/SOME y)))) s /\ h4/set__relation/finite__prefixes (h4/pred__set/GSPEC (h4/pair/UNCURRY (\x y. h4/pair/_2C (h4/pair/_2C x y) (?m n. h4/arithmetic/_3C_3D m n /\ f m = h4/option/SOME x /\ f n = h4/option/SOME y)))) s
% Assm: h4/llist/LTAKE__SNOC__LNTH: !n ll. h4/llist/LTAKE (h4/num/SUC n) ll = h4/option/option__CASE (h4/llist/LTAKE n ll) h4/option/NONE (\l. h4/option/option__CASE (h4/llist/LNTH n ll) h4/option/NONE (\e. h4/option/SOME (h4/list/APPEND l (h4/list/CONS e h4/list/NIL))))
% Assm: h4/option/THE__DEF: !x. h4/option/THE (h4/option/SOME x) = x
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/llist/LNTH__THM_c0: !n. h4/llist/LNTH n h4/llist/LNIL = h4/option/NONE
% Assm: h4/list/APPEND__11_c0: !l3 l2 l1. h4/list/APPEND l1 l2 = h4/list/APPEND l1 l3 <=> l2 = l3
% Assm: h4/list/APPEND0_c1: !l2 l1 h. h4/list/APPEND (h4/list/CONS h l1) l2 = h4/list/CONS h (h4/list/APPEND l1 l2)
% Assm: h4/list/APPEND0_c0: !l. h4/list/APPEND h4/list/NIL l = l
% Assm: h4/llist/LNTH__THM_c2: !t n h. h4/llist/LNTH (h4/num/SUC n) (h4/llist/LCONS h t) = h4/llist/LNTH n t
% Assm: h4/list/CONS__11: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/option/OPTION__MAP__DEF_c0: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (f x)
% Assm: h4/llist/LTAKE__THM_c2: !t n h. h4/llist/LTAKE (h4/num/SUC n) (h4/llist/LCONS h t) = h4/option/OPTION__MAP (h4/list/CONS h) (h4/llist/LTAKE n t)
% Assm: h4/llist/LTAKE__THM_c1: !n. h4/llist/LTAKE (h4/num/SUC n) h4/llist/LNIL = h4/option/NONE
% Assm: h4/option/OPTION__MAP__DEF_c1: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/llist/LTAKE0_c1: !n ll. h4/llist/LTAKE (h4/num/SUC n) ll = h4/option/option__CASE (h4/llist/LHD ll) h4/option/NONE (\hd. h4/option/option__CASE (h4/llist/LTAKE n (h4/option/THE (h4/llist/LTL ll))) h4/option/NONE (\tl. h4/option/SOME (h4/list/CONS hd tl)))
% Assm: h4/llist/LTAKE0_c0: !ll. h4/llist/LTAKE h4/num/0 ll = h4/option/SOME h4/list/NIL
% Assm: h4/llist/LNTH0_c0: !ll. h4/llist/LNTH h4/num/0 ll = h4/llist/LHD ll
% Assm: h4/llist/LTL__THM_c1: !t h. h4/llist/LTL (h4/llist/LCONS h t) = h4/option/SOME t
% Assm: h4/llist/LHD__THM_c0: h4/llist/LHD h4/llist/LNIL = h4/option/NONE
% Assm: h4/llist/LTL__THM_c0: h4/llist/LTL h4/llist/LNIL = h4/option/NONE
% Assm: h4/llist/LHD__THM_c1: !t h. h4/llist/LHD (h4/llist/LCONS h t) = h4/option/SOME h
% Assm: h4/llist/llist__CASES: !l. l = h4/llist/LNIL \/ (?h t. l = h4/llist/LCONS h t)
% Assm: h4/set__relation/range__def: !r. h4/set__relation/range r = h4/pred__set/GSPEC (\y. h4/pair/_2C y (?x. h4/bool/IN (h4/pair/_2C x y) r))
% Assm: h4/set__relation/domain__def: !r. h4/set__relation/domain r = h4/pred__set/GSPEC (\x. h4/pair/_2C x (?y. h4/bool/IN (h4/pair/_2C x y) r))
% Assm: h4/bool/EQ__EXPAND: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm: h4/set__relation/finite__prefixes__def: !s r. h4/set__relation/finite__prefixes r s <=> (!e. h4/bool/IN e s ==> h4/pred__set/FINITE (h4/pred__set/GSPEC (\e_27. h4/pair/_2C e_27 (h4/bool/IN (h4/pair/_2C e_27 e) r))))
% Assm: h4/pred__set/FINITE__COUNT: !n. h4/pred__set/FINITE (h4/pred__set/count n)
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/pred__set/SUBSET__FINITE: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/SUBSET t s ==> h4/pred__set/FINITE t)
% Assm: h4/pred__set/IN__IMAGE: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = f x /\ h4/bool/IN x s)
% Assm: h4/pred__set/IMAGE__FINITE: !s. h4/pred__set/FINITE s ==> (!f. h4/pred__set/FINITE (h4/pred__set/IMAGE f s))
% Assm: h4/pred__set/SUBSET__DEF: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm: h4/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% 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/set__relation/transitive__def: !r. h4/set__relation/transitive r <=> (!x y z. h4/bool/IN (h4/pair/_2C x y) r /\ h4/bool/IN (h4/pair/_2C y z) r ==> h4/bool/IN (h4/pair/_2C x z) r)
% Assm: h4/pred__set/IN__COUNT: !n m. h4/bool/IN m (h4/pred__set/count n) <=> h4/prim__rec/_3C m n
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% 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/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/arithmetic/NOT__LESS__EQUAL: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C n m
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/LESS__IMP__LESS__OR__EQ: !n m. h4/prim__rec/_3C m n ==> h4/arithmetic/_3C_3D m n
% 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/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/LESS__EQUAL__ANTISYM: !n m. h4/arithmetic/_3C_3D n m /\ h4/arithmetic/_3C_3D m n ==> n = m
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = 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/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/set__relation/antisym__def: !r. h4/set__relation/antisym r <=> (!x y. h4/bool/IN (h4/pair/_2C x y) r /\ h4/bool/IN (h4/pair/_2C y x) r ==> x = y)
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Goal: !x. h4/lbtree/optmin (h4/option/SOME x) h4/option/NONE = h4/option/SOME x
%   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_lbtrees_optminu_u_curriedu_u_def]: !x1 x. h4/lbtree/optmin x x1 = happ h4/lbtree/optmin__tupled (h4/pair/_2C x x1)
% Assm [h4s_lbtrees_optminu_u_defu_c0]: h4/lbtree/optmin h4/option/NONE h4/option/NONE = h4/option/NONE
% Assm [h4s_lbtrees_optminu_u_tupledu_u_primitiveu_u_def]: !_7. (!x y_27. happ (happ _7 x) y_27 = h4/combin/I (happ h4/option/SOME (h4/arithmetic/MIN x y_27))) ==> (!_6. (!v1 x. happ (happ _6 v1) x = h4/option/option__CASE v1 (h4/combin/I (happ h4/option/SOME x)) (happ _7 x)) ==> (!_5. (!y. happ _5 y = h4/combin/I (happ h4/option/SOME y)) ==> (!_4. (!v v1. happ (happ _4 v) v1 = h4/option/option__CASE v (h4/option/option__CASE v1 (h4/combin/I h4/option/NONE) _5) (happ _6 v1)) ==> (!_3. (!v. happ _3 v = happ _4 v) ==> (!_2. (!a. happ _2 a = h4/pair/pair__CASE a _3) ==> (!_1. (!optmin__tupled. happ _1 optmin__tupled = _2) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R) ==> h4/lbtree/optmin__tupled = h4/relation/WFREC (h4/min/_40 _0) _1)))))))
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_whiles_OLEASTu_u_INTRO]: !Q P. ((!n. ~happ P n) ==> happ Q h4/option/NONE) /\ (!n. happ P n /\ (!m. h4/prim__rec/_3C m n ==> ~happ P m) ==> happ Q (happ h4/option/SOME n)) ==> happ Q (h4/while/OLEAST P)
% Assm [h4s_lbtrees_optminu_u_ind]: !P. happ (happ P h4/option/NONE) h4/option/NONE /\ (!x. happ (happ P (happ h4/option/SOME x)) h4/option/NONE) /\ (!y. happ (happ P h4/option/NONE) (happ h4/option/SOME y)) /\ (!x y. happ (happ P (happ h4/option/SOME x)) (happ h4/option/SOME y)) ==> (!v v1. happ (happ P v) v1)
% Assm [h4s_whiles_OLEASTu_u_EQNSu_c0]: !_0. (!x n. happ (happ _0 x) n <=> n = x) ==> (!x. h4/while/OLEAST (happ _0 x) = happ h4/option/SOME x)
% Assm [h4s_llists_LFINITEu_u_HASu_u_LENGTH]: !ll. h4/llist/LFINITE ll ==> (?n. h4/llist/LLENGTH ll = happ h4/option/SOME n)
% Assm [h4s_whiles_OLEASTu_u_EQNSu_c1]: !_0. (!x n. happ (happ _0 x) n <=> x = n) ==> (!x. h4/while/OLEAST (happ _0 x) = happ h4/option/SOME x)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_relations_WFu_u_EMPTYu_u_REL]: h4/relation/WF h4/relation/EMPTY__REL
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_options_optionu_u_caseu_u_defu_c1]: !x v f. h4/option/option__CASE (happ h4/option/SOME x) v f = happ f x
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_options_SOMEu_u_11]: !y x. happ h4/option/SOME x = happ h4/option/SOME y <=> x = y
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = happ h4/option/SOME x)
% Assm [h4s_bools_NOTu_u_DEF]: !x. $not x <=> x ==> F
% Assm [h4s_bools_DISJu_u_COMM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_llists_LLENGTH0]: !_0. (!ll n. happ (happ _0 ll) n <=> h4/llist/llength__rel ll n) ==> (!ll. h4/llist/LLENGTH ll = h4/bool/COND (h4/llist/LFINITE ll) (happ h4/option/SOME (h4/min/_40 (happ _0 ll))) h4/option/NONE)
% Assm [h4s_bools_EXISTSu_u_REFL]: !a. ?x. x = a
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_options_optionu_u_caseu_u_ID]: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = happ h4/option/SOME x)
% Assm [h4s_whiles_OLEASTu_u_def]: !_0. (!P n. happ (happ _0 P) n <=> happ P n) ==> (!P. ?v. (v <=> (?n. happ P n)) /\ h4/while/OLEAST P = h4/bool/COND v (happ h4/option/SOME (h4/while/LEAST (happ _0 P))) h4/option/NONE)
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% 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_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. happ P x /\ Q) <=> (?x. happ P x) /\ Q
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% 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_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% 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_whiles_LEASTu_u_ELIM]: !Q P. (?n. happ P n) /\ (!n. (!m. h4/prim__rec/_3C m n ==> ~happ P m) /\ happ P n ==> happ Q n) ==> happ Q (h4/while/LEAST P)
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_DATATYPEu_u_TAGu_u_THM]: !x. h4/bool/DATATYPE x <=> T
% Assm [h4s_options_optionu_u_CLAUSESu_c9]: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm [h4s_options_datatypeu_u_option]: !option. h4/bool/DATATYPE (happ (happ option h4/option/NONE) h4/option/SOME)
% Assm [h4s_setu_u_relations_natu_u_orderu_u_isou_u_thm]: !_1. (!x f y. ?v. (v <=> (?m n. h4/arithmetic/_3C_3D m n /\ happ f m = happ h4/option/SOME x /\ happ f n = happ h4/option/SOME y)) /\ happ (happ (happ _1 x) f) y = h4/pair/_2C (h4/pair/_2C x y) v) ==> (!_0. (!f x. happ (happ _0 f) x = happ (happ _1 x) f) ==> (!s f. (!n m. happ f m = happ f n /\ ~(happ f m = h4/option/NONE) ==> m = n) /\ (!x. h4/bool/IN x s ==> (?m. happ f m = happ h4/option/SOME x)) /\ (!m x. happ f m = happ h4/option/SOME x ==> h4/bool/IN x s) ==> h4/set__relation/linear__order (h4/pred__set/GSPEC (h4/pair/UNCURRY (happ _0 f))) s /\ h4/set__relation/finite__prefixes (h4/pred__set/GSPEC (h4/pair/UNCURRY (happ _0 f))) s))
% Assm [h4s_llists_LTAKEu_u_SNOCu_u_LNTH]: !_1. (!l e. happ (happ _1 l) e = happ h4/option/SOME (h4/list/APPEND l (happ (h4/list/CONS e) h4/list/NIL))) ==> (!_0. (!n ll l. happ (happ (happ _0 n) ll) l = h4/option/option__CASE (h4/llist/LNTH n ll) h4/option/NONE (happ _1 l)) ==> (!n ll. h4/llist/LTAKE (h4/num/SUC n) ll = h4/option/option__CASE (h4/llist/LTAKE n ll) h4/option/NONE (happ (happ _0 n) ll)))
% Assm [h4s_options_THEu_u_DEF]: !x. h4/option/THE (happ h4/option/SOME x) = x
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_llists_LNTHu_u_THMu_c0]: !n. h4/llist/LNTH n h4/llist/LNIL = h4/option/NONE
% Assm [h4s_lists_APPENDu_u_11u_c0]: !l3 l2 l1. h4/list/APPEND l1 l2 = h4/list/APPEND l1 l3 <=> l2 = l3
% Assm [h4s_lists_APPEND0u_c1]: !l2 l1 h. h4/list/APPEND (happ (h4/list/CONS h) l1) l2 = happ (h4/list/CONS h) (h4/list/APPEND l1 l2)
% Assm [h4s_lists_APPEND0u_c0]: !l. h4/list/APPEND h4/list/NIL l = l
% Assm [h4s_llists_LNTHu_u_THMu_c2]: !t n h. h4/llist/LNTH (h4/num/SUC n) (h4/llist/LCONS h t) = h4/llist/LNTH n t
% Assm [h4s_lists_CONSu_u_11]: !a1_27 a1 a0_27 a0. happ (h4/list/CONS a0) a1 = happ (h4/list/CONS a0_27) a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c0]: !x f. h4/option/OPTION__MAP f (happ h4/option/SOME x) = happ h4/option/SOME (happ f x)
% Assm [h4s_llists_LTAKEu_u_THMu_c2]: !t n h. h4/llist/LTAKE (h4/num/SUC n) (h4/llist/LCONS h t) = h4/option/OPTION__MAP (h4/list/CONS h) (h4/llist/LTAKE n t)
% Assm [h4s_llists_LTAKEu_u_THMu_c1]: !n. h4/llist/LTAKE (h4/num/SUC n) h4/llist/LNIL = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c1]: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_llists_LTAKE0u_c1]: !_1. (!hd tl. happ (happ _1 hd) tl = happ h4/option/SOME (happ (h4/list/CONS hd) tl)) ==> (!_0. (!n ll hd. happ (happ (happ _0 n) ll) hd = h4/option/option__CASE (h4/llist/LTAKE n (h4/option/THE (h4/llist/LTL ll))) h4/option/NONE (happ _1 hd)) ==> (!n ll. h4/llist/LTAKE (h4/num/SUC n) ll = h4/option/option__CASE (h4/llist/LHD ll) h4/option/NONE (happ (happ _0 n) ll)))
% Assm [h4s_llists_LTAKE0u_c0]: !ll. h4/llist/LTAKE h4/num/0 ll = happ h4/option/SOME h4/list/NIL
% Assm [h4s_llists_LNTH0u_c0]: !ll. h4/llist/LNTH h4/num/0 ll = h4/llist/LHD ll
% Assm [h4s_llists_LTLu_u_THMu_c1]: !t h. h4/llist/LTL (h4/llist/LCONS h t) = happ h4/option/SOME t
% Assm [h4s_llists_LHDu_u_THMu_c0]: h4/llist/LHD h4/llist/LNIL = h4/option/NONE
% Assm [h4s_llists_LTLu_u_THMu_c0]: h4/llist/LTL h4/llist/LNIL = h4/option/NONE
% Assm [h4s_llists_LHDu_u_THMu_c1]: !t h. h4/llist/LHD (h4/llist/LCONS h t) = happ h4/option/SOME h
% Assm [h4s_llists_llistu_u_CASES]: !l. l = h4/llist/LNIL \/ (?h t. l = h4/llist/LCONS h t)
% Assm [h4s_setu_u_relations_rangeu_u_def]: !_0. (!r y. ?v. (v <=> (?x. h4/bool/IN (h4/pair/_2C x y) r)) /\ happ (happ _0 r) y = h4/pair/_2C y v) ==> (!r. h4/set__relation/range r = h4/pred__set/GSPEC (happ _0 r))
% Assm [h4s_setu_u_relations_domainu_u_def]: !_0. (!r x. ?v. (v <=> (?y. h4/bool/IN (h4/pair/_2C x y) r)) /\ happ (happ _0 r) x = h4/pair/_2C x v) ==> (!r. h4/set__relation/domain r = h4/pred__set/GSPEC (happ _0 r))
% Assm [h4s_bools_EQu_u_EXPAND]: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm [h4s_setu_u_relations_finiteu_u_prefixesu_u_def]: !_0. (!e r e_27. happ (happ (happ _0 e) r) e_27 = h4/pair/_2C e_27 (h4/bool/IN (h4/pair/_2C e_27 e) r)) ==> (!s r. h4/set__relation/finite__prefixes r s <=> (!e. h4/bool/IN e s ==> h4/pred__set/FINITE (h4/pred__set/GSPEC (happ (happ _0 e) r))))
% Assm [h4s_predu_u_sets_FINITEu_u_COUNT]: !n. h4/pred__set/FINITE (h4/pred__set/count n)
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_predu_u_sets_SUBSETu_u_FINITE]: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/SUBSET t s ==> h4/pred__set/FINITE t)
% Assm [h4s_predu_u_sets_INu_u_IMAGE]: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = happ f x /\ h4/bool/IN x s)
% Assm [h4s_predu_u_sets_IMAGEu_u_FINITE]: !s. h4/pred__set/FINITE s ==> (!f. h4/pred__set/FINITE (h4/pred__set/IMAGE f s))
% Assm [h4s_predu_u_sets_SUBSETu_u_DEF]: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm [h4s_predu_u_sets_GSPECIFICATION]: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = happ f x)
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% 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_setu_u_relations_transitiveu_u_def]: !r. h4/set__relation/transitive r <=> (!x y z. h4/bool/IN (h4/pair/_2C x y) r /\ h4/bool/IN (h4/pair/_2C y z) r ==> h4/bool/IN (h4/pair/_2C x z) r)
% Assm [h4s_predu_u_sets_INu_u_COUNT]: !n m. h4/bool/IN m (h4/pred__set/count n) <=> h4/prim__rec/_3C m n
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% 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_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_arithmetics_NOTu_u_LESSu_u_EQUAL]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C n m
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_LESSu_u_IMPu_u_LESSu_u_ORu_u_EQ]: !n m. h4/prim__rec/_3C m n ==> h4/arithmetic/_3C_3D m n
% 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_NOTu_u_LESS]: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_LESSu_u_EQUALu_u_ANTISYM]: !n m. h4/arithmetic/_3C_3D n m /\ h4/arithmetic/_3C_3D m n ==> n = m
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_LESSu_u_EQ]: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_setu_u_relations_antisymu_u_def]: !r. h4/set__relation/antisym r <=> (!x y. h4/bool/IN (h4/pair/_2C x y) r /\ h4/bool/IN (h4/pair/_2C y x) r ==> x = y)
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Goal: !x. h4/lbtree/optmin (happ h4/option/SOME x) h4/option/NONE = happ h4/option/SOME x
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_Q1288514,TV_Q1288510]: ![V_f, V_g]: (![V_x]: s(TV_Q1288510,happ(s(t_fun(TV_Q1288514,TV_Q1288510),V_f),s(TV_Q1288514,V_x))) = s(TV_Q1288510,happ(s(t_fun(TV_Q1288514,TV_Q1288510),V_g),s(TV_Q1288514,V_x))) => s(t_fun(TV_Q1288514,TV_Q1288510),V_f) = s(t_fun(TV_Q1288514,TV_Q1288510),V_g))).
fof(ah4s_lbtrees_optminu_u_curriedu_u_def, axiom, ![V_x1, V_x]: s(t_h4s_options_option(t_h4s_nums_num),h4s_lbtrees_optmin(s(t_h4s_options_option(t_h4s_nums_num),V_x),s(t_h4s_options_option(t_h4s_nums_num),V_x1))) = s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),h4s_lbtrees_optminu_u_tupled),s(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_options_option(t_h4s_nums_num),V_x),s(t_h4s_options_option(t_h4s_nums_num),V_x1)))))).
fof(ah4s_lbtrees_optminu_u_defu_c0, axiom, s(t_h4s_options_option(t_h4s_nums_num),h4s_lbtrees_optmin(s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none)).
fof(ah4s_lbtrees_optminu_u_tupledu_u_primitiveu_u_def, axiom, ![V_uu_7]: (![V_x, V_yu_27]: s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num))),V_uu_7),s(t_h4s_nums_num,V_x))),s(t_h4s_nums_num,V_yu_27))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_combins_i(s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),h4s_options_some),s(t_h4s_nums_num,h4s_arithmetics_min(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_yu_27))))))) => ![V_uu_6]: (![V_v1, V_x]: s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num))),V_uu_6),s(t_h4s_options_option(t_h4s_nums_num),V_v1))),s(t_h4s_nums_num,V_x))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_nums_num),V_v1),s(t_h4s_options_option(t_h4s_nums_num),h4s_combins_i(s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),h4s_options_some),s(t_h4s_nums_num,V_x))))),s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num))),V_uu_7),s(t_h4s_nums_num,V_x))))) => ![V_uu_5]: (![V_y]: s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),V_uu_5),s(t_h4s_nums_num,V_y))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_combins_i(s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),h4s_options_some),s(t_h4s_nums_num,V_y))))) => ![V_uu_4]: (![V_v, V_v1]: s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num))),V_uu_4),s(t_h4s_options_option(t_h4s_nums_num),V_v))),s(t_h4s_options_option(t_h4s_nums_num),V_v1))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_nums_num),V_v),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_nums_num),V_v1),s(t_h4s_options_option(t_h4s_nums_num),h4s_combins_i(s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))),s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),V_uu_5))),s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num))),V_uu_6),s(t_h4s_options_option(t_h4s_nums_num),V_v1))))) => ![V_uu_3]: (![V_v]: s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num))),V_uu_3),s(t_h4s_options_option(t_h4s_nums_num),V_v))) = s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num))),V_uu_4),s(t_h4s_options_option(t_h4s_nums_num),V_v))) => ![V_uu_2]: (![V_a]: s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),V_uu_2),s(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),V_a))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),V_a),s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num))),V_uu_3))) => ![V_uu_1]: (![V_optminu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),V_optminu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),V_uu_2) => ![V_uu_0]: (![V_R]: s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_bool)),V_R))) = s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_bool)),V_R))) => s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),h4s_lbtrees_optminu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_options_option(t_h4s_nums_num),t_h4s_options_option(t_h4s_nums_num)),t_h4s_options_option(t_h4s_nums_num))),V_uu_1)))))))))))).
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_whiles_OLEASTu_u_INTRO, axiom, ![V_Q, V_P]: (((![V_n]: ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n))))) => p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),V_Q),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))))) & ![V_n]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) & ![V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),V_Q),s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),h4s_options_some),s(t_h4s_nums_num,V_n)))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),V_Q),s(t_h4s_options_option(t_h4s_nums_num),h4s_whiles_oleast(s(t_fun(t_h4s_nums_num,t_bool),V_P)))))))).
fof(ah4s_lbtrees_optminu_u_ind, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool)),V_P),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none)))) & (![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool)),V_P),s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),h4s_options_some),s(t_h4s_nums_num,V_x))))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none)))) & (![V_y]: p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool)),V_P),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))),s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),h4s_options_some),s(t_h4s_nums_num,V_y)))))) & ![V_x, V_y]: p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool)),V_P),s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),h4s_options_some),s(t_h4s_nums_num,V_x))))),s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),h4s_options_some),s(t_h4s_nums_num,V_y))))))))) => ![V_v, V_v1]: p(s(t_bool,happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_options_option(t_h4s_nums_num),t_fun(t_h4s_options_option(t_h4s_nums_num),t_bool)),V_P),s(t_h4s_options_option(t_h4s_nums_num),V_v))),s(t_h4s_options_option(t_h4s_nums_num),V_v1)))))).
fof(ah4s_whiles_OLEASTu_u_EQNSu_c0, axiom, ![V_uu_0]: (![V_x, V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_x))),s(t_h4s_nums_num,V_n)))) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,V_x)) => ![V_x]: s(t_h4s_options_option(t_h4s_nums_num),h4s_whiles_oleast(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)),V_uu_0),s(t_h4s_nums_num,V_x))))) = s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),h4s_options_some),s(t_h4s_nums_num,V_x))))).
fof(ah4s_llists_LFINITEu_u_HASu_u_LENGTH, axiom, ![TV_u_27a]: ![V_ll]: (p(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_ll)))) => ?[V_n]: s(t_h4s_options_option(t_h4s_nums_num),h4s_llists_llength(s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),h4s_options_some),s(t_h4s_nums_num,V_n))))).
fof(ah4s_whiles_OLEASTu_u_EQNSu_c1, axiom, ![V_uu_0]: (![V_x, V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_x))),s(t_h4s_nums_num,V_n)))) <=> s(t_h4s_nums_num,V_x) = s(t_h4s_nums_num,V_n)) => ![V_x]: s(t_h4s_options_option(t_h4s_nums_num),h4s_whiles_oleast(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)),V_uu_0),s(t_h4s_nums_num,V_x))))) = s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),h4s_options_some),s(t_h4s_nums_num,V_x))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_relations_WFu_u_EMPTYu_u_REL, axiom, ![TV_u_27a]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_emptyu_u_rel))))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_options_optionu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_options_optionu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_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_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_options_NOTu_u_NONEu_u_SOME, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))).
fof(ah4s_bools_NOTu_u_DEF, axiom, ![V_x]: (p(s(t_bool,d_not(s(t_bool,V_x)))) <=> (p(s(t_bool,V_x)) => p(s(t_bool,f))))).
fof(ah4s_bools_DISJu_u_COMM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_bools_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
fof(ah4s_llists_LLENGTH0, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_ll, V_n]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_llists_llist(TV_u_27a),V_ll))),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_llists_llengthu_u_rel(s(t_h4s_llists_llist(TV_u_27a),V_ll),s(t_h4s_nums_num,V_n))) => ![V_ll]: s(t_h4s_options_option(t_h4s_nums_num),h4s_llists_llength(s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_bools_cond(s(t_bool,h4s_llists_lfinite(s(t_h4s_llists_llist(TV_u_27a),V_ll))),s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),h4s_options_some),s(t_h4s_nums_num,h4s_mins_u_40(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_llists_llist(TV_u_27a),V_ll))))))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))))).
fof(ah4s_bools_EXISTSu_u_REFL, axiom, ![TV_u_27a]: ![V_a]: ?[V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_a)).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_options_optionu_u_caseu_u_ID, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some))) = s(t_h4s_options_option(TV_u_27a),V_x)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_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_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))).
fof(ah4s_whiles_OLEASTu_u_def, axiom, ![V_uu_0]: (![V_P, V_n]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_fun(t_h4s_nums_num,t_bool),V_P))),s(t_h4s_nums_num,V_n))) = s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n))) => ![V_P]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n))))) & s(t_h4s_options_option(t_h4s_nums_num),h4s_whiles_oleast(s(t_fun(t_h4s_nums_num,t_bool),V_P))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),h4s_options_some),s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_fun(t_h4s_nums_num,t_bool),V_P))))))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none)))))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_bools_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_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_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_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_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_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_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_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_bools_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_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_whiles_LEASTu_u_ELIM, axiom, ![V_Q, V_P]: ((?[V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) & ![V_n]: ((![V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_Q),s(t_h4s_nums_num,V_n)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_Q),s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),V_P)))))))).
fof(ah4s_bools_LEFTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) & p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) & (p(s(t_bool,V_A)) | p(s(t_bool,V_C)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_DATATYPEu_u_TAGu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_bools_datatype(s(TV_u_27a,V_x))) = s(t_bool,t)).
fof(ah4s_options_optionu_u_CLAUSESu_c9, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some))) = s(t_h4s_options_option(TV_u_27a),V_x)).
fof(ah4s_options_datatypeu_u_option, axiom, ![TV_u_27b,TV_u_27a]: ![V_option]: p(s(t_bool,h4s_bools_datatype(s(TV_u_27b,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),TV_u_27b)),V_option),s(t_h4s_options_option(TV_u_27a),h4s_options_none))),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some))))))).
fof(ah4s_setu_u_relations_natu_u_orderu_u_isou_u_thm, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_x, V_f, V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_m, 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)))) & (s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),V_f),s(t_h4s_nums_num,V_m))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))) & s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),V_f),s(t_h4s_nums_num,V_n))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_y)))))) & s(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),V_uu_1),s(TV_u_27a,V_x))),s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),V_f))),s(TV_u_27a,V_y))) = s(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_pairs_u_2c(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_bool,V_v)))) => ![V_uu_0]: (![V_f, V_x]: s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),V_f))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),V_uu_1),s(TV_u_27a,V_x))),s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),V_f))) => ![V_s, V_f]: ((![V_n, V_m]: ((s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),V_f),s(t_h4s_nums_num,V_m))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),V_f),s(t_h4s_nums_num,V_n))) & ~ (s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),V_f),s(t_h4s_nums_num,V_m))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))) => s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)) & (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => ?[V_m]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),V_f),s(t_h4s_nums_num,V_m))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x)))) & ![V_m, V_x]: (s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),V_f),s(t_h4s_nums_num,V_m))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))) => p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))) => (p(s(t_bool,h4s_setu_u_relations_linearu_u_order(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),V_f))))))),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_setu_u_relations_finiteu_u_prefixes(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),V_f))))))),s(t_fun(TV_u_27a,t_bool),V_s))))))))).
fof(ah4s_llists_LTAKEu_u_SNOCu_u_LNTH, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_l, V_e]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a)))),V_uu_1),s(t_h4s_lists_list(TV_u_27a),V_l))),s(TV_u_27a,V_e))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),h4s_options_some),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_lists_cons(s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))))))) => ![V_uu_0]: (![V_n, V_ll, V_l]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a))))),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27a),V_ll))),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_llists_lnth(s(t_h4s_nums_num,V_n),s(t_h4s_llists_llist(TV_u_27a),V_ll))),s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_none),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a)))),V_uu_1),s(t_h4s_lists_list(TV_u_27a),V_l))))) => ![V_n, V_ll]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_llists_ltake(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_llists_ltake(s(t_h4s_nums_num,V_n),s(t_h4s_llists_llist(TV_u_27a),V_ll))),s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_none),s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a))))),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27a),V_ll)))))))).
fof(ah4s_options_THEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
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_llists_LNTHu_u_THMu_c0, axiom, ![TV_u_27a]: ![V_n]: s(t_h4s_options_option(TV_u_27a),h4s_llists_lnth(s(t_h4s_nums_num,V_n),s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_lists_APPENDu_u_11u_c0, axiom, ![TV_u_27a]: ![V_l3, V_l2, V_l1]: (s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l3))) <=> s(t_h4s_lists_list(TV_u_27a),V_l2) = s(t_h4s_lists_list(TV_u_27a),V_l3))).
fof(ah4s_lists_APPEND0u_c1, axiom, ![TV_u_27a]: ![V_l2, V_l1, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),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_l1))),s(t_h4s_lists_list(TV_u_27a),V_l2))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_lists_cons(s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
fof(ah4s_lists_APPEND0u_c0, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
fof(ah4s_llists_LNTHu_u_THMu_c2, axiom, ![TV_u_27c]: ![V_t, V_n, V_h]: s(t_h4s_options_option(TV_u_27c),h4s_llists_lnth(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27c),h4s_llists_lcons(s(TV_u_27c,V_h),s(t_h4s_llists_llist(TV_u_27c),V_t))))) = s(t_h4s_options_option(TV_u_27c),h4s_llists_lnth(s(t_h4s_nums_num,V_n),s(t_h4s_llists_llist(TV_u_27c),V_t)))).
fof(ah4s_lists_CONSu_u_11, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),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_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_lists_cons(s(TV_u_27a,V_a0u_27))),s(t_h4s_lists_list(TV_u_27a),V_a1u_27))) <=> (s(TV_u_27a,V_a0) = s(TV_u_27a,V_a0u_27) & s(t_h4s_lists_list(TV_u_27a),V_a1) = s(t_h4s_lists_list(TV_u_27a),V_a1u_27)))).
fof(ah4s_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) => ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_options_OPTIONu_u_MAPu_u_DEFu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27b)),h4s_options_some),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))).
fof(ah4s_llists_LTAKEu_u_THMu_c2, axiom, ![TV_u_27c]: ![V_t, V_n, V_h]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27c)),h4s_llists_ltake(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27c),h4s_llists_lcons(s(TV_u_27c,V_h),s(t_h4s_llists_llist(TV_u_27c),V_t))))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27c)),h4s_options_optionu_u_map(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_lists_list(TV_u_27c)),h4s_lists_cons(s(TV_u_27c,V_h))),s(t_h4s_options_option(t_h4s_lists_list(TV_u_27c)),h4s_llists_ltake(s(t_h4s_nums_num,V_n),s(t_h4s_llists_llist(TV_u_27c),V_t)))))).
fof(ah4s_llists_LTAKEu_u_THMu_c1, axiom, ![TV_u_27b]: ![V_n]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27b)),h4s_llists_ltake(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lnil))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27b)),h4s_options_none)).
fof(ah4s_options_OPTIONu_u_MAPu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)).
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_llists_LTAKE0u_c1, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_hd, V_tl]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a)))),V_uu_1),s(TV_u_27a,V_hd))),s(t_h4s_lists_list(TV_u_27a),V_tl))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),h4s_options_some),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_lists_cons(s(TV_u_27a,V_hd))),s(t_h4s_lists_list(TV_u_27a),V_tl))))) => ![V_uu_0]: (![V_n, V_ll, V_hd]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a))))),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27a),V_ll))),s(TV_u_27a,V_hd))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_llists_ltake(s(t_h4s_nums_num,V_n),s(t_h4s_llists_llist(TV_u_27a),h4s_options_the(s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27a)),h4s_llists_ltl(s(t_h4s_llists_llist(TV_u_27a),V_ll))))))),s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_none),s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a)))),V_uu_1),s(TV_u_27a,V_hd))))) => ![V_n, V_ll]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_llists_ltake(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27a),V_ll))),s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_none),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_llists_llist(TV_u_27a),t_fun(TV_u_27a,t_h4s_options_option(t_h4s_lists_list(TV_u_27a))))),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_llists_llist(TV_u_27a),V_ll)))))))).
fof(ah4s_llists_LTAKE0u_c0, axiom, ![TV_u_27a]: ![V_ll]: s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_llists_ltake(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),h4s_options_some),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))).
fof(ah4s_llists_LNTH0u_c0, axiom, ![TV_u_27a]: ![V_ll]: s(t_h4s_options_option(TV_u_27a),h4s_llists_lnth(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_llists_llist(TV_u_27a),V_ll))) = s(t_h4s_options_option(TV_u_27a),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27a),V_ll)))).
fof(ah4s_llists_LTLu_u_THMu_c1, axiom, ![TV_u_27b]: ![V_t, V_h]: s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27b)),h4s_llists_ltl(s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lcons(s(TV_u_27b,V_h),s(t_h4s_llists_llist(TV_u_27b),V_t))))) = s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27b)),happ(s(t_fun(t_h4s_llists_llist(TV_u_27b),t_h4s_options_option(t_h4s_llists_llist(TV_u_27b))),h4s_options_some),s(t_h4s_llists_llist(TV_u_27b),V_t)))).
fof(ah4s_llists_LHDu_u_THMu_c0, axiom, ![TV_u_27a]: s(t_h4s_options_option(TV_u_27a),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_llists_LTLu_u_THMu_c0, axiom, ![TV_u_27a]: s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27a)),h4s_llists_ltl(s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil))) = s(t_h4s_options_option(t_h4s_llists_llist(TV_u_27a)),h4s_options_none)).
fof(ah4s_llists_LHDu_u_THMu_c1, axiom, ![TV_u_27b]: ![V_t, V_h]: s(t_h4s_options_option(TV_u_27b),h4s_llists_lhd(s(t_h4s_llists_llist(TV_u_27b),h4s_llists_lcons(s(TV_u_27b,V_h),s(t_h4s_llists_llist(TV_u_27b),V_t))))) = s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27b)),h4s_options_some),s(TV_u_27b,V_h)))).
fof(ah4s_llists_llistu_u_CASES, axiom, ![TV_u_27a]: ![V_l]: (s(t_h4s_llists_llist(TV_u_27a),V_l) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lnil) | ?[V_h, V_t]: s(t_h4s_llists_llist(TV_u_27a),V_l) = s(t_h4s_llists_llist(TV_u_27a),h4s_llists_lcons(s(TV_u_27a,V_h),s(t_h4s_llists_llist(TV_u_27a),V_t))))).
fof(ah4s_setu_u_relations_rangeu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_r, V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_x]: p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),V_r))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),V_r))),s(TV_u_27a,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_y),s(t_bool,V_v)))) => ![V_r]: s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_range(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),V_r))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),V_r))))))).
fof(ah4s_setu_u_relations_domainu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_r, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_y]: p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_r]: s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_domain(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))))))).
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_setu_u_relations_finiteu_u_prefixesu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_e, V_r, V_eu_27]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(TV_u_27b,V_e))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))),s(TV_u_27a,V_eu_27))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_eu_27),s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_eu_27),s(TV_u_27b,V_e))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r))))) => ![V_s, V_r]: (p(s(t_bool,h4s_setu_u_relations_finiteu_u_prefixes(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r),s(t_fun(TV_u_27b,t_bool),V_s)))) <=> ![V_e]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_e),s(t_fun(TV_u_27b,t_bool),V_s)))) => p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(TV_u_27b,V_e))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_r)))))))))))).
fof(ah4s_predu_u_sets_FINITEu_u_COUNT, axiom, ![V_n]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_count(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_predu_u_sets_SUBSETu_u_FINITE, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_t]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_INu_u_IMAGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_s, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> ?[V_x]: (s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_IMAGEu_u_FINITE, axiom, ![TV_u_27b,TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_f]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_predu_u_sets_SUBSETu_u_DEF, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_GSPECIFICATION, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f)))))) <=> ?[V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_v),s(t_bool,t))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f),s(TV_u_27b,V_x))))).
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_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_setu_u_relations_transitiveu_u_def, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_setu_u_relations_transitive(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) <=> ![V_x, V_y, V_z]: ((p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) & p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_y),s(TV_u_27a,V_z))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))) => p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_z))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))).
fof(ah4s_predu_u_sets_INu_u_COUNT, axiom, ![V_n, V_m]: s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_count(s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_NOTu_u_LESSu_u_EQUAL, 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_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_LESSu_u_IMPu_u_LESSu_u_ORu_u_EQ, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
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_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_LESSu_u_EQUALu_u_ANTISYM, axiom, ![V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),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))))) => s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,V_m))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_LESSu_u_EQ, axiom, ![V_n, V_m]: s(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_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_setu_u_relations_antisymu_u_def, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_setu_u_relations_antisym(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) <=> ![V_x, V_y]: ((p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r)))) & p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_y),s(TV_u_27a,V_x))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))).
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(ch4s_lbtrees_optminu_u_defu_c1, conjecture, ![V_x]: s(t_h4s_options_option(t_h4s_nums_num),h4s_lbtrees_optmin(s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),h4s_options_some),s(t_h4s_nums_num,V_x))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_none))) = s(t_h4s_options_option(t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(t_h4s_nums_num)),h4s_options_some),s(t_h4s_nums_num,V_x)))).
