%   ORIGINAL: h4/hrat/hrat__QUOTIENT
% 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__ABS__def: !r. h4/hrat/hrat__ABS r = h4/hrat/hrat__ABS__CLASS (h4/hrat/trat__eq r)
% Assm: h4/hrat/hrat__REP__def: !a. h4/hrat/hrat__REP a = h4/min/_40 (h4/hrat/hrat__REP__CLASS a)
% 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__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\c. ?r. h4/hrat/trat__eq r r /\ c = h4/hrat/trat__eq r) rep
% Assm: h4/hrat/hrat__bijections_c0: !a. h4/hrat/hrat__ABS__CLASS (h4/hrat/hrat__REP__CLASS a) = a
% 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/TRAT__EQ__AP: !q p. p = q ==> h4/hrat/trat__eq p q
% 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/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/bool/TRUTH: T
% Assm: h4/hrat/TRAT__EQ__REFL: !p. h4/hrat/trat__eq p p
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/hrat/hrat__ABS__REP__CLASS_c1: !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
% 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/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% 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/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/hrat/hrat__ABS__REP__CLASS_c0: !a. h4/hrat/hrat__ABS__CLASS (h4/hrat/hrat__REP__CLASS a) = a
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% 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__MUL__SYM__EQ: !i h. h4/hrat/trat__mul h i = h4/hrat/trat__mul i h
% Assm: h4/quotient/EQUIV__def: !E. h4/quotient/EQUIV E <=> (!x y. E x y <=> E x = E y)
% 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/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/arithmetic/MULT__SYM: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% 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/hrat/TRAT__EQ__SYM: !q p. h4/hrat/trat__eq p q <=> h4/hrat/trat__eq q p
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% 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/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/prim__rec/PRE0_c1: !m. h4/prim__rec/PRE (h4/num/SUC m) = 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_c0: !m. h4/arithmetic/_2A h4/num/0 m = 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/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__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__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__MUL__SYM: !i h. h4/hrat/trat__eq (h4/hrat/trat__mul h i) (h4/hrat/trat__mul i h)
% 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__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__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__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__MUL__LID: !h. h4/hrat/trat__eq (h4/hrat/trat__mul h4/hrat/trat__1 h) h
% 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__NOZERO: !i h. ~h4/hrat/trat__eq (h4/hrat/trat__add h i) h
% 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/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/bool/FALSITY: !t. F ==> 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/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/arithmetic/num__CASES: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm: h4/arithmetic/MULT__CLAUSES_c1: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% 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/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/arithmetic/ADD__CLAUSES_c1: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% 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/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/hrat/trat__sucint0_c0: h4/hrat/trat__sucint h4/num/0 = h4/hrat/trat__1
% 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/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% 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/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_c2: !t. F /\ t <=> F
% 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_c4: !t. t /\ t <=> t
% 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__10: h4/hrat/trat__1 = h4/pair/_2C h4/num/0 h4/num/0
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = 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/IMP__CLAUSES_c3: !t. t ==> t <=> T
% 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/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/arithmetic/ADD1: !m. h4/num/SUC m = h4/arithmetic/_2B m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% 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/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% 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/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/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/SUB__EQ__0: !n m. h4/arithmetic/_2D m n = h4/num/0 <=> h4/arithmetic/_3C_3D m n
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% 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/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/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/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/quotient__pair/PAIR__REL__THM: !d c b a R2 R1. h4/quotient__pair/_23_23_23 R1 R2 (h4/pair/_2C a b) (h4/pair/_2C c d) <=> R1 a c /\ R2 b d
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/arithmetic/ADD__INV__0__EQ: !n m. h4/arithmetic/_2B m n = m <=> n = h4/num/0
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% 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/quotient__pair/SND__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!p1 p2. h4/quotient__pair/_23_23_23 R1 R2 p1 p2 ==> R2 (h4/pair/SND p1) (h4/pair/SND p2)))
% Assm: h4/quotient__pair/PAIR__MAP__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!R3 abs3 rep3. h4/quotient/QUOTIENT R3 abs3 rep3 ==> (!R4 abs4 rep4. h4/quotient/QUOTIENT R4 abs4 rep4 ==> (!f1 f2 g1 g2. h4/quotient/_3D_3D_3D_3E R1 R2 f1 f2 /\ h4/quotient/_3D_3D_3D_3E R3 R4 g1 g2 ==> h4/quotient/_3D_3D_3D_3E (h4/quotient__pair/_23_23_23 R1 R3) (h4/quotient__pair/_23_23_23 R2 R4) (h4/pair/_23_23 f1 g1) (h4/pair/_23_23 f2 g2)))))
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/quotient__pair/PAIR__QUOTIENT: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient__pair/_23_23_23 R1 R2) (h4/pair/_23_23 abs1 abs2) (h4/pair/_23_23 rep1 rep2))
% Assm: h4/pair/PAIR__MAP__THM: !y x g f. h4/pair/_23_23 f g (h4/pair/_2C x y) = h4/pair/_2C (f x) (g y)
% Assm: h4/quotient/FUN__REL: !g f R2 R1. h4/quotient/_3D_3D_3D_3E R1 R2 f g <=> (!x y. R1 x y ==> R2 (f x) (g y))
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/quotient/QUOTIENT__REP__REFL: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. R (rep a) (rep a))
% Assm: h4/quotient/QUOTIENT__REL: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. R r s <=> R r r /\ R s s /\ abs r = abs s)
% Assm: h4/quotient/QUOTIENT__ABS__REP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. abs (rep a) = a)
% Assm: h4/quotient/QUOTIENT__def: !rep abs R. h4/quotient/QUOTIENT R abs rep <=> (!a. abs (rep a) = a) /\ (!a. R (rep a) (rep a)) /\ (!r s. R r s <=> R r r /\ R s s /\ abs r = abs s)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~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/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Goal: h4/quotient/QUOTIENT h4/hrat/trat__eq h4/hrat/hrat__ABS h4/hrat/hrat__REP
%   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_ABSu_u_def]: !r. happ h4/hrat/hrat__ABS r = h4/hrat/hrat__ABS__CLASS (happ h4/hrat/trat__eq r)
% Assm [h4s_hrats_hratu_u_REPu_u_def]: !a. happ h4/hrat/hrat__REP a = h4/min/_40 (h4/hrat/hrat__REP__CLASS a)
% Assm [h4s_hrats_hratu_u_bijectionsu_c1]: !r. (?r0. happ (happ h4/hrat/trat__eq r0) r0 /\ r = happ h4/hrat/trat__eq r0) <=> h4/hrat/hrat__REP__CLASS (h4/hrat/hrat__ABS__CLASS r) = r
% Assm [h4s_hrats_hratu_u_TYu_u_DEF]: !_0. (!c. happ _0 c <=> (?r. happ (happ h4/hrat/trat__eq r) r /\ c = happ h4/hrat/trat__eq r)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep)
% Assm [h4s_hrats_hratu_u_bijectionsu_c0]: !a. h4/hrat/hrat__ABS__CLASS (h4/hrat/hrat__REP__CLASS a) = a
% 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_TRATu_u_EQu_u_AP]: !q p. p = q ==> happ (happ h4/hrat/trat__eq p) q
% Assm [h4s_hrats_tratu_u_eq0]: !y_27 y x_27 x. happ (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_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_hrats_TRATu_u_EQu_u_REFL]: !p. happ (happ h4/hrat/trat__eq p) p
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_hrats_hratu_u_ABSu_u_REPu_u_CLASSu_c1]: !c. (?r. happ (happ h4/hrat/trat__eq r) r /\ c = happ h4/hrat/trat__eq r) <=> h4/hrat/hrat__REP__CLASS (h4/hrat/hrat__ABS__CLASS c) = c
% Assm [h4s_hrats_TRATu_u_EQu_u_EQUIV]: !q p. happ (happ h4/hrat/trat__eq p) q <=> happ h4/hrat/trat__eq p = happ h4/hrat/trat__eq q
% 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_EQu_u_TRANS]: !r q p. happ (happ h4/hrat/trat__eq p) q /\ happ (happ h4/hrat/trat__eq q) r ==> happ (happ h4/hrat/trat__eq p) r
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% 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_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% 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_MULu_u_SYMu_u_EQ]: !i h. h4/hrat/trat__mul h i = h4/hrat/trat__mul i h
% 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_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_hrats_TRATu_u_ADDu_u_WELLDEFINED]: !r q p. happ (happ h4/hrat/trat__eq p) q ==> happ (happ h4/hrat/trat__eq (h4/hrat/trat__add p r)) (h4/hrat/trat__add 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_MULu_u_WELLDEFINED]: !r q p. happ (happ h4/hrat/trat__eq p) q ==> happ (happ h4/hrat/trat__eq (h4/hrat/trat__mul p r)) (h4/hrat/trat__mul q r)
% Assm [h4s_hrats_TRATu_u_EQu_u_SYM]: !q p. happ (happ h4/hrat/trat__eq p) q <=> happ (happ h4/hrat/trat__eq q) p
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_hrats_TRATu_u_SUCINTu_u_0]: !n. happ (happ h4/hrat/trat__eq (h4/hrat/trat__sucint n)) (h4/pair/_2C n h4/num/0)
% 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_primu_u_recs_PRE0u_c1]: !m. h4/prim__rec/PRE (h4/num/SUC m) = 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_c0]: !m. h4/arithmetic/_2A h4/num/0 m = 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_hrats_TRATu_u_ARCH]: !h. ?n d. happ (happ h4/hrat/trat__eq (h4/hrat/trat__sucint n)) (h4/hrat/trat__add h d)
% Assm [h4s_hrats_TRATu_u_INVu_u_WELLDEFINED]: !q p. happ (happ h4/hrat/trat__eq p) q ==> happ (happ h4/hrat/trat__eq (h4/hrat/trat__inv p)) (h4/hrat/trat__inv q)
% Assm [h4s_hrats_TRATu_u_ADDu_u_SYM]: !i h. happ (happ h4/hrat/trat__eq (h4/hrat/trat__add h i)) (h4/hrat/trat__add i h)
% Assm [h4s_hrats_TRATu_u_ADDu_u_WELLDEFINED2]: !q2 q1 p2 p1. happ (happ h4/hrat/trat__eq p1) p2 /\ happ (happ h4/hrat/trat__eq q1) q2 ==> happ (happ h4/hrat/trat__eq (h4/hrat/trat__add p1 q1)) (h4/hrat/trat__add p2 q2)
% Assm [h4s_hrats_TRATu_u_MULu_u_SYM]: !i h. happ (happ h4/hrat/trat__eq (h4/hrat/trat__mul h i)) (h4/hrat/trat__mul i h)
% Assm [h4s_hrats_TRATu_u_LDISTRIB]: !j i h. happ (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_ADDu_u_ASSOC]: !j i h. happ (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_SUCINTu_c0]: happ (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 (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_MULu_u_ASSOC]: !j i h. happ (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_MULu_u_LID]: !h. happ (happ h4/hrat/trat__eq (h4/hrat/trat__mul h4/hrat/trat__1 h)) h
% Assm [h4s_hrats_TRATu_u_MULu_u_WELLDEFINED2]: !q2 q1 p2 p1. happ (happ h4/hrat/trat__eq p1) p2 /\ happ (happ h4/hrat/trat__eq q1) q2 ==> happ (happ h4/hrat/trat__eq (h4/hrat/trat__mul p1 q1)) (h4/hrat/trat__mul p2 q2)
% Assm [h4s_hrats_TRATu_u_NOZERO]: !i h. ~happ (happ h4/hrat/trat__eq (h4/hrat/trat__add h i)) h
% Assm [h4s_hrats_TRATu_u_ADDu_u_TOTAL]: !i h. happ (happ h4/hrat/trat__eq h) i \/ (?d. happ (happ h4/hrat/trat__eq h) (h4/hrat/trat__add i d)) \/ (?d. happ (happ h4/hrat/trat__eq i) (h4/hrat/trat__add h d))
% Assm [h4s_hrats_TRATu_u_MULu_u_LINV]: !h. happ (happ h4/hrat/trat__eq (h4/hrat/trat__mul (h4/hrat/trat__inv h) h)) h4/hrat/trat__1
% Assm [h4s_bools_FALSITY]: !t. F ==> 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_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_arithmetics_numu_u_CASES]: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% 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_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(h4/num/SUC n = h4/num/0)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% 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_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_hrats_tratu_u_sucint0u_c0]: h4/hrat/trat__sucint h4/num/0 = h4/hrat/trat__1
% 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_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% 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_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_c2]: !t. F /\ t <=> F
% 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_c4]: !t. t /\ t <=> t
% 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_10]: h4/hrat/trat__1 = h4/pair/_2C h4/num/0 h4/num/0
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% 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_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% 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_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_arithmetics_ADD1]: !m. h4/num/SUC m = h4/arithmetic/_2B m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% 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_arithmetics_NOTu_u_LESS]: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% 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_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_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) ==> h4/prim__rec/_3C m (h4/arithmetic/_2B m n)
% 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_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_LESSu_u_IMPu_u_LESSu_u_ORu_u_EQ]: !n m. h4/prim__rec/_3C m n ==> h4/arithmetic/_3C_3D m n
% Assm [h4s_arithmetics_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_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_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_quotientu_u_pairs_PAIRu_u_RELu_u_THM]: !d c b a R2 R1. happ (happ (h4/quotient__pair/_23_23_23 R1 R2) (h4/pair/_2C a b)) (h4/pair/_2C c d) <=> happ (happ R1 a) c /\ happ (happ R2 b) d
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_arithmetics_ADDu_u_INVu_u_0u_u_EQ]: !n m. h4/arithmetic/_2B m n = m <=> n = h4/num/0
% 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_arithmetics_LESSu_u_LESSu_u_CASES]: !n m. m = n \/ h4/prim__rec/_3C m n \/ h4/prim__rec/_3C n m
% Assm [h4s_arithmetics_LESSu_u_ADDu_u_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 [h4s_quotientu_u_pairs_SNDu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!p1 p2. happ (happ (h4/quotient__pair/_23_23_23 R1 R2) p1) p2 ==> happ (happ R2 (h4/pair/SND p1)) (h4/pair/SND p2)))
% Assm [h4s_quotientu_u_pairs_PAIRu_u_MAPu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!R3 abs3 rep3. h4/quotient/QUOTIENT R3 abs3 rep3 ==> (!R4 abs4 rep4. h4/quotient/QUOTIENT R4 abs4 rep4 ==> (!f1 f2 g1 g2. h4/quotient/_3D_3D_3D_3E R1 R2 f1 f2 /\ h4/quotient/_3D_3D_3D_3E R3 R4 g1 g2 ==> h4/quotient/_3D_3D_3D_3E (h4/quotient__pair/_23_23_23 R1 R3) (h4/quotient__pair/_23_23_23 R2 R4) (h4/pair/_23_23 f1 g1) (h4/pair/_23_23 f2 g2)))))
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_quotientu_u_pairs_PAIRu_u_QUOTIENT]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient__pair/_23_23_23 R1 R2) (h4/pair/_23_23 abs1 abs2) (h4/pair/_23_23 rep1 rep2))
% Assm [h4s_pairs_PAIRu_u_MAPu_u_THM]: !y x g f. happ (h4/pair/_23_23 f g) (h4/pair/_2C x y) = h4/pair/_2C (happ f x) (happ g y)
% Assm [h4s_quotients_FUNu_u_REL]: !g f R2 R1. h4/quotient/_3D_3D_3D_3E R1 R2 f g <=> (!x y. happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_quotients_QUOTIENTu_u_REPu_u_REFL]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. happ (happ R (happ rep a)) (happ rep a))
% Assm [h4s_quotients_QUOTIENTu_u_REL]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. happ (happ R r) s <=> happ (happ R r) r /\ happ (happ R s) s /\ happ abs r = happ abs s)
% Assm [h4s_quotients_QUOTIENTu_u_ABSu_u_REP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. happ abs (happ rep a) = a)
% Assm [h4s_quotients_QUOTIENTu_u_def]: !rep abs R. h4/quotient/QUOTIENT R abs rep <=> (!a. happ abs (happ rep a) = a) /\ (!a. happ (happ R (happ rep a)) (happ rep a)) /\ (!r s. happ (happ R r) s <=> happ (happ R r) r /\ happ (happ R s) s /\ happ abs r = happ abs s)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~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_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Goal: h4/quotient/QUOTIENT h4/hrat/trat__eq h4/hrat/hrat__ABS h4/hrat/hrat__REP
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_Q1164566,TV_Q1164562]: ![V_f, V_g]: (![V_x]: s(TV_Q1164562,happ(s(t_fun(TV_Q1164566,TV_Q1164562),V_f),s(TV_Q1164566,V_x))) = s(TV_Q1164562,happ(s(t_fun(TV_Q1164566,TV_Q1164562),V_g),s(TV_Q1164566,V_x))) => s(t_fun(TV_Q1164566,TV_Q1164562),V_f) = s(t_fun(TV_Q1164566,TV_Q1164562),V_g))).
fof(ah4s_hrats_hratu_u_ABSu_u_def, axiom, ![V_r]: s(t_h4s_hrats_hrat,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_hrats_hrat),h4s_hrats_hratu_u_abs),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r))) = 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),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)),h4s_hrats_tratu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r)))))).
fof(ah4s_hrats_hratu_u_REPu_u_def, axiom, ![V_a]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_hrats_hrat,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_hrats_hratu_u_rep),s(t_h4s_hrats_hrat,V_a))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_mins_u_40(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)))))).
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),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)),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),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)),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_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),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)),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),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)),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_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_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_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),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)),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_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),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)),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_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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),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)),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_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_c1, axiom, ![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),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)),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),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)),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))).
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),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)),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),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)),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),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)),h4s_hrats_tratu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q))))).
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_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),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)),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),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)),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),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)),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_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_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_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_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_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_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_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_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),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)),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),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)),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_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_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),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)),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),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)),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_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),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)),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),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)),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_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_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),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)),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_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_primu_u_recs_PRE0u_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_MULTu_u_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_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_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_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),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)),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_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),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)),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),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)),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,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)),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_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),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)),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),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)),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),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)),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_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),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)),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_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),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)),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_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),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)),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_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),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)),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),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)),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_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),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)),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_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),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)),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_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),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)),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),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)),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),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)),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_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),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)),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_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),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)),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),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)),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),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)),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_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),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)),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_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => 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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
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_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_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_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_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_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_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_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_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_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_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_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_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_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,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_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_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_LESSu_u_IMPu_u_LESSu_u_ORu_u_EQ, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_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_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_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_quotientu_u_pairs_PAIRu_u_RELu_u_THM, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b,TV_u_27d]: ![V_d, V_c, V_b, V_a, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)),h4s_quotientu_u_pairs_u_23u_23u_23(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))),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(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),h4s_pairs_u_2c(s(TV_u_27c,V_c),s(TV_u_27d,V_d)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1),s(TV_u_27a,V_a))),s(TV_u_27c,V_c)))) & p(s(t_bool,happ(s(t_fun(TV_u_27d,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2),s(TV_u_27b,V_b))),s(TV_u_27d,V_d))))))).
fof(ah4s_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_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_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_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,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_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,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_quotientu_u_pairs_SNDu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_p1, V_p2]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),h4s_quotientu_u_pairs_u_23u_23u_23(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p1))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p2)))) => p(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_R2),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p1))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p2)))))))))).
fof(ah4s_quotientu_u_pairs_PAIRu_u_MAPu_u_RSP, axiom, ![TV_u_27e,TV_u_27f,TV_u_27g,TV_u_27h,TV_u_27a,TV_u_27b,TV_u_27c,TV_u_27d]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27e),V_abs1),s(t_fun(TV_u_27e,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27f),V_abs2),s(t_fun(TV_u_27f,TV_u_27b),V_rep2)))) => ![V_R3, V_abs3, V_rep3]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27c,t_fun(TV_u_27c,t_bool)),V_R3),s(t_fun(TV_u_27c,TV_u_27g),V_abs3),s(t_fun(TV_u_27g,TV_u_27c),V_rep3)))) => ![V_R4, V_abs4, V_rep4]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27d,t_fun(TV_u_27d,t_bool)),V_R4),s(t_fun(TV_u_27d,TV_u_27h),V_abs4),s(t_fun(TV_u_27h,TV_u_27d),V_rep4)))) => ![V_f1, V_f2, V_g1, V_g2]: ((p(s(t_bool,h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_fun(TV_u_27a,TV_u_27b),V_f2)))) & p(s(t_bool,h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27c,t_fun(TV_u_27c,t_bool)),V_R3),s(t_fun(TV_u_27d,t_fun(TV_u_27d,t_bool)),V_R4),s(t_fun(TV_u_27c,TV_u_27d),V_g1),s(t_fun(TV_u_27c,TV_u_27d),V_g2))))) => p(s(t_bool,h4s_quotients_u_3du_3du_3du_3e(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_bool)),h4s_quotientu_u_pairs_u_23u_23u_23(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27c,t_fun(TV_u_27c,t_bool)),V_R3))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27d),t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27d),t_bool)),h4s_quotientu_u_pairs_u_23u_23u_23(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27d,t_fun(TV_u_27d,t_bool)),V_R4))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_h4s_pairs_prod(TV_u_27b,TV_u_27d)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_fun(TV_u_27c,TV_u_27d),V_g1))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_h4s_pairs_prod(TV_u_27b,TV_u_27d)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27b),V_f2),s(t_fun(TV_u_27c,TV_u_27d),V_g2)))))))))))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_quotientu_u_pairs_PAIRu_u_QUOTIENT, axiom, ![TV_u_27c,TV_u_27a,TV_u_27d,TV_u_27b]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),h4s_quotientu_u_pairs_u_23u_23u_23(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(TV_u_27c,TV_u_27d)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))))))))).
fof(ah4s_pairs_PAIRu_u_MAPu_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27d,TV_u_27c]: ![V_y, V_x, V_g, V_f]: s(t_h4s_pairs_prod(TV_u_27b,TV_u_27d),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),t_h4s_pairs_prod(TV_u_27b,TV_u_27d)),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27d),V_g))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27c,V_y))))) = s(t_h4s_pairs_prod(TV_u_27b,TV_u_27d),h4s_pairs_u_2c(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_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_g),s(TV_u_27c,V_y)))))).
fof(ah4s_quotients_FUNu_u_REL, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f, V_R2, V_R1]: (p(s(t_bool,h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) <=> ![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_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(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_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y))))))))).
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_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_quotients_QUOTIENTu_u_REPu_u_REFL, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_a]: 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,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a)))))))).
fof(ah4s_quotients_QUOTIENTu_u_REL, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_r, V_s]: (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_r))),s(TV_u_27a,V_s)))) <=> (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_r))),s(TV_u_27a,V_r)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_s))),s(TV_u_27a,V_s)))) & 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_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_s)))))))).
fof(ah4s_quotients_QUOTIENTu_u_ABSu_u_REP, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![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))).
fof(ah4s_quotients_QUOTIENTu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) <=> (![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_a]: 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,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a)))))) & ![V_r, V_s]: (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_r))),s(TV_u_27a,V_s)))) <=> (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_r))),s(TV_u_27a,V_r)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_s))),s(TV_u_27a,V_s)))) & 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_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_s)))))))))).
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_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_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_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_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_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_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_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_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ch4s_hrats_hratu_u_QUOTIENT, conjecture, p(s(t_bool,h4s_quotients_quotient(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_hrats_tratu_u_eq),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_hrats_hrat),h4s_hrats_hratu_u_abs),s(t_fun(t_h4s_hrats_hrat,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_hrats_hratu_u_rep))))).
