%   ORIGINAL: h4/hrat/hrat__ABS__REP__CLASS_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/hrat/hrat__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\c. ?r. h4/hrat/trat__eq r r /\ c = h4/hrat/trat__eq r) rep
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% Assm: h4/hrat/hrat__bijections_c1: !r. (\c. ?r0. h4/hrat/trat__eq r0 r0 /\ c = h4/hrat/trat__eq r0) r <=> h4/hrat/hrat__REP__CLASS (h4/hrat/hrat__ABS__CLASS r) = r
% Assm: h4/hrat/hrat__bijections_c0: !a. h4/hrat/hrat__ABS__CLASS (h4/hrat/hrat__REP__CLASS a) = a
% Assm: h4/hrat/TRAT__EQ__EQUIV: !q p. h4/hrat/trat__eq p q <=> h4/hrat/trat__eq p = h4/hrat/trat__eq q
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/hrat/hrat__ABS__REP__CLASS_c0: !a. h4/hrat/hrat__ABS__CLASS (h4/hrat/hrat__REP__CLASS a) = a
% Assm: h4/hrat/TRAT__EQ__REFL: !p. h4/hrat/trat__eq p p
% Assm: h4/quotient/EQUIV__IMP__PARTIAL__EQUIV: !R. h4/quotient/EQUIV R ==> h4/quotient/PARTIAL__EQUIV R
% Assm: h4/quotient/PARTIAL__EQUIV__def: !R. h4/quotient/PARTIAL__EQUIV R <=> (?x. R x x) /\ (!x y. R x y <=> R x x /\ R y y /\ R x = R y)
% Assm: h4/quotient/EQUIV__def: !E. h4/quotient/EQUIV E <=> (!x y. E x y <=> E x = E y)
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/hrat/trat__eq0: !y_27 y x_27 x. h4/hrat/trat__eq (h4/pair/_2C x y) (h4/pair/_2C x_27 y_27) <=> h4/arithmetic/_2A (h4/num/SUC x) (h4/num/SUC y_27) = h4/arithmetic/_2A (h4/num/SUC x_27) (h4/num/SUC y)
% Assm: h4/bool/TRUTH: T
% Assm: h4/hrat/TRAT__EQ__TRANS: !r q p. h4/hrat/trat__eq p q /\ h4/hrat/trat__eq q r ==> h4/hrat/trat__eq p r
% Assm: h4/hrat/TRAT__EQ__AP: !q p. p = q ==> h4/hrat/trat__eq p q
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/hrat/TRAT__EQ__SYM: !q p. h4/hrat/trat__eq p q <=> h4/hrat/trat__eq q p
% Assm: h4/arithmetic/MULT__SYM: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm: h4/hrat/TRAT__ADD__WELLDEFINED: !r q p. h4/hrat/trat__eq p q ==> h4/hrat/trat__eq (h4/hrat/trat__add p r) (h4/hrat/trat__add q r)
% Assm: h4/hrat/TRAT__MUL__WELLDEFINED: !r q p. h4/hrat/trat__eq p q ==> h4/hrat/trat__eq (h4/hrat/trat__mul p r) (h4/hrat/trat__mul q r)
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/hrat/TRAT__MUL__SYM__EQ: !i h. h4/hrat/trat__mul h i = h4/hrat/trat__mul i h
% Assm: h4/hrat/TRAT__ADD__SYM__EQ: !i h. h4/hrat/trat__add h i = h4/hrat/trat__add i h
% Assm: h4/hrat/TRAT__INV__WELLDEFINED: !q p. h4/hrat/trat__eq p q ==> h4/hrat/trat__eq (h4/hrat/trat__inv p) (h4/hrat/trat__inv q)
% Assm: h4/hrat/TRAT__MUL__WELLDEFINED2: !q2 q1 p2 p1. h4/hrat/trat__eq p1 p2 /\ h4/hrat/trat__eq q1 q2 ==> h4/hrat/trat__eq (h4/hrat/trat__mul p1 q1) (h4/hrat/trat__mul p2 q2)
% Assm: h4/hrat/TRAT__ADD__WELLDEFINED2: !q2 q1 p2 p1. h4/hrat/trat__eq p1 p2 /\ h4/hrat/trat__eq q1 q2 ==> h4/hrat/trat__eq (h4/hrat/trat__add p1 q1) (h4/hrat/trat__add p2 q2)
% Assm: h4/hrat/TRAT__ADD__TOTAL: !i h. h4/hrat/trat__eq h i \/ (?d. h4/hrat/trat__eq h (h4/hrat/trat__add i d)) \/ (?d. h4/hrat/trat__eq i (h4/hrat/trat__add h d))
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/arithmetic/MULT__ASSOC: !p n m. h4/arithmetic/_2A m (h4/arithmetic/_2A n p) = h4/arithmetic/_2A (h4/arithmetic/_2A m n) p
% Assm: h4/hrat/trat__inv0: !y x. h4/hrat/trat__inv (h4/pair/_2C x y) = h4/pair/_2C y x
% Assm: h4/arithmetic/MULT__CLAUSES_c0: !m. h4/arithmetic/_2A h4/num/0 m = h4/num/0
% Assm: h4/arithmetic/MULT__CLAUSES_c4: !n m. h4/arithmetic/_2A (h4/num/SUC m) n = h4/arithmetic/_2B (h4/arithmetic/_2A m n) n
% Assm: h4/arithmetic/ADD__CLAUSES_c3: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/prim__rec/PRE0_c1: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/hrat/trat__mul0: !y_27 y x_27 x. h4/hrat/trat__mul (h4/pair/_2C x y) (h4/pair/_2C x_27 y_27) = h4/pair/_2C (h4/prim__rec/PRE (h4/arithmetic/_2A (h4/num/SUC x) (h4/num/SUC x_27))) (h4/prim__rec/PRE (h4/arithmetic/_2A (h4/num/SUC y) (h4/num/SUC y_27)))
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/OR__CLAUSES_c4: !t. t \/ t <=> t
% Assm: h4/arithmetic/ADD__CLAUSES_c1: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/MULT__CLAUSES_c5: !n m. h4/arithmetic/_2A m (h4/num/SUC n) = h4/arithmetic/_2B m (h4/arithmetic/_2A m n)
% Assm: h4/arithmetic/num__CASES: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/arithmetic/MULT__CLAUSES_c1: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/hrat/TRAT__MUL__SYM: !i h. h4/hrat/trat__eq (h4/hrat/trat__mul h i) (h4/hrat/trat__mul i h)
% Assm: h4/hrat/TRAT__MUL__LINV: !h. h4/hrat/trat__eq (h4/hrat/trat__mul (h4/hrat/trat__inv h) h) h4/hrat/trat__1
% Assm: h4/hrat/TRAT__ADD__SYM: !i h. h4/hrat/trat__eq (h4/hrat/trat__add h i) (h4/hrat/trat__add i h)
% Assm: h4/hrat/TRAT__ADD__ASSOC: !j i h. h4/hrat/trat__eq (h4/hrat/trat__add h (h4/hrat/trat__add i j)) (h4/hrat/trat__add (h4/hrat/trat__add h i) j)
% Assm: h4/hrat/TRAT__ARCH: !h. ?n d. h4/hrat/trat__eq (h4/hrat/trat__sucint n) (h4/hrat/trat__add h d)
% Assm: h4/hrat/TRAT__MUL__LID: !h. h4/hrat/trat__eq (h4/hrat/trat__mul h4/hrat/trat__1 h) h
% Assm: h4/hrat/TRAT__NOZERO: !i h. ~h4/hrat/trat__eq (h4/hrat/trat__add h i) h
% Assm: h4/hrat/TRAT__MUL__ASSOC: !j i h. h4/hrat/trat__eq (h4/hrat/trat__mul h (h4/hrat/trat__mul i j)) (h4/hrat/trat__mul (h4/hrat/trat__mul h i) j)
% Assm: h4/hrat/TRAT__LDISTRIB: !j i h. h4/hrat/trat__eq (h4/hrat/trat__mul h (h4/hrat/trat__add i j)) (h4/hrat/trat__add (h4/hrat/trat__mul h i) (h4/hrat/trat__mul h j))
% Assm: h4/hrat/trat__add0: !y_27 y x_27 x. h4/hrat/trat__add (h4/pair/_2C x y) (h4/pair/_2C x_27 y_27) = h4/pair/_2C (h4/prim__rec/PRE (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/num/SUC x) (h4/num/SUC y_27)) (h4/arithmetic/_2A (h4/num/SUC x_27) (h4/num/SUC y)))) (h4/prim__rec/PRE (h4/arithmetic/_2A (h4/num/SUC y) (h4/num/SUC y_27)))
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/arithmetic/MULT__SUC__EQ: !p n m. h4/arithmetic/_2A n (h4/num/SUC p) = h4/arithmetic/_2A m (h4/num/SUC p) <=> n = m
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/arithmetic/ADD__CLAUSES_c2: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/RIGHT__ADD__DISTRIB: !p n m. h4/arithmetic/_2A (h4/arithmetic/_2B m n) p = h4/arithmetic/_2B (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm: h4/hrat/TRAT__SUCINT__0: !n. h4/hrat/trat__eq (h4/hrat/trat__sucint n) (h4/pair/_2C n h4/num/0)
% Assm: h4/numpair/invtri0__tupled__primitive__def: h4/numpair/invtri0__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!a n. ~h4/prim__rec/_3C n (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) ==> R (h4/pair/_2C (h4/arithmetic/_2D n (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/pair/_2C n a)))) (\invtri0__tupled a_27. h4/pair/pair__CASE a_27 (\n a. h4/combin/I (h4/bool/COND (h4/prim__rec/_3C n (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/pair/_2C n a) (invtri0__tupled (h4/pair/_2C (h4/arithmetic/_2D n (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))))))
% Assm: h4/hrat/TRAT__SUCINT_c0: h4/hrat/trat__eq (h4/hrat/trat__sucint h4/num/0) h4/hrat/trat__1
% Assm: h4/hrat/TRAT__SUCINT_c1: !n. h4/hrat/trat__eq (h4/hrat/trat__sucint (h4/num/SUC n)) (h4/hrat/trat__add (h4/hrat/trat__sucint n) h4/hrat/trat__1)
% Assm: h4/hrat/trat__10: h4/hrat/trat__1 = h4/pair/_2C h4/num/0 h4/num/0
% Assm: h4/hrat/trat__sucint0_c0: h4/hrat/trat__sucint h4/num/0 = h4/hrat/trat__1
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/hrat/trat__sucint0_c1: !n. h4/hrat/trat__sucint (h4/num/SUC n) = h4/hrat/trat__add (h4/hrat/trat__sucint n) h4/hrat/trat__1
% Assm: h4/arithmetic/ADD1: !m. h4/num/SUC m = h4/arithmetic/_2B m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/arithmetic/LESS__LESS__CASES: !n m. m = n \/ h4/prim__rec/_3C m n \/ h4/prim__rec/_3C n m
% Assm: h4/arithmetic/LESS__ADD__1: !n m. h4/prim__rec/_3C n m ==> (?p. m = h4/arithmetic/_2B n (h4/arithmetic/_2B p (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% 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/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/arithmetic/ADD__INV__0__EQ: !n m. h4/arithmetic/_2B m n = m <=> n = h4/num/0
% Assm: h4/arithmetic/LEFT__ADD__DISTRIB: !p n m. h4/arithmetic/_2A p (h4/arithmetic/_2B m n) = h4/arithmetic/_2B (h4/arithmetic/_2A p m) (h4/arithmetic/_2A p n)
% Assm: h4/bool/NOT__CLAUSES_c0: !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__MONO__MULT: !p n m. h4/arithmetic/_3C_3D m n ==> h4/arithmetic/_3C_3D (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm: h4/arithmetic/RIGHT__SUB__DISTRIB: !p n m. h4/arithmetic/_2A (h4/arithmetic/_2D m n) p = h4/arithmetic/_2D (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% 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/SUB__ADD: !n m. h4/arithmetic/_3C_3D n m ==> h4/arithmetic/_2B (h4/arithmetic/_2D m n) n = m
% Assm: h4/arithmetic/SUB__EQ__0: !n m. h4/arithmetic/_2D m n = h4/num/0 <=> h4/arithmetic/_3C_3D m n
% Assm: h4/arithmetic/MULT__RIGHT__1: !m. h4/arithmetic/_2A m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm: h4/arithmetic/MULT__SUC: !n m. h4/arithmetic/_2A m (h4/num/SUC n) = h4/arithmetic/_2B m (h4/arithmetic/_2A m n)
% Assm: h4/arithmetic/LESS__ADD__NONZERO: !n m. ~(n = h4/num/0) ==> h4/prim__rec/_3C m (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/ONE: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/numpair/invtri0__def: !n a. h4/numpair/invtri0 n a = h4/bool/COND (h4/prim__rec/_3C n (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/pair/_2C n a) (h4/numpair/invtri0 (h4/arithmetic/_2D n (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm: h4/bool/EQ__EXPAND: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/numpair/invtri0__curried__def: !x1 x. h4/numpair/invtri0 x x1 = h4/numpair/invtri0__tupled (h4/pair/_2C x x1)
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/arithmetic/SUB__RIGHT__LESS: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2D m n) p <=> h4/prim__rec/_3C m (h4/arithmetic/_2B n p) /\ h4/prim__rec/_3C h4/num/0 p
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% 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/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/arithmetic/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/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 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/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/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = 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/prim__rec/WF__LESS: h4/relation/WF h4/prim__rec/_3C
% 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/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/relation/WF__inv__image: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Goal: !c. (?r. h4/hrat/trat__eq r r /\ c = h4/hrat/trat__eq r) <=> h4/hrat/hrat__REP__CLASS (h4/hrat/hrat__ABS__CLASS c) = c
%   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_hrats_hratu_u_TYu_u_DEF]: !_0. (!c. happ _0 c <=> (?r. happ (h4/hrat/trat__eq r) r /\ c = h4/hrat/trat__eq r)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep)
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% Assm [h4s_hrats_hratu_u_bijectionsu_c1]: !r. (?r0. happ (h4/hrat/trat__eq r0) r0 /\ r = h4/hrat/trat__eq r0) <=> h4/hrat/hrat__REP__CLASS (h4/hrat/hrat__ABS__CLASS r) = r
% Assm [h4s_hrats_hratu_u_bijectionsu_c0]: !a. h4/hrat/hrat__ABS__CLASS (h4/hrat/hrat__REP__CLASS a) = a
% Assm [h4s_hrats_TRATu_u_EQu_u_EQUIV]: !q p. happ (h4/hrat/trat__eq p) q <=> h4/hrat/trat__eq p = h4/hrat/trat__eq q
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_hrats_hratu_u_ABSu_u_REPu_u_CLASSu_c0]: !a. h4/hrat/hrat__ABS__CLASS (h4/hrat/hrat__REP__CLASS a) = a
% Assm [h4s_hrats_TRATu_u_EQu_u_REFL]: !p. happ (h4/hrat/trat__eq p) p
% Assm [h4s_quotients_EQUIVu_u_IMPu_u_PARTIALu_u_EQUIV]: !R. h4/quotient/EQUIV R ==> h4/quotient/PARTIAL__EQUIV R
% Assm [h4s_quotients_PARTIALu_u_EQUIVu_u_def]: !R. h4/quotient/PARTIAL__EQUIV R <=> (?x. happ (happ R x) x) /\ (!x y. happ (happ R x) y <=> happ (happ R x) x /\ happ (happ R y) y /\ happ R x = happ R y)
% Assm [h4s_quotients_EQUIVu_u_def]: !E. h4/quotient/EQUIV E <=> (!x y. happ (happ E x) y <=> happ E x = happ E y)
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_hrats_tratu_u_eq0]: !y_27 y x_27 x. happ (h4/hrat/trat__eq (h4/pair/_2C x y)) (h4/pair/_2C x_27 y_27) <=> h4/arithmetic/_2A (h4/num/SUC x) (h4/num/SUC y_27) = h4/arithmetic/_2A (h4/num/SUC x_27) (h4/num/SUC y)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_hrats_TRATu_u_EQu_u_TRANS]: !r q p. happ (h4/hrat/trat__eq p) q /\ happ (h4/hrat/trat__eq q) r ==> happ (h4/hrat/trat__eq p) r
% Assm [h4s_hrats_TRATu_u_EQu_u_AP]: !q p. p = q ==> happ (h4/hrat/trat__eq p) q
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_hrats_TRATu_u_EQu_u_SYM]: !q p. happ (h4/hrat/trat__eq p) q <=> happ (h4/hrat/trat__eq q) p
% Assm [h4s_arithmetics_MULTu_u_SYM]: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm [h4s_hrats_TRATu_u_ADDu_u_WELLDEFINED]: !r q p. happ (h4/hrat/trat__eq p) q ==> happ (h4/hrat/trat__eq (h4/hrat/trat__add p r)) (h4/hrat/trat__add q r)
% Assm [h4s_hrats_TRATu_u_MULu_u_WELLDEFINED]: !r q p. happ (h4/hrat/trat__eq p) q ==> happ (h4/hrat/trat__eq (h4/hrat/trat__mul p r)) (h4/hrat/trat__mul q r)
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_hrats_TRATu_u_MULu_u_SYMu_u_EQ]: !i h. h4/hrat/trat__mul h i = h4/hrat/trat__mul i h
% Assm [h4s_hrats_TRATu_u_ADDu_u_SYMu_u_EQ]: !i h. h4/hrat/trat__add h i = h4/hrat/trat__add i h
% Assm [h4s_hrats_TRATu_u_INVu_u_WELLDEFINED]: !q p. happ (h4/hrat/trat__eq p) q ==> happ (h4/hrat/trat__eq (h4/hrat/trat__inv p)) (h4/hrat/trat__inv q)
% Assm [h4s_hrats_TRATu_u_MULu_u_WELLDEFINED2]: !q2 q1 p2 p1. happ (h4/hrat/trat__eq p1) p2 /\ happ (h4/hrat/trat__eq q1) q2 ==> happ (h4/hrat/trat__eq (h4/hrat/trat__mul p1 q1)) (h4/hrat/trat__mul p2 q2)
% Assm [h4s_hrats_TRATu_u_ADDu_u_WELLDEFINED2]: !q2 q1 p2 p1. happ (h4/hrat/trat__eq p1) p2 /\ happ (h4/hrat/trat__eq q1) q2 ==> happ (h4/hrat/trat__eq (h4/hrat/trat__add p1 q1)) (h4/hrat/trat__add p2 q2)
% Assm [h4s_hrats_TRATu_u_ADDu_u_TOTAL]: !i h. happ (h4/hrat/trat__eq h) i \/ (?d. happ (h4/hrat/trat__eq h) (h4/hrat/trat__add i d)) \/ (?d. happ (h4/hrat/trat__eq i) (h4/hrat/trat__add h d))
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_arithmetics_MULTu_u_ASSOC]: !p n m. h4/arithmetic/_2A m (h4/arithmetic/_2A n p) = h4/arithmetic/_2A (h4/arithmetic/_2A m n) p
% Assm [h4s_hrats_tratu_u_inv0]: !y x. h4/hrat/trat__inv (h4/pair/_2C x y) = h4/pair/_2C y x
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2A h4/num/0 m = h4/num/0
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c4]: !n m. h4/arithmetic/_2A (h4/num/SUC m) n = h4/arithmetic/_2B (h4/arithmetic/_2A m n) n
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c3]: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_primu_u_recs_PRE0u_c1]: !m. h4/prim__rec/PRE (h4/num/SUC m) = 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_hrats_tratu_u_mul0]: !y_27 y x_27 x. h4/hrat/trat__mul (h4/pair/_2C x y) (h4/pair/_2C x_27 y_27) = h4/pair/_2C (h4/prim__rec/PRE (h4/arithmetic/_2A (h4/num/SUC x) (h4/num/SUC x_27))) (h4/prim__rec/PRE (h4/arithmetic/_2A (h4/num/SUC y) (h4/num/SUC y_27)))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c4]: !t. t \/ t <=> t
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c5]: !n m. h4/arithmetic/_2A m (h4/num/SUC n) = h4/arithmetic/_2B m (h4/arithmetic/_2A m n)
% Assm [h4s_arithmetics_numu_u_CASES]: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(h4/num/SUC n = h4/num/0)
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_hrats_TRATu_u_MULu_u_SYM]: !i h. happ (h4/hrat/trat__eq (h4/hrat/trat__mul h i)) (h4/hrat/trat__mul i h)
% Assm [h4s_hrats_TRATu_u_MULu_u_LINV]: !h. happ (h4/hrat/trat__eq (h4/hrat/trat__mul (h4/hrat/trat__inv h) h)) h4/hrat/trat__1
% Assm [h4s_hrats_TRATu_u_ADDu_u_SYM]: !i h. happ (h4/hrat/trat__eq (h4/hrat/trat__add h i)) (h4/hrat/trat__add i h)
% Assm [h4s_hrats_TRATu_u_ADDu_u_ASSOC]: !j i h. happ (h4/hrat/trat__eq (h4/hrat/trat__add h (h4/hrat/trat__add i j))) (h4/hrat/trat__add (h4/hrat/trat__add h i) j)
% Assm [h4s_hrats_TRATu_u_ARCH]: !h. ?n d. happ (h4/hrat/trat__eq (h4/hrat/trat__sucint n)) (h4/hrat/trat__add h d)
% Assm [h4s_hrats_TRATu_u_MULu_u_LID]: !h. happ (h4/hrat/trat__eq (h4/hrat/trat__mul h4/hrat/trat__1 h)) h
% Assm [h4s_hrats_TRATu_u_NOZERO]: !i h. ~happ (h4/hrat/trat__eq (h4/hrat/trat__add h i)) h
% Assm [h4s_hrats_TRATu_u_MULu_u_ASSOC]: !j i h. happ (h4/hrat/trat__eq (h4/hrat/trat__mul h (h4/hrat/trat__mul i j))) (h4/hrat/trat__mul (h4/hrat/trat__mul h i) j)
% Assm [h4s_hrats_TRATu_u_LDISTRIB]: !j i h. happ (h4/hrat/trat__eq (h4/hrat/trat__mul h (h4/hrat/trat__add i j))) (h4/hrat/trat__add (h4/hrat/trat__mul h i) (h4/hrat/trat__mul h j))
% Assm [h4s_hrats_tratu_u_add0]: !y_27 y x_27 x. h4/hrat/trat__add (h4/pair/_2C x y) (h4/pair/_2C x_27 y_27) = h4/pair/_2C (h4/prim__rec/PRE (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/num/SUC x) (h4/num/SUC y_27)) (h4/arithmetic/_2A (h4/num/SUC x_27) (h4/num/SUC y)))) (h4/prim__rec/PRE (h4/arithmetic/_2A (h4/num/SUC y) (h4/num/SUC y_27)))
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_arithmetics_MULTu_u_SUCu_u_EQ]: !p n m. h4/arithmetic/_2A n (h4/num/SUC p) = h4/arithmetic/_2A m (h4/num/SUC p) <=> n = m
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c2]: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_arithmetics_RIGHTu_u_ADDu_u_DISTRIB]: !p n m. h4/arithmetic/_2A (h4/arithmetic/_2B m n) p = h4/arithmetic/_2B (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm [h4s_hrats_TRATu_u_SUCINTu_u_0]: !n. happ (h4/hrat/trat__eq (h4/hrat/trat__sucint n)) (h4/pair/_2C n h4/num/0)
% Assm [h4s_numpairs_invtri0u_u_tupledu_u_primitiveu_u_def]: !_4. (!invtri0__tupled n a. happ (happ (happ _4 invtri0__tupled) n) a = h4/combin/I (h4/bool/COND (happ (happ h4/prim__rec/_3C n) (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/pair/_2C n a) (happ invtri0__tupled (h4/pair/_2C (h4/arithmetic/_2D n (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))))) ==> (!_3. (!invtri0__tupled n. happ (happ _3 invtri0__tupled) n = happ (happ _4 invtri0__tupled) n) ==> (!_2. (!invtri0__tupled a_27. happ (happ _2 invtri0__tupled) a_27 = h4/pair/pair__CASE a_27 (happ _3 invtri0__tupled)) ==> (!_1. (!invtri0__tupled. happ _1 invtri0__tupled = happ _2 invtri0__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!a n. ~happ (happ h4/prim__rec/_3C n) (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) ==> happ (happ R (h4/pair/_2C (h4/arithmetic/_2D n (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))) (h4/pair/_2C n a))) ==> h4/numpair/invtri0__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))
% Assm [h4s_hrats_TRATu_u_SUCINTu_c0]: happ (h4/hrat/trat__eq (h4/hrat/trat__sucint h4/num/0)) h4/hrat/trat__1
% Assm [h4s_hrats_TRATu_u_SUCINTu_c1]: !n. happ (h4/hrat/trat__eq (h4/hrat/trat__sucint (h4/num/SUC n))) (h4/hrat/trat__add (h4/hrat/trat__sucint n) h4/hrat/trat__1)
% Assm [h4s_hrats_tratu_u_10]: h4/hrat/trat__1 = h4/pair/_2C h4/num/0 h4/num/0
% Assm [h4s_hrats_tratu_u_sucint0u_c0]: h4/hrat/trat__sucint h4/num/0 = h4/hrat/trat__1
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_hrats_tratu_u_sucint0u_c1]: !n. h4/hrat/trat__sucint (h4/num/SUC n) = h4/hrat/trat__add (h4/hrat/trat__sucint n) h4/hrat/trat__1
% Assm [h4s_arithmetics_ADD1]: !m. h4/num/SUC m = h4/arithmetic/_2B m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_arithmetics_LESSu_u_LESSu_u_CASES]: !n m. m = n \/ happ (happ h4/prim__rec/_3C m) n \/ happ (happ h4/prim__rec/_3C n) m
% Assm [h4s_arithmetics_LESSu_u_ADDu_u_1]: !n m. happ (happ h4/prim__rec/_3C n) m ==> (?p. m = h4/arithmetic/_2B n (h4/arithmetic/_2B p (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% 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_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_arithmetics_ADDu_u_INVu_u_0u_u_EQ]: !n m. h4/arithmetic/_2B m n = m <=> n = h4/num/0
% Assm [h4s_arithmetics_LEFTu_u_ADDu_u_DISTRIB]: !p n m. h4/arithmetic/_2A p (h4/arithmetic/_2B m n) = h4/arithmetic/_2B (h4/arithmetic/_2A p m) (h4/arithmetic/_2A p n)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !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_MONOu_u_MULT]: !p n m. h4/arithmetic/_3C_3D m n ==> h4/arithmetic/_3C_3D (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm [h4s_arithmetics_RIGHTu_u_SUBu_u_DISTRIB]: !p n m. h4/arithmetic/_2A (h4/arithmetic/_2D m n) p = h4/arithmetic/_2D (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm [h4s_arithmetics_LESSu_u_IMPu_u_LESSu_u_ORu_u_EQ]: !n m. happ (happ h4/prim__rec/_3C m) n ==> h4/arithmetic/_3C_3D m n
% Assm [h4s_arithmetics_SUBu_u_ADD]: !n m. h4/arithmetic/_3C_3D n m ==> h4/arithmetic/_2B (h4/arithmetic/_2D m n) n = m
% Assm [h4s_arithmetics_SUBu_u_EQu_u_0]: !n m. h4/arithmetic/_2D m n = h4/num/0 <=> h4/arithmetic/_3C_3D m n
% Assm [h4s_arithmetics_MULTu_u_RIGHTu_u_1]: !m. h4/arithmetic/_2A m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm [h4s_arithmetics_MULTu_u_SUC]: !n m. h4/arithmetic/_2A m (h4/num/SUC n) = h4/arithmetic/_2B m (h4/arithmetic/_2A m n)
% Assm [h4s_arithmetics_LESSu_u_ADDu_u_NONZERO]: !n m. ~(n = h4/num/0) ==> happ (happ h4/prim__rec/_3C m) (h4/arithmetic/_2B m n)
% Assm [h4s_arithmetics_ONE]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% 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_numpairs_invtri0u_u_def]: !n a. h4/numpair/invtri0 n a = h4/bool/COND (happ (happ h4/prim__rec/_3C n) (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/pair/_2C n a) (h4/numpair/invtri0 (h4/arithmetic/_2D n (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/arithmetic/_2B a (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm [h4s_bools_EQu_u_EXPAND]: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_numpairs_invtri0u_u_curriedu_u_def]: !x1 x. h4/numpair/invtri0 x x1 = happ h4/numpair/invtri0__tupled (h4/pair/_2C x x1)
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_arithmetics_SUBu_u_RIGHTu_u_LESS]: !p n m. happ (happ h4/prim__rec/_3C (h4/arithmetic/_2D m n)) p <=> happ (happ h4/prim__rec/_3C m) (h4/arithmetic/_2B n p) /\ happ (happ h4/prim__rec/_3C h4/num/0) p
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% 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_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_arithmetics_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_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 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_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. happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = 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_primu_u_recs_WFu_u_LESS]: h4/relation/WF h4/prim__rec/_3C
% 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_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% 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_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% 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_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Goal: !c. (?r. happ (h4/hrat/trat__eq r) r /\ c = h4/hrat/trat__eq r) <=> h4/hrat/hrat__REP__CLASS (h4/hrat/hrat__ABS__CLASS c) = c
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_Q1164541,TV_Q1164537]: ![V_f, V_g]: (![V_x]: s(TV_Q1164537,happ(s(t_fun(TV_Q1164541,TV_Q1164537),V_f),s(TV_Q1164541,V_x))) = s(TV_Q1164537,happ(s(t_fun(TV_Q1164541,TV_Q1164537),V_g),s(TV_Q1164541,V_x))) => s(t_fun(TV_Q1164541,TV_Q1164537),V_f) = s(t_fun(TV_Q1164541,TV_Q1164537),V_g))).
fof(ah4s_hrats_hratu_u_TYu_u_DEF, axiom, ![V_uu_0]: (![V_c]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),V_c)))) <=> ?[V_r]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r)))) & s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),V_c) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),t_bool),V_uu_0),s(t_fun(t_h4s_hrats_hrat,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),V_rep)))))).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))) = s(TV_u_27a,V_r))))).
fof(ah4s_hrats_hratu_u_bijectionsu_c1, axiom, ![V_r]: (?[V_r0]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r0))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r0)))) & s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),V_r) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r0)))) <=> s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_hratu_u_repu_u_class(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_absu_u_class(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),V_r))))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),V_r))).
fof(ah4s_hrats_hratu_u_bijectionsu_c0, axiom, ![V_a]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_absu_u_class(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_hratu_u_repu_u_class(s(t_h4s_hrats_hrat,V_a))))) = s(t_h4s_hrats_hrat,V_a)).
fof(ah4s_hrats_TRATu_u_EQu_u_EQUIV, axiom, ![V_q, V_p]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q)))) <=> s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q))))).
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_hrats_hratu_u_ABSu_u_REPu_u_CLASSu_c0, axiom, ![V_a]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_absu_u_class(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_hratu_u_repu_u_class(s(t_h4s_hrats_hrat,V_a))))) = s(t_h4s_hrats_hrat,V_a)).
fof(ah4s_hrats_TRATu_u_EQu_u_REFL, axiom, ![V_p]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))))).
fof(ah4s_quotients_EQUIVu_u_IMPu_u_PARTIALu_u_EQUIV, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_quotients_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => p(s(t_bool,h4s_quotients_partialu_u_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ah4s_quotients_PARTIALu_u_EQUIVu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_quotients_partialu_u_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_x)))) & ![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_x))),s(TV_u_27a,V_y)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_x)))) & (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_y)))) & s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))) = s(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))))))))).
fof(ah4s_quotients_EQUIVu_u_def, axiom, ![TV_u_27a]: ![V_E]: (p(s(t_bool,h4s_quotients_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E)))) <=> ![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_E),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E),s(TV_u_27a,V_y)))))).
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_hrats_tratu_u_eq0, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_y))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_xu_27),s(t_h4s_nums_num,V_yu_27)))))) <=> s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_x))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_yu_27))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_xu_27))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_y))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_hrats_TRATu_u_EQu_u_TRANS, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r)))))).
fof(ah4s_hrats_TRATu_u_EQu_u_AP, axiom, ![V_q, V_p]: (s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q)))))).
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_hrats_TRATu_u_EQu_u_SYM, axiom, ![V_q, V_p]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q))) = s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p)))).
fof(ah4s_arithmetics_MULTu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_hrats_TRATu_u_ADDu_u_WELLDEFINED, axiom, ![V_r, V_q, V_p]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q)))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r)))))))).
fof(ah4s_hrats_TRATu_u_MULu_u_WELLDEFINED, axiom, ![V_r, V_q, V_p]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q)))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r)))))))).
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_hrats_TRATu_u_MULu_u_SYMu_u_EQ, axiom, ![V_i, V_h]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h)))).
fof(ah4s_hrats_TRATu_u_ADDu_u_SYMu_u_EQ, axiom, ![V_i, V_h]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h)))).
fof(ah4s_hrats_TRATu_u_INVu_u_WELLDEFINED, axiom, ![V_q, V_p]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q)))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_inv(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_inv(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q)))))))).
fof(ah4s_hrats_TRATu_u_MULu_u_WELLDEFINED2, axiom, ![V_q2, V_q1, V_p2, V_p1]: ((p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q2))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p1),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q1))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p2),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q2)))))))).
fof(ah4s_hrats_TRATu_u_ADDu_u_WELLDEFINED2, axiom, ![V_q2, V_q1, V_p2, V_p1]: ((p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q2))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p1),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q1))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p2),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q2)))))))).
fof(ah4s_hrats_TRATu_u_ADDu_u_TOTAL, axiom, ![V_i, V_h]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i)))) | (?[V_d]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_d)))))) | ?[V_d]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_d))))))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_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_arithmetics_MULTu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_hrats_tratu_u_inv0, axiom, ![V_y, V_x]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_inv(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_y))))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_y),s(t_h4s_nums_num,V_x)))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c4, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c3, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_primu_u_recs_PRE0u_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_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_hrats_tratu_u_mul0, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_y))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_xu_27),s(t_h4s_nums_num,V_yu_27))))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_x))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_xu_27))))))),s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_y))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_yu_27)))))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_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_ORu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c5, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(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,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_numu_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | ?[V_n]: s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))).
fof(ah4s_nums_NOTu_u_SUC, axiom, ![V_n]: ~ (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_hrats_TRATu_u_MULu_u_SYM, axiom, ![V_i, V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h))))))).
fof(ah4s_hrats_TRATu_u_MULu_u_LINV, axiom, ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_inv(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_1))))).
fof(ah4s_hrats_TRATu_u_ADDu_u_SYM, axiom, ![V_i, V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h))))))).
fof(ah4s_hrats_TRATu_u_ADDu_u_ASSOC, axiom, ![V_j, V_i, V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_j))))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_j))))))).
fof(ah4s_hrats_TRATu_u_ARCH, axiom, ![V_h]: ?[V_n, V_d]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_sucint(s(t_h4s_nums_num,V_n))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_d))))))).
fof(ah4s_hrats_TRATu_u_MULu_u_LID, axiom, ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_1),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h))))).
fof(ah4s_hrats_TRATu_u_NOZERO, axiom, ![V_i, V_h]: ~ (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h)))))).
fof(ah4s_hrats_TRATu_u_MULu_u_ASSOC, axiom, ![V_j, V_i, V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_j))))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_j))))))).
fof(ah4s_hrats_TRATu_u_LDISTRIB, axiom, ![V_j, V_i, V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_j))))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_i))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_h),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_j))))))))).
fof(ah4s_hrats_tratu_u_add0, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_y))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_xu_27),s(t_h4s_nums_num,V_yu_27))))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_x))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_yu_27))))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_xu_27))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_y))))))))),s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_y))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_yu_27)))))))))).
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_pairs_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_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_MULTu_u_SUCu_u_EQ, axiom, ![V_p, V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_p))))) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,V_m))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_RIGHTu_u_ADDu_u_DISTRIB, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(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))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_hrats_TRATu_u_SUCINTu_u_0, axiom, ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_sucint(s(t_h4s_nums_num,V_n))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))))))).
fof(ah4s_numpairs_invtri0u_u_tupledu_u_primitiveu_u_def, axiom, ![V_uu_4]: (![V_invtri0u_u_tupled, V_n, V_a]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_invtri0u_u_tupled))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_a))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_combins_i(s(t_h4s_pairs_prod(t_h4s_nums_num,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_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a),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_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_a))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_invtri0u_u_tupled),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a),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,V_a),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))))))) => ![V_uu_3]: (![V_invtri0u_u_tupled, V_n]: s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_invtri0u_u_tupled))),s(t_h4s_nums_num,V_n))) = s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_invtri0u_u_tupled))),s(t_h4s_nums_num,V_n))) => ![V_uu_2]: (![V_invtri0u_u_tupled, V_au_27]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_invtri0u_u_tupled))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_au_27))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_au_27),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_invtri0u_u_tupled))))) => ![V_uu_1]: (![V_invtri0u_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_invtri0u_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_invtri0u_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),V_R)))) & ![V_a, V_n]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a),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,V_a),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_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_a))))))))) => s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_numpairs_invtri0u_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),V_uu_1))))))))).
fof(ah4s_hrats_TRATu_u_SUCINTu_c0, axiom, p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_sucint(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_1))))).
fof(ah4s_hrats_TRATu_u_SUCINTu_c1, axiom, ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_sucint(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_sucint(s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_1))))))).
fof(ah4s_hrats_tratu_u_10, axiom, s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_1) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_hrats_tratu_u_sucint0u_c0, axiom, s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_sucint(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_1)).
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_hrats_tratu_u_sucint0u_c1, axiom, ![V_n]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_sucint(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_sucint(s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_hrats_tratu_u_1)))).
fof(ah4s_arithmetics_ADD1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,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_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
fof(ah4s_arithmetics_LESSu_u_LESSu_u_CASES, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n) | (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))) | 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_n))),s(t_h4s_nums_num,V_m))))))).
fof(ah4s_arithmetics_LESSu_u_ADDu_u_1, 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_n))),s(t_h4s_nums_num,V_m)))) => ?[V_p]: 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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))).
fof(ah4s_arithmetics_ADDu_u_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_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_ADDu_u_INVu_u_0u_u_EQ, 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,V_m) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_arithmetics_LEFTu_u_ADDu_u_DISTRIB, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(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_n))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_p),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_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_MONOu_u_MULT, 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,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))))).
fof(ah4s_arithmetics_RIGHTu_u_SUBu_u_DISTRIB, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_LESSu_u_IMPu_u_LESSu_u_ORu_u_EQ, 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_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_SUBu_u_ADD, 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)))) => s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2d(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_SUBu_u_EQu_u_0, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0) <=> 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_MULTu_u_RIGHTu_u_1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_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)).
fof(ah4s_arithmetics_MULTu_u_SUC, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(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,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_LESSu_u_ADDu_u_NONZERO, axiom, ![V_n, V_m]: (~ (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) => 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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_arithmetics_ONE, axiom, 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_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
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_numpairs_invtri0u_u_def, axiom, ![V_n, V_a]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_numpairs_invtri0(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_a))) = s(t_h4s_pairs_prod(t_h4s_nums_num,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_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a),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_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_a))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_numpairs_invtri0(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a),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,V_a),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_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_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_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_numpairs_invtri0u_u_curriedu_u_def, axiom, ![V_x1, V_x]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_numpairs_invtri0(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_x1))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_numpairs_invtri0u_u_tupled),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_x1)))))).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_arithmetics_SUBu_u_RIGHTu_u_LESS, axiom, ![V_p, 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_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,V_p)))) <=> (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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,V_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_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_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_arithmetics_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_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_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_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,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_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_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_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_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_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_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_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_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_bools_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_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_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_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_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(ch4s_hrats_hratu_u_ABSu_u_REPu_u_CLASSu_c1, conjecture, ![V_c]: (?[V_r]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r)))) & s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),V_c) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_tratu_u_eq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r)))) <=> s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_hrats_hratu_u_repu_u_class(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_absu_u_class(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),V_c))))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),V_c))).
