%   ORIGINAL: h4/hrat/TRAT__MUL__LID
% 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/prim__rec/num__Axiom: !f e. ?fn. fn h4/num/0 = e /\ (!n. fn (h4/num/SUC n) = f n (fn n))
% 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__EQ__AP: !q p. p = q ==> h4/hrat/trat__eq p q
% 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/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/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% 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/hrat/TRAT__EQ__REFL: !p. h4/hrat/trat__eq p p
% 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/arithmetic/MULT__SYM: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% 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/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__ADD__SYM__EQ: !i h. h4/hrat/trat__add h i = h4/hrat/trat__add i h
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% 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__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__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__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/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% 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/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% 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__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__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__ADD__SYM: !i h. h4/hrat/trat__eq (h4/hrat/trat__add h i) (h4/hrat/trat__add i h)
% Assm: h4/hrat/TRAT__EQ__SYM: !q p. h4/hrat/trat__eq p q <=> h4/hrat/trat__eq q p
% 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/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/arithmetic/ADD__CLAUSES_c1: !m. h4/arithmetic/_2B m h4/num/0 = m
% 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/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/MULT__CLAUSES_c0: !m. h4/arithmetic/_2A h4/num/0 m = h4/num/0
% Assm: h4/arithmetic/MULT__CLAUSES_c1: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm: h4/arithmetic/ADD__CLAUSES_c3: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/num__CASES: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm: h4/prim__rec/PRE0_c1: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm: h4/bool/OR__CLAUSES_c4: !t. t \/ t <=> t
% Assm: h4/hrat/trat__inv0: !y x. h4/hrat/trat__inv (h4/pair/_2C x y) = h4/pair/_2C y x
% 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/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% 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/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% 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/arithmetic/ADD__CLAUSES_c2: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/AND__CLAUSES_c4: !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/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% 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/AND__CLAUSES_c1: !t. t /\ T <=> t
% 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/rich__list/ZIP__COUNT__LIST: !n l1. n = h4/list/LENGTH l1 ==> h4/list/ZIP (h4/pair/_2C l1 (h4/rich__list/COUNT__LIST n)) = h4/list/GENLIST (\n0. h4/pair/_2C (h4/list/EL n0 l1) n0) (h4/list/LENGTH l1)
% Assm: h4/arithmetic/findq__eq__0: !n m a. h4/arithmetic/findq (h4/pair/_2C a (h4/pair/_2C m n)) = h4/num/0 <=> a = h4/num/0
% 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/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/rich__list/LENGTH__COUNT__LIST: !n. h4/list/LENGTH (h4/rich__list/COUNT__LIST n) = n
% Assm: h4/rich__list/EL__COUNT__LIST: !n m. h4/prim__rec/_3C m n ==> h4/list/EL m (h4/rich__list/COUNT__LIST n) = m
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/list/EL__ZIP: !n l2 l1. h4/list/LENGTH l1 = h4/list/LENGTH l2 /\ h4/prim__rec/_3C n (h4/list/LENGTH l1) ==> h4/list/EL n (h4/list/ZIP (h4/pair/_2C l1 l2)) = h4/pair/_2C (h4/list/EL n l1) (h4/list/EL n l2)
% Assm: h4/list/LENGTH__GENLIST: !n f. h4/list/LENGTH (h4/list/GENLIST f n) = n
% Assm: h4/list/LENGTH__ZIP: !l2 l1. h4/list/LENGTH l1 = h4/list/LENGTH l2 ==> h4/list/LENGTH (h4/list/ZIP (h4/pair/_2C l1 l2)) = h4/list/LENGTH l1 /\ h4/list/LENGTH (h4/list/ZIP (h4/pair/_2C l1 l2)) = h4/list/LENGTH l2
% Assm: h4/list/EL__GENLIST: !x n f. h4/prim__rec/_3C x n ==> h4/list/EL x (h4/list/GENLIST f n) = f x
% Assm: h4/list/LIST__EQ__REWRITE: !l2 l1. l1 = l2 <=> h4/list/LENGTH l1 = h4/list/LENGTH l2 /\ (!x. h4/prim__rec/_3C x (h4/list/LENGTH l1) ==> h4/list/EL x l1 = h4/list/EL x l2)
% Assm: h4/pair/CLOSED__PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/arithmetic/SUB__LEFT__LESS: !p n m. h4/prim__rec/_3C m (h4/arithmetic/_2D n p) <=> h4/prim__rec/_3C (h4/arithmetic/_2B m p) n
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% 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/arithmetic/SUB__LEFT__ADD: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2D n p) = h4/bool/COND (h4/arithmetic/_3C_3D n p) m (h4/arithmetic/_2D (h4/arithmetic/_2B m n) 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/arithmetic/LE__MULT__RCANCEL: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2A m n) (h4/arithmetic/_2A p n) <=> n = h4/num/0 \/ h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/NOT__SUC__LESS__EQ__0: !n. ~h4/arithmetic/_3C_3D (h4/num/SUC n) h4/num/0
% Assm: h4/arithmetic/MULT__EQ__0: !n m. h4/arithmetic/_2A m n = h4/num/0 <=> m = h4/num/0 \/ n = h4/num/0
% Assm: h4/arithmetic/findq__thm: !n m a. h4/arithmetic/findq (h4/pair/_2C a (h4/pair/_2C m n)) = h4/bool/COND (n = h4/num/0) a (h4/bool/LET (\d. h4/bool/COND (h4/prim__rec/_3C m d) a (h4/arithmetic/findq (h4/pair/_2C (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) a) (h4/pair/_2C m d)))) (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n))
% Assm: h4/arithmetic/LT__MULT__RCANCEL: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2A m n) (h4/arithmetic/_2A p n) <=> h4/prim__rec/_3C h4/num/0 n /\ h4/prim__rec/_3C m p
% Assm: h4/arithmetic/LESS__EQ__LESS__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/prim__rec/_3C n p ==> h4/prim__rec/_3C m p
% Assm: h4/arithmetic/NOT__LESS__EQUAL: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C n m
% Assm: h4/arithmetic/LESS__EQ__MONO: !n m. h4/arithmetic/_3C_3D (h4/num/SUC n) (h4/num/SUC m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/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/LT__ADD__LCANCEL: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2B p m) (h4/arithmetic/_2B p n) <=> h4/prim__rec/_3C m n
% Assm: h4/arithmetic/COMPLETE__INDUCTION: !P. (!n. (!m. h4/prim__rec/_3C m n ==> P m) ==> P n) ==> (!n. P n)
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/MULT__LEFT__1: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/ADD__COMM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/arithmetic/NOT__ZERO__LT__ZERO: !n. ~(n = h4/num/0) <=> h4/prim__rec/_3C h4/num/0 n
% Assm: h4/arithmetic/TWO: h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO) = h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/arithmetic/ONE: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm: h4/bool/RIGHT__FORALL__IMP__THM: !Q P. (!x. P ==> Q x) <=> P ==> (!x. Q x)
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/arithmetic/LESS__MONO__EQ: !n m. h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/DISJ__COMM: !B A. A \/ B <=> B \/ A
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Goal: !h. h4/hrat/trat__eq (h4/hrat/trat__mul h4/hrat/trat__1 h) h
%   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_primu_u_recs_numu_u_Axiom]: !f e. ?fn. happ fn h4/num/0 = e /\ (!n. happ fn (h4/num/SUC n) = happ (happ f n) (happ fn n))
% 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_EQu_u_AP]: !q p. p = q ==> h4/hrat/trat__eq p q
% 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_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_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% 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_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 [h4s_hrats_TRATu_u_EQu_u_REFL]: !p. h4/hrat/trat__eq p p
% Assm [h4s_hrats_TRATu_u_MULu_u_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 [h4s_arithmetics_MULTu_u_SYM]: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% 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_TRUTH]: T
% Assm [h4s_hrats_TRATu_u_EQu_u_TRANS]: !r q p. h4/hrat/trat__eq p q /\ h4/hrat/trat__eq q r ==> h4/hrat/trat__eq p r
% 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_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_hrats_TRATu_u_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 [h4s_hrats_TRATu_u_MULu_u_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 [h4s_hrats_TRATu_u_MULu_u_SYM]: !i h. h4/hrat/trat__eq (h4/hrat/trat__mul h i) (h4/hrat/trat__mul i h)
% Assm [h4s_hrats_TRATu_u_MULu_u_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 [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_hrats_TRATu_u_ADDu_u_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 [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% 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_ADDu_u_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 [h4s_hrats_TRATu_u_INVu_u_WELLDEFINED]: !q p. h4/hrat/trat__eq p q ==> h4/hrat/trat__eq (h4/hrat/trat__inv p) (h4/hrat/trat__inv q)
% Assm [h4s_hrats_TRATu_u_ADDu_u_SYM]: !i h. h4/hrat/trat__eq (h4/hrat/trat__add h i) (h4/hrat/trat__add i h)
% Assm [h4s_hrats_TRATu_u_EQu_u_SYM]: !q p. h4/hrat/trat__eq p q <=> h4/hrat/trat__eq q p
% Assm [h4s_hrats_TRATu_u_ADDu_u_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 [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(h4/num/SUC n = h4/num/0)
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2B m h4/num/0 = m
% 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_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_MULTu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2A h4/num/0 m = h4/num/0
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c3]: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_arithmetics_numu_u_CASES]: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm [h4s_primu_u_recs_PRE0u_c1]: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm [h4s_bools_ORu_u_CLAUSESu_c4]: !t. t \/ t <=> t
% Assm [h4s_hrats_tratu_u_inv0]: !y x. h4/hrat/trat__inv (h4/pair/_2C x y) = h4/pair/_2C y x
% 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_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% 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 (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. ~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_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% 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_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_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c4]: !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_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% 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_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% 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_richu_u_lists_ZIPu_u_COUNTu_u_LIST]: !_0. (!l1 n0. happ (happ _0 l1) n0 = h4/pair/_2C (h4/list/EL n0 l1) n0) ==> (!n l1. n = h4/list/LENGTH l1 ==> h4/list/ZIP (h4/pair/_2C l1 (h4/rich__list/COUNT__LIST n)) = h4/list/GENLIST (happ _0 l1) (h4/list/LENGTH l1))
% Assm [h4s_arithmetics_findqu_u_equ_u_0]: !n m a. h4/arithmetic/findq (h4/pair/_2C a (h4/pair/_2C m n)) = h4/num/0 <=> a = h4/num/0
% 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_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_richu_u_lists_LENGTHu_u_COUNTu_u_LIST]: !n. h4/list/LENGTH (h4/rich__list/COUNT__LIST n) = n
% Assm [h4s_richu_u_lists_ELu_u_COUNTu_u_LIST]: !n m. h4/prim__rec/_3C m n ==> h4/list/EL m (h4/rich__list/COUNT__LIST n) = m
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_lists_ELu_u_ZIP]: !n l2 l1. h4/list/LENGTH l1 = h4/list/LENGTH l2 /\ h4/prim__rec/_3C n (h4/list/LENGTH l1) ==> h4/list/EL n (h4/list/ZIP (h4/pair/_2C l1 l2)) = h4/pair/_2C (h4/list/EL n l1) (h4/list/EL n l2)
% Assm [h4s_lists_LENGTHu_u_GENLIST]: !n f. h4/list/LENGTH (h4/list/GENLIST f n) = n
% Assm [h4s_lists_LENGTHu_u_ZIP]: !l2 l1. h4/list/LENGTH l1 = h4/list/LENGTH l2 ==> h4/list/LENGTH (h4/list/ZIP (h4/pair/_2C l1 l2)) = h4/list/LENGTH l1 /\ h4/list/LENGTH (h4/list/ZIP (h4/pair/_2C l1 l2)) = h4/list/LENGTH l2
% Assm [h4s_lists_ELu_u_GENLIST]: !x n f. h4/prim__rec/_3C x n ==> h4/list/EL x (h4/list/GENLIST f n) = happ f x
% Assm [h4s_lists_LISTu_u_EQu_u_REWRITE]: !l2 l1. l1 = l2 <=> h4/list/LENGTH l1 = h4/list/LENGTH l2 /\ (!x. h4/prim__rec/_3C x (h4/list/LENGTH l1) ==> h4/list/EL x l1 = h4/list/EL x l2)
% Assm [h4s_pairs_CLOSEDu_u_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_LESS]: !p n m. h4/prim__rec/_3C m (h4/arithmetic/_2D n p) <=> h4/prim__rec/_3C (h4/arithmetic/_2B m p) n
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_arithmetics_SUBu_u_RIGHTu_u_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 [h4s_arithmetics_SUBu_u_LEFTu_u_ADD]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2D n p) = h4/bool/COND (h4/arithmetic/_3C_3D n p) m (h4/arithmetic/_2D (h4/arithmetic/_2B m n) p)
% 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_arithmetics_LEu_u_MULTu_u_RCANCEL]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2A m n) (h4/arithmetic/_2A p n) <=> n = h4/num/0 \/ h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_NOTu_u_SUCu_u_LESSu_u_EQu_u_0]: !n. ~h4/arithmetic/_3C_3D (h4/num/SUC n) h4/num/0
% Assm [h4s_arithmetics_MULTu_u_EQu_u_0]: !n m. h4/arithmetic/_2A m n = h4/num/0 <=> m = h4/num/0 \/ n = h4/num/0
% Assm [h4s_arithmetics_findqu_u_thm]: !_0. (!a m d. happ (happ (happ _0 a) m) d = h4/bool/COND (h4/prim__rec/_3C m d) a (h4/arithmetic/findq (h4/pair/_2C (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) a) (h4/pair/_2C m d)))) ==> (!n m a. ?v. (v <=> n = h4/num/0) /\ h4/arithmetic/findq (h4/pair/_2C a (h4/pair/_2C m n)) = h4/bool/COND v a (h4/bool/LET (happ (happ _0 a) m) (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n)))
% Assm [h4s_arithmetics_LTu_u_MULTu_u_RCANCEL]: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2A m n) (h4/arithmetic/_2A p n) <=> h4/prim__rec/_3C h4/num/0 n /\ h4/prim__rec/_3C m p
% Assm [h4s_arithmetics_LESSu_u_EQu_u_LESSu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/prim__rec/_3C n p ==> h4/prim__rec/_3C m p
% 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_LESSu_u_EQu_u_MONO]: !n m. h4/arithmetic/_3C_3D (h4/num/SUC n) (h4/num/SUC m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_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_LTu_u_ADDu_u_LCANCEL]: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2B p m) (h4/arithmetic/_2B p n) <=> h4/prim__rec/_3C m n
% Assm [h4s_arithmetics_COMPLETEu_u_INDUCTION]: !P. (!n. (!m. h4/prim__rec/_3C m n ==> happ P m) ==> happ P n) ==> (!n. happ P n)
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_MULTu_u_LEFTu_u_1]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_ADDu_u_COMM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_arithmetics_NOTu_u_ZEROu_u_LTu_u_ZERO]: !n. ~(n = h4/num/0) <=> h4/prim__rec/_3C h4/num/0 n
% Assm [h4s_arithmetics_TWO]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO) = h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_arithmetics_ONE]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_IMPu_u_THM]: !Q P. (!x. P ==> happ Q x) <=> P ==> (!x. happ Q x)
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_arithmetics_LESSu_u_MONOu_u_EQ]: !n m. h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_DISJu_u_COMM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Goal: !h. h4/hrat/trat__eq (h4/hrat/trat__mul h4/hrat/trat__1 h) h
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_Q1164282,TV_Q1164278]: ![V_f, V_g]: (![V_x]: s(TV_Q1164278,happ(s(t_fun(TV_Q1164282,TV_Q1164278),V_f),s(TV_Q1164282,V_x))) = s(TV_Q1164278,happ(s(t_fun(TV_Q1164282,TV_Q1164278),V_g),s(TV_Q1164282,V_x))) => s(t_fun(TV_Q1164282,TV_Q1164278),V_f) = s(t_fun(TV_Q1164282,TV_Q1164278),V_g))).
fof(ah4s_primu_u_recs_numu_u_Axiom, axiom, ![TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_e) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_nums_num,V_n))),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,V_n))))))).
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_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,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_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_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_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_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_eq0, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: (p(s(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_hrats_TRATu_u_EQu_u_REFL, axiom, ![V_p]: p(s(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_hrats_TRATu_u_MULu_u_WELLDEFINED, axiom, ![V_r, V_q, V_p]: (p(s(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,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_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_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_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,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,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,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_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_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_LDISTRIB, axiom, ![V_j, V_i, V_h]: p(s(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_MULu_u_WELLDEFINED2, axiom, ![V_q2, V_q1, V_p2, V_p1]: ((p(s(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,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,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_MULu_u_SYM, axiom, ![V_i, V_h]: p(s(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_ASSOC, axiom, ![V_j, V_i, V_h]: p(s(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_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_hrats_TRATu_u_ADDu_u_WELLDEFINED, axiom, ![V_r, V_q, V_p]: (p(s(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,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_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_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_ADDu_u_WELLDEFINED2, axiom, ![V_q2, V_q1, V_p2, V_p1]: ((p(s(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,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,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_INVu_u_WELLDEFINED, axiom, ![V_q, V_p]: (p(s(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,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_ADDu_u_SYM, axiom, ![V_i, V_h]: p(s(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_EQu_u_SYM, axiom, ![V_q, V_p]: s(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,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_hrats_TRATu_u_ADDu_u_ASSOC, axiom, ![V_j, V_i, V_h]: p(s(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_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_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_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = 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_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_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_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_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c3, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_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_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_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_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_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_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_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,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,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_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_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_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_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_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_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_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_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_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_richu_u_lists_ZIPu_u_COUNTu_u_LIST, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_l1, V_n0]: s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_uu_0),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_nums_num,V_n0))) = s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),h4s_pairs_u_2c(s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_n0),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_nums_num,V_n0))) => ![V_n, V_l1]: (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1))) => s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,V_n))))))) = s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_uu_0),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1)))))))).
fof(ah4s_arithmetics_findqu_u_equ_u_0, axiom, ![V_n, V_m, V_a]: (s(t_h4s_nums_num,h4s_arithmetics_findq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))) = s(t_h4s_nums_num,h4s_nums_0) <=> s(t_h4s_nums_num,V_a) = s(t_h4s_nums_num,h4s_nums_0))).
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_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_richu_u_lists_LENGTHu_u_COUNTu_u_LIST, axiom, ![V_n]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_richu_u_lists_ELu_u_COUNTu_u_LIST, 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)))) => s(t_h4s_nums_num,h4s_lists_el(s(t_h4s_nums_num,V_m),s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,V_m))).
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_lists_ELu_u_ZIP, axiom, ![TV_u_27a,TV_u_27b]: ![V_n, V_l2, V_l1]: ((s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27b),V_l2))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1))))))) => s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_lists_el(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2))))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(TV_u_27b,h4s_lists_el(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27b),V_l2))))))).
fof(ah4s_lists_LENGTHu_u_GENLIST, axiom, ![TV_u_27a]: ![V_n, V_f]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_lists_LENGTHu_u_ZIP, axiom, ![TV_u_27a,TV_u_27b]: ![V_l2, V_l1]: (s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27b),V_l2))) => (s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2))))))) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1))) & s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2))))))) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27b),V_l2)))))).
fof(ah4s_lists_ELu_u_GENLIST, axiom, ![TV_u_27a]: ![V_x, V_n, V_f]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_n)))) => s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_x),s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_x))))).
fof(ah4s_lists_LISTu_u_EQu_u_REWRITE, axiom, ![TV_u_27a]: ![V_l2, V_l1]: (s(t_h4s_lists_list(TV_u_27a),V_l1) = s(t_h4s_lists_list(TV_u_27a),V_l2) <=> (s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l2))) & ![V_x]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1)))))) => s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_x),s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_x),s(t_h4s_lists_list(TV_u_27a),V_l2))))))).
fof(ah4s_pairs_CLOSEDu_u_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_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_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_SUBu_u_LEFTu_u_LESS, axiom, ![V_p, 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,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_arithmetics_SUBu_u_RIGHTu_u_LESS, axiom, ![V_p, V_n, V_m]: (p(s(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,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,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_p))))))).
fof(ah4s_arithmetics_SUBu_u_LEFTu_u_ADD, 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_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2d(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_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_arithmetics_LEu_u_MULTu_u_RCANCEL, axiom, ![V_p, V_n, V_m]: (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_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_n)))))) <=> (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_p))))))).
fof(ah4s_arithmetics_NOTu_u_SUCu_u_LESSu_u_EQu_u_0, axiom, ![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,h4s_nums_0)))))).
fof(ah4s_arithmetics_MULTu_u_EQu_u_0, 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_nums_0) <=> (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_arithmetics_findqu_u_thm, axiom, ![V_uu_0]: (![V_a, V_m, V_d]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num))),V_uu_0),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_d))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_d))),s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,h4s_arithmetics_findq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_a))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_d))))))))) => ![V_n, V_m, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_nums_num,h4s_arithmetics_findq(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,h4s_bools_let(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num))),V_uu_0),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))))))))).
fof(ah4s_arithmetics_LTu_u_MULTu_u_RCANCEL, axiom, ![V_p, V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(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_p),s(t_h4s_nums_num,V_n)))))) <=> (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_LESSu_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_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
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_LESSu_u_EQu_u_MONO, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_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_LTu_u_ADDu_u_LCANCEL, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_p),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_arithmetics_COMPLETEu_u_INDUCTION, axiom, ![V_P]: (![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))))) => ![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_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_MULTu_u_LEFTu_u_1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_COMM, 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_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_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_arithmetics_NOTu_u_ZEROu_u_LTu_u_ZERO, axiom, ![V_n]: (~ (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) <=> p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_TWO, axiom, s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))) = s(t_h4s_nums_num,h4s_nums_suc(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_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_RIGHTu_u_FORALLu_u_IMPu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![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))))) <=> (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))))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_arithmetics_LESSu_u_MONOu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_nums_suc(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_primu_u_recs_LESSu_u_0, axiom, ![V_n]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_bools_DISJu_u_SYM, 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_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_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_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_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_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_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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(ch4s_hrats_TRATu_u_MULu_u_LID, conjecture, ![V_h]: p(s(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))))).
